Nanomagnetism: Ultrathin Films, Multilayers and Nanostructures

Nanomagnetism Ultrathin Films, Multilayers and Nanostructures

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Series: Contemporary Concepts of Condensed Matter Science Series Editors: E. Burstein, M.L. Cohen, D.L. Mills and P.J. Stiles

Nanomagnetism Ultrathin Films, Multilayers and Nanostructures

D.L. Mills Department of Physics and Astronomy University of California Irvine, CA 92697, USA

J.A.C. Bland Cavendish Laboratory University of Cambridge Cambridge CB3 0HE, UK

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

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r 2006 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., 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]. Requests may also be completed on-line via the Elsevier homepage (http:// www.elsevier.com/locate/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) 20 7631 5555; fax: (+44) 20 7631 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 the Publisher 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 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 2006 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data Nanomagnetism : ultrathin films, multilayers and patterned media. (Contemporary concepts of condensed matter science) 1. Magnetic films 2. Nanostructured materials 3. Ferromagnetic materials 4. Thin films - Magnetic properties I. Mills, D.L. II. Bland, J.A.C. (J. Antony C.), 1958-538.3 ISBN-10: 0444516808 ISBN-10: 0-444-51680-8 ISBN-13: 978-0-444-51680-0 ISSN: 1572-0934 (series)

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CONTENTS LIST OF CONTRIBUTORS

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SERIES PREFACE

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VOLUME PREFACE

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1. THE FIELD OF NANOMAGNETISM J. A. C. Bland and D. L. Mills

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2. FUNDAMENTAL PROPERTIES OF MAGNETIC NANOSTRUCTURES: A SURVEY Ruqian Wu

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3. EXCHANGE COUPLING IN MAGNETIC MULTILAYERS M. D. Stiles

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4. STATIC, DYNAMIC, AND THERMAL PROPERTIES OF MAGNETIC MULTILAYERS AND NANOSTRUCTURES R. E. Camley

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5. EXCHANGE ANISOTROPY A. E. Berkowitz and R. H. Kodama

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6. SPIN TRANSPORT IN MAGNETIC MULTILAYERS AND TUNNEL JUNCTIONS A. Fert, A. Barthe´le´my and F. Petroff

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7. ELECTRICAL SPIN INJECTION AND TRANSPORT IN SEMICONDUCTORS B. T. Jonker and M. E. Flatte´

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Contents

8. CURRENT-INDUCED SWITCHING OF MAGNETIZATION D. M. Edwards and J. Mathon

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AUTHOR INDEX

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SUBJECT INDEX

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LIST OF CONTRIBUTORS A. Barthe´le´my

Unite´ Mixte de Physique CNRS/Thales, Domain de Corbesville, 91404 Orsay, France

A. E. Berkowitz

Center for Magnetic Recording, University of California, La Jolla, CA 92093-0401, USA

J. A. C. Bland

Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

R. E. Camley

Department of Physics, University of Colorado, Colorado Springs, CO 80933-7150, USA

D. M. Edwards

Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 282, UK

A. Fert

Unite´ Mixte de Physique CNRS/Thales, Domain de Corbesville, 91404 Orsay, France

M. E. Flatte´

Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242-1479, USA

B. T. Jonker

Materials Science and Technology Division, Naval Research Laboratory, Washington, DC 20375-5343, USA

R. H. Kodama

Department of Physics, University of Illinois at Chicago, Chicago, IL 69607, USA

J. Mathon

Department of Mathematics, City University, London EC1V 0HB, UK

D. L. Mills

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

F. Petroff

Unite´ Mixte de Physique CNRS/Thales, Corbesville, 91404 Orsay, France

M. D. Stiles

Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8412, USA

R. Wu

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

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Domain

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SERIES PREFACE CONTEMPORARY CONCEPTS OF CONDENSED MATTER SCIENCE Board of Editors E. Burstein, University of Pennsylvania M. L. Cohen, University of California at Berkeley D. L. Mills, University of California at Irvine P.J. Stiles, North Carolina State University We introduce a new series of volumes devoted to the exposition of the concepts underlying the key experimental, theoretical and computational developments at the advancing frontiers of exciting, rapidly evolving sub-fields of condensed matter science. The term ‘‘condensed matter science’’ is central, because the boundaries between condensed matter physics, condensed matter chemistry and material science are disappearing. The overall goal of each volume in the series is to provide the reader with an intuitively clear discussion of the underlying concepts and insights that are the ‘‘driving force’’ for the high profile major developments at the advancing frontiers of the sub-field, while providing only the theoretical, experimental and computational detail, data, and results that would be needed for the reader to gain a conceptual understanding of the subject. Each section of a given volume will be devoted to a major development at the advancing frontiers of the sub-field. This will provide an opportunity for those in other areas of research, as well as those in the same area, to have access to the concepts underlying the major developments at the advancing frontiers of the sub-field. Each volume (250 printed pages) is to have a Preface written by the volumeeditor(s), that includes an Overview that highlights the exciting theoretical and experimental advances of the sub-field and their underlying concepts. It also provides an outline and brief summary of the major topics selected by the volumeeditor(s) and authored by key scientists recruited by the volume-editor(s), that highlight the most significant developments of the sub-field. The chapters are selfcontained—it should not be necessary to go to other sources to follow the presentation or the underlying science. The list of references will include the titles. The volume-editor(s) will interact closely with the chapter contributors to insure that the level and presentation of the material conform to the objective of the series. The volumes in this series will emphasize clear writing whose goal is to describe and to elucidate key developments in the sub-field focusing on the underlying concepts,. The model for this is a well-presented colloquium (not a seminar!) ix

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Series Preface

directed to those outside the specialized sub-field, which invites the audience to ‘‘come think with the speaker’’ and which avoids in-depth experimental, theoretical, and computational details. The overall goal of the series is ‘‘comprehension’’ rather than ‘‘comprehensive’’ and the goal of each volume is to provide an ‘‘overview, rather than a review’’ of the highlights of the sub-field. The audience for these volumes will have wide-ranging backgrounds and disparate interests (academic, industrial, and administrative). The ‘‘unique’’ approach of focusing on the underlying concepts should appeal to the entire community of condensed matter scientists, including graduate students and postdoctoral fellows. Further the audience will certainly include people not in the condensed science community that seek understanding of the sub-field.

VOLUME PREFACE During the past two decades, we have witnessed marvelous advances in our ability to synthesize nanoscale structures of all sorts, as well as the development of novel experimental methods that allow us to explore their physical properties. This is exciting for two reasons. First, new forms of matter with no counterpart in nature have been fabricated, and these have unique physical properties not found in bulk materials. This is so because either a large fraction of their atomic constituents reside in surface or interface sites of low symmetry, or their physical size is so small they are completely quantum dominated. Second, we have now realized nanostructures that open new avenues for the development of very small devices. This has already had a remarkable, qualitative impact on technology, as we will see from remarks in the next paragraph. It is our view that soon nanoscience and the nanotechnology derived from it will have an impact on human affairs comparable to the industrial revolution, when combined with the submicron technology of the past 10–15 years. We can appreciate this from the remarkable influence of modern information technology on our lives. This volume is devoted to the exposition of new structures and the associated physics in an important area of modern nanoscience. This is the field of nanomagnetism, where very small-scale structures such as ultrathin (few atomic layer) films of ferromagnetic material, often incorporated into superlattices or multilayer structures have been found to have magnetic and transport properties qualitatively and dramatically different than realized in bulk magnetic matter. More recently, we have seen a new generation of studies of the magnetism of patterned arrays ranging from micron-sized discs and wires down to dimers and single atoms adsorbed on substrates. Physicists have been intrigued by the new phenomena uncovered as these novel materials have been fabricated and their unique properties explored, materials scientists continue to present us with new structures, and by the time of this writing we have witnessed the enormous impact of ultra high-density magnetic data storage on computer technology. Here it is the remarkable phenomenon of giant magnetoresistance (GMR) of magnetic multilayers that has been exploited to increase the capacity of hard discs by over a factor of a hundred in a small number of years. Other exciting applications are envisioned, through the use of the systems and new concepts discussed in this volume. In this regard, we have in hand as well unique effects such as giant tunneling magnetoresistance (TMR), the phenomenon of the spin blockade, and other fascinating new effects that operate only in nanoscale magnetic systems. New methodologies developed by both theorists and experimentalists drive the field. Examples are the use of spin sensitive atomic force microscopy, and the development of large-scale computer simulations of real structures. The material xi

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in this volume is directed toward a broad audience of readers with backgrounds in condensed matter science who may not be experts in the field of nanomagnetism. It is our hope as well that the pedagogical nature of the discussions will also provide some experts with deeper insights in to the fundamental physics associated with areas of the field that have not been the focus of their own research. We comment next on the material discussed in the various chapters. The first chapter, written by the undersigned volume editors, contains a broad overview. We discuss the unique and special aspects of the magnetism of ultrathin ferromagnetic films. Spin ordering is a long-ranged phenomenon, and the excitations which control both the response characteristics and thermodynamics of ultrathin films are influenced by long ranged couplings as well. Thus, as we shrink magnetic structures down to nanometer length scales, we find fundamental differences in all aspects of their physics. In addition, a large fraction of the moment bearing ions sit in interface or surface sites, with qualitative consequences for both their magnetic and chemical properties. Ultrathin films are the ‘‘building blocks’’ of the magnetic multilayers, spin valves and related structures that have been explored and discussed intensively in recent years. We also introduce materials with lateral structure (nanodot, nanodisc and nanowire arrays), and then turn our attention to the experimental methods, which have proved central to the elucidation of the properties of very small magnetic structures. The nature of the magnetic moments found in ultrasmall structures can differ dramatically from their bulk counterparts, by virtue of the fact that a large fraction reside on surfaces, at interfaces as noted above. These also can be affected by the chemisorption of selected molecules. It follows that one realizes magnetic anisotropies one or two orders larger than that found in the bulk, and their strength and character are subject to design. Thus, we have spin engineering. R. Wu provides us with a broad survey of the ground state properties of diverse nanoscale magnetic structures. It is impressive to see the success of modern density functional theory in its ability to provide reliable quantitative accounts of the properties of these often-complex systems with low symmetry. When ultrathin ferromagnetic films are assembled into multilayers or superlattices, weak interactions of exchange character act between the constituent films. These are mediated by the spin polarization induced in nonferromagnetic spacer layers inserted between them. These weak exchange couplings, tunable both in sign and magnitude by varying structural details, lead us to magnetic entities whose underlying structure can be manipulated by very modest or weak applied fields, in contrast to bulk materials whose magnetic ions are tightly coupled by very strong interatomic exchange. This key property allows us to fabricate new, artificial materials whose magnetic properties differ qualitatively from those of bulk materials, and which can be varied over a wide range by design. M.D. Stiles provides us with discussions of the physical origin and nature of these interfilm exchange couplings. Arrays of ultrathin ferromagnetic films coupled by the exchange interactions just discussed display a rich variety of spin structures, where the total (macroscopic) magnetic moment of each ultrathin film plays the role of a large, and consequently fully classical ‘‘spin.’’ R.E. Camley discusses examples of

Volume Preface

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these structures, and the collective excitations they support. We have here the opportunity to develop new materials with microwave response tunable by the application of very modest magnetic fields. When one wishes to exploit magnetic multilayers in devices, the signal detected has its origin in the rotation of the magnetization vector in one selected film relative to that in a neighboring film, and the resulting influence on properties of the structure such as its electrical resistance, as discussed in the next paragraph. A commonly used structure is the ‘‘spin valve,’’ which consists of two ferromagnetic films separated by a nonmagnetic spacer which gives rise to the weak interfilm interactions described in the previous paragraph. A question is then how one may use an applied magnetic field to rotate the magnetization of one of the two films, while the second remains pinned in place. A phenomenon referred to as ‘‘exchange bias,’’ discovered many decades ago, allows one to selectively ‘‘pin’’ the magnetization of one layer. A.E. Berkowitz and R.H. Kodama introduce us to this central topic, whose origin is only very recently appreciated. Transport properties of magnetic multilayers have excited many researchers, since A. Fert and his colleagues reported the discovery of the astonishing phenomenon of GMR in 1988. In parallel with their work, P. Grunberg and collaborators observed this phenomenon as well at very close to the same time. The origin of GMR has stimulated efforts by many experimentalists and theorists for some years now, since it is not only a spectacular physical phenomenon, but has provided us with the basis for ultra high-density magnetic storage and its enormous impact on computer technology. A. Fert, A. Barthelemy and F. Petroff present us with a discussion of this most important phenomenon in their chapter, and cover tunneling magnetoresistance as well. Physicists, materials scientists and engineers actively discuss a new field called ‘‘spintronics,’’ wherein it is the spin of the electron rather than its charge that is exploited and manipulated. This has led to the exploration of new physics that must be understood before the field can prosper. As we have known for decades, we can inject electrons from semiconductors into metals. But if the electrons are highly spin polarized, is the spin polarization preserved or destroyed in the injection process, and if it is the latter how do we make structures that preserve spin polarization? This is a central issue that is fundamental to the new spintronics, and B.T. Jonker and M.E. Flatte provide us with an exposition of where we are at the moment, from both the experimental and theoretical perspectives. The injection of spin polarized electron currents into thin ferromagnetic films can also be used to manipulate and reorient the magnetization of the latter. D.M. Edwards and J. Mathon provide us with an understanding of the physics associated with this important process where the physics is again fascinating, and potential applications most important. As volume editors, we are very pleased indeed that leading figures in the field of nanomagnetism have come forward to provide us with their insights in the chapters summarized above. It is our hope that this volume will stimulate readers to acquire an active interest in this field, which continues to grow in size and impact. D. L. Mills and J. A. C. Bland

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Chapter 1 THE FIELD OF NANOMAGNETISM J. A. C. Bland and D. L. Mills ABSTRACT In this introductory chapter we introduce the field of nanomagnetism, briefly surveying the underlying key theoretical concepts and the experimental methods which have led to its recent rapid development and which, together, have led to key technological advances in computer read-head technology, sensors and magnetic memory. We begin by considering the magnetic length scales (e.g. spin wave length, exchange length), which define ‘‘small’’ in this context and introduce the concept of ‘‘spin engineering’’ that can be considered analogous to that of ‘‘band gap engineering’’, which has proved such a powerful concept in semiconductor physics. We describe the physics of ferromagnetism in the ultrathin film limit and consider, for example, the effect of magnetic anisotropies on the magnetic properties, such as magnetic ordering at finite temperature, the spin reorientation transition and spin wave excitations as well as interlayer coupling. We next describe the physics of magnetic nanostructures, describing the effect of finite size on the spin configurations and magnetization reversal processes in small elements. Finally, we review experimental techniques for highly sensitive magnetic measurements of nanostructures, including both static and dynamic techniques, imaging techniques with spin sensitivity and magnetoresistance measurements. Keywords: nanostructures, magnetic elements, arrays, dipole interactions, experimental techniques, BLS, MOKE, FMR, MFM, SPEELS, STM, Lorentz microscopy, spin - STM, PEEM, SEMPA, AFM, MFM, XMCD, magnetic particles, superparamagnetism, ferromagnetism, antiferromagnetism, dipolar effects, elements, anisotropies, giant magnetoresistance, tunnel magnetoresistance, magnetic random access memory, read heads, spin engineering, interlayer coupling, dynamic measurements, spin waves, magnetoresistance, spin reorientation transition, magnetic ordering, magnetic reading, fast magnetic switching, magnetic writing

Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01001-2

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1. INTRODUCTION Since the earliest days of condensed matter physics, the study of magnetically ordered materials has played a central role in establishing the fundamental principles and concepts of the field. An early example is provided by Bloch’s introduction of spin waves into physics, and his subsequent derivation of the T3/2 law for the initial decrease with increasing temperature of the magnetization of ferromagnets from the saturation value appropriate to zero temperature. This introduced the key notion that low-lying collective excitations dominate the thermodynamics of solid materials at low temperatures, one of the central differences between extended crystals with long-ranged order and small systems such as molecular entities. In the 1970s and 1980s, by virtue of the appearance of the scaling hypotheses and the related renormalization group method, theorists achieved a deep understanding of the previously challenging phenomenon of the thermodynamic phase transition. Experimental studies of the diverse forms of magnetism found in solids allowed experimentalists to verify the predictions of these theories with impressive quantitative accuracy. Perhaps the reason for the fundamental role played by magnetism in the development of fundamental concepts is that wide classes of materials can be described very well by Hamiltonians that are rather simple in structure, with few parameters in them. Thus, the theorist can apply sophisticated methodologies to these Hamiltonians, while the experimentalist can obtain an account of data with model forms that contain very few parameters. This allows quantitative tests of theory. In the current era, there is great excitement in the study of exotic quantum phenomena such as those realized in spin ladder compounds, Kagome lattices and frustrated systems. Magnetism is again playing the key role in the development of deep new fundamental concepts in condensed matter physics in this exciting new area. This volume is devoted to the topic of nanomagnetism and addresses the new physics and response characteristics encountered in magnetic materials when one or more of its linear dimensions is very small. We shall see that we encounter physics in this regime, qualitatively different from that realized in bulk magnetic materials. Our first task in this chapter is to define the sense in which the word ‘‘small’’ is used in the opening sentence of this paragraph. Broadly speaking, the new physics has its origins in two features one encounters as one or more linear dimensions of a sample are reduced. First, when one enters the nanometer regime, a large fraction of the magnetic moment bearing species resides in the surface or interface sites. For instance, if we have an ultrathin magnetic film five atomic layers in thickness, 40 percent of the magnetic ions reside in such sites. The key aspects of magnetism are very sensitive to local site symmetry, and thus the magnetism in such a film will differ qualitatively from a bulk crystal made with the same atomic constituents. One may vary magnetic properties of the ions that reside in such sites by varying the substrate or crystal face on which the film is grown and by chemisorbing material on the outer surface. One can thus create new ‘‘designer materials’’ with properties and response characteristics controlled by such features. We then have spin engineering, an

The Field of Nanomagnetism

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analog to the bandgap engineering of the semiconductor realm. In addition, an ultrathin film grown epitaxially on a substrate will have a lattice constant controlled by that of the underlying substrate and hence different from the corresponding bulk material. Also, the spacing between parallel atomic planes will differ as well, by virtue of expansion or contraction of the lattice constant parallel to the substrate surface. Hence, the site symmetries in the center of the film differ from the bulk material as well as from those on the surface or at an interface. We know very well that the fundamental interactions that control the nature of the ordered state are highly sensitive to both local geometry and interior spacings. New crystal structures not found in bulk matter are also realized in magnetic nanostructures. One may grow stable, thick epitaxial films of fcc cobalt on suitable substrates, for example, while the bulk form of this element is hcp. Second, bulk magnetic matter is characterized by fundamental length scales. The lattice constant, of course, is the smallest such length scale, but there are two other important lengths that play a central role. Each of these can be substantially larger than the lattice constant in the materials of current interest, as we shall see. When one or more sample dimensions are small or comparable to one of these fundamental lengths, we again enter a domain where the response characteristics, thermodynamic properties and other key aspects of the magnetism differ qualitatively from that found in bulk materials. One such length is the wavelength lT of thermally excited spin waves. Perhaps it is more relevant to discuss their wavevector kT ¼ 2p=lT : If L is any linear dimension of the sample of interest, then when kTLr1, we are in a regime where the fundamental magnetic properties of the material differ from those in the bulk and are affected by sample size. Since the time of Bloch, we know the dispersion relation ~ ¼ Dk2 for wavelengths long of spin waves in a ferromagnet is given by _oðkÞ compared to a lattice constant, where the parameter D is the exchange stiffness. Thus, kT
ðkB T=DÞ1=2 ; with kB being Boltzmann’s constant. We shall be interested primarily (but not exclusively) in the 3D transition metal ferromagnets in this volume. For ferromagnetic Fe, D
300 meV  A2 : Then at room temperature for Fe, we have a critical crossover length LT ¼ 1=kT
0:3 nm (nanometers), whereas if we cool down to liquid helium (He) temperature, LT
3 nm. If, then, we have an ultrathin film some two or three atomic layers in thickness, and we cool from room temperature to liquid He temperature, we cross over from a regime where the thermodynamics is three dimensional (room temperature) to a quasi-two-dimensional regime (He temperature). Similarly, if at He temperatures, we increase the thickness of the film, we can study the crossover from two to three-dimensional physics. Thus, ultrathin films (and other nanomagnetic structures) offer us the possibility of studying statistical mechanics as we make the transition from threedimensional physics to the physics of a lower dimensional world [1]. Another fundamental length is the exchange length, which for our purposes here we take to be Lex ¼ ðD=mB H A Þ1=2 ; where mB is the Bohr magneton and HA is a measure of the anisotropy field in the material. This may range from roughly 1 to 10 kG, or equivalently from 0.1 to 1 T in typical materials. The exchange length is the thickness of a domain wall of Bloch character in a bulk ferromagnet; for a Neel

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J. A. C. Bland and D. L. Mills

wall, one replaces HA by 4pMS where MS is the saturation magnetization. The magnitude of 4pMS is comparable to that of the anisotropy field. Typically, then, the exchange length lies in the range of 5–10 nm. In a ferromagnet, the cost in energy to create a non-uniform state characterized by a length scale small compared to Lex is very large. Thus, in ferromagnetic films whose thickness is small compared to this length, the magnetization is uniform in the direction normal to the surfaces. Such a film’s domain structure and hysteresis loop differ qualitatively from a bulk ferromagnet with its complex domain structure in the form of a three-dimensional network. Thus, in this volume our attention is directed toward the magnetism found in samples sufficiently small that a large fraction of the moment bearing ions reside in surface or interface sites, whose geometrical structure is controlled by the substrate on which it is grown or other aspects of the local environment, or whose linear dimensions are comparable to or smaller than the length scales discussed above. We conclude from the comments above that one or more of these conditions are satisfied when at least one dimension of the sample of interest is in the nanometer range. In Section 2, we focus on the specific case of the ultrathin ferromagnetic film to provide the reader with an introduction to the new physics encountered in the realm of small-scale magnetism. Nanoscale magnetism, as we shall see at various points in this volume, already has had a dramatic effect on computer technology and we expect more applications of nanoscale magnetic structures in the near future. Patterned media comprised of nanoscale elements are envisioned to be of increasing importance. Thus, this has become a very active research area in the past few years. Section 3 provides the reader with an introduction to this rapidly evolving area. The experimental study of nanoscale magnetic systems presents a challenge, since one wishes to probe a small number of magnetic moments located on a macroscopic substrate. We review the principal experimental methodologies in Section 4.

2. THE ULTRATHIN FERROMAGNETIC FILM In this section, we focus our attention on the unique aspects of ferromagnetism in ultrathin (few atomic layers) ferromagnetic films. Here we shall encounter dramatic and qualitative effects upon reducing one dimension of the sample to nanometerlength scales. It is the case as well that ultrathin ferromagnetic films such as explored here are the building blocks of many of the structures discussed later in the volume. In this brief chapter, our discussion will be somewhat schematic and sketchy. The interested reader may wish to consult two volumes [1] each of which contains chapters that explore in detail particular properties of ultrathin magnetic films and the means of probing them experimentally. A chapter written by one of the present authors may be viewed as an elaboration of the issues explored in the present section [2]. In few-atomic layer films, as discussed in Section 1, a large fraction of the magnetic moment bearing ions reside in sites of low symmetry, in the surface of the film

The Field of Nanomagnetism

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or at the interface between the film and the substrate upon which it is grown. This has a strong, qualitative influence of a key property of ferromagnets, their magnetic anisotropy. We begin by introducing the reader to the concept of magnetic anisotropy and its role. Suppose we begin by considering the simplest description of a crystalline array of magnetic moments provided by the well-known Heisenberg model, wherein pairs of spins interact by means of the exchange interaction whose ~2 : The total magnetic energy is found by summing ~1  S form may be written as J 12 S such interactions over all spins in the crystal. If the ground state is ferromagnetic, the only case considered in our brief remarks here, the energy of the spin array is left unchanged by a rigid rotation of all the spins in the system. Hence, the spontaneous magnetization in the ferromagnetic state may point in any direction it desires. It is not tied to the crystalline axes in any manner, for example. In real materials, there are terms in the Hamiltonian that select out particular directions as ‘‘easy axes’’ along which the magnetization prefers to be directed. One refers to such terms as the anisotropy energy. There are two interactions which enter centrally: the long-ranged magnetic dipolar couplings between the atomic magnetic moments, and spin–orbit interactions of the moment bearing electrons which allow the moments to sense the local crystalline axes, as we shall see below. We consider the role of magnetic dipolar interactions first. The magnetic field seen by a given moment with origin in a distant neighbor falls off as 1/r3, with r being the distance between them. Thus, the total magnetic dipole interaction energy of the system is proportional to Z Z 3 3 d rð1=r Þ  drr2 ð1=r3 Þ (1) V

V

where the integral is over the volume of the sample. The integral clearly diverges for a truly infinitely extended sample, and thus we must take due account of the finite sample size to evaluate this energy. When this is done, the overall shape of the sample controls the direction of the easy axis, if only dipolar anisotropy is present. For this reason, in the literature on magnetism, dipolar anisotropy is referred to often as ‘‘shape anisotropy.’’ We may appreciate the point just made for a ferromagnetic film, viewed in ~ S be canted out of the plane of macroscopic terminology. Let the magnetization M the film by the angle y. If z^ is the normal to the film surfaces, it is a matter of elementary magnetostatics to show that inside the film, one has a demagnetizing ~dip ¼ ^z4pM S sin y: This is antiparallel to the component of magnetization field H normal to the surface, and thus any canted, out-of-plane state of the magnetization is energetically unfavorable. Stated otherwise, the energy per unit volume associated with a state where the magnetization is canted out of plane is given by, again from elementary magnetostatics, þ2pM 2S sin2 y and the state of lowest energy has the magnetization in plane, with y ¼ 0: Thus, we conclude that if dipolar interactions are the only source of anisotropy in a ferromagnetic film, the shape anisotropy forces the magnetization to always lie parallel to the film surfaces. One may wonder if such macroscopic considerations apply to few atomic layer films. In this limit, one

J. A. C. Bland and D. L. Mills

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resorts to a lattice description, and interestingly, the macroscopic results prove rather accurate even close to the monolayer level [3]. As remarked above, the spin–orbit interaction allows a magnetic moment to sense the symmetry of its local neighborhood through interaction of the spins with crystalline electric fields associated with neighboring ions. This competes with shape anisotropy, and the resulting orientation of the magnetization is controlled by the one that is dominant. We may appreciate the role of spin–orbit interactions through a simple schematic argument. We introduce into the Hamiltonian the spin–orbit ~ with L ~  S; ~ being the orbital angular interaction, which we may write as lL momentum. The dependence of the energy of a local magnetic moment in a crystal as a function of its spin orientation may be found by the spin Hamiltonian method [4]. This approach notes that in the absence of spin–orbit coupling, which is a weak perturbation to the internal crystalline field for the 3d transition metal ions of primary interest to us, the wave function in the ground state is a product of the orbital wave function and the spin wave function. One generates an effective spin Hamiltonian through perturbation theory in which the coordinates of the electron are integrated out in the various matrix elements, but the spin operators are left standing to operate on the ground-state multiplet. Only even orders of perturbation ~ in any effective theory enter, since time-reversal symmetry forbids odd powers of S spin Hamiltonian. Thus, second-order perturbation theory leads to an effective spin Hamiltonian of the form X ð2Þ H ð2Þ K ab Sa Sb (1a) SO ¼ a;b

where K ð2Þ ab

¼l

2


X o0jLa
n4onjLb j04 n

E0  En

(1b)

Local site symmetry controls the structure of the right-hand side of Eq. (1a). Suppose we consider a bulk ferromagnet wherein each ion is positioned at a site of cubic ð2Þ symmetry, as in the case of Fe and Ni. Then the second-rank tensor K ð2Þ ab ¼ K dab ; 2 2 2 so the right-hand side of Eq. (1a) is proportional to S x þ S y þ Sz ¼ SðS þ 1Þ: There is thus no dependence on the direction of the spin, and we must turn to fourth-order terms, which for a cubic site have the form X ð4Þ 4 4 4 2 2 2 2 2 2 H ð4Þ ½K ð4Þ (2a) a fS x þ S y þ S z g þ K b fS x S y þ S y S z þ S z S x g SO ¼ a:::d

This may be rearranged to read 2 2 2 2 2 2 ð4Þ 2 ð4Þ H ð4Þ SO ¼ K a fSðS þ 1Þg þ DK fS x S y þ S y S z þ S z S x g ð4Þ

K ð4Þ b

2K ð4Þ a : ð4Þ

ð4Þ

(2b)

where DK ¼  If DK 40; the magnetization will align along a [100] direction, whereas if DK 40; the magnetization will align along a [111] direction. Even though the anisotropy energy per ion is very small, the order of 0.01 meV for bulk Fe and Ni, the moments are rigidly coupled together by the very strong exchange interactions, so the energy per unit volume associated with reorientation

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of the magnetization is substantial, the order of 106 erg/cm3. A prediction of the sign of DK(4) is challenging to theorists, since this depends on subtle details of the electronic structure, as one can appreciate from the complexity of the fourth-order perturbation theory. In bulk Fe, the [100] directions are the easy axes, while in bulk Ni, the [111] directions play this role. To return to our example of the film, for thick films of cubic ferromagnets where the bulk crystalline structure is realized throughout the film, the strong shape anisotropy drives the magnetization into the plane of the film, and the much weaker spin–orbit anisotropy controls its orientation in the plane. Now let us turn to an ultrathin film, where a very large fraction of the moment bearing ions reside in surface or interface sites. In such sites, cubic symmetry is strongly broken, and the normal to the surface, the z^ direction, is different from the ð2Þ 2 2 2 x^ and y^ directions. The form in Eq. (1a) then becomes K ð2Þ a fS x þ S y g þ K b S z ð2Þ 2 ð2Þ 2 ð2Þ ð2Þ or K a SðS þ 1Þ þ DK Sz where DK ¼ K a  K b : Thus, the low site symmetry realized at surface and interface sites activates the second-order terms in the spin– orbit energy, which are ‘‘silent’’ at cubic sites. The perturbation theoretic arguments sketched above tell us that the quadratic terms are larger than the quartic terms by roughly ðDE=lÞ2 ; with DE being a measure of energy level splittings in the d shell produced by crystalline electric fields. For the 3d transition metal ions, very roughly l0.1 eV whereas DE1 eV, so the second-order terms can be two orders of magnitude larger than the quartic terms. Hence, in an ultrathin film, the spin–orbit anisotropy activated by the low-symmetry sites at surfaces and interfaces can be two orders of magnitude larger than that found in bulk materials synthesized from the same atomic species. This is thus an enormous effect. In fact, the surface and interface anisotropy can often overwhelm the dipolar anisotropy discussed above. Thus, in films where DK ð2Þ o0; we can have films with magnetization perpendicular to the surface, quite in contrast to the conventional orientation. The sensitivity of the sign and magnitude of DK(2) to local environment means one can engineer ultrathin forms of magnetic materials to have anisotropy suited to a particular purpose by varying the substrate upon which the film is grown, or adsorbing material on its outer surface. It remains the case that in the materials of interest here, surface anisotropy remains small compared to the very strong interspin exchange couplings that drive the ferromagnetic order. Hence, as one probes anisotropy by using external magnetic fields to reorient the magnetization, it precesses rigidly. A consequence is that the classic signature of surface or interface anisotropy is its strength that varies inversely with the thickness of the film [5]. Thus, it is common in very thin films for surface anisotropy to dominate in the few-atomic layer limits, whereas as film thickness increases, one realizes a transition to shape anisotropy as the dominant source. One can then realize, with increasing film thickness, the interesting phenomenon of the spin reorientation transition (SRT). Given an ultrathin film where surface anisotropy orients the magnetization normal to the surfaces, as the thickness increases and one enters the regime where dipolar anisotropy dominates, the magnetization reorients parallel to the surface. In a particular film with fixed thickness, the reorientation transition can be realized in the form of a

J. A. C. Bland and D. L. Mills

8

thermodynamic phase transition driven by the difference in the temperature dependencies of the surface and shape anisotropies [6]. So far, in our discussion of ultrathin films, we have described two sources of anisotropy, the shape or dipolar anisotropy, and that associated with surface or interface sites where the low site symmetry activates the quadratic terms in the spin– orbit energy that are inoperative in bulk cubic materials. There is one more very important source of anisotropy in ultrathin films we must discuss, since it also plays an important role in films from materials whose bulk form is cubic. High-quality epitaxial films are grown typically on substrates whose lattice constant is close to that of the bulk crystalline form of the material from which the film is fabricated. However, the mismatch in lattice constant can still be appreciable, in the range of a few percent. If, for example, a film of a nominally cubic material is grown on a substrate whose lattice constant is a bit larger than that of the bulk material, then epitaxial growth requires the film to have an expanded unit cell in the plane. It follows that the spacing between the layers will contract somewhat, so the unit cell even in the center of the film will be tetragonal rather than cubic. This will then activate the quadratic terms in Eq. (1a), to produce uniaxial anisotropy similar to that found at the surface or interface sites. One may view the origin of this third form of ultrathin film anisotropy as having its origin in the phenomenon of magnetostriction, which is the dependence of the anisotropy energy on strain. Imagine assembling an ideal film in free space, where every atom is present in a site appropriate to the bulk crystalline form. Now bring such a film into contact with a substrate whose lattice constant is slightly mismatched. One must strain the ideal film for it to fit onto the substrate with the required perfect lattice match. The coefficients in Eq. (1a) may be regarded as a function of the strain tensor egd: X ð2Þ K ð2Þ ð@K ð2Þ ab ðfegd gÞ ¼ K dab þ ab =@egd Þ0 egd gd

The second terms lead to a contribution to the anisotropy energy with a form similar to that realized at the surface and interface sites. Since the strain is modest, the anisotropy energy per layer from this source will be considerably smaller than that from the interfaces, where the change in symmetry is so dramatic. In films that contain several layers, however, the magnetostrictive anisotropy associated with each layer can sum up to be substantial. An interesting example is the case of Ni films on Cu(100) [7]. Such films may be grown to appreciable thicknesses, in the tetragonally distorted shape. When the films are very thin, the interfacial anisotropy favors magnetization in plane, as does the shape anisotropy, which is rather weak here because of the small saturation magnetization of Ni. However, the magnetostrictive component favors out-of-plane alignment. Hence, when thickness is increased, the films undergo a reorientation transition from magnetization parallel to the surfaces to the perpendicular state, a behavior quite the opposite of that discussed earlier. The anisotropy energy controls the orientation of the magnetization in the ground state of the ultrathin film, as we have seen, and also the response of the film to applied DC or very low-frequency external magnetic fields. As remarked earlier, since the strength of the effective anisotropy field seen by any one particular spin is

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small compared to the interatomic exchange fields responsible for ferromagnetic alignment of the spins, the spins remain locked into ferromagnetic alignment in response to low-frequency applied magnetic fields, and the macroscopic magnet~ S precesses or rotates in response to such a field. The ability to ization vector M create ultrathin films with desired orientation of the moment in the ground state or desired response to external fields which vary slowly in time allows one to create ‘‘designer materials’’, particularly when this feature is combined with the ability to synthesize magnetic multilayers with the effective interfilm exchange couplings described in the M. Stiles chapter in the present volume. In the literature on ultrathin film and multilayer magnetism, the nature of the anisotropy realized in any structure and the ability to control it through design, or ‘‘spin engineering’’ of materials is a central issue. The reason is that a central role is played by magnetic nanostructures in current device technology, as various authors in this volume discuss. Such devices function by either switching the magnetization or rotating it through application of external magnetic field pulses of a suitable nature. The ability to do this with applied magnetic fields of magnitude suitable for device applications is thus of great significance, and as we have seen, the orientation of the magnetization is controlled by the various forms of anisotropy. As the magnetization is rotated from one easy direction to another, it must surmount a barrier associated with the ‘‘hard axes’’ along which the magnetic anisotropy is a maximum. The height of these barriers is a crucial issue. In the next section of this chapter, we introduce the reader to a number of issues of importance to device applications, and other means of structuring materials to achieve desired response characteristics. The above discussion explores only ground-state properties of the film, in the sense that the magnetization adiabatically follows the slowly varying fields mentioned in the previous paragraphs. The response of the film to an external probe of finite frequency is controlled by its spectrum of elementary excitations, the spin waves that enter the theory of ferromagnetism [8]. These are characterized by a ~ which describes the frequency of the wave as a function of dispersion relation OðkÞ; ~ which in the case of our film, will lie in the appropriate twoits wave vector k; dimensional Brillouin zone. The dispersion relation and character of these modes is influenced by the interatomic exchange interactions, and the anisotropy energies discussed above. The excitation spectrum of the ultrathin ferromagnetic film has unique features not found in bulk magnetic matter, and these influence the response of the film to microwave fields in a fundamental manner. As the spins precess after a spin wave is excited, their motion generates timedependent magnetic dipole fields, which react back on the spin system. These have a qualitative effect on the dispersion relation, in the limit of long-wavelength or small-wave vector, the dispersion relation depends on the direction of propa~ S : In bulk ferromaggation of the wave relative to the magnetization vector M ~ ¼ g½ðH 0 þ Dk2 Þ nets, the effect is well known, and the dispersion relation is OðkÞ 2 1=2 2 ðH 0 þ 4pM S sin yk þ Dk Þ ; where the term in 4pMS has its origin in the dipolar fields generated by the spin motions. Here H0 is an applied DC magnetic field, D is the exchange stiffness with origin in the interatomic exchange couplings and yk is the angle between the wave vector and the magnetization [9]. We have ignored

10

J. A. C. Bland and D. L. Mills

anisotropy, the focus of our earlier discussion, but this may be incorporated into H0 as an effective field. In the ultrathin ferromagnets, the long-ranged dipolar fields generated by the spin motions have the most peculiar influence on the dispersion relation at long wavelengths. They produce a term linear in wave vector at long wavelengths, with coefficient whose sign depends on yk. Thus, we have directions of propagation where the initial slope of the dispersion curve is negative. The influence of exchange, through terms such as the Dk2 in the formula quoted above, dominate at larger wave vectors, to produce a rather strange dispersion relation in which the minimum spin wave frequency occurs at a finite wave vector km(yk), for those directions for which the linear term in the dispersion relation is negative. This peculiar feature of spin waves in the ultrathin ferromagnets has the most important consequence. When one considers the damping of long-wavelength spin motions in the film, an extrinsic damping mechanism referred to as two-magnon damping is activated by this feature of the dispersion relation [10]. In real films, under a variety of circumstances, this can be the dominant source of damping. Thus, consideration of its role becomes most important in the design of devices, which exploit the rapid response of the magnetization. We refer the reader to a review that introduces this mechanism and describes several striking consequences of its influence on the response characteristics of ultrathin films [10]. As we move out to short wavelengths, interatomic exchange dominates and controls the dispersion relation and properties of the spin waves, again as one can see from the dispersion relation quoted earlier for waves in bulk materials. As we move into this domain, it is now clear that an issue ignored so far in this chapter asserts itself most strongly: the films of interest are metallic ferromagnets, or itinerant ferromagnets to employ another term. The moment bearing electrons move in energy bands of considerable width, approximately 4 eV for the d bands of the 3d metals. If we set this issue aside, and view our film as N layers of magnetic moments each of which is described as interacting with neighbors through Heisenberg ex~2 ; then one can see easily that for each wave vector k~ ~1  S change of the form J 12 S in the two-dimensional Brillouin zone, we have N spin wave frequencies, and hence N branches to the dispersion curve. In such a picture, each mode is undamped, with infinite lifetime. The lowest lying such branch, for small values of k~ is the lowfrequency mode discussed above, relevant to the microwave response of the film. However, theory of spin waves in truly itinerant ultrathin ferromagnetic films predicts that throughout much of the Brillouin zone, the spin wave modes are heavily damped by virtue of coupling to the spectrum of electronic excitations in the film/ substrate complex, to the point that only a single rather broad feature is evident in calculated frequency spectra [11] for a given wave vector. This feature moves with k~ in a manner quite similar to a single branch of a dispersion curve. Very interesting recent experiments provide us with our first data on short-wavelength spin waves in ultrathin films [12] through the method of spin-polarized electron energy loss spectroscopy (SPEELS), and this is precisely the behavior found. Calculations directed toward the system studied in the experiments provide a very good account of the data [13]. As metallic ferromagnetic structures of nanoscale dimensions are incorporated into devices, the strong damping revealed by the studies just cited will

The Field of Nanomagnetism

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become an issue, whenever spin motions with strong spatial gradients are excited. The issue will thus emerge as one of broad interest in the coming years. It is well known again since the time of Bloch that at finite temperatures, spin waves are present as thermal excitations, and drive the thermal fluctuations found in the system. The magnetization MS(T) then falls below the value MS(0) appropriate to the absolute zero of temperature. We may thus write M S ðTÞ=M S ð0Þ ¼ 1  DðTÞ where in bulk ferromagnets, we have the famous result of Bloch, wherein D(T)T3/2 [14]. Straightforward application of Bloch’s ideas to the ultrathin films leads to a disaster. In temperature units, the anisotropies and Zeeman energies discussed above are negligibly small, so the dispersion relation of the thermally excited spin waves is described very well by ignoring these small terms. Hence, we ~ ¼ Dk2 for calculating thermal properties of the ferromagnet well may take OðkÞ below the Curie temperature. In spin wave theory we have 8 9 <X = ~ =N DðTÞ ¼ nðkÞ : ~ ; k

~ ¼ 1=½expð_OðkÞÞ ~  1 is the where N is the number of spins in the system and nðkÞ Bose–Einstein occupation factor for the thermally excited spin waves. When the sum is converted to an integral for a two-dimensional film, a logarithmic divergence is found in the integral as k~ ! 0; so that at any small but finite temperature, one finds DðTÞ ¼ 1: Thermal fluctuations associated with very long- wavelength spin waves are so violent in the two-dimensional world that at any finite temperature, no matter how small, they disrupt the long-ranged order associated with ferromagnetism. The above argument of a ‘‘hand waving’’ nature is thus not rigorous. However, the same conclusion follows from a rigorous theorem in statistical mechanics known as the Mermin Wagner theorem [15]. The theorem states that long-ranged ferromagnetic order is absent in two dimensions, for any spin system described by an underlying Hamiltonian, which is form invariant under a rigid body rotation of all the spins. The theorem applies to the Heisenberg ferromagnet in two dimensions, and also equally well to itinerant materials as well, so long as the Hamiltionian contains only the kinetic energy term for the electron and the Coulomb interaction between them. Despite the Mermin Wagner theorem, it is an experimental fact that ultrathin films display ferromagnetism at finite temperature, many right down to the monolayer level. The Curie temperature may be depressed relative to that appropriate to the bulk material, but such materials are perfectly good ferromagnets so far as experiment is concerned. From the perspective of theory, the day is saved by the various forms of anisotropy discussed in the first portion of the present section. Quite clearly, if easy axes exist either by virtue of shape anisotropy or spin–orbit induced anisotropy, the underlying Hamiltonian is no longer form invariant under rigid rotation of the spins, and the Mermin Wagner theorem is violated. However, one’s first reaction is that the energies involved with these anisotropies are so very small compared to kBT that they cannot possibly result in Curie temperatures in the range observed for ultrathin films, which can be an appreciable fraction of the bulk

12

J. A. C. Bland and D. L. Mills

Curie temperature. In the 3d ferromagnets, Curie temperatures range from 500 to 1000 K or more. However, theory shows that even such small terms in the Hamiltonian can indeed produce substantial Curie temperatures. It has been shown [16]1 that the Curie temperature T ð2Þ C of a two-dimensional film subject to weak uniaxial anisotropy is related to the Curie temperature T ð3Þ C of its three-dimensional ð3Þ 2 counterpart through the relation T ð2Þ C ¼ 2T C = lnðp J=KÞ where J is the strength of the interspin exchange coupling, and K that of the uniaxial anisotropy. Even if ð2Þ K=J  0:01; we find T ð2Þ C  0:3T C ! Thus, very tiny anisotropies can drive the Curie temperature of the ultrathin films to substantial values. The two-dimensional Heisenberg ferromagnet is the most intriguing system. While thermal fluctuations break up its long-ranged order at any finite temperature, it really wants to order and even very weak symmetry breaking interactions such as the anisotropies discussed above can drive the Curie temperature of a monolayer film of 3d magnetic ions up to the vicinity of room temperature. Thus, anisotropy again enters as a central feature of ultrathin films, not only because of its role in devices, but because it is the key to the commonly observed ferromagnetic order in these systems. One may inquire how thick the film must be before it makes a transition from the two-dimensional behavior, just discussed, to quasi-three-dimensional magnetism. This has been addressed in Monte Carlo calculations of the Curie temperature as a function of film thickness [17], where it is found that by the time one has 10 layers or so, the Curie temperature evolves from its two-dimensional anisotropy-driven value to that appropriate to the three-dimensional limit. Thus, ultrathin ferromagnetic films are ideal systems for the study of the transition from the statistical mechanics of the two-dimensional world to that of the three-dimensional world we see around us. One may then ask how an experimentalist can interrogate properties of an ultrathin film as a function of its thickness, to examine the transition from two to three dimensions. The answer lies in the critical exponents of statistical mechanics. In the two-dimensional Heisenberg ferromagnet, theory [16 ] shows that the critical exponents should be those of the two-dimensional Ising model, even though the ratio K/J may be very small compared to unity. Thus, the critical exponent b should be found to have the Ising value of 0.125 ¼ 1/8 in the quasi-two-dimensional regime, whereas when the film becomes thick enough to display quasi-three-dimensional order, this exponent should rise to the three-dimensional value close to 1/3. In very interesting- studies of critical exponents in ultrathin Ni films, Yi and Baberschke [18] find precisely this behavior, and the crossover between two- and three-dimensional behavior indeed occurs at the few atomic layers’ level. This completes our brief discussion of the physics of ferromagnetism in ultrathin ferromagnetic films. We see that we encounter magnetism that differs qualitatively from that found in bulk magnetic matter. Furthermore, since the key property of anisotropy is controlled by atoms in surface or interface sites, one can fabricate ultrathin films with desired characteristics for device applications by ‘‘spin engineering’’, i.e. by varying the nature or crystal face on which the film is grown, by adsorbing material on the outer surface, and so on. In magnetic multilayers, nearby ultrathin films interact via weak exchange interactions mediated by the non-magnetic spacer layer which can be placed between them, as discussed in this volume by Stiles.

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This also occurs between nanoclusters or single spin-polarized atoms on appropriate metal surfaces. The strength and sign of this coupling can also be engineered through the choice of spacer material and its thickness, as we see from his discussion. We now have a very rich spectrum of new materials in hand with properties found nowhere in nature. Our focus here has been on the purely magnetic properties and response characteristic of ultrathin structures. Their transport properties also are spectacular, as we see from the discussion of giant magnetoresistance (GMR) presented by A. Fert and his colleagues in this volume. Applications of GMR to computer technology have had revolutionary consequences in the past 15 years. While GMR-based read heads have played the dominant role to date, we now see the tunnel magnetoresistance (TMR) heads beginning to be developed and we envision further applications such as magnetic random access memory (MRAM) becoming widespread in the very near future. The underlying physics is described in this volume in the chapter just mentioned. Beyond GMR and TMR we look to the field of spintronics, where the prospect of controlling the efficiency of electron spin injection and detection processes at the interfaces of heterogeneous materials offers, in principle at least, the prospect of achieving infinite magnetoresistance ratios – see the chapter by M. Flatte and B. Jonker. In applications of nanomagnetic structures, one manipulates the magnetization of a specific nanoelement. For instance, in an MRAM, a zero or one is represented by the orientation of the magnetization in very small structure. The act of writing consists of orienting the magnetization in the desired direction. Thus, the physics of magnetization reversals in nanoscale structures emerges as a central topic. Patterned arrays of nanosized magnetic elements have been intensively studied in recent years, with new applications in mind. One encounters fascinating physics in these structures as well. We turn next to a discussion of this area.

3. MAGNETIC NANOSTRUCTURES: SPIN CONFIGURATIONS AND MAGNETIZATION REVERSAL Nanoscale magnetic elements (typically 1–100 nm lateral size) and arrays are objects of intense current interest, in large measure due to their immense potential for applications in nanotechnology. These applications include memory arrays (e.g. MRAM), high-density storage media (the so-called patterned media), hard disc read heads, logic devices or miniaturized field sensors [19,20]. Since the effective fields which control the magnetic switching process (both static and dynamic) can be substantially controlled via the size, shape or density of the elements within the array, it is possible to precisely tailor the switching process for specific applications (e.g. fast writing, fast reading). Thus, we have one more feature that may be exploited in contemporary spin engineering. In general, the equilibrium magnetic states and reversal mechanisms are strongly determined by the interplay of magnetic anisotropies with the dipole fields which depend on the physical size and shape of the element, as we saw in Section 2. As the lateral size approaches 1 nm, the change in electronic structure significantly influences all key parameters such as the exchange constant,

14

J. A. C. Bland and D. L. Mills

magnetic anisotropy field (e.g. magnetocrystalline, strain, surface, atomic step, or edge induced) and spin polarization (and hence effective magnetization) as we have already noted. On this scale, the magnetic switching is profoundly modified in comparison with that of macroscopic elements, giving rise to fundamentally new phenomena. While in large structures the switching process is often well defined and fully repeatable, upon decreasing the lateral size of the element, the effective energy barrier for inducing switching (e.g. that associated with wall depinning or domain nucleation) is reduced, to the point where thermal fluctuations can induce stochastic switching [21]. Thermal effects start to become important in a single particle of volume V, and effective anisotropy Keff when K eff V  25kB T [22]. More subtly, even in micron-size elements, it is possible for a very small region of spins to control the magnetic switching, e.g. through the depinning of domain walls [23]. In this case, thermal fluctuations can influence the entire macroscopic switching process. One of the key challenges for device applications such as MRAM is in achieving sufficient control of the effective dipole field strength (which generally increases the effective switching field as the element size is reduced) and its dispersion (within an array) for a given element size. Spin engineering in multilayer elements offers a powerful strategy for controlling the dipole contribution to the switching field: in the case of a double ferromagnetic layer structure, for example, the two ferromagnetic layers can partially compensate each other’s external dipole field so they reduce the switching field without increasing the element size. This approach is already used in MRAM elements: for example, artificial antiferromagnets (comprising two exchange coupled magnetic layers separated by a thin non-magnetic layer) are used to exchange bias the hard layer (see the chapter by A. Berkowitz in this volume for a discussion of exchange bias); a double-layer structure can also be used to replace the single free layer (which responds to the external field) to achieve more reproducible switching. There is also intense current interest for applications in achieving controllable sub-nanosecond switching for ultrafast high-density writing/reading of data and therefore the dynamical switching processes in nanostructures have become a huge research field in themselves. Rapid advances in lithographic (e.g. optical – including X-ray, focused ion beam and e-beam) techniques and pattern transfer already make it possible to create extremely well-defined structures on the nanometer scale. Combined with advances in novel magnetic measurement methods, it is now possible to study the magnetic reversal process of isolated nanostructures corresponding to 100–1000 atomic spins (e.g. spin-polarized scanning tunneling microscopy (STM), ferromagnetic resonance (FMR), nano-SQUID and magneto-optical techniques) and techniques are being developed to combine high spatial resolution with fast temporal resolution to study switching events (e.g. pump-probe optical techniques) with sub-nanosecond resolution. At the same time, advances in computational techniques make it possible to accurately model the static spin configurations and magnetic reversal processes of nanostructures up to the few microns’ scale, provided the relevant magnetic parameters are known. However, a key challenge is satisfying the need to know magnetic parameters accurately as the nanometer size is reached and to include a proper description of defects, strain and chemically induced effects. For this reason, there is an intense ongoing research effort in studying well-defined nanostructures

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and comparing the experimentally observed magnetic switching behavior (both static and dynamic) with that predicted computationally. A large research effort is now being made to find the geometries that provide the simplest, fastest and most reproducible switching mechanism, which are essential prerequisites for device applications [24–28]. To control the magnetic switching precisely, one needs first to have a well-defined and reproducible remanent state (i.e. the magnetic state which develops on removing the applied field – ideally the single-domain state) and, second, the switching process itself must be simple and reproducible. In analogy with the ultrathin film limit, for a sufficiently small element – corresponding to typically a few nanometers’ lateral size in transition metals – exchange interactions give rise to a single-domain state, For larger structures, different geometries have been studied for this purpose, from simple circular discs [26,29] to more advanced needle-shaped elements [30]. In such topologically simple elements, the aim is to obtain a singledomain state and switching by coherent rotation of the magnetization [31]. However, due to the demagnetizing field induced by dipole interactions, the magnetization direction will almost always change close to the borders to form edge domains [25,29]. The magnetic configuration in these elements, therefore, is defined by the shape of the edges and is very sensitive to shape fluctuations and edge roughness. Indeed, the edge inhomogeneity in the magnetization will often dominate the magnetic switching properties; acting as nucleation sites for what is, in general, a complicated nucleation propagation switching process, where the magnetization passes through inhomogeneous intermediate states. The complex switching mechanism can lead to different remanent states depending upon the applied field history and hence influence the switching itself. One possible way to overcome these complications is to use circular elements. Circular structures have historically been of technological interest: for example, early computers used ferrite 3D rings as memory elements in a precursor to modern RAM. The perpendicular anisotropy circular disc manifests in particularly simple states: in addition to the uniform state (macrospin) stabilized in sufficient small discs, the so-called bubble bi-domain states appear for appropriate values of disc diameter and anisotropy strength [32]. In a disc with in-plane anisotropy on the other hand, one may realize the most intriguing fundamental state referred to as the vortex state, where the (in-plane) magnetization circulates in a closed loop. Vortices appear widely throughout physics and so the vortex state is of particular fundamental interest in magnetism. Magnetic force microscopy (MFM) has been used successfully to demonstrate the existence of such a state in discs with thicknesses much less than the diameters [33,34]. The magnetic flux is then closed in the element with the consequence that the edge roughness and edge domains play a minor role. The zero stray field in this state will also favor high-density storage for possible use in patterned storage media. A curious feature revealed by such experiments is that the sign of the magnetization singularity at the vortex core is uncorrelated with the vortex circulation. However, the vortex is only stable in discs for diameters typically above some 100 nm [26], depending on the thickness and the material, which limits the density achievable. Beneath this size, the exchange energy dominates and the vortex cannot form, while at very large diameters, a multidomain state again

16

J. A. C. Bland and D. L. Mills

occurs. Moreover, the vortex formation is complex and hard to control [26,28] in disc elements. As a solution to this problem, the vortex state can be made more stable if the highly energetic vortex core is removed by using the high-symmetry ring element [35], which has also been proposed for use in MRAM [19,36] and as a basis for nanoparticle sensors [20]. Apart from the improved stability, the vortex state in ring structures is also almost completely stray field free, whereas in discs, the magnetization in the vortex core points out of the plane of the film, creating a stray field. Circular elements represent only a sub-class – albeit an important one – of a veritable ‘zoo’ of magnetic elements that have been intensively studied using both magnetic imaging techniques such as MFM and computationally with the development of reliable micromagnetic codes. A key technological and fundamental issue is the effect of the dipolar coupling between elements on the reversal processes and switching fields. The challenge for studies of dipolar coupling is to define the element shape with sufficient precision to obtain meaningful results since any variation in edge definition or element shape can lead to a large variation in the interaction strength. This occurs even in the uniformly magnetized state since the dipole field at the edge varies according to the local geometry of the element boundary. In addition, the local spin configuration at the edge can change due to this variation, profoundly altering the resulting dipole field. The field due to the magnetostatic poles – equivalent to charges – created by the spin configuration in neighboring element (interelement magnetostatic coupling) contributes to the effective magnetostatic field. This term is negligible for large spacings, but becomes important for narrow spacings and so influences the switching field. With a view to using magnetic elements in applications, it is important to probe the minimum spacing where the elements can be switched independently (to obtain e.g. the maximum achievable storage density), while from a fundamental viewpoint, it is of particular interest to study the influence of domainwall coupling on distinct switching mechanisms [37]. In contrast to the interaction of single-domain particles and quasi-single-domain states in discs or squares, which have been widely investigated so far [38–41], in the case of rings, the reversal takes place by localized domain-wall processes and so the interaction between these welldefined domain walls governs the collective reversal process [42], and only manifests at small separations. A fundamental as well as practical issue of great current interest is that of the ultimate limits to spin reversal in a magnetic element. For a sufficiently small element, coherent switching of the macrospin (single-domain limit) is in principle achievable under conditions of precessional motion and so the switching time is determined by the applied field’s amplitude. In practice, precessional switching times appear to be limited (no matter how large the magnetic field pulse is) to an intrinsic time of the order of picoseconds, determined by thermal spin excitations [43], thus, significantly limiting the writing times in magnetic storage media. One intriguing possible solution to this problem has recently been proposed by Kimel et al. [44]. Whereas in a ferromagnetic material the Larmor precession frequency scales with the effective field amplitude Heff, in an antiferromagnet, this frequency scales as (HAHex)1/2 where Hex is the exchange field, HA is the effective anisotropy

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field and HexbHA. Consequently, switching times are much faster in antiferromagnetic materials. Kimel demonstrated that the effective anisotropy in an antiferromagnetic insulator such as the orthorhombic ferrites could be rapidly changed using a fast (picosecond) light pulse, which locally heats the material and induces a near-901 orientation of the spins in appropriate material. If a ferromagnetic element were to be exchange coupled to such an antiferromagnet, it can be expected that the ferromagnet can be made to precess spontaneously in the modified field created by the reoriented antiferromagnet, so offering a new route to rapid switching in magnetic elements.

4. EXPERIMENTAL TECHNIQUES Measurements of the magnetic properties of ultrathin structures are highly demanding in view of the small amount of magnetic material involved. A monolayer (ML) of Fe in a sample of area 1 cm2 has a total magnetic moment of less than 105 emu, for example, which corresponds to a signal close to the detection limit of a conventional vibrating sample magnetometer. It is therefore necessary to find methods that can probe such small signals routinely. The main challenge in using conventional magnetometric techniques in UHV for in situ measurements is the difficulty of meeting the sample–environment requirements (in particular, the ability to anneal the sample for cleaning single-crystal substrates). SQUID systems designed for the measurement of the magnetic moment of thin films can achieve sensitivities and accuracies that rival any other magnetometric technique, as exemplified by the pioneering work of Gradmann and coworkers [45] using a UHVcompatible SQUID. The advantage of in situ SQUID measurements for the case of overlayers grown on a solid substrate is that the relative bulk and magnetic thin film contributions to the total measured moment can be estimated by making thicknessdependent measurements using the same substrate. Wernsdorfer and coworkers [46,47] use micro-SQUID junctions to detect the switching fields of individual magnetic nanoparticles and have achieved sensitivities corresponding to 100–1000 spins. It is only relatively recently that the extraordinary sensitivity of the magnetooptic Kerr effect (MOKE) in probing the magnetic properties of magnetic films has been fully appreciated and now its use is widespread in the study of thin and ultrathin films [48–52]. A particular advantage of surface MOKE, pioneered by Bader and colleagues at Argonne [48], is that a polarized light beam can be readily passed through the windows of a UHV chamber and the reflected beam can also be easily analyzed using an external polarizer. Thus, in contrast with the relative complexity of polarized electron beam techniques, the MOKE can be readily adapted to the sensitive probing of the magnetic properties of ultrathin (typically 1–30 monolayer thickness for the case of Fe) films in situ. In such studies, the saturation and coercive fields can be accurately obtained as a function of the applied field orientation but the loop amplitude is frequently not analyzed. Many of these studies are made at fixed wavelength using a laser source, although a number of groups have now carried out spectroscopic measurements in which the Kerr

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rotation and ellipticity are studied as a function of wavelength. It is possible to obtain high-quality measurements using extremely simple arrangements. The most straightforward of these is based on DC detection using a high-quality photodiode and an intensity-stabilized laser source. Either the Kerr rotation can be measured by setting an analyzing polarizer close to extinction or the ellipticity can be measured by inserting a quarter-wave plate between the sample and analyzer. A great number of important insights into the origin of magnetism in ultrathin films have been obtained from in situ magnetic measurements using MOKE. Here we will briefly discuss just two illustrative examples of the use of MOKE to demonstrate how surface chemical effects can profoundly modify magnetic properties. A striking time-dependent evolution of the magnetic anisotropy of the Co/ Cu(1 1 0) system was first reported by Buckley et al. [53] using in situ MOKE. For a given deposition of Co in the thickness range 5–40 monolayers (ML), the adsorption of residual CO gas was found to cause the magnetic easy axis to switch 901 from the [0 0 1] direction to the [1–10] direction over a period of time which is dependent on the thickness of the initial Co film. This experimental study reinforces the arguments made above, where it was argued that in ultrathin materials where a large fraction of the atoms reside in surface and interface sites, the key property of anisotropy is subject to manipulation or modification through changes in the local site symmetry. We thus describe this study in some detail here to reinforce the point. Dosing experiments reveal that only a small fraction of an ML of CO is required in order to switch the easy axis of a 15ML Co film. The effect is chemically specific with oxygen, argon and hydrogen being unable to switch the easy axis. It was found that a 1 ML Cu overlayer deposited before the CO induced easy axis switch will stop the switch from occurring. This is thought to be the result of the Cu atoms occupying the specific adsorption sites preferred by the CO molecules. In support of this view, STM measurements reveal that the 3D Co growth proceeds via the formation of elongated Co island structures preferentially oriented along the [0 0 1] direction. This creates CO additional adsorption sites at the island edges. It is therefore likely that the effects seen are associated with CO preferentially adsorbing/desorbing at these specific sites. The question then arises as to whether or not it is possible to reverse the CO gas-induced 901 easy axis switch with Cu overlayers. It is found that by depositing sub-monolayer coverages of Cu onto the Co/Cu(1 1 0) system after the easy axis has been switched 901 from the [0 0 1] to the [1–10] direction by the residual CO in the UHV chamber, it is possible to fully reverse the effect of the CO, thereby switching the easy axis back to the [0 0 1] direction. Recent X-ray magnetic circular dichroism (XMCD) results [54] indicate that the effect of CO chemisorption on continuous ultrathin Co(1 1 1) films with perpendicular anisotropy is to suppress the Co orbital moment (but not the spin moment) within a restricted range of Co thickness. This results from a change in the Co d band filling due to electron back donation with the CO molecule and so is specific to CO. The effect of a change in the orbital moment is to profoundly modify (in this case to switch off) the surface anisotropy field so causing the abrupt switch in spin orientation. It is reasonable to infer that in the case of the Co island structures grown on Cu(1 1 0), a similar process is likely to be occurring at the edge of the islands,

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with the CO effectively switching off the edge-induced anisotropy. It should be recognized that the changes in effective field associated with the adsorption of CO are substantial since they compete with the dipolar fields. Electronic structure calculations [55] for CO/Co and CO/Ni [56] indicate that CO also induces a reduction in the Co or Ni moment according to the specific absorption site, but complete suppression of the spin moment is not predicted in agreement with the XMCD studies. These remarkable results illustrate the atomic-scale origin of magnetic properties in nanostructures and the striking results which can accompany subtle chemical changes which are not only site specific but involve a very small number of adsorption sites (as is clear from the low coverages involved). Such site and chemically selective effects are unique to the magnetism of nanoscale structures. Strain effects are also of great importance in the magnetism of nanostructures and chemisorption can have spectacular consequences for the magnetic anisotropy via the adsorbate-induced change in strain. Sander et al. [57] found a reversible switching of the easy axis of magnetization for Ni on Cu(0 0 1) from in-plane to outof-plane using MOKE by changing the partial pressure of hydrogen in the gas phase around the sample. A quantitative low-energy electron diffraction study of the diffracted intensity versus electron energy shows that the hydrogen-induced spin reorientation transition is accompanied by changes of the tetragonal distortion of the topmost Ni layer upon hydrogen adsorption. Surprisingly, the orientation switch perpendicular to the surface comes with a relaxation, i.e. reduction of the film’s tetragonal distortion rather than its amplification i.e. the clean Ni surface shows a strong inward relaxation, whereas upon H-coverage, the layer relaxes outward. Correspondingly, one has a large surface anisotropy in the first case, with magnetization in-plane, and a small surface anisotropy in the second case, with the magnetization out-of-plane. This behavior suggests that a structurally driven unquenching and quenching of the orbital moment occurs. This structural change is limited mainly to the top layer, and the average film structure does not change significantly. In support of this view, a surface X-ray diffraction study of Fe–Ni–W(1 1 0) did not detect any sizeable change of the average film structure upon Fe coverage and SRT from in-plane to out-of-plane [58]. We have here the second most interesting example of the chemical modification of the basic magnetic properties of ultrathin structures. In bulk magnetism, of course, the basic interactions are frozen into the crystal structure, and cannot be manipulated in this manner. MOKE techniques are also compatible with ultrafast magnetic switching measurements since the time resolution is determined by the response of the photodetector. This approach has been widely used in ultrafast (nanosecond and femtosecond range) measurements of the time-dependent reversal process in ferromagnetic elements as described above. Several groups have developed optical pump probe methods that are used in conjunction with strip lines, which generate RF field pulses at the sample position. The primary probe is used to close a photoconductive switch, which allows the field pulse to travel to the vicinity of the sample. A secondary pulse is then used to probe the magneto-optical response of the sample as a function of the time delay with respect to the initial applied field pulse. Alternately, a signal generator is used to generate the field pulse and the response is again

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monitored optically. The sample can also be placed in an external field and in this way FMR can be observed. In studies of the dynamical reversal processes, the separate vectorial components of the magnetization are directly measured in response to a finite-duration field pulse. In such measurements the sample is either combined with the strip line structure or the strip line is attached to a glass substrate which is then placed over the sample and light allowed to pass through the substrate to probe the magneto-optical signal. By either scanning the surface focusing position or using imaging optics in combination with a video camera and signal modulation, it is also possible to create movies of the spatially resolved magnetization across the sample with ultrafast time resolution. Such approaches have been used to investigate inhomogeneous modes of magnetic reversal in ferromagnetic elements and to compare the experimental results with the results of dynamic simulations. For recent reviews, see Choi and Freeman [59] and Hicken et al. [60]. RF techniques using FMR [61] and Brillouin light scattering (BLS) [62,63] have provided an important role in the quantitative determination of magnetic anisotropies and in the study of dynamical properties. In FMR the uniform mode of precession (gigahertz range) – the zero wavevector (infinite wave length) mode corresponding to the Larmor precession of the total sample moment in the effective field of the sample – is induced by an RF field applied in combination with an orthogonal static field of a strength sufficient to saturate the sample. The sample is usually placed in a resonant cavity connected via a waveguide to a Klystron source and the RF transmission strength measured as a function of the applied field strength (fixed-frequency FMR) or at fixed field as a function of frequency (swept field FMR). The resonant frequency observed at the transmission minimum is a direct measure of the effective local field (made up of the external field, dipole, effective anisotropy and exchange contributions) and by varying the applied field strength and orientation, the effective anisotropy field can be obtained directly. The line width provides a direct measure of a, the Gilbert damping constant. This parameter, which controls the strength of the damping term in the classic Landau–Lifshitz–Gilbert equation of motion for the magnetization (reference) enters importantly in the design of devices which exploit the dynamic response of the magnetization. However, at present it is largely treated as an empirical parameter with virtually no ab initio calculations available. In exchange-coupled trilayer structures two modes arise corresponding to the in-phase acoustic and anti-phase optical coupled uniform modes of the individual ferromagnetic layers. The optical mode provides a direct and accurate measure of the interlayer coupling strength [62]. Extremely high sensitivity can be achieved with single ML detection routine for samples of around 1 cm2 area. The technique has been ingeniously adapted to allow high-sensitivity spatially resolved measurements [64] by using frequency modulation techniques to probe the FMR signal via the mechanical resonance of a cantilever as in MFM. BLS uses a tandem Fabry–Perot interferometer to measure the inelastic light spectrum with very high contrast extremely close to the elastic beam. Typically, the free spectral range can be varied in the range 1–100 GHz making it ideal for the investigation of spin wave modes of thin film structures (both exchange type and dipole dominated). By focusing the incident beam it is possible to achieve high spatial

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resolution (around 1 mm) also. The use of a tandem Fabry–Perot interferometer makes extremely good rejection possible so that typically count rates of 0.01 Hz can be achieved within less than 1 GHz of the elastic beam. The technique can be readily combined with a UHV MBE chamber to allow in situ measurements during film growth. A number of groups have adopted this approach that has been shown to provide a powerful probe of the evolving dynamical properties [62,63]. Mode frequencies are identified from the increased photon energy gain or loss at a specific frequency 7o, with the intensity ratio termed the Stokes/anti-Stokes ratio. The inplane wavevector is varied by changing the angle of incidence and using a backscattering geometry so that ky ¼ 2k sin y, where y is the incidence angle and k the wavevector of the incident light. Owing to the relatively long wavelength of the Ar ion laser radiation typically used in BLS, the wavevector can only be varied up to around 105 cm1. However, this wavevector range is sufficient to observe dispersion effects, which characterize the modes and is particularly important for the study of nanostructures as we have already seen. In addition to the so-called Damon Eshbach modes localized at the film surfaces, the so-called exchange modes can be observed in which the film thickness corresponds to one or more half wavelengths of the spin wave spanning the depth of the film. In this case the exchange contributes to the effective field as in FMR and the spin wave amplitude varies across the thickness of the film. The mode frequency depends on the surface boundary conditions (usually the surface is assumed to correspond to a pure anti-node). Each exchange mode gives rise to a manifold of modes according to the value of the in-plane wavevector associated with it. However, the precise boundary conditions appropriate for nonellipsoidal nanostructures remain an open question [65]. The uniform mode of a single-domain magnetic element or particle [66] can be observed by both FMR and BLS. For example, the mode spectrum of single-crystal Fe nanoclusters have been studied using BLS [67] measurements during film growth. With increasing effective thickness, ferromagnetic (FM) order develops from the low-coverage superparamagnetic (SP) state. In the FM regime the spin wave peak line width broadens very close to dc, clearly indicating the influence of spin fluctuations in the vicinity of the phase transition. In the SP regime, where the Fe forms well-separated clusters, collective spin wave modes are still observed but with a frequency close to that of the uniform cluster mode oU By comparing the mode frequency calculated using the demagnetizing fields estimated for the average cluster dimensions determined from STM with that observed experimentally, it was possible to directly attribute the measured frequency in the SP phase to the uniform precessional modes of the clusters. It is found that the intercluster dipole coupling influences the propagation of the modes giving rise to phase coherence between the clusters (as evidenced by the observation of the Stokes/anti-Stokes ratio), but does not significantly change the measured frequency in this case. BLS techniques have been successfully used to study the way in which the spin excitations present in a thin film are modified in a magnetic element [68]. Mathieu et al. [69] were the first to report observations of discrete standing spin wave modes in laterally confined structures by this technique. The authors investigate micronsize wires and observe these modes when the transferred wave vectors are small (as

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compared to the wire dimensions) and oriented perpendicular to the wires. Such confined modes correspond to a hierarchy of dynamic states of increasing energy in analogy with the particle in a box mode found in quantum mechanics; in the case of spin waves the boundary conditions at the element edge are crucial in determining the mode structure (these again depend critically on e.g. edge anisotropies) Jorzick et al. [70] also investigated micron-size long stripes and rectangular elements with BLS. Due to the shape of these magnetic structures, the internal magnetic field is inhomogeneous (in contrast to the case of elliptical or circular dots) near the edges and there are even regions where H ¼ 0. These inhomogeneities create a potential well for propagating spin waves and consequently, modes were observed that are strongly spatially localized in these areas. A more general theoretical description of these modes has been given by Yu et al. [71]. Jorzick et al. [72] investigated micronsize circular magnetic dot arrays. Similar to the case of the wires, discrete standing spin wave modes are observed due to the in-plane confinement. In addition to that, a shift in spin wave frequency is observed for identical dot dimensions when the dot separation is varied. This finding is attributed to dipolar interdot coupling. The strength of this coupling is found to be stronger for modes with lower frequencies. Similar work on cylindrical dots has also been carried out in Gubbiotti’s group [73,74]. They investigate smaller dots (minimum radius 100 nm) than those investigated by Jorzick et al. but basically observe the same spin dynamic behavior as Jorzick et al., i.e. discrete laterally confined modes. As an alternative to BLS, Buess et al. [75] have successfully used time-resolved Kerr imaging to determine the modes of circular elements. Again, a hierarchy of modes is found, analogous to the vibrational modes of a drum, and excellent agreement between the observed and calculated spatial variations and frequencies is found. Extended antidot (i.e. hole) arrays offer the possibility of studying the effect of a periodic modulation of the local demagnetizing field on the transmission of extended spin wave modes. In the case of micron-scale square antidot arrays, for example, McPhail et al. [76] have shown that the long-wavelength (Damon Eshbach) singlefrequency mode found in a continuous film (derived from the uniform mode observed in FMR) is replaced by a broadened double mode according to the antidot size and wave propagation direction. Thus, we encounter particularly interesting mode structures in such patterned materials since the local effective field (which determines the mode frequency) varies according to the demagnetizing field spatial distribution, but the exchange field remains continuous. In contrast to the mode confinement seen in isolated elements, the two principal long-wavelength modes propagate almost independently across several repeats of the unit cell with distinct frequencies defined by the extremal regions of the demagnetizing field map. This behavior is believed to be a consequence of the specific size and separation of the antidots used in the experiment and, in principle, the degree of coupling between the principal modes can be expected to be modified according to the anti-dot dimensions. Demokritov et al. [77] described how spin waves propagate through a region of an inhomogeneous magnetic field. Spin waves are excited in a magnetized YIG film using a strip-line antenna and their propagation observed using time- and space-resolved BLS. Using a thin conductor mounted across the film, they create a

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local inhomogeneity of the magnetic field. If there are allowed spin wave states (defined by the frequency of the incoming wave and the dispersion relation) within the region of the inhomogeneous field, spin waves are transmitted through it and only change their k-vector depending on the local field. However, if there are no allowed states, the inhomogeneity acts as a tunnel barrier for spin waves. As in the case of quantum-mechanical tunneling of a small particle, the wave is partially reflected and transmitted (by tunneling), with the coefficients for each process depending on the tunnel barrier height and width. From the dependence of the transmission coefficient on the barrier width (which is non-exponential), the authors conclude that the spin waves are dominated by the long-range dipole–dipole interaction. SPEELS is capable of determining the entire surface spin wave dispersion throughout the Brillouin zone and has been successfully applied to the study of ultrathin fcc Co thin films [12]. However, in the case of Fe films, less well-defined modes are observed, possibly due to the surface modes entering the Stoner continuum. Several powerful high-sensitivity imaging techniques have been developed or advanced recently: for example, Atomic force microscopy (AFM) and MFM [78], scanning tunneling microscopy (STM) [79], spin-polarized STM [80], scanning Kerr microscopy, scanning electron microscopy with polarisation analysis (SEMPA) [81], photoelectron emission microscopy (PEEM) [82] and Lorentz microscopy [83]. The imaging of surfaces and of the magnetic configurations have provided valuable information about the surface morphology and the possible magnetic states either at remanence or during the switching of the magnetization process, respectively. This has led to a better understanding of both static and dynamic processes in magnetic films and elements. Probably, the most exhaustive review of magnetic domain imaging to date is that by Hubert and Scha¨fer [84]. Using X-ray optics, high resolution (around 100 nm) can be achieved with XMCD for studying magnetic domains with element specificity and the technique is unique in its ability to probe antiferromagnetic (AF) domains [85]. While there is insufficient space to describe all of these imaging methods in detail, we shall briefly discuss spin-polarized STM which is already making a profound impact on the field having taken almost 20 years to successfully develop. A successful approach to spin STM has been demonstrated by the Hamburg group and used for several studies of magnetic films and surfaces [21]. In their approach, a clean tungsten tip is coated in UHV with an ultrathin ferromagnetic film (e.g. Fe). The spin polarization of the tunneling current is then determined by the magnetic orientation of the magnetic atom or atoms at the tip. The method has also been shown to be compatible with magnetic field-dependent measurements [86]. Using I–V measurements, it is possible to correlate measurements of the electronic structure with the magnetic and physical information provided by STM [21,87]. When a spin split surface state is present, this provides a powerful means of distinguishing surface polarization effects from those of the bulk. Ravlic et al. [88] have described the correlation of structural, local, electronic and magnetic properties of Fe/Cr(0 0 1) using this method. Kawagoe et al. [89] have also used this approach to probe topological antiferromagnetic order on ultrathin Cr(0 0 1) film surfaces. One of the surprising results to have been obtained from the spin STM

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studies is the finding that in ultrathin Fe/W(1 1 0) films the domain walls are extremely narrow, indicating a very substantially reduced surface exchange parameter [21]. The Halle group [90] has developed an alternative approach to spin STM using a magnetically soft ferromagnetic tunneling tip. The tip magnetization can then be modulated at high frequency (which exceeds the bandwidth of the feedback system for the STM stabilization) using a miniature Helmholtz coil so that spin information can be probed using the AC contribution to the tunneling signal. This method has been applied to the study of magnetic domains in ultrathin Co films and again it is found that extremely narrow (2 nm) domain walls arise. Magnetoresistance measurement techniques have also been very effectively applied to the study of reversal processes in nanostructures and have the advantage over conventional techniques that the sensitivity does not decrease with the size of the sample (for a recent review on magnetoresistance, see [91]). Following the pioneering work of Berger [92,93] on current-induced magnetic switching, the huge research activity on magnetoresistance effects has led to a renewal of interest in the investigation of current-induced domain-wall motion [94–96]. This effect shows potential for novel memory and logic devices based on domain-wall propagation or magnetic reversal in nanoelements, as it could simplify designs by eliminating magnetic field-generating circuits as well as lowering the power requirements for switching. While field-induced domain-wall motion is well established, currentinduced domain-wall motion still lacks a thorough understanding. Several effects occur when large electrical currents flow across a domain wall, the most prominent ones being the action of the field created by the current itself (the so-called Oersted field) and the spin-transfer mechanism, also known as spin torque effect [97]. Since the original work by Berger, a number of different theories have been proposed that treat the interactions between the spin-polarized current and the magnetization, but the appropriate form of the spin-transfer contribution still is the subject of much debate. It is clear that this subject will continue to see an intense interplay between theory and experiment as our basic understanding of this fascinating effect develops – see the chapter by Mathon and Edwards which discusses fundamental models of current-induced switching.

NOTES 1. The final result in Ref. [16] for the two-dimensional ordering temperature should be multiplied by a factor of two, as pointed out in Ref. [17].

REFERENCES [1] Ultrathin Magnetic Structures, vols. 1 and 2, edited by B. Heinrich and J.A.C. Bland (Springer Verlag, Heidelberg, 1994). [2] D.L. Mills, Thermodynamic properties of ultrathin ferromagnetic films, in Ultrathin Magnetic Structures, vol. 1, edited by B. Heinrich and J.A.C. Bland (Springer, Heidelberg, 1994), chap. 3, p. 91. [3] H. Benson and D.L. Mills, Spin waves in thin films; dipolar effects, Phys. Rev. 178, 839 (1969).

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[4] A. Abragam and B. Bleaney (Eds), Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970), chap. 7. [5] B. Heinrich, K.B. Urquhart, A.S. Arrott, J.F. Cochran, K. Myrtle and S.T. Purcell, Ferromagnetic resonance study of ultrathin bcc Fe(1 0 0) films grown epitaxially on fcc Ag(1 0 0) substrates, Phys. Rev. Lett. 59, 1756 (1987); D.L. Mills, Ferromagnetic resonance in ultrathin film structures, in Ultrathin Magnetic Structures, vol. 2, edited by B. Heinrich and J.A.C. Bland (Springer, Heidelberg, 1994), chap. 3. [6] D.P. Pappas, K.-P. Ka¨mper and H. Hopster, Reversible transition between perpendicular and in-plane magnetization in ultrathin films, Phys. Rev. Lett. 64, 3179 (1990). [7] B. Schulz and K. Baberschke, Crossover from plane to perpendicular magnetization in ultrathin Ni/Cu(1 0 0) films, Phys. Rev. B 50, 13467 (1994). [8] C. Kittel, Introduction to Solid State Physics, 7th ed (Wiley, New York, 1996), p. 450. [9] C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963), Eq. (1 1 0), p. 67. [10] D.L. Mills and S. Rezende, Spin damping in ultrathin ferromagnetic films, in Spin Dynamics in Confined Magnetic Structures II edited by B. Hillebrands and K. Ounadjela (Springer, Heidelberg, 2003), chap. 2. [11] A.T. Costa, R.B. Muniz and D.L. Mills, Theory of spin excitations in Fe(1 1 0) multilayers, Phys. Rev. B 68, 22435 (2003). [12] R. Vollmer, M. Etzkorn, P.S. Anil Kumar, H. Ibach and J. Kirschner, Spin polarized electron spectroscopy of high-energy, large wave vector spin waves in ultrathin fcc Co films on Cu(1 0 0), Phys. Rev. Lett. 91, 147201 (2003). [13] A.T. Costa, R.B. Muniz and D.L. Mills, Theory of large wave vector spin waves in ultrathin ferromagnetic films, Phys. Rev. B 70, 054406 (2004). [14] See chap. 4 of Ref. [9]. [15] N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or twodimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966). [16] M. Bander and D.L. Mills, Ferromagnetism of ultrathin films, Phys. Rev. B 38, 12015 (1988). [17] R.P. Erickson and D.L. Mills, Anisotropy-driven long-range order in ultrathin ferromagnetic films, Phys. Rev. B 43, 11527 (1991). [18] Y. Yi and K. Baberschke, Dimensional crossover in ultrathin Ni (1 1 1) films on W(1 1 0), Phys. Rev. Lett. 68, 1208 (1992). [19] J.G. Zhu, Y. Zheng and G.A. Prinz, Ultrahigh density vertical magnetoresistive random access memory, J. Appl. Phys. 87, 6668 (2000). [20] M.M. Miller, G.A. Prinz, S.F. Cheng and S. Bounnak, Detection of a micron-sized magnetic sphere using a ring-shaped anisotropic magnetoresistance-based sensor, a model for a magnetoresistancebased biosensor, Appl. Phys. Lett. 81, 2211 (2002). [21] M. Bode, Spin-polarized scanning tunneling microscopy, Rep. Prog. Phys. 66, 523 (2003). [22] G. Bertotti, Hysteresis in Magnetism (Academic Press, Turin, 1998). [23] T. Hayward, T.A. Moore, D.H.Y. Tse, J.A.C. Bland, F.J. Castano and C.A. Ross, Exquisitely balanced thermal sensitivity of the stochastic switching process in macroscopic ring elements, Phys. Rev. B 72, 184430 (2005). [24] J. Rothman, M. Klaeui, L. Lopez-Diaz, C.A.F. Vaz, A. Bleloch, J.A.C. Bland, Z. Cui and R. Speaks, Observation of a bi-domain state and nucleation-free switching in mesoscopic ring magnets, Phys. Rev. Lett. 86, 1098 (2001). [25] Y. Zheng and J.G. Zhu, Switching field variation in patterned submicron magnetic film elements, J. Appl. Phys. 81, 5471 (1997). [26] R.P. Cowburn, D.K. Koltsov, A.O. Adeyeye, M.E. Welland and D.M. Tricker, Single-domain circular nanomagnets, Phys. Rev. Lett. 83, 1042 (1999). [27] G. Gubbiotti, L. Albani, G. Carlotti, M. De Crescenzi, E. Di Fabrizio, A. Gerardino, O. Donzelli, F. Nizzoli, H. Koo and R.D. Gomez, Finite size effects in patterned magnetic permalloy film, J. Appl. Phys. 87, 5633 (2000). [28] T. Pokhil, D. Song and J. Nowak, Spin vortex states and hysteretic properties of submicron-size NiFe elements, J. Appl. Phys. 87, 6319 (2000).

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[29] O. Fruchart, J.P. Nozieres, W. Wernsdorfer, D. Givord, F. Rousseaux and D. Decanini, Enhanced coercivity in submicrometer-sized ultrathin epitaxial dots with in plane magnetization, Phys. Rev. Lett. 82, 1305 (1999). [30] J. Yu, U. Ruediger, L. Thomas, S.S.P. Parkin and A.D. Kent, Micromagnetics of mesoscopic epitaxial (1 1 0) Fe elements with nanoshaped ends, J. Appl. Phys. 85, 5501 (1999). [31] E.C. Stoner and E.P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Phil. Trans. R. Soc. A 240, 599 (1948). [32] S. Komineas, C.A.F. Vaz, J.A.C. Bland and N. Papanicolaou, Bubble domains in disc-shaped ferromagnetic particles, Phys. Rev. B 71, 060405 (2005). [33] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono, Magnetic vortex core observation in circular dots of permalloy, Science 289, 930 (2000). [34] M. Hehn, K. Ounadjela, J.P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian and C. Chappert, Nanoscale magnetic domains in mesoscopic magnets, Science 272, 1782 (1996). [35] L. Lopez-Diaz, J. Rothman, M. Klaeui and J.A.C. Bland, Computational study of first magnetization curves in small rings, IEEE Trans. Magn. 36, 3155 (2000). [36] M. Klaeui, J. Rothman, L. Lopez-Diaz, C.A.F. Vaz, J.A.C. Bland and Z. Cui, Vortex circulation control in mesoscopic ring magnets, Appl. Phys. Lett. 78, 3268 (2001). [37] M. Klaeui, C.A.F. Vaz, J.A.C. Bland, T.L. Monchesky, J. Unguris, E. Bauer, S.H.S. Cherifi, A. Locatelli and L.J. Heyderman, Direct observation of spin configurations and classification of switching processes in mesoscopic ferromagnetic rings, Phys. Rev. B 68, 134426 (2003). [38] M. Natali, I.L. Prejbeanu, A. Lebib, L.D. Buda, K. Ounadjela and Y. Chen, Correlated magnetic vortex chains in mesoscopic cobalt dot arrays, Phys. Rev. Lett. 88, 157203 (2002). [39] V. Novosad, M. Grimsditch, J. Darrouzet, J. Pearson, S.D. Bader, V. Metlushko, K. Guslienko, Y. Otani, H. Shima and K. Fukamichi, Shape effect on magnetization reversal in chains of interacting ferromagnetic elements, Appl. Phys. Lett. 82, 3716 (2003). [40] C. Mathieu, C. Hartmann, M. Bauer, O. Buettner, S. Riedling, B. Roos, S.O. Demokritov, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, E. Cambril, A. Mu¨ller, B. Hoffmann and U. Hartmann, Anisotropic magnetic coupling of permalloy micron dots forming a square lattice, Appl. Phys. Lett. 70, 2912 (1997). [41] M. Natali, A. Lebib, Y. Chen, I.L. Prejbeanu and K. Ounadjela, Configurational anisotropy in square lattices of interacting cobalt dots, J. Appl. Phys. 91, 7041 (2002). [42] M. Kla¨ui, C.A.F. Vaz, J.A.C. Bland and L.J. Heyderman, Domain wall coupling and collective switching in interacting mesoscopic ring magnet arrays, Appl. Phys. Lett. 86, 032504 (2005). [43] I. Tudosa, C. Stamm, A.B. Kashuba, F. King, H.C. Siegmann, J. Sto¨hr, G. Ju, B. Lu and D. Weller, The ultimate speed of magnetic switching in granular recording media, Nature 428, 831 (2004). [44] A.V. Kimel, A. Kirilyuk, A. Tsvetkov, R.V. Pisarev and Th. Rasing, Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3, Nature 429, 850 (2004). [45] U. Gradmann, Magnetism in ultrathin transition metal films, in Handbook of Magnetic Materials, vol. 7, edited by K.H.J. Buschow Elsevier, Amsterdam, 1993), p. 1. [46] W. Wernsdorfer, K. Hasselbach, D. Mailly, B. Barbara, A. Benoit, L. Thomas and G. Suran, DC SQUID magnetization measurements of single magnetic particles, J. Magn. Magn. Mater. 145, 33 (1995). [47] W. Wernsdorfer, D. Mailly and A. Benoit, Single nanoparticle measurement technique, J. Appl. Phys. 87, 5094 (2000). [48] E.R. Moog, C. Liu, S.D. Bader and J. Zak, Thickness and polarization dependence of the magnetooptic signal from ultrathin ferromagnetic films, Phys. Rev. B 39, 6949 (1989). [49] J. Zak, E.R. Moog, C. Liu and S.D. Bader, Magneto optics of multilayers with arbitrary magnetization directions, Phys. Rev. B 43, 6423 (1991). [50] S.D. Bader, SMOKE, J. Magn. Magn. Mater. 100, 440 (1991). [51] S.D. Bader and J.L. Erskine, Magneto-optical effects in ultrathin magnetic structures, in Ultrathin Magnetic Structures, vol. II, edited by B. Heinrich and J.A.C. Bland Springer, Heidelberg, 1994), p. 297. [52] Z.Q. Qiu and S.D. Bader, Surface magneto optic Kerr effect (SMOKE), J. Magn. Magn. Mater. 200, 664 (1999).

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[53] M.E. Buckley, F.O. Schumann and J.A.C. Bland, Strong changes in the magnetic properties of ultrathin Co/Cu(0 0 1) films due to sub monolayer quantities of a non magnetic overlayer, Phys. Rev. B 52, 6596 (1995). [54] D. Matsumura, T. Yokoyama, K. Aremiya and T. Ohta, X-ray magnetic circular dichroism study of spin reorientation transitions of magnetic thin films induced by surface chemisorption, Phys. Rev. B 66, 024402 (2002). [55] S. Pick and H. Dreysse, Tight binding study of the CO chemisorption effect on cobalt magnetization, Phys. Rev. B 59, 4195 (1999). [56] Q. Ge, S.J. Jenkins and D.A. King, Localisation of adsorbate-induced demagnetization: CO chemisorbed on Ni(1 1 0), Chem. Phys. Lett. 327, 125 (2000). [57] D. Sander, W. Pan, S. Ouazi, J. Kirschner, W. Meyer, M. Krause, S. Mueller, L. Hammer and K. Heinz, Reversible H-induced switching of the magnetic easy axis in Ni/Cu(0 0 1) thin films, Phys. Rev. Lett. 93, 247203 (2004). [58] H.L. Meyerheim, D. Sander, R. Popescu, J. Kirschner, O. Robach and S. Ferrer, Spin reorientation and structural relaxation of atomic layers; pushing the limits of accuracy, Phys. Rev. Lett. 93, 156105 (2004). [59] B.C. Choi and M.R. Freeman, Nonequilibrium spin dynamics in laterally defined magnetic structures, Ultrathin Magnetic Structures III, edited by. J.A.C. Bland and B. Heinrich (Springer, Heidelberg, 2005), p. 211. [60] R.J. Hicken, N.D. Hughes, J.R. Moore, D.S. Schmool, R. Wilks and J. Wu, Magneto optical studies of magnetism on pico and femtosecond time scales, J. Magn. Magn. Mater. 559, 242–245 (2002). [61] B. Heinrich, Ferromagnetic resonance in ultrathin film structures, in Ultrathin Magnetic Structures II, edited by B. Heinrich and J.A.C. Bland (Springer, Heidelberg, 1994), p. 195. [62] B. Hillebrands and G. Gu¨ntherodt, Brillouin light scattering in magnetic superlattices, in Ultrathin Magnetic Structures II, edited by B. Heinrich and J.A.C. Bland (Springer, Heidelberg, 1994), p. 258. [63] G. Carlotti and G. Gubbiotti, Brillouin scattering and magnetic excitations in layered structures, Rivista del Nuovo Cimento 22, 1 (1999). [64] Z. Zhang, P.C. Hammel, M. Midzor and M.L. Roukes, Ferromagnetic resonance force microscopy on microscopic cobalt single layer films, Appl. Phys. Lett. 73, 2036 (1998). [65] K. Yu. Guslienko, S.O. Demokritov, B. Hillebrands and A.N. Slavin, Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes, Phys. Rev. B 66, 132402 (2002). [66] R. Arias and D.L. Mills, Magnetostatic modes in ferromagnetic nanowires II: A method for cross sections with very large aspect ratios, Phys. Rev. B 72, 104418 (2005). [67] S.J. Steinmu¨ller, M. Tselepi, G. Wastlbauer, V. Strom, D.M. Gillingham, A. Ionescu and J.A.C. Bland, Spin dynamics in ultrathin Fe film in the vicinity of the superparamagnetic/ferromagnetic phase transition, Phys. Rev. B 70, 024420 (2004). [68] J. Jorzick, C. Kra¨mer, S.O. Demokritov, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, E. Cambril, E. Søndergard, M. Bailleul, C. Fermon and A.N. Slavin, Spin wave quantization in laterally confined structures, J. Appl. Phys. 89, 7091 (2001). [69] C. Mathieu, J. Jorzick, A. Frank, S.O. Demokritov, A.N. Slavin, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux and E. Cambril, Lateral quantization of spin waves in micron size magnetic wires, Phys. Rev. Lett. 81, 3968 (1998). [70] J. Jorzick, S.O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K.Y. Guslienko, A.N. Slavin, D.V. Berkov and N.L. Gorn, Spin wave wells in nonellipsiodal micrometer size magnetic elements, Phys. Rev. Lett. 88, 047204 (2002). [71] K. Yu. Guslienko, S.O. Demokritov, B. Hillebrands and A.N. Slavin, Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes, Phys. Rev. B 66, 132402 (2002). [72] J. Jorzick, S.O. Demokritov, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux and E. Cambril, Spin wave quantization and dynamic coupling in micron size circular magnetic dots, Appl. Phys. Lett. 75, 3859 (1999). [73] G. Gubbiotti, G. Carlotti, T. Okuno, T. Shinjo, F. Nizzoli and R. Zivieri, Brillouin light scattering investigation of dynamic spin modes confined in cylindrical permalloy dots, Phys. Rev. B 68, 184409 (2003).

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[74] G. Gubbiotti, G. Carlotti, R. Zivieri, F. Nizzoli, T. Okuno and T. Shinjo, Spin wave modes in submicron cylindrical dots, J. Appl. Phys. 93, 7607 (2003). [75] M. Buess, T.P.J. Knowles, R. Hollinger, T. Haug, U. Krey, D. Weiss, D. Pescia, M.R. Scheinfein and C.H. Back, Excitations with negative dispersion in a spin vortex, Phys. Rev. B 71, 104415 (2005). [76] S. McPhail, C.M. Gu¨rtler, J.M. Shilton, N.J. Curson and J.A.C. Bland, Coupling of spin-wave modes in extended ferromagnetic thin film antidot arrays, Phys. Rev. B 72, 094414 (2005). [77] S.O. Demokritov, A.A. Serga, A. Andre, V.E. Demidov, M.P. Kostylev and B. Hillebrands, Tunneling of dipolar spin waves through a regions of inhomogeneous magnetic field, Phys. Rev. Lett. 93, 047201 (2004). [78] P. Grutter, H.-J. Mamin and D. Rugar, Springer Series in Surface Sciences, in Scanning Tunneling Microscopy II, vol. 28, edited by R. Wiesendanger and H.-J. Gu¨ntherodt (Springer, Berlin, 1992), p. 151. [79] H.-J. Gu¨ntherodt and R. Wiesendanger (Eds), in Scanning Tunneling Microscopy I, Springer Series in Surface Science, vol. 20 (Springer, Heidelberg, 1992). [80] R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy (Cambridge University Press, Cambridge, 1994). [81] M.R. Scheinfein, J. Unguris, M.H. Kelley, D.T. Pierce and R.J. Celotta, Scanning electron microscopy with polarization analysis (SEMPA), Rev. Sci. Instrum. 61, 2501 (1990). [82] A. Scholl, H. Ohldag, F. Nolting, J. Stohr and H.A. Padmore, X-ray photoemission electron microscopy; a tool for the investigation of complex magnetic structures, Rev. Sci. Instrum. 73, 1362 (2002). [83] J.N. Chapman, The investigation of magnetic domain structures in thin foils by electron microscopy, J. Phys. D 17, 623 (1984). [84] A. Hubert and R. Scha¨fer, Magnetic Domains (Springer, Berlin, 1998). [85] J. Sto¨hr, Exploring the microscopic origin of magnetic anisotropies with X-ray magnetic circular dichroism (XMCD) spectroscopy, J. Magn. Magn. Mater. 200, 470 (1999). [86] A. Kubetzka, O. Pietzsch, M. Bode and R. Wiesendanger, Spin polarized scanning tunneling microscopy study of 360 degree walls in an external magnetic field, Phys. Rev. B 67, 020401 (2003). [87] A. Kubetzka, O. Pietzsch, M. Bode and R. Wiesendanger, Determining the spin polarization of surfaces by spin polarized scanning tunneling microscopy, Appl. Phys. A 76, 873 (2003). [88] R. Ravlic, M. Bode and R. Wiesendanger, Correlation of structural, local electronic and magnetic properties of Fe/Cr(0 0 1) studied by spin-polarized scanning tunneling microscopy, J. Phys.: Cond. Matter 15, S2513 (2003). [89] T. Kawagoe, Y. Suzuki, M. Bode and K. Koike, Evidence of a topological antiferromagnetic order on ultrathin Cr(0 0 1) film surface studied by spin polarized scanning tunneling spectroscopy, J. Appl. Phys. 93, 6575 (2003). [90] W. Wulfhekel and J. Kirschner, Spin polarized scanning tunneling microscopy on ferromagnets, Appl. Phys. Lett. 75, 1944 (1999). [91] M.A.M. Gijs and G.E.W. Bauer, Perpendicular giant magnetoresistance of magnetic multilayers, Adv. Phys. 46, 285 (1997). [92] L. Berger, Prediction of a domain drag effect in uniaxial, nonocompensated ferromagnetic metal, Phys. Chem. Solids 35, 947 (1974). [93] L. Berger, Exchange interaction between ferromagnetic domain wall and electric current in very thin metallic films, J. Appl. Phys. 55, 1954 (1984). [94] J. Grollier, P. Boulenc, V. Cros, A. Hamzic, A. Vaures, A. Fert and G. Faini, Switching a spin valve back and forth by current induced domain wall motion, Appl. Phys. Lett. 83, 509 (2003). [95] M. Tsoi, R.E. Fontana and S.S.P. Parkin, Magnetic domain wall motion triggered by an electric current, Appl. Phys. Lett. 83, 2617 (2003). [96] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu and T. Shinjo, Real space observation of current driven domain wall motion in submicron magnetic wires, Phys. Rev. Lett. 92, 077205 (2004). [97] G. Tatara and H. Kohno, Theory of current driven domain wall motion; spin transfer versus momentum transfer, Phys. Rev. Lett. 92, 086601 (2004).

Chapter 2 FUNDAMENTAL PROPERTIES OF MAGNETIC NANOSTRUCTURES: A SURVEY Ruqian Wu 1. INTRODUCTION New breakthroughs in materials often have great impact on technological progress and can influence economies or even societies (e.g., the Bronze to Iron Age). While recent improvements in magnetic materials have been central to the notable development of much smaller motors and more efficient generators, it is in the technology of magnetic-data storage where huge improvements in materials have revolutionized the industry. The giant magnetoresistance (GMR) read-head [1–3], for example, has increased the recording density of computer hard discs by a factor 4100 and non-volatile magnetic random access memories (MRAM) are starting to be utilized in computer and communication devices. There is currently widespread belief that much more progress can be made as materials and devices are developed to operate at the nanometer length scale [4]. Applications such as spin filtering, sensing, and ultra-high density magnetic and magneto-optic recording are a few of the items on the horizon. It is known however, that the magnetism of ultra-small entities is not stable due to super-paramagnetism. Thermal fluctuations, the major obstacle for further size reduction of magnetic devices [5], have become an urgent issue, and ways to enhance anisotropy and the pinning of magnetic moments need attention. Simultaneously, the new field of spintronics is progressing [6–9]; here the spin and charge of the electron, rather than its charge alone, are manipulated. Highly spin polarized diluted magnetic semiconductors (DMS) form seamless hybrid structures on base semiconductors with minimal interfacial scattering and may hence provide high thermal stability and high spin injection rate. It is of great interest nowadays to make direct electrical or optical switching of magnetization in DMS for further integration of logic, display, and storage functions [10–16]. Nevertheless, the exploitation of DMS at ambient temperature requires better Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01002-4

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understanding in the mechanism of ferromagnetism and growth control. Furthermore, much needs to be done to understand new functional oxides and alloys for the exploitation of their exceptional properties. The application of innovative experimental tools and the desire to grow nanostructures and novel materials in a controlled manner have created an urgent need for highly reliable, robust, and predictive theoretical models and tools to achieve a quantitative understanding of their growth dynamics and the size, shape, and process dependences of their physical properties. Much of the basis of theoretical tools is founded on density-functional theory (DFT) [17] and related new methods being developed specifically for magnetic materials. It has been increasingly recognized in many fields of materials science that state-of-the-art ab initio electronic structure calculations based on the density-functional theory have been enormously successful, in both explaining existing phenomena and, more importantly, in predicting the properties of new systems. For example, the prediction of enhanced magnetic moments with lowered coordination number at clean metal surfaces and interfaces [18,19] has stimulated both theoretical and experimental investigations for new magnetic systems and phenomena in man-made nanostructures. Synergistic applications of theory and experiment, as have been demonstrated repeatedly in many areas of materials science, become a ‘‘must’’ to further advance our microscopic understanding in nanomagnetism. Although magnetism is probably one of the oldest branches of solid-state physics, studies of nanomagnetism are extremely vigorous and most problems are still very challenging. In nano- and subnano-structures, a large fraction of atoms are exposed to vacuum, which introduces new physics of a fundamental nature. For instance, the orbital magnetic moments can be larger than the spin counterpart. ‘‘Nonmagnetic’’ elements exhibit strong spin polarization, which, combined with their strong spin–orbit-coupling strengths, may enhance the magnetocrystalline anisotropy energy to an unanticipated level. Complex non-collinear magnetic structures are usually formed to compensate the exchange, magnetic anisotropy, and magnetostatic energies. The quantized magnetic transport through ultra-small entities offers new opportunities for fundamental studies and exploitations in new spintronics devices as well. Here we discuss these important magnetic quantities starting from the very basic concepts.

2. ENHANCEMENT OF MAGNETIZATION Magnetic materials are those in which atoms display spin polarization at finite temperature, measured in terms of spin and orbital magnetic moments. It is known that the magnetization stems from the alignment of electronic spins. As stated in Hund’s first rule, most free atoms adopt electronic configurations that maximize the spin moment, oS4, so as to gain the exchange energies. Nevertheless, very few solid materials in nature are magnetic, primarily because of the effects of crystalline fields along with the orbital hybridizations. The Stoner criterion, In(EF)41, is widely used to identify the instability toward spin polarization in itinerant magnetic

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systems. Here I and n(EF) denote the repulsive interaction between electrons with opposite spins and the value of density of states (DOS) at the Fermi level in the nonmagnetic state, respectively. While the value of I is rather stable [20] for a given element, n(EF) typically increases in nanostructures where the numbers of the nearest neighbors are reduced. Consequently, the Stoner criterion is satisfied in many small systems, even though the constituents are traditionally ‘‘nonmagnetic’’. The band narrowing triggers ‘‘revival’’ of magnetization of 4d and 5d elements and strong enhancement in magnetization of 3d elements in reduced dimensions. An additional contribution to magnetization is the orbital magnetic moment, oLz4, that stems from the relativistic spin–orbit coupling (SOC). To the order (v/c)2, the SOC Hamiltonian is expressed as * * _2 @V * * L
S ¼ x
L
S (1) 2 2 2m c r@r In magnetic solids such as the bulk Fe, Co and Ni, HSOC is much weaker than the crystal fields and thus their orbital magnetic moments are mostly quenched (o 0.1 mB). Intriguingly, the strength of HSOC becomes comparable to the bandwidth in nanostructures, especially when magnetized heavy elements are involved. This not only produces large oLz4, but also leads to formation of peculiar magnetic structures. Nowadays, both spin and orbital magnetic moments can be accurately determined through modern DFT calculations by solving the Kohn–Sham equation
1 2  2r þ V ext þ V c þ V ex c ¼
c (2)

H SOC ¼

Here, Vext and Vc are external potential and the Coulomb potential among electrons. Vex is the exchange-correlation potential that takes into account all the complexities of many-body interactions. The local spin density approximation (LSDA) and the more advanced generalized gradient approximation are adopted in practical calculations for Vex. The HSOC term is usually neglected to reduce the computational costs by treating two spin channels separately. Nevertheless, it becomes essential to invoke HSOC self-consistently for studies of magnetic properties of nanostructures. To demonstrate the concepts elaborated above, the spin and orbital magnetic moments for selected elements are presented in Fig. 1, obtained through all-electron full potential linearized augmented plane wave (FLAPW) calculations [21–23]. Enhancements of both spin and orbital magnetic moments in low-dimensional systems are obvious. Similar data have been reported in the literature for free and supported 3d, 4d, and 5d monolayers [24–30] and more recently for nanochains and wires. It is interesting to note that the spin magnetic moments of 4d and 5d elements follow those of their 3d counterparts well. In particular, the orbital magnetic moments can be larger than the spin counterparts in Ir and Pt wires. Nevertheless, magnetization of 4d and 5d atoms is very sensitive to the change in environments because of the large spatial extension of 4d and 5d wave functions. Strong hybridization occurs even with traditionally ‘‘inert’’ substrates such as C(0 0 0 1), on which most 4d and 5d elements are magnetically ‘‘dead’’ except Ru and Rh [31]. It was predicted that

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Fig. 1. The calculated spin and orbital magnetic moments of selected 3d, 4d, and 5d elements in their bulks, monolayers, and monatomic wires with an interatomic distance of 5.4 a.u.

Ru and Rh monolayers may possess large spin magnetic moments on MgO(0 0 1) (1.95 and 1.21mB) [32]. The strong magnetic enhancement predicted in DFT calculations has been examined extensively, using various experimental techniques [33,34]. While the magnetic enhancements of 3d films are mostly verified, corroboration for 4d and 5d magnetism is still controversial mostly due to the experimental impossibility to grow a perfectly uniform film. Using spin polarized secondary electron spectroscopy, Pfandzelter et al. [35] found an evidence for 4d ferromagnetism in Ru/C(0 0 0 1). In contrast, Chado et al. reported the absence of ferromagnetic order in ultrathin Rh deposited on gold [36]. A negative conclusion was also drawn by Beckmann et al. [37], using the anomalous Hall effect and weak localization, but they found a single Ru atom on Au or Ag surfaces possess a small fluctuating magnetic moments of 0.4mB.

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Nevertheless, sizeable magnetic moments have been observed in 4d clusters [38,39], for which the magnetic properties are easy to tailor through tunable parameters such as cluster size, alloying elements, geometry, and surroundings. Transition metal clusters, either in the freestanding form or on inert substrates, have been examined through DFT calculations from 1980s [40–46]. For example, Reddy et al. [47] found Rh13 has a magnetic moment as large as 21mB. Intermixing of small 4d and 5d clusters on Ag(0 0 1) with the substrate atoms can lead to an unexpected enhancement of the local moments [48]. Experimentally, Taniyama et al. [49] conducted dc magnetic measurements as a function of Pd particle size show that the magnetization increases rapidly with decreasing particle size. Clear evidence of the ferromagnetism of gas-evaporated Pd fine particles with a clean surface was also reported by Shinohara et al. [50]; they estimated that the magnetic moment of surface Pd atoms is 0.7570.31mB/atom. Using a molecular beam deflection technique, Cox et al. [51] found high magnetic moments, as high as 0.8mB per atom, in Rh969 clusters. More recently, very small thiol-capped gold clusters were also found to exhibit a localized permanent magnetism, in contrast to the metallic diamagnetism characteristic of bulk Au [52], primarily because of the creation of 5d-holes around the surface Au atoms through charge transfer to the capping molecules. Peculiar magnetic properties have also been found in clusters of 3d elements. Billas et al. found that for cobalt and nickel, convergence toward the bulk is fast while for iron clusters the magnetization stays well above the bulk value up to cluster sizes of several hundred atoms [53]. For Nin clusters (n ¼ 5–740), Apsel et al. [54] found magnetization minima for clusters with closed geometrical shells and maxima for relatively open clusters. Large orbital magnetic moments are also predicted for many clusters, including Ni [55]. One of the most powerful techniques for the quantitative determination of magnetic moments nowadays is soft X-ray magnetic circular dichroism (XMCD) [56]. Specially, XMCD provides element selectivity and resolves the spin and orbit magnetic moments separately through the powerful sum rules [57,58]. For transitions from p3/2 (the L3 edge) and p1/2 (the L2 edge) core levels, the sum rule can be stated as R R hLz i L3 sm ðEÞdE þ L2 sm ðEÞdE R R ¼ 2N h L3 st ðEÞdE þ L2 st ðEÞ dE R R (3) oS z 4 þ 7hT z i L3 sm ðEÞdE  2 L2 sm ðEÞdE R R ¼ 3N h L3 st ðEÞdE þ L2 st ðEÞdE Here sm and st are the difference and sum of the absorption spectra for the left- and right-circularly polarized soft X-ray beams. Tz is the magnetic dipole operator (Tz ¼ S (13cos2 y)/2). The number of valence holes, Nh, is usually obtained from integration over the unoccupied DOS from DFT calculations. The validity and applicability of these sum rules have been carefully examined [59,60], including the effects of technical aspects [61]. XMCD was applied to many low dimension magnetic systems and many interesting results have been obtained. For example, small

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magnetic moments on Cu atoms induced through contact with Fe or Co layers were uniquely measured through XMCD [62,63]. In addition, XMCD also allows exploration of element-specific hysteresis and complex magnetic ordering in various nanostructures [64]. More recently, XMCD was applied to detect smallinduced magnetization on semiconductor sites in DMS [65]. Our recent studies found that it is possible to furthermore use XMCD for the identification of the distribution of magnetic impurities in DMS [66]. Depth-resolved XMCD techniques were also developed recently to observe the magnetic coupling profiles in thin films [67]. Spin-polarized scanning tunneling microscope (SP-STM) is another powerful tool for studies of nanomagnetism, especially for properties in the ultra small scale down to a single atom [68–71]. Even non-spin polarized tips can detect magnetic signals: the presence of spin–split electronic states through scanning tunneling spectroscopy (STS) of electronic resonances [72]. In STS, the differential conductance (dI/dV) spectrum gives a measure of the local density of states (LDOS), which can be directly compared to results of DFT calculations. For 3d transition monomers on NiAl(1 1 0), for example, both DFT calculations and STS measurements reveal a pseudo gap for the bare NiAl(1 1 0) surface, which leads to a depletion zone in LDOS with a sudden onset at 2.5 eV. In the pseudo-band gap region, the sp-states of Mn, Fe, and Co monomers show remarkable resonances from which the exchange splittings can be extracted. The agreement in the spectroscopic features and exchange splittings (0.8, 0.5, and 0.2 eV for Mn, Fe, and Co respectively) given in Figs. 2a and b suggests the validity of DFT calculations for other

Fig. 2.

The calculated l-projected density of states and the measured dI/dV curves for Mn, Fe, and Co monomers on NiAl(1 1 0) (from Ref. [72]).

Fundamental Properties of Magnetic Nanostructures

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properties of such very small systems. The magnetic moments for Mn, Fe, and Co monomers on NiAl(1 1 0) are 4.4, 3.2, and 1.9mB, respectively.

3. MAGNETIC ANISOTROPY The magnetic anisotropy energy (MAE) determines the orientation of magnetization. In the bulk samples of cubic symmetry, the amplitude of MAE is merely a few or a few tenths meV/atom. In less than three dimensions, the MAE is strongly enhanced by a factor of 100–1000, with the leading term called uniaxial anisotropy (MAE ¼ Ksin2 y+K2sin4 y, where y is the polar angle away from the surface normal). For many thin films, K is positive and is sufficiently large to overcome the tendency of in-plane anisotropy caused by the magnetostatic interactions. Consequently, the easy axes are perpendicular to the film planes, a feature that is desired for the design of next generation high-density perpendicular magnetic recording media [73]. Studies on how to tailor MAE in bulk materials, alloys, and small entities through manipulating atomic structure, step edges, size, shape, and chemical environment are vigorous and fruitful [74–78]. Intriguingly, it was found recently [79] that bcc nickel, epitaxially grown on GaAs(0 0 1), has a positive MAE, opposite in sign from that in the natural fcc Ni. In principle, MAE is associated with two factors: the spin–orbit coupling (this contribution is called ‘‘magnetocrystalline anisotropy,’’ EMCA) and the magnetostatic dipole–dipole interaction (this contribution is called ‘‘shape anisotropy’’) [80–84]. The uniaxial MAE for magnetic films, K, can be determined through DFT calculations with satisfactory accuracy [82]; some of the calculations even gave atom-resolved contributions [85,86]. Fig. 3 illustrates theoretical and experimental results of MAEs for O/Ni/Cu(0 0 1) [87], plotted as a function of the inverse of the film thickness. Well-characterized experimental data allow unambiguous separation for contributions from different regions, namely, the y-intercept for bulk and the slope for surface/interface. It is interesting to note that the presence of O strongly

Fig. 3.

The experimental and theoretical results of magnetic anisotropy energies for Ni/Cu(0 0 1), Cu/Ni/Cu(0 0 1), and O/Ni/Cu(0 0 1).

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alters the surface anisotropy energy: contributions from the bulk Ni, the Cu/Ni and O/Ni interfaces are 27, 59, and 17 meV/atom, respectively. This leads to a shift of the spin-reorientation transition (SRT) from 10 Ni-MLs in the vacuum/Ni/Cu(0 0 1) to 5 Ni-MLs in O/Ni/Cu(0 0 1), as given by the intercept between K(1/d) line with the horizontal line of 2pM2 in Fig. 3a. The latter describes the shape anisotropy, which favors parallel alignment of the magnetization in the film. Similar phenomena have also been observed for Fe, Co, and Ni films covered by Cu, hydrogen or CO [88–91]. Theoretical calculations reproduced the trend of experimental data very well. Analyses in electronic structures furthermore attributed the O-induced change in EMCA to the new surface state with the dxz feature caused by the O adlayer. It is known that magnetism in wires or dots is unstable due to super-paramagnetism [92], driven by thermal fluctuation. How to restrain thermal fluctuations, the major obstacle for further bit reduction of both longitudinal and perpendicular recording media today [93], becomes an urgent issue for fundamental magnetism studies. Investigations of MAEs of magnetic clusters including molecular magnets are getting broad attention [94–97]. Using a new micro-Superconducting Quantum Interference Device (SQUID) setup, Jamet et al. studied magnetic anisotropy in a 3-nm cobalt cluster embedded in a niobium matrix and found the dominating role of the cluster surface [98,99]. Gambardella et al. reported observation of giant MAEs of Co particles grown on Pt(1 1 1) [100]. Ultra-small monatomic chains, the smallest possible magnetic recording units, have been grown on vicinal substrates [101] or built by STM tip manipulations [72,102]. If one can increases their blocking temperature, Tb, to room temperature, in principle one may realize a recording density up to 100–1000 (TB)/in2, a thousand times denser than the best hard drives that are available today. Monte Carlo N1 P* ! S i
S iþ1 þ simulations based on a classical model Hamiltonian, H ¼ J i¼1 N P S 2i;z indicate that this requires stringent conditions, e.g., J ¼ 3202420 meV K i¼1

and K ¼ 30250 meV=atom: Such a large K is exceptionally large but is found possible through model calculations. As listed in Table 1, 3d–5d, trimers, especially FeOsFe and FeIrFe, have large EMCA up to 108 meV. Calculations for longer FemIrn chains confirmed that EMCA scales almost linearly with the length. These results Table 1. The calculated K, spin and orbital magnetic moments distributed in different atoms. The easy axis for positive (negative) MAE is along (perpendicular to) the chain. System and magnetic ordering Fe–Ta–Fe (m k m) Fe–W–Fe (m k m) Fe–Re–Fe (m m m) Fe–Os–Fe (m m m) Fe–Ir–Fe (m m m) Fe–Pt–Fe (m m m)

MS (mB) 3d

ML (mB) 5d

3d

5d

MAE (meV)

2.99 2.93 3.22 3.32 3.36 3.34

1.96 2.85 2.94 3.21 1.58 0.69

0.16 0.10 0.03 0.60 0.00 0.20

0.30 0.13 0.04 0.57 1.53 0.34

20.5 30.8 31.2 75.0 108.5 26.6

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indicate that FemIrn chains are promising for exploitation in magnetic recording and spintronics devices. To this end, more theoretical and experimental investigations for magnetic chains and wires should be conducted in the near future [103–107].

4. MAGNETIC ORDERING The switch from antiferromagnetic (AFM) to ferromagnetic (FM) states in the spinvalve structures, which serve as the cores in GMR reading heads, has been studied thoroughly in the past decade [108]. Complex in-plane magnetic orderings and peculiar surface alloys have been investigated both theoretically and experimentally. Because of their exotic structural and magnetic properties, Mn, Cr, and fcc Fe are involved in most such systems. Mn and Cr have complex AFM magnetic phases in their bulks and show a strong tendency to adopt AFM ordering. For example, Mn takes on the c(2  2) magnetic configuration for the Mn monolayer on Fe(0 0 1) and Co(0 0 1) [109–112]. However, Mn and Cu forms a order c(2  2) FM surface alloy [113]. Mn trimers on NiAl(1 1 0), Mn512 clusters are also found to be ferromagnets through DFT and experimental studies [114–116]. The discovery of FM ordering in Mn-doped semiconductors has stimulated many endeavors to explore innovative spintronics materials for simultaneous exploitation of electron charge and spin. The d-doped DMS with doping profile approximately a d-function along the growth direction [117,118], circumvent complexities in 3D random doping cases and hence provide ideal model systems for comprehensive theoretical examinations. We investigated magnetic switching between the FM and AFM phases when the external carrier density r is manipulated for d-doped (Ga,Mn)As [119]. As shown in Fig. 4, a strong FM ground state can be found for thin GaAs spacers (d ¼ 5.56 and 11.31 A˚). The large magnitude of DEFMAFM, e.g., 35 meV/cell even for d ¼ 11.31 A˚, demonstrates the effectiveness of semiconductor spacers to mediate the exchange interaction between magnetic impurities. The carrier induced AFM-FM phase transition occurs for thick spacers. When, d ¼ 16.96 A˚, for example, the AFM configuration is the ground state without external carriers, and is further stabilized by electron doping. In contrast, the FM state gradually prevails in the hole doping side, especially when r41.0  1020 e/a.u.3 This result is in good accordance with experimental observations made by Ohno et al. [13,16] who found that both magnetization and coercive force of (Ga,Mn)As and (In,Mn)As can be manipulated by applying an electric field in the gated structure that subsequently change the effective carrier (hole) density in the DMS. This electrical control of the magnetization, previously inaccessible in other magnetic materials, opens up new opportunities for novel device applications. In reduced sizes, some interesting non-collinear magnetic patterns can be formed in order to ease the demagnetization energy [120]. For example, FM nanodots typically form curling magnetic structures [121–123]. The magnetization close to the center of the dot (Bloch point) turns up along the perpendicular orientation to the curling plane, as displayed in Fig. 5. Experimentally, the vortex core was inferred by magnetic force microscopy (MFM) in which the magnetization near the center

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Fig. 4. DEFMAFM (in meV per unit cell) versus the external carrier density r (in  1020 e/a.u.3) for the d-doped (Ga,Ms)As. The open symbols represent uniform carrier distributions, while the asterisks and crosses represent non-uniform carrier distributions.

Fig. 5.

In-plane and out-of-plan components of magnetization and the calculated magnetic patterns of Fe nanodots.

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Fig. 6. Direction of magnetization in the free Fe(1 1 0) monolayer and the bulk bcc Fe through non-collinear DFT calculations by Nakamura et al. The inset displays the domain structure of Fe/W(1 1 0) observed by Elmer et al.

assumes a perpendicular orientation, and the core structures were observed in a 4 or 5 nm in radius by SP-STM. Through DFT calculations for rod geometry, Nakamura et al. [124] reproduced the curling configuration and found that the formation energy of the vortex core, Evc, is 3 meV/atom. Interest in magnetic domain walls (DW) has been greatly increased in both basic and applied research – including the DW with a few nm widths in quantum spin interfaces and the role of the DW in the exchange bias and nanoscale geometrical structures. The DW, is a transition region within which the magnetization orientation changes from one easy axis to another, is known to be determined by a competition between the exchange energy and the MAE – as revealed in phenomenological continuum micromagnetic calculations. The exchange energy tends to favor a slow variation of the magnetization while the anisotropy energy favors a rapid change from one easy axis to another, which leads to a stable DW width of the order of 10–100 nm in the bulk. Pratzer et al. [125] studied Fe stripes on stepped W(1 1 0) and found extremely narrow DW, as thin as 6 A˚. Through the FLAPW calculations for a free Fe(1 1 0) monolayer with the freedom of non-collinear magnetization, Nakamura et al. [126] found that the moments rotate rapidly within approximately three lattice constants without any discontinuous changes, as shown in Fig. 6. The domain wall is only 6–8 A˚ thin in the simulations. The other important feature from STM observation is that domain walls in Fe strips are oriented along a certain crystallographic direction, regardless of the orientation of the wires. Monte Carlo simulations on a discrete lattice [127] found that the wall orientation is mainly determined by the atomic lattice and the resulting strength of an effective exchange interaction. The magnetic anisotropy and the magnetostatic energy play a minor role for the wall orientation in that system.

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5. MAGNETIC TRANSPORT A significant concern in nanoscience is the transport and magnetic transport properties through nano-entities. The use of molecules or molecular wires as elements in electrical circuits, for example, is an appealing idea. Molecules can have length scales smaller than 1 nm, and hence we would gain orders of magnitude in integration levels if we could assemble them onto a chip. Extensive experimental work has been done to measure junction transport utilizing mechanical break junctions, in the presence of a gating field. Some of these results feature molecular structures containing a single cobalt atom that acts, effectively, as an atomic quantum dot or atomic gate within the molecular structure. As mentioned above, the very narrow domain wall that forms in nanostripes and nanocontacts also has significant consequence on magnetic transport and leads to large room-temperature magnetoresistance (4100% at B ¼ 100 G) [128–131] . Magnetic transport through wires and molecules is still an unresolved issue and currently is under active exploration [132]. Here we discuss ballistic electron transport through monatomic chains suspended between contacting wires. The electrical conduction (G) is quantized according to the Landauer–Buttiker equation [133–135] in the linear response regime: G¼

G0 X T ns 2 ns

(4)

where G0 is the conductance quantum and is equal to 2e2/h or 12.9 kO1. Tns, the transmission coefficient for the nth channel and electron spin s, is either 1 or 0 for monovalent metals corresponding to an open or closed channel. Consequently, conductance measurements for metal nanocontacts display flat plateaus and abrupt drops during the elongation caused by atomic rearrangements. Quantum transport in atomic sized nanowires of various metals formed by mechanical means has been widely studied by several groups. For non-magnetic materials, (e.g., Au [136] and Ag [137]) one would expect that the two spin channels are degenerate; and G is thereby quantized with a unit of G0. If the wire is spontaneously magnetic, the unit for conductance jumps should be reduced to G0/2 (half a conductance quantum) even at room temperature, as long as the characteristic time for electrons conduction is longer than that of the spin rotation. This behavior is demonstrated in Fig. 7a by the plateau in the conductance histograms: in Fig. 7b by the pronounced peak at G0/2 for the count in the number of plateaus in different values of conductance of a Co wire [138]. Intriguingly, magnetic transport behavior was also found in Cu nanocontacts, formed by tapping macroscopic Cu wires together in air at room temperature [139]. With a small field of 5 mT, the G0/2 mode can be clearly seen and is highly repeatable [140]. In contrast, the G0/2 mode was not found in an experiment for Cu junctions done in ultrahigh vacuum [141]. First principles calculations revealed that 1D pure Cu nanowires are nonmagnetic within a wide range of Cu–Cu distances (4.4–6.0 a.u.). However, an alternating CuO chain has sizable magnetic moments in

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Fig. 7. Conductance measurements of Co wires at room temperature and without external magnetic field. (a) Typical electrical transport curves showing conductance plateaus. (b) Global histogram exhibiting the statistical conductance behavior of a sequence of nanowire generation (from Ref. [138]).

both Cu and O atoms when the contact is slightly stretched. The magnetic signal is hence not from the intrinsic properties of Cu wires but rather it is associated with the oxidation of Cu wires. Besides, the CuO wires display the half-metallic feature with the Fermi level lying in a wide band gap of the minority spin. In principle, this is potentially very important to make high efficient spin injection into nanodevices. Since most transition metal elements become magnetic in monatomic wires, one should observe conductance quantum of G0/2 as in Fig. 7b even for 4d and 5d contacts. Transport measurements thus provide a unique way to examine the status of magnetization and spin dynamics for different elements in the form of entities of a few atoms. Rodrigues found this behavior in suspended chains of Pd and Pt [138]. Nevertheless, this is not a settled issue yet since others reported featureless conductance histograms for Pd monatomic contacts in the ultra high vacuum environment [142]. Untiedt et al. [143] further denied the presence of quantization of

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conductance in Fe, Co, Ni, and Pt contacts and attributed the observed fractional conductance quantization to contamination. On the other hand, conductance plateaus at G0/2 were observed from carbon nanotubes, and the reason for this unexpected phenomenon is still unclear [144]. The electron conductance at finite temperature through ultra small entities will require understanding of strong correlations, another fertile ground for fundamental research [145,146].

6. SUMMARY AND FUTURE OUTLOOK Nanomagnetism is a cutting-edge research field, and may potentially revolutionize storage, sensing, spintronics, and optoelectronic technologies. In the near future demands on speed and memory capacity are expected to increase further in computing and communicating technologies and can be met only by further miniaturizing highly functionalized devices. Furthermore, as we have seen, one encounters new and fascinating fundamental physics in these studies. To this end, it is obvious that explorations in the nano- and subnano-scales are both challenging and rewarding. The field is evolving very rapidly and it is very difficult to make meaningful predictions for what will happen in the next 5 years or even shorter. We think several of the following directions deserve particular attention. On the side of theory, it is necessary to develop viable theories and robust software packages for the simulations of magnetic transport (I/V curves), the excitation spectrum and spin dynamics [147]. Accurate description for the excited states will be important since nanostructured ‘‘metals’’ have discrete energy spectra with gaps. The failure of DFT description for the Cr dimers is a good example in this aspect. The development and implementation of time-dependent DFT, dynamic mean field theory or other approaches to treat the correlation effects become crucial. Theoretical investigations of nanoentities usually involve ‘‘excessively large’’ systems for DFT calculations, especially when the environment such as substrates and chemical ligands has to be considered. Furthermore, it is still extremely tough to simulate the growth and post-growth molecular dynamics from first principles. Efficient simulation tools for multi-scale structural modeling, including structural response properties, structural optimization, and quantum dynamical simulations for nanomagnets are considered necessary. Currently, it is still very difficult to fabricate nanostructures in a controlled manner, and many observations may not explore intrinsic properties. Development or refinement of experimental procedures will be essential to study intrinsic properties. This type of research is increasingly dependent upon collaborative efforts among different disciplines, as well as effective partnerships between theory and experiment. Magnetic doping in nanotubes, nanowires, or junctions of semiconductor, for example, can introduce magnetic currents for spintronics manipulations. Detection of single-atom magnetization, spin dynamics and electronic spectrum using SP-STM will be very important to reveal new physics that is peculiar in nanostructures. For example, Heinrich et al. demonstrated the ability to measure the energy required to flip the spin of single adsorbed atoms [148]. Using STM tip

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manipulations, 1D magnetic chains can be built atom-by-atom, and measured hysteresis of single atom in small magnetic entities [149]. One would expect groundbreaking technical innovations as well as significant advances in science from these explorations under well-controlled conditions. On the other hand, for the exploitation of nanostructure in devices, it is equally important to search proper environments such as ligand molecules, hollowed materials, substrates and capping materials, which ideally allow the growth of self-assembled nanostructures and also provide other means to tune the magnetic properties through controlling the hybridization and charge transfer.

ACKNOWLEDGMENTS The author is indebted to simulative discussions with D.L. Mills and Prof. W. Ho at the University of California, Irvine. Collaborations with Profs. J.S. Hong, A.J. Freeman, K. Baberschke, J.A.C. Bland, and D.S. Wang on topics discussed above are highly appreciated. Research work on nanomagnetism was supported by the fund of UC Irvine and also by the Department of Energy.

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Chapter 3 EXCHANGE COUPLING IN MAGNETIC MULTILAYERS M. D. Stiles ABSTRACT In magnetic multilayers, the magnetizations of two ferromagnetic layers separated by a non-magnetic spacer layer are coupled by the electrons in the spacer layer. This coupling oscillates in sign as a function of the thickness of the spacer layer. Extensive research done on these systems has led to a simple model for this coupling and remarkable agreement between predictions of the model and measurements of the coupling. The model predicts that the periods of the coupling are determined by geometrical properties of the Fermi surface belonging to the spacer layer material. The oscillatory coupling is an instance of oscillations in metals caused by the existence of a Fermi surface. Keywords: magnetic multilayers, exchange coupling, quantum well, spin-dependent reflection, Fermi surface, biquadratic coupling, critical spanning vector, interface electronic structure PACS: 73.21.Ac, 75.30.Et, 75.70.Cn, 73.20.At, 72.25.Ba, 71.18.+y

1. INTRODUCTION Magnetic multilayers exhibit dramatic oscillations in their magnetic coupling as a function of the thickness of a metallic separation layer. These systems, see Fig. 1, consist of a sequence of thin films grown on top of each other, each film ranging in thickness from tenths of nanometers to tens of nanometers. Typically, two thin films of ferromagnetic material are separated from each other by a thin film of a non-magnetic material, referred to as a spacer layer. The magnetization directions of the ferromagnetic layers are coupled to each other through an exchange interaction. The sign of this coupling oscillates as the thickness of the spacer layer is Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01003-6

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M. D. Stiles

52 1 cm

1 cm

Au cap 2 nm 1 nm 1 nm

ping la

yer

Fe Au spa cer lay er Fe Ag buff er laye r

GaAs s

ubstrate

Fig. 1. Typical magnetic multilayer. Here, two Fe layers are separated by an Au spacer layer. An Au capping is grown on top to protect the multilayer from corrosion. The multilayer was grown on a GaAs substrate with a buffer layer of Ag to promote better growth of the Fe and Au films.

varied, with the best multilayer samples showing up to 30 periods of oscillation. This chapter gives a pedagogical description of the origin of the coupling and describes a few key measurements and calculations. In metals, the sharp cut-off in occupancy at the Fermi surface causes oscillatory phenomena. The oscillation of the induced magnetization as a function of applied magnetic field, called the de Haas-van Alphen (dHvA) effect [1], is a well-known example. Many other properties oscillate as well. These oscillations arise from fieldinduced oscillations in the density of states. The cross-sectional areas of the Fermi surface, a geometrical property, determine the oscillation periods. Spatially localized disturbances in metals lead to another type of oscillation. The oscillation of the electron density near surfaces of metals, known as Friedel oscillations, is an example. Another example is the oscillation in the interaction between two localized magnetic impurities in a metallic host. As the separation between the impurities is increased, the interaction between them oscillates between favoring parallel alignment and antiparallel alignment of the magnetic moments. This coupling of the moments is known as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [2]. A different geometrical property of the Fermi surface determines the spatial periods of the oscillations between the localized disturbances; in this case, a vector spanning the Fermi surface from one side to the other. Interlayer exchange coupling is a particularly dramatic instance of this type of oscillation. Observation of coupling in magnetic multilayers requires high-quality thin films. Early studies [3] were plagued by a number of extrinsic sources of coupling due to defects. One source of such coupling is the presence of pinholes, regions where the non-magnetic layer was not continuous. Pinholes give direct contact between the ferromagnetic layers, leading to coupling favoring parallel alignment of the magnetization directions, referred to as ferromagnetic coupling. In addition, correlated roughness of the films causes ferromagnetic ‘‘orange peel’’ coupling [4]. In 1986,

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Gru¨nberg et al. [5] demonstrated antiferromagnetic coupling between the magnetizations of two Fe layers separated by Cr and two groups demonstrated coupling in rare earth multilayers [6,7]. The observation of antiparallel alignment of the magnetizations meant that the magnetic layers had been grown well enough that the coupling due to any extrinsic coupling was much smaller than the intrinsic coupling. Once high-quality multilayers could be grown, interest in them started to explode. Two years later, Gru¨nberg’s group and Fert’s group discovered the giant magnetoresistance (GMR) effect [8,9]. Magnetoresistance refers to the dependence of a sample’s resistivity on an applied magnetic field. The field dependence may be indirect, for example, the resistance can depend on the magnetization direction, which can depend on the applied field. This is true for GMR; the resistance depends on the relative orientation of the magnetizations of the ferromagnetic layers. If the magnetizations of two layers are antiparallel for zero applied field because the layers are antiferromagnetically coupled, an applied field can overcome the coupling and bring the films into parallel alignment. This change in alignment leads to a change in resistance – GMR. It was immediately realized that the GMR effect could be a sensitive magnetic field detector, particularly for read heads in magnetic disk drives. In fact, read heads based on the GMR effect are now used in essentially all disk drives. To optimize the performance of the read heads, magnetic multilayers with different materials, layer thickness, growth conditions, and other parameters were studied. In 1990 [10], Parkin discovered oscillatory behavior of the dependence of the GMR on the thickness of the non-magnetic spacer layer. These oscillations were not due to variations in the transport properties, but rather variations in the coupling between the ferromagnetic layers. For some thicknesses the coupling was ferromagnetic, favoring parallel alignment of the magnetization directions. For these thicknesses, there was no change in the relative alignment of the magnetizations when a magnetic field was applied and hence the magnetoresistance was zero. The oscillations in the coupling as a function of the separation between two magnetic objects immediately brought to mind the RKKY interaction between magnetic impurities. There was one big puzzle however. The oscillation period, approximately 1.0 nm, was much longer than was expected from estimates based on the analogy with the RKKY interaction. The answer to the puzzle, as is described in detail below, is that it is possible to make much more quantitative comparisons between theory and experiment for magnetic multilayers than has been done for magnetic impurities. The rest of this article is devoted to describing what goes into this comparison. In the following section, I describe a simple model for the physics of the interlayer exchange coupling. Despite its simplicity, this model allows quantitative comparison between theory and experiment. I give this comparison as well as a comparison with more sophisticated models. Section 3 describes the experimental techniques that are used to prepare magnetic multilayers, characterize them, and measure the coupling. I discuss how the presence of defects affects the comparison between theory and experiment. Not only do these defects modify the interlayer exchange coupling, but they can also create effective couplings with different forms. Section 4 describes the most common such effective interaction – biquadratic coupling.

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2. QUANTUM WELL MODEL 2.1. Model for Transition Metal Ferromagnetism Developing a simple model for interlayer exchange coupling requires starting with a simple model for the electronic structure of ferromagnets. Unfortunately, the electronic structure of transition metal ferromagnets is quite complicated. Ferromagnetism in transition metals is driven by atomic-like exchange and correlation effects in the partially filled d-electron shells. The atomic-like effects suggest a localized description of this part of the electronic structure. However, the d orbitals are strongly hybridized with d orbitals on neighboring atoms and also with the s–p orbitals on neighboring atoms. The strong hybridization suggests an itinerant description of the electronic structure. Reconciling these aspects of the physics is an ongoing area of research, and the resulting models are not simple [11]. Simplifying the model requires approximations that favor one aspect of the physics over the other. In this article, I adopt a description favoring the itinerant aspects because the interlayer exchange coupling depends strongly on the properties of the electrons at the Fermi surface and a realistic description of the Fermi surface requires treating the itinerant nature of the d electrons. The local-spin-density approximation (LSDA) [12] accurately describes the itinerant aspects of the electronic structure while treating the atomic-like exchange and correlation effects in mean field theory. That is, all of the complicated electron– electron interactions are lumped into a local potential that depends on the local density and the local spin density. This approximation was derived for computing the ground state properties of materials. For transition metal ferromagnets, it works quite accurately for properties like the cohesive energy, equilibrium lattice constant, and the magnetic moment [13]. Formally, it is not justified for computing the electronic structure, but it is a good combination of simplicity and accuracy even for this case. A material becomes ferromagnetic when it is energetically favorable for a majority of electrons to align their spins parallel to one another. The electrons then split into those with spins parallel to the majority of the other spins (majority) and those antiparallel (minority). Spin–orbit coupling mixes these states, but is weak enough to ignore to a first approximation. Ignoring the spin–orbit coupling leads to a description of ferromagnets in terms of two separate sets of electrons, majority and minority, which have different properties. This description also holds in the non-magnetic layers in magnetic multilayers provided the magnetizations in the different ferromagnetic layers are all collinear with each other. When the magnetizations are not collinear, spin currents flow in the non-magnetic layers and exert torques on the ferromagnetic layers, as is described below. Figure 2 shows a band structure for Co in a face-centered cubic (fcc) structure calculated using the LSDA, highlighting the importance of the itinerant aspects of the electronic structure. The d-bands have a width in energy, approximately 5 eV, that is large compared to the exchange splitting between the majority and minority bands, about 2 eV. In addition, the hybridization between the d-bands and the s–p

Exchange Coupling in Magnetic Multilayers

55 Majority Fermi Surface

5

Majority bands Minority bands

"d" band width

Energy (eV)

0

Minority Fermi Surface unhybridized s-p band

-5

Exchange Splitting

s-d hybridization -10 L

Γ

X

Fig. 2. The band structure and Fermi surface of fcc Co. The solid (dashed) curves give the majority (minority) bands along two high-symmetry directions through the Brillouin zone center, G. The long dashed curve shows the s
p band if it were not hybridized with the d bands. The bars to the right of the bands show the width of the d bands and the shift between the majority and minority bands. The arrows in the band structure plots give the width of the gap due to the hybridization between the s
p and d bands of the same symmetry along the chosen directions. The Fermi surfaces are reproduced with permission from Choy et al. [14].

band is large enough to lead to a gap of about 3 eV where the bands would cross if they were not hybridized. Finally, Fig. 2 shows images of the Fermi surface for majority and minority electrons. Clearly, these are quite different, and the two sets of electrons have very different properties. There are two simplified models for the electronic structure of ferromagnets that have been used extensively in studies of magnetic multilayers. Each emphasizes different aspects. Both include free-electron-like bands. In the spin–split free-electron model, the free-electron-like bands for majority and minority electrons are shifted in energy relative to each other by a constant exchange shift. This model ignores the d electrons completely, but the Fermi surfaces for majority and minority electrons are different. I use this model for pedagogical purposes in much of the rest of this section. The other model, called the s–d model, emphasizes the atomic-like aspects of the d-orbitals by ignoring the d–d hybridization and treating the orbitals as completely localized. This model was originally used to treat isolated magnetic impurities in a non-magnetic host. For ferromagnets and magnetic multilayers, the model for isolated impurities is generalized to treat a dense set of such impurities. In this model, the s–p electrons are weakly hybridized with the d-electrons. In some limits, it is possible to map the s–d model onto a local moment model in which the d-electrons ~i ; where i labels the site. The hybridform a local moment with a fixed total spin S ization between the d and s electrons becomes an exchange interaction between the

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~i  ~ local moment and the s–p electron spin ~ s with the form
J S s: The s–d model and the local moment model have been used both to study interlayer exchange coupling and extensively to study transport in magnetic multilayers, i.e. the GMR, see the chapter by Fert in this volume. As long as the scattering rates for minority and majority electrons are different, these models capture much of the essential physics. However, care must be taken when the details of the band structure are important. 2.2. Spin-Polarized Quantum Well States In this section, I describe the properties of a magnetic multilayer using a spin–split free-electron model as described above [15]. In this model, interfaces are simply potential steps. Below, I generalize the results to more realistic models of the electronic structure. An interface between two materials acts on the electrons like a potential step; electrons that strike it have a transmission probability reduced from one. For a free electron going down a simple potential step of height V, the transmission probability is
 q 2k 2 T step ¼ (1) k kþq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, the incident wave vector is k ¼ 2mE=_2 and the transmitted wave vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is q ¼ 2mðE þ V Þ=_2 : The first factor accounts for the change in velocity on going over the step. The transmission probability is plotted in Fig. 3(a). If another step is introduced, the electron undergoes multiple reflection inside the quantum well that is formed (see Fig. 3(b)). If the steps are separated by a distance D, the transmission probability is  2   4eiqD kq  T well ¼  (2) 2 2 ðk þ qÞ
ei2qD ðk
qÞ  Note that the transmission probability goes to one whenever 2qD ¼ 2np; where n is an integer. At a fixed thickness, there is a series of transmission resonances at the energies E n ¼ ½_2 =ð2mÞ ðnp=DÞ2
V ; for integer n. At a fixed energy, there are 1

1 R

T(well)

T(step)

R T

T

D (a)

0 0

Fig. 3.

(b)

0 1 Energy

2

0

1 Energy

Electron transmission probabilities for a step and a quantum well.

2

Exchange Coupling in Magnetic Multilayers

57

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi resonances for D ¼ 2np=ð2qÞ with q ¼ 2mðE
V Þ=_2 Þ; these resonances are separated by a fixed increment in thickness. At the transmission resonances, the electrons have an increased probability to be in the quantum well; in other words, there is a peak in the density of states in the quantum well at the energy of the resonance. These peaks in the density of states are seen in magnetic multilayers using photoemission and inverse photoemission, for reviews see Refs. [16–19]. Photoemission is a technique in which photons of a particular energy, generally ultraviolet light or X-rays, are incident on a surface. When the photons are absorbed by the material, they excite electrons which can leave the surface and be measured. The energy of the electron in the solid can be inferred from the photon energy and the energy of the photoemitted electron. Peaks are observed in the photoemission spectrum where there is a large density of states in the material. Photoemission probes the near-surface region because the escape depth of the photoemitted electrons is on the order of a nanometer. In order to see the density of states peaks in the non-magnetic spacer layer, the top magnetic layer needs to be stripped off (or never deposited in the first place). In other words, the quantum well states studied in photoemission are not exactly the same quantum well states that are present in the complete magnetic multilayer. None the less, there is very good correspondence between these states and the related states in magnetic multilayers. Fig. 4 shows the photoemission intensity as a function of energy and spacer layer thickness. This figure shows the fixed spacing between peaks as a function of thickness and the quadratic variation of the peaks as a function of energy. There are some differences between what would be expected for a free electron model and what is observed. To understand how the free electron model generalizes to real materials, it is instructive to rewrite the transmission probability in terms of the transmission 0.0

E-EF (eV)

-0.5

-1.0

-1.5 10

20 30 Cu Thickness (Monolayers)

40

Fig. 4. Photoemission from a Cu overlayer on Co. White indicates high photoemission intensity and black low intensity. Reproduced with permission from Qiu and Smith [19].

M. D. Stiles

58

E

Unoccupied states

Antiparallel Alignment

EF

EF

e M Ferromagnet M

M

k

k Spacer

Parallel Alignment

M

M

M

M

EF e M

M

Fig. 5. Quantum wells used to compute interlayer exchange coupling. On the left, the two panels give typical band structures for free electron models of interlayer exchange coupling. On the right, the four panels give the quantum wells for spin up and spin down electrons for parallel and antiparallel alignment of the magnetization. The shaded regions designate the occupied states.

probability for a step and the reflection amplitude at an isolated step R ¼ ðk
qÞ=ðk þ qÞ: This substitution emphasizes the contribution made by multiple reflection inside the quantum well. One round trip through the well has the amplitude ei2qD R2 from reflecting from each step and propagating both directions through the well. The transmission probability is 2    2  1 X     n iqD iqD i2qD 2 1 ðe R Þ (3) T well ¼ T step e 1
ei2qD  2  ¼ T step e R   n¼0 The second form shows explicitly the coherent multiple scattering in the well. The basic physics of quantum well states in real materials is captured by replacing the wave vector for propagating through the spacer layer, q, by the appropriate value from the real band structure and by replacing the reflection amplitude and transmission probability by the values calculated for a realistic interface. If one of the materials is ferromagnetic, the potential step for the majority and minority electrons is different. In a multilayer with two magnetic layers, there are four possible quantum wells formed depending on the relative alignment of the magnetizations, see Fig. 5. The quantum well states for each of these are different because the potential steps, and hence reflection probability, is different for each quantum well. However, at a particular energy, like the Fermi energy, the quantum well states in all of the wells have the same periodicity as a function of the thickness of the spacer layer, because the periodicity only depends on the wavelength of the electron in the spacer layer at that energy. 2.3. Interlayer Exchange Coupling The interlayer exchange coupling can be described in terms of an energy that ^ i as depends on the magnetization directions of the two layers, m ^2 ^1 m E ¼
JAm

(4)

Exchange Coupling in Magnetic Multilayers

59

where A is the area of the two films, and Jo0 gives antiferromagnetic coupling favoring antiparallel alignment. The form of the coupling is called bilinear in contrast to the biquadratic coupling described in Section 4. For interlayer exchange coupling of the form given in Eq. (4), the strength of the exchange interaction is determined by the difference in energy between the quantum well with parallel magnetizations and that with antiparallel magnetizations. Computing the exchange interaction simply requires computing the energies of the quantum wells given in Fig. 5. For large spacer layer thicknesses, the result is   _vF Re ðR" R" þ R# R#
R" R#
R# R" Þei2kF D þ OðD
2 Þ 2pD _vF jR"
R# j2 cosð2kF D þ fÞ ð5Þ  2pD Here, kF is the Fermi wave vector in the spacer layer, vF ¼ _kF =me the Fermi velocity, me the electron mass, and R" and R# the reflection amplitudes for a majority and minority electron to reflect from the interface. Expanding the square of the reflection amplitudes gives four terms, one for each of the quantum wells in Fig. 5. The exchange coupling oscillates in sign with a period p/kF, the oscillation decays like D
1, and the amplitude of the oscillation is determined by the spin dependence of the reflection amplitudes. Physically, these oscillations arise because the quantum well resonances cross through the Fermi level as the thickness of the spacer layer is increased, see Fig. 4. Each time the thickness increases by p/kF, another resonance crosses through the Fermi level. The resonances for each of the quantum wells in Fig. 5 are all different, but they all have the same period because the period is determined by the Fermi wave vector in the spacer layer, and is independent of the properties of the magnetic material. J¼

2.3.1. Critical spanning vectors The expression for the interlayer exchange coupling, Eq. (5), is based on a onedimensional model. Generalizing to realistic three-dimensional systems is straightforward if the growth of the multilayer is coherent. For coherent growth, the inplane lattice of all of the layers is the same, so the in-plane momentum (wave vector) is conserved when the electrons scatter from the interfaces. In this case, electrons with different values of in-plane wave vector do not couple, so the quantum well states for different in-plane wave vectors are independent of each other. Including the full three dimensions of the multilayer then simply requires integrating the onedimensional result over the in-plane wave vector Z 2 h i J _ d K ~ ~ ~
R# ðKÞÞ ~ 2 ei2kz ðKÞD ¼ v ð KÞRe ðR ð KÞ (6) þ OðD
3 Þ F " A 2pD ð2pÞ2 This integral is simple in the limit of large D; there is a contribution whenever the ~ is constant as a function of K: ~ The vector spacer layer spanning vector, 2kz ðKÞ; connecting one sheet of the Fermi surface to another at this in-plane wave vector is called a critical spanning vector. In general, the oscillating contributions from

M. D. Stiles

60 Free Electron Fermi Surface

Copper Fermi Surface kmax

kx 0

q

c

k011 0

q q

0

kz

-kmax −π/d

L

q'L

S

0

k100

π/d

Fig. 6. Slices through a free electron and the Cu Fermi surfaces. For the spherical free electron Fermi surface, there is one critical spanning vector qc ¼ 2kF : For the Cu Fermi surface, there are two inequivalent critical spanning vectors, qL which is closely related to the free electron critical spanning vector, and qS which occurs in the necks of the Fermi surface. The Cu Fermi surface is shown within the Brillouin zone appropriate for Cu(001) multilayers. Indicated in gray are parts of the Fermi surface in the extended zone scheme. The critical spanning vector q0 L is equivalent to qL because of the discrete nature of the lattice.

different parts of the Fermi surface are out of phase and tend to cancel each other. ~ the However, when the spacer layer spanning vector is constant as a function of K; contributions from finite region of the Fermi surface near the critical point are in phase. This region gives a contribution to the integral that oscillates in thickness with a period determined by the critical spanning vector. As the thickness D becomes larger, the region of the Fermi surface that contributes in phase gets smaller so that the amplitude of the oscillation decreases with an additional power of D. For free electrons, the critical spanning vector is qcrit ¼ 2kF ; the vector that goes from one side of the sphere to the other, see Fig. 6. For a free electron model, the interlayer exchange coupling is i2 J _vF kF h ~ ~ ¼
R ð 0Þ
R ð 0Þ sinð2kF DÞ þ OðD
3 Þ " # A 16p2 D2

(7)

The in-plane wave vector dependence of the Fermi velocity and the reflection amplitudes is ignored to lowest order. The oscillation period only depends on the geometry of the spacer layer Fermi surface, but the strength of the oscillation depends on the Fermi surface through vF and kF and also the details of the electronic structure of the ferromagnet through R" and R# : The generalization to realistic materials is illustrated for Cu in Fig. 6. For a free electron model, the only critical spanning vector is one of length 2kF connecting one side of the Fermi sphere to the other. However, for realistic materials, the Fermi surface is more complicated than a sphere; in general, there are several critical spanning vectors and there is a contribution to the coupling from each. For thick spacer layers, the coupling is JðDÞ ¼

h i X _va ka a a 2 iqa? D iwa ? Re ðR
R Þ e e þ OðD
3 Þ " # 2 D2 16p a

(8)

Exchange Coupling in Magnetic Multilayers

61

For each critical point, qa? is the critical spanning vector, va? the Fermi velocity, a k the average radius of curvature of the Fermi surface, and eiw a phase that depends on whether the stationary point is a minimum, maximum, or saddle point. Eq. (8) is known as the asymptotic formula. It is valid for thick spacer layers; for thinner spacer layers, the O(D
3) represents preasymptotic corrections. Equation (8) shows that the oscillation periods observed in interlayer exchange coupling are determined by geometrical properties of the spacer layer Fermi surface. The first paragraph of this chapter mentioned that the dHvA effect measures the geometry of the Fermi surface. In 1991, Edwards et al. [20] pointed out the analogy between the oscillations in the interlayer exchange coupling and the oscillations seen in the dHvA effect. Also in 1991, Bruno and Chappert [21] showed that it was possible to use the Fermi surfaces determined in dHvA measurements to predict the periods that would be observed in measurements of magnetic multilayers. Table 1 shows the resulting comparison. The agreement between the two sets of periods is quite remarkable. I describe the experimental techniques used to measure the interlayer exchange coupling periods in the next section and discuss some of the possible reasons for the disagreement seen for Cu/Co(100). It is interesting to note that the oscillations in the interlayer exchange coupling and in the dHvA effect come from different critical features of the Fermi surface. In both cases, the oscillations are due to the abrupt change in occupancy at the Fermi energy. In the case of interlayer exchange coupling, there are oscillatory contributions to the coupling at all the energies and in-plane wave vectors of the occupied states. However, all of these oscillations tend to cancel out, except where the occupancy changes at the Fermi surface. Further, the oscillations at the Fermi surface all cancel except at points where two sheets of the Fermi surface become parallel to each other. There are similar oscillations in the de dHvA effect. All these also cancel except at the Fermi energy and where the area of the orbit in reciprocal space is stationary as a function of the wave vector along the magnetic field direction. a

Table 1. Comparison of oscillation periods measured in magnetic multilayers with those expected from the critical spanning extracted from Fermi surfaces measured in de Haas-van Alphen measurements (dHvA).

Ag/Fe(100) Au/Fe(100) Cu/Co(100)

Cr/Fe(100)

Cr/Fe(112)

Period (ML)

Period (ML)

Technique

Reference

5.58 5.7370.05 8.60 8.670.3 5.88 8.070.5 6.0
6.17 11.1 1271 12.5 14.4 15.4

2.38 2.3770.07 2.51 2.4870.05 2.56 2.6070.05 2.58
2.77

DHvA SEMPA DHvA SEMPA DHvA MOKE SEMPA DHvA SEMPA MOKE DHvA MOKE

[21] [22] [21] [23] [21] [24] [25] [26,27] [28] [29] [26] [29]

62

M. D. Stiles

In this respect, these oscillatory phenomena are physical examples of a simple mathematical effect. In a Fourier transform, a function is described as a superposition of oscillating functions with different frequencies. An approximate description results when the range of frequencies is finite, i.e. when there is an upper cutoff frequency. Whenever the function being described has a sharp feature, a kink or a step, for example, approximate descriptions with a finite cutoff oscillate close to the location of the sharp feature. These oscillations are known as Gibbs oscillations. For the interlayer exchange coupling and the dHvA effect, the sharp cutoff is the Fermi surface where the occupancy changes from one to zero. Another interesting comparison of the two sets of oscillations is found in the different conditions that are required to observe both. The dHvA oscillations come about when electrons in a magnetic field complete a circular orbit before they scatter. For typical fields, these orbits are large, requiring long mean free paths, that is, low temperatures and very high-quality crystals. As shown above, interlayer exchange coupling requires that electrons complete a round trip within the spacer layer before scattering. This path is much shorter than that required for dHvA oscillations, consistent with the fact that oscillatory interlayer exchange coupling is observed at temperatures even higher than room temperature. Even though the interlayer exchange coupling is less sensitive to the mean free path, it was discovered much later because growing sufficiently good multilayers requires deposition techniques that has only recently been developed. Figure 6 illustrates the resolution to the early doubts that interlayer exchange coupling was related to the RKKY interaction. These doubts were driven by the fact that the period expected from the free electron critical spanning vector is close to 0.3 nm, which is much shorter than the observed period of about 1.0 nm.Taking the actual Fermi surface into account does not immediately help. The Cu Fermi surface is a distorted sphere and the critical spanning vector qL is close to the free electron 2kF. The resolution of this discrepancy derives from the fact that Cu has a periodic lattice and in multilayers has layers of thickness d. Hence, the coupling can only be sampled at discrete values of the thickness nd for integer n. Since the oscillation corresponding to the free electron Fermi surface has a period shorter than d, the oscillation with wave vector qL is indistinguishable from an oscillation with wave vector q0 L ¼ ð2p=dÞ
qL : The period associated with q0 L is very close to what is seen experimentally. The equivalence of a discretely sampled rapid oscillation to as lower oscillation is referred to as aliasing. Aliasing is the reason that the Cu(100) critical spanning vectors are labeled the way they are in Fig. 6. The longer critical spanning vector qL gives a longer period oscillation than the shorter critical spanning vector qS gives. Without aliasing, it would be the other way around. 2.3.2. Coupling strength While the critical spanning vectors and other properties of the Fermi surface in Eq. (8) can be extracted from experiment, the reflection amplitudes cannot. For real multilayers, these can be calculated, but the resulting coupling strengths do not agree nearly as well with measured values as the periods do. Some of the disagreements are

Exchange Coupling in Magnetic Multilayers

63

Coupling Strength (mJ/m2)

due to experimental difficulties, which are discussed in the next section, but some disagreements are due to theoretical difficulties. The asymptotic form is an approximation to the difference in energy between the total energies of multilayers with parallel magnetizations and antiparallel magnetizations. Unfortunately, due to the complexities of the electron–electron interaction, it is currently impossible to compute these total energies without approximation. The best available approximation, as described in the beginning of this section, is the LSDA. This approximation works quite well for magnetic multilayers but with one caveat. The band structure is only approximate, so the Fermi surface deviates from its actual shape. Since the oscillation periods of the interlayer exchange coupling depend on the critical spanning vectors of the spacer layer Fermi surface, the periods computed using the LSDA will be wrong. This means that after a few oscillations, the calculated coupling is out of phase with the experimental coupling. The sign of the coupling may even be wrong. A direct comparison is then misleading because even if the physics is essentially correct, the agreement might be quite poor, see Fig. 7. There, the agreement is made even worse by the effect of disorder on the measured results. In Fig. 7, the calculated and measured periods of the oscillations disagree; the periods extracted from the experiment are 2.4870.05 monolayers and 8.670.3 monolayers and the critical spanning vectors of the theoretical Fermi surface would give periods of 2.65 monolayers and 8.03 monolayers. One way to compensate for the errors in the periods is to fit both theory and experiment to the asymptotic form, and compare the envelopes. The results of fits to the calculations and measurements in Fig. 7 are given in Table 2. The fits to experiment take into account measured thickness fluctuations, as discussed in the next section. Taking into account the thickness fluctuations leads to much better agreement between the strengths in Table 2 than in the raw comparison in Fig. 7. An additional source of disagreement that has not been accounted for is temperature. The calculations are done at zero temperature and the measurements at room temperature. While it is clear that 1.000

0.100

0.010 0.001 0

10 20 Thickness (monolayers)

30

Fig. 7. Calculated [30] and measured [31] coupling strengths for Fe/Au/Fe(100) multilayers. The solid curve is the best fit to the experimental data (symbols) when the measured thickness fluctuations are taken into account. The dashed curve is a linear interpolation between the coupling strengths (symbols) calculated for complete layers.

M. D. Stiles

64

Table 2.

Fit to experiment [31] Asymptotic calculation [34] Fit to full calculation [30]

Comparison of coupling strengths. J s/(1 nm)2

J L/(1 nm)2

1.2970.16 mJ/m2 2.0 mJ/m2 3.4 mJ/m2

0.1870.02 mJ/m2 1.1 mJ/m2 1.1 mJ/m2

accounting for the temperature will improve the comparison, the temperature scale for the correction [32,33] in this system is not known so it cannot be made quantitative. Measuring the coupling at low temperature would be the ideal solution. Possible reasons for any remaining disagreement are discussed in the next section. Also given in Table 2 are calculations done using the asymptotic form. Possible reasons for the imperfect agreement between calculations are discussed below. A difficulty with total energy calculations is that the total energies are typically many orders of magnitude larger than the difference in energy between parallel and antiparallel alignment of the magnetizations. In the LSDA, the density and the potential depend on each other and need to be determined self-consistently. Accurately computing energy differences makes the calculations quite time consuming. One way to simplify the process is to use the force theorem. The force theorem states that if the densities (or potentials) of two configurations are close to each other, the difference in total energies can be approximated by the difference in eigenvalue sums. The eigenvalue sum is the sum of the energies of all of the occupied states. The eigenvalue sum is more easily calculated, is much smaller than the total energy, and the calculation need not be self-consistent. The calculation shown in Fig. 7 used the force theorem. In a real sense, the ‘‘derivation’’ of the asymptotic form for the interlayer exchange coupling is based on the force theorem. The energy was assumed to be given as a sum of the energies of all of the occupied electronic states. It turns out that even for realistic systems such an approach can be a reasonable approximation. Having reduced the interlayer exchange coupling to eigenvalue sums, it is possible to derive the asymptotic formula, Eq. (8), for realistic multilayers [35,36]. This form is derived by ignoring the energy and wave vector dependence of the reflection amplitudes. In the thick limit, including the energy and wave vector dependence gives additional contributions to the coupling that are higher order in 1/D. These contributions are called preasymptotic corrections [37,38]. For Co/Cu(100), where these corrections have been studied in detail, they turn out to be quite important. Asymptotically, the strength of the long period coupling is weak, but for thinner layers, the corrections lead to substantial coupling. For the long period coupling, the wave vector dependence of the majority electron reflection amplitude gives the most important correction. On the other hand, the short period coupling is strong for thick layers, but the energy dependence of the phase of the majority reflection amplitude makes it weaker for thin layers. While Fe/Au(100) multilayers have not been analyzed for preasymptotic corrections, there are indications that they might be important. First, the coupling strengths for the short period coupling calculated using the asymptotic formula disagree with

Exchange Coupling in Magnetic Multilayers

65

the coupling strength extracted from the full calculation. Second, there are differences between the periods expected from the critical spanning vectors of the Fermi surface and those extracted from Fourier transforms of the calculated coupling [30]. 2.3.3. Torques and spin currents When the magnetizations of two adjacent layers are not collinear, the interlayer exchange coupling exerts a torque on both, given by the negative derivative of the energy, Eq. (4), with respect to the relative angle. Some of the first calculations of the interlayer exchange coupling [39,40] proceeded by directly computing the torque between the magnetization directions. Such calculations proceed analogously to the calculation sketched above, but it is necessary to match non-collinear spin states at the interfaces and to take great care in performing the energy integrals. It is interesting that the torque, which like the energy has contributions from all of the occupied states, can be cast in a form that depends only on the properties of the electrons at the Fermi energy. Since the interlayer exchange coupling is mediated by the electrons in the spacer layer, the torque is as well. The torque is associated with a spin current flowing in the spacer layer carrying angular momentum from one layer to the other. Angular momentum is extracted from one layer, effectively exerting a torque on the magnetization of that layer, and is deposited in the other layer, effectively exerting an opposite torque on the magnetization of second layer. This spin current differs from the spin current of interest in spintronics [41] because it is carried by all of the electrons in the spacer, not the electrons close to the Fermi level. It also exists independent of an applied voltage. Distinguishing spin currents due to quasi-equilibrium interactions and those related to transport [42] is important to understand possible spintronic devices and also current-induced torques [43,44].

3. MEASUREMENT OF INTERLAYER EXCHANGE COUPLING 3.1. Growth and Disorder Growing magnetic multilayers to compare measurements with calculations is quite difficult. Calculations are only tractable for systems that are close to ideal, requiring growth of systems equivalently close to ideal. The first requirement is that the lattices of the different materials need to be compatible. For example, when Co grows on Cu, it grows pseudomorphically, that is, it adopts the fcc structure of Cu, with a lattice constant that is very close to that of Cu. For a review of growth in this system see Ref. [45]. Another pair of metals with identical crystal structures and close lattice constants is Fe and Cr. Both of these pairs of materials can be grown with several different interface orientations. Some of these are shown in Table 1. Unfortunately, these are the only two pairs of materials with such similar crystal structures. The only other pairs that can be grown sufficiently well are Au or Ag and

M. D. Stiles

66

Fe, but only in the (0 0 1) interface orientation. It is somewhat surprising that these systems can be grown well because Fe has a body-centered cubic (bcc) structure, while Au and Ag have an fcc structure. In addition, the lattice constants are very different. However, it turns out that if the Fe lattice is rotated by 451 around the interface normal, there is very good in-plane lattice match for each of these pairs [46]. If the starting substrates are sufficiently flat, very good growth can be achieved. However, the presence of steps leads to extended defects through the layer because the Au/Fe growth is not pseudomorphic and the layer thicknesses are quite different. Interpretation of coupling through Cr spacer layers is complicated by the presence of spin density wave antiferromagnetism [47–49]. In bulk Cr, the transition temperature is close to room temperature, the temperature at which most measurements of the coupling are made. In very high-quality multilayers, the antiferromagnetism leads to a short period coupling, which is not well described by the model presented in this paper. However, in addition to the short period coupling associated with the antiferromagnetism, there is a long period oscillatory coupling, which appears to be well described by the model presented here. The properties of this long period coupling are given in Table 1. Unfortunately, the pairs of materials in Table 1 exhausts the systems that are well enough lattice matched to make high-quality comparisons between calculations and measurements. When a material is grown on a substrate that is not so well lattice matched, it assumes the in-plane lattice constant of the substrate on which it is deposited for a couple of monolayers. Then, as the thickness of the deposited layers increases, the strain energy associated with its modified lattice structure becomes too large and the film relaxes by introducing dislocations at the interface. Systems that have been studied in addition to those in Table 1 are reviewed in Refs [50,51]. The same quality of growth has not been achieved in these systems as in the systems with much smaller lattice mismatch. Even when the lattice mismatch is close to zero, real multilayers are still not perfect. The starting substrate is never perfectly flat and the growth is never perfectly layer-by-layer, so there are always variations in the thickness of the layers, typically called thickness fluctuations. Frequently, the lateral length scales of the thickness fluctuations are in an intermediate regime. The flat terraces are large enough that the ideal coupling is reasonably well defined over each terrace. On the other hand, the terraces are small enough that the magnetizations do not vary significantly on that length scale. In this intermediate regime, the comparison between calculated and measured coupling strengths is improved by averaging over measured variations of the spacer layer thickness. If the coupling for an ideal thickness of n layers is J(n) and the probability of having a thickness nd for a nominal deposited thickness of D is P(n,D), the effective coupling strength is X JðDÞ ¼ Pðn; DÞJðnÞ (9) n

If the width of the growth front is measured by scanning tunneling microscopy (STM), X-ray diffraction, or some other technique, theoretical coupling strengths can be averaged to compare with measured strengths. The curve in Fig. 7 that is a fit

Exchange Coupling in Magnetic Multilayers

67

to the experimental data includes the effect of the measured thickness fluctuations through Eq. (9). The coupling strengths with the effect of thickness fluctuations removed are given in Table 2. When the interlayer exchange coupling has rapid oscillations, the measured coupling is rapidly reduced by the thickness fluctuations. When the coupling strength is reduced by orders of magnitude, it is hard to extract meaningful coupling strengths, even if the growth front has been measured. For this reason, it is desirable to grow the sample in a layer-by-layer mode to keep the growth front as narrow as possible. Since layer-by-layer growth is determined by the competition between nucleation of islands and diffusion of deposited adatoms, it tends to require higher substrate temperatures during growth. Unfortunately, higher growth temperatures tend to promote interdiffusion at the interfaces. Interdiffusion, which gives rise to scattering centers, is more difficult to treat theoretically than thickness fluctuations. It also can be more difficult to measure. Interdiffusion can have an important and counterintuitive effect on the coupling. In the Fe/Cr(100) system, the measured interdiffusion [52–54] is believed [55] to cause the sign reversal in the measured short period coupling [28,56,57]. Extensions of the calculations discussed above for Fe/Au(100) [30], indicate that interdiffusion may be responsible for the difference between the calculated and measured coupling strengths in Table 2. Those values have already been corrected for the measured thickness fluctuations. It is peculiar that the short period coupling strengths agree much better than the long period strengths. Calculations that include interdiffused atoms at the interface indicate that interdiffusion reduces the long period coupling more than the short period. Since the interdiffusion is not measured for this case, it is possible that it explains the remaining discrepancy between theory and experiment. The choice of substrate plays a large role in the quality of the growth. The best measurements are made on substrates of one of the materials in the multilayer. Iron whiskers, which can be extremely flat [58], and copper single crystals give the best results. However, insulating substrates are necessary if the samples are also to be used for transport measurements. For these substrates, great care is required to get really high-quality growth. See Refs [59,60] for descriptions of the complexity of growing an Fe/Au multilayer on a GaAs substrate. One of the key advances that allowed accurate determination of the oscillation periods was the use of wedge-shaped spacer layers, see Fig. 6. Growing wedge samples simply involves moving a shutter between the sample and the evaporator during growth to expose different parts of the sample to different total fluxes. Such samples ensure that all thicknesses are grown under the same conditions because they are grown simultaneously on the same substrate. Wedge samples also make it easier to accurately determine the thickness of the samples. 3.2. Measurement Techniques Measurement of interlayer exchange coupling relies on two broad categories of measurements. One set determines the structure of the multilayer in as much detail as possible. The second determines the magnetic coupling. Common techniques for

68

M. D. Stiles

determining the structure of multilayers are reflection high-energy electron diffraction (RHEED), STM, X-ray scattering, and neutron scattering. Common techniques for determining the coupling are magneto-optical Kerr effect (MOKE), scanning electron microscopy with polarization analysis (SEMPA), Brillouin light scattering (BLS), and ferromagnetic resonance (FMR). The technique RHEED [61] is commonly used to determine the quality of a surface during growth. A high-energy electron beam is reflected from the surface at glancing angles. The resulting diffraction pattern is sensitive to the details of the surface, in particular the presence of steps. If the growth is layer-by-layer, there are fewer steps when layers are close to complete and more when the layer is half filled. In this case, the intensity of different spots in the RHEED pattern oscillate with a period of one layer. Since the oscillations have a period of a single layer, RHEED oscillations can be used to determine the total thickness of the film. For samples of uniform thickness, RHEED is used to monitor the thickness film during growth. For wedge samples, it is typically used after growth, when the RHEED beam is scanned along the wedge and the RHEED oscillations are monitored as a function of position to give the thickness at that position, see Fig. 8. Techniques used to measure the coupling fall into two broad classes. In the first class, the magnetic configuration is measured, frequently as a function of applied magnetic field. For example, Parkin et al. [10] first observed oscillatory interlayer exchange coupling using the GMR. Here, the resistance of the film in zero field was compared with the resistance in large field. If the coupling is ferromagnetic, there is no change, and if the coupling is antiferromagnetic, the change can be substantial. A commonly used technique to determine the magnetic configuration is MOKE [62]. The magneto-optic Kerr effect is the dependence of reflected light on the polarization of the light and the magnetization of the surface. Typically, the polarization of the light rotates through a small angle on reflection. Variations in the rotation can be measured to give the variations in the magnetization that cause them. For wedge samples, MOKE can be used in an imaging mode by scanning the focused spot of a laser across the surface or by imaging a wide area of illumination. It is not particularly surface sensitive and has the advantage that it is sensitive to the magnetic state of both layers. Using the sensitivity to both layers, MOKE images [63] have directly identified perpendicular alignment of two layers, see Section 4. MOKE was used to measure the coupling strengths in Fig. 7. An imaging technique that has been used to determine coupling periods is SEMPA [64]. When a high-energy electron beam scatters from a surface, it excites low-energy electrons, which leave the surface. These secondary electrons tend to maintain the polarization they had when in the surface. SEMPA measures the magnetization of a region of the sample’s surface by determining the polarization of the secondary electrons. Since this technique is based on measuring secondary electrons, which have low energy, it is generally not used with an applied field, limiting it to studies of the remnant state. On the other hand, it has greater spatial resolution than optical techniques like MOKE, and can be used on smaller wedges, requiring smaller areas of sample perfection. Since it can only measure the remnant state, SEMPA has not been used to measure coupling strengths, but it has been

Exchange Coupling in Magnetic Multilayers

69 2 mm

ayer e Overl Au Spacer Layer

Thin F

12 nm

Overlayer Magnetization

Fe Whisker Substrate

Magnetization

RHEED

0

20 40 Au Thickness (ML)

60

Fig. 8. Interlayer exchange coupling in a wedge-shaped Fe/Au/Fe trilayer measured by SEMPA and RHEED [22]. A schematic view of the wedge-shaped sample is shown at the top of the figure. The approximate dimensions give an indication of the very small slope of the wedge. Immediately below is a SEMPA image of the magnetization of the Fe overlayer. White and black indicate parallel and antiparallel alignment to the substrate, and hence ferromagnetic and antiferromagnetic coupling. Below that is a line scan through the image and then a measurement of the RHEED intensity along the wedge. The oscillations are used to determine the thickness of the spacer layer along the wedge. The wedge is slightly curved. The RHEED and the magnetization curves have been corrected for this curvature, but the image has not, hence the variation of the lines connecting the image with the line scan.

used to determine the sign of the coupling for enough oscillations of the coupling to allow high-precision measurements of the periods, see Fig. 8. In addition, the electron beam serves as a very high-resolution source for measuring RHEED so that both measurements can be done in situ. When the magnetic state is measured, the interlayer exchange coupling is inferred from the state rather than directly measured. SEMPA images are analyzed with the assumption that the interlayer exchange coupling dominates other energies so that the magnetization points in the direction of the coupling. The MOKE measurements in Fig. 7 are based on images like those shown for SEMPA in Fig. 8, but in the presence of an external applied field. In this case, it is assumed that the direction of the coupling is determined by the balance of the interlayer exchange coupling and the interaction with the external field. In both cases, magnetic hysteresis is ignored. Both of the analyses are simple examples of a more general approach. Usually, some magnetic property of a sample, like its hysteresis loop, is measured and the exchange coupling is inferred by comparing the measured property with a model. Some or all of the parameters of the model, including the interlayer coupling

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constant, are varied until the predictions of the model agree with the measurement. The reliability of the results depend on the accuracy of the model, in particular whether it includes all of the physics necessary to describe the experiment. The second class of measurements used to determine the coupling is based on determining the curvature of the free energy of the magnetization with respect to small variations in the magnetization direction. Two such techniques are FMR and BLS [65]. FMR is based on finding peaks in the microwave absorption of multilayers. The peaks identify resonance frequencies; in other words, the frequencies of the uniform modes of a layer. The coupling between the magnetizations of different layers gives rise to coupled oscillations. The different in-phase and out-of-phase resonance frequencies then give the size of the coupling. BLS is closely related. When light is scattered from the sample, there is a small amount of scattered light that has gained or lost energy by absorbing or exciting a mode of the system. Peaks in the absorption spectrum identify the modes of the system. The mode frequencies can then be analyzed to give the interlayer exchange coupling.

4. BIQUADRATIC COUPLING In almost all multilayers, there is a contribution to the coupling that favors perpendicular alignment of the magnetizations. In many multilayers, this contribution dominates, leading to actual perpendicular alignment of the magnetizations [63,66]. Phenomenologically, this alignment can be explained by a coupling of the form E ^1 m ^ 2 Þ2 ¼
J 2 ðm (10) A called biquadratic in contrast to the bilinear coupling discussed above. It is called biquadratic because it is quadratic in both of the magnetization directions. The fact that all measured values of J2 are negative, favoring perpendicular orientation of the two magnetizations, shows that biquadratic coupling does not have an intrinsic origin similar to the bilinear coupling, but, as Slonczewski [67,68] showed, has an extrinsic origin due to disorder. Thickness fluctuations lead to variations of the coupling strength on different terraces. To lowest order, the intralayer exchange coupling forces the magnetizations in each layer to be uniform so that the bilinear coupling gets averaged over the growth front as described in Eq. (9). To next order, the magnetization can fluctuate around its average direction. Over each terrace, the magnetization fluctuates in the direction that lowers the energy. The fluctuations lower the energy the most when the two magnetizations are perpendicular to each other. This is the origin of the effective interaction favoring perpendicular alignment between the magnetizations. Consider the simple model shown in Fig. 9, in which the spacer layer consists of parallel strips of width L with alternating thicknesses and hence coupling strengths J n and J n+1. The relative angle of the magnetizations is y ¼ y0+dysin(px/L), where dy is the size of the fluctuations. Over the region from 0 to L, where the coupling is J n, the energy change due to the fluctuations is proportional to J nsin(y0)dy. Over

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Upper Ferromagnet n

L

n +1

Spacer Layer

Lower Ferromagnet

Fig. 9. Thickness variations and biquadratic coupling. The thickness of the spacer layer varies periodically between n and n+1 layers, with ferromagnetic coupling for thicknesses n+1 and antiferromagnetic for thicknesses n. The heavy arrows show the local rotation in the magnetization direction into the direction that minimizes the coupling for each terrace.

the region from L to 2L, the sine function changes sign and the energy change due to the fluctuations is proportional to
J n+1sin(y0)dy. The net coupling energy per area due to the fluctuations is proportional to
DJsin(y0)dy, where DJ ¼ J n
j nþ1 : Fluctuations in the correct direction lower the energy of the system. The energy gain is balanced by the cost in intralayer exchange energy because the magnetization now varies spatially. Since the intralayer exchange coupling depends on the square of the gradient of the magnetization, for this simple model, it is proportional to (Aex/L)dy2, where Aex is the strength of the exchange interaction. Combining the changes due to the fluctuations for the interlayer exchange coupling and the intralayer exchange and finding the minimum with respect to the amplitude of the fluctuations gives dyp
sin(y0)DJ/(Aex/L). For this fluctuation amplitude, the change in the energy per area due to the fluctuations gives the strength of the biquadratic coupling J2 


ðDJÞ2 L Aex

(11)

While Eq. (11) is quite simple, it qualitatively describes the features of more realistic situations. In real systems, a characteristic length scale L of the arbitrary-shaped terraces replaces the width of the stripes. As this length scale increases, the biquadratic coupling strength increases because the fluctuations can get larger. Also, realistic growth fronts generally consist of more than two thicknesses, which introduces an effective DJ. The coupling increases as the difference in the coupling for the different terraces get larger. The differences tend to be largest when the coupling is oscillating rapidly, that is when there is short period coupling. The coupling gets weaker as the intralayer exchange interaction increases because exchange suppresses the fluctuations in magnetization direction that lower the energy.

5. SUMMARY I have attempted to pedagogically describe interlayer exchange coupling using a simple physical picture that has evolved over the last decade. Spin-dependent reflection from the interfaces in multilayers communicates the magnetic state of one

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ferromagnetic layer through a non-magnetic layer to a neighboring ferromagnetic layer. This communication leads to an exchange interaction between the layers. The exchange interaction has oscillatory contributions with periods determined by the critical spanning vectors of the spacer layer Fermi surface and strengths determined by the spin-dependent reflection at the interfaces. The oscillations in the exchange interaction arise from the spin-dependent quantum well states set up by the reflection. These quantum well states evolve in energy as the thickness of the spacer layer is varied. As these states pass through the Fermi energy, they fill or empty, changing the energy of the multilayer. These changes are periodic because the quantum well states cross the Fermi energy with a period determined by the Fermi surface of the spacer layer material. At critical points of this Fermi surface, many quantum well states have the same period giving a net oscillatory contribution to the energy. The energy difference between the parallel and antiparallel alignment of the magnetizations gives the interlayer exchange coupling. The oscillations in the interlayer exchange coupling are analogous to the oscillations in the dHvA effect. The periods of both are determined by critical geometrical properties of the Fermi surface. In fact, the Fermi surface geometries extracted from dHvA experiments can be used to predict the periods of the coupling. The agreement between these periods and the coupling periods, which have been measured to 3% accuracy in the best cases, provides strong support for our understanding of the coupling. The periods of the oscillatory interlayer exchange coupling are relatively insensitive to the presence of defects in the multilayer, but the coupling strengths are not. The measurements that have been done on defects and the calculations of their effect on the coupling give indications that our understanding of interlayer exchange coupling is correct. Improving our understanding will require measuring the coupling in systems that are as close to perfect as possible and then quantitatively measuring all the defects that remain. The necessary structural characterization will require multiple techniques to measure all of the defects. Then, calculations will need to explicitly treat the measured defects. The biquadratic coupling that is ubiquitous in magnetic multilayers is an example of an effective interaction that arises because of defects and frustration. Interfacial roughness gives rise to fluctuations in the strength of the bilinear coupling. The system can lower its energy by allowing the magnetization of the layers to fluctuate in response to the roughness-induced variations. The system can lower its energy the most when the two magnetizations are perpendicular to each other, giving an effective coupling that favors perpendicular alignment of the magnetizations. For more information on interlayer exchange coupling, there are a number of review articles. The series Ultrathin Magnetic Structures I–IV consists of review articles on the general topic of magnetic multilayers. Chapter 2 of Ultrathin Magnetic Structures II contains four review articles written around 1993 covering various aspects of interlayer exchange coupling [69]. Volume III contains yet another review article, written in 2002. Volume 200 of the Journal of Magnetism and Magnetic Materials consists of a series of review articles covering much of magnetism and includes many related to magnetic multilayers. The article on interlayer

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exchange coupling [50] focuses on the comparison between theory and experiment. Other general review articles on interlayer exchange coupling include [68,70–73]. For a compendium of theoretical and experimental results for specific systems, see [51]. The system Fe/Cr is sufficiently rich that it has generated three review articles on its own [47–49]. Most of the articles above focus on transition metal multilayers; for reviews of rare earth multilayers, see [74,75]. As mentioned above, photoemission studies of quantum well states in magnetic multilayers are reviewed in Refs [16–19]. Biquadratic coupling is reviewed in Ref. [76].

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Chapter 4 STATIC, DYNAMIC, AND THERMAL PROPERTIES OF MAGNETIC MULTILAYERS AND NANOSTRUCTURES R. E. Camley 1. INTRODUCTION Why have magnetic multilayers received so much attention in the last few years? There are many answers: (1) Multilayers represent a new class of materials where surface and interface effects fundamentally influence the properties of the entire material. For example, in a typical bulk material there might be 1015 surface atoms and 1023 atoms in the interior. Thus the ratio of surface to bulk atoms is 10
8. In contrast, a multilayer made up of films of alternating materials, where each film is five atomic layers thick, has a ratio of surface or interface atoms to bulk atoms of 40%. (2) A rich variety of structures and behaviors are possible. One can combine ferromagnets with nonmagnets, ferromagnets with ferromagnets, ferromagnets with antiferromagnets, ferromagnets with helical magnets, to name only a few possibilities. Each of these structures has unique and potentially useful properties. For example, the giant magnetoresistance effect [1–3]1 originally found in Fe/Cr multilayers is now being used in magnetic memory storage systems. In addition, one can obtain and design unusual behaviors including magnetization that increases with temperature and a thermal hysteresis that is tunable with a small magnetic field. (3) The importance of surface and interface effects may be controlled by the layering pattern. Adding more layers to each film emphasizes bulk effects and conversely thinner films as constituents in the multilayer emphasize interface effects. Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01004-8

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(4) The properties of magnetic multilayers can be substantially modified by small external fields. Because of the tunability discussed above, one can design multilayers that are sensitive to small, convenient magnetic fields. For example spinflop fields in antiferromagnets are often in the 100 kOe range. In contrast, a spin-flop in a magnetic multilayer can be designed to occur in the 100 Oe range. In this chapter, we will explore simple calculational methods for determining the equilibrium spin structures of multilayers and nanostructures at any magnetic field and at any temperature. The results of these calculations will then be compared with experimental measurements. In addition, we will develop simple methods for finding the dynamic behavior of magnetic multilayer structures and again compare the results with available experiments.

2. THEORETICAL TREATMENT OF MAGNETIC MULTILAYERS In bulk systems, one often employs the ‘‘mean field’’ approximation to understand the thermal behavior of magnetic materials. In this method, an individual spin feels an effective magnetic field produced by its near neighbors. From symmetry one expects, for example, that in a ferromagnet all spins will be parallel and equivalent and this assumption is used in calculating the average field. In a layered structure, it is necessary to modify this assumption and a reasonable supposition is that the spins within a single atomic layer are parallel and equivalent for a ferromagnet. For an antiferromagnet, one may assume that all spins on a given sublattice within a single atomic layer are equivalent. The orientation and thermal averaged magnitude of a spin in one atomic layer may be quite different from those in a neighboring layer. As a result of these considerations, the magnetic multilayer may often be treated as an effective one-dimensional system with one spin representing an entire layer of spins. We note that dealing with the system in a layer-by-layer fashion is particularly helpful in multilayer systems because surface/interface variations in exchange and anisotropy may easily be taken into account. A schematic illustration of the layer-by-layer structure is shown in Fig. 1. A spin in layer n, Sn, feels a number of fields which influence its motion. These include: (1) Local exchange fields from within a layer and exchange fields from neighboring layers (2) Local anisotropy fields (3) The external field, H0 (4) Demagnetizing fields caused by dipole motion. We first describe an iterative energy minimization method for the T ¼ 0 case. One starts with an assumed set of values for the angular position of each layer of spins. In many cases, the spins will lie in a plane parallel to the interfaces

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Local uniaxial anisotropy axis Layer n-1

Layer n Exchange coupling Jn,n+1

Layer n+1

Applied magnetic field

Fig. 1. Schematic illustration of the layered structure and interactions for a magnetic multilayer. The top layers have a uniaxial anisotropy. All the layers feel an external field and the exchange fields of the neighboring layers.

(this reduces the demagnetization energy) and for simplicity we will use this assumption. It is also assumed that all spins in each layer are equivalent. Due to exchange coupling, the energy of the spins in any given layer depends on the orientation and magnitude of the spins in its own layer and in the nearby layers. An arbitrary layer n, of spins, can be regarded as having an energy En (neglecting anisotropy) where X J nm Sn Snþm
gn mB H 0 S n (1) En ¼
m¼0;1;2

Here Jnm is the effective exchange coupling constant between spins in the nth and mth atomic layers. Interactions betweens spins in the same layer can be accounted for by a Jnn term. In terms of the angle yn that the spins in layer n make with the applied field: X J nm Sn Snþm cosðyn
ynþm Þ
gn mB Sn H 0 cosðyn Þ (2) En ¼
m¼0;1;2

For simplicity we have neglected anisotropy energy in this example, but it can be easily included, as we will see later. Thus at T ¼ 0 a single variable for each layer, yn, characterizes the magnetic structure. The initial set of angles described above will generally not be a ground state or a stable state of any kind. An iterative procedure can be used to allow the system to relax to a self-consistent stable state. First, the energy in an individual layer, n, is

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minimized by choosing the angle yn generated by the minimization condition @E n =@yn ¼ 0

(3)

This procedure can be viewed as rotating the spins in the nth layer to lie parallel to their effective field, a process which reduces the layer’s energy. We can then find a configuration of the entire structure as follows: a different layer is then randomly chosen and the spins in that layer are rotated to lie in the direction of the local effective field. The process is continued until one has a self-consistent, stable state where all the spins are aligned with their local fields. Since the method only minimizes the energy locally, different initial configurations may lead to different selfconsistent final states. The ground state is, of course, the stable state with the lowest energy for the entire structure. At finite temperatures, both the direction and thermal averaged magnitude of the spins in each layer must be specified. In this case the iteration procedure is slightly different. A layer of spins is first rotated into the direction of the effective field, and then the spin’s thermal averaged magnitude in that direction is found through the use of the Brillouin function hS n i ¼ S n Bsn ðxÞ

(4)

gn mB Sn H n kT

(5)

where x¼

Here Hn is the effective field acting on a spin in layer n and is defined explicitly in Eq. (7). The Brillouin factor is given by


 x ð2S þ 1Þ ð2S þ 1Þx 1 Bs ðxÞ ¼ (6) coth
coth 2S 2S 2S 2S Here oSn4 is the thermal average of the spins in the nth layer in the direction of the effective field and Hn the effective field acting on layer n. The effective field is now given by Hn ¼ H0 þ

X

 J nm  S nþm gm m¼0;1;2 n B

(7)

Note that in the effective field the spins in the neighboring layers are replaced by their thermal averaged magnitudes. Again the entire operation is iterated through all spins until a self-consistent state emerges. For a particular temperature and applied magnetic field the iteration procedure requires the order of 103 to 107 iterations per layer to reach a converged state. It generally takes a larger number of iterations to find a converged state near a phase transition. To speed up the calculation, one often uses the final state at one temperature to be the initial guess for the spin configuration at a nearby temperature. Of course, the final state may only be a local minimum and it is not guaranteed that the true ground state will always be found by this method. Nonetheless, if one takes

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a sufficient number of different initial sets of angles, the method seems to discover all the low-energy stable states. The stability of a particular state can be checked in a direct way. Once a final state is found, one picks a new spin configuration slightly perturbed from the final state. Variations of a few degrees in each layer are typical. The iterative process then begins with this new configuration and one checks whether the system relaxes back toward the original state or diverges away from it. We note in passing that this mean-field theory is really the simplest possible approximation that takes into account a layer-by-layer variation, spin orientation, and thermal averaged magnitude. Such methods have been used previously in thinfilm calculations [4]. Improvements would include Bethe–Peirels–Weiss methods or Monte Carlo techniques. However these methods involve significantly larger computational investment.

3. EXAMPLES OF MAGNETIC MULTILAYER STRUCTURES Among the most interesting of the magnetic superlattice structures are those that involve some antiferromagnetic coupling. In this case, the exchange energy favors some kind of antiparallel alignment while the Zeeman energy due to an external field is minimized if the magnetic moments are all parallel to the field. As a result, these structures can display a wide set of phase transitions, induced through variations of either the magnetic field or temperature, with the transition temperatures and fields controlled by changing the layering pattern. We, therefore, concentrate on multilayers with antiferromagnetic coupling. One of the ‘simplest’ layered systems with an interesting magnetic structure and a fascinating dependence on layering pattern is the Gd/Y superlattice. Although bulk Y is normally nonmagnetic, in the Gd/Y superlattice the Y layers mediate an effective exchange interaction between neighboring Gd films [5,6]. The application of a magnetic field in this system produces some interesting results. Because of the competition of interface exchange and Zeeman energies, the ground state is not a macroscopic antiferromagnetic state where the ferromagnetic films are strictly antiparallel. Instead, the equilibrium structure resembles a macroscopic spin-flop state that represents a compromise between exchange and Zeeman energies. The resulting configuration is illustrated in Fig. 2. An increase in the external field reduces the canting angle y and changes the net magnetic moment of the structure. Small fields can have large effects because the applied field acts on all spin of a given film, while the effective exchange between films is weak and acts only on the interface layers of the films. In contrast, spins in bulk materials see large exchange fields from nearest neighbors that are very large compared to laboratory Zeeman fields. The magnetization is thus very sensitive to applied fields and also to temperature, which plays a strong role in determining the average interlayer exchange energy. In Fig. 3, we present theoretical and experimental results [6,7] for the magnetization as a function of temperature for Gd/Y superlattices. We see an unusual

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Fig. 2. The spin configurations in the Gd/Y multilayer at low temperatures and high temperatures. Note that at high temperature the thermal averaged magnetic moments in Gd are smaller at the surface than in the bulk. From Ref. [7].

temperature dependence with a magnetization that increases as temperature is increased. This can be understood by looking at the exchange coupling of the Gd spins. The Gd spins on the interior are strongly coupled to each other compared to those on the exterior, which are only weakly coupled on the Y side via the RKKY interaction. When the temperature is increased, the Gd spins in the interior thus retain a larger thermal averaged magnitude than those on the exterior since they see a larger effective field. As a result, the interface exchange energy (which should scale as J1(SGd
exterior)2) is rapidly reduced and the competition between exchange and Zeeman energies now favors a state where the Gd moments point more closely along the external field. This is automatically included in the iterative calculation and the results are illustrated schematically in Fig. 2. Thus even though oSGd4 decreases as the temperature increases, the change in orientation of the Gd spins is sufficiently large such that the total magnetization of the structure increases. As T continues to increase the oSGd4 begin to decrease very quickly and the magnetization decreases (more or less as single Gd films). The field dependence of the magnetization is also striking. Here, we have an ordered state that displays a significant change in magnetization as a function of field for very modest fields as can be seen in Fig. 3. We now turn to the influence of the outermost surfaces of the magnetic structure of a finite multilayer system. The surface layer plays a special role in the magnetic structure simply because it is an exchange coupled to other spins on only one side. Thus in the competition between exchange and Zeeman energies, the influence of the exchange energy is reduced and it is easier for an external field to direct the surface spins. As the outermost layer of spins tries to turn toward the field direction, the remaining layers must adjust their orientations as well. Thus surface-induced

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Fig. 3. Magnetization as a function of temperature for different external fields for a Gd/Y multilayer. The top panel shows the theoretical calculation, Ref. [7], and the bottom shows the experimental results from Ref. [6].

phase transitions can occur at magnetic field strengths well below that required for the equivalent bulk phase transition [8–11]. Also the reconstructed state can have surprisingly large penetration-depths into the bulk of the superlattice. As the first example of a surface phase in a superlattice, we consider a finite Fe/ Cr- or Gd/Y-type structure. We assume that the temperature is low enough that all spins have a thermal averaged moment equal to their maximum value. Furthermore, we assume that the exchange coupling between atomic layers is sufficiently strong that the spins within a film are rigidly coupled together. These assumptions are reasonable for Fe/Cr or Co/Ru superlattices at room temperature, for film thicknesses under 10–20 nm.

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For an infinitely extended Fe/Cr-type structure in an external field one expects a canted configuration as discussed earlier. From energy considerations, it is easy to show that, in the absence of any anisotropy, the uniform canting angle is given by the equation cos a0 ¼ H 0 M=4J

(8)

where, M is the magnetization of a film and J the interface exchange energy. In a finite multilayer, however, this uniformly canted state is not stable [11]. As mentioned above, the reason for this is that the outer layers of the finite structure experience only half the exchange coupling of the interior layers and thus are easier to turn toward the applied field. Stable ground state configurations for finite Fe/Cr-type multilayers have been found [11] using the numerical method outlined above. These states are compared to the uniform canting state in Fig. 4. The structure in Fig. 4(b), the low-field case,

Fig. 4. Illustration of bulk and surface spin-flop configurations in an antiferromagnet with no anisotropy. From Ref. [11].

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is quite complex. The outermost spins are twisted into the direction of the field as expected, but the spins of the second layer are actually turned farther away from the field than they would be in the bulk configuration. This alternation continues as one progresses into the bulk of the multilayer, but the amplitude of the deviation decreases with increasing distance from the surface. The main difference between the configurations of Fig. 4(b) and (c) is that there is no alternation in the high-field configuration of Fig. 4(c). The surface twist states illustrated in Fig. 4(b) and (c) reduce the energy of the structure as compared to the uniform canting state by lowering the Zeeman energy. (As a result, the surface twisted states have a slightly higher magnetization than does the uniformly canted state.) However, the twist also results in a small increase of exchange energy. The resulting structure is as usual a compromise between exchange and Zeeman energies. It is worthwhile to compare these nonuniform canted structures discussed here to the very different case of nonuniform canting found in a domain wall. For example, the width of a domain wall in a ferromagnet is governed by the competition between the exchange and anisotropy energies. The exchange energy can be minimized by spreading the canting out over many layers of spins and thus favors a very extended wall. In contrast, the anisotropy energy is minimized when spins point in an easy direction, thus favoring a narrow wall. In the surface twist problem, however, the competition is between exchange and Zeeman energies, and there is also the additional freedom of having two sublattices. This allows new kinds of states to occur when minimizing the total energy. The alternating spin configuration seen in Fig. 4(b), for example, has the exchange energy between layers of spins, which alternately increases and decreases as one moves through the multilayer. This extra degree of freedom also allows the width of the nonuniform canting region to vary in an interesting manner as a function of applied field. At low fields the nonuniform region is very large, for moderate fields it is quite small, and then for larger fields the width of the nonuniform regions is again large. The considerations above apply to microscopic structures as well as multilayers. For example, one can use the same techniques as described above to examine the magnetic field and temperature behavior of ultra-thin antiferromagnetic films. A particularly interesting feature relates to the number of layers in the antiferromagnetic film. For an even number of layers, one of the outer layers must have its magnetization pointing opposite to the applied field and, as a result, a surface phase transition is nucleated at this surface. For an odd number of layers the magnetic moments at both surfaces will point parallel to the external field and no surface transition will be induced, and the change to a spin-flop structure must occur in the bulk. These configurations are illustrated in Fig. 5. Calculations for the fields required to cause a spin-flop have been done for several antiferromagnetic materials. An example is shown in Fig. 6, where the spin-flop field is shown as a function of the number of layers in an ultra-thin film of the antiferromagnet MnF2 [12]. For an even number of layers the spin-flop field is nearly independent of the number of layers. The reason for this is that there is always a surface spin-flop in this system. In contrast, when the number or atomic

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86 Even

Surface phase transition nucleates here

Odd

Applied Field

Fig. 5. Illustration of the difference between an even and odd number of layers in an antiferromagnetic film. For an even number of layers one of the surface layers always has its moments pointing oppositely to the applied field.

Fig. 6. Spin-flop field in MnF2 for an even and odd number of magnetic layers at T ¼ 0: When the number of layers is even the spin-flop field is smaller because a surface spin-flop occurs. From Ref. [12].

layers is odd, the spin-flop field is significantly larger for thinner films, in part due to the pinning effect of the outer layers. The spin-flop fields for MnF2 calculated above are at T ¼ 0: Finite temperature calculations show that the odd/even behavior persists at finite temperature as well, as can be seen in the phase diagrams of Fig. 7 [12] for and even number of atomic layers (N ¼ 10) and an odd number of layers (N ¼ 11). It is interesting to note that the differences between an odd number of atomic layers and an even number of atomic layers persists in temperature until the phase transition to a paramagnetic state. The even/odd considerations in the microscopic antiferromagnet are all important in the multilayer as well. The details of surface phase transitions in an Fe/ Cr(211) multilayer have been explored both theoretically and experimentally recently [13–15]. In these structures, the unit cell is rectangular in the plane of the surface and provides a uniaxial anisotropy.

Static, Dynamic, and Thermal Properties

Fig. 7.

87

Phase diagrams for an ultrathin antiferromagnetic film of MnF2 with an even (N ¼ 10) and odd (N ¼ 11) number of layers. From Ref. [12].

The even/odd behavior was demonstrated most dramatically in measurements of the magnetization and magnetic susceptibility in the Fe/Cr(211) multilayer system [13] as shown in Fig. 8. As seen in this figure, the magnetization of the 20-Fe film structure shows two rapid changes, one, at low field, associated with the surface spin-flop, and the other at higher field associated with a bulk spin-flop. In contrast, the 21-Fe film structure displays only one transition close to the bulk transition of the 20-Fe film sample. Spin-flop states are generally associated with substantially increased susceptibility. The reason for this is simple. Susceptibility measures the ability of the spin configuration to change due to an external field. In the aligned states only the magnitudes of the magnetic moments can change. In contrast, in the spin-flop state both the magnitude and orientation can change, leading to a large susceptibility. We will see an enhanced susceptibility used as an indication of a canted structure in other systems as well. The behavior of Fe/Cr(211) multilayers with an even number of Fe films has been shown to be particularly interesting [13–15]. If the system starts in the antiferromagnetic state at low field and the field is increased, a surface spin-flop occurs at fields well below that required for the bulk spin-flop. Essentially a domain wall is introduced near the surface. As the field is further increased, the domain wall migrates to the center of the film and then broadens in its spatial extent so that eventually the entire film is in a spin-flop configuration. These theoretical predictions were confirmed in detail in subsequent experimental work with polarized neutron scattering [14]. For example, it was shown that for the particular case measured the domain wall reached the center of the film a