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HANDBOOK of MAGNETIC MATERIALS VOLUME 15

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HANDBOOK of MAGNETIC MATERIALS

VOLUME 15 EDITED BY

K.H.J. BUSCHOW Van der Waals-Zeeman Institute University of Amsterdam Amsterdam The Netherlands

2003

Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

ELSEVIER B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands © 2003 Elsevier B.V. All rights reserved This work is protected under copyright by Elsevier and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-proﬁt educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made.

First edition 2003

ISBN: 0-444-51459-7 ISSN: 1567-2719 ∞The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).

Printed in The Netherlands.

PREFACE TO VOLUME 15

The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. When Peter Wohlfarth started the latter series, his original aim was to combine new developments in magnetism with the achievements of earlier compilations of monographs, producing a worthy successor to Bozorth’s classical and monumental book Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete cross-section of magnetism than Bozorth’s book. In the last few decades magnetism has seen an enormous expansion into a variety of different areas of research, comprising the magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason the Editor and Publisher of this Handbook series have carefully reconsidered the title of the Handbook series and changed it into Magnetic Materials. It is with much pleasure that I can introduce to you now Volume 15 of this Handbook series. Advanced ultra-high vacuum deposition methods, make it possible to manufacture highly perfect artiﬁcial layered magnetic materials. The investigations performed in the last two decades on nanometer-scale thin ﬁlm and artiﬁcial multilayers with well deﬁned layer thickness and interface ﬂatness have led to the discovery of novel and most interesting effects. A general overview of the giant magnetoresistance effect in magnetic multilayers was already presented in chapter 1 of volume 12 of this Handbook. A prominent role among the layered magnetic materials is played by the so-called exchange biased spin valves and their excellent magnetoresistive properties. The advent of the exchange biased spin valves has led to many sensor applications, including those in hard disk read heads and applications in position and velocity sensors. In chapter 1 of the present volume, an application-oriented overview is presented of the extensive research efforts made on spin valves during the last decade. This overview includes work dealing with the magnetoresistance ratio, the thermal and ﬁeld stability and the micromagnetic stability. The magnetic interactions and their interplay are discussed together with theoretical understanding and modeling of the magnetoresistance. Because of the high application relevance in spin valves and spin-electronic devices and because of the involved novel physics and materials science, special attention is paid to the exchange bias effect. Special emphasis is placed also on work dealing with magnetic tunnel junctions, which are presently considered as excellent candidates for storage elements in non-volatile magnetic random access memories. A further novel ﬁeld of interest in magnetism is that of transition metal nanostructures. It has largely kept pace with microelectronics, forming the core of information technology. Current research efforts include the preparation of thin ﬁlms for improved data storage, the exploitation of the electron spin rather than its charge for device switching (“spintronics”), v

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PREFACE TO VOLUME 15

and the development of new materials for lightweight and low-cost applications. Generally, there has been a need for adequate theoretical descriptions able to explain most of the experimental phenomena and results. The many-body aspect of magnetic systems makes the task of calculating low-energy conﬁgurations of spin ensembles a formidable one. Because a full quantum mechanical description is actually intractable, various approximations have been used. The concentrated effort and the enthusiasm of a large number of scientists have resulted in an impressive display of new ideas and truly new discoveries. Theoretical work has already played and still plays a most important role in the process of active feedback between theories and experiments which has helped and speeded up the occurrence of novel accomplishments. Indeed, all magnetic properties of a solid are attributable to its electrons. In a free atom, there are two contributions to the magnetic moment. First, every electron has intrinsic spin and its associated magnetic moment. Second, there is the magnetic moment associated with the electron’s orbital angular momentum. In a free atom these contributions are typically comparable in magnitude. For transition metals Hund’s rules predict the ground state conﬁguration, but the situation is quite different for solids in which a restricted number of these atoms have condensed into low-dimensional arrangements. In chapter 2 of the present volume a survey is given of the electronic structure of low-dimensional transition metals. It comprises not only thin ﬁlms and multilayers but also clusters of transition metal atoms and nanowires. Results of novel experimental techniques are discussed hand in hand with theoretical approaches proposed to describe the electronic structure of these low-dimensional systems. Diluted magnetic semiconductors (DMS) can be characterized as substitutional mixed crystals with some of the cations of the semiconductor host lattice replaced by magnetic ions such as Mn, Fe or Eu. These materials encompass a large number of different compounds. A review covering the ﬁeld of bulk II–VI compounds has been presented already in chapter 4 of volume 7 of the Handbook. In the last decade many new materials (e.g., IV–VI compounds) have been investigated. This is true in particular for low-dimensional quantum structures based on diluted magnetic semiconductors. Therefore, the present chapter is a logical extension of the earlier chapter presented in volume 7. It gives an overview of the research activities on low-dimensional structures of II–VI diluted magnetic semiconductors with manganese, including new DMS materials in which the magnetic components are different from Mn. Special emphasis is put on results obtained with IV–VI materials containing a magnetic component. In a way it can be regarded as a complement to the chapter on III–V Ferromagnetic Semiconductors that has appeared in volume 14 of the Handbook in 2002. Chapter 3 of the present volume, like the chapter in volume 14, will be of use in particular to the numerous scientists who have recently been attracted to the ﬁeld of DMS by the prospect of incorporating diluted magnetic semiconductors in “spintronic” devices including those for quantum information applications. The current interest in spintronics has given renewed impetus to studies of diluted magnetic materials. As a result, we are witnessing now a vast increase of the number of contributions to the ﬁeld of DMS, which has led to a new type of topical conference (Physics and Applications of Spin Related Phenomena in Semiconductors, or PASPS) organized already twice (in Sendai in 2000, and in Würzburg in 2002). High-Tc superconductors are prominent examples of novel materials that are not only interesting because of their surprisingly high superconducting transition temperatures but

PREFACE TO VOLUME 15

vii

also because of their unusual magnetic properties and the interplay between antiferromagnetism and superconductivity. After the discovery of the ﬁrst high-Tc superconductor La2−x Bax CuO4 by Bednorz and Müller in 1986 tremendous efforts have been spent world-wide in raising Tc even further and to interpret the rich phase diagrams displayed by the cuprates and nickelates for various doping levels. A thorough discussion of the two-dimensional antiferromagnetism of the cuprates was already presented in chapter 1 of volume 10. In chapter 4 of the present volume an account is given of the enormous progress made more recently. This is in particular true with regard to statical and dynamical stripes and the collective magnetic mode, the so-called resonance peak. Results of novel experimental techniques, like ARPES, STM and μSR are presented together with results obtained from neutron scattering, NMR and NQR. These and many other experimental result are discussed in the light of the corresponding theoretical framework. Magnetotransport properties of materials have become of quite substantial importance in the competitive market of technological devices. This is true in particular for devices dealing with the storage and reading of information in magnetic recording media. In the last decade we have seen concentrated efforts to search for new giant magnetoresistive (GMR) materials and to fully uncover its origin. Nowadays, GMR based devices are already a reality in commercial hard disks, and they are responsible for a considerable increase in the recording areal density. A chapter on Giant Magnetoresistance in Magnetic Multilayers has appeared already in volume 12 of the Handbook. Of almost equal technological importance is the so-called giant magnetoimpedance effect, GMI. Initially, its observation and the concomitant research accomplishments were received with only modest enthusiasm, probably because of the envisaged modest technological expectations and an apparent lack of intrinsically new magnetic effects related to its origin. Nevertheless, it soon became clear that its interpretation requires a deep understanding of the micromagnetic characteristics of soft magnetic materials and its dependence on dynamic magnetism. With the vast increase of the number of scientists all over the world investigating GMI and its technological applications, GMI has actually opened a new branch of research linking micromagnetics of soft magnets with classical electrodynamics. From the applications perspective, there exists already a wide range of prototypes of magnetic and magnetoelastic sensors and several devices have already penetrated the market. It is worth mentioning that the GMI-based devices are not intended to oust spin-valve-based heads from the magnetic recording market. However, GMI devices have reached a development stage that is mature enough for entering the relevant area of extremely sensitive magnetic ﬁeld sensoring. Indeed, in some systems, with additional advantages, the best characteristics of the wellestablished ﬂuxgate sensors were reached. Moreover, sensitivities as high as those found in sensors based on superconducting quantum interference devices (SQUIDs) are expected to be reached, great advantages being competitive price and operation at room temperature. The present chapter summarizes and updates the increasing number of information on the giant magnetoimpedance phenomenon. Volume 15 of the Handbook on the Properties of Magnetic Materials, as the preceding volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who wish to be introduced to a given topic in the ﬁeld of magnetism without the need to read the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 15 of the Handbook is

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PREFACE TO VOLUME 15

composed of topical review articles written by leading authorities. In each of these articles an extensive description is given in graphical as well as in tabular form, much emphasis being placed on the discussion of the experimental material in the framework of physics, chemistry and material science. The task to provide the readership with novel trends and achievements in magnetism would have been extremely difﬁcult without the professionalism of the North Holland Physics Division of Elsevier B.V., and I wish to thank Paul Penman for his great help and expertise. K.H.J. B USCHOW VAN DER WAALS -Z EEMAN I NSTITUTE U NIVERSITY OF A MSTERDAM , N ETHERLANDS

CONTENTS

Preface to Volume 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Contents of Volumes 1–14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

1. Giant Magnetoresistance and Magnetic Interactions in Exchange-Biased SpinValves R. COEHOORN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Electronic Structure Calculations of Low-dimensional Transition Metals A. VEGA, J.C. PARLEBAS and C. DEMANGEAT . . . . . . . . . . . . . . . . 3. II–VI and IV–VI Diluted Magnetic Semiconductors – New Bulk Materials and Low-Dimensional Quantum Structures W. DOBROWOLSKI, J. KOSSUT and T. STORY . . . . . . . . . . . . . . . . . 4. Magnetic Ordering Phenomena and Dynamic Fluctuations in Cuprate Superconductors and Insulating Nickelates H.B. BROM and J. ZAANEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Giant Magnetoimpedance M. KNOBEL, M. VÁZQUEZ and L. KRAUS . . . . . . . . . . . . . . . . . . .

1 199

289

379 497

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

649

Materials Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

657

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CONTENTS OF VOLUMES 1–14 Volume 1 1. 2. 3. 4. 5. 6. 7.

Iron, Cobalt and Nickel, by E. P. Wohlfarth . . . . . . . . . . . . . . Dilute Transition Metal Alloys: Spin Glasses, by J. A. Mydosh and G. J. Nieuwenhuys Rare Earth Metals and Alloys, by S. Legvold . . . . . . . . . . . . . Rare Earth Compounds, by K. H. J. Buschow . . . . . . . . . . . . . Actinide Elements and Compounds, by W. Trzebiatowski . . . . . . . . . Amorphous Ferromagnets, by F. E. Luborsky . . . . . . . . . . . . . Magnetostrictive Rare Earth–Fe2 Compounds, by A. E. Clark . . . . . . . .

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Volume 2 1. 2. 3. 4. 5. 6. 7. 8.

Ferromagnetic Insulators: Garnets, by M. A. Gilleo . . . . Soft Magnetic Metallic Materials, by G. Y. Chin and J. H. Wernick Ferrites for Non-Microwave Applications, by P. I. Slick . . . Microwave Ferrites, by J. Nicolas . . . . . . . . . . Crystalline Films for Bubbles, by A. H. Eschenfelder . . . . Amorphous Films for Bubbles, by A. H. Eschenfelder . . . Recording Materials, by G. Bate . . . . . . . . . . Ferromagnetic Liquids, by S. W. Charles and J. Popplewell . .

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Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, by U. Enz . . . . . . . . . . . . . . . . . . . . . 2. Permanent Magnets; Theory, by H. Zijlstra . . . . . . . . . . . . . . . . 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R. A. McCurrie . . 4. Oxide Spinels, by S. Krupiˇcka and P. Novák . . . . . . . . . . . . . . . 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure, by H. Kojima . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto . . . . . . . 7. Hard Ferrites and Plastoferrites, by H. Stäblein . . . . . . . . . . . . . . . 8. Sulphospinels, by R. P. van Stapele . . . . . . . . . . . . . . . . . . . 9. Transport Properties of Ferromagnets, by I. A. Campbell and A. Fert . . . . . . .

305 393 441 603 747

Volume 4 1. Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K. H. J. Buschow . . . . . 1 2. Rare Earth–Cobalt Permanent Magnets, by K. J. Strnat . . . . . . . . . . . . . . . . 131 3. Ferromagnetic Transition Metal Intermetallic Compounds, by J. G. Booth . . . . . . . . . . 211 xi

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4. Intermetallic Compounds of Actinides, by V. Sechovský and L. Havela . . . . . . . . . . . 309 5. Magneto-Optical Properties of Alloys and Intermetallic Compounds, by K. H. J. Buschow . . . . 493

Volume 5 1. Quadrupolar Interactions and Magneto-Elastic Effects in Rare-Earth Intermetallic Compounds, by P. Morin and D. Schmitt . . . . . . . . . . . . . . . . . . . . . . 2. Magneto-Optical Spectroscopy of f-Electron Systems, by W. Reim and J. Schoenes . . . . 3. INVAR: Moment-Volume Instabilities in Transition Metals and Alloys, by E. F. Wasserman . 4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P. E. Brommer and J. J. M. Franse . 5. First-Order Magnetic Processes, by G. Asti . . . . . . . . . . . . . . . . . 6. Magnetic Superconductors, by Ø. Fischer . . . . . . . . . . . . . . . . .

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1 133 237 323 397 465

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Volume 6 1. Magnetic Properties of Ternary Rare-Earth Transition-Metal Compounds, by H.-S. Li and J. M. D. Coey . . . . . . . . . . . . . . . . . . . . . . . . . 2. Magnetic Properties of Ternary Intermetallic Rare-Earth Compounds, by A. Szytula . 3. Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren . 4. Magnetic Amorphous Alloys, by P. Hansen . . . . . . . . . . . . . . . 5. Magnetism and Quasicrystals, by R. C. O’Handley, R. A. Dunlap and M. E. McHenry . 6. Magnetism of Hydrides, by G. Wiesinger and G. Hilscher . . . . . . . . . .

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Volume 7 1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann . . . . . . . . . . 2. Energy Band Theory of Metallic Magnetism in the Elements, by V. L. Moruzzi and P. M. Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides, by M. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . . . . . . 4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski . . . . . . . . . 5. Magnetic Properties of Binary Rare-Earth 3d-Transition-Metal Intermetallic Compounds, by J. J. M. Franse and R. J. Radwa´nski . . . . . . . . . . . . . . . . . . . 6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems, by M. Loewenhaupt and K. H. Fischer . . . . . . . . . . . . . . . . . . .

. . 139 . . 231 . . 307 . . 503

Volume 8 1. Magnetism in Artiﬁcial Metallic Superlattices of Rare Earth Metals, by J. J. Rhyne and R. W. Erwin . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics with Cobalt and Iron, by A. V. Andreev . . . . . . . . . . . . . . . . . . 3. Progress in Spinel Ferrite Research, by V. A. M. Brabers . . . . . . . . . . . . . 4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A. J. Moses . . . . 5. Magnetic Properties of Rare Earth–Cu2 Compounds, by Nguyen Hoang Luong and J. J. M. Franse . . . . . . . . . . . . . . . . . . . . . . . . . .

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Volume 9 1. Heavy Fermions and Related Compounds, by G.J. Nieuwenhuys . . . . . . . . . . . . . 2. Magnetic Materials Studied by Muon Spin Rotation Spectroscopy, by A. Schenck and F.N. Gygax . .

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3. Interstitially Modiﬁed Intermetallics of Rare Earth and 3d Elements, by H. Fujii and H. Sun . . . . 303 4. Field Induced Phase Transitions in Ferrimagnets, by A.K. Zvezdin . . . . . . . . . . . . 405 5. Photon Beam Studies of Magnetic Materials, by S.W. Lovesey . . . . . . . . . . . . . . 545

Volume 10 1. Normal-State Magnetic Properties of Single-Layer Cuprate High-Temperature Superconductors and Related Materials, by D.C. Johnston . . . . . . . . . . . . . . . . . . . . . . 1 2. Magnetism of Compounds of Rare Earths with Non-Magnetic Metals, by D. Gignoux and D. Schmitt 239 3. Nanocrystalline Soft Magnetic Alloys, by G. Herzer . . . . . . . . . . . . . . . . . 415 4. Magnetism and Processing of Permanent Magnet Materials, by K.H.J. Buschow . . . . . . . . 463

Volume 11 1. Magnetism of Ternary Intermetallic Compounds of Uranium, by V. Sechovský and L. Havela . . . 1 2. Magnetic Recording Hard Disk Thin Film Media, by J.C. Lodder . . . . . . . . . . . . 291 3. Magnetism of Permanent Magnet Materials and Related Compounds as Studied by NMR, by Cz. Kapusta, P.C. Riedi and G.J. Tomka . . . . . . . . . . . . . . . . . . . . 407 4. Crystal Field Effects in Intermetallic Compounds Studied by Inelastic Neutron Scattering, by O. Moze 493

Volume 12 1. Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy, A. Fert and F. Petroff 2. NMR of Thin Magnetic Films and Superlattices, by P.C. Riedi, T. Thomson and G.J. Tomka 3. Formation of 3d-Moments and Spin Fluctuations in Some Rare-Earth–Cobalt Compounds, by N.H. Duc and P.E. Brommer . . . . . . . . . . . . . . . . . . . . 4. Magnetocaloric Effect in the Vicinity of Phase Transitions, by A.M. Tishin . . . . .

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Volume 13 1. Interlayer Exchange Coupling in Layered Magnetic Structures, by D.E. Bürgler, P. Grünberg, S.O. Demokritov and M.T. Johnson . . . . . . . . . . . . . . . . . . 1 2. Density Functional Theory Applied to 4f and 5f Elements and Metallic Compounds, by M. Richter . 87 3. Magneto-Optical Kerr Spectra, by P.M. Oppeneer . . . . . . . . . . . . . . . . . . 229 4. Geometrical Frustration, by A.P. Ramirez . . . . . . . . . . . . . . . . . . . . 423

Volume 14 1. III-V Ferromagnetic Semiconductors, by F. Matsukura, H. Ohno and T. Dietl . . . . . . . 2. Magnetoelasticity in Nanoscale Heterogeneous Magnetic Materials, by N.H. Duc and P.E. Brommer 3. Magnetic and Superconducting Properties of Rare Earth Borocarbides of the Type RNi2 B2 C, by K.-H. Müller, G. Fuchs, S.-L. Drechsler and V.N. Narozhnyi . . . . . . . . . . . . 4. Spontaneous Magnetoelastic Effects in Gadolinium Compounds, by A. Lindbaum and M. Rotter .

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LIST OF CONTRIBUTORS

H.B. Brom, Kamerlingh Onnes Laboratory, Leiden University, NL 2300 RA, The Netherlands R. Coehoorn, Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands Eindhoven University of Technology, Department of Applied Physics, P.O. Box 513, 5600 MB Eindhoven, The Netherlands C. Demangeat, Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, 23, rue du Loess, F-67034 Strasbourg Cedex 02, France W. Dobrowolski, Institute of Physics of the Polish Academy of Sciences, Warsaw, Poland M. Knobel, Instituto de Física Gleb Wataghin (IFGW), Universidade Estadual de Campinas (UNICAMP), C.P. 6165, Campinas 13.083-970 S.P., Brazil J. Kossut, Institute of Physics of the Polish Academy of Sciences, Warsaw, Poland L. Kraus, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic J.C. Parlebas, Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, 23, rue du Loess, F-67034 Strasbourg Cedex 02, France T. Story, Institute of Physics of the Polish Academy of Sciences, Warsaw, Poland M. Vázquez, Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientíﬁcas (CSIC), 28049 Cantoblanco (Madrid), Spain A. Vega, Departamento de Física Teórica, Atómica, Molecular y Nuclear, Universidad de Valladolid, E-47011 Valladolid, Spain J. Zaanen, Lorentz Institute for Theoretical Physics, Leiden University, NL 2300 RA, The Netherlands

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chapter 1

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS IN EXCHANGE-BIASED SPIN-VALVES

R. COEHOORN Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands e-mail: [email protected] Eindhoven University of Technology, Department of Applied Physics, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Handbook of Magnetic Materials, edited by K.H.J. Buschow Vol. 15 ISSN: 1567-2719 DOI 10.1016/S1567-2719(03)15001-9

1

© 2003 Elsevier Science B.V. All rights reserved

CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1. Scope of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2. The GMR effect in magnetic multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.3. Structure and functioning of exchange-biased spin-valves . . . . . . . . . . . . . . . . . . . . . .

9

1.4. Experimental results for prototype simple spin-valves . . . . . . . . . . . . . . . . . . . . . . . .

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1.5. Dependence of the MR ratio on the layer thicknesses . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.6. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.7. Figures of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Structure and transport properties of conventional and advanced spin-valves . . . . . . . . . . . . . . .

29

2.1. Simple spin-valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.2. Dual spin-valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.3. Spin-valves with improved magnetic characteristics by additional layers outside the active part . .

53

2.4. Pseudo-spin-valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.5. Temperature dependence of the magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.6. Anisotropic magnetoresistance and anisotropy of the giant magnetoresistance . . . . . . . . . . .

61

2.7. Thermoelectric power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

2.8. Infrared optical properties – the magnetorefractive effect . . . . . . . . . . . . . . . . . . . . . . .

63

2.9. Deposition and microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.10. Thermal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3. Spin-polarized transport in spin-valves: theory and modelling . . . . . . . . . . . . . . . . . . . . . . .

77

3.1. Spin-dependent conductivity in ferromagnets – spin-dependent scattering . . . . . . . . . . . . .

78

3.2. Spin-dependent conductivity in ferromagnets – effect of spin-mixing . . . . . . . . . . . . . . . .

80

3.3. Spin-dependent scattering at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.4. Series resistor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.5. The Camley–Barnas semiclassical transport model . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.6. Applications of the Camley–Barnas model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

3.7. Extensions of the semiclassical Camley–Barnas model . . . . . . . . . . . . . . . . . . . . . . . . 103 3.8. Semiclassical models based on realistic band structures . . . . . . . . . . . . . . . . . . . . . . . 106 3.9. Quantum-mechanical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.10. Trends and future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4. Magnetic interactions and magnetization reversal processes . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.1. Applications of the Stoner–Wohlfarth model to spin-valves . . . . . . . . . . . . . . . . . . . . . 115 4.2. Deviations from the single-domain model – micromagnetics . . . . . . . . . . . . . . . . . . . . . 124 4.3. Frequency dependence of the magnetic response, magnetization ﬂuctuations and electronic noise 2

130

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3

4.4. Anisotropy and magnetostriction of the free layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.5. Interlayer coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.6. Exchange anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

1. Introduction 1.1. Scope of this chapter The use of advanced ultra-high vacuum deposition methods makes it possible to fabricate highly perfect artiﬁcial layered magnetic materials. In the past two decades studies of materials with nanometer-scale layer thicknesses, made with atomic-scale control of the layer thicknesses and interface ﬂatness, have led to the discovery of several very interesting and sometimes unanticipated effects (Bland and Heinrich, 1994). One of these developments was the discovery of the giant magnetoresistance (GMR) effect by Grünberg and co-workers (Binasch et al., 1989), and, independently, by Fert and co-workers (Baibich et al., 1988). For layered materials consisting of ferromagnetic (F) layers, separated by non-magnetic (NM) layers, a large change of the electrical resistance was observed upon the application of a magnetic ﬁeld. The relative resistance change can be more than 100% at room temperature. Some examples are given in section 1.2. The GMR effect is a consequence of the spin-polarization of the electrical conduction in the layer structure. In this chapter we focus on a speciﬁc class of layered GMR materials, called exchange-biased spin-valves (SVs), which are excellently suited for sensor applications. Less than a decade after the ﬁrst publication by Dieny and coworkers at IBM (Dieny et al., 1991a, 1991b, 1991c), exchange-biased SVs were already introduced as sensor materials in hard disk read heads (Tsang et al., 1998). Other (potential) applications include position, speed and velocity sensors, and electronic compasses. The purpose of this chapter is to give an overview of the extensive research on exchangebiased SVs that has taken place during the past 12 years. Section 1 contains a description of the basic structure of SVs, a phenomenological explanation of their functioning as a magnetoresistive material, and a brief overview of applications. Sections 2–4 contain indepth discussions on the various detailed subjects that have been studied in order to obtain a better understanding of the functioning of SVs and in order to improve their performance in applications. Section 2 provides an overview of the experimental results of studies of the resistance and magnetoresistance for conventional and advanced spin-valves. The theoretical understanding and modelling of the magnetoresistance is discussed in section 3. The magnetization reversal processes, the magnetic interactions and their interplay are reviewed in section 4. Section 5 contains a summary and outlook. An overall conclusion of this review is that important progress is still being made, leading, e.g., to ever larger magnetoresistances, better control of the magnetic interactions, and enhanced stability with respect to elevated temperatures and the application of large magnetic ﬁelds. Therefore, studies of SVs continue to open opportunities for novel innovative applications. Many of the developments that are described in this chapter are not only important for the advancement of SV applications, but have also an impact in the much broader ﬁeld 4

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5

of spin-electronics. Within this novel research ﬁeld materials and devices are studied of which the conduction can be inﬂuenced by the application of a magnetic ﬁeld, as a result of the spin-polarization of the electron transport in these devices. Besides devices based on GMR materials, the presently most intensively studied spin-electronic devices are magnetic tunnel junctions (MTJs, Moodera et al. (1995), Leclair (2003)). In MTJs, spin-polarized electron tunneling between two ferromagnetic electrode layers across an insulating spacer layer can give rise a high magnetoresistance, over 50% at room temperature. Apart from the presence of the insulating spacer layer, their layer structure can be quite similar to that of exchange-biased SVs (Lu et al., 1997). The physics and materials science of the magnetic interactions in SVs, discussed in section 4 of this chapter, is equally relevant for such MTJs. MTJs are presently considered as prime candidates for storage elements in nonvolatile Magnetic Random Access Memories (MRAMs) (section 1.6.3). Spin-electronic materials and devices can also be made of combinations of ferromagnetic metallic layers and semiconductors. The spin-polarized transport processes in these materials and devices, and the novel opportunities that are envisaged for applications in solid state memory, logic and sensor elements, have been reviewed by Prinz (1995), Wolf et al. (2001), Ziese and Thornton (2001), Gregg et al. (2002), and Awschalom et al. (2002). For a comprehensive review on the GMR effect we refer to chapter 1 in volume 12 of this Handbook series (Barthélémy et al., 1999). Other reviews, focused on the fundamental aspects, have been presented by Levy (1994), Fert and Bruno (1994), Parkin (1994b), Gijs and Bauer (1997), Tsymbal and Pettifor (2001), Shinjo (2002), and Levy and Mertig (2002). Earlier reviews focused on exchange-biased spin-valves and on sensor applications have been written by Dieny (1994), Kools (1996), Kools et al. (1998a, 1998b), Tsang et al. (1998), Coehoorn et al. (1998), Coehoorn (2000) and Sakakima (2002). The application of magnetoresistive materials in sensors has been reviewed by Tumanski (2001). 1.2. The GMR effect in magnetic multilayers The GMR effect in magnetic multilayers is the dependence of the resistance on the angles between the magnetization directions of the successive magnetic layers. The effect originates from the spin-dependence of the electrical conduction in ferromagnetic materials, and occurs for systems with nanometer-scale layer thicknesses. Prototype examples of thin ﬁlm materials showing a large GMR effect are antiferromagnetically (AF) coupled Co/Cu and Fe/Cr multilayers.1 At zero ﬁeld a strong antiferromagnetic interlayer exchange coupling (reviewed by Bürgler et al., 2001) gives rise to an antiparallel alignment of the magnetization directions of neighbouring ferromagnetic layers. The application of a magnetic ﬁeld leads to a transition to a parallel state, upon which the electrical resistance decreases strongly. The magnetoresistance (MR) ratio is deﬁned as ΔR RAP − RP ≡ , R RP

(1.1)

where RP and RAP are the resistances in the parallel and antiparallel states, respectively. 1 The notation ‘A/B multilayer’ is used to brieﬂy indicate a periodic layered material of the type (A/B) , formed n

by n 1 repetitions of a bilayer consisting of materials A and B.

6

R. COEHOORN

Fig. 1.1. Magnetoresistance curve for a sputtered epitaxial [110] oriented AF-coupled Co/Cu multilayer for the ﬁeld applied along the [011] direction in the plane of the superlattice. The insets show, schematically, the alignment of the magnetization directions in the Co95 Fe5 layers. From Parkin (2002).

At room temperature, the MR ratio can be as large as 65% for polycrystalline sputter deposited (0.8 nm Co/ 0.83 nm Cu)60 multilayers (Parkin et al., 1991a) and 42% for (0.45 nm Fe/ 1.2 nm Cr)50 superlattice structures grown by Molecular Beam Epitaxy (MBE) (Schad et al., 1994). At 4.2 K the MR ratios observed for the same systems are even larger, viz. 115% and 220%, respectively. The largest yet reported MR ratio at room temperature, 110%, has been observed for a sputter deposited epitaxial [110]-oriented AF-coupled Co95 Fe5 /Cu superlattice (Parkin, 2002). Fig. 1.1 shows the magnetoresistance curve. These experimental results were all obtained for the “Current In the Plane of the layers” (CIP) geometry. The current is then parallel to the plane of the layers. Standard four-point measurements can be carried out easily by applying needle-shaped current and voltage probes directly on the thin-ﬁlm specimen. For making devices, lithographic patterning is used for deﬁning the multilayer magnetoresistor stripes and low-resistive contact leads, as shown schematically in ﬁg. 1.2(a). Magnetoresistance studies of “Current Perpendicular to the Planes of the layers” (CPP) devices are technically much more demanding than CIP-GMR studies, but have revealed GMR ratio up to 300% at cryogenic temperatures (Gijs and Bauer, 1997). The geometry is shown schematically in ﬁg. 1.2(b). It is necessary to create samples with small cross-sectional areas, in order to obtain a resistance that is not unacceptably low, and to carefully consider the functioning of the contacts as proper equipotential planes. The term ‘giant’ MR effect was chosen because the MR ratio can be much larger than observed at room temperature for the magnetoresistance effects already known at the time of the discovery, the ordinary MR effect and the anisotropic (AMR) effect. The ordinary MR effect is the dependence of the resistance of a material on the magnetic induction (Bﬁeld), due to the effect of the Lorentz force on the trajectories of the conduction electrons

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

7

Fig. 1.2. Schematic device structures for measurements of the GMR ratio of a multilayer in (a) the Current-In-Plane (CIP) device geometry, and (b) the Current-Perpendicular-to-the-Plane (CPP) geometry. In order to eliminate the contact resistance, four-point CIP-measurements are carried out by making use of additional voltage probing contacts in between the current contacts (not shown). For an overview of methods for eliminating contact effects in the CPP-geometry we refer to Gijs and Bauer (1997).

(Jan, 1957). It occurs already in non-magnetic materials. The AMR effect is the dependence of the resistance of ferromagnetic materials on the angle between the current and the magnetization. It is of relativistic origin (Smit, 1951; Banhard and Ebert, 1995). The effect is relatively small, typically 1.5–3% in 10–30 nm thick permalloy layers (McGuire and Potter, 1975; Miyazaki et al., 1989; Mao et al., 1999a; Dieny et al., 2000b). Permalloy (Py) is a softmagnetic Ni1−x Fex alloy with x ≈ 0.20. AMR materials are presently used extensively in sensors (Tumanski, 2001), and have been used in hard disk read heads. However, in the latter application SVs have now fully replaced AMR materials. The GMR-effect arises as a result of the spin-dependent conductance in the ferromagnetic layers, and/or as a result of spin-dependent scattering at the interfaces. As proposed ﬁrst by Mott (1936, 1964), the conductance of ferromagnetic materials can be viewed as the sum of separate contributions from electrons with opposite spin directions when the spin quantum number of the conduction electrons is conserved in the most of the scattering processes. This so-called two-current model is a fair approximation for ferromagnets based on Fe, Co and Ni, at least at temperatures well below the Curie temperature. Experimental studies on these metals, and on dilute alloys based on these metals, have revealed that the conductance can be strongly spin-dependent (Campbell and Fert, 1982). Since the discovery of the GMR effect, it has become clear that also the transport through interfaces can be spin-dependent. Fig. 1.3(a) illustrates the origin of the CIP-GMR effect for a periodic F/NM multilayer. The ﬁgure depicts typical trajectories of diffusively scattered electrons with opposite spin directions.2 The conductance depends on the angles between the magnetization directions of the neighbouring magnetic layers if: (1) the conductance in the F layers is spin-dependent and/or if the scattering probability at the F/NM interfaces is spin-dependent, and if 2 When in this chapter a speciﬁc layer structure is considered the spin-directions are indicated as “+” and “−”,

as deﬁned with respect to an absolute axis direction. When a single F layer or an F/NM interface are considered, the spin directions are indicated as “↑” and “↓” (“up” and “down”), deﬁned as the majority and minority spin electrons, respectively, within the F layer. In ﬁg. 1.3, the “+” spin direction points to the right for systems with (e.g.) F = Py, Co or Co90 Fe10 and NM = Cu. The “+” spin direction points to the left for, e.g., Fe/Cr systems. Scattering in the bulk of the Fe layer and at the interfaces is then strongest for the majority spin electrons.

8

R. COEHOORN

Fig. 1.3. (a) Electron transport in F/NM multilayers for parallel and antiparallel alignments of the magnetizations. Typical trajectories of spin “+” and spin “−” electrons are indicated by dashed lines. (b) Effective resistor scheme that gives, within a simple model, the parallel and antiparallel resistances of an F/NM multilayer (see text in sections 1.2 and 3.4). (c) Electron transport in the active part of a spin-valve for parallel and antiparallel alignment of the magnetizations. Scattering at the outer boundaries is taken to be diffusive. In the case of multilayers or spin-valves based on Co, Ni, or on fcc Co–Fe–Ni binary or ternary alloys, and with NM = Cu, the majority-spin electrons have a lower scattering probability in the bulk of the F layers and at the interfaces, than the minority-spin electrons. The spin “+” and “−” directions then correspond in this ﬁgure to “right” and “left”.

(2) in the parallel aligned magnetic state the scattering probability upon traversal of one period is signiﬁcantly smaller than 1, for electrons of at least one spin-direction. In the ﬁgure the electrons for which in the parallel state the scattering probability is smallest are called the spin “+” electrons. For the parallel state the spin “+” contribution to the conductance is much larger than the spin “−” contribution. In contrast, in the antiparallel state, “+” as well as “−” electrons each scatter strongly in one of the F layers. This results in a relatively high antiparallel resistance. Each layer acts thus as a spin-selective valve: its magnetization direction determines whether it most easily transmits spin “+” or spin “−”

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

9

electrons. Therefore the GMR effect is sometimes, more appropriately, called the “spinvalve effect” (Dieny et al., 1991a). Within the simplest possible quantitative model of the GMR effect, the resistances RP and RAP in the parallel and antiparallel conﬁgurations can be obtained from the resistor schemes that are shown in ﬁg. 1.3(b). R+ and R− are the resistances in the parallel conﬁguration for spin “+” and spin “−” electrons. This series resistor model, which is further discussed in section 3.4, leads to an MR ratio equal to ΔR/R = (R+ − R− )2 /(4R+ R− ). The high relative resistance changes of the AF-coupled multilayer materials discussed above are only obtained in high magnetic ﬁelds. E.g., ﬁg. 1.1 reveals a saturation magnetic ﬁeld of approximately 2400 kA/m (corresponding to ≈ 30 kOe 3 ). For comparison, the amplitude of the earth magnetic ﬁeld is only ≈ 0.05 kA/m. For sensor applications, this is a disadvantageous situation. However, AF interlayer coupling is not a necessary requirement for obtaining the GMR effect. The GMR effect is observed in many other systems within which the relative alignment of the magnetization of nearby ferromagnetic regions can be changed by the application of a magnetic ﬁeld. Besides exchange-biased SVs, which are the subject of this chapter, other examples are multilayers comprising layers of which the magnetization switches via multidomain states (Dupas, 1990), granular systems (Xiao et al., 1992), so-called discontinuous layer systems (Hylton et al., 1993), and so-called hard–soft multilayers (Shinjo and Yamamoto, 1990; Yamamoto et al., 1991). Exchangebiased SVs have a layer structure in which the coupling between the F layers is very weak or even zero. The ﬁeld interval (around zero ﬁeld) within which the resistance change takes place can be as small as a few tenths of a kA/m (a few Oe). This is one of their advantageous properties for low-ﬁeld sensor applications. 1.3. Structure and functioning of exchange-biased spin-valves The layer structures of the three basic types of exchange-biased spin-valves (SVs) are shown schematically in ﬁg. 1.4. The structures (a) and (b) are so-called “simple SVs”. These consist essentially of a sandwich structure of two ferromagnetic (F) layers separated by a nonmagnetic (NM) spacer layer, and an antiferromagnetic (AF) layer that is in contact with one of the F layers. The magnetization of this F layer, the “pinned” or “reference” layer Fp , is held ﬁxed in a certain direction by the strong exchange interaction with the AF layer. Use is made of the “exchange anisotropy” effect. To a ﬁrst approximation, the AF/F exchange interaction acts as if a strong local magnetic ﬁeld, the so-called exchange bias ﬁeld, Heb , acts on the pinned layer. Therefore, the exchange anisotropy effect is also called the “exchange bias effect”. The preferred direction of the pinned layer is determined by the magnetization direction of the pinned layer during growth of the AF layer on top of it (in case of a top spin-valve), or during cooling of the system after heating the system to a temperature above the so-called “blocking temperature” (for top and bottom spinvalves). The blocking temperature, Tb , is the temperature above which Heb is zero. A more precise deﬁnition is given in section 4.6.4. The AF layer is often an antiferromagnetic Mncontaining alloy or compound, such as PtMn, or an antiferromagnetic oxide, such as NiO. 3 A magnetic ﬁeld of H = 1 kA/m (S.I.) corresponds to 12.57 Oe (c.g.s.). The corresponding magnetic induction

in vacuum is B = 1.257 mT (S.I.) or 12.57 G (c.g.s.).

10

R. COEHOORN

Fig. 1.4. Basic layer structures of exchange-biased spin-valves.

The pinned F layer can consist, e.g., of Co or permalloy. The other ferromagnetic layer, the “free layer” Ff , is effectively magnetically soft, and can consist, e.g., of permalloy. The NM spacer layer serves to magnetically decouple the F-layers. It consists usually of a 2–3 nm Cu layer. Sometimes, the thickness is even less. The thickness should be sufﬁcient to prevent direct ferromagnetic exchange coupling between the layers via “pinholes”. Even in the absence of pinhole coupling, indirect interlayer exchange coupling (due to a weak magnetic polarization of the NM layer by the exchange interaction with the F layers) and magnetostatic interactions contribute to a net coupling between the pinned and free F layers (see section 4.5). The AF layer can be deposited on top of the other layers (“top SV”, ﬁg. 1.4(a)) or on the bottom of the structure (“bottom SV” or “inverted SV”, ﬁg. 1.4(b)). The order of growth of the layers (top or bottom AF layer) affects the microstructure within the layers, the antiferromagnetic domain structure in the AF layer, and in some cases the effective ﬁeld that is sensed (e.g., due to the magnetic ﬁeld that is created by the sense current). However, the basic functioning of both structures is the same. Underlayers (sometimes called “buffer layers” or “seed layers”) are frequently used to inﬂuence the microstructure of the ﬁlm (e.g., the grain size, the preferential crystallite orientation and the interface ﬂatness) or to prevent interdiffusion with the substrate. For top spin-valves the underlayer must be nonmagnetic. For bottom spin-valves the underlayer may be ferromagnetic. Structures with two pinned layers are called “dual” or “symmetric” SVs (ﬁg. 1.4(c)). A thin cap layer (sometimes called “cover layer”) is often used to protect the structure from corrosion. In the early literature the term “spin-valve” was used to indicate multilayers in which the GMR effect arises upon a reversal of the magnetization of one or more soft magnetic layers that are not coupled to the other magnetic layers. However, it has become customary

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

11

Fig. 1.5. Schematic curves of the magnetic moment (a) and resistance (b) versus the applied magnetic ﬁeld for a simple SV. The magnetic moments per unit area of the free and pinned layers have been assumed to be equal. The top and bottom arrows indicate the magnetization directions of the pinned and free layer, respectively.

to use the term “spin-valve” more exclusively for exchange-biased spin-valves. We follow that convention. When a relatively small magnetic ﬁeld is applied, the free layer reverses whereas the magnetization direction(s) of the pinned layer(s) remain unchanged. Fig. 1.5(a) shows a schematic magnetization curve. The applied magnetic ﬁeld is parallel to the exchange bias ﬁeld. By deﬁnition, the applied ﬁeld is positive when its direction is the same as that of the exchange bias ﬁeld. For sufﬁciently large ﬁelds the free and pinned layers have parallel magnetizations. In a small ﬁeld interval around the coupling ﬁeld, Hcoupl , the magnetization of the free F layer reverses, whereas the magnetization of the pinned F layer remains ﬁxed. This deﬁnition of Hcoupl implies that it is negative or positive when the coupling is ferromagnetic or antiferromagnetic, respectively. Only upon the application of a large negative ﬁeld, the exchange bias interaction is overcome, and the pinned layer switches, too. Assuming ideal conditions, i.e., |Hcoupl| Heb , this happens when H ≈ −Heb . The complications that arise otherwise are discussed in section 4.1.2. Usually, the switching of the pinned layer is not fully reversible, leading to a certain hysteresis, as indicated in ﬁg. 1.5. For sensor applications, the magnetization of the free layer should reverse by a reversible, coherent rotation process. This can be realized by making use of a free layer

12

R. COEHOORN

Fig. 1.6. Schematic representation of the magnetization reversal processes in the F layers of a spin-valve with (a) crossed anisotropies and (b) parallel anisotropies. Ha and Heb are the anisotropy and exchange bias ﬁelds, respectively. Interlayer magnetic coupling is neglected.

with uniaxial magnetic anisotropy, with the easy magnetization axis perpendicular to the exchange bias direction. Within this crossed anisotropy (CA) conﬁguration the applied ﬁeld is along the hard axis direction of the free layer (ﬁg. 1.6(a)). Using the Stoner– Wohlfarth model (section 4.1) one ﬁnds that the free layer reverses in the ﬁeld interval [−Ha + Hcoupl, Ha + Hcoupl] (eq. (4.8)), where Ha is the magnetic anisotropy ﬁeld of the free layer. The switch ﬁeld range, ΔHsw , is equal to 2Ha . This is the situation that is depicted in ﬁg. 1.5(a). In actual devices the symmetry is slightly broken, in order to make one of the two otherwise equivalent reversal modes (clockwise or anti-clockwise) more favorable than the other. That favors real single-domain reversal (see sections 2.3, 4.1 and 4.2). The parallel anisotropy (PA) conﬁguration (easy axis of the free layer and exchange bias direction parallel) leads to hysteretic, irreversible switching of the free layer, via domain wall movement (ﬁg. 1.6(b)). This conﬁguration is useful when a spin-valve is to be applied as a memory element, with the two zero-ﬁeld states of the free layer corresponding to a digital “0” and “1”. The resistance versus applied ﬁeld (R(H )) curve, the transfer curve, is shown in ﬁg. 1.5(b). The angular variation of the resistance due to the GMR effect is to a good approximation given by R(θ ) = R(θ = 0) +

ΔRGMR (1 − cos θ ), 2

(1.2)

where θ is the angle between the magnetization directions of the free and pinned layers (Chaiken et al., 1990; Shinjo et al., 1992). It should be remarked that a study of a multilayer structure with a relatively high GMR ratio has revealed that the conductance, instead of the resistance, varies linearly with (1 − cos θ ) (Duvail et al., 1995). The difference between the two descriptions is of second order in the GMR ratio, and is therefore small in the case of a small GMR ratio. In the limit of weak coupling (|Hcoupl| Heb ) the transfer curve for

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

the crossed-anisotropy conﬁguration is then linear, and given by H − Hcoupl ΔRGMR R(H ) = R(H = ∞) + 1− 2 Ha

13

(1.3)

for |H − Hcoupl| Ha . We have used eq. (4.8) for cos θ . The resistance shows a steep slope around H = Hcoupl , where the magnetization of the free layer reverses. The linearity of the response is a very desirable property for sensors. For negative applied ﬁelds the resistance stays high until, at the exchange bias ﬁeld, the pinned layer switches, too. For CIP-SVs, the AF layers and underlayers may often be regarded as simple shunt resistors. Sometimes, the sheet resistances of these layers are so large that they can be neglected. The inner part of the stack, within which the detailed current density distribution depends on the alignment of the free and pinned layers, is called the “active layer” or “active part”. It is the part of the layer stack that is responsible for the GMR effect. The trajectories of electrons with opposite spin directions in the active part of the layer stack are shown schematically in ﬁg. 1.3(c). Spin-polarized transport through the bulk of the layer and/or spin-polarized scattering at the interfaces leads to a GMR effect when, for at least one of the spin-directions, the scattering probability upon traversal through the SV layer stack is signiﬁcantly smaller than 1. In this chapter, we do not discuss CPP-GMR studies of SVs. CPP-GMR spin-valve devices have only recently been studied at room temperature, and only recently possible applications of such devices have been considered. In all present applications the CIPgeometry is used. Experimental studies of spin-valves in the CPP-geometry have been carried out by Steenwyk et al. (1997), Gu et al. (2000), Nagasaka et al. (2001a, 2001b), Slater et al. (2001), Hosomi et al. (2002), Oshima et al. (2002), and Yuasa et al. (2002). Design studies of possible CPP-GMR hard disk read heads based on spin-valves have been carried out by Tanaka et al. (2002) and Takagishi et al. (2002). 1.4. Experimental results for prototype simple spin-valves The functioning of spin-valves is illustrated by ﬁg. 1.7. In the upper part, R(H ) curves are given for a series of (8 nm Ni80 Fe20 / tCu nm Cu/ 6 nm Ni80 Fe20 / 8 nm Fe50 Mn50) ﬁlms with varying Cu layer thickness.4 Fe50 Mn50 is an antiferromagnetic alloy with a Néel temperature of 230 ◦ C. These prototype materials were grown by sputter deposition in ultrahigh vacuum on a 3 nm Ta buffer layer on a Si(100) substrate, and covered by a 3 nm Ta cap layer (Rijks et al., 1994a). For tCu > 2.2 nm, the R(H ) curves have the ideal form given in ﬁg. 1.5. There is a well-deﬁned high-resistance plateau in the ﬁeld range in which the magnetizations of the free and pinned layers are antiparallel, as can be seen from the ﬁgures for tCu = 2.5 and 4.7 nm. The MR ratio increases with decreasing Cu thickness, to a maximum of almost 5% around tCu = 2.2 nm. Below that Cu layer thickness a perfectly antiparallel alignment is not obtained at any ﬁeld, because the coupling ﬁeld is then no longer much smaller than the exchange bias ﬁeld. As a result, the MR ratio decreases below tCu = 2.2 nm. At tCu = 1.3 nm, the magnetization directions of both layers remain almost parallel during the switching process. We will analyse the Cu thickness dependence of the MR ratio in sections 1.5 and 3. The shapes of the R(H ) curves will be analyzed 4 Throughout this chapter, the layers are given in the order of deposition.

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Fig. 1.7. Dependence of the room temperature magnetoresistance curves and the MR ratio on the Cu spacer layer thickness, for (8 nm Py/ tCu nm Cu/ 6 nm Py/ 8 nm Fe50 Mn50 ) spin-valves. The dashed line gives a ﬁt to the data using eq. (1.4), for the Cu thicknesses for which full antiparallel alignment is obtained. From Rijks et al. (1994a).

in more detail in section 4.1.2. The exchange-bias ﬁeld of these Fe–Mn based spin-valves shows an approximately linear decrease with increasing temperature, and vanishes around the blocking temperature, Tb ≈ 140 ◦ C. Fig. 1.8 shows the dependence of the MR ratio on the thickness and composition of the F layers in (F/Cu/F/Fe50 Mn50 ) SVs (Rijks, 1996a). For all F layers a rather broad maximum is found around tF = 6–8 nm. For spin-valves based on F = (Ni80 Fe20 )1−x Cox layers the MR ratio increases with increasing x, up to 8% at room temperature for F = Co. A comparison of ﬁgs 1.8(a) and 1.8(b) shows that the thermal stability of the MR ratio of Ni–Fe–Co based spin-valves increases with increasing Co content, at least in between 4.2 and 293 K. This is also observed above room temperature. We discuss this issue in section 2.5 in more detail. Spin-valves based on spacer layers other than Cu show lower MR ratios, Ag and Au being second-best choices. Fig. 1.9 shows the dependence of the GMR ratio on the layer thickness of Cu, Ag and Au spacer layers in simple spin-valves. In ﬁg. 1.10 we take a closer look at the transfer curves of (Py/Cu/Py/Fe50Mn50 ) SVs that are nominally equal to those discussed above (although from another batch), with tCu = 3 nm (Rijks et al., 1994b). All curves reveal a weak ferromagnetic coupling between the free and pinned layers, with Hcoupl ≈ −0.5 kA/m. Curves (a) and (b)–(d) show the transfer curves for systems with parallel and crossed-anisotropy conﬁgurations, respectively. The easy anisotropy axis of the free permalloy layer is parallel to the direction of the magnetization during its growth. This is due to a certain degree of pair ordering in the otherwise random permalloy alloy (see further section 4.4). The crossed-anisotropy conﬁguration can be obtained in situ during the growth by a 90◦ rotation of the applied ﬁeld just before the growth of the AF layer (curve (b)), or after the growth by an ex situ process of heating to 160 ◦ C and subsequent cooling in a magnetic ﬁeld that is parallel to the required

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15

Fig. 1.8. Dependence of the MR ratio on the free magnetic layer thickness for (F/Cu/F/Fe50 Mn50 ) spin-valves, grown on 3 nm Ta underlayers on Si(100) substrates, at (a) 293 K and (b) 5 K. Symbols: F = Co (squares), F = Ni66 Fe16 Co18 (triangles) and F = Ni80 Fe20 (plus signs). Layer thicknesses: tCu = 3 nm (but tCu = 2.5 nm for F = Ni80 Fe20 ), and tF,p = 5 nm (but tF,p = 6 nm for F = Ni66 Fe16 Co18 ). From Rijks (1996a).

bias direction (curve (c)). Both transfer curves are essentially free of hysteresis, in contrast to curve (a), as expected from the different switching processes (ﬁg. 1.6). Interestingly, the anneal process, which took approximately 10 minutes, is seen to reduce the anisotropy ﬁeld of the permalloy layer. Apparently, the degree of pair ordering is reduced. A similar effect takes place when the curve-(b) spin-valve is given an identical ﬁeld-cool treatment (curve (d)). It is important to be aware of such anneal effects on the transfer curve when designing sensors for high temperature applications, or when planning fabrication using high temperature processing steps (see section 4.4.2). A spin-valve would be electronically equivalent to an AF-coupled multilayer (with Flayers that are a factor of two thinner) if the scattering of electrons at the interfaces with the AF layer and with the cap layer would be specular. However, this is certainly not the case for the prototype (8 nm Ni80 Fe20 / 2.2 nm Cu/ 6 nm Ni80 Fe20 /Fe50 Mn50 ) spin-

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Fig. 1.9. Dependence of the room temperature MR ratio on the spacer layer thickness for spin valves with the layer structure (Si/ 7 nm Co/ tNM nm NM/ 4.7 nm Py/ 7.8 nm Fe50 Mn50 / 1.5 nm NM), with NM = Cu, Ag and Au. From Dieny et al. (1991c).

Fig. 1.10. Low-ﬁeld magnetoresistance curves of (8 nm Py/ 3 nm Cu/ 6 nm Py/ 8 nm Fe50 Mn50 ) SVs, with (a) parallel anisotropies of the free and pinned layer, (b)–(d) crossed anisotropy conﬁguration, obtained by rotating the applied ﬁeld during sputter deposition (b), heating the system to 160 ◦ C and subsequent cooling in a ﬁeld (c), and by the combination of these procedures (d). From Rijks et al. (1994b).

valves that were discussed above. In fact, for these systems the scattering at the Ta/Py and Py/Fe50 Mn50 interfaces is considered to be (almost) completely diffusive. This explains why the MR-ratio is quite small as compared to the MR-ratio for AF-coupled multilayers that are based on the same materials. For these spin-valves, ﬁg. 1.7 shows that ΔR/R is ≈ 5% at the optimum Cu thickness. It would not be much larger for spinvalves with smaller Cu layer thicknesses for which (hypothetically) still a full antiparallel alignment could be obtained (dashed curve, see section 1.4). In contrast, for (1.5 nm Ni80 Fe20 / 1.0 nm Cu)60 multilayers the MR ratio is approximately 20% (Inomata and Hashimoto, 1993). As can be seen from ﬁg. 1.8, the discrepancy is even larger for spin-

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17

valves with smaller permalloy-layer thicknesses. Diffusive boundary scattering limits the distance along the current direction between two consecutive scattering process of spin “+” electrons, thereby reducing the otherwise large spin “+” current in the parallel magnetization state. As a result, the GMR ratio is reduced. This situation is depicted schematically in ﬁg. 1.3(c). The GMR ratio of spin-valves with diffusively scattering outer boundaries can be increased by the introduction of a second pinned layer (dual spin-valves (ﬁg. 1.4(c)). The introduction of a second free layer would not be useful, because of the high spin-independent resistivity of AF layers. The GMR ratio can also be increased by modifying the structure of the outer boundaries in order to enhance the degree of specular reﬂection, e.g., by making use of an oxidic AF layer or of thin “nano-oxide layers” (NOLs). Both improvements have led to advanced spin-valves with GMR ratios that are larger than 20%. We discuss these technologically important developments in section 2. 1.5. Dependence of the MR ratio on the layer thicknesses For the CIP-geometry the physically relevant length scales are the spin-dependent electron mean free paths. The local current density is not proportional to the local conductivity at any point in a layer at which the distance to an interface is smaller than the local mean free path. Ohm’s law is then not applicable on a local scale. The layer thicknesses must therefore be sufﬁciently small. How do the resistance and magnetoresistance depend on the layer thicknesses? In section 3, we discuss a semi-classical transport model that provides an extremely useful starting point for the modelling of the CIP (magneto)conductivity of spin valves and other multilayer systems, developed by Camley and Barnas (CB) (1989). It employs the semi-classical Boltzmann transport equation, and uses the relaxation time approximation to describe scattering (Ashcroft and Mermin, 1976). The electronic structure in all layers is assumed to be identical and spin-independent, and is described within the nearly-free electron model. That implies that (superlattice) band structure effects are neglected. The GMR effect is then entirely due to spin-dependent scattering. The parameters that enter the model are the spin-dependent mean free paths in each of the layers and the probabilities for diffusive and specular scattering at the interfaces and outer boundaries. We will apply the model to spin-valves of various types, discuss its limitations, and discuss theories beyond the CB model. However, even without the application of this and other advanced models we can already learn something about the (spin-dependent) scattering in spin-valves. As shown in this section, useful insight is obtained by analyzing the layer thickness dependence of the MR ratio using some simple phenomenological expressions. 1.5.1. Spacer layer thickness dependence The GMR ratio of spin-valves decreases with increasing non-magnetic spacer layer thickness, tNM , as can be seen in ﬁgs 1.7 and 1.9. The spacer layer should ideally be thin as compared to the electron mean free path, λNM , in that layer. Otherwise the orientation of the magnetization in one of the F layers inﬂuences the current density in the other F layer only weakly, and the GMR ratio is small. The decrease of the GMR ratio with increasing spacer layer thickness is also due to the increasing current shunting by the spacer

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layer. These factors are taken into account by the following simple expression for the tNM dependence of the GMR ratio, ﬁrst proposed by Speriosu et al. (1991): ΔR exp(−tNM /NM ) =A× . R 1 + tNM /a

(1.4)

A(tF ), a(tF ) and NM should be viewed as phenomenological parameters. The factor exp(−tNM /NM ) may be viewed as the effective probability of electron transmission through the NM spacer layer without scattering. Dieny (1994) suggested that NM ≈ 12 λNM , where the factor 12 takes the distribution of electron velocity directions with respect to the current direction into account (see also section 3.6.1). The denominator expresses the shunting effect by the NM layer; a is of the order of the NM layer thickness at which the current through that layer is equal to the current through the F layers. Eq. (1.4) provides a reasonable ﬁt to the experimental data on Py/Cu/Py-based spinvalves in ﬁg. 1.7 (dashed curve), for Cu layer thicknesses for which the MR curves show that full antiparallel alignment has been reached. The ﬁt parameters are A = 7.7%, Cu = 15.8 nm and a = 4.6 nm (Rijks et al., 1994a). Assuming that Cu ≈ 12 λCu , it would follow that λCu ≈ 32 nm. This is close to value λCu ≈ 40 nm that is obtained from the room temperature resistivity of a single crystal using the Drude model (Ashcroft and Mermin, 1976). The small value of a as compared to the total thickness of the two permalloy layers (14 nm) indicates that the current density in the permalloy layers is small as compared to the current density in the Cu layer. This is supported by the results of model calculations within the CB-model (section 3.6.5), which further show that the current through the permalloy layers is not only small due to their large resistivity, but also to the diffusive scattering at the interfaces with the Ta and Fe50 Mn50 layers. 1.5.2. Magnetic layer thickness dependence The MR ratio of SVs, studied as a function of the F-layer thickness, shows a pronounced optimum (see ﬁg. 1.8). When scattering at the outer boundaries is diffusive, the optimum F-layer thickness is determined by the balance between the following two factors. On the one hand, the F layers should be sufﬁciently thick, so that spin-dependent scattering in the interior of the layer stack, and not the diffusive scattering at the outer boundaries, determines the overall conductance. On the other hand, the F layer thickness should not be much larger than the largest of the majority and minority spin electron mean free paths in that layer, λF,> . Otherwise, the orientation of the magnetization in one of the F layers does not inﬂuence the current density in the other F layer at a distance much larger than λF,> from the F/NM interface. The current density in the outer parts of the F layers then shunts the alignment-dependent conductance of the inner part of the trilayer, and the GMR ratio decreases with increasing F layer thickness. Dieny et al. (1992a) have proposed the following phenomenological expression for the dependence of the GMR ratio on the thickness tF,1 of one of the F layers: (1 − exp(−tF,1 /F )) ΔR =B × , R 1 + tF,1 /b

(1.5)

with F ≈ 12 λF,1> (Dieny, 1994). B and b depend on tNM and tF,2 . The denominator describes the shunting effect for thick F layers; b is of the order of the F1 layer thickness

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19

at which the current through that layer is equal to the current through the other layers. From room temperature studies of F/Cu/F/Fe50 Mn50 spin-valves, with F = Co, Py and Ni, Dieny et al. (1992a) obtained F = 7.2, 7.2 and 8.5 nm, respectively. For these materials, the majority-spin mean free paths are thus in the 10–20 nm range at room temperature. Mean free paths depend on the ﬁlm microstructure. However, these are representative values, also obtained from other studies (see section 3.6). In section 3 we show on the basis of theoretical considerations that in Co, permalloy and Ni the mean free path is largest for the majority-spin electrons. The “+” and “−” spin directions in ﬁg. 1.3(c) point then to the right and left, respectively. 1.6. Applications 1.6.1. Read heads The main present application of exchange biased SVs is in hard disk read heads. Fig. 1.11(a) shows the geometry of “shielded” read heads, which are commonly used in hard disks. The MR-element is present in between two soft magnetic shields that conﬁne the length of the region along a track to which the MR element is sensitive to essentially the bit length. For that purpose, the optimal distance between the shields is approximately two times the bit length. In addition to the MR material, the read gap contains two dielectric gap layers that electrically isolate the MR material from the metallic shields. The head is mounted on a slider, which ﬂies at a very small ﬁxed distance above the disk (typically 10 nm) due to the formation of an air ﬁlm that functions as an air bearing in between the head and the slider (Ashar, 1997). Fig. 1.12 shows the growth of the storage density since 1990. Recently, recording at an areal bit density beyond 100 Gbit/inch2 (≈ 150 bits per μm2) has been demonstrated. The bit lengths and track widths are then typically 35 nm and 200 nm, respectively (Zhang et al., 2002). An overview of selected publications on hard disk read heads containing exchange biased SVs is included in table 1.1. The ever smaller bit lengths, track widths and data rates pose a number of great challenges to the development of SV elements. (1) In order to retain a sufﬁcient signal-to-noise ratio (SNR), obtaining SVs with ever larger MR ratios would be very helpful. SVs with nano-oxide specular reﬂection layers (NOLs) and MR ratios above 20% hold, in this respect, a great promise (see section 2.1.7). (2) A second development that can lead to an increase of the SNR is the reduction of the Msat,f tf -product of the free layer. This can be understood as follows. The combination of the MR element and the shields functions as a ﬂux guide. Flux from a magnetic transition in the written track enters the head via the SV, and returns via the shields, or vice versa. The average ﬂux Φ through the SV element (per unit track width) increases with increasing free layer thickness, tf : the head becomes then more efﬁcient. However, the average ﬁeld in the free layer is proportional to Φ/(Msat,f tf ), where Msat,f is the saturation magnetization of the free layer. The net result of these two opposing effects is an increase of the average internal ﬁeld with decreasing Msat,f tf product (Bertram, 1995). The prototype SVs discussed in section 1.5, with tF,f = 8 nm, are in this respect already more advantageous than the 15–25 nm permalloy layer thickness employed in AMR read heads in the early 1990s. Important recent developments are the use of

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Fig. 1.11. Schematic views of the read principle for (a) a shielded head in a hard-disk recording system, and (b) a multitrack yoke-type head in a tape recording system. For the purpose of clarity the bit length shown in (b) is not to scale, and the MRE and contact metalization have in (b) not been drawn for one of the heads.

“spin-ﬁlter SVs” (section 2.1.3) or SVs with NOLs (section 2.1.7), within which tF,f can be as small as 1 to 3 nm, and for which at the same time the MR ratio can be large and the switch ﬁeld range small. (3) The ever decreasing space that is available for the MR material in between the shields restricts the possibilities of using thick AF exchange bias layers (see section 4.6) or other additional layers (see section 2.3).

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Fig. 1.12. Growth in the storage density for magnetic disk drives since 1990. Circles mark the areal bit densities attained in laboratory demonstrations. The target set for 2006 by the U.S. Information Storage Industry Consortium (INSIC), 1 Tbit/inch2 , is indicated by a square. In the late 1990s, commercial products with the same density were available within two years after the laboratory demonstration. In the early 1960s and 1980s, densities were less than 0.1 Mbit/inch2 and around 10 Mbit/inch2 , respectively. From Goss Levi (2002).

(4) A fourth challenge concerns the thermal stability during processing and operation. The ultimate performance of a given read head is determined by the maximum possible sense current. Element heating reduces the MR ratio (section 2.5) and reduces the lifetime as a result of thermally activated degradation processes (section 2.10). Progress on this issue is made by making use of novel exchange bias materials with function up to much higher temperatures than Fe50 Mn50 (section 4.6), by developing deposition methods that lead to an improved microstructure, and by creating an improved thermal environment around the MR element (Ju et al., 2002). The potential for further growth of the bit density, towards a projected target of 1 Tbit/inch2 for 2006 as indicated in ﬁg. 1.12, will depend on continued improvements on the MR materials, as well as on the resolution of challenging issues concerning the magnetic media (Moser et al., 2002). Read heads for tape recording can be shielded heads or so-called “yoke-type” heads (ﬁg. 1.11(b)). In the latter case the MR element is separated from the head-tape interface by a ﬂux guide. This protects the MR element against the mechanical wear and the thermal ﬂuctuations that occur due to the direct head-medium contact. In one type of tape recording, used, e.g., in the analogue Compact Cassette audio system, the tape moves linearly with respect to a stationary read head. Digital tape recording based on the same principle is generally used for backup purposes. As compared to hard-disk recording, the volume bit density is higher and the price per stored bit is lower. However, the areal density is much smaller, with track widths exceeding 10 μm, and the data rate per recording channel is generally much smaller. SV yoke-type heads for such applications have been demonstrated by Philips (see references in table 1.1), and give rise to much larger signal levels than AMRbased heads (Coehoorn, 2000). In so-called helical scan recording, used, e.g., within the analogue VHS video recording system, the tape is wound on a rotary drum on which the

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R. COEHOORN TABLE 1.1 Selected publications on applications of exchange-biased spin-valves

Application

Reference

(a) Read heads for hard disk recording Design, fabrication and testing

Head for perpendicular recording Reliability of SV deposition for read heads Process considerations of critical features Read head based on a dual SV Demonstration 1 Gbit/s head SV head technology Recording demonstration > 100 Gbit/inch2

Tsang et al. (1994, 1998, 1999), Yoda et al. (1996), Nakamoto et al. (1996) Kim et al. (1997) Gurney et al. (1997) Fontana (1999) Yan et al. (1999) Shi et al. (2000) Kanai et al. (2001) Zhang et al. (2002)

(b) Read heads for tape recording Yoke-type heads Modelling of yoke-type heads Shielded heads Shielded heads: comparison AMR and GMR Helical scan recording by shielded heads

Folkerts et al. (1994, 1995) Wei et al. (1997, 1999) Oliveira et al. (1999) Dee (2002) Ozue et al. (2002)

(c) Magnetic ﬁeld sensors Bridge sensor, pinning directions opposite in alternate branches Novel sensor principle using ac bias magnetic ﬁeld Linear displacement sensor Bridge sensor based on SVs with Sy-AF pinned layers with opposite layer thickness unbalance Thermally and magnetically robust bridge sensor Bridge sensor (2 elements inactive) Comparison SV and multilayer bridge sensors Bridge sensor, opposite antiparallel ﬁelds in alternate branches due to use of ﬂux guides Sensors for automotive applications Sensors for string instruments

Spong et al. (1996) Yamane et al. (1997) Miller et al. (1997) Marrows et al. (1999) Lenssen et al. (2000b) Freitas et al. (1999, 2000) Hill (2000) Prieto et al. (2001) Treutler (2001) Lenssen et al. (2002)

(d) SV-MRAMs Submicrometer MRAM cells Differential type MRAM cell MRAM with pinned layer as the storage layer

Tang et al. (1995) Yamane and Kobayashi (1998) Beech et al. (2000)

(e) Other applications Strain sensor Scanning magnetoresistance microscopy MEMS microbridge vibration sensor Biosensors

Mamin et al. (1998) Petrov et al. (1998) Li et al. (2002a) Graham et al. (2003)

heads are present. Digital helical scan recording up to an areal density of 11.5 Gbit/inch2 , using shielded SV heads, has been demonstrated by Sony (Ozue et al., 2002). SV heads are not yet employed in commercial tape recording systems.

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Fig. 1.13. Measurement of the rotation speed of an axis using a GMR sensor. The sensor is positioned between a soft-magnetic gear wheel and a permanent magnet (top part of the ﬁgure), with its sensitivity direction in the tangential direction. A varying position of the teeth of the wheel gives rise to a varying tangential component of the ﬁeld at the sensor position (lower part of the ﬁgure). The number of peaks per unit time of the approximately sinusoidal output signal is a measure for the rotation speed. From Lenssen et al. (2000b).

1.6.2. Magnetic ﬁeld sensors Magnetic ﬁeld sensors are used to measure the size and/or direction of an unknown magnetic ﬁeld, such as in electronic compasses. In another type of application, they are used to measure perturbations on a well-known magnetic ﬁeld, e.g., for contactless measurements of position, velocity and acceleration (see ﬁg. 1.13) and for non-destructive testing of magnetic materials with microcracks. Magnetic ﬁeld sensors should fulﬁll a combination of requirements that depends on the speciﬁc application. Sensors based on various physical principles are therefore in commercial use, including sensors containing Superconducting Interference Devices (SQUIDs), ﬂux-gate devices, Hall-elements, ordinary (semiconductor) MR-elements, AMR-elements, and GMR-elements (Lenz, 1990; Heremans, 1993; Popovic et al., 1996; Tumanski, 2001). Table 1.1 contains references to selected publications on sensors based on exchangebiased SVs. Several groups have developed sensors that consist of a Wheatstone bridge, using various methods for obtaining opposite signals in alternating branches of the bridge. One method is to use spin-valve stripes with pairwise opposite directions of the exchange bias ﬁeld (Coehoorn and van de Walle, 1995; Spong et al., 1996). This can, e.g., be realized by sequentially depositing the opposite bridge elements, in antiparallel ﬁelds, or by post-deposition local heating and subsequent ﬁeld cooling using integrated current leads underneath the sensor stripes. By making use of so-called synthetic AF pinned layers (see section 2.1.4). Lenssen et al. (2000a, 2000b) developed a robust sensor that can withstand high temperatures (>200 ◦ C) and large magnetic ﬁelds (>200 kA/m). Such speciﬁcations

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are relevant for applications in the commercially important market of automotive systems. Magnetic sensors provide more safety by their application in Antilock Brake Systems (ABS) and reduce fuel consumption and polluting gas emissions by their application in motor management systems. In the future they may even be used to replace mechanical by electronic connections (e.g., “steer-by-wire”, using steering actuator position sensors) (Schewe and Schelter, 1997; Treutler, 2001). 1.6.3. Magnetic Random Access Memories (MRAMs) MRAMs are non-volatile magnetoelectronic solid-state memory devices. A matrix of memory cells is deposited in a back-end process on top of a semiconductor CMOS integrated circuit, which is used for writing and readout. Advantages of MRAMs as compared to presently used semiconductor non-volatile ﬂash memories are operation at a low voltage, practically inﬁnite endurance, (in principle) very short (ns-scale) read and write times, and excellent radiation hardness. MRAM cells contain elements which can be reversibly switched between two (meta)stable magnetic states. Tang et al. (1995) have demonstrated MRAM cells based on exchange biased SVs. However, most work on GMR-MRAMs has been based on so-called pseudo-SV cells (Tehrani et al., 1999; Katti and Zhu, 2001). The functioning of SV and pseudo-SV MRAM cells is discussed in section 2.4. Present research on MRAMs focuses entirely on devices based on magnetic tunnel junctions (MTJs), using a matrix type architecture with at each cross point a selection transistor that is in series with a MTJ memory element. Successful demonstrators have been presented by Motorola (Durlam et al., 2002), IBM and Inﬁneon (Reohr et al., 2002), Samsung (Motoyishi et al., 2002) and Sony (Jeong et al., 2002). A review on the present state of the technology has been given by Kim et al. (2002a). MTJs can have a higher MR ratio (typically 50%) than SVs or pseudo SVs (section 2.4), and their relatively high resistance (as compared to that of equally large CIP-GMR elements) is better matched to the openchannel resistance of the selection transistor. 1.6.4. Other applications Stimulated by the research on MRAMs, the introduction of spin-valve GMR technology as a back-end process on top of Si-devices and circuitry is giving rise to various novel applications. Examples are: (1) MEMS (Micro Electro-Mechanical System) microbridge vibration sensors, in which a SV senses the oscillatory stray ﬁeld from a hard magnetic layer on a vibrating Si microbridge on a chip (Li et al., 2002a). (2) Magnetic biosensors. In magnetic biochips, SVs (Graham et al., 2003) or other GMR materials (Baselt et al., 1998; Edelstein et al., 1999; Miller et al., 2001) detect DNA or protein binding to capture molecules in a surface layer by measuring the stray ﬁeld from superparamagnetic label particles. (3) Magnetocouplers (magnetoresistive isolators), which serve for contactless (galvanically isolated) signal transmission between two parts of a circuit that operate at different voltage levels, or in sensitive equipment. Presently, optoelectronic couplers are used for such applications. However, MR isolators are expected to be more easily integrateable on a chip. GMR isolators were ﬁrst developed by Hermann et al. (1997), Fayﬁeld et al. (1999) and Daughton (1999a, 1999b, 2000). Recently they were realized

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

25

using SVs by Ganzer et al. (2003), who used a Wheatstone bridge circuit composed of four identical GMR elements whose resistances can be modiﬁed by the magnetic ﬁelds from currents ﬂowing in overlying strip lines that are not resistively connected. Such a bridge is insensitive to a homogeneous external ﬁeld, but yields an output voltage when the ﬁelds from the strip lines are antiparallel in adjacent legs of the bridge. Such devices can also be used as a transformer with a ﬂat frequency response down to dc. A generalized form of such a circuit, allowing for up to four independent input currents through the strip lines in the bridge, called a “transpinnor”, was made by Bae et al. (2002). Such multiterminal devices can in principle be used as logic gates (AND, OR, etc.) (Torok et al., 2002). SV read heads have been used as scanning probes in “scanning magnetoresistance microscopy” with sub-micrometer resolution of domain patterns in materials research (Petrov et al., 1998). SVs with a magnetostrictive free layer have been demonstrated to function advantageously as strain sensors (Mamin et al. (1998), see section 4.4.3). Some novel GMR device concepts which so far have only been demonstrated using other GMR materials, are expected to be feasible or even to be of more interest when SVs are used. Examples are magnetoelectronic logic (Boolean) gates (Johnson et al., 2000), contactless current detection sensors (Vieth et al., 2000), and bridge sensors for high magnetic ﬁeld measurements (Mancoff et al., 2000). 1.7. Figures of merit The optimal GMR material for a given application depends on a large number of factors. There is never a single ﬁgure of merit. This is illustrated by table 1.2, which gives a (nonexhaustive) overview of property requirements for the speciﬁc case of a magnetoresistive sensor that measures the size of a magnetic ﬁeld. References are given to the sections in this chapter where these properties are discussed. In this section we make a comparison between the sensitivities of various GMR materials (issue 1 in table 1.2), and introduce an “electronic” ﬁgure of merit (issue 5). From ﬁg. 1.14, the room temperature sensitivities of various unpatterned GMR materials can be compared. The ﬁgure gives the switch ﬁeld range as a function of the MR ratio. The switch ﬁeld range is deﬁned as ΔHsw ≡ ΔR/(∂R/∂H ), where ∂R/∂H is taken at the ﬁeld at which it is largest. The thin dashed lines in the ﬁgure connect points of equal sensitivity, deﬁned as S ≡ (1/R) × (∂R/∂H ). The datapoints 1–8 correspond to the following systems. (1) Conventional (Py/Cu/Py/Fe50Mn50 ) simple SVs (section 1.5). (2) State-of-the-art double specular simple SVs with Co90 Fe10 free layers in between a Cu spacer layer and a Cu back layer (section 2.1.7). The switch ﬁeld ranges indicated in the ﬁgure are typical lower limits. ΔHsw depends on the induced anisotropy ﬁeld, Ha , which can in principle be reduced to very low values by growth or post-deposition annealing in a rotating magnetic ﬁeld, and on lateral variations of the interlayer magnetic coupling (section 4.3). However, when Ha is too low, random lateral variations of the anisotropy ﬁeld lead to micromagnetic instabilities that deteriorate the sensor performance.

26

R. COEHOORN

TABLE 1.2 Required properties for spin-valves that are applied in sensors that measure the size of a magnetic ﬁeld. References are given to sections in which these issues are discussed Required property 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17.

High sensitivity S ≡ (1/R) × (∂R/∂H ) in the operating point. Good linear response in the operating point. Small coercivity. Switch ﬁeld range larger than the required dynamic range.a Optimal sheet resistance.b Small temperature dependence of the output signal. Good structural and magnetic thermal stability of the layer stack (during factory processing and lifetime). Good stability upon the application of large magnetic ﬁelds.c Well-controlled effect on the magnetic interactions and (micro)magnetic stability upon lithographic patterning: • Shape anisotropy free layer • Magnetostatic interactions free and pinned layers • Current-induced ﬁeld Good corrosion resistivity.d Low intrinsic electronic noise in the frequency range (bandwidth) of interest. Fast magnetic and electronic response to a change of the applied ﬁeld.e Appropriate direction of the axis of ﬁeld-sensitivity with respect to the sensor structure.f Industrially feasible deposition and patterning processes. Small total thickness of the layer stack.g Small effective Msat t-product of the sense layer.h Small sensitivity to electrostatic discharge (ESD).

18. Good stability under the application of a high sense current.

Reference 1.3, 1.4, sections 2–4 1.4 1.4, 4.2 1.3, 1.4 sections 2 and 3 2.5 2.10, 4.6 2.1.4, 4.1 4.2.2

table 4.7 4.3 4.3 4.1 2.9 1.6.1 1.6.1 Ohsawa et al. (1999), Inage et al. (2000) Gafron et al. (2000)

a The dynamic range is the ﬁeld range around the operating point within the requirements 2 and 3 are fulﬁlled to a sufﬁcient degree. b The optimal sheet resistance is determined by the balance between all factors that contribute to the signal and noise from the sensor, contact leads and ampliﬁer, and by the performance requirements (e.g., maximum sensitivity or minimal power consumption). See table 1.3, which gives the appropriate ﬁgure of merit for some situations of practical interest. c I.e., no irreversible changes after the application of a large magnetic ﬁeld. d This applies to applications where the bare sensor material cannot be sufﬁciently protected, such as hard disk read heads. e E.g., MR elements in future hard disk read heads should enable readout at data rates >1 Gbit/s. f For a stripe-shaped MR element the direction of sensitivity is usually transverse in-plane (as discussed in section 1.3). For certain applications, this may not be very suitable, so that spin-valves with non-standard properties are required. g This requirement applies speciﬁcally to MR materials in shielded read heads. The sensor layer, together with contact leads and insulation layers, should ﬁt in the ever narrower read gap. In today’s heads the gap length is already smaller than 100 nm. h This requirement applies speciﬁcally to MR materials in read heads (see section 1.6.1).

(3) A 20 nm Py AMR ﬁlm. (4) Weakly AF coupled (Py/Ag)n multilayers (Hylton et al., 1993), with a moderate MR ratio. An issue that hampers the application of these highly sensitive materials is that their fabrication requires extreme control of the temperature during an anneal treatment.

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

27

Fig. 1.14. Switch ﬁeld range of various magnetoresistive materials as a function of the magnetoresistance ratio. The thin lines are contours of equal sensitivity. The materials that correspond to the numbered data points have been discussed in the text. AMR-ﬁlms, AF-coupled GMR multilayers and hard–soft multilayers are found in the shaded part of the diagram.

(5) AF-coupled Fe/Cr multilayers (Schad et al., 1994). Like Co95 Fe5 /Cu multilayers (datapoint 6), these multilayers have a very high MR ratio but a relatively small sensitivity due to the high AF coupling ﬁeld. (6) Co95 Fe5 /Cu multilayers (Parkin (2002); ﬁg. 1.1). (7) AF coupled Co90 Fe10 /Cu multilayers (Wang et al., 1997a). The AF coupling is relatively weak due to the use of relatively thick (≈2.5 nm) Cu spacer layers. By varying the F and NM layer thicknesses and compositions, it is in practice possible to obtain AF-coupled multilayers with MR ratios, switch ﬁeld range and sensitivity that are intermediate between those of AMR layers and those of the strongly coupled GMRmultilayer systems 5 and 6. These multilayers are an example. See also Daughton and Chen (1993). (8) (Co/Cu/Py/Cu)n “hard–soft” multilayers (Shinjo and Yamamoto, 1990; Yamamoto et al., 1991). Ideally, these materials contain alternating magnetically hard (here Co) and soft (here permalloy) layers that are not or only very weakly coupled across the non-magnetic spacer layer. However, in practical systems the coercivity of electronically suitable hard magnetic layers (such as Co) is insufﬁcient. It is not possible to obtain sufﬁciently hard layers that function as a reference layer with a fully stable ﬁxed magnetization direction. In ﬁg. 1.14 the shaded area represents the part of the diagram within which single layer permalloy AMR ﬁlms, AF coupled periodic multilayer GMR ﬁlms, or hard–soft multilayers are found. It may be concluded that the sensitivity of state-of-the-art SVs is signiﬁcantly higher than that of such ﬁlms.

28

R. COEHOORN TABLE 1.3 Electronic ﬁgure of merit of thin ﬁlm MR elements

Boundary conditions

Figure of merit, η Thermal ﬂuctuation noise MRE dominates

Fixed sense current Fixed powera Fixed voltageb

1/2

(ΔR/R) × Rsh ΔR/R

−1/2 (ΔR/R) × Rsh

External noise dominates (ΔR/R) × Rsh

1/2

(ΔR/R) × Rsh ΔR/R

a In the case of a stripe-shaped MR element (MRE) from which the dissipated heat is transported transversely, perpendicular to the ﬁlm plane, this is the condition that leads to a ﬁxed maximum temperature rise in the element. b In the case of a stripe-shaped MRE from which the dissipated heat is transported longitudinally to the contact leads, this is the condition that leads to a ﬁxed maximum temperature rise. The in-plane thermal conductivity of the magnetoresistive material is assumed to be proportional to 1/Rsh , as expected from the Wiedemann–Franz law (Kittel, 1996).

A useful “electronic” ﬁgure of merit of MR materials for a certain given device application is obtained by considering the signal-to-noise (SNR) ratio for the case of maximal use of the dynamic range. We consider a stripe-shaped MR element with length L and width W . When the overall system noise is equal to the Johnson–Nyquist electrical thermal ﬂuctuation noise of the MR element (see section 4.3.3),5 the SNR ratio is given by L Isense ΔR ΔR VS Isense × = =√ Rsh , (1.6) VN 4kT W Δf R 4kRT Δf where VS and VN are the signal voltage and r.m.s. noise voltage, respectively, Δf is the band width, Isense is the sense current and Rsh is the sheet resistance. For a given element geometry and operating conditions (T , Δf ) and for a given ﬁxed sense current, an appro1/2 priate ﬁgure of merit is then η = (ΔR/R) × Rsh . Using the same approach, expressions for η have been deduced for ﬁxed power and ﬁxed voltage conditions. See table 1.3. The table also gives expressions for η for the situation in which the overall system noise voltage is entirely due to external sources (e.g., ampliﬁers) and is independent of the properties of the MR element. It may be concluded that one must not only consider the MR ratio but also the sheet resistance when judging the appropriateness of a SV material for a certain application. The following example explains how table 1.3 can be used. For hard disk read heads, the noise is predominantly due to the Johnson–Nyquist noise of the sensor element. Suppose that the sense current should be as large as possible, in order to maximize the signal-tonoise ratio, but that a certain maximum allowed temperature rise due to the dissipated power should not be exceeded. If the heat ﬂow is predominantly “transverse”, towards the shields, the appropriate boundary condition is that of a ﬁxed power dissipation. On the other hand, if the heat ﬂow is predominantly “longitudinal”, towards the contact leads, the 5 Eq. (1.6) applies only to high-frequency devices. For low-frequency devices 1/f noise is predominant (see

section 4.3.3).

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

29

appropriate expression for η coincides with that for the ﬁxed voltage boundary condition (see footnote b in table 1.3). Actual hard disk read heads are intermediate between these transverse and longitudinal cooling regimes (Guo and Yu, 1997). 2. Structure and transport properties of conventional and advanced spin-valves Since the introduction of simple top and bottom exchange-biased spin-valves with the “conventional” layer structures shown in ﬁg. 1.4 (a) and (b), several “advanced” layer structures have been introduced. We ﬁrst give overviews of the structure and transport properties of simple and dual SVs, of SVs with additional layers outside the active part in order to optimize the free layer hysteresis and the offset ﬁeld, and of so-called pseudospin-valves. Subsequently, speciﬁc properties are discussed, including the temperature dependence of the MR ratio, the superimposed AMR effect, the thermoelectric power and the infrared-optical properties. In the ﬁnal part of this section, brief overviews are given of the deposition methods and the resulting microstructure and thermal stability. An analysis of the differences between the magnetoresistances for the various types of SVs and an assessment of the potential for obtaining higher MR ratios will be given at the end of the next section (section 3.10). 2.1. Simple spin-valves The combination of various physically different ideas for optimizing the properties by introducing additional layers has led to investigations of a very large number of different stack sequences. In this subsection we discuss simple spin-valves with various modiﬁed layer sequences within the active part of the layer stack or at its boundaries. The modiﬁcations inﬂuence the current density distribution in the layer stack, and hence the transport properties. We have chosen to base the order of the discussion on the classiﬁcation scheme that is given in ﬁg. 2.1 (a)–(i). In the sequence with letter codes (a) to (i) each structure represents a novel approach. When in a spin-valve various approaches are combined, it belongs to various classes. It will then be discussed under the class given by the highestrank letter code. An overview of representative studies of the structures and properties of simple spin-valves is given in table 2.1. In addition to the MR ratio, ΔR/R, the table gives the sheet resistance, Rsh , and the change of the sheet resistance, ΔRsh . When in the text layer compositions are not given in detail, the reader is referred to the table. Unless stated otherwise, the structures discussed are deposited by sputter deposition. Our convention concerning the notation of chemical compositions is as follows. The composition of random substitutional alloys is indicated by subscripts (e.g., Fe50 Mn50, Co90 Fe10 ) if the composition is given in the original publication or if this (approximate) composition is with great certainty implied in the original publication (which is often the case when the notations FeMn or CoFe are used). The composition of ordered (nominally) 1:1 stoichiometric compounds is given without subscripts (e.g., PtMn). If the alloy composition is not given, or if a statement is made about an entire class of alloys, the constituting elements are separated by a hyphen (e.g., Fe–Mn). The latter notation is frequently used for Ir–Mn materials, because of the difﬁculty to precisely control the Ir:Mn ratio in many deposition systems.

30

R. COEHOORN

Fig. 2.1. Classiﬁcation scheme of conventional and advanced simple spin-valves.

2.1.1. Conventional spin-valves (ﬁg. 2.1(a)) The prototype of a conventional exchange biased spin-valve is the structure Py/Cu/Py/ Fe50 Mn50, introduced by Dieny et al. (1991a) and discussed extensively in section 1. Sputter deposition of Py, Co or Cu on a 2–3 nm Ta underlayer, on a substrate such as Si(100), SiO2 or glass, gives rise to a strong [111] texture. Fe50 Mn50, Cu and fcc-type Fe–Co–Ni alloys have nearly the same lattice constant. In spin-valves based on these materials, the layers grow therefore coherently. Typical average grain diameters are of the order of 15– 30 nm. The microstructure is affected by the sputter deposition conditions, and by the use of appropriate underlayers, as discussed in section 2.9 in more detail. Scattering at grain boundaries is reduced by increasing the average grain size, leading to an enlarged GMR ratio.

System (a) Conventional SVs free/ 2.2 Cu/ 4.7 Py/ 7.8 Fe50 Mn50 / 1.5 Cu

ΔR/R (%)

Rp,sh ( )

ΔRp,sh ( )

14.0 17.1 14.0

3.5 Ta/ 8 Py/ 2.2 Cu/ 6 Py/ 10 Fe50 Mn50 / 3.5 Ta 3.5 Ta/ 2 Py/ 7 Fe50 Mn50 / 6 Py/ 2 Cu/ 8 Py/ 3.5 Ta 5 Ta/ 5 Py/ 2.5 Cu/ 2.4 Co/ 15 Fe50 Mn50 / 5 Ta

4.5 3.5 1.5 9.5 6.5 5 2.5 2.5 4.5 1.4 0.3 0.2 3.0 4.3 3.8 9.0 5.0 4.7 5.2

3.5 Ta/ 8 Py/ 2 Cu/ 6 Py / 8 Fe50 Mn50 / 5 Ta

5.8

Ta/ free/ Cu/ pinned/ 8 Fe50 Mn50

Ta/ Co/ spacer/ Py/ 8 Fe50 Mn50 Ta/ Py/ spacer/ Py/ 8 Fe50 Mn50 5 Hf/ 5 Ni66 Fe16 Co18 / 2 Cu/ 5 Ni66 Fe16 Co18 / 5 Fe50 Mn50 / Hf 3 Ta/ 8 Py/ 2 Cu/ 6 Py/ 8 Fe50 Mn50 / 2 Ta 5 Ta/ 9 Cox Py1−x / 2.8 Cu/ 4 Cox Py1−x / 8 Fe50 Mn50 / 10 Ta

Remarks

Reference

0.63 0.60 0.21

free = 9 Co free = 9 Py free = 9 Ni free, pinned = Co, Co free, pinned = Co, Ni free, pinned = Py, Py free, pinned = Ni, Ni free, pinned = Fe, Py spacer = Au spacer = Ag spacer = Pt spacer = Pd

Dieny et al. (1991c)

14.6

0.63

22.0

1.1

17.5

0.91

12.5

0.72

See ﬁgs. 1.7 and 1.8 x = 0.0 x = 0.95 (full x-dependence) Study sputter conditions Study inverted structure Study thickness tolerances for read-head fabrication Study industrial scale deposition

Dieny (1994)a

Hoshino et al. (1994) Rijks et al. (1996a) Kitade et al. (1995) Kools (1995) Lenssen et al. (1996) Gurney et al. (1997)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

TABLE 2.1 Magnetoresistance and sheet resistance Rp,sh (in the parallel state) of conventional and advanced simple spin valves. The entries in the table are a selection from the literature. Within each category, they are given in chronological order. All materials have been made by sputter deposition, unless indicated otherwise. More details concerning the preparation and properties are given in section 2. Notation: UL = underlayer, BL = non-magnetic “back layer”, NOL = nano-oxide layer. “Ox” indicates than an oxidation step has taken place, A-B-C = binary (ternary) alloy containing elements A, B (and C) with an unspeciﬁed composition. Py = permalloy = Ni80 Fe20 (or an alloy with at most 3 at.% more or less Ni). When in a publication the non-scientiﬁc notations NiFe, CoFe, FeMn or IrMn have been used, these have been replaced by Py, Co90 Fe10 , Fe50 Mn50 and Ir–Mn, respectively (see also introduction to section 2.1). The composition of AF Ir–Mn alloys is generally close to Ir20 Mn80 . Details concerning an anneal treatment have been given if this treatment is not strictly necessary for obtaining the proper exchange bias ﬁeld (see section 4). The resulting properties are then given within square brackets. When for reference purposes structures have been made that deviate from the main structure, the corresponding data are given in between parentheses. It is likely that the experimental accuracy of Rp,sh and ΔRp is in some cases less than is suggested by the notation used. In part, this is due the possible inaccuracies involved when extracting these data from ﬁgures in the original publications. Also, it is likely that the experimental procedures used have in some cases not been optimized for determining these quantities with the precision that is suggested in the table

Schwartz et al. (1998) 31

(continued on next page)

32

TABLE 2.1 (continued) System 5 Ta/ 3.7 Co/ 2 Cu/ 3.2 Co/ 10 Fe50 Mn50 /5 Ta/ 0.5 Cu 5.2 Ta/ 4.4 Co/ 2.5 Cu/ 4.4 Co/ 10 Fe50 Mn50 / 5.2 Ta

ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks 11.2 10.0

11.3

1.13

Reference

Study sputter conditions Comparison sputter methods

Stobiecki et al. (2000) Langer et al. (2001)

x = 0.25 (x = 0) BL = Py, Co, Feb [After anneal at 250 ◦ C; low Hc , high sensitivity] F = Co90 Fe10 (F = Co) F = Co90 Fe10 [After 3 h anneal at 250 ◦ C]. F = (Co90 Fe10 )0.9 B0.1 [After 3 h anneal at 250 ◦ C].

Parkin et al. (1993)

(b) SVs with composite F layers (5.3 – x) Py/x Co/ 3.2 Cu/ x Co/ (2.2 – x) Py/ 9 Fe50 Mn50 / 1 Cu 5 Ta/ 4 Cu/ 8 Fe50 Mn50 / 5 Py/ 2.3 Cu/ 2 Py/BL/ 5 Ta SiO2 / 10 Co–Nb–Zr/2 Py/ 4.9 Co90 Fe10 / 2.8 Cu/ 4.9 Co90 Fe10 / 8 Fe50 Mn50 5 Ta/ 2 Py/5.5 F/ 3.2 Cu/ 5.5 F/15 Fe50 Mn50 / Ta

10 Co–Zr–Nb/2 Py/3 Co90 Fe10 / 3 Cu/ 2 Co90 Fe10 / 9 Ir–Mn/ 5 Ta 2 Ta/ 2 Py/2 Co90 Fe10 / 2.2 Cu/ 2.5 Co90 Fe10 / 5 Ir–Mn/ 2 Ta 5 Ta/ 5 F/2.5 Co90 Fe10 / 2.8 Cu/ 2.5 Co90 Fe10 /6 Ir–Mn/ 5 Ta 5 Ta/ 4 Ni–Fe–Ta/1 Py/1 Co/ 2.4 Cu/ 2 Co /20 PtMn/ 5 Ta 5 Ta/ 5 Py/1 Co/ 2.5 Cu/ 2 Co/20 PtMn/ 5 Ta Ta/ 4.5 Py/0.6 Co90 Fe10 / 2.4 Cu/ 3.2 Co/ AF

Ta/ 20 NiMn/ 2.4 Co90 Fe10 /2.4 Cu/ 1.2 Co90 Fe10 /2 Py/ Ta UL/ 3 Py/1 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 20 PtMn UL/ 2.5 Py/1.5 Co90 Fe10 / 2.8 Cu/ 2.5 Co90 Fe10 / 20 PtMn/ cap UL / 3.2 Py/0.5 Co90 Fe10 / 2.5 Cu/ 1 Co90 Fe10 / 1.9 Co60 Fe40 / 8 Ir–Mn/ 5 Ta

8.0 7.9 (6.2) 7.2 [4.8] 3.0 [4.9] 6.8 [8.5] 9.0 12.3 [9.5] (9.5) 7.5 (7.3) 10.9 9.4 8.9

23.3 18.7 [23.1] (14.7) 16.0 (15.1) 22.9 17.0 16.8

2.1 2.3 [2.2] (1.4) 1.2 (1.1) 2.5 1.6 1.5

9.2 12.0 14.7 11.5 8.0

16.7 16.3 17.4 18.7

2.0 2.4 2.0 1.5

[After 1 h anneal at 250 ◦ C]

Gurney et al. (1993) Kamiguchi et al. (1996) Kanai et al. (1996) Kanai et al. (1997)

Fuke et al. (1997)

Mao et al. (1999b) F = Py89 Ta11 Mizuguchi and Miauchi (1999) [After 5 h anneal at 280 ◦ C] (F = Py) Ni–Fe–Ta contains less Araki et al. (2000) than 15 at.% Ta. AF = 6 Ir–Mn Lin et al. (2000) AF = 25 NiMn AF = 25 PtMn Anneal treatment dependence Chen et al. (2000) Dieny et al. (2000a, 2000b) AlTiC/Al2 O3 substratesa , e Mao and Gao (2000a) UL = 4 Py60 Cr40 Childress et al. (2001) UL = 5 Ta (continued on next page)

R. COEHOORN

5 Ta/ 3.5 Py/4 F/ 3.2 Cu/ 4 F/ 10 Fe50 Mn50 / 10 Ta

5.5 (2.8)

TABLE 2.1 (continued) System

Reference

X = transition metal element; Marrows and Hickey (2001) δ < 0.1 nm; see ﬁg. 2.4.

4.5 for δ = 0

(c) SVs with NM back layer (“Spin-ﬁlter SVs”) 5 Ta/ 4 Cu/ 8 Fe50 Mn50 / 5 Py/ 2.3 Cu/ 2 Py/ BL / 5 Ta 5 Ta/ 2 Cu/ free/ 2 Cu/ 2 Co90 Fe10 / 7 Ir22 Mn78 / 5 Ta t1 Si/ 1.5 Cu/ 4 Py/ 0.5 Co/ 2.5 Cu/ 3 Co90 Fe10 / 8 Ir–Mn/ 3 Ru t2 Si/ 4 Py/0.5 Co90 Fe10 / 2.5 Cu/ 3 Co90 Fe10 / 8 Ir–Mn/ 3 Ru

9.3 8.4 9.8 (8.8) (7.1)

21.5 20.2 15.6 (18.2) (21.8)

2.0 1.7 1.5 (1.6) (1.5)

10.3

0.65

BL = Cu, Cu50 Au50 b Gurney et al. (1993) free = 1.6 Co90 Fe10 Fukuzawa et al. (1998, 2001a) free = 2 Py / 0.5 Co Study effect Si UL; t1 = 3 nm Carey et al. (2002) (t2 = 3 nm) (t2 = 1 nm)

(d) SVs with synthetic antiferromagnetic (Sy-AF) pinned layer 3.5 Ta/ 2 Py/ 10 Ir–Mn/ 4 Co90 Fe10 / 0.8 Ru/ 4 Co90 Fe10 / t Cu/ 0.8 Co90 Fe10 /5 Py/ 4 Ta 50 Fe2 O3 / 2 Co/ 0.7 Ru/ 3 Co/ 2 Cu/ 1 Co/5 Py/ 3 Ta 5 Ta/ 5 Py/ 1 Co/ 2.5 Cu/ 2 Co/ 0.8 Ru/ 2 Co/ 30 Cr45 Mn45 Pt10 / 5 Ta 5 Ta/ 2 Py/ 1.5 Co–Fe–B/ 3 Cu/ 2.5 Co–Fe–B/ 0.8 Ru/ 1.5 Co–Fe–B/ 10 Pt–Pd–Mn/ 6 Ta 5 Ta/ 2 Py/1.5 F/ 3 Cu/ 2.5 F/ 0.8 Ru/ 1.5 F/ 15 Pt32 Pd17 Mn51 / 6 Ta 2 Ta/ 2 Py/2 Co90 Fe10 / 2.2 Cu/ 2.5 Co90 Fe10 / 0.5 Ru/ 2.2 Co90 Fe10 / 5 Ir–Mn/ 2 Ta 3 Ta/ 5 Py/5 Co90 Fe10 / 2.6 Cu/ 2.3 Co90 Fe10 / 0.7 Ru/ 2 Co90 Fe10 / 5 Ir–Mn/ 2 Ta 5 Ta/ 5 Py/1 Co/ 2.5 Cu/ t Co/ 0.9 Ru/ 1.5 Co/20 PtMn/ 5 Ta 9.3 PtMn/ 1.5 Co/ 0.8 Ru/ 2.5 Co/ 2.2 Cu/ 0.3 Co/t Py/ 3 Ta Ta/ 20 NiMn/ 2.2 Co90 Fe10 / 0.9 Ru/ 2.2 Co90 Fe10 / 2.4 Cu/ 1.2 Co90 Fe10 /2 Py/ Ta Ta/ Py/ Co90 Fe10 / Cu/ 2 Co90 Fe10 / 0.6 Ru/ 2 Co90 Fe10 / Ir–Mn/ Ta UL/ 3 Py/1 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 0.8 Ru/ 2.3 Co90 Fe10 / 20 PtMn

7.2 6.3 9.0 6.9 4.0 7.3 8.1 8.5 6.9 6.2 7.6 8.8 6.0 7.7 7.9 9.0

17.8 19.7

1.3 1.6

16.1 14.5

1.0 1.1

15.5

1.4

t = 2.5 nm t = 3 nm Study magnetization reversal

Lenssen et al. (1999a) Lenssen et al. (2000a) Sugita et al. (1999) Meguro et al. (1999)

Study magnetization reversal

Noma et al. (1999)

F = Co–Fe–B Ion beam sputtering

Kanai et al. (1999) Mao et al. (1999b)

Huai et al. (1999) [After 10 h anneal at 250 ◦ C] t = 1.5 nm Araki et al. (2000) t = 3.0 nm t = 2.5 nm Saito et al. (1999), t = 1.3 nm Hasegawa et al. (2000) Chen et al. (2000) Study reversal processes Partial specular reﬂection at the Co90 Fe10 / Ru interface

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

5 Ta/ (2.5 – x) Co/δX/x Co/ 3 Cu/ x Co/ δ X/ (2.5 − x) Co/ 8 Fe50 Mn50 / 2.5 Ta

ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks

Tong et al. (2000) Dieny et al. (2000a, 2000b) 33

(continued on next page)

System 20 Fe2 O3 / 2 Co90 Fe10 / 0.7 Ru/ 2 Co90 Fe10 / 2 Cu/ 0.5 Co90 Fe10 /3 Py/ 3 Ta 1.5 Ta/ 2.5 Py/5 Ir22 Mn78 / 2 Co90 Fe10 / 0.8 Ru/ 2.3 Co90 Fe10 / 2.3 Cu/ 1 Co90 Fe10 /BL/ 3 Ta Ta/ UL/ PtMn/ F/ Ru/ F/ Cu/ F/ Py/ BL/ Ta 5.5 UL/ 1 Py/ 1 Co90 Fe10 / t Cu/ 1.3 Co90 Fe10 / 0.4 Ru/ 1.1 Co90 Fe10 / 7 Ir–Mn / 5 Ta 5.5 UL/ 1 Py/ 7 Ir–Mn/ 1 Co90 Fe10 / 0.4 Ru/ 1.3 Co90 Fe10 / t Cu/ 1 Co90 Fe10 /1 Py/ 5 Ta 3 Ta/ 3.2 Py/ 1.6 Co90 Fe10 / 2.8 Cu/ 3 Co90 Fe10 / 0.7 Ru/ 1.5 Co90 Fe10 /9 Ir–Mn

34

TABLE 2.1 (continued) ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks 9.4

25.5

2.4

7.9 (4.2) 9.5 9.5 9.6 11.9 11.7 9.0

22.8 (26.2) 15.7 28.4 22.9 35.3 26.5

1.8 (1.1) 1.5 2.7 2.2 4.2 3.1

Reference Kawawake et al. (2000)

BL = 1 Cu (no BL) F = Co-alloy Ultrathin Cu: t = 0.95 nm t = 1.6 nm Ultrathin Cu: t = 1.04 nm t = 1.8 nm Study magnetization reversal

Huai et al. (2000) Ueno et al. (2000) Jo and Seigler (2002a) Jo and Seigler (2002b) Park et al. (2002)

55 NiO/ 2.5 Co82 Fe18 / 0.6 Re/ 2.2 Co82 Fe18 / 2.2 Cu/ 3.7 Py/0.6 Re/1 Co82 Fe18 / 2 Ta 1.5 Ta/ 2.2 Py/0.6 Ru/2.5 Py/ 0.5 Co82 Fe18 / 2.5 Cu/ 2.5 Co82 Fe18 / 0.6 Ru/ 2.0 Co82 Fe18 / (4–10) Ir22 Mn78 / cap layer Al2 O3 / 9.3 PtMn/ 1.5 Co/ 0.8 Ru/ 2.5 Co/ 2.2 Cu/ 0.3 Co/3 Py/0.8 Ru/2 Py/ 3 Ta 5 Ta/ 2 Py/0.8 Ru/2 Py/1 Co/ 2.5 Cu/ 2 Co/ 20 PtMn/ 5 Ta 9.3 PtMn/ 1.5 Co/0.8 Ru/2.5 Co/ 2.2 Cu/ 0.3 Co/3 Py/0.8 Ru/2 Py/ 3 Ta

7.3

Veloso et al. (1999)

4.9

Veloso and Freitas (2000)

8.1 7.5 8.0

17.3

1.3

Study Sy-AF and Sy-F coupling Saito et al. (2000) ﬁeld and thermal stability Araki et al. (2000) Hasegawa et al. (2000), Saito et al. (2000)

(f) SVs with an oxidic AF 50 NiO/ 5 Py/ 2 Cu/ 5 Py 50 NiO/ 1 Py/ 4 Co/ 2 Cu/ 0.3 Co/5 Py 30 Ni50 Co50 O/ 3 Py/ 1.5 Co/ 2.2 Cu/ 1.5 Co/6 Py 75 NiO/ 3 Py/ 2 Co/ 1.7 Cu/ 2 Co/5 Py 50 NiO/ 2.5 Co/ 1.9 Cu/ 3 Co 50 NiO/ 2 Co/ 2 Cu/ 3 Co/ 1.2 Cu/ 10 NiO 50 NiO/ 2 Py/ 2 Cu/ 3 Py/ 1.2 Cu/ 10 NiO 50 NiO/ 5 Co/ 2.2 Cu/ 5 Co

4.3 7.5 6.0 13.5 16.2 (19.0) 15.0 5.5 15.0

Study coupling across Cu, interface “dusting” with Co

≈ 33 ≈ 36

≈ 4.9 ≈ 2.0

(O2 in chamber during growth) ΔR/R = 27% at 10 K ΔR/R = 14% at 10 K

Hoshiya et al. (1994) Lin et al. (1994a) Anthony et al. (1994) Egelhoff et al. (1996a, 1997a, 1997b) Swagten et al. (1996, 1998) Kitakami et al. (1996) (continued on next page)

R. COEHOORN

(e) SVs with synthetic ferromagnetic (Sy-F) free layer

TABLE 2.1 (continued) System

50 NiO/ 4 F / 3.2 Cu/ 2 F/5.5 Py/ 10 Ta 50 NiO/ 3 Co90 Fe10 / 2 Cu/ 0.8 Co90 Fe10 /4.2 Ni66 Fe16 Co18 / 1.5 Ta 50 Fe2 O3 / 2 Co/ 2 Cu/ 5 Co/ 0.4 Cu 50 NiO/2.5 Co/ 2 Cu/ 3 Co/ CL NiO/ 2.4 Co/ 2.5 Cu/ 4 Co/ 1 Ta NiO/ 2 F/ 2.6 Cu/ 1.6 F/ 5 nm Si–N 50 Fe2 O3 / 3 F/ 2.3 Cu/ 3.1 F/ 5 Au

7.6 10.0 4.5 6.0 [8.2] 14.0 18.0 16.0 16.7 (16.0) (14.0) 9.7 7.9 5.8 3.6 16.1 5.0

11.8 12.0

1.9 0.6

t = 1 nm t = 0 nm F = (Co90 Fe10 )0.9 B0.1 [After 3 h anneal at 250 ◦ C] Small free layer coercivity α-Al2 O3 (110) substrateg glass substrate CL = 0.5 ML oxygenc (CL = 0.4 nm Au) (no CLc ) Ion Beam Deposition F = Co90 Fe10 F = (Co90 Fe10 )0.95 B0.05 F = (Co90 Fe10 )0.90 B0.10 F = Cog F = Ni80 Fe20 g

Reference Hasegawa et al. (1996) Kanai et al. (1997) Kools et al. (1998b) Sugita et al. (1998), Kawawake et al. (1999) Egelhoff et al. (1999) Slaughter et al. (1999) Shirota et al. (1999) Bae et al. (2000a)

(g)–(i) SV with NOL in pinned layer and/or near free layer 5 Ta/ 1.5 Ru/ 1 Cu/ NOL1/ 1 Cu/ 2 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / NOL1/ 1.5 Co90 Fe10 / 7 Ir–Mn/ 5 Ta 5 Ta/ 2 Py/ 7 Ir–Mn/ 1 Co90 Fe10 F/ NOL1/ 1 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 1 Cu/ NOL2/ 2 Ta 3 Ta/ 1.5 Ru/ 1 Cu/ NOL1/ 1 Cu/ 2 Py/ 1 Co/ 2 Cu/ 2 Co / NOL3/ 1 Co/ 0.9 Ru/ 3 Co–Cr/ 10 PtMn/ 5 Ta 5 Ta/ 2 Py/ 7 Ir–Mn/ pinned layer + NOL1/ 2 Cu/ 2 Co90 Fe10 / 0.4 Co–Fe–O/ 0.4 Ta 7 Ta/ 5 Py/ 9 Ir–Mn/ 1.4 Co90 Fe10 // Ox// 2.5 Co90 Fe10 / 2 Cu/ 2.6 Co90 Fe10 // Ox// 2 Ta 3 Ta/ 15 PtMn/ 1–2 Co90 Fe10 / NOL/ 1–2 Co90 Fe10 / 2 Cu/ 1–2 Co90 Fe10 / NOL/ 3 Ta 3 Ta/ 15 PtMn/ 1–2 Co/ 0.7 Ru/ 1–2 Co/ NOL/ 1–2 Co/ 2 Cu/ 2 (Co90 Fe10 /Py)/ NOL/ 3 Ta

NOL1 = 1 nm Co–Fe–O

18.0 17.0

Kamiguchi et al. (1999a)

NOL2 = 1 nm Cu–Ta–O

17.0

NOL3 = 1 nm CoO Pinned layer and NOL1 not speciﬁed

16.0 13.0

14.6

1.9

14.5

17.9

2.6

thickness NOL = 1.4 nmd

11.3

25.7

2.9

thickness NOL = 1 nmd

Kamiguchi et al. (1999b) Veloso et al. (2000)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

50 NiO/ 5 Co/ 2.2 Cu/ 5 Py 100 Fe2 O3 / 6 − t Py/ t Co/ 2.2 Cu/ t Co/ 9 – t Py/ 5 Ta

ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks

Sakakima et al. (2000a, 2000b)

35

(continued on next page)

36

TABLE 2.1 (continued) System

UL/ Pd–Pt–Mn/ F/ Ru/ F/NOL/ F/ Cu/ 1.5 F/ Cu/ Al2 O3 UL/ Pd–Pt–Mn/ F/ Ru/ F/ 2 Cu/ 1.5 F/ Cu/ Al2 O3 60 NiO/ 5 Py/ 0.7 Co/ 2 Cu/ 7 Py// Ox 2 Ta/ 3 Py/ 7 Ir–Mn/ 2 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 TaOx 2 Ta/ NOL/ 2.5 Py/1 Co90 Fe10 / 2 Cu/ 3 Co90 Fe10 / 7 Ir–Mn/ 2 TaOx

Rp,sh ( )

ΔRp,sh ( )

12.0 (8.0) 11.8 (8.2) 13.0

14.2 (15.0) 15.2 (14.6) 37.0

1.7 (1.2) 1.8 (1.2) 4.8

10.0 (8.1) 13.8 (10.5) (7.7)

42.0 (40.0) 15.9 (16.2) (20.8)

4.2 3.2 2.2 (1.7) (1.6)

11.0 (8.0) 13.0

13.1

1.7

12.1

18.2

2.2

9.8 12.2

27.0 19.5

2.6 2.4

12.2 (8.5) 11.9

27.9 (29.4) 21.8

3.4 (2.5) 2.6

9.0 (5.6) 7.3 (4.9) 13.3

26.7

2.4

22.6

3.0

12.0 (8.6)

28.5 (29.5)

3.4 2.5

Remarks thickness NOL = 1.0 nmd (no NOL) NOL not speciﬁed. (No NOL)

Reference

Araki et al. (2000), Tsuchiya et al. (2000) Gillies et al. (2001)

Lin and Mauri (2001) (no Al2 O3 layers) F = 2.5 Py/ 1.5 Co90 Fe10 (No oxidation, F = 4 Py) Study thermal stability due to Ta/Ni–Fe–Cr underlayer. Ru–O functions as AF coupling layer and as NOL Ion Beam Deposition. Various oxidation methods. NOL = Co–Fe–O or Fe–O NOL = 1–2 nm Co90 Fe10 -O (no NOL) F = Co–Fe–B;a NOL not speciﬁed. (cap = Cu/Taa ) 80 days natural oxidation

Li et al. (2001a) Sugita et al. (2001)

Sant et al. (2001) Uhlig et al. (2001) Hong et al. (2001)

Kim et al. (2001) Gibbons et al. (2001)

NOL = 1–2 nm Co90 Fe10 -O (continued on next page)

R. COEHOORN

3 Ta/ 15 PtMn/ 2 Co90 Fe10 / NOL/ 2 Co90 Fe10 / 2 Cu/ 5 Co90 Fe10 / 3 Ta 5 Ta/ 2 Py/ NOL / 1 Py/2 Co90 Fe10 / 2.5 Cu/ 2 Co90 Fe10 / NOL/ 1 Co90 Fe10 / 10 Ru–Rh–Mn/ 5 Ta 3.5 Ta/ 2 Py/ 8 Ir–Mn/ 3 Co90 Fe10 // Ox// 3 Co90 Fe10 / 2.5 Cu/ 4 Co90 Fe10 // Ox 3.5 Ta/ 2 Py/ 8 Ir–Mn/ 3 Co90 Fe10 / 2.5 Cu/ 4 Co90 Fe10 // Ox (3.5 Ta/ 2 Py/ 8 Ir–Mn/ 3 Co90 Fe10 / 2.5 Cu/ 4 Co90 Fe10 / 3.5 Ta) 3 Al2 O3 / 3 Ni–Fe–Cr/ 1 Py/ 20 PtMn/ 2 Co90 Fe10 / 0.8 Ru/ 2.2 Co90 Fe10 / 2.2 Cu/ 0.9 Co90 Fe10 /2.7 Py/ 0.8 Cu/ 1 Al2 O3 / 6 Tah 3 Ta/ 20 PtMn/ 2 Co90 Fe10 / 0.8 Ru/ 2.2 Co90 Fe10 / 2.2 Cu/ 0.9 Co90 Fe10 /2.7 Py/ 0.8 Cu/ 6 Tah 2 Ta/ 3 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox// 2 Co90 Fe10 / 2.2 Cu/ 2 Co90 Fe10 /F// Ox// 4 Ta UL/ 15 PtMn/ 2 Co90 Fe10 / 0.7 Ru/ 2.5 Co90 Fe10 / 2.5 Cu/ 1 Co90 Fe10 /1 Py/ NOL/ 1 Cu/ 3 Ta UL/ 15 PtMn/ 2 Co90 Fe10 / 0.7 Ru// Ox// 3 Co90 Fe10 / 2.5 Cu/ 2 Co90 Fe10 / 1 Cu/ 3 Ta 2 Ta/ NOL/ 1 Py/2 Co90 Fe10 / 2.4 Cu/ 2 Co90 Fe10 / 6 Ir–Mn/ 2 Ta 5 Ta/ 2 Py/ 7 Ir–Mn/ 2 Co90 Fe10 // Ox// 2 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 1 Cu / 1 Ta 2 Ta/ NOL/ 2.5 Py/1 Co90 Fe10 / 2 Cu/ 3 Co90 Fe10 / 7 Ir–Mn/ 2 TaOx

ΔR/R (%)

TABLE 2.1 (continued) System

3.5 Ta/ 2 Py/ 8 Ir–Mn/ 3 Co90 Fe10 // Ox // 3 Co90 Fe10 / 2.5 Cu/ 4 Co90 Fe10 // Ox // Al 3 Ta/ 2 Py/ 10 PtMn/ 1.5 Co90 Fe10 / 0.8 Ru// Ox// 2 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 TaOx 5 Ta/ 2 Py/ 7 Ir–Mn/ 2 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 Ta UL/ 7 Ir–Mn/ Co90 Fe10 // Ox // 2 Co90 Fe10 / 2.2 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 Ta// Ox//

3 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 2 Cu/ Co90 Fe10 // Ox // 1 Ta 3 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 2 Cu/ 1 Co90 Fe10 /2 Py/1.2 Co90 Fe10 // Ox // 1 Cu/ 1 Ta 3 Ta/ 2 Py/ 6 Ir–Mn/ 3 Co90 Fe10 / 2 Cu/ 1 Co90 Fe10 /2 Py/1 Co90 Fe10 / 1 Cu/ 1 Ta 25 Fe2 O3 / 3 Co/ 2.3 Cu/ 3.1 Co/ 2.5 Ta2 O5 3.5 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2.2 Cu/ 1 Co90 Fe10 / 1 Cu/ Co90 Fe10 // Ox// 3 Ta 3.5 Ta/ 2 Py/ 6 Ir–Mn/ 3.5 Co90 Fe10 / 2.2 Cu/ 4 Co90 Fe10 / 3 Ta 2 Ta/ 2 Py1−x Crx / 12 PtMn/ 1.9 Co90 Fe10 / 0.8 Ru/ 1 Co90 Fe10 // Ox// 2 Co90 Fe10 / 2.1 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 Ta–O 6.7 Ta/ 4.2 Py/ 9 Ir–Mn/ 1.4 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 2.2 Cu/ 4 Co90 Fe10 // Ox // 3 Ta UL/ Co90 Fe10 // Ox // 2.5 Py/1 Co90 Fe10 / 1.9 Cu/ 3 Co90 Fe10/ 7 Ir–Mn/ 2 Ta Ta/ 2.5 Py/1 Co90 Fe10 / 1.9 Cu/ 3 Co90 Fe10 / 7 Ir–Mn/ 2 Ta

10.4 13.1 (6.5) (8.6) 14.0

23.1 (24.6)

14.9

22.1

15.5 (12.0) 17.0 (13.0) (12.9) (9.4) 15.3∗ 13.1∗ 14.2∗∗ (9.0)∗

2.4 (1.6)

20.0 (21.5) (20.2) (22.3)

32.0∗

3.3

3.4 (2.8) (2.6) (2.1)

4.5∗

16.0 14.5 (8.0) 16.1 15.0 (12.5)

22.7 20.1 (21.8)

3.7 3.0 (2.7)

12.5 (5.9) 12.5

28.3

3.5

(8.4)

(30.2)

(2.5)

Reference

Ion-beam sputtered, IBS Magnetron sputtered, MS (no oxygen exposure, IBS) (no oxygen exposure, MS) Al–O via solid state reaction; low-coercivity free layer. very low Hc

Lee et al. (2001)

Study oxidation methods. (No oxidation)

Kools et al. (2001)

Gillies et al. (2001) Mizoguchi and Kano (2001)

Fukuzawa et al. (2001b) (no NOL near free layerf ) (no NOL in pinned layer) (no NOLs)f * From ﬁg. 1 in reference.

Li et al. (2001b)

** From ﬁg. 2 in reference.

High coercivity. MRAM cell.g Study effect of variation thicknesses free layer and Cu back layer Plasma oxidation Natural oxidation No NOL in pinned layer. Study read heads. Study temperature dependence MR Study oxides of Ta, Co90 Fe10 , Py, Py1−x Crx , Cr, Cu, Ta, Nb, Al as NOL; varying Cu layer thickness

Bae et al. (2001a, 2001b) Lu et al. (2002a) Huai et al. (2002)

Sousa et al. (2002)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

3 Ta/ 1 Cu/ 8.5 Ir–Mn/ 1 Co90 Fe10 // Ox // 2.3 Co90 Fe10 / 2.4 Cu/ 1.6 Co90 Fe10 /2.2 Py/ 3.5 Ta

ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks

Mao et al. (2002)

37

(continued on next page)

38

TABLE 2.1 (continued) System 3 Ta/ Co90 Fe10 // Ox// 1.5 Co90 Fe10 / 2.2 Cu/ 3 Co90 Fe10 / 8 Ir–Mn/ 2 Ta 3 Ta/ 2 Py/ 6 Ir–Mn/ 2.5 Co90 Fe10 / 2.2 Cu/ 1.5 Co90 Fe10 /3 Py/ 2 Cu/ Al–O/ 1 Ta 2 Ta/ 2 Py–Cr/ 12 PtMn/ 1.9 Co80 Fe20 / 0.8 Ru/ 1 Co90 Fe10 / NOL/ 2 Co90 Fe10 / 2.1 Cu/ 2 Co90 Fe10 / 1 Cu/ 1 TaOx UL/ 10 PtMn/ (Sy-AF+NOL)/ Cu/ Co90 Fe10 / Cu/ Ta–O

PtMn/ F/ Ru/ F/ Cu/ F/ Cu/ NOL 3.2 Ni–Fe–Cr/ 0.8 Py/ 8 PtMn/ 1.6 Co90 Fe10 / 0.8 Ru/ NOL1/ 2.2 Co90 Fe10 / 1.9 Cu/ 2.2 Co90 Fe10 / 0.6 Cu/ NOL2/ 1 Ta UL/ 10 PtMn/ F/ 0.8 Ru/ 1.8 Co90 Fe10 / 1.9 Cu/ 2 Co90 Fe10 / NOL

10.8

33.3

3.6

8.4

27.4

2.3

16.0

23.1

3.7

17.8

18.5

3.3

15.0 (8.0) 10.1 10.4 14.2 (11.8) 20.5 20.5 20.0

20.1 17.6

4.1 3.6

Reference

Comparison oxides of Py, Al, Ta, Co90 Fe10 , Ta as NOL

Li et al. (2002b)

Plasma oxidation. Study thermal stability. Study head performance and reliability t = 1.5–4 nm. Study structure NOLs. Top NOL: Al–O via solid state reaction. Study NOL as a Mn diffusion barrier Study oxidation in mixed gasses NOL = 1.5 nm FeOx . MR from 0–300 K (No NOL)

Diao et al. (2002)

NOL1 = 0.6 nm air oxidized Co90 Fe10 . NOL2 = 1 nm plasma oxidized Al F layer (high resistivity) and NOL not speciﬁed

Hasegawa et al. (2002) Shen et al. (2002) Jang et al. (2002a) Jang et al. (2002b) Kato et al. (2002) Hong et al. (2002) Tsunekawa et al. (2002) Li et al. (2002c)

a No layer thicknesses given in paper. b Study spin-dependent electron mean free paths. Only ΔG given. sh c Submonolayer oxygen coverage results from partial oxygen pressure p(O ) = 5 × 10−9 Torr during sputter deposition. Reference system (“no cap”) produced in 2 p(O2 ) < 2 × 10−10 Torr. d NOL sputtered from magnetic oxide target. e No comments given in paper on possible role of Al O . 2 3 f 3 nm Ta cap layer. g α-Fe O does not give rise to an exchange bias, but to a large coercivity of the pinned layer. 2 3 h In the publication the thickness of the Cu back-layer is given as “8 nm”. It is assumed that 8 Å (0.8 nm) is the actual thickness.

R. COEHOORN

3.5 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2.2 Cu/ t Co90 Fe10 // Ox // 3 Ta 3.5 Ta/ 2 Py/ 6 Ir–Mn/ 3.5 Co90 Fe10 / 2.2 Cu/ 4 Co90 Fe10 / 3 Ta 5.0 Ta/ 2 Py/ 8 Fe50 Mn50 / 2 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2.6 Cu/ 1.5 Co90 Fe10 /4.5 Py/ 5 Ta 5.0 Ta/ 2.5 Py/ 7 Fe50 Mn50 / 2.5 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 2.6 Cu/ 1.6 Co90 Fe10 / 4.5 Py/ 5 Ta 3.0 Ta/ NOL/ 2 Py/ 1 Co/ 2.2 Cu/ 2 Co/ 15 PtMn/ 3 Ta

ΔR/R (%) Rp,sh ( ) ΔRp,sh ( ) Remarks

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

39

If grown with a strong [111] texture, the exchange bias ﬁeld by Fe50 Mn50 is optimal. Bottom spin-valves based on (111) Fe50 Mn50 can be made by making use of a thin Ta/Py buffer layer (Lenssen et al., 1996). A disadvantage of Fe50 Mn50 as an exchange biasing material is the relatively strong decrease with increasing temperature of the exchange bias ﬁeld. Heb becomes zero at blocking temperatures in the range Tb = 140–190 ◦ C (table 4.3). Several other AF materials show higher blocking temperatures. E.g., Ir20Mn80 random substitutional AF alloys give rise to Tb = 240–290 ◦ C. An advantageous property of this alloy is that, like Fe50 Mn50 , the exchange bias ﬁeld can already be obtained by deposition in a ﬁeld at room temperature. Ordered PtMn compounds, obtained after annealing the asdeposited SVs during 5–20 hours at typically 260 ◦ C, give rise to blocking temperatures in the range 350–400 ◦ C. The use of oxidic AF materials can give rise to an enhanced GMR ratio due to partially specular scattering, as discussed in section 2.1.6. An overview of the preparation and properties of AF exchange bias layers that are suitable for applications in spin-valves is given in section 4.6. A summary is provided by table 4.7. For Py pinned and free layers, the MR ratio obtained at the optimal layer thicknesses is about 5% at room temperature. Early studies in which Py was replaced by Co led to MR ratios around 9% (see table 2.1), however, at the expense of an increased coercivity. The larger MR ratio for Co-based systems, at room temperature and at 4.2 K (ﬁg. 1.8), results from the larger majority-spin mean free path in Co and the larger spin-dependence of the interface scattering (see section 3.6.5). Optimization of the sputter deposition conditions for each of the layers separately has led to an MR ratio above 11% for Co/Cu/Co/Fe50 Mn50 spin-valves (Stobiecki et al., 2000). The magnetoresistance of simple spin-valves in which the same basic structure is repeated is essentially equal to that of a single sequence (Dieny et al., 1991c), because the high resistivity of antiferromagnets such as Fe50 Mn50 (see table 4.7) leads to a very large probability of diffusive scattering of electrons in that layer. 2.1.2. Spin-valves with composite ferromagnetic layers (ﬁg. 2.1(b)) It is often advantageous to make use of composite F layers that are made of two or more strongly ferromagnetically exchange coupled F layers with different compositions. The most important example is the use of composite free layers that consist of a combination of a permalloy sublayer, in order to improve its magnetic softness, and a Co or Co90 Fe10 sublayer at the interface with the Cu spacer layer, in order to obtain an enhanced MR ratio and thermal stability. As demonstrated by Parkin (1993), a very thin Co “dusting” layer at the Py/Cu interfaces in Py/Cu/Py/Fe50Mn50 spin-valves can already lead to a drastic increase of the MR ratio. Fig. 2.2 shows that the MR ratio for spin-valves with the basic structure (5.3 nm Py/ 3.2 nm Cu/ 2.2 nm Py/ 9.0 nm Fe50 Mn50/ 1 nm Cu) can be increased from 2.9% at room temperature, without dusting layers, to 6.4% when only 0.6 nm Co replaces Py at the interfaces. Anthony et al. (1994) found that Co dusting of Py layers can even lead to ΔR/R = 9%, as high as the ratio observed for similar Co/Cu/Co/Fe50 Mn50 spin-valves. Adding Py dusting layers of only 0.4 nm to the Co/Cu interfaces of Co/Cu/Co/Fe50 Mn50 spin-valves was found to decrease the MR ratio from 6.8% to 3.9% (see Parkin (1993) for details of the layer structures). The beneﬁcial effect of Co was seen to be strongly localized at the interfaces. Upon varying the position of 0.5 nm Co layers that were buried in

40

R. COEHOORN

Fig. 2.2. Dependence of the room temperature MR ratio on the thickness of Co “dusting” layer at the interfaces between the F and NM layers, for SVs with the layer structure (Si/ (5.3 – tCo ) nm Py/ tCo nm Co/ 3.2 nm Cu/ tCo nm Co/ (2.2 – tCo ) nm Py/ 9 nm Fe50 Mn50 / 1 nm Cu). From Parkin (1993).

the Py layers no signiﬁcant increase of the MR ratio was found for distances to the Py/Cu interface larger than 0.6 nm. This experiment nicely demonstrated the crucial role played by the interfaces. The effect of Co is twofold (Coehoorn, 1993; Nicholson et al., 1994; Butler et al., 1995; see also sections 3.1 and 3.2). First, the diffusive minority spin scattering at imperfect Co/Cu interfaces is stronger than at equally rough Py/Cu interfaces, due to the larger mismatch of the minority spin d bands. The minority spin 3d-band-ﬁlling for Py and Co is 1.1 and 1.6 electron, respectively, less than for Cu. The scattering potential at Co/Cu interfaces is then larger, leading to a larger scattering probability. Second, one should take into account that magnetic moments at interfaces are not always equal to those in the bulk of the layers. As discussed in section 3.2, Ni magnetic moments at Ni/Cu interfaces are, already for sharp interfaces at T = 0, more strongly decreased by the presence of non-magnetic Cu neighbours than Co moments at Co/Cu interfaces. The negative effect of Cu neighbours on the Ni moments is enhanced by the relatively diffuse character of the interfaces, resulting from the good miscibility of Ni and Cu. Co and Cu are immiscible. In addition, at ﬁnite temperatures thermal ﬂuctuations are larger for Ni moments, reﬂecting the lower magnetic ordering temperature. Consistent with these considerations, Speriosu et al. (1993) observed a substantial decrease of the room temperature magnetization of permalloy near the interface with Cu, effectively corresponding to a “dead layer thickness” of 0.2 ± 0.016 nm. For Co/Cu interfaces, a dead layer thickness of only 0.1 ± 0.02 nm was observed. Theoretical and experimental evidence for a signiﬁcant moment reduction at the Py/Cu interface has also been given by Nicholson et al. (1994). The majority spin partial density of states of magnetic atoms at the interface has then 3d-character at the Fermi level, resulting in enhanced scattering of majority spin electrons at the interface. That reduces the GMR ratio. Gurney et al. (1993) used spin-valves with composite F layers to derive the spindependent mean free paths in Py and Co “back layers” (B) in structures of the type Fe50 Mn50/Py/Cu/Py/B. The Py part of the free layer fulﬁlls the role of a spin-ﬁlter layer. The results and the analysis are given in section 3.6.6. The method was also used for studying non-magnetic back layers. The presence of Co layers at the interface improves the thermal stability. Although that effect is enhanced by making the dusting layer thicker, one is limited by the increase of

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

41

Fig. 2.3. Magnetoresistance curves of (5 nm Ta/ 2 nm Py/ 5.5 nm F/ 3.2 nm Cu/ 5.5 nm F/ 15 nm Fe50 Mn50 ) SVs, with F = Co90 Fe10 (a) or F = Co (b). From Kanai et al. (1996).

the coercivity with increasing Co layer thickness. Kanai et al. (1996) showed that it is more favorable to use Co90 Fe10 instead of Co. As shown in ﬁg. 2.3, the resulting MR ratio is higher for an otherwise similar structure and, even more importantly, the coercivity is much smaller. The authors attributed the effect to the much smaller magnetostriction of Co90 Fe10 , and proved that it was essential that the Py part of the free layer induced a strong [111] texture in the Co90 Fe10 layer. (Co90 Fe10 )1−x Bx boron-containing alloys with x 0.2 are amorphous, and are therefore expected to be magnetically very soft. Attempts to obtain high MR ratios in spinvalves based on such alloys, or on other B-rich amorphous alloys, were unsuccessful (Wu et al., 1998; Feng and Childress, 1999). This is due to the high resistivity for both spin directions. A different approach was introduced by Kamiguchi et al. (1996), who used an amorphous Co–Nb–Zr alloy as a very soft but high resistive underlayer for enhancing the [111] texture of a composite Py/Co90 Fe10 /Cu/Co90 Fe10 /Fe50 Mn50 spin-valve, leading to a low-coercivity (0.01 kA/m) and high-sensitivity (17.8%/kA/m, or 1.4%/Oe). The highest MR ratios within this class of layer-stacks were obtained by Dieny et al. (2000a, 2000a) and Mao and Gao (2000a), viz. 12.0% and 14.7% for PtMn-based SVs. From a modeling study Dieny et al. explained the high MR ratio as the result of partial specular reﬂection (instead of diffusive scattering) at the Co90 Fe10 /PtMn interface (see section 3.6.5). The use of composite pinned layers can be of interest for enhancing the exchange bias ﬁeld, e.g., in the combinations (Co90Fe10 )80 B20 /Py/Fe50 Mn50 (Fujita et al., 1997) or Co90 Fe10 /Co60 Fe40 /Ir–Mn (Childress et al., 2001). 2.1.3. Spin-valves with a non-magnetic back layer (“spin-ﬁlter spin-valves”, ﬁg. 2.1(c)) Spin-valves of the type AF/F/NM/F/B with magnetic and non-magnetic “back layers” (B) were introduced by Gurney et al. (1993), with the purpose to derive the spin-dependent mean free paths in the back layer (see section 3.6.6). Replacement of a part of the free layer with an optimal layer thickness by a thin NM back layer does not lead to a decrease of the GMR ratio, provided that the scattering probability of minority-spin electrons that have crossed the spacer layer is 100% (due to strong interface scattering or due to bulk

42

R. COEHOORN

scattering when the layer thickness is larger than the minority spin mean free path). The F layer acts then as a perfect spin-ﬁlter for electrons that ﬁnally enter the back layer. This has led to the alternative term “spin-ﬁlter spin-valves” (SFSVs). In order to prevent an adverse effect on the GMR ratio by current shunting, typical Cu back layers have a thickness of only 1–1.5 nm. It has been recognized that the use of a NM back layer in spin-valves for sensor applications can give rise to several advantages (Fukuzawa et al., 1998, 2001a; Huai et al., 2000; Ueno et al., 2000): • SFSVs can show a large GMR effect at a very small effective magnetic thickness of the free layer (F layer thicknesses that would be suboptimal when no back layer is used), a property which is advantageous for read head applications (see section 1.6.1). • A very thin Co90 Fe10 free layer in between the Cu spacer layer and a Cu back layer is magnetically relatively soft. Making use of composite Py/Co90 Fe10 free layers for enhancing the softness would then not be necessary, leading to an enlarged thermal stability. • The presence of the Cu back layer shifts the center of the current density distribution away from spacer layer towards the center of the free layer, thereby reducing the sensecurrent induced ﬁeld that acts on the free layer. • When an oxidic specular reﬂection layer is used at the outer boundary (see section 2.1.7) the presence of a Cu back layer prevents oxidation of the free layer. NM back-layers have been studied in combination with a number of other modiﬁcations of conventional simple spin-valves. The experimental data can be found in parts (c), (d), (f), (g) and (h) of table 2.1. Several authors have interpreted observed increases of the MR ratio due to the presence of noble metal cap layers on the free layer in a bottom SV (Egelhoff et al., 1997a) or on an F/NM/F trilayer (Sakakima et al., 1998) as the result of specular reﬂection at these cap layers. However, Wang et al. (2000) criticized this point of view, based on in situ studies of the magnetoconductance of Co/Cu layer systems, and have explained the increase of the MR ratio as a back layer effect. 2.1.4. Spin-valves with a synthetic antiferromagnetic (Sy-AF) pinned layer (ﬁg. 2.1(d)) The magnetic stability of the pinned layer can be enhanced strongly by replacing the pinned layer by an antiferromagnetically coupled F/X/F bilayer, where X is a very thin interlayer that gives rise to strong antiferromagnetic interlayer exchange coupling. Van den Berg et al. (1996, 1997) proposed to replace the F/AF part of a spin-valve by a F/X/F trilayer, which they called “artiﬁcial antiferromagnet” (AAF). We will use the term “synthetic antiferromagnet” (Sy-AF), which has become more common. If the uniaxial magnetic anisotropy of at least one of the layers is strong, the magnetization of a Sy-AF is relatively stable against the application of a magnetic ﬁeld (Zhu, 1999). However, a sufﬁciently large ﬁeld can always be used to induce an irreversible switch between the two equivalent magnetic states of a Sy-AF with uniaxial anisotropy. Such a switch reverses the slope of the R(H ) curve near H = 0. As proposed by Heim and Parkin (1993) and others (see table 2.1), that problem can be solved by combining the Sy-AF with an AF exchange bias layer. In the remainder of this subsection we focus on such Sy-AF + AF spin-valves.

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Fig. 2.4. Layer stack of a SV with a Sy-AF pinned layer (a), its magnetoresistance curve at 298 K (b) and the temperature dependence of the effective exchange bias ﬁeld Heb,eff as compared to the exchange bias ﬁeld of an otherwise identical conventional SV with a single 4 nm Co90 Fe10 pinned layer (c). Figures (b) and (c) are taken from Lenssen et al. (1999b) and Lenssen et al. (2000a), respectively. (Although the pinned layer structures are nominally identical, the effective exchange bias ﬁelds from both publications are slightly different.)

F1 /NM/F2 trilayers in which the two F layers have different coercivities, e.g., by making use of a Sy-AF, are called pseudo-spin-valves. These are discussed in section 2.4. Fig. 2.4 shows some of the results obtained by Lenssen et al. (1999a, 1999b, 2000a). The layer stack is given in ﬁg. 2.4(a). The magnetoresistance curve (ﬁg. 2.4(b)) shows a sharp increase near zero ﬁeld, where the free layer reverses, followed at positive ﬁelds by a plateau where the 4 nm indirectly pinned Co90 Fe10 layer is antiparallel aligned with respect to the free layer. Above H ≈ 25 kA/m there is a wide ﬁeld interval in which the AF coupling across the Ru layer is gradually broken. The magnetization directions of the indirectly pinned “reference layer” and the directly pinned “keeper layer” (following the terminology used by Lin and Mauri (2001)) rotate towards the direction of the applied ﬁeld. The temperature dependence of the effective exchange bias ﬁeld, Heb,eff is given in ﬁg. 2.4(c). Heb,eff is deﬁned as the ﬁeld at which the resistance change has dropped to half the maximum value. For comparison, the ﬁgure also shows the exchange bias ﬁeld Heb,CSV for conventional SVs with a single 4 nm Co90 Fe10 pinned layer. Heb,CSV decreases approximately linearly with temperature to the blocking temperature Tb ≈ 280 ◦ C of the Ir–Mn exchange

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bias layer. Evidently, the use of the Sy-AF strongly increases the effective exchange bias ﬁeld at room temperature and even more at elevated temperatures. The enhanced ﬁeld and thermal stability make these spin-valves excellent candidates for robust magnetic ﬁeld and rotation sensors in, e.g., automotive systems (Lenssen et al., 2000b). A full analysis of the magnetization and transfer curves is given in section 4.1, where ﬁg. 4.6 provides the deﬁnitions of the various critical ﬁelds that can be distinguished. A qualitative explanation of the experimentally observed enhanced thermal stability follows already from the results that are obtained for the limiting case of a large AF interlayer exchange coupling energy, J , as compared to the exchange bias interaction energy, Jeb . J is by deﬁnition equal to half of the energy (per unit area) that is required to change the relative alignment of the magnetization directions of the coupled layers from parallel to antiparallel (eq. (4.6)). The coupling between the directly and indirectly pinned layers is antiferromagnetic, so J < 0. Jeb is equal to half of the energy (per unit area) that is required to change the relative alignment of the pinned layer magnetization and the exchange bias ﬁeld from parallel to antiparallel (eq. (4.4)). By deﬁnition, Jeb > 0. For a system with a fully compensated Sy-AF (two layers with equal Msat t products) and with negligible coupling across the Cu layer, one ﬁnds √ Heb,eff ∼ 2|J | 2/3 1 if |J | Jeb , − (2.1) = Heb,CSV Jeb 2 where Heb,CSV is the exchange bias ﬁeld for a conventional SV with a pinned layer that has the same Msat t product as that of the layers forming the Sy-AF. The enhancement of the width of the high resistance plateau (indicated in ﬁg. 2.4(b)) is in the strong coupling limit given by Hplat ∼ |J | 1/2 1 if |J | Jeb . − (2.2) = Heb,CSV Jeb 2 For the system studied by Lenssen et al. |J |/Jeb increases from ≈7 at room temperature to ≈20 at 200 ◦ C, because the decrease with temperature of Jeb (Co/Ir–Mn) is much stronger than that of J (Co/Ru/Co). The enhancement of the effective exchange bias ﬁeld is therefore even larger at elevated temperatures than at room temperature. In the usual “crossed anisotropy conﬁguration” (section 1.3) the magnetization direction of the pinned layer in a sensor stripe is perpendicular to the long axis. In conventional SVs, this gives rise to a demagnetizing ﬁeld that destabilizes the pinning and that gives rise to a shift of the transfer curve. In contrast, for SVs with a fully compensated Sy-AF pinned layer the demagnetizing ﬁeld is zero. Alternatively, the degree of compensation can be tuned by choosing directly and indirectly pinned layers with different Msat t products. For sufﬁciently narrow stripes this method can be used to obtain a certain desired positive or negative demagnetizing ﬁeld in order to compensate for the interlayer magnetic coupling with the free layer. In a SV with a Sy-AF pinned layer the current density through the directly pinned F layer is usually not determined by the magnetic alignment, because diffusive scattering in AF coupling layers such as Ru is very strong. If the microstructure of the active part is the same as that of a reference conventional spin-valve, the use of a Sy-AF leads thus to a slight decrease of the MR ratio. However, the comparison becomes less straightforward if

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specular scattering occurs at the AF bias layer in a CSV or at the AF coupling layer in the spin-valve with Sy-AF. Dieny et al. (2000a, 2000b) have found indications for both effects (see section 3.6.5). If required, the shunt conduction through the directly pinned F layer can be reduced by making use of an alloy layer with a higher resistivity than that of the reference layer (Li et al., 2002b). An intriguing novel opportunity for enhancing the change of the sheet resistance, ΔRsh , was demonstrated by Jo and Seigler (2002a, 2002b). They were able to fabricate SVs with a Sy-AF pinned layer in which down to Cu thicknesses well below 1 nm the coupling across the Cu spacer layer was predominantly due to the oscillatory interlayer exchange coupling. For bottom SVs with a Cu spacer layer thickness around 1.04 nm, where the coupling is weakly antiferromagnetic, they obtained an MR ratio of almost 12%, and ΔRsh = 4.2 . The results were obtained after an anneal treatment in a magnetic ﬁeld at 220 ◦ C for 2 hours. Excellent control of the layer thickness uniformity and a small interface roughness are likely explanations for the relatively small contributions of the ferromagnetic pin-hole and “orange-peel” coupling (see section 4.5). However, the growth conditions and underlayer structure that have made it possible to realize this result have not yet been disclosed. The AF coupling layer is selected on the basis of the coupling strength (see Parkin et al., 1990; Parkin, 1991b; Saito et al., 2000; Beach et al., 2000) and the thermal stability upon annealing. Most groups consider Ru layers with thicknesses in the range 0.7–0.9 nm as most suitable for the coupling of Co layers or Co90 Fe10 alloy layers. The interlayer coupling parameter J for a (Co90 Fe10 / 0.8 nm Ru/ Co90 Fe10 ) Sy-AF can be as large as −1.5 to −1.7 mJ/m2 (Nagai et al., 1999; Park et al., 2002), although most groups ﬁnd slightly lower values (e.g., J ≈ −0.95 mJ/m2 for (Co/0.8 nm Ru)n multilayers, Bloemen and van Kesteren, 1994). Jeb is typically 0.1 to 0.5 mJ/m2 (see table 4.7). The thermal stability of J has been investigated by Saito et al. (2000). Rh interlayers with a thickness of ≈0.8 nm give rise to large coupling (Parkin, 1991b; Saito et al., 2000), but the good miscibility of Co and Rh leads to a poor thermal stability (Manders et al. 1998). Re coupling layers have been used by Veloso et al. (1999) and Ir coupling has been investigated by Colis and Dinia (2002). 2.1.5. Spin-valves with a synthetic ferromagnetic (Sy-F) free layer (ﬁg. 2.1(e)) Speriosu et al. (1996) proposed to realize more sensitive read heads by using spin-valves with a free layer that consists of an “synthetic ferromagnet” of the form of a F/X/F trilayer. As in a Sy-AF, X is a thin layer such as Ru that couples the two F layers antiferromagnetically. Of course, the trilayer must be ferromagnetic, not antiferromagnetic, so that the free layer remains sensitive to the externally applied ﬁeld. However, by using a Sy-F the Msat t product can be reduced without reducing the physical thickness of the layer that is in contact with the spacer layer, i.e., without a large concurrent decrease of the MR ratio. This provides an alternative for the use of a non-magnetic back layer as a means to reduce the Msat t product (section 2.1.3). A second advantage is the possibility to completely compensate for the shape anisotropy of the free layer in a narrow stripe-shaped SV-element. This is accomplished by using a Sy-F free layer that is formed by F layers with a uniaxial in-plane anisotropy with the easy axes perpendicular to the stripe length (“transverse easy axes”, Coehoorn, 1999).

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2.1.6. Spin-valves with an oxidic antiferromagnet (ﬁg. 2.1(f)) Studies of the magnetoresistance of spin-valves that are based on an oxidic antiferromagnet, such as NiO or Fe2 O3 , have led to the insight that scattering at the outer boundaries is not necessarily diffusive, but can be (partially) specular. Full specular reﬂection at both outer boundaries would make a spin-valve effectively equivalent to a multilayer. The reduction of diffusive boundary scattering leads to an increase of the GMR ratio and to a shift of the optimal F layer thicknesses to lower values. Results of model calculations of these effects are discussed in section 3.6.3. The ﬁrst observations of a shift of the optimal F layer thickness to very low values, in combination with relatively high MR ratios, were obtained from studies by Anthony et al. (1994) of simple and dual NiO-based spin-valves. Egelhoff and coworkers were the ﬁrst to suggest that specular reﬂection could play a role. When studying the optimal F layer thicknesses for NiO/Co/Cu/Co bottom SVs, they found that the optimum GMR ratio was obtained for a slightly thinner bottom Co layer (see table 2.1). Such an effect is expected when scattering at the NiO is partially specular. Strong support for this hypothesis was provided by experimental and modelling studies by Swagten et al. (1996, 1998). They investigated back-layer spin-valves of the type NiO/Co/Cu/Co/Cu/NiO and made a comparison with “standard” all-metal Fe50 Mn50 based spin-valves. The Cu back layer had the function to magnetically separate the top NiO layer from the free Co layer. The NiO layer directly on top of the substrate was deposited at 200 ◦ C to ensure a [111] texture of the layer stack. The remainder of the stack was grown at room temperature to avoid interdiffusion between the separate layers. Fig. 2.5 shows the free layer thickness dependence of the GMR ratio for NiO based and Fe50 Mn50 -based spin-valves, measured at 10 K. Results of a calculation using the Camley–Barnas model (section 3.5), assuming fully reﬂective and fully diffusive scattering at the outer boundaries, are in fair agreement with the experimental data. The highest GMR ratio in this class, 18% at room temperature, has been reported by Sugita et al. (1998) for (Fe2 O3 /Co/Cu/Co/Cu) spin-valves. The 50 nm α-Fe2 O3 AF layer was epitaxially grown on a ﬂat polished (110) α-Al2 O3 single crystal substrate, which was argued to give rise to a very ﬂat Fe2 O3 /Co interface and a high probability of specular scattering. This work strongly emphasizes the potential beneﬁts of improving specular boundary scattering. As observed already by Hasegawa et al. (1996), Fe2 O3 layers do not give rise to an exchange bias. Instead, they induced a high coercivity of the pinned layer: Hc ≈ 100 kA/m in the experiments by Sugita et al. The magnetic structure is thus not stable with respect to high magnetic ﬁelds. Bae et al. (2000a, 2001b, 2001b) proposed to use the coercivity of the pinned layer in α-Fe2 O3 -based spin-valves for applications in magnetoelectronic devices, including an MRAM cell. The functioning of actual devices was demonstrated. Sakakima et al. (1999a) studied spin-valves based on AF perovskites of the type MFeO3 , with M = Y, La, Pr, Nd and Sm. Like α-Fe2 O3 , the perovskites induce coercivity, but no exchange bias. MFeO3 /Co/Cu/Co show MR ratios up to 11.4%. A disadvantage is the high substrate temperature (350 ◦ C) required during the deposition of the perovskite layers. CoO gives rise to a strong exchange bias ﬁeld at temperatures well below the Néel temperature, TN = 18 ◦ C. However, this precludes practical applications at and above room temperature. In principle, the use of an oxidic antiferromagnet is also advantageous because of the absence of current shunting by the AF layer. The advantage can be derived from the ratio

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Fig. 2.5. Closed circles: measured MR ratios of (50 nm NiO/ 2 nm Co/ 2 nm Cu/ tF2 nm Co/ 1.2 nm Cu/ 10 nm NiO) SVs, at 10 K. Closed squares: measured MR ratios of (10 nm Fe50 Mn50 / 2 nm Co/ 2 nm Cu/ tF2 nm Co/ 2 nm Cu) all-metal SVs, at 10 K. Solid curves: MR ratios calculated using the CB model for the system (oxide/ 2 nm Co/ 2 nm Cu/ 2 nm Co/oxide), as a function of the thickness of layer F2 , for specular reﬂection coefﬁcients R = 0, 0.5 and 1 at the interfaces between the F layers and the oxide layers. Dashed-dotted curve: MR ratios calculated using the CB model for the system (10 nm Fe50 Mn/ 2 nm Co/ 2 nm Cu/ tF2 nm Co/ 2 nm Cu), with diffusive scattering at the outer boundaries (R = 0). The mean free paths used in the calculations are λ↑ (Co) = 8 nm, λ↓ (Co) = 2 nm, λ(Cu) = 20 nm and λ(Fe50 Mn50 ) = 1.2 nm, and the interface transmission coefﬁcients ↑ ↓ used are TCo/Cu = 1 and TCo/Cu = 0.2. From Swagten et al. (1996).

of the sheet resistance of the total layer stack, which is usually in the range 15–25 (table 2.1), and the sheet resistance of the AF layer (see table 4.7). Let us assume that the sheet resistance of the active part is 20 , and that the sheet resistance of the AF layer is equal to the value RAF,sh,min = ρAF /tAF,min , given in table 4.7, for an AF layer with the minimal possible thickness for obtaining the exchange bias effect. It follows that shunting by, e.g., Ir–Mn, Fe50 Mn50 and NiMn decreases the MR ratio by typically 3, 11 and 21%, respectively. Early studies of spin-valves based on NiO were primarily motivated by the expected higher thermal stability of the exchange bias ﬁeld as compared that of Fe50 Mn50, in view of the higher Néel temperature (252 ◦ C versus 230 ◦ C, respectively). Indeed, the blocking temperature of NiO based systems was found to be higher than for Fe50 Mn50 based systems (180–230 ◦ C versus 140–190 ◦ C, see table 4.7). The excellent corrosion resistance is another advantage. Kools et al. (1998b) developed NiO based spin-valves with an optimized combination of a high MR ratio (14%), a high sensitivity of 11%/ (kA/m) and a small coercivity (Hc ≈ 0.1 kA/m) by making use of a compound free layer. Several groups have demonstrated hard disk read-heads based on NiO spin-valves (Hamakawa et al., 1996; Nakamoto et al., 1996; Pinarbasi et al., 2000). However, at present AF materials that are thermally even more stable, such as PtMn or Ir20Mn80 , are regarded as more suitable for that application.

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2.1.7. Spin-valves with nano-oxide layers (NOLs) in the pinned layer and/or near the free layer (ﬁg. 2.1(g)–(i)) An extremely fruitful novel concept was developed by Kamiguchi et al. (1999a, 1999b). They introduced very thin specularly reﬂecting oxide layers (“nano-oxide layers”, NOLs) inside the pinned layer and near the free layer of SVs and obtained for such “double specular” SVs enhanced MR ratios up to 18%. NOLs are formed by the oxidation of an already deposited magnetic layer, by the deposition of a (magnetic) oxide layer, or by a solid-state reaction (see below). Across NOLs in the pinned layer a strong exchange coupling should be retained, in order to retain a large effective exchange bias ﬁeld. NOLs near the free layer should preferably be not magnetic, or they should be separated from the free layer by making use of a non-magnetic back layer (ﬁg. 2.1(h)). Non-magnetic NOLs near the free layer can be made of non-magnetic oxides such as the oxides of Cu, Ta or Al. Table 2.1 gives an overview of the structures investigated, and references. A systematic investigation of the effect of NOLs on the MR ratio has been carried out by Fukuzawa et al. (2001b), for spin-valves with the structure underlayer/ Ir–Mn/ Co90 Fe10 / (Co–Fe–O)/ Co90 Fe10 / Cu/ Co90 Fe10 / Cu/ TaOx . The MR ratio for these double specular SVs was 17%. A reference system without NOLs showed an MR ratio of only 9.4%, and reference systems containing only a NOL in the pinned layer or near the free layer showed an MR ratio of only about 13%. Several authors have analysed the experimental resistance and magnetoresistance of SVs with NOLs using the semiclassical Camley–Barnas transport model (Sakakima et al., 2000a; Uhlig et al., 2001; Gibbons et al., 2001 and Lu et al., 2002a). From these analyses the probabilities of specular reﬂection at the NOLs used are found to be typically 0.6–0.85. One of these analyses is discussed in section 3.6.7. MR ratios in the range 17.8 to 20.5% have been reported by Hasegawa et al. (2002) in a regular scientiﬁc publication and by Kamiguchi et al. (1999a), Li et al. (2002c) and Tsunekawa et al. (2002) in a patent application and in conference digests and abstracts. The high value of 20% reported by Li et al. (2002c) for a SV with the structure (PtMn/ F/ Ru/ Co90 Fe10 / 1.9 Cu/ Co90 Fe10 / NOL), with F an unspeciﬁed high resistivity ferromagnetic layer, is remarkable. The system contains only one NOL. Its composition was not disclosed. At the time of writing this chapter, progress in the ﬁeld is fast and may be expected to lead to even higher MR ratios. Spin-valves based on NOLs can provide the combined advantages of the (partially) specular boundary scattering at the interface with an oxide, the high thermal stability of metallic antiferromagnets such as PtMn or Ir20Mn80 , the enhanced magnetic and thermal stability of a SV with a Sy-AF pinned layer and the very small effective free magnetic layer thickness and low free layer coercivity of SVs with a non-magnetic back layer. Moreover, NOLs improve the thermal stability in two additional ways. Firstly, NOLs give rise to a certain planarization of the grain structure, thereby slowing down the high-temperature diffusion of atoms along grain-boundaries (Kamiguchi et al., 1999a, 1999b). A decreased interface roughness of SVs containing NOLs has been observed by Veloso et al. (2000) and by Mizuguchi and Kano (2001). Secondly, NOLs act as a Mn diffusion barrier, as it effectively blocks Mn that diffuses out of Mn-containing AF layers (Kamiguchi et al., 1999a; Gillies et al., 2001; Jang et al., 2002a). That hinders diffusion of Mn atoms into the Cu spacer layer where Mn gives rise to strong diffusive scattering which reduces the GMR ratio. The use of NOLs affects the microstructure of the layer stack in a way that depends on

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the detailed composition and preparation method of the NOL (see, e.g., Mao et al., 2002). This affects the scattering in the bulk of the F layers, the coercivity of the free layer, and the interlayer magnetic coupling across the spacer layer (see, e.g., Lin and Mauri, 2001 and Li et al., 2001b). NOLs in the pinned layer are frequently made by the oxidation of a Co90 Fe10 layer. The preparation process that is used most often is thermal oxidation at room temperature, in highly pure oxygen, an oxygen/noble gas mixture, or in air. At a given exposure time, the end result is determined by the pressure. Oxidation by radicals (Kamiguchi et al., 1999a), by neutral atoms (remote plasma oxidation or atom beam oxidation, Veloso et al., 2000, Kools et al., 2001, Sant et al., 2001, Diao et al., 2002) and by ions (ion beam oxidation, Kools et al., 2001, Sant et al., 2001) have also been used. Alternatively, NOLs have been deposited by sputtering from a magnetic oxide target (Sakakima et al., 2000a). In contrast to results reported by Kamiguchi et al. (1999a), Gillies and Kuiper (2000) found that (at room temperature) oxides formed from pure Co do not couple the F layers in between which they are sandwiched. Oxidation of Co90 Fe10 layers was found to lead to more oxidized Fe than would be expected from the Co:Fe ratio in this alloy, suggesting that Fe segregates to the surface during oxidation. On the other hand, Vanhelmont (2003) found from an XPS study no signiﬁcant preferential oxidation for Co90 Fe10 -based NOLs. The precise composition and crystal structure of Co–Fe–O NOLs is not known. However, Gillies and Kuiper (2000), Uhlig et al. (2001) and Ventura et al. (2002) concluded from the observation of an exchange bias effect that such NOLs are antiferromagnetic below 100–200 K. In almost all cases deposition is followed by an anneal step. Information on the anneal process is not included in table 2.1. The reader is referred to the original publications. In general, the purpose of annealing is to induce crystallographic ordering in the AF layer (PtMn, e.g.), to set the direction of the exchange bias ﬁeld, to modify the microstructure by inducing grain growth, and/or to induce a solid state reaction involving the oxidized layers. An example of the latter situation has been described by Gillies et al. (2001), who created a NOL on top of a Co90 Fe10 sense layer in a bottom SV by ﬁrst oxidizing the Co90 Fe10 layer in air, by subsequently depositing 1.5 nm Al on top of the oxide, and by ﬁnally carrying out a 120 s anneal step at 300 ◦ C. Already upon deposition near room temperature, the Co–Fe–O layer is almost completely reduced by the Al layer. The anneal step was used to ensure that all oxygen is transferred into Al, forming Al2 O3 . Similarly, Shen et al. (2002) observed that a Ta layer on top of Co–Fe–O reduces that layer, giving rise to a Ta2 O5 -NOL. The optimal thickness of a NOL in the pinned layer is determined by a sometimes delicate balance between an increase of the probability of specular scattering with increasing thickness and the concurrent decrease of the effective exchange bias ﬁeld (Gillies et al., 2001; Li et al., 2001b; Huai et al., 2002). Co–Fe–O NOLs give already rise to a signiﬁcant specular reﬂection for ≈0.5–1 nm layer thicknesses. The exchange coupling between the metallic F layers at both sides of the oxide is then still sufﬁcient for obtaining a high resistance plateau in the R(H ) transfer curve. This trade-off is visible in ﬁg. 2.6, taken from Li et al. (2001b). For a double-specular spin-valve containing two NOLs the width of the high-resistance plateau is much smaller than that of a reference SV without NOLs. In this study, the largest MR ratios were found after in situ oxidation at room temperature in pure oxygen during one minute at a pressure of 10−1 Pa. Oxidation at a higher O2

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Fig. 2.6. Magnetoresistance curves for bottom SVs with the structure (3 nm Ta/ 2 nm Py/ 6 nm Ir–Mn/ 3 nm Co90 Fe10 , 2 nm Cu/ 1 nm Co90 Fe10 / 1 nm Cu/ 1 nm Ta) (curve (1)) and a double specular SV with a NOL with the structure (3 nm Ta/ 2 nm Py/ 6 nm Ir–Mn/ 1 nm Co90 Fe10 // Ox // 1.5 nm Co90 Fe10 / 2 nm Cu/ 4 nm Co90 Fe10 // Ox // 2 nm Ta) (curve (2)). From Li et al. (2001b).

pressure, for the same time, was observed to lead to a decrease of the MR ratio. A plateau region with full antiparallel alignment is then not anymore present. The insertion of NOLs is expected to be most effective for stack structures which have a high effective exchange bias ﬁeld in the absence of the NOL, e.g., for SVs with Sy-AF pinned layers. However, also for such SVs the plateau width decreases by the use of a NOL (see, e.g., Huai et al., 2001). For sensor applications, it is important that the free layer has a strong [111] texture to reduce the adverse effect of the cubic magnetocrystalline anisotropy on the coercivity. Gillies and Kuiper (2000) observed from Transmission Electron Microscopy (TEM) that 0.2–0.5 nm NOLs, prepared by thermal oxidation of Co90 Fe10 , do not interrupt the grain structure and thus preserve the crystallite texture. From CPP conduction experiments, Gillies and Kuiper found that such NOLs were incompletely closed discontinuous layers, containing metallic pinholes. A TEM study by Diao et al. (2002) of SVs with a NOL made by thermal oxidation conﬁrmed that such a NOL is indeed discontinuous, and further showed that NOLs made by plasma oxidation are much more continuous. 2.2. Dual spin-valves Dual (or symmetric) spin-valves consist of essentially three F layers, separated by two non-magnetic spacer layers (ﬁg. 1.4(c)). The magnetization of the two outer F layers is pinned by an AF exchange bias layer, whereas the inner F layer is free. In dual SVs diffusive scattering at the outer boundaries plays a less important role than in simple spin-valve with equal layer thicknesses. When scattering at the outer boundaries is (partially) diffusive, dual SVs will thus show a higher MR ratio than otherwise similar simple SVs (see section 3.6.2). This picture is conﬁrmed by the outcome of experimental studies. Table 2.2 gives an overview of selected results from the literature. Noguchi et al. (1994) were the ﬁrst to demonstrate that dual SVs can indeed give rise to a higher MR ratio than otherwise similar simple SVs. However, the MR ra-

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TABLE 2.2 Magnetoresistance and sheet resistance (in the parallel state) of dual spin valves. The entries in the table are a selection from the literature. The notation is identical to that used in table 2.1 ΔR/R (%)Rp,sh ( )ΔRp,sh ( ) Remarks System Reference 5 Hf/ 5 Cu/ 5 Fe50 Mn50 / 5 Ni–Fe–Co/ 2.5 Cu/ 5 Ni–Fe–Co/ 3.7 Noguchi et al. (1994) 2.5 Cu/ 5 Ni–Fe–Co/ 5 Fe50 Mn50 / 5 Cu 50 NiO/ 5 Ni–Fe–Co/ 2.5 Cu/ 5 Ni–Fe–Co/ 2.5 Cu/ 5.6 5 Ni–Fe–Co/ 5 Fe50 Mn50 / 5 Cu (2.2) 5 Hf/ 5 Cu/ 5 Fe50 Mn50 / 5 Ni–Fe–Co/ 2.5 Cu/ 5 Ni–Fe–Co/ 5 Cu Hoshiya et al. (1994) 50 NiO/ 4 Py/ 2 Cu/ 4 Py/ 2 Cu/ 4 Py/ NiO 7.0 ΔHsw = 0.8 kA/m Anthony et al. (1994) 75 NiO/ 3 Py/ 2 Co/ 3.4 Cu/ 4 Co/ 2.4 Cu/ 2 Co/ 3 Py/ 15 Fe50 Mn50 / 2 Py 13.3 50 NiO/ 2.5 Co/ 1.8 Cu/ 4 Co/ 1.8 Cu/ 2.5 Co/ 50 NiO 21.5 Large hysteresis Egelhoff et al. (1995) 50 NiO/ 2.5 Co/ 1.9 Cu/ 4 Co/ 1.9 Cu/ 2.5 Co/ 50 NiO 23.4 21.4 5.0 Egelhoff et al. (1996a, 1997a, 1997b) 24.8 O2 in chamber during 50 NiO/ 2.5 Co/ 3 Cu/ 4 Co/ 3 Cu/ 2.5 Co/ 50 NiO 16.0 22.0 Ff = Co Egelhoff et al. (1996d) 50 NiO/ 2.5 Co/ 1.9 Cu/ 4 Ff / 1.9 Cu/ 2.5 Co/ 50 NiO 17.0 Ff = Co95 Fe5 15.0 Ff = Co90 Fe10 12.0 Ff = Py 15.0 Ff = Co86 Fe10.5 Ni3.5 5.0 Ff = Co85 B15 5 Ta/ 5 Py/ 25 Pt–Pd–Mn/ 2.5 Co90 Fe10 / 2.5 Cu/2.5 Co90 Fe10 / 10.3 Tanaka et al. (1997, 1999) 2 Py/2.5 Co90 Fe10 / 2.5 Cu/ 2.5 Co90 Fe10 / 25 Pd–Pt–Mn 27.8 α-Al2 O3 (110) substrate Sugita et al. (1998), 50 α-Fe2 O3 / 2 Co/ 2 Cu/ 5 Co/ 2 Cu/ 2 Co/ 50 α-Fe2 O3 Sakakima et al. (1999b) UL/ Ir–Mn/ Sy-AF/ Cu/ free/ Cu/ Sy-AF/Ir–Mn/ Ta 13.0 13.8 1.8 UL = Ta/Py/Cu Tong et al. (1999) Sy-AF = Co90 Fe10 /Ru/Co90 Fe10 free = Co90 Fe10 /Py/Co90 Fe10 NiO/ 2.5 Co/ 2.5 Cu/ 4 Co/ 2.5 Cu/ 2.5 Co/ 10 Fe50 Mn50 / 1 Ta 12.1 Ion Beam Deposition Slaughter et al. (1999) 11.8 Lenssen et al. (2000a) 3.5 Ta/ 2 Py/ 6 Ir–Mn/ 4 Co90 Fe10 / 0.8 Ru/ 4 Co90 Fe10 / 2.5 Cu/ 1.6 Co90 Fe10 / 2.5 Cu/ 4 Co90 Fe10 / 0.8 Ru/ 4 Co90 Fe10 / 6 Ir–Mn/ 3.5 Ta 2 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 2 Co90 Fe10 / 2 Cu/ 18.5 Sant et al. (2001) 2.5 Co90 Fe10 / 2 Cu/ 2 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 6 Ir–Mn/ 2 Ta 20.5 Lee et al. (2001) 4 Ta/ 1.2 Cu/ 8.5 Ir–Mn/ 1 Co90 Fe10 // Ox // 2.3 Co90 Fe10 / 2.4 Cu/ (10.6) (no oxygen exposure) 3.0 Co90 Fe10 / 2.7 Cu/ 3 Co90 Fe10 / 7 Ir–Mn/ 3.5 Ta 21.8 19.3 4.2 Li et al. (2002a) 3 Ta/ 2 Py/ 6 Ir–Mn/ 1.5 Co90 Fe10 // Ox // 1.5 Co90 Fe10 / 2.2 Cu/ 1.0 Co90 Fe10 / 2.0 Cu/ 2 Co90 Fe10 // Ox// 1 Co90 Fe10 / 6 Ir–Mn/ 2 Ta 2.0 Dual SV read head. Tong et al. (2002) UL/ Ir–Mn/ t Co90 Fe10 / 0.6–0.8 Ru/ t Co90 Fe10 / 2–2.3 Cu/ t Co90 Fe10 / t = 1.5–2.0 nm 2–2.3 Cu/ t Co90 Fe10 / 0.6–0.8 Ru/ t Co90 Fe10 / Ir–Mn/ CL UL = CL = Ni–Fe–Cr.

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Fig. 2.7. Magnetoresistance curve of dual SVs with the layer structure α-Al2 O3 (110)/ 50 nm α-Fe2 O3 / 2 nm Co/ 2 nm Cu/ 5 nm Co/ 2 nm Cu/ 2 nm Co/ 50 nm α-Fe2 O3 . From Sugita et al. (1998).

tios obtained (at most 5.6%) were lower than for state-of-the-art (at that time) simple spin-valves, probably as a result of the use of relatively thick Cu underlayers and cap layers. For NiO-biased Co/Cu/Co/Cu/Co systems Egelhoff et al. (1995, 1996a) found MR ratios up to 23.4%, versus 17% for reference simple SVs. Growth in a 5 × 10−9 Torr partial oxygen pressure, even led to MR ratios up to 24.8% for dual SVs and 19.0% for reference simple SVs (Egelhoff et al., 1997b). Oxygen was argued to act as a surfactant, suppressing the formation of defects. As mentioned already in section 2.1.6 when discussing simple SVs, Swagten et al. (1996) provided support for the suggestion by Egelhoff et al. that the high observed MR ratio in NiO-biased systems is in part the result of specular scattering at the Co/NiO interfaces. The highest MR ratio reported so far for a simple or dual SV, 27.8%, has been obtained by Sugita et al. (1998) for α-Fe2 O3 -based Co/Cu/Co/Cu/Co systems. For comparable simple SVs the MR ratio was only 18% (see section 2.1.6). Fig. 2.7 shows the magnetoresistance curve. The α-Fe2 O3 layers do not give rise to an exchange bias ﬁeld, but to an enlarged coercivity. No analysis was made of the magnetoconductance. The authors emphasized the possible role of strong specular scattering at the very ﬂat interfaces that were observed from a TEM study. The speciﬁc growth method, sputter deposition on a saphire single crystal (see section 2.1.6), is regarded as being of crucial importance. Double specular dual SVs with two NOLs, with Ir–Mn exchange biased F layers and grown on a technologically relevant substrate, have so far been prepared with MR ratios up to 21.8% (Li et al., 2002a). Hard disk read heads based on dual SVs, with Ir–Mn AF layers and using Sy-AF pinned layers to obtain an enhanced exchange bias ﬁeld and thermal stability, have been demonstrated by Tong et al. (2002).

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Fig. 2.8. Overview of various types of SVs with improved magnetic characteristics by additional layers outside the active part. Dashed lines in ﬁgures (e)–(g) indicate the current stripes in which these materials are used. The current direction is not relevant for structures (a)–(d). Curved arrows in ﬁgs (f) and (g) indicate schematically the direction of the external magnetostatic ﬁeld. IEC = interlayer exchange coupling.

2.3. Spin-valves with improved magnetic characteristics by additional layers outside the active part In the two previous subsections we have discussed various methods for improving the GMR ratio and magnetic characteristics by making use of advanced layer structures that affect the current density distribution within the active part of the layer stack. In this subsection we focus on the application of additional layers outside the active part. Such layers can improve the magnetic characteristics by shifting the offset ﬁeld to zero, or by reducing the coercivity of the free layer. The offset ﬁeld, Hoffset , is the ﬁeld around which the free layer reverses. For a patterned micrometer or nanometer scale device structure Hoffset is equal to the macroscopic coupling ﬁeld, Hcoupl , plus the ﬁeld due to the sample-size dependent magnetostatic interactions with the other layers (see section 4.2.2). Various additional layer stacks have been proposed. An overview is given in ﬁg. 2.8, where for clarity the additional layers have been drawn separately from the basic structure. In the stacks

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Fig. 2.9. Full line: magnetoresistance curve of a SV with the layer structure (3.5 nm Ta/ 2 nm Py/ 10 nm Ir–Mn/ 4.5 nm Co90 Fe10 / 0.8 nm Ru/ 3.5 nm Co90 Fe10 / 3 nm Cu/ 1 nm Co90 Fe10 / 1 nm Py/ 1.6 nm Ta / 2 nm Co90 Fe10 / 0.8 nm Ru/ 1.5 nm Co90 Fe10 / 10 nm Ir–Mn/ 4 nm Ta). Dashed line: magnetoresistance curve for the same layer structure, but without the stabilizing top structure indicated by the italic font. From Lenssen et al. (2001).

shown in ﬁgures (a)–(e) use is made of exchange interactions, and in the stacks shown in ﬁgures (f) and (g) use is made of magnetostatic interactions (as indicated by the schematic ﬂux lines). In all cases, apart from the case shown by ﬁg. 2.8(e), the applied ﬁeld is parallel to the bias direction of the pinned layer in the active part. Compensation for the offset ﬁeld due to a coupling with the pinned layer can be realized by making use of an additional AF exchange bias layer (Lu et al., 2000). The AF layer is set to give rise to a “transverse” exchange bias ﬁeld, i.e., a ﬁeld perpendicular to the easy axis of the free layer (ﬁg. 2.8(a)). A problem is that the exchange bias ﬁeld must be very small, but still reproducible and temperature independent. It is difﬁcult to realize that combination of requirements. Lenssen et al. (2001) proposed to make use of weak interlayer exchange coupling between an exchange biased F layer and the free layer, across an interlayer such as Ta (ﬁg. 2.8(b)). As an even more stable alternative, they proposed to use a pinned Sy-AF (ﬁg. 2.8(c)). The coupling across 2 nm Ta between a (0.8 nm Co90 Fe10 / 3.5 nm Py/ 0.8 nm Co90 Fe10 ) free layer and a pinned Co90 Fe10 layer was shown to give rise to a reproducible offset compensation ﬁeld of about 1 kA/m. Identical layer stacks, but with the additional AF layer set to give rise to a “longitudinal” exchange bias ﬁeld, can be used to reduce the coercivity of the free layer (Lu et al., 2000; Lenssen et al., 2001). Without the effective longitudinal ﬁeld that is applied to the free layer, that layer could reverse upon a reversal of the applied ﬁeld by clockwise or counter-clockwise rotations of the magnetization. Both processes are equivalent, giving rise to the formation of domain walls. Already a small longitudinal ﬁeld is sufﬁcient for elimination of one of these processes, giving rise to perfect single-domain reversal processes. Barkhausen noise is thus suppressed. Fig. 2.9 shows the effect on the hysteresis of the use of a structure such as given in ﬁg. 2.8(d), taken from Lenssen et al. (2001). The authors also showed that hysteresis elimination and offset elimination can be realized simultaneously by setting the exchange bias direction of the top AF layer to an intermediate angle. It should be remarked that a longitudinal bias has also been obtained for conventional SVs in which the exchange bias ﬁeld had a certain longitudinal component (“canted pinning ﬁeld”, Suzuki and Matsutera (1998), Lu et al. (2002b)).

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In these cases the magnetostatic and interlayer coupling between the pinned and free layer was used, respectively. The use of an additional AF layer has the advantage that its functioning is independent of the presence of coupling between the pinned and free layer. Sankar et al. (1999) proposed a novel type of SV of the type AF/F/NM/F/AF, in which the two AF and F layers are identical. The AF-F exchange bias ﬁelds are weak, and directed under ±45◦ angles with respect to the ﬁeld that is to be sensed (ﬁg. 2.8(e)). This is accomplished by cooling from a temperature above the blocking temperature while a current ﬂows through the device, in the presence of an additional external ﬁeld parallel to the current direction. The current gives rise to opposite local ﬁelds in the two F layers, perpendicular to the current direction (see also section 4.2.2). The additional common external ﬁeld makes it possible to obtain the required ±45◦ angles of the local bias ﬁeld direction. The authors only demonstrated a system with ±90◦ angles of the two pinning directions with respect to the external ﬁeld, obtained by cooling in the presence of a current, but without an external ﬁeld. A ±90◦ doubly-biased spin-valve was also discussed by Ambrose et al. (1999). As for the structure given in ﬁg. 2.8(a), the difﬁculties to precisely control the size of the exchange bias ﬁeld seem to limit the applicability of this design. The offset ﬁeld due to the magnetization of the pinned layer can, in a patterned structure, be compensated by using the magnetostatic ﬁeld from an additional ferromagnetic “bias compensation layer”. That layer can be pinned by an AF layer, as show in ﬁg. 2.8(f). Opposite bias directions due to the two AF layers in the stack can be obtained by using AF materials with different blocking temperatures (Mao et al., 2000a). Alternatively, a soft magnetic layer can be used with the easy axis parallel to the longitudinal direction (Kanai et al., 1998; Tong et al., 1999). Such a layer can compensate at the same time for the ﬂux from the pinned layer and for the ﬁeld from the sense current. Childress et al. (2002) proposed to make use of magnetostatic biasing by a longitudinally pinned F layer on top of the element (ﬁg. 2.8(g)). This provides an alternative to the novel options discussed above for longitudinal biasing in order to reduce the coercivity and Barkhausen noise. In present submicrometer sensor elements in a hard disk read head longitudinal biasing is usually accomplished by adding hard magnet layers in the tail regions of the element, outside the track width and shunted by the current leads. 2.4. Pseudo-spin-valves Pseudo-spin-valves are F1 /NM/F2 trilayers within which the magnetic coupling across the NM spacer layer is small as compared to the effective magnetic anisotropy of at least one of the F layers. The F-layers can consist of a single material with different thicknesses in order to realize a different shape anisotropy in a patterned device, or can be composite stacks of strongly exchange coupled magnetic layers. There is no AF exchange bias layer, so that these SVs are outside the scope of this chapter. However, we include a brief discussion on their properties, in order to be able to make it possible to compare their suitability for sensor and memory applications with that of exchange biased SVs. A perfectly antiparallel state can be obtained by using the combination of a hard and a soft magnetic F layer. Such pseudo-SVs may be viewed as the n = 1 version of the hard–soft (Fhard /NM/Fsoft /NM)n multilayers introduced by Shinjo and Yamamoto (1990). Fig. 2.10 shows results of a study by Hütten et al. (2002) of such pseudo-SVs, with the layer structure (glass/ 3.6 nm Py/ 3.5 nm Co/ t Cu/ 3.9 nm Co). The permalloy-containing

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Fig. 2.10. Dependence of the room temperature magnetoresistance curves and the MR ratio on the Cu spacer layer thickness, for (3.6 nm Py/ 3.5 nm Co/ tCu nm Cu/ 3.9 nm Co) pseudo spin-valves. The dashed line gives a ﬁt to the data using eq. (1.4), assuming that for Cu thicknesses at which the MR ratio is peaked a full antiparallel alignment is reached. From Hütten et al. (2002).

bottom F-layer is magnetically softer than the Co F-layer on top of the structure. The MR ratio is maximal at Cu thicknesses t at which the oscillatory interlayer exchange coupling across the Cu layer is most strongly antiferromagnetic, i.e., near t = 1.1, 2.2 and 3.3 nm. Below t ≈ 3.3 nm, the interlayer exchange coupling is the dominant interaction. The MR ratio is only large when the coupling is antiferromagnetic. For t > 3.3 nm, the interlayer coupling is relatively weak as compared to the coercive ﬁeld of the Co layer, so that the MR ratio does not oscillate anymore with t. The trilayers may then be called pseudo-SVs. The MR ratio of approximately 11% for the t = 2.2 nm system is comparable with that of similar exchange-biased spin-valves. Mao et al. (2000b) reported MR ratios up to 14% for double specular pseudo-SVs with the layer structure 3 Cr/ Co–Fe–O/ 3 Co90 Fe10 / 2.3 Cu/ 1 Co90 Fe10 / 3 Ni–Co–Fe/ 1 Co90 Fe10 / TaOx containing two NOLs. The growth on a (110) bcc-Cr underlayer was used to induce a coercive ﬁeld of ≈10 kA/m in the hard bottom Co90 Fe10 layer. The pseudo-SVs discussed above would not be suitable for sensor applications. Firstly, a very moderate ﬁeld of only a 3–10 kA/m would already reverse the slope of the transfer curve near zero ﬁeld. Secondly, Hütten and co-workers found that minor R(H ) loops taken in a small ﬁeld range around H = 0 show a distinct coercivity, and that after repeated cycling through such loops the resistance change decreases irreversibly. This indicates that very stable 360◦ domain walls (walls within which the magnetization vector rotates over 360◦ , between domains with parallel magnetizations) had been formed. It is in practice not possible to obtain a uniaxial anisotropy and coercivity in electronically suitable com-

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positionally homogeneous reference layers that is sufﬁciently high for sensor applications. That would require the presence of a large defect density in order to pin domain walls, whereas at the same time the defect density should be small in order to obtain a large spindependence of the scattering. Obvious candidate materials would be Co1−x Ptx alloys with x ≈ 0.25, as Pt addition to Co is known to give rise to magnetic hardening. However, Irie et al. (1995) found that the coercive ﬁeld of such layers in (Co–Fe–Ni/Cu/Co0.75Pt0.25 /Cu)9 hard–soft multilayers is only 10 kA/m, and that the MR ratio of such multilayers is only about 3%. As discussed already in section 2.1.6, the magnetization of F-layers can be stabilized by making use of a synthetic antiferromagnet (Sy-AF) of the type F/X/F. At least one of the F layers that form the Sy-AF should have a large uniaxial in-plane anisotropy. Leal and Kryder (1998a, 1998b, 1999) studied pseudo-SVs of the type Co/Ru/Co/Cu/Co/Py, where Co/Ru/Co is the Sy-AF, and showed that uniaxial anisotropy in the bottom Co layers can be obtained by growth on a Cr underlayer. Seigler et al. (2002) and Wu et al. (2002) used Co– Cr–Pt permanent magnet underlayers to enhance the magnetic anisotropy of the adjacent magnetic layers in the Sy-AF, and Boeve et al. (2002) used Tb–Co underlayers for the same purpose. Seigler and coworkers pointed out that for some sensor applications the use of pseudo-SVs containing a Sy-AF pinned layer and permanent magnet layers such as Co–Pt–Cr can have the following advantages over an exchange biased SV with an AF bias material such as PtMn: better corrosion resistance, large switching ﬁeld at relatively high temperatures and no high temperature anneal to obtain the ordering that gives rise to the exchange bias ﬁeld. Hard–soft pseudo-SVs can also be used in non-volatile Magnetic Random Access Memories (MRAMs) cells (Tehrani et al., 1999; Katti and Zhu, 2001). The hard and soft layers have the functions of a storage and sense layer, respectively. The status of the bit (“0” or “1”) is determined by the magnetic state of the hard layer, and can be probed by measuring the element resistance as a function of the applied magnetic ﬁeld. As the ﬁelds that are generated by realistic write currents on a chip are small, the coercivity of the hard layer should be small but well deﬁned. A ﬁrst approach would be to make use of pseudo-SVs with a small intrinsic hard layer coercivity, i.e., the coercivity that is measured in an unpatterned sample and that is only determined by the composition and microstructure of the layers. Such pseudo-spin valves were developed by, e.g., Irie et al. (1995), Sakakima et al. (1996, 1997), Tsunashima et al. (1997) and Shirota et al. (1999). A second approach is to make use of pseudo-SV memory elements containing two F layers that are intrinsically soft, but that switch at different ﬁelds due to different shape anisotropies (Everitt et al., 1997). Let us suppose that the elements are smaller than a domain wall width, and that the magnetization reverses like a single domain. For permalloy, the Néel wall width is approximately 0.5–1 μm. For inﬁnitely long 5 nm thick and 1 μm wide permalloy stripes the shape anisotropy ﬁeld is already one order of magnitude larger than the ≈0.4 kA/m intrinsic uniaxial anisotropy ﬁeld. In stripe-shaped memory elements, with widths of a few tenths of a μm or less, the shape anisotropy is thus dominant. A schematic picture of a memory element that functions on the basis of this concept is shown in ﬁg. 2.11(a). Fig. 2.11(b) shows the transfer curve for a memory element based on a (5 nm Ta/ 6.5 nm Py/ 1.5 nm Co90 Fe10 / 3.7 nm Cu/ 1.5 nm Co90 Fe10 / 4.5 nm Py/ 20 nm Ta) pseudo-SV, patterned in the form of a 0.6 μm wide and approximately 2 μm long stripe with tapered ends (Katti et al., 2001).

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Fig. 2.11. (a) Device principle of a memory element based on a pseudo-SV. The full and dashed arrows indicate the magnetization directions in the two binary states. (b) Magnetoresistance curve of a (5 nm Ta/ 6.5 nm Py/ 1.5 nm Co90 Fe10 / 3.7 nm Cu/ 1.5 nm Co90 Fe10 / 4.5 nm Py/ 2 nm Ta) pseudo-SV 0.6 μm wide element. The element length is much longer than the width. The ﬁeld is parallel to the long axis. From Katti et al. (2001). (c) Device principle of a memory element based on an exchange-biased SV. The full and dashed arrows indicate the magnetization directions in the two binary states. (d) Magnetoresistance curves of 6 μm long exchange-biased SV elements with various widths. The SVs contain 9 nm F layers (composition not disclosed) and 2.2–2.5 nm Cu spacer layers. From Tang et al. (1995).

The ﬁeld is parallel to the stripe axis. The thicker bottom and thinner top (Co90 Fe10 /Py) layers function as storage layer and sense layer, respectively. The change of the resistance near H = 0 is due to a switch of the sense layer. The antiparallel state is stabilized by the magnetostatic interaction between both layers, due to the ﬁnite stripe length. The switching of sub-100 nm pseudo-SV elements was studied by Castaño et al. (2001). Micromagnetic modelling studies showed that precise control of the shape of the tapered ends is of crucial importance for obtaining reproducible switch ﬁelds and for preventing nonrepeatable switching by the formation of end domains (Gadbois et al., 1998; Fang and Zhu, 2000). For practical applications, the minimum element size will therefore be signiﬁcantly larger than the minimum feature size of the lithographic process. Alternatively, MRAM cells can be based on exchange-biased spin-valves. Fig. 2.11(c) shows the concept, which was ﬁrst proposed by Tang et al. (1995). The free layer functions as the storage layer. Bistability is obtained by making use of the shape anisotropy of stripeshaped elements width submicrometer widths. The magnetization of the pinned F layer is not changed during the write process. As shown in ﬁg. 2.11(d), the switch ﬁelds become more well deﬁned if the elements become smaller, i.e., when single-domain switching

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Fig. 2.12. Temperature dependence of the MR ratio of SVs biased by Fe50 Mn50 with different combinations of F layer compositions. The active parts of the SVs are: (9 nm Co/ 3 nm Cu/ 3.5 nm Co) (), (8 nm Co/ 3 nm Cu/ 5 nm Py) (), (7.5 nm Py/ 2.2 nm Cu/ 5 nm Py) () and (9 nm Ni/ 2.2 nm Cu/ 5 nm Py) (•). From Dieny et al. (1991d).

instead of switching via domain wall movement becomes dominant. At present, storage elements based on this magnetic concept, but made from magnetic tunnel junctions, are considered as the most promising candidates for MRAM applications (see references in section 1.6.3). 2.5. Temperature dependence of the magnetoresistance Extensive studies of the temperature dependence of the MR ratio of spin-valves have been carried out by Dieny et al. (1991d, 1992a). A very striking result is that for a variety of systems of the type F1 /NM/F2 /Fe50 Mn50 the MR ratio decreases (around room temperature) approximately linearly with increasing temperature. Fig. 2.12 shows some of their results for spin-valves with Cu spacer layers and with F = Ni, Py or Co. The MR ratio was found to extrapolate to zero at a temperature T0 that depends on the composition of the F materials at both sides of the interfaces, but not (or only weakly) on the thicknesses of the layers. T0 is smallest when one of the F layers is Ni, which has the lowest Curie temperature (627 K), and is highest when both F layers consist of Co, which has the highest Curie temperature (1388 K). This general picture has been conﬁrmed by later studies (see table 2.3). Szucs et al. (1997) observed that a 0.8 nm Co “dusting” layer in a Py/Co/Cu/Co/Py/ Fe50 Mn50 spin-valve enhances T0 from 470–500 K (for Py/Cu/Py systems) by approximately 100 K. Thicker (1.5 nm) Co dusting layers led rise to the same value of T0 as that of Co/Cu/Co systems, viz. approximately 700 K. Similarly, Lenssen et al. (2000a) observed for (Ir–Mn/ 4 nm Co90 Fe10 / 5 nm Cu/0.8 nm Co90 Fe10 /5 nm Py) SVs that the 0.8 nm Co90 Fe10 dusting layer led only to T0 ≈ 635 K, which is still signiﬁcantly smaller

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TABLE 2.3 Temperature dependence of the MR-ratio of spin-valves with Cu spacer layers. T0 is the temperature at which, from a linear ﬁt of the (ΔR/R)(T ) curve in the indicated temperature interval [Tmin − Tmax ], the MR ratio would become zero. The systems are ordered according to the combination of materials that is present at the interfaces with the Cu spacer layer. Only the active parts of the SVs are given. The layer thicknesses are in nm System

T0 (K)

Ni/Py 9 Ni/ 2.2 Cu/ 5 Py 420 2.5 Ni/ 2.2 Cu/ 5 Py 375 4.7–20 Ni/ 2.2 Cu/ 5 Py 430–445a Py/Py 7.5 Py/ 2.2 Cu/ 5 Py 470 12.5–41.5 Py/ 2.2 Cu/ 5 Py 500–515a 5 Py/ 3.3 Cu/ 2.5 Py 500 Py/Co 8 Co/ 3 Cu/ 5 Py 600 7.5–43 Co/ 2.2 Cu/ 5 Py 510–540 Py/Co90 Fe10 4–8 Py/2.2 Cu/2 Co90 Fe10 640–690 Co90 Fe10 /Co90 Fe10 2 Py/ 3 Co90 Fe10 / 3 Cu/ 2 Co90 Fe10 690 660 3 Py/ 1.5 Co90 Fe10 / 2.3 Cu/ 2.5 Co90 Fe10 635 4 Co90 Fe10 / 5 Cu/ 0.8 Co90 Fe10 / 5 Py 695 4 Py/ 1.5 Co90 Fe10 / 2.8 Cu/ 2 Co90 Fe10 /1.5 Py Co/Co 9 Co/ 3 Cu/ 3.5 Co 700 5 Py/ 8 Co/ 3.3 Cu/ 8 Co/ 2.5 Py 570 6 Py/ 1.5 Co/ 3.3 Cu/ 1.5 Co/ 3 Py 700 6 Co/ 2.2 Cu/ 5 Co 640 2.5 Py/ 1.5 Co/ 3 Cu/ 1.5 Co/ 5.5 Py 720 4.4 Co/ 2.2 Cu/ 4.4 Co 550(I), 685(II)b

Tmin (K)–Tmax (K) Reference 225–320 80–320 80–320

Dieny et al. (1991d) Dieny et al. (1992a)

200–320 80–320 150–300

Dieny et al. (1991d) Dieny et al. (1992a) Szucs et al. (1997)

80–320 80–320

Dieny et al. (1991d) Dieny et al. (1992a)

300–550

Tanoue and Tabuchi (2001)

300–430 300–450 300–500 300–570

Iwasaki et al. (1997) Mao et al. (2000c) Lenssen et al. (2000a) Zhang et al. (2000)

77–320 50–300 50–300 50–350 300–450 175–325

Dieny et al. (1991d) Szucs et al. (1997) Dieny et al. (1998) Mao et al. (1998) Stobiecki et al. (2000)

a From a second-order polynomial ﬁt. b Samples (I) and (II): different RF-power used for RF-magnetron sputter deposition of the Cu layers.

than the value of T0 ≈ 660–700 K that is commonly observed for SVs that contain thick Co90 Fe10 free and pinned layers. The value of T0 depends on the preparational conditions, as is evident from results obtained by Stobiecki et al. (2000). The approximately linear temperature dependence of the MR ratio around room temperature results from the combined effects an approximately linearly increasing sheet resistance, and a linearly decreasing change of the sheet resistance. See, e.g., Mao et al. (1998). In the limit to 0 K, the temperature dependence of the MR ratio goes to zero (Dieny et al., 1992c). A similar temperature dependence is observed for AFcoupled multilayers. For Co/Cu multilayers, e.g., Suzuki and Taga (1995) observed that below 100 K the temperature dependence of the resistivity is well described by a T 2 power law, as is expected when electron–electron and electron–magnon scattering dominate, and that the temperature dependence of the MR ratio is proportional to T 3/2 . Dieny et al. (1991d) proposed that the relatively small values of T0 as compared to the bulk Curie temperatures may be understood as a result of enhanced spin-ﬂip scattering at

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the interfaces due to the reduced Curie temperature of alloyed interface regions. It should be noted that even in the case of perfectly ﬂat interfaces thermal ﬂuctuations reduce the magnetization in the ﬁrst monolayers more strongly than in the bulk, due to the smaller number of magnetic nearest and next nearest neighbours. The effect of the temperature dependent scattering probabilities and of spin-mixing in the bulk of the layers and at the interfaces has been modelled by Duvail et al. (1995), and has been applied successfully to spin-valves by Dieny (1998) (see section 3.7.2). 2.6. Anisotropic magnetoresistance and anisotropy of the giant magnetoresistance Precise measurements show a small deviation from the cos θ angular response that would be expected from eq. (1.2), and from the linear response that would be expected from eq. (1.3) (Rijks et al., 1994a; Uehara et al., 1996; Sugawara et al., 1997; Johnson et al., 2002). This is due to a small contribution from the anisotropic magnetoresistance (AMR) of the free layer, whose magnetization direction changes with respect to the current direction. For stripe-shaped sensor elements, in which the direction of the magnetization of the pinned layer is perpendicular to the current direction, the resistance can to a good approximation be expressed as ΔRGMR (1 − cos θ ) + ΔRAMR × sin2 θ, (2.3) 2 with θ the angle between the pinned and free layer magnetization directions. For |H − Hcoupl| Ha the transfer curve is then given by: H − Hcoupl ΔRGMR R(H ) = R(H = ∞) + 1− 2 Ha 2 H − Hcoupl + ΔRAMR 1 − (2.4) . Ha R(θ ) = R(θ = 0) +

Schematic transfer curves for different ratios r ≡ ΔRAMR /ΔRGMR are given in ﬁg. 2.13. The AMR effect gives rise to a higher sensitivity when a sensor is used around an operating point Hcoupl < Hop < Hcoupl + Ha than when Hop = Hcoupl. However, it has the adverse effect of spoiling the linearity of the response. Typical GMR and AMR ratios for (8 nm Py/ 2.5 nm Cu/ 6 nm Py/ 8 nm Fe50 Mn50) spin-valves are 4.5% and 1.1%, respectively (Rijks et al., 1994a). It follows from ﬁg. 2.13 that the sensitivity of such spin-valves, as calculated for Hop = Hcoupl + 0.5Ha , is almost 50% higher than expected from the GMR effect only. The AMR/GMR ratio can be reduced to less then 10% by making use of thinner free layers and by replacing Py by Co90 Fe10 (Li et al., 2001c). Specular scattering at oxide layers increases the GMR ratio but also the AMR ratio (Dieny et al., 2000b). As far as I know, no reports are yet available on the AMR contribution to the transfer curves of double specular SVs. Rijks et al. (1993, 1995b) studied the interplay between the GMR and AMR effect in SVs on the basis of the semiclassical Camley–Barnas transport model (see section 3.5). The model was extended to include a microscopic model for the AMR effect in terms of spin and angle dependent electron mean free paths. Within a description using the twocurrent model the AMR effect in Ni-rich Ni–Fe alloys such as permalloy is predominantly

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Fig. 2.13. Dependence on the ratio r ≡ ΔRAMR /ΔRGMR of the magnetoresistance curves for an exchange-biased SV with crossed anisotropies.

due to a dependence of the mean free path of majority-spin electrons on the angle between the electron velocity vector and the local magnetization direction: λ⊥ > λ . The authors predicted that the GMR ratios that are obtained when the pinned layer magnetizations are perpendicular and parallel to the current direction, (ΔRGMR /R)⊥ and (ΔRGMR /R) , respectively, are then, in general slightly different. Eq. (2.3) is thus not entirely correct. Granovskii et al. (1995) arrived at the same conclusion using a less rigorous model. In view of the larger perpendicular mean free path, one would expect a negative anisotropy: (ΔRGMR /R) < (ΔRGMR /R)⊥ . For Py/Cu/Py-based spin-valves with small F and NM layer thicknesses Rijks et al. predicted indeed that (ΔRGMR, − ΔRGMR,⊥ )/RGMR,⊥ can be in the range −0.1 to −0.2%. Experimental observations by Dieny et al. (1996) for Ni/Cu/Py-based SVs provided support for this prediction, and for the predicted trends in the layer thickness dependence of the effect. A positive anisotropy that was predicted for large Py thicknesses was not observed. Miller et al. (1999) found a negative anisotropy for all systems in a large set of different SVs with Py and Co magnetic layers, studied at 4.2 K. The anisotropy of the sheet conductance change, (ΔGsh,GMR,⊥ − ΔGsh,GMR, ), was found to show a strong positive correlation with the sheet conductance change, ΔGsh,GMR, . From an analysis using the semiclassical transport model for the combined GMR and AMR effects that was introduced by Rijks et al. (1995b), Miller and coworkers concluded that the data provided support for a strong spin-dependence of the scattering in the bulk of the magnetic layers. 2.7. Thermoelectric power The thermoelectric power (TEP) of spin-valves is a property of fundamental interest, because its study can lead to further insight in the transport mechanism. It is also of practical interest, because a (time-dependent) temperature gradient over a device can give rise to an offset voltage and to additional noise. As far as is known to the author, there is only one report of a study of the TEP for spin-valves. Sato et al. (1998) studied the TEP of (50 nm NiO/ 3 nm Py/ 2.5 nm Cu/ 6 nm Py/ 1 nm Cu) bottom SVs (type I) and of (3 nm Ta/ 6 nm Py/ 2.5 nm Cu/ 3 nm Py/ 10 nm Fe50 Mn50 / 3 nm Ta) top SVs (type II), with

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MR ratios at room temperature of ≈4% and ≈2.5%, respectively. The TEP was found to vary approximately linearly with temperature T , and was at 300 K approximately equal to S = −10 μV/K for both systems. A magnetothermoelectric power (MTEP) effect was found when reversing the relative alignment of the free and pinned layers. The relative change of the TEP, deﬁned as ΔS = (SP − SAP )/SAP , was −4.5 and −8% for systems I and II, respectively. A temperature difference of 10 K along the sample would thus give rise to a thermoelectric voltage of 0.1 mV, and to a voltage change by the application of a ﬁeld that is one order of magnitude less. In devices that are operated using a small sense current, this contribution to the output voltage can be dominant. It is of interest to remark that qualitatively similar results have been obtained for AFcoupled Co/Cu multilayers, which have a negative thermoelectric power and a negative magnetothermoelectric power (Nishimura et al. (1994) and Shi et al. (1996)). Tsymbal et al. (1999) were able to successfully explain the experimental results for Co/Cu multilayers from ﬁrst principles electronic structure calculations, using the Mott formula: π 2 kB2 T ∂ ln σ . S =− (2.5) 3e ∂E E=EF Here EF is the Fermi energy and σ the conductivity. The elemental charge e is taken positive. Their method for calculating the conductivity is discussed in some detail in section 3.9. No analysis has yet been given for the TEP and MTEP of spin-valves. 2.8. Infrared optical properties – the magnetorefractive effect The spin-dependent conductivity that gives rise to the GMR effect gives also rise to an interesting magnetic ﬁeld effect on the infrared (IR) optical properties. The effect was discovered by Jacquet and Valet (1995), who observed that for hard–soft multilayers the IR transmission depends on the relative alignment of the magnetization directions of the layers. This was found to be a magnetorefractive effect, caused by a dependence of the complex refractive indices (or, equivalently, the εxx and εyy diagonal elements of the dielectric tensor, where x and y are the in-plane directions6 ) on the alignment. Van Driel et al. (2000a) observed the same effect for exchange-biased spin-valves. The relative transmission change for (Si(100)/ 40 nm NiO/ 4.5 nm Py/ t nm Cu/ 8 nm Py) spin-valves is shown in ﬁg. 2.14. The effect shows an interesting wavelength dependence with a layer thickness dependent minimum and a zero-crossing. The magnetorefractive effect can be understood as the result of a spin-dependence of the relaxation times in the F layers. The experimental results are well described within a semiquantitative two-current Drude model for the optical conductivity, introduced by Jacquet and Valet. The model predicts that in the limit of very long wavelengths, the relative transmission change will be positive, and equal to the CIP-GMR ratio. However, long wavelength experiments are very difﬁcult due to the small absolute transmission. In the limit of small wavelengths the intraband (Drude) contribution to the dielectric tensor becomes small, and is dominated by the larger contribution due to interband excitations. From the analysis, van Driel and coworkers obtained a spin asymmetry ratio α = τ↑ /τ↓ = 1.9 ± 0.15 6 The tensor elements ε xx and εyy are, independent of the alignment of the magnetic layers, equal when the

ﬁlm is isotropic in the ﬁlm plane.

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Fig. 2.14. Relative change of the infrared light transmission through (40 nm NiO/ 4.5 nm Py/ tCu nm Cu/ 8 nm Py) spin-valves, for various Cu thicknesses. The room temperature MR ratios are 6.4, 5.0, 4.1 and 2.7% for SVs with 1.6, 2.6, 3.6 and 6.0 nm Cu spacer layers, respectively. The lines are guides to the eye. From van Driel et al. (1999a, 2000b).

for the spin-dependent relaxation times in the permalloy layers. It may be expected that the IR relaxation times depend in a different way on the extrinsic scattering at, e.g., grain boundaries than the ratio of mean free paths that is deduced from the analyses of the CIPGMR effect of spin-valves (section 3.6), or than the ratio of resistivities that is deduced from CPP-GMR experiments (section 3.1). Studies of the IR magnetorefractive effect might therefore be used to provide complementary information about the spin-dependent scattering processes. Just as the magnetorefractive effect is related to the GMR effect, there is a linear magnetic dichroism effect that is related to the AMR effect. This was discovered by van Driel et al. (1999a, 2000a), who found that the transmission of linearly polarized IR light through spin-valves or single ferromagnetic layers depends on the polarization direction with respect to the magnetization direction of the layer(s). When the magnetization is along the x-axis, the εxx and εyy dielectric tensor elements are different. Like the AMR effect, this results from the anisotropy of the relaxation times. The effect depends also on the spin-dependence of the relaxation times. An analysis by van Driel and coworkers of the wavelength dependence of the transmission change for permalloy ﬁlms with variable layer thicknesses led to a spin-asymmetry ratio in the range α = 1.5–2.5. This is consistent with the result that was obtained independently from the magnetorefractive effect. 2.9. Deposition and microstructure Ideally, the microstructure of spin-valves should give rise to (1) A long mean free path for electrons of one type of spin (majority spin electrons for the commonly used fcc-type Ni–Co–Fe based SVs). (2) Strong scattering in the bulk of the F layers or at the interfaces for electrons with opposite spins.

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Fig. 2.15. Transmission Electron Microscopy cross-sectional image of a (Si(100)/ 3 nm Ta/ 8 nm Py/ 2.8 nm Cu/ 6 nm Py/ 10 nm Fe50 Mn50 / 5 nm Ta) spin-valve. From Rijks (1996a).

(3) Strongly specular scattering if specular reﬂection layers are present. (4) Optimized and well-controlled magnetic interactions. (5) A high thermal stability. The presence of defects in the bulk of the layers, such as vacancies, dislocations and grain boundaries, reduces the electron mean free paths, and gives potentially rise to a reduction of the energy barriers for processes that contribute to the thermally activated diffusion of atoms. The interface structure (ﬂat, rough or interdiffused) also inﬂuences the mean free paths, and moreover determines in part the strength of magnetostatic and exchange interactions between the layers. We discuss in this section various factors which inﬂuence the microstructure. 2.9.1. Lattice matching It is a fortunate situation that the F and NM materials that are most suitable from the point of view of their electronic properties, such as Co, fcc-type binary or ternary Co– Fe–Ni alloys, and Cu are very well lattice matched. The lattice constants are very similar, leading to coherent growth. This can be seen from the cross-sectional Transmission Electron Microscopy (TEM) micrograph shown in ﬁg. 2.15 for a Py/Cu/Py/Fe50 Mn50 spinvalve. The columnar grains extend through the Py, Cu and Fe50 Mn50 layers. The average grain size is 15–30 nm. The Si substrate and the Ta layers can be clearly distinguished, but the other layers are indistinguishable due to the small atomic number difference of the elements concerned. Grain-to-grain epitaxy in Py/Fe50 Mn50 bilayers had been observed earlier in a TEM study by Hwang et al. (1988). Lattice mismatch results in defects such as dislocations, at which scattering can take place. If in a structurally perfect system scattering (e.g., electron–phonon scattering) is already very strong for electrons of one spin, additional scattering at defects will (also) shorten the mean free path of the electrons of the opposite spin. The GMR ratio is then reduced, irrespective of the spin-dependence of the additional scattering (e.g., as a result of a spin-dependent density of ﬁnal states). Better lattice matching can therefore explain the observation that the GMR ratio of Co/Cu/Ni80 Fe20 -based spin-valves is larger than that of spin-valves containing the electronically very similar noble metals Ag or Au (ﬁg. 1.8).

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The mismatch between the cubic lattice constants of Ni80 Fe20 or Co with that of Cu is only approximately 1 and 2%, respectively. In contrast, the mismatch with Ag or Au is approximately 15%. However, also for spin-valves with Cu spacer layers scattering at grain boundaries cannot be neglected, as was concluded already by Dieny (1992b) and Dieny et al. (1993) (see section 3.6.5). 2.9.2. Layer thickness effect on the grain size The average grain size in thin ﬁlms tends to increase with increasing ﬁlm thickness. Grains with an energetically more favorable orientation tend to grow faster, at the expense of other grains. If that is the case, the effective mean free path increases with increasing ﬁlm thickness. Rijks et al. (1995a) studied this effect for Cu and permalloy ﬁlms, deposited on a Si(100) substrate covered by a 3 nm Ta underlayer. The sputter deposition conditions were typical for those used for fabricating spin-valves. The dependence of the conductance on the layer thickness was analysed within the semiclassical Boltzmann transport theory (see section 3.5). The results were analysed in terms of a layer thickness dependent mean −1 −1 free path, λ = (λ−1 i + λe ) . Here λi is the (temperature dependent) intrinsic mean free path and λe is the (temperature independent) extrinsic mean free path for defect scattering. For both materials, experiments carried out at 4.2 K and 300 K consistently revealed a monotonic increase of λe with increasing ﬁlm thickness, with λe proportional to the grains size as observed using TEM. This suggests that grain boundary scattering might be a major cause of the defect scattering (in the paper, the notation λgr , instead of λe , was therefore used). For Cu ﬁlms, λi (300 K) ≈33 nm was obtained, which is close to the value λi (300 K) ≈40 nm that would follow within the free electron Drude model. For ﬁlms in the 5–20 nm ↑ ↓ range, λe varied from 10 to 30 nm. For permalloy ﬁlms, (λi + λi ) ≈ 36 nm was obtained ↑ ↓ at 4.2 K, where λi and λi are the majority spin and minority spin intrinsic mean free paths, respectively. For ﬁlm thicknesses in the 5–20 nm range, the analysis yielded λe ≈ 3–10 nm. In conclusion, defect scattering can strongly affect the mean free paths in thin ﬁlms such as spin-valves. These results also imply that in studies of the layer thickness dependence of the transport properties of spin-valves, the possibility that the scattering parameters vary with the layer thickness should be considered (Rijks et al. 1996a). 2.9.3. Underlayers Underlayers are used for several purposes. They can act as a planarization layer or as a diffusion barrier between the spin-valve and the substrate (“buffer layer”). They can also be used as a specular reﬂection layer (see section 2.1). Here we focus on their effects on the microstructure. One of the most frequently used underlayers is a Ta layer with a thickness of a few nanometers. Its presence on top of various substrates such as Si(100), SiO2 , Si3 N4 or glass gives rise to a strong [111] texture of Cu and fcc-type Co–Fe–Ni alloys. X-ray diffraction of Ta/Py layers shows that a Ta layer thickness of only 3 nm is sufﬁcient to obtain the full effect (Duchateau et al. 1994). A strong [111] texture is often advantageous for the following reasons: (1) The [111] texture promotes magnetic softness of the free layer, because the magnetocrystalline energy of a cubic crystal such as permalloy, Co or Co90 Fe10 does not

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depend on the magnetization direction within the (111) planes, if the K2 anisotropy constant can be neglected (Jérome et al., 1994). The preferred in-plane magnetization direction is then fully determined by the uniaxial anisotropy that is induced by growth in a magnetic ﬁeld or by the shape anisotropy, and there are no grains with deviating easy-axis orientations that can act as nucleation or pinning centers of domain walls. (2) Growth on a 2 to 3 nm Ta underlayer is found to lead to a 10–15% larger MR ratio of Py/Cu/Py/Fe50Mn50 conventional spin-valves and a typically 20% lower sheet resistance (Duchateau et al. 1994). This may be attributed to a decreased defect scattering. The high sheet resistance of the thin Ta layer (resistivity typically 160 × 10−8 m) leads to negligible shunting. Gong et al. (2000) observed even a 20% increase of the AMR ratio of single permalloy layers by using 2–3 nm Ta underlayers. (3) The exchange bias ﬁeld due to AF layers such as Ir20Mn80 or Fe50 Mn50 is highest for [111] oriented systems. See section 4.1. However, for PtMn a strong [111] texture may not be ideal (Lee et al., 2002a). The Ta layer is amorphous or nanocrystalline (Holloway and Fryer, 1990; Galtier et al., 1994; Lefakis et al., 1996; Lenssen et al., 1996). Lenssen and coworkers concluded from transmission electron diffraction of MBE-grown and sputter-deposited permalloy layers on Ta underlayers that the [111] texture is correlated with a Ta layer consisting of nanometersize randomly oriented grains in a metastable cubic phase which is different from the stable bulk bcc-type α-Ta structure or from earlier-reported metastable β-Ta or fcc-Ta phases. The detailed crystallographic structure was not determined. For sputter-deposited ﬁlms the desired phase is observed up to tens of nanometers underlayer thickness, but for MBEgrown ﬁlms it is only present up to a thickness of 1 nm. Thicker underlayers have then the stable α-Ta structure which does not induce a [111] structure. These results suggest that the initial Ta structure gives rise to a larger mobility of the Ni and Fe atoms to form the energetically favorable [111] texture. Nakatani et al. (1994) studied the effect of other underlayer materials for Fe50 Mn50based SVs on Si(100) substrates. Strong [111] textures, relatively high exchange bias ﬁelds and relatively high MR ratios were found using Ta, Hf, Nb, Ti or Zr underlayers. Cu, Ag, Au and Cr underlayers, grown directly on the Si substrate, did not cause a signiﬁcant texture. Carey et al. (2002) studied the effect of thin Si underlayers. Several authors developed further improved SVs by making use of non-magnetic highresistivity fcc-type underlayers that give rise to an increased grain size in the active part of the layer stack, in order to enhance the GMR ratio and the thermal stability. Mizuguchi and Miauchi (1999) and Araki et al. (2000) found that the MR ratio of top SVs can be enhanced by making use of Ta/Py100−x Tax underlayers, with x < 15, instead of using only a Ta underlayer. For Ta/ 5 nm Py89 Ta11 / 2.5 nm Co90 Fe10 / 2.8 nm Cu/ 2.5 nm Co90 Fe10 / 6 nm Ir–Mn/ Ta spin-valves, Mizuguchu and Miyauchi obtained a very large GMR ratio (ΔR/R = 12.3%) and a change of the sheet resistance change that is about a factor of four larger than for conventional Py/Cu/Py/Fe50 Mn50 spin-valves (see table 2.1). Childress et al. (2001) studied Ta/ 4 nm Py60 Cr40 / 3.2 nm Py/ 0.5 nm Co90 Fe10 / 2.5 nm Cu/ 1 nm Co90 Fe10 / 1.9 nm Co60 Fe40 / 8 nm Ir–Mn/ 5 nm Ta

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spin-valves. The use of the Py60 Cr40 underlayer, instead of only Ta, was found to lead to an increase of the MR ratio from 8% to 11.5%. The observation of a decrease of the exchange bias ﬁeld upon inclusion of the Py–Cr underlayer was considered to be consistent with the assumed larger average grain size, because within some models for the exchange anisotropy effect Heb is expected to decrease with increasing grain size in the AF layer (see section 4.6.2). 2.9.4. Deposition methods Magnetron sputtering or ion beam deposition are found to be the most suitable techniques for depositing spin-valves with a high MR ratio and optimal magnetic properties and thermal stability. In a dc magnetron sputter deposition chamber ions from a noble gas plasma (usually Ar) that is present in between the sputter target and the sample substrate are accelerated towards the target. The atoms that are sputtered from the target have initially a high kinetic energy, but lose part of that energy due to collisions with the noble gas atoms in the chamber. The degree of thermalisation of the atoms that arrive at the substrate depends on the massmismatch between the sputtered atoms and the noble gas atoms, and on the (substrate-target distance) × (sputter gas pressure) product. The substrate-target distance is typically ﬁxed at 5 to 10 cm. The sputter gas pressure can be used to optimize the properties of the thin ﬁlm. The deposition rate is typically 0.1–0.3 nm/s. Kools (1995) investigated experimentally how the microstructure and MR ratio of Py/Cu/Py/Fe50Mn50 spin-valves depend on the Ar sputter pressure, and used the results of Monte Carlo calculations of the kinetic-energy distribution of atoms arriving at the substrate to interpret the observations. Fig. 2.16 shows the measured dependence of the MR ratio on the Cu thickness, for various sputter pressures. The full curves are a ﬁt using eq. (1.4) to the data above the critical Cu thickness below which there is no more a

Fig. 2.16. Dependence of the MR ratio on the Cu layer thickness and Ar sputter pressure, for SVs with the layer structure (Si(100)/ 3.5 nm Ta/ 8 nm Py/ tCu nm Cu/ 6 nm Py/ 10 nm Fe50 Mn50 / 3.5 nm Ta). The full lines represent a ﬁt to the data using eq. (1.4), for the Cu thicknesses for which a full antiparallel alignment is obtained. 1 mTorr corresponds to 0.131 Pa. From Kools (1995).

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high-resistance plateau in the transfer curve. The use of a higher Ar pressure has a positive effect on the parameter A (the MR ratio in the limit tCu = 0), which shows a gradual increase from 5.8% at 1.5 mTorr to 6.9% at 10 mTorr. (Note: in ﬁg. 2.16 the tCu scale starts at 1 nm, so this is not directly visible from the ﬁgure.) On the other hand, a high Ar pressure has a negative effect on the interlayer magnetic coupling across the Cu layer. Its increase with increasing pressure is evident from the larger decrease of the MR ratio at small Cu thicknesses. Deposition at intermediate pressures (3 to 5 mTorr) was found to be most optimal. These results were explained as follows. For low Ar pressures, a signiﬁcant fraction of the atoms arriving at the target was calculated to have a kinetic energy above 15 eV, which is a few times the atomic binding energy. This leads to atomic displacements from one layer to another over typically 2 to 3 atomic planes. The resulting large atomic mobility during the growth leads to a low probability of the formation of voids between the grains and to relatively ﬂat but strongly intermixed interfaces. For high Ar pressures, such collisional mixing occurs less frequently, leading to less intermixed interfaces. This explains the trend in (ΔR/R)0 , which is apparently quite sensitive to the interface quality. On the other hand, the small collisional mixing at high pressures gives rise to a large interface corrugation, and thereby to a strong magnetostatic interlayer coupling (“orange peel coupling”, see section 4.5.2). This picture was conﬁrmed by the observations that the 1.5 mTorr samples showed a smaller sheet resistance, smaller surface roughness (from atomic force microscopy) and larger effective magnetically “dead” Py/Cu interface layer thickness (from magnetization experiments) than the 10 mTorr samples. Park and Shin (1998) and Lu et al. (1999) conﬁrmed the trends observed by Kools, and showed that more optimal SV microstructures can be made by making use of a two-step sputter deposition process. The lower layers are deposited using a relatively low pressure, giving rise to dense layers with relatively large grains, whereas the upper layers are deposited at a much higher pressure in order to obtain less interdiffused interfaces. These studies were only carried out for SVs with permalloy or Ni66 Fe16 Co18 F-layers, resulting in fairly low MR ratios up to 5%. The beneﬁts of this approach to state-of-the-art SVs with much higher MR ratios, containing, e.g., Co or Co90 Fe10 F-layers, have yet to be demonstrated. Zhou and Wadley (1998) carried out a theoretical study of the deposition process, and concluded that a combination of sharp interfaces and dense bulk layers can best be obtained by a ﬂux of low energy adatoms at the beginning of each new layer, followed by a ﬂux of higher energy adatoms during the deposition of the remainder of the layer. The effect of the background pressure in the sputter chamber was studied by Mao et al. (2000d). They found that the MR ratio and magnetic properties of Ni–Fe– Co/Co/Cu/F/Fe50 Mn50 conventional SVs, with F = Co or Ni–Fe–Co, are suboptimal when the background pressure is larger than 5 × 10−7 Torr (≈ 7 × 10−5 Pa). However, in some cases pumping down to a lower background pressure is advantageous. Takahashi et al. (2000a, 2000b) found a beneﬁcial effect of a decrease of the background pressure in the chamber from 10−7 Torr to less than 10−10 Torr on the exchange bias interaction between Ir41 Mn59 and permalloy. Egelhoff et al. (1997b) found that the MR ratio of NiO/Co/Cu/Co bottom spin-valves is optimal at a partial O2 pressure of 5 × 10−9 Torr (5 × 10−7 Pa). The presence of such a small amount of oxygen gives rise to a relatively small ferromagnetic interlayer coupling and a decreased sheet resistance. It was argued that oxygen acts as a surfactant, leading to a decreased defect concentration, and that it enhanced the MR ratio

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by the formation of a specularly reﬂecting top layer. The effect of other surfactants, such as In, Pb and Ag was studied by Egelhoff et al. (1996b, 1996c) and Chopra et al. (2000a). The use of surfactants can give rise to smoother interfaces and hence to a reduced interlayer coupling (see section 4.5). It should be noted that in practice the partial pressure of contaminants such as water vapor can be less during deposition than just before deposition, as a result of the getting effect of deposited material (especially Ta). It can also be higher, when during the sputtering process contaminants are released from the chamber walls and from the targets by the impact of high energy atoms or ions, or when the sputter gas is insufﬁciently pure. Schwartz et al. (1998) investigated industrial scale dc magnetron sputter deposition, in order to assess the feasibility for applications such as read heads. They found that the thickness of nominally 2–3 nm Cu layers can be controlled within 0.05 nm. Variations of the sheet resistance and the MR ratio for a Py/Cu/Py/Fe50Mn50 top SVs over a 100 mm wafer were found to be only approximately ±1%, and the coupling ﬁeld Hcoupl was found to vary by ±0.04 kA/m (±0.5 Oe). DC magnetron sputtering led to a substantially larger MR ratio than rf diode sputtering using the same chamber and magnetron sources (5.5% versus 3.8% for layer stacks with tCu = 2.0 nm). Similarly, Langer et al. (2001) found that the MR ratio of dc magnetron sputtered Co/Cu/Co/Fe50 Mn50 top SVs is larger than that of rf sputtered samples (10.0% versus 7.1% for layer stacks with tCu = 2.5 nm). The interface roughness that results from dc magnetron sputter deposition has been studied using low-angle X-ray diffraction (XRD) by Huang et al. (1992) and Langer et al. (2001), and using High Resolution Transmission Electron Microscopy (HRTEM) by Bayle-Guillemaud et al. (1996). For F/Cu/F/Fe50 Mn50 top SVs with F = Py and Co, low angle XRD revealed a root-mean-square roughnesses of the bottom (top) Py/Cu and Co/Cu interfaces of 0.74 (0.62) nm (Huang and coworkers) and 0.41 (0.48) nm (Langer and coworkers). The HRTEM study by Bayle-Guillemaud and coworkers of similar SVs with F = Co revealed that the interfaces are locally diffuse or rough over two monolayers (≈0.42 nm). It should be noted that the lateral scale over which low-angle XRD averages the interface roughness is of the order of 100 nm, whereas the HRTEM result is an average over a 3–5 nm length scale. The latter length scale is more relevant when assessing the effect of the interface structure on the scattering probability. It should also be noted that neither of the two methods can make a distinction between compositionally sharp, but stepped interfaces, and interdiffused interfaces. The larger roughness for the Py/Cu interfaces may be due to the much larger miscibility of Ni and Cu than of Co and Cu. The different interface qualities can in part explain the much higher MR ratio for Co-based SVs (see also section 3.3). In an ion beam deposition (IBD) chamber, a separate ion source is used to sputter atoms from a target. For the deposition of spin-valves, IBD has a number of advantages and disadvantages as compared to magnetron sputtering (Slaughter et al., 1999; Hegde et al., 1999; Hylton et al., 2000). One of the most important differences is the low pressure in the deposition chamber. On the one hand, that makes deposition under cleaner conditions possible. On the other hand, the kinetic energy of ions that are reﬂected from the target as neutral atoms is less strongly reduced by collisions at the gas molecules in the chamber. They can arrive at the sample with a very large kinetic energy, which gives rise to strong collisional mixing and hence to a poor interface quality. The ﬂux of reﬂected neutrals decreases with increasing mass of the ions. IBD of spin-valves using Ar-ions led to

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disappointingly low MR ratios (Joo and Atwater, 1995; Wang et al., 1997b). The use of Xe-ions, which are heavier, has led to higher MR ratios due to less collisional mixing. However, the MR ratios still tend to be smaller than obtained by sputter deposition (Wang et al., 1997b; Slaughter et al., 1999; Hegde et al., 1999; Sant et al., 2001; Kools et al., 2001; Lee et al., 2001). E.g., Kools et al. (2001) studied Ta/Py/Ir–Mn/Co90Fe10 /NOL/Co90 Fe10 / Cu/Co90 Fe10 /Cu/Ta specular spin-valves. For samples made by sputter deposition and IBD, they obtained MR ratios up to 15.5% and 12%, respectively. Whereas for magnetron sputtering the kinetic energy of the arriving atoms can thus be chosen to be an optimal compromise between two extremes, that degree of freedom is not available for evaporation deposition. For the growth on single crystalline substrates, such as AF-coupled Fe/Cr multilayers with a high GMR ratio on GaAs or Ge, MBE is well suited, because a superlattice with a very small density of bulk defects can already be obtained as a result of the coherence with the substrate. However, there are no suitable and industrially feasible insulating single crystal substrates for the coherent MBE growth of the fcc-type materials of which spin-valves are composed. Lenssen et al. (1996) studied MBE grown Si(100)/Ta/Py/Cu/Py/Fe50Mn50 conventional SVs, and obtained MR ratios up to 4%. Sputter deposition yields typically 4.5% to 5%. The authors ascribed the slightly smaller MR ratio to the relatively small degree of [111] texture that could be obtained on the Ta underlayers. A later study of similar SVs by Huang et al. (1999) led to even lower MR ratios and a large coercivity of the free layer. The MBE deposition rate is typically one order of magnitude smaller than for sputter deposition and an ultralow (UHV) pressure is required. For all these reasons, MBE is not of interest for the industrial production of SV devices. 2.10. Thermal stability The thermal stability of spin-valves is determined by (i) the stability of the structure of the active part of the layer stack with respect to irreversible changes due to transformation or diffusion processes, and (ii) the stability of the exchange bias ﬁeld with respect to structural and magnetic changes of the AF layer. It is not always possible to strictly separate both aspects, e.g., because of diffusion of atoms from the AF layer into the active part. Here we focus on the stability of the active part. The thermal stability of the exchange bias ﬁeld is discussed in section 4.6. Irreversible thermal degradation of the structure can take place during the lithographic processing, or during the lifetime. In a read head fabrication process, SVs are exposed to 250–280 ◦ C for 2–10 hours. The working temperature of a hard disk read head during practical use is 120–150 ◦ C (Saito et al., 1998). Operational lifetimes are usually speciﬁed as >10 years. Automotive applications of SV sensors require stability during the lifetime up to 175–200 ◦ C. SVs that have been deposited in a back-end process on top of a CMOS semiconductor structure should be able to withstand short thermal treatments up to 400–450 ◦ C during processing. The diffusion processes that have been identiﬁed as most relevant are interface mixing and grain boundary diffusion. At the temperatures and timescales of interest, bulk diffusion is expected to play a less important role. We focus on the initial stages of degradation, before grain growth starts to give rise to a more drastic change of the microstructure (Meny et al., 1993).

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Fig. 2.17. Results of a study of the thermal stability of (Si(100)/ 5 nm Ta/ tf nm Py/ tCu nm Cu/ tp nm Py/ 11 nm Fe50 Mn50 / 5 nm Ta) SVs. Four consecutive anneal steps were performed at the temperatures given in (a), each lasting 6.5 hours. (a) Variation with the annealing temperature of the MR ratio, as a function of the Py free layer thickness tf , for tCu = 2.25 nm and tp = 5 nm. (b) Variation with the annealing temperature of the MR ratio, as a function of the Cu spacer layer thickness tCu , for tf = 7.5 nm and tp = 5 nm. (c) Variation with the annealing temperature of the effective interface roughness, δXRD , as determined from X-ray Diffraction, and of the effective “non-ferromagnetic layer thickness”, δNF , as determined from magnetization measurements. From Nozières et al. (1993).

2.10.1. Interface mixing Thermal deterioration by interface mixing occurs when the F and NM layers are miscible. Speriosu et al. (1993) and Nozières et al. (1993) concluded that this is the predominant initial degradation mechanism for Py/Cu/Py/Fe50Mn50 spin-valves. They studied (7.5 nm Py/ 2.2 nm Cu/ 5 nm Py/ 11 nm Fe50 Mn50) structures and observed a tenfold reduction of the GMR ratio after four sequential annealing steps at 240, 280, 320 and 360 ◦ C, each step lasting 6.5 hours. Fig. 2.17 (a), (b) shows the measured Py and Cu layer thickness dependences of the MR ratio after these annealing steps. From an analysis of the curves using eqs (1.4) and (1.5) it was argued that the MR ratio is mainly decreased due to enlarged scattering at the interfaces, and much less to enhanced scattering in the bulk of the layers. Evidence for the presence of an intermixed zone at the interfaces that increases with increasing anneal temperature was obtained from low-angle X-ray diffraction (XRD), which yielded the root-mean-square interface roughness, δXRD , and from magnetization measurements, which yielded the effective “non-ferromagnetic layer thickness” per interface, δNF . Both quantities are given in ﬁg. 2.17(c). As expected, δXRD is larger than δNF , because also the non-magnetic Cu atoms contribute to δXRD . A detailed low-angle XRD study of the annealing effects on all interfaces was carried out by Huang et al. (1993). The deterioration of the MR ratio that results from interface mixing can be avoided or reduced when no Ni, but only Co or Fe are present at the interface with the Cu spacer layers. Majority-spin scattering at rough or intermixed Co/Cu or Co–Fe/Cu interfaces is expected to be weaker than at Py/Cu interfaces (see section 3.3). In addition, Co and Fe are, at low temperatures, immiscible with Cu. It is not sufﬁcient to use Ni-containing alloys with a lower Ni-concentration than in permalloy. Kitade et al. (1995) observed a similarly strong thermal decrease of the MR ratio after annealing at 200–300 ◦ C for (F/Cu/F/Fe50 Mn50 ) spin-valves with Co-rich Py25 Co75 F-layers (as deposited ΔR/R = 7.5%), as for F = Py

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(as-deposited ΔR/R = 3.2%). The immiscibility of Co and Cu leads even to demixing upon a heat treatment, as was shown by Menyhard et al. (2000) from an Auger depth proﬁling study of a spin-valve. An as-deposited 4–5 monolayer thick intermixed Co/Cu interface layer was seen to be reduced to 2 monolayers after annealing during 2 hours at 260 ◦ C. The use of a composite free layer, consisting of a softmagnetic permalloy layer and a thin Co or Co90 Fe10 layer at the interface with the Cu spacer layer does not only enhance the MR ratio (section 2.1.2). It also gives rise to a signiﬁcant improvement of the thermal stability, as shown by Hwang et al. (1996) for (F/Cu)n multilayers and by Hamakawa et al. (1996) and Saito et al. (1998) for SVs. Hamakawa and coworkers concluded that insertion of a δ = 0.5 nm thick Co layer in (NiO/Py/ δ Co/Cu/Py) SVs was already sufﬁcient to obtain the full effect of that layer as a diffusion barrier (at 230 ◦ C). Zeltser et al. (1998) found that Co90 Fe10 interface layers, in combination with Py, give rise to a slightly better thermal stability than Co interface layers. Saito et al. (1998) investigated the structural changes that occur upon annealing using low-angle XRD, and found that, like Py/Cu interfaces, also Co90 Fe10 /Cu interfaces show a certain thermally induced interface roughening. Apparently, this is then less detrimental than in the case of Py/Cu interfaces. We note that, as remarked before, from low-angle XRD no distinction can be made between intermixing and an increase of the corrugation. Moreover, the technique averages over a lateral length scale as large as ≈100 nm. Portier et al. (1998) carried out a high-resolution TEM (HRTEM) study of annealed Ti/Co/Cu/Fe50 Mn50 /Ti SVs. The authors concluded that the observed decrease of the MR ratio, from 7.5% for as-deposited samples to 5.2% after 1 hour of annealing at 290 ◦ C, is not accompanied by a signiﬁcant change of the sharpness of the Co/Cu and Cu/Co interfaces. On a lateral scale of 3–5 nm, the interface remains mixed or corrugated over a thickness of only ≈2 monolayers. The decrease of the GMR effect was ascribed to alloying between the Ti underlayer and the Co layer directly on top of it, giving rise to a decrease of the effective free layer thickness. 2.10.2. Grain boundary diffusion of manganese from the AF layer Grain boundary diffusion affects the properties of SVs in various ways. Grain boundary diffusion of magnetic atoms into the Cu spacer layer can give rise to a change of the ferromagnetic magnetostatic and pin-hole coupling (Zeltser et al., 1998; Lin and Mauri et al., 2001). Grain boundary diffusion of Cu into the magnetic layers can lead to a decoupling of the grains, which for, e.g., Cu/Co/Cu trilayers has been shown to lead to an enhanced coercivity (Bensmina et al. 1999). We focus here on a third effect, viz. the reduction of the MR ratio due to diffusion of Mn from the AF layer into the active part of the layer. Mn-diffusion can give rise to a decrease of the GMR ratio because the presence of Mn impurities in Co, Co90 Fe10 or Cu layers gives rise to enhanced majority-spin electron scattering (see also section 3.6.9). The importance of Mn diffusion was discovered by Iwasaki et al. (1997), who observed that Ir–Mn based SVs are thermally more stable than otherwise identical Fe50 Mn50-based SVs. The different thermal stabilities of Ir–Mn and Fe50 Mn50 -based SVs, with the structure 10 nm Co–Nb–Zr/ 2 nm Py/ 3 nm Co90 Fe10 / 3 nm Cu/ 2.3 nm Co90 Fe10 / AF/ 5 nm Ta can be seen from ﬁg. 2.18, taken from Saito et al. (1998). The AF layers consisted of

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Fig. 2.18. Annealing time dependence of the normalized MR ratio at various temperatures for (10 nm Co–Nb–Zr/ 2 nm Py/ 3 nm Co90 Fe10 / 3 nm Cu/ 2.3 nm Co90 Fe10 / tAF AF/ 5 nm Ta) SVs, for (a) SVs with a 15 nm Fe50 Mn50 exchange bias layer, and (b) SVs with a 7 nm Ir22 Mn78 exchange bias layer. From Saito et al. (1998).

15 nm Fe50 Mn50 or 7 nm Ir–Mn, and the as-deposited MR ratios were 6–7% and 7.5– 8%, respectively. Takiguchi et al. (2000) found that Ir–Mn based SVs are also more stable than Rh17 Mn83 -based SVs. PtMn-based SVs, in turn, were found to be thermally more stable than otherwise identical Ir–Mn based SVs (Takiguchi et al., 2000; Anderson et al. 2000c). Furthermore, SVs based on (ordered) Pd29 Pt18 Mn53 are thermally more stable than Fe50 Mn50-based SVs (Aoshima et al., 1999). An explicit proof of the occurrence of a high concentration of Mn at grain boundaries after annealing was provided by Maesaka et al. (1998) from a HRTEM and energy dispersive X-ray spectroscopy (EDXS) with a resolution of 1 nm. Fig. 2.19 shows HRTEM crosssections of a Si(100)/ SiO2 / 5.2 nm Ta/ 3.8 Py/ 2 Co90 Fe10 / 2.4 nm Cu/ 3 nm Co90 Fe10 / 7.6 nm Ir–Mn/ 8 Ta SV, as-deposited (left) and after 1 hour annealing at 275 ◦ C (right).

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Fig. 2.19. High-resolution TEM images of (Ta/ 3.8 nm Py/ 2 nm Co90 Fe10 / 2.4 nm Cu/ 3 nm Co90 Fe10 / 7.6 nm Ir20 Mn80 / 8 nm Ta) SVs, (a) as-deposited, and (b) annealed at 275 ◦ C for 1 hour. The layer compositions at points A–D are discussed in the text. From Maesaka et al. (1998).

After the anneal, the resistance change had decreased by approximately 15%. The layer stack shows a strong [111] texture. The average grain size of 15 nm had not changed after annealing. The grain boundaries were seen to extend from bumps on the Ta surface, suggesting that the roughness of that layer determines the grain nucleation. The EDXS analysis revealed that before annealing only Co and Fe are present in the Co90 Fe10 layer and near a grain boundary in that layer, in points A and B, respectively, However, after annealing Ta, Ir and Mn were found to be present in that layer, with in particular a large Mn concentration at point D in the grain boundary. Takiguchi et al. (2000) came to similar conclusions from EDXS studies of the composition of the Cu spacer layer in annealed Rh17 Mn83 -based SVs. Results on PtMn and NiMn based systems are given in ﬁgs 2.20 and 2.21, respectively. Fig. 2.20 was taken from Lin and Mauri (2001), who studied PtMn based SVs with the layer structure as given in the ﬁgure caption, containing a Sy-AF pinned layer that can act as a Mn diffusion barrier. At short anneal times, the MR ratio increases as a result of the increase of the exchange bias ﬁeld when the PtMn structure orders. After an anneal of 2 hours at 280 ◦ C, the MR ratio was 13.8%. Longer annealing at higher temperatures leads to irreversible degradation of the layer stack. Up to 260 ◦ C no degradation of the MR ratio was observed until the end of the experiments after 14 hours, but changes of the coupling

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Fig. 2.20. Dependence of the MR ratio on the annealing time and temperature for (3 Al2 O3 / 3 Ni–Fe–Cr/ 1 Py/ 20 PtMn/ 2 Co90 Fe10 / 0.8 Ru/ 2.2 Co90 Fe10 / 2.2 Cu/ 0.9 Co90 Fe10 /2.7 Py/ 0.8 Cu/ 1 Al2 O3 / 6 Ta) SVs. From Lin and Mauri (2001).

Fig. 2.21. Dependence of the MR ratio on the annealing time and temperature for (Ta/ 4 nm Py/ 1.5 nm Co90 Fe10 / 2.8 nm Cu/ 2 nm Co90 Fe10 / 1.5 nm Py/ 25 nm NiMn/ Ta) SVs. From Zhang et al. (2000).

ﬁeld were already noticed at lower anneal temperatures. Other studies on the thermal stability of PtMn based SVs were carried out by Cool et al. (2000) and Kim et al. (2001a). Fig. 2.21 shows results reported by Zhang et al. (2000) on the thermal stability of the MR ratio of NiMn-based systems with the layer structure given in the ﬁgure caption, measured over a 10-day period. The samples had already been given a 7.5 hour postdeposition anneal at 275 ◦ C in order to obtain the exchange bias ﬁeld, after which the MR ratio was 6%. No systematic comparison of the stability of all relevant Mn-containing AF materials with respect to diffusion out of that layer has been made so far. If the thermodynamic sta-

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77

bility of the AF layer would be the determining factor, one would expect an increase of the stability of relevant AF materials in the order Fe50 Mn50 – Cr0.46 Mn0.46Pt0.08 – (Ru– Rh)20 Mn80 – NiMn – Ir20 Mn80 – Pd0.6Pt0.4 Mn – PtMn. This follows from the heats of formation for materials with these compositions as predicted from the Miedema model (de Boer et al., 1988). We have included the calculated heats of formation in table 4.7. An experimental indication that the thermodynamic stability is of importance was found by Anderson et al. (2000c), who observed a signiﬁcant slowing down of thermal degradation of PtMn based SVs after the occurrence of the fcc-fct disorder-order phase transition in PtMn. The observed increases of the stability in the series Fe50 Mn50 – Ir20Mn80 – PtMn, Rh17 Mn83 – Ir–Mn and Fe50 Mn50 – Pd29 Pt18 Mn53 are all consistent with the prediction given above. It is at present not possible to make further meaningful comparisons between the results from different groups, e.g., between Ir–Mn and NiMn or between NiMn and PtMn based systems, because of the sensitivity of the thermal stability on the layer thicknesses and ﬁlm microstructure. There are several ways for improving the thermal stability of SVs with respect to Mn diffusion. A ﬁrst possibility is to use a deposition process or an underlayer that leads to a microstructure for which grain boundary diffusion is slow. E.g., Maesaka et al. (1998) showed that growth on a MgO(111) substrate, or on a Co–Zr–Ta underlayer, leads to larger and smaller average grain sizes with respect to growth on Ta, respectively, leading to a better and worse thermal stability. A second option is to make use of a NOL in the pinned layer. NOLs act as a diffusion barrier, and give rise to planarization of the layer structure which makes grain boundary diffusion less fast (Kamiguchi et al., 1999a; Gillies et al., 2001; Jang et al., 2002a; see section 2.1.7). Thirdly, also a Ru layer acts as a diffusion barrier. Mn or Ni diffusion into the active part of the spin-valve can be slowed down by making use of a Sy-AF pinned layer of the form Co90 Fe10 /Ru/Co90 Fe10 , as shown by Anderson et al. (2000c) for Ir–Mn based SVs and by Huang et al. (2001) for NiMn based SVs. A fourth approach is to make use of ternary crystalline ferromagnetic (Co90 Fe10 )1−x Bx alloys (Kanai et al., 1997, 1999). Spin valves of the type Py/Co–Fe– B/Cu/(Co90 Fe10 )1−x Bx /Py/Fe50 Mn50 were found to show for x = 0 a considerable decrease of the MR ratio after annealing to 250–300 ◦ C during wafer fabrication. In contrast, the MR ratio of systems with x = 0.05 and x = 0.10 (see table 2.1(b)) was found to increase. An optimal MR ratio of 8.2% after annealing was found for NiO biased systems using (Co90 Fe10 )0.95B0.05 (table 2.1(f)). The B-containing layer may act as a barrier against Ni or Mn diffusion towards the Cu layer. B decreases the majority spin mean free path in the alloy, so the optimal composition is a compromise that depends on the required thermal stability. 3. Spin-polarized transport in spin-valves: theory and modelling In this section we ﬁrst discuss the microscopic origin of spin-dependent scattering in spinvalves, in the bulk of the layers and at the interfaces. Subsequently, we present analyses of the (magneto)conductance of various types of spin valves in terms of the semi-classical Camley–Barnas (CB) transport model, for the current-in-the-plane of the layers (CIP) geometry. Finally, we discuss brieﬂy some limitations of the CB model, some proposed extensions, and theories beyond the model.

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3.1. Spin-dependent conductivity in ferromagnets – spin-dependent scattering The spin-dependence of the conductivity in the bulk of the F layers is related to the spindependence of the electronic structure of the layers, and/or to the spin-dependence of the scattering potential. When the two-current model (already introduced in section 1.2) is applicable, the conductivity is given by σ = σ ↑ + σ ↓,

(3.1)

where σ↑ and σ↓ are the conductivities for the two spin-channels. The scattering spin asymmetry ratio α, deﬁned as, α≡

σ ↑ ρ↓ = σ ↓ ρ↑

(3.2)

is frequently used as a measure of the spin-dependence of the conductivity. Making the free-electron approximation, σ↑ and σ↓ are given by the Drude formula (Ashcroft and Mermin, 1976): e2 n↑(↓) τ ↑(↓) (3.3) m where e is the electron charge, n is the conduction electron density, τ is the electron momentum relaxation time and m is the electron mass. From the Fermi golden rule, the spindependent scattering rate is given by σ ↑(↓) =

2π ↑(↓) 2 ↑(↓) 1 = N (EF ), V h¯ sc τ ↑(↓) ↑(↓)

(3.4)

where |Vsc |2 is the (average) scattering potential and N ↑(↓) (EF ) is the density of states at the Fermi level. The spin-dependence of the conductivity is thus determined by factors that follow directly from the band structure of the unperturbed lattice (n, N(EF ) and m), and by the scattering potential, which arises from perturbations of the lattice periodicity. The spin-dependence of scattering in the elementary 3d-transition metals Fe, Co and Ni can be understood from the spin-dependent density of states (DOS), given in ﬁg. 3.1 (a)–(c). To a ﬁrst approximation, the DOS can be viewed as a superposition of a wide spband and a narrow d-band. The conductivity, which is determined predominantly by the contribution from the light sp-electrons, is decreased when d-states (to which scattering can take place) are present at the Fermi level. For Co and Ni the majority spin d-band is fully occupied, and is situated well below the Fermi level, whereas the minority spin 3d-band is only partially occupied. So when the scattering potential is spin-independent, the majority spin conductivity of Co and Ni is expected to be larger than the minority spin conductivity. Qualitatively, this picture is supported by analyses of the temperature dependence of the conductivity of dilute binary alloys (Loegel and Gautier, 1971; Fert and Campbell, 1976). For Co, analyses of the CPP magnetoresistance at 4.2 K of magnetic multilayers have yielded αCo = 3.1 (Yang et al., 1995; Pratt et al., 1996). For Fe, dilute alloy experiments have led to the conclusion that α is very close to 1 (Campbell and Fert, 1982). The much smaller polarization of the conductance for Fe, than for Co and Ni, can be understood from DOS, which shows that majority and minority spin

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79

Fig. 3.1. Densities of states (DOS) of the elemental metals Fe, hcp-Co, Ni and Cu, obtained from self-consistent band structure calculations using the Augmented Spherical Wave (ASW) method. The DOS of fcc-Co is well approximated by that of fcc-Ni, with the minority-spin DOS shifted upwards such that the arrow coincides with the Fermi-level. From Coehoorn (2000).

3d-states are both present at the Fermi level. Therefore, sp-type electrons with both spins can scatter to 3d-type ﬁnal states. Indeed, a study of the Fe layer thickness dependence of the GMR ratio of Fe/Cr multilayers has indicated that the spin-dependence of scattering in the bulk of the Fe layers is very weak, and that the high GMR ratio is mainly due to spin-dependent interface scattering (Schad et al., 1994). Much experimental work has been done on the spin-dependence of the conductivity of dilute Fe, Co or Ni based magnetic alloys. Using the two-current model, their residual resistivities have been obtained from the residual resistivities of ternary alloys (Dorleijn 1976; Campbell and Fert, 1982). The spin-dependence of the scattering potential due to impurities in a given host varies widely, giving rise to a wide range of observed spin asymmetry ratios. For dilute Co–Fe, Co–Ni, Ni–Fe or Ni–Co alloys, e.g., α 1, whereas for dilute Co–Cr, Co–V, Ni–Cr and Ni–V alloys, α 1 (Dorleijn 1976; Campbell and Fert, 1982). (The host metal is indicated by bold symbols.) We will use these results in section 3.6.9, where the magnetoresistance of SVs containing Co layers with transition metal impurity layers is discussed. For that purpose, the spin-polarized residual resistivities of transition metal impurities in Co are included in ﬁg. 3.11.

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Good qualitative understanding of these results has been obtained from Virtual Bound State theories (Campbell and Fert, 1982; Coehoorn, 1993). The spin-dependent residual resistivities are then considered to be related to the differences in the spin-dependent ﬁlling of the host metal and impurity atom 3d densities of states. Recently, results obtained from scalar-relativistic7 ﬁrst principles calculations of the spin-dependent residual resistivity of dilute magnetic alloys were for many alloys found to be in good quantitative agreement with experiment (Mertig, 1999). An important exception is the residual resistivity calculated for strong ferromagnets, such as Ni–Co and Ni–Fe alloys. In these alloys the ﬁlling of the majority-spin host and guest 3d-DOS is essentially 100%, reﬂecting that there is almost no difference between the host and guest electronic potentials. Majority-spin scattering is therefore calculated to be very weak, leading to spin-asymmetry ratios larger than 100 or even 1000. Experimentally, much lower values are found, e.g., αeff,exp = 13–30 for Ni–Co and αeff,exp = 7.3–20 for Ni–Fe (Campbell and Fert, 1982). A similar discrepancy is found for concentrated alloys that are more relevant to spin-valves, such as permalloy. In the next section, it is shown that a fully relativistic theory is required for explaining these results. 3.2. Spin-dependent conductivity in ferromagnets – effect of spin-mixing due to the spin-orbit interaction The picture that we have sketched in section 3.1 can be used to understand trends in the spin-dependence of the conductance. However, it has been found to be quantitatively incorrect for 3d-transition metal alloys in which the conductivity is strongly spin-dependent (Mertig, 1993; Butler et al., 1995; Banhart et al., 1997). The problem is that in such cases the spin-mixing due to the spin-orbit interaction (SOI) cannot be neglected. In the simple form presented above, the two-current model therefore fails, e.g., for bulk permalloy, which is frequently used in spin-valves. We consider the situation for permalloy in some detail. In permalloy, the Ni and Fe atoms are randomly or almost randomly distributed on the sites of an fcc lattice. The site disorder results in a ﬁnite bulk resistivity at 4.2 K of ρ = 4.7 × 10−8 m. Analyses of the CPP-GMR ratio at 4.2 K of magnetic multilayers have yielded ρ = 11.9 × 10−8 m, and αexp,CPP = 7 ± 2 (Steenwyk et al., 1997). Note that the resistivity of thin ﬁlms is generally larger than the bulk resistivity due to defect scattering. Scalar-relativistic calculations, which neglect the spin-orbit interaction (SOI), yield resistivities that are more than one order of magnitude smaller than the experimental value. Results reported by Butler et al. (1995) and Nicholson et al. (1997) are ρ↑ ≈ (0.18–0.3) × 10−8 m and ρ↓ ≈ 100 × 10−8 m. The conductivity would then be almost completely carried by the majority spin electrons, with an extremely large scattering spin-asymmetry ratio: α 100. Nicholson et al. (1997) found that the introduction of the SOI raised the resistivity to about 2.5 × 10−8 m. More recently, Blaas et al. (2001) obtained ρ ≈ 7 × 10−8 m for permalloy. These drastically increased values are in much better agreement with experiment. Banhart et al. (1997) performed calculations for 7 In scalar relativistic electronic structure calculations the relativistic mass-velocity and Darwin terms are taken into account, but the spin-orbit interaction is neglected.

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81

Fig. 3.2. Resistor scheme that represents the total resistivity (eq. (3.5)) after spin-mixing.

Ni1−x Fex and Ni1−x Cox alloys in wide concentration ranges, and found similarly large increases of the resistivity when the SOI was included. The introduction of the SOI has two effects: it mixes majority and minority spin states, and it leads to a spin-dependent energy shift of states. Ebert et al. (2000) showed explicitly that it is the mixing effect that is responsible for the increase of the resistivity. The majority spin electron states acquire some minority spin character, so that they can be scattered into the minority spin sub-band. In spite of the low degree of intermixing of spin character for light atoms such as Fe, Co and Ni, the additional scattering is strong because of the large minority spin density of ﬁnal states. The majority spin resistivity is then strongly increased. The enhancement of the resistivity by spin mixing due to the SOI can be treated in the same way as that by spin-mixing due to electron–magnon scattering. The resistivity is given by (Fert and Campbell, 1976; see also section 3.7.2) ↑ ↓

ρ=

↑

↓

ρ0 ρ0 + ρ ↑↓ (ρ0 + ρ0 ) ↑

↓

ρ0 + ρ0 + 4ρ ↑↓

,

(3.5)

where the indices “0” indicate the resistivities that would be obtained in the absence of the SOI and where ρ ↑↓ is called the spin-mixing resistivity. The total resistivity can be represented by the resistor scheme shown in ﬁg. 3.2. When discussing the resistivity enhance↓ ↑ ment for materials with α0 ≡ ρ0 /ρ0 > 1, it is convenient to make use of the dimensionless ↓ parameter x ≡ ρ ↑↓ /ρ0 . We show below that for the 3d-transition metal alloys and multilayers that are of interest for applications in spin-valves, x 1. It follows from eq. (3.5) that a signiﬁcant enhancement of the resistivity will only occur if the product α0 x is of the order of 1, or larger. In order to gain some understanding of the physical meaning of the parameter x, we consider the effect of spin mixing due to the SOI using a more microscopic model, in the limit α0 1. Due to the SOI, wavefunctions that have predominantly majority or minority spin character have the general form φ1 = aφ ↑ + bφ ↓ and φ2 = bφ ↑ + aφ ↓ , respectively, with b2 < a 2 . The normalized wavefunctions φ ↑ and φ ↓ and the parameters b and a = (1 − b2) are assumed to be representative for all electron states that contribute to the conduction. Neglecting terms containing V ↑ and N ↑ (EF ) (because α0 1), the scattering

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R. COEHOORN

rates of electrons in states |φ1 are from eq. (3.4) proportional to |bφ ↓ |V ↓ |aφ ↓ |2 . The ↓ resistivity for transport via the |φ1 channel is then equal to ρ1 = a 2b2 ρ0 . Similarly, the ↓ resistivity for transport via the |φ2 channel is equal to ρ2 = a 4 ρ0 . When the SOI is weak ↓ ↑ (b2 1) the total resistivity ρ ≈ ρ1 ≈ b2 ρ0 . Using eq. (3.5), with ρ0 = 0, it follows that ρ ↑↓ ↓

ρ0

≡x ∼ = b2

if α0−1 x 1.

(3.6)

This shows that x is determined by the degree to which the SOI induces wavefunction mixing. For a given material, x is proportional to the square of the relevant SOI parameter. In the case of 3d-transition metal systems, this is the spin-orbit parameter for the 3d states. The effect of the SOI parameter on the resistivity of permalloy has been studied by Banhart et al. (1996). Spin-mixing invalidates the two-current model. However, as suggested by the formal theory of spin-mixing (section 3.7.2) one may deﬁne a modiﬁed version of the two-current model, based on the effective resistivities ↑ ↓

↑

ρeff =

↑

↓

ρ0 ρ0 + ρ ↑↓ (ρ0 + ρ0 ) ↓

ρ0 + 2ρ ↑↓

,

(3.7)

.

(3.8)

and ↓ ρeff

↑ ↓

=

↑

↓

ρ0 ρ0 + ρ ↑↓ (ρ0 + ρ0 ) ↑

ρ0 + 2ρ ↑↓

These effective resistivities are physically meaningful quantities in following sense. Suppose that the resistivity is increased by adding a low concentration of additional scat↑ ↑ ↑ tering centers, giving rise to modiﬁed spin-dependent resistivities ρ0 = ρ0 + Δρ0 and ↓ ↓ ↓ ↑ ρ0 = ρ0 + Δρ0 , whereas ρ ↑↓ remains unchanged. For sufﬁciently small values of Δρ0 ↓ and Δρ0 an excellent approximation to the exact value of the modiﬁed overall resistivity, ↑ ↓ ↑ ↑ ↑ ρ , is then given by the resistivity ρ = (1/ρeff + 1/ρeff )−1 , with ρeff = ρeff + Δρ0 and ↓ ↓ ↓ ρeff = ρeff + Δρ0 , as if spin-mixing would play no role. The relative error, (ρ − ρ )/ρ , ↑ ↓ is to ﬁrst order in Δρ0 or Δρ0 equal to zero. It is thus possible to successfully predict the effect on the resistivity due to (sufﬁciently small) additional spin-polarized resistivities ↑ ↓ ↑ ↓ from a modiﬁed form of the two-current model that uses ρeff and ρeff instead of ρ0 and ρ0 . ↓ ↑ The effective scattering spin asymmetry ratio, αeff ≡ ρeff /ρeff , can be obtained from eqs (3.7) and (3.8). When α0−1 x 1, αeff is given by 1 1 +1∼ αeff ∼ (3.9) = = 2 + 1. 2x 2b It is then not determined by the ratio of spin-dependent conductivities that would follow from eqs (3.3) and (3.4), which depend on the (spin-dependent) scattering potentials, but by the degree to which the SOI mixes majority and minority states, an intrinsic property of the material. The detailed scattering mechanism then does not affect αeff , provided that it does not affect x.

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The application of this formalism to the theoretical results obtained for permalloy (see ↑↓ ↓ ↓ above) yields ρth ≈ ρth ρth,0 /(4ρth + ρth,0 ) ≈ 2.3 × 10−8 m, xth ≈ 0.023 and αeff,th ≈ 22. As expected from the low atomic masses of Ni and Fe, x is much smaller than 1. Application to the results obtained by Banhart et al. (1997) yields xth ≈ 0.018 for Ni– Co and Ni–Fe alloys in the Ni-rich limit, and a monotonous decrease to xth ≈ 0.004 and xth ≈ 0.002 in the Co and Fe-rich limits, respectively. This reﬂects, in part, the effect of the ≈25% and ≈50% decrease of the 3d-spin orbit parameter when replacing Ni by Co or Fe, respectively. Results obtained by analysing the CIP-magnetoresistance of spin-valves (section 3.6.5) and the CPP-magnetoresistance of multilayers (Steenwyk et al., 1997) yield values for αeff in the range 6 to 20. The (rough) theoretical estimate thus coincides with the upper limit of this range of experimental values. When making a comparison between theory and experiments at T = 0 it should be realized that in realistic alloy ﬁlms extrinsic scattering due to defects may be larger than the intrinsic scattering. Spin-mixing that is induced by the SOI will be less relevant when these additional scattering processes are dominant and relatively weakly spin-dependent, i.e., if for these processes α0 x 1. Effects of the microstructure of spin-valves on the effective mean free paths are discussed in section 3.6.5. However, a full understanding of ↑ ↓ microstructural effects on ρ0 , ρ0 and ρ ↑↓ is presently not available for the materials that are of interest in spin-valves. Binder et al. (2001) calculated the resistivity due to scattering at Cu impurities in Co, and obtained α ≈ 23. For fcc and hcp Co, Schulthess et al. (1997) have theoretically studied the effect of introducing structural defects on the spin-dependent conductivity. While neglecting the SOI, they found that stacking faults and twin boundaries lead to a resistivity rise which is largest for the minority spin channel. A qualitatively similar effect is expected for permalloy, which has a similar electronic structure. At ﬁnite temperatures, electron–phonon scattering may become dominant, and spin-mixing due to electron–magnon scattering may become more important than spin-mixing due to the SOI. 3.3. Spin-dependent scattering at interfaces The spin-dependence of the scattering potential plays an important role when considering scattering at the F/NM interfaces. Interfaces can be atomically ﬂat, rough but nonintermixed, or diffuse (forming an intermixed boundary layer). At atomically ﬂat interfaces, only specular reﬂection can take place. This occurs when, effectively, the potentials within the F and NM layers are different. At rough or diffuse interfaces diffuse scattering can take place. Both effects are relevant to the GMR effect in spin-valves. Fig. 3.3 shows a schematic picture of the “potential landscape” in the active layer of a simple spin-valve, indicating the different origins of scattering. Only deviations from the periodic atomic potentials are shown, as no scattering takes place in a regular periodic lattice. The size of the potential step and of the scattering potentials at imperfections of the interfaces depends on the degree of matching of the electronic structures of the F and NM materials in the bulk of the layers and near the interface (Itoh et al., 1993; Butler et al., 1995). In order to obtain a large GMR effect, good electronic structure matching is required for one spin direction, and bad matching for the opposite spin direction. Ideally, this is true for the electronic structures in the bulk of the layers, at atomically ﬂat parts of the interfaces, and even near imperfections of the interfaces.

84

R. COEHOORN

Fig. 3.3. “Potential landscape” inside the F/NM/F active part of a spin-valve, for the two spin channels in the parallel state (a), (b) and in the antiparallel state (c), indicating the different origins of scattering. Adapted from Vouille et al. (1999).

Excellent electronic structure matching explains, in part, the large GMR ratios for Fe/Cr and Co/Cu and systems. For Fe and Cr, matching is good for the minority spin electronic structures (Butler et al., 1995), also for atomically rough interfaces (Inoue and Maekawa, 1991; Coehoorn, 1995). For Co and Cu, good matching occurs for the majority-spin electronic structures in the bulk of the layers, as can be seen from the density of states given in ﬁg. 3.1. Also for binary and ternary Co–Fe–Ni alloys with a fully occupied majority spin 3d-band (such as Ni80 Fe20 ) matching of the bulk electronic structures with the band structure of Cu is excellent. Near Co/Cu interfaces, the magnetic moment of Co is hardly changed by the presence of Cu nearest neighbour atoms. The ﬁlling of the 3d-partial density of states of Co atoms near the interface is therefore not essentially different from that in bulk Co. The majority-spin 3d band is fully occupied, so that no or only very weak scattering of majority-spin electrons will occur at imperfections of the interface. In contrast, the local minority-spin electronic potential on a Co-atom at a rough interface is quite different from that on its Cu neighbours in the same atomic layer, giving rise to strongly diffusive scattering. Indeed, Binder et al. (2001) calculated the additional resistivity due to scattering at Cu impurities on an interface site, and found a scattering spin anisotropy ratio α ≈ 17. Quantifying the “spin-dependence of scattering at the interfaces” cannot be done without deﬁning the experiment that is used to probe the effect. CPP-GMR experiments can be analysed in terms of a series resistor model of spin-dependent bulk and interface resistances. For Co/Cu and Py/Cu interfaces, e.g., such analyses have led to interfacial spin-

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85

asymmetry ratios r↓ /r↑ = 7.3 and r↓ /r↑ = 5.7, respectively, at 4.2 K (Pratt et al., 1996, Steenwyk et al., 1997). Here r↑ and r↓ are the majority and minority spin interface resistances, respectively. These high ratios conﬁrm the remarks concerning electronic structure matching given above. However, in CIP-experiments the potential step at the interface plays a different role then in CPP-experiments. Within the semi-classical Camley–Barnas model (section 3.5), two spin-dependent parameters (the probabilities for diffusive scattering and for specular reﬂection) are introduced, which bear no formal relationship with the interfacial resistances measured in a CPP experiment. Nevertheless, the trends are generally the same: analyses of CIP-experiments on systems containing Co/Cu interfaces reveal a high (low) probability of minority (majority) spin-scattering at the interfaces (see section 3.6). For bulk Ni and Cu, electronic structure matching is very good for the majority spinelectrons (see ﬁg. 3.1). However, Ni-moments at interfaces with Cu are more sensitive to the presence of Cu neighbours than Co moments. Already at a perfect Ni/Cu interface, and even more at an imperfect interface, the Ni magnetic moments are smaller than the bulk value of approximately 0.6μB (Coehoorn, 1993; Nicholson et al., 1994). As a result, the majority-spin 3d-density of states is no longer zero at the Fermi level, so that there is no good majority spin electronic structure matching with the Cu spacer layer. This contributes to the decrease of the GMR ratio of spin-valves based on Ni–Co–Fe alloys with increasing Ni concentration, as shown in ﬁg. 1.8. For Ni/Cu/Ni/Fe50 Mn50 spin-valves the GMR ratio is only at most 2% at room temperature (see table 2.1). It should be noted that the solubility of Ni in Cu is larger than that of Fe or Co in Cu, giving rise to more diffuse interfaces, and that the thermal stability of Ni moments at interfaces with non-magnetic metals such as Cu is expected to be relatively small, in view of the relatively low Curie temperature of Ni. Both effects contribute to the low room-temperature GMR ratio of Ni/Cu/Ni/AF spinvalves. The role played by scattering at the interfaces in SVs containing Ni-rich F layers became very clear when studying the effect of thin Co “dusting” layers at the interfaces (see ﬁg. 2.2). A complicating effect is the possibility that the F moments near the interfaces are noncollinear. Oparin et al. (1999) have shown from ﬁrst-principles band structure calculations that this is expected to be the case for the Ni and Fe moments near interdiffused Py/Cu interfaces. The non-collinearity enhances the scattering for both spin-directions, thereby reducing the magnetoresistance. 3.4. Series resistor model A strongly simpliﬁed model of GMR effect in magnetic multilayers is obtained by making the (implicit) assumption that the current densities are homogeneous for both spin directions. If the resistances for the “+” and “−” channels in the parallel conﬁguration are R+ and R− , the resistances of the “+” and “−” channels in the antiparallel conﬁguration are equal to 12 (R+ + R− ). The parallel and antiparallel resistances are then given by Fert (1990) RP =

R+ R− R+ + R−

(3.10)

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R. COEHOORN

and R+ + R− , 4 and the giant magnetoresistance ratio is given by RAP =

(3.11)

ΔR (R+ − R− )2 (1 − α)2 = , = R 4R+ R− 4α

(3.12)

where α ≡ R− /R+ can be viewed as the effective scattering spin-asymmetry ratio for the multilayer. Effective resistor schemes that correspond to eqs (3.10) and (3.11) were already given in ﬁg. 1.3(b). The MR ratio is thus expected to be positive, as is observed indeed without exception for AF-coupled F/NM systems.8 As shown by Valet and Fert (1993), this series resistor model is valid for the CPP geometry, if spin-ﬂip can be neglected. It is then possible to express α in terms of the F and NM layer thicknesses, spin-polarized bulk resistivities and interface resistances. In spin-valves and other layered materials, the scattering probability when traversing a layer is usually close to 1 for electrons of at least one spin-direction. In the CIP-geometry, the current density is thus in general a non-uniform function of the position in the layer stack. Also diffusive scattering at the outer boundaries makes the current density nonuniform. Therefore, the series resistor model (eqs (3.10)–(3.12)) is generally not valid for the CIP-geometry (although it still has a certain didactical value). It would only be valid for CIP-systems in the weak scattering limit, when for both types of electrons the scattering probabilities upon traversal of one multilayer period is much smaller than one. In that case, the CIP and CPP-GMR ratios are equal. A semiclassical transport model for the CIPgeometry, within which the non-uniform current density is calculated, is discussed in the next section. 3.5. The Camley–Barnas semiclassical transport model A very useful model for describing the spin-polarized, inhomogeneous current densities in magnetic multilayers, studied in the CIP-geometry, is the semi-classical transport model proposed by Camley and Barnas (1989). Within this model, electrons are essentially regarded as pointlike particles (hence ‘classical’) although some consequences of quantum mechanics are taken into account, such as the use of quantum-mechanics to calculate the relation between the energy and momentum, Fermi–Dirac statistics and quantummechanically evaluated scattering probabilities (hence ‘semi’). Within the CB model the electronic structure in each of the layers is treated as a free-electron gas. The spin-dependent scattering probabilities in the bulk of the layers, at the interfaces, and at the outer boundaries, are free parameters. They can be varied in order to obtain good agreement with the observed (layer thickness dependent) (magneto)conductivities, or can be held ﬁxed if additional information is available. The conductivity is obtained from the electron distribution function f ↑(↓) (v, r), which gives the spin dependent probability that a state at position r with velocity v is occupied. A convenient 8 For (F /NM/F /NM) systems, with F layers having opposite signs of the spin-polarisations of the conducn 1 2

tance, an inverse GMR effect can be obtained (George et al., 1994; Renard et al., 1995).

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87

notation is ↑(↓)

f ↑(↓) ( v , r) = f0

( v , r) + g ↑(↓) ( v , r).

(3.13)

Here g ↑(↓) (v, r) is the deviation from the Fermi–Dirac equilibrium distribution function ↑(↓) f0 (v, r), in the presence of an electric ﬁeld E, for spin up (↑) and spin down (↓) electrons, at energy ε = 12 mv 2 . The electron distribution function is obtained by solving the Boltzmann transport equation in the relaxation time approximation: ↑(↓)

v ·

∂f (ε) g ↑(↓) ( v , r) v , r) ∂g ↑(↓) ( − eE · v 0 =− , ↑(↓) ∂ r ∂ε τ

(3.14)

in which e is the electron charge, m is the electron mass, and τ ↑(↓) is a spin-dependent and layer-dependent isotropic relaxation time. The spin-dependent current density follows from an integral over velocity space of the electron distribution function: 3 m j↑(↓) (r ) = e (3.15) v , r), d3 v vg ↑(↓) ( h where h is Planck’s constant. Solutions of the Boltzmann transport equation for a multilayer consisting of N layers, with the planes perpendicular to the z-axis and with the interfaces at positions {z0 , z1 , . . . , zN } have, in layer i, the following form: −(z − zi−1 ) ∂f0 (ε) v , z) = −eτi Ex vx g+,i ( (3.16) , 1 − F+,i exp ∂ε τi vz for vz > 0, and

∂f0 (ε) −(zi − z) g−,i ( v , z) = −eτi Ex vx 1 − F−,i exp , ∂ε τi |vz |

(3.17)

for vz < 0. The electric ﬁeld is along the x-direction. The spin (↑ and ↓) labels have been omitted from the relaxation time τ and the coefﬁcient A. The coefﬁcients F are determined from the boundary conditions at the interfaces or outer boundaries (transmission, reﬂection, diffusive scattering, see below). For the case of bulk metals, this model is discussed in many textbooks (see, e.g., Ashcroft and Mermin, 1976). It leads to the Drude formula (eq. (3.3)). The relaxation time τ can be expressed in terms of the electron mean free path λ and Fermi velocity vF as τ = λvF . Fuchs (1938) and Sondheimer (1952) have ﬁrst applied this model to the conductivity of homogeneous thin ﬁlms. For a ﬁlm with thickness t in between z = 0 and z = t, the coefﬁcients F in the g-functions, from which the current density follows via eq. (3.15), are obtained by the application of the boundary conditions g+ (z = 0) = pg− (z = 0) and g− (z = t) = pg+ (z = t). Here p is the probability for specular scattering at the outer surfaces. When p = 1, diffusive scattering at the outer boundaries leads to a decrease of the effective conductivity σ with decreasing t. The effect is signiﬁcant for t close to and smaller than λ. The ratio of the conductivity of the ﬁlm with respect to the bulk conductivity σ0 is given by (Fuchs, 1938; Fert, 1990) 3 ∞ 1 1

σ = 1 − (1 − p) − 5 1 − exp(−kx) dx, (3.18a) 3 σ0 2k 1 x x

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R. COEHOORN

where k = t/λ. A good approximation in the thick ﬁlm limit (t λ) is λ σ ∼ 3 = 1 − (1 − p) . σ0 8 t

(3.18b)

In that limit, the current density is close to the bulk value in the center of the ﬁlm, but lowered in a zone with thickness of the order λ near the outer boundaries. These two boundary zones overlap when the condition t λ is not satisﬁed, resulting in the more complicated general expression (3.18a). The current density in magnetic layered structures can be obtained after the introduction of two types boundary conditions (in addition to the specularity parameters p at the outer surfaces), which relate the g-functions at the interfaces. First, at the interfaces three possible processes are distinguished: transmission (probability Ti,s ), specular reﬂection (probability Ri,s ), or diffusive scattering (probability Di,s ). Specular reﬂection could arise in the case of a potential step at the interface between two layers (as depicted schematically in ﬁg. 3.3). These probabilities depend on the interface i and the spin s; T + R + D = 1. At the interface between two layers A and B the boundary conditions are: σ σ σ g± (B) = T σ g± (A) + R σ g∓ (B).

(3.19)

Second, systems with different magnetization directions in the different layers are dealt with using spin-transmission coefﬁcients T ↑↑ , T ↑↓ , T ↓↑ , and T ↓↓ , through a plane situated within the non-magnetic spacer layer between the magnetic layers (the precise position of this plane in the non-magnetic spacer layer does not affect the ﬁnal results). These transmission coefﬁcients determine the probability of an electron which has spin up or down with respect to the magnetization direction (quantization axis) in one magnetic layer to continue in the other magnetic layer as an electron with spin up or down with respect to the new quantization axis. The transmission coefﬁcients are given by T ↑↑ = T ↓↓ = cos2 (θ/2) and T ↑↓ = T ↓↑ = sin2 (θ/2),

(3.20)

where θ is the angle between both magnetization directions. For example, in the case of an antiparallel arrangement of the magnetization directions T ↑↑ = T ↓↓ = 0 and T ↓↑ = T ↑↓ = 1. A majority spin electron in one magnetic layer becomes then a minority spin electron in the other magnetic layer, after traversing a nonmagnetic layer in between. If the interfaces at which the spin-boundary conditions are applied are taken to coincide with interfaces between layers, the determination of the 2N coefﬁcients F∓ for N layers (and for each spin) involves solving a set of 2N linear equations, obtained by applying the boundary conditions. From the resulting g-functions, the z-dependent current densities in layer i can be calculated using eq. (3.15): e2 Ex m2 vF2 ↑(↓) π/2 −(z − zi−1 ) ↑(↓) ↑(↓) λ exp ji,x (z) = 2 − F i +,i ↑(↓) 8π 2 h¯ 3 β=0 λi cos β −(zi − z) ↑(↓) sin3 β dβ − F−,i exp ↑(↓) λi cos β

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

3 π/2 −(z − zi−1 ) ↑(↓) ↑(↓) = σ0 Ex × 2 − F+,i exp ↑(↓) 4 β=0 λi cos β −(zi − z) ↑(↓) sin3 β dβ. − F−,i exp ↑(↓) λi cos β

89

(3.21)

The second part of the equation is obtained by using the well-known relationship between the Fermi-velocity and the electron density (Kittel, 1996). The Drude formula is retained ↑(↓) for bulk systems, when F∓ = 0. For a layered system, the contribution from each spin current to the conductance follows after integration over the entire thickness of each layer, and summation of the contributions from all layers. The occurrence of the angular integral in eq. (3.21) prohibits the formulation of simple but physically transparent expressions for the sheet conductance. It is therefore sometimes useful to make an approximation, by replacing the cos(β) factor in the argument of the exponential functions by a ﬁxed, representative, average value, γ . The current density in each layer is then given by ↑(↓)

↑(↓)

ji,x (z) = σ0

↑(↓) F+,i −(z − zi−1 ) exp Ex × 1 − ↑(↓) 2 γλ i

↑(↓)

−

F−,i 2

−(zi − z) exp , ↑(↓) γ λi

and the contribution from layer i to the sheet conductance is zi 1 ↑(↓) Gi,sh ≡ j ↑(↓) (z) dz Ex zi−1 x γ λ↑(↓) ↑(↓) ti ↑(↓) ↑(↓) ∼ F+,i + F−,i 1− , 1 − exp − ↑(↓) = ti σ0 2 ti γλ

(3.22)

(3.23)

where ti = (zi − zi−1 ) is the thickness of the layer. Willekens (1997) has found that for realistic spin-valves quite an accurate approximation to eq. (3.21) is obtained when choosing γ = 1/2.9 3.6. Applications of the Camley–Barnas model 3.6.1. Model calculations for conventional simple spin-valves As a ﬁrst illustration of the use of the CB-model, we show in ﬁg. 3.4 the results of a calculation of the current density in a simple F/NM/F spin valve (Coehoorn, 2000). A similar result was presented by Prados et al. (1999). The current density distribution cannot be obtained directly by experiment, but is of practical importance for applications. In the case of a stripe-shaped magnetoresistive conductor the current leads to an effective magnetic ﬁeld in the magnetic layers, directed perpendicular to the current direction (section 4.2.2). 9 For a single layer with t/λ 1, the Fuch–Sondheimer formula, eq. (3.18b), is retained by taking γ = 3/8 (Willekens, 1997).

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R. COEHOORN

Fig. 3.4. Results of a model calculation of the current density per unit electric ﬁeld, in a (8 nm F/ 2 nm NM/ 8 nm F) sandwich system, using the Camley–Barnas model. The magnetization of the left layer is in all ﬁgures up, whereas the right layer is up (down) for parallel (antiparallel) alignment. Model parameters: λNM = 20 nm, ↑ ↓ ↑ ↓ λF = 12 nm, λF = 0.6 nm, TCo/Cu = TCo/Cu = 1, diffusive scattering at the outer boundaries for both spin

directions. Electron density: 8.45 × 1028 m−3 in all layers (as in bulk Cu). The calculated parallel and antiparallel sheet resistances are 6.82 and 7.57 , and the calculated GMR ratio is 11.0%. (a)–(b) Contributions to the current density by electrons with vz > 0 for parallel and antiparallel conﬁgurations. The contributions from electrons with vz < 0 follow from a reﬂection in the z = 9 nm plane. (c) Total current densities for the parallel and antiparallel conﬁgurations, for spin up and spin down electrons. (d) Total current density, after summation of the contributions from both spin directions. From Coehoorn (2000).

In the calculations, scattering is assumed to be diffusive at the outer boundaries. The NM spacer layer thickness is 2 nm and the F layer thicknesses are 8 nm. The conductivity ↑ ↓ is spin-dependent in the F layers, with mean free paths λF = 12 nm and λF = 0.6 nm; λNM = 20 nm. These are realistic values for, e.g., Co/Cu/Co spin-valves (section 3.6.2). For both spins the interface transmission coefﬁcients are equal to T = 1, and scattering is diffusive at the outer boundaries. For this speciﬁc example the calculated parallel and antiparallel sheet resistances are 6.8 and 7.5 , and the calculated GMR ratio is 11.0%. As remarked already by Dieny (1992b), the current density in the NM layers is, for the model assumptions used, the same for both alignments. This may be understood by noting that contributions to the current density for a given electron direction (positive or negative vz ) are the same in the ﬁrst F layer that is traversed, and in the NM layer. Only the current density in the second F layer depends on the relative alignment (see ﬁg. 3.4 (a), (b)). Upon

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91

switching from the parallel to the antiparallel alignment, the only change of the current density occurs in a thin region in the F layers, close to the interface with the NM layers. From these observations, it is not difﬁcult to understand the size of the change of the total sheet conductance. Let us ﬁrst consider an extreme case, with a vanishing spacer layer ↓ thickness, inﬁnite F layer thicknesses, and negligible λF . The thickness of the regions close to the interfaces in which the current density depends on the alignment is then expected to ↑ be of the order 12 λF . The factor 12 takes into account that electrons with a wide distribution of velocity directions contribute to the current. Only half of the spin ↑ electrons, viz. only those with a velocity direction away from the spacer layer, contribute to the change of the current density near a given interface. Effectively, each interface therefore contributes a ↑ change of the sheet conductance that is equal to 14 λF σ0,F , so that the total effect is ΔGsh ≈ ↑ ↓ 1 2 λF σ0,F . For systems with ﬁnite layer thicknesses and λF = 0, the change of the sheet conductance is (from eq. (3.23) with γ = 1/2) approximately equal to (Willekens, 1997): ∼ ΔGbulk sh =

↑ 2tNM 1 σ0,F × exp − 2 λ↑ λNM F

×

↑ λF

2 2tF 2tF ↓ . 1 − exp − ↑ − λF 1 − exp − ↓ λF λF

(3.24)

The label “bulk” indicates that only spin-dependent scattering (SDS) in the bulk of the ↓ F-layers has been considered. For tNM = 0, λF = 0 and tF → ∞, eq. (3.23) yields indeed ↑ ΔGsh → 12 λF σ0,F , in agreement with the estimate given above. The exponential NM layer thickness dependence agrees with the simple expression for the MR ratio given already in section 1 (eq. (1.4)). When the conductivity in the F layers is strongly spin-polarized, the F layer thickness dependence is proportional to [1 − exp(−2tF /λ> )]2 , where λ> is the larger of the two mean free paths in the F layer. This quadratic form reﬂects the dual polarizer/analyzer role played by the two F layers. It is of interest to investigate the sheet conductance change for the case of interface SDS. We consider (otherwise identical) spin-valves in which the transmission probability through the interfaces is spin-dependent (T ↑ = T ↓ ), but in which the F-layer conductivity ↑ ↓ is spin-independent. Taking λF = λF = λNM = λ, and assuming no specular reﬂection at the interfaces, Willekens (1997) has shown that the application of eq. (3.21) (again with γ = 1/2) leads to 2

↑ −2tNM −2tF int ∼ 1 ↓ 2 × exp . ΔGsh = σ0 λ T − T (3.25) × 1 − exp 4 λ λ Here σ0 is the total F (or NM) layer conductivity. For multilayers, a similar expression has been derived by Barthélémy and Fert (1991). It may be concluded from a comparison of eqs (3.24) and (3.25) that different mechanisms can give rise to layer thickness dependences of the magnetoconductance that are not very dissimilar. Studies of the layer thickness dependence of the magnetoconductance and the conductance are necessary to distinguish interface from bulk contributions to the CIP-magnetoconductance.

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R. COEHOORN

Fig. 3.5. Results of a model calculations of the MR ratio of (F/NM)N /F multilayers with tNM = 1 nm versus the F layer thickness, for various periods N . The model parameters used were identical to those given in the caption of ﬁg. 3.4. From Dieny (1992b).

3.6.2. Model calculations for dual and multilayer structures Dieny (1992b) have applied the CB model in order to investigate the effect on the GMR ratio of increasing the number of repetitions in a multilayer. For this purpose they considered (F/NM)N /F multilayers, with an NM layer thickness equal to 1 nm, and with variable F layer thicknesses. For N = 2, the calculations describe a dual spin valve. The same set of scattering conditions are used as employed in the example given in section 3.6.1 (ﬁg. 3.4). As shown in ﬁg. 3.5, the maximum GMR ratio increases with increasing number of repetitions, due to the decreasing importance of diffusive scattering at the outer boundaries. The shift with N of the maximum in the MR curves to lower F layer thicknesses reﬂects the strongly spin-dependent mean free paths in the F layers. For N = 1 (a simple spin-valve) the F layers should be relatively thick in order to make it possible for the spin ↑ electrons to contribute to the conduction before scattering diffusively at the outer boundaries. On the other hand, in the case of a large number of repetitions, scattering at the outer boundaries becomes relatively unimportant, and the F layer should simply be just thick enough ↓ to scatter the spin ↓ electrons (i.e., of the order of λF = 0.6 nm), without giving rise to appreciable scattering of spin ↑ electrons. 3.6.3. Model calculations of the effect of specular boundary scattering From the point of view of its magnetoresistance, a multilayer with an inﬁnite number of repetitions is equivalent to a simple spin valve with perfectly specular scattering at the outer boundaries (and with F layer thicknesses that are half as large). The effect on the F-layer thickness dependence of the GMR ratio, upon an increase of the probability of specular scattering at the outer boundaries, is thus (qualitatively) the same as the effect of increasing N . Swagten et al. (1996, 1998), Dieny (1998), Bailey (2000a) and Li et al. (2000a) employed the CB model to investigate the effect of variations of the specular reﬂection coefﬁcient R at the outer boundaries. This work was motivated by successful experimental

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Fig. 3.6. Results of model calculations of the MR ratio of (tF1 nm F1/ 2.5 nm NM/ tF2 nm F2) layer structures, for the case of a variable specular reﬂection coefﬁcient R at the top outer boundary (a)–(c) or at both outer boundaries (d)–(f). In ﬁgures (a)–(c), scattering at the bottom outer boundary is diffusive. Model parameters: ↑ ↓ ↑ ↑ ↓ ↓ λNM = λNM = λF = 10 nm, TF/NM = 1; λF and TF/NM are obtained by the expressions given in the top part of the ﬁgure. From Swagten et al. (1998).

demonstrations of enhanced GMR ratios in systems including NiO layers (Anthony et al., 1994; Egelhoff et al., 1995; Swagten et al., 1996) (see section 2.1.6). Fig. 3.6 shows the results of calculations by Swagten et al. (1998) of the GMR ratio for (8 nm F1 / 2.5 nm NM/ tF2 F2 ) spin-valves. The results shown in ﬁg. 3.6 (a)–(c) were obtained for the case of diffusive scattering at the “bottom” outer boundary (with the F1 layer), and variable reﬂectivity R at the “top” outer boundary. The mean free path in the NM layer, and λ↑ in the F layers are both equal to 10 nm. The other parameters used are given in the top part of the ﬁgure. It is seen that already partial reﬂection at the top boundary enhances the GMR effect strongly, in particular for low F2 layer thicknesses (well below the largest mean free path). The F2 layer thickness at which the GMR ratio is maximal, tmax , shifts to lower values. The precise effect depends on the origin of the spin-dependent scattering (interface (a), bulk (b) or both (c)). For these parameter values, the largest possible enhancement of the GMR ratio is slightly less than a factor of two. For specular scattering at both outer inter-

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faces (ﬁg. 3.6 (d)–(f)), the trends are analogous to the effects obtained in ﬁgs. 3.6 (a)–(c). However, a much larger enhancement and a much larger shift of tmax are predicted. 3.6.4. Angular dependence of the conductance For a simple spin-valve of the type F/NM/F, in which scattering at the outer boundaries is diffusive, and in which no specular scattering takes place at the interfaces, the assumptions made within the CB model lead to a conductance that is proportional to (1 − cos θ ) (de Jong, 1995). This result is only to ﬁrst order in the GMR ratio equivalent to the expression given by eq. (1.2), within which the resistance varies linearly with (1 − cos θ ). The difference will be only signiﬁcant for systems with relatively large GMR ratios. The resistance was found to be proportional to (1 − cos θ ) for exchange-biased Py/Cu/Py/Fe50 Mn50 systems (Dieny et al., 1991a)and for an AF-coupled Fe/Cr/Fe system (Chaiken et al., 1990), with GMR ratios of about 3 and 0.4%, respectively. So far, only for a certain multilayer system, with a relatively large MR ratio, a linear variation of the conductance with (1 − cos θ ) has been observed (Duvail et al., 1995). For simple spin valves a non-linear angular variation of the conductance with (1 − cos θ ) is expected in the case of non-zero specular reﬂection at the interfaces between the magnetic and non-magnetic layers (Vedyaev et al., 1994; Barnas et al., 1997; Sheng et al., 1997). Xu and Mai (1999) showed that a signiﬁcant non-linearity can also arise within the CB model, for the case of a very high probability of specular scattering at the outer boundaries. This indicates that the successful development of specularly reﬂecting layers (such as NOLs) at the outer boundaries, leading to a high GMR ratio, could have a disadvantageous effect on the linearity of the response of a sensor. 3.6.5. Determination of scattering parameters Pioneering studies, aimed at analyzing the conductance and magnetoconductance of simple spin valves in terms of the parameters that enter the CB model, have been carried out by Dieny (1992b), Dieny et al. (1993). They studied F/Cu/F/Fe50 Mn50 spin-valves at 1.5 K, with F = Fe, Co and Ni80 Fe20 . From preliminary analyses, they concluded that no satisfactory description of the resistance and the magnetoresistance was possible using the same set of parameters. Parameter sets describing correctly the layer thickness dependence of the conductivities were found to underestimate the MR ratio by about 20% in relative value. This was ascribed to anisotropic scattering due to the presence of grain boundaries (see also section 2.9.2). In the case of a columnar grain structure, stronger scattering is expected for electrons with velocities nearly parallel to the layers (which predominantly determine the resistivity), as compared to scattering for electrons with a substantial perpendicular velocity component (whose contribution to the magnetoresistance is larger). In order to incorporate anisotropic scattering, Dieny made use of an ad hoc extension of the semiclassical model, by using the following expression for the anisotropic mean free path for scattering in the bulk of the layers: λ(θ ) = λ sin2 θ + λ⊥ cos2 θ,

(3.26)

in which θ is the angle between the velocity vector and the ﬁlm-normal, and in which the parallel and perpendicular mean free paths are related by 1 1 1 = + . λ λ⊥ λgr

(3.27)

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TABLE 3.1 Resistivities and scattering parameters for (F/Cu/F/Fe50 Mn50 ) spin-valves at 1.5 K, as obtained by Dieny et al. (1992a, 1993) using the CB model. The parallel and perpendicular mean free paths are related to each other by eq. (3.27), with λgr = 40 nm ↑

↑

↓

↓

↑

Materials

ρ (10−8 m)

λ (nm)

λ⊥ (nm)

λ (nm)

λ⊥ (nm)

TF/Cu

Cu Ni80 Fe20 Co Fe

4.7 15.4 10.7 10.5

13 11 14 7

20 15 21 8.5

13 0.6 1 7

20 0.6 1 8.5

1 1 1

↓

TF/Cu 1 0.2 0.6

The parameter λgr is expected to be of the order of the grain diameter. Satisfactory ﬁts to the layer thickness dependent conductance and magnetoconductance were obtained when diffusive scattering was assumed at the outer boundaries, and taking λgr = 40 nm for the F as well as the Cu layer. For minority-spin electrons in Ni80 Fe20 or Co layers no anisotropy of the mean free path was taken into account. In view of the small values of their mean free paths, the ﬁt would be very unsensitive to such an effect. An overview of the scattering parameters that were obtained is given in table 3.1. The results indicate that for permalloybased systems spin-dependent bulk scattering is the predominant origin of the GMR effect, whereas for Co/Cu and Fe/Cu systems spin-dependent scattering at the interfaces plays an important role. For Fe, bulk scattering is seen to be almost spin-independent. An excellent example of the application of the CB model to the room temperature magnetoconductance of state-of-the-art spin-valves that are relevant to sensor applications has been presented by Dieny et al. (2000a). They studied simple PtMn-based spin-valves with a composite free layer: UL/ 3.0 nm Ni80 Fe20 / 1.0 nm Co90 Fe10 / 2.0 nm Cu/ x nm Co90 Fe10 /20 nm PtMn/Ta. The composition of the underlayer (UL) was not speciﬁed. For x = 2.0 nm, the GMR ratio is surprisingly high, viz. 12.0% (see also table 2.1). Note that the free and pinned layers are quite thin, as compared to the optimal F layer thicknesses for Fe50 Mn50 -based spin valves (ﬁg. 1.8). Based on numerous studies of a wide variety of spin-valve structures, the authors concluded that the effect must be due to a certain degree of specular scattering of electrons at the underlayer/Ni80Fe20 and Co90 Fe10 /PtMn interfaces. The ﬁnal results of their ﬁts are given in table 3.2. As compared to the mean free paths given in table 3.1 from a low-temperature study, the mean free paths in Cu and Ni80 Fe20 are somewhat smaller. Neglecting differences of the microstructure, this could be ascribed to increased electron– phonon scattering and (in the permalloy layers) electron–magnon scattering. Fig. 3.7 shows the effect of “switching on” specular scattering at the interface with the PtMn layer. Consistent with the results shown in ﬁg. 3.6, it leads to a high GMR ratio, even for small thicknesses of the pinned layer. The sheet resistance is slightly lowered. Analyses by the same authors of spin-valves with a synthetic AF pinned layer (Co90 Fe10 /Ru/Co90 Fe10 replaces the single Co90 Fe10 pinned layer in the structures studied above) led to the conclusion that specular reﬂection occurs also at Co90 Fe10 /Ru interfaces. The reﬂection coefﬁcient for the Co90 Fe10 /Ru interface is R = 0.3. Upon analysing the

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TABLE 3.2 Scattering parameters for (buffer/ 3.0 nm Ni80 Fe20 / 1.0 nm Co90 Fe10 / 2.0 nm Cu/ x nm Co90 Fe10 / 20 nm PtMn/Ta) spin valves at room temperature, obtained by Dieny et al. (2000a, 2000b) using the CB model. The composition of the high-resistivity buffer layer is not speciﬁed. The (spin-dependent) transmission coefﬁcients T and the (spin-independent) specular reﬂection coefﬁcient R correspond to the interfaces between the indicated layer and the previous layer in the stack. Specular reﬂection occurs only at the buffer/Ni80 Fe20 and Co90 Fe10 /PtMn interfaces Materials

λ↑ (nm)

λ↓ (nm)

T↑

T↓

R

buffer Ni80 Fe20 Co90 Fe10 Cu Co90 Fe10 PtMn Ta

0.2 7 9 12 9 0.2 0.4

0.2 0.7 0.9 12 0.9 0.2 0.4

0.6 1 1 1 0.5 1

0.6 1 0.5 0.5 0.5 1

0.3

0.3

Fig. 3.7. Experimental and theoretical sheet resistance (a) and MR ratio (b) of a SV with the composition (underlayer/ 3 nm Py/ 1 nm Co90 Fe10 / 2 nm Cu/ tp nm Co90 Fe10 / 20 nm PtMn), versus the thickness tp of the Co90 Fe10 pinned layer. Full line: model calculation using the CB model (table 3.2) assuming specular reﬂection at the Co90 Fe10 /PtMn interface (reﬂection coefﬁcient R = 0.3). Dashed line: model calculation assuming R = 0. From Dieny et al. (2000a).

CIP-magnetoconductance, the possibility that specular reﬂection occurs at metal/metal interfaces thus cannot always be disregarded. We emphasize that the mean free paths that are given in tables 3.1 and 3.2 should be considered as effective values, which could differ considerably from the values that would have been obtained in the absence of spin-mixing due to the spin-orbit interaction and/or due to electron–magnon interactions (see sections 3.1 and 3.7).

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3.6.6. Spin-valves with a back layer Spin-valves with a back layer (or “spin-ﬁlter spin-valves”) with the structure B/F/NM/F/ AF (with B the back layer, see ﬁg. 2.1(c)) have been studied in order to extract the scattering parameters in an alternative manner, as well as for their favorable combination of properties for applications in, e.g., read heads (section 2.1.3). They were introduced by Gurney et al. (1994), in order to study the spin-polarized mean free paths in the back layer. The authors observed that the variation of the sheet magnetoconductance as a function of the thickness tB of the back layer is to a very good approximation given by tB ΔGsh = ΔGsh,0 + ΔGsh,B 1 − exp − (3.28) . LB From room temperature experiments the following values were found for the ﬁt parameter LB : 19 nm (Cu), 2.2 nm (Au50Cu50 ), 4.6 nm (Ni80 Fe20 ), 5.5 nm (Co) and 1.5 nm (Fe). Gurney et al. associated this length scale to λ> , the largest of the two mean free paths in the back layer. An analysis using the CB model, using the approximation made in eq. (3.23) with γ = 1/2, has later revealed that for non-magnetic back layers the change of the sheet resistance is indeed expected to have the form given by eq. (3.28), but that LB is approximately equal to half the mean free path (Willekens, 1997). For Cu, the resulting value λ = 38 nm is in fair agreement with the “intrinsic” value λi ≈ 33 nm obtained by Rijks et al. (1995a) from single thin ﬁlm studies in the thick ﬁlm limit (see section 3.6.5). For ferromagnetic back layers in which λ↑ λ↓ , Rijks (1996a) has found from model studies using the CB-model that to a very good approximation ↓ T 1 ↑ λ↓ . LB ∼ (3.29) λ + = 2 T↑ In view of the parameter values given in table 3.1, a further approximation, LB ≈ 12 λ↑ , is expected to be very good. This leads for Ni80 Fe20 and Co to λ↑ ≈ 9 and 11 nm, respectively. These values are quite close to the results given in table 3.2 for Ni80 Fe20 and Co90 Fe10 . Less direct estimates of the mean free paths for the other spin direction yielded λ↓ 0.6 nm (Ni80 Fe20 ), λ↓ 1 nm (Co), and λ↓ = 2.1 ± 0.5 nm (Fe). Also these data thus provide evidence for a strong spin-dependence of the scattering in permalloy and Co, and a much weaker spin-dependence of the scattering in Fe. A different type of study of spin-valves with a back layer was carried out by Swagten et al. (1997) (see also Strijkers et al., 1997). Their aim was to investigate the importance of the interfacial contribution to the spin-polarized scattering that gives rise to the GMR effect. They noted that making a distinction between bulk and interface spin-dependent scattering (SDS) is difﬁcult when conventional spin-valves are used, because of the very similar forms of the F layer thickness dependences of the conductance change (within the approximation leading to eqs (3.24) and (3.25), these become even identical for bulk SDS with λ↑ λ↓ ). In contrast, for samples of the type F1 /NM/F2 /B, with a thick back layer, the F2 layer thickness dependences of the conductance change are expected to be very different for bulk and interface SDS. In the case of interface SDS a large magnetoconductance effect will even occur in the limit of a vanishing thickness of layer F2 , which is called the “probe layer”. In contrast, in the case of bulk SDS an F layer thickness of the order of the smallest of the mean free paths will be required to obtain an effect. Thinner layers are

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Fig. 3.8. Sheet conductance change from model calculations (a), (b) and experiment (c) for F/NM/F/B spin-valves (d), where B is a non-magnetic back layer. The CB model calculations (from Willekens (1997) are for (2.5 nm F/ ↑ ↓ ↑ ↑ 2.5 nm NM/ dPR nm F/ dB nm NM) ﬁlms, and were carried out for λNM = λNM = λF = 20 nm and TF/NM = 1. Figure (a) gives the sheet conductance change ΔGint assuming only interface spin dependent scattering (SDS), ↓ ↑ ↓ with λF = λF and with TF/NM = 0.4. Figure (b) gives the sheet conductance change ΔGbulk assuming only ↓

↓

bulk SDS, with λF = 2 nm and with TF/NM = 1. Figure (c) gives experimental results for (2.5 nm Co/ 3 nm Cu/ dPR nm Co/ dB nm Cu) SVs, from Swagten et al. (1997).

then not efﬁcient as a spin-ﬁlter. In ﬁg. 3.8 (a) and (b) results of model calculations are given for the change of ΔGsh for the cases of interface SDS and bulk SDS, respectively. In ﬁg. 3.8(c) experimental data, taken at 10 K, are given for a (7.5 nm Co/ 0.6 nm Ru/ 2.5 nm Co/ 3.0 nm Cu/ dpr nm Co/ dB nm Cu) spin-valve. The 2.5 nm reference Co layer is pinned by employing a synthetic AF structure. The ﬁnite ΔGsh at extremely small dpr is consistent with the occurrence of interfacial SDS. The dependence on the back layer thickness agrees, at least qualitatively, with the prediction given in ﬁg. 3.8(a). For permalloy, similar results were obtained. For Co and permalloy, the probe layer thickness required to obtain the full conductance change (called ξ ) was observed to be 0.1 to 0.3 nm at 10 K, and to increase to 0.2–0.7 at room temperature, depending on the details of the sample structure. This length scale was interpreted as the thickness which is required to form a (magnetically) continuous layer. A possible alternative explanation of the results, in terms of bulk SDS characterized by an extremely small minority spin mean free path, was argued to be less likely. In that case, ξ would be of the order of λ↓ . However, in contrast to the

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Fig. 3.9. Room temperature Cu layer thickness dependence of the MR ratio for (Ta/ 2 nm Ta/ NOL/ 2.5 nm Py/ 1 nm Co90 Fe10 / t nm Cu/ 3 nm Co90 Fe10 / 7 nm Ir–Mn/ 2 nm Ta) SVs with a NOL near the free layer, and for a non-oxide reference sample. The solid and dashed lines represent results of calculations using the CB model. The inset gives the results of sheet resistance measurements. For the model parameters, see the text of section 3.6.7. From Gibbons et al. (2001).

observations, it would then be expected to decrease with increasing temperature. For Co, the observed interface SDS agrees with the analysis by Dieny et al. (1992a, 1993) of data for conventional spin valves (table 3.1), but for permalloy, Dieny et al. did not detect any signiﬁcant interface SDS. 3.6.7. Spin-valves with a NOL near the free layer Gibbons et al. (2001) have analysed the effect of including a nano-oxide layer (NOL) near the free layer of spin-valves with the layer structure Substrate/ 2 Ta/ NOL/ 2.5 Py/ 1 Co90 Fe10 / t Cu/ 3 Co90 Fe10 / 7 Ir–Mn/ 2 TaOx . The experimental Cu layer thickness dependence of the MR ratio could be described within the CB model using the following mean free paths: λ↑ (Py) = 5.9 nm, λ↓ (Py) = 0.6 nm, λ↑ (Co90 Fe10 ) = 9 nm, λ↓ (Co90 Fe10 ) = 0.6 nm, λ↑ (Cu) = λ↓ (Cu) = 30 nm and λ↑ (Ta) = λ↓ (Ta) = 0.6 nm. For the non-oxide reference sample T = 0.5 and D = 0.5 are assumed at the Ta/Py interface. For the oxide sample fully specular reﬂection is assumed at the NOL/Py boundary. As shown in ﬁg. 3.9, the model provides a good description of the sheet resistance and magnetoresistance as a function of the Cu layer thickness. The discrepancy at small Cu thicknesses for the non-oxide sample can be attributed to the absence of a proper plateau region in the R(H ) transfer curve due to strong ferromagnetic coupling across the Cu layer (as in ﬁg. 1.7). Apparently, the use of a NOL suppresses this effect. Indeed, it has been observed that NOLs give rise to planarisation of the layer structure, so that ferromagnetic pin–hole coupling and “orange–peel” coupling are reduced (see section 2.1.7).

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3.6.8. Analysis of scattering lengths using spin-valves with Ru barrier layers A clear conclusion from the analyses by Dieny et al. and Gurney et al. of the magnetoconductance of spin-valves of several types (see sections 3.6.5 and 3.6.6) is that the conductance in Ni80 Fe20 , Co and Co90 Fe10 is strongly spin-polarized. The ratios λ↑ /λ↓ between the two mean free paths in the F-metals that follow from tables 3.1 and 3.2 are around 10, or even higher in some cases. Qualitatively, these results are consistent with theoretical considerations (section 3.1) and experimental results for dilute alloys and CPP-systems (section 3.2). However, in view of the severe simpliﬁcations made within the model concerning the electronic structure, it would be important to further investigate the consistency of the model. Parkin (1994a) has suggested to employ the Drude formula (eq. (3.3)) for this purpose. For a class of nearly free-electron metals with identical electron densities and electron masses, the ratio σ0 /λ> is expected to be twice as large for non-magnetic metals than for metals for which essentially only electrons with one type of spin contribute to the conductivity. Surprisingly, Parkin found (from a study of spin-valves, discussed below) that for a large number of magnetic and non-magnetic alloys, including Ni–Fe, Ni–Co, Co, Cu– Fe and Cu–Au alloys, the ratio σ0 /λ> is not signiﬁcantly different. This result (reproduced in the thesis of Willekens (1997)) would imply that scattering in these materials is spinindependent, a conclusion that raised a lot of controversy. Experimental results that are consistent with those of Parkin have been presented by Strijkers et al. (1996). They carried out an experimental study for the cases of Co and Cu. As in the study by Parkin, σ0 /λ> for the magnetic metal (Co) was deduced from the magnetoconductance of conventional spin-valves with a thin Ru layer in the free Co layer. The Ru layer is considered to function as a diffusive scattering blocking layer, so that λ> (Co) can be obtained from the dependence of the magnetoconductance on the distance of the Ru layer to the nearest Cu/Co interface. As would then be expected, the experimental data could be fairly well described by eq. (3.28). Making the assumption that λ↑ λ↓ (see section 3.6.5), λ> is then obtained using L ≈ 12 λ↑ . For Cu, σ0 /λ was deduced in a similar manner from the magnetoconductance of Co/Cu/Co/Cu spin-valves, containing a thin Ru blocking layer with a variable position in the Cu back layer. The conductivities were obtained from single layer studies. The results, obtained at different temperatures, are given in ﬁg. 3.10. For Cu, λ is very similar as obtained by Rijks et al. (section 3.6.5) and by Gurney et al. (section 3.6.6) for relatively thick ﬁlms. For Co, λ↑ is at low temperatures very similar to the values obtained by Dieny et al. (table 3.1). The decrease of λ↑ with increasing temperature is reasonably consistent with the concurrent decrease of the conductivity. The full line is a ﬁt through the data points. The dashed and dash-dotted lines have been obtained from the Drude model, using the electron density of Cu (one conduction electron per atom) and the free electron mass. For Cu, the measured ratio σ0 /λ is only slightly larger than the theoretical value for non-magnetic metals. However, for Co, σ0 /λ> is a factor 1.5–2.5 larger than expected when λ↑ λ↓ . Assuming that λ↑ λ↓ for Co, these results could be indicative of the necessity to extend the CB-model, or to go beyond it, in order to better take the real electronic structure into account. Modelling on the basis of realistic band structures (section 3.8) provides evidence for unequal effective values for Cu↑(↓) , Co↑ and Co↓ electrons of the quantity e2 n/m∗ that enters the Drude formula, and provides evidence for the possibility that strong

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Fig. 3.10. Temperature dependent conductivities of Co and Cu, as deduced from single layer studies, as a function of the majority spin mean free path, as deduced from analyses of the magnetoconductance of (Co/Cu/Co/Ru/Co) SVs and (Co/Cu/Co/Cu/Ru/Cu) SVs, respectively. The full line represents a linear ﬁt through the data. The dashed and dashed-dotted lines are based on the free electron Drude model, using the electron density of bulk Cu. From Strijkers et al. (1996).

specular scattering occurs for majority spin electrons in Cu at the Co/Cu interfaces. Alternatively, Dieny et al. (2000a, 2000b) have suggested that the controversy might be resolved by taking into account that partial specular reﬂection occurs at Co/Ru interfaces (see section 3.6.5) in contrast to the assumption made by Parkin and Strijkers et al. It is expected that L then becomes lower (see, e.g., ﬁg. 3.6(b)), so that λ↑ /L > 2. This suggestion is not contradictory with the observation by Strijkers et al. that thin Ru layers at Co/Cu interfaces drastically reduce the GMR ratio, because that could be related to strong diffusive scattering at the Cu/Ru interfaces. No modiﬁcation would then be required for the mean free path obtained for Cu. 3.6.9. Analysis of experiments with δ-doped magnetic layers Marrows and Hickey (2001) studied the magnetoresistance of Co/Cu/Co/Fe50 Mn50 spinvalves containing sub-monolayers (a few tenths of a monolayer, leading to “δ-doping” ) of 18 transition metal elements in the pinned and free Co layers. Fig. 3.11 shows the dependence of the GMR ratio on the distance x of the δ-layers from the Co/Cu interfaces. The GMR ratio for the Co reference system is, of course, independent of x. It is quite small (4%) due to the use of Co layers with a suboptimal thickness, d = 2.5 nm (compare ﬁg. 1.8). The effects of the δ-layers were argued to be of an electronic origin, because no effects on the microstructure were found. In the framework of the CB model, the simplest way to describe the effect of the δ-layer would be to treat it as an interface with spin-dependent specular transmission and diffusive scattering coefﬁcients, and to neglect possible specular reﬂection. A guideline for trends in the diffusive scattering coefﬁcients is obtained from the residual resistivities for impurities in Co (see section 3.2), given in ﬁg. 3.11. We consider two extreme cases. (1) Negligible majority-spin scattering and strong-minority spin scattering. The GMR ratio is then increased when x is smaller than the minority spin mean free path, λ↓ (Co), and the δ-layers will effectively enhance the spin-selectivity of interface scattering.

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Fig. 3.11. Dependence of the room temperature MR ratio of δ-doped (Si substrate/ 5 nm Ta/ (2.5 − x) nm Co/ X/ x nm Co/ 3 nm Cu/ x nm Co/ X/ (2.5 − x) nm Co/ 8 nm Fe50 Mn50 / 2.5 nm Ta) SVs, where X is an elemental metal layer with a submonolayer thickness, on the doping layer position x. At the top of each panel the residual majority and minority spin resistivities of dilute CoX systems are included, as given by Campbell and Fert (1982). From Marrows and Hickey (2001).

When x λ↓ (Co) almost no positive effect is expected, because minority spin electrons that have passed the Cu layer and the Co/Cu interface have then already been scattered in the bulk of the Co layer. Making use of the result λ↓ (Co) ≈ 1 nm (section 3.6), this picture can provide a qualitative explanation for the data for Fe and Ni. The result is consistent with the spin-dependence of scattering at Fe and Ni impurities in Co. (2) Strong majority-spin scattering. The effective thickness of the F layers is then reduced to x. The GMR ratio is zero for x = 0, and increases until x is of the order of the majority-spin mean free path, λ↑ . For the samples studied, such a saturation effect is not expected to be visible, because the Co layer thickness is much smaller than λ↑ (Co) ≈ 10–15 nm. One then expects a linear increase of the GMR ratio until x = d. For small x this provides a qualitative explanation for the data for Cr, Mo, Ru, Ta and W. This is consistent with the large majority-spin residual resistivity of dilute Coalloys containing these elements. When in the parent structure scattering at the outer boundaries is diffusive, one expects that the GMR ratio approaches the value for the parent system when x approaches d. For Ta δ-layers, the same material as the underlayer, this is seen to be a good approximation. The full GMR ratio is not quite reached when x = d, indicating that the thickness of the Ta δ-layer is in reality not entirely negligible. For many of the other systems larger deviations

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are found, indicating that improved descriptions of the structure, electronic structure and transport process are required. 3.7. Extensions of the semiclassical Camley–Barnas model 3.7.1. Overview Several effects that were not treated in the original version of the CB-model are taken into account in extended versions. In this section a brief overview is given. Dieny et al. (1992b, 1993) proposed to include anisotropic scattering due to grain boundaries, in a manner already described in section 3.6.5 Hood and Falicov (HF) (1992) proposed a generalization of the CB model, by introducing layer and spin-dependent effective masses and Fermi-velocities. The resulting potential steps at the interfaces lead to reﬂection and refraction at the interface, with a probability that depends on the angle of incidence. Potential steps at the interfaces would result in anomalies in the angular variation of the resistance (Vedyaev et al., 1994), which are so far not observed. Electronic structure calculations for Co/Cu/Co systems indicate that potential steps nevertheless can be important, but that their effect cannot simply be treated within a semi-classical model (see section 3.8). Litvinov et al. (1997) extended the HF-model by including geometrical interface roughness, characterized by two parameters: the roughness amplitude and the roughness lateral correlation length. It was shown that a spin-dependent electronic structure in the F layers, combined with interface roughness but without any other form of spin-dependent scattering, can lead to a considerable magnetoresistance. Vedyaev et al. (1992) and Zhang and Butler (1995) proposed to improve the CB model by excluding contributions to the conductivity due to electrons with velocities with respect to the x–y plane below a certain cutoff angle, given by θc = arccos(C · π/tkF ), with C = 1; t and kF are the layer thickness and the Fermi wave vector, respectively. This takes in an approximate way into account that in a thin ﬁlm the perpendicular wave vector is quantized due to the (partial) conﬁnement of states between the outer boundaries. In the CB “point particle” model electrons with velocities that are almost in-plane can travel long distances in the layer without “probing” the scattering potential at the outer boundaries, or, in a layered structure, in the other layers. This is forbidden by the quantum-mechanical position-wave vector uncertainty relationship. Rijks (1996a) argued that for thick ﬁlms such as spin valves C = 0.5 is more appropriate. When t is of the order of 10 nm, θc is of the order of a few degrees. Camblong and Levy (1992), Brataas and Bauer (1994), and Zhang and Butler (1995) argued that scattering at interfaces is expected to depend on the angle of incidence, even in the absence of potential steps. Camblong and Levy pointed out that this effect could be incorporated in the CB model by treating the interfaces as a thin additional layer instead of a mathematically sharp plane (see also Johnson and Camley, 1991). Interface and bulk scattering are then treated in the same way. In fact, it is quite natural to introduce thin interface layers, as for many systems of practical interest the scattering probabilities are non-bulk like in interface zones with a thickness of more than one atomic layer. In the limiting case of an interface layer with an inﬁnitesimal thickness ε, the effective specular transmission coefﬁcient is given by T (θ ) = exp(−ε/(λ cos θ )), in which λ is the mean free path and θ

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is the angle of incidence with respect to the ﬁlm normal. In this approach the ratio ε/λ is the only adjustable parameter. It should be noted that, similarly, specular reﬂection at the outer boundaries may be described more adequately using an angular dependent reﬂection parameter (Zhang and Butler, 1995). Rijks et al. (1993, 1995b) and Granovskii et al. (1995) argued that scattering within the magnetic layers is also anisotropic as a result of the anisotropic MR effect. This leads to an interference between the GMR and AMR effect: the GMR ratio may depend on the direction of the current with respect to the applied magnetic ﬁeld, and the AMR ratio may depend on the relative alignment of the magnetization of successive magnetic layers. Experimental proof of such an interference has been obtained by Dieny et al. (1995). See also section 2.6. Rijks et al. (1995a) pointed out that mean free paths are generally layer thickness dependent, due to a layer thickness dependent defect scattering (see section 2.9.2), and proposed to take this into account when analysing the layer thickness dependent conductivity of spin valves (1996a). Duvail et al. (1995) extended the CB model in order to take the effect of spin-mixing due to electron–magnon scattering into account. This provides a method for modelling the temperature dependence of the magnetoresistance. The approach is discussed in the next subsection. 3.7.2. Modelling spin-mixing – temperature dependence of the magnetoresistance Within the CB model, it is assumed that the electrical current is the sum of spin ↑ and spin ↓ currents in two independent spin-channels. This is the low-temperature limit of the two-current model. However, at ﬁnite temperatures spin-mixing by electron–magnon scattering couples the two spin-currents, because upon electron–magnon scattering momentum is transferred (Fert and Campbell, 1976; Fert et al., 1995). Spin-mixing by the spin-orbit interaction was already discussed in section 3.1. The resulting resistivity enhancement in the bulk of a material is given by eq. (3.5). In the case of a layered system, the electron distribution functions g ↑(↓) (v, z) are obtained by solving the following pair of spin-coupled Boltzmann transport equations (Duvail et al., 1994): ↑

vz ·

∂f g ↑ (g ↑ − g ↓ ) ∂g ↑ − eEx · vx 0 = − ↑ − ∂z ∂ε τ τ ↑↓

vz ·

∂f g ↓ (g ↓ − g ↑ ) ∂g ↓ − eEx · vx 0 = − ↓ − , ∂z ∂ε τ τ ↑↓

and ↓

(3.30)

where τ ↑↓ is the called the spin-mixing time, which is implicitly deﬁned by the expression ρ ↑↓ = m/(ne2 τ ↑↓ ) for the spin-mixing resistivity. Analytical solutions have been given by Chen and Hershﬁeld (1998), who, however, made the speciﬁc assumption that the effective intrachannel relaxation times are partly determined by the spin-mixing relaxation time. For a layer i in between z = a and z = b (b > a), the general solution is (omitting for simplicity indices i): −(z − a) −(z − a) ∂f0 ↑ ↑ ↑ 1 − A+ exp − B+ exp g+ = −eτ1 Ex vx ∂ε τ3 vz τ4 vz

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

and ↓

g+ = −eτ2 Ex vx

∂f0 −(z − a) −(z − a) ↓ ↓ 1 − A+ exp − B+ exp ∂ε τ4 vz τ3 vz

105

(3.31)

for the case vz > 0, with τ1 =

s + 2t , st + rs + rt

(3.32a)

τ2 =

r + 2t , st + rs + rt

(3.32b)

τ3 =

2 , r + s + 2t + (r − s)2 + 4t 2

(3.32c)

τ4 =

2 , r + s + 2t − (r − s)2 + 4t 2

(3.32d) ↑

↓

↓

↑

where r = 1/τ ↑ , s = 1/τ ↓ and t = 1/τ ↑↓ . B+ /A+ = C and B+ /A+ = −C, with −r + s + (r − s)2 + 4t 2 . C= (3.33) 2t As in eq. (3.16), the solutions for vz < 0 can be obtained from eq. (3.31) by replacing the argument of the exponential functions by −(b − z)/τ |vz |. Like the coefﬁcients F employed in eq. (3.16), the coefﬁcients A and B follow from the application of the boundary conditions discussed in section 3.5. The effective bulk spin ↑ and spin ↓ contributions to the conductivity are proportional to τ1 and τ2 , and eq. (3.28) follows straightforwardly. The B-terms are absent without spin-mixing, and become important when either vF τ3 or vF τ4 becomes smaller than the layer thickness. When λ↓ λ↑ , as for Co and permalloy, the minority-spin relaxation time will determine (for a given spin-mixing rate) whether spinmixing is important. Whereas low-temperature analyses of the (magneto)resistance most sensitively probe the longer of the two mean free paths, λ> , the temperature dependence due to spin-mixing is thus expected to be quite sensitive to λ< . Calculations of the temperature dependence of the CIP (magneto)conductance can then be carried out by expressing the resistance ρ σ (T ) in each spin channel σ as the sum of a 4.2 K residual resistance (which will generally depend on the microstructure) and an intrinsic temperature dependent contribution δρ σ (T ). The spin-mixing resistivity ρ ↑↓ (T ), δρ ↑ (T ) and δρ ↓ (T ), which are all zero at zero temperature, can be taken from the literature (e.g., from Loegel and Gautier (1971) for the case of Co). Without the introduction of additional parameters, one thus obtains temperature dependent scattering rates in the bulk of the layers. Duvail et al. (1994) introduced this formalism, and applied it successfully to describe the temperature dependence of the CIP resistance and magnetoresistance of Co/Cu multilayers. They treated scattering at interfaces by introducing thin interface layers. The introduction of a temperature dependent resistivity and spin-mixing in the bulk of the Co layers each could explain only 25 to 35% of the decrease of the magnetoresistance with temperature.

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Fig. 3.12. Results of model calculations of the temperature dependence of the MR ratio of SVs with the layer structure (tCo nm Co/ 2.2 nm Cu/ 5 nm Co/ 10 nm Fe50 Mn50 ), taking spin-mixing into account. The symbols are calculated results, the lines are linear extrapolations to T0 . From Dieny (1998).

The remaining effect could be explained by assuming a relatively large increase with temperature of the resistance for both spins of the interface layers. This is likely to be related to the smaller thermal stability of the Co magnetic moments at the interface with Cu, than in the bulk of the layers. Dieny (1998) used the method to successfully describe the temperature dependence of the magnetoresistance of Co/Cu/Co/Fe50 Mn50 spin-valves with a varying free layer thickness. As shown in ﬁg. 3.12, the formalism reproduces the almost linear temperature dependence of the magnetoresistance, and its extrapolation to a single characteristic temperature, T0,SV (see also section 2.5). From the results obtained by Duvail et al. for Co/Cu multilayers, T0,SV is expected to be determined by the strongly temperature dependent scattering at the interfaces. 3.8. Semiclassical models based on realistic band structures One of the weaknesses of the CB-model is that it uses the free-electron model for describing the electronic structure within each of the layers. Butler and coworkers (Butler et al., 1998, 2000; Zhang and Butler, 2000) developed a semi-classical model for the conductance of spin-valves and other layered materials that is based on realistic band structures. These are assumed to be uniform across the entire thickness of each layer. The deviations that occur in the very thin region near the interfaces (with a thickness of a few atomic layers) are thus neglected. Scattering in the bulk of the layers is parametrized by choosing spin-dependent relaxation times, τ ↑(↓) . For Cu↑(↓) , Co↑ and Co↓ , semi-classical calculations of the ratios σ/τ yield 8.3 × 1020 , 5.2 × 1020 and 4.2 × 1020 ( ms)−1 (Butler et al., 1998). For comparison, within the Drude model σ/τ = e2 n/m∗ = 1.2 × 1021 ( ms)−1 , for a density of 0.5 conduction electrons per atom per spin in Cu with m∗ = m0 . It follows from the differences between these σ/τ ratios that the quantitative values of the mean free paths that are obtained from analyses using the CB model in its original form (with identical electronic structures in all layers) should be used with care.

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

107

Fig. 3.13. (a) Schematic representation of channeling of majority spin electrons in a Cu spacer layer between two Co layers in a [111] textured SV. (b) Calculated spin-resolved current densities for parallel and antiparallel alignments of the Co layers in a (2.5 nm Cu/ 2 nm Co/ 2 nm Cu/ 2 nm Co) (111) spin-valve, from a semi-classical model using realistic electronic structures in each of the layers. The magnetization direction of the left (bottom) Co layer remains ﬁxed, whereas the right (top) layer switches. From Butler et al. (1998).

Scattering at the interfaces and outer boundaries is parametrized by choosing spindependent specularity parameters S and p, respectively. The coefﬁcients for specular transmission, specular reﬂection and diffusive scattering at the interfaces are then given by ST ∗ , SR ∗ and (1 − S). Here T ∗ and R ∗ = (1 − T ∗ ) are wave vector dependent specular transmission and reﬂection probabilities, respectively, that are obtained from the matching of the band structures at the perfect interfaces. The transmission T ∗ is generally different for opposite directions of incidence. The relevance of being able to treat the interfaces more properly than in the CB model has becomes evident from work of Stiles (1996) and Butler et al. (1996a, 1996b). Stiles showed from ﬁrst principles calculations that the transmission and reﬂection coefﬁcients for F/NM interfaces can be strongly spin-dependent, and proposed that this effect can give rise to a large GMR ratio, even when the all diffusive scattering in the system would be spin-independent. Butler and coworkers showed the importance of this effect for the GMR ratio of (111) Co/Cu/Co spin valves with perfectly ﬂat interfaces, from quantummechanical (Butler et al., 1996a, 1996b) and semi-classical (Butler et al., 2000) transport calculations based on realistic band structures. For majority spin electrons in the Cu layer the probability of specular reﬂection at the Co layers is, after averaging over all angles of incidence, much larger than for minority spin electrons. For a range of angles of incidence, majority spin electrons in Cu are even completely reﬂected from the Co layer. This happens when there are no states in the Co layer with the same parallel component of the wave vector. When the Co magnetizations are parallel, this internal reﬂection at the Co/Cu interfaces leads to channeling of majority spin electrons in the Cu layer (ﬁg. 3.13(a)). Their effective relaxation time is enhanced, because diffusive scattering at the outer boundaries is avoided. The conductance is thus increased. As this effect is absent in the antiparallel

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TABLE 3.3 Resistivities in the Cu and Co layers in spin-valves, as obtained from ﬁts to the experimental layer thickness dependence of the conductance and magnetoconductance at room temperature using a semi-classical model and a quantum-mechanical model based on realistic electronic structures. The systems studied are (I): Ta/Cu/Co/Cu/Co/Fe50 Mn50 /Ta spin-valves (Butler et al., 1998) and (II) Ta/Co/Cu/Co/Fe50 Mn50 /Ta spin valves (Butler et al., 1997). For Co in system II the conversion of the relaxation times (the primary variables in the calculations) to resistivities was not given explicitly. The approximate values given in the table are consistent with the total Co resistivity, and with the spin-dependence of the conversion factors given for system I. The assumptions made concerning scattering at the interfaces are discussed in the text Resistivity, ρ ↑(↓) (10−8 m)

Material

Cu↑(↓) Co↑ Co↓

System I Semi-classical model

System II Quantum-mechanical model

18 13.7 328

5.6 ≈ 15 ≈ 370

state, it leads to an increase of the GMR ratio. It should be emphasized that the channeling effect will be absent when scattering at the Co/Cu interfaces is fully diffusive (S = 0). Interestingly, this would imply that matching of the F and NM inner potentials is not the most ideal situation, in contrast to what was argued in section 3.2. Butler et al. (1998) used the semi-classical model to simultaneously analyse the Co layer thickness dependent conductance and magnetoconductance at 300 K of a set of [111] textured sputter deposited Ta/Cu/Co/Cu/Co/Fe50 Mn50 /Ta spin-valves with back layer. Good ﬁts were obtained when using the Cu and Co resistivities that are given in table 3.3, and when using S = 1 and S = 0.3 as the majority and minority spin specularity factors for the Co/Cu interfaces. The calculated current density distribution is shown in ﬁg. 3.13(b). Channeling leads to a large contribution of the Cu layer to the change of the current density upon switching the alignment from parallel to antiparallel, as can be seen from ﬁg. 3.13(b). This contribution would be zero when using the CB model with equal free electron band structures throughout the entire layer stack (see ﬁg. 3.4). A striking result is the high resistivity deduced for the Cu layers, more than a factor of 10 larger than for bulk Cu. The corresponding mean free path would be only about 3 nm, as can be deduced from analyses of the conductance in Cu ﬁlms by the same authors (Butler et al., 2000). This is more than a factor 3 lower than given in table 2.3 as the result from an analysis of the magnetoconductance of spin-valves using the CB model. The high Cu resistivity will partially compensate the increase of the conductance and GMR ratio due to channeling. We remark that analyses of experimental data using fully quantum-mechanical calculations by the same authors, discussed below, provide no clear evidence for the importance of channeling, and lead to a Cu resistivity that almost coincides with the result given in table 2.3. Experimental indications that electron channeling can indeed occur in layered magnetic structures were obtained by Dekadjevi et al. (2001) for Fe/Au (100) superlattices, fabricated by evaporation deposition. Their analysis has not yet been conﬁrmed by model calculations based on realistic band structures.

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

109

3.9. Quantum-mechanical models Within the semiclassical models discussed above electron transport is treated as if the electrons are point-like particles, and the electronic structure of the system is treated as if spin-valves are a combination of bulk systems that are in contact at the interfaces. Several authors have proposed theoretical models for the GMR effect based on a more complete quantum-mechanical description of the ground state electronic structure of the system and based on a quantum-mechanical treatment of electrical transport. Oguchi (1993) was the ﬁrst to demonstrate, from calculations of effective spin-dependent resistivities for Co/Cu multilayers using realistic multilayer band structures, that a GMR effect can (in principle) even arise when scattering is spin-independent. The effect then arises from the dependence on the magnetic alignment of the effective superlattice Fermi velocities. In theories based on the Boltzmann transport model, within which the state of the system is described in terms of a position and wavevector dependent electron distribution function, the phase of the electron states is neglected. Interferences between scattering processes are thus not taken into account. Quantum-mechanical theories, based on the Kubo–Greenwood formalism, take these effects into account. A comprehensive discussion of the theoretical work on quantum-mechanical theories is beyond of the scope of this chapter. We refer to the work by Levy et al. (1990), Camblong and Levy (1992), Vedyaev et al. (1992), Levy (1994), Zhang and Butler (1995), Butler et al. (1996a, 1996b), Schep et al. (1998), Barthélémy et al. (1999), Binder et al. (2000), Tsymbal and Pettifor (2001), Kokado (2001), and references therein. Full-relativistic model calculations were carried out by Blaas et al. (2001). In this section we focus on applications to spin-valves. Vedyaev et al. (1992) demonstrated that within a quantum-mechanical transport model, applied to a free-electron band structure, the experimental resistance and magnetoresistance of spin-valves may be described successfully within a single set of scattering parameters. Two differences with the semi-classical theory were regarded crucial: the effective cut-off angle (see section 3.7.1) and the treatment of scattering at the substrate/F layer interfaces as scattering at a very thin boundary layer (leading to an angular dependent specularity parameter). Both effects favour transport perpendicular to the ﬁlm plane. An analysis of the systems for which the scattering parameters as obtained from the CB model are given in table 3.1, led to only 10% higher mean free paths. Butler et al. (1997) analysed the room temperature experimental resistance and magnetoresistance of (glass/ 5 nm Ta/ t Co/2.1 nm Cu/ t Co/9 nm Fe50 Mn50 / 5 nm Ta) conventional (111) spin-valves and (glass/5 nm Ta/ 2.5 nm Cu/ t Co/ 2.1 nm Cu/ t Co/ 9 nm Fe50 Mn50 /5 nm Ta) (111) spin-valves with a back layer using a quantum-mechanical transport model, applied to a realistic electronic structure. The maximum GMR ratios were approximately 8%. Scattering at the outer boundaries was taken into account by placing a high-resistive monolayer at the Ta/Co, Ta/Cu and Co/Fe50 Mn50 interfaces. The scattering rates in the Cu and Co layers were constrained in order to yield the correct experimental resistance for thick ﬁlms, deposited under identical conditions as the spin-valves. Various scenarios were considered for the spin-dependence of scattering in the bulk of the Co layers and for the scattering at the Co/Cu interfaces. The best agreement with the experimental Co layer thickness dependent (magneto)conductance was obtained using a ratio of minority to majority spin bulk scattering rates that is equal to 20. No additional (diffusive) scattering at the interfaces was introduced, and the contribution from electron channeling

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R. COEHOORN

in the Cu layers (see section 3.8) was removed completely or almost completely from the calculated magnetoconductance. This removal procedure was possible because the speciﬁc transport model used allowed to make a distinction between the different processes that lead to the magnetoconductance. For conventional spin-valves containing 20 to 30 nm Co layers, channeling was predicted to lead to more than twice the magnetoconductance ΔG that was actually observed. The resistivities that are used to obtain the ﬁts are given in table 3.3. The Cu resistivity (ﬁxed by the constraint mentioned above) is much smaller than that assumed in the analysis of similar structures based on the semi-classical model (see section 3.8), and the corresponding mean free path of approximately 10 nm is close to the value given in table 3.2. The scattering spin asymmetry ratio for Co, which was the only free parameter in the scenario without additional interface scattering, is approximately equal to 25. It is very similar to the value obtained from the semiclassical model (table 3.3). In conclusion, the method is able to successfully describe the (magneto)conductance, but the analysis provides no clear proof of the importance of channeling. Brown et al. (1997) showed theoretically that the assumption of channeling could be tested by measuring the angular dependence of the conductance. Besides channeling, a second possible consequence of the occurrence of (partially) conﬁned states in a superlattice is the occurrence of oscillations of the magnetoconductance with the layer thicknesses, arising from the opening or closing of superlattice gaps in the band structure at the Fermi level upon a change of the alignment of magnetization directions. For magnetic layer structures clear examples of quantum-conﬁnement effects are the oscillatory exchange coupling between F layers across NM spacer layers (as reviewed by Bürgler et al. (2001)), and the oscillatory magneto-optical Kerr effect of metallic multilayers, sandwich structures and single layers (Bennett et al., 1990; Katayama et al., 1993). Theoretical studies in which various types of approximations are made indeed predict such effects in the magnetoconductance (Barnas et al., 1998; Schep et al., 1998; Tsymbal and Pettifor, 2001). However, as far as is known to the author, oscillatory CIP-magnetoresistance (interpreted as a superlattice effect) has only been reported for highly perfect AF-coupled Fe/Cr superlattice systems (Potter et al., 1994; Okuno and Inomata, 1994). No signs of oscillations of the CIP-GMR ratio with the layer thicknesses have so far been found for spin-valves. An explicit search for such effects was carried out by Speriosu et al. (1991), who investigated Co/Cu/Co-based spin-valves which do show oscillatory interlayer exchange coupling across the Cu layer. Tsymbal and Pettifor (1996, 2001) developed a quantum-mechanical transport model for metals and multilayers in which the electronic structure is treated in terms of a tight-binding model based on hybridized s, p and d orbitals. Scattering is obtained by adding random potentials with a Gaussian distribution to the electronic potentials on each lattice site. Interface disorder could be introduced by making use of a larger width of the distribution in monolayers at the interfaces. All applications of this work were carried out assuming a spin-independent width-parameter, γ . This corresponds to assuming spin-independent scattering potentials. As explained in section 3.1, it still gives rise to spin-dependent scattering. The scattering spin asymmetry ratio, α, is found to depend on γ , a feature that is absent within semiclassical models. In the weak scattering limit α ≈ 5 and α ≈ 0.55 for Co and Fe, respectively. For Co, this value is signiﬁcantly lower than the values obtained by analyses of experimental results for spin-valves (see, e.g., table 3.3), which shows that in

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

111

Fig. 3.14. Results of in situ measurements of the conductance of (NiO/ 2 nm Co/ tCu nm Cu/ Co) ﬁlms, carried out during the sputter deposition process. From Bailey et al. (2000b, 2000c).

real systems the scattering potential in Co is spin-dependent. However, this choice will still allow one to semi-quantitatively discuss trends. An interesting application of this work was reported by Bailey et al. (2000b, 2000c). For ion beam sputtered trilayers of the type (NiO/ 2 nm Co/ t Cu / 5 nm Co) they carried out in situ studies of the conductance G(t) during the growth of the ﬁlm, as a function of the Cu layer thickness t. The result is shown in ﬁg. 3.14. In the vicinity of interfaces, G(t) is strikingly non-linear. When Cu is deposited on top of Co, the upturn of the slope of G(t) that is expected because of the higher conductivity of Cu, sets in only after the deposition of approximately 0.5 nm Cu. When Co is deposited on top of Cu, the conductance shows initially a signiﬁcant drop. Independently, similar results were reported by Keavney et al. (1999) for magnetron sputtered samples. Bailey et al. modelled their results by describing scattering in the bulk of the Co and Cu layers by uniform values of γ , corresponding to resistivities of 5.9 × 10−8 m and 14.8 × 10−8 m as observed for thicker ﬁlms, and by assuming a 30% larger value of γ in the Co and Cu monolayers that are adjacent to the top Cu/Co interface. This indicates that the quality of the bottom and top interfaces are different. The asymmetry is due to the difference in the Co and Cu surface free-energies, which provides the driving force for segregation of Cu to the top of the ﬁlm. The shape of the conductance curve near the interfaces was interpreted as the result of hybridization between the Co-3d states and the conduction electron states in the Cu-layers adjacent to the interface. Near the bottom interface, this hybridization reduces the conductivity of the added Cu layers, leading to a decreased additional conductance. Near the top interface, already one added Co layer decreases the contribution to the conductance of the nearby Cu layers, leading to a decrease of the conductance. It was found to be impossible to model the experimental results using the semiclassical CB model (Bailey et al., 2000c). However, all observed features could successfully be understood from a semiclassical transport model based on an ab initio calculation of the electronic structure (Zahn et al., 2002).

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Chen and Fernandez-de-Castro (2001) used a similar model as that introduced by Tsymbal and Pettifor to calculate the magnetoresistance of several types of (111) Co/Cu/Cubased spin-valve structures. The following assumptions were made concerning scattering: (i) bulk scattering arises from a spin-independent scattering potential that is uniform across the spin-valve, (ii) the scattering probability at a certain position is proportional to the total local density of states, (iii) at the two atomic layers on each side of the Co/Cu interfaces an additional spin-dependent scattering potential is introduced in order to describe diffusive interface scattering, (iv) in two atomic layers near an insulating layer the bulk scattering potential is doubled (so that, effectively, the boundary scattering is only partially specular) and (v) metallic AF layers or metallic cap layers are replaced by three atomic layers with a very high scattering rate (leading to diffusive scattering; shunting by these layers is disregarded). Three types of SVs were studied: conventional simple SVs, simple spin valves with a Sy-AF, and dual spin valves. Table 3.4 gives a selection of the results. The numerical values of the two free parameters, the bulk and Co/Cu interface scattering potentials, were not given. However, in view of the fact that these were taken identical for all calculations, it is still of interest to look at the predicted trends. These are very similar to those observed from experimental studies, as discussed in section 2. E.g., dual SVs show a larger GMR ratio than simple SVs, but the sheet resistance change is not much different. SVs with a Sy-AF pinned layer show a decreased GMR ratio and a decreased change of the sheet resistance due to the shunting by the Sy-AF. The introduction of specularly reﬂecting layers increases the magnetoresistance ratio, and only slightly decreases the sheet resistance. The current density distribution and layer thickness dependence of the GMR ratio in Sy-AF pinned spin-valves, calculated using the same method, were given by Chen et al. (2000). 3.10. Trends and future prospects What are the prospects for further progress, towards SVs with higher MR ratios? In order to give an answer to this question, we ﬁrst formulate a simple model from which the highest MR ratios that have been observed so far for simple and dual SVs with and without specular scattering layers can be understood well. The model is based on the transport parameters that have been given in table 3.2. These were obtained by Dieny et al. (2000a, 2000b) from a successful analysis of the layer thickness dependence of the room temperature resistance and magnetoresistance of SVs using the free electron semiclassical transport model. The main conclusions are: (1) The effective majority spin mean free paths in F = Co90 Fe10 and NM = Cu layers are quite similar and both close to 10 nm, and the majority spin transmission probability through the F/NM interfaces is close to 1. (2) The effective minority spin mean free path in the F layers is at least one order of magnitude smaller than the effective majority spin mean free path, and the transmission probability through the F/NM interfaces is signiﬁcantly smaller than 1. Based on these results, it seems to be permitted to discuss trends and future prospects using a semiclassical free electron transport model with the extremely simpliﬁed parametrization ↑ ↓ ↑ ↓ λF = λNM = λ ≈ 10 nm, λF = 0, TF/NM = 1 and TF/NM = 0. The electron densities and effective masses in all layers will be assumed to be equal. For a given magnetization alignment and spin direction the conduction is then is equal to the sum of contributions from

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

113

Fig. 3.15. Calculated magnetoresistance ratio of various types of SVs (curves a–f, deﬁned in section 3.10), a function of the reduced thickness tF /λ of one of the magnetic layers. The model that has been used is explained in section 3.10. The reduced spacer layer thickness is t/λ = 0.2 (a) or t/λ = 0.1 (b); λ is the mean free path in the spacer layer and the majority spin mean free path in the F layers. The minority spin mean free path in the F layers is equal to zero.

one or more conducting slabs. The conductivity σ of these slabs is given by the Fuchs– Sondheimer formula (eq. (3.18)). The effective layer thickness that determines σ is equal to the nominal thickness, d, if scattering is perfectly diffusive at both boundaries. It is equal to 2d if scattering is specular at one of the boundaries, and it is equal to inﬁnity if scattering is specular at both boundaries. In ﬁg. 3.15 results of such model calculations are given for SVs with the following active layer stacks, as a function of the reduced F layer thickness tF /λ: (a) Conventional simple SVs: tF F/ t NM/ tF F, (b) Simple SVs with one specular reﬂection layer (I): I/ tF F/ t NM/ tF F, (c) Simple SVs with one specular reﬂection layer and with an optimized tF,2 layer thickness: I/ tF F/ t NM/ tF,2 F, (d) Dual SVs with an optimized tF,2 layer thickness: tF,2 F/ t NM/ tF F/ t NM/ tF,2 F, (e) Simple SVs with two specular reﬂection layers: I/ tF F/ t NM/ tF F/ I, (f) Dual SVs with two specular reﬂection layers and tF,2 = tF /2 (which optimizes ΔR/R): I/ tF,2 F/ t NM/ tF F/ t NM/ tF,2 F/ I. System (f) is equivalent to a (tF F/ t NM)n multilayer, and is (apart from a factor of 2 in the F layer thickness) equivalent to system (e). Fig. 3.15(a) gives the results for t/λ = 0.2. As λ is in practice typically 10 nm, this diagram corresponds to the situation for SVs with tCu ≈ 2 nm. When making a comparison with experimental results, the horizontal scale may be viewed to range from tF = 0 to 10 nm. The maximum possible MR ratio of conventional simple SVs is predicted to be just above 10%. This is in good agreement with the experimental results given in table 2.1 (parts (a) and (b)). The table reveals a highest value of 11.2% (Stobiecki et al., 2000) if, following Dieny et al. (2000a), the higher

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TABLE 3.4 Calculated GMR ratio, ΔRsh and Rsh for three types of model spin-valve structures with three types of boundary conditions at the outer surfaces (Chen and Fernandez-de-Castro, 2001). AFM = AF metal, AFI = AF insulator, M = metallic cap layer, I = insulating cap layer. The layer thicknesses are in nm. The free layer is indicated by italic and bold lettering, reﬂective layers are indicated by roman bold lettering Material (1) Simple spin-valves AFM/ 2.4 Co/2.4 Cu/ 0.6 Co/ 2.4 Ni/ M AFM/ 2.4 Co/2.4 Cu/ 0.6 Co/ 2.4 Ni/ I AFI/ 2.4 Co/ 2.4 Cu/ 0.6 Co/ 2.4 Ni/ I (2) Simple with Sy-AF pinned layer AFM/ 2.2 Co/1.0 Ru/ 2.2 Co/ 2.4 Cu/ 1.2 Co/ 2.0 Ni/ M AFM/ 2.2 Co/ 1.0 Ru/ 2.2 Co/ 2.4 Cu/ 1.2 Co/ 2.0 Ni/ I AFM/ 2.2 Co/1.0 Ru/ 1.0 Co/ I/ 1.2 Co/ 2.4 Cu/ 1.2 Co/ 2.0 Ni/ I (3) Dual spin-valves AFM/ 2.0 Co/ 2.0 Cu/ 3.0 Co/ 2.0 Cu/ 2.0 Co/ AFM AFM/ 2.0 Co/ 2.0 Cu/ 3.0 Co/ 2.0 Cu/ 2.0 Co/ AFI AFI/ 2.0 Co/ 2.0 Cu/ 3.0 Co/ 2.0 Cu/ 2.0 Co/ AFI

GMR ratio (%)

ΔRsh ( )

13.3 15.5 24.5

2.3 2.2 3.0

17.5 14.1 12.4

11.1 12.9

1.5 1.7

13.6 12.8

19.9

2.5

12.4

20.3 25.5 36.0

2.0 2.4 3.0

9.9 9.2 8.3

Rsh ( )

MR ratios (up to 14.7%) obtained for systems based on PtMn are ascribed to partially specular scattering at the PtMn exchange bias layer (section 3.6.5). The effect of using a single specular reﬂection layer is predicted to enhance the MR ratio to at best to ≈17%, provided that both F layer thicknesses are optimized. The optimum of curve c, at tF /λ ≈ 0.15, is obtained for tF,2 /λ ≈ 0.48. This is, again, consistent with the results reported so far. Further progress, to MR ratios well above 20%, can only be realized by making use of double specular simple or dual SVs. The record MR ratio of 27.8% for the double specular Co/Cu/Co/Cu/Co dual SVs studied by Sugita et al. (1998), with tF,f = 5 nm and tF,p = 2 nm, is consistent with the predicted value of approximately 29% that would follow from curve f when taking tF = 5 nm. We conclude that the predictions from the simple model that has been introduced above are in surprisingly good agreement with the highest MR ratios that have been obtained experimentally for various types of SVs. A comparison can also be made with the results of the quantum-mechanical calculations by Chen and Fernandez-de-Castro (2001), discussed in the previous section. E.g., for a double specular Co/Cu/Co/Cu/Co dual SV with tF,f = 3 nm, tF,p = 2 nm and tCu = 2 nm, the theoretical MR ratio is 36% (table 3.4), whereas curve f with tF = 3 nm (and tF,p = 1.5 nm) yields an MR ratio of approximately 34%. Fig. 3.15(a) suggests that there is room for a further increase of the MR ratio of double specular dual SVs by making use of thinner F layers, with an upper limit of about 40%. More could be gained by making use of thinner Cu layers. Such an approach was pioneered by Jo and Seigler (2002a, 2002b), see section 2.1.4, who studied SVs with tCu close to 1 nm. By precisely controlling the Cu layer thickness, the net effect of the various contributions to the interlayer magnetic coupling was a weak AF interaction. In principle, the use of such ultrathin Cu spacer layers can lead to strongly enhanced MR ratios, as show in ﬁg. 3.15(b). For λ = 10 nm, the ratio t/λ = 0.1 employed in this diagram corresponds

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to tCu = 1 nm. The highest possible MR ratios (around 87%) approach the value of 110% obtained for Co0.95 Fe0.05/Cu multilayers (ﬁg. 1.1). In conclusion, a simple model has been introduced that successfully explains the trends in the highest MR ratios observed experimentally for various types of SVs. Furthermore, it gives quantitative predictions on the prospects for further increases of the MR ratios. Of course, the model gives only upper limits. It does not deal, e.g., with the limitations that are imposed by the requirement that the magnetic interactions should be well-controllable and compatible with a speciﬁc application. This will be the subject of the next section. 4. Magnetic interactions and magnetization reversal processes In this section we discuss ﬁrst how the shape of the R(H ) transfer curves of SVs can be calculated from the Stoner–Wohlfarth (SW) model, and to what extent that model provides a good description of the micromagnetics of realistic systems. Subsequently, the time-dependence of the magnetic response and the electrical noise that is related to magnetization ﬂuctuations are discussed. In the ﬁnal three subsections an overview is given of the physics and materials science of the three most important magnetic interactions: magnetic anisotropy, interlayer coupling, and exchange anisotropy. 4.1. Applications of the Stoner–Wohlfarth model to spin-valves 4.1.1. Formalism The transfer curves of SVs are often calculated with considerable success using the Stoner–Wohlfarth (SW) model. The model assumes that all spins in each ferromagnetic layer remain rigidly coupled to each other during the entire reversal process. The reversal process is then coherent, in contrast to reversal via domain wall displacement. Originally, the model was developed to describe the reversal processes of small single domain ferromagnetic particles (Stoner and Wohlfarth, 1947, 1948). Slonczewski (1956) and Middelhoek (1961) used the model to describe the reversal of single layers with a uniaxial anisotropy under the application of a magnetic ﬁeld with an arbitrary direction. Their geometrical astroid model (see also Thiaville, 1998) can be used to analyse the hysteretic switching of memory elements based on SVs (Parker et al., 1995). Applications of the SW model to (F/NM)n magnetic multilayers with various types of magnetic anisotropies and F or AF interlayer exchange coupling were developed by Dieny et al. (1989), Dieny and Gavigan (1989), Folkerts (1991) and Folkerts and Purcell (1992). In this section we ﬁrst sketch the general formalism and then focus on applications for SVs. We consider systems that contain n ferromagnetic (F) layers with thicknesses ti (i = 1 to n) and saturation magnetizations Msat,i , and restrict ourselves to situations in which the applied ﬁeld and the magnetizations are directed in the plane of the ﬁlm. As the magnetization in each layer is assumed to be uniform, the magnetic state of the system is fully deﬁned by the angles αi between the magnetization direction of each of the F layers and the in-plane reference direction. The direction of the external ﬁeld H with respect to the reference direction is θ . See ﬁg. 4.1. The change of the magnetic state of the system that occurs when the external magnetic ﬁeld is changed, is calculated from the total energy per m2 , Etot(α1 , α2 , . . . , α2 , H ). Etot is

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Fig. 4.1. Deﬁnition of the angles θ and αi of the external ﬁeld and the layer-dependent magnetization directions with respect to the reference direction.

a sum of contributions from various interactions, as discussed below. A calculation starts with the choice of a stable or metastable initial state. For calculating the major hysteresis loop, the initial condition is that in which the magnetizations of all F layers are parallel to an inﬁnitely large magnetic ﬁeld. When H is then changed, the angles {αi } that describe the state of the system are assumed to vary in a continuous manner with H , as long as the system remains in a state of local (stable or metastable) equilibrium. Only if the system has arrived in a state in which it is no longer in a local energy minimum, it is allowed to slide via a discontinuous transition into a (meta)stable state of lower energy. That state is determined using the method of steepest descent. Such magnetization changes are irreversible, leading to a hysteretic magnetization curve. The initial conditions and the precise path along which the ﬁeld is varied are thus important. The equilibrium conﬁgurations at a ﬁeld H = (H0 , θ0 ) are solutions of the following set of n coupled equations: ∂Etot (4.1) = 0 for i = 1 to n. ∂αi H0 ,θ0 A solution is stable or metastable if all eigenvalues of the matrix M, deﬁned by ∂ 2 Etot Mij ≡ , ∂αi ∂αj H0 ,θ0

(4.2)

are positive. For a single layer the stability condition is ∂ 2 Etot/∂α 2 > 0. In the case of complicated situations for which no analytical solutions are available, the numerical solution method described by Tietjen et al. (2002) can be used. An alternative method is to numerically solve the Landau–Lifshitz torque equation for the magnetization dynamics in the static (inﬁnite damping) limit (Fidler and Schreﬂ, 2000; and references therein). The magnetic interactions that are most relevant to SVs are the exchange bias interaction Eeb due to the exchange coupling with AF layers, the uniaxial (two-fold) in-plane anisotropy energy of the layers, Ea , the Zeeman energy, EZeeman , due to the interaction of the external ﬁeld with the magnetization of each ferromagnetic layer, and the interlayer magnetostatic or exchange coupling between the F layers across non-magnetic layers, Ecoupl . The total energy of the system is then Etot =

n n (Eeb,i + Ea,i + EZeeman,i ) + Ecoupl,i,i−1 , i=1

i=2

(4.3)

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with Eeb,i = −Jeb,i cos(αi − ϕAF ),

(4.4)

Ea,i = Ki ti sin2 (αi − ϕa,i ),

(4.5)

Ecoupl,i,i−1 = −Ji,i−1 cos(αi − αi−1 ),

(4.6)

EZeeman,i = −μ0 Msat,i H ti cos(θ − αi ).

(4.7)

and

When layer i is not in contact with an AF layer, Eeb,i = 0. Eq. (4.4) is a phenomenological expression for the exchange bias effect due to the coupling of layer i to an AF layer (Meiklejohn and Bean, 1956, 1957). The angle φAF deﬁnes the preferred direction of the magnetization of the F layer. The exchange bias interaction parameter Jeb is therefore by deﬁnition positive. Eq. (4.4) is not valid if the internal sublattice magnetization directions in the AF layer are not completely frozen. This complication is discussed in section 4.6. In eq. (4.5) Ki is the uniaxial anisotropy energy density of layer i, with thickness ti , and the angle φa,i deﬁnes the direction of the easy axis. In eq. (4.6) positive and negative interlayer coupling parameters J describe ferromagnetic and antiferromagnetic coupling, respectively. The microscopic-scale physics of these interactions is discussed in sections 4.4 to 4.6. Here, we view Jeb , K and J as phenomenological parameters. 4.1.2. Conventional simple spin-valves – interplay between coupling and exchange biasing The transfer curve of a conventional simple F/NM/F/AF SV in the crossed anisotropy conﬁguration (see ﬁg. 1.6(a)), with the positive ﬁeld direction parallel to the magnetization direction of the pinned layer (the reference direction), is given by the generic form sketched in ﬁg. 1.5. In the notation introduced above, ϕAF = αp = 0 and θ = 0 or π . Fig. 1.5 is correct, provided that the exchange anisotropy interaction is much stronger than the coupling between the free and pinned layers, as will be shown below. It follows from the SW model that the reversal of the free layer is then described by cos αf =

H − Hcoupl , Ha,f

(4.8)

where αf is the angle between the free and pinned layer magnetization directions, where Ha,f is the anisotropy ﬁeld, Ha,f =

2Kf , μ0 Msat,f

(4.9)

and where Hcoupl is the coupling ﬁeld around which the reversal takes place: Hcoupl = −

J . μ0 Msat,f tf

(4.10)

When the anisotropy of the pinned layer and the coupling between the pinned and free layer can be neglected, the ﬁeld at which the pinned layer switches is equal to H = −Heb,0 ,

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where the exchange bias ﬁeld Heb,0 is given by: Heb,0 =

Jeb . μ0 Msat,p tp

(4.11)

The actual reversal of the pinned layer is often hysteretic, as can, e.g., be seen in ﬁg. 1.7. Xi et al. (1999) showed that this can be modelled fairly successfully by adding an effective pinned layer anisotropy energy Kp tp sin2 αp to the total energy. However, it was observed that the corresponding effective anisotropy ﬁeld Ha,p = 2Kp /(μ0 Msat,p ) can be different from the anisotropy ﬁeld of an isolated layer. An increase of the ferromagnetic coupling between the pinned and free layer leads to a narrowing and ﬁnal disappearance of the high-resistance plateau region, as was shown in ﬁg. 1.7 for a series of Py/Cu/Py/Fe50 Mn50 SVs with varying Cu spacer layer thicknesses. Similar results for (Ni16 Fe18 Co66 /Cu/Ni16 Fe18 Co66 /Fe50 Mn50) SVs were reported by Hoshino et al. (1994). This can be understood from the SW model, using the following expression for the total energy (Rijks et al., 1994a, 1996a): Etot = −Jeb cos(αp ) − μ0 Msat,pH tp cos(αp ) − μ0 Msat,f H tf cos(αf ) − J cos(αp − αf ).

(4.12)

The exchange bias interaction, the pinned layer and free layer Zeeman energies and the interlayer coupling are taken into account, but the anisotropies of the free and pinned layers are neglected. The ﬁeld axis is parallel to the direction of the exchange bias ﬁeld. By introducing the dimensionless quantities: j≡

J , Jeb

(4.13)

h≡

μ0 Mtp H H = Heb,0 Jeb

(4.14)

x≡

Mf tf , Mp tp

(4.15)

and

eq. (4.12) is transformed to etot ≡

Etot = −(1 + h) cos(αp ) − hx cos(αf ) − j cos(αp − αf ). Jeb

(4.16)

The solutions αf (h) and αp (h) depend thus only on the dimensionless parameter values j and x. Apart from the four trivial solutions (αf , αp ) = (0, 0), (0, π), (π, 0), and (π, π), the conditions (4.1) lead to the following non-trivial solutions: 1+h j j (1 + h) αf = arccos (4.17) − − , 2h2 x 2 2(1 + h) 2j j hx j hx − − . αp = arccos (4.18) 2(1 + h) 2hx 2j

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Fig. 4.2. (a) Calculated reduced ﬁeld dependence of the normalized resistance change (R − Rp )/(RAP − RP ) of a simple SV, for equal magnetic layer thicknesses of the free and pinned layers (x = 1) and for various values of the coupling parameter j ≡ J /Jeb = (for x = 1) − Hcoupl /Heb . (b) Field dependence of the magnetization directions in the free and pinned layers for j = 0.2 and j = 0.4. At the top of the ﬁgure the corresponding critical ﬁelds are indicated. From Rijks et al. (1994a).

Alternatively, when αf is already known, αp can be obtained from αf using −hx αp = arcsin sin αf . 1+h

(4.19)

Fig. 4.2 shows results for the case of ferromagnetically coupled free and pinned layers with equal thicknesses (x = 1 and j > 0). The magnetization reversal process is nonhysteretic. The resistance is calculated using eq. (1.2), with θ = αf − αp . Two regimes can be distinguished. In the weak coupling regime, j < 0.25, the ﬁeld ranges [h1 , h2 ] and [h3 , h4 ] in which the pinned and pinned layers switch, respectively, are broadened and shifted towards each other. However, there is still a high-resistance plateau region [h2 , h3 ] in which the layers are antiparallel. For very small j , the free layer reverses around H ≈ Hcoupl and the pinned layer reverses around H = Heb ≈ −Heb,0 − Hcoupl. In the strong coupling regime, for j > 0.25, the free and pinned layers reverse simultaneously in the ﬁeld interval [h1 , h4 ]. The angles αf and αp for the weak and strong coupling cases j = 0.2 and j = 0.4, respectively, are given in ﬁg. 4.2(a).

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Fig. 4.3. Phase diagram, indicating the ﬁeld dependence of the magnetic state of a pair of F (j > 0) or AF (j < 0) coupled layers of which one layer is pinned, close to the origin (a) and for a larger part of the phase space (b). The angles αf and αp are deﬁned in ﬁg. 4.2. Transitions take place in the shaded regions. The full solutions for j = 0.2, j = 0.4 and j = −7 (dashed lines) are given in ﬁgs 4.2 and 4.6, respectively.

For arbitrary x, the characteristic ﬁelds are given by (Beach et al., 2000; Dieny et al., 2000a): 1 x +1 1 x + 1 2 4j h1,4 = − ± j (4.20) ± ∓ , 1∓j 2 2x 2 x x x −1 1 x − 1 2 4j 1 ∓ . 1−j h2,3 = − + j (4.21) − 2 2x 2 x x √ From eq. (4.21), the critical coupling parameter is given by jcr = x/(1 + x )2 . The ﬁeld at which two very strongly free and pinned layers switch is h = −1/(1 + x). For x = 1, ﬁg. 4.3 gives the “phase diagram” that indicates the magnetic state of the system for any (j, h) combination. The ﬁeld ranges in which the transitions take place, with boundaries given by the critical ﬁelds h1 –h4 , are shaded. Fig. 4.3(a) focuses on a small region around the origin. The vertical dashed lines indicate the cross-sections for j = 0.2 and j = 0.4 for which the full solutions are given in ﬁg. 4.2. Fig. 4.3(b) gives the same diagram for larger j and h ranges. It has been included for later use. 4.1.3. Conventional simple spin-valves – interplay between coupling and anisotropy Kools (1996) and Labrune et al. (1997) used the SW model to study how the hysteresis loop of the free layer depends on the uniaxial anisotropy of the free layer and the interlayer coupling, for the case of a perfectly ﬁxed pinned layer direction. Arbitrary angles between the ﬁeld, the anisotropy axis and the pinning direction were considered. The authors identiﬁed three reversal modes, called A, B and C. In modes A and B the reversal has the character of a rotation – jump – rotation process, so that the magnetization curve is

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Fig. 4.4. Phase diagrams, indicating the mode of reversal (A, B or C, see text) of the free layer in a SV as a function of the angle θ between the applied ﬁeld and the exchange bias direction and as a function of the normalized coupling parameter J . (a) Crossed anisotropy conﬁguration (ε = 90◦ ). (b) Conﬁguration with a slightly misaligned anisotropy axis (ε = 80◦ ). From Labrune et al. (1997).

hysteretic. In mode C the reversal process is non-hysteretic, and occurs purely by rotation. The distinction between modes A and B is more subtle. It is related to the magnetization direction of the free layer with respect to the magnetic ﬁeld axis before and after the jump. Fig. 4.4(a) shows the calculated “phase diagram” for the angle ε = 90◦ between the easy axis and the exchange bias ﬁeld (crossed anisotropy conﬁguration), as a function of the angle θ between the magnetic ﬁeld and the exchange bias ﬁeld. When j ≡ −Hcoupl/Ha > 1 reversal is a pure rotation process for all applied ﬁeld directions. However, for j < 1 pure rotation occurs only when the ﬁeld is aligned precisely parallel to the bias direction (θ = 180◦ , dashed-dotted line in ﬁg. 4.4(a), see ﬁg. 4.5(a)). Although the exact crossed anisotropy conﬁguration eliminates coercivity, it is seen from the ﬁgure that non-zero coercivity is expected in the case of a small misalignment of the ﬁeld direction. Fig. 4.4(b) shows that also a misaligned anisotropy axis can lead to coercivity, for θ = 180◦ even up to inﬁnite coupling strength. In practice coercivity can even occur when the local easy axis and exchange bias directions are on average perfectly aligned, but show a certain lateral orientation distribution (“dispersion”) related to the detailed microstructure. Indeed, Labrune and coworkers observed some coercivity in the hysteresis loops of Py/Cu/Py/Fe50Mn50 SVs, studied in the crossed anisotropy conﬁguration. One remedy is to make use of a

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Fig. 4.5. (a) Crossed anisotropy conﬁguration, (b) slightly uncrossed anisotropy conﬁguration, and (c) nearly parallel anisotropy conﬁguration.

slightly uncrossed anisotropy conﬁguration (ﬁg. 4.5(b)). Fig. 4.2(b) gives the phase diagram for the case ε = 80◦ . In the presence of coupling, there is a range of angles θ around θ = 170 ◦ for which coherent rotation (mode C) is obtained. The free layer is then micromagnetically stabilized by the component of the coupling ﬁeld that is parallel to the easy axis. In a stripe-shaped sensor, this is the longitudinal direction, parallel to the length of the stripe. Other longitudinal stabilization methods were discussed in section 2.3. Parker et al. (1995), Nishioka et al. (1995) and Fujiwara et al. (1996) pointed out that a nearly parallel anisotropy conﬁguration (ﬁg. 4.5(c)) can give rise to a higher sensitivity ((∂R/∂H )/R) than the crossed anisotropy conﬁguration (for the same anisotropy ﬁeld), whereas the coercivity that would arise in the perfectly parallel anisotropy conﬁguration is reduced or even eliminated. The applied ﬁeld should be parallel to the easy axis of the free layer, but the exchange bias direction should deviate slightly from the easy axis in order to transversely bias the free layer via the coupling across the spacer layer. A disadvantage is the strong dependence of the sensitivity to the precise exchange bias ﬁeld direction, to the interlayer magnetic coupling strength, and to lateral variations of these parameters. It should be noted that when the anisotropy of a sensor stripe is determined by its shape anisotropy, the direction of high sensitivity for this conﬁguration is parallel to the stripe axis. 4.1.4. Spin-valves with a Sy-AF pinned layer – transfer curve The solution of the problem of the interplay between coupling and biasing, given in section 4.1.2, can also be used to describe the basic shape of the transfer curves of SVs with a Sy-AF pinned layer (ﬁg. 2.4 in section 2.1.4). The total energy for the AF/Fp1 /X/Fp2 part of such a spin-valve is thus given by eq. (4.12), with subscripts p1 instead of p, and p2 instead of f. The exchange bias direction is positive. The exchange coupling is now antiferromagnetic, and much stronger than the exchange bias interaction. The magnetization reversal process takes place in three stages: (1) Reversal of the directly pinned Fp1 layer at negative ﬁelds in the ﬁeld range h1 –h2 . (2) Reversal of the free layer at h = 0. (3) Reversal of the indirectly pinned Fp2 layer in the positive ﬁeld range h3 –h4 . As an example, ﬁg. 4.6 shows the magnetoresistance curve for the case j = −7 and x = 1. Using Heb,0 = 20 kA/m, these parameters provide a reasonable description of the shape of the experimental curve shown in ﬁg. 2.5(b).

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Fig. 4.6. (a) Calculated reduced ﬁeld dependence of the magnetization directions in the directly and indirectly pinned layers F1 and F2 in an (AF/F1 /X/F2 ) ﬁlm. X is a layer that AF-couples the two F layers. The dimensionless coupling parameter is j = −7; the magnetic layer thicknesses of the two F layers are equal (x = 1). (b) Corresponding normalized resistance change (R − RP )/(RAP − RP ) of a simple (AF/F1 /X/F2 /NM/Ff ) SV, for zero coupling across the spacer layer. The critical ﬁelds h1 − h4 are given by eqs (4.20) and (4.21).

The ﬁeld intervals within which reversal occurs for the case x = 1 are indicated as shaded areas in the (h − j ) diagram given in ﬁg. 4.3(b). The dashed line corresponds to the example given in ﬁg. 4.6. For x = 1, the width h3 of the high resistance plateau is given approximately by eq. (2.2), which follows straightforwardly from eq. (4.22). For some applications the width of the largest symmetric ﬁeld region [−h, h] which still falls in the range [h2 , h3 ] is of interest. Making use of slightly unequal F layer thicknesses in the Sy-AF, with x = −j/(1 − j ), leads to a plateau in the magnetization of the AAF that is symmetric around zero ﬁeld (h2 = −h3 ). In the limit of large coupling and for x = 1 the saturation ﬁelds are to a good approximation given by h1 ≈ −(1/2) − 2j and h6 ≈ −(1/2) + 2j . The reduced ﬁeld h5 at which the MR ratio has dropped to half its value (“the effective exchange bias ﬁeld”) can be found from eq. (4.17) with αp1 = π/2. When j 1 and x = 1 it is given approximately by eq. (2.1). Successful SW simulations of the transfer curves of SVs with a Sy-AF pinned layer were reported by Meguro et al. (1999), Noma et al. (1999), Dieny et al. (2000a), Beach et al. (2000), Strijkers et al. (2000), Park et al. (2002) and Tietjen et al. (2002). Tietjen and coworkers paid special attention to the dependence of the resistance at small rotating ﬁelds, taking also the uniaxial anisotropy and the coupling between the free and indirectly pinned layers into account. This is of interest for applications in rotation angle sensors.

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4.1.5. Spin-valves with a Sy-AF pinned layer – setting of the pinning direction The exchange bias direction of a SV with a Sy-AF can be deﬁned by cooling from a temperature above the blocking temperature in a magnetic ﬁeld that is sufﬁciently strong to saturate the directly pinned Fp1 layer, even in the presence of the strong AF coupling with indirectly pinned Fp2 layer. However, it may sometimes be of interest to be able to obtain pinning in a direction other than the applied ﬁeld during ﬁeld cooling, e.g., when using a second AF layer to create a stabilizing longitudinal bias ﬁeld by its interaction with the free layer (ﬁg. 2.8 (d), (g)). The pinning of this second AF layer should be orthogonal to that of the AF layer that deﬁnes the direction of the reference layer. Beach et al. (2002) developed a method for realizing this in a single ﬁeld-cool process and using a single material for both layers, using a Sy-AF layer with x ≡ Msat,2 t2 /Msat,1 t1 > 1. The total energy is given by eq. (4.12), with Jeb = 0. For small ﬁelds, layers 1 and 2 are parallel and antiparallel, respectively, to the applied ﬁeld. Beyond a certain critical “spin-ﬂop” ﬁeld HSF the magnetization of layer 1 starts to rotate towards the applied ﬁeld, until it is parallel to it at the saturation ﬁeld Hsat . The magnetization of layer 2 then rotates temporarily away from the applied ﬁeld, but aligns again for H = Hsat . The ﬁeld at which the magnetization of layer 1 is orthogonal to the applied ﬁeld is given by J 1 H⊥ = (4.22) 1− 2. μ0 Msat,2 x Interestingly, the angle between the magnetization direction of layer 2 and the applied ﬁeld is maximal at H = H⊥ . The required strength of the setting ﬁeld can therefore be determined experimentally using the GMR effect of an AF/Fp1 /X/Fp2 /NM/Ff spin-valve, measured above the blocking temperature. In the ﬁeld range HSF < H < Hsat , the resistance is maximal for H = H⊥ . Other experimental studies of spin-ﬂop magnetization rotation processes in SVs with a Sy-AF pinned layer were carried out by Tong et al. (2000) and Jang et al. (2002c). 4.2. Deviations from the single-domain model – micromagnetics 4.2.1. Continuous ﬁlms – Lorentz microscopy studies Magnetization reversal processes in realistic thin ﬁlms are often more complicated than predicted from the SW model. Reversal from a metastable state to the most stable state, which cannot proceed via a coherent process due to the presence of an energy barrier, does sometimes take place via a domain wall displacement process. Switching via wall displacement explains, e.g., why the coercive ﬁeld of a single thin ﬁlm with uniaxial anisotropy is generally observed to be smaller than the value Hc = Ha that is predicted by the SW model. The precise value depends on the ease of nucleation and propagation of domain walls, which is determined by the microstructure. The micromagnetic structure and the reversal processes in spin-valves have been studied by Lorentz transmission electron microscopy (LTEM). This technique, which is described in detail by Petford-Long et al. (1999) and by Chapman and Scheinfein (1999), permits performing time-resolved studies of reversal processes in the microscope with a submicrometer-scale resolution, even for complete current-carrying devices.

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LTEM micrographs of the free layer magnetization reversal processes in continuous (non-patterned) (Py/ t nm Cu/ Py/ Fe50 Mn50 ) SVs spin-valves, studied by Gillies et al. (1995a, 1996), are shown in ﬁg. 4.7. Figures (a) and (c) were obtained using the Fresnel mode, and ﬁgure (b) was obtained using the differential phase contrast (DPC) mode. The Fresnel mode is a defocus technique, which only gives rise to black/white contrast at positions in the sample where changes take place (such as domain walls), whereas the DPC mode makes use of a focused beam, which makes measurements of the magnetization direction with high spatial resolution possible (Gillies et al., 1995a). Fig. 4.7(a) shows the reversal process for the crossed-anisotropy conﬁguration, with the ﬁeld axis precisely parallel to the exchange bias ﬁeld. The experiment was carried out for a SV with a moderate interlayer coupling (t = 3.0 nm, and j ≡ −Hcoupl/Ha ≈ 2). As expected, in the absence of an applied ﬁeld the sample is single-domain. The contrast that is visible is a so-called ripple domain pattern, transverse to the local average magnetization direction. In isolated ﬁlms it results from the interplay between random components in the magnetic anisotropy of the grains, the intergrain exchange interaction and long-range magnetostatic interactions (Hoffmann, 1968; Berkov and Gorn, 1998). For permalloy ﬁlms, with thicknesses in the 10–100 nm range, the deviations in each ripple domain from the average magnetization direction (“the ripple angle”) is a few degrees when the applied ﬁeld is zero. The ripple angle diverges at the anisotropy ﬁeld, and goes to zero in high saturating ﬁelds (Hoffmann, 1968). The ripple contrast for a free layer in spin-valves is smaller than in equally thick isolated layers, which was ascribed to the interaction with the pinned layer (Gillies et al., 1995b; Gillies, 1996). The observation of ripple makes it possible derive the local magnetization axis in each of the domains. Fig. 4.7(a) shows that when a negative ﬁeld is applied, the ripple increases, followed by the formation of separate domains within which the magnetization rotates in a clockwise and anticlockwise sense. When saturation has been reached (H = −0.69 kA/m micrograph), these domains are separated by 360◦ domain walls which prevent the otherwise complete antiparallel alignment of the free and pinned layers. A subsequent decrease of the ﬁeld does not lead to an exactly reversed process. At zero ﬁeld the magnetization is almost everywhere back to the parallel state, but one loop-shaped 360◦ wall is still visible. For SVs with a 2 nm Cu layer and a larger coupling ﬁeld, the free layer reversal process was found to be similar. Gillies and coworkers observed that reversal by rotation in a perfectly coherent singledomain mode could be obtained by the application of a ﬁeld under a small offset-angle, |θ − 180◦ | > 4 ◦ . Fig. 4.7(b) shows the DPC micrographs for a SV with a 3.5 nm Cu layer, taken under these conditions. The authors explained their ﬁndings in the following manner. From the phase diagram given in ﬁg. 4.4(a), reversal by coherent rotation is expected for systems for which j > 1, as discussed above, and for all directions of the applied ﬁeld. Ideally, even the smallest deviation from the θ = 180◦ angle between the ﬁeld and the exchange bias direction would then already break the symmetry, giving rise to reversal in a single rotation direction. However, in practice small lateral variations of the local exchange bias direction and the anisotropy direction will give rise to rotation in both directions. The minimum required offset angle, 4◦ in this case, is then determined by the combined dispersions of the pinning and easy axis directions. For the parallel anisotropy (PA) conﬁguration, with the ﬁeld parallel to the exchange bias direction, reversal was found to occur by domain wall displacement. This can be seen

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Fig. 4.7. Lorentz TEM micrographs of the free layer reversal processes in continuous (non-patterned) (Py/ t nm Cu/ Py/ Fe50 Mn50 ) SVs. (a) Reversal process for SVs with the crossed anisotropy conﬁguration, for tCu = 3 nm. Fresnel mode. (b) Reversal process for SVs with the crossed anisotropy conﬁguration, for tCu = 3.5 nm, for a ﬁeld that is applied under a small offset angle with respect to the exchange bias direction. Differential Phase Contrast mode. (c) Reversal process for SVs with the parallel anisotropy conﬁguration, for tCu = 3 nm. Fresnel mode. The number quoted in each ﬁgure is the applied ﬁeld in kA/m. From Gillies et al. (1995a).

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from ﬁg. 4.7(c) for the case of a 3.0 nm Cu layer. Up to high ﬁelds, loop-shaped 360◦ domain walls separate inner and outer regions with magnetizations parallel to the external ﬁeld. The stability of such walls was ascribed to the occurrence of lateral variations of the exchange or magnetostatic interlayer coupling, e.g., due to the presence of pin–holes. Experiments for thicker Cu layers revealed a decreasing density of 360◦ walls, providing support for the analysis. Similar results were obtained for the PA conﬁguration by PetfordLong et al. (1999) for (Py/ 2.2 nm Cu/ Co/ Py/ Fe50 Mn50) SVs spin-valves. King et al. (1998, 1999) studied the reversal process as a function of the angle between the ﬁeld and the pinning direction, in order to investigate the validity of the predictions made by Labrune et al. (1997) (see section 4.1.2). Reversal processes that were closely resembling the postulated modes A, B and C were observed, but the distinction between these modes was not very sharp. Reversal by domain wall movement was frequently seen to occur in spite of substantial energy barriers. Lim et al. (2002a) performed a comparative LTEM study of spin-valves with Py/Co and Py/Co90 Fe10 free layers. For the CA conﬁguration reversal of the Py/Co free layers was seen to involve complex domain process with a high degree of irreversibility. In contrast, reversal of the Py/Co90 Fe10 free layers was usually by rotation, and described well by the SW-type model by Labrune and coworkers (see also Lim et al., 2002b). The difference was attributed to the much higher magnetostriction constant of Co, giving rise to much stronger spatial variations of the local anisotropy in the highly stressed ﬁlms. Marrows et al. (2001) studied SVs with a Sy-AF pinned layers. A Sy-AF pinned layer in a top-SV becomes during the deposition process in a ﬁeld micromagnetically disordered when the top F layer becomes thicker than the bottom F layer. A micromagnetically complex state is formed due to the random handedness of the reorientation of both layers, leading to an ill-deﬁned exchange bias ﬁeld and a low GMR ratio. The disorder decreases when the top (Fp1 ) layer becomes much thicker than the bottom (Fp2 ) layer. 4.2.2. Microstructured spin-valves – narrow stripes The internal magnetic ﬁeld in microstructured elements contains a contribution from the magnetostatic ﬁeld that arises due to the uncompensated magnetic pole density at the edges or within the element. This demagnetizing ﬁeld is non-uniform, so that the magnetization is also non-uniform. Generally, no analytical expressions are available for describing the magnetization. Numerical model calculations of the non-uniform magnetization in stripeshaped MR elements for sensors, read heads or memory elements have been carried out by Heim et al. (1994), Tsang et al. (1994), Yuan and Bertram (1994), Smith (1994), Bertram (1995), Folkerts et al. (1995), Miles and Parker (1996), Oti and Russek (1997), Takano et al. (1998a), Morinaga and Shiiki (1999), Zhu (1999) and Zhu et al. (2001). In this section we consider inﬁnitely long narrow stripes. Patterning a SV in the form of an inﬁnitely long stripe with a width w and with the exchange bias ﬁeld along the y-direction perpendicular to the stripe-axis gives rise to an effective AF coupling between the free and the pinned layer and to an effective uniaxial anisotropy (“shape anisotropy”). The easy axis is parallel to the stripe length. The magnetization of the free layer across the width of the stripe is non-uniform. Folkerts et al. (1995) and Folkerts and Kools (1998) argued that the magnetization at the edges is completely

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determined by the condition of continuity of ﬂux from the pinned layer, tp Msat,p My,f (y = ±w/2) =− , Msat,f tf Msat,f

(4.23)

and that the magnetization in the center of the free layer is, to a fair approximation, given by My (y = 0) ∼ H − = Msat,f Ha + wtf Msat,f

tp w Msat,p , tf π 2 Ha + w Msat,f

(4.24)

with H = Hext + Hic + Hcurr ,

(4.25)

where Hext is the externally applied ﬁeld, Hic is the interlayer coupling ﬁeld and Hcurr is the ﬁeld induced by the sense current (see below). Note that Hic = −Hcoupl, because Hic is the effective ﬁeld on the free layer due to the interlayer coupling, whereas Hcoupl (deﬁned in section 1.3) is the ﬁeld around which the free layer, as a result of this coupling, switches. The ﬁrst term in eq. (4.24) represents the effect of the shape anisotropy. Effectively, the shape anisotropy gives rise to an enhancement of the anisotropy ﬁeld by the shape anisotropy ﬁeld Ha,sh = (tf /w)Msat,f . The shape anisotropy is larger than the intrinsic anisotropy if w < tMsat /Ha . For t = 5 nm permalloy layers (Msat = 800 kA/m and, typically, Ha = 0.4 kA/m), this situation arises for stripes that are narrower than 10 μm. The second term represents the effect of the stray ﬁeld from the pinned layer. The free layer acts as a good ﬂux guide when its thickness and relative permeability are high (μr t w, with μr ≈ Msat /Ha ), so that no ﬂux from the pinned layer leaks out of the free layer. When H = 0, the magnetization in the free layer is then uniform, and given by eq. (4.23). When part of the ﬂux from the pinned layer leaks out of the free layer, the negative contribution to the magnetization due to the stray ﬁeld from the pinned layer is in the middle of the element smaller than at the edges. Fig. 4.8 shows results of ﬁnite-element calculations of the magnetization of a 10 μm wide stripe with 8 nm pinned and free layers, using Msat,f,p = 800 kA/m (permalloy) and Ha = 0.2 kA/m, in the absence of an external ﬁeld (Folkerts et al., 1995). The equal free and pinned layer thicknesses lead to saturation at the edges. The magnetizations in the middle of the layer without and with ferromagnetic coupling with the free layer (curves (a) and (b), respectively) are in reasonable agreement with the predictions from eq. (4.24) (crosses). Results of experimental studies on the reversal of narrow stripes and rectangular SV elements with large length: width aspect ratios (see, e.g., Tang et al., 1995; McCord et al., 1996; Chen et al., 1997; Russek et al., 1997) roughly support the scaling of the interaction ﬁelds with w that is expected from eq. (4.24). Hcurr can be expressed as the product of the ﬁeld that arises for the extreme situation within which the current density is conﬁned to the spacer layer, times a constant c < 1 that depends only on the actual current density distribution in the layer stack: Isense . (4.26) 2w Using Ampère’s law, c can be calculated straightforwardly from the current density across the layer thickness as obtained from a model calculation (see, e.g., ﬁgs 3.4 and 3.13). Hcurr = c

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Fig. 4.8. Results of ﬁnite-element calculations of the magnetization of a 10 μm wide stripe with 8 nm pinned and free permalloy layers (Msat = 800 kA/m and Ha = 0.2 kA/m), in the absence of an applied ﬁeld. (a) Magnetostatic (AF) interlayer coupling only. (b) Including a ferromagnetic interlayer coupling, with Hic = 0.48 kA/m (6 Oe). The two crosses indicate the magnetizations for both cases in the middle of the element, as calculated using eq. (4.24). The top part of the ﬁgure indicates schematically the two types of magnetic interactions that are involved. From Folkerts et al. (1995).

From an LTEM study of the effect of the current on the magnetization in a SV with the layer structure 5 nm Ta/ 8 nm Py/ 3 nm Cu/ 2 nm Co/ 6 nm Py/ 25 nm NiMn/ 5 nm Ta, c ≈ 0.67 was obtained (Petford-Long et al., 1999). In simple SVs with a conducing back layer (section 2.1.3), or in dual SVs with a fully symmetric layer structure, Hcurr is reduced or even equal to zero, respectively. Nearby soft magnetic layers (such as the shields in a hard disk head) can enhance or reduce Hcurr , depending on the precise geometry. When a stripe-shaped SV with the exchange bias direction perpendicular to the stripe axis is used in a sensor, edge saturation should be avoided in order to a retain micromagnetically stable element. A ﬁrst design rule for MR sensors is thus (Folkerts and Kools, 1998): Msat,p tp < Msat,f tf .

(4.27)

A second design rule states that at Hext = 0 the magnetization in the center of the element should be equal to zero (Heim et al., 1994; Gurney et al., 1997; Folkerts and Kools, 1998). The sensor is then at Hext = 0 in its micromagnetically most stable point, and the ﬁeld interval around H = 0 in which the transfer curve is sufﬁciently linear (“dynamic range”) is highest. The use of a Sy-pinned AF layer to reduce the magnetostatic ﬁeld from the pinned layer and the use of additional layers outside the active part of the sensor (section 2.3) can help realizing the optimal zero-ﬁeld magnetic state. 4.2.3. Microstructured spin-valves – rectangular elements The reversal modes of typically 10–100 μm2 micrometer rectangular SV elements with small aspect ratios have been studied using LTEM by Portier et al. (1997a, 1997b), Chapman et al. (1998) and Petford-Long et al. (1999). If made of single permalloy layers, such elements are still larger than the Néel wall width (≈0.5 μm), so that generally well deﬁned

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Fig. 4.9. Magnetization patterns upon reversal of 2 × 4 μm2 rectangular (5 nm Ta/ 8 nm Py/ 2.5 nm Cu/ 6 nm Py/ 6 nm Fe50 Mn50 / 5 nm Ta) SV elements, at sequential external ﬁelds. From a LTEM study using the Differential Phase Contrast mode. Ha = 0.16 kA/m (2 Oe) and Hcoupl = −0.56 kA/m (7 Oe). The ﬁeld and pinning directions are parallel (a) or perpendicular (b) to the long axis of the element. The ﬁelds are given in kA/m. From Chapman et al. (1998).

domains are formed to reduce the magnetostatic energy (Hubert and Schäfer, 1998). In the case of SVs, the zero ﬁeld state and the reversal processes are inﬂuenced by the coupling with the pinned layer. Reversal occurs by the formation of sometimes complicated domain structures, depending on the direction of the bias ﬁeld with respect to the short and long axes of the elements. As an example, ﬁg. 4.9 gives the magnetization patterns in 2 × 4 μm2 rectangular SV elements upon reversal at sequential external ﬁelds, deduced by Chapman and coworkers from imaging in the DPC mode. The ﬁeld and the pinning direction are perpendicular (ﬁg. 4.9(a)) or parallel (ﬁg. 4.9(b)) to the long axis of the element. The ﬁlm structure and the intrinsic interaction ﬁelds are given in the ﬁgure caption. In ﬁg. 4.9(a) narrow regions along the long edges are visible for a large positive ﬁeld (H = +4.8 kA/m). The magnetization is there rotated away from the ﬁeld direction, due to the stray ﬁeld from the pinned layer. Reversal by rotation occurs in the middle of the sample, but due to the local demagnetizing ﬁeld initially not near the short edges. In the antiparallel state at H = −4.8 kA/m, the free layer is completely single-domain. When the ﬁeld is parallel to the long edge (ﬁg. 4.9(b)), reversal by magnetization rotation starts from the edge regions, which expand towards the middle when decreasing the ﬁeld. This reversal mode is in fair agreement with a prediction by Oti and Russek (1997). 4.3. Frequency dependence of the magnetic response, magnetization ﬂuctuations and electronic noise 4.3.1. Dynamic susceptibility The time-dependence of the magnetization of a free layer in a SV can be obtained from the Landau–Lifshitz–Gilbert equation

∂M α ∂M (4.28) = −|γ | M × Heff + , M× ∂t Msat ∂t

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where γ is the gyromagnetic factor (γ = −μ0 ge/2m, with g ≈ 2 the spectroscopic splitting factor), α is the damping parameter, and where the effective internal ﬁeld is given by H eff ≡ ∂E/∂(μ0 M ), with E the energy density. The ﬁrst term expresses the torque on the magnetization due to the effective internal ﬁeld, and the second term expresses the effect of damping. After a sudden change of the effective ﬁeld direction, the magnetization starts to precess around the new ﬁeld axis, and relaxes due to the damping gradually towards the new ﬁeld direction. A different situation arises upon the application of an ultrashort inplane ﬁeld pulse along the hard axis (pulse width 1 ns), after which the external ﬁeld is switched off. In the absence of an external ﬁeld, Heff is then equal to the uniaxial anisotropy ﬁeld Ha , and the perturbed magnetization precesses around the easy axis with an angular frequency ω0 = |γ | Ha (Ha + Msat ). (4.29) For unpatterned permalloy layers, with Msat = 800 kA/m and typically Ha = 0.4 kA/m, the precession frequency f0 = ω0 /2π is approximately equal to 0.6 GHz. For narrow stripeshaped layers in sensor elements the resonance frequency can be larger than 1 GHz as a result of the shape anisotropy. Time-resolved “stroboscopic” magneto-optical Kerr-effect studies of the damped precession have shown that for thin permalloy layers α is of the order 0.01 (Gerrits et al., 2001 and references therein). The application of a small sinusoidal hard-axis ﬁeld gives rise to a resonant response of the magnetization at the frequency f0 . This is well known from ferromagnetic resonance (FMR) studies and from measurements of the frequency-dependence of complex hard-axis magnetic permeability, μ(ω). In the case of weak damping and a high dc permeability (α 1 and Msat /Ha 1) the relative permeability is given by 2 ] + i Msat αω/ω [1 − (ω/ω ) 0 0 Msat Ha μr (ω) ∼ (4.30) , =1+ Ha [1 − (ω/ω0 )2 ]2 + Msat [αω/ω0 ]2 Ha

as can be derived from eq. (5) in a paper of Van de Riet and Roozeboom (1997). Fig. 4.10 √ shows the shapes of the permeability curves for various values of the number p = α (Msat /Ha ). The inset in the ﬁgure gives the experimental permeability of (10 nm Py/2.5 nm Co90 Fe10 ) free layers in unpatterned SVs with a parallel anisotropy conﬁguration (Varga et al., 1999). The results were found to be described well by eqs (4.29) and (4.30), with α ≈ 0.015 and p slightly smaller than 1. Coupling with the pinned layer, leading to an effective ﬁeld Heff = Ha + |Hcoupl|, could account for the observation of a larger resonance frequency than the value f0 ≈ 1.0 GHz measured for otherwise identical isolated free layers. It may be concluded that the frequency up to which the hard-axis magnetoresistive response of exchange biased SVs is to a sufﬁcient approximation in phase with the applied magnetic ﬁeld is limited by the FMR precession mode of the free layer magnetization. For a given uncoupled free layer material the maximum frequency can be increased by enhancing Ha , e.g., by enhancing the shape anisotropy. However, this inevitably reduces the sensitivity. Barman et al. (2002) discussed the general problem of the resonant magnetic response of a SV with coherently precessing free and pinned layers. Strong interlayer coupling and a relatively small exchange bias ﬁeld can lead to two resonance frequencies that are of

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Fig. 4.10. Normalized relative real and imaginary relative permeabilities of a thin magnetic ﬁlm as a function of √ the reduced frequency, as obtained from eq. (4.30), for various values of the parameter p = α (Msat /Ha ). The inset gives experimental curves for (10 nm Py/2.5 nm Co90 Fe10 ) free layers in unpatterned SVs with a parallel anisotropy conﬁguration, measured by Varga et al. (1999).

the same order of magnitude, corresponding to in-phase and out-of-phase precessions of these layers. Time-resolved optical measurements of the two resonance frequencies for Py/Cu/Co/Ir–Mn SVs as a function of the applied magnetic ﬁeld were used to deduce the magnetic interaction ﬁelds Heb , Hcoupl and Ha . 4.3.2. Switching dynamics In applications in which SVs are used as a memory element, switching is induced by the application of a ﬁeld along the easy axis. Russek et al. (2000) and Kabos et al. (2000) studied the dynamic response of SV MRAM memory elements on the application of a ﬁeld, using a pump-probe technique that employed detection with the second-harmonic magneto-optic Kerr effect. For 0.8 × 4.8 μm2 SVs with (5 nm Py/ 1 nm Co) free layers, switching in less than 0.5 ns was observed. Quantitative modelling using the LLG equation revealed that the magnetization was far from uniform during the intermediate stages of the reversal process. Furthermore, small ﬂuctuations of the magnetization were found to persist for 1–2 ns due to a weak damping constant (α ≈ 10−2 ). During that time interval, the ﬁeld that was required for the reverse switching process was found to be well below the quiescent switching ﬁeld. This time-dependent write ﬁeld would enhance the minimum time between two subsequent write processes, and thereby limit the maximum write data rate. Schumacher et al. (2002a, 2002b) further addressed this issue in studies of the dynamic response of the resistance of 1.0 × 4.0 μm2 SV MRAM elements. They showed that the oscillations of the resistance after writing can be strongly reduced by using a short ﬁeld pulse for switching with a length that is precisely adjusted to the precession period.

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Gerrits et al. (2002a, 2002b) showed that elliptical permalloy 8.0 × 16.0 μm2 8 nm permalloy elements can be switched within about 200 ps, by making use of speciﬁcally shaped magnetic ﬁeld pulses. 4.3.3. Electronic noise The root-mean-square (rms) voltage noise across a √ conductor in a narrow frequency interval [f, f + Δf ] can formally be written as VN = (SV Δf ), where SV (f ) is called the voltage noise power spectral density. For spin-valves, SV (f ) can be expressed as SV = SV,J + SV,1/f + SV,mag ,

(4.31)

with SV,J the Johnson–Nyquist electrical thermal ﬂuctuation noise, SV,1/f the 1/f noise and SV,mag the magnetic thermal ﬂuctuation noise. In fact, SV,mag is due to only one speciﬁc type of magnetic ﬂuctuations, as will be discussed below. Magnetic ﬂuctuations also contribute to the 1/f noise. Johnson–Nyquist noise is a consequence of the random motion of the charge carriers (Kogan, 1996). The resulting ﬂuctuations of the voltage are even present at zero current, i.e., in equilibrium. For frequencies well below f = kB T / h ≈ 1013 Hz, the power spectral density is independent of the frequency (“white”) and is given by SV,JN = 4kB T R,

(4.32)

where R is the (ﬁeld and temperature dependent) element resistance. For today’s ≈0.2 × 0.5 μm2 SV sensor elements in hard disk read heads, which operate at frequencies in the 50–500 MHz range, this is the dominant noise source. For small sensor volumes, low frequencies and high sense currents 1/f noise becomes dominant. The effect is due to resistance ﬂuctuations, which, in the presence of a constant sense current, I , cause voltage ﬂuctuations. The voltage spectral density can be expressed in the form of the Hooge formula (Hooge, 1969) SV,1/f =

2 αH I 2 Rav , N fγ

(4.33)

where γ ≈ 1, Rav is the time-averaged resistance, N is the total number of charge carriers and αH is the Hooge constant. Of course, N is often not a well-deﬁned quantity. However, writing the prefactor in the form αH /N has been found useful because for a wide variety of conductive systems αH is found to be ∼10−3 , within one or two orders of magnitude (Hooge, 1969, Kogan, 1996). Eq. (4.33) expresses that the 1/f noise spectral density is inversely proportional to the sample volume: in smaller devices, the effect of ﬂuctuations is less effectively averaged out. For spin-valves, the inverse volume dependence is an oversimpliﬁcation that should not be taken literally. The layer thicknesses cannot be changed without affecting the magnetic and transport properties, and the width of a SV stripe cannot be changed without affecting the magnetic anisotropy and the micromagnetic stability. For long stripe-shaped SV elements, the prefactor is only strictly proportional to the inverse stripe length. The microscopic origin of the 1/f noise of non-magnetic conductors is not known. For some metal ﬁlms, a relationship with the density of structural imperfections has been found (Kogan, 1996).

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Fig. 4.11. Results of a study of the electronic noise at room temperature of a stripe-shaped 70 × 10 μm2 (5 nm Ta/ 8 nm Py/ 2.8 nm Cu/ 6 nm Py/ 10 nm Fe50 Mn50 / 10 nm Ta) SV element. The actual magnetic length of the element is larger than the 70 μm distance between the current contacts, in order to avoid edge domain effects. Crossed anisotropy conﬁguration. (a) Noise voltage spectral density at an applied transverse magnetic ﬁeld Ht = 0.5 kA/m, for a dc sense current Isense = 2.36 mA. The Johnson–Nyquist and 1/f noise contributions are indicated schematically by dotted lines. The dashed line gives the 1/f noise contribution for measurements in a ﬁeld that magnetically saturates the element. (b) R(H ) transfer curve. (c) Noise spectral density, extrapolated to 1 Hz, as a function of the applied transverse ﬁeld, measured for increasing ﬁeld. From van de Veerdonk et al. (1997).

Fig. 4.11 (a)–(c) shows experimental results for a (8 nm Py/ 2.8 nm Cu/ 6 nm Py/ 10 nm Fe50 Mn50) SV in the form of a 10 × 70 μm2 stripe (Van de Veerdonk et al., 1997). For magnetic ﬁelds for which the element is magnetically saturated, the 1/f noise is independent of the ﬁeld. Taking N equal to the number of atoms in the active part of the element

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(“one conduction electron per atom”), αH ≈ 7 × 10−3 . For small ﬁelds, within the switch ﬁeld range, the 1/f noise spectral density depends on the ﬁeld and is for the system studied largest at a ﬁeld H ≈ 0.5 kA/m where the slope of the transfer curve is steepest (compare ﬁgs 4.11 (b) and (c)). For that ﬁeld, and for the sense current used, the 1/f noise power was more than a factor 40 larger than the high-ﬁeld value. Below f = 104 Hz, it was larger than the Johnson–Nyquist noise (ﬁg. 4.11(a)). A magnetic ﬁeld dependent contribution to the 1/f noise was also found in other studies of SVs (Smith et al., 1997; Stokes et al., 1999; Nor et al., 2000) and for other types of GMR materials (Hardner et al., 1993, 1994; Gijs et al., 1996; Kirschenbaum et al., 1997; Maraner et al., 1997; Nor et al., 1998; Lhermet et al., 2000; Petta et al., 2000). The magnetic contribution to the 1/f noise is generally attributed to the occurrence of thermally induced ﬂuctuations of the laterally inhomogeneous magnetization of the free layer, e.g., due to displacements of domain walls that are weakly pinned. As expected, the 1/f noise in SVs is therefore stronger for the parallel anisotropy conﬁguration than for the crossed anisotropy conﬁguration (Nor et al., 2000). It is also stronger near the positive and negative saturation ﬁeld, where the ripple angle diverges (section 4.2.1), than near zero ﬁeld (Hill and Nor, 2001). Hardner et al. (1993) showed for AFcoupled Co/Cu multilayers that the magnetic contribution to the Hooge constant is proportional to the out-of-phase ac response of the resistance per unit applied ac magnetic ﬁeld, which is proportional to the out-of phase ac magnetic susceptibility, μ (f, H ). For SVs such a ﬂuctuation-dissipation relation was experimentally proven by Smith et al. (1997). The continuum of thermally excited ﬂuctuations that contributes to the magnetic 1/f noise of large systems can fall apart in a small number of dominant excitations when small systems are considered. The time-dependent voltage shows then distinct jumps, in some cases between only two states. This “random telegraph noise” has been has been observed for submicrometer spin-valve GMR heads (Xiao et al., 1999; Hardner et al., 1999; Li et al., 2000b; Nichols, 2002). The high-ﬁeld 1/f noise of small systems can be dominated by contact noise (Xiao et al., 2000). Magnetic thermal ﬂuctuation noise, SV,mag , is due to thermal excitations of the FMR mode of the free layer, discussed in section 4.3.1 (Smith and Arnett, 2001; Zhu, 2002; Zhou et al., 2002; Bertram et al., 2002; Jin et al., 2002; Jury et al., 2002). Using the ﬂuctuationdissipation theorem the noise spectral density can be shown to be proportional to 1/f times the out-of-phase (dissipative) component μr of the transverse magnetic susceptibility (see, e.g., Smith and Arnett, 2001). For a SV in the crossed anisotropy conﬁguration, with Hcoupl = 0 and a free layer for which α 1 and μr (0) 1, μr (f, H = 0) follows from eq. (4.30). For H = 0, SV,mag is then given by 1 (I ΔR)2 αkB T × SV,mag (f ) ∼ (4.34) = 2 2 2 μ0 Msat Vf γ Ha [1 − (f/f0 ) ] + MHsat [αf/f0 ]2 a with I the current, ΔR the difference between the parallel and antiparallel resistances, and Vf the volume of the free layer. Eq. (4.34) has been obtained from a more general expression given by Juraszek et al. (2002). For frequencies well below the resonance frequency, the magnetic thermal ﬂuctuation noise spectral density is white. Experimental studies of the noise of SV hard disk read heads have revealed the resonance peak that is expected from eq. (4.34). In some cases even multiple resonance peaks were found, which are indicative

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of a more complicated magnetic state of the free layer (Zhou et al., 2002; Jin et al., 2002; and Jury et al., 2002). Studies of the noise spectrum are regarded as a rapid diagnostic tool for SV heads. For sufﬁciently small free layer volumes and sufﬁciently large sense currents SV,mag becomes larger than the Johnson–Nyquist noise. This situation is expected to arise for future hard disk recording heads, when the bit density is larger than 100 Gbit/inch2 (Smith and Arnett, 2001). 4.4. Anisotropy and magnetostriction of the free layer 4.4.1. Anisotropy ﬁeld of as-deposited free layers The generally used method for obtaining a uniaxial in-plane magnetocrystalline anisotropy of the free layer of a SV is growth in a magnetic ﬁeld that is strong enough to magnetically saturate the layer. A ﬁeld of 15 kA/m is sufﬁcient if use is made of permalloy or ternary fcc type Ni–Fe–Co alloys. The induced anisotropy is the result of a small degree of directional pair order of the atoms in the otherwise random solid solution (Néel, 1953, 1954a; Taniguchi and Yamamoto, 1954; Tanuguchi, 1955; Chikazumi, 1956; Slonczewski, 1963; Smith, 1963). Growth with a strong [111] texture, by making use of a suitable underlayer, eliminates an undesired grain-orientation dependent contribution to the magnetocrystalline anisotropy due to the K1 cubic anisotropy term (see section 2.9.3). The induced anisotropy ﬁeld of a thin Ni–Fe–Co layer depends on its composition, layer thickness and the materials in between which it is sandwiched (Goto et al., 1986; Ueno and Tanoue, 1995; Rijks et al., 1997; Hung et al., 2000a). Fig. 4.12(a) shows experimental results for a series of dc magnetron sputter deposited (Ta/ Py1−x Cox / Ta) ﬁlms (Rijks et al., 1997). Ha increases with increasing Co concentration, and decreases for each composition with decreasing layer thickness. The results can be analysed within the Néel model for the effect of interfaces on the anisotropy (Néel, 1954b), within which the anisotropy energy density of the layer (per m2 ) is written as a sum of a bulk contribution K(t − td,m ), with td,m the “magnetically dead layer thickness”, and a contribution Kint from the two interfaces. The layer thickness dependent anisotropy ﬁeld is then given by Ha (t) = Ha (∞) +

2Kint . (t − td,m )μ0 Msat

(4.35)

Kint is the sum of the contributions from both interfaces. Rijks and coworkers analysed their data using eq. (4.35), and expressed the results in terms of the effective thickness td,a in which the magnetization is intact, but in which the local uniaxial anisotropy energy density is zero. The anisotropy ﬁeld is then given by td,a Ha (t) = 1 − (4.36) × Ha (t = ∞). t − td,m The full lines in ﬁg. 4.12(a) give the result of a ﬁt of the data to eq. (4.36). It is seen that this model provides a fair description of the experimental data. For Ta/Py/Ta ﬁlms similar results were found by Hung et al. (2000a). Rijks and coworkers observed that deposition of Py on Cu, instead of Ta, led to a 25% smaller value of Ha (∞). Replacement of the

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

137

Fig. 4.12. (a) Induced anisotropy ﬁeld of thin Py1−x Cox alloy layers for x = 0, 0.06, 0.12 and 0.18, sandwiched between Ta layers, as a function of the F layer thickness. The full lines give a ﬁt through the data using eq. (4.36). From Rijks et al. (1997). (b) Induced anisotropy ﬁeld of Cu/Co90 Fe10 /Cu and Ta/Co90 Fe10 /Ta ﬁlms, as a function of the Co90 Fe10 layer thickness. From Hung et al. (2000a).

Ta layer on top of the Py layer by Cu was found to have no effect on Ha (∞). In contrast, Hung et al. (2000a) did not ﬁnd a signiﬁcant difference between the layer thickness dependences of Ha for ion beam deposited Py ﬁlms sandwiched between Ta or Cu. The layer thickness dependence of Ha for Cu/Co90 Fe10 /Cu ﬁlms and Ta/Co90 Fe10 /Ta, as obtained by Hung et al. (2000a), is shown in ﬁg. 4.12(b). The effective dead layer thicknesses are 0.3 and 1.3 nm, respectively. Sandwiching between Cu leads to an anisotropy ﬁeld Ha ∼ = 0.5 kA/m that is only weakly layer thickness dependent. A similar result 0.64 kA/m) was obtained by Fukuzawa et al. (2001a) for a 2.5 nm Co90 Fe10 layer (Ha ∼ = in between Cu layers in a SV. Co90 Fe10 layers in between Cu layers show thus a fairly small anisotropy ﬁeld. Sandwiching thin Co90 Fe10 layers between Ta leads to much larger anisotropy ﬁelds, with a non-monotonous layer thickness dependence (ﬁg. 4.12(b)). Hung and co-workers suggested that this might be due to a magnetoelastic contribution to the anisotropy ﬁeld. The reversal of Co free layers is hysteretic. Combination with an additional Py layer can reduce but not eliminate the coercivity (see ﬁg. 2.4(b)). In elemental Co without defects, the

138

R. COEHOORN

mechanism of pair-ordering to obtain uniaxial anisotropy can obviously not be present. It is believed that the high coercivity is related to a high saturation magnetostriction coefﬁcient of Co ﬁlms (see e.g., Lim et al., 2002a, 2002b). Grain-to-grain variations of the stress lead then to grain-to-grain variations of the anisotropy size and direction, and thereby to nucleation and pinning sites for domain walls. In the absence of more systematic work, it is presently not possible to explain the interface contribution to the induced anisotropy. The effect might be related to: (i) interface intermixing, (ii) island growth in a very early stage of the growth (affecting the local magnetization direction during this stage of the growth, and thus the driving force towards ordered pair formation), (iii) a high defect density in the initial stages of growth of the magnetic layer on an underlayer, or (iv) the modiﬁcation of the electronic structure at the interfaces. Rijks et al. (1997) observed that for spin-valves with thin Cu spacer layers, the switch ﬁeld range ΔHsw is larger than the value 2Ha , obtained for thick Cu layers. This was attributed to a laterally varying coupling with the pinned layer. For (Ta/ Py1−x Cox / tCu Cu/ Py1−x Cox / Fe50 Mn50 ) SVs with tCu = 2 nm and 0 x 18, ΔHsw was typically about twice as large as the values for SVs with negligible interlayer coupling (t > 3 nm). Like the ripple effect, which is due to anisotropy dispersion and that is already present in single magnetic layers (section 4.2.1), this effect reduces the fraction of the switch ﬁeld range in which the transfer curve is to a good approximation linear. It may thus be concluded that a reduction of the spacer layer thickness, in order to increase the MR ratio, can in fact have an adverse effect on the sensitivity. Alternative methods for obtaining a uniaxial anisotropy in a magnetic thin ﬁlm are the growth on a stepped substrate (Encinas et al., 1999), growth on a morphologically textured Ta underlayer prepared by oblique deposition (McMichael et al., 2000; Fry et al., 2001), oblique sputter deposition of the magnetic ﬁlm itself (Oepts et al., 2000 and references therein), and post-deposition patterning of the free layer in order to create a surface morphology that gives rise to shape anisotropy (Morecroft et al., 2001). So far, these methods have not been explored for applications in exchange biased spin-valves. 4.4.2. Anisotropy ﬁeld of annealed free layers The anisotropy ﬁeld of a free layer with uniaxial anisotropy due to pair ordering can be changed by a post-deposition anneal treatment in a magnetic ﬁeld. The effect has been studied by Rijks et al. (1994a, 1994b) (see ﬁg. 1.9), Rijks et al. (1997) and Baril et al. (2001a, 2001b). If the applied ﬁeld during annealing is parallel or orthogonal to the easy axis, the anisotropy ﬁeld increases or decreases, respectively. In the latter case, the sign of the anisotropy energy density parameter Ka can eventually even reverse, so that the easy axis switches by 90◦ , and becomes parallel to the applied ﬁeld. For a SV in which the free layer already has an appreciable anisotropy in the as-deposited state, it is useful to describe the effect phenomenologically in terms of a parameter r, the time dependent “fraction of rotated pairs”: 1 Ka (t, Ta ) (4.37) − 2 2Ka (0) (Baril et al., 2001a). Here Ka (0) is the (positive) anisotropy energy density of the asdeposited sample, and Ta is the anneal temperature. Ka (t, Ta ) is measured at room temperature. Note that r can become larger than 1, because prolonged annealing could increase r(t, Ta ) ≡

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139

Fig. 4.13. Results analyses of the effect of annealing of SVs with (5 nm Py/ 0.6 nm Co90 Fe10 ) composite free layers with induced anisotropy, in a ﬁeld perpendicular to the as-deposited easy. The effect on the magnetic anisotropy is expressed as the “fraction of rotated pairs”, given as a function of the annealing time and temperature. From Baril et al. (2001a).

the absolute value of Ka to values above the original value. Fig. 4.13 shows experimental results from Baril et al. (2001a) for (5 nm Py/ 0.6 nm Co90 Fe10 ) composite free layers in a SV, with an as-deposited anisotropy ﬁeld of ≈0.6 kA/m. Switching of the easy axis occurs when r = 0.5. For Ta 160 ◦ C, r follows over a broad time interval an ln(t) law. For Ta = 120 ◦ C, two regimes are visible. The results shown in ﬁg. 4.13 can be explained within the framework of a model by Uchiyama et al. (1974) (see also Takayasu et al. (1974) and references in both publications). The change of the pair orientation distribution is viewed as the result of atomic diffusion via vacancy sites. The model predicts that the relaxation takes place in two subsequent stages, t1 < t < t2 , and t > t2 , preceded by a prelogarithmic stage (t < t1 ). In the ﬁrst stage, thermal relaxation takes place as a result of diffusion in the presence of an excess (non-equilibrium) vacancy concentration, as expected to be present for thin ﬁlms. A postdeposition heat treatment can be used to reduce the excess vacancy concentration, thereby reducing the anisotropy relaxation that can take place during this stage. When the anneal temperature is high (Ta > 300 ◦ C from the work of Takayasu et al. (1974)) a second stage can be reached in which the excess vacancies have been annihilated, so that much slower relaxation takes place due to diffusion via the equilibrium thermal vacancy concentration. Baril and coworkers viewed the data shown in ﬁg. 4.13 as corresponding to diffusion in the ﬁrst stage. For Ta = 120 ◦ C, the onset of the ﬁrst stage is visible at t1 ≈ 6000 s, whereas for the higher anneal temperatures chosen, the onset has already taken already place for t < 60 s. The onset time is equal to the average waiting time t1 = t1,0 exp(Ea /kTa ) before the ﬁrst atomic jump takes place, where t1,0 is the attempt time, and Ea is the activation energy for diffusion in the situation of the excess vacancy concentration. The data are well described using t1,0 = 10−13 s and Ea = 1.1 eV. For comparison, the activation energy in the second (thermal equilibrium) stage was found to be approximately 2.2 eV (Takayasu et al., 1974). It is concluded that for high-temperature applications of SVs the thermal stability of the induced anisotropy ﬁeld of a free layer may be an issue of serious concern. Signiﬁcant

140

R. COEHOORN

relaxation can already take place at temperatures well below 200 ◦ C, and on a time scale of 103 –104 s, when the processes that give rise to a reduction of the MR ratio (section 2.10) are not yet very effective. 4.4.3. Effect of stress – magnetostriction Magnetostriction affects the reversal of the free layer in two ways. First, almost inevitable non-uniform stress in the ﬁlm, varying from grain to grain or even within grains, leads via the inverse magnetostriction effect to lateral variations of the anisotropy. The resulting domain structure may give rise to nucleation and pinning centers of domain walls, and thereby to hysteretic magnetization reversal processes. This situation is encountered, e.g., for Co, as discussed in section 4.3.1. A clear indication that low magnetostriction is related to low coercivity was obtained by Kitade et al. (1995) from a systematic study of the coercivity of thin Co-rich Ni–Fe–Co ﬁlms. For sensor applications, the saturation magnetostriction coefﬁcient of the free layer, λs , should therefore be small. Second, a laterally uniform stress gives rise to an additional contribution to the anisotropy ﬁeld. When the stress, σ , is along the easy axis of the free layer, the contribution to Ha due to the inverse magnetostriction effect is Ha,ims =

3λs σ . μ0 Msat,f

(4.38)

In the case of tensile stress (σ > 0) and for λs > 0 the anisotropy ﬁeld becomes larger. Magnetostriction constants for thick Fe–Co–Ni alloy layers have been reported by Tolman (1967), Klokholm and Aboaf (1981), and Miyazaki et al. (1994). For thin layers, the magnetostriction constant can be different from the bulk or thick layer value, as has been revealed from numerous studies (Zuberek et al., 1988; Ounadjela et al., 1989; Valletta et al., 1991; Song et al., 1994; Kim and Silva, 1996; Gurney et al., 1997; Aoshima et al., 2000; Hung et al., 2000a; Gafron et al., 2001; Fukuzawa et al., 2002). An often used phenomenological Néel-type expression for the layer thickness dependence of the magnetostriction constant, analogous to eq. (4.35), is: λs (t) = λs,bulk +

λs,int (t − td,m )

.

(4.39)

We ﬁrst focus on permalloy. For thick Ni100−x Fex alloy layers, λs ≈ (x − 18) × (1.8 · 10−6) for concentrations within a few percent from the zero-magnetostriction point. Sandwiching Py in between Cu or Ta gives rise to a positive interface contribution to λs , as can be seen from ﬁg. 4.14(a), taken from Hung et al. (2000a). In good agreement with results reported earlier by Gurney et al. (1997), λs,int ≈ +7 × 10−15 m for Ta/Py/Ta ﬁlms. If required, the positive interface contribution to the magnetostriction can be compensated by using an alloy for which the bulk magnetostriction is negative. Microstructure or stress effects play a certain role, which follows from the difference between the interface contributions for Ta/Cu/Py/Cu/Ta and Cu/Py/Cu ﬁlms (both on Si(001)). Thick Co90 Fe10 layers have near-zero magnetostriction (Tolman, 1967) or a small negative magnetostriction (|λs | < 5 × 10−6 , Miyazaki et al. (1994)). Also for this alloy the magnetostriction of ultrathin layers is different from that of thick layers. Fig. 4.14(b), which gives results obtained by Hung et al. (2000a), shows that sandwiching Co90 Fe10

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

141

Fig. 4.14. Saturation magnetostriction constant of Py (a) and Co90 Fe10 (b), sandwiched in between Ta or Cu layers, as a function of the F layer thickness. From Hung et al. (2000a).

in between Ta or Cu gives rise to a positive interface contribution to λs . The bulk magnetostriction is negative, but the thin ﬁlm magnetostriction can be positive. The interfacial contribution is one order of magnitude larger than for Py, and eq. (4.39) is not valid for very thin ﬁlms. Fukuzawa et al. (2002) reported that the magnetostriction of 1 to 3 nm Co90 Fe10 layers depends sensitively on the Ru and Cu layer thicknesses in a composite Ru/Cu underlayer, which determines the bulk strain in the magnetic layer. By precisely controlling both layer thicknesses, it is possible to obtain ultrathin Co90 Fe10 layers with zero magnetostriction. The change of the anisotropy ﬁeld of the free layer in a SV due to the inverse magnetostriction effect (eq. (4.38)) has been used by Mamin et al. (1998) to obtain a spin-valve strain sensor. A (5 nm Ta/ 8 nm Ni84 Fe16 / 2.5 nm Cu/ 3 nm Co/ 15 nm Fe50 Mn50/ 5 nm Ta) SV was used, within which the free layer was chosen to have a large negative magnetostriction, λs ≈ −3 × 10−6 . The high sensitivity of the sensor was reported to make it competitive with piezoresistive Si cantilever strain sensors. Baril et al. (1999) presented an accurate method for deducing λs from the transfer curve of a SV structure that is deposited on a substrated that is bended. The sensitivity was found to be better than 1 × 10−7 . A quantitative analysis of the transfer curves of SVs with a Py/Co free layer conﬁrmed the increase of λs with decreasing Py layer thickness, as expected from the work discussed above on single layers (Gafron et al., 2001).

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R. COEHOORN

4.5. Interlayer coupling Three physically distinct effects contribute to the interlayer coupling: (1) Oscillatory interlayer exchange coupling (IEC). This is the indirect exchange coupling across NM layers that is used in AF-coupled multilayers to obtain an antiparallel alignment at zero ﬁeld. The amplitude decreases with increasing layer thickness. (2) Ferromagnetic magnetostatic Néel coupling. (3) Direct ferromagnetic exchange coupling via pinholes (ferromagnetic bridges across the NM layer). The latter two contributions decrease monotonically with increasing spacer layer thickness. As an example, ﬁg. 4.15 gives the coupling energy Jcoupl as obtained by Leal and Kryder (1996) for Ta/Py/Cu/Py/Fe50Mn50 SVs, sputter deposited on relatively ﬂat glass and Si substrates, and rough Si/Al2 O3 /Si3 N4 substrates. Fig. 4.15(a) shows two distinct regions in which the oscillatory coupling is AF, viz. around tCu = 1 and 2 nm. Fig. 4.15(b) shows that the ferromagnetic non-oscillatory contribution to the coupling (dashed and dotted-dashed

Fig. 4.15. (a) Interlayer coupling energy of (glass substrate/ 5 nm Ta/ 7 nm Py/ t nm Cu/ 5 nm Py/ 8 nm Fe50 Mn50 ) SVs. The dashed line gives the magnetostatic Néel coupling contribution, with h = 0.5 nm and L = 35 nm. The dashed-dotted line gives the IEC contribution, assuming an unstrained Cu spacer (so that Λ = 0.94 nm). The full line gives the magnetostatic + strained IEC contribution (with Λ = 1.35 nm). (b) Interlayer coupling energy of (substrate/ 4 nm Ta/ 6.5 nm Py/ t nm Cu/ 4 nm Py/ 8 nm Fe50 Mn50 ) SVs where the substrate is either bare Si or (Si/100 nm alumina/ 500 nm silicon nitride). The dashed line gives the magnetostatic coupling energy for h = 0.9 nm and L = 90 nm. The dashed-dotted line gives the magnetostatic coupling energy for h = 0.25 nm and L = 40 nm. From Leal and Kryder (1996).

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143

lines) is largest for the ﬁlms on the rougher Si/Al2 O3 /Si3 N4 substrate. In sections 4.5.1 and 4.5.2 the IEC and Néel-type coupling, respectively, are discussed in more detail. 4.5.1. Oscillatory interlayer exchange coupling In the idealized case of ﬂat interfaces, oscillatory interlayer exchange coupling (IEC) would be the only cause of coupling between the free and pinned layer. For F1 /NM/F2 trilayer systems, the effect can be understood as resulting from the quantum interference of electron states which accommodate electrons at the Fermi level that propagate across the spacer layer, reﬂect partially at the F1 layer and reﬂect subsequently partially at the F2 layer, so that a standing wave is formed. As the reﬂection coefﬁcients are spin-dependent, the resulting total energy of the system depends on the alignment of the F layer magnetizations (Bruno, 1995). The IEC effect is related to the oscillatory Ruderman–Kittel–Kasuya– Yoshida (RKKY) coupling between magnetic impurity atoms in a non-magnetic host metal (Kittel, 1996). The effect, which has been observed for a wide variety of spacer layer metals, has been reviewed by Fert and Bruno (1994), Stiles (1999) and Bürgler et al. (2001). To a ﬁrst approximation, the dependence of the coupling parameter J iec on the spacer layer thickness tNM can be written as JIEC = J0

sin(2π tNM Λ + φ) 2 tNM

,

(4.40)

where Λ is the period, φ is the phase, and J0 is a prefactor that (like φ) depends on the composition of the F layers (Coehoorn and Duchateau, 1993; Kudrnovsky et al., 1997), their thickness, and even on the thicknesses and compositions of more remote layers (such as a NM cap layer). This can be understood from the fact that the NM/F interfaces are partially transmissive, so that reﬂections at more remote interfaces modify (by interference) the effective size and phase of the reﬂection coefﬁcient at the NM/F interface. These effects are strongest for highly ordered layers and sharp interfaces. For a hypothetical structureless free electron metal spacer layer, the period Λ would be equal to λF /2, with λF the Fermi wavelength. For Cu, one would then expect that Λ ≈ 0.23 nm. However, after inserting this very short period in eq. (4.40) and plotting JIEC only for discrete spacer layer thicknesses that are an integral multiple of the interplanar distance d for the growth direction used, an “aliasing effect” becomes visible. The real period is thus orientation dependent and can be much larger than λF /2 (Coehoorn, 1991; Chappert and Renard, 1991; Deaven et al., 1991). For non-free electron spacer layer metals multiple oscillations are possible, with periods that follow from the detailed shape of the Fermi surface. For SVs with a strong [111] crystallite texture, a comparison with experimental and theoretical results on the coupling across Cu(111) spacer layers is most relevant. The theoretical period for coupling across a Cu(111) spacer layer is Λ ≈ 0.95 nm. MBE experiments on Co/Cu/Co trilayers, deposited on Cu(111) single crystals, have revealed a strong AF coupling peak around tAF1 = 0.9 nm, with J (tAF1 ) = −1.1 mJ/m2 (Johnson et al., 1992). However, no second AF peak has been observed for MBE grown systems. This has been attributed to a lack of lateral continuity in the Cu layers due to the formation of fcc twins (Camarero et al., 1994). The fact that in SVs a second AF peak is observed (ﬁg. 4.15) indicates that a more continuous spacer layer is formed, most likely due to the larger kinetic energy of the sputter deposited atoms that arrive at the substrate.

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R. COEHOORN

An overview of results of selected experimental studies on the IEC in SVs is given in table 4.1. The periods and coupling strengths that are given are in most cases only rough estimates, derived from data in a limited range of spacer layer thicknesses. If a second AF coupling peak is observed, it is situated in the range 1.8 nm < tAF2 < 2.3 nm. The period falls in the range 1.0 < Λ < 1.5 nm. The large width of this range, and the difference with the theoretical value given above, might be due to the extreme sensitivity of Λ to strain in the spacer layer (Leal and Kryder, 1996). The overall picture that emerges conﬁrms that the coupling strength is indeed extremely sensitive to the microstructure, as affected by the deposition conditions and the layer structure. Surfactants such as In, Pb, Au and Ag also affect the IEC, viz. by modifying the microstructure of the spacer layer (Egelhoff et al., 1996b, 1996c; Chopra et al., 2002). Note that even the largest observed coupling parameter in the second AF peak is a factor of ≈50 smaller than observed for MBE-grown Co/Cu/Cu trilayers in the ﬁrst peak, whereas from the 1/t 2 thickness scaling only a reduction by a factor ≈5 would be expected. A lack of systematic data and the strong effect of the interface structure and microstructure makes it at present difﬁcult to comment on the dependence on the composition of the F layers, on the thickness of the free layer, and on the presence of non-magnetic back layers (see, e.g., Hong et al. (2000) and Makino et al. (2001)). 4.5.2. Magnetostatic Néel coupling Kools (1995) explained the long range ferromagnetic coupling that is superimposed on the oscillatory IEC (see ﬁg. 4.15) in terms of the Néel model for magnetostatic coupling between non-ﬂat layers (Néel, 1962). Néel coupling (also called “orange peel coupling” or “topological coupling”) occurs in the case of a correlated waviness of the two NM/F interfaces. For the case of two semi-inﬁnite F layers, separated by a wavy NM layer with a uniform thickness, the effect can be understood from ﬁg. 4.16(a). When the magnetization directions in the F layers are uniform, there is an attractive interaction between the magnetic pole densities at the two NM/F interfaces for the case of parallel magnetization directions. In realistic SVs, this situation of a positively correlated waviness of the two interfaces arises in the case of a conformal coverage of a non-ﬂat bottom F layer by the NM layer. The lateral correlation length, L, is expected to be of the order of the average grain size. In the case of two semi-inﬁnite magnetic layers with saturation magnetizations Msat,1 and Msat,2 , separated by a NM layer with an average thickness t, of which the interfaces have a fully correlated two-dimensional sinusoidal waviness with a wavelength L and amplitudes h1 and h2 , the magnetostatic coupling energy per unit area is (Néel, 1962): √

1 JN = √ kh1 h2 μ0 Msat,1 Msat,2 exp −kt 2 , (4.41) 8 2 with k ≡ 2π/L. Note that JN is deﬁned according to the convention given by eq. (4.6). In agreement with a remark by Schulthess and Butler (2000), the expression given in the paper by Kools et al. (1995) has been corrected. Schulthess and Butler (2000) showed that eq. (4.41) is correct to second order in (h/L). For an amplitude h the average deviation from the mean height is Ra = (2/π)2 h ∼ = 0.405h. Ra can be measured by scanning probe methods such as atomic force microscopy.

System NiO/Py/Co/Cu/Co/Py NiO/Py/Co/Cu/Co/Py Ta/Py/Cu/Py/Fe50 Mn50 Ta/Py/Cu/Py/Fe50 Mn50 PtMn/Co90 Fe10 /Ru/Co90 Fe10 / Cu(1)/Co90 Fe10 /Py/Cu(2)/Ta Ta/PtMn/Co90 Fe10 /Ru/ Co90 Fe10 /Cu/Co90 Fe10 /Py/Ta Py–Cr/Py/PtMn/Co90 Fe10 /Ru/ Co90 Fe10 /Cu/Co90 Fe10 /Py/Cu/ Al2 O3 Ta/Py/Ir–Mn/ Co90 Fe10 /NOL/ Co90 Fe10 /Cu/Co90 Fe10 /Py/ Co90 Fe10 /NOL/Cu/Ta UL/PtMn/Co90 Fe10 /Cu/ Co90 Fe10 /Py/Ta

J0 /(tAF2 )2 (μJ/m2 ) ≈ 20

Λ (nm) 1.0

tAF2 (nm) 2.1

1.3 ≈ 1.0 1.35 –

1.8 2.0 2.0 2.2

≈ 13 ≈2 ≈7 ≈2

1.3

2.0

≈ 12

1.4

2.2

≈7

≈ 1.5

2.0

3

2.2 2.3 2.1 2.1

< 0.5 ≈2 ≈3 ≈5

1.15 1.16 1.13 1.17

Remarks

Reference

Proof that the GMR ratio is not related to the magnitude of the interlayer coupling.

Speriosu et al. (1991)

Pinhole coupling for tCu < 1.7 nm. Fig. 4.13(a) Oxygen exposure after Cu(1) deposition. Dependence on Py and Cu(2) thickness. Effect UL and cap layer on coupling

Anthony et al. (1994) Kools et al. (1995) Leal and Kryder (1996) Makino et al. (2001) Lin and Mauri (2001)

A reference structure without NOLs shows negligible IEC

Li et al. (2001b)

UL = Ta/Py–Cr, SDa , pAr = 1 mTorr UL = Ta/Py–Cr, PDa , pAr = 1 mTorr UL = Ta/Py–Cr, PDa , pAr = 0.5 mTorr UL = Py–Cr/Py, PDa , pAr = 0.5 mTorr

Kools et al. (2003)

a SD/PD = Static/Planetary Deposition (wide/small angular distribution of atoms arriving at the substrate).

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

TABLE 4.1 Results of selected studies on the oscillatory interlayer exchange coupling in SVs. The coupling parameter J0 and the period Λ are deﬁned by eq. (4.40), and tAF2 is the Cu layer thickness at which the second AF peak is located. UL = underlayer

145

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R. COEHOORN

Fig. 4.16. Schematic magnetic pole densities near the interfaces of layered structures with conformal interface waviness. (a) A spacer layer in between two semi-inﬁnite F layers, with ferromagnetic magnetostatic Néel coupling. (b) An F1 /NM/F2 system with ﬁnite F layer thicknesses. (c) An F1 /F2 /NM/F3 system with semi-inﬁnite F1 and F3 layers. (d) An AF/Fp1 /Fp2 /Fp3 /NM/Ff spin-valve with zero magnetostatic and Néel coupling. In (c) and (d) the thicknesses of the AF coupling layer(s) in the Sy-AF pinned layer are neglected.

In table 4.2 experimental data are given on the parameters L and h as deduced from microstructural analyses for SVs sputter deposited using a variable Ar pressure (Kools, 1995; Kools et al., 1995) and using various substrate/underlayer combinations (Leal and Kryder (1996), see ﬁg. 4.15). The use of a higher Ar pressure during sputtering leads to a smaller kinetic energy of the atoms that arrive at the substrate, and thereby to more corrugated interfaces (see also section 2.9.4). The theoretical interaction energy, as obtained using eq. (4.41), and the experimentally observed interaction energy show a good agreement. Further experimental studies on the relationship between the interface corrugation and the interlayer coupling were carried out by Park et al. (1996), Park and Shin (1997), Parks et al. (2000), Li et al. (2001b), Hong et al. (2001) and Kools et al. (2003). Li and coworkers and Hong and coworkers observed that the inclusion of a NOL below the spacer layer gives rise to a planarising effect, which reduces the Néel coupling. For layer stacks consisting only of metallic layers it is often observed that the roughness amplitudes increase from the bottom to the top of the layer structure (“cumulative waviness”). The coupling between two magnetic layers that each consist of one or more strongly coupled F layers is straightforwardly expressed as a sum of contributions of the form given by eq. (4.41). Each term corresponds to a pair of interfaces at opposite sides of the spacer layer, and has a sign that depends on the product of the phases (+ or −) of the pole densities at the two interfaces (Kools et al., 1999; Schulthess and Butler, 2000; Kim et al., 2001c). It is always possible to rewrite the resulting expression for JN as product of two factors that are determined exclusively by √ the internal layer structure of each of the two magnetic layers, times the factor exp(−kt 2 ). We discuss three simple examples with conformal

System Ta/Py/Cu/Py/Fe50 Mn50

Ta/Py/Cu/Py/Fe50 Mn50

exp

L (nm) 11 11 11

h (nm) 0.8 0.9 1.2

JNth (2 nm) (μJ/m2 ) 5 6 12

JN (2 nm) (μJ/m2 ) 5 6 12

40 50 75

0.4 0.2 0.9

2 0.2 4

3 1 5

Remarks

Reference

pAr = 1.5 mTorr pAr = 5 mTorr (optimum GMR ratio) pAr = 10 mTorr

Kools (1995), Kools et al. (1995)

Glass substrate (ﬁg. 4.13(a)) Si substrate (ﬁg. 4.13(b)) Si/Al2 O3 /Si3 N4 substrate (ﬁg. 4.13(b))

Leal and Kryder (1996)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

TABLE 4.2 Results of selected studies on the magnetostatic Néel coupling in SVs. L and h (= Ra /0.405, see text) are the lateral correlation length and roughness amplitude as exp deduced from microstructural analyses. JNth and JN are the coupling parameters obtained from L and h using eq. (4.41), and obtained from an analysis of experimental data, respectively

147

148

R. COEHOORN

waviness (i.e., equal amplitudes hi for all interfaces) and with F layers that all have the same saturation magnetization. See ﬁg. 4.16 (b)–(d). The coupling between two single F layers with ﬁnite thicknesses t1 and t2 (e.g., the free and pinned F layers of a conventional simple SV, ﬁg. 4.16(b)) √ is given by eq. (4.41), multiplied by a factor F (t1 )F (t2 ), with F (ti ) = (1 − exp(−kti 2 )). Kools et al. (1999) studied the Néel coupling for Py/Cu/Co/Fe50 Mn50 SVs, and indeed found the expected reduction of the coupling ﬁeld with decreasing Co pinned layer thickness. For a SV with a semi-inﬁnite free layer and a Sy-AF pinned layer, consisting of a semiinﬁnite layer Fp1 and an antiparallel layer Fp2 with thickness tF2 (see ﬁg. 4.16(c)), JN is √ √ given by eq. (4.41) multiplied by a factor [1 − (1 + exp(−kδ 2 )) × exp(−ktF2 2 )]. Here δ is the thickness of the AF coupling layer. If δ = 0, the pole density at the top interface of layer Fp2 is now doubled as compared to the situation shown by ﬁg. 4.16(b). AF Néel √ coupling is then obtained when tp2 is sufﬁciently small, viz. when ktp2 < ln(2)/ 2 ≈ 0.49. Such a situation was studied experimentally and theoretically by Kim et al. (2001c). The method discussed above could be used to eliminate the Néel coupling. However, a disadvantage is that the net magnetization of the Sy-AF pinned layer is then not equal to zero: the Néel coupling cannot be eliminated when tp1 = tp2 . The use of unequal pinned layers thicknesses leads to a reduced effective exchange bias ﬁeld, and (in a microstructured device) to magnetostatic coupling with the free layer. Wang et al. (2003) proposed to solve this by making use of an (AF/Fp1 /Ru/Fp2 /Ru/Fp3 /NM/Ff ) layer structure. The magnetizations of the Fp1 and Fp3 layers are antiparallel to that of the Fp2 layer. The layer structure is shown in ﬁg. 4.16(d). As above, we consider the case of equal saturation magnetizations of all F layers. For a given tp3 layer thickness the Néel coupling and the net magnetization √ √ √ are equal to zero when kt2 = −(1/ 2 ) ln[2 × cosh2 (−kδ/ 2 ) × exp(−kt3 2 ) − 1] and tp1 = tp2 − tp3 (Coehoorn, 2003). Note that there is only a solution if the argument of the natural logarithm is in between √ 0 and 1. When the Ru layer thickness δ can be neglected, this requires that ktp3 < ln(2)/ 2 ≈ 0.49. When tp3 is small (ktp3 1) and δ = 0, the solution is tp1 = tp3 and tp2 = 2tp3 . For SVs with a roughness correlation length L that is at least a factor of 15–20 larger than the Fp3 layer thickness, this novel approach may be expected to be practically feasible. 4.6. Exchange anisotropy Exchange anisotropy is a unidirectional magnetic anisotropy of an F layer that is exchange coupled to an AF layer. The effect was discovered already more than 40 years ago (Meiklejohn and Bean, 1956, 1957), and has been studied intensively during the past decade in view of its relevance to exchange-biased GMR spin-valves and magnetic tunnel junctions. In spite of that, the microscopic mechanisms that lead to the effect are not yet fully understood. The main obstacle is the difﬁculty to incorporate the complex interplay between the microstructure of the AF layer and the interfaces, and the magnetic structure in the AF layer. In this subsection we give a phenomenological description of the effect, we brieﬂy introduce some of the proposed models, and give a comprehensive overview of the recent advances on AF thin ﬁlm materials that are relevant to SV applications. Earlier reviews on the subject, with a stronger focus on the various proposed models and on experimental studies on model systems, have been written by Nogues and Schuller (1999), Berkowitz and Takano (1999), Stamps (2000) and Kiwi (2001).

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

149

Fig. 4.17. Contours of constant exchange-bias ﬁeld Heb (a) and coercive ﬁeld Hc (b) (both in kA/m) for (111) oriented Ni80 Fe20 /Fe50 Mn50 , plotted as a function of the thickness of both layers, measured using a double-wedge shaped sample. Although growth was performed by MBE on a Cu(111) single crystal, the results obtained are qualitatively very similar to those obtained by sputter deposition on, e.g., Si(100) using a 2 nm Ta underlayer. From Jungblut et al. (1994).

4.6.1. Phenomenological description The simplest possible phenomenological expression for the interaction energy between an AF and a pinned F layer is E = −Jeb cos αp ,

(4.42)

where αp is the angle between the magnetization direction of the pinned F layer and the exchange bias direction. This is eq. (4.4), under the condition that the spins in the AF layer are ﬁxed. If required, eq. (4.42) can be extended by including higher order terms in cos αp (Hu et al., 2002). The direction of the exchange bias ﬁeld is parallel to the pinned F layer magnetization during deposition or during ﬁeld cooling.10 From the SW model, the loop shift in a ﬁeld parallel to the exchange bias ﬁeld is given by Heb =

Jeb μ0 Msat,p tp

(4.43)

(which is eq. (4.11)). The exchange bias ﬁeld is thus proportional with the inverse of the pinned layer thickness. For sufﬁciently thick AF layer thicknesses, this is in good agreement with experiment, as can be seen, e.g., from ﬁg. 4.17(a). The ﬁgure shows the results of a systematic study of the F and AF layer thickness dependence of the exchange bias ﬁeld for (111) oriented permalloy, exchange biased by Fe50 Mn50 (Jungblut et al., 1994). For small AF layer thicknesses, Heb is seen to vanish. For Fe50 Mn50, this critical AF layer thickness, tAF,min , is ∼6 nm. This deviation from the simple description given above is observed for all AF/F bilayer systems. 10 An exception is observed for systems with (1) a very low Néel temperature, and (2) an antiferromagnetic AF/F

coupling at the interface. Field cooling in very high ﬁelds then leads to a “negative” exchange bias (Nogues et al., 1996). For the AF materials that are most relevant to SV applications (table 4.7), these conditions do not apply.

150

R. COEHOORN

When the applied ﬁeld is perpendicular to the exchange bias ﬁeld, the predicted ﬁelddependence of the magnetization of the pinned layer is given by H . M = Msat sin arctan (4.44) Heb This expression is well obeyed for samples that show a small hysteresis, Hc /Heb 1. However, often, the hysteresis upon reversal of the pinned F layer is much larger than observed for an isolated F layer, and is particularly large close to the critical thickness. For the case of Fe50 Mn50 , this can be seen from ﬁg. 4.17(b). The perpendicular ﬁeld that is required to saturate the magnetization then exceeds the exchange bias ﬁeld. In such cases, the full angular dependence of the M(H ) curves, the exchange bias ﬁeld and the coercive ﬁeld can be quite complicated (see, e.g., Ambrose et al., 1997; Xi and White, 2000c). These observations may be explained qualitatively by viewing the exchange bias ﬁeld as a result of the exchange interaction with stable AF regions (e.g., grains or domains that have a ﬁxed magnetic structure, independent of the F layer magnetization direction), and viewing the hysteresis as the result of the exchange interaction with unstable AF regions: their magnetization directions switch when the F layer magnetization is switched. In the latter case, the magnetic anisotropy of the AF layer, contributes, effectively, to the anisotropy of the F layer, and leads to hysteretic reversal. For some systems, the angular dependence of the hysteresis loops reveals a distribution (“dispersion”) of local exchange bias ﬁeld directions. For Fe–Mn based systems the dispersion is usually only a few degrees, but for Ir–Mn/Co–Fe bilayers, e.g., Hou et al. (2001) observed a dispersion of ∼30◦ of the pinning ﬁeld direction. The exchange bias ﬁeld decreases with increasing temperature, and vanishes at the blocking temperature, Tb . Fig. 4.18(a) shows experimental results for Fe50 Mn50 . For many materials, the blocking temperature increases with the AF layer thickness, until it saturates. In the case of Fe50 Mn50 , this happens at a thickness of about 10 to 12 nm. Fig. 4.18(b) shows experimental results for various AF materials. For all materials, the blocking temperature is signiﬁcantly lower than the Néel temperature (see the overview in table 4.7). The blocking temperature depends on the AF layer thickness, the AF grain size and other aspects of the microstructure of the AF layer, and in some cases on a thermal treatment that affects the magnetic structure in the AF layer. The dashed curve gives data for a heat treated (10 nm Ir–Mn/ 30 nm Py) bilayer. It shows that for different systems based on the same AF material a quite different temperature dependence of the exchange bias ﬁeld can be obtained. More detailed data on the exchange bias ﬁeld and the coercive ﬁeld for the case of Ir–Mn/Co90Fe10 bilayers are shown in ﬁg. 4.18 (c) and (d), respectively. A comparison of these two ﬁgures shows that the hysteresis is largest around the blocking temperature. The exchange bias ﬁeld does not always show a monotonic decrease with increasing temperature, but can peak at a certain temperature (Lin et al., 1994a; Hou et al., 2000; Nagasaka et al., 2000). This is found in particular for AF/F bilayers with TN (AF) > TC (F), such as NiMn, PtMn or (Pt–Pd)Mn in combination with, e.g., permalloy (TC ≈ 580 ◦ C). At least part of the effect can then be explained as a result of a stronger decrease of the saturation magnetization of the pinned layer with increasing temperature than that of Jeb (see eq. (4.43)).

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

151

Fig. 4.18. (a) Temperature and AF layer thickness dependence of the exchange bias ﬁeld for Fe50 Mn50 . From Coehoorn et al. (1998). (b) Temperature dependence of the exchange bias ﬁeld for (Si/ 5 nm Ta/ 5 nm Py/ 2 nm Co90 Fe10 / 2.5 nm Cu/ 2.2 nm Co90 Fe10 / AF/ 5 nm Ta) top SVs, with AF layers formed by 8 nm Fe–Mn, 5 nm Ir–Mn, 30 nm NiMn, 30 nm PtMn and 30 nm (Pt–Pd)Mn. The SVs based on NiMn, PtMn and (Pt–Pd)Mn were annealed after the deposition. From Anderson et al. (2000a, 2000b, 2000c). Dashed curve: (Py/ 10 nm Ir–Mn), from van Driel (1999). (c) Temperature dependence and AF layer thickness dependence of Heb of (Si/3.5 nm Ta/ 2 nm Py/ tAF nm Ir19 Mn81 / 20 nm Co90 Fe10 / 5 nm Ta) materials, measured during cooling in a ﬁeld from a temperature above Tb . From van Driel et al. (2000b). (d) Hc for the same materials as in (c).

The exchange bias ﬁeld displays magnetic relaxation. In a magnetic ﬁeld parallel to the preferred direction, Heb can increase or decrease, depending on the temperature, the preparation method and the thermal history. In a ﬁeld antiparallel to the preferred direction, Heb decreases and becomes even negative after some time. These effects depend on the direction of the magnetization of the F layer during the experiment; the size and direction of the ﬁeld are only important in as far they determine the F-layer direction. As an example, ﬁg. 4.19 shows the relaxation of the exchange bias ﬁeld for an Ir19Mn81 /Ni80 Fe20 bilayer. The relaxation becomes faster with increasing temperature. Similar results have been obtained for many other exchange bias materials (Fulcomer and Charap, 1972; Hempstead et al., 1978; Lin et al., 1995a; Van der Heijden et al., 1998a, 1998b; Fujikata et al., 1998a; Oshima et al., 1998; Nishioka, 1999; Carey et al., 2001). Application of the external ﬁeld under an angle with the preferred ﬁeld can lead to a rotation of the direction of the exchange bias ﬁeld (see, e.g., Hempstead et al., 1978; Lin and Mauri, 1999; Anderson et al., 2000a). As discussed in section 4.6.4, magnetic relaxation is due to thermally activated magnetiza-

152

R. COEHOORN

Fig. 4.19. Relaxation of the normalized exchange bias ﬁeld of (tAF nm Ir19 Mn81 / 6 nm Ni80 Fe20 ) bilayers, obtained from magneto-optical Kerr effect measurements. Measurement time per hysteresis loop: 12 s. At t = 0 the external ﬁeld is reversed, forcing the magnetization of the F layer to be switched from parallel to antiparallel with respect to the original exchange bias direction. Heb is normalized by Heb at t = 0. (a) Relaxation at T = 127 ◦ C, for tAF = 30 nm. After 2140 minutes the external ﬁeld is reversed again. (b) Relaxation curves at 77, 127 and 177 ◦ C, for tAF = 10 nm ﬁlms. From van Driel et al. (1999a, 1999b).

tion reversal processes in the AF layer. Obviously, this potentially limits the application of exchange-biased spin-valves at high temperatures. For improving the properties of SVs, it can be of interest to let the free layer interact weakly with a second AF layer (see section 2.3). The required small but well-controlled exchange bias ﬁeld can be obtained by making use of a thin NM spacer layer between the AF and F layers. The decay of the exchange bias ﬁeld across a NM layer has been studied by Gökemeijer et al. (1997), Thomas et al. (2000) and Mewes et al. (2000). Thomas and coworkers studied sputter deposited Ir22 Mn78/NM/Co84 Fe16 F systems, and observed an approximately exponential decay of the exchange bias ﬁeld with the spacer layer thickness. A decay length λ ≈ 0.2 nm was observed for NM = Cu, Au, Ru, Pd and Al, whereas for Ti, λ ≈ 0.1 nm and for Ag, λ ≈ 0.75 nm. 4.6.2. Models Eq. (4.42) would be a fully adequate phenomenological expression if the spin structure in the AF layer would remain ﬁxed when the F layer magnetization is rotated. A net magne-

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

153

Fig. 4.20. Magnetic structures near an AF/F interface within various models for exchange anisotropy. See the text for references. For clarity non-collinear structures in the AF layer are indicated by rotations of the spin-directions in the plane of the paper. However, in actual systems the AF magnetic moments could also have a component perpendicular to the plane of the paper. From Coehoorn et al. (2002).

tization of the surface layer of the AF ﬁlm (a layer with an uncompensated spin structure) would then give rise to a stable exchange bias ﬁeld. However, the AF layer thickness dependence of the effect, the difference between the blocking temperature and the Néel temperature, and the phenomenon of relaxation indicate that in reality the magnetic structure in the AF layer cannot be considered as completely ﬁxed and independent of the F layer magnetization direction. An additional indication of this is the observation that some AF/F bilayers show a “magnetic training” effect, i.e. a gradual reduction of the exchange bias ﬁeld with the number of hysteresis loop cycles (Schlenker and Paccard, 1967; Schlenker, 1968; Schlenker et al., 1986). Many of the models that have been proposed to understand the exchange anisotropy effect may be classiﬁed as either macroscopic, mesoscopic, or microscopic (Coehoorn et al., 2002). Fig. 4.20 gives schematics of the spin structures assumed in some of the proposed models within each category. Only recently, more integral approaches have been proposed. Within “macroscopic” models the spin-structure in the AF layer is assumed to be independent of the lateral (x, y) coordinate, the F/AF interface is assumed to be ﬂat and the AF interface layer is assumed to have an uncompensated spin structure (i.e. the net magnetic moment per atom is unequal to zero). Structural and magnetic mesoscopic scale or microscopic scale defects are neglected. The oldest example is the Meiklejohn–Bean (MB) model (Meiklejohn and Bean, 1956, 1957). The exchange bias ﬁeld is calculated using the SW model (inﬁnite exchange stiffness in the AF and F layers), assuming a ﬁnite magnetic anisotropy of the AF layer (anisotropy constant KAF ). In the limit of an inﬁnite AF layer thickness, the AF spins are then ﬁxed, leading to eq. (4.43) for Heb , but for ﬁnite AF layer thicknesses, a rotation of the F layer magnetization can induce a rotation of the AF sublattice magnetizations. A straightforward analysis shows that an exchange bias effect is only obtained above a critical AF layer thickness tAF,min =

Jeb . KAF

(4.45)

154

R. COEHOORN

For tAF > tAF,min the AF sublattice magnetizations are identical for pinned layer magnetization directions αp = 0 and π (see ﬁg. 4.20), leading to an exchange bias effect, but for tAF < tAF,min reversal of the pinned layer leads to reversal of the AF magnetic structure, and hence to a hysteretic magnetization loop centered around H = 0 (see, e.g., Tsunoda et al., 2000a). The MB model thus contains the two crucial ingredients for obtaining an exchange bias ﬁeld, Jeb and KAF , and can explain that a minimum AF layer thickness is required. A problem is that the use of well-known nearest-neighbour exchange interaction parameters for a model system such as Ni80 Fe20 /Fe50 Mn50 leads to a value of Jeb which is two orders of magnitude larger than the experimentally observed values (10 mJ/m2 versus 0.1 mJ/m2 ) (Stoecklein et al., 1988). A possible explanation is given within the Mauri model (Mauri et al. 1987a, 1987b, see also Xi and White, 2000a). It is a generalization of the MB model, which treats the reversal process beyond the SW model by taking the ﬁnite exchange stiffness of the F and AF layers into account. This would be important for thick AF layers, for which it is postulated that a “horizontal” domain wall (a wall parallel to the AF/F interface) will be formed upon reversal of the F layer (see ﬁg. 4.20(a), lower part). Jeb is then expected to be much smaller than as expected within the MB model, and to increase with increasing AF layer thickness, until it is much thicker than the AF wall width, δAF . However, for Fe50 Mn50, δAF ≈ 50 nm (Mauri et al., 1987a) whereas experimentally Heb is already almost independent from the AF layer thickness above tAF,min ≈ 6–8 nm (ﬁg. 4.17(a)). This implies that the Mauri model cannot adequately explain the experimental results. The MB and Mauri models can also for other reasons not fully explain the exchange bias effect. First, the surface plane of the AF layer is often not expected to be magnetically uncompensated. The bulk spin structure of Fe50 Mn50, e.g., is such that (111) planes are fully compensated. From the MB and Mauri models, no exchange bias would be expected for [111]-textured SVs whereas in practice this growth orientation gives rise to the highest exchange bias ﬁelds (see, e.g., Jungblut et al., 1994). Second, the assumption that the AF/F interfaces are perfectly ﬂat is in practice quite unrealistic. These issues are addressed in various mesoscopic models, in which the interface roughness and the ∼5–100 nm scale domain structure and/or the grain structure in the AF layer are taken into account. The ﬁrst mesoscopic model was formulated by Néel (1967a, 1967b), who explained the training effect as a result of the occurrence of a domain structure in the AF layer. Malozemoff (1987, 1988a, 1988b) proposed a model for AF/F systems with structurally perfect bulk layers but with random interface roughness. He argued that during the growth or ﬁeld cooling process the AF layers break up in domains with a stable AF magnetization, separated by “vertical” domain walls (see ﬁg. 4.20(b), upper part). In spite of the energy cost for creating the walls, AF domain formation would lead to a net energy gain due to the coupling between the F layer and the locally √ uncompensated AF surface. If there are N spins per AF domain, on the average ∼ N spins will be uncompensated. The domain sizes would be of the order of the AF domain wall width, δAF . For Fe50 Mn50 , δAF ≈ 50 nm and the interatomic distance is approximately ∼0.25 nm. Typically, there are thus of the order 104 surface spins per domain of which on the average typically 1 percent (102) are uncompensated. It can then be understood why, on the one hand, there is no complete averaging out of the interfacial exchange interaction, whereas, on the other hand, Jeb is a few orders of magnitude smaller than the value that is expected for the case of an uncompensated AF surface plane in which all spins are parallel. Takano, Berkowitz and coworkers

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

155

applied similar ideas to the (idealized) case of AF layers consisting of small single domain non-interacting grains (Takano et al., 1997, 1998b). This is shown schematically in ﬁg. 4.20(b), lower part, where the dashed line depicts a grain boundary. A consequence of this model is that the exchange bias energy Jeb is expected to decrease with increasing grain size, because of the random spin argument given above. This is indeed observed (see references in section 4.6.4). Within the Malozemoff model, the structure in the bulk of the AF layer is assumed to be perfect (no grain boundaries, e.g.). Miltenyi et al. (2000) (see also Nowak et al., 2001, 2002) criticized the model by arguing that interface roughness alone is then insufﬁcient to obtain an AF domain structure that is energetically favorable over a single domain state. Instead, they developed a model within which the formation of AF domains is the result of the presence of bulk (point) defects, leading to regions of low domain wall energy at which the walls can be pinned. The model was supported by experimental work on antiferromagnetic CoO based systems with defects of various types (Keller et al., 2002). Recently, various authors have started to address the more general problem of the relationship between the complex domain structure in the AF layer and the bulk and interface microstructure (Fujiwara et al., 2001; Suess et al., 2002, 2003). The proposed models take account of the effects of the grain size distribution, the exchange coupling across the grain boundaries, pinning of walls at grain boundaries or bulk (point) defects, the local magnetocrystalline anisotropy in relation to the crystallite orientation (random or textured ﬁlms), and the interface roughness. It has also been understood better how thermally activated relaxation processes affect the exchange bias ﬁeld (see section 4.6.4). Microscopic scale models, which deal with the detailed spin structures and electronic structures on the atomic scale, have been proposed by Koon (1997), Schulthess and Butler (1998) and Kiwi et al. (1999). One of the issues that is addressed in these studies is how the spins in a uniformly magnetized F layer couple to the spins in an AF surface layer with an (almost) compensated spin-structure (parallel or perpendicular magnetization directions?), and how the resulting interface magnetic structure changes when the F layer magnetization is rotated. Model studies at this level face many difﬁculties. A practical problem is the modelling of the structure of rough or intermixed interfaces, and of interfaces between layers that are not lattice matched. A more fundamental problem is the modelling of the bulk and interface anisotropy of the AF layer. Experimentally, not much is known on this subject. Theoretical work on this subject, for NiMn and NiMn/NiFe interfaces, has only recently started (Nakamura et al., 2000). Thirdly, the temperature dependence of the atomic-scale magnetic interactions in thin ﬁlms may be inﬂuenced by ﬁnite size effects (Parkin and Speriosu, 1990; van der Zaag et al., 2000). More integral mesoscopic/microscopic models, in which a grain structure in the AF layer as well as the detailed spin structure at the AF/F interface are taken account, have been presented by Stiles and McMichael (1999) and Li and Zhang (2001). So far, all models assume reversal of the pinned layer by a coherent rotation process. However, experimental studies using the magneto-optical Kerr effect and magnetization-induced second harmonic generation (NiO/Co; Kirilyuk et al., 2002) and Lorentz TEM (Ir–Mn/Co90Fe10 , Gogol et al., 2002), have shown that the reversal of the pinned layer proceeds in some cases by domain wall movement or is accompanied by the formation of a complex ripple domain structure. A more complete model of the exchange bias effect, which takes a non-uniform F layer magnetization into account, is required in such cases.

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R. COEHOORN

It may be concluded that exchange anisotropy is a complex phenomenon. The mesoscopic and microscopic scale atomic and magnetic structure in the AF layer plays a key role. In view of that, the detailed mechanism of exchange anisotropy is quite speciﬁc for each speciﬁc AF/F system. 4.6.3. Exchange bias materials This sections contains overviews of the properties of the AF exchange bias materials that have been most intensively studied in view of their application in spin-valves. For each material, a brief description is given of the structural and magnetic properties. The text is supplemented by tables 4.3–4.6, which contain references to selected publications on exchange bias interaction in AF/F bilayers and spin-valves containing various AF materials. The type of each study is indicated by letters A–D: A = AF/F bilayers, exchange anisotropy, B = AF/F bilayers, thermal relaxation of the exchange anisotropy and training effect, C = Spin valves, exchange anisotropy and magnetoresistance, D = Spin valves in devices (heads, sensors, etc.), exchange anisotropy and magnetoresistance. In all cases sputter deposition was used, unless indicated otherwise. As the interaction energy Jeb depends, in general, on the composition of the F layer and on the type of stack (top or bottom structure, see ﬁg. 1.4), this information is speciﬁed. In the case experiments have been done as a function of the AF layer thickness, the AF layer thickness tAF,min is given above which Jeb is essentially independent from the AF layer thickness. We note that the highest possible blocking temperature is generally obtained for AF layer thicknesses that are somewhat larger than tAF,min , and that optimizing Jeb do not always lead to an optimized Tb (see section 4.6.4). In the case of a study of Heb in SVs, the values deduced for Jeb are not corrected for a possible decrease of the apparent value of Jeb due to a large interlayer coupling (section 4.1.2). The notation of chemical compositions is as explained in section 2.1. Table 4.7 gives an overview of typical results for the AF materials that are most relevant to SV applications, based on the data given in tables 4.3–4.6. The given values of Jeb on annealed samples (indicated by an asterisk) should often be regarded with some caution, viz. when a possible change of the magnetic moment of the pinned layer after annealing has not been investigated. For the details of the anneal treatments the reader is referred to the original publications. The variation between the results obtained by different groups reﬂects the fact that Jeb , Tb and the coercivity (given as the fraction Hc /Heb ) depend on the layer thicknesses, composition and microstructures. The ratio ρ/tmin in table 4.7 gives the sheet resistance of the AF layer for the minimal AF layer thickness. This value should preferably be high as compared to the sheet resistance of the active part of the SV, to avoid shunting by the AF layer. The table also gives the heat of formation, ΔHform , in view of the relationship with the thermal stability (see section 2.10.2). 4.6.3.1. Fcc-type random substitutional X–Mn alloys. Table 4.3 contains selected references on exchange anisotropy due to fcc-type random substitutional Fe1−x Mnx alloys with x ≈ 0.50, Ir1−x Mnx alloys with x ≈ 0.80 and (Rh–Ru)1−x Mnx alloys with x ≈ 0.80. No

TABLE 4.3 Exchange anisotropy of systems based on metallic random substitutional fcc-type AF alloys. If not indicated otherwise, the Fe–Mn and Ir–Mn data given in the table either refer to compositions very close to Fe50 Mn50 and Ir20 Mn80 or have not been speciﬁed in the original paper. Data for bottom structures are indicated with (b). Otherwise, the data refer to top structures. * Sample annealed after deposition. See section 4.6.3 (introduction) for further explanations Reference

Study

tAF,min (tAF ) (nm)

Fe1−x Mnx (x ≈ 0.50) Hempstead et al. (1978)

A

120

(13)

Pinned layer Ni80 Fe20 Jeb (mJ/m2 ) 0.09 0.06 0.05 0.07 0.16∗ (b) 0.12 0.07∗ 0.16∗ (b) 0.10 0.13 0.07

Tsang et al. (1982) Tsang and Lee (1982) Parkin et al. (1990) Kung et al. (1991)

A

155

(25)

A A

140 155 190 (b)

Dieny et al. (1991a) Rijks et al. (1994a, 1994b) Nakatani et al. (1994)

AC AC C

8 8 8 (10) (8) (5)

Jungblut et al. (1994)

Lin et al. (1995a) Kanai et al. (1996)

A A A A ACD

6 10 5 7 (15)

0.10 0.07 0.06 0.09∗

Nishioka et al. (1996) Lenssen et al. (1997) Fujikata et al. (1998a) van der Heijden et al. (1998a, 1998b) Oshima et al. (1998) Lin and Mauri (1999) Devasahayam and Kryder (1999a) Nakagawa et al. (1999)

A ACD AB AB B AC ACD A

(15) (7) (15) (10)

0.01–0.08∗ 0.17∗ (b) 0.14∗

150

150 140 130 190 165

(15) 7.5 (20)

Hc /Heb

Remarks

Other F materials Jeb (mJ/m2 )

Hc /Heb x = 0.50 x = 0.40 x = 0.60 Temperature dependence

0.4

0.2 < 0.05

Study ﬁnite size scaling Non-monotonic AF layer thickness dependence of Jeb (b) First GMR spin valves

0.1 0.05 0.1

Effect underlayers on [111] texture and Jeb MBE on Cu(111); x = 0.57 MBE on Cu(100); x = 0.47 MBE on Cu(110); x = 0.59

< 0.05 0.4 1.4 < 0.05 0.13 (Co) 0.15 (Co90 Fe10 )

0.2 0.05

0.2–0.5

0.29 (Ni66 Co18 Fe16 ) 0.09∗ (Co) 0.07 0.11

0.1

0.15

Demonstration advantageous use of Co90 Fe10 instead of Co AF grain size dependence Heb

157

Annealing time dependence Heb Thermal relaxation Heb Creep of bias ﬁeld direction Field anneal effects Comparison bias materials Effect a-Si buffer layer (continued on next page)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

Tb (◦ C)

Reference

Study

Tb

tAF,min

(◦ C)

(tAF ) (nm)

Anderson et al. (2000a) Nozières et al. (2000a) Hughes et al. (2001)

AC 175 ABC 170 ABC

(8) (10) (15)

Hoshino et al. (1996)

A

130 120

Nakatani et al. (1997)

A

Fuke et al. (1997)

AC

150 250 260

Iwasaki et al. (1997) Devasahayam and Kryder (1999a, 1999b) Lederman (1999) van Driel et al. (1999b)

A ≈220 ACD 250

(63) (69) (57) (16) (14) (8) (8) (6) 8

Lin and Mauri (1999) Mao et al. (1999b) Mao et al. (1999c)

AC AC A

Fuke et al. (1999) Ro et al. (1999) Yagami et al. (1999, 2000), Takahashi et al. (2000a) Nozières et al. (2000a)

158

TABLE 4.3 (continued) Pinned layer Ni80 Fe20 Jeb (mJ/m2 )

Hc /Heb

Remarks

Other F materials Jeb (mJ/m2 ) 0.11 (Co90 Fe10 ) 0.06 (Ni66 Co18 Fe16 )

Hc /Heb

< 0.05 0.3

Thermal stability, diffusion Blocking temperature distribution Magnetic viscosity effects

Ir1−x Mnx (x ≈ 0.80)

0.19 (Co90 Fe10 ) 0.16∗ (Co90 Fe10 ) 0.15∗ (Co90 Fe10 ) 0.15

0.1 0.11∗ 0.1 0.09, 0.04∗ 0.07, 0.07∗ (b) 0.3

A

7 (10) (10) (12) (5) (4–15) (4–15) (6)/(12)

A AC

280

(20) 5

0.14 0.14

ABC 255

0.1

0.1

230 290 290 270 200 175 175 235/300

A AB

0.2

0.10

0.12∗ (Co90 Fe10 ) 0.09, 0.05∗ (Co90 Fe10 ) 0.09, 0.10∗ (b, Co90 Fe10 ) 0.05 (Co) 0.18 (Co90 Fe10 ) 0.15 (Co90 Fe10 ) 0.29 (b, Co90 Fe10 ) 0.15∗ /0.11∗ (Co90 Fe10 )

0.2 0.7 0.5 0.15

0.18 (Co90 Fe10 )

≈0.05

Annealing time dependence Heb Finite size scaling analysis Tb ; Comparison bias materials Comparison bias materials x = 0.82; Annealing dependence and thermal relaxation Field anneal effects

0.2 0.2

0.20 0.1

Relation grain size – O-content – Tb − Heb x = 0.76 x = 0.75. Dependence on composition and sputter deposition conditions Blocking temperature distribution (continued on next page)

R. COEHOORN

x = 0.80; Ion beam sputtering x = 0.70; Ion beam sputtering x = 0.90; Ion beam sputtering x = 0.78; Ion beam sputtering x = 0.78; RF sputtering

0.06 (b) 0.03 (b) 0.0 (b) 0.07 0.12

TABLE 4.3 (continued) Reference

Study

tAF,min (tAF ) (nm)

Pakala et al. (2000)

A

Anderson et al. (2000b)

AC

Chen et al. (2000) Zeltser et al. (2000) Van Driel et al. (2000b) Guo et al. (2001) Li et al. (2001a, 2001b, 2001c) Yagami et al. (2001)

A AD A AC AC A

Childress et al. (2001), Carey et al. (2001)

ABC

Childress et al. (2002) Tsunoda et al. (2002)

A A

250 250 250 290 295 290 323

235 < 230 290

(5) (5) 6 6 (20) 7.5 4 (6) 5 (7) (7) (8) (8) (8) (6.5) (5)

Pinned layer Ni80 Fe20 Jeb (mJ/m2 )

0.10

Hc /Heb

< 0.05

Remarks

Other F materials Jeb (mJ/m2 ) 0.22∗ (Co90 Fe10 ) 0.30∗ (b, Co90 Fe10 ) 0.16∗ (Co90 Fe10 ) 0.28∗ (b, Co90 Fe10 ) 0.17∗ (Co90 Fe10 ) 0.18∗ (b, Co90 Fe10 ) 0.23∗ (b, Co90 Fe10 ) 0.14∗ (Co75 Fe25 ) 0.30∗ (b, Co82 Fe18 ) 0.30 (b, Co90 Fe10 ) 0.39∗ (b, Co90 Fe10 ) 0.11∗ (Co90 Fe10 ) 0.06∗ (u, Co90 Fe10 ) 0.17∗ (u, Co60 Fe40 ) 0.34∗ (Co85 Fe15 ) 0.50∗ (b, Co70 Fe30 ) 0.18∗ (b, Co90 Fe10 ) 0.45∗ (b, Co50 Fe50 )

Hc /Heb

0.15 0.15 0.1

0.1

Dependence on underlayer and microstructure Top, bottom and dual SVs

Blocking temperature distribution x = 0.81; dependence on microstructure Effect Ga+ ion irradiation x = 0.76 x = 0.74. Ultraclean sputter deposition x = 0.75. Large-grain NiFeCr underlayer (u): lowers Jeb but enhances the MR ratio of top SVs Very high Jeb , low Tb Ta/Py/Cu underlayer. Ultraclean sputter deposition

(Ru1−y Rhy )1−x Mnx (x ≈ 0.80, 0 < y < 1) Veloso et al. (1998) Araki et al. (1998a), Araki et al. (1998b)

ACD ABC

Shimazawa et al. (1999)

AC

235

250

(17) 15 10 10 10 (8)

0.19 (Co90 Fe10 ) 0.13 0.11 0.15 0.12

0.12∗ (Co)

0.15

Rh0.22 Mn0.78 Rh0.14 Mn0.86 Ru0.23 Mn0.77 Ru0.04 Rh0.14 Mn0.82 Ru0.12 Rh0.08 Mn0.80 Ru0.03 Rh0.15 Mn0.82

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

Tb (◦ C)

159

160 TABLE 4.4 Exchange anisotropy of systems based on bct-type random substitutional (Cr50 Mn50 )1−x Ptx AF alloys. Only top structures were studied. * Sample annealed after deposition. See section 4.6.3 (introduction) for further explanations Reference

A ABCD

Tb ◦ ( C)

Pinned layer

tAF,min (tAF )

Ni80 Fe20

(nm)

Jeb (mJ/m2 )

Hc /Heb

(380)a 120 320

30 (50) 25

0.09 0.03 0.08 0.09∗

0.5

Hoshiya et al. (1997), Hamakawa et al. (1999) Soeya et al. (1997b)

A

300

20

Nishioka et al. (1998) Nishioka (1999)

ABC AB

320 260

30 (30)

Mao and Gao (2000b) Lu et al. (2000) Xi and White (2000b, 2000c)

AC AC A

320

25 (30) 25

a Obtained by linear extrapolation from H (T ) data up to 240 ◦ C. eb

0.16∗ 0.09∗ 0.04

0.4 0.1

Remarks

Other F materials Jeb (mJ/m2 )

Hc /Heb

0.07 (Co) 0.15∗ (Co) 0.34∗ (Co90 Fe10 )

0.3 0.1 0.2

0.16∗ (Co) 0.13∗ (Co)

0.1

0.22∗ (Co90 Fe10 ) 0.15 0.3

x

0.05 0 0.10 0.08 0.04

0.10 0.08 0.09

Anneal 1 h, 230 ◦ C. SV heads. Reliability NiO cap layer; anneal 3 h, 230 ◦ C Tb distribution Magnetic relaxation Cr36 Mn54 Pt10 Anneal 2 h, 230 ◦ C Anneal 1 h, 230 ◦ C

R. COEHOORN

(Cr50 Mn50 )1−x Ptx (x ≈ 0.07) Soeya et al. (1996b, 1997a)

Study

TABLE 4.5 Exchange anisotropy for systems based on CuAu-I (fct) type ordered metallic AF compounds. If not indicated otherwise, the Mn atomic concentration is x = 0.50, or has not been speciﬁed in the original paper. The composition of the (Pt–Pd)Mn systems is given by Pd1−x−y Pty Mnx . Data for bottom structures are indicated with (b). Otherwise, the data refer to top structures. All samples were annealed after deposition, to induce a phase transformation to the ordered fct phase. ** : Anneal treatment not (precisely) speciﬁed. See section 4.6.3 (introduction) for further explanations Study

Tb

tAF,min

(◦ C)

(tAF ) (nm)

Pinned layer Ni80 Fe20

Anneal

Other F materials Jeb (mJ/m2 )

Jeb (mJ/m2 )

Hc /Heb

Hc /Heb

0.4 0.5 0.5 0.7 1.4 0.8 0.5

40

0.27 0.30 0.30 0.29 0.32 (b) 0.21 (b, UL) 0.30 (b) 0.06 (b) 0.18 0.18 0.48 (b) 0.16

0.8 0.5 0.5

**

25

0.13

0.8

1 h, 250 ◦ C

NiMn Lin et al. (1994a, 1995a)

AD

Devasahayam and Kryder (1996)

A

240

(50) (50)

Mao et al. (1996,1998) Portier et al. (1997c) Fujikata et al. (1998a) Qian et al. (1999a, 1999b)

AC AC AB A

380

25 (25) (30) 18

Lederman (1999)

A

Loch et al. (1999)

AC

Anderson et al. (1999a)

AC

400

40

Yang et al. (2000a, 2001)

AC

300

(50)

0.24

0.1

10 h, 260 ◦ C

Nozières et al., (2000a, 2000b) Anderson et al. (2000a) Zhang et al. (2000) Han et al. (2000) Zhang et al. (2001) Rhee et al. (2001)

ABC AC ABCD AB AB AC

400 400 (425) 410

(25) (30) (25) (20) (25) (25)

0.30

0.8

**

0.28 0.17 0.18 0.29

0.8 0.7 1.0 0.5

> 450

360 430 > 375

380

25

30 h, 240 ◦ C 20 h, 280 ◦ C 6 h, 320 ◦ C 5× (5 h, 300 ◦ C)

x = 0.47; 6% Cr improves corrosion stability, lowers Heb x = 0.45–0.48. UL = Py underlayer

45 h, 280 ◦ C

x = 0.58. HR-TEM study Anneal time dependence Heb

**

15 h, 270 ◦ C 20 h, 280 ◦ C

0.38 (Co90 Fe10 )

Remarks

treatment

270 ◦ C

10 h, 250–270 ◦ C 7.5 h, 275 ◦ C **

0.5 h, 350 ◦ C 21 h, 220 ◦ C

AF/F interdiffusion above 300 ◦ C Optimization anneal treatment of SVs Anneal time not given

161

x = 0.55; anneal process dependence Tb distribution SV thermal stability Head thermal stability Thermal relaxation Rotational hysteresis x = 0.75 (continued on next page)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

Reference

162

TABLE 4.5 (continued) Reference

Study

Tb

(◦ C)

(tAF ) (nm)

AC A

380 230

12–16 (30)

A ABC AB ABC

310 380 400

350

Ni80 Fe20 Jeb Hc /Heb (mJ/m2 )

0.10 (b)

0.4

(30) (35) (20) (30)

0.08

0.2

(25) (20)

0.11 (b)

Mao et al. (2000a) Pokhil et al. (2001) Kim et al. (2001a, 2001b, 2002b)

AB A AC

Prakash et al. (2001) Sugita et al. (2001)

ABCD A

Lee et al. (2002b) (Pd–Pt)Mn Kishi et al. (1996)

AC AC

300

(25)

0.11

Tanaka et al. (1997, 1999)

ACD

280

20

0.16

Fujikata et al. (1998a) Lederman (1999) Anderson et al. (1999b) Shimizu et al. (1999) Hung et al. (2000b) Nagasaka et al. (2000) Yang et al. (2000b)

AB A AC AC AB AB AB

310 350 350

(30) 25 40

0.13 0.09

400 > 400 340 400

350 270

1.0

(30) (30) (15) (15) 13

35 (25) (25)

Anneal

Other F materials Jeb (mJ/m2 )

Hc /Heb

0.32 (b, Co)

0.6

0.18 (Co) 0.21 (Co90 Fe10 ) 0.29 (Co90 Fe10 ) 0.31 (b, Co90 Fe10 ) 0.35 (Co90 Fe10 )

0.7 1.2

0.17 (Co90 Fe10 ) 0.24 (Co90 Fe10 )

0.8

0.6

0.4 0.8

10 h, 250 ◦ C

0.28 (Co90 Fe10 ) 0.17 (b, Co90 Fe10 )

0.5 0.7

Sweep rate dependence Hc Micromagnetic study Annealing effects on MR ratio of SVs

5 h, 275 ◦ C 1.5–5 h, 280 ◦ C

SV heads, thermal stability Ta underlayer Ta/Ni–Fe–Cr underlayer x = 0.51. x-dependence

0.22 (Co90 Fe10 ) 0.19 (Co87 Fe9 B2 ) 0.37 (Co90 Fe10 )

0.6

3 h, 280 ◦ C ** **

0.5

Single and dual SVs x = 0.51, MBE on MgO x-dependence MBE on MgO Tb distribution Thermal relaxation SV thermal stability

4 h, 270 ◦ C 8 h, 260 ◦ C 10 h, 270 ◦ C

1 h, 230 ◦ C

0.2

0.6

** **

2.5 h, 280 ◦ C

0.24 (b, Co90 Fe10 )

0.17 (b) 0.11

4 h, 250 ◦ C growth at 200 ◦ C growth at 100 ◦ C

0.22 (Co90 Fe10 )

Remarks

treatment

10 h, 250 ◦ C 3 h, 280 ◦ C **

3 h, 280 ◦ C **

x = 0.50, y = 0.20. y-dependence x = 0.52, y = 0.18. UHV sputter deposition x = 0.51, y = 0.32 x = 0.50, y = 0.20 x = 0.52, y = 0.18 Tb distrib., training effect x = 0.52, y = 0.18 Magnetic viscosity

R. COEHOORN

PtMn Saito et al. (1996, 1999) Farrow et al. (1997), Krishnan et al. (1998) Shimoyama et al. (1999) Nozières et al. (2000a, 2000b) Han et al. (2000) Anderson et al. (2000a, 2000c)

Pinned layer

tAF,min

TABLE 4.6 Exchange anisotropy due to oxidic antiferromagnetic compounds. Data for bottom structures are indicated with (b). Otherwise, the data refer to top structures. * : Annealed sample Reference

Study

Tb

tAF,min

(◦ C)

(tAF )

NiO Carey and Berkowitz (1992) Soeya et al. (1993) Soeya et al. (1994) Lin et al. (1995a) Lai et al. (1995, 1996, 1997a, 1997b)

A A B A AC

190 200 230 200 200

(50) (100) (50) 30 (60)

Lin et al. (1995b) Hamakawa et al. (1996)

A ACD

180 180 230

(45) 30 (50)

Soeya et al. (1996a) Shen and Kief (1996) Michel et al. (1996) Everitt et al. (1996) Kitakami et al. (1996)

A A A AC AC

225

(100) (30) (50) (50) 25

Han et al. (1997a) Han et al. (1997b) Restorf et al. (1997) Van der Heijden et al. (1998a, 1998b)

A AC ABC AB

Lee et al. (1998) Kools et al. (1998b) Fujikata et al. (1998a) Fujikata et al. (1998b)

A AC AB ABC

190

190

(45) (45) (74) (60)

180 230

65 50 (50) (50)

Jeb (mJ/m2 )

Hc /Heb

0.05 (b) 0.06 (b) 0.06 (b) 0.06 (b) 0.03 (b) 0.02 (b) 0.06 (b) 0.05 (b) 0.05 (b) 0.03 0.05 (b) 0.06 (b) 0.10 (b) 0.05 (b) 0.04 0.05 0.06 (b) 0.03 (b) 0.03 (b) 0.05 (b)

1.0 0.3 0.6 0.6 6 6 0.8 0.6 1 0.2 0.6 0.6 0.6 0.2

Jeb (mJ/m2 )

Tb distribution NiO(111), MOCVD NiO(100), MOCVD NiO, sputter deposited Ni0.82 Fe0.18 O (MOCVD) Demonstration SV read head

0.04 (b) (Co)

0.2 0.4 0.1 1

0.4 0.12 (b) (Co90 Fe10 )

0.07 (b) 0.12∗ (b)

0.5

Hc /Heb Study Ni1−x Cox O/Py (0 x 1)

0.10 (b), (Ni66 Co18 Fe16 ) 0.04 (b)

Remarks

Other F materials

>1

Sputter conditions, structure Film texture, roughness Ion Beam Sputtering Reactive sputtering

Stress effects on Heb and Hc Small Hc by small roughness Magnetic viscosity effects Thermal relaxation

163

Ion beam sputtering SVs: high MR and low ΔHsw Thermal relaxation 2 nm Fe–O interface layer. Anneal 2 h, 250 ◦ C (continued on next page)

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

(nm)

Pinned layer Ni80 Fe20

Reference

Study

Tb

tAF,min

(◦ C)

(tAF ) (nm)

Cowache et al. (1998) Hwang. et al. (1998) Devasahayam and Kryder (1999b) Lin and Mauri (1999) Chopra et al. (2000b) Pinarbasi et al. (2000)

A A A ABC A ACD

190 173

(30) (35) 30 (40) (50) (43)

(40)

(40)

Farrow et al. (2000)

A

Fraune et al. (2000) Carey et al. (2001) Rezende et al. (2001)

A ABC A

200

A C A AC

100 90 150a 80

Pinned layer Ni80 Fe20

Remarks

Other F materials

Jeb (mJ/m2 )

Hc /Heb

0.05 (b) 0.06 (b) 0.08 (b)

0.7 0.4 0.5

0.04 (b) 0.07 (b) 0.11 (b)

0.9 0.9 0.9

Jeb (mJ/m2 )

Hc /Heb Fp = Co: no Heb , only Hc Effects of structure and texture Relationship Tb (tAF ) Thermal relaxation

0.002 (b) (Co)

0.04 (b) (Ni) 0.08 (b)

2

(50) (50) 20 (10)

0.09 (b) 0.11 (b) 0.09 (b) 0.06∗ (t)

0.4 0.2 0.3

(50) (100) (50) (50) (20)

0.02 (b) 0.03 0.06 0.01

13 0.7 1.5 23

0.3

Ni0.488 O0.512 Ni0.484 O0.516 Ni0.457 O0.543 Enhancement of Tb by a ≈ 3 nm Ni0.8 Fe2.2 O4 interface layer between NiO and Py Study NiO/Ni nanostructures Thermal relaxation Single crystal NiO(100)

Co0.5 Ni0.5 O Carey and Berkowitz (1992) Fujikata et al. (1995) Devasahayam and Kryder (1999b) Ambrose et al. (1999)

(CoO/NiO) superlattice Relationship Tb (tAF ) RF sputtering.

α-Fe2 O3 Hasegawa et al. (1996) Sano et al. (1998) Kawawake et al. (1999) Bae et al. (2000a, 2000b) Kawawake et al. (2000)

AC AC AC AC AC

250

250

0.07 (b, Co) 0.03 (b, Co)

Fe2 O2.92 tAF dependence; anneal effect 7 Sy-AF SVs

a This value of the T , far above the bulk value of T for Co Ni O, might indicate that a different ﬁlm composition has been obtained. N b 0.5 0.5

R. COEHOORN

210 230 232

164

TABLE 4.6 (continued)

AF material

TN

Tb

Typical anneal

(◦ C)

(◦ C)

treatment (hours/◦ C)

Pinned layer Ni80 Fe20 Jeb (mJ/m2 )

Random substitutional metallic fcc-type alloys (table 4.3) Fe50 Mn50 230 140–190 – 0.07–0.13 (t) 0.17 (b) 420 240–290 – 0.08–0.14 (t) Ir20 Mn80 0.07 (b)

tmin

(nm) (10−8 m)

Co90 Fe10

Hc /Heb

Jeb (mJ/m2 )

Hc /Heb

0.1 0.1 0.1 0.1

0.10–0.15 (t)

0.1

(360)m 240 – 0.10–0.15 (t) (Rh-Ru)20 Mn80 Random substitutional metallic bct-type alloys: Cr–Mn–Pt (table 4.4) Cr0.46 Mn0.46 Pt0.08 300–320 2/230 0.08∗ –0.16∗ (t) 0.1 Ordered metallic compounds (table 4.5) NiMn 800 360–400 10–40/280 0.18–0.30 (t) 0.5–0.8 0.20–0.50 (b) 0.5–0.8 PtMn 702 350–400 5–20/260 1.0 0.11 (b) (605)j 300–350 3/280 or 0.09–0.16 (t) 0.5 Pd0.6 Pt0.4 Mn 10/250 Oxides (table 4.6) NiO 252 190 – 0.04 (t) 0.2 0.04–0.08 (b) 0.5–0.9 0.12i (b) 0.5i 230i

0.10∗ –0.20∗ (t) 0.1–0.2 0.15∗ –0.35∗ (b) 0.1–0.2 0.5∗ (b, Co70 Fe30 ) 0.10–0.20 (t) 0.15 0.15∗ –0.34∗ (t) 0.38 (t) 0.18–0.30 (t) 0.32 (b) 0.20–0.40 (t) 0.17 (b)

ρ

0.1

ρ/tmin Corrosion ΔHform g ( )

resistance (kJ/(mole of atoms))

8

130a

160

–/–bc

0

6

325,b 200k

540

–bc

−16

10

200b

200

–b

−12

25

320,e 360b

140

0/–b

≈ −9

25

175a , 210b

0.4–0.8 14 156f 0.6 0.5 20–40 185,b 160d

30

Insulator

70–85 0/+,a –,b 0,c +h 60 +l

−12 −42

75

+b

−39

∞

++

−120

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

TABLE 4.7 Properties of exchange bias materials that are suitable for SV applications at and above room temperature. Data on random alloy samples that have been annealed after deposition have been indicated by an asterisk (*). NiMn, PtMn and (Pt–Pd)Mn based systems are always annealed. The speciﬁcations “t” (top SV) and “b” (bottom SV) do only apply to the given values for Jeb and Hc /Heb . See section 4.6.3 (introduction) for further explanations

0.12 (b) (continued on next page) 165

166

TABLE 4.7 (continued) AF material

α-Fe2 O3

TN

Tb

Typical anneal

(◦ C)

(◦ C)

treatment

675

250

Pinned layer Ni80 Fe20

Co90 Fe10

(hours/◦ C)

Jeb (mJ/m2 )

Hc /Heb

–

0.00–0.06 (b)

>1

Jeb (mJ/m2 )

tmin

ρ

ρ/tmin

Corrosion

ΔHform g

(nm)

(10−8 m)

( )

resistance

(kJ/(mole of atoms))

Hc /Heb >50

Insulator

∞

++

−165

ture effects and disregards the effects related to magnetic ordering. However, predicted trends are in good agreement with available experimental results. For oxides: experimental values. h Lin et al. (1994a). Sample with Cr addition. i With Fe-containing oxide layer at the interface between NiO and Py. j Linear interpolation between T for PdMn and PtMn. N k Lenssen et al. (2000a, 2000b). l The corrosion resistance is expected to be equal to that of Pd Pt Mn. 0.6 0.4 m Estimated from data in Sasao et al. (1999).

R. COEHOORN

a Lin et al. (1994a). b Lederman (1999). c Devasahayam et al. (1998). d Anderson et al. (1999b). e Soeya et al. (1996b). f Lin et al. (2000). g De Boer et al. (1988). For metallic AF materials: calculated of the free energy of formation using the Miedema model. The model disregards speciﬁc crystal struc-

GIANT MAGNETORESISTANCE AND MAGNETIC INTERACTIONS

167

lattice ordering is required to obtain an exchange bias ﬁeld. For top-SVs room temperature deposition in a ﬁeld is sufﬁcient. For bottom SVs, a ﬁeld cooling procedure is generally required, although, surprisingly, Ir–Mn based bottom SVs can already show an exchange bias ﬁeld after deposition at room temperature (Van Driel et al., 1999b). Fe–Mn was for a long time the most intensively studied exchange bias material. Although the required fcc-type γ phase is thermodynamically unstable at room temperature, ﬁlms with a thickness of a few tens of nanometers may be obtained by epitaxial growth on fcc conforming substrates such as Cu or permalloy, to which its lattice parameter is matched within 0.4% and 2.5%, respectively. Variations in the alloy stoichiometry, from 45 to 55 atomic percent Mn, do not strongly affect the exchange bias ﬁeld. From the point of view of robustness of the deposition process, this is an advantageous property. Growth on a permalloy underlayer leads to a [111] texture, for which Jeb is highest. If, in a bottom SV, the active part of the SV is grown on top of the Fe–Mn layer, the good lattice matching leads to a strong [111] ﬁbre texture of the free magnetic layer that is required from the point of view of good magnetic softness and a low scattering probability in the free layer. The relatively low blocking temperature, typically 140 to 190 ◦ C, would be adequate for many room temperature applications. However, it is too low for applications in hard disk recording and in high-temperature sensors for, e.g., automotive applications. Other disadvantages are the low stability with respect to Mn diffusion (see section 2.10.2) and the very poor corrosion resistance. Exchange biasing by Ir–Mn, generally in the 75 to 82 atomic percent Mn concentration range, was ﬁrst reported by Hoshino et al. (1996). As in the case of Fe–Mn, the exchange bias ﬁeld increases with increasing [111] texture (see, e.g., Van Driel et al., 2000b). For bottom SVs containing Co90 Fe10 pinned layers that received a short post-deposition anneal treatment values of Jeb up to 0.39 mJ/m2 have been reported. Li et al. (2001a, 2001b, 2001c) found that the anneal treatment increased the [111] texture. Tsunoda et al. (2002) observed that for annealed Ir25Mn75 /Co1−x Fex bilayers Jeb is maximal for x ≈ 0.3, and equal to approximately 0.5 mJ/m2 . This value exceeds the interaction constants for all other systems studied so far, with the exception of one similar result reported for NiMn/Py (see table 4.5). For this record system, Tb ≈ 290 ◦ C for tIr–Mn > 10 nm. As compared to Fe–Mn, biasing by Ir–Mn is more advantageous for a number of reasons: (i) much higher Tb (for Co90 Fe10 typically ∼270 ◦ C, but up to ∼290 ◦ C for 30 nm layers), (ii) larger Jeb , (iii) lower tAF,min (∼6 nm) and higher resistivity, leading to less (almost negligible) shunting by the metallic AF layer, (iv) better corrosion resistance, and (v) better stability with respect to Mn diffusion. Ir–Mn is an exchange biasing material that is excellently suitable for many SV applications. (Ru1−y Rhy )1−x Mnx AF layers, containing x = 77 to 86 atomic percent Mn and with 0 < y < 1, are another alternative to Fe–Mn. Blocking temperatures up to 250 ◦ C have been reported, intermediate between the highest values found for Fe–Mn and Ir–Mn based systems. Like Ir–Mn, the corrosion resistance is also better than for Fe–Mn. As Ir–Mn alloys perform at least as well on all properties, Ru–Rh–Mn exchange bias materials have received much less attention. Other random substitutional fcc-type alloys that were investigated as possible alternatives for Fe–Mn are Fe–Mn–Rh (Tb ≈ 150 ◦ C, Tong et al. (1997)), Pt10 Mn90 (Tb ≈ 160 ◦ C, Xi et al. (2000), Xi and White (2000a)), Ni25 Mn75 (Tb ≈ 150–175 ◦ C, Tsunoda et al. (1997, 1999)) and Os23 Mn77 (Tb ≈ 175 ◦ C, Parkin and Samant (1999)). The blocking tempera-

168

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tures of these systems are similar or only slightly larger than for Fe–Mn. This may explain why so far the research effort on these materials has remained relatively small. Os–Mn exchange biased magnetic tunnel junctions (MTJs) were found to be stable to temperatures exceeding 400 ◦ C (Parkin and Samant, 1999), signiﬁcantly higher than for, e.g., Ir–Mn exchange biased MTJs. Results for SVs have not been published. We note that the estimated enthalpy of formation of Os23 Mn77 alloys is not particularly large (ΔHform ≈ −9 kJ/(mole of atoms), compare table 4.7. The analysis given in section 2.10.2 seems therefore not to be applicable to the thermal stability of Os–Mn based systems. 4.6.3.2. Bct-type random substitutional Cr–Mn–X alloys. Cr50 Mn50 is a metastable random substitutional α-phase (bcc-type) AF alloy with a relatively high Néel temperature, TN ≈ 520 ◦ C (Adachi, 1986). At low temperatures, it segregates into an α-phase and a complex σ -phase. The exchange bias ﬁeld and blocking temperatures of Cr–Mn/F bilayers are very small. Soeya et al. (1996a, 1996b) discovered that Pt addition, with an optimum for 5–8 atomic percent Pt, leads to a drastic increase of Jeb and Teb , to Tb > 300 ◦ C (see table 4.4). Growth on an underlayer (such as Py) that stabilized the antiferromagnetic αphase was found to be required to obtain these results. Hoshiya et al. (1997) observed that a post-deposition anneal treatment (typically 1 hour at 230 ◦ C) led to even higher exchange bias ﬁelds for Cr–Mn–Pt on Co, but not on Py. The effect is attributed to an increase of a bct distortion of the α-phase, which to a lesser extent is already present in unannealed samples (Hamakawa et al., 1999). Such a distortion is assumed to give rise to a much larger AF magnetic anisotropy than that of the undistorted cubic phase, leading to an increase of Jeb (Soeya, 2002). KAF can be enhanced further by making use of the stress that results from the use of a NiO capping layer (Soeya et al., 1997b). As compared to Co/Ir–Mn, the blocking temperature of Co/Cr–Mn–Pt and the stability with respect to thermal relaxation are better (Hamakawa et al., 1999). For sensor applications only top structures are of interest, because in a bottom structure the bcc/bct AF layer does not give rise to the required [111] texture of the free layer. Top SVs, with an MR ratio of 14% were made by Mao and Gao (2000a, 2000b). For applications in heads for ultrashort bitlength hard disk recording, a disadvantage is the large required minimum layer thickness (∼25 nm, much larger than for Ir–Mn). The larger value of tAF,min leads in SV applications to more shunting than Ir–Mn (see table 4.7, column ρ/tAF,min). Like Pt, also Pd, Cu, Rh and Ir additions to Cr50 Mn50 lead to enhanced values of Jeb and Tb (Soeya et al., 1997a). Tb ≈ 380 ◦ C was obtained for ∼5% Pd or ∼10% Rh. However, Nozières et al. (2000a) obtained from a thermal relaxation study for Cr–Mn–Pd a much lower blocking temperature, Tb ≈ 275 ◦ C. This value is slightly lower than for Ir–Mn (see also ﬁg. 4.22). No other work was reported on these materials. 4.6.3.3. CuAu-I (fct) type ordered metallic compounds. NiMn, PtMn and PdMn are ordered AF intermetallic compounds with the tetragonal CuAu-I structure. The atoms reside on the sites of a slightly tetragonally distorted fcc lattice (with a c/a ratio close to 1), with alternating (001) planes containing the Mn and Ni (or Pt or Pd) atoms. Their high Néel temperatures, ranging from 540 ◦ C for PdMn to 800 ◦ C for NiMn, make these compounds excellent candidates for applications as an exchange bias material. As-deposited Ni–Mn,

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Fig. 4.21. Energy per grain as a function of the angle αAF between the magnetization of the AF sublattice that couples ferromagnetically to the F layer, for F layer magnetization directions αF = 0 and αF = π , within the Fulcomer–Charap model. The black dots symbolize an ensemble of AF grains, which are in a stable state when αF = 0, but become metastable when αF = π , leading to thermally activated relaxation towards the αF = π state (arrow in (b)).

Fig. 4.22. (a) Exchange bias interaction parameter for a (40 nm Py/ 20 nm Fe50 Mn50 ) bilayer upon heating to variable temperatures Ta (spheres) and after rapid cooling in a reverse ﬁeld to T0 = 77 K. From Speriosu et al. (1990). (b), (c) Areal fraction of unblocked grains and distribution of blocking temperatures for Fe50 Mn50 , Cr–Pd–Mn, Ir–Mn, PtMn and NiMn based AF/F bilayers, determined using the method depicted in (a). From Nozières et al. (2000a).

Pt–Mn and Pd–Mn ﬁlms are random substitutional fcc alloys that give only rise to a very small exchange bias ﬁeld. In a pioneering study, Lin et al. (1994a) succeeded in obtaining large exchange bias ﬁelds for Ni–Mn based systems by annealing as-deposited Ni50 Mn50 /F bilayers in order to induce a phase transformation to the ordered NiMn compound. The same method was used successfully in subsequent studies on PtMn and (Pt–Pd)Mn sys-

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tems, as reviewed below. For SVs, the optimal anneal conditions follow from a somewhat delicate balance between the requirement to maximize the volume fraction of transformed material, and the requirement not to deteriorate the spin-dependence of scattering due to, e.g., interdiffused interfaces or due to Mn diffusion into the SV. The thermal stability of SVs was discussed in section 2.10, and is illustrated by ﬁgs 2.20 and 2.21 for the case of PtMn and NiMn AF layers. In spite of the disadvantage of the unavoidable time-consuming anneal treatment (1 to more than 10 hours), these bias materials are of great interest in view of their high blocking temperatures, up to more than 400 ◦ C. Table 4.5 gives an overview of the results reported in selected publications. For NiMn, 10–40 hours of annealing at 280 ◦ C is typically required. The pinning of permalloy is strong, with Jeb typically 0.3 mJ/m2 . For Co90 Fe10 , Jeb can even be somewhat larger (Anderson et al. (1999a), see the table), but for Co the largest value reported so far is only Jeb ≈ 0.04 mJ/m2 (e-beam evaporation, Wang et al., 2001). Unlike Fe–Mn and Ir–Mn, Heb is only weakly temperature dependent near room temperature, and in some cases even increases slightly with increasing temperature (see ﬁg. 4.18). The high blocking temperature, 400 ◦ C or higher from many studies, leads to an excellent stability with respect to thermally activated magnetic relaxation (Nozières et al. (2000a, 2000b); see also ﬁg. 4.22). NiMn based SV hard disk read heads were demonstrated by Zhang et al. (2000). From a HR-TEM study, Wong et al. (1996) observed that Ni50 Mn50 grows epitaxially on permalloy (111), leading to a semi-coherent interface. Due to the stabilizing effect of the fcc permalloy, the phase transition is slowed down in a zone near the interface, which has a negative effect on the exchange bias ﬁeld. Near twin boundaries, the phase transition was observed to be much more complete. A good [111] texture leads to the smallest Hc /Heb ratio (Devasahayam et al., 1996), which is nevertheless quite high. This is indicative of the presence of a large fraction of relatively unstable AF grains of which the magnetization rotates when the F layer rotates. Exchange biasing by sputter deposited PtMn was ﬁrst demonstrated by Saito et al. (1996, 1999). For many SV applications, PtMn is more interesting than NiMn because of the shorter anneal treatment, typically 5–20 hours at 260 ◦ C. The values of Jeb for PtMn are very similar to those for NiMn, and the thermal stability is almost as good (see, e.g., ﬁg. 4.18). The minimum required AF layer thickness falls in the range 12–16 nm. In many studies a larger thickness is used in view of the resulting higher blocking temperature. However, recently enhanced thermal stability has been obtained for thinner PtMn layers (10 nm or even less) in bottom SVs, grown on underlayers (such as Py–Cr) which give rise to a large grain size. The relationship between the grain size and the thermal stability is discussed in section 4.6.4. Several examples of such studies can be found in table 2.1. E.g., the highest MR ratio reported so far for simple SVs, 20.5%, was obtained for double specular SVs containing a PtMn exchange bias with a thickness of only 8 nm (Tsunekawa et al., 2002). Several authors studied the compositional dependence of Heb and the phase transformations in Pt–Mn/F bilayers. For MBE deposited Pt1−x Mnx /Py bilayers Krishnan et al. (1998) found after annealing a sharp peak in Heb as a function of the Mn concentration at x = 0.51. For sputter deposited Pt1−x Mnx /Co90 Fe10 based SVs Lee et al. (2002a, 2002b) observed that a non-zero exchange bias ﬁeld was only obtained for 0.46 < x < 0.55, with a broad maximum around x = 0.51. On the other hand, Saito et al. (2001) reported for sputter

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deposited Pt1−x Mnx /Co90 Fe10 bilayers an optimum for a slightly Pt-rich alloy composition (x = 0.48). The same authors observed that an enhanced exchange bias ﬁeld can be obtained by inserting a thin (1–2 nm) Pt58 Mn42 layer at the interface between the Co90 Fe10 pinned layer and a ∼11 nm Pt50 Mn50 second AF layer. The Pt-rich interlayer was assumed to break the epitaxial relationship between the Co–Fe layer and the Pt–Mn layer, which would otherwise slow down the fcc-fct phase transformation (as observed for NiMn by Wong et al. (1996), see above). Maesaka et al. (2000) studied the mechanisms that limit the thermal stability of PtMn based systems (structural reconstructions, Mn diffusion), and observed that these depend on the PtMn grain size. Sato et al. (2002) performed a quantitative study of the relationship between the degree of ordering, tAF,min and Jeb . They concluded that in the limit of 100% ordering Jeb would be 0.26 mJ/m2 (for their samples) and tAF,min would be approximately 7 nm. Kishi et al. (1996) studied (Pd1−y Pty )0.5 Mn0.5 alloys, and discovered that after 1 hour of annealing at 230 ◦ C the largest exchange bias ﬁeld is obtained for compositions with y ≈ 0.4. The decrease of Heb towards more Pd-rich alloys may be viewed as being related to the decrease of the Néel temperature with decreasing Pt content (TN = 540 ◦ C and 702 ◦ C for PdMn and PtMn, respectively). The decrease of Heb towards more Pt-rich alloys may be viewed as being related to a shift of the optimal anneal time and temperature to higher values. For SV applications the anneal conditions are more favorable than for NiMn and PtMn. On the other hand, for some applications the lower blocking temperature (typically 300–350 ◦ C) is a disadvantage. An interesting option that is offered by these ordered AF materials is the possibility to obtain strong lateral variations of the exchange bias ﬁeld on a submicrometer scale by locally disordering the AF layer by ion irradiation through a resist mask (Mougin et al., 2001 and references therein, Juraszek et al., 2002). 4.6.3.4. Antiferromagnetic oxides The ﬁrst AF/F system for which an exchange bias effect was found was CoO/Co (Meiklejohn and Bean, 1956, 1957). At cryogenic temperature, large exchange bias ﬁelds can be obtained. Due to its low Néel temperature (TN = 18 ◦ C), CoO is not a suitable bias material for applications around and above room temperature. For NiO, TN is much higher, viz. 252 ◦ C. Carey and Berkowitz (1992) were the ﬁrst to explore its potential as an exchange bias material. They studied the exchange bias ﬁeld of (50 nm Co1−x Nix O/ Py) bilayers, with reactively sputtered polycrystalline (Co–Ni)O layers, and observed a linear increase of the blocking temperature with x, from Tb = 33 ◦ C for CoO (higher than the Néel temperature!) to 190 ◦ C for NiO. The highest exchange bias ﬁeld was obtained for x = 0.41. The interaction energy at room temperature is Jeb ≈ 0.08 mJ/m2 , which is similar to the values that are obtained typically for Fe–Mn/Py bilayers. This optimum is viewed as resulting from a compromise between the large KAF of CoO and the high TN of NiO. In view of the importance for device applications of a high blocking temperature, most subsequent work has been focused on NiO, and only few further studies have been done on the mixed oxides (see table 4.6). The work on NiO based SVs has led to the important discovery that specular reﬂection at an oxide can lead to a very high MR ratio (section 2.1.6). Other advantages of oxides are the absence of shunting by the AF layer and the excellent corrosion resistance. For some

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applications, the high coercivity and the large minimal layer thickness (∼30 nm) are disadvantages. Fujikata et al. (1998b) studied (NiO/ t nm Fe2 O3 /Py) systems, and discovered that inserting a 2 nm Fe2 O3 layer and annealing for 2 hours at 250 ◦ C leads to an increase of the blocking temperature from typically 190 ◦ C to 230–250 ◦ C, and of Jeb from typically 0.07 mJ/m2 to 0.12 mJ/m2 . Farrow et al. (2000) studied the same system, conﬁrmed the enhanced blocking temperature, and observed the presence of a thin Ni0.8 Fe2.2 O4 ferrite layer between the NiO and Py layers. Pinarbasi et al. (2000) showed that for Ni1−x Ox /Py bilayers Jeb depends strongly on the Ni:O stoichiometry, and obtained a highest value Jeb = 0.11 mJ/m2 for x = 0.543. NiO-based read heads have been demonstrated by, e.g., Hamakawa et al. (1996), Nakamoto et al. (1996) and Pinarbasi et al. (2000). Later developments on exchange biasing materials with higher blocking temperatures and on SVs with high MR ratios due to the use of NOLs have distracted some interest from NiO. The optical transparency of NiO was used by Ju et al. (1998, 1999), who demonstrated ultrafast modulation of the exchange bias ﬁeld for NiO/Py bilayers, on a time scale as short as 10−12 s. A sub-ps pump-pulse through the NiO layer was used to create electronic excitations at the AF/F interface, and a time-delayed probe pulse was used to detect magneto-optically the resulting precessional magnetic response of the pinned layer in an external ﬁeld antiparallel to exchange bias ﬁeld. The pump-pulse induced an increase of the interface spin temperature of the F layer that led to an almost instantaneous reduction of Heb , whereas the NiO bulk magnetic structure remained unchanged. α-Fe2 O3 has a high Néel temperature, TN = 675 ◦ C, but gives at best rise to only a small exchange biasing ﬁeld (Jeb < 0.06 mJ/m2 ) with a high Hc /Heb ratio (>1). The effect was ﬁrst reported by Bajorek and Thomson (1975). Other early work was done by Hempstead et al. (1978) and Cain et al. (1987). The latter authors reported Tb ≈ 300 ◦ C. Table 4.6 gives references to more recent work. α-Fe2 O3 based dual SVs presently hold the record of the largest MR ratio (27.8%, see ﬁg. 2.7), resulting from specular electron reﬂection at the AF/F interfaces. In this study, the α-Fe2 O3 layers actually induce a large coercivity, but no exchange bias. The exchange bias ﬁeld that is found in some studies is too small for applications in conventional SVs. The use of a Sy-AF pinned layer would make the properties of α-Fe2 O3 based SVs more acceptable for applications (Kawawake et al., 2000), but it seems unlikely that this will make these systems real competitors for applications in read heads. 4.6.4. Thermal relaxation The difference between the blocking temperature and the Néel temperature, and the gradual decrease of Heb in an antiparallel ﬁeld (ﬁg. 4.19), are due to thermal relaxation processes. The ﬁrst quantitative theory was proposed by Fulcomer and Charap (FC) (1972). They considered a continuous F layer that is in exchange contact with magnetically isolated AF particles (grains), and employed an Arrhenius-type model of thermally activated switching. The magnetization of the F layer and within each of the AF grains is assumed to be uniform. The total energy of the magnetic state in one AF grain is expressed as the sum of an interface exchange interaction energy (eq. (4.4)) and a bulk magnetic anisotropy energy: Egrain = KAF tAF AAF sin2 αAF − Jeb AAF cos(αF − αAF ),

(4.46)

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where αF and αAF are the F layer magnetization direction and the magnetization direction of one of the AF sublattices with respect to the exchange bias direction, where AAF is the surface area of the AF grain and where it is assumed that the AF easy axis is parallel to the exchange bias direction. In view of the ﬁnite volume of the AF grain, there is a ﬁnite probability that the AF grain is not in the stable ground state, but in a thermally excited metastable state. For F layer magnetizations parallel and antiparallel to the exchange bias directions, the corresponding energy diagrams are given in ﬁg. 4.21. Suppose that all AF grains are identical, that αF = 0 and that at t = 0 all AF grains are in the minimum energy state αAF = 0. When at t = 0 the F layer magnetization is reversed, the AF spin lattice is in a metastable state. The inverse relaxation time for thermal excitations to the new ground state is then given by 1 ΔE2 AAF ΔE1 AAF (4.47) + exp − , = ν0 exp − τ kB T kB T where ν0 is the reversal attempt rate, which from the theory of magnetic relaxation is of the order 109 s−1 (Fulcomer and Charap, 1972 and references therein). It follows straightforwardly from eq. (4.46) that the energy barriers ΔE1 and ΔE2 are ΔE1,2 = 2 /(4K t ) ± J . The relaxation of the exchange bias ﬁeld is described by KAF tAF + Jeb AF AF eb 2 t . Heb (t) = Heb (0) 1 − (4.48)

2Jeb AAF exp − τ 1 + exp − kB T

The actually observed thermal relaxation is often not well approximated by an exponential function. Initially, the relaxation is faster, and on longer time scales it is slower (see e.g., ﬁg. 4.19). Within the FC model, this can be explained by taking a distribution of grain areas into account. Even a narrow distribution can give rise to a very wide distribution of relaxation times. This implies that in general not all grains contribute to the exchange bias effect. The smallest grains, for which the relaxation time is smaller than the timescale on which a hysteresis loop measurement is carried out, do not contribute. The overall blocking temperature is the temperature at which, for the time-scale at which the measurement is done, the fraction of blocked grains extrapolates to zero. At that temperature, even the largest AF grains are no longer thermally stable. For grains of a certain size, the blocking temperature could be deﬁned as the temperature at which the relaxation time τ is equal to the measurement time tm . For the situation depicted in ﬁg. 4.20 (ΔE1 ΔE2 ), it follows that Tb =

1 ΔE2 AAF . kB ln(ν0 tm )

(4.49)

The important point to notice is that the blocking temperature is not a thermodynamically determined quantity, like the Néel temperature, but that it is determined by the reversal kinetics. This explains why generally Tb < TN . Tb depends on tm , an effect that is expected to be strongest for fast experiments. Note that in eq. (4.49) ΔE2 should be evaluated at T = Tb . The activation barrier is generally temperature dependent, because Jeb and KAF decrease with increasing temperature. This decreases the dependence of Tb on tm that would otherwise be expected.

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Nishioka et al. (1996) successfully used the FC model to explain the temperature dependence of the exchange bias ﬁeld and coercivity of (glass/ tCu Cu/7.5 nm Py/ 15 nm Fe50 Mn50) systems, as a function of tCu . Varying tCu between 0 and 32 nm led to a variation of the average grain size from ∼2 nm to ∼5 nm, and to an increase of Tb from 320 K to 420 K. This trend, and the shape of the Heb (T ) curves, were well described by calculated curves obtained by assuming a log-normal distribution of grain sizes. Nishioka et al. (1998) performed a similarly successful analysis for Co/Cr–Mn–Pt systems. The increase of Tb with the grain size has a positive effect on the exchange bias ﬁeld at elevated temperatures. On the other hand, for large grains the fraction of uncompensated spins at the surface is smaller than for small grains. On the basis of the random spin argument discussed −1/2 in section 4.6.2, in relation to the Malozemoff model, one expects that Jeb ∝ AAF . In that case, increasing the grain size is expected to lead to a lower value of Heb at low temperatures. These two opposing effects have indeed been observed in several studies, e.g., for Fe50 Mn50 by Kung et al. (1991) and Nishioka et al. (1996), and for Ir–Mn by Fuke et al. (1999) and Childress et al. (2001). The blocking temperature increases also with increasing AF layer thickness, tAF , because the energy barrier for reversal increases with tAF . Within a thermal ﬂuctuation model, the minimum AF layer thickness above which at a temperature T an exchange bias effect is obtained is given by the AF layer thickness at which the blocking temperature is equal to T . Thermally activated relaxation thus provides an additional explanation for the occurrence of a minimum AF layer thickness below which Heb is zero, beyond the explanation that follows from the MB model (eq. (4.45)). The observation that the increase of Tb with tAF saturates for high tAF cannot be explained from the FC model. A likely explanation is that the magnetization reversal in thick AF layers is not coherent throughout the entire ﬁlm thickness. At a temperature T for which the areal fraction fu of “unblocked grains” is smaller than one, the measured interaction constant Jeb,expt (T ) is smaller than the local value Jeb (T ) that would have been obtained if all grains would have contributed. This explains, in part, the decrease of Jeb,expt with increasing T . Generally, Jeb will also decrease with increasing temperature. Therefore, fu (T ) cannot be derived directly from Heb (T ). Fig. 4.22(a) shows how fu (T ) can be measured. First, the sample is heated rapidly in a ﬁeld parallel to the exchange bias ﬁeld from a very low temperature T0 at which no relaxation takes place to the temperature T . Second, the ﬁeld is reversed rapidly, after which the sample is kept at T during a well-deﬁned time t. Third, the sample is cooled rapidly (effectively instanta , neously) to T0 . When the exchange bias ﬁelds before and after this cycle are Heb and Heb 1 /H + 1). Results for ﬁve exchange bias materials are given respectively, fu (T , t) = 2 (Heb eb in ﬁg. 4.22(b). Unfortunately, the original publications do not give the time scale t, which is probably of the order of seconds to several minutes. Fig. 4.22(c) gives the derivatives of the fu curves in ﬁg. 4.10(b), which can be called the “distribution of blocking temperatures”. Fujikata et al. (1998a) performed experiments of the type depicted in ﬁg. 4.22(a) as a function of the anneal temperature and time, for Fe50 Mn50 , NiMn, (Pt–Pd)Mn and NiO based AF/Py bilayer systems, and for T0 = 298 K. The results are given in ﬁg. 4.23(a). The ﬁgure shows that, in the anneal time intervals studied, the relative decrease of the exchange bias ﬁeld is well described by ΔHeb ∼ = −a − b ln(ta ). Heb (0)

(4.50)

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Fig. 4.23. (a) Relative change of the exchange bias ﬁeld of (Py/ 15 nm Fe50 Mn50 ), (Py/ 30 nm NiMn), (Py/ 30 nm Pd17 Pt32 Mn51 ) and (50 nm NiO/ Py) bilayers, measured at room temperature after annealing in a reverse external ﬁeld (see ﬁg. 4.22(a)), as a function of the anneal time and temperature. From Fujikata et al. (1998a). (b) Schematic relaxation time distributions at room temperature and at the anneal temperature.

If ta is expressed in hours, −a is the relative decrease of the exchange bias ﬁeld after ta = 1 hour. The logarithmic time dependence can be understood by assuming that the grain size distribution in the AF layer is uniform between certain minimal and maximal values (Fujikata et al., 1998a), and that the range of anneal times falls well within the range of relaxation times. From eq. (4.47), and assuming Jeb KAF tAF , it follows that the distribution of the natural logarithm of the relaxation time, P (ln(τ ), Ta ), is uniform over the interval [ln(τmin ), ln(τmax )]. The distribution functions at the temperatures T0 and Ta are shown schematically in ﬁg. 4.23(b). Eq. (4.50) follows then from eq. (4.48) after integration over all relaxation times, with b=

2 ). ln(τmax /τmin

(4.51)

Here τmin is the relaxation time at Ta of the smallest grains that, at T0 , still contribute = τmin . However, if that is not the case, all to Heb . If at T0 all grains contribute, τmin grains which at room temperature have a relaxation time that is smaller than the hysteresis loop measurement time tm are irrelevant, so that τmin > τmin . Fig. 4.23 shows that for all materials studied b increases with increasing temperature, which implies that the ratio decreases. Qualitatively, this is as expected from eq. (4.47), when KAF /kB T τmax /τmin decreases with increasing temperature. We note that the conditions that lead to an ln(t) time dependence are in fact more general than discussed above. The only requirement is that P (ln(τ ), Ta ) does not vary much in the relaxation time interval that corresponds to the range of annealing times for which the experiment has been carried out (Street and Woolley, 1949; Hughes et al., 2001). Magnetic relaxation experiments can only probe the real relaxation time distribution if the range of anneal times and temperatures is sufﬁciently broad.

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The simple model discussed above predicts also that the decrease of Heb at a certain ﬁxed anneal temperature in an antiparallel ﬁeld is proportional to ln(t). However, Carey et al. (2001) studied the thermal relaxation of NiMn and Ir–Mn based SVs, for ta varying over almost three decades, and observed a clear deviation from the ln(t) dependence. They analysed the results in terms of a more realistic distribution of relaxation times, based on a log-normal distribution of AF grain areas. Other ingredients of their model are the −1/2 inclusion of a grain size dependent value of Jeb ∝ AAF (as explained above) and a temperature dependent anisotropy constant KAF . A realistic set of ﬁt parameters was found to lead to a good description of the experimental data. Another extension of the FC model was presented by Tsunoda and Takahashi (2000b), who treated the AF layer as a ensemble of monodisperse grains with random easy axis directions. Future improved models should address the assumption of non-interacting AF grains, which may not always be quite correct. An extreme example is the case of AF/F bilayers in the form of a coherent epitaxial superlattice (as studied, e.g., by Jungblut et al., 1994). The question which activation volumes in the AF layer then determine the thermal relaxation processes is still open. The decay rate of the exchange bias ﬁeld at a given temperature (parameter b in eq. (4.50)) does not necessarily increase with decreasing blocking temperature. E.g., ﬁg. 4.23(a) shows that for Fe–Mn the decay rate at 100 ◦ C is much smaller than for NiO, in spite of the lower blocking temperature for Fe–Mn. Carey et al. (2001) observed that (for small anneal times and for temperatures up to at least 175 ◦ C) the decay rate increases in the series (top Ir–Mn) – (bottom Ir–Mn) – (NiMn or PtMn) – NiO, whereas Tb (Ir– Mn) ≈ 250 ◦ C < Tb (NiMn or PtMn) ≈ 350 ◦ C. The authors explained the effect using the model discussed above, from a relatively high average activation energy for reversal of the magnetization of the Ir–Mn grains at low temperatures. In the analyses of thermal relaxation given above, it was assumed that at t = 0 all grains for which the relaxation time is larger than t m contribute positively to Heb . This is not always the case for actual samples. E.g., the increase of Heb of an as-deposited Ir–Mn/Py bilayer upon annealing in a parallel ﬁeld, shown in ﬁg. 4.19, indicates that during the anneal process an originally non-oriented fraction of grains with a high relaxation time is being ordered. Also from a ﬁeld cooling procedure a fully ordered state can sometimes not (easily) be created. Rather unexpectedly, cooling in a ﬁeld that is orders of magnitude larger than the exchange bias ﬁeld and coercive ﬁeld that are ﬁnally obtained at low temperatures does sometimes lead to a signiﬁcantly larger exchange bias ﬁeld than cooling in a ﬁeld that is only slightly larger than Heb (Ambrose and Chien, 1998; Van Driel et al., 2000b). At present, this is not well understood. Experimental work on ac ﬁeld cooling has been carried out by Tsang and Lee (1982) and Li et al. (2002) (see also references therein). Modelling of the ﬁeld cooling procedure could be carried out using the methods discussed above. We focus here on the simple case of a monodisperse AF layer, and neglect the practical constraints that can be imposed on the ﬁeld cooling procedure by the structural thermal stability of the system. With the help of ﬁg. 4.24, it can be understood that there is a single optimum ﬁeld cooling temperature Tfc < TN at which the sample should be kept during the entire ﬁeld cooling time tfc that is available, before cooling down to room temperature. The ﬁgure gives contours of equal degree of ordering, η (deﬁned in the ﬁgure caption), obtained after relaxation from a non-ordered high temperature state during times tfc = 1 s and 104 s, as a function of the dimensionless parameters KAF VAF /kB T

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Fig. 4.24. Degree of ordering, η, of the magnetization directions of monodisperse AF grains in an AF/F bilayer, obtained after a ﬁeld cooling procedure. It is assumed that the AF layer is initially magnetically disordered, as a result of heating to a temperature above the blocking temperature. Subsequently, it is kept for tfc seconds at a ﬁeld-cool temperature Tfc < Tb . As a result, fractions n+ and n− of the AF grains have sublattice magnetizations that contribute positively and negatively to the exchange bias ﬁeld, respectively. The diagram gives contours of equal η ≡ (n+ − n− )/(n+ + n− ) after tfc = 1 s (thin lines) and tfc = 103 s (thicker lines) as a function of the parameters KAF VAF and Jeb AAF at T = Tfc , as obtained from the Fulcomer–Charap model. The curves a, b and c connect (KAF VAF , Jeb AAF ) points at varying temperatures for three hypothetical materials. The maximum obtainable degree of ordering for these materials is discussed in the text.

and Jeb AAF /kB T . Ideally, at Tfc the thermodynamic driving force for ordering, Jeb AAF , is much larger than kB T , whereas, on the other hand, the relaxation time at that temperature is much smaller than tfc . The curves a, b and c give (KAF (T )VAF , Jeb (T )AAF ) curves for three hypothetical materials, assuming that both parameters increase with decreasing temperature. For curve a, tfc = 1 s is more than sufﬁcient for aligning almost all AF grains. For curve b, η can be at best ∼0.5 if tfc = 1 s. By ﬁeld cooling during 104 s it can be enhanced to η ∼ 0.8. For curve c, even for tfc = 104 s the best possible degree of alignment will be only ∼0.1. The experimental exchange bias parameter at a temperature T0 (e.g., room temperature) to which the sample is cooled down rapidly after the ﬁeld anneal procedure, is Jeb,expt(T0 ) = ηJeb (T0 ). It is assumed that at T0 the AF magnetic structure remains unaltered during a hysteresis loop measurement, i.e., that tm τ (T0 ). For AF materials with a grain size distribution, the ideal ﬁeld-cooling procedure would be a compromise that is determined by the (KV , J A) curves for the range of grain sizes that is present and by the required grain size dependence of the degree of ordering. In conclusion, the degree of ordering of the AF grains that results from a ﬁeld-cooling procedure is determined by the detailed time-temperature trajectory during that procedure. A more detailed understanding of the mechanism of exchange anisotropy is expected to emerge from magnetic viscosity experiments, within which Heb and Hc are measured as a function of the applied ﬁeld sweep rate, dH /dt. For NiO/Co bilayers that only showed coercivity (Heb = 0 because tAF = 25 nm < tAF,min ) such a study was carried out by

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Camarero et al. (2001). The authors measured Hc as a function of an applied ﬁeld sweep rate, over 10 decades of dH /dt, and deduced from that the thermal activation volume in the AF layer. Yang et al. (2000b) carried out such experiments for Ir–Mn/Py, Pd–Pt– Mn/Py, Ni–Mn/Py and Ni–Mn/Co90Fe10 bilayers, over 1–2 decades of dH /dt. They observed strong a increase of Hc with increasing sweep rate, but no effect on Heb . As far as is known to the author, studies of the dependence of Heb on the ﬁeld sweep rate and the temperature, yielding the relationship Tb (tm ), have so far not been done. 5. Conclusions and outlook In sections 2, 3 and 4 rather detailed discussions have been presented on a wide range of aspects of the physics and materials science of exchange-biased SVs. In this ﬁnal section we formulate some conclusions on the developments that have taken place and indicate possible directions for future research. The ﬁrst main development has been the increase of the MR ratio of SVs. In the conventional simple SVs developed (Dieny et al., 1991a, 1991b, 1991c) at IBM, diffusive scattering of electrons at the outer boundaries of the active layer limited the MR ratio to 9% or less at room temperature. Highlights of the research on SVs were the discovery that the MR ratio can be increased signiﬁcantly by making use of oxidic antiferromagnets at which (partially) specular scattering takes place (section 2.1.6), and that the same effect can be realized by making use of nano-oxide layers (NOLs, section 2.1.7). Recently “double specular” simple SVs with an MR ratio just above 20% at room temperature have been demonstrated (Tsunekawa et al., 2002). For dual SVs the highest MR ratio that has been obtained so far is 27.8% (Sugita et al., 1998). It may be clear from table 2.1 that these numbers have not realized by many groups. Typical MR ratios for simple SVs without and with NOLs, obtained by a number of leading groups, are 12% and 17%, respectively. In section 3.10, it was shown that the largest room temperature MR ratios that have been observed so far for simple and dual SVs without and with specularly reﬂecting boundary layers are consistent with the predictions from a surprisingly simple semiclassical transport model. The majority spin mean free paths in the F and NM layers are assumed to be both equal to approximately 10 nm, whereas the minority spin mean free path in the F layers is assumed to be zero. It follows from the model that double specular simple or dual SVs with a Cu spacer layer thickness of ≈2 nm could have an MR ratio of at most about 40%, in the limit of very thin F layers (ﬁg. 3.15(a)). Perhaps more realistically, the maximum possible MR ratio is about 31% when the F layer thickness is ≈2 nm. Of course, the model gives only upper limits. It does not deal, e.g., with the limitations that are imposed by the requirement that the magnetic interactions should be well-controllable and compatible with a speciﬁc application. Obtaining SVs with even larger MR ratios would require using thinner spacer layers. For tCu ≈ 1 nm, a maximum possible MR ratio of about 87% is predicted (ﬁg. 3.15(b)). First attempts to realize such SVs by precisely balancing the AF and F contributions to the magnetic coupling across the spacer layer, have been reported by Jo and Seigler (2002a, 2002b). So far, this has not led to higher MR ratios than for SVs with values of tCu more close to 2 nm (section 2.1.4).

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Analyses of the magnetoconductance using semiclassical or quantum-mechanical models with free electron or more realistic band structures, have led to a better understanding of the effects of, e.g., (i) thermal ﬂuctuations, (ii) scattering at bulk defects (grain boundaries) and at rough interfaces, and (iii) the inclusion of non-magnetic “back-layers” in “spin-ﬁlter SVs”, which can for various reasons be of great interest to applications (section 2.1.3). Electron channeling in the Cu layer has been predicted to affect the current density distribution and lead to an increased MR ratio (section 3.8). However, an analysis of the experimental conductance and magnetoconductance of SVs has not provided clear proof for the importance of the effect (section 3.9). Whereas intensive research efforts have been focused on the understanding of the magnetoresistance effect and on its enhancement by making use of novel layer stacks, comparatively little work has been done on the electronic noise of SVs (section 4.3.3). This may be explained by the fact that for present hard disk read heads, the most important application, Johnson–Nyquist noise predominates. The noise voltage density then simply follows from the element resistance. Recently, a novel type of high frequency noise, magnetic thermal ﬂuctuation noise, was discovered. It will probably be dominant in future hard disk read heads. Low-frequency applications of SVs are often limited by the 1/f noise of the element. A predominant magnetic contribution has been identiﬁed, which is related to thermally induced ﬂuctuations of the laterally inhomogeneous magnetization in the free layer. At present, there is no theory from which the size of even the order of magnitude of this effect could be predicted, and there is a lack of systematic experimental work on this subject. For the development of low-frequency applications of SVs, such studies would be of great importance. The second main development is the increase of the number of layers in SVs. Additional layers are not only used to obtain a higher MR ratio, but also to more precisely control and optimize the magnetic interactions and the thermal stability. The combination of the various improvement methods shown in ﬁg. 2.1 (b)–(i) has led to typical state-of-the-art layer stacks such as shown in ﬁgs 5.1(a) and 5.1(b). The bottom SV shown in ﬁg. 5.1(a) contains two underlayers, (e.g., Ta and Py–Cr, see section 2.9.3), a composite free F layer (e.g., Py and Co90 Fe10 , see section 2.1.2), a spacer layer (Cu), a Sy-AF pinned layer (containing, e.g., Co90 Fe10 , Ru and Co60 Fe40 , see section 2.1.2), an exchange bias layer (e.g., Ir–Mn), and a cap layer (e.g., Ta). The double specular spin-ﬁlter top-SV with a Sy-AF pinned layer, shown in ﬁg. 5.1(b), is equally complex. Optimizing the performance of the layer stacks shown in ﬁg. 5.1 requires the use of at least eight compositionally different layers, as compared to four (Ta, F, NM and AF) for the prototype simple SVs discussed in section 1. An even larger number of compositionally different layers can be required when using one of the methods explained by ﬁg. 2.8 for controlling the offset ﬁeld or for reducing the hysteresis of the free layer. This development emphasizes the versatility of the exchange-biased SV concept: “auxiliary” layers outside the active part can be used to realize a combination of properties that is optimally suited for a certain application. A potentially negative consequence is the increasing cost of sputter deposition equipment for SVs consisting of ever more compositionally different layers. This might disencourage starting up industrial production and even industrial R&D on a competitive level. The impact of this development on academic research, usually more focused on speciﬁc aspects and model systems, is expected to be less.

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Fig. 5.1. Two typical state-of-the-art spin-valve layer stacks.

A third and crucial development has been the discovery of exchange bias materials with much higher blocking temperatures than Fe50 Mn50 , which was the exchange bias material used in the ﬁrst SVs (section 4.6.3). This made it possible to apply SVs in hard disk read heads, at present still the most important application. However, the overview of properties given by table 4.7 shows that the ideal exchange bias material does not yet exist. The optimal choice will depend on the speciﬁc application. Research on novel and improved exchange bias materials will continue to be of great importance to the development of SV applications. One of the important issues is the relationship between the rate of thermal relaxation of the exchange bias ﬁeld and the microstructure and domain structure in the AF layer (section 4.6.4). A second important issue is the relationship between the AF layer composition and microstructure and the stability with respect to diffusion of Mn out of that layer into the active part of the layer stack. It would be of interest to systematically investigate the suggestion, proposed in section 2.10.2, that the stability with respect to Mn diffusion increases with increasing heat of formation of the AF exchange bias material. The research on SVs is strongly application driven. Much current research aims at meeting the challenging requirements of future hard read heads (section 1.6.1). Realization of the 1 Tbit/inch2 areal bit density target that has been set for 2006 (ﬁg. 1.12) would require SV elements that are sensitive to a read track width of only approximately 30 nm, and that are suitable for read out at data rates around 3 Gbit/s (Wood, 2000). Scaling of the present sensor elements would lead to stripe widths that are well below 100 nm. Patterning SV elements with such nanometer scale dimensions will become a challenge, and structural and magnetic edge effects and individual grain effects will become important. Operation at frequencies above 1 GHz will require shifting the magnetic resonance frequencies to higher values than observed for present read heads (section 4.3). Secondly, the

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application of SV sensors in automotive systems will stimulate further developments of even more thermally stable SVs (section 1.6.2). Finally, the introduction of SV technology as a back-end process on top of Si-devices and circuitry is expected to give rise to many novel applications (section 1.6.4), including, e.g., on-chip integrated current sensors, magnetocouplers, and magnetic biosensors. Acknowledgements I am very grateful to my colleagues of the Philips Research Laboratories and of the Eindhoven University of Technology, in particular to H. Boeve, J. Kohlhepp, K.-M.H. Lenssen, H. Swagten and F. Vanhelmont, for useful discussions and comments on the manuscript, and to J.C.S. Kools for drawing my attention to some relevant publications. References Adachi, K., 1986, in: Wijn, H.P.J. (Ed.), Landolt– Börnstein. In: New Series, Vol. 19a. Springer, Berlin, p. 300. Ambrose, T., Sommer, R.L., Chien, C.L., 1997, Phys. Rev. B 56, 83. Ambrose, T., Chien, C.L., 1998, J. Appl. Phys. 83, 7222. Ambrose, T., Lu, K., Chien, C.L., 1999, J. Appl. Phys. 85, 6124. Anderson, G., Huai, Y., Miloslavsky, L., 1999a. In: Mat. Res. Soc. Symp. Proc., San Francisco, Vol. 562, p. 45. Anderson, G.W., Juai, Y., Miloslavsky, L., Qian, C.X., 1999b, J. Appl. Phys. 85, 6109. Anderson, G.W., Huai, Y., Pakala, M., 2000a, J. Appl. Phys. 87, 5726. Anderson, G., Huai, Y., Miloslawsky, L., 2000b, J. Appl. Phys. 87, 6989. Anderson, G.W., Kakala, M., Huai, Y., 2000c, IEEE Trans. Magn. 36, 2605. Anthony, T.C., Brug, J.A., Zhang, S., 1994, IEEE Trans. Magn. 30, 3819. Aoshima, K., Kanai, H., Kane, J., Miyajima, T., 1999, J. Appl. Phys. 85, 5042. Aoshima, K., Hong, J., Kanai, H., 2000, IEEE Trans. Magn. 36, 3226. Araki, S., Omata, E., Sano, M., Ohta, M., Noguchi, K., Morita, H., Matsuzaki, M., 1998a, IEEE Trans. Magn. 34, 387. Araki, S., Sano, M., Ohta, M., Tsuchiya, Y., Noguchi, K., Morita, H., Matsuzaki, M., 1998b, IEEE Trans. Magn. 34, 1426. Araki, S., Sano, M., Li, S., Tsuchiya, Y., Redon, O., Sasaki, T., Ito, N., Terunuma, K., Morita, H., Matsuzaki, M., 2000, J. Appl. Phys. 87, 5377.

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Further reading Cheng, S.-F., Lubitz, P., 2000, J. Appl. Phys. 87, 4927.

Yu, S.Y., Lee, W.Y., Cyrille, M.-C., Fontana, R.E., Gurney, B.A., 2002, IEEE Trans. Magn. 38, 2259.

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chapter 2

ELECTRONIC STRUCTURE CALCULATIONS OF LOW-DIMENSIONAL TRANSITION METALS

A. VEGA Departamento de Física Teórica, Atómica, Molecular y Nuclear, Universidad de Valladolid, E-47011 Valladolid, Spain

J.C. PARLEBAS and C. DEMANGEAT Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, 23, rue du Loess, F-67034 Strasbourg Cedex 02, France

Handbook of Magnetic Materials, edited by K.H.J. Buschow Vol. 15 ISSN: 1567-2719 DOI 10.1016/S1567-2719(03)15002-0

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© 2003 Elsevier Science B.V. All rights reserved

CONTENTS List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Methodology: general aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Topics of this report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. List of some previous reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Extended contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quantum-mechanical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. All-electron DFT methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Pseudopotential DFT methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Semiempirical TB methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Dealing with the geometrical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Dealing with non-collinear magnetic arrangements . . . . . . . . . . . . . . . . . . . . . . . . 3. Thin ﬁlms and multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Iron based multilayered systems: general overview of a great variety of magnetic phenomena 3.3. Fe/Cr. A ferro/antiferromagnetic interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Fe/V. A controversial system. Induction of magnetic moments . . . . . . . . . . . . . . . . . . 3.5. Fe/Mn. A complex system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Fe/Ni. A ferromagnetic/ferromagnetic interface . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Fe/M interface (M = 4d, 5d transition metal) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Fe on Cu, Ag, Au. A self-surfactant system? . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Fe/graphite. Towards intrinsic non-collinear systems . . . . . . . . . . . . . . . . . . . . . . . 3.10. Fe/c-FeSi/Fe sandwiches and multilayers. A critical system . . . . . . . . . . . . . . . . . . . 3.11. Effect of light impurities (H, O) on magnetic multilayers . . . . . . . . . . . . . . . . . . . . . 4. Towards the nanoworld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Ni free-standing clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Co nanowires supported on Pd(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Transition metal clusters in contact with carbon . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. NiN embedded in Al. Lack of magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. CoN supported on noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Non-collinear magnetic conﬁgurations of V, Cr, Mn and Fe aggregates on Cu, Ag . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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201 202 202 204 207 210 211 211 212 213 213 214 216 217 217 218 221 224 228 229 234 237 241 244 248 250 250 254 258 261 266 267 269 273 277 278 288

List of abbreviations

AF, AFM AES ASA ASW BLS CPA DFT DLM DMFT DOS EAM FLAPW FM FP FP-LMTO GF GGA gs hff HOPG IEC KKR LCAO LDA LDOS LEED LMDAD (MLDAD) LSDA LSW MD ML MLD MTO MMD MO

antiferromagnetism Auger electron spectroscopy atomic sphere approximation augmented spherical waves Brillouin light spectroscopy coherent potential approximation density functional theory disordered local moment dynamic mean ﬁeld theory density of states embedded atom method full-potential linearized augmented plane waves ferromagnetic full potential full potential linearized mufﬁn-tin orbitals Green function generalized gradient approximation ground state hyperﬁne ﬁeld highly oriented pyrolic graphite interlayer exchange coupling Korringa–Kohn–Rostocker linear combination of atomic orbitals local density approximation local density of states low energy electron diffraction linear magnetic dichroism in the angular distribution local spin density approximation local spherical wave molecular dynamics monolayer magnetic linear dichroism mufﬁn-tin orbitals magnetic multilayered device molecular orbital 201

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MOKE NM NMR PAM PAW PDMEE PEEM PNR RW SCF SC TB SDW SEMPA SIESTA SPLEEM SPSEE SPSTM STM STS SQUID TC TK TN TB TB-LMTO TD TM VASP VSM XAS XC XMCD XPS UHV

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magneto-optic Kerr effect non-magnetic nuclear magnetic resonance periodic Anderson model projector augmented wave primary beam diffraction modulated electron emission photo electron emission microscope polarized neutron diffraction Rayleigh wave self-consistent ﬁeld self-consistent tight binding spin density wave scanning electron microscopy with polarization analysis Spanish initiative for electronic simulations with thousands of atoms spin polarized low energy electron microscopy spin polarized secondary electron emission spin polarized scanning tunneling microscopy scanning tunneling microscopy scanning tunneling spectroscopy superconducting quantum interference device Curie temperature Kondo temperature Néel temperature tight binding tight binding linear mufﬁn-tin orbital trilayer device transition metal Vienna ab initio simulation package vibrating sample magnetometry X-ray absorption spectroscopy exchange correlation X-ray magnetic circular dichroism X-ray photoelectron spectroscopy ultra high vacuum

1. Introduction 1.1. Methodology: general aspects The ﬁeld of transition metal nanostructures has largely kept pace with microelectronics, forming the workhorse of information technology. Current research efforts include the preparation of thin ﬁlms for improved data storage, the exploitation of electron spin rather than charge for device switching (“spintronics”), and the development of new materials for lightweight and low-cost applications (Osborne, 2001). The many-body aspect of magnetic systems makes the task of calculating a low-energy conﬁguration of the spin ensemble a

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formidable one. Because a full quantum mechanical description is actually intractable various approximations have been used so that the theory can give some explanation of most of the experimental phenomena and results. The concentrated effort and enthusiasm of a large number of scientists resulted in an impressive display of new ideas and truly new discoveries. Theoretical work played an immense role in the process and active feedback between theories and experiments which has helped and speeded up the ﬁeld. Indeed, all magnetic properties of a solid are attributable to its electrons. In a free atom, there are two contributions to the magnetic moment. First, every electron has an intrinsic spin s and its associated magnetic moment. Second, there is the magnetic moment associated with the electron’s orbital angular momentum l. In a free atom these contributions are typically comparable in magnitude. For transition metals Hund’s rules predict the ground state conﬁguration (Jensen, 1995) but the situation is quite different for solids. The very ﬁrst realistic description of the magnetic properties of a given element was based on the Heisenberg Hamiltonian. In this model the moment on a given atom i is postulated to be a vector quantity S i whose interactions Jij with S j are mostly considered as adjustable parameters. Within this approach, Mermin and Wagner (1966) proved that, for one- or two-dimensional systems with isotropic short-ranged interactions, there is no long range order at non-zero temperature (T ). Bruno (2001) has extended that result to systems with long-range interactions with particular emphasis on multilayers presenting oscillatory as well as decaying interactions. This Heisenberg Hamiltonian is a very general one and in many cases the best suitable approach because there are a priori no restrictions on the mutual direction between S i and S j . Therefore noncollinear magnetism is normally described in this model (Demangeat and Mills, 1976; Kawamura, 1998). Surface spin waves and surface spin correlations in the presence of magnetic surface reconstruction can be easily described within this Hamiltonian (Demangeat et al., 1977). This approach is however more suitable for non-conducting elements. For metallic nanostructures as those described in the present review there is a non-negligible overlap between the matrix elements of the Hamiltonian describing the system, so that, the magnetic moment on a given atom strongly depends on its surrounding. For this reason the Hubbard Hamiltonian has been shown to be a more suitable approach (Victora and Falicov, 1985). Similarly to the Heisenberg Hamiltonian, the Hubbard approach relies on a few number of parameters like the hopping and Coulomb terms. For transition metal elements it is however worthwhile to consider the extended Hubbard Hamiltonian which takes care of the various valence orbitals as well as the exchange integral. While the Heisenberg Hamiltonian is restricted to integral numbers of the magnetic moments, the Hubbard approach gives moments whose values are between 0 and 2 Bohr magnetons (μB ) per orbital. In general this description relies heavily on the recursion in the real space developed by Haydock (1980). However, most of these calculations were performed within a collinear approach which is only valid, for a small number of elements and for perfectly grown materials, i.e. materials with long-range periodicity which are difﬁcult to produce nowadays. Moreover, very perfect Cr and Fe in the fcc phase do present various types of spin-density waves. Therefore we have to point out that we strongly believe that the so-called constrained collinear magnetization is – most probably – a “very common” metastable conﬁguration which in very rare cases may be degenerate with the real ground state of the system under consideration. To be more precise it is – from a mathematical

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point of view – not possible to consider a “real” experimental system within a collinear approach because – in most cases – the symmetry of the system is broken by the presence of defects or impurities. Therefore the surrounding of a given atom is non-symmetric and up to now we have not found any mathematical theorem who can restore this symmetry. Thus it is highly necessary to point out that the magnetic moment of a given atom should always be considered as a vector whose modulus and direction should depend directly on its surrounding. Moreover, as discussed by Kawamura (1998) the ground state of 3d elements (like Cr) is intrinsically non-collinear when it adopts the triangular conﬁguration or when it is deposited on an hcp substrate. Up to now we are only aware of a few results obtained within a vector description of the electronic structure determined within a model Hamiltonian approach. A vector description of the magnetism has been described by Stoefﬂer and Cornea (1998), Cornea (1999), Cornea and Stoefﬂer (2000), Robles et al. (2003b). Moreover, non-collinear magnetism (Yartseva et al., 1998) can also be described within a Periodic Anderson Model (PAM, see Parlebas et al. (1986), Parlebas (1990) and references therein). The Anderson model continues to be fascinating (Gehring, 2002) because it can be incorporated in dynamic mean-ﬁeld theory (DMFT) and has a direct link with the Kondo Hamiltonian. Up to now we have restricted our discussion to semi-phenomenological Hamiltonians like Heisenberg, Hubbard or PAM, expressed in terms of parameters the exact values of which are difﬁcult to determine. However these Hamiltonians are very useful because they can give many interesting informations and explanations about very intricate and complex experimental results performed on magnetic nanostructures. To overcome the difﬁculty with the use of parameters, Kohn and Sham (1965) proposed to replace the calculation of the many-particle problem by a one-particle approach in the ﬁeld of the others. This is the so-called Density Functional Theory (DFT) which is the most suitable method when some kind of symmetry remains, so that, with the help of Bloch’s theorem the number of inequivalent atoms could be minimized. However, DFT remains an approach which also presents difﬁculties and drawbacks. The main difﬁculty is related to the exchangecorrelation term whose functional is not known and since 40 years many groups tried to ﬁnd the most suitable form. Another point of difﬁculty consists of the Born–Oppenheimer approach which separates the movements of the electrons from those of the nuclei. Via ab initio molecular dynamics, Car and Parinello (1985) tried to improve that difﬁculty via the determination of the electronic structure of a given assembly of atoms by the minimization of the forces acting on a free cluster (Oda et al., 1998). These points will be discussed in section 2. Before closing this ﬁrst introduction part we should point out DFT calculations of Cr or Mn monolayer on (111) triangular crystallographic surfaces of Cu or Ag. The unconstrained magnetism is derived (Hobbs and Hafner, 2000; Kurz et al., 2001; Wortmann et al., 2001) for the deposited atoms, which are postulated to stay in epitaxial positions. 1.2. Topics of this report Up to now we have reported mostly on the methods used to describe the electronic and magnetic structures of nanomaterials. We have also given some references concerning the very fast developing area of unconstrained surface and interface magnetism which, for elements like Cr and Mn, are very necessary for the derivation of the “real” ground state.

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We will now give a few speciﬁc examples for which the present authors have tried to reﬂect some of the excitement associated with this ﬁeld. One aspect of the excitement is related to a possible onset of magnetization in metallic systems which are not magnetic in their bulk form. Ru, Rh, Pd and V are such systems. Experimental observation of magnetism in Rh clusters was ﬁrst depicted by Cox et al. (1993) within a Stern–Gerlach device. Later on Goldoni et al. (1999) have seen “experimental evidence of magnetic ordering at the Rh(100) surface” within linear magnetic dichroism in the angular distribution (LMDAD) of the Rh-3d photoelectrons. The results of Goldoni et al. (1999), are clearly at odd with DFT-GGA (GGA: Generalized Gradient Approximation) calculations by Cho and Schefﬂer (1997) displaying a non-magnetic ground state for Rh(001). Later on, Goldoni et al. (2000) reinterpreted their LMDAD results by writing that “the two-dimensional ferromagnetic order of the Rh(100) surface could be extremely unstable or that the Rh(100) surface could be super-paramagnetic”. However, almost all theoretical calculations found some kind of magnetization for Rh nanostructures on various non-magnetic substrates. Among others, Bazhanov et al. (2000) investigated electronic and magnetic properties of one-dimensional Rh structures deposited on Ag. Later on Bellini et al. (2001) extended the calculations to 4d monoatomic rows on Ag vicinal surfaces. In section 3.7 we will mostly focus on the induced magnetization of Rh atoms on Fe substrates. Bouarab et al. (1990) were the ﬁrst to report that Pd ﬁlms could present a non-negligible magnetic moment. Within superconducting quantum interference device (SQUID) measurements, Suzuki et al. (2000) saw some magnetization in palladium-graphite multilayers. However it is difﬁcult to say, at present, if palladium nanostructures are magnetic or not. Indeed, Pd is a strange element which is non-magnetic in the atomic conﬁguration and in the bulk form. Therefore it is a priori unlikely that pure Pd in any form can present some kind of magnetic moment (Dreyssé and Demangeat, 1997). However as seen by Suzuki et al. (2000) and by Mikheenko et al. (2002) Pd in contact with graphite or certain bacteria could present a magnetic moment. Magnetization measurements by vibrating sample magnetometry (VSM) up to ﬁelds of 2T show a combination of ferromagnetic and paramagnetic behaviors associated by Mikheenko et al. (2002) to two distinct subsets of grains. Magnetization of Pd can be due to some kind of charge transfer between Pd and its surrounding so that the Pd conﬁguration may tend to that of Rh which is magnetic according to Cox et al. (1993). The onset of some kind of magnetism in Pd free-cluster or Pd in contact with non-magnetic elements is clearly not understood yet. On the contrary, Pd in contact with Fe can produce “giant moments”. Also Pfandzelter et al. (1995) experimentally showed within Spin-Polarized-Secondary-Electron-Emission that a monolayer of Ru on graphite presents a non-zero-in-plane polarization. Vanadium is a very speciﬁc example where, up to now, no real experimental proof concerning its magnetization was reported. We believe that, for example, a single V adatom on graphite may be in a high spin state in agreement with Duffy and Blackman (1998), but that the V supported clusters of Binns et al. (1992, 1999) are in a low spin state (or in an antiferromagnetic state) due to inter-atomic d–d hopping. From a theoretical point of view, however, most of the calculations obtained some kind of magnetization in the V nanostructured conﬁguration. Section 3.4 is devoted to a possible clariﬁcation about V magnetization. It will also be shown that V in contact with Fe acquires a magnetic moment. This fact, from a

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theoretical point of view, is trivial because the strong magnetization present in Fe induces (via d–d hybridization) a moment on the V atoms. However, experimentally, this point is not deﬁnitively settled and discussions are still present concerning the polarization range (long-range versus short-range; effect of intermixing at the interface; roughness effect and in some speciﬁc cases, non-collinear arrangements). A second very exciting point is that light impurities, like hydrogen, remaining (or introduced) in the high vacuum devices have a very drastic effect on the interlayer exchange couplings in Fe/V superlattices (Hjörvarsson et al., 1997) (see section 3.11 for details). Yonamoto et al. (2001) studied the magnetic behavior of a Mn ultrathin ﬁlm grown on a Co(100) surface with increasing oxygen exposure by using X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD). Oxygen preferably reacts with Mn and a MnO ﬁlm is formed, which is ferromagnetic with respect to the other Mn atoms but is antiferromagnetic to the Co substrate. DFT calculations within GGA approach by Pick and Demangeat (2003) conﬁrm the strong effect of oxygen on the Mn–Co coupling (section 3.11). In general, most of the calculations to derive magnetic maps of magnetic nanostructures have been performed within a constrained direction of the polarization, i.e. within the approximation of collinear magnetism. This approximation is very drastic and in many cases frustration can be minimized if non-collinearity is introduced in the Hamiltonian of the system. Besides, Taga et al. (2000) pointed out, using ﬁrst-principle theory, that material combinations such as Fe/V/Co multilayers can produce a non-collinear state in which the magnetization direction between Fe and Co layers differs by about π/2. Various aspects of non-collinear magnetization have been derived in the case of multilayered systems or superlattices, free standing clusters, overlayers (Oda et al., 1998; Hobbs et al., 2000; Robles et al., 2003b) or adsorbed clusters (Uzdin et al., 1999). Also Kurz et al. (2001) mainly reported about non-collinear magnetization of Mn monolayers on Cu(111) in connection with scanning tunneling microscopy and spectroscopy (STM/STS) results of Wiesendanger and Bode (2001). We have to point out here that a vectordescription of the magnetization map is of fundamental interest because it can lead to various new aspects in the magnetic phase diagram of the systems. Up to now only very rare results appeared in the literature because it remains a rather formidable task to obtain them properly. Most of the vector-description of nanostructures do not presently calculate the forces, i.e. the calculations are performed with a given position of the atoms in the aggregate or in the substrate. Moreover they do not yet consider a possible atomic exchange between the adsorbed atoms and the substrate as reported by Demangeat et al. (2001) for collinear magnetic conﬁgurations. Interlayer exchange coupling was discovered by Grünberg et al. (1986) in Fe/Cr multilayered systems and was the subject of a tremendous number of papers and review reports. Interlayer exchange coupling (IEC) is simply (in the case of Fe/Cr superlattices) the modiﬁcation of the direction of polarization of two adjacent Fe slabs versus the thickness of the Cr spacer. In this speciﬁc Fe/Cr case two periods of oscillations were obtained: one of short range of about 2 ML and a long-range one of roughly 18 Å. The long-range is measured in epitaxial as well as in sputtered superlattices and does not need any special requirement concerning the interfacial roughness. The observation of the short-period of 2 monolayers (ML) needs a very perfect and ﬂat interface, linked to special regimes of the epitaxial

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growth, which prevents the formation of steps and intermixing during the deposition (Unguris et al., 1991). Bruno (1995, 1999) and Stiles (1999) reported on phenomenological approaches used to derive the IEC period. Also Bürgler et al. (2001) gave an up to date review of this phenomenon. The short-period of the IEC in Fe/Cr superlattices was explained, beginning of the nineties by Herman et al. (1991), Stoefﬂer and Gautier (1990), and Vega et al. (1991). This short-period of the IEC is in agreement with practically all careful experiments (see, Heinrich et al. (1999) for discussion) but there is a π -phase-shift between theory and experiment, i.e. the 2 ML period is all-right but when theory predicts ferromagnetic (FM) coupling, experiment displays antiferromagnetic (AFM) coupling. Uzdin et al. (2002b) proposed to link this discrepancy to alloying effects at both Fe/Cr and Cr/Fe interfaces. Also renewed interest in this IEC has appeared in the case of heterostructures (Fe/Si for example and also many other cases). To report on all various aspects of transition metal magnetic multilayered materials would be a rather formidable task. Fortunately, up to now, a considerable number of review reports already appeared in the literature so that, in this present review, we will essentially focus on a few points which seem pertinent and exciting from our point of view. These points are generally rather unsolved problems, or at least, they present inherent theoretical problems (like the application of atomic sum rules in the XMCD approach, or more generally the connection between the many electron problem in atomic physics (see, for example, Uozumi et al., 2003) and the solid state effects, i.e. the interatomic interactions (see, for example, Kotani and Parlebas, 1988, and Pollini et al., 2001), or they present so many inequivalent atoms that a self-consistent procedure cannot be easily converged. Let us ﬁnally say a few words on a puriﬁcation algorithm for expanding the singleparticle density matrix in terms of the Hamiltonian that is simple, general, and with a computational complexity essentially independent of the system size even for very large metallic systems with a vanishing band gap. If the expansion is used together with a ﬁxed chemical potential, Niklasson (2002) shows that it is an asymmetric generalization of grand canonical McWeeny puriﬁcation. The algorithm is a substantial improvement of previous schemes and provides a framework for the understanding and optimization of puriﬁcation. 1.3. List of some previous reports Many years ago Gradmann (1993) reported about magnetism in ultrathin transition metal ﬁlms. He mostly discussed experimental results and his last sentence was: “ultrathin magnetic ﬁlms remain a promising ﬁeld for future research”. Similarly, Heinrich and Cochran (1993) discussed the magnetic anisotropies and exchange interactions. Moreover Bucher and Bloomﬁeld (1993) produced a review paper on the magnetism of free transition metal and rare earth clusters. Bucher (2003) updated the ﬁeld. Similarly, Bland and Heinrich (1994) published a two-volumes book concerning mainly multilayered systems. Also Bland (1997) reported on the growth and properties of ultrathin epitaxial layers. He pointed out that new impetus to this ﬁeld has been given by the application of scanning tunneling microscopy (STM). Later on Bland and Vaz (2003) reported on polarized neutron reﬂections studies on thin magnetic ﬁlms. It is worth mentioning the review by Himpsel et al. (1998) who put special emphasis on the relation between magnetism of nanostructures and the underlying electronic states, such as spin-split energy bands, sp versus d states,

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surface states and quantum well states. Zabel (1999) reviewed about the effect of spindensity-wave (SDW) magnetism of thin epitaxial Cr at surfaces, interfaces and in thin ﬁlms. Crommie (2000) discussed the utility of the STM as a spectroscopic probe of electronic properties for atomic-scale structures at metal surfaces. Actually STM spectroscopic imaging allows direct measurements of quantities, such as surface state scattering phase-shifts and transmission coefﬁcients that can be obtained in no other way. STM measurements of magnetic impurities provides direct, local measurement of electronic many-body effects that before could only be observed indirectly. Later, Wiesendanger and Bode (2001) discussed the results obtained on nano- and atomic-scale magnetism by using SP-STM/STS. Also Binns (2001) studied nanoclusters deposited on surfaces. Freeman and Choi (2001) reviewed advances in magnetic microscopy whereas Wolf et al. (2001) reported on spintronics: “A spin-based electronics vision for the future”. Bader (2002) gave selective examples to highlight three general areas of interest in the magnetism of low dimensionality: (i) characterization techniques, (ii) magnetic properties, and (iii) theoretical/simulational advances. Emerging directions including laterally conﬁned nanomagnetism and spintronics were also discussed. In the same issue number of Surface Science, Shen and Kirschner (2002) reviewed the most signiﬁcant progress in the last few years, in the effort of growing artiﬁcially structured magnetic materials (see also de Miguel and Miranda, 2002). Selforganization in magnetic materials is a currently considered topic. A good example can be found in the PhD thesis of Chado (2002) and in a paper entitled: “Absence of ferromagnetic order in ultrathin Rh deposits grown under various conditions on gold” (Chado et al., 2001). In their review, Bucher and Scheurer (2002) were primarily interested in magnetic particles that form spontaneously on surface during deposition of metal vapor. The synthesis and magnetic properties of organized metal hetero-structures on surfaces have been reviewed by those authors. In contrast to these static and quasi-static investigations, the wide range of dynamic properties of mesoscopic magnetic elements only starts to emerge with direct experimental evidence of spin dynamics in the time domain which allows to elucidate the switching mechanisms and the switching speeds on femtosecond timescales. The dynamics of small magnetic elements is expected to differ from that measured in single-layer magnetic ﬁlms due to large magnetostatic ﬁelds, shape and interface disorder and different magnetostatic excitation spectra (Hillebrands and Ounadjela, 2002). From a theoretical point of view, the ab initio description of magnetic multilayered ﬁlms has made tremendous progress since the development of the full-potential linearized augmented plane wave (FLAPW) method for thin ﬁlms proposed by Freeman et al. (1985). Applications of this method to interfaces, multilayers and thin-ﬁlm magnetism has been reviewed by Weinert and Blügel (1994), Blügel and Carbone (1996), Asada et al. (1999). On the other hand Dreyssé and Demangeat (1997) reported on transition-metal thin ﬁlms and nanostructures on semi-inﬁnite substrates by considering mainly semiempirical methods. Furthermore Binns et al. (1999) wrote a review paper, both experimental and theoretical, on the growth, electronic, magnetic and spectroscopic properties of transition metals on graphite. Wu and Freeman (1999) reported on the spin-orbit induced magnetic phenomena in bulk materials and their surfaces/interfaces. Later, Parlebas (2001) have focused on the electronic, magnetic and spectroscopic properties of manganese nanostructures. Calculations of the forces in the case of magnetic aggregates on substrates have been performed by Sander et al. (2002). In their review report entitled: “Stress, strain and

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magnetostriction in epitaxial ﬁlms” all Kohn–Korringa–Rostoker (KKR) calculations and quasi-molecular-dynamics calculation are performed within the constrained magnetic direction. The discovery of the sum rules has permitted the determination of the spin and orbital moments from the integrated XMCD spectra. XMCD can also be used to explore the magneto-crystalline anisotropy by determining the orbital moment anisotropy (Beaurepaire et al., 2001). Theoretically, the direction and magnitude of the magnetic moments are directly related to the spin-orbit component of the Hamiltonian. Reports on this subject are given by Wu and Freeman (1999), Alouani and Dreyssé (1999), Ebert (2000), Eriksson and Wills (2000), and Lazarovits et al. (2002) among others. The effect of temperature is important for the determination of the magnetic map of TM elements. In the bulk of Cr, Mn, Fe, Co and Ni, the magnetization is destroyed by temperature. Above the Curie (TC ), Curie–Weiss or Néel (TN ) temperatures the thermal ﬂuctuations of spins are so high that no magnetic moment can be measured. Nolting et al. (1995) and Vega and Nolting (1996) investigated the electronic quasiparticle structure of bcc Fe and fcc Co. Belonoshko et al. (2000) investigated the melting of hcp Fe at high pressure by employing molecular dynamics simulations in conjunction with full-potential linearized mufﬁn-tin orbitals (FP-LMTO). Also Lichtenstein et al. (2001) presented an ab initio quantum theory which combines the dynamical mean-ﬁeld theory (DMFT) with realistic electronic structure methods for the description of many-body features appearing in the derivation of the ﬁnite-temperature magnetism of Fe and Ni. In the case of surface or thin ﬁlms TC and TN decrease when the thickness decreases so that they deﬁne new values characteristic of the considered surface or thin ﬁlm and much differing from the bulk value. Various experiments and theoretical calculations tried to explain the connection between thickness and magnetic moment. Wilhelm et al. (2000) discussed the manipulation of the Curie temperature and the magnetic moments of ultrathin Ni and Co ﬁlms by Cu-capping. Pajda et al. (2000) constructed an effective two-dimensional Heisenberg Hamiltonian to estimate magnon dispersion laws, spin-wave stiffness constants and overlayer Curie temperatures. Also Razee et al. (2002) investigated the oscillation of the Curie temperature for thin fcc ﬁlms on and embedded in Cu(100) substrates. A ﬁrst principle formulation of the disordered local moment (DLM) picture was implemented by an adaptation of the selfconsistent ﬁeld KKR coherent potential approximation (SCF-KKR-CPA) method. A more simpliﬁed model has been proposed by Mokrani and Vega (2001) for the determination of the magnetic properties at ﬁnite temperature of Fe surfaces. Also Byczuk and Vollhardt (2002) derived the Curie–Weiss law in DMFT. Finally let us note the T dependence of Fe/Cu by Camley and Li (2000). At odds with this behavior are the magnetic properties of 3d impurities in noble metals or 3d adatoms (or small aggregates) on noble metals. In those cases there could appear a spin-rearrangement around the magnetic moment of the foreign 3d atom such as to kill the magnetic moment of it. Fluctuations have now an opposite role: they tend to recover the magnetization of the foreign atom so that above a Kondo temperature TK (Kondo, 1964), but below the Curie temperature TC , the system becomes magnetic. Within STM/STS many authors reported a strong variation of the local density of states (LDOS) near the Fermi level in connection with this Kondo effect. Variational theory developed for the description of small supported clusters, on the basis of the Coqblin–Schrieffer Hamiltonian (see, for example, Razaﬁmandimby et al., 1999), has been proposed by Kudasov and Uzdin (2002).

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This theory allows to take into account superposition of states in which some atoms of the cluster form Kondo singlets, whereas other conserve their magnetic moments. 1.4. Extended contents As we have seen in the Introduction, the magnetic multilayered systems are nowadays investigated by many experimental and theoretical groups. It is therefore a rather formidable task to report on all the results obtained and we need to restrict to a few points which are somewhat connected to the own work of the present authors. Before reporting on speciﬁc nanostructures we point out in section 2 the problems arising for a realistic description of the electronic and magnetic (scalar relativistic) properties of multilayers and clusters. We review on two types of approaches: (i) semiempirical methods for which the matrix elements of the Hamiltonian are given in terms of a few parameters. This method works in general in real space so that it is well adapted to nanostructures. (ii) Ab initio methods based on the DFT. Those methods need some kind of periodicity and are mainly restricted to perfect two-dimensional multilayered systems. However, these multilayered “perfect” systems cannot be grown easily and, in general, interdiffusion (at least one layer large) and roughness (steps or other imperfections) are typically present and should be included in any theoretical interpretation of the magnetic map of these systems. Semiempirical models are able to determine these “real” structures because they work in the real space. However they rely on parameters. A reasonable approach is therefore to perform an ab initio calculation on a periodic system with chemical and structural conﬁgurations very similar to the real one. Then we have to perform on the same system a semiempirical calculation the parameters of which are ﬁtted such as to recover the ab initio results. More and more semiempirical calculations are now done like that. We expect that the transferability of a set of parameters determined within a given chemical and structural conﬁguration towards another conﬁguration remains suitable. In section 3, after a short introduction concerning thin ﬁlms and multilayers, we report on Fe-based multilayered systems for various reasons: (i) the interlayer exchange coupling was ﬁrst detected in Fe/Cr superlattices and this system remains to be heavily studied. Results concerning brother systems like Fe/V and Fe/Mn are always compared to Fe/Cr when they are studied; (ii) the coupling of Fe with other elements varies from antiferromagnetic (for d elements at the beginning of the periodic table) towards ferromagnetic coupling for elements at the end of the d-series; (iii) Fe being a smaller atom as compared to Ag or Au, with surface energy larger than Ag or Au the well-known surfactant effect is present; (iv) Fe in its bcc conﬁguration is a good ferromagnet and when a non-ferromagnetic spacer is introduced between two Fe thick ﬁlms, the coupling between them can oscillate with the thickness of the spacer. However when Fe is itself a spacer between strong Ni or Co ferromagnets it stays antiferromagnetic-like and oscillations of the polarization of Ni (Co) through the fcc Fe take place; (v) Fe/Si systems are also interesting. In particular, Fe/c-FeSi thin ﬁlms and multilayers stand out among all such structures in which the spacer is semimetallic and are characterized by an unique behavior of the exchange coupling; (vi) interfacial roughness (interdiffusion, steps..) is discussed and reported. Finally we give a brief account on two effects: Hydrogen on the interlayer exchange coupling in Fe/V superlattices and oxygen on the Mn–Co coupling at the Mn–Co interface.

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Section 4 is devoted to the study of transition metals aggregates, mainly in a compact form (clusters) but also as nanowires. First section is devoted to an introduction concerning free and supported clusters on metallic or carbon substrates. For Cr and Mn the possible non-collinear magnetism showed up. Stern–Gerlach experiments by Bloomﬁeld et al. (2000) and Knickelbein (2001) displayed values of magnetic moments in-between classical antiferromagnetism and ferromagnetism. It appears that for both Cr and Mn the average magnetic moments are non-zero. A few words will be said on Kondo effect for adatoms on noble metals probed by Scanning Tunneling Spectroscopy (STS). The importance of selforganization is also pointed out. We report in section 4.2 on free-standing clusters. The remaining part of the chapter is concerned by the study of the magnetic map for an assembly of d atoms (from one to inﬁnity [in the case of nanowires]) adsorbed on (or embedded in) a substrate. Finally, in section 4.7 we say a few words concerning calculations of unconstrained spin magnetism in small Cr, Mn and Fe clusters deposited on non-magnetic metallic substrates. Section 5 is devoted to the conclusions and outlook. 2. Quantum-mechanical models 2.1. Introduction Achieving an accurate account of the electronic properties of low-dimensional transition metal (TM) systems requires a model precise enough to include the relevant quantummechanical features. At the same time, a large enough number of non-equivalent atoms must be considered to remove effects due to periodic boundaries. An overview of the quantum-mechanical models most suitable for electronic structure calculations in lowdimensional TM systems is presented in this chapter. These are one-particle methods based on Density Functional Theory (DFT) or semiempirical Tight-Binding (TB) approaches. Many of the DFT-based ab-initio methods that scientists use have been developed more than a decade ago. We can classify the DFT methods in two categories: (i) on one hand the full-potential methods with complete basis. Among the most accurate of such ab-initio DFT methods the Full-Potential Linearized Augmented Plane Waves (FLAPW) developed by Wimmer et al. (1981) has been widely used. However, they require a huge computational cost and their applicability is limited to periodic systems with few non-equivalent atoms. Therefore they cannot deal with the realistic systems that are experimentally produced nowadays. In the present review we will not focus on these ab-initio DFT methods mainly because many review reports have been published on this topic. On the other hand we consider in some detail those DFT methods with approximations at different levels (treatment of the core, potential and basis set mainly) and generally with some parameters, appropriately named “quasi ab-initio DFT methods”. The loss of some accuracy is overcome with the important gain in ﬂexibility. Methods like TB-LMTO, KKR-GF and SIESTA (Spanish Initiative for the Electronic Simulations with Thousands of Atoms) belong to this category and will be the topic of the two next subsections. DFT methods are continuously reﬁned and have beneﬁted from the increasingly availability of very fast computer facilities. It is not our purpose to describe the original formalism which can be found in the literature, but to focus on the approximations and features that make them suitable

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for the study of low-dimensional TM systems. The most common approximation is the use of a minimal basis set and a short range of interactions allowing to minimize the time spent on diagonalizing the Hamiltonian matrix which can be made sparse to a large extent. On the other side, the semiempirical TB method has proved to be of continuing interest for its combination of physical transparency, computational speed and surprising accuracy. Moreover, in its real space formulation it allows to study rather complex and realistic systems. As in the case of the DFT methods, the TB approach has been continuously reﬁned. The development of sophisticated quasi-ab-initio parameterizations for TB models have increased their transferability and accuracy. Although being yet one step behind “quasi abinitio DFT methods” from the point of view of accuracy, their computational cost is two or three orders of magnitude lower so that it is possible to treat complex systems which are out of the possibilities of “quasi ab-initio DFT”. The semiempirical TB method will be the topic of the fourth subsection. To summarize, we explore in the present section the state of the art of such scenario where “quasi-ab-initio DFT” and “quasi-ab-initio TB” methods coexist with the objective of describing low-dimensional TM systems, despite the fact of coming from opposite philosophies, accuracy and computational efﬁciency, respectively. 2.2. All-electron DFT methods The Linear Mufﬁn-Tin Orbitals method (LMTO) proposed in the seventies by Andersen (1975) has played a key role in the study of the electronic structure of periodic lowdimensional TM systems like surfaces, thin ﬁlms and multilayers, particularly in its Atomic Sphere Approximation (ASA). The mufﬁn-tin potential (Ziman, 1965) is approximated by a series of non-overlapping atomic-like spherical potentials, and a constant potential between the spheres. Schrödinger’s equation can be solved exactly in both regions. These solutions are matched at the boundaries of the spheres to produce mufﬁn-tin orbitals (MTOs). MTOs are used to construct a basis set which is energy independent, exact to linear order in energy, and rapidly convergent. Typically, nine standard LMTOs per site (spd basis set) give sufﬁcient accuracy for most transition metals (Andersen et al., 1986). This is a minimal basis set, which is what we would like. Unfortunately the MTOs are very long ranged and this makes the calculations slow. In 1984, Andersen and Jepsen introduced a localized LMTO basis set. A unitary transformation can be applied to the MTOs to render them short ranged. This new approach, called Tight-Binding LMTO (TB-LMTO) has given new impetus to the study of numerous physical properties of systems with large number of atoms. Later, a real-space formulation of the TB-LMTO-ASA method has been proposed for several studies (Spisak and Hafner, 2000a; Nogueira and Petrilli, 1999). The procedure is very similar to the usual reciprocal-space formalism, but for solving the eigenvalue problem in order to ﬁnd the local density of states, the k-space diagonalization is substituted by the real-space recursion method, thus allowing to deal with non-periodic systems. In the last decade, a variety of low-dimensional TM systems like impurities in bulk, adatoms or supported microclusters have been studied with models based on a developed Korringa–Kohn–Rostoker Green’s function method (KKR-GF) for defects at surfaces (Stepanyuk et al., 1994). Multiple scattering theory is applied to obtain the Green function in an angular-momentum representation from an algebraic Dyson equation. This Dyson equation contains the Green function matrix of the reference system (for example, the ideal surface if one is interested in adsorbed clusters, or the bulk if one is interested in impurities)

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plus a perturbation term that accounts for the changes associated with the modiﬁcation of the reference system. All angular momenta up to l = 3 are included. The potentials are assumed to be spherically symmetrical within the Wigner–Seitz spheres, although by multipole expansion the full charge density is taken into account. KKR-GF has been the ﬁrst all-electron DFT method used for the study of electronic structure and magnetic properties of supported microclusters. Readers interested in conceptual improvements of the KKR method can consult the review report by Papanikolaou et al. (2002). 2.3. Pseudopotential DFT methods SIESTA (Spanish Initiative for the Electronic Simulations with Thousands of Atoms) (Sánchez-Portal et al., 1997; Soler et al., 2002) is a self-consistent DFT method using standard norm-conserving pseudopotentials and a ﬂexible, numerical LCAO basis set, which includes multiple-zeta and polarization orbitals. The basis functions and the electron density are projected on a real-space grid. It uses a modiﬁed energy functional, whose minimization produces orthogonal wave-functions and the same energy and density as the Kohn–Sham energy functional, without the need of an explicit orthogonalization. The most basic approximations concern the treatment of exchange and correlation, and the use of pseudopotentials. Exchange and correlation (XC) are treated within Kohn–Sham DFT (Kohn and Sham, 1965), with either the local (spin) density approximation (LDA/LSDA) (Perdew and Zunger, 1981), or the gradient approximation (GGA) (Perdew et al., 1996). The method employs standard norm-conserving pseudopotentials (Hamann et al., 1979; Bachelet et al., 1982) in their fully non-local form (Kleinman and Bylander, 1982), including scalar-relativistic effects and nonlinear partial-core-corrections for XC in the core region (Louie et al., 1982). For TM, in particular when magnetism is concerned, the transferability of the pseudopotentials has to be carefully analyzed (Izquierdo et al., 2000). The basis is composed by conﬁned orbitals which are zero beyond a certain radius, as in the formalism of Sankey and Niklewski (1989), which is equivalent to put the atom into a slightly excited state. This keeps the energy strictly variational, thus facilitating the test of the convergence with respect to the radius of conﬁnement. 2.4. Semiempirical TB methods The semiempirical tight-binding method works by representing the Hamiltonian in an atomic-like basis, and replacing the exact many-body Hamiltonian operator with a parametrized Hamiltonian matrix. Thus, the basis set is not explicitly constructed. In general, only a small number of basis functions are used corresponding roughly to the orbitals present in the energy range of interest. For instance, when modeling 3d TM systems, only the 3d, 4s and 4p valence orbitals are usually considered in the calculation. It is possible to work in both orthogonal and non-orthogonal representations as well as in both the real and reciprocal-space, although the real-space formulation is much more appropriate for low-dimensional systems. The non-diagonal Hamiltonian matrix elements (three center hopping integrals) are obtained in terms of the two center hopping parameters following the key idea in the Slater and Koster (1954) approach. In this two-center approximation, the potential part of the Hamiltonian is replaced by the potential only due to the two atoms upon which the orbitals involved in the matrix element are located. The basis orbitals can

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be expanded as a sum of functions with well deﬁned angular momenta with respect to the axis between the atoms involved in the matrix element (in the hopping); for example, a p orbital can be expanded in pσ , pπ+ and pπ− . The value of the integral (at a given interatomic distance) between those expanded terms is a parameter. With these approximations, when compared with ab-initio methods, the self-consistent tight-binding method is two or three orders of magnitude faster, particularly when the Hamiltonian is solved using the recursion method (Haydock, 1980), and allows to deal with rather complex systems like atomic clusters supported or embedded in a host, systems with an extremely large number of inequivalent sites. The semi-inﬁnite systems can be simulated through the continued fraction of the recursion method with an appropriate terminator. However, the semiempirical tight-binding method suffers from some difﬁculty in the case of transferability relatively to the parametrization (see Lekka et al., 2002). In standard TB method, the bulk parametrization has been widely used. In this case, the two-center hopping parameters are obtained from a ﬁt to the ab-initio band structure of the bulk conﬁguration of the element under study (Papaconstantopoulos, 1986). The same holds for the Coulomb parameters. In the case of magnetic TM systems, the exchange integral is ﬁtted so as to give the bulk magnetic moment. For mixed systems this parametrization may not be adequate, in particular when interface effects are important or when one of the elements adopts a geometrical arrangement different from its bulk like in pseudomorphic growth. A ﬁt of the hopping integrals to an ab initio band structure of an hypothetical bulk of the same structure and interatomic distances as that adopted by the system under study constitutes an improvement over the standard TB. A great improvement in the transferability of the parametrization consists on ﬁtting the hoppings and Coulomb integrals to the TB-LMTO-ASA Hamiltonian (written in the TB form) of a well chosen system. This system must be chosen with two main conditions: (i) it has to be a system suitable to be treated within TB-LMTO-ASA method; (ii) it has to resemble the systems under study in the relevant aspects that have to be accurately accounted for. For example, for the study of Co clusters supported on Cu, one can ﬁt to the TB-LMTO-ASA results of Co monolayers on Cu (see Robles et al., 2002). This procedure leads to a set of parameters that implicitly include interface effects like Co–Cu hybridization, as well as surface effects. Another semiempirical approach called periodic Anderson model (PAM) has been used to describe both disorder at the interface between two TM metals (Kazansky and Uzdin, 1995) and unconstrained magnetization (Uzdin and Yartseva, 1998). PAM assumes that the electron spectrum may be described within a two-band approximation, i.e. with the itinerant sp and localized d subsystems. d electrons are usually represented by a Wannier approach whereas the delocalized electrons are described within a Bloch function representation. Details of the model can be found in the paper by Kazansky and Uzdin (1995). The method used comprises two key algorithms which are the random modeling of the epitaxial growth process and the iterative solution of the self-consistent system of equations for the occupation of both spin-polarized states for the atoms at each site. In section 4.7 we will say a few words on this non-collinear version of the PAM. 2.5. Dealing with the geometrical structure One of the most important characteristics intrinsically related to the electronic properties of TM systems is the geometrical structure of the material. It is well known for instance that

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the properties of itinerant 3d and 4d electrons, in particular the magnetism, are sensitive to changes in the position of the atoms within the system, so that spatial symmetry effects and variations of the local coordination number are ingredients to be taken into account in the calculation of electronic properties of low-dimensional TM systems. In principle, both the electronic and the geometrical properties should be determined together in the same calculation since they are not independent properties. There exist several DFT-based methods in which structural optimization has been performed in such a full self-consistent calculation. One example is the LCAO-MolecularOrbitals-DFT with pseudopotentials used by Reuse et al. (1995). Within the SIESTA code, forces and stresses can also be calculated accurately, thus allowing structural relaxation and molecular dynamics simulations (Diéguez et al., 2001). However, in the case of metals, DFT-based methods are limited to systems with very few non-equivalent sites due to the huge computational cost. In order to overcome this difﬁculty, much effort has been done in the last years for developing approaches able to scale linearly with the number (N) of atoms in the system under consideration (instead of the typical N 3 -scaling). A subﬁeld, appropriately named the order-N (O(N)) method, is thus emerging in the ﬁeld of computational condensed matter physics and material science. There are many different approaches to the construction of the O(N) procedures that are resumed in a report of Wu and Jayanthi (2002). Nevertheless, several problems still remain with the suitability of these O(N) methods for metals. It is well known that the variational O(N) methods produce large errors in the energy of metals (Bowler et al., 1997), since in both metals and narrow-gap semiconductors the density matrix has long-range correlations in real space compared to that of insulators with a wide gap (Ismail-Beigi and Arias, 1999). Therefore, in the case of metals, the development of efﬁcient O(N) methods can be considered still under investigation. One possible improvement to the ab initio computational algorithm permitting the treatment of small gap systems proposed by Goedecker (1999), is based on the Energy Renormalization Group method (Kenouﬁ and Polonyi, 2003). This improvement is linked to the projection methods developed by Feshbach in the sixties. In order to handle the complex low-dimensional TM systems usually produced in the experiments, there is little alternative but to perform semiempirical TB calculations. One possibility is to simplify the electronic description and combine it with a molecular dynamics algorithm (Andriotis et al., 1996). An expression for the repulsive term has to be proposed in this case (parametrized Born–Mayer form). When a spin-polarized calculation has to be performed, it is important to preserve the accuracy of the electronic description. Without simpliﬁcations in the electronic approach or in the self-consistency of the electronic calculation of a given atomic arrangement it is not feasible, even in semiempirical TB, to do molecular dynamics for complex systems. Here, a common and successful approximation is to separate the geometrical and electronic parts of the problem. The geometries can be previously determined through MD simulations using a many-body potential based in tight-binding theory whose suitability for the material under study has to be tested (Aguilera-Granja et al., 1998; Robles et al., 2002). Rather transferable many-body potentials based in TB theory for the geometrical optimization have been developed in the last years by ﬁtting their parameters to ab-initio calculations of a variety of systems different from the bulk or dimer (Levanov et al., 2000), in the same spirit as discussed in the previous subsection for the parametrization of the electronic part of the problem. This allows to

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perform very accurate studies of complex TM systems like clusters supported or embedded in a host (Robles et al., 2002). Finally, for some problems like ideal ﬁlms or multilayers, usually it is not necessary to perform a full geometrical optimization, but only to relax in a particular direction taking advantage of the high symmetry of the system. In such cases, the ab-initio full potential DFT methods have provided and still provide the most accurate results and serve as a test of the above mentioned approaches. 2.6. Dealing with non-collinear magnetic arrangements Spin-polarized calculations in low-dimensional transition metal systems have generally been conﬁned to a global quantization axis. This approximation provides successful results when it is justiﬁed as many materials exhibit collinear magnetic order, particularly those composed of strong ferromagnetic elements or adopting ideal structural conﬁgurations that prevent magnetic frustrations. However, there exist a variety of systems where the collinear magnetic arrangement cannot be the ground state or where it does not provide a satisfactory agreement with the experiments. A typical example is a frustrated system as a result of competing ferro- and antiferromagnetic interactions. Other candidates to noncollinear ground-state conﬁgurations are clusters of antiferromagnetic elements like Cr and Mn (Demangeat and Parlebas, 2002). Consequently, much interest has been directed towards the implementation of codes able to deal with non-collinear magnetism in both DFT and semiempirical TB methods. Within the DFT methods, both in all-electron and pseudopotentials approximations, the magnetization density has to be a continuous vector variable of position. Von Barth and Hedin (1972) local-spin-density theory implicitly allows for non-collinear spin arrangements, but Sandratskii and Guletskii (1986), and Kübler et al. (1988) were the ﬁrst to implement the non-collinear description in an electronic structure code. Sandratskii and Guletskii (1986) presented a generalization of the KKR method to non-collinear magnetism. Kübler et al. (1988) derived the effective single particle equations of non-collinear magnets based on density functional theory, which is expressed in terms of a 2 × 2 density matrix. This density matrix is transformed (using the Pauli spin matrices) in the equivalent magnetization vector (variable of position). Following Kübler et al. (1988), Hobbs et al. (2000) described how a vector magnetization density may be included in the all-electron projector augmented wave (PAW), based on a generalized local-spin-density theory. The algorithms described in their work have been implemented within a package called VASP (Vienna ab-initio simulation package) and are used to study non-collinear magnetic structures in small free-standing TM clusters. Pseudopotential DFT schemes for non-collinear magnetic structures have also been proposed. Oda et al. (1998) developed a plane-wave method that has been applied to small free-standing clusters. Non-collinear spin polarized systems can be also studied with SIESTA, where a local minimal basis is used. At present, non-collinear DFT calculations are limited to systems with few non-equivalent sites (Kohl and Bertsch, 1999; Hobbs et al., 2000). Non-collinear magnetic calculations of complex low-dimensional systems with many non-equivalent sites are performed currently in the framework of a semiempirical realspace TB method. Even if a system has periodicity from the structural point of view, a

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non-collinear magnetic conﬁguration imposes a much larger period (magnetic periodicity) and this is the reason why a method based on a transformation in the reciprocal space is not suitable for such complex systems. Stoefﬂer and Cornea (1998) and Robles et al. (2003b) developed a non-collinear real-space TB formalism. The recursion method is used for solving the Hamiltonian that contains a band term and an exchange term (magnetic part). The recursion method provides the partial densities of states projected on a local quantization axis through a rotation in the spin space. This method allows to describe accurately noncollinear magnetic structures in extended systems like multilayers, defects at surfaces or clusters supported on a host. Another non-collinear approach, based on the PAM model has been applied to the study of interfaces (Uzdin and Yartseva, 1998) and adsorbed clusters on noble-metal (Uzdin et al., 1999). A detailed description of this approach as well as applications to Cr, Mn and Fe trimers on Ag(001) is reported in section 4.7. 3. Thin ﬁlms and multilayers 3.1. Introduction The magnetic multilayered systems are so numerous that one cannot report on all of them in the present review. Fortunately there have been a good number of both experimental and theoretical reports in this ﬁeld (we have listed some of them in the Introduction) so that we can essentially restrict ourselves to Fe-based-transition-metal systems. Why Fe? There are of course several reasons to choose Fe: (i) Fe metal in its ground state has a bcc structure with a ferromagnetic conﬁguration. It can easily polarize many of the transition metal and rare-earth elements. This induced polarization could be either ferro- or antiferromagnetic. Since Grünberg et al. (1986) Fe based transition-metal multilayers have been studied by various groups and the results obtained can be extrapolated – in many cases – to other transition-metal based multilayered systems. A good knowledge of these Fe-based systems considered as prototypes are of utmost interest for people involved in the ﬁeld of magnetic multilayered systems; (ii) Besides its bcc ground state conﬁguration (Wang et al., 1985) Fe can be stabilized in fcc and hcp bulk phases. The fcc phase has clearly a non-ferromagnetic ground state but its exact magnetic conﬁguration is still under debate. However, calculations seem to converge towards non-collinear multiple spin density waves (Kakehashi et al., 2002). In contact with strong ferromagnetic metal like Co or Ni, Fe atoms in fcc phase present a ferromagnetic induced conﬁguration which is one monolayer thick (Hadj-Larbi et al., 2002); (iii) What about the hcp phase of Fe metal? Spisak et al. (2001) discussed the magnetism of hcp Fe and the induced polarization of Ru in Fe/Ru superlattices. Perjeru et al. (2000) investigated the properties of Fe/Re multilayers (hcp conﬁguration) with magneto-optic Kerr effect (MOKE) and XMCD. Zenia et al. (2002) analyzed this point via TB-LMTO. LDA may be critical in the description of magnetic ordering in Fe (Wang et al., 1985). In their paper Wang et al. (1985) found that there are fundamental deﬁciencies in the LDA, both quantitative and qualitative so that only GGA is able to obtain the experimental bcc ground state. The unstable fcc phase of Fe can be stabilized on Cu(100) (Biedermann et al., 2001; and references therein). Those authors studied the nucleation centers of bcc structure

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in ultrathin fcc Fe ﬁlms deposited at room temperature on Cu(100). These nucleation centers happen to be narrow and elongated bcc crystals (needles), which are commensurable with the fcc lattice and therefore substantially strained. Benedek et al. (2001) measured the Rayleigh wave (RW) dispersion curves for these thin fcc Fe ﬁlms of a few monolayers grown on Cu(100). The temperature dependence of RW frequency is considerably affected by the ferromagnetic transition and is used as a direct probe of the interplanar magnetic forces. This fcc-bcc transformation, called Bain path, was studied by Qiu et al. (2001) and by Friak et al. (2001) for tetragonal bulk Fe. These authors reported a complete bunch of various magnetic conﬁgurations versus c/a. Later Spisak and Hafner (2002a) demonstrated that along the Bain path, tetragonal Fe is unstable against monoclinic shear deformation producing a nearly bcc structure. However, in the limit of a monolayer of Fe on Cu(100), the epitaxial constrain suppresses the shear instability which takes place for a greater thickness of Fe. Before examining the magnetism of the various Fe surfaces, let us just mention that the relaxations and stabilities of (100), (110) and (111) surfaces of bcc Fe were calculated by Spencer et al. (2002) using DFT. Ohnishi et al. (1983) reported within FLAPW the ﬁrst realistic calculation of the magnetic polarization at the surface of Fe(100). Their magnetic moment result of 2.98μB is similar to the one obtained by Alden et al. (1992) within TB-LMTO (2.97μB ). For the other crystallographic faces with Low-Miller indices, lower moments were obtained: 2.57μB for Fe(110) by Alden et al. (1994) and 2.73μB for Fe(111) by Wu et al. (2000). It is clear from these results that the magnetism on the surface atoms depends not only on the coordination number ((100) and (111) surfaces have the same number of nearest neighbors) but also on other details of the atomic arrangement. In order to check this point Vega et al. (1992) reported on stepped Fe surfaces (103), (105) and (107). For these speciﬁc surfaces the coordination rule was respected. Later on Geng et al. (2001) wrote on the magnetization at the surface of Fe(310) within FLAPW. The spin magnetic moment of the Fe(310) surface and subsurface atoms is enhanced to 2.85μB and 2.65μB , respectively, from a bulk value of 2.23μB . The signiﬁcant enhancement of the moment in the subsurface (absent in the case of Fe(100) surface) is explained by its loss of two nearest neighbors during the formation of the surface. In section 3.2 we give an extended overview of the multilayered systems described in this report. In general we focus on Fe based multilayered systems. Beside Fe and, from a magnetic point of view, Cr and Mn are also very complex systems. The incommensurate spin density wave (SDW) magnetism of Cr fascinated many researchers since its ﬁrst discovery via neutrons scattering in 1959. An excellent and extensive review of the bulk properties of Cr was given by Fawcett (1988). Extension to thin ﬁlms was reported by Zabel (1999). Hobbs and Hafner (2001) determined all polymorphs of Mn. The long-range magnetic ordering at the surface of ultrathin ﬁlms of Cr, Mn and Fe on Fe(100) has been studied by electron capture spectroscopy. The observed spin polarization by Igel et al. (1999) indicates different magnetic orderings. 3.2. Iron based multilayered systems: general overview of a great variety of magnetic phenomena In this section we present a general overview of the most pertinent magnetic phenomena discussed in section 3. However, due to the rather formidable number of papers concerning the ﬁeld of Fe-based magnetic materials we will focus on a few types of interfaces:

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(i) Fe/Cr, (ii) Fe/V, (iii) Fe/Mn, (iv) Fe/Ni, (v) induced polarization of Rh, Pd and W as speciﬁc examples of 4d and 5d elements, (vi) a critical overview of self-surfactant systems: Fe on Cu, Ag and Au, (vii) effect of a triangular substrate on the magnetism of Fe: Fe on graphite, (viii) tunnel effect: Fe/Si interfaces, and (ix) effect of a metalloïd on the coupling of Fe with antiferromagnet and on the interlayer exchange coupling in Fe ﬁlms separated by non-magnetic material. Since the work of Grünberg et al. (1986), more than 1000 papers and many reports have focused on the Fe/Cr interfaces (Zabel, 1999; Pierce et al., 1999). Therefore, we will restrict here to the following speciﬁc aspects: (i) stepped interfaces and interfacial alloying; (ii) the minimization of the frustration of the moments of Cr atoms via a vector-description of the magnetism; effect of alloying on the π -phase-shift of the interlayer exchange coupling (IEC). The Fe/V system has gained considerable interest mainly since Hjörvarsson et al. (1997) showed that the IEC can be tuned by hydrogen. A more older interest goes back to Akoh and Tasaki (1997) whose measurements on V clusters could be explained via some kind of polarization. Therefore, before presenting the results obtained in the case of Fe/V interfaces we will point out the difﬁculties to reach any agreement about a possible magnetism at the surface of a semi-inﬁnite V slab. Discussions concerning the effect of H on Fe/V will appear in section 3.11. Mn is a very speciﬁc element, the electronic and magnetic description of which is difﬁcult. Results obtained by Hobbs and Hafner (2001) within GGA-DFT are of substantial interest. However, the strong correlations present in Mn cannot be really described by a classical band structure approach. There is clearly a lack of realistic description of Mn bulk. Also a controversy has appeared between various X-ray magnetic circular dichroism (XMCD) results about the sign of the polarization for a monolayer thick Mn on Fe(001) (see, for example, Bischoff et al., 2002a, 2002b). More details can be found in a review report by Demangeat and Parlebas (2002). The three previously considered Fe/Cr, Fe/V and Fe/Mn systems are predominantly of bcc (or bct) type. Now, for Fe/Ni interfaces, we mainly consider a rather thick Ni slab and a few number of Fe layers so that the system behaves predominantly like an fcc (or fct) system. Fe mostly remains of antiferromagnetic type but induced ferromagnetism is present at the Fe/Ni interface. The results concerning the interface between a fcc ferromagnet and Fe seem very scarce, theoretically as well as experimentally. More should be done on that subject in order to clarify the magnetic map of fcc Fe (Sjöstedt and Nordström, 2002). 4d and 5d elements present essentially a non-magnetic behavior. However, Rh and Pd can be singled out because, on one hand, Cox et al. (1993) have seen within Stern–Gerlach experiments that Rh clusters containing a small number of atoms are magnetic. A systematic study of the structural and magnetic properties of free-stang clusters can be found in Aguilera-Granja et al. (2002). On the other hand, Pd presents a high susceptibility so that it can be considered as a nearly magnetic element. Vogel et al. (1997) measured Pd L2,3 edges in Pd/Fe multilayers with various Pd interlayer thicknesses and found that, in the atomic layer adjacent to Fe, the Pd atoms present a local structure different from fcc. Also, the corresponding XMCD showed that the Pd atoms situated at the interface are strongly polarized with a total moment (mostly spin-moment) of about 0.4μB /atom. Finally the thickness dependence of XMCD spectra pointed out that Pd atoms carry a magnetic moment up to

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4-layers from the interface. W has been taken into account here because Wilhelm et al. (2001) claim that it violates Hund’s third rule. Considerable studies are also devoted to the growth and the determination of the magnetic properties of Fe on noble metals (Cu, Ag, Au), for various reasons. The ﬁrst reason is probably that these systems were considered as the realization of a nearly perfect twodimensional magnetism because of the (expected) small hybridization between the Fe 3d orbitals and the substrate sp electrons. This is probably not true because the surface energies of Fe is so different from noble metals that exchange of atoms between the adsorbed Fe atoms and the substrate can take place. Another interest is the fact that Fe grows mainly fct on these substrates so that the magnetization of fcc Fe can be studied without great perturbation from these non-magnetic atoms in the substrate. A very complex magnetic behavior of Fe is therefore expected and seen in the publications. However as shown in many theoretical calculations concerning the electronic and magnetic map of Fe on these substrates, the interfacial alloying between Fe and Cu, Ag or Au substrates is generally not taken into account. Therefore a comparison of these calculated values with the experimental results cannot be realistically performed. A few calculations incorporated such alloying but in a very approximate way. Much more detailed studies of these atomic exchange phenomena should therefore be performed until a realistic theory-experiment agreement could be obtained. The hcp phase of Fe is expected to take place on the (111) crystallographic surface of noble and transition fcc metals. However it looks to be more realistic to ﬁnd it when it grows on graphite (Binns et al., 1999) or in Fe/Re superlattices (Perjeru et al., 2000). Within DFT Zenia et al. (2002) showed that the Fe atoms present a ferromagnetic conﬁguration in hcp Fe/Re superlattices, in reasonable agreement with XMCD results of Perjeru et al. (2000). Binns et al. (1999) reported on theoretical and experimental aspects of the Fe/graphite interface. The graphite presents a surface with a triangular symmetry so that collinear antiferromagnetic conﬁguration cannot be stabilized because of topological frustrations. Besides this interesting topological frustration at the interface with carbone-graphite, other interfaces between magnetic transition metal (TM) elements and new forms of carbon could be very interesting too. In section 4.4 we will say a few words concerning aggregates of magnetic TM encapsulated in fullerenes and nanotubes. (Fe/c-FeSi) magnetic multilayer devices (MMD) stand out among all such structures as being MMDs in which the spacer is semimetallic. Indeed, experimental results of Briner and Landolt (1994), Mattson et al. (1993), Inomata et al. (1995), Chaiken et al. (1996), de Vries et al. (1997) and Strijkers et al. (2000) showed that the exchange coupling of these devices is always AFM. However the exchange coupling has a fairly large decay length, becoming negligible at spacer thicknesses more suitable for a metallic than for a semiconducting spacer. Gareev et al. (2001) found an oscillating behavior of the exchange coupling. This unique behavior of J (z) might also make MMD in this class strongly attractive for manufacturers interested in the design of spin polarized transport devices. Since the work of Hjörvarsson et al. (1997) showing that the interlayer exchange coupling in the Fe/V superlattices can be switched from AF to FM and vice versa upon introducing hydrogen into the V layers, renowned interest about the effect of residual metalloid in ultrahigh-vacuum chambers has come out. Andrieu et al. (1998) showed that the magnetic properties of Mn ultrathin ﬁlms grown on (001)Fe actually depend drastically on the

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oxygen contamination. Besides these speciﬁc examples of magnetic multilayered systems presented before and developed in section 3, there are also other Fe-metal systems. Fang and de Groot (2000) calculated Al/Fe interfaces. 3.3. Fe/Cr. A ferro/antiferromagnetic interface As pointed out in section 3.2, it is rather difﬁcult to be exhaustive concerning the work devoted to the study of the Fe/Cr interfaces. In this section we will report essentially on two points: (i) the interlayer exchange coupling (IEC) and (ii) the study of the interface alloying. In fact, it will be seen that the IEC depends strongly on the alloying present at the Fe/Cr interfaces. Most of the calculations were performed by supposing collinear magnetism. However, as pointed out by Fawcett (1988) bulk Cr presents spin-density wave (SDW) coupling with a rather long period so that it cannot survive in thin Cr spacers. Moreover the Cr moment (contrary to Mn moment) is not robust. It can be easily quenched by its neighboring atoms and its direction of magnetization is dramatically dependent on the symmetry of its environment. The Fe/Cr multilayers present clearly two periods of oscillation of its IEC. The long period of about 18 Å is measured in epitaxial and sputtered superlattices and does not need for observation any special requirement concerning the interface roughness. The short-period of 2 monolayers was observed only on samples grown on the Fe whiskers with very ﬂat interfaces (Unguris et al., 1991, 1992; Heinrich et al., 1999) and samples prepared with optimized temperature regimes of epitaxial growth (Schmidt et al., 1999). At ﬁrst sight the theoretical explanation of this short-range IEC oscillations seems to be related to the antiferromagnetic coupling between nearest Fe–Cr and Cr–Cr monolayers. Thus, for a Cr spacer with an odd number of layers, the magnetic moments of the Fe layers on its both interfaces will order parallel and the IEC will be ferromagnetic (FM). If the Cr spacer has now an even number of layers, the interlayer exchange coupling will therefore be antiferromagnetic (AF). This was proved in both ab initio (Herman et al., 1991) and semiempirical theories (Stoefﬂer and Gautier, 1990; Vega et al., 1991). Surprisingly, scanning electron microscopy with polarization analysis (SEMPA) (Unguris et al., 1991, 1992) and Brillouin light scattering (BLS) (Heinrich et al., 1999) showed that the phase of this short-wavelength oscillation is exactly opposite, i.e. FM coupling was observed for an even number of Cr layers whereas Fe layers separated by an odd number of Cr monolayers were coupled AF. The discrepancy between experiment and theory was not only restricted to this π phase-shift but also to the value of exchange itself, which in experiment was found to be much less than that obtained from ﬁrst principle calculations. Freyss et al. (1997) proposed to introduce some kind of alloying for Cr ﬁlms on Fe substrates. They show that, in the model with two mixed layers, the exchange of one quarter of a monolayer of Fe and Cr is enough to reverse the layer by layer AF structure of a 10 ML thick Cr ﬁlm. Similar results were obtained by Turek et al. (2001) within the semiempirical tight-binding method and ab initio TB-LMTO method in a CPA approach. However, introduction of alloyed layers at the second Fe/Cr interface in the Fe/Cr/Fe trilayer leads to an additional π -shift of the phase so that trilayers with intermixing on both interfaces and with ideally sharp interfaces demonstrated the same behavior (Freyss et al., 1997). The absence of interdiffusion for Fe deposited on a Cr substrate was postulated by Heinrich et al.

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(1999). To introduce alloying at this interface Heinrich et al. (1999) used codeposition of Cr with Fe for one ML at the Fe on Cr substrate, i.e. they replaced the Cr monolayer at the top of the Cr spacer by an Fe–Cr alloy. The results concerning the IEC measurements were quite surprising: they found that even a concentrations of Fe in the alloyed layer as small as 15% is enough to reverse the sign of the IEC. This is in contradiction with the theoretical calculations of Freyss et al. (1997) and Turek et al. (2001) showing that for Fe concentration less than 50% the Cr-alloy monolayer behaves like a pure Cr atomic layer. Alloying between Fe and Cr is present at both Fe/Cr and Cr/Fe interfaces. Direct observation by means of STS of Cr intermixing with a Fe substrate was reported by Davies et al. (1996). They showed that, in the initial stage of Cr deposition on an Fe whisker, only one from every four deposited atoms stays at the surface whereas the other three penetrate into the Fe substrate. Ab initio calculations based on a KKR-Green function method within LDA by Nonas et al. (1998) demonstrate a strong tendency for a direct site-exchange mechanism into the ﬁrst surface layer for Cr on an ideal Fe substrate in agreement with the results of Davies et al. (1996). For Fe deposited on a Cr spacer the situation is more complex. In Mössbauer experiments the distribution of hyperﬁne ﬁelds (hff) from samples with 2 ML thick 57 Fe-probe layers at the Fe/Cr and at the Cr/Fe interfaces were found essentially different (Shinjo and Keune 1999). It was interpreted as the suppression of intermixing at the Fe on Cr substrate. However more careful measurements together with calculations of magnetic moment distribution for particular structures does not conﬁrm this point of view (Uzdin et al., 2001a). A STM study of Fe on a Cr substrate (Uzdin et al., 1999a) also demonstrates strong alloying during the deposition of Fe on Cr. Moreover, analysis of the hyperﬁne ﬁelds (hff) on 57 Fe atoms shows the lack of symmetry between both sides of the interface (Uzdin et al., 2002c). Such asymmetry has been explained within a new scenario of the epitaxial growth (Uzdin, 2002), which presupposes the ﬂoating of some atoms up to the sample surface during the deposition of the next layer. This stochastic algorithm based on “ﬂoating” scenario was used to relate the phase of the IEC with the asymmetric alloying (Uzdin and Demangeat, 2002). Vega et al. (1994, 1995) discussed the onset of topological antiferromagnetism at the Cr vicinal surfaces (with terraces of one monoatomic height) and the corresponding interfaces with Fe through a Hubbard-like tight-binding model. These type of stepped interfaces are present in the typical wedge-shaped samples analyzed by the experimentalists in order to investigate the interlayer exchange coupling in Fe/Cr/Fe systems. For Cr vicinal surfaces, the magnetic arrangement is a double-cell conﬁguration which consists of monoatomic steps antiferromagnetically coupled to nearest-neighbor monoatomic steps (see ﬁg. 3.1). In this way, the average magnetization at the surface is zero although high local magnetic moments are present. For the Fe monolayer on vicinal Cr surfaces, a magnetic conﬁguration with the periodicity of two steps was also obtained. As a consequence, the sign of the magnetization at the Fe overlayer oscillates from step to step, as illustrated in ﬁg. 3.2. In this case, several magnetic solutions associated to magnetic frustration near the step were obtained. Therefore, this system is a natural candidate to exhibit a non-collinear magnetic conﬁguration. We have seen that the Fe–Cr coupling at Fe/Cr interfaces for thin ﬁlms of Fe (Cr) on Cr (Fe) substrates and Fe/Cr superlattices is, in general, of antiferromagnetic type. It is however of ferromagnetic type in the ordered B2 FeCr alloy which can be considered

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Fig. 3.1. Schematic view of the stepped (001) surface of Cr. The nearest-neighboring steps A and B present magnetic moments with opposite polarization.

Fig. 3.2. Local magnetic moments, in units of Bohr magnetons, for the most stable solution obtained by Vega et al. (1995) for the Fe overlayer on the stepped Cr(001) surface. The white circles represent the Cr atoms, whereas the dashed circles represent the Fe atoms. Notice that only a portion of the semi-inﬁnite system in its bcc structure is illustrated as a projection on the (010) plane. From Vega et al. (1995).

in the (001) direction as a Fen /Crm superlattice with n = m = 1. Amalou et al. (1999) have performed TB-LMTO-GGA calculations for the determination of a magnetic map at the (001) and (111) surfaces of B2 FeCr alloy. For both surfaces, non-ferromagnetic conﬁgurations are shown to be more stable than the ferromagnetic conﬁguration of the bulk alloy. The ferromagnetic polarization of Cr by Fe remains generally small whereas antiferromagnetic polarization could lead to very high moment (see, for example, KKRCPA results for Cr1−x Fex disordered alloys by Kulikov and Demangeat, 1997). Up to now the calculations have been performed with the constrain of collinear magnetism. However the calculations of magnetic order in collinear magnetism that take into account roughness and interdiffusion demonstrate the existence of a number of selfconsistent solutions close in energy. At the same time, the existence of frustrations even for relatively simple stepped interfaces indicates that the ground state of real systems is most probably non-collinear. Thus, the development of a theory for a vector description of the interface magnetism is a fundamental problem for the true interpretation of experimental data. Uzdin and Yartseva (1998) have proposed a method based on the periodic

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Anderson model (PAM) in the Hartree–Fock approximation within a real space recursion method. For the description of non-collinear magnetism they rewrite the PAM Hamiltonian in terms of spin quantization along a global z-axis, which is the same for all atoms. This leads to additional on-site hoppings with spin inversion, which depends on the angles between the magnetic moment and spin-quantization direction. The non-collinear magnetic structure for stepped Fe/Cr superlattices was obtained by Uzdin et al. (1999). The ground state is always non-collinear and the angle between the magnetic moments of the Fe layers depends on the thickness of the Cr and Fe layers. Also, Cornea and Stoefﬂer (2000) and Stoefﬂer and Cornea (2001) obtained non-collinear magnetic structures in Fe/Cr superlattices by considering atomic steps in the Cr layers. Finally, Lauter-Pasyuk et al. (2002) have obtained direct evidence of a non-uniformly canted state of the spin-ﬂop phase induced by a magnetic ﬁeld applied to Fe/Cr(100) superlattices, by polarized neutron reﬂectometry. It is the ﬁrst direct experimental evidence of a twisted ground-state conﬁguration realized in AF exchange coupled multilayers exposed to an in-plane external magnetic ﬁeld above the spin-ﬂop transition. 3.4. Fe/V. A controversial system. Induction of magnetic moments Vanadium, contrary to Cr (see section 3.3) is non-magnetic in the bulk. However, on one hand, Hattox et al. (1973) showed that V undergoes a transition from a paramagnetic (P) to a ferromagnetic (FM) conﬁguration as the lattice parameter is increased. On the other hand, Akoh and Tasaki (1997) found small V clusters to be magnetic. Also, Rau et al. (1986) through electron-capture spectroscopy concluded to the existence of ferromagnetic order at the V(001) surface at odds with FLAPW results of Ohnishi et al. (1985). The results of Ohnishi et al. (1985) were corroborated by other ab initio calculations until the work of Bryk et al. (2000) claiming a magnetic moment of 1.7μB at the surface of V. Blügel et al. (1989) conﬁrmed the ferromagnetic behavior of the V monolayer (Ohnishi et al., 1985) and found that an overlayer of V epitaxially grown on Ag(001) presents a stable in-plane AF solution in disagreement with the results of Fu et al. (1985). In order to clarify these points, Bouarab et al. (1992) performed a systematic study of V slabs from one to ﬁve planes. In the monolayer, a competition between in-plane antiferro and ferromagnetism is present, but a total-energy calculation versus the lattice parameter indicates the stabilization of the paramagnetic state. For thicker slabs, layered antiferromagnetic structures are present for the bulk lattice parameter for (001) and (111) crystallographic faces; for the (011) face, however, only in-plane antiferromagnetism is present. Also, V epitaxially grown on Ag(001) exhibits in-plane antiferromagnetism for one monolayer, but layered antiferromagnetic structures for thicker slabs. However, Finck et al. (1990) within MOKE failed to detect unambiguous manifestations of magnetism in ultrathin epitaxial V ﬁlms on Ag(001) substrates. This discrepancy may be explained by some “surfactant effect” (see section 3.8 for details) as usually shown when a 3d-element is deposited on Ag or Au. Later on Hamad et al. (2001) investigated the magnetization of a V monolayer on a stepped Mo surface by considering a perfect epitaxial growth. Large magnetic moments were obtained at odds with experimental investigations by Korenivski et al. (1997) displaying a low moment in Mo/V multilayers. The discrepancy may be related to many reasons like the neglect of relaxation in the calculation or a possible exchange between V and Mo atoms. To conﬁrm this point, Ponomareva et al. (2002) have studied by means of

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the ﬁrst-principles LMTO-GF-CPA and ﬁxed spin moment methods in conjunction with statistical Monte-Carlo method, the surface magnetism of vanadium-based disordered binary alloys V–Pd and V–Mo. Both systems are nonmagnetic in the bulk. However, the (100) surface of uniformly random alloys is magnetic, and magnetic moments may also be present in segregated V–Mo alloys. Before closing this section concerning V multilayered systems (ﬁlms, surface or supported thin V ﬁlms on non-magnetic substrates) let us focus in more detail on one of the main controversies, that concerning the possible onset of magnetization at the surface of V(001). Bryk et al. (2000) performed an ab-initio calculation for a seven layer V(001) ﬁlm in order to simulate the (001) surface, using a plane-waves method with pseudopotentials and the generalized gradient approximation (GGA). They obtained a magnetic moment of 1.7μB at the unrelaxed surface (outer plane) and 1.45μB when relaxed. Robles et al. (2001a, 2001b) studied the V(001) surface by using different LSDA and GGA approximations to the exchange correlation potential within two DFT methods: the all-electron TB-LMTO and the pseudopotentials LCAO code SIESTA. The surface was simulated with a N -layer V(001) ﬁlm (7 N 15). The LSDA approximation led to a non-magnetic V(001) surface with both theoretical models. The GGA within the pseudopotential method needs thicker slabs than the LSDA to yield zero moment at the central layer, giving a high surface magnetization (1.7μB ), in contrast with the non-magnetic solution obtained with the all-electron code. Other pseudopotential and FLAPW calculations by Batyrev et al. (2001) showed that both methods yield different surface relaxation and thus, different values for the magnetic moment. What is clear after all those works is that one has to be extremely careful when doing calculations of V systems. It is necessary to test in detail the pseudopotential. Results by Robles et al. (2003a) show, for instance, that treating the semicore states like 3p as valence states in the pseudopotential methods greatly improves the surface relaxation. Besides, a large enough ﬁlm has to be used for correctly simulate the surface in such supercell-based methods. Perhaps, the last experimental result related to this subject is the one of Bischoff et al. (2002a) showing with STS measurements that the clean V(001) surface is not magnetic. Less controversial appears the case where vanadium is in contact with a ferromagnet, as in Fe/V multilayered structures. Analysis of the distribution of the hyperﬁne ﬁeld of an Fe/V superlattice through a nuclear magnetic resonance (NMR) experiment (Takanashi et al., 1984) is compatible with an AF coupling between Fe and V at the interface, as obtained by Hamada et al. (1984) within LAPW. They found that the magnetic moment per atom of Fe at the interface is appreciably reduced from its bulk value, and that a small negative moment is induced on V at the interface. Conversion electron Mössbauer spectra of composition modulated Fe/V ﬁlms by Jaggi et al. (1985) do agree with the results of Hamada et al. (1984) concerning the decrease of the Fe magnetic moment at the Fe/V interface and the AF coupling between Fe and V at the interface. Vega et al. (1991) within the TB-approach conﬁrmed the strong AF coupling at the Fe/V interface for the (001) crystallographic interface. However, contrary to the results of Hamada et al. (1984) the polarization of the V layers are AF like. According to Hamada et al. (1984) and Vega et al. (1991) it was concluded that: (i) the magnetic moment of Fe at the Fe/V interface is diminished as compared to bulk Fe; (ii) an induced moment on the atoms is obtained; (iii) at the interface the coupling between Fe and V is AF. However, according to Hamada

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Fig. 3.3. Magnetic proﬁles of Fe5 Vn (n = 1, 3, 5, 7, 9) superlattices in the (100) and (110) orientations. The black bars correspond to the Fe monolayers and the white ones to the V monolayers. From Izquierdo et al. (1999).

et al. (1984) all the V atoms are polarized antiferromagnetically with Fe whereas Vega et al. (1991) display an AF polarization in the V spacer and in thin V ﬁlms on V(001). This point has been discussed speciﬁcally by Izquierdo et al. (1999) within TB-LMTO. For Fe5 Vn (n = 1, 3, 5, 7, 9) superlattices they obtained an induced short-range polarization antiferromagnetically coupled with Fe (see ﬁg. 3.3), in agreement with Hamada et al. (1984) and therefore at odds with Vega et al. (1991). For thin V ﬁlms on Fe(001) and Fe(110) Izquierdo et al. (1999) displayed however some kind of oscillatory behavior in the V layers in agreement with Vega et al. (1991) and with experimental results of Fuchs et al. (1996).

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The point concerning the short-range polarization of the V atoms in Fe/V superlattices has been probed by Scherz et al. (2002a). They reported on in situ XMCD experiments at the Fe and V L2,3 edges on Fe/V/Fe(110) trilayers. Their results show that the induced V magnetic moment, which couples antiparallel to the Fe one, decreases rapidly away from the interface indicating a short-range polarization in the V layers. This ﬁnding conﬁrms ab initio calculations by Izquierdo et al. (1999). From the careful XMCD result of Scherz et al. (2002a, 2002b) it could be said that there is some kind of agreement concerning the short-range polarization in V. However one point is not completely settled: it concerns a possible oscillatory polarization in the V layers. As seen by Fuchs et al. (1996) and conﬁrmed theoretically by Izquierdo et al. (1999) this oscillatory induced polarization appears when a thin V ﬁlm is adsorbed on Fe and generally not in the case of Fe/V multilayers. However, as shown by Vega et al. (1991) this oscillatory polarization seems to be present when a semiempirical TB approach is applied to Fe/V superlattices. Also this type of layered antiferromagnetic polarization has been found by Bouarab et al. (1992). We may discuss the suitability of this TB approach for the description of the V polarization but, it has been shown by Moruzzi and Marcus (1990) within ASW method that V bulk presents a second-order transition from the nonmagnetic ground state to the preferred antiferromagnetic state when the lattice parameter increases. Therefore it is clear that the antiferromagnetic state of V is preferred over the ferromagnetic state. On the other hand, as discussed by Izquierdo et al. (1999) in the particular case of Fe/V multilayers it seems that all the V layers are aligned antiferromagnetically with Fe. Thus, the V layers in the spacer of the Fe/V superlattices present a clear ferromagnetic-type of magnetization. It seems therefore that the induced magnetic polarization by Fe stabilizes the ferromagnetic state of V which is unstable against the antiferromagnetic state in bulk V. Fe and V are elements which are essentially miscible so that when V is grown thermally on Fe substrates, or Fe on V substrates, Fe–V alloy can form within a few layers. Growth and chemical composition of V ﬁlms on Fe(100) were studied for temperature between 370 and 620 K by Igel et al. (2000). They observed a pronounced temperature-dependent alloying. At some speciﬁc temperature an ordered B2 FeV alloy can form within a few layers. As a result the electronic structure and the magnetic map will be very different as compared to situations where a sharp interface can form. Indeed the FeV system is a substitutional alloy displaying a complete solubility at high temperature. The experimental phase diagram by Sanchez et al. (1996) and ﬁrst principle studies by Turchi et al. (1994) showed that FeV alloy at the equiatomic composition acquires an ordered CsCl (B2)-type structure around 823 K. Talanana et al. (2001) within TB-LMTO proved that the reconstruction at the surface of (001) and (111) may stabilize magnetic conﬁgurations unstable in bulk B2 FeV ordered alloys. Also Izquierdo et al. (2001a) discussed the origin of the dead magnetic Fe overlayers on V(110) observed within in situ polarized neutron-reﬂectometry and MOKE by Nawrath et al. (1997, 1998, 1999, 2000). Within TB-LMTO Izquierdo et al. (2001a, 2001b) showed that an ideal layer-by-layer growth of Fe on the V(110) surface leads to the ferromagnetic solution for every Fe thickness. However, if alloying is present at the interface, a strong decrease in the Fe magnetic moment is found. To be complete let us note the results of conversion electron Mössbauer spectroscopy (CEMS) to probe the inﬂuence of V and Fe thicknesses on the Fe magnetic exchange in Fe/V multilayers by Kalska et al. (2001b). Also, ﬁrst principle calculations of the magnetic

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proﬁles of the Fe–V multilayers by Le Bacq et al. (2001) as well as ﬁrst-principle calculations of the magnetic anisotropy energy of Fe–V multilayers by Le Bacq et al. (2002) are compared to SQUID measurements of in-plane magnetic anisotropy of Fe/V(001) superlattices by Broddefalk et al. (2002). Lindner et al. (2003) have addressed the question of the behavior of Fe/V(001) superlattices for a number of Fe layers less than 2. This system shows an unusual increase of the magnetization at low temperatures and a ﬁeld-dependent orbital magnetic moment. On the basis of combined ferromagnetic resonance and XMCD measurements these effects have been interpreted by an interplay between ferromagnetic and superparamagnetic regions. Also state-of-the-art XMCD measurements of V at the L2,3 edges with an excellent signal-to-noise ratio are analyzed for a Fe0.9 V0.1 disordered alloy and a Fe/V3 /Fe(110) trilayer which were prepared in UHV and measured in situ on a Cu(100) single crystal (Scherz et al., 2002b). The adsorption ﬁne structure and the magnetic dichroism are compared with theoretical results calculated using the SPR-KKR Green’s function method. 3.5. Fe/Mn. A complex system From the point of view of its structural and magnetic properties, Mn can be considered as the most complex of all metallic elements and considerable attention has been devoted both experimentally and theoretically to explain its bulk structure (see, for example, the paper by Hobbs and Hafner (2001)) as well as its nanostructure conﬁgurations (see the review paper of Demangeat and Parlebas (2002)). Non-collinear exchange coupling in Fe/Mn/Fe(001) superlattices was seen with STM by Pierce et al. (2000). Also Bischoff et al. (2002b) studied with STM/STS the submonolayer growth of Mn on Fe(001). Purcell et al. (1992) obtained an antiferromagnetic ground state for Mn in Fe/Mn multilayers within an ASW approach. Stoefﬂer and Gautier (1993) within semiempirical tight-binding approaches got an oscillatory behavior of the interlayer exchange coupling in Fe/Mn superlattices. Khan et al. (1995) derived the magnetic order of ultrathin Mn layers on Fe(001) growing in a distorted bct structure. The magnetic order was derived within a self-consistent tight-binding real space model within the unrestricted Hartree–Fock approximation applied to the Hubbard Hamiltonian. The calculations were performed with an exchange parameter JFe ﬁtted in order to recover the Fe bulk magnetic moment. Such ﬁtting procedure was not possible for Mn because of its very complicated magnetic structure. Therefore JMn was considered as a variable. For values of JMn lower than 0.5 eV the local moment on Mn was shown to present a low-spin and antiferromagnetic coupling with the Fe atoms. A ﬁrst-order transition between low-spin AF-conﬁguration and high-spin F-conﬁguration takes place at 0.5 eV (Khan et al., 1995; Bouarab et al., 1995). This type of calculation was extended to Fe-stepped surfaces by Elmouhssine et al. (1996). All these calculations were restricted to the simple p(1 × 1) ferromagnetic conﬁguration which, most probably, is not the ground state of the Mn overlayer on Fe(001) (Roth et al., 1995). Thus, within FLAPW, Wu and Freeman (1995) proposed to consider both p(1 × 1) as well as in-plane antiferromagnetic c(2 × 2) conﬁgurations. They show clearly that this c(2 × 2) conﬁguration is lower in energy. Later on, Vega et al. (1996) extended the case of Mn to V, Cr monolayers on Fe(001) within semiempirical calculations. However, within XMCD experiments, Andrieu et al. (1997) obtained a low-spin moment on Mn

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Fig. 3.4. Relative energy E − Ec(2×2) for 5 magnetic conﬁgurations at the Mn–Fe interfaces. From Taguchi et al. (1999).

atoms with a ferromagnetic coupling with Fe atoms of the substrate. Elmouhssine et al. (1997) explained this result by considering a new p(2 × 2) magnetic conﬁguration within a TB-LMTO approach. Later on proton-induced Auger-electron spectroscopy has been performed by Igel et al. (1998) for the topmost layer of epitaxial Mn ﬁlms on Fe(100). They found signiﬁcant interfacial alloying during the growth of the ﬁrst monolayer of Mn on Fe(100) while the second monolayer grows as a pure Mn ﬁlm. Alloy formation at the interface strongly affects the magnetic ordering, as shown by Taguchi et al. (1999) within TB-LMTO calculations. Actually Taguchi et al. (1999) were the ﬁrst to investigate the stability of a Mn–Fe ordered surface alloy on an Fe(100) substrate with respect to the previously considered perfect Mn ML on Fe(100). A 2 MLs thick ordered Mn 0.5 Fe0.5 surface alloy was shown to present an antiferromagnetic conﬁguration in the Mn sublattice. This conﬁguration called alloy II in ﬁg. 3.4 turned out to be the lowest state of the alloyed conﬁgurations. However its energy was found to be only 7 meV above the c(2 × 2) ground state obtained for one Mn ML on Fe(100). Therefore it is expected that alloying between Mn and Fe should be present at the MnFe interface in agreement with Igel et al. (1998). Let us recall here that Demangeat et al. (2001) reviewed the stabilization of the Mn–Fe and Mn–Co interfaces via ordered surface alloys. Let us conclude by pointing out that Spisak and Hafner (1997) performed real-space TB-LMTO calculations in order to investigate the frustrated exchange interactions at the Mn–Fe interfaces. Non-collinearity was introduced within the Hubbard-type exchange potentials including spin-orbit coupling. The non-collinear structure obtained at the Mn–Fe interface tends to reduce the frustrations between the in-plane antiferromagnetic state in the Mn overlayers and the ferromagnetic Fe substrate. This type of calculation may open the door to a more realistic fully relativistic calculation of the unconstrained description of the magnetic map for multilayers containing ferromagnetic as well as non-ferromagnetic components. 3.6. Fe/Ni. A ferromagnetic/ferromagnetic interface An interesting study concerns the interface between Fe which is in a ferromagnetic conﬁguration in its bcc ground state and another ferromagnet, but with fcc structure. Usually, as

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we have seen before, ferromagnetic Fe induces magnetism (generally ferromagneticallylike behavior) in transition metal elements in contact with it. This is clearly true at the Fe interfaces with Cr (section 3.3), V (section 3.4) or Mn (section 3.5). In all these cases Fe is clearly of bcc-type or marginally of bct-type. Also, if the Fe thickness is great enough it can modify the structure of a transition metal element which is clearly not of bcc type. As an example, Heinrich et al. (1988) synthesized metastable bcc Ni on Fe(001). The surface and interface magnetism of this metastable bcc Ni was investigated as an overlayer on Fe(001) within FLAPW by Lee et al. (1993). Two systems – a monolayer (1Ni/Fe) and a bilayer (2Ni/Fe) coverage on Fe(001) – were investigated. The Ni–Fe interlayer spacings were found to be relaxed downwards by 2.5% and 4% for 1Ni/Fe and 2Ni/Fe, respectively. In both overlayer systems the magnetic moments at the interface Ni and Fe sites are enhanced relative to their corresponding bulk values. However, as discussed in section 3.2, thin ﬁlms of Fe grown on fcc substrates like Cu, Co or Ni present an fcc conﬁguration which is clearly of non-ferromagnetic type. Small Fe ﬁlms on noble metal substrates of Cu, Ag and Au will be discussed in detail in section 3.8. In this particular case, and from a magnetic point of view, the substrate has a marginal effect on the Fe magnetization. On the contrary, Fe, in contact with strong ferromagnets like fcc Co (to obtain fcc Co experimentalists use a Cu seed layer) or Ni, should be strongly disturbed. Structural, electronic and magnetic properties of a Fe monolayer on Ni(111) were studies by Wu and Freeman (1992) within FLAPW method. A ferromagnetic coupling between Fe and Ni atoms is obtained. However, when Fe is grown on Ni(111) substrates, the evolution of the ﬁlm structure vs thickness is expected to show a transition from a pseudomorphic fcc to a bcc phase, which resembles, in a reversed way, the bcc-to-fcc martensitic transition occurring in bulk Fe vs temperature. Gazzadi et al. (2002) performed a structural study of Fe ultrathin epitaxial ﬁlms, grown at room temperature on Ni(111), in the 1.5–18 ML coverage range by anglescanned photoelectron diffraction. Interesting results concerned the Ni/Fe/Co/Cu(001) systems. Cu in these experiments is used as seed layer in order to produce fcc Co. Fe being mainly in a non-ferromagnetic conﬁguration, an interlayer exchange coupling between Ni and Co is shown to be present. Kuch et al. (2000) within photoelectron emission microscope (PEEM) and XMCD spectroscopies determined the magnetic phases of Fe in Ni/Fe/Co/Cu(001). Some kind of interlayer exchange coupling between Co and Ni through the Fe spacer has been obtained. A theoretical explanation of this IEC is somewhat complex because it needs a good knowledge of both Fe–Co and Fe–Ni interfaces. Later on Wu et al. (2002b) reported on the same system within PEEM. They found that the oscillatory IEC between Co and Ni layers results in an oscillation of the Ni layer Curie temperature. Such systems have not been investigated yet within ab initio approaches. In the remaining part of this section we ﬁrst discuss theoretical results concerning the Fe/Co interface, then we focus on ab initio calculations concerning the Fe/Ni superlattices in the fcc or fct conﬁgurations. Mokrani et al. (1999) investigated the magnetic conﬁgurations of n (n = 1–4) layers of Fe on Co(001) substrates within TB-LMTO-LDA calculations. In order to check a possible onset of complex magnetization in the Fe ﬁlm not only p(1 × 1)+ and p(1 × 1)- were considered but also the c(2 × 2) conﬁguration. For 1 ML of Fe on Co(001) the p(1 × 1)+ magnetic conﬁguration was clearly shown to be the ground state. The other conﬁgurations p(1 × 1)- and c(2 × 2) were clearly found to be metastable. Moreover for all thicknesses

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of the Fe ﬁlm a ferromagnetic coupling persists between Fe and Co at the Fe–Co interface. This ferromagnetic coupling is still present when gradient-corrections are taken into account (Spisak and Hafner, 2000b). For n = 2 the whole system remains globally ferromagnetic whereas for n = 3, only the interface coupling remains of ferromagnetic type. For n = 4, within LDA, Mokrani et al. (1999) obtained a clear layered-antiferromagnetic coupling between Fe layers whereas Spisak and Hafner (2000b) within GGA obtained a different coupling for the surface and subsurface Fe layers. As discussed by Spisak and Hafner (2000b) this is probably due to the breakdown of LDA approximation in the case of Fe bulk. This point has been already discussed in the introduction of this section. Mössbauer spectra (Kalska et al., 2001a) and XMCD (Dallmeyer et al., 2001) were obtained for Fe/Co superlattices. Dallmeyer et al. (2001) found some kind of aperiodic interlayer coupling between the Co slabs through the fcc non-ferromagnetic spacer. This aperiodic IEC may be due to the instability of the fcc phase of Fe when its thickness increases. Such system needs urgently very careful experimental and theoretical work in order to shed some light on this very complex system. Lu et al. (1989) studied the epitaxial growth of γ -Fe on Ni(001) at room temperature with LEED and Auger-electron spectroscopy. Their experiments indicate that Fe grows on the Ni(001) in strained fcc structure, but ﬁlms thicker than about 6 layers contain domains of the stable bcc phase. With a primary-beam diffraction modulated electron emission (PDMEE) and secondary electron imaging techniques, Gazzadi et al. (2000) precised that for Fe ﬁlms on Ni(001) substrate the transition to the bcc phase occurred through nucleation of bcc(110) domains. Fratucello et al. (2000) measured, by Mössbauer spectroscopy, a high hyperﬁne ﬁeld assigned to Fe atoms close to the Fe–Ni interface with high-spin ferromagnetic state, and a low hyperﬁne ﬁeld corresponding to the inside Fe atoms with low-spin antiferromagnetic state. With a SQUID magnetometer, Edelstein et al. (1990) measured a strong magnetic coupling between Ni and Fe layers at the interface which enhanced the magnetization of both Fe and Ni atoms. By using X-ray diffraction, Kuch and Parkin (1998) studied the structure of the Fe spacer in Fe/Ni(001) multilayers. They showed that, as the Fe layer thickness increased, its structure varied from a vertically expanded fcc structure to a nearly relaxed fcc phase, and ﬁnally to a bcc(011) phase. These three phases were respectively ferromagnetic, antiferromagnetic and ferromagnetic. Moreover, the experimental investigations by Luches et al. (1999), Fratucello and Prandini (1995) and Acharya et al. (1995) displayed a clear evidence of alloying at the interface. Hadj-Larbi et al. (2002) determined the electronic and magnetic structure in Fe/Ni(001) superlattices within a TB-LMTO approach and a generalized gradient approximation. As shown by Wang et al. (1985) LDA is not suitable because it does not give the bcc ground state of Fe. The lattice parameters of both fcc Ni and Fe in the bulk form have been obtained by minimization of their total energy. For all the Fen /Nim (n = 1 to 5; m = 3, 5) investigated, pseudomorphic growth has been considered, i.e. the in-plane interatomic distance of Fe is chosen equal to that of Ni whereas the Fe–Fe out-of-plane interatomic distance is determined according to the constant volume approximation. The Ni–Fe interface distance is chosen as the arithmetic mean value of the calculated Ni and Fe lattice parameter. In order to minimize the numerical errors Hadj-Larbi et al. (2002) have considered, in all calculations, two inequivalent atoms per plane and the antiferromagnetic supercell for both AF and FM IEC investigated. Thus, there appears no supplementary error resulting from the

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Fig. 3.5. Magnetic proﬁles (in μB ) of Fen /Ni5 (n = 1, 2, 3, 4, 5) superlattices for FM and AF interlayer exchange couplings between Ni slabs (left and right panels, respectively). Dark (open) bars represent the values of the magnetic moments of the Ni (Fe) atoms. Two magnetically inequivalent atoms per plane have been considered. From Hadj-Larbi et al. (2002).

differences in the irreducible Brillouin zones (IBZ) as we have the same IBZ and the same number of k-points in each system investigated. Fig. 3.5 reports the magnetic properties of Fen /Ni5 superlattices for n = 1 to 5, for AFM and FM IEC between Ni adjacent slabs. The FM coupling between Nim slabs leads to very similar results for m = 3 or 5 so that discussion will be limited to m = 5. However, for the AFM couplings, and for 1 ML of Fe in-between Ni slabs the in-plane antiferromagnetic ground state of the Fe plane tends to destroy the magnetic moments in the Ni3 slabs. This arises from the highly frustrated states in the Ni layer adjacent to Fe ML. The reason is that the Ni atoms have a strong tendency to couple ferromagnetically with Fe at nearest neighboring sites; the Fe ML being of in-plane AFM type, the Ni atom has two nearest Fe neighbors with spin in one direction and two other nearest Fe neighbors with spin in another direction. As usual the result is a strong frustration on the Ni sites which tends to kill its magnetic moment. This effect is however much less dramatic for the Fe1 Ni5 superlattices. It is difﬁcult to check this point by a comparison with experimental results because experimentally some kind of interdiffusion at the Fe–Ni interfaces is present. In ﬁg. 3.5 only the most stable solutions for ferromagnetic (left panels) and antiferromagnetic (right panels) IEC are reported. The magnetic moments on the Fe atoms at the interface with Ni are always enhanced as compared to the bulk fcc Fe. This enhancement depends on the Fe spacer thicknesses: it decreases from 2.56μB for Fe1 Ni5 to a quantity roughly equal to 1.95μB for n = 2 to 5. The magnetic moments of the Fe atoms decrease when their distance with the Ni slab increases. For FM-IEC the ground state displays a ferromagnetic coupling between Fe and Ni at the interface. Moreover, for

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Fig. 3.6. The energy differences (EF − EAF ) versus the number of Fe layers for Fen /Ni3 (dashed line) and Fen /Ni5 (solid line). EF and EAF are respectively the energy of the Fen /Ni3 and Fen /Ni5 superlattices in the case of a FM and AF interlayer exchange coupling between the Ni slabs. From Hadj-Larbi et al. (2002).

one and two Fe layers the superlattice remains totally ferromagnetic whereas for greater thickness the Fe layers tend to be antiferromagnetically coupled: the induced ferromagnetic polarization by the Ni atoms is destroyed by the intrinsic AFM coupling present in bulk Fe. Hadj-Larbi et al. (2002) obtained for the AFM-IEC of the Fen /Ni5 superlattices a c(2 × 2) solution in the central Fe layer for n odd. This c(2 × 2) magnetic conﬁguration is necessary for a minimization of the frustration. For Fe1 /Ni5 , however, the magnetic moments of the Ni atoms are nevertheless strongly depressed because the c(2 × 2) Fe conﬁguration is directly in contact with the Ni atoms whose moments are directly affected by this in-plane antiferromagnetic conﬁguration in the Fe layer; this is no more the case for greater thickness of Fe because the Fe–Ni interfaces remain ferromagnetically coupled. Nevertheless it is necessary to point out that, for n odd and greater than 1, Hadj-Larbi et al. (2002) obtained a zero magnetic moment on the central Fe layer if the input moment is zero; however, this solution is found metastable as compared with the c(2 × 2) conﬁguration reported in ﬁg. 3.5. The IEC between the Ni slabs through the Fe spacer is deﬁned by the difference of the total energies between FM and AFM conﬁgurations and the result, reported in ﬁg. 3.6 do not display a speciﬁc oscillation period like that shown in section 3.3 in the case of Fe/Cr superlattices. It is important to point out that in Ni/Fe superlattices, the Ni is the ferromagnetic slab whose direction of polarization depends on the Fe spacer which is of antiferromagnetic type whereas in Fe/Cr superlattices, Fe is now the ferromagnetic slab. Moreover, as discussed by Hadj-Larbi et al. (2002), interface alloying tends to stabilize the Fe/Ni superlattices and consequently the magnetic map and the IEC are strongly modiﬁed. However these authors have restricted their approach to some kind of ordered alloys at the Fe/Ni interfaces which may be not the ground state of the system. More precisely, van Schilfgaarde et al. (1999) implemented within their LMTO-LDA code an unconstrained vector approach for the magnetization and a random distribution of the Fe and Ni atoms. They showed that their approach is able to give the origin of the Invar

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effect in Fe–Ni alloys. It is clear that the calculation of Hadj-Larbi et al. (2002) concerning Fe/Ni superlattices misses this very important point of chemical disorder at the Fe/Ni interface, and non-collinearity in the magnetic map of this Fe–Ni alloy in-between Fe and Ni slabs. Such approach is computationally demanding but necessary in order to make a more careful contact with the experimental results. Before closing this section let us mention some TB-LMTO calculations by Ziane et al. (2003), concerning the relation between the IEC and the non-ferromagnetic behaviour of fcc-Fe in Co/Fe/Ni(001) superlattices. Similarly to Fe/Ni and Fe/Co interfaces were the Fe layer adjacent to Ni(Co) is ferromagnetically polarized, here also, the Fe layer adjacent to Ni, respectively to Co is ferromagnetically polarized by the strong Ni or Co ferromagnet. Also it is shown that Fe becomes non-ferromagnetic for thicknesses greater than 2 MLs. Finally the IEC is strongly dependent on the quality of both Fe/Co and Fe/Ni interfaces. 3.7. Fe/M interface (M = 4d, 5d transition metal) The induced polarization by the Fe layers in contact with non-magnetic elements of the 4d and 5d series has also attracted a great interest in the materials-science community. There are however less results concerning these Fe-4d and Fe-5d multilayered elements but still too much to be reported exhaustively here. Therefore we restrict ourselves to Rh and Pd for the 4d-series and to W for the 5d-series. The interest concerning Rh and Pd is connected to a possible onset of magnetism when such elements present reduced dimension. Of course, Rh clusters were found to present magnetic moments by Cox et al. (1993) whereas palladium-graphite multilayers display a magnetic behavior through SQUID magnetization (Suzuki et al., 2000). Also Mikheenko et al. (2001) proposed a biological route to obtain ferromagnetic palladium. The ferromagnetic Pd is obtained in the process of its biological recovery using the sulphate-reducing bacterium Desulfovibrio desulfuricans. The formation of ferromagnetic Pd takes place between the outer and cytoplasmic membranes and is attributed to the small size of nanocrystals grown in the periplasmic space. The Curie temperature in optimally prepared particles is above 550 K. Moreover, the presence of ferromagnetic Pd on the surface of bacteria cause orientation of the bacteria in a strong magnetic ﬁeld. Both results from Suzuki et al. (2000) and Mikheenko et al. (2001) deserve further studies. Because only macroscopic studies (SQUID, VSM) were performed it is, at present, not clear if the mechanism leading to ferromagnetism has a connexion with the size of the Pd clusters or with electronic charge transfer between Pd metallic atoms and their surrounding. There is presently a general agreement, both experimentally (Cox et al., 1993) as well as theoretically concerning the presence of magnetism in free-standing Rh clusters. For Rh surfaces or Rh nanostructures adsorbed on non-magnetic substrates no consensus has been obtained and a strong dispute between teams working in this ﬁeld is currently happening at magnetic conferences. Moreover, Rh being an fcc element its contact with bcc Fe induces strain so that its structural stabilization is complex. The magnetism of Pd is still under debate. What is granted by the community is the absence of magnetization in atomic Pd and in bulk metal. For Pd nanostructures, as reported by Dreyssé and Demangeat (1997) there are various clues showing that, in reduced dimensions, Pd could present some kind of magnetization (Bouarab et al., 1990). There have been some indication that Pd nanoclusters in contact with a substrate or embedded in elements like carbon or more generally

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polymers may display magnetization via MOKE or SQUID. It looks delicate to conclude anything from these macroscopic (and global) experiments because such experiments do not discriminate between the polarization of Pd and that of the “surrounding”. Therefore before looking for a theoretical explanation concerning the onset of magnetization on Pd atoms it would be suitable to use the dichroic devices which are chemical-speciﬁc. Those questions concerning a possible onset of magnetism in Pd nanostructures cannot be discussed here because there are not enough reliable (error-less) results. For this reason we restrict here our comments to the induced magnetization of Pd (Rh, W) in contact with Fe. Let us point out that Dahmoune et al. (2002) discussed a possible magnetic reconstruction at the (001) surfaces of B2 FePd and FeRh ordered alloys. Since the experimental work of Celinski et al. (1990) displaying induced Pd polarization at the Fe–Pd interfaces and some kind of interlayer exchange coupling, considerable attention has been focused on this system. DFT calculations showing induced polarization by an Fe ML on Pd were performed by Blügel et al. (1989) and Li et al. (1990). Later on, within semiempirical tight-binding calculations, Nait-Laziz et al. (1993) found a magnetic polarization for n (n = 1–4) Pd layers on a bcc Fe(001) substrate. Some kind of induced polarization takes place in the Pd overlayers with a striking effect for n = 4. In that case, a negative polarization appears on the Pd atoms which are not at the interface with Fe. Determination of the interlayer exchange coupling in Fe5 /Pdn superlattices displayed a ferromagnetic to antiferromagnetic transition of the IEC when n goes from 5 to 6. This trend is in qualitative agreement with the results of Celinski et al. (1990). Later on, Fullerton et al. (1995) reported an experimental and theoretical investigation of epitaxially strained Pd(001) thin ﬁlms on Fe(001) and in Fe/Pd/Fe(001) trilayers. Calculations were done with an ASW code on abrupt and alloyed Fe/Pd interfaces. Later, Boeglin et al. (1999) displayed a structural transition from fct to bct around 4 ML of Fe on Pd(001). Moreover, Cros et al. (2000) found a high-spin Fe phase in Fe/Pd(001) multilayers with a very thin thickness of an fcc-like Fe ﬁlm. Fe–Pd surface alloy appears to be present for ultrathin ﬁlms of Fe on Pd(001) (Lee et al., 2002) as well as on Pd(111) (Choi et al., 2001). Also, Lee et al. (2002) performed FLAPW-GGA calculations to determine the magnetic map and the total energy of Pd/Fe/Pd(001). They found that an inverted Fe monolayer is more stable as compared to a perfect Fe ML on Pd(001). This is in agreement with their LEED results displaying interdiffusion between Fe and Pd atoms. Contrary to Pd whose atomic and metallic bulk conﬁgurations are both in a nonmagnetic state, Rh does present some kind of intrinsic magnetism. It presents magnetism at the atomic level (like most of the other 4d transition metal elements) and also magnetism in its free-standing form (displayed within Stern–Gerlach experiments by Cox et al. (1993)) contrary to all other 4d transition metal elements. This magnetic behavior was found theoretically by Reddy et al. (1993) but, sadly enough, they obtained also onset of magnetism in Ru and Pd clusters which is at odds with all Stern–Gerlach experiments. Consequently, much reservation should be taken concerning many calculations displaying magnetization in free-standing Pd clusters. Moreover, Goldoni et al. (1999) displayed the presence of magnetic order at the Rh(100) surface within linear magnetic dichroism in angular distribution (LMDAD) at odds with most of the theoretical results. Later on, Goldoni et al. (2000) pointed out that LMDAD is present in the 3d core level photoemission spectra of Rh(100) upon reversing the magnetic ﬁeld in chiral experiments only when a small residual magnetic ﬁeld is left applied to the sample. The intensity of the LMDAD effect does not depend

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on the magnitude of the applied residual ﬁeld, saturating for very weak ﬁeld intensities (less than 3G). The conclusion of Goldoni et al. (1999) is that a two-dimensional ferromagnetic order of the Rh(100) surface could be extremely unstable or that the Rh(100) surface could be super-paramagnetic. Later, Goldoni et al. (2000) and Chado et al. (2001) reported on the growth and morphology of Rh on substrates presenting triangular symmetry. Goldoni et al. (2000) studied the growth mode and magnetic properties of ultrathin Rh layers deposited on highly oriented pyrolitic graphite using photoemission and STM. The possible presence of in-plane ferromagnetic ordering in these low-dimensional Rh systems has been checked by LMDAD. For submonolayer or monolayer Rh ﬁlms on HOPG, there is no evidence of in-plane long range magnetic ordering down to 150 K. Theoretically, Krüger et al. (1997b) showed that Rh on graphite may present a magnetic moment if the density of Rh atoms is rather low. The point is that the calculations are performed at zero K so that a direct comparison with experiment at 150 K is rather difﬁcult. Growth and magnetism of Rh on Au(111) have been investigated over a wide range of coverages and deposition temperatures by means of variable temperature ultra high vacuum STM (UHV-STM), Auger spectroscopy, and in situ Kerr effect measurements (Chado et al., 2001). Down to 30 K, no ferromagnetism was detected by Kerr effect. The contradiction between experiment and theory may be only apparent. In order to solve this point Ohresser (2002) is investigating the magnetic properties of a submonolayer of Rh grown on Au(111) at 4.2 K in a ﬁeld of 7 Teslas. Preliminary investigations at 10 K have given some indications of a low magnetic moment. This may bridge the gap between theoretical investigations displaying, in general, a clear onset of magnetism in a thin Rh layer on noble metal (or graphite) and experimental results displaying no such magnetic effect. Less controversial is the polarization obtained both experimentally and theoretically when Rh is in contact with a ferromagnetic metal. In that case the polarization at the Rh atoms is clearly due to 3d–4d hybridization effect. Kachel et al. (1992) obtained a Rh polarization at the interface with a semi-inﬁnite Fe(001) crystal through spin-resolved valence-band and core-level spectroscopy. Within an Hubbard-type Hamiltonian, Chouairi et al. (1993) determined the electronic and magnetic map of n (n = 1–6) Rh layers on Fe(001). Epitaxial growth with conservation of the atomic volume of Rh in bct and fct symmetries were considered. For both geometries a sizeable induced polarization of 0.6μB on the Rh atoms is obtained in the case of 1 ML of Rh on Fe(001). The sign of the Rh induced polarization in the Rh ML located at the interface with Fe is oscillatory. For 1 ML of Rh the coupling between Rh and Fe at the Fe/Rh interface is of ferromagnetic type whereas for 2 ML of Rh on Fe(001) its value decreases drastically and its polarization become of antiferromagnetic type. For bct and fct conﬁgurations the induced polarization remains short-ranged. Nevertheless, the Fe polarization at the Fe/Rh interface remains much higher in the bct crystallographic phase than in the fct structure. This is, however, not really surprising if we remember that fcc Fe is not a good magnetic system. The induced magnetization of Rh by Fe was conﬁrmed within XMCD experiments by Tomaz et al. (1997). Later, Hwang et al. (1999) and Hayashi et al. (2001) have grown Fe thin ﬁlms on Rh(001) but their MOKE and XMCD results display quite contradictory results. Up to six layers of Fe on Rh(001) were shown to be non-ferromagnetic by Hwang et al. (1999). Within FLAPWGGA calculations for 1 ML of Fe on Rh(001) they obtained an antiferromagnetic ground state which conﬁrms partly their MOKE ﬁndings. However, within XMCD, Hayashi et al. (2001) obtained sizeable moments and ferromagnetism for Rh ﬁlms starting from 3 ML.

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Fe polarizes also the 5d elements in contact with it as discussed by Demangeat (1975) for single impurities embedded in an Fe host. The orbital moment μL was determined too so that a comparison with Hund’s third rule was possible. μS and μL on 5d impurities were clearly opposite for Hf and Ta, whereas they display the same sign for Os, Ir and Pt: this is in full agreement with Hund’s rule (see section 4.2 of Demangeat (1975)). For W, Re the status was not fully understood because two solutions were obtained for a Re impurity whereas the spin-polarization on W was roughly zero. For Re the orbital and spin moments were shown to be parallel or antiparallel, depending on the considered solution. For W however, only one solution was obtained with a sizeable orbital moment but almost zero spin moment. The general trend follows clearly Hund’s rule, but for W and Re the semiempirical approach used was not able to report signiﬁcant result. XMCD measurements were performed at the beginning (W) and at the end (Ir, Pt) of the 5d series of the periodic table (Wilhelm et al., 2001). Considerable induced magnetic moments of about 0.2μB are obtained. For W the induced magnetic moment is polarized antiferromagnetically with the Fe atoms in agreement with DFT calculations (Qian and Hübner, 1999, 2001; Galanakis et al., 2000). For an Fe monolayer on vicinal W substrates, the magnetic map appears more complex (Hamad et al., 2002). Moreover μL and μS were found antiparallel on W within XMCD (Wilhelm et al., 2001) in agreement with the DFT results of Galanakis et al. (2000). Thus, Wilhelm et al. (2001) claimed that this result is a “violation of the third Hund’s rule”. It is difﬁcult to agree with this claim for simple reasons: (i) W, like Cr for the 3d transition metal series is in fact an element in the middle of the TM series; (ii) a complete calculation concerning all 5d transition metal impurities in Fe do show a “very complete” agreement with the third Hund’s rule, starting with “real elements” at the beginning of the series (Hf, Ta). Thus, before the conclusions of Wilhelm et al. (2001) could be taken for granted, XMCD results should be performed on other 5d elements. Also we can notice the determination of the spin-polarized unoccupied state band structure of Fe on W(110) by spin polarized low energy electron microscopy (SPLEEM) obtained by Zdyb and Bauer (2002). These results look as a good test to discriminate between the various ab initio calculations. Before closing we may notice some very interesting measurements concerning the Electric ﬁeld gradient (EFG) on 5d impurities in Fe and Ni by Seewald et al. (2002a). The EFG at the nuclear site is a direct measure of the noncubic charge distribution around the nucleus. In cubic ferromagnets the EFG is therefore directly connected to the spin-orbit coupling. The results of Seewald et al. (2002a) establish in particular that the effect depends in general considerably on the direction of the magnetization as pointed out by Demangeat (1975). Transparent treatment of the spin-orbit coupling for the determination of this EFG was done by Seewald et al. (2002b) within the tight-binding approximation. 3.8. Fe on Cu, Ag, Au. A self-surfactant system? The surface energy of Fe being considerably larger than those of noble metals Cu, Ag or Au, it is clear that the growth of Fe on these substrates should present some kind of atomic exchange between the Fe ad-atoms and the atoms of the substrate. Man et al. (2002) has given a very detailed report concerning the inﬂuence of the deposition rate upon the initial growth morphology and magnetism of ultrathin Fe ﬁlms on a Cu(100) surface. The dramatic changes in the ﬁrst diffraction intensity oscillation during growth can be explained

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in terms of kinetic and thermodynamic mechanisms that relate place exchange at the surface. Asada and Blügel (1997) performed total energy spectra of complete sets of magnetic states for fcc-Fe ﬁlms on Cu(100) within collinear FLAPW-GGA. Later, Qian et al. (2001) and Spisak and Hafner (2002a) discussed respectively a possible onset of SDW and shear instability in ultrathin ﬁlms of Fe on Cu(100). Investigations concerning Fe thin ﬁlms grown on Cu(111) were done by Rau et al. (1986) within electron capture spectroscopy. Long-ranged ferromagnetic order was obtained. Pierce et al. (2001) pointed out that magnetic Co capping layers can be used to generate a magnetic phase diagram for metastable Fe/Cu(100) ultrathin ﬁlms (see also Spisak and Hafner, 2002a). Their method could be generalized to other metastable thin ﬁlm systems. Within pulsed laser deposition Shen et al. (1998) demonstrated the possibility of growing γ -Fe on Cu(111) in a high-spin phase. Within TB-LMTO-GGA, Krüger et al. (2000) reported on the magnetism of 3d transition-metal monolayers on Cu(111) and Ag(111). Later on, Krüger (2001) extended the previous calculations in order to consider the total magnetic moment of Fe ﬁlm as a function of the coverage. His magnetic moment variation is in reasonable agreement with the experimental results of Shen et al. (1998). Moreover he pointed out that a SDW may be the ground state in agreement with Qian et al. (2001). Spisak and Hafner (2002b) reported results concerning the structural properties and magnetism of monoatomic Fe wires on densely stepped Cu(11n) surfaces for n = 3–11 by means of DFT calculations. The calculated equilibrium bond length of a freestanding wire is smaller by 12% than the corresponding bond length of a wire deposited on a Cu substrate. In a paper entitled: “Fe adatoms on Ag(100): site exchange and mobility”, Langelaar and Boerma (1998) presented a low-energy ion scattering study of the behavior of Fe adatoms on a Ag(100) surface. The Fe adatoms were found to exchange sites with Ag atoms from the top layer, starting at a temperature of 130 K. Also TB-LMTO calculations were performed for Cr, Mn and Fe on Ag(100) by Elmouhssine et al. (1998a, 1998b) and it was shown that, in the cases of Cr and Fe, the most stable state is respectively AFM and FM, whatever the nature is of the considered interface, i.e. an epitaxial ML, a buried one or two alloyed MLs (ﬁg. 3.7). Another interesting result (ﬁg. 3.8) is that a Cr, Mn or Fe epitaxial ML on Ag(100) is always unstable against alloying and the effects of the magnetic versus nonmagnetic solutions are then not really signiﬁcant. Moreover it was shown that a buried Cr, Mn or Fe ML on an Ag(100) substrate is the most stable state with respect to both alloying and epitaxial solutions (ﬁgs 3.9 and 3.10). This result was obtained for NM solutions but also for magnetic ones (FM and AFM) since magnetic ordering effects are not sufﬁcient to avoid the formation of a buried ML. By means of MOKE, Schaller et al. (1999) observed spin reorientations from in-plane to out-of-plane and vice-versa upon annealing thin Fe ﬁlms on Ag(001). Within fullyrelativistic full-potential DFT code, Lazarovits et al. (2002) determined the magnetic anisotropy energies of small Fe, Co, and Ni clusters on top of an Ag(100) surface. Blum et al. (1999) reported on “Fe thin-ﬁlm growth on Au(100): A self-surfactant effect and its limitations”. Deposition of only about 0.2 ML of Fe is sufﬁcient to lift the reconstruction of a clean surface. In the initial growth process, place exchange between Fe and Au atoms lead to an almost two-dimensional subsurface Fe ﬁlm growth with one layer of Au covering the entire ﬁlm. In this way, gold acts as “self-surfactant”. Stepanyuk and Hergert (2000)

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Fig. 3.7. Difference of energy between AFM and FM solutions for an equivalent Cr, Mn or Fe monolayer on Ag(100). Three structural conﬁgurations are considered: 1 ML epitaxially grown on Ag(100) noted EPI(1); ordered alloy TM0.5 Ag0.5 two layers thick on Ag(100) noted ALL(2); and one TM monolayer underneath the Ag(100) surface noted ENT(3). From Elmouhssine (1998).

Fig. 3.8. Difference of energy between EPI and ALL (notations in ﬁg. 3.7) structures of one equivalent TM-monolayer on Ag(100) for NM, FM and AFM solutions. From Elmouhssine (1998).

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Fig. 3.9. Difference of energy between buried TM monolayer (ENT) and ALL for NM, FM and AFM solutions. From Elmouhssine (1998).

Fig. 3.10. Difference of energy between buried TM monolayer (ENT) and EPI for NM, FM and AFM solutions. From Elmouhssine (1998).

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demonstrated within KKR-GF calculations that site exchanges between 3d transition metal adatoms on the Au(100) surface and Au atoms are energetically favorable. Epitaxial growth of Fe on a reconstructed Au(111) surface leads to the formation of self-organized fcc Fe dots and stripes with increasing coverage (Ohresser et al., 2001). Within XMCD they determined the changes in the magnetic properties in the low thickness range from 0.1% of 1 ML to 4 ML, covering the 1D coalescence around 0.3 ML and the 2D percolation limit around 1 ML. Let us also mention ﬁrst-principle description of the MOKE of ultra-thin surface layer systems based of the concept of non-local layer-resolved optical conductivity and the corresponding solution of the Maxwell equations. This powerful approach allowed a detailed investigation of the system Au(001)/nFe/mAu that shows quantum-conﬁnement effects (Huhne and Ebert, 2002). Hobbs and Hafner (2000) as well as Kurz et al. (2001) reported on the fully unconstrained magnetism in triangular Cr and Mn overlayers on Cu(111). While Cr and Mn monolayers on Cu(111) do present intrinsic non-collinear magnetism, it is less clear for Fe. Using PAM, Uzdin et al. (2001b) showed that trimers of Fe on Ag can present noncollinear magnetism only for speciﬁc distances between the Fe atoms. This will be discussed in details in section 4.7. 3.9. Fe/graphite. Towards intrinsic non-collinear systems In section 3.8 we reported on the magnetization of triangular Fe on the (111) face of Cu, Ag, and Au. Induced magnetization by Fe was found in touching metallic systems through contact. This magnetization was directly connected to the hybridization of the spinpolarized 3d orbitals of Fe with the respective 3d orbitals of Cu, 4d of Ag and 5d of Au and was mainly restricted to atoms in direct contact with Fe. Various growths were revealed but at room temperature the “self-surfactant-effect” is the most common way of growth. In fact this surfactant effect may help in the layer-by-layer growth. Nothing like that for Fe on graphite as reported in details by Binns et al. (1999). Growth in that case is deﬁnitively not layer-by-layer but bubbling. It is worth stating here that for Fe, as well as for the other 3d elements, the metal ﬁlms grow in the Volmer–Weber mode, i.e. islanded ﬁlms, and that all the 3d TM ﬁlms, including Fe, usually show electronic properties characteristic of nanoscale metal particles. This is more generally related to the fact that the 3d–3d interaction between Fe orbitals is stronger than the 3d–2p bonds (3d for Fe and 2p for carbon). Also magnetic linear dichroism (MLD) in the angular distribution (also called MLDAD) was performed by Baker et al. (2000) for islanded Fe ﬁlms on graphite. Let us just recall here that in photoelectron spectroscopy from a core level with non-zero angular momentum, if a chiral geometry of photon polarization (e), ˆ photoelectron wave number (k) and sample magnetization direction (M) are used, then reversing e, ˆ or more conveniently M, results in a change in the angular distribution of the photoelectrons. Thus if the spectrum is measured at a particular angle, reversing the magnetization produces a change in the core level lineshape as shown in ﬁg. 3.11 for an islanded Fe ﬁlm on graphite (Binns et al., 1999). The total dichroic signal (area in the difference spectrum) is proportional to the sample magnetization and MLDAD is thus a powerful atom speciﬁc probe of surface magnetization. From a theoretical point of view, and within TB-LMTO-ASA method, Krüger et al. (1998a) investigated the magnetism of hexagonal 3d TM MLs, epitaxially and ideally adsorbed on a graphite (0001) surface. They assumed a p(1 × 1) epitaxial structure (ﬁg. 3.12)

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Fig. 3.11. Photoemission spectra of Fe 3p core level taken with hω = 188 eV with two opposing magnetization directions from an islanded Fe ﬁlm on graphite showing the MLDAD. From Binns et al. (1999).

Fig. 3.12. The non-ferromagnetic conﬁgurations for one 3d TM monolayer upon graphite. The magnetic unit cells are indicated by dotted lines and the underlying honeycomb lattice of the graphite surface by full lines. Up (down) arrows correspond to positive (negative) magnetic moment. When one (out of two) positive moment of (b) unit cell cancel, because of frustration, conﬁguration (c) is produced. Conﬁgurations (a) and (b) are, respectively, denoted by row-by-row AFM and ferromagnetic (FIM) in the text. From Krüger et al. (1998b).

where each TM atom is placed 1.9 Å above the center of a graphitic hexagon. Actually this position, called six-fold ‘hollow’ position, is a very stable one for an Fe ML, according to non-spin-polarized FP-LMTO calculations (Peng et al., 1996). As a ﬁrst approximation this Fe position was also taken for the other 3d elements. For further technical details see Krüger et al. (1998a). The problem of AFM ordering on a 2D hexagonal lattice is a little subtle. This lattice cannot be divided into two inequivalent sublattices A and B such that all the nearest neighbors (NN) of an A site belong to the sublattice B, i.e. an AFM hexagonal lattice is necessarily partly frustrated on the NN level. The NN AFM Ising model has a highly degenerate ground state: see Ducastelle (1991). Among the degenerate ground state conﬁgurations Krüger et al. (1998b) have selected two, which are shown in ﬁg. 3.12 (a)

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Fig. 3.13. Local magnetic moments of unsupported 3d metal MLs in the FM (circles, full line), the AFM (squares, dotted line) and the FIM (triangles) conﬁgurations. Upwards triangles show the majority moments whereas downwards triangle show the absolute value of the minority moments. From Krüger et al. (1998b).

Fig. 3.14. Local magnetic moments of the 3d TM MLs adsorbed on graphite(0001) in the FM (circles, full line), the AFM (squares, dotted line) and the FIM (triangles) conﬁgurations. Upwards triangles show the majority moments whereas downwards triangles show the absolute value of the minority moments. From Krüger et al. (1998b).

¯ called ‘row-by-row’. Conﬁguration (b) has and (b). Conﬁguration (a) is an AFM [1010] twice more up than down spins and is referred as ‘Ferromagnetic’ (Fi) conﬁguration (also noted FIM in this review). When one (out of two) positive moments in unit cell (b) vanishes, because of frustration, conﬁguration (c) is produced (ﬁg. 3.12(c)). Results of Krüger et al. (1998b) are vizualized in ﬁgs 3.13–3.15. Looking ﬁrst at the result for free MLs (ﬁg. 3.13) we see that V is NM, Cr is FIM, Mn is AFM and Fe, as well as Co and Ni, are FM. Apart from NM, AFM solutions also exist for Cr, Fe and Co, but their moment is in each case smaller than the one corresponding to the most stable conﬁguration (see Krüger et al., 1998b). Especially, for the Cr and Mn ML, it is by no means clear that the lowest energy conﬁgurations are the ground states. Moreover, there is no reason why the ground state magnetic structure should be collinear. Actually the frustration of the AFM NN coupling can be reduced by allowing the spins to rotate (Kawamura, 1998). Fig. 3.12(c)

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Fig. 3.15. Magnetic energies per atom of unsupported 3d TM MLs (open symbols) and on graphite(0001) adsorbed ones (full symbols) in the FM (circles, full lines), the AFM (squares, dotted line) and the FIM (triangles) conﬁgurations. From Krüger et al. (1998b).

also provides an example of frustration effect when there is cancellation of one (out of two) positive moment (per unit cell) of the conﬁguration (shown in ﬁg. 3.12(b)). When the MLs are adsorbed on the graphite surface (ﬁgs 3.14 and 3.15), the magnetic moments become substantially reduced. The order of stability for the three various magnetic conﬁgurations is, however, in almost all cases maintained. The Fe, Co and Ni MLs are still FM and the Mn ML keeps preferring AFM NN coupling, i.e. AFM or FIM. For Cr ML, no magnetic solution can survive after adsorption on a graphite surface. This is not surprising since both magnetic solutions of the free ML are hardly more stable than the NM state and because electronic hybridization with the graphite 2p bands destabilizes them further (Parlebas, 2001). 3.10. Fe/c-FeSi/Fe sandwiches and multilayers. A critical system Magnetic multilayer devices (MMD) can be roughly classiﬁed according to the conducting nature of the spacer material, which leads to markedly different behavior of the exchange coupling constant between the magnetic layers, J , as a function of the thickness of the spacer, z. This can be understood in the framework of Bruno’s theory (1995). The ﬁrst class comprises those systems whose spacer is a metal. In this case, J (z) shows oscillatory behavior, whose period is typically of a few Å. Superimposed to these oscillations, J (z) also decays as 1/z2 , becoming negligible after several tens of Å. The second class is composed of those devices where the spacer is a semiconductor. Now, J (z) is frequently antiferromagnetic (AFM) and its magnitude decreases exponentially with a decay length of at most two or three Å. (Fe/c-FeSi) MMD stand out among all such structures as being a MMD in which the spacer is semimetallic. Indeed, experimental results of Briner and Landolt (1994), Mattson et al. (1993), Inomata et al. (1995), Chaiken et al. (1996), de Vries et al. (1997) and Strijkers et al. (2000) show that the exchange coupling constant of these devices is always AFM, but has a fairly large decay length, becoming negligible

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at spacer thicknesses more proper of a metallic than of a semiconducting spacer. Gareev et al. (2001) found an oscillating behavior of the exchange coupling. This unique behavior of J (z) might also make MMD in this class strongly attractive for manufacturers interested in the design of spin polarized transport devices. In (Fe/c-FeSi) MMD, the experiments of Chaiken et al. (1996) and de Vries et al. (1997) have shown that the (c-FeSi) spacer possesses the CsCl structure. It can be formed by interdiffusion of Fe and Si slabs of appropriate thicknesses grown epitaxially as observed by Fanciulli et al. (1999) and Strijkers et al. (2000). LEED and AES experiments of Strijkers et al. (2000), which have been applied successfully to study epitaxial Fe/Si/Fe(001), have shown that the perpendicular interlayer distance in this bcc-like structure remains constant at ≈1.43 Å, very close to the values for pure bulk bcc Fe. Pruneda and coworkers (2002) determined for the ﬁrst time the spin-polarized electronic structure of both Fe/c-FeSi/Fe(001) trilayer (TD) and Fe/c-FeSi multilayer devices as a function of the spacer thickness z using a scalar-relativistic version of the k-space TBLMTO method in the atomic spheres approximation. They ﬁrst made structural analyses to test that the experimental result of inter-diffusion of a thin Si ﬁlm into the Fe slabs, that creates the c-FeSi structure, can be understood in terms of the energetics of the different plausible atomic arrangements in the spacer. Besides, they performed molecular dynamics simulations, relaxing the atomic positions in the z-direction, for Fe9 /c-(Fe1 Si2 )/Fe9 and Fe7 /c-(Fe2 Si3 )/Fe7 TD using the SIESTA code developed by Sánchez-Portal et al. (1997) with a minimal basis set and the exchange-correlation functional of Perdew and Wang (1992). Pseudopotentials were generated with the Troullier and Martins (1991) method, with 4s 1 3d 7 and 3s 2 3p2 valence conﬁgurations for Fe and Si, respectively. The forces in the theoretical bcc-like conﬁguration just make the interstitial Fe atoms move slightly into the FeSi spacer, and expand a bit the Si–Si distances, the structure being very stable otherwise. We pass on now to discuss the behavior of J (z), deﬁned as the difference between the total energies of FM and AFM alignments of the Fe slabs, for thin TD with perfect interfaces. Fig. 3.16 shows the exchange coupling as a function of the number n of Si monolayers obtained by Pruneda et al. (2002). A positive J of the order of some mRy is always obtained, which decreases with z in a non-monotonic fashion. Experiments of de Vries et al. (1997) actually show that for z larger than ≈13 Å, J (z) follows the asymptotic behavior of a semiconductor, with a large decay length of about 3.6 Å. This overall theoretical behavior for thin trilayer devices, that was also obtained within the LSDA by Robles et al. (2001a, 2001b), is therefore in nice qualitative agreement with the experimental results by de Vries et al. (1997). The exchange constant obtained for multilayers follows the same trend as that found for trilayers, its magnitude being roughly a factor of two larger (see inset of ﬁg. 3.16). This fact can be qualitatively explained in terms of simple Heisenberg-like physics, making use of an analogy with spin chains: the energy to ﬂip a spin in a molecule composed of two atoms is half that required to do so in an inﬁnite chain. De Vries and coworkers (1997) observe FM-type contributions superimposed on top of the AFM-like behavior in their Kerr hysteresis loops for TD with spacer thicknesses smaller than about 6–7 Å. Pruneda and coworkers (2002) performed simulations of TD with a variety of atomic misconﬁgurations inside the spacer or at its interfaces with the iron slabs, in order to test whether they can induce such ferromagnetic coupling. For instance, a dense array of thin pinholes or rough interfaces for TD with n = 1, 2, and 3

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Fig. 3.16. Total energy difference J = EF − EAF as function of the number n of Si monolayers in the spacer for Fe7 /c-Fe(n−1) Sin /Fe7 (001) multilayers. The inset shows J for trilayers. Open squares denote values of J obtained using the theoretical lattice constant and the LMH functional. From Pruneda et al. (2002).

(e.g., up to about 8 Å) were considered. One can conclude that atomic misconﬁgurations generically give rise to very strong ferromagnetic couplings. A comparison with the experimental results by de Vries allow to infer some further conclusions about the structure of these sandwiches, namely: (a) it is very likely that for TD with thicknesses smaller than about 3 Å there is a process of diffusion of iron into the Si layer (or vice versa), so that the Si layer is disrupted; (b) for TD with spacer thicknesses larger than about 4 Å, diffusion takes place but leads to the formation of a c-FeSi spacer with interfaces of high quality. We come to comment now on the values found for the magnetic moments. First, we have that the c-FeSi spacer displays tiny magnetic moments of order 0.05μB at the interface with the iron slabs, with whom they couple antiferromagnetically. Second, the absolute values of these magnetic moments are almost identical for both alignments. Third, iron atoms in the vicinity of the interface with the spacer have reduced magnetic moments (∼1.46μB ) due to the hybridization of their orbitals with those belonging to c-FeSi atoms. Surface effects, on the other hand, induce enhanced magnetic moments at the Fe external layers, (M ∼ 2.97μB ). Finally, magnetic moments at the center of the Fe slabs slightly oscillate around the bulk value (∼ 2.20μB ). We ﬁnd similar behavior for multilayer devices, apart from the obvious fact that no surface effects exist for them. The asymptotic behavior of the exchange constant was analyzed by Pruneda et al. (2002) using Bruno’s (1995) theory. The exchange constant is usually evaluated at the saddle point, which is equivalent to evaluating its integrand exactly at the Fermi energy. A thorough study of the Fermi surface then allowed these authors to identify the relevant calipers and then determine the decay length of J (z). For a semimetallic spacer the Fermi surface

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Fig. 3.17. Details of the band structure (central panel) and total density of states in units of states/eV (right panel) of bulk c-FeSi. The band drawn with a dashed line has a band edge at the M point and a large ﬂat section around the X point. The two-dimensional Brillouin zone is plotted in the left panel. From Pruneda et al. (2002).

and its relevant calipers vanish, naively providing an inﬁnite decay length. Therefore, the integral for calculating the exchange constant was evaluated exactly for (Fe/c-FeSi) MMD. To do so they obtained, previously and independently, all band edges and effective masses from band structure calculations. We show in the middle panel of ﬁg. 3.17 a plot of the bands of the spacer in the neighbourhood of the Fermi energy, with the relevant one drawn as a dashed line. Such band has a van Hove singularity at the X point which gives rise to a strong peak in the density of states slightly above the Fermi energy. The piece of the Fermi surface related to the M point actually shrinks to zero, since the band has its edge located almost exactly at the Fermi energy. It gives rise to a three-dimensional band-edge in the density of states, which accounts for the semimetallic character of the spacer. De Vries et al. (1997) speculated that the main contribution to J (z) should come from the X point. But a brief inspection of the middle panel shows that such contribution must be oscillatory, due to the combined effect of the semiconducting character of the band and its negative effective mass at such point. This suspicion was indeed conﬁrmed after evaluating numerically the integral for the exchange constant. Having discarded the X point as the source of the behavior of J (z), Pruneda et al. (2002) subsequently turned their attention to the M point. The overall behavior of the exchange constant was in this case in excellent agreement with the experimental results of de Vries et al. (1997). They obtain exactly the same value of the decay length (3.6 Å) as in those experiments. The layout of ﬁg. 3.18 purposely mimics that of ﬁg. 5 in the work of de Vries et al. (1997) with which should be compared. However, the oscillatory behavior observed in these devices by Gareev et al. (2001) remains unexplained, additional studies being necessary in this respect. Up to now we have restricted the discussion to the Fe/Si interfaces. In principle, a convenient way to inject a spin-polarized current into a semiconductor is from ferromagnetic metals like Co and Fe which exhibit a signiﬁcant spin polarization even at room temperature. This control of the electron’s spin degree of freedom in semiconductors could pave the way to a new generation of electronic devices such as spin memories, spin transistors, and spin quantum computers. However, a fundamental problem must be solved before spins can be injected efﬁciently: the reactivity of transition metals with semiconductors can lead to magnetically dead layers which suppress the spin polarization across

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Fig. 3.18. Asymptotic behavior of the exchange constant as a function of the thickness of the spacer. Solid line is the result from our amended version of Bruno’s theory; dashed line is a ﬁt to e−z/3.6 . The inset shows the logarithm of J as a function of z to ease comparison with experiments. From Pruneda et al. (2002).

the interface by spin-ﬂip mechanism. Marangolo et al. (2002) investigated the magnetism of epitaxial ultrathin ﬁlms of Fe on ZnSe(001) by XMCD. In contrast to other metallic ferromagnet/semiconductor interfaces, no reduction of the Fe magnetic moment was found at the Fe/ZnSe(001) interface. This result is in good agreement with DFT calculations by Freyss et al. (2002). 3.11. Effect of light impurities (H, O) on magnetic multilayers The discovery by Grünberg et al. (1986) of oscillations of the inter-layer exchange coupling (IEC) between ferromagnetic slabs through a non-ferromagnetic spacer has opened a new ﬁeld in low-dimensional magnetism. The thickness of this spacer can be changed only by discrete values, which are a multiple of the lattice constant. Temperature-dependent interdiffusion modiﬁes essentially the oscillatory period. Reversible accumulation of hydrogen in Nb, V offers a new way for manipulation of IEC in magnetic superlattices via continuous variation of the spacer’s thickness (Klose et al., 1997; Hjörvarsson et al., 1997). A typical example of such a system is the Fe/V superlattice, where reversible, without hysteresis free switching of IEC from antiferromagnetic to ferromagnetic and vice versa with hydrogen was demonstrated via combination of magnetization and polarized neutron reﬂection (PNR) experiments (Hjörvarsson et al., 1997) as well as MOKE (Andersson et al., 1997; Labergie et al., 1999). The hydrogen atoms reside on interstitial positions in the V sublattice. Due to the restoring strain of the Fe layers, the expansion of the V spacer is restricted to the direction of the normal to the superlattice plane. The total thickness of the V layers can be changed reversibly at moderate hydrogen pressure without any memory effect. Electronic and magnetic structures of Fe/V superlattices with and without hydrogen in V spacer were investigated by Ostanin et al. (2000) using a FP-LMTO code. Hydrogen loading leads to the decrease of both the interface magnetic moments on V atoms and of the

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DOS at the Fermi energy. This low DOS can be one of the reasons for the increase of the superlattice resistance under hydrogenation and of the disappearance of the AFM-IEC for large hydrogen concentration. Due to the AFM coupling between Fe and V, the decrease of the induced V moments at the interface has to increase the total magnetic moment of the Fe/V superlattice with hydrogen uptake. This prediction was conﬁrmed by experiments with highly sensitive balance and in-situ loading with hydrogen (Labergie et al., 2001) but the variation of the total moment was found to be more than ten times larger than in ab initio calculations (Ostanin et al., 2000). Moreover, the value of the magnetic moment per Fe atom was shown to be very low. This disagreement was explained via calculations of the magnetic structure taking into account interface roughness and interdiffusion (Uzdin et al., 2002a). Intermixing introduced to the model by a developed random algorithm (Uzdin et al., 1999, 2001a) leads to better agreement with the experimental results. Hydrogen adsorption causes an enhancement and a phase change of the IEC in Ni/Cu/Ni trilayers on Cu(001) (Wu et al., 2000). Also, Marrows et al. (2000) saw that residual gases in the vacuum chamber caused damage to the IEC in Co/Cu multilayers. Chen et al. (1992) found the oxygen-induced magnetization reorientation from the perpendicular to in-plane orientation at the Fe bilayer deposited at Ag(001). The most striking fact was that the phenomenon occurred at an extremely low oxygen coverage. A similar effect (but with CO) was found by Hope et al. (1998) for Co ﬁlms at the Cu(011) substrate. Within a semiempirical tight-binding model, Pick and Dreyssé (2002) found indications of the tendency to surface–subsurface antiferromagnetic coupling (in Fe(001) bilayer), induced by oxygen and changing markedly the magnetic anisotropy. Nait-Laziz and Demangeat (1993) have already found a metastable antiferromagnetic coupling in the case of an Fe bilayer. The effect of oxygen seems to stabilize the antiferromagnetic conﬁguration of the Fe bilayer. The role of oxygen is also relevant in modifying the properties of a Co thin ﬁlm growth onto Fe(001) substrate (Duo et al., 2000). In particular the oxygen surfactant effect extend the stability range of the bct phase of Co up to 35 ML. Another important effect of oxygen is connected to the reversal of the Mn–Co ferromagnetic coupling observed by O’Brien and Tonner (1998). Indeed, O’Brien and Tonner (1998) as well as Choi et al. (1998) have shown that the Mn–Co coupling at the Mn/Co interface is of ferromagnetic type. This ferromagnetic coupling was obtained within TB-LMTO calculations by M’Passi-Mabiala et al. (2002). However, as seen by Pick and Demangeat (2003) within TB-LMTO, the effect of oxygen is to destabilize this ferromagnetic coupling between Mn and Co. A clear tendency towards an antiferromagnetic coupling between Mn and Co is obtained, in agreement with detailed experiments by Yonamoto et al. (2001). More generally the theoretical difﬁculties in the description of metal/oxide interfaces reﬂect the intrinsic complexity of the individual components. On one side is the oxide, which can have insulating, semiconducting or, in a few cases, even metallic character. On the other side is the metal, which can range from easily ionizable alkali metals to transition metals with complex band structure (Pacchioni, 2002; Mattson, 2002). Before closing this chapter let us notice the study of spin-polarized electrons in collisions of multicharged nitrogen ions with a magnetized Fe(001) surface by Pfandzelter et al. (2001). Also, Yang et al. (2002b) have applied atomic-scale SP-STM in the constant current mode to study the surface of Mn3 N2 and have clearly observed modulation of

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the normal row height proﬁle. The reproducible observations with different magnetic tips unambiguously demonstrate that only a spin-polarization effect can explain these results. Furthermore, this spin-polarization effect is a natural explanation in terms of the known bulk ordering of the magnetic moments of Mn3 N2 . These proﬁles are shown to agree with simulations based on FLAPW. One future direction of the study of magnetic multilayered systems is therefore the rather unexplored world of the interaction of metalloid with them and their drastic effect on the magnetic properties. 4. Towards the nanoworld 4.1. Introduction Clusters are an important state of matter, consisting of aggregates of atoms and molecules, that are small enough in order to have differing properties as bulk liquid or solid. Quantum states in clusters are size-dependent, leading to new electronic, optical, and magnetic properties (Brauman, 1996). Clusters offer attractive possibilities for innovative technological applications in ever smaller devices, and the ability to tune properties, especially in semiconductors, may produce novel electronic and magnetic capabilities. Exceptional progress is made in synthesis, analysis and theory in cluster research. With molecular beam measurements on ferromagnetic clusters, Billas et al. (1994) showed the evolution of ferromagnetism from atom to bulk. In section 4.2 we report on Ni free-standing clusters, a subject where nowadays theoretical and experimental results do agree to some extent. This agreement was not trivial to attain because some kind of spin-polarized quasi-ab initio molecular dynamics was necessarily to be included in the calculations. Up to know consensus has been reached only for the free-standing clusters of Fe, Co and Ni from the ﬁrst TM-series. There are some experimental indications concerning Cr and Mn: with the Stern–Gerlach technique, Bloomﬁeld et al. (2000) obtained magnetic moments averaging roughly from 0.5μB to 1.0μB per Cr atom for Crn clusters (n = 8–156). Later on, Knickelbein (2001) presented the results of magnetic deﬂection experiments which show that manganese clusters in the size range Mn11 –Mn99 display non-zero magnetic moments indicative of ferromagnetic ordering of spins, despite the fact that no known bulk phase of Mn displays such ordering. Hobbs et al. (2000) developed an all-electron projected augmented-wave (PAW) method for non-collinear structures, based on the generalized LSD theory. The method allows both the atomic and magnetic structures to relax simultaneously and self-consistently. The algorithms have been implemented within VASP (Vienna ab initio simulation package). A detailed study of small Fe and Cr clusters was done. However, no direct connection with the Stern–Gerlach results of Bloomﬁeld et al. (2000) was possible because the smallest cluster in the experiment started with 8 atoms. Therefore it looks like more should be done until a comparison between theory and experiment can be made. For Mn clusters we are presently not aware of any non-collinear calculation. Nevertheless let us point out an interesting DFT calculation by Rao and Jena (2002) concerning the stability and magnetic properties of small Mn clusters in the presence of nitrogen. Not only are their binding energies substantially enhanced, but also the coupling between the magnetic moments at Mn

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sites remains ferromagnetic irrespective of their size and shape. In addition, these nitrogendoped Mn clusters carry giant magnetic moments. Magnetic properties of free and supported vanadium clusters of up to four atoms have been calculated self-consistently using DFT, by Weber et al. (1997). While free clusters have ferromagnetic ground states, the supported V clusters in general prefer antiferromagnetic conﬁgurations. All these calculations were performed with minimal structural optimization. Taneda et al. (2001) optimized Vn free-standing clusters by employing a Tight-Binding-Molecular-Dynamic (TB-MD) method which used generally parametrized Hamiltonian matrix elements. For small n a self-consistent TB-MD (SCTB-MD) was also carried out. SCTB-MD takes account of charge-transfer effects in order to avoid unphysical large charge transfer which can occur in non-SCTB calculations. However, no spin-polarized calculations were performed by them. Within the GGA approach, Moseler et al. (2001) solved the Kohn–Sham equations by using the Born–Oppenheimer local-spindensity molecular dynamics method, with scalar-relativistic nonlocal pseudopotentials. They showed that some of the very small Pd clusters are magnetic. Underlying this behavior is the hybridization of atomic s and d states when clusters are formed that depletes local d contribution around each atom and leads to an open-shell-like behaviour. Later on, Moseler et al. (2002) reported on a ﬁrst-principles investigation of soft-landing of PdN clusters (N = 2–7 and 13) onto a MgO(001) surface containing an oxygen vacancy. The interaction with the surface quenches the spin of clusters with small value of N . Although the surface tends to reduce the spin of the adsorbed cluster, clusters larger than Pd3 remain magnetic at the surface, exhibiting several low-lying structural and spin isomers. The results of Moseler et al. (2002) provide the impetus for further investigations regarding the interplay of structural and magnetic states of supported clusters and their catalytic properties. For both V and Pd clusters the experimental situation looks particularly “obscure” and no clear onset of magnetization has been measured yet! To increase the confusion existing in this domain let us mention the calculation by Kumar and Kawazoe (2002) displaying speciﬁc Pd-clusters presenting magnetic moments as large as 0.61μB per atom. The remaining part of this chapter will focus on nanoclusters deposited on substrates like Co nanowires supported on Pd(110) (section 4.3), Co clusters supported on noble metals (section 4.6) and a non-collinear approach for V, Cr, Mn and Fe aggregates on Cu or Ag. We report also on a speciﬁc example of Ni clusters in Al. Finally we say a few words of a fast growing subject connected with the discovery, ten years ago, of carbon nanotubes. We try in section 4.4 to say a few words on transition metal clusters in contact with the various forms of carbon. The most interesting part in this section 4.4 is the present availability to encapsulate magnetic clusters in carbon nanotubes. The ﬁeld of adsorbed nanoclusters has been boosted by the rather formidable development, due to atom by atom manipulation via STM/STS (Wiesendanger and Bode, 2001; Crommie, 2000). These advances have allowed the observation of the Kondo effect (1964) for individual magnetic atoms. One hallmark of the Kondo effect is a strong temperatureinduced broadening of the Kondo resonance (Nagaoka et al., 2002). Jamneala et al. (2001) used atomic manipulation and STS to probe the local behaviour of frustrated, antiferromagnetic systems by fabricating and analyzing individual Cr trimers at the surface of gold. Also, with STS, Knorr et al. (2002) have investigated the Kondo effect of single Co adatoms on Cu surfaces and Susaki et al. (2002) did a similar study of Fe adatoms by using X-ray

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and ultraviolet photoemission. Finally, for a Co impurity on a Au(111) surface (Madhavan et al., 1998), the experimentally observed width and shift of the Kondo resonance were found in accord with those obtained from a combination of band structure and strongly correlated calculations (Ujsaghy et al., 2000). Similarly, atoms from the 3d series were adsorbed onto a Au(111) substrate and spectroscopically probed with ultrahigh vacuum STM at 6 K. Elements near the center of the 3d series (V, Cr, Mn and Fe) displayed featurless electronic structure over the energy range studied, while elements near the ends of the series (Ti, Co and Ni) showed narrow Kondo resonance near the Fermi energy (Jamneala et al., 2000). Chado et al. (2001) have grown Rh ﬁlms in the thickness range between 0.1 and 6 MLs on Au(111). Their investigations over a wide range of coverages and deposition temperatures were performed by means of variable temperature UHV-STM, Auger spectroscopy, and in situ Kerr effect measurements. Upon deposition of Rh on Au(111) at 300 K, STM images show that the Rh self-organize at the elbows of the gold reconstruction. By increasing the Rh coverage up to 0.8 ML, big Rh clusters are formed by coalescence. At 30 K, in the early stages of growth (0.006 ML), the formation of depressions and protrusions is attributed to a mixed Rh–Au interface layer, although these metals are known to be immiscible below 900 K in the bulk. No indication of magnetism was observed down to 30 K which looks striking because Rennert et al. (1993) as well as Bazhanov et al. (2000) did obtain sizeable magnetic moments for respectively Rh clusters and Rh chains on Ag(001). These semiempirical TB calculations were performed in real-space for the Rh cluster and in the k-space for the Rh chains. It is clear that the experimental results are at odds with the theoretical ones. Besides a possible intermixing between Rh adatoms and the substrate which can kill the magnetic moment it is clear that for these nanostructures the Curie temperature may be very low. Preliminary XMCD results by Ohresser (2002) seem to indicate some kind of magnetism at very low-temperature, but more should be done until a clear conclusion can be attained. The possible absence of magnetism could be caused by some kind of interdiffusion at the interface. Burnet et al. (2002) have displayed, within nuclear magnetic resonance (NMR), magnetism of small Rh particles on titanium-oxides. It seems therefore that the onset of magnetism in small supported Rh particles may depend on the non-magnetic substrate used. The magnetism of self-organized nanostructures and clusters is an active ﬁeld of research due to a rich variety of novel properties that are not found in the bulk (MeiwesBroer, 2000). Oepen and Kirschner (1999) discussed the actual concepts for fabricating nanostructures. The ﬁrst approach is to use conventional lithography to obtain quick access to nanostructures for studying their magnetic properties. The second approach uses self-assembled structures at surfaces and in thin ﬁlms to fabricate large arrays of nanostructures. The goal of this research is to ﬁnd new and easy methods for producing arrays of high periodicity with nanomagnets of perfect structure and narrow size distribution. In this context it is useful to see the effect of the coverage on the magnetic anisotropy. Perpendicular spin orientation in ultrasmall Fe islands on W(110) was observed by Röhlsberger et al. (2001) by nuclear resonant scattering of synchrotron radiation. Similarly Gambardella et al. (2002a) detected ferromagnetism in one-dimensional monoatomic metal chains of Co constructed on a Pt substrate. They found evidence that the monoatomic chains consist of thermally ﬂuctuating segments of ferromagnetically coupled atoms which, below a threshold

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temperature, evolve into a ferromagnetic long-range ordered state owing to the presence of anisotropy barriers. The Co chains are characterized by large localized orbital moments and correspondingly large magnetic anisotropy energies compared to two-dimensional ﬁlms and bulk Co. In this context we can notice the work by Pampuch et al. (2000) concerning the one-dimensional spin-polarized quantum-wire states in Au on Ni(110). Agreement between ab initio DFT calculations and photoemission spectra were obtained. Theoretical calculations for zero- and one-dimensional magnetism require very large periodic unit cells and thus very few such systems have been investigated. Stepanyuk et al. (1994) have studied the magnetism of 3d, 4d and 5d elements adsorbed as single atoms, in dimers and other chains, using the Korringa–Kohn–Rostocker approximation within the DFT-framework. As expected, the magnetization becomes enhanced when the coordination number decreases. One also observes that the magnetic moment increases by decreasing the charge density sampled by an adsorbate. Papanikolaou et al. (2000) presented ab initio calculations of scanning tunneling spectra for the Fe(001) surface and for 3d impurities in this surface. The calculations are performed by the FP-KKR-GF method and also partly by the FP-LAPW. Similarly, the magnetic states in mixed FeX (X = 3d) clusters on the Ag(001) surface have been determined by Stepanyuk et al. (1998) whereas, self-consistent ﬁeld molecular orbital theory and KKR-GF method are used to study the binding energies, electronic structure and magnetic properties of Fe, Co, and Ni dimers supported on Cu(001) surface (Nayak et al., 2001). Trioni et al. (2002) proposed a surface effective-medium approach for the determination of the local magnetic moment of 3d adatoms on Ag(100). First, by confronting ab initio results for an Fe atom, as a prototype of a 3d magnetic impurity on Al(100) and on Al-like jellium. They show how the electronic and magnetic properties of the realistic system are described very well by jellium. Next, they made it clear how the magnetic moment of the system can be simply related to the unperturbed local charge density of the metal surface at the adatom position. Fang et al. (2000) determined the electronic and magnetic structures of adatoms of O and Au on Fe(001) surface with the local-spherical-wave (LSW) method developed by Leuken et al. (1990). The LDOS of adatoms, steps, clean Fe surface and Fe atoms in the vicinity of a void are essentially different and could be used as a ﬁngerprint in STM measurements. The characteristic feature of the Fe(001) surface states has not been inﬂuenced signiﬁcantly by oxygen impurities. However, the inﬂuence of gold impurities on the Fe(001) surface states is signiﬁcant. Giant magnetic moments were observed in the anomalous Hall effect measurements for Fe and Co impurities on the surface and in the bulk of Cs ﬁlms (Beckmann and Bergmann, 1999). The deduced values of magnetic moments are about 7μB for Fe and 8μB for Co. These are unexpectedly large magnetic moments, as compared to bulk Fe and Co values. Kwon and Min (2000) have explored the origin of these giant magnetic moments using the relativistic LSDA approach. The deﬁciency of the LSDA has been removed by introducing Hubbard-like interaction terms for 3d electrons. They found that the orbital moment is not quenched as it is usually the case in Fe and Co metal. The conclusion of the work by Kwon and Min (2000) is that both Coulomb correlation and the spin-orbit interaction at the 3d site should be properly considered to obtain results consistent with experimental data. Giant moments have also been obtained for Fe and Co on and in rubidium and potassium ﬁlms (Hossain and Bergmann, 2002). X-ray absorption spectroscopy (XAS) spectra

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by Gambardella et al. (2002b) indicates that Fe, Co and Ni have localized atomic ground states with predominantly d 7 , d 8 , and d 9 character, respectively. XMCD by Gambardella et al. (2002b) showed that the localized impurity states possess large, atomic-like orbital moments. Therefore, experiments of Gambardella et al. (2002b) conﬁrm the results obtained by Bergmann’s group. Song and Bergmann (2002) determined the anomalous Hall resistance of thin quench-condensed ﬁlms of Na and K covered with 0.01 ML of vanadium. A magnetic moment of about 7μB was observed. We are awaiting XMCD measurements to conﬁrm the results of Song and Bergmann (2002). However V impurities do not couple usually ferromagnetically but antiferromagnetically as we have already discussed. Therefore one possibility is to perform LMDAD or add a magnetic ﬁeld to the system in order to destroy a possible antiferromagnetic conﬁguration. Indeed, the XMCD experiments of Gambardella et al. (2002b) were done in a ﬁeld of 7 T so that a possible antiferromagnetic (or more generally non-collinear) cannot survive. XMCD is therefore a suitable tool for measuring a possible magnetic moment on V. 4.2. Ni free-standing clusters Current Stern–Gerlach techniques based on the deﬂection of a cluster beam due to its interaction with an external inhomogeneous magnetic ﬁeld allow to determine the average magnetization per atom in free-standing clusters. Pioneering work along this line was performed by Billas et al. (1993, 1994, 1997). For the 3d ferromagnets, a non monotonic decrease of the average magnetization with increasing cluster size was observed (up to ≈700 atoms). Apsel et al. (1996), performed the most precise measurements of the average magnetic moment μ¯ in Ni clusters. These Stern–Gerlach deﬂection experiments were done for size selected clusters between Ni5 and about Ni700 . The non monotonic behavior was again clearly observed, in particular for the region below Ni100 . The decrease of the average magnetization with increasing cluster size is accompanied by oscillations displaying a strong reduction of the magnetization for certain sizes (sharp minima occur at Ni6 and Ni13 , and broad minima around Ni34 and Ni56 ) and a strong increase for other sizes (sharp maxima at Ni5 , Ni8 and Ni71 and broad maxima around Ni20 and Ni42 ). From the theoretical side, the most systematical study of free-standing Ni clusters, where both the global minimun structures and the electronic structure were calculated, is that of Bouarab et al. (1997) and Aguilera-Granja et al. (1998). They calculated the spinpolarized electronic structure of NiN clusters by solving self-consistently a tight-binding Hamiltonian for the 3d, 4s and 4p valence electrons and obtained a very good agreement with available experiments. The calculations were restricted to clusters of relatively small size, namely N 60. They used geometries determined by molecular dynamics simulations as well as a semiempirical many body potential whose functional form is derived from tight-binding ideas. Also Guevara et al. (1997) performed self-consistent calculations of the magnetic moments and other properties of Ni, Co and Fe clusters using a tightbinding Hamiltonian with s, p and d orbitals. The calculations for NiN were extended up to N = 177. However, the cluster geometries were restricted to be highly symmetric fragments of the fcc crystal lattice, whereas these authors admitted that this is not the structure suggested by the experimental results. A conclusion of the molecular dynamics study of NiN clusters is the pattern of icosahedral growth. This was present as early as for Ni7 , where the ﬁve fold symmetry is observed

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Fig. 4.1. Calculated average magnetic moments compared with the experimental results of Apsel et al. (1996) and Billas et al. (1993, 1994). Unit is the Bohr magneton (μB ). From Aguilera-Granja et al. (1998).

in its pentagonal bipyramid structure. The icosahedral growth for small Ni clusters is consistent with the saturation limits in experiments of reactivity with light molecules (Parks et al. 1994, 1995; Klots et al., 1990, 1991; Vajda et al., 1997), and this type of growth has been also studied theoretically by Montejano-Carrizales et al. (1996) using the embedded atom model (EAM) developed by Foiles et al. (1986) and Daw (1989). More details concerning the geometrical study can be found in these works and cited references. The magnetic moments per atom of NiN clusters, calculated by Bouarab et al. (1997) and Aguilera-Granja et al. (1998) with N up to 60, are compared in ﬁg. 4.1 with the experimental results. The experiments indicate an overall decrease of μ¯ with cluster size and this is due to the progressive increase of the average atomic coordination. The experimental decrease of μ(N) ¯ is non-monotonous and shows well deﬁned oscillations: sharp minima of μ¯ occur at Ni6 , Ni13 , and Ni55 –Ni56 , and another minimum, not so well deﬁned, is seen in the region around Ni34 . The key ideas for explaining most of the minima and maxima associated to particular sizes known as magnetic magic numbers are, ﬁrst of all, that the local magnetic moments decrease with the increase of the local atomic coordination. This idea was already used by Vega et al. (1995) to explain the magnetic moments at planar surfaces and surfaces with defects. Its validity can be seen from ﬁg. 4.2 which displays the local magnetic moments (μi ) versus the local coordination number (Zi ) for Ni20 and Ni45 in panels (a) and (b), respectively, as calculated by Aguilera-Granja et al. (1998). The second idea is that the average magnetic moment decreases when the interatomic distances decrease (the d band becomes wider). Ni13 is a compact cluster while Ni12 and Ni14 contain some low coordinated atoms. This leads to the minimum of μ¯ for Ni13 . On the other hand there is a sharp increase of the average nearest-neighbor distance dnn between Ni6 and Ni7 that produces the minimum of the μ¯ at Ni6 . In contrast, the oscillations of μ¯ for larger N appeared to be more difﬁcult to explain, linked to the increasing difﬁculties to be conﬁdent about the geometrical structures of the clusters as their size grows. Fig. 4.3 shows the separated contributions to μ¯ from sp and d electrons. The sp moment is important for small clusters, but for the rest it provides a very small contribution. In fact, the

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Fig. 4.2. Correlation between the local magnetic moment (μi ) and the local coordination number (Zi ), displayed here for Ni20 and Ni45 in upper and lower panels, respectively. The lines are linear ﬁts as an aid to the eye. From Aguilera-Granja et al. (1998).

pronounced oscillation of μ¯ in the range N = 5–9 is due to the oscillation of μ¯ sp , and μ¯ sp also contributes to the minimum of μ¯ at N = 13. Returning to ﬁg. 4.1, if we consider the whole range of sizes studied by Bouarab et al. (1997) and Aguilera-Granja et al. (1998), the calculations reveal a broad trend that can be characterized as an initial decrease of μ¯ for sizes up to about Ni28 followed by a weak increase of the μ¯ between N ≈ 28 and N = 60. ¯ This behavior is mainly related to the variation of the average coordination number Z. Superposed to the overall behavior just discussed, the calculations display some local anomalies of μ(N). ¯ First of all, the local minima at Ni6 and Ni13 , and the maximum at Ni8 , that have already been discussed and agree with the experimental features at the same sizes. Second, a local minimum at Ni55 . This minimum correlates with the minimum observed by Apsel et al. (1996) at N ≈ 56. Although the experimental minimum is broader and better

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Fig. 4.3. Separated sp (upper panel) and d (lower panel) contributions to the average magnetic moment as a function of the cluster size. From Aguilera-Granja et al. (1998).

deﬁned than the calculated one, the calculated magnetic moment for N = 55 is clearly below the corresponding moments for Ni54 and Ni56 . Clusters larger than Ni12 contain atoms with a full coordination of Z¯ = 12 in the innermost part of the cluster. The local magnetic moments of those atoms are however different from the bulk value, and do not tend smoothly to that limit. For example, the local magnetic moments for the innermost atoms with Z¯ = 12 in Ni20 , Ni33 , and Ni45 are in the range 0.45μB –0.47μB , 0.25μB –0.28μB and 0.42μB –0.60μB, respectively (see ﬁg. 4.2 for Ni20 and Ni45 ). The variation of the local moments is also large within a given cluster. For the same three clusters, the local magnetic moments vary between 0.46μB and 1.43μB for Ni20 , between 0.25μB and 1.33μB for Ni33 , and 0.43μB and 1.35μB for Ni45 . The local moments increase, evidently, from the inner to the outer part of the cluster: in other words, there is a rough correlation between the local magnetic moment μi and the local coordination number Zi that is displayed in ﬁg. 4.2 for Ni20 and Ni45 . Explaining the maxima observed by Apsel et al. (1996) was a more difﬁcult task. One possibility, suggested by the results of Guevara et al. (1997) on the magnetic moments of small fcc nickel clusters and by molecular dynamics simulations of Lathiotakis et al. (1996) using a tight-binding Hamiltonian, is that the structures are fcc instead of icosahedral in

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the regions corresponding to those maxima. In this context, we mention the experimental results by Parks et al. (1997) measuring the saturation coverage of Ni38 with different molecules (N2 , H2 , CO). Those results have been interpreted as indicating that the structure of this cluster is a truncated octahedron cut from a face-centered-cubic lattice. Motivated by this experimental result, Aguilera-Granja et al. (1998) made a detailed comparison between fcc and icosahedral structures for Ni38 and also for other NiN clusters with N = 13, 14, 19, 23, 24, 33, 36, 37, 39, 43, 44, 55, and 68. The sizes N = 36, 37, 39 were selected because those are near N = 38. On the other hand, for N = 13, 19, 43, 55, one can construct fcc clusters with ﬁlled coordination shells around a central atom, and for N = 14, 38, 68, clusters with ﬁlled coordination shells around an empty octahedral site of the fcc lattice. Finally, N = 23, 24, 44 are of interest for comparison with other authors. They obtained that the fcc isomer is more stable than the icosahedral one for Ni38 by 0.35 eV, a result that agrees with the reactivity experiments of Parks et al. (1997). For Ni38 they found μ¯ fcc = 0.99μB , that cuts the difference between the experimental and theoretical results to one half of the value of ﬁg 4.1. This moderate increase in the average magnetic moment with respect to its value for the icosahedral structure can be explained by the lower average coordination of the fcc geometry. In conclusion, the changes that the new calculations introduced on the theoretical results of ﬁg. 4.1 were negligible and the features derived from the model of icosahedral growth remain valid. It is interesting to compare the tight-binding results on the average magnetic moments of Bouarab et al. (1997) and Aguilera-Granja et al. (1998), with the LCAO density-functional calculations of Reuse et al. (1995) for small N ( 13). Those density-functional calculations predict substantially lower moments. 4.3. Co nanowires supported on Pd(110) Among the different supported nanostructures, the nanowires deserve a particular attention because they combine the properties typical of supported clusters with those related with the conﬁnement in one dimension. In this class of materials, several transport properties like the strong reduction of the scattering or the quantization of the conductivity have been found by Sakaki (1980) and Van Wees et al. (1988), respectively. However, from the magnetic point of view these promising systems are rather new. In this context, most of the progress made so far is related with the experimental techniques of their production at different size scales. Some of those techniques are described by Eigler and Schweizer (1990), Ehrichs et al. (1992) and McClelland et al. (1993). Figuera et al. (1995) showed the possibility of having a regular distribution of nanowires nucleated on the steps of vicinal surfaces. These studies show that it is possible to exploit the symmetry of the surface. This idea was used by the group of Bucher et al. (1994) for the growth of quasi 1D-wires of an fcc transition metal on a (110) surface of an fcc substrate. Concerning the magnetic properties of transition metal supported nanowires, several questions deserve to be investigated: magnetic order and magnetic coupling between wires as a function of inter-wire distance, average magnetic moment of the system, wire-substrate magnetic interaction, magnetic anisotropy, magneto-optics. For technological purposes one would like to assemble atomic wires optimizing their packing, but this packing may have a limit because if the wires are too close together, the intrinsic properties of the wires can be lost.

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Robles et al. (2000a, 2000b) presented the ﬁrst theoretical study of the electronic structure and related magnetic properties of monoatomic Co wires supported on Pd(110). The Co/Pd interfaces have attracted the attention of the Scientiﬁc Community during the last decade. In particular, Co/Pd multilayers have been studied since 1985 due to their perpendicular anisotropy which makes them suitable for magneto-optic devices, as shown by Carcia et al. (1985). The morphology of the system studied in the work of Robles et al. (2000a, 2000b) was modeled following the experimental results of Bucher et al. (1994) for Cu wires on Pd(110). The same atomic arrangement is expected for Co deposited on the same substrate. The spin-polarized electronic structure of the system was determined by self-consistently solving a semiempirical spd-band model Hamiltonian formulated in the real space and parametrized to ab-initio TB-LMTO calculations for the ideal Co monolayer supported on Pd(110). The system consists in a set of monoatomic Co-wires supported on the (110) semi¯ direction. This is the aligninﬁnite substrate of Pd. The wires are oriented along the (110) ment direction of the Cu wires grown and characterized by Bucher et al. (1994). The reason why the wires are aligned in this direction is that the (110) surface of an fcc system presents channels, where the wires are deposited (see ﬁg. 4.4). Therefore, the inter-wire distance can be characterized by the number of unoccupied channels between adjacent wires. When all the channels are occupied by wires one has the complete Co monolayer. This situation corresponds to the minimum inter-wire distance. The situation of “inﬁnitely” separated wires corresponds to that of an isolated supported Co wire. Due to the loss of neighbors, the magnetic moment of the wire (2.22μB per atom) is enhanced by about 8% with respect to the supported Co monolayer and much more (about 40%) with respect to the Co fcc bulk. The electronic localization in the wire compared with the complete overlayer and the bulk is reﬂected in the narrowing of the densities of states, accompanied by a larger degeneration at the Fermi level. Another interesting result is the spin-polarization induced by Co in the neighboring Pd atoms. It is well known that certain paramagnetic transition metals (V, Pd) can be magnetic in the presence of a strong ferromagnet. The induced spin-polarization in Pd is lower in the case of the supported wire than for the complete Co overlayer. The results of Robles

¯ direction where Fig. 4.4. View of the (110) surface of an fcc system. One can see the channels along the [110] the atomic wires are deposited. From Robles et al. (2000a, 2000b).

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et al. (2000a, 2000b) show that the Pd layer at the interface with the Co overlayer displays a magnetic moment of 0.48μB , whereas a much smaller moment of 0.22μB is on the nearest Pd neighbour of the Co wire. This is a consequence of the reduction of the Co–Pd hybridization when the Co coordination of the Pd sites decreases as one goes from the overlayer to the wire conﬁguration. Previous ab-initio KKR calculations of Oswald et al. (1986) for a dilute Co atom-impurity embedded in Pd bulk also showed this effect. In that case, the induced spin-polarization in Pd is even lower than for the supported wire system due to the extremely low Co–Pd hybridization (only one Co atom surrounded by Pd). Although the induced magnetic moment in Pd is relatively small, one open question concerns a possible magnetic interaction through the Pd substrate. In order to answer this question, Robles et al. (2000a, 2000b) calculated the magnetic moment distribution when the inter-wire distance is reduced and the Co-wires approach each other. Besides, they compared different possible magnetic conﬁgurations, in particular the adjacent Co wires with either parallel (P) or antiparallel (AP) magnetic coupling. Fig. 4.5 reports the magnetic moment distribution of both P and AP solutions for the supported Co wires separated by one channel. For this inter-wire distance, the resulting

Fig. 4.5. Local magnetic moment distribution (in units of μB ) for the P solution (upper half) and the AP solution (lower half) of the supported Co wires separated by one channel. From Robles et al. (2000a, 2000b).

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values for the local magnetic moment at the Co atoms were similar to those found in the case of the isolated wire. Furthermore, the absolute values in both P and AP conﬁgurations were also similar, an exception being those Pd sites in the AP conﬁguration located at the same distance of both the wire with positive magnetic moment and the wire with negative magnetic moment. Those Pd atoms are magnetically frustrated (they have 0μB ). This result indicates that at this inter-wire distance, the magnetic interaction between wires (both direct and indirect via the Pd substrate) is negligible. If magnetic interactions would take place, two behaviors would be expected. On the one hand, the local magnetic moments at the Co wires in the P solutions should reduce and converge to the corresponding value in the supported overlayer (2.05μB ). On the other hand, since Co is a strong ferromagnet, the AP solution should become less stable than the P solution. Neither of these effects were present for this inter-wire distance. The magnetic interaction between adjacent wires starts only when the intra-wire distance approaches 3.89 Å, that is, the inter-wire distance for the complete supported Co monolayer. This magnetic interaction is, however, small. The AP solution persists even in the case of the supported Co monolayer with absolute values for the local magnetic moments in the Co sites similar to those of the P solution. Besides, the energy difference per atom between the AP conﬁguration and the ground state P conﬁguration is about half the value of KT at room temperature. In view of the possible technological applications, these results show that it is possible to optimize an assembly of Co wires supported on Pd(110), since they preserve their intrinsic magnetic moment (nearly saturated) for inter-wire distances close to the monolayer regime. Besides, perpendicular magnetic anisotropy was predicted by Dorantes-Dávila and Pastor (1988) in a supported Co wire. All this opens the possibility of using these new systems for magneto-optic purposes. 4.4. Transition metal clusters in contact with carbon Experimental investigations of the interaction between 3d transition metal elements and C60 indicated a different behavior of the early 3d elements when compared to the late 3d transition elements (Nagao et al., 1998). Ferromagnetic nanoclusters have much better magnetic properties than bulk materials due to their single domain nature. However, application of Fe, Co, Ni, and their alloy nanoclusters is limited due to air-oxidation. The best way to protect metallic nanoclusters from air-oxidation is to encapsulate them with inert materials. Among inert materials, carbon appears ideal because it forms a variety of empty fullerenes such as onions and nanotubes to enclose metallic nanoclusters (Prados et al., 2002; Lee et al., 2002a). However this point seems to be at odds with STM studies by Zha et al. (2001) showing that Cr ﬁlling of carbon nanotubes results in strong electronic interaction between tube and ﬁlling materials. Rey et al. (2000) performed ab initio molecular dynamics simulations of Ni–C clusters using a fully self-consistent density-functional method that employs linear combinations of atomic orbitals as basis sets, standard norm-conserving pseudopotentials and GGA to treat exchange and correlation. Andriotis et al. (1999) and Froudakis et al. (2001) performed tight-binding molecular dynamics (TB-MD) as well as DFT based SIESTA calculations to investigate the magnetic conﬁgurations of small Nim Cn clusters and Ni atoms interacting with a graphite surface. The graphitic surface was simulated by a portion of a bulk graphitic plane. In the case of a Ni atom above the center of a hexagon the strong Ni–C interactions

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results in considerable distortions in the planar structure beneath the Ni atom. The value of the magnetic moment of the Ni atom is in quantitative agreement with the one obtained by Krüger et al. (1998b). Andriotis et al. (2000) presented a systematical theoretical study of the interaction of V and Ni atoms on graphite and C60 by considering all possible bonding sites. The calculation is performed with tight-binding molecular-dynamics (TB-MD) and Gaussian-98. The use of a minimal parameter basis set makes the scheme more transferable from one environment to another one. Moreover, TB-MD scheme is a suitable tool for the optimization of the unconstrained geometries. The graphite is simulated by a portion of a graphene sheet consisting of 128 carbon atoms. Molecular-dynamics relaxation resulted in stable bonding for V on graphite at hole (also called hollow) and atop sites. The adsorption of V is accompanied by a considerable distortion of the graphene sheet, especially for C atoms in the neighborhood of a V atom. The total energies for the fully relaxed geometries were obtained with Gaussian-98 and showed the ordering Ehole < Eatop , with the hole site being more stable by 0.50 eV. This is contrary to the results of Duffy and Blackman (1998) and the discrepancy is attributed to the neglect of relaxation effects. The same energetic ordering was obtained for Ni on graphite. At each adsorption site, the V atom exhibits a magnetic moment and there is appreciable charge transfer to or from the graphite atoms. The results of Andriotis et al. (2000) do not support the high-spin states for V on graphite reported by Duffy and Blackman (1998) and Krüger et al. (1999). Actually Duffy and Blackman (1998) calculated the stable positions and magnetic moments of the whole 3d-TM series of adatoms on graphite. Here we brieﬂy recall the case of the “hollow” position (Krüger et al., 1997b, 1998a) where the adatom sits above the center of a C6 ring at a distance h from it (C6v symmetry). In this model, detailed by Krüger et al. (1997b, 1998a), the C-cluster Hamiltonian was diagonalized by constructing appropriate molecular orbitals (MOs) which transformed as the irreducible representations of the symmetry group (ﬁg. 4.6). TM d orbitals and C MOs were then coupled through hybridization, which vanished for certain symmetry. By the construction of symmetrical C MOs, the number of non-vanishing hybridization integrals (between the TM atom and the C cluster) was minimized. The Hamiltonian was then resolved in the conﬁgurationinteraction approach; the values for Udd , Jdd , Udc and Jdc were kept the same as in the article of Parlebas et al. (1997), but now, the h-dependent hybridization is fully calculated. Also Peng et al. (1996) determined the equilibrium position of a single non-magnetic Fe atom adsorbed on the graphite surface by ab initio FLAPW total-energy calculation. They actually considered an extremely dilute Fe ML. The hollow Fe site 1.6 Å above the surface was found to be the stable position, in rather good agreement with Duffy and Blackman (1998). Krüger et al. (1999) extended the preceding study to the whole 3d-series for the three characteristic adsorption sites: hollow, on-top and bridge. In addition to the previously considered work of Peng et al. (1996), we should also recall that Duffy and Blackman (1998) and Duffy et al. (1998) calculated the stable positions and magnetic moments of 3d-TM adatoms on graphite, using LCAO molecular approach within DFT: they obtained that the early 3d-TM elements up to Mn occupy the on-top position with h = 2.1 Å while the late 3d-TM elements Fe, Co and Ni occupy the hollow site with h = 1.5 Å. Now, taking the preceding equilibrium positions and related h values, Krüger et al. (1999) found that the various 3d adatoms along the series are each in a high spin (Hund’s rule) ground state. Upon decreasing h, however, they observed in some cases, a ﬁrst-order spin

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Fig. 4.6. Scheme of the Hamiltonian when the adatom sits at a hollow position. Thick horizontal lines represent orbitals. TM core and d orbitals on the left and C cluster orbitals on the right. Thin solid lines indicate non-zero hybridization integrals. Dotted lines symbolize electron correlation. From Krüger et al. (1998a).

transition to a low spin state. These spin transitions are accompanied by electronic transitions between differing d orbitals and can be explained in an effective crystal ﬁeld model. The photoemission spectra (ﬁg. 4.7) obtained by Krüger et al. (1999) show intra-atomic exchange splitting as well as charge transfer satellites from ﬁnal state core hole screening by graphite π -electrons; the intensity of the charge transfer satellites increases with n0 , the number of d electrons, which reﬂects the increasing electron–electron correlation, from light to heavy 3d-elements. For the late 3d-elements, from Mn on, charge transfer and exchange satellites are strongly mixed and the spectra have essentially a three-peak structure. We conclude that for those elements, especially also for Fe, the observed splitting in the 3s XPS cannot be used as a reliable measure of the magnetic moment of the adatom. It seems that the next step in the calculation (in order to be realistically compared with most experiments) should concern TM islands on graphite. A straightforward extension could be to perform, for example, TB or PAM calculations. Unfortunately as discussed in this review paper a major difﬁculty cannot be solved satisfactorily by these types of methods: it is the determination of the distance h between the metal adatom and the graphite surface. The only realistic determination of this quantity was obtained by Peng et al. (1996) and Chen et al. (1997) in the case of 3d and 4d-TM elements on graphite, respectively. They both used full potential (FP) codes. However the geometrical conﬁguration used by Chen et al. (1997) was not compatible with the results of Pfandzelter et al. (1995). What is interesting in the calculations of Chen et al. is that the magnetic moment of the TM atoms are dramatically dependent on the metal-carbon distance. Therefore, only FP codes can be used to determine the magnetization of TM adsorbed clusters as obtained by Binns et al. (1992, 1999). Up to now we are not aware of any of such calculation. Duffy and

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Fig. 4.7. 3s XPS spectra of 3d TM adatoms on graphite (full lines) for the adsorption sites hollow (h), on top (t) and bridge (b). From Krüger et al. (1999).

Blackman (1998) performed very interesting calculations for 3d-TM adatoms and pairs on graphite within DFT. It is not trivial to extend such type of calculations to an arbitrary cluster conﬁguration on graphite because of considerable computing-time. What should be rewarding might be the determination by ab initio molecular dynamics of the growth process of TM adatoms on graphite. A few papers using molecular dynamics with empirical potentials have been published with interesting results. However, as said before, the TM-C distance is very critical so that only molecular dynamics in the Car–Parinello type of approach (1985) could give pertinent results. Using dichroism in X-ray absorption and photoemission, Binns et al. (2002) and Baker et al. (2002) studied the magnetic behavior of size-selected Fe nanoclusters adsorbed on a graphite substrate. Actually these clusters of sizes ranging from about 180 to 690 atoms did show enhanced orbital and spin moments. The total moment in the smallest 181-atom-

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cluster was found to be 2.4μB , i.e. about 10% larger than the bulk but still less than in a free cluster of comparable size (2.6μB ). Approximatively half of the enhancement was due to a large increase of the orbital contribution. The moments were decaying towards the bulk value with increasing cluster size or with increasing particle density on the surface. Also dense cluster monolayers develop a coherent in-plane anisotropy due to the overall shape of the ﬁlm. Finally coating the deposited Fe clusters with a Co ﬁlm substantially increased the spin moment of Fe as well as the total corresponding moment to a value close to that found in free clusters of the same size. The magnetization dependence of the quasiparticle spectrum and hysteresis in ferromagnetic metal nanoparticles was discussed by Cehovin et al. (2002). They used a microscopic Slater–Koster tight-binding model with short-range exchange and atomic spin-orbit interactions that realistically captures generic features of ferromagnetic metal nanoparticles to address the mesoscopic physics of magnetocrystalline anisotropy and hysteresis in nanoparticle–quasiparticle excitation spectra. Calculations show that the total energy as a function of magnetization direction per atom ﬂuctuate. The discrete quasiparticle excitation spectrum of a nanoparticle displays complex non-monotonic dependence on an external magnetic ﬁeld, with abrupt jumps when the magnetization direction is reversed by the ﬁeld, explaining spectroscopic studies of magnetic nanoparticles. For a VC60 cluster (Andriotis et al., 2000), V binds at hole, atop, and bridge sites, while the total energies satisfy the relation Ehole < Eatop < Ebridge . The hole site is more stable than atop and bridge sites by 0.37 and 0.95 eV, respectively. On the contrary, for Ni, the hole site is unstable while the atop site is the most stable. A comparison of the results for V on graphite and VC60 demonstrates that the substrate-curvature effect makes qualitative differences in the relative stability of the various adsorption sites. Furthermore, qualitative differences between V (early 3d-element) and Ni (late 3d-element) was pointed out in this study of the stable adsorption sites for V and Ni on graphite and C60 . V and Ni exhibit substantial magnetic moments and undergo signiﬁcant charge transfer process that depend on the detailed conﬁguration of their adsorption sites on both the graphite and the C60 . It is worth to mention that Billas et al. (1999) determined the structural and electronic changes of C60 when one or two atoms of carbon are replaced by Si atoms in order to form heterofullerenes like C59 Si and C58 Si. Their calculations were based on Car–Parinello’s method (1985) and then extended to paramagnetic Fe atoms (Billas et al., 2000). These preliminary results have to be completed by a study of the thermal stability of the considered structures in terms of possible cage destruction in metal-fullerene clusters (Tast et al., 1996). Finally let us mention the study of size-dependent magnetic properties of Ni/C60 granular ﬁlms by Zhao et al. (2002). Actually the authors considered Ni nanoparticles embedded in an amorphous C60 matrix with various nominal ratios of Ni atoms to C60 molecules (NNi /NC60 ) from 1.5 to 30. One of their most important results is that Ni nanoparticle as small as 3.3 nm still present ferromagnetic properties at room temperature. Since the discovery of carbon nanotubes, a particular interest has been devoted to the preparation of metal nanocluster ﬁlled carbon nanotubes for potential applications in nanoelectronic, catalytic or magnetic devices. Introduction of foreign elements inside the carbon tubules could give rise to peculiar effects. Cobalt ferrite nanowires with an average diameter of 50 nm and lengths up to several micrometers were synthesized by Pham-Huu et al. (2002) inside carbon nanotubes under mild reaction conditions using the conﬁnement

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effect provided by the carbon tubular template. The magnetic properties of these nanowires are presently studied by Mössbauer spectroscopy. Yang et al. (2002a) used VASP to study the binding energies and electronic structure of Ti, Al and Au chains on carbon nanotubes. Similarly, Bagci et al. (2002) used ultrasoft pseudopotentials for the determination of the structural and electronic properties of Al covered carbon nanotube. Extension to spinpolarized calculations seems presently lacking. 4.5. NiN embedded in Al. Lack of magnetization Over the last few years, the interaction of thin ﬁlms of transition metals with Al surfaces has attracted considerable attention because of the potential technological applications of transition metal aluminides as light-weight heat-resistant structural materials (see Smith, 1981) and as metalization layers on semiconductors (see Sands, 1988). Most of the work in this area has been experimental. In particular, Shutthanandan et al. (1993, 1996), Saleh et al. (1994, 1997) and Shivaparan et al. (1996, 1997) have used high-energy ion scattering, X-ray photoemission spectroscopy and low-energy electron diffraction to study the growth modes and interface structures of ultrathin Ni, Pd, Fe and Ti ﬁlms deposited on single crystal Al surfaces at room temperature. They reported that Ti grew as an epitaxial overlayer on both (110) and (001) surfaces, that Pd and Fe always formed surface alloys with the substrate, and that Ni formed an alloy at the (110) surface but tended to form an overlayer on the (001) surface. Their experimental results for Ni/Al(110) (see Shutthanandan et al., 1993) were reproduced in Monte Carlo simulations using the embedded atom model (EAM) potential that had been proposed for Ni–Al alloys by Voter and Chen (1987), noted “VC” hereafter. Robles et al. (2000a, 2000b) determined the ground-state conﬁgurations of small Ni clusters at the Al (001), (110) and (111) surfaces and calculated the spin-polarized electronic structure in order to see whether or not the magnetic moment of the Ni clusters was “deadened” by the nonmagnetic Al surface. The structural optimization was performed through quenched molecular dynamics (MD) simulations with the VC potential. It is worth pointing out that the structural features of small free Ni–Al clusters predicted using the VC EAM Ni–Al potential by Rey et al. (1996) agree with the results afforded by a self-consistent density-functional method by Calleja et al. (1999). For the electronic structure calculation, a self-consistent tight-binding (TB) model was used. The transferability of its parameterization was tested for Ni monolayers on or just below the Al(001) surface, showing that this method afforded results similar to those obtained using the ab initio tight binding linear mufﬁn tin orbital method (TB-LMTO) with the atomic sphere approximation. The validity of the strategy of ﬁrst using an n-body potential to determine ground-state structures and a TB model to obtain their magnetic moments had a precedent in the successful calculations of Bouarab et al. (1997) of the observed average magnetic moments of free-standing Nin clusters using the so-called Gupta (1981) potential for the structural description. The lowest-energy Ni clusters supported in Al are generally close-packed islands embedded in the substrate; in the second layer when the Al substrate has the (001) or (110) orientation, and in the ﬁrst layer for the Al(111) substrate. The only exceptions are Ni2 in Al(001) which was embedded in the third layer and Ni7 in Al(111) which was embedded in the second layer. The differences between the behavior at the different orientations can be attributed to Al(111) being a more close-packed surface than (001) and (110) and to

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its having a lower surface energy (Liu et al., 1991), both of which factors make mixing between Ni and Al atoms less favoured than at Al(110) and Al(001). Hence these results conﬁrm the tendency for Ni atoms to be embedded in the Al surfaces (see also Longo et al., 1999). Clusters deposited on the surface at room temperature (say) would eventually adopt the lowest-energy embedded structures if annealed slowly enough. The TB-LMTO results for the electronic structures of Ni monolayers supported on Al(001) or embedded below the surface, obtained by Robles et al. (2000a, 2000b) previously to the cluster calculations showed that the Ni atoms lose their magnetic moments due to hybridization with the sp states of Al. In fact, when the Ni monolayer is deposited on the Al surface, Ni–Ni distances are longer than in pure fcc Ni because the lattice parameter of Al (4.05 Å) is larger than that of Ni (3.52 Å), and therefore in the absence of Ni–Al hybridization the magnetic moments of the Ni atoms would be larger than in the bulk. In a study of free-standing Fe clusters using a TB model similar to that employed here, Vega et al. (1993) found that sp–d hybridization tended to reduce spin polarization because in these low-dimensional systems the resulting increase in the effective d bandwidth took the d band beyond the magnetic saturation limit. In the case of pure Ni systems, spin polarization comes mainly from the few d holes in the minority part of the density of states; the majority part is nearly saturated. Since the number of holes is very small, the magnetic moment of Ni systems is generally much smaller than in other magnetic transition metal systems, such as those composed of Fe. In the presence of the delocalized sp-type states of Al, the majority part of the Ni density of states is no longer saturated and magnetization consequently vanishes. The question then was whether or not the previously determined ground state Ni clusters in Al had a magnetic moment. It is well known that the free-standing Nin clusters are magnetic. The evolution of the average magnetic moment per atom with cluster size n has been measured by Billas et al. (1993) and by Apsel et al. (1996), both groups using a Stern–Gerlach deﬂection technique. The average magnetic moment decreases, although non monotonically, as the cluster size increases, reaching the bulk value of 0.61μB for a typical size of about 600 atoms. Fig. 4.8 shows the LDOS at some representative sites of the ground-state system obtained by Robles et al. (2000a, 2000b) for Ni6 embedded in Al(001). In this case there are two inequivalent Ni sites that have different LDOS because of their different local environments. The clusters were found to be nonmagnetic for the same reasons as the Ni monolayers: sp–d hybridization and the resulting increase in the effective d bandwidth, as reﬂected in the partial sp and d densities of states. The other Nin clusters (2 n 10) behaved similarly. The inﬂuence of the substrate is also reﬂected in the geometrical structures of the embedded clusters, which are different to those obtained by Bouarab et al. (1997) for the free-standing Ni clusters. All these results give further support to the strong environment dependence of the geometric and electronic properties of transition-metal systems. 4.6. CoN supported on noble metals In the context of supported magnetic clusters, when it is crucial to ensure the stability of the magnetization of each cluster while avoiding its interacting with other clusters, the best candidates as substrates are noble metals.

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Fig. 4.8. Local densities of electronic states (LDOS) at some representative inequivalent sites of the Ni6 /Al(001) system, as calculated using the self-consistent TB model by Robles et al. (2000a, 2000b) (only a portion of the model system is shown): Ni A, central Ni atoms; Ni B, Ni atoms at the cluster ends; Al C, top-layer Al atoms resting on four Ni atoms; Al D, third-layer Al atoms under four Ni atoms; Al E, displaced Al atoms on top of the top layer and laying over Ni B atoms. Orbital and spin projections are plotted. From Robles et al. (2000a, 2000b).

Particular attention has been focused on Co clusters and other Co structures deposited on Cu substrates. However, published data on the structures of these systems have been somewhat contradictory. Zimmermann et al. (1999) found that Co nanoparticles burrow into clean Cu(001) at 600 K. Atomic scale simulations by Stepanyuk et al. (2001) corroborated this burrowing and showed that it is promoted by the Co clusters becoming coated with Cu, which leads to high pressure at the interface; while Pentcheva and Schefﬂer (2000, 2002), using the full-potential linearized augmented plane wave (FP-LAPW) method, found that, on Cu(001), Co bilayers covered by a Cu layer are more stable than exposed Co bilayers, which is consistent with experimental observation of surface Cu atoms following annealing of Cu(001) covered with Co (Schmid et al., 1993). However, supported or partially embedded Co structures have also been observed. For instance, Nouvertné et al. (1999) used FP-LAPW calculations to interpret their scanning tunneling microscopy images as showing that Co atoms are incorporated in the Cu(001) surface layer, where they act as nucleation centers for the small Co islands that are observed in CO titration experiments; at the Cu(111) surface Pedersen et al. (1997) observed three-layer-high Co islands that had just one subsurface layer and were surrounded by a rim of Cu; and de la Figuera et al. (1993) observed two-layer-high triangular Co islands on top of the Cu(111) surface. Robles et al. (2002) computed the ground-state structures of Con clusters (n = 2–10) at the (001) and (111) surfaces of Cu using a many-body potential constructed on the basis

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of the second-moment approximation to the tight binding (TB) method and incorporating in the data set the binding energies of small Co clusters supported on the Cu(001) surface (calculated using the ﬁrst-principles Korringa–Kohn–Rostoker Green’s function method). In particular, magnetic effects are included implicitly by these energy calculations including spin-polarization effects. Using the cluster geometries and interatomic distances obtained as described above, these authors (Robles et al., 2002) computed the magnetic moment distribution of the Co/Cu systems using a self-consistent TB method similar to that employed by Robles et al. (2000b) in their study of the structural and electronic properties of Ni clusters at the Al(001) surface. The main difference is that in the Co/Cu work the parameters of the TB model (hopping and exchange integrals) were obtained from a single ﬁt to ab initio results for a Co monolayer embedded three layers below the Cu(001) surface that were obtained using the TB linear mufﬁn-tin orbital (LMTO) method (Andersen and Jepsen, 1984); an embedded monolayer was used so as implicitly to take into account hybridization between Co and Cu atoms and the inﬂuence of the surface, Co clusters having been found to embed themselves in the Cu substrates with three layers of Cu above the main cluster layer. These results show that the supported and partially embedded Co clusters observed at Cu surfaces were probably metastable. The balance between the pursuit of compactness of the embedded cluster and extension parallel to the surface appears to depend, reasonably enough, on proximity to the surface: although the presumably metastable clusters observed experimentally at the surface (Nouvertné et al., 1999; Pedersen et al., 1997; Figuera et al., 1993) were mainly two layers high, and Gómez et al. (2000) found that two-dimensional Co clusters deposited on Cu(111) transform into three-dimensional clusters above a certain critical size, Izquierdo et al. (2001b), in calculations explicitly including magnetic effects, found that two-dimensional Con clusters deposited on Cu(001) are stable up to n = 16. The electronic structure and magnetic properties of those embedded Con clusters at the (001) and (111) surfaces of Cu were calculated as a function of n. The average magnetic moment per atom is shown in ﬁg. 4.9. The clusters are ferromagnetic, with magnetic moments smaller than those found by a TB method for small Con clusters supported on Cu(001) (Izquierdo et al., 2001b), and considerably smaller than those of free Con clusters (Andriotis and Menon, 1998). The magnetic moment per atom of the Co clusters embedded in Cu decreased slightly and essentially monotonically as cluster size increased. This is attributed to the inﬂuence of the substrate lattice on coordination numbers and interatomic distances in the clusters, which as functions of cluster size cannot undergo the sudden changes that are responsible for “magnetic magic numbers” in series of free clusters. It seems likely that these conclusions are valid in general for ferromagnetic clusters embedded at noble metal surfaces. 4.7. Non-collinear magnetic conﬁgurations of V, Cr, Mn and Fe aggregates on Cu, Ag The DFT formalism for non-collinear states in bulk systems has been reviewed by Sandratskii (1997). This non-collinear magnetism seems to be the origin of the invar effect in Fe–Ni alloys (van Schilfgaarde et al., 1999). It was found that the magnetic structure is characterized by a continuous transition from the ferromagnetic state at low densities to a disordered non-collinear conﬁguration at high densities. This non-collinear behavior gives rise to an anomalous volume dependence of the binding energy. Also, Hobbs

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Fig. 4.9. Average magnetic moment per atom of embedded Con clusters at the (001) and (111) surfaces of Cu, as a function of n (squares and crosses, respectively). In the inset, data obtained by Andriotis and Menon (1998) for free Co clusters using a TB MD method. From Robles et al. (2002).

and Hafner (2001) showed that the non-collinear solution of Mn bulk is degenerate with classical non-ferromagnetic conﬁgurations. Cr and Mn are non-ferromagnetic elements of the ﬁrst transition metal series which can be ferromagnetically induced by contact with a strong ferromagnet like Fe (see section 3) or Co, or Ni (Demangeat and Parlebas, 2002). However, Cr or Mn atoms deposited on the (111) crystallographic face of a noble-metal substrate present typical non-collinear behavior (Hobbs and Hafner, 2000; Kurz et al., 2001). Topological non-collinear magnetism has been reviewed by Kawamura (1998) in the framework of the Heisenberg Hamiltonian. Within this approach it can be seen that in order to minimize the frustration on a triangular lattice, a non-collinear state with high local moment per atom is found but the total magnetic moment of the trimer is generally equal to zero. It is therefore of utmost interest to check if the ground state of any system containing Cr, Mn or Fe atoms do not present a non-collinear ground state. However, most of the reported calculations performed up to now consider only constrained collinear magnetism. Therefore those results cannot explain the experimental results because they only represent some kind of metastable magnetic conﬁguration. Uzdin et al. (1999) obtained the non-collinear magnetic structures of supported isosceles triangular Cr and Fe trimers within a model Hamiltonian approach. They used the real space recursion method for the periodic Anderson model (PAM) in the mean-ﬁeld approximation. This model is an extension of a previous scheme (Uzdin and Yartseva, 1998) devoted to non-collinear magnetism in bcc metals in bulk and at the interface between two semi-inﬁnite metals. Non-collinearity leads to the appearance of on-site hoppings with inversion of spin projection in addition to intersite hopping without spin inversion. For Cr trimers the ground state obtained displays a non-collinear magnetic conﬁguration with a

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Fig. 4.10. Dependence of the angle θ between magnetic moments localized on atoms 1 and 2 of the trimer versus hopping integral (a) v1 and (b) v2 . Solid squares, diamonds and circles correspond to the Cr, Mn and Fe trimers, respectively. Empty circles depict the metastable solution for the Fe trimer. From Uzdin et al. (2001b).

high magnetic moment on each atom. However, the total moment projection on the quantization axis is mainly zero. For Fe clusters the magnetic moments on each Fe atom do not differ drastically from those of Cr, but now, the projection on the quantization axis is practically the sum of the individual magnetic moments. However a metastable non-magnetic conﬁguration, a few meV above, is present with a nearly zero projection of the moments. These two solutions for Fe trimers are clearly the building blocks of respectively bcc Fe (ground state) and fcc Fe (metastable conﬁguration). Later on, Uzdin et al. (2001b) presented a complete description of Cr, Mn, and Fe trimers on the non-magnetic metal surface. The angles between the moments on the individual atoms depend strongly on the interatomic distances in the trimer. For isosceles trimers, atoms 2 and 3 are equivalent and, in the ground state, the angle θ between their magnetic moments and the moment of the atom 1 are equal. For these isosceles trimers the bonds

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Fig. 4.11. Modulus M of the total magnetic moment of Cr (black squares), Mn (black diamonds) and Fe (black circles) trimers as function of the hopping integrals v1 (a) and v2 (b) (noted V1 and V2 on the ﬁgure). Empty circles depict the metastable solution for the Fe trimer. From Uzdin et al. (2001b).

v1 between atoms 1–2 and 1–3 are equal but bond v2 between atoms 2 and 3 is different. Fig. 4.10 reports the value of θ versus v1 for v2 = 0.9. Small values of v1 and v2 correspond to large distances between atoms 1–2, 1–3 (d1 ) and 2–3 (d2 ). Consequently for Cr trimers θ decreases from π to π/2 versus d1 (ﬁg. 4.10(a)). The moments of the atoms 2 and 3 are

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antiparallel for small v1 and the moment of atom 1 is perpendicular to them. For large v1 there is a collinear structure. In the intermediate region all magnetic moments of Cr atoms are non-collinear and for equilateral Cr trimer (when v1 = v2 ) the angles between each pair of moments equal 2π/3. However, for Mn trimers there is a wide range of v1 values where the angle between moments is very close to 2π/3. Only in the limit of very small v1 , magnetic moments of atoms 2 and 3 become antiparallel. For an Fe trimer, in the region v1 < 1, two collinear sates are obtained, separated by an energy barrier. For both states the magnetic moments of the atoms 2 and 3 are parallel. For v1 > 1 only a single non-collinear state is observed. Fig. 4.10(b) reports θ as a function of v2 for v1 ﬁxed at 0.9. The modulus (M) of the total moment of Cr, Mn and Fe trimers are reported in ﬁg. 4.11, as a function of hopping integrals v1 (at v2 = 0.9) and v2 (at v1 = 0.9). For small values of v1 the magnetic moments of the Cr atoms 2 and 3 are compensated and the total magnetic moment coincides with the moment of atom 1 (ﬁg. 4.11(a)). The increase of v1 leads to non-collinearity of the moments localized on atoms 2 and 3 and consequently to the formation of a non zero magnetic moment for the dimer 2–3. The magnetic moment of this dimer is opposite to the moment of atom 1 and this decreases the total moment of the trimer. For the equilateral trimer the total moment equals zero and for larger values of v1 it is opposite to the moment of the atom 1. For a Mn trimer there is a wide range of values of v1 and v2 where the total magnetic moment is close to zero and the angle between each pair of moments equals 2π/3. For an Fe trimer two different collinear states corresponding to a high and low spin are shown. Jamneala et al. (2001) used atomic manipulation and STS to probe the local behavior of this type of systems by fabricating and analysing individual Cr trimers at the surface of gold. Cr atoms were deposited onto clean Au(111) at 7 K and manipulated with the tip of an STM to form artiﬁcial dimers and trimers. The low-energy excitation spectra of these structures were investigated using STM/STS. While single Cr atoms and Cr dimers show no structure in their excitation spectra, two distinct behaviors were observed for compact and triangular Cr trimers. One class of compact Cr trimer exhibits a featureless low-energy spectrum while the other displays a narrow resonance at the Fermi energy, signature of a Kondo response to a frustrated and antiferromagnetic cluster. Differences in electronic behaviors between these two classes of trimers have been explained through the switching between non-collinear magnetic states having zero and non-zero magnetic moments (Uzdin et al., 1999). 5. Conclusion In the present review we tried to provide some feeling concerning the state of the art in the electronic and magnetic calculations of low-dimensional transition metals. This report contains both aspects of the ﬁeld: (i) magnetic multilayered systems, mostly described presently by DFT calculations with usually a constrained collinear magnetization; (ii) a more semiempirical-like aspect related to the fundamental interest in considering not only the electronic and magnetic properties of a given geometrical conﬁguration but also to ﬁnd the ground state by minimization of the forces on a system where only the number of electrons are given. There is a critical link between the position of the atoms, their electronic behavior and consequently their vector-description of the magnetism. The situation

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is critical for V, Cr, Mn and Fe aggregates (free or adsorbed) but becomes non-tractable when in contact with a second entity, like in the case of free-standing Cr–N clusters or for adsorbates on Ag, Cu and Au. In this last case the geometry of the aggregate depends not only strongly on the crystallographic surface considered (Low-Miller index surface versus High-Miller index surface) but also on the quality of the surface obtained experimentally. A good example is the (111) surface of Au where reconstruction is present. The unconstrained vector magnetic description of aggregates on these substrates is intrinsically complex. They are even more complex if the surface tension of the substrate and that of the adsorbed cluster are very different: in that case an exchange of atoms at the interface will minimize the total energy of the system. However, at very low growth temperature this effect can be blocked by barriers. A complete chapter is devoted to Fe-based magnetic multilayered systems, a subject which has been under study at least since 15 years. A considerable bunch of results have been already obtained (electronic and magnetic maps, interlayer exchange couplings, anisotropic effects . . . ). Most of the calculations are now performed with ab initio codes based on the DFT calculations within the Kohn–Sham functional. Actually, static DFT is a ground-state theory in the form of an effective one particle equation. However in every spectroscopic experiment one perturbs the samples by incoming photons, electrons . . . and measures the response of the system to this perturbation. The system is therefore excited and it is, in general, not sufﬁcient to calculate ground state properties in order to interpret or predict results of various experiments (Onida et al., 2002). The simple one-particle picture is somewhat intrinsically inadequate for describing the process occurring in resonant photoemission, where core-valence absorption and valence electron Auger emission interfere. Many aspects of these experiments can, in principle, be described by the theoretical tools discussed by Onida et al. (2002). In contrast to these static and quasi-static investigations discussed in the present report, the wide range of dynamic properties of mesoscopic magnetic elements only start to emerge with direct experimental evidence of spin dynamics in the time domain which allows us to elucidate the switching mechanisms and the switching speeds on femtosecond timescales. These studies focus on the temporal limits of the magnetization reversal process induced by external ﬁeld pulses, the role of the damping parameter, the magnetic anisotropies as well as the system shape and size of this reversal. The dynamics of small magnetic particles differ from those seen in layered magnetic ﬁlms due to the large magnetostatic ﬁelds, shape and interface disorder and different magnetostatic excitation spectra (Hillebrands and Ounadjela, 2002). In this report, Zhang et al. (2002) describe the theoretical predictions and the experimental evidence of magnetization dynamics on the femtosecond timescale using femtosecond optical pulse techniques. These new ﬁndings open a new area of research called “femtomagnetism”. It ﬁnds its origin in the fact that the spin dynamics takes place through a mechanism that transfers efﬁciently the energy from the electrons to the spins on a time scale for which the lattice temperature has not yet been changed signiﬁcantly. Application was performed on CoPt3 by Guidoni et al. (2002). As discussed thoroughly in section 3 of this report, we have focused on Fe because this element has intrinsic properties which cannot be found in any of the other transition metal element. It can be found in the bcc, fcc and hcp form and the stabilization of the bcc ground state can only be obtained, in the Kohn–Sham formalism, if gradient corrections are

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included in the exchange-correlation functional. Within this formalism, bcc Fe is clearly of ferromagnetic type. Fcc and hcp phases are essentially metastable states appearing at high temperatures and do not present any onset of magnetization. Fe based bcc (or bct) magnetic multilayered systems have been the object of a huge number of experimental and theoretical studies so that the magnetic maps of many of these systems are presently well described. What is mainly remaining to be studied is the effect of metalloid on the electronic, magnetic and spectroscopic properties of these systems. The present focus is on non bcc Fe. The magnetic properties of fcc Fe are presently under consideration by many research groups (see, for example, Sjöstedt and Nordström, 2002; and references therein). Contrary to bcc Fe where, for interlayer exchange coupling, a non-ferromagnetic spacer is present between two ferromagnetic Fe blocks, fcc Fe is now the non-ferromagnetic spacer between Co or Ni blocks (Hadj-Larbi et al., 2002). Also, magnetization in hcp Fe could be stabilized when grown as Fe/Re superlattices (Perjeru et al., 2000; Zenia et al., 2002). The STM experiments of Lin et al. (2003) have found that mesoscopic strain induces magic Fe clusters on Cu(001). Relaxations of atoms in islands caused by mesoscopic misﬁt can lead to interplane lattice spacing variations. Several experiments on atom movement on and near islands in homoepitaxy suggested that strain relaxations in surface clusters strongly inﬂuence dynamics of adatoms. Lysenko et al. (2002) performed atomic scale simulations by a molecular static method. The system is modeled using many-body potentials developed in the framework of the second moment approximation to the tight-binding model. In mesoscopic islands the relaxation of edge atoms can be the dominating process. These atoms are relaxing in the direction of the center of the island and take other equilibrium positions with shorter bonds than in macroscopic systems. Therefore, a mesoscopic size-dependent mismatch between islands and the substrate in homoepitaxy exists and can locally affect the growth process (Lysenko et al., 2002) as well as the magnetic map. Other TM-elements appear to present somewhat interesting magnetic properties, intrinsically, or in the presence of bcc Fe. We have reported on V, Rh, and Pd for various reasons. The main reason is that these elements, non-magnetic in bulk conﬁguration, are at the border with magnetic elements in the periodic table (V near Cr, Pd near Ni and Rh near Co). From Stern Gerlach experiments it appears that only Rh clusters do present any indication of magnetism (Cox et al., 1993). Calculations by Dorantes-Davila et al. (1997) were able to reproduce these experimental results. Later on, Burnet et al. (2002) displayed incipient antiferromagnetism, below 80 K, for small supported Rh particles, by means of nuclear magnetic resonance. Induced Rh polarization is obtained in the case of Fe/Rh superlattices (Tomaz et al., 1997) whereas for Fe thin ﬁlms grown on Rh(001) surface studied by photoelectron spectroscopy (Hayashi et al., 2001) the magnetization depends strongly on the Fe thickness. A possible onset of magnetism in Pd and V nanostructures has already been discussed in Dreyssé and Demangeat (1997). Since then, onset of magnetization has been reported in palladium-graphite multilayers (Suzuki et al., 2000) and in free-standing palladium clusters (Vitos et al., 2000). Moreover, Kappler et al. (2003) have measured a small ﬁeld induced magnetic dichroism in Pd nanostructures. For V, Robles et al. (2001b) reported on a possible appearance of magnetism at the surface of V(001). Also, Gallani et al. (2003) were able to measure (at 4.2 K, in a ﬁeld of 7 T) a sizeable magnetic moment on two VO2+ derivatives: a vanadyl-bis-enaminoketone complex and vanadyl sulfate. It is clear that more studies have to be performed concerning the magnetization of those Rh, Pd and V based non-magnetic nanostructures.

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Spintronics, i.e. the use of spin rather than charge in electronics, calls for roomtemperature ferromagnetism in a semiconductor. This has been obtained by Theodoropoulou et al. (2002) by ion implantation of Mn ions into hole-doped GaP. Moreover, Hori et al. (2002) reported on the magnetic properties of Wurtzite GaN:Mn ﬁlms. The magnetic experiments on GaN:Mn revealed the coexistence of paramagnetic and ferromagnetic components. The highest Curie temperature of the ferromagnetism is about 940 K, with a clear hysteresis loop as high as 400 K. The transport properties exhibit a close relation between magnetism and impurity conduction. The double-exchange mechanism of the Mn impurity band was proposed to account for the high ferromagnetic transition temperature. Rao and Jena (2002) have suggested that the giant magnetic moments of Mnp N clusters may play a key role in the ferromagnetism of Mn-doped GaN which exhibit an high Curie temperature (Hori et al., 2002). Das et al. (2003) have carried out studies of magnetic coupling between Mn atoms in GaN by studying clusters of (GaN)p Mn2 as well as crystals of Mn-doped GaN. In the latter case, two Mn atoms were substituted at different Ga sites using a 32 atoms supercell. They found a ferromagnetic coupling both in clusters and crystals. Thus, Mn in GaN, whether forming clusters or substituted at Ga sites, tend to couple ferromagnetically. Another interesting ﬁnding concerns the experimental evidence of room temperature antiferromagnetic coupling between Fe layers across a very thin insulating MgO tunnel barrier (Faure-Vincent et al., 2002). In this case the spin information and the coupling are carried out across the spacer by equilibrium quantum tunneling of spin-polarized electrons. The explosion of demand for increased data-storage density is driving the exploration of new magnetic media. Li et al. (2002) used a modulated single/polycrystalline substrate surface to modify locally the magnetic anisotropy in subsequently deposited magnetic ﬁlms, which induces the desired artiﬁcial magnetic structure. Selective epitaxial growth introduces an alternation between single-crystal and polycrystalline structures in the ﬁlm, according to the substrate patterning. Epitaxial Ni/Cu(001) ﬁlms of appropriate thickness show perpendicular magnetization whereas the magnetization of polycrystalline Ni lies in the ﬁlm plane. The substrates used are GaAs(001). The spin-engineering of this magnetic medium opens up new avenues for controlling the spin structure. Nowadays, experimental teams are surrounded by many different and very precise equipment so that a cross-check can determine, in principle, the magnetic map of nanostructures. Bland and Vaz (2003) discussed the application of PNR on thin magnetic structures. Crommie (2000) has pointed out the versatility of the STM for manipulations at the atomic level. Theoretically the magnetic moment of a given atom depends drastically on its surrounding so that it is of utmost importance to have at least a good determination of the nearest neighboring shell of atoms. Ab initio molecular dynamics can be such a tool but presently it is not trivial to compute both the positions of the atoms and their magnetic map in TM-based systems. There is however a strong demand in this direction and such a versatile code could make a deﬁnitive progress towards the understanding of the very many experimental results. It is therefore of much importance to have a more stronger connection between theoretical and experimental teams. In the particular case of Cr/V interfaces, Hübener et al. (2002) found that Cr develops an incommensurate SDW state with a node at the V/Cr interface apparently at odds with the theoretical results of M’Passi-Mabiala et al. (1994), Boussendel and Haroun (1998), Bihlmayer et al. (2000),

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Hamad and Khalifeh (2001) displaying a clear antiferromagnetic coupling between V and Cr at the interface. This comparison is clearly misleading because as seen by Izquierdo et al. (2001a, 2001b) in the case of thin Fe ﬁlms on a V substrate, a dead layer appears because of alloying. Theoretical calculations concerning the Cr/V interfaces were all restricted to abrupt interfaces; introduction of alloying will therefore modify entirely the magnetic map and will most probably lead to similar results as those obtained by Izquierdo et al. (2001a, 2001b) in the case of thin Fe ﬁlms on V. Thus, an agreement between experiment and theory could be attained. It is however a long way to go but we strongly believe that this it is a realistic path to follow. In order to scientiﬁcally advance our understanding of frustrated itinerant magnets in low dimensions, Heinze et al. (2002) investigated well-deﬁned antiferromagnetic monolayers of Cr and Mn on a triangular lattice provided by the (111)-oriented surfaces of an Ag substrate. Guided by classical spin models they search on the basis of ﬁrst-principles vector spin-density calculations in the DFT for the magnetic ground states. This search includes 2D incommensurate non-collinear spin structures such as the spiral spin density wave (SSDW) states, and the 3D non-collinear states, e.g., the 3Q state. As magnetic ground state Heinze et al. (2002) found for Cr/Ag(111) a coplanar non-collinear periodic 2π/3 Néel structure, for Mn/Ag(111) a row-wise antiferromagnetic spin structure, and Fe/Ag(111) is ferromagnetic. They proposed the spin-polarized scanning tunneling microscope (SP-STM) operated in the constant-current mode as a powerful tool to investigate those complex atomic-scale magnetic structures. Their calculating results show that the predicted non-collinear magnetic ground state structure can clearly be discriminated from competing magnetic structures. More fundamentally, surface randomness modiﬁes the critical behavior near the surface (Hanke and Kardar, 2001), yet the common expectation is for the bulk properties to remain intact. Feldman and Vinokur (2002) pointed out that arbitrarily weak surface disorder destroys long-range order in the bulk of a system of continuous symmetry at the arbitrarily low temperature. The reason as to why surface impurities, however weak, break long-range bulk order is that the bulk contributes little to the energy of the long-wave Goldstone modes: the surface energy of long-wave excitations turns out to be greater than the corresponding bulk energy. As a result, the inhomogeneous state becomes favorable energetically. While long-range order breaks down, topological order survives and quasilong-range order emerges. This means that the correlation length is inﬁnite and that the correlation functions obey a slow logarithmic dependence of the distance (Feldman and Vinokur, 2002). In section 4 we have reported on several problems related to TM clusters, both freestanding or supported in a host. We believe that the ﬁeld of magnetic nanoparticles is still emerging and that a great advance is expected in the next years from both the experimental and theoretical sides. Acknowledgements This work was supported by the exchange program “PICASSO” between Spain and France, by the CICYT, Spain (Project MAT2002-04393-C02-01) and the Junta de Castilla y León, Spain (Grant VA073/02).

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chapter 3

II–VI AND IV–VI DILUTED MAGNETIC SEMICONDUCTORS – NEW BULK MATERIALS AND LOW-DIMENSIONAL QUANTUM STRUCTURES

WITOLD DOBROWOLSKI, JACEK KOSSUT, TOMASZ STORY Institute of Physics of the Polish Academy of Sciences Warsaw Poland

Handbook of Magnetic Materials, edited by K.H.J. Buschow Vol. 15 ISSN: 1567-2719 DOI 10.1016/S1567-2719(03)15003-2

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© 2003 Elsevier Science B.V. All rights reserved

CONTENTS 1. Introduction and scope of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 2. II-VI diluted magnetic semiconductors in low-dimensional environment . . . . . . . . . . . . . . . . . . 293 2.1. Preliminaries and historical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 2.2. The Faraday effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 2.3. Magnetic polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 2.4. Time resolved studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 2.5. Trions and combined optical resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 2.6. Dimension dependence of the sp–d exchange constants . . . . . . . . . . . . . . . . . . . . . . . . 303 2.7. Transport studies of a two-dimensional electron gas in DMS heterostructures with record mobility 303 2.8. Spin injection and spin aligners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 2.9. Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 2.10. Transfer of an excitation between carriers and localized moments . . . . . . . . . . . . . . . . . . 306 2.11. Quantum wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 2.12. DMS quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 2.13. Metallic ferromagnet/DMS quantum well hybrid structures . . . . . . . . . . . . . . . . . . . . . 309 2.14. Magnetic properties of DMS thin ﬁlms and quantum wells . . . . . . . . . . . . . . . . . . . . . . 310 2.15. New DMS II-VI materials – in search for ferromagnetism . . . . . . . . . . . . . . . . . . . . . . 311 3. Bulk crystals and low-dimensional structures of IV-VI DMS compounds . . . . . . . . . . . . . . . . . 312 3.1. Carrier concentration control of magnetic properties of IV-VI DMS with Mn . . . . . . . . . . . . 315 3.2. IV-VI DMS materials with Eu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 3.3. IV-VI DMS materials with Gd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 4. Magnetic-non-magnetic multilayers with II-VI and IV-VI semiconductors . . . . . . . . . . . . . . . . . 331 4.1. Ferromagnetic EuS-PbS and related multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4.2. Antiferromagnetic EuTe-PbTe, MnTe-ZnTe and MnTe-CdTe semiconductor superlattices . . . . . 342 4.3. Magnetic properties of MnTe-ZnTe and MnTe-CdTe superlattices . . . . . . . . . . . . . . . . . . 344 5. Other diluted magnetic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 5.1. Chromium and vanadium compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 5.2. Cobalt compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 5.3. Iron compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5.4. Narrow gap materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 5.5. Ti, Ni, and Sc impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

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1. Introduction and scope of the chapter The usual deﬁnition of diluted magnetic semiconductors – substitutional mixed crystals with some of the cations of a semiconductor host lattice replaced by “magnetic” ions such as Mn or Fe or Eu – encompasses a large number of diverse compounds. There exist several review articles or even collections of chapters covering the ﬁeld of bulk II-VI materials which have been most commonly studied in the past. Reviews on diluted magnetic semiconductors (DMS) have been published by Furdyna (1988), Kossut and Dobrowolski (1993), Dietl (1994), Kossut and Dobrowolski (1997) and Galazka et al. (1999). However, reviews concerning either low-dimensional quantum structures or new materials (e.g., IVVI compounds) are only few in number and they are of relatively limited scope (Heimbrodt and Goede, 1991; Luo and Petrou, 1997; Kossut, 2001). On the other hand, the last 15 years have seen a very rapid progress in these ﬁelds. The only possible exception to this rule is a rather complete and up-to-date review covering the new ﬁeld of III-V ferromagnetic DMS by Matsukura et al. (2002) and a special volume of Semiconductor Science and Technology (H. Ohno, 2002) partially devoted to this particular area. Therefore, the present chapter will attempt to give an overview of the literature concerning low-dimensional structures of II-VI DMS with manganese. It will also review the literature concerning new materials, typically with magnetic components other than Mn, and, with some emphasis, the literature pertaining to IV-VI materials with a magnetic component. Even with such a narrowing of the scope of our chapter we have to make yet another disclaimer. We will not attempt to present a completely exhaustive review of the existing literature. Some subjective choice of topics had to be made merely in view of the enormous number of papers that appeared in the past. For instance, apart from not covering the III-V materials, also studies of elastic properties of the materials in question, although important in the context of strained epitaxial structures, will not be properly represented here. There are more such omissions. Another example of those omissions are ferromagnetic properties of p-type doped II-VI structures (particularly those containing magnetic ions other than manganese) which are still at an early stage of development. Their representation in this chapter will be only fragmentary. Let us hope, nevertheless, that this review will be of use to those numerous researchers and readers who have been recently attracted to the ﬁeld by a new wave of hopes of incorporating diluted magnetic semiconductors in “spintronic” devices including those for quantum information applications. It has to be realized that the current interest in spintronics (which, of course, extends beyond the ﬁeld of DMS) represents a sudden boost to studies of diluted magnetic (or semimagnetic) materials. (A similar role has been played in relation to studies involved in a search for a blue laser based on II-VI materials, an activity that is practically abandoned now.) As a result, we are witnessing now a vast increase of the number of contributions to the ﬁeld of DMS. This has led to new topical conferences 291

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such as Physics and Applications of Spin Related Phenomena in Semiconductors (PASPS). These were organized already twice (in Sendai in 2000, and in Wuerzburg in 2002), to mention just one of many new initiatives. The proceedings of the former have already appeared (H. Ohno and Katayama-Yoshida, 2001). The part of this chapter devoted to II-VI DMS quantum structures is divided into several subsections. First, we shall give a brief historical introduction with necessary preliminaries. Some of the material here extends and updates the information from the previous review on DMS in this series (Kossut and Dobrowolski, 1993). We include in this part an overview of problems encountered when studying the DMS low-dimensional structures, such as the problem of interfaces, and we mention several new possibilities offered by the fact that the Zeeman splitting of the band states in DMS structures can be truly “giant”. In particular, we discuss formation of spin superlattices, changes of the type of the conﬁnement induced by a magnetic ﬁeld, an ingenious method of wave function mapping and studies of phonon propagation in the material (all based on optical studies of the spin splitting of low-dimensional excitons in these structures). Then, we cover objects of various dimensionalities: quantum wells and superlattices, quantum wires, and quantum dots. In the case of the quantum wells, we organize our review around certain topics, such as the Faraday rotation or magnetic polarons (in particular formation of free vs. bound magnetic polarons). Here we make a mention of time resolved optical studies, since they are of particular interest for the magnetic polaron formation times. Further, we shall describe how the variable g-factor, characteristic for DMS, can help in a study of charged excitons or trions, and we shall address an important issue whether or not the dimensionality affects the values of “fundamental” constants describing s–d and p–d interaction that plays the central role in the physics of DMS. Studies of transport properties, including quantum Hall effect in n-type modulation-doped structures, are brieﬂy described next. With spintronic applications in view, there were several recent studies of effectiveness of spin injection (either electrically or optically driven) from DMS spin aligners to other parts of the low-dimensional heterostructure. Similarly, spin dependent tunneling (and spin ﬁltering), typically between two quantum wells separated by a barrier, were studied and are to be discussed here. Further, we discuss structures having even lower dimensionality than the quasi two-dimensional quantum wells, i.e., quantum wires (the studies of which are not so abundant) and quantum dots, particularly self-assembled quantum dots. We ﬁnish the part devoted to low-dimensional structures of II-VI DMS with a brief discussion of hybrid structures and of Co- and Cr-containing structures, the study of which is only in the very initial stages. Finally we shall review the research concerning magnetic properties of epilayers and quantum wells. Magnetic properties of superlattices, in particular those related to interlayer coupling, are left until later sections in this chapter (namely, those devoted to IV-VI compounds) since many related important results on such systems were obtained on IV-VI superlattices. Therefore, it seemed proper to give a description of magnetic properties of all DMS superlattices in one single section. Further parts of this chapter are organized as follows. In section 3.1, an update is given for the carrier concentration induced ferromagnetism in Mn-based IV-VI DMS with a particular emphasis on Mn content and conducting hole concentration dependent paramagnetic–ferromagnetic–spin glass transitions in bulk crystals and thin layers of SnMnTe and related materials. In section 3.2, we discuss transport, magnetic, and optical

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properties of bulk crystals, thick epitaxial layers, and low-dimensional quantum structures of IV-VI DMS with Eu. Magnetic and electrical properties of IV-VI semiconductors with Gd are presented in section 3.3. In section 4.1, the magnetic and optical properties of allsemiconductor ferromagnetic multilayers built of ferromagnetic EuS and non-magnetic IV-VI (PbSe, PbS) semiconductors are presented. The magnetic and optical properties of closely related antiferromagnetic EuTe-PbTe and MnTe-ZnTe(CdTe) superlattices are discussed in section 4.2. The last section 5 of the present chapter describes DMS containing transition metals other than manganese. These materials never attracted as much attention as the ones containing Mn, mainly due to severe difﬁculties in their preparation, particularly when large percentages of the magnetic components were involved. However, in view of theoretical predictions of Blinowski and Kacman (1992a) that, depending on the occupation of the transition metal (TM) ion 3d shell, the p–d exchange interaction may evolve from an antiferromagnetic to a ferromagnetic one, those materials have become a very interesting object of study. The above stimulated considerable crystal growth efforts to obtain these novel DMS and to achieve magnetic ion concentrations high enough to enable systematic studies of the exchange interactions. Additionally, in this part we update the information from the previous review (Kossut and Dobrowolski, 1993) on cobalt- and iron-based DMS published since then. 2. II-VI diluted magnetic semiconductors in low-dimensional environment 2.1. Preliminaries and historical introduction Studies of low-dimensional structures of diluted magnetic semiconductors date back to the ﬁrst half of the nineteen eighties. It has been realized from the very beginning that

Fig. 1. Schematic representation of the DMS/non-DMS quantum structures. The solid lines in the lower part of the ﬁgure represent the band edges in the absence of an external magnetic ﬁeld, while the broken lines show spin-split band edges after application of a ﬁeld.

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incorporation of DMS in low-dimensional structures would give rise to a wealth of new phenomena (see, e.g., Datta et al. (1985), Kossut and Furdyna (1987a, 1987b)). All these new effects had their origin in a large difference between the Zeeman splitting in thin layers made of, respectively, either DMS or non-DMS layers, see ﬁg. 1. It was immediately realized that the giant Zeeman splitting, so characteristic for DMS compounds, could be comparable to that of the conduction and/or valence band discontinuities (that deﬁne the depth of the conﬁning potential) due to differences in chemical nature of constituents of a DMS quantum structure. Both conﬁgurations, i.e., with the lower band gap material representing a quantum well region, or, conversely, with the higher band gap (forming the barriers) made of DMS were conceivable. In particular, von Ortenberg (1982) has put forward an idea of a spin superlattice. The spin superlattice was to consist of alternating DMS and non-DMS layers whose chemical compositions were to be chosen in such a manner that in the absence of an external magnetic ﬁeld they would correspond to band edge proﬁles homogeneous in the entire sample. On the other hand, an artiﬁcial periodicity would appear after application of an external magnetic ﬁeld. It was due to a giant spin splitting occurring only in the DMS layers, which was to shift the band edges of a given spin species up (down – for the opposite spin component) in relation to those in the non-magnetic layers. The experimental realization of the spin superlattice took some time and effort. In fact, the experimental evidence of the spin superlattice formation was made not in the narrow gap HgSe/HgMnSe superlattice, as suggested in the original paper by von Ortenberg, but in rather wide gap structures involving ZnSe/ZnMnSe (X. Liu et al., 1989; Dai et al., 1991; Jonker et al., 1991), ZnSe/ZnFeSe (Chou et al., 1991; Warnock et al., 1995), and ZnMnSe/ZnBeMgSe (König et al., 1999). Later it was realized that of interest are also structures that are magnetically homogeneous, with the band edge proﬁles leading to a spatial conﬁnement effect achieved by modulating the content of a non-magnetic component as in Cd1−x−y Mgy Mnx Te/Cd1−x Mnx Te/Cd1−x−y Mgy Mnx Te heterostructures with x remaining the same throughout an entire structure. The reason for such interest is that in samples of that design one largely avoids problems with modiﬁcations of Mn–Mn exchange interactions at the heterointerfaces. Nearly all heterostructures involving DMS studied in the literature were prepared by molecular beam epitaxy (MBE). Various substrates were employed in view of the requirement of minimization of the internal strain due to the lattice parameter mismatch. Fortunately, ZnSe happens to be rather well lattice-matched to the most popular substrates made of GaAs, which greatly facilitated growth processes in the initial period. Soon it became clear that MBE is rather forgiving and one can grow on GaAs substrates DMS compounds with a considerable lattice mismatch, such as CdMnTe. The quality of such structures was shown to be quite good, allowing optical investigations (see, e.g., Waag et al. (1991)), particularly when thick buffer layers were deposited ﬁrst. Nevertheless, a number of studies were made on structures grown on other substrates such as InSb or ZnCdTe which were chosen with the reduction of the strain between the layers deposited epitaxially and the substrate in mind. As mentioned, the initial impetus of the DMS heterostructure investigations was closely related to attempts to develop blue light emitting laser diodes and other optoelectronic devices composed of II-VI semiconductors. Therefore, numerous pieces of information on DMS heterostructures can be found scattered in existing reviews on this topic (see, e.g.,

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Kolodziejski et al. (1986, 1991), Kobayashi et al. (1992); or more recent Luo and Furdyna (1995)). In fact, one of the breakthroughs in the ZnCdSe-based blue laser development was related to a successful growth by MBE of cubic form of CdMnSe (normally existing in the wurtzite structure) on GaAs substrates (Luo et al., 1991). In most of the cases the growth proceeded along the [100] crystallographic direction. Less frequent are studies of [111]-oriented heterostructures where one has to account additionally for piezoelectric ﬁelds which appears in strained structures (André et al., 1990; Bodin et al., 1995). Other directions of growth were employed sporadically. One of the examples is the study of samples grown along the [120] direction (Kutrowski et al., 1997) since it was expected that such growth direction will result in samples of, in this particular case, CdMnTe with a reduced number of nearest Mn-Mn neighbors. Since inter-manganese exchange interactions are antiferromagnetic (the nearest-neighbor interactions are at least by an order of magnitude stronger than in the case of further neighbors) they tend to reduce the magnetization of a sample. Therefore, they lead to a decrease of the spin splitting of the band states via sp–d interaction that is proportional to the magnetization. In other words, the antiferromagnetic Mn–Mn interactions make the most characteristic and distinguishing feature of diluted magnetic semiconductors less pronounced and, therefore, might be viewed as parasitic. It has to be mentioned that the above attempt to reduce the role of the antiferromagnetism of Mn ions was not successful. Molecular beam epitaxy made it possible to extend the accessible Mn molar fraction range compared to that in bulk DMS. In particular, it was possible to grow layers of cubic MnTe (Durbin et al., 1989; Han et al., 1991; Ando et al., 1992a, 1992b; Janik et al., 1995, 1998) as well as superlattices involving this phase. The latter led to several important results in neutron diffraction studies – see further in this chapter. The same growth technique enabled the growth of ternary compounds by a “digital” method, i.e., by incorporating, within a non-magnetic binary layer, several separate single monolayers (or even sub-monolayers) of, say, MnTe or MnSe. Such insertions produce a net effect on the band carriers that is qualitatively similar as if Mn ions were distributed evenly in the whole layer (say, quantum well), corresponding to the ternary material with a molar fraction given by a ratio of the total width of MnTe insertions and the non-magnetic layers (Crooker et al., 1995). Quantitatively, one has to account for the probability of ﬁnding the band carrier in the region of the quantum well where the insertions are introduced. Thus, the effective molar fraction that determines the giant spin splitting of such a digital alloy depends on the spatial positions of the insertions. A similar technique was used to produce quantum wells with graded proﬁle of the conﬁning potential, such as parabolic or triangular (Wojtowicz et al., 1996a, 1996b). Such samples proved to be very useful for precise determination of the valence band offsets – see further in this chapter. Even early investigations of DMS quantum well structures by optical methods revealed an important role of interfaces between magnetic and non-magnetic parts of the structure. In particular, the interdiffussion of magnetic and non-magnetic cations was shown to be responsible for such seemingly nonsensical observation of a Zeeman splitting of excitons in CdTe/CdMnTe quantum wells that exceeded the splitting in the bulk material of the same Mn content as the barriers. This is now understood well in terms of a combined action of the greater penetration into the barriers and a reduced antiferromagnetism (because of greater dilution of Mn) in the interfacial region due to intermixing of, say, Mn and Cd.

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There are many papers devoted to this issue and its detailed modeling (see, e.g., Harrison et al. (1993a, 1993b), Weston et al. (1993, 1998), Fatah et al. (1994), Jackson et al. (1994a, 1994b), Stirner et al. (1994a, 1994b), Crooker et al. (2000a), Semenov et al. (2001)). An extensive effort was made in the case of CdMnTe/CdTe quantum wells and the resulting quantitative model of an intermixed interface was developed that later was used for interface quality assessment by means of optical investigation of the Zeeman splitting (Gaj et al., 1994; Grieshaber et al., 1996; Lemaître et al., 1998). The model was tested by annealing induced diffusion at the interface (Tönnies et al., 1994; Kossacki et al., 1995, 1997; Wypior et al., 1998) and favorably checked directly by high resolution transmission electron microscopy of the interfaces (Charleux et al., 1998; Wypior et al., 1998). It remains consistent with independent measurement of diffusion of Mn in II-VI hosts (Shaw, 1988). In the same context the imperfections at the interfaces may act as effective localization centers of carriers or excitons enhancing magnetic polaron binding energy (Nurmikko et al., 1985; Akinaga et al., 1993). These effects may even lead to type-II excitons localized at the interfaces (Rossin et al., 1995, 1996; Streller et al., 1995). Proper understanding of the observed energy of optical transitions and of their Zeeman splitting in particular required, of course, precise knowledge of the band offsets. Extracting these from experiments proved to be even more complicated than in the case of non-magnetic counterparts, such as GaAs/GaAlAs structures where the controversy persisted for several years. Some of the reasons for this situation are related to the fact, already mentioned, that the magnetic interactions (particularly important for carriers whose wave functions are appreciable near the interfaces) introduce a considerable complication into the modeling. Other complications arose because of the strains (here an attempt was made to study structures with small differences in the composition of the well and the barrier material), the anisotropy (Kuhn-Heinrich et al., 1992, 1994), and the sensitivity of the magnetization (and, thus, of the effective band offsets) to the temperature (Zhu et al., 1999). In spite of these complications and uncertainties it is now commonly agreed that the CdTe/CdMnTe system represents a type-I structure (both electrons and holes conﬁned in the low band gap CdTe region), with the valence band offset being 35–40% of the total band gap difference between CdTe quantum wells and CdMnTe barriers (Jackson et al., 1994a, 1994b; Lebihen et al., 1997; Siviniant et al., 1999). This is very different from the original determination of the conduction to valence band offset ratio as 14 : 1 (S.-K. Chang et al., 1988) which was still compatible with the so-called common anion rule. Crucial in the proper determination was use of specially designed quantum well structures with the shape different from rectangular (D’Andrea and Tomassini, 1993; Lebihen et al., 1997; Wojtowicz et al., 1998; Kutrowski et al., 2000). The structures composed of Zn-containing materials may be (due to, e.g., the effect of internal strain) of the type-II (electrons and holes conﬁned in different spatial regions of the structure) since the valence band offset is determined to be much smaller in this case (10–20% of the band gap difference), see Klar et al. (1998a). The fact that the “chemical” band offsets may be comparable with the magnetic ﬁeldinduced energy shifts of various spin components of the conduction or valence band edges leads to the, so-called, type I-to-type II transition, i.e., a transition from a situation with both types of carriers (electrons and holes) forming an exciton residing in one (quantum well) region to a situation, which was predicted by Brum et al. (1986)

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to occur in CdTe/CdMnTe at high external magnetic ﬁelds, with the electron remaining conﬁned in the quantum well region. The spin polarized hole, on the other hand, is preferably localized in the DMS “barrier” layer. An unambiguous evidence of observation of such transition was difﬁcult in view of the role played by formation of polarons and modiﬁcations of the spin splitting due to the interface disorder and magnetic dilution, as well as obvious and large modiﬁcations of the exciton binding energy that takes place simultaneously with such a transition (Wasiela et al., 1992). Nevertheless, clear-cut observation of such a transition has been reported in two systems CdMnTe/CdTe (Deleporte et al., 1990a, 1990b; Delalande, 1992; Ribayrol et al., 1993; Cheng et al., 1997) and ZnSe/ZnCdMnSe (Deleporte et al., 1994; Yu et al., 1997a). As predicted theoretically by Brum et al. (1986), the transition is associated with modiﬁcations of the probability of the optical transitions involving various states of excitons. Such modiﬁcations were studied in detail theoretically and experimentally (Ivchenko et al., 1992; Pozina et al., 1992; Ribayrol et al., 1995). The same underlying principle that enabled an unambiguous interpretation of some features in the magneto-optical data as being signatures of spin superlattice formation (or the type I-to-type II transition, mentioned above), is responsible for a strong asymmetry of the spin splitting of excitons localized in spatially different parts of a structure (assuming that these parts are either magnetic or non-magnetic). Thus, by using two different circular light polarizations one could choose one given spin state of the exciton and follow its reaction to an external magnetic ﬁeld. If it were strong, then these excitons would predominantly be localized in the magnetic (DMS) layer constituting the superlattice, while when this dependence were only weak, one could conclude that the signal originated from the excitons predominantly localized in the non-magnetic part of the structure. The possibility of such spatial discrimination between magneto-optical signals, as coming from the magnetic or, alternatively, non-magnetic part of the structure, made possible yet another interesting observation. Namely, at higher energies of observation one could clearly identify signatures due to the resonant, above-the-barrier states (Luo et al., 1993; Dai et al., 1994). The same anisotropy of the spin splitting helped to identify spatially-direct from spatially-indirect excitons (Luo et al., 1993; Zhang et al., 1993; Deleporte et al., 1994; Syed et al., 2001; Yakovlev et al., 2001). Ultimately, this spatial sensitivity of the magneto-optical methods led to the wave function mapping where narrow magnetic insertions are placed within otherwise non-magnetic quantum wells. The sp–d interaction, and thus the degree of the Zeeman splitting, depends in this case on the overlap of the electron (or hole) wave functions with those narrow insertions. If this overlap is large (like in the case of centrally positioned insertion and the ground state of an electron conﬁned in a quantum well) then the splitting is also large. If the insertion is positioned closer to the quantum well edge, where the ground state wave function amplitude is small, then the observed splitting is also small (Lee et al., 1999). The mapping can be quite precise if monolayer-wide insertions are used (Prechtl et al., 2000). An interesting consequence of the strong magnetic ﬁeld dependence of the effective valence band offset is a theoretical possibility of moving excitons (electrically neutral objects) by a magnetic ﬁeld gradient: excitons in the lower spin-split state will ﬁnd it energetically more favorable to move to those regions of the sample where the ﬁeld is the strongest. For excitons that are mobile (not strongly localized by, e.g., composition or well width

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ﬂuctuations) one can think even of a condensation of these bosonic quasi-particles (Kavokin et al., 1998). With the Zeeman splitting in DMS being so large, the effect of realistic external magnetic ﬁeld gradients can be estimated to be quite sizable. An attempt to realize such a ﬁeld driven motion of excitons, unfortunately, was not successful due to strong pre-localization of excitons in the particular sample used for the experiment (Pulizzi et al., 2000). Of course, precursory effects, i.e., biexciton and electron–hole plasma formation were also studied in DMS low-dimensional structures (see, e.g., Kreller et al. (1995), Adachi et al. (2000), Mino et al. (2001)). In the context of optoelectronics one notices a study of polaritons in these systems (see Adachi et al. (2000)) as well as of microcavities with Bragg reﬂectors tunable by a magnetic ﬁeld (Ulmer-Tufﬁgo et al., 1997). Quantum well structures of II-VI DMS materials turned out to be suitable materials for studies of optical and magneto-optical anisotropies. We will ﬁrst deal with optical orientation. These effects were studied, e.g., by Kusrayev et al. (1995, 1996). Early investigations (Peyla et al., 1993; Suisky et al., 1998) showed that the anisotropy of the Zeeman effect, measured with the external magnetic ﬁeld pointing in various directions within the quantum well plane, is very signiﬁcant even in structures with symmetric proﬁles of the quantum well conﬁning potential. In such structures, the anisotropy related to a single interface is hidden and becomes evident only under special circumstances, e.g., when the interface does not have common anions as in the case of the ZnMnSe/BeTe interface (Yakovlev et al., 2002). The lack of common anions is not the only case where the hidden anisotropy becomes visible. The anisotropy observed by Kusrayev et al. (1999) in symmetric CdTe/CdMnTe quantum wells was so pronounced that in some cases the g-factor of holes measured in various directions showed a reversal of sign. This effect is probably associated with strain, which introduces low-symmetry perturbations that mix the heavy and light hole components of the valence band. Also in parabolic quantum wells the anisotropy was found to be particularly large (Kudelski et al., 2002). This effect still needs to be understood properly. Of course, in structures grown intentionally with an asymmetry of two heterointerfaces delimiting a quantum well (as in, e.g., CdTe quantum well surrounded by CdMnTe and CdMnMgTe barriers) one can expect a strong anisotropy. In fact, experiments do detect such anisotropy (Kudelski et al., 2002) which can lead even to a complete linear polarization of the light and heavy hole excitons when the Zeeman splitting between the corresponding spectral lines is decreased to a few meV’s. To ﬁnish this preliminary part devoted to II-VI DMS quantum well structures one has to mention a unique feature of these structures, namely that by monitoring the giant spin splitting of the excitonic optical transitions one can get access to other characteristics of the system. Accessing information on the magnetic state in this way is rather obvious since the giant spin splitting is proportional to the magnetization with the effective g-factor of electrons or holes given by M(B, T ) (1) μB B where the ﬁrst term describes the spin splitting due to spin–orbit interaction of the nonmagnetic host crystal and the second term describes the contribution due to the sp–d exchange interaction between the carriers and localized magnetic moments with M being the magnetic ﬁeld and temperature dependent magnetization of the localized moment subsystem. The magnetization can be in principle accurately calculated using phenomenological 0 ge(h) = ge(h) + Js(p)−d

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parametrization in terms of effective composition x0 and effective Curie–Weiss temperature T0 as determined for CdMnTe, e.g., by Grieshaber et al. (1996). The proportionality constant Js(p)−d is, in the case of conduction band electrons, traditionally denoted by Js−d = α, while in the case of the valence band holes by Jp−d = β (in low-dimensional structures these two quantities cease to be constants and may acquire values that vary with the width of quantum wells or spatial extensions of quantum dots – see later in this chapter). Since the second term often dominates over the ﬁrst one, the measurements of the exciton spin splitting give a direct access to the magnetization. In fact, by studying optical properties of CdMnTe/CdMgTe quantum well structures one could obtain reliable data concerning the dependence of the transition temperature to the spin glass state on the CdMnTe quantum well width (Yakovlev et al., 1999). The advantage of the optical method in studies of magnetic properties is that it is a very fast probe enabling temporal studies of evolution of, e.g., magnetic ordering process. Another phenomenon that could be studied indirectly via monitoring of the Zeeman splitting of exciton signatures in the luminescence spectra was propagation of nonequilibrium optical phonons in a CdTe/CdMnTe quantum well sample (Akimov et al., 1997; Yakovlev et al., 1999). In these studies phonons generated at one end of a sample reached a spot at which the exciton luminescence was monitored after some delay that could be measured precisely, thus, rendering information on the speed of the phonon propagation in the lattice. The phonon-induced changes of the exciton luminescence were, of course, due to changes of the local temperature which affected the Zeeman splitting via the temperature dependence of the magnetization. 2.2. The Faraday effect We single out the Faraday rotation in our review since it is the basis for one of the realistic (and real) applications of DMS materials (Ando, 1989; Zaets and Ando, 1999): optical isolators made of CdMnTe are produced from bulk crystals. Theoretically, the Faraday rotation, already a large effect in bulk DMS crystals, is expected to be even stronger in the quantum well conﬁguration (Nakamura and Nakano, 1990). The enhancement could be traced to a greater oscillator strength of the excitonic optical transitions in low-dimensional structures. The experimental studies of CdTe/CdMnTe quantum wells grown on sapphire substrates (since the Faraday effect is measured in the transmission conﬁguration, the substrates have to be either transparent or removed prior to measurements) by the same authors conﬁrmed their earlier expectations (Nakamura and Nakano (1992), see also Inukai and Ono (1992)). Later studies of the excitonic Faraday rotation on CdTe/CdMnTe samples grown on better lattice-matched substrates revealed that the rotation angles that could be reached in realistic magnetic ﬁelds were indeed very large and corresponded to a Verdet constant of 7 × 106 degrees/Tesla/cm. This is larger than that in the bulk by, approximately, two orders of magnitude (Buss et al., 1995, 1997; Leisching et al., 1996). Placing the DMS quantum wells in a resonant microcavity can enhance the Faraday rotation even further due to multiple passage of light through the quantum well region (Kavokin et al., 1997). However, the application of such structures as optical isolators is hindered by a strong absorption, occurring in the spectral region where the excitonic Faraday rotation is the greatest. This problem has been recognized already in the past in the context of Faraday rotation isolators fabricated from bulk material (Zaets and Ando, 1999). Unfortunately, in

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DMS quantum wells the excitonic absorption coefﬁcient is also enhanced compared to the bulk values. In quantum wells containing a quasi two-dimensional electron gas where the existence of negatively charged excitons could be detected, the Faraday effect related to these entities was found also. It was shown to be driven additionally by other mechanisms, apart from the usual ﬁeld-induced changes of the energy of the two circularly polarized transitions, namely by modiﬁcations of the oscillator strength and linewidth related to the variation of the external magnetic ﬁeld (Maslana et al., 2001). The related Kerr effect was studied in CdMnTe/CdTe quantum wells by Buda et al. (1994). The technique, augmented by a time resolving capability, led to the observation of new resonances (Teppe et al., 2002) whose origin is still being investigated. The Faraday effect or the Kerr effect, being an all-optical method of probing spin states in DMS materials, played a very important role in the determination of the dynamics of the spins since the time resolved techniques could be incorporated in various pump and probe conﬁgurations. Such measurements, starting as early as 1989 (Harwit et al., 1989) with a study of transient gratings, resulted in a series of important ﬁndings. Of those studies, the observation of the room temperature spin memory effect (Kikkawa et al., 1997) is probably the most outstanding. 2.3. Magnetic polarons Formation of magnetic polarons in bulk DMS is well documented (see Nawrocki et al. (1981), Suh et al. (1987)) and relatively well understood (see the chapter by Wolff in Furdyna and Kossut (1988)). However, all these studies pertain to bound (as contrasted with – free) magnetic polarons: the sp–d exchange interaction in real DMS systems turns out to be too weak to counterbalance the kinetic energy of a carrier. Therefore, as indicated in early studies of Benoit à la Guillaume (1993) and Kavokin and Kavokin (1991), the investigation of lower-dimensional systems seemed to be a good strategy to observe self-trapped, free magnetic polarons. Indeed, one of the ﬁrst papers that appeared on this subject (Yakovlev et al., 1990) reported that a selective excitation method made it possible to observe free magnetic polarons in a CdTe quantum well with DMS CdMnTe barriers by photoluminescence. However, it became clear soon that a part of the kinetic energy was taken care of by the potential energy of Coulombic binding that represents formation of the exciton. Also, the effect of interface trapping of the polarons was found to be an important effect (Mackh et al., 1993, 1994a, 1994b, 1995a, 1995b, 1995c, 1996; Aguekian et al., 1996) that in most, if not all, cases makes an unambiguous discrimination between free and bound polarons very difﬁcult if not impossible (Stirner et al., 1994a, 1999, 2001; Miao et al., 1996; Takeyama et al., 1999). Stirner et al. (1994a) even proposed to study the interface interdiffusion by means of observation of characteristics of the polarons. The role of the interfaces in polaron formation has been in fact recognized from the very beginning of investigations of DSM quantum well structures (Goncalves da Silva, 1986; Kavokin, 1994). Time studies are an important tool to study magnetic polarons and were soon employed to determine the formation time and the decay time of the spin polarization cloud formed around a center of the polaron (e.g., Dietl et al. (1995), Debnath et al. (1999), Oka et al. (1999), Pittini et al. (1999)). Of course, the polarons bound to impurities, common in n-type or p-type doped DMS samples, can also be found in

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lower dimensionalities (Boudinet and Bastard, 1992; Elangovan et al., 1994). Similarly, the observation of magnetic polarons is not limited to CdTe-based heterostructures. They were also commonly observed in those made of zinc compounds (Poweleit et al., 1994; Rossin et al., 1995). The studies of polarons, even in low-dimensional system are, unfortunately, not that systematic and will require further attention in the future. This is particularly tempting in view of the theoretical predictions by Bhattacharjee (1995) and by Bhattacharjee and Benoit à la Guillaume (1997) concerning polarons in quantum dots. 2.4. Time resolved studies Magnetic polarons in quantum wells brought into attention the problem of dynamics of their formation (Awschalom et al., 1991; Takahashi et al., 2000). Starting from these, many other aspects of time resolved studies were initiated resulting in a long series of important results. Some of them are also mentioned in other sections of this chapter. However, here we shall try to exemplify the diversity of the ﬁeld by mentioning some of those studies and results. First of all, the life times of excitons and magnetic polarons in low-dimensional structures were studied by many researches by means of time resolved photoluminescence (see, e.g., Jonker et al. (1993), Debnath et al. (1999), Pittini et al. (1999)). As already mentioned, time-resolved Faraday rotation experiments proved to be very useful (see, e.g., Baumberg et al. (1994), Kikkawa et al. (1997)). Other techniques included four-wave mixing used by Koch et al. (1993) and femtosecond near-ﬁeld microscopy (Levy et al., 1996), to mention only two examples. All these experiments determined various characteristic times of the DMS lowdimensional systems: electron and hole spin relaxation times, for example, were found to be 10 ps and 100–200 fs, respectively, at room temperature by Koopmans et al. (1999) in their magnetization modulation spectroscopy method in CdTe/CdMnTe quantum wells. All these characteristics are sensitive to the material quality, temperature, presence (or absence) of carriers, etc. Yet, a full picture is, at present, far from completion. 2.5. Trions and combined optical resonances Predicted already 40 years ago, the charged exciton complexes or trions were ultimately observed in CdTe quantum wells only relatively recently (Kheng et al., 1993). To form a trion, a negatively charged trion to be speciﬁc, one needs to create excitons in a sample that contains excess electrons, i.e., in n-type doped material. These excess electrons can bind to the excitons. The binding energy depends on the effective masses of the three particles forming the trion and on the dielectric function of the material. Moreover, because of Pauli’s exclusion principle, the ground state of the trion in the absence of an external magnetic ﬁeld is a spin-singlet with the two electrons having opposite spins. The choice of materials and the fact, that the trions were discovered in low-dimensional systems, were not coincidental: these factors maximized the binding energy of the second electron to an exciton and make the observation of trion signatures in optical spectra less difﬁcult. One of the methods of unambiguous identiﬁcation of the trion recombination peaks in the photoluminecence spectra is their very characteristic circular polarization behavior in the presence of an external magnetic ﬁeld. Therefore, the magnitude and the sign of the

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g-factor of electrons are of primary importance for such identiﬁcation. Here, DMS quantum wells with their variable g-factors (depending on Mn molar fraction, magnetic ﬁeld, and the temperature) proved to be particularly useful in studies of trions (Wojtowicz et al., 1998, 1999, 2000). In particular, one could observe a reversal of the polarization rules of the trions as the electronic g-factor changed its sign when the magnetic ﬁeld saturated the sp–d interaction induced part of the giant Zeeman splitting of the electrons without affecting, at the same time, the linear relationship between the ﬁeld and the usual Zeeman splitting (see, eq. (1) above). Another difﬁculty that was encountered in the search of trions was associated with the possible existence – at very similar energies – of donor bound excitons (Paganotto et al., 1998). A careful study of modulation doped and in-the-well doped samples helped to distinguish between these two possibilities. An important study of Huard et al. (2000) helped to establish experimentally the relationship between the trion binding energy and the Fermi level of the two-dimensional electron gas. The fact that in samples containing magnetic components the two electron spin species may have two distinctly different quasi-Fermi energies was helpful in this study. Negatively charged excitons consisting of one hole and two electrons were not the only type of trions observed in DMS quantum well structures. Also positively charged trions (two holes and one electron) were detected and studied by Kossacki et al. (1999a) who were able to trace the effects of phase space ﬁlling and its inﬂuence on the oscillator strength of both excitons and trions. Application of a magnetic ﬁeld to samples showing the existence of trions leads to a crossover of the energy levels of the latter with a triplet state becoming ultimately the lowest state of the tri-particle excitation, when – speaking in a simpliﬁed way – the magnetic ﬁeld aligns the spins of the two electrons comprising the trion. Calculations performed for non-magnetic quantum well structures (e.g. Redli´nski and Kossut (2002)) indicated that such a transition requires magnetic ﬁelds of considerable strength, not easy to ﬁnd in most laboratories. On the other hand, the giant spin splitting in DMS (if the quantum well layers are made of these magnetic materials) can reduce the ﬁeld required to induce this transition between singlet and triplet states to ﬁelds that are considerably smaller, with consequences that observation of the effect is not that difﬁcult as in non-magnetic samples (Yokoi et al., 2002). Investigations in lightly modulation-doped quantum well structures with DMS wells in the presence of an external magnetic ﬁeld made possible the observation of a series of optical resonances that were not easy to detect in other samples (Kochereshko et al., 1997; Yakovlev et al., 1997; Ossau et al., 2001). Namely, the usual trion and/or exciton optical resonances can be combined with transitions between two consecutive Landau levels giving rise to additional features at energies higher that those of simple excitons and trions (in contrast to shake-up resonances that occur at lower energies). Optical investigations of the modulation doped samples of DMS quantum wells revealed a wealth of other, sometimes not fully understood, features. In particular, studies in the presence of a magnetic ﬁeld in the quantum Hall regime showed a series of anomalies in the luminescence at both integer and fractional ﬁlling factors (see, e.g., Kunimatsu et al. (1998), Takeyama et al. (1998), Imanaka et al. (2001a, 2001b, 2002), Yokoi et al. (2002)). Some of these many features can be related to many-body effects (Teran et al., 1998), the origin of others remains still not clear. Also many-body elementary excitations were observable in samples containing slightly higher concentrations of a two-dimensional

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electron gas (Jusserand et al., 2001). Of course, the usual cyclotron resonance was also detected using extremely high magnetic ﬁelds (Imanaka et al., 2000, 2001b; Matsuda et al., 2000). The effective mass value of the electrons established in this manner was considerably higher that in bulk CdTe. This difference awaits yet a detailed understanding. The high optical quality of the CdMnTe/CdMgTe quantum wells made possible to study a wealth of various excitations by optical methods. Yet other quasiparticles detected in such samples are biexcitons thought to be of importance in devices leading to quantum entanglement. In particular, biexcitons in DMS quantum wells were objects of investigations of Takeyama et al. (1999) and Mino et al. (2001). Most of the results described in the present section were obtained on CdTe/CdMnTe or CdMgTe/CdMnTe heterostructure systems. In contrast to those, a study of ZnSe/ZnCdMnSe modulation doped samples by Crooker et al. (2000b) in the presence of high magnetic ﬁelds revealed considerably simpler spectra. Nevertheless, clear signatures due to trions and excitons were recorded. Possibly the singlet-to-triplet transition of the trion state was observed in these studies above the ﬁlling factor ν = 1. 2.6. Dimension dependence of the sp–d exchange constants It was suspected very early that the sp–d exchange constants, that are of vital importance for the behavior of DMS, might show variation when the dimensionality of the system is modiﬁed. However, probably as late as 1996, the ﬁrst experimental indications that this is indeed so were published in the literature (Mackh et al., 1996). These were soon followed by results of measurements by Yasuhira et al. (2002). The explanation of the effect, focusing on the former set of data, was offered by Merkulov et al. (1999) who pointed out that the reason why the kinetic exchange (which dominates the value of the p–d exchange in the bulk) is inoperative in the case of s–d exchange (i.e., involving the conduction electrons) ceases to be valid. Namely, the symmetry of the conduction electrons, particularly those occupying higher lying conﬁned states, is no longer purely s-like. Therefore, the hybridization, prohibited by symmetry in the purely s-like electrons, is now permitted leading to kinetic exchange entering into the picture again. As the calculations show, the kinetic exchange may, particularly in the case of very narrow quantum wells, even outweigh the direct s–d exchange with a resulting reversal of the sign of the g-factor of the conduction electrons (let us remind the reader that direct and kinetic exchange interaction are responsible for different signs of the exchange constants of electrons and holes, respectively). Even more pronounced effects are expected on the grounds of theoretical considerations in the case of DMS quantum dots (Bhattacharjee, 1998). In the case of quantum wires, a dependence of the exchange parameters on the wire width was reported by Z.H. Chen et al. (2001). The degree of the p–d hybridization, and thus, the strength of the kinetic exchange can also be affected by external factors such as hydrostatic pressure, as reported by Yokoi et al. (2000). 2.7. Transport studies of a two-dimensional electron gas in DMS heterostructures with record mobility Doped samples containing a two-dimensional electron gas were, of course, subject to transport studies, apart from optical investigations discussed so far. Indeed, the mobility of

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electrons achieved in ZnMnSe, using “digital” fractional monolayer insertions of MnSe inside ZnSe quantum wells (see Smorchkova et al. (1997, 1998)) and in modulation doped CdMnTe/CdMgTe heterostructures was sufﬁciently high to enable the observation of a pronounced fractional quantum Hall effect in samples that showed a large spin splitting of the Landau levels (Karczewski et al., 1997; Andrearczyk et al., 2002) leading to a complete spin polarization of the electron gas at suitably high magnetic ﬁelds. It was possible to study temperature and size scaling of the quantum Hall effect in those modulation doped samples (Jaroszy´nski et al., 2000a, 2000b). It is know now that n-type dopants (iodine, chlorine, indium) in CdTe, CdMnTe and CdMgTe give rise to pronounced persistent current effects upon illumination (Ray et al., 1999) and show a behavior similar to that of DX centres in GaAs (Wasik et al., 1995). Electrical properties of quantum wires obtained by electron beam lithography from doped CdMnTe layers were studied very intensively at very low temperatures. The thickness of these wires was in the submicron range making them mesoscopic in nature with the carriers remaining in the diffusive regime of transport. It was shown that the presence of magnetic impurities has a very strong effect on universal conductance ﬂuctuations (Jaroszy´nski et al., 1995; Dietl et al., 1996). The leading effect was traced to the giant Zeeman splitting and redistribution of the carriers between different spin sublevels. Conductance noise studies in those quantum wires showed additional irreversibilities and gave further support to the droplet model of the spin glass state that forms in DMS (Jaroszy´nski et al., 1998a, 1998b). 2.8. Spin injection and spin aligners Östreich et al. (1999) were probably the ﬁrst to demonstrate that DMS spin aligners can be very efﬁcient sources of spin-polarized particles (in this particular case, excitons) that can be injected (in this particular case, by diffusion) into a neighboring quantum well. This seminal paper preceded the ﬁrst reports of efﬁcient injection driven by electric ﬁeld into III-V quantum wells from II-VI DMS spin injectors composed of BeMnZnSe (Fiederling et al., 1999). This work was a considerable step forward in an effort to achieve spin injection into semiconductors since it proposed the use of a spin polarized semiconductor source. The conductivity mismatch between the ferromagnetic electrodes and the semiconductor layers used so far to inject the spins into semiconductor devices was shown to be ineffective for very simple physical reasons, provided that the injection was diffusive (as opposed to tunneling) in nature. The complicated composition of the spin aligner was motivated by the intention to make it lattice-matched to the underlying III-V LED structure which served as the spin detection device. In fact, such a precaution was shown (Jonker et al., 2000; Y.D. Park et al., 2000; Ghali et al., 2001, 2002) to be overly stringent since the spin injection process was proved to be very robust and forgiving as far as the presence of misﬁt dislocations is concerned. The greatest degree of injected spin polarization reported in structures using DMS spin aligners reached so far 90%. However, a certain caution has to be maintained since the exact estimate may depend on details of the spectral composition of the excitonic emission whose circular polarization is typically taken as a measure of the spin injection efﬁciency (Jonker et al., 2001). Further, it has been proposed that a study of trions, mentioned in the previous sections, might be an interesting alternative to a study of circularly polarized excitons used so far for

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detection and quantitative assessment of the spin injection into quantum wells (Ghali et al., 2002). The basis of this suggestion relies on Pauli’s exclusion principle which prefers one given spin direction of extra electrons to form trions, provided that the excitons that are to bind these extra excess electrons are already spin polarized as is the case in the presence of a magnetic ﬁeld. In other words, the basis of the proposition is that the stable ground state of a trion in moderate magnetic ﬁelds is a spin singlet. Of course, all these efforts to achieve spin injection from DMS II-VI spin aligners relied on the possibility of application of an external magnetic ﬁeld which induced a giant Zeeman splitting and, thus, led to a near perfect spin polarization of the exciton gas in the aligner. Apart from that, use of low temperatures was required. The ﬁrst requirement is not necessary in a conﬁguration which uses as a spin aligner a layer of ferromagnetic III-V material as in the design of Y. Ohno et al. (1999). Note also that even in the latter case, low temperature conditions were necessary. The possibility exists that this deﬁnite deﬁciency of II-VI DMS spin aligners will be overcome in a conﬁguration of hybrid structures with metallic ferromagnets deposited directly on top of II-VI DMS aligners (see further in this chapter). A design resembling a resonant tunneling diode was also proposed and explored experimentally as a spin injecting device (Gruber et al., 2001). It consisted of two non-magnetic barriers with a magnetic quantum well located between them. The entire double barrier structure was grown on a III-V light emitting diode serving as the spin detector. Such a device can also serve as a spin switch. 2.9. Tunneling The possibility of spin dependent tunneling has attracted the attention of researchers already early in the history of investigations of low-dimensional structures of DMS. Most of the works concentrated so far on the question of tunneling between double quantum wells separated by a single barrier (Goede et al., 1992; Heimbrodt et al., 1992; Lawrence et al., 1992, 1993, 1994). The emphasis in these works was put on the problem whether the tunneling of excitons, excited in one quantum well and whose recombination was observed in the other quantum well, involved tunneling of separate particles, either holes or electrons, or rather involved excitons as whole entities. It was found that the tunneling of the latter is in fact quite probable (Pier et al., 1994). The fact that one of the wells (or both) was made of DMS material was crucial in ascertaining the resonance conditions between the levels in the two quantum wells whose well widths were usually intentionally different (Smyth et al., 1992; Lee et al., 1996, 2000). Very helpful in these studies were time resolved methods in photoluminescence (see, e.g., Haacke et al. (1993), Kayanuma et al. (2001a, 2001b). Such a tunneling can be viewed as an example of spin injection provided that the tunneling time is shorter than the spin relaxation time, as found by Kayanuma et al. (2002). Obviously, the time constants deduced from this type of measurements were strongly dependent on particulars of the sample used for experiments. A theoretical analytical study of the tunneling lifetime in the double barrier quantum well was published by J.U. Kim and Lee (1998) within the general framework of the envelope function approximation. Another problem that was attacked in this context was vertical transport in a superlattice composed of alternating magnetic with non-magnetic layers, CdTe and CdMnTe, by Roussignol et al. (1993) who found that, depending on the energy of the ecitation, the transport along the superlattice axis can have a bipolar or hopping character.

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The conditions for the spin dependent tunneling across a DMS barrier were investigated theoretically by Semenov and Kirichenko (1996) who in particular considered the effect of spin-ﬂip that may occur under these conditions during the tunneling process. They showed that this led to additional features in the I –V characteristics of the tunneling. Spin-dependent tunneling is closely related to the problem of spin injection. In particular, spin transport of excitons between two asymmetric double quantum well heterostructures is an object of investigations of, e.g., Kayanuma et al. (2001a). Filtering of spins by passing them through a tunneling DMS barrier was studied theoretically by Carlos Egues (1998) and by Egues et al. (2001). So far, such ﬁltering effect has not been, unfortunately, demonstrated experimentally, in spite of the achievements in the spin injection from spin aligners. Also, only a theoretical study exists (K. Chang et al., 2002) of a ballistic transport in single and double barrier tunneling magnetoresistance structures. It is shown that, because of a large splitting of the Zeeman components of the conduction band, one may expect new oscillatory features in the conductivity as function of a magnetic ﬁeld. Finally, let us mention in this section, since it is of importance for tunneling, a study of Nicholas et al. (1994) of interband resonant polaron coupling in CdTe/CdMnTe quantum wells showing that the polaronic shift in such structures can be very large and has to be taken into account when designing samples for resonant tunneling. The particularly strong polaronic effect is tracked down by the authors to the fact that the polaron radius and the exciton radius can be made of very similar size. In fact, the magnetic polaron effect can be ﬁne-tuned by an external magnetic ﬁeld by making 1s and 2s states of conﬁned excitons to coincide energetically with the LO phonon energy (Yakovlev et al., 1995) which in itself represents a unique feature of the quantum well structures made of DMS. 2.10. Transfer of an excitation between carriers and localized moments Transfer of excitations between the holes and/or electrons and localized magnetic moments of Mn ions is an important issue from several points of view. First, it can be responsible for changes of the luminescence intensity (C.S. Kim et al., 2000b; Falk et al., 2002) due to, alternatively, excitonic transitions and intra-Mn transitions. Such transfer in these experimental studies was shown to be magnetic ﬁeld dependent. An important factor, that determines the effectiveness of the transfer, is the relative position of the exciton and the Mn d-levels. This could be “engineered” either by proper choice of the materials or by manipulations with the size quantization (e.g., by changing the width of the quantum wells). Another issue is heating of the carriers (or vice versa, of the localized moments) and can be of importance for spin dephasing that is a central issue in spintronic applications. These problems, and the related problem of spin–lattice relaxation, were studied by several authors (see, e.g., Kulakovskii et al. (1996a, 1996b); Shcherbakov et al. (1999, 2000)). In particular, it was found that the presence of excess carriers in doped samples speeds up such transfer of the excitation. This is particularly true in the case of valence band holes (König et al., 2000; Keller et al., 2001). The transfer of the excitation from the photoexcited carriers to the system of localized Mn magnetic moments, therefore, does depend on the intensity of the excitation. Interestingly, for high excitation power local hot and cold domains can be formed (Teppe et al., 2002) making the system of the localized magnetic moments spatially inhomogeneous as detected in time resolved Kerr effect measurements

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in pump and probe experiments. The spatial extensions of these domains are yet to be determined. A similar technique was used also by Akimoto et al. (1998) to study the Larmor precession of Mn moments in CdTe/CdMnTe quantum well structures and its changes induced by the presence of photo-excited carriers coupled to the former by sp–d interaction. 2.11. Quantum wires Quantum wires composed of diluted magnetic semiconductors are the least represented low-dimensional structures of these materials. Several types of methods of their fabrication have been so far attempted. First, there are quantum wires fabricated by lithography. Ray et al. (2000) studied the excitonic spin polarization in ZnSe/ZnCdMnSe quantum wires having width 20–80 nm deﬁned by electron beam lithography and etched by wet techniques. The authors report a strong spin polarization of the luminescence emission in the presence of a magnetic ﬁeld which they relate to the giant Zeeman splitting of the excitonic states in these wires. Electron beam lithography and chemical etching were used also by Takahashi et al. (2000) to fabricate wires of CdMnSe and of ZnSe/ZnMnSe by growth on patterned substrates (L. Chen et al., 2000). These proved to be of sufﬁciently high quality to allow optical investigations. The results of these investigations showed shifts of exciton luminescence and its polarization, that were interpreted by the authors as due to a combination of strain- and lateral-conﬁnement effects. The luminescence lines displayed a strong Zeeman splitting, characteristic for DMS. An ingenious method of preparing arrays of CdMnSe and CdMnSe wires by deposition in wire-like pores in silica hosts has been employed by L. Chen et al. (2000) who observed a strong blue shift due to the conﬁnement of excitons in those structures as well as a pronounced band gap which is thought to be due to the sp–d exchange interaction. Yet another method of fabrication of quantum wires was used by Marsal et al. (2001) who deposited them on vicinal surfaces of miscut substrates. Such a technique resulted in a lateral conﬁnement that shifted the energy of the excitonic features to the red compared to that in quasi two-dimensional structures. In the case of these studies the quantum wires were made of non-magnetic CdTe embedded in a DMS CdMnSe matrix. The results indicated that there is a very strong dependence of optical properties of such structure on the miscut angle. Cleaved-edge overgrowth was originally used to produce quantum wires of III-V compounds. In the case of II-VI materials this method was used with a limited success (Brinkmann et al., 1996; Cywi´nski et al., 1998a, 1998b). While additional features were observed in the microluminescence spectra, it was difﬁcult to study their polarization properties, which would prove their one-dimensional nature. Another method of wire fabrication was explored by Welsch et al. (2001) who made use of selective interdiffusion of Cd-based two-dimensional quantum structures in dependence whether or not there were SiO2 stripes lithographically deposited on top of such structures. Annealing of such structures resulted in regions that behaved in optical investigations as quasi one-dimensional objects. A similar method of fabrication was also employed to make quantum dots (see below). On the theoretical side, it was shown (Kyrychenko and Kossut, 2001) that the usual optical transition selection rules in diluted magnetic semiconductor quantum wires become quite different from their non-magnetic counterparts, and in fact magnetic ﬁeld-dependent,

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due to an sp–d coupling induced mixing of the heavy and light hole spin subbands. This fact makes an unambiguous identiﬁcation of the quasi one-dimensional excitons quite difﬁcult. 2.12. DMS quantum dots There are several methods of further reduction of the dimensionality down to zero, i.e., by fabrication of quantum dots. Among those, lithography and etching, growth of nanocrystals in dispersions and self-assembly during the epitaxy are the most popular methods. The vast majority of the works devoted to semiconductor self-assembled quantum dots made of IIVI compounds describe non-magnetic systems such as CdTe embedded in ZnTe. Only relatively few include studies of Mn incorporated either in the dots themselves or in the matrix that embeds them. The ﬁrst method of preparation was used by Ikada et al. (2002) who employed electron beam lithography to prepare ZnCdMnSe quantum dots by etching pillar-like structures, that could be studied by optical methods. Their size was 30 nm on the average with the width of the starting quantum well being 10 nm. The composition of the material could be determined by a well-calibrated study of the quantum well structures that were later used for lithography. The value of the Mn molar fraction obtained in this way was x = 0.08. The luminescence spectra of the dots in the absence of a magnetic ﬁeld revealed a shift of the emission compared to that from the quantum wells which the authors associate with strains introduced during the fabrication. The dot peak was only 8 meV wide and it showed a large shift in the presence of a magnetic ﬁeld to lower energies resembling that of the bulk material. However, the magnitude of this shift is smaller than that in the quantum well placed in the same magnetic ﬁeld. This may be due to modiﬁcations of either the exchange constants or the magnetization (fewer magnetic neighbor effect). At present it is not clear which of these possibilities is dominant. The electron beam lithography was used by Klar et al. (1996, 1998b) to fabricate ZnMnSe/ZnSe quantum discs with diameters in the range 100–200 nm. In this way dot-like structures were obtained in which the DMS material represented the conﬁning top and bottom barriers. Therefore an enhancement of the gfactors in those structures is solely due to possible diffusion of Mn into the ZnSe region and due to penetration of the wave functions of the conﬁned carriers into the barriers. Giant spin splitting, on the other hand, was observed in CdS nanocrystals containing on the average a single Mn atom per dot. Indeed, the splitting in those samples, prepared by precipitation of nanocrystals in a suitable solution, was observable in the presence of a magnetic ﬁeld but the authors claim that it also exists even in the absence of the ﬁeld. They estimate its magnitude as 3.2 meV at 2 K. Self-assembly driven by a large lattice mismatch is a very popular method of preparation of quantum dots. In the case of DMS materials it was used by S. Kuroda et al. (2000) and Terai et al. (2000) as well as by Ma´ckowski (2001, 2002a) to obtain CdMnTe quantum dots in a ZnTe matrix with typical sizes of 10–20 nm in diameter and 2–3 nm in height. In view of strong interdiffusion known to occur in those samples and revealed by analysis of high resolution transmission electron micrographs, it is difﬁcult to estimate the exact amount of Mn incorporated into those samples. The luminescence from those samples typically reveals a broad structure as is the case of non-magnetic CdTe dots. However, while in the latter case the broad structure breaks up into a series of very sharp lines when observed in a microluminescence apparatus, the Mn-containing dots in most of the cases did not

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possess such sharp spiky spectra. It is only after a prolonged search that spiky spectra in the microluminescence of CdMnTe quantum dots were observed (S. Kuroda et al., 2000; Ma´ckowski et al., 2002b). However, even then the full width at the half maximum of such spikes was by at least one order of magnitude larger than in the case of non-magnetic dots. It was suggested (Maksimov et al., 2000; Bacher et al., 2001; Brazis and Kossut, 2002) that this additional broadening is associated with ﬂuctuations of the spin degrees of freedom. Calculations based on a magnetic polaron model proved that those ﬂuctuations provide a correct order of magnitude for the width of the lines. Application of a magnetic ﬁeld can reduce the Mn spin ﬂuctuations which leads to narrowing of the luminescence lines of a single quantum dot (Maksimov et al., 2000). The source of this effect is in principle the same as that leading to “magnetic ﬁeld annealing” of the photoluminescence linewidth at the magnetization steps observed in ZnSe/ZnCdMnSe single quantum wells by Crooker et al. (1999). Such a suppression of spin ﬂuctuations requires very high magnetic ﬁelds unless the dot contains only a very minute number of magnetic atoms. Self-assembly was also used to fabricate CdZnMnSe dots by Reshina et al. (2001) who studied them by means of Raman scattering. In particular, they observed such scattering by magnetic excitations. They also studied non-magnetic dots surrounded by a DMS matrix. A similar arrangement (i.e., DMS matrix ZnMnSe with non-magnetic CdSe dots) was achieved by self-assembly during molecular beam epitaxy and studied by C.S. Kim et al. (2000a). The dots in this case showed an interesting inﬂuence of the luminescence intensity on the magnetic ﬁeld known also from other dimensionalities and related to spin-dependent selection rules for the excitation transfer from the exciton system to intra-Mn luminescent centers. Finally, let us note that the method of selective interdiffusion mentioned already in the context of the quantum wires, was successfully used to prepare quantum dots. Unfortunately, so far, only non-magnetic dots of CdTe/CdMgTe were fabricated in this fashion (Zaitsev et al., 2001). On the other hand, Kratzert et al. (2001) reported on fabrication of dot-like objects by thermally activated reorganization of an initially two-dimensional 3 monolayer-thick CdMnSe ﬁlm. The size of the dots obtained by the in situ annealing is estimated to be of the order of 10 nm. 2.13. Metallic ferromagnet/DMS quantum well hybrid structures Originally thought to serve as spin injectors, ferromagnetic contacts can also be used to induce further conﬁnement of particles in quasi two-dimensional quantum wells. This opportunity arises from a great sensitivity of the band edges on the magnetic ﬁeld, so that even fringe ﬁelds of a ferromagnet may have a sizable effect on, say, optical properties by trapping excitons in local minima of the potential. Such local minima may be viewed as a method of formation of dots or wires (depending on the shape of the ferromagnetic domain that is the source of a given fringe ﬁeld). There are relatively few examples of work in this direction in the case of DMS materials, although the idea of hybrid structures is quite vast and various conﬁgurations were considered in the past (Peeters and de Boeck, 2000). The underlying idea is to fabricate a small ferromagnetic metallic island with a proper shape in the immediate vicinity of a DMS quantum well, say, on the top surface of a structure with the quantum well buried shallowly under it. There are several factors that make this task difﬁcult: magnetically dead layers, strain introduced by lattice mismatch between

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the semiconductor and the metal, surface states formed during the epitaxy of the quantum well structure, etc. A ﬁrst attempt was made by Crowell et al. (1997) who studied the inﬂuence of an iron deposition on ZnMnSe quantum wells buried 500 Å beneath the surface. The observed changes of the microluminescence were ascribed to the strain effect rather than to the effects of the fringe ﬁeld. In samples made by using CdMnTe quantum wells buried only 300 Å beneath the surface of the sample there are indeed indications that some additional localization due to the fringe ﬁeld takes place (Kossut et al., 2001; Kudelski et al., 2001; Cywi´nski et al., 2002). Of course, the domain structure of the ferromagnetic island plays a very important role in these efforts (see Ferrand et al. (2001)). It is known that it becomes relatively simple when the rectangular ferromagnetic islands are made with width lower than 2 μm (assuming the length to be of the order of 10 μm), see Imada et al. (2000). However, more experimental studies are needed in order to optimize the design of such hybrid structures. 2.14. Magnetic properties of DMS thin ﬁlms and quantum wells Since the predominantly antiferromagnetic interactions between Mn ions are short-ranged and practically limited to the nearest-neighbor exchange it is extremely difﬁcult (if possible at all) to reach the limit of contiguous layer whose width is comparable to such characteristic interaction range. This is in contrast to the case of electrons and holes whose characteristic length (de Broglie’s wave length, for instance) can be easily of the same order or greater than the layer width in quantum heterostructures. Therefore, “two-dimensional magnetic systems interacting through a Heisenberg-like Hamiltonian” – systems that are the subject to the Mermin–Wagner theorem – are hardly possible to make when using DMS (although there are suggestions that it could have been the case, see Awschalom et al. (1987)). First, in these nearly always strained systems it is difﬁcult to ascertain isotropy strictly required by the theorem. Second, when depositing very thin layers one has to seriously worry about interdiffusion of magnetic and non-magnetic species, as shown by the study of interface sharpness in the case of the quasi two-dimensional quantum wells or by analysis of the chemical composition of self-assembled quantum dots and sub-monolayer thick depositions (C.S. Kim et al., 2000a). Therefore, the observed dependence of, e.g., spin glass transition temperatures on the width of CdMnTe quantum wells (Awschalom et al., 1987; Yakovlev et al., 1999) has to do with an effective dilution of the magnetic ion system rather than with the true reduction of the dimensionality in the spirit of the Mermin–Wagner theorem (Sawicki et al., 1995b). Nevertheless, there are features that make magnetic properties of superlattices made of DMS materials quite distinct and interesting in themselves. Here we shall limit our attention to thin ﬁlms and quantum wells of DMS leaving the interesting case of superlattices to the part devoted to IV-VI materials (see section 4.3). There we shall treat in one place the results found for both II-VI and IV-VI superlattice magnetic properties: magnetic order observed in those structures and an intriguing interlayer magnetic coupling across non-magnetic spacer layers. The ﬁrst interesting issue is simply related to the fact that molecular beam epitaxy enabled studies of magnetic properties of cubic forms of ZnMnTe and CdMnTe in the entire range of the Mn molar fraction, i.e., from x = 0 to x = 1. Such studies included measurements of the critical temperature of the transition to the antiferromagnetically ordered phase in MnTe-rich compounds, including cubic MnTe (Stachow-Wójcik et al., 2000). It

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turned out that remanent behavior of the magnetization persists even in the range of very high Mn molar fractions where antiferromagnetic order was known to set in. A detailed study of the dependence of the spin freezing temperature by Dahl et al. (1996) and by Sawicki et al. (1995a, 1995c, 1996) did not reveal vanishing of this temperature as the width of the ﬁlms were reduced even to three single monolayers. Clear evidences for the existence of magnon excitations were obtained by Raman scattering (Jouanne et al., 1998; Szuszkiewicz et al., 1997) and by neutron scattering (Hennion et al., 2002) where even the folding of the magnon dispersion relations could be revealed. 2.15. New DMS II-VI materials – in search for ferromagnetism Recent progress in growth of ferromagnetic III-V materials and quantum structures (for review, see Matsukura et al. (2002)) has stimulated parallel efforts in this direction within the family of II-VI DMS. In fact, traces of ferromagnetic behavior were observed in bulk ZnMnTe p-type doped either with phosphorus (Sawicki, 2002) or with nitrogen (Ferrand et al., 2001) as well as in CdMnTe quantum wells remotely doped with nitrogen (Haury et al., 1997). In the latter case, a certain degree of control of the ferromagnetism could have been exerted either by illumination of the samples with light or by application of an external voltage. These factors are believed to inﬂuence the surface states and, thus, to modify the hole concentration. It is commonly believed (Dietl et al., 2000) that the indirect coupling via holes (Zener mechanism) is responsible for the long ranging ferromagnetic interaction between Mn ions in these compounds, with the resulting paramagnetic–ferromagnetic phase transition observed by optical methods as the concentration of holes is increased. The quasi two-dimensional nature of the hole gas in these structures helps to increase (Dietl et al., 1997) the ferromagnetic coupling strength. Nevertheless, in the structures in question the ferromagnetic phase forms at temperatures in the range around 2 K. Therefore, materials containing other ions than Mn were investigated, such as Cr and Co containing II-VI compounds. The former were predicted theoretically to display ferromagnetic Cr–Cr interactions (Blinowski and Kacman, 2001), see also the review by Kacman (2001). Since these materials are described in further sections of this chapter let us only mention at this point that initial attempts to ﬁnd ferromagnetism in CdCrTe and ZnCrTe failed (Wojtowicz et al., 1997) but more recently there are reports (Saito et al., 2002) of observation of ferromagnetism in ZnCrTe (containing about 0.035 molar percentage of Cr). Similarly, there are very recent reports of ferromagnetism in ZnCrSe (Karczewski et al., 2003). These conﬂicting reports require further investigation. Particular caution is necessary in view of the known tendency of formation of ferromagnetic clusters independent of the semiconductor matrix. The common denominator of these observations of ferromagnetism was, again, its appearance at relatively low temperature in the liquid helium range. Surprising is also the ﬁnding of ferromagnetism in ZnCoO (Ueda et al., 2001) even in the vicinity of room temperature. The surprise is the greater, since the theoretical analysis (Blinowski and Kacman, 2001) did not predict a tendency to ferromagnetic interactions in cobalt substitutional alloys in II-VI hosts. However again, these ﬁndings are not supported by the study of excitonic Zeeman splitting of CdCoTe (Alawadhi et al., 2001) which indicates the existence of only antiferromagnetic coupling between cobalt ions, even stronger than in the case of the Mn-containing counterpart. Clearly, more studies are required to conﬁrm these new results and explain the mechanism of the ferromagnetic order that might occur in those new materials.

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3. Bulk crystals and low-dimensional structures of IV-VI DMS compounds In the IV-VI group of diluted magnetic (semimagnetic) semiconductors the best studied systems are materials with the 3d transition metal Mn (such as Pb1−x Mnx Te or Sn1−x Mnx Te) and materials with the 4f rare earth ion Eu, e.g., Pb 1−x Eux Te (for review see, e.g., Furdyna and Kossut (1988), de Jonge and Swagten (1991), Bauer et al. (1992), Kossut and Dobrowolski (1993), Dietl (1994), Galazka (1995), Story (1997a, 1997b), Galazka et al. (1999)). There exist also IV-VI DMS materials with another 3d ion, Cr (Akimov et al., 1989; Story et al., 1995; Grodzicka et al., 1996b), or with rare earth 4f ions, i.e., containing Gd (see section 3.1.3 of this chapter), Yb (Isber et al., 1995b, 1996; Grodzicka et al., 1996a; Mahoukou et al., 1996; Story, 1997b; Skipetrov et al., 1999, 2001; Ivanchik et al., 2000), or Ce (Gratens et al., 1997a, 2001; Fita et al., 1999). There are also materials with 5f ions, e.g., with uranium (Isber et al., 1995c). The solubility limit of Mn, Eu, and Yb ions in IV-VI materials grown under quasi-equilibrium thermodynamic conditions is typically about 10 at.%. For other 3d and 4f elements the solubility limit is usually below 1 at.% and corresponds rather to the limit of heavy doping than to the formation of DMS substitutional mixed crystals. The technological methods applied to grow IV-VI DMS involve the well-known Bridgman method for bulk crystals, as well as molecular beam epitaxy (MBE) and hot wall epitaxy (HWE) techniques for thin epitaxial layers and low-dimensional structures. In the case of bulk crystals grown by the Bridgman method one observes a strong change of the composition of the alloy along an ingot. TABLE 1 List of ternary IV-VI DMS semiconductors with rare earth ions. The composition range available in bulk crystals grown under quasi-equilibrium conditions is indicated. For some materials, a higher content of magnetic ions may be obtained (numbers marked with∗ ) using non-equilibrium growth techniques Material

Maximal reported composition (x)

Reference

Pb1−x Eux Te

0.06, 1.0∗

Pb1−x Eux Se

0.041, 0.11∗

Pb1−x Eux S

0.059, 0.20∗

Sn1−x Eux Te Pb1−x Gdx Te Pb1−x Gdx Se Pb1−x Gdx S Sn1−x Gdx Te Pb1−x Ybx Te

0.013 0.11 0.05 0.05 0.09 0.27, 0.25∗

Pb1−x Ybx Se Pb1−x Ybx S Pb1−x Cex Te Pb1−x Cex Se Pb1−x Cex S Sn1−x Cex Te

0.04 0.01 0.005 0.04 0.021 0.01

ter Haar et al. (1997) Krenn et al. (1999) Lambrecht et al. (1991) Bindilatti et al. (1996) Bindilatti et al. (1998) Ishida et al. (1988b) Gratens et al. (1998) Bruno et al. (1988) Kowalski et al. (1999) Kowalski et al. (1999) Górska et al. (1992) Mahoukou et al. (1996) Kowalski et al. (1999) Isber et al. (1996) Isber et al. (1996) Fita et al. (1999) Gratens et al. (1997a) Gratens et al. (2001) Fita et al. (1999)

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A list of IV-VI DMS materials with Mn is presented in the paper by Kossut and Dobrowolski (1993) (see also Galazka (1995), Galazka et al. (1999)). A list of IV-VI materials studied so far with rare earth ions is given in table 1. The solubility limits given in table 1 have been determined from X-ray powder diffraction and electron-probe microanalysis of bulk crystals grown under quasi-equilibrium conditions. In some cases, it is possible to considerably extend this limit using non-equilibrium growth techniques (e.g., in the MBE grown layers of Pb 1−x Eux Te, x 1, or in the polycrystalline bulk Sn1−x Mnx Te with x 0.4 and Ge1−x Mnx Te with x 0.5 obtained using rapid cooling techniques). There also exist quaternary materials frequently used to tailor the physical properties of the non-magnetic matrix. The important examples are carrier concentration control in Pb1−x−y Sny Mnx Te and energy band gap and refractive index engineering in Pb1−x Eux Te1−y Sey . Practically all IV-VI DMS compounds crystallize in the rocksalt crystal structure with magnetic ions randomly occupying the fcc lattice sites of the cation sublattice. The lattice parameter a0 changes linearly with the content of magnetic ions following Vegard’s law (see Galazka et al. (1999)). There exists no experimental evidence in IV-VI DMS for the low-temperature structural phase transitions from cubic to rhombohedral structure well-known, e.g., in SnTe and GeTe (Nimtz and Schlicht, 1983).

Fig. 2. Scheme of the band structure model used in the analysis of transport and optical properties of IV-VI DMS materials.

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Qualitatively, the electron band structure of IV-VI semimagnetic semiconductors is analogous to the band structure of the non-magnetic counterpart materials. The scheme of the band structure model providing a good basis for the analysis of both transport and optical effects in IV-VI semimagnetic semiconductors is presented in ﬁg. 2. The conduction band and the valence band as well as 4 other (far) bands at the L-point of the Brillouin zone are described within the frame of the Dimmock model. The energy dispersion relations of electrons and holes are non-parabolic and anisotropic. A detailed description of this model is given, e.g., by Nimtz and Schlicht (1983) and Bauer et al. (1992). The parameters describing the band structure in the model of Dimmock are: the band gap Eg , the transverse Pt and longitudinal Pl matrix elements of momentum, and the far bands contributions to the effective masses and to the effective g-factors of holes and electrons. The basic band structure parameters of IV-VI DMS with Mn are presented in the paper by Kossut and Dobrowolski (1993) (see also Bauer et al. (1992), Galazka et al. (1999)). The electron band structure parameters of materials with Eu are discussed in section 3.2.1 of this chapter. The electronic properties of p-type IV-VI semimagnetic semiconductors with very high concentration of carriers p 5 × 1019 cm−3 (e.g., Sn1−x Mnx Te or Pb1−x−y Sny Mnx Te) are inﬂuenced by the presence of the band of heavy holes (the so-called, Σ-band). The top of this band is located at the Σ-point of the Brillouin zone about EΣ = 0.2 eV below the top of the band of light holes (see ﬁg. 2). The important element of the electron structure of DMS materials is the location of the density of states (DOS) derived from 3d or 4f (magnetic) orbitals of the magnetic ions. The scheme of the situation encountered in IV-VI semiconductors with Mn and with Eu is presented in ﬁg. 3. The DOS labeled 3d and 4f corresponds to the half (spin-up) of the total DOS due to these orbitals. The other half (spin-down) of the total DOS is shifted up in energy scale by few eV’s and is expected to be located far above the bottom of the conduction band. The large energy difference

Fig. 3. Scheme of the density of states for Pb1−x Mnx Te (a) and Pb1−x Eux Te (b). The range of Fermi level positions encountered in these crystals depending on doping or annealing conditions is marked with EF .

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between both contributions is due to the strong on-site electron–electron interactions and correlations. In both cases the contributions due to magnetic orbitals are located far below the top of the valence band. The photoemission experiments providing direct access to the DOS give the location of the “center of gravity” of the Mn 3d orbitals about 3.5 eV whereas for the Eu 4f orbitals about 1 eV below the top of the valence band (Orlowski et al., 1994; Denecke et al., 1996). It means that magnetic orbitals are not contributing to the density of states in the energy range covered by the Fermi level positions encountered in IV-VI DMS even for heavily doped or annealed crystals. There are two main consequences of such location of DOS originating from magnetic orbitals with respect to the conduction and valence band edges. First, Mn and Eu are not expected to be electrically active and are not contributing to charge transport processes in DMS materials. Second, the magnetic ions should posses local magnetic moments similar to free ions. 3.1. Carrier concentration control of magnetic properties of IV-VI DMS with Mn 3.1.1. Mn content-carrier concentration (x–p) magnetic phase diagram of SnMnTe and PbSnMnTe Magnetic properties of IV-VI DMS with Mn depend strongly on the concentration of conducting holes. The crystals of PbMnTe, PbMnSe, and PbMnS in which the relatively low (typically n, p = 1018 cm−3 ) concentration of carriers is encountered are Curie–Weiss paramagnets exhibiting rather weak antiferromagnetic d–d nearest-neighbor exchange interaction of the order of JNN = −0.5 K (see, e.g., Górska et al. (1992), Galazka (1995)). In contrast, materials with very high concentration of conducting holes p (2–3) × 1020 cm−3 show a ferromagnetic transition. These are ternary and quaternary DMS systems based on SnTe and GeTe and their alloys with PbTe such as Sn1−x Mnx Te, Pb1−x−y Sny Mnx Te, Ge1−x Mnx Te, and Pb1−x−y Gey Mnx Te. The ferromagnetic properties of tin and germanium tellurides-based DMS materials with Mn were studied experimentally by measurements of the magnetic susceptibility, magnetization, magnetic speciﬁc heat, by neutron scattering and ferromagnetic resonance, and measurements of the thermoelectric power, and the anomalous Hall effect (Escorne and Mauger, 1979; Inoue et al., 1979; Story et al., 1993; Vennix et al., 1993; Łazarczyk et al., 1997a, 1997b, 1998; Radchenko et al., 2002). The effect of the carrier induced ferromagnetic phase transition observed in these compounds is discussed by Kossut and Dobrowolski (1993) and is described in detail by Story et al. (1986, 1990), Swagten et al. (1988), de Jonge and Swagten (1991) and Story (1997a). In this chapter, we will give an update of the magnetic properties of SnMnTe and PbSnMnTe in the entire Mn concentration range available (x 0.12) as well as discuss the magnetic phase diagram of these materials covering both carrier concentration induced paramagnetic–ferromagnetic and ferromagnetic–spin glass transitions. The detailed X-ray powder diffraction, electron-microprobe, and extended X-ray absorption ﬁne structure (EXAFS) experimental analysis of the crystal structure of IV-VI DMS with Mn showed that the maximal content of Mn in a single phase rocksalt PbSnMnTe crystal is about 16–18 at.% (Miotkowska et al., 1997; Iwanowski et al., 2001). In practice, good quality bulk crystals of PbSnMnTe are available up to 12 at.% of Mn. The carrier concentration induced ferromagnetic transition is observed in PbSnMnTe bulk crystals in the entire available composition range. Figures 4 and 5 illustrate this effect

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Fig. 4. Temperature dependence of the inverse magnetic susceptibility of Pb1−x−y Sny Mnx Te with x = 0.08 and y = 0.72 for various carrier concentrations. For crystals with p > 3 × 1020 cm−3 , the paramagnetic Curie temperature determined from these plots indicates ferromagnetic d–d exchange interactions.

in PbSnMnTe crystals with x = 0.03–0.12 and varying carrier concentration (Łazarczyk et al., 1997a). For samples with high carrier concentration, a Curie–Weiss-type, χ(T ) = C/(T − Θ), temperature dependence of the magnetic susceptibility is observed. The positive sign of the Curie–Weiss temperature Θ indicates the presence of the ferromagnetic d–d interactions. The strength of the ferromagnetic exchange interactions in IV-VI DMS materials depends on the concentration of carriers (see the shift of the parameter Θ in ﬁg. 4) which results also in the carrier concentration dependence of the ferromagnetic Curie temperature TC . The TC (p) dependence for crystals of Pb1−x−y Sny Mnx Te with Mn contents up to 12 at.% is presented in ﬁg. 5 (Łazarczyk et al., 1997a). A characteristic feature of the TC (p) dependence is the existence of a certain threshold carrier concentration p = pt ∼ = 3 × 1020 cm−3 , above which the IV-VI DMS materials show ferromagnetic properties. For carrier concentrations lower than the threshold value p < pt , the crystals are paramagnetic, similarly to the low carrier concentration materials like PbMnTe. The ferromagnetic d–d interspin exchange interactions in IV-VI DMS are due to the Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanism which proceeds via quasi-free carriers from the valence band. It originates from the spin polarization of the conducting carriers induced in a conducting hole gas by the local magnetic moments via sp–d exchange interaction. The RKKY interaction has long-range character and is known to result in spin– spin exchange interactions of different signs depending on the value of the characteristic parameter 2kF Rij , where kF is the Fermi wave vector and Rij is the interspin distance. The strength of the RKKY interaction depends on the concentration of carriers (via kF and the Fermi energy EF ), the sp–d exchange integral Jsd , and the effective mass of carriers m∗ as

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Fig. 5. Carrier concentration dependence of the ferromagnetic TC and paramagnetic Θ Curie temperature for Pb1−x−y Sny Mnx Te crystals (y = 0.72) with various carrier concentrations and Mn content.

given by the expressions Iij =

2 (k a )6 Jsd F 0 FRK (X)e−Rij /λ , 64EF π 3

FRK =

sin(X) − X cos(X) , X4

X = 2kF Rij

(2)

and EF = h¯ 2 kF2 /2m∗ ,

where λ is the carrier mean free path and a0 is the lattice parameter. In more detailed analysis of the RKKY mechanism in IV-VI DMS materials one has to take into account the anisotropic and multi-valley character of the valence band structure of PbSnMnTe. It results, in particular, in anisotropic (cubic) d–d interspin interactions (Story et al., 1992). Another important aspect of the realistic description of the RKKY mechanism in IV-VI DMS is the incorporation of the effects brought about by disorder present in both electronic and magnetic subsystems of IV-VI DMS. One of the consequences of the electronic disorder is that the exchange integrals are now statistical quantities characterized by their mean value and variance. The existence of a distribution of the values of the exchange integrals reﬂects the statistical distribution of the sources of electronic disorder

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(e.g., non-magnetic defects). A detailed analysis of this effect is presented by Eggenkamp et al. (1995). The application of the RKKY model to IV-VI DMS materials with Mn is, in particular, based on the assumption that one deals with Mn 2+ ions possessing spin-only magnetic moments of 5 Bohr magnetons (magnetic ground state with electron conﬁguration 3d5, S = 5/2 and L = 0). This assumption was positively veriﬁed experimentally by electron paramagnetic resonance, magnetic susceptibility, and magnetic speciﬁc heat measurements (Story et al., 1993, 1996b; Łusakowski et al., 2002). The strength of the RKKY interaction quadratically depends on the sp–d exchange integral Jsd . An analysis of this parameter in IV-VI DMS is given in the paper of Kossut and Dobrowolski (1993) (see also Furdyna and Kossut (1988), de Jonge and Swagten (1991), Bauer et al. (1992), Dietl (1994), Dietl et al. (1994), Galazka (1995), Story et al. (1996b)). The RKKY interaction is known to be responsible both for the ferromagnetic properties of the magnetic materials (like in rare earth metals) and the spin glass properties (like in diluted metallic alloys, e.g., Cu:Mn). The analysis of the RKKY interaction in IV-VI DMS shows that within the range of parameters available in these materials the effect of the

Fig. 6. Temperature dependence of the magnetization (a), the ac magnetic susceptibility (b), and the magnetic contribution to the speciﬁc heat (c) in Pb1−x−y Sny Mnx Te crystals with x = 0.02 and y = 0.72 for two carrier concentrations.

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Fig. 7. Temperature dependence of the ac magnetic susceptibility in samples of Sn1−x Mnx Te with x = 0.02 and varying carrier concentration.

Fig. 8. Temperature dependence of the neutron diffraction peak intensity (proportional to the square of the spontaneous magnetization) in Sn0.98 Mn0.02 Te crystals with various carrier concentrations given in the ﬁgure in units of 1020 cm−3 .

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transformation of an RKKY ferromagnet into an RKKY spin glass may be observed. This idea was realized experimentally in SnMnTe and PbSnMnTe. In ﬁg. 6 the basic magnetic properties are compared for two samples of PbSnMnTe with the same Mn content, x = 0.02, and with different carrier concentrations: the sample with p = 6.8 × 1020 cm−3 is ferromagnetic, whereas in the sample with p = 1.4 × 1021 cm−3 the ferromagnetic order breaks down and spin glass properties are observed. Another experimental evidence for the carrier induced transition from the ferromagnetic to spin glass state is presented in ﬁg. 7 for Sn0.98 Mn0.02 Te samples with three different concentrations of holes. For the ferromagnetic sample (p = 7.5 × 1020 cm−3 ) there is practically no frequency dependence of ac magnetic susceptibility in the range of 10 Hz–20 kHz whereas in the sample with very high carrier concentration (p = 2.4 × 1021 cm−3 ) a strong reduction of the magnetic susceptibility and a shift of the temperature of the characteristic cusp is clearly observed (Eggenkamp et al., 1993). The ﬁnal piece of experimental evidence is provided by neutron diffraction experiments showing the standard mean-ﬁeld-like temperature dependence of the spontaneous magnetization in the ferromagnetic sample and almost no spontaneous magnetic moment for the spin glass sample (see ﬁg. 8, after Vennix et al. (1993)). The simple model of this effect is based on the properties of the RKKY interaction. For the ferromagnetism to be observed it is required that the parameter 2kF Rij (evaluated for the interspin distance parameter Rij equal to its average value R) corresponds to the ferromagnetic region of the Ruderman–Kittel function. That leads to the condition R R0 , where the characteristic distance R0 ∼ 1/kF ∼ 1/p1/3 corresponds to the ﬁrst switch of the RKKY interaction from ferromagnetic to antiferromagnetic. In the other limit, R R0 , the oscillatory character of the RKKY interaction is expected to lead to spin glass order. An analysis of this effect is presented by de Jonge et al. (1992). The variety of magnetic properties of IV-VI DMS is summarized in form of the x–p magnetic phase diagram presented in ﬁg. 9. Depending on the concentration of carriers (p) and the content of Mn ions (x) the ferromagnetic (FM), paramagnetic (PM), spin glass (SG) and the mixed (so-called, re-entrant spin glass, RSG) phase may be observed.

Fig. 9. The x–p (Mn concentration–carrier concentration) magnetic phase diagram of SnMnTe and PbSnMnTe. FM – ferromagnets, PM – paramagnets, SG – spin glasses, RSG – re-entrant spin glasses. The lines correspond to the theoretical calculations in various models discussed by Eggenkamp et al. (1995).

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This magnetic phase diagram of the RKKY governed diluted magnetic system can be derived in a more rigorous way based on the analysis of the statistical distribution of the algebraic sum of interspin exchange couplings in a random substitutional magnetic alloy. The physical criterion for the ferromagnetic phase to be established is analogous to the well known Sherrington–Kirkpatrick condition. The mean value of the distribution of the sum of exchange integrals must be positive (ferromagnetic) and larger than the variance of the distribution. Physically, it corresponds to the dominance of ferromagnetic couplings in the spectrum of relevant exchange interactions. In the opposite case, ferromagnetic and antiferromagnetic interactions compete one with the other and a spin glass state is expected. A theoretical analysis of the x–p magnetic phase diagram of PbSnMnTe and SnMnTe crystals performed within the Sherrington–Kirkpatrick like model is given by Eggenkamp et al. (1995). 3.1.2. Thin layers of SnMnTe and GeMnTe A carrier concentration induced paramagnetic–ferromagnetic transition was recently demonstrated experimentally for about 1–2 micron thick epitaxial layers of Sn 1−x Mnx Te (x 0.04) grown by molecular beam epitaxy on BaF2 (111) substrate with a SnTe buffer layer. The control of the concentration of carriers in the range 5 × 1019 cm−3 p 2 × 1021 cm−3 was achieved by adjusting the molecular ﬂux from the Te effusion cell (additional to the standardly employed SnTe and Mn cells, see Nadolny et al. (1998, 2002)). Depending on the concentration of conducting holes both ferromagnetic and paramagnetic SnMnTe layers were grown. Figure 10 shows the results of the measurements of the magnetic susceptibility revealing the ferromagnetic and paramagnetic behavior of SnMnTe epilayers depending on the concentration of conducting holes. The ferromagnetic transition

Fig. 10. Temperature dependence of the ac magnetic susceptibility in Sn0.96 Mn0.04 Te epitaxial layers with varying concentration of conducting holes.

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in SnMnTe epitaxial layers was also studied with the ferromagnetic resonance technique (Nadolny et al., 2002). The analysis of the growth process of SnMnTe showed that, in contrast to II-VI and III-V DMS with Mn, the low-temperature MBE epitaxial growth of these layers is limited to about x = 4 at.%, i.e., the Mn content range is narrower than in the case of bulk crystals. The ferromagnetic transition in thin SnMnTe layers is somewhat broader (on the temperature scale) than in the case of bulk crystals indicating a possible non-perfect microscopic homogeneity of the layers. The strong inﬂuence of carrier concentration on the magnetic properties was also observed in thin layers of Ge1−x Mnx Te (x = 0.4) grown by the ionized-cluster beam technique on BaF2 (111) with GeTe buffer layer. Upon controlling GeTe, MnTe, and Te ﬂuxes the concentration of conducting holes was reduced from 2.6 × 1021 cm−3 down to 5.4 × 1019 cm−3 . Magnetization and magnetotransport measurements showed a reduction of the ferromagnetic Curie temperature and a substantial decrease of the saturation magnetization (Fukuma et al., 2002). A negative magnetoresistance effect is usually observed in ferromagnetic GeMnTe (as well as in SnMnTe and PbSnMnTe) but in Ge0.6Mn0.4 Te layers with very low concentration of holes it was found to be replaced by a common quadratic positive magnetoresistance (Fukuma et al., 2002). 3.2. IV-VI DMS materials with Eu 3.2.1. Bulk crystals and thick epitaxial layers of PbEuTe and related materials Lead chalcogenides (PbTe, PbSe, and PbS) form a range of substitutional solid solutions with the corresponding europium chalcogenides – a well known group of magnetic semiconductors. The crystals of Pb1−x Eux Te, Pb1−x Eux Se, and Pb1−x Eux S were grown in the form of bulk materials applying the Bridgman method as well as in the form of epitaxial layers deposited by molecular beam or hot wall epitaxy using effusion cells for the IV-VI compound and Eu and a separate Te (or Se, S) source. The composition range available of IV-VI DMS with Eu is shown in table 1. These DMS materials crystallize in the cubic rocksalt crystal lattice with Eu ions substituting Pb2+ ions in the fcc cation sublattice of the crystal. Europium incorporates into IV-VI semiconductors as Eu2+ ions possessing spin-only magnetic moments of 7 Bohr magnetons (electron conﬁguration 4f7 , total spin and orbital quantum numbers: S = 7/2 and L = 0) and being electrically neutral. This simple ionic picture (the substitution Eu2+ → Pb2+ ) is fully conﬁrmed by the various transport and magnetic measurements discussed below. However, as pointed out by Dietl et al. (1994), the microscopic situation is somewhat more complicated since one replaces Pb with its 4 valence electrons by Eu with 2 valence electrons. In the case of tin chalcogenides (e.g., SnTe) the solubility of Eu is much lower. The only crystals studied so far are Sn1−x Eux Te in which inclusions of a second crystal phase are observed already for Eu contents of only a few atomic percent (Errebbahi et al., 2002). The magnetic and charge state of the Eu ions in IV-VI semiconductors was studied in detail by the electron paramagnetic resonance technique. The characteristic 7-lines anisotropic spectrum is observed both in bulk crystals (Isber et al., 1995b, 1997b; Misra et al., 1995) and in epitaxial ﬁlms (Gratens et al., 1998). The standard theoretical analysis of the spectra, which is based on a model spin Hamiltonian for rare earth ions with a spin-only magnetic ground state, provides a satisfactory quantitative description of the

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experimental data. The observed cubic anisotropy of the EPR spectra proves octahedral symmetry of the lattice sites occupied by Eu ions in the lead chalcogenides (as expected for the rocksalt lattice). The g-factor of Eu ions was (at low temperatures) found to be gEu = 1.975 in PbS (Isber et al., 1997b), gEu = 1.982 in PbSe (Isber et al., 1995b; Misra et al., 1995), gEu = 1.98 in PbTe (Bacskay et al., 1969), and gEu = 1.980 in SnTe (Urban and Sperlich, 1975). It deviates from the g = 2.00 value expected for magnetic ions with spin-only magnetic ground state because of the higher order spin–orbit effects (admixture of excited states of the Eu2+ ion into the ground state). That mechanism is also responsible for the experimentally observed ﬁne structure of the EPR spectra and is directly related to the energy splitting of the magnetic ground state of the Eu ion in the crystal ﬁeld of neighboring ligand ions. The zero-ﬁeld splitting of the magnetic ground state is very small (e.g., in PbEuSe it is about 35 μeV, see Isber et al. (1995b)) and is relevant only for experiments performed at subkelvin temperatures. Since Eu is electrically neutral in PbEuTe and related materials, its primary effect on the transport properties of IV-VI DMS is lowering of the carrier mobility (Lambrecht et al., 1991; Krenn et al., 1999; Prinz et al., 1999). The incorporation of about 10 at.% of Eu into PbTe decreases the low-temperature electron mobility from about 106 cm2 /Vs down to 103 cm2 /Vs. The additional scattering mechanism brought about by the presence of the Eu ions is analyzed in terms of alloy scattering (Prinz et al., 1999). For Eu content x < 0.1–0.15, Pb1−x Eux Te bulk crystals and epitaxial layers show electrical properties qualitatively similar to the properties of the host non-magnetic materials (Nimtz and Schlicht, 1983). Depending on the stoichiometry and doping conditions both n- and p-type materials can be grown in a broad Eu composition range typically with carrier concentration n, p = 1017 –1018 cm−3 . Usually no freezing-out of carrier concentration is observed in IV-VI materials and metallic-type electrical conductivity is found down to helium temperatures. Further increase of the Eu content in PbEuTe (possible only in epitaxial layers) leads to a metal–insulator transition which is understood in terms of the Anderson disorderdriven transition related to the particularly strong alloy-scattering. A detailed analysis of the magnetotransport properties of PbEuTe is presented by Prinz et al. (1999). The scheme of the evolution of the band structure of Eu-based IV-VI DMS as a function of increasing Eu content is illustrated in ﬁg. 11 in the entire composition range 0 x 1 for the case of Pb1−x Eux Te (Krenn et al., 1999). For relatively low Eu concentration, x < 0.06, the fundamental electronic transitions are PbTe-like (see also ﬁg. 2), i.e., the direct transitions between the top of the valence band (located at the L-point of the Brillouin zone, L6+ symmetry) and the bottom of the conduction band (located at the L-point of the Brillouin zone, L6− symmetry). In this range of Eu compositions the energy gap of IV-VI DMS with Eu increases very rapidly with increasing content of the magnetic ions. This behavior is of practical importance (see next section) and is quantitatively characterized by the coefﬁcient dEg /dx which equals 4.48 eV, 3.0 eV, and 5.0 eV for PbEuTe, PbEuSe, and PbEuS, respectively (Goltsos et al., 1986; Ishida et al., 1988a, 1988b; Lambrecht et al., 1991; Iida et al., 1993; Geist et al., 1996, 1997; Ueta et al., 1997; Yuan et al., 1997; Maurice et al., 1998; Krenn et al., 1999). In most of IV-VI DMS materials the composition dependence of the energy gap is the dominant effect and the change of other band parameters can be neglected. This is the case of Pb1−x Eux Se (x < 0.07) layers in which the composition independent matrix elements of momentum

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Fig. 11. Fundamental electronic transitions in Pb1−x Eux Te epitaxial layers in the entire composition range of Eu (after Krenn et al. (1999)). Diamonds: PbTe-like transitions; circles: Eu 4f to PbTe conduction band transitions; triangles: lower valence bands to Eu 4f levels transitions; squares: EuTe-like transitions.

are experimentally observed: 2Pt2 /2m0 = 3.6 eV and Pt /Pl = 1.4 (Geist et al., 1997). In PbEuTe the matrix elements of momentum reveal some composition dependence: for PbTe, 2Pt2 /2m0 = 6.0 eV and Pt /Pl = 3.4, whereas for PbEuTe with 5 at.% of Eu, 2Pt2 /2m0 = 2.9 eV and Pt /Pl = 3.15 (Geist et al., 1997). Together with the energy gap, Eg , this parameter also determines other important band structures parameters such as, 2 /m E , and the e.g., the effective masses of the carriers at the band edge, m0 /m∗t,l = 2Pt,l 0 g density of electron states (Nimtz and Schlicht, 1983; Bauer et al., 1992). Similarly to nonmagnetic lead chalcogenides, the energy gap of IV-VI DMS with Eu increases with increasing temperature. For temperatures above about 70 K the energy gap increases linearly with a typical value of the temperature coefﬁcient dEg /dT = 40 meV/K (Ishida et al., 1988a; Maurice et al., 1998). The experimental investigations of the optical properties of IVVI DMS with Eu covered interband and intraband magneto-optical effects, Faraday rotation, magneto-optical Kerr effect, photoluminescence, photoconductivity, and coherent anti-Stokes Raman scattering (CARS) (Krost et al., 1985; Goltsos et al., 1986; Ishida et al., 1988a, 1988b; Lambrecht et al., 1991; Bauer et al., 1992; Karczewski et al., 1992; Iida et al., 1993; Yuan et al., 1993a, 1997; Geist et al., 1996, 1997; Ueta et al., 1997; Maurice et al., 1998; Krenn et al., 1999). For higher Eu content, 0.06 < x < 0.6, the dominant electronic transitions in PbEuTe are the transitions between the 4f states of Eu and the PbTe-like states of L6− symmetry at the bottom of the conduction band. In this range of Eu concentrations the energy gap of PbEuTe still increases linearly with increasing content of Eu but the coefﬁcient dEg /dx = 2.03 eV, i.e., it is about two times smaller. For Eu contents x > 0.8 the dominant optical feature is observed at 2.25 eV and corresponds to the EuTe-like electronic transitions, i.e., the transitions between the 4f states of Eu and the hybridized s–5d states at the bottom of the conduction band of EuTe (Krenn et al., 1999).

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In DMS materials, at low temperatures and in non-zero magnetic ﬁeld, the energy states of electrons in the conduction band and holes in the valence band are strongly inﬂuenced by the sp–d or sp–f exchange interaction that couples the magnetic moments of delocalized conducting carriers and the well localized magnetic moments of the magnetic ions (see Kossut and Dobrowolski (1993)). It results in a number of the well known DMS effects such as, e.g., giant Zeeman splitting of the conduction and valence band states which is responsible for the very strong magneto-optical phenomena (Furdyna and Kossut, 1988; Bauer et al., 1992; Kossut and Dobrowolski, 1993; Dietl, 1994; Galazka, 1995). The magnitude of these effects is proportional to the sp–f (or sp–d) exchange integral Jsp−f and to the magnetization of the DMS material. In IV-VI DMS with Eu the sp–f exchange integral values are among the smallest within the entire DMS family. In PbEuTe: Jpf = A = 26 meV (for the valence band) and Jsf = B = 5 meV (for the conduction band) whereas in PbEuSe: A = 25–35 meV and B = 7–20 meV. One may compare those values with values typical for II-VI DMS with Mn (Jpd = 1 eV for holes and Jsd = 0.2 eV for electrons) and IV-VI DMS with Mn (e.g., in PbMnTe: A = −(100–200) meV, B = −(30– 60) meV). The full list of exchange integral values for IV-VI DMS materials as well as a detailed discussion of the inﬂuence of the sp–d and sp–f exchange interactions on the band structure of these materials in given by Bauer et al. (1992), Kossut and Dobrowolski (1993), Geist et al. (1997) and Galazka et al. (1999). A microscopic discussion of the sign and magnitude of the sp–f exchange interactions in IV-VI DMS with Eu is presented by Dietl et al. (1994). In PbEuTe and PbEuSe the sp–f exchange interaction was studied by various magneto-optical methods. Particularly effective proved to be the measurements of coherent anti-Stokes Raman scattering (CARS) – an experimental technique providing direct access to various spin-dependent resonances (Bauer et al., 1992; Geist et al., 1997). In bulk crystals of SnEuTe the p–f exchange integral was estimated to be 8 times smaller than in bulk crystals of SnMnTe. This conclusion is based on the analysis of the Korringa effect, i.e., the linear increase of the magnetic ion EPR linewidth with increasing temperature (Urban and Sperlich, 1975). For almost the entire Eu composition range available the crystals of Pb1−x Eux Te, Pb1−x Eux Se, Pb1−x Eux S, and Sn1−x Eux Te are Curie–Weiss paramagnets revealing only very weak antiferromagnetic f–f exchange interactions between the magnetic moments of the Eu2+ ions. Only in thick Pb1−x Eux Te epitaxial layers with x > 0.8 antiferromagnetic transitions are observed below 10 K as it is very well known for the terminal (x = 1) EuTe compound (see section 4.2). The evolution of the magnetic properties of Pb1−x Eux Te as a function of Eu content is presented in ﬁg. 12. The antiferromagnetic exchange coupling between nearest magnetic neighbors is in IV-VI DMS with Eu (also with Mn or Gd) experimentally evidenced by the temperature dependence of the magnetic susceptibility which follows the Curie–Weiss law, χ(T ) = C/(T − Θ), with Θ ∝ xJff (Krenn et al., 1999); see also Goltsos et al. (1986), Górska and Anderson (1988), Anderson et al. (1990) and in the characteristic step-like behavior of the magnetic ﬁeld dependence of the magnetization, M(B), observed at low temperatures. The analysis of the magnetization steps of DMS materials with antiferromagnetically exchange-coupled small clusters (usually pairs and triples of magnetic ions) was pioneered in II-VI DMS (see section 5). The step-like feature in the M(B) dependence is observed at magnetic ﬁelds at which a Zeeman-split excited magnetic state of such a small cluster crosses the energy of the ground state leading, consequently, to a change of the magnetic moment of the ground state. That phenomenon

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Fig. 12. Dependence of the magnetic properties of Pb1−x Eux Te epitaxial layers on the concentration of Eu ions. Antiferromagnetic transitions are observed for x > 0.8 (after Krenn et al. (1999)).

provides a direct experimental measure of the strength of the nearest-neighbor exchange coupling. In the case of IV-VI DMS with Eu an additional contribution to this effect comes from the zero-ﬁeld crystal ﬁeld splitting of the ground state of single Eu ions as discussed above for the EPR spectra. The f–f exchange integral is small in all IV-VI DMS with Eu. For Pb1−x Eux Te: Jff = −0.26 K (magnetization step measurements, see ter Haar et al. (1997)), Pb1−x Eux Se: Jff = −0.24 K (magnetization steps, see Bindilatti et al. (1996), Isber et al. (1997a), Pb1−x Eux S: Jff = −0.23 K (magnetization steps, see Bindilatti et al. (1998)), and Sn1−x Eux Te: Jff = −0.29 K (magnetization measurements, see Errebbahi et al. (2002)). The microscopic mechanism of the f–f exchange interaction in these materials is due to the superexchange coupling proceeding via the relatively spatially extended 5d electron orbitals of Eu and the electron orbitals of anions. The mechanism of direct f–f exchange is negligible in IV-VI DMS with Eu due to the very strong localization of the 4f electron orbitals. The indirect exchange mechanism via spin polarization of conducting carriers (RKKY mechanism, see section 3.1) which is proportional to the square of the sp–f exchange integral is also negligible because of the very weak sp–f coupling encountered in IV-VI materials. 3.2.2. Low-dimensional PbEuTe-PbTe and related epitaxial structures Low-dimensional structures built of epitaxial layers of IV-VI DMS with Eu and their non-magnetic counterpart materials, such as Pb 1−x Eux Te-PbTe and Pb1−x Eux SePbSe superlattices and multiple quantum wells, are usually grown on BaF2 (111) substrates employing the technique of molecular beam epitaxy (Springholz et al., 1991; Springholz and Bauer, 1994; Ueta et al., 1997; Abramof et al., 2000). In the early studies of these epitaxial structures (also including Pb1−x Eux S-PbS multiple quantum wells)

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the technique of hot wall epitaxy was used (Ishida et al., 1989; Valeiko et al., 1991). The very good matching of the lattice parameters and low interdiffusion observed in these structures allow for the growth of layered low-dimensional structures of remarkably good structural and electronic perfection. The extensive investigations of various growth modes of IV-VI DMS with Eu on BaF2 (111) substrates are summarized in the form of an MBE growth diagram with an important parameter window (substrate temperature versus the ratio of molecular ﬂuxes of metallic and non-metallic elements) allowing for the layer-by-layer 2D-growth (Springholz and Bauer, 1992). These quantum structures can be very efﬁciently modulation-doped to n-type with Bi (Springholz et al., 1993a; Ueta et al., 1997). Most of the low-dimensional structures of PbEuTe-PbTe and PbEuSe-PbSe were grown with a relatively small (x < 0.07) content of Eu ions. These structures form type-I superlattices or multiple quantum wells with PbEuTe or PbEuSe layers constituting an electron barrier for both electrons and holes whereas PbTe or PbSe layers serve as quantum well materials. The band offsets in the conduction band, Ec , and in the valence band, Ev , are approximately equal, e.g., in PbEuTe Ec /Eg = 0.55 (Xu et al., 1994; Yuan et al., 1994). The band offset magnitude is determined by the content of Eu ions in the barrier material. Since the incorporation of Eu in IV-VI semiconductors increases their energy gap very rapidly (about 50 meV/at.% in PbEuTe and about 30 meV/at.% in PbEuSe), a broad range of barriers for electrons and holes is available up to barrier heights of the order of the energy gap of the material. The fundamental electronic transitions in PbEuTePbTe, PbEuSe-PbSe and PbEuS-PbS structures are observed in the near-infrared part of the spectrum. These are the direct electron transitions between the size-quantized energy states in the valence band and in the conduction band of PbTe, PbSe, and PbS, respectively. The band edges in IV-VI semiconductors are located at the L-point of the Brilloiun zone, i.e., at the boundary of the ﬁrst Brillouin zone in [111] direction. Therefore, in bulk cubic crystals one has 4 equivalent energy minima corresponding to 4 Fermi ellipsoids with their long axes directed along 4 equivalent [111] crystal directions. In the epitaxial structures grown on BaF2 (111) substrates, this degeneracy is reduced and one can distinguish the size-quantized energy states originating from the, so-called, longitudinal valley (with its long axis along the growth direction) and three other oblique valleys. An additional important factor is brought about by the thermal stress observed in these structures due to the difference between the thermal expansion coefﬁcients of the superlattice and the BaF2 substrate. In PbEuTe-PbTe superlattices this results in an in-plane tensile stress leading to about 5 meV shift (lowering) of the energy of electrons occupying the states in the longitudinal valley with respect to the oblique valleys. That results in different population of various valleys and is experimentally observed in optical and Shubnikov–de Hass experiments (Valeiko et al., 1991; Abramof et al., 2001). The quantitative description of the electron states in the quantum wells of IV-VI DMS with Eu is obtained by employing the well known method of envelope function calculations (Yuan et al., 1994; Abramof et al., 2001). The optical properties of PbEuTe-PbTe and related structures were extensively studied by near-infrared and far-infrared transmission, photoluminescence, interband magnetotransmission, and 2D-cyclotron resonance methods (Springholz et al., 1993b; Yuan et al. 1993b, 1994, 1997; Xu et al., 1994; Abramof et al., 2001; Aigle et al., 2001). Transport investigations of PbEuTe-PbTe quantum structures modulation-doped with Bi aimed

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at an analysis of quantum Hall effects (Springholz et al., 1993a) as well as at the inﬂuence of size-quantization on conductivity and thermoelectric effects (Hicks et al., 1996; Casian et al., 2000). IV-VI semiconductors are well known for their applicational potential as good optoelectronic materials for infrared detectors and lasers as well as important thermoelectric materials. Low-dimensional structures incorporating IV-VI DMS materials with Eu brought a new dimension in both of these ﬁelds of applications. The mid-infrared double heterostructure lasers fabricated on PbTe(100) substrates employing a PbTe active layer and Pb1−x Eux Te1−y Sey layers providing both electrical conﬁnement and optical wave guide effect, hold the record value of the working temperature for continuous wave operations of such a device in this spectral range (Feit et al., 1996). The successful design of these new optoelectronic structures explores the remarkably efﬁcient band gap (and band offset) as well as refractive index engineering practically realized by controlling the content of Eu in IV-VI DMS. Recently, new vertical-cavity surface-emitting (VCSEL) laser structures were fabricated based on the same PbTe-PbEuTe heterostructures incorporating very efﬁcient PbEuTe-PbTe Bragg mirrors (Schwarzl et al., 1999; Heiss et al., 2001c; Springholz et al., 2001a, 2001b). In the ﬁeld of thermoelectrics, the two-dimensional PbEuTe-PbTe epitaxial structures contributed to a re-newed active search for new thermoelectric materials with improved thermoelectric ﬁgure of merit, Z = S 2 σ/κ, where S is the thermoelectric power, σ the electrical conductivity, and κ the thermal conductivity. It was shown that by exploring the enhancement of the density of states observed in 2D-structures one can improve the thermoelectric parameters of the PbTePbEuTe multiple quantum well as compared to PbTe bulk material (Hicks et al., 1996; Casian et al., 2000). Finally, we mention the very successful work on self-assembled quantum dots made of IV-VI semiconductors with IV-VI DMS materials with Eu employed to control elastic properties of strained PbSe-PbEuTe hetero-interfaces (Springholz et al., 1998). 3.3. IV-VI DMS materials with Gd The list of IV-VI DMS materials with Gd is presented in table 1 (see also Galazka (1995), Galazka et al. (1999)). In contrast to materials with Mn or Eu, Gd is known to usually incorporate into IV-VI semiconductors as Gd3+ ion carrying a local magnetic moment of 7 Bohr magnetons (electron conﬁguration 4f7 , spin-only magnetic moment, S = 7/2, L = 0) and being electrically active as donor center of 3+ charge state substituting for Pb or Sn host cations acting as 2+ ions (see Averous et al. (1985), Bartkowski et al. (1985), Bruno et al. (1988), Lombos et al. (1989)). For example, in the crystals of Pb1−x Gdx Te and Pb1−x−y Sny Gdx Te an electron concentration exceeding n = 1020 cm−3 has been obtained (Averous et al., 1985; Story et al., 1997). This charge state of Gd in PbTe and related materials is conﬁrmed by the observation of an EPR spectrum with a characteristic 7lines ﬁne structure due to the effect of an octahedral crystal ﬁeld (Bartkowski et al., 1985; Gratens et al., 1997b). It is also supported by resonant photoemission studies of PbGdTe, PbGdSe and SnGdTe which provided information on the location of the density of states derived from Gd 4f7 orbitals (about 9.5 eV below the top of the valence band in all IV-VI materials) and the density of states derived from Gd 5d orbitals (in PbTe and PbSe above the bottom of the conduction band but in SnTe about 0.2 eV below the top of the valence

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band), see Kowalski et al. (1999). These conclusions are valid for the bulk of the available old and new experimental data. In some crystals of PbGdTe the presence of Gd in both 3+ and 2+ charge state was recently postulated based on EPR and Mössbauer spectroscopic studies (Zayachuk et al., 2001) pointing to the important role of complexes of defects such as, e.g., Gd–VTe (Te vacancy), see Zayachuk et al. (2001). The electron band structure of IV-VI DMS with Gd is generally expected to be similar to other IV-VI DMS materials although there are no reliable experimental optical data, e.g., demonstrating the evolution of the band structure parameters of IV-VI materials with increasing content of Gd. The strongly localized character of the 4f orbitals of Gd results in very weak direct and indirect f–f exchange interactions between the Gd ions as well as in a very weak sp–f exchange coupling between the magnetic ions and the free carriers. From the analysis of the Korringa effect in EPR of SnGdTe it was established that the Jsf carrier–magnetic ion exchange constant is related to the same exchange constant for Mn ions in the following way: Jsf (Gd)/Jsd (Mn) = 1/5 (Urban and Sperlich, 1975). That results in 1/25 reduction of the strength of the RKKY interaction for Gd-based IV-VI DMS making this interaction negligible in these materials. Therefore, similarly to Eu-based IV-VI DMS (but contrary to Mn-based materials), crystals of Sn1−x Gdx Te never reveal ferromagnetic properties despite the fact that the concentration of carriers encountered in these materials can be high enough and the Gd3+ ions form in IV-VI DMS systems of well deﬁned local magnetic moments. The f–f interspin exchange interaction in IV-VI DMS with Gd was investigated by measurements of the magnetization and magnetic susceptibility (Bruno et al., 1987; Górska et al., 1992). A theoretical analysis was performed by applying the well known model taking into account the contributions of single magnetic ions and small weakly antiferromagnetically coupled clusters (pairs and triples of Gd ions). The typical Gd–Gd exchange integrals are of the order of −0.5 K with a negative (antiferromagnetic) sign (for a detailed list, see (Galazka et al., 1999). An exceptional case is observed in certain Sn1−x Gdx Te crystals in which a new mechanism for the f–f exchange interaction between Gd ions is operating (Story et al., 1996a; Górska et al., 2001). Experimentally, it is observed as a strong resonance-like carrier concentration dependence of the antiferromagnetic f–f exchange integral (as measured by the paramagnetic Curie temperature normalized per Gd content, Θ/x, see ﬁg. 13). The inset in ﬁg. 13 shows the temperature dependence of the inverse magnetic susceptibility of Sn1−x Gdx Te crystals with Gd content x = 0.045 and different carrier concentrations. These “anomalous” magnetic properties of Sn1−x Gdx Te are observed in the Gd composition range 0.02 < x < 0.05 only for samples with a carrier concentration corresponding to the resonance position of the Fermi level with respect to the E0 level of Gd (see ﬁg. 14). The enhancement of both the carrier mobility and the f–f exchange interaction is related to the speciﬁc location of the magnetically relevant energy states of Gd (see ﬁg. 14). The microscopic model of the f–f exchange in SnGdTe involves an indirect f–d–d–f process based on intra-ion 4f–5d exchange and inter-ion 5d–5d coupling. This mechanism is expected to become resonantly enhanced for E0 ∼ = EF because at this position of the Fermi level the Gd 5d orbitals are expected to be populated by electrons and the exchange coupling may proceed via the spatially quite extended 5d orbitals (Story et al., 1996a; Górska et al., 2001).

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Fig. 13. Carrier concentration dependence of the normalized paramagnetic Curie temperature Θ/x in Sn1−x Gdx Te. Triangles show the behavior of crystals with x > 0.05. Dots show the result for samples with x < 0.05. The inset presents the temperature dependence of the inverse magnetic susceptibility of two samples of Sn0.955 Gd0.045 Te with different carrier concentrations.

In Sn1−x Gdx Te, the Gd ion is believed to form an electron E0 state (Gd2+/3+ level) resonant with the valence band (see the band structure model presented in the inset in ﬁg. 14). It strongly inﬂuences the electric properties of this material. For example, despite the presence of up to 9 at.% (i.e., about 1.5 × 1021 cm−3 ) of, as expected, Gd3+ donor centers, all the studied SnGdTe samples show metallic p-type conductivity. Moreover, an annealing processes of Sn1−x Gdx Te, performed in order to reduce the number of acceptor-like Sn vacancies, resulted in approximately the same hole concentration, e.g., p = (3 ± 0.5) × 1020 cm−3 for Sn1−x Gdx Te crystals with x = 0.045 (Story et al., 1996a). This lack of n-type conductivity is understood as a consequence of the fact that for the Fermi level location at or above the E0 level (see the inset in ﬁg. 14), the introduction of extra Gd ions will result in no effect on the Fermi level because all of them will be in the electrically neutral 2+ charge state. For Sn1−x Gdx Te crystals with x ∼ = 0.045 and a carrier concentration of p = 3 × 1020 cm−3 the factor of 5-enhancement of the hole mobility (as compared to the samples with different carrier concentration or different Gd contents) is also experimentally observed at low temperatures (see ﬁg. 14, after Story et al. (1996a)). That is another consequence of the location of the Fermi level in resonance

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Fig. 14. Carrier concentration dependence of the hole mobility for Sn1−x Gdx Te with x = 0.045 (triangles) and x = 0.057 (circles). The open symbols show the data at T = 4.2 K whereas the full symbols refer to T = 77 K. The inset presents a scheme of the band structure of Sn1−x Gdx Te with x < 0.05.

(EF = E0 ) with the energy level of Gd2+/3+ . The scattering on Gd3+ ions is expected to be effectively “switched-off” due to the absence of Gd ions in this charge state. 4. Magnetic-non-magnetic multilayers with II-VI and IV-VI semiconductors 4.1. Ferromagnetic EuS-PbS and related multilayers EuS belongs to the family of europium chalcogenides, the well known group of magnetic semiconductors including both ferromagnetic (EuO and EuS) and antiferromagnetic (EuTe) materials (Methfessel and Mattis, 1968; von Molnar and Kasuya, 1968; Wachter, 1979; Mauger and Godard, 1986). Bulk EuS orders ferromagnetically below its Curie temperature, TC = 16.6 K, and is one of just a few non-metallic ferromagnets with well deﬁned local magnetic moments – a model example of a Heisenberg ferromagnet. The exchange interactions in EuS are of short range with a dominant role of magnetic nearest-neighbor and next-nearest-neighbor interspin couplings. EuS usually shows semiinsulating electrical properties but can also be n-type doped, e.g., with gadolinium or with native defects (during non-stoichiometric growth). Both EuS and PbS crystallize in the same rocksalt crystal structure and their lattice parameters differ only by 0.5%. Therefore,

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employing various high vacuum epitaxial layer deposition techniques one can grow EuSPbS multilayers pseudomorphically up to an overall thickness of about 2000 Å (Kolesnikov and Sipatov, 1989). The non-magnetic (diamagnetic) spacer material in EuS-PbS multilayers is lead sulﬁde, PbS, a member of the family of IV-VI compound semiconductors, with its direct energy gap (Eg = 0.3 eV at low temperatures) corresponding to the fundamental optical transitions in the near-infrared. The EuS-PbS multilayers (trilayers and superlattices) discussed in this chapter were deposited in high vacuum on freshly cleaved monocrystalline KCl(100) and BaF2 (111) substrates. These substrates are insulating and optically transparent in the visible and nearinfrared. We also studied EuS-PbS multilayers deposited on conducting n-PbS(100) substrates cleaved from the bulk monocrystals grown by physical vapor transport techniques (Chernyshova et al., 2002, 2003). As the melting temperature of EuS is extremely high (above 2500 ◦ C) one has to employ an electron gun for the evaporation of EuS. In the case of PbS standard resistively heated sources are used. Typically, EuS-PbS superlattices and trilayers are grown with a thickness of the EuS layer of 30–70 Å and the thickness of the PbS spacer varying in the range 6–100 Å. As a semi-bulk reference we also studied thick (about 1000 Å) EuS-PbS bilayers. 4.1.1. Ferromagnetic Curie temperature and magnetic anisotropy The magnetic properties of EuS-PbS multilayers depend on the thickness of the magnetic layers and the in-plane strain present in these structures as well as on the interlayer exchange interaction that couples the magnetic moments of neighboring EuS lay-

Fig. 15. Temperature dependence of the magnetization of EuS-PbS multilayers with varying thickness t of the ferromagnetic EuS layers (indicated in the ﬁgure). The PbS spacer thickness is 15 nm (about 50 ML). The solid lines represent results of theoretical calculations of spin wave excitations performed by Swirkowicz and Story (2000).

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ers. The list of experimental techniques employed so far to investigate EuS-PbS multilayers includes magnetization (SQUID magnetometry), ac magnetic susceptibility, neutron diffraction, neutron reﬂectivity, ferromagnetic resonance, photoluminescence, intraband magneto-optics, and transport measurements. The ferromagnetic transition is observed in EuS-PbS multilayers even with ultrathin EuS layers of only 6 Å (for EuS: one monolayer corresponds to 2.99 Å for the [100] and 3.45 Å for the [111] crystal direction, respectively). The temperature dependence of the magnetization of a series of EuS-PbS layers presented in ﬁg. 15 illustrates the shift of the ferromagnetic transition temperature with decreasing thickness of the EuS layer (Stachow-Wójcik et al., 1997, 1999; Story et al., 2001). There practically is no effect of the thickness of the non-magnetic spacer on the Curie temperature of EuS-PbS. It suggests that the expected (see below) small change of the in-plane strain in these multilayers produces an effect negligible comparing to the much stronger effect of thermal stress (Stachow-Wójcik et al., 1999). The ferromagnetic Curie temperature of EuS-PbS multilayers gradually decreases with decreasing thickness t of the EuS layer from the value observed for thick semi-bulk reference layers: TC0 = 17.3 K for KCl(100), 17.0 K for PbS(100), and 13.6 K for BaF2 (111) substrates. For 2 ML thick layers, the Curie temperature is about 3/5 of these reference values (see ﬁg. 16). The experimentally observed TC (t) dependence can be described by a simple mean-ﬁeld model in which the Curie temperature is calculated from the total magnetic energy of the ferromagnetic ground state of the multilayer taking into account missing magnetic neighbors of magnetic ions located close to the interface. These model calculations (see the lines shown in ﬁg. 16) yield a simple expression for the thickness dependence of the Curie temperature: TC = TC0 (1 − c/t), where TC0 is the semi-bulk reference temperature given above

Fig. 16. Dependence of the ferromagnetic Curie temperature of EuS-PbS multilayers grown on various substrates on the thickness of the EuS layers. The solid lines show the predictions of a simple mean-ﬁeld model assuming sharp EuS-PbS interfaces whereas the broken lines present the cases of interfaces interdiffused over 2 ML.

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and c is a numerical coefﬁcient depending on the crystal direction and interdiffusion proﬁle (Stachow-Wójcik et al., 1997, 1999; Story et al., 2001). For a more detailed discussion see Stachow-Wójcik et al. (1999) where other theoretical models related to, e.g., ﬁnite size effects in magnetic systems are also analyzed. For thick EuS-PbS multilayers grown on KCl(100) and PbS(100) substrates the ferromagnetic transition is observed above the Curie temperature of the bulk crystals. This effect is attributed to the compressive strain present in these structures both due to the small lattice parameter mismatch (the lattice parameter aPbS = 5.94 Å is smaller than aEuS = 5.97 Å) and the difference in the thermal expansion coefﬁcients of the substrate and the multilayer. The last effect concerns only the layers grown on KCl and BaF2 substrates. It is particularly strong for multilayers grown on a KCl(100) substrate which has a large expansion coefﬁcient. For EuS-PbS/BaF2 the effect is smaller and it is expected to act in the opposite direction (tensile strain). The compressive biaxial in-plane strain (accompanied by Poisson expansion along the normal to the layer plane) results in an overall effect of a decrease of the distance between the nearest magnetic neighbors in EuS-PbS/KCl layers and leads to an increase of the exchange integrals and of the Curie temperature (Stachow-Wójcik et al., 1999). The effect of the thermal expansion induced strain on the ferromagnetic transition in EuS-PbS/KCl multilayers is observed experimentally as a decrease of the Curie temperature of these multilayers after removing the substrate by dissolving it in water, see ﬁg. 17 (Stachow-Wójcik et al., 1999; Story et al., 2001). Theoretically a more rigorous approach to the problem of the thickness and stress dependence of the ferromagnetic temperature of EuS-PbS/KCl multilayers

Fig. 17. Temperature dependence of ac magnetic susceptibility for a EuS(18 ML)-PbS(59 ML) superlattice grown on a KCl(100) substrate. The solid line shows the data for the (thermally strained) multilayer still being on the substrate whereas the dashed line represents the data for the free standing layer without the substrate (thermal stress absent).

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Fig. 18. Magnetic anisotropy constant K of EuS-PbS multilayers (with thick PbS spacer) as a function of the thickness t of the EuS layers. The solid triangles and the broken line describe layers grown on KCl substrate whereas the open triangles and the solid line represent data for layers grown on BaF2 substrate.

was developed by Swirkowicz and Story (2000) where the Green function formalism has successfully been applied to calculate the temperature dependence of the magnetization of thin strained layers of EuS. The magnetic anisotropy in EuS-PbS multilayers was studied by the ferromagnetic resonance technique (Stachow-Wójcik et al., 1999; Story et al., 2000). Figure 18 presents the dependence of the magnetic anisotropy constant K on the thickness t of the EuS layer. The linear Kt vs. t plot found in EuS-PbS structures shows that the K = Kv + 2Ks /t dependence, well known in metallic multilayers, is also observed in these structures. Here Kv is the volume and Ks is the surface contribution to the effective magnetic anisotropy. For EuSPbS structures grown on KCl(100) we found: Kv = −0.67 MJ/m3 and Ks = 0.06 mJ/m2 whereas for structures grown on BaF2 (111): Kv = −0.72 MJ/m3 and Ks = 0.10 mJ/m2 (Story et al., 2000). These results show that the surface contribution to the magnetic anisotropy is small in EuS-PbS multilayers. The dominant contribution is due to the shape anisotropy driven by dipolar interactions. It strongly prefers the magnetization vectors of EuS layers to stay in the plane of the layer. Our investigations show that the perpendicular anisotropy observed in many metallic systems could be obtained only for EuS with a thickness below 1 ML and in practice it is never observed. In the plane of the layer one observes only a very weak anisotropy with its strength (expressed in terms of the anisotropy ﬁeld) of the order of 5 mT. Apart from EuS-PbS structures, the ferromagnetic transition was also studied in the closely related EuS-PbSe multilayers in which the lattice parameter mismatch is 2.5%. For the equivalent set of EuS-PbSe/KCl multilayers the experimentally observed TC (t) dependence is of similar form but differs from the EuS-PbS/KCl case in two quantitative aspects: (1) for thick EuS-PbSe bilayers the Curie temperature is about 4 K lower than for similar

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EuS-PbS bilayers and (2) for very thin EuS layers the TC (t) dependence is much more pronounced indicating a possible stronger effect of interdiffusion in the anion sublattice (Story et al., 2001). Recently, the quantum size effects were also studied in transport and thermoelectric measurements of EuS-PbSe/KCl(100) bilayers (Rogacheva et al., 2002). Wosnitza et al. (1989) investigated the magnetization, magnetic susceptibility, and speciﬁc heat of EuS-SrS multilayers grown by MBE on Si(111) substrate. In contrast to EuSPbS multilayers in which EuS is an electron barrier material, in EuS-SrS the layers of EuS are expected to form electron quantum wells. Both EuS and SrS crystallize in the rocksalt structure and their lattice parameters match very well: for SrS a0 = 6.02 Å. Unfortunately, the epitaxial growth of EuS and SrS on Si(111) substrates could be achieved only for very high substrate temperatures (about 900◦ C) and these multilayers suffered from strong interdiffusion effects. 4.1.2. Interlayer exchange coupling Upon decreasing the thickness of PbS non-magnetic spacer layers the neighboring ferromagnetic layers of EuS may become coupled across the spacer by interlayer magnetic interactions either of exchange or of dipolar origin. Such interlayer coupling effects are well known in metallic multilayers. The microscopic mechanisms developed for these interactions in metallic systems rely on the presence of a very high concentration of quasi-free conducting carriers. In all-semiconductor magnetic multilayers with negligible (comparing to metals) concentration of carriers the interlayer coupling is experimentally observed in antiferromagnetic EuTe-PbTe, MnTe-CdTe, and MnTe-ZnTe (see section 4.1.3) as well as in ferromagnetic EuS-PbS superlattices and trilayers. In EuS-PbS multilayers the interlayer coupling was studied by neutron diffraction, neutron reﬂectivity, and magnetization measurements. These measurements provided a variety of experimental evidence for the presence of antiferromagnetic interlayer interactions in EuS-PbS multilayers grown on KCl(100) and PbS(100) substrates while they indicated the presence of ferromagnetic interactions in EuS-PbS/BaF2 (111) multilayers. The interlayer interactions in EuS-PbS/KCl(100) multilayers were investigated applying two neutron techniques: the standard diffraction method and the small angle reﬂectivity study (Giebultowicz et al., 2001; Kepa et al., 2001a, 2001b). The antiferromagnetic arrangement of the magnetization vectors of the individual EuS layers in the superlattice was directly observed as a characteristic set of magnetic diffraction peaks corresponding to a new periodicity that is doubled with respect to the chemical periodicity of the superlattice (see ﬁg. 19). The temperature dependence of the intensity of these peaks conﬁrmed their magnetic origin and provided independent information on the ferromagnetic transition temperature and the temperature dependence of spontaneous magnetization. The experiments carried out with an external magnetic ﬁeld applied in the plane of the structure revealed that a ﬁeld of the order of 100 Oe fully destroys diffraction peak in the double period position and creates one reﬂecting regular chemical periodicity. This indicates a switch of the magnetization vectors of the EuS layers from antiferromagnetic to ferromagnetic order. The external magnetic ﬁeld necessary to erase the antiferromagnetic peaks provides a quantitative measure of the energy of the exchange coupling and can be compared with results of other experimental methods (e.g., the analysis of saturation ﬁelds in magnetization hysteresis loops). Upon increasing the thickness of the non-magnetic spacer the strength of interlayer coupling decreases (see ﬁg. 20).

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Fig. 19. (a) Neutron diffraction spectrum of [EuS(6 nm)-PbS(2.3 nm)] × 15 superlattices grown on KCl(100) substrate. Open circles: data taken above the Curie temperature of EuS, full circles: data taken below the Curie temperature. (b) Magnetic contribution characteristic for a superlattice with antiferromagnetic interlayer coupling.

Fig. 20. Absolute value of the interlayer exchange energy (per unit area) as a function of the thickness of the non-magnetic spacer. Open squares: ferromagnetic EuS-PbS(100) multilayers; triangles: ferromagnetic EuS-PbS(111) multilayers; circles: antiferromagnetic EuTe-PbTe(111) multilayers; full squares: experimental results for EuS-PbS(100). After Kacman (2001).

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Fig. 21. Magnetic hysteresis loop of a EuS(4 nm)-PbS(0.8 nm)-EuS(4 nm) trilayer grown on KCl(100) substrate with a PbS buffer of 50 nm.

The antiferromagnetic interlayer coupling strongly modiﬁes the magnetization hysteresis loops of the multilayer. For the EuS-PbS structures with thick PbS spacer (thicker than about 2 nm) the hysteresis loops have a regular character with a large (typically about 70%) magnetic remanence and a coercive ﬁeld of the order of 20 Oe. On the contrary, hysteresis loops of EuS-PbS multilayers with thinner PbS spacer reveal nearly zero magnetic remanence and a characteristic plateau in the M(B) dependence. Figure 21 presents an example of such a loop observed in EuS-PbS/KCl structures (Kepa et al., 2001a; Kowalczyk et al., 2001; Chernyshova et al., 2003). Qualitatively similar magnetization hysteresis loops are also found in EuS-PbS structures with ultrathin PbS spacer grown on other (100)-oriented substrate, i.e., on PbS(100). In the case of multilayers grown on BaF2 (111) substrates the magnetization hysteresis loops are characterized by small coercive ﬁelds and very large remanences indicating either a lack of any interlayer coupling or the presence of ferromagnetic interlayer coupling (Kowalczyk et al., 2001). Another experimental effect brought about by the antiferromagnetic interlayer coupling in EuS-PbS structures is their characteristic temperature dependence of the magnetization. For EuS-PbS-EuS trilayers with PbS spacer thicker than about 2 nm we ﬁnd, for any applied external magnetic ﬁeld, a regular monotonic increase of the magnetization upon decreasing the temperature. For analogous structures but with much thinner spacer (about 1 nm) we ﬁnd, that at low external magnetic ﬁelds, the magnetization displays non-monotonic behavior upon decreasing the temperature revealing almost zero total magnetic moment at low temperatures, see ﬁg. 22 (Kepa et al., 2001a; Kowalczyk et al., 2001; Chernyshova et al., 2002, 2003). It strongly suggests that the magnetization vectors of the two EuS layers are aligned antiferromagnetically. The application of a stronger magnetic ﬁeld of the order of 50–100 Oe fully restores the monotonic M(T ) dependence. The M(T , B) dependence observed in EuS-PbS multilayers can be described by a simple model that takes into account three (temperature dependent) contributions to

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Fig. 22. Temperature dependence of (reduced) magnetization of a EuS(3 nm)-PbS(0.75 nm)-EuS(3 nm) trilayer grown on PbS(100) substrate. The inset presents the results of a theoretical model incorporating the antiferromagnetic interlayer coupling between EuS layers via the non-magnetic PbS spacer. The magnetic ﬁeld was applied in the plane of the structure with its strength indicated in the ﬁgure.

the total magnetic energy of the multilayer: the antiferromagnetic interlayer interaction energy, the Zeeman energy of the ferromagnetic layers in an external magnetic ﬁeld, and the in-plane magneto-crystalline anisotropy. The mutual orientation of the magnetization vectors of the magnetic layers is determined from the condition of minimal total magnetic energy for two antiferromagnetically coupled magnetic moments. The Weiss mean-ﬁeld theory is applied to calculate the temperature and magnetic ﬁeld dependence of the modulus of magnetization vectors of each of the layers. The results of these calculations for the M(T , B) dependence are presented in the inset in ﬁg. 22. The model successfully explains the characteristic features of the temperature dependence of the magnetization of our structures and allows for a quantitative estimate of the interlayer coupling energy. This effect is observed in EuS-PbS multilayers grown on both KCl(100) and PbS(100) substrates. For the structures grown on BaF2 (111) only the regular monotonic M(T ) dependence is found experimentally. Theoretical models proposed to explain the mechanism of the interlayer coupling in allsemiconductor multilayers considered the coupling being due to a quasi-free 2D electron gas in a non-magnetic quantum well as well as being due to electrons localized on shallow donor centers within their Bohr radius (Rusin, 1998; Shevchenko et al., 1998). Another proposed mechanism takes into account an interaction of the Bloembergen–Rowland type which involves electronic excitations across the energy gap of a semiconductor (Dugaev et al., 1999). However, all these models predict ferromagnetic interlayer coupling and cannot explain the experimental observations in EuS-PbS(100) structures. A theoretical model which correctly predicts both the antiferromagnetic sign and the order of magni-

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tude of the interlayer coupling in EuS-PbS multilayers was proposed by Blinowski and Kacman (2001), see also Kacman (2001). These authors show that due to the s–f, p–f, and s–d coupling between 4f and 5d electronic orbitals of Eu and the valence band states of the EuS-PbS superlattice the total electronic energy of a semiconductor multilayer depends on the magnetic conﬁguration of the multilayer. The energy minimum corresponds to an antiferromagnetic alignment of the magnetization vectors of the neighboring ferromagnetic EuS layers. In the case of antiferromagnetic EuTe-PbTe(111) superlattices (see section 4.2.1) the minimum corresponds to a, so-called, out-of phase arrangement in which the magnetization vectors of (111) Eu planes facing each other across the non-magnetic spacer are aligned antiferromagnetically (Blinowski and Kacman, 2001; Kacman, 2001). The predictions of this model are shown in ﬁg. 20. Apart from the electronic models discussed above one has also to consider the role of dipolar interactions which (for multidomain or morphologically non-perfects) multilayer may also lead to the antiferromagnetic arrangement of the magnetic layers. The theoretical calculations as well as the neutron analysis of the domain size in EuS-PbS multilayers suggest that dipolar interactions might be relevant only for relatively thick PbS spacers and are not likely to account for the antiferromagnetic interlayer coupling observed in EuS-PbS (Kepa et al., 2001a). Yet another theoretical approach to interlayer interactions in antiferromagnetic semiconductor superlattices is proposed in (Gomonay, 2001). It is shown that the magnetostriction in magnetically ordered layers may induce strain ﬁelds in both the non-magnetic spacer and the neighboring magnetic layers. Acting on the magnetic anisotropy part of the total energy of the multilayer the strain may be responsible for the correlated arrangement of the sublattice magnetization vectors. These correlations may extend over few monolayer thick non-magnetic spacers (Gomonay, 2001). 4.1.3. Optical properties The absorption edge of EuS is located at Ea = 1.6 eV and corresponds to transitions between 4f states of Eu and the bottom of the conduction band of s–5d symmetry. The energy gap between the top of the p-type valence band and the bottom of the conduction band exceeds 2 eV (see the model of the energy band structures of europium chalcogenides given by Methfessel and Mattis (1968)). PbS is a direct energy gap material with the edges of both conduction and valence bands at the L-point of the Brillouin zone. Photoluminescence studies of EuS-PbS structures showed that these multilayers form type-I multiple quantum wells, with non-magnetic PbS quantum wells and ferromagnetic EuS barriers, ﬁg. 23 (Kolesnikov et al., 1988; Kowalczyk et al., 1998; Stolpe et al., 2000). The model of the energy spectrum of EuS-PbS heterostructures proposed by several authors (Kolesnikov et al., 1988; Kowalczyk et al., 1998; Stolpe et al., 2000) indicates a very large band offset, of about 1.2 eV, in the conduction band and much smaller, 0.1 eV, band offset in the valence band. The fundamental optical transitions are observed in the near-infrared and correspond to the electronic transitions between the size-quantized states in the PbS quantum wells. The quantum size effect in the photoluminescence spectra of EuS-PbS/KCl(100) and EuS-PbS/BaF2 (111) multiple quantum wells was studied by Kolesnikov et al. (1988), Kowalczyk et al. (1998) and Stolpe et al. (2000). The transition energies observed in thick reference PbS/KCl(100) layers are by 80 meV lower than in the bulk PbS due to the strong in-plane compressing thermal strain

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Fig. 23. Model of the electronic band structure of EuS-PbS multilayers developed on the basis of an analysis of photoluminescent properties. The left panel corresponds to the case of ferromagnetically aligned magnetization vectors of both EuS barriers whereas the right panel shows the case of antiferromagnetic alignment. The potential barrier for electrons is very high (about 1 eV) and differs for electrons with different spin directions.

(see section 4.1.2). In EuS-PbS/BaF2 (111) layers this strain has the opposite (i.e., tensile) character and, correspondingly, the photoluminescence spectra are shifted by 26 meV to higher energies. These observations agree well with the well known property of IV-VI semiconductors, namely a decrease of the energy gap upon the application of hydrostatic pressure. The intensity of the photoluminescence spectra observed in EuS-PbS multilayers strongly depends on the quality (stoichiometry, purity) of the PbS source material used for the evaporation of the quantum well layers. In a number of cases, in especially prepared EuS-PbS/BaF2 (111) structures one observes a very bright stimulated emission exhibiting characteristic threshold dependence on the increasing laser excitation power. The photoluminescence measurements were performed with YAG:Nd laser excitation energy h¯ ω = 1.16 eV (i.e., below the absorption edge of EuS) with electron-hole pairs generated only in PbS wells and with 2h¯ ω = 2.33 eV (electron–hole pairs created in the entire structure). In the later case one observed pronounced changes in the two-line photoluminescence spectrum (transfer of intensity between the luminescent lines) suggesting an important role of the process of transfer of e–h pairs created in EuS (ferromagnetic) barriers into the PbS quantum well (Kowalczyk et al., 1998). Due to the large exchange splitting of the conduction band states in EuS below its ferromagnetic transition temperature (see ﬁg. 23) the EuS-PbS multilayers form an intriguing spintronic or spin-optoelectronic system with spin-dependent electron barriers serving, e.g., as a very efﬁcient spin ﬁlter (LeClair et al., 2002). The optical properties of EuS-PbS/BaF2 (111) multilayers were also studied by high ﬁeld intraband magneto-optics (Stolpe et al., 2000). The observed transitions were identiﬁed as the cyclotron resonance of the electrons in PbS quantum wells. The theoretical calculations of the size and magnetic ﬁeld (Landau) quantization of the electronic states of the conduction band in EuS-PbS multiple quantum wells were also performed both for the standard case of magnetic ﬁeld normal to the plane of the multilayer and for the case of magnetic

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ﬁeld parallel to the plane of the structures (with a complicated interplay between size and magnetic ﬁeld quantization in the growth direction, see Dugaev et al. (2000)). 4.2. Antiferromagnetic EuTe-PbTe, MnTe-ZnTe and MnTe-CdTe semiconductor superlattices 4.2.1. EuTe-PbTe superlattices EuTe is an antiferromagnetic member of the family of europium chalcogenides (Methfessel and Mattis, 1968; von Molnar and Kasuya, 1968; Wachter, 1979; Mauger and Godard, 1986). Bulk crystals of EuTe crystallize in a cubic (rocksalt) lattice with the lattice parameter aEuTe = 6.59 Å. The non-magnetic (diamagnetic) crystals of PbTe also grow in the rocksalt structure with the lattice parameter aPbTe = 6.46 Å. The mismatch of lattice parameters of these materials is 2.1% and the critical thickness for the pseudomorphic growth of EuTe on PbTe(111) is 44 ± 5 ML, i.e., about 160 Å (Koppensteiner et al., 1993; Springholz and Bauer, 1993a, 1993b; Springholz et al., 1994, 1995). Usually, EuTe-PbTe superlattices are grown by molecular beam epitaxy on freshly cleaved BaF2 (111) substrates employing the effusion cells for PbTe, Te, and Eu. The two-dimensional layer-by-layer heteroepitaxial growth is realized within a certain range of growth parameters as summarized in the growth phase diagram (Koppensteiner et al., 1993; Springholz and Bauer, 1993a, 1993b; Springholz et al., 1994, 1995) established on the basis of the in situ RHEED analysis of the growth mode as well as on the basis of post growth high resolution X-ray diffraction studies. EuTe-PbTe superlattices were also grown on PbTe(100) substrates (Kostyk et al., 1994). Stoichiometric EuTe is a semi-insulating material whereas PbTe is known for its tendency to grow nonstoichiometrically with electrically active vacancies producing a concentration of conducting holes (or electrons) of about 1017 –1018 cm−3 . In the MBE grown layers of PbTe p-type doping usually is due to Pb vacancies and is achieved by controlling additional Te ﬂux during the growth. For n-type doping of PbTe layers Bi donors are usually introduced from a Bi2 Te3 source. In EuTe-PbTe superlattices, PbTe layers possessing a narrow direct energy gap (Eg = 0.19 eV at low temperatures) are non-magnetic quantum wells whereas EuTe layers form electron barriers. The fundamental energy gap in EuTe crystals corresponds to the transitions between the top of the p-type valence band and the bottom of the 5d–6s conduction band and is observed in photoluminescence and transmission studies at 2.25 eV (Heiss et al., 2001a, 2001b). The absorption edge of EuTe is observed at lower energies, 1.5–1.6 eV, and is attributed to the transitions between localized 4f states and delocalized states at the bottom of the conduction band. In top quality epitaxial layers of EuTe very strong magnetic excitonic effects are observed with a large red-shift induced by a magnetic ﬁeld (Heiss et al., 2001a, 2001b). The very high structural and optical quality of EuTe-PbTe/BaF2 (111) multilayers allowed for a successful application of these structures in the construction of vertical cavity surface emitting lasers (VCSEL), with optical pumping, employing PbTe quantum wells, EuTe electron barriers, and EuTe-PbTe ultra-high-ﬁnesse Bragg mirrors (Schwarzl et al., 1999; Springholz et al., 2000, 2001b). One beneﬁts here, in particular, from the large contrast of the refractive indexes of EuTe and PbTe.

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EuTe orders in a type-II antiferromagnetic structure with the Néel temperature TN = 9.6 K. The dominant exchange interactions are due to the next-nearest magnetic neighbors (NNN, antiferromagnetic) and the nearest magnetic neighbors (NN, ferromagnetic but much weaker), see, for example, Methfessel and Mattis (1968), von Molnar and Kasuya (1968), Wachter (1979) and Mauger and Godard (1986). Along the [111] crystal direction the magnetic structures of EuTe can be viewed as consisting of (111) sheets of Eu2+ ions coupled ferromagnetically by in-plane NN exchange interactions. The magnetization vectors of adjacent (111) sheets are aligned antiferromagnetically by dominant out-of-plane NNN antiferromagnetic exchange interactions. In bulk crystals of EuTe there exist four symmetry-equivalent families of (111) antiferromagnetic domains. In contrast, in EuTePbTe/BaF2 (111) layers single (111)-domain magnetic order is observed and the ferromagnetic (111) sheets are parallel to the layer plane. This energetic preference is caused by the inﬂuence of lattice mismatch induced strain which, in particular, slightly changes the NN interspin distances and NN exchange interactions (Kepa et al., 1998; Nunez et al., 1998; Giebultowicz et al., 2001). Although strain selects just one of the (111) ferromagnetic planes, the multidomain state of EuTe-PbTe superlattices is still expected due to the sixfold symmetry of possible directions of sublattice magnetization vectors in the (111) plane. An antiferromagnetic transition is observed in EuTe-PbTe/BaF2 (111) and EuTePbTe/PbTe(100) multilayers even for ultrathin EuTe layers of only 2 ML as studied by magnetization (Heremans and Partin, 1988; J.J. Chen et al., 1994; Kostyk et al., 1994), ac magnetic susceptibility (Bergomi and Chen, 1997; Kepa et al., 1998), antiferromagnetic resonance (Wilamowski et al., 1997), and neutron diffraction techniques (Giebultowicz et al., 1995, 2001; Kepa et al., 1998; Nunez et al., 1998). For EuTe-PbTe multilayers with relatively thick non-magnetic spacers the Néel temperature depends on the thickness of the magnetic layer and the lattice mismatch induced strain. The ﬁrst effect has the same origin as in the related EuS-PbS system (see section 4.1.1) and can be understood by taking into account missing magnetic neighbors (Kostyk et al., 1994). The effect of compressing in-plane strain increases in the EuTe-PbTe structures with increasing ratio of the thickness of PbTe layer over the thickness of EuTe layer which (in free standing superlattices) determines the lattice mismatch strain acting in each of the layers. The Néel temperature of EuTe-PbTe/BaF2 (111) superlattices with various thicknesses of magnetic and non-magnetic layers are discussed in Kepa et al. (1998). In EuTe-PbTe superlattices with thin PbTe spacer the antiferromagnetic order in each of the layers is preserved but their sublattice magnetization vectors are arranged in a periodic (correlated) way with successive magnetic layers repeating the magnetic order of the previous layer. These correlations can be in-phase (with superlattice magnetic periodicity equal to the chemical one) or out-of-phase (superlattice magnetic periodicity doubled) which in ferromagnetic systems corresponds to ferromagnetic or antiferromagnetic interlayer coupling, respectively (Blinowski and Kacman, 2001; Giebultowicz et al., 2001; Kacman, 2001). The experimental evidences for these interlayer correlation in EuTePbTe superlattices are exclusively based on neutron diffraction measurements in which the change of the magnetic periodicity is detected as a characteristic diffraction spectrum with a central line (in-phase correlations) or without a central line but with two symmetric side lines (out-of-phase correlations). The interlayer interactions in EuTe-PbTe can be surprisingly long-range. The magnetically correlated state is observed in superlattices with

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non-magnetic PbTe spacer up 20 ML (Giebultowicz et al., 2001). Theoretical models proposed to explain these experimental observations are discussed in section 4.1.2. 4.3. Magnetic properties of MnTe-ZnTe and MnTe-CdTe superlattices Bulk crystals of the antiferromagnetic MnTe semiconductor crystallize in the hexagonal NiAs structure whereas in MnTe-ZnTe and MnTe-CdTe multilayers grown by atomic layer (or molecular beam) epitaxy on GaAs(100) substrates the layers of MnTe crystallize in the metastable cubic (zincblende) crystal structure with the lattice parameter aMnTe = 6.34 Å (Gunshor et al., 1990; Anno et al., 1992; Abramof et al., 1994). Below the Néel temperature (TN = 67 K) in these layers, type-III antiferromagnetic order is established. The lattice parameter of ZnTe, aZnTe = 6.10 Å, is smaller whereas the lattice parameter of CdTe, aCdTe = 6.48 Å, is larger than the parameter of cubic MnTe. The critical thickness for the pseudomorphic growth of MnTe on ZnTe(100) is 26 ML whereas for the growth of MnTe on CdTe(100) it is equal to 50 ML (Abramof et al., 1994). Therefore, in pseudomorphically grown epitaxial MnTe-ZnTe superlattices MnTe layers are under biaxial compressive stress whereas in the case of MnTe-CdTe structures the situation is the opposite – MnTe is subjected to tensile stress. This leads to a tetragonal distorsion of the unit cell as observed in X-ray and neutron diffraction measurements as well as concluded from an analysis of the effect of crystal ﬁeld on the ﬁne structure of EPR spectra of Mn2+ ions interdiffused into the non-magnetic layers (Gunshor et al., 1990; Anno et al., 1992; Klosowski et al., 1992; Giebultowicz et al., 1993, 1994; Abramof et al., 1994; Furdyna et al., 1995; Krenn et al., 1998). These tetragonal distorsions of the cubic lattice inﬂuence the magnetic ground state of the MnTe layers. In the MnTe-ZnTe superlattices the type-III antiferromagnetic structure is still observed but one of the (001) magnetic domains (with antiferromagnetic period doubling direction along the growth direction) is energetically preferred (Giebultowicz et al., 1993). In MnTe-CdTe superlattices a more complicated helical structure is found with the helix axis along one of the (100) in-plane crystal directions and a helix pitch being incommensurate with the in-plane lattice period (Giebultowicz et al., 1994). The parameters of the helix can be continuously varied by controlling the strain in the magnetic layers (e.g., by changing the thickness of the magnetic layer). Similar magnetic structures are also observed in closely related MnSe-ZnTe (3% lattice parameter mismatch, in-plane tensile stress, helical magnetic order) and MnSeZnSe (4% lattice parameters mismatch, in-plane compressive strain, type-III ordering) superlattices with zincblende antiferromagnetic MnSe semiconductor (Samarth et al., 1991; Giebułtowicz et al., 1992). Both MnTe-ZnTe and MnTe-CdTe structures are expected to form type-I superlattices with antiferromagnetic layers of MnTe being electronic barriers for both electrons and holes and non-magnetic CdTe and ZnTe layers being quantum well materials. The optical properties of zincblende MnTe layers were studied by magneto-optical Kerr effect (Han et al., 1991), photoluminescence and photoluminescence excitation spectroscopy (Kossacki et al., 1999b), optical reﬂectance (Pohlt et al., 1998), absorption measurements and X-ray photoelectron spectroscopy (Durbin et al., 1989). The fundamental energy gap of zincblende MnTe is 3.2–3.3 eV. For a more detailed analysis of the electronic structures and optical properties of these and related Zn1−x Mnx Te-ZnTe and Cd1−x Mnx Te-CdTe low-dimensional quantum structures see section 2 of this chapter.

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The antiferromagnetic order is preserved even in very thin MnTe layers consisting of just a few monolayers although the Néel temperature is considerably reduced in this case. In MnTe-CdTe and MnTe-ZnTe superlattices with relatively thick (10 or more ML) non-magnetic spacers the magnetic order in the individual layers is established independently and the parameters such as the pitch and phase of the helix or the direction of the sublattice magnetization vectors are uncorrelated in neighboring layers. In the case of thinner non-magnetic spacers the magnetic order in successive MnTe layers is correlated (spin helices are phase-synchronized in MnTe-CdTe superlattices). More details can be found in several reports (Durbin et al., 1989; Han et al., 1991; Samarth et al., 1991; Giebultowicz et al., 1992, 1994, 2001; Klosowski et al., 1992; Furdyna et al., 1995; Nunez et al., 1995; Kepa et al., 1998; Krenn et al., 1998; Lin et al., 1998; Pohlt et al., 1998; Rhyne et al., 1998, 2001; Kossacki et al., 1999b; Stumpe et al., 2000). Similarly to the EuTe-PbTe superlattices, the experimental evidences for these interlayer correlation in MnTe-ZnTe and MnTe-CdTe superlattices are exclusively based on neutron diffraction measurements in which the change of the magnetic periodicity is manifested as a characteristic diffraction spectrum. The intriguing phenomenon observed in these superlattices is the change of the interlayer correlations from the in-phase at low temperatures into the out-of-phase at higher temperatures (Rhyne et al., 2001). The range of these interlayer couplings is rather long and the correlated magnetic order is observed in MnTe-CdTe superlattices with CdTe spacer thicknesses up to 30 Å. In the case of MnTeZnTe the range of the coupling is smaller, about 15 Å, but can be changed, e.g., by doping with Cl both in the magnetic and in the non-magnetic part of the structure (Stumpe et al., 2000). Theoretical models developed for the interlayer exchange in these structures are analogous to those discussed in section 4.1.2. Particularly relevant are the models exploring the role of shallow defects in the non-magnetic quantum well as well as the role of strain ﬁelds as a coupling mechanism involving no exchange interactions. 5. Other diluted magnetic semiconductors DMS containing other than manganese transition metals (TM) never attracted as much attention as Mn-based materials. Among many reasons, the crucial ones are technological problems connected with a very poor solubility of most of TM ions in binary semiconductor compounds. Starting from manganese with its 3d shell half-ﬁlled, the solubility of TM in II-VI matrices strongly decreases with increasing ﬁlling of the 3d shell orbit. For instance, while the solubility limit of Mn in the majority of II-VI compounds is close to 1, the Fe solubility limit reaches not much more than 20–30 at.%, and, ﬁnally, the solubility limit of Ni is only 2–3 at.% (cf. Lawniczak-Jabłonska et al. (1996)). A detailed study of near edge soft X-ray absorption spectra of several Zn1−x Mx S mixed crystals (where M = Mn, Fe, Co, Ni) by Lawniczak-Jablonska et al. (1997) provided the explanation of this phenomenon. It was found that as the TM ion 3d orbit becomes more closed, the 3d electrons assume a more core-like character and their ability to participate in bonding decreases. It is known that the solubility decreases rapidly also when the number of 3d electrons becomes smaller than 5 (Galazka et al., 1999).

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Blinowski and Kacman (1992a, 1992b) predicted theoretically that, depending on the ﬁlling of the TM ion 3d shell, the p–d exchange interaction may evolve from antiferromagnetic to ferromagnetic. In particular Sc, Ti, V, and Cr containing materials were expected to show a ferromagnetic type of behavior. The above results stimulated technological efforts to obtain these novel DMS and to achieve magnetic ion concentrations high enough to enable a study of exchange interactions. 5.1. Chromium and vanadium compounds In general, the Hamiltonian describing non-interacting Cr2+ ions in a cubic crystal lattice in addition to the tetrahedral crystal ﬁeld splitting, spin–orbit interaction, and external magnetic ﬁeld effect, contains also a Jahn–Teller (JT) term (Vallin et al., 1970). The JT coupling leads to a static tetragonal distortion of the Cr2+ center. Based on this simple model Mac et al. (1994) obtained the scheme of Cr2+ energy levels in tetrahedral surrounding of Zn1−x Crx Se depicted in ﬁg. 24. In absence of JT interaction the low-energy structure of Cr2+ ion is different, in particular the ground state is a non-magnetic singlet which results in the Van Vleck type of paramagnetism observed, e.g., in Zn1−x Crx Te system. The structure of lowest lying energy levels of Cr2+ was investigated in Zn1−x Crx Se (x < 0.004) by Krevet et al. (1993) and in Zn1−x Crx S (x < 0.006) by Mac et al. (1995) by means of far-infrared laser absorption in magnetic ﬁelds up to 20 T. The description of the data, in particular of a strong anisotropy of absorption lines observed in both experiments, was based on a cubic crystal ﬁeld model with a static JT distortion included. Recently, a detailed EPR study of chromium-alloyed zinc chalcogenides Zn1−x Crx (S, Se, Te) in magnetic ﬁelds up to 20 T was reported by Boonman et al. (2000). Magnetic properties of Zn1−x Crx Se were investigated by Twardowski et al. (1993b). The authors performed an analysis of the speciﬁc heat and magnetization data taken along the [100], [110], and [111] crystallographic directions using a crystal ﬁeld model (including static JT distortion) for the Cr2+ ion. Magnetization and speciﬁc heat were also studied by Mac et al. (1994) for two materials: Zn1−x Crx Se and Zn1−x Crx S (x < 0.01). A representative ﬁeld dependence of the magnetization for a few temperatures and different ﬁeld orientations is shown in ﬁg. 25 for a Zn1−x Crx Se sample with x = 0.002. A strong anisotropy of magnetization is clearly visible. The magnetic behavior observed in Zn1−x Crx Se crystals is neither that of Brillouin paramagnets (characteristic of Mn- or Co-based DMS) nor that of Van Vleck systems (observed in DMS containing Fe). It results from a strong, static JT distortion present for the Cr ions. Pekarek et al. (1994) investigated magnetic properties of Zn1−x Crx Te crystal with x = 0.003. The heat capacity measurements showed a Schottky peak indicating an energylevel splitting of 3.1 K between the ground and ﬁrst excited states. The magnetization data revealed a small anisotropy (approximately 7%) with the [111] direction giving the largest value. The magnetization data were ﬁt with a model including a static JT distortion proposed previously in these materials (Vallin et al., 1970). A reasonable agreement was obtained between experimental data and theoretical calculations for a spin–orbit parameter of −59 cm−1 and a JT energy of 320 cm−1 . Two years later, the same group of authors reported results of magnetization measurements in single-crystalline samples Cd1−x Crx Te, x = 0.0036, and Zn1−x Crx Te, x = 0.0031 (Pekarek et al., 1996). Within the same theoretical model, good agreement was found with the data for effective spin–orbit

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(a)

(b) Fig. 24. Energy levels of the Cr2+ ion, (a) energies in the absence of magnetic ﬁeld (energy distances are not to scale), (b) the ﬁve lowest energy Cr2+ levels calculated as a function of magnetic ﬁeld B along [001], [111], and [100] directions. The static Jahn–Teller distortion is along [001]. The parameters used correspond to Cr2+ in ZnSe (after Mac et al. (1994)).

parameters of −49.9 cm−1 and −59.4 cm−1 and Jahn-Teller splittings of 370 cm−1 and 320 cm−1 for Cd1−x Crx Te and Zn1−x Crx Te, respectively. The key feature of both systems is an orbital singlet, S = 2, ground state with a small splitting (1.5 K for Cd1−x Crx Te and 12.4 K for Zn1−x Crx Te) of the ms states, being responsible for the speciﬁc magnetization behavior. McCabe et al. (1995) observed magnetization steps from non-interacting Cr2+ ions in ZnTe in a magnetic ﬁeld along the [001] direction. This phenomenon is due to the spe-

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Fig. 25. Anisotropy of the magnetization of Zn1−x Crx Se (x = 0.002) at various temperatures. The lines show the magnetization calculated according to the EQ (equilibrium) model. The concentration x was slightly adjusted to the value x = 0.0022 providing the correct absolute magnetization value for T > 10 K (after Mac et al. (1994)).

ciﬁc structure of the energy levels of isolated Cr2+ ions, displaying level crossings in high magnetic ﬁelds (see ﬁg. 26). In general, magnetization steps may arise from isolated spins whenever the spin Hamiltonian contains an axial anisotropy term and, in fact, they were observed in some Co-based DMS (see section 5.2.2). From the step magnitudes the authors drawn the conclusion that in Zn1−x Crx Te the distribution of JT distortions along the three equivalent [100] crystallographic directions is much closer to the situation when the distortions are free to change direction (so these three populations are governed by thermal equilibrium) than to the situation when the distortions are frozen. Nevertheless, a perfect equilibrium is not achieved. Blinowski et al. developed a theory of exchange interactions of localized Co2+ ions with band electrons as well as of spin–spin (d–d) exchange interactions between the Co2+ ions themselves (Blinowski et al. (1996a); see also Blinowski et al. (1996b)). The proposed model takes into account multielectron effects and provides an explanation of the origin of the ferromagnetic p–d interactions in zinc chalcogenides containing Cr ions. A ferromagnetic p–d exchange character was also predicted for cadmium chalcogenides doped with chromium. Within the model, the sign of the p–d exchange is determined by the energy of the charge transfer from the Cr2+ (d4 ) ion to the valence band. If that energy is positive, N0 β should be negative, otherwise N0 β is positive (i.e., ferromagnetic). The charge transfer energy values estimated from the donor energies for Zn1−x Crx S and Zn1−x Crx Se led to N0 β values very close to the experimental data. The spin exchange interaction between Cr2+ ions, calculated numerically for zinc chalcogenides, exhibits properties unique in DMS for it is also of the ferromagnetic type and depends on the relative orientations of the ionic Jahn–Teller distortions. The origins of this unusual behavior are considered in cited

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Fig. 26. Calculated energy levels for an isolated Cr2+ ion in ZnTe as a function of magnetic ﬁeld B. B is parallel to the Jahn–Teller distortion along the [100] direction. B1 and B2 denote the ﬁeld values at which magnetization peaks are expected due to energy level crossings (after McCabe et al. (1995)).

reports. The presented theory predicts that d–d exchange interactions in other Cr-based II-VI DMS should also be of the ferromagnetic type. To overcome the problem of the small solubility of Cr in II-VI compounds an attempt to grow Cd1−x Crx Te and Zn1−x Crx Te samples using MBE methods was undertaken by Wojtowicz et al. (1997). Unfortunately, the results turned out to be not satisfactory. Cd1−x Crx Se ﬁlms on fused silica and sapphire (001) substrates were successfully grown with the use of a vapor deposition technique by Adachi et al. (1998). Films retaining the host CdSe wurtzite structure were obtained in an extremely wide chromium composition range, 0 x 0.66. The saturation magnetization and magnetic susceptibility were observed to increase with increasing x. The initial magnetization curves saturated easily as compared to the Mn-based DMS systems. Since such a behavior is different from that established in most DMS materials in which the d–d interactions between the magnetic ions are antiferromagnetic, the authors concluded that d–d interactions of ferromagnetic type are likely to exist in Cd1−x Crx Se. Mac et al. (1993) investigated free exciton splitting and Faraday rotation of the Zn1−x Crx Se system for x 0.005, and found that the valence band splitting is reversed relative to the materials with Mn, Co, or Fe (see ﬁg. 27). This fact strongly indicates a ferromagnetic p–d exchange. That was the ﬁrst experimental observation of such a property in II-VI DMS conﬁrming theoretical predictions (Blinowski and Kacman, 1992a). The change of the sign of the exchange integral N0 β is most likely due to location of the d4 /d5 Cr level above the top of the valence band (contrary to the situation occurring in Mn, Co, and Fe-based DMS). The s,p–d exchange interaction in Cr-based DMS was studied by means of polarized magnetoreﬂectance and magnetization (Mac et al., 1996b).

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Fig. 27. Reﬂectance spectra of Zn1−x Crx Se and Zn1−x Mnx Se at T = 1.9 K. (Upper part) Zn0.996 Cr0.004 Se at B = 5 T; (Lower part) Zn0.95 Mn0.05 Se at B = 0.1 T. σ + , σ − denote circular right and left polarizations of light (after Mac et al. (1993)).

The exchange constant N0 β − N0 α for four different zinc chalcogenides (Zn1−x Crx Se, Zn1−x Crx Se0.95 S0.05, Zn1−x Crx S, and Zn1−x Crx Te) was derived from experimental data. The authors discussed the chemical trend in the N0 β variation assuming constant N0 α. The resulting N0 β values are listed in table 2 together with p–d exchange integrals found in other II-VI DMS. Magnetic circular dichroism (MCD) spectra of (Zn,Cr)Se, (Zn,Cr)Te, (Zn,Cr)S, and (Cd,Cr)S showed sharp excitonic MCD peaks reﬂecting the presence of a ferromagnetic p–d exchange interaction (Ando and Twardowski, 1996). A bound magnetic polaron (BMP) structure was observed in Cd1−x Crx S by means of spin-ﬂip Raman scattering experiments (Twardowski et al., 1996). Theoretical studies of the donor bound magnetic polaron in Cr-based DMS were performed by Herbich et al. (1998b). The Cr BMP generalizes the physical situation for Mn-based and Fe-based BMP due to the particular energy structure of the Cr2+ ions, exhibiting features typical for both Mn2+ and Fe2+ ions. The ground state of the Cr2+ ion is a semidoublet, yielding a permanent magnetic moment of the ion, similarly to Mn2+ or Co2+ ions. On the other hand, the ground state is followed by closely lying excited states, analogously to the situation for Fe2+ ions. The developed Cr BMP model recovers the characteristic behavior of both the Mn BMP and the Fe BMP: zero-ﬁeld spin-ﬂip energy (a ﬁngerprint of Mn BMP) and anticrossings of BMP states (typical for Fe BMP). Recently, a ferromagnetic p–d exchange interaction was also discovered in vanadium containing DMS. A study of magnetoreﬂectance and luminescence in magnetic ﬁelds up

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TABLE 2 s–d and p–d exchange constants in DMS Compound

N0 α (eV)

Cd1−x Fex Se

0.225 ± 0.005 0.258 ± 0.02

Cd1−x Fex Te

0.269 ± 0.017 0.3 ± 0.04 0.22 ± 0.04 0.25 ± 0.03

Zn1−x Fex Se Zn1−x Fex Te Cd1−x Cox S Cd1−x Cox Se

0.29 ± 0.09 0.179 ± 0.004 0.27 0.32 0.274 ± 0.006 0.279 ± 0.029

Cd1−x Cox Te < 0.020 0.2 ± 0.01

Zn1−x Cox Te Cd1−x Crx Se Zn1−x Crx S Zn1−x Crx Se0.95 S0.05 Zn1−x Crx Se Zn1−x Crx Te Cd1−x Vx S

0.31 ± 0.03 0.22 ± 0.02

0.2

N0 β (eV) −1.53 ± 0.05 −1.45 ± 0.1 −1.90 ± 0.15 −1.6 ± 0.1 −0.836 ± 0.029 −1.27 ± 0.08 −1.74 −1.76 ± 0.09 −1.9 ± 0.45 −1.5 ± 0.17 −2.25 −2.12 ± 0.05 −1.873 ± 0.042 −1.8 −2.331 ± 0.13 −1.6 ± 0.1 −1.82 ± 0.01 −2.2 ± 0.15 −2.93 ± 0.1 −3.03 ± 0.15 0.48 ± 0.05 0.62 0.93 0.95 4.25 1

Reference Heiman et al. (1988) Scalbert et al. (1990) Shih et al. (1990) Twardowski et al. (1990a) Twardowski et al. (1990b) Alawadhi et al. (2001) Testelin et al. (1991) Twardowski et al. (1990a) Twardowski et al. (1992) Mac et al. (1996a)a Testelin et al. (2000) Gennser et al. (1995) Adachi et al. (1995) Bartholomev et al. (1989) Gennser et al. (1995) Nawrocki et al. (1991) Abramishvily et al. (1997) Alawadhi et al. (2001) El Ouazzani et al. (1997) Shih et al. (1992)a Zielinski et al. (2001) Zielinski et al. (2001) Zielinski et al. (1996) Herbich et al. (1998a) Twardowski et al. (1996) Mac et al. (1996b)a Mac et al. (1996b)a Mac et al. (1996b)a Mac et al. (1996b)a Mac et al. (2000)b

a Estimated from (N α − N β) value by assuming N α = 0.2 eV. 0 0 0 b Estimation.

to 7 T performed for Cd1−x Vx S, x < 0.001, provided the evidence of the ferromagnetic sign of the p–d exchange constant (Mac et al., 2000). Herbich et al. (1999) performed magnetization measurements of the wurtzite crystals Cd1−x Vx S and Cd1−x Vx Se in magnetic ﬁelds (up to 6 T) applied parallel and perpendicular to the hexagonal crystal axis. A strong anisotropy of magnetization is observed for Cd1−x Vx S at low temperatures (2 K, 10 K), while for Cd1−x Vx Se the anisotropy is much weaker. The magnetization data may be well described within the crystal ﬁeld model of the V2+ center, taking account of a strong static Jahn–Teller effect of trigonal symmetry. The differences in the anisotropy of magnetization for Cd 1−x Vx S and Cd1−x Vx Se are interpreted as being due to different distributions of different V2+ centers for both systems.

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5.2. Cobalt compounds 5.2.1. Tetrahedral surrounding The energy level scheme of Co2+ in a II-VI semiconductor lattice for both tetrahedral (case of CdTe) and trigonal (ZnSe) crystal ﬁelds is presented in ﬁg. 14 in the review of Kossut and Dobrowolski (1993). Most of the investigations of Cd1−x Cox Te system were concentrated on the determination of the s–d and p–d exchange constants by means of magneto-optical measurements. Shih et al. (1992) studied the excitonic magnetoreﬂectance in ﬁelds up to 15 T in Cd1−x Cox Te sample with x = 0.004. The small Co2+ concentration allowed a clear observation of the direct Zeeman interaction and diamagnetic energy in addition to the s,p–d exchange splitting. From the optical and high ﬁeld magnetization data, the combined s,p–d exchange parameter (N0 β − N0 α) value was determined (see table 2). The paper of El Ouazzani et al. (1997) reported on a study of the magnetic ﬁeld induced coupling between originally ground (1s) and excited (3d) exciton states in Cd1−x Cox Te. In a certain magnetic ﬁeld the upper Zeeman pattern of the 1s state passes through the lower pattern of the 3s state, giving to the latter half of its oscillator strength. Such an anticrossing behavior has been observed in circularly polarized magnetoreﬂection spectra, which allowed the determination of both exchange coupling constants. In the work of Abramishvily et al. (1997) the magnetoreﬂectivity of the Cd1−x Cox Te crystal, x = 0.004, in the region of 1s exciton was investigated in magnetic ﬁelds up to 3.5 T and at temperature of 2 K. A giant spin splitting of excitonic transitions in σ + and σ − polarizations as well as two π transitions were observed. A strong asymmetry between the average energies of the exciton σ and π components was found. The theoretical description of the observed phenomena is based on the expanded carrier–ion exchange Hamiltonian which in addition to the usual Heisenberg-like term with exchange constant N0 β included also a biquadratic term with exchange constant N0 β1 . The best agreement with the observed exciton component splittings was obtained for the ﬁtting parameter value (x /x)N0 β = −1.8 eV and β1 /β = 0.26 with x being the active concentration of the magnetic ions and N0 the cation concentration. Some experimental data reported by other groups have been described within the proposed model using the same β1 /β ratio. Recently, the s,p–d exchange constants of Cd1−x Cox Te, x = 0.01, were determined from the huge excitonic Zeeman effect observed in wavelengthmodulated reﬂectivity spectra, supplemented with magnetization measurements (Alawadhi et al., 2001). To avoid complications connected with the observed asymmetry of the exciton splitting in magnetic ﬁelds, the authors analyzed only the outer Zeeman components. A novel interpretation of the observed asymmetry of the exciton Zeeman splitting was proposed by Zielinski et al. (2001). The authors performed a detailed study of the exciton Zeeman spectra in several samples of Cd1−x Cox Te with x < 0.5% by means of magnetoreﬂection and magnetoabsorption experiments in Faraday geometry at 1.6 K, supplemented with magnetization measurements performed in ﬁelds up to 5 T. Analysis of the σ ± reﬂectance spectra within the polariton model determines the transverse eigenenergies in different magnetic ﬁelds. The obtained energy pattern of the four Zeeman components shows unusual features which have been explained by taking into account the longitudinal– transverse splitting and the electron–hole exchange interaction. Very recently, Alawadhi et al. (2002) presented a study of spin–spin exchange interactions between Co2+ ions in Cd1−x Cox Te systems with relatively high Co concentrations,

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TABLE 3 Coupling constants of the Heisenberg interaction between nearest (J1 ) and next-nearest neighboring (J2 ) magnetic atoms in DMS −J1 /kB (K) CdFeSe CdFeTe ZnFeS ZnFeSe HgFeSe CdCoS CdCoSe CdCoTe ZnCoS

ZnCoSe

ZnCoTe

17.8 ± 3 8.8 ± 1.2 37 22 ± 1 13 ± 2 30.6 ± 1.7 30.3 ± 1.5 31 ± 2 25 ± 7 47.5 ± 0.6 47 ± 6 49.5 ± 1 54 ± 8 49.5 38 ± 2

−J2 /kB (K)

3 ± 1.5 2.25 ± 0.2a 3.04 ± 0.1 5.7 ± 0.6

Reference Twardowski et al. (1988) Testelin et al. (1992) Twardowski et al. (1993a) Twardowski et al. (1988) Soskic et al. (1997) Foner et al. (1995) Dahl et al. (1995) Foner et al. (1995) Alawadhi et al. (2002) Giebultowicz et al. (1990a) Lewicki et al. (1989) Shapira et al. (1990) Giebultowicz et al. (1990b) Lewicki et al. (1989) Shapira et al. (1990) Swagten et al. (1992) Giebultowicz et al. (1990b) Vu et al. (1992)

a This value is either J or J . 2 3

x 0.022, obtained by using the vertical Bridgman technique. Samples were characterized by magnetization and wavelength-modulated reﬂectivity measurements. Low-temperature magnetization and high-temperature magnetic susceptibility analyses were used to extract the exchange integrals for up to third-neighbor Co2+ pairs (see table 3). The reported values correspond well to the values obtained for other Co-based II-VI DMS and are a clear manifestation of the unusually large Co2+ –Co2+ antiferromagnetic interaction. The excitonic energy gap Eg (x) of Cd1−x Cox Te in the wavelength-modulated reﬂectivity shows a linear monotonic increase with increasing x and may be expressed as Eg = 1.597 + 1.095x (eV). A very interesting result was recently obtained by Ahn et al. (2001) for quaternary CdMnCoTe ﬁlms deposited on quartz glass substrates using the MBE equipment. At a temperature of 77 K, the Faraday rotation observed in the Cd0.647Mn0.34Co0.013Te ﬁlm was −0.49 deg/cm/G at 610 nm. This value is approximately two times larger than that of a Cd0.52 Mn0.48Te ﬁlm deposited on the same substrate. The origin of the enhancement of the Faraday rotation in CdMnCoTe ﬁlms has been discussed in terms of the magnetic susceptibility χ . The s,p–d exchange interaction in Zn1−x Cox Te single crystals (x 0.01) was investigated by means of correlated magneto-optical and magnetization measurements at T = 1.8 K (Zielinski et al. (1996); see also Zielinski et al. (2001)). The resonant magnetooptic Kerr effect was used to complement magnetoreﬂectance experiments. Correlated measurements of exciton Zeeman splittings in crystals of various Co compositions and of magnetization performed on the same samples made it possible to determine the exchange

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integrals for the Γ6 and Γ8 bands. The abnormally large value of the p–d exchange constant N0 β = −3.03 eV appeared not to vary with Co concentration. Low-temperature magnetization data obtained in ﬁelds up to 15 T were consistently interpreted in a model including the contributions of isolated ions and Co2+ –Co2+ pairs of second and third neighbors. Electronic Raman transitions due to the spin-ﬂip of the 3d electrons of Co2+ in Zn1−x Cox Te and Cd1−x Cox Te (x 0.01) were observed at hωPM = g(Co2+ )μB H with g(Co2+ ) = 2.295 ± 0.010 and 2.310 ± 0.002, respectively (Seong et al., 2001). The intensity of Raman electron paramagnetic resonance (Raman-EPR) showed strong resonant enhancement when the incident or scattered photon energy coincides with that of a Zeeman component of the free exciton. Under resonant conditions, the Raman spectra displayed “ZnTe-like” (or “CdTe-like”) and “CoTe-like” longitudinal optical (LO) phonons in combination with the spin-ﬂip transitions, a consequence of the Fröhlich interaction. The low-temperature speciﬁc heat of Zn1−x Cox Se (x < 0.05) was studied in magnetic ﬁelds up to 3 T (Swagten et al., 1992). The magnetic speciﬁc heat data as well as susceptibility and magnetization data are well described by a model that takes into account the nearest-neighbor exchange interaction (JNN = −49.5 K) as well as a long-range interaction (JLR = −30/R 6.3 K, where R is given in nearest-neighbor distance units). The photoluminescence (PL) associated with Co2+ ions was investigated in Zn1−x Cox Se epitaxial ﬁlms with x 3.7% (Bak et al., 1996). The authors presented temperature and pressure dependence of the photoluminescence performed in a diamond-anvil cell. Two sharp emission peaks at ∼2.36 eV showed a weak red shift under pressure with rapidly decreasing peak intensities. The PL data were discussed within the framework of conventional crystal ﬁeld theory based on the Racah and crystal ﬁeld parameters. Quantitative estimates for the enhancement in the p–d hybridization with pressure (evident in the lineshape proﬁles of the spectra) were deduced. Y.D. Kim et al. (1994) performed spectroscopic ellipsometry measurements for singlecrystalline ﬁlms of (001)Zn1−x Mx Se (M = Mn, Fe, Co; 0 x 0.144) grown by molecular-beam epitaxy on (001)GaAs (see also Ko et al. (1997)). In the 3.5–5.5 eV photon energy range, the dielectric function spectra displayed structures corresponding to the interband transitions at the L point of the Brillouin zone. A model including the effects of s,p–d hybridization on the L point band gap energies was developed and applied to describe their compositional dependence in all three materials. The EPR in the far-infrared region and high magnetic ﬁelds up to 40 T was studied by Martin et al. (1994) in Zn1−x Cox S, 0.005 x 0.17. For the x ≈ 0.05 sample, the transmission spectra exhibited very sharp satellite lines on both sides of the single magnetic ion resonance. The authors attributed these features to the exchange of coupled spin pairs. A calculation of the energy levels of the pairs, including Dzyaloshinskii–Moriya effects and symmetrical exchange terms of the interaction tensor, allowed to estimate the value of the anisotropic exchange constant, De = 0.06 K. 5.2.2. Trigonal surrounding In the wurtzite structure, due to the crystal ﬁeld axial term the four-fold degeneracy of the lowest isolated Co2+ ion level is partially lifted, see ﬁg. 14 in the review of Kossut and Dobrowolski (1993). Among the resulting two Kramers doublets, one corresponds to Sz = ±1/2 (being the ground state in Cd-based DMS) and the other to Sz = ±3/2. In the

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presence of a magnetic ﬁeld, the crossing of split levels leads to a change of the ground state, corresponding to a change of Sz from −1/2 to −3/2. At sufﬁciently low temperatures (i.e., in the dilution refrigerator temperature range) such a change gives rise to a step-like behavior of the magnetization versus magnetic ﬁeld curve. This phenomenon can be used to determine the single-ion axial anisotropy constant D. The magnetization steps arising from isolated Co2+ ions were observed in the Cd1−x Cox S system for low Co concentrations in a few experiments only. For example, direct magnetization measurements of a crystal with x = 0.005 at T = 30 mK yielded 2D = 132 ± 0.04 cm−1 , being very close to the value obtained employing EPR data for x = 0.016, i.e., 2D = 1.28 ± 0.04 cm−1 (Bindilatti et al., 1994). In the study of Vu et al. (1995) optical Faraday rotation measurements at 80 mK were used to detect a magnetization step in a crystal with x = 0.024. Analysis of the data gave D/kB = 0.96 ± 0.02 K. The cited values may be compared with the results of X.C. Liu et al. (1995) obtained from inelastic Raman light scattering. Direct observation of the transition associated with the lifting of the Kramers degeneracy (observed at zero magnetic ﬁeld) yielded 2D = 0.15 meV. The results obtained in magnetic ﬁelds up to B = 10 T enabled the authors to determine the value D/kB = 0.85 ± 0.03 K. The single-ion axial anisotropy parameter D for the Co2+ ion in Cd1−x Cox Se with x ≈ 0.005 was determined independently from the magnetization step due to isolated ions and using EPR measurements (Isber et al., 1995a). The magnetization step, observed at 30 mK, gave 2D = 0.95 ± 0.02 cm−1 . The EPR data, obtained with frequencies in the X band, yielded a very similar value, 2D = 0.936 ± 0.01 cm−1 . Foner et al. (1995) performed magnetization experiments at pumped helium temperatures and in pulsed magnetic ﬁelds up to 60 T for both the Cd1−x Cox Se and the Cd1−x Cox S systems with 0.036 x 0.064. The ﬁrst of the three expected steps in magnetization curve, due to the nearest-neighbor (NN) antiferromagnetic exchange J1 between Co2+ pairs, was observed in both materials. In contrast with earlier results on Cd1−x Mnx S, which revealed the magnetization steps in pulsed ﬁelds to be much narrower, the steps observed in the Co substituted materials were much broader, extending from about 35 to 55 T. In Cd1−x Cox S the broad magnetization step showed features due to the two inequivalent NN sites in the wurtzite structure. Several selected magnetic properties including s,p–d exchange energies, bound magnetic polarons, and magnetization in Cd1−x Cox Se and Cd1−x Cox S, 0.02 x 0.05, were investigated by means of spin-ﬂip Raman scattering, magnetization, and magnetoreﬂectivity measurements at liquid helium temperature (Gennser et al., 1995). The magnetization was adequately described by a modiﬁed Brillouin function for pure spin S = 3/2 ions. From spin-ﬂip Raman scattering of donor-bound electrons a large Zeeman splitting of approximately 4–10 meV in high magnetic ﬁelds was obtained as well as bound magnetic polaron energies of 0.2–0.4 meV for B = 0. Adachi et al. (1995) studied reﬂectance and luminescence of Cd1−x Cox Se, x ≈ 0.01, in steady magnetic ﬁelds up to 27 T. The bound and free exciton luminescence was clearly observed at 4.2 K. Magnetophotoluminescence under hydrostatic pressure up to 2 GPa was studied in Cd1−x Cox Se with x = 0.012 at liquid helium temperatures (N. Kuroda and Matsuda, 1996). It was found that pressure strengthens the exciton–Co and Co–Co exchange interactions. Further analysis in terms of the kinetic exchange theory showed that the on-site Coulomb energy and the charge-transfer energy are reduced markedly by pressure.

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Stankiewicz et al. (1994) investigated the photoconductivity in Cd1−x Cox Se, 0.011 x 0.041, at a temperature of 1.6 K and in magnetic ﬁelds up to 6 T. The photocurrent spectra showed two peaks near the band-gap energy which shifted towards lower energies as the magnetic ﬁeld intensity was increased. These lines were interpreted in terms of A and B free exciton dissociation (where A and B excitons denote excitons involving holes in two different valence bands split by the axial ﬁeld). The observed red shift of the photocurrent peaks arises from the s,p–d exchange interactions. Combining photoconductivity and magnetization data obtained on the very same samples allowed to estimate values of the exchange integrals in the material. 5.3. Iron compounds A detailed study of the magnetization in the Cd1−x Fex Te mixed crystals with x = 0.018 and 0.033 was presented by Testelin et al. (1992). Magnetization data were collected for ﬁelds up to 20 T applied along principal crystallographic directions, at liquid helium temperature. A theoretical computation involving the diagonalization of the Hamiltonian of an isolated Fe2+ ion subjected to crystal ﬁeld, spin-orbit, and Zeeman terms within the lowest 5 D manifold revealed the main features of the magnetization curves. In particular, the model predicted the magnetic anisotropy observed at low temperature for B > 5 T, but did not reproduce the low-temperature magnetization at high ﬁelds. The authors showed that a quantitative agreement can be achieved only if the Fe–Fe spin interaction is taken into account. Under the assumption of a random distribution of magnetic ions, a quantitative agreement with experiments was achieved for both the investigated compositions and the crystallographic directions studied. Alawadhi et al. (2001) determined the s,p–d exchange constants in Cd1−x Fex Te, x = 0.01, from the huge excitonic Zeeman effect observed in wavelength-modulated reﬂectivity spectra, supplemented with magnetization measurements. Below 5 K, both the magnetization and the excitonic Zeeman splittings recorded in magnetic ﬁelds up to 60 kGs clearly revealed the Van Vleck paramagnetism characteristic of Fe2+ ion in Cd1−x Fex Te. Mycielski et al. (1996) investigated the quaternary system (Cd1−x Fex )(Te1−y Sey ) containing two types of anions. In such a system, a part of the Fe2+ ions are located in crystal environments of a lowered symmetry. The energy level structure of these ions is substantially modiﬁed with respect to that of the Fe2+ ion in a one-anion (ternary) crystal. Regarding magnetic properties, the most essential modiﬁcation occurs when an Fe2+ ion is surrounded by three atoms of Te and one Se atom. Then, a doublet becomes the ground state of the magnetic ion, while in the case of the ternary crystal, the ground state of the Fe2+ ion is always a non-magnetic singlet, see ﬁg. 13 in the review of Kossut and Dobrowolski (1993). In consequence, for two-anion systems, a Curie-like paramagnetism is observed instead of an otherwise revealed temperature independent (T 10 K) Van Vleck paramagnetism. The results of photoconductivity measurements indicated that the Fe2+ ion ground energetic state can take several localizations in the band structure of the matrix crystal. A speciﬁc position depends on the conﬁguration of the nearest-neighbors of the Fe2+ ion, i.e., whether it is surrounded by four Te atoms, or three Te and one Se atom, etc. A similar behavior was also observed in Zn1−x Fex Te1−y Sey and Hg1−x Fex Te1−y Sey (see ﬁg. 28). One may note that while in Hg1−x Fex Te and Hg1−x Fex Se the Van Vleck paramagnetism is present, the mixed-anion crystals exhibit Curie-like paramagnetism.

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Fig. 28. Magnetic susceptibility of Hg1−x Fex Te1−y Sey at cryogenic temperature (after Mycielski et al. (1996)).

To close the paragraphs dealing with cadmium chalcogenides containing Fe let us mention surface photovoltage measurements performed in Cd1−x Fex Te by Kuzminski (1999), Mössbauer spectroscopy studies reported by Abramishvily et al. (1997), NMR experiments of Gavish et al. (1993), and millikelvin resistance investigations of Cd1−x Fex Se:In by Głód et al. (1994b). The exciton splitting induced by the s,p–d exchange interactions in Zn1−x Fex Te, x < 0.005, was studied by polarized magnetoreﬂectance (Mac et al., 1996a). Combining the results with magnetization data, a difference between the s–d and p–d exchange constants was obtained. Testelin et al. (2000) performed similar measurements. The magnetization of Fe2+ ions was quantitatively analyzed in the framework of the crystal ﬁeld and spin–orbit coupling model. The six Zeeman-like components of the free exciton spectrum were observed, in Faraday or Voigt conﬁguration. A comparison between the energy splittings and the magnetization data yielded the values of the conduction electron and valence electron exchange parameters. The same approach, i.e., complementary measurements of magnetization and excitonic magnetoreﬂectance in the region of the fundamental energy gap were used by Twardowski et al. (1992) to study the exchange interaction between band electrons and localized delectrons of Fe in cubic Zn1−x Fex Se (x < 0.06). The low-temperature magnetization was studied in detail in cubic Zn1−x Fex Se with x up to 0.007 by Fries et al. (1994). The data were collected in magnetic ﬁelds H along the [100] and [111] directions using a SQUID magnetometer for H 55 kOe and a vibrating sample magnetometer operating in a Bitter magnet for ﬁelds up to 200 kOe. At the lowest temperatures the magnetization at high magnetic ﬁelds is anisotropic. The difference between the values of magnetization for these two directions reaches a maximum value of 19% at the lowest temperatures and when H is about 150 kOe. As the temperature rises,

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the anisotropy of the high ﬁeld magnetization decreases gradually, becoming very small above 40 K. In low ﬁelds, H 10 kOe, the anisotropy is very small in the whole temperature range. These experimental results are generally in good agreement with theoretical calculations based on crystal ﬁeld theory and on the assumption that only isolated Fe2+ ions (singles) contribute to the magnetization. Twardowski et al. (1993a) measured the magnetic susceptibility of Zn1−x Fex S for x < 0.03 ﬁnding a typical Curie–Weiss behavior above 100 K. Based on the high-temperature series expansion developed for diluted magnetic semiconductors containing Fe, they estimated the exchange interaction constant between the nearest-neighbors, JNN = −37 K. Chih-Ming and Der (1999) presented results of a Raman scattering spectroscopy study of Zn1−x Fex Se within a wide range of composition, x < 0.66, under pressures up to 35 GPa. Preceding the semiconductor–metal phase transition, a visible anomaly of the TO Raman mode splitting was observed. For example, the Fe local mode, present between the pure ZnSe LO and TO modes, exhibited blue shift behavior before the metallization and disappeared as the pressure was higher than the metallization pressure. Mountziaris et al. (1996) reported epitaxial growth of high-quality Zn1−x Fex Se ﬁlms with Fe compositions x 0.22 using metalorganic vapor phase epitaxy (see also Peck et al. (1997)). The epilayers were characterized by X-ray diffraction, Raman, absorption, and X-ray photoelectron spectroscopies. Typical growth rates were from 3–4 μm/h, i.e., signiﬁcantly higher than those obtained by molecular beam epitaxy. Metastable excitons in ZnSe/Zn1−x Fex Se quantum-well structures were investigated by means of reﬂectance and photoluminescence experiments (Yu et al. (1997b); see also Warnock et al. (1995)). These states are associated with the type-II (−1/2, −3/2) ground state exciton of the system. The metastable excitons are formed because the Coulomb attraction of the mj = −1/2 electron tends to localize the mj = −3/2 hole in the vicinity of the electron. This complex breaks apart as the temperature increases, and the system returns to its ground state in which the electron and the hole are segregated. In 1991, in the Physical Review Letters 27 issue, vol. 67, two papers appeared reporting successful MBE growth and magneto-optical veriﬁcation of existence of a “spin superlattice” (see von Ortenberg (1982)). The ﬁrst paper by Dai et al. (1991) described the ZnSe/Zn1−x Mnx Se system. The other one, by Chou et al. (1991), dealt with ZnSe/Zn0.99Fe0.01Se superlattices with alternating non-magnetic and magnetic layers. In such a system a spin-dependent potential exists, tunable by an external magnetic ﬁeld. The authors showed that the ﬁeld-induced spin splittings in both valence and conduction bands can become much larger than the residual zero-ﬁeld potentials and that such systems exhibit spin superlattice behavior (see ﬁg. 29). Magnetoreﬂectance experiments at liquid helium temperature were used to investigate the nature of these structures, verifying through ﬁeld-dependent spin splitting and transition strengths that they are in fact true spin superlattices (see ﬁg. 30). Jonker et al. (1994) studied similar system – a quantum well in which the carrier–ion exchange interactions are determined by the spin of the carrier (see also Fu et al. (1993)). The samples consisted of alternating layers of Zn1−x Mnx Se and Zn1−y Fey Se. In zero ﬁeld, the carriers initially interact randomly with both transition metal species. When a magnetic ﬁeld is applied, the excitonic wave functions are increasingly localized in one or the other of the magnetic layers according to their spin state as the competing spin exchange interactions deﬁne the conﬁning potential. The spin components of the heavy hole exciton

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Fig. 29. Upper: Schematic diagram of the band alignment of a Zn0.99 Fe0.01 Se/ZnSe/Zn0.99 Fe0.01 Se quantum well. The zero ﬁeld conduction band offset is greatly exaggerated for the sake of clarity. Lower part: Calculated electron and heavy-hole wave functions for a single quantum well structure, plotted vs. distance along the growth direction. (a) B = 0, (b) B = 8 T, m = −1/2 electrons, m = −1/2 heavy holes, (c) B = 8 T, m = +1/2 electrons, m = +3/2 heavy holes (after Chou et al. (1991)).

Fig. 30. Energy of the heavy-hole exciton spin components of a single quantum well structure plotted vs. magnetic ﬁeld. Squares – (−1/2, −1/2)σ + transition; triangles – (+3/2, +1/2)σ − transition (after Chou et al. (1991)).

are subsequently dominated by different exchange interactions as revealed by their temperature and ﬁeld dependence: the behavior of the spin-down component (−3/2, −1/2) is described by exchange interactions of the carriers with the Mn2+ ions and exhibits Bril-

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louin paramagnetic behavior, while the spin-up component (+3/2, +1/2) is dominated by interactions with Fe2+ ions and exhibits Van Vleck paramagnetism. These structures are thus characterized by an initial competition and eventual coexistence of Brillouin- and Van Vleck-like paramagnetic behavior for the exciton. Chemical ordering was observed in Zn1−x Fex Se epilayers with x ≈ 0.5 as well as in nominal (ZnSe/FeSe) superlattices along the [001] growth direction and the [110] direction using transmission electron microscopy (K. Park et al. (1992), see also K. Park et al. (1996)). The ordered structure consists of alternating ZnSe and FeSe layers along the [001] and [110] directions. In nominal (ZnSe/FeSe) superlattices grown on (001) GaAs substrates, strain-induced interdiffusion between the layers takes place followed by ordering of the resultant Zn1−x Fex Se alloys. This ordered structure corresponds to the CuAu-I type structure. Computer simulated images for a Zn0.5Fe0.5 Se compound were obtained and compared with experimental images. To the authors’ knowledge, this was the ﬁrst observation of ordering in a II-VI alloy. 5.4. Narrow gap materials During the last decade, investigations of narrow gap II-VI DMS were practically limited to the studies of Hg1−x Fex Se and based on Hg1−x Fex Se quaternary crystals. The interest in Hg1−x Fex Se system is connected with the fact that a substitutional iron atom forms a resonant donor state whose energy is superimposed on the conduction band continuum. The resulting anomalous behaviors of the electron scattering rate, i.e., strong enhancement of the electron mobility (or drop of the Dingle temperature), which occur in Hg1−x Fex Se at low temperatures in a certain Fe concentration range are described in the review of Kossut and Dobrowolski (1993). In the review article of Dobrowolski (1996) the interested reader may ﬁnd information about investigations performed till the middle of this decade. Since then, only a few papers on this subject were published. Let us mention ﬁrst a paper of Zeitler et al. (1996) who measured perpendicular and parallel components of the magnetization of Hg1−x Fex Se in the strongly dilute limit (x < 10−3 ) in magnetic ﬁelds up to 20 T. Since in this compound the overall magnetization arises simultaneously from Fe3+ (Brillouin paramagnet), Fe2+ (Van Vleck paramagnet), and free electrons (diamagnetic de Haas–van Alphen effect), a torque magnetometer was used in order to determine individually the various contributions with their anisotropy. For very low iron content (x < 2.4 ×10−4), the Fe donors exclusively exhibit an isotropic Brillouin paramagnetism of noninteracting Fe3+ ions. For higher concentration Fe2+ also exist. Coexisting with the Brillouin paramagnetism of Fe3+ , the Fe2+ reveal an induced Van Vleck-type paramagnetism with a crystal ﬁeld-induced anisotropy. This anisotropy was analyzed by measuring the induced magnetic moment perpendicular to the magnetic ﬁeld when applying the ﬁeld in a nonsymmetric direction of the crystal. In contrast to higher concentration samples (see Dybko et al. (1995)), both the Brillouin paramagnetism of Fe3+ and the Van Vleck paramagnetism of Fe2+ can be attributed to the sum from individual Fe2+ donors with no magnetic interaction between them. Soskic et al. (1997) investigated low-ﬁeld ac magnetic susceptibility of Hg1−x Fex Se with x < 0.095 in the temperature range 12–150 K. Available data are compared with the extended nearestneighbor pair approximation calculations taking into account the long-range antiferromagnetic exchange interaction between Fe2+ ions in the form J (R) = JNN /R n . A good agreement was found between the theory and experiment with 11 KY |JNN | 15 K and

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n = 5. Hillberg et al. (1997) used zero and longitudinal ﬁeld μSR to examine the Fe2+ singlet ground state magnetism in Hg1−x Fex Se. For x 0.05, μSR experiments indicated a spin glass behavior which is limited to sample regions with locally enhanced Fe2+ content. Longitudinal ﬁeld spectra in the glassy state revealed a static local ﬁeld of ≈ 50 mT due to induced moments of Fe2+ . These were proposed to originate from a distribution of bound magnetic polaron energies. The ﬁrst successful epitaxial growth of Fe-doped HgSe layers was reported by Widmer et al. (1995a), see also Widmer et al. (1995b) and Schikora et al. (1995). The authors prepared pure HgSe and HgSe:Fe single layers and HgSe-HgSe:Fe microstructures on ZnTe/GaAs substrates. The structural quality of the layers was determined by high resolution X-ray diffraction. Electronic properties of HgSe and HgSe:Fe were studied by Hall effect and Shubnikov–de Haas (SdH) measurements. The low-temperature electron mobility in heavily Fe-doped HgSe appeared to be signiﬁcantly higher than in undoped layers revealing evidence for the well known correlation effects of Fe2+ donors in HgSe. The SdH-signal of HgSe-HgSe:Fe microstructures showed pronounced oscillations when the magnetic ﬁeld is oriented normal to the layers and a complete disappearance of the oscillations for the magnetic ﬁeld oriented parallel to the layers, indicating the existence of a new type of microstructure which we have called scattering superlattice. Infrared transmission in epitaxial Hg(Fe)Se layers in pulsed magnetic ﬁelds up to 150 T and static ﬁelds up to 12 T was investigated by Portugall et al. (1995). The dominant resonant structures were assigned to the cyclotron resonance of charge carriers in HgSe bulk states but also a new type of transitions not observed previously in bulk samples was detected. Szuszkiewicz et al. (2000) investigated MBE grown HgSe:Fe samples by X-ray diffraction and Raman spectroscopy. The crystal quality and the strain state of the epilayers were determined by X-ray diffraction measurements. Raman scattering spectra obtained for the layers appeared to be very similar to those taken for the Fe-doped HgSe bulk samples. The observation of a small frequency shift of the TO-phonon structure in various layers conﬁrmed the presence of strain effects. 5.5. Ti, Ni, and Sc impurities Ion implantation was successfully used to synthesize Zn1−x Nix Te (Ando et al., 1998). Ni ions were implanted into a ZnTe ﬁlm with acceleration energies from 30 to 390 keV to achieve a Ni density of 3×1020 cm−3 . Magnetic circular dichroism spectra showed that the Ni ions substitute for Zn ions and induce clear Zeeman splittings of the optical transitions. Analyses of the obtained spectra proved that the p–d exchange in Zn1−x Nix Te is antiferromagnetic. Cd1−x Nix Te thin ﬁlms were grown by Alvarez-Fregoso et al. (1996) using radio frequency sputtering from a target of CdTe and nickel compressed powders. The structural and electrical properties were studied as a function of the atomic nickel concentration in the ﬁlms with x = 0.05, 0.10, and 0.15. X-ray diffraction patterns showed a cubic CdTe parent structure with a (111) preferential orientation. From scanning electron microscopy micrographs, a ﬁne granular morphology with a random distribution of grain sizes in the ﬁlms was observed. Waldmann et al. (1993) investigated local magnetic behaviors of isolated Fe and Ni impurities in the two II-VI semiconductors ZnS and ZnTe directly after recoil implantation. From the measured magnetic properties, which were detected by the observation of the

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perturbed angular distribution, orbital contributions to the hyperﬁne ﬁeld as well as contributions from spin magnetism were found to be present at both impurities. An interpretation made on the basis of the intermediate ligand ﬁeld model suggested that the Fe and Ni ions are found to exist in 2+ and in 1+ states and that electronic excitations may play a signiﬁcant role even microseconds after the implantation. Finally, let us mention a paper by Głód et al. (1994a) in which the electronic structure of the ground state associated with a resonant Sc donor in CdSe and Cd1−x Mnx Se (see ﬁg. 8 in the review of Kossut and Dobrowolski (1993)) was investigated by the use of far-infrared reﬂectivity and millikelvin magnetoresistance measurements. Similarities in the results for samples doped either with Sc or with In, particularly in the spectral region of the 1s-2p excitations and near the metal–insulator transition, demonstrated that the ground state of the Sc impurity is hydrogenic. Thus, the Coulombic part of the potential of a resonant donor impurity can create hydrogenic states in the forbidden energy gap. The presented data provided support for the two-ﬂuid model of electronic states in the vicinity of the metal–insulator transition. Acknowledgements This work was partially supported by the State Committee for Scientiﬁc Research project No. 2 P03B 154 18 and within European Community program ICA1-CT-2000-70018 (Center of Excellence CELDIS). The authors are grateful to dr. M. Arciszewska for careful reading the manuscript and for her numerous remarks. References Abramishvily, V.G., Komarov, A.V., Ryabchenko, S.M., Savchuk, A.I., Semenov, Yu.G., 1997, Solid State Commun. 101, 397–402. Abramof, E., Faschinger, W., Sitter, H., Pesek, A., 1994, J. Cryst. Growth 135, 447–454. Abramof, E., Rappl, P.H.O., Ueta, A.Y., Motisuke, P., 2000, J. Appl. Phys. 88, 725–729. Abramof, E., de Andrada e Silva, E.A., Ferreira, S.O., Motisuke, P., Rappl, P.H.O., Ueta, A.Y., 2001, Phys. Rev. B 63, 085304-1-8. Adachi, N., Inoue, M., Mogi, I., Kido, G., 1995, J. Phys. Soc. Japan 64, 1378–1384. Adachi, N., Hirano, J., Yamazaki, T., Okuda, T., Kitazawa, H., Kido, G., 1998, Magnetization process in Cr substituted diluted magnetic semiconductor ﬁlms, in: Proc. High-Density Magnetic Recording and Integrated Magneto-Optics: Materials and Devices Symposium. Materials Research Society, Warrendale, pp. 633–638. Adachi, S., Takagi, Y., Takeyama, S., 2000, J. Cryst. Growth 214–215, 819–822.

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chapter 4

MAGNETIC ORDERING PHENOMENA AND DYNAMIC FLUCTUATIONS IN CUPRATE SUPERCONDUCTORS AND INSULATING NICKELATES

HANS B. BROM Kamerlingh Onnes Laboratory, Leiden University, NL 2300 RA, the Netherlands

JAN ZAANEN Lorentz Institute for Theoretical Physics, Leiden University, NL 2300 RA, the Netherlands

Handbook of Magnetic Materials, edited by K.H.J. Buschow Vol. 15 ISSN: 1567-2719 DOI 10.1016/S1567-2719(03)15004-4

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© 2003 Elsevier Science B.V. All rights reserved

CONTENTS Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 2. Microscopic theories of stripe formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 2.1. What causes stripes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 2.2. The early work: mean-ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 2.3. Improving on mean-ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 2.4. Stripes, valence bonds and superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 2.5. Considerations based on the t–Jz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 2.6. The “exact” numerical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 2.7. Frustrated phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 2.8. The role of the dynamical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 3. The magnetic resonance: theoretical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 3.1. The facts to be explained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 3.2. Symmetry (I): Spin modes and the coexistence phase . . . . . . . . . . . . . . . . . . . . . . . . . 413 3.3. Symmetry (II): SO(5) and the π particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 3.4. Restoring the fermions: the various viewpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 3.5. The resonance peak and spin-charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 3.6. The resonance and superconductivity

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3.7. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 4. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 4.1. NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 4.2. μSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 4.3. Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 4.4. ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 4.5. STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 4.6. Thermodynamic techniques, like susceptibility and heat capacity . . . . . . . . . . . . . . . . . . 442 5. The hole-doped single-layer 214-cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 5.1. Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 5.2. μSR and NMR/NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 5.3. Results of other magnetic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 5.4. Charge sensitive techniques – X-ray, conductance and ARPES . . . . . . . . . . . . . . . . . . . 453 5.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 6. Oxygen and strontium doped 214-nickelates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 6.1. Phases and structural changes under oxygen and Sr doping . . . . . . . . . . . . . . . . . . . . . 456 380

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6.2. Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 6.3. μSR and NMR/NQR and magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 6.4. Charge sensitive techniques – X-ray, conductance and ARPES . . . . . . . . . . . . . . . . . . . 461 6.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 7. The electron-doped single-layer compound Nd2−x Cex CuO4+y

. . . . . . . . . . . . . . . . . . . . . 463

7.1. Charge sensitive data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 7.2. Spin sensitive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 7.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 8. The hole-doped double-layer 123- and 124-compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 468 8.1. Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 8.2. μSR and NMR/NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 8.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 9. Other multilayered cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 9.1. Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 9.2. μSR and NMR/NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 9.3. Charge sensitive techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 9.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 10. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Appendix A. Static and dynamic properties of 2D Heisenberg antiferromagnets . . . . . . . . . . . . . . . 484 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Abbreviations 1D 2D 2DQHAF 123(-compound) 214(-compound) a† a ARPES B bi† bi Bi2212 † c¯iσ c¯j σ D DMFT DOS DRMG d-wave superconductivity δ δni,↑ Ea EFG F (q) F (ω) † fiα fiα FLEX g G χ0 χP χ χ χmolar

one-dimensional two-dimensional 2D quantum Heisenberg antiferromagnet (doped) YBa2 Cu3 O7 (doped) La2 CuO4 creation operator annihilation operator angular resolved photo emission spectroscopy magnetic ﬁeld (inductance) spinless boson creation operator boson annihilation operator Bi2 Sr2 CaCu2 O8 projected fermion creation operator projected fermion annihilation operator space dimension dynamic mean ﬁeld theory density of states density matrix renormalization group pairing in ground state with d-wave symmetry oxygen doping level change in occupation number operator for spin-up activation energy p64 crystal electrical ﬁeld gradient Form factor Frequency distribution function fermion creation operator fermion annihilation operator ﬂuctuation exchange (depending on the context) control parameter or the Landé g-factor differential tunneling resistance static spin-susceptibility Pauli spin-susceptibility real part of the spin susceptibility imaginary part of the spin susceptibility molar spin-susceptibility 382

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χFL χNAFL γe γn Γ HMM-10 HTC I IC J J j Jb Jz K kF LDA LDOS LTT LTO Mz MFL MMP model μ0 μB μSR ni,↑ , ni,↓ N NA NAFL-model NMR NQR O(N) ωSF QHAF Q RNG RVB ρi ρs S σ Sp(2N)

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susceptibility of a Fermi liquid susceptibility of a nearly-antiferromagnetic Fermi liquid electron-spin gyromagnetic ratio nuclear-spin gyromagnetic ratio center Brillouin zone Johnston in Handbook of Magnetic Materials 10 high-Tc superconductor nuclear spin angular momentum incommensurate exchange coupling constant exchange between spins on both sides stripe exchange constant between bilayers exchange between Cu in bilayer exchange term in the Ising hamiltonian Knight shift Fermi wave vector local density approximation p29 local density of states low temperature tetragonal low temperature orthorhombic magnetization along the z-axis marginal Fermi liquid model for the dynamic susceptibility by Millis, Monien and Pines (the same as the NAFL-model) vacuum permeability Bohr magneton muon spin rotation occupation number operator for spin-up and spin-down number of spins Avogadro’s number nearly antiferromagnetic Fermi liquid model (the same as the MMP-model) nuclear magnetic resonance nuclear quadrupole resonance Orthogonal symmetry group frequency relaxational spin mode quantum Heisenberg antiferromagnet antiferromagnetic wave vector real space renormalization group resonant valence bond site occupation density for both spin directions spin stiffness spin angular momentum spin label symplectic group

384

STM SQUID SU(N) s-wave superconductivity t T T0 T∗ Tc T1 T2 T2R T2G t–J model t–Jz model Tl2212 Tl2223 TC TN TS U U (N) V i Vxx vF W x xopt ξ z

H.B. BROM and J. ZAANEN

scanning tunneling microscopy superconducting quantum interference device special unitary group pairing in ground state with s-wave symmetry transfer integral temperature temperature where K starts to change temperature where 63 (T1 T )−1 peaks superconducting transition temperature nuclear spin–lattice relaxation time nuclear spin–spin relaxation time Redﬁeld contribution to T2 Gaussian part of the echo decay model using t and J as parameters model using t and Jz as parameters Tl2 Sr2 CaCu2 O8 Tl2 Sr2 Ca2 Cu3 O10 charge order temperature Néel ordering temperature spin order temperature on site Coulomb repulsion unitary group inter-site repulsion electrical ﬁeld tensor component Fermi velocity transition probability (often) Sr doping level optimal doping level correlation length (often) dynamical scaling exponent

1. Introduction The rich phase diagram of the so-called high-Tc superconductors (Bednorz and Muller, 1986), i.e., superconductors with surprisingly high superconducting transition temperatures Tc , as function of doping strongly stimulated the interest in the interplay between antiferromagnetism and superconductivity. The parent compounds La2 CuO4 and YBa2 Cu3 O6 of the high-Tc superconductors (HTC) La2−x Srx CuO4+δ and YBa2 Cu3 O6+δ , are twodimensional (2D) Heisenberg antiferromagnetic insulators. The spins reside mainly on the Cu ions and are antiferromagnetically aligned along the CuO-axes. The two-dimensional nature arises from the fact that the electronic spins responsible for the magnetism reside in the transition metal oxide layers, see ﬁg. 1.1, with a negligible coupling between the adjacent planes. The electric insulating character of La2 CuO4 (or 214-compound) and YBa2 Cu3 O6 (or 123-compound) has a different origin from that of conventional insulators. In ordinary

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Fig. 1.1. The CuO2 layers in the high-Tc compounds. The charges are mainly located on the oxygen sites (open circles). The electronic spins responsible for the magnetism reside mainly on the Cu ions (arrows) and are antiferromagnetically aligned along the CuO axes. In La2 CuO4 the antiferromagnetic CuO2 planes are sandwiched between two non-magnetic La2 O2 layers. In YBa2 Cu3 O6 , CuO2 double-layers share an yttrium site located in the center of the two Cu-squares. As in the single-layered cuprates the double layers are separated by non-magnetic layers of Ba2 O2 and a layer of Cu atoms (so-called Cu(1) sites, in contrast to the Cu(2) sites of the double layer). For La2−x Srx CuO4 the dimensions for the enlarged orthorhombic unit cell are 0.536, 0.540 and 1.316 nm (x = 0), and 0.350, 0.535, and 1.321 nm for x = 0.1. Above x = 0.21 the structure is tetragonal with a unit cell of 0.377 and 1.325 nm (HMM10). For YBa2 Cu3 O6+x the (quasi-)tetragonal unit cell dimensions are 0.385 and 1.183 nm for x = 0 and 0.388, 0.383 and 1.163 nm for x ∼ 1, with a distance between the two CuO2 layers of the double layer of 0.335 nm. These cell dimensions are T dependent: for x = 0.8 the c-axis changes from 1.182 to 1.178 nm, while the a, b axes shrink from 0.3851 to 0.3847 nm (Rossat-Mignod et al., 1991a, 1991b, 1992). For the Bi2 Sr2 Can−1 Cun O2n+4 family we only discuss the n = 2 compound Bi2 Sr2 CaCu2 O8 or Bi2212. The structure is simply related to Bi2201, by replacing the CuO2 monolayer by a CuO2 /Ca/CuO2 sandwich. The structure is pseudo-tetragonal and has a Tc of 85 K. Tc can be controlled by changing the oxygen concentration or by substituting Y or other rare-earth elements for Ca. As grown samples are close to optimal doping. Similar structures can be made with Tl in stead of Bi. The c-axis of the Bi2212 or Tl2212 compounds is 2.932 nm and for a triple layer, like Tl2223, 3.588 nm.

insulating systems, the electric conductivity is hindered by the Pauli exclusion principle. The motion of electrons is forbidden when the conduction band is ﬁlled, i.e., the contributing atomic orbitals all donate two electrons. Strongly correlated electron systems, of which the parent compounds of the HTCs are good examples, turn insulating due to a strong repulsion between electrons and are known as Mott–Hubbard insulators (Gebhard, 1997). The insulating and antiferromagnetic properties of the 214 and 123 parent-compounds disappear rapidly upon chemical doping. The doping is regarded as injecting holes or electrons into the two-dimensional antiferromagnetic layers. Holes are primarily located on the p orbitals of the four oxygens surrounding the central copper in the CuO2 plane and compensate the copper spins in so-called Zhang–Rice singlets (Zhang and Rice, 1988), see ﬁg. 1.2. In practice, hole doping is primarily achieved by either replacing some of the La3+ ions by (2+)-ions (Sr2+ or Ba2+ ), by adding excess oxygen to stoichiometric La2 CuO4 or by both methods.

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Fig. 1.2. Schematic diagram for the Zhang–Rice singlet that shows the hybridization of the Cu and O orbitals, after Zhang and Rice (1988). The four oxygen-hole states can form either a symmetric or antisymmetric state with respect to the central Cu-ion. The (+) and (−) signs represent the phase of the p- and d-states wave-functions. Both symmetric and antisymmetric states may combine with the d-wave Cu hole-state to form either singlet- or triplet-spin states. The symmetric state forms bonding and antibonding states, where the large binding energy in the singlet state is due to the phase coherence. The antisymmetric state is non-bonding. The energy of two O holes residing on the same square is much higher than the energy of two separated O holes: two holes feel a strong repulsion on the same square. Note that the localized states are not orthogonal to each other, because the neighboring squares share a common O-site. Zhang and Rice argue that the physics remains essentially unchanged if neighboring squares are taken into account.

Fig. 1.3 gives the phase diagram as function of doping for the 214 and 123 cuprates. The ﬁrst phase is the antiferromagnetic region, bounded by the Néel ordering temperature TN , and is theoretically well understood (Batlogg and Varma, 2000). A summary of the theory behind these Heisenberg antiferromagnets is given Johnston in the Handbook of Magnetic Materials, vol. 10 (Johnston, 1997), to which we refer to as HMM-10 (in Appendix A we brieﬂy repeat the most important aspects). At higher doping levels the system moves into a superconducting phase, for which at the optimal doping level xopt ≈ 0.15 the maximum Tc is achieved. The optimal level separates the underdoped and overdoped regimes respectively below and above xopt , see ﬁg. 1.3. The 214 and 123 compounds can be doped from 0 (antiferromagnetic phase) up to the overdoped regime. The pristine multilayered compounds with Bi, Tl, or Hg are often already close to optimally doped. Especially in the underdoped regime, subtle properties of the speciﬁc compound determine the particular phase at zero temperature (T = 0 K). The ground state at T = 0 K as function of the control parameter g and doping level x is sketched in ﬁg. 1.4, and is generic for hole-doped high-Tc superconducting compounds. The control parameter g will depend on the spin stiffness and coupling to the lattice. In the 2212 compounds doping levels are typically above 0.15 and samples go from a superconducting to a Fermi surface phase. The 214 and 123 compounds allow lower doping levels, where a coexistence phase of stripes and superconductivity is possible. Stripes, see ﬁg. 1.5, might be described as rivers of charge in an antiferromagnetic background. Only

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Fig. 1.3. Phase diagram for high-Tc compounds showing the transition temperatures as function of hole doping. In La2−x Srx CuO4+δ the doping level is due to doping by either Sr2+ , excess oxygen or both and is quantiﬁed as x + 2δ. The symbol Tc refers to the superconducting transition temperature. TN is the boundary for the Néel-ordered antiferromagnetic phase. The 214- and 123-compounds can be doped from 0 (antiferromagnetic phase) up to the overdoped regime. The doping levels of the multilayered compounds with Bi, Tl, or Hg are often in the optimally doped and overdoped regime, while strongly underdoped values are hard to realize. Especially in the underdoped regime, subtle properties of the speciﬁc compound determine the particular phase at T = 0 K, see also ﬁg. 1.4.

Fig. 1.4. Generic phase diagram at T = 0 K for hole-doped high-Tc superconducting compounds showing the ground state as function of a control parameter g (which will depend on the spin stiffness and coupling to the lattice) and doping level x. In the 2212 compounds doping levels are typically above 0.2 and samples go from a superconducting to a Fermi surface phase. The 123-compounds allow lower doping levels, where a coexistence phase of stripes and superconductivity is possible. Only in La2−x Srx CuO4+δ the stripe phase exists in a large x regime. At low doping levels the magnetic phase remains insulating. Spin and charge separation occurs as in the insulating nickelates (one might speak from classical stripes as the quantum character of strong stripe ﬂuctuations has become less important). Higher doping leads to a conducting stripe and superconducting coexistence phase, where the charges are more delocalized than in the nickelates. In the heavily doped regime signatures of stripes disappear. By adding Nd the low-temperature orthorhombic structure of the 214-compounds is transformed to low-temperature tetragonal and the pinning of stripes is promoted, while superconductivity is suppressed.

in La2−x Srx CuO4+δ the stripe phase exists in a large x regime. At low doping levels the magnetic phase remains insulating. Spin and charge separation occurs as in the insulating nickelates (we might speak of classical stripes). Higher doping leads to a conducting stripe

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Fig. 1.5. Possible stripe structures in the hole-doped 2-dimensional CuO2 layer. (Upper panel) Deﬁnition unit cell and directions. Open circles represent oxygen sites and ﬁlled circles 3d-ions. The unit cell in the CuO2 plane is formed by connecting the four nearest neighbor oxygens surrounding the Cu-ion (the Zhang–Rice conﬁguration, see ﬁg. 1.2). The drawn square on the right is the enlarged unit cell. (H, V) refers to the orientation of a horizontal or vertical domain wall. B refers to a bond-centered, D to a site-centered diagonal domain wall. (Lower panel) Schematic diagrams for an ideal diagonal static-striped-phase ground-state at 1/3 hole doping, as suggested theoretically by Zaanen and Gunnarson (1989) and observed later experimentally for hole-doped La2 NiO4 (Tranquada et al., 1995, 1997b). The charged domain walls can be either site-centered (left) or bond-centered (right). In the case of diagonal stripes with hole doping of 1/3 for site centered domains the domains are two spins wide, and the spins within the domain are equivalent and coupled by exchange interaction (J ), while spins on both sides of the stripe are aligned antiparallel with exchange J < J . For oxygen or bond centered stripes, the domains are 3 spins wide, and an uncompensated moment might appear. The magnetic moments on sites adjacent to a charge stripe are likely to be reduced in magnitude compared to those in the middle of the domain. Perfect compensation is not expected. In both cases the phase of the antiferromagnetic order shifts by π on crossing the domain wall, hence for a bond-centered stripes spins adjacent to the stripe are ferromagnetically aligned.

and superconducting coexistence phase, where the charges are more delocalized than in the nickelates. In the heavily doped regime signatures of stripes disappear. By adding Nd the low temperature orthorhombic structure of the 214-compounds is transformed to low temperature tetragonal phase and the pinning of stripes is promoted, but superconductivity is suppressed. HMM10 contains a thorough discussion of two-dimensional antiferromagnetism (brieﬂy summarized in Appendix A) and the properties of the following important two-dimensional Cu-monolayer compounds Sr2 CuO2 Cl2 , La2 CuO4 , Ca0.85 Sr0.15CuO2 , Nd2 CuO4 , the related linear chain compound (Ca, Sr)2 CuO3 and spin ladder compounds. At the end also the effects of phase separation and stripe formation after oxygen and Sr doping in La2 CuO4

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are presented. Especially in this area progress has been considerable during the last couple of years. We will update Johnston’s review by describing the fascinating phenomena in hole doped La2 CuO4 (section 5) and La2 NiO4 (section 6), and electron doped Nd2 CuO4 (section 7) regarding the stripe phases, which have appeared since 1997. We will also cover the CuO-multilayer compounds (sections 8 and 9), where the collective magnetic mode, the so-called Resonance Peak (Rossat-Mignod et al., 1991a, 1991b; Mook et al., 1993; Fong et al., 1995; Bourges et al., 1996), is the most intriguing topic from a magnetic point of view. Section 4 gives details of the experimental techniques, like ARPES (Angular Resolved Photo Emission Spectroscopy), STM (scanning tunneling microscopy), μSR (muon spin rotation), and especially NMR and NQR (resp. nuclear magnetic and quadrupole resonance) with relevant examples; details about neutron scattering are to be found in HMM-10. High-Tc superconductivity is an unsolved problem. The two theoretical sections 2 and 3 are dedicated to two aspects of the problem which have clearly to do with magnetism: the static stripes and the magnetic resonance. These sections are not intended as an exhaustive review of everything ‘magnetic’ of relevance to the problem of high-Tc superconductivity. We have just focused on the two aspects of the problem where we feel that in recent years a sense of consensus has emerged. In section 3, we deliberately focus on what can be called strong, static stripe order. In fact, the case can be made that the ‘strong’ stripe orders as they occur in nickelates, manganites and so on are in essence understood. In the cuprates this is less clear cut. In some regards, the static stripes in cuprates look quite like nickelate, etcetera, stripes but in other regards they are still mysterious. This has to be related to the fact they live in close proximity to the superconductors. Quantum effects are supposedly far more important and one can wonder if these are really understood. The next step is the idea of dynamical stripes. This corresponds with the notion that even in the fully developed superconductors correlations of the stripe kind persist, but only in some ﬂuctuating form. In ﬁrst instance, this notion should be viewed as an empiricism: some data are just quite suggestive in this regard and due attention will be paid to dynamical stripes in the experimental sections. However, for the theorists it is not an easy subject. Although quite a number of interesting ideas have emerged it is at present not easy to present them in a coherent fashion. It is the kind of subject matter which is more suited for monograph style reviews and several of these are right now under construction. Two such pieces are available: Zaanen et al. (2001), with an emphasis on the dynamical stripe issue, and Carlson et al. (2002) with a primary focus on how superconductivity and stripes hang together. Both reviews are largely complementary to what is presented here. High-Tc superconductivity is also a highly controversial subject. Considering (dynamical) stripes to be at one end of the controversy, the theoretical literature associated with the magnetic resonance is clearly at the opposite end. Much theoretical work has been devoted to the latter, and this will be reviewed in section 3. The resonance is a dynamical phenomenon which goes hand in hand with the superconductivity itself. While (dynamical) stripes are understood as a highly organized form of electron matter, having no evident relationship whatever with the Fermi-gas wisdoms of conventional metal physics, the resonance ﬁnds a surprisingly detailed explanation in terms of this fermiology language. Although the underlying interpretations can be quite different, the punch line is that the physics of the resonance can be successfully traced back to the existence of a Fermi-surface, undergoing

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a BCS instability into a d-wave superconductor. In fact, this section can also be read as a tutorial introduction of the main-stream theories of high-Tc superconductivity. The notion that the mechanism causing the superconductivity has to do with ‘something magnetic’ has been very popular all along, like in the approaches where magnetic ﬂuctuations behave like phonons, or in the more exotic SO(5) and spin-charge separation theories. In this mindset, the resonance appears as the “magnetic glue” becoming visible in experiment. We hope that by taking together sections 2 and 3 the reader acquires some understanding of the depth of the high-Tc mystery. Pending on the focal point of the experimental machinery, one seems to enter completely different universes (stripes, fermiology) which might appear even to be mutually exclusive. We have a strong suspicion that this has little to do with the incompetence of experimentalists or with theorists being carried away by their wild fantasies. Instead, we like to interpret it as the “theory of everything in highTc superconductivity” signaling to us that humanity has not quite ﬁgured out yet the real nature of the thing. Johnston (1997) lists relevant reviews about the magnetic and other properties of highTc superconductors, which have been presented before 1997. Of the new reviews that have appeared since then, we mention the work of Imada et al. (1998) about the metal-insulator transition. Starting from correlation effects, the authors describe the various theoretical approaches to the unusual metallic states and to the metal-insulator transition. The treatise contains a special section devoted to high-Tc cuprates, which includes a discussion of YBa2 Cu3 O7−y and Bi2 Sr2 CaCu2 O8+δ . Regarding NMR and NQR on stripes, the article of Hunt et al. (2001) might serve as a review as well. The reader likely will also enjoy the very accessible articles of Goss Levi (1998), Brooks (1999), Emery et al. (1999), Zaanen (1998, 2000b), Laughlin et al. (2000), Buchanan (2001), and Laughlin and Pines (2000). 2. Microscopic theories of stripe formation To avoid ambiguities, stripes should be in ﬁrst instance associated with the highly organized form of electron matter as encountered in for instance the nickelates (section 6) but also the manganites. These stripes should be appreciated as a generic order associated with the doped Mott-insulator, in the same sense that antiferromagnetism belongs to the Mottinsulating state itself. As it turns out, the cartoon picture of “rivers of charge” separated by Mott-insulating magnetic domains is surprisingly literal in these systems. Fully developed stripe order of this kind goes hand in hand with a strong insulating electrical behavior, and as such they are quite opposite to superconductors. The stripes in the cuprates, even the static ones, are a bit special. They live in close proximity, or even coexist with superconductivity, which has to imply that the stripe order as found in, e.g., the nickelates is to a further or lesser extent compromised in the “stripy” cuprates. In this regard we strongly recommend the newcomers to this ﬁeld to have a close look at the nickelate stripe systems ﬁrst, because they offer a simple setting to get used to the basic physics. It all starts out with the Mott-insulator itself, which is quite different from the band insulator. Even the problem of a single hole in the Mott-insulator is non-trivial, see Dagotto (1994) and references therein. When the hole moves through the dynamical spin system

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it causes severe spin frustrations, which in turn frustrates the free motion of the hole. In fact, the single hole can only propagate as a quasiparticle due to the presence of quantum spin ﬂuctuations repairing the damage in the spin background. However, when the density of holes is ﬁnite they can help each other repairing the spin damage by concerting their motions. This is in simple words the origin of the complex ordering phenomena called stripes. 2.1. What causes stripes? Stripes were discovered theoretically shortly after the discovery of superconductivity and long before they were seen experimentally, Zaanen and Gunnarson (1989) (this paper was delayed for ∼1.5 years by the refereeing process!). There is no such thing as a full proof theoretical method in this branch of science and one always relies on theoretical toy models which are treated in a more or less uncontrollable approximation. The original prediction was based on the standard Hubbard toy-models, treated in a most conventional fashion: Hartree–Fock electronic mean-ﬁeld theory. We will discuss this at some length in section 2.3 because it is conceptually interesting, while Hartree–Fock is actually quite successful in the description of the stripes of the nickelate variety. Subsequently, it became clear that various approximate methods designed to improve quantitatively on Hartree– Fock share the generic tendency towards stripe formation (section 2.3). Implicitly, Hartree– Fock and its descendants are controlled by large spin: it is designed to describe “classical” antiferromagnetic order, and the quantum ﬂuctuations in the spin system expected to be important for S = 1/2 are just neglected. Instead one can start in the limit dominated by these ﬂuctuations and this is theoretically achieved by generalizing the models to SU(N) (or Sp(2N)) symmetry, to consider the limit that N is large while S is small. Sachdev and coworkers (1999) discovered that also in this limit stripe like ordering phenomena occur, going hand in hand with superconductivity, and these will be discussed in section 2.4. In section 2.5 we will turn attention to the various attempts to acquire further insights using the hole- and spin language of the t–J model, to continue in section 2.6 with a discussion of the numerical “exact” results with an emphasis on the DMRG (density matrix renormalization group) work by Scalapino and White (1998). In the remaining sections we will focus on the bits and pieces of the physics which are ignored in the Hubbard- and t–J model for no good reasons: long-range Coulomb-interaction (section 2.7) and the electron–phonon interaction (section 2.8). Although initially perceived as offering competing explanations for stripe formation, more recently a consensus has emerged that these effects add up to the Hubbard model physics, to cause stripes to be even more ubiquitous. 2.2. The early work: mean-ﬁeld theory The idea that inhomogeneity could be intrinsic to strongly correlated electron systems has been around for some time, particularly in the Russian school (Kateev et al., 1998). However, these ideas were rather handwaving and stripes came into existence in the late 1980s, when Zaanen and Gunnarson (1989) discovered the stripes in the mean-ﬁeld theory of Hubbard models. It was directly realized that these “charged antiferromagnetic domain walls” should be viewed as close relatives of the Su–Schrieffer–Heeger type solitons (and holons) of one-dimensional physics (Zaanen and Gunnarsson, 1989). The simple mean-

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ﬁeld calculations got this novel and crucial aspect of “topological doping” right, following the terminology introduced by Kivelson et al. (1998). In hindsight this is not too surprising because Hartree–Fock is controlled by a powerful principle. It is often misinterpreted in the quantum-chemist” fashion, as a not so good quantitative approximation of the interaction term in a small system Hamiltonian. In physics it has a much stronger meaning: it is the theory telling how electrons collectively manage to cause “conventional” order, like spin density waves (antiferromagnets), charge density waves (Wigner crystals) and BCS superconductors. As an example, let us consider the well-known Hubbard hamiltonian † = −t aiσ aj σ + aj†σ aiσ + U ni↑ ni↓ , H (2.1) ij σ

i

describing a system of tight-binding fermions (creation and annihilation operators ai† resp. ai , spin label σ , and occupation number operator ni , all for site i) with a transfer integral t, subjected to a local repulsion ∼U . Except for the one- and inﬁnite-dimensional cases, this Hamiltonian cannot be solved exactly. However, in the case that electron systems break a symmetry of eq. (2.1) spontaneously, vacuum expectation values develop; broken (charge) translational symmetry means ρi = ni↑ + ni↓ = 0 and broken spin rotational invariance 2Siz = ni↑ − ni↓ = 0. Given that the order exists, the number operators appearing in the full hamiltonian turn into scalars, implying that the interaction term turns into a single particle self-consistent potential, thereby simplifying the problem to an extent that it becomes generally solvable. Write niσ = niσ + δniσ , and it follows that, U (2.2) ni↑ ni↓ = U (ni↑ ni↓ + ni↓ ni↑ − ni↑ ni↓ + Hﬂuct . i

i

This is just a rewriting of the interaction term, but the crucial observation is that Hﬂuct = U i δni↑ δni↓ describes the ﬂuctuations around the ordered state. The implication is that in the presence of order, the mean-ﬁeld hamiltonian obtained by neglecting Hﬂuct is a zeroth order Hamiltonian having the property that its ground state can be adiabatically continued to the true ground state: the perturbation theory in terms of Hﬂuct is converging. Therefore, mean-ﬁeld theory rests on the presence of order. According to a popular myth it would only be good for small interaction strength, U/t 1. This is nonsense. One can easily show that at half-ﬁlling and for large U one recovers the magnetic insulator: just compare the total energies of the antiferromagnet and the ferromagnet to ﬁnd that they differ by the superexchange J = t 2 /U , while the paramagnetic metal has a very high energy, ∼U/2. Schrieffer et al. (1989) demonstrated that one recovers linear spin-wave theory when Hﬂuct is evaluated on the Gaussian level. In fact, Hartree–Fock is controlled by the “smallness of h¯ ”, translating into the largeness of spin (S → ∞) in the Mott-insulators, as becomes particularly obvious in the ﬁeld-theoretic (path-integral) treatment, see Fradkin (1989). Energy is minimized in the half-ﬁlled system by the uniform antiferromagnet. However, there is no obvious reason for this to be the case in the doped systems. It is actually so that the uniform mean-ﬁeld solutions of the doped state are characterized by a negative compressibility indicating that these are unstable. This is not only true for the collinear magnets but also for the spin spiral states (see Dombre, 1990; Stojkovic et al., 1999) as suggested originally by Shraiman and Siggia (1989) which are also based on Hartree–Fock.

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Fig. 2.1. An electron stripe according to the early mean-ﬁeld calculations by Zaanen and Gunnarson (1989). These authors considered a three band Hubbard model, describing a Cu–O perovskite plane, including the oxygen 2p- and the Cu 3dx 2 −y 2 states. The arrows indicate the spin density on the Cu atoms in the insulating domains, while the circles represent the probability to ﬁnd the excess holes on the oxygens. For convenience, the (small) hole density on the Cu atoms is not shown. Notice the anti-phase boundariness: upon crossing the stripe the up spins move from, say, the A sublattice to the B sublattice.

As ﬁrst realized by Schrieffer et al. (1988a, 1989), even for the simple case of a single hole the mean-ﬁeld ground state is rather non-trivial: a self-trapped solution is found called the “spin bag”. Schrieffer’s conjecture was that the spin bag would give rise to a pairing mechanism, along the same lines as the bipolaron mechanism: two holes together can dig a deeper spin “hole” than two holes separately, and this acts like an attractive pairing force. However, as was discovered by Zaanen and Gunnarson (1989), energy is further minimized by forming many particle bound states: the stripes, see ﬁg. 2.1. These consist of linear Mott-insulating domains characterized by collinear Néel spin order, separated by lines of holes which are at the same time Ising type domain boundaries in the antiferromagnet: the staggered order parameter changes sign every time a charge stripe is crossed. Finally, every domain wall unit cell contains a net single hole. These are quite literally like the stripes of the nickelates (see section 6). Where is this coming from within this mean-ﬁeld “paradigm”? The key is that these solutions are topological (the domain wall property) and these stripes can be viewed as just two-dimensional versions of the well-known solitons associated with the semi-classical sector of one-dimensional physics, as realized in electron–phonon systems, see Heeger et al. (1988). As Mukhin (2000) and Matveenko and Mukhin (2000) demonstrate, in the one-dimensional “large-S limit” Hubbard-chains, the process of soliton formation is in essence the same as in the electron–phonon models. All one needs is a “parent” system

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Fig. 2.2. Explanation of the soliton (“holon”) found in the mean-ﬁeld solutions of the doped Hubbard chain, according to Matveenko and Mukhin (2000). The 1D version of the texture indicated in ﬁg. 2.1 translates into a kink in the staggered magnetization, giving rise to a mean-ﬁeld potential trapping the hole. The electronic structure (right panel) is characterized by upper- (UH) and lower (LH) Hubbard bands, associated with the electrons in the “magnetic domains”. The “holon” is associated with a state lying in the middle of the gap, being occupied by the excess hole.

which is insulating because of the potentials associated with a staggered order parameter: the Néel order parameter plays a role equivalent to the dimerization wave in polyacetylene. Matveenko and Mukhin (2000) consider an ansatz for the staggered order parameter mz = (−1)i Siz ∼ tanh(k0 x) describing a kink or soliton such that the order parameter is pointing in the opposite z directions away from the origin while it is vanishing at the origin (k0 determines the width), see ﬁg. 2.2. Diagonalizing the (linearized) mean-ﬁeld Hamiltonian for this potential yields a state centered at the middle of the gap. At half-ﬁlling this state is occupied by one electron, and it carries a net spin 1/2 quantum number (localized “spinon”). This is a relatively costly excitation. However, by adding one hole this near mid-gap state becomes unoccupied and the soliton becomes the ground state: the cheapest way to remove the hole is by taking it out of a state as close as possible to the middle of the gap, while at the same time a soliton proﬁle costs less order parameter gradient energy than a polaronic/“spin-bag” proﬁle. This state carries only a charge e quantum and no spin: the localized “holon”. Subsequently, Matveenko and Mukhin (2000) demonstrate that close to half-ﬁlling a periodic lattice is formed of these solitons, while at high doping and weak interaction strength a phase transition follows to a single harmonic Peierls wave. In summary, the mean-ﬁeld ground states of one-dimensional doped Mott insulator are soliton lattices which can be viewed as the 1D version of the ordered stripes. The 2D case is mathematically more complicated and no closed analytical solutions are available. However, by inspection of the numerical results the qualitative picture was already obtained in the early Zaanen–Gunnarsson paper. The one-dimensional solutions tell that much energy can be gained by binding the holes to the magnetic domain “walls”. However, there is clearly no space for the 1D “holons” in the 2D space, because these

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Fig. 2.3. According to mean-ﬁeld theory, one can view the stripes in two dimensions as an extension of the holons of the one-dimensional case, see Zaanen and Gunnarson (1989) and Zaanen and Ole´s (1996). In ﬁrst instance one should consider the motions perpendicular to the stripes and these form mid-gap states of the same kind as in 1D. However, since their energies are the same moving along the stripe, the holes are delocalized in this direction and a “mid-gap” band is formed with a dispersion depending on the momentum k// running along the stripe. In the case of “empty” stripes, this mid gap band is empty and stability is gained because the Fermi energy lies in the gap between the lower Hubbard band and the mid-gap. When the stripes is “half-ﬁlled”, the mid gap band is at quarter-ﬁlling and this might drive a on-stripe 4kF (ﬁg. 2.4) or 2kF 1D-like density wave instability. According to mean-ﬁeld, states of this kind are at least locally stable.

would cause inﬁnitely long strings of spin mismatches. However, this spin frustration is avoided by putting the “holons on a row”, thereby forming stripes. In terms of the electronic states, this means that the motion of the electrons perpendicular to the stripes causes midgap states (see Zaanen and Ole´s, 1996). However, the electrons can now also propagate along the stripes, and these mid-gap states hybridize in a “mid-gap band”: ﬁg. 2.3. The width of this band is less than the Mott gap and by having a density of one hole per domain wall unit cell, this mid gap band is empty. Hence, the thermodynamic potential is in the gap between the occupied lower Hubbard band and the empty mid-gap band and this new band gap is responsible for the stability of the stripe phase. These early papers triggered a number of follow ups, investigating these matters in further detail (Kato et al., 1990; Schulz, 1989, 1990; Poilblanc and Rice, 1989; Inui and Littlewood, 1991; Verges et al., 1991). Altogether, these studies made clear that the mean-ﬁeld stripes are quite robust and uniquely characterized by a number of properties: (a) their topological nature is at the heart of the phenomenon, (b) starting from the standard Hubbard-type problems, mean-ﬁeld stripe phases are insulating, (c) this insulating character goes in hand with a charge quantization rule stating that one hole stabilizes one domain wall unit cell (“empty” stripes), (d) it costs little energy to bend stripes, to reorient them from horizontal/vertical- to diagonal directions, etc. Of special interest is the work by

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Fig. 2.4. Like ﬁg. 2.3 except that now the stripe is half-ﬁlled (note that a ﬁlled stripe carries the same spins as in the domains and is charge neutral). As Zaanen and Ole´s (1996) pointed out, the stripe band is quarter ﬁlled and one obtains metastable solutions by invoking on stripe 4kF (ﬁgure) or 2kF (see ﬁg. 2.7) density waves causing a gap at EF inside the 1D like mid-gap band.

Viertio and Rice (1994), showing that the kinetic energy of kink motion can easily overcome the cost of creating a kink. Hence, it appears that “meandering” ﬂuctuations of stripes can be quite important for both the thermal- and quantum melting and this inspired studies on the nature of the dynamical stripes (e.g., Zaanen et al., 2001). Theory preceded the experimental fact and the above was quite instrumental in guiding Tranquada and coworkers (Tranquada et al., 1995) in the process which lead to the discovery of stripes in nickelates (section 6). However, nature came up with a surprise when the same group discovered the stripes in the La2−x−y Srx Ndy CuO4 system: at doping levels x < 1/8 charge appears quantized but now 1 hole is associated with two domain wall unit cells, while for x > 1/8 charge quantization seems to disappear altogether. This is often quoted as proving that mean-ﬁeld theory falls short fundamentally for the description of the cuprate stripes, and this is a too strong statement. As Zaanen and Ole´s (1996) showed, Hartree–Fock solutions exist corresponding with half-ﬁlled stripes (see also Bosch et al., 2001). Although “empty” stripes have a lower energy, these half-ﬁlled stripes are locally stable and they have quite low energies: one could argue that the problem is not so much in the approximation, but more in the choice of effective theory. More importantly, a commensurate “internal” (on-stripe) charge density wave is required for stability, causing a gap in the spectrum of mid-gap states, see ﬁg. 2.4. Hence, according to mean-ﬁeld also the half-ﬁlled stripes should be insulating which seems at ﬁrst sight in contradiction with the metal-like transport found in the striped cuprates, although it offers a natural explanation for the on-stripe charge quantization seen in the regime x < 1/8. Attempts to explain this with metallic stripes are not quite convincing (e.g., Nayak and Wilczek, 1997).

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2.3. Improving on mean-ﬁeld As we already discussed, straightforward mean-ﬁeld theory has in ﬁrst instance a qualitative use. To become quantitative one has to evaluate the ﬂuctuations with perturbation theory and this is a difﬁcult task. These ﬂuctuations come in two varieties: (a) the longwavelength ﬂuctuations which refer to the highly collective motions associated with the order itself, like spin ﬂuctuations/magnons but also the stripe meanderings, (b) the local ﬂuctuations which have to do with the problem that Hartree–Fock cannot be considered to be a faithful representation of the electron dynamics on the lattice scale. Eventually, the faith of stripes is decided on the microscopic scale and to arrive at meaningful quantitative statements one should go beyond Hartree–Fock. Several methods emerged to improve the short distance aspects. In order of increasing sophistication, these are the Gutzwiller meanﬁeld and the closely related local-ansatz, the Gutzwiller Ansatz and the modern dynamical mean ﬁeld-theory (DMFT). These methods have all been tried out on the stripe problem, conﬁrming that stripes are quite robust. The intrinsic weakness of Hartree–Fock is that it rests on a single Slater determinant description of the electronic wavefunction. The central ingredient of Hubbard type problems is that for large U the Hilbert space associated with the low energy dynamics is projected: conﬁgurations containing doubly occupied sites are at high energy. The only way to recover the correct Hilbert space within the limitations of a single determinant description is by having a large order parameter because the unwanted states are automatically projected out by the order parameter potential. Instead, one can attempt to directly attack this problem by brute force using the conﬁguration-interaction approach of quantum chemistry: see Louis et al. (2001) for an interesting recent attempt in the stripe context. In classical statistical physics (and bosonic quantum problems) mean-ﬁeld theory becomes always exact in the limit of inﬁnite space dimensions (D → ∞). Based on earlier work by Metzner and Vollhardt (1989), Georges and Kotliar (1992) discovered that in inﬁnite dimensions fermionic problems simplify. However, their “time axis stays active” and they are described in terms of an interacting impurity problem (Kondo-type problems) coupled self-consistently to a fermionic bath which is itself constructed from the solutions of the single impurity problem. In the dynamical mean-ﬁeld theory one uses the exact results of inﬁnite dimensions to calculate matters in ﬁnite dimensions: it amounts to an excellent treatment of the local physics while it inherits from its D → ∞ origin the complete neglect of the long wavelength ﬂuctuations. The Gutzwiller Ansatz, as well as the closely related local-ansatz can be considered as approximations to the DMFT. For a recent review we refer to Georges et al. (1996). The ﬁrst application of this methodology to the stripe problem is the work by Giamarchi and Lhuillier (1990) employing the Gutzwiller Ansatz. For recent extensions we refer to Yanagisawa et al. (2002), Seibold et al. (1998), Sadori and Grilli (2000), as well as Gora et al. (1999) for work based on the local Ansatz. The highlight is the work by Fleck et al. (2000, 2001) solving the DMFT equations for a large Hubbard model supercell, allowing full variational freedom with regard to the charge and spin ordering. It suggests a picture which is in some regards different from the Hartree–Fock outcomes. First, the doping dependence seems identical to the experiments: for x < 0.05 empty diagonal stripes are most stable, switching to half-ﬁlled horizontal stripes for x > 0.5 which extend up to x = 1/8. In the doping range 0.125 < x < 0.20 the stripe ordering wave vector does not change any

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longer, indicating that the charge density inside the stripe is no longer commensurate. Finally, stripes disappear in the overdoped regime, x > 0.20. Secondly, these DMFT stripes are charge-compressible: no energy gap is found in the spectrum and there is no tendency to form internal density waves on the stripes. Thirdly, Fleck et al. (2000, 2001) show that the DMFT self-energy shufﬂes the mean-ﬁeld spectrum considerably, and the result is a one electron spectral function which starts to resemble the experimental results. In summary, the DMFT calculations offer a striking similarity to experiment although it is not always clear where it is coming from. 2.4. Stripes, valence bonds and superconductivity Are stripes good for, or even causing the superconductivity, or is it that the stripes are hindering the superconductivity? At long wavelength, stripe and superconducting order do always compete in a restricted sense. As explained in section 3.2, they can coexist in the sense that the same electrons divide their time between the stripes and the superconductivity. However, stripe order also means charge order and this can only happen at the expense of the superconductor. Superﬂuid density is at maximum when all bosons can move freely and by partly crystallizing the bosons one takes away part of these free motions. Another issue is, however, could it be that stripes help the formation of these bosons, the Cooper pairs? Several theoretical groups came up with tentative answers to this question and these can be summarized as: stripes are a way to organize the resonating valence bond (RVB) mechanism of high-Tc superconductivity by Anderson (1987). Anderson conjectured that the natural quantum competitor of the antiferromagnet is a state which is constructed from bound pairs of S = 1/2 spins: the valence bonds or spin-dimers. One can cover the lattice with many conﬁgurations of these dimers and he envisaged that due to strong quantum ﬂuctuations these dimers would form a featureless quantum ﬂuid: the “resonating valence bond state”. Upon doping with holes, the spins turn into electrons which can move around, but because their spins are bound in pairs, the electrons are bound in Cooper pairs and the doped state is therefore a superconductor characterized by a large pairing energy ∼J . What is the relationship with stripes? This relationship has been at the center of the thinking of Kivelson and Emery and coworkers, and we refer to their review for a detailed exposition, see Carlson et al. (2002). Let us just discuss the idea of the “spin-gap proximity effect”, Emery et al. (1997). The charge stripes are assumed to be charge compressibility, and to zeroth order their internal electron systems are viewed as independent one dimensional Luttinger liquids. The next assumption is that the spin system living on the magnetic domains is a quantum paramagnet: the ground state is a singlet, separated by an absolute energy gap from the lowest triplet excitations. These two systems are not completely disconnected because the electrons conﬁned in the charge stripes can tunnel in and out the Mott-insulating domains. The tunneling probability will be higher for pairs of Luttinger liquid electrons, because these can enter the domains as pair singlets which directly “ﬁt” into the spin vacuum, while single electrons have to overcome the spin gap energy because they carry S = 1/2. Hence, since electron singlet pairs move more freely than single electrons, there is a pairing force at work of a kinetic origin, following the spirit of the RVB idea.

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Vojta and Sachdev (1999) and Vojta et al. (2000) arrived at yet another “RVB-ish” viewpoint. Although based on a limit (“large-N ”) which is not necessarily of physical relevance, it puts a strong emphasis on the quantum physics of the spin system. Let us review in some detail this perspective which should be regarded as complementary to the large-S limit. How to approach the limit where the quantum ﬂuctuations of the antiferromagnet are as important as can be imagined? One option is to increase the SU(2) spin symmetry to a larger symmetry group. Keeping the “spin” magnitude S small, the larger symmetry will cause stronger ﬂuctuations. For instance, one can consider SU(N) or the Sp(2N) generalizations of the Heisenberg model, both reducing to SU(2) when N = 2. It was discovered in the 1970s in high energy physics that in the limit N → ∞, keeping S ﬁnite, saddle points arise of quite a different nature than the large-S saddlepoints. This notion became later popular also in condensed matter physics, playing a prominent role both in heavy fermion physics, quantum magnetism and high-Tc superconductivity, see Auerbach (1994). In the large-N limit, at the moment that holes are introduced in the spin system superconductivity emerges. How does this work? The starting point is the well known t–J model, for present purposes extended with a nearest-neighbor repulsion, † Ht −J −V = J (2.3) Si · Sj + t c¯iσ c¯j σ + V ni nj ij

ij

ij

describing a system of electrons on a square lattice, under the constraint that there is either a hole or an electron present on every lattice site, while two electrons are forbidden to occupy the same site. The spins of neighboring electrons interact via a Heisenberg exchange (ﬁrst term) with strength J , while their charges interact via the (non-standard) repulsion V . In the presence of holes, these electrons can delocalize as described by the t term, under the constraint that no double occupancy occurs: the c¯† is not a normal fermion creation operator but takes care of this projection which amounts to a great complication. The large-N limit is now constructed as follows. The spins are described by fermions fα† , α = 1, 2N , transforming under the fundamental of Sp(2N). The holes are described by spinless bosons b such that the Sp(2N) “electron” annihilation operator becomes ciα = fiα bi† and the local constraint of the t–J model becomes † α fiα fi + bi† bi = N.

(2.4)

This is the usual operation in the “slave” theories, corresponding with the mathematical machinery underlying the idea of spin-charge separation (see also section 3.5). The constraint equation (2.4) acts locally and this can be enforced by adding a Lagrange multi † α plier term to the action ∼ i λi (fiα fi + bi† bi − N). Such a constraint is very hard to handle at ﬁnite N where it requires the introduction of U (1) gauge ﬁelds. However, in the limit N → ∞ the local constraint becomes global λi → λ. When this condition is fulﬁlled, it becomes straightforward to construct mean-ﬁeld theories. The “holons” (b’s) and spinons (f † ’s) become literal hard-core bosons and fermions. Nothing can prevent the bosons to condense (except crystallization) such that bi = 0. Since the bosons carry electrical charge this is a Meissner phase characterized by a large energy scale ∼xt (x is doping). Hence, high-Tc superconductivity arises naturally and since the original proposal by Baskaran et al. (1987) this has attracted much attention. The standard view on these matters is further discussed in section 3.5.

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Fig. 2.5. The phase diagram of the t–J –V model suggested by Vojta and Sachdev (1999). The y-axis should be interpreted qualitatively: large y is like the large-N (dashed line) where the calculations are performed and it is asserted that the spin-Peierls order (inset) changes into antiferromagnetism when N is reduced (M broken). The bars in the insets indicate two-spin singlets (“valence bond pairs”), condensing in columnar spin-Peierls at half-ﬁlling. As function of increasing doping δ the columnar spin-Peierls stays intact in the Mott-insulating domains, which are now separated by bond centered charge stripes which are build from local d-wave like pairs. These stripes are compressible and accordingly the state as a whole is a supersolid, showing charge order (“C broken”) and superconductivity. For increasing doping the stripes move closer together (compare the two insets on the right), to disappear at a high doping where the system changes into a uniform d-wave superconductor.

Besides the charge Bose-condensation, the spin system as encoded in the fermions † † (“spinons”) tends to BCS type (d-wave) pairing instabilities, fiσ fj σ = 0. The standard mean-ﬁeld phase diagram as discussed in section 3.5 is derived assuming spatial uniformity of both the holon superﬂuidity and this spinon d-wave pair condensate. This uniformity assumption is challenged by Vojta and Sachdev (1999): in the N → ∞ limit spatially non-uniform states have a lower energy and these states are quite similar to the stripes of the large-S limit! One should ﬁrst get the state of the half-ﬁlled insulator right. As Read and Sachdev (1989, 1990) showed, in the large-N limit energy is gained when the pair amplitudes break in addition translational invariance, relative to the uniform d-wave BCS-like states. These can be viewed as higher dimensional realizations of the spin-Peierls order known from one-dimensional physics. Sp(2N) (or SU(N)) singlets are formed from nearest-neighbor spins and these “dimers” or “valence bonds” are stacked in columns covering the lattice, see ﬁg. 2.5. Symmetry-wise, it is indistinguishable from a bond centered charge density wave, characterized by an energy (singlet-triplet) gap because it is commensurate with the lattice. Alternatively, it can be viewed as a “frozen out” form of Anderson’s RVB state. It appears that these spin-Peierls orders are not unique to the large-N limit. Sachdev and coworkers (Read and Sachdev (1990), and especially Sachdev and Park (2002)) demonstrate that these also merge starting from the large-S side. This is an elegant mechanism, rooted in the Berry phase. Although the Berry phase can be ignored in ordered co-linear

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antiferromagnets, it becomes active when the spins disorder quantum-mechanically. The “dynamics” of the Berry phase maps in this case on an effective electrodynamics which can be shown to describe spin-Peierls order. These considerations suggest that the spin-Peierls state is a generic competitor of collinear antiferromagnetic order, and it should therefore be also of relevance to frustrated quantum-antiferromagnets. Although still controversial, in several numerical studies on J1 –J2 Heisenberg models evidences are reported in favor of spin-Peierls order (du Croo de Jongh et al., 2000; Sushkov et al., 2001). What happens when such a spin-Peierls spin state is doped? Qualitatively, the situation is similar to the large-S case. The background state is protected by a commensuration gap, and since the holes tend to drive the system away from commensuration, these are expelled. The magnetic domains are now “built” from the dimers, and since these wants to stack in columns, stripes result according to the calculations by Vojta and Sachdev (1999), see ﬁg. 2.5. As compared to the large-S stripes there are still important differences. First, the high doping state which is locally realized inside the charge stripes corresponds now with a d-wave like superconductor. This can be viewed as a superposition of a pair of holes and a spin dimer, ∼U + Vfi† fj† bj bi . This intra-stripe superconductivity turns into an overall superconductivity due to inter-stripe Josephson coupling and one obtains an overall state which can be called a supersolid: it is a coexistence state of (charge) crystalline order (spin-Peierls and the stripe density modulation) and superconductivity (with s + d symmetry because the system is effectively orthorhombic). A second obvious difference is that these large-N stripes “live on the links”: the charge stripes are bond centered, while the insulating domains prefer a width corresponding with an even number of sites in order to maintain the spin-Peierls registry. As Vojta and Sachdev (1999) show, these ingredients add up to produce at least in an average sense an incommensurability which is proportional to the hole density in the underdoped (x < 1/8) regime, saturating at higher dopings caused by the diminishing strength of the stripe order. Summarizing, the large-N theory incorporates in a natural way superconductivity although it is not easy to account for the antiferromagnetism and the topological order. Instead, it rests on the spin-Peierls order which has never been seen in experiments. The superconductivity originates in the strong quantum spin-ﬂuctuations, being neglected in Hartree–Fock. Martin et al. (2001) showed recently that it is possible to stabilize stripesuperconductivity coexistence states in Hartree–Fock. However, such states are only realized in the presence of very large nearest-neighbor attractions. Nature is somewhere in the middle and it is quite signiﬁcant that the large-N and large-S stripes are quite similar: interpolating between the two does not require much imagination. Not so long ago, spin-charge separation and stripes were seen as opposing, mutually exclusive viewpoints, but in the light of the latest results this attitude can no longer be maintained. It makes clear that the tendency to form inhomogeneous structures supersedes even the differences between insulators and superconductors (charge sector) or antiferromagnets and valence bond phases (spin sector). The theoreticians should be aware that their preference for homogeneous states is in ﬁrst instance based on the ease of computation, and not on universal principle. The latest example might well be the ﬂux phases as advocated by for instance Wen and Lee (1996) and the closely related d-density wave order suggested by Chakravarty and Kee (2000). According to Chernyshyov and Wilczek (2002), stripes should also be formed upon doping an insulator of this kind and preliminary

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numerical results by Marston and Sudbö (2002, unpublished) suggest that such a kind of phenomenon indeed occurs in their ﬂux phase ladder systems, Marston et al. (2002). Finally, it even seems that spins are not necessary for electronic stripe formation: Hotta et al. (2001) showed recently that the stripes in manganites have likely more to do with orbital order than with spin order. 2.5. Considerations based on the t–Jz model Up to this point we have reviewed the outcomes of various mean-ﬁeld approaches. These theories are always controlled by limits and because nature is in the middle they are not quite trustworthy. Is it possible to see where stripes are coming from, without invoking mean-ﬁelds, i.e., in terms of the real motions of holes and spins? This is a very complicated task and only computer calculations (the subject of the next section) can be conclusive. Nevertheless, one can obtain quite some qualitative insights by considering simpliﬁed cases. A common denominator in these attempts is to consider in ﬁrst instance Ising spins instead of Heisenberg spins. This simpliﬁes matters considerably and the XY terms can subsequently be considered as quantum “corrections” changing the situation not too drastically. The most important contribution in this regard is the early one due to Prelovšek and Zotos (1993). Their considerations were motivated by a numerical study of a small t–J cluster, where they were the ﬁrst to ﬁnd indications for stripe correlations in the context of the t–J model. They proposed a simple picture for stripe formation in terms of holes moving through the Ising spin system, followed by a large scale numerical study suggesting that the basic mechanism is of relevance for the full model. In its bare form, this picture should be taken as a cartoon which is nevertheless most useful if it is handled with care. Let us ﬁrst consider what happens when a single hole hops through an Ising antiferromagnet; the Hamiltonian is just the t–J model, eq. (2.3) omitting the XY terms in the spin part (“t–Jz model”). The outcomes are well known (see Dagotto, 1994) one direction, a spin moves in the opposite direction and after a couple of hops one ﬁnds that the hole has left behind a string of misaligned spins, see ﬁg. 2.6 (a), (b). For every line element of this “magnetic string” an energy penalty ∼4Jz has to be paid and this implies conﬁnement: the hole cannot move far away from its origin, costing a considerable kinetic energy. It is believed that this magnetic string effect survives the quantum spin ﬂuctuations: the XY terms “destroy” the string by ﬂipping back the spins, thereby allowing the hole to propagate. However, this implies that the kinetic energy scale ∼J instead of the ∼t of the free particle, as conﬁrmed by numerical simulations of the one hole problem. Hence, the motions of an isolated hole are frustrated by the antiferromagnetic spin system. Prelovšek and Zotos (1993) suggested that at ﬁnite densities the holes can “help each other” to avoid these frustrations. The recipe is simple: take as a starting conﬁguration a domain wall occupied by holes, in other words a stripe. When a hole moves transversally to the stripe, the spin moves backwards meeting an antiferromagnetic-neighbor (ﬁg. 2.6(c)). Move the neighboring hole, and one sees two kinks appearing which propagate freely (∼t) when these processes are repeated (ﬁg. 2.6(d)). These kinks have in turn the effect of changing the position of the stripe in space and if they are freely created and annihilated the stripe as a whole will delocalize over the plane. The conclusion is that although the motions of a single hole are impeded by the spins, the stripe as a whole can meander rather

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Fig. 2.6. Cartoon pictures explaining the basic reason why stripes are formed according to the t–J model. When a hole is injected in a 2D S = 1/2 antiferromagnet (a), it leaves behind a string of ﬂipped spins when its delocalizes, the “magnetic string” (b). These violate the anti-parallel registry of the spins, thereby frustrating the delocalization of the hole. At ﬁnite hole densities, the holes can coordinate their motions in such a way that they help each other to avoid the frustrating magnetic strings. Start out with localizing the holes on a domain wall (c). An individual hole can now move sideways without causing a misoriented spin, and by repeating these hops one ﬁnds kinks propagating along the stripe (d) which will cause the stripe as a whole to move freely over the plane, unimpeded by the spin system.

freely. This means a gain in kinetic energy, although a price has to be paid because every hole has to “watch” its neighbors. One can view it in yet another way: stripes are “holons on a row”, but now interpreted in a more literal fashion as compared to the mean-ﬁeld case (section 2.2). In one dimensional physics spin-charge separation has a precise meaning and one can understand it using simple cartoons. Above all, this simple picture makes clear that at least in the language of the t–J model stripes are formed because quantum-mechanical delocalization energy is gained, analogous to the kinetic energy gain causing antiferromagnetic superexchange interactions between the spins in the Mott-insulator. The above consideration has in common with the mean-ﬁeld theories discussed in the previous sections that the stripes are caused by severe microscopic quantum ﬂuctuations. This is in marked contrast with the frustrated phase separation mechanism discussed in section 2.7: this is just based on potential energy. Consider the row in ﬁg. 2.6 (a), (b) where the hole moves, but now in isolation. One ﬁnds that a spin domain “wall” (or kink) is left at the origin while the object which moves is dressed up by the spin anti-kink. The spin-kink carries S = 1/2 and no charge, while the moving object carries charge but no spin. These turn out to be quite accurate representations of the spinons and the holons doing the real work in one dimensional physics. One could now view ﬁg. 2.6 (a), (b) as an illustration of the local tendency in 2D towards the 1D type spin-charge separation. However, the frustrations in the spin-system translate into forces “conﬁning” the

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holons in strings. The antiphase-boundariness of the stripes is then interpreted as the 2D manifestation of the topological nature of the 1D excitations. The question to what extent quantum physics is important for the formation of stripes is obviously important and it is still not completely settled. One immediately infers from the simple cartoon picture that the antiphase boundariness is at the heart of the quantummechanism: the kinetic energy is only released when the stripe is a domain wall. How to make it accessible for experimentalists? A quantitative measure for the importance of the quantum processes is the exchange interaction J between the spins located at opposite sides of the stripe. This interaction is surely mediated by the quantum motions of the holes; Pryadko et al. (1999) even proved that the antiphase boundary property cannot be caused by long wavelength physics. As the spin-wave calculations indicate (Zaanen and van Saarloos, 1997; Batista et al., 2001), a maximum in the spin-wave dispersions should occur at the (π, π) point at a scale set by J and both the data by Lee et al. (2002) on the nickelate and by Mook et al. (2002) in the underdoped 123 system indicate that this number is large, of order of the superexchange at half-ﬁlling. The picture by Prelovšek and Zotos (1993) inspired quite a number of other studies. It forms the starting point of a considerable effort aimed at the understanding of the dynamical stripes, which is beyond the scope of this review. Let us just mention the lattice string models constructed by Eskes et al. (1996, 1998) (see also Zaanen et al. (2001), Hasselmann et al. (1999) and Dimashko et al. (1999)) to study generic features of the kink-driven stripe delocalization. Another issue is, how to deal with half-ﬁlled or even metallic stripes in this language? The Prelov˘sek–Zotos picture addressed empty stripes, and it is much less clear how to think about “populated” domain walls. Various interesting ideas have emerged. An intriguing question was asked by Nayak and Wilczek (1996): accepting that empty stripes are bound states of holons, why is it so that half-ﬁlled stripes are bound states of electrons (i.e., equal amounts of charge e and S = 1/2 per unit length)? Zaanen et al. (1998) assumed the half-ﬁlled stripes in the underdoped regime (x < 1/8) to be insulating because of a 4kF internal density wave, to subsequently consider the possibility that this state is doped with additional holes for x > 1/8. They argued that this doped stripe should renormalize into a three component Luttinger liquid, characterized by additional bosonization ﬁelds associated with the transversal motions of the stripe, next to the usual spin- and charge ﬁelds. Bosch et al. (2001) suggested that these transversal kinks associated with the dopants could be responsible for the “Y-shift” discussed in section 5. Yet another approach has been followed by Tchernyshyov and Pryadko (2000) and Chernyshev et al. (2000). The starting point is the t–Jz model, but the focus is initially on low hole density: an Ising spin system is considered with a domain wall present and the dynamics of a single hole is investigated. When the “hole” moves along the domain wall it turns into a holon, dressed up by a transversal kink. It can only leave the domain wall by taking up spin to turn into the usual spin-polaron quasiparticle. Chernyshev et al. conjecture that one can take the single hole propagator of the dilute limit to calculate what happens at high on-wall hole concentrations. This is further substantiated by numerical (density matrix renormalization group) calculations by Chernyshev et al. (2002), arriving at the important conclusion that the superconductivity and quantum spin ﬂuctuations can be viewed as processes which are somewhat secondary compared to the “primary” t–Jz

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physics causing the stripes. We also mention the quantum Monte-Carlo study by Riera (2001), quantifying the scale where the antiphase boundariness sets in. Finally, Zachar (2000) challenges the assertion by Chernyshev et al. that domain walls are stabilized in the t–Jz model at low on-wall hole density. He is arguing instead that at very low hole concentrations in phase domain walls are more stable. 2.6. The “exact” numerical approaches The theory of strongly interacting electrons cannot be controlled by theoretical means and the only way to achieve deﬁnite results is often by numerical means. This is not an easy task either. One can attempt a direct diagonalization on a ﬁnite size cluster. The problem is that the size of the Hilbert space grows very fast (exponentially) with the system size and the largest system one can handle in this way is a t–J model deﬁned on a 20 site cluster. Although exact diagonalization studies have been quite important in the context of the problem of one hole in the antiferromagnet (see Dagotto, 1994), the largest exact diagonalization clusters are too small to address stripes meaningfully. The intrinsic length scales are of the same order as the linear dimension of the cluster and the physics gets lost in ﬁnite size effects. A breakthrough occurred some years ago due to White and Scalapino, using the density matrix renormalization group (DMRG) method invented by White (1992). “Renormalization group” is a bit of misnomer: this should be seen as a very efﬁcient method to isolate the states in Hilbert space which are of importance to the ground state. One-dimensional problems can be solved numerically to any desired accuracy. However, two-dimensional t–J type problems are at the frontier of this technology and the method can not be straightforwardly applied. Nevertheless, White and Scalapino (1998a, 1998b, 1999, 2000a, 2000b), presented a rather convincing case that the ground state of the t–J model in the physically relevant parameter regime is about stripes. As we will discuss in some length, these stripes are quite like the entities one would expect in between the large-S and large-N limits. With the DMRG method quite large systems can be handled and a typical example is shown in ﬁg. 2.7, with dimensions exceeding the stripe length scales (the stripe width, and separation) sufﬁciently. White and Scalapino (1998a, 1998b) discovered stripes accidently, in their exploration of the DMRG model ﬁrst on t–J ladders and later on the 2D t–J planes. Stripe patterns emerged spontaneously in these calculations. This refers in ﬁrst instance to the charge order; to ﬁnd antiferromagnetism they typically start out with a Néel order at one of the boundaries of the system which does not disappear when the DMRG is iterated to completion. This is at the same time indicating the limitation of the method. They consider a system with a ﬁnite size and in a ﬁnite volume symmetry breaking is not possible: the true ground state should be charge uniform and a spin singlet, characterized by a ﬁnite energy gap. However, one can argue that the symmetry restoring ﬂuctuations involve small numbers for the sizable systems they consider and these small numbers can get lost in the truncation procedure underlying the DMRG method. In a recent study it is shown how these problems can be avoided, see White et al. (2002). The results of the DMRG calculations are discussed at great length in a number of papers: White and Scalapino (1998a, 1998b, 1999, 2000a, 2000b). The (latent) d-wave superconductivity and the stripes are clearly competing, although both can be viewed as ordered

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Fig. 2.7. The stripes of the t–J model as calculated by White and Scalapino (1999) using the numerical DMRG method. These are quite like to what one would expect from the interpolation from the large N limit to the large-S limit. These “inherit” from large-S the antiferromagnetism of the magnetic domains while these stripes are also antiphase boundaries. At the same time, these stripes are bond centered while they are constructed from d-wave like pairs of holes. This is especially clear from this ﬁgure: it is clearly seen that there is a density wave on the stripe, which is like the 2kF density wave expected from the mean-ﬁeld theory for half-ﬁlled stripe: two holes – two electrons – two holes, etc. The “four hole plaquettes” just correspond with these pairs and all what remains to be done is to frustrate this internal density wave in order to obtain a superconductor.

states of the same d-wave pairs, which can be identiﬁed in the calculations. This competition can be illustrated using the DMRG calculations by extending the t–J model with a next-nearest-neighbor hopping t . Pending the sign one either helps the superconductor or the stripes; this should not be taken to the literal because the physical sign (t /t < 0) promotes the stripes. We refer in this regard to recent numerical studies by Martins and coworkers (2000, 2001), exploiting this to study the stripe tendency in more detail, adding further credibility to the notion that the driving force is the topological “holons on a row” mechanism. A controversial aspect is the competition between the stripes as found by White and Scalpino and the phase separation tendency in the t–J model, see also Carlson et al. (2002). For the two-dimensional system, Hellberg and Manouskasis (1997, 1999) have presented a case based on a variational Ansatz that instead of forming stripes the t–J model would be prone to phase separation in hole-rich and hole-poor matter at all values of t/J . This gets further support by the proof of Carlson et al. (1998), based on a largeD expansion for the t–J model, that phase separation will occur always in sufﬁciently large dimensions. There are several caveats. Although DMRG surely has its limitations, the Hellberg and Manouskasis work is also approximate because it does involve a ﬁxednode approximation to avoid the minus sign problem. This is a quite uncontrolled affair, and at a minimum one should investigate what happens when one starts with an Ansatz incorporating stripe order. In fact, this was accomplished in the paper where the lattice generalization of the ﬁxed-node Quantum Monte Carlo method was introduced. Van Bemmel et al. (1994) showed that although the stripe order of the Hubbard model was reduced as compared to the mean-ﬁeld solutions it still survived in their quantum Monte Carlo ground state. More seriously, it seems that the phase separation is interpreted as if the system falls apart in a half-ﬁlled insulator and a featureless Fermi-gas like hole-rich state.

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Given that the DMRG total energies and those calculated by Hellberg and Manouskasis are virtually the same, White and Scalapino (1999), this seems unlikely, because it would necessarily involve a highly accidental ﬁne tuning. The length scales associated with the White–Scalapino stripes are small, implying that sizable energies are involved in the stripe formation. How can these be nearly identical to those of a state which is microscopically completely different, i.e., the featureless Fermi-gas? It can be well imagined that the Hellberg–Manouskasis states involve strong stripe-like microscopic correlations which are hidden from the static properties The only way to settle these matters is by employing either the multiparticle correlation function as introduced by Prelovšek and Zotos (1993) or the non-local “topological” correlators introduced by Zaanen and van Saarloos (1997), discussed in more detail in Zaanen et al. (2001). Even when stripes are formed, one cannot avoid phase separation in the t–J model because of the attractive Casimir forces mediated by the spin-waves, as discussed by Pryadko et al. (1998) and Zaanen et al. (2001). These are however quite weak forces, which will be easily overwhelmed by the long range Coulomb interactions. The long-range Coulomb interaction should be included anyhow, and in doing so Arrigoni et al. (2002) arrived at the important observation that the long range repulsive actually enhance the superconducting correlations. The reason is likely that the stripes of the bare t–J form at least on the “DMRG level” internal charge density waves, see Bosch et al. (2001). The long range interactions will tend to destroy this on-stripe charge commensuration, with the effect that the d-wave pairs trapped by the stripes start to move more easily. 2.7. Frustrated phase separation Up to now we have focused on quantum theories for the origin of stripes. However, there is another way of viewing the stripes which seemed to be rooted in a completely different physics: the idea of frustrated phase separation. This emerged in roughly the same era as the Hartree–Fock stripes, with main advocates Emery and Kivelson (1987, 1988, 1990, 1993, 1995). Although initially perceived as competing explanations, in the course of time it became increasingly clear that complementary aspects of the physics are involved. As we already emphasized in the previous section, the Coulomb interaction has a long range tail which is just neglected in the standard models. Frustrated phase separation revolves around the idea that long range interactions can give rise to novel phenomena in highly correlated electron systems. The principle is of an elegant simplicity. The prevailing viewpoint is that high-Tc superconductivity is about pairing and therefore about rather strong, short range attractive forces. When the strength of the interactions becomes of order of the bandwidth, the Cooper instability protecting the pairs looses its inﬂuence, and instead the attractive interactions drive an inﬁnite particle bound state. The system phase separates. When these particles are electrically charged this in turn leads to a Coulomb catastrophe: the Coulomb energy would diverge, and the phase separation is “frustrated”. Hence, the system has to compromise between the long range repulsions and short range attractions and textures will appear with a characteristic scale set by this balance. Initially, Emery et al. (1990) focused on a ﬂuctuating inhomogeneity in the form of droplets because this is the generic outcome of frustrated phase separation in the continuum. However, studying a lattice gas model it was discovered by Löw et al. (1994) that

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commensuration effects change the physics drastically. This lattice gas model takes the shape of an Ising spin 1 model on a square lattice where Si = +1 and Si = −1 are interpreted as hole-rich and hole-poor phases, respectively, while S0 corresponds with charge neutrality. The Hamiltonian is, Si Sj H =K (2.5) Sj2 − L Si Sj + Q . rij j

ij

i<j

K corresponds with the thermodynamics while L, Q > 0 parameterize the short range attractions and the long range Coulomb repulsion, respectively. Löw et al. show that the ground states of this model are generically stripe-like, consisting of linear domains of hole rich and hole poor regions. More importantly, this model describes a surprisingly large variety of phases with nearly equal energies, and accordingly one ﬁnds quite a complex phase diagram as function of the parameters. In combination with the lattice, the competition between short range attractions and long range repulsions gives rise to dynamical frustration, and the physics is about an intrinsic tendency to form a glass instead of full order. With regard to the “stripe mechanism”, a consensus has emerged that the quantumaspects and the frustrated phase separation mechanism work in the same direction, involving different aspects of the physics which are likely both needed to explain the physics of the stripes. However, the frustrated phase separation adds an ingredient superseding this mechanism question: it suggests a mechanism for intrinsic glassiness. A central result is due to Nussinov et al. (1999) demonstrating that the O(N) continuum generalization of the lattice gas model equation (2.5) is characterized by an inﬁnite set of degenerate spiral-like ground states. This has the spectacular consequence that a jump occurs in the ordering transition temperature at inﬁnitesimal strength of the frustrating interaction (“avoided critical behavior”). This might well be a vital ingredient to the part of stripe physics in cuprates which matters most: it is apparently easy to quantum melt the stripe order, maintaining considerable microscopic stripe correlations. Two ingredients are needed to make possible the quantum melting of a crystalline state: sufﬁcient kinetic energy should be available (e.g., small mass density), while the crystal should be soft. It might well be that the frustrating inﬂuences coming from the long range interaction cause the stripes to be a particular soft form of crystalline matter. Alternatively, one might want to view the anomalous ordering dynamics of the static stripes as a manifestation of this intrinsic tendency towards glassiness, see Westfahl et al. (2001). 2.8. The role of the dynamical lattice The role of phonons in high-Tc superconductivity has been, and continues to be under intense investigation. Since the emphasis is here on magnetism, let us just restrict the discussion to some aspects of the electron–phonon coupling which are of direct relevance to the stripes. Bianconi and coworkers (2000) introduced “stripes” of a different kind, discussed in an electron–phonon language. We refer to the literature for a discussion of these stripes. On a gross level, this school of thought is focused on bipolarons being the cause of the pairing. The idea is that two carriers can gain extra energy by sharing the costs of the creation of a

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lattice deformation. However, also here one should worry about the absence of protection of the two particle channel when polaron binding energies become large; a-priori nothing can prevent the formation of an inﬁnite particle state, either in the form of phase separation or in the form of charge order. Although challenged by Alexandrov and Kornilovitch (2002), Kusmartsev et al. (2000), Kusmartsev (2000, 2001) addressed the physics of the Pekar–Fröhlich model (i.e., jellium electrons interacting via long range Coulomb, coupled to longitudinal-optical phonons) at a ﬁnite density. They demonstrate that in the case of a static lattice the electrons want to clump together in arbitrary numbers. However, the Coulomb repulsions again interfere and the net result is that string like droplets are formed. Effects of this kind would surely add up to the other mechanisms, rendering stripes to be further stabilized, see also Bussmann-Holder et al. (2000). The hard questions are related to the possible role of dynamical phonons. A much simpler issue is about the inﬂuence of static phonons/lattice deformations on the electronic stripes. The problems become of a quantitative nature; in these complicated structures many different deformations exist which might couple in quite distinct ways to the stripe charge order. Up to now, studies have appeared on strongly simpliﬁed Hubbard–Peierls type models addressing primarily the coupling of the electrons to the hard, metal-oxygen breathing type phonons. A recent exception is the work by Kampf et al. (2001) focused on the inﬂuence of the LTT buckling deformations on the stripe ordering. In going from cuprates via nickelates to manganites, it is clear from experiments that the magnitude of the lattice deformations dressing up the stripes is increasing drastically. Using the LDA+U method, Anisimov and Dederichs (1992) reproduce this trend which is explained by an increasing metal-oxygen covalency associated with the hole state. Zaanen and Littlewood (1994) presented mean-ﬁeld calculation based on the Peierls–Hubbard model, speciﬁcally aimed at the nickelate stripes. They arrived at the conclusion that for not to strong electron–phonon couplings the stripe driven electronic localization and the polaronic localization effects mutually enhance each other, adding further stability to the stripes. It was also argued that strong static lattice dressing of the stripes is detrimental for the superconductivity, because this will strongly increase the effective mass associated with the electronic matter. In this regard, the difﬁculty to observe cuprate stripes in lattice experiments might be seen as a direct consequence of the simple principle that a strong coupling to the (static) lattice would render the stripes to be quite classical, as seems to be the case in, e.g., the nickelates. This was taken up by Yi et al. (1998), extending it by a calculation of the mode spectrum (phonons, and electronic collective modes) around the stripe ground state. This is the starting point of a highly promising development. As discussed in the experimental sections, strong phonon-anomalies were (re)discovered in cuprates by McQueeney et al. (1999b), and Lanzara et al. (2001) linked these to the “kink” anomalies seen in photoemission (see also section 3.6). As it turns out, very similar phonon anomalies are seen in the nickelate, Tranquada et al. (2002). McQueeney et al. (2000) argued that similar anomalies appeared in the mode spectrum, calculated from the mean-ﬁeld stripes. Kaneshita et al. (2002) presented recently a detailed analysis, explaining the mechanism by which stripes induce phonon anomalies of the right kind. They show that these originate in mode couplings between the purely electronic collective charge modes of the stripes and the phonons. The charge modes become degenerate with the hard breathing phonons at momenta which are

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roughly half way the Brillouin zone, and because of this resonance condition a modest electron–phonon coupling sufﬁces to cause large anomalies. Several groups are at present extending this type of calculations to address the spin waves, the optical responses, etc., in stripe systems. Given the complexity of the collective mode spectra it is anticipated that these efforts will be quite instrumental in guiding the experiments. 3. The magnetic resonance: theoretical aspects More than anything else, the magnetic resonance is about the “magnetism” of the cuprates, but now in the regime where the superconductivity is at its best. There is no doubt that the ground state of these best superconductors (optimally doped 123 and 2212) is a spin singlet. Hence, by inelastic neutron scattering one obtains information on the spectrum of spin triplet excitations. In a conventional, weakly coupled BCS superconductor one expects a spectrum associated with the fermionic Bogoliubov quasiparticles. At energies lower than the superconducting gap the system has to maintain two particle singlet coherence. The Bogoliubov particle–hole excitations correspond with breaking up such pairs, liberating the S = 1/2 quantum numbers of the constituent fermions, which may form triplets showing up in the neutron dynamical form factor. In this case, one expects a incoherent excitation spectrum, starting at the BCS gap 2Δ. However, already a long time ago it was pointed out by Bardasis and Schrieffer (1961), that in the presence of any interaction triplet bound states will be formed in the gap because of the singularity in the BCS density of states. These bound states are the particle–particle analogues of the excitons occurring in the particle–hole channel, and can be viewed as triplet Cooper pairs propagating through the singlet vacuum. However, this binding is expected to be very weak in a conventional superconductor and nothing of the kind was ever observed until the arrival of the cuprate superconductors. 3.1. The facts to be explained This is a different affair in the cuprate superconductors. The spectrum of triplet excitations is astonishingly rich and the experimental study of the spectrum by neutron scattering (augmented by NMR and μSR) has been a fertile research. On the gross scale, one ﬁnds two rather distinct features, see ﬁg. 3.1. The magnetic excitation spectrum in the 214 system is dominated by magnetic ﬂuctuations occurring at incommensurate wavevectors (see section 5). A rather strong, mostly empirical case emerged that these have to do with one or the other quantum disordered form of the static stripes discussed in the previous section. We leave this discussion to the experimental sections 5 and 6. On the other hand, this spectrum looks quite different in the truly high-Tc superconductors like optimally doped 123 and 2212 (sections 8 and 9). Instead of the incommensurate ﬂuctuations one ﬁnds a peak which is both very sharp in energy and momentum, centered at the antiferromagnetic wave vector (π/a, π/a) at an energy ∼40 meV: the magnetic resonance (ﬁg. 3.1). In the optimally doped superconductors, the resonance disappears in the normal state while its weight increases with the strength of the superconducting order parameter. Upon decreasing doping, the energy of the peak decreases with doping concentration, suggesting that it would

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Fig. 3.1. Artist impression of the evolution of the magnetic dynamical formfactor of the cuprates at low temperatures on the energy interval 0 < ω < 50 meV, and wavevectors k along the (π/a, π/a) → (π/a, 0) direction in the vicinity of the (π/a, π/a) point. In La2−x Srx CuO4 (a) only incommensurate ﬂuctuations are found (hatched area’s) living at a wave vector which appears to be identical to the ordering wave vectors of the static stripe antiferromagnets. It seems that the main difference with the static stripes is that in the optimally doped superconductors (x 0.15, Tc 40 K) a small energy gap (∼5 meV) opens up at the low energy end of the triplet spectrum, suggestive of ﬂuctuating stripe order. The triplet spectrum of optimally doped YBa2 Cu3 O6+y with y 0.9 and Tc 90 K is completely dominated by an excitation which is sharp in energy and momentum: the “magnetic resonance, indicated by the black dot in (c). Upon reducing doping somewhat (y 0.7, Tc 60 K), “incommensurate” side branches appear, (c), suggesting that the resonance is part of a mode with an “inverted” dispersion. This ﬁnds an appealing explanation in the fermiology language of section 4.5 in terms of the Bogoliubov excitations of a BCS-like d-wave superconductor. However, upon further reducing doping to y 0.37 the recent data by Mook et al. (2002) indicate the onset of static stripe order. The spectrum is again dominated by the incommensurate ﬂuctuations of the 214 kind (b), although there is still a resonance discernible which seems to “interrupt” the incommensurate ﬂuctuations. To reconcile the stripe-like incommensurate fermions with the magnetic resonance is an important open problem.

end up at zero energy at zero doping. On the underdoped side, it persists in the normal state although it becomes quite broad at higher temperatures. Interestingly, in the presence of a magnetic ﬁeld it looses weight even above Tc . This suggests that there is still local superconducting order in the normal state of the underdoped superconductors, emphasizing once again that the resonance goes hand in hand with the superconductivity. There is a caveat. For technical reasons, the resonance has only been seen in the bilayer 123 and 2212 system where it occurs only in the odd c-axis momentum channel. It has been argued that it cannot be excluded that the resonance is a non-generic special effect associated with the presence of bilayers. It is not seen in the single-layer 214 and the experimental investigation of other single-layer cuprates is at present hotly pursued (He et al., 2002). Knowing only about optimally doped 123 and underdoped 214 one would be tempted to ascribe the resonance and the incommensurate “stripe” ﬂuctuations to completely different kinds of physics. Quite a confusing and controversial issue is that also in moderately underdoped 123 spin ﬂuctuations appear at incommensurate wavevectors at energies below the resonance. At ﬁrst sight these look quite like the stripe ﬂuctuations of the 214 sys-

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tem. However, these incommensurate ﬂuctuations and the resonance show quite a similar temperature dependence suggesting that they belong together. Adding to the confusion, it was shown very recently, Mook et al. (2002), that stripe order is present in strongly underdoped, barely superconducting 123 (YBa2 Cu3 O6.37 ). The neutron scattering shows very clear signatures of stripe-like incommensurate spin excitations, which is however interrupted by a small but discernible resonance structure (ﬁg. 3.1). We will come back to this deep problem at the end of this section: it makes one wonder that there is something not quite right regarding the present understanding of both the stripes and the resonance. In any other regard an appealing and quite detailed explanation of the resonance has emerged in terms of a relatively conventional fermiology language. The central assumption is that the physics is ultimately controlled by S = 1/2 fermionic quasiparticles, spanning up a big Fermi-surface and communicating with the superconductivity via a particle–particle potential. One can rest on the well developed theory of metal physics, routine calculations become possible and accordingly the subject is associated with a vast- and sometimes repetitive literature. In section 3.4 an overview of this literature will be found, emphasizing the differences in the interpretation of the physics behind the “metal like” spin excitations. In section 3.5 we will explain in more detail the interpretation of the resonance within the RPA framework. As an example we will discuss the recent work by Brinckmann and Lee (1999, 2001) in some detail. The reason for this choice is twofold: within the “metalphysics paradigm” it appears that the underlying spin-charge separation framework offers the most economic interpretation of the various aspects of the physics of the resonance. At the same time, on the technical level there is a considerable overlap with many other proposals and Brinckmann and Lee, being more or less the last word on the subject, present the most complete analysis. This work amounts to a possible answer to the question: “given that there is superconductivity, where is the resonance coming from”? Section 3.6 is devoted to the reciprocal question: “knowing about the resonance, what do we learn about superconductivity”? We will discuss the popular view that magnetic ﬂuctuations are responsible for the pairing, and pay particular attention to the ingenious sum rule constructions suggesting even deeper connections between antiferromagnetism and superconductivity. Before turning into speciﬁc explanations of the resonance, we will ﬁrst spend two sections addressing matters from a most general perspective. In sections 3.4, 3.5 the focus is on the question: how to create collective modes out of fermionic matter? Given that these collective entities exist one can subsequently change perspective, to ask the question: what can be said in general about the quantum physics associated with these collective entities? This is the theme of “ﬂuctuating-” or “competing orders”, which can be attacked with the mathematical language of quantum-ﬁeld theory. The resonance appears within this language as the signal that the superconductor is close to becoming an antiferromagnet as well. On this level symmetry is all what matters and in section 3.2 we will discuss the bare-bone version which seems particularly successful for the explanation of the competition between the stripe antiferromagnets and superconductivity in the 214 system, although Sachdev and coworkers also came up with some quite surprising ideas related to the resonance. In section 3.3 we will focus in on the deep ideas of Zhang and coworkers regarding the possibility that a larger symmetry (SO(5)) is at work.

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3.2. Symmetry (I): Spin modes and the coexistence phase In order to address the nature of the magnetic excitations in the superconductor one should ﬁrst address the more general issue how superconductivity and magnetism relate to each other on the level of their symmetries. Our focus will be on antiferromagnetism; ferromagnets break time reversal symmetry macroscopically and communicate thereby directly with the superconducting phase and this requires quite different considerations. It is often said that a (singlet to triplet) spin gap and superconductivity are one and the same thing. Although this identiﬁcation is literal in weak coupling, while a spin gap might be a good idea if one wants to optimize Tc , it is a rather meaningless statement in general. Except for the subtleties discussed in the next section, the fundamental fact is that the superconducting- and antiferromagnetic order parameter ﬁelds do not talk to each other at long wavelength. Only their amplitudes communicate, and pending numbers all combinations of superconducting- and antiferromagnetic (dis)orders have an equal right to exist. One should consider the Ginzburg–Landau–Wilson order parameter theory, resting entirely on symmetry principles. These considerations are so general that they apply equally well to the magnetic resonance as to the competition of stripe-antiferromagnetism and superconductivity. The ﬁrst question one should ask is, can a true coexistence state of antiferromagnetism and superconductivity exist, in the sense that the same electrons are responsible for both orders? Superconducting order breaks the phase symmetry associated with electrical charge (U (1)c ) while antiferromagnetism breaks internal spin rotational symmetry (SU(2)s ) and the overall symmetry is U (1)c × SU(2)s and this is all what one needs to conclude immediately that a coexistence phase is possible. Although discussed by various authors in the past (e.g., Balents et al. (1998) and Zaanen (1999b)), the subject got a new impetus by the recent observation of magnetic ﬁeld induced antiferromagnetism- (Lake et al., 2002, 2001) and stripe-charge (Hoffman et al., 2002a) order, see sections 4 and 9. We follow here Zhang et al. (2002), and Polkovnikov et al. (2002), but see also Sachdev and Zhang (2002), Kivelson et al. (2002), and Chen et al. (2002). Also notice that the considerations apply equally well to the coexistence of charge- and superconducting order. The antiferromagnetic order can be described in terms of a “soft-spin” vector ﬁeld Φ α and the superconductivity by the complex scalar ﬁeld Ψ , with the Euclidean GLW actions, in a standard notation, SΦ = dd x dτ (∂μ Φ α )2 + m2Φ |Φ|2 + wΦ |Φ|4 , (3.1) 2 d 2 2 2 4 μν SΨ = d x dτ (∂μ − iAμ )Ψ + m |Ψ | + wΨ |Ψ | + Fμν F Ψ

describing the (independent) dynamics of the superconducting and antiferromagnet orders. Given the U (1)c × SU(2)s symmetry, the lowest order allowed coupling between spin and charge is, Sint = g dd x dτ |Φ|2 |Ψ |2 . (3.2) Taking both m2Φ and m2Ψ strongly negative it follows immediately that a coexistence phase is present where both Ψ and Φ are condensed. Since eq. (3.2) is a pure amplitude coupling,

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this state carries a normal antiferromagnetic Goldstone boson (spin wave), as well as a normal Higg’s boson. Although the amplitudes of both order parameters communicate, their phase/direction is decoupled. The circumstance of relevance to the resonance is the one where m2Ψ 0 while m2eff = 2 real time this translates into a spin-triplet mode mΦ + g|Ψ |2 > 0. After continuation to with a mass gap, dispersing like ωq ∼

c2 q 2 + m2eff (c is the spin-wave velocity). The

correlation length associated with this spin gap ξM = hc/m2eff has the following meaning: at distances shorter than ξM and times shorter than h/m2eff the system rediscovers that it is actually in the coexistence phase. Obviously, this interpretation is quite different from what one gets from BCS theory. In the weak coupling limit the spin-gap is associated with the pairing gap, meaning that it corresponds with the energy to break up the singlet Cooper pair, while the characteristic (coherence) length ξc = hvF /(2Δ) is the size of the Cooper pair. Is there an experimental way to distinguish between both possibilities? This is the punch line of the recent vortex state experiments in 214. In the neighborhood of the vortices the superconductivity is suppressed and m2eff → m2Φ . Assuming that m2Φ is negative, while m2eff is positive merely because of the coupling to the superconducting amplitude, patches of antiferromagnetism will surround the vortices. Since the characteristic length scale ξM is large by default these will percolate in a long range ordered antiferromagnet already at a small density of vortices. It appears that along these lines the observations can be explained in a quantitative detail. Alternatively, when m2Φ is small but positive, the vortex lattice will translate into a potential seen by the triplet excitations and this causes states to appear in the spin-gap of the system in zero ﬁeld. This explains in quantitative detail the inelastic neutron-scattering observations in the vortex state of optimally doped (x = 0.15) 214. The above refers exclusive to the stripe antiferromagnetism of 214 and there is no a priori reason to interpret the magnetic resonance of 123 in terms of ﬂuctuating order. Although small as compared to 8J ∼ 1 eV, the magnetic resonance energy (∼40 meV) is much larger than the ∼5 meV gap found in 214 indicating that the correlation length is much smaller. One therefore expects much less, if any, response to magnetic ﬁelds of the type found in 214. However, Sachdev and coworkers discovered a most unexpected way to probe the collective nature of the physics behind the resonance. It was established by the Keimer group that already at a low concentration, Zn impurities are remarkably effective in broadening out the magnetic resonance. Zn impurities might in ﬁrst instance be considered as missing elementary spins. Sachdev et al. (1999) studied the effect of such magnetic defects when the spin system is close to the quantum phase transition from the quantum paramagnet (characterized by a magnetic resonance) to the ordered antiferromagnet. They demonstrated that in 2 space dimensions the physics of the impurities becomes universal. Among others, they show that the energy width of the triplet mode only depends on the impurity concentration ni , the spin-wave velocity c and the spin gap Δ as Γ = ni (h¯ c)2 /Δ and their estimate turns out to be remarkably close to the experimental outcome (see section 8). This theory only makes sense when the resonance energy is small as compared to the ultraviolet cut-off (i.e., the resonance scatters off impurities like a critical mode), in turn implying that the resonance signals ﬂuctuating order.

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3.3. Symmetry (II): SO(5) and the π particles The considerations in the previous subsection rest on the assumption that the U (1)c × SU(2)s symmetry is in charge. This is quite natural, because it is after all an exact symmetry of the Schrödinger equation. However, as the experience in, e.g., nuclear physics demonstrates, it might happen that symmetry is generated dynamically: the collective behavior of many particles might be more symmetric than that of a few particles. Zhang (1997) discovered a most natural generalization of this kind in the present context: SO(5), the smallest Lie group containing Uc (1) × SU(2)s as a subgroup. Only when this symmetry is exactly realized the macroscopic physics is truly different. Although there is no evidence whatever for this to be the case, a different matter is that the system can get quite close to it, while it eventually ﬂows back to U (1)c × SU(2)s at long distances due to a relatively small “anisotropy” energy. Compare it to a Heisenberg spin system subjected to a small Ising anisotropy. The long wavelength physics will be governed by Ising symmetry, but upon investing an energy exceeding the Ising anisotropy one will recover the Heisenberg spin dynamics. In the light of the present evidences such a role of SO(5) cannot be excluded: it could well be that the resonance as a relatively “high energy” excitation has to do with SO(5). Zhang and coworkers derived a phenomenology starting from mildly broken SO(5) which is quite similar from the U (1)c × SU(2)s case. In fact, the ﬁrst mention of antiferromagnetism associated with the superconducting vortex cores was inspired by SO(5), Arovas et al. (1997), and emerged from earlier ideas by Demler and Zhang (1995) regarding the nature of the resonance. As explained in the above, under the rule of U (1)c × SU(2)s superconducting- and antiferromagnetic order are just independent. The basic idea behind the SO(5) construction, Zhang (1997), is that they are just two manifestations of the same underlying unity. SO(5) is the symmetry associated with the rotations of a vector of ﬁxed length in a 5dimensional space. Ten different Euler angles are needed and so the SO(5) has ten different generators. The U (1)c × SU(2)s group provides 4 generators, and therefore there are six additional generators available and these correspond with the six ways to “rotate” the antiferromagnet in the superconductor and vice versa: the π modes. Imagine a vector (“superspin”) having two entries corresponding with the real and imaginary part of the d-wave superconducting order parameter, while the other three entries corresponds with the three (x, y, z) directions of staggered spin. The total charge Q is the generator of rotations of the superconducting phase. The generators of the antiferromagnet ordering at † wavevector Q are Sα = p cp+Q,i σijα cp,j with α = x, y, z. Consider now the six operators † † πα† = p g(p)cp+Q,i (σ α σ y )ij c−p,j , with g(p) = cos(px ) − cos(py ) being the d-wave form factor. It is easily checked that these in total 10 operators (Q, Sα , {πα† , πα }) commute like the generators of the SO(5) group, rotating the superspin n in its 5-dimensional space. Imagine now that the long wavelength dynamics could be described in terms of strongly renormalized quasi-fermions, such that the Hamiltonian could be written exclusively in terms of combinations of the 10 SO(5) operators. The dynamical algebra of the problem would become SO(5) and the spectrum of collective modes would change as compared to the simple U (1)c × SU(2)s case. For instance, consider the spin singlet superconductor. Under the rule of U (1)c × SU(2)s one would ﬁnd a single massless phase mode and

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a triplet of massive spin excitations. In SO(5) one would ﬁnd besides the phase Goldstone mode a total of 9 massive excitations corresponding with linear combinations of spin- and π -excitations. If one of these modes would soften, a phase transition would follow to a state where a π mode is condensed. Since π modes rotate the superconductor into the antiferromagnet (and vice versa) this state corresponds with a superconductingantiferromagnetic coexistence phase. Finally, when all SO(5) operators would appear on the same footing in the Hamiltonian so that the system becomes invariant under SO(5) rotations, the higher symmetry would be fully realized. Zhang (1997)speculated that the cuprates could approach this symmetric point sufﬁciently closely that one can proﬁt from the inherent simpliﬁcations associated with the larger symmetry. The problem has been all along in the microscopic deﬁnition of the SO(5) operators. In the original proposal, the fermions were taken to be literal electron operators, while the antiferromagnet was associated with the Mott-insulator. It was immediately recognized that this does not make sense, Greiter (1997) and Baskaran and Anderson (1998): how to connect smoothly a Mott-insulator with an energy gap of 2 eV with a superconductor being charge-wise inﬁnitely soft? However, soon thereafter numerical evidences appeared in favor of a SO(5) algebra at work in the t–J model. By studying the spectral functions of the π operators, Meixner et al. (1997), and Eder et al. (1998) demonstrated that the excitation spectra of small clusters could be categorized in terms of approximate SO(5) multiplets. This puzzle was at least in part resolved by work on t–J ladders, Scalapino et al. (1998). Instead of constructing the operators in momentum space one should use a strong coupling, real space representation. The spin operators are associated with the electrons of half-ﬁlling † σγαδ ci,δ . The Cooper turning into on-site spins due to the Mott-constraint: Si,α = γ δ ci,γ pairs, on the other hand, cannot exist on a single site because of the strong repulsions and αβ † † cjβ , having instead one can consider “link” or nearest-neighbor pair operators: Lij = ciα a ﬁnite overlap. With the operators used in the 2D t–J model calculations, Scalapino et al. demonstrated that by choosing a Hamiltonian such that these link pairs are formed on the rungs of the ladder it becomes easy to construct a manifestly SO(5) symmetric Hamiltonian. As it turns out, to construct such a Hamiltonian one has to allow for single site double occupancy (associated with the pair ﬂuctuations) thereby violating the single occupancy constraint. As Zhang et al. (1999) showed, this problem could be overcome by a Gutzwiller projection afterwards, leaving SO(5) in essence intact (“projected” or p − SO(5)). Matters are still not quite resolved in the two-dimensional case. Interpreted in terms of bare electron operators, one ﬁnds that the set of link-pair and spin operators do not close into a SO(5) algebra, as related to the fact that operators associated with different links can share a site. This problem can be overcome by invoking long range interactions as shown by Rabello et al. (1998), Henley (1998) and Burgess et al. (1998), but this leads into Hamiltonians which have not much to do with electrons in solids. Van Duin and Zaanen (2000) discovered a way to avoid the above difﬁculties by tailoring a special two-dimensional lattice structure. A similar construction involves plaquette-, instead of link degrees freedom, Dorneich et al. (2002). The idea is to start out with a square lattice, to subsequently pull apart the sites in “site-plaquettes”. The link-pairs live on the links of the original lattice while the links of the site-plaquettes only mediate ferromagnetic exchange interactions between the spins of the electrons residing on the same small plaquette. The link operators can

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be combined in a (projected) SO(5) and subsequently a class of Hamiltonians can be constructed illuminating various aspects of the physics. One arrives at a quite versatile model, describing a large variety of possible phases, including the antiferromagnet Mott insulators, singlet d-wave superconductors, but also antiferromagnetism-d-wave coexistence, s-wave and triplet superconductors, etc., and even a tricritical point where p − SO(5) is exactly realized. Quite signiﬁcantly, one ﬁnds that matters reduce to U (1)c × SU(2)s , unless one incorporates pair hopping processes which frustrate the antiferromagnet. This might well be a quite general feature: at the moment the motion of the Cooper pairs start to frustrate the antiferromagnetic correlations in the spin system one cannot avoid an admixture of π modes in the excitation spectrum. Let us now return to the discussion of the magnetic resonance. SO(5) started out with the assertion of Demler and Zhang (1995) (see also Demler et al. (1998)) that the magnetic resonance was the experimental manifestation of the π rotation, with its energy measuring “the distance” between the superconducting ground state and the (commensurate) antiferromagnet. This suggestion was dismissed by Greiter (1997) and Tchernyshyov et al. (2001) emphasizing that the resonance has to be a massive spin wave associated. On closer inspection one ﬁnds, however, that these arguments are based on quantitative considerations regarding particular models having no fundamental status. In fact, the debate makes not much sense. The issue is that π modes and spin waves are not mutually exclusive: it is by principle a matter of degree. In a singlet superconductor there are two ways of exciting a massive spin triplet at (π, π): either by spin ﬂips or by a π operator. Except for the SO(5) symmetric point, the spin waves and the π modes are symmetry wise indistinguishable, meaning that these modes are always coupled, see for instance van Duin and Zaanen (2000). The strength of this mode coupling is governed by microscopic dynamics and it is therefore impossible to make decisive theoretical statements. The only way to decide the issue is by experiment. The information on the spin content, as derived from neutron scattering, does not sufﬁce. One needs to know in addition the spectral function associated with the π operators Sπ , and this information is at present not available, see Bazaliy et al. (1997). 3.4. Restoring the fermions: the various viewpoints The focus in the above two sections has been entirely on the collective ﬁelds. However, eventually these collective ﬁelds are made out of electrons and it is a-priori unclear if the energy scale where the electrons are “eaten” by the order parameter ﬁelds can be taken to be inﬁnitely large. To the contrary, photoemission and tunneling measurements demonstrate that degrees of freedom having a ﬁnite overlap with the bare electron exists all the way down to zero energy: the nodal fermions. Since the electron carries S = 1/2 this implies the existence of low energy spin degrees of freedom “derived” from the nodal states. In addition, in the overdoped regime a reasonable case can be made that a conventional BCS type behavior is re-established, such that the onset of superconductivity goes hand in hand with the formation of the electron pairs. Last but not least, just observing the empirical facts pertaining to the resonance itself, it is clear that one misses important pieces of the physics if one takes only into account the workings of the collective ﬁelds. Why is the resonance disappearing above Tc , at high dopings? How to explain the incommensurate “lobes” of the resonance seen at energies below the resonance moving away from (π, π)

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with decreasing energy (ﬁg. 3.1)? This dispersion is rather the opposite of what one would expect for a “normal” massive spin wave. These facts do ﬁnd a qualitative explanation in a rather conventional fermiology language. The fundamental assumption is that the vacuum is controlled by a Fermi-surface, at least associated with the S = 1/2 spin excitations, undergoing a weak coupling instability into a d-wave superconductor. This is the working horse, producing a successful phenomenological interpretation, leaving still room for various further interpretations. The most conservative version is to take it very literally, meaning that the cuprates are interpreted as Fermi-liquids all along (including the normal state) undergoing a weak coupling BCS instability, while the excitations can be calculated from the leading order in perturbation theory controlled by the weakness of the residual interactions (random phase approximation, RPA), see Maki and Won (1994), Lavagna and Stemmann (1994), Mazin and Yakovenko (1995), Blumberg et al. (1995), Bulut and Scalapino (1996), Salkola and Schrieffer (1998), Abrikosov (1998), and Norman (2001). Given the anomalies of the normal state, and especially the pseudo-gap behavior in the underdoped regime (including the persistence of the resonance) this is generally perceived as an oversimpliﬁcation. At a next step, one can push the perturbation theory to a higher order and the most successful work of this kind is based on the ﬂuctuation exchange (FLEX) approximation, where the magnetic excitations are calculated from a single electron propagator, dressed with a self-energy which is self-consistently calculated from the scattering against the collective ﬂuctuations (Pao and Bickers, 1995; Dahm et al., 1998; Takimoto and Moriya, 1998; Manske et al., 2001). Although not much changes qualitatively at low temperatures, FLEX gives a better account of the physics at high temperatures. It describes damping of the resonance, while at least a sense of pseudo-gapping is recovered in the underdoped regime. Although more phenomenological, the work by Littlewood et al. (1993), based on the marginal Fermi-liquid (MFL) phenomenology by Varma et al. (1989), has a similar attitude. The RPA bubbles are calculated using single particle propagators dressed up with marginal Fermi-liquid self-energies. Calculations of this kind have been presented, aimed at an explanation of the anomalous frequency- and temperature-dependences of the incommensurate spin ﬂuctuations in the normal state of 214. Although this has not been pursued for the superconducting state, one would expect quite similar answers as for the “bare” RPA given that the anomalous self-energy is assumed to disappear below the superconducting gap. As an alternative, the work by Onufrieva and Rossat-Mignod (1995), based on a Hubbard operator technique should be mentioned. It is widely believed that on a microscopic scale the electrons in the cuprate planes have to do with a single component electron system subjected to strong on-site interactions (i.e., Hubbard model and extensions). Given that the system starts out to be very strongly interacting, it is a-priori unclear if perturbation theory in terms of the bare interactions can be pushed far enough to produce meaningful answers. Instead, one can assert that this difﬁcult microscopic dynamics renormalizes into some effective theory at intermediate scales which can subsequently be used to address the low energy phenomena. This underlies the idea of Pines and coworkers of the “nearly antiferromagnetic Fermi-liquid”, see for instance Monthoux et al. (1991). It is asserted that besides the fermion system a sector of collective, overdamped spin excitations is generated which interacts relatively weakly with the bath of fermions. Upon lowering temperature, this spin system tends to antiferromagnetic order. In this framework, the resonance is interpreted as the antiferromagnetic mode

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becoming underdamped because of the opening of the superconducting gap in the fermion spectrum, while its decreasing energy as function of decreasing doping is interpreted with the softening associated to the approach to the antiferromagnetic instability, Barzykin and Pines (1995), Morr and Pines (1998, 2000a, 2000b). Taking this spin–fermion model as a starting point, Chubukov (1997) has made the case that one can control the perturbation theory with unconventional means, making it possible to evaluate the theory even in the strongly coupled regime. The idea is that under the condition of overdamping of the spin ﬂuctuations the inverse of the number of hot spots (i.e., the number of crossings (8) where the magnetic Umklapp surface intersects the Fermi surface) can be used as small parameter to control the theory. The end result of this strong coupling theory, Abanov and Chubukov (1999) looks much like the phenomenological modelling based on lowest order perturbation theory. On a side, multicomponent models like the spin–fermion model, but also the bosonfermion model asserting that fermions coexist with preformed Cooper pairs (see Domanski and Ranninger (2001) and references therein), have to be considered as Ansatzes, ill founded in the microscopy of the electrons in the cuprate planes. However, quite recently Altman and Auerbach (2002) reported on a novel type of real space renormalization group (RNG) procedure which they applied to the t–J model. Remarkably, according to this RNG a separation between a fermionic sector and a bosonic sector is dynamically generated. The bosonic sector takes care of the “ﬂuctuating order’ aspects both involving the superconductivity and the antiferromagnetism along the lines discussed in sections 3.2 and 3.3, which is in turn relatively weakly coupled to the fermions. Finally, there is a class of theories having in common a more radical departure of the physics of conventional metals and superconductors. These can be grouped in two subfamilies: the “hidden order” theories, and the spin-charge separation gauge theories. The hidden order theories ﬁnd their inspiration in the interpretation that the anomalies of the cuprates are in ﬁrst instance governed by a form of order present in the underdoped regime, of a kind which is hard to detect by existing experiments. This order is then assumed to disappear at a zero-temperature quantum phase transition, governing the physics of the optimally doped superconductors. A typical example of a possible order is based on the idea that some regular pattern of spontaneous orbital currents could be present: the staggered ﬂux phase or d-density wave, breaking time-reversal and translational symmetry, Chakravarty et al. (2001), and the “charge transfer” ﬂux phase by Varma (1999) which is just breaking time-reversal symmetry. These phases have their own form of “magnetism”: the spontaneous orbital currents give rise to a pattern of small magnetic moments which should be in principle observable. In Varma’s version these orbital moments compensate each other within the unit cell and he suggested that this should lead to observable consequences in the polarization dependence of the angular resolved photoemission. His predictions seem to be conﬁrmed in a recent experiment of this kind, Kaminski et al. (2002). The staggered ﬂux phase is more easy to detect because it also breaks translational symmetry, and the orbital moments should be observable in conventional magnetic probes. Mook et al. (2001) identiﬁed a weak antiferromagnetism which appeared to be of the right kind. However, follow up experiments by Sidis et al. (2001) showed that the associated moments lie in the planar directions, consistent with “conventional” spin magnetism and hard to explain in terms of orbital currents, expected to cause moments pointing in a perpendicular

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direction. As was argued by Tewari et al. (2001), the persistence of the magnetic resonance in the underdoped normal state is easily explained in this framework: the d-density wave order persists above Tc and this causes a gap in the quasiparticle spectrum which is sufﬁciently similar to the superconducting gap to cause a resonance along the lines explained in the next section. Besides the ﬂux phases, other suggestions for hidden order invariably involve the even more exotic physics associated with gauge theories. A most recent idea by Zaanen et al. (2001) further elaborated by Zhang et al. (2002), and Polkovnikov et al. (2002) is based on the notion that the domain-wall property of the stripes can persist into the otherwise featureless, spin- and charge-disordered superconductor. At some doping also this “topological order” should be destroyed, and using symmetry arguments it can be demonstrated that this “stripe fractionalization” transition is governed by the conﬁnement transition of Ising gauge theory. This theory is not yet understood to an extent that meaningful statements regarding the resonance can be made. This brings us ﬁnally to the gauge theories of spin-charge separation being the subject of the next subsection. 3.5. The resonance peak and spin-charge separation In the spin-ﬂuctuation theories as discussed in the previous section, the basic outlook is that the zero-temperature ground state is much like a conventional BCS superconductor. However, the underlying Fermi-liquid is strongly interacting in the sense that interaction effects switch on much more rapidly when one goes up in energy or temperature, and the magnetic resonance is a signature of the importance of these interactions. Upon raising temperature, the same interactions start to take over and their main effect is that they “mess up” the Fermi-liquid, causing propagators to become overdamped. The normal state is perceived as a rather structureless affair. Given its remarkable regularities, one could wonder if there is not more behind the physics of the normal state. Given that the superconducting state is derived from this normal state, a hidden structure in the latter could also be quite consequential for the nature of the mechanism which causes the anomalously large stability of the superconductor. In the hidden order theories discussed at the end of the previous section it is just assumed that some other order is around which is just hard to see. The gauge theories of spin-charge separation are conceptually more subtle: there is only a single superconductor, but the competitors of the superconducting state are exotic and they leave their mark in the normal state. Mathematically, these matters are controlled by gauge principle and these states are then believed to carry elementary excitations corresponding with pieces of the electrons. The idea was introduced early on by Baskaran et al. (1987), and it evolved in the course of time to a high level of sophistication (e.g., the Z2 version of Senthil and Fisher (2000), and the SU(2) theory by Wen and Lee (1996)). The magnetic resonance was already addressed in this language quite some time ago by Tanamoto et al. (1991), followed by a number of other studies: Zha et al. (1993), Tanamoto et al. (1993, 1994), Stemmann et al. (1994), Lercher and Wheatley (1994), Liu et al. (1995), and Normand et al. (1995), culminating in the most complete recent work by Brinckmann and Lee (1999, 2001). The last work is especially quite insightful regarding the kinematics behind the RPA which is also of relevance to the theories discussed in the previous section.

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Fig. 3.2. The “classic” phase diagram of the high-Tc superconductors, as suggested by the spin-charge separation theories, after Brinckmann and Lee (2001). The electrons fractionalize in pure spin 1/2 excitation (spinons) and charge −e excitations (holons). The holons are hard-core bosons occurring at a density x, which are therefore Bose-condensing at a temperature TBE ∼ x. The spinons behave like fermions subjected to attractive interactions in the d-wave channel, and they undergo a BCS instability at Td , and this temperature is at maximum at half-ﬁlling to go down with doping. Below Td a spin gap opens with d-wave symmetry and the doping dependence of the size of this gap (Δ0 (x)/J ) is indicated in the inset. In addition, on the RPA level an instability towards antiferromagnetism is found (dotted line), supposedly strongly weakened by higher order corrections, such that only antiferromagnetism is found at very low dopings. Only when both charge and spin are condensed one is dealing with a normal superconductor. Since Td > TBE in the underdoped regime, a spin gap is opening up (going hand in hand with the appearance of the resonance) at a temperature (much) higher than Tc : the explanation of the “pseudo-gap phase”. It is noticed that in the regime where the theory can be truly controlled (large-N ) the present phase diagram is superseded by the phase diagram of Vojta and Sachdev (1999), see ﬁg. 2.5.

In the remainder of this section we will discuss this the Brinckmann–Lee version in more detail. We introduced the idea of spin-charge separation already in section 2.3. In ﬁg. 3.2 the well known phase diagram is reproduced. The basic idea is that the electrons fall, at least approximately, apart in bosons carrying the charge of the electrons and fermions representing the spins of the electrons. The former (“holons”) just behave like hard-core bosons occurring at a density proportional to the chemical doping x, and they Bose condense at a temperature TBE ∼ x. The fermions, on the other hand, span up a large Fermi surface which is thought to represent the spin system and its “spinon” excitations. These are subjected to a d-wave BCS like interaction which causes an instability at a temperature Td into a “spinon superconductor”. As in a normal superconductor, the ground state is a spinsinglet and the nodal states now correspond with pure S = 1/2 excitations. This spinon condensation temperature would be at maximum at half-ﬁlling, if the competing antiferromagnetic order would not interfere, and it decreases with increasing doping because of the decreasing volume enclosed by the spinon Fermi-surface. The superconductor corresponds

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with the state where both spinons and holons are condensed: such a state is symmetry wise indistinguishable from a BCS superconductor. This produces clearly a maximum of Tc at an “optimal” doping, see ﬁg. 3.2. However, in the underdoped regime the spinons condense at a temperature which is higher than Tc and this corresponds with the opening of a d-wave like spin gap. This is the simple interpretation of the pseudogap phase in this framework. From a phenomenological point of view it is quite appealing. Because the pseudo-gap has a great effect on the magnetic properties and the single electron properties, while its effects on charge transport are quite moderate, one could directly conclude that a spin-charge separation should be at work of this kind. The starting point is the t–J model and the slave formalism which we already discussed in the large-N context in section 2.4. There is no doubt that the correct saddlepoints in the large-N limit are the ones discussed by Vojta and Sachdev (1999), Vojta et al. (2000), with the main difference that the spin system becomes a spin-Peierls state instead of the massless d-wave state. However, one could argue about the relevance of these results when N = 2. The only way to proceed is by treating exactly the gauge ﬂuctuations, needed to describe the locality of the constraint and this is a very difﬁcult task. Instead one can in ﬁrst instance take a phenomenological position and assert that a saddle point which is metastable at large N might become relevant for N = 2 for no other reason than that it looks more like the experiments. Given the present state of the gauge theory, the status of the phase diagram (ﬁg. 3.2) is no more than an educated guess. Omitting these gauge ﬂuctuation-difﬁculties, the bottom line is that according to the “classic” spin-charge separation scenario the magnetic properties can be derived from a system of weakly interacting spin-full fermions, characterized by a big Fermi-surface with a volume ∼1 − x and a tendency to form a d-wave “superconductor”. This is the same basic problem underlying most of the theories discussed in section 3.4. The difference is that the “spin-physics” of the d-wave superconductor sets in at the pseudogap temperature which is unrelated to the superconducting transition temperature. In the magnetic ﬂuctuation scenario’s discussed in section 3.4, Tc and the pseudo-gap temperature are one and the same thing on the saddle-point level, and they have to be “disconnected” by tedious perturbation theory. Let us now turn to the details of the RPA analysis. RPA (or “bubble summation”, “time dependent Hartree–Fock”) has the same status as the use of classical equations of motion to derive spin waves, but now applicable to fermion systems. The well known result for the dynamical magnetic susceptibility is, χ( q , ω) =

χ0 ( q , ω) , 1 + J ( q )χ0 ( q , ω)

(3.3)

where J ( q ) is the effective interaction (becoming here 2J [cos(qx ) + cos(qy )], where J is the superexchange constant) while χ0 is the Lindhard function associated with the fermions, see also eq. (4.8). In the presence of a BCS pair potential, this becomes f (s E ) − f (sE ) 1 εk εk+ q + Δk Δk+ q q k+ k 1 + ss χ0 ( q , ω) = , E 8 E E ω + sE − s + i0+ k+ q q k k k+ k s,s =±1 (3.4)

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where f is the Fermi-function, Δk the gap function, Ek = ε2 + Δ2 the dispersion of k

k

the Bogoliubov quasiparticles and εk the band dispersion. The imaginary part χ0 of χ0 represents the density of states of particle–hole excitations at energy ω and momentum q, weighted in a BCS superconductor by coherence factors. The real part χ0 can be obtained from the imaginary part by Kramers–Kronig transformation. In the normal metallic state, χ0 is a rather uninteresting function both of momentum and energy. In order to cause it to be interesting one has to be close to a nesting condition (or van Hove singularity), causing strong enhancements of χ0 at particular momenta q0 . This causes in turn χ0 ( q0 , 0) to become large and negative. χ0 appears in the denominator of eq. (3.3), and it might become large enough such that J ( q0 )χ0 ( q0 , 0) → −1. A singularity appears signaling the instability towards order at wave vector q0 . In the approach to the instability, the real part of the denominator is decreasing, and the peak in χ0 will be further enhanced in the full susceptibility χ . This is the conventional view on spin ﬂuctuations in Fermi-liquid metals. In the context of the normal state of the cuprates, it was pointed out by several groups that it is not unreasonable to expect that the Fermi-surface of the 214 family shows sufﬁcient pseudo-nesting to explain the incommensurate spin ﬂuctuations, see Littlewood et al. (1993), Si et al. (1993), and Zha et al. (1993). In the mean-time this is taken√ less serious because a Fermi-surface derived incommensurability should behave like ε ∼ x as function of doping (because the Fermi surface volume ∼1 − x) while the experiments show ε ∼ x for x < 1/8, to become doping independent at larger dopings, more consistent with a stripe interpretation. It was also argued that because of differences in the Fermi-surface shape a broad commensurate response should be expected in 123. In the mean-time this interpretation is superseded by the knowledge regarding the resonance. All together, signs of these “Fermi-surface spin ﬂuctuations” are completely lacking in the normal state at temperatures above the pseudo-gap temperature. The situation changes drastically when the superconducting gap opens up. By considering carefully the kinematics, one will generically ﬁnd that in the presence of the superconducting gap, χ0 ( q , ω) will disappear discontinuously at some ω0 ( q ). This should be obvious for a system of normal fermions with a k dependent gap. As an added difﬁculty one has to account for the coherence factors in the superconducting state and as it turns out these are just of the right kind in a d-wave superconductor, while they in fact smooth out the singularities of the s-wave superconductor. Such a discontinuity in χ0 gives rise q )χ0 ( q , ω) has two zero crossings, to a singularity in χ0 . This in turn implies that 1 + J ( q ) < ω0 ( q ). Hence, χ develops a pole inside the one of which occurs at an energy ωres ( gap at ωres ( q ), corresponding with a bound state (sharp peak) inside the gap. This is the basic explanation for the magnetic resonance in this RPA language. The dispersion of the resonance ωres ( q ) and its q dependent spectral weight now become a matter of detailed consideration of the kinematics, depending in turn on the details of the Fermi-surface and the gap function. Using the spinons one ﬁnds precisely the right Fermi-surface evolution as function of doping to explain the doping dependence of the magnetic resonance. Let us ﬁrst consider the “magnetic” response at q = (π, π). At half-ﬁlling, the spinon Fermi surface exactly coincides with the (π, π) Umklapp surface, and accordingly the nodal points lie on the

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Fig. 3.3. Illustration of the “dynamical nesting” condition giving rise to the singularities in the Lindhard function of the d-wave superconductor, responsible for the binding of the resonance (after Brinckmann and Lee, 2001). The square indicates the magnetic Umklapp surface (the arrows correspond with q = (π/a, π/a)). At half-ﬁlling the nodes of the d-wave Bogoliubov quasiparticles lie on the Umklapp surface according to the spin-charge separation scenario and the Lindhard function has weight all the way to zero-energy. On the RPA level, the resonance “hits the zero of energy” causing an antiferromagnetic instability. However, upon doping the nodal points move towards the center of the Brillioun (the dots in the ﬁgure). One should now imagine what happens when one could at will raise the Fermi-energy: this energy cost corresponds with the minimal energy needed to ﬁnd particle–hole excitations. This “dynamical Fermi surface” corresponds with the ellipses in the ﬁgure centered around the nodal points. At the moment that they hit the Fermi-surface χ0 becomes non-zero. In the inset the energies of the particle–hole excitations are shown in the upper right corner of the Brillioun zone, with the minima ω0 (“hot spots”) as dots while the contours refer to higher energies. It is noticed that due to a special choice of the bare dispersions the nodal “ellipses” turn into “banana’s” with a ﬂat outer face being parallel to the (π, π ) Umklapp surface. This strengthens the singularity in χ0 signiﬁcantly and Brinckmann and Lee argue that this dynamical nesting effect is required for the binding of the resonance.

Umklapp surface. It follows immediately that χ0 has ﬁnite weight all the way to zero energy and a bound resonance cannot be formed (instead the system is unstable towards antiferromagnetism). Upon doping the volume of the Fermi-surface shrinks ∼ 1 − x with the effect that the nodal points move away from the Umklapp surface and χ0 is zero until one reaches an energy ω0 which is roughly twice the energy of the Bogoliubov particles at the hot spots, where χ0 switches on discontinuously, see ﬁg. 3.3. Upon calculating the full χ one ﬁnds the resonance with an energy ωres (π, π) (ﬁg. 3.4), which grows roughly proportional to x and a spectral weight which is steadily decreasing with x. Finally, at large dopings the peak start to shift downwards: here ωres follows the gap maximum, and according to the slave theory this maximum shifts down in the overdoped region. The “inverted” dispersion, i.e., peaks moving away from (π, π) upon lowering energy (ﬁg. 3.1), is quite natural in the RPA framework. The results obtained by Brinckmann and Lee (2001) are shown in ﬁg. 3.4. At energies above the resonance the inﬂuence of the

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Fig. 3.4. The imaginary part of the full dynamical magnetic susceptibility χ ( q , ω), after Brinckmann and Lee (2001). These are plotted as function of wave vector in the two directions as indicated around the (π/a, π/a) point. Top row: sequence of energies ω/ωres = 1.0–1.4, at and above the resonance; the response broadens out rapidly at energies above the resonance because the featureless particle–hole continuum of the metal takes over. Middle row: ω/ωres = 1.0–0.7; the incommensurate side branches become visible. Bottom row: energies far below ωres ; the signal becomes very weak (notice change of scale), and notice the ﬂip from the “horizontal” to the “diagonal” orientation of the incommensurate peaks.

superconducting gap rapidly diminishes and the χ0 starts to look like the featureless susceptibility of the normal state. At energies below the resonance the superconducting gap is of course in charge, but the wave numbers associated with the singularities shift away from (π, π). Recall ﬁg. 3.3; the ω = 0 nodal points are connected by a “2kF ” vector which is smaller than (π, π), and at very low energy one would expect very weak features in χ at momenta (π ± δn , π ± δn ) with δn being the difference between the nodal spanning vectors and (π, π): one expects a preferential orientation of the “incommensurate peaks” along the zone diagonals, inconsistent with the experiments. This “ﬂip” from the diagonal to the hor-

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izontal/vertical direction is explained by Brinckmann and Lee invoking a specialty of the quasifermion dispersions. In the direction perpendicular to the Fermi-surface, the dispersion is more steep for momenta k > kF than for k < kF turning the contours in momentum space of the quasifermion dispersions at a given energy from the generic ellipsoids into “banana’s” (ﬁg. 3.3) having a ﬂat face oriented parallel to the magnetic Umklapp surfaces. This corresponds with a dynamical nesting condition giving rise to incommensurate peaks with a horizontal/vertical orientation. This completes the discussion of the principles behind the RPA theories of the magnetic resonance: upon ignoring the spinon aspect, the Brinckmann and Lee work is representative for this school of thought. In detail the different versions do look different, and the outcomes do depend quite a bit on the details of the bandstructure, effective interactions and so on. In order to arrive at a convincing comparison with experiment ﬁne-tuning is involved and one should always be aware of the possibility that RPA could be no more than a form of curve ﬁtting. We already alluded to the ﬁne tuning in the Brinckmann–Lee calculations required to get the incommensurate peaks at the right momenta. Even worse, in their case it turns out that the next-nearest-neighbor hopping t has to be ﬁne tuned to change the nodal “ellipsoids” in the “banana’s” (ﬁg. 3.3), in turn needed to ﬁnd a magnetic resonance all together Let us ﬁnally discuss the striking dependence of the magnetic resonance on the momentum in a direction perpendicular to the planes. According to the experiments, the resonance appears exclusively in the odd (qz = π/c) channel, while in the even (qz = 0) channel only a incoherent background is found starting at a large energy (∼90 meV). Initially, it was suspected that the presence of bilayers could be essential for the appearance of the resonance, but given the recent demonstration by He et al. (2002) for the presence of a resonance in a single-layer thallium compound this seems no longer to be an issue. Considering the overall energy scales, the situation is not that different from what is found in the half-ﬁlled antiferromagnetic insulator YBa2 Cu3 O6 , characterized by a massless acoustic magnon in the odd channel, while the spectrum in the even channel is characterized by an optical magnon associated with a counter-precession of the spins in the two layers having an energy ∼100 meV, set by the interlayer exchange. Given the shear size of this energy, this is about a short distance physics and it can be imagined that this even–odd difference of the spin system of the insulator can persist up to rather high dopings. Millis and Monien (1996) discuss the interlayer spin physics in this spirit. In the RPA view, the physical picture is quite different. Intrinsically the physics at high energies is that of free fermions. To explain the exclusive occurrence of the resonance in the even channel, initially suggestions appeared referring to specialties of the interlayer pairing, see Mazin and Yakovenko (1995), and Yin et al. (1997). However, this is not at all necessary. We again refer to Brinckmann and Lee (2001): for simple reasons the interaction J ( q ) is substantially weaker in the even channel as compared to the odd channel with the effect that the even channel susceptibility looks much more like the Lindhard function. The resonance is missing and instead a hump is found in the even susceptibility around 100 meV. This is just caused by scattering involving the singularities at the gap maximum, which is reﬂected in the bare susceptibility.

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3.6. The resonance and superconductivity As we already discussed, an intriguing aspect of the resonance is its close relationship with superconductivity: its weight grows with the strengthening of the superconductivity, and it is suppressed by magnetic ﬁelds. The RPA considerations suggest a simple interpretation. The superconducting gap is needed for the singularities in the bare susceptibility, as required for the formation of a bound state. In this view, the resonance is just a passive side effect of the superconductivity. Could there be more to it? Much work on the resonance is motivated by the spin-ﬂuctuation exchange mechanism for superconductivity. The idea is that instead of phonons, the spin ﬂuctuations can be taken as the bosonic glue mediating the attractive interactions between the electrons. This old idea acquired quite some popularity when it became clear that these are d-wave superconductors: using general arguments based on the strong momentum dependence of the interaction vertex between spin ﬂuctuations and electrons, d-wave symmetry is natural, see Bickers et al. (1989), and Monthoux and Pines (1992). In this mindset, advocated by Morr and Pines (1998, 2000a) and Abanov and Chubukov (1999) the magnetic resonance acquires a special signiﬁcance: it is the long-sought for pairing glue becoming directly visible in experiments. This should not be taken too literally. The energy of the magnetic resonance is less than the superconducting gap maximum and it would therefore act by itself as a pair breaking mode. Instead, it is envisioned that the magnetic modes responsible for the pairing are strongly overdamped in the normal state and thereby hard to detect. However, when the superconductivity sets in a back reaction follows on the spin ﬂuctuations causing a propagating piece to emerge at the low end of their spectrum – the magnetic resonance. Given that the magnetic resonance exists and assuming that it has to do with the spin– fermion model, it should be that the quasiparticles of the superconductor are scattered by the resonance. Since the resonance is sharp in energy and momentum, while it is showing strong temperature and doping dependences, it should be possible to ﬁnd back the effects of the scattering in the quasiparticle spectra as measured by photoemission and tunneling. Subsequently, from these residual effects one could attempt to deduce the properties of the bare spin–fermion model. Although still controversial, strong claims along these lines are found in the literature. The ARPES spectra of the superconductors are characterized by a peak-dip-hump structure at the anti-nodes, Dessau et al. (1992), which has been explained in terms of a strong scattering to the resonance ∼40 meV, Norman and Ding (1998) and Abanov and Chubukov (1999). Much attention was drawn to the observation of a kink in the quasiparticle dispersion along the (π, π) direction, Bogdanov et al. (2000). The quasiparticle velocity increases suddenly upon passing an energy ∼50 meV, and this is reminiscent of well known mass renormalization effects by optical phonons in conventional metals: at energies less than the mode energy the quasiparticle has to carry around the lattice polarization, decreasing its velocity, while at higher energies the phonon cannot follow the quasiparticle. Eschrig and Norman (2000) claim that this kink follows closely the resonance both with regard to it doping- and its temperature dependence. However, this interpretation was challenged by Lanzara et al. (2001), especially so by their demonstration that the same kink occurs in the 214 system where the magnetic resonance is absent. Lanzara et al. argued that it has to do instead with the anomalous Cu–O breathing phonon which was discussed in section 2.8.

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Recently, an interesting argument started regarding the numbers in the game. Kee et al. (2002) observe that in absolute units the resonance carries quite a small spectral weight. q , ω) d2 q dω/(8π 3 ) turns out to be only a few percent of the The total weight I0 = S( 2 local moment sum rule h¯ S(S + 1) = h¯ 2 (3/4). Assuming a spin–fermion coupling of the form g S · ψ † σ ψ, the leading contribution to the electron self-energy is of order I0 g 2 /ωres , less by a factor I0 as compared to a naive estimate. This is just a pole strength effect: during 1 − I0 of its time the magnetic resonance is not behaving like the bare magnetic mode. As a consequence, in order to have a substantial inﬂuence on the quasiparticles the coupling g has to very large. This coupling should be estimated using independent information, and Kee et al. arrive at g 10 meV, while Abanov et al. (2002) ﬁnd g = 1 eV starting from different assumptions! Much of the above is based on model assumptions which cannot be claimed to be universally valid. A different matter is the recent discovery of a number of sum rules associated with the properties of the superconductors, having a much more fundamental status. These connect properties of the superconducting state to changes occurring in the dynamical magnetic susceptibility, with the magnetic resonance playing a central role. A ﬁrst sum rule is due to Scalapino and White (1998). They observed that if one can split off a magnetic interaction J ij Si · Sj in the effective Hamiltonian describing the physics at intermediate length scales (e.g., the t–J model), one can relate the gain in magnetic energy ΔEJ associated with the superconducting order to the difference between the dynamical form factors of the zero temperature normal state (SN , e.g., above Hc2 ) and the superconducting state (SS ) by, π/a ∞

d(h¯ ω)

3 cos(q ΔEJ = J d2 q ) + cos(q ) S ( q , ω) − S ( q , ω) . x y N S 2 −π/a 8π 3 0 (3.5) According to the spin–fermion school of thought, the inﬂuence of the superconductivity on the dynamical form factor is mostly due to a redistribution of spectral weight and explicit calculations by Abanov and Chubukov (2000) demonstrate that ΔEJ is of order 10 K, in fact already comparable to the condensation energy. As Demler and Zhang (1998) argued, this sum rule acquires a more direct signiﬁcance if the magnetic resonance corresponds with a pure π mode. The normal state has to be viewed in the SO(5) language as a state where the SO(5) superspin has disappeared due to amplitude ﬂuctuations. Accordingly, the spectral weight of the resonance has disappeared as well. Below the superconducting transition antiferromagnetic correlations appear which were absent in the normal state, and these manifest themselves through the weight of the magnetic resonance/π mode, emerging “from nowhere”. Hence, one should take the total weight of the magnetic resonance into account in the sum rule equation (3.5), and this leads to a δEJ 18 K which is again of the right order. Chakravarty and Kee (2000) arrived at a different sum rule which has a more fundamental status. Without invoking any assumption, they demonstrate that for a singlet d-wave superconductor, 3 ΔS(π/a, π/a) = λM , 2

(3.6)

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∞ where ΔS( q ) is the change in the instantaneous structure factor S( q ) = −∞ dω S( q , ω) due to the presence of the off-diagonal long range order, while λM is the condensate fraction corresponding with the fraction of the electrons participating in the pairing. In a weak coupling BCS superconductor λM = N(0)Δ, expected to be unobservable given the resolution of neutron scattering. Assuming that ΔS equals the spectral weight of the resonance one arrives instead at a condensate fraction of a couple of percents, an estimate which seems quite reasonable for a high-Tc superconductor. 3.7. Concluding remarks In this section we have reviewed theories dealing with the magnetic resonance, and in the process we have covered much of the theoretical main stream. The reader might have perceived it as a bit confusing. It reﬂects the state of the ﬁeld: the problem of high-Tc superconductivity is not solved. Instead, there is a patchwork of theoretical ideas, devised to explain some aspects of the data successfully, failing badly in other regards. A large majority of the theorists will agree that spins/magnetism are important, if not at the heart of the problem. Next, the school of thought ﬁnding its inspiration in the idea that one can at least start out with a conventional fermiology is quite inﬂuential. Indeed, only knowing about the optimally doped superconductors at low temperatures, it looks like as if all the work is done. However, it appears to us that this is too easy. This apparent success involves a subtle conceptual ﬂaw. On closer inspection, the fermiology philosophy rests on the quite reasonable notion that at large lengths and long times the high-Tc superconductors behave quite like conventional, Fermi-liquid derived BCS superconductors. This is not quite the case at the short times and short lengths where the pairs are formed, as exempliﬁed by the anomalous, non Fermi-liquid behaviors found at high temperatures in the normal state. It is then implicitly taken for granted that one can get away with the free quasiparticles of the long wavelength limit, getting increasingly “shaken and stirred” by the effects of the interactions. After a moment of thought, one should come the conclusion that this cannot quite work. On the lattice scale there is no such thing as a free fermion, as should be clear to anybody having familiarity with the t–J model. The spins and holes span up a Hilbert space of a completely different nature than that of the Fermi-gas, and the dynamics of these spins and holes is much richer (or “much more complicated”) than the empty and silent world of the Fermi-gas. This is probably best illustrated with the considerations in the previous section regarding stripe formation. The conclusion is that the scaling ﬂow is irreversible. Starting from a very complex physics at short distances matters apparently can simplify to something as simple as the BCS superconductor, believing for the time being that the guess for the ﬁxed point is correct. However, this is only possible because information content gets lost, and it is impossible to recover this information by starting at the low end, and trying to climb upward on the energy ladder. More than anything else, stripes symbolize this problem. We believe the basic understanding of the nature of static stripes as discussed in section 2 should be correct. However, experiments suggests another entity called “dynamical stripes”. It is quite well established in the 214 system, and it refers to experiments demonstrating that on rather short time and length scales correlations exist in the cuprate electron system which are quite like those found in static form when stripes are condensed. These will be discussed at various

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instances in the remainder of this text. However, despite these dynamical stripe complexities, 214 does not look all that different from the other superconductors at long lengthand time scales. It carries nodal fermions, it is thermodynamically quite similar, and Tc can still be quite high. At the same time, we already discussed the convincing evidence for strong stripe correlations in heavily underdoped 123. We already draw attention to the fact (ﬁg. 3.1) the dynamical factor of this “stripy” 123 does not look that different from those at higher dopings which seem so effortlessly explained with RPA. Although weak, there is still a resonance, but now the “incommensurate side branches” have clearly to do with the stripes. Somehow, this stripy affair transforms in a very smooth fashion into the RPA-like physics of the optimally doped superconductor. Another disturbing fact is the one-dimensional polarization of the incommensurate peaks in the single domain 60 K 123 superconductor discovered by Mook et al. (2000). Very strange things have to happen with Fermi-surfaces to make this possible in RPA, while it is trivially explained with stripes, see Zaanen (2000a). The theorists are in the unpleasant situation that nobody seems to have a clue how to unify the BCS-ish features with the stripy side. Starting from the static stripes, nodal fermions should not be present. The resonance is equally hard to explain. The only proposal in this regard, Batista et al. (2001), suggests that the resonance has to do with a van Hove singularity in the stripe spin-wave spectrum, and this is clearly falsiﬁed by the experiments on strongly underdoped 123. Starting from the weak coupling side, there is little hope for ﬁnding back the slightest sign of a stripe correlation employing manageable Feynman graphs. The conclusion is that the main result of 15 years of research in high-Tc superconductivity has made clear that the mystery is far deeper than initially expected. 4. Experimental techniques Thermodynamic methods like susceptibility or magnetization measurements are well known to yield reliable information about bulk magnetism, while speciﬁc heat data are essential for the analysis of the entropy content in (magnetic) phase transitions. Reﬁnements of the possible spin structures require neutron scattering, μSR, and magnetic resonance methods. In stripe systems, where both charges and spins are involved, angular resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM) and X-ray play equally important roles. Below we illustrate the merits of these techniques by recent examples. We will start by a more extensive explanation of NMR/NQR as the technique is sensitive for both charges and spin and is complementary to those of neutrons regarding time scales. Various aspects of NMR/NQR are also important for μSR. In HMM10 the reader will ﬁnd a similar concise treatise of neutron scattering. 4.1. NMR Nuclear relaxation rates, line positions, linewidths and line intensities are the main NMR/NQR parameters, see Abragam (1961), Winter (1970), Slichter (1991), and van der Klink and Brom (2000). For the application to high-Tc ’s various introductions and reviews have appeared. Of the more recent ones we mention those of Mehring (1992), Asayama et al. (1996), Berthier et al. (1996), and Brom (1998). Here we update these reviews with emphasis on the work on stripes.

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4.1.1. The NMR method A standard technique in pulse NMR is the echo method. By application of a π/2 pulse the nuclear magnetization component along the direction of the static ﬁeld B0 (the z-axis) is rotated to the xy plane ⊥ B0 . After a delay time τ a second (π ) pulse, twice the duration of the π/2 pulse, is applied that refocuses the magnetization in the xy-plane after a time 2τ . The height of the echo is then proportional to the magnetization originally along the zaxis, Mz . By slightly different procedures two important relaxation rates can be measured. The measuring cycle of the nuclear Zeeman relaxation rate (T1−1 ), which is a measure for the rate of energy exchange of the nuclear spins with the surrounding, usually starts by a comb of π/2-pulses, by which Mz is destroyed. Thereafter one measures the recovery of Mz via the echo-method. Study of the nuclear relaxation rate is often a convenient method to ﬁnd details about the electron spin dynamics. In a determination of the transverse or spin–spin relaxation rate (T2−1 ) the memory loss in the refocusing process in the xy plane is measured by varying the delay time τ in the echo method. Every cycle starts with the same rotation of Mz to the xy-plane. T2 is a measure for the time correlation of the local ﬁeld ﬂuctuations, which is often determined by the interaction between the nuclear spins. If the echo in the time domain is Fourier transformed to the frequency domain, the position and width of the resonance curve can be analyzed. 4.1.1.1. Knight shift and line width. In general the relative shift of the resonance line with respect to a suitable reference, K = ΔB/B, contains an orbital (Korb ), a diamagnetic (Kdia ) and a spin (Kspin) contribution: K = Korb + Kdia + Kspin . In simple s-electronmetals the hyperﬁne interaction between a nuclear spin (I ) and electronic spins (S) Hhf = i I·Ai ·Si is often dominated by the scalar Fermi contact term A/γn γe h¯ 2 ≈ (8π/3)|uk (0)|2 EF , with |uk (0)|2 EF the density of electrons at the Fermi energy at the nuclear site (Korringa, 1950; Slichter, 1991). The spin contribution (usually called Knight shift) is then given by A AS ΔB = (4.1) = χ0 , B γn h¯ gμB hγ ¯ n with χ0 the spin susceptibility (in SI units χ0 = χmolar /NA μ0 with NA Avogadro’s number), g = 2 the Landé g-factor, μB = γe h¯ /g the Bohr magneton, and A the already introduced scalar Fermi contact interaction term. The electron spin gyromagnetic ratio γe is typically a factor 103 larger than the nuclear spin gyromagnetic ratio γn . One contribution to the linewidth stems from the T2 processes mentioned above and is called the homogeneous broadening. A distribution of static (on the time scale of the experiment) magnetic ﬁelds gives an inhomogeneous contribution. Kspin =

4.1.1.2. Spin–lattice relaxation. Fluctuating magnetic ﬁelds with a frequency component (spectral density) at the frequency of the nuclear Zeeman splittings (in an applied magnetic ﬁeld B) cause transitions between the levels involved. Often the spectral density originates in electronic spins (S) linked to the lattice (the thermal reservoir). The transition probabilities W that determine the Zeeman relaxation rates of the various nuclear spins α I are then functions of the imaginary part χ of the electronic susceptibility χ . For Bz, α Wz is given by (Zha et al., 1996; Moriya, 1963): 2 kB T α α (4.2) Wz = 2 2 F⊥ (q) χ⊥ (q, ω → 0). 4h¯ μB ω q

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In this equation q denotes the electron wave vector, μB the Bohr magneton. The real and imaginary parts of the electronic susceptibility are given by χ and χ . The form factors F⊥ (q) in eq. (4.2) denote the components of the spatial Fourier transforms of the hyperﬁne tensor A in the plane perpendicular to the z-axis. The hyperﬁne interaction between the nuclear spin (I ) and an electronic spin (S) at a particular site n is now written as Hhf = y y Axx Inx Snx + Ayy In Sn + Azz Inz Snz . For spins I = 1/2, T1−1 equals 2W . For I > 1/2 – 63 Cu and 65 Cu have I = 3/2 and 17 O has a spin I = 5/2 – electric ﬁeld gradients at the nucleus give a splitting in zero magnetic ﬁeld. If the zero-ﬁeld splittings are large enough (for 63 Cu in the high-Tc compounds they are of the order of 20–30 MHz) the relaxation rates and Knight shifts can be determined not only by NMR but also by zero-ﬁeld NQR, see, e.g., MacLaughlin (1976). The (in general non-exponential) recoveries measured by NQR and NMR are related to the same transition probability W , Narath (1967). 4.1.1.3. Spin–spin relaxation. The signal induced in a coil due to the rotating in-plane nuclear spin components, decays due to spin dephasing T2 and T1 processes according to Slichter (1991): 2t 2 −2t − 2 , I (t) ∝ M(2t) = M(0) exp (4.3) T2R T2G −1 = (2 + R)/(3T1 ) is the Redﬁeld term and T2G is the Gaussian part of the echo where T2R decay assuming static neighbors. Here, R is the T1 anisotropy ratio, measured in and out of the plane. For the Cu-sites in the high-Tc compounds with Bc the Gaussian component of the transverse relaxation rate T2G takes the form (Zha et al., 1996): 2 2 1 0.69 1 −2 eff 4 eff 2 , (4.4) Fz (q) χz (q, 0) − F (q) χz (q, 0) T2G = N q z 128h¯ 2 μ4B N q

where Fzeff are effective form factors. In such cases T2 is a probe for χ (q, 0). The transverse relaxation or echo decay time T2 is sensitive for motions that are typically a few orders of magnitude slower than seen in T1 , and the ﬁeld ﬂuctuations that cause the relaxation are along the external ﬁeld direction. In solids in the absence of conduction electrons, T2 is often determined by the direct nuclear dipole–dipole interaction between like and unlike nuclear spins, see Abraham (1961), Slichter (1991), Walstedt and Cheong (1995). In the presence of conduction electrons the nuclear interaction can be enhanced by the polarization of the electrons. 4.1.1.4. NMR-NQR signal intensity. In a standard NMR-NQR experiment the amplitude of the detected voltage in the coil is proportional to the nuclear magnetization and depends on the waiting time τ after the π/2 pulse as: 2τ Nγ 2 I (I + 1)Beff I ∝ M(2τ ) = M0 exp − (4.5) . M0 = T2 3kB T The extrapolated M(2τ = 0) = M0 , integrated over the frequency window and corrected for T (Curie law) and T2 effects has to be temperature independent.

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4.1.2. Metals, superconductors and magnetic materials On the CuO2 layered compounds, like doped La2 CuO4 , magnetic, metallic, and superconducting features are present. Below we look in more detail into these particular cases. 4.1.2.1. Magnetic materials. Let us start by illustrating the power of NMR-NQR to study antiferromagnetism with the energy levels of 139La in La2 CuO4 . 139La has a nuclear spin I = 7/2 and a nuclear quadrupole moment. The line position and shape of the NMR−→ i =− NQR spectrum are determined by the following hamiltonian: H h¯ γi Iˆi .Beff + i i i 2 2 i 2 2 i i ˆ ˆ ˆ ˆ ¯ v /6](3I − I + η (I − I )) where v = [eQi Vzz ]/[4h¯ Ii (2Ii − 1)] and η = i [h Q

iz

i

ix

iy

Q

i − V i ]/[V i ]. The index i runs over all nuclear species with spin Iˆ , a nuclear gy[Vxx i yy zz i is the effective magnetic romagnetic ratio γi and an electric quadrupole moment Qi . Beff i , Vi ﬁeld at the nuclear site i that determines the magnetic properties of the spectrum. Vzz xx i are the electric ﬁeld gradient (EFG) tensor components at the nuclear site i. At and Vyy i (3Iˆ2 − Iˆ2 )/6 and for = i hv zero ﬁeld and η = 0, the hamiltonian is simpliﬁed to H ¯ Q iz i vQ = 6.0 MHz leads to three distinct resonance frequencies at 6, 12, 18 MHz, respectively. When a static local magnetic ﬁeld is present at the 139 La nuclear site, the full hamiltonian has to be considered. Diagonalization of the hamiltonian gives a splitting for each of the three lines into more than one line as a function of the growing magnetic ﬁeld. The magnitude of the splitting depends on the magnetic ﬁeld value and its orientation with respect to Vzz . Fig. 4.1 shows the splitting as a function of the development of internal ﬁeld in the x–y plane, see also MacLaughlin et al. (1994), and Borsa et al. (1995). Matsumura et al. (1997)

Fig. 4.1. Transition frequencies for the ﬁrst two zero-ﬁeld transitions, if quadrupole interaction and Zeeman energies (e.g., due to the development of an internal ﬁeld) are both relevant. The illustration is made for 139 La with I = 7/2. It is assumed that 139 La has zero ﬁeld splittings of 6.0, 12.0, and 18.0 MHz. These values are close to those typically found in the cuprates and nickelates. The ﬁeld splitting depends on the angles θ and φ, which ﬁx the orientation of the magnetic ﬁeld with respect to the electrical ﬁeld gradient EFG. Here the internal ﬁeld is assumed to develop in the x–y plane – i.e., the angle θ with the main component of the electrical ﬁeld gradient (Vzz ) equals π/2 – and φ = 0. The anisotropy parameter η is taken to be η = 0.02. The 18.0 MHz transition is not shown, after Abu-Shiekah et al. (2001).

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used 63 Cu(2) NQR to study the nature of the ordered two-dimensional antiferromagnetic spins in this same compound. Relaxation rates. The dynamics of the 2-dimensional quantum Heisenberg antiferromagnet (2DQHAF) is conventionally probed by neutron scattering (NS) and NMR-NQR techniques. Let us consider here how to make use of the relaxation times discussed in Appendix A to extract information about the dynamic properties of 2DQHAF. When → scaling conditions are used, the generalized susceptibility has the form χ (− q , ωR ) = S(S + 1)ξ z f (qξ, ω/ξ z )/(3kB T ) (Rigamonti et al., 1998), where ξ is the magnetic correlation length and z is the dynamical scaling exponent which depends on the properties of the spin dynamics. function f is given by f (qξ ) = β/1 + q 2 ξ 2 The scaling 2 2 2 2 where 1/β = [ξ /(4π )] d q /(1 + q ξ ) is a normalization factor that has been introduced to preserve the sum rule q |Sqα |2 = εNS(S + 1)/3. ε is the reduction factor for the spin ﬂuctuations amplitude in a 2DQHAF, Chakravarty et al. (1989). The correla63 to tion length ξ can be related √ to 1/T21 (see eq. (4.2)). For Cu in La2 CuO4 it reduces z 1/T1 ≈ 4200(ξ/a) /[ln(2 π ξ/a)] . The value of ξ(T ) extracted from the 1/T1 63 Cu NQR data in La2 CuO4 (Caretta et al., 1999) can be compared to the theoretical predictions of Chakravarty et al. (1989), and Hasenfratz and Neidermayer (1991) for the renormalized classical regime. The ﬁtting is excellent when z = 1 is used and is in full agreement with the correlation lengths extracted from neutron scattering. Furthermore, the correlation length can be related to the Gaussian part of the spin–spin relaxation time 1/T2G (Rigamonti et al., 1998; Caretta et al., 1999; Berthier et al., 1996). In the quantum critical regime (z = 1 and ξ a), T1 T /T2G is temperature independent. Signal intensity. Close to the ordering temperature the magnetic ﬂuctuations slow down that much that detection of the nuclei by NMR is no longer possible. Especially due to the work on the 214 cuprates and nickelates, it has become clear that the region where the signal starts to disappear (the so-called wipe-out regime) gives detailed information about the spin dynamics, which is otherwise almost impossible to get. In a pulsed NMR-NQR experiment, the total ∞ signal intensity is proportional to the nuclear magnetization M given by: M(2t) = M0 0 [exp(−2t/T2 (ω))]F (ω) dω, where F (ω) is a normalized frequency distribution function which determines the line shape. t is a waiting time after the π/2 pulse, which is usually set to be t = td as a minimum experimental “dead” time. M(0)T should be temperature independent, as the nuclear magnetization follows the Curie law. This basic NMR concept seems to be violated in strongly underdoped high-Tc cuprates, where the signal is often lost far above the magnetic freezing or ordering temperature. A description for the experimental observation of this wipe-out effect is discussed below. Curro et al. (2000), Suh et al. (2000) and Teitel’baum et al. (2000a) showed that wipeout effects in the cuprates are due to a distribution of short dephasing times (caused by a distribution in correlation times (τ ) or activation energies, because τ = τ∞ exp(Ea /T ). The distribution of the activation energies Ea is not known beforehand and might be, e.g., half Gaussian or Gaussian. Due to the distribution in activation energies M becomes: ∞ ∞

M(2t) = exp −2t/T2 (ω, Ea ) F (ω) dω P (Ea ) dEa . (4.6) NM0 0 0 If the signal is governed by various magnetization recovery rates, the resulting time dependence can often be approximated by a stretched exponential of the form M(0) ∝

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Fig. 4.2. (a) Numerical solution of the time dependence of the signal intensity I according to eq. (4.6) at f = 38.2 MHz at 120 K (drawn), 60 K (dotted) and 30 K (dashed). Due to the rf pulse at t the echo appears at time 2t. The parameters Δ and E0 are 28, 60 K. Iex is the value at t = 0, extrapolated from the decay after the dead time 2td . The ﬁgure shows that an extrapolation of the magnetization data M(t) to time zero (t = 0) with the apparent T2 will always be less than the real magnetization due to the non-zero dead time of the set-up (the magnetization M and intensity I are proportional to each other). The higher the spin dephasing rate the larger this error will be. (b) Numerical solution for eq. (4.7) with the same parameters as in a, E2 /T = 1 and E1 /T = 8, after Abu-Shiekah (2001). The reappearance at low temperatures has the same explanation as the wipe-out at high temperature. The slowing down of the electron spin ﬂuctuations lead to very short dephasing times in a certain temperature window. At high T the ﬂuctuations are too fast, at low T too slow. In real systems the spin dynamics changes when entering the low temperature phase and the low temperature recovery cannot be described by the high temperature physics, see, e.g., next section.

exp[−(2t/T2 )α ]. In the usual echo pulse sequence π/2-t-π -t, t can never be smaller than the dead time td of the set-up. Fig. 4.2(a) gives a numerical plot for the signal decay at three temperatures as a function of 2t. It shows that an extrapolation of M(t) to t = 0 with the apparent T2 will always be less than M0 due to the ﬁnite dead time. In case of a Gaussian distribution of activation energies, the signal extrapolated to time zero, is given by Teitel’baum et al. (2000a, 2000b): E2 M(0, ω) Ea − E0 2 −2td exp − √ dEa exp

d = T2 (ω, Δ) 2Δ NM0 exp T2t(ω) E0 2

+

∞ E1

−2td Ea − E0 2 exp dEa . exp − √ T2 (ω, Δ) 2Δ

(4.7)

The ﬁrst term in eq. (4.7) represents the signal decay, while the second term is responsible for the recovery. E1 is related to E2 by E2 − E1 = T ln(τ2 /τ1 ), where the cut-off τi follows from the hyperﬁne coupling and the shortest measurable dephasing time (Teitel’baum et al., 2000a). A typical numerical solution for eq. (4.7) is shown in ﬁg. 4.2(b). In real systems the spin dynamics changes when entering the low temperature phase and the low temperature recovery cannot be described by the high temperature physics, see sections 5 and 6. 4.1.2.2. Strongly correlated metals. In simple metals, where the Fermi contact term of the s-electrons dominates, the susceptibility is of the Lindhard form (Shastry and Abrahams, 1994):

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1 q 2 , χL (q, ω) = χP 1 − 3 2kF

χL (q, ω) =

πχP ω

θ |ω − vF q| 2 vF q

(4.8)

to leading order in q, ω (the spin susceptibility χ0 in eq. (4.1) now equals χL (q, 0)). Here χP is the Pauli susceptibility, θ the step function and vF the Fermi velocity (Ashcroft and Mermin, 1976). Eq. (4.8) leads to the well-known Korringa relation (Korringa, 1950), 2 1 μB 2 K T1 T = (4.9) =S π h¯ kB γn with S the Korringa constant. This equation links the relative Knight shift (K) to the nuclear-spin lattice relaxation rate (T1−1 ). Although different independent electronic contributions to the Knight shift might cancel each other, they always add in the relaxation rate. In the presence of an effective on-site repulsive Coulomb interaction U the susceptibility will be Stoner enhanced (Moriya, 1963; Shastry and Abrahams, 1994; van der Klink and Brom, 2000) and the simple Korringa relation has to be modiﬁed. Below we give more details about the NMR approach in the strongly correlated cuprates. We use the 123compounds, see section 8, as example. The Mila–Rice–Shastry hamiltonian. Mila and Rice (1989), see also Shastry (1989), derived the effective electron-nuclear interaction for the various nuclei in YBa2 Cu3 O7−δ . The nuclear spin of 63 Cu at the lattice site ri (63 I (ri )) appears to interact not only with the electron spin at ri but to have also a non-negligible transferred hyperﬁne coupling to its four neighboring Cu2+ spins: 63 NN Iα (ri ) Aα,β Sβ (ri ) + B Sα (rj ) , Hhf = (4.10) α

β

j

where Aα,β is the tensor for the direct, on-site coupling of the 63 Cu nuclei to the Cu2+ spins, and B is the strength of the transferred hyperﬁne coupling. Similar expressions can be written down for the hyperﬁne interactions of the nuclear spins of 17 O (17 I ) and 89 Y (89I ) with Cα,β resp. Dα,β the transferred hyperﬁne couplings of the 17 O resp. 89 Y nuclear resp. Dα,β its coupling to the nextspins to its nearest-neighbor (NN) Cu2+ spins, and Cα,β 2+ nearest-neighbor (NNN) Cu spins (C and D are scalars in the SMR form). The C -term is not present in the original SMR hamiltonian and is introduced by Zha et al. (1996). Knight shift. The expression for the Cu(2) Knight shift contains Ac , Aab and B as three free parameters (the static spin susceptibility χ0 is in principle known from experiment). As the experiment gives only the Knight shift in the c and ab direction, Kc and Kab resp. 63

Kc =

(Ac + 4B)χ0 , γe63 γn h¯ 2

63

Kab =

(Aab + 4B)χ0 , γe63 γn h¯ 2

(4.11)

there remains one parameter undetermined. The unknown parameter is often chosen to be the ratio αr between Aab and 4B (Millis et al., 1990; Millis and Monien, 1992). For the oxygen sites in the CuO2 -plane the Knight shift is given by 17 Kβ = [2(Cβ + Cβ )χ0 / γe17 γn h¯ 2 . For yttrium a similar equation holds. The description of the Knight shift in La2−x Srx CuO4 and YBa2 Cua3 O7−δ for various values of x or δ with the same parame-

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ter set works well and is a strong point for the one spin model, see Millis et al. (1990), Alloul et al. (1988), Monien et al. (1991), Takigawa et al. (1991). In underdoped compounds χ0 decreases below some temperature T0 well above Tc , see ﬁg. 1.3, which can be seen as the opening of a pseudogap (see pseudogap) (Pines, 1997; Berthier et al., 1996). Relaxation rates. In contrast to the Knight shifts the T -dependencies of the relaxation rates for the various nuclei differ considerably. For the Cu nuclei 1/T1 T has a maximum at (in case of optimal doping) or above (for underdoped samples) Tc , and follows often a Curie–Weiss like behavior (Asayama and Kitaoka, 1996). For oxygen and yttrium T1 T is Korringa-like (due to the inaccuracy in K it is experimentally hard to distinguish whether T1 T K or T1 T K 2 is constant, see Berthier et al. (1996)). Enhancement of the dynamic susceptibility due to antiferromagnetic (AF) ﬂuctuations in combination with ﬁltering due to the form factors can explain these results. This idea is behind the nearly antiferromagnetic Fermi liquid (NAFL) model, proposed by Millis, Monien and Pines (MMP) (1990). A normal Fermi-liquid and an antiferromagnetic part (nearly antiferromagnetic Fermi liquid) make up the susceptibility: χ(q, ω) = χFL (q, ω) + χNAFL(q, ω).

(4.12)

The Fermi liquid like dynamic susceptibility has the form χFL (q, ω) = χq /(1 − iω/Γq ). Γq = is comparable to the bandwidth and χq ∼ χ0 . The real and imaginary part of χFL are χFL ωπχ0 h¯ /Γ and χFL = χ0 . The antiferromagnetic part dominates close to Q = (π/a, π/a), and has a Lorentzian dependence on the correlation length ξ : χNAFL = χQ /[1 + (Q − q)2 ξ 2 − iω/ωSF ] with the static susceptibility at the antiferromagnetic wave vector χQ ∝ ξ 2 . MMP propose that the frequency of the relaxational mode, ωSF , depends on ξ as ωSF ∝ ξ −2 . Adding χFL and χNAFL gives (for low ω) (ξ/a)4 πχ0 h¯ ω 1+β , χ (q, ω) = Γ [1 + (q − Q)2 ξ 2 ]2 (4.13) 2 (ξ/a) χ (q, ω) = χ0 1 + β . 1 + (q − Q)2 ξ 2 The real and imaginary part of the susceptibilities are linked via the Kramers–Kronig relations. Apart from the magnetic correlation length ξ(T ) and the broadening Γ the proposed form for χ has as an additional parameter β, which gives the relative strength of the AF ﬂuctuations to the zone center ﬂuctuations. The form factors F (q), see eq. (4.2), follow from the hyperﬁne interaction hamiltonian (the hyperﬁne ﬁelds act coherently, see Berthier et al. (1996), Millis et al. (1990)). For sufﬁciently large values of ξ (ξ/a > 2), 63 (T1 T )−1 varies as (ξ/a)2 , while 1/63T2G ∝ (ξ/a), see eq. (4.4). 4.1.2.3. Superconductors. As superconductivity means shielding of applied magnetic ﬁelds, at ﬁrst sight it seems surprising that NMR remains possible in the superconducting state. In type I superconductors conventional NMR experiments are indeed hampered in their sensitivity for three reasons: (1) the applied static magnetic ﬁeld is shielded, which means that only the nuclei within the penetration depth λ feel a (very inhomogeneous) static ﬁeld, (2) the rf-pulses cannot penetrate the sample except for this outer layer, and (3) the

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magnetization of the nuclei which leads to the ﬁnal NMR signal is shielded again. In type II superconductors the situation is different as the external ﬁeld penetrates the sample via vortices and gives a more homogeneous ﬁeld distribution. Also the rf pulses are not so strongly shielded from the interior as they bend the vortex pattern. NMR on powders and single crystals with the right geometry is possible, albeit the line intensity is much reduced with respect to the normal state (Moonen and Brom, 1995). Due to spin pairing the spin susceptibility χ0 and hence the Knight shift (K ∝ χ0 ) is expected to decrease with T according to the Yoshida function: ∞ K ∝ μ2B NBCS (E) ∂f (E)/∂E dE Δ

with the density of excited states NBCS (E) = N(0)|E|/(E 2 − Δ2 )1/2 . The opening of the superconducting gap increases the density of excited states just above the gap (Schrieffer, 1988; Tinkham, 1975), while also the coherence factors will be important (MacLaughlin, 1976). As a result the nuclear relaxation can show the so-called Hebel–Slichter enhancement below Tc . In ﬁelds close to Bc2 , the vortices are closely packed and the nuclei in the “normal” vortex cores can form an additional relaxation channel (Weger, 1972). 4.2. μSR In neutron spectroscopy the complex susceptibility χ(q, ω) can be measured as a function of q, where NMR is only sensitive for certain q-values (depending on the location of the NMR nuclei). The energy ranges of neutrons and NMR are almost complementary, neutrons covering typically the range from 0.1 meV up to eVs, while the ﬂuctuations measured by NMR typically extend from 1011 s−1 to orders of magnitude below 1 Hz. By extrapolation NMR and neutron data can be compared (Berthier et al., 1996; Rossat-Mignod et al., 1991a, 1991b). The dephasing on the initial aligned moments of the muons is governed by the same ﬂuctuating ﬁelds that determine the NMR T1 . The muon-data (counts of positrons being the decay product of the muon, which has a life time of 2.2 μs; the positrons are preferentially emitted along the direction of the muon spin before decaying) are usually taken in low ﬁelds (<0.1 T) compared to NMR (>0.1 T), and are sensitive for spin ﬂuctuation rates in the range 104 –1011 s−1 . μSR (Brewer, 1995) and NMR are also both sensitive for local magnetic order. Representative zero-ﬁeld μSR spectra for La1.8−x Eu0.2 Srx CuO4 are shown in ﬁg. 4.3. Klauss et al. (2000) performed μSR measurements for 0 x 0.2 and found a striking non-monotonic Sr concentration dependence for the static magnetic order. In the data for x = 0 and x = 0.014 a clear precession of the time evolution of the muon spin polarization is visible which proves the presence of a magnetic ﬁeld at the muon site. This precession is absent for higher doping 0.02 x 0.08. For x = 0.02 a strong decay of the muon spin polarization signals the proximity of spin freezing, whereas there is no indication for static magnetic order at 10 K for x = 0.04 and x = 0.08. However, clear signatures of static magnetic order reappear for higher Sr concentrations. A strong decay and a precession is visible in the spectrum for x = 0.1 and the signatures of magnetic order are even more pronounced for x = 0.12 and x = 0.15. Further increasing the doping suppresses the magnetic order and for x = 0.2 the authors ﬁnd only an onset of spin freezing at T = 8 K. For further discussions see the section about 214-compounds.

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Fig. 4.3. Representative zero-ﬁeld μSR spectra at T = 10 K for La1.8−x Eu0.2 Srx CuO4 , after Klauss et al. (2000). Klauss et al. performed μSR measurements for 0 x 0.2 and found a striking non-monotonic Sr concentration dependence for the static magnetic order. In the data for x = 0 and x = 0.014 a clear precession of the time evolution of the muon spin polarization is visible which proves the presence of a magnetic ﬁeld at the muon site. This precession is absent for higher doping 0.02 x 0.08. For x = 0.02 a strong decay of the muon spin polarization signals the proximity of spin freezing, whereas there is no indication for static magnetic order at 10 K for x = 0.04 and x = 0.08. However, clear signatures of static magnetic order reappear for higher Sr concentrations. A strong decay and a precession is visible in the spectrum for x = 0.1 and the signatures of magnetic order are even more pronounced for x = 0.12 and x = 0.15. Further increasing the doping suppresses the magnetic order and for x = 0.2 the authors ﬁnd only an onset of spin freezing at T = 8 K.

4.3. Neutron scattering Elastic and inelastic neutron scattering have played and still play an essential role in our understanding of the normal and superconducting properties of the high-Tc cuprates. Due to its spin sensitivity detailed information about static and dynamical stripes have been obtained. HMM-10 gives a concise review of the relevant scattering theory. In ﬁg. 4.4 we reproduce the data of Mook et al. (2000) on YBa2 Cu3 O6.6 , showing the one dimensional nature of the magnetic ﬂuctuations in underdoped 123-compounds. The inelastic neutron data show a four-fold pattern of incommensurate points around the magnetic (1/2, 1/2) reciprocal lattice position. The interpretation of such measurements had been unclear, with both striped phases and nested Fermi surfaces being possible explanations. Because the stripe phase is expected to propagate along a single direction, only a single set of satellites around the antiferromagnetic position should be observed. For La1.48 Sr0.12Nd0.4CuO4 Tranquada et al. (1995) suggested that the four-fold symmetry might arise from stripes that alternate in directions as the planes are stacked along the c-axis, in accordance with the tilt of the octahedra. In the 123-system, the a and b directions are inequivalent in the CuO2 planes, due to the presence of the chains (orthorhombic structure), and it is expected that in a single orthorhombic domain stripes preferentially align along one of the chains. Mook et al. (2000) took scans of the magnetic incommensurate scattering for the twinned and partially detwinned sample at 10 K. The scan centers of scan 1 is at (0.55, 0.55, 2) in reciprocal lattice units (rlu), while the center of scan 2 is (0.45, 0.45, 2) rlu, both along the b∗ -direction, connecting the IC peaks. Before detwinning the intensity ratio’s of the satellites along the a ∗ and b∗ direction are 1.05 ± 0.05. After detwinning 1.95 ± 0.21, which is the same as the population of the domains for the two directions. The results support one-dimensional striped excitations.

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Fig. 4.4. One-dimensional nature of the magnetic ﬂuctuations in YBa2 Cu3 O6.6 , after Mook et al. (2000). Inelastic neutron data show a four-fold pattern of incommensurate points around the magnetic (1/2, 1/2) reciprocal lattice position. The interpretation of these measurements has been unclear, with both striped phases and nested Fermi surfaces being possible explanations. Because the stripe phase is expected to propagate along a single direction, only a single set of satellites around the antiferromagnetic position should be observed. Scans of the magnetic incommensurate scattering for the twinned and partially detwinned sample are taken at 10 K. The scan centers of scan 1 is at (0.55, 0.55, 2) in reciprocal lattice units (rlu), while the center of scan 2 is (0.45, 0.45, 2) rlu, both along the b∗ -direction, connecting the IC peaks. Before detwinning the intensity ratios of the satellites along the a ∗ - and b∗ -direction are 1.05 ± 0.05. After detwinning 1.95 ± 0.21, which is the same as the population of the domains for the two directions. The results support one-dimensional striped excitations.

4.4. ARPES In angular resolved photoemission spectroscopy (ARPES) photons are absorbed by the material and electrons are ejected. The control parameters are the frequency and the polarization of the incident photons and the measured quantities are the kinetic energy and the angle of emergence (θ, φ) of the outgoing electrons relative to the sample normal. During the high-Tc era the wave vector and energy resolution of the ARPES technique has improved by orders of magnitude, making the technique one of the major sources of insight. In ARPES the component of k parallel to the surface is conserved as the electron emerges from the sample. In the quasi-two-dimensional cuprates this allows an unambiguous determination of the momentum of the hole state from the measurement of the photoelectron

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Fig. 4.5. One-dimensional electronic structure in La1.28 Nd0.6 Sr0.12 CuO4 according to ARPES, after Zhou et al. (1999). The cross deﬁned by the drawn lines in (a) deﬁne the regions, where the spectral weight is mainly concentrated. For comparison the conventional 2D Fermi surface is indicated by the dashed lines. The Fermi surface expected from two perpendicular 1D stripe domains in the 1D interpretation form a cross (b), that resembles the cross in (a) closely. Γ is the center of the Brillouin zone.

momentum. The most accurate ARPES data come from Bi2212, where the natural cleavage plane lies in between the BiO bilayer, which results in extremely smooth surfaces with minimal charge transfer. These conditions are crucial for ARPES, since it is a surface sensitive technique due to the short escape depth of about 1 nm of the outgoing electron. One main difﬁculty in precisely determining the Fermi-level crossing or the Fermi wave vector (kF ) by ARPES lies in the fact that ARPES probes only the occupied states. This procedure works well when the band dispersion is experimentally well deﬁned. In more difﬁcult cases one might plot the ARPES intensity around EF as function of k – however it is important to account well for the inﬂuence of the matrix elements (Kipp et al., 1999; Bansil and Lindroos, 1999; Borisenko et al., 2000, 2001; Fretwell et al., 2000; Lindroos et al., 2002). In ﬁg. 4.5 the one-dimensional electronic structure in La1.28Nd0.6Sr0.12 CuO4 is illustrated, according to the ARPES data of Zhou et al. (1999). The dotted lines deﬁne the regions, where the spectral weight is mainly concentrated. The Fermi surface expected from two perpendicular 1D stripe domains in the 1D interpretation form a cross, that resembles the dotted lines closely. Γ is the center of the Brillouin zone. 4.5. STM Recently scanning tunneling microscopy (STM) has been used to image the electronic structure of Bi2212 and revealed stunning results. It appears that in optimally doped Bi2212 the vortex cores are surrounded by a four unit cell periodic pattern of quasi-particle states, see Hoffman et al. (2002a) and Sachdev and Zhang (2002), which is related to a localized spin modulation. In underdoped Bi2212 even without a magnetic ﬁeld STM reveals an apparent segregation of the electronic structure into superconducting domains of a few nm in size, located in an electronically distinct background, see Lang et al. (2002) and Zaanen (2002). We discuss the experiment of Hoffman et al. (2002a) in more detail.

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Fig. 4.6. A schematic of the power spectrum of the Fourier transform of the function S(E2 , E1 )(x, y, B) = E2 E1 [LDOS(E, x, y, B) − LDOS(E, x, h, 0)] dE with E1 = ±1 V and E2 = ±12 V in Bi2212, after Hoffman et al. (2002a). Peaks due to the atoms at (0, ±1) and (±1, 0) are labelled A. Peaks due to the supermodulation are observed at B. The four peaks at C occur only in a magnetic ﬁeld and represent the vortex-induced effects at k-space locations (0, ±1/4) and (±1/4, 0).

First the surface is imaged in the usual topographic mode, where the tip current is kept constant by adjusting the height of the tip above the surface. The displacements of the tip are translated to the structure of the scanned surface. Hoffman et al. (2002a) used in addition a recently developed technique to measure the low-energy quasi-particle density at the vortices (Pan et al., 2000) by taking differential tunneling conductance spectra as function of sample bias voltage. Making a Fourier transform gives the charge density, n(q, ω), at the measured energy interval as function of q. Before applying the magnetic ﬁeld, ﬁrst the differential tunneling resistance (G = dI /dV ) is measured in zero-ﬁeld. Because G(V ) is proportional to the local density of states (LDOS) at E = eV , this results in a two-dimensional LDOS map. Thereafter a similar scan is made with the magnetic ﬁeld (B). The two spectra, integrated over all additional spectral density induced by the B ﬁeld between the energies E1 and E2 at each location x, 2 y are subtracted: S(E2 , E1 )(x, y, B) = E E1 [LDOS(E, x, y, B) − LDOS(E, x, h, 0)] dE with E1 = ±1 V and E2 = ±12 V. A schematic image of the two-dimensional Fourier transform of the power spectrum in 5 T is shown in ﬁg. 4.6. Scans as function of the energy interval show that the data are linked to the strong inhomogeneity in the samples, which leads to scattering events that are sensitive to the peculiar gap structure. 4.6. Thermodynamic techniques, like susceptibility and heat capacity Although the principles of thermodynamic techniques are known for a long time, the application to high-Tc compounds is far from trivial. In the speciﬁc heat data precise correction for the lattice background appeared to be crucial (Loram et al., 1993), and in the analysis the role of ﬂuctuations appears to be a source of confusion, see van der Marel et al. (2002). In the susceptibility data often only by taking derivatives relevant structures appear. In ﬁg. 4.7 we sketch the susceptibility data of Lee and Cheong (1997) on La5/3Sr1/3 NiO4 , where the effects of charge and spin order are made visible by taking the T derivative

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Fig. 4.7. The magnetic susceptibility in La5/3 Sr1/3 NiO4 , after Lee and Cheong (1997). The effects of charge and spin order are made visible by taking the T derivative d(χ T )/dT of χ T (units on the vertical scale are emu/mole). Often these effects are less pronounced. There are three different phases denoted by A, B, and C with decreasing T . Spin order occurs in phases B and C, but not in A. The anomalies establish that the charge-spin correlations in the three phases are static on a time scale more than a few seconds. Charge correlations are much shorter in A than in B or C.

d(χT )/dT of χT . In samples with the same nominal composition, these effects are often less pronounced. In this particular case there are three different phases denoted by I, II, and III with decreasing T . Spin order occurs in phases II and III, but not in I. The anomalies establish that the charge-spin correlations in the three phases are static on a time scale more than a few seconds. Charge correlations are much shorter in I than in II or III. 5. The hole-doped single-layer 214-cuprates Compared to other high temperature superconductors, the phase diagram of hole-doped La2 CuO4 (ﬁg. 5.1 gives the unit cell), or more speciﬁcally of La2−x−y Srx (Nd, Eu)y CuO4+z is especially rich, see also the introduction. It involves the undoped antiferromagnetic phase, the stripe phase, the d-wave superconducting and the overdoped, possibly Fermi-surface phase. The antiferromagnetic phase has been reviewed in HMM10. Figs 5.2 and 5.3 show the results of the elastic neutron scattering experiment of Tranquada et al. (1995), that spurred the interest in stripes. It appeared possible to capture the typical features of a modulated charge and spin state in quasi-elastic neutron scattering not only in oxygen doped non-superconducting nickelates (next section), but also in La1.48Sr0.12 Nd0.4CuO4 . The data are well explained by the presence of doped stripes of holes with carry no spin and serve as antiphase boundary for domains of antiferromagnetically aligned spins that have no charge. Since 1997, the properties of the stripe phase have been reﬁned by a variety of techniques, which not only include the closely related spin sensitive spectroscopy techniques of neutron scattering, magnetic resonance and muon spin rotation, but also via charge sensitive methods like electric transport, ARPES, and X-ray (see experimental techniques). Due to the chemical phase separation in samples with excess oxygen, we conﬁne our discussion to samples with z = 0. References to neutron, NMR and other magnetic work on doped 214 before 1997 can be found in HMM10. The structural properties are brieﬂy

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Fig. 5.1. The unit cell of La2−x Srx CuO4 , in the tetragonal phase a = 0.378 nm, c = 1.329 nm, spacegroup I 4/mmm. The (x, y, z) coordinates for (La, Sr) (gray), Cu (black), O1 and O2 (open circles) are resp. (0.000, 0.000, 0.3606), (0.000, 0.000, 0.000), (0.000, 0.500, 0.000) and (0.000, 0.000, 0.1828), see HMM10.

Fig. 5.2. The modulated spin and charge wave vector in La1.6−x Nd0.4 Srx CuO4 with x = 0.12. The magnetic peaks are characterized by the two-dimensional wave vectors (1/2 ± ε, 1/2, 0) and (1/2, 1/2 ± ε, 0) with ε ≈ x, which corresponds to one hole for every 2 Cu sites. If stripes are pinned by the atomic displacement pattern in the low temperature tetragonal (LTT) phase within a layer, they must rotate by π/2 from one layer to another. Indeed the stripes are essential two-dimensional with modulation wave vectors (±2ε, 0, l) and (0, ±ε, 0), after Tranquada et al. (1995).

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Fig. 5.3. Idealized diagram of the spin (arrows) and charge (circles) stripe pattern within a CuO2 plane of hole-doped La2 CuO4 with nh = 1/8. Only the Cu atoms are represented – oxygen atoms that surround the Cu atom in a square array are omitted. The arrows indicate the orientation of the magnetic moments of the metal atoms, which are locally antiparallel. The spin direction rotates by π (relative to a simple antiferromagnetic structure) on crossing a domain wall (antiphase boundary). Holes are located at the anti-phase domain boundaries, which are indicated by the rows of circles. A ﬁlled circle denotes the presence of one hole, centered on a metal site (the hole density is assumed to be uniform along a domain wall), after Tranquada et al. (1995).

mentioned in section 1 and the caption to ﬁg. 5.1. The theoretical background is given in section 2. 5.1. Neutron scattering After extensive neutron research on other nickelates and cuprates, Tranquada et al. (1999) reexamined the degree of stripe order in La1.48Sr0.12Nd0.4 CuO4 and La1.45 Sr0.15Nd0.4CuO4 , in which static stripe order was promoted by substituting 40% of the La-sites by Nd. For x = 0.12 the width of the elastic magnetic peak saturates below 30 K to a value that corresponds to a spin–spin correlation-length of 20 nm. Previously measured data on chargeorder peaks and these new data can be consistently explained by disorder in stripe spacing. Above 30 K the width of the elastic peak increases, which is characteristic for slowly ﬂuctuating spins, see ﬁg. 5.4. This phenomenon will be discussed later together with the NMR and μSR data. Inelastic scattering measurements show that incommensurate spin excitations survive at and above 50 K, where the elastic signal is negligible. For the x = 0.12 sample, it appears that the low temperature tetragonal (LTT) phase, which is believed to be favorable for pinning the stripe phase, see also μSR, and the low-temperature orthorhombic (LTO) phase coexist between 40 and 70 K. For x = 0.15 the coexistence region of both phases is only 7 K. The sharp Bragg peaks in the LTT phase of the latter sample indicate that the domain size exceeds 100 nm. In slightly doped La1.55Nd0.4Sr0.05 CuO4 Wakimoto et al. (2001b) observed diagonal stripe correlations with the modulation direction only along the orthorhombic b axis, just as in Nd-free La1.95Sr0.05 CuO4 , see below. Also in pristine La2−x Srx CuO4 signatures of static stripes have been observed. These signatures are most developed close to x = 1/8, where the periodicity of the stripe phase is commensurate with the lattice and might be pinned more easily. Suzuki et al. (1998) observed magnetic superlattice peaks in neutron-diffraction measurements on orthorhombic

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Fig. 5.4. The growth of the correlation length in La1.64 Nd0.4 Sr0.12 CuO4 , seen by Tranquada et al. (1999) in elastic neutron scattering. (a) Integrated intensity of the magnetic peaks in a.u. The resolution width is much less than the measured width. The elastic signal seen by neutron scattering appears to survive up to approximately 50 K, while in μSR a loss of order was already detected near 30 K. In μSR the local hyperﬁne ﬁeld is considered to be static on a time scale of a few μs, while neutron measurements integrate over ﬂuctuations within a Gaussian resolution window with a full width at half maximum of approximately 0.2 THz. (b) Peak half-width κ, showing the restricted size of the correlation length at low temperatures. Taking the inverse of κ at low temperature gives a correlation length of 20 nm. Above 30 K κ starts to increase and the data can be ﬁtted to κ = κ0 + A exp[−B/T ], with κ0 = 0.00050 nm−1 , A = 0.06 nm−1 and B ∼ 200 K. The second term on the right side has the form predicted for the 2D quantum antiferromagnetic Heisenberg model in the renormalized regime, see text.

La1.88Sr0.12CuO4 , and magnetic broadening of the 136 La NMR-line below 45 K and softening of the longitudinal sound waves along the [110] direction in the same single crystal. These features suggest that the dynamical incommensurate spin correlation is pinned by a lattice instability, which develops when approaching the low-temperature tetragonal phase. Kimura et al. (1999) noted that the magnetic peak intensity ﬁrst appears at the onset of superconductivity. The static magnetic correlation length exceeds 20 nm. Zn-substitution degrades the magnetic order. Katano et al. (2000) have investigated the effects of a strong magnetic ﬁeld on the superconductivity and the static antiferromagnetic correlations. If the ﬁeld is applied perpendicular to the CuO2 plane superconductivity is severely suppressed at 10 T. The intensity of the incommensurate magnetic peaks around the (π, π) point is increased by as much as 50% of that at 0 T. This enhancement can be explained by the suppression of the low energy spin ﬂuctuations. Lake et al. (2002) performed magnetic neutron diffraction to detect the spin ordering in single crystals of underdoped La2−x Srx CuO4 with x = 0.10. The magnetic ﬁeld was aligned perpendicular to the CuO2 -planes. Like in x = 0.12, in zero ﬁeld incommensurate elastic peaks appeared in the superconducting state, which are absent above the superconducting transition temperature of 30 K. The external ﬁeld markedly enhanced the amplitude of the signal. In the ﬁeld of 14.5 T the magnetic in-plane correlation length of 40 nm is much greater than the superconducting coherence length and the intervortex spacing, which implies that superconductivity and antiferromagnetism coexist throughout the bulk of the material. The ﬁnding seems to point to the existence of a magnetic quantum critical point very close to the superconductor in the phase diagram. The neutron data combined with resistivity measurements also strongly suggest that in a magnetic ﬁeld above the critical ﬁeld Hc2 the sample becomes an incommensurate antiferromagnetic insulator.

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Fig. 5.5. Sr-doping dependence of the incommensurability δ, deﬁned as half of the peak splitting between the two peaks at (1/2, 1/2 ± δ, 0) indexed in the tetragonal plane (h, k, 0), of the spin ﬂuctuations seen in inelastic neutron scattering by Yamada et al. (1998) for La2−x Srx CuO4 (open circles). Closed circles are data of Wakimoto et al. (1999, 2000) on La2−x Srx CuO4 for 0.03 x 0.07, obtained by elastic neutron scattering. The relation between δ and hole concentration appears linear up to 0.12. At low hole concentration the modulated structure changes form horizontal (vertical) stripes (b) to diagonal (a) as in the nickelates, see inserts. Measurements on a single twin of x = 0.05 show that the modulation wave vector is only along the b∗ -axis, demonstrating that these crystal have one-dimensional rather than two-dimensional static diagonal spin modulation at low temperatures (comparable to the nickelates).

Aeppli et al. (1997) used polarized and unpolarized neutron scattering to measure the wave vector- and frequency-dependent magnetic ﬂuctuations in the normal state of La1.86Sr0.14 CuO4 . They found nearly diverging amplitude and length scales and ω/T scaling, suggesting a nearby quantum critical point with a dynamical critical exponent z = 1. The doping dependence of the low lying excitations over the hole range of dopants in La2−x Srx CuO4 has been systematically studied by Yamada et al. (1998). A compilation of their results with those of other groups allows to follow the doping dependence of the incommensurability δ of the spin ﬂuctuations, see ﬁg. 5.5. The incommensurability δ is deﬁned as half of the peak splitting between the two peaks at (1/2, 1/2 ± δ, 0) indexed in the tetragonal reciprocal plane (h, k, 0). The position (1/2, 1/2, 0) corresponds to an antiferromagnetic Bragg peak position in the long-range-ordered antiferromagnetic phase. It appeared also possible to add points to the static magnetic phase diagram at very low doping concentrations. Wakimoto et al. (1999, 2000) performed neutron-scattering experiments on lightly doped La2−x Srx CuO4 single crystals in both the insulating (x = 0.03, 0.04, 0.05) and superconducting (x = 0.06) regions. Elastic magnetic peaks appeared at low temperature in all samples. Incommensurate peaks are observed only at x = 0.05, the position of which are rotated by 45◦ in reciprocal space about (π, π) from those observed for x = 0.06. Reﬁned measurements on a single twin of x = 0.05 show that the modulation wave vector is only along the b∗ -axis. It demonstrates that La1.95 Sr0.05 CuO4 has a one-dimensional rather than two-dimensional static diagonal spin modulation at low temperatures. The x = 0.04 crystal has the same IC structure as the x = 0.05 crystal. δ follows the relation δ = x up to x = 0.14. Fujita et al. (2002a) conﬁrmed the one-dimensional spin modulation along the orthorhombic b axis for the insulating compounds with x = 0.04 and

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x = 0.053. Just inside the superconducting phase x = 0.06 two pairs of incommensurate peaks are additionally observed corresponding to the spin modulation along the tetragonal axis. In ﬁg. 5.5 the data of elastic and inelastic neutron scattering of Yamada et al. (1998), Wakimoto et al. (1999, 2000) and Fujita et al. (2002a) are summarized. The relation between δ and hole concentration appears linear up to 0.12, see section 2. At high doping, not only at x = 1/8 but also at x = 1/5 static stripe ordering might be promoted. Koike et al. (2001) report resistivity, neutron scattering and μSR data for La2−x Srx Cu1−y Zny O4 with y = 0.01. At 2 K dynamical stripe correlations of spins and holes seem to become static. The ordered magnetic moment μ for the same series (Wakimoto et al., 2001a) varies form 0.18 μB /Cu (x = 0.03) to 0.06 μB /Cu (x = 0.07). No signiﬁcant anomaly is observed at the insulator-superconducting boundary (x = 0.055). The elastic neutron scattering is enhanced in the vicinity of x = 0.12, where the apparent magnetic and superconducting transitions coincide. The ordered phases in samples with 0.03 x 0.07 have a small correlation length ξ = 0.2 nm, while for x = 0.12 the peak is resolution limited, i.e., ξ 20 nm. When looked at the oxygen lattice vibrations by inelastic neutron scattering, McQueeney et al. (2001) do observe an abrupt development of new vibrations near the doping-induced metal-insulator transition, which correlates with the electronic susceptibility measured by photoemission. The electron–lattice coupling can be regarded as a localized one-dimensional response of the lattice to short-ranged metallic charge ﬂuctuations. Bo˜zin and coworkers (2000) resolved the atomic pair distribution function for 0 < x < 0.3 at 10 K. The in-plane Cu–O bond distribution broadens as a function of doping up to optimal doping. Thereafter the peak abruptly sharpens. The peak can well be explained by a local microscopic coexistence of doped and undoped material. It suggests a crossover from a charge inhomogeneous state at and below optimal doping to a homogeneous charge state above optimal doping. This effect correlates with the disappearance of the normal-state pseudogap. Also in an oxygen doped predominantly stage 4 La2 CuO4+y single crystal (onset temperature of 42 K) the low temperature magnetic scattering shows the same incommensurability as the Sr-doped samples (Lee et al., 1999). The simultaneous appearance of the elastic magnetic peaks and superconductivity suggest that the two phenomena are correlated. The incommensurate wave vector appears not precisely aligned with the Cu–O–Cu tetragonal direction, but rotated by 3.3◦ with respect to the Cu–O–Cu direction (Y-shift), see also section 2. Mason et al. (1996) measured the effect of superconductivity on the magnetic response near (π, π) and at intermediate frequencies (6 < h¯ ω < 15 meV). As the total moment sum rule and the singlet nature of superconductivity suggest, they found that the suppression of the low energy magnetic response in the superconducting state is accompanied by an increase in the response at higher energies. For h¯ ω just above the energy, where superconductivity begins to enhance magnetic scattering, the spectral weight added by superconductivity is extraordinary sharp, implying a new and long length scale. Lake et al. (1999) used inelastic neutron scattering to determine the wavevector-dependence of the spin pairing. The spin gap is wavevector independent, even though superconductivity signiﬁcantly alters the wavevector dependence of the spin ﬂuctuations at higher energies. It shows that the spin

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excitations do not parallel the charge excitations in the superconducting state, which have d-wave character (note that the charge excitations seen in photon electron spectroscopy are not hole states as they are seen via the emitted electrons). Lake et al. (1999) note the similarity with Luther–Emery liquids (Luther and Emery, 1974; Rokhsar and Kivelson, 1988), materials with gapped (triplet) spin excitations and gapless spin-zero charge excitations. According to Morr and Pines (2000a) this effect is a direct consequence of changes in the damping of incommensurate antiferromagnetic spin ﬂuctuations due to the appearance of a d-wave gap in the fermionic spectrum. In La1.875Ba0.125−x Srx CuO4 (x = 0.05, 0.06, 0.075 and 0.085) Fujita et al. (2002b) studied the competition of charge- and spindensity-wave order and superconductivity. CDW and SDW order develop simultaneously and the order is stabilized in the low temperature tetragonal phase and low-temperature less-orthorhombic phase (the latter appears at Sr-doping levels below 0.08 and has a different symmetry (Pccn) than the low temperature orthorhombic phase (Bmab)) and severely suppresses superconductivity. Study of the magnetic properties around the vortex core appears to be very informative. Lake et al. (2001) found that the vortex state can be regarded as an inhomogeneous mixture of a superconducting spin ﬂuid and a material containing a nearly ordered antiferromagnet. Demler et al. (2001) argue this to be a sign of the proximity to a phase with co-existing superconductivity and spin-density-wave order. Zhu et al. (2002) explain the observations by solving self-consistently an effective hamiltonian including interactions for both antiferromagnetic spin-density-wave and d-wave superconductivity orderings. No signs of the magnetic resonance have been seen in doped La2−x Srx CuO4 , or more generally in single-layered cuprates, with one notable exception. Very recently the neutron data of He et al. (2002) showed such a resonance to be present in the single-layer compound Tl2 Ba2 CuO6+δ with a superconducting transition temperature of 90 K. This ﬁnding restricts the theoretical models for the origin of the resonant mode, discussed in the theoretical section. 5.2. μSR and NMR/NQR Already in early μSR and NMR experiments signs of patterned charge localization were seen, see HMM10. For example, Borsa et al. (1995) analyzed the staggered magnetization in La2−x Srx CuO4 , as measured by NQR and μSR for 0 < x < 0.02, invoking microsegregation of holes in domain walls, separating spin-rich domains. Most of the experiments discussed below are intended to characterize the stripe phase in more detail. For that reason samples are selected, where already charge and spin ordering were seen by neutron scattering. μSR. μSR data conﬁrm the results of the neutron and X-ray studies, and in addition allow a more detailed picture of the homogeneity of the ordered or frozen spin state. When Nd/Eu is substituted for La in La2−x Srx CuO4 , a well developed magnetic phase develops for x = 0.12 which involves the whole sample, while for x = 0.15 the magnetic volume has decreased to about half, leaving the remaining volume fraction non-magnetic down to 2 K (Kojima et al., 2000). The line shape of the x = 0.12 sample is well accounted for by a stripe model. Also in the excess oxygen sample of Lee et al. (1999), the magnetic site fraction amounted to about 0.4 (Savici et al., 2000). Data of Nachumi et al.

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Fig. 5.6. Phase diagram of La1.8−x Eu0.2 Srx CuO4 , after Klauss et al. (2000). The dashed line denotes the structural transition from low temperature orthorhombic (LTO) to low temperature tetragonal (LTT); in the upper right corner the high temperature tetragonal (HTT) phase is just visible. In the gray area with increasing doping values magnetic interactions lead to long range antiferromagnetism, short range antiferromagnetism, and static stripe phases. The solid line around x = 0.20 marks the region with diamagnetic signals due to the superconductivity. In the stripe regions the diamagnetic signal is weak.

(1998) on La2−x−y Srx Ndy CuO4 (x = 0.125, 0.15, 0.2) and La1.875 Ba0.125−y Sry CuO4 (y = 0.025, 0.065) show for all samples with dopant concentrations x + y 0.15 a similar static magnetic order with 0.3μB Cu-moment. Superconductivity and magnetic order coexist in x = 0.15. The phase diagram of La2−x−y Srx Euy CuO4+z has been investigated by Klauss et al. (2000), see also experimental section. In the system a low temperature tetragonal structure is present in the entire range of doping. Following the evolution from the long range antiferromagnetic state at x = 0 to the static magnetic stripes, they ﬁnd a non-monotonic change of the Néel temperature with increasing x. The obtained magnetic phase diagram of the LTT phase resembles the generic phase diagram of the cuprates where the superconductivity is replaced by a second antiferromagnetic phase (striped antiferromagnet), see ﬁg. 5.6. Also for La2−x Srx CuO4 (and for Y1−x Cax Ba2 Cu3 O6 ), being samples without additional Nd or Eu, a spin glass or spin/charge-ordered state has been deduced for doping values between 0.03 and 0.07 by Niedermayer et al. (1998). Even for overdoped samples with x = 0.21 Watanabe et al. (2000a) ﬁnd a slowing down of the Cu-spin ﬂuctuations below about 0.98 K in zero ﬁeld, which is a strong indication of the existence of spin/chargeordered state. NMR/NQR. Although the results of all mentioned techniques are very consistent in the detection of striped phases, there are differences where it comes to precise values. For example, the magnetic freezing or ordering temperature seen in neutron data is always much higher than seen in μSR. The obvious explanation is that time scales matter. It is especially in this respect that nuclear quadrupole data appear to be useful. As explained in the experimental section, slowing down of the electronic spin ﬂuctuations speeds up the spin dephasing (and spin-lattice relaxation) rate. At magnetic phase transitions this process normally leads to a loss of nuclear signal intensity in the experiment. However, the corresponding nuclear magnetization or signal, when extrapolated to time zero using

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Fig. 5.7. Wipe-out of 139 La and 63 Cu NQR-signals in La1.48 Nd0.4 Sr0.12 CuO4 , after Teitel’baum et al. (2000a). The labels m = 7/2, 5/2, and 3/2 refer respectively to the (±7/2, ±5/2), (±5/2, ±3/2), and (±3/2, ±1/2) 139 La transitions. For both Cu isotopes wipe out starts around 70 K, while for the 3 satellites of 139 La this temperature is around 40 K. Drawn lines are ﬁts with the model discussed in the experimental section with numerical constants that are found from the relaxation data. The value for the mean activation energy of 143 ± 5 K is consistent with the value found by Tranquada et al. (1999), see ﬁg. 5.4, of 200 ± 50 K, and points to a strongly reduced spin stiffness (an order of magnitude smaller than of the pure antiferromagnet). This implies that the spin system should be in the vicinity of a quantum phase-transition to a disorder state. TC and TS are the charge- and spin-ordering temperatures as revealed by neutron scattering. Tm is the magnetic ordering temperature according to μSR. Below the magnetic freezing temperature around 30 K the high temperature description of the wipe-out feature (drawn line) breaks down. In the spin frozen state apparently ﬂuctuations which much smaller activations energies play a role. Below 3 K slowing down of the Nd moments dominate the intensity loss.

the measured T2 , keeps its non-reduced value. Hunt et al. (1999) and Singer et al. (1999) were the ﬁrst to report a signal loss in connection to the formation of the stripe phase in La2−x−y Srx (Nd, Eu)y CuO4 . Typically, the normalized signal intensity (corrected for the apparent T2 ) started to drop around 150 K and only recovered some of its intensity below 10 K. Suh et al. (2000) and Curro et al. (2000) demonstrated that a distribution in dephasing times could account for this effect, which was further worked out by Teitel’baum et al. (2000a). Teitel’baum et al. (2000a) argued that in La2−x−y Srx Ndy CuO4 the data of T2 and wipe-out of Cu and La signals (which don’t happen at the same temperatures) could be well explained by the same spin dynamics with some spread in spin stiffness. Fig. 5.7 shows the wipe-out in La and Cu NQR in La1.48 Nd0.4 Sr0.12 CuO4 . For both Cu isotopes wipe out starts around 70 K, while for the 3 satellites of 139 La this temperature is around 40 K. Drawn lines are ﬁts to the equation discussed in the experimental section with numerical constants that are consistent with the nuclear relaxation data. The value for the mean activation energy of 143 ± 5 K is similar to the value found by Tranquada et al. (1999), see ﬁg. 5.4, of 200 ± 50 K, and points to a strongly reduced spin stiffness (an order of magnitude smaller than in the pure antiferromagnet). This implies that the spin system should be in the vicinity of a quantum phase-transition to a disorder state. In ﬁg. 5.4 TC and TS are the charge- and spin-ordering temperatures as revealed by neutron scattering. In μSR magnetic freezing occurs at a much low temperature Tm ≈ 30 K. Below Tm the

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Fig. 5.8. Phase diagram of the 214-compounds as function of hole doping. The construction is for an important part based on wipe-out and recovery features seen in nuclear resonance (NR). The darker gray tones indicate increasingly slow ﬂuctuation time scales. The superconducting boundary is indicated by the lower dashed line. The other dashed line, that connects the charge-order temperature and the Cu wipe-out inﬂection points is a guide to the eye. This plots is the result of a compilation of many data points obtained on a variety of samples by various groups and techniques, after Hunt et al. (2001) and Ichikawa et al. (2000): (1) onset of the Cu recovery in La1.8−x Eu0.2 Srx CuO4 , (2) onset of the La recovery in La1.8−x Eu0.2 Srx CuO4 , (3) onset of the μSR coherent precession in La1.6−x Nd0.4 Srx CuO4 , (4) onset of La wipeout in La1.8−x Eu0.2 Srx CuO4 , (5) long range spin order detected by neutron spectroscopy in La1.6−x Nd0.4 Srx CuO4 , (6) temperature where 139 La 1/T1 T = 0.05 s−1 K−1 in La1.8−x Eu0.2 Srx CuO4 , (7) copper wipe-out inﬂection point in La1.6−x Nd0.4 Srx CuO4 , (8) onset temperature for charge order in La1.6−x Nd0.4 Srx CuO4 according to neutron and X-ray scattering, (9 and 10) onset Cu wipe-out in La1.6−x Nd0.4 Srx CuO4 and La1.8−x Eu0.2 Srx CuO4 , and (11) upturn in the (ab)-plane resistivity in La1.6−x Nd0.4 Srx CuO4 . The NQR and diffraction data for x < 0.12 are not necessarily in conﬂict, since NQR is an inherently local probe, whereas the diffraction measurements require substantial spatial correlations of the charge order to get detectable peaks.

high temperature description of the NQR wipe-out features (drawn lines) breaks down. In the spin frozen state apparently ﬂuctuations which much smaller activations energies play a role. Below 3 K slowing down of the Nd moments dominate the intensity loss (Teitel’baum et al., 2000a). That spin ﬂuctuations are directly linked to the wipe-out features was also concluded by Julien et al. (2000, 2001). Based on similar observations, Hunt et al. (2001) constructed a phase-diagram for the magnetic spin ﬂuctuations as function of hole doping, see ﬁg. 5.8. In this ﬂuctuation diagram it is also seen that down to x = 1/8, charge ordering in stripes slows down the spin ﬂuctuations and starts the wipe-out process for the nuclear Cu spins. The data of Hunt et al. (1999), Teitel’baum et al. (2000a, 2000b) and Hunt et al. (2001) also reveal that the signal recovery at low temperatures is only partial. In line with the explanation at high temperatures, slow spin ﬂuctuations associated with typical features of the stripes might well be the reason for this process. Also the featureless low temperature lineshapes (Hunt et al., 2001) are suggestive for a motional averaging process – see also the next section about the nickelates, where the line shape could be ﬁtted by static stripes (Abu-Shiekah et al., 2001). Also Suh et al. (1998, 1999) interpret the increase in the low temperature NQR 139 La linewidth in lightly hole-doped La 1.8−x Eu0.2 Srx CuO4 and La2 Cu1−x Lix O4 in terms

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of local spin structures, and suggest stripes (antiphase domain walls) in the form of mobile loops. In these experiments the structural transition from low temperature orthorhombic to low temperature tetragonal is seen to modify the spin state. Early NMR and NQR experiments showed the hole concentration in the CuO2 -planes of the high-Tc superconductors to be inhomogeneous, leading to the presence of two Cusites with different electrical ﬁeld gradients. Haase et al. (2000) using an ingenious NMR method found a substantial temperature dependent spatial modulation of the O and Cu parameters of La1.85 Sr0.15 CuO4 between 10 K and 300 K with a length scale of the modulations of only a few lattice distances. Measurements on other cuprates indicate universality of this phenomenon. Singer et al. (2002) performed 63 Cu NQR on samples enriched with 63 Cu isotope with 0.04 x 0.16. The extent of the spatial variation Δx hole of the local hole concentration is reﬂected in the relaxation rate. It is seen that the inhomogeneous distribution starts below 500–600 K and reaches values as large as 0.5 below 150 K. The length of the spatial variation in xhole is estimated to be larger than 3 nm. Although these features point to phase separation, their connection to stripes is not yet clear. 5.3. Results of other magnetic techniques In the superconducting compounds the most pronounced effect in the magnetic susceptibility, which overshadows often all other subtleties, is the diamagnetism and ﬂux properties of the superconducting phase. Above Tc the magnetization exhibits two-dimensional Heisenberg antiferromagnetism, e.g., Huh et al. (2001). However, by scanning SQUID microscopy of La2−x Srx CuO4 thin ﬁlms, Iguchi et al. (2001) observed inhomogeneous magnetic domains which persist up to 80 K, well above the superconducting transition temperature of 18–19 K. The result suggests the existence of diamagnetic regions that are precursors to the Meissner state far above the superconducting transition temperature and can be seen as an intrinsic tendency toward electronic inhomogeneity. The suppression of the superconducting transition temperature in La2−x Srx CuO4 by non-magnetic Zn ions was investigated with ESR by Finkelstein et al. (1990). The data showed the appearance of a new resonance line, due to the creation of magnetic moments, localized on the Cu ions. In samples without impurities, Kochelaev et al. (1997) found a broad but well deﬁned single EPR line, which they attribute to a three spin polaron, consisting of two Cu2+ ions and one p hole. Kataev et al. (1998) analyzed the EPR spectra in La2−x−y Srx Euy CuO4 and found that in the low temperature tetragonal phase, the frequency of the spin ﬂuctuations of the Cu ions considerably slows down in the samples with strongly suppressed superconductivity. The effect is especially pronounced for x close to 1/8. For x > 0.17, where superconductivity fully develops, these signatures vanish. These effects are further discussed in section 8. 5.4. Charge sensitive techniques – X-ray, conductance and ARPES Neutron scattering, muon spin rotation and nuclear quadrupolar or magnetic resonance are obvious techniques to address magnetic properties. Because of the intimate connection between charges and spins in striped systems (see the preceding theoretical and experimental sections), the results obtained by techniques that are especially sensitive for the charges, like ARPES and charge transport, are relevant as well and are mentioned below.

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Zhou et al. (1999), see also Zaanen (1999a), performed ARPES experiments on La1.28Nd0.6Sr0.12 CuO4 . The electronic features, see ﬁg. 4.5 in the experimental section, are consistent with other cuprates, such as a ﬂat band at low energy near the Brillouin zone face. However the frequency integrated spectral weight is conﬁned inside one-dimensional segments in the momentum space (deﬁned by horizontal momenta |kx | = π/4 and vertical momenta |ky | = π/4). This departure from the two-dimensional Fermi surface persists to very high energy scale. In La1.85Sr0.15CuO4 Zhou et al. (2001) found the same straight segments near (π, 0) and (0, π) antinodal regions, but also identiﬁed the existence of spectral weight along the [1, 1] nodal direction. This observation reveals the dual nature of the electronic structure of stripes. On the one hand, the electrons seem to move in the strong potentials due to the stripes. At the same time, the ARPES in the nodal directions ﬁnds an explanation in terms of a rather “conventional” d-wave BCS superconductor. This seemingly paradoxical behavior still waits for a theoretical explanation. The ARPES patterns are not only due to electronic interactions alone. From their ARPES data on a variety of high-Tc superconductors including La2−x Srx CuO4 , Lanzara et al. (2001) conclude to the ubiquitous presence of strong coupling between electrons and phonons, see also Allen (2001). From X-ray experiments on La2−x−y Srx (Nd, Eu)y CuO4 von Zimmermann et al. (1998) found for x = 0.12 also a sinusoidal modulation along the c∗ -axis due to long range Coulomb interaction between the stripes. In a superconducting x = 0.15 sample, Niemoller et al. (1999) observed the disappearance of stripe order above 62 K. Data of Saini et al. (2001) reveal a stepwise increase of the local lattice ﬂuctuations below the charge order. A rapid decrease in the magnitude of the Hall coefﬁcient at low temperatures in La2−x−y Ndy Srx CuO4 has been reported by Noda et al. (1999), see also Zaanen (1999a). The presence of this effect for x < 1/8 and its absence for x > 1/8 indicate a cross-over from one- to two-dimensional charge transport taking place at x = 1/8. An explanation within the t–J model has been put forward by Prelovšek et al. (2001). Indications of a 1D to 2D cross-over in the Hall coefﬁcient of La1.48Nd0.4 Sr0.12CuO4 as function of pressure were found by Arumugam et al. (2002). Ando et al. (2001), see ﬁg. 5.9, have presented new in-plane resistivity data on a high quality single crystals of La2−x Srx CuO4 and ﬁnd that the hole mobility at 300 K changes intriguingly similar to that of the inverse antiferromagnetic correlation length. In a stripe picture the data show that the charge transport is inﬂuenced by the rigidity of the magnetic correlation in the magnetic domains, which probably means that the transverse ﬂuctuations of stripes in the antiferromagnetic environment must be signiﬁcant, consistent with the electronic liquid crystal put forward by Kivelson et al. (1998). Ichikawa et al. (2000) analyzed the in-plane resistivity of La2−x−y Ndy Srx CuO4 . They found that the transition temperature for local charge ordering decreases monotonically with x, and hence conclude that the local antiferromagnetic order is uniquely correlated with the anomalous depression of superconductivity at x = 1/8. This result supports theories where superconductivity depends on the existence of charge-stripe order. 5.5. Summary Static and dynamic properties of charge and spins in La2−x−y Ndy Srx CuO4 are well documented. The magnetic resonance is not seen. Charges and spins are short range ordered and

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Fig. 5.9. x dependence of the inverse mobility 1/μ = nh eρab (open circles – left axis) as derived from in-plane resistivity data on high quality single crystals of La2−x Srx CuO4 crystals at 300 K, after Ando et al. (2001). The dashed line shows the x dependence of the antiferromagnetic correlation length ξAF (right axis), which is reported √ to be 0.38/ x nm from neutron experiments of Ino et al. (1998). The change of the hole mobility with x is similar to that of the inverse antiferromagnetic correlation length. In a stripe picture the data show that the charge transport is inﬂuenced by the rigidity of the magnetic correlation in the magnetic domains, which probably means that the transverse ﬂuctuations of stripes in the antiferromagnetic environment must be signiﬁcant, consistent with the electronic liquid crystal put forward by Kivelson et al. (1998).

stripes and superconductivity coexist. However, some issues remain to be solved. In ﬁg. 5.8 we reproduced the compilation made by Hunt et al. (2001) and Ichikawa et al. (2000) for La1.6−x Nd0.4 Srx CuO4 . The charge and magnetic order temperatures are determined by various techniques, like NQR, X-ray and neutron diffraction studies and magnetic susceptibility and are linked to the time scale of the experiment. It is clear that below x < 0.12 the NQR and diffraction data lead to different interpretations. These data are not necessarily in conﬂict, since NQR is an inherently local probe, whereas the diffraction measurements require substantial spatial correlations of the charge order to get detectable peaks. Also for the origin of the spin ﬂuctuations below 1 K, which makes the recovery of the NQR signal only partial and removes clear antiferromagnetic signs in the line shape, a more thorough theoretical understanding is required. 6. Oxygen and strontium doped 214-nickelates In the elastic neutron-scattering experiments of Tranquada et al. (1995) static stripes were seen in La2 NiO4+δ with δ = 1/6, i.e., at hole doping of 1/3 (the unit cell is shown in ﬁg. 6.1). For that reason a comparison of the superconducting cuprates and the hardly conducting isostructural nickelates might shed light on the role played by superconductivity and charge localization in the formation of the striped phase. In the striped phase charge and spin separate. In the cuprates the charged stripes have the tendency to be half ﬁlled and the stripes run horizontally or vertically. The Cu3+ ions in the stripes carry no spin, while the Cu2+ ions in the magnetic domains have S = 1/2. In the nickelates the charge spin separation is different – ﬁlled charged stripes are formed of Ni3+ , which ion carries a spin S = 1/2. The spin domains have Ni2+ -ions with S = 1. The stripes run diagonally.

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Fig. 6.1. The unit cell of La2−x Srx NiO4+δ , in the tetragonal phase a = 0.378 nm, c = 1.329 nm, spacegroup I 4/mmm. The (x, y, z) coordinates for (La, Sr) (gray), Ni (black), O1 and O2 (open circles) are resp. (0.000, 0.000, 0.3606), (0.000, 0.000, 0.000), (0.000, 0.500, 0.000) and (0.000, 0.000, 0.1828), see HMM10. For δ = 0.18 the interstitial excess oxygen is located at (0.183, 0.183, 0.217) and equivalent positions (small dots) – when slowly cooled the interstitial oxygens might form a periodic pattern. Although the structure of the nickelate is similar to La2−x Srx CuO4 , the occupation of the 3dz2 orbital of the Ni ion gives a much stronger hyperﬁne coupling between the Ni electron and the La nuclear spin (via the apical oxygen) than between Cu and La in La2−x Srx CuO4 .

The presence of the domainwall spins is an advantage for spin sensitive studies, like NMR. Compared to the phase diagram of the oxygen doped cuprates, the oxygen doped nickelates have much larger regions of uniform phases. While holes can be introduced by Sr or oxygen doping, the properties of oxygen and Sr doped nickelates are not identical. Below we summarize the main results obtained in the last few years on both types of nickelates. Before doing so, we ﬁrst show details about the phase diagram under doping. 6.1. Phases and structural changes under oxygen and Sr doping Several groups have investigated the location of the excess oxygen site, possible staging and uniformity of the ﬁnal phase in La2 NiO4+δ . In ﬁg. 6.2 we reproduce the results of an Xray study of Rice and Buttrey (1993). Pure and mixed phases are found to alternate, but uniform phases are clearly more frequent than in La2 CuO4 . Similar or slightly different phase diagrams were obtained via neutron scattering by Hosoya et al. (1992) and Tranquada et al. (1994), and via X-ray diffraction by Tamura et al. (1996). Poirot et al. (1998) traced the role

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Fig. 6.2. Evolution of the √ ratio√of the basal plane lattice constants and the c-lattice constant (2c/a + b) in La2 NiO4+δ (using the 2 × 2 × 1 cell) with oxygen stoichiometry at room temperature. Data from the low-cooled ceramic specimen with δ = 0.168 and from results of Jorgensen et al. (1989) form the basis for the proposed high δ biphase region (after Rice and Buttrey, 1993). It is seen that with oxygen doping pure and mixed phases alternate, but uniform phases are clearly more abundant than in La2 CuO4 . Similar, sometimes slightly different phase diagrams are reported by other authors.

of oxygen stoichiometry in their samples (0.015 δ 0.17) by magnetic resonance. The location of the excess oxygen has been reﬁned by Mehta and Heaney (1994). In La2 NiO4.18 oxygen is located at interstitial sites equivalent to (0.183, 0.183, 0.217). Various levels of oxygen staging were deduced from the heat capacity data of Kyomen et al. (1999) in the temperature range 160–240 K. The time dependence of the oxygen ordering was followed in a neutron study by Lorenzo et al. (1995). The structural effects of Sr-substitution in La2−x Srx NiO4 reveal three distinct regimes (Heaney et al., 1998). For 0 x 0.2 the compounds are orthorhombic. For 0.2 x 0.6 the symmetry lowers to monoclinic. For 0.6 x 1 diffraction patterns suggest phase immiscibility between a Sr-poor and a Sr-rich phase. Tang et al. (2000) compared the crystal symmetry and the electrical properties of La2−x Ax NiO4 (A = Ca, Sr and Ba) and linked the transfer of electron density from the dx 2 to the dx 2 −y 2 band to the increase in the unit cell axis a and the decrease of c. 6.2. Neutron scattering The experimental study of stripes started by the discovery of static stripes in the oxygen doped nickelate La2 NiO4.125 by Tranquada et al. (1994). Using single-crystal neutron diffraction, they observed not only the magnetic ﬁrst and third harmonic Bragg peaks linked to the incommensurate magnetic ordering, but also second harmonic peaks associated with charge ordering. The magnitude of the incommensurate splitting appeared strongly T dependent. Presumably the competition between the ordering of the hole stripes and a lattice modulation due to ordering of the interstitial oxygens leads to a devil’s staircase of ordered

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Fig. 6.3. Low-energy spin-wave spectrum dispersions measured on crystals of La2 NiO4+δ with δ = 0, 0.105 and 0.133. Left inset indicates directions in the (h, k, l) plane along which the dispersion has been characterized. Main panel shows dispersions along A; results for scans along B for δ = 0.133 are shown in the right inset. Bars indicate measured peak widths. The lines through the data are spin-wave dispersion curves with the spin-wave velocities (h¯ c) of 7 ± 0.8 for δ = 0.105 and 20 ± 2 for δ = 0.133. For δ = 0 the value is 34 meVnm. The two curves for each δ correspond to in-plane and out-of-plane spin-wave modes, which have different anisotropy gaps. On increasing δ from 0 to 0.105, the spin-wave velocity decreases by a factor of 5. This trend is abruptly reversed on entering the stripe-ordered phase found at δ = 0.133 (after Tranquada et al., 1997a).

phases. In subsequent studies similar stripe features were seen also at other oxygen concentrations. Hole doping of 1/4 and 1/3 (oxygen doping of 1/8 resp. 1/6) have a special meaning due to the commensurability of the stripes with the underlying lattice (Tranquada et al., 1995, 1997a, 1997b; Wochner et al., 1998). In Tranquada et al. (1999) charge order has been observed up to a temperature of at least twice the value of the magnetic transition around 110 K. The charge stripes were all oxygen centered at T > Tm , with a shift towards Ni centering at T < Tm , where Tm is the magnetic ordering temperature. In the same sample spin-wave excitations were measured by inelastic neutron scattering, Tranquada et al. (1997a), see ﬁg. 6.3. On increasing δ from 0 to 0.105, the spin-wave velocity decreases by a factor of 5. This trend is abruptly reversed on entering the stripe-ordered phase found at δ = 0.133. The spin-wave excitations propagating parallel to the stripes is around 60% of that in the pristine sample, and three times greater than the velocity for δ = 0.105. In contrast, excitations perpendicular to the stripes appear to have a greater damping. The results exhibit an energy dispersion and broadening that is qualitatively similar to that found in the cuprate system. Charge modulation in Sr-doped nickelates was seen in electron diffraction studies by Chen et al. (1993). Sachan et al. (1995) reported neutron-scattering experiments that revealed coupled incommensurate peaks arising from magnetic as well as charge ordering at

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low temperatures (below 100 K) in stoichiometric La2−x Srx NiO4+δ crystals. The modulation wave vector appears to be a sensitive function of the net hole concentration p = x + 2δ. The magnetic correlation length increases with doping and the incommensurability follows the simple relationship ε ≈ p. Stripe ordering does not occur exclusively in commensurate samples but also for x = 0.225, Tranquada et al. (1996). The commensurability effects were further studied by Lee and Cheong (1997) for x = 1/3, see also the experimental section. Upon cooling the systems undergoes three successive transitions associated with quasi-two-dimensional commensurate charge and spin stripe ordering in the NiO2 planes. The two lower temperature phases are stripe lattice states with quasi-long-range in-plane charge correlation. When the lattice of the 2D charge stripes melts, it goes through an intermediate glass state before becoming a disordered liquid state. This glass state shows short-range charge order without spin order, and resembles the hexactic/nematic state in 2D melting. The linear relationship between hole doping and magnetic modulation was extended up to x = 0.5 by Yoshizawa et al. (2000). By using polarized neutron diffraction on samples with x = 0.275 and x = 1/3, Lee et al. (2001) were able to show that the spins in the ordered phase are canted in the NiO2 plane away from the charge and spin stripe direction. In later work of Lee et al. (2002) on the x = 0.275 sample the existence of a stripe-liquid phase could be demonstrated. The sample was selected because the spin- and charge-ordering wave vectors do not coincide. The incommensurate magnetic ﬂuctuations evolve continuously through the charge-ordering temperature of 190 K. The effective damping decreases abruptly when cooling through the transition. The energy and momentum dependence of the data can be effectively parameterized with a damped harmonic-oscillator model describing overdamped spin waves associated with the antiferromagnetic domains deﬁned instantaneously by the charge stripes. Signatures of strong electron–phonon coupling were seen in the inelastic neutron data of McQueeney et al. (1999a). Tranquada et al. (2002) noticed the similarities of the bond-stretching-phonon anomalies in La1.69Sr0.31 NiO4 with those in the cuprates. The absence of a collective signature makes it likely that local interactions between charge and lattice ﬂuctuations dominate. 6.3. μSR and NMR/NQR and magnetic susceptibility In the oxygen doped samples Odier et al. (1999) correlated the (dc and optical) conductivity and the magnetic susceptibility on samples with well-known oxygen stoichiometries. The susceptibility data were modelled by a Curie–Weiss law plus a cluster contribution to account for the nearly constant T dependence between 200 and 800 K. Hasegawa et al. (1996) found the magnetization to be anisotropic and suggest a spin glass state below 20 K. In La2−x Srx NiO4 close to the conducting composition weakly temperature-dependent magnetic susceptibilities and resistivities were found, similar in magnitude to those in the isostructural metallic La2−x Srx CuO4 (Cava et al., 1991). The magnetic data of Lee and Cheong (1997) on La5/3Sr1/3 NiO4 were given in the experimental section as an example, see ﬁg. 4.7, and are also mentioned under neutron scattering. Ramirez et al. (1996) presented ultrasonic and speciﬁc heat measurements on La1.67Sr0.33 NiO4 to characterize the thermodynamic transition observed at Tc = 240 K. They conclude that the possibility of independent hole-spin behavior is consistent with the observed entropy and measured anomalies in the resistivity, susceptibility and ultrasound. Also in Raman spectroscopy on La5/3 Sr1/3NiO4 (Blumberg et al., 1998), the formation of a superlattice and the opening

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Fig. 6.4. Doping dependence of the magnetic freezing or ordering temperature of La2−x Srx NiO4+δ according to μSR data of Jestädt et al. (1999). Tm ﬁrst drops with increasing x, reaches a minimum at x ∼ 0.15 and then rises again to a maximum of 205 K for x = 0.33 (different Tm values for a given x are associated with different values of δ). For larger x values the transition temperature drops again falling to a value of 45 K at x = 1. By checking Tm at the same p = x + 2δ value, p appeared not to be the appropriate parameter for the characterization of the order temperature. The interstitial accommodation of oxygen requires a local lattice distortion while quite different structural consequences result from Sr substitution. Both these structural effects are likely to have signiﬁcant, but dissimilar effects on the magnetic properties.

of a pseudogap in the electron–hole excitation spectra as well as two types of double spin excitations (within the antiferromagnetic domain and across the domain wall) were observed below the charge ordering temperature. The evolution of the two-dimensional antiferromagnetic spin correlations in the charge-ordered state even below the spin-ordering temperature was also seen by Yamamoto et al. (1998). μSR. Chow et al. (1996) reported μSR measurements on a series of La2−x Srx NiO4+δ with x + 2δ > 0.4. A composition dependent magnetic transition temperature Tm is found in all samples. Below Tm clear precession signals are observed in zero applied magnetic ﬁeld indicating the existence of at least short-range magnetic order on a time scale greater than 10−8 s. Above Tm the correlation times decrease by several order of magnitude. Jestädt et al. (1999) performed μSR on samples with Sr levels 0 < x < 1. For x = 0.33 they ﬁnd peaks in both Tm and the zero-temperature staggered magnetization. The observation is attributed to the higher degree of localization of the holes at this doping level, see also ﬁg. 6.4. NMR/NQR. Wada et al. (1993) and Furukawa and Wada (1994) followed the successive magnetic phase transitions in La2−x Srx NiO4+δ by 139 La NQR. In the magnetic phase the internal ﬁeld at the 139 La site was found to be as large as 1.8 T. Furukawa and Wada (1994) conclude that the doped holes occupy the Ni 3dz2 orbital up to x 1.0. By 139 La NMR Yoshinari et al. (1999) examined the domain formation in La5/3 Sr1/3NiO4 . Below the charge-order temperature their single crystal showed two magnetically distinct sites, the ﬁrst located in the domain walls and the second in the hole-free domains. Although the two regions are spatially proximate and strongly interacting, their static and dynamic magnetic properties are quite different. The NMR data provide speciﬁc evidence in support of

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the stripe-glass hypothesis and reveals pronounced and unusual spin disorder arising from stripe defects. A hall mark of glassy systems is sensitivity of the transition temperature to measurement time scale. Yoshinari et al. (1999) ﬁnd that the onset of the spin ordering (160 K) is 30 K less than seen in neutron scattering as a consequence of this difference in time scale (μs compared to ps for neutron scattering). At low temperatures where charge order is well developed, the static spin order exhibits a continuous distribution of moment magnitudes and in-plane orientations. For B ⊥ c the lineshape demonstrates that the ordered moments rotate in response to the applied ﬁeld such that the most probable spin orientation is perpendicular to the applied ﬁeld. For this particular sample, NMR seems to rule out the motion of the domain walls below the charge order. A δ = 1/6 sample (hole doping similar to x = 1/3) was investigated by Abu-Shiekah et al. (1999, 2001). Compared to the quenched potential in the Sr-doped samples the oxygen ordered samples might be more “clean”. Abu-Shiekah et al. found that the nickelate exhibits a ﬂuctuation spectrum which closely parallels the ﬂuctuations of the cuprate stripes: although neutron-scattering experiments in La2 NiO4.13 show charge and spin freezing at 220 K resp. 110 K, the NMR data indicate that the spins in the stripes become static only at a temperature of 2 K. These slow spin ﬂuctuations play a much smaller role in the analogous Sr-doped sample. As in the cuprates, the spin stiffness is strongly renormalized, suggesting strong quantum spin ﬂuctuations due to the proximity of a spin disordering transition, see the introduction and theoretical section. The signal recovery is only partial (even at 0.3 K) indicating that spin dynamics is still at work at temperatures as low as 0.3 K. AbuShiekah et al. (2001) demonstrated that in nickelate stripe systems the nature of the stripe order can be deduced in detail from NMR (this in contrast to the 214-cuprates, see Hunt et al. (2001)). They ﬁnd that the stripe structure is strongly solitonic, with sharply deﬁned charge stripes with a width which is not exceeding the lattice constant by much. Surprisingly, they look quite like the site centered stripes predicted by mean ﬁeld calculations for the nickelate system (also in La1.775Sr0.225NiO4 the location of the charges appeared to be on the Ni-sites as inferred from phonon and neutron data, see Pashkevich et al. (2000a)). Fig. 6.5 shows the spectra measured for Bab and the simulations. Below the charge ordering the line clearly has contributions from two sites (A and B). The spectra were simulated with the same parameters as for Bc. The internal ﬁeld leads to a splitting of the lines and hence can be determined precisely. While the ﬁeld on the B-site is less than 0.3 T, the ﬁeld on the A-site is about 2 T, similar to the ﬁeld seen in undoped antiferromagnetically ordered samples. 6.4. Charge sensitive techniques – X-ray, conductance and ARPES Several studies compare the magnetic and conductance properties. Cheong et al. (1994) found clear indications of a mutual connection in the phase transitions at hole doping of 1/3. In the oxygen doped samples Odier et al. (1999) correlated the conductivity (dc and optical) and the magnetic susceptibility on samples with well-known oxygen stoichiometries. The relatively low hopping energies deduced from conductivity data in La2 NiO4.125 were ascribed to the role of the transferred exchange interaction between the Ni2+ 3dz2

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Fig. 6.5. NMR spectra measured in La2 NiO4.17 for Bab at high (a) and low (b) temperatures and their simulations (after Abu Shiekah et al., 2001). Below the charge ordering the spectra for Bc clearly have contributions from two sites (A and B). Fitting gives precise values for the electrical ﬁeld gradients but are not very sensitive for the strength of the internal ﬁeld. The spectra for Bab were simulated with the same parameters as for Bc. The internal ﬁeld leads to a splitting of the lines and hence can be determined precisely. While the ﬁeld on the B-site is less than 0.3 T, the ﬁeld on the A-site is about 2 T, similar to the ﬁeld seen in undoped antiferromagnetically ordered samples. The variation in hyperﬁne coupling and the values of the internal ﬁeld are strong indications for site-ordered stripes, see also the theory section.

orbital and the La3+ 6s orbital through the 2pz orbital of the apical O2− (Iguchi et al., 1999). A higher hole doping diminishes the number of Ni2+ ions and hence decreases the exchange. Isaacs et al. (1994) used synchrotron X-ray scattering from La1.8Sr0.2 NiO4 and La1.92Sr0.075CuO4 to establish a direct relationship between the carriers due to Sr-doping and strong diffuse scattering. In the nickelate the scattering is peaked at the fourfold symmetric satellite positions (±δ, ±δ, l), where the basal-plane coordinate δ varies with the out-of plane coordinate l of the momentum transfer. A similar scattering pattern is observed in the cuprate. The observations point to charged domain walls. Using high resolution X-ray scattering the existence of quenched disordered charge stripes were established for La5/3Sr1/3 NiO4 . The two-dimensional nature appears from the correlation

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lengths ξa ≈ 18.5 nm, ξb ≈ 40 nm and ξc ≈ 2.5 nm and by the critical exponents of the charge stripe transition. The charge stripe ordering did not develop long-range order even at low temperatures. Charge stripes are disordered and the length scale of the disorder is quenched. Using near-edge X-ray diffraction, Pellegrin et al. (1996) saw evidence for the Zhang–Rice character of the doped carriers and of considerable overlap between the polaron states. From hard X-ray diffraction on La1.775Sr0.225NiO4 Vigliante et al. (1997) found good agreement between the charge density modulation peaks seen in these data and in neutron data. In a recent ARPES study (Satake et al., 2000) a large downward shift of the chemical potential with hole doping in the high doping regime x > 0.33 is seen, while the shift is suppressed at lower values. The suppression is attributed to segregation of doped holes. Pashkevich et al. (2000b) conclude that the strong Fano antiresonance in the optical conductivity in striped La1.775Sr0.225NiO4 is caused by Ni–O stretching motion along the stripes. The antiresonance results from electron–phonon coupling and provides evidence for ﬁnite conductivity along the stripes at optical frequencies. 6.5. Summary The 214 nickelates and 214 cuprates differ greatly in conductance: below x = 1/3 the dc conductivity of the nickelates becomes very small, while in the cuprates at low temperatures superconductivity appears. However, the two materials show a great similarity in magnetic properties. Over a large doping regime stripes are formed of which the features are well characterized by a variety of techniques. Like in the Sr-cuprates spin freezing in the oxygen doped samples has a surprisingly large temperature regime of slow dynamics. Even at 1 K the size of the stripe domains is limited to about 10 nm. The spin stiffness is strongly renormalized due to the nearby presence of a quantum critical point. In NMR there are differences between Sr-doped and oxygen doped samples with a similar hole count. The quenched Sr hole-donors might be less screened than in the analogous cuprates while oxygen diffusion in the nickelates allows the oxygen ions to form more regular structures. At low temperatures neutron and NMR data are in favor for site centered stripes with a strong solitonic character, at least in the 1/6 oxygen doped samples. 7. The electron-doped single-layer compound Nd2−x Cex CuO4+y Compared to the hole-doped cuprates, less attention has been paid to electron-doped cuprates, like Nd2−x Cex CuO4 partly because of their difﬁcult chemistry. The unit cell differs from La2−x Srx CuO4 in the location of the oxygen atoms outside the CuO2 plane, see ﬁg. 7.1. The antiferromagnetism in undoped Nd2 CuO4 has been described in HMM10. In systems doped by Ce, which acts as an electron donor, the ordering temperature decreases with increasing amount of dopant and disappears around a doping level of 0.15. The magnetism is that of a spin-diluted Heisenberg antiferromagnet, see Vajk et al. (2002). The low temperature ordered moment decreases almost linearly with the ordering temperature from 0.66μB /Cu in La2 CuO4 with TN ∼ 340 K, to 0.5μB /Cu for TN ∼ 250 K in Nd2 CuO4 and less than 0.2μB /Cu for Ce doped Nd2 CuO4 with transition temperatures below 150 K.

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Fig. 7.1. The unit cell of Nd2−x Cex CuO4 , in the tetragonal phase a = 0.378 nm, c = 1.329 nm, spacegroup I 4/mmm. The (x, y, z) coordinates for (Nd, Ce) (gray), Ni (black), O1 and O2 (open circles) are resp. (0.0000, 0.0000, 0.3606), (0.0000, 0.0000, 0.0000), (0.0000, 0.5000, 0.0000) and (0.0000, 0.0000, 0.1828), see HMM10. The location of the oxygens outside the CuO2 plane is different from La2−x Srx CuO4 .

Electron- and hole-doped cuprates both have copper-oxygen planes separated by rareearth layers. Hence, from symmetry and dimensionality arguments similar superconducting properties might be expected. However, in the transition metal oxides correlation effects are proven to be extremely important (Zaanen et al., 1985), and these are expected to be more dominant for electron than hole conductors, although in a more conventional band approach (King et al., 1993) these difference are less outspoken. Experimentally, asymmetry in the electronic properties has indeed been observed. In the hole-doped samples, already a doping level of 0.05 is sufﬁcient to kill the magnetic state, while, e.g., in Nd2−x Cex CuO4 the magnetic state survives doping up to 0.15. Connected to this, in Nd2−x Cex CuO4 the region where superconductivity occurs is restricted to 0.15 x 0.20 only, see ﬁg. 7.2. Still, like in the hole doped case, also in electron-doped cuprates convincing indications for d-wave pairing have been obtained. Using a scanning superconducting quantum interference device microscope Tsuei and Kirtley (2000) observed the half ﬂux quantum effect, and the dx 2−y 2 anisotropy of the superconducting gap was conﬁrmed by the ARPES data of Sato et al. (2001), and reﬁned in the low energy polarized electronic Raman scattering experiments of Blumberg et al. (2002). In the overdoped state the d-wave pairing is likely changed to s-wave pairing (Biswas et al., 2002; Skinta et al., 2002).

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Fig. 7.2. Magnetic and superconducting transition-temperatures as function of doping for electron and hole-doped 214-cuprates. In the hole-doped samples already a doping level of 0.05 is sufﬁcient to kill the magnetic state, while e.g., in Nd2−x Cex CuO4 the magnetic state survives doping up to 0.15. Connected to this, in Nd2−x Cex CuO4 the region where superconductivity occurs is much more restricted: only between 0.15 and 0.20.

7.1. Charge sensitive data The presence of charge-spin structures is not clear. Theoretically, arguments for the formation of charge stripes were given in an unrestricted Hartree–Fock calculation by Sadori and Grilli (2000). Experimentally, a charge-order instability was suggested by Onose et al. (1999, 2001) from the temperature variation of the optical conductivity and Raman spectra for oxygenated antiferromagnetic and reduced superconducting crystals of Nd1.85Ce0.15 CuO4+y . The presence of the apical oxygens as impurities plays an important role in the realization of the superconducting or antiferromagnetic phase. In the reduced crystal the spectra change little with T . In the spectra of the oxygenated crystal a pseudogap structure evolves around 0.3 eV and activated infrared and Raman Cu–O phonon modes grow in intensity below 340 K. The origin is linked to a charge-order instability induced by a minute amount of interstitial oxygen, which seems also responsible for the absence of superconductivity, see ﬁg. 7.3. Here a strong analogy exists with the oxygen doped nickelates, where charge order is also linked to the excess oxygens. In a later study, using crystals with various amounts of Ce dopant, Onose et al. (2001) found that below a characteristic temperature T ∗ a notable pseudogap Δ opens in the optical conductivity for metallic but non superconducting samples with Δ = 10kB T ∗ . The Drude-like component is seen to evolve concomitantly with the pseudogap – a situation, which is reminiscent of the spin density wave gap in Cr metal. Armitage et al. (2001) reported high-resolution photoemission spectra on superconducting Nd1.85 Ce0.15 CuO4+y . They observe regions around the Fermi surface where the near-EF intensity is suppressed and the spectral features are broad in a manner reminiscent of the high-energy pseudogap in the underdoped hole-doped cuprates. However, instead of occurring near the (π, 0) region, as in the p-type materials this pseudogap falls near the intersection of the underlying Fermi surface with the antiferromagnetic Brillouin zone boundary. In a second paper, Armitage et al. (2002), the evolution of the ARPES spectra from the half ﬁlled Mott insulator to the Tc = 24 K superconductor was followed. At low doping, the Fermi surface is an electron-pocket centered around (π, 0)

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Fig. 7.3. Temperature variation of the change in spectral weight for Nd1.85 Ce0.15 CuO4 (in units of effective number of electrons) in the region of 0.145–0.330 eV (open circles) due to the electronic pseudogap formation, and in the region of 0.030–0.046 eV (closed circles) due to the activated phonon modes (Onose et al., 1999, 2001). The data at 390 K are taken as reference. Above this temperature the conductivity data show no T dependence. The quantities vary with temperature in an almost parallel manner (the dashed line is a guide to the eye). It is clear that the pseudogap formation and the lattice distortion begin at about 340 K, which is much higher than the antiferromagnetic transition (120–160 K) and the resistivity upturn (around 200 K). The authors rule out the possibility of a conventional spin-density-wave transition for the pseudogap formation, as the energy scale of the pseudogap (about 0.3 eV) is too high, and propose charge ordering or its ﬂuctuation, which is induced by a minute amount of apical oxygen as an alternative.

with volume ∝ x. Further doping leads to the creation of a new hole-like Fermi surface (volume ∝ (1 + x)) centered at (π, π). Harima et al. (2001) followed the evolution of the chemical potential μ as function of doping via core-level photoemission spectra. The shift can be deduced from these data because the binding energy of each core level is measured relative to the chemical potential. The result shows that μ monotonically increases with x. If the suppression of μ with doping in La2−x Srx CuO4 is attributed to strong stripe ﬂuctuations, the increase of μ as function of doping in the n-type 214-compounds is consistent with the absence of stripe ﬂuctuations. Infrared reﬂectance measurements of Singley et al. (2001) signal global similarities of the cuprate phase diagram, like the presence of the pseudogap and carrier localization. 7.2. Spin sensitive results The magnetic order and spin correlations for a Ce-doped as grown crystal with a magnetic transition temperature of 125 K were measured by Matsuda et al. (1992), see ﬁg. 7.4. The staggered moments of the Cu spins grow very gradually below 125 K, while the correlation length saturates to about 10 nm. The T dependence above 125 K can be ﬁtted to renormalized classical behavior with a renormalized spin stiffness ρ = 300 meV, compared to 800 meV in the undoped compound. Yamada et al. (1999) did not ﬁnd a well-deﬁned incommensurate spin modulation in a sample with a Tc of 18 K. The magnetic signal appears at (π, π) with a q-width, which is broader than in the as-grown antiferromagnetic phase. For both systems a static magnetic order was observed in the superconducting state. In NMR signal wipe-out effects due to slow spin dynamics occur, which are closely linked to the formation of stripes in the 214-cuprates. Here the link to stripes is only sug-

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Fig. 7.4. Staggered moments of the Cu spins (drawn line – left axis) and the inverse correlation length (dashed line – right axis) for an as grown Nd1.85 Ce0.15 CuO4 crystal with a transition temperature of 125 K (Matsuda et al., 1992). The smooth magnetization data of the raw data show that for this sample no spin reorientation occurs (as observed in some other samples). The magnetization intensity is decomposed according to a contribution coming from the Nd and the Cu moments. Using data for the (101) and (103) direction the staggered Cu component is deduced. Above 100 K the correlation length starts to decrease signiﬁcantly. The dashed line corresponds to renormalized classical behavior with a renormalized spin stiffness ρs = 300 meV, compared to 800 meV in the undoped compound.

gestive. Fig. 7.5 shows the T dependence of the signal wipe-out in Nd1.85Ce0.15 CuO4 . The intensity is corrected for temperature and spin–spin relaxation effects. In the superconducting (oxygen reduced) sample the intensity, which is constant at high temperatures, starts to decrease below 120 K and looses about half of the intensity at 4 K. The wipe-out of the signal is not dependent on frequency and direction of H with respect to crystallographic axes or size of the crystal. In the oxygenated sample the wipe-out of the Cu NMR signal starts at 200 K and is complete at 75 K. Below 25 K the signal starts to reappear again. In hole-doped compounds the Cu wipe-out is due to a distribution in slowing-down of the Cu-spin ﬂuctuations rates (Suh et al., 1999; Curro et al., 2000; Hunt et al., 1999; Teitel’baum et al., 2000a, 2000b; Hunt et al., 2001). In the electron-doped material wipe-out can be explained for the same reason taking into account that Cu NMR line is much narrower in electron-doped compounds (typically by a factor 103 ), Bakharev et al. (2002). The internal ﬁeld distribution in the oxygen rich non-superconducting sample is peaked around 0 (15%) and 2 T (65%). Such a clear difference in ﬁeld distribution in that ratio is expected if the doped sites carry no spin. The ﬁeld distribution around 2 T is indicative of a disordered spin density wave. At lower doping levels the spin structure becomes more disordered, showing patches of spins that are antiferromagnetically coupled with a large spread in spin direction. 7.3. Summary The magnetic properties in the electron-doped 214-cuprates are more resistant against doping than their hole-doped analogues. Dopants seem to dilute the magnetic system in a random way. Stripes, if present, appear hard to detect. Only around x = 0.15 a pattern seems

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Fig. 7.5. T dependence of the 63 Cu signal wipe-out in Nd1.85 Ce0.15 CuO4 , after Bakharev et al. (2002). The T dependence of the normalized integrated signal intensity is corrected for temperature and spin–spin relaxation effects. In the superconducting oxygen-reduced (red.) sample the intensity, which is constant at high temperatures, starts to decrease below 120 K and looses about half of the intensity at 4 K. This effect does not originate from skin-effects. The wipe-out of the signal is not dependent on frequency, direction of B with respect to crystallographic axes or size of the crystal. In the oxygenated (oxy.) sample the wipe-out of the Cu-NMR signal starts at 200 K and is complete at 75 K. Below 25 K the signal reappears again. The recovered signal has a strong resemblance with that in the nickelates. This feature, the presence of a non-magnetic and magnetic site seen in the line proﬁle and the distribution of the internal ﬁelds are seen as evidence for the presence of similar stripes.

to develop which is reminiscent of a spin density wave. Apical oxygens seem to play an important role in the balance between superconductivity and magnetism around x = 0.15. 8. The hole-doped double-layer 123- and 124-compounds YBa2 Cu3 O6+x (unit cell shown in ﬁg. 8.1) remains nonmetallic and tetragonal for 0 x 0.4, around which value the symmetry changes to orthorhombic and superconductivity sets in. Especially for 0.4 x 0.6 the way oxygen ﬁlls the CuO-chains makes the results dependent on the preparation procedure. The magnetism in undoped YBa2 Cu3 O6 with the CuO2 double layer is analogous to that found in the single-layered La2 CuO4 . The spin dynamics is well described as a two-dimensional Heisenberg antiferromagnet. The Néel temperature of the insulating phase is as high as 400 K. More precise, the compound should be regarded as a double-layer antiferromagnet. The in-plane Cu–Cu exchange constant J (as in HMM10 we follow the J in stead of the 2J -convention) is around 100 meV. The ratio between J and the exchange constant between bilayers J is J /J 105 . The interaction between the planes of the bilayer Jb , which is probably of the direct exchange type, is much stronger than J : 10−2 < Jb /J < 10−1 , and also the in plane anisotropy is very weak ΔJ /J 10−4 . Due to quantum ﬂuctuations the ordered moment is reduced to 0.64μB . The magnetic excitations are well described by spin wave theory. Although the XY anisotropy is very small, it affects the critical behavior. The staggered magnetization follows a power law ∝ [1 − (T /TN )]β with β 0.25 (Rossat-Mignod et al., 1991a; Jurgens, 1990). As in

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Fig. 8.1. The unit cell of YBa2−x Srx Cu3 O7−δ , in the orthorhombic phase a = 0.382 nm, b = 0.388 nm, c = 1.168 nm, spacegroup Pmmm. The (x, y, z) coordinates for Y (light gray), (Ba, Sr) (dark gray), Cu1, Cu2 (black), O1, O2, O3 and O4 (open circles) are resp. (0.5000, 0.5000, 0.5000), (0.5000, 0.5000, 0.1843), (0.0000, 0.0000, 0.0000) and (0.0000, 0.0000, 0.3556), (0.000, 0.5000, 0.0000), (0.5000, 0.0000, 0.3779), (0.0000, 0.5000, 0.3790) and (0.0000, 0.0000, 0.1590), see HMM10.

the 214 compounds in slightly doped, but still magnetic materials, the coupling constants will be renormalized. In the 123-compounds the amount of hole doping in the CuO2 planes can be varied via the oxygen content in the chains. For the analysis of the results the various oxygen structures in the chains in the underdoped materials form a complication. In YBa2 Cu4 O8 this complication is avoided as double instead of a single Cu–O chains run parallel to the b axis, for which the oxygen content is thermally stable up to 850 ◦ C (Karpinski et al., 1988). The orthorhombic c-axis is approximately twice as long with two rare-earth ions in the chemical unit cell. The samples behave similarly to slightly underdoped 123-samples. In the following they will not be addressed separately. In ﬁg. 8.2 we reproduce the phase diagram of the 123-compounds. For references to the original papers we refer to Johnston (1991). Here we will concentrate on the two major issues of this contribution to the handbook: the presence and magnetism of stripes and the so-called magnetic resonance peak. 8.1. Neutron scattering Inelastic neutron scattering gives direct information about the spin and charge ﬂuctuations in the material. For the 123-compounds the spectra contain several important features,

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Fig. 8.2. Phase diagram of YBa2 Cu3 O6+x , after Rossat-Mignod et al. (1991a). Five typical regimes can be deﬁned: (i) The pure AF-state without donated holes in the Cu(2)O2 plane (up to x = 0.2). (ii) The doped AF-state for 0.20 < x < 0.40. (iii) The weakly-doped metallic state 0.40 < x < 0.50 which results in an insulator-metal transition. (iv) The strongly-doped metallic state which develops at low temperatures two superconducting states with Tc ≈ 60 K for 0.5 x 0.7 and Tc ≈ 90 K for 0.8 x 1.

including a gap in the superconducting state, a pseudogap in the normal state, and the magnetic resonance peak (Dai et al., 1999; Fong et al., 2000). In addition the scattering experiments have revealed a pattern of incommensurate spin ﬂuctuations, similar to those in the 214-compounds, consistent with the stripe picture (Mook et al., 2000). By this resolution of the broad neutron spot at (π, π ) into the incommensurate peaks the long standing puzzling asymmetry in the neutron data between the 214- and 123-compounds has disappeared. At low doping incommensurate static charge ordering is found and the magnetic pattern is complex with a resonance and incommensurate structure at low temperature (Mook et al., 2002). Below we start with the discussion of stripes. Stripes. In the experimental section we reproduced the inelastic neutron-scattering data of Mook et al. (2000), as an illustration of the one-dimensional nature of the magnetic ﬂuctuations in YBa2 Cu3 O6.6 . The data show a four-fold pattern of incommensurate points around the magnetic (1/2, 1/2) reciprocal lattice position. By comparing the results of twinned and partially detwinned samples, the authors conclude that the stripe excitations are one-dimensional. In the inelastic data of Arai et al. (1999) the spin dynamics of underdoped YBa2 Cu3 O6.7 (Tc ≈ 67 K) revealed an incommensurate wave vector dependence with “pillars” in the dispersion relation at the position (1/2 ± d, 1/2, 0) and (1/2, 1/2 ± d, 0), which is the same symmetry found in the 214-compounds. The value of the incommensurability d is about 1/8, as expected for the hole concentration in a stripe domain structure. Antiferromagnetic ordering in superconducting YBa2 Cu3 O6.5 was inferred by Sidis et al. (2001) from their polarized and unpolarized elastic neutron-scattering data. The magnetic peak intensity exhibits a marked enhancement at the superconducting transition temperature. From μSR it appears that the staggered magnetization in this sample ﬂuctuates on a nanosecond time scale and seems to suggest an unusual spin density wave coexisting with superconductivity. In their high-ﬁeld NMR study of the vortices in the superconducting state, Mitrovi´c et al. (2001) saw strong AF ﬂuctuations outside the core, whereas inside

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Fig. 8.3. The T dependence of the neutron intensity at the AF wave vector near the resonance energy of hω ¯ = 40 meV and Q = (0.5, 0.5, 5.2) in YBa2 Cu3 O6.97 , after Bourges et al. (1996). In agreement with previous data by Rossat-Mignod et al. (1991a, 1991b) and Mook et al. (1993), the resonance peak undergoes a clear change at Tc , above which temperature the scattering remains constant within the error bars, at least up to 190 K. The line is a guide to the eye.

the electronic states seem to have a mini-gap of about 5 meV, much sharper than the variation of the density of states outside the vortex core. The AF features might be seen as evidence for the presence of an underlying spin density wave or magnetic stripes even in the superconducting state (Sachdev and Zhang, 2002), see also the STM experiment of Hoffman et al. (2002a) in La2−x Srx CuO4 . In YBa2 Cu3 O6.35 Mook et al. (2002) measured stripes of holes lying along every eighth row of copper atoms. Using inelastic neutron scattering on the same crystal it appeared possible to observe the incommensurate scattering even in the region of the magnetic resonance. Only at the lowest energy (just above the gap energy of 10 meV) magnetic incommensurate scattering has the correct spacing of 1/16 reciprocal lattice units to result from antiphase boundaries (every fourth row). Incommensurate ﬂuctuations and the resonance, see below, could be even observed at the same time at the resonance energy, be it that the accuracy does not allow a quantitative description. Resonance peak. Already in the beginning of high-Tc inelastic neutron data on YBa2 Cu3 O7 (Rossat-Mignod et al., 1991a, 1991b, 1992; Mook et al., 1993; Fong et al., 1995; Bourges et al., 1996) showed the presence of a sharp magnetic collective mode, the socalled (magnetic) resonance peak, in the superconducting state, see ﬁg. 8.3. Since then this peak has also been observed in underdoped 123-compounds (Dai et al., 1996, 1998; Fong et al., 1997; Bourges et al., 1997), see ﬁg. 8.4. As the doping decreases the peak frequency ωr decreases, while both the peak width and its integrated intensity increase. In the underdoped systems, a considerably broadened peak at ωr is also observed in the normal state (Fong et al., 1997; Bourges et al., 1997). The evolution of the resonance and incommensurate spin ﬂuctuations in superconducting YBa2 Cu3 O6+x as function of x has been further studied by Arai et al. (1999), see ﬁg. 8.5, and Dai et al. (2001). It is conﬁrmed that these are general features of the spin dynamical behavior for all oxygen doping levels. Furthermore the resonance and incommensurate ﬂuctuations appear to be intimately connected. In some theoretical studies (see theoretical section) the resonance is interpreted as the consequence of the d-wave gap symmetry of the cuprate superconductors, while other stress the importance of the Coulomb correlations or view the resonance as a pseudo Goldstone

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Fig. 8.4. T dependence of the magnetic resonance peak in YBa2 Cu3 O6.6 , after Bourges et al. (1997). The major problem encountered in the inelastic neutron-scattering experiments is the extraction of the magnetic contribution from the scattering that arises from the nuclear lattice. There is no unambiguous determination of the energy dependence of Imχ over a wide range of energies. The use of polarized neutrons has even led to spurious effects like the persistence of the resonance peak in the normal state. The major criteria to determine the magnetic signal are the dependence of the measured intensities on the wave vector as well as the temperature. Bourges et al. (1997) determined the dynamic spin ﬂuctuations in YBa2 Cu3 O6.6 . The magnetic resonance, that occurs around 40 meV in overdoped samples, is shifted to 34 meV. The ratio between Tc and Er is found to be almost independent of doping level. This ﬁnding supports the idea that the resonance energy is proportional to the superconducting gap. The wave vector and temperature dependencies demonstrate the magnetic origin of this signal.

mode, see theoretical section. At the low doping side the neutron-scattering experiments in YBa2 Cu3 O6.35 Mook et al. (2002) were able to see stripes and resonance features with even better resolution. Impurities. The effect of small amounts of doping in the CuO2 might reveal more about the interactions in this plane crucial for superconductivity and magnetism. Most of the neutron scattering and NMR experiments (discussed separately below) use the S = 0 Zn2+ or Li+ and the S = 1 Ni2+ ions to replace the S = 1/2 Cu2+ sites. Sidis et al. (2000) compared YBa2 (Cu0.97Ni0.03)3 O7 with Tc = 80 K and YBa2 (Cu0.99Ni0.01)3 O7 with Tc = 78 K. In the pure system the magnetic resonance peak is at Er = 40 meV. In the Nisubstituted system, the peak shifts to lower energy with a preserved Er /Tc ratio, while the shift is much smaller upon Zn substitution. By contrast Zn, and not Ni, restores signiﬁcant the spin ﬂuctuations around 40 meV in the normal state. The fact that the ratio Er /Tc for Ni-doping remains constant, suggests that the collective mode and the superconducting gap are renormalized in the same way, which is consistent with the spin-exciton scenario (see theoretical section). According to Polkovnikov et al. these experiments show that Zn creates a net spin S = 1/2 at the nearest neighbor Cu ions. Small moments. Mook et al. (2001) identiﬁed recently a new magnetic scattering feature: small magnetic moments that increase in strength when the temperature is reduced below

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Fig. 8.5. Sketch of the energy dependence of the incommensurate peaks projected along Q2D in YBa2 Cu3 O6.7 (with a transition temperature of 67 K), after Arai et al. (1999). The ﬁgure is obtained by making constant energy slices at 80, 130 and 180 meV, extracted from the two-dimensional maps of the measured inelastic scattering at 20 K. The separation Δq is about 0.16 rlu in the low-energy region, shrinks at Ec and then gradually increases above Ec . The error bars in q (full width at half the peak height) are typically around 0.1 rlu. The resonance peak is seen as a single peak in (Q, E) space and is centered on (1/2, 1/2, 0) at an energy Ec ∼ 41 meV. The peak might be a feature, which is not connected with the incommensurate legs, being the bottom piece of the broad dispersive feature above Ec .

T ∗ and further increase below Tc . The moments are antiferromagnetic between the Cu–O planes with a correlation length longer than 20 nm in the ab-plane and about 4 nm along the c-axis. These data might be explained as being due to orbital currents (Chakravarty et al., 2001), see also discussion in section 3. However, care is required, as small moments have been seen before in nuclear magnetic resonance experiments in 123-samples, where they were shown to be due to the inﬂuence of moisture (Dooglav et al., 1999). 8.2. μSR and NMR/NQR From a NMR point of view YBa2 Cu3 O7−δ has been very accessible as 89 Y, 17 O and 63,65Cu have non-identical locations and are good nuclei (samples have to be enriched in oxygen-17 as its natural abundance is only 3.7 × 10−2 %) to perform NMR, see experimental section for more details. Consistent data have been obtained by several groups. For references to the early work we refer to the introductions and reviews of Mehring (1992), Asayama et al. (1996), Berthier et al. (1996), and Brom (1998). Here we concentrate on the strongly underdoped regime, where magnetism is most pronounced. The low frequency (compared to the neutron data) magnetic response, as probed by the spin–lattice relaxation rates (χ (ω)) and the spin–spin relaxation rates (χ (ω)), and the uniform susceptibility (probed by the Knight shift) appeared to be quite unusual for a metallic system. In normal metals the susceptibility is T independent and the relaxation rate of the nuclei follow the Korringa relation 1/T1 T K 2 is constant (see experimental section). The most prominent features of the underdoped superconductors are the almost featureless decrease of the Knight shift with temperature starting at some temperature T0 well above Tc , see ﬁg. 8.6, which behavior is the same for all nuclei (Alloul et al., 1988, 1989; Monien et al., 1991). In contrast to the Knight shift, the T dependences of the relaxation rates of Cu on the one hand and oxygen and yttrium on the other differ considerably,

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Fig. 8.6. The T -dependence of the normalized Knight shift of 69 Y, 63 Cu, 17 O in underdoped YBa2 Cu3 O6.63 with Tc = 62 K. The values for the static spin susceptibility per Cu(2) is given on the right-hand scale. Below T0 ≈ 300 K K and χspin decrease with decreasing T . Note that the Knight shifts have exactly the same common temperature dependence as χ0 , after Takigawa et al. (1991) and Alloul et al. (1988, 1989).

see ﬁg. 8.7 (Takigawa et al., 1991). For the Cu nuclei 1/T1 T has a maximum at (in case of optimal doping) or above (for undoped samples) the superconducting transition (Imai et al., 1988, 1989; Horvati´c et al., 1989; Warren et al., 1989; Takigawa et al., 1991), and even follows a Curie–Weiss-like behavior (Asayama et al., 1996). Although the Cu2+ spins become itinerant upon doping and the material becomes superconducting, the system remains to a remarkable extent well described by the physics of localized spins. Millis et al. (1990) presented a phenomenological model for the explanation of the early NMR data in YBa2 Cu3 O7−δ , which is based on the Mila–Rice–Shastry hamiltonian, presented in the experimental section. The form factor plays a major role: oxygen and yttrium sites hardly or do not feel the antiferromagnetic correlations, while the Cu nuclei do. The cancellation on the oxygen sites becomes problematic if the antiferromagnetic correlation has a wave vector that is not exactly at π, π , see experimental techniques and also Pennington et al. (2001). The deviations from Korringa behavior of 63 T1 , already seen in the early experiments (Imai et al., 1989), established the doping dependence of the antiferromagnetic correlations. The strong deviations from a constant Pauli susceptibility in the Knight shift data are a sign of a change in the relevant density of states, and hence might be interpreted as the opening of a pseudogap. However, it is important to realize that growth of short range spin order alone causes a reduction of χ (q = 0) without having any gaps (Singer and Imai, 2002). By performing 89 Y NMR and 63 Cu NQR experiments on Y1−z Caz Ba2 Cu3 O7−δ Singer and Imai (2002) conﬁrm that a spin gap appears as the oxygen concentration is increased from 6.0 to 6.5 (at ﬁxed z). The NMR phase-diagram. The decrease in the Knight shift below T0 > Tc in the underdoped compounds is related to χ(0, 0), see ﬁg. 8.6. Pines (1997) thinks in terms of a precursor to a spin density wave (indications of which might also be seen in ARPES data), and hence favors a magnetic scenario, see ﬁg. 8.8. There will be in general two distinct groups of quasiparticles on the Fermi surface. Hot quasiparticles are located in the vicinity

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Fig. 8.7. Sketch of the T -dependence of (T1 T )−1 for planar Cu(2) sites (solid lines) and O(2, 3) sites (dashed lines) in underdoped YBa2 Cu3 O6.63 with Tc = 62 K, based on the data of Takigawa et al. (1991). In the underdoped compound (63 T1 T )−1 has a maximum around T ∗ ≈ 150 K. The Cu-relaxation rate is strongly inﬂuenced by magnetic correlations. The clearly different T dependencies of the Cu and O relaxation rates are explained by the symmetry of the form factors, that ﬁlter out the antiferromagnetic correlations at the oxygen site.

Fig. 8.8. Model of a Fermi surface in the cuprates (solid line) and the magnetic Brillouin zone boundary (dashed line), after Pines (1997). The intercept of the two lines marks the center of the hot spots on the Fermi surface. These are regions near (π, 0), that can be connected by the wave vector Qi and hence can be strongly scattered into each other. This can be seen as the precursor to a spin density wave and favors a magnetic scenario for the explanation of the pseudogap. Cold quasiparticles are found elsewhere, where the interaction is as in normal Fermi liquids.

of hot spots, i.e., those regions in momentum space where the magnetic Umklapp surface intersects the Fermi surface. Because of the kinematic mismatch far away from these Umklapp surfaces, the quasi-particles do not interact with the magnetic ﬂuctuations and behave therefore more normally (cold spots), Williams et al. (1997) compared the Knight shift data of many superconducting cuprates having widely different Tc,max values and

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Fig. 8.9. T dependence of the antiferromagnetic correlation length for YBa2 Cu3 O6+x for x = 0.63 and x = 1 and for YBa2 Cu4 O8 , after Barzykin and Pines (1995). The correlation times are determined from 63 T2G , the Gaussian part of the spin dephasing time of 63 Cu, see the discussion in the experimental section. In the 214-compounds the results obtained via NMR and via neutron scattering compare well. For that reason the analysis is believed to give reliable results in 123 as well.

different hole concentrations. The Knight shift data were analyzed using the standard ex∞ pression χs = μB ∞ N(E)[−∂f (E)/∂E] dE, with the density of states N(E) = δ(E − E(k)) dk2 /2π 2 with the quasiparticle energies E(k) = [ε(k)2 + ΔT (k)2 ]1/2 . The equation for the total gap below Tc is supposed to be a superposition: ΔT (k)2 = Δ(k)2 + Eg (k)2 , with Δ(k) being the superconducting and Eg (k) being the normal-state gap. A ﬁt to the data with both Δ(k) = Δ cos 2θ and Eg (k) = Eg cos 2θ is clearly superior to ﬁts with an s-wave pseudo-gap. The Eg /kB Tc,max values obtained from the NMR data are consistent with those found from ARPES measurements (Marshall et al., 1996). The Knight shift pseudo-gap is therefore ascribed to charge in stead of spin excitations (Williams et al., 1997; Berthier et al., 1996). In underdoped samples the maximum in (63 T1 T )−1 , see ﬁg. 8.7, at temperature T ∗ with −1 T0 > T ∗ > Tc is followed by a simultaneous decrease of (63 T1 T )−1 and increase of 63 T2G with decreasing T . This suppression of the low frequency spectral weight around the antiferromagnetic wavevector is a manifestation of the opening of a pseudo gap in the spin excitations (Pines, 1997; Berthier et al., 1996). In the MMP theory outlined in the experimental section, the NMR relaxation data are sensitive probes for the magnetic correlation length ξ , and the characteristic frequency of the spin ﬂuctuations ωSF . The third undetermined parameter is α, which is a scale factor (units of states/eV), that relates χQ to ξ 2 : the height of each of the four incommensurate peaks is χQi = (α/4)ξ 2 μ2B . For YBa2 Cu3 O6+x with x = 0, 0.63 and 1, 63 T1 T and 1/63 T2G are resp. given by [135 (sK/eV2 )ωSF /α, 301 (eV/s)αξ ], [145 (sK/eV2 )ωSF /α, 293 (eV/s)αξ ] and [126 (sK/eV2 )ωSF /α, 310 (eV/s)αξ ]. The α value for x = 0.63 equals 8.34 and for x = 1 it is 14.8. The antiferromagnetic correlation length, as deduced from the 63 T2G measurements, is shown in ﬁg. 8.9. Based on the NMR data Pines (1997) proposed the phasediagram of ﬁg. 8.10. In the terminology of MMP (Pines, 1997), above T0 χNAFL displays mean ﬁeld behavior with 2 is ωSF and ξ −2 varying linearly with T : ωSF ∼ ξ −2 (T ) ∝ a + bT , and hence 63 T1 T /T2G ∗ −1 63 constant. Between T0 and T , ωSF ∼ ξ (T ) ∝ c + dT , leading to T1 T /T2G is constant.

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Fig. 8.10. Phase diagram for the cuprate superconductors, after Pines (1997). If the NMR data are analyzed in the context of the MMP model, the Cu-data close to (π, π ) are dominated by the antiferromagnetic correlations described by a nearly antiferromagnetic Fermi-liquid susceptibility χNAFL (see the experimental section), while far away the dynamic susceptibility is Fermi-liquid like. The Cu spin-lattice relaxation-rate goes with T /ωSF and T2G ∝ 1/ξ . The relaxation data show that χNAFL varies dramatically with doping and temperature through changes in χQ , ωSF and ξ , and the dependence of ωSF on ξ . Above Tcr χNAFL displays mean ﬁeld or RPA behavior with ωSF and ξ −2 varying linearly with T . Between Tcr and a second cross over temperature T ∗ , χ displays z = 1 dynamic scaling behavior. The phase between Tcr and T ∗ is called pseudo-scaling because the scaling behavior is not universal. Below T ∗ one enters the pseudogap phase, in which the antiferromagnetic correlations become frozen, while ωSF after reaching a minimum increases rapidly as T is further decreased. Pseudogap denotes the quasiparticle gap-like behavior found between T ∗ and Tc , a behavior not accompanied by long-range AF order. The two cross-over temperatures Tcr and T ∗ are also seen in the magnetic susceptibility. Detailed analysis of NMR and INS data makes it possible to obtain a criterion for Tcr in terms of the strength of the AF correlations ξ at Tcr : in PS ξ 2a, while in MF ξ 2a.

The scaling between T0 and T ∗ is not universal (“pseudoscaling”). Below T ∗ one enters the pseudogap phase, where ξ is frozen and ωSF increases rapidly as the temperature is decreased. At Tcr one ﬁnds ξ(Tcr ) 2a. With μSR Sonier et al. (2001) found that the presence of small magnetic ﬁelds of electronic origin is intimately related to the pseudogap transition. For optimal doping these weak static magnetic ﬁelds appear well below the superconducting transition temperature. Stripes. In the 214-compounds below the charge-order temperature the stripes lead to slow spin dynamics, and strong wipe-out features. As static charge order has not been seen in the 123 compounds, see also the 89 Y NMR results of Bobroff et al. (2002b), it is no surprise that such wipe-out features are absent for δ < 0.5. The wipe-out seen at lower doping values is linked to the inhomogeneous distribution of the localized carriers. Combined with the additional neutron data of Mook et al. (2002) this might be seen as evidence of stripes (Kobayashi et al., 2001). Magnetic structures that possibly are linked to spin density waves or stripes have also been seen in and around the vortex cores in 123 in the NMR experiments of Mitrovi´c et al. (2001). Also in the experiments performed by Haase et al. (2000) the spatial modulation of the NMR properties in 214 and 123 cuprates, although still puzzling, are possibly connected to stripes.

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By NQR on the Cu(2) nuclei (see experimental section) Krämer and Mehring (1999) conﬁrmed the existence of a λ-like peak in the spin–spin relaxation rate in the superconducting state of highly doped YBa2 CuO7−δ . They also observed an increasing quadrupolar broadening of the Cu(2) NQR line below 35 K. This new feature conﬁrmed the quadrupolar nature of the mechanism involved in the 35 K anomaly, which is therefore connected with some type of charge redistribution. By comparing Cu(2) and Cu(1) NQR data Grévin et al. (2000) conclude that the CDW develops in the chains and that these in-chain CDW correlations are strongly involved in the appearance of an in-plane modulated structure below Tc . Impurity induced structures. In a series of experiments using 89 Y NMR by Alloul et al. (1991), Mahajan et al. (1994, 2000) together with SQUID susceptibility data (Mendels et al., 1999) and 7 Li resonance results (Bobroff et al., 1999) and also by 63 Cu NMR experiments by Julien et al. (2000) the effects of impurities in the CuO2 -plane have been investigated. It is has been shown that each Zn or Li impurity, despite having no spin, induces a local, unpaired S = 1/2 moment on the Cu ions in its vicinity at intermediate energy scales (see also the discussion between Bobroff et al. (2002a) and Tallon et al. (2002). A similar conclusion was reached by Finkelstein et al. (1990) from their EPR data in Zn doped La2−x Srx CuO4 . In the underdoped regime this can be understood by the conﬁning property of the host antiferromagnet, in which the impurity is a localized “holon” which binds the momentum of a S = 1/2 “spinon”. More generally, the features can be understood by taking into account the exchange interactions between a local S = 1/2 near the Zn/Li site with the fermionic S = 1/2 excitations of a d-wave superconductor (Polkovnikov et al., 2001). 8.3. Summary The magnetic properties of YBa2 Cu3 O6 and heavily underdoped YBa2 Cu3 O7−δ have many aspects in common with 214-cuprates, but there are differences. Regarding stripes, incommensurate one-dimensional features have been seen in inelastic and elastic neutron scattering, be it only at low doping levels. It suggests that the time scale of stripe excitations at higher doping concentrations is much shorter in the 123 than in the 214 cuprates. The difference in time scale can also be concluded from the NMR data. In NMR some features reminiscent of slowly ﬂuctuating stripes have been seen only in very underdoped materials, while the more heavily doped materials are reasonably well described in terms of a homogenous delocalized conducting magnetic system. The presence of a small spatial structure in the NMR properties is however beyond suspicion – but its origin is still under investigation. Another difference is the presence of the magnetic resonance in the double layered YBa2 Cu3 O7−δ and as we will see later on, also in the bismuthates, but not in La2−x Srx CuO4 . The various explanations are discussed in the theoretical section. 9. Other multilayered cuprates High-Tc cuprates with additional oxide layers of bismuth, thallium or mercury are located on the optimally or overdoped side of the phase diagram, and display no magnetic order, see the introduction. Below we brieﬂy give the main results for Bi-2212, one of the best

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investigated structures due its stable surface properties and well deﬁned cleaving plane. The unconventional form of superconductivity in this high-Tc superconductor was recently convincingly demonstrated by the optical measurements of Molegraaf et al. (2002), which showed spectral-weight transfer from high to low energy when cooling down through the superconducting transition temperature. This behavior contradicts that expected from traditional models, where pairing is due to a lowering of the potential energy and goes together with a small increase in the kinetic energy. Information about the thallium compounds can be found in “Thallium-based high-temperature superconductors” (Hermann and Yakmi, 1993). Like in the 123 and 214 systems, in Bi-2212 a magnetic resonance peak has been observed in neutron scattering in the superconducting state. 9.1. Neutron scattering The average structure of Bi2212 is the centrosymmetric space group Bbmb with the unitcell parameters a = 0.541 nm, b = 0.541 nm, and c = 0.309 nm, see ﬁg. 9.1. The incommensurate one-dimensional modulation is given by the wave vector qs = 0.21b∗ + c∗ . Etrillard et al. (2000) performed elastic neutron scattering on a high quality single crystal Bi2 Sr2 CaCu2 O8+δ . The data show that the previous picture is too simple and the incommensurate structure to be made up out of two interpenetrating subsystems. In a subsequent study (Etrillard et al., 2001)investigated the structural dynamic properties. The data indicate two acoustic-like longitudinal phonon branches along the incommensurate direction with a weak interaction between these two subsystems. 9.2. μSR and NMR/NQR After reports that stripes might have been seen in ARPES data by Shen et al. (1998) – see below – Brom et al. (2000) performed NMR measurements on a low doped sample with Tc of 60.5 K to look for possible features of slow stripe ﬂuctuations. These NMR data do not differ essentially from those of Ishida et al. (1998) obtained on higher doped samples and compare well with 123-samples with higher doping levels. It means that antiferromagnetic correlations are present, but no static or slow ﬂuctuating spin structures develop. No wipeout effects were observed. Watanabe et al. (2000b) applied μSR on Bi2 Sr2 Ca1−x Yx (Cu1−y Zny )2 O8+δ over a wide range of hole concentrations. At a hole doping concentration of 1/8 there is a singularity in the magnetic correlation between the Cu spins. The muon spin depolarization rate of the Zn-substituted sample with hole doping of 1/8 (x = 0.3125 and y = 0.025) at 0.3 K goes almost with B −1/2 for the magnetic ﬁeld B applied along the initial muon spin direction. From these data they conclude that the magnetically excited state of Cu spins moves in a one-dimensional direction, which supports the evidence of stripes, see ﬁg. 9.2. 9.3. Charge sensitive techniques By using scanning tunneling microscopy (STM) Hoffman et al. (2002a) were able to determine the spatial structure of the low energy quasiparticle states associated with quantized vortices in Bi2 Sr2 CaCu2 O8+δ , see experimental techniques. These states break continuous translational and rotational symmetries and exhibit a checkerboard pattern with a periodicity of four unit cells and an orientation parallel to the Cu–O bonds. Their existence may be

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Fig. 9.1. The tetragonal unit cell of Tl2 Ba2 CaCu2 O8 , which closely resembles that of orthorhombic Bi2 Sr2 CaCu2 O8 , with a = 0.3855 nm, and c = 29.318 nm, spacegroup I 4/mmm. The (x, y, z) coordinates for Tl (dark gray), Ba (middle gray), Ca (light gray), Cu (black), O1, O2, O3 and O4 (open circles) are resp. (0.5000, 0.2285, 0.0522), (0.0000, 0.2537, 0.1409), (0.5000, 0.2500, 0.2500), (0.5000, 0.2498, 0.1967), (0.7500, 0.0000, 0.1950), (0.2500, 0.5000, 0.2020), (0.5000, 0.2800, 0.1220), and (0.000, 0.1500, 0.0530), see HMM10.

related to the spin structure seen in the neutron-scattering experiments of Lake et al. (2001) on the spins in the vortices of La2−x Srx CuO4 , which is explained by a pinned spin density wave (SDW) (Demler et al., 2001) of wavelength λ. In stripes such a spin density wave is associated with a charge modulation of the same orientation and spatial extent, but which wavelength λ/2, as observed in the STM experiment, see also Sachdev and Zhang (2002). Lang et al. (2002) report STM data on underdoped Bi2212 that reveal an apparent segregation of the electronic structure into superconducting domains that are ∼3 nm in size, located in an electronically distinct background. Ni-impurity resonances are used as markers. The resonances are present in the superconducting, but not in the pseudogap regime (see also Zaanen, 2002). Scans as function of the energy interval show that the data are linked to the strong inhomogeneity in the samples, which leads to scattering events that are sensitive to the peculiar gap structure. In their zero-ﬁeld STM study on nearly optimally doped

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Fig. 9.2. Longitudinal-ﬁeld dependences of the muon-spin depolarization rate λ for Bi2 Sr2 Ca1−x Yx · (Cu1−y Zny )2 O8+δ with x = 0.3125 and y = 0.025 at 0.30 K (after Watanabe et al., 2000b). The solid line is the best ﬁt with λ ∝ B −0.63(3) , which is close to 0.5 (dotted line). From these data the authors conclude that the magnetically excited state of Cu spins moves in a one-dimensional direction, which gives evidence for stripes.

Bi2212 Howald et al. (2003) show the existence of static charge-density modulation of quasiparticle states both in the normal and superconducting state. The modulation is aligned with the Cu–O bonds, with a periodicity of four lattice constants, and exhibits features of a two-dimensional system of line objects. According to Hoffman et al. (2002b) quasiparticle interference, due to elastic scattering between characteristic regions of momentum-space, provides a better explanation for the conductance modulation without appeal to another order parameter. According to the authors these scattering processes might be a potential explanation for some other incommensurate phenomena in the cuprates as well. STM studies have also revealed that the vortex core center has an enhanced density of states, with a decay length of about 2 nm (Pan et al., 2000). The STM technique has also been used to shed light on the inﬂuence magnetic (Ni) and non-magnetic (Zn) impurities have on the high Tc superconductors, see also the 123section. Hudson et al. (2001), see also Flatté (2001), observe at each Ni site two d-wave impurity states of apparently opposite spin direction, whose existence indicates that Ni retains a magnetic moment in the superconducting state. However, the quasi-particle spectrum is predominantly non-magnetic. While the non-magnetic Zn impurity destroys superconductivity, the superconducting energy gap and correlations are not affected by Ni. These ﬁndings are explained by several models. Polkovnikov et al. (2001) invoke the exchange interactions between a local S = 1/2 near the impurity site with the fermionic s = 1/2 excitations of a d-wave superconductor. Martin et al. (2002) add that the images of resonant states are the result of quantum interference of the impurity signal coming from distant paths, while stripe pinning is seen as another important ingredient by Smith et al. (2001). In break-junction tunneling data of Zasadzinski et al. (2001) in Bi2 Sr2 CaCu2 O8+δ (see ﬁg. 9.1) over a wide range of hole concentrations the conductance exhibits sharp dips at a voltage /e measured with respect to the superconducting gap. While phonons are collective excitations of the lattice and therefore exist in the normal state, the collective modes relevant in this experiment seem to develop only below Tc . The dip strength is maximal at optimal doping and scales with 4.9kTc over the entire doping range, which

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Fig. 9.3. Momentum-dependent spectral-weight change along (0, 0) to (π, 0) for Bi2 Sr2 CaCu2 O8+δ (after Shen et al., 1998). The quantity Asc − Anorm , where A is the k resolved single particle spectral weight, when normalized to the normal state value Anorm , gives the difference of the occupation probabilities in the two states [nk (s) − nk ]/nk . nk and nk (s) represent the occupation probabilities of the normal and superconducting states, respectively. The spectral weight is transferred from one momentum to another, with a transfer vector Q broadly peaking between 0.4π and 0.5π . This Q value is close to the expected momenta of charge and spin ordering in the charge stripes.

is close to that of the resonance spin excitation energy seen in 123 compounds by neutron scattering by, e.g., Fong et al. (1997), Dai et al. (1999), and He et al. (2002). ARPES data in Bi2212, like in the other high-Tc ’s, show a sudden change in the energy dispersion curves along the (0, 0)–(π, π) direction. Lanzara et al. (2001) see this as evidence for the ubiquitous presence of strong electron–phonon interaction. According to Johnson et al. (2001) the features in Bi2212 are associated with a coupling to a resonant mode, see also Eschrig and Norman (2000). The data for Bi2 Sr2 CaCu2 O8+δ of Shen et al. (1998) show changes in the single particle excitations that strongly depend on q when T is lowered from above to below Tc , see ﬁg. 9.3. These changes extend up to an energy of about 0.3 eV or 40kB Tc . The data suggest an anomalous transfer of spectral weight from one momentum to another, involving a sizable momentum transfer Q = (0, 0.45π, 0). There are several possible interpretations of the data. The observed q dependence can be readily explained if the system has collective excitations with real-space periodicities corresponding to Q, that are enhanced or developed below Tc . It is intriguing that Q is close to the expected momenta of charge and spin ordering in the charge stripes that were ﬁrst observed in neutron scattering from La1.46Nd0.4Sr0.12 CuO4 . Taking a nominal doping of 0.18 near optimal doping and using the saturated value of x = 1/8, q = (±0.5π, 0) or (0, ±0.5π) and q = (π, π ± 0.12π) or (π ± 0.12π, π), indeed close to the observations. Using underdoped, optimally doped and overdoped samples, Chuang et al. (1999) ﬁnd electron-like portions of the Fermi surface centered around the Γ point and a depletion of spectral weight around M. The ﬂat bands observed at other photon energies may indicate the presence of two electronic components. This is in agreement with the data of Feng et al. (2001) on heavily overdoped Bi2212. They found the long sought bilayer splitting. The data favor the bilayer Hubbard model over LDA calculations. It shows that the bilayer splitting in the superconducting state is reduced to 23% of the splitting in the normal state. The presence of bonding and

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antibonding states explains also the detection of a “peak-dip-hump” structure in the normal state of heavily overdoped samples. Also Kordyuk et al. (2002) stress the superposition of spectral features originating from different electronic states to explain this feature. 9.4. Summary Under the multilayered compounds the cleavage plane of Bi-2212 makes it very attractive for ARPES and other surface sensitive techniques. Via ARPES it has been possible to show the closeness of the superconducting and stripes phases. Recent STM data are an even stronger argument for the competition between superconductivity and spin/charge order in this compound. Till now due to the small size of the crystals the presence of a magnetic resonance is only indirectly demonstrated. 10. Concluding remarks In HMM10 Johnston noticed that high-Tc cuprates and parent compounds provide a remarkably rich variety of normal state magnetic behaviors, including short-range static and dynamic antiferromagnetic ordering of conventional spin-glass and cluster spin-glass types, and long-range three-dimensional order. Since 1997, when the review of HMM10 was written, these remarks have only be proven to be correct. Especially more details have been obtained about the so-called stripes, that have been seen in inelastic neutron scattering and ARPES data in single and double-layered compounds and in techniques which much longer time scales, like elastic neutron scattering, NMR/NQR, μSR, susceptibility, and transport. The so-called Resonance Peak has an even longer experimental history than stripes. While static stripes in the 214 cuprates were reported in 1995, the resonances in 123 were seen already in the neutron data almost from the start (1988). Also here the experimental development has been impressive. It has been possible to follow the resonance from optimally doped to strongly underdoped 123, and as latest development even in a single-layer cuprate (with Tl). Stripes and resonance peaks form the leading theme for the discussion of the new magnetic features in the singly and multi-layered cuprates and the related nickelates presented in this contribution. The data obtained by a variety of techniques are in good agreement and when different results are mentioned they usually can be traced back to differences in sample preparation, e.g., leading to a difference in oxygen content or surface morphology. In the theoretical section we have covered much of the theoretical main stream. There is a patchwork of theoretical ideas, devised to explain some aspects of the data successfully, but failing badly in other regards. We believe the basic understanding of the nature of the static stripes in 214 as discussed in section 2 should be correct. Regarding the dynamical stripes and the resonance peak the situation is more complicated. By inelastic neutron scattering both phenomena have now been seen together in some sample as a weak resonance together with incommensurate side branches. How these phenomena have to be explained is still not settled. As mentioned in section 3, it is fair to say that the main result of 15 years of research in high-Tc superconductivity has made it clear that the mystery is far deeper than initially expected.

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Acknowledgements HBB gratefully acknowledges the contributions of Oleg Bakharev, Gregory Teitel’baum, Evgenii Nikolaev, Oscar Bernal and especially Issa Abu-Shiekah during their stay in Leiden. This work was supported by FOM/NWO. Appendix A. Static and dynamic properties of 2D Heisenberg antiferromagnets An overview of the theory of the Heisenberg antiferromagnets is given by Johnston in HMM10. Here we brieﬂy summarize the main ideas as far as relevant for those quantities that are measured experimentally. The physical properties of 2DQHAF follow from the nearest neighbor Heisenberg hamiltonian (the model is an idealization as it is known that other interactions are present as well, see, e.g., Katanin and Kampf (2002) for the ring exchange, and Lavrov et al. (2002) for the strong coupling between the magnetic subsystem and the lattice): =J H (A.1) Sj , Si . ij

where ij refers to the summation over pairs of nearest spin neighbors. J > 0 is the antiferromagnetic exchange coupling constant. The spin operator Si is deﬁned as: 1 † † c c c σα ↑ , (A.2) c↓ 2 ↑ ↓ where σα s are the Pauli spin matrices: 0 1 0 −i ; σy = ; σx = 1 0 i 0

1 0 σz = . 0 −1

(A.3)

Many physical and mathematical methods have been applied to handle the above hamiltonian in order to extract quantitative parameters that match the measured quantities (Auerbach, 1994). The most popular method is the non-linear sigma model (NLσ M), which in combination with spin-wave theory successfully predicts the properties of La2 CuO4 spin dynamics (Chakravarty et al., 1989). The extracted parameters that can be veriﬁed experimentally include the zero-temperature staggered magnetization M(0), the Néel temperature (TN ) and the spatial correlation length ξ(T ) in the paramagnetic state. The staggered magnetization is designated as the ordering parameter of 2DQHAF. The temperature dependence of the correlation length has three distinct regimes depending on the strength of the spin–spin interaction, which is related to the spin stiffness ρs (Chakravarty et al., 1989; Zaanen, 1998). The starting point in the analysis is to deﬁne a dimensionless coupling constant g in order to construct a generic phase diagram for the 2DQHAF. The constant g is increased upon disorder and doping and has the form: g=

hcΛ ¯ d−1 , ρs

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Fig. A.1. Schematic phase diagram for the 2DQHAF system as a function of the coupling strength g and temperature T , after Chakravarty et al. (1989). The Néel ordered state at T = 0 is where the long-range antiferromagnetic order sets in. The centrum quantum critical region is controlled by the T = 0 ﬁxed point at gc . In this region the magnetic coherence length ξ ∝ 1/T (at low temperature) and hence diverges for T → 0. Also in the renormalized classical regime ξ diverges as we approach zero Kelvin. For the quantum disordered region ξ becomes independent of T for T → 0. For g > gc , the effect of ﬁnite temperature will be felt, when one exceeds the crossover temperature Tc ≈ Δ/kB , with Δ the energy gap in the spin excitations spectrum. For g < gc the crossover temperature occurs when the length scale of the short range critical ﬂuctuations equals that of the long range antiferromagnetic magnons.

√ where c is the spin-wave velocity and Λd−1 = 2π/a for d = 2 is a cut-off wave vector. At 0 K and a critical gc , a crossover occurs between the three different regimes as shown in ﬁg. A.1 (Chakravarty et al., 1989). The ﬁrst phase to consider is the renormalized classical (RC) regime for g < gc . RC is the only phase that has an ordered Néel ground state at T = 0 K and has been experimentally realized in many systems like the cuprates (Chakravarty et al., 1989)and nickelates (Nakajima et al., 1995). The most important outcome of the work of Chakravarty et al. (1989) is the prediction of the temperature dependence for the ﬁnite spin-correlation length ξ(T ) for T > TN and g < gc : 2πρs ξ(T ) ∝ exp (A.4) . T The correlation length diverges when T → 0 K as: 2πρs hc ¯ exp , ξ ≈ 0.9 kB T T

(A.5)

where ρs is the actual spin stiffness renormalized by quantum ﬂuctuations. Later Hasenfratz and Neidermayer (1991) calculated the prefactor and an extra correction term for the exponential temperature dependence of eq. (A.4): 2 2πρs T T e h¯ c exp +O 1− . ξ(T ) = (A.6) 8 2πρs kB T 2πρs 2πρs

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The second regime is the quantum critical regime, which is controlled by the T ≈ 0 K ﬁxed critical coupling gc . Exactly at g = gc the system will behave like a three-dimensional classical spin system for short length scales, before it breaks up by two-dimensional quantum ﬂuctuations on larger length scales. At low temperatures the correlation length equals: ξ(T ) ≈

h¯ c . kB T

(A.7)

The third phase with g > gc is the quantum disordered regime, which is temperature independent as T → 0 K. In this regime the correlation length is short and a gap is opened in the spin excitations spectrum of the order of Δ ≈ h¯ c/ξ (Chakravarty et al., 1989). In the renormalized classical regime the constants ρs and c are not free ﬁtting parameters, but are calculated quantities from spin-wave theory. This implies that eq. (A.6) predicts the absolute value for ξ and not only its temperature dependence. The values of √ c and ρs are determined from ρs = Zρ (S)J S 2 and c = Zc (S)2 2 J Sann with J the microscopic nearest neighboring exchange coupling and ann the nearest neighboring distance (Igarashi, 1992). The functions Zρ and Zc are respectively estimated within the spin-wave theory approximation as: 3 0.235 (0.041 ± 0.03) 1 + O , − Zρ (S) = 1 − (A.8) 2S 2S (2S)2 3 0.1580 0.0216 1 − Zc (S) = 1 + (A.9) + O . 2 2S 2S (2S) The predictions made above are in excellent agreement with experimental results extracted from systems with S = 1/2 such as La2 CuO4 (Chakravarty et al., 1989) and Sr2 CuO2 Cl2 (Greven et al., 1994). The deviations in systems with higher spin values (S 1) will be discussed below. Before the work of Chakravarty et al. (1989), Haldane (1983) had already conjectured that systems with integer spins behave differently than half-integer spins. Based on topological arguments, it was shown that integer spin systems have an energy gap (nowadays called Haldane gap) between the non-magnetic singlet state and the ﬁrst magnetic triplet state, while half-integer spin systems are gapless. According to Chakravarty et al. (1989), a Haldane gap is ruled out at low energies in the g parameter region where there is a Néel ordered state at T = 0 K. At high temperatures the work of Chakravarty et al. (1989) does not distinguish between half integer and integer spins. In the renormalized classical regime, one would expect that such an approach will be more reliable as S is increased to higher values, the classical limit. However, the experimental data deviate from the theoretical predictions in systems with spin S = 1 like K2 NiF4 (Greven et al., 1995) and La2 NiO4 (Nakajima et al., 1995), where the experimental value for ρs differs about 20% from that calculated from spin-wave theory. The discrepancy becomes larger for systems with S = 5/2 like Rb2 MnF4 (Lee et al., 1998). In order to understand the origin of this discrepancy, Elstner et al. (1995) calculated the high-temperature series expansion for the Fourier transform of the spin–spin correlationz z function S−q S−q for all spin values in the range of 1/2 S 5/2. The results lead to a

MAGNETIC ORDERING PHENOMENA AND DYNAMIC FLUCTUATIONS

scaling law for the correlation length given by eq. (A.6) as: 2πZρ T /J S 2 2 eZc Sξ(T ) T /J S 2 = √ exp + O 1 − . ann T /J S 2 4πZρ 4πZρ 16 2πZρ

487

(A.10)

When plotted in this form, the data for systems of different spin values fall on the same curve. It was suggested by the authors that the origin for the deviation between the experimental results and the predictions of eqs (A.4) and (A.6), is due to a series of energy scale crossovers observed for systems with S > 1/2. The crossovers start from renormalized classical [T J S 2 , eq. (A.6)], to classical [J S T J S 2 ], and ﬁnally to Curie–Weiss [T J S 2 ] regimes. Finally, the correlation length diverges when a three-dimensional long range ordering sets in. The long range order is due to the coupling between the adjacent planes, which is about a factor of 10−3 –10−5 less than the exchange coupling J between the nearest neighbors. The staggered magnetization at T ∼ 0 K is much reduced due to the two-dimensional quantum ﬂuctuations. As an example the maximum predicted value for the ordered magnetic moment for the Cu spins in La2 CuO4 is about 0.6μB (Chakravarty et al., 1989) which is slightly larger than the experimentally measured value of 0.54μB . References Abanov, Ar., Chubukov, A.V., 1999, Phys. Rev. Lett. 83, 1652. Abanov, Ar., Chubukov, A.V., 2000, Phys. Rev. B 62, R787. Abanov, Ar., Chubukov, A.V., Eschrig, M., Norman, M.R., Schmalian, J., 2002, Phys. Rev. Lett. 89, 177002. Abraham, A., 1961. Principles of Nuclear Magnetism. Clarendon Press, Oxford. Abrikosov, A.A., 1998, Phys. Rev. B 57, 8656. Abu-Shiekah, I.M., Bernal, O.O., Menovsky, A.A., Brom, H.B., Zaanen, J., 1999, Phys. Rev. Lett. 83, 3309. Abu-Shiekah, I.M., Bakharev, O., Brom, H.B., Zaanen, J., 2001, Phys. Rev. Lett. 87, 237201. Abu-Shiekah, I.M., 2001, Ph.D. thesis, Leiden. Aeppli, G., Mason, T.E., Hayden, S.M., Mook, H.A., Kulda, J., 1997, Science 278, 1432. Alexandrov, A.S., Kornilovitch, P.E., 2002, J. Phys. Cond. Mat. 14, 5337. Allen, P.B., 2001, Nature 412, 494. Alloul, H., Mendels, P., Monod, P., 1988, Phys. Rev. Lett. 61, 746. Alloul, H., Ohno, T., Mendels, P., 1989, Phys. Rev. Lett. 63, 1700. Alloul, H., Mendels, P., Casalta, H., Marucco, J.F., Arabski, J., 1991, Phys. Rev. Lett. 67, 3140. Altman, E., Auerbach, A., 2002, Phys. Rev. B 65, 104508.

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chapter 5

GIANT MAGNETOIMPEDANCE

M. KNOBEL Instituto de Física Gleb Wataghin (IFGW) Universidade Estadual de Campinas (UNICAMP), C.P. 6165 Campinas 13.083-970 S.P. Brazil

M. VÁZQUEZ Instituto de Ciencia de Materiales Consejo Superior de Investigaciones Cientíﬁcas (CSIC) 28049 Cantoblanco (Madrid) Spain

L. KRAUS Institute of Physics Academy of Sciences of the Czech Republic 182 21 Prague 8 Czech Republic

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© 2003 Elsevier Science B.V. All rights reserved

CONTENTS List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 2. Giant magnetoimpedance: overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 2.1. Impedance and skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 3. Phenomenology of GMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 3.1. Very low frequency regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 3.2. Low and intermediate frequency regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 3.3. High frequency regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 4. Theory of GMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 4.1. Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 4.2. Landau–Lifshitz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 4.3. Description of theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 5. Analysis of selected experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 5.1. Data in novel materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 5.2. GMI as a research tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 5.3. Asymmetric GMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 5.4. High frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 5.5. A short summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 6. Applications of GMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 6.1. Magnetic ﬁeld sensor devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 6.2. Applications derived form ﬁeld sensing: current and position sensors, magnetic signatures . . . . . 552 6.3. Stress-impedance applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 6.4. RF and microwave applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 6.5. GMI sensor heads: magnetic requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 7. Present trends and ﬁnal remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

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List of symbols Section 3 f Iac δ a νT νL νR Li Rdc ω Hex μt μrot t μDW t R X Heff hac |Z|

frequency driving current skin depth transversal dimension AC voltage (total voltage) inductive voltage (imaginary part of νT ) resistive voltage (real part of νT ) internal inductance DC resistance angular frequency (= 2πf ) external DC magnetic ﬁeld transverse magnetic permeability rotational contribution domain wall contribution resistance (real part of Z) reactance (imaginary part of Z) effective static ﬁeld AC magnetic ﬁeld modulus of Z [|Z| = (R 2 + X2 )0.5 ]

Section 4 Z R X ω = 2πf Iac Uac e ρ j L q S Rdc C

complex impedance resistance (real part of Z) reactance (imaginary part of Z) angular frequency amplitude of driving current amplitude of measured voltage AC electric ﬁeld resistivity AC current density length of sample cross-section area surface of sample DC resistance contour of area q 499

500

l hφ ζˆ n Le D E J B H M μ0 H0 h M0 m δ0 μ R k δ γ Ms H eff α Ha A τ d u U1,2 β K HK χt dw χt rot χ0 τ P c a χˆ (r) nK θ0 ψ

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length of the contour circumferential component of AC ﬁeld surface impedance tensor normal vector of the surface “external” self-inductance electric induction electric ﬁeld current density magnetic induction magnetic ﬁeld magnetization permeability of free space DC internal ﬁeld AC internal ﬁeld DC magnetization AC magnetization nonmagnetic skin depth permeability conductor diameter propagation constant skin depth gyromagnetic ratio saturation magnetization effective ﬁeld Gilbert damping parameter anisotropy ﬁeld exchange stiffness constant Bloch–Bloembergen relaxation time domain width displacement of domain walls free energies of domains domain wall pinning parameter anisotropy constant anisotropy ﬁeld transverse domain wall susceptibility transverse rotational susceptibility transverse domain wall susceptibility for ω = 0 phenomenological relaxation time power width of bamboo domain wire radius local susceptibility tensor unit vector of easy direction angle between M 0 and H 0 angle between M 0 and nK

GIANT MAGNETOIMPEDANCE

Ω, Δ b hext X Z

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AC component of B applied AC ﬁeld effective susceptibility tensor impedance tensor

Abstract The giant magnetoimpedance (GMI) effect consists of the huge change of both real and imaginary parts of the impedance upon the application of static magnetic ﬁeld. The relative change of impedance can reach ratios up to around 700%, with extremely large sensitivities in the very low ﬁeld region. It is the aim of the present work to summarize and update the increasing amount of information accumulated about the giant magnetoimpedance phenomenon. After a short historical review, an overview of GMI is given, including a brief analysis of the phenomenology of GMI, where three frequency regimes can be roughly evidenced and studied applying somewhat different approaches. The work includes a detailed theoretical framework of GMI, together with the current topics of basic investigation in this ﬁeld. Several novel experimental results are shown, including the extensive use of giant magnetoimpedance as a tool to investigate intrinsic and extrinsic magnetic properties of soft magnetic materials. With the theoretical and experimental developments in mind, the applications of GMI are detailed, going from interesting proposed prototypes to devices which are already on the market. A clear comparison of GMI sensors with other type of sensors is given. GMI opens a new branch of research by combining classical electrodynamics and micromagnetism of soft ferromagnetic materials. 1. Introduction Magnetotransport properties of materials are of remarkable importance in the competitive market of technological devices. Owing to its economical and social repercussions, one of the most important areas of research and innovation concerns the magnetic recording development, including the storage and reading of information in magnetic recording media. It explains the great excitement that arose in 1988 and subsequent years, with the discovery of the giant magnetoresistance (GMR), which led to new perspectives in the magnetic reading technologies employed in computer hard disks. A concentrated research effort was soon established to search for new GMR materials and to fully uncover its origin. Nowadays, giant magnetoresistive based devices are already a reality in commercial hard disks, and they are responsible for a considerable increase in the recording areal density. Another boom occurred in 1994, with the discovery of manganites exhibiting the phenomenon labelled as colossal magnetoresistance, CMR, from then on. Although very stimulating from the fundamental physics viewpoint, the CMR effect is still far from any real application, and its interest is essentially related to the novelty and the involved magnetic and structural phenomena. On the other hand, the so-called giant magnetoimpedance effect, GMI, is running in a parallel way in the ﬁeld of novel magnetotransport phenomena. Initially, its observation

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and research was received with less enthusiasm, probably because of the envisaged lower technological expectations and of the apparent lack of intrinsically new magnetic effects related to its origin. Nevertheless, it soon became clear that its interpretation requires a deep understanding of the micromagnetic characteristics of soft magnetic materials and its dependence on dynamic magnetism. With the rapid increase of the number of teams all over the world investigating GMI and its technological applications, GMI is actually opening a new branch of research combining the micromagnetics of soft magnets with classical electrodynamics. From the applications perspective, there are already a wide range of prototypes of magnetic and magnetoelastic sensors, and there are even several devices that were put on the market in the last few years. It is worth mentioning that the GMIbased devices are not intended to oust spin-valve-based heads from the market of magnetic recording. However, GMI devices are achieving a development stage that is mature enough to enter in the relevant area of extremely sensitive magnetic ﬁeld sensoring. Indeed, some particular proposed systems have achieved, with some advantages, the best characteristics of the well-established ﬂuxgate sensors. Furthermore, sensitivities as high as the ones found in sensors based on superconducting quantum interference devices (SQUIDs) are expected to be reached, with the great advantage of competitive price and operation at room temperature, among others. It is the aim of the present chapter to summarize and update the increasing amount of information accumulated about the giant magnetoimpedance phenomenon. Some reviews on the subject were already published in the last few years (Kamruzzaman et al., 2001; Knobel, 1998; Knobel et al., 2003; Knobel and Pirota, 2002; Ripka and Kraus, 2001; Vázquez et al., 1997; Vázquez, 2001; Kraus, 2003a; International Workshop on Magnetic Wires, 2002). In order to better organize the content, research on GMI is here reviewed from three main viewpoints: (i) Theory: from the theoretical viewpoint, the research on GMI began with phenomenological models developed to understand some basic aspects found in experimental data, such as the frequency and ﬁeld dependence of the effect. Afterwards, the research on GMI evolved to more accurate descriptions, based on the close relation of GMI and ferromagnetic resonance. After proper adaptations of the geometrical conﬁguration and boundary conditions, it is now possible to have a more precise description of the phenomenon. This fact opened a completely new perspective in the study of GMI, but brought together more intricate mathematics to deal with. (ii) Additional tool to investigate soft magnetic materials: a deeper understanding of the mechanism behind GMI allows one to predict some expected behaviours, under particular assumptions, and to use the GMI as an additional tool to investigate some intrinsic and extrinsic magnetic properties of novel artiﬁcially grown soft magnetic materials. (iii) Applications: after early observation of GMI in soft magnetic amorphous wires and ribbons, the effect has been studied in several systems, including commercial materials with different shapes (thin ﬁlms, sandwich structures, glass-covered microwires, electrolytically coated tubes), and different structural character (polycrystals, amorphous alloys, nanocrystalline materials, and single crystals). Each particular system displays peculiar properties, being interesting for several practical applications, many of them already proposed and tested in laboratory prototypes, and some of them already on the market. The chapter is organized as follows: An overview of the phenomenon known as giant magnetoimpedance, including a short historical review, is given is section 2. Section 3

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presents a brief analysis of GMI phenomenology, where three frequency regimes can be roughly evidenced and studied using different approaches. The detailed theoretical framework of GMI, together with the current topics of investigation in this ﬁeld, is shown in section 4. Section 5 presents several novel experimental results, including the extensive use of giant magnetoimpedance as a tool to investigate intrinsic and extrinsic magnetic properties of soft magnetic materials. With the theoretical and experimental developments in mind, section 6 focuses on the applications of GMI. A comparison of GMI sensors with other type of sensors is given. Finally, a short discussion on the present trends of research and ﬁnal remarks are given in section 7. Throughout the work, the SI unit system is used. For the vector quantities, which consist of an AC component and a DC component, the following notation is used: (a) a capital letter with the subscript 0 is used for the DC component, (b) a small letter means the AC component and (c) the total is denoted by capital letter (without subscript). For example, electric ﬁeld E = E 0 + e consists of the DC (E 0 ) and AC (e) components. 2. Giant magnetoimpedance: overview The magnetoimpedance phenomenon is observed in soft magnetic metals, and phenomenologically consists of the change of the AC impedance, Z = R + iX (where R is the real part, or resistance, and X is the imaginary part, or reactance), when submitted to a static magnetic ﬁeld, H0 . Alternatively, changes in impedance induced by an applied mechanical stress are labelled stress-impedance. The GMI ratio is usually deﬁned as ΔZ/Z = {|Z(H0)| − |Z(H0 max)|}/|Z(H0 max), where |Z| is the impedance modulus (|Z|2 = R 2 + X2 ) and H0 max is the maximum measuring ﬁeld, at which the sample is considered to be magnetically saturated. For example, maximum GMI ratios up to 600% have been reported for amorphous microwires, at frequencies around 1 MHz for maximum applied ﬁelds H0 max of the order of ten thousand Am−1 (hundreds of Oe) (Pirota et al., 2000). Maximum ﬁeld sensitivities up to about 10%/Am−1 can be achieved in the very low ﬁeld region (typically less than 100 Am−1 ) (Costa-Krämer and Rao, 1995). That makes magnetoimpedive materials quite suitable as sensing elements in many devices. Before proceeding further, it is worth noticing some peculiarities of impedance in ferromagnetic metals. According to the usual deﬁnition, the complex impedance of a linear electronic element at the circular frequency ω is given by Z(ω) = Uac /Iac = R + iX,

(2.1)

where Iac is the harmonic current with frequency ω ﬂowing through the element and Uac is the harmonic voltage of the same frequency, measured between its terminals. First, it should be mentioned that the deﬁnition (2.1) is not fully applicable to ferromagnetic conductors because usually such materials are not linear. This occurs because Uac is generally not proportional to Iac and it is not a harmonic function of time (it contains higher order harmonics) (Antonov et al., 2001; Beach et al., 1996; Kurlyandskaya et al., 2002a). As will be shown later, under certain circumstances the ferromagnetic conductor can be considered as a linear element. Then the procedure for the calculation of complex impedance, which we call the “linear approximation”, can be used.

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Second, although widely used, the deﬁnition of the GMI ratio ΔZ/Z, mentioned above, may be useful for quantifying the huge attained variations of impedance, but in fact it is not very much appropriate for physics. Even though the ratio ΔZ/Z linearly depends on |Z|, there are several reasons to adopt another deﬁnition: (i) the information about the phase shift is lost; (ii) it depends on the ambiguously chosen H0 max (although the sample might be apparently magnetically saturated it does not mean that GMI is also saturated); (iii) the ratio ΔZ/Z is rather sensitive to how much of the measuring circuit is included in Z(H0 max). From this point of view the deﬁnition by means of the ratio Z/Rdc (where Rdc is the DC resistance of the sample) should be more appropriate. Apart from the applied DC ﬁeld, the main parameter determining GMI is the frequency of the driving current (which generates the circular AC driving magnetic ﬁeld). Depending on the frequency, approximately three main regions can be roughly deﬁned, which will be thoroughly discussed in section 3. The ﬁrst reports dealing with what is now labelled as GMI are nearly as old as Heisenberg’s and Landau’s basic works on ferromagnetism (Harrison et al., 1935, 1937). Then it reappeared in 1991, few years before its present deﬁnition, and dealt with the developing of magnetic sensor application (Makhotkin et al., 1991). Other early works (Machado et al., 1993; Mandal and Ghatak, 1993) performed on amorphous ribbons and wires with vanishing magnetostriction were ﬁrst interpreted in terms related to GMR possibly induced by the outstanding impact created by that phenomenon. But GMR takes very modest values in GMI materials (Barandiarán et al., 2000; Vázquez et al., 1995). Although the right interpretation by the skin effect was already introduced by Harrison et al. (1935) it was proposed again simultaneously by Panina and Mohri (1994) and Beach and Berkowitz (1994a). Shortly afterwards, and continuously in the following years, a race started all over the world to report on new GMI materials with different geometries and structures: rapidly solidiﬁed amorphous wires (Blanco et al., 2001; Brunetti et al., 2001; Knobel et al., 2003; Vázquez, 2001) and ribbons (Ahn et al., 2001; Pirota et al., 1999c; Sommer and Chien, 1995; Tejedor et al., 1996), ﬁlms obtained by sputtering (Panina et al., 1995; Sommer and Chien, 1995), layered ﬁlms with insulator (Antonov et al., 1997; Morikawa et al., 1996a; Yu et al., 2000) or conductor (Zhou et al., 2001), permalloy ﬁbres (Ciureanu et al., 1996b; Vázquez et al., 1998a), electroplated microtubes (Beach et al., 1996; Kurlyandskaya et al., 1999; Sinnecker et al., 2000b), nanocrystalline alloys (Chen et al., 1996; Guo et al., 2001; Knobel et al., 1995a; Tejedor et al., 1998a), glass-coated microwires (Chiriac et al., 1998, 1999b; Kraus et al., 1999; Pirota et al., 2000; Vázquez et al., 1998b), thin etched wires (Yoshinaga et al., 1999), Mumetal (Nie et al., 1999), pure Fe wires (Hu and Qin, 2001b) and ﬂakes (Hu and Qin, 2002), Si steels (Carara et al., 2000), among others. It is worth noting that the general characteristic of GMI materials is their outstanding soft magnetic behaviour when magnetized in the circular (transverse) direction. The amplitude of GMI as well as its ﬁeld proﬁle can be properly tailored by means of speciﬁc thermal treatments, which lead to induced magnetic anisotropies. As most of the magnetoimpedive samples are amorphous ferromagnets, the main parameter conditioning the magnetic softness is the saturation magnetostriction constant (Sartorelli et al., 1997). Their domain structure is then controlled by the magnetoelastic anisotropy induced by fabrication. The typical GMI material is a Co94 Fe6 -base amorphous alloy with vanishing but negative magnetostriction (−0.1 ppm), which displays a large value of the

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effective permeability. In the case of wires, the domain structure is characterised by a multidomain inner nucleus with roughly axial easy direction, and an outer shell with transverse domains (Antonov et al., 1999; Chiriac and Óvári, 1996; Squire et al., 1994; Vázquez and Chen, 1995). Electroplated microtubes and sputtered ﬁlms show anisotropy determined by the growth and shape (Garcia et al., 2001b; Panina et al., 1999). As mentioned before, large GMI ratios can be also observed in amorphous ribbons and ﬁlms, as well as in soft crystalline materials of any shape. However, the geometry of the system has a strong inﬂuence on the overall GMI behaviour, as will be further explained in section 4. In particular, the cylindrical shape of wires and tubes minimizes the formation of magnetic poles at surfaces and closure domain structures, naturally favouring the softness of the circumferential magnetization process (Chiriac and Óvari, 2002; Vázquez, 2001). 2.1. Impedance and skin effect The skin effect, which is responsible for GMI at medium and high frequencies, is a phenomenon well described by the classical electrodynamics (Landau and Lifshitz, 1975) many years ago. As a consequence of induced eddy currents, the high frequency AC current is not uniformly distributed √ in the conductor volume but is conﬁned to a shell close to the surface, with depth δ = 2ρ/ωμ (where ω is the circular frequency, ρ the resistivity and μ the magnetic permeability). Let us assume an inﬁnite straight conductor (satisfying the Ohm’s law e = ρj , where e is the AC electric ﬁeld vector, ρ is the resistivity and j is the AC current density vector) with a uniform cross-section of the area q (see ﬁg. 1). The impedance Z measured between two points at the distance L is given by the formula ez (S) dz jz (S) dz Uac L Z= (2.2) = ρ L , = Iac q jz dq q jz dq where jz and ez = ρjz are the amplitudes of longitudinal components of AC current density and electric ﬁeld, respectively. The symbol S refers to the surface of conductor. If the current density jz is independent of coordinate z one gets from eq. (2.2) Z jz (S) = , Rdc jz q

(2.3)

Fig. 1. Scheme to the deﬁnition of impedance.

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where Rdc = ρL/q is the DC resistance and q denotes the average value over the crosssection q. As can be seen, the ratio Z/Rdc is given by the ratio of the current density at the surface to its average value. Using the Ampere’s law I = C h dl, where C is the contour of the area q, the total current is given by Iac = lhφ (S), where hφ (S) is the circumferential component (tangential to the surface and perpendicular to z) of AC magnetic ﬁeld on the surface and l the length of the contour C. In metals the relationship between the tangential components of e and h at the surface can be described by the 2 × 2 surface impedance tensor ζˆ (Landau and Lifshitz, 1975) et (S) = ζˆ n × ht (S),

(2.4)

where n is the normal vector of the surface (directed outside the conductor). Using eqs. (2.2) and (2.4) one gets the relation between Z and the surface impedance hz L Z= (2.5) . ζzz − ζzφ l hφ In the case of special symmetry, when the surface impedance tensor is diagonal or the axial component of h is zero, the second term on the right hand side vanishes and the impedance is proportional to the surface impedance ζzz . The off-diagonal component ζzφ is responsible for “cross-magnetization” and asymmetric GMI, as will be discussed in Section 5.3.2. The impedance shown above was derived under the assumption of an inﬁnitely long conductor. In a closed measuring circuits the “external” self-inductance Le of the conductor, which is related to the magnetic energy stored in the circuit (outside the conductor volume) (Landau and Lifshitz, 1975), should be added to the impedance Z. The self-inductance Le , however, is not an intrinsic property of the conductor and depends on the particular geometry of the measuring circuit. If it is properly determined, it can be included into the impedance of the measuring circuit itself. In the linear approximation the impedance of a magnetic conductor (Z) can be calculated either from eqs. (2.3) or (2.5). Both formulas are practically equivalent because the current density j and magnetic ﬁeld h are related through Maxwell’s equation j = rot h. However, it should be mentioned that the assumption of current density jz independent of z, which was used in formula (2.3) need not always to be fulﬁlled. This happens, for example, at microwave frequencies when the electromagnetic wavelength in the surrounding medium, corresponding to frequency ω, becomes comparable or smaller than the sample length L. Then the radiation effects are important and the displacement currents through the medium surrounding the conductor are not negligible. The current Iac and the electric ﬁeld ez (S) are not constant along the z-axis. The deﬁnition of Z, according to eq. (2.2), looses its standard form, because it may depend on the geometry of the measuring circuit. Then it is better to refer to the surface impedance tensor ζˆ and to ﬁnd its relation to the transmission or reﬂection coefﬁcients of the particular measuring cell. Some measuring cells are discussed, for example, by Knobel et al. (2003). The very essence of GMI lies in the fact that in extremely soft magnets the skin effect appears at frequencies several orders of magnitude lower than those expected for a nonmagnetic conductor with equivalent conductivity. Besides that, GMI requires a high efﬁciency of the static ﬁelds to modify the AC permeability, the skin depth and consequently

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the impedance. Moreover, in ferromagnetic metals the AC permeability is not isotropic. It depends on the orientation of both the AC and DC magnetic ﬁelds as well as on the magnetic and shape anisotropies of the sample. Therefore the AC permeability, which takes part in GMI, is the effective transverse permeability μt . In short, to observe GMI, the transverse permeability μt (H, f ) (or circumferential permeability μφ in the case of cylindrical geometry) has to be large enough and must be signiﬁcantly modiﬁed by the static ﬁeld. 3. Phenomenology of GMI Figure 2 shows the frequency and ﬁeld dependence of the impedance for an as-quenched amorphous soft magnetic ribbon. If one pays attention to the saturation value of the impedance, the frequency dependence of Z follows the expected dependence with the squareroot of the frequency (Beach and Berkowitz, 1994a; Machado et al., 1995). It is worth noting the evolution of the double-peak structure observed in the proximity of the anisotropy ﬁeld, showing a clear displacement towards higher ﬁelds and a broadening as the frequency increases, as will be further explained in section 4. As previously mentioned, owing to the high magnetic permeability of the soft magnetic materials, the frequency dependence of the impedance is shifted towards lower frequency values, and the skin effect becomes considerable in frequencies much lower than the ones observed in nonmagnetic metallic samples. Depending on the frequency f of the driving AC current Iac ﬂowing through the sample, the giant magnetoimpedance can be roughly separated into three different regimes:

Fig. 2. Impedance of Fe73.5 Cu1 Nb3 Si13.5 B9 as-quenched amorphous ribbon. Notice the f 1/2 dependence of the saturation value and the interesting evolution of the whole ﬁeld dependent proﬁle. Courtesy of R.L. Sommer.

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(i) In the very low frequency range of 1–10 kHz the sharp voltage peaks measured between the sample’s ends are mainly due to large Barkhausen jumps in the circular magnetization reversal (Kraus et al., 1994b; Mohri, 1992; Velázquez et al., 1994). The effect is observed for sufﬁciently high amplitudes of driving current and is highly nonlinear. It is also labelled as magnetoinductive effect and has been extensively studied by Mohri and co-workers (Mohri et al., 1993). In this case, skin depth is always larger than the radius or half thickness of the sample (very weak skin effect). Then changes of impedance are due to a circular magnetization process exclusively, and might not be considered properly as GMI if one accepts that skin effect is the mechanism associated to GMI. (ii-a) In low frequencies, from 10–100 kHz to 1–10 MHz, one has the earliest and intensively studied giant magnetoimpedance, originating basically from variations of the magnetic penetration depth due to strong changes of the effective magnetic permeability caused by a static magnetic ﬁeld (Beach and Berkowitz, 1994b). In this case both domain walls and magnetization rotation contribute to changes in the circular permeability and consequently to the skin effect. (ii-b) Intermediate frequencies (from 1–10 MHz to 100–1000 MHz depending on the geometry of the sample): in this regime, the skin effect is also the origin of GMI, but domain walls are strongly damped. Only magnetization rotation contributes to GMI (Garcia and Valenzuela, 2000). (iii) At very high frequencies, of the order of GHz, the magnetization rotation is strongly inﬂuenced by the gyromagnetic effect and GMI peaks are shifted to static ﬁelds where sample is magnetically saturated. Strong changes of the sample’s impedance occur owing to the ferromagnetic resonance (FMR) (Yelon et al., 1996). Before proceeding, it must be ﬁrst clear that the criterion used for dividing frequency into different regions is merely used for the sake of clarity, and, to some extent it is rather rough and arbitrary. Besides the material properties, which were chosen as the main criterion for the frequency ranges deﬁned above, other rules may be also used. For example, these may be based on: • Experimental techniques used for measurements. This is probably one of the worst criteria, because different techniques can be used for the same frequency region and vice versa, the same technique can be used for more than one frequency region. • The ratio of skin depth to transversal dimensions (δ/a). This criterion is used by some authors, for example, by Makhnovskiy et al. (2001). In this case, there is a commonly used terminology: δ/a 1 indicates a weak skin effect regime, while δ/a 1 indicates a strong skin effect. Generally, weak skin effect is observed at lower frequencies. However, δ/a depends on many other parameters, such as sample dimensions, material properties, magnetic ﬁeld, etc. Therefore, this criterion also does not seem to be appropriate for the distinction among different frequency ranges. Another point that it is important to emphasize is that although different regimes can be roughly separated, the underlying physical mechanism is always the same, and it is not possible to distinguish between the skin effect and FMR mechanisms. In fact, the skin effect is also responsible for FMR absorption in ferromagnetic metals (Ament and Rado, 1955).

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3.1. Very low frequency regime In the very low frequency range (i), the driving current simply generates a circumferential time dependent magnetic ﬁeld. Such ﬁeld causes a circular magnetic ﬂux change and generates a longitudinal electric ﬁeld that, in turn, gives rise to an inductive voltage across the sample. In other words, the inductive voltage is determined by the internal inductance Li that, in turn, depends on the spatial distribution of the transverse permeability within the sample. For example, when a time varying current is ﬂowing through a ferromagnetic wire, an AC voltage Uac appears between the ends of the wire. This total voltage is the complex sum of a resistive voltage UR and an inductive voltage UL : Uac = UR + iUL = Rdc Iac + iωLi Iac ,

(3.1)

where Rdc is the DC resistance of the sample. When an external DC magnetic ﬁeld H0 is applied, both the circular component of magnetization and the circular permeability change, giving rise to a large change in Uac . The complex impedance of the sample is deﬁned as the ratio between Uac and the driving current Iac , i.e., Z=

Uac = Rdc + iωLi . Iac

(3.2)

It can be concluded that, at very low frequencies, the ﬁeld dependence of impedance is attributed to its inductive part, which is simply proportional to the circumferential permeability μφ (I, Hext , f ). Therefore, at very low frequencies the change of the material’s impedance is exclusively ascribed to the magnetoinductive effect arising from the circular magnetization process (Hernando and Barandiarán, 1978; Mohri et al., 1993). It should be noted that because of large Barkhausen jumps in the domain wall motion the inductive voltage UL can be very far from the harmonic waveform (especially for high amplitudes of driving current), which allows to distinguish easily the resistive and inductive components of the total voltage Uac (Mohri, 1992). In the range of frequencies typical of the magnetoinductive effect a very simple experimental setup can be used. It is even possible to use the regular four probe method with an AC current source for the probe current and a measurement performed with a conventional lock-in ampliﬁer. If phase information is not necessary, a simple AC voltmeter or oscilloscope can be used to measure the voltage drop across the magnetoinductive element. 3.2. Low and intermediate frequency regimes The changes of complex impedance in the moderate frequency range (10 kHz up to a few hundred MHz), induced by magnetic ﬁeld, were ﬁrst identiﬁed as the GMI effect. The phenomenon was soon explained in terms of classical skin effect in magnetic conductors with a large effective permeability and its strong dependence on the magnitude of external DC magnetic ﬁeld (Beach and Berkowitz, 1994b). Therefore, the explanation of GMI response of a particular sample is equivalent to the understanding of the dependence of its permeability on the external magnetic ﬁeld and frequency. In the classical description of skin effect the permeability is considered as a scalar quantity. In real ferromagnetic materials, however, the situation is much more intricate. Magnetic induction B and magnetic ﬁeld H are usually not parallel and the relationship

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between them is non-linear. Therefore, the AC permeability is generally a complex tensor, which depends not only on the frequency f and magnetic ﬁeld H , but also on some other parameters, such as the amplitude of the AC magnetic ﬁeld associated to the driving current, as well as on the anisotropies, stress distribution and the particular domain structure in the sample. Usually, both the domain wall motion and magnetization rotation contribute to the effective transverse permeability (Panina et al., 1995): μt = μt rot + μt dw ,

(3.3)

where μt rot and μt dw are the corresponding contributions to the effective transverse permeability. At relatively low frequencies (less then about 1 MHz), both contributions are effective in the transverse magnetization process (low frequency range (ii-a)). At higher frequencies, the domain wall motion is strongly damped by eddy currents and the magnetization rotation dominates the process (intermediate frequency range or RF range (ii-b)). Both contributions for the magnetization process can be accurately separated by means of the complex permeability formalism, employed by Valenzuela et al. in soft magnetic amorphous wires (Valenzuela, 2001, 2002; Valenzuela and Betancourt, 2002). Regarding the measurement techniques in the moderate frequency range, special care must be taken for the correct measurement of the driving current amplitude. There is a need for impedance match over all connections. As the frequency is increased, this impedance match becomes critical to ensure the power delivery to the magnetoimpedance element. Special care must be taken for the sample leads because they can produce the unbalancing of the whole impedance. Another possibility which works for a limited range of frequencies is to use a relay to measure a voltage drop across a series resistor, and adjust the current ﬂowing through the sample for each frequency and ﬁeld value (Sinnecker et al., 1998a). In this case, a perfect impedance match is less important once the actual current value is measured. However, cables with a higher characteristic impedance (e.g. oscilloscope probes) are needed. The careful choice of the relay is crucial. There are some RF relays available in the market that should be preferred. Also, another possibility to measure the actual current ﬂowing through the sample is the use of an external current sensor designed for the frequency range used in this kind of experiment. The circuitery details are indeed very important to determine relevant parameters of GMI (Raposo et al., 2003). In particular, LC resonances of the circuit may give rise to apparent maximum in the GMI ratio. Employing a LCZ bridge balancing parasite impedances, a shifting of the applied ﬁeld to observe maximum GMI is found. For example, the ﬁeld for maximum GMI in a (Fe0.06Co0.94)72.5 Si12.5B15 amorphous wire, that spontaneously shows a double-peak behavior, is found to increase from 0.4 to 0.8 Oe and remains practically constant at that position when frequency increases up to 30 MHz. 3.2.1. Low frequency range Figure 3 shows a typical giant magnetoimpedance response, obtained for an as-quenched Co-rich amorphous wire of nominal composition (Fe0.06Co0.94)72.5 Si12.5B15 . Saturation magnetostriction of this as-quenched wire is negative and quite small (λs ∼ = −1 × 10−7 ). This fact makes it to behave as a very soft magnetic material, thus exhibiting large magnetoimpedance. The measurement was made using a lock-in ampliﬁer and constant driving

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Fig. 3. Field dependence of the real (R) and imaginary (X) parts of the impedance (Z), for an amorphous wire (Fe0.06 Co0.94 )72.5 Si12.5 B16 with driving current amplitude Iac = 1 mA and frequency f = 500 kHz.

current Iac of amplitude 1 mA and frequency 500 kHz, as described elsewhere (Sinnecker et al., 1998a). Notice that both the real (R) and imaginary (X) parts of the impedance (Z) strongly decrease when a rather small external magnetic ﬁeld, parallel to the wire axis, is applied to the sample. In the case shown in the ﬁgure the behaviour of total impedance is mainly determined by the resistance R, owing to the relative low value of the reactance X, mainly at higher ﬁeld values. As pointed out in section 2, it is useful to normalize the impedance signal, in order to compare different samples and systems. The ratio |Z|/Rdc is shown in ﬁg. 4, and reaches a maximum value of almost 5 for the present as-quenched amorphous wire. If one plots the relative variation of the impedance, considering the value at saturation as reference, one obtains a maximum GMI ratio of around 360%, as shown in the inset of ﬁg. 4. It is worth mentioning that such huge GMI ratio is obtained in the sample as-prepared, without any additional treatment. Furthermore, the effect occurs for relatively low values of the applied magnetic ﬁeld, a fact that causes the enormous expectations regarding applications. When a tensile stress σ is applied to the negative magnetostriction wire, the magnetoelastic coupling makes the wire axis a hard magnetic direction, with anisotropy ﬁeld HK = −3λs σ/Js , and the domain structure essentially converts into circumferential, bamboo-like, domains. Consequently the circular magnetization process is strongly modiﬁed (Knobel et al., 1997c). When the DC ﬁeld H 0 is applied along the longitudinal direction (the hard magnetic axis), the magnetization in each domain rotates towards the applied ﬁeld. The circumferential permeability (and so the impedance Z, see section 4) increases with the increase of H0 , reaching a maximum when H0 balances the transverse anisotropy ﬁeld HK . A further increase of H0 leads to the decrease of effective permeability towards

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Fig. 4. Typical magnetoimpedance curve of an as-cast low magnetostriction amorphous wire of composition (Fe0.06 Co0.94 )72.5 Si12.5 B16 , with driving current of 1 mA (500 kHz). The impedance modulus is normalized to the sample’s DC resistance Rdc . The inset shows the common representation of the GMI ratio, ΔZ/Z, which, in this case, reaches values up to 360%. This proﬁle is known as single-peak structure.

a constant and very low value. Fig. 5 displays the effect of applied stress on the impedance response. The two maxima observed in the vicinity of the anisotropy ﬁeld are typical for samples with transversal magnetic anisotropy, as will be discussed in detail in section 4. This proﬁle is known as the double-peak behaviour, while the previous one, displayed on ﬁg. 4 is known as the single-peak one (Vázquez et al., 1999c). For the single-peak GMI the magnetic easy-direction is parallel to the wire axis (longitudinal magnetic anisotropy), and the transversal magnetization is always due to rotational processes. Consequently, the impedance displays a monotonous decrease from H = 0, as shown in ﬁg. 4. It is worth noting that the reactance X shown in ﬁg. 3 already shows some traces of a double-peak structure. The ﬁne double-peak structure appearing in typical single-peak behaviour has been ascribed to the axial magnetization process, whose coercivity is correlated to that local GMI maximum (during axial magnetization reversal, circular permeability also shows a local maximum) (Vázquez et al., 1999c). By applying an increasing tensile stress it is possible to continuously increase the induced anisotropy of the samples. An example, using the (Fe0.06 Co0.94)72.5Si12.5 B15 as-cast wire, is shown in ﬁg. 6, for tensile stresses up to 133 MPa. It is instructive to observe the gradual appearance of the double-peak structure, which reﬂects the increasing anisotropy ﬁeld HK . Furthermore, the total GMI ratio decreases with increasing applied stress, also indicating that the induced anisotropy contributes to the decrease the maximum circular

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Fig. 5. Magnetoimpedance ratio as for the results given in ﬁg. 4, but when the same wire is submitted to an applied tensile stress of 133 MPa. Note that in this case a double-peak behaviour is observed, with peaks located in the proximity of the anisotropy ﬁeld. This proﬁle is known as double-peak structure.

permeability. Nevertheless, permanent anisotropies can be induced by means of speciﬁc thermal treatments as will be further discussed in section 5. 3.2.2. Intermediate frequency (RF) range At frequencies from few MHz up to a few hundred MHz the skin effect is very strong. In this range, the GMI maximum is still not shifted to high enough ﬁelds, where the sample is fully saturated (in contrast to the case of very high frequencies, see section 3.3 below). Therefore, the domain structure is still present, and can play a role in the GMI behaviour. Because the domain wall motion is effectively suppressed by induced eddy currents, the transverse permeability is mainly due to magnetization rotations (Panina et al., 1995). The RF stray ﬁelds, which can appear due to magnetic charges on the domain walls (the perpendicular component of m does not need to be continuous through the wall) complicate very much the theoretical analysis (it is equivalent to ferromagnetic resonance in samples with domain structures) and all the theoretical approaches are trying to avoid this situation by means of some simplifying assumptions. Also, because the GMI peak is not much broadened by ferromagnetic relaxation or by the exchange-conductivity effect, all the ﬂuctuations of effective ﬁeld (magnetic anisotropy dispersion, magnetization ripples, etc.) can play important roles in the magnetization process and therefore in the impedance behaviour (see discussion by Yelon et al. (2002) and Barandiarán et al. (2002)).

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Fig. 6. Evolution of the GMI behaviour with tensile applied stress in a negative magnetostriction amorphous wire: note the transition from single-peak to double-peak behaviour.

3.3. High frequency regime Although ferromagnetic resonance (FMR) is usually studied in saturated samples placed in a cavity subjected to a microwave excitation at about 9 GHz (X-band) or higher, it is nowadays accepted that the GMI effect is a ﬁngerprint of FMR, even in the low frequency ranges (Kraus, 1999a; Pirota et al., 1999a; Yelon et al., 1996). Thus, the description of magnetoimpedance should take into account the dynamical effects on the magnetization, related to FMR. The basic conditions to obtain the resonance are: (a) the presence of an effective static ﬁeld H eff 0 that ﬁxes the spin orientation and (b) the presence of a component of AC magnetic ﬁeld h perpendicular to the static ﬁeld. h may exhibit either planar symmetry, for ﬁlm conductors, or circumferential symmetry, for wires. Close to the resonance, h supplies energy to compensate the losses associated to the interaction of the rotating magnetic moments with the surrounding medium, thus maintaining the magnetization precession. The FMR regime results in drastic changes of magnetic permeability with frequency and/or ﬁeld, this behaviour being reﬂected in the impedance as well. At resonance, a small change in the static magnetic ﬁeld results in a large change of the impedance. At frequencies above 10 MHz, where the electromagnetic radiation becomes important, the use of microwave lines and cavities is recommended. The classical designs have been employed in very few cases (Loﬂand et al., 1999). In most works, in order to facilitate the experimental methods, the coaxial line technique (for wires) (Ciureanu et al., 1996a; Menard et al., 1997) and the stripline cavity technique (for ribbons and ﬁlms) (Acher et al., 1994; Viegas et al., 2001) are used. In the case of coaxial lines, the magnetoimpedance element is placed as the central wire of a short circuited coaxial cavity designed to have a

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known characteristic impedance. Once the wavelength of the RF signal is not less than 4L, where L is the sample length, the electromagnetic wave can be considered as independent of the position in the sample. The unbalance produced by the sample in the cavity can be measured by means of the reﬂected signal by a network analyzer for each value of frequency and ﬁeld (Ciureanu et al., 1998; García-Miquel et al., 2001; Malliavin et al., 1999). The same procedure may be used for the stripline cavity, where the sample is placed between two grounded planes. Once the sample width is much smaller than the cavity width, the calculation of the characteristic impedance is straightforward. The cavity has to be designed to present the characteristic cavity impedance that will be unbalanced by the sample (ﬁlm + substrate). A careful calibration is needed in order to ensure that the sample substrate is not responsible for most of the unbalancing. The theoretical description of GMI in the high frequency regime is given in section 4. It is based on the simultaneous solution of the Landau–Lifshitz and Maxwell equations. It was shown (Kraus, 1999a) that the maximum theoretical GMI ratio is determined by the minimum skin depth, which is achieved for the FMR resonance condition. The measurements of GMI in the high frequency region (up to 5 GHz) (Menard et al., 2000) show that experimental results can be excellently ﬁtted by the theory for frequencies above 3 GHz. At lower frequencies, however, Ménard et al. found a discrepancy between the predicted and the experimental GMI ratios. The discrepancy increases with decreasing frequency. Four types of mechanisms, which were not included in the theoretical models and could possibly explain the discrepancy, were proposed by Yelon et al. (2002). They are: (a) surface conditions, (b) damping mechanism, (c) anisotropy dispersion and (d) magnetization nonuniformity. The ﬁrst two can, in principle, be readily incorporated into the theory (Melo, 2003). It is not, however, quite clear how the other two could be theoretically treated. 4. Theory of GMI In order to fully understand the experimental data, several authors (Atkinson and Squire, 1997; Machado and Rezende, 1996) initially presented models based on the minimization of the free energy for some particular domain structure. Although these kinds of models (they are called “quasistatic”) do not consider the dynamic effects related to the rapid motion of the magnetization, they are useful to explain the experimental results at low frequencies, and to obtain important magnetic parameters. Panina and Mohri theoretically investigated the inﬂuence of micro-eddy current damping of domain wall motion on GMI effect (Panina et al., 1994). Their model is valid in the low frequency regime, where the circular magnetization by domain wall motion is still signiﬁcant. At higher frequencies, when the magnetization rotation completely dominates the magnetization process, the dynamic characteristics play an important role, and a more rigorous model must be based on a simultaneous solution of Maxwell equations and the Landau–Lifshitz equation of magnetization motion. This procedure is well known from theories of ferromagnetic resonance (Akhiezer et al., 1968). To achieve this task, the effective dynamic permeability μeff has been derived for a wire with circumferential or helical anisotropy and for a thin ﬁlm with in-plane uniaxial anisotropy (Panina et al., 1995). This model, and others (Morikawa et al., 1996a; Usov et al., 1998; Dong et al., 2002), neglect the exchange interaction in the skin layer,

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and therefore they are just approximations for ferromagnetic metals. Yelon et al. (1996) showed that the theory of ferromagnetic resonance in metals, developed more than forty years ago, and which takes explicitly into account the exchange effect, completely agrees with the observed GMI effect at rather high frequencies. After such elucidation, a number of papers using this approach appeared to explain the effect in many different situations. Menard et al. (1998) solved the problem for an isotropic axially saturated wire and, later on, for a non saturated anisotropic wire (Menard and Yelon, 2000). Kraus (1999b) applied the approach to a planar geometry, considering a ﬁlm with uniaxial anisotropy. Britel et al. (2000) reported the observation of both ferromagnetic resonance and antiresonance in a magnetic metal using a GMI technique. 4.1. Maxwell equations Because the skin depth as well as the characteristic correlation length in a magnetic structure (e.g., domain width) in soft magnetic materials are much larger than the interatomic distances, the classical electrodynamics of continuous media can be used for the description of GMI effects. In ferromagnetic metals, where the displacement currents can be neglected (D˙ = 0) and the material relations E = ρJ and B = μ0 (H + M) hold, Maxwell’s equations can be written as rot H = J , μ0 ˙ rot J = − (H˙ + M), ρ

(4.1)

div(H + M) = 0.

(4.3)

(4.2)

Applying the rot operator to eq. (4.1) and substituting for rot J and div H from eqs. (4.2) and (4.3) one gets the ﬁrst basic equation of the problem ∇ 2H −

μ0 ˙ μ0 ˙ H= M − grad div M. ρ ρ

(4.4)

To obtain the solution, the relation between M and H must be known. The more realistic the relation used, the more accurate are the obtained results. Separating the vectors H and M into their DC and AC components H = H 0 + h, M = M 0 + m and assuming h, m ∝ eiωt (the linear approximation) from eq. (4.4) one gets for the AC components 2i 2i (4.5) h = 2 m − grad div m, 2 δ0 δ0 √ where δ0 = 2ρ/ωμ0 is the nonmagnetic skin depth. Using the simplest possible material relation b = μ0 (h+m) = μh, with a constant scalar permeability μ, a well known solution for the skin effect in nonmagnetic metals (Landau and Lifshitz, 1975) can be found. For example, for a cylindrical conductor with diameter R ∇ 2h −

Z J0 (kR) = kR Rdc 2J1 (kR)

(4.6a)

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where J0 and J1 are the Bessel functions of the ﬁrst kind. For an inﬁnite ﬁlm of thickness t one has t Z t (4.6b) = k cot k . Rdc 2 2 The propagation constant k in eqs. (4.6a) √ and (4.6b) is given by the relation, k = (1 − i)/δ, with the classical skin depth δ = 2ρ/ωμ. In the case of strong skin effect (i.e., when the skin depth δ is much smaller than the transversal dimensions of the conductor) much simpler formula can be obtained. Assuming jz (r) in eq. (2.2) in the form of an exponentially decaying function of the distance from the surface one gets for a conductor as shown in ﬁg. 1 1+i q Z . ≈ (4.7) Rdc δ l Though derived with the oversimpliﬁed assumption (b = μh), eqs. (4.6) and (4.7) describe rather well many characteristic features of GMI and have been frequently used to interpret the experimental results. Many theoretical models were devoted to the attempt to ﬁnd the dependence of permeability μ on the frequency, applied DC ﬁeld and some other parameters, which would best describe the behaviour of a speciﬁed material under given experimental conditions. Some of these models will be discussed later. We wish to emphasize here that the magnitude of the GMI effect does not depend only on the angular frequency ω and material properties (μ and ρ) but also on the transversal dimensions of the sample. Thus larger GMI can be obtained if a thicker conductor is made of the same material. On the other hand, the GMI effect diminishes if the transversal dimensions are smaller than the minimum skin depth δ. Because δ cannot be lower than the theoretical limit, which is for the known soft magnetic metals of the order 10−7 m (Kraus, 1999b), GMI elements of submicron dimensions cannot be principally realized. 4.2. Landau–Lifshitz equation To get a more accurate theory of GMI the proper relation between M and H , which covers the whole frequency range from 0 to a few GHz, should be used. The dynamic behaviour of a ferromagnetic continuum can be best described by the phenomenological Landau– Lifshitz equation of motion ˙ = γ M × H eff − α M × M, ˙ M (4.8) Ms where γ is the gyromagnetic ratio, Ms the saturation magnetization, H eff the effective ﬁeld and α the Gilbert damping parameter. This equation can describe all magnetizing processes starting from quasistatic domain wall movements or magnetization rotations at low frequencies up to, for example, spin waves exited at the ferromagnetic resonance experiments. The effective ﬁeld H eff , which includes all the macro- and microscopic forces acting on the magnetic system, can be calculated from an appropriate free energy density. The most important parts of the free energy are: the Zeeman energy, the magnetostatic energy, the anisotropy energy and the exchange energy. Then H eff = H + H a +

2A 2 ∇ M, μ0 Ms

(4.9)

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where H is the internal magnetic ﬁeld (the sum of applied ﬁeld and the demagnetizing ﬁeld), H a the anisotropy ﬁeld and A the exchange stiffness constant. Other types of interactions, such as the magnetoelastic coupling in magnetostrictive materials, can be also included in H eff . In eq. (4.8) the Gilbert form of damping term is used, which for small damping is mathematically equivalent to the original Landau–Lifshitz damping term (Bertotti, 1998). These phenomenological damping terms conserve the length of the magnetization vector M, as required for ideal ferromagnets. In real materials, however, relaxation processes, which do not conserve |M| are possible. Such relaxation can be described, for example, by the addition of the Bloch–Bloembergen type of damping −(M − M 0 )/τ . If the vectors M and H eff are divided into their DC and AC components from eq. (4.8) one gets the equation M 0 × H eff 0 = 0,

(4.10)

which is equivalent to Brown’s micromagnetic equation (Bertotti, 1998) for the static magnetization conﬁguration M 0 (r) and the equation m ˙ = γ m × H eff + γ M 0 × heff −

α M × m, ˙ Ms

(4.11)

for the AC component of magnetization m. Equations (4.11) and (4.4) together with the boundary conditions represent the general equations of GMI theory. Their exact analytical solution is, however, impossible. Some approximate solutions will be described in the following. Usually the Maxwell equations (eqs. (4.1)–(4.3)) and the Landau–Lifshitz equation (4.8) are nonlinear. Their solution can, however, substantially be simpliﬁed if the linear approximation is used. For small deviations of the magnetization vector M from its DC value M 0 (i.e. |m| |M 0 |) we have also |heff | |H eff 0 |. The second order terms of m and heff in eq. (4.11) then can be omitted. Using m ˙ = iωm the linearized equation of motion ω αω i m = m × H eff 0 + i (4.12) M 0 + M 0 × heff γ γ Ms is obtained. When the modiﬁed Bloch–Bloembergen damping term is introduced into eq. (4.8) ω on the left hand side must be replaced by ω − i/τ . The linearized equations (4.5) and (4.12), originally used in the FMR theory, are also frequently used in the theory of GMI. Because such conditions are usually justiﬁed for small amplitudes of driving current, this approach is also called the “low signal approximation”. It should, however, be noted that in contrast to the FMR experiment the condition |m| |M 0 | can be easily violated in the GMI measurement, mainly near the GMI curve maxima. 4.3. Description of theoretical models Different approaches of different authors can be classiﬁed according to different criteria, such as sample geometry (e.g., cylindrical wire, inﬁnite ﬁlm, nonmagnetic wire with magnetic layer on the surface, sandwich ﬁlms, etc.), model domain structures, mathematical procedures used, etc. Here we divide the theories into three categories, which can be closely related to the three frequency ranges discussed in section 3.

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4.3.1. Quasistatic models If the frequency of driving current is so small that the system can reach the equilibrium state at every moment, the quasistatic approach can be used. Then eqs. (4.6) or (4.7) are used for the impedance with the permeability μ in the classical skin depth replaced by the effective transverse permeability μt (see end of section 3). The transverse permeability can be calculated from eq. (4.11), where the derivative m ˙ is set equal to zero. This procedure is equivalent to the minimization of the free energy. Different model domain structures with selected sets of variable parameters were used to minimize the free energy. The quasistatic approach will be illustrated here on a ferromagnetic ﬁlm with a uniaxial in-plane anisotropy and the periodic bar-domain structure as shown in ﬁg. 7. θ1 and θ2 are the magnetization orientations in domains 1 and 2, 2d is the period of domain structure and u the displacement of domain walls from their equilibrium positions. The free energy can be written as (Squire, 1990): U = U1 + U 2 + U W ,

(4.13)

where U1 and U2 are the free energies of the domains 1 and 2 and UW the domain wall energy 2 u 1 . UW = β (4.14) 2 d It is assumed that the walls are pinned in identical parabolic potential wells. The wall bowing is neglected and the walls are considered to move as a whole. Only H 0 perpendicular or parallel to the easy axis will be considered, for simplicity. A uniform AC ﬁeld h is assumed to lie in the ﬁlm plane perpendicularly to the DC ﬁeld H 0 . (a) H 0 perpendicular to the easy axis (H 0 x). In this case, one gets 1 u

U1,2 = (4.15) ± −μ0 Ms H0 cos θ1,2 + K cos2 θ1,2 ∓ μ0 Ms h sin θ1,2 . 2 d The demagnetizing ﬁelds induced by magnetic charges on the walls, which appear when θ1 = θ2 , are neglected. The equilibrium orientation of magnetization M 0 in the domains is given by cos θ1 = cos θ2 = cos θ0 = H0 /HK (for H0 HK ), where the anisotropy

Fig. 7. Domain structure of a uniaxial ﬁlm with the easy direction along the y-axis. θ1 and θ2 are the magnetization orientations in domains 1 and 2, 2d is the period of domain structure and u the displacement of domain walls from their equilibrium positions.

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ﬁeld HK = 2K/μ0 Ms . By minimizing U with respect to the parameters θ1 , θ2 and u the transversal permeability can be calculated. The details of calculation can be found in the reports of Machado and Rezende (1996) and Atkinson and Squire (1998). Here we show only the limit cases of weak and strong domain wall pinning. If the walls are free to move, then the domain wall displacement dominates and the magnetization rotation can be neglected. By keeping θ1 = θ2 = θ0 constant and minimizing the free energy with respect to u, one obtains u/ h = 2dμ0Ms sin θ0 /β and the transverse susceptibility H02 2u 4μ0 Ms2 χt dw = (4.16) 1− 2 . Ms sin θ0 = dh β HK The ﬁeld dependence of χt dw is shown in ﬁg. 8 (curve a). On the other hand, if the walls are completely pinned (u = 0) the transverse susceptibility χt rot can be obtained by minimizing the free energy (eq. (4.13)) with respect to θ1 and θ2 . Here we shall, however, use the solution of the linearized equation of motion (4.16) in the quasistatic limit (ω = 0) m × H eff 0 − heff × M 0 = 0.

(4.17)

The effective ﬁelds in the domains 1 and 2 are given by H eff i = −(1/μ0 )∂U/∂M i , i.e. H eff 1,2 = H 0 ± HK sin θ0 ey + h +

HK m1,2y ey . Ms

(4.18)

Fig. 8. Transverse susceptibility calculated for a uniaxial ﬁlm: (a) H 0 ⊥ easy axis – domain wall movement, (b) H 0 ⊥ easy axis – magnetization rotation, (c) H 0 easy axis.

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Using the condition (mi · M 0,i ) = 0 from (4.17) one obtains m1y = m2y = χt rot h, where χt rot =

Ms cos2 θ0 H0 cos θ0 − HK cos 2θ0

(4.19)

(see, for example, the report of Panina et al. (1995)). The ﬁeld dependence of the rotational susceptibility (see curve b in ﬁg. 8) exhibits singularities at H0 = ±HK , where the torque of anisotropy on magnetization is compensated by the DC ﬁeld and the free rotation of magnetic moment is allowed. The singularities are the consequence of the linear equation (4.11). In fact, the transverse magnetization cannot exceed Ms which implies χt rot Ms / h. Then the singularities appear only in the limit h → 0. If the damping of magnetization rotation (the last term in eq. (4.11)) is taken into account the singularities completely disappear. (b) H 0 parallel to the easy axis (H 0 y). Because the AC ﬁeld is perpendicular to the domain walls it does not act on them and the transversal magnetization takes place only by magnetization rotations. The transverse susceptibility can be obtained from eqs (4.17) and (4.18) H02 M0 H0 Ms χt rot = (4.20) 1− 1− 2 , HK Ms HK HK where M0 = 2Ms u/d is the total DC magnetization. The ﬁeld dependence of the transverse susceptibility, calculated for the hysteresis loop M0 (H0 ) shown in the upper part of ﬁg. 8 (dotted line), is shown by the curve c in the lower part of the ﬁgure. For ﬁelds H0 well above the coercive force, where the sample is saturated, χt rot = 2Ms /(H0 + HK ). The hysteresis, at lower ﬁelds, is reﬂected also on the GMI curves. Although this model cannot explain the frequency dependence and other GMI properties, it can well describe some basic features of GMI curves not only in ﬁlms but also in ribbons and wires. Notice the similitude of the observed behaviours with the experimental curves displayed in ﬁgs. 4 and 5. Both characteristic proﬁles, the single and double peak structures, are straightforwardly obtained using the quasistatic approximation. As can be seen, for anisotropy transverse to the conductor axis the shape of GMI curves depends on the character of the magnetization process. For domain wall movement a single-peak behaviour is observed with a broad maximum at H0 = 0 and zero contribution for H0 > HK , where the sample is saturated. The rotational process results in a double-peak behaviour with zero contribution at H0 = 0 and sharp maxima at H0 = ±HK . In real materials the sharp peaks can be smeared out by the dispersion of HK values and the local easy axis directions (Atkinson and Squire, 1998; Panina et al., 1995) or by magnetization non-uniformity (Yelon et al., 2002). If the easy direction is parallel to the conductor axis the domain walls do not contribute to GMI. When the hysteresis is neglected, the GMI curves show a single maximum at H0 = 0 and its height decreases with increasing HK . Atkinson and Squire (1998) investigated the effect of oblique anisotropy on the rotational susceptibility. They have shown that with increasing the angle between the easy direction and the conductor axis the shape of GMI curve evolves from the single- to double-peak behaviour.

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4.3.2. Eddy current damping of domain wall movement Because the domain wall movements in metallic ferromagnets are heavily damped, the quasistatic models can be used only for very low frequencies, where the magnetoimpedance is very small and uninteresting for practical applications. If the role of domain wall displacements should be properly considered, the eddy current damping of domain wall movement must be taken into account. The rough estimate of the inﬂuence of driving current frequency on GMI can be obtained by varying the domain wall pinning parameter β in eq. (4.14). It was shown that with increasing β a continuous change from singleto double-peak behaviour takes place for the case of transversal anisotropy (Atkinson and Squire, 1998). Machado and Rezende (1996) used a viscous friction force to describe phenomenologically the damping of domain wall motion. Then the frequency dependence of transversal susceptibility is given by χ0 (4.21) , 1 − iτ ω where the phenomenological relaxation time τ is the ﬁtting parameter. A more rigorous treatment can be found in the reports of (Panina et al., 1995; Panina and Mohri, 1994), where the effective medium approximation was used for the calculation of circumferential permeability for a periodic bamboo like domain structure in cylindrical wires. The same formula as eq. (4.21) was obtained with the relaxation time τ = bχ0 /ρ, where the proportionality constant b depends only on the wire diameter and domain structure period. An exact solution of the eddy current problem in a cylindrical wire with periodic bamboo like domain structure for H0 = 0 was obtained by Chen et al. (1998). The authors calculated the internal magnetic ﬁeld induced by oscillating domain walls and from eq. (4.1) the electric ﬁeld on the wire surface. Using the Poynting theorem they obtained the power P entering the domains and the corresponding impedance Z = P /I 2 . The frequency dependence of Z was numerically calculated for different domain period to wire diameter ratios c/a. For ﬁne domain structures (c/a 1) and higher frequencies both the real and imaginary parts of Z obey the ω1/2 dependence, which is in good agreement with experimental observations (Beach and Berkowitz, 1994b). For larger c/a the frequency dependence decreases. It has been theoretically predicted (Panina et al., 1995) and experimentally proved (Melo and Santos, 1999), that the domain wall motion in soft magnetic materials with conductivities typical for crystalline, nanocrystalline and amorphous metals practically stops in the frequency region of a few hundreds of kHz to a few MHz. Therefore the contribution of domain walls to the GMI effect is important mainly for bulk conductors, such as Mumetal or Permalloy strips, melt-spun amorphous wires, and other materials where the large GMI effect appears in the 100 kHz frequency region. In tiny conductors, such as ribbons, thin ﬁlms, glass-coated microwires, the maximum GMI appears in the MHz and higher frequency regions. In such cases the domain wall contribution can be neglected and only the magnetization rotation must be considered. χt dw =

4.3.3. High frequency models of GMI At high frequencies, where only magnetization rotation takes place, the procedure known from the ferromagnetic resonance theory is suitable for the solution of the Maxwell and the Landau–Lifshitz equations (Yelon et al., 1996). The standard FMR theory, which

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uses the linearized equations (4.5) and (4.12), can be used for GMI in the low signal approximation. Two phenomena, which were neglected in the quasistatic models, become quite important at high frequencies. They are the ferromagnetic relaxation and the gyromagnetic effect. The ferromagnetic relaxation, which is described by the last term in eq. (4.11), causes damping of the magnetization rotation and determines the minimum skin depth possible in a ferromagnetic metal. The gyromagnetic effect, which is related to the inertia momentum of magnetization, is described by the term on the left side of eq. (4.12). It causes the precessional character of magnetization movement at high frequencies and the shift of the permeability maximum to higher applied DC ﬁelds (ferromagnetic resonance). In the following the high frequency models are divided into two categories according to whether they take into account the exchange term in the effective ﬁeld (eq. (4.9)) or not. 4.3.3.1. Electromagnetic models. These models neglect the exchange interaction. We call this category “electromagnetic” because, as will be shown, this approach neglects the spin-wave roots and takes into account only the electromagnetic roots of the secular equation. If the exchange interaction is neglected the linearized Landau–Lifshitz equation, eq. (4.12), can be easily solved and the magnetization on the right hand side of the linearized equation (4.5) can be substituted by m = χ(r)h, ˆ where χ(r) ˆ is the local susceptibility tensor. The tensor usually depends on many factors, including the DC ﬁeld, local anisotropy, actual domain structure, among others. In real materials χˆ (r) can be a very complex function of the coordinates, which makes the modelling very difﬁcult. The problem is twofold: ﬁrst, the calculation of the magnetic structure M 0 (r) even for ideal (spatially uniform) samples is far from trivial; second, even if the right magnetic structure were known the susceptibility tensor cannot be easily calculated, especially if the effect of long-range dipolar ﬁelds is taken into account. To simplify the solution, the spatial average of the susceptibility tensor over the sample volume was taken in the effective medium approximation (Panina et al., 1995). This approximation is, however, justiﬁed only if the spatial variations of χ(r) ˆ (for example, the correlation length of material inhomogeneities or the period of domain structure) are much ﬁner than the skin depth itself. In nonsaturated soft magnetic samples, where large magnetic domains exist, this assumption is seldom fulﬁlled. Since in real magnetic structures the accurate solution of eq. (4.5) is difﬁcult, simple magnetic structures (free of magnetic domains) and conductors of simple geometrical shapes (circular cylinders or inﬁnite planar ﬁlms) are usually considered. A simple illustration of the electromagnetic model will be shown on a single domain inﬁnite planar ﬁlm (ﬁg. 9). Both the DC ﬁeld H 0 and the easy direction of a uniaxial magnetic anisotropy are assumed to lie in the ﬁlm plane. For the easy direction along the unit vector nK the effective anisotropy ﬁeld H a , appearing in eq. (4.9), is Ha =

HK nK (nK · M). Ms

(4.22)

From eq. (4.12) one gets the equation for the equilibrium angle θ0 H0 sin θ0 − 12 HK sin 2ψ = 0

(4.23)

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Fig. 9. Coordinate systems in a single domain planar ﬁlm. The unit vector nK deﬁnes the easy axis.

and the DC component of effective ﬁeld H eff 0 = Heff 0 M 0 /Ms Heff 0 = H0 cos θ0 + HK cos2 ψ.

(4.24)

Using eq. (4.11) the susceptibility tensor χˆ can be obtained. In the coordinate system (x, y , z ) related to the equilibrium direction of M 0 it has the form χxy 0 χxx χˆ = −χxy χy y 0 , (4.25) 0 0 0 where χxx =

Ms (Ω + HK cos 2ψ), Δ

χy y =

χxy = −i

ω Ms , γ Δ

Ms

Ω + HK cos2 ψ Δ

(4.26)

and ω Ω = H0 cos θ0 + iα , γ 2

ω Δ = Ω + HK cos2 ψ (Ω + HK cos 2ψ) − . γ

(4.27)

Substituting m in eq. (4.5) by χh ˆ and using (4.22) the second order differential equation for internal ﬁeld h is obtained. Solving this equation with appropriate boundary conditions and then using eq. (4.1) the current density j in the ﬁlm can be obtained. Eq. (4.5) can be simpliﬁed if h and m are assumed to depend only on the x-coordinate, perpendicular to the ﬁlm. Then the internal AC ﬁeld satisfying eq. (4.3) and the surface boundary conditions (continuity of the tangential components of h and the normal component of b) is h = hext − mx n,

(4.28)

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where hext is the external AC ﬁeld (due to the driving current) and n a vector normal to the surface. Because of the precessional movement of M (due to the gyromagnetic effect) the AC demagnetizing ﬁeld −mx n appears even if hext is conﬁned to the ﬁlm plane. The which relates the AC magnetization to the external tensor of the effective susceptibility X, AC ﬁeld (m = Xhext ), is then χxx χxy 0 1 −χxy χy y (1 + χxx ) + χ 2 0 = X (4.29) xy 1 + χxx 0 0 0 and the effective transverse susceptibility χt = Xyy = cos2 θ0 Xy y (see Panina et al. (1995)) is χt =

Ms cos2 θ0 (Ω + Ms + HK cos2 ψ) . (Ω + HK cos 2ψ)(Ω + Ms + HK cos2 ψ) − (ω/γ )2

(4.30)

The angle θ0 , between the static magnetization and the DC ﬁeld, can be obtained by solving eq. (4.24) with the condition θ0 + ψ = constant. It can be seen that for ω = 0 and nK ⊥H 0 eq. (4.30) is identical to eq. (4.19), obtained in the quasistatic limit. Panina et al. (1995) used this transverse susceptibility for the skin depth calculation and eq. (4.6a) for the magnetoimpedance of a cylindrical wire with helical anisotropy. A more rigorous solution of this problem can be found in the reports of Makhnovskiy et al. (2001) and Usov et al. (1998). Though it has been shown that at high frequencies the origins of both GMI and FMR are identical (Yelon et al., 1996), this fact is still a point of controversies (see, e.g., Makhnovskiy et al. (2001)). The relation of the two phenomena can be well demonstrated using eq. (4.30). This equation was derived from the linearized Landau–Lifshitz equation with the only assumption that m and h depend only on the coordinate x perpendicular to the ﬁlm plane. The actual shape of the functions m(x) and h(x) was not signiﬁcant. It is therefore applicable to any type of AC ﬁeld excitation, which satisﬁes this assumption. It does not matter if it is a practically homogeneous ﬁeld of a microwave cavity (used in the FMR experiment) or a highly inhomogeneous ﬁeld produced by the AC current passing through the sample (in the GMI experiment). The transverse susceptibility χt shows the typical resonance behaviour, i.e. a sharp maximum of χt and a change of sign of χt at the resonance ﬁeld. The FMR resonance condition can be obtained by putting the real part of the denominator on the right hand side of eq. (4.30)) equal to zero. For weak ferromagnetic damping (α 1) one gets the well known formula

(ω/γ )2 = (H0 cos θ0 + HK cos 2ψ) H0 cos θ0 + Ms + HK cos2 ψ . (4.31) The resonance frequency vs. ﬁeld, calculated with material parameters typical for low magnetostrictive amorphous alloys, is shown in ﬁg. 10. If the resonance condition (4.31) is satisﬁed χt reaches its maximum value and the corresponding skin depth its minimum (Kraus, 1999b): αρ δm = (4.32) . γ μ0 Ms

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Fig. 10. Resonance frequencies of a single domain ﬁlm with an in-plane uniaxial anisotropy. Calculated for Js = 0.7 T, HK = 500 A/m and g = 2.09. ψ + θ0 is the angle between the in-plane DC ﬁeld H 0 and the easy axis.

If an additional Bloch–Bloembergen damping term with the relaxation time τ is used in the Landau–Lifshitz equation, α in eq. (4.8) should be replaced by α + 2/γ Ms τ (Kraus, 1999a). The minimum theoretical skin depth δm is frequency independent (for constant damping parameter α) and in the usual soft magnetic metals is of the order of few tenths of μm. Thus the electromagnetic model predicts the maximum theoretical GMI ratio |Z|/Rdc independent of driving frequency with magnitudes up to 104 for conductors with transversal dimensions about 1 mm. The actual GMI ratios, however, are frequency dependent and much smaller than the theoretical value. This discrepancy can be easily explained. As can be seen from ﬁg. 10, for frequencies less than 100 MHz, where GMI is usually measured, the resonance condition cannot be easily satisﬁed. It is fulﬁlled for H0 ≈ HK , but only if the anisotropy axis is very precisely perpendicular to the DC ﬁeld direction. Any small deviation or ﬂuctuation of easy direction infringes the resonance condition and substantially reduces the GMI ratio at lower frequencies. Also the Bloch–Bloembergen type of damping or the exchange-conductivity effect reduce the maximum theoretical GMI at lower frequencies (Kraus, 1999b). With increasing driving frequency, however, these effects become less important and in the GHz region the experimentally observed GMI ratios well satisfy the theoretical prediction (Britel et al., 1999; Menard et al., 2000). derived here for a planar ﬁlm (eq. (4.29)), can The effective susceptibility tensor X, be used also for cylindrical wires with the anisotropy easy axis tangential to the surface (Makhnovskiy et al., 2001) and also for other shapes of conductors in the case of strong skin effect approximation, where the skin layer can be considered as a planar ﬁlm. Procedures similar to those described above were used for calculations of magnetoimpedance of cylindrical wires with different types of magnetic anisotropy (Makhnovskiy et al., 2001; Usov et al., 1998) and planar or cylindrical conductors with more complicated structures (Makhnovskiy et al., 1998; Menard et al., 2000; Usov et al., 1999). Only the most important results of these works will be summarized here.

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Usov et al. have shown that in a wire with axial or circumferential anisotropy formula (4.6a), with δ calculated from the effective transverse susceptibility (eq. (4.30)), can be used only if θ0 = 0, i.e. when M 0 H 0 . For the circumferential anisotropy and H0 < |HK |, where χφz = χzφ = 0, eq. (4.5) for the components hφ and hz are coupled and their so Its lution is much more complicated. Then the impedance is a two dimensional tensor Z. longitudinal component Zzz is given by eq. (2.5) and the transverse component Zφz is deﬁned as L eφ L hz Zφz = (4.33) = . ζφz − ζφφ l hφ l hφ An analytical solution of eq. (4.5) was given for the strong skin effect approximation. The exact solution was obtained in the form of power series of r with recursion formulas for the expansion coefﬁcients. It was concluded that two independent electromagnetic modes exist in the wire. Their relative amplitudes were determined from the electromagnetic boundary conditions assuming that the general solution outside the wire is a combination of electromagnetic TE (transversal electric) and TM (transversal magnetic) cylindrical waves. The transverse component Zφz of magnetoimpedance causes that the driving AC current induces also an axial component of magnetization, which can be measured by a pick-up coil wound around the wire (Antonov et al., 1997; Usov et al., 1998). This phenomenon is an analogue of the inverse Wiedemann effect (Barandiarán and Hernando, 1980). Another method of solution of Maxwell’s equations was reported by Makhnovskiy et al. (2001). The tensor of surface impedance ζˆ was calculated for cylindrical wires with circumferential and helical anisotropy. In addition to the classical GMI effect the inﬂuence of bias DC current or axial AC magnetic ﬁeld on magnetoimpedance was also investigated. Equation (4.5), with m = χh ˆ and χˆ according to eq. (4.25), was solved by means of the asymptotic-series expansion method. Numerical simulations for a wire with helical anisotropy have shown that with increasing DC bias current considerable asymmetry appears in the ﬁeld dependence of both ζzz and ζzy components. Magnetoimpedance of a planar bi-layer ﬁlm with crossed anisotropy, which is qualitatively similar to the wire with helical anisotropy, was investigated by Makhnovskiy et al. (1998). In contrast to the wire, this planar geometry allows simple analytical solution of eq. (4.5). In the local coordinate system (x, y , z ) the solution for each layer has the form hy = A sin kx + B cos kx,

(4.34) hz = C sin k0 x + D cos k0 x, √ where k0 = (1 − i)/δ0 and k = k0 1 + χt . Thus two electromagnetic waves with different wave vectors (k0 and k) can generally be excited in each layer. The eight coefﬁcients at the trigonometric functions (four for each layer) are determined from the electromagnetic boundary conditions on the two surfaces and the continuity condition at the interface. The effect of antisymmetric transverse DC bias ﬁeld Hy (H 0 = H z ± H y ) on the surface impedance tensor ζˆ was investigated. Similarly as in the previous case (Makhnovskiy et al., 2001) the asymmetrical magnetoimpedance was observed for increasing Hy . The maximum ﬁeld sensitivity of MI was obtained for Hy slightly higher than HK cos φ, where φ is the deviation of the easy direction from the z-axis.

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Usov et al. (1997) studied a cylindrical wire composed of a nonmagnetic inner core and an axially magnetized outer shell with longitudinal anisotropy. They have found that for certain ratio of the shell thickness to the core diameter the GMI effect can be substantially increased if the conductivity of the core is much larger than that of the shell. It is caused by the reduced Rdc (DC current ﬂows mainly through the core) but practically unchanged Z max if the skin depth is smaller than the shell thickness (AC current ﬂows only in the shell). Such results were also tested and conﬁrmed using ﬁnite elements methods, where in principle more complex structures and geometries can be simulated (Muñoz et al., 2002; Sinnecker et al., 2002). Cylindrical wires having axial anisotropy in the inner core and circumferential anisotropy in the outer shell were investigated by Usov et al. (1999). Numeric modelling shows that the axially magnetized inner core slightly increases the ratio |Z|/Rdc . 4.3.3.2. Exchange-conductivity models. The models mentioned above describe qualitatively well the basic features of GMI and can be used to interpret most of the experimental data. There are, however, some aspects that cannot be explained in the framework of the electromagnetic approximation and require more rigorous theoretical treatment. To include the exchange-conductivity effects, the importance of which is well known from the ferromagnetic resonance in metals, the exchange term in the effective ﬁeld (the last term in eq. (4.9)) must not be neglected. When the exchange ﬁeld is taken into account the permeability tensor χˆ is no longer independent of the spatial variation of the AC magnetization m(r). The Maxwell and Landau–Lifshitz equations then provide a system of six partial differential equations for the vector components of m and h and their solution becomes more complicated. Some authors (for example, Makhnovskiy et al. (1998)) suggest that the exchange-conductivity effect plays an important role only for very highfrequencies, where the skin depth becomes comparable with the exchange length δex = 2A/μ0Ms2 . This argument, however, is not quite adequate because it is the exchange-conductivity that weakens the discrepancy between the skin depth, theoretically predicted by the electromagnetic model (see eq. (4.32)), and the experimental data at low and intermediate frequencies (Kraus, 1999b). The importance exchange-conductivity effect in metallic ferromagnets was ﬁrst recognized by Kittel and Herring (1950). It is caused by the interplay between the skin effect and the exchange interaction. Because within the skin depth the amplitude of AC magnetization m decreases from its surface value to zero in the bulk, the magnetization is inhomogeneous and exchange energy increases. Therefore the exchange interaction weakens the skin effect and enhances the skin depth. In other words, the inhomogeneous AC ﬁeld excites spin waves with wavelengths of the order of the skin depth, which enhance the energy consumption in ferromagnetic metals by eddy currents. It may be interpreted as an apparent increase of resistivity. Yelon et al. (1996) have realized that the calculation of skin depth in GMI is equivalent to the procedure used in the theory of ferromagnetic resonance of metals. They suggested that many results of former FMR calculations could be directly applied to GMI. For a brief review of FMR theory in metals see, for example, the treatise of Frait and Fraitová (1988). Though the analogy between FMR and GMI is quite evident there are still some differences between both of them. Ferromagnetic resonance is usually

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measured at GHz frequencies, where the resonance ﬁelds, owing to the gyromagnetic effect, are so high that the samples are well saturated. This assumption, which substantially simpliﬁes the theoretical analysis, need not be always fulﬁlled in case of GMI. Also the low signal condition, which is well satisﬁed in the low power FMR experiments, can be easily infringed in the GMI measurements. Moreover, the difference in the symmetry of external AC ﬁeld hext may be important in tiny conductors, where different standing spin wave modes can be excited (Sukstanskii and Korenivski, 2001). Before describing the individual models, let us ﬁrst mention the basic outcomes of the FMR theory, which are valid also for GMI. When the exchange term is introduced into the effective ﬁeld H eff the linearized equation of motion (4.12) cannot be expressed by means of an independent susceptibility tensor χˆ . Then eqs. (4.5) and (4.12) must be solved simultaneously. For a conductor with planar geometry the general solution is assumed as a superposition of plane waves e−ikr propagating perpendicularly to the surface (Frait and Fraitová, 1988). For a general direction of wave propagation with respect to M 0 the secular equation for propagation vectors is quartic in k 2 (Sooho, 1960), leading to four pairs of waves −k 0 , k 0 , −k 1 , k 1 , −k 2 , k 2 , −k 3 , k 3 of mixed spin-wave and electromagnetic character. The amplitudes of individual waves can be obtained from the boundary conditions, which besides the usual electromagnetic boundary conditions include also the surface spin-pinning, which in the case of the simplest symmetry of surface anisotropy has the form (Sooho, 1960) ∂m Ks − m = 0, ∂n A

(4.35)

where Ks is the surface anisotropy constant and ∂/∂n the derivative in the direction normal to the surface. In case of parallel resonance (M 0 ⊥ n and r n) the biquartic secular equation decomposes to k0 = (1 − i)/δ0 , for longitudinally polarized nonmagnetic waves, and the bicubic equation k 6 + c1 k 4 + c2 k 2 + c3 = 0

(4.36)

for three pairs of transversally polarized magnetic waves (Ament and Rado, 1955). By polarization we understand the orientation of h with respect to M 0 . The coefﬁcients ci are functions of ω, H0 and other parameters. Their particular forms depend on the type of magnetic anisotropy and the orientation of anisotropy axes. Following Patton (1976), the three transversally polarized wave branches will be called “Larmor electromagnetic” (LE), “Larmor spin wave” (LS), and “anti-Larmor spin wave” (AS). The fourth “anti-Larmor electromagnetic” branch corresponds to the nonmagnetic wave with wave vector k0 . At very high frequencies the individual branches can be well distinguished. But below certain, so called “crossover”, frequency the two Larmor branches LE and LS become strongly mixed near the resonance ﬁeld. It is just the region where the exchange-conductivity effect is observed. Procedures similar to that described above were used to model GMI in planar ﬁlms (Kraus, 1999a) and circular cylinders (Menard et al., 1998; Menard and Yelon, 2000). Kraus (1999a) investigated GMI in a planar ﬁlm with in-plane anisotropy (see ﬁg. 9). By the simultaneous solution of eqs. (4.5) and (4.12), in the coordinate system (x, y , z ),

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the secular equation

Ms Ωk + Ms + HK cos2 ψ 2

ω i 2 2 2 (Ωk + HK cos 2ψ) Ωk + Ms + HK cos ψ − =0 + 1 − δ0 k 2 γ (4.37) was obtained. Here Ωk = Ω + 2Ak 2/μ0 Ms . This equation is equivalent to eq. (4.36) but it is better suited for a comparison of the exchange-conductivity model with the electromagnetic approximation described in the previous section. As can be seen, when the exchange term is neglected (A = 0) it reduces to the quadratic equation i 2 2 δ k = 1 + Xy y , 2 0

(4.38)

(see eq. (4.29)). Its roots where Xy y is the component of effective susceptibility tensor X correspond to the propagation vectors of the Larmor electromagnetic branch. The spin wave solutions are omitted (that is why the models described in previous section were called the “electromagnetic” ones). It was shown (Kraus, 1999a) that the two dimensional is diagonal in the (x, y , z ) coordinates and the impedance for an impedance tensor Z arbitrary angle β between the driving current and DC magnetization M 0 is Z = Zy y sin2 β + Zzz cos2 β.

(4.39)

Zy y is the nonmagnetic component of impedance and is given by eq. (4.6b), where k = k0 . An analytic formula for the longitudinal magnetic component Zzz was obtained from the boundary conditions for free surface spins (Ks = 0). If the exchange is neglected the formula for Zzz reduces to eq. (4.6b) with k = k0 1 + Xy y . GMI in an isotropic cylindrical wire (HK = 0) was investigated by Menard et al. (1998). The axially symmetric solution was assumed in the form of cylindrical waves ﬁnite at r = 0 (Bessel functions of the ﬁrst kind). An expression for the surface impedance Zs was found with boundary conditions for free surface spins. When the exchange was ignored the impedance reduced to eq. (4.6a) with k corresponding to the Larmor electromagnetic wave. It was pointed out that for the strong skin effect the surface curvature can be neglected and the solution for the planar plate can be used. Magnetic anisotropy in cylindrical wires was taken into account by Menard and Yelon (2000). The surface impedance tensor of a wire with helical anisotropy was calculated. In the local coordinates (x, y , z ) the surface impedance tensor is diagonal with ζy y = Z0 and ζzz = ZM , where Z0 corresponds to nonmagnetic (anti-Larmor) electromagnetic wave (k0 ) and ZM to the mixture of the three other magnetic waves. The tensor ζˆ in cylindrical coordinates (r, φ, z) was obtained by rotation of local coordinates (x, y , z ) over the angle θ0 around the x-axis. The surface impedance Zs was then given by eq. (2.5). For hz = 0 Ménard and Yelon obtained Zs = Z0 sin2 θ0 + ZM cos2 θ0 , which is equivalent to (4.39). They suggest that this is, however, only an approximate solution for |Z0 |, |ZM | 1 because if θ0 = 0 one has hz = 0 even if no AC ﬁeld is explicitly applied along the z-axis. hz was calculated from the electromagnetic boundary conditions assuming, as done by Usov (Usov et al., 1998), that outside the wire the solution was given by a combination of

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electromagnetic TE and TM cylindrical waves. To ensure the regular behaviour of electromagnetic waves for r → ∞ their ﬁnite wavelength along the z-axis had to be considered, which resulted in hz = 0. Then the exact formula obtained for the surface impedance was 1 + Z0 ZM + (1 − Z0 ZM )2 − (Z0 − ZM )2 sin2 2θ0 Zs = (4.40) . 2(Z0 cos2 θ0 + ZM sin2 θ0 ) Z0 and ZM were calculated from the boundary conditions with free surface spins. To simplify the rather complicated formula for ZM an approximate solution of the secular equation (4.36) was found using the procedure introduced in the FMR theory (see for example (Fraitová, 1983)). Because the penetration depth of the anti-Larmor spin wave (called also “surface spin wave”) is very small and only weakly depends on applied ﬁeld its contribution to GMI can be neglected. Moreover, by eliminating its approximate root (k32 ≈ −c1 ) from eq. (4.36) a biquadratic equation for the Larmor electromagnetic and Larmor spinwave branches is obtained. The analysis shows that below the “cross-over frequency” ωc = 4α 2 γ 2 AMs /ρ there is a strong mixing of LE and LS branches near the resonance ﬁeld, which leads to the exchange-conductivity effect. When the damping was neglected (α = 0), an approximate formula for the effective minimum skin depth was obtained: 1/4 Aρ δmin ≈ (4.41) ωμ20 Ms2 for frequencies well below the crossover frequency. In typical soft magnetic amorphous metals ωc /2π is of the order 100 MHz. Thus, as a consequence of the exchangeconductivity effect, for strong skin effect and medium frequencies the maximum theoretical GMI ratio scales as (ω)−1/4 . Above ωc the theoretical limit of GMI is given by eq. (4.32). The role of exchange interaction and surface spin pinning on magnetoimpedance of thin planar ﬁlms and sandwich ﬁlms (ferromagnetic/nonmagnetic/ferromagnetic) was investigated by Sukstanskii and Korenivski (2001). Though in this work the DC component of the external magnetic H 0 is assumed to be zero, the results can be used also for H 0 parallel to the easy anisotropy axis (considering Ha = H0 + HK in their formulas). The difference between the surface impedance for the symmetric (FMR) and asymmetric (GMI) external ﬁeld hext , perpendicular to M 0 , was systematically investigated. It was shown that for strong skin effect the surface impedance was independent of the symmetry of hext . There is however a substantial difference in surface impedance for symmetric and antisymmetric excitation in very thin ﬁlms, where the skin depth is larger then the ﬁlm thickness and the waves propagating from the two opposite surfaces can interfere in the middle. The inﬂuence of surface anisotropy on the surface impedance is similar for both excitation symmetries: with increasing spin pinning the intensity of higher order spin wave modes increases. In the case of asymmetric spin pinning (different surface anisotropies on the two surfaces) the surface impedances of the two surfaces are different. With increasing asymmetry of pinning the main magnetoimpedance peak decreases and the intensity of higher order spin-wave modes increases. In sandwich ﬁlms the situation is much more complicated. But qualitatively, when the conductivity of the central nonmagnetic layer is much larger than that of the outer magnetic layers the exciting ﬁeld in magnetic layers is more homogeneous and the result is more similar to the symmetric excitation (FMR) in a single layer ﬁlm.

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5. Analysis of selected experimental results It is extremely difﬁcult to make a complete review of all published experimental data on GMI, because a large number of works have been published as a result of intense research in the last years. Generally speaking, conventional GMI measurements are performed to test novel materials and the validity of some theoretical models under speciﬁc conditions, while different geometries and techniques are employed to shed new light on some still obscure points. Furthermore, a great quantity of investigations deal with different kinds of materials subjected to a broad variety of annealings, which are performed in order to induce speciﬁc magnetic anisotropies, to tailor the GMI response for speciﬁc needs. In this context, some experimental works were selected, mainly focusing on novel effects and relevant aspects found in different kinds of materials. The use of GMI as an additional research tool was also considered. 5.1. Data in novel materials Since its discovery, GMI has been studied in a wide range of systems. Firstly, GMI was observed in single phase bulk structures, like soft magnetic amorphous wires (Beach and Berkowitz, 1994a; Brunetti et al., 2002; Chiriac and Óvari, 2002; Knobel et al., 1997c; Mohri et al., 2002b, 2002a; Panina and Mohri, 1994; Vázquez, 2001) and ribbons (Machado et al., 1995; Medina et al., 1996; Sommer and Chien, 1995). It was soon realized that the ﬁrst most important characteristic of a good GMI element was the magnetic softness. Therefore some studies were focused also on soft nanocrystalline ferromagnets (Guo et al., 2001; Kim et al., 2002b; Knobel et al., 1995a, 1997b; Zhao et al., 2002). A novel GMI effect was observed in granular Fe-Ag systems, with Fe grains of the order of few nanometers (Soares et al., 2002). Such effect has been attributed to the large transverse susceptibility present on the studied samples (Soares et al., 2002). Giant magnetoimpedance has also been extensively investigated in glass covered amorphous microwires (Antonov et al., 2001; Brunetti et al., 2000; Chiriac et al., 1999b;

Fig. 11. GMI ratio in Co67.05 Fe3.85 Ni1.4 B11.55 Si14.5 Mo1.65 glass-coated microwires with different ratio, ρ (diameter of metallic core to total diameter). The different response was ascribed to different magnetoelastic anisotropy introduced by coating (Zhukov, 2002) (courtesy of A. Zhukov).

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Kraus et al., 1999, 2003; Vázquez et al., 1998b), where the stress due to the glass coat can induce very strong magnetoelastic anisotropies (Chiriac et al., 1999b; Antonov et al., 2001). This family of amorphous wires are fabricated by a quenching and drawing method (Taylor–Ulitovsky) (Chiriac and Óvári, 1996). This is an alternative technique compared to the in-rotating-water quenching method. It allows one to obtain wires with more reduced diameter but which are additionally covered by an insulating protective coating (Chiriac and Óvári, 1996; Vázquez and Zhukov, 1996). While the radius of their metallic core can range from around 1 to 30 μm, the Pyrex-like coating is typically 2 to 10 μm thick. The presence of the coating, in addition to the quenching and drawing stresses, induces large stresses (up to around 400 MPa) due to the different thermal expansion coefﬁcient of core and coating. In comparison to in-waterquenched amorphous wires, their comparatively more reduced permeability and radius cause the GMI phenomenon to be observed at higher frequencies (at the order of MHz). Fig. 11 shows the GMI ratio for Co67.05Fe3.85Ni1.4 B11.55 Si14.5 Mo1.65 glass-coated microwires having different ratio, ρ (diameter of metallic core to total diameter). As observed, maximum magnetoimpedance response (in fact, among the largest reported values in a microwire) is obtained for the microwire with the thinest glass-cover (Zhukov, 2002; Zhukova et al., 2002). Figure 12 shows the GMI response of a glass-covered amorphous Co68.25Fe4.5 Si12.25B15 microwire, measured at 15 MHz with an AC current of 1 mA. The microwire was previously Joule-heated for 10 minutes with 70 mA, in order to induce speciﬁc anisotropies (Kraus et al., 2003). The inset shows the dependence of maximum GMI on annealing current (Ia ) for two different driving current frequencies. It is worth noticing that, after being

Fig. 12. GMI curve of a glass-covered amorphous Co68.25 Fe4.5 Si12.25 B15 microwire, measured at 15 MHz with an AC current of 1 mA. The sample was Joule-heated for 10 min with 70 mA. The inset shows the dependence of maximum GMI on annealing current Ia (Kraus et al., 2003).

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Fig. 13. Relative variation of the total impedance modulus and the corresponding real and imaginary components of a CoP microtube electroplated on Cu, after Sinnecker et al. (2000b). Courtesy of J.P. Sinnecker.

properly annealed, the glass covered microwires can exhibit large and sensitive giant magnetoimpedance ratios. The shape of the sample has a strong inﬂuence on the giant magnetoimpedance behaviour. Novel geometries of sandwiched thin ﬁlms have been tested in order to tailor the impedance response for speciﬁc purposes (Kurlyandskaya et al., 2002b; Zhou et al., 2001). Samples with ring-shaped geometry, such as microtubes, display enhanced GMI ratios and ﬁeld sensitivities (Garcia et al., 2001b; Óvari et al., 2001) (as an illustration, see ﬁg. 13). It is worth noting that the surface roughness of the samples and its associated variations in local magnetic anisotropy can play an important role in the GMI behaviour. Systematic studies on chemically thinned and polished samples have revealed that superior values of GMI ratios are found in micropatterned samples made of polished foils (Amalou and Gijs, 2001). Later, it was clearly shown that the geometrical dimensions of samples and measuring conditions play a very important role when reporting numerical values in MI studies (Vázquez et al., 2002). In this way, it is important to mention an experimental fact which normally no attention is paid to: the place where electrical contacts are ﬁxed. It is known that closure domain structures appear especially at the ends of elongated samples (e.g. wires or ribbons) in order to reduce the otherwise very large magnetostatic energy. These structures modify locally the permeability so that signiﬁcantly different GMI values can be reported for nominally the same sample. This effect is especially important in very short wires and in principle should be avoided (Vázquez et al., 2002).

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Although GMI seems promising for applications (see section 6), there are nevertheless some drawbacks that must be overcome in order to design a material for speciﬁc devices. For example, GMI presents relaxation due to the aftereffect of the permeability (Knobel et al., 1997a; Sartorelli et al., 1997). This GMI aftereffect is undesiderable for technological applications, but can be strongly reduced through appropriate annealings (Pirota et al. 1999c, 1999b). Another undesiderable effect for some applications of GMI is the hysteretic behaviour that is correlated with the irreversibilities of the magnetization process (Kim et al., 2001a; Sinnecker et al., 1998b). Like the GMI aftereffect, hysteresis can be strongly reduced through adequate thermal treatments (Vázquez et al., 1999c). It is worth noticing that magnetoimpedance has been used to investigate other magnetic systems, such as La0.7 Ca0.3 MnO3 (Hu and Qin, 2001a), Cu-Co multilayers (Rausch et al., 1999), and metal-insulator transition of V2 O3 sintered samples containing micron-size Fe precipitates (Kale et al., 2000). It has also been used at low frequencies (up to 5 kHz) to study the ferromagnetic phase transition in Heusler alloys Pd2 MnSn and Pd2 MnSb. The sharp decrease of magnetoimpedance on the transition to paramagnetic state can be used as a tool for the investigation of critical phenomena in metallic ferromagnets (Fraga et al., 2002). Finally, it is interesting to mention results connecting giant magnetoimpedance and the “wire media” in negative index materials (Pendry, 2001). Composites with engineered or tunable dielectric response have been prepared made from diluted magnetic inclusions as glass-coated microwires. Negative real permittivity is observed in lattices of conducting magnetic wires with an axial magnetized core. The underlying physics involved in the mechanism has been correlated to the GMI (Reynet et al., 2002; Acher et al., 2003). 5.2. GMI as a research tool Once the GMI phenomenon is now becoming better understood, it is possible to predict some expected trends, and also use it as an additional tool to investigate some intrinsic and extrinsic magnetic properties of novel artiﬁcially grown soft magnetic materials. This section gives an overview of some works that make use of GMI to extract relevant parameters of the samples. 5.2.1. On the circular magnetization process The study of the giant magnetoimpedance effect comprises the application of an alternating circular magnetic ﬁeld to evaluate changes in the impedance. A direct correlation between the circular magnetization process induced by that ﬁeld and the GMI is then clear, as has been previously mentioned. Then, GMI can additionally serve as a technique to extract information on such magnetization process. The potential application of GMI effect as a research tool was already noticed in one of the initial works, where a ﬁt of the impedance versus frequency curve gave as a result the effective circular permeability for a FeCoSiB wire (Beach and Berkowitz, 1994b, 1994a), a parameter which is difﬁcult to obtain using conventional magnetic techniques. It was later shown that even at high frequencies it is possible to extract information about the magnetic permeability of the material (Melo et al., 2002). By means of simple approximations, it was possible to develop a useful model to visualize the GMI effect in amorphous and

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Fig. 14. GMI ratio for different driving current (circular magnetic ﬁeld) amplitudes. An evolution from double-peak to single-peak behaviour is observed. The wire exhibit a clear transverse magnetic anisotropy (bamboo-like domains). After Freijo (2001).

nanocrystalline wires, which leads to rough estimations of the magnetic penetration depth δ and also the circular permeability μφ , using one single measurement instead of the whole frequency dependence (Knobel et al., 1995a, 1995b). Later, the high sensitivity of giant magnetoimpedance was used to investigate the gradual reduction of magnetic coupling among magnetic nanoparticles in soft ferromagnetic nanocrystalline alloys when the temperature is increased (Zeng et al., 2002). The thermal dependence of GMI effect clearly reveals the enhancement of the Curie temperature of the amorphous matrix (Zeng et al., 2002), in agreement with previous reports that investigated the temperature dependence of the coercive ﬁeld (Hernando and Kulik, 1994). Also, the circular magnetization process has been studied by analysing the real component of complex inductance (Betancourt and Valenzuela, 2002). An important parameter is the amplitude of the circular magnetic ﬁeld created by the electrical current. See, for example, ﬁg. 14, which displays a clear change in the GMI proﬁle when the driving current amplitude is increased. The experiment was performed at 100 kHz in a Co-based conventional wire which was previously thermally treated to exhibit transverse anisotropy. One possible explanation is that for very low values of the driving current only fully reversible circular magnetization processes are activated. When the current is large enough, so that the coercivity of circular magnetization process would be overcome, notable changes in the giant magnetoimpedance would be observed. However, one cannot exclude the fact that large current amplitudes can also lead to highly anharmonic responses. In fact, it was noted that extremely high and sensitive magnetoimpedance effect can be found by studying higher harmonics of the voltage signal of the GMI element (Duque et al., 2003). In the case of FeCoNi magnetic tubes electroplated onto CuBe nonmagnetic wire,

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GMI ratios up to 800% were found at current frequencies of around 1 MHz, with huge sensitivities of the order of tens of thousands %/Oe (Kurlyandskaya et al., 2001b). The higher harmonics (especially the second one) of the impedance signal display larger variations with the external magnetic ﬁeld than the fundamental one (Kurlyandskaya et al., 2001a). The results are qualitatively explained in terms of a strong variation of the transverse susceptibility, that is extremely high at the points of orientational phase transitions, giving rise to strong nonlinear effects (Kurlyandskaya et al., 2002a). Similar experimental results were presented by Antonov et al. (2001) in Co-based amorphous microwires. The authors have found that the even harmonics dominate the frequency spectra of the voltage signal, if the AC current amplitude exceeds some threshold value, which strongly depends on the applied magnetic ﬁeld. The experimental data is described in terms of the Faraday law and the quasi-static Stoner–Wohlfarth model (Antonov et al., 2001). Such strong variation of highorder harmonics in the non-linear regime can be very useful in ultra-sensitive magnetic ﬁeld sensors, with an enormous perspective in several applications (Antonov et al., 2002). The formalism of correlation between circular permeability, inductance and impedance has been extensively considered by Chen et al. in several works (Chen et al., 2000b; Chen and Muñoz, 1997). Depending on the anisotropy easy axis, the magnetoimpedance is calculated considering micromagnetic models for the magnetization processes. The circular permeability and circular magnetization processes are also derived from inductance measurements (Chen et al., 2000b). An alternative approach was developed by Valenzuela and co-workers, who investigated the GMI using equivalent circuits (Sánchez et al., 1996; Valenzuela et al., 1995). In this impedance spectroscopy formalism it was possible to show that a CoFeBSi wire can be approximated by a series Rs Ls arrangement, connected to a parallel Lp Rp arm. Ls and Lp are inductors associated with the rotational and domain wall contributions to μφ , respectively. Rp is a resistive element related to wall damping and R s accounts for all resistances in the circuit like the wire itself, contacts and so on. This methodology allows one not only to evaluate the circumferential permeability as a function of the applied ﬁeld but also to estimate the respective rotational and domain wall contributions to permeability (Carrasco et al., 1999; Valenzuela, 2002). Later, a novel approach has been introduced to study the GMI effect (Gómez-Polo et al., 2001a). In this method, the mean value of the circumferential permeability and the circular hysteresis loops can be estimated by means of a Fourier analysis of the time derivative of the magnetization calculated from a simple quasistatic rotational model, minimizing the free energy equation. The mean value of the circumferential permeability is obtained through Fourier analysis of the time derivative of the estimated circular magnetization. Moreover, the existence of a second harmonic component of the GMI voltage is also experimentally detected. Its amplitude sensitively evolves with the axial DC magnetic ﬁeld and its appearance is associated to an asymmetry in the circular magnetization process (Gómez-Polo et al., 2001b). The simple model also explains quite well the experimental results for a FeCoSiB joule heated wire biased by a DC current (asymmetric response), and it is being tested in several soft magnetic systems (Pirota et al., 2002). 5.2.2. Magnetoelasticity and magnetostriction Although typically GMI is observed in materials characterised by a reduced value of saturation magnetostriction, λs , (less than 1 p.p.m.) the application of mechanical stresses,

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σ , can give rise to very signiﬁcant changes in the magnetization process of the material through the magnetic anisotropy term Eσ = 3/2λs σ . The magnetoelastic anisotropy induced by applied stress competes by enhancing or balancing intrinsic anisotropies induced by fabrication (e.g. internal stresses quenched during preparation of amorphous materials, columnar growth in electrodeposited tubes or ﬁlms, etc.). Two types of mechanical stresses are usually applied to ribbons or wires: tensile and torsional stresses. Tensile stress induces magnetoelastic easy axis promoting the generation of domains with longitudinal magnetization in the case of positive magnetostriction materials. As analysed in a previous section (section 4) the GMI response is then characterised by a single-peak structure, and consequently the magnetoimpedance ratio, ΔZ/Z, decreases continuously with applied DC ﬁeld. In the case of negative magnetostriction materials, tensile stresses induce transverse easy axis (for example, giving rise to a bamboo-like domain structure in GMI wires). Thus, the typical GMI response becomes double-peak like with a maximum around the transverse anisotropy ﬁeld as discussed above (see also ﬁg. 6). The modiﬁcation of the GMI ratio by applied tensile stress, in a given static magnetic ﬁeld, has given rise to a new effect labelled as Stress-Impedance, SI. Studies of GMI as a function of applied tensile stress (Knobel et al., 1995c, 1997c; Mandal et al., 2000) or torsion angle (Blanco et al., 1999; Bordin et al., 2000; Sánchez et al., 2002) have additionally revealed that magnetoimpedance can be a suitable technique to build sensitive stress or torsion sensors, as will be discussed in section 6. Indeed, using the stress-impedance it is possible to study the magnetoelastic behaviour through impedance measurements. The inﬂuence of applied tensile stress on GMI effect has been investigated in amorphous wires (Atkinson and Squire, 1997; Kim et al., 2001b; Knobel et al., 1995c, 1997c) and ribbons (Tejedor et al., 1998b). The stress dependence of GMI can be easily applied to estimate the saturation magnetostriction constant (λs ) of negative magnetostriction samples (Knobel et al., 1996). The strength and the easy axis of magnetic anisotropy has been further derived by studying the magnetoimpedance behaviour in the microwave region (García-Miquel et al., 2001). One question that has not yet been fully solved is related to the distribution of intrinsic magnetoelastic anisotropy in GMI amorphous materials. It is known that the fabrication process of amorphous wires induces a radial distribution function of mechanical stresses and consequently of magnetoelastic anisotropies. It leads to a radially distributed circumferential permeability that can pose some problems when trying to fully understand the GMI theoretically (especially when the skin effect is not so large that the penetration depth is comparable to the radius of the sample). Similar inhomogeneity of internal stress appears in rapidly quenched amorphous ribbons. The magnetoelastic anisotropy induced by torsional stress in a wire induces a helical easy magnetization direction at 45◦ with the longitudinal axis of the sample. Torsion, τ , can be decomposed into tensile stresses along the positive helical direction plus compressive stresses along the negative helix for positive magnetostriction samples (see ﬁg. 15). For negative saturation magnetostriction, the opposite or negative helix becomes the easy axis. The induced magnetoelastic anisotropy can be expressed as Eτ = 3/2λs τ = 3/2λs Gξ r, where G is the shear modulus, ξ is the angular displacement per unit length, and r the distance to the axis of the wire. Note that in the case of torsion, the magnetoelastic anisotropy is inhomogeneous and that leads to a maximum value at the surface of the wire or ribbon.

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(b)

Fig. 15. The application of torsional stress induces helical magnetoelastic anisotropy and can be considered as the addition of tensile and compressive stresses (a). Applied torsion changes the impedance, as shown for CoFeSiB wire submitted to previous current annealing treatments (no static ﬁeld was applied during measurements) (b). Courtesy of A. Zhukov (Blanco et al., 2001).

The inﬂuence of torsion has been investigated also by Tejedor et al. (1998a). An example for torsion-impedance is given in ﬁg. 15(b), where the impedance ratio is plotted as a function of applied torsion in a (Fe0.06Co0.94)72.5 Si12.5B15 amorphous wire (Blanco et al., 2001). As can be seen, the impedance can clearly be modiﬁed by the application of stresses even when no static ﬁeld is applied. 5.2.3. GMI and induced magnetic anisotropies As a consequence of the correlation between magnetic permeability and impedance, GMI is very sensitive to intrinsic or induced anisotropies that can modify the magnetization process. In this regard, induction of particular magnetic anisotropies can be very convenient for obtaining particular GMI response. GMI can be improved by suitable thermal treatments. Besides the induction of anisotropies, thermal treatments may give rise to very signiﬁcant changes in the structure of the materials as is the case of the partial devitriﬁcation of FeSiB base amorphous alloys with small additions of Cu and Nb. Nanocrystalline alloys may result with vanishing magnetocrystalline and magnetoelastic anisotropies with outstanding GMI behaviour (Guo et al., 2001; Knobel et al., 1995a, 1997b; Knobel, 1998). Sometimes thermal treatment is employed just to relax the intrinsic magnetic anisotropies generated during fabrication with the consequence of enhancing the transverse permeability. But in other cases, a particular anisotropy may be desirable to achieve a given GMI behaviour. Indeed, one of the most interesting characteristics of the GMI elements is the possibility to control the induction of anisotropies, fact that allows one to suitably tailor GMI responses for speciﬁc purposes. By combining thermal treatments (either in furnaces or by Joule heating), with the application of external magnetic ﬁelds, tensile and/or torsional stresses, it is possible to obtain larger GMI ratios, extremely linear regions, ﬁeld

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asymmetries, and a broad variety of responses that can suitably be applied in different devices or experiments. Slight transverse anisotropy can be induced by annealing in the presence of a transverse magnetic ﬁeld that is obtained via current-annealing (making use of the circular magnetic ﬁeld created by the current). This allows the achievement of very large GMI response. Thermal treatments in the presence of applied tensile stress and both stress plus ﬁeld induce transverse anisotropies with stronger amplitude. This may improve the achievement of a bamboo-like domain structure in wires but the maximum GMI ratio is decreased because the transverse permeability reaches not so high values. The magnetic domain structure determined by the induced anisotropy in a system (its strength, direction and distribution) strongly affects the GMI behaviour (Kraus, 1999a; Makhnovskiy et al., 2001), as already pointed out in section 4. An example of the effect of speciﬁc thermal treatments is shown in ﬁg. 16, where an amorphous ribbon (10 mm × 22 μm) with the nominal composition Co67 Fe4 Cr7 Si8 B14 was investigated (Kraus et al., 1998). The ribbon was stress-annealed for one hour at 380 ◦ C with different applied stress. The anisotropy ﬁeld HK and the transverse demagnetizing factor were determined from DC hysteresis loops measured along and transverse to the ribbon axis, respectively. The transverse permeability μt as a function of longitudinal bias ﬁeld H0 was measured using a simple inductive method. The impedance Z of the ribbon was investigated at several frequencies in the range 10–120 kHz with the bias ﬁeld applied either parallel or transverse to the ribbon axis. The longitudinal hysteresis loops (ﬁg. 16(a)) were linear with a sharp knee at the anisotropy ﬁeld HK . The anisotropy was proportional to the stress σa applied during annealing. The transversal hysteresis loops were also linear but with the knee (at about 1060 A/m) independent of σa . From its position the transverse demagnetizing factor Ny = 2.37 × 10−3 was calculated. The relative change of transverse permeability μt (H0 ), measured at the frequency 75 Hz with hex = 12 A/m, is shown in ﬁg. 16(b). As can be seen, μt (H0 ) is nearly constant for |H0 | < HK , which indicates that domain wall movement dominates in this ﬁeld region. The observed hysteresis may be explained by some domain structure changes with applied bias ﬁeld H0 . The GMI-effect measured at the frequency of 100 kHz with an AC current of 10 mA is shown in ﬁg. 16(c). The results clearly show that the transverse permeability and the GMI-effect are closely related to the transverse magnetic anisotropy, which is controlled by the stress applied during annealing. This conclusion is in a good qualitative agreement with the quasi-static models described in section 4. When, however, the magnitudes of experimental and theoretical GMI curves are compared the agreement becomes much worse. For example, using eq. (4.32) with the known parameters (Ms = 4.5 × 105 A/m and ρ = 1.5 × 10−6 m) and an estimated value of Gilbert damping parameter (α = 10−2 ) one obtains |ΔZmax |/Rdc = 32.5, which is almost three orders of magnitude higher than the experimental value. The discrepancy between the theory and experiment may be explained by the dispersion of HK or local easy axes, as already pointed out in section 4. Besides the illustration given above, many kinds of annealing have been reported in order to induce speciﬁc anisotropies: Field annealing (Yoon et al., 2000), stress-annealing (Hernando et al., 2001; Kraus et al., 1998; Pirota et al., 2001), torsion-annealing (Blanco et al., 2000b; Kawashima et al., 1999), laser annealing (Ahn et al., 2001; Kawashima et al., 1999) and Joule heating (Appino et al., 2001; Cho et al., 2000; Kraus et al., 1999; Li et al., 2002) and their combinations (Blanco et al., 2000a; Garcia et al., 2001c; He et

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Fig. 16. (a) Longitudinal hysteresis loops, (b) transverse permeability and (c) GMI effect in stress annealed amorphous Co67 Fe4 Cr7 Si8 B14 ribbons. The annealing stress σa is given in (a). After Kraus et al. (1998).

al., 2001). Furthermore, induced anisotropies play an important role in crystalline Mumetal strips (Cho et al., 2000; Nie et al., 1999), sputtered thin ﬁlms (Mendes et al., 1996; Sommer and Chien, 1995), sandwich structures (Xiao et al., 1999; Zhou et al., 2001) or microtubes (Sinnecker et al., 2000a; Vázquez et al., 2000). An important issue besides the relative orientation of the DC ﬁeld and the easy magnetization axis, is the dispersion of the local anisotropy axis. As it is well known, the materials that have exhibited the largest GMI ratios were soft magnetic amorphous materials, which are microscopically usually taken as random anisotropy materials. A macroscopic anisotropy can, in general, be induced in these materials by stress or ﬁeld annealing, thus enhancing the permeability and the GMI ratio. However, amorphous materials exhibit always some degree of dispersion of the local anisotropy axes with respect to the induced macroscopic anisotropy axis described by Kef . Pirota and co-workers (Pirota et

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al., 1999a) have performed a detailed calculation of the effect of the angle θ between the easy axis and the magnetizing ﬁeld, taking into account the demagnetization tensor of the ribbons. By introducing a distribution of the easy axes, Pirota et al. were able to reproduce most of the experimental results, among them, the strong dependence of the peak positions on the GMI vs. H curves as function the angle between the applied ﬁeld and the ribbon axis (Pirota et al., 1999a). These authors also pointed out that the effect of the dispersion is more dramatic than usually considered. In fact, this behaviour can be properly used to experimentally determine the dispersion of anisotropy in amorphous materials. Carara et al. (2000) showed that relevant information about the magnetization dynamics of soft magnetic materials can be obtained from GMI measurements. Also, the GMI technique was applied to evaluate the evolution of anisotropy ﬁeld of FeNbB amorphous ribbons as a function of iron content (Ryu et al., 2000), while GMI was combined with magnetic force microscopy to characterize the domain structure of electrodeposited CoP microtubes (García et al., 2001a). 5.3. Asymmetric GMI Much attention has been paid to the asymmetric GMI effect, which can be very promising for a new generation of sensor devices. To build a linear ﬁeld sensor two asymmetrical GMI elements are needed. When the voltages induced across of the two oppositely asymmetric elements are subtracted nearly linear characteristic (in some ﬁeld range) can be obtained (Mohri, 1994). In what has been discussed in previous sections the GMI effect was mainly symmetric with respect to the applied ﬁeld direction. The asymmetric characteristic can be achieved with a symmetric GMI element by applying a dc bias ﬁeld produced by a coil or by permanent magnets put around the sample, which shifts the GMI curve along the ﬁeld coordinate. It can be called “GMI asymmetry due to DC bias ﬁeld”. Sometimes this procedure can be troublesome and power consuming. Therefore other mechanisms of asymmetric giant magnetoimpedance (AGMI) are requested for the development of autobiased linear ﬁeld sensors. An example is given in ﬁg. 17 where AGMI proﬁles are shown at various frequencies f of AC driving current for an CoFeSiB ribbon previously annealed at Ha = 240 A/m (3 Oe), ta = 8 hr and Ta = 380 ◦ C (Jang et al., 2000). AGMI was ﬁrst announced by Kitoh et al. (1995) for Co-based amorphous wires. When the wire was twisted and a DC bias current Idc was applied together with an AC driving current Iac a large asymmetry of the GMI curve was obtained. Later on AGMI was reported also for DC current-biased as-quenched amorphous CoFeSiB ribbons (Machado et al., 1999) or as-quenched (Song et al., 2000b) and Joule heated (Gómez-Polo et al., 2001a) amorphous CoFeSiB wires. Field-annealing in air or moisture atmosphere with weak longitudinal or circumferential magnetic ﬁelds also produces AGMI in amorphous CoFeNiBSi ribbons (Kim et al., 1999a) and CoFeSiB wires (Song et al., 2000b). Gunji et al. (1997) and by Makhnovskiy et al. (2000) reported another method of producing asymmetrical GMI characteristics, utilizing an axial AC bias ﬁeld. An interesting asymmetric response was also found in Co-rich melt-extracted wires at frequencies of 10 MHz (Ciureanu et al., 2002). The negative magnetostriction wire was submitted to tensile stresses, and magnetoelastic effects produce a very steep AGMI response with respect to the applied DC magnetic ﬁeld. Thus, a melt-extracted wire under stress could in principle be used as an internally biased sensing element (Ciureanu et al., 2002).

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Fig. 17. Asymmetric GMI proﬁles in a CoFeSiB ribbon annealed with Ha of 240 A/m for various measuring frequencies. The line and symbols (◦) are decreasing and increasing ﬁeld, respectively (courtesy of C.G. Kim).

Reviews on asymmetrical GMI can be found in (Panina, 2002, 2003; Kraus, 2003a). Though the mechanisms of AGMI may be different in different cases and sometimes the origin is even unknown, three different mechanism of AGMI can be now recognized: (a) asymmetry due to DC bias current, (b) asymmetry due to AC bias ﬁeld, (c) asymmetry due to exchange bias. The principal features and the origins of these mechanisms are brieﬂy discussed below. 5.3.1. AGMI due to DC bias current This type of asymmetry was observed on as-quenched wires (Kitoh et al., 1995; Song et al., 2000b) and ribbons (Machado et al., 1999) as well as on various heat-treated wires (Gómez-Polo et al., 2001b; Kitoh et al., 1995). Without bias current (Idc = 0) a symmetric double-peak GMI curve is usually observed. When Idc increases one peak enhances and the other diminishes, depending on Idc orientation (Machado et al., 1999). The positions of the peaks remain practically unchanged. When the direction of the bias current is reversed the asymmetry also reverses. With increasing frequency the asymmetry ﬁrst increases and then decreases showing a maximum at a certain frequency. The asymmetry as a function of Idc , measured at a constant frequency, also shows a maximum. For higher frequencies the maximum shifts towards higher currents, it broadens and its height decreases (Song et al., 2000b). The origin of this type of asymmetry has been attributed to the combination of helical magnetic anisotropy with circumferential DC ﬁeld produced by the bias current

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(Kitoh et al., 1995). Such explanation seems to be adequate even though the helical anisotropy had been intentionally induced neither in the as-quenched ribbons (Machado et al., 1999) nor in the as-quenched and Joule heated wires (Gómez-Polo et al., 2001a; Song et al., 2000b). It was, however, proved that in as quenched amorphous wires a spontaneous helical anisotropy already exists (Blanco et al., 2001). This spontaneous helical component of anisotropy is responsible for the spontaneous observation of Matteucci and inverse Wiedemann effects in as-cast amorphous wires (Velázquez et al., 1991). The same can apply for the as-quenched ribbons. A theoretical analysis, based on the electromagnetic model, was given by Panina and coworkers (Makhnovskiy et al., 2001; Panina et al., 1999). The theory is rather complicated because for helical anisotropy eq. (4.5) (see section 4) leads to two coupled differential equations for the AC ﬁeld components hφ and hz , which must be solved by approximate methods. Panina et al. (1999) used an iteration method, while Makhnovskiy et al. (2001) employed an asymptotic-series-expansion method. These authors showed that the ζzz component of the surface impedance, which is responsible for the GMI effect, becomes asymmetric when a circumferential DC magnetic ﬁeld Hφ is added. The GMI asymmetry is related to the corresponding asymmetry of the axial hysteresis loop. Numerical simulations showed that for Hφ slightly higher than HK sin α, where α is the angle between the circumferential and the easy directions (spiral angle), the hysteresis disappears and a large GMI asymmetry is observed. A straightforward explanation (Kraus, 2003a) of the asymmetry due to DC bias current can be given for the case of strong skin effect, when the wire can be approximated by a thin ﬁlm on its surface. Using the resonance condition (4.31) with ω → 0 the approximate condition for maximum GMI can be obtained. When a DC bias current Idc is passing through the wire it produces the circumferential ﬁeld Hφ = Idc /2πa on the surface. Then the total DC ﬁeld on the surface is H02 + Hφ2 and makes an angle arctan(Hφ /H0 ) with the wire axis. Then the resonance condition for the total DC ﬁeld requires H0 ≈ HK cos α

and Hφ ≈ −HK sin α.

(5.1)

When the spiral angle of helical anisotropy α = 0 the resonance condition can be satisﬁed only for one orientation of H 0 and therefore AGMI appears. If the parameters HK and α of the helical anisotropy are known, eq. (5.1) can be used for the estimation of the optimum bias current Idc . 5.3.2. AGMI due to AC bias ﬁeld This type of asymmetry was ﬁrst reported by Gunji et al. (1997) for a cold drawn and stress-annealed CoFeSiB wire. The wire was fed by a pulse current, which provided both the DC and AC components of a circumferential magnetic ﬁeld. An axial component of the AC ﬁeld hz was produced by an exciting coil wound around the wire and connected in parallel to the wire. Later the phenomenon was investigated on an as-quenched CoFeSiB wire using two independent sources of Iac and Idc (Makhnovskiy et al., 2000). In this case the asymmetry of GMI comes from the combination of a helical magnetization with the axial AC ﬁeld. Using eq. (2.5) one can obtain the voltage along the wire Iac Uac = ZIac = L ζzz (5.2) − ζzφ hz . l

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The ﬁrst term on the right hand side is related to the circumferential magnetization process (mφ − hφ ) and corresponds to the ordinary magnetoimpedance effect. The second term, on the other hand, describes the “cross-magnetization” process (mφ − hz ) which is equivalent to the well known Matteucci effect (Kraus et al., 1994a). As has been shown in section 4.3.3.1, the off-diagonal element ζzφ is nonzero when θ0 = 0. This happens in the case of helical anisotropy or in the case of circumferential anisotropy, when H0 < HK . It was shown (Makhnovskiy et al., 2000) that while ζzz is a symmetric function of H0 , ζzφ is anti-symmetric. Thus only the Matteucci component of Uac is responsible for the asymmetry. It should be mentioned that the left-hand and right-hand helically magnetized domains contribute with opposite signs to the total Matteucci voltage. Therefore some small circumferential DC bias ﬁeld should be used to unbalance the two contributions, and to get the asymmetrical response. The Matteucci component of impedance can be controlled also by the ratio hz / hφ , i.e. by the number of turns of the driving coil. When this ratio is high only the Matteucci effect contributes to the impedance and the asymmetry is the highest. Finally, it is worth mentioning that the “asymmetry due to an AC bias ﬁeld” should, in fact, be excluded from the GMI effects, because its real origin is the Matteucci effect. Moreover the maximum asymmetry, which is required for sensor applications, is obtained when the AC current passes only through the exciting coil (Makhnovskiy et al., 2000). This conﬁguration should better be called the “Matteucci element”. 5.3.3. AGMI due to exchange bias The third, and yet not well understood, type of AGMI was ﬁrst announced by Kim et al. (1999b). They observed a large asymmetry in CoFeNiBSi ribbons, which were ﬁeldannealed in air at 380 ◦ C with a weak magnetic ﬁeld (4–240 A/m) applied along the ribbon (see ﬁg. 17). Because the samples annealed in vacuum did not show the asymmetry, the phenomenon was attributed to oxidation assisted surface crystallization. This type of heat treatment is known to produce asymmetric hysteresis loops in amorphous ribbons due to the exchange interaction of amorphous bulk with the magnetically harder crystalline phase on the surface of the ribbon (Shin et al., 1992). When the oxidation takes place below, but close to, the crystallization temperature of the amorphous alloy, B and Si preferentially diffuse to the surface to form the oxides. The crystallization temperature of the underlayer, which becomes depleted in B and Si, decreases and the surface crystallization starts. In the presence of a weak magnetic ﬁeld uniaxial magnetic anisotropies are induced in both the crystalline layer and the amorphous bulk. Because the crystalline phase is magnetically much harder it remains magnetically ordered, along the direction of the ﬁeld applied during the annealing, in relatively large range of magnetic ﬁelds. The exchange interaction between the amorphous bulk and the surface crystallites produces an effective unidirectional surface anisotropy, which is believed to be responsible for AGMI (Kim et al., 1999a). The exchange biased AGMI does not require any additional bias (current or ﬁeld). The asymmetry in ribbons annealed in a longitudinal ﬁeld Ha depends on its magnitude and direction with respect to the measuring ﬁeld H0 . For Ha = 0 a nearly symmetric double-peak GMI curve is observed. With increasing Ha , the peak located in the same direction of longitudinal ﬁeld H0 increases, while the opposite one decreases. For Ha > 40 A/m a very large asymmetry is observed at low frequencies (about 100 kHz) with a step-like change at H0 ≈ 0. That is why this phenomenon was called the “GMI

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valve” (Kim et al., 1999b). The asymmetry decreases at higher frequencies, where the rotational contribution to GMI becomes dominant. Similar AGMI was reported for amorphous CoFeSiB wires ﬁeld-annealed in a moist atmosphere with a weak circumferential ﬁeld produced by DC current though the wire (Song et al., 2000a). A profound asymmetry was observed for currents above 4.5 mA. Again the asymmetry was large for 100 kHz and decreased with increasing frequency, which may indicate that the domain wall motion contribution to the circumferential susceptibility is responsible for the asymmetry. An attempt to give a theoretical explanation of the exchange-biased asymmetry was made by Kim et al. (1999a, 2000, 2001a) using a quasistatic model for the rotational transverse susceptibility of a single domain ﬁlm with an in-plane anisotropy. The unidirectional exchange anisotropy was accounted for by introducing an effective bias ﬁeld H b . Chen et al. (2000a) criticized this model. Besides the computational errors they claimed that it could not explain the asymmetries of the hysteresis loops and of the GMI observed experimentally with physically reasonable magnitudes and orientations of H b and the anisotropy ﬁeld. It is evident that H b could be formally included into H 0 and the formulas of section 4.3.3.1. could be used. Then the exchange bias would be equivalent to the DC ﬁeld bias, which, however, cannot explain the observed behaviour. The electromagnetic model, taking into account the exchange boundary condition at the interface of the soft magnetic bulk and the hard magnetic surface layer (Kraus, 2003b), can quatitatively explain the experimentally observed exchange biased AGMI. 5.4. High frequency Although the GMI has been studied and even used as a characterization tool at low and intermediate frequencies, the great potential of GMI as a characterization tool of soft ferromagnetic metals is manifested at higher frequencies, following the enormous advances in theoretical models. A clear correspondence has been established by Menard et al. (1998) between GMI and magnetization curves which has been used to investigate and model the domain structure in glass-covered amorphous wires (Loﬂand et al., 1999, 2002; Óvari et al., 2000). From the position of the peaks on the GMI curves, which correspond to the FMR resonance ﬁeld (as discussed in section 4.3.3.1), it is possible to estimate parameters such as the saturation magnetization Ms , or the anisotropy ﬁeld, HK , in good agreement with the values directly measured using conventional techniques (Ciureanu et al., 1998). From the resonance condition (4.31), with θ0 = 0 and H0 , HK Ms , a linear dependence of ω2 vs. H0 with the proportionality constant equal to γ 2 Ms is obtained: (ω/γ )2 = Ms (H0 + HK ).

(5.3)

Knowing the value of the γ -factor, the saturation magnetization and the anisotropy ﬁeld can be determined from the f 2 vs. H0 plot. In ﬁg. 18(a), the square of scattering parameter, |S11 |2 , (proportional to the absorbed power that, in turn, is proportional to the real part of impedance) is plotted as a function of frequency for a range of applied ﬁelds, H0 , for the Co72.5 Si12.5B15 glass-coated microwire. The resonance frequency, fr = ω/2π , is determined from the maximum of absorbed power. Figure 18(b) shows the

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(a)

(b)

(c) Fig. 18. The square scattering parameter, |S11 |2 , as a function of frequency for a range of applied ﬁelds, H0 , for the Co72.5 Si12.5 B15 glass-coated microwire (a), and the square resonance frequency, (fr )2 , as a function of applied ﬁeld, H0 , (b) for selected alloys of the (Co100−x Fex )72.5 Si12.5 B15 microwire series. M indicates the percentage of Fe (c). Courtesy of H. García-Miquel.

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square of resonance frequency as a function of applied ﬁeld. According to eq. (5.3), the anisotropy ﬁeld, HK , can be obtained by extrapolating the linear dependence to fr = 0, and the saturation magnetization from the slope of the dependence (García-Miquel, 1999; García-Miquel et al., 2001; Yoshinaga et al., 1999). Similar results were reported on amorphous microwires, where the impedance was investigated as a function of frequency from 1 MHz to 1.8 GHz using a coaxial microwave cavity (da Silva et al., 2002a). In this work, the effects of Joule heating (from 1 up to 50 mA for 10 min) on induced anisotropy and crystallization were investigated (da Silva et al. 2002a, 2002b). A similar analysis was performed also by Araújo et al. in Co70.4Fe4.6 Si15 B10 amorphous ribbons, where the GMI was measured from 2 to 12 GHz (Araújo et al., 2001). Because the minimum skin depth is determined by the damping parameter α (see eq. (4.32)) the peak value of GMI is closely related to the ferromagnetic relaxation. In the GHz frequency range, where the exchange-conductivity effect is already negligible and the GMI ratio is only little affected by anisotropy ﬂuctuations or magnetization non-uniformity the peak value of GMI can be used for investigation of the damping parameter (Britel et al., 1999; Menard et al., 1999). The studies made in Yelon’s group lead to the observation of both ferromagnetic resonance and antiresonance in an amorphous NiCo-rich soft-magnetic wire using the GMI technique from which it is possible to ﬁt the ﬁeld dependence of the resonance and antiresonance frequencies, and to obtain the gyromagnetic ratio, anisotropy ﬁeld and the saturation magnetization of the sample in good agreement with parameters obtained through conventional measurements (Britel et al., 2000). Similar results were also reported for amorphous microwires (Bhagat et al., 2002). In some particular cases, especially when intrinsic anisotropy is strong enough, the natural ferromagnetic resonance can be observed (zero-ﬁeld microwave absorption, or H0 = 0 in eq. (5.3)). This is the case, for example, of amorphous glass-coated microwires where even though the magnetocrystalline anisotropy is absent, they exhibit a strong magnetoelastic anisotropy (Baranov et al., 1989). The same principle has been considered to interpret the absorption peaks in arrays of nanowires electrodeposited to ﬁll nanoporous membranes (Encinas et al., 2001, 2002). In such case, ferromagnetic resonance is shown to be tunable with magnetic ﬁeld, anisotropy properties, or even just adjusting the dipolar interaction among nanowires (Ramos et al., 2003). 5.5. A short summary Such diversity of experiments clearly demonstrates that the giant magnetoimpedance indeed provides a powerful tool for characterizing the intrinsic properties of magnetic metals. Owing to its extreme sensitivity and easiness of use, GMI can be compared, including several advantages, with traditional techniques such as transverse susceptibility and ferromagnetic resonance. Many novel materials and structures are currently being investigated through giant magnetoimpedance. Such materials, allied with a broad variety of possible thermal treatments, open enormous perspectives in the future of research in this emerging area. Table 1 displays a brief summary of some of the investigations previously mentioned, indicating the studied material, the main properties that were studied using the GMI effect, and the respective reference.

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TABLE 1 Some selected works that use GMI as a research tool to investigate other materials characteristics Material

Property

Reference

Comment

FeCoSiB wire

Scalar circular permeability

Beach and Berkowitz (1994b)

Fit of impedance vs frequency curve

Nanocrystalline FeSiBCuNb wire

Magnetic penetration depth and circular permeability

Knobel et al. (1995a)

Real and imaginary components of impedance as functions of ﬁeld

CoFeSiB wire

Rotational and domain wall contribution to μφ

Valenzuela et al. (1995)

Equivalent circuits approach

CoFeSiB wire

Saturation magnetostriction constant

Knobel et al. (1996)

Negative magnetostrictive samples

CoFeSiB wire

Domain structure

Menard et al. (1998)

Glass-covered and after glass removal

Polycrystalline NiFeMo wires, amorphous NiCo- and CoFe-rich wires

Saturation magnetization (Ms )

Ciureanu et al. (1998)

f02 –H0 plot at resonance frequency and ﬁeld (straight line)

Co-based ribbon

Easy axes distribution function

Pirota et al. (1999a)

Angular dependence of GMI

FeCoSiB wire

Landau–Lifschitz damping parameter

Menard et al. (1999)

Peak value of the wire impedance at resonance

NiCo-rich wire

Anisotropy ﬁeld, saturation magnetization, gyromagnetic ratio

Britel et al. (2000)

Ferromagnetic resonance and antiresonance

(110)[001]FeSi3%

Magnetization dynamics (reversible and irreversible parts)

Carara et al. (2000)

Low frequency range experiment (100 Hz– 100 kHz)

FeNbB ribbons

Anisotropy ﬁeld as a function of Fe content

Ryu et al. (2000)

HK decreases with Fe content

CoP microtubes electrodeposited on Cu wires

Domain structure

García et al. (2001a)

Combined with magnetic force microscopy

Fe and Co base microwires

Magnetic anisotropy

García-Miquel et al. (2001)

Ferromagnetic resonance

FeCoSiB amorphous ribbons

Anisotropy ﬁeld, saturation magnetization

Araújo et al. (2001)

Ferromagnetic resonance

FeSiBCuNb nanocrystalline ribbons

Magnetic coupling among nanoparticles

Zeng et al. (2002)

Temperature dependence of GMI

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6. Applications of GMI In addition to fundamental aspects related to micromagnetics and to magnetization dynamics, the main interest of giant magnetoimpedance effect certainly lies in the large number of possibilities that it offers to technical researchers to employ this phenomenon as sensing principle in novel sensor devices (Robbes et al., 2001; Steindl et al., 2000). As reviewed in the previous sections, under particular suitable conditions (ultrasoft magnetic character, adequate magnetic anisotropy, and adequate geometry), the GMI material undergoes modiﬁcations in its impedance in the presence of external agents such as static magnetic ﬁeld and mechanical stress. Consequently, this variation of impedance will be the measurement principle to sense the correlated changes in magnetic ﬁeld strength, stress or torsion (Hauser et al., 2001). Topics of application for sensors based on GMI are very broad (Mohri et al., 2002b, 2002a): automobile industry, industrial measurements and automation, computers & information technology, biomagnetics & health, environmental sensors, power electronics and energy, security and safety, scientiﬁc and academic measurements. In the following we discuss different types of sensor applications: magnetic ﬁeld sensors (wire and thin ﬁlm technologies); current, position and rotation sensors (applications that derive from ﬁeld sensing); stress sensors; microwave applications; and ﬁnally, the particular characteristics necessary for the materials to be employed in GMI applications. 6.1. Magnetic ﬁeld sensor devices The largest developments on GMI based sensors have been carried out in Japan around the group of Prof. Kaneo Mohri (Mohri, 1994; Mohri et al., 1997, 2002b, 2002a). Most technological applications are related with microsensor applications based on the static ﬁeld dependence of impedance. 6.1.1. Wire technology Magnetic sensors with small dimensions are demanded for a wide spectrum of applications in establishing advanced intelligent measurements and control systems. Most of the advances in GMI based microsensors have been reached using short wire (around 1– 2 mm in length) as sensing element. Low-ﬁeld sensitivity (by change of voltage or impedance) can reach up to 1.25%/Am−1 (100%/Oe) at frequencies of several MHz and ﬁelds of the order of dozens of Am−1 (less than one Oe) (Mohri et al., 1997). Resolution is down to 10 mAm−1 (10−6 Oe) at full scale of 160 Am−1 (2 Oe) with maximum sensitivity of around 1 nT. In fact, 1D and 3D magnetic ﬁeld sensors are already commercially available by Aichi Micro Intelligent Corporation (2002) having full scale of 160 Am−1 (2 Oe) and 80 k Am−1 (1 kOe) (in combination with Hall sensor) (Aichi, 2002; Uchiyama et al., 1997). Concerning technical electronic arrangements, micro magnetic-ﬁeld sensors have been developed with a GMI head installed in a Colpits oscillation circuit or in a multivibrator circuit followed by a demodulator detector, differential ampliﬁer and negative-feedback loop circuit. They have sensitivities of the order of 10 mAm−1 , and cut-off frequency of about 1 MHz (Mohri et al., 2002a).

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TABLE 2 Rough comparison of several types of magnetic sensors (adapted from (Mohri et al., 2002a)) Sensor principle

Head length (m)

Resolution/full scale (Am−1 )

Hall 10 ∼ 100 × 10−6 40/ ± 8 × 104 Magnetoresistance (MR) 10 ∼ 100 × 10−6 8 / ± 8 × 103 Giant Magnetoresistance (GMR) 10 ∼ 100 × 10−6 0.8/ ± 1.6 × 103 Fluxgate 10 ∼ 20 × 10−3 8 × 10−5 / ± 2.4 × 102 SQUID 10 ∼ 20 × 10−3 50 × 10−12 / ± 1 × 10−6 8 × 10−5 / ± 2.4 × 102 Magnetoimpedance 1 ∼ 2 × 10−3 Stress impedance 1 ∼ 2 × 10−3 0.1 Gal/30 Gal

Response Power speed (Hz) consumption (W) 106 106 106 5 × 103 5 × 103 106 104

10−2 10−2 10−2 1 − 5 × 10−3 5 × 10−3

A CMOS multivibrator type circuit, with very low power consumption, has been developed where a sharp pulse current ﬂows in the power source line at the CMOS switching. The MI sensor has a pair of 2 mm long amorphous wires with the pulse bias circuit in the balanced circuit. A pulse current with rising time of about 10 ns is generated through a RC differential circuit, where the induced pulse voltage at the amorphous wire is rectiﬁed through a peak voltage holder R and C for generation of the sensor output voltage (Panina and Mohri, 2000). In table 2 (after Mohri et al., 2002a), some important characteristics of sensors based on different principles are summarized. Advantages of GMI based sensors (two last lines of the table) appear when comparing the resolution. Also, it is worth noting that the head length of GMI sensors can be up to 20 times smaller than the corresponding ﬂuxgate sensor head length, a fact that strongly increases the resolution for detection of localized magnetic-pole ﬁelds (Mohri et al., 2002b; Shen and Mohri, 2000). The other parameters of the sensors are very equivalent. For example, thin ﬁlm magnetoresistance sensors can detect ﬁelds in the range 10−3 to 104 Am−1 (of the order of 10−5 to 102 Oe), and they can be produced by means of photolithography, thus allowing smaller size compared to typical GMI sensors. However, when one compares MR sensors with GMI sensors one notices that in the later case, the relative change ratio is only of 2% typically for ﬁelds of 2400 Am−1 . In fact, nominal relative ratios can be up to 50%, but the saturation ﬁelds are quite large and sensitivity does not exceed 1%/Oe, with some hysteresis and temperature problems. 6.1.2. Thin ﬁlm technology The widest variety of sensors has been developed using a wire shaped sensing element likely because of its earlier developing as GMI materials. Nevertheless, nowadays more efforts are focusing on thin ﬁlm technology in order to integrate this family of sensors into semiconductor or high density magnetic recording microtechnologies. Two types of materials can be considered: single layer ﬁlms, and sandwich-like structures (Kanno et al., 1997). In both cases, GMI materials are mostly fabricated by sputtering techniques and they differ from wires and ribbons by two main characteristics: the smaller thickness (of the order of few microns), and the somehow harder magnetic character, both affecting the GMI response. In single layer ﬁlms, being electrically uniform, MI originates purely from the change of skin depth with transverse susceptibility (skin-effect). Non-magnetostrictive amorphous

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Fig. 19. Schematic view of a sandwich-like GMI based sensor. Driving current ﬂows along the Cu layer, and the ﬂux lines in the magnetic layer are indicated.

single layer ﬁlms exhibit lower GMI ratios (up to 10%) than soft wires or ribbons owing to their smaller transverse dimension and larger anisotropy induced by fabrication. The working frequency is of the order of GHz due to their relative small thickness (Uchiyama et al., 1995). A 3D position sensor has been designed on small magnetic marker using CoNbZr 3 μm thick ﬁlm (Yamaguchi et al., 2000). Alternatively, an integrated thin ﬁlm MI sensor head has been developed using a plating process (Takayama et al., 1999). A Ni-Fe ﬁlm core is wound by helically shaped Cu coils using microplating technology. Sandwich-like multilayer ﬁlms, F/M/F, consist of two soft ferromagnetic ﬁlms (nonmagnetostrictive CoFeSiB amorphous or permalloy) and a conductive inner lead (Cu or Ag). Figure 19 shows schematically the typical conﬁguration of a sandwich-like sensor based on GMI. Although full technological application of these sandwiches is still at an early stage the expectations are quite large. GMI can change signiﬁcantly: e.g. 400% GMI for 1 μm thick Cu lead at 10–100 MHz (maximum ﬁeld: 1600 A/m, or 20 Oe), or 20% GMI for 0.1 μm at 400 MHz (Panina and Mohri, 2000). An improved sensor has been proposed by Morikawa et al. (1996a) where an insulating SiO2 layer is introduced between the Cu conductor and the layers so preventing the driving current to penetrate into the soft magnetic layer. A large GMI ratio is obtained: at 10 MHz the skin depth is calculated to be 4 μm for a soft layer thickness of 2 μm. 6.2. Applications derived form ﬁeld sensing: current and position sensors, magnetic signatures Quite a large number of sensors for particular applications have been designed and developed using the GMI working principle. Current sensors are based on evaluating the magnetic ﬁeld created by a current to be measured. A family of this type of sensors was proposed by Valenzuela et al. (1997) where the DC ﬁeld created by the current circulating along a plate is detected through the changes in impedance on an amorphous wire wound around the plate. The device has a small size (2 × 2 × 1 cm3 ), and linearization between current and GMI voltage can be achieved by geometrical arrangements. Linearity cover

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Fig. 20. Schematic view of a DC current sensor based on detecting the magnetic ﬁeld created by the current (a). Linearity of the signal can be suitably modiﬁed (up to 12 A in this case) (b) (Valenzuela et al., 1997).

low (tenths of A) and high (tens of A) current ranges. A schematic view of this type of DC current sensor is shown in ﬁg. 20(a), while ﬁg. 20(b) indicates the response of a particular prototype where linearity reaches 12A. Various families of position sensors have been proposed for different purposes (Mohri et al., 1997; Vázquez et al., 1997). In most cases detection of the position is performed by sensing the magnetic ﬁeld created by a magnet ﬁxed to a mobile object the position of which is to be detected. One particular application is to record the magnetic signature of an automobile either at parking places (Valenzuela et al., 1996) or moving vehicles (Uchiyama et al., 2000). In this particular regard, interesting achievements have been reached in Japan to control both trafﬁc and its density (Honkura, 2002). In a different case, the control of position of pneumatic piston is done by attaching a small magnet to the moving piston and locating two GMI based sensors at the ends of the device (Vázquez et al., 1999a). Particular position sensors have been also developed to control the circular position or the rotation velocity in rotary encoders (Mohri et al., 1997). Another type of sensor is used to localize the position of cancer tumours (e.g. in the brain) (Atkinson et al., 2000). A GMI sensing head is used to search clusters of magnetite particles ﬁxed to tumour, and the resolution of the sensor is 10 μG (see ﬁg. 21). The control of the magnetic signature of ships has important consequences in defense applications. An array of discrete GMI wires has been developed for surveillance of the magnetic signature of ships (Tejedor et al., 2000). The length of GMI wires (up to a few meters) is small compared to dimensions to be covered, and the output signals of the array are combined by hardware and software to give an integrated ﬁeld measurement. 6.2.1. Application of GMI in magnetic non-destructive analysis Non-destructive analyses (NDA) are mostly required when a system (or part of it) needs to be inspected for the presence of ﬂaws, such as metal-loss due to corrosion in pipelines, for example. NDA allows the system to be inspected even when it is being used. It is being used in building inspection, geology, stainless steel industry, and pipeline networks, just to list few applications. In health care NDA is also used and it is commonly named noninvasive analysis. NDA use techniques based on several physical principles: the response of the tested material to gamma- or X-rays, ultrasound, optics, eddy currents, thermal,

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Fig. 21. Prototype of magnetic ﬁeld sensor based on GMI (a) developed by Mohri et al., which can be employed in neurosurgery to detect the presence of small magnetic particles for brain tumor positioning (b). Courtesy of K. Mohri.

chemical, magnetic resonance, and magnetic ﬂux leakage. A very important application of NDA is the inspection of pipeline networks in the oil industry: corrosion yields metalloss in the pipeline wall, for example. Inspection in pipeline networks is hard to be made because they have hundreds of kilometers and most of the time it is difﬁcult to reach them. Magnetism, on the other hand, is being used in detecting metal-loss in pipelines because they are made of stain-less steel that can be magnetized by applying a uniform magneticﬁeld parallel to the pipeline wall. Near a ﬂaw the magnetic ﬁeld leaks-out from the material in both inside and outside the pipeline. The stray ﬁeld can be measured using inductive coils, and Hall and SQUID sensors, for instance. Each of these sensors presents positive points and drawbacks that need to be balanced for a given application. Because of the high sensitivity of the GMI at low magnetic ﬁelds, magnetic sensors based on the GMI are being developed for this application (Machado, 2002). Preliminary results showed that the GMI sensors are at least one order of magnitude more sensitive than the Hall sensors. They are also better than the inductive ones because the inductive signal depends on the velocity of the moving pick-up coil with respect to the defect. SQUID sensors are also difﬁcult to be used because they are operating at cryogenic temperatures. Other advantages of the use of GMI sensors are that the corresponding devices are easy to build and the strength of the magnetic ﬁeld required to yield the stray-ﬁeld is smaller than the ones needed for the other techniques. The inspection is made by assembling the magnetic sensors in a PIG (“pipeline inspection gizmo”) that travels inside the pipeline pushed by the oil. A set of permanent magnets establishes the magnetic ﬁeld inside the pipeline wall and a computer based data acquisition system records the stray magnetic ﬁelds through out the line. At the end of the run the PIG is collected and the data analyzed. Even though is not difﬁcult to detect the existence of ﬂaws through the measurement of the stray magnetic ﬁeld, it is not so easy to invert the problem, e.g., to ﬁnd the shape of the ﬂaw based on the magnetic ﬁeld proﬁle. However this problem does not depend on the kind of magnetic sensor used and several research groups are working it out.

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The same principle can be applied for the control of magnetic imperfections of thin steel sheets (detection of pin holes). In the quality control process, a steel sheet runs typically at a velocity of 10 m/s (Mohri et al., 1997). 6.3. Stress-impedance applications The stress dependence of magnetoimpedance can be used for sensing stresses applied to a soft magnetic material. This principle has been successfully employed on a stress annealed CoFeSiB amorphous ribbon achieving stress sensitivity of 214 MPa/Oe at 1 MHz frequency, and for measuring small loads on membranes with annealed glass-coated lowmagnetostrictive microwire (Cobeño et al., 2002; Shen et al., 1997). In the case of microwires, the DC electrical resistance is quite large so that GMI results in large variations in voltage induced by mechanical loading. Figure 22 shows a schematic view of the sensor device using Co-base low-magnetostrictive microwire. The change of GMI (at 10 MHz frequency) with loading and the modiﬁcation of pick up voltage (about 5 V with load of 3 g) are also shown (Cobeño et al., 2001). It should be noted that improved stress sensitivity is achieved for the amplitude of applied DC ﬁeld (denoted by the arrow in ﬁg. 22(b)). Strain-gauges using a stress-impedance element can achieve gauge factors of around 4000, which can be compared with the gauge factors of 2 and 140 for resistive wires and semiconductors, respectively (Mohri et al., 2002b).

Fig. 22. Schematic view (a) of a magnetoelastic sensor based on the stress dependence of GMI effect (b) in a Co-base glass-coated microwire. Final voltage response as a function of loading (c), for a static ﬁeld of 155 A/m (Cobeño et al., 2001). Courtesy of A. Zhukov.

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A magnetoelastic signature pen based on stress impedance effect has been proposed recently to identify and authenticate signatures (Vázquez et al., 2003). The magnetoelastic signature voltage spectrum corresponds to the impedance changes caused by stresses acting on the sensing wire element. A sensitive stress sensor based on CMOS IC circuit was developed by Kusumoto et al. (1999) with in gage factor of 4000. Chiriac et al. (1999a) developed a ﬁnger-tip blood vessel pulsation sensor for diagnosis of blood circulation and health conditions. Also, as reported by Atkinson et al. (2000), mechano-encephalogram based on stress impedance detects small skin deformation at a forehead centre reﬂecting brain activity. Remote interrogation of the stress state of GMI material can be performed by a emisor/receptor antenna system. Various materials with microwire or thin ﬁlm shape have been successfully employed to evaluate the material undergoing stress for frequencies in the range 10 MHz to 1 GHz. 6.4. RF and microwave applications An interesting application of GMI for novel sensor elements (transponders) was proposed by Hausleitner et al. (2002) to develop a cordless, batteryless wheel mouse. In their work they developed sensor elements from surface acoustic wave devices for wireless identiﬁcation systems (ID tags), by using the magnetic-ﬁeld variation caused by the wheel rotation and key-click functionality of a computer mouse. Such sensors do not need any power supply and may function wirelessly. Interdigital transducers are used as loadable reﬂectors, while a ﬁxed reﬂector is used as reference to compensate cross-sensitivity for temperature and mechanical stress, for example. The load of the reﬂectors inﬂuences amplitude and phase of the reﬂected wave, and its impedance is changed through the GMI effect in a nonmagnetostrictive amorphous wire (30 μm diameter). A mouse key click or the rotation of the mouse wheel inﬂuences in a controlled way the magnetic ﬁeld, which, in turn, varies the impedance of the GMI wire element. Consequently, the load of the reﬂector inﬂuences the reﬂected acoustic wave and the response signal of the radio sensor. In this way, it was possible to develop a SAW transponder to operate as reliable passive cordless interrogable sensor, without the need of a battery and almost unlimited lifetime (Hauser et al., 2002). In the speciﬁc application presented, the authors found that the sensing of all PC-mouse functionalities are covered using only one SAW transponder with four impedance loaded IDTs (Hausleitner et al., 2002). The combination of such device with a GMI wire as sensing element results in an excellent quick-response microsized magnetic ﬁeld sensor, offering new perspectives in easy handling man-machine interface units. 6.5. GMI sensor heads: magnetic requirements The GMI elements used as head in sensor devices usually require particular magnetic conditions to optimize GMI response. In particular, thermal treatments to relax the amorphous state or to induce magnetic anisotropies (when performed at the presence of magnetic ﬁeld or stress) are helpful. The main requirements are summarized in the following. Most sensors developed by the groups in Japan utilize wires with vanishing magnetostriction (alloy composition CoFeSiB), fabricated by the in-rotating-water quenching

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technique. The typical wire diameter, around 120 μm, is somehow inconvenient for applications and otherwise its magnetic behaviour should be stabilized. Normally, wires are ﬁrst cold drawn in different steps to reduce their diameter to about 30 μm. As this treatment deteriorates strongly the soft magnetic properties, additional stress thermal treatment is performed for a softening and to induce circumferential anisotropy (Kawashima et al., 1999). Pieces as small as 1 to 2 mm in length are then suitable as sensing heads. Asymmetric GMI response is important in some cases to facilitate the measuring procedure. As discussed in section 5.3, asymmetric GMI can be achieved by inducing helical anisotropy by annealing at the presence of torsion or just by applying torsional stress. In both cases an additional DC current must be superimposed to the driving AC current (Morikawa et al., 1996b). Asymmetric GMI produced by weak ﬁeld-annealing in air (Kim et al., 2002a), which does not require the current biasing may be quite promising for such applications. In the case of using thin ﬁlms or sandwiches as sensor heads uniaxial magnetic anisotropies are required. This is achieved by annealing in the presence of a magnetic ﬁeld along the requested easy axis. When stress impedance is used as measuring principle, a thermal pre-treatment is typically performed with the presence of tensile stress. In this way a homogeneous transverse anisotropy is induced which improves the GMI response (Tejedor et al., 1998c). Finally, there are some GMI characteristics that must be avoided as much as possible. The ﬁrst one is the disaccommodation effect, when the magnetic response is shown to be dependent on time (Knobel et al., 1997a). This is an undesired phenomenon for sensing elements, and can be strongly reduced by means of speciﬁc thermal treatments (Pirota et al., 1999b). On the other hand, it has been shown by (Vázquez et al., 1999b), that GMI displays hysteretic behaviour especially in the low-ﬁeld range. Again this is unsuitable for some sensor applications. This hysteresis is related to irreversible domain wall movements due to the applied DC ﬁeld or stress and can also be notable reduced by suitable heat treatments. 7. Present trends and ﬁnal remarks An updated review on giant magnetoimpedance was presented. Although the GMI phenomenon has been known for more than 65 years, only recently its enormous possibilities in both basic and applied research were recognized. Owing to the rapid development of theories and applications, both supported by systematic experimental investigations, some sensors based on this phenomenon are already available in the market, and it is possible to predict a successful future to the appropriate use of this effect. Once the theories become more and more accurate, it will be possible to further use the GMI effect as an additional tool for the basic study soft magnetic materials, including the distribution of quenched-in and induced anisotropies, magnetoelastic behaviour, ferromagnetic resonance and anti-resonance, and other intrinsic and extrinsic magnetic properties. Furthermore, the progress on models and experimental data will certainly lead to improved materials from the applications viewpoint, mainly extremely-sensitive magnetic ﬁeld sensors. However, there are still many unsolved problems and questions that remain to be clariﬁed during this challenging road to understand and develop better application oriented soft magnetic materials.

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AUTHOR INDEX

aan de Stegge, J., see Johnson, M.T. 143 aan de Stegge, J., see Jungblut, R. 149, 154, 157, 176 Abad, H., see Jonker, B.T. 358 Abad, H., see Warnock, J. 294, 358 Abanov, Ar. 419, 427, 428 Abell, S., see Borisenko, S.V. 441 Abernathy, C.R., see Theodoropoulou, N. 276 Aboaf, J.A., see Klokholm, E. 140 Abraham, A. 430, 432 Abrahams, E., see Shastry, B.S. 435, 436 Abrahams, E., see Varma, C.M. 418 Abramishvily, V.G. 351, 352, 357 Abramof, E. 326, 327, 344 Abrikosov, A.A. 418 Abrikosov, I.A., see Ponomareva, A.V. 224 Abrikosov, I.A., see van Schilfgaarde, M. 233, 269 Abu-Shiekah, I.M. 433, 435, 452, 461, 462 Abu-Shiekah, I.M., see Bakharev, O.N. 467, 468 Abu-Shiekah, I.M., see Brom, H.B. 479 Abu-Shiekah, I.M., see Teitel’baum, G.B. 434, 435, 451, 452, 467 Acharya, B.R. 231 Acher, O. 514, 535 Acher, O., see Malliavin, M.J. 515 Acher, O., see Reynet, O. 535 Adachi, H., see Sakakima, H. 35, 46, 48, 49 Adachi, K. 168 Adachi, N. 349, 351, 355 Adachi, S. 298 Adachi, S., see Mino, H. 298, 303 Adachi, S., see Takeyama, S. 300, 303 Adachi, T., see Koike, Y. 448 Adachi, T., see Watanabe, I. 450 Adamczewska, J., see Nadolny, A.J. 321 Adams, J.B., see Liu, C.L. 267 Adelerhof, D.J., see Lenssen, K.-M.H. 22, 23, 44, 166 Adenot, A.-L., see Acher, O. 535 Adenot, A.-L., see Reynet, O. 535

Aeppli, G. 447 Aeppli, G., see Dai, P. 470, 482 Aeppli, G., see Isaacs, E.D. 462 Aeppli, G., see Kee, H.Y. 428 Aeppli, G., see Lake, B. 413, 446, 448, 449, 480 Aeppli, G., see Littlewood, P.B. 418, 423 Aeppli, G., see Mason, T.E. 448 Aeppli, G., see Mook, H.A. 389, 471 Afﬂeck, I., see White, S.R. 405 Aggarwal, R.L., see Shih, O.W. 351, 352 Agosta, C.C., see Vu, T.Q. 353 Agrestini, S., see Bianconi, A. 408 Agrestini, S., see Saini, N.L. 454 Aguekian, V.F. 300 Aguiar, F.M., see Araújo, A.E.P. 548, 549 Aguilera-Granja, F. 215, 219, 254–258 Ahn, J.Y. 353 Ahn, S.J. 504, 540 Ahn, S.J., see Kim, C.G. 557 Ahuja, R., see Belonoshko, A.B. 209 Aigle, M. 327 Aigle, M., see Springholz, G. 328, 342 Aitchison, P.R., see Chapman, J.N. 129, 130 Aitchison, P.R., see Marrows, C.H. 127 Aiura, Y., see Chuang, Y.-D. 482 Ajima, T., see Miyazaki, T. 7 Akhiezer, A.I. 515 Akimoto, R. 307 Akimov, A.V. 299 Akimov, A.V., see Shcherbakov, A.V. 306 Akimov, B.A. 312 Akinaga, H. 296 Akoh, H. 219, 224 Akoshima, M., see Koike, Y. 448 Akoshima, M., see Watanabe, I. 450 Aksay, I.A., see Fong, H.F. 389, 470, 471, 482 Aksay, I.A., see Sidis, Y. 472 Aksyanov, I.G., see Kusrayev, Yu.G. 298 Akyel, C., see Britel, M.R. 526, 548 Akyel, C., see Ciureanu, P. 514, 515, 546, 549 565

566 Al-Jibouri, A., see Shen, F. 38, 49 Al-Jumaily, G., see Yang, Z. 162, 178 Alawadhi, H. 311, 351–353, 356 Alawadhi, H., see Seong, M.J. 354 Albrecht, M., see Moser, A. 21 Alden, M. 218 Alemany, M.M.G., see Calleja, M. 266 Alemany, M.M.G., see Diéguez, O. 215 Alemany, M.M.G., see Rey, C. 261 Aleszkiewicz, M., see Cywi´nski, G. 310 Aleszkiewicz, M., see Kudelski, A. 310 Alexandrov, A.S. 409 Alexopoulos, P.S., see Ounadjela, K. 140 Ali, M., see Barman, A. 131 Ali, S., see Jackson, S. 296 Allancon, Ch., see Poirot, N.J. 456 Allen, E., see Lee, W.Y. 37, 51, 71 Allen, J.W., see Zaanen, J. 464 Allen, P.B. 454 Alloul, H. 437, 473, 474, 478 Alloul, H., see Bobroff, J. 477, 478 Alloul, H., see Dooglav, A.V. 473 Alloul, H., see Mahajan, A.V. 478 Alloul, H., see Mendels, P. 478 Almeida, B., see Veloso, A. 35, 48, 49 Almeida, N., see Freitas, P.P. 22 Alnot, M., see Andrieu, S. 220 Alonso, J.A., see Aguilera-Granja, F. 215, 254–258 Alonso, J.A., see Bouarab, S. 254–256, 258, 266, 267 Alonso, J.A., see Montejano-Carrizales, J.M. 255 Alouani, M. 209 Alouani, M., see Galanakis, I. 237 Alouani, M., see Ostanin, S. 248, 249 Altkemper, U., see Kachel, T. 236 Altman, A., see Lu, Y. 5 Altman, E. 419 Altman, M.S., see Man, K.L. 237 Altoé, M.V.P., see Mendes, K.C. 541 Alvarez-Fregoso, O. 361 Amalou, F. 223, 534 Amalou, F., see Talanana, M. 227 Amano, E., see Carrasco, E. 537 Ambai, M., see Kobayashi, Y. 477 Ambrosch, K.E., see Krost, A. 324 Ambrose, T. 55, 150, 164, 176 Ambrose, T., see Gökemeijer, N.J. 152 Ambrose, T., see Wu, X.W. 41 Amemiya, K., see Yonamoto, Y. 206, 249 Ament, W.S. 508, 529 Amin, N., see Mao, S. 60, 161 Ammerahl, U., see Curro, N.J. 434, 451, 467

AUTHOR INDEX Ammerahl, U., see Suh, B.J. 434, 451 An, G., see van Bemmel, H.J.M. 406 Andersen, N.H., see Niemöller, T. 454 Andersen, N.H., see Sidis, Y. 419, 470 Andersen, N.H., see Vigliante, A. 463 Andersen, N.H., see Zimmermann, M.v. 454 Andersen, O.K. 212, 269 Anderson, A., see Perjeru, F. 217, 220, 275 Anderson, G. 151, 159, 161, 170 Anderson, G., see Pakala, M. 159 Anderson, G.W. 74, 77, 151, 158, 161, 162, 166 Anderson, G.W., see Huai, Y. 33, 34, 42 Anderson, J., see Wang, D. 27 Anderson, J.A., see Beech, R.S. 22 Anderson, J.R. 325 Anderson, J.R., see Górska, M. 312, 315, 325, 329 Anderson, P.E., see Seigler, M.A. 57 Anderson, P.W. 398 Anderson, P.W., see Baskaran, G. 399, 416, 420 Anderson, P.W., see Fong, H.F. 389, 471 Anderson, P.W., see Yin, L. 426 Andersson, G. 248 Andersson, G., see Hjörvarsson, B. 206, 219, 220, 248 Ando, K. 295, 299, 350, 361 Ando, K., see Akimoto, R. 307 Ando, K., see Herbich, M. 351 Ando, K., see Janik, E. 295 Ando, K., see Saito, H. 311 Ando, K., see Zaets, W. 299 Ando, Y. 454, 455 Ando, Y., see Chuang, Y.-D. 482 Ando, Y., see Lavrov, A.N. 484 André, R. 295 André, R., see Bodin, C. 295 André, R., see Ulmer-Tufﬁgo, H. 298 Andrearczyk, T. 304 Andrearczyk, T., see Jaroszy´nski, J. 304 Andrieu, S. 220, 228 Andriotis, A. 215 Andriotis, A.N. 261, 262, 265, 269, 270 Andriotis, A.N., see Froudakis, G.E. 261 Andriotis, A.N., see Lathiotakis, N.N. 257 Angelakeris, M., see Wilhelm, F. 220, 237 Animisov, A.N., see Lindner, J. 228 Anisimov, A.N., see Bindilatti, V. 355 Anisimov, A.N., see Isber, S. 355 Anisimov, V.I. 409 Anno, H. 344 Anno, T., see Shinjo, T. 12 Anthony, T.C. 34, 39, 46, 51, 93, 145 Anthony, T.C., see Bayle-Guillemaud, P. 70

AUTHOR INDEX Anthony, T.C., see Egelhoff, W.F. 51, 52, 93 Anthony, T.C., see Egelhoff Jr., W.F. 34, 51, 52 Anthony, T.C., see Lai, C.-H. 163 Anthony, T.C., see Petford-Long, A.K. 124, 127, 129 Anthony, T.C., see Portier, X. 73, 129, 161 Antonov, A. 504, 505, 527, 537 Antonov, A., see Makhnovskiy, D.P. 526–528 Antonov, A., see Usov, N. 526–528 Antonov, A.S. 503, 532, 533, 537 Antonov, A.S., see García, J.M. 542, 549 Antonov, A.S., see Ménard, D. 514 Antonov, A.S., see Usov, N. 515, 525–527, 530 Aoki, M., see Bowler, D.R. 215 Aoki, Y., see Sato, H. 62 Aoshima, K. 74, 140 Aoshima, K., see Hong, J. 144 Aoshima, K., see Kanai, H. 32, 33, 35, 41, 77, 157 Aoshima, K., see Noma, K. 33, 123 Aoyama, M., see Koike, Y. 448 Aoyama, M., see Watanabe, I. 450, 479, 481 Appino, C. 540 Apsel, S.E. 254–257, 267 Arabski, J., see Alloul, H. 478 Aragoneses, P., see Vázquez, M. 504, 533 Arai, K.I., see Yamaguchi, M. 552 Arai, M. 470, 471, 473 Arai, R., see Hamakawa, Y. 47, 73, 163, 172 Arai, R., see Soeya, S. 160, 168 Araki, S. 32–34, 36, 67, 159 Araki, S., see Inage, K. 26 Araki, S., see Sano, M. 164 Araki, S., see Shimazawa, K. 159 Araki, S., see Tsuchiya, Y. 36 Araújo, A.E.P. 548, 549 Araújo, A.E.P., see Soares, J.M. 532 Araújo, J.H., see Soares, J.M. 532 Arauzo, A.B., see Gratens, X. 312, 322 Araya-Pochet, J., see Finck, F.L. 224 Arcade, P., see Andrieu, S. 228 Arcas, J., see Vázquez, M. 504, 533 Archambault, V., see Feit, Z. 328 Arciszewska, M., see Dybko, K. 360 Arciszewska, M., see Łazarczyk, P. 315, 316 Arciszewska, M., see Łusakowski, A. 318 Arciszewska, M., see Mycielski, A. 356, 357 Arciszewska, M., see Nadolny, A.J. 321, 322 Arciszewska, M., see Stachow-Wójcik, A. 333–335 Arciszewska, M., see Story, T. 328–330, 333–336 Arciszewska, M., see Twardowski, A. 351, 353 Arias, T., see Ismail-Beigi, S. 215

567

Arifuku, M., see Fukuma, Y. 322 Arko, A.J., see Chuang, Y.-D. 482 Arko, A.J., see Dessau, D.S. 427 Arko, A.J., see King, D.M. 464 Armitage, N.P. 465 Armitage, N.P., see Feng, D.L. 482 Armstrong, T., see Mook, H.A. 389, 471 Arnett, P., see Smith, N. 135, 136 Arnoult, A., see Haury, A. 311 Arnoult, A., see Huard, V. 302 Arnoult, A., see Kossacki, P. 302 Arovas, D.P. 415 Arrigoni, E. 407 Arrigoni, E., see Chen, H.-D. 413 Arrigoni, E., see Dorneich, A. 416 Arrigoni, E., see Zhang, S.-C. 416 Arrio, M.-A., see Gallani, J.L. 275 Arrot, A., see Katti, R.R. 57, 58 Arrott, A.S., see Celinski, Z. 235 Arrott, A.S., see Heinrich, B. 230 Artacho, E., see Calleja, M. 266 Artacho, E., see Izquierdo, J. 213 Artacho, E., see Onida, G. 274 Artacho, E., see Sánchez-Portal, D. 213, 245 Artacho, E., see Soler, J.M. 213 Arumugam, S. 454 Asada, H., see Fukuma, Y. 322 Asada, T. 208, 238 Asada, T., see Bihlmayer, G. 276 Asayama, K. 430, 437, 473, 474 Asayama, K., see Ishida, K. 479 Asenjo, A., see García, J.M. 505, 534, 542, 549 Asenjo, A., see Sinnecker, J.P. 504, 534, 541 Ashar, K.G. 19 Ashcroft, N.W. 17, 18, 78, 87, 436 Ashenford, D.E., see Aguekian, V.F. 300 Ashenford, D.E., see Cheng, H.H. 297 Ashenford, D.E., see Goede, O. 305 Ashenford, D.E., see Jackson, S. 296 Ashenford, D.E., see Klar, P.J. 308 Ashenford, D.E., see Kusrayev, Yu.G. 298 Ashenford, D.E., see Nicholas, R.J. 306 Ashenford, D.E., see Östreich, M. 304 Ashenford, D.E., see Peyla, P. 298 Ashenford, D.E., see Ribayrol, A. 297 Ashenford, D.E., see Suisky, D. 298 Ashenford, D.E., see Wasiela, A. 297 Ashenford, D.E., see Weston, S.J. 296 Askenazy, S., see Martin, J.L. 354 Astakhov, G.V., see Ossau, W. 302 Atalay, S., see Squire, P.T. 505 Atkinson, D. 515, 520–522, 538, 553, 556 Atkinson, D., see Squire, P.T. 505

568

AUTHOR INDEX

Atlan, D., see Schmid, A.K. 268 Atwater, H.A., see Joo, H.S. 71 Auerbach, A. 399, 484 Auerbach, A., see Altman, E. 419 Auerbach, A., see Zhang, S.-C. 416 Auffret, S., see Cowache, C. 164 Averback, R.S., see Zimmermann, C.G. 268 Averous, M. 328 Averous, M., see Bindilatti, V. 312, 326, 355 Averous, M., see Bruno, A. 312, 328, 329 Averous, M., see Gratens, X. 312, 322, 328 Averous, M., see Isber, S. 312, 322, 323, 355 Averous, M., see Lombos, B.A. 328 Averous, M., see Mahoukou, F. 312 Averous, M., see Maurice, T. 323, 324 Averous, M., see Misra, S.K. 322, 323 Averous, M., see ter Haar, E. 312, 326 Awschalom, D.D. 5, 301, 310 Awschalom, D.D., see Baumberg, J.J. 301 Awschalom, D.D., see Crooker, S.A. 295, 296, 303, 309 Awschalom, D.D., see Crowell, P.A. 310 Awschalom, D.D., see Harwit, A. 300 Awschalom, D.D., see Kikkawa, J.M. 300, 301 Awschalom, D.D., see Levy, J. 301 Awschalom, D.D., see Ohno, Y. 305 Awschalom, D.D., see Ray, O. 307 Awschalom, D.D., see Smorchkova, I.P. 304 Awschalom, D.D., see Smyth, J.F. 305 Awschalom, D.D., see Wolf, S.A. 5, 208 Axe, J.D., see Tranquada, J.M. 388, 396, 439, 443–445, 455, 458, 459 Azevedo, A., see Rezende, S.M. 164 Baberschke, K., see Lindner, J. 228 Baberschke, K., see Scherz, A. 227, 228 Baberschke, K., see Wilhelm, F. 209, 220, 237 Babucke, H., see Streller, U. 296 Bachelet, G.B. 213 Bacher, G. 309 Bacher, G., see Maksimov, A.A. 309 Bacher, G., see Tönnies, D. 296 Bacher, G., see Welsch, M.K. 307 Bacher, G., see Zaitsev, S. 309 Bachman, H.N., see Mitrovi´c, V.F. 470, 477 Baclet, P., see Acher, O. 514 Bacskay, G.B. 323 Bader, S.D. 208 Bader, S.D., see Mattson, J.E. 220, 244 Bae, S. 25, 35, 37, 46, 164 Bae, S., see Torok, E.J. 25 Bagci, V.M.K. 266 Baghat, S.M., see Loﬂand, S.E. 514, 546

Bagus, P.S., see Mauri, D. 154 Baibich, M.N. 4 Baibich, M.N., see Carara, M. 504, 542, 549 Bailey, W.E. 92, 111 Bailey, W.E., see Lai, C.-H. 163 Bailey, W.E., see Wang, S.-X. 42, 71 Baj, M., see Wasik, D. 304 Bajorek, C.H. 172 Bak, J. 354 Bak-Misiuk, ˛ J., see Janik, E. 295 Bak-Misiuk, ˛ J., see Szuszkiewicz, W. 361 Baker, S.H. 241, 264 Baker, S.H., see Binns, C. 205, 208, 220, 241, 242, 263, 264 Bakharev, O., see Abu-Shiekah, I.M. 433, 452, 461, 462 Bakharev, O., see Brom, H.B. 479 Bakharev, O., see Teitel’baum, G.B. 434, 435, 451, 452, 467 Bakharev, O.N. 467, 468 Bakharev, O.N., see Julien, M.-H. 452, 478 Baksalary, O., see Barna´s, J. 94 Bal, K., see Gerrits, Th. 131, 133 Balatsky, A.V., see Batista, C.D. 404, 430 Balatsky, A.V., see Martin, I. 401, 481 Balatsky, A.V., see Monthoux, P. 418 Balatsky, A.V., see Smith, C.M. 481 Balbás, L.C., see Izquierdo, J. 213 Balbás, L.C., see Robles, R. 225, 245, 275 Balbás, L.C., see Vega, A. 207, 218, 221, 222, 225–227, 267 Balci, H., see Biswas, A. 464 Baldwin, D.A., see Hylton, T.L. 70 Balents, L. 413 Ballentine, C.A., see Finck, F.L. 224 Ballentine, C.A., see Song, O. 140 Baltrunas, D., see Zayachuk, D. 329 Bamming, M., see Nouvertné, F. 268, 269 Bangert, E., see Kuhn-Heinrich, B. 296 Banhart, J. 7, 80, 82, 83 Banhart, J., see Ebert, H. 81 Bansil, A. 441 Bansil, A., see Lindroos, M. 441 Bansmann, J., see Röhlsberger, R. 252 Bär, L., see Gerrits, Th. 133 Baraldi, A., see Goldoni, A. 205, 235, 236 Barandiarán, J.M. 504, 513, 527 Barandiarán, J.M., see Hernando, A. 509 Barandiarán, J.M., see Kraus, L. 545 Barandiarán, J.M., see Kurlyandskaya, G.V. 504, 534 Barandiarán, J.M., see Muñoz, J.L. 528 Barandiarán, J.M., see Pirota, K.R. 504, 535, 557 Barandiarán, J.M., see Sartorelli, M.L. 504, 535

AUTHOR INDEX Barandiarán, J.M., see Vázquez, M. 553 Baranov, S.A. 548 Barcz, A., see Karczewski, G. 304 Barczewski, M., see Portugall, O. 361 Barczewski, M., see Widmer, T. 361 Bardasis, A. 410 Bardorf, S.R., see Harrison, P. 296 Baril, L. 138, 139, 141 Barman, A. 131 Barnaba, M., see Goldoni, A. 205, 235, 236 Barna´s, J. 94, 110 Barna´s, J., see Camley, R.E. 17, 86 Barna´s, J., see Jouanne, M. 311 Barna´s, J., see Szuszkiewicz, W. 311 Barnett, R.N., see Moseler, M. 251 Barradas, N.P., see Veloso, A. 35, 48, 49 Barrett, N., see Orlowski, B.A. 315 Bartenlian, B., see Renard, J.-P. 86 Barth, J.V., see Lin, N. 275 Barthel, J., see Qian, D. 238 Barthel, J., see Shen, J. 238 Barthélémy, A. 5, 91, 109 Barthélémy, A., see Duvail, J.L. 12, 61, 94, 104 Barthélémy, A., see George, J.M. 86 Barthélémy, A., see Vouille, C. 84 Bartholomev, D.U. 351 Bartkowski, M. 328 Bar’yakhtar, V.G., see Akhiezer, A.I. 515 Barzykin, V. 419, 476 Barzykin, V., see Zha, Y. 431, 432, 436 Baselt, D.R. 24 Baselt, D.R., see Edelstein, R.L. 24 Baskaran, G. 399, 416, 420 Baskes, M.I., see Foiles, S.M. 255 Basov, D.N., see Singley, E.J. 466 Bass, J., see Gu, J.Y. 13 Bass, J., see Pratt, W.P. 78, 85 Bass, J., see Steenwyk, S.D. 13, 80, 83, 85 Bass, J., see Yang, Q. 78 Bassani, F., see Kheng, K. 301 Bassat, J.M., see Poirot, N.J. 456 Bastard, G., see Boudinet, P. 301 Bastard, G., see Brum, J.A. 296, 297 Bastard, G., see Deleporte, E. 297 Batholomew, D.U., see Suh, E.-K. 300 Batista, C.D. 404, 430 Batlogg, B. 386 Batlogg, B., see Cava, R.J. 459 Batlogg, B., see Cheong, S.-W. 461 Batyrev, I.G. 225 Bauer, E., see Zdyb, R. 237 Bauer, G. 312, 314, 318, 324, 325 Bauer, G., see Aigle, M. 327 Bauer, G., see Chen, J.J. 343

569

Bauer, G., see Denecke, R. 315 Bauer, G., see Dietl, T. 318, 322, 325 Bauer, G., see Geist, F. 323–325 Bauer, G., see Giebułtowicz, T.M. 343–345 Bauer, G., see Głód, P. 362 Bauer, G., see Kepa, H. 343, 345 Bauer, G., see Koppensteiner, E. 342 Bauer, G., see Krenn, H. 312, 323–326, 344, 345 Bauer, G., see Krost, A. 324 Bauer, G., see Nunez, V. 343, 345 Bauer, G., see Pohlt, M. 344, 345 Bauer, G., see Prinz, A. 323 Bauer, G., see Springholz, G. 326–328, 342 Bauer, G., see Ueta, A.Y. 323, 324, 326, 327 Bauer, G., see Yuan, S. 323, 324, 327 Bauer, G.E.W., see Brataas, A. 103 Bauer, G.E.W., see Gijs, M.A.M. 5–7 Bauer, G.E.W., see Schep, K.M. 109, 110 Bauer, M., see Schumacher, H.W. 132 Baumberg, J.J. 301 Baumgart, B., see Dieny, B. 4, 16, 31, 39, 178 Baumgart, P., see Dieny, B. 18, 19, 59, 60, 95, 99 Bayle-Guillemaud, P. 70 Bayle-Guillemaud, P., see Portier, X. 73 Bayliss, S.C., see Binns, C. 205, 263 Bayreuther, G., see Ganzer, S. 25 Bazaliy, Y.B. 417 Bazaliy, Y.B., see Pryadko, L.P. 404 Bazhanov, D., see Sander, D. 208 Bazhanov, D.I. 205, 252 Bazhanov, D.I., see Izquierdo, J. 227, 269, 277 Bazhanov, D.I., see Levanov, N.A. 215 Bazhanov, D.I., see Lin, N. 275 Bazhanov, D.I., see Stepanyuk, V.S. 268 Beach, R.S. 45, 120, 123, 124, 503, 504, 507–509, 522, 532, 535, 549 Beach, R.S., see Sankar, A. 55 Bean, C.P., see Meiklejohn, W.H. 117, 148, 153, 171 Beatrice, C., see Appino, C. 540 Beaurepaire, E. 209 Beaurepaire, E., see Guidoni, L. 274 Beaurepaire, E., see Zhang, G. 274 Beauvillain, P., see Dupas, C. 9 Beauvillain, P., see Renard, J.-P. 86 Bebenin, N.G., see Kurlyandskaya, G.V. 503, 537 Becker, C.R., see Bacher, G. 309 Becker, C.R., see Maksimov, A.A. 309 Becker, C.R., see Welsch, M.K. 307 Becker, C.R., see Zaitsev, S. 309 Becker, J.A., see Billas, I.M.L. 254, 255, 267 Becker, W.M., see Kolodziejski, L.A. 295

570

AUTHOR INDEX

Beckmann, H. 253 Becla, P., see Shih, O.W. 351, 352 Bednorz, J.G. 384 Beech, R.S. 22 Beenakker, C.W.J., see Van Wees, B.J. 258 Beliën, P., see Gijs, M.A.M. 135 Beliën, P., see Potter, C.D. 110 Beliën, P., see Schad, R. 6, 27, 79 Beliën, P.J.L., see Van de Veerdonk, R.J.M. 134 Bell, R.F., see Warren Jr., W.W. 474 Bellet, D., see Bodin, C. 295 Bellini, V. 205 Bellini, V., see Freyss, M. 248 Bellouard, C., see Faure-Vincent, J. 276 Belonoshko, A.B. 209 Benaissa, M., see Lefakis, H. 67 Benakki, M., see Amalou, F. 223 Benakki, M., see Bouarab, S. 228 Benakki, M., see Dahmoune, C. 235 Benakki, M., see Khan, M.A. 228 Benakki, M., see Talanana, M. 227 Bencok, P., see Marangolo, M. 248 Bendjaballah, X.L., see Zhao, Z.J. 532 Benea, D., see Scherz, A. 227, 228 Benedek, G. 218 Benneman, K.H., see Manske, D. 418 Bennett, B.R., see Johnson, M. 25 Bennett, B.R., see Jonker, B.T. 304 Bennett, B.R., see Park, Y.D. 304 Bennett, W.R. 110 Bennington, S.M., see Arai, M. 470, 471, 473 Benoit à la Guillaume, C. 300 Benoit à la Guillaume, C., see Bhattacharjee, A.K. 301 Benoit à la Guillaume, C., see Scalbert, D. 351 Bensmina, F. 73 Berger, H., see Borisenko, S.V. 441 Berger, H., see Kordyuk, A.A. 483 Berger, J.D., see Kavokin, A.V. 299 Bergmann, G., see Beckmann, H. 253 Bergmann, G., see Hossain, M. 253 Bergmann, G., see Song, F. 254 Bergomi, L. 343 Berkov, D.V. 125 Berkowitz, A.E. 148 Berkowitz, A.E., see Beach, R.S. 503, 504, 507–509, 522, 532, 535, 549 Berkowitz, A.E., see Carey, M.J. 163, 164, 171 Berkowitz, A.E., see Egelhoff, W.F. 51, 52, 93 Berkowitz, A.E., see Egelhoff Jr., W.F. 34, 35, 51, 52, 69, 70, 144 Berkowitz, A.E., see Sankar, A. 55 Berlanga-Ramirez, E.O., see Aguilera-Granja, F. 219

Berlinsky, A.J., see Arovas, D.P. 415 Bernal, O.O., see Abu-Shiekah, I.M. 461 Bernal, O.O., see Brom, H.B. 479 Bernardo, J., see Freitas, P.P. 22 Bernel, S., see Reuse, F.A. 215, 258 Bernhard, C., see Niedermayer, Ch. 450 Bernhard, C., see Sidis, Y. 419, 470 Bernhard, T., see Pfandzelter, R. 249 Berroir, J.M., see Deleporte, E. 297 Berry, J.J., see Ray, O. 307 Bertacco, R., see Duo, L. 249 Berthier, C. 430, 434, 437, 438, 473, 476 Berthier, C., see Horvati´c, M. 474 Berthier, C., see Julien, M.-H. 452, 478 Berthier, Y., see Berthier, C. 430, 434, 437, 438, 473, 476 Berthier, Y., see Grévin, B. 478 Berthier, Y., see Horvati´c, M. 474 Bertin, F., see Malliavin, M.J. 515 Bertotti, G. 518 Bertram, H.N. 19, 127, 135 Bertram, H.N., see Jin, Z. 135, 136 Bertram, N.W., see Yuan, S.W. 127 Bertsch, G.F., see Kohl, C. 216 Berzhanskii, V.N., see Baranov, S.A. 548 Berzigiarova, N.S., see He, H. 411, 426, 449, 482 Beschoten, B., see Keller, J. 155 Beschoten, B., see Miltenyi, P. 155 Beschoten, B., see Nowak, U. 155 Beschoten, B., see Ohno, Y. 305 Besenbacher, F., see Pedersen, M.O. 268, 269 Bessho, K., see Motoyoshi, M. 24 Betancourt, I. 536 Betancourt, I., see Carrasco, E. 537 Betancourt, I., see Valenzuela, R. 510 Bettac, A., see Röhlsberger, R. 252 Bhagat, S.M. 548 Bhagat, S.M., see Kale, S. 535 Bhagat, S.M., see Loﬂand, S.E. 546 Bhattacharjee, A.K. 301, 303 Bian, B., see Xi, H. 167 Bianconi, A. 408 Bianconi, A., see Kusmartsev, F.V. 409 Bianconi, A., see Saini, N.L. 454 Bianconi, G., see Bianconi, A. 408 Bianconi, G., see Kusmartsev, F.V. 409 Bickers, N.E. 427 Bickers, N.E., see Pao, C.-H. 418 Bicknell, R.N., see Suh, E.-K. 300 Bicknell-Tassius, R.N., see Ivchenko, E.L. 297 Bicknell-Tassius, R.N., see Pozina, G.R. 297 Bicknell-Tassius, R.N., see Waag, A. 294

AUTHOR INDEX Bicknell-Tassius, R.N., see Yakovlev, D.R. 300 Biedermann, A. 217 Biermann, K., see Heiss, W. 328 Bigenwald, P., see Kavokin, A.V. 298 Bigot, J.-Y., see Guidoni, L. 274 Bigot, J.Y., see Zhang, G. 274 Bihlmayer, G. 276 Bihlmayer, G., see Asada, T. 208 Bihlmayer, G., see Heinze, S. 277 Bihlmayer, G., see Kurz, Ph. 204, 206, 241, 270 Bihlmayer, G., see Pampuch, C. 253 Bihlmayer, G., see Wortmann, D. 204 Billas, I.M.L. 250, 254, 255, 265, 267 Billas, I.M.L., see Tast, F. 265 Billinge, S.J.L., see Bo˜zin, E.S. 448 Binasch, G. 4 Binder, J. 83, 84, 109 Bindilatti, V. 312, 326, 355 Bindilatti, V., see Gratens, X. 312 Bindilatti, V., see Isber, S. 355 Bindilatti, V., see McCabe, G.H. 347, 349 Bindilatti, V., see Shapira, Y. 353 Bindilatti, V., see Shih, O.W. 351 Bindilatti, V., see ter Haar, E. 312, 326 Bindilatti, V., see Vu, T.Q. 353 Bindley, U., see Rhyne, J.J. 345 Bindley, U., see Stumpe, L.E. 345 Binns, C. 205, 208, 220, 241, 242, 263, 264 Binns, C., see Baker, S.H. 241, 264 Birch, J., see Korenivski, V. 224 Birgeneau, R.J., see Elstner, N. 486 Birgeneau, R.J., see Greven, M. 486 Birgeneau, R.J., see Kimura, H. 446 Birgeneau, R.J., see Lee, Y.S. 448, 449, 486 Birgeneau, R.J., see Matsuda, M. 466, 467 Birgeneau, R.J., see Nakajima, K. 485, 486 Birgeneau, R.J., see Savici, A.T. 449 Birgeneau, R.J., see Wakimoto, S. 445, 447, 448 Birgeneau, R.J., see Yamada, K. 447, 448, 466 Birthwisle, K., see Nor, A.F. 135 Bisaro, R., see Maurice, T. 323, 324 Bischoff, M.M.J. 219, 225, 228 Bischoff, M.M.J., see Fang, C.M. 253 Bishop, A.R., see Bussmann-Holder, A. 409 Bishop, A.R., see Chernyshev, A.L. 404 Bishop, A.R., see Martin, I. 401 Bishop, A.R., see McQueeney, R.J. 409 Bishop, A.R., see Stojkovi´c, B.P. 392 Bishop, A.R., see Verges, J.A. 395 Bishop, A.R., see Yi, Y.S. 409 Bishop, A.R., see Zhu, J.-X. 449 Bishop, D.J., see Ramirez, A.P. 459 Biswas, A. 464

571

Bitter, R.H.J.N., see Swagten, H.J.M. 34, 46, 92, 93 Blaas, C. 80, 109 Blachard, N., see Bobroff, J. 478 Black, W.C., see Hermann, T.M. 24 Blackman, J.A., see Duffy, D.M. 205, 262–264 Blamire, M.G., see Morecroft, D. 138 Blamire, M.G., see Prieto, J.L. 22 Blanchard, N., see Bobroff, J. 477 Blanchard, N., see Dooglav, A.V. 473 Blanchard, N., see Mendels, P. 478 Blanco, J.M. 504, 538–540, 544 Blanco, J.M., see Cobeño, A.F. 555 Blanco, J.R., see Chen, L. 263 Bland, J.A.C. 4, 207, 276 Bland, J.A.C., see Choi, B.Ch. 249 Bland, J.A.C., see Fullerton, E.E. 235 Bland, J.A.C., see Hope, S. 249 Bland, J.A.C., see Li, S.P. 276 Blank, H.-R., see Zeltser, A.M. 159 Blank, H.-R., see Zhang, Y.B. 60, 76, 161, 170 Blasius, T., see Niedermayer, Ch. 450 Blesse, P.A., see Popovic, R.S. 23 Bleuse, J., see Ulmer-Tufﬁgo, H. 298 Blinkin, V.A., see Pashkevich, Yu.G. 461, 463 Blinowski, J. 293, 311, 340, 343, 346, 348, 349 Blinowski, J., see Giebułtowicz, T.M. 336, 343–345 Blinowski, J., see Kepa, H. 336, 338, 340 Bloemen, P.J.H. 45 Bloemen, P.J.H., see Johnson, M.T. 143 Bloemen, P.J.H., see Swagten, H.J.M. 34, 46, 47, 52, 92, 93 Blomquist, P., see Broddefalk, A. 228 Blomquist, P., see Kalska, B. 227, 231 Blomquist, P., see Lindner, J. 228 Bloom, S.H., see Heiman, D. 351 Bloom, S.H., see Shih, O.W. 351 Bloomﬁeld, I.A., see Bucher, J.P. 207 Bloomﬁeld, L.A. 211, 250 Bloomﬁeld, L.A., see Apsel, S.E. 254–257, 267 Bloomﬁeld, L.A., see Cox, A.J. 205, 219, 234, 235, 275 Blügel, S. 208, 224, 235 Blügel, S., see Asada, T. 208, 238 Blügel, S., see Bihlmayer, G. 276 Blügel, S., see Heinze, S. 277 Blügel, S., see Kachel, T. 236 Blügel, S., see Kurz, Ph. 204, 206, 241, 270 Blügel, S., see Pampuch, C. 253 Blügel, S., see Weinert, M. 208 Blügel, S., see Wortmann, D. 204 Blum, V. 238 Blumberg, G. 418, 459, 464

572

AUTHOR INDEX

Blundell, S.J., see Chow, K.H. 460 Blundell, S.J., see Jestädt, Th. 460 Bobroff, J. 477, 478 Bobroff, J., see Dooglav, A.V. 473 Bobroff, J., see Mendels, P. 478 Bode, J.P., see Choi, B.Ch. 249 Bode, M., see Wiesendanger, R. 206, 208, 251 Bodin, C. 295 Bodin, C., see Lawrence, I. 305 Bodin-Deshayes, C., see Gaj, J.A. 296 Boeglin, C. 235 Boerma, D.O., see Langelaar, M.H. 238 Boero, M., see Billas, I.M.L. 265 Boeve, H. 57 Bogdanov, P., see Armitage, N.P. 465 Bogdanov, P., see Feng, D.L. 482 Bogdanov, P.V. 427 Bogdanov, P.V., see Lanzara, A. 409, 427, 454, 482 Bogdanov, P.V., see Zhou, X.J. 441, 454 Bonanni, A., see Prechtl, G. 297 Bonevich, J.E., see Fry, R.A. 138 Bonevich, J.E., see McMichael, R.D. 138 Bonﬁm, M., see Camarero, J. 178 Bönicke, I.A., see Pedersen, M.O. 268, 269 Bonn, D.A., see Sonier, J.E. 477 Boom, R., see de Boer, F.R. 77, 166 Boonman, M.E.J. 346 Boonman, M.E.J., see Mac, W. 346 Bordin, G. 538 Borgers, J.A., see van der Zaag, P.J. 155 Borisenko, S.V. 441 Borisenko, S.V., see Kordyuk, A.A. 483 Borisov, O.V., see Antonov, A. 505 Borisov, V.T., see Antonov, A. 505 Borowski, R., see Kataev, V. 391, 453 Borsa, F. 433, 449 Borsa, F., see Carretta, P. 434 Borsa, F., see Huh, Y.M. 453 Borsa, F., see Julien, M.-H. 452 Borsa, F., see Rigamonti, A. 434 Bosch, M. 396, 404, 407 Boscher, C., see Malliavin, M.J. 515 Bossy, J., see Bourges, P. 471, 472 Bossy, J., see Fong, H.F. 470 Bossy, J., see Rossat-Mignod, J. 385, 389, 438, 471 Bossy, J., see Sidis, Y. 472 Bottger, T., see Rossin, V.V. 296 Böttner, H., see Lambrecht, A. 312, 323, 324 Böttner, H., see Springholz, G. 326 Bouarab, S. 205, 224, 227, 228, 234, 254–256, 258, 266, 267

Bouarab, S., see Aguilera-Granja, F. 215, 254–258 Bouarab, S., see Dahmoune, C. 235 Bouarab, S., see Hadj-Larbi, A. 217, 231–234, 275 Bouarab, S., see Khan, M.A. 228 Bouarab, S., see M’Passi-Mabiala, B. 276 Bouarab, S., see Nait-Laziz, H. 235 Bouarab, S., see Pruneda, J.M. 245–248 Bouarab, S., see Rennert, P. 252 Bouarab, S., see Robles, R. 225, 245 Bouarab, S., see Talanana, M. 227 Bouarab, S., see Vega, A. 207, 221, 225–228 Bouarab, S., see Zenia, H. 217, 220, 275 Bouarab, S., see Ziane, A. 234 Boudinet, P. 301 Bounnak, S., see Miller, M.M. 24 Bourges, P. 389, 471, 472 Bourges, P., see Etrillard, J. 479 Bourges, P., see Fong, H.F. 470 Bourges, P., see He, H. 411, 426, 449, 482 Bourges, P., see Rossat-Mignod, J. 385, 389, 438, 471 Bourges, P., see Sidis, Y. 419, 470, 472 Bourgognon, C., see Ferrand, D. 310, 311 Bourret, A., see Charleux, M. 296 Boussendel, A. 276 Bovensiepen, U., see Wilhelm, F. 209 Bowler, D.R. 215 Bo˜zin, E.S. 448 Bozovic, I., see Marshall, D.S. 476 Braden, M., see Tranquada, J.M. 409, 459 Brandão, D.E., see Fraga, G.L.F. 535 Branz, W., see Billas, I.M.L. 265 Brataas, A. 103 Bratkowski, M.A., see Averous, M. 328 Brauman, J.I. 250 Brazis, R. 309 Brendel, B., see Arrigoni, E. 407 Brennert, G.F., see Warren Jr., W.W. 474 Breton, G., see Gratens, X. 312, 322 Breton, G., see Maurice, T. 323, 324 Brewer, J.H. 438 Brewer, J.H., see Sonier, J.E. 477 Briaire, J., see Gijs, M.A.M. 135 Brieler, F., see Chen, L. 307 Brinckmann, J. 412, 420, 421, 424–426 Briner, B. 220, 244 Brinkmann, D. 307 Brinkmann, D., see Matsumura, M. 433 Britel, M., see Ciureanu, P. 504, 514, 515, 546, 549 Britel, M., see Ménard, D. 514, 516, 529, 530, 546, 548, 549

AUTHOR INDEX Britel, M., see Yelon, A. 508, 514, 516, 522, 525, 528 Britel, M.R. 516, 526, 548, 549 Britel, M.R., see Ménard, D. 515, 526 Brivio, G.P., see Trioni, M.I. 253 Broddefalk, A. 228 Brodsky, M.N., see Grünberg, P. 206, 217, 219, 248 Brom, H.B. 430, 473, 479 Brom, H.B., see Abu-Shiekah, I.M. 433, 452, 461, 462 Brom, H.B., see Bakharev, O.N. 467, 468 Brom, H.B., see MacLaughlin, D.E. 433 Brom, H.B., see Moonen, J.T. 438 Brom, H.B., see Teitel’baum, G.B. 434, 435, 451, 452, 467 Brom, H.B., see van der Klink, J.J. 430, 436 Brookes, N., see Cros, V. 235 Brookes, N.B., see Binns, C. 264 Brookes, N.B., see Lindner, J. 228 Brookes, N.B., see Marangolo, M. 248 Brookes, N.B., see Ohresser, P. 241 Brooks, M. 390 Brote, J.M., see Baibich, M.N. 4 Broto, J.M., see Bruno, A. 329 Brown, H., see Cool, S. 76 Brown, H.J., see Chopra, H.D. 70 Brown, L.B., see Speriosu, V.S. 45 Brown, R.H. 110 Browning, N.D., see Kim, C.S. 309, 310 Browning, V., see Edelstein, A.S. 231 Brug, J.A., see Anthony, T.C. 34, 39, 46, 51, 93, 145 Brug, J.A., see Bayle-Guillemaud, P. 70 Brug, J.A., see Egelhoff, W.F. 51, 52, 93 Brug, J.A., see Egelhoff Jr., W.F. 34, 51, 52 Brug, J.A., see Petford-Long, A.K. 124, 127, 129 Brug, J.A., see Portier, X. 73, 129 Brum, J.A. 296, 297 Brun, G., see Averous, M. 328 Brunetti, L. 504, 532 Bruno, A. 312, 328, 329 Bruno, F., see Gazzadi, G.C. 230 Bruno, P. 143, 203, 207, 244, 246 Bruno, P., see Fert, A. 5, 143 Bruno, P., see Pajda, M. 209 Bruno, P., see Renard, J.-P. 86 Brunthaler, G., see Prinz, A. 323 Bruynseraede, Y., see Barna´s, J. 110 Bruynseraede, Y., see Potter, C.D. 110 Bruynseraede, Y., see Schad, R. 6, 27, 79 Bryk, T. 224, 225 Bubber, R., see Schwartz, P.V. 31, 70 Buchanan, M. 390

573

Bucher, B., see Karpinski, J. 469 Bucher, J.P. 207, 208, 258, 259 Bucher, J.P., see Chado, I. 208, 236, 252 Buchina, M., see Li, H. 22, 24, 51, 52 Buchmeier, M., see Gareev, R.R. 220, 245, 247 Büchner, B., see Curro, N.J. 434, 451, 467 Büchner, B., see Kataev, V. 391, 453 Büchner, B., see Klauss, H.-H. 438, 439, 450 Büchner, B., see Suh, B.J. 434, 451, 452, 467 Büchner, B., see Teitel’baum, G.B. 435, 452, 467 Buda, B. 300 Budnick, J.I., see Niedermayer, Ch. 450 Buhrman, R.A., see Wolf, S.A. 5, 208 Bulou, H., see Boeglin, C. 235 Bulou, H., see Ohresser, P. 241 Bulut, N. 418 Buoni, M.J., see Pennington, C.H. 474 Buragohain, C., see Sachdev, S. 391, 414 Burchard, A., see Seewald, G. 237 Burgess, C.P. 416 Bürgler, D.E. 5, 110, 143, 207 Bürgler, D.E., see Gareev, R.R. 220, 245, 247 Bürgler, D.E., see Schaller, D.M. 238 Bürgler, D.E., see Schmidt, C.M. 221 Burke, K., see Perdew, J.P. 213 Burkel, E., see Röhlsberger, R. 252 Burkett, S.L., see Gafron, T.J. 26, 140, 141 Burlet, P., see Bourges, P. 471, 472 Burlet, P., see Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Burnet, S. 252, 275 Bushida, K., see Panina, L.V. 504, 510, 513, 515, 521–523, 525 Bushida, K., see Shen, L.P. 555 Busolt, U., see Vajda, S. 255 Buss, C. 299 Buss, C., see Leisching, P. 299 Bussmann-Holder, A. 409 Butaud, P., see Horvati´c, M. 474 Butler, W.H. 40, 80, 83, 84, 106–109 Butler, W.H., see Brown, R.H. 110 Butler, W.H., see Nicholson, D.M.C. 40, 80, 85 Butler, W.H., see Oparin, A.B. 85 Butler, W.H., see Schulthess, T.C. 83, 144, 146, 155 Butler, W.H., see Zhang, X.-G. 103, 104, 106, 109 Buttino, G., see Bordin, G. 538 Buttner, H., see Bussmann-Holder, A. 409 Buttrey, D.J., see Isaacs, E.D. 462 Buttrey, D.J., see Lee, S.-H. 404, 459 Buttrey, D.J., see Lorenzo, J.E. 457 Buttrey, D.J., see Pashkevich, Yu.G. 461, 463

574

AUTHOR INDEX

Buttrey, D.J., see Rice, D.E. 456, 457 Buttrey, D.J., see Sachan, V. 458 Buttrey, D.J., see Tranquada, J.M. 388, 456–458 Buttrey, D.J., see Vigliante, A. 463 Buttrey, D.J., see Wochner, P. 458 Buznikov, N.A., see Antonov, A. 537 Buznikov, N.A., see Antonov, A.S. 503, 532, 533, 537 Byczuk, K. 209 Bylander, D.M., see Bryk, T. 224, 225 Bylander, D.M., see Kleinman, L. 213 Caballero, J.A., see Slater, R.D. 13 Cabral, F.A.O., see Soares, J.M. 532 Cabral, J.M.S., see Graham, D.L. 22, 24 Cadeville, M.C., see Sanchez, J.M. 227 Cai, B.C., see Yu, J.Q. 504 Cai, B.C., see Zhou, Y. 504, 534, 541 Cai, C.M., see Mohri, K. 532, 550, 551, 555 Cain, W.C. 172 Caldeira, A.O., see Dimashko, Y.A. 404 Calleja, M. 266 Camarero, J. 143, 178 Camarero, J., see Gómez, L. 269 Camblong, H.E. 103, 109 Camilleri, C., see Teppe, F. 300, 306 Camley, R.E. 17, 86, 209 Camley, R.E., see Johnson, B.L. 103 Campana, A., see Julien, M.-H. 452 Campbell, I.A. 7, 78–80, 102 Campbell, I.A., see Fert, A. 78, 81, 104 Campo, J., see Stankiewicz, J. 355, 356 Campuzano, J.C., see Fretwell, H.M. 441 Campuzano, J.C., see Kaminski, A. 419 Canali, C.M., see Cehovin, A. 265 Canﬁeld, P.C., see MacLaughlin, D.E. 433 Capelli, R., see Gazzadi, G.C. 230, 231 Capponi, S., see Chen, H.-D. 413 Car, R. 204, 264, 265 Car, R., see Oda, T. 204, 206, 216 Carara, M. 504, 542, 549 Carbone, C., see Blügel, S. 208 Carbone, C., see Dallmeyer, A. 231 Carbone, C., see Gambardella, P. 252, 254 Carbone, C., see Kachel, T. 236 Carbone, C., see Pampuch, C. 253 Carcia, P.F. 259 Cardoso, S., see Fraune, M. 164 Carey, K., see Childress, J.R. 32, 41, 67, 159, 174 Carey, M., see Beach, R.S. 45, 120, 123 Carey, M., see Lee, W.Y. 37, 51, 71, 162, 170 Carey, M., see Pinarbasi, M. 47, 164, 172

Carey, M.J. 33, 67, 151, 159, 163, 164, 171, 176 Carey, M.J., see Childress, J.R. 32, 41, 55, 67, 159, 174 Carey, M.J., see Farrow, R.F.C. 164, 172 Carey, M.J., see Ju, G. 172 Carlos Egues, J. 306 Carlson, E.W. 389, 398, 406 Carrasco, E. 537 Carretta, P. 434 Carretta, P., see Borsa, F. 433, 449 Carretta, P., see Julien, M.-H. 452 Carretta, P., see Rigamonti, A. 434 Carroll, D.L., see Zha, F.-X. 261 Carter, S.A., see Cheong, S.-W. 461 Cartier, M., see Camarero, J. 178 Cartier, M., see Cowache, C. 164 Casalta, H., see Alloul, H. 478 Casian, A. 328 Castaño, F.J. 58 Castaño, F.J., see Óvári, T.A. 534 Castellani, C., see Seibold, C. 397 Castro Neto, A.H., see Chernyshev, A.L. 404 Castro Neto, A.H., see Hasselmann, N. 404 Castro Neto, A.H., see Smith, C.M. 481 Castro Neto, A.H., see Stojkovi´c, B.P. 392 Cava, R.J. 459 Cava, R.J., see Warren Jr., W.W. 474 Ceballos, G., see Scherz, A. 227 Ceballos, G., see Wilhelm, F. 209 Cebollada, A., see Cros, V. 235 Cecchetti, A., see Bordin, G. 538 Cederström, A.F., see Hunt, A.W. 390, 452, 455, 461, 467 Cederström, A.F., see Singer, P.M. 451 Cehovin, A. 265 Celino, M., see Fanciulli, M. 245 Celinski, Z. 235 Celinski, Z., see Fullerton, E.E. 235 Celotta, R.J., see Davies, A. 222 Celotta, R.J., see McClelland, J.J. 258 Celotta, R.J., see Pierce, D.T. 219, 228 Celotta, R.J., see Unguris, J. 207, 221 Cerjan, C., see Mao, M. 37, 49, 56 Cerjan, C., see Mao, S. 69 Cernogora, J., see Scalbert, D. 351 Cha, Y., see Lee, S.K. 235 Chado, I. 208, 236, 252 Chai, C.L., see Lu, Z.Q. 69 Chai, C.L., see Yang, T. 161 Chaiken, A. 12, 94, 220, 244, 245 Chaiken, A., see Michel, R.P. 163 Chakravarty, S. 401, 419, 428, 434, 473, 484–487 Chakravarty, S., see Tewari, S. 420

AUTHOR INDEX Chakravarty, S., see Yin, L. 426 Chambers, S.A., see Saleh, A.A. 266 Chaminade, J.P., see Horvati´c, M. 474 Chandesris, D., see Boeglin, C. 235 Chandra, G., see Acharya, B.R. 231 Chandra, P., see Ramirez, A.P. 459 Chang, C.R., see Wang, Y.H. 170 Chang, H.-C., see Lee, W.Y. 37, 51, 71 Chang, K. 306 Chang, L.L., see Awschalom, D.D. 310 Chang, L.L., see Deleporte, E. 297 Chang, L.L., see Harwit, A. 300 Chang, S.-K. 296 Chang, S.-K., see Nurmikko, A.V. 296 Chang, Y., see Misra, S.K. 322, 323 Chapman, J., see Li, H. 36, 159, 167 Chapman, J.N. 124, 129, 130 Chapman, J.N., see Gillies, M.F. 125, 126 Chapman, J.N., see Gogol, P. 155 Chapman, J.N., see King, J.P. 36, 127 Chapman, J.N., see Lim, C.K. 127, 138 Chapman, J.N., see Marrows, C.H. 127, 249 Chapman, J.N., see Rijks, Th.G.S.M. 136–138 Chappert, C. 143 Chappert, C., see Dupas, C. 9 Chappert, C., see Renard, J.-P. 86 Chappert, C., see Schumacher, H.W. 132 Charap, H., see Fulcomer, E. 151, 172, 173 Charar, S., see Bindilatti, V. 312, 326 Charar, S., see Errebbahi, A. 322, 326 Charar, S., see Gratens, X. 312, 322, 328 Charar, S., see Isber, S. 312, 322, 323, 326 Charar, S., see Mahoukou, F. 312 Charar, S., see Maurice, T. 323, 324 Charar, S., see Misra, S.K. 322, 323 Charar, S., see ter Haar, E. 312, 326 Charleux, M. 296 Châtelain, A., see Billas, I.M.L. 250, 254, 255, 267 Chaussy, J., see Dieny, B. 62 Chayes, L.N., see Nussinov, Z. 408 Chazelas, J., see Baibich, M.N. 4 Chen, A.P., see Blanco, J.M. 504, 539, 540, 544 Chen, C. 504 Chen, C., see Chow, K.H. 460 Chen, C.H. 458 Chen, C.H., see Cheong, S.-W. 461 Chen, C.T., see Pellegrin, E. 463 Chen, D.-X. 522, 537, 546 Chen, D.-X., see Vázquez, M. 504, 505, 534 Chen, D.-X., see Velázquez, J. 508 Chen, E., see Tehrani, S. 24, 57 Chen, E.Y. 128 Chen, E.Y., see Slaughter, J.M. 35, 51, 70, 71

575

Chen, F.-R., see Huang, R.-T. 50, 77 Chen, G., see Zeng, L. 536, 549 Chen, H., see Shi, S. 22 Chen, H.-D. 413 Chen, J. 32, 33, 104, 112, 114, 159, 249 Chen, J., see Giebułtowicz, T.M. 343 Chen, J., see Hou, C. 150 Chen, J., see Mao, S. 55, 60, 162 Chen, J.J. 343 Chen, J.J., see Bergomi, L. 343 Chen, L. 263, 307 Chen, P.J., see Bae, S. 25, 35, 37, 46, 164 Chen, P.J., see Chopra, H.D. 70, 144, 164 Chen, P.J., see Egelhoff Jr., W.F. 34, 35, 42, 51, 52, 69, 70, 144 Chen, P.J., see Fry, R.A. 138 Chen, P.J., see McMichael, R.D. 138 Chen, P.J., see Parks, D.C. 146 Chen, P.J., see Torok, E.J. 25 Chen, S., see Dong, C. 515 Chen, S.P., see Voter, A.F. 266 Chen, W., see Jamneala, T. 252 Chen, W., see Madhavan, V. 252 Chen, Y., see Li, S.P. 276 Chen, Y., see Shi, S. 22 Chen, Y., see Tong, H.C. 51, 52 Chen, Y.J., see Daughton, J.M. 27 Chen, Z., see Ikada, H. 308 Chen, Z., see Kayanuma, K. 305 Chen, Z.H. 303 Cheng, H.H. 297 Cheng, S.F., see Miller, M.M. 22 Cheng, S.F., see Restorf, J.B. 163 Cheng, Z.H., see Guo, H.Q. 504, 532, 539 Cheong, H.D., see Jonker, B.T. 304 Cheong, S.-W. 461 Cheong, S.-W., see Blumberg, G. 459 Cheong, S.-W., see Chen, C.H. 458 Cheong, S.-W., see Isaacs, E.D. 462 Cheong, S.-W., see Lee, S.-H. 404, 442, 443, 459 Cheong, S.-W., see MacLaughlin, D.E. 433 Cheong, S.-W., see Ramirez, A.P. 459 Cheong, S.-W., see Walstedt, R.E. 432 Cheong, S.-W., see Yoshinari, Y. 460, 461 Chernova, N.A., see Skipetrov, E.P. 312 Chernyshev, A.L. 404 Chernyshova, M. 332, 338 Chernyshova, M., see Kowalczyk, L. 338 Chernyshova, M., see Story, T. 333, 334, 336 Chernyshyov, O. 401 Chevrier, F., see Andrieu, S. 220, 228 Chiang, C., see Hamann, D.R. 213 Chiang, W.C., see Pratt, W.P. 78, 85

576

AUTHOR INDEX

Chiba, A., see Ando, K. 361 Chiba, K., see Suzuki, T. 445 Chien, C., see Gibbons, M.R. 36, 48, 99 Chien, C., see Mao, M. 56 Chien, C.L., see Ambrose, T. 55, 150, 164, 176 Chien, C.L., see Gökemeijer, N.J. 152 Chien, C.L., see Sommer, R.L. 504, 532, 541 Chien, C.L., see Strijkers, G.J. 123 Chien, C.L., see Viegas, A.D.C. 514 Chien, C.L., see Wu, X.W. 41 Chien, C.L., see Xiao, J.Q. 9 Chien, L., see Diao, Z. 38, 49, 50 Chih-Ming, L. 358 Chikazumi, S. 136 Childress, J.R. 32, 41, 55, 67, 159, 174 Childress, J.R., see Carey, M.J. 151, 159, 164, 176 Childress, J.R., see Feng, T. 41 Chin, T.K., see Zeltser, A.M. 159 Chin, T.K., see Zhang, Y.B. 60, 76, 161, 170 Chiriac, H. 504, 505, 532, 533, 556 Chiriac, H., see Brunetti, L. 504, 532 Chiriac, H., see Kraus, L. 504, 533, 540 Chiriac, H., see Óvári, T.A. 534, 546 Chiriac, H., see Pirota, K.R. 503, 504, 540 Chizhik, A., see Blanco, J.M. 504, 539, 544 Chizhik, A., see Zhukova, V. 533 Cho, J.-H., see Batyrev, I.G. 225 Cho, J.H. 205 Cho, J.H., see Borsa, F. 433, 449 Cho, J.H., see Carretta, P. 434 Cho, W.S. 540, 541 Choi, B., see Hope, S. 249 Choi, B.C., see Freeman, M.R. 208 Choi, B.Ch. 249 Choi, J.H. 235 Choi, K.R., see Hwang, D.G. 73 Choi, S.J., see Hwang, D.G. 73 Choi, Y., see Ro, J. 158 Chong, T., see Li, K. 37, 38, 45, 49, 50, 145, 146, 159, 167 Chong, T.C., see Li, K. 92 Chopra, H.D. 70, 144, 164 Chou, F.C., see Borsa, F. 433, 449 Chou, W.C. 294, 358, 359 Chou, W.C., see Jonker, B.T. 294, 301 Chou, W.C., see Luo, H. 295 Chouairi, A. 236 Chouairi, A., see Vega, A. 222, 223, 255 Chow, K.H. 460 Chow, K.H., see Jestädt, Th. 460 Chowdhury, A.J.S., see Chow, K.H. 460 Christensen, N.B., see Lake, B. 413, 446 Christensen, N.E., see Fanciulli, M. 245

Christianen, P.C.M., see Ma´ckowski, S. 309 Christianen, P.C.M., see Pulizzi, F. 298 Christianen, P.C.M., see Yakovlev, D.R. 302 Christidis, T.C., see Errebbahi, A. 322, 326 Chtchelkanova, A.Y., see Wolf, S.A. 5, 208 Chu, S.N.G., see Theodoropoulou, N. 276 Chu, V., see Li, H. 22, 24, 51, 52 Chuang, Y.-D. 482 Chubukov, A. 419 Chubukov, A.V., see Abanov, Ar. 419, 427, 428 Chubukov, A.V., see Tchernyshyov, O. 417 Chung, J., see Choi, J.H. 235 Cibert, J., see André, R. 295 Cibert, J., see Bodin, C. 295 Cibert, J., see Buss, C. 299 Cibert, J., see Dietl, T. 311 Cibert, J., see Ferrand, D. 310, 311 Cibert, J., see Gaj, J.A. 296 Cibert, J., see Grieshaber, W. 296, 299 Cibert, J., see Haury, A. 311 Cibert, J., see Kossacki, P. 302 Cibert, J., see Lawrence, I. 305 Cibert, J., see Leisching, P. 299 Cibert, J., see Maslana, W. 300 Cibert, J., see Teppe, F. 300, 306 Ciccaci, F., see Duo, L. 249 Ciorneiu, B., see Hylton, T.L. 70 Ciraci, S., see Bagci, V.M.K. 266 Ciureanu, P. 504, 514, 515, 542, 546, 549 Ciureanu, P., see Britel, M.R. 516, 526, 548, 549 Ciureanu, P., see Chiriac, H. 504, 532, 533 Ciureanu, P., see Duque, J.G.S. 536 Ciureanu, P., see Melo, L.G. 535 Ciureanu, P., see Ménard, D. 514–516, 526, 529, 530, 546, 548, 549 Ciureanu, P., see Robbes, D. 550 Ciureanu, P., see Yelon, A. 508, 513–516, 521, 522, 525, 528 Clausen, K.N., see Lake, B. 413, 449, 480 Clauss, W., see Zha, F.-X. 261 Clemens, B.M., see Mancoff, F.B. 25 Clemens, H., see Springholz, G. 326 Clemens, W., see van den Berg, H.A.M. 42 Clemens, W., see Vieth, M. 25 Cline, J.M., see Burgess, C.P. 416 Cobeño, A.F. 555 Cobeño, A.F., see Blanco, J.M. 504, 539, 540, 544 Cochran, J.F., see Celinski, Z. 235 Cochran, J.F., see Heinrich, B. 207, 221, 222, 230 Cochrane, R.W., see Britel, M.R. 516, 526, 548, 549

AUTHOR INDEX Cochrane, R.W., see Ciureanu, P. 515, 546, 549 Cochrane, R.W., see Ménard, D. 515, 526, 548, 549 Coehoorn, R. 5, 21, 23, 40, 45, 79, 80, 84, 85, 89, 90, 143, 148, 151, 153, 184 Coehoorn, R., see de Vries, J.J. 220, 244, 245, 247 Coehoorn, R., see Duchateau, J.P.W.B. 66, 67 Coehoorn, R., see Folkerts, W. 22, 127–129 Coehoorn, R., see Johnson, M.T. 143 Coehoorn, R., see Jungblut, R. 149, 154, 157, 176 Coehoorn, R., see Kools, J.C.S. 5, 144–147 Coehoorn, R., see Kudrnovsky, J. 143 Coehoorn, R., see Lenssen, K.-M.H. 31, 39, 67, 71 Coehoorn, R., see Oepts, W. 138 Coehoorn, R., see Purcell, S.T. 228 Coehoorn, R., see Rijks, Th.G.S.M. 13, 14, 16, 18, 61, 62, 66, 97, 104, 118, 119, 136–138, 157 Coehoorn, R., see Van de Veerdonk, R.J.M. 134 Coehoorn, R., see van Driel, J. 63, 64, 151, 152, 158, 159, 167, 176 Coffee, K.R., see Hylton, T.L. 9, 26 Cohen, J.M., see Liu, C.L. 267 Cohen, M.L., see Louie, S.G. 213 Coisson, M., see Appino, C. 540 Coisson, M., see Brunetti, L. 532 Colesniuc, C.N., see Chiriac, H. 556 Colis, S. 45 Collin, G., see Bobroff, J. 477, 478 Collin, G., see Dooglav, A.V. 473 Collin, G., see Grévin, B. 478 Collin, G., see Julien, M.-H. 452, 478 Collin, G., see Mahajan, A.V. 478 Collin, G., see Mendels, P. 478 Collin, G., see Sidis, Y. 472 Colocci, M., see Roussignol, Ph. 305 Colton, R.J., see Baselt, D.R. 24 Colton, R.J., see Edelstein, R.L. 24 Colton, R.J., see Miller, M.M. 24 Comelli, G., see Goldoni, A. 205, 235, 236 Conde, J.P., see Li, H. 22, 24, 51, 52 Conklin, J.B., see Hattox, T.M. 224 Connolly, J., see Lathiotakis, N.N. 257 Cool, S. 76 Cooper, A.S., see Chen, C.H. 458 Cooper, B.R., see Peng, S.S. 242, 262, 263 Cooper, J.R., see Loram, J.W. 442 Cooper, S.L., see Kim, Y.D. 354 Cooper, S.L., see Ko, Y.D. 354 Coquillat, D., see El Ouazzani, A. 351, 352 Coquillat, D., see Paganotto, N. 302 Coquillat, D., see Ribayrol, A. 297

577

Coquillat, D., see Siviniant, J. 296 Cornea, C. 204, 224 Cornea, C., see Stoefﬂer, D. 204, 217, 224 Cornut, B., see Britel, M.R. 516, 548, 549 Costa, J.L., see Freitas, P.P. 22 Costa, L., see Freitas, P.P. 22 Costa-Krämer, J.L. 503 Couach, M., see Bourges, P. 471, 472 Couach, M., see Horvati´c, M. 474 Cowache, C. 164 Cowache, C., see Dieny, B. 62, 104 Cowache, C., see Vedyaev, A. 94, 103 Cox, A.J. 205, 219, 234, 235, 275 Cox, P., see Lee, C. 67, 170 Cox, R.T., see Huard, V. 302 Cox, R.T., see Kheng, K. 301 Crawford, M., see Marrows, C.H. 127 Crawford, M.K., see Nachumi, B. 449, 450 Creeth, G.L., see Marrows, C.H. 127 Crespo, P., see Prados, C. 261 Creuzet, G., see Baibich, M.N. 4 Crommie, M.F. 208, 251, 276 Crommie, M.F., see Jamneala, T. 251, 252, 273 Crommie, M.F., see Madhavan, V. 252 Crommie, M.F., see Nagaoka, K. 251 Cronin, S.B., see Rogacheva, E.I. 336 Crooker, B.C., see Pekarek, T.M. 346 Crooker, S.A. 295, 296, 303, 309 Crooker, S.A., see Baumberg, J.J. 301 Cros, V. 235 Cros, V., see Marangolo, M. 248 Cros, V., see Vogel, J. 219 Cross, R.W., see Russek, S.E. 128 Crow, J.E., see Twardowski, A. 351, 357 Crowell, P.A. 310 Crozat, P., see Schumacher, H.W. 132 Crozier, P.A., see Yang, Z. 162, 178 Cuchet, R., see Lhermet, H. 135 Cullen, W.G., see Kaminski, A. 419 Cundiff, S.T., see Koch, M. 301 Currie, J.F., see Ciureanu, P. 504 Curro, N.J. 434, 451, 467 Cyrille, M.-C., see Ju, Y.S. 21 Cywi´nski, G. 307, 310 Cywi´nski, G., see Jusserand, B. 303 Cywi´nski, G., see Kossut, J. 310 Cywi´nski, G., see Kudelski, A. 298, 310 Cywi´nski, G., see Kutrowski, M. 295 Cywi´nski, G., see Wojtowicz, T. 295 Czeczot, M., see Cywi´nski, G. 310 Czeczot, M., see Kossut, J. 310 Czerw, R., see Zha, F.-X. 261 Czyzyk, M.T., see Leuken, H.v. 253

578

AUTHOR INDEX

da Silva, R.B. 548 Dabbagh, G., see Warren Jr., W.W. 474 Dabrowski, B., see Jorgensen, J.D. 457 D’Addato, S., see Fratucello, G.B. 231 D’Addato, S., see Gazzadi, G.C. 231 Daemen, J.T.F., see Rijks, Th.G.S.M. 13, 14, 18, 61, 118, 119, 138, 157 Dagotto, E. 390, 402, 405 Dagotto, E., see Hotta, T. 402 Dagotto, E., see Martins, G.B. 406 Dahl, M. 311, 353 Dahl, M., see Buda, B. 300 Dahlberg, E., see Miller, B.H. 62 Dahm, T. 418 Dahmani, F., see Kools, J.C.S. 37, 49, 71 Dahmoune, C. 235 Dai, N. 294, 297, 358 Dai, N., see Luo, H. 297 Dai, N., see Zhang, F.C. 297 Dai, P. 470, 471, 482 Dai, P., see Mook, H.A. 404, 411, 412, 419, 430, 439, 440, 470–472, 477 Dai, Y.Y., see Xiao, S.Q. 541 Daimon, H., see Imada, S. 310 Dakroub, H., see Jin, Z. 135, 136 Dallmeyer, A. 231 Dallmeyer, A., see Gambardella, P. 252 Damascelli, A., see Armitage, N.P. 465 Damascelli, A., see Feng, D.L. 482 Damm, T., see Tsunoda, M. 159, 167 D’Andrea, A. 296 Dang, L.S., see Brinkmann, D. 307 Dang, L.S., see Ulmer-Tufﬁgo, H. 298 Darlington, C.N.W., see Mikheenko, I.P. 234 Das, G.P. 276 Dashevsky, Z., see Casian, A. 328 Dass, R.I., see Tang, J.P. 457 Datta, S. 294 Datta, S., see Kolodziejski, L.A. 295 Datta, S., see Nurmikko, A.V. 296 Daughton, J.M. 24, 27 Daughton, J.M., see Beech, R.S. 22 Daughton, J.M., see Everitt, B.A. 57, 163 Daughton, J.M., see Wang, D. 27, 148 Daughton, J.M., see Wolf, S.A. 5, 208 Dauguet, P., see Dieny, B. 62 Davies, A. 222 Davies, A.D., see Pierce, D.T. 228 Davies, J.J., see Klar, P.J. 296, 308 Davis, J.C., see Hoffman, J.E. 413, 441, 442, 471, 479, 481 Davis, J.C., see Hudson, E.W. 481 Davis, J.C., see Lang, K.M. 441, 480 Davis, J.C., see Pan, S.H. 442, 481

Daw, M.S. 255 Daw, M.S., see Foiles, S.M. 255 de Aguiar, F.M., see Rezende, S.M. 164 de Andrada e Silva, E.A., see Abramof, E. 327 de Andrade, A.M.H., see da Silva, R.B. 548 de Andrade, A.M.H., see Viegas, A.D.C. 514 de Araújo, A.E.P., see Duque, J.G.S. 536 de Araújo, A.E.P., see Machado, F.L.A. 542–544 de Boeck, J., see Peeters, F.M. 309 de Boer, F.R. 77, 166 de Boer, F.R., see van Driel, J. 63, 64, 151, 152, 158, 159, 167, 176 de Gronckel, H., see Teitel’baum, G.B. 435, 452, 467 de Groot, R.A., see Fang, C.M. 221, 253 de Groot, R.A., see Leuken, H.v. 253 de Heer, W.A., see Billas, I.M.L. 250, 254, 255, 267 de Jong, M.J.M. 94 de Jong, M.J.M., see Rijks, Th.G.S.M. 61, 62, 104 de Jonge, W.J.M. 312, 315, 318, 320 de Jonge, W.J.M., see Chernyshova, M. 332, 338 de Jonge, W.J.M., see de Vries, J.J. 220, 244, 245, 247 de Jonge, W.J.M., see Eggenkamp, P.J.T. 318, 320, 321 de Jonge, W.J.M., see Grodzicka, E. 312 de Jonge, W.J.M., see Kepa, H. 336, 338, 340 de Jonge, W.J.M., see Koopmans, B. 301 de Jonge, W.J.M., see LeClair, P. 341 de Jonge, W.J.M., see Oepts, W. 138 de Jonge, W.J.M., see Rijks, Th.G.S.M. 13, 14, 16, 18, 61, 62, 66, 97, 104, 118, 119, 136–138, 157 de Jonge, W.J.M., see Stachow-Wójcik, A. 333–335 de Jonge, W.J.M., see Story, T. 315, 317, 318, 335 de Jonge, W.J.M., see Strijkers, G.J. 97, 100, 101, 220, 244, 245 de Jonge, W.J.M., see Swagten, H. 97, 98 de Jonge, W.J.M., see Swagten, H.J.M. 34, 46, 47, 52, 92, 93, 315, 353, 354 de Jonge, W.J.M., see Twardowski, A. 353 de Jonge, W.J.M., see Van de Veerdonk, R.J.M. 134 de Jonge, W.J.M., see van der Heijden, P.A.A. 151, 157, 163 de Jonge, W.J.M., see Vennix, C.W.H.M. 315, 320 de Jonge, W.J.M.M., see van der Heijden, P.A.A. 151, 157, 163

AUTHOR INDEX de la Figuera, J., see Gómez, L. 269 de la Huerta-Garnica, M.A., see Figuera, J. 258 de la Prieto, J.E., see Figuera, J. 268, 269 de Lozanne, A.L., see Ehrichs, E.E. 258 de Melo, M.A.C., see Hillberg, M. 361 de Miguel, J.J. 208 de Miguel, J.J., see Camarero, J. 143 de Miguel, J.J., see Gómez, L. 269 de Nooijer, M.-C., see McCord, J. 128 de Nooijer, M.C., see Folkerts, W. 22, 127–129 de Nooijer, M.C., see Kools, J.C.S. 5 de Nooijer, M.C., see Van de Veerdonk, R.J.M. 134 de Rooy, J.C.J.M., see MacLaughlin, D.E. 433 de Veirman, A.E.M., see Kools, J.C.S. 144–147 de Veirman, A.E.M., see Lenssen, K.-M.H. 31, 39, 67, 71, 157 de Veirman, A.E.M., see Rijks, Th.G.S.M. 66, 97, 104 de Visser, A., see Ivanchik, I.I. 312 de Vries, J.J. 220, 244, 245, 247 de Vries, J.J., see Johnson, M.T. 143 Deaven, D.M. 143 Debnath, M.C. 300, 301 Debnath, M.C., see Chen, Z.H. 303 Debnath, M.C., see Kayanuma, K. 305, 306 Debska, U., see Bartholomev, D.U. 351 Dederichs, P.H., see Anisimov, V.I. 409 Dederichs, P.H., see Bellini, V. 205 Dederichs, P.H., see Blügel, S. 224, 235 Dederichs, P.H., see Freyss, M. 248 Dederichs, P.H., see Levanov, N.A. 215 Dederichs, P.H., see Nonas, B. 222 Dederichs, P.H., see Oswald, A. 260 Dederichs, P.H., see Papanikolaou, N. 213, 253 Dederichs, P.H., see Stepanyuk, V.S. 212, 253 Dederichs, P.H., see Weber, S.E. 251 Dee, R.H. 22 Degiorgi, L., see Pashkevich, Yu.G. 461, 463 DeHerra, M., see Tehrani, S. 24, 57 Dekadjevi, D.T. 108 Dekker, M.J., see Coehoorn, R. 153 Delalande, C. 297 Delalande, C., see Deleporte, E. 297 Delalande, C., see Lebihen, T. 296 Delalande, C., see Roussignol, Ph. 305 Deleporte, E. 297 Deleporte, E., see Lebihen, T. 296 Deleporte, E., see Roussignol, Ph. 305 Delin, A., see Le Bacq, O. 228 Demand, M., see Encinas, A. 548 Demangeat, C. 203, 206, 216, 219, 228, 229, 237, 270 Demangeat, C., see Amalou, F. 223

579

Demangeat, C., see Bazhanov, D.I. 205, 252 Demangeat, C., see Binns, C. 205, 208, 220, 241, 242, 263 Demangeat, C., see Bouarab, S. 205, 224, 227, 228, 234 Demangeat, C., see Chouairi, A. 236 Demangeat, C., see Dahmoune, C. 235 Demangeat, C., see Dorantes-Dávila, J. 275 Demangeat, C., see Dreyssé, H. 205, 208, 234, 275 Demangeat, C., see Elmouhssine, O. 228, 229, 238 Demangeat, C., see Hadj-Larbi, A. 217, 231–234, 275 Demangeat, C., see Hamad, B.A. 224, 237 Demangeat, C., see Izquierdo, J. 226, 227, 277 Demangeat, C., see Khan, M.A. 228 Demangeat, C., see Krüger, P. 242–244, 262 Demangeat, C., see Kulikov, N.I. 223 Demangeat, C., see Mokrani, A. 230, 231 Demangeat, C., see M’Passi-Mabiala, B. 249, 276 Demangeat, C., see Nait-Laziz, H. 235, 249 Demangeat, C., see Ostanin, S. 248, 249 Demangeat, C., see Parlebas, J.C. 262 Demangeat, C., see Pick, S. 206, 249 Demangeat, C., see Rennert, P. 252 Demangeat, C., see Taguchi, M. 229 Demangeat, C., see Talanana, M. 227 Demangeat, C., see Uzdin, S. 241, 271, 272 Demangeat, C., see Uzdin, V. 207 Demangeat, C., see Uzdin, V.M. 206, 217, 222, 224, 249, 270, 273 Demangeat, C., see Vega, A. 207, 218, 221–223, 225–228, 255 Demangeat, C., see Yartseva, N.S. 204 Demangeat, C., see Zenia, H. 217, 220, 275 Demangeat, C., see Ziane, A. 234 Demianiuk, M., see Bindilatti, V. 355 Demianiuk, M., see Boonman, M.E.J. 346 Demianiuk, M., see Foner, S. 353, 355 Demianiuk, M., see Gennser, U. 351, 355 Demianiuk, M., see Herbich, M. 351 Demianiuk, M., see Isber, S. 355 Demianiuk, M., see Krevet, R. 346 Demianiuk, M., see Liu, X.C. 355 Demianiuk, M., see Mac, W. 346–351, 357 Demianiuk, M., see McCabe, G.H. 347, 349 Demianiuk, M., see Twardowski, A. 346, 350, 351, 353, 357, 358 Demler, E. 415, 417, 428, 449, 480 Demler, E., see Bazaliy, Y.B. 417 Demler, E., see Meixner, S. 416

580

AUTHOR INDEX

Demler, E., see Rabello, S. 416 Demler, E., see Zhang, Y. 413, 420, 471, 480 Demler, E.A., see Pryadko, L.P. 404 Demokritov, S.O., see Bürgler, D.E. 5, 110, 143, 207 Demokritov, S.O., see Mewes, T. 152 den Broeder, F.J.A., see de Vries, J.J. 220, 244, 245, 247 Denecke, R. 315 Deng, J., see Apsel, S.E. 254–257, 267 Deng, J., see Bloomﬁeld, L.A. 211, 250 Denier van de Gon, A.W., see Shutthanandan, V. 266 Dennis, B.S., see Blumberg, G. 464 Dennis, C., see Gregg, J.F. 5 Deportes, J., see Gratens, X. 312 Deportes, J., see Nawrocki, M. 351 Deportes, J., see Zielinski, M. 351, 353 Deprot, S., see Reynet, O. 535 Der, S., see Chih-Ming, L. 358 Derbyshire, H.S., see Binns, C. 205, 263 Desousa Meneses, D., see Odier, P. 459, 461 Dessau, D.S. 427 Dessau, D.S., see Chuang, Y.-D. 482 Dessau, D.S., see King, D.M. 464 Dessau, D.S., see Marshall, D.S. 476 Devasahayam, A. 157, 158, 164 Devasahayam, A., see Hegde, H. 70, 71 Devasahayam, A., see Lee, C. 67, 170 Devasahayam, A.J. 161, 166, 170 Devasahayam, A.J., see Kools, J.C.S. 145, 146 Devine, P., see Jackson, S. 296 Devreese, J.T., see Nicholas, R.J. 306 Devreese, J.T., see Swagten, H.J.M. 315 Dey, S., see Shi, S. 22 Dey, S., see Tong, H.C. 33, 51, 55, 124 Dhese, K.A., see Harrison, P. 296 Dhesi, S.S., see Baker, S.H. 264 Dhesi, S.S., see Binns, C. 264 Dhesi, S.S., see Gambardella, P. 254 di Bona, A., see Luches, P. 231 Di Castro, C., see Seibold, C. 397 Di Castro, D., see Bianconi, A. 408 Di Castro, D., see Kusmartsev, F.V. 409 Di Castro, D., see Saini, N.L. 454 Diao, Z. 38, 49, 50 Diao, Z.T., see Huai, Y. 49 DiCarlo, J., see King, D.M. 464 Diduszko, R., see Nadolny, A.J. 321, 322 Diéguez, O. 215 Diéguez, O., see Longo, R.C. 267 Diéguez, O., see Rey, C. 261 Diekhöner, L., see Knorr, N. 251

Dieny, B. 4, 5, 7, 9, 16, 18, 19, 30–33, 39, 41, 45, 59–62, 66, 90, 92, 94–96, 99, 101, 103, 104, 106, 112, 113, 115, 120, 123, 157, 178 Dieny, B., see Camarero, J. 178 Dieny, B., see Cowache, C. 164 Dieny, B., see Li, M. 61, 159, 167 Dieny, B., see Nozières, J.P. 72 Dieny, B., see Speriosu, V.S. 18, 40, 72, 110, 145 Dieny, B., see Vedyaev, A. 94, 103, 109 Dietl, T. 291, 300, 304, 311, 312, 318, 322, 325 Dietl, T., see Andrearczyk, T. 304 Dietl, T., see Ferrand, D. 310, 311 Dietl, T., see Głód, P. 357, 362 Dietl, T., see Haury, A. 311 Dietl, T., see Jaroszy´nski, J. 304 Dietl, T., see Karczewski, G. 311 Dietl, T., see Matsukura, F. 291, 311 Dietl, T., see Prinz, A. 323 Dietl, T., see Sawicki, M. 310, 311 Dimashko, Y., see Hasselmann, N. 404 Dimashko, Y.A. 404 Dimitriev, A., see Lin, N. 275 Dimitrov, D.V., see Li, Y.F. 176 Dimitrov, D.V., see Prados, C. 89 Ding, H., see Norman, M.R. 427 Ding, J., see Han, D. 161, 162 Dinia, A., see Bensmina, F. 73 Dinia, A., see Colis, S. 45 Dite, A.F., see Kulakovskii, V.D. 306 Dłu˙zèwski, P., see Ghali, M. 304, 305 Dłu˙zèwski, P., see Ma´ckowski, S. 308 Dobrowolska, M., see Dai, N. 294, 297, 358 Dobrowolska, M., see Kim, C.S. 306, 309, 310 Dobrowolska, M., see Lee, S. 297, 305 Dobrowolska, M., see Luo, H. 295, 297 Dobrowolska, M., see Ma´ckowski, S. 308 Dobrowolska, M., see Syed, M. 297 Dobrowolska, M., see Zhang, F.C. 297 Dobrowolski, W. 360 Dobrowolski, W., see Dugaev, V.K. 339, 342 Dobrowolski, W., see Grodzicka, E. 312 Dobrowolski, W., see Hillberg, M. 361 Dobrowolski, W., see Kossut, J. 291–293, 312–315, 318, 325, 352, 354, 356, 360, 362 Dobrowolski, W., see Łusakowski, A. 318 Dobrowolski, W., see Mycielski, A. 356, 357 Dobrowolski, W., see Nadolny, A.J. 321, 322 Dobrowolski, W., see Stachow-Wójcik, A. 333–335 Dobrowolski, W., see Stolpe, I. 340, 341 Dobrowolski, W., see Story, T. 312, 328–330, 333–336 Dobrowolski, W., see Zeitler, U. 360 Dobrowolski, W.D., see Ivanchik, I.I. 312

AUTHOR INDEX Do˘gan, F., see Dai, P. 470, 471, 482 Do˘gan, F., see Fong, H.F. 389, 471 Do˘gan, F., see Mook, H.A. 404, 411, 412, 419, 430, 439, 440, 470–472, 477 Dohnomae, H., see Yamamoto, H. 9, 27 Dolabdjian, C., see Robbes, D. 550 Domanski, T. 419 Dombre, T. 392 Dominguez, M., see Loﬂand, S.E. 514, 546 Domukhovski, V., see Chernyshova, M. 332, 338 Domukhovski, V., see Łusakowski, A. 318 Domukhovski, V., see Nadolny, A.J. 321, 322 Domukhovski, V., see Radchenko, M.V. 315 Donahue, M.J., see Russek, S.E. 132 Dong, C. 515 Dong, J., see Sheng, L. 94 Donkers, J.J.T.M., see Lenssen, K.-M.H. 31, 39, 43, 67, 71, 157 Dooglav, A.V. 473 Doole, R.C., see Portier, X. 129 Doran, A., see Wu, Y.Z. 230 Dorantes-Dávila, J. 261, 275 Dorantes-Dávila, J., see Vega, A. 207, 218, 221, 225–227, 267 Dorleijn, J.W.F. 79 Dorneich, A. 416 Dos Santos, O., see Mahoukou, F. 312 Dovek, M., see Gurney, B.A. 22, 31, 129, 140 Dovek, M.M., see Spong, J.K. 22, 23 Dragon, T., see Guo, H.Q. 504, 532, 539 Drakaki, M., see Chen, J. 249 Drchal, V., see Kudrnovsky, J. 143 Drchal, V., see Pajda, M. 209 Dresselhaus, G., see Chen, J.J. 343 Dresselhaus, M.S., see Chen, J.J. 343 Dresselhaus, M.S., see Giebułtowicz, T.M. 343 Dresselhaus, M.S., see Hicks, L.D. 328 Dresselhaus, M.S., see Rogacheva, E.I. 336 Drewes, J., see Katti, R.R. 57, 58 Dreyssé, H. 205, 208, 234, 275 Dreyssé, H., see Alouani, M. 209 Dreyssé, H., see Bouarab, S. 205, 224, 227, 228, 234 Dreyssé, H., see Chouairi, A. 236 Dreyssé, H., see Dorantes-Dávila, J. 275 Dreyssé, H., see Freyss, M. 221, 222 Dreyssé, H., see Galanakis, I. 237 Dreyssé, H., see Izquierdo, J. 226, 227 Dreyssé, H., see Nait-Laziz, H. 235 Dreyssé, H., see Ostanin, S. 248, 249 Dreyssé, H., see Pick, S. 249 Dreyssé, H., see Rennert, P. 252 Dreyssé, H., see Turek, I. 221, 222

581

Dreyssé, H., see Vega, A. 207, 218, 221–223, 225–228, 255 Drittler, B., see Blügel, S. 224, 235 du Croo de Jongh, M.S.L. 401 Ducastelle, F. 242 Duchateau, J.P.W.B. 66, 67 Duchateau, J.P.W.B., see Coehoorn, R. 143 Duffy, D.M. 205, 262–264 Dugaev, V.K. 339, 342 Dugaev, V.K., see Kowalczyk, L. 340, 341 Dugaev, V.K., see Litvinov, V.I. 103 Dugaev, V.K., see Stolpe, I. 340, 341 Dumas, J.F., see Bruno, A. 312, 328, 329 Dumelow, T., see Soares, J.M. 532 Dunlap, B.I., see Reddy, B.V. 235 Dunn, J.H., see Mancoff, F.B. 25 Dunsiger, S.R., see Chow, K.H. 460 Dunsiger, S.R., see Sonier, J.E. 477 Duo, L. 249 Dupas, C. 9 Dupas, C., see Renard, J.-P. 86 Dupree, R., see Williams, G.V.M. 475, 476 Duque, J.G.S. 536 Duque, J.G.S., see Knobel, M. 502, 504, 506 Dura, J.A., see Hjörvarsson, B. 206, 219, 220, 248 Durbin, S.M. 295, 344, 345 Durbin, S.M., see Han, J. 295, 344, 345 Durlam, M. 24 Durlam, M., see Chen, E.Y. 128 Durlam, M., see Tehrani, S. 24, 57 Dürr, C., see Borisenko, S.V. 441 Dutcher, J.R., see Heinrich, B. 230 Duvail, J.-L., see Fert, A. 104 Duvail, J.L. 12, 61, 94, 104, 105 Duvail, J.L., see George, J.M. 86 Dwight, K., see Dahl, M. 353 Dwight, K., see Fries, T. 357 Dwight, K., see Shapira, Y. 353 Dwight, K., see Shih, O.W. 351 Dwight, K., see Vu, T.Q. 353, 355 Dybko, K. 360 Dybko, K., see Łazarczyk, P. 315 Dybko, K., see Mycielski, A. 356, 357 Dynowska, E., see Hennion, B. 311 Dynowska, E., see Janik, E. 295 Dynowska, E., see Jouanne, M. 311 Dynowska, E., see Miotkowska, S. 315 Dynowska, E., see Sawicki, M. 311 Dynowska, E., see Stachow-Wójcik, A. 310 Dynowska, E., see Szuszkiewicz, W. 311, 361 Dynowska, E., see Wojtowicz, T. 295 Ebels, U., see Encinas, A. 548

582

AUTHOR INDEX

Ebels, U., see Mougin, A. 171 Eberhardt, W., see Dallmeyer, A. 231 Eberhardt, W., see Gambardella, P. 252 Eberhardt, W., see Kachel, T. 236 Eberhardt, W., see Pampuch, C. 253 Ebert, H. 81, 209 Ebert, H., see Banhart, J. 7, 80, 82, 83 Ebert, H., see Huhne, T. 241 Ebert, H., see Scherz, A. 227, 228 Echer, C.J., see Krishnan, K. 162, 170 Eckert, D., see Borisenko, S.V. 441 Eckstein, J.N., see Marshall, D.S. 476 Eddrief, M., see Marangolo, M. 248 Edelstein, A.S. 231 Edelstein, A.S., see Miller, M.M. 22 Edelstein, R.L. 24 Edelstein, R.L., see Miller, M.M. 24 Eder, R. 416 Edmonds, K.W., see Baker, S.H. 241, 264 Edmonds, K.W., see Binns, C. 264 Egami, T., see Arai, M. 470, 471, 473 Egami, T., see McQueeney, R.J. 409 Egelhoff, W.F. 51, 52, 93 Egelhoff, W.F., see Bae, S. 25, 35, 37, 46, 164 Egelhoff, W.F., see McMichael, R.D. 138 Egelhoff Jr., W.F. 34, 35, 42, 51, 52, 69, 70, 144 Egelhoff Jr., W.F., see Bae, S. 164 Egelhoff Jr., W.F., see Bennett, W.R. 110 Egelhoff Jr., W.F., see Chopra, H.D. 70, 144, 164 Egelhoff Jr., W.F., see Fry, R.A. 138 Egelhoff Jr., W.F., see Parks, D.C. 146 Egelhoff Jr., W.F., see Rezende, S.M. 164 Egelhoff Jr., W.F., see Torok, E.J. 25 Eggenkamp, P., see Twardowski, A. 346 Eggenkamp, P.J.T. 318, 320, 321 Eggenkamp, P.J.T., see de Jonge, W.J.M. 320 Eggenkamp, P.J.T., see Mac, W. 346–348 Eggenkamp, P.J.T., see Story, T. 315, 317, 318 Eggenkamp, P.J.T., see Swagten, H.J.M. 353, 354 Eggenkamp, P.J.T., see Vennix, C.W.H.M. 315, 320 Egorov, A.V., see Dooglav, A.V. 473 Eguchi, S., see Nagasaka, K. 150, 162 Eguchi, S., see Tanaka, A. 13 Egues, J.C. 306 Ehresmann, A., see Juraszek, J. 135, 136, 171 Ehresmann, A., see Mougin, A. 171 Ehret, G., see Pham-Huu, C. 265 Ehrichs, E.E. 258 Eigler, D.M. 258 Eisaki, H., see Armitage, N.P. 465 Eisaki, H., see Arumugam, S. 454 Eisaki, H., see Chuang, Y.-D. 482

Eisaki, H., see Feng, D.L. 482 Eisaki, H., see Hoffman, J.E. 413, 441, 442, 471, 479, 481 Eisaki, H., see Howald, C. 481 Eisaki, H., see Hudson, E.W. 481 Eisaki, H., see Ino, A. 455 Eisaki, H., see Kojima, K.M. 449 Eisaki, H., see Lanzara, A. 409, 427, 454, 482 Eisaki, H., see Noda, T. 454 Eisaki, H., see Pan, S.H. 442, 481 Eisaki, H., see Pellegrin, E. 463 Eisaki, H., see Zhou, X.J. 441, 454 El Ouazzani, A. 351, 352 Elangovan, A. 301 Elefant, D., see Tietjen, D. 116, 123 Elmouhssine, O. 228, 229, 238–240 Elmouhssine, O., see Izquierdo, J. 226, 227 Elmouhssine, O., see Mokrani, A. 230, 231 Elmouhssine, O., see Taguchi, M. 229 Elsaki, H., see Lang, K.M. 441, 480 Elschner, B., see Kochelaev, B.I. 453 Elstner, N. 486 Emery, V.J. 390, 398, 407 Emery, V.J., see Carlson, E.W. 389, 398, 406 Emery, V.J., see Kivelson, S.A. 392, 454, 455 Emery, V.J., see Löw, U. 407 Emery, V.J., see Luther, A. 449 Emery, V.J., see Pryadko, L.P. 404 Emmert, J.W., see Apsel, S.E. 254–257, 267 Emmert, J.W., see Bloomﬁeld, L.A. 211, 250 Encinas, A. 138, 548 Enders, A., see Sander, D. 208 Endoh, Y., see Arai, M. 470, 471, 473 Endoh, Y., see Greven, M. 486 Endoh, Y., see Hosoya, S. 456 Endoh, Y., see Kimura, H. 446 Endoh, Y., see Koike, Y. 448 Endoh, Y., see Lee, Y.S. 448, 449 Endoh, Y., see Matsuda, M. 466, 467 Endoh, Y., see McQueeney, R.J. 409 Endoh, Y., see Nakajima, K. 485, 486 Endoh, Y., see Wakimoto, S. 447, 448 Endoh, Y., see Yamada, K. 447, 448, 466 Engel, D., see Juraszek, J. 135, 136, 171 Engel, D., see Mougin, A. 171 Enomoto, H., see Iida, M. 323, 324 Eremenko, V.V., see Pashkevich, Yu.G. 461, 463 Eremin, I., see Manske, D. 418 Eriksson, O. 209 Eriksson, O., see Broddefalk, A. 228 Eriksson, O., see Le Bacq, O. 228 Eriksson, O., see Taga, A. 206 Erler, F., see Zahn, P. 111

AUTHOR INDEX Ernult, F., see Camarero, J. 178 Ernzerhof, M., see Perdew, J.P. 213 Errebbahi, A. 322, 326 Erskine, J.L., see Chen, J. 249 Erskine, J.L., see Finck, F.L. 224 Erwin, E.W., see Lee, Y.S. 448, 449 Erwin, R., see Kimura, H. 446 Erwin, R., see Wakimoto, S. 447, 448 Erwin, R.W., see van der Zaag, P.J. 155 Eschrig, M. 427, 482 Eschrig, M., see Abanov, Ar. 428 Eschrig, M., see Mitrovi´c, V.F. 470, 477 Escorcia, O., see Hylton, T.L. 70 Escorne, M. 315 Eskes, H. 404 Estournès, C., see Pham-Huu, C. 265 Etgens, V.H., see Marangolo, M. 248 Etienne, P., see Baibich, M.N. 4 Etrillard, J. 479 Evangelakis, G.A., see Lekka, Ch.E. 214 Everett, R.K., see Edelstein, A.S. 231 Everitt, B.A. 57, 163 Everitt, B.A., see Qian, Z. 161 Evers, J., see Lambrecht, A. 312, 323, 324 Evetts, J.E., see Prieto, J.L. 22 Fabricius, K., see Löw, U. 407 Faini, G., see Mougin, A. 171 Falco, C.M., see Keavney, D.J. 111 Falicov, L.M., see Hood, R.Q. 103 Falicov, L.M., see Parlebas, J.C. 204 Falicov, L.M., see Victora, R.H. 203 Falk, H. 306 Fanciulli, M. 245 Fang, C.M. 221, 253 Fang, M., see Orlowski, B.A. 315 Fang, T.-N. 58 Fantner, E.J., see Krost, A. 324 Farrow, R.F.C. 162, 164, 172 Farrow, R.F.C., see Ju, G. 172 Farrow, R.F.C., see Krishnan, K. 162, 170 Faschinger, W., see Abramof, E. 344 Faschinger, W., see Giebułtowicz, T.M. 344, 345 Faschinger, W., see Krenn, H. 344, 345 Faschinger, W., see Nunez, V. 345 Faschinger, W., see Pohlt, M. 344, 345 Fassbender, J., see Juraszek, J. 135, 136, 171 Fassbender, J., see Mougin, A. 171 Fatah, J.M. 296 Fau, C., see Averous, M. 328 Fau, C., see Errebbahi, A. 322, 326 Fau, C., see Gratens, X. 328 Fau, C., see Isber, S. 312, 322, 323

583

Fau, C., see Lombos, B.A. 328 Fau, C., see Mahoukou, F. 312 Fau, C., see Misra, S.K. 322, 323 Faure-Vincent, J. 276 Fauster, Th., see Nolting, W. 209 Fawcett, E. 218, 221 Fayﬁeld, R. 24 Faymonville, R., see Krost, A. 324 Fedorenko, A.I., see Kolesnikov, I.V. 340 Fedorov, A.V., see Johnson, P.D. 482 Fehér, T., see Julien, M.-H. 452, 478 Feiguin, A., see Hotta, T. 402 Feiguin, A., see Martins, G.B. 406 Feiner, L.F., see van der Zaag, P.J. 155 Feit, Z. 328 Feldhaus, J., see Iwanowski, R.J. 315 Feldman, D.E. 277 Feldman, J., see Koch, M. 301 Feng, D.F., see Lanzara, A. 409, 427, 454, 482 Feng, D.L. 482 Feng, D.L., see Armitage, N.P. 465 Feng, D.L., see Shen, Z.-X. 479, 482 Feng, T. 41 Fensham, P.J., see Bacskay, G.B. 323 Ferhat, M., see Gratens, X. 328 Fernandes, J., see Veloso, A. 159 Fernández, P., see Bucher, J.P. 258, 259 Fernandez-de-Castro, J., see Chen, J. 32, 33, 112, 114, 159 Fernandez-de-Castro, J., see Nakamura, K. 155 Ferrand, D. 310, 311 Ferrand, D., see Dietl, T. 311 Ferrand, D., see Kossacki, P. 302 Ferrand, D., see Sawicki, M. 311 Ferré, J., see Schumacher, H.W. 132 Ferreira, H.H., see Graham, D.L. 22, 24 Ferreira, M., see Veloso, A. 159 Ferreira, S.O., see Abramof, E. 327 Ferrer, J., see Pruneda, J.M. 245–248 Ferrer, J., see Robles, R. 225, 245 Ferrón, J., see Gómez, L. 269 Fert, A. 5, 78, 81, 85, 87, 104, 143 Fert, A., see Baibich, M.N. 4 Fert, A., see Barna´s, J. 94 Fert, A., see Barthélémy, A. 5, 91, 109 Fert, A., see Campbell, I.A. 7, 78–80, 102 Fert, A., see Duvail, J.L. 12, 61, 94, 104, 105 Fert, A., see George, J.M. 86 Fert, A., see Levy, P.M. 109 Fert, A., see Valet, T. 86 Fert, A., see Vouille, C. 84 Fettar, F., see Camarero, J. 178 Feuillet, G., see Bodin, C. 295 Feuillet, G., see Gaj, J.A. 296

584

AUTHOR INDEX

Feuillet, G., see Lawrence, I. 305 Fidler, J. 116 Fidler, J., see Suess, D. 155 Fiederling, R. 304 Fiederling, R., see Gruber, Th. 305 Fiederling, R., see Kutrowski, M. 295, 296 Fiederling, R., see Wojtowicz, T. 295 Figuera, J. 258, 268, 269 Filin, A.I., see Kulakovskii, V.D. 306 Fillion, G., see Bruno, A. 312, 328 Finazzi, M., see Andrieu, S. 228 Finazzi, M., see Cros, V. 235 Finazzi, M., see Gallani, J.L. 275 Finck, F.L. 224 Finetti, P., see Binns, C. 264 Fink, J., see Borisenko, S.V. 441 Fink, J., see Kordyuk, A.A. 483 Finkelstein, A.M. 453, 478 Finnemore, D.K., see Huh, Y.M. 453 Fischer, F., see Mackh, G. 300 Fischer, H., see Andrieu, S. 220, 228 Fischer, M., see Pashkevich, Yu.G. 461, 463 Fisher, M.P.A., see Balents, L. 413 Fisher, M.P.A., see Senthil, T. 420 Fishman, G., see Brinkmann, D. 307 Fishman, G., see Ferrand, D. 310, 311 Fishman, G., see Marsal, L. 307 Fishman, G., see Nawrocki, M. 300 Fisk, Z., see MacLaughlin, D.E. 433 Fisk, Z., see Suh, B.J. 452 Fisk, Z., see Takigawa, M. 437, 474, 475 Fita, P. 312 Fjaerestad, J.O., see Marston, J.B. 402 Flack, F., see Crowell, P.A. 310 Flanagan, J.A., see Popovic, R.S. 23 Flatté, M.E. 481 Fleck, M. 397, 398 Flevaris, N.K., see Wilhelm, F. 220, 237 Flores, A.G., see Raposo, V. 510 Flytzanis, C., see Buss, C. 299 Flytzanis, C., see Leisching, P. 299 Foiles, S.M. 255 Folkerts, W. 22, 115, 127–129 Folkerts, W., see Kools, J.C.S. 5 Folkerts, W., see Rijks, Th.G.S.M. 14, 16, 138, 157 Follath, R., see Kordyuk, A.A. 483 Foner, S. 353, 355 Foner, S., see Dahl, M. 353 Foner, S., see Shapira, Y. 353 Fong, H.F. 389, 470, 471, 482 Fong, H.F., see Sidis, Y. 472 Fontaine, A., see Cros, V. 235 Fontaine, A., see Lawniczak-Jabło´nska, K. 345

Fontaine, A., see Vogel, J. 219 Fontaine, J.M., see Acher, O. 514 Fontana, R.E. 22 Fontana, R.E., see Childress, J.R. 55, 159 Fontana, R.E., see Ju, Y.S. 21 Fontana, R.E., see Lin, T. 151, 157, 161, 163 Fontana, R.E., see Spong, J.K. 22, 23 Fontana, R.E., see Tsang, C. 22, 127 Fontana Jr., R.E., see Gurney, B.A. 22, 31, 129, 140 Fontana Jr., R.E., see Heim, D.E. 127, 129 Fontana Jr., R.E., see Tsang, C.H. 4, 5, 22 Forchel, A., see Bacher, G. 309 Forchel, A., see Kulakovskii, V.D. 306 Forchel, A., see Maksimov, A.A. 309 Forchel, A., see Tönnies, D. 296 Forchel, A., see Welsch, M.K. 307 Forchel, A., see Zaitsev, S. 309 Forró, L., see Borisenko, S.V. 441 Fournier, P., see Biswas, A. 464 Fournier, P., see Blumberg, G. 464 Fournier, P., see Marshall, D.S. 476 Foxon, C.T., see Van Wees, B.J. 258 Foy, E., see Andrieu, S. 220 Fradkin, E. 392 Fradkin, E., see Kivelson, S.A. 392, 413, 454, 455 Fraga, E., see Kraus, L. 545 Fraga, G.L.F. 535 Frait, Z. 528, 529 Frait, Z., see Kraus, L. 533 Fraitová, D. 531 Fraitová, D., see Frait, Z. 528, 529 Frank, N., see Geist, F. 323, 324 Frank, N., see Springholz, G. 342 Frank, S., see Tast, F. 265 Frankel, R.B., see Story, T. 315 Fratucello, G.B. 231 Fraune, M. 164 Freeman, A.J. 208 Freeman, A.J., see Fu, C.L. 224 Freeman, A.J., see Geng, W.T. 218 Freeman, A.J., see Lee, J.I. 230 Freeman, A.J., see Li, C. 235 Freeman, A.J., see Nakamura, K. 155 Freeman, A.J., see Ohnishi, S. 218, 224 Freeman, A.J., see Wimmer, E. 211 Freeman, A.J., see Wu, R. 208, 209, 228, 230 Freeman, M.R. 208 Freeman, M.R., see Awschalom, D.D. 301 Freijo, J.J. 536 Freijo, J.J., see Valenzuela, R. 552, 553 Freijo, J.J., see Vázquez, M. 553

AUTHOR INDEX Freitas, P., see Fraune, M. 164 Freitas, P.P. 22 Freitas, P.P., see Graham, D.L. 22, 24 Freitas, P.P., see Li, H. 22, 24, 36, 51, 52, 159, 167 Freitas, P.P., see Oliveira, N.J. 22 Freitas, P.P., see Schumacher, H.W. 132 Freitas, P.P., see Sousa, J.B. 37 Freitas, P.P., see Veloso, A. 34, 35, 45, 48, 49, 159 Freitas, P.P., see Ventura, J.O. 49 Frello, T., see Niemöller, T. 454 Frello, T., see Vigliante, A. 463 Frello, T., see Zimmermann, M.v. 454 Fretwell, H.M. 441 Fretwell, H.M., see Kaminski, A. 419 Frey, R., see Buss, C. 299 Frey, R., see Leisching, P. 299 Freyss, M. 221, 222, 248 Freyss, M., see Elmouhssine, O. 228 Freyss, M., see Turek, I. 221, 222 Friak, M. 218 Friederich, A., see Baibich, M.N. 4 Fries, T. 357 Fries, T., see Foner, S. 353, 355 Fries, T., see Gennser, U. 351, 355 Fries, T., see Twardowski, A. 346, 353, 358 Frikkee, E., see Vennix, C.W.H.M. 315, 320 Fritzsche, H., see Nawrath, T. 227 Froba, M., see Chen, L. 307 Fromherz, T., see Głód, P. 362 Fronc, K., see Cywi´nski, G. 310 Fronc, K., see Kossut, J. 310 Fronc, K., see Kudelski, A. 310 Fronc, K., see Ma´ckowski, S. 309 Froudakis, G.E. 261 Froudakis, G.E., see Andriotis, A.N. 261, 262, 265 Froyen, S., see Louie, S.G. 213 Fry, R.A. 138 Fryer, P.M., see Holloway, K. 67 Fu, C.L. 224 Fu, C.L., see Freeman, A.J. 208 Fu, C.L., see Lee, J.I. 230 Fu, C.L., see Li, C. 235 Fu, C.L., see Ohnishi, S. 224 Fu, L.P. 358 Fu, L.P., see Jonker, B.T. 301, 358 Fu, Q., see Durbin, S.M. 295, 344, 345 Fuchs, K. 87 Fuchs, P. 226, 227 Fudamoto, Y., see Kojima, K.M. 449 Fudamoto, Y., see Nachumi, B. 449, 450 Fudamoto, Y., see Savici, A.T. 449

585

Fujii, T., see Takayasu, M. 139 Fujii, T., see Uchiyama, S. 139 Fujikata, J. 151, 157, 161–164, 172, 174, 175 Fujimori, A., see Harima, N. 466 Fujimori, A., see Imada, M. 390 Fujimori, A., see Ino, A. 455 Fujimori, A., see Lanzara, A. 409, 427, 454, 482 Fujimori, A., see Satake, M. 463 Fujimori, A., see Zhou, X.J. 454 Fujimoto, F., see Nakamoto, K. 22, 47, 172 Fujita, M. 41, 447–449 Fujita, M., see Kato, M. 395 Fujita, M., see Wakimoto, S. 447, 448 Fujita, T., see Saini, N.L. 454 Fujiwara, H. 122, 155 Fujiwara, H., see Hou, C. 150 Fujiwara, H., see Nishioka, K. 122, 157, 174 Fujiwara, H., see Parker, M.R. 115, 122 Fujiwara, H., see Zhang, K. 161 Fujiwara, S., see Yokoi, H. 302 Fujiyasu, H., see Ishida, A. 312, 323, 324, 327 Fukamachi, T., see Kobayashi, Y. 477 Fukamichi, K., see Sasao, K. 166 Fukase, T., see Katano, S. 446 Fukase, T., see Suzuki, T. 445 Fuke, H.N. 32, 158, 174 Fuke, H.N., see Fukuzawa, H. 37, 48 Fuke, H.N., see Saito, A.T. 71, 73, 74 Fukuda, S., see Ozue, T. 22 Fukui, H., see Hamakawa, Y. 160, 168 Fukui, H., see Hoshiya, H. 160, 168 Fukui, H., see Meguro, K. 33, 123 Fukui, H., see Soeya, S. 160, 168 Fukuma, Y. 322 Fukumoto, C., see Motoyoshi, M. 24 Fukuyama, H., see Normand, B. 420 Fukuyama, H., see Tanamoto, T. 420 Fukuzawa, H. 33, 37, 42, 48, 137, 140, 141 Fukuzawa, H., see Kamiguchi, Y. 35, 48, 49, 77 Fulcomer, E. 151, 172, 173 Fullerton, E.E. 235 Fullerton, E.E., see Mattson, J.E. 220, 244 Fullerton, E.E., see Moser, A. 21 Fulthorpe, B.D., see Dekadjevi, D.T. 108 Funada, S., see Huai, Y. 33 Funada, S., see Hung, C.-Y. 136, 137, 140, 141, 162 Funada, S., see Mao, M. 158 Funada, S., see Tong, H.C. 33, 124 Funayama, T., see Ohsawa, Y. 26 Funayama, T., see Takagishi, M. 13 Furdyna, J.K. 291, 300, 312, 318, 325, 344, 345 Furdyna, J.K., see Awschalom, D.D. 301

586

AUTHOR INDEX

Furdyna, J.K., see Bartholomev, D.U. 351 Furdyna, J.K., see Baumberg, J.J. 301 Furdyna, J.K., see Dai, N. 294, 297, 358 Furdyna, J.K., see Datta, S. 294 Furdyna, J.K., see Giebułtowicz, T.M. 343–345, 353 Furdyna, J.K., see Karczewski, G. 324 Furdyna, J.K., see Kim, C.S. 306, 309, 310 Furdyna, J.K., see Klosowski, P. 344, 345 Furdyna, J.K., see Kossut, J. 294 Furdyna, J.K., see Lee, S. 297, 305 Furdyna, J.K., see Lewicki, A. 353 Furdyna, J.K., see Lin, J. 345 Furdyna, J.K., see Luo, H. 295, 297 Furdyna, J.K., see Ma´ckowski, S. 308 Furdyna, J.K., see Nunez, V. 345 Furdyna, J.K., see Rhyne, J.J. 345 Furdyna, J.K., see Samarth, N. 344, 345 Furdyna, J.K., see Smyth, J.F. 305 Furdyna, J.K., see Stumpe, L.E. 345 Furdyna, J.K., see Syed, M. 297 Furdyna, J.K., see Zhang, F.C. 297 Furis, M., see Jonker, B.T. 304 Furis, M., see Park, Y.D. 304 Furukawa, A., see Makino, E. 144, 145 Furukawa, S., see Yoshinaga, T. 504, 548 Furukawa, Y. 460 Furukawa, Y., see Wada, S. 460 Furuno, K., see Uchiyama, T. 552 Fuyama, M., see Hamakawa, Y. 47, 73, 160, 163, 168, 172 Fuyama, M., see Hoshiya, H. 160, 168 Fuyama, M., see Meguro, K. 33, 123 Fuyama, M., see Nakamoto, K. 22, 47, 172 Fuyama, M., see Soeya, S. 160, 163, 166, 168 Gabay, M., see Mendels, P. 478 Gadbois, J. 58 Gafron, T.J. 26, 140, 141 Gaines, J.M., see van der Zaag, P.J. 155 Gaj, J.A. 296 Gaj, J.A., see Grieshaber, W. 296, 299 Gaj, J.A., see Kossacki, P. 296, 302, 344, 345 Gaj, J.A., see Kudelski, A. 298, 310 Gaj, J.A., see Lemaître, A. 296 Gaj, J.A., see Mac, W. 349, 350 Gaj, J.A., see Maslana, W. 300 Gaj, J.A., see Scalbert, D. 351 Gaj, J.A., see Wypior, G. 296 Galanakis, I. 237 Gałazka, ˛ R.R. 291, 312–315, 318, 325, 328, 329, 345 Gałazka, ˛ R.R., see Kepa, H. 336, 338, 340

Gałazka, ˛ R.R., see Kowalczyk, L. 340, 341 Gałazka, ˛ R.R., see Łazarczyk, P. 315, 316 Gałazka, ˛ R.R., see Łusakowski, A. 318 Gałazka, ˛ R.R., see Nadolny, A.J. 321, 322 Gałazka, ˛ R.R., see Nawrocki, M. 300 Gałazka, ˛ R.R., see Orlowski, B.A. 315 Gałazka, ˛ R.R., see Stachow-Wójcik, A. 333–335 Gałazka, ˛ R.R., see Story, T. 315, 328–330, 335 Gałazka, ˛ R.R., see Swagten, H.J.M. 315 Gale, J.D., see Soler, J.M. 213 Gallagher, W.J., see Lu, Y. 5 Gallani, J.L. 275 Gallego, J.M., see Blum, V. 238 Gallego, L.J., see Calleja, M. 266 Gallego, L.J., see Diéguez, O. 215 Gallego, L.J., see Longo, R.C. 267 Gallego, L.J., see Rey, C. 261, 266 Gallego, L.J., see Robles, R. 214–216, 259, 260, 266–270 Galli, F., see Carretta, P. 434 Galtier, P. 67 Galtier, P., see Encinas, A. 138 Galtier, P., see Jérome, R. 67 Gambardella, P. 252, 254 Gambino, R.J., see Hegde, H. 70, 71 Gammel, J.T., see Yi, Y.S. 409 Gammel, P.L., see Ramirez, A.P. 459 Gamo, K., see Imada, S. 310 Gancedo, R., see Prados, C. 261 Gandit, P., see Dieny, B. 62 Gandra, F.G., see Kraus, L. 540, 541 Gandra, F.G., see Medina, A.N. 532 Gangopadhyay, S., see Mao, S. 161 Gangopadhyay, S., see Nishioka, K. 122 Gangopadhyay, S., see Szucs, J. 59, 60 Ganzer, S. 25 Gao, Z., see Han, D. 161, 162 Gao, Z., see Hou, C. 150 Gao, Z., see Mao, S. 32, 41, 60, 160, 168 García, A., see Soler, J.M. 213 García, D., see Barandiarán, J.M. 504 García, D., see Kurlyandskaya, G.V. 504 García, J.M. 505, 534, 542, 549 García, J.M., see Blum, V. 238 García, J.M., see García-Miquel, H. 515, 538, 548, 549 García, J.M., see Sinnecker, J.P. 504, 534, 541 García, J.M., see Vázquez, M. 541 Garcia, K.L. 508, 540 Garcia, K.L., see Carrasco, E. 537 Garcia, R., see Crooker, S.A. 295 García-Arribas, A., see Barandiarán, J.M. 513 García-Arribas, A., see Kurlyandskaya, G.V. 534 García-Arribas, A., see Muñoz, J.L. 528

AUTHOR INDEX García-Arribas, A., see Sinnecker, J.P. 541 García-Beneytez, J.M., see Garcia, K.L. 540 García-Beneytez, J.M., see García-Miquel, H. 515, 538, 548, 549 García-Beneytez, J.M., see Loﬂand, S.E. 514, 546 García-Beneytez, J.M., see Tejedor, M. 504, 539 García-Beneytez, J.M., see Vázquez, M. 504, 533, 541 García-Miquel, H. 515, 538, 548, 549 García-Miquel, H., see Bhagat, S.M. 548 García-Miquel, H., see Kurlyandskaya, G.V. 537 García-Miquel, H., see Loﬂand, S.E. 546 García-Rodeja, J., see Rey, C. 266 Gardner, J.S., see Sonier, J.E. 477 Gardonio, S., see Gambardella, P. 254 Gareev, R.R. 220, 245, 247 Garifullin, I.A., see Hübener, M. 276 Garrett, A., see Arai, M. 470, 471, 473 Garrison, J., see Kale, S. 535 Gaspar, J., see Li, H. 22, 24, 51, 52 Gassen, H.J., see Lenssen, K.-M.H. 22, 23, 44, 166 Gat, I.M., see Kojima, K.M. 449 Gat, I.M., see Savici, A.T. 449 Gatt, R., see Fretwell, H.M. 441 Gauthier, J., see Britel, M.R. 526, 548 Gauthier, J., see Ciureanu, P. 514 Gautier, F., see Loegel, B. 78, 105 Gautier, F., see Stoefﬂer, D. 207, 221, 228 Gautier-Picard, P., see Sidis, Y. 472 Gavigan, J.P., see Dieny, B. 115 Gavish, M. 357 Gazza, C., see Martins, G.B. 406 Gazzadi, G.C. 230, 231 Gazzadi, G.C., see Luches, P. 231 Gebhard, F. 385 Gebhardt, W., see Wypior, G. 296 G˛ebicki, W., see Cywi´nski, G. 307 Gedik, Z., see Bagci, V.M.K. 266 Gehanno, V., see Oliveira, N.J. 22 Gehring, G.A. 204 Gehring, P.M., see Fujita, M. 447, 448 Gehring, P.M., see Ichikawa, N. 452, 454, 455 Gehring, P.M., see Vajk, O.P. 463 Gehring, P.M., see Wakimoto, S. 445, 447, 448 Geiss, R.H., see Hwang, C. 65 Geist, F. 323–325 Geist, F., see Springholz, G. 327 Geng, W.T. 218 Genin, J.B., see Vedyaev, A. 94, 103 Gennser, U. 351, 355 Gennser, U., see Liu, X.C. 355 George, J.M. 86

587

George, J.M., see Marangolo, M. 248 George, P.K., see Qian, Z. 161 Georges, A. 397 Gerhard, T., see Dahl, M. 311 Gerhard, T., see König, B. 294 Gerrits, Th. 131, 133 Geurts, J., see Merkulov, I.A. 303 Geurts, J., see Wojtowicz, T. 302 Gewinner, G., see Elmouhssine, O. 238 Ghali, M. 304, 305 Ghatak, S.K., see Mandal, K. 504 Ghijsen, J., see Kowalski, B.J. 312, 329 Giamarchi, T. 397 Gibbons, M.R. 36, 48, 99 Gibbons, M.R., see Yan, X. 22 Gibbs, D., see Vigliante, A. 463 Gibbs, D., see Zimmermann, M.v. 454 Gibbs, H.M., see Kavokin, A.V. 299 Gibbs, M.R.J., see Squire, P.T. 505 Gibson, J.M., see Zimmermann, C.G. 268 Gider, S., see Baril, L. 138, 139 Gider, S., see Farrow, R.F.C. 162 Giebeler, C., see Lenssen, K.-M.H. 54 Giebułtowicz, T.M. 336, 343–345, 353 Giebułtowicz, T.M., see Kepa, H. 336, 338, 340, 343, 345 Giebułtowicz, T.M., see Klosowski, P. 344, 345 Giebułtowicz, T.M., see Lin, J. 345 Giebułtowicz, T.M., see Nunez, V. 343, 345 Giebułtowicz, T.M., see Rhyne, J.J. 345 Giebułtowicz, T.M., see Samarth, N. 344, 345 Gielkens, O., see Gerrits, Th. 131, 133 Gieres, G., see van den Berg, H.A.M. 42 Gierlings, M., see Miltenyi, P. 155 Giesbers, J.B., see Gijs, M.A.M. 135 Gijs, M.A.M. 5–7, 135 Gijs, M.A.M., see Amalou, F. 534 Gijs, M.A.M., see Van de Veerdonk, R.J.M. 134 Gil, B., see Kavokin, A.V. 298 Giles, N.C., see Suh, E.-K. 300 Gill, H., see Pinarbasi, M. 47, 164, 172 Gill, H.S., see Petrov, D.K. 22, 25 Gilles, J., see Imada, S. 310 Gilles, J., see Kuch, W. 230 Gillies, M. 36, 37, 48–50, 77 Gillies, M.F. 125, 126 Gillies, M.F., see Chapman, J.N. 129, 130 Gillies, M.F., see Gogol, P. 155 Gillies, M.F., see King, J.P. 36, 127 Gillies, M.F., see Rijks, Th.G.S.M. 66, 97, 104, 136–138 Giriat, W., see Giebułtowicz, T.M. 353 Giriat, W., see Heiman, D. 351

588

AUTHOR INDEX

Giriat, W., see Lewicki, A. 353 Głód, P. 357, 362 Głód, P., see Twardowski, A. 351, 357 Gnezdilov, V.P., see Pashkevich, Yu.G. 461, 463 Godard, C., see Mauger, A. 331, 342, 343 Godlewski, M., see Wojtowicz, T. 295 Goebel, E.O., see Koch, M. 301 Goebel, E.O., see Mackh, G. 300 Goede, O. 305 Goede, O., see Heimbrodt, W. 291, 305 Goede, O., see Pier, Th. 305 Goedecker, S. 215 Gogol, P. 155 Gogol, P., see Li, H. 36, 159, 167 Goiran, M., see Bindilatti, V. 355 Goiran, M., see Martin, J.L. 354 Goiser, A.M.J., see Hausleitner, C. 556 Goka, H., see Fujita, M. 449 Gökemeijer, N.J. 152 Gołacki, Z., see Anderson, J.R. 325 Gołacki, Z., see Bindilatti, V. 312, 326 Gołacki, Z., see Errebbahi, A. 322, 326 Gołacki, Z., see Fita, P. 312 Gołacki, Z., see Górska, M. 312, 315, 329 Gołacki, Z., see Gratens, X. 312, 328 Gołacki, Z., see Isber, S. 312, 322, 323 Gołacki, Z., see Kowalski, B.J. 312, 329 Gołacki, Z., see Lawniczak-Jabło´nska, K. 345 Gołacki, Z., see Mahoukou, F. 312 Gołacki, Z., see Martin, J.L. 354 Gołacki, Z., see Misra, S.K. 322, 323 Gołacki, Z., see Nawrocki, M. 351 Gołacki, Z., see Story, T. 329, 330 Gołacki, Z., see ter Haar, E. 312, 326 Goldberg, B.B., see Springholz, G. 327, 328 Golden, M.S., see Borisenko, S.V. 441 Golden, M.S., see Kordyuk, A.A. 483 Goldman, K.I., see Kepa, H. 343, 345 Goldman, K.I., see Nunez, V. 343 Goldoni, A. 205, 235, 236 Gollop, H., see Harrison, E.P. 504 Golnik, A., see Kudelski, A. 298 Golnik, A., see Maslana, W. 300 Golnik, A., see Niedermayer, Ch. 450 Goltsos, W.C. 323–325 Golubev, A.V., see Skipetrov, E.P. 312 Gómez, L. 269 Gomez, R.D., see Parks, D.C. 146 Gómez-Polo, C. 537, 542–544 Gómez-Polo, C., see Knobel, M. 504, 532, 536, 538, 539, 549 Gómez-Polo, C., see Pirota, K.R. 537 Gómez-Polo, C., see Vázquez, M. 553 Gomonay, H. 340

Goncalves da Silva, C.E.T. 300 Gong, H. 67 Gonsalves, J., see Durbin, S.M. 295, 344, 345 González, J., see Blanco, J.M. 504, 538–540, 544 González, J., see Cobeño, A.F. 555 González, J., see Garcia, K.L. 540 González, J., see Zhukova, V. 533 González, J.M., see Prados, C. 261 Gonzalez, M.T., see Carrasco, E. 537 Gooding, R.J., see Borsa, F. 433, 449 Goodwin, J.P., see Stirner, T. 296, 300 Gopalakrishnan, J., see Kale, S. 535 Gora, D. 397 Gore, J., see Atkinson, D. 553, 556 Gorecka, E., see Gallani, J.L. 275 Goringe, C.M., see Bowler, D.R. 215 Gorman, G., see Valletta, R.M. 140 Gorman, G.L., see Kung, K.T.-Y. 157, 174 Gorman, G.L., see Lin, T. 34, 150, 161, 166, 169 Gorn, N.L., see Berkov, D.V. 125 Gorny, K.R., see Pennington, C.H. 474 Goronkin, H., see Chen, E.Y. 128 Górska, M. 312, 315, 325, 329 Górska, M., see Anderson, J.R. 325 Górska, M., see Łusakowski, A. 318 Górska, M., see Story, T. 328–330 Gosele, U., see Ramos, C. 548 Goss Levi, B. 21, 390 Goto, M. 136 Goto, M., see Nachumi, B. 449, 450 Goto, T., see Suzuki, T. 445 Gould, C., see Egues, J.C. 306 Goulon, J., see Vogel, J. 219 Gourgon, C., see Brinkmann, D. 307 Gozar, A., see Blumberg, G. 464 Grabecki, G., see Dietl, T. 304 Grabecki, G., see Jaroszy´nski, J. 304 Grabecki, G., see Karczewski, G. 311 Grabecki, G., see Prinz, A. 323 Grabner, F., see Mao, M. 56 Grabner, F., see Mao, S. 69 Gradmann, U. 207 Graham, D.L. 22, 24 Graham Jr., C.D., see Shin, K.H. 545 Grange, W., see Cros, V. 235 Granovskii, A.B. 62, 104 Granovsky, A., see Dieny, B. 104 Granovsky, A.B., see Antonov, A. 537 Granovsky, A.B., see Usov, N. 526–528 Grasza, K., see Chernyshova, M. 332, 338 Grasza, K., see Kutrowski, M. 295 Gratens, X. 312, 322, 328

AUTHOR INDEX Gratens, X., see Bindilatti, V. 312, 326 Gratens, X., see Isber, S. 312, 322, 323, 326 Gray, K.E., see Zasadzinski, J.F. 481 Grazioli, C., see Gambardella, P. 254 Green, M.A., see Chow, K.H. 460 Green, M.A., see Jestädt, Th. 460 Green, S.M., see Górska, M. 312, 315, 329 Greene, R.L., see Biswas, A. 464 Greene, R.L., see Blumberg, G. 464 Greene, R.L., see King, D.M. 464 Gregg, J.F. 5 Gregory, T., see Heimbrodt, W. 305 Greibe, T., see Skinta, J.A. 464 Greiter, M. 416, 417 Grenier, B., see Mendels, P. 478 Greven, M. 486 Greven, M., see Armitage, N.P. 465 Greven, M., see Elstner, N. 486 Greven, M., see Howald, C. 481 Greven, M., see Kimura, H. 446 Greven, M., see Lee, Y.S. 486 Greven, M., see Nakajima, K. 485, 486 Greven, M., see Yamada, K. 447, 448 Greven, P.K., see Vajk, O.P. 463 Grévin, B. 478 Griesche, J., see Streller, U. 296 Grieshaber, W. 296, 299 Grieshaber, W., see Dietl, T. 300 Grieshaber, W., see Gaj, J.A. 296 Grigorov, S.N., see Rogacheva, E.I. 336 Grilli, M., see Sadori, A. 397, 465 Grilli, M., see Seibold, C. 397 Grimberg, R., see Eskes, H. 404 Grimsditch, M., see Mattson, J.E. 220, 244 Grinstein, G., see Awschalom, D.D. 310 Grobert, N., see Prados, C. 261 Grobis, M., see Nagaoka, K. 251 Grodzicka, E. 312 Grodzicka, E., see Stachow-Wójcik, A. 333, 334 Grodzicka, E., see Story, T. 312, 328–330 Gromko, A.D., see Chuang, Y.-D. 482 Gronbech-Jensen, N., see Stojkovi´c, B.P. 392 Grössinger, R., see Knobel, M. 532, 539 Grove, M., see Pashkevich, Yu.G. 461, 463 Gruber, Th. 305 Gruber, Th., see Keller, D. 306 Grünberg, P. 206, 217, 219, 248 Grünberg, P., see Binasch, G. 4 Grünberg, P., see Bürgler, D.E. 5, 110, 143, 207 Grünberg, P., see Gareev, R.R. 220, 245, 247 Grünberg, P., see Potter, C.D. 110 Gu, E., see Hope, S. 249 Gu, G., see Bogdanov, P.V. 427 Gu, G.D., see Johnson, P.D. 482

589

Gu, G.D., see Shen, Z.-X. 479, 482 Gu, J.Y. 13 Gudat, W., see Kachel, T. 236 Gudat, W., see Pampuch, C. 253 Guerrero, F., see Loﬂand, S.E. 514, 546 Guevara, J. 254, 257 Guha, S., see Deleporte, E. 297 Guidoni, L. 274 Guillen, V., see Bobroff, J. 477 Guillot, C., see Orlowski, B.A. 315 Guillot, M., see Mycielski, A. 356, 357 Guillot, M., see Scalbert, D. 351 Guillot, M., see Testelin, C. 351, 353, 356 Guinea, F., see Louis, E. 397 Guinea, F., see Verges, J.A. 395 Guletskii, P.G., see Sandratskii, L.M. 216 Gullikson, E.M., see Lawniczak-Jabło´nska, K. 345 Gülseren, O., see Bagci, V.M.K. 266 Gümlich, H.-E., see Goede, O. 305 Gümlich, H.-E., see Heimbrodt, W. 305 Gümlich, H.-E., see Pier, Th. 305 Gunji, T. 542, 544 Gunnarson, O., see Zaanen, J. 388, 391, 393, 395 Gunshor, R.L. 344 Gunshor, R.L., see Chang, S.-K. 296 Gunshor, R.L., see Datta, S. 294 Gunshor, R.L., see Durbin, S.M. 295, 344, 345 Gunshor, R.L., see Han, J. 295, 344, 345 Gunshor, R.L., see Kobayashi, M. 295 Gunshor, R.L., see Kolodziejski, L.A. 295 Gunshor, R.L., see Nurmikko, A.V. 296 Güntherodt, G., see Fraune, M. 164 Güntherodt, G., see Keller, J. 155 Güntherodt, G., see Miltenyi, P. 155 Güntherodt, G., see Nouvertné, F. 268, 269 Güntherodt, G., see Nowak, U. 155 Güntherodt, G., see Pashkevich, Yu.G. 461, 463 Güntherodt, H.J., see Schaller, D.M. 238 Güntherodt, H.J., see Schmidt, C.M. 221 Guo, H.Q. 504, 532, 539 Guo, H.Q., see Chen, C. 504 Guo, H.Q., see He, J. 540 Guo, Y. 29 Guo, Z., see Li, K. 37, 49, 50, 145, 146, 159, 167 Guo, Z., see Wu, Y. 57 Guo, Z.B. 159 Gupta, A., see Petrov, D.K. 22, 25 Gupta, A.K., see Pan, S.H. 442, 481 Gupta, J.A., see Ray, O. 307 Gupta, J.P., see Petrov, D.K. 22, 25 Gupta, R.P. 266 Gurney, B., see Baril, L. 141

590

AUTHOR INDEX

Gurney, B., see Lee, W.Y. 37, 51, 71 Gurney, B., see Pinarbasi, M. 47, 164, 172 Gurney, B.A. 22, 31–33, 40, 41, 97, 129, 140 Gurney, B.A., see Bensmina, F. 73 Gurney, B.A., see Butler, W.H. 108, 109 Gurney, B.A., see Carey, M.J. 33, 67, 151, 159, 164, 176 Gurney, B.A., see Childress, J.R. 32, 41, 55, 67, 159, 174 Gurney, B.A., see Dieny, B. 4, 9, 16, 18, 19, 30, 31, 39, 59, 60, 66, 94, 95, 99, 103, 157, 178 Gurney, B.A., see Heim, D.E. 127, 129 Gurney, B.A., see Huang, T.C. 70, 72 Gurney, B.A., see Ju, G. 172 Gurney, B.A., see Ju, Y.S. 21 Gurney, B.A., see Lefakis, H. 67 Gurney, B.A., see Mamin, H.J. 22, 25, 141 Gurney, B.A., see Meny, C. 71 Gurney, B.A., see Nicholson, D.M.C. 40, 85 Gurney, B.A., see Nozières, J.P. 72 Gurney, B.A., see Speriosu, V.S. 18, 40, 45, 72, 110, 145 Gurney, B.A., see Tsang, C. 22, 127 Gurney, B.A., see Tsang, C.H. 4, 5, 22 Gustavsson, F., see Marangolo, M. 248 Guthmiller, G., see Valletta, R.M. 140 Gutierrez, J., see Barandiarán, J.M. 504 Gutierrez, J., see Kurlyandskaya, G.V. 504 Gutierrez, J., see Pirota, K.R. 504, 535, 557 Gutierrez, J., see Sartorelli, M.L. 504, 535 Gutjahr-Loser, Th., see Sander, D. 208 Guziewicz, E., see Kowalski, B.J. 312, 329 Guziewicz, M., see Kossacki, P. 296 Gyorffy, B.L., see Razee, S.S.A. 209 Ha, J.K., see Chernyshova, M. 332, 338 Ha, J.K., see Kowalczyk, L. 338 Ha, J.K., see LeClair, P. 341 Ha, K., see Kepa, H. 336, 338, 340 Ha, T., see Egelhoff, W.F. 51, 52, 93 Haacke, S. 305 Haacke, S., see Lawrence, I. 305 Haas, H., see Seewald, G. 237 Haase, J. 453, 477 Hadj-Larbi, A. 217, 231–234, 275 Hadjipanayis, G.C., see Prados, C. 89 Hafner, J., see Hobbs, D. 204, 206, 216, 218, 219, 228, 241, 250, 269, 270 Hafner, J., see Spisak, D. 212, 217, 218, 229, 231, 238 Haga, A., see Yamaguchi, M. 552 Hagele, D., see Östreich, M. 304 Hagerott, M., see Han, J. 295, 344, 345

Häggström, L., see Kalska, B. 227, 231 Hagn, E., see Seewald, G. 237 Hagston, W.E., see Fatah, J.M. 296 Hagston, W.E., see Harrison, P. 296 Hagston, W.E., see Jackson, S. 296 Hagston, W.E., see Kusrayev, Yu.G. 298 Hagston, W.E., see Miao, J. 300 Hagston, W.E., see Stirner, T. 296, 300 Hagston, W.E., see Weston, S.J. 296 Hahn, E., see Bucher, J.P. 258, 259 Hahn, H., see Johnson, A. 61 Hai, G.Q., see Nicholas, R.J. 306 Haines, E.M., see Williams, G.V.M. 475, 476 Häkkinen, H., see Moseler, M. 251 Haldane, F.D.M. 486 Halperin, B.I., see Chakravarty, S. 434, 484–487 Halperin, W.P., see Mitrovi´c, V.F. 470, 477 Hamad, B.A. 224, 237, 277 Hamada, H. 225, 226 Hamakawa, Y. 47, 73, 160, 163, 168, 172 Hamakawa, Y., see Hoshiya, H. 160, 168 Hamakawa, Y., see Meguro, K. 33, 123 Hamakawa, Y., see Nakamoto, K. 22, 47, 172 Hamann, D.R. 213 Hamann, D.R., see Bachelet, G.B. 213 Hamberger, P., see Koppensteiner, E. 342 Hamdani, F., see Nawrocki, M. 351 Hammar, P.R., see Johnson, M. 25 Hammel, P.C., see Curro, N.J. 434, 451, 467 Hammel, P.C., see MacLaughlin, D.E. 433 Hammel, P.C., see Suh, B.J. 434, 451, 452, 467 Hammel, P.C., see Takigawa, M. 437, 474, 475 Hammel, P.C., see Yoshinari, Y. 460, 461 Hammer, L., see Blum, V. 238 Han, D. 161, 162 Han, D., see Yang, Z. 162, 178 Han, D.-H. 163 Han, G., see Li, K. 37, 49, 50, 145, 146, 159, 167 Han, G., see Wu, Y. 57 Han, J. 295, 344, 345 Han, J., see Durbin, S.M. 295, 344, 345 Han, W., see Li, D. 540 Han, W.K., see Lee, S.K. 235 Han, X.-X., see Zhu, Z.-M. 296 Hanbicki, A.T., see Jonker, B.T. 304 Handschuh, S., see Asada, T. 208 Hanf, M.C., see Elmouhssine, O. 238 Hanke, A. 277 Hanke, W., see Arrigoni, E. 407 Hanke, W., see Dorneich, A. 416 Hanke, W., see Eder, R. 416 Hanke, W., see Meixner, S. 416 Hanke, W., see Scalapino, D.J. 416 Hanke, W., see Zhang, S.-C. 416

AUTHOR INDEX Hansen, H.J., see Li, C. 235 Hansen, L., see Sawicki, M. 311 Hansen, L., see Yakovlev, D.R. 297, 298 Hao, Y., see Castaño, F.J. 58 Hao, Y.G., see Peng, S.S. 242, 262, 263 Happ, M., see Klar, P.J. 296, 308 Happ, M., see Suisky, D. 298 Harasawa, A., see Hayashi, K. 236, 275 Haratani, S., see Castaño, F.J. 58 Harbecke, B., see Krost, A. 324 Hardner, H.T. 135 Hardy, W.N., see Sonier, J.E. 477 Harig, J.C., see Kim, Y.-K. 22 Harima, N. 466 Harju, A.P., see Arrigoni, E. 407 Harman, T.C., see Hicks, L.D. 328 Haroun, A., see Boussendel, A. 276 Harp, G.R., see Perjeru, F. 217, 220, 275 Harp, G.R., see Tomaz, M.A. 236, 275 Harper, R.L., see Suh, E.-K. 300 Harrison, E.P. 504 Harrison, P. 296 Harrison, P., see Fatah, J.M. 296 Harrison, P., see Jackson, S. 296 Harrison, P., see Stirner, T. 296, 300 Harrison, P., see Weston, S.J. 296 Hartman, J.M., see Jouanne, M. 311 Hartman, J.M., see Szuszkiewicz, W. 311 Hartmann, M., see Charleux, M. 296 Harwit, A. 300 Hase, T.P.A., see Marrows, C.H. 249 Hasegawa, K., see Nishimura, K. 63 Hasegawa, M. 459 Hasegawa, N. 33–35, 38, 46, 48, 164 Hasegawa, N., see Kuroda, S. 308, 309 Hasegawa, N., see Saito, M. 33, 34, 45, 162, 170 Hasenfratz, P. 434, 485 Hashimoto, A., see Kanai, H. 33, 77 Hashimoto, S., see Inomata, K. 16 Hashimoto, S., see Kamiguchi, Y. 35, 48, 49, 77 Hashimoto, T., see Tsunoda, M. 154, 159, 167 Haslar, V., see Kraus, L. 540, 541 Hasselmann, N. 404 Hasselmann, N., see Dimashko, Y.A. 404 Hattox, T.M. 224 Hauch, J., see Ganzer, S. 25 Haury, A. 311 Haury, A., see Dietl, T. 311 Haury, A., see Grieshaber, W. 296, 299 Hauser, H. 550, 556 Hauser, H., see Hausleitner, C. 556 Hauser, H., see Steindl, R. 550 Hausleitner, C. 556 Hausleitner, C., see Hauser, H. 556

591

Hausleitner, Ch., see Steindl, R. 550 Hayashi, A., see Tamura, H. 456 Hayashi, K. 236, 275 Hayashi, K., see Fujikata, J. 151, 157, 161–164, 172, 174, 175 Hayashi, K., see Sugawara, N. 61 Hayashi, M., see Katayama, T. 110 Hayden, S.M., see Aeppli, G. 447 Hayden, S.M., see Dai, P. 470, 482 Hayden, S.M., see Lake, B. 448, 449 Hayden, S.M., see Mason, T.E. 448 Haydock, R. 203, 214 Hayes, A., see Hegde, H. 70, 71 Hayes, W., see Chow, K.H. 460 Hayes, W., see Jestädt, Th. 460 Hazelton, T., see Fayﬁeld, R. 24 He, H. 411, 426, 449, 482 He, H.F., see Etrillard, J. 479 He, J. 540 He, K.Y., see He, J. 540 Heaney, P.J. 457 Heaney, P.J., see Mehta, A. 457 Hebard, A.F., see Theodoropoulou, N. 276 Heberle, A.P., see Haacke, S. 305 Hedgcock, F.T., see Bartkowski, M. 328 Hedin, L., see von Barth, U. 216 Hedstrom, J., see Lee, W.Y. 162, 170 Heeger, A.J. 393 Heffner, R.H., see Nachumi, B. 449, 450 Heffner, R.H., see Sonier, J.E. 477 Heffner, R.H., see Takigawa, M. 437, 474, 475 Hegde, H. 70, 71 Hegde, H., see Mao, M. 32, 33, 158 Hegde, H., see Mao, S. 7 Hehn, M., see Faure-Vincent, J. 276 Heim, D.E. 42, 127, 129 Heim, D.E., see Gurney, B.A. 22, 31, 129, 140 Heim, D.E., see Tsang, C. 22, 127 Heiman, D. 351 Heiman, D., see Dahl, M. 353 Heiman, D., see Foner, S. 353, 355 Heiman, D., see Gennser, U. 351, 355 Heiman, D., see Liu, X.C. 355 Heiman, D., see Shapira, Y. 353 Heiman, D., see Twardowski, A. 350, 351 Heiman, D., see Vu, T.Q. 355 Heiman, N., see Tsang, C. 157 Heimbrodt, W. 291, 305 Heimbrodt, W., see Chen, L. 307 Heimbrodt, W., see Falk, H. 306 Heimbrodt, W., see Goede, O. 305 Heimbrodt, W., see Klar, P.J. 296, 308 Heimbrodt, W., see Östreich, M. 304

592 Heimbrodt, W., see Pier, Th. 305 Heimbrodt, W., see Suisky, D. 298 Heinebrodt, M., see Billas, I.M.L. 265 Heinebrodt, M., see Tast, F. 265 Heinke, H., see Mackh, G. 300 Heinonen, M.H., see Iwanowski, R.J. 315 Heinrich, B. 207, 221, 222, 230 Heinrich, B., see Bland, J.A.C. 4, 207 Heinrich, B., see Celinski, Z. 235 Heinrich, B., see Fullerton, E.E. 235 Heinrich, B., see Schmid, A.K. 268 Heinz, K., see Blum, V. 238 Heinz, K., see Camarero, J. 143 Heinze, S. 277 Heinze, S., see Asada, T. 208 Heinze, S., see Papanikolaou, N. 253 Heinze, S., see Wortmann, D. 204 Heiss, W. 328, 342 Heiss, W., see Ma´ckowski, S. 308 Heiss, W., see Prechtl, G. 297 Heiss, W., see Schwarzl, T. 328, 342 Heiss, W., see Springholz, G. 328, 342 Heitmann, S., see Hütten, A. 55, 56 Hellberg, C.S. 406 Hellmann, R., see Koch, M. 301 Hellmann, R., see Mackh, G. 300 Helm, D.E., see Tsang, C.H. 4, 5, 22 Hempel, T., see Hütten, A. 55, 56 Hempstead, R.D. 151, 157, 172 Hendorfer, G., see Widmer, T. 361 Hendorfer, G., see Widmer, Th. 361 Henley, C. 416 Henneberger, F., see Kratzert, P.R. 309 Henneberger, F., see Kreller, F. 298 Henneberger, F., see Rossin, V.V. 296, 301 Henneberger, F., see Streller, U. 296 Henning, T., see Klar, P.J. 308 Henninger, B., see Pier, Th. 305 Hennion, B. 311 Hennion, B., see Sidis, Y. 472 Henry, J.Y., see Bourges, P. 471, 472 Henry, J.Y., see Horvati´c, M. 474 Henry, J.Y., see Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Henry, L.L., see Pratt, W.P. 78, 85 Henry, L.L., see Yang, Q. 78 Herbich, M. 350, 351 Herbich, M., see Łazarczyk, P. 315 Herbich, M., see Mac, W. 351, 357 Herbst, W., see Geist, F. 323–325 Herbst, W., see Krenn, H. 312, 323–326 Herbst, W., see Pohlt, M. 344, 345 Heremans, J. 23, 343 Heremans, J., see Karczewski, G. 324

AUTHOR INDEX Hergert, W., see Bazhanov, D.I. 205, 252 Hergert, W., see Izquierdo, J. 227, 269, 277 Hergert, W., see Lysenko, O.V. 275 Hergert, W., see Nayak, S.K. 253 Hergert, W., see Rennert, P. 252 Hergert, W., see Stepanyuk, V.S. 238, 253, 268 Hergert, W., see Weber, S.E. 251 Herman, F. 207, 221 Herman Jr., D.A., see Speriosu, V.S. 169 Hermann, A.M. 479 Hermann, T.M. 24 Hernan, O.S., see Blum, V. 238 Hernando, A. 509, 536 Hernando, A., see Barandiarán, J.M. 527 Hernando, A., see Chen, D.-X. 522, 546 Hernando, A., see Knobel, M. 504, 511, 532, 536, 538, 539, 549 Hernando, A., see Kraus, L. 508, 545 Hernando, A., see Mandal, K. 538 Hernando, A., see Óvári, T.A. 534, 546 Hernando, A., see Prados, C. 261 Hernando, A., see Sánchez, M.L. 537 Hernando, A., see Sinnecker, J.P. 535 Hernando, A., see Valenzuela, R. 537, 549, 552, 553 Hernando, A., see Vázquez, M. 504, 533, 553 Hernando, A., see Velázquez, J. 508, 544 Hernando, B. 540 Hernando, B., see Sánchez, M.L. 538 Hernando, B., see Tejedor, M. 504, 538, 539, 553, 557 Herr, U., see Zimmermann, C.G. 268 Herres, N., see Lambrecht, A. 312, 323, 324 Herring, C., see Kittel, C. 528 Herrmann, M., see Marrows, C.H. 249 Hershﬁeld, S., see Chen, J. 104 Herzer, G., see Tejedor, M. 504, 539 Hesse, D., see Langer, J. 32, 70 Heuvelmans-Wijdenes, S.S.J., see van Driel, J. 63, 64 Hewer, V., see Harrison, P. 296 Hicken, R.J., see Barman, A. 131 Hickey, B.J., see Barman, A. 131 Hickey, B.J., see Dekadjevi, D.T. 108 Hickey, B.J., see Marrows, C.H. 22, 33, 101, 102, 127, 249 Hicks, L.D. 328 Hieke, K., see Goede, O. 305 Hieke, K., see Heimbrodt, W. 305 Hieke, K., see Pier, Th. 305 Hieke, K., see Weston, S.J. 296 Higemoto, W., see Koike, Y. 448 Higemoto, W., see Watanabe, I. 450

AUTHOR INDEX Hikami, F., see Nagai, H. 45 Hikami, F., see Ueno, M. 34, 42 Hill, E.W. 22, 135 Hill, E.W., see Nor, A.F. 135 Hillberg, M. 361 Hillberg, M., see Klauss, H.-H. 438, 439, 450 Hillebrands, B. 208, 274 Hillebrands, B., see Juraszek, J. 135, 136, 171 Hillebrands, B., see Mewes, T. 152 Hillebrands, B., see Mougin, A. 171 Hillebrecht, F.U., see Roth, Ch. 228 Hillebrecht, F.U., see Uzdin, V.M. 222 Hilpert, M., see Mackh, G. 300 Himpsel, F.J. 207 Hindmarch, A.T., see Barman, A. 131 Hinks, D.G., see Haase, J. 453, 477 Hinks, D.G., see Johnson, P.D. 482 Hinks, D.G., see Jorgensen, J.D. 457 Hinks, D.G., see Zasadzinski, J.F. 481 Hirai, K., see Kurz, Ph. 204, 206, 241, 270 Hiraka, H., see Fujita, M. 447, 448 Hirano, J., see Adachi, N. 349 Hirota, E., see Sakakima, H. 42, 46 Hirota, K., see Kimura, H. 446 Hirota, K., see Koike, Y. 448 Hirota, K., see Wakimoto, S. 447, 448 Hitti, B., see Chow, K.H. 460 Hjörvarsson, B. 206, 219, 220, 248 Hjörvarsson, B., see Andersson, G. 248 Hjörvarsson, B., see Labergie, D. 248, 249 Hjörvarsson, B., see Uzdin, V.M. 249 Ho, J., see Parks, E.K. 255 Ho, M.K., see Childress, J.R. 32, 41, 55, 67, 159, 174 Hobbs, D. 204, 206, 216, 218, 219, 228, 241, 250, 269, 270 Hochst, H., see Kaminski, A. 419 Höck, K.H., see Kübler, J. 216 Hoffman, J.E. 413, 441, 442, 471, 479, 481 Hoffman, J.E., see Lang, K.M. 441, 480 Hoffmann, H. 125 Hoffmann, N., see Streller, U. 296 Hofmann, W., see Springholz, G. 327 Hofstetter, W., see Knobel, M. 532, 539 Hogg, J.H.C., see Harrison, P. 296 Hogg, J.H.C., see Jackson, S. 296 Hogg, J.H.C., see Weston, S.J. 296 Hohlfeld, J., see Gerrits, Th. 131, 133 Holl, S., see Kepa, H. 343, 345 Holl, S., see Krenn, H. 344, 345 Holloway, K. 67 Holody, P., see Duvail, J.L. 12, 61, 94, 104 Holody, P., see Pratt, W.P. 78, 85 Holody, P., see Yang, Q. 78

593

Holy, V., see Springholz, G. 328 Holzinger, A., see Springholz, G. 326 Hommet, J., see Boeglin, C. 235 Honda, H., see Uchiyama, T. 550 Hone, D.W., see Pryadko, L.P. 407 Hong, J. 36, 38, 144, 146 Hong, J., see Aoshima, K. 140 Hong, J., see Kanai, H. 22 Hong, J.M., see Awschalom, D.D. 310 Hong, J.M., see Deleporte, E. 297 Hong, J.M., see Harwit, A. 300 Hong, S.C., see Hwang, C. 236 Hong, S.C., see Lee, J.I. 230 Hong, S.C., see Lee, S.K. 235 Honkura, Y. 553 Honkura, Y., see Mohri, K. 532, 550, 551, 555 Hood, R.Q. 103 Hooge, F.N. 133 Hope, S. 249 Hori, H. 276 Hori, H., see Kappler, J.-P. 275 Hormes, J., see Lawniczak-Jabło´nska, K. 345 Horng, C., see Dieny, B. 7, 32, 33, 41, 45, 61, 95, 96, 101, 112, 113, 120, 123 Horng, C., see Li, M. 38, 48, 61, 159, 167 Horsﬁeld, A.P., see Bowler, D.R. 215 Horvati´c, M. 474 Horvati´c, M., see Berthier, C. 430, 434, 437, 438, 473, 476 Horvati´c, M., see Julien, M.-H. 452, 478 Hoshino, K. 31, 118, 158, 167 Hoshino, K., see Nakatani, R. 67, 157, 158 Hoshino, K., see Noguchi, S. 50, 51 Hoshiya, H. 34, 51, 160, 168 Hoshiya, H., see Hamakawa, Y. 47, 73, 160, 163, 168, 172 Hoshiya, H., see Hoshino, K. 31, 118, 158, 167 Hoshiya, H., see Meguro, K. 33, 123 Hoshiya, H., see Nakatani, R. 158 Hoshiya, H., see Soeya, S. 160, 166, 168 Hosoito, N., see Takanashi, K. 225 Hosomi, M. 13 Hosomi, M., see Makino, E. 144, 145 Hosoya, S. 456 Hosoya, S., see Nakajima, K. 485, 486 Hosoya, S., see Wada, S. 460 Hosoya, S., see Yamada, K. 447, 448 Hossain, M. 253 Hossain, S., see Parker, M.R. 115, 122 Hotta, T. 402 Hou, C. 150 Hou, C., see Nishioka, K. 157, 174 Hou, C., see Zhang, K. 161

594

AUTHOR INDEX

Hou, C.-H., see Fujiwara, H. 122 Hou, J.G., see Zhao, Z. 265 Hoving, W., see Purcell, S.T. 228 Howald, C. 481 Howard, J.K., see Gong, H. 67 Howard, J.K., see Hwang, C. 65 Howard, J.K., see Hylton, T.L. 9, 26 Howard, J.K., see Lin, T. 34, 150, 151, 157, 161, 163, 166, 169 Howard, J.K., see Mauri, D. 154 Howard, K., see Schlenker, C. 153 Hricovini, K., see Andrieu, S. 228 Hricovini, K., see Orlowski, B.A. 315 Hsu, F.Y., see Vouille, C. 84 Hsu, S.Y., see Steenwyk, S.D. 13, 80, 83, 85 Hsu, T.Y., see Dong, C. 515 Hu, J. 504, 535 Hu, J.-G. 37, 149 Hu, J.-P., see Chen, H.-D. 413 Hu, J.P., see Feng, D.L. 482 Hu, J.P., see Zhang, S.-C. 416 Hu, Q., see Borsa, F. 433, 449 Huai, Y. 33, 34, 42, 49 Huai, Y., see Anderson, G. 151, 159, 161, 170 Huai, Y., see Anderson, G.W. 74, 77, 151, 158, 161, 162 Huai, Y., see Diao, Z. 38, 49, 50 Huai, Y., see Mao, S. 7 Huai, Y., see Pakala, M. 159 Huai, Y., see Tong, H.C. 51, 55 Huang, J.C.A. 71 Huang, R.-T. 50, 77 Huang, T.C. 70, 72 Huang, T.C., see Nozières, J.P. 72 Huang, T.C., see Speriosu, V.S. 40, 72 Huard, V. 302 Hübener, M. 276 Hubert, A. 130 Hubert, A., see McCord, J. 128 Hubner, J., see Östreich, M. 304 Hübner, W., see Qian, X. 237 Hübner, W., see Zhang, G. 274 Hücker, M., see Curro, N.J. 434, 451, 467 Hücker, M., see Kataev, V. 391, 453 Hücker, M., see Klauss, H.-H. 438, 439, 450 Hücker, M., see Suh, B.J. 434, 451, 452, 467 Hudson, E.E., see Lang, K.M. 441, 480 Hudson, E.W. 481 Hudson, E.W., see Hoffman, J.E. 413, 441, 442, 471, 479 Hudson, E.W., see Pan, S.H. 442, 481 Hue, S.S., see Mao, S. 60 Huebner, J., see Falk, H. 306 Hughes, A.E., see Vallin, J.T. 346

Hughes, T. 158, 175 Huh, S.H., see Lee, G.H. 261 Huh, Y.M. 453 Huhne, T. 241 Hui, S., see Hermann, T.M. 24 Hulpke, E., see Benedek, G. 218 Hults, W.L., see Pennington, C.H. 474 Humbert, P., see Bensmina, F. 73 Humbert, P., see Dieny, B. 18, 19, 59, 60, 95, 99 Humbert, P., see Lefakis, H. 67 Humbert, P., see Meny, C. 71 Humbert, P., see Speriosu, V.S. 18, 110, 145 Hung, A., see Spencer, M.J.S. 218 Hung, C.-Y. 136, 137, 140, 141, 162 Hung, C.-Y., see Huai, Y. 33 Hung, C.-Y., see Mao, M. 32, 33, 158 Hünnefeld, H., see Niemöller, T. 454 Hunt, A.W. 390, 451, 452, 455, 461, 467 Hunt, A.W., see Singer, P.M. 451, 453 Hunt, R.D., see Dai, P. 470, 471, 482 Hunt, R.D., see Mook, H.A. 430, 439, 440, 470 Hurben, M.J., see Hardner, H.T. 135 Hurst, A., see Gadbois, J. 58 Hussain, Z., see Bogdanov, P.V. 427 Hussain, Z., see Lanzara, A. 409, 427, 454, 482 Hussain, Z., see Zhou, X.J. 441, 454 Hussey, N.E., see Lake, B. 413, 449, 480 Hütten, A. 55, 56 Huynen, I., see Encinas, A. 548 Hwang, C. 65, 236 Hwang, C., see Lin, T. 34, 150, 161, 166, 169 Hwang, C., see Ounadjela, K. 140 Hwang, D.-G., see Kim, J.-K. 162 Hwang, D.G. 73, 164 Hwang, D.G., see Rhee, J.-R. 161 Hwang, H.Y., see Cheong, S.-W. 461 Hwang, M., see Castaño, F.J. 58 Hylton, T.L. 9, 26, 70 Hylton, T.L., see Spong, J.K. 22, 23 Iakubov, I.T., see Antonov, A. 504, 527, 537 Iakubov, I.T., see Antonov, A.S. 503, 532, 533, 537 Ichikawa, N. 452, 454, 455 Ichikawa, N., see Nachumi, B. 449, 450 Ichikawa, N., see Niemöller, T. 454 Ichikawa, N., see Tranquada, J.M. 445, 446, 451, 458, 459 Ichikawa, N., see Zimmermann, M.v. 454 Ichinokawa, T., see Schmid, A.K. 268 Ichioka, M., see Kaneshita, E. 409 Ide, Y., see Hasegawa, N. 33, 34 Ide, Y., see Saito, M. 34, 45, 170

AUTHOR INDEX Igarashi, J.-I. 486 Igel, T. 218, 227, 229 Iguchi, E. 462 Iguchi, I. 453 Ihninger, G., see Springholz, G. 327, 328 Iida, M. 323, 324 Ijiri, Y., see van der Zaag, P.J. 155 Ikada, H. 308 Ikarashi, K., see Hasegawa, N. 35, 38, 46, 48, 164 Ikeda, H., see Bogdanov, P.V. 427 Ikeda, H., see Shen, Z.-X. 479, 482 Ikeda, Y., see Moser, A. 21 Ilbnouelghazi, E., see Averous, M. 328 Imada, M. 390 Imada, R., see Shirota, Y. 35, 57 Imada, S. 310 Imada, S., see Kuch, W. 230 Imada, Y., see Tsunashima, S. 57 Imagawa, T., see Nishioka, K. 160, 174 Imagawa, T., see Soeya, S. 163, 168 Imai, T. 474 Imai, T., see Hunt, A.W. 390, 451, 452, 455, 461, 467 Imai, T., see Singer, P.M. 451, 453, 474 Imamura, M., see Ahn, J.Y. 353 Imanaka, Y. 302, 303 Inage, K. 26 Inden, G., see Sanchez, J.M. 227 Ingall, L.M., see Childress, J.R. 32, 41, 67, 159, 174 Ingram, D.C., see Tomaz, M.A. 236, 275 Iñiguez, J.I., see Raposo, V. 510 Iñiguez, M.P., see Aguilera-Granja, F. 215, 254–258 Iñiguez, M.P., see Bouarab, S. 254–256, 258, 266, 267 Iñiguez, M.P., see Montejano-Carrizales, J.M. 255 Ino, A. 455 Inomata, K. 16, 220, 244 Inomata, K., see Nishimura, K. 63 Inomata, K., see Okuno, S.N. 110 Inoue, J. 84 Inoue, J., see Itoh, H. 83 Inoue, M. 315 Inoue, M., see Adachi, N. 351, 355 Inui, M. 395 Inukai, T. 299 Irie, Y. 57 Irie, Y., see Sakakima, H. 57 Isaacs, E.D. 462 Isaacs, E.D., see Heiman, D. 351 Isaev, E.I., see Ponomareva, A.V. 224

595

Isber, S. 312, 322, 323, 326, 355 Isber, S., see Bindilatti, V. 312, 326, 355 Isber, S., see Errebbahi, A. 322, 326 Isber, S., see Gratens, X. 312, 322, 328 Isber, S., see Misra, S.K. 322, 323 Isberg, P., see Andersson, G. 248 Isberg, P., see Broddefalk, A. 228 Isberg, P., see Hjörvarsson, B. 206, 219, 220, 248 Isella, G., see Duo, L. 249 Ishida, A. 312, 323, 324, 327 Ishida, K. 479 Ishida, K., see Asayama, K. 430, 437, 473, 474 Ishihara, K., see Fujikata, J. 164 Ishii, S., see Maesaka, A. 171 Ishii, S., see Makino, E. 144, 145 Ishii, S., see Takiguchi, M. 74, 75 Ishikawa, T., see Onose, Y. 465, 466 Ishio, S., see Miyazaki, T. 140 Ishizaka, K., see Onose, Y. 465, 466 Ishizone, M., see Hasegawa, N. 38, 48 Ismail-Beigi, S. 215 Isshiki, M., see Lake, B. 448, 449 Itabashi, M., see Maesaka, A. 74, 75, 77 Ito, N., see Araki, S. 32–34, 36, 67 Itoh, H. 83 Itoh, H., see Motoyoshi, M. 24 Itoh, H., see Schmid, A.K. 268 Itoh, M., see Kyomen, T. 457 Itoh, T., see Uchiyama, T. 553 Itskos, G., see Jonker, B.T. 304 Itskos, G., see Park, Y.D. 304 Ivanchik, I.I. 312 Ivanov, A., see Fong, H.F. 470 Ivanov, A., see Sidis, Y. 472 Ivanov, S.V., see Reshina, I.I. 309 Ivanov, V., see Karczewski, G. 311 Ivchenko, E.L. 297 Ivchenko, E.L., see Yakovlev, D.R. 298, 306 Iwamura, E., see Lai, C.-H. 163 Iwanowski, R.J. 315 Iwanowski, R.J., see Lawniczak-Jabło´nska, K. 345 Iwasaki, H. 60, 73, 158 Iwasaki, H., see Fuke, H.N. 32, 158, 174 Iwasaki, H., see Fukuzawa, H. 33, 37, 42, 48, 137, 140, 141 Iwasaki, H., see Hasegawa, N. 38, 48 Iwasaki, H., see Kamiguchi, Y. 32, 35, 41, 48, 49, 77 Iwasaki, H., see Kools, J.C.S. 37, 49, 71 Iwasaki, H., see Saito, A.T. 71, 73, 74 Iwasaki, H., see Sant, S. 36, 49, 51, 71 Iwasaki, H., see Takagishi, M. 13

596

AUTHOR INDEX

Iwasaki, H., see Yoda, H. 22 Iwasaki, H., see Yuasa, H. 13 Iwata, S., see Kato, T. 38 Iwata, S., see Shimoyama, K. 162 Iwata, S., see Shirota, Y. 35, 57 Izquierdo, J. 213, 226, 227, 269, 277 Izquierdo, J., see Robles, R. 225, 245, 259, 260, 266–268, 275 Jackson, H.E., see Kim, C.S. 309, 310 Jackson, S. 296 Jackson, S., see Goede, O. 305 Jackson, S., see Harrison, P. 296 Jacobs, K., see Streller, U. 296 Jacquart, P.M., see Acher, O. 514 Jacquet, J.C. 63 Jaffres, H., see Kirilyuk, A. 155 Jaggi, N.K. 225 James, E.M., see Kim, C.S. 309, 310 James, P., see Le Bacq, O. 228 James, P., see Taga, A. 206 Jamison, K., see Rau, C. 224, 238 Jamneala, T. 251, 252, 273 Jamneala, T., see Madhavan, V. 252 Jamneala, T., see Nagaoka, K. 251 Jan, J.P. 7 Jang, K.J. 542 Jang, K.J., see Kim, C.G. 535, 542, 545, 546 Jang, K.J., see Yoon, S.S. 540 Jang, S.H. 38, 48, 77, 124 Jang, S.H., see Kim, K.Y. 146, 148 Jang, S.H., see Kim, M.J. 162 Jang, W.-J., see Hasegawa, M. 459 Jang, W.Y., see Loram, J.W. 442 Janik, E. 295 Janik, E., see Ghali, M. 304, 305 Janik, E., see Hennion, B. 311 Janik, E., see Prechtl, G. 297 Janik, E., see Sawicki, M. 311 Janik, E., see Stachow-Wójcik, A. 310 Jantsch, W., see Prechtl, G. 297 Jantsch, W., see Widmer, T. 361 Jantsch, W., see Widmer, Th. 361 Jantsch, W., see Wilamowski, Z. 343 Jaouen, N., see Kappler, J.-P. 275 Jardine, D.B., see Morecroft, D. 138 Jaren, S., see Nozières, J.P. 158, 161, 162, 168–170 Jaren, S., see Zeltser, A.M. 159 Jaren, S., see Zhang, Y.B. 60, 76, 161, 170 Jaroszy´nski, J. 304 Jaroszy´nski, J., see Andrearczyk, T. 304 Jaroszy´nski, J., see Dietl, T. 304

Jaroszy´nski, J., see Ferrand, D. 310, 311 Jaroszy´nski, J., see Karczewski, G. 304 Jaroszy´nski, J., see Takeyama, S. 302 Jaroszy´nski, J., see Teran, F.J. 302 Jay, J.P., see Meny, C. 71 Jayanthi, C.S., see Wu, S.Y. 215 J˛edrzejczak, A., see Łazarczyk, P. 315 J˛edrzejczak, A., see Łusakowski, A. 318 J˛edrzejczak, A., see Miotkowska, S. 315 Jeffers, F., see Beach, R.S. 503, 504 Jena, P., see Das, G.P. 276 Jena, P., see Nayak, S.K. 253 Jena, P., see Rao, B.K. 250, 276 Jena, P., see Weber, S.E. 251 Jensen, H.J.F. 203 Jeong, H.S. 24 Jeong, J.W., see Lee, G.H. 261 Jepsen, O., see Andersen, O.K. 212, 269 Jepsen, O., see Kakehashi, Y. 217 Jérome, R. 67 Jérome, R., see Galtier, P. 67 Jestädt, Th. 460 Jestädt, Th., see Chow, K.H. 460 Jian, S., see Johnson, P.D. 482 Jiang, J.S., see Viegas, A.D.C. 514 Jiang, J.S., see Xiao, J.Q. 9 Jibouri, A.A., see Lu, Z.Q. 37, 48 Jilek, E., see Karpinski, J. 469 Jimbo, M., see Shirota, Y. 35, 57 Jimbo, M., see Tsunashima, S. 57 Jin, G.-J., see Hu, J.-G. 37, 149 Jin, X.F., see Qian, D. 238 Jin, X.F., see Wu, Y.Z. 230 Jin, X.F., see Zhao, Z. 265 Jin, Z. 135, 136 Jin, Z., see Bertram, H.N. 135 Jo, S. 34, 45, 114, 178 Johansson, B., see Alden, M. 218 Johansson, B., see Belonoshko, A.B. 209 Johansson, B., see Le Bacq, O. 228 Johansson, B., see Ponomareva, A.V. 224 Johansson, B., see Taga, A. 206 Johansson, B., see van Schilfgaarde, M. 233, 269 Johansson, B., see Vitos, L. 275 Johnson, A. 61 Johnson, A.B., see Lim, C.K. 127, 138 Johnson, B.L. 103 Johnson, L.E., see Michel, R.P. 163 Johnson, M. 25 Johnson, M., see Deaven, D.M. 143 Johnson, M.T. 143 Johnson, M.T., see Bürgler, D.E. 5, 110, 143, 207

AUTHOR INDEX Johnson, M.T., see Jungblut, R. 149, 154, 157, 176 Johnson, M.T., see Lenssen, K.-M.H. 31, 39, 67, 71 Johnson, M.T., see Purcell, S.T. 228 Johnson, P., see Mao, M. 32, 33, 158 Johnson, P., see Mao, S. 7 Johnson, P.D. 482 Johnson, R.L., see Kowalski, B.J. 312, 329 Johnston, D.C. 386, 390, 469 Johnston, D.C., see Borsa, F. 433, 449 Johnston, D.C., see Carretta, P. 434 Johnston-Halperin, E., see Crooker, S.A. 303 Jona, F., see Lu, S.H. 231 Jonas, K.L., see Röhlsberger, R. 252 Jones, R.E., see Wei, Y. 22 Jones Jr., R.E., see Wei, Y. 22 Jonker, B.T. 294, 301, 304, 358 Jonker, B.T., see Bak, J. 354 Jonker, B.T., see Chou, W.C. 294, 358, 359 Jonker, B.T., see Fu, L.P. 358 Jonker, B.T., see Kim, Y.D. 354 Jonker, B.T., see Ko, Y.D. 354 Jonker, B.T., see Liu, X. 294 Jonker, B.T., see Park, K. 360 Jonker, B.T., see Park, Y.D. 304 Jonker, B.T., see Poweleit, C.D. 301 Jonker, B.T., see Warnock, J. 294, 358 Jonker, B.T., see Yu, W.Y. 297, 358 Joo, H.-W., see Kim, J.-K. 162 Joo, H.S. 71 Jorgensen, J.D. 457 Jouanne, M. 311 Jouanne, M., see Szuszkiewicz, W. 311, 361 Jouguelet, E., see Gregg, J.F. 5 Jouneau, P.H., see Bodin, C. 295 Joyce, J., see Chuang, Y.-D. 482 Ju, G. 172 Ju, K., see Dieny, B. 7, 32, 33, 41, 45, 61, 95, 96, 101, 112, 113, 120, 123 Ju, K., see Li, M. 38, 48, 61, 159, 167 Ju, Y.S. 21 Juai, Y., see Anderson, G.W. 162, 166 Judy, J., see Bae, S. 25, 37, 46 Judy, J.H., see Bae, S. 35, 46, 164 Judy, J.H., see Egelhoff, W.F. 51, 52, 93 Judy, J.H., see Egelhoff Jr., W.F. 34, 51, 52, 69, 70, 144 Judy, J.H., see Han, D.-H. 163 Judy, J.H., see Lin, C.-L. 163 Judy, J.H., see Qian, Z. 161 Judy, J.H., see Torok, E.J. 25 Julien, M.-H. 452, 478 Julien, M.-H., see Huh, Y.M. 453

597

Julien, M.H., see Berthier, C. 430, 434, 437, 438, 473, 476 Jung, M., see Mougin, A. 171 Jung, R., see Imada, S. 310 Jungblut, R. 149, 154, 157, 176 Jungblut, R., see de Vries, J.J. 220, 244, 245, 247 Jungblut, R.M., see Coehoorn, R. 153 Junquera, J., see Izquierdo, J. 213 Junquera, J., see Soler, J.M. 213 Juraszek, J. 135, 136, 171 Jurgens, M.J.G.M. 468 Jusserand, B. 303 Kabos, P. 132 Kaburagi, M., see Wada, S. 460 Kachel, T. 236 Kachel, T., see Pampuch, C. 253 Kachniarz, J., see Miotkowska, S. 315 Kachniarz, J., see Nadolny, A.J. 321 Kacman, P. 311, 337, 340, 343 Kacman, P., see Blinowski, J. 293, 311, 340, 343, 346, 348, 349 Kacman, P., see Giebułtowicz, T.M. 336, 343–345 Kacman, P., see Kepa, H. 336, 338, 340 Kadmon, Y., see Egelhoff, W.F. 51, 52, 93 Kadowaki, K., see Fretwell, H.M. 441 Kai, J.-J., see Huang, R.-T. 50, 77 Kai, T., see Fujiwara, H. 155 Kai, T., see Zhang, K. 161 Kai, W., see Huang, R.-T. 50, 77 Kaiser, G., see Robbes, D. 550 Kaiser, H., see Rhyne, J.J. 345 Kaiser, H., see Stumpe, L.E. 345 Kaiser, S., see Wypior, G. 296 Kajimoto, R., see Yoshizawa, H. 459 Kajitani, T., see Wada, S. 460 Kaka, S., see Kabos, P. 132 Kaka, S., see Russek, S.E. 132 Kakahiri, Y., see Saito, M. 162, 170 Kakala, M., see Anderson, G.W. 74, 77, 151, 162 Kakehashi, Y. 217 Kakeshita, T., see Zhou, X.J. 454 Kakizaki, A., see Hayashi, K. 236, 275 Kakudate, Y., see Semenov, Yu.G. 296 Kakudate, Y., see Yokoi, H. 302 Kaldis, E., see Karpinski, J. 469 Kale, S. 535 Kaliteevski, M.A., see Kavokin, A.V. 299 Kalitsov, A.V., see Granovskii, A.B. 62, 104 Kallin, C., see Arovas, D.P. 415 Kalska, B. 227, 231 Kamiguchi, Y. 32, 35, 41, 48, 49, 77

598

AUTHOR INDEX

Kamiguchi, Y., see Fuke, H.N. 32, 158 Kamiguchi, Y., see Fukuzawa, H. 33, 37, 42, 48, 137, 140, 141 Kamiguchi, Y., see Saito, A.T. 71, 73, 74 Kamiguchi, Y., see Yuasa, H. 13 Kamimori, T., see Goto, M. 136 Kami´nska, E., see Andrearczyk, T. 304 Kami´nska, E., see Dietl, T. 304 Kami´nska, E., see Jaroszy´nski, J. 304 Kaminski, A. 419 Kaminski, A., see Fretwell, H.M. 441 Kamiyama, T., see Sato, T. 464 Kampf, A.P. 409 Kampf, A.P., see Katanin, A.A. 484 Kamruzzaman, Md. 502 Kanai, H. 22, 32, 33, 35, 41, 55, 77, 157 Kanai, H., see Aoshima, K. 74, 140 Kanai, H., see Hong, J. 144 Kanai, H., see Noma, K. 33, 123 Kanai, H., see Tanaka, A. 13, 51, 162 Kanai, H., see Uehara, Y. 61 Kanamine, M., see Kanai, H. 32, 33, 35, 41, 55, 77, 157 Kanarov, V., see Hegde, H. 70, 71 Kane, J., see Aoshima, K. 74 Kane, J., see Hong, J. 36, 144, 146 Kane, J., see Kanai, H. 32, 35, 41, 77, 157 Kane, J., see Noma, K. 33, 123 Kane, M.H., see Kim, Y.-K. 22 Kane, S.N., see Kraus, L. 504, 533, 540, 545 Kanehisa, M.A., see Jouanne, M. 311 Kanehisa, M.A., see Szuszkiewicz, W. 311 Kaneko, N., see Armitage, N.P. 465 Kaneko, N., see Howald, C. 481 Kaneshita, E. 409 Kang, S., see Imada, S. 310 Kang, S.S., see Kuch, W. 230 Kang, T., see Jang, S.H. 38, 48, 77, 124 Kang, T., see Kim, K.Y. 146, 148 Kang, T., see Kim, M.J. 162 Kania, H., see Hong, J. 36, 38, 146 Kanno, T. 551 Kano, H., see Mizuguchi, T. 37, 48 Kano, H., see Motoyoshi, M. 24 Kano, H., see Sugawara, N. 61 Kany, F., see Ulmer-Tufﬁgo, H. 298 Kao, A.S., see Li, Z. 135 Kapitulnik, A., see Dessau, D.S. 427 Kapitulnik, A., see Howald, C. 481 Kapitulnik, A., see Marshall, D.S. 476 Kappler, J.-P. 275 Kappler, J.-P., see Gallani, J.L. 275 Kappler, J.P., see Beaurepaire, E. 209 Kappler, J.P., see Cros, V. 235

Kappler, J.P., see Vogel, J. 219 Karasawa, T., see Takeyama, S. 300, 303 Karczewski, G. 304, 311, 324 Karczewski, G., see Andrearczyk, T. 304 Karczewski, G., see Cywi´nski, G. 307 Karczewski, G., see Dahl, M. 311 Karczewski, G., see Dietl, T. 304 Karczewski, G., see Imanaka, Y. 302, 303 Karczewski, G., see Janik, E. 295 Karczewski, G., see Jaroszy´nski, J. 304 Karczewski, G., see Jouanne, M. 311 Karczewski, G., see Jusserand, B. 303 Karczewski, G., see Kochereshko, V.P. 302 Karczewski, G., see König, B. 306 Karczewski, G., see Kossacki, P. 296, 344, 345 Karczewski, G., see Kunimatsu, H. 302 Karczewski, G., see Kusrayev, Yu.G. 298 Karczewski, G., see Kutrowski, M. 295, 296 Karczewski, G., see Lemaître, A. 296 Karczewski, G., see Mackh, G. 300 Karczewski, G., see Ma´ckowski, S. 308, 309 Karczewski, G., see Maslana, W. 300 Karczewski, G., see Merkulov, I.A. 303 Karczewski, G., see Mino, H. 298, 303 Karczewski, G., see Ossau, W. 302 Karczewski, G., see Prechtl, G. 297 Karczewski, G., see Pulizzi, F. 298 Karczewski, G., see Sawicki, M. 310, 311 Karczewski, G., see Semenov, Yu.G. 296 Karczewski, G., see Shcherbakov, A.V. 306 Karczewski, G., see Stachow-Wójcik, A. 310 Karczewski, G., see Stirner, T. 300 Karczewski, G., see Story, T. 315 Karczewski, G., see Szuszkiewicz, W. 311 Karczewski, G., see Takeyama, S. 300, 302, 303 Karczewski, G., see Teran, F.J. 302 Karczewski, G., see Wasik, D. 304 Karczewski, G., see Wojtowicz, T. 295, 296, 302, 311, 349 Karczewski, G., see Wypior, G. 296 Karczewski, G., see Yakovlev, D.R. 299, 310 Karczewski, G., see Yokoi, H. 302, 303 Kardar, M., see Hanke, A. 277 Karpinski, J. 469 Kasahara, N., see Shimazawa, K. 159 Kasprzak, J., see Kudelski, A. 298 Kastner, M.A., see Greven, M. 486 Kastner, M.A., see Kimura, H. 446 Kastner, M.A., see Lee, Y.S. 448, 449 Kastner, M.A., see Matsuda, M. 466, 467 Kastner, M.A., see Savici, A.T. 449 Kastner, M.A., see Wakimoto, S. 447, 448 Kastner, M.A., see Yamada, K. 447, 448

AUTHOR INDEX Kasuya, T., see von Molnár, S. 331, 342, 343 Kataev, V. 391, 453 Kataev, V.E., see Finkelstein, A.M. 453, 478 Katanin, A.A. 484 Katano, S. 446 Kataoka, Y., see Shimizu, Y. 162 Katayama, T. 110 Katayama-Yoshida, H., see Ohno, H. 292 Kato, H., see Takayama, A. 552 Kato, M. 395 Kato, T. 38 Kato, T., see Shimoyama, K. 162 Katsnelson, A., see Levanov, N.A. 215 Katsnelson, A.A., see Bazhanov, D.I. 205, 252 Katsnelson, A.A., see Stepanyuk, V.S. 268 Katsnelson, M.I., see Lichtenstein, A.I. 209 Katsufuji, T., see Satake, M. 463 Katsufuji, T., see Yamamoto, K. 460 Katsufuji, T., see Yoshizawa, H. 459 Katti, R.R. 24, 57, 58 Kavokin, A.V. 298–300 Kavokin, A.V., see El Ouazzani, A. 351, 352 Kavokin, A.V., see Ivchenko, E.L. 297 Kavokin, A.V., see Paganotto, N. 302 Kavokin, A.V., see Pozina, G.R. 297 Kavokin, A.V., see Ribayrol, A. 297 Kavokin, K.V., see Kavokin, A.V. 300 Kavokin, K.V., see Kusrayev, Yu.G. 298 Kawabe, T., see Hamakawa, Y. 47, 73, 163, 172 Kawabe, T., see Nakamoto, K. 22, 47, 172 Kawaga, K., see Sugawara, N. 61 Kawaguchi, K., see Takanashi, K. 225 Kawai, M., see Susaki, T. 251 Kawai, T., see Ueda, K. 311 Kawamata, T., see Koike, Y. 448 Kawamata, T., see Watanabe, I. 450 Kawamura, H. 203, 204, 243, 270 Kawamura, M., see Hasegawa, N. 38, 48 Kawana, T., see Ozue, T. 22 Kawana, Y., see Kools, J.C.S. 37, 49, 71 Kawashima, K. 540, 557 Kawashima, K., see Mohri, K. 508, 509 Kawato, Y., see Nakamoto, K. 22, 47, 172 Kawawake, Y. 34, 35, 164, 172 Kawawake, Y., see Irie, Y. 57 Kawawake, Y., see Kato, T. 38 Kawawake, Y., see Sakakima, H. 35, 42, 48, 49, 51, 57 Kawawake, Y., see Sugita, Y. 33, 35, 36, 46, 51, 52, 114, 162, 178 Kawazoe, Y., see Kumar, V. 251 Kawazoe, Y., see Taneda, A. 251 Kay, E., see Mauri, D. 154 Kaya, K., see Nagao, S. 261

599

Kayanuma, K. 305, 306 Kazansky, A.K. 214 Keavney, D.J. 111 Kee, H.-Y., see Tewari, S. 420 Kee, H.Y. 428 Kee, H.Y., see Chakravarty, S. 401, 428 Keen, A.M., see Baker, S.H. 241 Keim, M., see Durbin, S.M. 295, 344, 345 Keim, M., see Gruber, Th. 305 Keim, M., see König, B. 294 Keimer, B., see Borisenko, S.V. 441 Keimer, B., see Etrillard, J. 479 Keimer, B., see Fong, H.F. 389, 470, 471, 482 Keimer, B., see Greven, M. 486 Keimer, B., see He, H. 411, 426, 449, 482 Keimer, B., see Sidis, Y. 419, 470, 472 Kekeshita, T., see Yoshizawa, H. 459 Kellar, S.A., see Bogdanov, P.V. 427 Kellar, S.A., see Lanzara, A. 409, 427, 454, 482 Kellar, S.A., see Zhou, X.J. 454 Keller, A., see Merkulov, I.A. 303 Keller, A., see Wojtowicz, T. 302 Keller, D. 306 Keller, J. 155 Keller, J., see Miltenyi, P. 155 Keller, J., see Nowak, U. 155 Keller, N., see Pham-Huu, C. 265 Keller, S.A., see Zhou, X.J. 441, 454 Kellock, A., see Lee, W.Y. 162, 170 Kellock, A.J., see Krishnan, K. 162, 170 Kellock, A.J., see Thomas, L. 152 Kelly, P.J., see Schep, K.M. 109, 110 Kelm, M., see Fiederling, R. 304 Kempnyk, V., see Zayachuk, D. 329 Kendziora, C., see Johnson, P.D. 482 Kendziora, C., see Zasadzinski, J.F. 481 Kendziora, C.A., see Blumberg, G. 464 Kenouﬁ, A. 215 Kepa, H. 336, 338, 340, 343, 345 Kepa, H., see Giebułtowicz, T.M. 336, 343–345 Kepa, H., see Nunez, V. 343 Keren, A., see Nachumi, B. 449, 450 Kern, K., see Gambardella, P. 252 Kern, K., see Knorr, N. 251 Kern, K., see Lin, N. 275 Kerns, K.P., see Parks, E.K. 258 Kershaw, R., see Dahl, M. 353 Kershaw, R., see Fries, T. 357 Kershaw, R., see Shapira, Y. 353 Kershaw, R., see Shih, O.W. 351 Kershaw, R., see Vu, T.Q. 353, 355 Kerszykowsky, G., see Tehrani, S. 24, 57 Kes, P.H., see Brom, H.B. 479

600

AUTHOR INDEX

Kes, P.H., see Molegraaf, H.J.A. 479 Ketterson, J.B., see Jaggi, N.K. 225 Keune, W., see Shinjo, T. 222 Keune, W., see Uzdin, V.M. 222, 249 Khalifeh, J., see Elmouhssine, O. 228 Khalifeh, J.M., see Hamad, B.A. 224, 237, 277 Khalil, I., see Ciureanu, P. 542 Khan, M.A. 228 Khan, M.A., see Bouarab, S. 228 Khandozhko, O., see Zayachuk, D. 329 Khanna, S.N., see Reddy, B.V. 235 Khanna, S.N., see Reuse, F.A. 215, 258 Kheng, K. 301 Khitrova, G., see Kavokin, A.V. 299 Khokhlov, D.R., see Ivanchik, I.I. 312 Ki, J., see Mao, S. 60 Kido, G., see Adachi, N. 349, 351, 355 Kido, G., see Anderson, J.R. 325 Kido, G., see Górska, M. 312, 315, 329 Kido, G., see Imanaka, Y. 302, 303 Kido, G., see Kuroda, S. 308, 309 Kief, M.T., see Hou, C. 150 Kief, M.T., see Qian, Z. 161 Kief, M.T., see Shen, J.X. 163 Kief, M.T., see Zhang, K. 161 Kieﬂ, R.F., see Chow, K.H. 460 Kieﬂ, R.F., see Sonier, J.E. 477 Kikkawa, J.M. 300, 301 Kikkawa, J.M., see Levy, J. 301 Kikkawa, J.M., see Smorchkova, I.P. 304 Kikuchi, H., see Kitade, Y. 31, 72, 140 Kim, B., see Lee, S.K. 235 Kim, C., see Armitage, N.P. 465 Kim, C., see Edelstein, A.S. 231 Kim, C., see Feng, D.L. 482 Kim, C., see Shen, Z.-X. 479, 482 Kim, C.-S. 24 Kim, C.G. 535, 538, 542, 545, 546, 557 Kim, C.G., see Ahn, S.J. 504, 540 Kim, C.G., see Jang, K.J. 542 Kim, C.G., see Ryu, G.H. 542, 549 Kim, C.G., see Song, S.-H. 542–544 Kim, C.G., see Song, S.H. 546 Kim, C.G., see Yoon, S.S. 540 Kim, C.O., see Cho, W.S. 540, 541 Kim, C.O., see Kim, C.G. 535, 546, 557 Kim, C.S. 306, 309, 310 Kim, D.Y., see Kim, C.G. 542, 545, 546 Kim, H., see Aigle, M. 327 Kim, H.C., see Kim, C.G. 545, 546, 557 Kim, H.C., see Song, S.H. 546 Kim, H.J., see Jang, S.H. 38, 48, 77, 124 Kim, H.J., see Kim, K.Y. 146, 148 Kim, H.J., see Kim, M.J. 162

Kim, J.-K. 162 Kim, J.-V., see Suess, D. 155 Kim, J.S., see Lee, S.K. 235 Kim, J.U. 305 Kim, J.Y., see Choi, J.H. 235 Kim, K.-S., see Song, S.-H. 542–544 Kim, K.H., see Edelstein, A.S. 231 Kim, K.S. 532 Kim, K.Y. 146, 148 Kim, K.Y., see Jang, S.H. 38, 48, 77, 124 Kim, K.Y., see Kim, M.J. 162 Kim, M., see Geng, W.T. 218 Kim, M., see Kim, C.S. 306, 309, 310 Kim, M., see Nakamura, K. 155 Kim, M.-S., see Skinta, J.A. 464 Kim, M.J. 162 Kim, M.Y., see Hwang, D.G. 73 Kim, M.Y., see Rhee, J.-R. 161 Kim, T.K., see Kordyuk, A.A. 483 Kim, W., see Choi, J.H. 235 Kim, Y., see Semenov, Yu.G. 296 Kim, Y., see Yokoi, H. 302, 303 Kim, Y.-K. 22, 76, 162 Kim, Y.D. 354 Kim, Y.D., see Ko, Y.D. 354 Kim, Y.J., see Yamada, K. 447, 448 Kim, Y.K. 140 Kim, Y.K., see Kim, K.S. 532 Kim, Y.K., see Kirschenbaum, L.S. 135 Kim, Y.K., see Michel, R.P. 163 Kim, Y.K., see Park, J.-S. 34, 45, 123 Kim, Y.K., see Russek, S.E. 128 Kimura, A., see Hayashi, K. 236, 275 Kimura, H. 446 Kimura, H., see Koike, Y. 448 Kimura, H., see Suzuki, T. 445 Kimura, H., see Yamada, K. 447, 448 Kimura, N., see Kakehashi, Y. 217 Kimura, T., see Imada, S. 310 Kimura, T., see Ino, A. 455 Kinder, L.R., see Moodera, J.S. 5 Kindo, K., see Hori, H. 276 King, D.M. 464 King, J.P. 36, 127 Kinkhabwala, A., see Kojima, K.M. 449 Kioseoglou, G., see Jonker, B.T. 304 Kioseoglou, G., see Park, Y.D. 304 Kioussis, N., see Chen, L. 263 Kipp, L. 441 Kirichenko, F.V., see Semenov, Yu.G. 306 Kiriliyk, A., see Manders, F. 45 Kirilyuk, A. 155 Kirk, K.J., see Chapman, J.N. 129, 130

AUTHOR INDEX Kirschenbaum, L.S. 135 Kirschner, J., see Imada, S. 310 Kirschner, J., see Kuch, W. 230 Kirschner, J., see Lysenko, O.V. 275 Kirschner, J., see Oepen, H.P. 252 Kirschner, J., see Qian, D. 238 Kirschner, J., see Sander, D. 208 Kirschner, J., see Schmid, A.K. 268 Kirschner, J., see Shen, J. 208, 238 Kirschner, J., see Wu, Y.Z. 218, 249 Kirschner, M., see Suess, D. 155 Kirtley, J.R., see Petrov, D.K. 22, 25 Kirtley, J.R., see Tsuei, C.C. 464 Kirtley, J.R., see van der Marel, D. 442 Kishi, H. 162, 171 Kishi, H., see Kitade, Y. 31, 72, 140 Kishi, H., see Tanaka, A. 51, 162 Kishio, H., see Lanzara, A. 409, 427, 454, 482 Kishio, K., see Bogdanov, P.V. 427 Kishio, K., see Ino, A. 455 Kishio, K., see Ishida, K. 479 Kisker, E., see Kurlyandskaya, G.V. 503, 537 Kisker, E., see Roth, Ch. 228 Kisker, E., see Uzdin, V.M. 222 Kita, E., see Hardner, H.T. 135 Kita, E., see Shen, L.P. 555 Kita, E., see Shi, J. 63 Kita, T., see Marsal, L. 307 Kitade, Y. 31, 72, 140 Kitade, Y., see Kishi, H. 162, 171 Kitakami, O. 34, 163 Kitaoka, Y., see Asayama, K. 430, 437, 473, 474 Kitaoka, Y., see Ishida, K. 479 Kitayama, K., see Kyomen, T. 457 Kitazawa, H., see Adachi, N. 349 Kitoh, T. 542–544 Kittel, C. 28, 89, 143, 528 Kivelson, S., see Heeger, A.J. 393 Kivelson, S.A. 392, 413, 454, 455 Kivelson, S.A., see Arrigoni, E. 407 Kivelson, S.A., see Carlson, E.W. 389, 398, 406 Kivelson, S.A., see Emery, V.J. 390, 398, 407 Kivelson, S.A., see Kee, H.Y. 428 Kivelson, S.A., see Löw, U. 407 Kivelson, S.A., see Nussinov, Z. 408 Kivelson, S.A., see Pryadko, L.P. 404, 407 Kivelson, S.A., see Rokhsar, D.S. 449 Kiwi, M. 148, 155 Klaassen, K.B., see Xiao, M. 135 Klar, P.J. 296, 308 Klar, P.J., see Chen, L. 307 Klar, P.J., see Falk, H. 306 Klar, P.J., see Östreich, M. 304 Kläsges, R., see Pampuch, C. 253

601

Klaua, M., see Qian, D. 238 Klaua, M., see Shen, J. 238 Klauss, H.-H. 438, 439, 450 Klauss, H.H., see Hillberg, M. 361 Kleeman, Th., see Roth, Ch. 228 Klein, B.M., see Wang, C.S. 217, 231 Klein, M.V., see Blumberg, G. 418, 459 Klein, M.V., see Kim, Y.D. 354 Klein, M.V., see Ko, Y.D. 354 Kleinman, L. 213 Kleinman, L., see Batyrev, I.G. 225 Kleinman, L., see Bryk, T. 224, 225 Klemmer, T.J., see Gong, H. 67 Kleyna, R., see Seewald, G. 237 Klokholm, E. 140 Klopotowski, L., see Ghali, M. 304, 305 Klose, F. 248 Klose, F., see Nawrath, T. 227 Klosowski, P. 344, 345 Klosowski, P., see Giebułtowicz, T.M. 344, 345, 353 Klosowski, P., see Samarth, N. 344, 345 Klots, T.D. 255 Kmruglyak, V.V., see Barman, A. 131 Knabben, D., see Uzdin, V.M. 222 Knickelbein, M.B. 211, 250 Knobel, M. 502, 504, 506, 511, 532, 535, 536, 538, 539, 549, 557 Knobel, M., see Duque, J.G.S. 536 Knobel, M., see Gómez-Polo, C. 537, 542–544 Knobel, M., see Kraus, L. 504, 533, 540, 541 Knobel, M., see Medina, A.N. 532 Knobel, M., see Nie, H.B. 504, 541 Knobel, M., see Pirota, K.R. 503, 504, 514, 535, 537, 540–542, 549, 557 Knobel, M., see Sartorelli, M.L. 504, 535 Knobel, M., see Sinnecker, J.P. 504, 510, 511, 528, 534 Knobel, M., see Tejedor, M. 504 Knobel, M., see Valenzuela, R. 537, 549 Knobel, M., see Vázquez, M. 502, 553 Knobel, R., see Crooker, S.A. 303 Knorr, N. 251 Knupfer, M., see Borisenko, S.V. 441 Knupfer, M., see Kordyuk, A.A. 483 Ko, Y.D. 354 Koayashi, T., see Uchiyama, T. 550 Kobayashi, K., see Ino, A. 455 Kobayashi, K., see Kishi, H. 162, 171 Kobayashi, K., see Kitade, Y. 31, 72, 140 Kobayashi, K., see Satake, M. 463 Kobayashi, M. 295 Kobayashi, M., see Durbin, S.M. 295, 344, 345

602

AUTHOR INDEX

Kobayashi, M., see Gunshor, R.L. 344 Kobayashi, M., see Han, J. 295, 344, 345 Kobayashi, M., see Kayanuma, K. 305 Kobayashi, M., see Yamane, H. 22 Kobayashi, S., see Akimoto, R. 307 Kobayashi, T., see Wong, B.Y. 170, 171 Kobayashi, T., see Yoda, H. 22 Kobayashi, Y. 477 Kobayashi, Y., see Sato, H. 62 Koch, M. 301 Kochelaev, B.I. 453 Kochereshko, V.P. 302 Kochereshko, V.P., see Akimov, A.V. 299 Kochereshko, V.P., see Ivchenko, E.L. 297 Kochereshko, V.P., see Kutrowski, M. 296 Kochereshko, V.P., see Ossau, W. 302 Kochereshko, V.P., see Pozina, G.R. 297 Kochereshko, V.P., see Yakovlev, D.R. 298, 302 Koeppe, P.V., see Smith, N. 135 Kogan, Sh. 133 Kogan, V.G., see Huh, Y.M. 453 Koh, H., see Choi, J.H. 235 Kohl, C. 216 Kohlhepp, J., see de Vries, J.J. 220, 244, 245, 247 Kohlhepp, J.T., see Coehoorn, R. 153 Kohlhepp, J.T., see LeClair, P. 341 Kohlhepp, J.T., see Strijkers, G.J. 220, 244, 245 Kohn, W. 204, 213 Kohno, H., see Demler, E. 417 Kohno, H., see Normand, B. 420 Kohno, H., see Rabello, S. 416 Kohno, H., see Tanamoto, T. 420 Kohzawa, T., see Mohri, K. 508, 509 Koi, K., see Fukuzawa, H. 33, 37, 42, 48, 137, 140, 141 Koi, K., see Kools, J.C.S. 37, 49, 71 Koi, K., see Sant, S. 36, 49, 51, 71 Koi, K., see Takagishi, M. 13 Koi, K., see Yuasa, H. 13 Koike, F., see Hasegawa, N. 33–35, 38, 46, 48, 164 Koike, F., see Saito, M. 33, 34, 45, 162, 170 Koike, S., see Yanagisawa, T. 397 Koike, Y. 448 Koike, Y., see Watanabe, I. 450, 479, 481 Koitzsch, A., see Blumberg, G. 464 Kojima, H., see Matsuda, M. 466, 467 Kojima, K.M. 449 Kojima, K.M., see Nachumi, B. 449, 450 Kojima, K.M., see Savici, A.T. 449 Kojima, K.M., see Wakimoto, S. 445 Kokado, S. 109 Kokko, K., see Bazhanov, D.I. 205, 252

Kokko, K., see Stepanyuk, V.S. 253 Kolb, E., see Renard, J.-P. 86 Kole´snik, S., see Ferrand, D. 310, 311 Kole´snik, S., see Sawicki, M. 311 Kolesnikov, I.V. 332, 340 Kolesnikov, N.N., see He, H. 411, 426, 449, 482 Kollar, J., see Vitos, L. 275 Kolodziejski, L.A. 295 Kolodziejski, L.A., see Chang, S.-K. 296 Kolodziejski, L.A., see Gunshor, R.L. 344 Kolodziejski, L.A., see Nurmikko, A.V. 296 Komarov, A.V., see Abramishvily, V.G. 351, 352, 357 Komeda, T., see Susaki, T. 251 Komiya, S., see Ando, Y. 454, 455 Komiya, S., see Lavrov, A.N. 484 Komiyama, K., see Tsunashima, S. 57 Komuro, M., see Hamakawa, Y. 160, 168 Komuro, M., see Hoshiya, H. 34, 51 Kondo, J. 209, 251 Kondo, M., see Ozue, T. 22 Kondo, R., see Tanaka, A. 13 Kondo, R., see Varga, L. 131, 132 König, B. 294, 306 König, B., see Keller, D. 306 König, B., see Kutrowski, M. 296 König, B., see Welsch, M.K. 307 König, B., see Wojtowicz, T. 302 König, B., see Yakovlev, D.R. 297 König, B., see Zaitsev, S. 309 Konishiike, I., see Hosomi, M. 13 Konoto, M., see Tsunoda, M. 167 Konvicka, C., see Bischoff, M.M.J. 219, 225 Kools, J., see Mao, M. 37, 49 Kools, J., see Sant, S. 36, 49, 51, 71 Kools, J.C.S. 5, 31, 35, 37, 47, 49, 68, 71, 120, 144–148, 163 Kools, J.C.S., see Chapman, J.N. 129, 130 Kools, J.C.S., see Coehoorn, R. 5, 151 Kools, J.C.S., see Duchateau, J.P.W.B. 66, 67 Kools, J.C.S., see Folkerts, W. 22, 127–129 Kools, J.C.S., see Gillies, M.F. 125, 126 Kools, J.C.S., see King, J.P. 36, 127 Kools, J.C.S., see Labrune, M. 120, 121, 127 Kools, J.C.S., see Lenssen, K.-M.H. 31, 39, 67, 71 Kools, J.C.S., see McCord, J. 128 Kools, J.C.S., see Oepts, W. 138 Kools, J.C.S., see Rijks, Th.G.S.M. 14, 16, 66, 97, 104, 136–138, 157 Kools, J.C.S., see Schwartz, P.V. 31, 70 Kools, J.C.S., see Swagten, H.J.M. 34, 46, 92, 93 Kools, J.C.S., see Van de Veerdonk, R.J.M. 134

AUTHOR INDEX Kools, J.C.S., see van der Heijden, P.A.A. 151, 157, 163 Koon, N.C. 155 Koopmans, B. 301 Kopalko, K., see Cywi´nski, G. 307 Kopalko, K., see Wojtowicz, T. 295 Kopinga, K., see Vennix, C.W.H.M. 315, 320 Kopmann, W., see Klauss, H.-H. 438, 439, 450 Kopp, Th., see Heimbrodt, W. 305 Koppensteiner, E. 342 Kordyuk, A.A. 483 Kordyuk, A.A., see Borisenko, S.V. 441 Korenivski, V. 224 Korenivski, V., see Sukstanskii, A.L. 529, 531 Kornilovitch, P.E., see Alexandrov, A.S. 409 Korringa, J. 431, 436 Korte, U., see Nouvertné, F. 268, 269 Koshizuka, N., see Johnson, P.D. 482 Koshizuka, N., see Shen, Z.-X. 479, 482 Kossacki, P. 296, 302, 344, 345 Kossacki, P., see Maslana, W. 300 Kossacki, P., see Wojtowicz, T. 295 Kossacki, P., see Wypior, G. 296 Kossut, J. 291–294, 310, 312–315, 318, 325, 352, 354, 356, 360, 362 Kossut, J., see Brazis, R. 309 Kossut, J., see Cywi´nski, G. 307, 310 Kossut, J., see Dahl, M. 311 Kossut, J., see Dietl, T. 304 Kossut, J., see Dybko, K. 360 Kossut, J., see Furdyna, J.K. 300, 312, 318, 325 Kossut, J., see Gałazka, ˛ R.R. 291, 312–314, 325, 328, 329, 345 Kossut, J., see Ghali, M. 304, 305 Kossut, J., see Imanaka, Y. 302, 303 Kossut, J., see Janik, E. 295 Kossut, J., see Jaroszy´nski, J. 304 Kossut, J., see Jouanne, M. 311 Kossut, J., see Karczewski, G. 304 Kossut, J., see Kim, C.S. 306 Kossut, J., see Kochereshko, V.P. 302 Kossut, J., see König, B. 306 Kossut, J., see Kossacki, P. 344, 345 Kossut, J., see Kudelski, A. 310 Kossut, J., see Kunimatsu, H. 302 Kossut, J., see Kusrayev, Yu.G. 298 Kossut, J., see Kutrowski, M. 295, 296 Kossut, J., see Kyrychenko, F.V. 307 Kossut, J., see Lemaître, A. 296 Kossut, J., see Mackh, G. 300 Kossut, J., see Ma´ckowski, S. 308, 309 Kossut, J., see Maslana, W. 300 Kossut, J., see Merkulov, I.A. 303 Kossut, J., see Mino, H. 298, 303

603

Kossut, J., see Ossau, W. 302 Kossut, J., see Pulizzi, F. 298 Kossut, J., see Redli´nski, P. 302 Kossut, J., see Sawicki, M. 310, 311 Kossut, J., see Semenov, Yu.G. 296 Kossut, J., see Shcherbakov, A.V. 306 Kossut, J., see Stachow-Wójcik, A. 310 Kossut, J., see Stirner, T. 300 Kossut, J., see Syed, M. 297 Kossut, J., see Szuszkiewicz, W. 311 Kossut, J., see Takeyama, S. 300, 302, 303 Kossut, J., see Wasik, D. 304 Kossut, J., see Wojtowicz, T. 295, 296, 302, 311, 349 Kossut, J., see Wypior, G. 296 Kossut, J., see Yakovlev, D.R. 299, 310 Kossut, J., see Yokoi, H. 302, 303 Koster, G.F., see Slater, J.C. 213 Kostyk, D. 342, 343 Kosuge, K., see Imai, T. 474 Kotani, A. 207 Kotani, A., see Krüger, P. 236, 241, 242, 262–264 Kotani, A., see Parlebas, J.C. 262 Kotani, A., see Uozumi, T. 207 Kotliar, G., see Georges, A. 397 Kotliar, G., see Lichtenstein, A.I. 209 Kotsugi, M., see Imada, S. 310 Koudinov, A.V., see Aguekian, V.F. 300 Koudinov, A.V., see Kusrayev, Yu.G. 298 Koui, K., see Kamiguchi, Y. 35, 48, 49, 77 Kouwenhoven, L.P., see Van Wees, B.J. 258 Kowalczyk, L. 338, 340, 341 Kowalczyk, L., see Anderson, J.R. 325 Kowalczyk, L., see Chernyshova, M. 332, 338 Kowalski, B.J. 312, 329 Kowalski, B.J., see Orlowski, B.A. 315 Koyanagi, T., see Anno, H. 344 Koyanagi, T., see Fukuma, Y. 322 Kozanecki, A., see Kowalski, B.J. 312, 329 Krajewski, J.J., see Cava, R.J. 459 Krakauer, H., see Wang, C.S. 217, 231 Krakauer, H., see Wimmer, E. 211 Krämer, S. 478 Krasemann, V., see Shivaparan, N.R. 266 Kratzert, P.R. 309 Kraus, L. 502, 504, 508, 514–517, 525, 526, 528–530, 533, 540, 541, 543–546 Kraus, L., see Hauser, H. 550 Kraus, L., see Pirota, K.R. 503, 504, 514, 540–542, 549 Kraus, L., see Ripka, P. 502 Kraus, L., see Sinnecker, J.P. 528

604

AUTHOR INDEX

Kräußlich, J., see Langer, J. 32, 70 Krauth, W., see Georges, A. 397 Krebs, J.J., see Chaiken, A. 12, 94 Krebs, J.J., see Jonker, B.T. 294 Krebs, J.J., see Miller, M.M. 22 Krebs, O., see Kudelski, A. 298 Kreijveld, M.W., see Stachow-Wójcik, A. 333–335 Kreller, F. 298 Krembel, C., see Elmouhssine, O. 238 Krenn, H. 312, 323–326, 344, 345 Krenn, H., see Kepa, H. 343, 345 Krenn, H., see Springholz, G. 326, 327 Krenn, H., see Yuan, S. 323, 324, 327 Kresse, G., see Hobbs, D. 206, 216, 250 Kret, S., see Kutrowski, M. 296 Kret, S., see Ma´ckowski, S. 308 Krevet, R. 346 Krevet, R., see Mac, W. 346 Kriechbaum, M., see Springholz, G. 327 Kriechbaum, M., see Yuan, S. 327 Krill, G., see Andrieu, S. 220, 228 Krill, G., see Beaurepaire, E. 209 Krill, G., see Vogel, J. 219 Krishnan, K. 162, 170 Krishnan, R., see Acharya, B.R. 231 Krishnan, R., see Zuberek, R. 140 Kroeker, M., see Vieth, M. 25 Kroha, J., see Ujsaghy, O. 252 Kronenberger, A., see Juraszek, J. 135, 136, 171 Krongelb, S., see Hempstead, R.D. 151, 157, 172 Kronmüller, H., see Guo, H.Q. 504, 532, 539 Krost, A. 324 Kroto, H.W., see Prados, C. 261 Krowczynski, A., see Gallani, J.L. 275 Krüger, P. 236, 238, 241–244, 262–264 Krüger, P., see Elmouhssine, O. 228 Krüger, P., see Parlebas, J.C. 262 Kruis, H.V., see Zaanen, J. 389, 396, 404, 407, 420 Krusin-Elbaum, L., see Petrov, D.K. 22, 25 Kryder, M., see Xi, H. 118 Kryder, M.H., see Cain, W.C. 172 Kryder, M.H., see Devasahayam, A. 157, 158, 164 Kryder, M.H., see Devasahayam, A.J. 161, 166, 170 Kryder, M.H., see Leal, J.L. 57, 142, 144–147 Kryder, M.H., see Wei, Y. 22 Kryder, M.H., see Xiao, M. 135 Kübler, J. 216 Kuboki, K., see Tanamoto, T. 420 Kuch, W. 230, 231 Kuch, W., see Imada, S. 310

Kudasov, Yu.B. 209 Kudelski, A. 298, 310 Kudrnovsky, J. 143 Kudrnovsky, J., see Pajda, M. 209 Kuhn, S., see Lambrecht, A. 312, 323, 324 Kuhn-Heinrich, B. 296 Kuhns, P., see Julien, M.-H. 452 Kuhns, P., see Mitrovi´c, V.F. 470, 477 Kuiper, A.E.T., see Gillies, M. 36, 37, 48–50, 77 Kuiper, A.E.T., see Lenssen, K.-M.H. 22, 23, 33, 43, 44, 51, 54, 59, 60, 166 Kuiper, A.E.T., see van Driel, J. 152, 158, 167 Kukovitskii, E.F., see Finkelstein, A.M. 453, 478 Kula, W., see Kools, J.C.S. 146, 148 Kulakovskii, V.D. 306 Kulakovskii, V.D., see Bacher, G. 309 Kulakovskii, V.D., see Maksimov, A.A. 309 Kulakovskii, V.D., see Zaitsev, S. 309 Kulda, J., see Aeppli, G. 447 Kulik, T., see Hernando, A. 536 Kulikov, N.I. 223 Kumar, S., see Mattson, J.E. 220, 244 Kumar, V. 251 Kume, M., see Fujita, M. 41 Kung, K.K., see Tang, D.D. 22, 24, 58, 128 Kung, K.T.-Y. 157, 174 Kunimatsu, H. 302 Kunimatsu, H., see Takeyama, S. 302 Kuoksan, H., see Kim, Y.-K. 22 Kurahashi, K., see Sato, T. 464 Kurahashi, K., see Singley, E.J. 466 Kurahashi, K., see Yamada, K. 447, 448, 466 Kurikawa, T., see Nagao, S. 261 Kuriyama, T., see Hasegawa, N. 33, 34 Kuriyama, T., see Saito, M. 33, 34, 45, 162, 170 Kurlyandskaya, G.V. 503, 504, 534, 537 Kurlyandskaya, G.V., see Barandiarán, J.M. 504, 513 Kurlyandskaya, G.V., see Muñoz, J.L. 528 Kurlyandskaya, G.V., see Sinnecker, J.P. 535 Kurlyandskaya, G.V., see Vázquez, M. 512, 535, 557 Kurnosov, V.S., see Pashkevich, Yu.G. 461 Kuroda, N. 355 Kuroda, S. 308, 309 Kuroda, S., see Akinaga, H. 296 Kuroda, S., see Matsuda, Y.H. 303 Kuroda, S., see Terai Jr., Y. 308 Kuroda, S., see Yasuhira, T. 303 Kuroda, T., see Kuroda, S. 308, 309 Kurz, P., see Heinze, S. 277 Kurz, Ph. 204, 206, 241, 270 Kurz, Ph., see Asada, T. 208

AUTHOR INDEX Kurz, Ph., see Wortmann, D. 204 Kusmartsev, F.V. 409 Kusrayev, Yu.G. 298 Kusrayev, Yu.G., see Aguekian, V.F. 300 Kusumoto, D. 556 Kutner-Pielaszek, J., see Kepa, H. 336, 338, 340 Kutrowski, M. 295, 296 Kutrowski, M., see Karczewski, G. 304 Kutrowski, M., see Kochereshko, V.P. 302 Kutrowski, M., see König, B. 306 Kutrowski, M., see Kossacki, P. 296 Kutrowski, M., see Sawicki, M. 311 Kutrowski, M., see Wojtowicz, T. 295, 296, 302 Kuzminski, S. 357 Kwei, G.H., see Bo˜zin, E.S. 448 Kwon, S.K. 253 Kyomen, T. 457 Kyrychenko, F., see Kudelski, A. 310 Kyrychenko, F.V. 307 Kyrychenko, F.V., see Kossut, J. 310 Kyrychenko, F.V., see Siviniant, J. 296 Labergie, D. 248, 249 Labergie, D., see Uzdin, V.M. 249 Labrune, M. 120, 121, 127 Lacour, D., see Kirilyuk, A. 155 Lacroix, J.M., see Lombos, B.A. 328 Ladd, T., see Petta, J.R. 135 Lægsgaard, E., see Pedersen, M.O. 268, 269 Lagar’kov, A.N., see Antonov, A. 504, 527 Lagar’kov, A.N., see Antonov, A.S. 503, 532, 533, 537 Lagar’kov, A.N., see Makhnovskiy, D.P. 526–528 Lagar’kov, A.N., see Usov, N. 515, 525–528, 530 Lai, C.-H. 163 Lai, C.H., see Wang, Y.H. 170 Lai, W., see Lu, Z. 54, 160 Lai, W., see Lu, Z.Q. 69 Lai, W.Y., see Shen, F. 38, 49 Lai, W.Y., see Yang, T. 161 Laidler, H., see Hughes, T. 158, 175 Lairson, B.M., see Kim, Y.-K. 22 Lairson, B.M., see Stokes, S.W. 135 Lake, B. 413, 446, 448, 449, 480 Lamberti, V.E., see Heaney, P.J. 457 Lambeth, D.N., see Gong, H. 67 Lambrecht, A. 312, 323, 324 Lambrecht, W.R.L., see Yang, H. 249 Landau, L. 505, 506, 516 Landman, U., see Moseler, M. 251 Landolt, M., see Briner, B. 220, 244

605

Landolt, M., see Fuchs, P. 226, 227 Landwehr, G., see Akimov, A.V. 299 Landwehr, G., see Bacher, G. 309 Landwehr, G., see Ivchenko, E.L. 297 Landwehr, G., see Koch, M. 301 Landwehr, G., see Kochereshko, V.P. 302 Landwehr, G., see König, B. 294 Landwehr, G., see Kuhn-Heinrich, B. 296 Landwehr, G., see Kulakovskii, V.D. 306 Landwehr, G., see Mackh, G. 300, 303 Landwehr, G., see Maksimov, A.A. 309 Landwehr, G., see Merkulov, I.A. 303 Landwehr, G., see Ossau, W. 302 Landwehr, G., see Pozina, G.R. 297 Landwehr, G., see Reshina, I.I. 309 Landwehr, G., see Shcherbakov, A.V. 306 Landwehr, G., see Tönnies, D. 296 Landwehr, G., see Waag, A. 294 Landwehr, G., see Wojtowicz, T. 302 Landwehr, G., see Yakovlev, D.R. 297–300, 302, 306, 310 Lang, K.M. 441, 480 Lang, K.M., see Hoffman, J.E. 413, 441, 442, 471, 479, 481 Lang, K.M., see Hudson, E.W. 481 Lang, P., see Stepanyuk, V.S. 212, 253 Langelaar, M.H. 238 Langer, J. 32, 70 Lanzara, A. 409, 427, 454, 482 Lanzara, A., see Bogdanov, P.V. 427 Lanzara, A., see Feng, D.L. 482 Lanzara, A., see Saini, N.L. 454 Lanzara, A., see Zhou, X.J. 454 Lapertot, G., see Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Lardoux, S., see Kools, J.C.S. 5, 35, 47, 163 Larin, V., see Cobeño, A.F. 555 Larin, V., see Malliavin, M.J. 515 Larin, V., see Zhukova, V. 533 Larin, V.S., see Baranov, S.A. 548 Larkin, M., see Nachumi, B. 449, 450 Larkin, M.I., see Kojima, K.M. 449 Larkin, M.I., see Savici, A.T. 449 Larson, B.E., see Samarth, N. 344, 345 Larson, W., see Katti, R.R. 57, 58 Lascaray, J.P., see Bruno, A. 312, 328, 329 Lascaray, J.P., see El Ouazzani, A. 351, 352 Lascaray, J.P., see Nawrocki, M. 351 Lascaray, J.P., see Paganotto, N. 302 Lascaray, J.P., see Ribayrol, A. 297 Lascaray, J.P., see Siviniant, J. 296 Lascialfari, A., see Borsa, F. 433, 449 Lashkarev, G.V., see Łazarczyk, P. 315 Lashkarev, G.V., see Radchenko, M.V. 315

606 Lathiotakis, N.N. 257 Lathiotakis, N.N., see Andriotis, A. 215 Latrach, M., see Reynet, O. 535 Lau, B.K., see Shapira, Y. 353 Laughlin, D.E., see Wong, B.Y. 170, 171 Laughlin, D.E., see Xi, H. 167 Laughlin, R.B. 390 Laughlin, R.B., see Chakravarty, S. 419, 473 Lauter, H.J., see Lauter-Pasyuk, V. 224 Lauter-Pasyuk, V. 224 Lavagna, M. 418 Lavagna, M., see Stemmann, G. 420 Lavrov, A.N. 484 Lavrov, A.N., see Ando, Y. 454, 455 Law, B., see Mao, M. 56 Law, B., see Mao, S. 69 Lawless, M.J., see Nicholas, R.J. 306 Lawniczak-Jabło´nska, K. 345 Lawniczak-Jabło´nska, K., see Fita, P. 312 Lawniczak-Jabło´nska, K., see Iwanowski, R.J. 315 Lawrence, I. 305 Łazarczyk, P. 315, 316 Łazarczyk, P., see Story, T. 328 Lazarovits, B. 209, 238 Le, S., see Tang, D.D. 22, 24, 58, 128 Le, T., see Carey, M.J. 33, 67 Le Bacq, O. 228 Le Bacq, O., see Broddefalk, A. 228 Le Cann, X., see Boeglin, C. 235 Le Fevre, P., see Boeglin, C. 235 Le Si Dang, see André, R. 295 Le Si Dang, see Bodin, C. 295 Le Van Khoi, see Sawicki, M. 311 Leal, J.L. 57, 142, 144–147 Lebihen, T. 296 Lebihen, T., see Deleporte, E. 297 LeClair, P. 5, 341 Lederman, D., see Nogues, J. 149 Lederman, D., see Tomaz, M.A. 236, 275 Lederman, M. 158, 161, 162, 166 Lederman, M., see Tong, H.C. 51, 55 Ledieu, M., see Acher, O. 535 Ledoux, M.J., see Pham-Huu, C. 265 Lee, C. 67, 170 Lee, C.-G., see Fry, R.A. 138 Lee, C.-L., see Kools, J.C.S. 145, 146 Lee, C.G., see McMichael, R.D. 138 Lee, C.H., see Yamada, K. 447, 448 Lee, D.-H., see Hoffman, J.E. 481 Lee, D.H., see Kivelson, S.A. 413 Lee, G.H. 261 Lee, G.U., see Baselt, D.R. 24 Lee, G.Y., see Rhee, J.-R. 161

AUTHOR INDEX Lee, H., see Cho, W.S. 540, 541 Lee, H., see Ro, J. 158 Lee, H.B., see Kim, K.S. 532 Lee, H.H., see Kim, J.U. 305 Lee, J.I. 230 Lee, K., see Tsang, C. 157, 176 Lee, K.-A., see Kim, J.-K. 162 Lee, K.A., see Hwang, D.G. 73 Lee, M.-H. 163 Lee, N.I., see Rhee, J.-R. 161 Lee, P.A., see Brinckmann, J. 412, 420, 421, 424–426 Lee, P.A., see Wen, X.G. 401, 420 Lee, S. 297, 305 Lee, S., see Kim, C.S. 306, 309, 310 Lee, S., see Lee, M.-H. 163 Lee, S., see Ma´ckowski, S. 308 Lee, S., see Rhyne, J.J. 345 Lee, S., see Stumpe, L.E. 345 Lee, S.-H. 404, 442, 443, 459 Lee, S.-H., see Ichikawa, N. 452, 454, 455 Lee, S.-H., see Kimura, H. 446 Lee, S.-H., see Lee, Y.S. 448, 449 Lee, S.-H., see Wakimoto, S. 445, 447, 448 Lee, S.-R., see Kim, Y.-K. 76, 162 Lee, S.-R., see Park, J.-S. 34, 45, 123 Lee, S.-S., see Kim, J.-K. 162 Lee, S.H., see Fujita, M. 447, 448 Lee, S.H., see Wakimoto, S. 447, 448 Lee, S.K. 235 Lee, S.R., see Mattson, J.E. 220, 244 Lee, S.S., see Hwang, D.G. 73, 164 Lee, S.S., see Rhee, J.-R. 161 Lee, W.Y. 37, 51, 71, 162, 170 Lee, W.Y., see Ju, Y.S. 21 Lee, Y.H., see Hwang, D.G. 73 Lee, Y.H., see Yoon, S.S. 540 Lee, Y.S. 448, 449, 486 Lee, Y.S., see Kimura, H. 446 Lee, Y.S., see Savici, A.T. 449 Lee, Y.S., see Wakimoto, S. 447, 448 Lefakis, H. 67 Lefakis, H., see Dieny, B. 18, 19, 59, 60, 95, 99 Lefakis, H., see Gurney, B.A. 22, 31–33, 40, 41, 97, 129, 140 Lefakis, H., see Huang, T.C. 70, 72 Lefakis, H., see Meny, C. 71 Lefakis, H., see Nozières, J.P. 72 Lefakis, H., see Ounadjela, K. 140 Lefakis, H., see Speriosu, V.S. 18, 40, 45, 72, 110, 145 Lefmann, K., see Lake, B. 413, 446, 448, 449, 480

AUTHOR INDEX Legge, M., see Welsch, M.K. 307 Leggett, A.J., see van der Marel, D. 442 Legner, S., see Borisenko, S.V. 441 Leibbrandt, G.W.R., see Gillies, M. 36, 37, 48, 49, 77 Leisching, P. 299 Leisching, P., see Buss, C. 299 Leisner, T., see Vajda, S. 255 Lekka, Ch.E. 214 Lemaître, A. 296 Lemaître, A., see Jusserand, B. 303 Lemaître, A., see Zielinski, M. 351, 353 Lemberger, T.R., see Skinta, J.A. 464 Lemmens, L.F., see Story, T. 317 Lemmens, P., see Pashkevich, Yu.G. 461, 463 Lemor, W., see Kochelaev, B.I. 453 Leng, Q., see Mao, S. 7 Lenssen, K.-M.H. 22, 23, 31, 33, 39, 43, 44, 51, 54, 59, 60, 67, 71, 157, 166 Lenssen, K.-M.H., see Coehoorn, R. 5, 151 Lenssen, K.-M.H., see Folkerts, W. 22, 127–129 Lenssen, K.-M.H., see van Driel, J. 64, 151, 152, 158, 159, 167, 176 Lenz, J.E. 23 Leotin, J., see Martin, J.L. 354 Lercher, M.J. 420 Leszczy´nski, M., see Janik, E. 295 Leuken, H.v. 253 Leung, C.W., see Morecroft, D. 138 Leupold, O., see Röhlsberger, R. 252 Levanov, N.A. 215 Levin, K., see Liu, D.Z. 420 Levin, K., see Si, Q.M. 423 Levin, K., see Zha, Y. 420, 423 Levin, Yu.K., see Makhotkin, V.E. 504 Levy, J. 301 Levy, J., see Crooker, S.A. 295 Levy, P. 5 Levy, P.M. 5, 109 Levy, P.M., see Blaas, C. 80, 109 Levy, P.M., see Camblong, H.E. 103, 109 Lew, W.S., see Li, S.P. 276 Lewicki, A. 353 Lewicki, A., see Alawadhi, H. 352, 353 Lewicki, A., see Twardowski, A. 353 Ley, L., see Denecke, R. 315 Lhermet, H. 135 Lhuillier, C., see Giamarchi, T. 397 Li, C. 235 Li, D. 540 Li, D., see Camley, R.E. 209 Li, D., see Durbin, S.M. 295, 344, 345 Li, G.-H., see Zhu, Z.-M. 296 Li, H. 22, 24, 36, 51, 52, 159, 167

607

Li, J., see Li, D. 540 Li, J., see Uhlig, W.C. 36, 48, 49 Li, K. 37, 38, 45, 49, 50, 92, 145, 146, 159, 167 Li, K., see Wu, Y. 57 Li, K.B., see Guo, Z.B. 159 Li, L., see Vázquez, M. 504 Li, M. 38, 48, 61, 159, 167 Li, M., see Dieny, B. 7, 32, 33, 41, 45, 61, 95, 96, 101, 112, 113, 120, 123 Li, M., see Molegraaf, H.J.A. 479 Li, Q., see Johnson, P.D. 482 Li, Q., see Lee, S.-H. 404, 459 Li, S., see Araki, S. 32–34, 36, 67 Li, S., see Oliveira, N.J. 22 Li, S., see Tsuchiya, Y. 36 Li, S.P. 276 Li, Y.F. 176 Li, Y.F., see Chen, D.-X. 537 Li, Y.F., see Vázquez, M. 534 Li, Y.H., see Kale, S. 535 Li, Y.S., see Lu, S.H. 231 Li, Z. 135, 155 Li, Z., see Kaminski, A. 419 Li, Z.G., see Parkin, S.S.P. 6 Li, Z.Y., see King, D.M. 464 Liang, B., see Borisenko, S.V. 441 Liang, B., see Etrillard, J. 479 Liang, R., see Sonier, J.E. 477 Liao, S., see Li, M. 38, 48 Liao, S., see Zhu, J.-G. 127 Liao, S.-H., see Li, M. 61, 159, 167 Liao, S.H., see Dieny, B. 7, 32, 33, 41, 45, 61, 95, 96, 101, 112, 113, 120, 123 Lichtenstein, A.I. 209 Lichtenstein, A.I., see Fleck, M. 397, 398 Lifshitz, E.M., see Landau, L. 505, 506, 516 Lim, C.K. 127, 138 Lim, S.H., see Park, J.-S. 34, 45, 123 Lim, W.Y., see Song, S.H. 546 Lin, C.-L. 163 Lin, C.-L., see Egelhoff, W.F. 51, 52, 93 Lin, C.-L., see Egelhoff Jr., W.F. 34, 51, 52, 70, 144 Lin, C.T., see Borisenko, S.V. 441 Lin, C.T., see Etrillard, J. 479 Lin, H.-J., see Pellegrin, E. 463 Lin, H.Q., see Emery, V.J. 407 Lin, J. 345 Lin, J., see Rhyne, J.J. 345 Lin, N. 275 Lin, T. 32, 34, 36, 43, 49, 73, 75, 76, 145, 150, 151, 157, 158, 161, 163, 164, 166, 169 Lin, T., see Baril, L. 138, 139

608

AUTHOR INDEX

Lin, T., see Carey, M.J. 151, 159, 164, 176 Lin, T., see Kools, J.C.S. 146, 148 Lin, T., see Perjeru, F. 217, 220, 275 Lin, T., see Tsang, C. 22, 127 Lin, T., see Tsang, C.H. 4, 5, 22 Lindemer, T.B., see Dai, P. 471 Lindner, J. 228 Lindroos, M. 441 Lindroos, M., see Bansil, A. 441 Ling, W.L., see Man, K.L. 237 Linville, E., see Pokhil, T. 162 Lischka, K., see Schikora, D. 361 Lischka, K., see Widmer, T. 361 Lischka, K., see Widmer, Th. 361 List, R.S., see Dessau, D.S. 427 Litterst, F.J., see Hillberg, M. 361 Litterst, F.J., see Klauss, H.-H. 438, 439, 450 Littlewood, P.B. 418, 423 Littlewood, P.B., see Inui, M. 395 Littlewood, P.B., see Varma, C.M. 418 Littlewood, P.B., see Zaanen, J. 409 Litvinov, D., see Gong, H. 67 Litvinov, V.I. 103 Litvinov, V.I., see Dugaev, V.K. 339, 342 Litvinov, V.I., see Eggenkamp, P.J.T. 318, 320, 321 Litvinov, V.I., see Kolesnikov, I.V. 340 Litz, T., see Dahl, M. 311 Litz, T., see Kuhn-Heinrich, B. 296 Litz, T., see Mackh, G. 300 Liu, C.L. 267 Liu, D.Z. 420 Liu, F., see Shi, S. 22 Liu, F., see Tong, H.C. 33, 51, 52, 55, 124 Liu, G.D., see Xiao, S.Q. 541 Liu, G.L., see Zhao, Z. 265 Liu, H., see Katti, R.R. 57, 58 Liu, M.T., see Bindilatti, V. 312, 326 Liu, M.T., see Twardowski, A. 350, 351 Liu, X. 294 Liu, X., see Jonker, B.T. 294 Liu, X.C. 355 Liu, X.C., see Gennser, U. 351, 355 Liu, Y.H., see Chen, C. 504 Liu, Y.H., see Xiao, S.Q. 541 Liu Jr., M.T., see Bindilatti, V. 312, 326 Lizzit, S., see Goldoni, A. 205, 235, 236 Llois, A.M., see Guevara, J. 254, 257 Lo, C.K., see Wang, Y.H. 170 Loch, C. 161 Loch, C., see Stobiecki, F. 32, 39, 60, 113 Lodder, A., see Leuken, H.v. 253 Loegel, B. 78, 105 Loeser, A.G., see King, D.M. 464

Loeser, A.G., see Marshall, D.S. 476 Löfﬂer, A., see Brinkmann, D. 307 Loﬂand, S.E. 514, 546 Loﬂand, S.E., see Bhagat, S.M. 548 Loﬂand, S.E., see Kale, S. 535 Loidl, A., see Kochelaev, B.I. 453 Loiseau, A., see Zha, F.-X. 261 Loloee, R., see Duvail, J.L. 12, 61, 94, 104 Loloee, R., see Gu, J.Y. 13 Loloee, R., see Slater, R.D. 13 Loloee, R., see Steenwyk, S.D. 13, 80, 83, 85 Loloee, R., see Vouille, C. 84 Loloee, R., see Yang, Q. 78 Lombos, B.A. 328 Lombos, B.A., see Averous, M. 328 Lomdahl, P.S., see Verges, J.A. 395 Longo, R.C. 267 Longo, R.C., see Robles, R. 214–216, 259, 260, 266–270 Lopatin, V.A., see Makhotkin, V.E. 504 López, M.J., see Aguilera-Granja, F. 215, 254–258 López, M.J., see Bouarab, S. 254–256, 258, 266, 267 López, M.J., see Montejano-Carrizales, J.M. 255 Lopez-Diaz, L., see Li, S.P. 276 Loram, J.W. 442 Loram, J.W., see Tallon, J.L. 478 Loram, J.W., see van der Marel, D. 442 Lorenz, R., see Spisak, D. 217 Lorenzo, J.E. 457 Lorenzo, J.E., see Sachan, V. 458 Lorenzo, J.E., see Tranquada, J.M. 456, 457 Loss, D., see Awschalom, D.D. 5 Lottis, D., see Duvail, J.L. 104, 105 Lottis, D.K., see Szucs, J. 59, 60 Louch, S., see Baker, S.H. 264 Louch, S.C., see Binns, C. 264 Louderback, J.G., see Cox, A.J. 205, 219, 234, 235, 275 Louie, L.K., see Kung, K.T.-Y. 157, 174 Louie, S.G. 213 Louis, E. 397 Louis, E., see Verges, J.A. 395 Lounis, S., see Dahmoune, C. 235 Loup, J.P., see Poirot, N.J. 456 Lovett, B.W., see Jestädt, Th. 460 Löw, U. 407 Lowisch, M., see Kreller, F. 298 Lowther, J.E., see Andriotis, A.N. 261 Lu, D.H., see Armitage, N.P. 465 Lu, D.H., see Feng, D.L. 482 Lu, E.D., see Bogdanov, P.V. 427

AUTHOR INDEX Lu, E.D., see Lanzara, A. 409, 427, 454, 482 Lu, E.D., see Zhou, X.J. 454 Lu, J.P., see Si, Q.M. 423 Lu, J.P., see Yang, C.-K. 266 Lu, K., see Ambrose, T. 55, 164 Lu, S.H. 231 Lu, Y. 5 Lu, Y., see Katti, R.R. 57, 58 Lu, Z. 54, 160 Lu, Z., see Li, D. 540 Lu, Z.Q. 37, 48, 54, 69 Lu, Z.Q., see Shen, F. 38, 49 Luan, K.Z., see Chen, C. 504 Lubitz, P., see Miller, M.M. 22 Lucena, M.A., see Rezende, S.M. 164 Luches, P. 231 Luches, P., see Fratucello, G.B. 231 Luches, P., see Gazzadi, G.C. 231 Luke, G.M., see Kojima, K.M. 449 Luke, G.M., see Nachumi, B. 449, 450 Luke, G.M., see Savici, A.T. 449 Luning, J.E., see Pekarek, T.M. 346 Lunn, B., see Aguekian, V.F. 300 Lunn, B., see Cheng, H.H. 297 Lunn, B., see Goede, O. 305 Lunn, B., see Heimbrodt, W. 305 Lunn, B., see Jackson, S. 296 Lunn, B., see Klar, P.J. 308 Lunn, B., see Kusrayev, Yu.G. 298 Lunn, B., see Nicholas, R.J. 306 Lunn, B., see Östreich, M. 304 Lunn, B., see Peyla, P. 298 Lunn, B., see Pier, Th. 305 Lunn, B., see Ribayrol, A. 297 Lunn, B., see Roussignol, Ph. 305 Lunn, B., see Suisky, D. 298 Lunn, B., see Wasiela, A. 297 Lunn, B., see Weston, S.J. 296 Luo, H. 291, 295, 297 Luo, H., see Awschalom, D.D. 301 Luo, H., see Baumberg, J.J. 301 Luo, H., see Dai, N. 294, 297, 358 Luo, H., see Furdyna, J.K. 344, 345 Luo, H., see Giebułtowicz, T.M. 344, 345 Luo, H., see Klosowski, P. 344, 345 Luo, H., see Lee, S. 305 Luo, H., see Samarth, N. 344, 345 Luo, H., see Smyth, J.F. 305 Luo, H., see Zhang, F.C. 297 Luo, Y., see Lin, T. 32, 166 Łusakowska, E., see Chernyshova, M. 332, 338 Łusakowska, E., see Kutrowski, M. 295 Łusakowska, E., see Nadolny, A.J. 321, 322 Łusakowski, A. 318

609

Łusakowski, A., see Dybko, K. 360 Łusakowski, A., see Górska, M. 329 Łusakowski, A., see Mycielski, A. 356, 357 Łusakowski, A., see Story, T. 328–330 Luther, A. 449 Luther, S., see Portugall, O. 361 Luther, S., see Schikora, D. 361 Luther, S., see Widmer, Th. 361 Lyngnes, O., see Kavokin, A.V. 299 Lynn, J.W., see Vajk, O.P. 463 Lyo, S.K., see Crooker, S.A. 309 Lysenko, O.V. 275 Ma, H., see Qiu, S.L. 218 Ma, Y.-Q., see Hu, J.-G. 37, 149 Maan, J.C., see Boonman, M.E.J. 346 Maan, J.C., see Ma´ckowski, S. 309 Maan, J.C., see Pulizzi, F. 298 Maan, J.C., see Yakovlev, D.R. 302 Maan, J.C., see Zeitler, U. 360 Maas, T.F.M.M., see van der Heijden, P.A.A. 151, 157, 163 Maaß, W., see Langer, J. 32, 70 Maass, W., see Loch, C. 161 Maass, W., see Stobiecki, F. 32, 39, 60, 113 Mac, W. 346–351, 357 Mac, W., see Boonman, M.E.J. 346 Mac, W., see Herbich, M. 351 Mac, W., see Janik, E. 295 Mac, W., see Krevet, R. 346 Mac, W., see Łazarczyk, P. 315 Mac, W., see Maslana, W. 300 Mac, W., see McCabe, G.H. 347, 349 Mac, W., see Stachow-Wójcik, A. 310 Mac, W., see Story, T. 333, 334, 336 Macaskie, L.E., see Mikheenko, I.P. 205, 234 MacDonald, A.H., see Cehovin, A. 265 MacDonald, S.A., see Childress, J.R. 32, 41, 67, 159, 174 MacFarlane, W.A., see Bobroff, J. 478 MacFarlane, W.A., see Dooglav, A.V. 473 Machado, F.L.A. 504, 507, 515, 520, 522, 532, 542–544, 554 Machado, F.L.A., see Araújo, A.E.P. 548, 549 Machado, F.L.A., see Mendes, K.C. 541 Machado, F.L.A., see Soares, J.M. 532 Machel, G., see Schikora, D. 361 Machel, G., see Widmer, T. 361 Machida, K., see Kaneshita, E. 409 Machida, K., see Kato, M. 395 Mack, A., see Mao, S. 60 MacKenzie, R., see Burgess, C.P. 416 Mackh, G. 300, 303

610

AUTHOR INDEX

Mackh, G., see Kutrowski, M. 295 Mackh, G., see Wojtowicz, T. 295 Ma´ckowski, S. 308, 309 Ma´ckowski, S., see Cywi´nski, G. 310 Ma´ckowski, S., see Kudelski, A. 310 Ma´ckowski, S., see Lemaître, A. 296 Ma´ckowski, S., see Prechtl, G. 297 Ma´ckowski, S., see Teran, F.J. 302 MacLaren, J.M., see Brown, R.H. 110 MacLaren, J.M., see Butler, W.H. 40, 80, 83, 84, 106–109 MacLaren, J.M., see Nicholson, D.M.C. 40, 80, 85 MacLaren, J.M., see Schulthess, T.C. 83 MacLaughlin, D.E. 432, 433, 438 MacLaughlin, D.E., see Nachumi, B. 449, 450 Madhavan, V. 252 Madhavan, V., see Hoffman, J.E. 413, 441, 442, 471, 479 Madhavan, V., see Hudson, E.W. 481 Madhavan, V., see Jamneala, T. 251, 252, 273 Madhavan, V., see Lang, K.M. 441, 480 Madsen, J., see Vigliante, A. 463 Madsen, J., see Zimmermann, M.v. 454 Maeda, A., see Fujita, M. 41 Maekawa, S., see Inoue, J. 84 Maekawa, S., see Itoh, H. 83 Maekawa, S., see Prelovšek, P. 454 Maesaka, A. 74, 75, 77, 171 Magnan, H., see Boeglin, C. 235 Mahajan, A., see Bobroff, J. 477 Mahajan, A.V. 478 Mahalingam, K., see Luo, H. 295 Mahenthiran, D., see Lee, C. 67, 170 Maher, M.J., see Baker, S.H. 264 Maher, M.J., see Binns, C. 264 Mahnke, H.E., see Waldmann, H. 361 Mahoukou, F. 312 Mahoukou, F., see Maurice, T. 323, 324 Mai, M., see Gibbons, M.R. 36, 48, 99 Mai, Z., see Xu, M. 94 Maier, M., see Springholz, G. 326 Maiti, K., see Dallmeyer, A. 231 Maiti, K., see Gambardella, P. 252 Majewski, J.A., see Blinowski, J. 348 Majkrzak, C.F., see Hjörvarsson, B. 206, 219, 220, 248 Majkrzak, C.F., see Kepa, H. 336, 338, 340, 343, 345 Majkrzak, C.F., see Kimura, H. 446 Majkrzak, C.F., see Lee, S.-H. 459 Majkrzak, C.F., see Nunez, V. 343 Mak, C.-L., see Bak, J. 354 Mak, P., see Feit, Z. 328

Makhnovskiy, D.P. 508, 525–528, 540, 542, 544, 545 Makhnovskiy, D.P., see Panina, L.V. 505, 544 Makhotkin, V.E. 504 Maki, K. 418 Makino, A., see Hasegawa, N. 35, 46, 164 Makino, E. 144, 145 Makino, E., see Hosomi, M. 13 Makino, E., see Takiguchi, M. 74, 75 Maksimov, A.A. 309 Maksimov, A.A., see Bacher, G. 309 Malagoli, M.C., see Gambardella, P. 252 Malajovich, I., see Ray, O. 307 Maletta, H., see Klose, F. 248 Maletta, H., see Nawrath, T. 227 Mali, M., see Matsumura, M. 433 Malinovski, N., see Billas, I.M.L. 265 Malinovski, N., see Tast, F. 265 Maljuk, A., see Borisenko, S.V. 441 Malliavin, M.J. 515 Malozemoff, A.P. 154 Mamin, H.J. 22, 25, 141 Man, K.L. 237 Mancoff, F.B. 25 Mandal, K. 504, 538 Manders, F. 45 Mang, M., see Vajk, O.P. 463 Mang, P.K., see Armitage, N.P. 465 Mangkorntong, N., see Lake, B. 413, 446, 449, 480 Mankey, G.J., see Himpsel, F.J. 207 Manousakis, E., see Hellberg, C.S. 406 Manske, D. 418 Manske, D., see Dahm, T. 418 Manthiram, A., see Tang, J.P. 457 Mao, M. 32, 33, 37, 49, 56, 158 Mao, M., see Hegde, H. 70, 71 Mao, M., see Hung, C.-Y. 136, 137, 140, 141, 162 Mao, M., see Kools, J.C.S. 37, 49, 71, 145, 146 Mao, M., see Lee, C. 67, 170 Mao, M., see Sant, S. 36, 49, 51, 71 Mao, M., see Uhlig, W.C. 36, 48, 49 Mao, S. 7, 32, 41, 55, 60, 69, 160–162, 168 Mao, S., see Chen, J. 32, 33, 112, 159 Mao, S., see Han, D. 161, 162 Mao, S., see Hou, C. 150 Mao, S., see Huang, R.-T. 50, 77 Mao, S., see Pokhil, T. 162 Mao, S., see Qian, Z. 161 Mao, S., see Szucs, J. 59, 60 Mao, S., see Yang, Z. 162, 178 Mapps, D.J., see Makhnovskiy, D.P. 508, 525–527, 540, 542, 544, 545

AUTHOR INDEX Maraner, A. 135 Marangolo, M. 248 Marassi, L., see Gazzadi, G.C. 231 Marassi, L., see Luches, P. 231 Marchukov, P.Yu., see Makhotkin, V.E. 504 Marco, J.F., see Prados, C. 261 Marcus, P.M., see Moruzzi, V.L. 227 Marcus, P.M., see Qiu, S.L. 218 Margulies, D.T., see Moser, A. 21 Mariette, H., see Brinkmann, D. 307 Mariette, H., see Haacke, S. 305 Mariette, H., see Jouanne, M. 311 Mariette, H., see Lawrence, I. 305 Mariette, H., see Marsal, L. 307 Mariette, H., see Szuszkiewicz, W. 311 Marín, P., see Knobel, M. 504, 532, 536, 539, 549 Marín, P., see Vázquez, M. 504, 533 Marinescu, C.S., see Chiriac, H. 504, 532, 533 Marinescu, C.S., see Óvári, T.A. 534 Marks, R.F., see Farrow, R.F.C. 162, 164, 172 Marks, R.F., see Ju, G. 172 Marks, R.F., see Krishnan, K. 162, 170 Marley, A., see Lu, Y. 5 Marley, A.C., see Farrow, R.F.C. 162 Marrows, C.H. 22, 33, 101, 102, 127, 249 Marrows, C.H., see Barman, A. 131 Marsal, L. 307 Marschner, G., see Ueta, A.Y. 323, 324, 326, 327 Marshall, D.S. 476 Marshall, D.S., see King, D.M. 464 Marston, J.B. 402 Martien, D., see Egelhoff Jr., W.F. 35 Martin, I. 401, 481 Martin, I., see Zhu, J.-X. 449 Martin, J.L. 354 Martin, T.P., see Billas, I.M.L. 265 Martin, T.P., see Tast, F. 265 Martinez, E., see Robles, R. 204, 206, 217 Martinez-Pastor, J., see Roussignol, Ph. 305 Martins, C.S., see Machado, F.L.A. 507, 532 Martins, G.B. 406 Martins, J.L., see Troullier, N. 245 Marucco, J.-F., see Bobroff, J. 477, 478 Marucco, J.-F., see Julien, M.-H. 452, 478 Marucco, J.F., see Alloul, H. 478 Marucco, J.F., see Mahajan, A.V. 478 Marucco, J.F., see Mendels, P. 478 Maruyama, B., see Edelstein, A.S. 231 Masaryk, K., see Lee, C. 67, 170 Maslana, W. 300 Mason, T.E. 448 Mason, T.E., see Aeppli, G. 447

611

Mason, T.E., see Lake, B. 413, 446, 448, 449, 480 Mason, T.E., see Mook, H.A. 389, 471 Masri, P., see Bindilatti, V. 312, 326 Masri, P., see Isber, S. 326 Masri, P., see Maurice, T. 323, 324 Massobrio, C., see Billas, I.M.L. 265 Massobrio, C., see Levanov, N.A. 215 Mathet, V., see Isber, S. 312, 322, 323 Matsonashvili, B.N., see Valeiko, M.V. 327 Matsubara, K., see Anno, H. 344 Matsuda, M. 466, 467 Matsuda, M., see Fujita, M. 449 Matsuda, M., see Greven, M. 486 Matsuda, Y., see Kuroda, N. 355 Matsuda, Y.H. 303 Matsuda, Y.H., see Yasuhira, T. 303 Matsukura, F. 291, 311 Matsukura, F., see Dietl, T. 311 Matsukura, F., see Karczewski, G. 311 Matsukura, F., see Ohno, Y. 305 Matsumara, D., see Yonamoto, Y. 206, 249 Matsumura, M. 433 Matsuno, J., see Harima, N. 466 Matsushita, H., see Kimura, H. 446 Matsutera, H., see Suzuki, T. 54 Matsuura, A.Y., see Marshall, D.S. 476 Matsuura, S., see Ishida, A. 323, 324 Matsuyama, H., see Lai, C.-H. 163 Matsuzaki, M., see Araki, S. 32–34, 36, 67, 159 Matsuzaki, M., see Sano, M. 164 Matsuzaki, M., see Shimazawa, K. 159 Matsuzaki, M., see Takano, K. 127 Matsuzaki, M., see Tsuchiya, Y. 36 Mattens, W.C.M., see de Boer, F.R. 77, 166 Mattheis, R., see Boeve, H. 57 Mattheis, R., see Langer, J. 32, 70 Mattis, D.C., see Methfessel, S. 331, 340, 342, 343 Mattocks, P.G., see Mountziaris, T.J. 358 Mattocks, P.G., see Peck, J. 358 Mattson, A.E. 249 Mattson, J.E. 220, 244 Matuzono, A., see Makino, E. 144, 145 Matveenko, A.V., see Valeiko, M.V. 327 Matveenko, S.I. 393, 394 Mauger, A. 331, 342, 343 Mauger, A., see Escorne, M. 315 Mauger, A., see Mycielski, A. 356, 357 Mauger, A., see Scalbert, D. 351 Mauger, A., see Testelin, C. 353, 356 Mauri, D. 154 Mauri, D., see Baril, L. 138, 139 Mauri, D., see Beach, R.S. 124

612

AUTHOR INDEX

Mauri, D., see Dieny, B. 4, 9, 30, 94, 157, 178 Mauri, D., see Kools, J.C.S. 146, 148 Mauri, D., see Lee, W.Y. 37, 51, 71, 162, 170 Mauri, D., see Lin, T. 32, 34, 36, 43, 49, 73, 75, 76, 145, 150, 151, 157, 158, 161, 164, 166, 169 Maurice, T. 323, 324 May, U., see Nouvertné, F. 268, 269 Maylin, M.G., see Atkinson, D. 553, 556 Mayo, E., see Tomaz, M.A. 236, 275 Mayur, A.J., see Aigle, M. 327 Mazin, I.I. 418, 426 McCabe, G.H. 347, 349 McCabe, G.H., see Bindilatti, V. 312, 326 McCabe, G.H., see Foner, S. 353, 355 McCabe, G.H., see ter Haar, E. 312, 326 McCann, P.J., see Yuan, S. 323, 324, 327 McCarron, E.M., see Nachumi, B. 449, 450 McCartney, M., see Yang, Z. 162, 178 McClelland, J.J. 258 McClure, M.T., see Hylton, T.L. 70 McCord, J. 128 McCord, J., see Baril, L. 138, 139 McCord, J., see Beach, R.S. 124 McDonald, A., see Bacher, G. 309 McDonald, A., see Maksimov, A.A. 309 McDonald, M., see Feit, Z. 328 McElfresh, M., see Alawadhi, H. 311, 351–353, 356 McElroy, K., see Hoffman, J.E. 481 McGee, N.W.E., see Johnson, M.T. 143 McGee, N.W.E., see Purcell, S.T. 228 McGuire, T.R. 7 McMichael, R.D. 138 McMichael, R.D., see Egelhoff, W.F. 51, 52, 93 McMichael, R.D., see Egelhoff Jr., W.F. 34, 35, 42, 51, 52, 69, 70, 144 McMichael, R.D., see Fry, R.A. 138 McMichael, R.D., see Stiles, M.D. 155 McMorrow, D.F., see Lake, B. 413, 448, 449, 480 McMorrow, P., see Lake, B. 413, 446 McNiff Jr., E.J., see Bindilatti, V. 312, 326 McNiff Jr., E.J., see Fries, T. 357 McNiff Jr., E.J., see Kostyk, D. 342, 343 McNiff Jr., E.J., see McCabe, G.H. 347, 349 McNiff Jr., E.J., see Shapira, Y. 353 McNiff Jr., E.J., see ter Haar, E. 312, 326 McNiff Jr., E.J., see Vu, T.Q. 353 McQueeney, R.J. 409, 448, 459 McQueeney, R.J., see Tranquada, J.M. 409, 459 McVitie, S., see Chapman, J.N. 129, 130 McVitie, S., see Marrows, C.H. 249 Medina, A.N. 532 Medina, A.N., see Kraus, L. 540, 541

Meguro, K. 33, 123 Meguro, K., see Soeya, S. 160, 168 Mégy, R., see Renard, J.-P. 86 Mehring, M. 430, 473 Mehring, M., see Krämer, S. 478 Mehta, A. 457 Mehta, A., see Heaney, P.J. 457 Mei, L.M., see Chen, C. 504 Meigs, G., see Pellegrin, E. 463 Meiklejohn, W.H. 117, 148, 153, 171 Meiklejohn, W.H., see Cain, W.C. 172 Meinhaldt, A.D., see Carcia, P.F. 259 Meisinger, F., see Schaller, D.M. 238 Meisinger, F., see Schmidt, C.M. 221 Meiwes-Broer, K.H. 252 Meiwes-Broer, K.H., see Röhlsberger, R. 252 Meixner, S. 416 Mejía-García, C., see Geist, F. 323–325 Mejia-Lopez, J., see Kiwi, M. 155 Melo, L.G. 535 Melo, L.G., see Britel, M.R. 516, 548, 549 Melo, L.G., see Ciureanu, P. 542 Melo, L.G.C. 515, 522 Melo, L.G.C., see Ménard, D. 515, 526 Melo, L.G.C., see Yelon, A. 513, 515, 521 Melo, L.V., see Freitas, P.P. 22 Melo, L.V., see Veloso, A. 34, 45 Menant, M., see Mycielski, A. 356, 357 Menant, M., see Testelin, C. 351, 357 Menant, M., see Zielinski, M. 351–353 Ménard, D. 514–516, 526, 529, 530, 546, 548, 549 Ménard, D., see Britel, M.R. 516, 526, 548, 549 Ménard, D., see Chiriac, H. 504, 532, 533 Ménard, D., see Ciureanu, P. 504, 515, 546, 549 Ménard, D., see Yelon, A. 508, 513–516, 521, 522, 525, 528 Mendels, P. 478 Mendels, P., see Alloul, H. 437, 473, 474, 478 Mendels, P., see Bobroff, J. 477, 478 Mendels, P., see Dooglav, A.V. 473 Mendes, K.C. 541 Mendoza-Alvarez, J.G., see Alvarez-Fregoso, O. 361 Menéndez, J.L., see Cros, V. 235 Menke, D.R., see Durbin, S.M. 295, 344, 345 Menke, D.R., see Han, J. 295, 344, 345 Menon, M., see Andriotis, A. 215 Menon, M., see Andriotis, A.N. 261, 262, 265, 269, 270 Menon, M., see Froudakis, G.E. 261 Menon, M., see Lathiotakis, N.N. 257 Menovsky, A.A., see Abu-Shiekah, I.M. 461

AUTHOR INDEX Menovsky, A.A., see Bakharev, O.N. 467, 468 Menovsky, A.A., see Brom, H.B. 479 Meny, C. 71 Menyhard, M. 73 Menyhard, M., see Zeltser, A.M. 73 Mergler, A., see Dahl, M. 311 Merkulov, I.A. 303 Merkulov, I.A., see König, B. 306 Merkulov, I.A., see Wojtowicz, T. 302 Merle d’Aubigné, Y., see Dietl, T. 300, 311 Merle d’Aubigné, Y., see Gaj, J.A. 296 Merle d’Aubigné, Y., see Grieshaber, W. 296, 299 Merle d’Aubigné, Y., see Haury, A. 311 Merle d’Aubigné, Y., see Kheng, K. 301 Merle d’Aubigné, Y., see Kossacki, P. 302 Merle d’Aubigné, Y., see Peyla, P. 298 Merle d’Aubigné, Y., see Wasiela, A. 297 Mermin, N.D. 203 Mermin, N.D., see Ashcroft, N.W. 17, 18, 78, 87, 436 Merrin, J., see Nachumi, B. 449, 450 Mertig, I. 80 Mertig, I., see Binder, J. 83, 84, 109 Mertig, I., see Levy, P. 5 Mertig, I., see Zahn, P. 111 Meservey, R., see Moodera, J.S. 5 Mesot, J., see Fretwell, H.M. 441 Methfessel, S. 331, 340, 342, 343 Metin, S., see Dieny, B. 4, 16, 31, 39, 59, 60, 178 Metin, S., see Pinarbasi, M. 47, 164, 172 Mettler, A.H.M., see Strijkers, G.J. 97 Metzger, R.D., see Nishioka, K. 157, 174 Metzger, S.W., see Baselt, D.R. 24 Metzner, W. 397 Mewes, T. 152 Mewes, T., see Juraszek, J. 135, 136, 171 Meza-Aguilar, S., see Demangeat, C. 206, 229 Meza-Aguilar, S., see Krüger, P. 238 Meza-Aguilar, S., see M’Passi-Mabiala, B. 249 Miao, J. 300 Miao, J., see Stirner, T. 300 Miao, J., see Weston, S.J. 296 Miauchi, T., see Mizuguchi, T. 32, 67 Michaelian, K., see Aguilera-Granja, F. 219 Michalak, R., see Williams, G.V.M. 475, 476 Michalke, W., see Boeve, H. 57 Michel, R., see Cool, S. 76 Michel, R.P. 163 Michel, R.P., see Chaiken, A. 220, 244, 245 Michelini, F., see Marsal, L. 307 Middelhoek, S. 115 Miedema, A.R., see de Boer, F.R. 77, 166 Miglio, L., see Fanciulli, M. 245

613

Mikheenko, I.P. 205, 234 Mikheenko, P.M., see Mikheenko, I.P. 205, 234 Mila, F. 436 Miles, J.J. 127 Milius, D.L., see Fong, H.F. 470, 471, 482 Milivojevi´c, D., see Soskic, Z. 353, 360 Millburn, J.E., see Chow, K.H. 460 Millburn, J.E., see Jestädt, Th. 460 Miller, B.H. 62 Miller, M., see Hegde, H. 70, 71 Miller, M., see Hung, C.-Y. 136, 137, 140, 141, 162 Miller, M., see Mao, M. 32, 33, 158 Miller, M., see Mao, S. 7 Miller, M.M. 22, 24 Miller, M.M., see Edelstein, R.L. 24 Miller, M.M., see Johnson, M. 25 Miller, R.I., see Sonier, J.E. 477 Miller, W., see McMichael, R.D. 138 Milling, C.T., see Haase, J. 453, 477 Millis, A.J. 426, 436, 437, 474 Millius, D.L., see Sidis, Y. 472 Mills, D.L., see Demangeat, C. 203 Miloslavski, L., see Hung, C.-Y. 136, 137, 140, 141, 162 Miloslavski, L., see Tong, H.C. 33, 51, 55, 124 Miloslavsky, L., see Anderson, G. 161, 170 Miloslavsky, L., see Anderson, G.W. 162, 166 Miloslavsky, L., see Mao, M. 32, 33, 56, 158 Miloslavsky, L., see Mao, S. 7 Miloslavsky, L., see Pakala, M. 159 Miloslawsky, L., see Anderson, G. 151, 159 Miltat, J., see Schumacher, H.W. 132 Miltenyi, P. 155 Miltenyi, P., see Keller, J. 155 Miltenyi, P., see Nowak, U. 155 Min, B.I., see Kwon, S.K. 253 Min, H.G., see Lee, S.K. 235 Min, K.-I., see Kim, Y.-K. 76, 162 Min, K.I., see Park, C.-M. 146 Minami, F., see Kuroda, S. 308, 309 Minar, J., see Scherz, A. 227, 228 Ming-Le, see Brom, H.B. 479 Mino, H. 298, 303 Miotkowska, S. 315 Miotkowska, S., see Alawadhi, H. 311, 351–353, 356 Miotkowska, S., see Seong, M.J. 354 Miotkowski, I., see Alawadhi, H. 311, 351–353, 356 Miotkowski, I., see Głód, P. 357, 362 Miotkowski, I., see Pekarek, T.M. 346 Miotkowski, I., see Seong, M.J. 354

614

AUTHOR INDEX

Miranda, R., see Blum, V. 238 Miranda, R., see Camarero, J. 143 Miranda, R., see de Miguel, J.J. 208 Miranda, R., see Figuera, J. 258, 268, 269 Miranda, R., see Gómez, L. 269 Mirbt, S., see Alden, M. 218 Mirlin, D.N., see Reshina, I.I. 309 Mirza, K.A., see Loram, J.W. 442 Misra, A., see Nowak, U. 155 Misra, R.D.K., see Egelhoff, W.F. 51, 52, 93 Misra, S.K. 322, 323 Misra, S.K., see Gratens, X. 328 Misra, S.K., see Isber, S. 312, 322, 323, 326 Missell, F.P., see Mendes, K.C. 541 Mita, J., see Yamane, H. 22 Mito, T., see Ishida, K. 479 Mitrovi´c, V.F. 470, 477 Mitsu, H., see Debnath, M.C. 300, 301 Mitsu, H., see Oka, Y. 300 Mitsu, H., see Pittini, R. 300, 301 Mitsumara, C., see Wong, B.Y. 170, 171 Mitsuoka, K., see Hoshiya, H. 34, 51 Mitsuoka, K., see Soeya, S. 163 Mitzi, D.B., see Dessau, D.S. 427 Miura, N., see Akinaga, H. 296 Miura, N., see Kunimatsu, H. 302 Miura, N., see Matsuda, Y.H. 303 Miura, N., see Nicholas, R.J. 306 Miura, N., see Portugall, O. 361 Miura, N., see Takeyama, S. 302 Miura, N., see Yasuhira, T. 303 Miya, S., see Sato, H. 62 Miyajima, K., see Nagao, S. 261 Miyajima, T., see Aoshima, K. 74 Miyakawa, N., see Zasadzinski, J.F. 481 Miyake, Y., see Kishi, H. 162, 171 Miyashita, K., see Kato, T. 38 Miyashita, T., see Kobayashi, Y. 477 Miyazaki, H., see Kayanuma, K. 305 Miyazaki, M., see Yanagisawa, T. 397 Miyazaki, T. 7, 140 Mizokawa, T., see Ino, A. 455 Mizokawa, T., see Satake, M. 463 Mizoshita, Y., see Kanai, H. 32, 35, 41, 77, 157 Mizuguchi, T. 32, 37, 48, 67 Mizukami, K., see Ueno, M. 34, 42 Mizuno, M., see Ishida, A. 323, 324 Mogi, I., see Adachi, N. 351, 355 Mohan, Ch.V., see Shen, J. 238 Mohri, K. 508, 509, 532, 542, 550, 551, 553, 555 Mohri, K., see Gunji, T. 542, 544 Mohri, K., see Kanno, T. 551 Mohri, K., see Kawashima, K. 540, 557 Mohri, K., see Kitoh, T. 542–544

Mohri, K., see Kusumoto, D. 556 Mohri, K., see Panina, L.V. 504, 505, 510, 513, 515, 521–523, 525, 532, 544, 551, 552 Mohri, K., see Shen, L.P. 551, 555 Mohri, K., see Takayama, A. 552 Mohri, K., see Uchiyama, T. 550, 552, 553 Mohri, K., see Yoshinaga, T. 504, 548 Mokrani, A. 209, 230, 231 Mokrani, A., see Amalou, F. 223 Mokrani, A., see Bouarab, S. 205, 224, 227, 234 Mokrani, A., see Dorantes-Dávila, J. 275 Mokrani, A., see Elmouhssine, O. 228 Mokrani, A., see Nait-Laziz, H. 235 Mokrani, A., see Uzdin, V.M. 206, 217, 224, 249, 270, 273 Mokrani, A., see Vega, A. 207, 218, 221, 225–227 Molegraaf, H.J.A. 479 Molenkamp, L.W., see Bacher, G. 309 Molenkamp, L.W., see Egues, J.C. 306 Molenkamp, L.W., see Fiederling, R. 304 Molenkamp, L.W., see Karczewski, G. 311 Molenkamp, L.W., see Keller, D. 306 Molenkamp, L.W., see Maksimov, A.A. 309 Molenkamp, L.W., see Sawicki, M. 311 Molenkamp, L.W., see Welsch, M.K. 307 Molenkamp, L.W., see Yakovlev, D.R. 297, 298 Molenkamp, L.W., see Zaitsev, S. 309 Monchesky, T., see Heinrich, B. 207, 221, 222 Monfort, Y., see Robbes, D. 550 Monien, H. 437, 473 Monien, H., see Littlewood, P.B. 418, 423 Monien, H., see Millis, A.J. 426, 436, 437, 474 Monod, P., see Alloul, H. 437, 473, 474 Montaigne, F., see Encinas, A. 138 Montaigne, F., see Faure-Vincent, J. 276 Montarroyos, E., see Machado, F.L.A. 504 Montejano-Carrizales, J.M. 255 Montejano-Carrizales, J.M., see Aguilera-Granja, F. 215, 254–258 Montenegro, F.C., see Mendes, K.C. 541 Montes, L., see Furdyna, J.K. 344, 345 Monthoux, P. 418, 427 Moodenbaugh, A., see Niedermayer, Ch. 450 Moodenbaugh, A.R., see Johnson, P.D. 482 Moodenbaugh, A.R., see Tranquada, J.M. 388, 458 Moodera, J.S. 5 Mook, H.A. 389, 404, 411, 412, 419, 430, 439, 440, 470–472, 477 Mook, H.A., see Aeppli, G. 447 Mook, H.A., see Dai, P. 470, 471, 482 Mook, H.A., see Mason, T.E. 448

AUTHOR INDEX Moonen, J.T. 438 Morais-Smith, C., see Dimashko, Y.A. 404 Morais-Smith, C., see Hasselmann, N. 404 Moraitis, G., see Elmouhssine, O. 229, 238 Moraitis, G., see Mokrani, A. 230, 231 Morales, F., see Alvarez-Fregoso, O. 361 Moran, T.J., see Nogues, J. 149 More, N., see Parkin, S.S.P. 45, 157 Morecroft, D. 138 Morecroft, D., see Prieto, J.L. 22 Morel, R., see Duvail, J.L. 12, 61, 94, 104 Morhange, J.F., see Jouanne, M. 311 Morhange, J.F., see Szuszkiewicz, W. 311, 361 Mori, H., see Motoyoshi, M. 24 Môri, M., see Arumugam, S. 454 Morikawa, T. 504, 515, 552, 557 Morinaga, A. 127 Morita, H., see Araki, S. 32–34, 36, 67, 159 Morita, H., see Sano, M. 164 Morita, H., see Shimazawa, K. 159 Morita, H., see Tsuchiya, Y. 36 Moriya, T. 431, 436 Moriya, T., see Takimoto, T. 418 Moriyama, K., see Motoyoshi, M. 24 Morozov, A.V., see Ivanchik, I.I. 312 Morr, D.K. 419, 427, 449 Morr, D.K., see Chakravarty, S. 419, 473 Morris, G.D., see Sonier, J.E. 477 Moruzzi, V.L. 227 Moschalkov, V.V., see Barna´s, J. 110 Moschalkov, V.V., see Potter, C.D. 110 Moschalkov, V.V., see Schad, R. 6, 27, 79 Moseler, M. 251 Moser, A. 21 Mosser, A., see Pollini, I. 207 Motisuke, P., see Abramof, E. 326, 327 Motoyoshi, M. 24 Mott, N.F. 7 Mougin, A. 171 Moulton, W.G., see Julien, M.-H. 452 Moulton, W.G., see Mitrovi´c, V.F. 470, 477 Mountziaris, T.J. 358 Mountziaris, T.J., see Peck, J. 358 M’Passi-Mabiala, B. 249, 276 M’Passi-Mabiala, B., see Demangeat, C. 206, 229 Mpondo, F.E., see Vouille, C. 84 Mühlhäuser, M., see Froudakis, G.E. 261 Muir, W.B., see Celinski, Z. 235 Muirhead, C.M., see Mikheenko, I.P. 205, 234 Mukhamedshin, I.R., see Dooglav, A.V. 473 Mukhin, S.I. 393 Mukhin, S.I., see Matveenko, S.I. 393, 394 Mulheran, P.A., see Duffy, D.M. 262

615

Muller, D., see Bensmina, F. 73 Müller, E., see Fanciulli, M. 245 Muller, K.A., see Bednorz, J.G. 384 Müller, S., see Blum, V. 238 Mulloy, M., see Renard, J.-P. 86 Munakata, F., see Iguchi, E. 462 Munekata, H., see Deleporte, E. 297 Muñoz, J.L. 528 Muñoz, J.L., see Barandiarán, J.M. 504, 513 Muñoz, J.L., see Chen, D.-X. 522, 537 Muñoz, J.L., see Kurlyandskaya, G.V. 534 Munson, J., see Lee, C. 67, 170 Murayama, A., see Kayanuma, K. 305 Murdock, E., see Mao, S. 55, 60, 161, 162 Murdock, E., see Szucs, J. 59, 60 Murdock, E.S., see Qian, Z. 161 Murooka, Y., see Tsunekawa, K. 38, 48, 170, 178 Mycielski, A. 356, 357 Mycielski, A., see Dybko, K. 360 Mycielski, A., see Nadolny, A.J. 321, 322 Mycielski, A., see Scalbert, D. 351 Mycielski, A., see Testelin, C. 351, 353, 356, 357 Mycielski, A., see Twardowski, A. 351 Mycielski, A., see Zeitler, U. 360 Mycielski, A., see Zielinski, M. 351–353 Nachumi, B. 449, 450 Nachumi, B., see Tranquada, J.M. 459 Nadimi, S., see Dieny, B. 4, 178 Nadolny, A.J. 321, 322 Nagahara, S., see Marsal, L. 307 Nagai, H. 45 Nagai, M., see Tsunekawa, K. 38, 48, 170, 178 Nagamine, K., see Koike, Y. 448 Nagamine, K., see Watanabe, I. 450, 479, 481 Nagamine, Y., see Tsunekawa, K. 38, 48, 170, 178 Nagao, S. 261 Nagaoka, K. 251 Nagasaka, K. 13, 150, 162 Nagasaka, K., see Oshima, H. 13 Nagasaka, K., see Tanaka, A. 13, 51, 162 Nagasaka, K., see Varga, L. 131, 132 Nagengast, D., see Klose, F. 248 Nahm, T.U., see Choi, J.H. 235 Nait-Laziz, H. 235, 249 Nait-Laziz, H., see Bouarab, S. 224, 227, 228 Nait-Laziz, H., see Chouairi, A. 236 Nait-Laziz, H., see Khan, M.A. 228 Nait-Laziz, H., see M’Passi-Mabiala, B. 276 Naito, M., see Skinta, J.A. 464

616

AUTHOR INDEX

Nakada, M., see Fujikata, J. 151, 157, 161–163, 172, 174, 175 Nakada, M., see Oshima, N. 151, 157 Nakada, M., see Sato, H. 62 Nakagawa, S. 157 Nakahara, N., see Ishida, A. 312, 323, 324, 327 Nakajima, A., see Nagao, S. 261 Nakajima, D., see Tsunekawa, K. 38, 48, 170, 178 Nakajima, K. 485, 486 Nakajima, K., see Hosoya, S. 456 Nakajima, K., see Tranquada, J.M. 409, 459 Nakamoto, K. 22, 47, 172 Nakamoto, K., see Hamakawa, Y. 47, 73, 160, 163, 168, 172 Nakamura, F., see Saini, N.L. 454 Nakamura, K. 155, 299 Nakamura, K., see Chuang, Y.-D. 482 Nakamura, K., see Kossut, J. 310 Nakamura, M., see Zhou, X.J. 454 Nakamura, S., see Kamiguchi, Y. 32, 41 Nakamura, Y., see Han, J. 295, 344, 345 Nakamura, Y., see Tranquada, J.M. 388, 396, 439, 443–445, 455, 458, 459 Nakanishi, H., see Kato, M. 395 Nakano, H., see Nakamura, K. 299 Nakashima, K., see Uchiyama, T. 553 Nakatani, R. 67, 157, 158 Nakatani, R., see Hoshino, K. 31, 118, 158, 167 Nakatani, R., see Hoshiya, H. 160, 168 Nakatani, R., see Noguchi, S. 50, 51 Nakatani, R., see Shi, J. 63 Nakatsugawa, H., see Iguchi, E. 462 Nakayama, Y., see Ishida, K. 479 Nakazawa, Y., see Hasegawa, N. 33, 34, 38, 48 Nannarone, S., see Fratucello, G.B. 231 Nannarone, S., see Gazzadi, G.C. 230, 231 Nannarone, S., see Luches, P. 231 Naoe, M., see Nakagawa, S. 157 Narath, A. 432 Narisawa, H., see Motoyoshi, M. 24 Narishige, S., see Nishioka, K. 160, 174 Narishige, S., see Soeya, S. 163 Naruse, Y., see Kusumoto, D. 556 Nasedkin, K.A., see Rogacheva, E.I. 336 Nashchekina, O.N., see Rogacheva, E.I. 336 Natali, M., see Li, S.P. 276 Natesan, M., see Baselt, D.R. 24 Navaneethakrishnan, K., see Elangovan, A. 301 Navrotsky, A., see Heaney, P.J. 457 Nawrath, T. 227 Nawrocki, M. 300, 351 Nawrocki, M., see Teppe, F. 300, 306 Nayak, C. 396, 404

Nayak, C., see Balents, L. 413 Nayak, C., see Chakravarty, S. 419, 473 Nayak, C., see Tewari, S. 420 Nayak, S.K. 253 Need, O.U., see Gurney, B.A. 32, 33, 40, 41 Need, O.U., see Speriosu, V.S. 45 Néel, L. 136, 144, 154 Neerinck, D.G., see Rijks, Th.G.S.M. 66, 97, 104 Negoro, N., see Sugawara, N. 61 Neidermayer, F., see Hasenfratz, P. 434, 485 Nelson, C., see Krishnan, K. 162, 170 Nelson, D.R., see Chakravarty, S. 434, 484–487 Nenkov, K., see Borisenko, S.V. 441 Nenkov, K.A., see Kordyuk, A.A. 483 Neugebauer, F., see Suisky, D. 298 Ney, A., see Wilhelm, F. 209 Ng, K.-W., see Pan, S.H. 442, 481 Nguyen The Khoi, see Kossacki, P. 296, 344, 345 Nguyen The Khoi, see Lemaître, A. 296 Nguyen The Khoi, see Mac, W. 349–351, 357 Nguyen The Khoi, see Wojtowicz, T. 295 Nguyen The Khoi, see Wypior, G. 296 Nguyen Van Dau, F., see Baibich, M.N. 4 Nguyen Van Dau, F., see Encinas, A. 138 Nguyen Van Dau, F., see Kirilyuk, A. 155 Nicholas, R.J. 306 Nicholas, R.J., see Cheng, H.H. 297 Nicholls, J.E., see Goede, O. 305 Nicholls, J.E., see Harrison, P. 296 Nicholls, J.E., see Jackson, S. 296 Nicholls, J.E., see Pier, Th. 305 Nicholls, J.E., see Suisky, D. 298 Nicholls, J.E., see Wasiela, A. 297 Nicholls, J.E., see Weston, S.J. 296 Nichols, M. 135 Nicholson, D.M., see Butler, W.H. 40, 80, 83, 84, 107, 109 Nicholson, D.M.C. 40, 80, 85 Nicholson, D.M.C., see Brown, R.H. 110 Nicholson, D.M.C., see Butler, W.H. 108, 109 Nicholson, D.M.C., see Oparin, A.B. 85 Nicholson, D.M.C., see Schulthess, T.C. 83 Nicilics, J., see Steindl, R. 550 Nicolics, J., see Hauser, H. 556 Nie, H.B. 504, 541 Niedermayer, C., see Sidis, Y. 419, 470 Niedermayer, Ch. 450 Nielsch, K., see Ramos, C. 548 Nieman, G.C., see Parks, E.K. 258 Niemöller, T. 454 Niemöller, T., see Ichikawa, N. 452, 454, 455 Niemöller, T., see Zimmermann, M.v. 454

AUTHOR INDEX Niessen, A.K., see de Boer, F.R. 77, 166 Nigam, A.K., see Acharya, B.R. 231 Nihijima, T., see Arai, M. 470, 471, 473 Nikitin, V., see Crowell, P.A. 310 Nikitin, V., see Levy, J. 301 Niklasson, A.M.N. 207 Niklewski, D.J., see Sankey, O.K. 213 Nikolaev, E.G., see Brom, H.B. 479 Nimtz, G. 313, 314, 323, 324 Nir, J., see Egelhoff, W.F. 51, 52, 93 Nishibe, Y., see Morikawa, T. 504, 515, 552, 557 Nishida, H., see Ueno, M. 34, 42 Nishikawa, K., see Tsunoda, M. 159, 167 Nishimura, K. 63 Nishimura, K., see Nakagawa, S. 157 Nishina, Y., see Anderson, J.R. 325 Nishioka, K. 122, 151, 157, 160, 174 Nishioka, K., see Fujiwara, H. 122 Nishizima, Y., see Ishida, A. 327 Niu, C.-M., see Shapira, Y. 353 Noda, M., see Panina, L.V. 504, 510, 513, 515, 521–523, 525 Noda, T. 454 Noda, T., see Arumugam, S. 454 Noda, T., see Lanzara, A. 409, 427, 454, 482 Noda, T., see Zhou, X.J. 441, 454 Noguchi, K., see Araki, S. 159 Noguchi, K., see Li, M. 38, 48 Noguchi, K., see Sano, M. 164 Noguchi, S. 50, 51 Noguchi, S., see Hoshino, K. 31, 118 Noguchi, S., see Nakatani, R. 67, 157 Nogueira, R.N. 212 Nogues, J. 148, 149 Nohara, M., see Lake, B. 413, 446, 448, 449, 480 Nolting, W. 209 Nolting, W., see Vega, A. 209 Noma, K. 33, 123 Noma, K., see Hong, J. 36, 144, 146 Noma, K., see Kanai, H. 22, 33, 77 Nomura, Y., see Shirota, Y. 35, 57 Nonas, B. 222 Nonas, B., see Papanikolaou, N. 253 Nonomura, Y., see Morikawa, T. 504, 515, 552, 557 Nor, A.F. 135 Nor, A.F., see Hill, E.W. 135 Nordblad, P., see Broddefalk, A. 228 Nordlund, K., see Zimmermann, C.G. 268 Nordman, C., see Wang, D. 148 Nordström, L., see Sjöstedt, E. 219, 275 Nordström, L., see Taga, A. 206 Norman, M.R. 418, 427 Norman, M.R., see Abanov, Ar. 428

617

Norman, M.R., see Eschring, M. 427, 482 Norman, M.R., see Fretwell, H.M. 441 Norman, M.R., see Tchernyshyov, O. 417 Normand, B. 420 Norris, C., see Baker, S.H. 241 Norris, C., see Binns, C. 205, 263 Nørskov, J.K., see Pedersen, M.O. 268, 269 Northcott, D.J., see Bartkowski, M. 328 Norton, P., see Springholz, G. 326 Nossov, A., see Dieny, B. 62 Nouvertné, F. 268, 269 Nowak, U. 155 Nowak, U., see Keller, J. 155 Nowak, U., see Miltenyi, P. 155 Nowikov, J., see Nawrath, T. 227 Noyes, G.I., see Kim, Y.-K. 22 Nozières, J.-P., see Dieny, B. 60, 66, 94, 95, 99, 103 Nozières, J.-P., see Gurney, B.A. 32, 33, 40, 41 Nozières, J.-P., see Huang, T.C. 70, 72 Nozières, J.-P., see Zhang, Y.B. 60, 76, 161, 170 Nozières, J.P. 72, 158, 161, 162, 168–170 Nozières, J.P., see Meny, C. 71 Nozières, J.P., see Speriosu, V.S. 40, 72 Nozières, J.P., see Zeltser, A.M. 159 Nugroho, A.A., see Bakharev, O.N. 467, 468 Nugroho, A.A., see Brom, H.B. 479 Nunez, V. 343, 345 Nunez, V., see Giebułtowicz, T.M. 343–345 Nuñez-Regeiro, J., see Kools, J.C.S. 37, 49, 71 Nurmikko, A.V. 296 Nurmikko, A.V., see Chang, S.-K. 296 Nurmikko, A.V., see Durbin, S.M. 295, 344, 345 Nurmikko, A.V., see Goltsos, W.C. 323–325 Nurmikko, A.V., see Gunshor, R.L. 344 Nurmikko, A.V., see Han, J. 295, 344, 345 Nurmikko, A.V., see Ju, G. 172 Nurmikko, A.V., see Kobayashi, M. 295 Nurmikko, A.V., see Kolodziejski, L.A. 295 Nussinov, Z. 408 Nussinov, Z., see Carlson, E.W. 406 Nussinov, Z., see Zaanen, J. 389, 396, 404, 407, 420 O’Barr, R., see Cowache, C. 164 O’Brien, T., see Szucs, J. 59, 60 O’Brien, W.L. 249 O’Brien, W.L., see Tomaz, M.A. 236, 275 Ocal, C., see Figuera, J. 258, 268, 269 Ocker, B., see Langer, J. 32, 70 Ocker, B., see Loch, C. 161 Ocker, B., see Stobiecki, F. 32, 39, 60, 113 Oda, T. 204, 206, 216

618 Odier, P. 459, 461 Odier, P., see Poirot, N.J. 456 O’Donnell, D.O., see Lim, C.K. 127, 138 O’Donnell, K., see Lim, C.K. 127, 138 Oepen, H.P. 252 Oepts, W. 138 Ofﬁ, F., see Imada, S. 310 Ofﬁ, F., see Kuch, W. 230 Ogale, S.B., see Kale, S. 535 Oganesyan, V., see Kivelson, S.A. 413 Ogasawara, I., see Kawashima, K. 540, 557 O’Grady, K., see Hughes, T. 158, 175 Oguchi, T. 109 Oguchi, T., see Fu, C.L. 224 Oguni, M., see Kyomen, T. 457 Oh, S.J., see Choi, J.H. 235 O’Handley, R.C., see Song, O. 140 Ohashi, M., see Suzuki, T. 445 Ohdaira, H., see Yamaguchi, M. 552 Ohira, S., see Koike, Y. 448 Ohira, S., see Watanabe, I. 450, 479, 481 Ohkawara, S., see Hosomi, M. 13 Ohnesorge, B., see Deleporte, E. 297 Ohnishi, S. 218, 224 Ohnishi, S., see Freeman, A.J. 208 Ohno, H. 291, 292 Ohno, H., see Dietl, T. 311 Ohno, H., see Matsukura, F. 291, 311 Ohno, H., see Ohno, Y. 305 Ohno, T., see Alloul, H. 473, 474 Ohno, Y. 305 Ohresser, P. 236, 241, 252 Ohresser, P., see Gambardella, P. 254 Ohresser, P., see Shen, J. 238 Ohresser, Ph., see Kappler, J.-P. 275 Ohsawa, Y. 26 Ohshima, A., see Uchiyama, T. 550 Ohta, M., see Araki, S. 159 Ohta, M., see Sano, M. 164 Ohta, T., see Yonamoto, Y. 206, 249 Ohtsuka, Y., see Kanai, H. 32, 41, 157 Ohuchi, J., see Uchiyama, T. 553 Oitmaa, J., see Shevchenko, P. 339 Oitmaa, J., see Sushkov, O.P. 401 Oka, K., see Chuang, Y.-D. 482 Oka, Y. 300 Oka, Y., see Chen, Z.H. 303 Oka, Y., see Debnath, M.C. 300, 301 Oka, Y., see Ikada, H. 308 Oka, Y., see Kayanuma, K. 305, 306 Oka, Y., see Pittini, R. 300, 301 Oka, Y., see Takahashi, N. 301, 307 Okabe, A., see Maesaka, A. 74, 75, 77, 171 Okabe, A., see Sugawara, N. 61

AUTHOR INDEX Okabe, A., see Takiguchi, M. 74, 75 Okada, T., see Hamakawa, Y. 160, 168 Okamura, T., see Ishida, A. 312, 323, 324, 327 Okuda, T., see Adachi, N. 349 Okuda, T., see Ando, K. 295 Okuno, S.N. 110 Okuyama, T., see Shinjo, T. 12 Okuyama, T., see Yamamoto, H. 9, 27 Ole´s, A.M., see Fleck, M. 397, 398 Ole´s, A.M., see Gora, D. 397 Ole´s, A.M., see Zaanen, J. 395, 396 Oliveira, A.B., see Rezende, S.M. 164 Oliveira, N.J. 22 Oliveira, N.J., see Freitas, P.P. 22 Oliveira Jr., N.F., see Bindilatti, V. 312, 326, 355 Oliveira Jr., N.F., see Gratens, X. 312 Oliveira Jr., N.F., see Isber, S. 355 Oliveira Jr., N.F., see McCabe, G.H. 347, 349 Oliveira Jr., N.F., see ter Haar, E. 312, 326 Oliveira Jr., N.J., see Veloso, A. 159 Olligs, D., see Gareev, R.R. 220, 245, 247 Olson, C.G., see Kaminski, A. 419 Olver, M.M., see Springholz, G. 327, 328 Omata, E., see Araki, S. 159 Omata, T., see Hosoya, S. 456 Onda, N., see Fanciulli, M. 245 O’Neill, M., see Harrison, P. 296 O’Neill, M., see Pier, Th. 305 O’Neill, M., see Weston, S.J. 296 Onida, G. 274 Ono, K., see Inukai, T. 299 Ono, T., see Wakimoto, S. 445 Onodera, S., see Ozue, T. 22 Onose, Y. 465, 466 Onose, Y., see Armitage, N.P. 465 Onose, Y., see Harima, N. 466 Onufrieva, F. 418 Oomori, T., see Miyazaki, T. 140 Oparin, A.B. 85 Ordejón, P., see Calleja, M. 266 Ordejón, P., see Diéguez, O. 215 Ordejón, P., see Izquierdo, J. 213 Ordejón, P., see Onida, G. 274 Ordejón, P., see Sánchez-Portal, D. 213, 245 Ordejón, P., see Soler, J.M. 213 Orera, V.M., see Isber, S. 355 Orgad, D., see Carlson, E.W. 389, 398, 406 Orlowski, B.A. 315 Orlowski, B.A., see Kowalski, B.J. 312, 329 Ormston, M., see Marrows, C.H. 249 Ortega, J.E., see Blum, V. 238 Ortega, J.E., see Himpsel, F.J. 207 Ortiz, G., see Batista, C.D. 404, 430

AUTHOR INDEX Ortiz, G., see Martin, I. 401 Osborn, R., see McQueeney, R.J. 448, 459 Osborne, I.S. 202 Oshiki, M., see Shimizu, Y. 162 Oshiki, M., see Tanaka, A. 51, 162 Oshima, H. 13 Oshima, H., see Tanaka, A. 13 Oshima, N. 151, 157 Osinniy, V., see Łusakowski, A. 318 Osinniy, V., see Nadolny, A.J. 321, 322 Osinniy, V., see Radchenko, M.V. 315 Osman, O.Y., see Eskes, H. 404 Osman, O.Y., see Zaanen, J. 389, 396, 404, 407, 420 Ossau, W. 302 Ossau, W., see Akimov, A.V. 299 Ossau, W., see Fiederling, R. 304 Ossau, W., see Gruber, Th. 305 Ossau, W., see Keller, D. 306 Ossau, W., see Kochereshko, V.P. 302 Ossau, W., see König, B. 294, 306 Ossau, W., see Kuhn-Heinrich, B. 296 Ossau, W., see Kutrowski, M. 295, 296 Ossau, W., see Mackh, G. 300, 303 Ossau, W., see Merkulov, I.A. 303 Ossau, W., see Shcherbakov, A.V. 306 Ossau, W., see Waag, A. 294 Ossau, W., see Welsch, M.K. 307 Ossau, W., see Wojtowicz, T. 295, 302 Ossau, W., see Yakovlev, D.R. 297–300, 302, 306, 310 Ossau, W., see Zaitsev, S. 309 Ostanin, S. 248, 249 Ostenson, J.E., see Huh, Y.M. 453 Östreich, M. 304 Östreich, M., see Falk, H. 306 Oswald, A. 260 Otagiri, M., see Kitade, Y. 31, 72, 140 Oti, J.O. 127, 130 Oti, J.O., see Russek, S.E. 128 Otsuka, N., see Durbin, S.M. 295, 344, 345 Otsuka, N., see Gunshor, R.L. 344 Otsuka, N., see Han, J. 295, 344, 345 Otsuka, N., see Kobayashi, M. 295 Otsuka, N., see Kolodziejski, L.A. 295 Otsuka, N., see Luo, H. 295 Otsuka, N., see Samarth, N. 344, 345 Ott, K.C., see Takigawa, M. 437, 474, 475 Ouazi, S., see Bobroff, J. 477 Ouazi, S., see Sander, D. 208 Ounadjela, K. 140 Ounadjela, K., see Fullerton, E.E. 235 Ounadjela, K., see Hillebrands, B. 208, 274 Ouse, M., see Kamiguchi, Y. 32, 41

619

Ousset, J.C., see Bruno, A. 329 Óvári, T.A. 534, 546 Óvári, T.A., see Brunetti, L. 532 Óvári, T.A., see Chiriac, H. 504, 505, 532, 533, 556 Overberg, M.E., see Theodoropoulou, N. 276 Oyanagi, H., see Saini, N.L. 454 Ozaki, H., see Iida, M. 323, 324 Ozue, T. 22 Ozyuzer, L., see Zasadzinski, J.F. 481 Paccard, D., see Schlenker, C. 153 Pacchioni, G. 249 Padovani, S., see Ohresser, P. 241 Paganotto, N. 302 Pagliuso, P.G., see McQueeney, R.J. 448 Pagliuso, P.G., see Pirota, K.R. 514, 541, 542, 549 Paik, S.Y., see Man, K.L. 237 Pailhès, S., see He, H. 411, 426, 449, 482 Paja, A., see Stobiecki, F. 32, 39, 60, 113 Pajda, M. 209 Pakala, M. 159 Pakala, M., see Anderson, G.W. 151, 158, 161, 162 Pakala, M., see Huai, Y. 34, 42 Pakhomov, A.B., see Nie, H.B. 504, 541 Pakula, K., see Twardowski, A. 351 Palacio, F., see Stankiewicz, J. 355, 356 Palm, E.C., see McClelland, J.J. 258 Palosz, W., see Chernyshova, M. 332, 338 Palstra, T.T., see Cava, R.J. 459 Pampuch, C. 253 Pan, G., see Lu, Z.Q. 37, 48, 54 Pan, G., see Shen, F. 38, 49 Pan, S.H. 442, 481 Pan, S.H., see Hoffman, J.E. 413, 441, 442, 471, 479 Pan, S.H., see Hudson, E.W. 481 Pang, Y., see Grünberg, P. 206, 217, 219, 248 Panina, L. 543 Panina, L.V. 504, 505, 510, 513, 515, 521–523, 525, 532, 543, 544, 551, 552 Panina, L.V., see Gunji, T. 542, 544 Panina, L.V., see Makhnovskiy, D.P. 508, 525–528, 540, 542, 544, 545 Panina, L.V., see Mohri, K. 532, 550, 551, 553, 555 Panina, L.V., see Uchiyama, T. 552 Panissod, P., see Meny, C. 71 Pankoke, R., see Buss, C. 299 Pankoke, R., see Leisching, P. 299 Pao, C.-H. 418

620

AUTHOR INDEX

Paolucci, G., see Goldoni, A. 205, 235, 236 Papaconstantopoulos, D.A. 214 Papaconstantopoulos, D.A., see Lekka, Ch.E. 214 Papanicolaou, N.I., see Lekka, Ch.E. 214 Papanikolaou, N. 213, 253 Papanikolaou, N., see Bellini, V. 205 Papanikolaou, N., see Freyss, M. 248 Papanikolaou, N., see Zahn, P. 111 Papis, E., see Andrearczyk, T. 304 Papis, E., see Jaroszy´nski, J. 304 Papp, J., see Vu, T.Q. 353 Paramonov, V.P., see Ménard, D. 514 Paranjpe, A.P., see Schwartz, P.V. 31, 70 Pareek, A., see Luo, H. 295 Parinello, M., see Billas, I.M.L. 265 Parinello, M., see Car, R. 204, 264, 265 Parisi, F., see Guevara, J. 254, 257 Park, C.-H., see Marshall, D.S. 476 Park, C.-M. 69, 146 Park, C.-M., see Kim, J.-K. 162 Park, C.G., see Ahn, S.J. 504, 540 Park, C.H., see King, D.M. 464 Park, C.M., see Hwang, D.G. 73, 164 Park, C.M., see Rhee, J.-R. 161 Park, G.-S., see Kim, Y.-K. 76, 162 Park, J.-S. 34, 45, 123 Park, J.H., see Ko, Y.D. 354 Park, J.M., see Bartkowski, M. 328 Park, K. 360 Park, O., see Sachdev, S. 400 Park, S.-K., see Keavney, D.J. 111 Park, W., see Gu, J.Y. 13 Park, Y.D. 304 Park, Y.D., see Jonker, B.T. 304 Parker, F.T., see Mattson, J.E. 220, 244 Parker, M., see Nishioka, K. 122 Parker, M., see Pinarbasi, M. 47, 164, 172 Parker, M.A., see Hylton, T.L. 9, 26 Parker, M.R. 115, 122 Parker, M.R., see Miles, J.J. 127 Parker, M.R., see Nor, A.F. 135 Parkin, S.S.P. 5, 6, 27, 32, 39, 40, 45, 100, 155, 157, 167, 168 Parkin, S.S.P., see Dieny, B. 4, 9, 16, 30, 31, 39, 94, 157, 178 Parkin, S.S.P., see Farrow, R.F.C. 162 Parkin, S.S.P., see Hardner, H.T. 135 Parkin, S.S.P., see Heim, D.E. 42 Parkin, S.S.P., see Kuch, W. 231 Parkin, S.S.P., see Lu, Y. 5 Parkin, S.S.P., see Schlenker, C. 153 Parkin, S.S.P., see Shi, J. 63 Parkin, S.S.P., see Speriosu, V.S. 45

Parkin, S.S.P., see Thomas, L. 152 Parks, C., see Luo, H. 297 Parks, D., see Egelhoff Jr., W.F. 35 Parks, D.C. 146 Parks, D.C., see Chopra, H.D. 164 Parks, E.K. 255, 258 Parks, E.K., see Klots, T.D. 255 Parlebas, J.C. 204, 208, 244, 262 Parlebas, J.C., see Binns, C. 205, 208, 220, 241, 242, 263 Parlebas, J.C., see Demangeat, C. 206, 216, 219, 228, 229, 270 Parlebas, J.C., see Elmouhssine, O. 228, 229, 238 Parlebas, J.C., see Kotani, A. 207 Parlebas, J.C., see Krüger, P. 236, 241–244, 262–264 Parlebas, J.C., see Pollini, I. 207 Parlebas, J.C., see Razaﬁmandimby, H. 209 Parlebas, J.C., see Taguchi, M. 229 Parlebas, J.C., see Uozumi, T. 207 Parmigiani, F., see Gallani, J.L. 275 Parsons, F.G., see Miller, M.M. 22 Partin, D.L., see Goltsos, W.C. 323–325 Partin, D.L., see Heremans, J. 343 Partin, D.L., see Karczewski, G. 324 Pascard, H., see Zha, F.-X. 261 Pascher, H., see Aigle, M. 327 Pascher, H., see Bauer, G. 312, 314, 318, 324, 325 Pascher, H., see Dietl, T. 318, 322, 325 Pascher, H., see Geist, F. 323–325 Pascher, H., see Krenn, H. 312, 323–326, 344, 345 Pascher, H., see Pohlt, M. 344, 345 Pascher, H., see Springholz, G. 327, 328, 342 Pascual, L., see Chen, D.-X. 537, 546 Pashkevich, Yu.G. 461, 463 Pasquali, L., see Dallmeyer, A. 231 Pasquali, L., see Gazzadi, G.C. 230, 231 Pasquarelo, A., see Oda, T. 204, 206, 216 Pastalan, J.Z., see Springholz, G. 327, 328 Pastor, G.M., see Dorantes-Dávila, J. 261 Pastor, G.M., see Vega, A. 267 Paszkowicz, W., see Iwanowski, R.J. 315 Patchett, A.J., see Fratucello, G.B. 231 Pattenden, P.A., see Chow, K.H. 460 Patton, C.E. 529 Pavarini, E., see Fleck, M. 397, 398 Pawlowska, Z., see Andersen, O.K. 212 Pearton, S.J., see Theodoropoulou, N. 276 Peck, J. 358 Peck, J., see Mountziaris, T.J. 358

AUTHOR INDEX Peck, W.F., see Cava, R.J. 459 Pedersen, M.O. 268, 269 Peeters, F.M. 309 Peeters, F.M., see Chang, K. 306 Peeters, F.M., see Nicholas, R.J. 306 Pei, S., see Jorgensen, J.D. 457 Pekarek, T.M. 346 Pelekanos, N., see Durbin, S.M. 295, 344, 345 Pelekanos, N., see Haacke, S. 305 Pelekanos, N., see Han, J. 295, 344, 345 Peletminskii, S.V., see Akhiezer, A.I. 515 Pellegrin, E. 463 Pendry, J. 535 Peng, J.L., see King, D.M. 464 Peng, S.S. 242, 262, 263 Pennec, Y., see Camarero, J. 178 Pennington, C.H. 474 Pentcheva, R. 268 Pentcheva, R., see Nouvertné, F. 268, 269 Pentek, K., see Menyhard, M. 73 Pentek, K., see Nozières, J.P. 158, 161, 162, 168–170 Pentek, K., see Prakash, S. 162 Pentek, K., see Zeltser, A.M. 73, 159 Pentek, K., see Zhang, Y.B. 60, 76, 161, 170 Pépin, C., see Stemmann, G. 420 Perdew, J.P. 213, 245 Pereira, L.G., see Dieny, B. 60, 61, 92, 104, 106 Pereira, L.G., see Duvail, J.L. 104, 105 Pereira, L.G., see George, J.M. 86 Pereira, L.G., see Mendes, K.C. 541 Perenboom, J.A.A.J., see Dybko, K. 360 Perera, R.C.C., see Lawniczak-Jabło´nska, K. 345 Perez, I., see Twardowski, A. 351 Perjeru, F. 217, 220, 275 Permogorov, S., see Kayanuma, K. 305, 306 Pernambuco-Wise, P., see Twardowski, A. 351, 357 Perring, T.G., see Dai, P. 470, 482 Persat, N., see Manders, F. 45 Pesek, A., see Abramof, E. 344 Petej, I., see Gregg, J.F. 5 Peterson, D.T., see Dieny, B. 4, 178 Petford-Long, A.K. 124, 127, 129 Petford-Long, A.K., see Bayle-Guillemaud, P. 70 Petford-Long, A.K., see Marrows, C.H. 249 Petford-Long, A.K., see Portier, X. 73, 129, 161 Petkov, V., see Misra, S.K. 322, 323 Petrilli, H.M., see Nogueira, R.N. 212 Petroff, F., see Baibich, M.N. 4 Petroff, F., see Barthélémy, A. 5, 109 Petroff, F., see Cros, V. 235 Petroff, F., see Duvail, J.L. 12, 61, 94, 104 Petroff, F., see George, J.M. 86

621

Petroff, F., see Marangolo, M. 248 Petroff, F., see Vogel, J. 219 Petrou, A., see Chou, W.C. 294, 358, 359 Petrou, A., see Fu, L.P. 358 Petrou, A., see Heiman, D. 351 Petrou, A., see Herbich, M. 350 Petrou, A., see Jonker, B.T. 294, 301, 304, 358 Petrou, A., see Liu, X. 294 Petrou, A., see Luo, H. 291, 295 Petrou, A., see Mountziaris, T.J. 358 Petrou, A., see Park, Y.D. 304 Petrou, A., see Peck, J. 358 Petrou, A., see Warnock, J. 294, 358 Petrou, A., see Yu, W.Y. 297, 358 Petrov, D.K. 22, 25 Petrov, Y., see McQueeney, R.J. 409 Petta, J.R. 135 Pettifor, D.G., see Bowler, D.R. 215 Pettifor, D.G., see Tsymbal, E.Y. 5, 109, 110, 181 Pettifor, D.G., see Tsymbal, E.Yu. 63, 110 Pettit, K., see Shi, J. 63 Peyla, P. 298 Peyla, P., see Dietl, T. 300 Peyla, P., see Lawrence, I. 305 Peyla, P., see Wasiela, A. 297 Pfandzelter, R. 205, 249, 263 Pfandzelter, R., see Igel, T. 218, 227, 229 Pham-Huu, C. 265 Picard, P.G., see Dooglav, A.V. 473 Pichler, T., see Borisenko, S.V. 441 Pick, S. 206, 249 Piecuch, M., see Andrieu, S. 220, 228 Pier, Th. 305 Pier, Th., see Goede, O. 305 Pier, Th., see Heimbrodt, W. 305 Pierce, D.T. 219, 228 Pierce, D.T., see Davies, A. 222 Pierce, D.T., see Unguris, J. 207, 221 Pierce, J.P. 238 Pierron-Bohnes, V., see Sanchez, J.M. 227 Pinarbasi, M. 47, 164, 172 Pinarbasi, M., see Beach, R.S. 45, 120, 123 Pinczolits, M., see Springholz, G. 328 Pines, D. 437, 474–477 Pines, D., see Barzykin, V. 419, 476 Pines, D., see Laughlin, R.B. 390 Pines, D., see Millis, A.J. 436, 437, 474 Pines, D., see Monien, H. 437, 473 Pines, D., see Monthoux, P. 418, 427 Pines, D., see Morr, D.K. 419, 427, 449 Pines, D., see Zha, Y. 431, 432, 436 Pintschovius, L., see Tranquada, J.M. 409, 459

622

AUTHOR INDEX

Piorek, T., see Fatah, J.M. 296 Piotrowska, A., see Andrearczyk, T. 304 Piotrowska, A., see Dietl, T. 304 Piotrowska, A., see Jaroszy´nski, J. 304 Piramanayagam, S.N., see Acharya, B.R. 231 Piraux, L., see Encinas, A. 548 Pirota, K., see Ramos, C. 548 Pirota, K.R. 503, 504, 514, 535, 537, 540–542, 549, 557 Pirota, K.R., see Gómez-Polo, C. 537, 543 Pirota, K.R., see Knobel, M. 502, 504, 506 Pirota, K.R., see Kraus, L. 533 Pirota, K.R., see Sinnecker, J.P. 504, 528, 534 Pittini, R. 300, 301 Pittini, R., see Oka, Y. 300 Pizzini, S., see Camarero, J. 178 Pizzini, S., see Lawniczak-Jabło´nska, K. 345 Planel, R., see Nawrocki, M. 300 Platonov, A.V., see Yakovlev, D.R. 298 Platt, C.L., see Beach, R.S. 503, 504 Plummer, E.W., see Pierce, J.P. 238 Podloucky, R., see Bischoff, M.M.J. 219, 225 Pohl, A., see Hauser, H. 556 Pohl, A., see Hausleitner, C. 556 Pohl, A., see Steindl, R. 550 Pohlt, M. 344, 345 Pohm, A., see Wang, D. 148 Pohm, A.V., see Beech, R.S. 22 Pohm, A.V., see Everitt, B.A. 57 Poilblanc, D. 395 Poirot, N.J. 456 Poirot, N.J., see Odier, P. 459, 461 Pokhil, T. 162 Pokhil, T., see Hou, C. 150 Polaczyk, C., see Nawrath, T. 227 Polkovnikov, A. 413, 420, 478, 481 Pollini, I. 207 Polonyi, J., see Kenouﬁ, A. 215 Polyhach, Ye., see Zayachuk, D. 329 Ponomareva, A.V. 224 Popova, E., see Faure-Vincent, J. 276 Popovic, R.S. 23 Popp, M., see Kuhn-Heinrich, B. 296 Poppa, H., see Man, K.L. 237 Poppe, S., see Juraszek, J. 135, 136, 171 Poppe, S., see Mougin, A. 171 Poppi, M., see Bordin, G. 538 Popple, T., see Fayﬁeld, R. 24 Portier, X. 73, 129, 161 Portier, X., see Petford-Long, A.K. 124, 127, 129 Portugal, R.D., see Kiwi, M. 155 Portugall, O. 361 Portugall, O., see Stolpe, I. 340, 341 Posina, G.R., see Ivchenko, E.L. 297

Postma, L., see Folkerts, W. 22, 127–129 Potemski, M., see Ghali, M. 304, 305 Potemski, M., see Kutrowski, M. 296 Potemski, M., see Teran, F.J. 302 Potemski, M., see Wojtowicz, T. 302 Potter, C.D. 110 Potter, C.D., see Schad, R. 6, 27, 79 Potter, R.I., see McGuire, T.R. 7 Poulopoulos, P., see Lindner, J. 228 Poulopoulos, P., see Scherz, A. 227 Poulopoulos, P., see Wilhelm, F. 209, 220, 237 Pourovskii, L.V., see Ponomareva, A.V. 224 Poweleit, C.D. 301 Powell, C.J., see Egelhoff, W.F. 51, 52, 93 Powell, C.J., see Egelhoff Jr., W.F. 34, 35, 42, 51, 52, 69, 70, 144 Powroznik, W., see Stobiecki, F. 32, 39, 60, 113 Pozdnyakov, V.A., see Antonov, A. 505 Pozina, G.R. 297 Prados, C. 89, 261 Prakash, S. 162 Prakash, S., see Wong, B.Y. 170, 171 Prandini, A.M., see Fratucello, G.B. 231 Prasad, S., see Acharya, B.R. 231 Pratt, F.L., see Chow, K.H. 460 Pratt, F.L., see Jestädt, Th. 460 Pratt, W.P. 78, 85 Pratt, W.P., see Steenwyk, S.D. 13, 80, 83, 85 Pratt Jr., W.P., see Gu, J.Y. 13 Pratt Jr., W.P., see Slater, R.D. 13 Pratt Jr., W.P., see Yang, Q. 78 Prechtl, G. 297 Prechtl, G., see Heiss, W. 342 Prechtl, G., see Ma´ckowski, S. 308 Prelovšek, P. 402, 404, 407, 454 Presura, C., see Molegraaf, H.J.A. 479 Prida, V.M., see Hernando, B. 540 Prida, V.M., see Sánchez, M.L. 538 Prida, V.M., see Tejedor, M. 504, 538, 539, 553, 557 Prieto, J.E., see Blum, V. 238 Prieto, J.E., see Figuera, J. 258 Prieto, J.L. 22 Prieto, J.L., see Morecroft, D. 138 Prikhodko, M., see Yang, H. 249 Prinz, A. 323 Prinz, G.A. 5 Prinz, G.A., see Chaiken, A. 12, 94 Prinz, G.A., see Crowell, P.A. 310 Prinz, G.A., see Jonker, B.T. 294 Prinz, G.A., see Liu, X. 294 Prinz, G.A., see Miller, M.M. 22 Prokoshin, A.F., see Antonov, A. 505, 537

AUTHOR INDEX Prost, J.B., see Testelin, C. 351, 357 Provencio, M., see Vázquez, M. 556 Pruneda, J.M. 245–248 Pruneda, J.M., see Robles, R. 225, 245 Pryadko, L.P. 404, 407 Pryadko, L.P., see Tchernyshyov, O. 404 Przybytek, J., see Wasik, D. 304 Puça, A.A., see Machado, F.L.A. 542–544 Puerta, S., see Mandal, K. 538 Puhlmann, N., see Portugall, O. 361 Puhlmann, N., see Stolpe, I. 340, 341 Pulizzi, F. 298 Pulizzi, F., see Ma´ckowski, S. 309 Puls, J., see Kratzert, P.R. 309 Puls, J., see Kreller, F. 298 Puls, J., see Rossin, V.V. 296, 301 Purcell, S., see Folkerts, W. 115 Purcell, S.T. 228 Purcell, S.T., see Heinrich, B. 230 Pureur, P., see Fraga, G.L.F. 535 Qadri, S.B., see Edelstein, A.S. 231 Qadri, S.B., see Luo, H. 295 Qazilbash, M.M., see Biswas, A. 464 Qazzaz, M., see Furdyna, J.K. 344, 345 Qian, C., see Hung, C.-Y. 136, 137, 140, 141, 162 Qian, C., see Mao, M. 32, 33, 158 Qian, C., see Mao, S. 7 Qian, C., see Tong, H.C. 33, 51, 52, 55, 124 Qian, C., see Yan, X. 22 Qian, C.X., see Anderson, G.W. 162, 166 Qian, D. 238 Qian, X. 237 Qian, Z. 161 Qian, Z., see Wang, D. 148 Qin, H., see Hu, J. 504, 535 Qiu, J., see Li, K. 37, 38, 45, 49, 50, 145, 146, 159, 167 Qiu, J., see Wu, Y. 57 Qiu, J.J., see Guo, Z.B. 159 Qiu, S.L. 218 Qiu, Z.Q., see Man, K.L. 237 Qiu, Z.Q., see Wu, Y.Z. 230 Quinn, A.J., see Bischoff, M.M.J. 219, 225, 228 Ronnow, H.M., see Lake, B. 413, 446 Rabe, M., see Kratzert, P.R. 309 Rabello, S. 416 Radchenko, M.V. 315 Radchenko, M.V., see Łazarczyk, P. 315 Rader, O., see Dallmeyer, A. 231 Rader, O., see Pampuch, C. 253

623

Rado, G.T., see Ament, W.S. 508, 529 Raffy, H., see Kaminski, A. 419 Rahman, I.Z., see Kamruzzaman, Md. 502 Rahman, M., see Lim, C.K. 127, 138 Rahman, M.A., see Kamruzzaman, Md. 502 Rakhmanov, A.L., see Antonov, A. 537 Rakhmanov, A.L., see Antonov, A.S. 503, 532, 533, 537 Rakotomahevitra, A., see Krüger, P. 242–244, 262 Ram-Mohan, L.R., see Dai, N. 297 Ram-Mohan, L.R., see Lee, S. 297, 305 Ramdas, A.K., see Aigle, M. 327 Ramdas, A.K., see Alawadhi, H. 311, 351–353, 356 Ramdas, A.K., see Bartholomev, D.U. 351 Ramdas, A.K., see Luo, H. 297 Ramdas, A.K., see Seong, M.J. 354 Ramdas, A.K., see Suh, E.-K. 300 Rameev, B., see Kataev, V. 391, 453 Ramesha, K., see Kale, S. 535 Ramirez, A.P. 459 Ramirez, A.P., see Cava, R.J. 459 Ramirez, R., see Kiwi, M. 155 Ramos, C. 548 Rampe, A., see Nouvertné, F. 268, 269 Rana, P., see Huai, Y. 33 Randeria, M., see Fretwell, H.M. 441 Ranninger, J., see Domanski, T. 419 Rao, B.K. 250, 276 Rao, B.K., see Das, G.P. 276 Rao, B.K., see Weber, S.E. 251 Rao, D., see Yan, X. 22 Rao, K.V., see Costa-Krämer, J.L. 503 Rao, K.V., see Korenivski, V. 224 Rao, K.V., see Kossacki, P. 296 Raposo, V. 510 Rappl, P.H.O., see Abramof, E. 326, 327 Rasing, T.H., see Manders, F. 45 Rasing, Th., see Gerrits, Th. 131, 133 Rasing, Th., see Kirilyuk, A. 155 Rath, Ch., see Blum, V. 238 Ratner, E.R., see King, D.M. 464 Rau, C. 224, 238 Rau, C., see Pfandzelter, R. 205, 263 Rausch, T. 535 Ravot, D., see Errebbahi, A. 322, 326 Ray, O. 304, 307 Ray, R., see Burgess, C.P. 416 Razaﬁmandimby, H. 209 Razee, S.S.A. 209 Read, N. 400 Rebouillat, J.P., see Dieny, B. 115 Reddoch, A.H., see Bartkowski, M. 328

624

AUTHOR INDEX

Reddy, B.V. 235 Redinger, J., see Bischoff, M.M.J. 219, 225 Redli´nski, P. 302 Redon, O., see Araki, S. 32–34, 36, 67 Regan, T.J., see Lai, C.-H. 163 Regensburger, H., see Wu, Y.Z. 218, 249 Regi´nski, K., see Ghali, M. 304, 305 Regnault, L.P., see Bourges, P. 389, 471, 472 Regnault, L.P., see Fong, H.F. 470 Regnault, L.P., see He, H. 411, 426, 449, 482 Regnault, L.P., see Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Regnault, L.P., see Sidis, Y. 419, 470, 472 Rehm, Ch., see Klose, F. 248 Reilly, A., see Vouille, C. 84 Reilly, A.C., see Gu, J.Y. 13 Reimann, K., see Heiss, W. 328 Reinders, A., see Coehoorn, R. 153 Reinders, A., see de Vries, J.J. 220, 244, 245, 247 Reinders, A., see Jungblut, R. 149, 154, 157, 176 Reinders, A., see Lenssen, K.-M.H. 31, 39, 67, 71 Reinhard, L., see Turchi, P.E.A. 227 Reining, L., see Onida, G. 274 Reiss, G., see Hütten, A. 55, 56 Reiter, G., see Emery, V.J. 407 Renard, D., see Dupas, C. 9 Renard, J.-P. 86 Renard, J.P., see Chappert, C. 143 Renard, J.P., see Dupas, C. 9 Reneerkens, R.O., see Rijks, Th.G.S.M. 136–138 Rennert, P. 252 Rennert, P., see Bazhanov, D.I. 205, 252 Rennert, P., see Stepanyuk, V.S. 253 Reohr, W. 24 Repetski, E.J., see Chopra, H.D. 70 Reshina, I.I. 309 Restorf, J.B. 163 Rettori, C., see Pirota, K.R. 514, 541, 542, 549 Reuscher, G., see Fiederling, R. 304 Reuscher, G., see Gruber, Th. 305 Reuse, F.A. 215, 258 Revcolevschi, A., see Curro, N.J. 434, 451, 467 Revcolevschi, A., see Huh, Y.M. 453 Revcolevschi, A., see Julien, M.-H. 452 Revcolevschi, A., see Suh, B.J. 434, 451 Rey, C. 261, 266 Rey, C., see Calleja, M. 266 Rey, C., see Diéguez, O. 215 Rey, C., see Longo, R.C. 267 Rey, C., see Robles, R. 214–216, 268–270 Reyes, A.P., see Julien, M.-H. 452 Reyes, A.P., see MacLaughlin, D.E. 433

Reyes, A.P., see Mitrovi´c, V.F. 470, 477 Reyes, A.P., see Takigawa, M. 437, 474, 475 Reynet, O. 535 Reynet, O., see Acher, O. 535 Rezende, S.M. 164 Rezende, S.M., see Araújo, A.E.P. 548, 549 Rezende, S.M., see Machado, F.L.A. 507, 515, 520, 522, 532 Rezende, S.M., see Mendes, K.C. 541 Reznik, D., see Fong, H.F. 389, 471 Rhee, J.-R. 161 Rhee, J.R., see Hwang, D.G. 73 Rho, H., see Kim, C.S. 309, 310 Rhyne, J.J. 345 Rhyne, J.J., see Giebułtowicz, T.M. 344, 345, 353 Rhyne, J.J., see Klosowski, P. 344, 345 Rhyne, J.J., see Lin, J. 345 Rhyne, J.J., see Samarth, N. 344, 345 Rhyne, J.J., see Stumpe, L.E. 345 Ri, H.-C., see Lee, G.H. 261 Ribayrol, A. 297 Rice, D.E. 456, 457 Rice, P., see Lee, W.Y. 37, 51, 71 Rice, P.M., see Childress, J.R. 55, 159 Rice, P.M., see Farrow, R.F.C. 164, 172 Rice, T.M., see Mila, F. 436 Rice, T.M., see Poilblanc, D. 395 Rice, T.M., see Viertio, H.E. 396 Rice, T.M., see Zhang, F.C. 385, 386 Richter, G., see Egues, J.C. 306 Richter, M., see Duo, L. 249 Rickel, D.G., see Crooker, S.A. 309 Rieger, G., see Ganzer, S. 25 Riera, J.A. 405 Rietjens, G.H., see van Driel, J. 63, 64, 152 Rigamonti, A. 434 Rigamonti, A., see Carretta, P. 434 Rigamonti, A., see Julien, M.-H. 452 Rigaux, C., see Jusserand, B. 303 Rigaux, C., see Lemaître, A. 296 Rigaux, C., see Mycielski, A. 356, 357 Rigaux, C., see Testelin, C. 351, 353, 356 Rigaux, C., see Zielinski, M. 351–353 Rijks, Th.G.S.M. 13–16, 18, 31, 61, 62, 65, 66, 97, 103, 104, 118, 119, 136–138, 157 Rijks, Th.G.S.M., see Coehoorn, R. 5, 151 Rijks, Th.G.S.M., see Folkerts, W. 22 Rijks, Th.G.S.M., see Kools, J.C.S. 144–147 Riley, S.J., see Klots, T.D. 255 Riley, S.J., see Parks, E.K. 255, 258 Ripka, P. 502 Ripka, P., see Hauser, H. 550

AUTHOR INDEX Rishton, S.A., see Lu, Y. 5 Ritchie, I.M., see Bacskay, G.B. 323 Ritter, M.B., see Harwit, A. 300 Rivero, G., see Kraus, L. 545 Ro, J. 158 Robbes, D. 550 Roberts, S., see Vallin, J.T. 346 Robles, R. 204, 206, 214–217, 225, 245, 259, 260, 266–270, 275 Robles, R., see Izquierdo, J. 227, 277 Robles, R., see Pruneda, J.M. 245–248 Roche, K.P., see Dieny, B. 4, 178 Roche, K.P., see Parkin, S.S.P. 45, 157 Rodrigues, A.R., see Machado, F.L.A. 542–544 Rodriguez, S., see Bartholomev, D.U. 351 Rodriguez-Lopez, J.L., see Aguilera-Granja, F. 219 Roesler, A., see Zhou, Y. 135, 136 Rogacheva, E.I. 336 Rogalev, A., see Kappler, J.-P. 275 Rogalev, A., see Vogel, J. 219 Rogalev, A., see Wilhelm, F. 220, 237 Rogers, C.T., see Kirschenbaum, L.S. 135 Röhlsberger, R. 252 Rokhsar, D.S. 449 Rokhsar, D.S., see Deaven, D.M. 143 Röll, K., see Loch, C. 161 Röll, K., see Stobiecki, F. 32, 39, 60, 113 Romaine, S., see Springholz, G. 327, 328 Romashev, L., see Lauter-Pasyuk, V. 224 Ronning, F., see Armitage, N.P. 465 Ronning, F., see Feng, D.L. 482 Rook, K., see Kools, J.C.S. 37, 49, 71, 145, 146 Roos, B.F.P., see Mewes, T. 152 Roozeboom, F., see Kools, J.C.S. 5, 35, 47, 163 Roozeboom, F., see Lenssen, K.-M.H. 33, 43 Roozeboom, F., see Van de Riet, E. 131 Roozeboom, F., see van der Heijden, P.A.A. 151, 157, 163 Rosciszewski, K., see Gora, D. 397 Rosengaard, N.M., see Alden, M. 218 Rosenkranz, S., see Kaminski, A. 419 Ross, C.A., see Castaño, F.J. 58 Ross, J., see Matsumura, M. 433 Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Rossat-Mignod, J., see Onufrieva, F. 418 Rosseinsky, M.J., see Chow, K.H. 460 Rosseinsky, M.J., see Jestädt, Th. 460 Rossin, V.V. 296, 301 Rossnagel, K., see Kipp, L. 441 Roth, Ch. 228 Roth, S., see Zha, F.-X. 261 Rouabhi, M., see Britel, M.R. 516, 526, 548, 549 Rouabhi, M., see Ciureanu, P. 515, 546, 549

625

Rouabhi, M., see Ménard, D. 515, 526, 548, 549 Roukes, M.L., see Wolf, S.A. 5, 208 Rouse, N., see Prieto, J.L. 22 Roussignol, Ph. 305 Roussignol, Ph., see Deleporte, E. 297 Rouviere, J.L., see Charleux, M. 296 Rowe, H., see Harrison, E.P. 504 Rozenberg, M.J., see Georges, A. 397 Ruan, J.Z., see Zeng, L. 536, 549 Ruban, A., see Pedersen, M.O. 268, 269 Rubio, A., see Onida, G. 274 Rubio, A., see Vega, A. 207, 218, 221, 225–227 Rubo, Yu.G., see El Ouazzani, A. 351, 352 Ruckenstein, A.E., see Varma, C.M. 418 Rüdiger, U., see Fraune, M. 164 Rudkowska, G., see Ciureanu, P. 504, 514 Rudkowski, P., see Ciureanu, P. 504, 514, 515, 542, 546, 549 Rudkowski, P., see Ménard, D. 514 Rudnick, J., see Nussinov, Z. 408 Rüdt, C., see Lindner, J. 228 Ruechle, W.W., see Falk, H. 306 Ruff, R.N., see Bacskay, G.B. 323 Rüffer, R., see Röhlsberger, R. 252 Ruhle, W.W., see Haacke, S. 305 Ruhle, W.W., see Lawrence, I. 305 Ruhle, W.W., see Östreich, M. 304 Ruigrok, J.J.M., see Folkerts, W. 22, 127–129 Rukhadze, Z.A., see Valeiko, M.V. 327 Rupp, G., see Boeve, H. 57 Rupp, G., see van den Berg, H.A.M. 42 Rupp, G., see Vieth, M. 25 Rupp, L.W., see Cava, R.J. 459 Rupp, L.W., see Cheong, S.-W. 461 Rupprecht, R., see Geist, F. 323–325 Rupprecht, R., see Krenn, H. 344, 345 Rusiecki, S., see Karpinski, J. 469 Rusin, T.M. 339 Russek, S.E. 128, 132 Russek, S.E., see Gafron, T.J. 26, 140, 141 Russek, S.E., see Kabos, P. 132 Russek, S.E., see Kirschenbaum, L.S. 135 Russek, S.E., see Oti, J.O. 127, 130 Rüster, C., see Karczewski, G. 311 Ryabchenko, S.M., see Abramishvily, V.G. 351, 352, 357 Ryabchenko, S.M., see König, B. 306 Ryabova, L.I., see Akimov, B.A. 312 Ryan, P.A., see Dekadjevi, D.T. 108 Ryan, P.J., see Yang, Z. 162, 178 Ryu, G.H. 542, 549 Ryzhanova, N., see Dieny, B. 66, 94, 95, 99, 103, 104

626

AUTHOR INDEX

Ryzhanova, N., see Vedyaev, A. 94, 103, 109 Sachan, V. 458 Sachan, V., see Lorenzo, J.E. 457 Sachan, V., see Tranquada, J.M. 456, 457 Sachan, V., see Wochner, P. 458 Sachdev, S. 391, 400, 413, 414, 441 Sachdev, S., see Demler, E. 449, 480 Sachdev, S., see Polkovnikov, A. 413, 420, 478, 481 Sachdev, S., see Read, N. 400 Sachdev, S., see Vojta, M. 399–401, 421, 422 Sachdev, S., see Zhang, Y. 413, 420, 471, 480 Sadori, A. 397, 465 Sadowski, J., see Kowalczyk, L. 340, 341 Sadowski, J., see Nadolny, A.J. 321, 322 Sadowski, M.L., see Teran, F.J. 302 Saez, S., see Robbes, D. 550 Safonov, V.L., see Bertram, H.N. 135 Saha, R., see Shi, S. 22 Sahashi, M., see Fuke, H.N. 32, 158, 174 Sahashi, M., see Fukuzawa, H. 33, 37, 42, 48, 137, 140, 141 Sahashi, M., see Hasegawa, N. 38, 48 Sahashi, M., see Iwasaki, H. 60, 73, 158 Sahashi, M., see Kamiguchi, Y. 32, 35, 41, 48, 49, 77 Sahashi, M., see Kools, J.C.S. 37, 49, 71 Sahashi, M., see Ohsawa, Y. 26 Sahashi, M., see Saito, A.T. 71, 73, 74 Sahashi, M., see Sant, S. 36, 49, 51, 71 Sahashi, M., see Takagishi, M. 13 Sahashi, M., see Yoda, H. 22 Sahashi, M., see Yuasa, H. 13 Sahrakorpi, S., see Lindroos, M. 441 Sainctavit, Ph., see Marangolo, M. 248 Saini, N.L. 454 Saini, N.L., see Bianconi, A. 408 Saito, A., see Kamiguchi, Y. 35, 48, 49, 77 Saito, A.T. 71, 73, 74 Saito, A.T., see Iwasaki, H. 60, 73, 158 Saito, H. 311 Saito, K., see Fuke, H.N. 32, 158, 174 Saito, K., see Kamiguchi, Y. 32, 41 Saito, M. 33, 34, 45, 162, 170 Saito, M., see Fujikata, J. 163, 172 Saito, M., see Hasegawa, N. 33, 34 Saito, T., see Ikada, H. 308 Saito, Y., see Inomata, K. 220, 244 Saito, Y., see Nishimura, K. 63 Saitoh, Y., see Imada, S. 310 Saitou, T., see Chen, Z.H. 303 Sakai, M., see Inage, K. 26

Sakaki, H. 258 Sakakima, H. 5, 35, 42, 46, 48, 49, 51, 57 Sakakima, H., see Irie, Y. 57 Sakakima, H., see Kato, T. 38 Sakakima, H., see Kawawake, Y. 34, 35, 164, 172 Sakakima, H., see Sugita, Y. 33, 35, 36, 46, 51, 52, 114, 162, 178 Sakata, H., see Ohsawa, Y. 26 Sakata, J., see Morikawa, T. 504, 515, 552 Sakurai, J., see Nishimura, K. 63 Salamanca-Riba, L., see Park, K. 360 Salamon, M.B., see Hardner, H.T. 135 Salamon, M.B., see Shi, J. 63 Salamon, M.B., see Tsymbal, E.Yu. 63 Salcedo, A., see Valenzuela, R. 552, 553 Saleh, A.A. 266 Saleh, A.A., see Shutthanandan, V. 266 Salem, N.M., see Borsa, F. 433, 449 SalemSugui, S., see Medina, A.N. 532 Salgueiro, M.A., see Ventura, J.O. 49 Salgueiro da Silva, M.A., see Sousa, J.B. 37 Salib, M.S., see Yu, W.Y. 297 Salkola, M.I. 418 Samant, M.G., see Parkin, S.S.P. 167, 168 Samarth, N. 344, 345 Samarth, N., see Awschalom, D.D. 5, 301 Samarth, N., see Baumberg, J.J. 301 Samarth, N., see Crooker, S.A. 295, 296, 303, 309 Samarth, N., see Crowell, P.A. 310 Samarth, N., see Dai, N. 294, 358 Samarth, N., see Giebułtowicz, T.M. 344, 345 Samarth, N., see Kikkawa, J.M. 300, 301 Samarth, N., see Klosowski, P. 344, 345 Samarth, N., see Levy, J. 301 Samarth, N., see Luo, H. 295, 297 Samarth, N., see Ray, O. 304, 307 Samarth, N., see Smorchkova, I.P. 304 Samarth, N., see Smyth, J.F. 305 Samarth, N., see Zhang, F.C. 297 Saminadayar, K., see André, R. 295 Saminadayar, K., see Huard, V. 302 Saminadayar, K., see Kheng, K. 301 Samwer, K., see Zimmermann, C.G. 268 Sanchez, J.M. 227 Sánchez, M.L. 537, 538 Sánchez, M.L., see Hernando, B. 540 Sánchez, M.L., see Knobel, M. 504, 511, 532, 536, 538, 539, 549 Sánchez, M.L., see Tejedor, M. 504, 538, 539, 553, 557 Sánchez, M.L., see Vázquez, M. 502, 553 Sánchez-Portal, D. 213, 245

AUTHOR INDEX Sánchez-Portal, D., see Calleja, M. 266 Sánchez-Portal, D., see Izquierdo, J. 213 Sánchez-Portal, D., see Soler, J.M. 213 Sancho, M.P.L., see Louis, E. 397 Sander, D. 208 Sanders, I.L., see Speriosu, V.S. 169 Sandratskii, L.M. 216, 269 Sands, T. 266 Sankar, A. 55 Sankey, O.K. 213 Sano, M. 164 Sano, M., see Araki, S. 32–34, 36, 67, 159 Sano, M., see Inage, K. 26 Sano, M., see Li, M. 38, 48 Sano, M., see Shimazawa, K. 159 Sano, M., see Tsuchiya, Y. 36 Sant, S. 36, 49, 51, 71 Sant, S.B., see Kools, J.C.S. 37, 49, 71 Santos, A.D., see Melo, L.G.C. 522 Santos, C., see Freitas, P.P. 22 Santos, C., see Suisky, D. 298 Sarosi, G., see Heaney, P.J. 457 Sarrao, J.L., see McQueeney, R.J. 448, 459 Sarrao, J.L., see Suh, B.J. 452 Sartorelli, M.L. 504, 535 Sartorelli, M.L., see Knobel, M. 535, 557 Sartorelli, M.L., see Pirota, K.R. 504, 535, 557 Sartorelli, M.L., see Sinnecker, J.P. 510, 511 Sas, C., see Yakovlev, D.R. 297, 298 Sasagawa, S., see Uchiyama, T. 553 Sasagawa, T., see Ino, A. 455 Sasagawa, T., see Lake, B. 413, 446 Sasaki, F., see Akimoto, R. 307 Sasaki, S., see Akinaga, H. 296 Sasaki, S., see Nicholas, R.J. 306 Sasaki, T., see Araki, S. 32–34, 36, 67 Sasaki, T., see Hori, H. 276 Sasao, K. 166 Sase, Y., see Ishida, A. 312, 323, 324, 327 Sassik, H., see Knobel, M. 532, 539 Satake, M. 463 Sato, F., see Miyazaki, T. 7, 140 Sato, H. 62 Sato, M., see Katano, S. 446 Sato, M., see Kobayashi, Y. 477 Sato, T. 171, 464 Sato, T., see Chen, Z.H. 303 Sato, T., see Debnath, M.C. 300, 301 Sato, T., see Fretwell, H.M. 441 Sato, T., see Ikada, H. 308 Sato Turtelli, R., see Knobel, M. 532, 539 Satoh, H., see Iguchi, E. 462 Satomi, M., see Kawawake, Y. 34, 35, 164, 172

627

Satomi, M., see Sakakima, H. 35, 46, 48, 49, 51, 57 Satomi, M., see Sugita, Y. 33, 35, 36, 46, 51, 52, 114, 162, 178 Satomi, Y., see Irie, Y. 57 Saurenbach, F., see Binasch, G. 4 Savage, H.T., see Velázquez, J. 544 Savchuk, A.I., see Abramishvily, V.G. 351, 352, 357 Savici, A.T. 449 Savinkov, A.V., see Dooglav, A.V. 473 Sawada, M., see Hayashi, K. 236, 275 Sawasaki, T., see Ueno, M. 34, 42 Sawatzky, G.A., see Zaanen, J. 464 Sawicki, M. 310, 311 Sawicki, M., see Dahl, M. 311 Sawicki, M., see Dietl, T. 304 Sawicki, M., see Głód, P. 357 Sawicki, M., see Jaroszy´nski, J. 304 Sawicki, M., see Karczewski, G. 311 Scalapino, D.J. 391, 416, 428 Scalapino, D.J., see Bickers, N.E. 427 Scalapino, D.J., see Bulut, N. 418 Scalapino, D.J., see Kampf, A.P. 409 Scalapino, D.J., see White, S.R. 405–407 Scalbert, D. 351 Scalbert, D., see Herbich, M. 350 Scalbert, D., see Paganotto, N. 302 Scalbert, D., see Siviniant, J. 296 Scalbert, D., see Teppe, F. 300, 306 Schad, R. 6, 27, 79 Schad, R., see Potter, C.D. 110 Schäfer, M., see Potter, C.D. 110 Schäfer, P., see Schikora, D. 361 Schäfer, R., see Hubert, A. 130 Schäfer, R., see Potter, C.D. 110 Schaller, D.M. 238 Schaller, D.M., see Schmidt, C.M. 221 Schattke, W., see Kipp, L. 441 Schefﬂer, M., see Cho, J.H. 205 Schefﬂer, M., see Nouvertné, F. 268, 269 Schefﬂer, M., see Pentcheva, R. 268 Scheinfein, M., see Yang, Z. 162, 178 Scheinfein, M.R., see Chapman, J.N. 124 Schelp, L.F., see da Silva, R.B. 548 Schelter, W., see Schewe, H. 24 Schelter, W., see van den Berg, H.A.M. 42 Schenk, H., see Yakovlev, D.R. 302 Schep, K.M. 109, 110 Schep, K.M., see Van de Veerdonk, R.J.M. 134 Scherrer, H., see Casian, A. 328 Scherz, A. 227, 228 Scherz, A., see Lindner, J. 228 Scherz, A., see Wilhelm, F. 209, 220, 237

628

AUTHOR INDEX

Schetzina, J.F., see Suh, E.-K. 300 Scheurer, F., see Beaurepaire, E. 209 Scheurer, F., see Bucher, J.P. 208 Scheurer, F., see Chado, I. 208, 236, 252 Scheurer, F., see Ohresser, P. 241 Schewe, H. 24 Schieffer, P., see Elmouhssine, O. 238 Schikora, D. 361 Schikora, D., see Portugall, O. 361 Schikora, D., see Widmer, T. 361 Schikora, D., see Widmer, Th. 361 Schinagl, F., see Kepa, H. 343, 345 Schinagl, F., see Ueta, A.Y. 323, 324, 326, 327 Schindler, A.I., see Lewicki, A. 353 Schlegel, H., see Krost, A. 324 Schlenker, C. 153 Schlesinger, M., see Rausch, T. 535 Schlicht, B., see Nimtz, G. 313, 314, 323, 324 Schlüter, M., see Bachelet, G.B. 213 Schlüter, M., see Hamann, D.R. 213 Schmalian, J., see Abanov, Ar. 428 Schmalian, J., see Laughlin, R.B. 390 Schmalian, J., see Westfahl, H. 408 Schmeusser, S., see Waag, A. 294 Schmid, A.K. 268 Schmid, M., see Biedermann, A. 217 Schmid, M., see Bischoff, M.M.J. 219, 225 Schmidt, C.M. 221 Schmidt, C.M., see Schaller, D.M. 238 Schmidt, G., see Camarero, J. 143 Schmidt, G., see Fiederling, R. 304 Schmidt, G., see Gruber, Th. 305 Schmiedel, T., see Warnock, J. 294, 358 Schmitt-Rink, S., see Varma, C.M. 418 Schmoranzer, H., see Juraszek, J. 135, 136, 171 Schmoranzer, H., see Mougin, A. 171 Schneider, C., see Rau, C. 224, 238 Schneider, C.M., see Tietjen, D. 116, 123 Schneider, J.R., see Ichikawa, N. 452, 454, 455 Schneider, J.R., see Niemöller, T. 454 Schneider, J.R., see Vigliante, A. 463 Schneider, J.R., see Zimmermann, M.v. 454 Schneider, M.A., see Knorr, N. 251 Schneider, T., see Hung, C.-Y. 136, 137, 140, 141, 162 Schneider, T., see Mao, M. 158 Schoenmaker, J., see Pirota, K.R. 504, 535 Scholl, A., see Wu, Y.Z. 230 Scholl, D., see Mauri, D. 154 Scholten, R.E., see McClelland, J.J. 258 Scholtz, W., see Suess, D. 155 Schomig, H., see Bacher, G. 309 Schomig, H., see Welsch, M.K. 307 Schomig, H., see Zaitsev, S. 309

Schönfeld, R., see Portugall, O. 361 Schonmaker, J., see Sinnecker, J.P. 510, 511 Schreﬂ, T., see Fidler, J. 116 Schreﬂ, T., see Suess, D. 155 Schreiber, R., see Gareev, R.R. 220, 245, 247 Schreiber, R., see Grünberg, P. 206, 217, 219, 248 Schrieffer, J.R. 392, 393, 438 Schrieffer, J.R., see Bardasis, A. 410 Schrieffer, J.R., see Heeger, A.J. 393 Schrieffer, J.R., see Salkola, M.I. 418 Schröder, A., see Lake, B. 413, 448, 449, 480 Schröder, A., see Mason, T.E. 448 Schroeder, P.A., see Duvail, J.L. 12, 61, 94, 104 Schroeder, P.A., see Pratt, W.P. 78, 85 Schroeder, P.A., see Vouille, C. 84 Schroeder, P.A., see Yang, Q. 78 Schrör, H., see Uzdin, V.M. 222, 249 Schuhl, A., see Encinas, A. 138 Schuhl, A., see Faure-Vincent, J. 276 Schuller, I.K., see Nogues, J. 148, 149 Schulthess, T.C. 83, 144, 146, 155 Schulthess, T.C., see Brown, R.H. 110 Schulthess, T.C., see Butler, W.H. 107–109 Schulz, H.J. 395 Schulzgen, A., see Streller, U. 296 Schumacher, H.W. 132 Schwartz, L.H., see Jaggi, N.K. 225 Schwartz, P.V. 31, 70 Schwartzacker, W., see Bennett, W.R. 110 Schwartzendruber, L., see Chopra, H.D. 70 Schwarzl, T. 328, 342 Schwarzl, T., see Heiss, W. 328 Schwarzl, T., see Springholz, G. 328, 342 Schweizer, E.K., see Eigler, D.M. 258 Schwickert, M.M., see Perjeru, F. 217, 220, 275 Sciacca, M.D., see Aigle, M. 327 Scott, J.C., see Schlenker, C. 153 Sebastian, L., see Kale, S. 535 Seewald, G. 237 Segawa, K., see Ando, Y. 454, 455 Ségransan, P., see Horvati´c, M. 474 Ségransan, P., see Julien, M.-H. 452, 478 Seibold, C. 397 Seigle, H., see Johnson, A. 61 Seigler, M., see Jo, S. 34, 45, 114, 178 Seigler, M.A. 57 Seki, H., see Saito, M. 33, 162, 170 Semenov, Yu.G. 296, 306 Semenov, Yu.G., see Abramishvily, V.G. 351, 352, 357 Semenov, Yu.G., see Siviniant, J. 296 Semenov, Yu.G., see Teppe, F. 300, 306

AUTHOR INDEX Senthil, T. 420 Senz, S., see Langer, J. 32, 70 Senz, V., see Röhlsberger, R. 252 Seo, J., see Lee, S.K. 235 Seong, M.J. 354 Serpa, G., see Egelhoff Jr., W.F. 35 Severino, A.M., see da Silva, R.B. 548 Seyama, Y., see Nagasaka, K. 13 Seyama, Y., see Oshima, H. 13 Seyama, Y., see Tanaka, A. 13 Sham, L.J., see Kohn, W. 204, 213 Shapira, Y. 353 Shapira, Y., see Bindilatti, V. 312, 326, 355 Shapira, Y., see Foner, S. 353, 355 Shapira, Y., see Fries, T. 357 Shapira, Y., see Gennser, U. 351, 355 Shapira, Y., see Gratens, X. 312 Shapira, Y., see Herbich, M. 351 Shapira, Y., see Isber, S. 355 Shapira, Y., see Kostyk, D. 342, 343 Shapira, Y., see Mac, W. 346–348, 351, 357 Shapira, Y., see McCabe, G.H. 347, 349 Shapira, Y., see Shih, O.W. 351 Shapira, Y., see ter Haar, E. 312, 326 Shapira, Y., see Twardowski, A. 346, 350, 351, 353, 358 Shapira, Y., see Vu, T.Q. 353, 355 Shastry, B.S. 435, 436 Shaw, D. 296 Shcherbakov, A.V. 306 Shcherbakov, A.V., see Akimov, A.V. 299 Shcherbakov, A.V., see Keller, D. 306 Sheehan, P.E., see Edelstein, R.L. 24 Sheehan, P.E., see Miller, M.M. 24 Sheenan, P.E., see Baselt, D.R. 24 Shelton, W.A., see Brown, R.H. 110 Shelton, W.A., see Nicholson, D.M.C. 80 Shelton, W.A., see Oparin, A.B. 85 Shen, B.G., see Chen, C. 504 Shen, B.G., see Guo, H.Q. 504, 532, 539 Shen, B.G., see He, J. 540 Shen, F. 38, 49 Shen, J. 208, 238 Shen, J., see Oka, Y. 300 Shen, J., see Pierce, J.P. 238 Shen, J.X. 163 Shen, J.X., see Debnath, M.C. 300, 301 Shen, J.X., see Pittini, R. 300, 301 Shen, J.X., see Takahashi, N. 301, 307 Shen, K.M., see Armitage, N.P. 465 Shen, K.M., see Feng, D.L. 482 Shen, L.P. 551, 555 Shen, L.P., see Kanno, T. 551 Shen, L.P., see Kusumoto, D. 556

629

Shen, L.P., see Mohri, K. 532, 550, 551, 555 Shen, Z.-X. 479, 482 Shen, Z.-X., see Armitage, N.P. 465 Shen, Z.-X., see Dessau, D.S. 427 Shen, Z.-X., see Feng, D.L. 482 Shen, Z.-X., see King, D.M. 464 Shen, Z.-X., see Marshall, D.S. 476 Shen, Z.-X., see Zhou, X.J. 441, 454 Shen, Z.X., see Bogdanov, P.V. 427 Shen, Z.X., see Lanzara, A. 409, 427, 454, 482 Sheng, L. 94 Shephard, L., see Bae, S. 25, 37, 46 Shephard, L.E., see Torok, E.J. 25 Shevchenko, P. 339 Shi, J. 63 Shi, J., see Tehrani, S. 24, 57 Shi, J., see Tsymbal, E.Yu. 63 Shi, J.M., see Nicholas, R.J. 306 Shi, S. 22 Shi, X., see Tong, H.C. 33, 51, 52, 55, 124 Shibata, K., see Chen, Z.H. 303 Shibata, K., see Ikada, H. 308 Shibuya, M., see Matsuda, Y.H. 303 Shigematsu, S., see Nishioka, K. 160, 174 Shih, O.W. 351, 352 Shimada, Y., see Kitakami, O. 34, 163 Shimazawa, K. 159 Shimizu, S., see Hori, H. 276 Shimizu, T., see Iida, M. 323, 324 Shimizu, T., see Imai, T. 474 Shimizu, T., see Taneda, A. 251 Shimizu, Y. 162 Shimizu, Y., see Hou, C. 150 Shimizu, Y., see Nagasaka, K. 13, 150, 162 Shimizu, Y., see Nakagawa, S. 157 Shimizu, Y., see Oshima, H. 13 Shimizu, Y., see Tanaka, A. 13, 51, 162 Shimoyama, J., see Ishida, K. 479 Shimoyama, J.-I., see Bogdanov, P.V. 427 Shimoyama, J.-I., see Lanzara, A. 409, 427, 454, 482 Shimoyama, K. 162 Shin, K.H. 545 Shin, K.H., see Kim, K.Y. 146, 148 Shin, K.H., see Park, C.-M. 69, 146 Shinjo, T. 5, 9, 12, 27, 55, 222 Shinjo, T., see Nishimura, K. 63 Shinjo, T., see Takanashi, K. 225 Shinjo, T., see Yamamoto, H. 9, 27 Shinkai, M., see Uchiyama, T. 550 Shinoda, T., see Suzuki, T. 445 Shinohara, K., see Ishida, A. 327 Shinomori, S., see Onose, Y. 465, 466

630

AUTHOR INDEX

Shiohara, Y., see Arai, M. 470, 471, 473 Shirado, E., see Debnath, M.C. 300, 301 Shirado, E., see Kayanuma, K. 305, 306 Shirado, E., see Takahashi, N. 301, 307 Shirai, Y., see Akinaga, H. 296 Shirane, G., see Fujita, M. 447, 448 Shirane, G., see Greven, M. 486 Shirane, G., see Kimura, H. 446 Shirane, G., see Lee, Y.S. 448, 449, 486 Shirane, G., see Matsuda, M. 466, 467 Shirane, G., see McQueeney, R.J. 409 Shirane, G., see Sachan, V. 458 Shirane, G., see Wakimoto, S. 447, 448 Shirane, G., see Yamada, K. 447, 448, 466 Shirokova, N.A., see Akimov, B.A. 312 Shirota, Y. 35, 57 Shivaparan, N.R. 266 Shivaparan, N.R., see Saleh, A.A. 266 Shivaparan, N.R., see Shutthanandan, V. 266 Shoenmaker, J., see Knobel, M. 532, 539 Shoenmaker, J., see Pirota, K.R. 535, 557 Shoenmaker, J., see Sartorelli, M.L. 504, 535 Shoji, H., see Takahashi, M. 69 Shraiman, B.I. 392 Shringi, S.N., see Acharya, B.R. 231 Shukh, A.M., see Seigler, M.A. 57 Shurukhin, B.P., see Makhotkin, V.E. 504 Shutthanandan, V. 266 Shutthanandan, V., see Saleh, A.A. 266 Shutthanandan, V., see Shivaparan, N.R. 266 Shyamala, D., see Elangovan, A. 301 Si, Q., see Zha, Y. 420, 423 Si, Q.M. 423 Sichelschmidt, J., see Kochelaev, B.I. 453 Sides, P.J., see Devasahayam, A.J. 166 Sidis, Y. 419, 470, 472 Sidis, Y., see Bourges, P. 389, 471, 472 Sidis, Y., see Fong, H.F. 470 Sidis, Y., see He, H. 411, 426, 449, 482 Siefert, F., see Hausleitner, C. 556 Siegman, H.C., see Mauri, D. 154 Siggia, E.D., see Shraiman, B.I. 392 Sigmund, E.E., see Mitrovi´c, V.F. 470, 477 Silva, B.L., see Machado, F.L.A. 504 Silva, F., see Freitas, P.P. 22 Silva, F.C.S., see Sinnecker, J.P. 510, 511 Silva, T.J., see Kabos, P. 132 Silva, T.J., see Kim, Y.K. 140 Simon, A., see Bussmann-Holder, A. 409 Simon, P., see Odier, P. 459, 461 Simon, P., see Poirot, N.J. 456 Sin, K., see Lee, M.-H. 163 Singer, P.M. 451, 453, 474

Singer, P.M., see Hunt, A.W. 390, 451, 452, 455, 461, 467 Singh, R.R.P., see Elstner, N. 486 Singleton, E., see Mao, S. 60 Singley, E.J. 466 Sinnecker, E.H.C.P., see Sinnecker, J.P. 535 Sinnecker, J.P. 504, 510, 511, 528, 534, 535, 541 Sinnecker, J.P., see García, J.M. 505, 534, 542, 549 Sinnecker, J.P., see Knobel, M. 532, 535, 539, 557 Sinnecker, J.P., see Vázquez, M. 504, 512, 535, 541, 557 Sipatov, A.Y., see Chernyshova, M. 332, 338 Sipatov, A.Yu., see Chernyshova, M. 332, 338 Sipatov, A.Yu., see Kepa, H. 336, 338, 340 Sipatov, A.Yu., see Kolesnikov, I.V. 332, 340 Sipatov, A.Yu., see Kowalczyk, L. 338, 340, 341 Sipatov, A.Yu., see Stachow-Wójcik, A. 333–335 Sipatov, A.Yu., see Stolpe, I. 340, 341 Sipatov, A.Yu., see Story, T. 333–336 Sipatow, A., see Stachow-Wójcik, A. 333, 334 Sirenko, A.A., see Ray, O. 307 Sitter, H., see Abramof, E. 344 Sitter, H., see Giebułtowicz, T.M. 344, 345 Sitter, H., see Nunez, V. 345 Sivertsen, J.M., see Egelhoff, W.F. 51, 52, 93 Sivertsen, J.M., see Egelhoff Jr., W.F. 34, 51, 52, 70, 144 Sivertsen, J.M., see Han, D.-H. 163 Sivertsen, J.M., see Lin, C.-L. 163 Sivertsen, J.M., see Qian, Z. 161 Siviniant, J. 296 Siviniant, J., see Paganotto, N. 302 Sji, J., see Uhlig, W.C. 36, 48, 49 Sjöstedt, E. 219, 275 Skibowski, M., see Kipp, L. 441 Skinta, J.A. 464 Skipetrov, E.P. 312 Skipetrova, L.A., see Skipetrov, E.P. 312 Sko´skiewicz, T., see Dietl, T. 304 Sko´skiewicz, T., see Jaroszy´nski, J. 304 Sko´skiewicz, T., see Sawicki, M. 310, 311 Skriver, H.L., see Alden, M. 218 Slack, G.A., see Vallin, J.T. 346 Slater, J.C. 213 Slater, J.C., see Hattox, T.M. 224 Slater, R.D. 13 Slaughter, J.M. 35, 51, 70, 71 Slaughter, J.M., see Keavney, D.J. 111 Slaughter, J.M., see Tehrani, S. 24, 57 Slichter, C.P. 430–432 Slichter, C.P., see Haase, J. 453, 477

AUTHOR INDEX Sliwa, C., see Dietl, T. 318, 322, 325 Slonczewski, J.C. 115, 136 Slutzky, C., see Gómez, L. 269 Slyn’ko, E., see Zayachuk, D. 329 Slyn’ko, E.I., see Grodzicka, E. 312 Slyn’ko, E.I., see Ivanchik, I.I. 312 Slyn’ko, E.I., see Skipetrov, E.P. 312 Slyn’ko, V.I., see Ivanchik, I.I. 312 Smelbidl, P., see Lake, B. 413, 446 Smirnov, V.A., see Kowalczyk, L. 340, 341 Smit, J. 7 Smith, A.R., see Yang, H. 249 Smith, C., see Beach, R.S. 503, 504 Smith, C.M. 481 Smith, D.J., see Farrow, R.F.C. 164, 172 Smith, D.O. 136 Smith, H.I., see Castaño, F.J. 58 Smith, J.L., see Pennington, C.H. 474 Smith, L.M., see Kim, C.S. 309, 310 Smith, L.M., see Poweleit, C.D. 301 Smith, N. 127, 135, 136 Smith, N., see Carey, M.J. 151, 159, 164, 176 Smith, N., see Childress, J.R. 32, 41, 67, 159, 174 Smith, N.V., see Wu, Y.Z. 230 Smith, R.J., see Saleh, A.A. 266 Smith, R.J., see Shivaparan, N.R. 266 Smith, R.J., see Shutthanandan, V. 266 Smith, S.J., see Parkin, S.S.P. 6 Smith, W.F. 266 Smith, W.F., see Ehrichs, E.E. 258 Smits, C.J.P., see Chernyshova, M. 332, 338 Smoli´nski, K., see Fita, P. 312 Smolyaninova, V.N., see Biswas, A. 464 Smorchkova, I.P. 304 Smorchkova, I.P., see Kikkawa, J.M. 300, 301 Smorchkova, I.P., see Ray, O. 304 Smyth, J.F. 305 Snook, I.K., see Spencer, M.J.S. 218 Soares, J.C., see Veloso, A. 35, 48, 49 Soares, J.M. 532 Sob, M., see Friak, M. 218 Sob, M., see Kudrnovsky, J. 143 Soda, Y., see Ozue, T. 22 Soeya, S. 160, 163, 166, 168 Soeya, S., see Hoshiya, H. 160, 168 Sokol, A., see Elstner, N. 486 Soler, J.M. 213 Soler, J.M., see Calleja, M. 266 Soler, J.M., see Izquierdo, J. 213 Soler, J.M., see Onida, G. 274 Soler, J.M., see Sánchez-Portal, D. 213, 245 Solterbeck, C., see Kipp, L. 441 Somers, G.H.J., see Folkerts, W. 22, 127–129

631

Somers, G.H.J., see Kools, J.C.S. 5 Somers, G.H.J., see Lenssen, K.-M.H. 22, 23, 44, 166 Sommer, R.L. 504, 532, 541 Sommer, R.L., see Ambrose, T. 150 Sommer, R.L., see Carara, M. 504, 542, 549 Sommer, R.L., see da Silva, R.B. 548 Sommer, R.L., see Knobel, M. 502, 504, 506 Sommer, R.L., see Viegas, A.D.C. 514 Sommers, C., see Blaas, C. 80, 109 Son, J., see Hylton, T.L. 70 Sondheimer, E.H. 87 Song, F. 254 Song, O. 140 Song, S.-H. 542–544 Song, S.A., see Kim, Y.-K. 76, 162 Song, S.H. 546 Sonier, J.E. 477 Sonobe, Y., see Moser, A. 21 Sonoda, S., see Hori, H. 276 Sooho, R.F. 529 Sooryakumar, R., see Bak, J. 354 Soskic, Z. 353, 360 Sotomayor Torres, C.M., see Ribayrol, A. 297 Souma, I., see Debnath, M.C. 300, 301 Souma, I., see Ikada, H. 308 Souma, I., see Kayanuma, K. 305, 306 Souma, I., see Oka, Y. 300 Souma, I., see Takahashi, N. 301, 307 Sour, R.L.H., see Rijks, Th.G.S.M. 66, 97, 104 Sousa, J.B. 37 Sousa, J.B., see Li, H. 36, 159, 167 Sousa, J.B., see Veloso, A. 35, 48, 49 Sousa, J.B., see Ventura, J.O. 49 Souw, V., see Alawadhi, H. 311, 351, 352, 356 Souza, R.C., see Schumacher, H.W. 132 Sowers, C.H., see Mattson, J.E. 220, 244 Sowers, H., see Grünberg, P. 206, 217, 219, 248 Spanger, B., see Lambrecht, A. 312, 323, 324 Spellmeyer, B., see Waldmann, H. 361 Spencer, M.J.S. 218 Spendeler, L., see Camarero, J. 143 Speriosu, V., see Baril, L. 141 Speriosu, V., see Spong, J.K. 22, 23 Speriosu, V.S. 18, 40, 45, 72, 110, 145, 169 Speriosu, V.S., see Bensmina, F. 73 Speriosu, V.S., see Butler, W.H. 108, 109 Speriosu, V.S., see Dieny, B. 4, 9, 16, 18, 19, 30, 31, 39, 59, 60, 66, 94, 95, 99, 103, 157, 178 Speriosu, V.S., see Gurney, B.A. 22, 31–33, 40, 41, 97, 129, 140 Speriosu, V.S., see Heim, D.E. 127, 129 Speriosu, V.S., see Huang, T.C. 70, 72

632

AUTHOR INDEX

Speriosu, V.S., see Lefakis, H. 67 Speriosu, V.S., see Mamin, H.J. 22, 25, 141 Speriosu, V.S., see Meny, C. 71 Speriosu, V.S., see Nicholson, D.M.C. 40, 85 Speriosu, V.S., see Nozières, J.P. 72, 158, 161, 162, 168–170 Speriosu, V.S., see Ounadjela, K. 140 Speriosu, V.S., see Parkin, S.S.P. 155 Speriosu, V.S., see Tang, D.D. 22, 24, 58, 128 Speriosu, V.S., see Tsang, C. 22, 127 Speriosu, V.S., see Zeltser, A.M. 159 Speriosu, V.S., see Zhang, Y.B. 60, 76, 161, 170 Sperlich, G., see Urban, P. 323, 325, 329 Spicer, W.E., see Dessau, D.S. 427 Spicer, W.E., see King, D.M. 464 Spicer, W.E., see Marshall, D.S. 476 Spisak, D. 212, 217, 218, 229, 231, 238 Spitzer, R., see Torok, E.J. 25 Spong, J.K. 22, 23 Springelkamp, F., see Leuken, H.v. 253 Springholz, G. 326–328, 342 Springholz, G., see Aigle, M. 327 Springholz, G., see Chen, J.J. 343 Springholz, G., see Denecke, R. 315 Springholz, G., see Geist, F. 323–325 Springholz, G., see Giebułtowicz, T.M. 343 Springholz, G., see Heiss, W. 328, 342 Springholz, G., see Kepa, H. 343, 345 Springholz, G., see Koppensteiner, E. 342 Springholz, G., see Krenn, H. 312, 323–326 Springholz, G., see Nunez, V. 343 Springholz, G., see Prinz, A. 323 Springholz, G., see Schwarzl, T. 328, 342 Springholz, G., see Ueta, A.Y. 323, 324, 326, 327 Springholz, G., see Wilamowski, Z. 343 Springholz, G., see Yuan, S. 323, 324, 327 Squire, P.T. 505, 519 Squire, P.T., see Atkinson, D. 515, 520–522, 538, 553, 556 Stachow-Wójcik, A. 310, 333–335 Stachow-Wójcik, A., see Janik, E. 295 Stachow-Wójcik, A., see Kowalczyk, L. 340, 341 Stachow-Wójcik, A., see Łazarczyk, P. 315 Stachow-Wójcik, A., see Story, T. 335 Stamps, R.L. 148 Stamps, R.L., see Suess, D. 155 Stankiewicz, J. 355, 356 Stanley, F.E., see Marrows, C.H. 22, 127 Stantero, A., see Brunetti, L. 532 Staud, N., see Lin, T. 34, 150, 161, 166, 169 Staunton, J.B., see Razee, S.S.A. 209 Steenwyk, S.D. 13, 80, 83, 85

Steenwyk, S.D., see Gu, J.Y. 13 Steierl, G., see Pfandzelter, R. 205, 263 Steindl, R. 550 Steindl, R., see Hauser, H. 556 Steindl, R., see Hausleitner, C. 556 Steiner, K., see Xu, J. 327 Steinhögl, W., see Benedek, G. 218 Stelmashenko, N.A., see Morecroft, D. 138 Stemmann, G. 420 Stemmann, G., see Lavagna, M. 418 Stensgaard, I., see Pedersen, M.O. 268, 269 Stepanyuk, V., see Robles, R. 214–216, 268–270 Stepanyuk, V.S. 212, 238, 253, 268 Stepanyuk, V.S., see Bazhanov, D.I. 205, 252 Stepanyuk, V.S., see Izquierdo, J. 227, 269, 277 Stepanyuk, V.S., see Levanov, N.A. 215 Stepanyuk, V.S., see Lin, N. 275 Stepanyuk, V.S., see Lysenko, O.V. 275 Stepanyuk, V.S., see Nayak, S.K. 253 Stepanyuk, V.S., see Sander, D. 208 Stepanyuk, V.S., see Weber, S.E. 251 Stephens, P.W., see McQueeney, R.J. 448 Steren, L., see George, J.M. 86 Steren, L.B., see Duvail, J.L. 12, 61, 94, 104 Stern, R., see Haase, J. 453, 477 Sternlieb, B.J., see Tranquada, J.M. 388, 396, 439, 443–445, 455, 458 Sticht, J., see Herman, F. 207, 221 Sticht, J., see Kübler, J. 216 Stiles, M.D. 107, 143, 155, 207 Stiles, M.D., see Egelhoff, W.F. 51, 52, 93 Stiles, M.D., see Egelhoff Jr., W.F. 34, 42, 51, 52, 69, 70, 144 Stiles, M.D., see Pierce, D.T. 219 Stirner, T. 296, 300 Stirner, T., see Fatah, J.M. 296 Stirner, T., see Harrison, P. 296 Stirner, T., see Jackson, S. 296 Stirner, T., see Miao, J. 300 Stirner, T., see Weston, S.J. 296 Stobiecki, F. 32, 39, 60, 113 Stobiecki, T., see Stobiecki, F. 32, 39, 60, 113 Stocks, G.M., see Nicholson, D.M.C. 80 Stocks, G.M., see Oparin, A.B. 85 Stocks, G.M., see Turchi, P.E.A. 227 Stoecklein, W. 154 Stoefﬂer, D. 204, 207, 217, 221, 224, 228 Stoefﬂer, D., see Cornea, C. 204, 224 Stoefﬂer, D., see Freyss, M. 221, 222 Stoefﬂer, D., see Fullerton, E.E. 235 Stoefﬂer, D., see Robles, R. 204, 206, 217 Stoefﬂer, D., see Turek, I. 221, 222 Stoev, K., see Shi, S. 22

AUTHOR INDEX Stoev, K., see Tong, H.C. 51, 52 Stoji´c, B.B., see Soskic, Z. 353, 360 Stoji´c, M., see Soskic, Z. 353, 360 Stojkovi´c, B.P. 392 Stojkovi´c, B.P., see Blumberg, G. 418 Stojkovi´c, B.P., see Laughlin, R.B. 390 Stojkovi´c, B.P., see Miller, B.H. 62 Stokes, S.W. 135 Stolpe, I. 340, 341 Stoltz, S., see Mountziaris, T.J. 358 Stoltz, S., see Peck, J. 358 Stoltz, S., see Yu, W.Y. 358 Stoner, E.C. 115 Story, T. 312, 315, 317, 318, 328–330, 333–336 Story, T., see Chernyshova, M. 332, 338 Story, T., see de Jonge, W.J.M. 320 Story, T., see Eggenkamp, P.J.T. 318, 320, 321 Story, T., see Gałazka, ˛ R.R. 291, 312–314, 325, 328, 329, 345 Story, T., see Górska, M. 329 Story, T., see Grodzicka, E. 312 Story, T., see Kepa, H. 336, 338, 340 Story, T., see Kowalczyk, L. 338 Story, T., see Łazarczyk, P. 315, 316 Story, T., see Łusakowski, A. 318 Story, T., see Miotkowska, S. 315 Story, T., see Nadolny, A.J. 321, 322 Story, T., see Radchenko, M.V. 315 Story, T., see Stachow-Wójcik, A. 333–335 Story, T., see Swirkowicz, R. 332, 335 Strasser, T., see Kipp, L. 441 Street, R. 175 Streller, U. 296 Strijkers, G.J. 97, 100, 101, 123, 220, 244, 245 Strijkers, G.J., see Swagten, H. 97, 98 Strijkers, G.J., see Swagten, H.J.M. 34, 46, 47, 52, 92, 93 Ström-Olsen, J.O., see Ciureanu, P. 504, 514, 515, 546, 549 Ström-Olsen, J.O., see Ménard, D. 514 Stronach, C.E., see Sonier, J.E. 477 Stroscio, J.A., see Davies, A. 222 Stroscio, J.A., see Pierce, D.T. 228 Strycharczuk, A., see Jaroszy´nski, J. 304 Stumpe, L.E. 345 Stumpe, L.E., see Rhyne, J.J. 345 Su, W.P., see Heeger, A.J. 393 Sudbö, A., see Marston, J.B. 402 Suess, D. 155 Suga, K., see Hori, H. 276 Suga, S., see Imada, S. 310 Suga, S., see Kuch, W. 230 Sugano, S., see Yagami, K. 158 Sugawara, N. 61

633

Sugawara, N., see Hosomi, M. 13 Sugawara, N., see Maesaka, A. 74, 75, 77 Sugimoto, A., see Iguchi, I. 453 Sugita, Y. 33, 35, 36, 46, 51, 52, 114, 162, 178 Sugita, Y., see Hamakawa, Y. 47, 73, 163, 172 Sugita, Y., see Hoshino, K. 31, 118, 158, 167 Sugita, Y., see Hoshiya, H. 34, 51, 160, 168 Sugita, Y., see Kato, T. 38 Sugita, Y., see Kawawake, Y. 34, 35, 164, 172 Sugita, Y., see Nakamoto, K. 22, 47, 172 Sugita, Y., see Nakatani, R. 67, 157, 158 Sugita, Y., see Noguchi, S. 50, 51 Sugita, Y., see Sakakima, H. 35, 48, 49, 51 Suh, B.J. 434, 451, 452, 467 Suh, B.J., see Curro, N.J. 434, 451, 467 Suh, E.-K. 300 Suh, E.-K., see Bartholomev, D.U. 351 Suh, S., see Ro, J. 158 Suisky, D. 298 Sukstanskii, A.L. 529, 531 Sulyok, A., see Menyhard, M. 73 Sulyok, A., see Zeltser, A.M. 73 Sulzer, G., see Waldmann, H. 361 Sun, Q., see Trioni, M.I. 253 Sun, S., see Moser, A. 21 Sun, X., see Hicks, L.D. 328 Sun, X.F., see Ando, Y. 454, 455 Suna, A., see Carcia, P.F. 259 Sundgren, J.E., see Korenivski, V. 224 Sungki, O., see Durbin, S.M. 295, 344, 345 Sur, I., see Casian, A. 328 Sürgers, C., see Wang, S.-X. 71 Suris, R.A., see Kochereshko, V.P. 302 Suris, R.A., see Yakovlev, D.R. 302 Surma, M., see Kossacki, P. 296 Surma, M., see Kutrowski, M. 295 Surma, M., see Wojtowicz, T. 295 Susaki, T. 251 Sushkov, O.P. 401 Sussiau, M., see Duvail, J.L. 12, 61, 94, 104 Sussiau, M., see Encinas, A. 138 Sutjahja, I.M., see Bakharev, O.N. 467, 468 Sutter, Ch., see Labergie, D. 248 Suzuki, I.S., see Suzuki, M. 205, 234, 275 Suzuki, M. 60, 205, 234, 275 Suzuki, T. 54, 445 Suzuki, T., see Katano, S. 446 Suzuki, T., see Tsunoda, M. 167 Suzuki, T., see Yang, T. 161 Suzuki, Y., see Hamakawa, Y. 47, 73, 160, 163, 168, 172 Suzuki, Y., see Katayama, T. 110 Suzuki, Y., see Nakamoto, K. 22, 47, 172

634

AUTHOR INDEX

Svalov, A.V., see Kurlyandskaya, G.V. 534 Svane, A., see Fanciulli, M. 245 Svrcek, V., see Wilamowski, Z. 343 Swagten, H. 97, 98 Swagten, H.J.M. 34, 46, 47, 52, 92, 93, 315, 353, 354 Swagten, H.J.M., see Chernyshova, M. 332, 338 Swagten, H.J.M., see de Jonge, W.J.M. 312, 315, 318, 320 Swagten, H.J.M., see Eggenkamp, P.J.T. 318, 320, 321 Swagten, H.J.M., see Grodzicka, E. 312 Swagten, H.J.M., see Kepa, H. 336, 338, 340 Swagten, H.J.M., see LeClair, P. 341 Swagten, H.J.M., see Litvinov, V.I. 103 Swagten, H.J.M., see Mac, W. 346–348 Swagten, H.J.M., see Stachow-Wójcik, A. 333–335 Swagten, H.J.M., see Story, T. 315, 318, 335 Swagten, H.J.M., see Strijkers, G.J. 97, 100, 101, 220, 244, 245 Swagten, H.J.M., see Twardowski, A. 346, 353 Swagten, H.J.M., see Vennix, C.W.H.M. 315, 320 Swan, A.K., see Hwang, C. 236 ´ atek, Swi ˛ K., see Nadolny, A.J. 321, 322 ´ Swierkowski, L., see Shevchenko, P. 339 ´ Swierkowski, L., see Story, T. 315 ´ Swirkowicz, R. 332, 335 ´ Swirkowicz, R., see Story, T. 333, 334, 336 Swüste, C.H.W., see Eggenkamp, P.J.T. 318, 320, 321 Swüste, C.H.W., see Stachow-Wójcik, A. 333–335 Swüste, C.H.W., see Story, T. 315, 317, 318, 335 Syed, M. 297 Syoji, M., see Makino, E. 144, 145 Szadkowski, A., see Mycielski, A. 356, 357 Szczepa´nska, A., see Ma´ckowski, S. 308 Szczerbakow, A., see Chernyshova, M. 332, 338 Szczerbakow, A., see Story, T. 315, 318 Szczurek, T., see Rausch, T. 535 Szucs, J. 59, 60 Szunyogh, L., see Blaas, C. 80, 109 Szunyogh, L., see Lazarovits, B. 209, 238 Szunyogh, L., see Razee, S.S.A. 209 Szunyogh, L., see Ujsaghy, O. 252 Szuszkiewicz, W. 311, 361 Szuszkiewicz, W., see Hennion, B. 311 Szuszkiewicz, W., see Janik, E. 295 Szuszkiewicz, W., see Jouanne, M. 311 Szymczak, H., see Zuberek, R. 140 Szymczak, R., see Mycielski, A. 356, 357 Szytula, A., see Soskic, Z. 353, 360

Tabat, N., see Hardner, H.T. 135 Tabata, H., see Ueda, K. 311 Tabbal, M., see Errebbahi, A. 322, 326 Tabuchi, K., see Tanoue, S. 60 Tabuchi, K., see Ueno, M. 34, 42 Tacke, M., see Geist, F. 323–325 Tacke, M., see Lambrecht, A. 312, 323, 324 Tacke, M., see Xu, J. 327 Tadokoro, S., see Soeya, S. 160, 163, 166, 168 Taga, A. 206 Taga, Y., see Morikawa, T. 504, 515, 552, 557 Taga, Y., see Suzuki, M. 60 Taguchi, M. 229 Taguchi, M., see Demangeat, C. 206, 229 Taguchi, M., see Krüger, P. 236, 238, 241, 242, 262, 263 Taguchi, M., see Parlebas, J.C. 262 Taguchi, M., see Razaﬁmandimby, H. 209 Taguchi, Y., see Armitage, N.P. 465 Taguchi, Y., see Harima, N. 466 Taguchi, Y., see Onose, Y. 465, 466 Tailor, R.H., see Dieny, B. 60, 61, 92, 106 Tajima, S., see Arai, M. 470, 471, 473 Takabayashi, J., see Oka, Y. 300 Takabayashi, K., see Takahashi, N. 301, 307 Takada, J., see Tsunekawa, K. 38, 48, 170, 178 Takagi, H., see Bo˜zin, E.S. 448 Takagi, H., see Lake, B. 413, 446, 448, 449, 480 Takagi, H., see Nachumi, B. 449, 450 Takagi, Y., see Adachi, S. 298 Takagi, Y., see Takeyama, S. 300, 303 Takagishi, M. 13 Takagishi, M., see Yuasa, H. 13 Takahashi, K., see Ando, K. 295 Takahashi, K., see Takayasu, M. 139 Takahashi, M. 69, 158 Takahashi, M., see Arai, M. 470, 471, 473 Takahashi, M., see Pittini, R. 300, 301 Takahashi, M., see Sato, T. 171 Takahashi, M., see Tsunoda, M. 154, 159, 167, 176 Takahashi, M., see Yagami, K. 158, 159 Takahashi, N. 301, 307 Takahashi, N., see Ikada, H. 308 Takahashi, N., see Oka, Y. 300 Takahashi, T., see Fretwell, H.M. 441 Takahashi, T., see Hasegawa, N. 38, 48 Takahashi, T., see Sato, T. 464 Takamasu, T., see Imanaka, Y. 302, 303 Takamasu, T., see Kuroda, S. 308, 309 Takanashi, K. 225 Takano, K. 127, 155

AUTHOR INDEX Takano, K., see Berkowitz, A.E. 148 Takano, K., see Egelhoff, W.F. 51, 52, 93 Takano, K., see Egelhoff Jr., W.F. 34, 51, 52, 69, 70, 144 Takano, K., see Moser, A. 21 Takashima, H., see Arumugam, S. 454 Takashima, H., see Kitakami, O. 34, 163 Takayama, A. 552 Takayasu, M. 139 Takayasu, M., see Uchiyama, S. 139 Takei, H., see Hasegawa, M. 459 Takeshita, N., see Arumugam, S. 454 Takeuchi, M., see Morikawa, T. 504, 515, 552, 557 Takeyama, S. 300, 302, 303 Takeyama, S., see Adachi, S. 298 Takeyama, S., see Kossut, J. 310 Takeyama, S., see Kunimatsu, H. 302 Takeyama, S., see Mino, H. 298, 303 Takeyama, S., see Semenov, Yu.G. 296 Takeyama, S., see Stirner, T. 300 Takeyama, S., see Yokoi, H. 302, 303 Takezawa, M., see Yamaguchi, M. 552 Takigawa, M. 437, 474, 475 Takigawa, M., see Monien, H. 437, 473 Takiguchi, M. 74, 75 Takiguchi, M., see Sugawara, N. 61 Takimoto, T. 418 Takita, K., see Akinaga, H. 296 Takita, K., see Kuroda, S. 308, 309 Takita, K., see Matsuda, Y.H. 303 Takita, K., see Terai Jr., Y. 308 Takita, K., see Yasuhira, T. 303 Talanana, M. 227 Talanana, M., see Dahmoune, C. 235 Talanana, M., see Izquierdo, J. 227, 277 Taliashvili, B., see Nadolny, A.J. 321, 322 Tallon, J.L. 478 Tallon, J.L., see Williams, G.V.M. 475, 476 Tamanaha, C.R., see Edelstein, R.L. 24 Tamanaha, C.R., see Miller, M.M. 24 Tamasaku, K., see Ino, A. 455 Tamura, H. 456 Tanabe, M., see Inoue, M. 315 Tanabe, T., see Satake, M. 463 Tanabe, T., see Yamamoto, K. 460 Tanabe, T., see Yoshizawa, H. 459 Tanaka, A. 13, 51, 162 Tanaka, A., see Hou, C. 150 Tanaka, A., see Kishi, H. 162, 171 Tanaka, A., see Nagasaka, K. 13, 150, 162 Tanaka, A., see Oshima, H. 13 Tanaka, A., see Shimizu, Y. 162 Tanaka, A., see Varga, L. 131, 132

635

Tanaka, K., see Hasegawa, N. 33, 34 Tanaka, K., see Saito, M. 34, 45, 170 Tanaka, M., see Ahn, J.Y. 353 Tanake, I., see Matsuda, M. 466, 467 Tanamoto, T. 420 Tananaeva, O.I., see Akimov, B.A. 312 Taneda, A. 251 Tang, D.D. 22, 24, 58, 128 Tang, J.P. 457 Tange, H., see Goto, M. 136 Tani, T., see Akimoto, R. 307 Taniguchi, S. 136 Tanner, B.K., see Dekadjevi, D.T. 108 Tanner, B.K., see Marrows, C.H. 249 Tanone, S., see Ueno, M. 136 Tanoue, H., see Ando, K. 361 Tanoue, S. 60 Tanuma, T., see Fujita, M. 41 Tarhan, E., see Aigle, M. 327 Tasaki, A., see Akoh, H. 219, 224 Tast, F. 265 Tast, F., see Billas, I.M.L. 265 Tatarchenko, A.F., see Stepanyuk, V.S. 253 Tatarenko, S., see André, R. 295 Tatarenko, S., see Ferrand, D. 310, 311 Tatarenko, S., see Haury, A. 311 Tatarenko, S., see Huard, V. 302 Tatarenko, S., see Kheng, K. 301 Tatarenko, S., see Kossacki, P. 302 Tatarenko, S., see Maslana, W. 300 Tatarenko, S., see Teppe, F. 300, 306 Tatsukawa, T., see Inoue, M. 315 Tavazza, F., see Fanciulli, M. 245 Tavrina, T.V., see Rogacheva, E.I. 336 Taylor, R.H., see Cowache, C. 164 Tchernyshyov, O. 404, 417 Tchernyshyov, O., see Nachumi, B. 449, 450 Tedenac, J.C., see Averous, M. 328 Tedenac, J.C., see Errebbahi, A. 322, 326 Tedenac, J.C., see Gratens, X. 328 Tedoldi, F., see Carretta, P. 434 Tehrani, S. 24, 57 Tehrani, S., see Chen, E.Y. 128 Tehrani, S., see Slaughter, J.M. 35, 51, 70, 71 Teitel’baum, G.B. 434, 435, 451, 452, 467 Teitel’baum, G.B., see Brom, H.B. 479 Teitel’baum, G.B., see Finkelstein, A.M. 453, 478 Tejedor, M. 504, 538, 539, 553, 557 Tejedor, M., see Hernando, B. 540 Tejedor, M., see Sánchez, M.L. 538 ten Haaf, D.F.B., see van Bemmel, H.J.M. 406 Teppe, F. 300, 306

636

AUTHOR INDEX

ter Haar, E. 312, 326 ter Haar, E., see Bindilatti, V. 312, 326 ter Haar, E., see Gratens, X. 312 Terai, Y., see Kuroda, S. 308, 309 Terai Jr., Y. 308 Terakura, K., see Hamada, H. 225, 226 Teran, F., see Ghali, M. 304, 305 Teran, F.J. 302 Teran, F.J., see Kutrowski, M. 296 Teran, F.J., see Wojtowicz, T. 302 Terekhov, A.A., see Ivanchik, I.I. 312 Terki, F., see Errebbahi, A. 322, 326 Terrones, M., see Prados, C. 261 Terunuma, K., see Araki, S. 32–34, 36, 67 Terunuma, K., see Inage, K. 26 Terunuma, K., see Li, M. 38, 48 Tessier, M., see Zuberek, R. 140 Testelin, C. 351, 353, 356, 357 Testelin, C., see Jusserand, B. 303 Testelin, C., see Lemaître, A. 296 Tewari, S. 420 Tewordt, L., see Dahm, T. 418 Theodoropoulou, N. 276 Thiaville, A. 115 Thiaville, A., see Katayama, T. 110 Thiaville, A., see Labrune, M. 120, 121, 127 Thieme, C.L.H., see Shapira, Y. 353 Thomas, L. 152 Thompson, D.A., see Hempstead, R.D. 151, 157, 172 Thompson, J.D., see Suh, B.J. 452 Thompson, J.D., see Takigawa, M. 437, 474, 475 Thompson, T., see Carey, M.J. 33, 67 Thomson, D.A., see Bajorek, C.H. 172 Thomson, J.D., see MacLaughlin, D.E. 433 Thornton, M.J., see Ziese, M. 5 Thornton, S.C., see Baker, S.H. 241, 264 Thrush, C.N., see Karczewski, G. 324 Thurber, K.R., see Hunt, A.W. 451, 452, 467 Tian, D., see Lu, S.H. 231 Tiberto, P., see Appino, C. 540 Tiberto, P., see Brunetti, L. 504, 532 Tiberto, P., see Maraner, A. 135 Tiberto, P., see Sinnecker, J.P. 535 Tietjen, D. 116, 123 Tikhonov, D., see Hübener, M. 276 Tinkham, M. 438 Tiusan, C., see Faure-Vincent, J. 276 Toda, J., see Kanai, H. 32, 35, 41, 55, 77, 157 Todd, N.K., see Prieto, J.L. 22 Tohyama, T., see Prelovšek, P. 454 Tokunaga, Y., see Ishida, K. 479 Tokura, Y., see Armitage, N.P. 465 Tokura, Y., see Harima, N. 466

Tokura, Y., see Imada, M. 390 Tokura, Y., see Onose, Y. 465, 466 Tokura, Y., see Satake, M. 463 Tokura, Y., see Yamamoto, K. 460 Tokura, Y., see Yoshizawa, H. 459 Tolman, C.H. 140 Tomassini, N., see D’Andrea, A. 296 Tomaz, M.A. 236, 275 Tomimoto, K., see Arai, M. 470, 471, 473 Tomita, H., see Fukuzawa, H. 37, 48 Tomita, H., see Hasegawa, N. 38, 48 Tomkowicz, Z., see Soskic, Z. 353, 360 Tondra, M., see Wang, D. 148 Toney, M.F., see Lee, W.Y. 37, 51, 71, 162, 170 Tong, H., see Shi, S. 22 Tong, H.-C., see Mao, M. 32, 33, 158 Tong, H.-C., see Mao, S. 7 Tong, H.C. 33, 51, 52, 55, 124, 167 Tong, H.C., see Hung, C.-Y. 136, 137, 140, 141, 162 Tong, H.C., see Yan, X. 22 Tong, R.Y., see Li, M. 61, 159, 167 Toninato, F., see Maraner, A. 135 Tonner, B.P., see O’Brien, W.L. 249 Tönnies, D. 296 Toperverg, B.P., see Lauter-Pasyuk, V. 224 Torgeson, D.R., see Borsa, F. 433, 449 Torija, M.A., see Pierce, J.P. 238 Torkunov, A.V., see Baranov, S.A. 548 Torkunov, A.V., see Zhukova, V. 533 Torok, E.J. 25 Torok, E.J., see Bae, S. 25, 37, 46 Toropov, A.A., see Reshina, I.I. 309 Totland, K., see Fuchs, P. 226, 227 Toyama, F., see Wu, Y.Z. 230 Tozer, S., see Yokoi, H. 303 Tozer, S.W., see Semenov, Yu.G. 296 Tran, S., see Huai, Y. 33 Tran, T.T., see Saleh, A.A. 266 Tran Anh, T., see Szuszkiewicz, W. 361 Tranquada, J.M. 388, 396, 409, 439, 443–446, 451, 455–459 Tranquada, J.M., see Emery, V.J. 390 Tranquada, J.M., see Ichikawa, N. 452, 454, 455 Tranquada, J.M., see Lee, S.-H. 404, 459 Tranquada, J.M., see Lorenzo, J.E. 457 Tranquada, J.M., see Niemöller, T. 454 Tranquada, J.M., see Pashkevich, Yu.G. 461, 463 Tranquada, J.M., see Sachan, V. 458 Tranquada, J.M., see Vigliante, A. 463 Tranquada, J.M., see Wakimoto, S. 445 Tranquada, J.M., see Wochner, P. 458 Tranquada, J.M., see Zimmermann, M.v. 454

AUTHOR INDEX Traverse, A., see Lawniczak-Jabło´nska, K. 345 Treger, D.M., see Wolf, S.A. 5, 208 Treutler, C.P.O. 22, 24 Trickey, S., see Hattox, T.M. 224 Trigui, F., see Dupas, C. 9 Trioni, M.I. 253 Trotter, S., see Zeltser, A.M. 159 Trotter, S., see Zhang, Y.B. 60, 76, 161, 170 Trouilloud, P.L., see Lu, Y. 5 Troullier, N. 245 Troyer, M., see Dorneich, A. 416 Trullinger, S.E., see Demangeat, C. 203 Tsang, C. 22, 127, 157, 176 Tsang, C., see Childress, J.R. 32, 41, 67, 159, 174 Tsang, C., see Heim, D.E. 127, 129 Tsang, C., see Lin, T. 151, 157, 161, 163 Tsang, C., see Pinarbasi, M. 47, 164, 172 Tsang, C.H. 4, 5, 22 Tsang, C.H., see Childress, J.R. 55, 159 Tsao, C.H., see Huang, J.C.A. 71 Tsapenko, V.V., see Pashkevich, Yu.G. 461, 463 Tsen, S.Y., see Yang, Z. 162, 178 Tsivline, D.V., see Stepanyuk, V.S. 268 Tsu, I.-F., see Huang, R.-T. 50, 77 Tsuchiya, Y. 36 Tsuchiya, Y., see Araki, S. 32–34, 36, 67, 159 Tsuchiya, Y., see Li, M. 38, 48 Tsuchiya, Y., see Shimazawa, K. 159 Tsuchiya, Y., see Tsunoda, M. 154, 167 Tsuei, C.C. 464 Tsukamoto, Y., see Oshima, N. 151, 157 Tsunashima, S. 57 Tsunashima, S., see Hoshino, K. 158, 167 Tsunashima, S., see Kato, T. 38 Tsunashima, S., see Shimoyama, K. 162 Tsunashima, S., see Shirota, Y. 35, 57 Tsunekawa, K. 38, 48, 170, 178 Tsunoda, M. 154, 159, 167, 176 Tsunoda, M., see Sato, T. 171 Tsunoda, M., see Takahashi, M. 69, 158 Tsunoda, M., see Yagami, K. 158, 159 Tsutai, A., see Iwasaki, H. 60, 73, 158 Tsutai, A., see Yoda, H. 22 Tsymbal, E.U., see Petford-Long, A.K. 124, 127, 129 Tsymbal, E.Y. 5, 109, 110, 181 Tsymbal, E.Yu. 63, 110 Tsymbal, E.Yu., see Bailey, W.E. 111 Tuchscherer, T., see Zhang, Y.B. 60, 76, 161, 170 Tufﬁgo, H., see Lawrence, I. 305 Tulchinsky, D.A., see Crooker, S.A. 295 Tulchinsky, D.A., see Pierce, D.T. 228 Tumanski, S. 5, 7, 23

637

Turchi, P.E.A. 227 Turek, I. 221, 222 Turek, I., see Freyss, M. 248 Turek, I., see Pajda, M. 209 Turney, G.L., see Harrison, E.P. 504 Twardowski, A. 346, 350, 351, 353, 357, 358 Twardowski, A., see Ando, K. 350 Twardowski, A., see Boonman, M.E.J. 346 Twardowski, A., see Fries, T. 357 Twardowski, A., see Gennser, U. 351, 355 Twardowski, A., see Herbich, M. 350, 351 Twardowski, A., see Janik, E. 295 Twardowski, A., see Kepa, H. 336, 338, 340 Twardowski, A., see Kostyk, D. 342, 343 Twardowski, A., see Krevet, R. 346 Twardowski, A., see Liu, X.C. 355 Twardowski, A., see Mac, W. 346–351, 357 Twardowski, A., see McCabe, G.H. 347, 349 Twardowski, A., see Stachow-Wójcik, A. 310, 333–335 Twardowski, A., see Story, T. 333–336 Twardowski, A., see Swagten, H.J.M. 353, 354 Tworzydlo, J., see Zaanen, J. 389, 396, 404, 407, 420 Tyazhlov, M.G., see Kulakovskii, V.D. 306 Uchida, K., see Kunimatsu, H. 302 Uchida, K., see Takeyama, S. 302 Uchida, K., see Yasuhira, T. 303 Uchida, S., see Arumugam, S. 454 Uchida, S., see Hoffman, J.E. 413, 441, 442, 471, 479, 481 Uchida, S., see Hudson, E.W. 481 Uchida, S., see Ichikawa, N. 452, 454, 455 Uchida, S., see Ino, A. 455 Uchida, S., see Kojima, K.M. 449 Uchida, S., see Lang, K.M. 441, 480 Uchida, S., see Lanzara, A. 409, 427, 454, 482 Uchida, S., see Nachumi, B. 449, 450 Uchida, S., see Niemöller, T. 454 Uchida, S., see Pan, S.H. 442, 481 Uchida, S., see Pellegrin, E. 463 Uchida, S., see Tranquada, J.M. 388, 396, 439, 443–446, 451, 455, 458, 459 Uchida, S., see Wakimoto, S. 445 Uchida, S., see Zhou, X.J. 441, 454 Uchida, S., see Zimmermann, M.v. 454 Uchida, S.-I., see Noda, T. 454 Uchida, S.I., see Chuang, Y.-D. 482 Uchiyama, S. 139 Uchiyama, S., see Takayasu, M. 139 Uchiyama, T. 550, 552, 553 Uchiyama, T., see Kanno, T. 551

638

AUTHOR INDEX

Uchiyama, T., see Kusumoto, D. 556 Uchiyama, T., see Mohri, K. 532, 550, 551, 553, 555 Uchiyama, T., see Shen, L.P. 555 Uchiyama, T., see Takayama, A. 552 Udovic, T.J., see Giebułtowicz, T.M. 353 Udovic, T.J., see Hjörvarsson, B. 206, 219, 220, 248 Ueda, K. 311 Ueda, S., see Imada, S. 310 Ueda, Y., see Imai, T. 474 Ueda, Y., see Tamura, H. 456 Uefuji, T., see Singley, E.J. 466 Uehara, Y. 61 Uehara, Y., see Hong, J. 38 Ueki, S., see Wakimoto, S. 447, 448 Ueki, S., see Yamada, K. 447, 448 Uematsu, Y., see Kanai, H. 33, 77 Uemura, Y.J., see Kojima, K.M. 449 Uemura, Y.J., see Nachumi, B. 449, 450 Uemura, Y.J., see Savici, A.T. 449 Uenhara, Y., see Kanai, H. 33, 77 Ueno, H., see Kanai, H. 33, 77 Ueno, M. 34, 42, 136 Ueno, M., see Nagai, H. 45 Ueno, S., see Kawashima, K. 540, 557 Uesugi, T., see Tsuchiya, Y. 36 Ueta, A.Y. 323, 324, 326, 327 Ueta, A.Y., see Abramof, E. 326, 327 Ueta, Y., see Geist, F. 323–325 Ueta, Y., see Krenn, H. 312, 323–326 Ueta, Y., see Prinz, A. 323 Ueta, Y., see Yuan, S. 323, 324, 327 Uhlig, W.C. 36, 48, 49 Ujsaghy, O. 252 Ulmer-Tufﬁgo, H. 298 Ulmer-Tufﬁgo, H., see Lawrence, I. 305 Ulrich, C., see He, H. 411, 426, 449, 482 Ulrich, C., see Sidis, Y. 419, 470 Umehara, T., see Takayama, A. 552 Underwood, J.H., see Lawniczak-Jabło´nska, K. 345 Uneyama, K., see Takahashi, M. 69, 158 Uneyama, K., see Tsunoda, M. 167 Unguris, J. 207, 221 Unguris, J., see Pierce, D.T. 219, 228 Uozumi, T. 207 Uraltsev, I.N., see Ivchenko, E.L. 297 Uraltsev, I.N., see Pozina, G.R. 297 Uraltsev, I.N., see Yakovlev, D.R. 300 Urban, P. 323, 325, 329 Urban, R., see Heinrich, B. 207, 221, 222 Urquhart, K.B., see Heinrich, B. 230 Usadel, K.D., see Keller, J. 155

Usadel, K.D., see Miltenyi, P. 155 Usadel, K.D., see Nowak, U. 155 Ushiyama, T., see Kitoh, T. 542–544 Ushiyama, T., see Panina, L.V. 504, 510, 513, 515, 521–523, 525 Usof, Z., see Johnson, P.D. 482 Usov, N. 515, 525–528, 530 Usov, N., see Antonov, A. 505 Ustinov, V., see Lauter-Pasyuk, V. 224 Uzdin, S. 241, 271, 272 Uzdin, V. 207 Uzdin, V., see Uzdin, S. 241, 271, 272 Uzdin, V.M. 206, 214, 217, 222–224, 249, 270, 273 Uzdin, V.M., see Kazansky, A.K. 214 Uzdin, V.M., see Kudasov, Yu.B. 209 Uzdin, V.M., see Ostanin, S. 248, 249 Uzdin, V.M., see Yartseva, N.S. 204 Vaidya, S., see Mao, S. 69 Vajda, S. 255 Vajk, O.P. 463 Valeiko, M.V. 327 Valenzuela, R. 510, 537, 549, 552, 553 Valenzuela, R., see Betancourt, I. 536 Valenzuela, R., see Carrasco, E. 537 Valenzuela, R., see Garcia, K.L. 508, 540 Valenzuela, R., see Sánchez, M.L. 537 Valenzuela, R., see Vázquez, M. 502, 553 Valeri, S., see Gazzadi, G.C. 231 Valeri, S., see Luches, P. 231 Valet, T. 86 Valet, T., see Fert, A. 104 Valet, T., see Galtier, P. 67 Valet, T., see Jacquet, J.C. 63 Valet, T., see Jérome, R. 67 Validov, A., see Kataev, V. 391, 453 Valla, T., see Johnson, P.D. 482 Valletta, R.M. 140 Vallin, J.T. 346 van Bemmel, H.J.M. 406 van Bentum, P.J.M., see Boonman, M.E.J. 346 Van de Riet, E. 131 Van de Veerdonk, R.J.M. 134 van de Vin, C.H., see LeClair, P. 341 van de Walle, G.F.A., see Coehoorn, R. 23 van den Berg, H., see Vieth, M. 25 van den Berg, H.A.M. 42 van den Berg, H.A.M., see Boeve, H. 57 van den Berg, H.A.M., see Gerrits, Th. 131, 133 van den Berg, H.A.M., see Manders, F. 45 van den Broek, J.J., see Lenssen, K.-M.H. 23, 33, 43, 51, 59, 60, 166

AUTHOR INDEX van der Heijden, P.A.A. 151, 157, 163 van der Heijden, P.A.A., see Zeltser, A.M. 159 van der Heijden, P.A.A., see Zhang, Y.B. 60, 76, 161, 170 van der Klink, J.J. 430, 436 van der Klink, J.J., see Burnet, S. 252, 275 van der Linden, P.J.E.M., see Dybko, K. 360 van der Marel, D. 442 van der Marel, D., see Molegraaf, H.J.A. 479 van der Marel, D., see Van Wees, B.J. 258 van der Rijt, R.A.F., see Lenssen, K.-M.H. 23, 33, 43, 51, 59, 60, 166 van der Zaag, P.J. 155 van der Zaag, P.J., see van der Heijden, P.A.A. 151, 157, 163 van Driel, J. 63, 64, 151, 152, 158, 159, 167, 176 van Driel, J., see Lenssen, K.-M.H. 43 van Duin, C.N.A. 416, 417 van Est, J.W., see Gijs, M.A.M. 135 van Houten, H., see Van Wees, B.J. 258 Van Kampen, M., see Koopmans, B. 301 van Kempen, H., see Bischoff, M.M.J. 219, 225, 228 van Kempen, H., see Fang, C.M. 253 van Kesteren, H.W., see Bloemen, P.J.H. 45 van Leeuwen, J.M.J., see du Croo de Jongh, M.S.L. 401 van Leeuwen, J.M.J., see van Bemmel, H.J.M. 406 van Loon, A., see Lenssen, K.-M.H. 23, 33, 43, 51, 59, 60, 166 van Peppen, J.C.L., see Xiao, M. 135 van Saarloos, W., see Bosch, M. 396, 404, 407 van Saarloos, W., see du Croo de Jongh, M.S.L. 401 van Saarloos, W., see Eskes, H. 404 van Saarloos, W., see van Bemmel, H.J.M. 406 van Saarloos, W., see Zaanen, J. 404, 407 van Schilfgaarde, M. 233, 269 van Schilfgaarde, M., see Herman, F. 207, 221 Van Wees, B.J. 258 van Zon, J.B.A.D., see Lenssen, K.-M.H. 22, 23, 44, 166 Vandamme, L.K.J., see Gijs, M.A.M. 135 Vanhelmont, F. 49 Vanhelmont, F.W.M., see Gogol, P. 155 Varga, L. 131, 132 Varga, L., see Nagasaka, K. 13, 150, 162 Varga, P., see Biedermann, A. 217 Varga, P., see Bischoff, M.M.J. 219, 225 Varma, C.M. 418, 419 Varma, C.M., see Batlogg, B. 386 Varma, C.M., see Kaminski, A. 419

639

Vas’kovkiy, V.O., see Kurlyandskaya, G.V. 504, 534, 537 Vaudaine, M.-H., see Lhermet, H. 135 Vavra, I., see Springholz, G. 328, 342 Vavra, W., see Gadbois, J. 58 Vaz, C.A.F., see Bland, J.A.C. 207, 276 Vaz, C.A.F., see Li, S.P. 276 Vázquez, M. 502, 504, 505, 512, 532–535, 541, 553, 556, 557 Vázquez, M., see Barandiarán, J.M. 504 Vázquez, M., see Chen, D.-X. 522 Vázquez, M., see García, J.M. 505, 534, 542, 549 Vázquez, M., see Garcia, K.L. 540 Vázquez, M., see García-Miquel, H. 515, 538, 548, 549 Vázquez, M., see Gómez-Polo, C. 537, 542–544 Vázquez, M., see Hernando, B. 540 Vázquez, M., see Kim, C.G. 538 Vázquez, M., see Knobel, M. 504, 511, 532, 536, 538, 539, 549 Vázquez, M., see Kraus, L. 508, 545 Vázquez, M., see Kurlyandskaya, G.V. 504, 537 Vázquez, M., see Loﬂand, S.E. 514, 546 Vázquez, M., see Mandal, K. 538 Vázquez, M., see Óvári, T.A. 534, 546 Vázquez, M., see Ramos, C. 548 Vázquez, M., see Raposo, V. 510 Vázquez, M., see Sánchez, M.L. 537, 538 Vázquez, M., see Sinnecker, J.P. 504, 534, 535, 541 Vázquez, M., see Song, S.-H. 542–544 Vázquez, M., see Tejedor, M. 504, 538, 539, 553, 557 Vázquez, M., see Valenzuela, R. 537, 549, 552, 553 Vázquez, M., see Velázquez, J. 508, 544 Vázquez de Parga, A.L., see Blum, V. 238 Vázquez de Parga, A.L., see Gómez, L. 269 Vedyaev, A. 94, 103, 109 Vedyaev, A., see Dieny, B. 66, 94, 95, 99, 103, 104 Vedyaev, A.V., see Granovskii, A.B. 62, 104 Veenstra, K.J., see Gerrits, Th. 131, 133 Veenstra, K.J., see Manders, F. 45 Vega, A. 207, 209, 218, 221–223, 225–228, 255, 267 Vega, A., see Aguilera-Granja, F. 215, 219, 254–258 Vega, A., see Bouarab, S. 254–256, 258, 266, 267 Vega, A., see Elmouhssine, O. 228 Vega, A., see Izquierdo, J. 213, 226, 227, 269, 277

640

AUTHOR INDEX

Vega, A., see Mokrani, A. 209 Vega, A., see Nolting, W. 209 Vega, A., see Pruneda, J.M. 245–248 Vega, A., see Robles, R. 204, 206, 214–217, 225, 245, 259, 260, 266–270, 275 Vega, S., see Gavish, M. 357 Veillet, P., see Dupas, C. 9 Veillet, P., see Renard, J.-P. 86 Veith, M., see van den Berg, H.A.M. 42 Velázquez, J. 508, 544 Velázquez, J., see Knobel, M. 538 Velázquez, J., see Vázquez, M. 504 Velikov, Yu.Kh., see Ponomareva, A.V. 224 Veloso, A. 34, 35, 45, 48, 49, 159 Veloso, A., see Sousa, J.B. 37 Veloso, A., see Ventura, J.O. 49 Vélu, E., see Dupas, C. 9 Vélu, E., see Renard, J.-P. 86 Venkataramani, N., see Acharya, B.R. 231 Venkateswaran, U.D., see Bak, J. 354 Vennix, C.W.H.M. 315, 320 Vennix, C.W.H.M., see Eggenkamp, P.J.T. 320 Ventura, J.O. 49 Ventura, J.O., see Sousa, J.B. 37 Verbanck, G., see Potter, C.D. 110 Verbanck, G., see Schad, R. 6, 27, 79 Verges, J.A. 395 Verges, J.A., see Louis, E. 397 Verheijden, M.A., see van der Zaag, P.J. 155 Vermeulen, J.L., see Acher, O. 514 Vernes, A., see Banhart, J. 80, 82, 83 Vernes, A., see Ebert, H. 81 Verschueren, G.L.J., see Swagten, H. 97, 98 Verteletskii, P.V., see Akimov, B.A. 312 Vescovo, E., see Kachel, T. 236 Vettier, C., see Bourges, P. 389, 471, 472 Vettier, C., see Rossat-Mignod, J. 385, 389, 438, 468, 470, 471 Vevtukov, R., see Hegde, H. 70, 71 Victora, R.H. 203 Victora, R.H., see Parlebas, J.C. 204 Viegas, A., see da Silva, R.B. 548 Viegas, A., see Knobel, M. 502, 504, 506 Viegas, A.D.C. 514 Viertio, H.E. 396 Vieth, M. 25 Vietkin, A., see Huh, Y.M. 453 Vietkin, A., see Julien, M.-H. 452 Vieux-Rochaz, L., see Lhermet, H. 135 Vigliante, A. 463 Vigliante, A., see Zimmermann, M.v. 454 Vila, L., see Encinas, A. 548 Vinai, F., see Appino, C. 540 Vinai, F., see Brunetti, L. 504, 532

Vinattieri, A., see Roussignol, Ph. 305 Vinokur, V.M., see Feldman, D.E. 277 Vitale, S., see Maraner, A. 135 Vitek, V., see Friak, M. 218 Vithayathil, J.P., see MacLaughlin, D.E. 433 Vitos, L. 275 Vladimirova, M.R., see Kavokin, A.V. 299 Vogel, J. 219 Vogel, J., see Camarero, J. 178 Vogel, J., see Cros, V. 235 Vögeli, B., see Castaño, F.J. 58 Vogt, T., see Katti, R.R. 57, 58 Voisen, P., see Kudelski, A. 298 Vojta, M. 399–401, 421, 422 Vojta, M., see Martins, G.B. 406 Vojta, M., see Polkovnikov, A. 413, 420, 478, 481 Vojta, M., see Sachdev, S. 391, 414 Vollhardt, D., see Byczuk, K. 209 Vollhardt, D., see Metzner, W. 397 Vollmer, R., see Wu, Y.Z. 218, 249 Volobuev, V., see Kepa, H. 336, 338, 340 Volobuev, V.V., see Chernyshova, M. 332, 338 Volobuev, V.V., see Kowalczyk, L. 338, 340, 341 von Barth, U. 216 von Känel, H., see Fanciulli, M. 245 von Löhneysen, H., see Wosnitza, J. 336 von Molnár, S. 331, 342, 343 von Molnár, S., see Wolf, S.A. 5, 208 von Ortenberg, M. 294, 358 von Ortenberg, M., see Hillberg, M. 361 von Ortenberg, M., see Krevet, R. 346 von Ortenberg, M., see Mac, W. 346 von Ortenberg, M., see Portugall, O. 361 von Ortenberg, M., see Schikora, D. 361 von Ortenberg, M., see Stolpe, I. 340, 341 von Ortenberg, M., see Szuszkiewicz, W. 361 von Ortenberg, M., see Widmer, T. 361 von Ortenberg, M., see Widmer, Th. 361 von Truchsess, V., see Buda, B. 300 von Zimmermann, M. 454 von Zimmermann, M., see Vigliante, A. 463 Voos, M., see Brum, J.A. 296, 297 Vorderwisch, D.F., see Lake, B. 413, 446 Vos, K.J.E., see Borsa, F. 433, 449 Voss, M., see Seewald, G. 237 Voter, A.F. 266 Voter, A.F., see Liu, C.L. 267 Vouille, C. 84 Vu, T.Q. 353, 355 Vu, T.Q., see Dahl, M. 353 Vu, T.Q., see Foner, S. 353, 355 Vu, T.Q., see Fries, T. 357

AUTHOR INDEX Vu, T.Q., see Gennser, U. 351, 355 Vu, T.Q., see Kostyk, D. 342, 343 Vu, T.Q., see Liu, X.C. 355 Vu, T.Q., see Shapira, Y. 353 Vu, T.Q., see Shih, O.W. 351, 352 Vygranenko, Yu.K., see Grodzicka, E. 312 Vygranenko, Yu.K., see Skipetrov, E.P. 312 Waag, A. 294 Waag, A., see Buda, B. 300 Waag, A., see Dahl, M. 311 Waag, A., see Fiederling, R. 304 Waag, A., see Ivchenko, E.L. 297 Waag, A., see Keller, D. 306 Waag, A., see Koch, M. 301 Waag, A., see König, B. 294 Waag, A., see Kuhn-Heinrich, B. 296 Waag, A., see Kulakovskii, V.D. 306 Waag, A., see Kutrowski, M. 296 Waag, A., see Mackh, G. 300, 303 Waag, A., see Merkulov, I.A. 303 Waag, A., see Pozina, G.R. 297 Waag, A., see Reshina, I.I. 309 Waag, A., see Sawicki, M. 311 Waag, A., see Shcherbakov, A.V. 306 Waag, A., see Tönnies, D. 296 Waag, A., see Wojtowicz, T. 302 Waag, A., see Yakovlev, D.R. 297–300, 302, 306, 310 Wachter, P. 331, 342, 343 Wachter, P., see Pashkevich, Yu.G. 461, 463 Wada, J., see Yamada, K. 447, 448 Wada, S. 460 Wada, S., see Furukawa, Y. 460 Wadley, H.N.G., see Zhou, X.W. 69 Wagener, W., see Hillberg, M. 361 Wagener, W., see Klauss, H.-H. 438, 439, 450 Wagner, H., see Mermin, N.D. 203 Wahl, P., see Knorr, N. 251 Wakabayashi, T., see Uchiyama, T. 550 Wakimoto, S. 445, 447, 448 Wakimoto, S., see Fujita, M. 447, 448 Wakimoto, S., see Lee, Y.S. 448, 449 Wakimoto, S., see Yamada, K. 447, 448 Waldmann, H. 361 Wales, D.J., see Vajda, S. 255 Walf, H., see Klauss, H.-H. 438, 439, 450 Wall, M.A., see Chaiken, A. 220, 244, 245 Walstedt, R.E. 432 Walstedt, R.E., see Warren Jr., W.W. 474 Walter, J., see Suzuki, M. 205, 234, 275 Walterfang, M., see Uzdin, V.M. 222, 249 Walton, R.M., see Prados, C. 261

641

Walz, U., see Wosnitza, J. 336 Wang, B., see Zhao, Z. 265 Wang, C.S. 217, 231 Wang, D. 27, 148 Wang, D., see Everitt, B.A. 163 Wang, G., see Li, Z. 135 Wang, H., see Mao, S. 60 Wang, H., see Yang, Z. 162, 178 Wang, H., see Zhao, Z. 265 Wang, J., see Hegde, H. 70, 71 Wang, J., see Mao, M. 32, 33, 158 Wang, J., see Mao, S. 7 Wang, J.S., see Wang, Y.H. 170 Wang, P.K., see Tang, D.D. 22, 24, 58, 128 Wang, S.-X. 42, 71 Wang, S.X., see Bailey, W.E. 111 Wang, Y., see Li, D. 540 Wang, Y., see Nicholson, D.M.C. 80 Wang, Y., see Oparin, A.B. 85 Wang, Y., see Perdew, J.P. 245 Wang, Y.H. 170 Wang, Z., see Li, H. 36, 159, 167 Wang, Z.-P., see Zhu, Z.-M. 296 Wang, Z.D., see Sheng, L. 94 Wang, Z.H., see Chen, J.J. 343 Wang, Z.Q., see Lu, S.H. 231 Wäppling, R., see Broddefalk, A. 228 Wäppling, R., see Kalska, B. 227, 231 Wäppling, R., see Lindner, J. 228 Warmenbol, P., see Swagten, H.J.M. 315 Warnock, J. 294, 358 Warnock, J., see Chou, W.C. 294, 358, 359 Warnock, J., see Fu, L.P. 358 Warnock, J., see Jonker, B.T. 294, 301, 358 Warnock, J., see Liu, X. 294 Warnock, J., see Yu, W.Y. 297, 358 Warren Jr., W.W. 474 Wasiela, A. 297 Wasiela, A., see Ferrand, D. 310, 311 Wasiela, A., see Gaj, J.A. 296 Wasiela, A., see Grieshaber, W. 296, 299 Wasiela, A., see Haury, A. 311 Wasiela, A., see Kossacki, P. 302 Wasiela, A., see Lawrence, I. 305 Wasiela, A., see Leisching, P. 299 Wasiela, A., see Marsal, L. 307 Wasiela, A., see Peyla, P. 298 Wasik, D. 304 Watanabe, I. 450, 479, 481 Watanabe, I., see Koike, Y. 448 Watanabe, K., see Hamakawa, Y. 160, 168 Watanabe, K., see Meguro, K. 33, 123 Watanabe, N., see Tsunekawa, K. 38, 48, 170, 178

642

AUTHOR INDEX

Watanabe, T., see Hjörvarsson, B. 206, 219, 220, 248 Watanabe, T., see Saito, M. 162, 170 Waters, G., see Hylton, T.L. 70 Webb, P., see Beach, R.S. 124 Webb, W.E., see Parker, M.R. 115, 122 Weber, S.E. 251 Wecker, J., see Boeve, H. 57 Wecker, J., see van den Berg, H.A.M. 42 Wecker, J., see Vieth, M. 25 Weckesser, J., see Lin, N. 275 Weger, M. 438 Wei, P., see Veloso, A. 35, 48, 49 Wei, Y. 22 Weidiger, A., see Klose, F. 248 Weihong, Z., see Sushkov, O.P. 401 Weinberger, P., see Blaas, C. 80, 109 Weinberger, P., see Kudrnovsky, J. 143 Weinberger, P., see Lazarovits, B. 209, 238 Weinberger, P., see Turek, I. 221, 222 Weinert, M. 208 Weinert, M., see Freeman, A.J. 208 Weinert, M., see Ohnishi, S. 218 Weinert, M., see Wimmer, E. 211 Weissman, M.B., see Hardner, H.T. 135 Weissman, M.B., see Petta, J.R. 135 Weissmann, M., see Guevara, J. 254, 257 Wells, B.O., see Dessau, D.S. 427 Wells, B.O., see Johnson, P.D. 482 Wells, B.O., see King, D.M. 464 Wells, B.O., see Lee, Y.S. 486 Welsch, M.K. 307 Welsch, M.K., see Bacher, G. 309 Welsch, M.K., see Zaitsev, S. 309 Wen, X.G. 401, 420 Wen, X.G., see Schrieffer, J.R. 392, 393 Wende, H., see Lindner, J. 228 Wende, H., see Scherz, A. 227, 228 Wende, H., see Wilhelm, F. 209, 220, 237 Werckmann, J., see Lefakis, H. 67 Wergert, W., see Levanov, N.A. 215 Westerholt, K., see Hübener, M. 276 Westerholt, K., see Labergie, D. 249 Westerholt, K., see Uzdin, V.M. 249 Westfahl, H. 408 Weston, S.J. 296 Weston, S.J., see Pier, Th. 305 Weyer, G., see Fanciulli, M. 245 Wheatley, J.M., see Lercher, M.J. 420 White, P.J., see Shen, Z.-X. 479, 482 White, R., see Wei, Y. 22 White, R.L., see Lai, C.-H. 163 White, R.L., see Mancoff, F.B. 25 White, R.M., see Xi, H. 118, 150, 154, 160, 167

White, S.R. 405–407 White, S.R., see Bickers, N.E. 427 White, S.R., see Chernyshev, A.L. 404 White, S.R., see Kampf, A.P. 409 White, S.R., see Scalapino, D.J. 391, 428 Whitman, L.J., see Edelstein, R.L. 24 Whitman, L.J., see Miller, M.M. 24 Widmer, T. 361 Widmer, T., see Portugall, O. 361 Widmer, Th. 361 Widmer, Th., see Schikora, D. 361 Wiedmann, M., see Duvail, J.L. 12, 61, 94, 104 Wiesendanger, R. 206, 208, 251 Wilamowski, Z. 343 Wilamowski, Z., see Grodzicka, E. 312 Wilamowski, Z., see Story, T. 312 Wilczek, F., see Chernyshyov, O. 401 Wilczek, F., see Nayak, C. 396, 404 Wildberger, K., see Nonas, B. 222 Wildberger, K., see Stepanyuk, V.S. 212, 253 Wildberger, K., see Weber, S.E. 251 Wilhelm, F. 209, 220, 237 Wilhelm, F., see Kappler, J.-P. 275 Wilhelm, F., see Lindner, J. 228 Wilhoit, D., see Baril, L. 141 Wilhoit, D.R., see Dieny, B. 4, 9, 16, 30, 31, 39, 60, 94, 157, 178 Wilhoit, D.R., see Gurney, B.A. 22, 31–33, 40, 41, 97, 129, 140 Wilhoit, D.R., see Mamin, H.J. 22, 25, 141 Wilhoit, D.R., see Speriosu, V.S. 45 Willekens, M.M., see Strijkers, G.J. 97, 100, 101 Willekens, M.M.H. 89, 91, 97, 98, 100 Willekens, M.M.H., see Grodzicka, E. 312 Willekens, M.M.H., see Litvinov, V.I. 103 Willekens, M.M.H., see Swagten, H. 97, 98 Willekens, M.M.H., see Swagten, H.J.M. 34, 46, 47, 52, 92, 93 Williams, A.R., see Kübler, J. 216 Williams, G.V.M. 475, 476 Williams, G.V.M., see Tallon, J.L. 478 Williams, H., see Isaacs, E.D. 462 Williams, M.L., see Heim, D.E. 127, 129 Williams, M.L., see Tsang, C. 22, 127 Williams, M.L., see Tsang, C.H. 4, 5, 22 Williams, S.A., see Duffy, D.M. 262 Williamson, F., see Fayﬁeld, R. 24 Williamson, J.G., see Van Wees, B.J. 258 Willis, R.F., see Himpsel, F.J. 207 Wills, J., see Eriksson, O. 209 Wills, J.M., see Ostanin, S. 248, 249 Wills, P., see Nozières, J.P. 158, 161, 162, 168–170

AUTHOR INDEX Wilson, R.G., see Theodoropoulou, N. 276 Wilson, R.J., see Childress, J.R. 32, 41, 67, 159, 174 Wilson, W.L., see Stokes, S.W. 135 Wimmer, E. 211 Wingreen, N.S., see Madhavan, V. 252 Winter, B.J., see Klots, T.D. 255 Winter, H., see Igel, T. 218, 227, 229 Winter, H., see Pfandzelter, R. 249 Winter, I., see Lawniczak-Jabło´nska, K. 345 Winter, J. 430 Wise, P., see Twardowski, A. 351 Wissmann, H., see Szuszkiewicz, W. 361 Witkowska, B., see Grodzicka, E. 312 Witkowska, B., see Iwanowski, R.J. 315 Witkowska, B., see Łusakowski, A. 318 Witkowska, B., see Radchenko, M.V. 315 Witkowska, B., see Story, T. 312, 328 Wittlin, A., see Boonman, M.E.J. 346 Wittlin, A., see Dybko, K. 360 Wittlin, A., see Mac, W. 346 Wittlin, A., see Zeitler, U. 360 Wochner, P. 458 Wochner, P., see Tranquada, J.M. 388, 458 Wochner, P., see Zimmermann, M.v. 454 Wohlfarth, E.P., see Stoner, E.C. 115 Wojtowicz, T. 295, 296, 302, 311, 349 Wojtowicz, T., see Andrearczyk, T. 304 Wojtowicz, T., see Cywi´nski, G. 307, 310 Wojtowicz, T., see Dahl, M. 311 Wojtowicz, T., see Dietl, T. 304 Wojtowicz, T., see Hennion, B. 311 Wojtowicz, T., see Imanaka, Y. 302, 303 Wojtowicz, T., see Janik, E. 295 Wojtowicz, T., see Jaroszy´nski, J. 304 Wojtowicz, T., see Jouanne, M. 311 Wojtowicz, T., see Jusserand, B. 303 Wojtowicz, T., see Karczewski, G. 304 Wojtowicz, T., see Kochereshko, V.P. 302 Wojtowicz, T., see König, B. 306 Wojtowicz, T., see Kossacki, P. 296, 344, 345 Wojtowicz, T., see Kossut, J. 310 Wojtowicz, T., see Kudelski, A. 310 Wojtowicz, T., see Kunimatsu, H. 302 Wojtowicz, T., see Kusrayev, Yu.G. 298 Wojtowicz, T., see Kutrowski, M. 295, 296 Wojtowicz, T., see Lemaître, A. 296 Wojtowicz, T., see Mackh, G. 300 Wojtowicz, T., see Ma´ckowski, S. 308 Wojtowicz, T., see Maslana, W. 300 Wojtowicz, T., see Merkulov, I.A. 303 Wojtowicz, T., see Mino, H. 298, 303 Wojtowicz, T., see Ossau, W. 302 Wojtowicz, T., see Pulizzi, F. 298

643

Wojtowicz, T., see Sawicki, M. 310, 311 Wojtowicz, T., see Semenov, Yu.G. 296 Wojtowicz, T., see Shcherbakov, A.V. 306 Wojtowicz, T., see Stachow-Wójcik, A. 310 Wojtowicz, T., see Stirner, T. 300 Wojtowicz, T., see Szuszkiewicz, W. 311 Wojtowicz, T., see Takeyama, S. 300, 302, 303 Wojtowicz, T., see Teppe, F. 300, 306 Wojtowicz, T., see Wasik, D. 304 Wojtowicz, T., see Wypior, G. 296 Wojtowicz, T., see Yakovlev, D.R. 299, 310 Wojtowicz, T., see Yokoi, H. 302, 303 Wold, A., see Dahl, M. 353 Wold, A., see Fries, T. 357 Wold, A., see Shapira, Y. 353 Wold, A., see Shih, O.W. 351 Wold, A., see Vu, T.Q. 353, 355 Wolf, R.M., see van der Zaag, P.J. 155 Wolf, S., see Vajda, S. 255 Wolf, S.A. 5, 208 Wolff, P.A., see Story, T. 315 Wolfman, J., see Prieto, J.L. 22 Wolters, C., see Górska, M. 329 Wolverson, D., see Klar, P.J. 296, 308 Wolynes, P.G., see Westfahl, H. 408 Won, C., see Wu, Y.Z. 230 Won, H., see Maki, K. 418 Wong, B.Y. 170, 171 Wong, H.K., see Jaggi, N.K. 225 Wong, T.M., see Moodera, J.S. 5 Wood, R. 180 Woods, R.J., see Feit, Z. 328 Woolley, J.C., see Street, R. 175 Wortmann, D. 204 Wortmann, D., see Heinze, S. 277 Wosnitza, J. 336 Wöste, L.W., see Vajda, S. 255 Woynes, P., see Laughlin, R.B. 390 Wróbel, J., see Andrearczyk, T. 304 Wróbel, J., see Cywi´nski, G. 310 Wróbel, J., see Dietl, T. 304 Wróbel, J., see Jaroszy´nski, J. 304 Wróbel, J., see Kossut, J. 310 Wróbel, J., see Kudelski, A. 310 Wróbel, J., see Ma´ckowski, S. 309 Wrotek, S., see Chernyshova, M. 332, 338 Wu, J.-W., see Chang, S.-K. 296 Wu, R. 208, 209, 228, 230 Wu, R., see Chen, L. 263 Wu, R., see Trioni, M.I. 253 Wu, S.Y. 215 Wu, X.W. 41 Wu, Y. 57

644

AUTHOR INDEX

Wu, Y., see Li, K. 37, 38, 45, 49, 50, 92, 145, 146, 159, 167 Wu, Y.H., see Guo, Z.B. 159 Wu, Y.Z. 218, 230, 249 Wun-Fogle, M., see Restorf, J.B. 163 Wun-Fogle, M., see Velázquez, J. 544 Wypior, G. 296 Xavier, J.C., see Martins, G.B. 406 Xi, H. 118, 150, 154, 160, 167 Xia, J.B., see Chang, K. 306 Xiao, G., see Lu, Y. 5 Xiao, J.Q. 9 Xiao, J.Q., see Li, Y.F. 176 Xiao, M. 135 Xiao, S.Q. 541 Xie, H., see Li, K. 37, 49, 50, 145, 146, 159, 167 Xin, S.H., see Furdyna, J.K. 344, 345 Xin, Y., see Kim, C.S. 309, 310 Xing, D.Y., see Sheng, L. 94 Xing, G., see Rau, C. 224, 238 Xiong, W., see Kools, J.C.S. 37, 49, 71 Xu, D., see Yu, J.Q. 504 Xu, J. 327 Xu, M. 94 Xu, Q.Y., see Shen, F. 38, 49 Yagami, K. 158, 159 Yagami, K., see Tsunoda, M. 167 Yagi, H., see Inoue, M. 315 Yagi, T., see Kanno, T. 551 Yakabchuk, H., see Kurlyandskaya, G.V. 503, 537 Yakmi, J.V., see Hermann, A.M. 479 Yakovenko, V.M., see Mazin, I.I. 418, 426 Yakovlev, D.R. 297–300, 302, 306, 310 Yakovlev, D.R., see Akimov, A.V. 299 Yakovlev, D.R., see Ivchenko, E.L. 297 Yakovlev, D.R., see Keller, D. 306 Yakovlev, D.R., see Koch, M. 301 Yakovlev, D.R., see Kochereshko, V.P. 302 Yakovlev, D.R., see König, B. 294, 306 Yakovlev, D.R., see Kulakovskii, V.D. 306 Yakovlev, D.R., see Kutrowski, M. 295, 296 Yakovlev, D.R., see Mackh, G. 300 Yakovlev, D.R., see Merkulov, I.A. 303 Yakovlev, D.R., see Ossau, W. 302 Yakovlev, D.R., see Pozina, G.R. 297 Yakovlev, D.R., see Shcherbakov, A.V. 306 Yakovlev, D.R., see Waag, A. 294 Yakovlev, D.R., see Wojtowicz, T. 302 Yakunin, A.M., see Antonov, A. 537 Yakunin, A.M., see García, J.M. 542, 549

Yamada, K. 447, 448, 466 Yamada, K., see Fujikata, J. 164 Yamada, K., see Fujita, M. 447–449 Yamada, K., see Hosoya, S. 456 Yamada, K., see Kanai, H. 32, 41, 55, 157 Yamada, K., see Katano, S. 446 Yamada, K., see Kimura, H. 446 Yamada, K., see Koike, Y. 448 Yamada, K., see Lee, S.-H. 404, 459 Yamada, K., see Lee, Y.S. 448, 449 Yamada, K., see Matsuda, M. 466, 467 Yamada, K., see Nakajima, K. 485, 486 Yamada, K., see Sato, T. 464 Yamada, K., see Singley, E.J. 466 Yamada, K., see Suzuki, T. 445 Yamada, K., see Uehara, Y. 61 Yamada, K., see Wakimoto, S. 447, 448 Yamada, K., see Wang, S.-X. 42 Yamada, T., see Bischoff, M.M.J. 219, 228 Yamada, Y., see Wada, S. 460 Yamadera, H., see Morikawa, T. 504, 515, 552, 557 Yamagata, S., see Saito, H. 311 Yamagishi, Y., see Kanai, H. 33, 77 Yamaguchi, M. 552 Yamaguchi, T., see Iguchi, I. 453 Yamaguchi, Y., see Chuang, Y.-D. 482 Yamaguchi, Y., see Suzuki, T. 445 Yamaji, K., see Yanagisawa, T. 397 Yamakawa, I., see Kossut, J. 310 Yamamoto, H. 9, 27 Yamamoto, H., see Fujikata, J. 151, 157, 161–164, 174, 175 Yamamoto, H., see Sato, H. 62 Yamamoto, H., see Shinjo, T. 9, 12, 27, 55 Yamamoto, K. 460 Yamamoto, M., see Mohri, K. 532, 550, 551, 555 Yamamoto, M., see Taniguchi, S. 136 Yamamoto, S.Y., see Cowache, C. 164 Yamamoto, S.Y., see Dieny, B. 60, 61, 92, 106 Yamamoto, Y., see Hori, H. 276 Yamanaka, N., see Takano, K. 127 Yamane, H. 22 Yamano, K., see Fujita, M. 41 Yamauchi, H., see Sasao, K. 166 Yamauchi, R., see Sasao, K. 166 Yamazaki, T., see Adachi, N. 349 Yan, X. 22 Yan, X., see Nie, H.B. 504, 541 Yanagisawa, J., see Imada, S. 310 Yanagisawa, T. 397 Yanase, A., see Hamada, H. 225, 226 Yang, C.-K. 266 Yang, D.L., see Smith, N. 135

AUTHOR INDEX Yang, D.P., see Zhao, Z.J. 532 Yang, D.X., see Chopra, H.D. 144, 164 Yang, F., see Bindilatti, V. 355 Yang, F.Y., see Strijkers, G.J. 123 Yang, G., see Borisenko, S.V. 441 Yang, G., see Furdyna, J.K. 344, 345 Yang, G.L., see Dai, N. 297 Yang, G.L., see Syed, M. 297 Yang, H. 249 Yang, H.-S., see Kim, Y.-K. 76, 162 Yang, Q. 78 Yang, Q., see Pratt, W.P. 78, 85 Yang, T. 161 Yang, X.L., see Zeng, L. 536, 549 Yang, X.L., see Zhao, Z.J. 532 Yang, Z. 162, 178 Yang, Z., see Trioni, M.I. 253 Yaoi, T., see Sugawara, N. 61 Yartseva, N.S. 204 Yartseva, N.S., see Uzdin, V. 207 Yartseva, N.S., see Uzdin, V.M. 206, 214, 217, 223, 224, 249, 270, 273 Yasuhira, T. 303 Yasuoka, H., see Imai, T. 474 Yasuoka, H., see Takanashi, K. 225 Ye, W., see Kools, J.C.S. 37, 49, 71 Yeadon, M., see Zimmermann, C.G. 268 Yelon, A. 508, 513–516, 521, 522, 525, 528 Yelon, A., see Britel, M.R. 516, 526, 548, 549 Yelon, A., see Ciureanu, P. 504, 514, 515, 542, 546, 549 Yelon, A., see Duque, J.G.S. 536 Yelon, A., see Melo, L.G. 535 Yelon, A., see Ménard, D. 514–516, 526, 529, 530, 546, 548, 549 Yethiraj, M., see McQueeney, R.J. 409 Yethiraj, M., see Mook, H.A. 389, 471 Yi, J.H., see Rhee, J.-R. 161 Yi, Y.S. 409 Yi, Y.S., see McQueeney, R.J. 409 Yildirim, T., see Bagci, V.M.K. 266 Yin, L. 426 Yiu, V., see Uhlig, W.C. 36, 48, 49 Yoda, H. 22 Yoda, H., see Kamiguchi, Y. 35, 48, 49, 77 Yoda, H., see Ohsawa, Y. 26 Yogi, T., see Speriosu, V.S. 169 Yokoi, H. 302, 303 Yokoi, H., see Semenov, Yu.G. 296 Yonamoto, Y. 206, 249 Yonezawa, T., see Burnet, S. 252, 275 Yoon, S.S. 540 Yoon, S.S., see Jang, K.J. 542

645

Yoon, S.S., see Kim, C.G. 535, 538, 542, 545, 546, 557 Yoon, S.S., see Ryu, G.H. 542, 549 Yooyama, T., see Yonamoto, Y. 206, 249 Yoshida, H., see Mohri, K. 508, 509 Yoshida, K., see Ishida, K. 479 Yoshida, T., see Lanzara, A. 409, 427, 454, 482 Yoshida, T., see Zhou, X.J. 454 Yoshikawa, M., see Fuke, H.N. 158, 174 Yoshikawa, M., see Takagishi, M. 13 Yoshikawa, M., see Yuasa, H. 13 Yoshimura, K., see Imai, T. 474 Yoshinaga, T. 504, 548 Yoshinari, Y. 460, 461 Yoshinari, Y., see Suh, B.J. 452 Yoshiwaka, M., see Kamiguchi, Y. 35, 48, 49, 77 Yoshizaki, R., see Bogdanov, P.V. 427 Yoshizaki, R., see Shen, Z.-X. 479, 482 Yoshizawa, H. 459 You, D., see Guo, Z.B. 159 You, H., see Kaminski, A. 419 Young, D.K., see Ohno, Y. 305 Yu, C.C., see Huang, J.C.A. 71 Yu, G.H., see Shen, F. 38, 49 Yu, H.Y., see Edelstein, A.S. 231 Yu, J.Q. 504 Yu, J.Q., see Zhou, Y. 504, 534, 541 Yu, K., see Guo, Y. 29 Yu, S., see Pennington, C.H. 474 Yu, S.-C., see Ryu, G.H. 542, 549 Yu, S.-C., see Song, S.-H. 542–544 Yu, S.-C., see Song, S.H. 546 Yu, S.C., see Ahn, S.J. 504, 540 Yu, S.C., see Jang, K.J. 542 Yu, S.C., see Kim, K.S. 532 Yu, W.Y. 297, 358 Yu, W.Y., see Fu, L.P. 358 Yu, W.Y., see Jonker, B.T. 301, 358 Yu, W.Y., see Mountziaris, T.J. 358 Yu, W.Y., see Warnock, J. 294, 358 Yu, Z.G., see McQueeney, R.J. 409 Yu, Z.G., see Stojkovi´c, B.P. 392 Yu, Z.G., see Yi, Y.S. 409 Yuan, S. 323, 324, 327 Yuan, S., see Springholz, G. 327 Yuan, S.W. 127 Yuasa, H. 13 Yuasa, H., see Kamiguchi, Y. 35, 48, 49, 77 Yubero, F., see Andrieu, S. 228 Yuguchi, A., see Takayama, A. 552 Yunovich, A.E., see Kolesnikov, I.V. 340 Yusu, K., see Inomata, K. 220, 244

646

AUTHOR INDEX

Zaanen, J. 388–391, 393, 395, 396, 404, 407, 409, 413, 420, 430, 441, 454, 464, 480, 484 Zaanen, J., see Abu-Shiekah, I.M. 433, 452, 461, 462 Zaanen, J., see Bakharev, O.N. 467, 468 Zaanen, J., see Bosch, M. 396, 404, 407 Zaanen, J., see Brom, H.B. 479 Zaanen, J., see Eskes, H. 404 Zaanen, J., see Littlewood, P.B. 418, 423 Zaanen, J., see Martin, I. 481 Zaanen, J., see Pellegrin, E. 463 Zaanen, J., see Teitel’baum, G.B. 434, 435, 451, 452, 467 Zaanen, J., see van Duin, C.N.A. 416, 417 Zabel, H. 208, 218, 219 Zabel, H., see Hübener, M. 276 Zabel, H., see Labergie, D. 248, 249 Zabel, H., see Uzdin, V.M. 249 Zacchigna, M., see Gallani, J.L. 275 Zachar, O. 405 Zachar, O., see Emery, V.J. 398 Zaets, W. 299 Zahn, P. 111 Zahn, P., see Binder, J. 83, 84, 109 Zaitsev, S. 309 Zakharchenya, B.P., see Aguekian, V.F. 300 Zakharchenya, B.P., see Kusrayev, Yu.G. 298 Zakrzewski, A., see Janik, E. 295 Zakrzewski, A., see Sawicki, M. 311 Zamir, D., see Gavish, M. 357 Zangrando, M., see Gallani, J.L. 275 Zasadzinski, J.F. 481 Zasavitskii, I.I., see Valeiko, M.V. 327 Zawadowski, A., see Ujsaghy, O. 252 Zawadzki, W., see Bauer, G. 312, 314, 318, 324, 325 Zayachuk, D. 329 Zayets, V., see Saito, H. 311 Zazo, M., see Raposo, V. 510 Zdyb, R. 237 Zech, E., see Seewald, G. 237 Zehnder, U., see König, B. 294 Zehnder, U., see Kutrowski, M. 295 Zehnder, U., see Wojtowicz, T. 295 Zehnder, U., see Yakovlev, D.R. 299, 310 Zeitler, U. 360 Zeitz, W.D., see Waldmann, H. 361 Zelaya-Angel, O., see Alvarez-Fregoso, O. 361 Zeller, R., see Bellini, V. 205 Zeller, R., see Blügel, S. 224, 235 Zeller, R., see Freyss, M. 248 Zeller, R., see Nonas, B. 222 Zeller, R., see Oswald, A. 260 Zeller, R., see Papanikolaou, N. 213, 253

Zeller, R., see Stepanyuk, V.S. 212, 253 Zeller, R., see Weber, S.E. 251 Zeltser, A., see Nozières, J.P. 158, 161, 162, 168–170 Zeltser, A.M. 73, 159 Zeltser, A.M., see Menyhard, M. 73 Zeltser, A.M., see Smith, N. 135 Zeltser, A.M., see Zhang, Y.B. 60, 76, 161, 170 Zeltzer, A., see Nozières, J.P. 161, 162, 170 Zeng, L. 536, 549 Zenia, H. 217, 220, 275 Zha, F.-X. 261 Zha, Y. 420, 423, 431, 432, 436 Zha, Y., see Liu, D.Z. 420 Zha, Y.Y., see Si, Q.M. 423 Zhang, F.C. 297, 385, 386 Zhang, F.C., see Dai, N. 294, 297, 358 Zhang, F.C., see Luo, H. 295, 297 Zhang, G. 274 Zhang, H., see Bloomﬁeld, L.A. 211, 250 Zhang, H., see Li, D. 540 Zhang, H.W., see He, J. 540 Zhang, J., see Huai, Y. 33, 34, 42, 49 Zhang, K. 161 Zhang, K., see Fujiwara, H. 155 Zhang, K., see Hou, C. 150 Zhang, L., see Xiao, S.Q. 541 Zhang, S., see Anthony, T.C. 34, 39, 46, 51, 93, 145 Zhang, S., see Levy, P.M. 109 Zhang, S., see Li, Z. 155 Zhang, S.-C. 415, 416 Zhang, S.-C., see Chen, H.-D. 413 Zhang, S.-C., see Demler, E. 415, 417, 428 Zhang, S.-C., see Dorneich, A. 416 Zhang, S.-C., see Rabello, S. 416 Zhang, S.-C., see Sachdev, S. 413, 441 Zhang, S.-C., see Scalapino, D.J. 416 Zhang, S.C., see Arovas, D.P. 415 Zhang, S.C., see Bazaliy, Y.B. 417 Zhang, S.C., see Eder, R. 416 Zhang, S.C., see Meixner, S. 416 Zhang, S.C., see Schrieffer, J.R. 392, 393 Zhang, W., see Zhu, Z.-M. 296 Zhang, X.-C., see Nurmikko, A.V. 296 Zhang, X.-G. 103, 104, 106, 109 Zhang, X.-G., see Brown, R.H. 110 Zhang, X.-G., see Butler, W.H. 40, 80, 83, 84, 106–109 Zhang, X.-G., see Nicholson, D.M.C. 40, 80, 85 Zhang, X.-G., see Oparin, A.B. 85 Zhang, X.-G., see Schulthess, T.C. 83 Zhang, X.X., see Nie, H.B. 504, 541

AUTHOR INDEX Zhang, Y. 413, 420, 471, 480 Zhang, Y., see Demler, E. 449, 480 Zhang, Y., see Prakash, S. 162 Zhang, Y., see Vojta, M. 399, 422 Zhang, Y.B. 60, 76, 161, 170 Zhang, Y.B., see Nozières, J.P. 158, 161, 162, 168–170 Zhang, Y.B., see Zeltser, A.M. 159 Zhang, Z. 19, 22 Zhang, Z., see Shen, F. 38, 49 Zhao, J., see Yang, C.-K. 266 Zhao, J.G., see Chen, C. 504 Zhao, M., see Huai, Y. 33 Zhao, T., see Fujiwara, H. 155 Zhao, T., see Zhang, K. 161 Zhao, X.L., see Zhou, Y. 504, 534, 541 Zhao, Z. 265 Zhao, Z.J. 532 Zhao, Z.J., see Zeng, L. 536, 549 Zheng, G.-q., see Asayama, K. 430, 437, 473, 474 Zheng, W.J., see Bogdanov, P.V. 427 Zheng, Y., see Li, M. 61, 159, 167 Zheng, Y., see Lu, Z. 54, 160 Zheng, Y., see Lu, Z.Q. 37, 48, 54 Zheng, Y., see Zhu, J.-G. 127 Zhmodikov, A.L., see Akimov, A.V. 299 Zhong, L., see Miller, M.M. 24 Zhong, L., see Nakamura, K. 155 Zhou, H.P., see Ribayrol, A. 297 Zhou, S., see Li, D. 540 Zhou, S.M., see Strijkers, G.J. 123 Zhou, S.X., see Xiao, S.Q. 541 Zhou, X.J. 441, 454 Zhou, X.J., see Bogdanov, P.V. 427 Zhou, X.J., see Lanzara, A. 409, 427, 454, 482 Zhou, X.W. 69 Zhou, Y. 135, 136, 504, 534, 541 Zhou, Y., see Shin, K.H. 545 Zhou, Y., see Yu, J.Q. 504

647

Zhu, J.-G. 42, 127, 135 Zhu, J.-G., see Fang, T.-N. 58 Zhu, J.-G., see Gadbois, J. 58 Zhu, J.-G., see Han, D.-H. 163 Zhu, J.-X. 449 Zhu, J.G., see Wei, Y. 22 Zhu, J.G., see Zhou, Y. 135, 136 Zhu, L., see Parks, E.K. 255 Zhu, T., see Chen, E.Y. 128 Zhu, T., see Katti, R.R. 24, 57, 58 Zhu, Z.-M. 296 Zhukov, A. 532, 533 Zhukov, A., see Blanco, J.M. 504, 539, 540, 544 Zhukov, A., see Cobeño, A.F. 555 Zhukov, A., see Garcia, K.L. 540 Zhukov, A., see Vázquez, M. 504, 533, 541, 556 Zhukov, A., see Zhukova, V. 533 Zhukov, A.P., see Blanco, J.M. 538 Zhukova, V. 533 Ziane, A. 234 Zielinski, M. 351–353 Zielinski, M., see Testelin, C. 351, 357 Ziese, M. 5 Zigone, M., see Haacke, S. 305 Ziman, J. 212 Zimmermann, C.G. 268 Zinn, W., see Binasch, G. 4 Zinn, W., see Wosnitza, J. 336 Zlomanov, V.P., see Akimov, B.A. 312 Zoll, S., see van den Berg, H.A.M. 42 Zotos, X., see Prelovšek, P. 402, 404, 407 Zotov, S.K., see Baranov, S.A. 548 Zou, Z., see Baskaran, G. 399, 420 Zschack, P., see Isaacs, E.D. 462 Zuberek, R. 140 Zukhov, A., see Vázquez, M. 502, 553 Zunger, A., see Perdew, J.P. 213 Zurn, S., see Bae, S. 25, 37, 46 Zurn, S., see Torok, E.J. 25

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SUBJECT INDEX

Brillouin zone boundary in cuprates 475 bulk crystals of diluted magnetic semiconductors 312

AF-coupled multilayers 16 AGMI due to bias current 543 AGMI due to bias ﬁeld 544 AGMI due to exchange bias 545 anisotropic (AMR) effect 6 antiferromagnetic correlation length in cuprates 454, 455, 476 antiferromagnetic Goldstone boson in cuprates 414 antiferromagnetic spin correlations of nickelates 460 antiferromagnetism in cuprates 421, 450 antiphase domain boundaries in cuprates 445 antiphase-boundariness in cuprates 404 apical oxygen of cuprates 466 application of exchange biased SVs 19 applications of GMI 550 ARPES experiments in cuprates 454 ARPES in Bi2212 482 ARPES of cuprates 440, 464 ARPES of nickelates 461 ARPES spectra of cuprates 427, 465 asymmetric giant magnetoimpedance (AGMI) 542 asymmetric GMI effect 542 atomic pair distribution function for cuprates 448 Atomic Sphere Approximation for low-dimensional TM systems 212

Camley–Barnas model extensions 103 characteristic features of GMI 517 charge and spin order in nickelates 443 charge compressibility in cuprates 398 charge density modulation in nickelates 463 charge density wave in cuprates 396 charge localization in nickelates 455 charge modulation in nickelates 458 charge order in cuprates 454, 455 charge order in nickelates 461 charge ordering in nickelates 457 charge sensitive techniques 479 charge stripe ordering in cuprates 454 charge stripe ordering in nickelates 463 charge stripes in nickelates 461 charge wave vector in cuprates 444 charge- and superconducting order in cuprates 413 charge-ordering temperature in nickelates 459 charged domain walls in cuprates 388 charged domain walls in nickelates 462 circular magnetization process in GMI materials 535 coexistence phase in cuprates 413 collective ﬁelds 417 complex impedance 509 complex susceptibility in cuprates 438 conductivity of nickelates 461 control parameter g for cuprates 387 correlation length in cuprates 446 Coulomb catastrophe in cuprates 407 coupling between electrons and phonons in cuprates 454 coupling constants of the Heisenberg interaction in diluted magnetic semiconductors 353 coupling strength 485 crossover temperature 485 CuO2 layers in high-Tc compounds 385

band structure in DMS with Eu 323 band structure of diluted magnetic semiconductors 313, 314 BCS instability in cuprates 421 BCS type (d-wave) pairing instabilities in cuprates 400 Berry phase in cuprates 400 Bloch–Bloembergen damping in GMI materials 518, 526 blocking temperature 9 bond centered charge stripes in cuprates 400 649

650

SUBJECT INDEX

CuO2 plane 388 Curie temperature for Pb1−x−y Sny Mnx Te crystals 317 current and position sensors based on GMI 552 “Current In the Plane of the layers” (CIP) geometry 6 d-wave BCS-like states in cuprates 400 d-wave like superconductor 401 d-wave superconductor 422, 427 damping of domain wall motion in GMI materials 522 damping of the resonance in cuprates 418 DC current sensor based on GMI 553 Density Functional Theory for low-dimensional TM systems 211 density matrix renormalization group (DMRG) – calculations for cuprates 404 – method for cuprates 405 DFT-based ab-initio methods for low-dimensional TM systems 211 diagonal static-striped-phase in cuprates 388 diagonal stripes in cuprates 397 dispersion of the resonance in cuprates 423 DMFT stripes in cuprates 398 DMS materials with Eu 322 DMS quantum dots 303, 308 DMS quantum wells 303 DMS semiconductors with rare earth ions 312 DMS spin aligners 304 DMS/non-DMS quantum structures 293 domain wall motion in GMI materials 513 domain wall movement in GMI materials 522 doped Hubbard chain in cuprates 394 doped Mott-insulator 390 double-peak behaviour in GMI materials 512 double-peak structure in GMI materials 513 dual spin-valves 50 dynamical Fermi surface in cuprates 424 dynamical magnetic susceptibility in cuprates 422, 425 dynamical mean ﬁeld-theory (DMFT) for cuprates 397 dynamical stripes in cuprates 396, 429 effective g-factor of electrons in diluted magnetic semiconductors 298 electrical ﬁeld gradients of nickelates 462 electromagnetic models of GMI 523 electromagnetic radiation in GMI materials 514 electron band structure in DMS with Gd 329 electron band structure of diluted magnetic semiconductors 314

electron stripe in cuprates 393 electron–phonon coupling 463 electronic band structure of EuS-PbS multilayers 341 electronic ﬁgure of merit of thin ﬁlm MR elements 28 electronic structure calculations – in low-dimensional TM systems 211 – of 3d adatoms on graphite 262 – of Co nanowires 258 – of CoN supported on noble metals 267 – of Cr clusters 250 – of Fe clusters 250 – of Fe ﬁlm on Cu, Ag, Au 237 – of Fe ﬁlm on graphite 241 – of Fe/c-FeSi/Fe multilayers 244 – of Fe/Mn multilayers 228 – of Fe/Ni superlattices 230 – of Fe/Pd superlattices 235 – of Mn clusters 251 – of multilayers 217 – of Ni free-standing clusters 254 – of Ni–C clusters 261 – of NiN embedded in Al 266 – of non-collinear magnetic conﬁgurations 269 – of Pd clusters 251 – of Rh clusters 252 – of Rh layers on Fe 236 – of thin ﬁlms 217 – of V clusters 251 – of V multilayered systems 225 electronic structure of stripes in cuprates 454 electronic transitions in DMS with Eu 327 “empty” stripes in cuprates 395 energy of the magnetic resonance in cuprates 427 energy spectrum of EuS-PbS heterostructures 340 ESR in cuprates 453 excess oxygen site in nickelates 456 exchange anisotropy effect 9 exchange anisotropy of spin valve systems 157, 160, 161, 163 exchange bias effect 9 exchange bias materials 156, 165 exchange bias thermal relaxation 172 exchange constants – in diluted magnetic semiconductors (DMS) 303, 351 – of Cd1−x Cox Te 352 exchange interaction – in DMS with Eu 325 – in DMS with Gd 329

SUBJECT INDEX – in Zn1−x Cox Te 353 – in Zn1−x Fex Te 357 exchange-biased GMR spin-valves 148 exchange-biased spin-valves 9, 22 exchange-conductivity models of GMI excitons in diluted magnetic semiconductors

ground state of cuprates 387 Gutzwiller Ansatz for cuprates

302

Fano antiresonance in nickelates 463 Faraday rotation in diluted magnetic semiconductors 299 Fe/Cr interfaces 221 Fermi surface in cuprates 475 fermionic Bogoliubov quasiparticles in cuprates 410 ferromagnetic III-V materials 311 ﬁeld annealing of GMI materials 540 ﬁeld dependence of the impedance 507 ﬁgures of merit of GMR materials 25 ﬂuctuation exchange (FLEX) approximation for cuprates 418 formation of stripes in cuprates 404 frequency dependence of the impedance 507 frustrated phase separation in cuprates 407 Full-Potential Linearized Augmented Plane Waves for low-dimensional TM systems 211 g-factor of electrons in diluted magnetic semiconductors 302 giant magnetoimpedance (GMI) effect 501 giant magnetoresistance, anisotropic 61 “giant” MR effect 6 giant Zeeman splitting in diluted magnetic semiconductors 302, 304, 307 Gilbert damping parameter 517 glass-coated microwire 546 glass-covered amorphous wires 546 GMI – in an isotropic cylindrical wire 530 – in magnetic non-destructive analysis 553 – ratio 503 – sensor heads 556 – theory 515, 518 GMI as a research tool 549 GMI based sensors 553 GMR effect – applications of the Camley–Barnas model 89 – Camley–Barnas semiclassical transport model 86 – in magnetic multilayers 5 – quantum-mechanical models 109 – semiclassical models based on realistic band structures 106 – series resistor model 85

651

397

half-ﬁlled stripes in cuprates 396 Hall coefﬁcient in cuprates 454 hard disk read heads 19 Hartree–Fock electronic mean-ﬁeld theory for cuprates 391 Heisenberg 2D antiferromagnetic insulators 384 Higg’s boson in cuprates 414 high frequency models of GMI 522 high-resolution photoemission spectra of cuprates 465 high-Tc superconductivity in cuprates 407 hole mobility for Sn1−x Gdx Te 331 hole mobility in cuprates 455 hole-doped single-layer 214-cuprates 443 holons in cuprates 394, 399, 403 horizontal or vertical domain wall in cuprates 388 horizontal stripes in cuprates 397 hot spots on the Fermi surface of cuprates 475 Hubbard hamiltonian for cuprates 392 Hubbard–Peierls type models for cuprates 409 hybridization of the Cu and O orbitals in cuprates 386 hyperﬁne coupling in nickelates 462 hyperﬁne ﬁeld in cuprates 446 hyperﬁne interaction in cuprates 432 II-VI diluted magnetic semiconductors 293 impedance modulus 503 impedance of a magnetic conductor 506 impurities in cuprates 414 impurity induced structures in cuprates 478 in-plane resistivity data for cuprates 454 incommensurate ﬂuctuations 411, 471 – in cuprates 471 incommensurate magnetic ordering of nickelates 457 incommensurate peaks in cuprates 473 incommensurate side branches in cuprates 425 incommensurate spin excitations in cuprates 412 incommensurate spin ﬂuctuations in cuprates 418, 423, 470 incommensurate static charge ordering in cuprates 470 induced magnetic anisotropies in GMI materials 539 inelastic neutron data for cuprates 440 inelastic neutron scattering – of cuprates 447, 469 – of nickelates 458

652

SUBJECT INDEX

infrared optical properties of GMR materials 63 infrared reﬂectance measurements of cuprates 466 inhomogeneous magnetic domains in cuprates 453 interlayer coupling in EuS-PbS multilayers 336 interlayer coupling in EuS-PbS structures 338 interlayer exchange coupling in Fe based multilayers 219 interlayer interactions in EuTe-PbTe 343 internal ﬁeld in nickelates 462 interstitial excess oxygen in nickelates 456 inverse correlation length of cuprates 467 Ising type domain boundaries in cuprates 393 Josephson coupling in cuprates Joule heating of GMI materials

401 540

Kerr effect in diluted magnetic semiconductors 300 kink-driven stripe delocalization in cuprates 404 KKR-GF for low-dimensional TM systems 213 Knight shift in cuprates 431, 436, 474 Korringa–Kohn–Rostoker Green’s function method for low-dimensional TM systems 212 Landau–Lifshitz equation of motion for GMI materials 517 laser annealing of GMI materials 540 lattice constants of nickelates 457 lattice gas model for cuprates 407 LCAO-Molecular-Orbitals-DFT for low-dimensional TM systems 215 lines of holes in cuprates 393 local lattice ﬂuctuations in cuprates 454 low-dimensional quantum structures of diluted magnetic semiconductors 289 low-dimensional structures of diluted magnetic semiconductors 312 low-energy spin-wave spectrum of nickelates 458 luminescence in diluted magnetic semiconductors 308 Luttinger liquid electrons in cuprates 398 μSR in cuprates 438, 446, 449, 473, 477, 479 μSR in nickelates 459 μSR spectra in cuprates 439 magnetic biosensors 24 magnetic correlation length – in cuprates 434, 437 – in nickelates 459

magnetic domain walls in cuprates 394 magnetic domains 390 magnetic dynamical formfactor in cuprates 411 magnetic ﬁeld induced antiferromagnetism in cuprates 413 magnetic ﬁeld sensor 23 – based on GMI 554 – devices based on GMI 550 magnetic ﬂuctuations in cuprates 447 magnetic layer thickness dependence of GMR 18 magnetic ordering temperature of nickelates 458 magnetic phase diagram – of diluted magnetic semiconductors 321 – of SnMnTe and PbSnMnTe 320 magnetic polarons in diluted magnetic semiconductors 300, 301 magnetic properties – of diluted magnetic semiconductors 315 – of DMS thin ﬁlms 310 – of EuS-PbS multilayers 332 – of iron based multilayered systems 218 – of MnTe-ZnTe and MnTe-CdTe superlattices 344 – of Pb1−x Eux Te 326 – of Zn1−x Crx Se 346 – of Zn1−x Crx Te 346 Magnetic Random Access Memories (MRAMs) 24, 57 magnetic resonance in cuprates 410, 411 magnetic resonance peak in cuprates 472 magnetic spin ﬂuctuations in cuprates 452 magnetic string in cuprates 403 magnetic susceptibility – of cuprates 453 – of nickelates 443, 459 – of Pb1−x−y Sny Mnx Te 316 magnetic transition in nickelates 458 magnetic transition temperature of cuprates 465 magnetic transition temperature of nickelates 460 magnetic tunnel junctions 5, 148 magnetization – in Cd1−x Cox Se and Cd1−x Cox S 355 – in the Cd1−x Fex Te 356 – of Hg1−x Fex Se 360 – of Zn1−x Crx Se 348 magneto-optical effects in DMS with Eu 324 magnetocouplers 24 magnetoelasticity in GMI materials 537 magnetoimpedance of thin planar ﬁlms 531 magnetoimpedance phenomenon 503 magnetoimpedive materials 503 magnetorefractive effect in GMR materials 63

SUBJECT INDEX magnetoresistance – and sheet resistance of spin valves 51 – and sheet resistance Rp,sh of spin valves 31 – anisotropic 61 – temperature dependence of 59 magnetoresistance (MR) ratio 5 magnetostriction in GMI materials 537 magnons in cuprates 397 marginal Fermi-liquid (MFL) in cuprates 418 mean-ﬁeld calculations for cuprates 393 mean-ﬁeld stripe phases in cuprates 395 MEMS microbridge vibration sensors 24 Mermin–Wagner theorem 310 metallic ferromagnet/DMS quantum well hybrid structures 309 mid-gap band in cuprates 395 mid-gap state in cuprates 394, 395 Mila–Rice–Shastry hamiltonian for cuprates 436 Mott–Hubbard insulators 385 Mott-insulators 390 – in cuprates 392 multilayered compounds 386 multilayered cuprates 478 multilayers with II-VI and IV-VI semiconductors 331 muon-spin depolarization rate in cuprates 481 nano-oxide layers (NOLs) 17 narrow gap II-VI DMS 360 nearly antiferromagnetic Fermi liquid (NAFL) model in cuprates 437 Néel coupling in spin valves 144, 147 Néel order parameter of cuprates 394 Néel ordered state 484, 485 Néel spin order in cuprates 393 Néel temperatures of EuTe-PbTe/BaF2 (111) superlattices 343 Néel-ordered antiferromagnetic phase in cuprates 387 neutron intensity in cuprates 471 neutron scattering in cuprates 439, 445, 469, 479 neutron scattering in nickelates 457 Ni–O stretching motion in nickelates 463 NMR – in cuprates 430, 432, 449, 466, 473, 479 – in nickelates 459 – spectra of nickelates 462 NMR phase-diagram for cuprates 474 nodal fermions in cuprates 417, 430 non-destructive analyses based on GMI 553 NQR in cuprates 432, 449, 451, 473, 479 NQR in nickelates 459 nuclear Zeeman relaxation in cuprates 431

653

one-dimensional electronic structure in cuprates 441 one-dimensional striped excitations in cuprates 439, 440 optical conductivity of cuprates 465 optical conductivity of nickelates 463 optimal dopeding for cuprates 387 optical measurements of cuprates 479 optical resonances in diluted magnetic semiconductors 302 orbital order in cuprates 402 ordering parameter of 2DQHAF 484 ordering temperature of nickelates 460 orthorhombic domain stripes in cuprates 439 oscillatory interlayer exchange coupling in spin valves 143, 145 overdoped regime in cuprates 387, 417, 424 overdoped state in cuprates 464 oxygen and Sr doping of nickelates 456 oxygen lattice vibrations in cuprates 448 oxygen ordering in nickelates 457

p–d exchange in diluted magnetic semiconductors 303 parallel anisotropy (PA) conﬁguration 12 Peierls–Hubbard model for cuprates 409 phase diagram – as function of hole doping for cuprates – for cuprate superconductors 477 – for the 2DQHAF system 485 – of cuprates 386 – of high-Tc superconductors 421 – of La1.8−x Eu0.2 Srx CuO4 450 – of the 123-compounds for cuprates – of YBa2 Cu3 O6+x 470 phase separation in cuprates 453 phases of nickelates 456 photoemission spectra of cuprates 466 photoluminescence of EuS-PbS structures 340 pseudo-gap formation in cuprates 466 pseudo-gap phase in cuprates 421 pseudo-nesting in cuprates 423 pseudo-spin-valves 55

quantum critical region 485, 486 quantum disordered region 485, 486 quantum dots in diluted magnetic semiconductors 308 quantum Hall effect in diluted magnetic semiconductors 304 quantum Hall effects in DMS with Eu 328

654

SUBJECT INDEX

quantum Monte Carlo method for cuprates 406 quantum phase transition in cuprates 414, 419, 451 quantum spin ﬂuctuations in cuprates 391, 401 quantum wells in diluted magnetic semiconductors 302 quantum wires of diluted magnetic semiconductors 304, 307 quasistatic models of GMI 519 Raman scattering experiments of cuprates 464 Raman spectra of cuprates 465 Raman spectroscopy of nickelates 459 random phase approximation (RPA) for cuprates 418 read heads for tape recording 21 real space renormalization group (RNG) procedure for cuprates 419 renormalized classical regime 486 resistivities in the Cu and Co layers in spin-valves 108 resistivities of nickelates 459 resonance in cuprates 425 resonance peak in cuprates 389, 420, 471, 473 resonating valence bond (RVB) mechanism for cuprates 398 RF and microwave applications of GMI 556 RKKY interaction in diluted magnetic semiconductors 316 RPA theories of the magnetic resonance in cuprates 426 Ruderman–Kittel–Kasuya–Yoshida (RKKY) coupling in spin valves 143 s–d exchange in diluted magnetic semiconductors 303 s-wave superconductor 423 scanning tunneling microscopy (STM) of cuprates 441, 479 scattering parameters for spin valves 96 short range antiferromagnetism in cuprates 450 short-range magnetic order in nickelates 460 site-ordered stripes in nickelates 462 skin effect in GMI materials 505 soliton in cuprates 393, 394 sp–d exchange integral in diluted magnetic semiconductors 318 spacer layer thickness dependence of GMR 17 spatial correlation length 484 speciﬁc heat for cuprates 442 speciﬁc heat measurements of nickelates 459 spectral-weight in cuprates 482

spin dependent tunneling across a DMS barrier 306 spin dependent tunneling of diluted magnetic semiconductors 305 spin dephasing in diluted magnetic semiconductors 306 spin excitations 460 spin ﬂuctuations 472 – in cuprates 397, 447 spin freezing in nickelates 463 spin frustrations in cuprates 391 spin gap in cuprates 421 spin glass order in diluted magnetic semiconductors 320 spin injection of diluted magnetic semiconductors 304 spin modes in cuprates 413 spin order in cuprates 402 spin ordering of nickelates 461 spin stiffness 485 spin susceptibility in cuprates 438 spin valves – anisotropy and magnetostriction of the free layer 136 – applications of the Stoner–Wohlfarth model 115 – deposition and microstructure 64 – deviations from the single-domain model 124 – electronic noise 130 – exchange anisotropy 148 – exchange bias ﬁeld 149 – frequency dependence of magnetic response 130 – interlayer coupling 142 – magnetic interactions 115 – magnetization ﬂuctuations 130 – magnetization reversal processes 115 – switching dynamics 132 – thermal stability 71 spin–lattice relaxation in cuprates 431 spin–lattice relaxation rates in cuprates 473 spin–spin correlation 486 spin–spin correlation-length in cuprates 445 spin–spin relaxation in cuprates 432 spin–spin relaxation rate in cuprates 431, 473 spin-bag in cuprates 394 spin-charge separation gauge theories for cuprates 419 spin-charge separation in cuprates 401, 412, 420, 422 spin-dependent conductivity in ferromagnets 78, 80 spin-dependent scattering at interfaces 83

SUBJECT INDEX spin-ﬂuctuation exchange mechanism for cuprates 427 spin-gap proximity effect in cuprates 398 spin-Peierls order in cuprates 400 spin-polarized transport in spin-valves 77 spin-valves – conventional 30 – experimental results for prototype 13 – with a non-magnetic back layer 41 – with a synthetic antiferromagnetic (Sy-AF) pinned layer 42 – with a synthetic ferromagnetic (Sy-F) free layer 45 – with an oxidic antiferromagnet 46 – with composite ferromagnetic layers 39 – with improved magnetic characteristics 53 – with nano-oxide layers (NOLs) 48 spin-wave excitations of nickelates 458 spin-wave modes in nickelates 458 spin-wave theory for cuprates 392 spinon 394 spinon Fermi surface in cuprates 423 spinon superconductor 421 spinons in cuprates 399, 403 staggered ﬂux phase in cuprates 419 staggered magnetization 484 staggered moments of the Cu spins in cuprates 466, 467 staggered order parameter in cuprates 393, 394 static spin susceptibility of cuprates 436, 474 static stripe phases in cuprates 450 static stripes in cuprates 429, 430 stress dependence of GMI 538 stress-annealing of GMI materials 540 stress-impedance applications of GMI 555 stress-impedance in GMI materials 538 stripe and superconducting order in cuprates 398 stripe charge order in cuprates 409 stripe density modulation in cuprates 401 stripe domains in nickelates 463 stripe formation 390 stripe fractionalization in cuprates 420 stripe meanderings in cuprates 397 stripe order in cuprates 390 stripe pattern in cuprates 445 stripe structure in cuprates 388 stripe structure in nickelates 461 stripe-charge order in cuprates 413 stripe-liquid phase in nickelates 459 stripe-ordered phase in nickelates 458 striped phase in nickelates 455 stripes 401 – in cuprates 470, 477 – in nickelates 396

655

structural changes of nickelates 456 structure of Bi2212 479 Su–Schrieffer–Heeger type solitons 391 SU(2) spin symmetry in cuprates 399 sum rule in cuprates 428 superconducting transition-temperatures of cuprates 465 superconducting-antiferromagnetic coexistence phase in cuprates 416 superconductivity of nickelates 455 superexchange coupling in DMS with Eu 326 surface impedance tensor, 506, 527 – in GMI materials 531 susceptibility for cuprates 442 synchrotron X-ray scattering of nickelates 462 t–J model 406 – for cuprates 391, 399 t–J –V model for cuprates 400 t–Jz model 404 – for cuprates 402 temperature dependence of the MR-ratio of spin-valves 60 thermal treatments of GMI materials 540 thermoelectric power (TEP) of spin-valves 62 thin ﬁlm technology for GMI materials 551 three band Hubbard model for cuprates 393 Tight-Binding LMTO for low-dimensional TM systems 212 time resolved studies of diluted magnetic semiconductors 301 torsion-annealing of GMI materials 540 transferred exchange interaction in nickelates 461 transferred hyperﬁne coupling in cuprates 436 transition temperatures as function of hole doping in cuprates 387 transport studies of diluted magnetic semiconductors 303 transverse ﬂuctuations of stripes in cuprates 454 trions in diluted magnetic semiconductors 302 triplet Cooper pairs in cuprates 410 triplet excitations in cuprates 410 two-current model 7 Umklapp surface in cuprates 424 underdoped regime in cuprates 387, 421 uniform d-wave superconductor 400 unit cell – of La2−x Srx NiO4+δ 456 – of Nd2−x Cex CuO4 464 – of Tl2 Ba2 CaCu2 O8 480 – of YBa2−x Srx Cu3 O7−δ 469

656 vortex

SUBJECT INDEX 442

weak coupling BCS instability in cuprates 418 wipe-out of the Cu NMR signal in cuprates 467 wire technology for GMI materials 550

X-ray experiments in cuprates 454 X-ray scattering of nickelates 462 Zhang–Rice singlets 385, 386 Zn-substitution for cuprates 446

MATERIALS INDEX

123 superconductor 430 123-compounds 387 123-system 439 214 nickelates 463 214 system 412, 427 2212 compounds 387 2212 system 411 amorphous glass-coated microwires amorphous microwires 532 amorphous ribbons 504, 532 amorphous wires 504, 532, 533

CoFeSiB 539, 542–544, 556 CoP 534 Cu-Co multilayers 535 CuO-multilayer compounds 389 double-layer 123- and 124-compounds electrodeposited CoP microtubes electron-doped cuprates 463 EuS-PbS 332 EuS-PbS/BaF2 (111) 341 EuS-PbSe/KCl 336 EuS-SrS 336 EuTe-PbTe 342 EuTe-PbTe/BaF2 (111) 343 EuTe-PbTe/PbTe(100) 343

548

Bi2212 385, 441, 442, 478 Bi2 Sr2 Ca1−x Yx (Cu1−y Zny )2 O8+δ 479, 481 Bi2 Sr2 Can−1 Cun O2n+4 385 Bi2 Sr2 CaCu2 O8 385, 480 Bi2 Sr2 CaCu2 O8+δ 390, 479, 481, 482 bilayer 123 411 (Ca, Sr)2 CuO3 388 Ca0.85 Sr0.15 CuO2 388 Cd1−x Cox S 351, 355 Cd1−x Cox Se 351 Cd1−x Cox Te 351, 352 Cd1−x Crx Se 351 Cd1−x Fex Se 351 Cd1−x Fex Te 351, 356 Cd1−x Mnx Te-CdTe 344 Cd1−x Nix Te 361 Cd1−x Vx S 351 Cd1−x Vx Se 351 CdMnMgTe 298 CdMnTe/CdMgTe 299, 303 CdTe/CdMnTe 299 Co-based amorphous microwires 537 Co-based conventional wire 536 (Co100−x Fex )72.5 Si12.5 B15 547 Co67.05 Fe3.85 Ni1.4 B11.55 Si14.5 Mo1.65 Co67 Fe4 Cr7 Si8 B14 540 Co68.25 Fe4.5 Si12.25 B15 533 Co72.5 Si12.5 B15 546

468

542

(Fe0.06 Co0.94 )72.5 Si12.5 B15 512 (Fe0.06 Co0.94 )72.5 Si12.5 B16 511 Fe/Cr multilayers 219, 221 Fe/Cr superlattices 222 Fe/Cr/Fe trilayer 221 Fe/Mn multilayers 219 Fe/Ni multilayers 219 Fe/V multilayers 219 Fe/V superlattice 225 FeCoNi magnetic tubes 536 Ge1−x Mnx Te 313 GeMnTe 321 glass covered amorphous microwires glass-coated microwires 533 Heusler alloys 535 Hg1−x Fex Se 360 Hg1−x Fex Te1−y Sey K2 NiF4

532

356

486

La0.7 Ca0.3 MnO3 535 La1.28 Nd0.6 Sr0.12 CuO4 441, 454 La1.45 Sr0.15 Nd0.4 CuO4 445 La1.48 Nd0.4 Sr0.12 CuO4 451, 454 657

532

658

MATERIALS INDEX

La1.48 Sr0.12 Nd0.4 CuO4 439, 443, 445 La1.55 Nd0.4 Sr0.05 CuO4 445 La1.6−x Nd0.4 Srx CuO4 444, 452, 455 La1.64 Nd0.4 Sr0.12 CuO4 446 La1.67 Sr0.33 NiO4 459 La1.775 Sr0.225 NiO4 461, 463 La1.8−x Eu0.2 Srx CuO4 438, 439, 452 La1.85 Sr0.15 CuO4 453, 454 La1.86 Sr0.14 CuO4 447 La1.875 Ba0.125−x Srx CuO4 449 La1.88 Sr0.12 CuO4 446 La1.8 Sr0.2 NiO4 462 La1.92 Sr0.075 CuO4 462 La1.95 Sr0.05 CuO4 447 La2 Cu1−x Lix O4 452 La2 CuO4 384, 385, 388, 443, 445, 463 La2 NiO4 389, 486 La2 NiO4+δ 455, 457, 458 La2 NiO4.125 457, 461 La2 NiO4.13 461 La2 NiO4.17 462 La2−x Ax NiO4 457 La2−x Srx Cu1−y Zny O4 448 La2−x Srx CuO4 385, 411, 436, 445–447, 449, 450, 453–455, 459, 463, 471 La2−x Srx CuO4+δ 384, 387 La2−x Srx NiO4 459 La2−x Srx NiO4+δ 459, 460 La2−x−y Ndy Srx CuO4 454 La2−x−y Srx Euy CuO4 453 La2−x−y Srx Euy CuO4+z 450 La2−x−y Srx (Nd, Eu)y CuO4 451, 454 La2−x−y Srx (Nd, Eu)y CuO4+z 443 La2−x−y Srx Ndy CuO4 396, 450 La5/3 Sr1/3 NiO4 442, 443, 459 manganites 402 MnTe-CdTe 342 MnTe-ZnTe 342 Mo/V multilayers 224 Nd1.85 Ce0.15 CuO4 466–468 Nd1.85 Ce0.15 CuO4+y 465 Nd2 CuO4 388, 463 Nd2−x Cex CuO4 463, 465 nickelates 387, 390, 443 oxygen doped nickelates Pb1−x Cex S 312 Pb1−x Cex Se 312 Pb1−x Cex Te 312 Pb1−x Eux S 312, 322 Pb1−x Eux Se 312, 322

456

Pb1−x Eux Te 312–314, 322, 324 Pb1−x Eux Te1−y Sey 313 Pb1−x Gdx S 312 Pb1−x Gdx Se 312 Pb1−x Gdx Te 312 Pb1−x Mnx Te 314 Pb1−x Ybx S 312 Pb1−x Ybx Se 312 Pb1−x Ybx Te 312 Pb1−x−y Sny Mnx Te 313, 318 PbEuTe 322 PbEuTe-PbTe 326 PbSnMnTe 315, 317 Pd2 MnSb 535 Pd2 MnSn 535 single-layer 214 411 Sn0.96 Mn0.04 Te 321 Sn1−x Cex Te 312 Sn1−x Eux Te 312 Sn1−x Gdx Te 312 Sn1−x Mnx Te 313, 319 SnMnTe 315, 321 Sr-doped nickelates 458 Sr2 CuO2 Cl2 388 strontium doped 214-nickelates Tl2212 Tl2223

455

385 385

underdoped 123-compounds 471 underdoped 123-samples 469 V2 O3 sintered samples

535

Y1−x Cax Ba2 Cu3 O6 450 Y1−z Caz Ba2 Cu3 O7−δ 474 YBa2 Cu3 O6+y 411 YBa2 (Cu0.97 Ni0.03 )3 O7 472 YBa2 (Cu0.99 Ni0.01 )3 O7 472 YBa2 Cu3 O6 384, 385, 426 YBa2 Cu3 O6+δ 384 YBa2 Cu3 O6+x 385, 468, 471, 476 YBa2 Cu3 O6.6 439, 440, 472 YBa2 Cu3 O6.63 474, 475 YBa2 Cu3 O6.7 470, 473 YBa2 Cu3 O7 471 YBa2 Cu3 O7−δ 436, 473 YBa2 Cu3 O7−y 390 YBa2 Cu4 O8 469 YBa2 CuO7−δ 478 Zn1−x Cox Se 354 Zn1−x Cox Te 351, 354

MATERIALS INDEX Zn1−x Crx S 346, 351 Zn1−x Crx Se 346, 350, 351 Zn1−x Crx Se0.95 S0.05 351 Zn1−x Crx Te 346, 351 Zn1−x Fex Se 351 Zn1−x Fex Te 351 Zn1−x Fex Te1−y Sey 356 Zn1−x Mx S, M = Mn, Fe, Co, Ni

345

Zn1−x Mnx Se 350, 358 Zn1−x Mnx Te-ZnTe 344 Zn1−x Nix Te 361 ZnCdMnSe 308 ZnCoO 311 ZnMnSe/BeTe 298 ZnSe/Zn1−x Fex Se 358 ZnSe/ZnMnSe 307

659

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