High-Temperature Cuprate Superconductors: Experiment, Theory, and Applications (Springer Series in Solid-State Sciences, 166)

Springer Series in solid-state sciences 166 Springer Series in solid-state sciences Series Editors: M. Cardona P. ...

Author: Nikolay Plakida

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Springer Series in

solid-state sciences

166

Springer Series in

solid-state sciences Series Editors: M. Cardona P. Fulde K. von Klitzing R. Merlin H.-J. Queisser H. St¨ormer The Springer Series in Solid-State Sciences consists of fundamental scientif ic books prepared by leading researchers in the f ield. They strive to communicate, in a systematic and comprehensive way, the basic principles as well as new developments in theoretical and experimental solid-state physics.

Please view available titles in Springer Series in Solid-State Sciences on series homepage http://www.springer.com/series/682

Nikolay Plakida

High-Temperature Cuprate Superconductors Experiment, Theory, and Applications

With 161 Figures

123

Professor Nikolay Plakida Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research Joliot Curie 6, 141980 Dubna Moscow Region, Russia E-mail: [email protected]

Series Editors: Professor Dr., Dres. h. c. Manuel Cardona Professor Dr., Dres. h. c. Peter Fulde∗ Professor Dr., Dres. h. c. Klaus von Klitzing Professor Dr., Dres. h. c. Hans-Joachim Queisser Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany ∗ Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨othnitzer Strasse 38 01187 Dresden, Germany

Professor Dr. Roberto Merlin Department of Physics, University of Michigan 450 Church Street, Ann Arbor, MI 48109-1040, USA

Professor Dr. Horst St¨ormer Dept. Phys. and Dept. Appl. Physics, Columbia University, New York, NY 10027 and Bell Labs., Lucent Technologies, Murray Hill, NJ 07974, USA

Springer Series in Solid-State Sciences ISSN 0171-1873 ISBN 978-3-642-12632-1 e-ISBN 978-3-642-12633-8 DOI 10.1007/978-3-642-12633-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010933960 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The beginning of 1986 marked the inauguration of the cuprate superconductor epoch in the search for high-temperature superconductivity. The discovery by Karl Alex M¨ uller and Johan Georg Bednorz of the occurrence of superconductivity in the lanthanum and barium copper oxides at temperatures up to 35 K caused an unprecedented wave of scientiﬁc activity in the study of superconductivity. In early 1987, the replacement of La by Y in the “Z¨ urich” compounds raised the superconductivity onset to 90 K. Within the next few years, new copper oxide compounds containing bismuth, thallium, or mercury were discovered such that the maximum transition temperature at ambient pressure was raised to 136 K. K.A. M¨ uller and J.G. Bednorz, at that time research associates at the IBM Research Division, Z¨ urich Research Laboratory, were awarded the Nobel Prize in Physics in 1987. During the elapsed two decades, their discovery has also opened new doors in solid state physics, in particular in the physics of strongly correlated systems. The recent discovery of superconductivity in the ferropnictide compounds at temperatures up to 55 K points to the existence of alternate high-temperature candidates and revives the hope of ﬁnding even room-temperature superconductors in the future. As a result of an enormous research eﬀort of a large number of physicists, chemists, and material scientists, high-quality samples of cuprate superconductors have been manufactured and their generic physical properties have been studied with high precision by applying various experimental methods. It has emerged that these compounds possess a number of unusual normal state and superconducting properties due to a complicated interplay of electronic, spin, and lattice degrees of freedom. In view of the complicated character of the interplay, any theory of cuprate superconductors encounters a number of diﬃculties. Despite the powerful modern methods of statistical physics, the study of various microscopic models has not so far resulted in a commonly accepted interpretation of all the physical phenomena and the mechanism for formation of the superconducting state. Presently, more than a hundred of thousands of papers are published on the problem of high-temperature superconductivity in cuprates in the form of

VI

Preface

journal articles and reports at numerous conferences. A number of excellent reviews and monographs have been already published in which the results of studies obtained in separate ﬁelds or on the basis of particular experimental methods are discussed. At the same time, there are only a few publications, where the essential properties of high-temperature cuprate superconductors are reviewed on the background of their theoretical interpretation. The purpose of the present book is to achieve this aim in a form accessible to a wide circle of researchers in the ﬁeld of cuprate superconductivity, both beginners and experts interested in a general overview. As it follows from the contents of the book, the main physical properties of the cuprate superconductors are discussed in a concise form in Chap. 2–6, and the essential theoretical models are considered in Chap. 7. Several examples of successful technical applications are discussed only brieﬂy in Chap. 8, since this ﬁeld constitutes a separate large branch of high-temperature cuprate superconductor studies. In the Appendix, the superconductivity theory is formulated within the equation of motion method for the Green functions. The discussions are given at the text-book level. The author expresses his deep gratitude to a large number of colleagues for valuable discussions. Among them are Gh. Adam, S. Adam, V.L. Aksenov, A. Alexandrov, L. Anton, A. Avella, N.A. Babushkina, A. Balagurov, A. F. Barabanov, S. Bariˇsi´c, S. Borisenko, Ph. Bourges, A. Chubukov, S.-L. Drechsler, H. Ebisawa, I. Eremin, L. Falkovsky, G. Fink, Ø. Fischer, R. Hayn, J.E. Hirsch, P. Horsch, D. Ihle, V.A. Ivanov, Yu.A. Izyumov, G. Jackeli, V. Kabanov, Yu.m. Kagan, D.I. Khomskii, Yu.V. Kopaev, M. Knupfer, M.L. Kuli´c, A. Kurdyuk, A.I. Lichtenstein, V.M. Loktev, S. Maekawa, E.G. Maksimov, L.A. Maksimov, F. Mancini, K. Matho, V.A. Moskalenko, A.S. Moskvin, K.A. M¨ uller, Y. Nagaoka, V. Oudovenko, S.G. Ovchinnikov, E.A. Pashitskii, L. Pintschovius, P. Prelovˇsek, A. Ramˇsak, J.-L. Richard, M.V. Sadovskii, V.S. Shakhmatov, Yu.M. Shukrinov, I.V. Stasyuk, A.N. Taldenkov, V.V. Val’kov, V.S. Vysotskii, V. Yushankhai, R.O. Zaitsev. Unfortunately, the author was unable to mention all the important publications in the ﬁeld and apologizes to colleagues whose works have not been cited in the book; those are unintentional omissions. I am very thankful to Professor Peter Fulde for his hospitality during my stay at the Max-Planck Institute for the Physics of Complex Systems where a major part of this book was prepared. The name of academician N.N. Bogolubov, my teacher, should be also mentioned, who supported our research activities in the ﬁeld of superconductivity for many years. My special thanks are to Gh. Adam who read the manuscript and made a lot of valuable suggestions and V. Oudovenko who helped to prepare the book for publication. The support of Dr. C.E. Ascheron, Springer Verlag, is highly appreciated. Last, but not least, I am very thankful to my wife for her patience and allowance to work on the book at the expense of family time. Dubna – Dresden May 2010

Nikolay Plakida

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problem of High-Temperature Superconductivity . . . . . . . . . . . . 1.2 Discovery of High-Temperature Superconductors . . . . . . . . . . . . 1.3 Generic Properties of Cuprate Superconductors . . . . . . . . . . . . .

1 1 4 6

2

Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Structure of Ba1−x Kx BiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Structure of La2−x Mx CuO4−y . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Structural Phase Transitions in La2−x Mx CuO4 . . . . . . . . 2.2.2 Theory of Structural Phase Transitions . . . . . . . . . . . . . . . 2.2.3 Copper-Oxide Ladder Compounds . . . . . . . . . . . . . . . . . . . 2.3 Nd2−x Cex CuO4 Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 YBaCuO-Based Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Structure of YBa2 Cu3 O7−y . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Modiﬁcations of the YBCO Structure . . . . . . . . . . . . . . . . 2.4.3 Rutheno-Cuprates Magneto-Superconductors . . . . . . . . . 2.5 Bi-, Tl- and Hg-Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 High-Pressure Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 15 17 19 25 28 32 33 34 38 39 40 45 49

3

Antiferromagnetism in Cuprate Superconductors . . . . . . . . . . 3.1 Magnetic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Antiferromagnetism in La2−x Mx CuO4 Compound . . . . . . . . . . . 3.2.1 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Theory of Magnetic Phase Transitions . . . . . . . . . . . . . . . 3.2.4 Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds . . . . . . . . . . . 3.3.1 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Resonance Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Antiferromagnetism in REBa2 Cu3 O6+x . . . . . . . . . . . . . .

51 52 55 55 59 63 68 80 80 84 92 98

VIII

Contents

3.4 Nuclear Magnetic Resonance Studies . . . . . . . . . . . . . . . . . . . . . . . 100 3.4.1 The Knight Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.4.2 Spin–Lattice Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.3 Spin Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4

Thermodynamic Properties of Cuprate Superconductors . . . 121 4.1 Anisotropic Ginzburg–Landau Model . . . . . . . . . . . . . . . . . . . . . . 121 4.2 Speciﬁc Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2.1 Low-Temperature Electronic Speciﬁc Heat . . . . . . . . . . . . 126 4.2.2 Pseudogap in Electronic Speciﬁc Heat . . . . . . . . . . . . . . . . 130 4.2.3 Fluctuation Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.3.1 Vortex Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.3.2 Critical Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3.3 Magnetic Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . 167 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5

Electronic Properties of Cuprate Superconductors . . . . . . . . . 177 5.1 Electronic Structure: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.1.1 Crystal Chemistry and Bands . . . . . . . . . . . . . . . . . . . . . . . 178 5.1.2 Eﬀects of Impurity Substitution . . . . . . . . . . . . . . . . . . . . . 183 5.2 Photoemission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.2.1 High-Energy Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2.2 Angle-Resolved Photoemission Spectroscopy . . . . . . . . . . 214 5.3 Optical Electron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 5.3.1 Dynamical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 5.3.2 Normal-State Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . 256 5.3.3 Superconducting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.3.4 Electronic Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . 290 5.4 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 5.4.1 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 5.4.2 Hall Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.4.3 Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 5.4.4 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.5 Superconducting Gap and Pseudogap . . . . . . . . . . . . . . . . . . . . . . 327 5.5.1 Gap Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 5.5.2 Tunneling Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 5.5.3 Phase-Sensitive Experiments . . . . . . . . . . . . . . . . . . . . . . . . 342

6

Lattice Dynamics and Electron–Phonon Interaction . . . . . . . 349 6.1 Neutron Scattering Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 6.1.1 Doping Dependence of Phonon Spectra . . . . . . . . . . . . . . . 351 6.1.2 Phonon Renormalization in Superconducting State . . . . 356

Contents

6.2 6.3 6.4 6.5

IX

Optical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Isotope Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

7

Theoretical Models of High-Tc Superconductivity . . . . . . . . . . 377 7.1 Electronic Structure of Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . 378 7.1.1 Band-Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . 378 7.1.2 Model Hamiltonians for CuO2 Plane . . . . . . . . . . . . . . . . . 382 7.2 Electron Excitations in the Normal State . . . . . . . . . . . . . . . . . . . 393 7.2.1 Single-Particle Electron Spectrum . . . . . . . . . . . . . . . . . . . 393 7.2.2 Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 7.3 Magnetic Mechanism of Superconductivity . . . . . . . . . . . . . . . . . . 428 7.3.1 Unconventional Ground State . . . . . . . . . . . . . . . . . . . . . . . 428 7.3.2 Spin-Fluctuation Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 7.3.3 Models with Strong Correlations . . . . . . . . . . . . . . . . . . . . 439 7.4 Electron–Phonon Superconducting Pairing . . . . . . . . . . . . . . . . . . 455 7.4.1 Anisotropic Electron–Phonon Interaction . . . . . . . . . . . . . 456 7.4.2 Van Hove Singularity Scenario . . . . . . . . . . . . . . . . . . . . . . 459 7.4.3 Polaron and Bipolaron Superconductivity . . . . . . . . . . . . 460 7.5 Charge Fluctuation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 7.5.1 Plasmon Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 7.5.2 Exciton Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 7.5.3 Coulomb Repulsion Pairing . . . . . . . . . . . . . . . . . . . . . . . . . 472 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

8

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 8.1 Electric Power Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 8.1.1 Superconducting Tapes and Cables . . . . . . . . . . . . . . . . . . 480 8.1.2 Fault Current Limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 8.1.3 Superconducting Rotating Machines . . . . . . . . . . . . . . . . . 483 8.2 Electronic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 8.2.1 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 8.2.2 Passive Microwave Devices . . . . . . . . . . . . . . . . . . . . . . . . . 488 8.2.3 Active Microwave Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 489 8.2.4 Superconducting Quantum Interference Devices . . . . . . . 491 8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

A Thermodynamic Green Functions in Superconductivity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 A.1 Thermodynamic Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 496 A.1.1 Green Function Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . 496 A.1.2 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 A.1.3 Sum Rules and Symmetry Relations . . . . . . . . . . . . . . . . . 498

X

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A.2 Eliashberg Equations for Fermion–Boson Models . . . . . . . . . . . . 499 A.2.1 Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 A.2.2 Noncrossing Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 501 A.3 Superconductivity in the Hubbard Model . . . . . . . . . . . . . . . . . . . 503 A.3.1 Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 A.3.2 Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 505 A.3.3 Self-Energy Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 A.4 Superconductivity in the t–J Model . . . . . . . . . . . . . . . . . . . . . . . . 510 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

1 Introduction

Ever since 1911 when Heike Kamerlingh Onnes ﬁrst discovered superconductivity, physicists have been interested to ﬁnd out why the temperature of the transition to the superconducting state Tc is so low compared to the temperatures of other phase transitions. The temperatures of the transitions to ferromagnetic or antiferromagnetic (AF) states in metals are hundreds of kelvin, while for conventional superconductors Tc does not exceed 10–20 K. This fact seems surprising, since in the both cases the phase transitions take place in the electron subsystem of the crystals and are ultimately due to electron–electron interaction. To answer this question, we shall discuss in this chapter the history of the copper-oxide superconductor discovery by Bednorz and M¨ uller [118, 120]. We also consider the generic properties of these compounds that distinguish them from the conventional superconductors. It is assumed that the reader is familiar with the conventional superconductivity theory, which is presented in a number of textbooks, for example, Parks [963], Ketterson and Song [598], and Buckel and Kleiner [185].

1.1 Problem of High-Temperature Superconductivity It is natural, with respect to applications, to refer a superconductor as a high-temperature superconductor (after Ginzburg [393]) if its transition temperature Tc exceeds the boiling temperature of liquid nitrogen, Tb,N2 = 77.4 K, that is, if the superconducting state can be obtained by cooling in liquid nitrogen. A large number of papers (see, e.g., [392]) are devoted to the general problem of designing high-temperature superconductors. To discuss the factors that inﬂuence the temperature of the superconducting transition, we shall consider the pairing theory of Bardeen, Cooper, and Schrieﬀer (BCS) [104]. There are three speciﬁc parameters that characterize the interaction of electrons: the density of the electron states at the Fermi level N (0) (per spin direction), an eﬀective attraction coupling constant V , and an energy shell h ¯ω around the Fermi surface, over which this coupling is nonzero. In the weak

2

1 Introduction

coupling limit λ = N (0)V 1, the transition temperature according to the BCS theory is kTc ¯ hω exp (−1/λ). (1.1) This expression determines the temperature below which the normal state of electrons becomes unstable with respect to the formation of a condensate of electron (Cooper) pairs with opposite spins and zero orbital momentum (singlet s-wave pairing). The nature of the forces responsible for the eﬀective attraction bears no relevance to the derivation of (1.1). The BCS theory and the relation (1.1) can therefore be used to treat superconductivity in the case of other, that is, nonphononic mechanisms of pairing. In these cases, one speaks of a generalized BCS pairing theory. It is interesting to note that, besides the superconducting electron pairing, an instability with respect to the formation of electron–hole pairs below a certain temperature T0 is possible, which results in the charge density waves (CDW) and the metal–insulator transition or in the spin density waves (SDW) and ferromagnetic or antiferromagnetic order. In certain cases (e.g., when the electron (hole) Fermi surface has perfect nesting), this temperature is determined by the same relation (1.1). In this case, the attraction V between an electron and a hole in the energy shell h ¯ ω may be of purely Coulomb nature. For typical Coulomb energies h ¯ ω 1 eV, according to (1.1) we obtain T0 400 K even for a weak coupling λ 0.3. These phase transitions can occur at higher temperatures and often hinder the system from entering the superconducting state, especially for low dimensions: quasi-oneor two-dimensional metals (see, e.g., [392]). The electron attraction that is responsible for superconductivity in conventional metals is caused by the retarded electron–phonon interaction. It is related to the phonon energy and comes into play only in a narrow energy shell of the order of ¯hω/kB ≤ 400 K near the Fermi surface. The coupling constant itself depends on the phonon spectrum (see, e.g., [48]): 1 λ = N (0) g 2 − μ∗ . (1.2) M ω2 The ﬁrst term is determined by the square of the matrix element g for the electron–ion interaction averaged over the Fermi surface and by the averaged static lattice susceptibility 1/M ω 2 , that is, the inverse lattice rigidity 1/Φ, which does not depend on the mass of lattice ions M . The second term μ∗ describes the renormalized Coulomb repulsion of electrons [150, 857]: μ∗ =

VC , 1 + VC ln(EF /¯ hω)

(1.3)

where VC is the bare Coulomb repulsion, which may be comparable or even larger than the electron–phonon interaction – the ﬁrst term in (1.2). In conventional superconductors, the parameter μ∗ is usually small, μ∗ 0.1 − 0.2, ¯ ω. due to the large Fermi energy EF in comparison with the phonon energy h

1.1 Problem of High-Temperature Superconductivity

3

A direct conﬁrmation of the electron–phonon mechanism of pairing is the isotope eﬀect, that is, the dependence of Tc on the mass M of the lattice ions, Tc ∝ M −α . The exponent α in this dependence is given by the expression α = −d log Tc /d log M .

(1.4)

According to (1.1) and (1.2), for conventional superconductors α 1/2 since the phonon frequency in (1.1) ω ∝ M −1/2 and the coupling constant λ (1.2) does not depend on the mass (if we neglect μ∗ ). Only for superconductors with low Tc , the value of α can be considerably lower due to large μ∗ , which depends on the cut-oﬀ frequency ω in (1.3). Before the discovery of cuprate superconductors by Bednorz and M¨ uller, much eﬀort in boosting Tc was devoted to synthesizing materials with a high density of states, which can increase signiﬁcantly the coupling constant (1.2). There is, however, a limit to the increase of the coupling constant λ, since a large value in metals leads to a strong renormalization of the phonon spectrum, that is, to its softening and to structural instability (see, e.g., [392]). The greatest success in this direction has been attained for intermetallic compounds of transition metals with the A15 structure of the A3 B type, where A = Nb, V and B = Sn, Si, Ge, etc. (see, e.g., [1323]). Despite immense research eﬀorts, the maximum temperature Tc 23.2 K obtained for the compound Nb3 Ge in 1973 was not pushed up till the discovery of Bednorz and M¨ uller [118]. Soon after Bednorz and M¨ uller’s discovery, another idea of a strong enhancement of Tc proved to be fruitful. In the early 1960s, a high Tc was predicted for superconductors with light atomic mass such as metallic hydrogen. A high phonon frequency ω in the relation (1.1), of the order of several thousand kelvin, even for a weak coupling can provide high Tc . In the absence of metallic hydrogen, attention was focused on compounds with light elements like carbides and nitrides. On this road, quite high values of Tc 40 K have been attained in recent years in fulleride compounds Rx C60 [420, 447] and recently in magnesium diboride MgB2 by Akimitsu and co-workers [878]. In fullerides, the high density of electronic states in narrow bands and the strong coupling to high-frequency C60 molecular vibration modes result in strong enhancement of Tc . In the magnesium diboride, the record Tc for conventional electron–phonon superconductors was reached due to strong electron–phonon coupling g 2 at a modest density of electronic states and high-frequency phonons of light boron atoms. The quasi-two-dimensional character of electronic spectra and the two-band nature of superconductivity in MgB2 further enhance Tc (for a review see [197]). Possibly, in these compounds it has been achieved the highest Tc 40 K mediated by the electron–phonon coupling mechanism, as it has been generally believed for long time. An electronic mechanism of pairing like AF exchange in cuprates (see Sect. 7.3.2) could produce a really high superconducting temperature if instead of a phonon frequency ω ≤ 0.1 eV in the prefactor in the BCS formula (1.1) there would be an electronic energy, such as the Fermi energy, EF 0.5 eV, that results in Tc ≥ 200 K even for weak coupling, λ 0.3. Therefore, it

4

1 Introduction

is tempting to speculate that only an electronic pairing mechanism can compete with other instabilities (SDWs or CDWs) and promote high-temperature superconductivity. Concerning the general problem of high-temperature superconductivity, we should also mention another scenario of attaining high Tc based on the Bose– Einstein condensation (BEC) of local electron pairs proposed by Schafroth [1108, 1109]. In this approach, contrary to the BCS pairing theory, the formation of singlet electron pairs – bipolarons – is assumed at some higher temperature, while the superconducting transition occurs at the temperature of BEC, Tc = TBEC . Since the correlation length in cuprates is much smaller in comparison with conventional superconductors, this scenario is considered to be relevant to high-temperature superconductivity (for reviews, see Micnas et al. [827], Alexandrov and Mott [29], Alexandrov [43,44]). A simple estimate for the BEC temperature of the three-dimensional ideal Bose gas, TBEC

3.3 ¯h2 n2/3 m 2.9 × 10−11 ∗ n2/3 K, m∗ kB m

(1.5)

for the eﬀective mass of bipolaron m∗ 10m, where m is the electron mass, and the density of bipolarons n 1021 cm−3 gives quite a high Tc 300 K. While in the underdoped region a pseudogap is observed in cuprate materials at T ∗ Tc , which can be considered as caused by the preformed bipolaron pairs, in the optimal doped or overdoped regions no preformed pairs were detected. Bipolarons, the Bose quasiparticles, have no Fermi surface, which has been unambiguously detected in cuprates (see Sect. 5.2.2). Therefore, a crossover from the BEC at low doping to the Cooper pairing of fermions in the overdoped region should occur if we adopt the bipolaron scenario of the high-temperature superconductivity in cuprates (see Sect. 7.4.3).

1.2 Discovery of High-Temperature Superconductors One might try to achieve Tc in compounds with large value of the high electron–ion interaction g 2 . However, in conventional metals with high electron density, the matrix element of the electron–ion interaction g is considerably weakened due to strong screening and cannot attain high values. Taking into account this circumstance and the absence of progress in studying compounds of transition metals, in 1983 Bednorz and M¨ uller addressed their attention to another class of compounds, namely, the oxide superconductors. The high polarizability of oxygen ions and the poor screening of the electron–ion Coulomb interaction due to a low density of carriers could result in a strong electron–phonon coupling in these compounds. By that time, conducting oxides with relatively high transition temperatures Tc 13 K at very low densities of electron states were already known. The most interesting was a perovskite Ba(PbBi)O3 discovered by Sleight et al. [1176]. At a suﬃciently

1.2 Discovery of High-Temperature Superconductors

5

low concentration of carriers (n = 4 × 1021 cm−3 , i.e., two orders of magnitude smaller than in transition metals) and therefore a small value of N (0), the high value of Tc could be accounted for in the frame of the electron–phonon model (1.2) only, if one assumes a large value of the electron–ion interaction. However, attempts to raise Tc in this compound by increasing the density of states N (0) by varying the ratio Pb to Bi failed. With increasing density of states, the compound underwent a metal–insulator transition with the formation of a CDW (see, e.g., [1281]). The search for new oxide superconductors undertaken by Bednorz and M¨ uller was based on the idea of creating conducting oxides containing so-called Jahn–Teller ions. Such ions, for example, Ni3+ or Cu2+ , are characterized by a strong interaction of electrons with local distortions of a crystal lattice. The distortion considerably decreases the electronic energy of the ion because of a lifting of the degeneracy of electron levels. (The Jahn–Teller eﬀect for Cu2+ ions is considered in Sect. 5.1.1.) The strong interaction of electrons with displacements of surrounding ions can result in the formation of polarons whose BCS type pairing or BEC of bipolarons can also lead to high-temperature superconductivity as discussed above (1.5). The study of nickel oxide compounds, however, did not yield encouraging results. In 1985, Bednorz and M¨ uller turned to compounds of copper oxides. Among them, lanthanum and barium copper oxides with metallic conductivity were known (see [1139]). On varying the ratio La3+ to Ba2+ in these compounds, it was easy to control the valence of copper and the concentration of carriers. In January 1986 when performing measurements of conductivity in compounds with various concentrations of barium, Bednorz and M¨ uller discovered a dramatic fall of the resistivity in some samples at temperatures below 35 K. The results of the measurements were published in the September issue of Zeitschrift f¨ ur Physik [118] The ﬁnal conﬁrmation of the superconducting nature of the phase transition in these samples was obtained after a veriﬁcation of the Meissner eﬀect [119]. The publication of this discovery attracted the attention of many scientists who, in a short period of time, conﬁrmed the occurrence of superconductivity in the ceramics La–M–Cu–O, where M = Ba, Sr, Ca [120]. Later on, it became clear that the oxide superconductors of this type have a layered perovskite structure La2−x Mx CuO4 (LMCO) (see Sect. 2.2.1). Still higher superconducting transition temperatures were reached in January 1987 by the group of C.W. Chu at the University of Houston in collaboration with the group of M.-K. Wu at the University of Alabama. Having replaced La by Y, they obtained Tc = 90 K in a multiphase ceramic sample [1371]. The superconducting phase in this compound has the layered perovskite structure YBa2 Cu3 O7−y (YBCO) with a deﬁcit in oxygen (see Sect. 2.4.1). Thus, within one year, the temperature of the superconducting transition increased several times as compared to the value Tc = 23 K, the record known in 1973. It is very important that Tc in the new copper-oxide superconductors exceeds the boiling

6

1 Introduction

point of liquid nitrogen which is a criterion deﬁning a true high-temperature superconductor. Further active search for new compounds with higher values of Tc led to the discovery of superconductivity in the systems Bi–Sr–Ca–Cu–O [767] and Tl–Ba–Ca–Cu–O [450, 1147] in which Tc reached 110–120 K. A new class of mercury compounds was discovered by Putilin et al. [1034]. A maximal value of Tc 135 K was found in the three-layer mercury compound HgBa2 Ca2 Cu3 O8+δ at ambient pressure [1035, 1112] and 164 K at 30 GPa [375]. The synthesis of the compound (K–Ba)BiO3 [212, 813] with Tc = 30 K considerably exceeding the Tc = 13 K in Ba(Pb–Bi)O3 was important for understanding the mechanisms of superconductivity in oxide compounds. The absence of copper ions with spin 1/2 and the large isotope eﬀect exclude, for these compounds, magnetic mechanism of superconductivity proposed for copper-oxide superconductors. The above oxide superconductors have hole-type conductivity. Therefore, the discovery of superconductivity in the electronically doped compounds Nd2−x Cex CuO4 [1211, 1248] with Tc = 20 K points to the existence of a presumably general mechanism of high-temperature superconductivity in copper-oxide compounds.

1.3 Generic Properties of Cuprate Superconductors At present, there are more than 150 superconducting compounds with Tc higher than the record of 23 K for conventional intermetallic superconductors [232]. However, only cuprates can be called the true high-temperature superconductors since only they have Tc above the liquid-nitrogen boiling point and many of them even have Tc > 100 K.1 The composition of these compounds and their transition temperatures Tc are given in Table 1.1. It is convenient to subdivide them into several classes: the La2−x Mx CuO4 (LMCO) type, the YBa2 Cu3 O6+x (YBCO) type, and Bi-, Tl-, and Hg-type compounds described by a general formula Am M2 Can−1 Cun Ox , where A = Bi, Tl, Hg and M = Ba, Sr. These are the hole-doped materials. The Nd-based compounds Nd2−x Cex CuO4 have an electronic conductivity. It should be mentioned that the value of Tc strongly dependents on the concentration of oxygen or other doping ions and on various types of disorder as impurities, cation nonstoichiometry, etc. (see Sect. 5.1.2). In Table 1.1, the maximum values of Tcmax are shown at the optimal doping and at ambient pressure. A comprehensive list of Tcmax was presented by Eisaki et al. [304]. All cuprate superconductors have layered structure with a stacking sequence of CuO2 planes and charge-reservoir blocks. In stable, one-phase compounds, the number n of CuO2 planes is usually less then four. Under high 1

A claimed hole superconductivity with Tc = 117 K in C60 /CHBr3 by Sch¨ on et al. [1160] was later retracted.

1.3 Generic Properties of Cuprate Superconductors

7

Table 1.1. Representative classes of cuprate superconductors Superconducting compounds

Tcmax (K)

LMCO-type compounds (LMCO) M = Ba, Sr, Ca La2−x Mx CuO4 La2 CuO4+y Ca2−x Nax CuO2 Cl2 R2−x Mx CuO4 (RMCO) (electronically doped cuprates) R = Pr, Nd, Sm, Eu, M = Ce, Th, Ce+Sr Sr14−x Cax Cu24 O41 (ladder compound – superconducting under high pressure at x = 13.6)

39 45 26 24 12

YBCO-type compounds RBa2 Cu3 O6+x (x > 0.4) (R-123) R = Y, La, Ca, RE; RE = Pr, Nd, Sm, Eu, Gd, Dy, Ho, Er, Tm, Yb, Lu (Y-124) YBa2 Cu4 O8 YBa2 Cu3.5 O8−y (Y-247) A = R +Sr, R + Ca Pb2 Sr2 ACu3 O8+y , La2−x Srx CaCu2 O8 RuSr2 GdCu2 O8−δ (Ru-1212) (superconducting ferromagnet)

93 80 87 80 60 46

Bi-, Tl-, Hg-type compounds Bi-22(n − 1)na (n = 1–3) Bi2 Sr2 Can−1 Cun O2n+4+δ Bi2 Sr2 CuO6+δ Bi2 Sr2 Ca2 Cu3 O10+δ Tlm Ba2 Can−1 Cun O2n+m+2+δ Tl-m2(n − 1)na (m = 1, 2; n = 1–4) Tl2 Ba2 CuO6+δ Tl1 Ba2 Ca2 Cu3 O9+δ Tl2 Ba2 Ca2 Cu3 O10+δ HgBa2 Can−1 Cun O2n+2+δ Hg-12(n − 1)na (n = 1–5) HgBa2 CuO4+δ HgBa2 Ca2 Cu3 O8+δ HgBa2 Ca2 Cu3 O8+δ (under pressure of 30 GPa) a

10 110 93 133 125 98 135 164

The sequential numbers represent the composition ratio of the cations

pressure, it is possible to synthesize compounds with larger n. With increasing number of CuO2 planes, a certain increase in Tc is observed, with maximum Tc usually attained for n = 3. The values range from Tc = 38 K in La2−x Srx CuO4 with a single CuO2 plane to Tc = 134 K in compounds of Hg-1223 with three CuO2 planes. The highest Tc = 164 K was obtained in Hg-1223 by applying external pressure P = 30 GPa [375]. It was suggested that the mercury-based compounds have the highest Tc (even the one-layer compound Hg-1201 has Tc = 97 K) because their copper–oxygen layers have the most ideal tetragonal structure without pronounced buckling as compared with other cuprate materials. They have also the largest apical Cu–O distance and therefore the apex oxygen produces only a small perturbation on the electronic structure of a CuO2 plane (see Sect. 2.5).

8

1 Introduction

By applying the high-pressure technique, new types of cuprate superconductors have been synthesized. The ladder type compound Sr0.4 Ca13.6 Cu24 O41 , which consists of two-leg Cu2 O3 ladders and edge-sharing CuO2 chains, undergoes a superconducting transition below Tc 12 K under high pressure P = 3–4 GPa [1287]. Under pressure, a crossover from one-dimensional to two-dimensional electronic structure occurs that enforces superconductivity in copper–oxygen planes (see Sect. 2.2.3). A novel superconducting state has been observed in rutheno-cuprates RuSr2 GdCu2 O8−δ [130] and Ru2 Sr2 (Gd0.7 Ce0.3 )2 Cu2 O10−δ [328], where superconductivity occurs below the transition to a ferromagnetic state at Tm > Tc . The Ru-sublattice is responsible for the unusual magnetic properties though the real magnetic structure is not precisely known (see Sect. 2.4.3). Despite the great diversity of the cuprate superconductor compounds, they have a common structural element – the CuO2 planes. It is generally believed that the high values of superconducting Tc and the anomalous normal state physical properties of cuprate materials are determined by the unique electronic and magnetic structure of the CuO2 plane. It allows us to single out these materials into a class of cuprate superconductors. Even the ﬁrst studies of cuprate superconductors have shown that they exhibit many properties in common with the conventional superconductors. In particular, measurements of the Shapiro steps in the Josephson eﬀect, observation of a ﬂux lattice in a magnetic ﬁeld, and a direct measurement of the ﬂux quantum φ0 = hc/2e have shown that Cooper pairs with charge 2e occur in the superconducting state. Tunneling experiments unambiguously indicate the formation of a superconducting gap in the spectrum of the charge carriers below Tc , which also conﬁrms the picture of Cooper pairs. The decrease in the Knight shift in the superconducting state and the temperature dependence of the penetration depth of a magnetic ﬁeld point to the singlet nature of pairing as in the usual BCS scenario. However, the cuprate compounds have revealed a number of anomalous physical properties by which they essentially diﬀer from the conventional metals [175]. Here, we discuss these properties only brieﬂy, since they will be considered in more detail in the subsequent chapters. As structural studies show, the cuprate superconductors with a general chemical formula Am M2 Rn−1 Cun O2n+m+2 have a layered structure: n (Cu– O2 )-layers interleaving with n − 1 R-layers deﬁne the active conducting block, while [(MO)(AO)m (MO)]-layers form the charge-reservoir block. As a result, a high anisotropy of the electronic and, in particular, the superconducting properties are speciﬁc for the quasi-two-dimensional copper-oxide superconductors. The physical properties and the superconducting Tc are strongly inﬂuenced by the concentration of charge carriers, which is regulated by variation of the charge-reservoir block composition. Generally, the superconducting transition temperature Tc for copper-oxide superconductors has a parabolic dependence on the concentration of charge carriers p with a maximum at an optimal doping popt . As suggested by Presland et al. [1028], a universal formula for Tc (p)

1.3 Generic Properties of Cuprate Superconductors

9

Fig. 1.1. Generic phase diagram of the cuprate superconductors

can be proposed:

Tc (p) = Tc,max [1 − β (p − popt )2 ],

(1.6)

where the parameters β and popt have the constant values, β = 82.6, popt = 0.16 for a large number of compounds [1221]. In the underdoped region, p < popt , the cuprate superconductors exhibit anomalous physical properties below a characteristic temperature T ∗ when the so-called pseudogap in the electronic spectrum is opened. Therefore, the underdoped region is called a “strange” metal. In the overdoped region, p > popt , “normal” metal properties are regained. The generic phase diagram in the temperature T and hole concentration p (per CuO2 layer) coordinates is shown in Fig. 1.1. The crystal structure and the phase diagram for cuprate compounds are discussed in Chap. 2 A unique feature of the parent copper-oxide materials is an AF long-range ordering of spins, which are almost localized at the copper sites, in the CuO2 planes in the insulating Mott–Hubbard phase. The exchange energy for the copper 1/2 spins is extremely large, J 1,500 K, which would result in the record N´eel temperature for AF ordering, TN 1,500 K, if the spin lattice had a three-dimensional structure. Due to the layered structure of copper oxide materials, it is much lower, TN 300–500 K, and is rapidly suppressed if holes are doped in the CuO2 plane (see Fig. 1.1). However, strong dynamical short-range AF ﬂuctuations survive in the metallic phase. A resonant peak around 40 meV in the dynamical spin susceptibility at the AF wave vector was observed below Tc in YBCO, Bi-2212 and Ta-2212 compounds even at optimal doping [170, 1168]. The AF correlations strongly aﬀect the properties of the cuprate compounds in the normal phase and they are believed to be the source of nonphononic mechanisms of superconductivity. Chapter 3 is devoted

10

1 Introduction

to the description of the AF phase transitions in LSCO and YBCO compounds and neutron inelastic scattering and nuclear magnetic resonance experiments, which conﬁrm the existence of strong dynamical AF ﬂuctuations. Studies of the thermodynamic properties of copper-oxide superconductors evidenced a number of peculiarities of the temperature dependence of the critical magnetic ﬁelds, the penetration depth, and the speciﬁc heat. Due to a low density of charge carriers, the cuprate superconductors have a large magnetic penetration depth λ and a small correlation length ξ. This results in a very large value of the Ginzburg–Landau parameter κ = λ/ξ 1, which means that the cuprates are strong type II superconductors with an extremely large upper critical magnetic ﬁeld Hc2 . The quasi-two-dimensional character of the electronic structure leads to a large anisotropy of the penetration depth and the critical magnetic ﬁelds in-plane and perpendicular to the copper– oxygen plane. The small pinning energy and high transition temperature Tc with large anisotropy result in a very complicated magnetic phase diagram (Fig. 1.2). We notice the occurrence of a phase transitions from the normal state to the vortex liquid at the upper critical magnetic ﬁeld Hc2 , then a transition to the vortex solid with pinned vortices, and at the lower ﬁrst critical magnetic ﬁeld Hc1 the transition occurs to the Meissner phase (see Sect. 4.3.1). The large anisotropy of the electronic structure leads also to anisotropic correlation lengths for the order parameter. The estimation of the correlation length shows that its in-plane value equals several lattice constants, while in the direction perpendicular to the plane it is approximately equal to or even smaller than the lattice constant in this direction. Such small values of the correlation length show that the number of electrons (holes) ns in the

Fig. 1.2. Magnetic phase diagram of the cuprate superconductors

1.3 Generic Properties of Cuprate Superconductors

11

Cooper pair is several orders of magnitude smaller than those in conventional superconductors, in which ns = 104 –106 . A small number of electrons in Cooper pair results in considerable ﬂuctuations eﬀects. The anomalous thermodynamic properties of the cuprate superconductors, in particular, speciﬁc heats, penetration depths and critical magnetic ﬁelds, are discussed, within the anisotropic Ginzburg–Landau theory, in Chap. 4. The large anisotropy of the electronic properties of the copper-oxide superconductors was predicted by the very ﬁrst theoretical calculations of their electronic band structure (Sect. 7.1). It was shown that the main contribution to the states near the Fermi surface is made by the strongly bonded 3d electron states of copper Cu2+ and 2p states of oxygen O2− in the CuO2 planes. Rather accurate experimental investigations have conﬁrmed this picture at a qualitative level only. They have found a considerable contribution of electron single-site correlations at copper ions, which were not taken into account in the ﬁrst theoretical calculations. In particular, metal–insulator transition with a charge transfer gap and the formation of the AF state in the undoped compounds, the considerable localization of the spin density at copper sites, the appearance of excitations related to the p–d charge transfer, and a number of other phenomena observed in electron spectroscopy can be accounted for only by taking into consideration strong electronic correlations. Experimental studies of electronic properties in the normal and superconducting phases are discussed in Chap. 5. The unconventional pairing of the d-wave symmetry is speciﬁc for the cuprate superconductors. In the conventional superconductors, the electron–phonon pairing mechanism is veriﬁed by large values of the isotope eﬀect (1.4). In the copper-oxide compounds, the isotope eﬀect is suppressed, α = 0.2–0.05, although in some cases it reveals an anomalous growth for small values of Tc . In this respect, studies of phonon spectra of oxide superconductors and the observation of manifestations of electron–phonon interaction are important for clarifying the mechanisms of high-temperature superconductivity. The results of these studies are considered in Chap. 6. To explain such unusual properties of the cuprate high-temperature superconductors, various types of theoretical models, ranging from the standard models of the Fermi-liquid with strong electron–phonon coupling to rather exotic models of quantum spin liquids with unusual ground states, have been proposed. Chapter 7 is devoted to the discussion of basic models and mechanisms of high-temperature superconductivity. After the discovery of high-temperature superconductivity in cuprates, many laboratories all around the world hoped to develop useful applications of these materials. However, mechanical (brittleness) and electric (high anisotropy, d-wave gap symmetry) properties of these materials, which are not conventional metals, considerably hindered the technological progress in the development of their applications. We consider several successful applications of the high-temperature superconductors in Chap. 8.

12

1 Introduction

Now, there are more then 100,000 publications devoted to the problem of the high-temperature superconductivity in cuprates. The main results of studies of high-temperature superconductors, materials, and mechanisms of superconductivity are presented in the proceedings of various International Conferences, in particular, “Proceedings of the International Conference on High-Temperature Superconductors and Materials and Mechanisms of Superconductivity” [754–761]. Early results of studies of the high-temperature superconductors are reported in the monographs edited by Ginsberg [384– 388] and also can be found in the books edited by Fukuyama et al. [367], by Bednorz and M¨ uller [121], and by Maekawa and Sato [768]. There are many excellent and detailed reviews devoted to the theoretical and experimental studies of the high-temperature cuprate superconductors published in last years in review journals and as separate monographs, which will be cited in the subsequent chapters.

2 Crystal Structure

It is well known that, in solids, the atomic structure determines the character of chemical bonding and a number of other related physical properties. Even small changes in structure can considerably change electronic properties of a solid, for example, as in the Peierls metal–insulator phase transitions. Therefore, the investigation of the crystal structure and its dependence on temperature, pressure, and composition plays an important role in studying high-temperature superconductors. This is important both for understanding the mechanisms of high-temperature superconductivity and in predicting possible ways to synthesize new superconducting compounds. Many topologically diﬀerent types of crystal structure of layered copperoxide superconductors have been studied (for a review, see [444]). These various structures can be divided into several families depending on the type of packing of a small number of structure elements, that is, perovskite-like copper–oxygen CuO2 layers and buﬀer rock salt or ﬂuorite blocks [1250,1251]. It should be pointed out that besides the type of crystal structure, that is, the long-range order, the properties of cuprate superconductors also depend strongly on the short-range atomic order, which determines the local charge distribution in the crystal [214, 564]. The structure of quaternary copper-oxide compounds with the general formula (Ln1−x Mx )n+1 Cun O3n+1−m (where Ln is a trivalent rare-earth ion RE or Y, and M is a divalent alkaline ion, Ba, Sr, or Ca) can be characterized by the packing of CuO6 octahedrons and an ordered system of oxygen vacancies. The number of copper–oxygen planes is deﬁned by the quantity n = 1, 2, . . .. For n = 1, we have the layered perovskite structure of K2 NiF4 , while for n → ∞ we get a cubic perovskite ABO3 (see Fig. 2.1). The number of oxygen vacancies is characterized by the quantity m describing the multiplicity of copper coordination. Since copper readily allows the four-fold coordination CuO4 in a plane and the ﬁve-fold coordination CuO5 in a pyramid besides the six-fold coordination CuO6 in an octahedron, a large number of perovskitelike structures with oxygen deﬁciency appear. In these compounds, the copper

14

2 Crystal Structure

Bi

Ba or K

Fig. 2.1. The structure of (K–Ba)BiO3 in the cubic phase (reprinted with permisc 1990) sion by APS from Pei et al. [970],

is usually in the state Cuv+ with a formal valence 2 ≤ v ≤ 2.4, so that the number m turns out to be related to the concentration x of divalent ions M. Among various copper-oxide structures, only few of them that have regular CuO2 planes with a particular degree of oxidation become superconducting [1254]. The most studied compounds are La1−x Mx CuO4 and YBa2 Cu3 O7−y and its modiﬁcation (see, Table 1.1). The structure of the quinary compounds in Table 1.1, with a general formula Am M2 Can−1 Cun Ox , where A is Bi, Tl, or Hg and M is Sr or Ba, can be also represented as a number n copper– oxygen planes: CuO2 –(CuO2 –Ca)(n−1) , coupled by the buﬀer blocks (MO)– (AO)m –(MO), which deﬁne the oxidation level of the planes. We consider these structures below in detail. Here, we should also mention the excess-oxygen-doped La2 CuO4+δ compound (see, e.g., [1347]). While in LMCO compounds the M2+ doping ions are essentially immobile, the excess oxygen dopants remain mobile down to 200 K. Therefore, the disorder produced by the randomly substituted M2+ ions is “quenched,” while the disorder caused by the mobile intercalated oxygen is “annealed.” In the latter case, we should observe a much weaker inﬂuence of the disorder on the electronic and other properties of cuprates. However, the phase diagram of this oxygen-doped LCO compound appears to be very complicated due to phase separation into oxygen-rich and oxygen-poor regions with the modulation along the c-axis in the form of stages: The stage-n compound has a periodicity of n CuO2 layers. The highest superconducting Tc 45 K was observed for the n = 2 stage compound. By applying a high-pressure synthesis technique, a number of Sr–Cu–O phases was produced. Among them are the so-called inﬁnite-layer structure (Sr1−x Cax )1−y CuO2 (Tc = 80–110 K) [91, 460] and Sr1−x Ndx CuO2 (Tc = 40 K) [1180], which contain the CuO2 planes with the Sr (Ca) or Nd layers between them. Hiroi et al. [459] reported also that the Sr2 CuO3+δ and Sr3 Cu2 O5+δ compounds show superconductivity below Tc = 70, 100 K, respectively. However, these results were questioned in later publications. According to Shaked et al. [1137], the inﬁnite-layer phase is not superconducting and the superconductivity observed in previous publications is due to other phases,

2.1 The Structure of Ba1−x Kx BiO3

15

Srn+1 Cun O2n+1+δ included as impurity. In a detailed study of the series of compounds Sr2 Can−1 Cun Oy , Kawashima et al. [592] suggested that the n = 1 member of the series, Sr2 CuO3+δ is not superconducting. But they observed superconductivity for the phases with n = 2, 3, 4 in the series at Tc = 70, 109, 83 K, respectively. Another interesting family of cuprates without apical oxygen is cupric oxychlorides Sr2 CuO2 Cl2 (SCOC) and Ca2 CuO2 Cl2 (CCOC). They have similar crystal structure to LSCO (also the tetragonal I4/mmm space group) with a replacement of the La2 O2 buﬀer layer by the Sr(Ca)2 Cl2 layer. The SCOC crystal structure and AF ordering of copper spins are shown in Fig. 3.1b (in the orthorhombic (Bmba) notation). By applying a high-pressure synthetic route, Hiroi et al. [461] have achieved superconductivity in CCOC material doped with sodium, Ca2−x Nax CuO2 Cl2 (Na-CCOC). The compound becomes superconducting at x > 0.08 with a maximum Tc = 28 K at optimal doping, x ∼ 0.20. New ladder compounds with the general chemical formula Srn−1 Cun+1 O2n were synthesized by Hiroi et al. [458] by creation lines of oxygen vacancies ordered in the ideal ﬁlled copper–oxygen plane. Superconductivity in the ladder compounds was observed in a more complicated system with the chemical formula Sr14−x Cax Cu24 O41 . By applying external pressure, it was possible to increase the metallic conductivity [881] and to discover a superconducting transition at Tc 12 K at a pressure close to 3 GPa [1287]. We discuss the ladder compounds in Sect. 2.2.3. Observation of the coexistence of ferromagnetic and superconducting properties in the ruthenate–cuprate compound RuSr2 GdCu2 O8−δ (Ru-1212) by Bernhard et al. [130] attracted the interest of many researches. The structure of the Ru-1212 compound can be viewed as a modiﬁcation of the Y-123 structure in which the CuO chains are replaced by the RuO2−δ sheets. We consider the ruthenate–cuprate compounds in Sect. 2.4.3. By applying external pressure, one can change the lattice parameters of a crystal, which provides useful information for our understanding of the superconducting phase transition. The inﬂuence of high pressure on crystal structure and superconducting Tc is discussed in Sect. 2.6. We begin with the consideration of the crystal structure of the simplest three-dimensional oxide compound BaBiO3 , which has no copper ions and only a modest Tc ≤ 40 K but its classical perovskite CaTiO3 structure shares many features with the superconducting cuprates.

2.1 The Structure of Ba1−xKxBiO3 The crystal structures of BaBiO3 -based compounds under replacing Ba by K or Bi by Pb can be described as small distortions of the original perovskite cubic phase P m¯3m shown in Fig. 2.1 [970]. In this ﬁgure, the thermal factors are shown as ellipsoids whose dimensions characterize the thermal ﬂuctuations

16

2 Crystal Structure

of the ions in the lattice. For the oxygen ions, the strongly anisotropic thermal factors with large vibration amplitude in the plane of the cube face implies that the lattice is predisposed to structural phase transitions related to the rotations of the BiO6 octahedrons. In fact, the freezing out of the rotations of the octahedrons (in antiphase to neighboring cells) around the cube axes [001] or [110] gives rise to the tetragonal I4/mcm or orthorhombic Ibmm phases, √ respectively. The lattice constant increases by a factor of 2 in the basal x− y plane, while it gets twice larger along the z-axis. An additional condensation of the breathing phonon mode (variation in the lengths of the Bi–O bonds) in the orthorhombic phase Ibmm decreases its symmetry to monoclinic I2/m [970]. In the latter case, the periodic variation in the lengths of the Bi–O bonds gives rise to a similar periodic variation in the charge on the Bi ions, which can be described as a charge density wave (CDW). This monoclinic phase I2/m is just observable in the pure compound BaBiO3 , which turns out to be an insulator with an optical gap of ∼2 eV due to the formation of a CDW. A replacement of Ba by K suppresses the CDW. At a potassium concentration x > 0.1, nonequivalent positions of Bi are no longer observed. The long-range CDW disappears and the symmetry increases to Ibmm. However, metallic conductivity does not occur up to the transition into the cubic phase (at low temperatures for x > 0.37). Then, in the cubic phase, superconductivity with a transition temperature Tc = 30 K is observed. As the concentration increases up to x = 0.5, which is the solubility limit for potassium ions in a solid solution, the value of the transition temperature decreases. Under doping, the lattice constant of the pseudo-cubic lattice apc smoothly decreases. Its concentration dependence at room temperature is given by the formula apc = 4.3548 − 0.1743x (˚ A). Since the ion radius of K+ and Bi2+ have similar values, 1.64 and 1.61 ˚ A, respectively, the decrease in the volume of the primitive cubic cell can only be related to the decrease in the radius of the bismuth ion as the degree of its oxidation increases. The ionic radii of Bi3+ and Bi5+ are 1.03 and 0.76 ˚ A, respectively. A similar sequence of structural phase transitions is observed in another perovskite compound BaBi1−x Pbx O3 when Bi is replaced by Pb. At x > 0.05, the symmetry of the monoclinic phase I2/m increases to orthorhombic, although the transition to the metallic state takes place only at x > 0.65 when the symmetry increases to tetragonal I4/mcm. In the metallic phase, superconductivity is observed with a maximum value Tc = 13 K at x = 0.75. Under further increase of the lead concentration, the transition temperature Tc decreases and at x = 1 a nonsuperconducting orthorhombic phase Ibmm is found with typical metallic properties. Recently, a new family of bismuth-oxide-based superconductors was discovered, based on the SrBiO3 compound. A partial substitution of potassium or rubidium for strontium induces superconductivity with Tc 12 K for Sr1−x Kx BiO3 (x = 0.45–0.6) and Tc 13 K for Sr1−x Rbx BiO3 (x = 0.5) [593]. The crystal structure of the strontium-based compounds also changes

2.2 The Structure of La2−x Mx CuO4−y

17

with doping from the less symmetric, monoclinic P 21 /n for SrBiO3 , to the tetragonal I4/ncm in the superconducting phase, as in BaBi1−x Pbx O3 . Thus, despite the relative simplicity of the structure of the cubic perovskites, the compounds Ba1−x Kx BiO3 , Sr1−x Kx BiO3 , and BaBi1−x Pbx O3 reveal several phases with various crystal symmetry depending on the concentration x of the doping ions and on the temperature. While the semiconducting properties of the original compound at x = 0 are readily accounted for by the formation of the CDW in the monoclinic phase, their persistence in orthorhombic phases, where no nonequivalent positions of Bi are observed, cannot be explained so simply. A possible reason why the semiconducting gap in the electronic spectrum should remain in these phases may be the existence of local (or incommensurate) CDWs with a small coherence length that obscure their observation in diﬀraction experiments. A speciﬁc feature of BaBiO3 compounds is the occurrence of superconductivity near to the metal–insulator transition with decrease of the superconducting transition temperature Tc , as the number of charge carrier in the region of normal metallic properties increases. Such a nonmonotonic dependence of Tc on the concentration of carriers and the suppression of superconductivity in the normal metallic phase are the features speciﬁc to the copper-oxide superconductors also. It should be also pointed out that the highest Tc in bismuth-based compounds occurs in the phase with the highest three-dimensional crystal symmetry, cubic in the Ba1−x Kx BiO3 , as the highest Tc in cuprates is observed in compounds with the most regular copper–oxygen planes as discussed below.

2.2 The Structure of La2−xMxCuO4−y Let us consider the ﬁrst copper-oxide superconductor La2−x Mx CuO4−y (LMCO or 124), where M = Sr, Ba, or Ca. In the high-temperature tetragonal (HTT) phase, these compounds have a structure of the K2 NiF4 type, 17 that is, a body-centered tetragonal lattice (I4/mmm − D4h ). The tetragonal unit cell of this lattice, which has two formula units, is shown in Fig. 2.2a [561]. The primitive unit cell that has one formula unit and is determined by the vectors a1 , a2 , and a3 is shown in Fig. 2.2b. The corresponding Brillouin zone in the tetragonal body-centered phase and in the folded Brillouin zone of the face-centered-orthorhombic phase are shown in Fig. 2.3. The structural parameters of LMCO crystals for diﬀerent compositions and their temperature dependence are discussed by Hazen [444]. Typical values of the lattice constants in the tetragonal phase are at = 3.78 ˚ A, ct = 13.2 ˚ A. The distance ˚ Cu–O1 in the plane is given by at /2 = 1.89 A and the distance Cu–O2 to the apical oxygen O2 is 2.42 ˚ A. The large anisotropic thermal factors for the oxygen ions are noteworthy. They indicate a large vibration amplitude of these ions in the tilting type mode for the CuO6 octahedron. As the temperature decreases, a structural phase transition from the tetragonal to the

18

2 Crystal Structure

a

b

a1

a2 La/M O2 O1

Cu a3

Fig. 2.2. (a) The structure of La2−x Mx CuO4 in the tetragonal phase (after [561]). (b) The tetragonal body-centered I4/mmm unit cell of La2 CuO4 with the primitive unit cell deﬁned by √ the vectors a1 , a2 , and a3 and the orthorhombic unit cell Cmca with ao co at 2, bo ct (see Fig. 3.1) KZ Z

L X

X

Ky

Kx

Fig. 2.3. The Brillouin zone in the tetragonal body-centered phase (solid lines) and the folded Brillouin zone in the face-centered-orthorhombic phase (dashed lines). The X point is folded back to Γ , while the X point is folded back to Z (after [981]) 18 low-temperature orthorhombic (LTO) phase (Cmac − D2h ) takes place, with the doubling of the unit cell volume as shown in Fig. 2.2b. The directions of the orthorhombic axes (ao , bo , co )Cmac are chosen along the directions [¯110], [001], and [110] of the HTT phase I4/mmm, respectively, and are √ related to the initial tetragonal phase axes as ao co at 2 = 5.35 ˚ A and bo ct . Another space group, Bmab, with the lattice parameters

2.2 The Structure of La2−x Mx CuO4−y

19

(ao , bo , co )Bmab = (ao , −co , bo )Cmac is often used for the characterization of the orthorhombic phase. In this case, the orthorhombic axes are related to √ those of the tetragonal phase as ao bo at 2 and co ct (see Fig. 3.1). 2.2.1 Structural Phase Transitions in La2−x Mx CuO4 The study of the phonon spectrum (see [137, 590]) shows that the structural phase transition HTT → LTO is due to the condensation of a soft tilting mode at the X-point of the Brillouin zone boundary. In the LTO phase, two domains that are related to the rotation of the CuO6 octahedron around the tetragonal axes [110] and [¯ 110], respectively, appear (see Figs. 2.7 and 3.1a). Under the rotation of an octahedron in the orthorhombic phase, up to 5◦ at low temperatures [444], the change in the Cu1–O1 bond lengths in the plane is negligible (less than 0.01%). An accompanying orthorhombic deformation increases the lattice constant along the direction perpendicular to the rotation axis, that is, co > ao under the rotation around the ao -axis in Fig. 3.1a. In the LTO phase, the compressibility of the La2 CuO4 lattice turns out to be anisotropic. At room temperature, for the directions ao , bo , and co (Cmac phase, co > ao ) it equals 1.8, 1.5, and 4.1(×10−4 kbar−1 ), respectively [496]. The temperature of the HTT → LTO structural phase transition T0 in La1−x Mx CuO4−y rapidly decreases with increasing concentration x of the doping divalent M ions. A typical dependence T0 (x) for the case M = Sr is shown in Fig. 2.4 [137], where the phases with the long-range antiferromagnetic order (the N´eel state), the superconducting state, the boundary between the insulating and metallic phases, and the region of frozen spin states (the spin glass) at low temperatures are also shown. An additional study of the dependence of T0 on the concentration of the oxygen vacancies y shows that T0 decreases as the value x − 2y increases, that is, as the total number of charge carriers in a CuO2 plane grows up. In fact, the replacement of La3+ by M2+ increases the number of holes by one, while the formation of a vacancy of O2− increases the number of electrons by two [604, 858]. Under applied pressure, T0 (x) goes down and the orthorhombic phase disappears at pressures above the critical pc (x) (e.g., pc = 15 kbar for x = 0.12 in La2−x Srx CuO4 [604,858]). More careful studies of the structure of La2−x Bax CuO4 have revealed another phase transition in the region of low temperatures and concentration x ∼ 0.12 from the orthorhombic phase to the low-temperature tetragonal 16 (LTT) phase, P 42 /ncm (D4h ) [90,677,678,1093,1204,1205]. The LTO → LTT phase transition is driven by condensation of the soft optical phonon mode at the Z-point of the Brillouin zone. The temperature of the superconducting transition in the LTT has a minimum around the concentration x 0.12. The sharp decrease of Tc in the LTT phase is correlated with anomalous changes in other electronic properties such as the conductivity, the Hall eﬀect, the thermoelectric power [677,678,1093,1204,1205]. A freezing of the copper magnetic moments in the LTT phase was also observed by the μSR-method.

20

2 Crystal Structure 600 Tetragonal

Temperature (K)

500

400 Orthorhombic

300 Neel state 200 Insulator 100 Spinglass

Metal

Superconductor

0 0

0.1

0.2 0.3 Sr concentration, x

0.4

Fig. 2.4. The temperature–concentration phase diagram of La2−x Srx CuO4 (after [137])

A much weaker decreasing of Tc was observed also in the La2−x Srx CuO4 compound at x ∼ 1/8 = 0.125, which may be explained by incipient LTO–LTT structural transformation. The latter reveals as the softening of the Z-point phonon in the LTO phase [137]. In the subsequent inelastic neutron scattering study of the soft Z-point phonon in a single crystal of La2−x Srx CuO4 , it was shown that at concentrations x = 0.15, 0.18 the softening of the Z-point phonon breaks at Tc , while for lower concentrations, x = 0.10, 0.12, the softening continues below Tc [607]. This observation revealed a certain competition between the LTO–LTT transition and the appearance of superconductivity. Kimura et al. [607] also observed an incommensurate splitting of the central peak along the [1, 1, 0] direction in the HTT phase at temperatures much higher than the HTT–LTO transition Ts1 = 240 (125) K at x = 0.12 (0.18). The central peak originates from atomic displacements in the HTT phase and it is usually considered as a precursor of short-range order of the LTO phase. Its incommensurate splitting 2 ∼ 0.24 r.l.u. (reciprocal lattice units) implies that incipient lattice modulation appears at very high temperature which may be coupled with the incommensurate spin modulation (see Sect. 3.2.4). In a subsequent series of elastic neutron scattering measurements by Fujita et al. [363] on 1/8-hole doped La1.875 Ba0.125−x Srx CuO4 (LBSCO) single crystals with x = 0.05, 0.06, 0.075, and 0.085, it has been shown that the CDW order in the LTT phase is responsible for the suppression of superconductivity.

2.2 The Structure of La2−x Mx CuO4−y

21

Fig. 2.5. (a) T –x phase diagram for La1.875 Ba0.125−x Srx CuO4 of the superconducting (Tc – closed circles) and the structural phase transitions: LTO–LTT (Td2 – open circles) and LTT–LTO2 (LTLO). Concentration dependence of the normalized integrated peak intensities for (b) the structural phase transition LTO–LTT, (c) CDW, and (d) SDW transitions. The data at x = 0.12 are referred to La1.88 Sr0.12 CuO4 (after [363])

Figure 2.5a shows the phase diagram of the superconducting (Tc – closed circles) and structural phase transitions (Td2 – open circles) in the LBSCO compound. The superlattice peak intensities describing the LTO–LTT, CDW and spin-density-wave (SDW) transitions are shown in Figs. 2.5b–d, respectively. In some interval of Sr-ion concentration x > 0.05, an intermediate phase, the second low-temperature orthorhombic phase LTO2 (denoted in Fig. 2.5a as LTLO – low-temperature less-orthorhombic), space group P ccn, is observed. It also originates from the oxygen octahedron rotation as we will describe below. While in the LTO phase the superconducting Tc does not depend on the Sr concentration, it abruptly drops in the LTO2 phase below x < xc ∼ 0.09 and remains low in the LTT phase. This anomalous behavior in the LTT and LTO2 phases can be explained by appearance of a CDW order in these phases as shown in Fig. 2.5c. The static CDW (and SDW) order appears at a temperature close to the structural phase transition into the LTT phase at Td2 . The neutron scattering study of a pure LBCO single crystal, La1.875 Ba0.125 CuO4 , by Fujita et al. [365] conﬁrmed the simultaneous appearance of static CDW and SDW but at temperature Tcs = 50 K, a little bit lower than the temperature Td2 = 60 K of the LTO–LTT structural phase transition. A substantial oxygen isotope eﬀect on the LTO–LTT transition was observed in LSBCO by Wang et al. [1339]. A substitution of 16 O by 18 O resulted in an increase of the structural phase transition temperature and

22

2 Crystal Structure

in suppression of the superconducting transition temperature Tc . This shows that the static CDW stripe phase pinned by the LTT phase competes with the superconducting transition (see Sect. 6.3). The incommensurate charge order in LBSCO compounds was investigated later by Kimura et al. [608] with much higher precision in the synchrotron X-ray diﬀraction study. Two single LBSCO crystals were investigated with x = 0.05 in the LTT phase and with x = 0.075 in the LTO2 phase. The superlattice peaks show an incommensurate CDW for both of the crystals with the propagating wave vectors Qch = ±0.24 ∓ η (r.l.u.) with η = 0 (0.007) for x = 0.05 (0.075) and two-fold charge periodicity along the c-axis. Here, the reciprocal lattice units (r.l.u.) are deﬁned in the HTT (I4/mmm) symmetry. In the second orthorhombic LTO2 phase, the propagating wave vector is shifted, which indicates that the pattern of the charge order is pinned by the crystal structure. The high resolution synchrotron X-ray diﬀraction enables one to measure the correlation lengths for the CDW. The CDW shows a two-dimensional character with the correlation length in the plane A and ξb = 110, (70) ˚ A for the x = 0.05 (0.075), where a ξa 130 (120) ˚ and b are the directions along and perpendicular to the propagation vector Qch . The correlation length along the c-axis is much smaller, ξc ∼ 9 ˚ A. These studies unambiguously show that the long-standing “1/8 problem” of anomalous suppression of Tc in La-124 compounds at this hole concentration can be explained by appearance of the static CDW, which is pinned by the LTT or LTO2 crystal structure. This explanation is conﬁrmed by a more detailed study of structural phase transitions in La1−y−x REy Srx CuO4 compounds in which the La3+ ions are replaced by the rare-earth ions RE3+ = Nd, Sm, Eu, Gd of a smaller radius [183,184]. The phase diagram for RE = Nd compound at y = 0.4 [253,254] and RE = Eu at y = 0.2 [617] are shown in Fig. 2.6. For the RE-doped compounds, the LTT phase exists in a broad region of the Sr concentrations, which enables one to study this phase at diﬀerent concentration of doped holes. Figure 2.6a shows the Tc (x) dependence for Nd-compound in the LTO phase, Tc (LTO), (at y = 0) and in the LTT phase, Tc (LTT) (at y = 0.4). Similar to the LBCO compound, Tc (x) has a minimum at x = 0.12, which is deeper in the LTT phase. For a small Sr concentration, there appears the second low-temperature orthorhombic phase P ccn, LTO2, as in the LBSCO compound. A detailed neutron scattering study of CDW and SDW in the LTT phase of La1.6−x Nd0.4 Srx CuO4 compound was performed by Tranquada et al. [1259– 1261]. They discovered a cooperative spin and charge order in which doped holes spatially segregates into stripes that separate antiphase AF domains. The incommensurate spin ﬂuctuations were observed also in the excessoxygen-doped La2 CuO4+y single crystal [1347]. We discuss these results in Sect. 3.2.4. in connection with study of spin correlation in La-copper-oxide compounds. According to B¨ uchner et al. [182–184], the bulk superconductivity in the LTT phase exists only if the tilting angle Φ < Φc 3.6◦ . At ﬁxed concentration

2.2 The Structure of La2−x Mx CuO4−y

23

Fig. 2.6. T –x phase diagrams for La2−y−x REy Srx CuO4 . (a) For RE = Nd at y = 0.4 18 16 the phases LTO (D2h ), LTO2 (P ccn), and LTT (D4h ) are shown. The superconducting temperatures Tc (x) are given in the LTO phase, Tc (LTO), at y = 0 and in the LTT phase, Tc (LTT), at y = 0.4 (after [253]). (b) For RE = Eu at y = 0.4 the phases HTT, LTO, and LTT are shown. Depending on Sr concentration x, longrange (AF LR), short-range (AF SR), and static-stripe (AF SS) antiferromagnetic (AF) phases are found. Bulk superconductivity (SC) in the LTT phase is observed only for x > 0.18 (after [617])

of holes (Sr2+ ), the superconducting transition temperature Tc (y) smoothly decreases in the sequence of phase transitions LTO → LTO2 → LTT. In the intermediate LTO2 phase (P ccn), the oxygen ion displacements can be represented as a sum of the displacements in the two domains in the LTO phase but with diﬀerent amplitudes. Starting from one domain of the LTO phase and smoothly increasing the contribution for the oxygen displacements from the second domain, one can get a continuous phase transition from LTO to LTT phase through the P ccn phase, while a direct LTO → LTT phase transition is of the ﬁrst order. The continuous phase transition LTO → LTO2 → LTT can be described as a smooth change in the direction of the rotation axis for the CuO6 octahedron from [110] in the LTO phase to [100] in the LTT phase (in the tetragonal HTT phase notations). Since the tilting angle of rotation does not change notably at the LTO → LTT phase transition, Crawford et al. [254] have pointed out that just the change of the rotation direction should explain the suppression of Tc at ﬁxed concentration of holes (Sr2+ ) in the considered sequence of phase transitions. Strong dependence of Tc on the direction of the CuO6 octahedron rotation axis was conﬁrmed in studies of the phase diagram for Eu-doped LSCO material investigated by Klauss et al. [617] with μSR technique shown in Fig. 2.6b. It appears that the temperature Td2 130 K of the structural phase transition LTO → LTT at Eu concentration 0.2 only weakly varies with Sr concentration, while the magnetic order changes drastically from the AF long-range

24

2 Crystal Structure

order to a modulated phase (AF static stripes) depending on Sr concentration. Superconductivity in the LTT phase is strongly suppressed and it can be observed as a bulk property only for large Sr concentration, x > 0.18, when the tilting angle Φ becomes smaller then the critical value, Φc 3.6◦ [183, 184]. The electronic property changes in the LTT phase can be related to the appearance of nonequivalent positions of the O1 oxygen ions in the Cu–O plane. In the orthorhombic phase, under the rotation of CuO6 octahedron around the ao -axis (Fig. 3.1) (or co for the other domain) all the four O1 oxygen ions move out of the plane, as is shown in Fig. 2.7a,b. In this case, the variation of the crystal ﬁeld potential, which is proportional to the square of the displacement of the O1 ion is the same for all the four O1 ions. In the LTT phase, which can be represented as a sum of the displacements in the two domains of the LTO phase shown in Fig. 2.7a,b, the rotation of the octahedron occurs around the tetragonal axes, that is, the x-axis in one of the CuO2 planes and the y-axis in the neighboring planes. Only two of the four O1 ions move out of the plane as shown in Fig. 2.7c. This gives rise to a variation in their crystal ﬁeld potentials and to a local charge redistribution. In this case, the formation of CDW [105] and a gap in the electronic spectrum [983] occur, which should induce considerable changes in the electronic properties observed in the LTT phase. So, one may conclude that the static CDW formation in the LTT (or LTO2) phase close to the hole doping x = 1/8 is responsible for strong suppression of superconductivity and appearance of SDW. We consider in Sect. 3.2.4 the relevance of the incommensurate static and dynamic SDW

Fig. 2.7. The displacement of the O1 oxygen ions in the two domains of the orthorhombic phase (a, b) and in the low-temperature tetragonal phase (c)

2.2 The Structure of La2−x Mx CuO4−y

25

(or stripes) in La-124 compounds to their electronic properties and, in particular, to mechanism of high-temperature superconductivity in cuprates. It was suggested that dynamical stripes should exist in other copper-oxide compounds at small doping and play an essential role in cuprate superconductors (see, e.g., [615, 1079]). 2.2.2 Theory of Structural Phase Transitions Studies of the La2 CuO4 -based copper-oxide superconductors reveal considerable anomalies in the elastic characteristics of the crystals under structural phase transitions [828, 1372]. In these compounds, structural phase transitions may also inﬂuence their electronic and magnetic properties. In this respect, we shall consider a phenomenological theory of structural phase transitions based on a symmetry analysis and the Landau expansion for the free energy [24, 999, 1001]. The interplay of structural phase transitions and antiferromagnetic ordering in La2 CuO4 is discussed in Sect. 3.2.3. The sequence of structural phase transitions in La2 CuO4 from the HTT 17 18 phase (D4h ) to the LTO phase (D2h , LTO2 – P ccn) and to the LTT phase 16 (D4h ) can be described as a series of successive condensations of a twocomponent order parameter C1 , C2 . Namely, C1 = 0, C2 = 0 in the LTO phase, C1 > C2 = 0 in the LTO2 phase and C1 = C2 = 0 in the LTT phase. Figure 2.7 shows that the condensation of a soft mode related to the rotation of CuO6 octahedron C1 ∝ R1 (kx (1)) ,

C2 ∝ R2 (kx (2))

(2.1)

corresponds to a two-component order parameter. In (2.1), R1,2 = Rx ∓ Ry is the rotation of the octahedron around the [¯110] or [110] axis for the wave vectors kx (1, 2) = (π/a)(±1, 1, 0), respectively. The wave vectors kx (1, 2) form a two-arm star of the wave vector at the X-point of the Brillouin zone of the body-centered tetragonal lattice (Fig. 2.3). The primitive cell of the body-centered tetragonal phase, which contains one LMCO formula unit, is given by the translation vectors a1 = (−τ, τ, τz ),

a2 = (τ, −τ, τz ),

a3 = (τ, τ, −τz ),

where 2τ = at = a and 2τz = ct are the lattice parameters of the bodycentered unit cell shown in Fig. 2.2b. The reciprocal-lattice vectors are determined by the relations 1 1 1 1 1 1 b1 = π 0, , , 0, , ,0 . , b2 = π , b3 = π τ τz τ τz τ τ With this notation, kx (1) =

1 π b3 = (1, 1, 0), 2 a

kx (2) =

1 π (b1 − b2 ) = (−1, 1, 0). 2 a

(2.2)

26

2 Crystal Structure

The sum of the two vectors, kx (1) + kx (2) = b1 − kz , is equivalent to the vector kz = (π/τz )(0, 0, 1) within the reciprocal-lattice vector b1 . Thus, the two-component order parameter determines two domains. Each of them is related to the irreducible representation X3+ for the wave vectors kx (1) or kx (2), respectively. The expansion of the free energy in terms of the order parameter (OP) can be constructed as a function of the corresponding invariants, I1 = (C12 + C22 ) and I2 = C12 C22 . It is convenient to write the Landau expansion in the form Fc =

1 1 1 r(C12 + C22 ) + uC12 C22 + v(C14 + C24 ) + · · · , 2 2 4

(2.3)

where r = a(T − T0 ), u, and v are phenomenological constants. The thermodynamic potential (2.3) for the two-component order parameter describes a wide class of structural transitions (see, e.g., [180]). In the tetragonal phase, the strain contribution to the free energy is 1 C11 (21 + 22 ) + C12 1 2 + C13 (1 3 + 2 3 ) 2 1 1 1 (2.4) + C33 23 + C44 (24 + 25 ) + C66 26 , 2 2 2 where Voigt’s notations are used for the strain tensor μ and the elastic coeﬃcients Cμν of the crystal. The symmetrized square of the irreducible representation X3+ contains the invariants (C12 + C22 ) and (C12 − C22 ). This determines the interaction of the order parameter with the strains in the form F =

FC = [α(1 + 2 ) + β3 ](C12 + C22 ) + γ6 (C12 − C22 ),

(2.5)

where higher-order terms are omitted. We point out that the interaction of the type λ6 C1 C2 is forbidden since the product C1 C2 belongs to the irreducible representation with wave vector kx (1) + kx (2) = kz = 0. For second-order phase transitions, the equilibrium values of the order parameter and the strains μ are found from the minimum of the Landau free energy ∂F ∂F = 0, = 0, ∂Ci ∂μ where the full free energy is equal to the sum of the contributions (2.3)–(2.5). An analysis of these equilibrium conditions also enables one to determine the jumps in the elastic coeﬃcients under a structural phase transition. We shall brieﬂy summarize the results of calculations [999, 1001]. The phase transition HTT → LTO occurs when (v − 4γ 2 )/C66 < u. Under this condition two domains appear, C1 = 0, C2 = 0 or C2 = 0, C1 = 0. In the orthorhombic phase, the elastic coeﬃcients then undergo the jumps ΔC11 = ΔC12 = −2α2 /v, ΔC13 = −2αβ/v, ΔC33 = −2β 2 /v ΔC66 = −2γ 2 /v, ΔC44 = 0,

(2.6)

2.2 The Structure of La2−x Mx CuO4−y

27

while the equilibrium strains 1 = 2 , 3 , and 6 are proportional to the square of the order parameter, for example, 6 = −(γ/C66 )C 2 . The phase transition HTT → LTT occurs when (v − 4γ 2 )/C66 > u. Under this condition, C1 = C2 = C = 0. The strains 1 = 2 and 3 are proportional to the square of the order parameter C 2 , while the elastic coeﬃcients experience the jumps ΔC11 = ΔC12 = −4α2 /(v + u), ΔC13 = −4αβ/(v + u), ΔC33 = −4β 2 /(v + u), ΔC66 = 0, ΔC44 = 0.

(2.7)

By comparing (2.6) and (2.7) with experimental data of the measured velocity of sound in the LMCO crystals and their jumps under structural phase transitions (see, e.g., [828, 1372]), one can determine the coupling constants α, β, and γ in (2.5) (see [1003]). The phase transition LTO → LTT occurs as a ﬁrst-order transition since they are not coupled by space subgroup relationships. The transition temperature is determined by the equality of the free energies F in the LTO (at C1 = 0, C2 = 0) and LTT phases (at C1 = C2 = 0). In this case, higher powers of the invariants in the expansion of the free energy (2.3) start to play a role. For example, Ishibashi [528] proposed a model that contains the order parameter in sixth power in the form ΔF = w(I12 − 4I2 )I1 = w(C12 − C22 )2 (C12 + C22 ).

(2.8)

In the LTO phase, this contribution diﬀers from zero, while in the LTT phase at C1 = C2 , it vanishes. At suﬃciently low temperatures when equilibrium values of the order parameters become large, the LTT phase can thus be energetically more favorable than the LTO phase at w > 0, since the latter contains a large positive contribution (2.8). Ishibashi [528] has successfully described the temperature–concentration phase diagram for La2−x Bax CuO4 represented in Fig. 2.5 by assuming a certain dependence of the coeﬃcients u and v on the concentration x of the doping ions. In other models, this phase diagram was described by assuming a temperature dependence for the coeﬃcients u and v (see, e.g., [1204, 1205]). However, it is diﬃcult to justify such an assumption. To consider the phase transition to the LTO2 (P ccn) phase, one should also consider higher order terms in the free energy (2.3) (see, e.g., [224]). The phenomenological expansion of the free energy (2.3), (2.5) can be obtained by calculating the free energy on the basis of a microscopic theory. Such a model microscopic theory has been proposed by Flach et al. [340] and Plakida et al. [1003]. In the model, anharmonic vibrations of oxygen ions in a soft tilting mode and their interaction with acoustic phonons are described by the model Hamiltonian in terms of the local normal modes R1,2 = Rx ∓ Ry . The parameters of the anharmonic model were estimated from the calculation of the ground state energy for the La2 CuO4 crystal by the density-functional

28

2 Crystal Structure

0 –20 – 40 –4

–4

40

–2

–2 0

ΔE (mev / 7 atoms)

0

20

2

2 4

4

0 Pccn –20 LTO LTT

–40

–60 –6

–4

–2

0

2

4

5

Q

Fig. 2.8. Variation of the ground state energy of the La2 CuO4 crystal under a rotation of the octahedron in the orthorhombic LTO, LTO2 (P ccn), and low-temperature tetragonal (LTT) phases as a function of the normal mode amplitude Q (after [983])

method [983]. In the latter calculations, one derives the dependence of the ground state energy on the normal mode amplitude Q related to the rotation of the CuO6 octahedron in the LTO and LTT phases as shown in Fig. 2.8. It turned out that the LTT phase has the lowest energy. The sequence of phase transitions HTT → LTO → LTT can be described as a freezing of the soft tilting mode R1,2 = Rx ∓ Ry , (2.1). In the LTO phase, only one component of the soft mode R1 or R2 is condensed, while in the LTT phase both the components are condensed, which results in the rotation along x or y tetragonal axis. In the limit of strong anharmonicity of lattice vibrations in the two-well potential, one can consider the phase transitions as being the order–disorder type connected with the ordering of rotations of CuO6 octahedron in the four minima shown in Fig. 2.8. According to Pickett et al. [983], the instability of the tetragonal phase with respect to the rotations of the octahedron is due to a competition between repulsive forces in the CuO2 plane and the long-range Coulomb forces determined by the Madelung energy. 2.2.3 Copper-Oxide Ladder Compounds A new modiﬁcation of copper-oxide compounds was synthesized by Hiroi et al. [458] under high-pressure by creating oxygen vacancy lines ordered in the ideal

2.2 The Structure of La2−x Mx CuO4−y

29

Fig. 2.9. (a) The two-leg ladder SrCu2 O3 and (b) the three-leg ladder Sr2 Cu3 O5 (after [92])

copper–oxygen plane. These materials were called ladder compounds with the general chemical formula Srn−1 Cun+1 O2n . The ﬁrst member of the series with n = 3 comprises the two-leg ladder SrCu2 O3 , the n = 5 compound Sr2 Cu3 O5 is the three-leg ladder, etc. In Fig. 2.9 [92], we show the two- and three-leg ladders, where the number of legs is deﬁned by the number of copper chains strongly coupled by 180◦ pdσ Cu–O–Cu bonds. The ladders are coupled by the 90◦ Cu–O–Cu bonds, which are weak due to the orthogonality of the px , py oxygen orbitals at that bonds. This results in some kind of the electronic and magnetic isolation of diﬀerent ladders, which enables one to study low dimensional copper–oxygen compounds. Hiroi [462] also synthesized another material, LaCuO2.5 , which contains weakly coupled two-leg ladders. It turned out that even- and odd-leg ladders have quite diﬀerent physical properties (for review, see [264, 265]). The ladders with even number of legs show a large spin gap Δs of the order of the antiferromagnetic (AF) exchange interaction J, namely, Δs /J 0.5, 0.2, 0.05 for 2-, 4-, and 6-leg ladders, respectively. For instance, direct magnetic susceptibility measurements and 63 Cu NMR measurements by Azuma et al. [92] revealed a spin gap of about 420 K in the two-leg material, SrCu2 O3 , while for the three-leg ladder Sr2 Cu3 O5 a gapless excitation spectrum was observed. The spin gap in the magnetic excitation spectrum brings about the spin-liquid ground state, where only short range AF spin correlations are observed with the correlation length decaying exponentially with the distance. Properties of odd-leg ladders are much close to that of a single spin-1/2 chain with AF coupling. The spectrum of the magnetic excitation of the odd-leg ladders is gapless and this results in long-range spin correlations. They can evolve into a true AF long-range order as for the copper–oxygen plane at zero temperature. These quite diﬀerent magnetic properties were proved by various experiments such as NMR, neutron scattering, μSR technique. These simple ladder compounds are insulators and it is diﬃcult to make them conducting by doping. Hiroi [462] has synthesized another material,

30

2 Crystal Structure

LaCuO2.5 , which contains weakly coupled two-leg ladders. This compound can be hole doped replacing La by Sr, which bring about metallic conductivity at Sr concentration x 0.20. The spin gap seems to disappear in metallic phase but superconductivity has not been observed. The theoretical investigation predicted quite a diﬀerent behavior of doped ladder compounds (for a review, see [189]). While even-leg ladders have shown pair correlations for doped holes and superconductivity, odd-leg ladders should be much closer to a non-Fermi liquid of the Luttinger type. The reasons why, contrary to the theoretical predictions, superconductivity was not observed in the metallic ladder compound La1−x Srx CuO2.5 are unclear (see [265]). Doping of ladder compounds was successful for the more complicated system with the chemical formula (Sr,Ca)14 (CuO2 )10 (Cu2 O3 )7 , which show that it consists of 10 chains (CuO2 ) and 7 two-leg ladders (Cu2 O3 ). There are six holes per formula unit: ﬁve of which approximately are in the chains and one in the ladders [910]. With increasing of Ca ions concentration (the radii of which are smaller than those of the Sr ions) or under application of external pressure, it was possible to increase metallic conductivity [881] and to discover a superconducting transition at Tc 12 K at pressure of 3–4 GPa [1287]. It was observed also that the spin gap, which is quite large at ambient pressure, Δs 250 K, is vanishing at high pressure, P 3 GPa [817]. Experiments with a larger single crystal, however, have shown that an activation-type component in spin susceptibility that is characteristic for the spin-gap system survives at high pressure in the superconducting state [366]. They have also observed a superconducting coherence peak (Hebel–Slichter peak), which implies that the superconducting gap has no nodes at the Fermi surface. The large spin gap in the two-leg ladder SrCu2 O3 is caused by the strong 180◦ pdσ Cu–O–Cu bond along the rung. If the angle of the bond deviates substantially from 180◦, the rung coupling decreases and becomes comparable with the interladder interaction along the stack in c-direction. As a consequence, the spin gap can disappear, while the AF long-range order will take place. This was observed indeed in the pseudo-ladder compound CaCu2 O3 in which small Ca ions produce bending of the ladder and decreasing of the Cu–O–Cu bond angle up to 123◦ (see, e.g., [614]). The crystal structure of the CaCu2 O3 compound can be viewed as corner-shared CuO2 zigzag chains running along the b-axis. They are tilted by nearly 29◦ from a straight Cu–O–Cu bond with the neighboring zigzag chains forming this way positively and negatively buckled ladders with “kinked” rungs in a-direction. Magnetic susceptibility and neutron-diﬀraction studies reveal incommensurate AF long-range order below TN = 25 K with strong AF exchange interaction along the chains, J 0.17 eV, and much weaker coupling in the rung, J⊥ 0.09 eV. Experimental (polarized dependent X-ray absorption) and theoretical (LDA band structure calculation) investigations reveal also a signiﬁcant coupling perpendicular to the ladders, in c-direction [606]. This material can be therefore considered as an anisotropic bilayer system and a candidate for high-temperature superconductivity at hole doping away from the AF

2.2 The Structure of La2−x Mx CuO4−y

31

insulating state investigated so far. Theoretical estimate by Plakida et al. [1016] shows that strong AF interaction along the chains can produce quite high Tc 50 K though much smaller, due to large anisotropy of the system, in comparison with ideal copper–oxygen layer materials as the mercury cuprates. As a consequence, this low-dimensional copper–oxygen material is very interesting low-dimensional magnetic system where quantum ﬂuctuations play an essential role but strong anisotropy of the material precludes attaining high temperature of the superconducting transition. The role of low dimensionality and the interlayer coupling between the CuO2 planes in superconductivity in cuprates were elucidated in studies of the artiﬁcial YBa2 Cu3 O7 /PrBa2 Cu3 O7 (YBCO/PrBCO) lattices (see, e.g., [397, 714, 1266, 1267] and the references therein). With the aid of a special layer-by-layer deposition technique, it was possible to obtain ﬁlms made up by alternating layers of (YBCO)n and (PrBCO)m . In this case, it turns out that the transition temperature depends both on the thickness of the layer (YBCO)n and on the distance between the layers, which is deﬁned by the layer (PrBCO)m . Since the PrBCO compound is an insulator, it plays the role of an insulating layer separating the superconducting layers (YBCO)n . As was detected by Triscone et al. [1266], if the thickness of the Pr-layer exceeds 96 ˚ A or m = 8, one can neglect the connection between the superconducting Y-layers. It was revealed that even a YBCO layer consisting of two unit cells (24 ˚ A) resulted in Tc > 50 K, while for a one-cell layer (12 ˚ A), Tc 10 K. The gradual suppression of Tc as the number of unit cells decreases is explained by the growth of ﬂuctuations of the phase of the superconducting order parameter on crossover to a quasi-two-dimensional system, where a phase transition of the Berezinsky– Kosterlitz–Thouless type should be observed. At the same time, it should be noted that Tc for a superlattice with an equal number of Pr/Y layers proves much higher than for the solid solution Pr0.5 Y0.5 BCO. For instance, Tc 50 (60) K for the superlattices with n = m = 1 (2) as compared to Tc ∼ 20 K for the 0.5/0.5 solid solution [1266]. These sublattices, when subject to external magnetic ﬁelds, demonstrate rather unconventional strongly anisotropic properties [1266, 1267]. These investigations again conﬁrm the local nature of superconductivity in copper-oxide compounds, which have a small correlation length of the order parameter. However, a “giant proximity eﬀect” observed in cuprate superconductors, which reveals a supercurrent running through very thick barriers in artiﬁcial layers, is in conﬂict with the standard theoretical picture of small correlation lengths (see, e.g., [174] and the references therein). Interesting results concerning the coexistence of ferromagnetism and superconductivity were obtained for ferromagneticsuperconducting bilayer structures as in the La2/3 Ca1/3 MnO3 /YBa2 Cu3 O7−δ bilayers (see, e.g., [1127, 1181] and references therein), which partly may be due to a very large magnetic penetration depth in cuprates (see Table 4.6).

32

2 Crystal Structure

2.3 Nd2−xCexCuO4 Compounds The crystal structure of the Nd2 CuO4 -based superconducting compounds with electron conductivity is similar to that of the lanthanum compounds. It is described by the same space group I4/mmm but with the displaced oxygen ions O2 from their apex positions to sites on the faces of the tetragonal cell [1248, 1249]. In Fig. 2.10, the tetragonal cell for the compounds Nd(Ce)-124 (T phase), La(Sr)-124 (T phase), and the mixed compounds Nd(Sr, Ce)-124 (T∗ phase) are represented for comparison. In the T∗ phase, the apex oxygen ions are maintained only in the layer Nd–Sr, while in the layer Nd–Ce the oxygen ions are shifted to the faces. This reconstruction of the lattice in the T phase brings about a corresponding variation in its parameters compared to the T phase. The length of the tetragonal axis increases by about 4%, at = 3.94 ˚ A, while the c-axis decreases by 8%, ct = 12.1 ˚ A. The lattice parameters in the T∗ phase assume intermediate average values between those in the T and T phases. The variation in packing of the O2 ions in the T and T∗ phases compared to the T phase can be related to the diﬀerence in the sizes of the La and Nd ions. A primitive cell of the T∗ phase contains two formula unit and coincides with the tetragonal cell in Fig. 2.10 (space group 17 P 4/mmm − D4h ). The phase diagram of the Nd2−x Cex CuO4 and Pr2−x Cex CuO4 compounds are shown in Fig. 2.11. It is qualitatively similar to the phase diagram of the LMCO compounds (Fig. 2.4). At x = 0, the compound displays an AF insulating phase (TN 240 K), which is destroyed by doping at much higher concentration of Ce ions (x 0.12) than in the LMCO compounds (x 0.02).

Fig. 2.10. The tetragonal unit cells of the T , T, and T∗ phases of the electron-doped (Nd-Ce-Sr)2CuO4 compounds (after [1249])

2.4 YBaCuO-Based Compounds 500

b TN

20

TC

5

10

SC

Nd Pr TC

SC

Metallic

10

Semiconducting

AF

30

T(K)

T (K)

100 50

Metallic

a

33

0

1 0

0.1

0.2 x (Ce)

0.3

0

0.10

0.15

0.20

x (Ce)

Fig. 2.11. The temperature–concentration phase diagram for (a) Nd2−x Cex CuO4 (after [1282]) and (b) for (Nd-Pr)2−x Cex CuO4 (after Takagi et al. [1211])

The superconducting phase (SC) appears in the vicinity of the AF phase. It exists, however, inside a narrower interval of concentrations than in the LMCO compounds, 0.13 ≤ x ≤ 0.18 (see Fig. 2.11b), and has a lower transition temperature, Tc ≤ 24 K. In the T and T∗ phases, no structural transitions related to displacements of O1 ions in the tilting type modes have been observed. This can be related to the absence of complete CuO6 octahedron in these compounds. The absence of the apex oxygen O2 in the T phase manifests itself in a number of the electronic properties of Nd–Ce compounds, for example, in a peculiar dependence of Tc on pressure in T and T∗ phases (see Sect. 2.6). As studies of mercury compounds show, the role of the apex oxygen in copperoxide layered compounds is negligible, due to the quite large distance of the apex oxygen to the plane (see Sect. 2.5). This is in strong contrast with the YBa2 Cu3 O7−y compound in which the electronic properties of the CuO2 plane correlate with the apex oxygen distance to the plane since the charge transfer from chains to planes occurs through the apex oxygen ions.

2.4 YBaCuO-Based Compounds A vast literature is devoted to the study of the compound YBa2 Cu3 O7−y (YBCO or Y-123) and its various modiﬁcations (see, e.g., [444]). It was the ﬁrst found high-temperature superconductor with a transition temperature exceeding the boiling point of nitrogen (see Chap. 1). By varying the oxygen content, the physical properties of this compound can be changed over a wide range without any signiﬁcant changes in its structure. A whole class of compounds ABa2 Cu3 O7−y with A = Y, La, Nd, Sm, Eu, Gd, Ho, Er, and Lu was discovered with similar physical properties and Tc ∼ 90 K (see, e.g., [485, 788]). The exception is Pr-123 compound, which shows neither metallic nor superconducting behaviour (see Sects. 3.2.4, 5.1.2, and 5.2.1). As with the

34

2 Crystal Structure

layered copper-oxide compounds, YBCO allows a certain modiﬁcation of its structure through variation of the coupling between copper–oxygen layers. In particular, compounds with two chains YBa2 Cu4 O8 (124) or Y2 Ba4 Cu7 O15 (247), and compounds of type Pb2 (Y-Ca) Sr2 Cu3 O8+y (2123) have been synthesized via the modiﬁcation of Cu–O chain layer into a more complicated structure Pb2 CuO2 . Replacing Cu–O chain layer by RuO2 sheet results in the magneto-superconductor RuSr2 GdCu2 O8−δ (see Sect. 2.4.3). All these features of YBCO have brought about a wide interest in the study of its structure and other properties. 2.4.1 Structure of YBa2 Cu3 O7−y The original compound YBa2 Cu3 O7−y can be synthesized in two structural modiﬁcations depending on temperature and oxygen content y. The ﬁrst is 1 the orthorhombic phase P mmm(D2h ). The second is the tetragonal phase 7 P 4/mmm(D4h ). The elementary cells of these structures with one formula unit are shown in Figs. 2.12a,b [562]. The main structural parameters in the orthorhombic phase (Fig. 2.12b) (in ˚ A) at room temperature and y 0 are the following. The lattice constants are a = 3.828, b = 3.888, c = 11.65. The lengths of the bonds are Cu1–O1 Cu1–O4 = 1.94, Cu2–O2 = 1.92, Cu2–O3 = 1.96, Cu2–O4 = 2.3. The length of the four Cu–O bonds for the four oxygen ions nearest to the copper both in the plane Cu2–O2, O3 and in the chains Cu1–O1, O4 are approximately the same and correspond to the lengths of bonds in the CuO2 plane for LMCO compounds. The distance between the copper ion in the plane and the apex oxygen, Cu2–O4, (as the

b

a

Cu2

Cu2 Y

O2 Ba O4 O1 Cu1

Y O3

O2 Ba O4 O1 Cu1

Fig. 2.12. The structure of YBa2 Cu3 O7−y in (a) the tetragonal and (b) the orthorhombic phases (after [562])

2.4 YBaCuO-Based Compounds

35

lattice constant c) varies strongly, as the oxygen content decreases under the transition into the tetragonal phase (see Fig. 2.14). In the tetragonal phase (Fig. 2.12a), oxygen positions O2 and O3 become equivalent with equal bond length Cu2–O2 and Cu2–O3, while remaining structural parameters are close to those in the orthorhombic phase. The orthorhombic phase is observed at low temperatures for an oxygen content x = 7 − y ≥ 6.4. The transition to the tetragonal phase occurs at temperatures T ≥ 500◦ C when oxygen content starts to decrease together with disordering of oxygen ions in the Cu1–O1 plane (Fig. 2.12a). It is seen that YBCO has a typical layered perovskite-like structure with two CuO2 planes separated by an oxygen free layer of Y ions which are coupled by the buﬀer layers Ba–O4–Cu1–O1–Ba–O4. The oxygen O2 and O3 are strongly coupled with Cu2 in the CuO2 planes, unlike the weakly coupled oxygen O1 in the Cu1–O1 chains. Upon heating above 500◦ C, the latter oxygen ions diﬀuse away from the sample. This enables the oxygen content to be smoothly varied from x = 7 (y = 0) to x = 6 (y = 1), when all oxygen ions O1 in chains have been removed out of the compound. In the latter case, the tetragonal phase persists up to low temperature. At intermediate values of x, the structure of the compound depends on the way in which oxygen is removed [409]. Quenching from the HTT phase at x ≤ 6.5 preserves this tetragonal phase with disordered O1 positions in the Cu1–O1 plane. If the samples are prepared by the lower-temperature Zr-gettered annealing technique, the orthorhombic phase can be maintained up to x 6.2. In this case, several modiﬁcations of the orthorhombic phase occur. Besides the OI phase shown in Fig. 2.12b when x = 7 and all the Cu–O1 chains are ﬁlled, an ordered phase OII at x = 6.5 can occur when alternate chains in the Cu1–O1 plane turn out to be empty. More complicated phases at intermediate √ √ values of x are also observed. For example, at x 6.35 a phase 2 2a × 2 2a occurs when half-ﬁlled chains alternate with chains that are one-quarter ﬁlled [1183] Formation of chains of a ﬁnite length was detected in NMR and NQR experiments (see, e.g., [750, 1385]). Theoretical calculations of the x–T phase diagram and studies of the oxygen ordering in OII phase were performed within a lattice-gas model described by the asymmetric next-nearest-neighbor Ising (ASYNNNI) model [272,435] and by band structure calculations [982]. The physical properties of YBa2 Cu3 O7−y -based compounds depend considerably on the oxygen content. The highest value of superconducting temperature, Tc = 92 K, is attained at the optimal doping in metallic phase at x = 7 − y = 6.92. With decreasing oxygen content, Tc goes down and the metallic phase transforms into the semiconducting phase at y 0.6. In the latter phase, long-range AF order appears with a maximum N´eel temperature TN 500 K at y = 1 (see Sect. 3.2.1). The way in which Tc depends on x = 7 − y is determined by the type of sample preparation. In Fig. 2.13a, the curve Tc (x) is plotted. The dots correspond to high-temperature quenching and the crosses to the lower-temperature Zr-gettering (solid line) [409]. In the latter case, two plateau are observed at Tc = 90 K (for 6.85 < y < 7.0) and

36

2 Crystal Structure

Fig. 2.13. (a) The dependence of Tc on oxygen content x in YBa2 Cu3 Ox for samples with oxygen removed by high-temperature quenching (dots) and by lowertemperature Zr-gettering (solid line) (after [409]) and (b) the relation between Tc and the eﬀective copper valence (after [215])

Tc = 60 K (for 6.45 < y < 6.65), which are related to the aforementioned two orthorhombic phases OI and OII. Thus, the short-range order in the Cu1–O1 chain has an essential eﬀect on the electronic properties of the superconductor. This points to a local nature of the doping of the conducting CuO2 plane due to charge (hole) transfer from the Cu1–O1 chains [215, 563]. The dependence of Tc and the lattice parameters of the YBCO compound on the oxygen content has been studied in detail by Cava et al. [215] by applying the lower-temperature Zr-gettering annealing technique. This technique has permitted a number of low-temperature equilibrium phases to be obtained by varying the oxygen content and has revealed a correlation between changes in electronic properties, in particular, Tc , and structural parameters. Figure 2.14a plots the dependence of the lattice parameters a, b, and c in the orthorhombic phase on the oxygen content x = 7 − y [215]. At x 6.4, under the transition from the orthorhombic (O) to the tetragonal (T) phase, a considerable increase in the c-axis lattice constant is observed. The length of the out-of-plain bond Cu2–O4 (O4 is the apex oxygen in the CuO5 pyramid, see Fig. 2.12, is denoted as O1 in Fig. 2.14b) undergoes equally strong variation, while only a slight variation of the in-plane bond lengths, Cu2–O2, O3, occurs as shown in Fig. 2.14b. In this work, to ﬁnd the correlation between the change in structural parameters and electronic properties, a bond valence sum V = i exp[(R0 − Ri )/Bi ] has been calculated for various ions. The sum determines an eﬀective

2.4 YBaCuO-Based Compounds

b

a

37

2.5

c, Å

11.8 2.4 bond length, Å

c

11.7

11.6 b

a,b, Å

3.9

7

x in YBa2Cu3 Ox

2.3

2.0

Cu2 - O3

1.9

a 3.8

Cu2 - O1

Cu2 - O2 6

1.8 7

x in YBa2Cu3 Ox

6

Fig. 2.14. The dependence of (a) lattice constants and (b) copper–oxygen bond lengths (Cu2–O2, O3), (Cu2–O4) in YBa2 Cu3 Ox on oxygen content x (after [215])

valence of a given ion speciﬁed by the parameters Ri and Bi . The eﬀective valence of copper ions in Cu1 chains has turned out to be linear in x. It varies from V = 2.5 at x = 7 to V 1.3 at x = 6. The eﬀective valence of Cu2 undergoes a downwards jump at the transition from the orthorhombic to tetragonal phase, which is accompanied by a correspondingly sharp variation in the length of the Cu2–O4 bond. Figure 2.13b demonstrates a correlation between Tc and the eﬀective copper valence. The similar behavior of these functions indicates that Tc is determined by the value of the eﬀective charge in the CuO2 plane. The decrease of Tc from 92 to 60 K is due to the transfer of (negative) charge of about 0.03 e from the chains to the plane. The disappearance of superconductivity at x = 6.45 is connected with a further transfer of charge of about 0.05 e to the plane. As the oxygen content further decreases to x = 6, the eﬀective charge of the copper in the plane remains constant. The charge transfer from the chains to the plane under oxygen doping was conﬁrmed later in the photoemission experiments (see Sect. 5.2.1, Fig. 5.14). Thus, detailed structural studies of YBa2 Cu3 Ox compounds have unambiguously demonstrated the local nature of the charge transfer from CuO chains to CuO2 planes and revealed drastic changes in the electronic properties of the system, including superconductivity, related to the transfer. Cava et al. [215] have noted that for samples with the same oxygen content, the transition temperature Tc can vary considerably depending on the oxygen ordering in the Cu1–O1 chains. The positive charge transfer (holes) from chain to plane can occur only when two oxygen positions O1, nearest to a copper site Cu1 in the chain O1–Cu1–O1, are ﬁlled by oxygen ions. Then, the formal valence

38

2 Crystal Structure

of the Cu1 becomes v > 2, which produces a hole transfer from the plane to oxygen in chains. It explains why the short-range order of oxygen ions in chains is so important for proper hole doping of CuO2 planes. 2.4.2 Modifications of the YBCO Structure In synthesizing single crystals of YBCO in the orthorhombic phase, polydomain samples with a twin plane of the (110) type usually appear. In such twinned crystals, the anisotropy of physical properties in the (a, b) plane of the orthorhombic lattice cannot be studied. In this respect, the synthesis of untwinned single crystals YBa2 Cu4 O8 (or 124) was of a great interest [444]. In Fig. 2.15, the structures of three compounds of the Y2 Ba4 Cu6+n O14+n family are shown for comparison. The original YBCO-123 structure corresponds to n = 0. The structure YBCO-124 corresponds to n = 2. At n = 1, an intermediate structure YBCO-247 occurs. The most important feature of the new modiﬁcations is the appearance of double chains instead of the single Cu1–O1 chains in the 123 compound. The oxygen coordination in the chains increases from two (Cu1–O1–Cu1) to three, which stabilizes the entire structure. Due to this strengthening of the oxygen binding in the chains, the YBCO-124 compound can be heated to much higher temperatures (of order 800◦ C) without any signiﬁcant loss of oxygen. In the YBCO-124 and -247 compounds, the temperature of the superconducting phase transition attains the values Tc = 80 K and Tc = 87 K, respectively [406]. The lower Tc observed in these modiﬁcations compared to Tc = 92 K in YBCO-123 is usually attributed to the decrease of the eﬀective charge of copper in the CuO2 plane [406] so these compounds correspond to the underdoped YBCO-123 material. The synthesis of untwinned YBCO-124 and -247 crystals has enabled a number of interesting studies of their physical properties to be carried out [181, 579].

Y Ba Cu

Fig. 2.15. The crystal structure of the Y2 Ba4 Cu6+n O14+n compounds at n = 0 (YBCO-123), n = 1 (YBCO-247), and n = 2 (YBCO-124) (after Kaldis et al. [579])

2.4 YBaCuO-Based Compounds

39

There exist several modiﬁcations of the YBCO structure obtained by means of transformations of the three layers Ba–O4, Cu1–O1, and Ba–O4 into a more complicated structure, while keeping the bilayer CuO2 –R–CuO2 intact. One such modiﬁcation has the general formula Pb2 ASr2 Cu3 O8 with Tc = 70 K, where A = Y, R, Ca [213]. In this structure, the main 123 unit ASr2 Cu2 O6 is preserved upon the replacement of Ba by Sr. The connection between these units is no longer provided by Cu–O chains. It is now due to a more complicated structure Pb2 CuO2 , which consists of two PbO layers separated by a layer of copper with two-fold coordination. The structure of this compound is described by an orthorhombic unit cell (space group Cmmm) with the parameters a = 5.40 ˚ A, b = 5.43 ˚ A, and c = 15.8 ˚ A [213]. The small distortion of the tetragonal cell is accounted for by an ordering of oxygen in the PbO plane in noncentrosymmetric positions. In quenching from the high-temperature phase at 500◦ C, the tetragonal structure P 4/mmm is observed. A modiﬁcation of the Y-123 structure, which does not contain any Cu–O chains, is investigated by Roth et al. [1071]. This is the compound RSr2 GaCu2 O7 in which the main element of the Y-123 structure, that is, the CuO2 – R–CuO2 bilayer, is preserved. However, instead of the copper chains, the bilayers are now bounded by GaO4 octahedron, which also form chains. This considerable change in the structure results in quite diﬀerent electronic properties. Having the stoichiometric composition with respect to oxygen O7 , this compound possesses semiconducting properties. Substitution of the trivalent ions R = Y, Yb, Er by divalent ones, for example, Ca, which can be performed only under high oxygen pressure, results in metallic properties and superconductivity at Tc = 40–50 K in multiphase samples [216]. 2.4.3 Rutheno-Cuprates Magneto-Superconductors In the conventional superconductors, long-range ferromagnetic ordering and superconductivity usually exclude each other. Therefore, the observation of ferromagnetic and superconducting properties in the ruthenate–cuprate compound RuSr2 GdCu2 O8−δ (Ru-1212) by means of zero-ﬁeld muon-spin rotation and dc magnetization measurements [130] attracted the interest of many researches. It has been established that the material exhibits ferromagnetic order of the Ru moments (μRu ∼ 1μB ) on a microscopic scale below TC = 133 K and becomes superconducting at a lower temperature Tc = 16 K, which can reach 46 K depending on the sample preparation. Coexistence of superconductivity and ferromagnetism was also reported in the more complicated structure RuSr2 (Gd0.7 Ce0.3 )2 Cu2 O10−δ (Ru-1222) [328]. The structures of both Ru-1212 and Ru-1222 compounds can be also viewed as modiﬁcations of Y-123 structures. In Ru-1212, the CuO2 –R–CuO2 bilayer (R = Gd) is preserved but the charge reservoir of CuO1−δ chains in Fig. 2.12 are replaced by RuO2−δ sheets. In Ru-1222, the single oxygen free R-layer between the CuO2 planes is replaced by a three-layer ﬂuorite-type

40

2 Crystal Structure

(R,Ce)–O2 –(R,Ce) block. The Ru-1212, Ru-1222 compounds have tetragonal structures (space group P 4/mmm and I4/mmm respectively) with lattice parameters: a = b = 3.822 ˚ A, c = 11.476 ˚ A for Ru-1212 and a = b = 3.834 ˚ A, ˚ c = 27.493 A for Ru-1222 crystal. Though the bulk ferromagnetism in RuO2 layers of Ru-1212 compound on the microscopic scale was conﬁrmed by the μSR-method and ESR studies, the exact long-range magnetic structure is still controversial. While neutron diﬀraction experiments show antiferromagnetic ordering, magnetization studies indicate ferromagnetic component. Speciﬁc heat measurements conﬁrm the bulk nature of superconducting transition, while at the magnetic phase transition only hump is seen in CP , which indicates inhomogeneous magnetic order. It should be pointed out that the physical properties of these compounds strongly depend on the sample preparation conditions and there are still many controversial results in the literature. A critical review of physical properties of these magneto-superconductors is given by Awana et al. [89] (see also [232]). In theoretical studies of coexistence of superconductivity and ferromagnetism, usually the Fulde–Farrell–Larkin–Ovchinnikov (FFLO) phase is considered [370, 690], where superconducting or (and) ferromagnetic order parameter develops spatial variation to decrease the total free energy of the system. By applying the local density approximation and its generalization for electron structure calculation, Pickett et al. [984] have concluded that coexistence of superconductivity and ferromagnetism is possible within the FFLO phase. This result can be explained by several speciﬁc properties of the layered type compounds: small magnetization of RuO2 layer, a certain degree of isolation of the RuO2 and CuO2 sublattices due to weak chemical coupling of the relevant atomic orbitals for Cu, O, and Ru ions in the layers, while these layers are thin enough to allow three-dimensional coupling for both order parameters. So, this quite complicated crystal structure could aﬀord coexistence of two competing ordering on the microscopic scale, which now is well established on the macroscopic scale for artiﬁcial sandwich structures of thin ferromagnetic and superconducting ﬁlms (see, e.g., [545]).

2.5 Bi-, Tl- and Hg-Compounds Soon after Bednorz and M¨ uller’s discovery, a new class of quinary copperoxide compounds were discovered: the bismuth Bi2 Sr2 CaCu2 O8+δ (Bi-2212) [767] and the thallium Tl2 Ba2 CaCu2 O8+δ (Tl-2212) [1147] compounds with superconducting transition temperatures Tc above 100 K. These were major achievements. Even more exciting was the discovery of mercury cuprate compounds HgBa2 Can−1 Cun O2n+2+δ (Hg-12(n − 1)n) [1034, 1035, 1112] with a record Tc = 134 K (n = 3) which can be further enhanced up to Tc = 164 K under an external pressure P 30 GPa [375]. These compounds can have various numbers of copper–oxygen planes and are described by the general

2.5 Bi-, Tl- and Hg-Compounds

41

TI2 Can–1 Ba2 Cun O2n+4 n=1

n=2

n=3

Fig. 2.16. The ideal crystal structure of the Tl2 Ba2 Can−1 Cun O2n+4+δ compounds for n = 1, 2, 3 copper–oxygen layers (reprinted with permission by APS from Parkin c 1988) et al. [962],

formula Am B2 Can−1 Cun Ox , where A = Bi, Tl, or Hg and B = Sr for Bi and B = Ba for Tl and Hg compounds. For bismuth m = 2, mercury m = 1, while the thallium compounds can be synthesized with one or two Tl layers: m = 1, 2 [962]. Figure 2.16 shows the ideal pseudo-tetragonal unit cell of the 17 Tl-22(n − 1)n compounds (space group (I4/mmm − D4h ). The structure of the Bi-22(n − 1)n compounds is the same as that of the Tl-compounds shown in Fig. 2.16. The pseudo-tetragonal unit cell of Bi- and Tl-compounds has the dimensions 3.9×3.9 ˚ A in the basal plane. The lattice constant along the c-axis depends on the number of copper-oxide planes n. It equals to c = 24.4, 30.8, 37.1 ˚ A and c = 23.2, 29.4, 36 ˚ A for n = 1, 2, 3 for Bi- and Tl-compounds respectively. The actual structure of the Bi compounds has orthorhombic√distortions. 23 ) has the dimensions ao bo at 2 5.4 ˚ A, The unit cell (F mmm − D2h co ct . Moreover, an incommensurate modulation in the Bi–O layer with the wave vector q = 0.21b∗ is observed, which can be roughly approximated in a supercell orthorhombic model with bo = 5ao . The average Cu–O distances in the plane have typical value 1.9 ˚ A. The Cu–O distance along the c-axis in the CuO5 pyramid is 2.6 ˚ A, which is larger than the corresponding Cu2–O4 distance of about 2.3 ˚ A in the YBCO compound. A weak coupling between Bi–O layers due to the large distance ∼3 ˚ A between them is typical for Bi mica-like compounds. The crystal is easily cleaved between the Bi–O layers producing a clean surface. Although the modulation of the structure within Bi–O layers does not inﬂuence superconductivity, this structure irregularity in Bi compounds greatly complicates the study of the single crystal properties. To suppress the distortions of the crystal structure, a mixed Bi–Pb compound

42

2 Crystal Structure

was synthesized, Bi2−x Pbx CaSr2 Cu2 O8+δ . Its orthorhombic structure with lattice parameters a 5.40, b 5.38, c 30.78 ˚ A for a large concentration of lead x ≥ 0.4 shows modulations only for Pb displacements in one direction with much a larger period, which is characterized by the wave vector q = 0.12b∗ [551]. This conﬁguration allows more regular distances between neighboring Bi–O layers and that considerably simpliﬁes the studies of single crystal properties. The structure of the Tl compounds undergoes much less distortion from the ideal tetragonal lattice due to the relatively strong coupling between Tl–O layers. The distance between them is ∼2 ˚ A, while the lattice constant is smaller than in the Bi compounds. The length of the Cu–O bonds in the CuO2 planes is 1.92 ˚ A, while the copper–apical oxygen distance along the c-axis reaches 2.7 ˚ A, which is sensibly larger than the corresponding distance in YBCO. More detailed studies of the structure of Tl compounds have shown, however, a considerable statistical disorder in the Tl–O layers [451]. The Tl and O ions in the Tl–O layers have been found to be displaced from there centrosymmetric positions in the layer in such a way that they form Tl–O chains with a shorter bond. However, this structure possesses only short-range order so that it maintains the tetragonal symmetry on the average [284]. The superconducting transition temperature in the Bi- and Tl-compounds depends on the number of the copper–oxygen planes n = 1, 2, 3 and takes the values Tc 10, 85, 110 K for the Bi compounds, and Tc 80, 100, 125 K for the Tl compounds, respectively, with the highest Tc for three layer compounds [444]. This trend is general for all cuprate superconductors as discussed below for mercury compounds. The crystal structures of mercury compounds with one, two, and three copper layers are shown in Fig. 2.17. Their tetragonal structure P 4/mmm is characterized by close values of the lattice parameter a and Cu–O distances in the plane a/2 1.94 − 1.93 ˚ A with, however, a well resolved decreasing on the number of planes n (see Fig. 2.18). The c-lattice parameter depends on A. The hole doping of these the number of CuO2 planes: c 9.5 + 3.2(n − 1) ˚

Fig. 2.17. Crystal structures of mercury compounds Hg-1201, Hg-1212 and Hg-1223 (after [77])

2.5 Bi-, Tl- and Hg-Compounds

43

Fig. 2.18. The dependence of superconducting Tc on the lattice parameter a for the Hg-12(n − 1)n series for n = 1, 2, 3, 4, and 5 (after [77])

compounds is achieved by annealing them in oxygen atmosphere, which gives excess oxygen ions with concentration δ in the buﬀer layers Hg–Oδ , as shown in Fig. 2.17. The structure of the TlBa2 Can−1 Cun O2n+3+δ (Tl-12(n − 1)n) compounds is similar to that of the compounds Hg-12(n − 1)n structure. The main diﬀerence comes from the number of oxygen ions in the Hg and Tl layers: while for Hg2+ only a small content δ ≤ 0.2–0.4 of doping oxygen (depending on the number n of Cu-planes) is needed to obtain the optimal concentration of doped holes, for Tl3+ ions the number of oxygen ions in the Tl–O planes equals to 1 + δ, which produces distorted octahedron coordination for Tl ions. The characteristic feature of the Hg-based compounds is a stable dumbbell Hg2+ coordination with two strong covalent Hg–O bonds with the nearest oxygen ions in Ba–O layers. This results in a weak coupling with nonstoichiometric oxygen ions in the Hg–Oδ layers and an ideal tetragonal structure. The mercury compounds show the largest distance between the CuO2 plane and the apical oxygen of about 2.77–2.8 ˚ A. As a result, the buﬀer layers exert a very weak inﬂuence on the copper–oxygen planes, which results in the smallest buckling of the planes among the copper-oxide materials. It is inferred that just this ideal ﬂat structure of CuO2 plane results in the highest Tc in cuprates [77]. The stable structure of the Hg-based cuprates enables a detail investigation of the dependence of the superconducting temperature on the number n of copper–oxygen planes [77]. The behavior of Tc,max with the number of layers is similar to that observed in the Bi- and Tl-compounds. It ﬁrst increases, Tc 96, 127, 135 K for the number of Cu layers n = 1, 2, 3, but then decreases: Tc 127, 110, 107 K for n = 4, 5, 6. The decrease of Tc for the number of layers n > 3 can be explained by underdoping of inner Cu layers. The eﬀective concentration of holes in these layers appeared to be lower than the optimal

44

2 Crystal Structure

Fig. 2.19. Superconducting transition temperature for Hg-1201 as a function of oxygen or ﬂuorine concentration (after Abakumov et al. [1])

value of xopt 0.16 per one CuO2 unit cell in the plane. Figure 2.18 plots the Tc dependence on the lattice parameter a for n = 1, 2, 3, 4, 5 along the HgBa2 Can−1 Cun O2n+2+δ homologous series. The functional dependence of Tc as a function of the concentration δ of doped oxygen in the Hg–Oδ layer shows parabolic form similar to that of (1.5), Tc (δ) = Tc,max 1 − q (δ − δopt )2 , (2.9) where for Hg-1201, Tc,max 97 K, q 52, δopt 0.128. These parameters can vary from one family of compounds to another but the general parabolic form (2.9) is retained. Interesting results in this respect were obtained for mercury compounds by using ﬂuorine instead of oxygen for hole doping. Figure 2.19 plots Tc as a function of oxygen and ﬂuorine ion concentration in the Hg-1201 compound [1]. The maximum Tc value is reached at ﬂuorine ion (F− ) concentration δF 0.26 twice as large as the oxygen ion (O2− ) concentration δO 0.13. Taking into account the ratio of the ion valences, this results in the same optimal hole doping. If we assume that every doped oxygen ion O2− transfers two holes into the CuO2 plane, then the optimal concentration of doped holes xopt 0.26 appears to be larger than for other cuprates: xopt 0.16 in (1.5). The apparent discrepancy originates in the distribution of the doped charge over the CuO2 layer and Hg–O2 dumb-bell, which results in transferring only 60% of the doped oxygen charge into the CuO2 plane. The comparison of the lattice parameters for oxidized and ﬂuorinized samples has revealed that the maximal Tc at optimal doping strongly depends on the in-plane Cu–O bond length, while it does not depend on small variations of the copper– apical oxygen distance. Figure 2.20 shows the variation of maximal Tc for mercury compounds as a function of the lattice parameter a in the plane [737].

2.6 High-Pressure Eﬀects

45

Fig. 2.20. Superconducting transition temperature for mercury compounds at optimal doping as a function of the lattice parameter a in the CuO2 plane for n = 1, 2, 3 copper–oxygen materials. Hg-1223F and Hg-1201F are ﬂuorinated compounds (after [737])

While for the Hg-1201 compound the ﬂuorination does not change the lattice parameter a and Tc (but changes the copper–apical oxygen distance), the lattice parameter a decreases and Tc increases up to 138 K in the ﬂuorinated Hg-1223 sample. The rate of Tc enhancement with the decrease of a appears to be very large, dTc /da 1,300 K/˚ A. It is worthwhile to note that the same coeﬃcient deﬁnes the Tc dependence on the lattice parameter a for all the three mercury compounds, as shown in Fig. 2.20. From these experiments, we can draw the general conclusion that the Cu–Oapical distance (at small variation) is irrelevant for the superconducting Tc value, while the compression of the in-plane Cu–O distance can greatly enhances Tc . This conclusion holds true provided that other parameters of the copper–oxygen bond, in particular the Cu–O–Cu bond angle in the plane, do not change. In mercury compounds, this angle is close to 180◦ and does not change under ﬂuorination, while the application of an external pressure signiﬁcantly reduces the angle which results in buckling of the CuO2 plane (e.g., at the pressure of 2 GPa the angle reduces to 175.0◦ from 178.4◦ for oxygenated [177.3◦ for ﬂuorinated] Hg-1223 compound [737]). Therefore, the chemical compression of the structure without changing the buckling angle is much more eﬃcient in enhancing Tc than by isotropic external pressure.

2.6 High-Pressure Eﬀects High-pressure studies of cuprate superconductors play an essential role in our understanding of both the normal state and the superconducting properties of these compounds. It should be mentioned that the idea of enhancing superconducting Tc by “chemical pressure” resulted in the discovery of YBCO superconductor by Wu et al. [1371] and later on in achieving the record

46

2 Crystal Structure

Tc = 164 K under external pressure P 30 GPa [375]. The application of external pressure enables a smooth variation of the lattice parameters without uncontrollable side eﬀects, which usually accompany chemical substitution. Therefore, the study of the Tc dependence on the pressure provides an important approach to the elucidation of the mechanism of high-temperature superconductivity. There are several reviews on the studies of high-pressure eﬀects in high-temperature superconductors (see, e.g., [1111, 1118, 1119, 1213, 1358]). Here, we discuss only the most important results concerning the pressure dependence of Tc . In studies of the conventional electron–phonon superconductors, it was observed that Tc decreases with the pressure P in most cases. This feature can be explained if, by using the BCS formula (1.1) for Tc , we write d ln Tc ∂ ln ω 1 ∂ ln λ ∂ ln ω 1 ∂ ln η ∂ ln ω = + + −2 . (2.10) dP ∂P λ ∂P ∂P λ ∂P ∂P Here, the interaction in (1.2) is written as λ = η/M ω 2 , where the Hopﬁeld parameter η = N (0)g 2 . For conventional s–p superconductors, both the density of states N (0) and the electron–phonon interaction g weakly depend on pressure due to the high electron density. The most important contribution to the variation of Tc with P originates in the second (negative) term in (2.10) owing to the increase of the lattice stiﬀness with the pressure: ∂ ln ω/∂P > 0. Therefore, a behavior dTc /dP < 0 is expected to be found in the s–p metals, a feature conﬁrmed, for example, by studies of the MgB2 superconductor: dTc /dP −1.1 K GPa−1 [1118]. Only for narrow d-band metals in which the density of states N (0) may strongly vary with pressure, the second term in (2.10) can be positive, ∂ ln η/∂P > 2 ∂ ln ω/∂P , and Tc would increase with the pressure. Concerning the copper-oxide superconductors, we should take into account that Tc strongly depends on the density of charge carriers, as given by the universal formula (1.5). Therefore, we should distinguish two contributions, the ﬁrst one due to intrinsic eﬀects, as changing the parameters of the pairing interaction, and the second one due to charge transfer x under pressure: ∂Tc ∂x ∂Tc dTc = . (2.11) + dP ∂P intrinsic ∂x ∂P At optimal doping, x = xopt , we have dTc /dP = dTc,max /dP , which gives the intrinsic pressure eﬀect since the second term according to the parabolic Tc dependence in (1.5) gives no contribution. As many experiments show, contrary to the conventional superconductors, in cuprates dTc,max /dP > 0 [1359]. For “normal” cuprate superconductors, dTc,max /dP 1.5 K GPa−1 [1118, 1119]. For underdoped and overdoped hole cuprate superconductors, the Tc dependence on pressure has opposite signs: dTc dTc > 0, < 0. (2.12) dP underdoped dP overdoped

2.6 High-Pressure Eﬀects

47

In these regions, the charge transfer term ∂x/∂P in (2.11) becomes important and it deﬁnes the pressure dependence. Under pressure, the distance between the negatively charged copper–oxygen CuO2 layer and positively charged doping buﬀer blocks decreases, which results in positive charge transfer to the copper–oxygen layer with the maximum rate of ∂x/∂P 0.02 GPa−1 [1359]. Owing to the parabolic Tc (x) in (1.5), for hole doped cuprates the dependence (2.12) is realized, while for electron doped cuprates it should be opposite. This general trend for Tc dependence on pressure in cuprates is conﬁrmed by many experiments [1359]. In LaSr-124 compounds, Tc ﬁrst increases under pressure reaching a certain maximum but then decreases at higher pressure. In electron doped Nd(Ce)-124 compounds, Tc appears to be pressure independent, while in mixed compounds Nd(Sr, Ce)-124 quite a strong pressure dependence is observed [789]. This difference in the Tc (P ) dependence can be related to a special role of the apex oxygen: in the T-phase of La(Sr)-124 compounds, there are two apex oxygen ions inside the complete CuO6 octahedron, while in the T -phase of Nd(Ce)124 compounds the apex oxygen ion is absent (see Fig. 2.10). In the T∗ -phase of Nd(Sr, Ce)-124, there is only one apex oxygen in the CuO5 pyramid. Therefore, the mechanism of charge transfer under pressure from the buﬀer blocks La–Sr–O or Nd–Ce–O is quite diﬀerent. In some cases, an extremely large pressure dependence is observed for cuprates in the underdoped region, as for YBa2 Cu3 O7−y for small oxygen content, y 0.5, or for YBCO-124, where Tc 80 K at ambient pressure increases up to 105 K under a pressure of 10 GPa [1359]. Strong Tc dependence on the pressure was also obtained close to structural phase transformations, as in La1−x Bax CuO4 system in the vicinity of x = 1/8, where Tc is strongly suppressed due to transformation to the LTT phase (see Fig. 2.6) and dTc /dP 12 K GPa−1 [1111]. This strong Tc dependence on pressure in the LTT phase was conﬁrmed by Takeshita et al. [1215], who have found a highly anisotropic pressure dependence in the (a, b) plane for La1.64 Eu0.2 Sr0.16 CuO4 system: (dTc /dP )[110] 2.5 K GPa−1 and much weaker in [100] direction. In Eu-doped LSCO-124-compound, the LTT structure occurs in a wide range of Sr concentrations (see Fig. 2.6). As discussed in Sect. 2.2, bulk superconductivity in the LTT phase exists only if the tilting angle Φ of the octahedron rotation is smaller than the critical one, Φc 3.6◦ [183, 184]. A uniaxial pressure along the [110] direction decreases the amplitude of the octahedron rotation in the LTT phase (and CDW-stripe amplitude) and this enhances Tc . The most accurate studies of Tc dependence on the pressure were performed in Hg-based superconductors. Due to a stable crystal structure over a wide range of doping, it was possible to investigate pressure eﬀects both in the underdoped and overdoped regions with high precision (see, e.g., [77,96,231]). For all optimally doped Hg-12(n − 1)n compounds, Tc increases with the pressure with a rate dTc /dP 1 − 2 K GPa−1 up to 5 GPa. Further increase of pressure leads to saturation and at some speciﬁc value, depending on

48

2 Crystal Structure

the number n of copper–oxygen layers, Tc decreases. For compounds with n ≥ 3, the inequivalent inner and outer layers have diﬀerent doping levels and the charge transfer under external pressure can be important in enhancing Tc . Therefore, the studies of Tc and structure under pressure in Hg-1201 compound with only one copper–oxygen layer by Balagurov et al. [96] were important for the elucidation of the intrinsic pressure eﬀects. By using a highresolution neutron diﬀraction under pressure in a wide region of oxygen, Oδ , doping 0.06 < δ < 0.19, the authors were able to ﬁnd out a correlation between Tc and structure changes. In the underdoped δ = 0.06 and optimally doped δ 0.13 regions, the compression of the structure is uniform and the charge transfer plays a minor role. However, in the overdoped region δ = 0.18, a large compression of the apical Cu–O2 and Ba–Oδ distances is observed. These results explain why at low and optimal doping Tc increases with pressure (the intrinsic eﬀect in cuprates), while in the overdoped region a charge transfer occurs under pressure, which leads to Tc decreasing according to the parabolic law (2.9). The copper-oxide materials are strongly anisotropic systems and, therefore, uniaxial deformations are required to distinguish between the charge transfer (which is sensitive to c-axis compression) and the intrinsic pressure eﬀects in the conducting CuO2 plane caused by (a, b) compression. These quite diﬀerent pressure eﬀects were conﬁrmed in direct measurements of the changes of Tc induced by uniaxial compression along the a, b, and c axes of an untwinned crystal of YBa2 Cu3 O7 by Welp et al. [1350]. It was found that the pressure derivatives are large but have opposite signs for compression along diﬀerent axes (in K GPa−1): dTc −2.0, dPa

dTc 1.9, dPb

dTc −0.3. dPc

(2.13)

Under hydrostatic pressure, only a small pressure eﬀect is observed in YBCO at optimal doping due to signiﬁcant cancellation of these opposite in sign derivatives. This experiment also proves that Tc increases by diminishing the orthorhombicity b/a of the crystal: dTc /dPi have opposite signs along the a and b directions. Similar results were obtained for the Bi2 Sr2 CaCu2 O8+x single crystal [822]. The uniaxial-pressure dependencies of Tc were calculated from the amplitudes of the measured expansivity anomalies. The pressure derivatives occurred also large but of opposite signs for compression along diﬀerent axes (in K GPa−1): dTc 1.6, dPa

dTc 2.0, dPb

dTc −2.8. dPc

(2.14)

The crystal structure of the Bi2 Sr2 CaCu2 O8+x compound is close to the orthorhombic one and compressions along a and b axes both increase Tc , while compression along c axes decreases Tc . These results agree quite well with the values found in uniaxial stress experiments on single crystals by

2.7 Conclusion

49

Watanabe et al. [1341]: dTc /dPa,b 1.5, dTc /dPc −4.5 (K GPa−1 ). Under hydrostatic pressure, a large cancellation of the uniaxial pressure derivatives occurs, which results in a small pressure eﬀect on Tc . Interesting results were obtained for large Tc enhancement in underdoped LSCO ﬁlms under compressive epitaxial strain. As shown by Bozovic et al. [173], the underdoped LSCO ﬁlms are extremely sensitive to oxygen intake and, therefore, a strong increase of Tc observed in some experiments in these ﬁlms may be caused by oxygen doping. However, they obtained the record Tc = 51 K for LSCO ﬁlm on the LaSrAlO4 substrates which produce compressive epitaxial strains. Summarizing the high-pressure investigations, we can draw the general conclusion that, even for an optimally doped cuprate superconductor, Tc can be enhanced by compression of the Cu–O bond length in the plane, which is an intrinsic pressure eﬀect. As pointed out by Schilling [1118], for optimally doped cuprates Tc is approximately proportional to the inverse square of the area of the CuO2 plane: d ln Tc /d ln a −4.5, where a is the lattice parameter in the plane. This value should be compared with the results for Hg-based compounds considered in the previous section, Figs. 2.18 and 2.20. Application of chemical pressure by ﬂuorination gives an order of magnitude stronger dependence: d ln Tc /d ln a −40. As explained above, under chemical pressure in Hg-1223, the Cu–O–Cu bond angle in the plane is not changed, while application of an external pressure causes the buckling of the CuO2 plane. The pressure eﬀects in cuprates should be also compared with those in electron– phonon superconductors, where, for instance, for MgB2 d ln Tc /d ln a +16 [1118]. Therefore, if we try to explain the large increase of Tc in cuprates with applied pressure within the BCS electron–phonon mechanism of pairing in (2.10), an extremely strong pressure dependence of the interaction should be invoked to overcome the negative contribution from the stiﬀening of the lattice. For an electronically driven pairing mechanism with an electronic energy as a prefactor in BCS-like formula (1.1), ω EF , weakly dependent on pressure and the interaction λ increasing with pressure (e.g., the antiferromagnetic exchange interaction), one can easily explain large increasing of Tc with compression. In Sect. 7.3.3, we consider the antiferromagnetic exchange pairing mechanism that gives a Tc (a) dependence close to experiments.

2.7 Conclusion We now emphasize the most important results obtained in the study of the crystal structure of copper-oxide superconductors: 1. The structure of all the copper-oxide superconductors is of a block nature. The main unit determining the metallic and superconducting properties of a compound is the CuO2 plane, which is a square lattice formed by copper ions bound to each other through oxygen ions. Each copper ion can additionally be coordinated to oxygen ions in the apex positions of

50

2 Crystal Structure

an octahedron; this gives rise to several possible coordinations of copper ions equal to four, ﬁve, and six. The eﬀective charge of the CuO2 complex is determined by the buﬀer blocks binding them in a crystal structure. Only structures with regular CuO2 planes show superconductivity, which appears in the range of the formal copper valence v equal to positive values 2.05 ≤ v ≤ 2.25 for hole-doped and 1.8 ≤ v ≤ 1.9 for electron-doped compounds. Superconductivity in complex copper-oxides with three-dimensional network of Cu–O bonds have not yet been detected. 2. The highest Tc was obtained for the mercury compounds due to the ideal square and ﬂat structure of CuO2 plane with the smallest buckling among the copper-oxide materials and the largest copper-apex oxygen distance. The maximal Tc,max in the cuprates increases under external pressure (contrary to the conventional electron–phonon superconductors) due to the compression of the Cu–O bond distance in the plane, d ln Tc /d ln a −4. An order of magnitude larger increase of Tc,max is observed in the mercury cuprates under chemical pressure, which preserves the Cu–O–Cu bond angle close to 180◦ . 3. However, the crystal structure of copper-oxide compounds does not completely determine their superconducting properties, which crucially depend on structural defects and short-range order. The structural instability and tendency to a short-range ordering were observed in the Bi- and Tlcompounds. The arrangement and local ordering of doping atoms in a lattice signiﬁcantly inﬂuence the superconducting properties of the system, as has been clearly proved in YBCO, due to the local nature of charge transfer from the buﬀer layer to the CuO2 plane. 4. A peculiar property of the hole-doped cuprates is the formation of incommensurate static or dynamic CDWs, which results in an inhomogeneous charge (and spin) distribution in crystals in the form of stripes. They were explicitly detected in the LTT phase of La-Ba-124 and La-Nd(Eu)-124 compounds at hole doping close to x = 1/8. The strong suppression of superconductivity at this “magic” hole concentration may be explained by a localization of charge carriers caused by the CDW formation.

3 Antiferromagnetism in Cuprate Superconductors

A generic feature of the copper–oxide compounds is the occurrence of strong antiferromagnetic (AF) copper spin correlations in the CuO2 planes. In the undoped compounds, the in-plane copper ions are in the state Cu2+ (3d9 ) with one hole of spin S = 1/2 in the 3d shell. A strong superexchange interaction (via oxygen ions) of the order of 1,500 K between the copper spins gives rise to a three-dimensional (3D) AF long-range order in La2 CuO4 and YBa2 Cu3 O6 compounds with high N´eel temperatures TN = 300–500 K (see the reviews by Birgeneau et al. [137], Rossat-Mignod et al. [1069], Johnston [560], Kastner et al. [590], Furrer [372], Bourges et al. [170], Tranquada [1264]). Although the long-range order disappears in the metallic and superconducting phases, strong short-range dynamical spin correlations are observed even at temperatures above 100 K. While the normal state spin ﬂuctuation spectra of YBCO, Tl– and Bi–copper–oxide compounds are rather broad, their superconducting phases are characterized by the presence of strong resonance peaks at energies close to 40 meV, at the AF wavevector. This fact resulted in a number of hypotheses on the possible electron pairing in copper–oxide compounds via magnetic degrees of freedom (Sect. 7.3.2). The two-dimensional (2D) character of the copper spin–1/2 interactions in the copper–oxygen planes results in strong quantum ﬂuctuations. In this respect, besides exploring the interplay of antiferromagnetism and superconductivity, the study of the quantum 2D antiferromagnet is of great interest in its own right (see [783]). The ﬁrst indications of the existence of AF order in the La2 CuO4 compound was obtained in macroscopic susceptibility measurements of by Johnston et al. [557, 558]. However, the determination of the spin structure in the N´eel state became possible only with the aid of neutron diﬀraction [351,843,1294]. Later on, a detailed information concerning both the magnetic structure and the dynamical spin correlations was obtained in neutron scattering studies. Since these experiments require large single crystals, most of the results have been obtained for the compounds La2 CuO4 and YBa2 Cu3 O6+x which can be synthesized as large crystals of high quality. Later on, large single crystals of Bi-2212 and Tl-2201 were prepared by assembling them

52

3 Antiferromagnetism in Cuprate Superconductors

from oriented small single crystals. It permitted to perform inelastic neutron scattering experiments and to observe the AF resonance peak in the superconducting phases of these compounds as in YBCO. In this chapter, the magnetic properties of LMCO and YBCO compounds: magnetic structure, spin correlations, and magnetic excitations, are discussed in detail. Spin dynamics in the superconducting phase is also responsible for variation in the spin–lattice relaxation rates of nuclear spins. The main results obtained by the nuclear magnetic resonance (NMR) and the nuclear quadrupole resonance (NQR) methods are presented at the end of the chapter.

3.1 Magnetic Neutron Scattering Experimental studies by means of neutron scattering play an essential role in the investigation of both the magnetic phase diagram and the spin dynamics of copper–oxide compounds. Later we give a short introduction to the theory of magnetic neutron scattering (for more detail see, e.g., [538, 792]). The differential cross section of the magnetic neutron scattering is directly coupled to the Fourier transform of the spin pair correlation function: d2 σ k = (γre )2 |F (q)e−W (q) |2 (δαβ − qˆα qˆβ )S αβ (q, ω), dΩdE k0

(3.1)

αβ

where (γre )2 0.29 · 10−24 cm2 is the magnetic cross section (γ = −1.9, re = e2 /me c2 ), F (q) is the (anisotropic) magnetic form factor1 of an atom with spin S, and exp[−W (q)] is the Debye–Waller factor. We denote the incoming and the outgoing neutron momentum and energy by k0 and k, and by E0 and E, respectively. They deﬁne the momentum q = k0 − k and the energy h ¯ ω = E0 − E transferred to the system by the scattered neutron. The angular dependence of the magnetic scattering is deﬁned by the unit vector ˆ = q/ | q |. The Fourier transform of the spin pair correlation function q ∞ 1 S αβ (q, ω) = dte−iωt S α (−q, 0)S β (q, t), (3.2) 2π¯ hN −∞ determines the space–time ﬂuctuations of the spin density eiq·n Snα (t), S α (q, t) =

(3.3)

n

where the summation is performed over all N lattice sites n. For a system with long-range magnetic order, for example, AF ordering with a wave vector qAF ≡ Q, the expectation value of the spin (3.3) is diﬀerent 1

The q-dependence of the form factor F (q) for the Cu3dx−y2 electron state was considered by Shamoto et al. [1138].

3.1 Magnetic Neutron Scattering

from zero:

S α (q, t) = N Δ(q − Q)Sα .

53

(3.4)

This gives rise to magnetic Bragg peaks in the cross section (3.1) dσ (1 − qˆα )2 Sα2 , ∝ N Δ(q − Q)δ(ω) dΩdE α

(3.5)

where the function Δ(k) equals unity if k is zero or a reciprocal-lattice vector, and equals zero otherwise. By measuring the dependence of the cross-section intensity on the direction of the wavevector qα one can determine the direction of spin ordering. When integrated over all scattering energies (at ﬁxed q), the function (3.2) determines a static spin correlation function 1 iq·(m−n) α α αα e Sn Sm . (3.6) S (q) = d(¯hω)S αα (q, ω) = N m,n In the paramagnetic phase, at T > TN , the spin correlations decay exponentially as the spin separation r increases S α (0)S α (r) ∝

1 −κr e . r

(3.7)

In this case, the scattering cross section is speciﬁed by the function S αα (q) ∝

q˜2

1 . + κ2

(3.8)

˜ = q − Q enables Its measurement at various values of the scattering vector q one to determine the AF correlation length ξ = 1/κ. It is convenient to perform the measurement of the integral scattering intensity (3.6) with the aid of a two-axis spectrometer, where only the direction of outgoing neutrons is ﬁxed. This has the eﬀect of integrating over the scattering energy ω. Inelastic magnetic scattering study by using a three-axis spectrometer enables one to measure the spin-ﬂuctuation energy spectrum. Generally, the spectrum of spin-1 excitations is determined by the dynamical spin susceptibility − ω , (3.9) χ+− (q, ω) = −Sq+ |S−q written in terms of the thermodynamic retarded Green function related to the operator of spin-density ﬂuctuations Sq± = Sqx ± iSqy (see Appendix A.1). The dynamical structure factor (3.2) is related to the dynamical spin susceptibility by the equation S αβ (q, ω) = [1 + n(ω)]

1 Imχαβ (q, ω + i0+ ), π

(3.10)

where n(ω) = [exp(¯ hω/kB T ) − 1]−1 is the Bose function. For the neutron scattering with creation of a magnetic excitation, E < E0 , the energy transfer

54

3 Antiferromagnetism in Cuprate Superconductors

¯ ω > 0 and the scattering intensity (3.10) is proportional to [1 + n(ω)], which h is nonzero even for T → 0. For the scattering with annihilation of a magnetic excitation E > E0 , the energy transfer ω < 0 and the scattering intensity is proportional to the average number of the excitations n(|ω|). In experiments, the dynamical spin susceptibility is usually measured in (gμB )2 units where g 2.2 is the Land´e factor for copper spins. In that case in the relation (3.10) one should write χ(q, ω)/(gμB )2 . In a paramagnetic state, the spin susceptibility is isotropic and summation over the components in (3.1) gives: (δαβ − qˆα qˆβ )S αβ (q, ω) = 2S αα (q, ω). αβ

Therefore, the scattering cross section (3.1) for an isotropic system is k d2 σ 2 = (γre )2 |F (q)e−W (q) |2 [1 + n(ω)] Imχαα (q, ω). dΩdE k0 π

(3.11)

The total spectral weight of magnetic scattering is restricted by the momentum and energy sum rule for the spin pair correlation function: 1 +∞ Snα Snα = d(¯hω)S αα (q, ω) = S(S + 1), (3.12) N −∞ α α q which gives the sum rule for the dynamical spin susceptibility for spin S = 1/2: hω ¯ 1 1 1 +∞ 1 d(¯hω) coth Imχαα (q, ω) = S(S + 1) = , N q π 0 kB T 3 4

(3.13)

This sum rule is useful for the estimation of contributions from diﬀerent parts of momentum or (and) energy spectrum of spin ﬂuctuations to the spin scattering intensity. The real, Reχ(q, ω), and imaginary, Imχ(q, ω), parts of the dynamical spin susceptibility are even and odd functions of ω, respectively, and they are coupled by the dispersion relation (see A.6) 1 ∞ Imχ(q, ω ) . (3.14) dω Reχ(q, ω) = π −∞ ω − ω In particular, the static spin susceptibility is determined by the equation 2 ∞ Imχ(q, ω ) dω , (3.15) χ(q) = Reχ(q, ω = 0) = π 0 ω which can be used in the evaluation of the static spin susceptibility from inelastic magnetic neutron scattering.

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

55

3.2 Antiferromagnetism in La2−xMxCuO4 Compound 3.2.1 Magnetic Phase Diagram The stoichiometric undoped La2 CuO4 (LCO) compound is an antiferromagnet with N´eel temperature TN 325 K. The phase diagram of La2−x Mx CuO4 in Fig. 2.4 shows that the AF state occurs in the orthorhombic (LTO) phase (Sect. 2.2). Figure 3.1a shows the tetragonal unit cell of La2 CuO4 in the AF phase [589]. The Cmac orthorhombic unit cells are determined by the lattice parameters a 5.34, b 13.1, and c 5.42 ˚ A at 5 K. The spins S = 1/2 of the Cu2+ ions are directed along the orthorhombic c-axis and the AF modulation is along the a-axis with the wavevector QAF = (1, 0, 0). Here and below we use the notation for the wavevectors in the orthorhombic Brillouin zone, Fig. 2.3 (right panel), with the reciprocal-lattice vectors a∗ = 2π/a = 1.17 ˚ A−1 , b∗ = 2π/b = 0.48 ˚ A−1 , and c∗ a∗ . Neutron diﬀraction measurements show that the ordered magnetic moment in stoichiometric LCO compound at a copper site is μ 0.6μB [137]. For the copper ion Cu2+ with spin S = 1/2, the ordered magnetic moment should be equal to μ = gSμB 1.1μB where the Land´e factor for Cu2+ ion g 2.2. The smaller observed value of the magnetic moment is due to quantum spin ﬂuctuations

a

b

La2CuO4 Sr2CuO2CI2

b

c

c

a a

Cu2+

Cu2+

O2–

O2–

O2– La3+

CI– Sr2+

b

Fig. 3.1. Crystal and magnetic structure of (a) La2 CuO4 (LCO) (after Kastner et al. [589]) and (b) Sr2 CuO2 Cl2 (SCOC) (reprinted with permission by APS from c 1998). The orthorhombic axes (a, b, c) shown for the Cmac Kastner et al. [590], (a) and Bmab (b) space groups. The spin conﬁguration of Cu2+ ions is indicated by black arrows. The tilting rotation for the central octahedron of oxygen ions in (a) is shown by open arrows

56

3 Antiferromagnetism in Cuprate Superconductors

and to a less extent, to the covalency of the Cu–O bond. In the case of a 2D Heisenberg magnet with spin S = 1/2, the quantum spin ﬂuctuations reduce the magnetic moment to 0.62 of its static value, i.e., to 0.68μB for Cu2+ [783]. This value is quite close to the experimentally observed one which proves that the quantum spin ﬂuctuations are very important for spin dynamics in LCO. The cupric oxychloride Sr2 CuO2 Cl2 (SCOC) reveals AF structures similar to LCO [1293]. The SCOC tetragonal (I4/mmm) crystal structure diﬀer from the LCO one with a replacement of the La2 O2 buﬀer layer by the Sr2 Cl2 layer. For a comparison, Fig. 3.1b shows the crystal and magnetic structure of the SCOC in the tetragonal phase with the Bmab orthorhombic unit cell speciﬁed by the vectors (a, b, c). Due to a perfect tetragonal structure, without distortions or modulations, the crystals represent an ideal 2D AF compound with a single CuO2 layer with only a small anisotropy and an extremely weak coupling between the layers due to frustration of the nearest-neighbor interlayer exchange. For SCOC, αxy 1.5 × 10−4 , αDM = 0, and J 125 meV in (3.17) and the N´eel temperature is TN 251 K. In spite of many eﬀorts to dope SCOC with carriers, the material did not show any superconductivity. By applying a high-pressure synthetic route, Hiroi et al. [461] managed to achieve superconductivity in the similar material Ca2 CuO2 Cl2 (CCOC). The Na-doped Ca2−x Nax CuO2 Cl2 (Na–CCOC) compound becomes superconductive at x > 0.08 with a maximum Tc = 28 K at optimal doping, x ∼ 0.20. Single crystals of Na–CCOC are easily cleaved between the weakly coupled double Ca–Cl layers producing a clean surface which makes possible to use them in high-precision ARPES studies (see Sect. 5.2.2). A peculiar feature of the AF spin ordering in LCO is the occurrence of a weak ferromagnetic moment in the CuO2 planes which is directed perpendicularly to the plane, along the b-axis, and has opposite directions in neighboring planes [1237, 1238]. The ferromagnetic moment has a small value of μ1 = 2 · 10−3 μB per copper ion. It results from the canting of copper spins by a small angle ( 0.17◦) out of the basal (a, c) plane due to the Dzyaloshinsky–Moriya (DM) antisymmetric exchange interaction. In an external magnetic ﬁeld along the b-axis, a spin reorientation transition occurs. In this transition, the weak ferromagnetic moment changes its direction along the ﬁeld in those planes where it is directed opposite to the ﬁeld. It results from the rotation of the spins by 180◦ in the basal (a, c) plane so that the wavevector of the AF structure changes its direction from [100] (along the a-axis) to [001] (along the c-axis). This magnetic structure can be represented by Fig. 3.1a if the spin of the copper ion in the center of the cell is rotated in the opposite orientation so that it reaches the same direction as the spins in the (a, b) plane. Kastner et al. [589] have directly observed the disappearance of the magnetic Bragg peaks from the (100) plane and the appearance of a new Bragg peak (201) as the external magnetic ﬁeld exceeds the critical value, at H > Hcr . The critical magnetic ﬁeld Hcr (T ) depends on temperature, with Hcr 5 T for T → 0. Surprisingly, it was observed that in spite of the very small weak ferromagnetic moments, their ordering along b-axis at

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

57

H > Hcr (T ) was accompanied by a strong decrease (by a factor of two) of the resistance which points to a strong interaction of the holes with the spins [1237]. A strong spin–charge coupling was detected in electron-doped materials as well. The Pr2 CuO4 (PCO) compound has a noncollinear AF structure (TN = 250–285 K) in the tetragonal lattice (space group I4/mmm) with spin orientations along the [100] and [010] directions in the adjacent planes. Application of a strong magnetic ﬁeld in the plane results in a spin-ﬂop transition which converts the noncollinear structure into a collinear one with the spins aligned perpendicular to the magnetic ﬁeld. Measurements of the magnetoresistance in Ce-doped Pr–copper oxide compounds reveal an increase of both the in-plane, ρab , and the out-of-plane, ρc , resistivity up to 30% at the spinﬂop transition [695]. Such a strong change of the resistivity induced by the modiﬁcation of the spin structure in the adjacent CuO2 planes is consistent with a strong electron–spin coupling. Antiferromagnetic long-range order in the low-temperature tetragonal (LTT) structure (see Fig. 2.7) has been studied in the rare-earth (RE)-doped compounds, RE = Nd, Sm, and Eu, where the LTT phase is stabilized in a broad range of RE concentrations (see Sect. 2.2.1). It appears that the structural phase transition from LTO to LTT phase has no noticeable inﬂuence on the N´eel temperature TN and the weak ferromagnetism. In particular, neutron scattering studies by Keimer et al. [596] on La2−y Ndy CuO4 (LNCO) in the LTO2 (P ccn) phase, that is close to the LTT phase, revealed DM spin canting with Cu spins being perpendicular to the octahedron rotation axis. It was even found that DM interaction at low temperature, T ∼ 10 K, in the LNCO sample is larger than in the LTO phase of the LCO sample (see later). Microscopic calculations of super-exchange interaction gave controversial results. Koshibae et al. [648] predicted Cu-spin canting in the LTT phase, while Stein et al. [1187] claimed that the Cu spins should be directed along the octahedron rotation axis in the LTT phase without spin canting. The experimental ﬁnding of spin canting in Nd-doped compound was attributed by Stein et al. [1187] to the magnetic interaction between Nd and Cu spins. To exclude the latter interaction, the Eu-doped, La2−y−x Euy Srx CuO4 (LECO), compound was investigated where the LTT structure was stabilized in a broad range of Sr concentrations (see, e.g., [502, 617]). It appears that the N´eel temperature TN (x) dependence on the Sr concentration x < 0.02 in the Eu-doped compound is close to that in the pure LSCO-124 crystal and the phase transition LTO → LTT has no appreciable eﬀect on TN (x) [617]. Careful magnetization measurements by H¨ ucker et al. [502] have proved the existence of the DM antiferromagnetic phase with canted Cu spins directed perpendicular to the octahedron rotation axis in the LTT phase. Moreover, the canted DM-moment in Eu-doped crystals was about 50% larger than in the LTO phase of the pure LCO crystal, which was attributed to the larger octahedron tilt angle in the LECO sample. However, the spin-ﬂip transition

58

3 Antiferromagnetism in Cuprate Superconductors

observed at H > Hcr was not well deﬁned due to the frustrated character of interlayer coupling in the LTT phase. The temperature of the AF phase transition in the doped compound, La2−x Mx CuO4−y , turns out to be very sensitive both to the concentration of divalent ions M = (Ba, Sr), that replace the trivalent La ions, and to the concentration of oxygen vacancies y. The phase diagram in Fig. 2.4 shows that already at a small concentration x = 0.02, the long-range AF order disappears. Only a spin-glass phase, i.e., the phase of frozen spins at copper sites, remains in the region of low temperatures. This phase has been observed most distinctly in μSR experiments. In these experiments, the time evolution of the muon spin polarization is measured. This time evolution is described by the formula 1 2 1 2 Gz (t) = cos(γμ Bμ t + φ) exp − (γμ ΔBμ t) + exp(−λt), (3.16) 3 2 3 where γμ = 851.4 MHz/T is the gyromagnetic ratio of the muon. The average internal magnetic ﬁeld Bμ deﬁnes the oscillation frequency of the muon spin, while ΔBμ characterizes the ﬂuctuating transverse ﬁeld. The dynamic spin–lattice relaxation rate is given by the parameter λ. The μSR method is sensitive to spin ﬂuctuations within a time window of 10−9 –10−6 s. The magnetic phase diagram measured by Niedermayer et al. [897] by the μSR method is shown in Fig. 3.2 for La2−x Srx CuO4 (open symbols) and Y1−x Cax Ba2 Cu3 O6.02 (solid symbols). The upper panel (a) shows the hole concentration dependence of the N´eel temperature TN , the spin-glass transition temperature Tg , the freezing transition temperature of doped holes Tf , and the superconducting Tc . The panels (b) and (c) show the average magnetic ﬁeld and its rms deviation, respectively, at T < 1 K. The results imply that the long-range AF order (region I) disappears at p 0.02, while the oscillation which indicates the static magnetic ﬁeld Bμ at the muon site (at the time scale τ < 10−6 s) is observed in the regions II and III over a wide range of hole concentrations p < 0.09. In this region a spin-glass transition occurs at low temperature which extends in the superconducting phase at p > 0.06. It proves the coexistence of superconductivity and frozen AF correlations. A coexistence of AF short-range order and superconductivity below the spin freezing temperature Tf ∼ 10 K was found by the μSR method also in pure YBa2 Cu3 O6+x polycrystalline samples with 0.37 < x < 0.39 by Sanna et al. [1091]. Similar results were obtained in μSR studies of the phase diagram for the Zn-doped La2−x Srx CuO4 and Y-doped Bi-2212 systems by Panagopoulos et al. [957]. A critical slowing down of spin ﬂuctuations and disappearance of the spin-glass transition were observed. Both the temperature Tf of slowing down of spin ﬂuctuations (at the time scale ∼ 10−9 s) and the temperature Tg of the spin-glass freezing (at the time scale ∼ 10−6 s) vanishes at the hole concentration p = 0.19 which coincides with the opening of the normal state pseudogap (see Sect. 4.2.1). Since no long-range order in the normal

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

59

Fig. 3.2. (a) Doping dependence of TN , Tf , Tg , and Tc (see text), (b) the average local ﬁeld Bμ , and (c) its rms deviation ΔBμ measured by the μSR method in La2−x Srx CuO4 (open symbols) and Y1−x Cax Ba2 Cu3 O6.02 (solid symbols) (reprinted c 1998) with permission by APS from Niedermayer et al. [897],

state is observed at this critical concentration, an occurrence of a quantum critical (glass) transition at pcrit = 0.19 with onset of short-range magnetic correlations at p < pcrit has been suggested. Thus, the authors proposed to divide the phase diagram of cuprates into two distinct regions: underdoped region below pcrit of weak superconductivity with nonzero temperatures Tf and Tg , and the overdoped region above pcrit of strong superconductivity with suppressed spin correlations. 3.2.2 Microscopic Models To describe the magnetic phase diagram of LCO, a Heisenberg model with antisymmetric exchange interaction of the Dzyaloshinsky–Moriya (DM) type has been proposed by Thio et al. [1237]. It is convenient to write the model in the form: b c b c H=J Si Sj − αxy Sib Sjb + αij (3.17) DM (Si Sj − Sj Si ) + α⊥ Si Sj ,

60

3 Antiferromagnetism in Cuprate Superconductors

where < i, j > denotes summation over pairs of nearest-neighbor spins coupled by the superexchange interaction Jnn = J. Spin–orbit coupling results in an exchange anisotropy of XY symmetry (easy plane) which is given by the parameter αxy = (J − J bb )/J > 0. The rotation of the CuO6 octahedron in the orthorhombic phase of LCO allows an antisymmetric exchange interaction ij J αij DM of the DM type, which is given by the parameter αDM = |αDM | = J bc /J. For a small value of the DM interaction the spins are in the (b, c) plane canting by a small angle θ αDM /2 out of the basal (a, c) plane (in the Cmac phase notation, Fig. 3.1a). The last term α⊥ J = J⊥ gives the interlayer AF coupling between the nearest neighbors along the b-axis. The largest nearest neighbor exchange interaction J produces strong AF correlations in the plane, while the AF interlayer exchange interaction and an anisotropy brings about the 3D long-range AF order as shown in Fig. 3.1. A microscopic derivation of the Heisenberg model (3.17), in particular the DM interaction, can be found in a number of theoretical works, as e.g., Coﬀey et al. [242], Koshibae et al. [648], and Stein et al. [1187]. In the early experiments, the following value of the exchange interaction was obtained: Jnn 116 meV, while the anisotropic parameters appeared to be small: αxy ∼ 4 × 10−5, αDM ∼ 5 × 10−3 [137]. The average value of the interplane coupling was estimated as J⊥ 0.002 meV. The values of the parameters show that the system of copper spins in La2 CuO4 is well described by the 2D Heisenberg model with a small anisotropy. Later on, these parameters were inferred more accurately by means of inelastic neutron scattering in studies of the spin-wave spectrum. In fact, the exchange AF interaction J in the plane and the anisotropic parameters can be deduced from the dispersion of 2D AF spin-waves. For the isotropic 2D Heisenberg model the energy of AF spin excitations is given by the equation (see, e.g., [783]): ω(k) = 4SZc J 1 − γ 2 (k), where spin S = 1/2, the quantum correction parameter Zc 1.17, and γ(k) = (1/2)(cos kx a + cos ky a) (a is the square lattice constant in the plane). In the isotropic 2D Heisenberg model there are no gaps neither at the Brillouin zone (BZ) center, k = 0, nor at the AF wave vector, k = QAF = (π, π). Close to these points, the excitation energy reveals √ a linear dispersion ω(q) = cq which deﬁnes the spin velocity c = 2SZc J 2a. The anisotropy terms in the model (3.17) bring about gaps in the spin-wave spectrum which depend on the polarization. For the wave vector k close to the AF wavevector, k = QAF +q, the out-of-plane and in-plane polarized spin waves have the following dispersion (see, e.g., [242]): ωout (q ) = 4SZc J 2αxy + (1/2)(aq )2 , ωin (q ) = 4SZc J

α2DM + (1/2)(aq )2 ,

(3.18)

where q2 = qx2 + qy2 . The early measurements by Peters et al. [978] gave rather small values for the gaps in a crystal with low N´eel temperature, TN = 195 K, apparently due to excess oxygen. Measurements of the spin gaps

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

61

in the deoxygenated LCO compound (TN = 325 K) and the LNCO compound close to the LTT phase were reported by Keimer et al. [596]. It has been found that the gaps, both for the LCO and LNCO compounds, are close at temperature T = 100 K. At low temperature T = 10 K the out-of-plain gaps Δout = 2Zc J 2αxy are also comparable: Δout 5.0 ± 0.8 (6.0 ± 0.8) meV for the LCO (LNCO) compound. The in-plane gaps Δin = 2Zc JαDM are diﬀerent: Δin 2.3 ± 0.5 meV and Δin 4.5 ± 0.5 meV for the LCO and LNCO compounds, respectively. Since the in-plane gap is proportional to αDM , the larger value in the LNCO compound should be due to a larger tilting angle as observed in LECO compound [502]. The corresponding estimates for the anisotropic parameters of the LCO compounds with the exchange interaction J 135 meV are: αxy = 1.3 × 10−4 , and αDM = 7.3 × 10−3, which are larger than the one measured earlier [137]. The parameters determined from magnetic susceptibility measurements by Thio et al. [1239] are the following: J = 110 meV, J bc 1 meV, and J⊥ = 1.6 µeV. In a number of neutron scattering experiments the velocity of AF spin-wave √ c = Zc J 2 a was measured with high precision. In Table 3.1, we present the results a systematic inelastic neutron scattering study by Bourges et al. [168] for the spin velocity c, the AF exchange interaction J, the N´eel temperature TN , and the 2D momentum averaged spin susceptibility χ˜2D for the three monolayer cuprates. By comparing these values, the authors concluded that the exchange interaction does not exhibit a monotonous behavior versus the Cu–O–Cu bond distance but extremely sensitive to the bonding angle. For instance, the La2 CuO4 has a smaller lattice constant in comparison with the Nd2 CuO4 crystal but a lower value of J. A distortion of the CuO2 plane (“buckling”) in the LTO phase of LCO results in deviation of the Cu–O–Cu bonding angle from 180◦ . The Nd2 CuO4 compound has the T’ phase with the linear Cu–O– Cu bonding (see Fig. 2.10) and thus the higher values of J. The low value of J in Pr2 CuO4 (also having T’ phase) can be explained by its large bonding distance. The 2D momentum averaged (local) spin susceptibility for the mea

sured spin-wave spectrum χ ˜2D = dq2D Imχ(q, ω)/ dq2D appears to be almost energy independent. Its theoretical estimation by the function χ ˜2D = S(gμB )2 Zχ /2J ∼ μ2B /2J, where Zχ = 0.51 is the quantum correction factor Table 3.1. Spin velocity c, exchange interaction J, and the N´eel temperature TN in monolayer cuprates [168]

c (eV˚ A) ± 0.02 J (meV) ± 3 TN (K) χ ˜2D (μ2B /eV) ± 0.4

La2 CuO4

Nd2 CuO4

Pr2 CuO4

0.85 133 320 2.7

1.02 155 243 1.8

0.80 121 247 2.3

62

3 Antiferromagnetism in Cuprate Superconductors

for 2D Heisenberg model [783], is about two times larger than the experimental values in Table 3.1. High-resolution inelastic neutron scattering by Coldea et al. [247] revealed a substantial interaction beyond the nearest-neighbor Heisenberg model (3.17). In Fig. 3.3a the spin-wave dispersion along the symmetry direction in 2D BZ (inset C) are shown at T = 10 K (open symbols) and at T = 295 K (solid symbols). Figure 3.3b demonstrates wave vector dependence of the spin-wave intensity at T = 295 K. Along the 2D antiferromagnetic BZ, given by the equation | cos kx a + cos ky a| = 0, the energy of spin-wave excitation should

Fig. 3.3. (a) Dispersion relation for the 2D spin-waves in La2 CuO4 along the symmetry directions, shown in the inset (c), at T = 10 K (open symbols) and at T = 295 K (solid symbols). (b) wave vector dependence of the spin-wave intensity c 2001) at T = 295 K (reprinted with permission by APS from Coldea et al. [247],

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

63

be constant, ω(kx , ky = kx − π) = Zc 2J if one takes into account only the nearest-neighbor exchange interaction J. However, Coldea et al. [247] observed quite a noticeable dispersion along the AF BZ which demanded an introduction of interaction beyond the nearest-neighbor one. The result of the ﬁts to the dispersion relation at T = 10 K (T = 295 K) is shown in Fig. 3.3 by the solid (dashed) lines. The nearest-neighbor and the second-neighbor exchange interactions, extracted from the dispersion curves, are J = 112 ± 4 meV, J = −11.4 ± 3 meV at T = 295 K and J = 104 ± 4 meV, J = −18 ± 3 meV at T = 10 K. While the nearest-neighbor interaction J is antiferromagnetic, the second-neighbor exchange interaction J , across the diagonal of square plaquette for four Cu2+ -spins, is ferromagnetic, which contradicts theoretical predictions. To avoid this diﬃculty, the authors proposed to take into account a ring exchange interaction coupling of four Cu2+ spins at a square plaquette (clockwise) which is described by the Hamiltonian Hc = Jc {(Si Sj )(Sk Sl ) + (Si Sl )(Sk Sj ) − (Si Sk )(Sj Sl )}. (3.19)

To calculate the parameters of the exchange interaction, one can consider the Hubbard model speciﬁed by a hopping energy t between the nearest-neighbor Cu sites and the single-site Coulomb energy U . By applying a perturbation expansion in terms of the small parameter (t/U ), we get the nearest-neighbor exchange interaction, Jnn ∝ t2 /U , and three higher-order spin coupling terms, J , J , and Jc ∝ t4 /U (see, e.g., [763, 1212]). By ﬁtting the dispersion and intensities of the spin-wave excitations the authors obtained the following values of the corresponding parameters: J = 138±4 meV, Jc = 38±8 meV, and J = J = 2 ± 0.5 meV (at T = 295 K). It appears that the relative magnitude of the cyclic exchange interaction is quite large, Jc /J 0.3, and close to the estimates given by other methods [247]. 3.2.3 Theory of Magnetic Phase Transitions The N´ eel Transition Usually, the theoretical description of magnetic properties in undoped copper– oxide compounds, in particular, the N´eel transition, is based on the consideration of a spin-1/2 2D quantum Heisenberg model (2DQHM) [590, 783]. This approach is justiﬁed, if we take into account that the values of the parameters of the Heisenberg model (3.17) speciﬁed in the previous section ﬁt experimental data. A very strong nearest-neighbor exchange interaction J ∼ 1, 500 K, small anisotropy ∼ 10−4 J, and extremely weak interlayer coupling J⊥ ∼ 10−5 J show that in the ﬁrst approximation we should take into account only the strong nearest-neighbor exchange interaction in a single copper–oxygen layer where quantum ﬂuctuations due to low dimensionality

64

3 Antiferromagnetism in Cuprate Superconductors

and low spin are essential. Therefore, later we present the results of theoretical calculations within the 2DQHM relevant for further discussion of neutron scattering experiments in cuprates. Despite great eﬀorts, the exact solution of the ground state energy of 2DQHM for spin 1/2 was not found. At ﬁnite temperature, according to Mermin et al. [824], irrespective of the spin value, a 2D isotropic Heisenberg model cannot exhibit long-range order due to the gapless spin excitations. Motivated by the lack of rigorous analytical results, extensive numerical calculations for ﬁnite lattices and studies by the renormalization-group technique have been undertaken. They have shown that the spin-1/2 2DQHM exhibits an AF longrange order at zero temperature over a certain range of the eﬀective coupling constant. The most convincing results were obtained for the quantum nonlinear 2D sigma model (QNLSM) on which the 2DQHM may be mapped in the longwavelength limit. In particular, Chakravarty et al. [217, 218] (see also [433]) have shown that the renormalization-group equations for the QNLSM at T = 0 have a nontrivial ﬁxed point for an eﬀective coupling constant g = gc which describes the quantum phase transition with the critical indices of the classical 3D Heisenberg model. The N´eel order exists for the coupling constant g < gc , where the renormalized classical behavior is observed. In this region the correlation length diverges exponentially as T → 0: ξ(T ) kB T 2π ρs Cξ exp 1+O , (3.20) a kB T J where a is the lattice constant, Cξ = (e/8)(¯hc/a)/(2πρs ), ρs a spin stiﬀness constant (a parameter of the QNLSM), and c is the spin-wave velocity. For the spin-1/2 2DQHM these parameters are: ¯hc 1.66Ja, 2πρs 1.13J, and Cξ 0.5 [117]. In the classical Heisenberg model, the correlation length also diverges exponentially: ξ(T ) ∝ exp (2πJ/kB T ), but quantum ﬂuctuations strongly renormalize the exponential factor: J → ρs 0.21J which makes the correlation length much shorter, though it is still very large at low temperatures. Quantum ﬂuctuations in 2DQHM for spin 1/2 below 3D N´eel temperature TN can be estimated within the renormalized spin-wave theory [783]. They are signiﬁcant in the suppression of the staggered magnetization: m∗ 0.62m and give a noticeable renormalization of the spin-wave velocity: c∗ = Zc c, Zc 1.17. The disordered quantum phase with excitation gap (quantum paramagnet) arises for g > gc . In the disordered quantum region, the correlation length becomes independent of temperature and is given by −ν ξ(T = 0) g −1 , (3.21) a gc where ν 1. For g = gc (quantum critical regime) the N´eel order disappears even for zero temperature, while the inverse magnetic correlation length

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

65

ξ(T )−1 ∝ T for all temperatures. An estimate of the coupling constant for La2 CuO4 gives the following value: g/gc 0.685, which shows that this compound should be in the regime of renormalized classical critical ﬂuctuation of the N´eel order parameter for T > TN . To explain a quite large ﬁnite N´eel temperature TN = 325 K in the undoped La2 CuO4 compound within the 2DQHM one should take into account small anisotropy and weak interlayer coupling in the Heisenberg model (3.17) which describes the real LCO compound. These weak interactions become increasingly important as the correlation length grows with lowering temperature. The long-range AF order could occur when the number of spins within the correlation length in the plane (ξ/a)2 multiplied by the anisotropic and interlayer coupling equals kB TN [590]: kB TN = (ξ/a)2 J(z αxy + z⊥ α⊥ ) = (ξ/a)2 Jαeﬀ ,

(3.22)

where z and z⊥ are the in-plane and out-of-plane coordination numbers, respectively. The mean-ﬁeld estimate (3.22) for the N´eel temperature: kB TN 4πs / ln(1/αeﬀ ), for αeﬀ 10−4 gives a reliable value for cuprate materials. The weak logarithmic dependence of the N´eel temperature on the interlayer coupling results in close values TN in monolayer cuprate compounds. In the tetragonal materials, such as Sr2 CuO2 Cl2 , Sm2 CuO4 , and Pr2 CuO4 , the N´eel temperatures are lower than that in the orthorhombic phase of La2 CuO4 due to frustration of the interlayer AF exchange interaction α⊥ J (see Table 3.1). Phenomenological Theory Studies of the AF phase transition in La2 CuO4 and in the related compounds La2 NiO4 and La2 CoO4 (see [137]) show that the magnetic ordering of spins is connected with structural transitions. In view of this, we consider a phenomenological theory of magnetic phase transitions taking into account the theory of structural transitions discussed in Sect. 2.2.1 [1001, 1004]. The phenomenological theory is based on the symmetry analysis and does not use any microscopic model. The mean-ﬁeld theory of magnetic phase transition in La2 CuO4 was given by Thio et al. [1237, 1239]. As shown in Sect. 2.2.1, the structural transitions from the hightemperature tetragonal (I4/mmm – HTT) to orthorhombic (Cmac – LTO) and low-temperature tetragonal (P 42 /ncm – LTT) phases are speciﬁed by a two-component order parameter (C1 , C2 ) related to the irreducible representation X3+ on the two-arm star of the wavevector kx (1, 2) (2.2). The order parameter describes the condensation of the soft rotational mode (2.1) R1,2 = Rx ∓ Ry for the wavevectors kx (1, 2), respectively. The AF phase transition in La2 CuO4 is speciﬁed by a two-component order parameter (S1 , S2 ) related to freezing-in of spin ﬂuctuations at copper sites S1 [kx (1)] ∝ (Sx − Sy ),

S2 [kx (2)] ∝ (Sx + Sy ),

(3.23)

66

3 Antiferromagnetism in Cuprate Superconductors

which also have symmetry X3 . The weak ferromagnetic moment in the CuO2 planes is speciﬁed by a secondary-order parameter, namely, the spin component Sz at the copper sites with the wavevector kz = π(0, 0, 1/τz ) in the notation of Sect. 2.2.1. A symmetry analysis of the spin–lattice interaction in the tetragonal phase, which is chosen as the basic phase in describing the sequence of structural and magnetic phase transitions, enables one to write down a complete expansion for the free energy in the order parameters [1001, 1004]: F = Fc + F + Fc + Fs + Fs + Fsc .

(3.24)

Here, the ﬁrst three terms are the lattice part of the free energy. They are determined by the expansion (2.3)–(2.5). The magnetic term Fs and the magneto–deformation part Fs have the same functional form as the expressions (2.3) and (2.5) with the substitution Ci → Si and by taking into account the secondary-order parameter Sz also. For example, Fs =

1 1 1 1 rN (S12 + S22 ) + uN S12 S22 + vN (S14 + S24 ) + rz Sz2 , 2 2 4 2

(3.25)

where rN = aN (T − TN ) and rz speciﬁes the anisotropic interaction which does not depend on the temperature. The interaction of the magnetic and structural order parameters is described by the expansion Fsc = (αs (S12 + S22 ) + βs Sz2 )(C12 + C22 ) + γs (S12 − S22 )(C12 − C22 ) + λ(C1 S2 Sz − C2 S1 Sz ),

(3.26)

where the mixed invariant of the third-order occurs since the sum of the wavevectors kx (1) + kx (2) + kz is equal to the reciprocal lattice vector b1 (see (2.2)). This invariant is directly related to the antisymmetric Dzyaloshinsky– Moriya interaction in the microscopic model (3.17), J bc ∝ λC1 or λC2 . The analysis of the complete expansion of the free energy (3.24) enables one to study the phase diagram in the space of the order parameters of the structural and magnetic phase transitions. In particular, in the LTO phase at T < TN < T0 , the solution C1 = 0, S2 = 0, Sz = 0 at C2 = S1 = 0 is possible for one domain. A similar solution under the interchange of indices 1 and 2 is possible for the second domain. This solution describes a noncollinear AF structure in La2 CuO4 , where the direction of the spin-order parameter S2 (S1 ) is perpendicular to the wavevector kx (2)(kx (1)) (see (3.23)) and the axis of rotation of the octahedron R1 (R2 ) (see (2.1)). To deﬁne the weak ferromagnetic moment Sz perpendicular to the plane, we consider the DM energy: 1 FDM = r˜z Sz2 + λ(C1 S2 Sz − C2 S1 Sz ), (3.27) 2 where r˜z = rz + 2βs (C12 + C22 ) is the renormalized anisotropy parameter. The equilibrium value of the ferromagnetic moment Sz is deﬁned from the condition ∂FDM /∂Sz = 0. In the LTO phase, under nonvanishing-order parameters

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

67

C1 = 0, S2 = 0, we get the solution: Sz = −

λ C1 S2 , r˜z

FDM = −

λ2 (C1 S2 )2 . 2˜ rz

(3.28)

The nonzero solution for the ferromagnetic moment Sz decreases the free energy due to DM interaction. The moment Sz points in opposite directions for neighboring CuO2 planes, since it is determined by the wavevector kz = π(0, 0, 1/τz ). In the LTT phase for the domain with the structural-order parameter C1 = C2 = C ∝ Rx = 0(Ry = 0) we have two solutions for the spin-order parameter: S1 = S2 ∝ Sx = 0 (Sy = 0) or S1 = −S2 ∝ Sy = 0 (Sx = 0). The ﬁrst solution for the AF spin Sx = 0 along the rotation axis Rx gives zero equilibrium value of the ferromagnetic moment: Sz ∝ λRx (S1 − S2 ) = 0 and no contribution for the DM energy FDM (3.27). The second solution for the AF spin S1 ∝ Sy = 0 gives nonzero solution for Sz : Sz = −λ

1 Rx S y , r˜z

FDM = −

λ2 (Rx Sy )2 . 2˜ rz

Therefore, the solution with the AF spin directed perpendicular to the rotation axis in the LTT phase and the corresponding nonzero weak ferromagnetic moment Sz will give the lower free energy for r˜z > 0. We would like to point out that this conclusion is based on the phenomenological theory which does not depend on a speciﬁc microscopic model. Both the neutron diﬀraction studies by Keimer et al. [596] and the magnetization measurements by H¨ ucker et al. [502] have revealed the Dzyaloshinsky–Moriya AF phase with canted Cu spins which are directed perpendicular to the octahedron rotation axes in the LTT phase. The magnetic ordering of spins in La2 NiO4 and La2 CoO4 is related to another irreducible representation X5 , the pseudovector basis functions of which are transformed as spin components S1 [kx (1)] ∝ (Sx + Sy ),

S2 [kx (2)] ∝ (Sx − Sy ).

(3.29)

In this case, the direction of the spins S1 (S2 ) coincides with that of the wavevector kx (1) (kx (2)). The expansion of the free energy in terms of the order parameters (S1 , S2 ) has the same form as (3.25) and (3.26) except for the invariant of the third order in (3.26) which becomes F3 = λ(C1 S1 Mz − C2 S2 Mz ),

(3.30)

where Mz is a ferromagnetic moment along the tetragonal axis z. The collinear AF structure in La2 NiO4 and La2 CoO4 in the orthorhombic phase is described by the solution S2 = 0, S1 = Mz = 0 for the domain C1 = 0, C2 = 0 and a similar solution (under permutation of indices 1 and 2) for the second domain. The magnetic structure of La2 CuO4 in an external magnetic ﬁeld Hz > Hcr due to a spin-reorientation transition is also described by the basis

68

3 Antiferromagnetism in Cuprate Superconductors

functions (3.29). Under the spin-reorientation transition in the orthorhombic phase, the AF structure with order parameter S2 = 0, Sz = 0 for domain C1 = 0 transforms into a structure with the order parameters S1 = 0, Mz = 0 when the energy of the external ﬁeld Mz Hz becomes larger than the energy diﬀerence F (Si ) − F (Si ). The AF vector kx (2) for the parameter S2 (3.23) transforms into the AF vector kx (1) for the parameter S1 (3.29), while the weak ferromagnetic moment in the plane Sz ∝ (λ/rz )C1 S2 transforms into the homogeneous moment Mz ∝ (λ/rz )C1 S1 . The above phenomenological theory also allows one to analyze possible magnetic structures in the LTT phase P 42 /ncm of La2 CoO4 [137,1001,1004]. 3.2.4 Spin Dynamics Nonsuperconducting State, x < 0.05 In view of the 2D character of the spin interaction in the model (3.17) for La2 CuO4 , quasi-2D spin correlations should be expected in this compound. In this case, the diﬀuse scattering intensity described by the quantity S αα (q) (3.6) should have the form of rods connecting the reciprocal lattice sites in the (a∗ , c∗ ) plane along the b∗ direction, perpendicular to the plane (Cmac notation, see Fig. 3.1a). In fact, this pattern has been already observed in early experiments [137], which reveled for T > TN a novel 2D magnetic state labeled as a quantum spin-liquid (QSL) state [1158]. In these experiments, a special scattering geometry was chosen, where the momentum of scattered neutrons k was parallel to the reciprocal lattice vector b∗ : k = (0, ζ , 0). In this case, for a ﬁxed momentum vector of the incoming neutrons k0 = (ν, ζ, 0) in the plane (a∗ , b∗ ) of the reciprocal space, the momentum transfer q = k0 − k = (ν, ζ − ζ , 0) has a constant value ν for the component q in the plane (a∗ , c∗ ) independent of the value of k. This means that in experiments with a two-axis spectrometer, where only the direction but not the energy of the scattered neutrons is ﬁxed, the eﬀective energy integration is carried out. Therefore in this geometry, it is possible to determine the correlation function S αα (q ) (3.6) describing the 2D spin correlations in the CuO2 plane. A detailed study of the diﬀuse magnetic scattering was carried out in a number of experiments [137, 590]. Measurement of the function S αα (q ) (3.6) in La2 CuO4 has shown that scattering from spin ﬂuctuations is observed far above the N´eel temperature. The magnetic spin correlation length ξ in the plane increases from 40 ˚ A at 500 K to 400 ˚ A as the temperature goes down to TN (see Fig. 3.4). The temperature dependence for the correlation length ξ (T ) in a wide temperature range from TN up to ∼ 800 K shows an excellent agreement with the theoretical prediction (3.20) for the renormalized classical spin ﬂuctuation behavior [139]. No evidence of the crossover to the quantum critical region was seen. These experiments explain the origin of the QSL state in La2 CuO4 .

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

69

Fig. 3.4. Temperature dependence of the inverse magnetic correlation length ξ −1 in La2−x Srx CuO4 for small concentrations of doped holes x ≤ 0.04 (after Keimer et al. [595])

The integral intensity of the diﬀuse scattering rises slightly as the temperature decreases and approaches TN . However, below the temperature of magnetic ordering, the intensity goes down gradually and tends to zero as T → 0. At the same time, the intensity of the magnetic Bragg peak (100) increases and approaches its maximum value at T = 0 [1158]. Thus, a transformation of the diﬀuse scattering into the Bragg peak takes place. This is observed in conventional 2D antiferromagnets, for example, in K2 NiF4 . In La2 CuO4 , however, this transformation is smooth, while in K2 NiF4 it is rather sharp and occurs close to TN over a small temperature interval of about 2% of TN . This reﬂects the fundamental diﬀerence in the nature of 3D ordering in these planar antiferromagnets. In K2 NiF4 , which is an Ising type magnet, the transformation to long-range order has essentially a 2D character. In La2 CuO4 , 3D long-range order is formed due to an interplane interaction. Below the N´eel temperature, well-deﬁned spin waves are observed (see Fig. 3.3). These are described by the Heisenberg model (3.17) (with a small correction (3.19)) as discussed in Sect. 3.2.2. In the isomorphic magnets La2 NiO4 and La2 CoO4 which possess atomic spins S = 1 for Ni and S = 3/2 for Co, respectively, the phase transition to long-range order is of the Ising type and is similar to that in K2 NiF4 . Let us now proceed to doped compounds La2−x Srx CuO4 . Figure 2.4 shows that Sr-doped compounds becomes metallic at x 0.05. In the region x ≤ 0.05 a spin-glass phase is observed (see Fig. 3.2). A comprehensive study of the evolution of the magnetic correlations from a pure La2 CuO4 to a weakly doped compound La2−x Srx CuO4 in the spin-glass regime was carried out by Keimer et al. [595]. This study revealed a crossover from a spin correlation region dominated by classical ﬂuctuations (3.20) at x = 0 to a region with a temperature-independent correlation length dominated by doped holes. The temperature dependence of the measured inverse correlation length was

70

3 Antiferromagnetism in Cuprate Superconductors

represented as a sum of two contributions: κ(x, T ) = κ(x, 0) + κ(0, T ).

(3.31)

Here the temperature-independent contribution κ(x, 0) corresponds to correlation lengths of 150, 65, and 42 ˚ A for x = 0.02, 0.03, 0.04, respectively, while κ(0, T ) is described by (3.20) for the correlation length with the ﬁtted value of 2π ρs = 150 meV. This simple formula provides a good description of the experimental data as shown in Fig. 3.4. For high temperatures, T > 300 K, all correlation lengths exhibit exponential temperature dependence characteristic for a pure system, while at lower temperature they are dominated by holes. This study also revealed quite a diﬀerent mechanism of suppression of the long-range N´eel order by impurities. A substitution of Cu2+ by Zn2+ (3d10 state with S = 0), which introduces static vacancies into the spin-1/2 system, shows a slow percolation type suppression of TN as in the electron-doped system Pr2−x Cex CuO4 [803]. A mobile hole doping by Sr2+ destroys the N´eel order much faster: this disappears at a very low concentration x 0.02. The strong inﬂuence of the doped holes at the oxygen sites has been explained by Aharony et al. [21] within a frustration model. In the model, a strong exchange interaction between the localized spin σ of an oxygen hole and two neighboring copper spins S1 and S2 , H = Jσ σ(S1 +S2 ), is assumed. Irrespective of the sign of Jσ , this coupling gives rise to an eﬀective ferromagnetic interaction of the spins S1 and S2 which frustrates the AF exchange Jnn in the Heisenberg model (3.17). An alternative mechanism of destroying the N´eel order proposed within the t-J model, where the doped holes deﬁne Zhang–Rice singlets [1419]. At low concentration, the holes create propagating spin–polaron quasiparticles which strongly perturb the AF background (see Sect. 7.2.1). Actually, the hopping kinetic energy of the bare holes ∼ t is much larger than the exchange interaction Jnn 0.3t. Therefore, delocalization of the holes accompanied by the destruction of the long-range AF order may turn out to be energetically more favorable. A certain compromise between the lowering of the hole kinetic energy and preserving the AF order can be achieved by the stripe formation as discussed later. Extensive inelastic neutron scattering measurements by Keimer et al. [595] on La1.96 Sr0.04 CuO4 showed a peculiar spin dynamics in the region of low energy, 0.75 < ω < 45 meV. They observed inelastic magnetic scattering close to the AF wavevector ((1, 0, 0) in the orthorhombic Cmac notation or (1/2, 1/2) in the square lattice notation). This observation was explained as an evidence for the short-range AF spin order with the low-temperature correlation length of ∼ 40 ˚ A. As we discuss in the next section, studies of highquality single crystals with much higher resolution by using the three-axis spectrometer have revealed incommensurate spin correlations (see Fig. 3.9). A remarkably simple scaling behavior of the q-integrated (local) spinﬂuctuation susceptibility χ (ω) was observed as shown in Fig. 3.5, where the

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

71

a

5

I(IωI, 0) (arb. units)

Fig. 3.5. Normalized q–integrated spin susceptibility as a function of (ω/T ). Various symbols refer to diﬀerent energies (2 meV≤ ω ≤45 meV). The solid curve is the function (3.32) (after Keimer et al. [595])

4

b

La1.96Sr0.04CuO4

Width (r.l.u.)

3 2 1 0 0

10

20 30 ω (meV)

40

50

0.06 0.04 0.02 0

0

10

20 30 40 ω (meV)

50

Fig. 3.6. (a) Energy dependence of the q-integrated intensity at T = 10 K. (b) The intrinsic width of the scattering proﬁle in reciprocal lattice units (r.l.u.) (after Keimer et al. [595])

solid curve is the ﬁtting function:

ω 3 2 ω + a2 χ (ω) = d2 qχ (q, ω) = I(|ω|, 0) arctan a1 , π T T

(3.32)

and a1 = 0.43, a2 = 10.5. The low-temperature intensity I(|ω|, 0) measured at T = 10 K is represented in Fig. 3.6a. For higher energy the width of the scattering proﬁle increases as shown in Fig. 3.6b, which reﬂects a crossover to propagating spin waves for ω = cq > 20 meV. One can expect that at short wave-lengths, of the order of the AF correlation length, the spin dynamics corresponding to high excitation energies should not be qualitatively diﬀerent from that of the undoped system. Contrary to the high-frequency region, the low-frequency spin dynamics is rather peculiar – at low temperature a narrow quasielastic peak appears which may stem from a transition to the spin-glass state. A canonical three dimensional spin-glass transition at 7 K was observed in La1.96 Sr0.4 CuO4 in

72

3 Antiferromagnetism in Cuprate Superconductors

magnetization measurement by Chou et al. [229]. The spin-glass transition in low-doped cuprates is clearly observed in μSR experiments (see Fig. 3.2) and NQR studies. Incommensurate Spin Correlations: Stripes A peculiar property of the AF spin correlations in the hole-doped cuprates is the appearance of incommensurate (IC) spin and charge density waves (SDW and CDW), or stripes, which have attracted much attention in the recent years. It was argued that the state of the electrons in cuprates cannot be described as a gas of quasiparticles within the Fermi-liquid theory but rather as a collective quantum state in the form of ﬂuctuating dynamical stripes [308,1400]. At present, the “stripology” is a well developed branch of research of cuprates and other systems with strong electron correlations as manganites (for review see, e.g., [211, 615, 616, 930, 1079, 1264]). Initially, the IC dynamical spin correlations were observed in inelastic neutron scattering experiments in LSCO compounds with Sr concentration x > xc 0.05 at the onset of superconductivity [138, 223, 802, 804, 1393]. In a systematic low-energy neutron-scattering study, Yamada et al. [1383] have shown that the low-energy inelastic peak at 2–3.5 meV at low temperatures shifted at x > 0.05 from the AF position (1/2, 1/2) to the IC wavevectors qδ = (1/2 ± δ, 1/2), (1/2, 1/2 ± δ) (in the square lattice notation). It was found that the incommensurability δ is proportional to the eﬀective doping charge xeﬀ . In the region of Sr concentration, 0.06 < x < 0.12 the incommensurability behaves like δ x but it saturates at larger x at the value δ 1/8. A surprising result pointed out by Yamada et al. [1383] was that the superconducting transition temperature scales linearly with δ up to the optimal doping xopt 0.15. Figure 3.7 shows the results of Yamada et al. [1383] and several later investigations [1327] for LSCO crystals. In the panel (a) the hole concentration dependence of the incommensurability δ is shown, while in the panel (b) the neutron-scattering linewidth κ (HWHM – half width at half maximum) of the magnetic peaks around (1/2, 1/2, −0.3) is given. The closed symbols represent the HWHM of the elastic components (in the region of x < 0.05 – the results of Keimer et al. [595], see Fig. 3.4). The open symbols are the results of Yamada et al. [1383]. The incommensurability δ depends only on the eﬀective hole concentration produced either by Sr2+ doping or oxygen reduction and is not altered by small concentration of Zn impurities which suppresses Tc . The IC peaks of the inelastic scattering are rather sharp, which means that the low frequency IC spin excitations have large dynamical coherence length. At the hole concentration close to x = 1/8 the dynamical peaks are extremely sharp, while in the overdoped region they become rather broad. The static spin correlation length ξ2D for x > 0.04 measured in early double-axis (energy integrating) experiments appeared to be quite short, of √ A) [1242]. the order of the hole spacing in the CuO2 plane: ξ2D = 3.8/ x(˚

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

73

b 0.08

a

K (Å–1)

La2-xSrxCuO4

δ (r.l.u.)

0.15 0.10

k tet

0 0

O

1/2 O O 2δ

0.05

O

0.06 0.04 0.02

1/ 2 h tet 0.05

0.10

0.15

0.20

Hole concentration x

0

0

0.05

0.10

0.15

0.20

Hole concentration x

Fig. 3.7. Hole concentration dependence of (a) the incommensurability δ of the spin ﬂuctuations, and (b) the neutron-scattering linewidth κ of the magnetic peaks in La2−x Srx CuO4 crystals. The open circles are the data of Yamada et al. [1383] and the closed symbols are elastic components (after Wakimoto et al. [1327])

In subsequent investigations of high-quality single crystals with much higher resolution narrow (quasi)elastic IC peaks were discovered which were inferred to originate in the IC static spin-density waves (SDW) with a large correlation length. As we discussed in Sect. 2.2.1, the anomalous suppression of superconducting Tc at the speciﬁc hole concentration x ∼ 1/8, observed at ﬁrst in LBCO compound in the LTT phase, may be explained by the occurrence of the IC static CDW and SDW (see Fig. 2.5). Earlier, a detailed neutron scattering study of IC CDW and SDW in the LTT phase of La1.6−x Nd0.4 Srx CuO4 (LNSCO) compound was performed by Tranquada et al. [1259–1261]. They proposed a model for cooperative spin and charge order in which doped holes (one for two copper sites) spatially segregate into 1D stripes which separate antiphase AF domains. The model for stripe order of holes and spins within the copper–oxygen layers in the LTT phase at x = 1/8 is shown in Fig. 3.8 [1260]. The holes are delocalized along the stripe and can jump perpendicular to the stripe between two AF antiphase domains. The magnetic peaks were observed at the superlattice point (1/2 − δ, 1/2, 0), while the charge-order peaks at (2 + 2δ, 0, 0) with δ = 0.118 ± 0.001 ∼ x. The doubling of the magnetic order modulation, λm = 2λch = a/δ is caused by the antiphase AF order in the spin domains. The IC modulation which occurs along the Cu–O bonds was called the parallel incommensurate modulation. With decreasing temperature, ﬁrst the LTO–LTT phase transition occurs near 70 K, then the charge-order peaks appear at ∼ 60 K and ﬁnally, the magnetic peaks emerge at lower temperature ∼ 50 K. This sequence of transitions is the same as in the LBCO compound (see Sect. 2.2.1, Fig. 2.5) which proves that the magnetic ordering is driven by the freezing of charges in the LTT phase. The spin–spin correlation length

74

3 Antiferromagnetism in Cuprate Superconductors

Fig. 3.8. Model for the stripe order in the LTT phase of LNSCO compound at hole doping x ∼ 0.12. (a) Bragg peaks are shown by ﬁlled circles, while superlattice peaks from two diﬀerent domains are shown by open circles and squares (for IC SDW) and by diamonds and triangles (for IC CDW). (b) Orientation of stripe patterns in neighboring planes of the LTT phase. (c) Stripe order for holes (filled circles – one hole per two Cu sites) and spins (black and open arrows for two antiphase domains) c 1996) (reprinted with permission by APS from Tranquada et al. [1260],

reaches its maximum value of ∼ 200 ˚ A below 30 K [1262]. Studies of LNSCO at diﬀerent Sr concentrations have demonstrated that superconductivity coexists with charge-stripe order, though the order parameters compete with each other [517, 1261]. The 1D character of the IC modulation was conﬁrmed by Wakimoto et al. [1329] who observed the diagonal incommensurate modulation (see below) along the orthorhombic b-axis in the LTO2 phase (P ccn) of the “single-domain” crystal of La1.55 Nd0.4 Sr0.05 CuO4 . In the stripe phase of the La1.875 Ba0.125 CuO4 compounds the inelastic neutron-scattering study at T = 12 K (> Tc ) by Tranquada et al. [1263] revealed high-energy spin excitations at h ¯ ω > 40 meV and a low energy branch of incommensurate spin excitations which were treated within the quantum two-leg-ladder spin model. The La2−x Srx CuO4 compound remains in the orthorhombic LTO phase up to the lowest temperatures (for x < 0.2). Therefore, the discovery of the static IC SDW in La1.88 Sr0.12 CuO4 crystals [609, 1207] at the same superlattice points (1/2 ± δ, 1/2, 0) as in the LNSCO compounds shows that the LTT phase may be important for the formation of IC CDW and SDW but it is not necessary. The black diamond in Fig. 3.7b shows the results of Kimura et al. [609] who found sharp elastic magnetic peaks in La1.88 Sr0.12 CuO4 at the magnetic transition temperature Tm equal to Tc 31 K. However, we remind here the results of Kimura et al. [607], discussed in Sect. 2.2.1, who observed

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

75

the softening of the Z-point phonons in LSCO compound as a precursor of the phase transition to the LTT phase. The observation of the IC spin ﬂuctuations in the excess-oxygen-doped La2 CuO4+y single crystals by several groups was important in establishing the universality of this phenomenon among the La-124 cuprates. In these crystals, the interstitial oxygen ions segregate into oxygen-poor and oxygen-rich regions. The latter forms a structure of periodic planes, known as staging, with n CuO2 layers between the interstitial oxygen planes [1347]. Inelastic neutron scattering study of the stage-6 La2 CuO4.055 single crystal with Tc ≈ 32 K by Wells et al. [1347] have revealed the low-energy IC spin ﬂuctuations at low temperature at the same IC wavevector as in experiments in LSCO by Yamada et al. [1383]. Elastic magnetic IC peaks in the stage-4 La2 CuO4+y orthorhombic single crystal with Tc 41 K at hole concentration nh ∼ 0.15 were discovered by Lee et al. [705]. The magnetic transition to the SDW appears at the same temperature Tm as the superconducting transition. This shows that SDW and superconductivity coexist. There were found four elastic IC magnetic peaks close to the AF wavevector: (1 ± δH , ∓δK , L) (in the Bmab notation for the reciprocal space (H, K, L)) originated from one domain and other four peaks (1 ± δH , ±δK , L) from the twin-related domain. The incommensurability direction was slightly diﬀerent along H and K directions: δH 0.114, δH 0.128 which correspond to the deviation of the IC wavevector by 3.3◦ from the copper–oxygen bond direction. To account for the four IC magnetic peaks, two models of stripes were proposed. In the ﬁrst model, two magnetic twin domains of parallel IC stripes in orthogonal directions were assumed, while in the second, the grid model, a rectangular array of stripes running along the orthorhombic axis was considered. The SDW coherence length in the CuO2 plane was quite large, of the order of 400 ˚ A. However, this was much shorter along the direction perpendicular to the plane, about two or three interplane distances. High resolution neutron scattering studies revealed existence of IC SDW in the lightly doped La2−x Srx CuO4 single crystals, x < 0.06, as well. In one of the ﬁrst experiments devoted to this topic, Wakimoto et al. [1327] discovered an elastic IC peak (the double circle in Fig. 3.7) in LSCO at x = 0.05 which appears at low temperature, T < 15 K. However, its position, observed at (1/2 ± δ, 1/2 ∓ δ), δ 0.06, is rotated by 45◦ from the IC peak positions at x ≥ 0.06. This pattern of SDW at x = 0.05 corresponds to the diagonal incommensurate modulation pattern (i.e., running along the diagonal of the Cu–O bond directions) which was found previously in insulating La2−x Srx NiO4+y [439]. In a subsequent publication [1328], it was shown that in a single domain of LSCO at x = 0.04, 0.05 only two satellites are observed ∗ along the orthorhombic √ b -axis, namely, at (1, ±, L) in the orthorhombic Bmab notation, = δ 2. Therefore, the IC modulation in the static diagonal spin stripes has a 1D character and it runs along the spin direction which is orthogonal to the AF wavevector (1, 0, L) and the rotation axes of CuO6 octahedron in the LTO phase (see Fig. 3.1a).

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3 Antiferromagnetism in Cuprate Superconductors

Later on, the whole region of the insulating spin-glass phase in LSCO, 0.01 < x < 0.06, was studied with high resolution and in good quality single crystals. Inelastic neutron scattering has revealed that the dynamical magnetic correlations change from diagonal incommensurate (stripe) pattern to commensurate for energy higher than ∼ 7 mev and with increasing temperature at T > 70 K [805]. In the low doping region, 0.01 < x < 0.02, the coexistence of the 3D long-range AF order and spin-glass state with the diagonal IC spin modulation was observed at low temperature, below T ∼ 20–30 K, [806]. In this transition region, the ordered magnetic moment in AF phase decreases with doping, while the fraction of the spin-glass phase increases. The incommensurability parameter δ did not change with doping, which indicated a microscopic spin-charge separation with ﬁnite size regions of the stripe phase and the AF phase. The spin correlation lengths of the AF stripes at x < 0.02 were large: ξa > 500 ˚ A, ξb ∼ 200 ˚ A, ξc ∼ 10 ˚ A(a, b, and c are the orthorhombic axes in Bmab notation and the stripes run along a-axis with a modulation along b-axis). A detailed study of variation of the IC spin modulation upon crossing the phase boundary between the spin-glass and superconducting states at x ∼ 0.055 by Fujita et al. [364] has revealed a transition from the diagonal to the parallel IC modulation. Figure 3.9 shows the variation with Srconcentration of the incommensurability δ (a) and the angle α (b). The angle α here deﬁnes the rotation from the diagonal (α = 90◦ ) to the parallel (α = 45◦ ) IC modulation in the polar coordinates around the (h, k) = (1/2, 1/2) reciprocal lattice point in the tetragonal notation. The solid and the open circles represent the diagonal IC and parallel IC modulation patterns, respectively. As shown in Fig. 3.9, the incommensurability δ does not change at the transition point x 0.055, but only the angle shows the rotation of the IC modulation. This suggests a coexistence of the diagonal IC and parallel IC modulation at

Fig. 3.9. Sr-concentration dependence of (a) the incommensurability δ and (b) the angle α (see the text). Solid and open circles represent the diagonal IC and parallel IC modulation patterns depictured in panel (a) (after Fujita et al. [364])

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77

the transition from the spin-glass to the superconducting phase or a microscopic phase separation of these phases. Large width of the magnetic peaks in the transition region also points to a phase separated state (see Fig. 3.7b). These very precise measurements revealed a linear hole doping dependence of the incommensurability, δ = x, in the whole range of Sr concentrations. Fujita et al. [364] suggested also, after Yamada et al. [1383], that the superconducting Tc ∝ x and therefore the superconducting phase transition at x 0.055 should be of the ﬁrst order type. To conclude the discussions of the experimental studies of static and dynamic IC spin modulation in the LMCO compounds, we brieﬂy comment on theoretical explanations of the phenomena. The driving mechanism of IC CDW (stripe) formation is the lowering of the kinetic energy of charge carriers. There are basically two models for CDW formation. The ﬁrst one is based on the Fermi-liquid picture where the CDW arises from the nesting phenomena which causes instability of the Fermi-surface. Below the transition temperature gaps or pseudogaps open at the Fermi-surface which leads to decreasing of the electron energy. This mechanism can be realized only in weak-coupling systems where well-deﬁned quasiparticles exist at the Fermi-surface. Another mechanism can be realized in systems with strong Coulomb correlations like the Mott–Hubbard insulators. The lowering of the kinetic energy of doped holes and at the same time preserving strong AF spin correlations in the parent insulator can be achieved by a real-space micro-phase separation in the form of metallic stripes between the antiphase domain walls of the AF order. The ﬁrst mechanism was studied within various analytical methods like the Hartree–Fock approximation, as by Zaanen et al. [1399] who in fact have predicted the stripe phase. Studies of models appropriate for strongly correlated systems were mostly based on numerical methods for ﬁnite clusters, like Monte Carlo, exact diagonalization and density-matrix renormalization group approach (for discussion see [615] and [211]). Most of the experiments in the cuprates support the strong-coupling mechanism of the stripe formation. Existence of the insulating state of parent undoped compounds with the strong superexchange interaction J ∼ 1,500 K between the localized 1/2 copper spins results in strong AF correlations even at the temperatures much higher than the N´eel temperature. The values of the local magnetic moments at the copper sites only weakly depends on doping. Contrary to that, in the case of the SDW formation caused by the nesting phenomena both the insulating gap and the itinerant magnetic moments should be strongly suppressed with doping, and no well-deﬁned spin correlations should be seen above the N´eel temperature (for a more detailed discussions see [615]). The neutron scattering experiments discussed above, point to the occurrence of an evident correlation between the spin (charge) modulation, superconductivity and the underlying crystal structure. This issue has been already discussed in Sect. 2.2.1. As shown in Fig. 2.5, there is an obvious relation between the structural phase transition to the LTT phase and appearance of the CDW and SDW. It can be suggested that the coupling between the

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3 Antiferromagnetism in Cuprate Superconductors

structural and AF order parameters is caused by the Dzyaloshinsky–Moriya antisymmetric exchange interaction (see Sect. 3.2.3, (3.27)). It favors, locally, the orthogonal directions between the Cu-spin and the CuO6 octahedron rotation axis (see Fig. 3.1a). In the LTT phase, the octahedrons rotate around the x or y axes (tetragonal notation) which are the directions of the copper–oxygen bonds and the spin directions are also along the copper–oxygen bonds. Therefore, in the LTT phase the parallel stripes may be pinned by the rotation pattern, namely, by the corrugation of the CuO2 planes which runs along the bonds. In the LTO phase, the octahedrons coherently rotate around the diagonal of the copper–oxygen bonds and the spin directions in this case coincide with the diagonal of the copper–oxygen bonds. The diagonal stripes may be pinned by the corrugation pattern of the CuO2 planes which runs along the diagonal of the bonds. However, the charge localization is much weaker in the LTO phase in comparison with the LTT phase as discussed in Sect. 2.2.1. The above considerations are supported by the neutron scattering study of electron-doped Nd–Ce cuprate superconductors. The Nd–Ce compound has the tetragonal T crystal structure (see Fig. 2.10) where the apex oxygen ions are absent and so no structural transitions related to the tilting type modes have been observed. The elastic and inelastic neutron scattering study of Nd2−x Cex CuO4 single crystals (x = 0.15, Tc = 18, 25 K) by Yamada et al. [1384] revealed only commensurate magnetic ﬂuctuations at the AF wavevector, (1/2, 1/2). For the in-plane and the out-of-plane static correlation lengths they obtained the values: ξab ∼ 150 ˚ A and ξc ∼ 80 ˚ A. A spin gap in the magnetic ﬂuctuation energy spectrum opens in the superconducting phase at T < Tc and increases with decreasing temperature up to ∼ 4 meV at T ∼ 2 K. Spin Gap and Magnetic Excitation Spectra Here we brieﬂy summarize the results of the inelastic neutron scattering study of the spin-ﬂuctuation spectrum in LSCO. Depending on doping, we can single out three typical regions of magnetic excitation spectra in LMCO compounds: undoped, weakly doped nonsuperconducting, and metallic superconducting regions. As shown in Fig. 3.3, in the undoped compounds the well-deﬁned 2D spin waves with a very high spin-wave velocity, c ∼ 0.8 eV·˚ A, two times larger than the sound velocity, are observed (see Table 3.1). The q-integrated density of the spin-wave excitations is given by the imaginary part of the spin susceptibility χ (ω). It exhibits a broad featureless energy spectrum up to ¯hω 2J ∼ 300 meV showing a weak temperature dependence. In the nonsuperconducting state, at the hole doping x < 0.05, the density of spin excitations χ (ω) increases at low frequencies and low temperatures as shown in Fig. 3.6. With increasing doping and entering the superconducting region, the spectrum of magnetic excitations becomes rather broad without well-deﬁned spin waves, though the maximal excitation energy is still high, of the order of 280 meV even for the optimal doping [440, 1381]. Contrary to the

3.2 Antiferromagnetism in La2−x Mx CuO4 Compound

79

nonsuperconducting region, in LSCO crystals at optimal doping a decrease of the density of spin ﬂuctuations at low energy ∼ 10 meV and low temperature was observed [804] pointed thus to a spin gap formation. Later on, in experiments with high-quality single LSCO crystals the opening of the spin gap was conﬁrmed in the spin-ﬂuctuation spectrum at low temperature. In particular, the spin gap of 3.5 meV for the IC spin excitations at Qδ = (1/2 ± δ, 1/2) was detected by Yamada et al. [1382] for LSCO crystal with x = 0.15, and Tc = 37 K. Quite a sharp spin gap of the value ∼ 6.7 meV was observed in superconducting LSCO crystal with Tc = 38.5 K by Lake et al. [686]. Subsequent experimental investigations of large and high-quality LSCO crystals at various doping, 0.10 < x < 0.25, have shown that a well-deﬁned spin energy gap ∼ 6 meV in the IC spin ﬂuctuations is observed in the superconducting state only for samples close to the optimal doping, x ∼ 0.15 [697]. No energy gap was detected in the underdoped, x = 0.10, and heavily overdoped, x = 0.25, samples though they show bulk superconductivity at Tc = 29 K and 15 K, respectively. The absence of the energy gap in these samples may be caused by heavily disordered states of the underlying incommensurate spin structures which have small correlation length in comparison with the optimally doped samples at x ∼ 0.15 (see Fig. 3.7). This observation was further supported in a comprehensive study of the magnetic excitations in normal state LSCO crystals [698]. The energy dependence of χ (ω) at T = 8 K and at the superconducting transition temperature is shown in Fig. 3.10 for LSCO crystals with Sr-concentration x = 0.15, 0.18, and 0.20. At low temperature, Fig. 3.10a, a clear spin gap structure is observed for all samples. At T = Tc only the sample with x = 0.18, Fig. 3.10c, exhibits a peak at the energy ∼ 6 meV which can be coupled with a pseudogap formation in the normal state (see Sect. 3.3.2). The peak disappears at higher temperatures, T ∼80–150 K. In other samples, only a broad maximum showing the linear ω-dependence of χ (ω) at ω → 0 was observed. In conclusion, these studies revealed the existence of s spin gap in the dynamical spin susceptibility at low temperature for LSCO crystals close to the optimal doping. In the excess-oxygen-doped La2 CuO4+y single crystals no spin gap at low energy h ¯ ω ∼ 2 meV was detected [705]. In this crystal below Tm Tc the static SDW with large coherence length appears which may result in spin-wave like excitations at low-energy and constant χ (ω). sectionmarkAntiferromagnetism in La2−x Mx CuO4 Compound In the next section, we consider the spin-ﬂuctuation spectra in the YBCO compound where the opening of the spin gap in the superconducting state and a strong enhancement of χ (QAF , ω) (resonance peak) below Tc were already observed in the very ﬁrst neutron scattering experiments.

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3 Antiferromagnetism in Cuprate Superconductors

Fig. 3.10. q-integrated energy spectrum χ (ω) of spin ﬂuctuations in La2−x Srx CuO4 at low temperatures (a) and at Tc (b, c, d) (reprinted with c 2003) permission by APS from Lee et al. [698],

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds 3.3.1 Magnetic Phase Diagram Early dc magnetic susceptibility measurements on YBa2 Cu3 O6+x samples did not show clear evidences for the AF phase transition, contrary to the pronounced peak in the susceptibility of the La2 CuO4+y compound at the N´eel temperature [558]. The 3D AF ordering with high N´eel temperature TN ∼ 400 K in YBa2 Cu3 O6+x with x ∼ 0 was discovered in magnetic neutron scattering experiments [712,1065,1066,1255]. Further neutron-scattering measurements on large single crystals of YBCO in a wide range of oxygen content, 0 < x < 1, have yielded a great deal of detailed information on the AF ordering, spin correlations, and spin dynamics. In the insulating phase of YBa2 Cu3 O6+x , at x < 0.4 below the N´eel temperature, the AF ordering of the magnetic moments at Cu2 sites occurs in CuO2 planes as shown by the large ﬁlled and open circles in A and C layers in Fig. 3.11 [1138]. The AF ordering in the tetragonal phase of YBa2 Cu3 O6+x is described by the wavevector QAF = (1/2, 1/2, l), which corresponds to

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

81

Fig. 3.11. The antiferromagnetic spin structure of YBa2 Cu3 O6+x with x ∼ 0. Filled and open circles indicate antiparallel spins at Cu2 sites oriented perpendicular to the c-axis. (a) At high temperature the Cu1 ions in the chains (shown by cross-hatched circles) have no magnetic moments. (b) At low temperature (T < 15 K for x = 0.15) small magnetic moments appear at Cu1 sites (shown by small filled and open circles) c 1993) (reprinted with permission by APS from Shamoto et al. [1138],

√ √ a magnetic unit cell with the parameters (a 2, a 2, c) where a and c are the lattice constants of the tetragonal unit cell (see Fig. 2.12a). Usually, in the YBCO crystal with a homogeneous distribution of doped oxygen only integral values of l in the magnetic Bragg peaks (1/2, 1/2, l) are observed which shows that the magnetic moments at Cu2 ions lie in the basal plane and there are no magnetic moments on the Cu1 sites in the chains shown by the cross-hatched circles in B layers in Fig. 3.11a. However, the direction of the spins, due to the tetragonal symmetry, is not ﬁxed in the CuO2 planes (contrary to the LSCO compounds where the spin direction is deﬁned by the DM interaction). The absence of scattering with l = 0 points to antiferromagnetically ordered magnetic moments in the bilayer Cu–Y–Cu. The value of the magnetic moment on the Cu2 ions in the region x < 0.2 is μ 0.6μB , the same as the magnetic moment in La2 CuO4 . From the temperature dependence of the magnetic Bragg peak (1/2, 1/2, l), ﬁtted by a simple power law I ∝ (T − TN )2β , the critical exponent of the order parameter (sublattice magnetization) can be determined: β = 0.27 which corresponds to a quasi-2D XY model [1138]. A second magnetic transition at low temperature, T < 15 K, in the YBCO crystal with x = 0.15 was observed by Shamoto et al. [1138]. Below this temperature the intensity of magnetic Bragg peaks (1/2, 1/2, l) decreases, while new peaks (1/2, 1/2, l + 1/2) appear. The new peaks show a doubling of the AF structure along the c-axis as shown in Fig. 3.11b where the small ﬁlled and open circles show magnetic moments on Cu1 sites. In this structure, a ferromagnetic coupling between bilayers mediated by the layers with chains

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3 Antiferromagnetism in Cuprate Superconductors

occurs contrary to the AF coupling in the simple AF structure in Fig. 3.11a. As we discuss in the next section, this type of the AF structure with nonzero magnetic moments on Cu1 ions was ﬁrst observed in NdBa2 Cu3 O6+x [752]. In the YBCO samples with small oxygen content, 0 < x < 0.2, isolated magnetic Cu2+ ions can occur in the chains, the ordering of which may result in the new AF phase [1138]. As the oxygen content x in the Cu–O1 chains increases, both the value of the ordered magnetic moment μ and the N´eel temperature TN decrease so that μ(x) ∝ TN (x). However, in contradistinction to La2−x Srx CuO4 the value of the local moment on the Cu2 ions does not change appreciably. Figure 3.12 shows the phase diagram for YBa2 Cu3 O6+x with the N´eel temperature TN (x) and superconducting Tc (x) dependence on the oxygen content [1067, 1069]. The dependence μ(x) and TN (x) can be subdivided into three regions [1066, 1312]. At x < 0.20 these quantities are roughly constant. For x 0.35 they decrease gradually, and afterwards they rapidly tend to zero as x tends to a critical oxygen content x → xc = 0.41, at which the long-range AF order vanishes. The corresponding critical hole concentration at xc = 0.41 is equal to nh ∼ 0.02, similar to La2−x Srx CuO4 (see Fig. 2.4). This speciﬁc variation of TN (x) in YBCO is accounted for by a more complicated mechanism of charge transfer from Cu–O1 chains to CuO2 planes, as the oxygen content in YBa2 Cu3 O6+x increases. As already noted in Sect. 2.4.1, the eﬀective valence of copper Cu2 in the planes is almost constant up to an oxygen content x ∼ 0.4 (Fig. 2.13b). In this region, only charge redistribution at Cu1 ions (Cu+1 → Cu+2 ) and the formation of holes at oxygen ions O1 in the chains take place. Therefore, only a smooth decrease of TN can be expected in this regime due to weakening of the AF coupling between the Cu2–Y–Cu2 bilayers with the same strong AF correlations within each bilayer. As the hole

Fig. 3.12. Phase diagram of YBa2 Cu3 O6+x : N´eel temperature TN and superconducting Tc as a function of oxygen content x (after Rossat-Mignod et al. [1067])

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

83

Fig. 3.13. Temperature dependence of the magnetic susceptibility in YBa2 Cu3 O6+x for various oxygen content corrected for a Curie-like contribution Cg /T (after Tranquada et al. [1256])

concentration increases x → xc , the number of holes in the bilayer rapidly increases as shown in Fig. 2.13b. Similar to LSCO, the appearance of holes in the CuO2 layers suppresses 2D AF correlations. Figure 3.13 shows the temperature dependence of the static magnetic susceptibility, corrected for a Curie-like contribution from impurity phases, for various values of oxygen content x [1256]. At low oxygen concentration, x ≤ 0.4, the susceptibility shows a broad maximum at high temperatures, T ≥ 800 K, of the order of the exchange interaction. In this region, the spin susceptibility is increasing with temperature as thermal spin ﬂuctuations diminish spin correlations. A more detailed study revealed a weak bump in χ(T ) at the N´eel temperature which has been overlooked in early experiments since a well-pronounced peak at TN , observed in La2 CuO4 due to DM interaction, has been expected. In the metallic phase, with increasing hole concentration, the susceptibility becomes more temperature independent and Pauli-like. Thus, under increasing oxygen content, the susceptibility shows a smooth transition from 2D AF-type behavior for localized spins to Pauli-type behavior characteristic to itinerant electron systems. At the hole concentration nh > nc ∼ 0.02 per Cu2 ion, the long-range AF order disappears. However, the short-range 2D AF correlations persist and a transition to a spin-glass state occurs at low temperature. As μSR experiments show, a coexistence of superconductivity and frozen AF correlations in YBCO occurs for hole concentrations up to nh ∼ 0.09 (see region III in Fig. 3.2). Several studies reported the coexistence of superconductivity and AF order in YBCO crystals with x ∼ 0.5 or in crystals where Cu ions in the chains (Cu1 sites) were substituted by Co, Fe, or Li ions. The most extensive studies were done on YBa2 Cu3−y Coy O6+x (Co-YBCO) compounds since the Co ions may be easily substituted for Cu up to y = 1. For instance, elastic neutron scattering in a fully oxygenated Co-YBCO single crystal (x = 1 + δ) with a small concentration of Co ions, y = 0.04, reveals coexistence of the AF order with N´eel temperature TN ∼ 330 K and superconductivity with high Tc ∼ 93 K

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3 Antiferromagnetism in Cuprate Superconductors

[474]. Reduced value of the ordered magnetic moment, μ ∼ 0.14 μB , and a lower intensity of the resonance peak, in comparison with the Co-free sample, suggest an inhomogeneous state where the AF phase with the correlation length of the order of ∼ 200 ˚ A coexists with the superconducting phase. The Co ions produce a disorder in the Cu1 chains that results in a “granular” type superconductivity, as suggested by the authors. For YBCO crystals with oxygen content around x ∼ 0.5 a certain disorder is produced by magnetic Cu2+ ions in the chains at Cu1 positions which also can results in a coexistence of AF order and granular superconductivity. Contrary to single crystal results, in a subsequent neutron diﬀraction experiment in Co-YBCO powder samples no long-range AF order was detected in the superconducting phase [816]. By varying the Co concentration over a wide range, y = 0.04 − 0.72, the phase diagram for YBa2 Cu3−y Coy O7 was established. The superconducting state was found to disappear at Co concentrations y > 0.42 where the long-range N´eel state appears. At high Co concentration, x ∼ 0.84, large magnetic moments at Cu2 sites, μ ∼ 0.8μB , and high N´eel temperature TN ∼ 400 K were detected. The AF order in CoYBCO compound persists up to a much higher hole concentration than in the pure YBCO compound which may be explained by hole localization at Co sites. The average magnetic moment at Cu1 sites is small, μ ∼ 0.1μB , which may be caused by disorder produced by Co ions. However, the Cu1 magnetic moments are ordered below TN and the AF structure of the second type shown in Fig. 3.11b is realized. The best ﬁt to the powder diﬀraction pattern was reached for directions of both the Cu1 and Cu2 moments parallel to the c-axis, contrary to the in-plane direction of Cu2 moments in pure YBCO. The authors concluded that there is no clear evidence of the AF long-range order in the superconducting phase of the Co-YBCO powder samples, while the observation of coexistence of these two phases in some single crystals are due to a “granular” character of the phases. 3.3.2 Spin Dynamics Nonsuperconducting Region Spin-wave excitations in the AF phase of YBa2 Cu3 O6+x at low oxygen content, x < 0.4, were evidenced in a number of inelastic neutron scattering experiments (see, e.g., [441,1053,1066,1067,1138,1257]). Spin waves are most distinctly observed in the region x < 0.2 where the hole concentration in CuO2 planes are negligible and the spin-wave excitations are well deﬁned. In comparison with monolayer compounds of the La2 CuO4 type, there are two Cu2 ions in the unit cell of the bilayer YBCO compound which results in the splitting of the spin wave excitations into two branches of acoustic and optic modes. The acoustic branch corresponds to odd-parity excitations in the bilayer, while the optic branch corresponds to even-parity excitations. Therefore, the total spin

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

susceptibility can be decomposed into two parts [347]: dqz dqz χac (q, ω) + cos2 χop (q, ω), χ (q, ω) = sin2 2 2

85

(3.33)

where the intensities of acoustic and optic contributions are given by modulation factors depending on the wave vector qz perpendicular to the plane, d = 3.3 ˚ A is the intrabilayer distance. By taking into account this qz dependence, it is possible to distinguish two types of excitations in neutron scattering experiments. To describe the spin excitation spectrum in the bilayer YBa2 Cu3 O6+x compound, we can use the model Hamiltonian [1257]: z z Si,n Sj,n − αxy Si,n H= J Sj,n n=1,2

+ J⊥

i

Si1 Si2 + J⊥1

Si,1 Si+c,2 ,

(3.34)

i

where < i, j > denotes summation over pairs of nearest-neighbor spins and n = 1, 2 refers to the two Cu ions in the bilayer (A and C planes in Fig. 3.11). Here J denotes the nearest-neighbor superexchange interaction in the plane, αxy > 0 deﬁnes the anisotropy of the exchange interaction for the easy plane (XY ) Heisenberg model. Intrabilayer direct AF coupling between two Cu2 ions are given by J⊥ , while a much weaker coupling between bilayers via an O–Cu1–O bridge along c-axis is given by J⊥1 . Detailed calculations of the spectrum of spin waves for the model (3.34) in the linear spin-wave approximation have found four magnon modes. Two original modes for the Cu–O bilayer system, acoustic and optical, split into pairs of in-plane and out-of-plane modes caused by the anisotropy of the exchange interaction αxy J [1069, 1138]. The anisotropy gap energy at the AF zone center, (1/2, 1/2, l) has the value 2J 2αxy , while the gap at the zone √ boundary along the qz direction is given by 2J J⊥1 . If we neglect these small interactions, then the dispersion relations for the 2D acoustic and optical magnons read [1053]: 1/2 ω(ac/op) (qx , qy ) = 2J 1 − γ 2 (q) + (J⊥ /2J) [1 ∓ γ(q)] ,

(3.35)

where the minus sign is for the acoustic mode and γ(q) = (1/2)(cos qx a + cos qy a). The spin-wave √ velocity for the acoustic magnons in this approximation is given by c = J 2 a (up to the quantum renormalization Zc discussed in Sect. 2.2.1). Early inelastic neutron-scattering measurements on single crystals of YBCO with x = 0.15, TN = 415 K by Rossat-Mignod and co-workers [1067, 1312] have found a rather large in-plane spin-wave velocity c = 1 eV·˚ A which resulted in an exchange energy value J = 150 meV. The other parameters in the model (3.34) were estimated as: J⊥ /J ∼ 10−2, αxy ∼ 10−4,

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3 Antiferromagnetism in Cuprate Superconductors

and J⊥1 /J ∼ 10−5. In the subsequent neutron-scattering measurements on YBCO, single crystal of the same composition, x = 0.15 and TN = 415 K, Shamoto et al. [1138] obtained the following results for these parameters: J = 120 ± 20 meV, αxy ∼ 7 × 10−4 , and J⊥1 ∼ 0.04 meV. However, the incident neutron energy was not high enough to resolve the optic mode and only a lower limit for the interbilayer interaction was suggested: J⊥ ∼ 8 meV. The direct observation of optical magnons in YBCO crystal in AF phase became possible later, in high-energy inelastic neutron scattering experiments. The measurements by Reznik et al. [1053] of the optical magnon branch in the YBCO crystal with x = 0.2 revealed the minimum of the optical magnon energy at 65–70 meV that gives J⊥ ∼ 10 meV for the exchange energy J = 120 meV. Similar results were obtained by Hayden et al. [441] for an YBCO crystal with x = 0.15: J = 125±5 meV, J⊥ ∼ 11±2 meV. The maximal energy of spin excitations were detected close to 2J value at ∼ 250 meV. It should be pointed out quite a large value of the AF coupling in the bilayer, J⊥ /J ∼ 0.09, which shows a substantial inter-bilayer overlap of the Cu2 atomic wave functions. In comparison with La2 CuO4 , the superexchange interaction in the CuO2 layers of YBCO determined from the spin-wave velocity, J ∼ 120 meV (with quantum corrections, J/Zc ∼ 105 meV), is smaller than in the monolayer cuprates (see Table 3.1). However, there is no gap in the acoustic in-plane mode in YBCO which appears in LCO due the DM interaction (see (3.18)). Therefore, strong 2D ﬂuctuations should persist in the spin dynamics of YBCO which may be well described within the XY bilayer Heisenberg model (3.34) for spins S = 1/2. Like in LSCO, the appearance of holes in the CuO2 layers drastically changes the spin dynamics. In the region x < 0.4 there are still observed spinwave-like excitations though with a large damping and a small correlation length. According to Rossat-Mignod and co-workers [1067, 1312], in YBCO crystal with x = 0.37 and TN = 180 K, the correlation length extends over a few lattice spacing only, ξ 7.5a. The hole concentrations evaluated from the correlation length, nh = (a/ξ)2 ∼ 0.018. At low temperature the holes are localized producing static disorder in the form of spin polarons. They strongly disturb the propagation of in-plane spin excitations and reduce the spin stiﬀness so that the spin-wave velocity decreases up to one-half of the original value, c 0.45 eV·˚ A. At the critical hole concentration, nh ∼ 0.02, when the long-range AF order disappears, the spin stiﬀness should vanish and instead of propagating acoustic spin waves close to the AF wavevector, a diﬀuse type of spin excitations should appear. However, due to strong AF short-range correlations spin-wave like excitations survive in the metallic phase at high excitation energy as described in the next section.

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

87

The Metallic Region: Spin Gap The metallic region of YBa2 Cu3 O6+x from the weakly doped (x = 0.45−0.55) to the heavily doped (x = 0.6 − 0.9) and the overdoped (x = 1) regimes [167, 1046,1047,1068,1070,1094,1258] was studied extensively. In the metallic region the inelastic magnetic scattering observed around AF wave vector exhibited several peculiar phenomena: energy gap in the superconducting phase and pseudogap in the normal state, incommensurate modulation, resonance mode behavior. These studies have attracted much attention concerning the role of spin ﬂuctuations in the mechanism of high-temperature superconductivity. The increase of hole concentration nh in the CuO2 layers and their delocalization at the transition to the metallic phase strongly suppresses the magnetic correlation length ξ2D in the plane. Its value, deﬁned by the q-width of the inelastic scattering across the magnetic rods at the AF wave vector (QAF = 1/2, 1/2, l), is almost temperature independent and is determined √ only by the doping level: ξ2D ∼ a/ nh [1068]. In spite of the short-range character of the magnetic correlations, intensive dynamic AF spin ﬂuctuations persist for wavevectors near the AF wave vector. In the studies of YBCO crystal with x = 0.45 and Tc 35 K RossatMignod and co-workers [1067] have discovered an energy spin gap in the magnetic excitation spectrum at low energy h ¯ ω ∼ 2 meV which is opened below Tc . Subsequent studies of samples at various doping in metallic phase have proved the existence of the spin gap for the acoustic mode in the odd spin susceptibility channel which retains the characteristic qz modulation of magnetic scattering in (3.33) at all dopings. The spectrum of spin ﬂuctuations deﬁned by the imaginary part of the spin susceptibility at the AF wavevector as a function of energy for diﬀerent oxygen contents is shown in Fig. 3.14

11

c (q=kAF,w) (arbitrary units)

350 x=0.5 x=0.52 x=0.83 x=0.92 x=1

YBa2 Cu3 O6+x

300

T=2-5k

250 250 150 100 50 0 0

10

20 30 Energy (mev)

40

50

Fig. 3.14. Spin ﬂuctuation spectrum χ (QAF , ω) for various oxygen contents in the superconducting state (reprinted with permission by Elsevier from Regnault et al. c 1995) [1047],

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3 Antiferromagnetism in Cuprate Superconductors

[1047]. The energy gap EG is clearly seen at low temperature in the superconducting phase for all samples, including the overdoped one with x = 1. The value of the gap (deﬁned as the ﬁrst inﬂection point of the χ (QAF , ω) curve) does not depend on the temperature and increases with doping. The gap is roughly proportional to Tc : in the weakly doped state, EG ≈ kB Tc , while in the heavily doped regime for Tc > 60 K, EG ≈ 3.8kB Tc [167, 270, 1046]. The spin gap may be washed out by sample inhomogeneities (in particular, oxygen disorder in chains in YBCO) or by impurities, as it has been shown in Zn-substituted YBCO by Sidis et al. [1166]. A comprehensive polarized and unpolarized neutron scattering study of spin-ﬂuctuation spectra in a broad range of energy up to 120 meV in the underdoped YBa2 Cu3 O6+x crystals with x = 0.5, 0.7, and 0.85, and Tc (K) = 52, 67, and 87 respectively, was performed by Fong et al. [347]. It appears that the interbilayer coupling J⊥ is still strong in the doped samples which permits to distinguish two modes, odd and even, in the dynamical spin susceptibility as in the AF state (3.33). While the odd, acoustic-like mode shows a gap behavior in the superconducting state, as described above, the even mode, which is an order of magnitude weaker, appears at energy larger than 50 meV and only weakly depends on the temperature and the oxygen content. The intensity of the odd spin susceptibility shows a strong energy redistribution with the gap opening below Tc : the low-energy part below the gap decreases, while the intensity enhances at energy higher than the gap as in the highly doped samples where the resonance peak appears (see the next section). The intensity of the even channel shows no appreciable variation at Tc . At high energies above 50 meV a large broadening of the q -width in the plane is observed for the magnetic response in the underdoped samples. This can be related with the dispersion of these excitations which resembles the dispersion of magnons in AF phase. The dispersion for the even excitations in YBa2 Cu3 O6.5 crystal was 2 ﬁtted by a simple parabolic law: ωop (q) = ω02 +c2 q2 with the gap ω0 ∼ 55 meV and the dispersion c ∼ 420 meV·˚ A [169]. However, a large damping parameter comparable to the gap should be used which shows that the spin ﬂuctuations propagate only over a small in-plane distance. In studying the in-plane q dependence of the magnetic excitations in the metallic phase, several publications have reported an incommensurate spin modulation at low energy (see, e.g., [270, 454, 455, 1190, 1191] and references therein). Contrary to the LSCO crystal, only a weak dynamic spin modulation with a small value of the incommensurability δ ≤ 0.1 were registered in the orthorhombic phase of YBCO. The most distinctly incommensurate spin modulation was detected in the underdoped samples, as in the ﬁrst observation in YBa2 Cu3 O6.6 crystal by Mook et al. [854]. They have revealed four maxima for spin excitations with energy ∼ 24 meV at low temperature which were shifted from the AF position (1/2, 1/2) by δ ∼ ±0.1 as in the LSCO system (see Fig. 3.7). In the subsequent publication by Mook et al. [856] a onedimensional character of the dynamical magnetic ﬂuctuations in the partially

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

Intensity (counts/5.5 min)

600 400

a

b

33.5 mev

c

d

31 meV

e

f

28 meV

89

200 0 –200 600 400 200 0 –200 400 200 0 –200 –400

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 H (r.l.u.) K (r.l.u.)

Fig. 3.15. Comparison of the constant-energy scans for the YBa2 Cu3 O6.6 crystal along the a∗ -axis (left panel) and the b∗ -axis (right panel) at various energies (after Hinkov et al. [454])

detwinned YBa2 Cu3 O6.6 crystal was revealed. The authors suggested a stripe picture for the charge and spin separation as in LSCO crystals discussed in Sect. 3.2.4. However, later on Hinkov et al. [454] reported neutron scattering study on “untwinned” (95%) YBa2 Cu3 O6+x crystals with x = 0.6 and 0.85 which demonstrated that the geometry of the IC magnetic ﬂuctuations is 2D. Figure 3.15 shows constant-energy scans for YBa2 Cu3 O6.6 crystal along a∗ axis ((a), (c), and (e) panels at the left) and along b∗ -axis ((b), (d), and (f) panels at the right) in the reciprocal space at various energies. The plots show subtractions of the intensities at T = 5 and 250 K to emphasize the magnetic signal. The incommensurate peaks are observed both along a∗ and b∗ directions at three measured energies which points to the 2D character of the IC spin ﬂuctuations. A similar picture was obtained for a crystal with x = 0.85. As evidenced from Fig. 3.15, the amplitude and the width of the incommensurate peaks exhibit an in-plane anisotropy, which was explained by the authors as a result of a possible liquid-crystalline stripe phase instead of a rigid stripe arrays proposed by Mook et al. [856]. A comprehensive study of spin ﬂuctuations in the oxygen-ordered and partially detwinned ortho-II phase (see Sect. 2.4.1) of YBa2 Cu3 O6.5 (Tc = 59 K) was performed by Stock et al. [1190]. The hole doping in each CuO2 plane was estimated as nh ∼ 0.09 which corresponds to the underdoped region. Due to a perfect oxygen order in the chains of the YBCO crystal in the ortho-II phase, it was possible to remove the eﬀects of impurity scattering and diﬀerent type of structural disorder on the spin dynamics.

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3 Antiferromagnetism in Cuprate Superconductors

It was found that the low-energy spin ﬂuctuations have a modulation only along the a∗ -axis, perpendicular to the chains, with the incommensurability δ ∼ 0.06. To describe the intensity of incommensurate dynamic spin ﬂuctuations, the authors used the overdamped spin-wave model of Chou et al. [228]. They obtained for the spin velocity c ∼ 300 meV·˚ A, while the A due to dynamic correlation length appeared to be small, ξ0 = c/Γ ∼ 20 ˚ a large damping parameter Γ for the spin-waves. The low-energy incommensurate modulation along a∗ was observed in the normal state up to ∼ 120 K, while in superconducting state they persisted up to the energy of ∼ 25 meV. The overall intensity of low-energy (¯hω < 16 meV) spin ﬂuctuations was described by a scaling function of the same type (3.32) as that for the LSCO compounds: χ (ω, T )/χ (ω, 0) = (2/π) arctan(ω/a1 T ) with a1 ∼ 0.9. The authors suggested that the low-energy scaling behavior of the one-dimensional incommensurate spin ﬂuctuations could be described within the stripe model, while the quantum critical point scenario did not explain the scaling. In the superconducting state, the intensity of the spin ﬂuctuations below the energy of 16 meV is suppressed, while at energies larger than 24 meV is enhanced. The crossover energy 16 meV scales with 3kB Tc . The authors observed a spin gap formation in the ortho-II phase of YBa2 Cu3 O6.5 though they did not detect a full gap. The most remarkable result, is the occurrence in the spin ﬂuctuation spectrum in the superconducting phase of an intense and well-deﬁned commensurate resonance peak at the energy 33 meV which ﬁrst appears in the normal state in the form of an abroad and weak feature. We discuss the resonance peak phenomenon in superconducting phase of cuprates in the next section. In a subsequent publication by Stock et al. [1191], a high-energy part of the spin-ﬂuctuation spectrum in the same oxygen-ordered ortho-II phase of YBa2 Cu3 O6.5 was reported. While the low-energy excitations below the resonance energy at 33 meV show a one-dimensional character, as discussed above, the high-energy excitations reveal an isotropic spin-wave character both for the acoustic and optic modes. Figure 3.16 shows the dispersion of the acoustic (a) and optic (b) modes close to the AF wave vector at T = 6 K. The solid lines represent the dispersion measured by Hayden et al. [441] on the insulating YBa2 Cu3 O6.15 crystal. The dashed lines are ﬁts of the highenergy dispersion to a linear spin-wave theory. The acoustic dispersion from the resonance energy ωr ∼ 33 meV to high energy up to 120 meV was ﬁtted by the function: ωac (q) = (ωr2 + c2 q 2 )1/2 where the spin-wave velocity c ∼ 365 meV·˚ A. This value is about half of the value found in the insulating compound YBa2 Cu3 O6.15 . The optic mode was observed up to the energy of 20–25 meV, much lower than the optic gap in the insulating compound ∼ 70 meV. The low-energy mode becomes very broad in q-space as shown by Δqoptic in Fig. 3.16b. This suggests that inter-bilayer spin correlations are strongly suppressed by hole doping.

The momentum integrated (local) susceptibility χ (ω) = d2 qχ (q, ω), for the acoustic mode is shown in Fig. 3.17 [1191]. The low-energy spectrum,

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

91

Fig. 3.16. The dispersion of the acoustic (a) and optic (b) modes near the (π, π) point at T = 6 K in the ortho-II phase of YBa2 Cu3 O6.5 crystal. The solid lines represent the dispersion for the insulating YBa2 Cu3 O6.15 crystal. The dashed lines are ﬁts of the high-energy dispersion to a spin-wave theory (after Stock et al. [1191])

Fig. 3.17. Energy dependence of the momentum integrated susceptibility for the acoustic mode of the oxygen-ordered crystal of YBa2 Cu3 O6.5 in ortho-II phase at T = 6 K. The dashed line is the data for YBa2 Cu3 O6.15 [441] (after Stock et al. [1191])

measured with the aid of a triple-axis spectrometer (open circles), is dominated by a resonance peak at the energy ∼ 33 meV. The high-energy part, measured with the aid of a time-of-ﬂight technique (ﬁlled circles), is nearly constant and close to the spectrum of the insulating compound YBa2 Cu3 O6.15 is shown by the dashed line [441]. These results lead the authors to suggest that there are two distinct regions of spin ﬂuctuations: a low-energy part, below the resonance energy ∼ 33 meV, which can be described as incommensurate stripe-like collective

92

3 Antiferromagnetism in Cuprate Superconductors

spin excitations for the acoustic mode, and a high-energy part, which has a spin-wave character. The high-energy part of the spin ﬂuctuations has an isotropic in-plane dispersion and cannot be described within the stripe model or a static spin-ladder model. Two branches of spin excitations merge at the AF wavevector QAF = (π, π) to form a resonance mode. Another interpretation of the resonance mode was proposed by Fong et al. [343] as the triplet spin exciton inside the gap of a particle–hole continuum ω < Ωc (q) ≤ |Δd (kF )| + |Δd (kF + q)| in the d-wave superconducting state. The gap Ωc (q) has a maximum at q = QAF and the downward dispersion shown in Fig. 3.16 is related to the q-dependence of the superconducting gap Δd (q) (see Sect. 7.2.2). The diﬀerent nature of the two parts of the spectrum is also revealed in their temperature dependence: the low-energy acoustic part of the spectrum is strongly inﬂuenced by the superconducting transition which induces the appearance of the resonance peak, while the high-energy part of the spectrum does not change appreciably with the temperature up to 85 K. Quite a diﬀerent spin dynamics was observed in a strongly underdoped YBa2 Cu3 O6.353 at the superconducting boundary with Tc = 18 K (hole density p ∼ 0.06) by Stock et al. [1192, 1193]. In contrast to more heavily doped samples, no spin gap or a resonance mode was found below Tc , while two distinct time scales of spin ﬂuctuations were revealed. One was related to damped in-plane spin ﬂuctuations at low energy ∼ 2.5 meV and another to an intense central peak with an energy resolution limited width. However, the total spectral weight integrated over energy and momentum is conserved and is close to that in the ortho-II YBa2 Cu3 O6.5 superconductor. The central peak which shows the 3D short-range spin correlations can be ascribed to occurrence of slowly ﬂuctuating spin clusters (spin glass) which are a precursor of the AF order similar to the central peak phenomena observed at structural phase transitions (see, e.g., [180]). A continuous transfer of the spectral weight from the inelastic spin relaxation mode to the central peak on cooling without any anomaly at Tc suggests a single phase in which short-range spin-glass-like state and damped spin ﬂuctuations coexist with superconductivity. 3.3.3 Resonance Mode In the superconducting state, the spin excitation spectrum of the high-Tc cuprates is dominated by a sharp magnetic excitation at the planar AF wavevector QAF = (π, π) which was called the resonance mode (RM). This was discovered by Rossat-Mignod et al. [1068] in the optimally doped YBCO at the energy Er = 41 meV. Later on, the RM was detected in other cuprates as well. Figure 3.18 [1168] shows the intensities of the RMs for several cuprate compounds: (a) YBa2 Cu3 O6.95 (Tc = 93 K) [343, 344]; (b)YBa2 (Cu1−y Niy )3 O7 (Tc = 80 K) [1167]; (c) Tl2 Ba2 CuO6+δ (Tc ∼ 90 K) [446]; (d) Bi2 Sr2 CaCu2 O8+δ (Tc = 91 K, Er = 42.4 meV) [345]. The resonance spin excitation in an overdoped Bi2 Sr2 CaCu2 O8+δ (Tc = 83 K) compound was observed at the lower energy Er = 38 meV [445]. The energy of the RM

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

93

Fig. 3.18. Resonance peak at the wavevector (π, π) for (a) YBa2 Cu3 O6.95 (Tc = 93 K); (b) YBa2 (Cu1−y Niy )3 O7 (Tc = 80 K); (c) Tl2 Ba2 CuO6+δ (Tc ∼ 90 K); (d) Bi2 Sr2 CaCu2 O8+δ (Tc = 91 K) given by the diﬀerence of the neutron intensities at low temperatures. The energy resolution is indicated by the solid bars (reprinted c 2004) with permission by Wiley-VCH from Sidis et al. [1168],

in YBCO and Bi-2212 scales with the superconducting temperature, Er ∼ 5.3 kB Tc as shown in Fig. 3.19. The observation of the RM at Er = 47 meV ∼ 6 kB Tc in the single-layer Tl2 Ba2 CuO6+δ compound by He et al. [446] has demonstrated that the resonance peak is a generic magnetic excitation in the superconducting state of the cuprates with high-Tc ∼ 90 K. A possible interrelation between the RM and the incommensurate spin ﬂuctuations at low energy was discussed in a number of publications. A large increase in intensity and narrowing in the q-width of the resonance peak at the energy Er = 41 meV below Tc = 89 K was observed in the YBa2 Cu3 O6.85 crystal by Bourges et al. [171]. The appearance of the resonance peak was accompanied by weak incommensurate peaks with downward dispersion: E(q) = (Er )2 − (cq)2 where q˜ = |q − QAF| and c = (125 ± 15) meV·˚ A,

94

3 Antiferromagnetism in Cuprate Superconductors

Fig. 3.19. Doping dependence of the magnetic resonance peak energy, Er , (measured by inelastic scattering experiments) and twice the maximum of the single particle gap, 2Δm , (determined from ARPES or SIS tunneling) (reprinted with c 2004) permission by Wiley-VCH from Sidis et al. [1168],

which shows that the incommensurability depends continuously on the energy. A plausible explanation of the RM and its dispersion can be given within the spin triplet exciton model as discussed above. Two RMs of odd and even symmetries were observed in a slightly overdoped Y0.9 Ca0.1 Ba2 Cu3 O7 compound (Tc = 85.5 K) by Pailh`es et al. [943]. Their energies were equal to Er, odd = 36 meV and Er, even = 43 meV, while the intensity of the even mode is three time less than the odd one. In a subsequent publications by Pailh`es et al. (2006) [945,946], two commensurate RMs around AF wave vector were detected in the crystal YBa2 Cu3 O6.85 . In addition to the resonant mode of odd symmetry at Er, odd = 41 meV, the resonant mode of even symmetry at higher energy, Er, even = 53 meV, was detected. The even mode appears six times weaker than the odd one which explains why it has not been observed in earlier experiments. It was also found that the RM of odd symmetry actually has two branches, with upward and downward dispersions which can be ﬁtted by the relations: Er,±odd (q) = (Er,o )2 ± (c(±) q)2 , A, c(+) = 110 meV·˚ A. These where Er,o = 40.5 meV and c(−) = 125 meV·˚ two incommensurate branches are separated by a “silent” band close to the wave vector Q0 = (0.8π, 0.8π) where the scattering intensity vanishes. Two branches of spin excitations close to the resonance mode dispersing downward and upward was called an “hourglass” structure. Detailed measurements of the dispersion of the magnetic RM in optimally doped YBa2 Cu3 O6.95 crystal (Tc = 93 K) by Reznik et al. [1054] also revealed two magnetic resonance branches with upward and downward dispersion

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

95

which merge at the AF wavevector QAF = (π, π) to form the resonance peak at the energy Er = 41.2 meV. Their dispersions were presented by the model: E(−) = Er − aq 2 , with a = 191 meV·˚ A2 for the low-energy branch up to the A2 for spin-gap energy of 33 meV, and E(+) = Er + bq 2 with b = 75–100 meV·˚ the high-energy branch up to the energy of 55 meV. The downward dispersion at small q coincides with the results of Bourges et al. [171] given above: A2 , while the upward dispersion is much smaller. a c2(−) /2Er,o = 193 meV·˚ The intensity ratio of the upper branch to the lower one is of the order of 0.6. The low-energy incommensurate mode approximately has circular symmetry which cannot be reconciled with the picture of the one-dimensional dynamical stripes. So the authors have concluded that the resonance peak and incommensurate magnetic scattering with upward and downward dispersion represent the same physical phenomenon induced by the superconducting transition. (see Sect. 7.2.2). In the early investigations, the RM was detected only in optimally or overdoped YBCO crystals, while in underdoped compounds usually broad peaks were observed as shown in Fig. 3.14. Further investigations suggested that the appearance of broad and symmetric peaks may be due to an oxygen disorder in the Cu–O chains of the YBCO crystals with low oxygen content. This hypothesis was conﬁrmed by the observation of a well deﬁned but asymmetric resonance peak in the underdoped YBa2 Cu3 O6.5 crystal in the oxygen-ordered ortho-II phase by Stock et al. [1190]. The resonance peak at the energy Er ∼ 33 meV (∼ 6.5kB Tc ) appears in the odd (acoustic) mode at the commensurate wavevector QAF = (π, π). It has high intensity at low temperature as shown in Fig. 3.20a in the superconducting phase at T = 6 K. But it is still observed in the normal pseudogap state with much less intensity as shown in Fig. 3.20b for T = 85 K. This suggests that the RM can occur as the spin exciton inside the particle–hole continuum in the normal state pseudogap similar to the spin exciton in superconducting state. A more plausible explanation can be given by consideration of the damping of spin excitations at the QAF point. With lowering of the temperature, the damping decreases and the

Fig. 3.20. Resonance peak for the oxygen-ordered crystal of YBa2 Cu3 O6.5 in orthoII phase (Tc = 59 K) for temperatures T = 8 K (a) and T = 85 K (b). The energy resolution is indicated by the solid bar (after Stock et al. [1190])

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3 Antiferromagnetism in Cuprate Superconductors

peak becomes more sharp, particularly with opening of the superconducting gap below Tc (see Sect. 7.2.2). Later on, a comprehensive study of the RM in the underdoped twin-free YBa2 CU3 O6.6 crystal (hole concentration ph = 0.12 and Tc = 61 K was performed by [455, 456] where an analytical model functions for the magentic susceptibility Imχ(q, ω) in absolute units were derived at T = 5 K and 70 K. It is interesting to compare the energy-integrated spectral weight of the RM with the total magnetic scattering weight. The total spectral weight integrated over all energies and momenta is given by the total momentum sum rule for the dynamical spin susceptibility (3.13). In particular, for the YBCO crystal with two spins of Cu2+ ions per formula unit it reads (see, e.g., [1190]): hω ¯ 2 1 +∞ Imχαα (q, ω) = μ2B g 2 S(S + 1), (3.36) d(¯hω) coth I= πN q 0 kB T 3 where the spin susceptibility is measured in μ2B g 2 units. For S = 1/2 and g ∼ 2 the total sum rules gives I ∼ 2 μ2B . The integral scattering weight of the RM per formula unit for YBa2 Cu3 O6.5 appears to be rather small, Ires ∼ 0.052 μ2B , i.e., only 2.6% of the total spinﬂuctuation intensity [1190]. For a larger oxygen concentration, in the optimally doped and overdoped regions, the intensity of the RM further decreases. For instance, for YBa2 Cu3 O6+x crystals with x = 0.5, 0.7, 0.85, and 1 studied by Fong et al. [347], the following values for the energy and momentum integrated spectral weight of the resonance peak were obtained: Ires (μ2B ) ∼ 0.022, 0.018, 0.022, and 0.014, respectively (divided by a factor of π to compare with the results of Stock et al. [1190]). The small value of the RM intensity suggests that it may be not so important in the superconducting pairing in cuprates and the high-energy spin ﬂuctuations, which observed up to ∼ 2J 250 meV, could play the role of bosons which mediate the pairing. By integrating the spin ﬂuctuation spectrum from 0 to 120 meV in both the momentum and energy over the optic and acoustic modes shown in Fig. 3.16, one ﬁnds Io+a,Σ ∼ 0.3 μ2B or about 13 % of the total weight (3.36). The momentum integrated intensity of spin ﬂuctuations at the RM energy is large, Imχ(Er ) ∼ 30μ2B /eV, while at high energies it weakly depends on the energy and is approximately 4–5 µ2B /eV which is close to the value in insulating YBCO compounds (see Fig. 3.17). In a number of neutron scattering experiments, the inﬂuence of the impurities on the RM was considered. While in the pure YBa2 Cu3 O7 crystal the resonance peak is very sharp, in the system doped by a small amount of nonmagnetic impurities (0.5% of Zn), the peak is substantially broadened [346]. Moreover, contrary to a pure system, where the peak is observed only below Tc , in the Zn-substituted system it persists up to a temperature T ∼ 250 K, much higher than Tc ∼ 87 K. The imaginary part of the dynamical susceptibility in the Zn-doped sample exhibits only a weak inﬂection point near Tc which shows a very small inﬂuence of superconductivity on the spin excitations. The energy-integrated spectral weight at the wavevector QAF = (π, π) for the Zn-doped sample (∼ 2.2μ2B ) appears even larger than for the pure

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

97

YBCO system (∼ 1.6μ2B ) [346]. A comparative study of the spin excitation spectra in Ni- and Zn-substituted optimally doped YBa2 Cu3 O7 crystals by Sidis et al. [1167] unveiled a diﬀerent inﬂuence of these impurities on the RM. While both types of the impurities result in the broadening of the resonance peak (see Fig. 3.18b), Ni impurities (with spin S = 1) do not enhance the normal state intensity of the RM, contrary to the Zn impurities, which produce a considerable increase of the scattering in the normal state. The RM has not been clearly seen in the LSCO crystals. Only a redistribution of the intensity of spin ﬂuctuations at the superconducting phase transition and the opening of the spin gap below Tc were observed, as discussed in Sect. 3.2.4. Instead of a sharp RM, a broad spectrum of spin ﬂuctuations is characteristic for the LSCO systems. One of the possible explanations of this diﬀerence is a much higher disorder in the LSCO crystals produced by Sr-doping in comparison with the oxygen-doped YBCO crystals. Only in the stripe phase of the La1.875 Ba0.125 CuO4 a broad resonance-type spin excitations at h ¯ ω ∼ 50 meV at the AF wavevector (π, π) was detected by Tranquada et al. [1263]. It was accompanied by a low energy branch of incommensurate spin excitations with downward dispersion in the hourglass fashion as in YBCO. Similar results were observed by Fujita et al. [365] in the LBCO crystal in the LTT phase. Dispersive incommensurate magnetic excitations have been observed also in the optimally doped La2−x Srx CuO4 (x = 0.16, Tc = 38.5 K) by Christensen et al. [230]. They can be characterized by a broad continuum at the incommensurate wavevector Qδ (ω) which weakly depends on energy ω. In the superconducting phase, the spectral weight shifts from below the spin gap at 7 meV to higher energies producing a broad maximum at ∼ 12 meV which is reminiscent of the RM in YBCO compounds. A high-resolution neutron scattering study on the same LSCO crystal revealed two energy scales in the spin excitations [1313]. The low-energy part was related to the incommensurate excitations with a weak inward dispersion described above. The intensity of these excitations rapidly dropped down around 20 meV. The highenergy part was peaked at the wavevector (π, π) for energy 40–50 meV and then showed a broad response at higher energies with a long tail up to the highest measured energy transfer of 155 meV. The authors ascribed the lowenergy incommensurate part of the spectrum to electron–hole pair excitations, while the high-energy part was related to the residual AF interaction with the eﬀective exchange coupling J ∼ 80 meV. An observation of the RM in the electron-doped superconductor PLCCO (Pr0.88 LaCe0.12 CuO4−δ ) by Wilson et al. [1364] suggests that the magnetic resonance in the superconducting state is a common feature of high-Tc cuprates. The energy of the resonance Er ∼ 11 meV is much lower than that in the hole-doped cuprates but it also scales with Tc = 24 K: Er ∼ 5.8kB Tc . However, contrary to the hole-doped superconductors, the RM lies close or above the particle–hole continuum Ωc and does not show dispersion being conﬁned in the vicinity of the AF wave vector (π, π). This peculiar properties of the RM in electron-doped cuprates question an interpretation of the magnetic resonance as a spin exciton (see [665]).

98

3 Antiferromagnetism in Cuprate Superconductors

The relationship between a system of localized spins at copper sites in the CuO2 plane with strong AF exchange coupling, and a system of doped holes in the same CuO2 plane, is not yet clear. The observation of two diﬀerent types of spin ﬂuctuation spectra at high and low-energy suggests their diﬀerent nature. The low energy spin ﬂuctuations below the resonance energy may be considered as a collective spin excitations of strongly correlated doped holes, which form a spin exciton in the superconducting state inside the particle– hole continuum or a dynamic stripe structure at low doping. The high-energy spin ﬂuctuations observed up to energies of ∼ 2J should be controlled by the short-range AF exchange interaction of the localized copper spins which play essential role even for optimal-doped samples. A theoretical interpretation of the spin ﬂuctuation spectra and in particular an origin of the RM is discussed in Sect. 7.2.2. Nuclear magnetic and quadrupole resonance studies, which probe the lowenergy part of the spin-ﬂuctuation spectrum, show that the picture of a single one-component spin liquid looks the most plausible. The data on the rate of spin–lattice relaxation obtained in these experiments qualitatively agree with the intensity of low-energy spin ﬂuctuations measured by neutron experiments. We consider NMR and NQR studies in greater detail in Sect. 3.3 where a general magnetic phase diagram concerning a pseudogap crossover is also discussed. 3.3.4 Antiferromagnetism in REBa2 Cu3 O6+x Soon after the discovery of the YBCO system, it was found that the replacement of Y by trivalent rare-earth (RE) ions RE = Nd, Sm, Eu, Gd, Dy, Ho, Er, Tm, and Yb preserves the crystal structure and only slightly aﬀects Tc despite the large RE atomic magnetic moment [485, 788]. This phenomenon indicates a weak coupling of the 4f electrons of the RE ions with the holes in adjacent CuO2 planes, although the distance RE–(O2, O3) does not exceed 2.4 ˚ A . The magnetic subsystem of 4f electrons can therefore be considered as uncoupled to the strong AF correlations of the copper spins in CuO2 planes. Indeed, at very low temperatures, an AF ordering of RE ions was observed. The N´eel temperatures for the rare-earth ions RE = Yb, Nd, Er, Dy, and Gd turned out to be TNR (K) = 0.35, 0.5, 0.5, 1.0, and 2.2, respectively. The AF ordering temperatures of the RE sublattice is so low because the localized moments of 4f shells are weakly coupled to the electrons at the Fermi surface, and TNR are determined primarily by a weak dipole–dipole interaction. The increase of TNR for Er, Dy, and Gd ions may be related only to the increase of their magnetic moments μ = 4.9, 7.2, and 7.4 μB , respectively. Neutron experiments demonstrate that the magnetic moments of RE ions form a simple AF structure with unit cell dimensions doubled in all three directions. Within the compounds with RE = Dy, Gd, and Nd, the magnetic moments are oriented along the c-axis, while for RE = Er compounds one observes a chain AF structure with the moments along the a (or b) axis [753].

3.3 Antiferromagnetism in YBa2 Cu3 O6+x Compounds

99

It is important to note that TNR does not depend on the oxygen content x, which again, conﬁrms a weak coupling of the 4f electrons to the holes in CuO2 planes. Among REBa2 Cu3 O6+x compounds, the most detailed study of the magnetic structure by means of neutron scattering has been performed for NdBa2 Cu3 O6+x (see e.g., [715, 752, 753]). In this compound, the existence of two N´eel temperatures has been found, TN1 > TN2 , which characterize the AF ordering of Cu spins. At the smaller temperature TN2 , one observes the disappearance of the magnetic Bragg peaks (1/2, 1/2, l) with integral values of l and the appearance of new ones with l = 1/2 and 3/2. This implies a doubling of the period of the magnetic lattice along the c axis. It has been interpreted by the authors as ordering of the magnetic moments at Cu1 ions in the chains. The AF coupling of spins in the bilayer CuO2 –RE–CuO2 is maintained, while the coupling between bilayers becomes ferromagnetic due to the AF ordering of the Cu1 copper spins in the chains and in adjacent CuO2 planes as shown in Fig. 3.11b. As the oxygen content x increases, both TN1 and TN2 decrease. For instance, Lynn et al. [753] observed that in NdBa2 Cu3 O6+x , the N´eel temperature TN1 = 430 K for x = 0.1 and TN1 = 230 K for x = 0.35. The N´eel temperature of the ordering of Cu1 moments in the chains was TN2 = 80 K and 10 K, respectively. At low temperatures for an x = 0.1 sample, the saturated ordered Cu magnetic moments for the chains and the planes were μ ∼ 0.35μB and μ ∼ 0.8μB , respectively. At temperatures T > TN2 , the Cu1 spins in the chains become disordered, while the in-plane Cu2 magnetic moments still remain quite high, μ ∼ 0.65μB . As suggested by the authors, the increase of the ordered magnetic moment of Cu2 ions in the planes below TN2 may be explained by the suppression of the quantum spin ﬂuctuations in the plane due to AF coupling between Cu spins in the planes and the chains. However, the Cu chain magnetic moment ordering behavior appeared to be dependent on the sample preparation. It was observed in single crystals with ordered oxygen in the chains and usually was absent in powder samples with disordered oxygen positions in the chains as in the pure YBa2 Cu3 O6+x for x = 0.15 [1138], as discussed in Sect. 3.2.1. Presumably, this unusual behavior of the spins at Cu1 ions can be related to a chemical inhomogeneity of the sample [137]. The ordered magnetic moment at Nd ions at T < TNR = 0.5 K is only μ = 0.38μB and it should not inﬂuence the spin ordering at copper ions. A much higher temperature TNR = 17 K of AF ordering of the magnetic moments of RE ions has been found in PrBa2 Cu3 O7+x (Pr-123) compounds [713]. Unlike in the other compounds with RE ions, the replacement of Y by Pr results in nonmetallic and nonsuperconducting state. With increasing Pr concentration x in the mixed Y1−x Prx Ba2 Cu3 O7−y compounds, Tc monotonically decreases from Tc 92 K at y 0.1, x = 0 and disappears at x = 0.61, while the temperature of AF ordering of Pr magnetic moments TNR increases with and reaches its maximum of 17 K at x = 1 [594]. Taking into account the small value of the ordered magnetic moment μ = 0.24μB and the large value

100

3 Antiferromagnetism in Cuprate Superconductors

of the N´eel temperature TNR , it was suggested that a strong hybridization of the 4f electrons of Pr with the electrons in the copper–oxygen planes should exist. A large value of the Sommerfeld constant γ in the low-temperature electronic speciﬁc heat, comparable to the one for systems of heavy fermions: γ = C/T 200 (mJ/mol K2 ) for x = 0.4 [594], also indicates a strong coupling of the localized 4f electrons of Pr with the holes in the CuO2 plane. The properties of Pr-123 diﬀer drastically from those of other compounds with RE ions due to a much smaller localization of the 4f electrons in the Pr ion which, after Ce, has the smallest charge in the group of 4f elements. Magnetic ordering of Pr spins was also observed in double-chain PrBa2 Cu4 O8 (Pr-124) compound below 17 K [716]. Contrary to Pr-123 compound, Pr-124 retains metallic conductivity but reveals no superconductivity down to 4.2 K. The role of Pr substitution for Y in YBCO compounds is discussed in more detail in Sects. 5.1.2 and 5.2.1 (see Fig. 5.15).

3.4 Nuclear Magnetic Resonance Studies As it was discussed in the earlier sections, in the copper–oxide compounds, the superconductivity occurs in the metallic phase on the background of the spin quantum liquid characterized by strong dynamic AF correlations. A complementary contribution to the understanding of the nature of the AF spin ﬂuctuations observed by neutron scattering experiments has been made in studies of nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR). These methods yield both static (the Knight shift) and dynamic (nuclear spin–lattice relaxation rates) characteristics for a given ion and its nearest neighbors. The most complete data have been obtained by NMR and NQR for the nuclei 63 Cu, 65 Cu, 17 O, and 89 Y in YBCO and LSCO compounds (see, e.g., the reviews by Pennington et al. [974], Alloul et al. [52], Berthier et al. [132], Asayama et al. [81], Slichter [1179]). By studying samples with various oxygen content, one can investigate the variation of the electron and spin characteristics in these compounds during the transition from the AF insulating phase to the metallic state and superconducting phase. The investigations have proved the presence of AF spin ﬂuctuations over the whole range of doping levels which can be described within a one-component spinliquid model. The pseudogap phenomenon in the spin ﬂuctuation spectra in the underdoped cuprates was ﬁrst revealed in the spin–lattice relaxation rate experiments. There are only few studies of another magnetic resonance phenomena – electron paramagnetic resonance (EPR) on Cu2+ spin 1/2. The EPR line is very broad due to very fast relaxation rates of Cu spins and it is diﬃcult to observe the resonance. A distinct EPR signal was discovered on impurity ions like Mn2+ by using a bottleneck eﬀect [624]. In this regime a collective motion of the total magnetic moment of the Mn2+ and Cu2+ spin system can be measured due to the strong isotropic Mn–Cu exchange interaction and a

3.4 Nuclear Magnetic Resonance Studies

101

much faster relaxation rate between the magnetic moments of the Mn and Cu than their relaxation rates to the lattice. Since the Mn spin–lattice relaxation is rather slow, the EPR signal reveals quite a sharp line which can be measured (see also [1149]). Let us now discuss the main physical parameters measured by the NMR method. Under a dc magnetic ﬁeld H0 , the NMR frequency in a transverse radio-frequency ﬁeld is determined by the Zeeman splitting energy hω0 = h ¯ ¯ γn H0 (1 + K),

(3.37)

where γn is the nuclear gyromagnetic ratio. K is the total shift of the NMR frequency due to the interaction of the nuclear spin with electron spins. The total shift is represented as a sum K = K D + K L + K S where K D is the diamagnetic contribution due to inner electron shells, K L the orbital Van Vleck contribution, and K S is the Knight shift in metals due to the spin paramagnetism of the conduction electrons. The diamagnetic term is usually small and together with the orbital shift K L (for s-electrons, K L = 0) it determines a temperature-independent chemical shift. The paramagnetic term K S is due to the ﬁnite density of the s-electron states at the nucleus. It is described by a contact, anisotropic in general, hyperfine interaction Hhf = Aαα Iiα Siα . (3.38) α,i

Here Aαα is the hyperﬁne coupling constant (in energy units) of the nuclear spin Ii with the electron spin Si at the lattice site i. The value of the Knight shift is determined by the static paramagnetic susceptibility χαα 0 KαS = Aαα

χαα 0 , h2 γ γn ¯

(3.39)

where γ is the gyromagnetic ratio for an electron, ¯hγ = 2μB and μB = e¯h/2mc is the Bohr magneton. For a gas of noninteracting electrons, the paramagnetic (Pauli) susceptibility is isotropic and is given by 2 χαα 0 = 2μB N (0),

(3.40)

where N (0) is the density of electronic states per atom per spin direction at the Fermi energy. The width of the NMR line is determined by the longitudinal spin–lattice relaxation rate 1/T1 of the initial magnetization M0 (t) ∝ exp(−t/T1 ) in the external ﬁeld H0 . In metals, the relaxation rate is mainly determined by the interaction of the nuclear spins with the conduction electrons and it may be written as kB T 1 2 Im χ+− (q, ω0 ) 1 = A⊥ , (3.41) T1 h N q ¯ (2μB )2 ¯h ω0

102

3 Antiferromagnetism in Cuprate Superconductors

where A⊥ is the transverse component with respect to M0 ∝ H0 of the hyperﬁne interaction tensor in (3.38). The dynamical spin susceptibility (3.9) χ+− (q, ω0 ) for a circularly polarized radio-frequency ﬁeld describes the spinﬂip process. In a paramagnetic phase for an isotropic case χ+− (q, ω0 ) = 2χαα (q, ω0 ). For a gas of noninteracting electrons, the dynamical spin susceptibility reads (2μB )2 f (p−q ) − f (p ) χ+− (q, ω) = , (3.42) N hω − p−q + p ¯ p where f (p ) = {exp[(p −μ)/kB T ]+1}−1 is the Fermi distribution function, p the energy of the electron with momentum p, and μ is the chemical potential. The static paramagnetic susceptibility (3.40) follows from (3.42) in the static +− (q → 0, ω = 0). uniform limit: χαα 0 = (1/2) Re χ Since in the NMR experiments the inequality h ¯ ω0 kB T usually holds, the integration in (3.41) can be performed in the limit ω0 → 0. Using (3.42) for a noninteracting electron gas, we obtain πkB T A2⊥ ∞ df 1 πkB T A2⊥ 2 = d N 2 ()(− ) = (3.43) N (0). T1 h ¯ d ¯h −∞ If we express the constant of the hyperﬁne interaction in terms of the Knight shift (3.39), we get the Korringa relation 1 4π kB = S 2 h ¯ T1 T (K⊥ )

γn γ

2 Λ.

(3.44)

We introduced here the coeﬃcient Λ to take into account the corrections due to interactions in the electron gas. For noninteracting electrons, Λ = 1. Apart from universal constants, the Korringa relation contains only quantities which can be directly measured in experiments such as the Knight shift K S and the spin–lattice relaxation time T1 . It is therefore convenient for experimental data analysis. A deviation from the Korringa law is usually related to additional mechanisms of relaxation of the nuclear spins. At the transition to the superconducting state, the electrons form Cooper pairs and a gap occurs at the Fermi surface. In the case of a singlet pairing, the electrons with opposite spins in a Cooper pair do not contribute to the paramagnetic susceptibility. Therefore, for the singlet pairing, as T → 0, the electronic Knight shift (3.39) should vanish: KαS (T = 0) = 0. A rapid decrease in the Knight shift at T < Tc points to the singlet electron pairing. In the superconducting phase, due to the formation of a gap Δ at the Fermi surface, the density of the electronic states (DOS) decreases. For the conventional s-wave pairing, when the gap is nonzero everywhere at the Fermi surface, for the DOS in the superconducting state at zero temperature we have: Ns √ (| |< Δ) = 0, while Ns (| |≥ Δ) N0 (0)(| | / 2 − Δ2 ). In view of (3.43), the appearance of a peak in the DOS at | |≥ Δ gives rise to a peak

3.4 Nuclear Magnetic Resonance Studies

103

in the rate of the spin–lattice relaxation at T ≤ Tc . This is referred to as the Hebel–Slichter peak [974]. For the conventional s-wave pairing the relaxation rate for T → 0 should fall exponentially: (1/T1 ) ∝ exp(−Δ/kT ). In the case of unconventional pairing, the energy gap Δ(k) has nodes in some directions in the k-space, as e.g., in the case of d-wave pairing along the directions kx = ky , the spin–lattice relaxation rate does not show the Hebel–Slichter peak at T ≤ Tc and, instead of the exponential decay, a T -power-law decay for the Knight shift and the spin–lattice relaxation rate is to be observed. For nuclear spins I > 1/2 the nuclear quadrupole resonance (NQR) can be studied. The NQR is due to the interaction of the quadrupole nuclear moment Q with the electric ﬁeld gradient (EFG) deﬁned by the surrounding charge distribution [132, 974]: hω Q ¯ I(I + 1) η 2 2 2 HQ = (3.45) Iz − + (Ix − Iy ) , 2 3 3 where ωQ = 3eQVzz /2I(2I − 1)¯ h is the quadrupole frequency and Vαα are the components of the EFG tensor along the principal axes (α = x, y, z). It is assumed that |Vzz | ≥ |Vyy | ≥ |Vxx |, where η = (Vxx − Vyy )/Vzz is the asymmetry parameter. The NQR can be studied in the absence of an external magnetic ﬁeld. This allows to investigate the spin–lattice relaxation avoiding the inﬂuence of magnetic vortices in the mixed state of superconducting phase. Information about the AF exchange interaction between electronic spins can be obtained from measurements of the nuclear spin–spin relaxation rate T2G . It appears that in cuprates the T2G relaxation rate mostly depends on the indirect nuclear spin–spin interaction mediated by the electronic spin susceptibility χ(q ∼ QAF ) [973, 975]. Therefore, the study of the temperature dependence of T2G provides useful information concerning the static part of the staggered spin susceptibility Re χ(q ∼ QAF , ω = 0). 3.4.1 The Knight Shift A considerable contribution to the understanding of the nature of AF spin ﬂuctuations has been brought by early studies of the Knight shift and the spin– lattice relaxation rate in the compounds YBa2 Cu3 O7−y and La2−x Srx CuO4 (see [132,974] and references therein). In view of the variety of the local lattice symmetry at the sites occupied by the nuclei of copper, oxygen, and yttrium in these crystals, AF ﬂuctuations of copper spins contribute diﬀerently to the measured quantities. In this way, it was possible to study the interplay of holes at copper and oxygen sites in the CuO2 plane and clarify whether the holes have independent spin degrees of freedom, i.e., whether they form a two-component spin-liquid, or the hybridization between them is so strong that they should be treated as a one-component spin-liquid. As ﬁrst reported by Alloul et al. [51], the temperature dependence of the Knight shift of yttrium in YBCO scales linearly the macroscopic susceptibility. Figure 3.21 shows the temperature dependence of the 89 Y Knight

104

3 Antiferromagnetism in Cuprate Superconductors

Fig. 3.21. Temperature dependence of the 89 Y Knight shift for various oxygen content 0.35 ≤ x ≤ 1 in YBa2 Cu3 O6+x (after Alloul et al. [51])

b

a 1.3

KII Tc

0.2

K

% 0.5

17K,

63K, %

0.6

Tc

T

Kc

1.2

0.1 Kc

0.4

Kab

0

0.3 0

100

200 T.K

300

0

100

200

300

T.K

Fig. 3.22. The temperature dependence of the Knight shifts: (a) 63 K for Cu2 ions for H c (Kc ) and H ⊥ c (Kab ) and (b) 17 K for oxygen ions O2 and O3 for an external ﬁeld along (K ) and perpendicular (K⊥ ) to the Cu–O bond and along the c-axis (Kc ), in YBa2 Cu3 O6.63 . The results in YBa2 Cu3 O7 are shown by solid line for 63 K reported by Barret et al. [108] and for 17 K⊥ reported by Takigawa et al. c 1991) [1217] (reprinted with permission by APS from Takigawa et al. [1218],

shift at diﬀerent oxygen contents x in YBa2 Cu3 O6+x . This looks similar to the macroscopic susceptibility deﬁned by CuO2 planes (Fig. 3.13). Since the main contribution to the yttrium Knight shift comes from the coupling with the nearest-neighbor eight oxygen ions, this points to a one-component spinliquid. Later on, detail studies of copper and oxygen Knight shifts in YBCO compound by diﬀerent groups have conﬁrmed the one-component spin-liquid picture. Let us consider the results of the Knight shift measurements by Takigawa et al. [1218] in oriented (along the c-axis) powders of YBa2 Cu3 O7−y at y 0, Tc = 90 K and y = 0.37, Tc = 62 K. Figure 3.22a shows the temperature

3.4 Nuclear Magnetic Resonance Studies

105

dependence of the total Knight shift at 63 Cu nuclei for copper ions in the CuO2 plane for the external ﬁeld H c (Kc ) and H ⊥ c (Kab ). The solid line represents these quantities for a sample with y = 0 according to data of Barret et al. [108], while the dots correspond to y = 0.37. In Fig. 3.22b the temperature dependence of the Knight shift 17 O is shown for oxygen ions O2 and O3 for the components along (K ) and perpendicular (K⊥ ) to the Cu–O bond and along the c-axis (Kc ). The solid line represents the results for K⊥ for a sample with y = 0 according to data of Takigawa et al. [1217]. First of all, we see quite diﬀerent temperature dependence of the Knight shifts 63 Kab and 17 K⊥ for overdoped (y = 0) and underdoped (y = 0.37) samples. This result was observed in many experiments also for other nuclei as, e.g., for the yttrium (Fig. 3.21). In the ﬁrst case (the solid line) the dependence K(T ) is typical for conventional superconductors. The Knight shift K S is constant in the normal state and rapidly decreases in the superconducting phase. In samples with smaller oxygen content, y = 0.37, a smooth variation of K(T ) is observed over the entire range of temperature without any remarkable change at Tc . In view of the general expression for the paramagnetic contribution to the Knight shift (3.39), the variation of K(T ) can be related to the temperature dependence of the static paramagnetic susceptibility χ0 (T ) which is shown in Fig. 3.13. For samples with y = 0 the susceptibility is of metallic nature, with the main contribution coming from the temperatureindependent paramagnetic Pauli susceptibility (3.40). The sharp decrease in K(T ) and, therefore, in χ0 (T ) at T < Tc unambiguously indicates the formation of singlet Cooper pairs. In the underdoped sample (y = 0.37), the decrease of K(T ) and the corresponding susceptibility χ0 (T ) with decreasing temperature can be accounted for by the suppression of the AF ﬂuctuations caused by establishing a short-range AF order. A pseudogap formation in the electronic spectrum may also be responsible for this behavior (see Sect. 4.2.1). Assuming that in the superconducting phase K S (T → 0) vanishes, one can estimate the orbital term K L . According to measurements by Barret et al. [108], performed on single crystals of YBCO at y = 0, the orbital 63 KαL and spin 63 KαS terms in the normal phase for the Cu2 nucleus are equal to: KaL = KbL = 0.28, KcL = 1.3, KaS = KbS = 0.3, and KcS = −0.02 (%) (for comparison, see Fig. 3.22a). Measurements of the Knight shift in the same sample for the nucleus Cu1 in the chains, have shown that the paramagnetic term 63 KαS is approximately isotropic KaS = 0.24, KbS = 0.29, KcS = 0.32 and it is accompanied by the same anisotropy of the orbital term KaL = 1.08 (perpendicular to the chain), KbL KcS = 0.27, as for the Cu2 nucleus. To account for the considerable anisotropy of the Knight shift for Cu2, S (KcS 0, Kab 0.3), the nature of the hyperﬁne interaction of the nuclear spins with electron spins should be considered in greater detail. As was proposed by Mila et al. [833] and Shastry [1141], the Hamiltonian of the hyperﬁne interaction in copper–oxide superconductors is largely determined by the polarization of the s-electrons of the corresponding atoms due to their hybridization with holes in the 3d(x2 − y 2 ) states at planar copper sites.

106

3 Antiferromagnetism in Cuprate Superconductors

Therefore, the hyperﬁne interaction for copper nuclei should be written as a sum of two terms. The ﬁrst one is an anisotropic contact interaction Aα due to a spin–orbital coupling of 3d holes and a dipole interaction of the nucleus at the same copper site. The second term is an isotropic transferred hyperﬁne interaction B due to the polarization of 4s copper shells by 3d spins at four neighboring copper sites. Taking into account only the transferred hyperﬁne interaction for other nuclei in the crystal (oxygen and yttrium in YBCO), we arrive to the Mila–Rice–Shastry model Hamiltonian: 63 α 63 α Ii Aα Siα + Ii B Sjα Hhf = iα

+

α

17 α Ii

Cα Sjα +

α

89 α Ii

Dα Sjα ,

(3.46)

α

where < ij > denotes the summation over the nearest-neighbor (n.n.) lattice sites, Cα is the transferred hyperﬁne interaction for the planar oxygen from two n.n. copper sites, and Dα is the transferred hyperﬁne interaction for the yttrium from eight n.n. copper sites. This model is commonly used in theoretical description of the NMR data, as e.g., in the phenomenological theory of Millis et al. [834, 835] applied to the analysis of the NMR data in YBCO [847] and LSCO [848] samples. In view of (3.39), for the copper Knight shift we obtain 63

KαS =

Aα + 4B 2 χ0 , 63 γ γ ¯ n h

(3.47)

where α = c or α = a, b for an external magnetic ﬁeld along or perpendicular to the axis c of the YBCO crystal, respectively. Since KcS = 0 regardless of the oxygen content y (Fig. 3.22a), the condition Ac + 4B = 0 should hold. This mutual compensation of the direct and transferred hyperﬁne interactions for planar copper ions is also observed in La2−x Srx CuO4 [848] but it does not hold for Tl-compounds [580, 1431, 1432], Bi-compounds [531, 1332] and Hg-compounds [565, 770]. The hyperﬁne interaction for oxygen nuclei in the CuO2 plane (O2, O3) is determined, in fact, also by the spin density for the 2p orbitals. The hyperﬁne ﬁeld from these orbitals gives rise to an anisotropic dipole interaction Ap that determines the axial contribution to the Knight shift 17

S Kax =

1 (K − K⊥ ) = 3

Ap χ0 . h2 n γ¯

17 γ

(3.48)

The isotropic part of the Knight shift for oxygen nuclei due to the polarization of the 2s oxygen states by 3d spins at two neighboring copper sites reads 17

S Kiso =

1 (K + K⊥ + Kc ) = 3

2C χ0 , h2 nγ¯

17 γ

(3.49)

3.4 Nuclear Magnetic Resonance Studies

107

S Since the anisotropic part 17 Kax > 0 for samples y = 0 [1217] and with y = 0.37 [1218], i.e., K > K⊥ , the spin density at ions O2 and O3 should be concentrated at 2pσ orbitals along the Cu–O bond. The Knight shift for yttrium in YBa2 Cu3 O7−y after the summation over the eight nearest copper sites is given by 89

S Kiso =

8D χ0 . h2 nγ¯

(3.50)

89 γ

In fact, the transferred hyperﬁne interaction D is related to the interaction of the 89 Y nucleus with the eight nearest oxygen sites, i.e., with the spin density of the O2p orbitals, while the macroscopic susceptibility χ0 is on the whole determined by 3d spins at Cu2 sites. In writing (3.47)–(3.50), we have implicitly assumed that the static susceptibility at copper and oxygen sites is described by a one-component paramagnetic susceptibility χ0 (T ), i.e., we have accepted the model of a onecomponent spin-liquid. The most important result of Takigawa et al. [1218] is the experimental check of the validity of the one-component, or the single spin-liquid model. Figure 3.23 shows the temperature dependence of the Knight shift 63 Kab related to the susceptibility at copper sites χd0 and 17 Kα related to the susceptibility at oxygen sites χp0 for a sample with y = 0.37. The dependence ﬁts a single temperature-dependent function χ0 (T ) under a corresponding choice of scale (to take into account the constants of the hyperﬁne interaction) and shift of the origin (due to the orbital terms). The temperature-independent values of the Knight shifts for the YBCO sample at y = 0 are also plotted by vertical lines. Within the experimental accuracy, this 0.20

0.25 0.05

0.6

2.5 0.15

1.5 Tc

17K c 17K ax 17 Kiso 63K ab

0.05 0.02

0.05 0.3

0 100

200

1.0

x0, (states/ev)

0.03

0.10

0.10

17K

c,

%

%

0.15

17K , ax

% 17K , iso

63K

0.4

2.0

0.04

0.5

ab,

%

0.20

0.5 0 300

Fig. 3.23. Reduced temperature dependence of the various components of the Cu and O Knight shifts for YBa2 Cu3 O6.63 with T -independent values for the YBa2 Cu3 O7 sample shown by dashed line (reprinted with permission by APS from c 1991) Monien et al. [847],

108

3 Antiferromagnetism in Cuprate Superconductors

gives for the susceptibility in the normal phase, χ0 /μ2B = 2N (0) 2.6 eV−1 [847]. As shown in Fig. 3.21, the temperature dependence of the yttrium Knight shift 89 K [51] at various oxygen contents in YBa2 Cu3 O6+x , is scaling linearly the macroscopic susceptibility, in agreement with the picture of a onecomponent spin quantum liquid. Subsequent investigations in Tl-compounds [580, 1431, 1432], Bi compounds [531, 1332] and Hg compounds [565, 770] supported the one-component spin-liquid model in cuprates. 3.4.2 Spin–Lattice Relaxation Already early experiments in copper–oxide superconductors revealed an anomalous nuclear relaxation rate at the copper sites in CuO2 plane that strongly deviated from the Korringa relation (3.44) and was much faster than at the oxygen sites in the same CuO2 plane [489, 1340, 1387]. For the planar oxygen O2, O3, and Y nuclei in YBCO, instead of the Korringa law (3.44) the relation T1 T K S ∼ const. was observed. Further investigations have shown that the temperature dependence of the spin–lattice relaxation rates noticeably depend on the doping and have rather complicated temperature dependence. These results were considered as a manifestation of AF spin ﬂuctuations which were later analyzed within the model of the transferred hyperﬁne interaction (3.46). Let us discuss these results in more detail. To obtain accurate results, usually powder samples with a high degree of c-axis orientation produced by magnetic ﬁeld alignment were studied. Figure 3.24 shows the spin–lattice relaxation rates in the YBa2 Cu3 O7−y ori-

a

b 0.4

y = 0.37

6 ( 63 T1, T)–1, (sek. k) –1

y=0

(17T1T)–1,(sek. k)–1

7

0.3

0.2

y = 0.37

0.1

Tc

5 y=0

4 3 2 1

0

Tc

0

100

200 T.K

300

0 0

100

200

300

T.K

Fig. 3.24. Temperature dependence of the spin–lattice relaxation (T1 T )−1 at the O2 (O3) sites (a) and at the Cu2 sites (b) in the ﬁeld H c in YBa2 Cu3 O7−y (y = 0.37). The data for y = 0 (open circles) are from [424] (reprinted with permission by APS c 1991) from Takigawa et al. [1218],

3.4 Nuclear Magnetic Resonance Studies

109

ented powder sample at the planar oxygen and copper sites at y = 0.37 measured by Takigawa et al. [1218]. The temperature dependence of the relaxation rates, namely the function (T1 T )−1 , for 17 O at sites O2 (O3) in a ﬁeld H c is plotted in the left panel (a) and the same dependence for 63 Cu at Cu2 sites is shown in the right panel (b). The results for y = 0 by Hammel et al. [424] are shown by open circles also. If we compare the temperature dependence of the relaxation rate and the Knight shift for oxygen nuclei (Fig. 3.22b), we ﬁnd that for the sample with y = 0 the Korringa relation (3.44) holds with constant Λ = 1.4. In view of the temperature dependence of the static susceptibility shown in Fig. 3.23, the Korringa relation is no longer valid for the sample with y = 0.37. However, one has instead the relation 17 S T1 T 17 K⊥ ∼ const. This linear scaling of the relaxation rate (17 T1 T )−1 at S the O(2,3) site with the Knight shift 17 K⊥ was conﬁrmed for YBCO single crystals by Horvati´c et al. [490, 491]. The same relation, 89 T1 T 89 K ∼ const., applies also to 89 Y nuclei over a wide range of values 0 < y < 0.6 [51, 913]. At the same time, there is no simple relation between the relaxation rate and the Knight shift for 63 Cu nuclei. An attempt to compare these quantities on the basis of the Korringa relation (3.44) for the sample with y = 0 yields Λ = 11 at T = 100 K [424]. This result indicates the existence of an additional mechanism of the relaxation at Cu2 nuclei as compared to the O2 (O3) and Y nuclei. For underdoped compounds, with decreasing temperature, the function (63 T1 T )−1 at ﬁrst increases, passes through a maximum at some temperature T ∗ > Tc , and then decreases with T showing no appreciable anomaly at Tc . In all the cases the ratio of the copper and oxygen relaxation rates (63 T1 )−1 /(17 T1 )−1 1 (∼20–40) and increases with decreasing temperature up to T ∼ Tc [1252]. The diﬀerent behavior of the relaxation rates at Cu2 and at O2 (O3) sites can be accounted for if a nonlocal nature of the transferred hyperﬁne interaction in copper–oxide materials is taken into consideration. According to the model (3.46), the hyperﬁne interaction contains not only a contact contribution but also a transferred hyperﬁne interaction induced by 3d spins at neighboring Cu2 sites. As a result, in the calculation of the spin–lattice relaxation rate (3.41), instead of the local coupling constant A⊥ one needs to use the wave vector dependent coupling constants 63 17 89

Aα (q) = Aα + 2B (cos qx a + cos qy b), Aα (q) = 2Cα (cos qx a/2),

(3.51) (3.52)

Aα (q) = 8Dα cos(qx a/2) cos(qy b/2) cos(qz c/2),

(3.53)

where q is measured from the zone center. The Knight shifts are determined by the values of these functions at q → 0 and by the uniform static susceptibility χ(q → 0, ω = 0) as given by (3.47), (3.49), and (3.50). At the same time, AF spin ﬂuctuations give the largest contribution to the susceptibility at the planar AF wave vector QAF = (π, π) when the spins at the neighboring Cu2 sites have opposite directions. Since the sites O2 (O3) and Y are arranged in

110

3 Antiferromagnetism in Cuprate Superconductors

a symmetric way with respect to the Cu2 sites, their net contribution to the AF ﬂuctuations vanishes: 17 Aα (QAF ) = 89 Aα (QAF ) = 0. Thus, the AF spin ﬂuctuations of 3d spins are ﬁltered out at the sites O2 (O3) and Y but they give a ﬁnite contribution at the Cu2 sites: 63 Aα (QAF ) = Aα − 4B. To provide a quantitative description of the role of AF spin ﬂuctuations in the spin–lattice relaxation rate (3.41), Millis, Monien, and Pines (MMP) have proposed a phenomenological model for the dynamic spin susceptibility [834,835]. A similar phenomenological model was developed independently by Moriya group (see [860, 1286]). The model takes into account both contributions from the quasi-particles and the AF spin ﬂuctuations as a sum of two terms: (ξ/ξ0 )2 1 χ(q, ω) = χ0 (T ) . (3.54) + 1 − iπω/Γ 1 + ξ 2 (QAF − q)2 − iω/ωsf The quasi-particle term is determined by two parameters, namely, the uniform static susceptibility χ0 (T ) and the speciﬁc electronic energy ¯hΓ/π which is of the order of the Fermi energy EF . The contribution from the AF ﬂuctuations is described by the AF static susceptibility χ(QAF ) = χ0 (ξ/ξ0 )2 and the typical energy of AF ﬂuctuations h ¯ ωsf = (¯ hΓ/π)(ξ0 /ξ)2 , where ξ(T ) is the correlation length of AF spin ﬂuctuations. It is assumed that (ξ/ξ0 )2 1, and therefore χ(QAF ) χ0 and Γ/π ωsf . The Knight shift is determined by the uniform static susceptibility (3.55) χ0 (T ) = χ(0, 0) = χ0 (T )[1 + β/2π 2 ]. Here the contribution of AF ﬂuctuations, deﬁned by the parameter β = (a/ξ0 )4 ∼ 10, appears to be small, of the order of 16 %. Therefore, temperature dependence of all the Knight shifts is determined by a single function which is the quasi-particle static susceptibility χ0 (T ). Now let us consider the temperature dependence of the spin–lattice relaxation rate (3.41) which is determined by the function kB T Imχ(q, ω + iδ) hω ¯ ⎡ 2 ⎤ 2 (ξ(T )/a) kB T ⎦. (3.56) χ0 (T ) ⎣1 + β =π hΓ ¯ 1 + ξ 2 (T )(QAF − q)2

S(q, ω → 0) = lim

ω→0

The contribution coming from AF ﬂuctuations is determined by the second term which is small at q = 0, being of the order β/4π4 0.03. However, it becomes the major term at q = QAF , of the order β(ξ/a)2 1. Therefore, the spin–lattice relaxation rate at Cu2 sites depends essentially on the AF spin ﬂuctuations since, according to (3.51), the major contribution comes from the region of q ∼ QAF . Contrary to Cu2 sites, the region q ∼ QAF does not contribute, according to (3.52), (3.53), to the relaxation rate at the sites O2 (O3) and Y. Therefore it does not exhibit any essential dependence on AF

3.4 Nuclear Magnetic Resonance Studies

111

ﬂuctuations. This accounts for the very diﬀerent temperature behavior of the relaxation rates at the nuclei Cu2 and O2(O3) or Y. The calculation of the Knight shifts (3.47), (3.49), and (3.50) on the basis of the representation (3.55) enables the temperature dependence χ0 (T ) to be determined and the constants of the hyperﬁne interaction in (3.51)–(3.53) to be evaluated. The remaining three parameters of the MMP model, i.e., (ξ(T )/a), Γ , and β = (ξ0 /a)4 , can be determined by comparing calculations with experimental data for the relaxation rates for Cu2, O2, and Y nuclei. In particular, the temperature dependence of the correlation length (ξ(T )/a) can be obtained from the ratio of the relaxation rates 2 2 B β ξ(T ) + ... , (3.57) R1 = (63 T1−1 )/(17 T1−1 ) ∝ 2 1 + 1.7 2 C π a where only the leading term is written. The experimental values of (3.57) at temperature T = 100 (300) K for y = 0.37 YBCO sample are R1 = 43 (16). For the correlation length in (3.57) then we get: (ξ/a) ∼ 3.8 (2.2), respectively. For the y = 0 YBCO sample, R1 ∼ 16 and the correlation length (ξ/a) ∼ 2.3 at T = 100 K has a weak temperature dependence [847]. The MMP theory was also used by Monien et al. [848] to discuss the results of NMR experiments in the LSCO compound. Thus, according to the MMP theory, a complicated temperature dependence of the spin–lattice relaxation rate at the Cu2 sites is accounted for by an interplay of temperature dependence of the static susceptibility χ0 (T ) shown in Fig. 3.23 and the correlation length ξ(T ). In particular, the maximum of the 1/(T1 T ) curve for 63 Cu at y = 0.37 (Fig. 3.24b) in the region T ∼ 150 K is due to an increase in ξ(T ) and a decrease of χ0 (T ), with decreasing temperature. A detailed comparison by Millis and Monien [835] of experimental data with calculations on the basis of the MMP theory revealed that, taking into account the weak temperature dependence of the parameter ¯hΓ = 0.4–0.5 eV, the model parameter β = (ξ0 /a) ∼ π2 can be taken for a constant. In this case, the spin–lattice relaxation rates at O2 (O3) nuclei and Y are determined only by the temperature-dependent static susceptibility so that they satisfy the relation (T1 T )−1 ∼ K S . A quantitative comparison between the NMR and inelastic neutron scattering (INS) experiments in YBCO single crystals was performed by Horvati´c et al. [490] and Berthier et al. [131]. The temperature dependence of the spin– lattice relaxation rates at Cu sites, which is dominated by the dynamical spin susceptibility, Im χ(QAF , ω → 0), has been evaluated from the spinﬂuctuation spectrum measured in INS experiments for a ﬁnite energy transfer ¯hω ∼ 10 meV. It appeared that NMR and INS results for Cu2 sites agreed quite well, while the spin–lattice relaxation rates for O(2,3) sites evaluated from INS revealed a much stronger contribution from AF spin ﬂuctuations (due to a large q-width of INS) than measured in NMR experiments. The discrepancy of the results at the oxygen sites may be explained either by an

112

3 Antiferromagnetism in Cuprate Superconductors

uncertainty in the measurement of the q-width of spin excitations in INS, or a failure of the one-component spin-liquid model. A similar discrepancy for the spin–lattice relaxation rates at the in-plane oxygen from those calculated from the INS results was evidenced by Walstedt et al. [1333] in the LSCO compound. They concluded that the Knight shift and relaxation rate of the planar oxygen nuclei are not driven by the copper-site susceptibility. These results question the validity of the one-component spin-liquid model. To reconcile the NMR results and the inelastic neutron scattering (INS) data on both LSCO and YBCO, Zha et al. [1418] have introduced a modiﬁed MMP model for the dynamical spin susceptibility χ(q, ω) =

χ0 (T ) χQi 1 + , 4 i 1 + ξ 2 (Qi − q)2 − iω/ωsf 1 − iπω/Γ

(3.58)

where Qi are four incommensurate (IC) wavevectors of the low-energy AF spin ﬂuctuations found in INS experiments in the LSCO (see Sect. 3.2.4, Fig. 3.7) and in the underdoped YBCO crystals (see Sect. 3.2.2, Fig. 3.15). It is assumed that χQi = (αξ 2 ) χ0 as in the model (3.54). The IC AF spin ﬂuctuations in (3.58) give a substantial contribution to the 17 T1 spin–lattice relaxation rate within the Mila–Rice–Shastry model (3.46) and bring about an anomalous temperature dependence of the 17 T1 relaxation rate which is not seen in experiments. To resolve this problem, a transferred hyperﬁne interaction Cα between the 17 O nuclei and their next-nearest neighbor Cu2+ spins in the model (3.46) was introduced. This interaction changes the wave vector dependence of the coupling constant (3.52) and for a fraction of r = Cα /Cα = 0.25 eﬀectively screens out a contribution from the IC AF spin ﬂuctuations seen by 17 O nucleus. At the same time, by assuming IC AF spin ﬂuctuations in the underdoped YBCO compounds, it was possible to explain within this model temperature dependence of the planar anisotropy of 17 T1 observed in NMR experiments in YBCO [800]. To explain the doping dependence of the anisotropy of the 63 T1 relaxation rate, a variation with doping of the transferred hyperﬁne interaction B between a Cu2+ spin and its nearest-neighbor 63 Cu nucleus in the model (3.46) was suggested. Thus, by a modiﬁcation of the MMP model (3.58) and by introducing a transferred hyperﬁne interaction Cα for the next-nearest neighbors for the oxygen coupling constant (3.52), it was possible to reconcile the NMR experiments with the observed in the INS experiments IC AF ﬂuctuations within the one-component spin-liquid model. Concerning the spin–lattice relaxation at T < Tc , we note two very important facts. The ﬁrst one is the absence of a Hebel–Slichter peak at T < Tc for 17 O nuclei in YBCO at y = 0 (Fig. 3.24b). The second is a T -power-like decrease of the relaxation rates at T < Tc in comparison with the exponential decay in conventional superconductors. These features, observed already in early experiments, were later explained as originating in the d-wave symmetry of the superconducting pairing. For instance, the study of the spin–lattice relaxation time T1 by Gippius et al. [394] in a superconducting HgBa2 CuO4+δ

3.4 Nuclear Magnetic Resonance Studies

113

sample with Tc = 96 K revealed that the dx2 −y2 -wave symmetry of the superconducting gap with the ratio 2Δ/kB Tc = 7 ﬁtted quite well the experimental data. The dx2 −y 2 -wave symmetry of the gap was proved unambiguously in phase-sensitive experiments (see Sect. 5.5). 3.4.3 Spin Pseudogap NMR studies of the temperature dependence the Knight shift (3.39) related to the static uniform spin susceptibility χ0 and the spin–lattice relaxation (63 T1 T )−1 on copper sites determined by the low-frequency spin susceptibility in the vicinity of the AF wavevector have revealed speciﬁc crossover lines in the magnetic phase diagram of the cuprate superconductors. The crossover lines can be related to a pseudogap phenomenon observed in other physical properties as speciﬁc heat (Sect. 4.2.2), electronic spectra measured in ARPES experiments (Sect. 5.2.2), optic properties (Sect. 5.3), transport properties (Sect. 5.4), and tunneling spectra (Sect. 5.5.2) (for a review see [1244]). Inelastic neutron scattering experiments (Sects. 3.2.4 and 3.2.2) have revealed a spin gap in the spin-ﬂuctuation spectrum in LSCO and YBCO compounds at low temperatures, in particular, at T < Tc (see Figs. 3.10 and 3.14). However, a certain redistribution from low to high energy parts of the AF spinﬂuctuation spectrum in the underdoped compounds starts at a temperature Ts higher than Tc . This phenomenon, called later as a spin pseudogap formation, in fact, was ﬁrst discovered in NMR experiments in underdoped YBCO [489, 1340]. It was observed that the spin lattice relaxation (63 T1 T )−1 with temperature lowering passed through a maximum at temperature Ts ∼120– 150 K and then decreased (see Fig. 3.24b). Yasuoka et al. [1387] pointed out that this behavior is explained by a suppression of the low-energy part of the AF spin-ﬂuctuation spectrum caused by opening a spin pseudogap. In comparison with the inelastic neutron scattering experiments, where the spin-ﬂuctuation spectrum Im χ(q, ω) is measured for energies h ¯ ω ≥ 1 meV, in the spin–lattice relaxation the quantity Im χ(QAF , ω0 ) ∂ Im χ(QAF , ω) ∼ lim ω→0 ω0 ∂ω is determined since the NMR energies are extremely small: ¯hω0 ∼ µeV. Therefore, a change of the slope of the dynamical spin susceptibility at ω → 0 due to the spectral weight transfer of AF spin ﬂuctuations from low to high energies accounts for the pseudogap phenomenon in NMR experiments. The spin pseudogap can be accurately determined by studying the variation of the ratio of the spin–spin and spin–lattice relaxation. The nuclear spin–spin relaxation rate 1/T2G measured in spin-echo experiments enables to estimate indirect electronic spin–spin interaction proportional to Reχ(q ∼ QAF , ω = 0) [973,975]. The ratio of the spin–spin to the spin–lattice relaxation

3 Antiferromagnetism in Cuprate Superconductors

(T1T)–1 / (A - 4B)2 (a.u.)

a

b

2

3 (T2g)–1 / (A c– 4B)2 (a.u.)

114

LSCO x= 0.075

2

0.15 0.24 1

0

Y124 Y123 TI-1 0

100

Y1236.63 Y124 Y1236.92 1 Y1236.94 TI-3 TI-1

TI-3 200 300

0

0

100

200

300

Temperature (K)

Temperature (K)

Fig. 3.25. Temperature dependence of (a) the NMR relaxation rate (63 T1 T )−1 and (b) the nuclear spin–spin relaxation rate (63 T2G )−1 for various cuprates: La2−x Srx CuO4 (LSCO, x = 0.075, 0.15, 0.24), YBa2 Cu3 Oy (Y123y ), YBa2 Cu4 O8 (Y124), Tl-2201 (Tl-1), and Tl-2223 (Tl-3) (after Berthier et al. [132])

rates is given by the expression: χ(QAF ) (63 T2G )−1 ∝ = 63 −1 ( T1 T ) Imχ(QAF , ω0 )/ω0

dω Imχ(QAF , ω )/πω , Imχ(QAF , ω0 )/ω0

(3.59)

where we have used (3.15) for deﬁnition of the static spin susceptibility χ(QAF ). When the spectral weight transfer of AF spin ﬂuctuations from low to high energies occurs at the pseudogap formation, the integral over the energy does not change too much, while the amplitude Imχ(QAF , ω0 )/ ω0 at low energy decreases rapidly. Figure 3.25 compares (a) the NMR relaxation rate in dimensionless (arbitrary) units (63 T1 T )−1 /(A − 4B)2 and (b) the nuclear spin–spin relaxation rate (63 T2G )−1 /(Ac − 4B)2 for various cuprates. For the underdoped YBa2 Cu4 O8 (Y-124) compound, it is clearly seen that the maximum of (63 T2G )−1 appears at a temperature T ∼ Tc which is lower than the maximum of (63 T1 T )−1 at T = Ts ∼ 150 K. Therefore, the ratio (3.59) will show a sharp drop at the pseudogap temperature Ts . A large pseudogap temperature was observed in the underdoped three-layered mercury cuprate HgBa2 Ca2 Cu3 O8+δ by Julien et al. [565], Ts ∼ 230 K, which correlates with high Tc = 115 K. The general conclusion inferred from NMR and NQR experiments is that the spin pseudogap temperature Ts in underdoped cuprates decreases with doping and at the optimal doping approaches Tc [132]. An anomalous decrease of the Knight shift below a certain temperature T ∗ also points to a spin pseudogap formation. At this temperature the Knight shift reaches its maximum and at lower temperature decreases. Sometimes, T ∗ is deﬁned as the temperature where the Knight shift changes behavior from being independent of temperature to the linear T -dependence (see, e.g.,

3.4 Nuclear Magnetic Resonance Studies

115

[1244]). Since the Knight shift (3.39) is related to the static uniform spin susceptibility χ0 which is determined by the average density of electronic states (DOS), a sharp decrease of the Knight shift implies a pseudogap opening in DOS. At the same time, the static uniform spin susceptibility χ0 (T ) scales with the Sommerfeld constant γ(T ) in the electronic speciﬁc heat which also shows a pseudogap behavior in underdoped cuprates below a certain crossover temperature T ∗ (see Sect. 4.2.1). Therefore, the spin pseudogap detected by the Knight shift and the pseudogap in electronic DOS should have a similar origin and therefore we have used them for the same notation T ∗ (in literature, a variety of notation is used for the spin (pseudo)gap and the pseudogap crossover temperatures). Contrary to the pseudogap temperature T ∗ related to the static susceptibility χ0 (q = 0), the spin pseudogap temperature Ts is related to a gap in low-energy spin excitations at the AF wavevector QAF determined by the dynamic susceptibility χ(QAF , ω → 0). Diﬀerent origin of these crossover temperatures is conﬁrmed by studies of the Ni- and Zn-impurity doping eﬀects. Antiferromagnetic spin ﬂuctuations and Ts are signiﬁcantly suppressed by Zn but not by Ni impurities, whereas T ∗ is insensitive to Zn impurities but decreasing roughly linearly with hole concentrations. As we discussed in Sects. 3.2.2 and 3.2.3, Zn impurities strongly aﬀect the AF correlations: they wash out the spin gap and broaden the resonance peak at q ∼ QAF even at small concentrations, of the order of 1%, and completely suppress the spin pseudogap (see, e.g., [1430, 1433]). In this respect, it is interesting to point out that the spin pseudogap at Ts is not well pronounced in the LSCO compounds. Apart from the incommensurability of AF spin ﬂuctuations discussed in Sect. 3.2.4, a possible origin of the weak spin pseudogap may be due to a disorder produced by Sr-doping which is equivalent to a disorder resulted from the Zn-doping. As shown by Ohusugi et al. [916] in the 63 Cu NQR and NMR study in LSCO compound, the full width at half maximum of the Cu NQR peak is about one order of magnitude larger than in YBCO compounds. These authors explained it by the random distribution of Sr content in LSCO. To explain crossover phenomena observed in the magnetic phase diagram of cuprates and, in particular, the pseudogap phenomenon, several phenomenological models have been proposed (for a review see [901]). In the ﬁrst class of the models, the pseudogap revealed in the static spin susceptibility is related to the spin-singlet formation at T ∗ as in the resonating valence bond (RVB) scenario of Anderson [61]. In that scenario, the spin singlets are incoherent preformed pairs which become superconducting only at a higher doping. In the second class of models, the pseudogap phase reﬂects the onset of strong AF spin correlations, a spin-liquid without long-range order (see, e.g., [109, 110, 1121] and references therein). In the second scenario, the pseudogap formation below T ∗ competes with the superconducting pairing of the Fermi-like quasiparticles at Tc < T ∗ . There are also scenarios where a true quantum phase transition (at zero temperature) is suggested, which is related to some “hidden” order parameter below the superconducting “dome”. Below

116

3 Antiferromagnetism in Cuprate Superconductors

we consider in more detail the AF scenario which is based on analysis of NMR studies. A microscopic model of the pseudogap caused by short-range AF spin ﬂuctuations will be considered in Sect. 7.2.1. A uniﬁed magnetic phase diagram for LSCO and YBCO compounds was proposed by Barzykin et al. [109,110] by considering the scaling behavior both for the in-plane static susceptibility χ0 (T ) probed by the 63 Cu Knight shift and the NMR relaxation rate 63 T1 T . Figure 3.26a shows the scaling function χ(T )/χmax normalized to its maximum χmax = χ(T ∗ ) at temperature T ∗ for the YBCO compounds at various doping. Earlier, the scaling behavior for the bulk magnetic susceptibility χ(T ) in LSCO compounds was found by Johnston [559] and was reexamined by Nakano et al. [882] using samples of high purity. It was shown that the T -dependent part of the susceptibility χ(T )/χmax plotted as a function of the reduced temperature T /Tmax follows the universal curve regardless of hole concentration. It is remarkable, that the scaling behavior for χ(T )/χmax in Fig. 3.26a for YBCO compounds is also well described by Nakano et al., scaling function is shown by solid line. At the same time, Nakano et al. scaling behavior shown by dots in Fig. 3.26b perfectly ﬁts to the Monte Carlo calculations for the 2D Heisenberg model by Makivi`c et al. [777] shown by the solid line. This observation suggests that the temperature dependence of the static susceptibility in a broad region of hole concentration can be determined by the AF spin ﬂuctuations within the 2D quantum Heisenberg model (2DQHM) of localized spins, while the quasiparticle component of doped holes gives no temperature-dependent contribution. Barzykin et al. [109] have suggested that at T ∗ a crossover occurs from the conducting Fermi liquid to a “pseudogap matter” with the quantum critical point (QCP) at zero temperature close to the doping level xc ∼ 0.22. Therefore, at T < T ∗ a two-component description can be suggested where insulating spin liquid coexists with Fermi liquid. In this case, the low-frequency dynamic spin susceptibility can written in the

Fig. 3.26. (a) Scaling for the 63 Cu Knight shift in YBCO-compounds at various doping (symbols). (b) The scaling function obtained by Nakano et al. [882] for the bulk spin susceptibility for LSCO compound (dots) (for details see the text). (After Barzykin et al. [109])

3.4 Nuclear Magnetic Resonance Studies

117

form: χ(q, ω) = f (x) χSL (q, ω) + [1 − f (x)] χFL (q, ω).

(3.60)

where f (x) and [1 − f (x)] are the fractions of spin liquid (SL) and Fermi liquid (FL), respectively. The spin-liquid fraction in the static long wavelength limit reveals the scaling behavior shown in Fig. 3.26: f (x)χSL (T ) = χmax χ(T ˜ /T ∗ (x)), while the Fermi bulk susceptibility χFL is doping and temperature independent. The crossover temperature T ∗ (x) in this model is determined by the eﬀective exchange interaction T ∗ (x) = J(1 − x/xc ). It should be pointed out, that a similar doping dependence has been proposed for the pseudogap Eg in the DOS at the Fermi energy deduced from the electronic speciﬁc heat measurements (see Sect. 4.2.2, Fig. 4.4). In this model, the localized spin ﬂuctuations can be described within the 2DQHM or by the quantum nonlinear 2D sigma model (QNLSM). As discussed in Sect. 3.2.3, a quantum phase transition takes place for an eﬀective coupling constant g = gc . In the quantum critical (QC) regime the inverse magnetic correlation length ξ(T )−1 is proportional to temperature T for all T . The N´eel order exists for g < gc where the renormalized classical behavior is observed (see (3.20)), while for g > gc a quantum disordered (QD) phase appears with an excitation gap (quantum paramagnet) and the temperatureindependent correlation length (see (3.21)). Based on these results, Barzykin et al. [109] have assumed that below the temperature T ∗ (x) a crossover occurs from mean-ﬁeld behavior with a dynamical critical exponent z = 2 to a QC regime with a dynamical critical exponent z = 1 which is followed by a crossover to a QD state below Ts . The QC regime in YBCO and LSCO has been conﬁrmed by NMR measurements of the spin–lattice relaxation rate (63 T1 )−1 and the spin–spin relaxation rate (63 T2G )−1 . In particular, a linear temperature dependence was observed for the inverse correlation length and 63 T1 T : ξ −1 (T, x) ∝ T /T ∗ + a(x),

63

T1 T ∝ ωs ∝ T /T ∗ + A(x).

(3.61)

The scaling of the 63 T1 T relaxation rate for LSCO compound is shown in Fig. 3.27a (a similar scaling is observed for YBCO). The oﬀset A(x) depends linearly on x and tends to zero suggesting a QCP to a magnetically ordered state at xc ∼ 0.05 as shown in Fig. 3.27b. (The oﬀset a(x) in (3.61) is also tends to zero at xc ). The sharp upturn in 63 T1 T points to a crossover from QC to QD regime with a gap in the spin excitation spectrum, Δ0 = c/ξ0 (x) where c ∝ J ∼ T ∗ is the spin-wave velocity. In the pseudogap phase 0.05 < x < 0.22, a dynamical phase separation for hole-rich and insulating AF ordered regions can be realized as in the ﬁrstorder phase transition. The inhomogeneous states can be viewed as either in the form of a stripe-like pattern (see Sect. 3.2.4, Fig. 3.8) or as a coexisting metallic and incommensurate magnetic phases [402,403]. The second scenario has been supported by EPR studies of weakly doped La2−x Srx Cu0.98 Mn0.02 O4 polycrystalline samples with 0 < x < 0.06 by Shengelaya et al. [1149]. They

118

3 Antiferromagnetism in Cuprate Superconductors

Fig. 3.27. (a) Scaling function of the NMR relaxation rate 63 T1 T in LSCO at various doping. (b) Doping dependence of the oﬀset function A(x) (after Barzykin et al. [109])

observed two EPR signals. A narrow line which has low intensity at high temperatures but increases exponentially below 150 K was ascribed to a hole reach (metallic) regions. An opposite temperature dependence of the intensity of another broad line which becomes unobservable at low temperatures (presumably due to AFM ordering) was related to hole-poor regions. Therefore, metallic properties of the system can be related to a Fermi-liquid component with a concentration proportional to the hole concentration x. In the overdoped region x > 0.22, the spin-liquid component disappears and a uniform Fermi liquid evolves from a mixture of two phases.

3.5 Conclusion To summarize the study of magnetic properties of copper–oxide compounds, we point out the following most important results. 1. In the insulating phase of the parent La- and Y-cuprate compounds, a longrange AF order with a high N´eel temperature TN 300–500 K is observed. The AF state is induced by a strong superexchange interaction J 1,500 K between localized spins on the copper sites in the CuO2 plane. The eﬀective magnetic moment at copper sites is μ 0.6μB and weakly depends on doping which rules out the SDW scenario for the AF insulating state. The spectrum of magnetic excitations is determined by quasi-2D AF spin waves with high in-plane velocity c 0.8–1 eV·˚ A decreasing with doping. The experimental data can be fairly well described within a spin-1/2 quasi-2D quantum Heisenberg model. 2. Under hole doping, the long-range AF order is destroyed at a low hole concentration, nh ∼ 0.02, while in electron-doped cuprates the corresponding

3.5 Conclusion

3.

4.

5.

6.

119

electron concentration is much larger, ne ∼ 0.15. However, strong 2D shortrange AF spin correlations are maintained with a static AF correlation length of the order of several lattice constants. At low temperature a spinglass state is observed up to hole concentration nh 0.08, as evidenced from μSR experiments. Inelastic neutron scattering experiments prove the occurrence in the metallic phase of AF dynamic spin ﬂuctuations close to the AF planar wave vector QAF = (π, π) with a wide spectrum of excitations up to the energy of 2J ∼ 250 meV, even in optimally doped samples. In the superconducting state of YBCO-123 compounds, a spin gap in the low-energy part of the spectrum is observed, EG ∼ 3.8kB Tc , and a resonance mode (RM) at the energy Er ∼ 5.3kB Tc ∼ 40 meV. The RM was also detected in the Bi-2212 compound, the single-layered Tl-2201 system, and in the electron-doped PLCCO crystal. The RM in the hole-doped cuprates at the AF wavevector QAF and its downward dispersion away from QAF can be explained by a formation of the triplet spin exciton inside the gap of the particle– hole continuum ω ≤ Ωc (q) ∼ 2Δd in the d-wave superconductor. The high-energy spin-ﬂuctuations in YBCO crystals resemble the isotropic inplane spin-wave excitations. In the LSCO compounds the spin gap is rather small, ∼3–5 meV, while the RM is revealed only as a broad maximum in the superconducting state. This diﬀerence with YBCO system may be explained by a disorder introduced by Sr doping. This conclusion is in agreement with observation of the spin gap suppression and washing out of the RM by Zn-impurity substitution for the in-plane copper sites in YBCO. In YBCO, both the acoustic (odd) and optic (even) type modes of spin excitations caused by the bilayer coupling are detected, even for optimally doped samples. In the LSCO (LBCO) compounds the low-energy spin ﬂuctuations at low temperatures are incommensurate (IC) and forms a one-dimensional dynamical stripe-like pattern characterized by a modulation vector Q = π(1 ± δ) where the incommensurability δ is proportional to the hole concentration. Static stripes are observed in the low-temperature tetragonal phase where they are pinned by lattice distortion. In YBa2 Cu3 O6+x the stripe-like dynamical pattern with small incommensurability, δ ∼ 0.01, is detected in the ordered ortho-II phase at x = 0.5. At larger doping, a 2D character of the incommensurate modulation is found. The Knight shifts K S at diﬀerent nuclei in NMR experiments can be described within a single paramagnetic spin susceptibility χ0 (T ) for the CuO2 plane. The scaling behavior of the bulk spin susceptibility χ0 (T )/ χmax (T ∗ ) versus T /T ∗ for various doping suggests a two-component model for the spin susceptibility with a temperature-dependent spin-liquid component of localized spins and a temperature-independent quasiparticle Pauli-like component below the pseudogap temperature T ∗ . The spin–lattice relaxation rates T1−1 in the NMR and NQR experiments at copper sites are of an order of magnitude faster than at planar oxygen

120

3 Antiferromagnetism in Cuprate Superconductors

(or yttrium in YBCO) sites which proves an existence of AF short-range correlations. In the underdoped cuprates the relaxation rates T1−1 reveal an emergence of a spin pseudogap at T < Ts where Ts > Tc but decreases with doping and at an optimal doping Ts ∼ Tc . 7. In the superconducting phase, a sharp decrease of the Knight shift in optimally doped cuprates indicates singlet pairing at Tc . The absence of the Hebel–Slichter peak and the T -power law like decrease in the spin–lattice relaxation rate at T ≤ Tc are in agreement with the d-wave symmetry of the superconducting pairing.

4 Thermodynamic Properties of Cuprate Superconductors

The study of thermodynamic properties of cuprate superconductors provides important data on macroscopic parameters of these compounds, which yields certain boundary conditions for the microscopic theories. The measurements of thermodynamic quantities, such as critical magnetic ﬁelds and related critical currents, are also important for the development of high-temperature superconductor applications. The description of the thermodynamic properties of superconductors near the phase transition is usually done in the frame of the phenomenological Ginzburg–Landau (GL) theory. Its generalization to the anisotropic case is considered in Sect. 4.1. The measurement of the speciﬁc heat C(T ) in cuprates allows one to evaluate the low-temperature electronic speciﬁc heat Ce = γT and the corresponding electronic density of states (DOS). Studies of the lowtemperature speciﬁc heat C(T, H) in the external magnetic ﬁeld H in cuprate superconductors enables the investigation of the ﬂuctuation eﬀects at phase transition. These results are discussed in Sect. 4.2. Experimental data on the critical magnetic ﬁelds Hc1 and Hc2 and an estimate of the related parameters – the correlation length ξ(T ) and the penetration depth of magnetic ﬁeld λ(T ) – are considered in Sect. 4.3.

4.1 Anisotropic Ginzburg–Landau Model In the phenomenological GL theory [389], the superconducting state is described by the complex scalar order parameter Ψ (r) = |Ψ (r)|exp[iΦ(r)].

(4.1)

The modulus of the order parameter is usually normalized to the concentration of superconducting electron pairs: |Ψ (r)|2 = ns /2, while the phase of the order parameter is related to the superconducting current. Thus, in the GL theory, the superconducting state is described by a two-component (n = 2) order parameter.

122

4 Thermodynamic Properties of Cuprate Superconductors

The equilibrium properties of a superconductor are determined by a functional of the free energy, which depends on the order parameter and the external magnetic ﬁeld (see [722]). A microscopic derivation of the GL functional was given by Gor’kov [400]. Anisotropy is usually taken into account by the introduction of an anisotropic eﬀective mass of the superconducting pairs. In this approximation, we arrive at the following expression for the free-energy functional (see for example, [188]): 2 1 B + a|Ψ |2 + b|Ψ |4 F (Ψ ) = Fn0 + dV 8π 2 2e + (4m)−1 h∇α − Aα )Ψ |2 , (4.2) α |(−i¯ c α where Fn0 is the free energy in the normal-state, B = curlA is the magnetic induction, 2m−1 α are the principal values of the tensor of the inverse masses for a superconducting pair of electrons with the charge 2e. Near the superconducting phase transition at temperature Tc , in the GL theory it is assumed that the parameter a = α(T − Tc ) ≡ αTc τ , while the parameter b and the eﬀective mass mα do not depend on temperature. In this approximation, the free-energy functional describes the second-order phase transition from the normal to the superconducting state at Tc with the mean-ﬁeld critical indices. Equilibrium values of the order parameters and the superconducting current in an external ﬁeld H = 0 are determined by the minimum of functional (4.2) under its variation with respect to Ψ (r) and A(r) 2 2e 1 (4.3) −i¯ h∇α − Aα Ψ + aΨ + b|Ψ |2 Ψ = 0, 4mα c ie¯ h 2e2 |Ψ |2 Aα . (Ψ ∗ ∇α Ψ − Ψ ∇α Ψ ∗ ) − (4.4) jα = − 2mα mα c In the absence of a magnetic ﬁeld, the homogeneous equilibrium value of the order parameter at T < Tc is a α ns |Ψ0 |2 = = − = (Tc − T ). (4.5) 2 b b Under transition from the superconducting to the normal state, the speciﬁc heat undergoes a jump ΔC = Cs − Cn = α2

Tc . b

(4.6)

The thermodynamic critical ﬁeld Hc (T ) determined by the condition that the free energy in the normal and the superconducting states is equal to each other is given by the relation 1/2 1/2 4πa2 4πα2 Hc0 = = (Tc − T ) . (4.7) b b

4.1 Anisotropic Ginzburg–Landau Model

123

The jump in the speciﬁc heat (4.6) is related to the derivative of the critical ﬁeld by the Rutgers formula 2 1 ∂Hc ΔC = . (4.8) Tc 4π ∂T T =Tc The penetration depth of an external magnetic ﬁeld is determined by the corresponding screening current. To calculate the penetration depth, it is necessary to solve two coupled equations for the current given by the GL theory (4.4) and by the Maxwell equation j = (c/4π)curlB. Let us consider, for example, an external ﬁeld Hc parallel to c-axis and calculate the current ja and magnetic induction Bc in the plane ac. The solution of this set of equations gives the following dependences ja (y) = ja (0)exp(−y/λa ),

Bc (y) = Bc (0)exp(−y/λa ),

(4.9)

where the coordinate y runs along b-axis inside the superconductor perpendicular to its surface, where y = 0. The screening of the magnetic induction by the current ja (y), which is perpendicular to the induction Bc , is determined by the penetration depth λa . It deﬁnes the exponential decay of the current and the induction inside the superconductor. In anisotropic case, the magnetic penetration depth is given by the equation λ2α (T ) =

λ2 (0) mα c 2 mα c2 b = α . = 2 2 4πe ns 8πe |a| |τ |

(4.10)

It is convenient to represent the penetration depth in the form of a tensor with principal values λα = λ(mα /m)1/2 , where m = (ma mb mc )1/3 and λ = (mc2 /4πe2 ns )1/2 . In the quasi-two-dimensional cuprate superconductors, ma ∼ mb mc and λa ∼ λb λc since the superconducting current (4.4) in the (a, b) plane is much larger than along the c-axis. Another special length in the GL theory is the correlation length ξ(T ). It determines a speciﬁc distance, within which the order parameter is coherent Ψ (r)Ψ (0) ∝

1 exp(−r/ξ), r

where the average . . . is calculated on the basis of the GL functional. In the anisotropic case (4.2), the correlation length is determined by the relation ξα (T ) =

¯2 h 2mα |a|

1/2 =

ξα (0) . |τ |1/2

(4.11)

The behavior of superconductors in an external magnetic ﬁeld is determined by the GL parameter κ = λ/ξ. In the case of superconductors of the second √ type (κ > 1/ 2), there are two critical ﬁelds Hc1 < Hc0 < Hc2 . The cuprate superconductors have small correlation lengths ξ λ, and therefore are superconductors of the second type with a large parameter κ 1. The upper critical

124

4 Thermodynamic Properties of Cuprate Superconductors

ﬁeld Hc2 (T ) determines the stability limit of the superconducting phase. In the anisotropic case, it is equal to c = Hc2

√ φ0 2 = κc 2Hc0 , 2πξab

ab Hc2 =

√ φ0 = κab 2Hc0 , 2πξab ξc

(4.12)

where φ0 = hc/2e = π¯ hc/e = 2.07 · 10−7 Gs·cm2 is the quantum of magnetic ﬂux. Due to the deﬁnition (4.7) for the thermodynamic critical ﬁeld Hc0 =

1 φ0 √ , 2π 2 ξα λα

(4.13)

the relations (4.12) yield the following expressions for the anisotropic GL parameters √ 1/2 1/2 λab λab λc mc c b , κab = = mab = κc . (4.14) κc = ξab 2πe¯h ξab ξc mab The lower critical ﬁeld Hc1 (T ) determines the stability limit of the homogeneous Meissner superconducting phase without magnetic vortices. The formation of vortices at H > Hc1 becomes proﬁtable when the energy of a vortex becomes smaller than the energy of the external ﬁeld Hc1 φ0 /4π. In the limit κ 1, calculations for the anisotropic case yield (see for example, [241]) φ0 lnκc √ Hc0 , 2 lnκc = 4πλab κc 2 φ0 lnκab √ Hc0 . = lnκab = 4πλab λc κab 2

c = Hc1 ab Hc1

(4.15)

The temperature dependence of all critical ﬁelds (4.12) and (4.15) is determined by the function Hc0 (T ). According to (4.7), they should vanish following α a linear law as T → Tc . Therefore, within the GL mean-ﬁeld theory, Hc1(c2) ∝ (Tc − T ). As we see later, critical ﬂuctuations of the order parameter change this dependence close to Tc to a nonlinear one. The measurement of the upper critical ﬁelds (4.12) enables one to estimate the correlation lengths ξα (T ). The parameter κα can be found from the relation α Hc1 lnκα α = 2κ2 . Hc2 α

(4.16)

The penetration depth can be determined from (4.14). Measurement of the critical ﬁelds in cuprate superconductors turns out to be a complicated problem, which we consider in Sect. 4.3. The description of superconductors on the basis of the phenomenological GL theory in the approximation of anisotropic masses (4.2) is reasonable only when the correlation length (4.11) satisﬁes the relations ξα (T ) d, where d

4.1 Anisotropic Ginzburg–Landau Model

125

is a speciﬁc interatomic distance of the order of the lattice constant. In the cuprate superconductors, this relation may be violated due to the occurrence of a small correlation length and a strong anisotropy. For example, in compounds of bismuth or thallium, the interplane correlation length is rather small, r = ξc (0)/d 1. In this case, the GL functional should be written for the order parameter ψn (x, y) in the layer n. Then the Josephson coupling between adjacent layers (n, n + 1) should be taken into account to describe a three-dimensional superconducting phase transition (see [188]). This theory predicts a more complicated picture of the penetration of magnetic ﬁeld into a superconductor at H > Hc1 and a more complicated temperature dependence of the critical ﬁelds and the vortex-lattice. Another possible violation of the applicability of the GL theory is connected with critical ﬂuctuations, which become important close to the phase transition temperature at τ = (T /Tc − 1) → 0. As with any mean-ﬁeld theory of the second-order phase transition, the GL theory fails in the region where the ﬂuctuation of the order parameter becomes larger than its equilibrium value. The size of the critical region according to the Ginzburg criterion is estimated by the dimensionless temperature – the Ginzburg number: 2 2 1 1 kB 1 kB Tc = , (4.17) τGi = 2 (0)ξ ξ ξ 2 4πΔC ξa ξb ξc 2 Hc0 a b c where ΔC is the speciﬁc heat jump and ξα = ξα (0) is the correlation length (4.11) at zero temperature. In the second equation, the speciﬁc heat jump (4.6) is written in terms of the thermodynamic critical ﬁeld (4.7) at zero temperature. In the temperature region τ < τGi , ﬂuctuation corrections become important. In the conventional quasi-isotropic superconductors, the ﬂuctuation region τGi is extremely small due to a small ratio of the thermal energy at the critical temperature kB Tc to the superconducting condensation energy 2 ∝ Hc0 (0) in a large coherence volume (ξa ξb ξc ). The mean-ﬁeld approximation then applies to the whole region of temperatures, which can be reached experimentally. In the cuprate superconductors with high values of superconducting Tc and a small coherence volume due to short coherence lengths, the region of critical ﬂuctuations is much broader, of the order of τGi ∼ 0.01 or ∼1 K in YBCO. For the quasi-two-dimensional cuprates, such as Bi- or Tl-based superconductors, with the interlayer coherence length ξc less than the interlayer distance d, the region of critical ﬂuctuations is deﬁned by the 2D Ginzburg number 2 kB Tc 1 2 φ0 2D 2 , ε0 = τGi = √ = Hc0 (0)ξab , (4.18) 4πλab 2 2ε0 d where we used the expression (4.13) for the thermodynamic critical ﬁeld. The 2D ∼ 0.1 or region of 2D ﬂuctuations (4.18) is quite large, of the order of τGi ∼10 K for Bi-2212 and the mean-ﬁeld behavior characteristic to the secondorder phase transition, e.g. the expected speciﬁc heat jump is absent at all. We discuss the critical ﬂuctuations observed in the speciﬁc heat measurements in Sect. 4.2.3.

126

4 Thermodynamic Properties of Cuprate Superconductors

4.2 Specific Heat Studies of the speciﬁc heat make possible the investigation of the bulk properties of solids, primarily integral characteristics of the excitation spectra of electronic, phononic, and magnetic degrees of freedom. In particular, the observation of a speciﬁc heat jump occurring at the superconducting phase transition has conﬁrmed the bulk nature of the high-temperature superconductivity in cuprates. The speciﬁc heats of the YBCO and LMCO compounds have been best studied, while that of the Bi, Tl, and Hg compounds have been investigated to a less extent (for reviews, see [338,567,569]). Below we discuss the results that are the most important for understanding the nature of the high-temperature superconductivity in cuprates. 4.2.1 Low-Temperature Electronic Speciﬁc Heat Investigations of the electronic speciﬁc heat Cel (T ) in the high-temperature cuprate superconductors face a number of diﬃculties. First of all, due to a high value of the critical temperature Tc , the phononic contribution to the speciﬁc heat is large and it is diﬃcult to single out the electronic part of the speciﬁc heat. In particular, the speciﬁc heat anomaly at the superconducting phase transition ΔCel (Tc ) = Cel,s − Cel,n , which characterizes the electronic contribution is very small. Even in the LSCO and YBCO crystals with moderate anisotropy, it amounts no more than 2–4% of the phononic background. The anomaly of the speciﬁc heat at Tc in highly anisotropic Bi, Tl, and Hg compounds, due to the quasi-two-dimensional character of the phase transition, turns out to be even smaller, ∼0.5%. Moreover, during the preparation of compounds like YBCO, it is diﬃcult to avoid the appearance of impurity phases. In some of them, such as the Y2 Cu2 O5 or BaCuO2+x phases, the low-temperature speciﬁc heat is 10 − 100 times larger than the electronic speciﬁc heat in the YBCO compounds [567]. To obtain any quantitative results, high-accuracy measurements are therefore required. The speciﬁc heat measurements enable the estimation of the electronic DOS at the Fermi energy, which is one of the important parameters for a superconductor. The DOS is related to the Sommerfeld constant γ entering the low-temperature electronic speciﬁc heat by the equation Cel (T → 0) = γT,

γ=

2π 2 2 k N (0). 3 B

(4.19)

Here, the DOS at the Fermi energy, N (0), is deﬁned per atom and per one spin direction. The speciﬁc heat is usually measured in the units of (J/mol) or (J/gat), where a mole of substance contains NA = 6.02 · 1023 formula units and a gram-atom (gat) contains NA atoms. Therefore, for instance, in YBa3 Cu3 O7 one mole is equal to 13 gat. The density of states N (0) measured in the unit of (states/eV·atom·spindirection) is coupled to the Sommerfeld constant in (4.19) by the following relation: N (0)(states/eV · atom ·

4.2 Speciﬁc Heat

127

Table 4.1. Sommerfeld constant γn (0)(mJ/K2 ·mole), the speciﬁc heat jump (Cs − Cn )/Cn at Tc for slightly overdoped samples, the band structure DOS (0) 2N (0) (0) (state/eV·cell) [981], the corresponding γn (mJ/K2 ·mole) and the mass (0) renormalization, m∗ /m = γn /γn Material

γn

YBCO7 YBCO6.92 YBCO7 (Y0.8 Ca0.2 )BCO Bi-2212 Tl-2201 (La1.86 Sr0.14 )CO (La1.78 Sr0.22 )CO

18.2 18 15 25 15 7.5 7.5 14

(0)

ΔC/C 2N (0) (0) γn 2.5 2.6 4 1.5 1.3 1.47 1.4 0.9

5.6 – 5.6 – 3.0 1.24 2.1 –

13.2 – 13.2 – 7 3 5.0 –

m∗ /m 1.4 – 1.14 – 2.14 2.5 1.5 –

References Loram et al. [741] Janod et al. [552] Wang et al. [1334] Loram et al. [743] Loram et al. [744] Wade et al. [1326] Matsuzaki et al. [811] Matsuzaki et al. [811]

spin direction) = 0.212γ(mJ/K2 · gat). By comparing the experimentally measured DOS N (0) given by (4.19) with N (0) (0) in the band structure calculations (see Sect. 7.1.1), we can evaluate a many-body renormalization of the eﬀective electronic mass m∗ /m = N (0)/N (0) (0) = (1 + λ) and estimate an eﬀective interaction parameter λ (see Table 4.1). In a superconducting state, depending on the symmetry of the gap, the electronic speciﬁc heat Cel (T ) shows a particular temperature dependence. For instance, lines of nodes in k-space for the d-wave gap symmetry cause a linear increase of the DOS at low energy, N (E) ∝ |E|, which results in the quadratic speciﬁc heat temperature dependence and linear dependence for the Sommerfeld constant: Cel (T )/T = γ(T ) = α T

at T Tc ,

(4.20)

while for a conventional superconductor with a nodeless gap Δ0 we have Cel (T )/T ∝ exp(−Δ0 /T ). Measurements of electronic speciﬁc heat Cel (T ) at low temperatures as a function of the magnetic ﬁeld conﬁrm the d-wave symmetry of the superconducting gap as a bulk property. In a magnetic ﬁeld, in the mixed state of a superconductor, the quasiparticle energy receives a Doppler shift due to supercurrents around a vortex core [1321]. At a superﬂuid velocity vs , the quasiparticle energy E(k) is shifted by (k · vs ). The shift at high ﬁelds, being proportional to vs ∝ H 1/2 , is comparable with the width of the superconducting gap close to the nodes. Therefore, the DOS at the Fermi level is ﬁeld sensitive, N (E = 0) ∝ H 1/2 . Depending on whether the thermal energy kB T is larger or smaller than the Doppler energy, at low temperatures and for magnetic ﬁelds Hc1 H Hc2 , one has either Cel ∝ T 2 or Cel ∝ T H 1/2 behaviors, respectively. This crossover in a magnetic ﬁeld was reported in a number of publications (see e.g., [846, 1369]). In a later publication by Wang et al. [1334], a smooth crossover from Cel (T Tc , H = 0) = αT 2 at zero ﬁeld to Cel (T ∼ 0, H Hc2 ) = Ac T H 1/2 at zero

128

4 Thermodynamic Properties of Cuprate Superconductors

temperature in a high-quality YBa2 Cu3 O7 crystal was observed. The values of the coeﬃcients α 0.2mJ/K3 mol and Ac 1.34 mJ/K2 T 1/2 mol are in a qualitative agreement with other measurements. For a conventional pairing with a nodeless gap Δ0 , the asymptotic behavior Cel (T Tc , H = 0) ∝ exp(−Δ0 /T ) and Cel (T ∼ 0, H Hc2 ) ∝ H should be observed which are quite diﬀerent from the results in the YBCO crystals. Studies of temperature and doping (x) dependence of the Sommerfeld constant γ(T, x) = Cel (T, x)/T enable one also to estimate another important thermodynamic parameter – the entropy in the normal Sn and superconducting Ss states:

T

Sn,s (T, x) =

γn,s (T , x)dT ,

Sn (Tc ) = Ss (Tc ),

(4.21)

0

where the last equation follows from the constraint on the second-order phase transition. The entropy diﬀerence integrated from zero temperature to T = Tc deﬁnes the superconducting condensation energy U (0) =

Tc

(Sn (T ) − Ss (T ))dT,

U(BCS) (0) 0.24γn Tc2 ,

(4.22)

0

where the last equation gives the condensation energy for a BCS superconductor with d-wave pairing. In a number of experiments, the spin susceptibility χ(T ) has been studied in addition to measurements of the electronic speciﬁc heat (4.19). The static spin susceptibility contains several contributions (diamagnetic, orbital, or antiferromagnetic ﬂuctuations) as discussed in Sect. 3.3.1. The paramagnetic Pauli contribution is directly related to the DOS. For an ideal Fermi gas, one has χS =

μ2B 2N (0)

γ = , aW

aW

γ 1 = = χS 3

πkB μB

2 ,

(4.23)

where we introduced the Wilson ratio aW . Deviation of the measured Wilson ratio from the universal value (4.23) shows that other contributions to the spin susceptibility as e.g., antiferromagnetic ﬂuctuations in the uderdoped region, are important. As shown in Sect. 3.3.1 (Figs. 3.21 and 3.22), only for the optimally doped or overdoped region, the Knight shift reveals the Pauli temperature-independent spin susceptibility. The electronic density of states estimated from the spin susceptibility data on YBCO compounds at y = 0 (Fig. 3.23) equals 2N (0) = χS /μ2B 2.7 (states/eV·Cu atom), a value which agrees with speciﬁc heat measurements for overdoped systems. In Table 4.1, we compare values of the Sommerfeld constant in the normal-state extrapolated to zero temperature, γn (0)(mJ/K2 ·mole) for different materials with the results of the band-structure calculations [981], (0) γn (mJ/K2 ·mole) 2.36 · 2N (0) (0) (state/eV·cell). Although the LDA band,

4.2 Speciﬁc Heat

129

structure calculations fail in the underdoped region, they can give reasonable values in the overdoped region, where a Fermi-like behavior outside the pseudogap region is observed. For the YBCO7 and LSCO samples, we get a suﬃciently small eﬀective mass renormalization: m∗ /m ≤ 1.4 − 1.5. A modest renormalization is observed for the Bi-2212 and Tl-2201 materials, m∗ /m ∼ 2. In Table 4.1, we also report the speciﬁc heat jump (Cs − Cn )/Cn at Tc , which can be compared with the universal constant A = 1.43 in the weak-coupling limit of the BCS theory. It should be stressed, however, that ﬂuctuation eﬀects which are discussed in Sect. 4.2.3 below, strongly enhance the anomaly of the speciﬁc heat at Tc . This makes more diﬃcult the accurate estimation of the mean-ﬁeld contribution to the speciﬁc heat jump. Numerous measurements of the low-temperature speciﬁc heat in copperoxide superconductors revealed a ﬁnite value of the so-called residual Sommerfeld constant, γr = γ(T → 0). In the LSCO, γr ∼ 1.5 (mJ/K2 ·mole) and in the YBCO and Bi-2212 γr ∼ 4 (mJ/K2 ·mole), which are a signiﬁcant fraction of the normal-state Sommerfeld constants [567]. However, these values depend on the sample preparation and have the lowest values for the “best” samples, where the contributions to the speciﬁc heat due to doping phases, structural defects, dopants, and other “external” sources are reduced to a minimum. For instance, it was found that in the YBa2 Cu3 O7−δ crystals, the linear-T term decreases with decreasing twin boundaries and oxygen vacancies and for the best detwinned sample γr ∼ 1.2 (mJ/K2 ·mole) at δ = 0.01 [846]. A similar small value of γr of unknown origin was detected in the fully oxidized YBa2 Cu3 O7 crystal by Wang et al. [1334]. Therefore, the residual speciﬁc heat can hardly be characterized as a speciﬁc phenomenon for the superconducting phase of copper-oxide superconductors. It is known that “linear” residual speciﬁc heat is often observed in conventional doped metallic superconductors. As stressed by Phillips et al. [979], there is no evidence for an intrinsic contribution to γr . The high sensitivity of the electronic contribution to structural defects, such as impurities, atomic disorder, and paramagnetic impurities (produced, e.g., by uncoupled Cu spins), is another important feature which makes it difﬁcult to measure the low-temperature speciﬁc heat. This is due to the small value of the correlation length ξ which is comparable to the interatomic distance and therefore is sensitive to a short-range disorder. This disorder occurs in all high-temperature superconductors, since they are many-component solid solutions. These extrinsic perturbations produce a strong inﬂuence on the lowtemperature speciﬁc heat. In particular, the residual Sommerfeld constant under these perturbations can increase by an order of magnitude and the speciﬁc heat jump can be completely smeared out. For example, under 2% Fe doping of YBa2 (Cu1−x Fex )3 O7−y , the speciﬁc heat jump falls to almost one third of its initial value. At x = 4%, the jump is no longer observed. The total suppression of superconductivity, Tc = 0, occurs only at x = 7% [567]. A similar eﬀect is produced by a decrease of the oxygen content or by doping with nonmagnetic Zn. The speciﬁc heat jump is halved at x = 1% of Zn impurities,

130

4 Thermodynamic Properties of Cuprate Superconductors

practically disappears at x = 2.5%, while the critical temperature vanishes at a Zn concentration x = 7% [567]. The reason that the speciﬁc heat jump decreases so sharply and the phase transition gets smeared is presumably the small value of the correlation length ξ (4.11) in copper-oxide superconductors. The critical temperature is determined locally by the energy of condensation averaged over volume (ξ 3 ). When the distance between structural defects becomes comparable to ξ, a considerable ﬂuctuation of the local temperature of the superconducting transition occurs and the speciﬁc heat jump becomes smeared out. A remarkable decrease of the Meissner eﬀect, i.e., the decrease of the fraction of superconducting volume in a sample under doping, is also indicative of this mechanism. 4.2.2 Pseudogap in Electronic Speciﬁc Heat The most accurate measurements of the electronic speciﬁc heat were performed for the LSCO and YBCO compounds. Let us start from consideration of speciﬁc heat measurements for the LSCO compound. Figure 4.1 shows temperature dependence of the speciﬁc heat ratio C/T for the La2−x Srx CuO4 compound measured for the normal and the superconducting states for various Sr concentrations x = 0.10, 0.14 and 0.22 [811]. The residual value γr (T = 0) of unknown origin is shown by crosses on the vertical axis. It has the largest value for the underdoped sample, γr (x = 0.1) ∼ 1 mJ/K2 ·mole, and disappears for the overdoped one (x = 0.22). The superconducting Tc is shown by arrows and the temperature region of superconducting ﬂuctuations is marked by Tsf . The speciﬁc heat jump is clearly seen at the superconducting transition, A = (Cs − Cn )/Cn (Tc ) 1.5, 1.4 and 0.9, respectively. At x = 0.10, 0.14, these ratios are close to the universal constant A = 1.43 in the weak-coupling

Fig. 4.1. Temperature dependence of the speciﬁc heat ratio C/T for the La2−x Srx CuO4 compound for various Sr contents (after Matsuzaki et al. [811])

4.2 Speciﬁc Heat

131

Fig. 4.2. Doping dependence of the Sommerfeld constant in the normal-state γn (x) for the La2−x Srx CuO4 compound (after Matsuzaki et al. [811])

limit of the BCS theory. The electronic contribution to the speciﬁc heat in the LSCO strongly depends on doping as shown for the Sommerfeld constant γ(x) extrapolated to zero temperature from the normal-state in Fig. 4.2 [811]. Here, the black and open squares are the results of the present speciﬁc heat measurement, the open rhombuses represent the results for the DOS measured in the angle-integrated photoemission spectroscopy, and the open circles represent the results of the Knight shift measurements below Tc (see [811]). The values of γn = 7.5 − 14 mJ/K2 ·mole for x = 0.14 − 0.2 can be compared with the results of the band-structure calculations 2N (0) (0) ∼ 2.1 (state/eV·cell) at x = 0.15 close to the Van Hove singularity [981]. This value gives for (0) the Sommerfeld constant γn = 2.36 · 2N (0) (0) 5 mJ/K2 · mole, which show a modest many-body renormalization of the eﬀective electronic mass m∗ /m = N (0)/N (0) (0) = 1.5 − 3. A sharp decrease of the Sommerfeld constant (and the DOS) in the region x < 0.2 is caused by suppression of the DOS in the electronic spectrum. This conclusion is supported by the calculation of the condensation energy (4.22), which rapidly decreases for x < 0.2 and strongly deviates from the BCS value. To adjust the doping dependence of the gap function Δ0 with the superconducting Tc in this region, the authors proposed to introduce an eﬀective superconducting gap, Δeﬀ = 4.5xΔ0 , which scales with the BCS 0 relation for the d-wave superconductor: 2Δeﬀ 0 ∼ 5.3kB Tc (see e.g., [1367]). The eﬀective gap is deﬁned, according to the authors, by a coherent part of the superconducting gap, which is formed over the nodal Fermi arcs of the trancated Fermi surface (see Sect. 5.2.2). A comprehensive study of the electronic speciﬁc heat in the YBCO compounds has been performed by several groups (see e.g., [567,569,740–743,745]). Let us consider the studies by Loram et al. who have presented convincing results for a normal-state pseudogap occurrence in the electronic spectrum of the underdoped cuprates. To single out the electronic contribution, they have used as reference samples the YBCO compounds doped with a small amount of Zn impurities. The Zn2+ ions substitute the in-plane Cu2+ ions, which does not aﬀect the hole doping and has a weak inﬂuence on the phononic

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4 Thermodynamic Properties of Cuprate Superconductors

contribution to the speciﬁc heat. At the same time, a small concentration of Zn impurities, ∼2% strongly suppresses superconductivity which allows highly accurate measurements of the electronic speciﬁc heat anomaly at the superconducting transition. To study both the underdoped and overdoped regions, the Ca-doped YBCO compounds were used. Temperature dependence of the Sommerfeld constant γ(δ, T ), entropy S, and S/T are shown in Fig. 4.3 for the overdoped (a) and underdoped (b) Y0.8 Ca0.2Ba2 Cu3 O7−δ compound [743]. The maximum Tc ∼85 K in this compound occurs at the optimal doping δopt = 0.32. The temperature behavior of these quantities in the overdoped and underdoped regions is strikingly diﬀerent. While for the overdoped region well-pronounced peaks in γ(Tc ) and the doping independent entropy for T > Tc are observed, a

b (mJ/g-at.K2)

4 3

0.29 0.32

4

0.37 0.41

3 0.44 0.50 0.55 0.60

2

1

1

0

0

200

100

0.040 0.247

0.040 0.054 0.078 0.125 0.192 0.195 0.247

0 2 0.040

1

0 0

0.67

0.192 0.195 0.247 0.285 0.323 0.369 0.408 0.439 0.501 0.552 0.673

300 200 100

0.323

0.192

0.673

0

0.247

0.040 0.054 0.078 0.125 0.192 0.195 0.247

20 40 60 80 100 120 140 160 T(K)

S / T (mJ/g-at.K2)

S/T (mJ/g-at.K2)

5

γ

2

S (mJ/g-at.K)

S (mJ/g-at.K)

γ

(mJ/g-at.K2)

5

0.25 0.29 0.20 0.19 0.13 0.08 0.06 δ=0.04

0.323(δopt)

2 0.192

1

0

0.673

0

40

80

120 160 T(K)

200 240

Fig. 4.3. Temperature dependence of the Sommerfeld constant γ(δ, T ), entropy S and S/T for overdoped (a) and underdoped (b) Y0.8 Ca0.2 Ba2 Cu3 O7−δ (δopt = 0.32) c 1998) (reprinted with permission by Elsevier from Loram et al. [743],

4.2 Speciﬁc Heat

133

Fig. 4.4. (a) Doping dependence of the pseudogap energy Eg , the condensation energy U (0) and U (0)/γn Tc2 for Y0.8 Ca0.2 Ba2 Cu3 O6+x . (b) The normal-state pseudogap energy Eg /kB dependence on the planar hole density p for LSCO-, Bi-2212-, and YBCO-doped with Ca (0%, 20%) or 2% of Zn. The superconducting gap Δs (0) is shown for YBCO (20% Ca) (reprinted with permission by Elsevier from Loram c 2001) et al. [745],

in the underdoped region the peaks in γ(Tc ) collapse and the entropy drastically decreases in the normal-state with the oxygen removal. The behavior changes abruptly at the critical value of oxygen concentration δcrit ∼ 0.25, which corresponds to the hole doping pcrit ∼ 0.18 per CuO2 in-plane unit. The shoulder in the function S/T , which is proportional to the thermally averaged over ∼3kB T DOS close to the Fermi level, clearly demonstrates a loss of the spectral weight in the normal phase at p < pcrit . The latter is conﬁrmed by the negative value of the entropy S extrapolated to zero temperature in the underdoped region, which suggests an appearance of a pseudogap in the normal-state quasiparticle spectrum. The shoulder in S/T occurs at a characteristic pseudogap temperature T ∗ , which reﬂects the width of the pseudogap Eg /kB ∼3T ∗. To explain the data, the authors proposed a model for the normal-state pseudogap Eg in the DOS at the Fermi energy in a triangular-like form, gpg (E)∼gn · |E − EF |/Eg for |E − EF | < Eg where gn is the DOS per the CuO2 cell outside the pseudogap region. This states-non-conserving pseudogap explains permanent loss of the entropy at a rate ∼kB per hole, which is not regained at higher temperatures up to ∼3kB T . The doping dependence of the gap energy is ﬁtted by the equation Eg ∼J(1 − p/pcrit) (Fig. 4.4a). The value of the parameter J∼1,300 − 1,500 K of the order of the in-plane AF exchange energy suggests a magnetic origin of the pseudogap. With doping, a new orbital per one hole appears on the ﬂanks of the pseudogap, which results in shrinking of the gap as Eg (p)∼J − p/gn . Therefore, the pseudogap closes at pcrit ∼ gn J. For an averaged DOS in cuprates, gn ∼ 1.5 (states/eV·Cu atom) we get the universal pcrit ∼0.18. As Loram et al. [745] pointed out, it is remarkable that the pseudogap only exists when J exceeds the kinetic energy ∼p/gn of the doped holes.

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4 Thermodynamic Properties of Cuprate Superconductors

This scenario is further strengthend by calculations of the doping dependence, the condensation energy U (0) (4.22), and its ratio to the BCS-like one, U (0)/γn Tc2 , for Y0.8 Ca0.2 Ba2 Cu3 O6+x shown in Fig. 4.4a [745] . The condensation energy U (0), being close to the BSC value for x > xcrit ∼ 0.8, sharply decreases at x < xcrit , which is caused by an opening of the pseudogap. An abrupt crossover from metallic behavior and strong superconductivity to a semiconducting-like leads the authors to the conclusion that “Eg (p) reﬂects the energy scale of correlated holes and spins, and is not due to superconducting ﬂuctuations or additional competing interactions” [743]. This conclusion is supported by studies of other compounds: the LSCO, YBa2 Cu3 O6+x [740–742], Tl-2201 [1326], and Bi-2212 [744]. A universal behavior of the normal-state pseudogap energy Eg on the hole concentration p for diﬀerent cuprate materials is demonstrated in Fig. 4.4b [745] for LSCO-, Bi-2212-, and YBCO-doped with Ca (0%, 20%) or 2% of Zn. The superconducting gap Δs (0) is shown for YBCO (20% Ca). Let us discuss the results for Bi-2212 [744, 745]. Figure 4.5a shows Tc and the anomaly height Δγ(Tc ) dependence on the planar hole density p. To cover a broad region of doping, 0.095 < p < 0.22, the undoped Bi-2212 sample (black triangles), doped with 20% Pb (empty squares), and doped with 15% Y (crosses) were used. For the overdoped samples, the speciﬁc heat jump anomaly height ΔCel (Tc )/Cel (Tc ) = Δγ(Tc )/γ(Tc ) ∼ 1.5 is close to the BCS value 1.43, while for p < pcrit ∼ 0.19 the anomaly height decreases rapidly as for the YBCO compound (Fig. 4.3). The pseudogap energy for the Bi-2212 Eg /kB ∼ 980(1 − p/pcrit) K, the condensation energy U (0) given by (4.22), the Bose condensation energy U (0)BE , and the free-energy diﬀerence ΔF (H) = Fs (H) − Fs (0) are shown in Fig. 4.5b. The condensation energy U (0) in the Bi-2212 reveals a rapid fall for p < pcrit from the BCS-like value

Fig. 4.5. The planar hole density p dependence for Bi-2212 for (a): Tc , the anomaly height Δγ(Tc ); (b): the pseudogap energy Eg /kB , the condensation energy U (0), the Bose condensation energy U (0)BE , and the free-energy increase ΔF (13T) ∝ ρs (0) c 2001) (reprinted with permission by Elsevier from Loram et al. [745],

4.2 Speciﬁc Heat

135

at p > pcrit as in the YBCO compounds. Although the Bose condensation energy, estimated as U (0)BE ∼ 0.28(p/2)kBTc /CuO2 , is close to the calculated one U (0) for the Bi-2212 at pcrit , it does not reproduce a rapid fall below pcrit and Tc2 dependence above pcrit. The free-energy diﬀerence ΔF (H) = Fs (H) − Fs (0) (calculated by integration of the entropy ΔS(H, T ) = Ss (H, T ) − Ss (0, T ) over the temperature from T = Tc to T = 0 for the ﬁxed magnetic ﬁeld H = 13 T) in Fig. 4.5b also shows a sharp change at pcrit as the condensation energy U (0). Within the London model, ΔF (H) is proportional to the superﬂuid density ρs (0)the sharp decrease of which at p < pcrit demonstrates a strong inﬂuence of the normalstate pseudogap on the superconducting properties. Moreover, while a strong magnetic ﬁeld does not change appreciably the Sommerfeld constant in the normal-state Δγn (13T) = γn (13T) − γn (H = 0) and therefore the pseudogap, it produces a modest change in Δγn (H)/γn ∼ H/Hc2 at T ≤ Tc . Therefore, the pseudogap cannot be connected with superconducting ﬂuctuations above Tc as suggested in the models of a precursor superconducting pairing. Further justiﬁcation of the normal-state pseudogap occurrence at p < pcrit comes from a comparison of the behavior of the Sommerfeld constant γ(p, T ) with that of the magnetic susceptibility χ(p, T ). Similar temperature and doping dependence of the function S/T (the T -averaged Sommerfeld constant γav ) and the bulk spin susceptibility χ(p, T ) were observed for the LSCO, YBCO, and Bi-2212 cuprates [745]. Both the entropy S(p, T ) and the function T χ(p, T ) increase linearly with the hole doping. Their ratio results in a 2 diﬀerential Wilson ratio (dS/dp)/(T dχ/dp) ∼ kB /0.3μ2B close to the value of a free electron gas (4.23): aW = (π 2 /3)(kB /μB )2 . An additional evidence that the pseudogap origin is independent of the superconducting pairing was obtained in studies of disorder eﬀects on the superconducting Tc and characteristic pseudogap temperature T ∗ (p). While the former is strongly depressed by impurity substitutions, in particular, by Ca in Bi-2212 [1223] and Zn [742, 885], the pseudogap energy scale Eg , as shown in Fig. 4.4b, does not change appreciably with a disorder (up to a certain level). The pseudogap temperature T ∗ (p) persists below the superconducting Tc0 of the pure sample and does not merge with it extrapolating to zero as the hole concentration p → pcrit 0.19. Studies of the oxygen isotope eﬀect by Williams et al. [1362] did not reveal isotope shift for the normal-state pseudogap energy, while a distinct shift was observed for Tc , which also points against the precursor superconducting pairing scenario for the pseudogap. A more detailed discussion of an interrelation between the superconductivity and the pseudogap is given by Tallon et al. [1224, 1226]. Evidences of the pseudogap occurrence in the quasiparticle spectra observed in angleresolved photoemission experiments, tunneling and transport measurements are discussed in Chap. 4. Theoretical models for a pseudogap are considered in Sect. 7.2.1.

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4 Thermodynamic Properties of Cuprate Superconductors

4.2.3 Fluctuation Eﬀects Studies of the ﬂuctuation eﬀects at phase transitions in the high-temperature superconductors provide valuable information concerning the mechanisms responsible for superconductivity and enable the derivation of certain constrains for the microscopic theory. Due to the small values of coherence length and the high anisotropy ζ = ξab /ξc = mc /mab 1, the temperature region of critical ﬂuctuations (4.17) in the cuprate superconductors appears to be large and can be studied in detail. A survey of phase transitions and critical phenomena in cuprate superconductors was given by Schneider et al. [1123]. Universal critical properties of n- and p-type cuprates, which undergo a crossover from the 3D-XY universality class to a quantum critical point in 2D, were discussed later by Schneider [1124]. Earlier results concerning ﬂuctuation eﬀects in high-temperature superconductors were reviewed by Salamon [1086]. Here, we discuss only the most important results concerning speciﬁc heat measurements in cuprates in magnetic ﬁelds near Tc , which have allowed the elucidation of the nature of the phase transition in the hightemperature superconductors. These investigations were reviewed by Junod et al. [570, 571]. In general, depending on the correlation length ξ, three diﬀerent types of critical behaviors at a superconducting phase transition can be distinguished. For conventional superconductors with a large number of Cooper pairs in the coherence volume, the dimensionless parameter ξkF 1 where kF ∼ π/a is the Fermi wavenumber. For these superconductors, the mean-ﬁeld BCS theory or the G–L model of the second-order phase transition is applicable. Within the mean-ﬁeld theory, at Tc , the second-order derivatives of the free energy F , as, for instance, the speciﬁc heat C/T = ∂ 2 F/(∂T )2 , show jumps with a negligible region of thermal ﬂuctuations. At a crossover, for ξkF ∼ 1, the temperature region τGi (4.17) becomes large and ﬂuctuations of the superconducting order parameter can be revealed at τ < τGi . The thermal ﬂuctuations of the electronic speciﬁc heat and the correlation length ξ(τ ) are described by the critical exponents α and ν: A± −α C C ± (τ ) ∼ − |τ | , ξ(τ ) ∼ ξ(0)|τ |−ν , (4.24) = T T background α where τ = (T /Tc − 1) and (+, −) are referred to τ > 0 and τ < 0, respectively. The critical exponents for a D-dimensional system are coupled by the hyperscaling relation α + Dν = 2. The mean-ﬁeld jump of the speciﬁc heat is deﬁned by ΔC/T ∝ (A− − A+ )/α. In the region of weak ﬂuctuations, the mean-ﬁeld G–L theory for the two-component order parameter (4.1) can be applied. Within the theory, a speciﬁc heat jump (4.6) and thermal ﬂuctuations near Tc of Gaussian type with the critical exponents α = 2 − D/2 = 1/2 for D = 3 and ν = 1/2 should be observed. In fact, the magnetic penetration depth (4.10) and the correlation length

4.2 Speciﬁc Heat

137

(4.11) at τ → 0 are proportional to τ −1/2 . In the region of strong ﬂuctuations, the critical behavior is characterized by the 3D-XY model with the following universal parameters: A+ /A− 1.054, α −0.013, ν 0.671. The speciﬁc heat anomaly at Tc shows a ﬁnite cusp, which looks like a quasilogarithmic divergence. In fact, in the limiting case of |α ln |τ || 1, the temperature dependence (4.24) for the speciﬁc heat can be approximated by a logarithmic-type divergence: ⎧ + at T > Tc , ⎨ A (− ln |τ |) ± C (τ ) C ∼ (4.25) − = T T background ⎩ A+ (A − ln |τ |) at T < Tc , where A = (A− /A+ − 1)/α 4 is a universal dimensionless parameter for the 3D-XY model. The speciﬁc heat anomaly looks like a λ-peak at Tc which is observed at the superﬂuid phase transition in 4 He at 2.18 K. Finally, in the region of small correlation lengths ξkF 1, the Bose– Einstein condensation (BEC) of local pairs takes place at Tc . In a 3D ideal Bose gas, the heat capacity reveals a triangular peak without jump or divergence. For T < Tc , the heat capacity C = 1.925kB (T /Tc )3/2 , while for T > Tc it decreases from the value 1.925 kB reaching the Dulong Petit value C = 1.5kB . The BEC belongs to the universality class of the spherical model with the critical exponents in 3D equal to α = −1, ν = 1. The type-II superconductors at large value of the GL parameter, κ 1, belong to the 3D-XY universality class. Therefore, the cuprate superconductors should show the 3D-XY model critical behavior at τ < τGi . However, for strongly anisotropic systems, ζ = mc /mab 1, the 3D-XY critical behavior can be observed only at temperatures lower than the crossover temperature τcross ∼ ζ −3/2 , while at τ > τcross a 2D-XY critical behavior should occur. Estimates for the anisotropy parameters ζ and the crossover temperatures τcross are given in Table 4.2 for the representative classes of cuprates [1123]. It follows from these estimates that only in the YBCO one can really observe a 3D phase transition in the vicinity of Tc . The largest temperature region of the 3D behavior appears in the overdoped YBa2 Cu3 O7 [1334] of the order |T − Tc | ≤ 0.08Tc ∼ 7 K. By taking into account that inhomogeneities and impurities in a sample induce ﬁnite size eﬀects and smear out (broaden) the singular behavior in the close vicinity of Tc , in highly anisotropic compounds, Table 4.2. Anisotropy parameters ζ = mc /mab and the 2D→3D crossover temperatures τcross ∼ ζ −3/2 for cuprates [1123]

Tc (K) ζ τcross ∗

YBCO∗

YBCO

LSCO

Hg-2201

Tl-2201

Bi-2212

87.8 5.3 0.08

91.7 9 0.037

35 14 0.02

94 27 0.007

87.6 117 0.0008

84.5 ∼250 0.00025

YBa2 Cu3 O7 [1334]

138

4 Thermodynamic Properties of Cuprate Superconductors

such as Bi- or Tl-based cuprates, only 2D-type critical ﬂuctuations are available for observation. The region of 2D-type critical ﬂuctuations deﬁned by the Ginzburg number (4.18) is much broader, of the order of ∼ 10 K. In the underdoped cuprates, the anisotropy increases and a crossover from the 2D to the 3D critical behavior should occur much closer to Tc . To study the critical behavior of the heat capacity at the superconducting phase transition, besides preparing a high-quality sample to suppress ﬁnite size eﬀects and broadening of the phase transition, one needs a high-resolution method to measure a small anomaly at the phase transition on the large phononic background. By applying a high magnetic ﬁeld it is possible to suppress the superconducting anomaly and, by subtracting the remaining background, to study the critical behavior. First observations and analyses of the critical ﬂuctuations, as opposed to Gaussian, were done by the Urbana group [521, 1085, 1087] for YBCO crystals of high quality. speciﬁc heat measurements in single YBCO crystals in magnetic ﬁelds normal and parallel to the CuO2 plane match each other and can be ﬁtted with one curve if scaled by a factor representing the anisotropy of the critical ﬁelds, ab c /Hc2 = ζ = mc /mab [552, 1073]. Below we discuss more recent results Hc2 obtained by Junod et al. [570, 571]. Figure 4.6 shows the temperature dependence of the speciﬁc heat ratio C/T near Tc for the optimally doped YBa2 Cu3 O6.92 (Tc = 93 K, ζ 7) (a) and the slightly overdoped Bi2.12 Sr1.9 Ca1.06 Cu1.96 O8+x (Tc = 85 K) (b) crystals in magnetic ﬁelds up to 14 Tesla [571]. By subtracting the results obtained in the largest available magnetic ﬁeld of 14 Tesla as background, it is possible to reveal the critical anomalies in the speciﬁc heat more distinctly. Figure 4.7 demonstrates the temperature dependence of the speciﬁc heat diﬀerence ΔC(T, B)/T = [C(T, B) − C(T, B = 14T)]/T near Tc for the above-mentioned optimally doped YBCO (a) and Bi-2212 (b) crystals in magnetic ﬁelds [570].

Fig. 4.6. Temperature dependence of the speciﬁc heat ratio C/T near Tc for (a) the optimally doped YBa2 Cu3 O6.92 and (b) for the slightly overdoped Bi2.12 Sr1.9 Ca1.06 Cu1.96 O8+x crystals in magnetic ﬁelds up to 14 Tesla (after Junod et al. [571])

4.2 Speciﬁc Heat

139

Fig. 4.7. Temperature dependence of the speciﬁc heat diﬀerence ΔC(T, B)/T = [C(T, B) − C(T, B = 14T)]/T near Tc for (a) the optimally doped YBa2 Cu3 O6.92 and for (b) the slightly overdoped Bi2.12 Sr1.9 Ca1.06 Cu1.96 O8+x crystals in magnetic ﬁelds up to 14 Tesla (after Junod et al. [570])

A striking diﬀerence is observed in the shape of the speciﬁc heat anomaly at Tc . While the speciﬁc heat anomaly in the YBCO sample shows a λ-peak structure characteristic to the critical behavior of the 3DXY model, the anomaly in the Bi-2212 crystal reveals a symmetric triangular peak as in the BEC phase transition. Moreover, the magnetic ﬁeld shifts the speciﬁc heat anomaly in the YBCO to lower temperatures, while in the Bi-2212 only the peak height is reduced without changing its position. The asymmetric peak in the YBCO crystal in zero ﬁeld is caused by a mean-ﬁeld jump with the ﬂuctuation contribution about one-half of the peak height. Contrary to the YBCO crystal, the jump in Bi-2212 at Tc appears to be negligible and the transition is deﬁned by ﬂuctuation contribution alone. The position of the peak does not change due to an extremely large derivative ∂Hc2 /∂T for the quasi-two-dimensional Bi-2212 compound. A logarithmic critical divergence and the symmetric shape of anomalies at Tc were also observed in the high-resolution thermal expansion measurements in the Bi-2212 single crystal by Meingast et al. [822]. The critical behavior in the YBCO crystals is reproduced quite well by the logarithmic-type divergence (4.25) with the constants A 3.6 for the optimally doped sample, and A ∼ 6, and A = 2 − 3 for the overdoped and underdoped samples, respectively [570]. Larger values of the parameter A, which is proportional to the speciﬁc heat jump for the overdoped YBCO crystals with corresponding decrease of the anisotropy parameters ζ, indicate that the ﬂuctuations in these crystals are less important and they are close to the BCS-type superconductors. Thus, the speciﬁc heat anomaly in the quasi3D YBCO crystals demonstrates the 3D-XY -model critical behavior, with some deviations from the perfect ﬁtting observed for the optimally doped crystal. It is interesting to point out that the onset of the superconducting transition in the YBCO crystals does not change with the magnetic ﬁeld. This results from a compensation of the decrease of the mean-ﬁeld Tc and simultaneous increase of the ﬂuctuation region in the magnetic ﬁeld.

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4 Thermodynamic Properties of Cuprate Superconductors

An attempt to ﬁt the Bi-2212 anomaly with the logarithmic-type divergence (4.25) results in extremely small values of A = 0.2 − 0.3, which shows that the phase transition does not belong to the 3D-XY universality class. The same behavior was observed for other strongly anisotropic cuprates such as Tl-2201 [836], and Hg-based crystals. Much better ﬁt is obtained with the general formula (4.24), which yields for the critical exponents α −0.67, ν 1.06 [570]. These values are close to the BEC parameters: α = −1, ν = 1. Thus, the critical behavior of the quasi-two-dimensional cuprates such as Bi-2212 is better described by the Bose condensation than the 3D-XY phase transition [1123]. Alexandrov et al. [33] have explained the critical behavior of the speciﬁc heat within a model of a weakly interacting charged Bose-gas (see Sect. 7.4.3). The scaling properties of the speciﬁc heat in magnetic ﬁelds were reported in a number of publications. In large ﬁelds, the eﬀective dimensionality of a system is reduced to one, which strongly enhances ﬂuctuations. Instead of the Ginzburg criterion for the size of the critical region in 3D case (4.17), in the magnetic ﬁeld H this is given by the dimensionless temperature: 2/3 2/3 kB H H = 2(τGi )1/3 . (4.26) τGi (H) = ΔC φ0 ξc Hc2 The critical region τGi (H) = |T /Tc(H) − 1| in magnetic ﬁelds of 15 T appears approximately ten times larger than (4.17), which aﬀord to study critical behavior in detail [568]. Two models of critical behavior in magnetic ﬁelds were proposed: the lowest Landau level (3D-LLL) and the 3D-XY model. It is believed that the 3D-LLL approximation is valid close to Hc2 , where high magnetic ﬁeld conﬁnes the Cooper pairs in the lowest Landau level, while around Tc and H = 0 the critical behavior is better described by the 3D-XY model. The critical behavior of the free energy in the 3D-LLL model is described by the scaling function [93, 1360]: T − Tc (H) , (4.27) F (T, H) = (HT )4/3ϕ3D (HT )2/3 while in the 3D-XY model the singular part of the free energy in magnetic ﬁeld is given by [93, 1087]: Hξ 2 (τ ) Q ± kB T G Fs (T, H) = 3 , (4.28) ξ (τ ) φ0 where G(y) is an unknown scaling function. In the layered superconductors, 2 2 ξc and Hξ 2 = Hc ξab or Hab ξab ξc for the magnetic we should take here ξ 3 = ξab ﬁeld H c or H ⊥ c, respectively. Taking the ﬁrst derivative with respect to H from (4.28), we obtain the scaling law for the magnetization ﬂuctuations: Mab,c = −

∂Fs (T, H) Q± kB T = − 3/2 H 1/2 Γab,c , ∂Hab,c φ 0

(4.29)

4.2 Speciﬁc Heat

141

2 where Γc = ζG (y)/y 1/2 and Γab = ζ −1/2 G (y/ζ)/(y/ζ)1/2 , and y = Hξab (0)/ 2ν φ0 τ . Thus, in the critical region of the 3D-XY model (ν = 2/3) we should observe a scaled M/T H 1/2 versus H/τ 4/3 behavior for magnetization for the both directions of the magnetic ﬁeld. A good ﬁt of the reversible magnetization to 3D XY scaling law (4.29) for the twinned YBa2 Cu3 O7−δ crystals with various oxygen contents δ in the region of the reduced temperatures 0.07 ≤ τ ≤ 0.2 was reported by Babi´c et al. [93]. Figure 4.8 demonstrates the 3D XY scaling of the measured magnetization for the YBCO crystal with δ = 0.03(Tc = 91.3 K) for the magnetic ﬁeld (a) H c and (b) H ⊥ c. For the YBCO crystal with 0.08 ≤ δ ≤ 0.35, the scaling curves for the magnetic ﬁeld H c are shown in the plots: (c) δ = 0.35 (Tc = 61.4 K), (d) δ = 0.21 (Tc = 83.7 K), (e) δ = 0.15 (Tc = 91.0 K), and (f) δ = 0.08 (Tc = 93.2 K). Various curves refer to temperatures T corresponding to the reduced temperatures τ = (1−T /Tc ): (a) 0.025 < τ < 0.157, (b) 0.014 < τ < 0.047, (c) 0.088 < τ < 0.23, (d) 0.068 < τ < 0.164, (e) 0.054 < τ < 0.21, and (f) 0.013 < τ < 0.077. Analyzing these scaling dependence, the authors have inferred that the dominant ﬁeld-dependent contribution to the scaling function is G(y) ∝ 1/y, which results in scaling laws of the free energy (4.28) and the magnetization (4.29) in the forms:

H 1 kB T kB T Fs (T, H) ∝ 2 , Mc ∝ − 2 . (4.30) ξ (τ ) φ0 2ξ (τ ) Hφ0

Fig. 4.8. 3D XY scaling of the magnetization for the twinned YBa2 Cu3 O7−δ crystals with δ = 0.03(Tc = 91.3 K) in the magnetic ﬁeld (a) H c and (b) H ⊥ c. The scaling for the magnetic ﬁeld H c for various doping δ is shown in the plots (c) – (f ) (for details, see the text) (after Babi´c et al. [93])

142

4 Thermodynamic Properties of Cuprate Superconductors

Table 4.3. Fitting parameters for the melting line Bm (T ) = Bm0 (1 − T /Tc )n [1074] (ﬁrst 3 columns, n 1.33) and for the irreversibility line Hirr (T ) = H ∗ (1 − T /Tc )4/3 [93] (last 4 columns) for singlecrystals of YBa2 Cu3 Ox with the anisotropy ratio ζ = mc /mab x=

7.0

6.96

6.94 6.97

6.92 6.85

6.65

Tc (K) ζ Bm0 (T) H ∗ (T) Hc2 (0) (kOe)2

87.8 5.3 135

89.7 6 137

92.6 7 103

93.2 91 6.5 7.2

61.4 9.5

91.3 5

122.5 100 120 110

43 83

8 14

By using these scaling laws, the authors estimate the coherence length ξab (0) 12.6 ˚ A at δ = 0.03, which should increase by a factor of 2.5 at δ = 0.35. A ﬁxed point for the irreversibility line, which is appeared on the plots in Fig. 4.8, is discussed in Sect. 4.3.1 (see Table 4.3). In several studies, a crossover from the 3D-XY critical behavior at low magnetic ﬁeld to the 3D-LLL model at a sample-dependent ﬁeld was observed (see e.g., [937, 938] and references therein). In the later publication by Roulin et al. [1073], it was argued that both descriptions are valid in the same range of ﬁelds. The main problem in the deﬁnitions of the ﬂuctuation contribution is the identiﬁcation of a singular part, which is only a few percent of the total speciﬁc heat. Therefore, depending on the model of the regular part, diﬀerent conclusions could be reached. To overcome the problem, measurements of the derivatives of the total speciﬁc heat versus ﬁeld or temperature may be performed. Since the regular part of the speciﬁc heat does not depend on magnetic ﬁelds and weakly depends on the temperature, contrary to the singular part, more reliable results can be achieved. By using this approach in studies of the YBa2 Cu3 O6.93 crystals in magnetic ﬁelds up to 16 T, Roulin et al. [1073] have shown that the low-ﬁeld 3D-XY model is valid for high ﬁelds of ∼ 1 T as well. The 3D-LLL scaling is also applied for these ﬁelds if one assumes that the temperature dependence of the upper critical ﬁeld has a ﬁnite curvature: Hc2 ∝ |T − Tc |4/3 , as opposed to the linear dependence (4.7) in the G–L theory where Hc2 ∝ |T − Tc |. These results were veriﬁed by independent measurements of the reversible magnetization. A “gauge theory” scenario of critical ﬂuctuations in extreme type-II superconductors in a magnetic ﬁeld was proposed by Te˘sanovi´c [1235,1236] and compared with the 3D-XY and 3D-LLL models.

4.3 Magnetic Properties Studies of macroscopic magnetic properties of copper-oxide superconductors reveal a number of novel phenomena, the interpretation of which is still controversial. The most exciting one was the discovery of a “vortex matter”, which

4.3 Magnetic Properties

143

shows a variety of unconventional magnetic, thermal, and transport properties (for review, see [141,176,251,382,779,887]). Already in the ﬁrst magnetic measurements in the high-temperature superconductors performed by M¨ uller et al. [871] to conﬁrm the Meissner eﬀect in the La–Ba–Cu–O compound, an anomalous behavior - the irreversibility line in the H-T phase diagram was ∗ (H) below which the ﬁeldfound. This is speciﬁed by the temperature Tirr cooled susceptibility, χFC , and the zero-ﬁeld-cooled susceptibility, χZFC , take diﬀerent values. The irreversibility line was described by the formula Hirr (T ) = H ∗ (1 − T /Tc∗ )n ,

(4.31)

where H ∗ = 1.17 T, Tc∗ = 23 K, and the exponent n ∼ 3/2. Such irreversible phenomena are observed in spin glasses, which are viewed as metastable thermodynamical systems. In this respect, a hypothesis has been suggested that ∗ a metastable state, a superconducting glass, arises at T < Tirr (H) . This state was observed earlier in granular superconductors with a weak-link structure. Later on, it was shown that the irreversibility line is close to the vortex-lattice melting line in the mixed state of the high-temperature superconductors as shown in Fig. 1.2 and discussed below in detail. Presently, the study of the vortex matter in cuprates is a very rich, complex, and the most rapidly advancing branch of the physics of the high-temperature superconductors. These studies are motivated not only by their importance for potential applications of high-temperature superconductors but also by their role in developing fundamental physics of phase transitions and dynamic behavior of elastic strings in a random environment [141, 176]. 4.3.1 Vortex Matter Within the mean-ﬁeld G–L theory, Sect. 4.1, the H-T phase diagram of the type-II superconductors comprises three phases: the homogeneous Meissner phase at H < Hc1 (T ), the mixed (or Shubnikov) phase at Hc1 (T ) < H < Hc2 (T ), and the normal, nonsuperconducting phase at H > Hc2 (T ). In the mixed phase, the magnetic ﬁeld penetrates the superconductor in the form of ﬂux lines or vortices, which make up the Abrikosov triangular vortex-lattice [6]. Under the external electric current density j, the vortices experience the Lorentz force FL = (1/c)[j × B] but do not move until the force is less than the pinning force Fpin . This deﬁnes the critical current jc = cFpin /B (for the current j perpendicular to the ﬁeld B). Under thermal ﬂuctuations, the pinning potential becomes weaker, which reduces the critical current. When the thermal ﬂuctuations of the vortex positions become comparable to the coherence length, the depinning of vortex lines occurs. The depinning is estimated to occur at the magnetic induction Bdp (T ) 8τGi Hc2 (0)(T /Tc )2 [141, 176]. The thermal depinning is a continuous crossover, which is observed as a ﬂux-creep phenomenon in the type-II superconductors. In general, a very complicated picture of a creep-type motion of the vortex system emerges (see e.g.,[141]).

144

4 Thermodynamic Properties of Cuprate Superconductors

The pinning energy can be estimated as a value of the condensation energy that depends on the thermodynamic critical ﬁeld Hc0 and the coherence 2 3 2 length: U (0) ∼ Hc0 ξ /8π where ξ 3 = ξab ξc is a coherence volume. In conventional superconductors with large coherence length, the pinning energy is rather large and thermal ﬂuctuations do not destroy the Abrikosov vortexlattice. Therefore, the lower Hc1 (T ) as well as the upper Hc2 (T ) critical ﬁelds are well deﬁned, and they can be measured with high precision. In hightemperature superconductors, due to small value of the coherence length, the pinning energy is small (in the YBCO compound U (0) ∼ 20 meV), while the superconducting temperature Tc is high. Therefore, the thermal energy at high-transition temperature is comparable to the pinning energy: √ kB Tc ∼ 8π 2τGi ∼ 3, (4.32) 2 3 Hc0 (0)ξ /8π if we take τGi ∼ 0.01 for the Ginzburg number (4.17) for the YBCO-type compound. Large thermal ﬂuctuations results in the giant ﬂux creep discovered already in the ﬁrst measurements of resistivity (see [779]). For instance, the current-independent thermally activated electrical resistivity was observed at all temperatures below Tc and low critical ﬁelds in the mixed phase in Bi2.2 Sr2 Ca0.8 Cu2 O8+δ single crystal [948, 949]. The resistivity has been described by the Arrhenius law, ρ(T, H, φ) = ρ0 exp[−U0 (H, φ)/kB T ],

(4.33)

where the activation energy depends only on the direction φ and the value of magnetic ﬁeld, U0 (H) ∼ 1/H α , α = 0.15 − 0.5, which is relatively small, U0 (H, φ) = 300 − 3,000 K. The prefactor ρ0 is magnetic ﬁeld independent and is three orders of magnitude larger than the normal-state resistivity. This ﬂux-creep motion results in a large broadening of the resistive transition, which precludes accurate measurements of Tc (H) by this method (see e.g., Fig. 4.14b). In the layered Bi-type compounds, the thermal ﬂuctuation region, estimated by the 2D Ginzburg number (4.18), is even larger due to a much higher anisotropy ζ = ξab /ξc ∼ 250. In the 2D-type compounds, an interaction between the layers is mediated by a weak Josephson coupling and the ﬂux lines for Hc consist of a stack of 2D point vortices, which are called pancake vortices (see e.g., [141, 176]). Figure 4.9 shows a schematic H-T phase diagram for the high-temperature superconductors: the left panel for the anisotropic YBCO-type cuprate and the right panel for the strongly layered Bi-type cuprate [141]. The drawings in the upper parts (a) in the Fig. 4.9 are not to scale, while the lower parts (b) showing the melting curves Bm (T ) are to scale. The following parameters were A , London used: Hc1 (0) 750(650) G, Hc2 (0) 230(100)T, ξBCS 16(25) ˚ magnetic penetration depth λ 1, 400 ˚ A, and the anisotropy parameters ζ 2 = mc /mab 25(2, 500) for the YBCO (Bi-2212 type) model systems.

4.3 Magnetic Properties

145

a vortex liquid fluctuations

B2D

a

H

vortex liquid

vortex lattice

fluctuations

Bm(T)

Hc2(T)

vortex lattice

H Hc1(T)

Hc (T) 2

b

Bm(T)

O

Hc (T)

b

O

B Hc1(O)

Bm

B Hc1(O)

10–3 O

10–3

O O

Hc1

Bm

1

T

Tc

B* O

T* 2D Tm

T

Tc

Fig. 4.9. H-T phase diagram for high-temperature superconductors: the anisotropic YBCO-type cuprate (left panel ), and the strongly layered Bi-type cuprate (right panel ). Hc1 (T ), Hc2 (T )− lower and upper critical ﬁelds and Bm (T )− vortex melting c 1994) line (reprinted with permission by APS from Blatter et al. [141],

In the case of large thermal ﬂuctuations, the upper critical ﬁeld line Hc2 (T ) determines only a crossover region where the modulus of the order parameter rapidly increases, which results in the appearance of a large diamagnetic susceptibility and a speciﬁc heat anomaly (see Fig. 4.6). However, the phase of the order parameter experiences large ﬂuctuations, which can be described as a vortex liquid of ﬂexible, entangled vortex lines [889]. This region can occupy a substantial portion of the H-T phase diagram. In the vortex-liquid phase, a linear resistivity is observed as in a normal metal. At some lower temperature speciﬁed by the magnetic induction Bm (T ), a phase transition to the vortex-lattice phase occurs with ﬁxed positions of vortex lines. Within the Lindemann melting theory, the ﬁrst-order melting transition is described by the function Bm (T ) (5.6c4L /τGi )Hc2 (0)(1 − T /Tc)2 [141]. The Lindemann parameter cL ∼ 0.2 deﬁnes the amplitude of the thermal ﬂuctuations of the vortex positions at melting: u2 (Tm ) c2L a20 where a0 = φ0 /B is the vortex-lattice constant. The vortex-liquid phase in strongly anisotropic Bi- or Tl-based compounds is much broader in comparison with the 3D anisotropic YBCO compound. The two-dimensional character of these layered compounds manifests itself in the appearance of a speciﬁc ﬁeld B2D (πφ0 /Λ2 ) ln(Λ/ξ), where Λ = dζ denotes

146

4 Thermodynamic Properties of Cuprate Superconductors

the Josephson screening length in the layered system with the interlayer separation d. In the low-ﬁeld region of the phase diagram, Hc1 B < B2D , the well-deﬁned lattice of vortices undergoes a 3D-type melting transition. At higher ﬁelds, B2D < B Hc2 , the melting transition has a quasi2D two-dimensional character with the ﬁeld independent temperature Tm

√ ε0 d/4 3π ∼ 25 K for the Bi-type compound (ε0 is deﬁned in (4.18)). At this decoupling transition, the coherence between the layers is lost and a weakly coupled sheets of pointlike pancake vortices arises. The vortex-lattice can melt not only due to large thermal ﬂuctuations but also due to a weak magnetic ﬁeld as shown by the lower melting lines in Fig. 4.9. In this region, above the lower critical ﬁeld Hc1 (T ), interaction between the vortices is weak due to a large distance a0 = φ0 /B between them, larger than the London penetration length λL , which results in the vortex-lattice melting. However, the width of the vortex-liquid phase close to Hc1 is rather narrow, of the order of 1 G, and the experimental observation of this reentrant melting phase transition presents a problem. In real systems, the vortex lines interact with diﬀerent type of crystal imperfections: isolated impurities (of a “zero” dimension), dislocations and twin boundaries (2D objects), and specially produced by ion irradiation columnar defects (3D correlated disorder). An interplay between thermal ﬂuctuations and quenched disorder further complicates the H-T phase diagram as shown in Fig. 4.9. As ﬁrst pointed out by Larkin [691], even a weak disorder in the dimensions D < 4 destroys the translational long-range order of the Abrikosov vortex-line lattice. Later, a theory of collective pinning, which takes into account the nonlocal character of the elastic medium, was developed by Larkin et al. [692]. This approach, valid for small-scale displacements, was generalized by Feigel’man et al. [325] to arbitrary distances to describe the large-scale behavior relevant to the cuprate superconductors. Thus, pinning of vortices due to lattice imperfections destroys the long-range correlations of the vortex-lattice replacing it with a disordered vortex solid, a “vortex glass” phase [339]. While the phase transition between the crystal and liquid phases is always of the ﬁrst order due to diﬀerent symmetry, the melting phase transition between the vortex liquid and vortex glass can be continuous. It was suggested also that a “hexatic vortex glass” can occur in the disordered type-II superconductors, which has no long-range translational order but has a ﬁnite orientational long-range order, which is characterized by correlations of the local orientation of the vortex-lattice [240]. In the case of weak disorder, a thermodynamically stable glassy phase with quasi-long-range order with perfect topological order (without defects as dislocations) was proposed by Giamarchi et al. [383]. This phase is called the Bragg glass, which is characterized by algebraically divergent Bragg peaks observed in small angle neutron scattering. A detailed theoretical description of an emerging H-T phase diagram caused by a complicated interplay of thermal ﬂuctuations and quenched disorder of various types is given by Blatter et al. [141] (see also [176, 382, 887]). To elucidate the

4.3 Magnetic Properties

147

nature of the phase transitions in the mixed state of high-temperature superconductors, numerical studies of statics and dynamics in various microscopic models were also carried out (see e.g., [1076] and references therein). Extensive magnetic, resistivity, and thermal measurements in cuprate superconductors have supported this general picture of a complicated H-T phase diagram. Generally, the vortex matter phase diagram should have three distinct phases: an ordered vortex-lattice at low ﬁelds, a highly disordered solid at high ﬁelds, and a vortex ﬂuid phase at high temperatures. The vortexlattice melting transition was ﬁrst detected by resistivity measurements in single crystals of YBCO [684, 685, 1082, 1083]. A distinct “kink” was observed in the magnetoresistance ρ(T, H) at a certain temperature Tm , which was considered as a melting transition temperature. The ﬁrst-order type transition was proved by the observation of a magnetic hysteresis loop at Hm in the resistivity curve at ﬁxed temperature. Later on discontinuous jumps of the magnetization were detected along the melting curve in clean untwinned YBCO single crystals (see e.g., [717, 1351]). Using the Clausius–Clapeyron equation, the entropy change ΔS at the melting transition and the latent heat L per unit volume were evaluated: ΔS = −

ΔB dHm , 4π dT

L = Tm ΔS.

(4.34)

Figure 4.10 shows SQUID magnetization measurements in an untwinned YBa2 Cu3 O7−δ single crystal (Tc = 92.9 K) for H = 4.2 T parallel to c-axis [1351]. The inset shows the jumps in the magnetization (top) and the temperature dependence of the magnetoresistance (bottom) in several ﬁelds. The locations of the magnetization jumps and the resistive drops coincide, which demonstrates the existence of the ﬁrst-order melting transition. The entropy jump estimated from (4.34) is ΔS ∼ 0.65 kB per vortex per CuO2 double layer at T ∼ 88 K. The melting line as determined from resistivity and magnetic measurements is well described by a power law Bm (T ) = Bm0 (1 − T /Tc )n ,

(4.35)

where Bm0 = 99.7 T and the exponent n 1.33 = 2ν. The exponent ν = 2/3 describes the 3D-XY critical behavior for the correlation length ξ ∝ τ −ν as discussed in Sect. 4.2.3. The scaling behavior of the melting transition according to Bm (T ) ∝ 1/ξ 2 demonstrates the importance of large ﬂuctuation eﬀects in cuprates, though the ﬁrst-order transition, in fact, should not obey the scaling theory of critical phenomena. The calorimetric measurements of the latent heat in the untwinned YBa2 Cu3 O7−δ single crystal (Tc 91.9 K) unambiguously conﬁrmed the ﬁrst-order phase transition (FOT) along the melting line Bm (T ) [1114]. Later on, in the same crystal an angular dependence of the melting magnetic ﬁelds and the latent heat were measured [1115,1116]. It was found that the melting magnetic

148

4 Thermodynamic Properties of Cuprate Superconductors 0

H = 4.2 T IIc

1

–10

4π(M-Ms) (G)

0.8 5.6

4.2 2.9 1.8 T

H IIc

0.6

4π M(G)

0.4 0.2 0 60

–20

–30

ρ(μΩ.cm)

50 40

Js = 0.65 A/cm2 H IIc

30 20 10 5.6 0 80

–40 80

90

4.2 2.9 1.8 85

T(K)

1 0.4 0.2 0.1

T(K) 90

100

95

110

Fig. 4.10. Temperature dependence of the magnetization in untwinned YBa2 Cu3 O7−δ crystal for H = 4.2 T parallel to c axis. The inset shows the jumps in the magnetization (top) and the temperature dependence of the magnetoresistance (bottom) in several ﬁelds (reprinted with permission by APS from Welp et al. [1351], c 1996)

ﬁeld at ﬁxed temperature T scales with the varying angle θ of the magnetic ﬁeld direction according to the equation: ζ , Bm (T, θ) = Bm0 (T, 0)) sin2 (θ) + ζ 2 cos2 (θ)

(4.36)

where θ = 0 corresponds to Hc, ζ = mc /mab . A power-law ﬁt (4.35) with Bm0 91.5 T, ζ = 7.71, n 1.26 perfectly accommodates all the data [1116]. At the same time, the discontinuity of the entropy at melting depends only on the temperature for all magnetic ﬁeld orientations with a typical speciﬁc heat diﬀerence ΔC/T = Δ(∂S/∂T )H ∼ 1.5 mJ/mole K2 at T = 82 − 88 K. The anisotropy of the magnetization discontinuity at vortex-lattice melting was also studied by the magnetic torque measurement in the high-quality untwinned YBCO single crystal with Tc 92.3 K [1117]. This experiment has conﬁrmed the calorimetric results concerning the anisotropy scaling (4.36) for the melting ﬁelds up to T /Tc = 0.99 and the θ-independence of the speciﬁc heat discontinuity ΔC/T at the melting transition. It was also shown that the jump of the magnetization ΔM at the transition is always parallel to the magnetization M. Figure 4.11 shows the discontinuity in entropy at vortex-lattice melting ΔS both from magnetic (triangles) and thermal (stars) experiments

4.3 Magnetic Properties

a

b 2.0

12 10

1.5

μ0Hm0 (T)

ΔS (mJ/mole K)

149

1.0 Tcrit,up

0.5

8 6 4 2

0 0.75

0.80

0.85 0.90 T/Tc

0.95

1

0 0.75

0.80

0.85 0.90 T/Tc

0.95

1

Fig. 4.11. (a) The entropy jumps ΔS(T /Tc ) at the vortex-lattice melting both from magnetic (triangles) and thermal (stars) experiments for the untwinned YBa2 Cu3 O7−δ single crystal for Hc. (b) The melting ﬁelds Hm (T /Tc ) for the same crystal. The upper critical point occurs for T /Tc = 0.78 and μ0 Hm0 = 12.5 T c 2000) (reprinted with permission by APS from Schilling et al. [1117],

(a) and the melting ﬁelds Hm (T ) for Hc (b) [1117]. The dashed line is a power-law (4.35) ﬁt to the data below the magnetic ﬁeld Bm = 6.5 T. The discontinuity in entropy ΔS vanishes slightly below the superconducting temperature Tc (the FOT is still detectable at Bm = 89 mT and T /Tc = 0.994). At lower temperatures and high-magnetic ﬁelds, the power law (4.35) deviates from the experimental data, while the discontinuity in entropy ΔS extrapolates to zero, which have been interpreted by the authors as occurrence of the upper critical point at Bcr = 12.5 T and T /Tc = 0.78. Calorimetric transition on the vortex-lattice melting line as a function of the oxygen concentrations x in twinned YBa2 Cu3 Ox crystals was studied by Roulin et al. [1074]. In the overdoped crystal (x = 7), speciﬁc heat peaks were observed at all magnetic ﬁelds (up to 16 T) at the melting transition. However, in crystals with lower oxygen concentration (x = 6.96, 6.94), these peaks shrank and at higher ﬁelds only speciﬁc heat steps were revealed. Therefore, the ﬁrst-order transition line Bm (T ) (speciﬁc heat peaks) ends at a critical point Bcr above, which it turns into a second-order transition (speciﬁc heat jumps) from the vortex liquid to the vortex glass. The latent heat, the slope of the Bm (T ) and Bcr increase with x, e.g., Bcr = 9.5 T for x = 6.94 and Bcr > 16 T for x = 6.96. For the sample with x = 6.94, no transition was observed for B > Bend = 11.5 T. In Table 4.3 in the ﬁrst 3 columns ﬁtting parameters for the melting line (4.35) are given for various oxygen concentrations x [1074]. The exponent n 1.33 for B < Bcr . The function Bm (T, x) scales with 1/ζ where the anisotropy ratio ζ = mc /mab increases with oxygen concentration decrease. The value of the critical magnetic ﬁeld also depends on the level of crystal imperfections. In YBCO crystals, a strong pinning is produced by twins and oxygen vacancy disorder, which suppress the ﬁrst-order melting transition and drive the vortex-lattice into a vortex glass state.

150

4 Thermodynamic Properties of Cuprate Superconductors

The technique of small-angle neutron scattering is a powerful tool in studies of ﬂux lattice in superconductors since it provides direct evidence on temperature dependence, magnetic ﬁeld, and symmetry of the ﬂux line lattice in bulk samples. It was also used in oxide superconductors in a number of studies (for a review see [348]). Here, we mention only a few of them. The vortex-lattice melting in a heavily twinned YBa2 Cu3 O7−x crystal (Tc 92 K) with imperfections caused by nonsuperconducting inclusions was studied by Aegerter et al. [19]. They observed a continuous decrease of the diﬀraction intensity which was described by the scaling function (4.35) with an exponent n varying from n 1.5 for small ﬁelds up to n 1 for strong applied ﬁelds B = 4 − 5 T. In the region of the linear T -dependence, the slope of the melting ﬁeld was dBm /dT ∼ −0.6 T/K. It was also found that the vortex structure strongly depends on the magnetic ﬁeld direction relative to twin boundaries: for H c the diﬀraction pattern revealed a fourfold symmetry, while for H applied at 51◦ to the c axis two distorted hexagonal lattices were found. However, the melting line was only slightly aﬀected by the change of the magnetic ﬁeld direction and therefore by the twin boundaries. A change of the symmetry of the vortex structure in the overdoped YBa2 Cu3 O7 single crystal (Tc = 86 K) in magnetic ﬁeld was also observed by Brown et al. [179]. At a high-magnetic ﬁeld, B ∼ 11 T, the ﬂux lattice structure changed smoothly from distorted triangular to square coordination. The latter is expected for the d-wave pairing and high density of ﬂux lines at overlapping vortex cores. A transition from the Bragg glass phase to a more disordered vortex glass state in La1.9 Sr0.1 CuO4 crystal was detected by Divakar et al. [283] from small-angle neutron scattering and μSR measurements. The possibility of reentrant melting transition close to the lower critical ﬁeld Hc1 , as shown in Fig. 4.9, was suggested by Pissas et al. [996] in studies of vortex-lattice melting at low-magnetic ﬁeld close to Tc . A sharp drop in the local permeability measured by local Hall probe ac-susceptibility was attributed to the melting transition described by the power-law (4.35) with n = 2ν 4/3 at the magnetic ﬁelds above H1c ∼ 232 Oe. Below this ﬁeld, the melting curve Hm (T ) bends down from the scaling law (4.35) to lower temperatures, which points to the existence of a lower critical point in the melting transition or a reentrant behavior. The coupling between the crystal lattice and the vortex melting in the untwinned YBa2 Cu3 Ox single crystal (Tc = 92.4 K) was demonstrated by Lortz et al. [747]. Distinct discontinuity in the thermal expansion of the crystal lattice was observed at the vortex-lattice melting at Tm 81.7 K, which were explained by strong pressure dependence of the electronic structure and Tc . The uniaxial pressure coeﬃcients for Tm were similar to that ones for Tc : dTm /dPi 0.9dTc /dPi (see (2.23) in Sect. 2.6). In particular, these coefﬁcients for the melting transition in magnetic ﬁeld H = 6 T are equal to dTm /dPi (K/GPa) −2.1, +1.8, 0±1 for pressure Pi along the axes i = a, b, c, respectively.

4.3 Magnetic Properties

151

In the strongly anisotropic layered Bi-type cuprate, the melting transition occurs at much lower magnetic ﬁelds. A clear evidence of an ordered vortexlattice with the hexagonal symmetry in the low-ﬁeld region was obtained by using small-angle neutron scattering in the Bi2.15 Sr1.95 CaCu2 O8+x crystal by Cubitt et al. [255]. A diﬀraction signal seen at low ﬁelds and temperatures disappeared rapidly under increasing temperature or magnetic ﬁeld. It was interpreted as a melting transition of the vortex-lattice, which was conﬁrmed by the appearance of ﬁnite resistance of the vortex liquid within the superconducting state. The temperature of the melting transition at low ﬁelds coincided with the irreversible magnetic behavior. However, no diﬀraction signal was observed at the lowest temperature for the “crossover” magnetic ﬁeld B2D ∼ 650 G, which was explained by the decoupling of the 3D ﬂux lines into two-dimensional pancake vortices at high ﬁelds. The melting transition in Bi2 Sr2 CaCu2 O8 was conﬁrmed by observation of a discontinuous step of local magnetization [1411]. The entropy change evaluated from the Clausius–Clapeyron equation (4.34) was quite high, ΔS ∼ 2 kB per vortex, per layer, slightly below Tc = 90 K but then vanished at lower temperature, T ∼ 40 K. Later on, the ﬁrst-order melting transition was studied in Bi2 Sr2 CaCu2 O8 crystals with diﬀerent oxygen stoichiometry by Khaykovich et al. [602] as shown in Fig. 4.12a. With increasing anisotropy from overdoped (Tc = 83.5 K) to optimallydoped (Tc = 89 K) crystals, the melting ﬁeld Bm (T ) decreases. The authors observed also a second magnetization peak at

a

b

700 600 500

1000

as-grown

400

Bm (T)

300 200 Bsp (T) 100

B [G]

B [G]

1500

Td

Tsb

Tx

over-doped C

B +

A

Bsp

500 D

E

TFOT

optimally-doped

0 20

30

40

50 60 T [K]

70

80

90

0

20 30 40 50 60 70 80 90 T [K]

Fig. 4.12. (a) First-order vortex-lattice melting transition Bm (T ) (empty symbols) and the second transition Bsp (T ) (filled symbols) for Bi2 Sr2 CaCu2 O8 crystals with diﬀerent oxygen stoichiometry. The arrows indicate the position of critical point where a step of local magnetization vanishes (reprinted with permission by APS c 1996). (b) Phase diagram for Bi2 Sr2 CaCu2 O8 single from Khaykovich et al. [602], crystal in dc magnetic ﬁeld parallel to the c axis. In phases A and D, vortices are immobile. Phase B has an intermediate ordered vortex structure between the vortexlattice in phase E and a pancake gas in phase C. TFOT indicates the ﬁrst-order melting transition as in the left panel Bm (T ). Tsb shows a crossover from uniform to the surface barrier dominated current ﬂow (reprinted with permission by APS from c 1998) Fuchs et al. [358],

152

4 Thermodynamic Properties of Cuprate Superconductors

lower temperatures, which they connected with a second transition Bsp (T ). The two lines seem to form one continuous transition of the vortex matter, though at the melting transition the vortex-lattice transforms into a state with reduced vortex pinning, while above Bsp (T ) a new phase shows enhanced vortex pinning. Taking into consideration the neutron diﬀraction measurements discussed above, it was suggested that the Bm (T ) line is a simultaneous melting and decoupling transition into a pancake liquid, whereas Bsp (T ) corresponds to a transition to a decoupled pancake solid. The ﬁrst-order melting transition in the Bi2 Sr2 CaCu2 O8 crystal at elevated temperatures was conﬁrmed by simultaneous observation of the magnetization step and of a sharp resistive onset by Fuchs et al. [356]. In a later publication by Fuchs et al. [358], a more complicated vortex matter phase diagram for the Bi2 Sr2 CaCu2 O8 crystal (Tc 88 K) in magnetic ﬁeld parallel to the c-axis was reported. The studies were based on a special technique (discussed below) of measuring the distribution of the transport current across the sample [359], which enabled to investigate the vortex activation over the surface barrier in the presence of driving force. Figure 4.12b shows this diagram where several new phases are depicted in comparison with usually considered three phases: an ordered vortex-lattice at low ﬁelds and temperatures, a disordered vortex solid at high ﬁelds, and a vortex liquid at high temperatures. At high ﬁelds, H ∼ 1,000 G, there are three forms of the vortex transport current. At high temperature, T > Tsb , in the pancake gas phase the vortex drift velocity is determined by bulk viscosity or pinning. In the wide region of temperatures Td < T < Tsb the vortices are mobile but their ﬂow is determined by a low hopping rate over the surface barrier. Below the depinning temperature Td in the phase A, the bulk vortex pinning occurs. At low-magnetic ﬁelds, a FOT is observed at temperature TFOT above which the bulk pinning is very weak. However, the vortices are mobile also below TFOT in the phase E, though the resistivity in this phase is approximately two orders of magnitude lower, which is usually registered as the liquid-melting transition. The vortices become immobile only at lower temperatures below a new line Td , which separates the strongly disordered Josephson glass phase D from the weakly disordered elastic Bragg glass phase E. At the Bsp line, as in Fig. 4.12a, a second magnetization peak is detected. The transport measurements reveal also a new Tx line, which merges with the TFOT line at low ﬁelds. The line denotes the transition from weak surface barrier at T > Tx to much stronger one in the phase B. It was suggested that the phase B has a structure with an intermediate order between that of the phases E and C and represents a disentangled liquid of vortex lines. The phase B has a very weak shear modulus and it can be viewed as a soft solid with a hexatic order or a supersolid, which represents an aligned stack of ordered 2D pancake layers. Some kind of ordering in the B phase was conﬁrmed by the observation of a weak Bragg peak shown by the cross at 900 G and T = 45 K in Fig. 4.12b. Based on these transport measurements, the authors suggested that the ﬁrst-order transition at TFOT line is a decoupling or softening transition rather

4.3 Magnetic Properties

153

than melting. This suggestion was conﬁrmed by other experiments. The resistive transitions for currents applied both parallel and perpendicular to the (ab) plane are consistent with a ﬁrst-order phase transition, which demonstrates that melting and decoupling transitions occur simultaneously as a sublimation of the vortex solid directly into a vortex gas of pancakes [357]. Studies of the vortex thermal ﬂuctuations using Josephson plasma resonance by Colson et al. [248] revealed that the ﬂuctuations soften the Josephson coupling between the layers, which results in dissociation of the vortices at the ﬁrst-order phase transition. The new Tx line then can be considered as a melting of a supersolid, and the Bsp line as describing a disorder-induced decoupling rather than disorder-induced solid entanglement. Subsequent studies should determine which of these transition lines, observed in the transport measurements, represent generic thermodynamic phase transitions. A common origin of the irreversibility line Hirr (T ) and the lattice-vortex melting line Hm (T ) was suggested in a number of studies (e.g., [250]). Above the irreversibility line where the diamagnetic signal is still large, the magnetization curves M (H) are reversible and the resistivity is ﬁnite similar to the region above the melting line. Below the irreversibility line, the ﬁeldcooled susceptibility, χFC , and the zero-ﬁeld-cooled susceptibility, χZFC , take diﬀerent values and the resistivity tends to zero. The ﬁrst measurements of the irreversibility line performed by M¨ uller et al. [871] in the La–Ba–Cu–O compound discovered the power law dependence (4.31) with an exponent n = 3/2. By using diﬀerent methods for magnetization measurements (magnetic ﬁeld and temperature sweeps) on twinned YBa2 Cu3 O7−δ crystals with 0.03 ≤ δ ≤ 0.35, Babi´c et al. [93] observed a power law scaling for the irreversibility line at τ = (1 − T /Tc) ≤ 0.2, similar to that for the melting line, (4.35): Hirr (T ) = H ∗ (1 − T /Tc)n ∝ ξ −2 (T ), (4.37) with n = 2ν = 4/3 for small values of δ. Deviations from (4.37) close to Tc was observed for oxygen-depleted crystals. In particular, at δ = 0.35, an expoc nential behavior Hirr (T ) = H0 exp(−T /T0 ) described better the experimental curve. The values Hirr (T ) obtained from the magnetization measurements of the χFC and χZFC by slow temperature sweeps were 20−40% lower than the melting line Bm0 (T ) and better described by the power law (4.37) with the exponent n = 3/2 as in (4.31). It was attributed to the ﬂux creep eﬀects during the slower temperature sweeps at these measurements. It was also found that the magnetic ﬁeld along the c-axis scales with that one in c ab the plane, Hirr (T ) Hirr (T )/ζirr , where the ﬁtting anisotropy parameter ζirr ζ = mc /mab . The ﬁtting parameters in (4.37) for various values of δ are presented in Table 4.3 in the last four columns. If we compare the magnetic ﬁeld H ∗ in (4.37) with Bm0 in (4.35), we can conclude that the melting transition is actually an upper bound for the irreversibility line, Hirr (T ) < Hm (T ), the two being close at small oxygen depletion [93].

154

4 Thermodynamic Properties of Cuprate Superconductors

The results of the 3D XY scaling of the reversible magnetization for the twinned YBa2 Cu3 O7−δ crystals have been discussed in Sect. 4.2.3. It is remarkable that the irreversibility ﬁeld Hirr (T ) scales to the same point on the 3D XY plot as shown in Fig. 4.8, which demonstrates the important role of thermodynamic ﬂuctuations above and around Hirr . This conclusion is supported by consideration of the condensation energy deﬁned by the thermodynamic critical ﬁeld Hc2 (0), which is also given in the last four columns of Table 4.3. As the authors remark, the ratio of the condensa2 tion energy in the coherence volume, ξ 3 (0) = ξab ξc , to the thermal energy, 2 3 K = (Hc0 (0)ξ /8πkB Tc ), appears to be small, in the range 0.15 − 0.23. This ratio is, in fact, depending only on the Ginzburg number (4.17) as given √ by (4.32): K = (8π 2τGi )−1 1/3, which is small due to a large value of τGi ∼ 0.01 in YBCO crystals. Another interesting point is a similar strong suppression of the irreducible magnetic ﬁeld H ∗ and the condensation energy Hc2 (0) with oxygen depletion. According to Table 4.3, they decrease by more than a factor of 10, while the anisotropy ζ changes only by a factor of 2 when the oxygen content decreases from x = 6.97 to x = 6.65. Therefore, the suppression of H ∗ is not caused by the deterioration of the interlayer coupling with the anisotropy increase but by the diminishing of the condensation energy Hc2 (0), which measures the strength of superconductivity and also the resistance of the vortex-lattice to melting. The slope of the irreversibility line being proportional to H ∗ also strongly decreases with oxygen depletion, e.g., dHirr /dT ∼ −0.7 T/K for δ = 0.03, while dHirr /dT ∼ −0.1 T/K for δ = 0.35 (at the temperature T 0.8Tc ). A similar evaluation holds for the slope of the melting line Bm (T ) (4.35). As discussed in Sect. 4.2.2, a strong reduction of the condensation energy is associated with the opening of a normal-state pseudogap as shown in Figs. 4.4 and 4.5. Thus, we conclude that a competition between the thermodynamic ﬂuctuations and the condensation energy deﬁnes the melting and the irreversibility lines in the H-T phase diagram of the cuprate superconductors. Studies of the irreversibility line in the Bi2 Sr2 CaCu2 O8 crystals disclosed a dimensional crossover from three- to quasi-two-dimensional vortex ﬂuctuations at low temperatures. In the vicinity of Tc , the irreversibility line Hirr (T ) can be described by a function similar to (4.37) with an exponent n ∼ 2 and H ∗ ∼ 1,000 G. However, at T /Tc ≤ 0.5 the irreversibility line shows upturn and grows exponentially with T −1 [1113]. This crossover shown in Fig. 4.9 is a characteristic feature of the highly anisotropic compounds consisting of weakly coupled layers such as Bi- and Tl-based materials. Local vortex dynamics studied in Bi2 Sr2 CaCu2 O8 single crystals by Zeldov et al. [1412] disclosed various mechanisms of irreversible behavior. At high temperatures, it is due to geometrical barriers, whereas at intermediate temperatures the irreversibility is determined by surface barriers. The bulk vortex pinning deﬁnes the irreversibility line only at low temperatures. This mechanism was later conﬁrmed in studies of transport properties in Bi2 Sr2 CaCu2 O8 single crystals by Fuchs et al. [359]. By measuring the self-induced magnetic

4.3 Magnetic Properties

155

induction Bac of the transport current Iac across the sample, it was possible to study the local magnetization and to obtain a valuable information about the transport current distribution across the sample. The measurements disclosed that at elevated temperatures the transport current ﬂows uniformly across the width of the sample reﬂecting bulk vortex properties. At intermediate temperatures, the current ﬂows at the edges of the sample is due to the presence of surface barriers and it characterizes the properties of the barriers rather than the bulk properties. At low temperatures, the bulk pinning is observed when the current ﬂows only at the edges of the sample. Thus, the commonly used description of thermally activated resistivity by the Arrhenius law (4.33) actually reﬂects the thermal activation over the surface barrier and not the bulk pinning. Similarly, a sharp drop of the resistivity observed at the ﬁrstorder phase transition in the vortex-lattice (see Fig. 4.10) does not probe the true bulk vortex dynamics at the transition. Above, we have considered the vortex structures in the ﬁeld applied perpendicular to the layers. In this conﬁguration, the magnetic ﬁeld penetrates inside the superconductor in the form of pancake vortices coupled by the Josephson and magnetic interactions. If the ﬁeld is applied parallel to the layers, the magnetic ﬁeld penetrates inside the layered superconductor Bi2 Sr2 CaCu2 O8 in the form of Josephson vortices. The latter has no normal cores, but wide nonlinear cores located between two central layers. Much more complicated vortex structures are observed in the case of tilted magnetic ﬁelds, such as a kinked lattice, a tilted vortex-lattice, and coexisting lattices of pancake, and Josephson vortices (see e.g., [647] and references therein). Figure 4.13 demonstrates a set of crossing lattices, which contains a sublattice of Josephson vortices (JV) generated by the magnetic ﬁeld component parallel to the layers and a sublattice of pancake vortex stacks (PVs) generated by the perpendicular component of the ﬁeld [411]. Depending on

Fig. 4.13. Schematic view of the structure of a pancake vortex stack (PVs) intersected by a Josephson vortex (JV) stack in a tilted magnetic ﬁeld (reprinted with c 2005) permission by APS from Grigorenko et al. [411],

156

4 Thermodynamic Properties of Cuprate Superconductors

the in-plane London penetration length and the anisotropy parameter, various types of structures of Josephson vortices and pancake vortex-lattices can evolve. Investigation of a very rich phase diagram of the vortex matter in the multidimensional space of temperature, magnetic ﬁeld and its orientation, and sample anisotropy constitutes a separate ﬁeld of extensive studies in the physics of layered superconductors. 4.3.2 Critical Magnetic Fields As discussed in the previous section, large thermodynamic ﬂuctuations in the layered cuprate superconductors hinder the critical magnetic ﬁeld measurements. Strong thermal ﬂuctuations broaden the superconducting phase transition in magnetic ﬁelds and the upper critical ﬁeld Hc2 (T ) shows itself only as a crossover line from the normal-state to the superconducting state deﬁned by a nonzero diamagnetic signal. Since below the superconducting transition there appears a vortex liquid with ﬁnite resistivity, the Hc2 (T ) line cannot be determined from resistivity measurements. A ﬁnite drop of the resistivity occurs at the vortex-lattice melting line much below the Hc2 (T ) line (see Fig. 4.10). The electronic speciﬁc heat measurements in the optimally doped or overdoped YBCO crystals reveal a ﬁnite jump, which can be used for evaluation of the Hc2 (T ) dependence close to Tc . In the quasi-two-dimensional Bi-, Ta-, or Hg-type cuprates, the mean-ﬁeld speciﬁc jump in the heat capacity is washed out, while the ﬂuctuation anomaly at Tc in magnetic ﬁelds only broadens without a detectable shift (see Fig. 4.6). Recently, a new method to determine Hc2 (T ) by measurement of the Nernst signal in magnetic ﬁeld was proposed, which we discuss at the end of the section. The much smaller lower critical magnetic ﬁeld Hc1 (T ) can be measured easier since the thermal ﬂuctuations are not so important at the transition from the Meissner phase to the vortex-lattice. Below, we consider several studies where the critical magnetic ﬁelds have been estimated. Early results on critical magnetic ﬁeld measurements in various high-temperature superconductors are reported by Malozemoﬀ [779]. Upper Critical Magnetic Field Hc2 A commonly used method in the earlier studies devoted to determination of the upper critical ﬁeld was based on measurement of the dc reversible magnetization M (T ). In this method, a well-deﬁned onset of diamagnetism at Tc and a linear behavior with temperature outside the critical region are observed, which allows to determine critical ﬁeld slopes ∂Hc2 /∂T . Figure 4.14 shows the dependence of the zero-ﬁeld-cooled (ZFC) and the ﬁeld-cooled (FC) magnetization M (T ) (a) and resistivity R(T ) (b) in external ﬁelds H c up to 5 T in two twinned single crystals of YBCO with Tc = 92.5 and 92.2 K, respectively [1348]. As the ﬁeld H increases, the magnetization curves M (T ) shift toward lower temperatures, while remaining linear in temperature. This

4.3 Magnetic Properties

a

0.2 10 G ZFC 1 T ZFC 2 T ZFG 2 T FC 4 T FC 4 T ZFC 5 T ZFC

0.0 – 0.2 M( Gauss )

157

– 0.4 – 0.6 – 0.8 – 1.0 – 1.2 – 1.4 – 1.6

R ( mW- cm )

b

150 0T 1T 2T 4T 5T

100

50

0 78

80

82

84

88 86 T(K)

90

92

94

Fig. 4.14. The temperature dependence of the magnetization (a) and the resistivity (b) for YBCO single crystals in magnetic ﬁelds H c, 0 ≤ H ≤ 5 Tesla c 1989) (reprinted with permission by APS from Welp et al. [1348],

enables the determination of Tc (H) as the intercept of a linear extrapolation of the magnetization with the normal-state baseline. At the same time, as the external ﬁeld is switched on, the dependence R(T ) is drastically smeared out being pulled into the region of low-temperatures. The temperature dependence of Hc2 (T ) near Tc in this twinned YBCO crystal obtained from the magnetization measurements is shown in Fig. 4.15 [1348]. The upper critical ﬁeld slopes estimated from the linear part of the curves c ab are: ∂Hc2 /∂T −1.9 T/K and ∂Hc2 /∂T −10.5 T/K. The region of nonlinear dependence of Hc2 (T ) in low magnetic ﬁelds occurring in the twinned sample (see the inset in Fig. 4.15) was not observed in the untwinned crystal [1349], while the upper critical ﬁeld slopes were close to the twinned sample ones. The dashed lines in Fig. 4.15 show the resistive zero points in magnetic ﬁelds, which deﬁne the ﬂux-lattice melting line as discussed in Sect. 4.3.1. (see Fig. 4.11b). Since direct results of measurements of the huge upper critical ﬁelds at low temperatures in the high-temperature superconductors are not accessible, the Werthamer–Helfand–Hohenberg (WHH) formula

158

4 Thermodynamic Properties of Cuprate Superconductors 6 B ⎜⎜ab –10.5T/K

B⎜⎜c –1.9T/K

5

B(T)

4 3 2 1 0 85

86

87

88

89 T(K)

90

91

92

93

Fig. 4.15. The temperature dependence of the upper critical ﬁeld Hc2 (T ) in the twinned YBCO crystal for H c and H ⊥ c (solid lines). The resistive zero points in magnetic ﬁelds (R = 0) is shown by the dashed line (reprinted with permission c 1989) by APS from Welp et al. [1348],

Hc2 (0) 0.7 Tc

∂Hc2 ∂T

,

(4.38)

Tc

obtained for the BCS model with weak coupling is often used to estimate this c,ab quantity. The values of Hc2 of this formula, the (0) calculated on the basis ab c anisotropy parameter ζ = mc /mab = ξab /ξc = Hc2 /Hc2 , and the corresponding coherence lengths within the anisotropic GL model (4.12) are given in Table 4.4. Comparison of the speciﬁc heat jump calculated from the slopes of the magnetization curves according to the Rutgers formula (4.8) which reads in the anisotropic case as c,ab 2 ∂Hc2 1 ΔC = , (4.39) Tc 8πκ2c,ab ∂T T =Tc

with the jump value obtained in calorimetric experiments, can serve as an additional test for the accuracy of the measurement of Hc2 (T ). The calculations of the speciﬁc heat jump (4.39) by Welp et al. [1348] for the YBCO crystal with the G–L parameters κc = 52 − 60 and κab = 350 − 420 yield ΔC/Tc 42 mJ/mol K2 and 30mJ/mol K2 for the upper critical ﬁelds parallel to the c and a, b axes, respectively. These values are in a reasonable agreement with the experiments discussed in Sect. 4.2.1 (see Table 4.1). The upper critical magnetic ﬁeld in YBCO crystals was also studied by the calorimetric method. speciﬁc heat measurements in magnetic ﬁelds up to 14 T in the YBa2 Cu3 O6.92 single crystal (Tc = 92.4 K) by Janod et al. [552] allowed c /∂T −3.2 T/K, which to determine the slope of the upper critical ﬁeld: ∂Hc2 is larger than those discussed above. The upper critical ﬁeld evaluated by the c (0) 212 T and the corresponding coherence WHH formula (4.38) was Hc2

4.3 Magnetic Properties

159

Table 4.4. Upper critical magnetic ﬁelds Hc2 (a) single crystals of YBa2 Cu3 O7−δ Tc

c ∂Hc2 /∂T

ab ∂Hc2 /∂T

c ab Hc2 (0) Hc2 (0)

(K)

(T/K)

(T/K)

(T)

(T)

92.5 92.4 62.2

−1.9 −3.2 −2.0

−10.5

122 212 87

674

−8.7

380

ζ= mc /mab

ξc (0) ξab (0) (˚ A) (˚ A)

5.5 7±1 4.4

3

References

16.4 12.4 20

4.5

[1] [2] [3]

[1] Welp et al. [1348] [2] Janod et al. [552] [3] Vandervoort et al. [1301] (b) polycrystals of Y0.8 Ca0.2 Ba2 Cu3 O7−δ [749] δ=

0.13

0.21

0.3

0.36

70 220

80 310

84 190

83 140

Tc (K) Hc2 (0) (T)

(c) single crystals of Bi2 Sr2 CaCu2 O8+x [710] Samples

Bi-UD Bi-OP Bi-OD

Tc

Tc (0)

c ∂Hc2 /∂T

(per Cu)

(K)

(K)

0.10 0.16 0.20

65.0 87.3 77.0

69.6 90.1 78.4

ph

(T/K)

ξab (0) (˚ A)

λab (0) (˚ A)

−0.82 −2.0 −1.2

25 13 12

2,690 1,850 1,680

length was ξc (0) 12.4 ˚ A. The anisotropy ratio for the studied crystal was ab c estimated as ζ = Hc2 /Hc2 = 7 ± 1. A dc-magnetization study of the critical magnetic ﬁelds in the oxygendeﬁcient YBa2 Cu3 O7−δ single crystal (Tc 62 K, δ 0.35) was reported by Vandervoort et al. [1301]. The temperature dependence of the upper Hc2 (T ) and lower Hc1 (T ) critical magnetic ﬁelds for H c and H ⊥ c is shown in Fig. 4.16. The slopes of the upper critical magnetic ﬁelds evaluc /∂T −2 T/K and ated from the linear ﬁt to the high-ﬁeld behavior, ∂Hc2 ab ∂Hc2 /∂T −8.7 T/K, are similar to those found for the 90-K material (see c,ab Table 4.4). The corresponding values of Hc2 (0) calculated on the basis of the WHH formula (4.38) are lower than in the optimally doped YBCO crystals c (0)) are larger due to a lower Tc , while the coherence lengths (ξab (0) ∝ 1/ Hc2 (see Table 4.4). The anisotropy parameter evaluated from the upper critical ab c magnetic ﬁelds, ζ = Hc2 /Hc2

4.36, is astonishingly small for the oxygendeﬁcient YBCO crystal, where it is usually much larger than in the optimally doped ones (see e.g., Table 4.3). This discrepancy may be explained by a nonlinear temperature dependence of the upper critical ﬁelds in the underdoped sample, which precludes a reliable estimation of the ﬁelds.

160

4 Thermodynamic Properties of Cuprate Superconductors

Fig. 4.16. The temperature dependence of the upper critical ﬁeld Hc2 (a) and the lower critical ﬁeld Hc1 (b) for H c and H ⊥ c in an oxygen-deﬁcient YBCO single crystal (Tc 62 K) (after Vandervoort et al. [1301])

The temperature dependence of the upper critical ﬁelds in polycrystals of Y0.8 Ca0.2 Ba2 Cu3 O7−δ with various oxygen content δ was determined by Luo et al. [749] from the speciﬁc heat measurements in magnetic ﬁelds up to 14 T. The critical ﬁeld Hc2 (T ) was calculated from the ﬁeld dependence of the free energy, which was obtained by integrating the speciﬁc heat over temperature in the superconducting state. By using the thermodynamic equation for the diﬀerence of the free energy density between the normal and superconducting states, fn − Fs = Hc2 (T )/8π, and the equation for the superconducting free energy in magnetic ﬁeld, it is possible to determine Hc2 (T ) in a polycrystal. The calculated temperature dependence of the upper critical ﬁeld showed approximately the WHH behavior with a negative curvature at low temperature. The values of Hc2 (0) deduced from an extrapolation of the Hc2 (T ) curves to T → 0 for various oxygen contents are presented in Table 4.4b. In the Y0.8 Ca0.2 Ba2 Cu3 O7−δ sample, the optimum doping occurs at δ = 0.3, which allows to study both the underdoped (δ ≥ 0.3) and the overdoped (δ ≤ 0.3) regions in the YBCO material. The largest Hc2 (0) ∼ 310 T occurs for the overdoped sample, δ = 0.21, while at the maximum Tc 84 K the critical ﬁeld drops down, which is attributed by the authors to the opening of the normal-state pseudogap in the electronic spectrum as discussed in Sect. 4.2.2 (see Fig. 4.4). Thus, the opening of the pseudogap decreases the superﬂuid density and suppresses critical magnetic ﬁelds. Direct measurements of the upper critical ﬁeld in highly anisotropic Tland Bi-based superconductors are prevented by the strong thermal ﬂuctuations which transform the Hc2 (T ) line into a crossover line. However, just by studying the critical ﬂuctuations in the vicinity of the Hc2 (T ) line, it is possible to obtain useful information concerning the phase transition at Hc2 (T ). The studies of the reversible magnetization in the magnetic ﬁeld Hc as a function of temperature in the high-quality single crystals of Bi2 Sr2 CaCu2 O8+x by Li et al. [710] allowed them to estimate the slopes of the upper critical magnetic ﬁeld close to Tc and to calculate other relevant parameters of the

4.3 Magnetic Properties

161

samples. It happens that all the magnetization curves M (T ) in diﬀerent magnetic ﬁelds intersect at the same crossing point (M ∗ , T ∗ ), which reveals the 2D nature of the critical ﬂuctuations in this layered compound. By using √ √ the 2D scaling relation for M/ T H versus [T − Tc (H)]/ T H, it is possic c ble to estimate ∂Hc2 /∂T from the relation: Tc (H) = Tc (0) + H/(∂Hc2 /∂T ). In Table 4.4c, the results of these calculations for the underdoped (Bi-UD), optimally doped (Bi-OP), and overdoped (Bi-OD) Bi-samples are given. The number of holes ph per Cu ion was evaluated from the empirical relation (1.5) for Tc (p) given by Presland et al. [1028]. To estimate the coherence c 2 length ξab (0), the G–L relation (4.12) was used: (∂Hc2 /∂T )Tc0 φ0 /2πξab . The authors also calculated the penetration depth λab by using the meanﬁeld London approximation for the reversible magnetization, which provides the relation λ−2 ab (T ) ∝ dM (T )/d(ln H). In Table 4.4c, the penetration depth for a d-wave superconductor is reported (for results of other models, see Li et al. [710]). A linear dependence for 1/λ−2 (0) versus the hole number ph is observed as in the G–L relation (4.10). Comparison of the slopes for the upper critical ﬁeld in Bi-compounds and YBCO compounds reported in Table 4.4 c /∂T ∼ 2 (T/K) in the optishows that they are close to each other, −∂Hc2 mally doped samples and much larger than the slopes of the melting and c c irreversible lines: −∂Hm /∂T −∂Hirr /∂T = 0.1 − 0.6 (T/K), which points to a diﬀerent origin of these phenomena. An anomalous temperature dependence of the resistive upper critical magnetic ﬁeld was observed in overdoped single crystals of Tl2 Ba2 CuO6+δ . (Tc 20 K) by Mackenzie et al. [764] and in Bi2 Sr2 CuOy thin ﬁlms by Osofsky et al. [932]. The same puzzling behavior was registered for thin ﬁlms of the underdoped YBa2 (Cu0.97 Zn0.03 )3 O7−δ compound with Tc (K) = 9.4, 12.5 [1331]. The measurements show a sharp drop in the resistance R(T < Tc∗ ) and a parallel shift of the transition at Tc∗ with increasing magnetic ﬁeld, contrary to a broadening of the resistive transition in magnetic ﬁeld usually found in other cuprate superconductors (see Fig. 4.14b) . The resistive upper critical magnetic ﬁeld shows an upward curvature without saturation down to very low temperatures, T /Tc 10−3 . A similar behavior for the out-of-plane resistivity in perpendicular magnetic ﬁeld was detected in Bi2 Sr2 CaCu2 O8 by Alexandrov et al. [31] and Zavaritsky et al. [1407]. A compelling parallel shift of the resistive transition at Tc∗ with increasing magnetic ﬁeld, as in conventional superconductors, may suggest that the resistive upper critical magnetic ∗ ﬁeld Hc2 is the real upper critical magnetic ﬁeld Hc2 above, which there is a normal-state phase. At the same time, the optimally doped or slightly overdoped Tl2 Ba2 CuO6+δ compounds with higher Tc (K) = 62 K and 28 K show a ﬁeld-induced broadening of the resistive transition as in other cuprate superconductors [206]. Therefore, the anomalous resistive transition in Tl-compounds is speciﬁc only to the overdoped region outside the pseudogap state. The speciﬁc heat measurements in the single crystal Tl2 Ba2 CuO6+δ with low Tc 15 K in magnetic ﬁeld parallel to the c-axis by Carrington et al. [207] demonstrated behavior

162

4 Thermodynamic Properties of Cuprate Superconductors

similar to other strongly anisotropic cuprates. Instead of shifting toward lower temperatures with the applied magnetic ﬁeld, the speciﬁc heat anomaly at the superconducting transition is washed out as shown in Figs. 4.6 and 4.7. Magnetization measurements by Bergemann et al. [127] in the strongly overdoped Tl2 Ba2 CuO6+δ sample with Tc = 15 K revealed the remnant superconducting order, which was characterized by a linear diamagnetic response above the irreversibility line Hirr (T ). However, the magnetization above the irreversibility line, which displayed a conventional upward curvature, demonstrated a significant departure from London vortex liquid behavior: M ∼ ln(Hc2 /H). These measurements suggest that the observed resistive upper critical magnetic ﬁeld ∗ Hc2 (T ) may be an irreversibility line with an upward curvature above, which a remnant superconducting order and a ﬂux liquid phase persist. This conjecture was supported by Wen et al. [1353], who studied the superconducting transition in overdoped Bi2 Sr2−x Lax CuO6+δ by measuring the diamagnetic moment. They observed two superconducting transitions at Tc1 and Tc2 , which showed quite a diﬀerent behavior in the magnetic ﬁeld. The high temperature Tc1 did not shift appreciably with the ﬁeld, while the transition at low temperature Tc2 demonstrated the same upward curvature ∗ behavior with the magnetic ﬁeld as the resistive transition at Hc2 (T ). The transition at Tc1 was described within the conventional critical ﬂuctuation c theory with a very steep slope (∂Hc2 /∂T ) −1 (T/K) like that for the underdoped Bi-2212 compound (see Table 4.4c). To explain the magnetic ﬁeld dependence of the second transition temperature Tc2 , the authors used the Geshkenbeim et al. [380] theory. The theory is based on a suggestion that the samples under studies are inhomogeneous superconductors consisting of small grains with high-transition temperature Tg , which reveal the bulk superconducting phase coherence only at a lower temperature below the irreversibility line Tc∗ . Due to a very low resistivity of the 2D vortex liquid at temperatures Tg > T > Tc∗ (and large normal-state resistivity), the ﬁrst transition at Tg was diﬃcult to register in resistivity measurements. The bulk superconductivity is established by the Josephson coupling between the grains and is deﬁned by the phase-ordering ﬁeld Bg (T ) = B0 (Tc1 /T ) exp(−T /T0) where T0 ∼ 1 K. This relation ﬁts quite well the magnetic ﬁeld dependence of the second transition temperature Tc2 measured by Wen et al. [1353] with physically reasonable parameters, as e.g., T0 ∼ 1.85 K. However, the theory fails to describe the vanishing of the critical magnetic ﬁeld above Tc1 since the exponential dependence predicts the nonzero phase-ordering ﬁeld Bg (T ) at any temperatures (see [1407]). Several other theoretical explanations for the observed upward curvature of the resistivity upper critical magnetic ﬁeld have been proposed. Among them is the Abrikosov theory [10] based on a model of extended saddle-point singularities in the electron spectrum in cuprates with a long-range phononmediated electron attraction. In the model, a positive curvature of the upper critical magnetic ﬁeld along c-axis was caused by a quasi-one-dimensional electronic spectrum. Alexandrov [28] has proposed an explanation of the upward

4.3 Magnetic Properties

163

curvature of Hc2 based on the BEC of charged bosons. In particular, a model of charged bosons scattered by impurity in a magnetic ﬁeld predicts two anomalies in the speciﬁc heat [33, 1408]. At the lower temperature which coincides with the resistive upper critical magnetic ﬁeld, the BEC is destroyed and the normal-state of preformed pairs – bipolarons occur. Therefore, in this theory ∗ . no vortex liquid exists above the resistive upper critical magnetic ﬁeld Hc2 The second anomaly is the normal-state feature characteristic for an ideal charge Bose-gas, which is hardly shifted by the magnetic ﬁelds as discussed in Sect. 4.2.3. Anomalous Nernst Eﬀect A new approach for the estimation of the upper critical magnetic ﬁeld Hc2 by measuring the Nernst signal was proposed by Princeton group. Initially, an anomalously large Nernst signal of several (µV/K) was found above superconducting Tc in the strongly underdoped LSCO compound [1380] and later in the Bi-type compounds [1335]. It was suggested that the vortex excitation state persists at temperatures much higher than the superconducting Tc and the upper critical ﬁeld Hc2 extends above this temperature. These conclusions were based on observation of a large positive Nernst signal well above Tc at the “onset temperature” Tonset in high magnetic ﬁelds (for a review, see [1338] and [711]). The Nernst signal is deﬁned by the ratio of the electric ﬁeld Ey to the temperature gradient ∂x T , which appears in the perpendicular magnetic ﬁeld Bz : ey (T, B) = Ey /|∂x T | (see Sect. 5.4). A ﬁeld-enhanced diamagnetic signal detected by the torque magnetometry in intense magnetic ﬁeld was observed by Wang et al. [1336] in the underdoped, optimally doped, and overdoped samples of Bi2 Sr2 CaCu2 O8+δ below the same temperature Tonset as the Nernst signal. Figure 4.17 shows the phase diagram with the “Nernst” region between Tc and Tonset in LSCO (a) and Bi-2212 (b) compounds [1338]. The positive vortex Nernst signal diﬀers from the negative quasipartical one by a strong nonlinear dependence on magnetic ﬁeld in the form of a “tilted hill” proﬁle, which makes it possible to discriminate between the signals. Initially, the Nernst signal increases with magnetic ﬁeld, shows a maximum, and at higher ﬁelds falls rapidly toward zero. It was suggested to determine the upper critical magnetic ﬁeld Hc2 by a linear extrapolation of the highﬁeld Nernst signal to zero. This method applied to the La2−x Srx CuO4 single crystal will result in the maximum value of Hc2 ∼ 70 T at x ∼ 0.12 and Hc2 ∼ 50 T for the optimally doped Bi-2201 crystal (Tc 28 K). Contrary to conventional superconductors, the upper critical magnetic ﬁeld Hc2 measured by this method weakly depends on temperature and does not vanish above Tc up to T ∼ 1.5 Tc . As shown in Fig. 4.17, the Tonset temperature, for both the LSCO and Bi-2212 compounds, is well below the pseudogap temperature T ∗ , but at the same time it correlates with the Tc curve: at Tc = 0 at the underdoped and overdoped ends Tonset = 0 as well. Therefore, there is an intimate relation

164

4 Thermodynamic Properties of Cuprate Superconductors

Fig. 4.17. The phase diagram of LSCO (a) and Bi-2212 (b). The Nernst region is shown by the shaded area between Tonset where the Nernst signal and the ﬁeld-enhanced diamagnetic signal appear and superconducting Tc . The pseudogap temperature is T ∗ (after Wang et al. [1338])

between the 3D superconducting transition at Tc and the Nernst region below Tonset . It appears that the maximum Tonset ∼ 130 K is similar for diﬀerent types of compounds, LSCO, Bi-2212, and YBCO. In the electron-doped NCCO compounds, the vortex Nernst signal was weaker than in hole-doped cuprates and was not observed above Tc . The upper critical magnetic ﬁeld at low temperature Hc2 10 T and goes to zero linearly with temperature, Hc2 ∝ (Tc − T ). To explain anomalously large positive Nernst signal observed well above Tc , it was suggested that a vortex-liquid state appears above Tc . According to the “phase ﬂuctuation” scenario by Emery et al. [309], in the pseudogap region the phase coherence θ for the superconducting order parameter Ψ (r) = |Ψ (r)| exp iθ(r) is destroyed, exp iθ(r) = 0, but its amplitude |Ψ (r)| remains ﬁnite (see Sect. 7.3.1). In strongly anisotropic cuprates, such as Bi- or Tl-compounds or underdoped LSCO, YBCO, in a large temperature region above the 3D transition temperature Tc a short range 2D order of the Berezinskii–Kosterlitz–Thouless (BKT) type in the CuO2 planes with a large 2D correlation length could appear below Tonset (see [126, 649]). At the BKT transition, the vortex-antivortex unbinding destroys long-range phase coherence and superﬂuidity. In zero-magnetic ﬁeld, the number of vortices n+ and antivortices n− is equal but in external magnetic ﬁeld a magnetic induction B ∝ (n+ − n− ) will appear and the Nernst signal eN proportional to the vorticity (n+ − n− ) can be detected. Also, a weak diamagnetic signal can be observed due to local supercurrents on the length-scale of the order

4.3 Magnetic Properties

165

of average vortex spacing a0 ∼ φ0 /B. The lack of the vortex Nernst signal in the electron-doped compounds NCCO may be explained by a much larger phase rigidity in comparison with the hole-doped cuprates (see [309]). The pseudogap region was not also unambiguously detected in those compounds. There are several alternative to the vortex-liquid scenario explanations of the anomalous Nernst eﬀect. Tan et al. [1229] associated the phenomenon with preformed pairs of uncorrelated bosons, which may appear at a temperature T ∗ larger than Tc as a quantum ﬂuctuation ﬁeld. A microscopic theory of the giant Nernst signal due to ﬂuctuating Cooper pairs in a twodimensional superconductor beyond the upper critical ﬁeld was developed by Serbyn et al. [1136]. It was shown that the Nernst coeﬃcient may be large even far beyond the superconducting transition exceeding by orders of magnitude, the Fermi liquid terms. An explanation for the anomalous Nernst eﬀect is due to a speciﬁc electronic structure was proposed by Alexandrov et al. [41]. A model of a disordered conductor with the chemical potential near the mobility edge was considered where an interference of itinerant and localized-carrier contributions to the thermomagnetic transport could occur. In longitudinal transport, the localized-carrier contribution adds to the contribution of the itinerant carriers, which results in a large Nernst signal. At the same time, the contribution of the localized-carrier to the transverse transport is small and therefore the thermopower and the Hall coeﬃcient are reduced. An anomalous diamagnetism at T > Tc was explained as the Landau diamagnetism of nondegenerate carriers above the mobility edge. Within this approach, the Nernst eﬀect, semiconducting-like resistivity and diamagnetism were explained in the underdoped cuprates as normal-state phenomena (for a review, see [43]). Lower Critical Magnetic Field Hc1 Measurement of the lower critical magnetic ﬁeld Hc1 in cuprate superconductors allows to estimate the GL parameters κc,ab (4.14) from (4.16) as the ratio of the lower and upper critical ﬁelds. Several methods are applied in the studies of the lower critical ﬁelds. A frequently used method is based on the measurement of the magnetic entry ﬁeld when a vortex penetrates into the sample which determines the transition from the Meissner phase to the mixed phase at H > Hc1 . This is detected by a deviation of the magnetization M (H) from the linear law M (H) = χH, which holds for H < Hc1 (see e.g., [1291]). The occurrence of a sharp kink on the curve M (H) at Hc1 , a speciﬁc feature of conventional superconductors of the second type, is absent in the high-temperature superconductors. Only a smooth deviation from a straight line for the magnetization M (H) is observed, which makes it diﬃcult to ensure suﬃcient accuracy for the determination of Hc1 . Moreover, small deviations of the magnetization M (H) depend on the geometry of the crystals, e.g., sharp corners of the crystals or edge barriers strongly inﬂuence the vortex entry

166

4 Thermodynamic Properties of Cuprate Superconductors

⊥ Fig. 4.18. Temperature dependence of the lower critical ﬁeld Hc1 and Hc1 in single crystals of YBCO (Tc ∼ 91 K) (after Krusin–Elbaum et al. [666])

Table 4.5. Lower critical magnetic ﬁelds Hc1 in single crystals of YBa2 Cu3 O7−δ Tc (K) c Hc1 ab Hc1

Hc0 κc κab

(G) (G) (T)

Reference

92 690 120

92 70∗ − 210 0.85∗ − 1.55

44 230

540∗ − 300

[1]

[2], [5]

∼ 91

88.2

62

530 180

850 250

83 25 0.38 160 700

[4]

[5]

[3] ∗

[1] Umezawa et al. [1291], [2] Umezawa et al. [1292] ( − untwinned), [3] Krusin-Elbaum et al. [666], [4] Wu et al. [1373], [5] Vandervoort et al. [1301].

energy and the measured Hc1 (T ). To overcome this problem, a temperaturedependent rather than ﬁeld-dependent magnetization measurements of the Hc1 (T ) was proposed by Krusin–Elbaum et al. [666]. The authors studied the temperature dependence of the dc magnetization in several high-quality YBCO crystals with Tc 91 K, which enabled them to measure Hc1 (T ) with a reasonable accuracy for temperatures T > Tc /2 as shown in Fig. 4.18. The results for Hc1 (T = 0) obtained by the low-temperature extrapolation is given in Table 4.5. The important role of the twin barriers in YBCO crystals in the vortex pinning was pointed out by Umezawa et al. [1292]. They observed that the deviation of the magnetization curve from linearity depends on the orientation of magnetic ﬁeld with respect to the twin boundaries. A larger value of the ﬂux entry ﬁeld was detected for a ﬁeld orientation perpendicular to the twin boundaries, which was explained by increase of the vortex creation energy when it intersects a twin boundary. For an untwinned YBCO single crystal, the lower critical ﬁeld in the (a, b)−plane was much lower than for a twinned crystal as shown in Table 4.5.

4.3 Magnetic Properties

167

Results of several other measurements of the lower critical ﬁeld in YBCO crystals are presented in Table 4.5. The critical ﬁeld Hc1 was determined by Wu et al. [1373] by measuring the penetration depth of a radio frequency ﬁeld into a sample. The temperature dependence of Hc1 for both directions parallel and perpendicular to the c− axis showed the conventional behavior with saturation at low temperatures. The estimates for Hc1 (T ) were based on calculations within the BCS theory. The measured anisotropy of the lower ⊥

3.4 showed no temperature dependence. critical ﬁelds Hc1 /Hc1 The temperature dependence of the lower critical ﬁeld in a strongly underdoped (twinned) YBa2 Cu3 O7−δ crystal with Tc 62 K (δ 0.35) is shown c in the right panel of Fig. 4.16 [1301]. The obtained values for both the Hc1 ab and the Hc1 critical ﬁelds are much smaller than for 90 K samples as shown in Table 4.5. The substantial drop of Hc1 and the corresponding condensa2 tion energy ∝ Hc0 can be explained by a lower density of electronic states at the Fermi level which is due, as discussed in Sect. 4.2.2, to the opening of the pseudogap. The corresponding GL parameters for this crystal are rather large, while the anisotropy parameter evaluated from the lower critical c ab ﬁelds, ζ = (Hc1 /Hc1 )(ln κab / ln κc ) 4.3, is surprisingly small for the oxygendeﬁcient YBCO crystal as we have mentioned already in discussing the upper critical ﬁelds for this crystal (see Table 4.4). c (0) show The data summarized in Table 4.5 for the lower critical ﬁelds Hc1 that their average values for YBCO crystals close to the optimal doping lie in c ab the interval Hc1 = 550 − 700 G and Hc1 = 100 − 200 G. For these values of c the lower critical ﬁelds and the upper critical ﬁelds in Table 4.4, Hc2

120 T, ab Hc2 670 T, we obtain κc = 55 − 60, κab = 300 − 400. The above values of the G–L parameters are only crude estimates, because the accuracy of the measurement of Hc1 (0) and calculations of Hc2 (0) according to the WHH formula (4.38) is not suﬃciently high to produce reliable results. However, these estimates prove that the cuprate superconductors undoubtedly belong to strong type-II superconductors. 4.3.3 Magnetic Penetration Depth Besides the measurement of the critical magnetic ﬁelds, important information on the nature of the superconducting pairing can be obtained from measurements of the magnetic penetration depth λα (T ) for the external magnetic ﬁeld (4.10), which determines the superﬂuid density: ρα s (T ) =

c2 λ2α (T )

=

4πe2 ns (T ) 2 = ωs,α , mα

(4.40)

where ωs,α is the plasma frequency of charge carriers in the condensate. The temperature dependence of the penetration depth is directly related to the excitation spectrum of the superconducting pairs in a manner similar to the electronic speciﬁc heat in a superconducting state discussed in

168

4 Thermodynamic Properties of Cuprate Superconductors

Sect. 4.2.1. In the case of the conventional s-wave pairing with a nonzero gap Δ(T ) over the entire Fermi surface, the temperature dependence of the penetration depth at low temperatures, T → 0, should be of activation nature, i.e., λ(T ) ∝ exp(−Δ(0)/T ). For an unconventional pairing, like the d-wave pairing with a line of nodes in the k-space, a DOS at low energy N (E) ∝ |E| and a linear temperature dependence of the of λ(T ) is expected. Actually, as shown by Annett et al. [75] on the basis of point-group symmetry analysis (see Sect. 5.5.1), in clean superconductors with tetragonal or orthorhombic lattice symmetry, a linear temperature dependence Δλab (T ) = λab (T ) − λab (0) ∝ T should be observed for all types of pairing apart from the s-wave singlet pairing. While early experiments usually demonstrated quadratic temperature dependence, Δλab (T ) ∝ T 2 , the linear temperature dependence in highquality single crystals of various types of copper-oxide superconductors was conﬁrmed later. Results of these studies are summarized in several reviews (see e.g., [1269, 1375]). Below we consider several experiments on the penetration depth measurements and the results of which are presented in Table 4.6. There are several techniques for measuring the penetration depth. A commonly used one is the ac magnetic susceptibility measurement, which enables one to determine λα (T ) both parallel and perpendicular to the copper-oxygen planes and to evaluate the low-temperature limit λα (0). Several results of these measurements by Panagopoulos et al. [954] are reported in Table 4.6. Studies of the temperature and doping dependence of the penetration depth λab and λc in YBa2 Cu3 O7−δ at diﬀerent oxygen concentrations, δ = 0, 0.3, 0.43, demonstrated for the in-plane penetration depth λab a linear low-temperature dependence in all samples, while λc obeys a T 2 law at low temperature for the underdoped samples δ = 0.3, 0.43 [956]. As was shown by Xiang et al. [1374], for a pure sample with low anisotropy and the tetragonal structure, an intrinsic T 5 law should be observed for the out-of-plane penetration depth λc . This behavior is caused by a characteristic wavevector dependence of the hopping integral along the c-direction, t⊥ (kx , ky ) ∝ (cos kx − cos ky )2 in the layered cuprate superconductors, which results in the low-energy DOS N (E) ∝ E 5 for the c-axis coupling in a d-wave superconductor. Impurities and other defects (as well as chains in YBCO crystals) wash out to some extent this coherent hopping along the c-direction and λc (T ) ∝ T 5 law changes to T 3 or T 2 law as observed in most experiments. With decreasing hole concentrations, δ = 0.3, 0.43, the penetration depths and their anisotropy, λc /λab , quickly increase. At Zn-doping, the penetration depth also increase but the anisotropy decreases [952] and the linear T -dependence of the in-plane penetration depth changes to a quadratic one. A linear low-temperature dependence for λab was observed also for the mercury compounds HgBa2 Can−1 Cun O2n+2+δ , n = 2, 3, while the λc varies as T 5 and T 3 for n = 1, and 3, respectively [953, 955]. Their low temperature values λab (0), λc (0) are given in Table 4.6. Later, detailed studies of the doping dependence of the superﬂuid density (4.40) in two monolayer cuprates La2−x Srx CuO4 and HgBa2 CuO4+δ by Panagopoulos et al. [958] conﬁrmed the

4.3 Magnetic Properties

169

Table 4.6. Magnetic penetration depth in cuprate superconductors Compounds Hole-doped YBa2 Cu3 O7−δ δ = 0.05 δ = 0.005 δ δ δ δ

= 0.0 = 0.3 = 0.43 = 0.48

YBa2 (Cu1−x Znx )3 O7 x = 0.02 x = 0.03 x = 0.05 YBa2 Cu4 O8 La1.85 Sr0.15 CuO4 La1.8 Sr0.2 CuO4 HgBa2 CuO4+δ HgBa2 Ca2 Cu3 O8+δ Bi2 Sr2 CaCu2 O8 Bi2 Sr2 CaCu2 O8+δ Bi1.6 Pb0.4 Sr2 Ca2 Cu3 O10 Tl2 Ba2 CuO6+δ Tl2 Ba2 CaCu2 O8 Tl2 Ba2 Ca2 Cu3 O10 Electron-doped La1.9 Ce0.1 CuO4 La1.92 Ce0.08 CuO4 Pr1.855 Ce0.145 CuO4−y [1] et [6] [9]

Tc (K)

93.5 89 92 66 56 56

68 55 46 80 37.7 36 93 134.5 89.5 91 104 78 99.3 109 30 25 24

λab (0) (µm)

λa = 0.16 λb = 0.103 λa = 0.103 λb = 0.080 0.14 0.21 0.29 λa = 0.20 λb = 0.14 0.26 0.30 0.37 λa = 0.20 λb = 0.08 0.26 0.196 0.171 0.177 0.31 0.26 0.232 0.165 0.221 0.196 0.22 0.38 0.18

λc (0) (µm)

References

[1] [2] 0.635 1.26 4.53 7.17

[3] [3] [3] [2]

7.50 1.42 1.55 1.64

[4] [4] [4] [1]

4.0 2.35 1.36 6.14

[5] [5] [6] [6] [7] [8] [7] [9] [7] [7]

40

[10] [10] [11]

Basov et al. [113], [2] Pereg-Barnea et al. [977], [3] Panagopoulos al. [956], [4] Panagopoulos et al. [952], [5] Panagopoulos et al. [958], Panagopoulos et al. [955], [7] Schilling et al. [1110], [8] Jacobs et al. [549], Broun et al. [178], [10] Pronin et al. [1029], [11] Skinta et al. [1175]

strong dependence of both penetration depths λab and λc on the number of charge carriers. Figures 4.19 and 4.20 show the doping dependence of 1/λ2ab (0) and 1/λ2c (0) for LSCO and Hg-1201, respectively. The plots demonstrate a quick increase of the superﬂuid density with the doping and its saturation (or a peak) for the hole concentration p > pc 0.2. As discussed in Sect. 4.2.1, at p < pc the appearance of the pseudogap in the electronic state strongly suppresses the condensation energy (see Figs. 4.4 and 4.5). For both compounds, the anisotropy of the penetration depth quickly decreases with the doping and at p ∼ pc gets close value λc (0)/λab (0) 12.

40 30

Tc(K)

20 20 10 10 0 0.05

b

30

La-214

0.1

0.15

0.2

0.2

La-214

1/λ 2c (0) [μm–2]

a

4 Thermodynamic Properties of Cuprate Superconductors

1/λ2ab(0) [μm–2]

170

0.15 0.1 0.05

0 0.25

0 0.05

0.1

holes/planar

0.15

0.2

0.25

holes/planar Cu

Fig. 4.19. Doping dependence: (a) of Tc and inverse square of the in-plane penetration depth 1/λ2ab (0) and (b) of the out-of-plane penetration depth 1/λ2c (0) for La2−x Srx CuO4 (reprinted with permission by APS from Panagopoulos et al. [958], c 2003)

a

b Hg-1201

100 Hg-1201

Tc(K)

35

60

40 0.05

0.1

0.15

0.2

holes /planar Cu

25 0.25

1/λ2ab(0) [μm–2]

45 80

1/λ2c (0) [μm–2]

0.4

0.2

0 0.05

0.1

0.15

0.2

0.25

holes/planar Cu

Fig. 4.20. Doping dependence (a) of Tc and inverse square of the in-plane penetration depth 1/λ2ab (0) and (b) of the out-of-plane penetration depth 1/λ2c (0) for HgBa2 CuO4+δ (reprinted with permission by APS from Panagopoulos et al. [958], c 2003)

Results of penetration depth measurements in Bi- and Tl-type compounds are given in Table 4.6. Data on penetration depth in Bi2 Sr2 CaCu2 O8 single crystals were also mentioned in Table 4.4c. Close to each other results were reported by Schilling et al. [1110]: λab (0) 0.21 µm, (Tc = 89.7 K) and by Zeldov et al. [1411]: λab (0) 0.16 µm (Tc = 92.2 K). Measurements of the magnetic penetration depth in strongly underdoped Bi2 Sr2 CaCu2 O8+δ crystal with Tc 70 K by Colson et al. [248] using Josephson plasma resonance gave the following values: λc (0) 240 µm, λab (0) 0.4 µm and the anisotropy ζ = λc (0)/λab (0) 600. A more precise method for studying temperature dependence of the penetration depth is the microwave technique, which measures the surface impedance Zs (T ) = Rs (T ) + iXs (T ) and the complex conductivity σ1 + iσ2 in

4.3 Magnetic Properties

171

radiofrequency ﬁelds in the THz region (see Sect. 5.3.3). The surface resistance Rs is proportional to the loss of the microwave power caused by normal carriers. The reactance Xs is largely determined by the nondissipating response of the superconducting carriers. In the superconducting state, the latter dominates over the dissipative part Rs , and for σ1 σ2 at low frequency one gets (see e.g., [156, 1269]): Rs (T )

1 2 2 ω μ0 σ1 (ω, T )λ3 (T ), 2

Xs (T )

ωμ0 σ2 (ω, T )

1/2 = ωμ0 λ(T ).

(4.41) (in MKS units, μ0 = 4π × 107 H/m). Measurement of the temperature dependence of the reactance Xs (T ) provides quite accurate data on the temperature dependence of penetration depth λab (T ) and less accurate data on λc (T ). Unfortunately, the absolute values of λα (0) cannot be evaluated by this method. Measurement of the real part of the conductivity σ1 in (4.41) provides information on quasiparticle scattering rates in superconducting state, which we discuss in Sect. 5.3.3. The ﬁrst unambiguous results revealing a linear temperature dependence of Δλ(T ) = λ(T ) − λ(0) ∝ T at low temperatures were obtained by this method [428]. Later on, it was proved that the quadratic temperature dependence of the penetration depth observed in microwave measurements on ﬁlms is caused by impurities or other defects of the samples [12, 156]. Figure 4.21 plots the temperature dependence of the penetration depth in the a and b directions of the YBa2 Cu3 O6.95 untwinned crystal from 1.3 K to 100 K [1421]. The temperature dependence of Δλ(T ) = λ(T ) − λ(0) in both directions below 15 K is linear in T with slopes 4.7 ˚ A/K for Δλa and 3.6 ˚ A/K for Δλb . To plot the temperature dependence of the superﬂuid density (4.40), ρs (T )/ρs (0) = λ2 (0)/λ2 (T ), in Fig. 4.21 two parameters were used: λa (0) = 0.16 µm, and λb (0) = 0.103 µm, taken from infrared data [113]. A novel microwave technique based on a broadband zero-ﬁeld ESR absorption spectroscopy of Gd3+ ions in the Gdx Y1−x Ba2 Cu3 Oy single crystals was used to measure the London penetration depth λ(T → 0) [977]. This technique provides an accurate determination of absolute values of λ(0) in the three crystallographic directions, a, b, and c. In comparison with other methods, these measurements shown in Table 4.6 for two YBCO crystals, y = 6.995, 6.52 in the ordered ortho-I and ortho-II phases, respectively, give smaller values for the penetration depths, though quite close to other results. The linear dependence Δλ(T ) ∼ T reported in the ﬁrst experiments on YBCO crystals was later conﬁrmed for other cuprate superconductors when high-quality Bi-, Tl-, and Hg-based single crystals had become available (see e.g., [178, 549, 704, 939, 1269]). Thus, these results provide reliable evidence for an unconventional pairing with nodes for a superconducting gap in the hole-doped cuprates.

4 Thermodynamic Properties of Cuprate Superconductors 1.0

100

0.8

80

0.6

60

0.4

40

0.2

20

0

0

20

40

60

80

Δλ(nm)

λ2(0)/ λ2(T)

172

0 100

T(K)

Fig. 4.21. Temperature dependence of the penetration depth Δλ(T ) = λ(T ) − λ(0) and λ2 (0)/λ2 (T ) in the a direction (filled symbols) and b direction (open symbols) of the untwinned single crystal YBa2 Cu3 O6.95 (after Zhang et al. [1421])

Concerning the electron-doped crystals, the results are controversial. For instance, microwave measurements of the temperature dependence of Δλ(T ) in the Pr2−x Cex CuO4−δ (PCCO) and Nd2−x Cex CuO4−δ (NCCO) single crystals by Kokales et al. [630] showed a power-law behavior for T < Tc /3, which suggested nodes in the superconducting gap. Similar results were obtained for PCCO and NCCO single crystals by Prozorov et al. [1032]. At the same time, measurements of the temperature dependence of 1/λ2 (T ) in several highquality Pr1.855 Ce0.145 CuO4−y ﬁlms by a low frequency inductance method demonstrate an exponential temperature dependence, which is consistent with a nodeless gap [1175]. Studies of the La2−x Cex CuO4 ﬁlms by a quasioptical method at millimeter wavelengths by Pronin et al. [1029] showed the occurrence of signiﬁcant variations of the temperature dependence of the penetration depth with the doping: in the underdoped sample λ(T ) ∝ T 2 , while for the optimally doped sample the temperature dependence is close to an exponential. To explain their results, the authors suggested a change in the gap anisotropy or symmetry with the doping. The absolute values of the penetration depth in these superconductors at low temperature also depend on the doping as shown in Table 4.6. The infrared spectroscopy with polarized light can be used for the measurement of the absolute values of the diagonal components λa , λb , λc of the penetration depth in a single (untwinned) crystal. By using the Kramers– Kronig relations, the real and imaginary parts of the complex conductivity σ(ω) = σ1 (ω) + iσ2 (ω) can be obtained from reﬂectance measurements (see Sect. 5.3.3). In the superconducting state, the real part of the conductivity reveals a delta function peak, σ1 (ω) = πe2 (ns /m∗ )δ(ω), caused by superconducting carriers. Therefore, the superﬂuid density (4.40) can be calculated from the oscillator strength sum rule (see Sect. 5.3.3):

4.3 Magnetic Properties

c2 =8 λ2

∞ [σ1n (ω) − σ1s (ω)]dω

173

(4.42)

0

where σ1n and σ1s are the real components of the conductivity in normal and superconducting states. The imaginary part of the conductivity in a superconducting state, as given by (4.41), directly deﬁnes the penetration depth: c2 = 4πωσ2 (ω). (4.43) λ2 The results of the penetration depth measurements by this method by Basov et al. [113] in the single untwinned YBa2 Cu3 O6.95 crystal with Tc = 93.5 K and the YBa2 Cu4 O8 crystal with Tc = 80 K are given in Table 4.6. A strong a-b anisotropy in the London penetration depth reported in this infrared measurements, e.g., (λa /λb )2 = (ns /m∗ )b /(ns /m∗ )a 2.4 for the YBCO crystal, is consistent with the resistivity measurements (see Sect. 5.4.1) and microwave data reported above [1421]. The lower value for the London penetration depth along the chain direction b proves that the charge carriers in the chains of YBCO also condense in the superconducting state. Results of the London penetration depth measurements by infrared spectroscopy in various types of cuprates are reported also by Homes et al. [479, 480]. The muon-spin-rotation (μSR) technique was also used for penetration depth measurements in type-II superconductors and was extensively applied in studies of cuprate superconductors (for reviews, see [124,1182]). The rate of the muon spin depolarization depends on the ﬂuctuating transverse magnetic ﬁelds ΔBμ : Gz (t) ∝ exp[−(1/2)(γμΔBμ t)2 ] (see (3.15) in Sect. 3.2.1). In high magnetic ﬁelds in the vortex phase, Hc1 < H < Hc2 , when the separation between the vortices is smaller than the penetration depth λ, these ﬁelds are deﬁned by the distribution of the local ﬁelds in vortex cores, which is proportional to 1/λ2 . Therefore, measurements of the muon spin depolarization rate enables the evaluation of the superﬂuid density (4.40). Already the ﬁrst studies of penetration depth by μSR technique in various copper-oxide superconductors revealed a certain correlation between the increase of Tc and the decrease of the penetration length with doping [1288]: Tc ∝ ΔBμ ∝

1 ns ∝ ∗. λ2 m

(4.44)

This scaling relation known as the “Uemura plot” has been studied later in greater detail by many groups. Since within the BCS theory, there is no direct proportionality between Tc and the density of superconducting carriers ns (see (1.1)), an unconventional mechanism of superconductivity in copper-oxide compounds was invoked. In particular, this linear relation was considered as a strong evidence of the BEC scenario where Tc ∝ ns (in two-dimensional systems, see (1.4)). However, further μSR studies of the penetration depth in overdoped cuprates, as Tl2 Ba2 CuO6+δ [896, 1289], showed a strong decrease

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4 Thermodynamic Properties of Cuprate Superconductors

of the measured depolarization rate and the corresponding superﬂuid density ns /m∗ with increase of the normal-state carrier concentration n with the doping. In the overdoped region, Tc decreases with the doping as well and this results in a reﬂex loop type dependence for Tc versus (ns /m∗ ). This behavior was explained by the reduction of the condensate energy due to a pair-breaking, which was ﬁtted by the equation [896]: ns = 0.5n/(1 + ξ/lpb ), where ξ is the a − b plane coherence length and lpb is the pair-breaking scattering length lpb ∼ 60 ˚ A (which should not be confused with the transport mean free path). As discussed in Sect. 4.2.2, the pseudogap in the electronic spectrum in underdoped cuprates strongly reduces the condensation energy and the corresponding values of 1/λ2 (0) as explicitly demonstrate Figs. 4.4 and 4.5. Emphasizing the importance of the normal-state pseudogap phenomenon, a quasi-linear scaling relation was proposed by Tallon et al. [1225]: Tc /Δ0 ∝ λ−2 (0) where Δ0 is the maximum spectral gap, which depends both on the normal-state pseudogap and the superconducting gap. In the underdoped region Δ0 is weakly dependent on doping and Tc follows the original Uemura quasilinear relation (4.44). For hole concentration p > pcr 0.2, the pseudogap closes and Tc is proportional to a superconducting gap Δ0 in this region. Therefore, instead of the reﬂex loop type dependence for Tc versus λ−2 (0), a quasilinear relation occurs. The importance of the normal pseudogap inﬂuence on the superﬂuid density was conﬁrmed by scaling λ−2 (0) with S(Tc )/Tc S/T Tc ∼ N (E), where S/T Tc and N (E) are average electronic entropy and the density of states over temperature from 0 to Tc . Another scaling relation inferred from a comparison of the normal-state conductivity σdc with the superﬂuid density ρs (4.40), both measured by infrared spectroscopy technique, was suggested by Homes et al. [479, 480]. The authors obtained the scaling relation ρs 35σdc Tc for a wide variety of superconductors (here ρs = ωs2 is measured in cm−2 units, σdc in cm−1 , and T in cm−1 = 1.44 K). It is remarkable that this relation holds both for the ab-plane and c-axis penetration depths. Since the dc conductivity in ab plane equals σdc = ne2 τ /m∗ , this relation couples the transport scattering time τ to Tc : ¯h/τ 2.7kB Tc if we assume ns n at T = 0. The scaling behavior can be explained by considering conventional superconductivity in the dirty limit or by assuming the Josephson coupling either along the c-axis in layered systems or in strongly inhomogeneous CuO2 planes [1228]. Summarizing the data on the penetration depth in Table 4.6 measured by various techniques, we can argue that they agree quite well and give a reliable evaluation for the superﬂuid density in cuprate superconductors. The data demonstrate also similar temperature and doping dependences. In the underdoped region, the corresponding superﬂuid density ρs (4.40) is rather small. This can be explained by a large normal pseudogap detrimental to superconducting condensate. It rapidly increases with the doping and gets saturated (or reaches a maximum) at the critical concentration p = pcr 0.2, where the

4.4 Conclusion

175

pseudogap closes. The anisotropy of the penetration depth λc /λab is large in the underdoped region while strongly reduced in the overdoped region. The linear temperature dependence of the λab (T ) at T → 0 unambiguously proves a superconducting gap with nodes in hole-doped superconductors. Various phenomenological scaling relations proposed for superﬂuid density ρs appear not to be universal laws and have not resulted in any particular conclusion concerning a pairing mechanism in cuprate superconductors.

4.4 Conclusion To summarize the study of the thermodynamic properties of the copper-oxide compounds, we point out the following most important results. 1. Measurements of the Sommerfeld constant in the normal-state show a modest renormalization of the electronic eﬀective masses. Temperature and magnetic ﬁeld dependence of the electronic speciﬁc heat Cel (T, H) conﬁrms the occurrence of superconducting gap with nodes as a bulk property of the hole-doped cuprates. 2. Studies of doping dependence of the electronic speciﬁc heat reveal an opening of a normal-state pseudogap below a critical hole concentration p < pcr 0.2. The pseudogap properties do not support its precursor pairing origin in quasiparticle spectra but rather point to an independent origin, most probably caused by antiferromagnetic exchange interaction. The pseudogap strongly reduces the superﬂuid density and competes with superconductivity. 3. Large thermal ﬂuctuations close to the phase transition Tc (H) line are observed in the speciﬁc heat measurements in cuprates. In YBCO-type compounds, a critical behavior can be explained within the 3D-XY critical scaling. In quasi-two-dimensional Tl- and Bi-type compounds, the 3D critical region is extremely narrow and the 2D-XY-type scaling behavior can be inferred from magnetization measurements, while a critical behavior of the speciﬁc heat in Bi-2212 compound is better described by the Bose-condensation type phase transition. 4. The H-T phase diagram for the cuprate superconductors shows several phases of the vortex matter: an ordered vortex-lattice at low ﬁelds, α α Hc1 < H Hc2 , a highly disordered vortex solid at high ﬁelds and low temperatures (various types of vortex glasses), and a vortex liquid at high temperatures. The vortex-lattice melting line Bm (T ) and the irreversibility line Birr (T ) ≤ Bm (T ) show a complicated behavior depending on doping, anisotropy, lattice disorder, and temperatures. α α 5. Measurements of critical magnetic ﬁelds Hc1 (T ), Hc2 (T ) demonstrate α extremely large slopes ∂Hc2 /∂T , which result in huge upper critical ﬁelds, c small values of the correlation length ξab ∝ Hc2 , and high values of the

176

4 Thermodynamic Properties of Cuprate Superconductors

α /H α . In overdoped Tl-2201 and simiGL parameter κα ∝ λα /ξα ∝ Hc2 c1 lar type compounds, an anomalous upward curvature of the resistive upper ∗ critical magnetic ﬁeld Hc2 (T ) was found. To explain a sizable Nernst signal and diamagnetic response in underdoped LSCO and Bi-type compounds in a broad temperature range, Tc < T < Tonset ∼ T ∗ /2, a vortex-liquid state was suggested to exist above Tc . 6. A linear temperature dependence of the in-plane magnetic penetration depth at T → 0 in high-quality samples unambiguously proves a superconducting gap with nodes in the hole-doped superconductors. In the underdoped region, the corresponding superﬂuid density ρs ∝ λ−2 is rather small but rapidly increases with doping and gets saturated (or reaches a maximum) at a critical concentration p = pcr 0.2 when the normal-state pseudogap closes. The anisotropy of the penetration depth λc /λab is large in the underdoped region while strongly reduced in the overdoped region.

5 Electronic Properties of Cuprate Superconductors

The superconducting properties of metals are principally deﬁned by their electronic structure in the normal state. Therefore, it is essential to study the latter state to elucidate the mechanism of superconductivity in the cuprate superconductors. Intensive experimental studies of their electronic structure by photoelectron and optic spectroscopy, by transport measurements and by tunneling spectroscopy reveal an unconventional behavior which strongly deviates from a traditional Fermi-liquid description of normal metals. There is still no clear understanding of the normal state properties of cuprate superconductors, which precludes the development of a consistent theory of the high-temperature superconductivity. To discuss the unconventional electronic properties of the cuprate compounds, we brieﬂy consider their basic electronic structure ﬁrst (crystal chemistry and band structure essentials), while a more detailed theoretical description and corresponding models will be given in Sect. 7.1. Eﬀects of impurity substitution, which probe the electronic structure locally, are also considered. Then we consider experimental investigations of the electronic structure by diﬀerent spectroscopy methods: high-energy photoemission and electron spectroscopy to elucidate the ionic structure, angle-resolved photoemission studies which play essential role in parameterizations of one-electron excitations close to the Fermi energy, and optical methods which give information on the electronic charge excitations. We consider also studies of the transport properties – the electrical resistivity, the Hall eﬀect, the heat conductivity, and the thermopower which provide additional information on the electronic interactions in the metallic phase. At the end of the chapter, we discuss results of the tunnelling spectroscopy which provide direct information about the superconducting gap and the pseudogap. Phase-sensitive experiments which have proved the d-wave symmetry of the superconducting gap are also considered.

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5 Electronic Properties of Cuprate Superconductors

5.1 Electronic Structure: Overview 5.1.1 Crystal Chemistry and Bands With respect to the electronic structure, the copper-oxide compounds may be related to the class of ionic semiconductors, in which the metallic conductivity appears upon the change of stoichiometry. Their electronic structure is deﬁned by a complicated interaction of localized and itinerant electronic states with strong Coulomb correlations. The electronic structure is also rather sensitive to the short-range order in the atomic positions and to impurities. By taking into account the ionic character of bonding in the undoped compounds, one can choose as a basis, in discussing the electronic structure, an ionic model in which the atomic states are described by a certain degree of oxidation, or a formal valence z [1102,1177,1178]. In this case, the actual charge of the ions in the buﬀer layers, such as Y2+ , La3+ , Ba2+ , Sr2+ , rare-earth ions RE=Nd3+ , Eu3+ and others in the La2−x Mx CuO4 (LMCO), RBa2 Cu3 O7−y (RBCO), and in the Bi-, Tl-, and Hg-based compounds proves to be close to their formal valences since their ionic energies are much higher then the oxygen O2− energy level. At the same time, the value of Cu 3d atomic energy level reaches the minimum in the series of 3d elements which brings the Cu2+ energy level close to the O2− energy level in crystal ﬁelds. This results in a strong covalent bonding of the copper 3d9 states with the oxygen 2p6 states in copper–oxygen planes, which is a distinctive feature of cuprates. Therefore, the charges of Cu2+ and O2− ions diﬀer signiﬁcantly from their formal valence and depend on the degree of doping. As a result, there appear mixed 3d–2p states in which the copper ions with the formal valence z = +2, +3 are represented in the form Cu2+ → α|3d9 2p6 + β|3d10 2p5 , Cu3+ → α1 |3d8 2p6 + β1 |3d9 2p5 .

(5.1)

The coeﬃcients β and β1 determine the degree of transfer of the positive charge (hole) from the 3d shell of copper to the ﬁlled 2p shell of an oxygen ion O2− . When La2−x Mx CuO4−y is doped with divalent ions of M = Ba2+ , Sr2+ , Ca2+ , the formal valence of copper becomes z = 2 + x − 2y. Due to the two nonequivalent positions of copper in the RBa2 Cu3 O7−y compounds (in-plane Cu2 and in-chain Cu1), a more complicated doping dependence of the copper formal valence is observed as discussed in Sect. 2.4. Doping of the Bi-, Tl-, and Hg-based compounds occurs due to the presence of nonstoichiometric oxygen ions Oδ in the buﬀer layers. As a next step in discussing the electronic structure of the CuO2 plane, the crystal ﬁeld splitting of the copper 3d levels and oxygen 2p levels is to be taken into account. In Fig. 5.1, we outline schematically the formation of the in-plane electronic structure, taking as an example La2 CuO4 . For a copper atom in a spherically symmetric ﬁeld, the 3d levels are degenerate in energy. In a crystal ﬁeld of cubic symmetry Oh , for a proper octahedron

5.1 Electronic Structure: Overview

179

z σ∗

x,y eg 3d9

x2-y2 3z2-r2 xy

π π

t2g

σ

xy,yz Cu2+ atomic level

Crystal field

2p6

σ Covalency

Crystal field

O2– atomic level

Fig. 5.1. Formation of the electronic structure in CuO2 planes, accounting for the splitting of 3d and 2p levels in the crystal ﬁeld and their covalent bonding (after [335])

CuO6 , the ﬁve 3d levels split into a doublet eg = {d(x2 − y 2 ), d(3z 2 − r2 )} and a triplet t2g = {d(xy), d(xz), d(yz)}. The value of this splitting is 1–2 eV [1102]. Upon the decrease of symmetry to tetragonal D4h , a further splitting of the 3d levels into singlets b1g {d(x2 − y 2 )}, a1g {(3z 2 − r2 )}, b2g {d(xy)} and a doublet eg = {d(xz), d(yz)} occurs. The degenerate atomic 2p oxygen levels p(x), p(y), and p(z) split in the crystal ﬁeld of D2h site symmetry into three levels: (pπ ), (pπ⊥ ), and (pσ). The π-type states correspond to the in-plane orbitals p(x) or p(y) (π ), or to the out-of-plane states p(z) (π⊥ ), which are directed perpendicular to the Cu–O bonds. The hybridization of the π-type states with the Cu orbitals is weak. This results in narrow π-bands. The σ-type states are formed by the oxygen in-plane orbitals p(x) or p(y), which are directed along the Cu–O bonds and the d(x2 − y 2 ) copper orbital. The linear combination of the four oxygen σ-type orbitals of the b1g symmetry around a copper site experiences the strongest covalent bonding with the d(x2 − y 2 ) copper orbital which gives rise to broad bonding (σ) and antibonding (σ∗ ) bands of hybridized pdσ-states. The other conﬁguration of the four oxygen σ-type orbitals of a1g symmetry does not couple to the copper d(x2 −y 2 ) orbital and results in a narrow nonbonding oxygen band (for details see Sect. 7.1.1). The three σ-type bonding conﬁguration in the CuO2 plaquette are shown schematically in the upper part of Fig. 5.2 [271]. The shaded and empty lobes of the Cud(x2 − y 2 ) orbitals and the in-plane O2p(x, y) orbitals denote the phases (positive and negative) of the wave functions at the (π, π) point of the Brillouin zone. The tetragonal distortion of the octahedron in La2 CuO4 for copper in the 3d9 state is due essentially to the Jahn–Teller eﬀect. In this eﬀect, an electronic energy decreases due to the partial removal of degeneracy of the electronic states in a symmetric crystal ﬁeld, in particular, by a reduction

180

5 Electronic Properties of Cuprate Superconductors

Fig. 5.2. Upper panel: phases for bonding, antibonding and non-bonding hybridized wave functions of Cu d(x2 − y 2 ) and O 2p(x, y) orbitals in the CuO2 plaquette at (π, π) point of the 2D Brillouin zones (BZ). Lower panel: 2D projected BZ for various copper oxides with the irreducible symmetry units (shaded areas) and conventional notations for the wave vectors at the BZ boundary (reprinted with permission by c 2003) APS from [271],

in the octahedron symmetry from the cubic Oh to the tetragonal D4h one. Indeed, in the case of singly occupied levels d(x2 − y 2 ) and doubly occupied levels d(3z 2 − r2 ), the electronic energy of the system will decrease if they split. For Cu3+ or Ni2+ in the 3d8 triplet (S = 1) state with one electron on the level d(x2 − y 2 ), and one on the level d(3z 2 − r2 ), the Jahn–Teller eﬀect is absent since the splitting of two singly occupied levels does not change the electronic energy. This conclusion is supported by comparing the structurally equivalent lattices La2 NiO4 and La2 CuO4 . The CuO6 octahedron is stretched along the z-axis under an increase of the length of the Cu–O2 bond up to 2.4 ˚ A as compared to the in-plane Cu–O1 distance 1.9 ˚ A . The stretching along the z-axis of the octahedron NiO6 is twice as small as that of CuO6 . The typical energy of the Jahn–Teller distortion EJ−T 0.5 eV is much higher than the corresponding phonon energy and therefore this static distortion could not give a contribution to the dynamical coupling of the electrons with the lattice. From the picture of the formation of the electronic structure of La2 CuO4 sketched in Fig. 5.1, it follows that according to the band theory, this compound should be a metal with a half-ﬁlled antibonding pdσ band. We qualitatively arrive at the same conclusion when this scheme is applied to the electronic structure of YBa2 Cu3 O6 or to other undoped copper-oxides. However, these conclusions contradict the experiments, which demonstrate that the stoichiometric compounds La2 CuO4 and YBa2 Cu3 O6 are antiferromagnetic insulators with a moderately wide energy gap of 1–2 eV. This discrepancy is related to the fact that, in the band scheme described, one neglects the Coulomb single-site repulsion of the 3d electrons. The typical value of the

5.1 Electronic Structure: Overview

181

Coulomb correlation energy is Ud 8–10 eV. This is much larger than the typical width of the pdσ antibonding band W ∼ 3 eV, thus leading to the splitting of this band into two one- and two-hole subbands, Cu2+ and Cu3+ in the notation of (5.1). The possibility of a correlation splitting and a metalinsulator transition was ﬁrst noted by [869, 870] and investigated in the framework of a simple model by [499, 500]. In the Mott–Hubbard insulator model, the charge ﬂuctuations of the type dni dnj ↔ dn−1 dn+1 between two i j n di ions of charges n at lattice sites i, j, involve the large d–d Coulomb interaction Ud > W . In this insulating state, a separation into low-energy spin ﬂuctuations and high-energy charge ﬂuctuations occurs. Taking into account the separation of spin and charge degrees of freedom in correlated electronic systems, [58, 60] has developed a theory of the superexchange in transition metal compounds. In the Mott–Hubbard model, it is also assumed that Ud < Δ, where Δ is the energy of an anion–cation charge transfer (in our case, Δpd = Ed − Ep ). Therefore, the anion energy states can be neglected if one considers only the narrow d-band with a direct d–d exchange. However, the opposite situation, U > Δ > W , is also possible. In that case, the insulator correlation gap is deﬁned by the energy of the charge transfer Δ of charge-ﬂuctuations of the L where L denotes a hole in the anion valence band. This type type: dni ↔ dn+1 i of insulator was called charge-transfer insulator [1398]. It is just this situation which is realized in cuprates where the hole transfer energy Δpd = εp −εd 3– 4 eV from Cu2+ to Cu3+ in (5.1) is smaller than the correlation energy Ud of two holes: Ud > Δpd > W and therefore the copper oxide materials belong to the class of the charge-transfer insulators. The diﬀerent types of electronic structures discussed above are illustrated in Fig. 5.3 [271]. For a p–d model with three bands, bonding (B), nonbonding (NB) and the half-ﬁlled antibonding (AB) band, the following type of electronic structure can be realized, depending on the Coulomb repulsion Ud at the d-site, the charge-transfer gap Δpd and the hybridization bandwidth W : (a) a metallic state of AB band for Ud = 0, (b) a Mott–Hubbard insulator for Δpd > Ud > W, (c) a charge-transfer insulator for Ud > Δpd > W . If one takes into account the strong d–d correlations and the p–d hybridization for two-hole states in the CuO2 plaquette, then the two-hole p–d band splits into a triplet (S = 1) and a singlet (S = 0) bands [1419] as shown in Fig. 5.3d. By taking into account that in a copper–oxygen plane an electronic structure similar to that shown in Fig. 5.3c, d is realized, we can discuss qualitatively a doping-induced metal-insulator transition. In the insulating phase, there is a ﬁlled hybridized (p–d) band, primarily of O2p-type, which is the valence band, and an empty hybridized (d–p) conduction band, primarily of 3d type; these bands are separated by a charge transfer gap Eg ≤ Δpd . The conduction band, which is frequently said to be the upper Hubbard band (UHB) – singly occupied (one hole Cu2+ ) band, is separated by a large Coulomb correlation energy Ud from the two-hole (Cu3+ ) lower Hubbard band (LHB). There are several models describing the change in electronic structure

182

5 Electronic Properties of Cuprate Superconductors

Fig. 5.3. Illustration of the electronic structure of the p–d model with three bands, bonding (B), nonbonding (NB), and antibonding (AB): (a) a metallic state at halfﬁlling of AB band for U = 0, (b) a Mott–Hubbard insulator for Δ > U > W, (c) a charge-transfer insulator for U > Δ > W, and (d) – the same as (c) but with the two-hole p–d band splitted into the triplet (T,S = 1) and the Zhang–Rice singlet c 2003) (ZRS, S = 0) bands (reprinted with permission by APS from [271],

of this insulating phase under doping by electrons (n) or holes (p) (see e.g., [1309]). In the rigid band approximation, under hole doping O2p-type holes appear at the top of the valence band, while under electron doping – 3d-type electrons at the bottom of the conduction band. In other models, it is assumed that under doping an impurity band of p- or n-type appears, or else the doping band gradually ﬁlls the gap from the Fermi level, lying close to the top of the valence band or to the bottom of the conduction band under p- or n-doping, respectively. For high carrier concentrations, a wide metallic band arises, which is also predicted by the band-structure calculations. However, photoemission and inverse photoemission studies gave evidence against this “impurity band” scenario. Instead, under hole doping a shift of the chemical potential through the valence band occurs in combination with a spectral weight transfer from high- to low-energy states as discussed in Sect. 5.2.1. In the model schematically shown in Fig. 5.3d, two-hole p–d states appear in the Zhang and Rice (ZR) singlet band [1419]. This scenario is discussed in Sect. 7.1.2. Concerning the Brilloun zone (BZ) structure and the position of the Fermi level in the BZ under doping, the situation is much more complicated, as will be discussed in Sect. 5.2.2. Due to the quasi-two-dimensional character of the electron structure of layered copper-oxides, usually it is suﬃcient to consider only 2D projected Brilloun zones, which are shown at the lower part in Fig. 5.2 for several compounds with conventional notations for diﬀerent symmetry points in the BZ.

5.1 Electronic Structure: Overview

183

5.1.2 Eﬀects of Impurity Substitution Electronic properties of copper-oxide compounds are strongly aﬀected by impurities and various types of disorder in the materials. The impurity substitutions, on the other hand, can be used as a local probe to study the electronic structure of cuprates. In early experiments, mostly macroscopic parameters as the bulk superconducting Tc , the speciﬁc heat, the static magnetic susceptibility, etc., were measured to study impurity averaged properties of doped systems. Later on, synthesis of high quality single crystals and development of the NMR technique and the scanning tunneling microscopy/spectroscopy (STM/STS) enabled of one the measurements of local properties, such as the local density of electronic states g(r, ω) at the impurity site, and the investigation of the range of spatial perturbation produced by an impurity. There are three distinct types of impurity substitutions in superconducting cuprates, which result in diﬀerent eﬀects: (1) altervalent substitutions in the buﬀer layers which change the charge carrier concentration in the CuO2 planes and Tc ; (2) isovalent substitutions in the buﬀer layers which in most cases weakly aﬀect the superconducting properties; and (3) substitutions at copper sites in the CuO2 planes which strongly aﬀect both the normal and the superconducting states and usually rapidly suppress Tc . In all cases, the chemical disorder produced by impurities or nanoscale electronic inhomogeneity (stripes in LSCO, YBCO compounds or patch-shaped inhomogeneity in Bi-based compounds) result in suppressing Tc (see e.g., [304, 980] and references therein). Most of the experiments were performed on LMCO and YBCO compounds which can be manufactured as high quality single crystals or ﬁlms. Results of early experiments on these compounds were reported by [409, 790, 886]. More recently, Bi-based compounds were extensively studied by the angleresolved photoemission spectroscopy (ARPES) and STM methods (see Sects. 5.2.2 and 5.5.2). Let us consider ﬁrst the substitutions in the buﬀer layers the inﬂuence of which may be strong for impurities in the layers close to the CuO2 planes. Altervalent substitutions in buﬀer layers are usually used to change the number of charge carriers, as discussed in Chap. 2. In the La2−x Mx CuO4−y compounds substitution of La3+ by divalent ions M2+ ⇒ Ba2+ , Sr2+ , Ca2+ leads to the appearance of metallic conductivity and superconductivity for hole concentration 0.05 < ph < 0.3, ph = x−2y. The metallic state and superconductivity may also be obtained by oxygen intercalation which increase the oxygen content above the stoichiometric value in La2 CuO4+δ with δ > 0. In these crystals, the interstitial oxygen ions segregate into oxygen-poor and oxygen-rich regions. The latter form a structure of periodic planes, known as staging, with n CuO2 layers between the interstitial oxygen planes [1347]. While the substitution of La3+ located next to the appical oxygen by a divalent ion M2+ produces a random Coulomb potential and a strong perturbation in the CuO2 planes, intercalated oxygen ions give much weaker perturbations

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5 Electronic Properties of Cuprate Superconductors

which results in higher maximal Tc ∼ 45 K in the latter case. As discussed in Sect. 2.2.1, various types of structural disorder, such as stripes, charge density waves, etc., strongly suppress Tc . The charge carrier concentration in the YBCO-based, Bi-, Tl-, and Hgbased compounds is controlled by the excess oxygen atoms doped in the buﬀer layers which have no close connection with the appical oxygen ions. In this case, the CuO2 planes experience weaker perturbations which results in higher Tc . The most convincing examples are the single-layered cuprates Tl2 Ba2 CuO6+δ and HgBa2 CuO4+δ with the highest values of Tc = 93–96 K, which are believed to be due to a weak perturbation of the CuO2 planes produced by the excess oxygen Oδ located far away from those planes (see Sect. 2.5). In the two-layered YBCO compounds, Tc strongly depends on the oxygen disorder in the Cu1–O1 chains but this is caused by a complicated character of charge transfer from chains to planes as discussed in Sect. 2.4.1. Under isovalent substitution of La in LMCO compounds by the rare-earth ions RE ⇒ Nd, Sm, Eu, Gd, a smooth decrease of Tc with decreasing ionic radius in the series of these ions occurs [84, 409]. At the same time, the value of the magnetic moment of the RE ion has no eﬀect on Tc , which indicates that the Cooper pairs in the CuO2 plane are weakly coupled to the magnetic moments of ions in the La–O layers. An analogous situation is observed for the electronic superconductors Nd2−x Cex CuO4 under the substitution of Nd by the rare-earth ions Pr, Sm, Eu. As shown by [84] in the studies of the series of (Ln1−x Mx )2 CuO4 superconductors, Tc is very sensitive to lattice strains produced by cation disorder at the A-sites in the perovskite structure ABO3 . These sites are adjacent to the apex oxygen (see Fig. 2.1) and the random potential produced by the disorder may strongly disturb the electronic structure of the nearby CuO2 planes. It was found that Tc decreases linearly with increasing A-site disorder, which was characterized by the variance in the distribution of the cation radii. The Tc decrease caused by local lattice distortions caused by the ionic radius mismatch was conﬁrmed by [304] who studied a series of Ln3+ substituted Bi2 Sr1.6 Ln0.4 CuO6+δ single crystals with Ln = La, Pr, Nd, Sm, Eu, Gd, Bi. The ionic radius in this series of Ln ions monotonically decreases, from 1.14 ˚ A (La3+ ) to 0.96 ˚ A (Bi3+ ), which correlates with monotonic decrease of the superconducting temperature from Tc = 33 K for Ln = La to Tc = 12 K for Ln = Gd and the absence of the superconductivity down to 1.8 K for Ln = Bi. The authors suggested that the chemical disorder produced by the ionic radius mismatch in the Sr–O layers containing apical oxygen atoms strongly aﬀect the electronic structure of the CuO2 planes, which results in Tc suppression. A strong inﬂuence on Tc of Bi:Sr nonstoichiometry and the corresponding chemical disorder in the Sr–O layer in Bi-based compounds was conﬁrmed by studies of the two-layer Bi2+x Sr2−x CaCu2 O8+δ crystals [304]. An analogous suppression of Tc caused by substitution of Tl for Ca in the three-layer Tl1223 compounds was demonstrated by [651], who managed to increase the

5.1 Electronic Structure: Overview

185

standard value Tc 125 K up to 133 K by minimizing the chemical disorder in the compound. Eﬀects of impurity substitution were extensively studied in the YBCO systems because these compounds allow a much larger variation of their composition [409, 790, 886]. Shortly after the discovery of superconductivity with Tc = 90 K in the YBCO compound, a large class of RBCO compounds with similar high-Tc values was synthesized. They were obtained by substituting yttrium by La, Nd, Sm, Eu, etc. (see Table 1.1). For them, no suppression of Tc in the compounds with rare-earth ions having a large magnetic moment was observed, which is a hint to their weak coupling with the in-plane holes. This was also conﬁrmed by the consideration of magnetic properties of the RBCO compounds (see Sect. 3.2.4). The only exceptions were Pr, Ce, and Tb substitutions. With increasing Pr concentration x in the mixed Y1−x Prx Ba2 Cu3 O6.9 compounds, Tc monotonically decreases and disappears at x = 0.61 [594]. Pr substitution in other rare earth compounds, R1−x PrxBa2 Cu3 O7−y with R = Nd, Eu, Gd, Dy, Y, Er, and Yb, also monotonically suppresses Tc . However, at ﬁxed Pr concentration x, Tc decreases approximately linearly with the increasing ionic radius of the rare earth ion [1379]. Many experiments have demonstrated that the YBCO compounds with Ce and Tb cannot form single-phase samples. Only in YBCO thin ﬁlms, Fincher et al. [333] managed to substitute Ce and Tb for Y and demonstrated that Ce impurities suppress Tc as Pr, while Tb has little inﬂuence on the transition temperature. The Pr substitution for Y into YBCO compound was most extensively studied. They can be synthesized as single crystals with lattice parameters close to those of the pure YBCO system. In spite of the structure similarities, the electronic properties of Pr-123 diﬀer drastically from those of other compounds with RE ions. This can be explained by a much smaller localization of the 4f electrons in the Pr ion which, after Ce, has the smallest Coulomb charge in the group of 4f elements. The anomalous magnetic properties of Pr-123 compounds have been discussed in Sect. 3.2.4 and here we consider only one of the numerous experiments investigating the Tc dependence on Pr content. A more detailed discussions of the electronic structure of Pr-YBCO compounds revealed by high-energy spectroscopy will be given in Sect. 5.2.1 (see Fig. 5.15). In the frequently used hole depletion model, it is suggested that the hole concentration in the CuO2 planes is reduced under Pr doping, carrying a valence larger than 3+ . The magnetic pair-breaking eﬀect which may occur due to magnetic scattering of holes in CuO2 planes on the less localized 4f moments of Pr, μ 2.67μB , has been also suggested. To test this hypothesis, the system Y1−x−y Cay Prx Ba2 Cu3 O7−δ was studied by [891] and the Tc (x, y) dependence was investigated. Figure 5.4 demonstrates the Tc (y) dependence for various Pr concentrations x (a) and Tc (x) at y = 0 (b). The authors have supposed that the substitution of Pr with the formal valence 4+ for Y decreases the number of in-plane holes, while the substitution of Ca2+ has the opposite eﬀect, which allows one to investigate separately the dependence of

186

5 Electronic Properties of Cuprate Superconductors

a

b 96 0 % Pr 92

TC(K)

88 84 80

10 % Pr 0 % Ca 15 % Pr

76 20 % Pr 72 –0.1

0.1 0.0 Ca (y)

0.2 0

0.1 Pr (x)

0.2

Fig. 5.4. Tc dependence (a) in Y1−x−y Cay Prx Ba2 Cu3 O6.95 on the concentration y of Ca at various contents x of Pr; (b) in Y1−x Prx Ba2 Cu3 O6.95 on the concentration x of Pr. The dashed line in (b) conjectures the Tc dependence on the concentration of holes without “magnetic scattering” (after [891])

Tc on the concentration of holes p ∝ (y − x) and on the suggested magnetic scattering on the localized moments of Pr. Indeed, it follows from Fig. 5.4a that Tc (y) exhibits a conventional dependence Tc (p) that reaches a speciﬁc maximum at an optimal hole concentration popt for each concentration x of Pr ions. The location of these maxima shifts to larger values of y under the increase of Pr content, which proves directly that the concentration of itinerant holes p decreases with Pr doping. At the same time, a decrease of the maximum value of Tc (x, y) at optimal popt is observed, which indicates a decrease of Tc due to some other mechanism which the authors ascribed to pair breaking by magnetic scattering. Figure 5.4b plots the maximum of Tc (x, y = 0) under changes of the hole concentration p together with the suggested magnetic scattering (solid curve), and the presumed variation of Tc under changes of p ∝ −x alone (dashed line). The hole depletion model was supported by studies of Pr0.5 Ca0.5 Ba2 Cu3 O7−δ thin ﬁlms by [905] who have found the superconducting transition in thin ﬁlms at Tc ∼ 35 K which excludes the possibility of percolative superconductivity due to a YBa2 Cu3 O7−δ phase. In Sect. 5.2.1 we discuss other models which predict the decrease of Tc with Pr doping due to hole concentration decrease in the singlet band in CuO2 planes of the YBCO compounds. An investigation of the eﬀect of substituting the Ba in YBCO by the rareearth ions has been carried out for the Ln(Ba2−x Lnx )Cu3 O7−y compounds, where Ln = La, Nd, Sm, Eu, Gd [790]. All these lanthanides have been found to produce a similar decrease in Tc with respect to the impurity concentration x.

5.1 Electronic Structure: Overview

187

This points to a weak sensitivity of the superconducting transition to the appearance of magnetic moments at the Ba sites. The authors also observed a phase transition from the orthorhombic to the tetragonal phase (at x 0.2–0.3), which, however, did not have a signiﬁcant eﬀect on the superconducting properties. In view of an often uncontrolled increase of the oxygen content under the substitution of Ln for Ba and a complicated charge rearrangement in the layers Ba–O4, Cu1–O1, Cu2–O(2,3) in these experiments, it is diﬃcult to reach an unambiguous conclusion regarding the hole concentration in CuO2 planes at these substitutions. In any case, the above-mentioned evidence points to the fact that the chemical disorder in the Ba–O4 layer produced by Ln substitutions is partly responsible for the Tc decrease. Isovalent substitutions of copper by the of 3d metal ions have much stronger eﬀects on Tc . At concentrations of Ni, Fe x = 5–7%, and at x = 2–3% for Zn ions, the superconductivity in LMCO disappears (see [409, 790]). The same strong suppression of Tc for copper substitution was observed in YBCO compounds. Perhaps, the most peculiar eﬀects originating in such substitutions have been noticed in the case of nonmagnetic Zn ions with 3d10 closed shell. The most unexpected result observed already in earlier experiments (see e.g., [886]) was a strong suppression of superconductivity without changing the carrier concentration in low Zn-doped samples. Figure 5.5 [1222] demonstrates the Tc dependence on the hole concentration p for various values of the Zn substitution in (a) La2−x Srx Cu1−y Zny O4 and (b) Y0.8 Ca0.2 Ba2 (Cu1−y Zny )3 O7−δ (the Zn concentration is denoted in percentage (%)). Tc dependence on hole concentration p = x/2 for the pure Y1−x Cax Ba2 Cu3 O7−y compound is shown by solid squares. It is remarkable that for both LSCO and YBCO compounds the curves show a more rapid suppression of Tc (y) with increasing y toward the underdoped side together with the collapse around the pseudogap line falling to zero at p = 0.19. This peculiar Zn impurity concentration eﬀect on the behavior of Tc can be explained by the inﬂuence of the normal-state pseudogap (see Sect. 4.2.2). The experimental values for the initial slope [dTc /dy]y=0 show a constant value ∼ 8(K/% Zn)

a

b

50

80

30

0

20

1 2

10 0

TC(K)

TC(K)

40

0.2

0.1 p

40 1

20

3 0

0

60

0.3

0

2 0

3

0.1

0.2

0.3

p

Fig. 5.5. Tc dependence on hole concentration p for various values of Zn substitution y (in %) for (a) La2−x Srx Cu1−y Zny O4 and (b) Y0.8 Ca0.2 Ba2 (Cu1−y Zny )3 O7−δ compounds (after [1222])

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5 Electronic Properties of Cuprate Superconductors

for hole doping p ≥ pc 0.19, while it rapidly increases in the underdoped region as the normal-state pseudogap increases. Theoretical calculations by [1363] based on the anisotropic model for the pseudogap, Eg (k) = Eg | cos(2θ)|, where the amplitude depends on the hole concentration but not on the Zn content, Eg ∝ (1 − p/pc ), describe quite well the experimental Tc (y) dependence within the unitary limit of the potential scattering. The calculations suggest that the much stronger impurity suppression of Tc in the underdoped region, which was observed in many experiments, is caused by the pseudogap-induced depression of the density of state near the Fermi level. Generally, the studies reveal a more complicated picture in the YBCO compounds than in LMCO compounds. First, in the YBCO compounds there are two nonequivalent copper positions: Cu1 in chains and Cu2 in planes (Fig. 2.12), whose substitutions by impurities have diﬀerent eﬀects on the electronic structure and the superconductivity. Second, some impurities, as e.g., Fe and Co aﬀect the oxygen content and the short-range order in the Cu1–O1 layer, which may change the number of carriers in the Cu2–O(2,3) planes. For instance, the neutron-diﬀraction study of YBa2 Cu2.7 Zn0.3 O6+y compounds by the isotope contrast method [95] demonstrated that Zn2+ ions (at suﬃciently high concentration, 10%) are distributed proportionally between the two copper sites: 1/3 at the Cu1 positions and 2/3 at the Cu2 positions. It was also observed that the substitution of Zn2+ and Ni2+ ions for copper results in a small positive charge transfer to the CuO2 plane: estimated from the bond-valence sum it equals ∼ 0.06 10% Zn [95]. The three-valent cations such as Ga, Al, Fe, and Co ions, substitute for Cu1 sites in the chains. Therefore, they reduce the positive charge transfer from chains to planes resulting thus in a decrease of Tc , though their inﬂuence, if the hole concentration is kept constant, is much weaker than that of Zn2+ and Ni2+ impurities [409]. It was found that although the Zn impurities are nonmagnetic, they strongly disturb the antiferromagnetic spin correlations in the CuO2 planes. An appreciable reduction in the N´eel temperature TN by Zn substitution was detected both in La2 Cu1−y Zny O4 [1376] and YBa2 (Cu1−y Zny )3 O6+x compounds [53]. In La2 Cu1−y Zny O4 compound, TN → 0 at Zn concentration y 0.055, which is much smaller than the site percolation threshold of 0.41 for a square lattice [1376]. The reduction in the N´eel temperature by Zn impurities increases with the doping [53]. At the same time, formation of an induced magnetic moment by Zn substitution in cuprates was found out in several studies. Measurement of the static magnetic susceptibility revealed a Curie law C/T behavior, where a static magnetic moment μ ∝ C of the order of 1μB per Znsite was observed in La2−x Srx Cu1−y Zny O4 for the underdoped region which decreased with Sr doping [1376, 1377]. Similar results were obtained by [823] in SQUID magnetic measurements of Ni and Zn substituted YBa2 Cu3 O6+x . Static magnetic moments were detected both for Zn, μeﬀ 0.36μB , and Ni, μeﬀ 1.5μB, for fully oxydized samples and a larger moment, μeﬀ 0.85μB , for Zn impurity in an underdoped sample, x = 0.6. Local magnetic moments can be also induced by other nonmagnetic substitutions with closed shells like

5.1 Electronic Structure: Overview

189

Al3+ and Ga3+ as found by Xiao et al. [1377] and Ishida et al. [532] for the LSCO compound. NMR experiments in YBa2 (Cu1−y Zny )3 O6+x revealed that the induced local magnetic moments occur on the nearest neighbour Cu orbitals [53, 771]. Furthermore, the symmetric character of the line broadening and its speciﬁc temperature dependence of the 17 O and 63 Cu NMR lines were accounted for by the staggered spin density oscillations induced by Zn and Ni impurities evolved from the short-range antiferromagnetic spin correlations in cuprates (see e.g., [144, 145, 566] and references therein). It was shown that the uncompensated sum of the staggered moments agrees with the local moments detected by the static susceptibility measurements on Zn and Ni doped YBCO compounds discussed above. In the NMR and NQR studies of the YBa2 (Cu1−y My )3 O7 compound doped with M = Zn, Ni, quite a diﬀerent behavior of Zn and Ni impurities was revealed, as compared to that observed in static susceptibility measurements, by Ishida et al. [530]. In the case of Zn, a local suppression of antiferromagnetic spin correlations near Zn impurities and a gapless superconductivity occurred, whereas in the case of Ni impurities with local paramagnetic moments, only a small decrease of Tc was detected. Similar contrasting behavior was observed by Nakano et al. [883] in studies of Ni- and Zn-substitution eﬀects in the La2−x Srx Cu1−y My O4 compound, which has only one copper site in the plane. Figure 5.6 shows the Tc dependence on the impurity concentration y for M = Zn and M = Ni at two-hole concentrations: p = 0.16 and p = 0.22. The authors explained the more rapid decrease of Tc in the case of Zn in comparison with Ni substitution by a much weaker disruption of the 3d-spin correlations on the nearest Cu-sites around the Ni-impurity. Contrary to other measurements, which detected an eﬀective magnetic moment induced by Ni impurity, no Curie term was found in La2−x Srx Cu1−y My O4 compound at small Ni concentration, y < y0 , (y0 ∼ 0.02 for x ∼ 0.14). To explain the absence of an uncoupled magnetic moment at Ni site, the authors suggested that, at low concentration, the Ni atoms are substituted as Ni3+ ions with a spin in the 3d(x2 − y2 ) orbital, which can couple antiferromagnetically to neighboring copper spins. In that case, the Ni3+ ions preserve the local nature of the 3d(x2 −y 2 )-spin correlations, while nonmagnetic Zn2+ impurity strongly disturbs the spin correlations on several neighboring Cu-sites. On the basis of these studies, the authors concluded that the 3d-spin correlations within the CuO2 plane are essential for high temperature superconductivity in cuprates. The important role of the 3d-spin correlations in cuprates for achieving high-Tc was conﬁrmed in NMR studies of YBCO compounds doped with Zn and Li impurities by Bobroﬀ et al. [146]. The Li1+ ion is nonmagnetic like Zn2+ whereas it has a diﬀerent valence which enables one to compare the charge and spin perturbation eﬀects on Tc . Figure 5.7 demonstrates the Tc dependence in the optimally doped YBa2 (Cu1−y My )3 O7−ε and in underdoped YBa2 (Cu1−y My )3 O6.6 compounds on the concentration of impurity yplane (per CuO2 layer) for M = Zn and M = Li. By measuring the intensities of the

190

5 Electronic Properties of Cuprate Superconductors

Fig. 5.6. Tc dependence on the impurity concentration y for the La2−x Srx Cu1−y My O4 (LSCMO) compound: M = Zn (open symbols) and M = Ni (closed symbols), for two-hole concentrations: p = 0.16 (circles) and p = 0.22 (squares) (after [883])

Fig. 5.7. Tc dependence on the concentration of the impurity yplane (% per CuO2 layer) in the optimally doped YBa2 (Cu1−y My )3 O7−ε (circles) and in the underdoped YBa2 (Cu1−y My )3 O6.6 (triangles) compounds for M = Zn (open symbols) and M = Li (closed symbols) (after [146])

NMR lines for 89 Y and 7 Li, the authors concluded that about half of the Li impurities substitutes in plane, while all Zn impurities substitute in plane. A similar reduction in Tc with impurity concentration yplane both for Zn and for Li indicates that the itinerant hole scattering on spinless impurity is mostly responsible for the Tc decrease, while the charge perturbation plays a minor role. At the same time, 89 Y NMR measurements revealed the presence of Li1+ -induced impurity magnetic moments on the nearest neighbor copper sites similar to the Zn impurity. The local magnetic susceptibility measured by the 7 Li NMR showed a Curie-type behavior in the underdoped samples

5.1 Electronic Structure: Overview

191

YBa2 Cu3 O6+x , x ≤ 0.85, which changed to a C/(T + θ) law with the θ constant increasing with doping. As the authors pointed out, this screening for a large doping of the Li1+ -induced local moment in the correlated hole band in CuO2 plane is similar to the Kondo screening of local magnetic impurity by conduction electrons. Detailed studies of various NMR spectra in the Li-doped YBa2 Cu3 O6+x compound by Ouazi et al. [933] enabled them to determine the shape and the magnitude of the staggered spin polarization induced by a nonmagnetic impurity. The correlation length of the staggered polarization ξimp showed a pronounced temperature dependence in the underdoped sample, x = 0.6, which increased from ξimp ∼ 1.5a at T = 160 K up to ξimp ∼ 6a at T = 80 K, while a weak temperature dependence was observed in the overdoped sample, x = 1, with ξimp ∼ 3a at T = 80 K (a is the lattice constant). Measurements of 7 Li NMR spectra in YBa2 Cu3 O6+x below Tc by Bobroﬀ et al. [143] registered the persistence of the local magnetic moment induced by nonmagnetic Li impurity, which was conﬁned primarily to the impurity ﬁrst nearest neighbor copper sites. This observation conﬁrms that the short range AF correlations do survive in the superconducting state. The Kondo-like screening of the local susceptibility detected in the normal state for optimally doped sample, as discussed above, was strongly reduced below Tc , while the Curie-type susceptibility in the underdoped sample, x = 0.6, was not aﬀected by the superconducting transition. Similar results were obtained for Zn and Ni impurity in YBa2 Cu3 O7 below Tc in measurements of the 17 O NMR spectra by Ouazi et al. [934]. A staggered paramagnetic polarization with extension of about three lattice spacings was revealed for both Zn and Ni impurities in the superconducting state in continuity with the normal state. An additional local density of states (LDOS) near the Fermi energy was detected at low temperature for Zn but not for Ni impurities. It is to be stressed that the Zn impurities are detrimental not only for the superconductivity but they also strongly suppress the spin-ﬂuctuation spectrum. A strong suppression of the spin gap and a rapid decrease of Tc with Zn concentration were observed by Williams et al. [1361] in 93 Y NMR measurements in the YBa2 (Cu1−y Zny )4 O8 compound. The spin gap was “ﬁlled in” and completely suppressed when the adjacent Zn atoms shared the same next-neighbor at the concentration y = 6.25%. The range of the spin-gap suppression around the Zn atoms appeared to be short, about 1.4a where a is the lattice parameter. Important details of the evolution of the the spin-ﬂuctuation spectra induced by Zn substitution in YBa2 (Cu1−y Zny )3 O6+x have been evidenced by inelastic neutron scattering studies. In the underdoped sample with x = 0.6, the quasi-gap behavior in spin ﬂuctuations at the antiferromagnetic wave vector (π, π) in the pure, superconducting sample disappears upon Zn doping [578]. In the slightly overdoped compound with x = 0.97, the spin-ﬂuctuation spectrum below Tc changes drastically [1166]. In the zinc-free sample, it exhibits a gap below 35 meV and a resonance region around 41 meV. In the

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5 Electronic Properties of Cuprate Superconductors

Fig. 5.8. Spectrum of spin ﬂuctuations deﬁned by χ (Q, ω), [Q = (0.5, 0.5, 5.2)] in YBa2 (Cu1−y Zny )3 O6.97 at T = 4.5 K for Zn-free sample (y = 0, crosses) and at T = 10.5 K for y = 2% (squares and circles – hatched area) (after [1166])

sample with Zn concentration y = 0.02, the gap is closed and a broad spectrum in the low energy range appears with a much lower intensity in the resonance region as shown in Fig. 5.8. The most detailed picture of how a single impurity of Zn or Ni atoms perturbs the surrounding electronic environment in the CuO2 plane of the cuprate superconductors was obtained in studies of the STM conductance spectra in Bi2 Sr2 CaCu2 O8+δ compounds. Figure 5.9a shows the STM spectra for the Zn-doped Bi-2212 sample [950]. A zero-bias diﬀerential tunneling conductance map of the cleaved Bi-O surface of the sample taken at 4.2 K revealed bright and dark regions which indicated high and low quasiparticle local density of states (LDOS). The ﬁrst ones are associated with the high tunneling conductance above the position of the Zn impurity, which strongly suppresses the superconducting coherence peaks. At the same time, a very strong resonance peak appears at the energy Ω = 1.5 meV. The dark regions represent the Zn-free superconducting regions with low conductance due to superconducting gap as shown by the ﬁlled dots with the superconducting coherence peaks indicated by arrows. The resonance peak was attributed to the quasiparticle scattering at the Zn impurity atom, which strongly changes the LDOS. In particular, the LDOS at the resonance energy does not decay monotonically with distance from the impurity but shows local maxima at the positions of eight second nearest neighbors (nn) and the third nn copper atoms. Contrary to the NMR results which detected large magnetic moments on the four nn copper atoms, the LDOS in the STM spectra has no local maxima at these sites. Thus, this STM study has conﬁrmed a strong suppression of superconductivity within the range of approximately 15 ˚ A around the Zn impurity and the d-wave symmetry of the superconducting gap, which was revealed by a fourfold symmetric cloud of the LDOS. The results of similar STM measurements for the Ni-doped Bi-2212 sample are shown in Fig. 5.9b [503]. The spectra are shown for four positions around Ni impurity. The superconducting coherence peaks are indicated by arrows. Two local impurity states manifest themselves as the resonance peaks at the

5.1 Electronic Structure: Overview

193

Fig. 5.9. STM spectra for Bi2 Sr2 CaCu2 O8+δ superconductors doped with (a) Zn impurities (reprinted by permission from Macmillan Publishes Ltd. from [950], c 2000) and (b) Ni impurities (reprinted by permission from Macmillan Publishes c 2001). In (a), the spectrum of the Zn-free superconducting region Ltd. from [503], is shown by filled dots with the superconducting coherence peaks indicated by arrows, while the spectrum above the Zn atom site is shown by open circles with the resonance peak at the energy Ω = 1.5 meV. In (b), the spectra are shown for four positions around Ni impurity (from top to bottom): above the Ni atom site (solid circles), ﬁrst (open circles) and second (open squares) nearest neighbor Cu atom positions, and at a distance of 30 ˚ A from the impurity site. The superconducting coherence peaks are indicated by arrows, while the resonance peaks are indicated by the energies Ω1 ±9.2 meV and Ω2 ±18.6 meV. The bottom panel in (b) shows the average conductance spectrum

energies Ω1 ±9.2 meV and Ω2 ±18.6 meV. The bottom panel shows the conductance spectrum averaged over the whole impurity state which demonstrates almost perfect particle-hole symmetry. The spectra reveal a much weaker inﬂuence of the magnetic Ni impurity (which is assumed to be in the Ni2+ (3d8 ) state) on the superconducting properties in comparison with the

194

5 Electronic Properties of Cuprate Superconductors

Zn doped sample. First of all, the coherence peaks are only weakly depressed, even above the Ni site, and the gap magnitude (∼ 28 meV) does not change appreciably with the distance from the impurity site. Therefore, contrary to the Zn case, the Ni impurity does not locally disrupt superconductivity. The appearance of two impurity resonance peaks for opposite spin polarizations indicates that the Ni magnetic moment survives in the superconducting state which lifts the spin degeneracy of the localized state. The analysis of the scattering parameters for the resonance peaks (discussed below) has shown that the quasiparticle scattering at the Ni impurity can be well described by a potential scattering with a phase shift δ0 0.36π lying close to the unitary limit (π/2) occuring at the Zn scattering, while the magnetic part gives a minor contribution. This observation helps to elucidate the diﬀerence between Ni- and Zndoped superconducting cuprates. The phenomena sensitive to impurity scattering, like resistivity, should show similar behavior since both impurities demonstrate strong potential scattering. However, the superconducting properties, like Tc , the superﬂuid density, the penetration depth (see Table 4.6), etc., are strongly disrupted by the spinless Zn impurities which destroy the 3d electron correlations and superconductivity within a radius of 3–5 lattice constants. This correlates with the strong suppression of the antiferromagnetic short-range order and the dynamical spin ﬂuctuations suggesting a magnetic mechanism of pairing. Contrary to Zn, the Ni impurity, due to a nonzero spin in the 3d(x2 − y 2 ) orbital, preserves the delicate electron correlations in the Cu3d9 −O2p6 band and does not disrupt the d-wave superconductivity. In the case of s-wave superconductivity, the situation would be opposite as we will discuss below. To explain the strong inﬂuence of the Zn2+ impurities on THE density of states (DOS), Tc , and other superconducting properties in the cuprate superconductors, and the appearance of STM experiment detected local states at a magnetic and a nonmagnetic impurity, several theoretical models were proposed (see the review by Balatsky et al. [97] and references therein). We start with the description of a general approach to the calculation of the averaged DOS in superconductors at low concentration of impurities and then discuss the LDOS near the impurity. According to Anderson [59], in conventional superconductors with an isotropic s-wave gap, the electron scattering on nonmagnetic impurities should not destroy the superconductivity. However, as shown by Abrikosov and Gor’kov [7], the scattering of the Cooper pairs on magnetic impurities results in pair-breaking. As a result, a ﬁnite density of states appears within the superconducting gap even at zero temperature (“gapless superconductivity”), which suppresses Tc . For the d-wave superconductors, the scattering on nonmagnetic impurities averages out of the k-wave vector dependence of the gap. This disrupts the phase assignment and generates a ﬁnite density of quasiparticle states at the Fermi energy [401]. Therefore, similar to the s-wave superconductors, it results in Tc reduction and to describe the potential scattering

5.1 Electronic Structure: Overview

195

on nonmagnetic impurities for a d-wave superconductor we can apply the Abrikosov–Gor’kov theory. We consider a simple model of the potential s-wave impurity scattering characterized by the phase shift δ. The density of quasiparticle states is determined by the single-particle Green function. In the T -matrix approximation, it is deﬁned by the equation: ˆ r , ω) = G ˆ 0 (r − r , ω) + G ˆ 0 (r, ω)Tˆ0 (ω)G ˆ 0 (−r , ω), G(r,

(5.2)

ˆ 0 (r, ω) is the Green function of the clean system and the matrix Tˆ 0 (ω) where G describes the scattering on impurities. In the superconducting state, these are matrices in the Nambu particle-hole-spinor space. The solution for the averaged over the impurities Green function can be written in the conventional form ˜ τ0 + ξ˜k τ3 + Δ˜k τ1 ˆ ω) = ω , (5.3) G(k, 2 ˜k |2 ω ˜ 2 − ξ˜k − |Δ where τˆi are the Pauli matrices. The impurity scattering is taken into account by the renormalization of the frequency ω ˜ = ω − Σ0 (ω), the single-particle energy ξ˜k = ξk + Σ3 (ω) and the gap function Δ˜k = Δk + Σ1 (ω). Neglecting a weak renormalization of the single-particle energy ξ˜k , the impurity self-energy in the T -matrix approximation is given by (see e.g., [165]): Σi (ω) = Γ

Gi , D

D = c2 + G21 − G20 ,

Gi (ω) =

1 ˆ Tr(ˆ τi G), 2πN0

(5.4)

k

where the scattering rate parameter Γ = ni /(πN0 ) depends on the impurity concentration ni and the density of states at the Fermi energy N0 . The strength of a single impurity scattering is deﬁned by the phase shift δ: c = cot δ. The Born limit corresponds to δ 1, c 1, while the unitary limit is given by δ → π/2, c → 0. Therefore in the Born limit we have Γ/D Γ/c2 = Γ tan2 δ = ΓN – the scattering rate in the normal state, while in the unitary limit Γ/D Γ/(G21 − G20 ). For the d-wave superconductors, the oﬀ-diagonal self-energy Σ1 (ω) gives no contribution since the sum over ˜ky ,kx vanishes, the k-vectors for the d-wave gap function Δ˜kx ,ky = −Δ G1 (ω) =

˜k Δ 1 = 0, 2 2πN0 ˜ 2 − ξ˜k − |Δ˜k |2 k ω

(5.5)

and therefore the gap function is not renormalized, Δ˜k = Δk . Contrary to the d-wave superconductors, in anisotropic s-wave superconductors the oﬀdiagonal contribution (5.5) is nonzero and the gap function is renormalized by a positive shift independent of the k-vector. This distinction brings about quite a diﬀerent behavior of the density of states and Tc (ni ) for the s- and d-wave superconductors.

196

5 Electronic Properties of Cuprate Superconductors

Fig. 5.10. Normalized density of states N (ω)/N0 for s- and d-wave gap symmetry for various potential scattering rates ΓN /Δ0 in the Born approximation (a) and in the unitary limit, c = cot δ = 0 (b) (after [165])

Figure 5.10 shows the renormalized density of states (DOS) N (ω)/N0 for a superconductor with the s- and d-wave gap symmetry for various potential scattering rates ΓN /Δ0 [165]. The results in the Born approximation is presented in the panel (a) and in the unitary limit c = cot δ = 0 – in the panel (b). The authors have solved the Dyson equation (5.3) together with a BCS-type gap equation with a phenomenological interaction for the order parameter in the form: Δk = Δd,s 0 Φd,s (k) with Φd,s (k) = cos 2φ, | cos 2φ| for the d- and s-symmetry, respectively. In the pure case, ni = 0, DOS is linear at small frequency for both symmetries as shown by the dashed lines. However, with impurity doping, the DOS shows quite a diﬀerent behavior. Whereas in the case of the d-symmetry the gap is ﬁlled with excitations, in the case of s-symmetry in the limit of strong scattering the gap becomes isotropic, Δk ⇒ Δavg with the BSC density of states: N (ω) = Re [ω/(ω 2 − Δ2avg )1/2 ]. Therefore, an activated behavior in thermodynamic and transport properties should be observed in the case of dirty anisotropic s-wave superconductors, whereas power laws are predicted for both clean and dirty d-wave superconductors. The Tc dependence on the impurity scattering is described by the conventional Abrikosov–Gor’kov (1960) formula, both the s- and d-wave cases (see e.g., [324]): Tc0 1 1 α ln =a Ψ −Ψ + , (5.6) Tc 2 2πTc 2 where α = Γ/(1 + c2 ) is the pair-breaking parameter and Ψ is the digamma function. For the d-wave symmetry, the prefactor a = 1 and the critical concentration at which Tc vanishes is given by the conventional formula: nc = π 2 (c2 + 1)N0 Tc0 /2eγ , γ is the Euler constant. For the s-wave symmetry, a = 1/4 and Tc is nonzero at any impurity concentration ni . For the d-wave superconductors, due to the vanishing of the oﬀ-diagonal contribution (5.5)

5.1 Electronic Structure: Overview

197

in the denominator of the scattering matrix: D = c2 − G20 , even a weak scattering in the unitary limit, c = 0, results in a strong variation of the DOS and a corresponding suppression of Tc . To study the localized states in a superconductor, one should analyze the equation for the Tˆ-matrix in (5.2) the poles of which determine the location of the bound states. In the case of an s-wave superconductor, the bound states appear close to the gap edge and are irrelevant for the physical properties. For the d-wave pairing and the particle-hole symmetry of the energy spectrum in the normal state, the Tˆ -matrix has only diagonal components due to (5.5): Tˆ (ω) = (G0 (ω) τˆ0 + c τˆ3 )

c2

1 . − G0 (ω)2

(5.7)

The resonance or bound states are given by the poles of the T -matrix: G0 (Ω) = ±c. The Green function for a clean system G0 (ω) with a model d-wave superconducting gap Δ(ϕ) = Δ0 cos 2ϕ in 2D case is given by the equation (see e.g., [1088]): G0 (ω) = −

2 [ sgn(ω) K( 1 − 2 ) + i K() ], π

(5.8)

where = |ω|/Δ0 < 1 and K() is the elliptic integral of the second kind. In the case of strong scattering, c 1, the solution for the poles of the scattering matrix Ω = ∓Ω0 − iΓ determines the resonance state with the parameters [1088]: π/2 π/2 Ω0 cΔ0 , Γ |Ω0 | , (5.9) ln(8/π|c|) ln(8/π|c|) where Ω0 and Γ are the energy and the inverse life-time of the resonance. For the d-wave superconductor, the density of state is nonzero inside the gap and varies linearly with the energy at the Fermi level (see Fig. 5.10), which results in the ﬁnal width Γ of the resonance (5.9). In the unitary limit c → 0, an extremely sharp resonance emerges at the Fermi energy. Using (5.9), Pan et al. [950] estimated the phase shift δ 0.48π for the Zn-doped Bi-2212 compound which conﬁrmed that scattering by Zn impurity is close to the unitary limit π/2. The analysis of the r-dependence of the LDOS N (r, ω) = −(1/π) ImG11 (r, r, ω + iε) shows that the intensity of the impurity resonance state should have the form of a cross with long tails extended along the nodes of the d-wave gap and decays as a power law ∼ 1/r2 (for a detailed discussion, see [97]). If an impurity has a magnetic moment, then the spin degeneracy of the resonance is lifted and one observes two asymmetric peaks at the energies Ω± given by the same (5.9) but with the the scattering parameter c± = 1/(πN0 (w ± U ) where w and U characterize the strength of the magnetic and the potential parts of the interaction with the impurity (see e.g., [1089]). Using this formula for the resonance peaks Ω1,2 detected in the Ni-doped Bi-2212 (Fig. 5.9b), Hudson at al. [503] estimated the parameters of the scattering,

198

5 Electronic Properties of Cuprate Superconductors

N0 U = 0.67, |N0 w| = 0.14, which shows that the potential scattering is the dominant contribution produced by the Ni-impurity with a phase shift δ = tan−1 (πN0 U ) 0.36π to only a small extent weaker than for the Zn impurity. The experimental results for the resonance peaks observed in Zn-doped and Ni-doped Bi-2212 discussed above (Fig. 5.9) do not fully agree with the theoretical predictions. While the energy and the width of the impurity induced resonance can be well described by the theory, the spatial distribution of intensity does not ﬁt the theoretical models. In particular, in the unitary limit the scattering produces a node at the impurity site, while in experiments the strongest intensity is observed at the impurity site which decays much faster along the nodal direction than along the gap maxima contrary to the theoretical prediction. To explain these discrepancies, it was proposed to take into account that the STM experiment in Ba-2212 probes the Bi-O layer above the CuO2 plane and therefore tunneling between these layers (some kind of a “ﬁlter function”) is involved [799]. It should be also taken into account that Zn and Ni impurities in fact are vacancies at the Cu site and therefore the perturbation potential includes also interaction with neighboring sites. Such a nonlocal perturbation results in the appearance of cluster resonance states of diﬀerent symmetry (s-, p-, d-types) the wave functions of which are linear combinations of the quantum states at the neighboring sites. A microscopic t– J model allowing for the nonlocal perturbation produced by Zn impurity and the corresponding cluster resonance states was considered by Kovaˇcevi´c et al. [656]. For a quantitative comparison of the STM/STS experimental results with the theory, the self-consistent calculations of the realistic electronic structure and the superconducting order parameter should be performed beyond the simple model approach discussed above (see e.g., [341, 342]). Calculations of the local density of states around Zn, Ni and Cu-vacancy impurities, within the density functional theory [1337] have shown that the eﬀective scattering potentials are short-range, of order 1 ˚ A, which support models of impurity as point-like potential scatters. At the same time, the eﬀective impurity potential has weak long-range tails which oscillates and strongly depends on the wave functions of the particular impurity. Several other problems and recent developments in the studies of electronic properties induced by impurities were discussed by Balatsky et al. [97]. Resume To summarize the studies of impurity substitution eﬀects in cuprate superconductors, we note the following results: 1. Nonisovalent substitutions in the buﬀer layers control the charge carrier concentration in the CuO2 planes and Tc , as described in Chap. 2. However, substitutions in the layers adjacent to the CuO2 planes, as the substitution of La3+ located next to the apical oxygen by divalent ions M2+ in LSCO compounds, produces a random Coulomb potential and a strong

5.2 Photoemission Spectroscopy

199

perturbation in the CuO2 planes, which strongly suppresses Tc . The weakest perturbation of the electronic structure occurs under oxygen doping in Tl- and Hg-based compounds which have the highest Tc . 2. Isovalent substitutions in the buﬀer layers or cation disorder at the Asites in the perovskite structure produce lattice strains caused by ionic radius mismatch which strongly disturb the electronic structure of the nearby CuO2 planes and suppress Tc . However, no suppression of Tc in the compounds with rare-earth ions having a large magnetic moments was observed (except Pr and Ce ions), which indicates their weak coupling with the in-plane holes. 3. The strongest suppression of superconductivity occurs upon impurity substitution (Zn, Li, Ni, etc.) for copper in CuO2 planes. Among them, the eﬀects produced by the nonmagnetic Zn2+ ion with a ﬁlled 3d atomic shell are the most peculiar ones: Zn impurity reveals a static magnetic moment of the order of 1μB , it strongly disturbs the antiferromagnetic spin correlations in the CuO2 planes exhibiting the staggered spin density oscillations within 3–5 lattice spacing. STM/STS experiments discovered strong resonance peaks at low temperature in the vicinity of Zn and Ni impurities induced by strong potential scattering with phase shifts close to the unitary limit. However, the magnetic Ni impurity shows only a small suppression of the superconducting coherence peaks contrary to Zn impurity which destroys superconductivity within a radius of several lattice spacings. It suggests that a Ni impurity, due to a nonzero spin in the 3d(x2 −y 2 ) orbital, preserves the delicate electron correlations in the Cu3d9 −O2p6 band and does not strongly suppress the d-wave superconductivity.

5.2 Photoemission Spectroscopy Direct information on the electronic structure of solids may be obtained using the electron spectroscopy. Photoemission or photoelectron spectroscopy (PES) is based on the phenomenon of the photoelectric eﬀect discovered by Hertz in 1887 (for a review, see [505]). Presently, PES has received further development in studies of the X-ray absorption ﬁne structure XAFS (see the review by [1048]), the resonant inelastic X-ray scattering spectra (RIXS) (see the review by [650]). The most important results for cuprate superconductors were obtained with the aid of the angle-resolved photoemission spectroscopy (ARPES). Results of extensive studies of strongly correlated systems and copper-oxides are reviewed in several articles and books, see e.g., [196, 271, 316, 362, 506, 751, 1143, 1436]. There are several techniques in the photoemission spectroscopy. In the direct PES, a beam of monochromatized radiation with energy h ¯ ω is incident on a sample which emits electrons in the vacuum with kinetic energy Ekin . By measuring the energy of the emitted electrons, Ekin , one can determine the binding energy EB , which is given by the energy conservation law:

200

5 Electronic Properties of Cuprate Superconductors

|EB | = h ¯ ω − Ekin − φ,

(5.10)

where φ is the work function of the metal under study. The measurement of the angle of the emitted photoelectron enables one√to determine its momentum: the modulus is given by the kinetic energy p = 2mEkin and the components parallel and perpendicular to the sample surface are deﬁned by the polar θ and the azimutal ϕ angles. The parallel component of the momentum p = √ ¯hk = 2m Ekin sin θ is conserved at the electron transmission through the surface into the vacuum, which enables one to study the in-plane dispersion of the electron energy in the sample. The perpendicular component p⊥ is not conserved across the sample surface and therefore only some averaged values over this direction of the binding energy (electron dispersion) in the sample can be obtained. To investigate the valence (occupied) electronic states in the solids, various incoming photon energies in photoionization are used. The photon energy in the ultraviolet regime (Ei = 5–100 eV) (ultraviolet photoemission spectroscopy – UPS) is used to study low energy electronic states close to the Fermi energy in metals. High energy (few mega electron volt) and momentum (∼ 1 %(π/a)) resolutions achieved in recent years at photoemission beamlines on high-ﬂux synchrotron facilities enables one to obtain electron dispersion curves in metals with high precision in ARPES (for a review, see [271]). In the X-ray regime (Ei > 1, 000 eV) (X-ray photoemission spectroscopy – XPS), one can study only electron density of states due to the low momentum resolution. Recently, a new low-photon regime around 6 eV was accessed with the help of laser-based ARPES [636]. Studies of Bi2 Sr2 CaCu2 O8+δ crystals with the new technique demonstrated a much better resolution and a low background. The low-energy laser ARPES is signiﬁcantly more bulk sensitive since the electron mean free paths at this energy increase up to 16 ˚ A in comparison with ∼ 6 ˚ A in standard ARPES studies with 52 eV photons. To study unoccupied electronic states, the inverse photoemission spectroscopy (IPES) is used. In the IPES, an electron with the initial kinetic energy Ei is incident on a sample and being captured at an empty ﬁnal state Ef emits Bremsstrahlung which is detected. The technique that uses a ﬁxed incident electron energy is called IPES, while the technique in which photons with a ﬁxed energy are detected is called Bremsstrahlung isochromat spectroscopy (BIS) (see e.g., [505]). The information on the unoccupied electronic states can be also obtained with the aid of the X-ray absorption spectroscopy (XAS). In XAS technique, electrons from deep core levels are promoted to the unoccupied states. The site-selected unoccupied density of state can be also measured by the transmission electron energy-loss spectroscopy (EELS). Polarization dependent XAS and orientation dependent EELS measurements on single crystals can provide valuable information on the symmetry of the hole states in cuprates (for a review, see [336]). Auger spectroscopy (AES – Auger electron spectroscopy) makes it possible to study more complicated two-hole states in the valence band.

5.2 Photoemission Spectroscopy

201

With the aid of the photoemission spectroscopy, it is possible to investigate only surface layers of a sample since the electron escape depth for UPS is of the order 5–16 ˚ A and for XPS – 20–50 ˚ A. This fact complicates the analysis of the data because the surface properties of a material may diﬀer from those of the bulk, and moreover the measurement results depend on the quality of the surface and the changes of its properties under irradiation. Therefore, measurements by the EELS technique of fast electron spectra on transition through ∼ 1, 000 ˚ A ﬁlms are preferable in studies of the bulk electronic density of states. Measurements of the near edge X-ray absorption ﬁne structure in the ﬂuorescence yield mode also enable one to obtain information about bulk properties (probing depths ∼ 600 ˚ A) and to avoid the surface problems. The intensity of the photoemission current is characterized by the transition probability wf i for an excitation of electrons from the initial state i of the system of N electrons with wave function ΨiN and the energy Ei to the ﬁnal state f with wave function ΨfN and the energy Ef (N − 1 electrons in the system and an emitted electron in vacuum). In the Born approximation, it reads (see e.g., [505]): wf,i ∝

2π N | Ψi |(e · ∇)| ΨfN |2 δ(Ef − Ei − ¯hω), h ¯

(5.11)

where ¯hω and e are the energy and polarization of the incident photon. In (5.11), the dipole approximation was used which can be justiﬁed for photons with a wavelength much larger than the atomic size. In this approximation, only the dipole transitions i → f changing the angular momentum by Δl = ±1 (e.g., s → p, p → d transitions) are permitted. In the simplest (sudden) approximation, the initial and the ﬁnal wave functions are written as a product of single-electron states φi(f ),k (in a one-band approximation) and the wave N −1 function of the remaining system ψi(f ),k : N −1 Ψi = Cφi,k ψi,k ,

N −1 Ψf = Cφf,k ψf,k ,

where k denotes the wave vector of the photoelectron with the energy Ekin . The factorization of the wave functions permits to write the intensity of the photocurrent in a simple form: wf,i ∝ |M (k, ω)|2 A(k, E) f (E), (5.12) I(k, ω) = i,f

where the single-electron matrix element M (k, ω) ∝ φi,k |e∇)|φf,k depends on the photon energy h ¯ ω and the state k of the removed electron. This dependence of the matrix element M (k, ω) on h ¯ ω may be important and should be taken into account in analyzing the experimental data (see e.g., [100]). The single-electron spectral function A(k, E) describes the probability of removing an electron with energy E and momentum ¯hk from the system of N electrons. The Fermi function f (E) takes into account that direct PES probes only the

202

5 Electronic Properties of Cuprate Superconductors

occupied electronic states. As mentioned above, only the component parallel to the surface h ¯ k of the electron momentum h ¯ k in the sample can be measured. In the case of quasi-2D systems as the copper-oxide compounds with a small dispersion perpendicular to the layers, averaging over the momentum ¯hk⊥ in many cases can be neglected (for a discussion of 3D eﬀects in cuprates see [99]). In the EELS experiments, the partial diﬀerential cross-section is measured, which determines the fraction of incident electrons with energy Ei scattered into a solid angle dΩ with an energy between Ef and Ef + dE (see e.g., [336]):

1 d2 σ 1 ∝ 2 Im − , (5.13) dΩdE q ε(q, ω) where ε(q, ω) is the dielectric function. The inverse dielectric function in (5.13) is related to the charge density response function χ(q, ω) = ρq |ρ†q ω by the equation ε−1 (q, ω) = 1 + v(q) χ(q, ω) where v(q) is the Coulomb interaction. Here the scattering vector q = ki − kf where ki and kf are the incident and scattered electron wave vectors and ¯hω = Ei − Ef is the electron energy loss. For small energy transfer h ¯ ω ≤ 10 eV, the main contribution to the scattering (5.13) comes from the collective excitations of electrons, the plasma oscillations with frequency ωp . For higher energy excitations, the EELS is used to study core excitations as considered below. In both XAS and EELS techniques, electron excitations between two quantum states of N electrons in a solid are studied, like charge excitations in optic measurements (see Sect. 5.3.1), while in PES the ionization processes N → (N − 1) are investigated which are determined by single particle excitations. 5.2.1 High-Energy Spectroscopy In early photoemission experiments on copper oxide superconductors mainly the binding energies of core levels, like O 1s or Cu 2p, were studied. These experiments, though performed on ceramic samples of LSCO and YBCO, nevertheless, revealed several important features of the electronic structure of cuprates. First of all, it was detected that the undoped compounds are semiconductors with strong electron correlations in the Cu3d band, in view of which the two-hole states of Cu3+ (3d8 ) appeared to be shifted by about 9 eV to the higher binding energy. The weak dependence of the copper spectra on doping and pronounced changes in the oxygen spectra conﬁrmed that under doping, holes appear in the O 2p band which conﬁrmed the charge transfer insulator ground state of cuprates (see e.g., [360, 361, 906]). In this case, the electronic structure is described by Fig. 5.3c. The inﬂuence of electron–electron correlations on the electronic structure of cuprates was studied by comparing the PES spectra of the simplest model compounds Cu2 O and CuO. While Cu2 O is a semiconductor with a gap of 2.17 eV due to the ﬁlled Cu 3d10 shell of Cu1+ ions, CuO compound should be a metal according to the band theory

5.2 Photoemission Spectroscopy

203

since Cu2+ ions have open 3d9 shell. However, CuO is an antiferromagnetic semiconductor with the gap of about 1.4 eV. Detailed studies of photoemission spectra of CuO and CuO2 compounds and cluster calculations have given the ﬁrst estimates for the parameters of their electronic structure. These values (in electronvolt) are the following: the Coulomb correlation energies Udd = 9.3 and Upp = 4.6, the transfer integrals: Tpdσ (b1 ) = 2.3, Tpp = 1.25, and the charge-transfer gap Δpd = 2.2 [318, 381]. These parameters are similar to those ones obtained for cuprate superconductors as discussed in Sect. 7.1.2 (see Table 7.1). The most direct information on the symmetry of doped hole states in copper oxides came from the EELS and XAS studies (for a review, see [336]). In EELS, measurement of absorption spectra of fast electrons with incident energy E = 170 keV with high resolution, ΔE 0.4 eV, in ﬁlms of thickness ∼ 1, 000 ˚ A has allowed investigations of the local density of hole states. Under a small momentum transfer, the main contribution arises from dipole transitions with Δl = ±1, and therefore the studies of the absorption edge under excitations of O 1s level (EB = 528.5 eV) or Cu2p3/2 level (EB = 931 eV) allows the investigation of hole states in O 2p and Cu 3d bands, respectively. The polarization dependence and the dipole selection rule of XAS enable one to obtain similar information on the symmetry of hole states in this technique. Below we present several examples of applying EELS and XAS techniques in studies of hole states in copper oxides under excitation of the O 1s and Cu 2p core levels. In Fig. 5.11a the results of the EELS measurements by [1057] on polycrystalline La2−x Srx CuO4+δ sample are shown. The stoichiometric compound (x = 0, δ 0) gives the lowest curve. The absorption peak at Ec = 530.2 eV is ascribed to a small admixture of O 2p states in the upper Hubbard band (UHB) (see Fig. 5.3d). According to (5.1), the wave function of Cu2+ has an admixture (β) of the ligand O 2p hole states in this band, determining the intensity of the order β 2 of this absorption peak in the conduction band (C). Appearance of a hole in the O 2p band (due to excess oxygen x = 0, δ > 0, or doping by Sr, x > 0) leads to a shift of the absorption edge to the lower energy domain with a well-pronounced peak V near EF = 528.7 eV, corresponding to the Fermi energy. This energy lies 1.5 eV below the conduction band EC , which gives an estimate for the charge transfer gap. With increase of Sr concentration, the intensity of the peak near the valence band V at EV 529 eV gradually increases, and that of the peak at EC gradually decreases. At the same time the insulator-metal transition at x = 0.06 occurs smoothly (within the energy resolution ∼ 0.4 eV), which may be related to forming a metallic state due to delocalization of holes at x > 0.06. The correlation eﬀects are strong even at large doping: the UHB at x = 0.3 still has approximately 60% of its spectral weight observed at x = 0. Thus, the O 2p hole band gradually emerges under doping near the lower edge of the charge-transfer p–d gap. At larger concentrations of Sr, x ≥ 0.3, the system transfers to a state with normal metallic properties where superconductivity no longer occurs.

204

5 Electronic Properties of Cuprate Superconductors

a La2-xSrxCuO4+δ

b

V

→

→

E ⊥c

C

E ||c 0.3

0.30 0.15 0.11 0.07 0.05

La2-xSrxCuO4+δ

0.2 Intensity (arb. units)

Intensity (arb. units)

x=

0.3

0.15

0.2

0.1

0.15

0

0.1

0.02 0

0

x=0, d ~ 0 527 528 529 530 531 Energy (eV)

526

528

530

532 526 528 Energy (eV)

530

532

Fig. 5.11. O 1s absorption spectra of La2−x Srx CuO4+δ measured by (a) EELS on a polycrystal (after [1057]) and (b) polarization-dependent XAS on a single crystal (after [971]) for various contents x of Sr

The polarization dependent XAS studies by [971] on the bulk La2−x Srx CuO4+δ single crystal revealed important details of the hole distribution between diﬀerent orbitals in the CuO2 plane. The O (1s) XAS absorption spectra are shown for the polarization E ⊥ c and for E a in Fig. 5.11b and (c) for various doping from x = 0 (AF insulating state) to x = 0.3 (nonsuperconducting metallic state). These studies have conﬁrmed that doped holes mainly come into the valence band of O 2px,y symmetry (pre-edge peak at 528.5 eV). The hole concentration is roughly proportional to x up to x ≤ 0.2 (for larger x an oxygen deﬁciency, δ < 0, may occur). Due to the strong pdσ hybridization, approximately 15% of the O 2px,y states at x = 0 are in the UHB given by the pre-edge peak at 530.2 eV. With doping the intensity of this peak is decreasing, as in the polycrystalline sample, while in addition to O 2px,y states, O pz states appear in the valence band as seen in the E c polarization. The fraction of unoccupied O 2pz states from the apical O sites of the total number of empty O 2p states does not change much with doping and is about 8% for x < 0.15. Measurements of Cu 2p3/2 polarized dependent XAS have shown that most of the 3d holes, independent on the Sr concentration

5.2 Photoemission Spectroscopy

205

x, have 3d(x2 − y 2 ) symmetry. Only a low fraction of unoccupied states having symmetry Cu 3d(3z 2 − r2 ) was detected in measurements for polarization E c. With respect to the total number of unoccupied Cu 3d states in the UHB states near the Fermi level, this fraction was less than 5% for all doping range and no correlation of it with superconducting Tc was observed. The symmetry of hole states appearing under doping in Bi2 Sr2 CaCu2 O8 (Bi-2212) and YBa2 Cu3 O7 (YBCO-7), YBa2 Cu3 O6 (YBCO-6) was also studied using the EELS method by [907]. The absorption edge at O 1s and Cu 2p3/2 levels was measured on ﬁlms cut from single crystals. It is important that the EELS in transmission is not a surface-sensitive method as the PES. By taking into account the absorption dependence on the orientation of the momentum transfer q, it was possible to study diﬀerent types of orbitals. For q directed along the c-axis the transitions of the core electrons to unoccupied hole states with orbital orientation along the z-axis, namely of O 2pπ (z) or Cu 3d(3z 2 −r2 ) type, can be examined. For q lying in the (a, b) plane the states with orbitals oriented in this plane, namely of the type O 2p(x, y) or Cu 3d(x2 − y 2 ), can be probed. In Figs. 5.12 and 5.13, the EELS absorption spectra for exciting the core levels O 1s (a) and Cu 2p3/2 (b) are shown for Bi-2212 and YBCO-7, respectively [907]. The simplest case is Bi-2212. The absorption edge of O 1s for q a, b was detected at the energy 527.8 eV, which may be compared with the mean binding energy of the O 1s levels for two diﬀerent O sites EB = 528.7, 529.5 eV observed in XPS experiments. The absence of absorption spectral weight at q c between these energies (see Fig. 5.12a) indicates the absence of hole states of O 2pπ (z) type. The absorption at q a, b is related to hole states in CuO2 planes (and possibly BiO layers) with the symmetry of 2p(x, y). The nature of bonding, σ or π, cannot be found from this experiment. Investigations of absorption under excitations of the Cu 2p3/2 level for q a,b and q c (see Fig. 5.12b) demonstrate that besides the in-plane hole states, 3d(x2 − y 2 ), there is a small admixture of the order of 10–15%, of out-of-plane states, most probably of 3d(3z 2 − r2 ) symmetry. Various polarization dependent XAS studies have conﬁrmed the results of the EELS experiment (see [336]). Similar results were obtained in the EELS studies of Tl2 Ba2 CaCu2 O8 [1058] and polarization-dependent XAS measurements on Tl2 Ba2 CaCu2 O8 and Tl2 Ba2 Ca2 Cu3 O10 compounds [972], which have crystal structures close to the Bi-based compounds (see Sect. 2.5). In the case of Tl-2223 compounds with three CuO2 layers, the unoccupied O 2px,y density of states close to the Fermi level per unit cell as measured by XAS was 1.5 times higher than in Tl-2212 compound. However, the average number of doped holes per CuO2 layer was found close to the optimal doping in other p-type cuprates. Contrary to Bi-2212, the O 1s absorption edges for O pz orbitals were observed both in EELS at q c and in XAS for polarization parallel to the c-axis. But the edges were detected at higher energy of 529 eV and ascribed to the O pz

206

5 Electronic Properties of Cuprate Superconductors

b

a

Bi2Sr2CaCu2 08+x Bi2Sr2CaCu2 08+x

Cu 2p3/2

01s →

→→

→

→

q || c

INTENSITY (arb. units)

INTENSITY (arb. units)

q || a ,b

→

→→

→

→

q || a ,b

q || c

525

530 535 ENERGY (eV)

540

930

935 ENERGY (eV)

Fig. 5.12. Electron energy-loss spectra at the O (1s → 2p) (a) and Cu (2p3/2 → 3d) (b) absorption edges for the momentum transfer q parallel to the c-axis and in the a, b-plane of Bi2 Sr2 CaCu2 O8 single crystal (reprinted with permission by APS c 1989) from [907],

orbitals in the Tl-O layers according to band structure calculations (see e.g., [981]). An interpretation of absorption spectra for YBCO-7 shown in Fig. 5.13 is more complicated as both the CuO2 planes and the CuO3 chains (ribbons) contribute to the density of states near the Fermi level. The absorption O 1s spectra for q c show the absorption edge at 527.2 eV, which is lower than for q a, b near 528 eV. This may be explained by the variation of the crystal ﬁeld energy of O 2p levels for diﬀerent sites O(1), O(2), O(3), and O(4) (see Fig. 2.12). The pronounced peaks at these energies are absent in the O 1s absorption spectra for YBCO-6 which conﬁrms that the O 2p states observed at the Fermi level are created by the hole doping in YBCO-7. The contribution to the absorption spectrum at O 1s from the 2pσ orbitals for q a, b comes from 2pσ (x, y) orbitals on O(2, 3) oxygen ions in the CuO2 planes and 2pσ (y) orbitals on in-chain oxygen ions O(1). The absorption spectrum for q c is deﬁned by contribution from the 2pσ (z) orbital on the O(4) ions. If one takes into account only O 2pσ orbitals, then the ratio for the in-plane and out-of-plane intensity should be equal to Ix,y /Iz 2, which is similar to

5.2 Photoemission Spectroscopy

a

207

b Y Ba2 Cu3 07 01s

→

→→

→

→

Y Ba2 Cu3 07 Cu 2p3/2

q || c

INTENSITY (arb. units)

INTENSITY (arb. units)

q || a ,b

→

→→

→

→

q || a ,b

q || c 525

530 535 ENERGY (eV)

540

930

935 ENERGY (eV)

Fig. 5.13. Electron energy-loss spectra at the O (1s → 2p) (a) and Cu (2p3/2 → 3d) (b) absorption edges for the momentum transfer q parallel to the c-axis and in the a, b-plane of YBa2 Cu3 O7 single crystal (reprinted with permission by APS from c 1989) [907],

what has been obtained from the comparison of the absorption intensities for YBCO-7 and YBCO-6 samples at energies E < 931 eV. Based on this result, the authors concluded that the main contribution to the O 1s absorption comes from the O 2pσ hole states on O(2, 3) oxygen in the planes or O(1) oxygen in the chains, though some contribution from pπ (x, y) orbitals in the plane could not be excluded. However, the existence of holes on O 2p orbitals of the pπ (z) type perpendicular to CuO2 planes, or the 2pπ (x) orbitals perpendicular to the CuO3 ribbons, is not supported by the experimental data [907]. Absorption spectra for the Cu 2p3/2 level in YBCO-7 (Fig. 5.13b) demonstrate that an approximately equal density of holes is located on σ-orbitals 3d(x2 − y 2 ) for in-plane Cu(2), and 3d(z 2 − y 2 ) for in-chain Cu(1). A small asymmetry of the spectrum (and a shoulder for q c) relates to the transition to the state Cu 2p5 3d10 L, when the screening of the core Cu 2p5 hole due to the charge transfer from the ligand oxygen L to 3d10 states occurs. This asymmetry and the shoulder for q c disappear in the spectrum of YBCO-6,

208

5 Electronic Properties of Cuprate Superconductors

which signals the absence of these states. At the same time, in YBCO-6 the abrupt fall of intensity for q c and the appearance of a peak at EB = 934 eV, which is characteristic of Cu1+ were observed. This indicates that removal of in-chain oxygen results in Cu(1) transition to a monovalent state 3d10 with a small number of Cu2+ (3d9 ) states, mostly of the 3d(3z 2 − r2 ) type. A more detailed study of the hole occupation numbers on O(2,3) oxygen in planes and O(1,4) oxygen in chains as a function of the oxygen content in single-domain YBa2 Cu3 Ox crystals, 6 < x < 7, was performed using polarization-dependent XAS by [909]. The largest concentration of doped holes was detected on the O(1) oxygen in the chains ∼ 0.34, while only approximately 0.2 holes per CuO2 unit cell were observed in the planes in YBa2 Cu3 O7 . Under doping, the spectral weight in the valence band appears at x ∼ 6.35 and increases roughly proportional to x saturating at x = 7, while the spectral weight in the UHB decreases but remains high even at x = 7, of the order of 60% of its value at x = 6. Therefore, due to the low hole occupations in planes, the on-site correlations play an essential role for the electronic structure of the planes, whereas the high level of chain doping results in their more normal metallic conductivity. The authors also observed a strong correlation between the in-plane hole concentration and Tc . Important results concerning the hole concentration dependence on doping by oxygen O(1) sites in chains or by Ca2+ substitution for Y3+ sites were obtained by near-edge X-ray absorption ﬁne structure studies on the detwinned Y1−x Cax Ba2 Cu3 O7−y single crystals by [826]. Using the polarization dependence of the XAS, the spectra with E a, b, c axes were studied, which enabled them to investigate the doping dependence of various unoccupied oxygen orbitals in the CuO2 planes and in the chains. In Fig. 5.14, the absorption spectra of O 1s for polarization Ea (a) and of Cu (2p) for polarizations Ec (b) and Ea (c) are shown. By changing the Ca concentration, x = 0.03, 0.12, 0.23, of the oxygen-depleted samples, y = 1, it was possible to observe a smooth transition from insulator to metallic state as shown in Fig. 5.14a for the O 1s absorption intensity for the O 2px orbitals (E a) in the plane. For insulating sample at x = 0.03, y = 1 a distinct upper Hubbard band (UHB) is observed at about 930 eV as in LSCO compound (see Fig. 5.11). With increasing Ca concentration, holes are doped in the sample and the intensity of the UHB decreases while a new peak at a lower energy ∼ 528.5 eV in the valence band (VB) appears. The authors ascribe this peak to the Zhang–Rice (ZR) singlet states in the CuO2 plane (see Fig. 5.3d). Comparison of the results of Ca doping at y = 0 with the oxygen doping, y = 0.5, 0.09 at x = 0, and y = 0.09, x = 0.10 shows that the spectral weight transfer from the UHB to the VB occurs in a similar way for the two doping mechanisms. This picture was conﬁrmed by measurements of the Cu 2p XAS spectra shown in Fig. 5.14 for polarizations Ec (b) and Ea (c). Independent of doping, for the polarization Ea a strong Cu2+ peak ascribed to the Cu 3d9 → Cu 2p3d10 transition at Cu(2) 3d(x2 − y 2 ) is observed at 931.7 eV. At the same time, a shoulder due to ligand holes L in O 2p(x2 − y 2 ) orbitals at the

5.2 Photoemission Spectroscopy

b

a

4 E||a

c

UHB

0 x = 0.03, y = 1.0 x = 0.23, y = 1.0 x = 0.00, y = 1.5 x = 0.00, y = 0.09 x = 0.10, y = 0.09

E||a 12

2 x = 0.03, y = 1.0 x = 0.12, y = 1.0 x = 0.23, y = 1.0 x = 0.00, y = 0.5 x = 0.00, y = 0.09 x = 0.10, y = 0.09

1

528

530 Energy (eV)

532

σ (Mbarn / unit cell)

3

0

E||c

2

ZR

4

σ (Mbarn / unit cell)

209

10

Ligand holes

8 6

Cu2+

4 Cu1+ 2 0

930

932 934 Energy (eV)

936

Fig. 5.14. Near-edge absorption spectra of the detwinned Y1−x Cax Ba2 Cu3 O7−y single crystals for O (1s) for polarization Ea (a) and Cu (2p) for polarizations Ec (b) and Ea (c) for various oxygen content, 7 − y, and Ca concentrations, x c 1998) (reprinted with permission by APS from [826],

transition Cu 3d9 L → Cu 2p3d10 L appears with doping. This proves that doped holes appear at ZR singlet states. The monovalent Cu1+ peak at about 934 eV decreases with oxygen doping (lower curves) but remains unaﬀected at Ca doping (two upper curves) as also revealed by the spectra for Ec in the (b) panel. On the basis of these observations, and also on detailed studies of other XAS spectra, the authors concluded that Ca doping introduces holes exclusively in the CuO2 planes, while the oxygen doping in the chains increases the hole concentration at the O(1) and apex O(4) sites in chains. As discussed in Sect. 5.1.2, substitution yttrium by Pr ions in YBCO crystals exceptionally strongly suppresses Tc in comparison with other rare-earth ions: at Pr concentration x = 0.61 the system becomes nonmetallic and nonsuperconducting, while the AF ordering of Pr magnetic moments occurs with the maximum value of TNR = 17 K at x = 1, much higher than in other rare earth compounds (see Sect. 3.2.4). At the same time, as shown in Fig. 5.4, this suppression can be compensated to some extent by hole doping with Ca. To explain the Tc suppression by Pr substitution in YBCO crystals, several models have been proposed based on diﬀerent experimental observations (see e.g., [825], and references therein). Frequently used is the hole depletion model which suggests the reduction of hole concentration in CuO2 planes due to the Pr valence larger than 3+ (a mixture of Pr3+ and Pr4+ oxidation states). The model explains the nonmetallic behavior at large Pr concentration, which can

210

5 Electronic Properties of Cuprate Superconductors

be compensated by Ca substitution. A magnetic pair-breaking eﬀect or hole localization can be suggested if one takes into account the strong hybridization of the localized Pr 4f electrons with the O (2p) states in the planes. Also some charge redistribution between the planes and chains under Pr doping was suggested. There are several theoretical models in which the electronic structure of PrBaCu3 O7 compound was calculated by taking into account the hybridization Pr 4f electrons with 3d–2p states in the planes. As discussed above, a smooth transition from insulating to metallic and superconducting state occurs when doped holes ﬁll in the valence band of the pdσ ZR singlet states in the CuO2 planes, which is accompanied by the spectral weight transfer from the UHB. To explain a hole depletion under Pr doping, it is natural to suggest that a new band, caused by hybridization of Pr 4f electrons with O 2p states in the planes, appears which grabs holes from the ZR singlet states. Fehrenbacher and Rice (FR) [323] have proposed such a model where the formation of a low energy hole states was suggested as a mixture of 4f 1 and 4f 2 L conﬁgurations (FR model). Here, the Pr ligand states L are O 2pπ planar orbitals with z(x2 − y 2 ) symmetry on the eight oxygen sites pointing to the central Pr ion. These O 2pπ planar orbitals being perpendicular to the Cu–O bonds can be visualized as rotated by an angle 45◦ out of plane [825]. The hybridized Pr 4fz(x− y2 ) –O 2pπ states (FR band) may be energetically more favorable than the singlet pdσ states for speciﬁc values of the 4f electron energy level and the hybridization parameter tpf . In that case, a transfer of holes from the ZR singlet band to the planar O 2pπ orbitals in the FR band will result in a nonmetallic and nonsuperconducting state. The holes in the chains were assumed to be localized due to disorder and their density was kept constant, nch ≈ 0.5. Instead of the localized FR band, Liechtenstein and Mazin [720] have proposed a model with a dispersive band (LM model), which was studied within LDA+U calculations for ABa2 Cu3 O7 compounds (A = Y, Pr, Nd). The Hubbard parameter U was introduced to take into account the local Coulomb correlation on the rare earth (RE) sites for RE = Nd and Pr. Depending on the position of the RE 4f energy level with respect to the Fermi level, the system may be metallic or insulating. Within a simple three-band tight-binding model for O 2p(x, y) and 4f orbitals, the authors could explain a smooth hole transfer from the ZR singlet band to the FR band and describe a gradual transition to the insulating state with RE doping. In the case of Nd, the additional FR band remains completely ﬁlled, while for Pr this band crosses the Fermi level grabbing holes from the ZR singlet band. To elucidate this problem and to decide among various models, Merz et al. [825] studied the polarization-dependent O 1s near-edge X-ray absorption of the Prx Y1−x Ba2 Cu3 O7−y detwinned single crystals. In Fig. 5.15, the absorption spectra for the polarizations Ea are compared for the oxygenated (y ≈ 0.1) samples at various Pr contents, x = 0.8, 0.4, 0.0 (a), and the oxygen depleted (y ≈ 0.9) and the oxygenated (y ≈ 0.1) samples at large Pr doping, x = 0.8 (b). The pure crystal (x = 0) is a superconductor with

5.2 Photoemission Spectroscopy

a 5

b

Ella

5

Ella

4 σ (Mbarn/unit cell)

σ (Mbarn/unit cell)

4

211

UHB ZR

3

2

UHB

3

2 FR 1

1

0

526

530 532 528 Photon energy (eV)

534

0

526

532 528 530 Photon energy (eV)

534

Fig. 5.15. O(1s) near-edge absorption spectra for polarizations Ea of the detwinned single crystals (a) Prx Y1−x Ba2 Cu3 O6.91 at various Pr content, x = 0.8 (closed circles), x = 0.4 (open crossed diamonds), and x = 0 (open squares), and (b) Pr0.8 Y0.2 Ba2 Cu3 O7−y at y ≈ 0.1 (closed circles), y = 0.95 (open circles), and the diﬀerence spectrum (open triangles) revealing the FR band (after [825])

Tc 93 K, while the crystal with x = 0.8 is an antiferromagnetic insulator. At x = 0.4, the compound is metallic and still superconducting with Tc ≤ 15 K. In the pure crystal, the well pronounced ﬁrst peak at 528.5 eV with the width of ΔE 1.2 eV is attributed to the ZR singlet state. With the Pr content increase, the intensity of the ZR peak is strongly reduced, while the intensity of the second peak at 529.5 eV ascribed to the UHB increases. The same weight transfer from the ZR singlet state to the UHB was observed for pure YBCO crystals with decreasing oxygen content as shown in Fig. 5.14a. The analysis of the peak intensity revealed that the number of holes on both the O(2) and the O(3) sites per CuO2 unit cell decreases from na 0.2 in the pure sample to na 0.15 (0.12) for Pr content x = 0.4 (0.8), respectively. This means that the hole concentration in the ZR band decreases with Pr doping and the Fermi level moves up to the top of the hole ZR band. Studies of the absorption spectra for Eb and the diﬀerence spectra, Eb − Ea, which gives approximately the contribution from the O(1) 2py orbitals in the chains, have shown a small decrease of hole occupation numbers in the chains with Pr doping, from nchain 0.24 in the pure sample to nchain 0.17 for the sample with Pr content x = 0.8. To conﬁrm the charge transfer from the ZR singlet pdσ band to FR states with Pr doping, a comparison of Ea spectra for oxygen-deﬁcient and oxygenrich Pr0.8 Y0.2 Ba2 Cu3 O7−y samples was performed as shown in Fig. 5.15b. Both spectra reveal a strong UHB peak at 529.5 eV, which means that the

212

5 Electronic Properties of Cuprate Superconductors

concentration of Cu 3d(x2 −y 2 ) holes in the UHB in Pr reach samples does not depend on the oxidation state. Therefore, under Pr doping, the hole ZR singlet states in the oxygen-rich sample, which in pure crystal would capture the 3d(x2 − y 2 ) states from the UHB, are pushed out keeping the UHB unaﬀected. They should move to some other more energetically favorable states on the low-energy side of the UHB. The diﬀerence spectrum denoted as FR peak in Fig. 5.15b demonstrates the spectral density of these states which the authors ascribe to the FR band. After more detailed studies of the peak intensities at diﬀerent polarizations and taking into account the structural transformations under Pr doping of YBCO crystals, the authors came to the following conclusions. The models involving only hole depletion or charge transfer between planes and chains are incompatible with the experimental data. However, the data support the picture based on the FR and LM approaches of a Pr 4f -O(2, 3) 2pπ hybridization. The best ﬁtting was obtained for an intermediate angle of O(2,3) 2pπ orbital rotation out of plane, 20–25◦, in comparison with 45◦ in FR model and purely planar (0◦ ) LM model. The data for Pr0.8 Y0.2 Ba2 Cu3 O6.91 sample suggest the following distribution of doped hole concentration: about napex + nchain ≈ 0.5 holes occupy chain sites, nPr ≈ 0.2 holes transferred to the Pr ion, and nFR ≈ 0.3 holes are moved into the FR states. Thus, instead of the magnetic scattering suggested by Neumeier et al. [891] in Fig. 5.4, a hole transition into the FR states can be involved, which cannot be recovered by Ca2+ substitution. These studies have conﬁrmed that in the electronic structure of cuprates the most important role is played by the ZR pdσ singlet band which determines the metallic and superconducting properties of the cuprate superconductors. A transfer of states from the singlet band to some other states, even close to the Fermi level, results in the transition to a nonmetallic and nonsuperconducting phase. While the picture of charge distribution under hole doping as described above seems to be well established, results of electron doping in R2−x Cex CuO4 cuprates are more controversial. EELS studies by Alexander et al. [25] of absorption spectra of R2−x Mx CuO4−δ (R = Pr, Nd, Sm; M = Ce, Th) systems revealed that in all cases the O 1s absorption edges observed at E 528.5 eV were almost the same and doping independent. These peaks were assigned to contribution of O 2p hole states to the UHB, as in LSCO compounds. While their admixture in the UHB was doping independent, the intensity of the Cu 2p3/2 loss peak under doping by Th4+ ions decreased by approximately 14% at x = 0.15, indicating a fall of 3d holes concentration under electron doping. The XPS studies by Suzuki et al. [1206] of eﬀects of Ce substitution in Nd2−x Cex CuO4−δ polycrystalline samples have shown that the Fermi edge is formed by the electron doping into Cu 3d10 states. As suggested by Suzuki et al. [1206], inconsistencies among the PES studies of Nd-Ce systems could be explained by diﬀerent methods of the sample surface preparation. Polarization-dependent XAS studies of the O 1s and Cu 2p edges on single crystals R2−x Cex CuO4−δ (R = Nd, Sm) by Pellegrin et al. [971] conﬁrmed the

5.2 Photoemission Spectroscopy

213

EELS results. The O 1s absorption edges were doping independent and the polarization analysis indicated that most of the O 2p states have in-plane 2px,y symmetry. The Cu 2p3/2 absorption edges demonstrated that Cu 3d holes have mostly in-plane 3d(x2 − y 2 ) symmetry as O 2p holes. Upon Ce-doping in Nd-system the intensity of these peaks decreased pointing to a ﬁlling of the Cu 3d states in the UHB. The polarization-dependent studies by Pellegrin et al. [971] of the “inﬁnite layer” compound Ca0.86 Sr0.14 CuO2 (see Introduction to Chap. 2) have shown similar results: a peak at 528.3 eV in the O 1s absorption edge for the O 2px,y state in the UHB and the Cu 2p3/2 absorption edge for Cu 3d(x2 − y 2 ) holes. XAS studies of the ladder compounds were discussed in Sect. 2.2.3. EELS studies of low-dimensional cuprates revealed a more complicated picture of electronic excitations in these strongly correlated systems (see e.g., [293, 623]). Resume Experimental studies with the aid of high-energy electron spectroscopy (XPS, XAS, EELS) have elucidated several important features of the electronic structure of high-Tc superconductors. 1. The observation of strong Coulomb correlations in the XPS Cu2p spectra, characterized by the Coulomb energy Ud = 8–10 eV, explained the insulating state of undoped compounds and the band splitting into the upper Hubbard band (UHB) and the lower Hubbard band (LHB). 2. The studies of O1s absorption spectra disclosed the charge-transfer character of insulating state. At p-type doping the holes ﬁll in the top of the O2p valence band, which appears between the UHB and LHB: Ud > Δpd = p − d as shown in Fig. 5.3d. Results on n-doped Nd2−x Cex CuO4 cuprates point to electron ﬁlling of the bottom of the Cu3d UHB. 3. A spectral weight transfer from UHB at hole doping to the O2p band revealed the correlation character of the valence states, which comprise the pdσ ZR singlet band as shown in Fig. 5.3c. Strong correlations remain important even in the overdoped systems since the spectral weight of the UHB, of approximately 60% of its value in the undoped state, is clearly seen there. As studies of Pr-YBCO compounds have shown, grabbing holes from this band results in a nonmetallic and nonsuperconducting state. 4. The in-plane character of the pdσ hole states was proved by the polarization analysis of the EELS and XAS spectra, which showed only a small fraction of the Opz and Cu3d3z 2 −r 2 states in the O 2p and Cu3d spectra for the CuO2 planes. 5. The experimental studies presented were also important in ﬁxing the parameters of the pdσ eﬀective model Hamiltonian. A more detail picture of how the Fermi surface evolves from the insulating phase was obtained with the aid of ARPES considered below.

214

5 Electronic Properties of Cuprate Superconductors

5.2.2 Angle-Resolved Photoemission Spectroscopy The angle-resolved photoemission spectroscopy (ARPES) plays a major role in the study of the electronic structure of the cuprate superconductors since it provides direct information on the single-particle excitations. Owing to improved sample quality and recently achieved high resolution both in energy (5–10 meV) and in scattering angles (∼ 0.1◦ ), within this method valuable information concerning the Fermi surface (FS) mapping in the momentum space, the electronic dispersion curves close to the FS, many-body eﬀects in the electronic excitations, and values of the superconducting gap and the pseudogap have been obtained. A weak point in ARPES studies is the surface sensitivity of the method since the electron escape depth for ultraviolet photon energies used in ARPES amounts 5–10 ˚ A, which is about the size of one unit cell in copper oxides. Therefore, early ARPES study was performed on samples with low-quality surfaces frequently produced unreliable results. In this respect, the Bi-2212 single crystals provide the best surfaces when cleaved between the Bi-O layers. This explains why the majority of high-precision measurements by the ARPES method were performed on Bi-based systems. Later on, the Na-doped cupric oxycloride Ca2−x Nax CuO2 Cl2 (Na-CCOC)compound was extensively studied by the ARPES method also. The Na-CCOC crystal, similar to Bi-2212, could be easily cleaved between the weakly coupled double Ca–Cl layers producing a clean surface [461]. A number of important results were obtained for the LSCO compound which has the advantage of covering the whole range of the phase diagram in a single system from the insulator to the overdoped nonsuperconducting state. The YBCO compounds, otherwise the most extensively studied by various techniques, are diﬃcult for the ARPES investigations due to a poor quality of the cleaved surface, surface states, and natural twinning. Only recently reliable results for ARPES spectra have been achieved (see e.g., [164] and references therein). Below we discuss several examples of ARPES studies which are important for understanding the nature of the high-temperature superconductivity of cuprates. There are several extensive reviews of recent results in the ﬁeld where the reader can ﬁnd more detailed information (see e.g., [196, 271, 316], and references therein). According to (5.12), the photoemission intensity is proportional to the single-particle spectral function A(k, ω) for a quasiparticle (QP) with the momentum ¯hk and the energy h ¯ ω. This is deﬁned by the imaginary part of the retarded single-particle Green function, A(k, ω) = −(1/π) Im G(k, ω +iδ), as follows (see Appendix A.1): A(k, ω) =

1 − Σ (k, ω) , π [ω − (k) − Σ (k, ω)]2 + [Σ (k, ω)]2

(5.14)

5.2 Photoemission Spectroscopy EDC

215

MDC

I(k,w)

b. units)

2000

Intensity (a

1500 1000 500 0

Bi

0.4 0.2 0.0 Binding energy (eV)

nd 0.4 in 0.2 g en er 0.0 gy (e V)

0.2 0.3 –1 ) 0.5 (Å m 0.6 ntu e 0.7 om 0.3 0.4 0.5 0.6 0.7 M Momentum (Å–1) 0.4

Fig. 5.16. Middle panel: photocurrent I(k, ω) vs. binding energy and momentum along the Γ (0, 0) → (π, π) direction at 120 K in an underdoped Pb-Bi-2212 crystal. Left panel: EDC at ﬁxed momentum values, right panel: MDC at ﬁxed energy values. Data courtesy of Borisenko et al. [160]

where (k) denotes the bare (band) dispersion of QP with (k0F ) = 0. The self-energy Σ(k, ω + iδ) = Re Σ(k, ω) + i ImΣ(k, ω) ≡ Σ (k, ω) + i Σ (k, ω) takes into account the many-body eﬀects of the QP interactions. Its real part, Re Σ(k, ω), describes the renormalization of the bare dispersion, while the imaginary part, Γ (k, ω) = −ImΣ(k, ω) > 0, characterizes the scattering rate of the QP. For a ﬁxed wave vector, k = q, the function A(q, ω) determines the energy distribution curve (EDC) Iq (ω) and for a ﬁxed energy, ω = E, the function A(k, E) determines the momentum distribution curve (MDC) IE (k). Figure 5.16 demonstrates the photocurrent intensity I(k, ω) as a function of the binding energy and the momentum transfer along the symmetry direction Γ (0, 0) → (π, π) at 120 K in an underdoped Pb-Bi-2212 crystal [160]. In the left panel, the EDC as parallel intensity proﬁles corresponding to ﬁxed momentum values and in the right panel the MDC as parallel intensity proﬁles corresponding to ﬁxed energy values are shown. In the vicinity of the FS, which is given by the equation ω = (kF ) + Σ (kF , 0) = 0 where Σ (kF , 0) determines the renormalization of the chemical potential, the real part of the self-energy can be approximated as Σ (k, ω) − Σ (k, 0) ω (∂Σ /∂ω)|ω=0 = −ω λk . This enables the transformation of the EDC given by the spectral function (5.14) into a sum consisting of a coherent

216

5 Electronic Properties of Cuprate Superconductors

QP part and an incoherent background caused by many-particle excitations: A(k, ω) Zk

γ(k, ω)/π + Ainc (ω), [ω − η(k)]2 + [γ(k, ω)]2

(5.15)

where the QP weight Zk = (1 + λk )−1 , the renormalized QP energy η(k) = Zk [(k) + Σ (k, 0)], and the scattering rate γ(k, ω) = Zk Γ (k, ω)). Here we should keep the frequency dependence in the scattering rate γ(k, ω) to satisfy the condition γ(kF , ω = 0) = 0 and limω→∞ γ(k, ω) → 0 which should be hold at zero temperature for a pure system. In this limit, the spectral function representation (5.14) as the sum of coherent and incoherent parts usually used in ARPES experiments should be taken with caution: at ﬁrst one should take γ(kF , ω → 0) → 0 which gives for the coherent part Acoh (k, ω) Zk δ[ω − η(k)] and then consider the incoherent contribution. Quite a general representation for the spectral function (5.14) can be obtained by using the continued fraction method [443]. For a suﬃciently large quasiparticle life-time, 1/τk = γ(k) η(k), the function (5.15) has a maximum at ω = η(k) which deﬁnes the energy of the QP excitations above the FS at η(k) > 0 and below the FS at η(k) < 0. The intensity of this maximum determines the QP weight Zk . Usually in ARPES experiments in the cuprate superconductors, the EDC shows a maximum at wave vectors close to kF superimposed on a broad background in the range of several tenth of electron volt as seen in the left panel of Fig. 5.16. The background has a complicated character and is caused by various processes such as multiple scattering, 3D eﬀects, surface eﬀects, etc. A certain contribution can be related to incoherent excitations – the “shakeoﬀ spectrum” which arises with the abrupt removal of an electron at the photoionization process [1101]. Generally speaking, we can refer to a QP excitation when its energy is larger than the scattering rate, |η(k)| ≥ γ(k). A clear evidence of QP excitations in cuprate superconductors was obtained in laser-based ARPES studies of the near-optimally doped Bi2 Sr2 CaCu2 O8+δ crystal at low temperatures where a signiﬁcant reduction in the background was observed [636]. The QP peaks also appear in the superconducting state where the QP scattering is suppressed due to an opening of a superconducting gap in the excitation spectrum. While the EDC given by (5.14) at a ﬁxed k has a complicated shape as discussed above, the MDC (5.14) at ﬁxed ω reveals a simpler structure. It shows a maximum at km (ω) determined by the equation: ω − (km ) − Σ (km , ω) = 0. Close to the FS, the dispersion can be linearized: (km ) − (k) = vα (km − k)α where vα = ∂(k)/∂kα is the bare Fermi velocity in the α-direction. Therefore, in the denominator of (5.14) we can write ω − (k) − Σ (k, ω) = vα (km − k)α + Σ (km , ω) − Σ (k, ω) vα (km − k)α if we neglect k-dependence of Σ (k, ω). This approximation results in a Loretzian curve for the spectral function (5.14) at ﬁxed energy:

5.2 Photoemission Spectroscopy

A(k, ω)MDC

W (ω)/π 1 . v (km − k)2 + W (ω)2

217

(5.16)

where the half width at half maximum W (ω) = −(1/v) Σ (km , ω). The positions of the maxima and their widths determine the dispersion km (ω) and the scattering rates W (ω) of single-particle excitations, respectively. Using the dispersion relations for the self-energy (the Kramers–Kronig relations), the real part of the self-energy Σ (ω) can be calculated from the the imaginary part Σ (ω) = −v W (ω) as discussed below. The Loretzian proﬁle of the MDC is clearly seen in the right panel of Fig. 5.16, which conﬁrms a weak k-dependence of the imaginary part of the self-energy. Many important results in ARPES studies concerning the QP dispersion and the gap function were obtained in studies of the spectral function in the superconducting state. To describe the superconducting state, a 2 × 2 matrix Green function (GF) for the electron-hole and the pair excitations should be considered. The electron GF can be written as follow (see Appendix A.2): ω + η(k, −ω) , [ω + η(k, −ω)][ω − η(k, ω)] − | Φ(k, ω) |2

Gel (k, ω) =

(5.17)

where the renormalized electron energy η(k, ω) = (k) + Σ(k, ω) and the pairing function Φ(k, ω) are deﬁned by the corresponding self-energy functions. By introducing odd and even parts of the self-energy Σ(k, ω) 1 1 Z˜k (ω) = 1 − [Σ(k, ω) − Σ(k, −ω)] = , 2ω Zk (ω) ξk (ω) = [Σ(k, ω) + Σ(k, −ω)],

(5.18)

we write the representation for the GF in the superconducting state as Gel (k, ω) =

ω Z˜k (ω) + ((k) + ξk (ω)) . (ω Z˜k (ω))2 − ((k) + ξk (ω))2 − | Φ(k, ω) |2

(5.19)

The spectral function in the superconducting state determined by the imaginary part of the single-electron GF (5.19) has a complicated form since the imaginary parts in the renormalization functions in (5.18) and the pairing function Φ(k, ω) should be taken into account. Close to the Fermi level, a quasiparticle (QP) approximation for the GF (5.19) can be used. Then the spectral function in terms of the Bogoliubov QPs representing coherent particle-hole excitations can be written as 1 vk2 Γk /π u2k Γk /π Asc (k, ω) = − ImGel (k, ω) Zk + , π (ω − Ek )2 + Γk2 (ω + Ek )2 + Γk2 Ek =

η˜(k)2

+

|Δ(k)|2 ,

u2k

= 1−

vk2

1 = 2

η˜k . 1+ Ek

(5.20)

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5 Electronic Properties of Cuprate Superconductors

Here, the renormalized electron dispersion η˜(k) = ((k) + ξk )Zk and the gap function Δ(k) = Φ(k)Zk with Zk = Z˜k−1 = (1 + λk )−1 are determined close to the the FS at ω 0, and Γk is an eﬀective scattering rate for the QPs. The coherence factors uk = u−k and vk = −v−k determine the particlehole mixing in the Bogoliubov QPs: α†σk = uk a†σk + v−k a−σ,−k . The two peaks in (5.20) determine the hole excitation for ω = Ek > 0 proportional to u2k that is large above the FS, η˜(k) > 0, and the electron excitation for ω = −Ek < 0 proportional to vk2 that is large below the FS, η˜(k) < 0. At the Fermi level EkF = |Δ(kF )| and the spectral function (5.20) shows two maxima at the gap energy ω = ±Δ(kF ). Since the intensity of the ARPES spectra is proportional to the spectral function (5.20) multiplied by the Fermi function f (ω) = [exp(ω/kB T ) + 1]−1 , at low temperatures the peak at ω = −Δ(kF ) proportional to vk2 gives a signiﬁcant contribution to the EDC, while the hole contribution ∝ u2k is much weaker. Whereas in the normal state the peak of the spectral function (5.15) at QP energy ω = ηk disperses toward the Fermi energy with increasing k and crosses it, in the superconducting state the spectral peak in (5.20) at ω = −Ek for k > kF disperses back, moving away from EF with decreasing intensity. The second peak at ω = Ek of lower intensity disperses in the opposite direction. This peculiar Bogoliubov QPs dispersion has been observed in the superconducting cuprates also (see e.g., [193, 807]), though the EDC and MDC have revealed a more complicated energy and wave vector dependence than follows from the QP approximation (5.20). The many-body eﬀects and strong QP renormalization appear to be important and therefore the general representation for the GF (5.19) is more appropriate for the data analysis. A pseudogap in QP spectra persisting in the underdoped cuprates further hampers ARPES data analysis in the superconducting state. Fermi Surface The low-energy electron excitations close to the Fermi surface (FS) determine many physical properties of metals and therefore the knowledge of the FS topology is very important for the characterization of the copper-oxide materials. Numerous investigations were devoted to studies of the shape and the doping dependence of the FS of cuprate superconductors. The corroboration of the experimental evidence by means of diﬀerent techniques, ARPES, positron two-dimensional angular correlation of annihilation radiation (2D-ACAR), angular magnetoresistance oscillations (AMRO), conﬁrms the existence of an FS in cuprate superconductors in agreement with theoretical band structure calculations. The recent improvement in the performance of ARPES measurements, and especially in getting high angle resolution which enables to provide high quality momentum distribution maps, were decisive in clarifying many controversial results of early experiments.

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Bi2 Sr2 CaCu2 O8+δ System The most detailed studies of the excitation spectra near the Fermi surface have been performed for single crystals of Bi2 Sr2 CaCu2 O8 , which have a stable surface. The quasi-two dimensional nature of the electronic spectrum in Bi-based compounds, due to a weak dispersion along the kz -direction, substantially simpliﬁes the measurements of the photoemission spectra as a function of the in-plane momentum components (see [99]). However, the study of the FS of the Bi-2212 compounds is complicated by several secondary features besides those related to primary electronic structure. These are the diﬀraction replicas or umklapp bands originating from the incommensurate modulation of the BiO layers, the shadow bands due to the superstructure formation, and the bilayer band splitting caused by the two CuO2 planes in the Bi-2212. The incommensurate modulation of the BiO layers can be suppressed using Pbdoped Bi-2212 samples as discussed in Sect. 2.5. To disentangle the primary contributions to the electronic spectra from secondary features, the energy and polarization dependency of the ARPES matrix element |M (k, ω)| in (5.12) was exploited as discussed, e.g., by Asensio et al. [82]. Below we consider the most important results concerning these issues. A detailed and systematic ARPES investigation of the doping dependence of the normal-state FS of the Pb-Bi-2212 crystals were performed by Kordyuk et al. [637]. Six samples from the underdoped (UD, Tc = 76 K) to the overdoped (OD, Tc = 69 K) crystals were studied (for the hole concentrations x ∼ 0.1 − 0.22 as estimated from the empirical Tc (x) relation). Figure 5.17 shows the momentum distribution maps (MDM) for these two samples in (a) and (b) plots. The plot (c) is the same as (b) with the ﬁrst Brillouin zone (BZ) shown by the white square and the main and the shadow Fermi surfaces emphasized by the black and white lines, respectively [628]. The maxima of the MDM (the bright regions), which reﬂect the maxima of the MDCs measured at the Fermi energy, correspond to the FS. In addition to the main FS represented by the large hole-like rounded square or a “barrel” around (π, π) point in the BZ, the much weaker shadow bands shifted from the main FS by the wave vector (π, π) (the white arrows in the plot c) are also depicted. With increasing hole concentration x, the size of the FS barrels increases as can be seen in the decrease of the interbarrel separation at the (π, 0) point in the (b) plot in comparison with the (a) plot. The estimation shows that the two-dimensional FS area scales with (1 + x) in the region 0.1 < x < 0.2. The bilayer splitting is not resolved in the MDM in Fig. 5.17 but reveals as the doping independent width of the FS curves being approximately proportional to sin2 ϕ where ϕ is the angle away from the nodal direction. This dependence agrees with the bilayer energy splitting predicted in the band structure calculations (see below). With doping, the FS shape changes from being quite rounded in the UD sample (a) to the form of a square with rounded corners in the OD sample (b) but no change in the FS topology was observed within the doping range studied.

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Fig. 5.17. (color online) The normal-state (T = 300 K) Fermi surface map of Bi(Pb)-2212: underdoped (UD, Tc = 76 K) (a), overdoped (OD, Tc = 69 K) (b) (after [637]). Panel (c) is the same as (b) with the ﬁrst Brillouin zone shown by the white square and the main and the shadow Fermi surfaces emphasized by the black and white lines, respectively (after [628])

The FS shape changes in Fig. 5.17 are accompanied by the MDM intensity variation from the maximum values in the nodal directions (0, 0) → (π, π) at the FS barrel in the UD sample to a uniform distribution over the FS with faint maxima at antinodal (π, 0) points in the OD sample. This reﬂects the variation of the density of electronic states (DOS) at diﬀerent parts in the k-space of the FS with doping. We discuss this phenomenon later in more detail in connection with the ARPES experiments on LSCO-type compounds. Bilayer Band Splitting in Bi-2212 One of the central issues of the ARPES study of the Fermi surface topology in the Bi-2212 crystals was the detection of the bilayer band splitting caused by the intracell coupling t⊥ between the two adjacent CuO2 layers in a same unit cell. The band structure calculations predict the occurrence of an anisotropic bilayer energy splitting in the Bi-2212 as well as in YBCO crystals described approximately by the function 2t⊥ (k) = (t⊥ /2)(cos kx − cos ky )2 , which reaches the maximum value 2t⊥ 300 meV at the antinodal point (π, 0) (see e.g., [57]). Therefore, a direct observation of the bilayer splitting in the ARPES experiments on the overdoped Bi-2212 compounds by Feng et al. [329] and Chuang et al. [233] was very important in elucidating several contradictory statements concerning the origin of the diﬀerent shapes of the EDC along the nodal and the antinodal directions. In the superconducting state, the EDC near the (π, 0) point was characterized by a sharp peak close to the FS, followed at larger binding energy by a dip and a broad hump, while in the antinodal direction only one sharp peak was observed.

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This peak-dip-hump (PDH) structure was ascribed to the many-body eﬀects arising from the interaction of the electrons with a collective mode of wave vector (π, π), most probably of magnetic origin (see e.g., [194] and references therein). The observation of the bilayer splitting has questioned this explanation suggesting that the PDH structure is due to the bilayer splitting. This suggestion was conﬁrmed in the high-resolution ARPES measurements in a broad range of incident photon energies h ¯ ω = 18–65 eV by Kordyuk et al. [638]. They have shown that the two features in the PDH structure are in fact two diﬀerent single-particle spectral functions: the sharp peak is related to the antibonding band, while the broad hump is formed by the bonding band. The intensity of these peaks strongly depends on the incident photon energy, which results in producing various shapes of the observed EDCs, at some energies having no PDH structure at all. The energy diﬀerence between these two bands was found as about 140 meV in agreement with the band splitting 2t⊥ 110 meV observed in the normal state by Chuang et al. [233]. These values are smaller than in the band structure calculations, ∼ 300 meV, but larger than the value of ∼ 40 meV predicted for the bilayer Hubbard model by Liechtenstein et al. [721]. A detailed study of the doping dependence of the bilayer splitting in the Bi-2212 crystals was performed by Chuang et al. [234] by exploiting the strong dependence of the ARPES matrix element for the bonding (B) and antibonding (A) bands on the energy and the polarization of the incident photons (see [1084] and references therein). It was shown that the B and A bands posses similar ARPES cross sections at the (π, 0) point for photons with energy around 22 eV, while at a special energy of 47 eV the cross section for the A band is much larger than that for the B band. This enhancement of the A band cross section was helpful to deconvolute the contributions of the two bands even in the underdoped samples where the broadening of the EDC precludes direct measurements of the bilayer splitting. Figure 5.18 in the left panel shows ARPES spectra for two diﬀerent photon energies (In the original paper: 22 eV – red line and 47 eV – blue line) for the EDCs (a), (b), (c) at the (π, 0) point and the MDCs (d), (e), (f) at EF for the wave vectors (π, kx ) for: (a), (d) – the overdoped (OD, Tc = 55 K), (b), (e) – the optimally doped (OpD, Tc = 91 K), and (c), (f) – the underdoped (UD, Tc = 78 K) Bi-2212 single crystals. At the top, the FS plot in the ﬁrst Brillouin zone is shown. While for the OD sample two bands labeled A (for antibonding) and B (for bonding) are clearly seen in the EDCs and MDCs, for the OpD and UD samples only broad peaks are observed. Nevertheless, the peaks at 47 eV (mostly due to A band) are much sharper than at 22 eV, which suggests the presence of two bands in the spectra at 22 eV, which are formed by the contributions of the both bands. These two bands are clearly seen in the right panel of Fig. 5.18 where the results of ﬁtting various spectra are shown. The normal-state dispersion curves ﬁtted the 47 eV data (open circles – from EDCs, full circles – from MDCs)

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Fig. 5.18. (color online) Left panel : ARPES spectra for two diﬀerent photon energies (22 eV – upper curve and 47 eV – lower curve) for the EDC at the (π, 0) point for: (a) – the overdroped (OD), (b) – the optimally doped (OpD), and (c) – the underdoped (UD) Bi-2212 single crystals. Right panel : the corresponding normalstate dispersion curves ﬁtted the 47 eV data for the antibonding (A) bands (upper curves) and the bonding (B) bands (after [234])

are plotted for the y A bands (upper curves) and the bonding (B) bands (lower curves) for OD (d), OpD (e), and UD (f) samples, respectively. The authors managed to ﬁt the dispersion deduced from the EDCs and the MDCs (which usually give diﬀerent curves) by the same parabolas with the energy separation of ∼ 70 meV (at the (kx , ky ) ≈ (0.15π, 1.27π) point for the OD sample). They were only shifted in energy depending on doping: 30 meV shift when going from OD to OpD sample and further 10 meV shift from OpD to UD. The intracell hopping parameter deduced from this ﬁt t⊥ ≈ 57 meV was independent of doping and close to previous measurements. Due to the bilayer splitting, the antibonding band has its van Hove singularity (vHs) at the (0, π) point in Fig. 5.18g very close to the Fermi energy and for a strong overdoping the vHs may cross the Fermi level changing a holelike topology to an electronlike. In several publications, it was suggested that two FS pieces exist: one has a holelike topology (the bonding band), while the other one resembles an electronlike topology (the antibonding band) close to the vHs at the (π, 0) point (see e.g., [148]). Both bands are holelike at optimal doping with the vHs being situated quite below the Fermi surface, around 70(125) meV in Fig. 5.18h for the A (B) band. Studies of the single layer overdoped Bi2 Sr2 CuO6 compounds by Sato et al. [1096] indicated a holelike FS centered at the (π, π) point,

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while a transition to the electronlike topology in a strongly overdoped sample (Tc < 2 K) was reported by Takeuchi et al. [1216]. The detailed study of the bilayer splitting and its observation in the OpD and UD Bi-2212 compounds were very important in proving the coherent character of the quasiparticles in bilayer cuprates. However, Chuang et al. [234] emphasize that the large intracell hopping parameter t⊥ controls only the bilayer splitting while the c-axis dispersion depends on the much smaller intercell hopping parameter t⊥ . The latter restricts electron propagation in the c-axis direction and the corresponding ρc resistivity which may be incoherent if the scattering rate Γc t⊥ while the bilayer hopping is coherent at t⊥ Γc . Further ARPES investigations of the Fermi surface maps with high momentum resolution by Kordyuk et al. [640] revealed a bilayer splitting in the nodal direction of Bi-2212 as well. The measurements were performed at T = 25 K below Tc for various OD, OpD and UD Bi-2212, and Pb-Bi2212 samples. The bilayer splitting manifested itself in the asymmetric MDCs in Bi-2212 samples in contrast to the single-layer Bi-2201 compound where the line shape of the MDCs was symmetric and excitation energy independent. Since in the nodal direction the superconducting gap vanishes, it was possible to extract from the MDCs the dispersion curves η(k) vF (k − kF ) close to the Fermi energy for both the bonding and the antibonding bands. The estimated band splitting ∼ 48 meV (for the Fermi velocity vF = 4.0 eV ˚ A) was in agreement with the electronic band structure calculations where the interplane hopping integral tpp between O2pσ orbitals as well as transfer via Ca3d orbitals were taken into account. A smaller nodal bilayer splitting ∼ 14 meV (for the Fermi velocity vF = 1.9 eV ˚ A) was found in OpD Bi-2212 crystal (Tc = 86 K) by Yamasaki et al. [1386]. By using low-energy photons (hν = 7.57 eV), high momentum and energy (4 meV) resolution was obtained that enabled them to measure separately the scattering rates for the bonding and antibonding bands. At low energy, the scattering rates linearly depend on energy and increase above Tc . Shadow Fermi Surface In early Fermi surface map measurements of Bi-2212, besides a holelike FS, a weak (2 × 2) superstructure, the so-called shadow Fermi surface (SFS), was observed which has been ascribed to the existence of short range antiferromagnetic (AF) correlations [18]. The SFS looks like a centered at the wave vector (π, π) replica of the main FS, as shown in Fig. 5.17c by the white lines. The electrons coupled to the short range antiferromagnetic correlations characterized by the wave vector (π, π) may indeed produce the SFS as it has been proposed in a number of theoretical publications (see [628] and references therein). A structural mechanism producing a (2×2) lattice superstructure has been also suggested [1173], which has been supported in a low energy electron diﬀraction study [1195]. The ARPES study of the shadow and main bands along the nodal direction (0, 0) − (π, π) in pristine and Pb-doped Bi-2212 and

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Bi-2201 compounds by Koitzsch et al. [628] did not support the AF scenario. The authors observed that the width of the MDCs and band dispersions for the shadow and main bands were identical. No signiﬁcant temperature dependence of the intensity ratio of the SFS and main FS was detected which should occur in the case if the short-range AF ﬂuctuations are responsible for the SFS. On the basis of these studies, the authors come to the conclusion that the SFS in Bi-based compounds has a structural origin. Metal-Insulator Transition Although the ARPES studies of Bi-based compounds accumulated valuable information on the electronic structure of the cuprate superconductors in the metallic state, they could not clarify the key issue of how the electronic structure of a parent insulating compound evolves to the metallic state with doping. To answer this question, one has to investigate ARPES in the vicinity of the Mott insulating state and the metal-insulator transition (MIT) in strongly underdoped compounds. Contrary to the Bi-based compounds which are difﬁcult to prepare in the deeply underdoped region, the La2−x Srx CuO4 (LSCO) crystals enable such investigations. By changing the Sr concentration, it is possible to study the MIT transition from the insulating antiferromagnetic state in La2 CuO4 at x = 0 to metallic state at x ∼ 0.06 (see Sects. 2.2 and 3.2.1). The cupric oxyclorides Sr2 CuO2 Cl2 (SCOC) and Ca2 CuO2 Cl2 (CCOC) are other systems suitable MIT studies in cuprates. The stoichiometric parent compounds are antiferromagnetic Mott insulators (TN = 256 K in SCOC) that are chemically stable and posses a simple tetragonal structure without distortions or modulations with a single CuO2 layer like the La2 CuO4 crystal (see Fig. 3.1b). The Na-doped Ca2−x Nax CuO2 Cl2 (Na-CCOC) compound becomes superconducting at x > 0.08 with a maximum Tc = 28 K at optimal doping [461]. Single crystals of Na-CCOC are easily cleaved between the weakly coupled double Ca-Cl layers producing a clean surface. Below we consider recent ARPES studies of LSCO and Na-CCOC which were helpful to elucidate the MIT in cuprates. La2−x Srx CuO4 Compounds In early ARPES studies of lightly doped LSCO compounds, a “twocomponent” electronic structure was observed in the “antinodal” (AN) region close to the (π, 0) point of BZ: a broad “hump” at a large binding energy around −0.5 eV and a weak QP peak close to EF which appears at x ∼ 0.05. With doping, a weight transfer occurred from the broad hump to the QP peak at EF , while in the nodal (N) direction, close to the (π/2, π/2) point, the QP peak can be clearly resolved only at moderate doping x ∼ 0.15 (see [271]). This behavior was explained as reﬂecting the evolution of the in-gap states in the Mott insulator under doping with the chemical potential pinned

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Fig. 5.19. Energy distribution curves for La2−x Srx CuO4 at k = kF at various dopings: along the nodal direction ∼ (π/2, π/2) (a) and at ∼ (π, 0) (b). Panel (c): doping dependence of the nodal QP spectral weight ZNQP (open circles), integrated at EF spectral weight nPES (filled circles), and the hole concentration nHall (filled squares) evaluated from the Hall coeﬃcient RH = 1/e nHall (reprinted with permission by c 2003) APS from [1391],

inside the Mott gap. However, this “impurity band” scenario contradicts the high-energy PES and EELS studies discussed in Sect. 5.2.1 where a shift of the chemical potential through the valence band was observed under hole doping in combination with a spectral weight transfer from high- to low-energy states (see Fig. 5.14). Later ARPES studies of a lightly doped LSCO sample by Yoshida et al. [1391] have revealed a QP peak in the N direction as well (in the second BZ), which crosses the FS in the form of four “arcs” (similar to Fig. 5.21 below). Figure 5.19 shows the dependence with the doping of the EDCs for La2−x Srx CuO4 at k = kF in the vicinity of the N point (π/2, π/2) (a) and at the AN point (π, 0) (b) of the BZ. The QP peak in the N direction is already visible at x = 0.03 and its intensity increases with the doping. In contrast, the AN EDCs show pseudogap behavior for the doping region x ≤ 0.15. The panel (c) demonstrates the doping dependence of the nodal QP spectral weight ZNQP (open circles) deﬁned by the EDC peak intensity and the spectral weight nPES (ﬁlled circles) integrated at EF over the arc region. The functions ZNQP and nPES monotonically increase with the doping, similar to the hole concentration nHall (ﬁlled squares) evaluated from the Hall coeﬃcient RH = 1/e nHall. This observation suggests that the metallic behavior in the underdoped LSCO is related to the carrier number n ∼ x deﬁned by the QP states at the FS arcs. This assumption was conﬁrmed by the comparison of the experimental results for the thermodynamic and transport properties of underdoped cuprates with the similar ones deduced from the spectral weights

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observed in ARPES studies [1392] and the infrared absorption measurements (see Sect. 5.3.2). In the low temperature measurements (at T = 20 K) by Yoshida et al. [1391] the intensity, but not the length of the arc, increases with the doping up to x ≤ 0.15 when the full FS is formed. A similar arc formation in the N direction induced by the pseudogap at low temperature, coined as a destruction of the FS, was observed in an underdoped Bi-2212 samples (see [898] and references therein). The d-wave node at (π/2, π/2) in the superconducting state became a gapless arc above Tc . The arc length increased with the temperature to form the full FS at T ≥ T ∗ ∼ 180 K. In the pseudogap model considered in Sect. 4.2.2, the doping dependence of the gap energy was ﬁtted by the function Eg J(1 − p/pc ) with J 0.13 eV and pc ∼ 0.19. To reconcile this behavior with the ARPES results, d-wave type symmetry of the pseudogap had to be assumed. ARPES studies of the QP peaks in the underdoped LSCO by Zhou et al. [1435] detected sharp nodal QP peaks at several doping levels (x = 0.063, Tc = 12 K and x = 0.09, Tc = 28 K), in contrast to the antinodal peaks, which were resolved only for optimally and overdoped samples. The disappearance of the sharp peaks occurs abruptly outside the nodal (arc) segments of the FS. The quite diﬀerent behavior of low energy excitations, below 70 meV, observed along the N and AN directions, the “nodal-antinodal dichotomy,” was explained by the authors by the scattering of the QPs across the almost parallel AN segments of the FS close to the (π, 0) points. These scattering could be also responsible for the pseudogap formation and suppression of the sharp QP peaks in the AN direction. As discussed above, the hole doped cuprates reveal large hole-like FS centered at the (π, π) point of the BZ in the underdoped and optimally doped regions. For the overdoped region, the topological transition to an electronlike FS centered at Γ (0, 0) point of the BZ can occur in agreement with the band structure calculations for the LSCO compounds at x > 0.17 [981]. A clear evidence for an electronlike FS centered at the (0, 0) point was obtained by Yoshida et al. [1390] in the overdoped LSCO at (x = 0.22). The tightbinding ﬁt for the shape of the FS and the band dispersions agrees with the band-structure calculations, but with narrower bands corresponding to the mass enhancement by a factor of ∼ 3. Ca2−x Nax CuO2 Cl2 Compounds High-precision ARPES investigation of the doping dependence of the electronic structure and the chemical potential shift at the MIT in the Ca2−x Nax CuO2 Cl2 (Na-CCOC) compound were reported by Shen et al. [1145]. To study the dependence of the chemical potential μ on the doping, the authors measured the shifts of the valence bands at low energies (O 2pz and nonbonding O 2pπ at E < 5 eV) for various doping levels, x = 0.0, 0.05, 0.10, 0.12. The latter two samples were superconductors with Tc = 13 K and 22 K, respectively. While the change of μ from the insulating sample at x = 0 to the

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Fig. 5.20. Evolution of an insulator–metal transition in Ca2−x Nax CuO2 Cl2 : EDCs for x = 0.0, 0.05, 0.10, 0.12 in the nodal direction, (0.2π, 0.2π) → (0.72π, 0.72π). The hump A positions are marked by open symbols and the spectra at kF are c 2004) shown in bold (reprinted with permission by APS from [1145],

ﬁrst conducting one at x = 0.05 was quite large, Δμ = −0.20 eV, in the conducting state it was much smaller, (∂μ/∂x) −1.8 eV, comparable to estimates from band structure calculations ∼ −1.3 eV/hole [814]. This suggests a jump of the chemical potential at the MIT from a position inside the charge-transfer gap at x = 0 to the lower Hubbard band (Zhang–Rice singlet band, see Figs. 5.3c and 7.2), with a further smooth decrease with hole doping. Figure 5.20 shows the MIT evolution of the in Na-CCOC where the EDCs are represented at various doping levels in the nodal direction, (0.2π, 0.2π) → (0.72π, 0.72π). The chemical potential shifts μ at diﬀerent doping levels are depicted by the dotted lines relative to the underdoped case, E = 0. The broad hump around −0.5 eV at the A positions marked by open symbols is attributed by the authors to the incoherent shake-oﬀ polaronic excitation spectrum within the Franck–Condon theory. This dispersive band ascribed as QP (overdamped) excitations was detected by early ARPES experiments in the insulating undoped SCOC [295,603,1346] and CCOC [1059,1060] compounds as well. Theoretical calculations by Mishchenko et al. [837, 839] within the t–J model with a modest electron–phonon coupling have explained the unexpectedly broad dispersive QP peak at large binding energy (sometimes called as a “large pseudogap”) as caused by a polaronic many-body eﬀect in a system with strong electron correlations in the Mott insulating state. The weight of the polaronic band appears to be quite large, while the true QP peak for a doped hole at the FS has a vanishingly small intensity (see Sect. 7.2.1). A similar broad polaronic side-band about −0.5 eV binding energy reported in another ARPES study in the undoped La2 CuO4 crystal was described within a model with strong electron–phonon coupling [1064].

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Fig. 5.21. ARPES spectra for Ca2−x Nax CuO2 Cl2 for x = 0.05, 0.10, 0.12: (a–c) the momentum distribution maps of the spectral weight within ±10 meV around EF ; (d–f) the FS contours; (e) MDC along the (0, 0) → (π, π) and (π, 0) → (π, π) directions are shown by shaded areas (reprinted with permission by AAAS from c [1146], 2005)

The spectra at kF shown in bold curves in Fig. 5.20 demonstrate that the true nodal QP peak at low doping is hidden within the tail of the spectral intensity and becomes visible only at x ≥ 0.10. The study of the low-energy nodal QP excitations close to the FS shows a linear dispersion with the Fermi velocity vF ∼ 1.8 eV ˚ A, close to the band structure results, and a linear dependence of the Fermi wave vector on the chemical potential: ΔkF ∼ Δμ/vF . This study has resolved the long-standing problem of the chemical potential behavior at MIT of whether it is being pinned in midgap or it is shifting with doping to the valence or conduction band. It conﬁrmed the second scenario in agreement with the PES and EELS studies presented in Sect. 5.2.1. Similar ARPES results for the Na-CCOC compound at dopings x = 0.05, 0.10, 0.12 were reported by Shen et al. [1146]. Figure 5.21 shows the momentum distribution maps of the spectral weight within ±10 meV around EF in the ARPES spectra. The arc features are clearly seen in the panels (a–c). The elongation with the doping of the FS arcs toward the antinodes does not reﬂect their real size since it is only the intensity that depends on doping and not the shape of the angular distribution of the spectral weight. The FS contours estimated from the maximal position on the MDC at EF are shown in the panels (d–f). In the (e) panel, the MDC along the nodal (N) (0, 0) → (π, π) and antinodal (AN) (π, 0) → (π, π) directions are depicted by the shaded areas. The data demonstrate a strong momentum anisotropy of the electronic states when coming from N to AN directions which should be due to some strongly anisotropic interaction as it was observed in the underdoped

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LSCO [1435]. The MDC study revealed a dispersion of the sharp nodal QP excitations and the absence of an energy dispersion for the AN excitations. The authors conclude that the latter cannot be considered as true QPs and that the weak AN segments of the FS do not represent a real FS of a normal metal. The broad incoherent pseudogapped excitations can be explained within the Franck–Condon theory as discussed above. Another peculiarity of the broad AN states is that they are coupled by a nesting wave vector |q| ∼ 2π/4a0 at the AN segments of the FS. The authors suggest that the large nested AN segments of the FS can be associated with the dichotomy between the sharp nodal QPs and broad antinodal states and the real space “checkerboard” pattern 4a0 × 4a0 of the two-dimensional charge order observed by scanning tunneling microscopy (see Sect. 5.5.2). The doping evolution of the electronic structure, the Fermi surface, and the arc features similar to Na-CCOC were found by Hashimoto et al. [434] in ARPES studies of the single-layer cuprate Bi2 Sr2−x Lax CuO6+δ from underdoped to overdoped region. By measuring the chemical potential shift with doping, it was suggested that the Fermi surface and arcs evolve continuously from the top of the lower Hubbard band (LHB) like in Na-CCOC, while in the LSCO compound the Fermi surface develops away from the LHB. The two diﬀerent kind of the electronic structure evolution may be material dependent, e.g., due to disorder eﬀects in LSCO. Overdoped Tl2 Ba2 CuO6+δ Compound Contrary to the strongly underdoped region of the cuprate phase diagram where the non-Fermi liquid behavior due to strong electron correlations is assumed, the heavily overdoped region should reveal Fermi-liquid-type physical properties. One of the convenient materials to study this region is the Tl2 Ba2 CuO6+δ (Tl-2201) compound, which can be prepared from the optimal to the highly overdoped region. This compound has a well-ordered crystal structure of body-centered tetragonal symmetry with perfect ﬂat CuO2 planes and with one of the highest Tc,max = 93 K among the single-layer materials (see Sect. 2.5). However, it is diﬃcult to undertake ARPES investigations of the Tl-2201 compound due to the low quality of the cleaved surfaces. Plat´e et al. [1020] managed to prepare high quality overdoped single Tl-2201 crystals with Tc = 63 and 30 K and to perform an extensive ARPES study of the low-energy electronic structure of the samples. Measured EDCs resolved QP peaks with a pronounced dispersion in the nodal (N) direction and a shallower band near the antinodal (AN) (π, 0) point with the van Hove singularity at the binding energy ∼ −39 meV close to the Fermi level. The ARPES data were ﬁtted by the tight-binding dispersion which reproduced the measured FS shown in Fig. 5.22a by black lines. For comparison, the results of the band structure calculations (LDA) of the FS for two diﬀerent hole concentrations corresponding to the hole FS volumes VFS = 50% (dashed) and VFS = 63% (solid) of the BZ are shown in Fig. 5.22b. These results agree with previous

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Fig. 5.22. FS for the overdoped Tl-2201: (a) ARPES FS of the overdoped Tl2201 (Tc = 30 K) along with a tight-binding ﬁt of the data (black lines); (b) LDA band structure calculations for FS volumes VFS = 50% (dashed) and VFS = 63% (solid) c 2005) (reprinted with permission by APS from [1020],

LDA calculations (e.g., [1172]) and predict a small electron pocket of the Tl–O band at the Γ (0, 0) point shown by the dashed line at the half-ﬁlled case (nonoxygenated, δ = 0 Tl-2201) which, however, has not been observed in the ARPES study. The Bi–O pockets around the (π, 0) point have been also predicted by the LDA calculations for the Bi-compounds (see e.g., [981]) but have never been observed in the ARPES experiments. As shown by Lin et al. [727], the cation-derived bands (with the hole pockets) are lifted above the Fermi surface when hole doping eﬀects are properly included in the LDA calculations. In the overdoped Tl-2201, the EDC shows an unexpectedly sharp QP peak in the antinodal (AN) region close to the (π, 0) point, which becomes much broader when moving to the nodal (N) region at (π/2, π/2) point [1020]. With increasing doping, the nodal QP peak becomes sharper but remains broader than the antinodal peak up to the optimal doping. The 3D eﬀects due to the kz dispersion cannot explain such a behavior since the interlayer coupling vanishes at both the antinodal and the nodal directions according to the band structure calculations for the b.c.t. crystal structure. The same peculiar behavior for the QP peaks was observed in the overdoped LSCO sample at x = 0.22 [1435], which could be explained by electron scattering on incommensurate spin ﬂuctuations found in LSCO. Large FS in the overdoped Tl-2201 crystal has been conﬁrmed by the polar angular magnetoresistance oscillation (AMRO) measurements [509]. For a highly anisotropic metal which has an open quasi-2D cylindrical FS with a small k⊥ modulation (perpendicular to the 2D plane), the interlayer resistivity ρ⊥ oscillations should occur as the external magnetic ﬁeld H rotates in a polar plane relative to the 2D plane. The AMRO was detected in the

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overdoped Tl-2201 compound at T = 4.2 K in a magnetic ﬁeld of 45 T in a sample with Tc = 20 K. The AMRO were observed at diﬀerent azimuthal angles ϕ relative to the in-plane Cu–O–Cu bond direction thus enabling the reconstruction of the 3D FS for Tl-2201. The (a, b)-plane FS projection has revealed a slightly rounded square with the center at the X(π, π) point of the BZ and the volume around 62%, which corresponds to a hole doping of p = 0.24 (i.e., the total number of holes per CuO2 unit cell equals to 1.24). The coherent c-axis normal state transport in spite of a high resistive anisotropy: ρ⊥ /ρab ∼ 103 is remarkable. Thus, these measurements unambiguously conﬁrmed the existence of a coherent 3D FS in the overdoped Tl-2201 compound with still a high Tc = 20 K. As will be discussed in Sect. 5.4.2, a direct observation by Vignolle et al. [1314] of the quantum oscillations in the interlayer R⊥ magnetoresistance (the Shubnikov–de Haas eﬀect) and in the magnetic torque (the de Haas–van Alphen eﬀect) in the ﬁeld B ∼ 50 T have unambiguously conﬁrmed an existence of the large FS in the Tl-2201 crystal. 2D-ACAR Technique Complementary to the ARPES technique, the positron two-dimensional angular correlation of annihilation radiation (2D-ACAR) can be used to study the Fermi surface (FS) in metals (see e.g., [128]). When entering a solid, the positron is quickly thermalized (∼ 10−12 s) and within ∼ 10−10 s remains delocalized until it annihilates (predominantly via 2-γ annihilation) with an electron in the solid. The pair of emergent gamma quanta with the energies of approximately 0.5 MeV carry away the momentum and the energy of the (e+ − e − ) pair. The 2D-ACAR technique measures the projection Nn (px , py ) = ρ2γ (p)dpz of the e+ − e− momentum density ρ2γ (p) on a chosen crystallographic direction n. In the independent particle model,

ρ2γ (p) = const. nj (k) | dr eipr ψ + (r)ψj,k (r)|2 . (5.21) j,k

Here, ψ + (r) and ψj,k (r) are the Bloch wave functions of the positron and the electrons and the summation is carried out over all the occupied electron states nj (k) in bands j. Thus, the FS signatures can be observed as jumps in the 2D projection Nn (px , py ) onto the plane normal to n. To reproduce the electronic density of states in the momentum space, a single crystal should be measured in several crystallographic directions. In contrast to ARPES, the 2D-ACAR spectroscopy is less sensitive to surface eﬀects and has proved useful in analyzing FS of several transition metals but has encountered diﬃculties in interpretation of the results for copperoxide superconductors (see e.g., [981]). A modulation of the positron wave function in the crystal ﬁeld of the ionic lattice of copper oxides contributes signiﬁcantly to the momentum distribution function (5.21). For instance, due to a low density of conduction electron states in YBCO crystals, in the course

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5 Electronic Properties of Cuprate Superconductors

of 2D-ACAR experiments, only a small diﬀerence in electron momentum density for insulator and metallic states of YBCO has been observed. The study of small changes in the momentum density (5.21) caused by a small concentration of doped holes in the metallic phase of YBCO requires high statistical precision measurements and the proper analysis methods for processing the measured spectra (see e.g., [14, 15]). Adequate elimination of the spurious eﬀects coming from positron annihilation on electron sites at crystal impurities and imperfections (more than 40%) has to be done to resolve details of the Fermi surface. The YBCO system was studied most extensively. Additional diﬃculty for YBCO is due to the preferential concentration of positrons near the Cu1–O1 chains with the highest density of states, which hinders measuring the electronic density of states in the CuO2 planes. Thus, the quasi-one-dimensional ridge going along the Γ − X direction over the whole ﬁrst Brillouin zone (BZ) was identiﬁed through its various Umklapp signatures (e.g., [13,14,421]). The pillbox around the S-point of the ﬁrst BZ was clearly resolved at 400 K [1163] and afterward at room temperature [14,15]. However, the two-hole FS sheets, originating in the Cu–O planes were not resolved. ARPES, as well as 2DACAR, studies of the FS in YBa2 Cu3 O6.9 have been performed by Campuzano et al. [192]. Comparing the results of these two studies conﬁrms the existence of a large hole Fermi surface in the CuO2 plane. Combined 2D-ACAR and positron life-time studies were useful in the clariﬁcation of the FS behavior in the RBa2 Cu3 O7−y compounds, as, e.g., the study of the temperature dependence of the FS in YBCO by Shukla et al. [1163]. The 2D-ACAR investigation of PrBa2Cu3 O7−y by Hoﬀmann et al. [478] and positron life-time investigation of Y1−x Prx Ba2 Cu3 O7−y by Zhao et al. [1425] provided hints for a hole depletion mechanism in CuO2 planes in these compounds as discussed in Sect. 5.2.1 (see Fig. 5.15). The 2D-ACAR technique was successively used in studies of the FS in other cuprate superconductors as well (see e.g., [529, 1189]). Electron-Doped Superconductors ARPES studies of electron-doped (n-type) superconductors oﬀer a good opportunity to elucidate a crucial problem of the interplay between the antiferromagnetic (AF) correlations and the high-temperature superconductivity since these two phases are in close proximity to each other in the n-type superconductors (see Fig. 2.11), contrary to the hole-doped materials. Several ARPES studies of the Nd2−x Cex CuO4 (NCCO) at various doping levels 0 ≤ x ≤ 0.15 revealed a strong inﬂuence of the AF long- or short-range order on the QP spectra and the Fermi surface in the n-type superconductors. Below we consider several examples of these studies. A clear evidence of the QP spectra renormalization caused by the AF correlations was detected in the ARPES study of the Nd2−x Cex CuO4 crystal located at the phase boundary between the AF and the superconducting phase at x = 0.13 by Matsui et al. [808]. The AF N´eel state below TN ∼ 110 K

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Fig. 5.23. ARPES spectra (upper panel) and intensity plot (lower panel) of Nd1.87 Ce0.13 CuO4 along several cuts parallel to (0, 0) → (π, π) direction crossing AF BZ (0, π) − (π, 0) from ∼ (0.16π, 0.84π) (a) to (0.5π, 0.5π) (f). Peak positions are shown by bars and dots (see the text) (reprinted with permission by APS from c 2005) [808],

was observed in the sample by the μSR and the magnetic susceptibility measurements, while a superconducting phase was noticed below 20 K. Figure 5.23 shows the ARPES spectra (EDCs – upper panel) and the intensity plot (lower panel) along several cuts parallel to the (0, 0) → (π, π) direction crossing AF BZ (0, π) − (π, 0) from (kx , ky ) ∼ (0.16π, 0.84π) (a) to (0.5π, 0.5π) (f). The peak positions are shown by bars on the EDCs and dots on the plot. In the original paper, the ﬁgure is given in color where the EDCs at the Fermi surface are shown by blue curves, which correspond to the second (or third) curves with bars from the top. There are three distinct regions of the k-wave vectors, which show diﬀerent QP spectra closely connected with the three pieces of the FS map shown in Fig. 5.24c [78]. In the ﬁrst region, cuts (a–c), two separated QP band dispersions are observed: a very steep one below 0.1 eV and a ﬂat band close to EF . The latter yields the bright region of the FS map in Fig. 5.24c with high intensity of the low energy excitations close to the (0, π) points of the BZ. The gap between these two bands increases in moving from cut (a–c) up to ∼ 120 meV as also seen in the “peak-dip-hump” (PDH) structure in the EDCs. In the second region, cut (d), close to the dark region of the FS map, the QP peak at EF disappears and an energy gap of approximately 100 meV opens. Further moving to the (π/2, π/2) point from cut (d–e), the gap decreases and at cut (f) eventually closes revealing the Fermi-edge-like

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Fig. 5.24. Schematic representation of the Fermi surface (FS) (bold curve) and AF BZ (thin straight line) for (a) hole- and (b) electron-doped cuprate superconductors. Arrows show the AF scattering vector Q = (π, π), which couples the hot spots (open circles) (after [809]). (c) FS intensity map in a quadrant of BZ for Nd1.85 Ce0.15 CuO4 . The AF BZ is shown by the dashed line (see the text) (after [78])

structure in EDCs for the steep QP band. This third region corresponds to the bright part of the FS map close to the nodal direction in Fig. 5.24c. The peculiar behavior of the QP dispersion in these regions can be explained by the strong AF correlations at the AF BZ. The latter is shown by dashed line on the FS map in Fig. 5.24c or schematically, by a thin straight line in Fig. 5.24b where the FS is depicted by bold curves [809]. The AF BZ crosses the underlying FS at “hot spots” (open circles) coupled by the AF wave vector Q = (π, π). Quasiparticles at the intersection points experience a strong scattering due to the AF correlations, which results in the splitting of the QP band and the appearance of a gap at the FS as in cut (d). The intersecting points below the FS are close to the antinodal point (0, π) which results in energy gaps below the FS as in cuts (a–c), while these points are above the FS at the nodal point (0.5π, 0.5π) which recovers the steep QP band at the FS as in cut (f). The studies of the temperature dependence of the heavy QP where the PDH structure is observed, cuts (a–c), conﬁrm the AF correlation origin of the heavy QP mass renormalization. At low temperature below TN , a well-deﬁned QP peak at EF and a broad hump at lower binding energy ∼ 190 meV (“large pseudogap”) are observed. With increasing temperature, at T ∼ 130 K, the QP peak is washed out. At the much higher temperature T ∼ 250 K, the pseudogap is ﬁlled in, suggesting the disappearance of the AF correlations around this temperature [809]. In the absence of the long-range AF order, the short-range AF correlations are still playing an essential role in the QP dispersion. As shown for the sample at x = 0.15 in Fig. 5.24c, a strong suppression of the spectral weight close to the FS near (0.65π, 0.3π) and (0.3π, 0.65π) occurs due to a pseudogap formation at the hot spots [78]. In Fig. 5.24a, the FS for the hole-doped superconductors is also shown where the intersection points – the hot spots, are closer to the antinodal points (0, π) because of the larger FS. This explains the distinction of the FS maps and QP dispersion (EDCs) in the hole-doped materials (a pseudogap at the antinodal

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points and a high DOS at the arcs in the nodal direction) as compared to the electron-doped superconductors discussed above. A doping-dependent ARPES study of the FS evolution from the insulating state to the superconducting state in the NCCO single crystals at various doping levels (0.0 ≤ x ≤ 0.15) was reported by Armitage et al. [80]. In the insulating state, at x = 0, a broad dispersive feature at low binding energy ∼ 1.3 eV was detected which was ascribed to the charge-transfer gap between the conduction band at EF and the valence band. Under doping, a weak QP peak of increasing intensity appears close to the Fermi level near (π, 0.3π) point and a nondispersive broad spectral weight appears at the higher energy ∼ 0.4 eV, in close similarity with the results of the studies of the Na-CCOC compound (see Fig. 5.20). Taking into account the occurrence of a similar behavior of this two-component EDC structure with the doping at the MIT transition in NCCO and Na-CCOC compounds, we can suggest a common origin for the broad hump at larger binding energy as caused by polaronic many-body eﬀects in a system with strong electron correlations in the Mott insulating state [837]. At large doping, x = 0.15, the k-dependence of the EDCs is similar to that considered above for the NCCO at x = 0.13. The FS map constructed by integrating the EDC in a small energy window about EF demonstrates a continuous transition from small FS “patches” close to (0, π) points of the BZ at x = 0.04 to a full FS at x = 0.15 with the regions of suppressed intensity near (0.65π, 0.3π) points as shown in Fig. 5.24c. The authors suggest that the evolution of the FS from small electron pockets close to (0, π) points at low doping into a large hole-like FS around the (π, π) point at optimal doping can explain the electron-like transport properties at low doping and hole-like – at large doping.

Resume Summarizing the ARPES and other studies of the Fermi surface (FS) in cuprate superconductors, we can point out the following results. 1. Optimal-doped and overdoped cuprates reveal a large FS in the form of a rounded square or a “barrel” centered at the (π, π) point of the BZ with the volume proportional to (1 + x). Coherent quasiparticle states on the 3D FS were found at these doping level in the hole cuprates. 2. Strongly underdoped cuprates show a “truncated” FS in the form of arcs, for hole-doped located at the nodal points (π/2, π/2) and for electrondoped – close to the (0, π) point. The QP weights and density of states at the arcs are proportional to the concentration of the charge carriers x, while a dispersing band with much higher intensity at larger binding energy 0.5–1 eV was ascribed to polaronic eﬀects. 3. At the insulator-metal transition, a shift of the chemical potential is observed from the midgap state in the charge-transfer gap to the valence (conduction) lower (upper) Hubbard subband at hole (electron) doping.

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5 Electronic Properties of Cuprate Superconductors

4. A strong inﬂuence of the AF correlations on the quasiparticle dispersion and FS is detected, in particular for the electron-doped cuprates where the AF phase is adjacent to the superconducting state. Many-Body Eﬀects High-resolution ARPES studies of the quasiparticle dispersion close to the Fermi surface were important in elucidating the mechanism of superconducting pairing in cuprates, in particular, in attempts to characterize the type of bosons: phonons or spin-ﬂuctuations, which are believed to be responsible for the electron pairing. ARPES measurements of the EDC A(k, ω)EDC (5.15) in the vicinity of the FS allow a study of the renormalization of the QP dispersion, η(k) = Zk (k) Zk vF (k − kF ) and the scattering rate γ(k) = −Zk Σ (k, η(k)) where vF is the bare (band) Fermi velocity and Zk = (1 − (∂Σ /∂ω))−1 = (1 + λk )−1 deﬁnes the quasiparticle weight and the eﬀective interaction λk at the FS. The studies of the MDC A(k, ω)MDC (5.16) give more accurate information concerning the scattering rate deﬁned by the width of the MDC W (ω) = −(1/v)Σ (km , ω). At large binding energies (greater than the boson energy responsible for the interaction), the self-energy eﬀects vanish and the electron dispersion should return to the bare value, giving a sharp bend, the so-called “kink” in the electron dispersion. The amplitude of the kink and the energy scale where it occurs are related to the strength of the electron–boson interaction and the boson energy, respectively. Therefore, the investigations of the temperature and doping dependence of the QP dispersion for various wave vectors k can reveal the origin of the many-body interactions responsible for the QP renormalization. In the early ARPES studies, the self-energy eﬀects were measured mostly along the nodal (N) direction, (0, 0) → (π, π), where well-deﬁned QP peaks could be easily resolved, while measurements at the antinodal (AN) region close to the (π, 0)-type points demand much better resolution which was achieved in the later ARPES studies. ARPES studies of the spectral function (5.20) in a superconducting state have revealed QP peaks in the AN region as well which demonstrated a particularly strong coupling of electrons with bosonic (spin-ﬂuctuation or phonon) modes. Nodal Kink In ARPES studies of electron spectra close to the Fermi energy by Lanzara et al. [688], a pronounced dispersion renormalization in three diﬀerent families of copper-oxide s: Bi-2212, Bi-2201 and LSCO was found. An abrupt change of the electron dispersion, a kink, in the nodal direction at 50–80 meV of the binding energy was detected. The value of the renormalization did not change appreciably with the temperature, it only slightly decreased at overdoping. From a simple estimation of the renormalized Fermi velocity, v˜F vF /(1 + λ), the eﬀective coupling constant was evaluated as λ ∼ 1 which increased less

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than by a factor of two for the antinodal direction (π, 0). The universal kink energy close to the in-plane oxygen stretching longitudinal optical phonon mode around 55 meV and the weak temperature and doping dependence have led the authors to the conclusion that a strong electron–phonon coupling should be responsible for the kink phenomenon. In several other publications, a correlation of the electron renormalization eﬀects with behavior of the magnetic resonance mode (see Sect. 3.2.3) was found. For instance, Johnson et al. [556] have revealed a strong doping dependence of the self-energy corrections measured in the nodal direction in Bi-2212 samples. The renormalization coupling constant λ in the superconducting state decreased with doping in a similar manner as the intensity of the resonance mode. Moreover, the kink in the dispersion curves and the magnetic resonance mode displayed the same temperature dependence. The characteristic energies of the maximum of the renormalization, the kink position in the EDCs, scaled linearly with Tc similar to the energy of the resonance mode. These studies provided strong evidence for a spin-ﬂuctuation origin of the lowenergy electron dispersion renormalization, which was supported by further investigations discussed below. Further ARPES studies of the nature of electron–lattice interaction by using the isotope eﬀect (IE) at the substitution 16 O→18 O have revealed a complex behavior of the electron spectra in the optimally doped Bi-2212 [417,418]. Whereas the width and the dispersion of the broad high-energy “incoherent” part (IP) of the EDC around 100–300 meV have shown a strong IE, only a small IE has been detected for the dispersion and the width of the quasiparticle “coherent” part (CP) of the spectrum below the kink energy ∼ 70 meV. The IE for the antinodal excitations was larger in comparison with the nodal ones and increased below Tc with increasing of the superconducting gap. The quasiparticle part of the spectrum shifted to the lower energies at the isotope substitution whereas the IP of the spectrum revealed the sign-reversed IE. This unusual isotope eﬀect is diﬃcult to explain within the standard electron– phonon Migdal–Eliashberg theory and suggests an important contribution from the the polaronic many-body eﬀects in a system with strong electron correlations. The multiphonon excitations, the “shake-up” eﬀect can explain the high energy IP of the spectrum and a large IE at high binding energies [839]. Further ARPES studies have revealed that the IE is very sensitive to doping: in slightly overdoped Bi-2212 samples the IE is strongly suppressed or completely disappears [292, 419]. A surprisingly weak doping dependence of the renormalized nodal electron velocity close to the Fermi level, a “universal” nodal Fermi velocity, in comparison with its strong variation at the binding energy higher than ∼ 70 meV in the LSCO compounds for 0 < x ≤ 0.30 was discovered by Zhou et al. [1434]. This can hardly be accepted as originating in the electron coupling with a particular phonon mode. To perform an accurate estimate of the many-body eﬀects within the general QP spectral function representation (5.14), the knowledge of the bare

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electron dispersion (k) is essential. The evaluation of the bare dispersion from band structure calculations may be misleading due to the inadequate treatment of the strong correlation eﬀects as e.g., in the LDA. A self-consistent approach for the calculation of the bare dispersion and complex self-energy was proposed by Kordyuk et al. [642]. Starting from the general representation (5.14) and using the dispersion relation for the real Σ (ω) and imaginary Σ (ω) parts of the self-energy (Kramers–Kronig (KK) transformation), the authors managed to extract both the bare band dispersion and the selfenergy functions from the ARPES spectra. The method was demonstrated for ARPES spectra measured in the nodal direction of the antibonding band at normal states of three samples: underdoped Bi(Pb)-2212 (UD77), overdoped Bi(Pb)-2212 (OD75), and optimally doped Bi(La)-2201 (OP32). (Here and what follows in this section we use the notations UDTc , OPTc , and ODTc for the underdoped, optimally doped, and overdoped samples with the corresponding superconducting temperature Tc ). It was observed that the difference between the measured and the calculated from KK transformations self-energy parts is within the resolution of the ARPES measurements, which demonstrates the reliability of the self-consistent calculations. Figure 5.25a shows the real parts Σ (ω) of the self-energy for UD77 and OD75 samples at 130 K. Two bold lines are the ﬁtting functions for the lower

b UD76 OD73

0.2

a

∑′ (eV)

UD77 OD75

0.10

∑′ (eV)

UD76 - FL OD73 - FL

0.1

0.05

0

0.0 0

– 0.1

– 0.2 ω (eV)

– 0.3

0.00

– 0.10 – 0.20 ω (eV)

Fig. 5.25. (a) The real part Σ (ω) of the self-energy in the underdoped UD77 (bold squares) and the overdoped OD75 (open circles) Bi(Pb)-2212 samples (after [642]). (b) The imaginary part Σ (ω) of the self-energy of nodal QPs in the underdoped UD76 (bold squares) and the overdoped OD73 (open circles) Bi(Pb)-2212 samples. The solid parabola represents the electron–electron scattering contribution, while the shaded areas at the bottom are the additional contributions for the corresponding samples (UD76-FL, OD73-FL) (after [641])

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energy parts of Σ (ω) (for ω ≤ 0.17 eV in UD77 and ω ≤ 0.12 eV in OD75 samples). The arrows mark the Σ (ω) maxima. The dashed line denotes the 70 meV “kink” energy. Comparison of the data for diﬀerent samples revealed the doping dependence of the bare and renormalized Fermi velocities: vF 3.82 (3.87) eV·˚ A and v˜F 2.04 (2.46) eV·˚ A for the UD77 (OD75) samples, which are characterized by diﬀerent coupling constants λ 0.87 (0.57). The kink energy, deﬁned as the second derivative of the low energy part of Σ (ω), for the UD77 sample ωk ∼ 63 meV is slightly larger than for the OD75 sample. The imaginary part of the self-energy reaches the maximum at an energy ωc ∼ ω0 /2 where the real part changes the sign. Here ω0 ∼ 0.9 eV is the energy at the bottom of the bare band. Both ω0 and ωc are slightly larger for the UD77 sample than for the OD75 sample. The obtained results show the existence of well-deﬁned quasiparticles along the nodal direction even in the underdoped sample in the pseudogap region: the QP spectral weight is ﬁnite, Zk = (1 − (∂Σ /∂ω)|ω=0 )−1 0.54 for the UD77 sample, while at small ω the imaginary part of the self-energy −Σ (ω) ∝ C + ω 2 . The nonzero value of −Σ (ω = 0) ∝ C is caused by impurity scattering and ﬁnite temperature eﬀects. A detailed analysis of the doping dependence of the scattering rate of the nodal QPs in Bi-2212 compounds by Kordyuk et al. [641] makes it possible to distinguish between two diﬀerent scattering channels. By studying the frequency dependence of the MDC width for Bi(Pb)-2212 compounds at various dopings and temperatures, the authors observed that the main contribution was doping independent and it could be ﬁtted by a parabola which was ascribed to the electron–electron scattering within the conventional Fermi-liquid model. An additional contribution, much smaller in magnitude, strongly temperature dependent and changing with doping was also found. Figure 5.25b shows the imaginary part of the self-energy of nodal QPs in the Bi(Pb)-2212 underdoped UD76 and the overdoped OD73 samples at 25 K. The solid parabolic ﬁtting line represents the doping independent electron– electron scattering contribution. The ﬁlled squares and open circles are the total contributions measured on the UD76 and OD73 samples, respectively. The diﬀerences between these data and the parabola shown by the shaded areas at the bottom demonstrate a strong doping dependence of the additional scattering. In the UD76 samples, the scattering rate is much higher and shows a “scattering rate kink” around 100 meV in comparison with the OD73 sample where this contribution is rather weak. In a later publication by Kordyuk et al. [643], this suggestion of the two constituents in the kink in the cuprate superconductors was conﬁrmed. The quantitative analysis of the QP self-energy along the nodal direction in the UD77, OP92 and OD75 Bi-2212 samples (doping levels are x = 0.11, x = 0.16, x = 0.20, respectively) has allowed to conclude that the main channel caused by the electron–electron scattering does not depend on temperature and the doping and is characterized by a coupling strength λ1 0.43 and a scattering cutoﬀ ωc1 0.35 eV. The secondary channel revealed quite

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5 Electronic Properties of Cuprate Superconductors

a diﬀerent behavior: the coupling constant increased with underdoping from λ1 0.1 for OD75 to λ1 0.4 for UD77 samples at low temperature but vanished at the higher temperature T ∗ (x), with T ∗ (x = 0.21) < 200 K < T ∗ (x = 0.16). This channel is also characterized by a certain energy scale ∼60 meV deﬁning the energy of the kink position, which increases with doping at low temperatures. It is diﬃcult to describe such a strong doping and temperature dependence of coupling constant and the energy scale for the second channel by the interaction with a phonon mode, while the electron scattering on spin ﬂuctuations, due to both the resonance mode and the broad continuum (see Sect. 3.2.3), provides a rather plausible explanation. Taking into account the temperature and the doping dependence of the second scattering channel, the authors concluded that it should be of a magnetic origin and could have an intimate relation with the mechanism of superconductivity in the cuprates. Antinodal Kink The study of the electron dispersion anomalies close to the (π, 0) region of the BZ where the superconducting pairing, the pseudogap phenomena, and non-quasipartical-like behavior are observed, is of a particular interest for the cuprate superconductors. However, just these phenomena and the large bilayer splitting in Bi-2212 compounds make ARPES studies in the antinodal direction much more diﬃcult in comparison with the nodal ones. In several ARPES studies, an accurate deconvolution of the bilayer splitting as well as superstructure eﬀects in Bi-2212 compounds was achieved and reliable data for the electron dispersion and renormalization eﬀects related to the “antinodal kink” were obtained which are discussed below. Detailed high-resolution ARPES measurements on Bi-2212 compounds at various doping levels by Gromko et al. [413] revealed signiﬁcant diﬀerences between the nodal (N) and the antinodal (AN) electron dispersion renormalization. By studying the dispersion in the N direction extracted from the MDCs in underdoped UD75 and overdoped OD64 samples, a weak temperature dependence of the strength of the N kink, even across Tc was detected. The kink strength increased with underdoping while its energy around 70 meV slightly softened similar to other studies. EDCs and MDCs ARPES measurements for the AN kink showed quite a diﬀerent behavior in comparison with the N kink. Figure 5.26 shows the ARPES intensity maps close to the (π, 0) point of the BZ for the OD58 sample in the normal state at T = 85 K (a) and below Tc at T = 10 K (b), and for the OD71 (c) and OP91 (d) samples at T = 10 K. The band splitting between the bonding (B) and antibonding (A) bands is clearly seen. The electron dispersions determined by the MDC peak positions in the bands are shown by lines (black for A and gray (red) for B bands). While in the normal state, panel (a), the dispersions for both bands are featureless, in the superconducting state the B-band dispersion shows strong renormalization in the panel (b). A strong B-band dispersion renormalization is also observed for the OD71 and OP91 samples in panels

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241

Fig. 5.26. (color online) ARPES intensity maps close to (π, 0) point for the Bi2212: (a) OD58 sample at T = 85 K and (b) at T = 10 K, (c) OD71 and (d) OP91 samples at T = 10 K (after [413])

(c) and (d). The EDC peaks are rather broad in the normal state and become sharp and intense only in the superconducting state for energies smaller than the kink energy. There is also a diﬀerence between the MDC-derived dispersions for the N and AN direction in the superconducting state. While the N dispersion shows only a kink, the AN spectra revealed an S-like shape in the vicinity of the kink. Detailed analysis of the temperature and doping dependence of the (π, 0) kink energy and its strength for the B-band resulted in the following conclusions. The AN kink is much stronger than the N one and dramatically depends on temperature: its strength estimated by the ReΣ(ω = ωk ) is proportional to the superconducting gap which disappears above Tc . The kink energies ωk being close at the optimal doping, ωk ∼ 50 meV, are diﬀerent at overdoping: ωk,AN ≤ 40 meV and ωk,N ∼ 70 meV. The AN kink intensity is strongly depressed outside the (π, 0) point of the BZ: at (π, 0.7π) point, it is barely visible. Similar results were reported in other publications. A much stronger coupling at the AN point with an S-like shape of the MDC-derived dispersion was observed in the superconducting state for Bi-2212 and Bi-2223 compounds, while in Bi-2201 no signiﬁcant temperature dependence, even across Tc , was detected [1100]. A modest doping independent coupling λ ∼ 1 for both the bonding and the antibonding bands was found in the normal state

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5 Electronic Properties of Cuprate Superconductors

at the AN point in Bi(Pb)-2212, while λ was found to get strongly enhanced in the superconducting state, especially for the underdoped samples [162]. A detailed comparison of the ARPES data for the optimally doped (Pb,Bi)-2212 crystal with theoretical results for the fermion–boson models enables to estimate the eﬀective coupling constant λ = −(∂ReΣ/∂ω)|ω=0 in the superconducting and normal state [334]. Both the coupling of the charge carriers at the AN region to a single bosonic mode and to a collective mode with a continuum spectrum (e.g., spin ﬂuctuations) were considered. In the superconducting state, a resonance mode evolves from the gapped continuum spectrum and therefore two models give close results. The main contribution comes from a strong coupling to a single boson (resonance) mode, λb 2, while the coupling to the continuum spectrum is smaller, λf 0.7, that yields a large total coupling constant λt 2.7. By taken into account strong dependence of the renormalization eﬀects on temperature, doping, and an anisotropy of the coupling strength (nodal vs. antinodal direction), the authors suggested that sizeable renormalization eﬀects of the charge carriers found in the superconducting state in the AN region should be assigned to a strong coupling to the magnetic resonance mode. ARPES studies of the MDCs by Kaminski et al. [581] revealed that the scattering rate in the underdoped and optimally doped Bi-2212 samples around the Fermi surface in the normal state could be written in the form a + bω with a strongly anisotropic elastic term a and an isotropic inelastic contribution b. The anisotropic term, which correlates with the pseudogap, disappears in the overdoped samples. The high anisotropy (nodal vs. antinodal direction) and strong enhancement below Tc of the AN kink, in particular for the underdoped region, have led many authors to the conclusion that the N and AN kinks could be of diﬀerent origin: while a strong electron–phonon coupling may cause the N kink, electron interaction with spin-ﬂuctuations (in particular with the resonance mode which becomes sharp below Tc ) should be deﬁnitely the origin of the AN kink. A further support for an electronic origin of the QP renormalization in the cuprate superconductors was found in the high-resolution ARPES studies of several untwinned YBCO crystals (OD90, UD61, UD35) by Borisenko et al. [164]. A large, independent of doping bilayer splitting in the nodal direction was observed in accordance with the LDA calculations. A clear evidence for quasiparticle excitations near EF was obtained for the bonding band, which made it possible to study the electron dispersion in detail. For the strongly underdoped sample (UD35K), two kinks were revealed at energies around 48 meV and 150 meV in both the dispersion curves and the scattering rates. Upon doping the renormalization decreased, while the position of the lowenergy kink shifted to higher energy up to ∼ 78 meV in the OD90K sample. Although the absolute value of the kink energy did not match the resonance peak observed in the inelastic neutron scattering on the same YBCO crystals (see Fig. 3.19), its doping dependence followed those one of the resonance peaks. The authors revealed also a substantial interband scattering which

5.2 Photoemission Spectroscopy

243

is characteristic to the spin-ﬂuctuation origin of the quasiparticle renormalization. Observation of the kinks in both electron dispersion and scattering rate and strong interband scattering in the bilayer YBCO crystals implies a universality of these features ﬁrst observed in Bi-2212 compound. A more detailed studies by Zabolotnyy et al. [1402] of energy-momentum intensity maps of the untwinned YBa2 Cu3 O7−δ crystals, nearly optimally doped (δ = 0.15, Tc ≈ 90 K) and underdoped (δ = 0.6, Tc ≈ 35 K) ones, have conﬁrmed a two-component nature of the ARPES spectra in YBCO crystals. One of the components arises from the bilayer nearest to the cleavage, presumably located between CuO and BaO layers, which is strongly overdoped even in the UD sample. This component shows no signature of the superconductivity. It was possible to remove the overdoping eﬀect by Ca-substitution in the YBCO sample. The other component comes from the inner CuO2 layer which doping level should be close to the bulk. This component shows contributions from the bonding and antibonding bands and reveals the superconducting gap consistent with a d-wave symmetry. A substantial doping and wave vector dependence of the renormalization found for the superconducting component suggests a magnetic origin of the electron spectra renormalization, similar to that one observed in the Bi-2212 compounds. Another evidence for a magnetic origin of the renormalization eﬀects in Bi2212 compound was provided by observation of the “magnetic isotope eﬀect,” i.e., a strong inﬂuence on the charge carrier renormalization upon substitution of Cu by Zn. In ARPES studies of the 1% Zn- and 2% Ni-doped Bi-2212 samples, the EDCs were measured both at the antinodal (π, 0) point and in the nodal direction along the BZ diagonal [1401]. While in the pure Bi-2212 sample a well-resolved peak for the antibonding band with a dip at a higher binding energy was observed at the antinodal point, the dip smeared out in the Zn-doped sample and only slightly decreased in the Ni-doped sample. The appearance of the dip in the EDC can be explained by QP coupling to a suﬃciently narrow bosonic mode which broadening results in washing out the dip structure. As discussed in Sect 3.2.3, a sharp resonance mode in the pure YBa2 Cu3 O7 crystal is substantially broadened in the system doped by a small amount of Zn impurities (see also Fig. 5.8), which may explain the vanishing of the dip in the EDC of the Zn-doped sample. Studies of the QP renormalization in the nodal direction revealed a reduction in the kink coupling constant approximately to 15% and 30% for Ni- and Zn-doped samples, respectively, which also correlates with a much stronger inﬂuence on the spin dynamics of Zn than Ni impurities. At the same time, a small concentration of Zn or Ni impurities shows no apparent changes in phonon spectra, which could produce the observed anomalies of the QP renormalization. Similar weakening of the oﬀ-nodal kink in the Zn- and Ni-substituted samples of Bi-2212 crystal (Tc = 91 K) was obtained by Terashima et al. [1234]. Comparison between the temperature dependence of the electron selfenergy measured in these samples and the intensity of the AF resonance mode in pure and doped by Zn and Ni impurities YBCO crystals (see Sect. 3.2.3)

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5 Electronic Properties of Cuprate Superconductors

has revealed their similar behavior. This similarity demonstrates an important role of electron interaction with AF spin ﬂuctuations in the renormalization of electron dispersion at the antinodal region. An abrupt change in the high-energy dispersion around a binding energy of 340 meV was observed by ARPES experiments in Bi-2212 and LBCO cuprate superconductors, which was termed a “waterfall” anomaly [1297]. The anomaly was found to be anisotropic and weakly dependent on hole doping. However, high-resolution ARPES studies of the momentum and photon energy dependence of the high-energy dispersion down to 1 eV have revealed that the waterfall anomaly may be related to the matrix element eﬀects in ARPES intensity and therefore may be not an intrinsic feature of the spectral function [523]. The dispersion anomalies and unusual lineshapes of the EDCs and the MDCs were theoretically explained by a strong scattering of electrons close to the AN points by the AF spin ﬂuctuations (see e.g., [238, 314, 315]). The coupling of electrons close to the Van Hove singularity at (π, π) points with the sharp spin-1 resonance mode Eres ∼ 40 meV in the superconducting state (with the gap ΔAN at the AN point) results in strong cusps or kinks in the lowenergy dispersion and a dip in the EDC at the energy Ωk ∼ Eres + ΔAN . This interaction explain also the S-like shape of the MDCs in the superconducting state at the AN point. Therefore, the position of the kink and the dip in the EDC, which depends only on the energy of the electrons with a ﬂat dispersion in the AN region, do not change appreciably along the FS. By contrast, the kink intensity for an electron with the wave vector k depends on the intensity of the resonance mode at the scattering wave vector q = k± kAN . It results in strong variation of the kink intensity along the FS with a maximum in the AN region for q (π, π). With overdoping, the intensity of the resonance mode decreases which suppresses the AN kink intensity. A broad spin-ﬂuctuation continuum (see Sect. 3.2.2) determines the electron dispersion renormalization at binding energies higher than the kink energy. A certain contribution to the dispersion renormalization at low energy should come also from the electron–phonon interaction (for a review see [1436]). For instance, Cuk et al. [256] have explained the highly anisotropic coupling by the electron–phonon interaction with the B1g bond-buckling phonon mode at the antinode. The coupling constant for this mode near 40 meV can reach the values of λ ∼ 2.8 at the (π, 0) point with quite a small value for the averaged over the whole BZ constant λ ∼ 0.3 [279]. However, a strong coupling for those phonons arises only for CuO2 planes with a large buckling as in the YBCO crystals, while in Bi-2212 this coupling should be small. At least, in the mercury compounds which show the highest Tc and the smallest buckling of the planes among the cuprate superconductors (see Sect. 2.5) such type interaction seems to be irrelevant for superconducting pairing. As inelastic neutron scattering shows, the in-plane Cu–O stretching mode along the Cu–O bonds (in the [1, 0] direction, see Sect. 6.2) reveals a strong renormalization at the transition to the superconducting state

5.2 Photoemission Spectroscopy

245

and therefore can be a good candidate for strong electron–phonon coupling. As estimations show [279], the coupling with this mode may give a substantial contribution to the electron renormalization in the nodal direction (for a review see [258]). A detailed review of experimental and theoretical investigations of the dispersion anomalies in cuprates was given by Eschrig [316]. Resume To conclude, studies of renormalization of the electron dispersion at 40– 70 meV below the Fermi level, the “kink” phenomenon, and electron scattering rates for the underdoped and optimally doped cuprates have proved a strong anisotropy of the interaction responsible for the kink. The doping, temperature, and k-dependence revealed two channels, fermionic and bosonic, in the scattering. The latter contribution correlates with the evolution of the spin-ﬂuctuation spectrum which suggests its magnetic origin. A certain contribution to the electron renormalization, evidenced in the polaronic eﬀects, should come from the electron–phonon interaction as well. Superconducting Gap and Pseudogap ARPES investigations have played an important role in conﬁrming the existence of an anisotropic superconducting gap (SG) and a normal state pseudogap (PG) in the cuprate superconductors (see [271]). In ARPES only, the absolute value of the gap |Δ(k)| in the spectral function Asc (k, ω) (5.20) can be measured. The k-wave vector dependence of the gap on the 2D FS is often ﬁtted by the dx2 −y 2 -like function: Δ(kx , ky ) = Δ0 (cos kx − cos ky )/2, or in terms of the angle φ on the FS in the (kx , ky ) plane: Δ(φ) = Δ0 cos 2φ where Δ0 is the maximal value of the gap. Symmetry aspects of the SG are discussed in Sect. 5.5.1. In the simple quasiparticle approximation (5.20), the EDC should show a maximum at the binding energy ω = −|Δ(k)| at the Fermi wave vector in the superconducting state, though the PG persisting below Tc hampers an unambiguous gap determination. The bilayer splitting in the two-layer systems such as Bi-2212 hampers further analysis of ARPES spectra. Already in the early studies, a shift of the EDC below the Fermi level – the appearance of a leading edge gap (LEG) was observed below the superconducting transition in the Bi-2212 compounds. The LEG was identiﬁed with the opening of the SG (see e.g., [1142] and references therein). A clear d-wave momentum dependence | cos kx − cos ky | for the gap function along the FS was obtained by Ding et al. [282]. A normal state PG was detected by the observation of the LEG in the underdoped samples (see e.g., [793]). Diﬀerent doping dependencies of the PG and superconducting Tc in the underdoped region have been revealed: while Tc vanishes with decreasing the hole concentrations, the

246

5 Electronic Properties of Cuprate Superconductors

PG persists well above Tc [430]. Both the SG and PG demonstrated similar wave vector dependence with a maximum close to the (π, 0) point of the BZ and a minimum in the nodal direction (0, 0) → (π, π), which suggested a d-wave like dependence. Contrary to the underdoped region, in the optimally doped and overdoped Bi-2212 compounds, it was only the SG was observed below Tc [1356]. A similar behavior was discovered for the single-layer Bi-2201 compounds, which conﬁrmed the persistence of the superconducting state and pseudogap phenomenon in weakly coupled CuO2 planes [431]. Further ARPES studies of SG and PG at much better resolution reported a scaling behavior of the SG and PG with Tc in all three modiﬁcations of the Bi-crystals: Bi2 Sr2 Can−1 Cun O2n+4 (n = 1 − 3) [330, 1097, 1099]. Below Tc a coherentce peak and a dip in the EDCs emerged which were ascribed to the SG. The following values for the maximum SG Δ0 (at the (π, 0) point of the BZ) were found from the EDC data ﬁtted by the BCS spectral function: Δ0 = 10–15 meV for n = 1 and Δ0 = 40–50 meV for n = 2, 3, while the observed LEGs were about half of the SGs [1099]. Moreover, the relative weight of the coherence peak – the superconducting peak ratio (SPR), which measures the superﬂuid density, was found to scale with Tc also [330]. Therefore, the authors have concluded that the strength of the superconducting pairing (proportional to Δ0 ) and the phase stiﬀness (estimated by the SPR) are both important in reaching high-Tc in the cuprate superconductors. With increasing temperature above Tc , the SG smoothly evolved into PG which was gradually ﬁlled in and disappeared at some temperature T ∗ [1099]. The determination of the LEG, which is the lowest binding energy at which the kF -EDC reaches half of its maximum intensity – the leading edge midpoint, does not yield the absolute value of the gap. As numerical simulations have shown [639], the LEG depends on many factors: the energy and momentum resolutions, the Fermi velocity, the self-energy, but mostly on the temperature. At low temperatures, a ratio between the LEG and the real gap Δ can be roughly estimated by the relation d LEG(Δ)/d Δ ∼ 0.5, i.e., the measured LEG is close to 1/2 of the real SG Δ. Another method for determination of the SG is based on ﬁtting the EDC by using a model spectral function, e.g., the BCS one (5.20), which gives a peak close to the gap value Δ(k), though the results are model dependent. After observation of the bilayer splitting in ARPES studies of the Bi2212 materials (see above), it was natural to investigate how the SG and the PG behave on the two Fermi-surface sheets. By exploiting the strong dependence of the ARPES matrix element for the bonding and antibonding bands on the incident photon energy, Borisenko et al. [161] have been able to provide independent derivations of the superconducting LEGs corresponding to the bonding and antibonding sheets of the FS. It has been found that the LEGs on both Fermi-surface sheets have close to each other values that proved similar superconducting energy gaps for the both bands. Taking into account the occurrence of an intricate and model dependent relation between the LEG and the real gap, the authors did not determine absolute values of

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247

the gaps. The d-wave-like symmetry of the gaps was conﬁrmed, though some deviation was detected from the simple dependence | cos kx − cos ky | in the form of a strong ﬂattening out of the gap function around the nodal direction: a U -shape instead of a V -shape was observed. Possible explanations of the observed behavior are the presence of higher harmonics in the pairing potential which results in pairing beyond the nearest neighbor sites, or a momentumdependent scattering from impurities which could also produce an extended gapless regions close to the nodes of the pure d-wave superconducting gap. The same deviation was observed for the PG in the underdoped Bi-2212 in the normal state, which was modeled by the phase ﬂuctuation of the gap [300]. The ARPES investigation of the overdoped Bi-2223 crystal (Tc = 108 K) in the superconducting state by Matsui et al. [807] unambiguously proved a particle-hole mixing and the Bogoliubov-quasiparticle nature of the sharp peaks emerging in the superconducting state. The observed at T = 60 K dispersion curves, both for the particle and for the hole branches below and above the FS, showed the typical back-bending eﬀect at the Fermi energy. By measuring the spectral intensity of these energy dispersions, the authors succeeded to determine independently the coherence factors u2k and vk2 in the spectral function (5.20). Their values are in agreement with (5.20) and satisﬁes the sum rule: u2k + vk2 = 1. It is noteworthy that the the energy gap 2Δ 40 meV measured at the wave vector kF near the antinodal point is close to that one expected for a pure d-wave superconductor: 2Δ0 4.3 kB Tc [1367]. This study is important in establishing the BCS pairing scenario of the superconducting transition in the overdoped cuprates. Early ARPES studies of superconductivity in YBCO crystals were hampered by a poor quality of cleaved surfaces, surface states, and natural twinning. EDC measurements in the untwinned YBa2 Cu3 O7−δ single crystal of high quality by Lu et al. [748] revealed a strong in-plane anisotropy of the observed “peak-dip-hump” structure below Tc = 89 K. This was ascribed to the superconducting gap anisotropy caused by the presence of CuO chains along b-axis in the orthorhombic YBCO crystal. As discussed above, the superconducting gap consistent with a d-wave symmetry was observed in studies of energy-momentum intensity maps of the untwinned nearly OpD YBCO crystal by Zabolotnyy et al. [1402]. The gap was detected in the ARPES spectra arising from the inner CuO2 layers, while no signature of the superconductivity was found in the strongly overdoped bilayer nearest to the surface. In several ARPES studies in the LSCO crystals, an SG was detected, as e.g., by Ino et al. [522], who estimated the gap values from the LEG as Δ = 10–15 meV on the FS at (π, 0.2π) point in the underdoped and optimally doped LSCO crystals. A normal state PG at EF with the size of 30–35 meV was observed by Sato et al. [1095] on the basis of angle-integrated photoemission experiments in the optimally doped La1.85 Sr0.15 CuO4 crystal at 16–200 K. With increasing temperature, the PG was ﬁlling in by a state transfer from the higher energy region but did not change the size as was observed in the thermodynamic measurements (see Sect. 4.2.2). The SG value was estimated

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5 Electronic Properties of Cuprate Superconductors

as Δ ∼ 8 meV below Tc = 38 K, which is much smaller than the PG and both gaps do not seem to be smoothly connected to each other, contrary to the Bi-2212 crystals. Therefore, the authors suggested a diﬀerent origin of the SG and PG, the latter probably is caused by the short-range AFM correlations which decreasing with temperature suppresses the PG intensity. An anisotropic dx2 −y 2 -like SG was also revealed in the electron-doped superconductors. An evidence for an anisotropic SG of the value 1–2 meV near (π, 0) point of the BZ was obtained by Armitage et al. [79] in the Nd1.85 Ce0.15 CuO4 crystal. In high-resolution ARPES studies of the Nd1.85 Ce0.15 CuO4 crystal (Tc = 22 K), Sato et al. [1098] observed a leading edge shift about 2–3 meV at the FS crossing in the direction (π, 0) → (π, π), while no shift was detected along the diagonal of the BZ, (0, 0) → (π, π). The value of the SG obtained from the ﬁtting procedure within the BCS-like formula was equal to 5±1 meV, about twice the LEG value, which gives a ratio 2Δ/kB Tc ∼ 5. Studies of the EDCs in the Pr0.89 LaCe0.11 CuO4 crystal (Tc = 26 K) at different wave vector points on the FS [809] have revealed an anisotropic SG with unconventional angular dependence. The authors ﬁtted the angle dependence of the LEG by a function Δ(φ) = Δ0 [B cos 2φ + (1 − B) cos 6φ] with Δ0 = 1.9 meV and B = 1.43 where (φ) is the FS angle measured relative to the antinode from the center of the hole FS at (π, π) (see Fig. 5.27). The presence of higher harmonics shifts the maximum of the gap from the (π, 0) point to the hot spot where Δmax = 2.5 meV. As discussed above (see Fig. 5.24), the hot spots (coupled by the AF wave vector) are moved from the (π, 0) point toward the zone diagonal due to shrinking of hole-like FS for the electrondoped superconductors. The authors concluded that the observed correlation between the hot spot and the maximum of the gap suggests the spin-mediated pairing mechanism in the electron-doped cuprate superconductors. One of the most intriguing problems in studies of the high-temperature cuprate superconductors is an elucidation of a relation between the SG and PG: whether there is one gap and the SG is a coherent continuation of the PG caused by preformed pairs, or there are two independent gaps of diﬀerent origin (for a review, see [504, 1244]). To observe an appearance of the energy gap in the ARPES spectra, usually the temperature dependence of the spectra at the nodal (N) and antinodal (AN) regions is compared. While in the N direction no gap is observed (LEG is zero), a large energy gap Δp evolves near the AN region at a much higher temperature T ∗ . At low temperature T Tc , the energy gap Δp shows the dx2 −y2 symmetry with four nodes on the FS. Both the magnitude and the symmetry of the gap do not reveal an essential temperature dependence below Tc whereas the superconducting state is indicated by a coherent peak-dip structure in the EDC only. ARPES studies of the UD Bi-2212 samples have shown that in crossing Tc , the magnitude of the gap Δp has not changed, while a sharp transformation of the nodal points on the FS into ﬁnite Fermi arcs has been detected [585, 586]. A linear scaling of the Fermi arcs length with the dimensional temperature t = T /T ∗ above Tc was found. The spectral weight of the gapped state diminished with increasing

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249

Fig. 5.27. (color online) Schematic illustrations of the temperature dependence of the gap function and the Fermi arc in the k-space at diﬀerent doping in Bi-2212 crystals: (a) underdoped UD75 and (b) overdoped OD86. The temperature increases from T Tc (upper curve) to T < Tc (middle curve) and T > Tc (lower curve). Fermi arcs are shown by the thick lines close to the nodal point (45◦ ) (after [701])

T by ﬁlling in the PG and above T ∗ a full FS was restored. This observation has been considered as a prove of the one-gap scenario, according to which the PCG is a precursor to the SG but lacks pair phase coherence. However, it was argued that insuﬃcient momentum-space sampling in the ARPES measurements has precluded observing two diﬀerent gaps. Using a much higher energy and momentum resolution, it was possible to discover near the nodal direction the SG in the UD Bi-2212 crystals which opens below Tc [1231]. Contrary to the PG in the AN region, the SG decreases with underdoping and scales with Tc . A more detailed study by Lee et al. [701] has revealed the canonical BCS-type temperature dependence and the coherent nature of the excitations in the superconducting state by observation of the upper branch of the Bogoliubov quasiparticles. The temperature dependence of the SG near the nodal region of a slightly underdoped Bi-2212 crystal UD92 (Tc = 92 K) clearly shows an existence of two diﬀerent gaps: the SG which opens at Tc and PG which is observed below and above Tc . Figure 5.27 sketches a gap function evolution with temperature for two Bi-2212 crystals at diﬀerent doping: (a) underdoped UD75 (Tc = 75 K) and (b) overdoped OD86 (Tc = 86 K) samples. At 10 K above Tc (lower curve), gapless Fermi arc regions near the node (thick line) and a large PG |Δk | near the AN are seen. With doping, the PG eﬀect is suppressed and the length of the Fermi arcs increases, while the PG |Δk | decreases. At T < Tc a d-wave like SG starts to open near the nodal region (middle curve) and at T Tc the full dx2 −y2 like symmetry gap extends over the entire FS (upper curve). Thus, two gaps revealing a diﬀerent temperature dependence has been observed, which proves their diﬀerent origins. Similar results have been obtained for the optimally doped Bi-2201 crystal (Tc = 35 K) where two distinct gaps have been found [632]. While a small SG around the node closes at Tc , a large PG in the AN region persists below and above Tc . The authors have suggested that the PG competes with the SG by suppressing the spectral density in the AN region. In a subsequent publication by Kondo et al. [633], clear evidence for two diﬀerent gaps, SG and

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5 Electronic Properties of Cuprate Superconductors

PG, have been found. By examining the temperature and momentum dependence of the EDC measured by ARPES in three Bi-2201 crystals, UD23K, OP35K, and OD29K, the PG has been observed in the AN region of the BZ, while the SG appears close to the N region on the arcs of the FS. Whereas coherent peaks in the EDC appeared only below Tc , incoherent peaks were revealed in the temperature range Tc < T < T ∗ . It is remarkable that suppression of the coherent peak in the AN region is observed even in the OD sample and becomes stronger with overdoping. A clear unticorrelation between the PG and SG in the momentum space found in the experiment unambiguously shows their diﬀerent microscopic origins and excludes the scenario of the PG formation owing to the Cooper pairing above Tc . The two-gap scenario has been supported recently by electronic Raman scattering experiments (see Sect. 5.3.4) and by scanning tunneling spectroscopy measurements (see Sect. 5.5.2). Resume To summarize the ARPES studies of electronic spectra in the superconducting state, it is important to emphasize a coherent character of single-particle excitations below Tc as Bogoliubov-type quasiparticles, which proves the pairing scenario of high-temperature superconductivity in cuprates. A strong anisotropy of the superconducting gap with zeros along the BZ diagonal was found which points to a d-wave character of the pairing. The normal state pseudogap in the underdoped region was detected as a leading edge energy shift in the normal state below a certain temperature T ∗ . Two gaps of different origin have been suggested, a superconducting gap and a pseudogap which coexist below Tc and a pseudogap above Tc which is ﬁlling in at much higher T ∗ . A detailed momentum and temperature study of ARPES spectra points to a two-gap scenario of diﬀerent microscopic origin. Conclusion High-resolution ARPES studies in the cuprate superconductors have provided answers to several important problems concerning the electronic structure of high-Tc superconductors. Here, we brieﬂy summarize these results. 1. A large Fermi surface (FS) in the form of a rounded square or a barrel centered at the (π, π) point of the BZ with the volume proportional to (1 + x) was observed for cuprates at the hole (electron) concentrations x ≥ xopt . In strongly underdoped cuprates, the measurements suggest a “truncated” FS in the form of arcs. The QP weight and the density of states at the arcs are proportional to x, while a dispersing band with much higher intensity at a larger binding energy of 0.5–1 eV was ascribed to a polaronic incoherent band.

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2. The many-body eﬀects found in the renormalization anomalies, the “kink” in the dispersion below the Fermi energy and a dip in the EDCs, can be subdivided into two channels: a Coulomb-induced electron–electron scattering and an interaction with a sharp boson mode. The strong anisotropy, doping and temperature dependencies of the bosonic channel suggest its spin-ﬂuctuation origin. 3. ARPES studies conﬁrmed the appearance of the anisotropic superconducting gap (SG) below Tc of a dx2 −y 2 -type symmetry and a coherence character of the paired state. The leading edge shift near the antinodal region in EDCs observed in the normal state points to the pseudogap (PG), both in the hole- and n the electron-doped cuprates. The SG in the vicinity of the nodal direction and the PG at the antinodal region show diﬀerent temperature dependence. The SG scales with Tc , while the PG in the antinodal region increases with underdoping, Epg ∝ (xc − x) which suggests their diﬀerent origin.

5.3 Optical Electron Spectroscopy Unlike photoemission spectroscopy which probes single-particle states, optical studies of electronic spectra provide information on the spectrum of collective electron-hole pair excitations. Due to its high precision, symmetry selection rules and possibility of performing experiments on single crystals of small size, light scattering method has received considerable attention in performing rigorous quantitative investigations. The Hamiltonian of electron–light interaction reads H=

e e2 A2 − (ˆ pA + Aˆ p) , 2 2mc 2mc

(5.22)

→ − where p ˆ = −i¯h ∇ is the electron momentum operator and A is a vector potential of the external electromagnetic ﬁeld. First-order processes of light scattering are used in infrared and optic absorption to study longwavelength odd parity phonons or electronic density ﬂuctuations within the energy from tenth of electron volt to several electron volt near the Fermi surface. Second-order processes when a photon absorbed and reemitted in the Raman scattering provide information about even parity phonons and electronic density ﬂuctuations which, however, strongly depends on the electronic band structure through the Raman vertices. The early experiments carried out on ceramic samples led to contradictory results, due to strong anisotropy of conductivity in copper-oxide compounds and inadequate surface quality (for a review, see [1233,1243]). In recent years, both the quality of samples and the experimental technique were greatly improved which provides a detailed experimental picture of electronic structure and excitations in cuprate superconductors. For a review of more recent

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5 Electronic Properties of Cuprate Superconductors

studies of the electromagnetic response using infrared (IR) and optical spectroscopy, see [116, 931]. Discussion of recent developments in the theory of electronic Raman scattering (RS) and experimental results can be found in Sherman et al. [1151] and [280]. Below we consider several representative investigations by IR and optic technique and RS studies which are important for our understanding of the electronic structure and the pairing mechanism of cuprate superconductors. 5.3.1 Dynamical Conductivity Infrared and optic spectroscopy methods probe excitations and collective modes by measuring reﬂectance R(ω) on single crystals or transmission T (ω) in thin-ﬁlm samples. These measurements enable one to study the complex dielectric function in the long-wave limit (q = 0) which determines the complex refractive index N = n + iκ (see e.g., [1243]): (ω + i0+ ) = (N )2 = 1 (ω) + i2 (ω) = 1 + i (4π/ω) σ(ω),

(5.23)

where 1 (ω) and 2 (ω) are the real and imaginary parts and σ(ω + i0+) = σ1 (ω)+iσ2 (ω) is the dynamical complex conductivity. Since the infrared measurements are usually performed within the intraband region of frequencies, ¯hω ≤ 2 eV 16, 000 cm−1 , one should take into account interband electronic transitions by introducing in (5.23) the dielectric constant ∞ at a high frequency and to write the conductivity as σ(ω) = −i (ω/4π) ((ω) − ∞ ).

(5.24)

In cuprates, ∞ ∼ 4. The real part of the conductivity σ1 (ω) = (ω/4π) 2 (ω) determines absorption of radiation at the frequency ω, while the imaginary part σ2 (ω) = −(ω/4π) (1 (ω) − ∞ ) is useful in studies of low-frequency response of superconductors. The real and imaginary parts of the conductivity and dielectric functions are coupled by the Kramers–Kronig (KK) relations, which can be written as follow: 1 σ2 (ω) = P π

∞ −∞

σ1 (z) dz, ω−z

∞ 1 (ω) = 1 + 8P 0

σ1 (z) dz, z2 − ω2

(5.25)

where P is the principal value of the integral and in the second equation we take into account that σ1 (z) = σ1 (−z). The KK relations allow one to calculate both the real and the imaginary parts of (ω) or σ(ω) from the raw experimental data. In the ellipsometric technique, the real and imaginary parts of the complex dielectric function can be measured independently. In an orthorhombic crystal, the optical coeﬃcients are tensors with three different components, e.g., a , b and c , with principal axes along a, b, and c crystallographic axes. Due to the layered structure of cuprates, the in-plane

5.3 Optical Electron Spectroscopy

253

a, b-components diﬀer signiﬁcantly from the out-of-plane c-component, while the in-plane a, b-components are usually close to each other due to a small orthorhombic distortion observed in cuprate superconductors. The total absorption coeﬃcient obeys the so-called f -sum rule,

∞

1 σ1 (ω)dω = ωp2 , 8

ωp2

e2 = 3π 2 ¯ h

v · dSF =

4π ne2 , m

(5.26)

0

where the plasma frequency ωp is deﬁned by the integral of the electron velocity v projected on the Fermi surface SF . For a spherical Fermi surface ωp given by the last equation depends on the averaged electron density n, the charge e and the mass m of a free electron. To study separate components of the absorption spectrum, for example due to only intraband transitions of free carriers, the partial sum rule is frequently used. It determines an eﬀective number of carriers in a primitive cell of volume v0 : 2mv0 Neﬀ (ω) = πe2

ω σ1 (z)dz,

(5.27)

0

which participate in optical transitions with frequencies less than ω. In particular, accounting in (5.26) only for free carriers contributing to the lowfrequency absorption, ω < ωg , the eﬀective optical plasma frequency can be found: 4πne2 4πe2 Neﬀ (ωg ) = , (5.28) ωp∗ 2 = m∗ m v0 where m∗ is an eﬀective (optical) mass of free carriers. For the analysis of experimental data, a certain model for the conductivity σ(ω) is required. The Drude–Lorentz oscillator model σ(ω) =

2 ωpj ωp2 1 ω + , 4π 1/τ − iω 4π j ω/τj − i(ω 2 − ωj2 )

(5.29)

describes both relaxation of free charge carriers with the scattering rate Γ = 1/τ – the ﬁrst Drude term, and a response of bound charges characterized by the vibration frequency ωj , the scattering rate 1/τj and the oscillator 2 strength ωpj . To account for the life-time dependence on the frequency and the renormalization of mass of free carriers due to many-body eﬀects, the extended Drude model is frequently used: σ(ω) =

ωp2 m 1 , 4π m∗ (ω) 1/τ ∗ (ω) − iω

(5.30)

where m∗ (ω) and 1/τ ∗ (ω) = (m/m∗ (ω)) (1/τ (ω)) are the renormalized mass and the scattering rate, respectively.

254

5 Electronic Properties of Cuprate Superconductors

The Drude-type formula (5.30) can be derived within the linear response theory by using the memory function representation [395]: σ(ω) =

ωp2 i , 4π ω + M (ω)

M (ω) = ω λ(ω) + iΓ (ω),

(5.31)

where we introduced the real and imaginary parts of the retarded memory function M (ω + i0+ ) = M (ω) + iM (ω) which determine the renormalized optic mass m∗ (ω)/m = 1 + λ(ω) and the scattering rate 1/τ (ω) = Γ (ω). These functions are coupled by the KK relations: λ(ω) =

2 P π

∞ 0

Γ (z) dz . z2 − ω2

(5.32)

Therefore, measurement of the scattering rate 1/τ (ω) in a broad region of frequencies allows one to calculate the optic mass renormalization. In the low frequency limit, we get from (5.32): λ(0) = (∂M (ω)/∂ω)|ω=0 > 0. The renormalized transport life-time in (5.30) equals to τ ∗ (ω) = τ (ω) [1 + λ(ω)]. These parameters are expressed in terms of the real and imaginary parts of the inverse conductivity: ωp2 1 1 = Re , τ (ω) 4π σ(ω)

ωp2 m∗ (ω) 1 = 1 + λ(ω) = − Im . m 4πω σ(ω)

(5.33)

By applying the Mori projection technique, the memory function in (5.31) can be written in terms of the so-called force–force correlation function, Mα (ω) = ((Fα |Fα∗ ))ω /χ0 , where the force Fα = i (dJα /dt) = (1/¯h) [Jα , H] is the time derivative of the charge current Jα in α-direction and ((A|B))ω is the Fourier component of the retarded Mori correlation function. The static current–current susceptibility is given by χ0 = (Jα , Jα ) = ωp2 /4π. For a given model of charge carriers described by the Hamiltonian H, the forces can be readily calculated and the memory function and conductivity σ(ω) (5.31) can be evaluated (see e.g., [395, 1008, 1011]). To study many-body eﬀects in cuprates and, in particular, the nature of bosonic modes which are responsible for the self-energy eﬀects observed in ARPES, it is important to compare the parameters (5.33) in the extended Drude model measured in the optic conductivity with the self-energy Σ(k, ω) for quasiparticles (QPs) in (5.14) extracted from the ARPES. Since the optic conductivity is a two-particle property, it involves integration over QP energies and therefore the optical parameters are determined by the averaged over the wave vectors QP self-energies: Σ(ω) = Σ (ω) + iΣ (ω). For a brief discussion, we consider here the zero temperature limit. Within this approximation, the QP scattering rate for a general electron–boson interaction is determined by the equation:

ω 1 = −Σ (ω) = π dΩα2 F (Ω), (5.34) τQP (ω) 0

5.3 Optical Electron Spectroscopy

255

where α2 F (Ω) is averaged over the Fermi surface the bosonic spectral density multiplied by the square of the coupling constant α2 (ω) and the QP density of states N (0). This determines the dimensionless coupling constant

∞

λQP = 2

dΩ 0

α2 F (Ω) . Ω

(5.35)

Usually, for the optical conductivity a Kubo-type expression in terms of the current–current correlation function is used (see e.g., [201,1140,1164]). In the lowest order of the scattering, ω 1/τ , the optic scattering rate at zero temperature can be written in the form [46]: 2π 1 = τ (ω) ω

0

ω

dΩ (ω − Ω) α2tr F (Ω),

(5.36)

where the transport coupling function α2tr determines the momentum relaxation of the QPs. The optical mass m∗ (ω)/m = 1 + λ(ω) can be calculated from the the KK relations (5.32). From (5.36), simple relations follows: 1 ω d =2 , dω τ (ω) τQP (ω)

ω 1 d2 = α2 F (ω), 2π dω 2 τ (ω)

(5.37)

if we assume that α2tr α2 . Then from the ﬁrst equation it follows that the both scattering rates should have the same frequency dependence: 1/τQP (ω) = (z + 1)/2τ (ω) for 1/τ (ω) ∝ ω z . This is diﬃcult to justify, since the momentum relaxation rate usually has a higher power of ω-dependence than the QP one. As an approximation, the relationship (5.36) can be also used at low temperatures, kB T ¯ hω. At higher temperatures, due to the retardation nature of electron–boson interaction, quite a diﬀerent frequency and temperature dependence for 1/τ (ω) and 1/τQP (ω) can be found [1164]. A more accurate comparison between the QP and optic relaxation rates can be achieved by a direct calculation of these functions for a speciﬁed model of the electron–boson interaction. For example, numerical calculations by Carbotte et al. [201] within the strong-coupling Eliashberg theory for a general electron–boson interaction α2 F (ω) (with phonons or spin-ﬂuctuations) have revealed a much broader spectrum for the optical coupling constant ωλ(ω) in comparison with the QP self-energy Σ (ω). If the spectral density of bosons F (ω) has a sharp peak at ω = ωE which results at low temperature in a maximum at ωE in the QP self-energy Σ (ω) (a logarithmic divergence) and a jump in the 1/τQP (ω) = −Σ (ω) for ω > √ ωE , the optical coupling constant ωλ(ω) shows a broad maximum at ω 2ωE and a smoothly increased E ∼ 1/τ (ω) ∝ (1 − ωE /ω) for ω > ωE . Therefore, the authors proposed to use a model for the QP scattering rate 1/τ QP (ω) = d[ω/τ (ω)]/dω which ﬁts much better the frequency dependence of the QP scattering rate 1/τQP (ω) than 1/τ (ω) (see (5.37)). In a superconducting state, due to energy dependence of

256

5 Electronic Properties of Cuprate Superconductors

the QP density of state, correlation between the spectral density of bosons and the optical parameters is diﬃcult to establish. A similar problem occurs for a pseudogap state with an energy-dependent density of QP state close to the Fermi energy (see [1140]). A method for extracting the spectral function α2 F (ω) from IR or ARPES data based on the inverse theory was proposed by Dordevic et al. [291]. It was shown that to ﬁt the IR data, the boson spectral function in addition to a strong peak at ω ≤ 500 cm−1 should contain high-frequency contributions extending up to several tenth of electron volt. Extensive model calculations revealed that ﬁne details of the spectral functions using this method is diﬃcult to extract from IR or ARPES data and therefore an unambiguous comparison between these data is diﬃcult. Nevertheless, important conclusions on the electron–boson coupling function, in particular, its spectral density, can be derived from optical spectra (see e.g., [1303], as discussed below). Whereas in optic studies only homogeneous charge-ﬂuctuation response functions can be measured, the dielectric function (5.23) or the conductivity (5.24), in the EELS experiments one can investigate collective charge excitations at ﬁnite wavevectors (see Sect. 5.2.1). In particular, the dispersion of plasma oscillations ω(q) can be found by measuring the EELS scattering cross-section (5.13) in the region of a small energy transfer h ¯ ω ≤ 10 eV. For example, the transmission EELS measurements by N¨ ucker et al. [908] on crystalline ﬁlms of Ba-2212 (Tc 87 K) revealed an anisotropic in-plain plasmon dispersion: ω(q) = ωp + h ¯ 2 /mαhk q 2 where ωp 1 eV and α10 = 0.63, α11 = 0.35. The Brillouin zone (BZ) of the tetragonal body-centered Bi2212 crystal is shown in Fig. 5.2 where the crystal direction [1, 0]([0, 1]) runs along the diagonal of the Cu–O bonds. The plasmon dispersion is determined by the density response function and in the long-wavelength limit is given by the average over the FS of the function (q · v)4 FS /q 2 . Therefore, the anisotropy of the in-plain dispersion is explained by the anisotropy of the Fermi velocity, which is larger in the [1, 0] direction than in the [1, 1] (Cu–O bond) direction. It is worth while to point out that only a conventional q 2 plasmon dispersion was found for the in-plane momentum transfer q ≥ 0.05 ˚ A, which questions an acoustic plasmon model proposed for the superconducting pairing in cuprates (see Sect. 7.5.1). 5.3.2 Normal-State Optical Spectra Doping Dependence of the Optical Conductivity Studies of frequency dependence of infrared reﬂection on single crystals and ﬁlms of the cuprate superconductors have provided complementary to the ARPES studies information on transition from a charge-transfer insulator to a metallic state under hole or electron doping. Figure 5.28 shows the doping dependence of optical conductivity σ1 (ω) for single crystals of (a) La2−x Srx CuO4 and (b) Nd2−x Cex CuO4−y for an electric ﬁeld parallel to the

5.3 Optical Electron Spectroscopy

a

257

1.5

0.20

1.0

b

x=0

0.15

s (ω) (103 Ω –1cm–1)

s (ω) (103 Ω –1cm–1)

0.34

0.02 0.06 0.10

0.10

0.5

0.15 0.20

0.06

0.34 0.02

0 0

0

1

2 – ω ( eV ) h

3

4

1.6 x=0y=0

1.2 0.8

x=0.17 x=0.10

0.4 x=0yπ0

0

1

2 hω (eV)

3

4

–

Fig. 5.28. Optical conductivity σ1 (ω) of La2−x Srx CuO4 (a) and of Nd2−x Cex CuO4−y (b) as a function of doping (reprinted with permission by APS c 1991) from [1283],

conducting planes CuO2 [1283]. In the insulator phase (x = 0), absorption appears only at photon energies ¯hω > 1 eV, which indicates the existence of the insulator optical gap h ¯ ω ∼ 1.5 eV. In accordance with the general structure of the electronic spectrum shown in Fig. 5.3c, d, this gap is caused by a charge transfer from the ﬁlled O 2p-type band to the upper Hubbard Cu 3d subband. Under doping (p-type in LSCO and n-type in NCCO compounds), the intensity of this absorption decreases, but there appears an absorption in the mid-infrared (MIR) region, h ¯ ω < 1 eV, which increases more rapidly than the concentration of carriers. An absorption characteristic of metals in the form of Drude peak in (5.29) also appears near ω = 0. These results suggest a spectral weight transfer from the upper Hubbard band to the p–d singlet states under hole doping as was found in the EELS spectra for LSCO (see Fig. 5.11) or ﬁlling in the upper Hubbard band under electron doping. A strong dependence of the optical conductivity on doping, as shown in Fig. 5.28, was obtained only for the absorption in the CuO2 plane – for the polarization of light E c the absorption spectrum preserves its insulator nature even in the overdoped region [1284]. In Fig. 5.29, the integrated spectral weights (SW) Neﬀ (ω) (5.27) are compared for the IR absorption for the ab c light polarization in plane (Neﬀ ) and perpendicular to the plane (Neﬀ ). In the left panel, the results are shown for the energy h ¯ ω = 0.25 eV which is presumed to represent the Drude-type contribution of free charge carries. In the underdoped region, the SW along the c-axis is much smaller than the c /NDab and the band theoretical calcuin-plane SW. In the inset the ratio Neﬀ 2 2 lation for the squared plasma frequency ratio ωp,c /ωp,ab [49] are shown. The theoretical ratio for the eﬀective number of charge carriers, Neﬀ ∝ ωp2 , is by orders of magnitude larger and much stronger depends on x than the optical

258

5 Electronic Properties of Cuprate Superconductors

at 2eV 0.2

0.3

Neff

0.4

0.2

ab

Neff

nh(Px,z)

ab N ceff, ND

c

Neff

0.1

nh(Pz)

ab N ceff / Neff

N ab D

0.1

N ceff X100 (0.25eV)

0 0

0 0.3

0.3

0.2

0.2

0.1

0.1

0

0.1

0.2 X

0.3

0

0.1

0.2 X

0.3

nh(Pz)/nh(Px,y)

a

O2p Hole Density (arb. unit)

b

0

Fig. 5.29. Doping dependence of the eﬀective spectral weights in La2−x Srx CuO4 : c ab (x) at 0.25 eV and ND (x) at ∼ 0.2 eV (in-plain Drude weight) and (b) (a) Neﬀ c ab (x) and Neﬀ (x) at 2 eV (solid curves). The XAS results for hole numbers nh (px,y ) Neﬀ and nh (pz ) are shown by the dashed curves in (b) (after [1284])

data of SW ratio. This observations prove a diﬀerent character of in-plane and out-of-plane charge dynamics for free carriers. In the right panel of Fig. 5.29, the SWs are shown for the energy ¯hω = 2 eV, which includes both the Drude and the MIR contributions to the spectral weight transfer into the low-energy region of the optical conductivity. The eﬀective number of in-plane charge carab riers linearly increases with doping, Neﬀ 2x, up to x ∼ 0.15 which reveals a composite character of doped charge carriers: for one hole doped in the CuO2 plane a doubly occupied singlet state is created in the pdσ band. The XAS results (see Fig. 5.11) for the hole numbers nh (px,y ) and nh (pz ) are shown by the dashed curves. The in-plane XAS doping dependence is similar to the optic ones, while the out-of-plane XAS data show a diﬀerent behavior at low doping, x ≤ 0.15. Theoretical studies of the optical conductivity in the model with strong electron correlations have revealed results close to experimental observations. For instance, exact diagonalization calculations for the Hubbard model by Nakano et al. [884] have shown a complicated character of the electronic structure restructuring upon doping into the Mott insulator. At low doping x < 0.1, a quick grows of the MIR incoherent part of the conductivity spectrum occurred due to the spectral weight transfer from above the Mott gap, while the Drude weight increased as D ∝ x2 . The eﬀective carrier density determined by both contributions was proportional to the doping, Neﬀ ∝ x. At larger doping, the MIR part did not change much and the further increase

5.3 Optical Electron Spectroscopy

259

of the conductivity with doping was due to the Drude weight growth, which saturated at overdoping. So a complicated behavior of the optical conductivity with doping can be accounted for by strong electron correlation eﬀects. Studies of the in-plane charge dynamics in the LSCO crystals within the extended Drude model (5.30) revealed quite a diﬀerent behavior of the energy-dependent scattering rate 1/τ (ω) and eﬀective mass m∗ (ω)/m in the underdoped and the overdoped regions [1283]. Therefore, the authors suggested a two-component description (5.29) in the underdoped region with the Drude-type component for coherent motion of doped holes at ω ∼ 0 and a broad component in the MIR region due to incoherent hole motion or the states created in the gap region. In the moderately doped region, 0.1 < x < 0.25, the separation of the IR absorption into the Drude-type and MIR band becomes ambiguous and a description within the general Drude formula can be used. The out-of-plane charge dynamics in the underdoped region show a typical of insulators or semiconductors behavior, which does not change much with doping. The metallic in-plane and insulating out-ofplane charge dynamics demonstrate that the low-energy charge excitations are conﬁned to the copper–oxygen plane. A similar dependence of the optical conductivity on concentration of carriers was observed in other cuprate materials. A detailed study of temperature and frequency dependence of the optical conductivity in a broad frequency range 30–20,000 cm−1 (0.1 eV= 806.5 cm−1 ) in the several twinned YBa2 Cu3 O6+x single crystals by Orenstein et al. [929] revealed in the insulator sample (x = 0.2) an optical gap, h ¯ ωg 1.75 eV, determined by the energy of charge transfer O2p →Cu3d. With an increase of the oxygen content x absorption appeared in the MIR region, accompanied by the Drude absorption peak at ω ∼ 0. These two regions were clearly separated by a strong suppression of σab (ω) at ω ∼ 500 cm−1 clearly seen at low temperatures, in the underdoped crystals even above Tc . The analysis of frequency and temperature dependence of σab (ω) at diﬀerent concentrations of carriers have led to the conclusion that two components exist in the infrared absorption. A rather narrow Drude component in (5.29) with a typical width Γ kT aﬀects only a small part of the eﬀective number of free carriers (5.27) participating in absorption. The second component, which also contributes to Neﬀ , is related to absorption in the MIR region, kB T < ¯hω ≤ 1 eV, whose intensity, like that of the Drude peak, rapidly increases with the number of doped carriers. Additional conﬁrmation of the two-component model was obtained when trying to describe the frequency and temperature dependence of σab (ω) within the extended Drude model (5.30). To describe the low frequency Drude component, a suﬃciently weak coupling, λ(0) 0.4 in (5.33) at the eﬀective plasma frequency (5.28) h ¯ ωp∗ 1.5 eV should be assumed which results in the absorption essentially less than the experimental value in the frequency range h ¯ ω ≥ 50 meV. The ﬁtting in the high frequency range with λ(0) ∝ 2 and h ¯ ωp∗ 2.15 eV gives a much larger value of σab (ω) in the low frequency domain. From this analysis, it was concluded that the two components of the

260

5 Electronic Properties of Cuprate Superconductors

optical absorption σ1 (ω) have a diﬀerent nature. The low-frequency, Drude part of the absorption was ascribed to a relaxation of coherent, translational motion of quasiparticles with the eﬀective mass m∗ /m = 2–3. Absorption in the MIR region was suggested to be induced by incoherent excitations of bound-type charge carriers. Using the untwinned crystal of YBa2 Cu3 O7 , Rotter et al. [1072] were able to measure independently the conductivities σa (ω) and σb (ω) for polarization E parallel to the a and b axes. The conductivity along the a-axis indicated a sharp decrease at ω ∼ 500 cm−1 only below the superconducting transition temperature Tc 90 K, which was ascribed to the opening to the superconducting energy gap. At the same time, the conductivity along the b-axis displayed a suppression in this frequency range even above Tc as it was observed by Orenstein et al. [929] on the twinned crystals. Taking into account a diﬀerent behavior of the a- and b-axes conductivity, the authors have concluded that the suppression of the conductivity above Tc is due to the absorption by the in-chain charge carriers. Since the in-plane conductivity σa (ω) did not demonstrate a noticeable structure in the MIR region, its frequency dependence was described on the basis of the one-component extended Drude model (5.30). A nearly linear ω-dependence of 1/τ ∗ (ω) with the eﬀective mass m∗ (ω)/m = 2 − 3 at ω ≤ 0.1 eV was observed as in the later experiments for overdoped YBCO samples. A detailed analysis of the infrared conductivity in the untwinned single crystals of YBa2 Cu3 Oy for ten diﬀerent doping levels, 6.28 < y < 7.00, reported by Lee et al. [699] has helped to unveil the evolution of the infrared conductivity with doping from the antiferromagnetic (AF) phase in strongly underdoped region to the superconducting phase at optimal doping. The doping dependence of the a-axis optical spectra σ1 (ω) is shown in Fig. 5.30 (upper panel) for low temperatures: at T = 10 K for the nonsuperconducting samples

b

2000

6000

900

y = 6.75 1000

4000

600

2000

300

T*(K)

3000

ωmid (CM–1)

σ1(ω) (Ω–1 cm–1)

a

6.30 0

100

10000

1000 –1)

Wavenumber (cm

0 6.2

6.4

6.6

0 6.8

y

Fig. 5.30. (a) Optical conductivity σa (ω) of YBa2 Cu3 Oy single crystals under various doping 6.30 < y < 6.75 and (b) the peak position of mid-IR absorption ωmid (solid squares) with the pseudogap onset temperature T ∗ (open squares) (after [699])

5.3 Optical Electron Spectroscopy

261

(x ≤ 6.35) and at T ∼ Tc for the superconducting samples (x ≥ 6.40). In the inset, the σ1 (ω) is shown up to the photon energy ¯hω = 3.75 eV, which demonstrates the charge-transfer gap at ∼ 1.5 eV. Two components are clearly seen in the conductivity at low temperature, a narrow Drude-like peak at the energies below ∼ 30 meV and a broad MIR band with a maximum at ωmid , which decreases with doping similar to the pseudogap onset temperature T ∗ , h ¯ ωmid /kB T = 7 − 9, as shown in the bottom panel. The most striking result of the study is an observation of a two-component behavior with a Drude-like metallic peak and MIR broadband in the pseudogap region at low temperatures, even in the samples with AF order, as in the y = 6.35 sample with TN = 80 K. The Drude peak narrows at low temperatures and reveals a strong temperature dependence, while the MIR absorption is essentially T -independent. This behavior is demonstrated in Fig. 5.31 for two crystals: (a) y = 6.35 and (c) y = 6.65 where the two components are displayed by the dotted and dot-dashed lines in the left panels. A one-component analysis with the aid of the extended Drude model (5.30) shown in the right panel results in the frequency-dependent relaxation rate (5.33) 1/τ (ω) = (ωp2 /4π) Re[1/σ(ω)].

a

1500

12000

b

σ1(ω) (Ω −1 cm−1)

1000 6000 500

1/τM (ω) (cm –1)

ωc = 4200 cm–1

High T Low T 0

c

d ωc = 1600

6000

cm–1

2000 High T

3000

Low T

0

100

10000 0 1000 Wavenumber (cm–1)

2000 Wavenumber (cm–1)

1/τM (ω) (cm –1)

σ1(ω) (Ω −1 cm−1)

0 4000

0 4000

Fig. 5.31. Left panels: Optical conductivity σa (ω) of the YBa2 Cu3 Oy single crystals for (a) y = 6.35 and (c) y = 6.65 at high (low) temperature shown by the thick (thin) lines. Right panels: Model scattering rates 1/τM (ω) for (b) y = 6.35 and (d) y = 6.65 c 2005) (reprinted with permission by APS from [699],

262

5 Electronic Properties of Cuprate Superconductors

The relaxation rate for the y = 6.65 sample in the (d) panel shows a linear frequency dependence at high T ∼ 293 K and a strong suppression at low T ≤ Tc = 65 K with a threshold around ω ∼ 500 cm−1 as was reported in earlier studies. Contrary to this sample, the relaxation rate for the y = 6.35 sample in the (b) panel reveals a peak structure which indicates a breakdown of a single-component analysis within (5.30) model. These studies have disclosed that the two-component structure in the optical spectra σ1 (ω) occurs below the pseudogap temperature, T < T ∗ . This is distinctly seen in the underdoped samples for y < 6.5, while at T > T ∗ a broad spectrum emerges extending from low frequencies to the charge-transfer gap which can be well described by the single-component model (5.30). The same type crossover from a two-component to a single-component response was observed for the LSCO compounds as discussed above. The authors have suggested that observation of the coherent Drude component at low temperature in the underdoped region reveals a Fermi liquid nature of charge carriers which, according to the ARPES studies, should be located on the Fermi surface arcs in the CuO2 plane (see Fig. 5.19). This state of charge carriers was termed as the nodal metal. Further justiﬁcation of occurrence of a nodal metal state under the hole doping was obtained by Padilla et al. [942] by using a combination of dc transport and infrared spectroscopy in the LSCO and YBCO crystals. They have shown that the optical eﬀective mass remains constant throughout the entire phase diagram from the AF state in the underdoped region up to the optimally doped region and reveals only a small renormalization, m∗ /m = 3 − 4. The optical eﬀective mass of mobile holes in the underdoped region was evaluated coh as m∗opt = (nH /Neﬀ ), where nH is the hole density evaluated from the the coh is the eﬀective number of coherent Hole number RH = 1/enH , and Neﬀ carriers in the Drude peak (5.27) below the energy ¯hω ∼ 80 meV. The coherent coh contribution Neﬀ is approximately 20% of the total spectral weight below the charge transfer gap, both in the LSCO and the YBCO crystals, and reveals a similar doping dependence as the nH in the underdoped region in Fig. 5.29. In the moderately doped region, where the coherent component in the Drudelike peak cannot be unambiguously separated from the MIR contribution, the eﬀective optical mass was evaluated according to (5.33) within the extended Drude model: m∗ (ω)/m = −(ωp2 /4πω) Im(1/σ(ω)). It was found that two methods produced similar values of the eﬀective optical mass. These remarkable results imply that the Mott transition from the insulating to the metallic state occurs through an increase in the density of holes nh contrary to early prediction of the mass divergence or strong enhancement at nh → 0 in the AF state. Quasiparticle Interaction To elucidate the origin of the quasiparticle relaxation rate and the mass renormalization, intensive studies of the doping and temperature dependence of the

5.3 Optical Electron Spectroscopy

263

IR conductivity were performed on YBCO, Bi and Ta compounds and compared with the data obtained in the ARPES experiments (see Sect. 5.2.2). Although a direct evaluation of the single-particle relaxation rate from the optical data is diﬃcult as discussed at the end of Sect. 5.3.1, a certain information on the origin of the quasiparticle interaction can be obtained which may help to unveil the origin of bosonic modes responsible for the superconducting pairing in cuprates. Detailed studies of the doping and temperature dependence of the infrared conductivity of YBCO, Bi and Ta crystals by Puchkov et al. [1033] have revealed a systematic modiﬁcation of the optical scattering rate and the eﬀective mass renormalization (5.33) in the whole doping range from the underdoped (UD), pseudogap state to the overdoped (OD) region. While in the c-axis conductivity of UD samples, a gap-like suppression was observed below the pseudogap temperature T ∗ , in the in-plane conductivity an increase in coherence of the electronic state was found which showed up in a narrowing of the Drude peak at low frequency. The in-plane scattering rate 1/τ (ω) in the pseudogap region displays a strong suppression at low frequencies (ω < 800 cm−1 ) and a linear increase above this threshold without visible temperature dependence. The eﬀective mass below T ∗ at low frequency reveals an enhancement up to m∗ /m ∼ 4. At the superconducting transition at Tc < T ∗ , the conductivity in the gap region shows only a smooth changes. With doping and transition to the OD region, the suppression of the scattering rate becomes weaker and the threshold structure in the OD samples reveals only below Tc . The high-frequency dependence of 1/τ (ω) remains linear but, contrary to the UD region, shows a noticeable temperature dependence. The eﬀective mass renormalization with doping also decreases. Zinc substitution in the UD YBa2 Cu4 O8 enhances the overall scattering rate, while weakens the threshold structure below T ∗ as if suppressing the pseudogap state. The latter may be explained also as that Zn-impurities inﬂuence the inelastic scattering rate by changing the spin-ﬂuctuation spectrum (see Fig. 5.8). A temperature independent scattering rate above the threshold in the UD samples also suggests a spin-ﬂuctuation origin of inelastic scattering as will be discussed below (see Fig. 5.33). Later on, a more detailed comparison of the optic and ARPES studies has been done. As an example of these studies, in Fig. 5.32 the doping and temperature dependence of the optical scattering rate 1/τ (ω) = −2Σ2op (ω) (left panels) and the corresponding real part of the optical self-energy −2Σ1op(ω), (right panels) for several samples of the Bi-2212 crystals are shown [511]. According to (5.33), these parameters can be directly evaluated from the real and imaginary parts of the complex dynamical conductivity σ(ω). The optical single-particle self-energy deﬁned by the authors as follow Σ op (ω) = Σ1op (ω) + iΣ2op (ω) in terms of the memory function M (ω) in (5.31) reads: 2Σ op (ω) = −M (ω). This results in the following deﬁnitions for the scattering rate 1/τ (ω) = −2Σ2op(ω) and the eﬀective optical mass 2Σ1op (ω) = −ωλ(ω) = ω(1 − m∗ (ω)/m). The experimental results demonstrate a strong doping and

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Fig. 5.32. (color online) The doping and temperature dependence of the optical scattering rate (left panels) and the corresponding real part of the optical self-energy (right panels) of the Bi-2212 crystals: (a, d) UD67, (b, e) OP96, and (c, f) OD60. The lowest curve at the left (right) panel corresponds to the lowest (highest) temperature c 2004) (reprinted by permission from Macmillan Publishes Ltd. from [511],

temperature dependence of the the optical self-energy. Below a characteristic frequency of about 700 cm−1 , a strong suppression of the relaxation rate with the temperature decrease was found, which accompanied by the enhancement of the real part of the optic self-energy. In earlier experiments, the suppression of the relaxation rate sometimes was ascribed to the opening of the superconducting gap. However, since this suppression starts at temperatures higher than Tc it was concluded later that this phenomenon should be due to a coupling to a boson mode or to opening of the pseudogap at T ∗ . The temperature dependence is particularly strong for the underdoped sample UD67, panels (a, e), which decreases for the overdoped sample OD60, panels (d, h), similar to behavior of the quasiparticle self-energy in ARPES experiments, e.g., by Johnson et al. [556]. Since for the overdoped sample

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with still high Tc = 60 K, the renormalization eﬀects disappears the authors concluded that both the magnetic resonance peak and the phonons cannot be responsible for the superconducting pairing. At the same time, excitations in the broad background observed in all cuprate superconductors may be good candidates for the “glue” that binds the electrons. As was later commented by Cuk et al. [257], this conclusion based on the optical data which provide only momentum averaged optical self-energy cannot be a sensitive probe for strong k-dependent bosonic modes as the magnetic resonance mode at k = (π, π). A strong quasiparticle renormalization was observed in ARPES experiments for the antinodal quasiparticles even for the overdoped samples, as shown, e.g., in Fig. 5.26. A direct evaluation of the coupling constant from the reﬂectance data was performed by Hwang et al. [512] for the Bi-2212 crystals in a broad range of doping. The measurements at T = 300 K have revealed that the dimensionless coupling constant varies smoothly from λ = 0.93 for the underdoped sample to λ = 0.52 for the overdoped one for the hole concentration p = 0.103 and 0.226, respectively. These results are in qualitative agreement with the coupling constant decrease with doping observed in ARPES experiments (see Sect. 5.2.2). An interesting comparison of the optical spectra with magnetic neutron scattering data on the detwinned single crystal of YBa2 Cu3 O6.5 in the ortho-II phase has been done by Hwang et al. [513]. The measurements of the a-axis optical conductivity were performed over a wide frequency range (70–42,000 cm−1 ) for temperatures from 28 K (below Tc = 59 K) up to 290 K. In Fig. 5.33, the temperature dependence of the optical scattering b

a

2500

5000

1/τ(ω) (cm–1)

28 K 67 K 100 K 126 K 147 K 171 K 200 K 244 K 295 K

3000 2000 1000 0

0

500 1000 1500 2000 2500 3000 –1

Frequency (cm )

–2Σ1op(ω) (cm–1)

2000

4000

1500 1000 500 0 –500

0

500

1000 1500 2000 2500 3000 Frequency (cm–1)

Fig. 5.33. The temperature dependence of (a) the optical scattering rate 1/τ (ω) = −2Σ2op (ω) and (b) the real part of the optical self-energy −2Σ1op (ω) of the a-axis optical conductivity of the detwinned ortho-II YBa2 Cu3 O6.5 single crystal. The lowest curve in (a)-panel ((b)-panel) correspond to the lowest (highest) temperature (after [513])

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5 Electronic Properties of Cuprate Superconductors

rate (left panel) and the real part of the optical self-energy (right panel) are shown. These parameters were calculated from the extended Drude model (5.30) using the deﬁnition (5.33) for the optical scattering rate 1/τ (ω, T ) = −2Σ2op (ω, T ) = (ωp2 /4π) Re[1/σ(ω, T )] and the corresponding real part of the optical self-energy, ωλ(ω, T ) = −2Σ1op (ω, T ) = −(ωp2 /4π) Im[1/σ(ω, T )] − ω. Here, ωp = 15,110 cm−1 1.88 eV is the plasma frequency and ∞ = 3.63 is the dielectric constant in (5.24). The scattering rate increases noticeably with temperature at low frequencies, ω < 400 cm−1 , while it shows a quasilinear frequency dependence at higher frequencies, ω > 850 cm−1 , with a much weaker temperature dependence. In the superconducting state at T = 28 K (the lowest curve), a steplike increase of the scattering rate at frequency 400 cm−1 is observed, which is washed out at higher temperatures. In the normal state, there are no welldeﬁned excitations at all temperatures since the scattering rate is larger than their energy, h ¯ /τ > ¯ hω. In the real part of the self-energy, a triangular peak at frequency ∼ 600 cm−1 is observed at low temperatures. This peak on top of a broader background can be resolved only below T 155 K and therefore has been ascribed to a particular temperature-dependent bosonic mode. The similar peak was observed in the underdoped Bi-2212 crystals as discussed above (see Fig. 5.32). Using the second relation in (5.37), an eﬀective bosonic density of state W (ω) = α2 F (ω) was calculated from the scattering rate shown in Fig. 5.33. In the normal state, two broad peaks were detected, an intensive one at 350 cm−1 which decreases with temperature and a much weaker one at 800 cm−1 without noticeable temperature dependence. In the superconducting state, the lower peak shifts to higher frequencies, while in the region of the higher peak a negative spectral function W (ω) develops as predicted theoretically (see e.g., [2]). To obtain a qualitative description of charge carrier coupling to bosonic ﬂuctuations, a general expression for the scattering rate (after [1140]) was used with a ﬁtting bosonic density of state α2 F (ω). The latter was represented by two contributions: a Gaussian peak at ω = 248 cm−1 ( 31 meV) with a temperature-dependent intensity and a temperature independent Loretzian background with maximum at ω = 320 cm−1 ( 40 meV). Also, a model pseudogap was introduced to take into account a suppression of the density of electronic states N (ε) in the underdoped sample at low temperatures. As was pointed out by Sharapov et al. [1140], the decrease of N (ε) in the pseudogap state reduces the scattering rate at low frequencies which enables to choose the energy of the bosonic peak in α2 F (ω) equal to the magnetic resonance mode of 31 meV in the ortho-II YBCO [1190]. For a constant N (ε) at T = 0, the bosonic spectral function α2 F (ω) would show a peak at ω = 350 cm−1 . The parameters of the background were ﬁxed at high temperature T = 295 K when the peak contribution could be neglected. Then keeping these parameters, the temperature dependence of the peak intensity was ﬁtted. In this way, it was possible to extract the frequency and temperature dependence of

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the peak and background contributions to the scattering rate. The dimensionless coupling constants (5.35) appeared to be quite large: λQP = 3.3 for the background contribution, while for the peak contribution it strongly decreased with temperature from λQP = 2.2 at T = 67 K to λQP = 0 at T = 295 K. This procedure revealed that the temperature dependence of the peak contribution followed the energy integrated intensity of the magnetic resonance mode in the ortho-II YBa2 Cu3 O6.5 single crystal measured by neutron scattering ([1190], see Sect. 3.2.2, Fig. 3.17). The peak contribution to the scattering rate decreased with temperature opposite to the background contribution that resulted in a strong temperature dependence at low frequencies and essentially temperature independent behavior above ∼ 500 cm−1 . Thus, the two-component bosonic model speciﬁed by the temperature dependent bosonic resonance mode and the ﬁxed background is able to reproduce a complicated temperature and frequency dependence of the optic scattering rate. The similar temperature dependence of the bosonic mode and the magnetic resonance mode observed in neutron scattering suggests their common origin and spin-ﬂuctuation nature of bosons responsible for charge carrier scattering in cuprates. A substantial doping and temperature dependence of the optical selfenergy were found in the Tl-2201 crystals by Ma and Wang [762]. The Tl-2201 crystals can be synthesized in the whole overdoped region up to Tc = 0, which makes it possible to investigate a crossover from a marginal Fermi liquid (MFL) regime to a Fermi-liquid behavior (see Sect. 5.2.2). The authors studied three samples: nearly optimally doped with Tc = 89 K (OP79) and two overdoped ones with Tc = 70 K and 15 K (OD79, OD15). The overall spectral weight within the charge-transfer gap ∼ 1 eV does not change with doping, which suggested that the eﬀective number of charge carriers Neﬀ (5.27) and the plasma frequency ωp = 1.875 eV did not increase with overdoping. The scattering rate 1/τ (ω) decreases as the temperature is lowered, while a sharp peak in the real part of the optic self-energy is found at low temperature in the OP79 sample, which is disappeared in the OD15 sample. The scattering rate for the OD15 sample shows a power-law behavior which is close to the Fermi-liquid-like dependence 1/τ (ω, T ) ∝ (kB T )2 in the region up to the frequency 1, 600 cm−1 = 0.2 eV. At the same time, the optimally doped sample reveals an approximately linear T - and ω-dependence, which can be characterized as the MFL behavior. A strong suppression of the scattering rate with overdoping can be explained by weakening of electron interaction with a bosonic mode, which is believed to be responsible for strong scattering in the optimally doped region. For overdoped samples, in a large region of temperatures the scattering rate appears to be smaller than the excitation energy, 1/τ (ω) ≤ ω, which reveals a coherent quasiparticle-like behavior characteristic for the Fermi liquid. As discussed above, most of optical experiments indicate a nearly linear temperature and frequency dependence of the optical relaxation rates in the cuprates near optimal doping in the normal phase, which contradicts

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5 Electronic Properties of Cuprate Superconductors

to the Fermi-liquid theory and has been described by the phenomenological MFL model. To explain this anomalous behavior, various theoretical scenarios have been proposed. In particular, a quantum phase transition has been suggested close to the optimal doping, under the superconducting dome, which is characterized by a universal power-law behavior of correlation functions. Such a universal behavior has been observed by van der Marel et al. [1298] who have measured the optical conductivity in a wide frequency range in the optimally doped Bi-2212 single crystal by using the relation (5.24) with ∞ = 4.5. A linear frequency dependence of the scattering rate for ω > 500 cm−1 showed a saturation at around ω ∼ 5, 000 cm−1 . This dependence was ﬁt by a power law 1/τ (ω) ∝ ω 0.65 which resulted in a universal scaling law for both the real and the imaginary parts of the conductivity |σ(ω)| ∝ ω −0.65 with a constant phase angle: arctan(σ2 /σ1 ) 60◦ for frequencies between kB T and 7, 500 cm−1 . Such behavior was suggested as evidence for quantum-critical scaling. However, in a later publication Norman et al. [902] have demonstrated that the ﬂattering of the scattering rate 1/τ (ω) can be obtained within a model of electrons interacting with a broad spectrum of bosons with an upper cutoﬀ of the order of 300 meV. Within this approach, it was possible to reproduce the scaling law for |σ(ω)| and a constant phase angle observed by van der Marel et al. [1298]. The energy of the cutoﬀ, much higher than the phonon frequencies, suggests an electronic origin of bosons. Since the spin-ﬂuctuations have an upper bound of spectra of the order of 2J ∼ 300 meV and have suﬃcient intensity even at the optimal doping but disappear with overdoping (see Sect. 3.2.2), they could be a possible origin of these bosons. This consideration supports magnetic scenario of superconducting pairing in cuprates. Further evidence for importance of electronic contribution to the spectral function of electron–boson interaction, the “glue function” Π(ω), was obtained by van Heumen et al. [1303]. They analyzed the IR optical conductivity of several cuprate superconductors within the memory function formalism (5.31) and evaluated the temperature and doping dependence of Π(ω). The self-energy of the single-electron Green function in the strongcoupling theory, both in the normal and in the superconducting states, is determined as a convolution of Π(ω) with a kernel depending on the Fermiand Bose-distribution functions (see Appendix (A.26) and (A.27)). Although the inverse problem of extracting Π(ω) from experimental data for the memory function M (ω) can be done with a limited accuracy, as discussed in Sect. 5.3.1, deﬁnite conclusions concerning the spectrum of Π(ω) have been obtained. It was found that the spectrum consists of two main features: a temperature independent peak around 55 meV and a doping-dependent broad continuum extending up to 300 meV. Moreover, the coupling constants (5.35) both for the peak contribution, λpeak 0.7, and the for the continuum part of the spectrum, λcont 1.2, were estimated. It was concluded that the contribution of electron–phonon coupling to the superconducting pairing, presumably originated from the peak feature, is too small to account for high transition

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temperature and the contribution from the continuum of an electronic origin (interaction with spin- or charge ﬂuctuations) is essential in superconducting pairing in cuprates. Electron-Doped Cuprates Similar results were obtained for the optical conductivity in the electrondoped cuprates. Below we consider a systematic study of the optical spectra for the Nd2−x Cex CuO4 single crystals (NCCO) at doping 0 ≤ x ≤ 0.15 by Onose et al. [925]. Figure 5.34 shows evolution of the conductivity spectra σ1 (ω) with doping and temperature variation. In the insulating sample at x = 0, a charge-transfer gap of 1.5 eV is observed which is ﬁlled in with doping (see also Fig. 5.28). Several phonon peaks are also seen at low doping. In the underdoped samples, x < 0.15, and low temperatures a partial gap in the conductivity spectra, a pseudogap (PG), in the energy region Eg = 0.2– 0.4 eV evolves concomitantly with the Drude peak at ω ∼ 0 with doping.

Fig. 5.34. Doping and temperature dependence of the in-plane optical conductivity spectra for crystals Nd2−x Cex CuO4 (0 ≤ x ≤ 0.15) (after [925])

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5 Electronic Properties of Cuprate Superconductors

A strong temperature dependence is observed in the pseuodgap region (see panel (b–f)), while for large doping in the normal metal state, e.g., at x = 0.15, panel (b), the spectrum does not show temperature variation. The eﬀective number of electron-type carriers Neﬀ (ω) (5.27) in the Drude component at ω ≤ 0.03 eV increases linearly with doping. This value was found to be close to the electron density measured by the inverse value of the Hall coeﬃcient ne = 1/|RH | at 10 K [924]. With increasing temperature, the Drude peak loses its spectral weight, which is transferred to the PG region. The PG energy Eg decreases with electron doping and scales with the PG onset temperature: Eg ∼ 10 kB T ∗ . In the superconducting sample, x = 0.15, Tc = 25 K, the characteristic PG dip at low temperature disappears. Similar results were obtained for the Pr2−x Cex CuO4 ﬁlms by Zimmers et al. [1437] where a PG, detected up to optimal doping x = 0.15, extrapolates to zero at x = 0.17. A strong suppression (a kink) of the optical scattering rate 1/τ (ω) accompanied by a large mass enhancements, m∗ (ω)/m0 ∼ 4, (see (5.33)) at ω ∼ 0.07 eV for the x = 0.10–0.15 crystals was found at low temperatures [925]. Since the suppression of 1/τ (ω) did not show a doping dependence (contrary to the hole-doped cuprates, see Fig. 5.32), the authors ascribed the kink to a coupling with optical phonon modes. However, spin-ﬂuctuation spectra should not depend distinctly on doping at low temperatures since in the underdoped region of NCCO the long-range antiferromagnetic (AF) order persists up to x ∼ 0.14. Therefore, electron scattering on spin-ﬂuctuations could be also responsible for appearance of the kink. A much more pronounced suppression of 1/τ (ω) in the AF state in comparison with the optimally doped superconducting state in NCCO detected by Singley et al. [1174] supports the magnetic origin of the kink in the scattering rate. In the underdoped region the PG onset temperature scales with the AF temperature, T ∗ ∼ 2TN , which suggests that the AF spin correlations in the metallic state are the origin of the PG. This conjecture was supported by studies of the two-magnon B1g Raman scattering spectra of the underdoped samples, x = 0.05, 0.10 which revealed a rapid increase of the intensity of the Raman spectra below the PG temperature T ∗ [925]. At the same time, a much larger value of the PG energy Eg in comparison with the superconducting gap rules out the superconducting ﬂuctuation origin of the PG in electron-doped cuprates. As we discussed in Sect. 5.2.2, ARPES spectra in the NCCO crystals also revealed a PG formation at the intersection points of the Fermi surface with the AF Brillouin zone (BZ) (see Figs. 5.23 and 5.24). The ARPES study by Armitage et al. [80] has shown a continuous transition from small electron pockets close to (0, π) points of the BZ and a large PG at around (π/2, π/2) region at x = 0.04 to a large FS at x = 0.15. Therefore, the small Drude weight Neﬀ (ω) proportional to x in the underdoped region corresponds to the electrons doped at around to (0, π) points of the BZ. These “patches” of the FS in the underdoped region can be also considered as an “antinodal” metallic state at low temperatures, as compared with the “nodal metal” in the

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hole-doped cuprates. Regions of the suppressed intensity in the ARPES spectra near (0.65π, 0.3π) points of the large FS at the optimal doping in Fig. 5.24c are not observed in the optical spectra. This may be due to their small contribution to the conductivity which measures charge response averaged over the whole k-space. An AF origin of the PG was conﬁrmed by measurements, the temperature dependence of ARPES spectrum of NCCO (x = 0.13) by Matsui et al. [808]. They have shown that with increasing temperature the QP peak at the FS weakens and disappears at around TN , while the AF pseudogap, deﬁned by the hump structure in the EDCs, is observed up to much higher temperatures. We may identify the latter with the PG onset temperature T ∗ in the optical studies below which the short-range AF correlations evolve. The interlayer, in the c-direction, normal-state optical conductivity in the electron-doped cuprates resembles that one of the hole-doped cuprates. As in the LSCO compounds (see Fig. 5.29), strong phonon modes are observed in the low-frequency region, while the electron conductivity σ1 (ω) is extremely small without any clear evidence of the Drude peak [1174]. The latter proves strongly incoherent interlayer transport. Studies of the optical conductivity in the superconducting state reveal an anomalous behavior of the superﬂuid condensate which we discuss below. Resume In infrared and optic studies of the normal state of cuprate superconductors, we can single out the following results. 1. Undoped materials show a charge-transfer gap of the order of 1.5–1.75 eV. A spectral weight transfer is observed from the upper Hubbard band to the p–d singlet states under hole doping or ﬁlling in the upper Hubbard band under electron doping as was found in the EELS and PES spectra. The metallic in-plane and insulating out-of-plane charge dynamics demonstrate that the low-energy charge excitations are conﬁned to the copper–oxygen plane. 2. In the hole-doped cuprates, below the pseudogap temperature, T < T ∗ , two components in the optical absorption σ1 (ω) were found: a low-frequency Drude part and the mid-infrared (MIR) region of incoherent excitations. Above T > T ∗ and in the optimally doped region, a broad spectrum extending from low frequencies to the charge-transfer gap was observed, which can be well described by a single-component model. The low-frequency, Drude part of the absorption was ascribed to relaxation of coherent, translational motion of quasiparticles (QPs). Throughout the entire phase diagram, from the AF state in the underdoped region up to the optimally doped region, the QP mass renormalization is rather small, m∗ /m = 3–4, which imply that the Mott transition from the insulating to the metallic state occurs through an increase in the density of holes nh . A Fermi liquid nature of the QPs at low temperatures in the pseudogap phase suggested that the QPs

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5 Electronic Properties of Cuprate Superconductors

should be conﬁned to the Fermi surface arcs in the CuO2 plane detected in ARPES. This state of charge carriers was denoted as the “nodal metal.” 3. The optical scattering rate 1/τ (ω) revealed a nearly linear temperature and frequency dependence in the optimally doped samples characteristic to the marginal Fermi liquid model, while in the overdoped region a Fermi-liquid like behavior was observed. A strong suppression of the optical scattering rate at low temperatures below a characteristic frequency of approximately 500 cm−1 can be described by a charge carrier coupling to a bosonic mode. A strong temperature dependence of the mode and the coupling constant decrease with doping suggest its spin-ﬂuctuation origin. Relaxation in the MIR region can be modeled by a coupling to a broad background of incoherent spin excitations. An upper cutoﬀ of the bosonic spectrum of the order of 300 meV can explain a saturation of the optical scattering rate at high frequency ω ∼ 5,000 cm−1 . An analysis of the IR optical spectra has shown a two component structure of the spectral function of electron– boson interaction: a robust peak around 55 meV and a broad continuum extending up to 300 meV. 4. In the electron-doped cuprates, optical studies revealed a pseudogap (PG) formation. Scaling of the onset temperature with the N`eel temperature, T ∗ ∼ 2TN , and large values of the PG, Eg 10 kB T ∗ , suggest the AF spin correlations in the metallic state as the origin of the PG. A small Drude weight Neﬀ (ω) in the underdoped region proportional to the electron concentration x can be related to formation of small electron pockets close to (0, π) points of the BZ as observed in ARPES. As in the hole-doped cuprates, a strong suppression of the scattering rate was observed below 70 meV caused by electron coupling to a bosonic mode. 5. The interlayer normal state optical conductivity in the hole-doped and electron-doped cuprates revealed strong phonon modes in the low-frequency region, while the electron conductivity σ1c (ω) is extremely small without any clear evidence of the Drude peak. The latter proves strongly incoherent interlayer transport. 5.3.3 Superconducting State Microwave Spectroscopy The microwave spectroscopy provides a powerful technique for studying the superconducting state, in particular, the superﬂuid density and the lowenergy charge dynamics. By measuring the surface impedance Zs (T ) = Rs (T ) + i Xs (T ), the magnetic penetration depth λ(T ) can be evaluated directly from the reactance Xs (T ) = μ0 ωλ(T ), while the real part of the conductivity σ1 (ω, T ) can be extracted from the surface resistivity Rs (T ) (1/2) ω 2 μ20 σ1 (ω, T ) λ3 (T ) (see (4.41)). In Sect. 4.3.3, the results of the magnetic penetration depth measurements were reviewed. Below we discuss the quasiparticle dynamics in the superconducting state.

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Already early microwave studies revealed several common features of the quasiparticle relaxation spectra close to the nodes of the d-wave superconducting gap (see e.g., [157,158,1269]). In the normal state for a normal skin eﬀect, we have Rs (T ) = Xs (T ). Opening of a gap below Tc in the superconducting state results in a collapse of the surface resistivity Rs (T ), which becomes several orders of magnitude smaller than the reactance, Rs (T ) Xs (T ). With further temperature decrease, a quasiparticle inelastic scattering rate Γin falls down and the real part of the conductivity substantially increases reaching at low frequencies much higher values in comparison with the normal state. Whereas the quasiparticle scattering rate reaches its minimal value, the elastic scattering rate Γel , the density of thermally excited quasiparticles decreases with temperature. A competition between rapid increase of the transport scattering time and decrease of the number of quasiparticles results in a broad peak at intermediate temperatures in optimally doped high-quality YBCO crystals, whereas in the tetragonal Bi-2212 and Tl-based crystal no well-pronounced peak has been observed. With underdoping, the conductivity peak in the YBCO crystals decreases and a crossover from an anisotropic 3D metallic to the 2D Drude-type conductivity in the CuO2 planes with the tunneling conductivity between the planes occurs [1270]. The linear temperature dependence both of the surface resistance Rs (T ) and of the reactance Xs (T ) was revealed at low temperatures, T Tc , which conﬁrmed the d-wave pairing symmetry. In all cuprate superconductors, no coherence peak below Tc , characteristic to the conventional s-wave superconductors, has been detected which corroborates with NMR studies (see Sect. 3.3.2). A sharp peak in σ1 (T ) usually seen at Tc is caused by the inhomogeneous broadening of the superconducting transition δTc and ﬂuctuation eﬀects. Using the untwinned YBa2 Cu3 O6.95 single crystals, Zhang et al. [1421] have measured the ab plane anisotropy of the surface impedance. The anisotropy of the magnetic penetration depth was discussed in Sect. 4.3.3 (see Fig. 4.21). For the anisotropy of the conductivity, they have found the ratio σ1b /σ1a ∼ 2 through the entire temperature range below Tc . This largely can be explained by a higher quasiparticle density along the chains, (n/m∗ )b /(n/m∗ )a ≈ 2.4, inferred from the penetration depth anisotropy, λ−2 (0) ∝ n/m∗ . The ab plane anisotropy of the conductivity which is consistent with the dc resistivity anisotropy indicates a similar quasiparticle dynamics in the a and b directions in the superconducting state of the optimally doped YBCO crystal. In recent years, extensive studies of the surface impedance of high-quality YBCO, Bi-2212 and Tl-based single crystals and ﬁlms provided detailed and much more accurate information concerning the quasiparticle relaxation spectra in the low-frequency GHz region and low temperatures. To discuss these results in more detail, we consider the microwave measurements by Vancouver group of high-purity detwinned single crystals YBa2 Cu3 Ox with x = 6.993, Tc = 89 K and x = 6.5, Tc = 56 K [432]. At oxygen concentration

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5 Electronic Properties of Cuprate Superconductors

x = 6.99, larger than the optimal one 6.93, the crystal has nearly ﬁlled chain oxygen sites which provides a defect-free sample in the ordered ortho-I phase with a sharp superconducting transition, while the underdoped single crystal with x = 6.5 represents the ortho-II phase of YBCO in which the CuO chains alternate between the completely ﬁlled and empty (see Sect. 2.4.1). Figure 5.35 shows the frequency dependence of the a-axis microwave conductivity σ1a (ω, T ) at diﬀerent temperatures for YBa2 Cu3 Ox crystals: in the ortho-I phase (a) and in the ortho-II phase (b). The temperature dependence of the a-axis conductivity at various frequencies is shown in Fig. 5.36 in the ortho-I phase (a) and in the ortho-II phase (b).The b-axis microwave conductivity σ1b (ω, T ) reveals an additional very broad contribution due to the quasi-one-dimensional band of holes in the chains. In the ortho-I phase, this contribution does not change notably the conductivity and σ1b (ω, T ) ≈ σ1a (ω, T ), while in the ortho-II phase the b-axis conductivity is larger than the a-axis one: σ1b (ω, T ) ≈ 2.5 σ1a (ω, T ) for T ≥ 10 K (for details, see [432]). The frequency dependence of the conductivity in the ortho-I phase demonstrates a Drude-like behavior of thermally excited quasiparticles (except at the lowest temperature), while in the ortho-II phase the conductivity shows a cusplike relaxation spectrum in Fig. 5.35. In the the ortho-I phase, there are clearly visible in Fig. 5.36a linear temperature dependence of the conductivity at low temperatures T ≤ 20 K and broad peaks at higher temperatures, whose positions depend on the frequency. In the ortho-II phase in Fig. 5.36b, the linear temperature dependence and less pronounced peaks are revealed at lower temperatures. Sharp peaks at Tc caused by ﬂuctuation eﬀects are also seen in both samples with a more broad one in the ortho-II phase.

Fig. 5.35. The frequency dependence of the a-axis microwave conductivity at diﬀerent temperatures for YBa2 Cu3 Ox crystals: (a) in the ortho-I phase at x = 6.993 and (b) in the ortho-II phase at x = 6.5 (after [432])

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Fig. 5.36. The temperature dependence of the a-axis microwave conductivity at various frequencies for YBa2 Cu3 Ox crystals: (a) in the ortho-I phase at x = 6.993 and (b) in the ortho-II phase at x = 6.5 (after [432])

To describe the conductivity in the superconducting state, a phenomenological two-ﬂuid model for the charge carriers was used: σ(ω, T ) = σsc (ω, T ) + σqp (ω, T ), The ﬁrst term is determined by the superﬂuid conductivity ns (T ) e2 ns (T ) e2 i i , = σsc (ω, T ) = π δ(ω) + P m∗ ω + i0+ m∗ ω

(5.38)

(5.39)

where superﬂuid spectral weight is related to the magnetic penetration depth: ns (T )e2 /m∗ = c2 /4π λ2 (T ) (see (4.40)). The second term due to thermally excited quasiparticles (QP) may be described by the Drude-type formula (5.29): nn (T ) e2 1/τ , (5.40) σ1,QP (ω, T ) = m∗ (1/τ )2 + ω 2 where nn (T ) is the normal-ﬂuid density of quasiparticles and 1/τ (T ) is the energy averaged quasiparticle transport relaxation rate. However, instead of the Lorentzian form given by (5.40), the conductivity σ1 (ω, T ) in Fig. 5.35 at ω → 0 shows a cusplike behavior. To take into account this deviation from (5.40), a more general frequency dependence for the 2D quasiparticale conductivity has been proposed [1280]: σ12D (ω, T ) =

σ0 (T ) , 1 + [ωΛ(T )]y(T )

(5.41)

The Drude formula follows from (5.41) for the power y = 2 with the inverse spectral width Λ(T ) = τ (T ). It should be pointed out that the nonanalytic conductivity spectra (5.41) have no microscopic justiﬁcation since it

276

5 Electronic Properties of Cuprate Superconductors

violates the symmetry condition: σ1 (ω) = σ1 (−ω) and does not show a proper asymptotic behavior at high frequencies as 1/ω 2. Nevertheless, the formula (5.41) appears to give quite a good ﬁt to the a-axis conductivity experimental data that motivates its application. For the b-axis conductivity, an additional frequency independent term σ11D (T ) was added which took into account a contribution from the quasi-one-dimensional chain electronic band. The superﬂuid and quasiparticle spectral weights, according to the sum rule (5.26) for the real part of the conductivity (5.38), are coupled by the Ferrel–Tinkham–Glover equation: nn (T ) e2 ns (T ) e2 n e2 + = , m∗ m∗ m∗

(5.42)

where n e2 /m∗ is the total temperature independent spectral weight for charge carriers in all conduction bands. Moreover, the Kramers–Kronig relation (5.25) for the real and imaginary parts of the conductivity was used to check consistency of the calculations. By ﬁtting the model (5.41) to the absorption spectra of the samples in the low-temperature region, a weak temperature-dependent and nearly (a, b) isotropic exponents y(T ) were found: y 1.75, 1.45 for the ortho-I and the ortho-II samples, respectively. The spectral width 1/Λ(T ) extracted from the ﬁtting reveals quite a diﬀerent temperature behavior in the ortho-I and ortho-II samples. In the ortho-I sample, the 1/Λ(T ) is found to be roughly (a, b) isotropic which in the low temperature region below T ∼ 20 K shows a saturation at the limit of 1/Λ(0) 4.6 × 1010 s−1 . This value is close to the scattering rate 1/τ estimated with the help of the Drude model (5.40) that gives extremely long quasiparticle elastic scattering lifetime and a very long mean free path of the order of 4 µm for the Fermi velocity vF ∼ 2 × 107 cm s−1 [493]. At higher temperatures, the relaxation rate rapidly increases due to an inelastic scattering of quasiparticles and can be ﬁt in the region 4–40 K by the function 1/Λ(T ) ∝ (T /Tc)β with β ∼ 4.6 or even by an exponential dependence. In the ortho-II sample, the 1/Λ(T ) shows (a, b) anisotropy and demonstrates temperature dependence down to 1.2 K. The spectral width 1/Λ(T ) in a-direction roughly is twice as large in comparison with b-direction at low temperatures. The data for the conductivity show an unusual frequency-temperature scaling in the ortho-II phase, σ1 (ω, T ) = σ1 (ν), ν = ω/(T + T0 ) with T0 = 2.0 K [1280]. The spectral weight of the 2D quasiparticle conductivity obeys the sum rule (5.26) and for the phenomenological spectrum (5.41) is given by the relation:

∞ 2 2σ0 (T ) nn (T ) e2 = σ12D (ω, T )dω = . (5.43) ∗ m π y(T )Λ(T ) sin[π/y(T )] 0

The temperature dependence of the superﬂuid spectral weight can be derived directly from measurements of the London penetration depth as given by

5.3 Optical Electron Spectroscopy

277

(4.40): ns (T ) e2 /m∗ = c2 /4πλ2 (T ) (see Sect. 4.3.3). Therefore, the Ferrel– Tinkham–Glover sum rule (5.42) can be tested using the ﬁtting parameters y(T ), Λ(T ) in (5.43) and the independently measured London penetration depth. It was observed that the superﬂuid spectral weight and the 2D quasiparticle spectral weight both vary linearly with T as it should be for the superconducting gap with line nodes. To ﬁx the penetration depth values λ(T → 0), the results of a broadband zero-ﬁeld ESR absorption spectroscopy of Gd3+ ions in the Gdx Y1−x Ba2 Cu3 Oy single crystals were used [977]. By comparing the superﬂuid spectral weight derived from these measurements and the 2D quasiparticle spectral weight in the limit T → 0, a residual uncondensed quasiparticle oscillator strength was found from (5.43). The fraction of the residual weight was approximately equal to 0.5% and 3% of the total spectral weight for the ortho-I and the ortho-II samples, respectively. The sharp distinction between the isotropic conductivity in the ortho-I phase, with approximately equal spectral weights and widths in the a- and b-directions, and the strong anisotropic conductivity in the ortho-II phase, is puzzling. A diﬀerent ratio of hole concentrations in the Cu–O1 chains and the Cu-(O2,O3) planes in the underdoped and overdoped YBCO samples, as discussed in Sect. 5.2.1 (see [909]) may be responsible for this contrast. A qualitatively similar results for the microwave conductivity of the ¨ Bi2 Sr2 CaCu2 O8+x and Tl2 Ba2 CuO6+δ single crystals were obtained by Ozcan et al. [939]. In particular, a ﬁnite residual normal ﬂuid fraction of the order of 10–15% at T → 0, the linear temperature dependence for the transport relaxation rate 1/τ at low temperatures, and a strong inelastic scattering at higher temperatures were revealed as in the ortho-II phase of YBCO. The relaxation rates observed in the samples, due to a cation nonstoichiometry, are an order of magnitude higher than in the YBCO single crystals but still are low enough to show broad peaks in the conductivity at intermediate temperatures. On the basis of these similarities, the authors suggested a universal picture of quasiparticle relaxation in cuprates. Several theoretical models were proposed to explain transport properties observed in experiments for the d-wave superconductors and, in particular, the microwave conductivity (see e.g., [296,472,703,1038,1039,1107] and references therein). Lines of nodes in the k-space for the Fermi surface in the d-wave superconductors lead to the linear energy dependence of the DOS at low energy, N (E) ∼ NF (|E|/Δ0 ) where NF is the normal state DOS on the FS and Δ0 is the maximum value of the superconducting gap in a clean system (see Sect. 4.2.1). In real crystals with defects or impurities, the DOS transforms to a band of quasiparticle states bound to impurities at E → 0. In the T -matrix approximation (see (5.2)–(5.4) in Sect. 5.1.2 and Fig. 5.10), for a strong scattering, close to the unitary limit, c = cot δ = 0, the impurity band has a ﬁnite width γ ∝ (Γ Δ0 )1/2 , while for a weak Born-type scattering, c 1, δ 1, this impurity band is exponentially small, γ ∝ Δ0 exp(−Δ0 /ΓN ), where ΓN /¯h is the impurity scattering rate in the normal state (see e.g., [472]).

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5 Electronic Properties of Cuprate Superconductors

Consequently, in the gapless region for low temperatures and frequencies, (kB T, ¯hω) γ, the transport properties are determined by the impurity band. In the limit of unitary scattering, a universal residual conductivity σ00 has been predicted by Lee [703], which is independent of impurity concentration in the lowest order. In the low-temperature limit, the conductivity is given by [472]: 2 π 2 kB T n e2 ¯h , σ00 = . (5.44) σ1 (ω → 0, T ) = σ00 1 + 12 γ πm Δ0 This relation is a consequence of the cancelation of two opposing eﬀects: an increase in the impurity-induced DOS Nimp ∼ NF (ΓN /Δ0 ) and an decrease of the impurity scattering life-time τ ∼ ¯ h/ΓN in the conductivity. This results in the independent of the impurity scattering rate contribution (¯ h/Δ0 ) in σ00 . The observed low-temperature microwave conductivity appears to be much larger than the universal residual conductivity (5.44), which is usually an order of magnitude smaller than the normal state conductivity σ1 (0, Tc ). An accord between theory and experiment can be improved if one takes into account vertex corrections βVC and Fermi-liquid corrections αFL which strongly enhance σ00 calculated in the simple bubble approximation in (5.44): σ0 = σ00 βVC α2FL [296]. The quadratic temperature behavior in the low temperature region has been observed for conductivity σ1 (T ) and λ−2 (T ) in ﬁlms and in Zn-doped YBCO crystals, where Zn impurities reveal the unitary scattering behavior [156]. In the Born scattering limit, the gapless region is exponentially small that precludes its experimental observation. Usually, the conductivity in highquality single crystals do not show T 2 behavior that presumably points to a weak scattering in these materials. At high temperatures, kB T γ, a clean limit should be considered with a linear energy dependence of the DOS at low energies. In the clean limit, a simple expression for the node quasiparticle conductivity may be written in a Drude-like form [472]: nn (T ) e2 1 Im (5.45) σ1,QP (ω, T ) = m∗ ω − i/τ (T, ω ) ω where . . .ω denotes the frequency average over DOS Ns (ω ) for a clean superconductor close to the Fermi energy. In the hydrodynamic limit ω τ 1, the conductivity in the pure limit reduces to σ1,qp (T ) ≈ nn (T ) e2 τ /m∗ . By taking into account a strong energy dependence of the scattering rate, 1/τ (ω) ∝ 1/ω or 1/τ (ω) ∝ ω in the limit of strong or weak scattering, respectively, for the conductivity, one obtains an asymptotic behavior σ1,qp (T ) ∝ T 2 , or σ1,qp (T ) → σ0 in the corresponding scattering limits. The linear temperature dependence observed in experiments for the quasiparticle conductivity in pure crystals would appear for a constant averaged scattering time τ

5.3 Optical Electron Spectroscopy

279

with nn (T ) ∝ T . At higher temperatures, inelastic scattering processes, e.g., by spin-ﬂuctuations, become important which results in (T /Tc )3 dependence for the scattering rate [472, 1038, 1039]. Consequently, the predicted T 2 dependence of the quasiparticle microwave conductivity in the low temperature region (or (T /Tc )3 at higher temperatures) is in disagreement with the experimentally observed linear one in the pure YBCO crystals discussed above. To overcome these inconsistencies, several other models were considered. The in-plane microwave impedance of YBCO ﬁlms was described within the nested Fermi-liquid model by Hensen et al. [449] where the spin-ﬂuctuation exchange scattering was taken into account for the inelastic electron scattering. By ﬁtting the scattering phase shift δ, a reasonably good agreement was obtained in a broad temperature range. The importance of the umklapp processes for the d-wave superconductors, which produce an activated exponential-like temperature dependence of the relaxation rate, was stressed by Walker et al. [1330]. By taking into account a strong energy dependence of the quasiparticle scattering rate, a cusplike behavior of the conductivity and a ﬁnite residual quasiparticle oscillator strength, observed in the ortho-II phase of YBCO, were found by Schachinger et al. [1107] for an intermediate impurity scattering parameters between the weak and strong limits. To ﬁt the experimental data for the surface impedance in YBCO and Bi-2212 crystals, a phenomenological two-ﬂuid model was proposed [1269]. In the model the quasiparticle density below Tc is assumed to be proportional to T , as considered above, while the quasiparticle scattering rate is determined by the sum 1/τ = 1/τi + 1/τin of the elastic impurity scattering 1/τi and the inelastic one 1/τin induced by strong electron–phonon interaction. The latter is described by the Bloch–Gr¨ uneisen formula. In the limit of weak impurity scattering, the quasiparticle scattering rate in the low temperature limit increases as T 5 and at high temperatures becomes proportional to T . Within this phenomenological model, the temperature dependence of the resistivity below Tc of YBCO crystals at various doping was described with a reasonable accuracy [1270]. Resume Summarizing, we can conclude that a complicated interrelation between impurity scattering, temperature dependence of the quasiparticle density, and k-dependent inelastic scattering induced by spin-ﬂuctuations (or phonons) produces a rather complex picture of temperature- and frequency-dependent microwave absorption. Nevertheless, the d-wave symmetry of the gap has to be assumed to ﬁt the distinctly observed linear low temperature dependence of the conductivity and 1/λ2 (T ) in pure samples. The microwave experiments, as a whole, have conﬁrmed a picture of a nodal quasiparticle Fermi liquid below Tc with a peculiar density of states and an energy-dependent relaxation rate.

280

5 Electronic Properties of Cuprate Superconductors

These quantities essentially depend on the strength of the impurity scattering. In high-quality single crystals, a weak Born-type elastic scattering of nodal quasiparticles is realized, while in less clear samples an intermediate impurity scattering phase should be invoked to explain the low temperature conductivity. Below Tc , a collapse of inelastic scattering occurs due to a superconducting gap opening. This results in a broad peak in the conductivity σ1 (ω, T ) at intermediate temperatures. However, there is still no comprehensive microscopic theory which is able to explain self-consistently all experimental data in the microwave absorption. Superconducting Gap In conventional s-wave superconductors, the infrared conductivity σ1 (ω) below Tc shows an onset of absorption for frequency ω > 2Δ, which reaches the normal state conductivity at frequencies several times 2Δ. It was hoped to observe a similar onset in the infrared absorption experiments at the superconducting transition in the copper-oxide compounds. In early experiments, a conductivity minimum detected around ω ∼ 350 cm−1 in YBCO single crystals was ascribed to the energy gap formation (see e.g., [1072]). However, the minimum persisted also above Tc and its position did not scale with Tc . As discussed above, a decrease of the inelastic relaxation rate 1/τ below a characteristic frequency ω ∼ 500 cm−1 results in a minimum of the conductivity, which is strongly suppressed below Tc (see Figs. 5.30–5.33). Moreover, since the mean free path of the quasiparticles is large compared to the superconducting correlation length ξ0 , the cuprate superconductors is in the clean limit where only a small change of optical properties occurs below Tc . To illustrate the salient features of the observed conductivity spectra in the superconducting state, we present in Fig. 5.37 the in-plane complex conductivity σ(ω) of the underdoped La2−x Cex CuO4 ﬁlm (x = 0.081,

Fig. 5.37. Complex conductivity σ1 (ω) and σ2 (ω) of the La2−x Cex CuO4 ﬁlm (x = 0.081, Tc = 25 K) on a logarithmic scale of photon energy (after [987])

5.3 Optical Electron Spectroscopy

281

Tc = 25 K) measured by two methods: the infrared reﬂectivity technique and submillimeter-wave transmission spectroscopy below 40 cm−1 [1029]. At low temperature, a strong suppression of the real part of the conductivity σ1 (ω) occurs below ω ∼ 100 cm−1 , which can be represented in the Drude-like form (5.39) with a frequency-dependent relaxation rate as discussed in the previous sections. The eﬀective relaxation rate in (5.39) reveals a strong decrease by an order of magnitude just below Tc . The imaginary part of the conductivity below this frequency clearly demonstrates a superﬂuid behavior in form given by (5.40): σ2 (ω) = (c2 /4π λ2 ω), where the London penetration depth λ(0) = 0.38 µm. The observed frequency and temperature dependence of the conductivity in the electron-doped superconductor is quite similar to the holedoped cuprates such as YBCO crystals and provides evidence for the d-wave superconductivity (see Sect. 4.3.3). To explain the peculiar frequency dependence of the infrared conductivity in the superconducting state, Quinlan et al. [1039] have considered a BCS model with a dx2 −y2 gap Δk , where the quasiparticle life-times were determined by impurity elastic scattering and inelastic scattering by spin ﬂuctuations. The proposed calculations is a direct extension to the infrared region of the theory developed for the microwave absorption considered above [472]. In the clean limit, the theory predicts a minimum of the quasiparticle conductivity σ1 (ω) in the superconducting state at ¯hω ∼ Δ0 for T Tc with a broad peak at 4Δ0 . The minimum is caused by a strong suppression of the wave vector dependent inelastic scattering rate which at low energy 1/τ (k, ω) ∝ (ω − Δk )3 , crossing over to a linear frequency dependence at higher energy. Since in the clean limit the inelastic scattering processes, due to the momentum conservation, can occur only for a four-quasiparticle ﬁnal state, an onset of the quasiparticle absorption appears close to 4Δ0 . For higher frequency, the absorption in the superconducting and normal states becomes equal. So, the theory qualitatively explains the frequency dependence of the quasiparticle conductivity below Tc , though it fails to describe the minimum of the conductivity observed also at higher temperatures and does not show a suﬃcient spectral weight in the mid-infrared region. Similar results were obtained by Carbotte et al. [204] within the strong-coupling theory for a model, which include both an inelastic and elastic impurity scattering for a superconductor with nodes in the gap. While in the frequency-domain spectroscopy discussed above it is rather diﬃcult to distinguish between diﬀerent excitations which determine the optical response, in the time-domain spectroscopy one can separate contributions from various components by their diﬀerent relaxation dynamics. The real-time measurements of the quasiparticle (QP) relaxation dynamics in the pump-probe experiments have given valuable information concerning the superconducting gap and the pseudogap dependence on temperature and doping (see e.g., [274, 297, 574] and references therein). In these experiments, the relaxation of electron-hole pairs exited by a laser pulse in the superconducting state was studied by measuring the time dependence of the photoinduced

282

5 Electronic Properties of Cuprate Superconductors

change in reﬂectivity ΔR/R ∼ 10−4 . The nonequilibrium carriers accumulated in quasiparticle states above the energy gap displayed, due to a bottleneck eﬀect, slow relaxation which was studied by a second probe laser pulse. Two distinct relaxation times were observed in various cuprate crystals, which were ascribed to two diﬀerent gaps: the superconducting one and the pseudogap. In particular, a slower decay with the relaxation time τA ∼ 3 ps were observed in YBCO crystals which revealed a strong temperature dependence and disappeared above Tc . A faster one with the relaxation time τB ∼ 0.5 ps was temperature independent but showed dependence on the carrier concentration x. The second one was observed even above Tc up to some T ∗ > Tc [274]. This suggests that the slow QP relaxation is determined by the superconducting gap Δc (T ), while the fast one is due to the pseudogap Δps (x) ∝ 1/x. A coexistence of the two distinct gaps was found in the overdoped and optimally doped YBCO crystals, in YBCO-124 [297] and other cuprate compounds. A similar temperature dependence of the peak amplitude ΔR/R and the magnetic resonance mode (see Sect. 3.2.3) observed in the MIR reﬂectivity measurements [574] supports an important role of a coupling between charge and spin excitations in cuprates. Optical Spectral Weight Transfer at Superconducting Transition In conventional superconductors, the pairing of quasiparticles (QP) into Cooper pairs results in an increase in the kinetic energy since the electrons forming the pair have to occupy states outside the Fermi sea. This increase of the kinetic energy is overcompensated by the lowering of the potential energy determined by the attractive potential. The change of the kinetic energy is of the order of (Δ/EF )2 and in conventional superconductors with the large Fermi energy EF and a small gap Δ is extremely small, ∼ 10−6 –10−8. In the cuprate superconductors, with much larger superconducting (SC) gap and the lower Fermi energy, this ratio can be of the order of 10−2 . Therefore, it may be possible to distinguish diﬀerent scenarios of high-temperature superconductivity by studying the inﬂuence of SC transition on the intraband optical spectral weight, which can be correlated with the change of the electron (hole) kinetic energy (see e.g., [1299, 1300]). For instance, in several theoretical models a “kinetic-energy-driven” scenario of the high-temperature superconductivity has been proposed when the lowering of the energy in the SC phase is due to a decrease of the electronic kinetic energy opposite to the BCS theory (see Sect. 7.5.3). Below we brieﬂy give a theoretical background and discuss several experimental results in studies of the optical spectral weight at SC transition. The partial sum rule (5.27) for the intraband transition below the cutoﬀ frequency Ωc ∼ W/¯ h of the order of the bandwidth W determines the optic plasma frequency (5.28) in terms of the spectral weight (SW)

Ωc A(Ωc ) = 8

σ1 (ω)dω = 0

4πne2 4π χ0 , = ∗ m V

(5.46)

5.3 Optical Electron Spectroscopy

283

where V is the system volume and χ0 is the static current–current correlation function. Within the Kubo theory of conductivity, χ0 is determined by the dispersion relation for the current–current susceptibility χJJ (ω):

1 Ωc dω i (5.47) ImχJJ (ω) = [Jx , Px ], χ0 = π −Ωc ω ¯h where Ωc should be large enough to take a proper account of contributions within the bandwidth energy. Here [Jx , Px ] is an equilibrium average of the commutator of the current Jx and polarization Px operators of the electronic = dPx /dt is deﬁned as a time system. The electronic current operator Jx derivative of the polarization operator Px = e i Rix Ni where Rix is the x-axis component of an electron coordinate and Ni is the number operator. The global f -sum rule (5.26) follows from (5.46) in the limit Ωc → ∞: A(∞) = (∞) (4π/V ) χ0 = ωp2 = 4π ne2 /m where n is the density of electrons with the bare mass m. To calculate χ0 , let us consider a single-band model speciﬁed by the kinetic energy HK = − tij c†i,σ cj,σ = ε(k) c†k,σ ck,σ , (5.48) i =j,σ

k,σ

where tij is the electron hopping parameter between lattice sites i = j, which determines the electron dispersion ε(k) in the band. c†i,σ , cj,σ are creation and annihilation operators for electrons with spin σ. In this notation, Px = h)[Px , H] = (ie/¯h) i =j,σ tij [Rix −Rjx] c†i,σ cj,σ e i,σ Rix c†i,σ ci,σ and Jx = −(i/¯ and we obtain the following equation for the static correlation function: χ0 =

e2 e2 ∂ 2 ε(k) tij [Rix − Rjx ]2 c†i,σ cj,σ = 2 nk,σ , 2 ¯h i =j,σ h k,σ ∂kx2 ¯

(5.49)

where nk,σ = c†k,σ ck,σ is the electron occupation number. For a model with the nearest neighbor hopping only, tij = t δj,i+a and [Rix − Rjx ]2 = a2 , one can write the static correlation function (5.49) in terms of the average kinetic energy (5.48). This results in the following relation for the SW (5.46): A(Ωc ) = 8

Ωc 4πe2 a2 HK . σ1 (ω)dω = − 2 h V ¯

(5.50)

0

Thus, ∞ neglecting interband transitions determined by the excess SW ΔA(Ωc ) = 8 Ωc σ1 (ω)dω we can correlate the intraband SW A(Ωc ) with the average kinetic energy for a model with the nearest neighbor hopping. For a more complicated band structure model, the SW (5.46) determined by the static correlation function (5.49) may strongly deviate from the average kinetic energy (5.48). In particular, quite a diﬀerent behavior of the intraband SW (5.46) in comparison with the average kinetic energy was found by

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5 Electronic Properties of Cuprate Superconductors

Marsiglio et al. [797] close to a van Hove singularity in electronic spectrum. Moreover, the simple version of the partial sum rule (5.50) does not apply to multilayer cuprates such as YBCO and Bi-type compounds (see e.g., [874]). Therefore, to draw any conclusion concerning the inﬂuence of the SC transition on the kinetic energy from optical SW studies one should exclude the band structure eﬀects. In an SC state, condensation of Cooper pairs results in a delta function peak in the real part of the conductivity, which can be written as reg (ω). From the sum rule (5.26) applied σ1,s (ω) = π e2 (ns /m∗ ) δ(ω) + σ1,s for the normal and SC states allowing for the superﬂuid density (4.41) 2 2 2 ∗ ρα s = c /λα = 4πe ns /mα , we can derive the Ferrel–Tinkham–Glover (FTG) sum rule [332, 1245]:

∞ ρs = 8

reg [σ1n (ω) − σ1s (ω)] dω,

(5.51)

0+ reg where σ1n and σ1s are the conductivity in the normal and SC states, respectively, at ω > 0. In conventional superconductors, the SW transfer from the normal state to the delta-function peak in SC state occurs within the energy range of the order of h ¯ Ωc ∼ 4Δ0 . Therefore, integration over frequency in (5.51) can be performed only within the low frequency region Ωc so that ρs An (Ωc ) − As (Ωc ) (see bottom panel in Fig. 5.38). More generally, we can write the “partial” FTG sum rule referring to the intraband transition in (5.50) in the form:

ρs = An (Ωc ) − As (Ωc ) + α [Kn − Ks ],

(5.52)

h2 V and Kn,s = HK n,s is the average kinetic energy in where α = 4π e2 a2 /¯ the normal and SC states, respectively. In conventional superconductors, the contribution from the last term is negligibly small and all contribution to ρs emerges from the ﬁrst low-energy term. In cuprate superconductors, the diﬀerence An (Ωc ) − As (Ωc ) provides only for a part of the SC condensate within the energy scale of several gaps: ¯hΩc ∼ 0.15 eV, and one should suggest an SW transfer at SC transition from much higher energies, up to the mid-infrared region of 0.5–1 eV or even from the interband transitions. This “violation” of the low-energy FTG sum rule according to (5.52) can be accounted for by a decrease of the kinetic energy at the SC transition as proposed by several groups (see e.g., [64, 67, 219, 221, 465, 466] and references therein). This change of the kinetic energy is caused by an SW transfer from higher energies as follows from the global (5.51) and “partial” (5.52) FTG sum rule:

∞ 8 Ωc

reg [σ1n (ω) − σ1s (ω)] dω = α [Kn − Ks ].

(5.53)

5.3 Optical Electron Spectroscopy 0

25

meV 75

50

100

285

125

1.0

[Nn(ω)-Ns(ω )] ρs

0.5

Tl2Ba2CuO 6+x

0.0 1.0

La2-x SrxCuO4

0.5 0.0 1.0

YBa2Cu3O6.6

0.5 0.0

0

500

250

750

1000

–1

500

Wave numbers (cm ) 1000 0 500

1.0 Dirty Limit Superconductor 0.0

0

2

4

6 0 ω/Δ

1000 1.0

T= Tc T= 0 2

4

6

0

σ1(ω) /σdc

0

Fig. 5.38. Normalized spectral weight diﬀerence N (ω) = (Nn (ω) − Ns (ω))/ρs for the interlayer conductivity in the Tl2 Ba2 CuO6+x , La2−x Srx CuO4 , and YBa2 Cu3 O6.6 cuprates. Bottom: results for a conventional superconductor in the dirty limit with the scattering rate Γ = 20Δ0 (left panel) and spectra of σ1 (ω)/σdc (right panel). Data courtesy of [114]

The lowering of kinetic energy means a decrease of the eﬀective mass, i.e., electrons or holes become lighter (theory of hole “undressing” in the SC state, see Sect. 7.5.3). The anomalous behavior of the SW transfer at the SC transition in cuprates most distinctly was registered in the interlayer infrared conductivity. The c-axis optic measurements revealed a strongly incoherent and extremely weak infrared absorption in a broad energy range without a clear evidence for a Drude peak (for a review see see [116]). An incoherent single-particle tunneling in the normal state changed to a coherent Josephson tunneling of pairs in the SC state, which manifests itself as a Josephson plasma resonance (see e.g., [1299]). In particular, studies of the interlayer infrared conductivity in the Tl-2201, LSCO and YBCO compounds revealed a strong violation of the partial FTG sum rule as shown in Fig. 5.38 [114]. The superﬂuid density determined independently from the imaginary part of the conductivity, ρs = 4πω σ2 (ω), far exceeds the SW accumulated from the normal-state infra-red conductivity.

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5 Electronic Properties of Cuprate Superconductors

Fig. 5.39. Normalized spectral weight NT c (see the text) vs. the static conductivity σdc for the interlayer conductivity in the Tl2 Ba2 CuO6+δ , La2−x Srx CuO4 , and YBa2 Cu3 O6+δ cuprates (after [115])

The normalized diﬀerence between the spectral weights in the normal state Nn (ω) at T ∼ Tc and SC state Ns (ω) at T Tc deﬁned as N (ω) = (Nn (ω) − Ns (ω))/ρs is of the order of N (ω) ∼ 0.5 for h ¯ ω up to 0.12 eV (here and in Figs. 5.38 and 5.39 the SW A(ω) (5.46) is denoted as N (ω)). The most striking observation is that the SW diﬀerence N (ω) is quite large within the whole infrared region. Contrary to conventional superconductors with the pairing energy of the order of phonon frequencies, in cuprates the SC condensate partly stems from much higher energies, which suggests unconventional pairing mechanism. A systematic studies of the doping dependence of the partial sum rule have shown that in the overdoped (OD) crystals the partial FTG sum rule violation is less pronounced in comparison with underdoped (UD) samples. Figure 5.39 shows normalized spectral weights NT c obtained from the integration of the interlayer conductivity N (ω) up to the cutoﬀ energy 0.1 eV as a function of the static conductivity σdc along the c-axis at T ≈ Tc for various cuprates [115]. For relatively well conductive crystals, like optimally or OD YBCO crystals with a modest anisotropy, the FTG sum rule is fulﬁlled: the ratio NT c ∼ 1. For the crystals with a weak interlayer conductivity like Tl-2201 and UD YBCO crystals, a strong violation of the partial FTG sum rule is observed: NT c ∼ 0.5, that points to a large kinetic energy change at the SC transition. A similar dependence was found in the c-axis optical studies in LSCO crystal [682] where NT c ∼ 0.2 and ∼ 0.8 for UD and OD samples, respectively, for ω ∼ 0.3 eV. These studies have shown that the superﬂuid spectral weight is accumulated from lower energies when the c-axis conductivity has a more coherent, delocalized character.

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287

Based on this observation, an interlayer tunneling model (ILT) of hightemperature superconductivity has been proposed [64,67,219,221]. The theory suggests that in the normal state charge carriers are conﬁned within the copper–oxygen layers and a coherent single-particle tunneling between the layers is blocked. In the SC state, Josephson pair tunneling becomes available and the energy gain in SC state is just the Josephson energy EJ , which should be equal to the condensation energy U (0) = ηEJ with η ∼ 1. An estimation of the kinetic energy change below Tc can be performed on the basis of the sum rule (5.52) from the interlayer optic studies or by measuring the Josepson interlayer plasma frequency ωJ , which determines the Josephson energy EJ and the corresponding interlayer kinetic energy. The condensation energy U (0) can be calculated from the speciﬁc heat measurements (see (4.22) in Sect. 4.2.1). These studies revealed that for YBCO and Bi-2223 cuprate superconductors the lowering of kinetic energy evaluated from the sum rule (5.52), with a proper account of the multilayer structure of the compounds, is comparable with condensation energy [874]. However, for the single-layer Tl- and Hg- compounds the value of kinetic energy saving was found much smaller than the condensation energy. For instance, the Josephson energy EJ calculated from the measured Josephson interlayer plasma frequency ωJ in Tl2 Ba2 CuO6 crystals was found much smaller than the condensation energy: η = EJ /U (0) 0.002 [1279], which could not be reconciled with the prediction of the ILT model. Certain arguments in support of the ILT model were proposed by Chakravarty et al. [219], who pointed out that accurate determination of the condensation energy in Tl-compound from heat capacity characterized by strong ﬂuctuations at the SC transition is diﬃcult, while the c-axis penetration depth measurements are controversial. Thus, in multilayer cuprate superconductors a signiﬁcant contribution to the condensation energy emerges from an increase of interlayer kinetic energy, ΔK/U (0) ∼ 0.5, while in single-layer Tl- and Hg-based compounds this contribution appears to be much smaller, though the interlayer tunneling can be considered as an enhancement for an original in-plane pairing mechanism [221]. It should be pointed out that a strong violation of the FTG sum rule is observed in those compounds where the pseudogap state develops at T ∗ > Tc in the underdoped region. The interlayer conductivity probes electronic states close to the Brillouin zone boundary due to a characteristic wave vector dependence of the hopping integral along the c-direction, t⊥ (kx , ky ) ∝ (cos kx − cos ky )2 (see Sect. 4.3.3). As discussed in Sect. 5.2.2, the electronic excitations in this (“antinodal”) region show essentially incoherent character in the pseudogap state in the underdoped region. Therefore, it was suggested that the pseudogap state in UD samples is responsible for the anomalous c-axis conductivity with a large decrease of the kinetic energy at SC transition [115]. Studies of the c-axis magnetic ﬁeld dependence of the SC condensate revealed a large interlayer optic SW transfer back to the gap region under applied magnetic ﬁeld [629]. It was found that the transferred

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5 Electronic Properties of Cuprate Superconductors

SW far exceeded the total c-axis SC condensate SW. This was explained as a pseudogap SW transfer simultaneous with the SC SW. This provides evidence of a competing nature of the pseudogap and SC states. Similar studies of the in-plane SC-induced optical SW transfer have demanded much more accurate measurements since the SW change at the SC transition is only a small fraction ∼ 10−3 of the large in-plane SW with a well-pronounced Drude peak of coherent excitations. Both the infrared reﬂection and ellipsometric measurements in a broad energy range have been used to extract this tiny change of the SW at the SC transition. In particular, Molegraaf et al. [845] have reported a blue shift of the plasma frequency below Tc with the SW transfer from the interband to the infrared SW in the optimally (OP) and UD single crystals of Bi-2212. This was explained by the increase of the kinetic energy at the SC transition as has been predicted by Hirsch (see [469] and references therein). Ellipsometric measurements by Boris et al. [159] on the OP doped detwinned YBa2 Cu3 O6.9 single crystal (Tc = 92.7 K) and for the slightly underdoped Bi-2212 crystal in a broad region of energy also revealed a T -dependent SW shift from high to low energy. While a conservation of the total spectral weight was observed in the normal state with temperature lowering, at the SC transition the total SW loss was inferred from the temperature dependence of ε1 (ω). The intraband SW loss at the SC transition was considered as an increase of the kinetic energy, while a blue shift of the dielectric constant ε1 (ω = 0) was explained by the narrowing of the Drude peak. Similar experimental data of the ellipsometric and temperature dependence measurements were obtained by Kuzmenko et al. [683], but a diﬀerent analysis of the data has provided evidence for the SC-induced transfer of the interband to the intraband SW for the OP doped Bi-2212 crystal and corresponding decrease of the electronic kinetic energy. Later on, the SW increase at SC transition was conﬁrmed by SantanderSyro et al. [1092] in measurements on high-quality Bi-2212 thin ﬁlms. A clear violation of the FGT sum rule was found in the UD sample up to the high energy of 1 eV. The corresponding lowering of the kinetic energy according to (5.52) was estimated of the order of 0.5–1 meV per copper site, which is much larger than the condensation energy. Further studies [199,200,276] have shown that a crossover is occurred from the unconventional behavior in the UD samples, with decrease of the kinetic energy at the SC transition, to the conventional BCS-like in the OD samples, with increase of the kinetic energy at SC transition. This crossover correlates with the doping dependence of the interlayer optic SW transfer and a transition from incoherent electronic spectra in the UD samples to a quasiparticle-like in the OD region as seen in ARPES. It should be pointed out that the in-plane conductivity is dominated by more coherent nodal electronic excitations in comparison with the antinodal, incoherent one in the c-axis conductivity (in the hole-doped cuprates). This diﬀerence explains the contrasting behavior seen in the in-plain and c-axis optical data.

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The problem of the SW transfer and a possible kinetic energy shift at the SC transition were considered in a number of theoretical studies. In particular, it was shown that the SW dependence on the quasiparticle (QP) scattering rate may explain the partial sum rule violation. A simple model with a frequency-dependent scattering rate was considered by Norman et al. [899]. It was argued that in conventional superconductors an increase of the kinetic energy is due to the particle-hole mixing below Tc , which results in a broadening of the momentum distribution nk of well-deﬁned QPs in the normal state. In the cuprate superconductors, the formation of QPs occurs only in the SC state which, contrary to the conventional superconductors, sharpens the momentum distribution nk resulting in a decrease of the kinetic energy (see also [798]). By taking into account optical data for the scattering rates in the Bi-2212 samples for various doping [1033], the in-plane kinetic energy shift below Tc was calculated. The results were close to the experimental data. In particular, no low frequency sum rule violation was observed in the overdoped region with well-deﬁned in-plane QPs in the normal state. By considering the temperature dependence of the relaxation rate within a one-band electron–phonon model, Karakozov et al. [588] have shown that the sum rule may show up a various temperature behavior depending on the strength of the electron–phonon interaction, while a contribution from the SC gap is negligible. By changing the parameters of the model, it was possible to explain an experimentally observed temperature dependence of the sum rule. Another approach of analyzing the SW transfer at the SC transition is based on consideration of models with strong electron correlations. In particular, the t–J and Hubbard models which is believed to be relevant for the cuprate superconductors have been studied (see Sect. 7.1.2). The energy scales in these models are diﬀerent. In the Hubbard model, the kinetic energy is determined by the hopping parameter t, both for the low energy intraband transitions in one subband and for the interband transitions with high-energy of the order of the on-site Coulomb energy U between singly and doubly occupied subbands. In the t–J, model the kinetic energy is determined by the hopping parameter t only in one-band, while the interband transitions are projected out by the canonical transformation, which results in the exchange interaction J ∼ t2 /U . Therefore, in comparison of experimental data with the theoretical results, it is important to take into account that depending on the cutoﬀ frequency Ωc in the SW (5.46) one probes either intraband transitions for Ωc < t or interband transitions for Ωc ∼ U in the Hubbard model. In the t–J model, only low energy excitations Ωc < t give a contribution to the kinetic energy, while the exchange energy plays the role of BCS-like attraction. In this context, there is no diﬀerence between the SC transition driven by the kinetic energy lowering in the Hubbard model and the SC transition driven by the potential energy decrease in the t–J model. As was pointed out by Chakravarty et al. [221], this is “a semantic problem” which depends on the low-energy eﬀective Hamiltonian employed in calculations.

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5 Electronic Properties of Cuprate Superconductors

This diﬀerence in the energy scales of the models has been conﬁrmed by Wrobel et al. [1370] in detailed studies of the conductivity in the t–J and Hubbard models. Using the spin polaron approximation in the t–J model at low hole doping, they obtained a lowering of the kinetic energy and the potential energy (the exchange interaction) of holes. The calculations of the kinetic energy in the Hubbard model have shown its decrease at the SC transition. Similar results for the Hubbard model were obtained within the dynamical cluster approximation by Maier et al. [772] who found a lowering of the total kinetic energy below Tc for various doping levels, including the strongly overdoped region. At the same time, studies of the t–J model within the cluster dynamical mean ﬁeld theory revealed a lowering of the hole kinetic energy in the UD regime, while it increases above the optimal doping, as observed in the experiments discussed above [200, 436]. However, the kinetic energy variation at the SC transition appeared to be rather small in comparison with the lowering of the superexchange energy. In the overdoped regime, the gain in the exchange energy exceeds the increase of the kinetic energy as in the BCS-like system. Resume To summarize the studies of the optical spectral weight (5.46) at the SC transition, we can emphasize the following results. A distinct violation of the low-energy FTG sum rule (5.52) was observed in the c-axis weak incoherent infrared optical conductivity in a broad energy range. This can be accounted for by a decrease of the kinetic energy at the SC transition. The violation is particularly strong in the UD regime while in the OD regime it becomes less evident (see Fig. 5.39). Measurements of the SW transfer for the in-plane conductivity at the SC transition is less conclusive since the diﬀerence of the SW amounts only ∼ 10−3 of a much more intensive and more coherent infrared in-plane conductivity. Nevertheless, a certain trend was detected from a noticeable decrease of kinetic energy in the UD region to its increase in the OD region. This corroborates with transition from a non-Fermi-liquid like behavior in the UD region to a more coherent quasiparticle-like in the OD region observed in ARPES and other experiments in cuprates. It is important to stress that the SW transfer at the SC transition comes from much higher energies than in conventional superconductors, up to the mid-infrared region or even from the interband transitions. This suggests a certain contribution to the pairing energy from the interband transition in cuprates, of the order of the charge-transfer gap ∼ 1.5 eV. However, there is no commonly accepted theoretical model, which can explain all the experimental data. 5.3.4 Electronic Raman Scattering Raman scattering (RS) has become an eﬃcient technique in studies of electronic excitations close to the Fermi energy and many-body eﬀects in the

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strongly correlated electronic systems as cuprates. In contrast to the infrared spectroscopy, the symmetry selection rules for RS enable to measure momentum dependent transport properties by projecting out diﬀerent regions of the Brillouin zone (BZ). In particular, observation of a strong anisotropy in Raman spectra in high-temperature superconductors has given an additional conﬁrmation for the d-wave symmetry of the superconducting gap in cuprates [278]. A general theory of phonon and electron RS and experimental results are reviewed recently by Sherman et al. [1151], Falkovsky [321], and Devereaux et al. [280]. Magnetic light scattering experiments also give valuable information concerning magnetic interactions in cuprates [405, 707]. Below we give a brief introduction to the theory of the RS and present several examples of experimental investigations of the electronic RS in cuprates, while the phonon RS will be discussed in Sect. 6.2. In the RS experiments, one measures the light scattering intensity as a function of the frequency transfer Ω = ωI − ωS of incident (I) and scattered (S) light with the frequency ωI and ωS and the polarization eI and eS , respectively. Since the light wavelength is much larger than the lattice constants, the scattering wave vector q = kI − kS is close to zero and in the ﬁrst order RS only phonons at the Γ -point of the BZ and long-wavelength electronic ﬂuctuations can be studied. However, second-order processes in RS make it possible to study electronic excitations in a broad region of energy and wave vectors. In RS experiments, one studies a response of charge ﬂuctuations in solids induced by both phonons and electron density ﬂuctuations. The electronic RS spectral density is given by the dynamical structure function for the electron polarization ﬂuctuations: 1 S(q, Ω) = − [1 + n(Ω)] Im χρ˜ ˜ρ (q, Ω), π

(5.54)

ρ(q)|˜ ρ(−q)Ω where n(Ω) is the Boze–Einstein distribution and χρ˜ ˜ρ (q, Ω) = ˜ is the electron density–density susceptibility which determines the spectral density of ﬂuctuations. Comparison of the intensities of the energy loss (Stocks, Ω > 0) and energy gain (anti-Stocks, Ω < 0) spectra can be used to determine the temperature of the sample. Since the optical conductivity σ(Ω) is determined by the current–current susceptibility, a general relation between the optical conductivity and the RS spectral density (neglecting anisotropy of RS) can be inferred: S(Ω) ∝ Ωσ(Ω) which shows that S(Ω → 0) → 0. For the electron–light interaction (5.22), the polarization ﬂuctuations is determined by the eﬀective electron density ﬂuctuations induced by light scattering: ρ˜(q) = γ(k, q) c†k+q,σ ck,σ , (5.55) k,σ

where c†k,σ , ck,σ are the creation and annihilation operators for an electron with the wave vector k and spin σ. The Raman vertex γ(k, q) depends on the

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5 Electronic Properties of Cuprate Superconductors

polarization of the incident and scattered photons and can be written in the form (see e.g., [280]): β eα γ(k, q) = I eS γα,β (k, q), α,β

1 k + q| pˆβ |kν kν | pˆα |k m Ek − Ekν + ¯hωI kν k + q| pˆα |kν kν | pˆβ |k , + Ek+q − Ekν − ¯ hωS

γα,β (k, q) = δα,β +

(5.56)

where pˆα is the electron momentum operator and Ek , Ekν are electron energies. The ﬁrst term comes from the ﬁrst-order two-photon scattering ∝ (A)2 in (5.22) and gives contribution only for nonorthogonal polarizations of the incident and scattered photons. The second term arises from the single-photon scattering ∝ (ˆ pA) in (5.22) in the second order via intermediate states kν , which include both intraband and interband transitions. In the limit of small q and low photon energies, ¯hωI,S |Ek −Ekν |, an eﬀective mass approximation can be used: h2 ¯ ∂ 2 Ek γα,β (k, q → 0) = . (5.57) m ∂kα ∂kβ For the incident or scattered photon energy close to the electron interband transition energy, h ¯ ωI,S ∼ |Ek − Ekν |, resonant RS can occur which strongly enhances the scattering intensity. Depending on the light polarization, electronic density ﬂuctuations of different symmetry can be probed by RS. In particular, for the square CuO2 lattice for the light propagating perpendicular to the plane, q zˆ, and polarized in the (x, y) plane the Raman tensor in (5.56) can be presented as the sum of three terms related to the irreducible representations A1g , B1g , and A1g B1g B2g + γα,β + γα,β . These contributions are determined by the B2g : γα,β (k) = γα,β functions A

B

B

γk 1g = γxx (k) + γyy (k), γk 1g = γxx (k) − γyy (k), γk 2g = γxy (k).

(5.58)

The corresponding selection rules related to these irreducible representation result in the following dependence of the Raman spectral function on the light polarization: S(A1g ) ∝ (eI eS )2 ,

S(B1g ) ∝ (exI exS − eyI eyS )2 ,

S(B2g ) ∝ (exI eyS + eyI exS )2 .

Usually, the following polarization geometries in (x, y) plane are used: (i) (XX) eI eS x ˆ, (ii) (X X ) eI eS , eI x ˆ , ˆ, (iv) (X Y ) eI ⊥ eS , eI x ˆ , (iii) (XY ) eI ⊥ eS , eI x √ where x ˆ = (ˆ x + yˆ)/ 2 and xˆ, yˆ are directed along the unit cell axes ax , ay . Depending on the geometry, diﬀerent components of the Raman tensor can

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be measured: the A1g and B1g symmetries in the (XX) geometry, the A1g and B2g symmetries in the (X X ) geometry, the B2g symmetry in the (XY ) geometry, and B1g symmetry in the (X Y ) geometry. Thus, to extract RS scattering of the A1g symmetry one should perform measurements in several geometries. In the eﬀective mass approximation (5.57), the functions (5.58) for a tightA B binding model can be estimated as follows: γk 1g ∝ t (cos kx a+cos ky a), γk 1g ∝ B2g t (cos kx a − cos ky a), and γk ∝ t (sin kx a sin ky a) where t and t are the nearest [±ax , ±ay ] and the next-nearest [±(ax ± ay )] hopping integrals (the lattice constants ax = ay = a). Therefore, RS of the B1g symmetry probes the charge excitations close to the antinodal (π, 0) regions of the BZ, the B2g symmetry – close to the nodal (π/2, π/2) regions, while for the A1g symmetry the charge excitations in the whole BZ give contributions (excluding the AF BZ boundary: |kx ±ky | = π/a). In cuprates, |t /t| is usually small and therefore scattering of the B2g symmetry should be the weakest. However, many-body eﬀects may strongly renormalize the hopping parameters, especially in the underdoped region (see Appendix (A.48)). For a rigorous calculation of the Raman vertex (5.56), a detailed knowledge of the electronic band structure in the BZ is needed with a proper account of Coulomb correlations and screening eﬀects. A simple eﬀective mass approximation (5.57) can be used only for low energy RS in a one-band model. The excitation spectrum of charge ﬂuctuations at low energies for the Fermi liquid model is linear in q and the Raman shift Ω ∼ vF q where vF is the Fermi velocity. The RS in metals in the clean limit can be observed for the wave vector transfer q ∼ 1/δ, where δ is the light penetration depth. For cuprates with low charge concentration and large δ, this results in a featureless Raman spectrum in the low energy region up to h ¯ Ω ∼ 10 meV. In contrast to this, in all cuprates broad spectral functions for electronic RS are observed as can be seen in Figs. 5.40 and 5.41. It can be suggested that strong manybody interactions result in the intensive incoherent contribution as in ARPES experiments (see Sect. 5.2.2). To explain these observations, a relaxation model for anisotropic charge ﬂuctuations was proposed by Zawadowski et al. [1409]. The RS spectral function in the model is described by the Drude-like function S(Ω) = [1 + n(Ω)] N (0) |γL |2

ΩτL , 1 + Ω 2 τL2

(5.59)

where N (0) is the electron density of states (DOS) at the Fermi energy and 1/τL is an eﬀective relaxation rate caused by impurities for the density ﬂuctuations speciﬁed by a quantum number L (e.g., an angular momentum or light polarization). However, the simple impurity scattering model (5.59) fails to give a quantitative description of experimental results for cuprates and various theoretical models for quasiparticle relaxation have been proposed. In particular, strong anisotropy of quasiparticle interaction and anisotropy of the Fermi surface (FS) in cuprates (e.g., Van Hove singularities, nesting eﬀects)

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5 Electronic Properties of Cuprate Superconductors

Fig. 5.40. Temperature dependence of the electronic Raman spectrum B1g symmetry in Bi-2212 crystal below Tc = 89 K (after [1151]) 0.4

a

b

0.3

0.4 0.3

0.2 χ (Ω) (arb. units)

0.5

∼0.4 cm–1

0.1

0.2 0.1

0.0 0.4

0.0

0.3

0.6

0.2

0.4

0.1

0.2

0.0

0.8

0

200 400 600 800 0 200 400 600 800 Raman shift (cm–1)

0.0

Fig. 5.41. Electronic Raman spectra of the B1g (a) and A1 g (b) symmetries for the photon energies h ¯ ω = 1.95 eV (upper panels) and ¯ hω = 2.7 eV (lower panels) in the Bi-2212 crystal at T = 10 K (solid lines – upper curves) and T = 90 K (dashed lines – lower curves). The hatched areas in panels (b) represent the 2Δ peaks (after [841])

were taken into account and diﬀerent channels of electron scattering induced by impurities, spin ﬂuctuations, electron–phonon interaction were considered (see [280, 1151]). As a result, a relaxation-like behavior for the RS spectral function (5.59) was obtained with the frequency and temperature-dependent anisotropic relaxation rate 1/τL (Ω, T ) as in the extended Drude model (5.30) for conductivity. In the normal state, the slope of the Raman spectrum (5.59)

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at Ω → 0 determines the relaxation time τL (T ). The values of the relaxation rates ΓL = 1/τL (Ω, T ) observed in the optimal doped Bi-2212 compounds at 300 K are in a reasonable agreement with that one found for the infrared conductivity. A substantial dependence of the RS spectral function on the photon energy was found in cuprates, as shown in Fig. 5.41 for Bi-2212 crystal [841]. A complicated band structure in the region of few electron volt in the vicinity of the FS in cuprates gives rise to resonance eﬀects in the Raman vertex (5.56). Electron correlations also aﬀect the temperature and frequency dependence of the RS intensity. Speciﬁcally, the resonant scattering caused by particle-hole excitations close to the Hubbard (charge-transfer) energy results in a strong enhancement of the nonresonant scattering and changes frequency dependence of the spectral density (see e.g., [1165]). One of the unsolved problems is a large intensity in the fully symmetrical A1g polarization, which should be suppressed due to Coulomb screening in comparison with other polarizations. The resonant interband scattering may be partly responsible for the large intensity of the A1g scattering. RS experiments in the high-frequency region Ω ∼ 3,000 cm−1 have been used to study magnetic scattering in the AF insulating parent compounds of the superconducting cuprates and samples in the underdoped region. In particular, two-magnon scattering enables one to measure the AF exchange energy J and its dependence on doping. At the photon-induced two-magnon scattering, a superexchange of two spins on the nearest neighbor Cu sites occurs. The energy of two-magnon scattering can be estimated as ¯hΩ 3J, or more accurately, by taking into account renormalization of the magnon energy, h ¯ Ω 2.7J (see e.g., [235]). Since the superexchange is mediated by the Cu3d–O2p electronic transitions, two-magnon scattering should show a resonant enhancement when the energy of the incident (or scattered) photon ¯hωI(S) is close to the charge-transfer energy εp − εd . Two-magnon scattering has been observed in various hole-doped cuprates in the B1g symmetry with resonance enhancement for photon energy in the region h ¯ ωI(S) ∼ 3 eV (see e.g., [142,1197]). With doping, the two-magnon scattering peak broadens, weakens, and shifts to lower frequency due to reduction in the exchange energy by doped carriers. In the hole-doped cuprates, the two-magnon scattering can be detected up to the optimal doping, while in the electron-doped cuprates the scattering vanishes at electron doping x ∼ 0.15 (see e.g., [925]). Generally, the magnitude of the exchange energy measured by two-magnon scattering agrees well with those one obtained from inelastic neutron scattering experiments (see Sects. 3.2.4 and 3.2.2). Experimental results of two-magnon scattering studies are discussed in several reviews (see e.g., [405, 707]). To explain an asymmetric proﬁle of the two-magnon scattering peak and its intensity dependence on the incident photon energy in antiferromagnetic cuprates, a theory of resonance scattering within the Hubbard model was considered by Chubukov et al. [235].

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5 Electronic Properties of Cuprate Superconductors

In the superconducting state, a gap Δ is opened in the electronic spectrum, which makes it possible to observe RS in the clean limit even at q = 0. For energy h ¯ Ω > 2Δ breaking of Cooper pairs with q = 0 by incoming photon gives rise to a coherent “pair-breaking peak” in the Raman spectrum as has been shown by Abrikosov and Falkovsky [8]. For a superconductor with a ¯ Ω < 2Δ0 ﬁnite energy gap Δ0 on the FS, the RS intensity should vanish for h as in fact has been observed for the conventional s-wave superconductors. However, in cuprates the RS was found to be ﬁnite at low frequencies below Tc as shown in Fig. 5.40. This points to a superconducting gap with nodes. For a general case of an anisotropic gap Δ(k), the spectral density of the RS can be written in the form [618]: 4|γ(k)|2 Δ(k)2 S(Ω, T → 0) = N (0) Re , (5.60) hΩ (¯hΩ)2 − 4Δ(k)2 FS ¯ where . . .FS denotes the Fermi surface average. The pair-breaking peak appears at 2Δ in the superconducting state as a result of the redistribution of electronic continuum at low energy. The intensity and the position of the coherent pair-breaking peaks depend on the light polarization due to interplay of the polarization dependent Raman vertex and the gap symmetry. The B Raman vertex for the B1g channel γk 1g ∝ (kx2 − ky2 ) vanishes at the d-wave gap nodes and averaging over the FS projects out the maxima of the gap close to the nodal (N) (π, 0) regions of the BZ. Therefore, the B1g pair-breaking peak should have the highest energy position for the d-wave gap [278]. For the B2g and A1g channels, the gap nodes give contribution to the spectral intensity (5.60) and these peaks should have lower energy and weaker intensity. The low-frequency line shape, due to the same symmetry considerations, also depends on the light polarization. The d-wave gap nodes yield a linear Ω dependence for the density of states which results for the B2g and A1g symmetries in the linear dependence of the Raman spectral function (5.60) in the limit Ω → 0. For the B1g symmetry, the Raman vertex vanishes at the gap nodes. This gives additional Ω 2 dependence which results in the Ω 3 frequency dependence in (5.60) at low frequencies. These general features have been observed for various types of the holedoped cuprates close to the optimal doping which has conﬁrmed the d-wave pairing symmetry. Figure 5.40 shows the pair-breaking 2Δ peak for nearly optimal doped Bi-2212 crystal with Tc = 89 K for the B1g polarization [1151]. It is clearly seen that below the superconducting transition the pair-breaking peak appears. The low frequency part of the spectrum for the B1g polarization shows the Ω 3 dependence, whereas the B2g and the A1g spectra are linear in frequency. The pair-breaking peaks reveal resonance properties as shown in Fig. 5.41 for the B1g (a) and A1 g (b) symmetries for the photon energy h ¯ ω = 1.95 eV (upper panels) and h ¯ ω = 2.7 eV (lower panels) in the Bi-2212 crystal with Tc = 89 K [841]. The maxima and interception points for the B1g peak (a)

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shift to higher energies for higher photon energy as shown by the dashed lines. The 2Δ peaks for the A1 g symmetry (b) shown by the hatched areas are also changed. A noticeable redistribution of the low-energy part of the electronic spectrum depending of the photon energy is also observed, while sharp phonon-induced peaks are unchanged. An important role of the interband transitions in multilayer cuprates was revealed in studies of the A1g Raman spectra at diﬀerent photon energies in Bi-, Hg-, and Ta-based compounds by Limonov et al. [724]. In particular, they observed for Bi-2223 compound a crossover in the A1g spectrum from a broad pair-breaking peak at h ¯ Ω = 6kB Tc to a sharp peak at h ¯ Ω = 8kB Tc for a small change of the photon excitation energy, from 2.54 eV to 2.18 eV. The Raman vertex is subjected to renormalization by the many-body interactions. In particular, Chubukov et al. [239] have predicted that electron interaction with spin-ﬂuctuations may result in appearance of a resonance in the B1g scattering channel which will strongly enhance the pair-breaking peak and shift it to lower frequency. Contrary to the magnetic interaction, the electron–phonon coupling suppresses the pair-breaking peak and shifts it to higher frequency. The authors argue that the investigation of the RS in B1g channel can allow to distinguish between phonon-mediated and spin-ﬂuctuation mechanisms of superconducting d-wave pairing in cuprates. Studies of the doping and temperature dependence of the pair-braking peaks have revealed their weak temperature dependence and a speciﬁc behavior with doping as shown in Fig. 5.42 [280]. Close to the optimal doping p = 0.16, the pair-breaking peaks are clearly seen for all three Raman active symmetries. The energy position of the peaks conﬁrm the d-wave pairing with the ratio of the superconducting gap maximum to the superconducting temperature 2Δ0 /kB Tc ∼ 8 in the hole-doped cuprates. The B2g peak follows

Fig. 5.42. The ratio h ¯ Ωpeak /kB Tcmax of the coherent peak energy to the maximum max for various cuprates vs. doping p. The positions of the B1g peaks are shown Tc by diamonds and the B2g peaks – by squares (for references see [280]). The data of ARPES [195] are shown by circles and of the tunneling density of states – by triangles [1405]. The dashed line represents 6Tc (p)/Tcmax (see (1.5)) (after [280])

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5 Electronic Properties of Cuprate Superconductors

Tc in the whole doping region, while the position of the B1g peak decreases near linearly with doping. In the underdoped region, the B1g peak has much higher energy: h ¯ Ω ∼ 15kB Tc , while in the overdoped region the ratio is lower: ¯hΩ ∼ 6kB Tc . Higher energy of the B1g peak in the underdoped region is due to the contribution from the pseudogap in the antinodal (AN) (π, 0) region. This contribution vanishes in the N direction probed by the B1g peak. In certain RS experiments, a scaling of the position of the A1g peak with the energy of the magnetic resonance mode (see Sect. 3.2.3) was reported. The polarization dependence of the RS enables also to study the pseudogap in diﬀerent parts of the BZ in the underdoped cuprates in the normal state. Measurements of RS at the B2g polarization on the underdoped YBCO and Y-doped Bi-2212 crystals by Nemetschek et al. [890] have revealed a reduction in the spectral weight in the frequency range 250–700 cm−1 below ∼ 200 K. At the same time, a metallic character of conductivity at small frequency suggests an opening of the pseudogap away from the N point probed by the B2g polarization. The energy scale for the pseudogap was estimated as comparable to the exchange interaction J, which led the authors to the conclusion that the pseudogap has a magnetic origin. Further RS studies of the dynamical scattering rate in these crystals have conﬁrmed a strong in-plane anisotropy [928]. The relaxation rate in the B2g channel (in the N direction) agreed quite well with transport data and IR low-frequency results and did not vary signiﬁcantly with doping. Contrary to that, the B1g relaxation rate showed a strong doping dependence. The pseudogap and the superconducting gap revealed a completely diﬀerent behavior with doping also. In the optimally doped region, the pseudogap takes up only a small part of the FS in the k-space and the superconducting gap is well deﬁned over the entire FS. In the underdoped region, the pseudogap is extended over a large part of the FS and the superconducting gap can be detected only close to the nodes. A reduction of the low-frequency spectral weight and an appearance of a sharp RS peak at ∼ 600 cm−1 in the underdoped Bi-2212 crystals above Tc in the B1g polarization were reported by Blumberg et al. [142]. The peak was ascribed to the pseudogap in the AN region of the BZ probed by the B1g polarization. Below Tc , the peak intensity showed a strong enhancement caused by transition to a coherent superconducting state. In the region of high energy, the two-magnon RS was observed in the B1g polarization. For the AF insulating compound (Y-doped Bi-2212), the two-magnon band was seen with the maximum at Ω ∼ 2,850 cm−1 . This yielded an estimate J ≈ 125 meV for the exchange energy. With doping the two-magnon scattering weakens and shifts to lower energy, while a short-range AF order persists up to the optimal doped region. Similar dichotomy between nodal and antinodal charge ﬂuctuations probed by RS has been conﬁrmed in several studies. Doping dependence of the B1g and the B2g polarization scattering in LSCO crystals in a wide range of doping reported by Naeni et al. [875] revealed a strong anisotropic quasiparticle interaction. The RS intensity in the B1g channel sharply decreased below

5.3 Optical Electron Spectroscopy

299

optimal doping signaling of the pseudogap opening in the AN region of the BZ, while the spectral weight in the B2g polarization weakly depended on doping. A large energy of the pseudogap estimated as Eg ∼ J ∼ 700 cm−1 implies that the origins of the pseudogap and superconducting gap are diﬀerent. An abrupt crossover near the optimal doping for the Raman relaxation rate evaluated from the slope of the RS spectra (5.59) was detected by Venturini et al. [1310] in Bi-2212 crystals. It was suggested that the electrons at the “hot” (AN) region exhibit a crossover from the metallic to the insulating state below the optimal doping. At the same time, the electrons at the “cold” (N) region show metallic behavior for all doping levels as has been detected in the infrared conductivity, which is determined mainly by the nodal electrons (see Sect. 5.3.2). Systematic RS studies of several cuprate superconductors, LSCO, YBCO, Bi-2212, and Bi-2201 from the underdoped to the overdoped region by Sugai et al. [1197] have revealed a crossover of the coherent peak position in k-space from (π/2, π/2) (B2g peak) to (π, 0) (B1g peak) at the transition from the underdoped to the overdoped region. This crossover is accompanied by the change of the electron DOS near the Fermi energy estimated by the low-energy part of the Raman spectra. The crossover occurs close to the optimal doping p ∼ 0.16 though depends on the material. In the underdoped region, the electron DOS is extremely low in the AN (π, 0) region which can be ascribed to the pseudogap formation as discussed above. At the optimal doping, the pseudogap vanishes and the B1g peak becomes more intensive than the B2g peak. However, the peak intensity variation is more rapid than the DOS changes. Studies of the two-phonon scattering intensity by the authors on the same samples have shown that the strength of the electron–phonon interactions also changes with doping. In the underdoped region, the strongest electron– phonon interaction comes from the LO breathing mode at (π, π), while in the overdoped region the interaction with the LO half-breathing mode at (π, 0) becomes stronger. At the same time, the magnetic interaction mediated by the AF exchange interaction measured by the two-magnon RS smoothly decreases with doping from J ∼ 120 meV in the AF insulating state and underdoped region to J ∼ 60 meV in the optimal doped region. The authors suggest that electron–phonon interactions should play a certain role in the superconducting d-wave pairing, enhancing or suppressing the superconducting pairing mediated most likely by AF spin ﬂuctuations, as suggested by Shen et al. [1144]. A clear identiﬁcation of diﬀerent energy scales for the pseudogap and the superconducting gap was obtained in the electronic RS study of the HgBa2 CuO4+δ single crystals in a broad region of doping, 0.09 < p < 0.25 [416,709]. (The hole concentration p was determined by using the universal formula (1.5) for Tc proposed by Presland et al. [1028]). The single-layer mercury cuprate superconductor Hg-1201 has some advantages over other cuprates for RS studies. A pure tetragonal symmetry, an ideal ﬂat CuO2 plane enables to measure B2g (N) and B1g (AN) responses without mixing caused by the

300

5 Electronic Properties of Cuprate Superconductors

lattice modulation, bucling, bilayer splitting observed in YBCO and Bi-type compounds (see Sect. 2.5). Studies of the doping and temperature dependence of the B1g and B2g spectra have revealed quite a diﬀerent behavior of the normal and superconducting properties in the OD, p > pc , and the UD p < pc , regions where the critical hole concentration pc = 0.16. In the OD region, the pair-breaking peaks in the B1g and B2g spectra are observed at a close energy, EAN EN which follows Tc (p) dependence: EAN 6 Tc . The gap energy EAN = 2ΔAN in this region strongly depends on temperature and follows that one expected for the BCS d-wave superconductor [416]. In the overdoped region, the AN energy gap EAN increases with underdoping in contrast to the N energy gap EN which decreases and follows Tc (p) as depicted in Fig. 5.42. An opposite temperature dependence of the AN B1g RS scattering intensity for the frequency shift 100–700 cm−1 was observed in the OD and UD regions in the normal state. In the OD sample, it decreases with increasing temperature as in conventional metals, while in the UD sample the intensity increases showing the “ﬁlling in” and closing at T ∗ ∼ 140 K the pseudogap [416]. These studies unambiguously show existence of two diﬀerent energy gaps in the UD region, which merge into one superconducting gap in the OD region. The nature of the normal state in these region is also distinctly diﬀerent. In the electron-doped superconductors, the pair-breaking peaks are also well pronounced in the optimal doped region for all three symmetry channels. However, contrary to the hole-doped cuprates, the highest intensity appears in the B2g symmetry pair-breaking peak, while the B1g symmetry contribution shows much weaker intensity. A detailed study of RS in the Pr2−x Cex CuO4−δ (PCCO) and Nd2−x Cex CuO4−δ (NCCO) compounds at various doping reported by Qazilbash et al. [1037] have revealed a complicated temperature, doping and magnetic ﬁeld dependence of the superconducting properties of these electron-doped compounds. It was observed that the coherence peak in the B2g symmetry persists for all doping levels, while the B1g and A1g peaks have much weaker spectral weight and can be well resolved only at optimal doping. The ratio 2Δ0 /kB Tc ∼ 4.5 (at optimal doping) is consistent with the mean-ﬁeld BCS values for the d-wave superconductors in the weak coupling limit in contrast to the strong coupling in the hole-doped cuprates. A depletion of the low-frequency intensity below Tc suggests that the gap has nodes on the FS. These ﬁndings can be explained by taking into account the ARPES experiments (see Sect. 5.2.2), which have shown a change of the FS topology in NCCO from electron-like pockets close to the (π/4, π)-like regions of the BZ at small doping to the hole-like pockets appearing in the nodal (π/2, π/2) regions at the optimal doping (see Fig. 5.24). Since the B2g symmetry channel originated from the N regions, it was concluded that the main contribution to the superconducting pairing comes from the charge carriers in the hole-like pockets on the FS, while the electron-like parts of the FS give a weaker contribution. This conclusion is corroborated by the peculiar angular dependence of the superconducting gap function deduced from ARPES (see Fig. 5.27).

5.4 Transport Properties

301

Studies of the RS in magnetic ﬁelds enable to measure the upper critical mag∗ 2Δ netic ﬁelds Hc2 at which the superﬂuid stifness vanishes and Hc2 at which 2Δ ∗ the coherent 2Δ peak is suppressed. It was found that Hc2 > Hc2 which suggests that a region of local pairing with phase ﬂuctuations should exist between these ﬁelds. Due to low values of the critical ﬁelds, a large superconA for the optimally doped and ducting coherence length was found: ξsc ∼ 60 ˚ ξsc ∼ 220 ˚ A for the overdoped samples. Resume To summarize, the polarization-sensitive electronic RS experiments have provided valuable information concerning the charge transport in the normal and superconducting states. 1. In the normal state, generally a broad spectrum of incoherent excitations is observed for all symmetries in the electronic RS due to strong quasiparticle scattering by impurities, spin- and charge-ﬂuctuations. 2. A large anisotropy of the charge transport is found. In the hole-doped cuprates, metallic conductivity is revealed in the nodal direction (B2g symmetry RS) for all doping, while in the antinodal direction (B1g symmetry RS) a crossover occurs from the metallic to the insulating state below the optimal doping. This conﬁrms opening of the pseudogap on the FS in the antinodal region. 3. In the superconducting state, the anisotropic pair-breaking coherent peak evolves from low-frequency excitations conﬁrming d-wave symmetry of the energy gap. In the overdoped region, the nodal B2g and antinodal B1g energy gaps show a similar temperature and doping dependence as in the d-wave BCS superconductors. In contrast, in the underdoped region energy gaps demonstrate quite a diﬀerent nature: the nodal gap scales with Tc (p), while the antinodal gap scales with the pseudogap temperature increasing with decreasing hole concentration p. 4. Similar results have been obtained for the electron-doped cuprates but with diﬀerent momentum dependence in agreement with ARPES data. The superconducting pairing of the charge carriers in the hole-like pockets on the FS in the nodal direction has been suggested in the weak coupling limit, contrary to the strong coupling for the hole-doped cuprates in the antinodal regions on the FS.

5.4 Transport Properties After the discovery of high-temperature superconductivity in cuprates the transport properties of the materials, i.e., conductivity, Hall eﬀect, thermal conductivity and thermopower, have been extensively studied. These studies have helped to elucidate the nature of electronic excitations near the

302

5 Electronic Properties of Cuprate Superconductors

Fermi surface. Earlier experimental data on the transport properties of hightemperature superconductors were reviewed by Ong [917], Iye [537], Uher [1290] and more recent results are discussed by Hussey [510]. Theoretical studies of the transport properties within the electronic band-structure calculations were reported by Allen et al. [49] and Pickett [981]. In the last years, a wealth of high precision experimental data on high quality cuprate single crystals has been obtained. Below we consider only a few examples of these studies which shed light on the anomalous electronic structure of the cuprate superconductors. For further discussion, it is convenient to consider brieﬂy a general theory of the transport properties in solids (see e.g., [83]). Consider a conductor under the action of an external electric ﬁeld Ej , magnetic ﬁeld Bk , and thermal gradient ∂l T . In the linear approximation, the electric current density Ji and the thermal current density Jiq in the system are described by the equations Ji = σij Ej + αil (−∂l T ),

Jiq = α ˜ ij Ej + κil (−∂l T ),

(5.61)

where summation over the repeating tensor indexes is assumed. Here, σij is the conductivity tensor, α ˜ ij = T αij are the thermoelectric (Peltier conductivity) tensors related by the Onsager reciprocity, and κil is the thermal conductivity tensor. We choose the coordinate axes i = (x, y, z) along the crystallographic axes a, b, c when the tensors in (5.61) have simple symmetry relations determined by the point group of the crystal. Hall Eﬀect Under the external electric ﬁeld Ex orthogonal to the magnetic ﬁeld Bz = B and zero thermal gradient, ∂l T = 0, the current density Jx and the current density Jy , induced by the Lorentz force, are determined by the equations: Jx = σxx Ex + σxy Ey , Jy = σyy Ey + σyx Ex , where the transverse conductivity σxy = −σyx = Rxyz B. The Hall coeﬃcient RH and the Hall angle ΘH are deﬁned by the transverse electric ﬁeld Ey for Jy = 0: RH =

1 σxy Ey 1 = , ≈ Jx B B σxx σyy nH e c

tan ΘH =

Ey σxy = = ωc τ. Ex σyy

(5.62)

The last formulas for RH , and tan ΘH in (5.62) are written in the isotropic relaxation time τ approximation for the diagonal conductivity: σxx = σyy = (ne2 /m)τ . The oﬀ-diagonal conductivity then takes the form: σxy = ωc τ σxx where the cyclotron frequency ωc = e B/mc. In this one-band and isotropic relaxation time approximations nH = (RH ce)−1 determines the density of free carriers, electrons for RH < 0 (with charge e < 0) or holes for RH > 0. Thermoelectric Eﬀects In the absence of a magnetic ﬁeld and under the action of the temperature gradient (−∂x T ) parallel to the external electric ﬁeld Ex , the electric current

5.4 Transport Properties

303

density Jx and the thermal current density Jxq in (5.61) are given by the equations: Jx = σxx Ex + αxx (−∂x T ), Jxq = α ˜ xx Ex + κxx (−∂x T ). The ﬁrst equation under the condition Jx = 0 determines the electric ﬁeld induced by the temperature gradient: Ex = −Sxx (−∂x T ),

Sxx = αxx /σxx ,

(5.63)

where Sxx is the thermoelectric power or Seebeck coeﬃcient. The second equation determines the thermal conductivity under the condition of zero electric current ﬂow: Jxq = Kxx (−∂x T ),

Kxx = κxx − T (α2xx /σxx ) ≈ κxx .

(5.64)

where we neglect the last term of the order of (kB T /EF )2 , which is small in metals. Within the Boltzmann theory for the Fermi liquid, the thermal conductivity is proportional to the electric conductivity which is known as the Wiedemann–Franz law: κxx = L0 T σxx ,

L0 = (π 2 /3) (kB /e)2 ,

(5.65)

where the Lorenz number L0 = 2.72 × 10−13 esu/deg2 is a universal constant. In the absence of a temperature gradient, the thermal current is produced by the electric current: Jxq = Πxx Jx ,

Πxx = α ˜ xx /σxx = T Sxx .

(5.66)

where Πxx is the Peltier coeﬃcient. Thermomagnetic Eﬀects In a system under the action of the magnetic ﬁeld Bz orthogonal to the temperature gradient ∂x T transverse Nernst and Ettingshausen eﬀects can be observed. The Nernst signal eN is determined by the transverse electric ﬁeld Ey produced by the temperature gradient ∂x T . The Ettingshausen eﬀect is reciprocal to the Nernst eﬀect and is determined by the temperature gradient ∂x T created by the transverse current Jy : eN (B, T ) =

Ey , |∂x T |

QE (T ) =

|∂x T | . Jy B

(5.67)

The relation between Ey and ∂x T is determined by the equations for the electric current Jx and the transverse (Hall-type) current Jy induced by the magnetic ﬁeld: Jx = σxx Ex + σxy Ey + αxx (−∂x T ), Jy = σyy Ey + σyx Ex + αyx (−∂x T ), where αyx = −αxy is the oﬀ-diagonal Peltier conductivity. (Here, we neglected the transverse thermal Hall conductivity which produces the temperature gradient (−∂y T ) ∝ κyx (−∂x T )). The diﬀerence between the Hall current σyx Ex and the oﬀ-diagonal Peltier current αyx (−∂x T ) results in a weak Nernst eﬀect. Solution of these equations for Ey under the boundary

304

5 Electronic Properties of Cuprate Superconductors

condition Jx = 0 and Jy = 0 results in the following equation for the Nernst signal: Ey αxy αxy σxx − αxx σxy ≈ − S tan Θ (5.68) eN = = xx H . 2 (−∂x T ) σxx σyy + σxy σyy 2 In the last formula, we neglected in the denominator the term σxy of the higher order in B and introduced the thermopower coeﬃcient (5.63) and the Hall angle (5.62). The Ettingshausen coeﬃcient in (5.67) can be found from the two equations for the thermal current Jxq and the electric current Jy in (5.61): Jxq = α ˜ xx Ex + α ˜ xy Ey + κxx (−∂x T ), Jy = σyy Ey + σyx Ex + αyx (−∂x T ) under the boundary condition Jxq = 0, Ex = 0. The solution of these equations for the current Jy = (σyy κxx /α ˜ xy − αyx ) (∂x T ) gives the following expression of the Ettingshausen coeﬃcient:

QE (T ) =

1 |∂x T | = Jy B B

σyy κxx − αyx α ˜ xy

−1 ≈

α ˜ xy B σyy κxx

(5.69)

where in the approximate formula the nondiagonal contribution (˜ αxy αyx ) of the higher order in B is neglected. In conventional metals, the Nernst signal (5.68) is rather small – within the Botzman theory for a parabolic one-band model of a metal the “Sondheimer cancelation” of the two terms in (5.68) takes place. For a more complicated band structure, the cancelation does not occur and a ﬁnite Nernst signal can be observed (see e.g., [41]). In the superconducting state, thermogalvanomagnetic phenomena are determined by the ﬂux ﬂow in type II superconductors (see [605]). In particular, in the liquid vortex phase a large positive Nernst signal can be detected which is caused by vortex motion under the action of a temperature gradient. The vortex motion occurs nearly parallel to the temperature gradient producing a transverse Josephson voltage VJ via phase slips. This voltage is detected as the Nernst signal in the superconducting state. By neglecting the small Hall contribution ∝ σyx Ex related to vortex diﬀusion in the equation for the charge current Jy = σyy Ey + αsyx (−∂x T ), we ﬁnd for the Nernst signal esN = Ey /(−∂x T ) = αsxy /σyy (see e.g., [1338]). Here, the oﬀ-diagonal Peltier conductivity αsxy = −αsyx refers to the vortex motion. As discussed in Sect. 4.3.2, a large positive Nernst signal has been detected in the hole-doped cuprates in a large region of the phase diagram above Tc (see Fig. 4.17), which was ascribed to a vortex liquid state. 5.4.1 Resistivity A typical feature of the copper-oxide compounds is the strong anisotropy of their transport properties caused by the layered structure of cuprates. The

5.4 Transport Properties

305

b 1200

T(K)

rab (mW -cm)

rc / rab

40

1.0 0.8

20 I 0

SC 0.1

x

M

x = 0.08

0.2 0.3 x = 0.12

0.6

0.2

800

8000 x = 0.08

6000 4000

600 x = 0.22 400

2000

200

0

x = 0.15

1

10 T(K)

100

x = 0.15

x = 0.17

0.2 0.0

x = 0.17

x = 0.22

0

100

200 T(K)

x = 0.12

x = 0.13

0.1

0.4

10000

1000

0.3

1.2

rc (W-cm)

a

x = 0.22

300

0.0

0

100

200

300

T(K)

Fig. 5.43. In-plane ρab (a) and out-of-plane ρc (b) resistivity of La2−x Srx CuO4 at various doping: solid lines in zero magnetic ﬁeld and symbols in magnetic ﬁeld 60 T. The inset (a) shows the Tc vs. Sr concentration and (b) displays the temperature dependence of the anisotropy ratio ρc /ρab (reprinted with permission by APS from c 1996) [147],

most clearly the anisotropy manifests itself in the measurement of the resistivity. Generally, in all cuprate superconductors the out-of-plane resistivity ρc across the blocking layers is always large with respect to the in-plane resistivity ρab along the CuO2 planes. Figure 5.43 shows the in-plane ρab and perpendicular to the plane ρc resistivity of La2−x Srx CuO4 at various doping [147]. The anisotropy of the resistivity ρc /ρab increases with underdoping and temperature decrease (see the inset in panel (b)). However, in the nonsuperconducting samples, 0.01 < x < 0.05, Komiya et al. [631] have found that at moderate temperatures, T ≥ 100 K, the anisotropy weakly depends on doping which has been explained by a hole segregation into stripes of hole-rich regions (see Sect. 3.2.4, Fig. 3.8). At lower temperatures, a maximum in the resistivity anisotropy was found caused by strong localization eﬀects which suppress the in-plane conductivity σab = 1/ρab and reduce the anisotropy ρc /ρab . Generally, at room temperature for the underdoped LSCO samples the anisotropy is quite large, of the order of ρc /ρab 1,000, which decreases for the optimally doped samples, ρc /ρab 200. It is possible to study the normal state resistivity below the superconducting Tc and the metal–insulator (MI) transition by applying a magnetic ﬁeld larger than the upper critical ﬁeld. In Fig. 5.43, the resistivity ρab and ρc are displayed by symbols in the low temperature region measured in a pulsed (61-T) magnetic ﬁeld [147]. In the underdoped samples, the resistivity

306

5 Electronic Properties of Cuprate Superconductors

displays an insulating behavior dρ/dT < 0 characterized by the logarithmic resistivity divergence, ρ(T ) ∝ log(1/T ), while in the overdoped samples the resistivity reveals a metallic behavior, dρ/dT > 0, down to the lowest temperatures. A crossover from insulating to metallic behavior occurs close to the optimal doping at hole concentration p x 0.16 when the mean free path ! is still rather large, kF ! 13 (kF is the Fermi wave vector). This parameter can be estimated from the in-plane resistivity, kF ! = hc0 /ρab e2 where c0 is the interlayer distance (6.5 ˚ A in LSCO). The magnitude of kF ! at the MI crossover in LSCO is anomalously large as compared to conventional metals where the insulating behavior induced by disorder occurs at kF ! < 1. Similar MI crossover was observed in the Bi-based compound, Bi2 Sr2−x Lax CuO6+δ (BSLCO) [919]. A distinct resistivity divergence, ρab ∝ log(1/T ), was found at low temperatures as in LSCO. The MI crossover occurred inside the underdoped regime at the lower than in LSCO hole concentration p ∼ 0.12 but the magnitude of the mean free path kF ! ∼ 12 was found to be close to that one in LSCO. MI crossover near optimal doping was also found in the electron-doped Pr2−x Cex CuO4 compounds [349]. In YBCO compounds, the anisotropy is moderate, ρc /ρab 30 (see e.g., [536]), while in the Ta- and Bi-based compounds the resistivity anisotropy is extremely large, ρc /ρab = 103 –104 . Systematic studies of the Bi2.1−x Pbx Sr1.8 CaCu2 Oy single crystals have revealed a drastic decrease of the anisotropy with increasing Pb content [868]. The resistivity anisotropy parameter ρc /ρab at 100 K decreases from 8.5 × 103 , (ρb /ρa = 2.0) in the Pb-free crystal to 1.2 × 103 , (ρb /ρa ∼ 1) in the crystal with x = 0.6 concentration of Pb. The authors have suggested that the ρc /ρab anisotropy decrease is caused by the enhancement of conductivity (1/ρc ) across the blocking layers by Pb doping, which also results in a less anisotropic in-plane crystal structure and more isotropic in-plane conductivity. Generally, the LSCO compounds can be considered as strongly anisotropic conductors in comparison with the YBCO compounds, which show a three-dimensional anisotropic conductivity. The conductivity in the Ta-, Bi-based compounds is close to a two-dimensional type behavior. Another peculiar behavior of the resistivity in cuprates is its quasi-linear temperature dependence over a wide range without saturation at high temperatures as ﬁrst was found in ceramic samples [415]. In many transition metals, the resistivity at high temperatures shows a saturation: a reduction of the slope of the T -linear resistivity. This occurs when the mean free path ! becomes comparable to the interatomic distance d. This limit ! ∼ d is referred to as the Ioﬀe–Regel criterion who have pointed out that in this limit the semiclassical theory based on the Boltzmann equation should fail. The experimental magnitude of the resistivity in cuprates at high temperatures has been found to be much larger than the Ioﬀe–Regel limit considered within the conventional transport theory. Gunnarson et al. [414] have shown how the Ioﬀe–Regel criterion can be derived quantum mechanically for various types of systems by employing the

5.4 Transport Properties

307

f -sum rule for the intraband optical conductivity (5.26) and its relation to the average kinetic energy |HK | as given by (5.50). Assuming a smooth frequency variation of the optical conductivity σ(ω) at high temperatures within the electronic bandwidth W , they obtained the following estimation for the upper bound of the static conductivity γ¯ h σmax (0) W

∞ σ(ω)dω =

γ πe2 a2 |HK |, W 2¯hV

(5.70)

0

where γ ∼ 2 depends on the shape of the σ(ω) spectrum. For strongly correlated systems, as cuprates the kinetic energy is greatly reduced which results in a decrease of the upper bound for the conductivity. In the case of the cuprates particular, assuming that the kinetic energy of a hole within the t–J model is proportional to x(1 − x), where x is the doped hole concentration, they have found for the saturation resistivity ρsat ∼ (1/σmax (0)) 0.4/x(1 − x) mΩ cm. This value is much larger for small x in comparison with the Ioﬀe–Regel limit ρsat 0.7 mΩ cm within the conventional transport theory. This large value of the ρsat explains why the resistivity in cuprates does not show saturation at high temperatures. A universal electronic temperature–hole concentration, T –p, phase diagram was proposed by Ando et al. [71] from a mapping of the in-plane resistivity curvature (d2 ρab /dT 2 ) of the LSCO, YBCO, and BSLCO crystals. For all the samples, the T -linear resistivity (d2 ρab /dT 2 = 0) was found only in a narrow region of temperatures near the optimal doping. This is at variance with the prediction of a “funnel-shaped” region in the T –p phase diagram at a quantum phase transition (see [1079]). The linear temperature dependence of the resistivity is a characteristic feature of the phenomenological “marginal” Fermi-liquid (MFL) model [1306]. In the overdoped region, a sharp crossover to a positive curvature was detected, which is usually considered as a crossover to the Fermi-liquid (FL) behavior. The latter can be related to a change of the FS from an arc-type to a large FS observed in ARPES experiments (see Fig. 5.21). A linear behavior in the high quality untwinned single YBCO crystals at optimal doping was found by Friedmann et al. [355] along all three axes, ρα , α = a, b, c. At room temperature, the in-plane anisotropy is ρa /ρb 2 which is close to the optic conductivity ratio σb /σa in the infrared frequency range [113]. The out-of-plane anisotropy was estimated as ρc /ρa 35, ρc /ρb 75. Another characteristic crossover line found in the T –p phase diagram is the pseudogap temperature Tpg (p) determined from the inﬂection point of the resistivity, d2 ρab /dT 2 = 0. The Tpg (p) temperature decreases approximately linearly with p and terminates near the optimal doping popt 0.16. In the underdoped samples, below Tpg (p) a positive curvature d2 ρab /dT 2 > 0 was observed, while above Tpg it was negative. At temperatures close to Tc the superconducting ﬂuctuation region arises below Tscf (p) > Tc characterized by a negative curvature which precludes Tpg (p) determination. To disentangle

308

5 Electronic Properties of Cuprate Superconductors

Tpg (p) and Tscf (p) temperatures, Naqib et al. [885] have studied the doping dependence of the in-plane resistivity in Y1−x Cax Ba2 (Cu1−x Zny )3 O7−δ thin ﬁlms at various concentration of Ca and Zn substitutions. By applying magnetic ﬁeld and changing the Zn concentration which both suppress Tc and Tscf (p) but has no inﬂuence on Tpg (p), they have managed to show that the pseudogap region persists below Tc on the overdoped side and Tpg (p) extrapolates to zero at pc 0.19. This suggests that Tpg (p) is similar to the pseudogap temperature T ∗ (p) inferred from thermodynamic data (Sect. 4.2.2) and the spin pseudogap temperature T ∗ (x) found in the Knight shift measurements in NMR experiments (Sect. 3.3.3). A crossover from the MFL at the optimal doping to the FL behavior in the overdoped region can be studied comprehensively in Tl-compounds, which can be synthesized in the entire overdoped region until superconductivity disappears. The crossover has been observed by Kubo et al. [668] for Tl-2201 and Tl-1212 crystals where a smooth variation of the resistivity from a linear T -dependence in optimally doped samples to a quadratic T -dependence for strongly overdoped samples with zero Tc has been found. Similar results have been obtained by other groups. Figure 5.44a shows the in-plane temperaturedependent resistivity of Tl2 Ba2 CuO6+δ (Tl-2201) crystals at various doping: nearly optimally doped with Tc = 89 K (OP89) and three overdoped (OD) ones with Tc = 70, 59 K and 15 K (OD79, OD59, OD15) [762]. We have already discussed in Sect. 5.3.2 the optical studies of these crystals which show a power law frequency dependence for the scattering rate from a linear ω-dependence near optimal doping to a shape with upward curvature for the OD samples. A similar power-law temperature dependence was found for the resistivity in all samples, ρ(T ) ∼ T β , which is shown in the logarithmic plot log(ρ(T ) − ρ0 ) ∝ β in the (b) panel. The OP89 sample displays a linear MFL behavior, β = 1, while in the OD samples the slope β increases with overdoping: for the OD15 sample the power parameter reaches the value β = 1.78 close to β = 2 expected in the canonical FL. As the authors pointed out, a

b

600 TI-2201

– 0.5

Tc = 89 K Tc = 70 K Tc = 59 K Tc = 15 K

400

Log ( ρ– ρ0)

Resistivity (μΩ cm)

a

200

m

– 1.0 – 1.5 – 2.0 – 2.5

0

0

100 200 Temperature (K)

300

.0)

L (1

al F

in arg

K (1 ) 89 7) 52 .0 1 ( (1. K K 70 59 8) 1.7 ( 5K 1 = Tc

1.2

1.5

)

.0

) .00

L

F al

(2

nic

no

ca

2.1 1.8 Log (T)

2.4

Fig. 5.44. (a) Temperature dependence of the resistivity of Tl-2201 single crystals with Tc = 89 K, 70 K, 59 K and 15 K (from top to bottom). (b) The log − log plot for the resistivity shown in (a) (after [762])

5.4 Transport Properties

309

smooth variation of the properties of Tl compounds with doping suggests that both the superconducting and the normal metallic phases should be described on the same basis. The MFL-FL crossover has been related to the doping dependence of the anisotropy of the transport scattering rate by Abdel-Jawad et al. [3] who have studied the magnetotransport in the OD TlBa2 CuO6+δ single crystals. By measuring the temperature and magnetic ﬁeld dependence of the angular magnetoresistance oscillation (AMRO) (see Sect. 5.2.2), the momentum and temperature dependence of the in-plane transport life-time τ (k, T ) was determined. A two-component inelastic scattering rate was suggested which includes an isotropic scattering term proportional to T 2 and an anisotropic T -linear scattering term: 1 = A + BT 2 + CT cos2 2ϕ, τ (k, T )

(5.71)

where A is the impurity scattering contribution. The anisotropic scattering rate 1/τan ∝ (1 + cos 4ϕ) = 2 cos2 2ϕ vanishes at the nodes (ϕ = ±π/4) and has a maximum near the antinodes (ϕ = 0, ±π/2), where it gives quite a large contribution: at T = 55 K, the ratio CT /BT 2 ∼ 2. In the OD nonsuperconducting samples (p ∼ 0.30), the anisotropic contribution seems to be absent which points to a relation between the anisotropic term and the T -linear dependence of the resistivity ρab and anomalous behavior of the Hall coeﬃcient RH in the Tl-2201 compounds [765]. While the T 2 isotropic component in (5.71) can be related to the electron–electron scattering as in conventional metals, the anisotropic component is speciﬁc for cuprates and it is believed to be due to scattering by spin ﬂuctuations or other bosonic modes. In a subsequent publication, it was reported that the anisotropic scattering term scales linearly with Tc suggests that the anisotropic scattering is related to the superconducting pairing mechanism in cuprates [4]. We should mention that a two-component scattering rate was also found in ARPES experiments in Bi-2212 compounds (see Fig. 5.25). However, the magnitude of the quasiparticle scattering rate seen by ARPES in Tl-2201 compounds is approximately ten times larger than estimated by AMRO (for discussions see [969]). Partly it may be explained if we assume strong smallangle scattering which gives a large contribution to the single-particle decay studied in ARPES, while it is insigniﬁcant for the transport life-time [675] (see Sect. 7.4.1). In any case, the increase of anisotropic scattering with underdoping results in a transformation of the normal metal into a “nodal” metallic state in which the quasiparticles are observed only on the Fermi “arcs” where scattering is strongly suppressed. This picture has been conﬁrmed in studies of the Hall eﬀect in cuprate materials.

310

5 Electronic Properties of Cuprate Superconductors

5.4.2 Hall Eﬀect Studies of the the Hall eﬀect (5.62) provide valuable information concerning the Fermi surface (FS) topology, the anisotropy of the scattering rate and the carrier density. As shown by Ong [918], the Hall conductivity σxy in a two-dimensional metal has a simple geometrical interpretation in terms of the “scattering path length” vector l = τk vk where τk is the scattering time and vk = h ¯ (∂εk /∂k). In this representation, σxy equals (in units of e2 /h) twice the number of the ﬂux quanta φ0 threading the l curve in the lx − ly space, which helps to discriminate between the contributions to the Hall conductivity caused by the the FS curvature vk and the anisotropy of the scattering time τk . The Hall coeﬃcient RH of the conventional metals is independent of temperature. However, early experiments in the cuprate superconductors evidenced a strong temperature dependence of the Hall coeﬃcient and the Hall angle [537, 917]. In particular, studies of the Hall eﬀect and of the resistivity of the Zn-doped single crystals YBa2 Cu3−x Znx O7−δ by Chien et al. [227] have shown a peculiar “scattering rate separation” into the longitudinal transport relaxation rate 1/τtr and the transverse relaxation rate 1/τH . The observed T -linear dependence of the resistivity, ρ = aT + c ∝ 1/τtr , suggests that 1/τtr ∝ T . At the same time, the Hall angle reveals a T 2 -dependence which only shifts under Zn doping proportionally to the concentration x: cot ΘH = A T 2 + βx. Since cot ΘH = 1/τH ωc , this implies that 1/τH ∝ T 2 . According to the Anderson theory [63], the observation of two distinct relaxation rates is explained by the fact that only the spinons participate in the Hall (transverse) current. Their interaction determines the quadratic temperature dependence of 1/τH . The longitudinal current is determined by the relaxation of spinless holes (holons), which are not aﬀected by the magnetic ﬁeld. The observed temperature dependence for resistivity and the Hall angle results in a complicated function of temperature for the Hall coeﬃcient (5.62): RH =

1 aT + c ρxx . ∝ Bz coth ΘH AT 2 + C

(5.72)

Depending on the parameters, various temperature behaviors of RH can emerge. In particular, for high temperatures and/or pure samples (residual contributions C and c are small) RH ∝ 1/T , while at low temperatures or samples with defects (with large values of C and c) RH ∼ const. should be seen. These limiting cases have been observed in many experiments. Figure 5.45a shows the temperature dependence of the in-plane Hall coeﬃcient RH for untwinned YBa2 Cu3 Oy single crystals measured in the magnetic ﬁeld along c-axis at various oxygen content y in a semilog plot [1130]. Under decreasing y, the Hall coeﬃcient sharply increases indicating a strong decrease of the hole concentrations. A pronounced temperature dependence is observed for optimally doped (OP) and slightly underdoped (UD) samples, while a much weaker dependence is found for the nonsuperconducting samples with low hole concentrations, p ∼ 3%, (y = 6.30 − 6.35). In the high temperature

5.4 Transport Properties

311

Fig. 5.45. Temperature dependence of (a) the Hall coeﬃcient RH and (b) the Hall pl for YBa2 Cu3 Oy at various oxygen content (after [1130]) angle cot ΘH(b)

region, most of the samples reveal increase RH with lowering of temperature, while the slightly UD samples with y = 6.70 − 6.80 show a sharp decrease of the Hall coeﬃcient below ∼ 120 K. Using untwinned samples, the authors meas managed to extract from the measured values of RH shown in Fig. 5.45a the Hall coeﬃcient related to the CuO2 plane only. For the electric current along the chain direction b, this component, which is assumed to be isotropic, was pl(iso) meas = RH (ρa /ρb ). For strong UD samples, the diﬀerence calculated as RH is small since the chains become insulating and ρb ∼ ρa , while for OD samples pl(iso) meas /RH = 2–3 due to the large anisotropy of the in-plane conductivity. RH The temperature dependence of the Hall angle is also unconventional. Figure 5.45b shows the temperature dependence of the cotangent of the Hall angle for the electric current along the chain direction b. In this geometry, this can be pl pl meas determined from measurable properties only: cot ΘH(b) = ρpl b /ρyx = ρb /ρyx . 2 For the samples close to the OP doping, y > 6.8, the T dependence shown by dotted lines is obvious. For UD samples with y < 6.75, the data are ﬁtted by the function AT α + C (shown by solid lines) with 2 < α < 2.5. The residual component C vanishes for y > 6.6 but strongly increases for lower values of the oxygen content concomitant with a sharp decrease of the amplitude A.

312

5 Electronic Properties of Cuprate Superconductors

Similar results have been obtained for the La2−x Srx CuO4 single crystals in a wide doping range from the lightly doped to OD samples, 0.02 ≤ x ≤ 0.25 by Ando et al. [72]. Figure 5.46 shows the temperature dependence of the Hall coeﬃcient RH (a) and the Hall angle cot ΘH (b), (c) for various doping rates. The temperature dependence is less pronounced in comparison with the YBCO system. This may be due to a stronger impurity scattering, but the general behavior is similar. In the lightly doped region, 0.02 < x < 0.05, the temperature dependence is rather weak below 300 K, while for OD samples it becomes quite noticeable. The T 2 temperature dependence of cot ΘH holds for doping 0.02 < x < 0.14, which is shown by solid lines. A comparison of the temperature dependence of the Hall conductivity and the in-plane resistivity suggests that in the lightly doped region a Fermi-liquid-like transport takes place related to holes on the Fermi “arcs” in the nodal direction seen in ARPES experiments (see Fig. 5.19). At higher doping, a contribution from the emerging large FS with ﬂat parts in the antinodal region of the BZ becomes important. In heavily OD La2−x Srx CuO4 thin ﬁlms, a smooth change of the Hall coeﬃcient from positive to negative between x = 0.28 and

Fig. 5.46. (color online) Temperature dependence of the Hall coeﬃcient RH (a) and the Hall angle cot ΘH (b) and (c) for La2−x Srx CuO4 at various doping rates c 2004) (reprinted with permission by APS from [71],

5.4 Transport Properties

313

x = 0.32 was observed [1278], which suggests a crossover from hole-type to electron-type FS. The important role of the diﬀerent type charge carriers in the Hall conductivity has been conﬁrmed by further studies of LSCO single crystals up to 1,000 K for the entire doping range [920]. The Hall coeﬃcient of the insulating parent La2 CuO4 compound was also measured and it revealed a typical semiconductor behavior. The temperature dependence of RH was described by the thermal activation of two groups of charge carriers. These are the impurity states with a small concentration ∼ 0.57% located above the valence band at 0.087 eV and the hole states of high density separated from the valence band by the charge transfer gap ΔCT 0.89 eV. In the lightly doped region for x = 0.01−0.05, along with the previous thermally activated charge carriers over the reduced charge transfer gap ΔCT 0.53 eV, a temperature independent contribution from the doped holes ∼ x appears. In the superconducting doping range, a complicated temperature dependence was observed which was described by a two-component model for holes from the Fermi “arcs” (or hole pockets) and the large FS. Thus, it was stressed that a complicated temperature dependence of the Hall coeﬃcient to some extent can be explained by thermal activation of charge carriers even at moderate temperature. In this respect, the scaling behavior of the Hall coeﬃcient RH ∝ f (T /T ∗ ) found in LSCO [514] may be related to the freezing of the thermal creation of charge carriers. It has been shown that a strong temperature dependence occurs below a characteristic temperature T ∗ similar to the pseudogap temperature deduced from the Knight shift scaling (see Fig. 3.26), The corresponding Hall angle studies have revealed that T 2 dependence of cot ΘH is observed only below 300–400 K in the superconducting doping region. Therefore, a peculiar “scattering-time separation” into the longitudinal and transverse relaxation times discussed above develops only at intermediate temperatures. In this region, the anisotropy of the inelastic scattering time (5.71) may be responsible for this phenomenon. The Hall eﬀect studies in polycrystalline HgBa2 CaCu2 O6+δ samples (Tc ∼ 124 K) have shown a steep 1/T temperature dependence of the Hall coeﬃcient and a T 2 dependence of cot ΘH at moderate temperature, similar to those in YBCO compounds [429]. The small magnitude of the oﬀset terms observed in both temperature-dependent functions indicates a low-impurity scattering in the mercury compound. A peculiar temperature dependence of the normal-state Hall eﬀect in the Tl2 Ba2 CuO6+δ single-crystals has been observed by Mackenzie et al. [765]. The distinct longitudinal and transverse relaxation rate behavior was found in the OD samples for T ≥ 30 K but in the low (millikelvin) temperature region only a single scattering rate of lower power in T was observed. To explain the anomalous transport properties and the peculiar behavior of the upper critical ﬁeld Hc2 at low temperatures (see Sect. 4.3.2), the authors suggested the existence of a very low energy scale, which is important in determining the normal-state properties.

314

5 Electronic Properties of Cuprate Superconductors

b

15

R H(T) / R H (2/3T0)

-R11 (10 –3 cm 3/C)

a

x= 0.05 0.075

10

0.10 5

1.5

1

0.5 X = 0.05 0.075 0.10 0.125

0.125 0.15

0

0

200 T (K)

400

0

0

1 T/ T0

2

Fig. 5.47. (a) Temperature and doping dependence of the Hall coeﬃcient RH in the underdoped Nd2−x Cex CuO4 crystal and (b) its T –x scaling behavior (after [924])

Studies of the Hall eﬀect in the electron-doped cuprates have conﬁrmed the complicated FS transformation with doping observed in ARPES experiments (see Sect. 5.2.2, Figs. 5.23 and 5.24). Figure 5.47a shows temperature and doping dependence of the Hall coeﬃcient RH in the underdoped Nd2−x Cex CuO4 (NCCO) crystal [924]. The concentration dependence of RH in Pr2−x Cex CuO4 (PCCO) at low temperature, T = 0.35 K, was studied by Dagan et al. [260]. The Hall coeﬃcient in the underdoped NCCO is negative, while in PCCO it changes the sign from negative at low Ce concentrations to positive close to x = 0.155 suggesting a transition from an electron-like to a hole-like-type FS. It was observed also a crossover from a strong doping dependence of RH to a weak one with increasing Ce concentrations. A concomitant sharp change of the resistivity data at x ∼ 0.165 suggests a quantum phase transition, which may be related to a signiﬁcant reorganization of the FS. As discussed by Lin et al. [726], the rearrangement of the FS caused by the onset of the spindensity wave close to this concentration can qualitatively explain the Hall coeﬃcient doping dependence. As we have already discussed in Sect. 5.3.2 (see Fig. 5.34), studies of the charge dynamics in NCCO have revealed an opening of a pseudogap below a characteristic temperature T ∗ (x) ∼ 2TN . Similar to LSCO [511], the scaling behavior of the RH was found for NCCO. The T -dependence of the Hall coeﬃcients in NCCO at various doping can be described by a single scaling function f (T /T0 ) = RH (T, x)/RH (2T0 /3, x) for T ≥ 0.4T0 as shown in Fig. 5.47b [924]. The scaling temperature T0 (x) ∼ 2TN (x) and almost coincides with the pseudogap temperature T ∗ inferred from the optic measurements. The origin of

5.4 Transport Properties

315

the pseudogap formation and the scaling behavior of the Hall coeﬃcient in the electron-doped cuprates is most probably due to strong AF correlations which are observed well above the N´eel temperature TN . Their important role in the quasiparticle spectra have been proved by ARPES experiments. Quantum Oscillations In conventional metals, studies of the quantum oscillation eﬀects – the magnetoresistance oscillations, Shubnikov–de Haas (SdH) eﬀect, or the magnetic susceptibility oscillations, de Haas–van Alphen (dHvA) eﬀect, play an essential role in establishing the existence of the Fermi surface (FS) in metals (see e.g., [1161]). The oscillations arise from the Landau quantization of the electronic levels in strong magnetic ﬁelds and probe the bulk properties of the quasiparticles on the FS. The amplitude of the oscillations is proportional to the Digle factor RD = exp(−π/ωc τ ), which impose the restriction ωc τ 1 on the cyclotron frequency ωc = eB/m∗ c and the scattering time τ . This condition is diﬃcult to realize in cuprates due to the large scattering rates 1/τ and high critical magnetic ﬁelds Hc2 . Only recently the quantum oscillations have been discovered in two underdoped cuprate systems: the YBa2 Cu3 O6.5 (Y123-II) crystals in the ortho-II phase and the YBa2 Cu4 O8 (Y-124) crystals. These crystals have well ordered CuO-chains that results in weak disorder scattering and a large orbitally averaged mean free path lSdH . A low critical magnetic ﬁelds, Bc2 ∼ 50 T enables the observation of quantum oscillations close to the normal state and above the irreversibility line Birr (T ) in the magnetic susceptibility (see Sect. 4.3). The SdH oscillations of the Hall resistance Rxy in the Y-123-II crystal with Tc = 57.5 K and the hole doping level p = 0.1 was discovered by DoironLeyraud et al. [285]. The measurements were performed in a magnetic ﬁeld B of up to 62 T perpendicular to the CuO2 at temperatures T = 1.5–4.2 K. The oscillations ΔRxy as a function of 1/B revealed a frequency F = 530 ± 20 T. From the Onsager relation: F = (¯ hc/2πe) Ak , the following value for the Fermi surface area Ak = πkF2 = 5.1 nm−2 was obtained which is only a fraction of ∼ 1.9% of the entire Brillouin zone (BZ) area of 4π2 /ab. This suggests a small FS in the form of pockets the locations of which in the BZ cannot be deﬁned from the SdH eﬀect. Negative sign of the the Hall resistance Rxy implies that it should be an electron FS. The cyclotron mass estimated from the oscillation amplitude was equal to m∗ = (1.9±0.1) m0 where m0 is the free electron mass. This remarkable discovery of a small FS was conﬁrmed by the observation of the dHvA eﬀect in the torque measurements on a similar Y-123-II crystal by Jaudet et al. [553]. The measured oscillation frequency 540 ± 4 T and the eﬀective mass m∗ = (1.76 ± 0.07) m0 were in close agreement with the SdH data. For the cyclotron parameter ωc τ = 0.7 ± 0.2 at B = 35 T, the mean free path was estimated as lSdH = 160 ˚ A. The angular dependence of the SdH frequency as a function of the tilting angle θ: F ∝ 1/ cos θ, conﬁrmed a quasi-two-dimensional FS in the (a, b)-plane. Thus, these measurements have

316

5 Electronic Properties of Cuprate Superconductors

proved the existence of a coherent closed FS in underdoped YBCO crystals at low temperatures, apparently in the pseudogap phase. Similar results were obtained for the Y-124 crystals. The SdH oscillations were observed for both the longitudinal and the Hall magnetic resistance in pulsed magnetic ﬁeld up to 61 T for two Y-124 crystals with Tc = 81 K and 82 K by Bangura et al. [98]. The frequency of oscillations F = 660 ± 30 T yields a small FS area Ak , which corresponds only to ∼ 2.4% of the average area of the BZ. For the eﬀective mass and the mean free path, the following estimations were obtained: m∗ = (2.7 ± 0.3) m0 and lSdH = 90 ± 30 ˚ A, respectively. Using a tunnel-diode oscillator technique, the quantum oscillations of the resonant frequency in Y-124 was found by Yelland et al. [1388]. The oscillation frequency F = 660 ± 15 T and the quasiparticle mass m∗ = (3 ± 0.3) m0 are close to the SdH data. Comparison of the heat capacity proportional to the cyclotron mass m∗ : γ = C/T = 1.48 (m∗ /m0 ) = 4.44 (mJ mol−1 K−2) with the upper experimental value γ = 9 mJ mol−1 K−2 in the normal state suggests that this closed two-dimensional FS is the only conducting sheet in this compound at low temperature. The observation of similar small FS pockets in the diﬀerent 123 and 124 YBCO compounds are diﬃcult to describe within the standard band structure calculations, e.g., as originated from CuO-BaO bands (see e.g., [208,305]). To explain the appearance of electron pockets, the reconstruction of the FS at a critical hole concentration p ∼ 0.14 caused by some kind of density wave instability was suggested [696]. Studies of the Hall resistance Rxy in the underdoped Y-123 and Y-124 crystals at low temperature have shown that below a speciﬁc temperature (e.g., T0 = 30 K for Y-123-II) Rxy becomes negative. It was suggested that the Hall mobility in the small electronic pockets considered above is high enough at low temperatures to overweight the positive contribution from other, hole-like region of the FS. Further high-resolution ARPES studies are important to elucidate this problem of small FS pockets in the undoped YBCO compounds. Remarkable results was reported by Vignolle et al. [1314] who observed the quantum oscillations in the overdoped Tl2 Ba2 CuO6+δ (Tl-2201) with Tc ∼ 10 K. As discussed above (see Sect. 5.2.2, Fig. 5.22), a large FS has been found in the overdoped Tl-2201 crystal by the ARPES experiment [1020] and by the polar angular magnetoresistance oscillation measurement [509]. A direct observation of the quantum oscillations in the interlayer R⊥ magnetoresistance (the SdH eﬀect) and in the magnetic torque (the dHvA eﬀect) in the ﬁeld B ∼ 50 T oriented close to the c axis has unambiguously conﬁrmed an existence of the large FS. The measured dHvA oscillation frequency 18,100 ± 50 T corresponds to the cross-section area of the FS A 172.8 ± 0.5 nm−2 , which is approximately 65% of the Brillouin zone and scales as (1+p) for the doping level p = 0.30. The corresponding average Fermi wave vector is kF = 7.42±0.05 nm−1 . The temperature dependence of the SdH eﬀect gives the cyclotron eﬀective mass m∗ = 4.1 ± 1 m0 and the ﬁeld dependence of the dHvA oscillation estimates the mean free path as lSdH = 320 ˚ A.

5.4 Transport Properties

317

The value of the eﬀective mass m∗ corresponds to the Sommerfeld coeﬃcient in the electronic speciﬁc heat in the normal phase γn = 6.0 ± 1 (mJ K−2 ·mole), which is in agreement with the speciﬁc heat measurements (see Sect. 4.2.1, Table 4.1). Thus, these measurements have conﬁrmed an applicability of a generalized Fermi liquid picture on the overdoped side of the phase diagram for the cuprate superconductors. It should be noted, however, that strong electron correlations are still important even in the overdoped region which should be taken into account to explain a large mass renormalization or the linear temperature dependence of the electrical resistivity which contradicts to the square temperature dependence, a characteristic of the conventional Fermi liquid (see Sect. 5.4.1, Fig. 5.44). Resume In conclusion, the complicated temperature and doping dependence of the Hall eﬀect in cuprates can be related to a topological FS transformation with increasing doping from the small FS, of “arc-” or hole-pocket-type, to a large FS above the pseudogap temperature T ∗ (x). While in the low-doping regime strong correlations and non-Fermi-liquid behavior are evident, e.g., in a rapid doping dependence of the Hall coeﬃcient |RH (x)| ∝ 1/x, in the OD region the Hall coeﬃcient dependence |RH (x)| ∝ 1/(1 + x) at low temperatures points to a large FS. This transformation is most clearly seen in the electrondoped cuprates where strong AF correlations are evident in a broad doping region. They are responsible for the quantum phase transition at T = 0 to the AF long range order below a critical concentration xc close to the optimal doping. Observation of quantum oscillations, the SdH and dHvA eﬀects, in underdoped YBCO compounds revealed small electron-like FS pockets the origin of which has not yet found an unambiguous explaination. 5.4.3 Heat Transport Thermal Conductivity Thermal conductivity studies in cuprates have proved to be a powerful probe of low-energy quasiparticle (QP) excitations providing complimentary information to the electric conductivity data. In particular, the electronic thermal conductivity of the d-wave superconductors in the low-temperature limit reveals a universal behavior similar to the electric conductivity (5.44). As discussed in Sect. 5.1.2, the lines of nodes on the 3D FS in the d-wave superconductors with impurities results in a gapless region ∼ γ in the DOS. Hence, the transport properties at low temperatures and frequencies, (kB T, ¯hω) γ, in the d-wave superconductors should be determined by QPs in the impurity band γ. In the vicinity of the four gap nodes along the kx = ±ky directions on the 2D FS, the QP excitation spectrum takes a cone-like form with respect to the local wave vectors (k1 , k2 ):

318

5 Electronic Properties of Cuprate Superconductors

Ek =

ε2k + Δ2k = h ¯

2 k2, vF2 k12 + vΔ 2

(5.73)

where vF = (1/¯h)(∂εk /∂k1 ) and vΔ = (1/¯ h)(∂Δk /∂k2 ) are the QP velocities along directions normal (k1 ) and tangential (k2 ) to the FS at each node. Calculations of electric and heat currents induced by the QP excitations (5.73) give for the electric σ0 and thermal κ0 /T conductivity the following results in the universal clean limit (kB T γ Δ0 ) [296]: 2 kB n vF e 2 n vF κ0 vΔ 2 , (5.74) = βVC αFL , + σ0 = hπ2 d vΔ ¯ T 3¯h d vΔ vF where n/d is the staking density of the CuO2 planes. In the expression for the electric conductivity, βVC and αFL are vertex and Fermi-liquid corrections, respectively, to a simple bubble approximation in (5.44). The vertex correction takes into account that back scattering is more important than the small-angle scattering in the current relaxation, which is usually accounted for by the 1 − cos θ term in the normal-state conductivity. These corrections are absent in the thermal conductivity κ0 /T , which enables one to measure unambiguously the slope of the gap vΔ at the node. For a simple d-wave gap of the form Δ(ϕ) = Δ0 cos 2ϕ we get 2Δ0 = h ¯ kF vΔ . Thus, the thermal conductivity provides a bulk measure of the gap unlike the surfacesensitive techniques (ARPES, optical conductivity, tunneling). Veriﬁcation of the Wiedemann–Franz (WF) law (5.65) for the thermal and electric conductivity, κxx /T σxx = L0 , allows one to check whether the cuprates may be considered as a Fermi-liquid [407]. Early studies of the thermal conductivity of high-temperature superconductors are reviewed by Uher [1290]. Later on, intensive low-temperature thermal conductivity studies of high-quality single crystals revealed its complicated behavior in the cuprate superconductors. In the optimally doped (OP) and overdoped (OD) YBCO, Bi-2212 and Tl-2201 compounds, a universal character of the heat transport was conﬁrmed in a number of studies and the data obtained for the quasiparticle spectrum were in agreement with those measured by ARPES and inferred from the thermodynamic properties (see e.g., [226, 438]). However, the breakdown of the universal behavior of the thermal conductivity in the OP doped and UD YBCO, LSCO and Bi-2212 systems induced by electronic inhomogeneity of the samples (e.g., created by Zn impurities) was suggested by Sun et al. [1201]. Below we consider several examples of these studies. Figure 5.48a shows temperature dependence of the thermal conductivity κ(T ) of YBa2 Cu3 Oy and La2−x Srx CuO4 single crystals normalized to Tc . In the YBCO crystal of high quality (y = 6.99, Tc = 89 K, grown in a the BaZrO3 (BZO) crucible – ﬁlled symbols), a high peak in the thermal conductivity is observed at T ∼ 0.3Tc . In the less perfect YBCO crystal (y = 6.95, Tc = 93.5 K, grown in Y2 O3 -stabilized ZrO2 (YSZ) crucibles – open symbols) the peak is much lower. The peak nearly vanishes for a Zndoped sample and is not seen in optimally doped LSCO crystals. Although

5.4 Transport Properties

319

Fig. 5.48. Temperature dependence of the thermal conductivity: (a) κ(T ) in YBa2 Cu3 Oy and La2−x Srx CuO4 vs (T /Tc ) and (b) κ(T )/T in YBa2 Cu3 Oy vs T 2 at various doping (after [1202])

the phonon contribution is about seven times larger than the electronic one at these temperatures, it is believed that most of the peak in the thermal conductivity is due to the electronic contribution, similar to that observed in the microwave conductivity (see Fig. 5.36). The peak is the result of the competition between the quasiparticle life-time increase below Tc and the decrease of the quasiparticle density with the temperature lowering [473]. The impurity and the structure defects suppress the quasiparticle life-time and wash out the peak for the Zn-doped YBCO and the LSCO crystals where dopant Sr atoms produce strong scattering. The low-temperature thermal conductivity κ/T (along the a-axis) of the detwinned single YBa2 Cu3 Oy crystals at diﬀerent doping and crystal perfection are shown in Fig. 5.48b against T 2 [1202]. In the limit T → 0 the phonon contribution in the boundary-scattering regime κph ∝ T 3 , vanishes more faster than the electronic one, κel ∝ T . This allows to extract the latter as the intercept of the κ/T plot vs T 2 with the T = 0 axis. The magnitude of κ/T steadily decreases with underdoping and for a fully deoxygenated sample with y = 6 vanishes. Similar plots were obtained for the La2−x Srx CuO4 single crystals for Sr doping x = 0.06–0.20, though the data interpretation are less conclusive. The corresponding anisotropy ratio vF /vΔ extracted from (5.74) are shown in Fig. 5.49a against carrier concentration p. Close values of vF /vΔ for the YBCO BZO and YSZ grown crystals, in spite of an order of magnitude diﬀerence in their purity level, conﬁrm that κ/T reaches universal value in the clean limit for these samples. At OP doping, the magnitude of vF /vΔ for LSCO, YBCO, and Bi-2212 [226] is comparable. At the same time, an extremely small magnitude of κ/T measured in the lightly doped LSCO samples (x = 0.06), less than the minimum allowed value at vF = vΔ , suggests the breakdown of the universal thermal conductivity behavior given by (5.74). As discussed in Sect. 5.4.1, localization eﬀects in LSCO for x ≤ 0.17

320

5 Electronic Properties of Cuprate Superconductors

Fig. 5.49. (a) Doping dependence of the anisotropy ratio vF /vΔ for the YBCO, Bi-2212 and LSCO single crystals (after [1202]); (b) Thermal conductivity κ/T of the Tl-2201 crystals: close to OP doping (Tc = 84 K) and OD sample (Tc = 26 K) (after [438])

are observed in the resistivity measurements in magnetic ﬁelds (see Fig. 5.43), which may be responsible for the peculiar behavior of the thermal conductivity in LSCO. It is interesting to study the doping dependence of the d-wave supercon¯ kF vΔ /2, where the wave vector kF at nodes on the FS ducting gap Δ0 = h weakly depends on the doping p. Figure 5.49b shows the thermal conductivity κ/T of the Tl-2201 crystals close to OP doping (Tc = 84 K, p 0.19) and of an OD sample (Tc = 26 K, p 0.25) [438]. The linear behavior of κph /T vs. T , instead of T 2 as in the boundary-scattering regime for the phonon contribution, is explained by a strong electron–phonon scattering in the OD Tl-2201 compounds with high carrier concentration. The corresponding magnitude for the gap calculated from the anisotropy ratio in the OD region is close to the d-wave weak-coupling BCS value Δ0 = 2.14kB Tc and reaches a value twice as large near OP doping. Similar results hold for the YBCO and Bi-2212 crystals [1202]. In the UD region Δ0 (p) behavior inferred from the κ/T measurements follows the pseudogap dependence: it increases nearly linearly with underdoping while Tc decreases. This observation lets us infer that the full gap in the superconducting state in the underdoped region has the nodes along the QP cone-like excitation spectrum (5.73) of the charge carriers. Thus, in general, the gap function Δ0 for the OP doped and OD cuprates estimated from the slope of the gap at the nodes at very low energy is in agreement with the spectroscopic measurements of the gap at antinodes. At the same time, the interpretation of the thermal conductivity studies in the UD region is less conclusive. As suggested by Sutherland et al. [1203], at hole concentration p < psc , below the boundary of the superconducting transition at psc ∼ 0.05, there exist delocalized fermionic excitations at nodes in the YBCO compound and the normal ground state of the system is a “thermal”

5.4 Transport Properties

321

metal, while in the LSCO compound the normal state is insulating. However, it was argued by Sun et al. [1200] that localization eﬀects in the lightly doped cuprates, revealed by a ln(1/T ) resistivity divergence both in the LSCO and in the YBCO systems, point to the localization of the nodal QP in the normal state in both compounds which precludes appearance of the “thermal” metal. According to this scenario, the superconducting state at p > psc evolves from the insulating normal state with appearance of the nodal QPs exhibiting themselves by a ﬁnite thermal conductivity. Veriﬁcation of the WF law (5.65) in the cuprate superconductors has conﬁrmed a nearly Fermi-liquid-like behavior in the OD region, while a departure from the WF law in the vicinity of the metal-insulator transition points to a peculiar normal state properties in the pseudogap phase. Due to the low upper critical ﬁeld Hc2 ∼ 13 T in the OD Tl-2201 crystal (Tc 15 K, p ∼ 0.26), it was possible to measure the residual resistivity ρ0 and the thermal conductivity κ0 /T at T → 0 in the normal state of the crystal by applying a magnetic ﬁeld (Hc) larger than Hc2 [1030]. The calculated Lorentz ratio appeared to be in a perfect agreement with the WF law: L = ρ0 κ0 /T = 0.99±0.01L0. This points to the existence of an electronic Fermi-liquid in the OD cuprates, which does not support a spin-charge separation scenario or a 2e charge-boson condensation in the superconducting state (see Sect. 7.4.2). Similar results have been obtained for the overdoped LSCO and PCCO systems. Detailed studies of the heat transport and resistivity in the normal state of several Bi-2201 compounds by Proust et al. [1031] have conﬁrmed the WF law in the overdoped side of the phase diagram but have revealed an increase of the Lorenz number at the OP doping p ∼ 0.16, which becomes quite large in the UD region, L/L0 = 2–3. The departure from the WF law correlates with the disorder increase in the samples with underdoping and occurs close to the metal-insulator transition (in Bi-2201 at p ∼ 0.12) as discussed in Sect. 5.4.1. Thermopower Unconventional electronic properties of the cuprate superconductors have also been found in studies of the thermoelectric power (TEP) (5.63) (for a review see [575]). The TEP coeﬃcient S is deﬁned by the ratio of the voltage drop ΔV to the temperature diﬀerence ΔT on the sample boundaries, S = ΔV /ΔT . The voltage drop arises due to the change of the diﬀusion rate of the electrons along and opposite to the temperature gradient (diﬀusion contribution Sd ), and also due to electron drag by a current of nonequilibrium phonons (phonondrag contribution Sg ) (see e.g., [140]). The diﬀusion electronic contribution can be represented in the form [49]: kB 1 Sd = e σ

∞ −∞

ε kB εav ∂f (ε) σ(ε) ≡ , dε − ∂ε kB T e kB T

(5.75)

322

5 Electronic Properties of Cuprate Superconductors

where σ(ε) is the energy ε dependent conductivity and f (ε) is the Fermi distribution function. If the average energy of the charge carriers εav is equal to the thermal energy kB T , then Sd kB /e = 87 µV/K. In metals, the average thermal electron energy near the FS μF is determined by the parameter εav (kB T )2 /μF . In common metals (kB T /μF ) 1 and the TEP is rather small: Sd ∼ (kB /e) (kB T /μF ). For a free electron gas, the Mott formula 2 follows from (5.75): Sd,el = (π 2 kB T /3e)(∂ ln σ/∂ε)μF , which shows a linear temperature dependence of the electronic TEP. In metals having complicated electronic spectra, the TEP may have a nonlinear temperature dependence and an arbitrary sign. The phonon-drag contribution Sg is determined by the electron–phonon interaction and has an essential nonlinear temperature dependence: in conventional metals it rises as T 3 below T ∼ 0.1ΘD and falls as 1/T above T ∼ 0.5ΘD where ΘD is the Debye temperature. In the cuprate superconductors, both in ceramic samples and in single crystals, a more complicated temperature dependence was found. In the superconducting phase, S = 0, since the voltage drop is zero, ΔV = 0, but certain thermoelectric eﬀects remain. Figure 5.50a shows a typical temperature dependence of the TEP denoted by Σ(T ) in units of (µV/K) for the polycristalline samples of Y0.8 Ca0.2Ba2 Cu3 O6+x (YCBCO) at various oxygen concentrations from the UD samples at x < 0.20 (Tc = 38 K) to the OD sample at x = 0.96, (Tc = 46 K) with the maximal Tc,max = 85.5 K at the OP doping x = 0.62 [249]. The substitution of Y3+ by Ca2+ introduces additional hole carriers into the CuO2 planes, which allows the preparation of YCBCO samples with optimally doped CuO2 planes but strongly deoxygenated CuO chains (x < 0.70). In this case, the “chain” b

70

TI-2201

0.10 0.17 0.22

50 Σ (mv/ K)

10

X

60 40

0.23

30

0.4

20

0.45 0.5 0.56 0.63 0.71 0.96

10 0 –10 0

100

200 T(K)

300

Bi-2212

100

5

TI-2223

S (µV/K)

a

0 –5

10

Ca-123

–10

123 TI-1212

1

0.0

0.1

0.2

–15 0.3

p

Fig. 5.50. (a) Temperature dependence of the thermoelectric power Σ(T ) for Y0.8 Ca0.2 Ba2 Cu3 O6+x (YCBCO) at various x (reprinted with permission by Elsevier c 2000); (b) Thermoelectric power S(T = 290) K vs. hole concentration from [249], p for various cuprates: YBCO (123), YCBCO (Ca-123), etc., in a logarithmic scale (left) for the underdoped cuprates and in a linear scale (right) for the overdoped cuprates (after [1221])

5.4 Transport Properties

323

contribution is almost excluded from the TEP (for details, see [129]). For the entire UD to OP doped region, the TEP is positive for Tc < T < 290 K with a maximum at some temperature T ∗ which is close to Tc at OP doping but shifts to higher temperatures with underdoping. In the UD region, S(T ∗ ) is quite large, of the order of kB /e but quickly decreases as the carrier concentration grows. For T > T ∗ , the TEP steadily decreases with increasing temperature reaching values of 1–2 µV/K at optimal doping at 290 K. In the OD region, the TEP becomes negative with a weak temperature dependence in YBCO. The latter is due to the cancelation of a negative slope due to the plane contribution and a positive slope due to the chain contribution (Bernhard et al. 1996). A scaling behavior of the TEP Σ(T )/Σ(T ∗) vs. (T /T ∗ ) was found in the UD region for x < 0.63 where the crossover temperature T ∗ ∼ 2.5Eg correlates with the pseudogap energy Eg found in the speciﬁc heat measurements [249] (see Sect. 4.2.2, Fig. 4.4). The in-plane TEP Sa in the untwinned YBCO crystals [246,1196] shows a negative slope for the entire OD region for T > T ∗ ∼ Tc . Similar temperature and doping dependence of the TEP have been observed for other cuprate superconductors such as Bi-2212, Tl-2201 excluding the LSCO compound where the TEP is positive in the whole doping region. The nonlinear temperature dependence observed in the UD region below the crossover temperature T ∗ may be related to the pseudogap phenomena [249], though phonon drag eﬀects (see e.g., [1268]) may be also responsible for the strong T -dependence of the TEP. The “universal” negative linear T -dependence is found for higher temperatures, T > T ∗ , as observed in conventional metals. An intriguing universal correspondence of the TEP at room temperature, S(290) = S(T = 290 K), with the in-plane hole concentration p (and Tc (p)) over the whole range of doping was discovered by Obertelli et al. [911] (OCT law) shown in Fig. 5.50b [1221]. In the OD region for p > popt ∼ 0.16 where Tc < Tc,max (popt ) the TEP S(290) depends quasi-linearly on the hole concentration: S(290) (24.2−139 p) µV/K, while in the UD region the dependence is close to the exponential. It has been suggested by the authors that the universal behavior of the TEP for various families of the cuprate superconductors conﬁrms a single-band picture for the charge carriers (holes) in the CuO2 planes in the cuprates. The OCT law represented by Fig. 5.50b can be used for an estimation of the hole concentration in a sample by the magnitude of the TEP S(290 K) similar to the universal Tc (p)/Tc,max dependence (1.5) proposed by Presland et al. [1028]. However, in certain cases a departure from the OCT law has been observed (see e.g., [294]). 5.4.4 Theoretical Models A number of theoretical models have been proposed to explain the unconventional transport properties in cuprates. In the earliest studies, the bandstructure theory has been applied to calculate temperature dependence of the

324

5 Electronic Properties of Cuprate Superconductors

resistivity, the Hall coeﬃcient and the thermoelectric power (TEP) in the OPdoped YBCO and LSCO crystals [49, 981]. In these studies, it was supposed that the main contribution to the transport current relaxation comes from the scattering of electrons by phonons. For this, the following estimation for the transport relaxation rate at high temperatures was used h2 ω 2 ¯ h ¯ = 2πλtr kB T 1 − + ... . (5.76) τtr 12(kB T )2 The eﬀective electron–phonon coupling constant λtr was determined by the function α2tr F (ω) (compare with (5.36)). Here, α2tr is the transport matrix element of electron–phonon interaction and F (ω) is the experimentally measured phonon density of states for the LSCO and YBCO crystals, which determines also the average of the square phonon frequency ω 2 . Calculations within the relaxation rate approximation (5.76) results in the resistivity ρ ∝ 1/(τtr ωp2 ) with linear T -dependence over a broad temperature range. However, the absolute values of the resistivity, or of the slope dρ/dT ∝ λtr /ωp2 , prove to be several times smaller than the experimental values and 5–8 times smaller anisotropy ρc /ρab . As noted by the authors, this discrepancy between theory and experiment could be improved by using larger values of λtr = 1.5–2 instead of the calculated λtr = 0.65–0.32 for optimally doped LSCO and YBCO, or by assuming the theoretical plasma frequencies ωp smaller by a factor two to three. Similar calculations of the Hall coeﬃcient (5.62) for the OP-doped LSCO and YBCO compounds yield results which agree with experimental data both in absolute magnitude and in sign of the transverse conductivity σxy = Rxyz Bz : Rxyz > 0, Ryzx < 0, Rzxy < 0. The band-structure calculations also predict the sign change of Rxyz in LSCO as the concentration of Sr increases. Thus, the band-structure calculations of the transport coeﬃcients in a model with a strong electron–phonon interaction have demonstrated a qualitative agreement with the experimental data in the OP-doped and OD regions. However, in the UD (pseudogap) region these calculations fail to give even a qualitative description, e.g., a strong temperature dependence of the Hall coeﬃcient and the RH ∝ (1/x) dependence on the hole concentration x. An explanation of the strong temperature dependence of the RH with the occurrence of a pronounced maximum in the UD YBCO crystal (see Fig. 5.45a) has been achieved within a model of preformed pairs of bipolarons by Alexandrov et al. [41, 42]. To explain the large resistivity in cuprates with a quasi-linear T -dependence and an anomalous T -dependence of the Hall coeﬃcient and the Hall angle, several models have been proposed (for a brief review, see [1194]). In particular, the spin-ﬂuctuation scattering mechanism has been considered in a number of studies of transport coeﬃcients (see e.g., [860, 861, 1194] and references therein). This interaction provides a suﬃcient intensive and anisotropic scattering of the charge carriers, as has been discussed in previous sections concerning the QP life-time observed in

5.4 Transport Properties

325

ARPES experiments (Sect. 5.2.2) and the scattering rates in optic experiments (Sect. 5.3.2). Calculations of the resistivity within the nearly antiferromagnetic Fermi liquid (NAFL) model (see (3.53)) have reproduced the main features of the T -dependent resistivity and the Hall coeﬃcient [849, 850, 1194]. In particular, a strong anisotropy of scattering rates around the FS within the NAFL theory, with a “hot” region at the antinodes (strongly coupled QPs) and a “cold” region at the nodes (weakly coupled QPs), plays a crucial role in explaining the anomalous temperature dependence of the Hall coeﬃcient and the quadratic temperature dependence of the Hall angle. The importance of the vertex corrections to the current resulting in the anomalous behavior of the Hall eﬀect was stressed by Kontani et al. [635]. A model of “cold spots” in which the QP life-time is short everywhere on the Fermi surface except near the BZ diagonal was considered by Ioﬀe et al. [525]. To take into account the strong Coulomb correlations at the copper sites, an eﬀective Hubbard p–d model for the CuO2 plane was used in calculations of the resistivity within the memory function method (see (5.31)) [518]. Contrary to the NAFL and other spin-fermion models, in the Hubbard model no ﬁtting parameters are required to describe the electron scattering by the spin ﬂuctuations determined by the kinematic interaction related to the hopping matrix elements. Both the quasi-linear temperature dependence of the resistivity in a broad temperature range (up to a maximal spin-ﬂuctuation energy ∼ 2J ∼ 3,000 K) and the large magnitude of the resistivity observed in experiments have been reproduced. Numerical methods of studying charge dynamics in ﬁnite correlated systems have proven to be useful in elucidating the role of strong electron correlations in the transport properties of the cuprate superconductors (for a review, see [550]). The weakness of these methods is the smallness of systems being studied. To explain the temperature and doping dependences of the TEP, several theoretical models have been proposed. The band-structure theory was used to calculate the TEP in the OP-doped YBCO and LSCO crystals [49]. Three diﬀerent models of the energy-dependent conductivity σ(ε) in (5.75) were considered which produced negative in-plane TEP for YBa2 Cu3 O7 : Sxx < Syy < 0, which decrease with increasing temperature. A signiﬁcant quantitative diﬀerence in the magnitude of |Sαα (290 K)| = 10–70 µV/K depending on the σ(ε) model was observed. Similar results were obtained for the LSCO model. In the OD region, like in the Bi-2201 samples [634], the saddle point van Hove singularity (narrow-band eﬀects) may result in a nonlinear T -dependence [893]. The nonlinear T -dependence of the TEP can be also explained by strong phonon drag eﬀects assuming a strong electron–phonon interaction (see e.g., [245, 294, 1268]). In the high temperature region, the phonon drag contribution Sg may be large but at low temperatures it vanishes. The competition between a negative diﬀusion contribution Sd and a positive drag contribution

326

5 Electronic Properties of Cuprate Superconductors

Sg , which depends on doping, results in a strong temperature variation. The peculiar temperature dependence can be caused by freezing out of the “Umklapp” scattering processes at low temperatures [245]. These processes give a negative contribution to the drag term Sg since in this case the charge carriers are dragged in the direction opposite to the phonon diﬀusion. Thus, quite complicated processes of the charge carrier diﬀusion and their scattering by phonons govern the TEP dependence on doping and temperature. Resume In conclusion, studies of the transport properties in cuprates have conﬁrmed the complicated evolution of the metallic properties in cuprates with the temperature and doping evidenced by the spectroscopic data. 1. At low-hole concentration, p < 0.05, localization eﬀects are observed at low temperatures in the normal state resistivity. After crossing the superconducting boundary, the “universal” charge and thermal conductivities are emerging. They are induced by nodal QPs on the FS in the impurity band of the d-wave superconductor. The superconducting d-wave gap magnitude estimated from the gap curvature at nodes are close to a pseudogap gap at antinodes. At high temperatures, a metallic-like in-plane conductivity is restored. 2. In the UD and OP-doped regions, approximately in the interval of hole concentrations 0.05 < p < 0.20, the pseudogap in the electronic DOS shows up in the charge dynamics, which exhibits a series of unusual transport properties: the resistivity varies linearly with the temperature in a broad temperature range, the Hall coeﬃcient and the Hall angle reveal anomalous temperature dependence (in clean samples), the WF law is violated. Transformation of the transport properties with doping and temperature suggests a topological phase transition from an “arc,” or pocket-type FS to a large FS. Observation of quantum oscillations (the SdH and DHvA eﬀects) in the underdoped YBCO (123 and 124) compounds suggests existence of small electron-like FS pockets, while this conﬁrms a large FS in the overdoped Tl-2212. 3. In the OD region p > 0.20, a crossover to the Fermi-liquid like behavior is found, which is seen in a nearly T 2 temperature-dependence of the resistivity, a weakly temperature dependent Hall coeﬃcient (at moderate temperatures), the fulﬁllment of the WF law. 4. The anomalous charge dynamics in cuprates suggests a highly anisotropic transport relaxation rate with a “hot” region at the antinodes (strongly coupled QPs) and a “cold” region at the nodes (weakly coupled QPs) on the FS, which is supported by the magnetoresistance study in the Tl-2201 compound. The anisotropy and its doping dependence can be explained by electron scattering on spin ﬂuctuations.

5.5 Superconducting Gap and Pseudogap

327

5.5 Superconducting Gap and Pseudogap The superconducting gap studies of the quasiparticle excitation spectrum provide crucial information concerning the mechanism of high temperature superconductivity. The symmetry of the gap and its temperature dependence is directly related to the superconducting pairing mechanism. We have already discussed studies of the superconducting gap and pseudogap by thermodynamic methods in Sects. 4.2.2 and 4.3.2, by ARPES and microwave spectroscopy in Sect. 5.2.2, by optic spectroscopy in Sects. 5.3.3 and 5.3.4 and the inﬂuence of the gap on the transport properties in Sect. 5.4.3. However, in these studies only the absolute value of the superconducting gap can be found. The phase of the superconducting order parameter, which is a two-component function can be determined by the phase-sensitive experiments. These have directly proved the d-wave symmetry of the gap as discussed by van Harlingen [1302] and Tsuei and Kirtley [1276]. A comprehensive analysis of temperature and doping dependence of the gap and pseudogap magnitude were obtained by the scanning tunneling microscopy (STM) and spectroscopy (STS), which are reviewed in depth by Fischer et al. [337]. These studies have evidenced also a nanoscale spatial heterogeneity in the electronic structure of cuprates which is important for the general understanding of the high-temperature superconductivity. In this section, we consider the results obtained by the STM, STS and the phase-sensitive experiments. 5.5.1 Gap Symmetry Already in the ﬁrst experiments with cuprates an occurrence of superconducting pairs at T < Tc was found. These experiments performed ﬂux quanta measuring in superconducting rings of ceramic YBCO under random ﬂuctuations of magnetic ﬂux [404]. The value of the ﬂux quanta φ0 = hc/2e has unambiguously shown that the superconducting current is transferred by electron pairs having the charge 2e. This result was conﬁrmed later in observations of a ﬂux lattice, measuring the Shapiro steps in the nonstationary Josephson eﬀect at the bias voltage V = h ¯ ω/2e, etc. The existence of Cooper pair condensate with a zero total momentum has been conﬁrmed by Andreev (or Andreev–Saint–James) reﬂection experiments (for a review, see [275]). In the general pairing theory, the superconducting order parameter is represented by the pair correlation function: Δαβ (k) ∝ cα,k cβ,−k

(5.77)

where cα,k is the annihilation Fermi operator for an electron (hole) with the wave vector k and the spin α and . . . denotes the statistical average in the superconducting state. The pair correlation function should be antisymmetric with respect to the permutation of the two Fermi operators. By writing the

328

5 Electronic Properties of Cuprate Superconductors

spin indices (α, β) dependence in (5.77) as a 2 × 2 matrix, the k-dependence of the gap can be represented as a sum of two terms (see e.g., [1320]): ˆ ] σy Δαβ (k) = i [Δ(k) I + d(k) · σ

(5.78)

where I is the unit matrix and σ ˆ = (σx , σy , σz ) are the Pauli matrices. The one-dimensional representation, which is even in the k-wave vector, is given by the scalar function Δ(k) = Δ(−k), which describes the singlet pairing (the spin of the Cooper pair S = 0). The three-dimensional representation, which is odd in the k-wave vector, is given by the vector function d(k) = −d(−k), which determines the triplet pairing (with S = 1). The suppression of the Knight shift below Tc in cuprate superconductors has conﬁrmed the singlet pairing with spin S = 0 (see Sect. 3.3.1, Fig. 3.22). Therefore, we consider below only the singlet pairing. In the discussion of the superconducting gap symmetry, it is customary to distinguish the conventional pairing from the unconventional one. For the conventional pairing, which is frequently called s-wave pairing the gap function Δ(k) has the same symmetry as the Fermi surface (FS). For the unconventional pairing, the symmetry of Δ(k) is lower than that of the FS. General aspects of the gap symmetry and possible irreducible representations of the gap in the tetragonal lattice (D4h point group) and the orthorhombic lattice (D2h point group) are discussed in several reviews, e.g., Annett [74], Annet et al. [76], Sigrist et al. [1169]. For simplicity, we consider a square lattice in the CuO2 plane with a period a. Then the conventional pairing is given by the A1g representation, which can be speciﬁed by the function Δs (k) = ψ0s + ψ1s (cos akx + cos aky ),

(5.79)

where ψ0s is the (complex) order parameter for the conventional s-pairing (with an isotropic gap), and ψ1s is the order parameter for the so-called “extended” s-wave pairing. In the latter case, the gap is zero along the lines determined by the equation | kx | + | ky |= π/a in the Brillouin zone (BZ). The unconventional pairing is described by order parameters with a lower symmetry. The d-wave pairing with orbital l = 2 can be described by the functions Δ1d (k) = ψ1d (cos akx − cos aky ),

Δ2d (k) = ψ2d sin akx sin aky .

(5.80)

The (complex) order parameters Δ1d (k) and Δ2d (k) correspond to the B1g and B2g irreducible representations, respectively, which are usually called as the dx2 −y2 and dxy gaps. For the d-wave pairing, the gaps (5.80) change the sign on the FS and have four nodes on the 2D FS (four lines on 3D FS) where the lines (0, 0) → (±π, ±π) in the 2D BZ cross the FS for the Δ1d (k) or the lines (0, 0) → (0, ±π), (±π, 0) for the Δ2d (k). The d-wave gaps violate the tetragonal FS symmetry characterized by the fourfold axis Cz4 . For the orthorhombic lattice with D2h symmetry, as in YBCO crystals the “extended” s-wave pairing of the A1g symmetry can be written in the form

5.5 Superconducting Gap and Pseudogap

Δs (k) = ψa cos akx + ψb cos bky ,

329

(5.81)

where a, b are the lattice constants. In this case, the symmetry of the gap preserves the orthorhombic FS symmetry characterized by the twofold axis Cz2 at any values of the coeﬃcients. For instance, we can take ψa = α ψ1s + ψ1d and ψb = α ψ1s − ψ1d which looks like a superposition of the s-wave pairing ψ1s with the amplitude α and the “d-wave pairing” ψ1d . For a small orthorhombic distortion (b − a) a, we can assume α 1. In this case, the gap (5.81) reveals the “d-wave symmetry” with a small “admixture” of the “s-wave symmetry.” The gap Δ(k) = (1 + α) cos akx − (1 − α) cos aky vanishes along the lines |kx | ≈ |ky +2α cot aky | which FS crossing results in nodes lying close to the nodes for the d-wave pairing Δ1d (k) (5.80) in a tetragonal lattice. For a three-dimensional (3D) lattice, a gap on out-of-plane electron orbitals crossing the 3D FS may appear. This gap may have either s-wave symmetry (A1g representation) or d-wave symmetry (B1g or B2g ) representations which violate the FS symmetry with respect to the rotation around x or y axes. 5.5.2 Tunneling Experiments The tunneling experiments provide unique information regarding changes of electronic density of states close to the Fermi energy EF at the superconducting phase transition. In the case of a one-particle tunneling from a superconductor through an insulator layer to a normal metal (SIN junction), a break of Cooper pairs occurs, and the voltage dependence of the tunneling conductance measures the densities of quasiparticle states below and above EF in the superconductor. Direct observation of the superconducting gap in the tunneling spectra on conventional superconductors has played a major role in conﬁrming the BCS pairing theory. The invention of the scanning tunneling microscopy (STM) and the development, on the basis of this technique, of the scanning tunneling spectroscopy (STS) allows to measure the local density of states (LDOS) with a spatial resolution down to atomic scale and the energy resolution of a few mega electron volt. In recent years, high precision data have been accumulated by STM and STS for the cuprate superconductors and this greatly contributed to our understanding of the superconductivity in these materials (for a review, see [337]). In the STS, the tunneling conductance as a function of the bias voltage V is measured by a tip at a constant distance z of approximately 10 ˚ A above the surface of a sample. By changing the tip position r = (x, y)a parallel to the sample surface, variation of the LDOS can be studied. Within the tunneling Hamiltonian formalism, the bias voltage dependence of the conductance is determined by the thermally smeared LDOS at the Fermi energy of the sample:

dI g(r, V ) = ∝ dω [−∂f (ω − eV )/∂ω]N (r, ω), (5.82) dV

330

5 Electronic Properties of Cuprate Superconductors

where f (ω) is the Fermi distribution and N (r, ω) is the LDOS. In deriving (5.82), it is assumed that the DOS of the tip and the tunneling matrix element do not essentially depend on the energy (or the wave vector) within the measured bias potential V . For a positive bias voltage V applied to the sample electrons tunnel into unoccupied sample states, while for a negative bias they tunnel out of occupied sample states (or holes tunnel into the sample). Similar to the ARPES, the STM and STS are surface-sensitive experiments as only layers close to the surface are probed by the tunneling current. The ARPES and STS experiments give complimentary information. In ARPES, the wave vector dependent single-particle spectral function A(k, ω) averaged over the light spot on the sample surface is measured (see Sect. 5.2.2). In the STS, the tunneling current yields the LDOS at a point r on the sample surface averaged over the angle perpendicular to the tip direction (in the (a, b) plane for a tip mounted perpendicular to the CuO2 plane). In planar junctions, the averaged over a sample surface DOS N (ω) = k A(k, ω) can be measured. In the superconducting state, the spectral function Asc (k, ω) in the quasiparticle approximation (5.20) has two peaks at the energy of Bogoliubov excitations: ω = ±Ek , Ek = η˜(k)2 + |Δ(k)|2 where η˜(k) is the electron dispersion in the normal state, η˜(kF ) = 0. For a conventional pairing with a k-independent gap Δ0 , the conductance (5.82) at T Tc vanishes for |eV | < Δ0 and has a U-shape with two peaks at |eV | ≥ Δ0 . In the case of the d-wave pairing (5.80), the gap Δ(k) has nodes which results in the linear dependence of the DOS at the Fermi energy: N (ω) ∝ |ω|. Because of this, the conductance (5.82) reveals a V-shape and two coherent peaks at the bias voltage |V | near the maximal value of the gap Δmax . These coherent peaks below Tc have been observed in the tunneling experiments and, in particular, in STS which has conﬁrmed the pairing scenario of high-temperature superconductivity in cuprates. Below we consider several examples of these studies. Superconducting Phase The most reliable STM and STS measurements were obtained, similar to the ARPES experiments, for the Bi-compounds, where atomically ﬂat and clean surfaces can be prepared by cleaving between adjacent BiO layers. STM images of the BiO surface have revealed the bismuth lattice, which suggests that the tunneling into the BiO surface probes the underlying CuO2 layer. To observe reproducible tunneling spectra, homogeneous samples should be used, which can be prepared at low temperatures in ultra high vacuum. Figure 5.51a, b shows diﬀerential conductance spectra on Bi-2212 sample close to the optimal doping (Tc = 92.3 K) [1049]. The variation of the spectrum with the tip position along a path of 34 nm (plot a) is rather small and the spectra plotted on top of each other (plot b) show a unique curve which proves the high quality of the tunnel junction. The panel (c) of Fig. 5.51 demonstrates the doping dependence of the tunneling spectra in the Bi-2212 crystals from the

5.5 Superconducting Gap and Pseudogap

a

c

0

74.3K 68meV

0 2.0

92.2K 83meV

1.5 1.0 0.5 0 –200 –150 –100 –50

0

50 100 150 200

VSample[mV]

dI /dV [Normalized]

–1

dI /dV[GΩ ]

b

Tc = 56.0K 2Δp= 42meV

x[nm]

dI /dV[GΩ –1]

34

2

331

2

83.0K 88meV

1 0 –300

– 150

0

150

300

VSample[mV]

Fig. 5.51. Diﬀerential conductance spectra on Bi-2212 at T = 4.8 K: (a) as a function of the tip position along a path of 34 nm and (b) the spectra plotted on top c 1995). (c) Doping of each other (reprinted with permission by APS from [1049], dependence of the tunneling spectra on Bi-2212 at T = 4.2 K with increasing oxygen concentration in the samples from the slightly underdoped sample (Tc = 83 K – at the bottom) to highly overdoped one (Tc = 56 K – at the top), 2Δp is the energy between two coherent peaks (after [1050]

underdoped (UD) (Tc = 83 K) to overdoped (OD) (Tc = 56 K) samples at low temperature [1050]. The spectra reveal intense and sharp peaks of comparable heights at the gap edges ±Δp with a ﬂat background conductance below ±500 meV. A dip feature at −2Δp is seen in this spectrum at the negative sample bias only, while nearly symmetric dip features (in normalized conductance) have been found in other STS experiments (see e.g., [1406]). The observation of the peak-dip structure corroborates with the ARPES energy dispersion curves (see Sect. 5.2.2) where this structure has been explained by a bilayer splitting. Comparison of the ARPES results with the scanning tunneling spectra on Bi-2212 has revealed the important role of the strong coupling of the quasiparticles with a collective mode in explaining the dip feature, while the asymmetry of the spectrum and its doping dependence have been ascribed to the bilayer splitting of the van Hove singularities below the Fermi energy [484]. At the same time, a better ﬁtting of the low-energy part of the spectrum was obtained with an isotropic tunneling matrix element. The essential role of the strong-coupling of the electrons with a bosonic mode was stressed by Zasadzinski et al. [1406]. By using the Eliashberg d-wave strong-coupling theory, they have performed a quantitative ﬁtting of the tunneling spectrum and, in particular, the symmetric dip features in the Bi-2212 normalized spectrum. The asymptotic V-like shape of the low-energy DOS, which does not change notably with doping, is characteristic to the d-wave superconducting

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5 Electronic Properties of Cuprate Superconductors

gap. The ﬁnite quasiparticle density of states at zero-bias may be ascribed to an impurity scattering as discussed in Sect. 5.1.2 (see Fig. 5.10) where the occurrence of local impurity states inside the gap was also considered (see Fig. 5.9). The gap Δp decreases with doping and shows a reduction of the ratio 2Δp /kB Tc from ∼ 12 in the UD sample to ∼ 8.7 for the OD sample in Fig. 5.51c. The ratio is much larger than it would be expected for a pure d-wave superconductor 2ΔSC /kB Tc = 4.3. The increase of the gap Δp in the UD region with a simultaneous decrease of Tc is a behavior speciﬁc to the pseudogap which scales with the crossover temperature T ∗ as observed in other experiments (see Fig. 5.42). Similar results were obtained for tunneling spectra on Bi-2223 [672], while the STS on Bi-2201 showed certain peculiarities, which may be related to its single-layer structure. The tunneling spectra on YBCO samples revealed a more complicated structure characterized by a large ﬁnite conductance at zero bias voltage, multiple coherence peaks at the gap edges, etc. These features can be partially related to the presence of the CuO conducting chains in the YBCO crystal structure, which give a nonvanishing contribution to the tunneling current. The tunneling experiments on electron-doped cuprates as NCCO gave controversial results: most of them point to the s-wave gap symmetry, while several measurements were more consistent with the d-wave symmetry. These controversial results observed in other experiments also, e.g., in ARPES and optic measurements, can be partially explained by an unconventional angular dependence of the gap with the maximum away from the antinodal (π, 0) points [809]. Normal State Pseudogap Studies of the temperature dependence of tunneling spectra have conﬁrmed the occurrence of a normal state pseudogap in cuprate superconductors, which has been also revealed by various experiments as discussed in previous sections (for a review, see [504, 1244]). Figure 5.52 shows the temperature dependence of the tunneling spectra on the Bi-2212 samples: UD (Tc = 83 K) (a) and OD (Tc = 56 K) (b) [1050]. The plots clearly demonstrate a pseudogap structure above Tc both for the UD and OD samples. The sharp coherence peaks observed at low temperatures diminish as the temperature increases and above Tc transform to broad maxima. The dip structure characteristic to the superconducting state vanishes above Tc also. Thus, the superconducting gap evolves continuously into a normal state pseudogap above Tc . With further increase of the temperature, the pseudogap is “ﬁlling up” by excitations, the zero-bias conductance increases and the pseudogap levels oﬀ at a crossover temperature T ∗ . However, the gap magnitude does not change appreciably with temperature so that the pseudogap scales with the superconducting gap as shown in Fig. 5.52b: the 4.2 K spectrum with Δ69 = Δ4.2 thermally smeared to 69 K (dashed line) ﬁts the measured curve better than the similar spectrum with a smaller gap Δ69 = 0.8Δ4.2 (dotted line). In the

5.5 Superconducting Gap and Pseudogap

333

Fig. 5.52. Temperature dependence of the tunneling spectra (a) on the UD Bi-2212 (Tc = 83.0 K) and (b) on the OD Bi-2212 (Tc = 74.3 K) samples (after [1050])

OD sample, the gap is smaller than in the UD one and closes more rapidly at a lower temperature T ∗ . A similar behavior was observed in the OD single-layer Bi-2201 compound (Tc = 10 K) [671]. The coherent peaks at the gap edges Δp = ±12 meV and the dip structure vanish at Tc , while the pseudogap is washed out at much higher temperature of T ∗ 68 K. The reduced gap magnitude 2Δp /kB Tc 28 in Bi-2201 is much larger than in Bi-2212 but the ratio 2Δp /kB T ∗ 4.3 is the same as in other cuprate compounds. It is believed that a low magnitude of Tc in Bi-2201 is due to strong ﬂuctuations of the superconducting order parameter in this single-layer compound. In the optimally doped YBCO and electron-doped compounds, no signature of the pseudogap was observed: the coherence peak and the gap vanish at the bulk Tc (for details, see [337]). Remarkable results have been obtained in STM studies of vortices in superconductors in magnetic ﬁelds. In particular, it was possible to establish a direct connection between the pseudogap inside the vortex core at low temperatures and that one observed in the normal state. By changing the tip position across the core, a smooth variation of the spectrum occurs: coherence quasiparticle peaks outside the core evolve into a pseudogap-like structure inside the core. This behavior is similar to the temperature-dependent evolution of the gap considered above. Figure 5.53 shows the tunneling spectra in the UD Bi-2212 (a) and OD Bi-2212 (b) samples measured at 4.2 K at the center of a vortex core (solid line) and between vortices (dashed line) [1051]. The spectra above Tc measured in zero ﬁeld (solid lines) are similar to the spectra at the center of the core at 4.2 K smeared to the indicated temperatures (dotted lines). These results demonstrate a correspondence between the pseudogap in the vortex core and in the normal state.

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5 Electronic Properties of Cuprate Superconductors

b

3.5 Tc = 74.3K

dI/dV [GΩ–1]

3.0 2.5

4.2 K

2.0 1.5 81.9 K

1.0 0.5 0 –300

–200

–100

0

100

200

300

VSample [mV]

Fig. 5.53. Tunneling spectroscopy of (a) UD Bi-2212 (Tc = 83.0 K), and (b) OD Bi-2212 (Tc = 74.3 K) measured at 4.2 K at the center of a vortex core (solid line) and between vortices (dashed line). The spectra measured in zero ﬁeld at high temperatures above Tc (solid lines) reﬂect the pseudogap structure. They are compared with the spectra at the center of a core at 4.2 K smeared to the denoted temperatures (dotted lines) (after [1051])

Thus, the main result of the pseudogap studies considered above is the observation of the pseudogap Δp in both the underdoped and the overdoped compounds and its scaling with the superconducting gap Δsc . Although the pseudogap is much larger than the superconducting temperature, it scales with the crossover temperature: 2Δp /kB T ∗ ≈ 4.3 which corresponds to the BCS-like relation for the d-wave pairing with Tc replaced by T ∗ . These results suggest a common origin of the superconducting gap and pseudogap and support a preformed pair scenario of superconductivity in cuprates [337]. However, this conclusion is at odds with other experiments. In particular, the superconducting gap measured by Andreev reﬂection is diﬀerent from the gap found in STS [277]. Recent ARPES results have revealed a distinct two-gap structure with a pseudogap observed in the antinodal region of the BZ and a superconducting gap in the vicinity of the nodal points on the FS arcs ([586, 633, 1231] – see Sect. 5.2.2, Fig. 5.27). The superconducting gap scales with Tc and closes in the UD region, while the pseudogap scales with the crossover temperature T ∗ as observed in STS. Two diﬀerent gaps are also observed in the intrinsic Josephson junction tunneling experiments in Bi-2212, which will be considered below. To reconcile the STS results with other experiments, Boyer et al. [172] have proposed to study a normalized diﬀerential conductance gN (E, r, T ) = g(E, r, T )/g(E, r, TN ),

(5.83)

where g(E, r, TN ) is the conductance measured at some reference temperature TN above Tc . This is a common procedure in the conventional superconductors where the normalized DOS NS (ω)/N0 (ω) is studied. In this way, it is possible to exclude the inﬂuence of the normal state background. In the cuprate superconductors where the LDOS strongly depends on a surface disorder and the

5.5 Superconducting Gap and Pseudogap

335

normal state pseudogap, the normalization is crucial to separate the two contributions. The STM studies of the Pb-Bi2 Sr2 CuO6+x sample (Tc = 15 K) by Boyer et al. [172] have revealed gap maps nearly independent of temperature with a broad distribution of gaps Δp = 16±8 meV similar to other STM experiments. However, the normalized conductance (5.83) has shown homogeneous gap maps with the small superconducting gap Δ 6.7 meV, which closes below Tc . Since the temperature dependence of the pseudogap is determined by a large energy (of the order of the exchange energy J ∼ 1, 500 K as found in thermodynamic studies, see Sect. 4.2.2), a weak temperature dependence of the gap maps is observed in STS measurements. Thus, in this approach a coexistence of a sharp homogeneous superconducting gap below Tc superimposed on a large inhomogeneous pseudogap has been found which supports the two independent gap scenario. A local pairing on the atomic scale has been observed in the OD and OpD Bi-2212 samples in STM-STS measurements by Gomes et al. [396]. A local gap Δp in the spectra was observed above Tc at a temperature Tp , which was determined by the criterion dI/dV (V = 0) ≥ dI/dV (V > 0). In spite of a large variation of the Δp over the surface of the sample, the gap followed the relation 2Δp /kB Tp 8 as observed in other STS experiments. It has been suggested that the superconductivity occurs due to the proliferation of the local gap. However, in the UD Bi-2212 samples two energy scale was found, one related to a pairing gap Δp and a larger one to a pseudogap. The lack of the momentum resolution in the STS studies precludes disentangling these two gaps observed in ARPES as mentioned above. Intrinsic Josephson Junctions The strong anisotropic high temperature superconductors (HTSC), such as Bi2 Sr2 CaCu2 O8+δ (Bi-2212), form natural superconducting multilayers where the superconducting order parameter is periodically modulated along the c-axis. The modulation is so strong that the adjacent superconducting layers are only weakly coupled by the Josephson eﬀect that leads to interlayer tunneling between neighboring CuO2 planes [620]. Every pair of adjacent superconducting layers, together with the intervening nonsuperconducting layer, behaves like a superconductor-insulator-superconductor (SNS) intrinsic Josephson junction (IJJ), and the whole crystal acts as a vertical stack of IJJ. This behavior has been directly observed in c-axis transport measurements (for a review, see [1395]). If all junctions of an N junction stack can be switched into the resistive state individually, the total current–voltage (I–V ) characteristics consists of N +1 branches diﬀering by the number of junctions in the resistive state. As an example, Fig. 5.54 shows the ﬁrst three branches of the I–V -characteristic of an IJJ stack with a total number of 130 junctions for the Tl2 Ba2 Ca2 Cu3 O10+δ (Tl-2223) crystal (a) and for comparison the complete I–V -characteristic of an artiﬁcial threefold Nb Josephson junction stack (b), both measured at

336

a

5 Electronic Properties of Cuprate Superconductors 40 20

I (µA)

0 –20

c

–40 –100

0 U (mV)

50

100

4.2 K 20 38 K

2

dI/dV (mS)

b

–50

25

1 I 0 (mA)

15 50 K 10

74 K

5

–1

92 K –2 –10

–5

0 U (mV)

5

10

0 –3

–2

–1 0 1 2 V/10 (10 mV)

2

3

Fig. 5.54. (a) First three branches of the I–V characteristic of a Tl-2223 step stack with a total number of 130 junctions and (b) I–V characteristics of an artiﬁcial threejunction Nb/Al-AlOx /Nb stack at T = 4.2 K. Arrows indicate voltage switching (after [1120]).(c) Temperature dependence of dI/dV for the intercalated (HgBr2 )Bi2212 crystal with Tc = 66 K as a function of the voltage per one mesa (after [1394])

T = 4.2 K [1120]. After exceeding the critical current of an individual junction in the stack, the I–V characteristic exhibits a voltage jump Vc . For the Nb tunnel junctions, the jump is 2.5 mV for each junction which is approximately 2Δ/e as expected for an SIS tunnel junction between two s-wave superconductors. The IJJ stacks in Bi-2212 and Tl-2223 exhibit multiple branched I–V characteristics. The voltage jump is approximately 27 mV in the Tl-2223 and about 22 mV in Bi-2212 material. These values are close to Δ/e which was found for the c-axis tunneling between two dx2 −y2 superconductors [1230]. The current rises in the quasiparticle branch of the Nb tunnel junctions nearly vertically at the gap voltage, while in the I–V characteristic of the IJJ for the dx2 −y2 -wave superconductors the current shows a gradual increase. The intrinsic tunneling spectroscopy is a bulk technique (“a look from the inside at a superconductor”) in comparison with the STS and ARPES methods which probe the surface of the sample. Extensive studies of the current–voltage characteristics in Bi- and Tl- based compounds have presented clear evidence for coexistence of the SG and the PG in cuprate superconductors (see e.g., [661, 1208] and references therein). Figure 5.54c demonstrates temperature dependence of the dynamic conductance dI/dV in the intercalated (HgBr2 )Bi-2212 crystal as a function of the voltage V /10 per one mesa [1394]. Insertion of inert HgBr2 -molecules in between adjacent BiO-layers slightly suppresses the superconducting transition temperature Tc 75 K

5.5 Superconducting Gap and Pseudogap

337

of the pristine crystal to Tc = 62–65 K but decreases the c-axis critical current Ic that results in signiﬁcant reduction of the Joule heating. Two sharp quasiparticle peaks corresponding to a superconducting gap edge and dips at higher voltage are seen below Tc . A PG-like feature in the form of a weakly temperature-dependent wide maximum is observed up to 270 K. The symmetric form of the dI/dV curves is noteworthy in comparison with the asymmetric conductance observed in STS measurements. Magnetic ﬁeld studies of the intrinsic tunneling characteristics have enabled to distinguish two gaps: while the superconducting gap (SG) closes at Hc2 (T ), the pseudogap (PG) does show neither temperature nor magnetic ﬁeld dependence [660]. This implies that the SG and PG represent diﬀerent coexisting phenomena. The inﬂuence of the heating eﬀect on the I–V -characteristic of the IJJ stack has been analyzed by Krasnov [662], who has shown how to distinguish between the nonlinear eﬀects due to overheating in small mesas and the quasiparticle I − −V characteristics observed in the IJJ. Thus, based on diﬀerent temperature, doping and magnetic ﬁeld dependence of the SG and PG, the intrinsic tunneling spectroscopy provides strong evidence for independent and competing origins of these two gaps. Inhomogeneity and Superstructures The STM/STS studies have revealed atomic-scale spatial variations of the LDOS correlated with nanometer scale inhomogeneity in the cuprate superconductors. Measurements of the LDOS (5.82) as a function of the tip position on the underdoped Bi-2212 surface have shown a strong variation of the tunneling spectrum from a superconducting-like with well-deﬁned coherent peaks to a pseudogap-type with broad maxima. These variations occurred on a short-length scale of several nanometers of the order of the superconducting phase coherence ξ. The LDOS inhomogeneity may be either an intrinsic property of cuprate superconductors related to the electronic phase separation (see e.g., [494, 980]) or an extrinsic one resulting from the stoichiometric inhomogeneity, e.g., due to a local variation in oxygen concentrations [951]. As discussed above, high-quality tunnel junctions can be prepared, which demonstrate quite homogeneous LDOS (see Fig. 5.51), and therefore the large gap inhomogeneity should be of an extrinsic origin (see also [483, 672]). Further studies have proved a strong correlation between the impurity state location and the nanoscale disorder in electronic spectra [820]. It has been found that close to the dopant defect cluster, identiﬁed by a local impurity level, a suppression of the superconducting peak occurs while a strong superconductivity develops between them. It should be pointed out also that thermodynamic measurements and NMR data indicate that the bulk electronic state is quite homogeneous [746]. This supports the conclusion that the strong inhomogeneity discussed above is due to extrinsic surface eﬀects. At the same time, weak low-energy LDOS modulations were found by studying the Fourier-transform (FT) g(q, ω) of the diﬀerential tunneling

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5 Electronic Properties of Cuprate Superconductors

conductance g(r, ω) (5.82). In this FT-STS method, the g(r, V ) is measured at diﬀerent bias voltages V on a nanometer-size (∼ 50 nm) square ﬁeld of view (FOV) on a sample surface. If a periodic charge-density wave appears in the sample, the FT g(q, E) shows a maximum for the wave vector qi associated with the period of the modulation λi = 2π/qi . Another origin of a weak modulation in the LDOS is the quantum interference of quasiparticles (QPs) in the presence of elastic scattering. These results in electronic standing ways revealed as maxima of the FT g(q, E) at characteristic scattering vectors qi (E) for QPs with the energy E(qi ). Thus, the FT-STS method enables to determine the QP energy dispersion relations. The QP interference and periodic spatial modulations have been discovered in FT-STS studies of the low-energy part of the tunneling spectra. Initially, the modulation in the form of a “checkerboard-pattern” with a periodicity of four lattice spacing 4a0 oriented along the copper–oxygen bonds in Bi-2212 compound was observed in magnetic ﬁeld around the center of a vortex core [476]. Subsequently, similar pattern of the charge modulations were found without magnetic ﬁeld in the superconducting phase of Bi-2212 [495]. It was argued that the observed stripe-like modulations coexist with the superconductivity, which suggests their common origin. However, the bond-oriented, energy-independent modulations with incommensurate periodicity ∼ 4.7a0 were detected in the normal pseudogap region also [1311]. This points to a correlation between the electronic ordering phenomenon and the pseudogap formation in the low-energy LDOS. Similar energy-independent spatial modulations at low energy were found for the lightly doped Ca2−x Nax CuO2 Cl2 (Na-CCOC) cuprates with x = 0.08, 0.10, 0.12 [425]. All the tunneling spectra showed the V -shape gap for E < 100 meV with very low conductance at E = 0. The FT g(q, E) of the conductance map revealed nondispersive wave vectors, which corresponds to the strong “checkerboard-pattern” modulations with periodicity 4a0 and 4a0 /3. It was suggested that the electronic modulations, consistent with a crystalline order, are associated with pseudogap formation. Much weaker modulations ascribed to QP interference were discovered at low temperatures in Bi-2212 compounds close to the optimal doping. In FT-STS studies, Hoﬀman et al. [477] have found two appreciable maxima in the FT g(q, E) at the incommensurate wave vectors qA,B dispersing with energy. Their dispersion was explained by considering the interference of Bogoliubov QPs with the d-wave gap energy E = |Δ(k)| on the FS. Detailed studies of atomic-scale spatial modulations in a slightly overdoped Bi-2212 crystal by McElroy et al. [818] revealed several modulation vectors qi (E). It was suggested that the largest contributions to the QP interference at a constant energy E should give the scattering wave vectors qi (E) = km (E) − kn (E) connecting eight equivalent points, an octet ks , at the ends of the Fermi surface arcs (“bananas”) determined by the d-wave gap energy E = |Δ(ki )|. By analyzing the energy dependence of several scattering wave vectors qi (E), the location of the octet elements ks was found. The loci of

5.5 Superconducting Gap and Pseudogap

339

Fig. 5.55. FT-STS study of Bi-2212 crystal: (a) The loci of scattering vectors ks (E) on the FS (open circles). The gray line is the FS measured by ARPES; (b) Energy gap dispersion Δ(Θk ) derived from the FT-STS at negative bias (open circles), positive bias (open triangles), and ARPES data (filled circles with bars) (after [818])

the octet ends ks (E) on the FS for diﬀerent energy E are shown in Fig. 5.55a by open circles. Their location perfectly agrees with the FS measured by ARPES (gray line) [282]. By determining the energy E = Δ(k) associated with kn , the energy gap dispersion was derived. Figure 5.55b shows the angle dependence Δ(Θk ) obtained from the FT-STS at the negative (open circles) and the positive biases (open triangles). A good agreement is seen with the ARPES data shown by ﬁlled circles with bars [282]. The similarity between the results obtained from the momentum space structure of the unoccupied states (positive bias) and the one from the occupied states (negative bias) proves that the superconducting state is associated with Bogoliubov QPs, which are a superposition of particle and hole states. An almost identical dispersion near the gap node was found in FTSTS studies of the nearly optimally doped Na-CCOC (Tc = 25–28 K) by Hanaguri et al. [426]. To detect the particle-hole interference of the Bogoliubov QPs on the FS, the authors examined the ratio map deﬁned by Z(r, E) = g(r, +E)/g(r, −E). In this way, it was possible to suppress the strong “checkerboard” modulation which dominated the conventional LDOS g(r, ±E). The QP interference in Na-CCOC was observed close to the nodes only and in narrower, as compared to Bi-2212 energy (E < 15 meV) and momentum ranges, which correlate with a large diﬀerence of their Tc . This suggests that the gap dispersion in the nodal region does not exclusively determine the energy scale of Tc and the energy region where coherent QPs exist is also of importance. It may be concluded that only coherent states on the FS arcs around the node determine the superconducting pairing and Tc . Then antinodal incoherent states on the FS can be assigned to the pseudogap and the “checkerboard” charge modulations seen in the FT-STS maps.

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5 Electronic Properties of Cuprate Superconductors

Fig. 5.56. (a) The average spectrum g(E) in Bi-2212 crystal associated with a particular gap magnitude Δ on the gap map. (b) The characteristic spectra from the two regions: Δ < 65 meV with the coherent peaks and Δ > 65 meV of the c 2005) PG-like type (reprinted with permission by APS from [819],

The coexistence of coherent QP excitations at low energies and incoherent excitations at higher energies in the pseudogap region was found in the gap map studies for four Bi-2212 samples: OD89, UD79, UD75, and UD65 by McElroy et al. [819]. The gap map was derived from the FT-STS measurements of the LDOS on a nanometer-size (∼ 50 nm) square ﬁeld of view (FOV). The map shows a distribution of the gaps Δ(r) = [Δ+ (r) − Δ− (r)]/2 on the FOV. Here, Δ+ (r) (Δ− (r)) is the positive (negative) energy of the ﬁrst maximum in LDOS above (below) the Fermi level at E = 0. Analysis of the gap maps revealed speciﬁc changes of the average LDOS and energy gaps with doping. Figure 5.56a shows the average spectrum g(E) associated with a particular gap magnitude Δ on the gap map. At small values of the gap, Δ < 50 meV, the spectra reveal coherence peaks (curves 1–4). The spectra for larger gap values exceeding ∼ 65 meV show a pseudogap-like behavior without well-deﬁned peaks (curve 6). For the OD89 sample, the average gap value is small, Δ 33 meV, and the majority of the tunneling spectra on the gap map are similar to the curves 1 and 2. With underdoping, the average gap value increases and the majority of the tunneling spectra look similar to curve 6. The characteristic spectra in Fig. 5.56b from the regions Δ < 65 meV show coherent peaks, while from the regions Δ > 65 meV the PG-like spectra are observed. However, the spectra at low energies, below ∼ Δ/2 in between the two vertical arrows in Fig. 5.55a, demonstrate for all doping studies a similar behavior with coherent peaks. This shows that there are two coexisting contributions to the tunneling spectra. The ﬁrst one originates from the low-energy coherent quasiparticles at the nodal points of the d-wave superconductor. The second contribution, at higher energies, originates from the incoherent excitations of the PG state, predominantly at the antinodal regions. The only

5.5 Superconducting Gap and Pseudogap

341

probability of ﬁnding diﬀerent types of spectra depends on doping. Further FT-STS studies of the low-energy LDOS agree with this conclusion. The analysis of the weak, incommensurate energy-dispersive modulations within the octet model, described above, has shown that Bogoliubov QP interference occurs at all dopings in the low-energy region on the FS arcs. The incoherent spectra appear at small doping, below p ∼ 0.14, in the antinodal regions at higher energies, with an emergence of the strong “checkerboard” charge modulations with a period of ∼ 4.5a0 . A new type of intrinsic nanometer scale inhomogeneity was found by Kohsaka et al. [627]. By using an atomic-resolution tunneling-asymmetry (TA) imaging, they observed a bond-centered electronic glass phase in two lightly hole-doped cuprate compounds: Na-CCOC (x = 0.12) and Bi-2212 doped by Dy. In the TA-imaging, the ratio of the tunneling currents at positive and negative biases was studied: R(r, V ) = I(r, +V )/I(r, −V ). Measurements of the R(r, V )-maps for the tunneling current have revealed the short-range 4a0 × 4a0 periodic correlations in the form of unidirectional domains. Both the translational and the rotation invariance (speciﬁed by the fourfold axes C4 ) were violated in the glass state. Moreover, the spatial variations in the electronic states on oxygen sites within the Cu–O–Cu bonds were detected. Since similar domain structures were found in two diﬀerent types of cuprates: the single layer Na-CCOC and the double-layer Bi-Dy-2212 compounds, it was argued that the observed bond-centered electronic glass is an intrinsic and universal phenomenon in the underdoped cuprates. A strong asymmetry of the tunneling conductance found at low doping with a much larger conductance for a negative bias for holes with respect to a positive bias for electrons was explained by strong correlation eﬀects for electrons caused by no-double occupancy restriction for them [69]. Resume 1. The scanning tunneling spectroscopy (STS) in superconducting state reveals intense and sharp peaks at the superconducting gap edges ±Δs with a ﬂat background conductance at higher energies. A smooth transition from the sharp peaks below Tc to broad maxima at the pseudogap energy ±Δp at Tc < T < T ∗ suggests a common origin of the gaps. However, studies of the normalized diﬀerential conductance have shown a coexistence of a sharp homogeneous superconducting gap below Tc superimposed on a large inhomogeneous pseudogap, which supports the two independent gap scenario. 2. The intrinsic Josephson tunneling spectroscopy provides strong evidence for independent and competing origins of the superconducting gap and pseudogap, which show diﬀerent temperature, doping and magnetic ﬁeld dependence. 3. The STS/STM studies have revealed several types of inhomogeneity in cuprate superconductors. The large gap inhomogeneity observed in direct

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5 Electronic Properties of Cuprate Superconductors

LDOS measurements can be considered as the extrinsic eﬀect caused by a chemical disorder. The much weaker inhomogeneity observed at low energies in the Fourier transform maps of the LDOS shows two type of modulations. The ﬁrst one is due to the Bogoliubov QP interference eﬀects close to the d-wave nodes in the superconducting state. The second one is a nondispersive modulation at higher energies, which can be related to the incoherent pseudogap states at the antinodes. They manifest themselves at low doping as a short-range local charge ordering with periods close to four lattice spacing in the form of the square “checkerboard” or unidirectional domains. These two modulations coexist in the superconducting state but compete with each other: with underdoping the local charge order suppresses the low-energy region on the FS arc where the interference of coherent QPs occurs. 5.5.3 Phase-Sensitive Experiments The phase-sensitive experiments exploit the interference eﬀects of the quantum-mechanical phases in Josephson tunnel junctions which enable to determine the phase ϕ of the superconducting order parameter Δ(k) = |Δ| exp(iϕ). For the unconventional d-wave pairing, the orbital moment l = 2 of the Cooper pair and the phase ϕ of the order parameter are quenched by the lattice potential. This enables to detect a change of the gap sign in (kx , ky )plane and to reveal the unconventional d-wave pairing in cuprates. Below we consider experiments where a direct observation of quantum interference eﬀects in a dc SQUID and a Josephson junction [1302] and the detection of a half-ﬂux quantum Φ0 /2 = hc/4e in a frustrated geometry [1276]. The supercurrent Is of Cooper pairs tunneling through a barrier in a Josephson junction (JJ) between two superconductors, left (L) and right (R), is determined by the expression Is = Ic sin γ,

γ = γL − γR + 2 π

Φ , Φ0

(5.84)

where Ic is the Josephson critical current and the phase γ depends on the left γL and right γR phases of superconducting order parameters and the R magnetic ﬂux Φ = L A · dl. This is determined by the vector potential A integrated across the barrier between the left (L) to the right (R) surfaces of the JJ (for detail, see Sect. 8.2.1). The critical current Ic depends on the properties of the tunneling barrier and the symmetry of the order parameters. In the phase-sensitive experiments, the changes of the sign of the critical current Ic is probed avoiding problems of determination of the magnitude of the current, which depends on many experimental parameters. While the sign of Is in a particular JJ is not deﬁned since the phases γL , γR on the both sides of the junction are not ﬁxed, in a closed loop with a Josepson weak link the phase coherence must be maintained resulting in a phase constraint. In

5.5 Superconducting Gap and Pseudogap

343

conventional s-wave superconductors, the phase shift at the junction interphase is proportional to 2π n and this junction is called the conventional, or the zero junction. For the unconventional pairing like d-wave, a frustrated geometry may occur when the phase shift is π(2n + 1) and this junction is called the π-junction. Formally, for the π-junction, the critical current is negative, Is = |Ic | sin(γ + π) = −|Ic | sin γ. At ﬁrst, a π phase shift was proposed for a JJ where a spin-ﬂip scattering by magnetic impurities may occur [187] and was predicted theoretically for a frustrated geometry for a ring of three JJs involving unconventional superconductors as heavy fermions [377, 378] or cuprate superconductors [1170]. The critical current in a JJ can be calculated by minimizing the total free energy of the system including the interface term determining the coupling between the order parameters (see e.g., [1169, 1171]). Depending on the orientation of the crystallographic axes in the L and R d-wave superconductors (and their quenched order parameters) with respect to the interface of the JJ, the Josephson current density Js across the interface is given by the expression [1170]: (5.85) Js = Cs (θL , θR ) sin γ, where the orientations are speciﬁed by the angles θL , θR for the L and R superconductors, respectively. The angle-dependent function Cs in the case of an ideal and clean interface is given by the expression: Cs (θL , θR ) = As cos(2θL ) cos(2θR ). For the case of a strongly disordered junction interface, it still depends on the crystal orientations: Cs (θL , θR ) = As cos 2(θL + θR ). Therefore, the Josephson eﬀect for the d-wave superconductors appears direction sensitive that allows to probe the phase of the order parameter. For instance, let us consider a single superconducting loop with one junction of the zero or π type. Minimization of the free energy for the loop shows that the current Is ﬂowing in the loop depends on the external ﬂux Φex quite diﬀerently for zero and π junctions [1170]. Depending on the dimensionless parameter β = 2π LIc /Φ0 c where L is the self-inductance of the loop and Ic is the critical current, there are two regimes: for small β < 1 the current I is a one-valued periodic function of Φex and for a weak external ﬁeld I reveals the diamagnetic response (I ∝ −Φex ) for the zero junction but the paramagnetic response (I ∝ Φex ) for the π-junction. For large values of the parameter β > 1, the function I(Φex ) is many-valued near Φex = (2m + 1) Φ0 /2 with a stable ground state with Φ = 0 for the zero junction and near Φex = m Φ0 with a doubly degenerate ground state with Φ ≈ ± Φ0 /2 (for β 1) for the π-junction. The phase gradient 2πΦ/Φ0 ≈ ±1/2 in the ground state of the π-junction means a ﬁnite spontaneous current, which generates a ﬂux Φ ≈ ±Φ0 /2. These predictions have been veriﬁed in several experiments. Initially, the π junctions were detected in the paramagnetic Meissner eﬀect (the Wohlleben eﬀect) in weak external magnetic ﬁelds in granular Bi-2212 compounds (for a review, see [1171]). It was explained by the paramagnetic response of a π ring which induces a ﬂux with the same sign as applied ﬂux

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5 Electronic Properties of Cuprate Superconductors

as discussed above. The π junction was clearly detected in phase-sensitive experiments in a YBCO-Pb dc SQUID (for a review, see [1302]). Two SQUID geometry were used: a corner SQUID made between Pb thin ﬁlms and two orthogonality oriented ac- and bc-plane faces at the corners of the single YBCO crystal and an edge SQUID with two junctions on the same ac or bc face of the crystal. The critical current was measured as a function of the external magnetic ﬂux Φex through the SQUID and then extrapolated to zero. As discussed above, the critical current I(Φex ) should have a maximum at Φex = 0 for a zero ring, but a minimum for a π ring. It was found that for the corner SQUID the extrapolated values of the ﬂux varied from 0.3 to 0.6 Φ0 , while those for the edge SQUID were around zero value. Despite a number of complications concerning the twinning of the YBCO crystal, a possible ﬂux-trapping at the corner SQUID, etc. (see [1302]), this was the ﬁrst phase-sensitive experiment which demonstrated the π phase shift of the order parameter measured in the kx (ac face) and ky (bc face) directions in the YBCO crystal, as expected for the d-wave superconductors. Reﬁned subsequent experiments conﬁrmeed these results. In particular, the observation of the critical current modulation of a single YBCO-Au-Pb junction in magnetic ﬁeld demonstrated the sign change of the order parameter [1366]. The critical current in a “short” junction with a uniform Josephson current should show the Fraunhofer pattern: Ic (Φ) = I0 | sin(πΦ/Φ0 )/(πΦ/Φ0 )| as a function of the ﬂux Φ threading the zero-type junction. For a π type junction, the critical current should show another interference pattern Ic (Φ) = I0 | sin2 (πΦ/Φ0 )/(πΦ/Φ0 )|. Therefore, for an edge junction with either s-wave or d-wave symmetry and for a corner junction with s-wave symmetry, the conventional Fraunhofer pattern should be observed with a maximum critical current in zero ﬁeld. Contrary to that, the corner junction built of a d-wave superconductor with an opposite sign of the order parameter in the a and b directions, the critical current should have a minimum at zero applied ﬁeld. Figure 5.57 shows the magnetic

b

100

Critical current (μA)

Critical current (μA)

a

80 60 40 20 0

–200

–100 0 100 200 Applied magnetic field (mG)

200 150 100 50 0

–1000

–500

0

500

1000

Applied magnetic field (mG)

Fig. 5.57. Magnetic ﬁeld dependence of the critical current for (a) an edge and for (b) a corner YBCO-Au-Pb junctions (reprinted with permission by APS from c 1995) [1366],

5.5 Superconducting Gap and Pseudogap

345

0)

a

(10

(01

0)

(01

0) (10 30°

0)

60°

b

0.5

Pickup loop flux (φ/φ0)

ﬁeld dependence of the critical current for an edge (a) and for a corner (b) YBCO-Au-Pb junctions (the inset shows the geometry of the junctions). The interference patterns agree with the expected dependence: at zero applied ﬁeld the maximum current is observed for the edge junction, while the minimum is seen for the corner junction. A ﬁnite critical current at zero magnetic ﬁeld in the corner junction may be assigned to asymmetries in the current densities in the junctions but not to the asymmetry of the d-wave order parameter (“s + d mixing”) in the orthorhombic YBCO crystal (see [1302]). In a series of experiments, a spontaneous generation of a half-ﬂux quantum was observed in the frustrated geometry in the multicrystal ﬁlms of cuprate superconductors (for a review, see [611,1276]). The ﬁrst experiment was done by Tsuei et al. [1274] on a YBCO tricrystal sample. Tricrystal (100) SrTiO3 substrates with controlled orientation of crystalline axes were fabricated, which were used for deposition of c-axis-oriented cuprate ﬁlms. The tricrystal geometry is shown in Fig. 5.58a with polar plots of assumed dx2 −y2 order parameters aligned along the crystalline axes in (a, b) plane. Several cuprate rings were patterned on the surface of the tricrystal: the three-junction ring located at the tricrystal meeting point, and control two-junction rings and a ring with no junction. In this geometry according to (5.85), the three-junction ring is a π-ring due to the frustrated geometry which should reveal a half-ﬂux quantum in zero applied magnetic ﬁeld. The two-junction rings and the ring without junction should behave as a zero-ring. By using a specially fabricated scanning SQUID microscope, magnetic ﬂux threading of the rings was measured. For a sample cooled in zero magnetic ﬁeld, no ﬂux was detected in the

0.4

B = 3.7mG

60°

0.3 0.2 0.1

(010)

B =0 0.0 –320 (100)

–160

0

160

320

Position(mm)

Fig. 5.58. (a) Tricrystal geometry with polar plots of dx2 −y 2 order parameters aligned along the crystalline axes in (a, b) plane. (b) Flux measured by the scanning SQUID microscope through the tricrystal point on the Bi-2212 epitaxial ﬁlm in zero ﬁeld and applied external ﬁeld B = 3.7 mG (reprinted with permission by APS from c 2000) [1276],

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5 Electronic Properties of Cuprate Superconductors

control rings. But the central three-junction ring revealed a ﬂux of the magnitude of which was estimated by diﬀerent methods as Φ = 0.57 ± 0.1Φ0. This proves the spontaneous generation of a half-ﬂux quantum, which is consistent with the dx2 −y 2 symmetry of the order parameter in YBCO. In subsequent experiments, a half-ﬂux quantum was observed in other cuprates as well: in tetracrystal Tl-2201 ﬁlms [1275], in tricrystal Bi-2212 ﬁlms [612]. Figure 5.58b shows the ﬂux measured by the SQUID microscope along the line crossing the tricrystal point (at zero) on the c-axis-oriented Bi-2212 ﬁlm deposited on a tricrystal SrTiO3 substrate with the geometry shown in Fig. 5.58a. In zero magnetic ﬁeld, only a vortex at the tricrystal point was observed (lower curve), while in a ﬁeld of 3.7 mG a series of vortices were found in the grains and along the grain boundaries with a weak vortex at the tricrystal point (upper curve). The ﬂux-imaging data processing has shown that the vortex at the tricrystal point has a total ﬂux of Φ0 /2, while the ﬂux value of Φ0 can be ascribed to all other single vortices. The data analysis of the tricrystal experiments has shown that the main contribution is due to the d-wave symmetry, and the s term in the orthorhombic crystals is rather weak. Studies of the temperature dependence of the spontaneous half-ﬂux quantum eﬀect in a YBCO ﬁlm have shown its persistence at the tricrystal meeting point from low temperatures to Tc , which suggests a robust d-wave pairing symmetry in cuprates [613]. The angle-resolved, phase-sensitive technique was used to determine the in-plane anisotropy of the gap in the optimally doped YBCO crystal [610]. It has been found that the gap along the b-axis direction is approximately 20% larger than that along the a-axis, while the imaginary components idxy or ip, if exist, must be much smaller, less than a few percent, in comparison with d-wave component. The spontaneous generation of a half-integer Josepson vortex at the tricrystal point was detected in the electron-doped NCCO and PCCO cuprates also [1277]. Thus, the d-wave pairing symmetry of the order parameter is a common feature of the cuprate superconductors. It should also be noted several c-axis tunneling experiments where a significant s-wave component in the order parameter was found. Early experiments on c-axis tunneling between the s-wave Pb superconductor and the YBCO (a, b) plane revealed a conventional Fraunhofer interference pattern [1199]. Later experiments have shown the important role of the twinning in the YBCO crystals for the c-axis tunneling. In particular, in the c-axis Josepson tunneling experiment with a magnetic ﬁeld applied in the plane of the junction, a strong dependence of the Fraunhofer-like pattern on the angle of the magnetic ﬁeld was observed [655]. The results were explained by considering the mixed s + d symmetry of the order parameter where the s-component changes its sign across the twin boundary in the YBCO crystal. The pairing symmetry was also tested by using the junctions with varying misorientation of the (a, b) plains between two identical cuprate crystals. For a c-axis tunneling conserving, the in-plane momentum of the order parameter, for two

5.5 Superconducting Gap and Pseudogap

347

d-wave superconductors the critical current should depend on the misorientation angle ϕ as Ic ∝ cos 2ϕ. However, in experiments with planar Bi-2212 Josephson junctions, no angle dependence was found (for a review, see [622]). A much stronger angle dependence was observed in the Bi-2212 cross-whisker junctions [1214]. To explain these experiments, several problems should be resolved concerning the incoherent contribution to c-axis tunneling processes (which wash out the angle dependence), an angle dependence of the tunneling matrix element as well as the junction quality, self-ﬁeld eﬀects, etc. A supplementary information concerning the symmetry and the magnitude of the superconducting gap in cuprates provides the Andreev or Andreev– Saint–James (ASJ) reﬂections [275]. The ASJ reﬂections occurs when an electron from a normal metal (N) with an energy less than the superconducting gap tunnels into a superconductor (S) as a Cooper pair via reﬂection of a hole of the opposite wave vector. This results in an enhancement of the conductance of the N–S contact which can be measured (ASJ spectroscopy). The coherent processes involved in the reﬂections imply the electron-hole mixing in the Bogoliubov quasiparticles. Observation of ASJ reﬂections in all cuprates proves the occurrence of a Cooper pairing mechanism similar to that in conventional metals. The energy dependence of the ASJ reﬂections shows that the energy of the coherent excitations determined by the superconducting gap scales with the superconducting Tc contrary to the energy dependence of the single-particle tunneling related to the pseudogap in the normal state [277]. This proves a diﬀerent origin of superconducting pairing in comparison with the pseudo gap formation as discussed in previous sections. The occurrence of low-energy, surface-bound states in cuprates, revealed by ASJ spectroscopy, has conﬁrmed the d-wave pairing symmetry since this eﬀect is caused by the sign change of the pair potential in the d-wave superconductors. Resume 1. Observation of the π-junction in the SQUID measurements and the halfﬂux quantum in tricristal geometry experiments unambiguously conﬁrms the robust d-wave pairing in the cuprate superconductors with a small admixture of the s-wave component in the orthorhombic crystals. 2. Occurrence of Andreev (Andreev–Saint–James) reﬂections in cuprates proves the Cooper pairing mechanism. An energy dependence of the reﬂections reveals a diﬀerent origin of the superconducting gap and the pseudogap.

6 Lattice Dynamics and Electron–Phonon Interaction

Studies of phonon spectra in conventional superconductors were essential in developing the microscopic theory based on the electron–phonon pairing mechanism. Phonon dispersion anomalies and peculiar temperature dependence of phonon frequencies at the superconducting phase transition have conﬁrmed the involvement of phonons in the cooper pairing. In the theory, to calculate the superconducting Tc , both the phonon density of states (PDOS) F (ω) and the electron–phonon coupling function α2 (ω) should be speciﬁed (for a review see [48, 202]). Whereas the phonon PDOS can be determined directly by inelastic neutron scattering experiments, the estimation of the electron–phonon coupling α2 (ω) is a much more diﬃcult problem. As discussed in Sect. 5.2.2, ARPES studies have revealed a substantial electron spectrum renormalization in cuprates caused by electron (hole) interaction with bosonic modes (a “kink” phenomenon) (for a review see [258]). However, it is diﬃcult to separate the phonon and spin-ﬂuctuation contributions in the electron spectrum renormalization. Therefore, studies of the phonon spectra of cuprate superconductors, their doping and temperature dependence, may help to elucidate the phonon role in the superconducting pairing. The electron–phonon interaction results in the renormalization of the bare ionic frequencies of the lattice vibrations Ωqj according to the following relation (see, e.g., [47]): 2 2 = Ωqj + 2Ωqj Πqj (ω = ωqj ), ωqj

(6.1)

where q, j is the phonon wave vector and polarization index, respectively. Here, ωqj is the renormalized phonon frequency and Πqj (ω) is the polarization operator. To the second order in electron–phonon interaction in the normal phase, it may be written as fm (p) − fn (p + q) Πqj (ω) = |gj (p, m; p + q, n)|2 , (6.2) εm (p) − εn (p + q) + ω p,m,n where gj (p, m; p + q, n) is the matrix element of the electron–phonon interaction, fm (p) the Fermi distribution function for electrons of energy εm (p),

350

6 Lattice Dynamics and Electron–Phonon Interaction

p the wave vector and m is the band index. The real part of the polarization operator (6.2) determines a softening of the ionic phonon frequencies due to the electron–phonon interaction: Δωqj = ωqj − Ωqj = ReΠqj (ωqj ) < 0, and its imaginary part determines the phonon linewidth γqj = −ImΠqj (ωqj + i0+ ) > 0. The latter function is directly related to the Eliashberg function α2 (ω)F (ω) =

γqj 1 δ(ω − ωqj ), h N (0) q,j ωqj

(6.3)

where N (0) is the density of electronic states (per atom and per spin direction) on the Fermi surface. This relation allows to determine the coupling constant λ in the BCS theory ∞ λ=2 0

γqj α2 (ω)F (ω) 2 dω = 2 , ω h N (0) q,j ωqj

(6.4)

if the phonon linewidth γqj due to electron–phonon interaction is known. Thus, the magnitude of the electron–phonon interaction in copper-oxide superconductors can be estimated by investigations of the phonon dispersion and their linewidth. In Sect. 6.1, the results of inelastic neutron scattering experiments are reviewed. Optical studies relevant to observation of electron–phonon interaction and, in particular, of phonon spectrum changes at the superconducting transition, are considered in Sect. 6.2. The isotope eﬀects on the superconducting transition temperature and the magnetic penetration depth are discussed in Sect. 6.3. Theoretical models of lattice dynamics and electron–phonon interaction are presented in Sect. 6.4.

6.1 Neutron Scattering Studies Inelastic neutron scattering (INS) technique has a great advantage over optic measurements since by this method the phonon dispersion relations can be determined in the whole Brillouin zone (BZ) for a broad range of energies, 4–100 meV1 . However, for INS studies large single crystals which have restricted INS investigations mostly to the La2−x Srx CuO4 (LSCO), Nd2−x Cex CuO4 (NCCO), and YBa2 Cu3 O6+x (YBCO) cuprate superconductors are required. Results of these investigations were reviewed by Pintschovius et al.[990–992] with a particular emphasis on the electron–phonon coupling in the review article [994]. Local lattice eﬀects induced by electron–phonon interaction can be studied by measuring the pair distribution function by pulsed neutron scattering technique. These experiments were discussed by Egami et al. [303]. 1

Conversion factors for energies: 1 meV = 8.07 cm−1 = 0.2418 THz.

6.1 Neutron Scattering Studies

351

6.1.1 Doping Dependence of Phonon Spectra In early experiments on polycrystalline samples, only the generalized PDOS G(ω), equal to the sum of the partial contributions weighted with the neutron scattering cross sections of separate atoms could be measured [991, 992]. This allows one to determine a general shape of the PDOS F (ω) in the Eliashberg function (6.3) and to consider speciﬁc models of the electron–phonon interaction. Measurements on polycrystalline samples of LSCO have shown only a moderate change of G(ω) with doping, while in YBCO a strong deformation of phonon spectra was observed with changing the oxygen content from x = 0 in the undoped sample to x = 1 in the fully doped sample. The observed softening of the low-frequency part of the phonon spectra at 15– 20 meV in YBCO with ﬁlling the O1 positions along the chains was explained by structural eﬀects: increasing of the number of bonds for the Cu1–O1 and O1–O4 atoms. However, the softening of the phonon spectra in the range of 40–60 and 80 meV, caused by the in-plane vibrations of the O2, O3, and Cu2 atoms suggested a strong renormalization of the oxygen bond-bending and bond-stretching modes due to a strong electron–phonon coupling. A similar deformation of phonon spectra was observed at substitution Cu by Zn or Y by Pr. Since in this case the structural changes are insigniﬁcant, the change of phonon spectra should be caused by the electron–phonon interaction. Later, after synthesizing large single crystals suﬃciently, the phonon dispersion curves were measured throughout the BZ for the main crystallographic directions for the LSCO, NCCO, and YBCO crystals. The phonon dispersion curves of single crystals of LSCO have been studied in great detail (see [990– 992]). In particular, in one of the ﬁrst experiments the tilt soft mode of the symmetry Σ4 at the structural phase transition from tetragonal to orthorhombic phase was found [155]. Measuring the scattering intensity conﬁrmed the soft mode model as a rigid rotation of the CuO6 octahedra. In general, both the temperature dependence of the soft mode frequency above and below To and the critical scattering near the transition temperature To looked typical of the structural phase transitions in perovskites with a soft rotation mode. Further studies of the soft mode conﬁrmed the early results [991]. A strong interaction of the soft mode with the acoustic vibrations was found. As we noted in Sect. 2.2.2, the latter are due to tilting of the CuO6 octahedron in a strongly anharmonic two-well potential (see Fig. 2.8). The data of acoustic measurements [828], in particular, which found an essential softening of the elastic coeﬃcient C66 (by 30–40%) at T < To have conﬁrmed the strong coupling of the acoustic modes with the anharmonic soft mode. However, the displacement pattern of the tilt mode suggests a moderate electron–phonon coupling only. Comparative studies of phonon dispersion curves in dielectric and metallic samples of LSCO, NCCO, and YBCO have shed some light on the role of the electron–phonon interaction in these crystals. In the undoped insulating state, the phonon dispersions look typical of ionic crystals. In particular, an

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6 Lattice Dynamics and Electron–Phonon Interaction

essential splitting of the longitudinal optical (LO) and transverse optical (TO) modes in the zone center was observed, which is caused by long-range Coulomb forces. Upon the transition to metallic state this splitting disappears due to the screening of the Coulomb forces in the CuO2 planes by free charge carriers. On the other hand, vibrations of ions perpendicular to the CuO2 planes preserve their ionic nature the in view of the small screening of these displacements. In most cases, quite satisfactory theoretical description of phonon modes was obtained within the framework of the conventional shell model with a shortrange repulsive potential and an anisotropic screening in the metallic phase (see, e.g., [222, 993]). However, the behavior of the high-frequency Cu–O longitudinal bondstretching (“breathing” type) modes both in LSCO (ν 20 THz) and in YBCO (ν 18 THz) in the transition from the insulating to the metallic phase was unusual. A symmetric stretching of the four in-plane Cu–O bonds occurs for the full breathing mode propagating in the (1, 1, 0)-direction while only two bonds are stretched for the “half-breathing” mode propagating in the (1, 0, 0) or (0, 1, 0)-directions. According to the conventional nonorthogonal tight binding theory of lattice dynamics [1342], the electron–phonon interaction should strongly renormalize the frequency of the full breathing mode at the zone boundary caused by the Fermi surface nesting in LSCO. Contrary to this prediction, only a nonessential renormalization of this mode was found in experiments, while the “half-breathing” mode reveled a strong doping-induced softening at the zone boundary near the wave vector q = (0.5, 0, 0) = (π/a, 0, 0) as shown in Fig. 6.1a. Similar doping-induced

Fig. 6.1. Doping-induced frequency changes of the longitudinal plane polarized bond-stretching mode in the (100)-direction (in units of 2π/a) in: (a) LSCO for the Cu–O bonds and (b) in NCCO for the Cu–O bonds (filled symbols) and the Nd–O bonds (open symbols) (reprinted with permission by Wiley-VCH from Pintschovius c 2005) [994],

6.1 Neutron Scattering Studies

353

softening of this mode was observed in the electron-doped cuprate NCCO having T structure (see Sect. 2.3). Figure 6.1b shows the diﬀerence of the Cu–O and Nd–O bond-stretching mode frequencies between the undoped and optimally doped NCCO. The anticrossing of these modes shifts the maximum of the softening from the zone boundary at q = (0.5, 0, 0) to q ∼ (0.25, 0, 0). At the zone center, q = (0, 0, 0), both modes change appreciably reﬂecting the insulator–metal transition which suppresses the LO–TO splitting. But a substantial renormalization at q ∼ (0.25, 0, 0) of the Cu–O bond contrary to the Nd–O bond-stretching mode points to a strong electron–phonon coupling for the Cu–O mode. Such a diﬀerence in the behavior of these modes may be explained by the peculiarities of the electronic structure of the copper-oxide compounds (see Sect. 5.1.1). Strong Coulomb correlations on copper sites hinder the charge transfer from oxygen to copper. In the full “breathing” mode the charge transfer should occur between the Cu3d and O2p bands since electrons on all four oxygen ions retain the same energy in the crystal ﬁeld. In this case, the polarization operator (6.2) yields a small contribution to the phonon renormalization because of the large denominator – the charge transfer gap between the Cu3d and O2p bands. At a shift of only two oxygen ions, the charge transfer occurs between the two nonequivalent oxygen positions, and the contribution of (6.2) may be large since the denominator is now small – there is no gap in the oxygen-like Zhang–Rice (ZR) singlet band. A similar diﬀerence in electron–phonon interaction is observed for the tilt soft modes at a rotation of the CuO6 octahedrons in LSCO. A variation of the electronic spectrum for the rotation around the [100] axis (or [010]) is larger than for the rotation around the [110] axis since the latter retains the equal electron crystal ﬁeld energies at the oxygen positions in the [100] and [010] directions (see Sect. 2.2). Whereas the doping-induced phonon frequency changes reﬂect just the screening eﬀects at the transition from the insulating to the metallic state, the doping dependence of the linewidth γqj is directly related to the electron– phonon coupling in (6.3). Figure 6.2a shows the doping dependence of the phonon linewidth for the Cu–O bond-stretching LO modes in LSCO. In the underdoped compound (x = 0) the phonon linewidth is quite small (of the order of the experimental resolution shown by the line). The linewidth increases with the doping (full symbols for x = 0.1, 0.15) but in the overdoped sample (open symbols for x = 0.30) it decreases. Similar large phonon linewidth was found for the Cu–O bond-stretching LO mode in the optimally doped NCCO shown in Fig. 6.2b. At the same time, no essential softening or the large linewidths were found for the transverse Cu–O bond stretching mode. An anomalously large softening was found for the symmetrical apexoxygen-breathing mode, the so-called “Ozz mode”, related to apex oxygen displacements in the (0, 0, 1) (z)-direction at the Z-point at the zone boundary [994]. Under the transition from the insulating to the optimally doped state, the Ozz mode frequency changes from ν = 17 to ν = 11.5 THz and acquires a

354

6 Lattice Dynamics and Electron–Phonon Interaction

Fig. 6.2. Phonon linewidth for the Cu–O bond-stretching modes in (a) LSCO at various doping and (b) in NCCO at optimal doping (reprinted with permission by c 2005) Wiley-VCH from Pintschovius [994],

large linewidth of the order of 3 THz which suggests a strong electron–phonon interaction (electron–plasmon coupling, see Sect. 6.4). The studies of the Cu–O bond-stretching modes in YBCO are complicated by the interaction of the phonon modes having the same symmetry but diﬀerent polarizations [993, 994]. In general, the doping dependence of the LO bond-stretching modes are similar to those observed in LSCO. A sizeable softening is observed both in the (1, 0, 0) and (0, 1, 0)-directions and a less pronounced one in the (1, 1, 0)-direction. In the orthorhombic phase, the Cu–O bond-stretching mode propagating along the chains, the (0, 1, 0)-direction, has a lower frequency in comparison with the mode in (1, 0, 0)-direction. Studies of the lattice dynamics of the Hg-1201 by inelastic X-ray scattering have also shown a strong softening of the bond-stretching modes towards the zone boundary [1285]. Therefore, the anomalous behavior of the bondstretching modes with doping seems to be a universal eﬀect in the cuprate superconductors which points to a sizeable electron–phonon coupling. It was suggested that a strong softening and a large linewidth near the optimal doping may be explained by the phonon coupling to the dynamical charge ﬂuctuations (dynamical stripes) [994]. In particular, apart from the softening,

6.1 Neutron Scattering Studies

355

the phonon dispersion of the LO Cu–O bond-stretching modes in LSCO (around x = 1/8) and underdoped YBCO (for x = 0.6) show a very steep slope around q ∼ (0.25, 0, 0) leading to a strong deviation from the conventional sinusoidal shape of the dispersion [994]. To ﬁnd out whether these anomalies are induced by a coupling to charge ﬂuctuations, the Cu–O bond-stretching modes have been studied by INS in single crystals of La1.48 Nd0.4 Sr0.12 CuO4 (LNSCO) and La1.875 Ba0.125 CuO4 (LBCO) [1055]. In these crystals a static charge density wave (CDW) characterized by the wave vector qco = (0.25, 0, 0) has been found at low temperatures (in LBCO, at T < 60 K – see Figs. 2.5 and 3.8). The strong anomaly in the INS scattering intensity for the bondstretching phonon branch found in LBCO near qco was ﬁtted by a two-peak structure. This suggests a splitting of the phonon branch into a normal one with a monotonic cosine-like downward dispersion and another one with highly anomalous dispersion. The dispersion of the anomalous branch drops down by ∼10 meV around qco and then rises back to merge with the normal dispersion. The linewidth of the anomalous branch was also extremely large. The anomaly was large at low temperature (∼10 K) but decreased at higher temperature (∼200 K). At the same time the dispersion along the [110] direction did not show strong dispersion anomaly. Such a behavior points to a coupling of the anomalous phonon branch to the CDW at low temperatures – stripes along the Cu–O bonds. A similar anomalous dispersion was observed in the LNSCO sample. The doping dependence of the anomalous dispersion was studied by INS measurements on the LSCO crystals at diﬀerent Sr concentrations: x = 0.07, 0.15, and 0.3. The anomalous behavior was found for superconducting samples at x = 0.07 and 0.15 while the nonsuperconducting sample (x = 0.3) showed a normal cosine-like dispersion. Later on, highquality INS studies on the overdoped LSCO crystal (x = 0.3) have conﬁrmed these results [995]. In particular, the doping-induced frequency renormalization of the bond-stretching modes, both in the (1, 0, 0) or (0, 1, 0)-directions, increases with overdoping, while the linewidth decreases. Moreover, the large linewidth observed close to qco in the optimally doped LSCO was not found in the overdoped sample. This indicates a weakening of the electron–phonon coupling in the latter case, which may be due to the suppression of the charge ﬂuctuations which are strong in the x = 1/8 anomaly region (see Sect. 2.2.1). The similar strong coupling of the B2u and B3u planar oxygen modes in (0, 1) direction with the charge ﬂuctuations was found in YBa2 Cu3 O7−x by Mook et al. [855]. The large line widths for phonons with the energy around 45 meV were observed in the sample with the oxygen content 6.35 at the wave vector q0 = 0.1 (r.l.u.) and in the sample with the oxygen content 6.6 at the wave vector q0 = 0.2 (r.l.u.). The authors suggested the dynamical charge stripe formation with the wave vectors q0 that are twice of the wave vectors found for the incommensurate spin ﬂuctuations.

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6 Lattice Dynamics and Electron–Phonon Interaction

6.1.2 Phonon Renormalization in Superconducting State It is expected that in the superconducting state the linewidths of the phonons with the energies less than the energy gap 2Δ should decrease because the energy gap opening in the electronic spectrum reduces the number of available decay channels. Contrary to that, phonons with energies greater than 2Δ can decay via pair-breaking mechanism and therefore their linewidth should increase in the superconducting state. The phonon linewidth and the corresponding phonon frequency shift are determined by the imaginary and real parts of the polarization operator which in the superconducting state has a more complicated form than (6.2) (for the d-wave gap see, e.g., Nicol et al. [895], Jiang et al. [554]). The superconductivity-induced phonon renormalization was detected by INS in conventional superconductors. In the cuprate superconductors, the anomalous phonon behavior below Tc revealed a strong doping dependence and was observed only for particular modes. These eﬀects were ﬁrst found in Raman scattering (RS) experiments (see next Sect. 6.2) and later conﬁrmed in INS studies. Whereas the RS experiments probe the phonon modes at the center of the BZ only, the INS studies enable the measurement of the q-dependence of the phonon excitations overall inside the BZ. The superconducting pairing, and in particular, the d-wave pairing, depends on the electron–boson interaction throughout the whole BZ and therefore studies of the q-dependence of the polarization operator are essential for elucidating the mechanism of the pairing. Strong superconductivity-induced eﬀects were found in YBCO for the bond-bending type mode with frequency 340 cm−1 (42 meV) for the ﬁrst time in RS studies (see, e.g., [659] and references therein). In this mode, the c-polarized motion of the in-plane oxygen O2 and O3 atoms occurs outof-phase at q = 0, both in the single CuO2 plane and in the two nearest neighbor CuO2 planes as shown in Fig. 6.3 (left panel). The symmetry of the 7 mode is B1g in the tetragonal (D4h ) phase and Ag in the orthorhombic phase 1 (D2h ) phase (see Sect. 2.4.1). The antiphase motion of the nearest neighbor O2 and O3 oxygen ions may result in a strong interaction of the electrons with the phonons, having as consequence the induction of a charge transfer in the oxygen-like ZR singlet band as discussed earlier for the bond-stretching modes in LSCO. Figure 6.3 (right panel) shows the phonon shifts at the superconducting phase transition in YBa2 Cu3 O6+x . The shifts in the fully oxygenated sample at x = 1 (ﬁlled dots) of the order of 0.2% is larger than at x = 0.92 (open dots). The INS measurement conﬁrmed the RS results at the zone center (shown by squares) and revealed a sizeable variation of the shifts with the wave vector. In particular, the phonon softening at the zone boundary vanishes suggesting a decrease of the electron–phonon coupling. In later studies both the frequency shifts and changes of the linewidth were measured with better resolution below Tc throughout the BZ as shown in Fig. 6.4. The much smaller superconductivity-induced phonon shifts in the [1, 1, 0] direction in comparison with the [1, 0, 0] one may be due to diﬀerent reasons: coherent

6.1 Neutron Scattering Studies

357

Δv (THZ)

0

–0.1

RA 06.92 07.0

–0.2

0.5 0.3 0 (0, 0, )

0.1 0.2 0.3 0.4 ( , 0, 0) (0, , 0)

0.5

peak position (meV)

a

43 42.8 42.6 42.4 42.2 42 41.8 0.6

100K 50K 10K 0.4

0.2 [1,1,0]

b

3.2

width (meV)

Fig. 6.3. Displacement pattern of the in-plane oxygen atoms in the B1g mode at 340 cm−1 in YBa2 Cu3 Ox (left) and the phonon frequency shift Δν = ν(50 K)−ν(100 K) for this mode at the superconducting transition (right) at x = 6.92 and x = 7. The Raman data (RA) from Krantz et al. [659] are shown by squares (after Pyka et al. [1036])

2.8

100 K 50 K

2.4 2

0 q

0.2 0.4 [1,0,0]

0.6

1.6 –0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 q [1,0,0]

Fig. 6.4. Phonon energy (a) and linewidth (b) of the bond-bending B1g -type mode in the BZ at diﬀerent temperatures in YBa2 Cu3 O7 (reprinted with permission by c 1995) APS from Reznik et al. [1052],

eﬀects caused by the d-wave gap symmetry, the Fermi surface geometry, or the q-dependence of the electron–phonon interaction [1052]. A superconductivity-induced phonon softening by ∼0.2 meV was also found in YBa2 Cu3 Oy for the B1u mode with frequency 193 cm−1 (24 meV) by Harashina et al. [427]. The displacement pattern in the B1u mode is similar to B1g mode in one CuO2 plane, but the oxygen O2 and O3 atoms in the

358

6 Lattice Dynamics and Electron–Phonon Interaction

bilayer move in-phase. Contrary to the B1g phonon mode with the energy of 42 meV, the B1u mode with lower energy reveals a decrease of the linewidth below Tc . The renormalization eﬀects decreased with underdoping: they were larger for y = 7 in comparison with the underdoped sample at y = 6.7. An anomalous behavior of the B1u mode was observed below T = 200 K also which was ascribed to spin-gap opening. No appreciable anomaly was found for the 18 meV mode in which the in-phase motions of the O2 and O3 atoms within the CuO2 -plane do not induce a charge transfer between them. Resume The following results of investigations of phonon spectra by INS should be singled out. 1. Phonon dispersion measured by INS shows that ionic and covalent bonding of atoms gives a major contribution to the lattice force constants. At the transition to a metallic state with doping, electron–phonon interaction is screened in the CuO2 planes but the ionic character of the vibrations perpendicular to the planes survives. 2. The INS studies revealed a notable q-dependence of the phonon polarization operator. Generally, the bond-stretching type phonon modes associated with the in-plane asymmetric oxygen displacements and the bondbending out-of-plane and out-of-phase (B1g -type) modes have shown a strong doping dependence of their frequency and linewidth which may be related to a strong electron–phonon interaction at optimal doping. 3. Appreciable superconductivity-induced phonon anomalies were observed for several phonon modes at frequencies comparable with the energy gap 2Δ which points to an electron–phonon coupling of superconducting quasiparticles.

6.2 Optical Investigations Unlike neutron scattering, optical spectroscopy allows one to observe the phonon modes only in the center of the BZ, at q = 0. However, the existence of symmetry selection rules for the light scattering and an increase of the precision of measurements by an order of magnitude together with the possibility of carrying out experiments on samples of small size, render the optic methods unique for rigorous quantitative investigations. The results of the phonon optic studies on high-Tc superconductors have been summarized and discussed in several review papers. A general symmetry analysis of optically active phonons and early Raman results are reviewed by Feile [326], Thomsen [1241], and Evarestov et al. [320]. The IR studies are discussed by Litvinchuk et al. [734]. The theory of RS and, in particular, the anomalous

6.2 Optical Investigations

359

increase of the Raman intensity in the superconducting state are considered by Sherman et al. [1151]. In this section, we consider several optic experiments which have revealed the important role of the electron–phonon interaction in copper-oxide compounds. These concern observation of the Fano eﬀect, superconductivity-induced changes of phonon frequencies and linewidths and polaronic eﬀects in light scattering. Certain indications regarding the possibility of strong electron–phonon interaction for particular optical modes have been obtained in observations of the Fano eﬀect (see, e.g., [1241]). The Fano eﬀect results from the interference of the scattering from discrete and continuous excitations and shows itself in an asymmetry of the lines for a discrete scattering spectrum. In this case, the scattering intensity as a function of frequency ω is proportional to (Fano proﬁle) (q + )2 . (6.5) f (ω) = πρ() Tel2 1 + 2 In the approximation of a constant density of states in the continuous spectrum ρ() = ρ, the dimensionless frequency = (ω − ωp )/Γ and the lineshape parameter q = Tph /πρV Tel deﬁne the proﬁle of the line. Here Tel and Tph are the scattering matrix elements for the continuous (electronic) spectrum and the discrete excitation (phonon of frequency ωp ), V is their coherent interaction, Γ is the total linewidth. In the noninteracting case, V → 0, we have q → ∞ and the line shape becomes Lorentzian. At q = 1 the line shape is the most asymmetric. At the frequency ωmin = ωp − Γ q an antiresonance is observed when f (ωmin ) = 0. The Fano eﬀect is well pronounced for the B1g mode (illustrated in Fig. 6.3, left) at the frequency ν 340 cm−1 in YBCO, where it was discovered. Figure 6.5 compares the RS intensity for (a) superconducting at y = 0, and (b) semiconducting at y = 1, samples of YBa2 Cu3 O7−y . In the latter case, the form of the scattering line is symmetric, which corresponds to q → ∞ in (6.5). For a ﬁnite coupling constant V Tel = 0 it is natural to assume that, in dielectric samples, the density of the electronic states shows a continuous spectrum with ρ = 0, while in metallic samples ρ = 0, and the Fano eﬀect appears at a ﬁnite value q −4. It is interesting to note the antiresonance behavior of the Ag -mode of the Ba vibrations at the frequency ν = 122 cm−1 , which indicates a strong interaction of these ionic vibrations with the in-plane charge carriers. The Fano eﬀect remains present for the Ag -mode when the temperature falls below Tc , which indicates a ﬁnite density of the quasiparticle excitations in this energy range, below the superconducting gap energy 2Δ, as it should be for the d-wave pairing. Quantitative estimates of the electron–phonon interaction constant were obtained in studies of the superconductivity-induced optical phonon renormalization. These eﬀects were investigated in a series of RS experiments on YBCO-type compounds. The electron–phonon coupling constant was estimated in these measurements by comparison of the phonon frequency and linewidth changes in the superconducting state with results of the theory

360

6 Lattice Dynamics and Electron–Phonon Interaction

a

b 344

Fano Intensity (arb. units)

337

Fano 107

112

100

200

300

400

100

200

300

400

–1 Frequency (cm )

Fig. 6.5. Asymmetric (Fano eﬀect) (a) and symmetric (b) Raman spectra for (a) superconducting, and (b) semiconducting samples of YBCO (after Thomsen [1241])

developed by Zeyher and Zwicknagel [1414,1415] (ZZ theory). The calculation of the polarization operator at the center of the BZ was performed within the strong-coupling Eliashberg theory for the s-wave pairing with a gap Δ0 and for a simpliﬁed model of the Fermi surface. The ZZ theory predicts that the renormalization due to electron–phonon interaction for phonons of frequency h ¯ ω > 2Δ0 should increase their frequency (leads to hardening), and for phonons of frequency ¯hω < 2Δ0 it should reduce the frequency (leads to softening). An increase in the density of states under the superconducting transition in the energy range h ¯ ω ≥ 2Δ0 also leads to an increase in the relaxation rate and linewidth γν . A generalization of the theory for the polarization operator to q = 0-phonons was given by Zeyher [1416]. The temperature dependence of the phonon frequency and linewidth were studied for modes of A1g (ω 440 cm−1 ) symmetry and B1g (ω 340 cm−1 ) symmetry, in which c-polarized vibrations of the O2 and O3 ions occur inphase and out-of-phase, respectively. Varying the composition of the RBCO compounds (R = Eu, Dy, Er, Tm, Y) and doing the isotopic exchange 16 O→18 O it was possible to vary the frequencies of B1g and A1g modes in a certain range, keeping the superconducting transition temperature Tc and the gap Δ intact. An external magnetic ﬁeld H, which reduces Tc simultaneously reduces the phonon softening temperature Ts , dTc /dH ∝ dTs /dH, which conﬁrms the relation of the phonon anomalies to the superconducting transition. A notable softening of the B1g -mode and hardening of the A1g -mode at T < Tc in fully oxygenated RBCO compounds with Tc 90 K were

6.2 Optical Investigations

361

observed by Thomsen et al. [1240]. This led them to the conclusion that the superconducting gap magnitude should be between the two energies of these phonon excitations. Subsequent studies of the phonon linewidth shifts by Friedl et al. [353] conﬁrmed this conclusion. The relative phonon frequency and linewidth changes, Δ ων /ων and Δ 2γν /ων , for a phonon mode ν were compared with the results of the ZZ theory for the polarization operator Δ Πν = Δ ων − iγν : Δ Πν /ων = λν F (ων /2Δ0 , T /Tc).

(6.6)

Here F (˜ ω , T˜) is a universal function of dimensionless variables, | F |∼ 1, and λν is a dimensionless coupling constant λν = 2N (0) |gν,k,k |2 FS /ων ,

(6.7)

where N (0) is the density of state per spin, · · · FS denotes averaging of the matrix element gν,k,k of the electron–phonon interaction (in one band model) over the Fermi surface. A good agreement of the theoretical results (6.6) with the experimental data was obtained for the coupling constants λν (B1g ) = 0.02 and λν (A1g ) = 0.01 at 2Δ0 316 cm−1 , and 2Δ0 /kB Tc 5 [353]. It is remarkable that the coupling constant λν found in the experiments coincided with the results of the calculations in the local-density approximation in the frozen-phonon method [1056]. However, the eﬀective coupling constant (6.4) estimated as a sum over all the phonon modes of an averaged, 3r λeﬀ = ν=1 λν ∼ 0.8 (r is the number of atoms in the unit cell in YBCO: 3r = 39), appears to be too small to obtain a high superconducting temperature. A much larger coupling constant λ ∼ 2.9 in the ZZ theory is required to reach Tc = 91 K. Similar results were obtained for other vibration modes: the A1g -mode for apical oxygen O4 (ω 500 cm−1 ) which is related to the stretching of the Cu–O bonds along the c-axis of the crystal, and the B2g , B3g modes related to the bending of these bonds [354]. In this case, the electron–phonon interaction coupling constants appear to be small as well: λν (A1g ) 0.01, λν (B3g ) < 0.003, λν (B2g ) < 0.02. Further studies of the superconductivity-induced phonon renormalization of the Raman modes in YBCO have shown controversial results. In particular, it was found that the linewidth of the 340 cm−1 mode exhibited a very diﬀerent behavior in comparison with previous results (see [54] and references therein). A systematic study of temperature dependence of the Raman phonons at 340, 440, and 500 cm−1 in YBCO at diﬀerent oxygen concentrations by Altendorf et al. [54] revealed that the behavior of the RS modes strongly depends on the impurity concentration (e.g., a disorder in the YBCO chains) and the hole concentration as reported previously by Krantz et al. [659]. Figure 6.6 shows the superconductivity-induced frequency shift (upper panel) and linewidth shift (bottom panel) between 15 and 100 K for the 340 cm−1 phonon mode as a function of the oxygen concentration y. The phonon linewidth and shift of the 340 cm−1 mode depend

362

6 Lattice Dynamics and Electron–Phonon Interaction

a

8

Δωv (cm–1)

340 4 0 –4 –8

Δ2γv (cm–1)

b

6.7

6.8 6.9 Oxygen concentration Y

7.0

6.7

6.8 6.9 Oxygen concentration Y

7.0

6 4

340

2 0 –2 –4 –6

Fig. 6.6. Superconductivity-induced frequency shift (a) and linewidth shift (b) for the 340 cm−1 phonon mode as a function of oxygen concentration in YBa2 Cu3 Oy (after Altendorf et al. [54])

sensitively on the variation of the oxygen concentration near y = 6.95 in YBa2 Cu3 Oy at a small variation of Tc 92 K. In particular, subsequent to the broadening observed at y = 7, the phonon mode demonstrates a hardening at y < 7. A rapid variation of the superconducting gap was also found: the gap increase from 2Δ0 /kB Tc 5.4 for samples with y = 7.0 to 2Δ0 /kB Tc 7.0 for y = 6.9. Phonon modes at 440 and 500 cm−1 revealed a smaller superconductivity-induced variation of phonon shift and linewidth and a weaker oxygen concentration dependence which may be due to the coupling of these phonon modes to a diﬀerent electron continuum than does the B1g 340 cm−1 phonon mode. Studies of the superconductivity-induced renormalization of the oddsymmetry phonon modes by the infrared (IR) spectroscopy have shown similar results. Although the interaction of the odd IR phonons with electrons involves the interband transitions, the superconductivity-induced changes of the polarization operator estimated from the ZZ theory revealed a weak electron–phonon interaction λν = 0.01 − 0.02 as well [733, 734]. A systematic investigation of the impurity eﬀects on the B1u phonon polarization operator in YBa2 Cu4 O8 (Y-124) compound has shown a weakening of the phonon renormalization with suppression of the superconducting gap by Zn and Pr

6.2 Optical Investigations

363

Fig. 6.7. Superconductivity-induced changes of the real (a, c) and imaginary (b, d) parts of the polarization operator Π(ω). Results for a BCS model (a, b) are shown for the s-wave gap Δ0 by solid lines and for the d-wave gap by dashed lines and compared with experimental data (solid points) of Thomsen et al. [1240] and Friedl et al. [353]. Results within the strong coupling theory for the d-wave pairing at various impurity concentrations (see the text) are shown in panels (c) and (d) (after Jiang et al. [554])

substitutions of the Cu and Y ions, respectively [735]. At the same time, a replacement of Ba by Sr did not change Tc and phonon anomaly at the superconducting transition. In underdoped samples, the Y-124 or oxygen-deﬁcient RBCO compounds, phonon frequency renormalization was observed above Tc , at a temperature T ∗ of the order of the spin-gap opening temperature. This points to a coupling of phonons to spin ﬂuctuations. To elucidate these controversial results, a more general theory of the phonon polarization operator for a d-wave superconductor was developed by Nicol et al. [895] for a BCS model and by Jiang et al. [554] within the strong-coupling Eliashberg theory. Figure 6.7 shows the results of numerical calculations for the diﬀerence of the real (a) and imaginary (b) parts of the polarization operator Π(ω). Positive shifts of the real (imaginary) part correspond to a hardening (narrowing) of the phonons, while the negative shifts to a softening (broadening). Experimental data of Thomsen et al. [1240] and Friedl et al. [353] are shown by solid points and the theoretical results of Nicol et al. [895] for a BCS model for the s-wave constant gap Δ0 (solid line) and for the d-wave gap Δk = (Δ0 /2)(cos kx − cos ky ) (dashed line) are shown as a function of the dimensional frequency ω/2Δ0 . The sharp structures in the polarization operators observed for the s-wave gap are smeared for the dwave gap which improves agreement with experimental data. Strong-coupling

364

6 Lattice Dynamics and Electron–Phonon Interaction

and impurity eﬀects further smoothed the curves as shown for the real (c) and the imaginary (d) parts of the polarization operator as a function of frequency ω/Tc0 . The concentration of impurity is speciﬁed by the ratio of the reduced superconducting temperature Tc to Tc0 of a pure sample: Tc /Tc0 = 1.0 (solid line), 0.95 (dotted line), 0.8 (short dashed line), and 0.6 (long dashed line). The results are given for low temperature T /Tc = 0.1 for the weak impurity scattering (Born limit). Similar results were obtained in the strong (unitary) scattering limit. With increasing impurity concentration, the superconductivity-induced eﬀects become weaker and at large concentration the broadening of the phonon linewidth may change to the narrowing in the low-frequency region which qualitatively agrees with the experimental data for the 340 cm−1 mode shown in Fig. 6.6. To give a quantitative description of the experimental data for Raman phonons a more accurate theory should be developed with a proper account of the complicated band structure of the cuprates and the strong Coulomb correlations. In a number of optic experiments local distortions of polaronic type and anharmonic vibrations have been observed. For instance, a strongly anharmonic nature of the apical oxygen vibrations was found in studies of photo-induced Raman scattering [830]. Exciting a small number of carriers with the aid of laser radiation leads to an increase of the frequency of the A1g -vibrations of apical oxygen in semiconducting samples of YBa2 Cu3 O6.3 and Tl2 Ba2 Ca1−x Gdx Cu2 O8 to the same extent as doping. The analysis of the frequency dependence of the photo-induced conductivity demonstrates that the carriers form polaronic states due to a strong coupling to the lattice [829]. The hypothesis was also put forward that the conductivity in the midIR region (see Sect. 5.3.2, Fig. 5.30) may be related to multiphonon excitation processes which arise when the free carriers loose their phonon polarization cloud. Resume The most important results of phonon spectra investigations by the Raman and infrared scattering can be summarized as follows. 1. Observation of the Fano eﬀect suggests a suﬃciently strong interaction of certain optical modes with doped charge carriers. 2. A distinct superconductivity-induced phonon frequency and linewidth shifts are observed for RS and IR modes with frequencies close to the energy gap 2Δ0 . However, the estimated electron–phonon coupling constants for these modes λν ≤ 0.02 and the eﬀective total coupling constant λeﬀ ∼ 0.8 are too small to explain the high superconducting temperature in cuprates. Strong dependence of phonon renormalization on the wave vector q observed by INS studies should be taken into account for a more reliable electron–phonon coupling estimate. 3. Polaronic type local distortions found in RS experiments suggest a strong local coupling of charge carriers to the lattice.

6.3 Isotope Eﬀect

365

6.3 Isotope Eﬀect The observation of the isotope eﬀect (IE) in conventional superconductors, i.e., the dependence of Tc on the mass M of the lattice ions, Tc ∝ M −α , with an isotope exponent αc = −d ln Tc /d ln M 0.5 (see (1.3)), was a direct evidence of the electron–phonon pairing mechanism in these materials. Comparable values αc 0.4 were found in the three-dimensional oxide superconductors BaPb1−x Bix O3 and Ba1−x Kx BiO3 which again proved the electron–phonon mechanism [555]. A notable isotope eﬀect in MgB2 superconductor (αc 0.32) also points to an electron–phonon pairing in this “high-Tc” superconductor with Tc 40 K [457]. However, already the ﬁrst measurements on isotope shift in copper-oxide superconductors have shown only a slight change in the transition temperature, ΔTc = 0.5 − 1 K, for the oxygen isotope eﬀect (OIE) observed at isotopic substitution of 16 O by 18 O which suggested a non-phononic pairing mechanism [50]. Due to the large mass diﬀerence of the oxygen isotopes, the OIE shows a pronounced isotope exponent αc 8ΔTc /Tc for superconductors with low Tc , as in the LSCO compounds: αc ∼ 0.2 − 0.4, but quite a negligible values for optimally doped YBCO and Bi-, Tl- based compounds: αc ∼ 0.05. Only a weak isotope shift with αc ≤ 0.05 was found in the electron-doped superconductor Nd1.85 Ce0.15 CuO4 . A comprehensive review of the isotope eﬀect studies in the cuprate superconductors was given by Franck [350] and Zhao et al. [1429]. In certain cases a strong variation of αc with the doping and impurity substitution was found. For instance, close to the x = 1/8 anomaly in La2−x Srx CuO4 , a large value of αc 0.8 was observed [252, 1428]. Similar increase of the OIE was found in La2−x Srx Cu1−y My O4 with 3d metal impurities M = Ni, Zn, Fe. As discussed in Sect. 5.1.2, these impurities strongly suppress Tc and increase, depending on the type and concentration of the impurities, the isotope exponent up to αc ∼ 1 [94, 940]. Studies of the siteselective OIE in YBCO compounds have revealed that the planar oxygen ions give the major contribution (∼80 %) to the total isotope shift at all doping levels [1410, 1426]. A value comparable to the OIE was found to the copper IE in the (Y-Pr)BCO compounds. Here we should remind that the large OIE observed by Gweon et al. [417, 418] in ARPES experiments on Bi-2212 samples appears to be very sensitive to doping: in slightly overdoped samples the IE is strongly suppressed or completely disappears [292, 419] (Sect. 5.2.2). Controversial results have been obtained for the OIE on the pseudogap. Williams et al. [1362] found a negligible OIE on the pseudogap Eg in YBa2 Cu4 O8 studied by high-resolution 89 Y NMR. The OIE exponent was estimated as αEg = −d ln Eg /d ln M < 0.01, while a sizeable OIE on Tc was found: αc = 0.076. A large OIE on the charge-stripe ordering temperature was reported in La1.94 Sr0.06 CuO4 by Lanzara et al. [689]. By using the X-ray absorption near-edge spectroscopy they observed a change of the local structure at a temperature Tst which increased upon replacing 16 O by 18 O

366

6 Lattice Dynamics and Electron–Phonon Interaction

from about 110 to 180 K. This was ascribed to the dynamical charge-stripe formation at Tst and appearance of the pseudogap. A large OIE on Tc and on the structural LTO–LTT transition (see Sect. 2.2.1) was found in La1.6−x Nd0.4 Srx CuO4 by Wang et al. [1339]. Due to a static stripe formation in this compound (see Sect. 3.2.4, Fig. 3.8), the superconducting temperature was quite low: Tc (K) 11, 7.9, 17.8, 25, while the isotope exponent was extremely large: αc = 1.24, 1.89, 0.98, 0.33 for the samples with x = 0.10, 0.125, 0.15, and 0.175, respectively. At the same time, the temperature of the structural phase transition to the LTT phase TLT T (K) 67, 73, 80 for the samples with x = 0.10, 0.15, 0.175 increased at the replacement of 16 O by 18 O; the corresponding exponents were estimated as αT LT = −0.32, −0.20, −0.17. Assuming that the LTT phase acts as a pinning potential for charge-stripes, it was argued that the observed correlation of the OIE on Tc and the stripe formation temperature Tst TLTT gives a clear evidence for competing between stripe phase and superconductivity. Whereas the OIE on the structural transition conﬁrms a major role of the lattice dynamics in the stripe formation in LMCO compounds, the pseudogap found in various types of cuprates may be of a diﬀerent, e.g., magnetic, origin and unrelated to stripes. In particular, the charge-stripe order measured by the wipe-out eﬀect of 63 Cu nuclear quadrupole resonance in La2−x Srx CuO4 was found at x < 0.125 [508] while the pseudogap phase was observed up to xcrit ∼ 0.19 (see Sect. 4.2.2, Fig. 4.1). An unusual IE on the magnetic penetration depth λα or the superﬂuid density ρs was detected in cuprate superconductors (for references see, e.g., Zhao et al. [1429], Khasanov et al. [600], Keller [597]). It was found for the oxygen isotope substitution in the powder samples of LSCO (see, e.g., [1427, 1428]) and later directly on the in-plane penetration depth λab (0) in single microcrystals Hofer et al. [475]. At the isotope substitution, the superﬂuid 2 2 2 ∗ density ρα s = c /λα = 4πe ns /mα (4.10) can vary either due to a change of the eﬀective mass of the superconducting charge carriers m∗α or of their concentration ns . It has been argued that the change of ns is negligibly small at the isotope substitution [1429] and therefore, the IE on the zero temperature ∗ superﬂuid density ρα s (0) gives the direct information on the IE on mab which is characterized by the exponent: βs = −

d ln λ−2 d ln m∗ab d ln ρab s (0) ab =− = . d ln M d ln M d ln M

(6.8)

Figure 6.8 shows the oxygen isotope shift for the magnetic penetration depth Δ λab (0)/λab (0) and the exponent βO versus −ΔTc/Tc and the exponent αO for several cuprate materials measured by diﬀerent techniques. In particular, open triangles show the results of magnetic torque measurements on the underdoped microcrystals of La2−x Srx CuO4 for the in-plane pen−2 etration depth −Δ λ−2 ab (0)/λab (0) = 2Δ λab /λab 0.1 and 0.08 and the corresponding exponents αO 0.47, 0.40 of the OIE on Tc for x = 0.080 and 0.086, respectively [475]. The OIE on the in-plane magnetic penetration depth

6.3 Isotope Eﬀect

367

Fig. 6.8. The oxygen isotope shift Δ λab (0)/λab (0) and the exponent βO versus −ΔTc /Tc and the exponent αO for La2−x Srx CuO4 at x = 0.080, 0.086 (open triangles), x = 0.15 (closed triangle); Y1−x Prx Ba2 Cu3 O7−δ (closed circles), optimally-doped YBa2 Cu3 O7 (diamonds and stars) (reprinted with permission by c 2004) IOP Publishing Ltd. from Khasanov et al. [600],

λab (0) in optimally doped YBa2 Cu3 O7 and La1.85 Sr0.15 CuO4 , and in slightly underdoped YBa2 Cu4 O8 and Y1−x Prx Ba2 Cu3 O7 was studied by means of muon-spin rotation technique [600, 601]. While in the underdoped region a proportionality of the both OIE was observed, Δ λab (0)/λab (0) |ΔTc /Tc | (dotted line in Fig. 6.8), in the optimally doped region the OIE on Tc is extremely small, αO = 0.02 − 0.1, as discussed earlier, but the OIE on the in-plane magnetic penetration depth λab (0) remains still substantial, βO 0.4. There are several studies of the OIE on magnetic properties of cuprate superconductors. In particular, a lowering of the N´eel temperature TN 313 K in the undoped La2 CuO4 compound by about 1.9 K was observed at the replacement of 16 O by 18 O [1424]. The isotope shift characterized by a small exponent αN = −d ln TN /d ln M 0.05 shows a certain inﬂuence of the oxygen dynamics (presumably, zero-point vibrations) on the exchange interaction J which determines the antiferromagnetic ordering of the copper spins below the N´eel temperature (see Sect. 3.2.3). Since the latter is proportional to J (see (3.21)) we can estimate the OIE on the exchange interaction J by the same exponent αJ = −d ln J/d ln M 0.05. Measurements of the OIE in doped compounds did not reveal a distinct isotope shift of TN . Also a negligible OIE

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on the magnetic resonance mode at 41 meV was found in the nearly optimally doped YBCO where an OIE with ΔTc ∼= −1.2 K was detected [945]. This suggests a weak spin–phonon coupling in cuprate superconductors near the optimal doping. It should be mentioned that the magnetic properties of cuprates strongly depend on the oxygen (hole) concentration in the sample which makes IE experiments very diﬃcult. In certain cases, the OIE can be strongly enhanced by using a sample with magnetic impurities. For instance, a huge OIE on the spin glass transition temperature Tg was found in La2−x Srx Cu1−z Mnz O4 for x = 0.03, z = 0.02 characterized by the exponent αg = −d ln Tg /d ln M −6, while for the pure sample (z = 0) the OIE was negligible [1148]. To explain the unconventional IE in cuprates, several scenarios were suggested. In the earlier studies a substantial OIE on Tc and m∗ab in underdoped compounds was taken for an evidence for the polaronic character of the charge carriers condensing into Cooper pairs (see, e.g., [1427, 1428]). This scenario follows from the polaronic theory of the superconductivity in cuprates developed by Alexandrov (see Sect. 7.4.2). In this theory a polaron formation, i.e., of a quasiparticle dressed by lattice distortions, caused by a strong electron–phonon coupling is considered. The eﬀective mass of a polaron m∗ = mb exp (gp2 ) where the dimensionless polaron coupling constant gp2 = γEp /ω depends on a characteristic phonon frequency ω and the polaron binding energy Ep (here γ ≤ 1, while mb is the band mass). For a frequency ω ∝ M −1/2 the exponent of the IE on m∗ is given by βs = d ln m∗ /d ln M = (1/2) ln(m∗ /mb ). In conventional metals where nonadiabatic or polaronic eﬀects are weak, the renormalization of the band mass is small, m∗ /mb 1, and there is a negligible isotope eﬀect on m∗ (as e.g., in MgB2 , see Di Castro et al. [281]). The IE on Tc and m∗ab within the (bi)polaronic theory of superconductivity was considered by Alexandrov [27]. In later publications a numerical analysis of the IE on the polaronic band structure in doped insulators was given by Kornilovich et al. [644] and by Hague et al. [422]. A detailed studies of the IE within the polaronic model and comparison with experimental data were performed by Bussmann-Holder et al. [191]. A two-component model of hole-rich metallic regions for the Fermi-like charge carriers and hole-poor AF regions for charge carriers with strong correlations described by the t–J model was proposed. Both charge carriers interact with phonons that results in their coupling. There are two superconducting gaps in the model, one of the d-wave and another of the s-wave symmetry referred to the two charge carrier components, respectively. By taking into account polaronic renormalization of the hopping parameters between the nearest and next neighbors, it was possible to reproduce experimental results in the underdoped region, Δ λab (0)/λab (0) |ΔTc /Tc |. However, the complicated two-component model can be justiﬁed in the strongly underdoped region only (in the stripe phase of LSCO, as discussed by M¨ uller [873]). Therefore the

6.3 Isotope Eﬀect

369

obtained results do not provide a convincing proof of the polaronic scenario for the whole superconducting phase. To explain the increase of the IE with decreasing Tc , a model of the electronic DOS N (ε) with a van Hove singularity (vHs) was proposed by Tsuei et al. [1273] (see Sect. 7.4.2). Depending on the position of the chemical potential with respect to the vHs, a rapid variation with doping of Tc and a corresponding isotope shift was obtained. Anharmonic eﬀects at the structural phase transition in LSCO (see, e.g., [983, 1000]) can also enhance the isotope exponent αc as observed in experiments. In the case of the impurity substitution, as in the La2−x Srx Cu1−y My O4 compounds or Y1−x Prx Ba2 Cu3 Oy , a strong increase of the isotope exponent can be explained by pair-breaking eﬀects [203]. These eﬀects result in the suppression of the initial transition temperature Tc0 according to the Abrikosov–Gor’kov theory [7]: ln(Tc0 /Tc ) = Ψ (ρ + 1/2) − Ψ (1/2),

(6.9)

where Ψ (x) is the digamma function and ρ = h ¯ /[2πτp (1 + λ)kTc ], τp is the relaxation time for the scattering on paramagnetic impurities. For a nonzero isotope exponent in a pure system α0 = −d ln Tc0 /d ln M , for the system with paramagnetic scattering one gets from (6.9): αc = α0 /[1 − ρΨ (ρ + 1/2)].

(6.10)

In the limit of weak scattering when τp → ∞, and ρ → 0, we have αc = α0 /(1 − 0.7τpc /τp ) where τpc is the critical value of the relaxation time at which Tc → 0. In the latter limit, αc 0.24α0 (Tc0 /Tc )2 diverges. The relation (6.10) agrees suﬃciently well with the dependence of αc in the LSCO compounds on the concentration of the 3d impurities (except Fe) for small initial values of α0 ∼ 0.1 [940]. The combined eﬀect of the nonadiabaticity of certain phonon modes and magnetic impurities was studied by Bill et al. [136]. If a charge transfer between the reservoir and the conducting layers involves nonadiabatic phonons, e.g., apex oxygen vibrations, the charge-carrier density depends on the ionic masses, n = n(M ). This results in the isotope shifts both of Tc and the magnetic penetration depth λab . In particular, the nonadiabatic (na) part (na) of the isotope exponent αc ∼ (n/Tc)dTc /dn. This vanishes at optimal doping, dTc /dn = 0, but may give a large contribution in the underdoped region. However, the charge-carrier density ns (M ) dependence on the ionic masses has not been observed by Zhao et al. [1429]. A model of singlet–triplet pseudo-Jahn–Teller (PJT) centers was considered by Moskvin et al. [866] to explain the IE related to the vibronic reduction of the transfer integral for charge carriers. It has been suggested that a doped hole in a system of CuO4 clusters can occupy the nearly degenerate molecular terms 1 A1g (Zhang-Rice singlet) or the singlet 1 E1u (triplet 3 E1u ) state given by the conﬁgurations (B1g )2 or B1g eu , respectively (see Sect. 7.5.2). The

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polar oxygen eu -centers couple with active local displacements of the corresponding symmetry resulting in the PJT eﬀect. As a result, the doped hole motion can be considered as a local boson hopping in a lattice of singlet–triplet overhole-centers. The transfer integral for bosons tBB ∝ Keh depends on the √ lap integral for local oscillatory states for hole and boson Keh ∝ exp(−A M ) which relation to the atomic masses M determines the IE. A qualitative agreement of theoretical estimates for the IE with experiments for YBCO-type compounds was observed. Tallon et al. [1227] have developed a theory of the unconventional IE on Tc and ρs based on consideration of the impurity scattering eﬀects and the pseudogap formation in cuprates. According to the theory, in the overdoped region without the pseudogap the IE on ρs should disappear in clear samples without impurity scattering, βs ∼ 0, while the OIE on Tc should be small: αc 0.06. However, in the underdoped region both the normal-state pseudogap and the impurity scattering suppress Tc and ρs in a similar way and could result in large values of the IE. These conclusions were compared with experimental studies of the OIE in the La2−x Srx Cu1−y Zny O4 compounds at various Zn doping (impurity eﬀects) and Sr doping (pseudogap variation). An overall agreement between the theory and the experiment was achieved except at the hole doping x ∼ 0.12, close to the 1/8 singularity. There a strong anomaly in OIE was observed which was ascribed to a stripe formation. It was concluded that stripes and pseudogap correlations are diﬀerent phenomena and both compete with superconductivity. It should be mentioned, however, that a ﬁnite value of βO 0.4 for the OIE on λab (0) in optimally doped YBCO shown in Fig. 6.8 is at odds with the theoretical prediction of βO ∼ 0. Usually it is claimed that the OIE cannot be described within the magnetic mechanism of pairing. However, an observation of the OIE on the exchange interaction, αJ = −d ln J/d ln M 0.05, provides an explanation for an OIE on Tc [1013]. In the theory of the antiferromagnetic exchange pairing in cuprates, the coupling constant λs ∝ J that results in the OIE on Tc with the exponent αc = (d ln Tc /d ln J) αJ (1/λs ) αJ (see Sect. 7.3.3). For a weak coupling λs ∼ 0.1 and low Tc in the underdoped region the exponent αc ∼ 0.5, while for a stronger coupling at optimal doping it decreases, e.g., for λs ∼ 0.5 we have αc ∼ 0.1. Polaron eﬀects can enhance OIE of the magnetic scenario. Resume Thus, studies of the isotope eﬀects in cuprate superconductors have not provided unambiguous evidence for a phonon mechanism of the pairing. Large values of the exponents αc and βs can be obtained even in models where the electron–phonon interaction gives only a small contribution to the intrinsic Tc0 . At the same time, the rather small values of α0 ∼ 0.01 in the optimally doped samples with high Tc ∼ 100 K suggest that the lattice eﬀects are not decisive and only indirectly involved in determining the superconducting state, e.g., by

6.4 Theoretical Models

371

polaron renormalization of the charge carriers, particularly in the underdoped region.

6.4 Theoretical Models The studies of the phonon spectra in cuprate superconductors are complicated by the large number of atoms in the unit cell of the crystal. By applying symmetry analysis of lattice vibrations, it is possible to classify various phonon modes with respect to the irreducible representations of the space group for a given wave vector k. This procedure permits to determine polarization vectors for the phonon modes under study which simpliﬁes the assignment of diﬀerent phonon modes observed in INS and optic experiments as discussed above. However, to compute the vibration frequencies and polarization vectors for separate atoms one has to adopt a certain model of atomic interaction in the crystal. The simplest one is the force constant model, in which one speciﬁes the harmonic coupling constants of the interatomic interaction for several coordination spheres. The assessment of the constants is performed on the basis of a best ﬁt to experimental data. Due to the phenomenological character of the method, it gives no insight into the nature of the coupling forces in the lattice. An approach based on a model of atom–atom potentials seems to be more consistent. For ionic crystals a shell model is often used. In the model, besides the direct Coulomb interaction of rigid ion cores and a short-range repulsive potential between them, a coupling of the valence electronic shell with the ionic core are introduced. On the basis of this model, the lattice dynamics of the main types of copper-oxide superconductors has been computed (see, e.g., [657, 676]). The same shell model, with an inclusion of anisotropic screening in the metallic phase demonstrated quite a satisfactory theoretical description of the phonon dispersion curves measured by INS as discussed in Sect. 6.1.1 [222,993]. A rather good agreement between the experimental and theoretical results for these calculations, with a small number of adjustable parameters in the ionic shell model, have conﬁrmed ionic nature of the interatomic forces in the cuprate superconductors. By taking this into account, it is suitable to discuss the main features of the electron–phonon interaction (EPI) within the framework of an ionic model for the CuO2 plane [106] . In this case, accounting for lattice vibrations leads to ﬂuctuations of the model parameters – the hopping integral tij and the site energies of the p, d states in the crystalline ﬁelds ΔTij tij (uij /d) + · · · ,

(6.11)

δi αi V (ui /d) + βi V (ui /d) + · · · . 2

(6.12)

where ui is a displacement of the type i ion and uij = ui − uj , d is the interatomic distance. In the conventional metals with broad bands, the main

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contribution comes from the ﬁrst term (6.11), since due to a strong screening the long-range Coulomb ﬁelds V e2 /εd are small (the dielectric function ε is large), and small is the interaction (6.12). In the copper-oxide compounds, due to the tight localization of wave functions of valence electrons, their overlap is small and therefore the transfer integral tij is small and the interaction (6.11) is weak. At the same time, due to the low concentration of the charge carriers and the poor screening (especially for directions perpendicular to the CuO2 planes), the Coulomb energy is large, V tij , and the second contribution (6.12) is greater than the ﬁrst one (6.11). Estimates of the dimensionless parameters αi and βi for several typical displacements of the ions Cu2, Ba, O2, O3, and O4 in the YBCO crystal in Raman-active modes, and for the ions Cu and O for the soft mode in La2 CuO4 have been made by Bariˇsi´c [106]. In YBCO, due to a buckling of the CuO2 plane, there is a linear EPI with αA ∼ 2 in (6.12) for Ag type modes. In the tetragonal phase of LSCO the linear term in (6.12) for oxygen displacements in the soft mode vanishes, αSM = 0, while βSM ∼ 1. As a result, for the Coulomb interaction Vpd 1 − 2 eV, for the A), deformation potential we obtain the estimates gA ∼ αA V /d ∼ 1 − 2 (eV/˚ gSM ∼ βV (u/d2 ) ∼ 0.1 (eV/˚ A) at d 2 − 3 ˚ A and equilibrium displacements of oxygen ions in the soft mode u ∼ 0.2 − 0.3 ˚ A. The renormalization of the phonon frequencies for a given vibration mode ν according to (6.1) and (6.2) is determined by the relations Δ ων ∼ |gν |2 χν . Thus, apart from the matrix element gν proportional to the deformation potential, the electronic susceptibility χν is of importance in evaluating the electron–phonon interaction. As noted by Bariˇsi´c [106], the susceptibility χpd related to the charge transfer through the gap Δpd = εp −εd ∼ 3 eV between p and d states appears to be much smaller than the susceptibility χpp related to the charge transfer between oxygen ions inside the unit cell of CuO2 . Therefore one should anticipate a rather strong renormalization of frequencies for those vibration modes in which the equivalence of oxygen ion positions is violated and the phonon shift Δ ων is determined by the susceptibility χpp , as in the “half-breathing” mode or the in-plane Jahn–Teller Q2 -type mode related to the deformation of the CuO2 unit cell. The renormalization of frequencies determined by the p–d charge transfer should be weak due to the small susceptibility χpd , e.g., for the full “breathing” mode. An important role of the long-range Coulomb interaction and strong local electron correlations in the EPI have been emphasized by Kuli´c [673, 674]. He has stressed that a large contribution to the EPI arises from the variation of the Madelung energy at the lattice vibrations due to the poor screening in the cuprates and the strong anisotropy of the interaction in respect to the scattering momentum of electrons, q = p − p in (6.2). Due to the Coulomb renormalization, the strong forward scattering peak in the EPI arises which results in the small electron–phonon scattering observed in the transport experiments related to the electronic momentum relaxation but produces the strong pairing interaction at the small momentum transfer (for detail see Sect. 7.4.1).

6.4 Theoretical Models

373

A quantitative description of the lattice dynamics and EPI have been achieved by computation of the electronic energies in a crystal within the density functional theory (DFT). Here we should mention the earlier studies of lattice structure and dynamics for the metallic La2−x Mx CuO4 and YBa2 Cu3 O7 componds on the basis of the linear augmented plane wave (LAPW) method [243,658,981]. It proves possible to perform suﬃciently precise calculations of the ground state energy of the crystal, and, using the frozen-phonon approach, to ﬁnd the frequency and the polarization vector for certain modes in high symmetry points of the Brillouin zone. In particular, it was shown that the La2 CuO4 lattice (in metallic phase) is stable with respect to the condensation of the “breathing” mode – its frequency is high at the X-point of the Brillouin zone, but it is unstable with respect to the tilting soft mode (see Fig. 2.19) [243, 983]. Similar calculations of a series of optical frequencies for YBa2 Cu3 O7 have conﬁrmed the high reliability of this method [244]. For some optical Ag -modes suﬃciently large coupling constants λν ∼ 1 − 2 were obtained, which signiﬁcantly exceed the corresponding values computed in the rigid muﬃn-tin approximation. In the latter, the ionic contribution due to changes of the Madelung potential at the displacement of atoms is disregarded. These ﬁrst-principle calculations have conﬁrmed the essential inﬂuence of the Coulomb forces on the lattice dynamics and EPI. Extensive band structure calculations of YBa2 Cu3 O7 compounds by the linearized muﬃn-tin orbital (LMTO) method were performed by Andersen et al. [55] with special emphasis on the EPI constants. As discussed in Sect. 6.2, the calculations by Rodriguez et al. [1056] were used to evaluate the phonon shift and linewidth in the strong coupling approximation (6.6). The estimate of the averaged constants leads to the conclusion that the magnitude of the coupling constant is moderate in YBCO compounds, λ ≤ 1. There, an irregular dependence of the susceptibility χ(q,ω) and linewidth γ(q,ω) ∼ g 2 Imχ(q, ω) on the wave vector q inside the Brillouin zone was noticed. In particular, the existence of saddle points on the Fermi surface 15–25 meV below the Fermi level leads to a strong mixing of the lattice vibrations with quasiparticle excitations and to a weaker energy dependence of their linewidth γν ∼ (k −μF )3/2 than in the conventional Fermi liquid. A systematic study of the phonon dispersion curves and the EPI in YBa2 Cu3 O7 within the DFT by using a mixed-basis pseudopotential method was performed by Bohnen et al. [153]. Satisfactory agreement between theoretical calculations of phonon frequencies and INS and RS experiments was obtained. It was also possible to calculate the generalized PDOS and the Eliashberg function α2 F (ω) (6.3). For the eﬀective EPI coupling constant (6.4) an estimation λ = 0.27 was obtained. On the basis of this estimation for λ, only a very low superconducting transition temperature Tc was found. However, a weak EPI coupling should result in a small phonon linewidth which is at odds with the INS data, at least for particular modes, e.g., the bond-stretching one (see Fig. 6.2).

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6 Lattice Dynamics and Electron–Phonon Interaction

As discussed in Sect. 5.1.1 (see also Sect. 7.1.1), the Coulomb correlations are not properly taken into account in the band-structure calculations within the DFT and therefore, the description of the electronic spectrum and lattice dynamics may be unreliable. In particular, this method cannot be applied to study the lattice dynamics of the insulating parent compounds as La2 CuO4 and YBa2 Cu3 O6 . A more general approach within a microscopic model by using a linear response theory was proposed by Falter and coworkers (see the review by Falter [322]). To take into account the strong on-site Coulomb correlations, the local part of the electronic charge response was calculated within an ab initio rigid-ion model (RIM). Eﬀective ionic charges reduced by covalent bonding were found in the tight-binding approximation for the singleparticle band structure. The nonlocal part of the electronic density response and EPI were described by charge and dipole ﬂuctuations localized on the outer shells of the ions. This enables to consider the strong local Coulomb correlations, covalent coupling of the ions, and the charge ﬂuctuations selfconsistently. It was suggested that a textured “two-component” electronic structure exists with an insulator-like sublattice of strongly localized Cu 3d states and a sublattice of the metalic-like O 2p states. With hole doping, delocalization occurs via ﬁlling O 2p states and more delocalized Cu 4s, 4p states, while at electron doping the Cu 3d, 4s states are ﬁlled in. This explains the electron–hole asymmetry of the cuprate compounds. It was shown also that in the underdoped cuprate superconductors a strongly nonlocal EPI is observed which becomes more local in the overdoped region. Within this approach phonon dispersion curves were calculated for the La2 CuO4 (LCO) crystals in the metallic and insulating phases which were in a good overall agreement with INS data. In particular, a softening of the B1g bond-stretching mode with doping was properly reproduced. The anomalous phonon softening and the anticrossing eﬀects observed in INS experiments in YBa2 Cu3 O7 were explained within the theory also. In considering dielectric and infrared properties of LCO, a much larger static dielectric tensor ε0 than the macroscopic high-frequency one ε∞ was found, as e.g., for the in-plane εxx = εyy tensors: ε0 ∼ 33, ε∞ ∼ 6.7. The infrared- and Raman-active c-polarized modes were computed for the Bi-based cuprate superconductors. The large softening of the apex-oxygen breathing mode OzZ observed in INS was explained by the nonadiabatic mixing of the phonon mode with the low-lying intrabandplasmon generated by interlayer coupling. Concerning the mechanism of the superconducting pairing, it was argued that the magnetic degree of freedom at Cu sites could be coupled to the charge ﬂuctuations and act synergetically with the strong nonlocal EPI in the doped cuprates. It would be interesting to calculate within the developed theory the phonon linewidth and estimate the EPI constant as given by (6.3), (6.4). As discussed in Sect. 5.2.2, in explaining the substantial electron spectrum renormalization observed in ARPES studies (a “kink” phenomenon) a strong and highly anisotropic EPI coupling was suggested by Devereaux et al. [279].

6.5 Conclusion

375

It was found that the coupling constant for the buckling B1g mode at the (π, 0) point could reach a value of λ ∼ 2.8, while quite a small λ ∼ 0.3 was observed for the constant averaged over the whole BZ . A strong coupling for in-plane Cu–O breathing modes was found for electrons in the nodal direction. The large anisotropy was explained by a highly anisotropic EPI matrix elements g(k, q) and the anisotropy of the electronic 2D band structure with Fermi energy close to the van Hove singularity. Although the phonon dispersion could be well described within the “ﬁrst principle” density functional theory, as discussed above, the EPI appears to be too small to explain the large phonon linewidth observed in INS experiments. Therefore, it was suggested that eﬀects of strong Coulomb correlations may be important in the EPI. These eﬀects were studied within the t–J model describing the hole motion as the Zhang–Rice singlet states. In the model, the EPI was induced by the modulation of the hopping integral tij and the exchange interaction J [1061]. In spite of a weak EPI in this case, a hole coupling to the antiferromagnetic ﬂuctuations leading to the formation of magnetic polarons can greatly enhance the phonon renormalization [837, 1063]. A much larger renormalization of the half-breathing mode in the (1, 0)-direction in comparison with the full breathing mode in the (1, 1)-direction was predicted by Khaliullin et al. [599] within the slave-boson treatment of the t–J model. The strong softening of the mode is due to the phonon interaction with the low energy charge ﬂuctuations observed in the t–J model [487, 488]. Exact diagonalization studies for the t–J model with EPI conﬁrmed these results and correctly reproduced the strong softening with doping of the half-breathing mode and only a weak softening of the full-breathing mode [1061]. At the same time, studies of the three-band p–d model with EPI in the mean-ﬁeld Hartree–Fock approximation have shown a much weaker dependence of the phonon softening on doping [1062]. Thus, many-body eﬀects giving rise to lowenergy excitations in the t–J model reinforce the EPI in strongly correlated systems. The important role of the coupling between the spin and phonon degrees of freedom in cuprates was pointed out in a number of publications (see, e.g., [533, 904, 985, 1090]).

6.5 Conclusion To summarize the studies of lattice dynamics of cuprate superconductors the following results should be pointed out: 1. Experimental studies by inelastic neutron scattering and optic methods revealed the ionic character of the lattice dynamics where the Coulomb forces play the major role both in insulating and metallic phases. In the latter case a highly anisotropic screening occurs which is weak in the c-direction and strong in the basal CuO2 plane. 2. Investigations of the doping dependence of the phonon spectra show a weak electron–phonon interaction in general, except for particular modes

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6 Lattice Dynamics and Electron–Phonon Interaction

related to oxygen vibrations. A strong symmetry-dependent coupling of these modes to charge ﬂuctuations is suggested to explain their large phonon energy and linewidth renormalization. 3. The oxygen isotope eﬀect (OIE) is rather small in optimally doped cuprates. This suggests that the electron–phonon interaction is involved in determining the superconducting state only indirectly, in particular, through polaron eﬀects observed by the OIE on the magnetic penetration length. 4. “First principle” band structure calculations in the local density approximation generally predict correct phonon dispersion in the metallic state but fail to describe the doping dependence of certain phonon modes, in particular, at the transition from the insulating to metallic phase. Manybody eﬀects caused by strong Coulomb correlations and phonon coupling to the spin ﬂuctuations are essential in enhancing the electron–phonon interaction which may contribute to the superconducting pairing.

7 Theoretical Models of High-Tc Superconductivity

The Bardeen–Cooper–Schrieﬀer (BCS) pairing theory in conventional superconductors is based on the picture of a Fermi liquid with a properly determined spectrum of single-electron excitations, “quasiparticles” near the Fermi surface (see Sect. 1.1). In cuprate superconductors, the normal state is unconventional and many electronic properties show anomalous behavior as discussed in Chap. 5. Therefore, a model that works perfectly well for a system of welldeﬁned quasiparticles in conventional metals may be inapplicable to a system of strongly correlated electrons or at least the pairing theory should be modiﬁed to take into account the peculiarity of the normal state in cuprates. In this connection, in this chapter, we shall ﬁrst discuss the problem of electronic structure and spectrum of single-electron excitations in the normal phase of cuprates. In discussing the mechanism of high-temperature superconductivity (HTSC) in cuprates, a large number of models have been proposed since its discovery by Bednorz and M¨ uller [118]. From the very beginning, two diﬀerent approaches have been considered. A conventional electron–phonon pairing mechanism in a Fermi-liquid-type metal and, in particular, a strong-coupling of polarons advocated by K.A. M¨ uller was the ﬁrst one (for a review, see [873]). Another “doped Mott insulator” approach, based on the occurrence of strong electron correlations in cuprates, and in particular, the antiferromagnetic (AF) insulating state in the undoped compounds, was put forward by Anderson [61]. Later on, when the d-wave symmetry of Cooper pairs has been ﬁrmly established by phase-sensitive experiments (see Sect. 5.5.3), an electronic pairing mechanism due to spin (or charge) ﬂuctuation exchange has been accepted by many researches. On the whole, there are many models of HTSC but a generally accepted theory of HTSC is lacking (see e.g., the special issue of the Nature Physics [888]). It is even argued that the quantum criticality hidden under the superconducting dome Tc (x) in the cuprate superconductors has been masking the true nature of the “hidden” phase emerging at a quantum critical point xc , which may be important in our understanding

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of the mechanism of HTSC [693]. In the next Sections, we consider in some detail only the basic models of HTSC in cuprates.

7.1 Electronic Structure of Cuprates A general overview of the electronic structure of the cuprate materials within the model of doped ionic semiconductor was given in Sect. 5.1. Below we consider several theoretical approaches to band structure calculations of cuprate compounds. At ﬁrst, we brieﬂy mention results of the conventional density functional method and then discuss eﬀective models, which take into account strong electron correlations. 7.1.1 Band-Structure Calculations A very eﬃcient approach for investigating the electronic structure of solids is based on the density functional theory (DFT), which has been used in early studies of the electronic properties of copper-oxide compounds (for a review, see [981]). Within DFT, self-consistent calculations of the electronic charge distribution and the form of the potential inside the primitive cell are performed on the basis of one-electron functions for the Bloch states in a crystal. This method allows “ﬁrst principle” – without any ﬁtting parameters – calculations of the ground state electronic energy of the crystal, as well as an investigation of its dependence on deformations and the displacements of ions. In particular, on the basis of the “frozen phonon” method, one can determine force constants in the dynamical matrix of the crystal and electron–phonon matrix elements (see Sect. 6.4). At the same time, in the framework of the DFT the many-body Coulomb interactions are usually treated in the local-density approximation (LDA). The latter approximation may be justiﬁed only for s–p metals with broad bands and strong screening. In transition metals with narrow d-bands, an important role is played by Coulomb on-site correlations, and therefore the one-particle wave functions and the spectrum of one-electron excitations as computed in the frame of the LDA are not directly related to the excitations of the many-particle electron system. Nevertheless, on the basis of the DFT, one can get a general picture of the electronic structure of the crystal, ground state charge density distribution, hybridization parameters, and others estimated to an accuracy of several tenth of electron volt. In the very ﬁrst band-structure calculations for La2−x Mx CuO4 compounds, it has been found that the most important contribution to the electronic density of states near the Fermi surface arises from the pdσ-band, originated from d(x2 − y 2 ) states of Cu2+ and pσ (x, y) states of O2− (see Fig. 5.1). Figure 7.1a shows the electronic energy spectrum in the body-centered tetragonal La2 CuO4 (LCO) crystal along symmetry lines (shown in the inset) calculated within the linear-augmented-plane-wave (LAPW) method by Mattheiss [812]. The electronic density of states (DOS), both the total one and

7.1 Electronic Structure of Cuprates

b

6.0 2

Energy (eV)

12.0

4

AB

0

εF

AB

–2 –4 B

–6 –8

Γ

Δ

y Γ

U ZΛΓ

X U Z

B XS

3.5

Cu

0.0 1.0

Oxy

0.0

0.0 –8 Z

Tot

0.0

2.0

XS Δ

D.O.S. (States/ eV)

a

379

Oz

–6

–4 –2 E (eV)

0

2

Fig. 7.1. (a) Energy band structure and (b) total and projected DOS in the tetragonal phase of La2 CuO4 (after Mattheiss [812])

its partial components on separate ions is represented in the panel (b) (after Pickett [981]). Of the total of 17 bands formed by Cu(3d)–O(2p) states (ﬁve 3d states on copper ions, and three 2p-states on each of the four oxygen ions in the primitive cell of LCO), only two have a large dispersion: bonding (B) and antibonding (AB) pdσ type bands, constructed from the orbitals d(x2 − y 2 ) and pσ (x, y). The Fermi surface crosses the AB-type band, and the other bands lie far below the Fermi energy (EF = 0). Since pdσ-band electrons are conﬁned in the CuO2 planes, which are located far from each other, this band is an essentially two-dimensional (2D) one – the dispersion along the z-axis is small (see the line Λ from the center of the band Γ to the point Z(0, 0, 1/2)). By taking into account only p–d hybridization the two-dimensional B-AB bands in the tight-binding approximation can be described by the following formula: 1/2 1 1 kx a ky a + sin2 EB,AB (k) = (Ep + Ed )± (Ep − Ed )2 +(4t)2 sin2 , 2 2 2 2 (7.1) √ where Ep Ed = −3.2 eV and t = ( 3/2) Vpdσ , Vpdσ = −1.8 eV, accord√ ing to this calculations. The total width of the B-AB band W = 4t 2 9 eV appears to be too large in comparison with that observed in ARPES experiments (Sect. 5.2.2). Nevertheless, a useful information concerning the high-energy electronic level positions can be deduced from this computations. The La levels are weakly coupled to the states in Cu–O bands: the 5d level of La lies ∼1 eV above, and the 5p-level ∼ 15 eV below the Fermi level. Therefore, La may be viewed as an isolated ion with the charge +3, which, when replaced by a rare-earth ion with the same charge, has a minor eﬀect on

380

7 Theoretical Models of High-Tc Superconductivity

the electronic properties. In particular, the magnetic scattering of pdσ-band electrons on the magnetic moment of rare-earth ions is small and does not inﬂuence superconducting properties. The strong anisotropy of the pdσ-band leads to a quasi-two-dimensional nature of the electronic Fermi surface in the form of the square-like cylinder centered at the Γ (0, 0) point of the Brillouin zone (BZ). The full potential LAPW band-structure computations of the La2−x Mx CuO4 (LMCO) compounds by Xu et al. [1378] have shown that the Fermi surface touches the boundary of the BZ at (0, ±π/a) or (±π/a, 0) points (in notation for the 2D BZ) at a ﬁnite concentration x 0.17 of divalent ions M = Sr, Ba. This leads to a strong 2D van Hove saddle point singularity (vHs) in the electronic spectrum and the logarithmic-type divergence in the DOS N (ε). In the vHs scenario of high-Tc superconductivity, the maximal value of Tc is related to this maximum of DOS (see Sect. 7.4.3). However, the averaged density of states on the Fermi surface (per two spin directions) proves to be small: 2N (0) 1.2, or 1.9 states/eV at x = 0 and 0.16, respectively. In other studies, similar values of the density of states have been obtained [981]. The existence of ﬂat regions on the Fermi surface provides “nesting” – a high degree of congruence of the Fermi surface under a translations by the vectors | kx (1, 2) |= 2kF , where kF is the Fermi momentum and kx (1, 2) = (π/a)(±1, 1, 0) are the wave vectors at X points of the BZ. As suggested by Mattheiss [812], the nesting should lead to an instability of the LCO lattice with respect to freezing of the breathing mode and formation of a charge density wave (CDW). As a result, a redistribution of charge on neighboring copper ions appears, forming a CDW with the wave vector kx (1) (or kx (2)). This leads to a gap opening of in the electron spectrum at the Fermi surface which explains, according to Mattheiss [812], the semiconducting behavior of LCO. However, the experimentally observed structural phase transitions in LMCO from the tetragonal (HTT) to the orthorhombic (LTO) or the low temperature tetragonal (LTT) phases are related to the condensation of tilting-type modes with the wave vectors kx (1) (or kx (2)) (see Sects. 2.2.2 and 6.1.1). On the basis of the total electronic energy calculation and the frozen phonon method, it was found that the breathing mode in LMCO is stable and has a high frequency [983]. The dependence of the crystal energy on the ion displacements under rotations of the CuO6 octahedrons is described by a double-well potential (Fig. 2.9), which suggests a possibility of phase transitions to the LTO- or LTT-phases. The instability of the lattice induced by soft tilting modes is of ionic nature of forces in LMCO compounds typical to perovskite structure and has no connection with FS nesting. In the LTOphase with the displacements of all the four O1 ions in the CuO6 octahedron, the gap in the electronic spectrum does not appear, while in the LTT-phase, with only two O1 ion displacements (see Fig. 2.8), there appears a gap at the M-points: (0, 1, 0) or (1, 0, 0) [983]. In the opinion of the authors, the formation of this gap and the almost twofold decrease of the density of states at the Fermi surface in the LTT-phase can explain the anomalous behavior of several

7.1 Electronic Structure of Cuprates

381

electronic properties (dielectric properties, suppression of superconductivity, appearance of local magnetic moments). Similar results for electronic band-structure calculations within the DFT have been obtained for other copper-oxide compounds also (see [981] and references therein). These calculations conﬁrm a high degree of hybridization – strong covalent coupling of Cu3d and O2p states with the formation of a broad pdσ-band. The other electronic states have a weak hybridization, and the atoms out of the CuO2 planes may be described in the ionic coupling approximation. In particular, in YBa2 Cu3 O7−y (YBCO) the Fermi surface crosses two antibonding bands (corresponding to two CuO2 planes) and one quasi-one-dimensional band of the Cu1–O1 chains for y = 0 as discussed in Sect. 5.2.2. The removal of oxygen O1 ions from the chains (y = 1) leads to the destruction of the latter band with an insigniﬁcant deformation of the pdσ-bands in the CuO2 planes. Similar to the LCO case, according to these calculations, YBCO should be a metal (including y = 1) with suﬃciently broad pdσ-bands and a rather small density of states: 2N (0) 5.5 states/eV (per cell, per two spin directions). The density of states as related to one copper ion is the same as that obtained for LCO (see Table 4.1). Extensive band-structure LDA calculations for the metallic YBa2 Cu3 O7 crystal by Andersen et al. [56, 57] revealed an extended saddle-point anomaly about 30 meV below the Fermi level near (kx a/π, ky b/π) = (1, 0) and (0, 1) points of the BZ. This “ﬂatband”-type anomaly has been observed in ARPES experiments in Bi-2212 crystals (see Sect. 5.2.2, Fig. 5.18). The anomaly is due to a bifurcation of the saddle-points in the antibonding pdσ-band and caused by dimpling of the CuO2 plane. It was pointed out that besides the Cu3d and O2p orbitals participating in the pdσ-band other orbitals such as Cu4s and O2pz are crucial in building up the band-structure near the Fermi level (see next section). The band-structure of Bi2 Sr2 CaCu2 O8 (Bi-2212) looks more complicated. Besides the broad pdσ-band of Cu3d–O2p states, which is typical of the CuO2 planes, a rather broad ppσ-band of Bi 6p(x, y) and O 2p(x, y) states crosses the Fermi surface too. The total density of states per cell is about 3 states/eV, and, if related to one Cu atom, after subtracting the contributions of Bi–O states, it reduces to 1 states/eV. The band-structure of Tl-based compounds diﬀers from that of Bi compounds only by the fact that at the Fermi surface, instead of the Bi–O band, a band originated from Tl-6s-levels and O2p-levels appears (see Fig. 5.22). The average density of states per copper atom is also not large, of the order 1 state/eV, independent of the number of CuO2 planes. Thus, electronic band-structure calculations within the DFT predict the existence of a broad quasi-2D pdσ-band and the small density of states at the Fermi surface. These calculations with the Coulomb correlations accounted for only in the one-particle mean-ﬁeld approximation, fail to obtain an antiferromagnetic ground state with a suﬃciently large insulator gap and signiﬁcant local magnetic moments at copper sites in the CuO2 planes [981]. To overcome this drawbacks of the DFT theory and, in particular the LDA, which ignores

382

7 Theoretical Models of High-Tc Superconductivity

the strong orbital polarization of the local 3d-states originating in Coulomb correlations, more eﬃcient methods were developed such as the LDA+U approximation (for a review see [73]) and the LDA+DMFT method – a combination of the LDA and the dynamical mean ﬁeld theory (DMFT). The problem of the metal-insulator transitions caused by strong electron correlations and corresponding theoretical models are considered in depth by Imada et al. [520]. Our main attention will be devoted to speciﬁc models used in the theory of the electronic spectra in cuprate superconductors. Several examples of relevant single-particle excitation spectra within these models will be given in Sect. 7.2.1.

7.1.2 Model Hamiltonians for CuO2 Plane The problem of the many-body eﬀects caused by strong electron correlations in solids is one of the most diﬃcult ones, which has not yet found a comprehensive solution. Strong electron correlations are usually treated within speciﬁc models in which only few relevant electronic states are taken into account (for a review see [371]). In cuprate materials, the two most important features of the electronic structure – a large hybridization of O2p and Cu3d states in the pdσ-band and a strong local Coulomb repulsion on Cu3d states – may be accounted for by constructing an eﬀective Hamiltonian for the CuO2 plane. To deﬁne the parameters of the Hamiltonian, one can use results of the band-structure calculations or quantitative computations for Cun Om clusters as considered below. The reduced Hamiltonian depends on the basis of electronic states used in the model. To take into account the local character of the Coulomb correlations, it is convenient to use the lattice site representation for electronic states. In particular, the atomic wave functions of the Cu3d and O2p states can be used which, however, do not form a basis of orthogonal states in the crystal. Therefore, these functions cannot be used in writing down the eﬀective Hamiltonian in the second quantization representation, which requires an orthogonalized basis of one-electron states. In the standard approach to the derivation of an orthogonalized basis for local states in a crystal, the Wannier functions (linear combinations of the Bloch functions) are often used. But because of their essential delocalization, the Wannier functions are inadequate for the localized 3d-states. In this context, a method of sequential orthogonalization of atomic wave functions is more appropriate. It has been used by Bogoliubov [149] in developing the polar model of metals initially proposed by Shubin and Vonsovsky [1162]. In this approach, the Heitler–London theory of metals is generalized by taking into account charged (polar) atomic states which enables to consider rigorously the strong Coulomb correlations in charged atomic states (e.g., doubly occupied electronic orbitals). The wave function Ψiλ for a localized state λ on the i-th lattice site in this theory is written as a linear combination of

7.1 Electronic Structure of Cuprates

383

the atomic function χiλ and the functions on the nearest lattice sites j: λβ Sij χjβ . (7.2) Ψiλ = χiλ + jβ λβ are chosen from the orthogonality condition Ψiλ | Ψjβ = The coeﬃcients Sij δij δλβ . This method proves eﬀective for small overlaps of functions on neighboring sites, | χiλ | χjλ | 1 when only the nearest neighbors can be taken into account. Using this orthogonalized set of the single-particle states on lattice sites an eﬀective Hamiltonian in the second quantization representation can be derived as, e.g., for 3d(x2 − y 2 ) states on Cu sites and 2pσ (x, y) states on O sites in the CuO2 plane [152]. In a more general approach, one can account for the other states: Cu3d(3z 2 − r2 ), Cu4s, O2pπ (x, y), O2pπ (z) as well. The “ﬁrst principle” calculation of the parameters of the eﬀective Hamiltonian given by the matrix elements of the kinetic energy operator and Coulomb interaction for the wave functions (7.2) is a diﬃcult problem (see [149]). Alternatively, these parameters can be ﬁxed with the aid of the LDA and cluster computations or from comparison with ARPES data. In the simplest approximation, one can consider a one-band model and take into account only the single-electron hopping matrix element t between the nearest neighbors and the largest single-site Coulomb energy U . In this approximation, the complicated resulting polar model Hamiltonian is reduced to the so-called Hubbard model [498]: † H = −t ciσ cjσ + U ni↑ ni↓ , (7.3) i=jσ

i

where c†iσ (ciσ ) are the creation (annihilation) operators for electrons of spin σ = (↑, ↓) at the lattice site i and niσ = c†iσ ciσ is the electron occupation number. The model (7.3) allows the consideration of the cases of both weak correlations, U W , and strong correlations, U W , where W = 2zt is the bandwidth (z is the number of nearest neighbors, z = 4 for a square lattice). In the weak correlation limit a metallic state is observed, while in the strong correlation limit the one-band model in (7.3) splits into two Hubbard subbands. For the half-ﬁlled case (characterizing by an average occupation electron number n = 1) the model describes a Mott–Hubbard insulating state, which becomes an unconventional metal under the hole doping (n < 1) of the lower Hubbard subband (LHB), or under the electron doping (n > 1) of the upper Hubbard subband (UHB). Much attention has been devoted to studies of the Hubbard model (see e.g., [190, 1106]). Despite its simplicity, only in some limiting cases a rigorous solution for the model has been proposed. In particular, the Hubbard model can be exactly solved in the one-dimensional (1D) case [719], and in the limit of inﬁnite dimensions, D → ∞, within the eﬀective dynamical mean-ﬁeld theory (DMFT) (for references see Sect. 7.2.1). A ferromagnetic ground state was found rigorously by Nagaoka [879] for an

384

7 Theoretical Models of High-Tc Superconductivity

almost half-ﬁlled band with few doped electrons in several cubic lattices in the limit of strong correlations (“Nagaoka theorem”). Eﬀective p–d Model Hamiltonian In the theoretical investigation of the many-body eﬀects in copper-oxide superconductors, it is most frequently used a model more complicated than (7.3), namely, the three-band p–d eﬀective Hamiltonian proposed by Emery [307] and Varma et al. [1305]. A more general many-band model was considered by Gaididei et al. [374] (see also [738]). In the p–d model, to take into account the charge-transfer character of the insulating state, the copper and oxygen states in the CuO2 plane are considered explicitly: H=

εi niσ +

iσ

tij c†iσ cjσ +

i=jσ

1 1 Ui niσ ni−σ + Uij niσ njσ . (7.4) 2 iσ 2 i=jσσ

Here, c†iσ (ciσ ) are the creation (annihilation) operators for holes of spin σ at the copper or oxygen sites i of the square CuO2 lattice, niσ = c†iσ ciσ , εi = (εp , εd ) are the energies of the O2pσ (x, y)-states and Cu3d(x2 − y 2 )states, respectively, tij = (±tpd , ±tpp ) are the transfer integrals for p–d and p–p states on nearest Cu–O and O–O sites, respectively; Ui = (Ud , Up ) are the on-site Coulomb repulsion energies for 3d and 2p states, and Uij = (Upd , Upp ) are the intersite Coulomb interactions. The Hamiltonian is written in the hole representation for the vacuum state deﬁned by ﬁlled Cu3d10 and O2p6 states. Figure 7.2 shows Cudx2 −y 2 and bonding Opx , Opy in-plane orbitals in the CuO4 cluster. The energy levels are sketched in the right panel: the local onehole εd state and the two-hole εd + εp and 2 εd + Ud states. The singlet two p-hole state with higher energy 2 εp is not shown. The ligand tpd hybridization results in a repulsion of both the one-hole energy levels and the two-hole + p y − − − −px

+

+

tpp dx2-y2 +

−

tpd

+ − −py

−

+

px

2εd+Ud εd+ εp ε2 εd

Δ p-d ε1

Fig. 7.2. Cu3dx2 −y2 and bonding Opx , Opy orbitals in the CuO4 cluster and copper εd and oxygen εp hole energy levels

7.1 Electronic Structure of Cuprates

385

singlet energy levels as shown in the right part for tpp = 0. Since the Coulomb repulsion Ud shifts up the energy of the d-hole singlet, the tpd hybridization between three singlet states, p–p, d–d and p–d, leads to decrease of the lowest p–d singlet state energy. Because the energy of the p–d triplet state does not change, the singlet and triplet two-hole p–d states are split out. Therefore, the lowest two-hole state becomes the p–d singlet state ε2 – the Zhang–Rice singlet (ZR) [1419]. The excitation energy for transition from the one-hole d-like state ε1 to the two-hole ZR singlet state ε2 determines the charge transfer gap Δpd εp −εd . The p-hole hybridization tpp splits p-levels and the charge-transfer gap reduces to Egap = Δpd − tpp (see e.g., [318]). There are several studies reporting values of the parameters of the Hamiltonian (7.4) by using modiﬁed-LDA band-structure calculations or cluster computations. One of the earliest calculations of the model parameters in (7.4) was performed by McMahan et al. [821] within the local-density-functional studies of the LCO band structure. The Coulomb energies (Ui , Uij ) were found by considering the total electronic energy E(n) dependence on the occupation number n for an isolated orbital. The parameters of the eﬀective Hamiltonian (7.4) obtained within this approach are summarized in the ﬁrst column of Table 7.1. To take into account the strong Coulomb correlations, an Anderson impurity model was proposed for the 3d-states imbedded in the broad O2p band. On the basis of the impurity model, it has been found that the ground state is magnetic and insulating with a correlation gap of the order of 1 − 2 eV. One hole in the ground state is occupied by the antibonding Cu3d–O2p hybrid orbital of the x2 − y 2 symmetry. The ﬁrst ionization state is a two-hole singlet state (with ≈80% of oxygen character) while an electron added state is mostly of the Cu character (≈ 60%). Similar parameters were obtained by Hybertsen et al. [515]. The dependence of the total energy E(ni ) on the local charge states at the Cuand O-sites was calculated microscopically within the constrained densityfunctional theory for a 2×2 supercell in the basal plane of LCO. Comparison of the energy E(ni ) with the mean-ﬁeld solution for the model Hamiltonian (7.4) enabled the determination both the hopping parameters and the Coulomb Table 7.1. Parameters of the eﬀective Hamiltonian (7.4) Parameters (eV)

[1]

[2]

[3]

[4]

[5]

εp − εd tpd tpp Ud Up Upd

– 1.6 0.65 8.5 4.1 − 7.3 0.6 − 1.3

3.6 1.3 0.65 10.5 4 1.2

2.75 2.5 1.0 7.8 – –

3.5 1.3 0.65 8.8 6 1<

3.5 1.3 0.65 9 4 1.3

[1] McMahan et al. [821], [2] Hybertsen et al. [515], [3] Eskes et al. [318], [4] Eskes et al. [317], [5] Feiner et al. [327]

386

7 Theoretical Models of High-Tc Superconductivity

(screened) energy, which are listed in the second column of Table 7.1. Whereas for a free copper ion the Coulomb energy Ud 17 eV, in the lattice it is much smaller, Ud = 8 − 10 eV, due to the screening eﬀects. Extensive studies of the LDA band structure have been used by Andersen et al. [57] for the derivation of a model Hamiltonian of the bi-layer compound YBa2 Cu3 O7 by systematic integrating out the high-energy part of the LDA spectra. Using the LMTO down-folding technique, two tight-binding twodimensional Hamiltonian with eight and four orbitals, respectively, per CuO2 plane were derived. In the eight-orbital Hamiltonian, H8+ and H8− for even and odd bands of the bi-layer, four σ-character orbitals, Cudx2 −y 2 , O(2)px , O(3)py , and Cu4s, and four π-character orbitals, Cu(dxz , dyz ), O(2)pz , O(3)pz , were taken into account (O(2) and O(3) are the in-plane oxygen ions in the YBCO structure, see Fig. 2.12). The four-σ-band Hamiltonian was characterized by the parameters: εs − εd = 6.5 eV, εp − εd = −0.6 eV, tpd = 1.6 eV, and tsp = 2.3 eV. Although the energy of Cu4s orbital is much higher than that of εd and εp levels, it has a strong coupling with O(2)px , O(3)py orbitals and determines the hopping between distant in-plane neighbors. Moreover, the Cu4s orbital provides the coupling between the planes in the bi-layer, t⊥ ∼ 0.25 eV·[1/2(cos bky − cos akx )]2 . The π-orbitals are responsible for the bifurcation of the saddle-points for dimpled planes [56]. The four-band model was further reduced to the three-band model (7.4) by integrating out the high-energy bands. In particular, exclusion of the Cu4s orbital in the four-band model brings about hopping between the oxygen ions, tpp t2sp /(εs − ε) ∼ 1.1 eV and increases the charge transfer gap Δpd = εp − εd ∼ −3 eV where εp = εp − 2tpp . The single-band model obtained numerically by diagonalizing the eight-orbital Hamiltonian was characterized by non-negligible hopping parameters up to nine neighbors, with at least third neighbors giving the essential contribution to the in-plane electron energy dispersion (k) = − 2t (cos kx + cos ky ) + 4t cos kx cos ky − 2t (cos 2kx + cos 2ky ) + · · · ,

(7.5)

where t ∼ 0.35 eV, t /t ∼ 0.27, t /t ∼ 0.18, t /t ∼ 0.05, etc. An interesting correlation between the maximal Tc,max at optimal doping in various cuprate superconductors and the parameters of the model Hamiltonian was found [968]. It was shown that the axial orbitals, a hybrid between Cu4s, Cud3z2 −r2 and apical oxygen Oc (pz ), play an important role providing the in-plane hopping between distant neighbors. Within the four-orbital model, the hopping parameters in (7.5) can be estimated as t ∼ rt, t ∼ t /2 where r = 1/[2(1 + s(ε))] with s(ε) = (εs − ε)(ε − εp )/(2t2sp ). This shows that the range r of the in-plane hopping is determined by the structure and the chemical composition perpendicular to the CuO2 plane: t increases with diminishing εs − εF and increasing tsp hybridization. Calculations of the parameter r ∼ t /t for various single-layer cuprate superconductors have

7.1 Electronic Structure of Cuprates

387

shown that it increases with the distance between Cu and the apex oxygen Oc , which correlates with increasing Tc,max; for instance, t /t ∼ 0.2 for the LSCO compound with Tc,max ∼ 40 K, while t /t ∼ 0.4 for the single-layered Tl- and Hg-compounds with Tc,max ∼ 90 K. These band structure studies of the YBCO compound have proved the limited character of the three-band model (7.4) after the Cu4s orbital integration out. Whereas in conventional LDA band-structure calculations the strong correlation eﬀects are treated in the mean-ﬁeld approximation, studies of electronic spectra for a system of several atoms by the cluster method enables to consider the many-particle correlations rigorously. In particular, the ZR singlet band resulting from a large Ud in (7.4) does not appear in LDA calculations. A certain problem for the cluster method is accounting for surface eﬀects and the eﬀects of long-range Coulomb forces. In copper-oxide compounds of ionic nature, the Madelung potential plays an important role in determining the crystal ﬁeld eﬀects. Electronic spectra in the clusters of CuO6− and CuO10− accounting for 4 6 enlarged set of orbitals for Cu3d states with the symmetry a1 = d3z 2 −r2 , b1 = dx2 −y2 , b2 = dxy , eg = {dxz , dyz }, and the O2p states of the same symmetry have been studied by Eskes et al. [318]. The calculated electronic spectra were compared with photoemission data on CuO compound which permitted to optimize the values of the model parameters, one set of which is given in the third column of Table 7.1. A “standard set” of parameters for the model (7.4) (see fourth column of Table 7.1) was used by Eskes et al. [317] in studies of the eﬀective transfer integrals for singlets in the clusters Cu2 O7 and Cu2 O8 . The eﬀect of the apex oxygen in CuO5 pyramids and in CuO6 octahedrons on the electronic spectrum has been investigated by Ohta et al. [914] for clusters with two copper ions, Cu2 O9 and Cu2 O11 . It was found that the stability of the ZR singlet state for the doped in-plane hole strongly depends on the energy diﬀerence Δ εA = ε(pz ) − ε(pσ ), between the energy levels of the apex ε(pz ) and planar ε(pσ ) oxygen states. If these levels come closer, the hybridization increases between the one-hole states on the apex 2pz and the in-plane 2pσ oxygen ions and the singlet state is destroyed. The energy diﬀerence Δ εA depends on the splitting energy 2t for the ground (A1g ) and excited (B1u ) states of the system of a singlet | Φ(2)S and a Cu 3d hole | Φ(1)σ on the nearest site. Comparison of Tc,max with Δ εA (or 2t) for diﬀerent cuprates shows a certain correlation: superconductivity appears at Δ εA ≥ −1 eV (or t ≥ 0.22 eV), and Tc attains the maximum values for Δ εA 5 eV (or t = 0.3 − 0.35 eV). A much smaller eﬀect on the stability of the singlet is caused by the change of the charge transfer gap Δpd and Δ εd = εd (3z 2 − r2 ) − εd (x2 − y 2 ) for multiplet 3d states. In the LDA calculations discussed above, Tc,max increase with the Cu – apex O distance was explained by enhancing the in-plane hopping parameters t /t due to the lowering of the εs energy. For analytical investigation of the many-body eﬀects, the three-band model (7.4) still looks very complicated. The essential physics of hole-doped and

388

7 Theoretical Models of High-Tc Superconductivity

electron-doped cuprates can be studied within an eﬀective Hubbard model with only two lowest bands, namely the singly occupied d-like band and the doubly occupied ZR singlet band as shown in the right part of Fig. 7.2. To reduce the three-band model (7.4) to a two-band one, the pdσ hybridization should be properly taken into account. As follows from estimates of the model parameters in Table 7.1, the hybridization is very strong in comparison with the charge transfer gap, 2tpd ∼ Δpd , and therefore a direct perturbation theory over tpd /Δpd is unjustiﬁed. A proper reduction in the three-band model (7.4) to the eﬀective Hubbard model can be performed within a cell-cluster perturbation method (see [327, 442, 1397], and references therein). As a ﬁrst step, the symmetric a1 ∝ (px + py − p−x − p−y ) and antisymmetric b1 ∝ (px − py − p−x + p−y ) combinations of Opx , py orbitals around the copper ion (with a proper consideration of the wave function phases) in one unit cell (see Fig. 7.2) are transformed to oxygen Wannier orbitals, which are orthogonal to each other in diﬀerent unit cells, i = j. In the simplest approximation for tpp = 0, the a1 and b1 oxygen bands are decoupled and only two strongly coupled by tpd bands are left: the oxygen band of the b1 symmetry and the dx2 −y 2 band. Neglecting for simplicity the Coulomb interaction terms Up , Upd , the resulting two-band p–d model in the site representation can be written as H= Hi0 + Vij ( d†iσ bjσ + H.c.), (7.6) i

Hi0

=

σ

i=j,σ

[εp npσ + εd ndσ + V0 [(d†iσ biσ + H.c.) ] + Ud nd↑ nd↓

Here, εp , εd are energies and npσ = b†iσ biσ , ndσ = d†iσ diσ – occupation numbers for the bonding p-holes and d-holes, respectively. The hybridization parameters for the d-hole and p-hole in Wannier state is given by Vij = 2tpd λij where the local unit cell hybridization is strong, λ0 = λii 0.96, while the hybridization in the neighbor cells are rather weak, λ1 = −0.14 for the nearest neighbor (n.n.) (i − j) = ±(ax , ay ), λ2 = −0.02 for the next n.n. (i − j) = ±(ax ± ay ) cells, etc. (for detail see e.g., [327, 442]). Now we can ﬁnd the eigenstates of the unit cell Hamiltonian: Hi0 |iα = εα |iα and then express the hopping inter-cell part in (7.6) in terms of these eigenstates |iα. In the limit Ud → ∞, the results for the renormalized one-hole ε1 and singlet two-hole ε2 energies can be written explicitly ε1 = (1/2)(εp + εd ) − (1/2) Δ2pd + 4V02 , (7.7) ε2 = (1/2)[(εp + εd ) + 2εp ] − (1/2) Δ2pd + 8V02 . The pd hybridization does not change the energy of the two-hole triplet state ε2,T = (εp + εd ) that results in a splitting between triplet and singlet state: ε2,T − ε2 8t2pd /Δpd in the limit tpd Δpd [1419].

7.1 Electronic Structure of Cuprates

389

It is convenient further to introduce the Hubbard operators [501] Xiαβ = |iαiβ| acting in the space of the four eigenstates |α: the vacuum state |0, one-hole state |σ with spin σ = (↑, ↓) ≡ (±1) and doubly occupied singlet state |2 = | ↑↓. The eﬀective two-subband model in terms of the Hubbard operators reads [1007]: Xiσσ + (ε2 − 2μ) Xi22 H = (ε1 − μ) +

i,σ σ0 0σ t11 ij Xi Xj

i

+

2σ σ2 t22 ij Xi Xj

2¯ σ 0σ + σt12 ij (Xi Xj + H.c.) , (7.8)

i=j,σ

where σ ¯ = −σ and hopping integrals tαβ ij = Vij Kαβ for two Hubbard subbands, the singly occupied, α = 1, and the doubly occupied, α = 2. The coeﬃcients Kαβ depend on the hybridization parameter tpd /Δpd . We introduce also the chemical potential μ, which depends on the average number of holes n = 1 + δ = Ni , Ni = Xiσσ + 2Xi22 , (7.9) σ

where . . . denotes the statistical average. The Hubbard operators obey the completeness relation Xiσσ + Xi22 = 1, (7.10) Xi00 + σ

which rigorously preserves the local constraint of no double occupancy of any quantum state |α at each lattice site i. Thus, in the eﬀective p–d Hubbard model (7.8) only two lowest energy states are taken into account, the one-hole d-like state ε1 and two-hole singlet p–d-like state ε2 . In this respect, the model (7.8) is equivalent to the one-band Hubbard model (7.3) where the Coulomb repulsion is replaced by the charge transfer gap Ueﬀ = Δpd = ε2 −ε1 . Although the energy Δpd = 3−4 eV is much lower then the original Coulomb energy Ud ∼ 8 eV, the eﬀective hoping parameters are also reduced in comparison with the original hybridization parameter tpd , e.g., for the nearest neighbors |tαβ ij | ∼ 2tpd λ1 ∼ 0.4 eV, which is close to the LDA results in (7.5). Since the eﬀective two-dimensional bandwidth W ∼ 8|tαβ ij | < Ueﬀ , the model (7.8) describes a strongly correlated electronic system, and a perturbation expansion over |tαβ ij |/Δpd can be justiﬁed. A detailed investigation in the reduction of the three-band p–d model (7.4) to the two-subband Hubbard model (7.8) was given by Feiner et al. [327] and Yushankhai et al. [1397]. It has been found that the model is robust to the variation of parameters tpp , Ud and Upd , Up in a wide range which only results in small variation of the hopping parameters tαβ ij and their asymmetry for oneand two-hole bands. A “standard” set of parameters for the three-band p–d model (7.4) used by Feiner et al. [327] is given in the last column of Table 7.1. A more general ﬁve-band p–d model, allowing for the apical oxygen pz orbital

390

7 Theoretical Models of High-Tc Superconductivity

and the out-of-plane d3z2 −r2 copper orbital, was considered by Raimondi et al. [1040]. It was found that hybridization of the oxygen pz and in plane px , py orbitals is detrimental to the ZR singlet and may strongly renormalize the hopping parameters t for the second neighbors in the singlet subband as discussed above. To conclude, the eﬀective p–d Hubbard model (7.8) in which essential electron interactions in CuO2 plane are taken into account may be considered as an appropriate model for the investigation of many-body eﬀects in cuprates. The model parameters can be ﬁtted from experiments for each particular compound. One-Band t–J Model In the limit of a strong Coulomb repulsion, U t, when the one-band in the Hubbard model (7.3) splits into two subbands with a correlation gap Ecr ∼ (U −W ) one can consider explicitly only the subband, which crosses the Fermi level. The other subband can be taken into account by a perturbation theory over t/U 1. This results in a further reduction in the two-subband model (7.3) to a one-subband t–J model. The important role of a strong Coulomb repulsion in superconducting cuprates was ﬁrst pointed out by Anderson [61] who proposed to study superconductivity in La2 CuO4 compounds within the t–J model. The model has been derived earlier by a number of researchers (for review see e.g., [543, 912]). A simple derivation of the t–J model is based on writing down the original Fermi operators in (7.3) as the linear combination of the Hubbard operators c†iσ = Xiσ0 + σXi2¯σ ,

Xiσ0 = c†iσ (1 − ni¯σ ),

ciσ = Xi0σ + σXiσ¯ 2 , Xi2¯σ = σ c†iσ ni¯σ .

(7.11)

Here the sign of the spin σ = ±1 allows for the order of spins in the doubly occupied state: |2 = | ↑, ↓ ≡ |+, −. Using the representation (7.11), the model (7.3) can be written in the form of the two-subband model (7.8) with equal hopping integrals: tαβ ij = −t and ε2 = 2ε1 + U where ε1 is a singleparticle energy level. Therefore, we consider below the derivation of the t–J model for the more general two-subband Hamiltonian (7.8) which we write in the form 2¯ σ 0σ σ0 σ ¯2 σ t12 (7.12) H = H0 + H1 + V, V = ij (Xi Xj + Xj Xi ). i=j,σ

Here H0 is the single-site terms and H1 is the hopping term for the singly 22 (∝ t11 ij ) and doubly (∝ tij ) occupied subbands in (7.8). By considering the hybridization term V ≡ T21 + T12 in (7.12) as a small perturbation in comparison with the energy gap ε2 − 2ε1 , we can exclude the hybridization by the

7.1 Electronic Structure of Cuprates

391

canonical transformation: ˜ = eS He−S = H0 + H1 + V + [S, H0 ] + [S, V ] + 1 [S, [S, H0 ] ] + · · · , (7.13) H 2 where the operator of the transformation S = λ(T21 −T12 ) = −S + . To exclude the linear inter-subband terms, we ﬁx the parameter λ from the condition V + [S, H0 ] = 0 or T21 = λ[H0 , T21 ] = λ(ε2 − 2ε1 )T21 , which gives λ = (ε2 − 2ε1 )−1 . Therefore, in the second order of the inter-subband hopping, the transformed Hamiltonian (7.13) takes the form ˜ = H0 + H1 + HJ , H

HJ = λ [T21 , T12 ].

(7.14)

Direct calculations of the commutator [T21 , T12 ] shows that the exchange term 21 is proportional to the three-site interaction, HJ ∝ t12 ij tjn . By taking into account only the two-site contribution, n = i, we obtain for the exchange term the following representation: (2)

HJ =

1 Jij (Xiσ¯σ Xjσ¯ σ − Xiσσ Xjσ¯ σ¯ + Xi20 Xj02 + Xi22 Xj00 ), 4

(7.15)

i=j,σ

with the exchange interaction for the Hubbard model (7.3) Jij = 4t2 /U and 2 for the eﬀective p–d model (7.8) Jij = 4(t12 ij ) /Δpd . By considering only one subband, e.g., the singly occupied one and projecting out the doubly occupied states |2 the resulting t–J model can be written in the conventional form 1 1 † tij cˆiσ cˆjσ + Jij Si Sj − ni nj . (7.16) Ht−J = 2 4 i=j,σ

i=j

Here in the hopping term, instead of the Hubbard operators (7.11), we introduced the projected fermion operators Xiσ0 = c†iσ (1−ni¯σ ) ≡ cˆ†iσ . The exchange term (7.15) is

written in terms of the spin Siα operators and the number operator ni = σ ni,σ which for the singly occupied subband read: σ Xiσσ , ni,σ = Xiσσ = cˆ†iσ cˆiσ . (7.17) Siσ = Xiσ¯σ , Siz = (1/2) σ

Sometimes, the summation in the exchange term in (7.16) is performed only over pairs of the nearest neighbors denoted as ij or i < j. In that case the factor (1/2) before the sum over i, j should be dropped. Originally, the t–J model for the CuO2 layer was derived by Zhang et al. [1419] directly from the three-band p–d model (7.4). It was found that the energy diﬀerence between the singlet and triplet two-hole p–d states in a cluster shown in Fig. 7.2 is much larger than the hopping integral and therefore only the lowest singlet state of a doped oxygen hole can be taken into account. It was further observed that the hopping of a singlet from site j to i simultaneously leads to the hopping of a Cu hole in the opposite direction, i.e.,

392

7 Theoretical Models of High-Tc Superconductivity

from the site i to j. Therefore, one can consider either motion of singlets as 2σ σ2 given by the term t22 or motion of d-holes described by the term ij Xi Xj 11 σ0 0σ tij Xi Xj in the general model (7.14) where the vacuum state |0 denoting an empty hole state is considered as the singlet state. The exchange term (7.15) or those one in (7.16) describes the interaction between the Cu spins. In this way, it was shown that the low-energy physics of copper-oxygen system can be described by the single-band t–J Hamiltonian as in the case of the Hubbard model considered by Anderson [61]. In the derivation of the t–J model from the p–d Hamiltonian, it is important to start from the local basis for the oxygen states in the cluster CuO4 (Fig. 7.2) with a proper account of their phase coherence as in the cellperturbation theory. If one considers the p–d hybridization in the CuO2 plane for the Bloch oxygen wave functions, the singlet-triplet splitting for the twohole p–d states does not appear since in the k-space the p–d hybridization disappears at k = 0. Parameters of the t–J model were evaluated by many researches both from ﬁnite cluster computations and from reduction in the p–d model to a oneband eﬀective Hamiltonian. We present several parameter sets in Table 7.2 in the hole notation for a single-particle dispersion both for the hole- and electron-doped cases (signs of t, t should be reversed in the latter case for the electron notation as given in (7.5)). For the undoped case, the exchange interaction Jnn between two nearest-neighbor d-holes can be estimated within the perturbation theory for the p–d model by the expression [317]: Jnn = 4

t4pd Δ2pd

1 1 + , Ud Δpd + Up /2

(7.18)

which gives for the standard parameter set (see Table 7.2) Jnn ∼ 0.25 eV. Computations for the hole-doped Cu2 O7 and Cu2 O8 clusters by Eskes et al. [317] for a standard set of the p–d parameters (for Upd = 0 in Table 7.1) resulted in the hopping parameters shown in the ﬁrst column of Table 7.2. It appears that Jnn is reduced by ﬁnite values of Upd and larger values of Ud and Up as follows from (7.18). Results of computations for large clusters Table 7.2. Parameters of the t–t –J model (7.16) (hole notation) Parameters (eV)

[1]

[2]

[3]

[4]

thh thh tee tee Jnn

0.44 −0.17 0.40 −0.10 0.24

0.44 −0.06 0.41 −0.07 0.13

0.22 −0.12 0.30 −0.06 0.15

0.5 −0.06 0.32 −0.01 –

[5]

[6]

0.5 −0.04 – – 0.23

– – 0.35 −0.1 –

[1] Eskes et al. [317], [2] Hybertsen et al. [516], [3] Tohyama et al. [1246], [4] Feiner et al. [327], [5] Yushankhai et al. [1397], [6] Andersen et al. [57]

7.2 Electron Excitations in the Normal State

393

of Cun Om , up to n = 5, m = 16, by Hybertsen et al. [516] are presented in the second column of Table 7.2. The clusters Cu2 O11 and Cu2 O12 including the hybridization between the Cud3z2 −r2 and the apical oxygen Opz orbital but neglecting the Cu4s one were studied by Tohyama et al. [1246]. Their parameters of the t–J model are given in the third column of Table 7.2. They found a negligibly small next-nearest neighbor exchange interaction Jnnn ∼ 0.07Jnn , in agreement with other studies. Parameters obtained within the cell-perturbation theory for CuO2 plane are given in the fourth [327] and the ﬁfth [1397] columns of Table 7.2. Generally, the values of the t–J model parameters in various calculations shown in Table 7.2 are similar, though the t parameter varies substantially. As pointed out in the LDA band-structure calculations, the hopping matrix elements for distant neighbors are determined by the hybridization of the Cu4s orbital with the oxygen in-plane orbitals. Therefore, disregarding this contribution may strongly underestimate the hopping parameter t (and to more distant neighbors, see (7.4)) as follows from the comparison of the LDA results given in the sixth column of Table 7.2 with other data in the table. In conclusion, we can argue that the model Hamiltonians for the CuO2 plane considered above, either the three-band p–d model (7.4) with parameters speciﬁed in Table 7.1 or the reduced two-band eﬀective Hubbard model (7.8), contains enough physics to explain the many body-eﬀects in cuprate superconductors caused by strong electron correlations. To study low-energy physics close to the Fermi level, a simpliﬁed eﬀective one-band t–t –J model with parameters suggested in Table 7.1 may be used. However, for a quantitative description of the electronic spectra in cuprates more general multi-orbital models should be considered with proper allowance made for strong electron correlations.

7.2 Electron Excitations in the Normal State 7.2.1 Single-Particle Electron Spectrum The spectrum of the single-particle electron excitations is determined by the Green function (GF), which can be written in a conventional form as Gσ (k, ω) = ckσ |c†kσ ω =

1 , ω − σ (k) − Σσ (k, ω)

(7.19)

where we used the notation for the retarded GF (see Appendix A.1) for the creation c†kσ and annihilation ckσ operators for an electron with the wave vector k and spin σ. Here, the band electron energy σ (k) is taken as a zero-order approximation, while the many-body eﬀects on the electron excitations are described by the self-energy operator Σσ (k, ω). As discussed in Sect. 5.2.2, the single-particle spectral function A(k, ω) (5.14) directly measured in ARPES experiments is determined by the imaginary part of the GF (7.19). In the

394

7 Theoretical Models of High-Tc Superconductivity

vicinity of the FS (ω → 0) within the quasiparticle (QP) approximation, the spectral function takes the simple form (5.15) AQP (k, ω) Zk

γ(k, ω)/π , [ω − η(k)]2 + [γ(k, ω)]2

(7.20)

where the renormalized QP energy η(k) = Zk [ (k)+Σ (k, 0)] and the scattering rate γ(k, ω) = Zk Γ (k, ω) are determined by the real and imaginary parts of the self-energy, respectively: Σ (k, ω) = Re Σ(k, ω), Γ (k, ω) = −Im Σ(k, ω + i0+ ), Γ (k, ω) > 0 . The QP weight Zk = (1 + λk )−1 depends on the eﬀective coupling constant λk = −(∂Σ /∂ω)|ω=0 and is related to the jump of the momentum distribution function at the FS at zero temperature Z(kF ) = [n(kF − δ) − n(kF + δ)]δ→0 .

(7.21)

Coherent propagation of QP is possible only for a ﬁnite value of Z(k). For the Fermi liquid model at T → 0, we have −ImΣ(kF , ω) ∝ ω 2 and ReΣ(k, ω) ∝ ω, which gives a ﬁnite value to the QP weight Z(kF ). However, in certain cases, as in the one-dimensional (1D) interacting Fermi systems, the quasiparticles are unstable and gapless elementary excitations occur as bosonic collective charge and spin ﬂuctuations (for reviews see [273, 1308, 1318]). Their diﬀerent velocities entails the spin-charge separation. The QP spectral weight vanishes at the Fermi points ±kF : Zk ∝ |k − kF |α , while the momentum distribution function is continuous: n(k) = n(kF ) − C|k − kF |α sgn(k − kF ), though its slope diverges for k → kF : ∂n(k)/∂k → ∞ where α < 1 is an interaction dependent exponent. The single-electron Green function has a branch cut at T = 0 instead of poles. This peculiar properties of the 1D Fermi liquid have been rigorously proved for the Luttinger model, which appears to be closely related to the 1D Hubbard model. An exact solution of the model via Bethe ansatz has been obtained by Lieb and Wu [719] who found the phase diagram and thermodynamic properties. At halfﬁlling, n = 1, the 1D Hubbard model is a Mott insulator at any ﬁnite value of the Coulomb repulsion U > 0, while away from half-ﬁlling the model reveals metallic properties the low-energy asymptotic behavior of which is described by the Luttinger model. Thus, the Luttinger, or Tomonaga–Luttinger, model presents the universality class of the gapless 1D quantum systems and it is usually called the Luttinger liquid. The QP weight is characterized by the degree of an overlap of the oneparticle state c†kσ Ψ0 (N ) created by adding a particle to the N -particle system and the true many-particle state Ψkσ (N + 1) in the N + 1-particle system Z ∝ c†kσ Ψ0 (N ) | Ψkσ (N + 1). The unique properties of the 1D system are related to the nontrivial dependence of the phase shift caused by adding or removing a single particle in the system. In these processes, a phase shift of the wave function of the whole

7.2 Electron Excitations in the Normal State

395

system Ψkσ (N + 1) occurs (unrenormalizable Fermi-surface phase shift), while in the 3D or higher dimensions the phase shift can be compensated by a renormalization of all the QP energies [62]. Therefore, in 1D systems the overlap of these states vanishes, the QP Z → 0 contrary to systems of higher dimensions with a ﬁnite Z. As suggested by Anderson [62], electrons in the two-dimensional Hubbard model with strong interaction may behave like a Luttinger liquid with a separation of spin and charge degrees of freedom. An intermediate behavior between the Landau Fermi liquid and the Luttinger liquid was proposed by Varma et al. [1306] within the “marginal” Fermi liquid (MFL) model. The model was formulated for explanation of anomalous normal state properties of cuprate superconductors. The theory is based on a single hypothesis that there exist excitations over a wide range of the wavenumber q depending only on the ratio ω/T . They determine the spin and charge polarizability in the form: N (0) (ω/T ) for |ω < T | , (7.22) − ImP (q, ω) ∝ N (0) sign ω for |ω > T | where N (0) is the one-particle density of states at the FS. The interaction of the electrons with these excitations leads to the self-energy Σσ (ω) = g [ω ln(x/ωc ] − i(π/2) x signω),

(7.23)

where g is a coupling constant, ωc is a cutoﬀ energy and x = max(| ω |, T ). For the QP weight, in this case we obtain 1/Z = 1 − (∂Σ /∂ω) = 1 + g ln(y/ωc ),

(7.24)

where y = (max(| Ek − μ |, T ). Near the FS, Ek → μ, and for T → 0 the QP weight vanishes logarithmically, violating the QP picture in the conventional Fermi liquid only marginally. Spectral function in the MFL has a well pronounced maximum at the Fermi energy, which can be considered as the FS observed in ARPES experiment. To develop a microscopic theory of the electronic spectrum in cuprate materials, one should take into account the many-body eﬀects caused by various types of electron interactions. Whereas electron–phonon interaction can be treated by the conventional diagram technique or within a more sophisticated polaron theory which will be discussed in Sect. 7.4, the strong electron–electron correlations comprise a much more intricate problem. Generally, there are two diﬀerent approaches for the treatment of the many-body electron interactions. In the weak-correlation limit one starts from a Fermi-liquid model with strong interaction with the bosonic modes, the spin or charge ﬂuctuations. In this approach, anomalous normal state properties in the underdoped region and, in particular, the pseudogap state, are explained by closeness to some kind of a continuous quantum phase transition, a quantum critical point

396

7 Theoretical Models of High-Tc Superconductivity

(QCP) at zero temperature, which is “hidden” near the optimal doping below the superconducting dome Tc (δ). At ﬁnite temperatures above the QCP, a non-Fermi-liquid behavior should be observed as in the pseudogap region (for review see [1077, 1078, 1322]). In particular, the critical behavior near the AF QCP was considered in a nearly antiferromagnetic Fermi liquid phenomenological model developed by Pines and coworkers (see Sect. 3.3.2). In a more general spin-fermion model both the critical behavior of the spin susceptibility and the single-particle electron spectrum were studied self-consistently (for reviews, see [2,316,900]). A quantum critical behavior due to the charge-density wave instability has been also proposed [209]. We discuss the weak-coupling models in more detail in relation to the superconducting pairing in Sects. 7.3.2 and 7.5.2. In the strong-correlation limit, the theory starts, similar to the Hubbard model (7.3), with a model of doped Mott insulator with a strong Coulomb repulsion U . In this case, the simple one-particle approximation for interacting electrons fails. This demands new approaches beyond the Fermi-liquid description. Among them one should mention extensive numerical studies of ﬁnite clusters by various methods: quantum Monte Carlo, exact diagonalization, the renormalization group method, etc. (see [190,261,550,769,1106]). Even though these unbiased methods treat the strong correlations rigorously and can give a general picture of the excitation spectrum within a considered model, the results of these studies are diﬃcult to apply to real systems due to poor energy and momentum resolutions, ﬁnite size eﬀects, and other drawbacks as the sign problem in the QMC method. At the same time, unbiased numerical methods allow one to estimate the accuracy of analytical calculations. Various analytical methods, like the dynamical mean ﬁeld theory (DMFT) and its generalization based on the cluster approach, variational methods will be discussed later, in relation to studies of the Hubbard model. To capture a new physics in the strongly correlated electrons, Anderson [61] has proposed an unconventional, resonating valence bond (RVB) ground state. A spin-charge separation in this state was suggested in which electrons are considered as composite quasiparticles determined by the convolution of a fermion with spin 1/2 without charge (spinon) and a spinless charged boson (holon). This model is usually treated within a slave boson (or a slave fermion) formulation of the Hubbard or the t–J model (for a detailed review and references, see [700, 912]). We discuss the slave-particle representation and the variation approach in the RVB theory in more detail in Sect. 7.3.1. A more general concept of a “hidden Fermi liquid” for HTSC cuprates has been proposed by Anderson [70]. It was argued that the non-Fermi liquid behavior emerging in cuprates is due to strong Hubbard correlations which cannot be considered as the perturbative continuation from a Fermi system of free electrons. The quasiparticle states in the hidden Fermi liquid can be coupled to the conventional Fermi liquid by the full Gutzwiller projector which eliminates all doubly occupied states. We consider this variational-based approach in more detail in Sect. 7.3.1. Alternatively, to describe quasiparticles

7.2 Electron Excitations in the Normal State

397

in the two Hubbard subbands, one should use projected electron operators, which are not single-particle operators, as discussed below. An exact representation for the composite quasiparticles as projected fermion-like excitations is given by the Hubbard operators (see (7.8)). There are several techniques which are using the original Hubbard operator kinematics. Among them, there are a diagram technique for Hubbard operators (see [539]) and the equation of motion method for the Green functions (see Appendix). The latter method will be mostly used below in the discussion of several representative results for a single-particle electronic spectrum in the normal state calculated within the basic models of strongly correlated electrons. Spin-Polaron Model We start with the simplest problem of the spectrum of a hole moving on in antiferromagnetic (AF) background. The problem has a long history and has attracted attention of many researches (for references see [102, 103, 225, 543, 877]). A major role of spin correlations for a hole in AF insulators was ﬁrst recognized by Bulaevskii et al. [186]. It was shown that a hole moving in a two-dimensional lattice with AF spin ordering (N´eel state) generates a chain of ﬂipped spins, a string, the creation of which costs an energy proportional to the length of the path. As a result, the hole is trapped in a potential with a linear dependence on the distance. The spectrum of excitations of this quasioscillatory state turns out to be discrete. The contribution of the exchange interaction J⊥ to the transverse spin components leads to string relaxation, which makes possible a coherent motion of the hole. The ﬁrst evaluation of its eﬀective mass was given by Nagaev [876]. The spectrum of incoherent excitations of the hole in the limit J → 0 was obtained by Brinkman et al. [177]. Later, this purely theoretical problem appeared to be relevant in studies of ARPES spectra in the AF undoped Sr2 CuO2 Cl2 (SCOC) and weakly doped Ca2−x Nax CuO2 Cl2 (Na–CCOC) compounds (see Sect. 5.2.2, Fig. 5.20). Let us consider the motion of a hole within the t–J model (7.16). Generally, two types of magnetic polarons can be considered. In the limit of the small exchange interaction t J, a ferromagnetic ordering of the spins around the hole is favorable since it decreases the energy of the system by increasing the kinetic energy of the hole. This type of ferromagnetic polaron, Nagaev’s “ferron” [877], is realized in the Nagaoka limit for t ∼ 50J [1357]. However, if the AF exchange interaction J is comparable with the hoping energy t, then the ferromagnetic spin ordering leads to a large increase of the magnetic energy and the string-type polaron considered above is realized. In cuprate superconductors, the AF exchange interaction is extremely large, J ∼ 0.3t and a polaron dressed by AF spin ﬂuctuations is energetically more favorable than a ferron. The motion of a hole in the Heisenberg antiferromagnet (HAF) can be described in the simplest form within a spin-polaron model derived from the

398

7 Theoretical Models of High-Tc Superconductivity

t–J model (7.16) [584, 1122]. In the model, the two-sublattice representation for the HAF with spin up (i ∈↑) and spin down (i ∈↓) is introduced and hole spinless fermion operators for two AF sublattices are deﬁned: + ˜i↓ = h+ c˜i↑ = h+ i , c i Si (i ∈↑);

c˜i↓ = fi+ , c˜i↑ = fi+Si− (i ∈↓)

(7.25)

where Si+ , Si− are spin operators on the corresponding sublattices. In the linear spin-wave approximation (LSWA), the exchange part of the t–J model (7.16) can be written as (see e.g., [736, 801]): HJ =

q

+ J ωq (α+ q αq + βq βq ) + E0 ,

(7.26)

where E0J is the ground-state energy. The magnon creation (annihilation) + operators α+ q (αq ) and βq (βq ) are coupled to the spin-lowering operators on + two sublattices Si ai , (i ∈↑), Si+ b+ i , (i ∈↓) by the Bogoliubov canonical + transformation: ak = vk αk + uk β−k , bk = vk βk + uk α+ u = [(1 + −k , with

k 1/2 1/2 , vk = −sign(γk )[(1 − νk )/(2νk )] . Here, νk = 1 − γk2 , γk = νk )/(2νk )] (1/2)(cos akx + cos aky ). The spin-wave energy is given by ωk = SzJ(1 − δ)2νk where δ is the hole concentration and z = 4 is the number of the nearest neighbors. In the two-sublattice representation (7.25) for holes and within the LSWA for the spin operators, the hopping part of the model (7.16) becomes + (h+ Ht = k fk−q [g(k, q)αq + g(q − k, q)β−q ] + H.c.) k,q

+

( k − μ)(h†k hk + fk† fk ),

(7.27)

k

where g(k, q) = (zt/ N/2)(uq γk−q + vq γk ) is the spin–hole interaction deﬁned by the nearest neighbor (n.n.) hopping parameter t > 0. Here, we consider the t–t –J model where the next n. n. hopping parameter t determines the kinetic energy of a hole hopping on one sublattice: k = 4t cos akx cos aky . The summation over wave vectors in (7.26), (7.27) and below is restricted to N/2 points in the AF Brillouin zone (BZ). To consider the model at ﬁnite hole concentration δ, we introduce the chemical potential μ which is determined by an averaged hole concentration at temperature T from the equation: + δ = h+ i hi + fi fi . To study the hole spectrum and pairing induced by the spin–hole interaction, we calculate the single-hole thermodynamic Green function (GF) G(k, t − t ) = Ψk (t)|Ψk† (t ) by using the equation of motion method. The matrix GF for the two sublattices in terms of the Nambu operators Ψk and Ψk† in ω-representation is determined as (see Appendix A.2) G(k, ω) =

Ψk |Ψk† ω

=

h†k |hk ω

f−k |hk ω

† h†k |f−k ω

† f−k |f−k ω

.

(7.28)

7.2 Electron Excitations in the Normal State

399

As described in Appendix A2.1, we can obtain the following Dyson equation for the GF (7.28) [1009]: {G(k, ω)}−1 = ωˆ τ0 + ( k − μ)ˆ τ3 − Σ(k, ω),

(7.29)

where the self-energy operator is given by the multi-particle GF as deﬁned in (A.25). Using the noncrossing diagram approximation (NCA) or the selfconsistent Born approximation (SCBA) for the spin-hole interaction as in (A.26), we derive the following equations for the self-energy matrix components in the Matsubara frequencies ωn = πT (2n + 1) representation Σhh (k, iωn ) = −T Ghh (q, iωm )λ11 (k, k − q | iωn − iωm ), q,m

Σhf (k, iωn ) = −T

Ghf (q, iωm )λ12 (k, k − q | iωn − iωm ).

(7.30)

q,m

The interaction functions are λ11 (k, q | iων ) = g 2 (k, q)D(q, −iων ) + g 2 (q − k, q)D(−q, iων ), λ12 (k, q | iων ) = g(k, q)g(q − k, q){D(q, −iων ) + D(−q, iων )}. (7.31) (0) Here, the diagonal magnon GF D(q, ω) = αq | α+ (q, ω) = (ω − q ω ≈ D −1 ωq ) . The renormalization of the magnon spectrum ωq caused by the spin– hole interaction is given by the polarization operator Πα,β (q, ω), which can be calculated in NCA as described by Plakida et al. [1005,1009]. At a low hole concentration, the renormalization is small, ∼ δ, and can be neglected. Let us ﬁrst discuss the spin-polaron spectrum in the normal state. It is interesting to consider the solution of (7.29), (7.30) in special limiting cases. By neglecting the kinetic energy due to the next n.n. hopping, k = 0, the + equations for the GF G(k, ω) = h+ k | hk ω = fk | fk ω in (7.29) and the self-energy Σhh (k, ω) (7.30) at zero temperature can be written as: −1 g 2 (k, q) G(k − q, ω − ωq ) (7.32) G(k, ω) = ω − q

The self-consistent equation (7.32) for a hole GF was originally derived by Schmitt–Rink et al. [1122] and numerically solved in one-dimension. The twodimensional case has been extensively studied analytically by Kane et al. [584] in the dominant-pole approximation. Later on, the hole spectrum within (7.32) was investigated by a number of researchers (see e.g., [736, 795, 801]). A particularly simple solution can be obtained for the Ising model, when the transverse spin interaction is absent, J⊥ = 0. In this case, the spin excitation energy does not depend on the wave vector, ω0 = 2Jz , the k-independent self-energy is obtained Σ(ω) =

zt2 . ω − ω0 − Σ(ω − ω0 )

(7.33)

400

7 Theoretical Models of High-Tc Superconductivity

The iterative solution of (7.33) leads to a discrete spectrum with the distance between the levels of the order t(J/t)2/3 obtained by Bulaevskii et al. [186]. In the model with zero exchange interaction, Jz = 0, the solution of (7.33) results in the spectrum E(ω) = (1/2)(ω + (ω 2 − 4zt2 )1/2 ) and the GF√(7.32) locates the spectrum of incoherent excitations inside the band ±4t z. The spectrum of the incoherent excitations obtained by Brinkman et al. [177] within the retraceable-path approximation has a narrower band, ±4t(z − 1)1/2 . To obtain this spectrum in the spin-polaron model, certain conditions excluding unphysical states should be taken into account [801]. The self-consistent solution of (7.32) has shown that the spectrum consists of the quasiparticle (QP) peak near the bottom of the band at ω = −zt and the incoherent spectrum at higher energies up to ω ∼ zt (see e.g., [801]). The QP spectrum has an absolute minimum (for the hole dispersion) at the (±π/2, ±π/2) points (“hole pockets”) in the BZ and can be approximated by the function E(k) −ε0 + t2 (cos kx + cos ky )2 + t3 (cos 2kx + cos 2ky ) with t2 ∼ J/2, t3 ∼ 0.1J . Such a spectrum corresponds to the tight-binding approximation with the transfer integrals t2 , t3 between the next n.n., i.e., the hole motion occurs on one sublattice of the N´eel AF. The eﬀective QP masses in parallel m ∝ t−1 and perpendicular m⊥ ∝ (t2 + t3 )−1 directions 3 to the AF BZ at the (π/2, π/2) point have a large anisotropy, m /m⊥ ∼6 for J = 0.4. It was found that the imaginary part of the self-energy for QPs with wave vectors (π/2, π/2) is zero and shows narrow oscillations only above the QP energy. Therefore, the spin-polarons in this region are well-deﬁned QPs coherently propagate with an inﬁnite life-time. The QP weight Z(k) ∝ J α , α ∼ 0.7 for J < t and the bandwidth of the QP states is of the order of the exchange energy, W ∼ 1.5 J 0.8 . It is remarkable that the spectral function calculated within the NCA from the self-consistent equation (7.32) is in close agreement with the results of the exact diagonalization for the original t–J model [801]. This shows that the NCA is a reliable approximation for the self-energy and the vertex corrections neglected in the NCA are not essential. Indeed, consideration of the vertex corrections by Liu et al. [736] has revealed that the leading two-loop crossing diagram vanishes identically, while the next three-loop crossing diagram gives a small contribution to the self-energy. In the region of small J/t values, string excitations above the QP peaks, characteristic to the Ising model, were found [736, 784]. It is interesting to calculate the QP spectrum at ﬁnite temperatures and hole concentrations which allows one to study the Fermi surface (FS) and the momentum distribution of doped holes. Below we present results of the numerical solution of the system of equations for the normal GF Ghh (k, ω) (7.29) and the self-energy Σhh (k, ω) (7.30) [1005, 1009]. The hole spectral function A(k, ω) and the density of states A(ω) 1 A(k, ω) = − Im hk | h+ k ω+i , π

A(ω) =

1 A(k, ω) N k

(7.34)

7.2 Electron Excitations in the Normal State

401

Fig. 7.3. The hole density of states in the spin-polaron model (left panel) and the QP dispersion curves (right panel) [1009]

were calculated for hole concentrations 0.03 ≤ δ ≤ 0.3 for J = 0.4 and t = 0, ±0.1 (all energies are measured in units of t). Figure 7.3 (left panel) shows the hole density of states A(ω) with renormalized (solid line) and unrenormalized (dashed line) magnon spectra for δ = 0.06 which are very close to each other that demonstrates a small eﬀect of the magnon renormalization. The QP peak is clearly seen close to energy ω = 0 (measured from the chemical potential μ), while string-like multispin excitations give contribution for higher energy up to ω ∼ 7t. At low hole concentration, a weak incoherent background below the chemical potential at ω < 0 is observed the intensity of which is determined by the hole concentration [1005]. The QP energy dispersion deﬁned as E(k, 0) = (k) + Re Σ(k, 0) is shown in Fig. 7.3 (right panel) along the symmetry directions Γ (00) → X(π, 0) → M (π, π) → S(π/2, π/2) → Γ (00) in the BZ for various hole concentrations at t = −0.1t. The shape of the QP spectrum does not change much with the doping but the band width increases substantially. The rigid band approximation for the QP dispersion cannot be therefore used. For t = 0 or t = −0.1, the minimum of the dispersion curves is at the points (±π/2, ±π/2) in the BZ that results in a 4-pocket like form of the FS at low hole concentration. With increasing hole concentration, a transition from the 4-pocket like FS to a large one occurs quite sharply. However, for t = +0.1 the minimum of the dispersion curves shifts to points of the type (0, ±π) at the BZ boundary similar to the electron doping case. The corresponding FS appears to be large even at small concentration of doped holes. The temperature dependence of the momentum distribution for holes N (k) = 2h+ k hk has shown that the FS in the form of four pockets at low temperatures is washed out at higher temperature of the order of Td 0.6 δ t which is much larger than the temperature T ≈ 0.01t considered in the numerical calculations of the QP dispersion [1005].

402

7 Theoretical Models of High-Tc Superconductivity

The general picture of the hole spectrum obtained within the framework of the polaron model (7.26), (7.27) was conﬁrmed by numerical calculations for the t–J model by the exact diagonalization and the quantum Monte Carlo methods for ﬁnite clusters (for review, see [261, 550, 769]). It was shown that at small excitation energies h ¯ ω ≤ J, the hole can move in a coherent way since the presence of quantum spin ﬂuctuations in the ground AF state and of dynamic ﬂuctuations due to the interaction of the transverse spin components, J⊥ , permit preservation of a “polaron dressing” of the hole QP. In this case, the hole QP dispersion can be ﬁtted by the function E(k) −ε0 + t2 cos kx cos ky + t3 (cos 2kx + cos 2ky ) , which describes the hole hopping in one sublattice over the next n.n. in the bandwidth of the order 2J [262]. At large energies, h ¯ ω ∼ t J, the hole loses this spin deformation cloud and the spectrum becomes incoherent in a broad band W ∼ 7t. The QP spectrum found in the t–J model at moderate hole doping, δ ≥ 0.1, changes dramatically. Calculations of the single-particle GF by means of diagonalization technique have shown that the QP dispersion can be approximated by a function E(k) = ε0 − teﬀ (cos kx + cos ky ), which corresponds to a n. n. tight-binding dispersion [486]. However, the eﬀective hopping parameter is still proportional to the exchange interaction, teﬀ ∼ J, which gives the total bandwidth of the order of 4J. In this case, a large electron FS arises in the form characteristic to noninteracting particles. At the same time, the incoherent part of the spectrum with a bandwidth W = (5 − 7) t is preserved and it indicates the presence of strong correlations between the charge and spin degrees of freedom. The analytic investigation of the excitation spectrum in the region of moderate hole concentrations is complicated, since the AF correlation length becomes comparable with the correlation length of two holes, and a simple polaron model turns out to be inadequate. Exact diagonalization studies of the t–t –t –J model have shown that details of the QP dispersion, in particular the electron–hole asymmetry, critically depend on the second- and third-neighbor hopping parameters t , t . To explain the energy dispersion of the holes observed in ARPES experiments in the AF undoped Sr2 CuO2 Cl2 (SCOC) and weakly doped Ca2−x Nax CuO2 Cl2 (Na–CCOC) compounds one should take into account the t , t parameters (see [1247] and references therein). However, for a quantitative comparison of the spin-polaron model theoretical results for with ARPES data one should take into account the electron–phonon coupling as discussed in Sect. 5.2.2. In particular, it was found that the QP mass renormalization caused by electron– phonon coupling substantially increases in the presence of spin-polarons. A coherent slow motion of the polaron enhances the eﬀect of the lattice distortion and the QP renormalization [1041]. At modest electron–phonon coupling in the t–J model, a lattice polaron is formed which shows broad dispersive QP peak at large binding energies as discussed in Sect. 5.2.2 (see [837, 839, 1063]). The electron–phonon coupling plays an important role in the higher-energy string-like excitations in the t–J model as well [587].

7.2 Electron Excitations in the Normal State

403

Extensive studies of the spin-polaron spectra have been performed for a small radius polaron inside an AF matrix with frustration caused by the AF exchange interaction both for the n.n., J1 > 0, and the next n.n., J2 > 0, copper sites (see [103] and references therein). Assuming that the increase of the hole concentration leads to the frustration of the AF coupling of spins in the lattice, the authors have calculated the dependence of the QP spectrum of the holes on the degree of frustration on the basis of a variational method. It has been found that the topology of the FS changes from a hole type one with pockets in the neighborhoods of the (±π/2, ±π/2) points in the BZ to a large electronic FS. Investigations of the spin-fermion model obtained by projection from the three-band p–d Emery model [101, 681] have conﬁrmed the important role of the coupling between the oxygen holes and the copper spins which results in local spin-polaron formation (the Zhang–Rice singlet). The QP spin-polaron spectrum demonstrates vanishing damping close to the (π/2, π/2) point and a much lower boundary energy in comparison with the bare hole QP spectrum in the Emery model (see e.g., [572]). In comparison with the results for the spin-polaron model for spinless fermions (7.26) in an AF background, the spinrotation invariant GFs approach in the spin-fermion Emery model revealed several new features of the polaron spectral functions. In particular, a strong suppression of the QP weight at the Γ point and the asymmetry of the spectral function with respect to the AF BZ were found in agreement with the ARPES data for the SCOC crystals.

t–J Model Although it was believed that spin polarons dressed by AF spin ﬂuctuations continue to be the relevant QPs even at ﬁnite hole concentrations for the AF correlation lengths larger than the polaron size, numerical studies of the 2D t–J model at moderate doping δ ∼ 0.1 mentioned above have questioned this picture. Therefore, a theory of the electron spectrum and superconducting pairing for the original t–J model (7.16) in the paramagnetic state should be developed. Here we present a theory based on the projection technique for the GFs in terms of Hubbard operators [1012]. A similar theory was developed by Prelovˇsek [1022] (see also [1023, 1150, 1152, 1413] and references therein). The Dyson equation for the single-electron GF for the t–J model is derived in Appendix A.4. In the normal state, it is convenient to introduce for the GF 0σ σ0 (A.64) the notation G11 σ (k, ω) = Xk |Xk ω = Q G(k, ω) where Q = (1 − n/2) is the weight of the lower Hubbard subband. Then the Dyson equation for the GF in the imaginary Matsubara frequency representation can be written as 1 G(k, iωn ) = . (7.35) iωn − Ek − Σ(k, iωn )

404

7 Theoretical Models of High-Tc Superconductivity

Here the QP dispersion Ek in the MFA is determined by the expression (A.65) and the self-energy in the NCA according to (A.67) is deﬁned as Σ(k, iωn ) =

T 2 g (q, k − q) χ(+) sc (k − q, iωn − iωm ) G(q, iωm ), N q,m

(7.36)

where g(q, k − q) = [ t(q) − (1/2)J(k − q)]. Here, as in (7.5), we introduced: t(q) = −4t γ(q)−4t γ (q), γ(q) = (1/2)(cos qx +cos qy ), γ (q) = cos qx cos qy , and J(q) = 4J γ(q). The chemical potential μ is determined by the average electron concentration n=1−δ =

2Q N (k), N

N (k) =

k

∞ 1 T G(k, iωn ), + 2 N n=−∞

(7.37)

k

where δ is the hole concentration. From (7.37), it follows that n/2Q ≤ 1 and therefore n ≤ 1. The self-consistent system of equations for the GF (7.35) and the self-energy (7.36) should be solved in the frame of strong-coupling Migdal–Eliashberg theory. For numerical calculations, we should specify the spin-charge susceptibility (+) χsc (q, ω) in (7.36) given by (A.62). In the limit of strong correlations, the charge ﬂuctuations with large excitation energies can be neglected and only the spin ﬂuctuation contribution needs to be taken into account. As described in Sects. 3.2.4 and 3.2.2, a broad spin-ﬂuctuation spectrum in the normal state is observed for energies up to ωs ∼ J. The resonance mode has a weak intensity and it may be important only in the superconducting state. Therefore, for the spin susceptibility χs (q, ω) = 3χzz (q, ω) = −Sq |S−q ω , we can use the model suggested in numerical studies [550]: Im χs (q, ω) = χ(q) χ (ω) =

χ0 ξ 2 tanh(ω/2T ) , 2 2 1 + ξ (q − Q) 1 + (ω/ωs )2

(7.38)

The q-dependence is deﬁned by the static susceptibility χ(q) with a sharp maximum for large AF correlation lengths ξ at the AF wave vector Q = (π, π). The frequency dependence determined by χ (ω) is characterized by a broad spin-ﬂuctuation spectrum with a cutoﬀ energy ωs ∼ J. Whereas the two ﬁtting parameters, ξ (measured in the unit of the lattice spacing a = 1) and ωs , determine only the shape of the susceptibility (7.38) in the (q, ω)-space, the strength of the spin-ﬂuctuation interaction χ(Q) = χ0 ξ 2 is ﬁxed by the normalization condition (see (3.35)): S2i

+∞ 3 1 1 ω = (1 − δ). = χ(q) dω χ (ω) coth N q π 2T 4

(7.39)

0

This leads to the deﬁnition χ0 = (2/ωs ) C(ξ) where the coeﬃcient C(ξ) = S2i {(1/N ) p 1/(ξ −2 + p2 )}−1 ∝ 1/ ln ξ. The static spin correlation functions which determine the renormalization of the MFA QP dispersion Ek in

7.2 Electron Excitations in the Normal State

405

(A.65) can be calculated within the same parameterization (7.38) from Cq = Sq S−q = C(ξ)/([ξ −2 +(q − Q)2 ]. The doping dependence of the AF correlation length ξ(δ) at low temperature was evaluated from the comparison of the

spin correlation function for the nearest neighbors C1 = (1/N ) q γ(q) Cq (see (A.49)) with the numerical results of the exact diagonalization for ﬁnite clusters [154]. It was found that ξ = 3−2 for δ = 0.1−0.2 , accordingly. Using the model for the spin dynamical susceptibility (7.38), the spin-ﬂuctuation contribution to the self-energy in (7.36) is written as χs (k − q, iων ) = χ(k − q) Fs (ων ), ∞ 1 x ωs 2xdx 1 tanh Fs (ων ) = . 2 2 2 π 0 x + (ων /ωs ) 1 + x 2T

(7.40)

b

6.0

δ=0.1

Γ

3.0

0.0 – 3.0 – 6.0 w

M

A(k,w)

A(k,w) 0.5 1.0 1.5 2.0

a

0.5 1.0 1.5 2.0

Several results of the self-consistent solution of the equations of the GF (7.35) and the self-energy (7.36) for the dynamical spin susceptibility parameterization (7.38) are given below [1012]. The exchange interaction we taken J = 0.4 t and all energies were measured in units of t 0.4 eV. For small hole concentrations, δ = 0.1, quite narrow QP peaks for the single-electron spectral function A(k, ω) = −(1/π) Im G(k, ω) are observed at the wave vectors crossing the FS along the nodal direction M (π, π) → Γ (0, 0) (Fig. 7.4a). In addition to the QP dispersion, a band of incoherent excitations with large dispersion below the Fermi energy, ω < 0, is also seen. The incoherent band is caused by the self-energy contribution peaked at the AF wave vector (“shadow bands”). The dispersion of the QP band increases with increasing hole concentration and the intensity of the QP peaks are enhanced as shown in Fig. 7.4b for δ = 0.4, ξ = 1. At the same time, the intensity of the incoherent excitations is suppressed. This strong doping dependence of the dispersion can be explained even by considering the QP dispersion in MFA (A.65): E(k) ≈ −4teﬀ γ(k) − 4teﬀ γ (k). At low doping, the n.n. hopping teﬀ is blocked by the strong short range AF correlations and the dispersion is determined mostly by the next n.n. hopping teﬀ . For instance, at δ = 0.1 and ξ = 3 , the AF correlations functions C1 −0.23, C2 0.13 and teﬀ 0.13 t,

δ=0.4

Γ

6.0

3.0

0.0 – 3.0 – 6.0 w

M

Fig. 7.4. Electron spectral density A(k, ω) for (a) δ = 0.1, ξ = 3 and (b) δ=0.4, ξ = 1 in the t–J model [1012]

6.0

b δ=0.1

Γ

3.0

0.0 –3.0 –6.0 ω

M

–ImS(k,w) 0.5 1.0 1.5 2.0

–ImΣ(k,w)

a

7 Theoretical Models of High-Tc Superconductivity 0.5 1.0 1.5 2.0

406

δ=0.3

Γ 6.0

3.0

0.0 –3.0 –6.0 w

M

Fig. 7.5. Imaginary part of the electron self-energy for (a) δ = 0.1, ξ = 3 and (b) δ = 0.3, ξ = 1 in the t–J model [1012]

while the next n.n. hopping is enhanced: teﬀ 0.8 t (in comparison with the “Hubbard I” approximation, teﬀ = [(1 + δ)/2] t , for details see [1012]). These conclusions are supported by studies of the imaginary part of the self-energy Im Σ(k, ω) (7.36) shown in Fig. 7.5 for δ = 0.1, ξ = 3 (a) and for δ = 0.3, ξ = 1 (b). With increasing hole concentration and decreasing AF correlation length ξ, the self-energy decreases due to the weakening of the electron scattering on the spin-ﬂuctuations. It is interesting to note that for the underdoped region, δ ≤ 0.1, and T ≤ ω ≤ J the self-energy Im Σ(k, ω) is approximately proportional to ω, while for the overdoped region, δ ≥ 0.3, for small ω we have Im Σ(k, ω) ∝ ω 2 . However, the (k, ω) resolution was not high enough to evidence a transition from the marginal Fermi liquid to the Fermi-liquid behavior with doping. Our results for the electron spectral function [1012] show good overall agreement with the calculations reported by Prelovˇsek [1022] at zero temperature and with the self-consistent calculations by Sherman et al. [1150, 1152] at ﬁnite temperatures. The comparison of our spectral function results with those obtained within the exact-diagonalization technique for ﬁnite clusters (see [550]), concerning both the coherent QP dispersion and the incoherent band, shows reasonably good agreement as well. In particular, we observe a similar large asymmetry of the imaginary part of the self-energy in Fig. 7.5 of smaller absolute values,however, having as a result a less pronounced incoherent part of the spectra below the FS. Partly, this can be explained by the underestimate of the electron scattering on spin ﬂuctuations. In a more rigorous approach, one should perform self-consistent calculations of both the hole and the spin-ﬂuctuation spectra as done in a certain approximation by Sherman et al. [1150, 1152]. Remarkable results were obtained for the electron occupation numbers (7.37). In Fig. 7.6, the momentum distribution function N (k) is shown for two diﬀerent hole concentrations: (a) δ = 0.1, ξ = 3, and (b) δ = 0.2, ξ = 1. The shape of the FS changes from the hole-like around M (π, π) point of BZ at small doping to the electron-like around Γ (0, 0) point of BZ at large doping, δ = 0.4. However, the drop of N (k) at the FS is quite small, especially at

7.2 Electron Excitations in the Normal State

a

b

δ = 0.2

N(k) 0.5

2

kx

2.4

8 0.

3.2

1.

3. 2 4

0.0

2.

6

1.6

3.

ky

0.8

2.

0.8

6

1.6

kx

8

2.4

0.

1.

y

4

0.0

k

N(k) 0.5

1.0

1.0

δ = 0.1

407

3.2

Fig. 7.6. Electron momentum distribution function N (k) at (a) δ = 0.1, ξ = 3 and (b) δ = 0.2, ξ = 1 [1012]

small doping, which is a speciﬁc feature of the strongly correlated electronic systems. Large occupation numbers throughout the BZ are due to the incoherent contribution to the spectral function A(k, ω) under the Fermi level. The maximal occupation numbers for electrons, nk = (1 − n/2) N (k) ≤ 0.55 for δ = 0.1, agree with the results of the exact-diagonalization technique for ﬁnite clusters [1188]. The FS crosses the (±π, 0), (0, ±π) points of BZ at δ 0.3. The volume of the FS at small doping is proportional to the hole concentration δ, while according to the Luttinger theorem it should be proportional to n/2 = (1 − δ)/2. This violation of the Luttinger theorem was observed in a number of studies of the strongly correlated electronic systems described within projected bases of electron operators. In the self-consistent calculations of the self-energy (7.36), the electron scattering on the short-range paramagnon-like spin ﬂuctuations at q, q˜ > 1/ξ and the longer-range longitudinal spin ﬂuctuations at q˜ < 1/ξ were considered on equal footing within the model (7.38) for overdamped spin ﬂuctuations ˜ = q − Q). A more detailed analysis has suggested that the electron (here q interaction with the paramagnon-like transverse spin ﬂuctuations Sq± is more appropriate to describe in a way similar to the spin-polaron model, while the scattering on the longitudinal spin components Sqz can be considered within the model (7.38) [1023]. In this approach, the paramagnon-like contribution to the self-energy determines the hole QP spectrum as in the spin-polaron model, while the contribution from the longitudinal spin ﬂuctuations leads to the emergence of the pseudogap at low doping. It was shown that for large AF coherence lengths ξ the pseudogap close to the AF BZ boundary, ∝ (cos kx + cos ky ) = 0, can be found with the d-wave-like symmetry: ΔPG k |2J − 4t cos2 kx | [1023,1024]. Therefore, the shape of the pseudogap crucially depends on the value of the next n. n. hoping parameter t as has been revealed also in the exact diagonalization studies [1413]. We consider the AF scenario of pseudogap formation in more detail for the Hubbard model below.

408

7 Theoretical Models of High-Tc Superconductivity

Hubbard Model Various methods have been proposed to the study of the electronic structure within the Hubbard model (7.3) covering both the weak Coulomb correlations, U ≤ t, and the strong correlations, U t, limits. For weak correlations, a spin-fermion model was extensively used to describe electronic spectra of cuprates and, in particular, the subtle features of the QP spectra observed in ARPES (see Sect. 5.2.2) as e.g., the pseudogap and the FS arcs, “kinks” in the electron dispersion (see e.g., [2, 316, 900, 1121] and references therein). In the limit of strong correlations, extensive numerical studies of electronic spectra of ﬁnite size clusters have been done as discussed at the beginning of this section. Several analytical methods are based on perturbation expansion as the diagram technique for Hubbard operators (see [539]) and a cumulant expansion based diagram technique [865], which are very complicated. A technically simpler approach is the equation of motion method for the Green functions (GFs) that allows, within the Mori-type projection technique, the derivation of an exact Dyson equation for the single-particle GF (see Appendix, Sect. A.3). However, in the theory often the MFA is used, as in the equations of motion method for composite operators [782] or in the generalized tight-binding method [645, 935, 936]. In the MFA, the QP interaction with dynamical spin and charge ﬂuctuations is neglected and therefore it is not possible to describe the above-mentioned peculiar eﬀects caused by the self-energy contribution. In the dynamical mean ﬁeld theory (DMFT) used in describing the MottHubbard transition (for reviews see [376, 448, 546, 653, 654]), the self-energy is treated in the single-site approximation which is also unable to describe wave vector dependent phenomena. To overcome this ﬂaw of the DMFT, various dynamical cluster approximation (DCA) approaches have been developed (for reviews see [773, 1186, 1265]). In particular, the cellular dynamical mean-ﬁeld theory (C-DMFT) has been proposed and is currently a subject of active investigation (see e.g., [437] and references therein). In the C-DMFT, the exact solution for a ﬁnite number of particles in a cluster is found which is then “embedded” into the lattice. Whereas in the DMFT, the same self-energy is assigned to all the electronic states at the FS, in the C-DMFT a diﬀerentiation between the coherent nodal QPs and antinodal incoherent pseudogap states at the FS is possible. Since a cluster embedded in the lattice violates its periodicity, a periodization problem emerges within the cluster theories. Generally, certain restrictions on the wave vector and energy resolutions, depending on the size of the clusters, exist in the DCA, while the physical interpretation of the origin of an anomalous electronic structure in the numerical methods is not straightforward. Below we consider studies of the single-particle electron spectrum in the eﬀective Hubbard model within a microscopic theory based on the equation of motion method for GFs in terms of Hubbard operators [16, 86, 87, 680, 1007, 1017]. In this approach, the eﬀects originating in the strong Coulomb

7.2 Electron Excitations in the Normal State

409

correlations are rigorously taken into account by the Hubbard operator kinematics, while a self-consistent calculation of the self-energy from the Dyson equation enables to consider an energy dependent QP spectrum renormalization caused by spin and charge ﬂuctuations beyond the MFA. We emphasize that in the theory no phenomenological coupling constants are introduced since the electron scattering by dynamical AF spin ﬂuctuations is induced by the kinematic interaction determined by the hopping integrals. Similar results for the single-particle excitation spectrum were obtained within the self-consistent projection operator method by Kakehashi et al. [576, 577] and operator projection method by Onoda et al. [921–923]. The Dyson equation for the eﬀective two-subband Hubbard model (7.8) for holes is derived as described in Sect. A.3.1. For the normal components of ˆ ijσ (ω) = X ˆiσ | X ˆ † ω which are deﬁned in the matrix single-particle GF G jσ ˆ † = (X 2σ X σ¯ 0 ), ˆ iσ and X terms of the two-component Hubbard operators X i i iσ the Dyson equation reads: −1 Q2 0 ˆ ˆ ˆ ˆ G(k, ω) = ωˆ τ0 − εˆ(k) − Σ(k, ω) Q, Q = , (7.41) 0 Q1 where the weight factors of the Hubbard subbands Q2 = Xi22 + Xiσσ = n/2 and Q1 = Xi00 + Xiσ¯ σ¯ = 1 − Q2 depend on the average occupation number of holes n given by (7.9). The normal state QP spectrum in MFA εˆ(k) is deﬁned by (A.47), (A.48). In the lowest order approximation for the ˆ self-energy matrix Σ(k, ω) only the diagonal components for the two-hole 22 Σ (k, ω) and single-hole Σ 11 (k, ω) subbands are taken into account where, e.g., Σ 22 (k, ω) = M 22 (k, ω)/Q2 in the NCA is determined by (A.59). This enables us to write the diagonal components of the full GF (7.41) in the form ˆ 11(22) (k, ω) = Q1(2) {[1 − b(k)] G1(2) (k, ω) + b(k) G2(1) (q, ω)}, G

(7.42)

were the hybridization parameter b(k) (ε2 (k) − ω2 (k))/(ε2 (k) − ε1 (k)). In the numerical solution, the Matsubara imaginary frequency representation for the subband GFs is used G1(2) (k, iωn ) =

1 . iωn − ε1(2) (k) − Σ(k, iωn )

(7.43)

The self-energy has a similar form for the both subbands: Σ(k, iωn ) =

T |t(q)|2 χ(+) Gα (q, iωm ). (7.44) sc (k − q, iωn − iωm ) N q,m α=1,2

The self-consistent system of equations for the GFs (7.43) and the self-energy (7.44) has been solved by using the model spin-susceptibility χs (k − q, iων ) (7.40) along the approach developed for the t–J model. Before discussing the results of the numerical solution of this system of equations, it is worthwhile to make some general remarks concerning our

410

7 Theoretical Models of High-Tc Superconductivity

approach in comparison with other methods. In the theory based on the HO technique, we start from the two-subband representation for the GF (7.41), which rigorously takes into account the strong electron correlations. This results in the Mott (charge-transfer) gap at large eﬀective Coulomb energy Ueﬀ = Δpd , similar to the DMFT. On the contrary, the electron scattering by the spin (charge) dynamical ﬂuctuations induced by the kinematic interaction is responsible for the pseudogap formation in the underdoped region and the strong renormalization of the QP spectra. As discussed in Sect. 3.3.3, there are several scenarios of the pseudogap origin (for a review, see [901]). Here, we consider the AF scenario which follows from the present theory and was originally proposed by Kampf et al. [582] in the spin-fermion model and later extensively studied by Pines and coworkers (see [2, 1121] and references therein). The AF scenario within the Hubbard model was considered in the two-particle self-consistent approach (TPSC) by Vilk et al. [1315] (for review, see [1265]) and studied within a generalized dynamical mean-ﬁeld theory approach by Sadovskii et al. [1081] and Kuchinskii et al. [670] (for reviews, see [669, 1080]). In the underdoped region, a spin-liquid without long-range order emerges, which can be characterized by a large AF spin correlation length ξ and lowfrequency dynamical spin ﬂuctuations, ωs J. Therefore, in this region we can consider the classical limit for the spin susceptibility (7.38), ωs T , by taking into account only the zero Matsubara frequency in (7.44): ωs χ0 δn,m χs (q), χs (q) = 2 χs (q, iωn − iωm ) . (7.45) T κ + p2 In the limit of large AF correlation lengths, κ = ξ −1 1, the static spin susceptibility χs (q) shows a sharp peak close to the AF wave vector Q = (π, π) and it may be expanded over the small wave vector p = q − Q. For further qualitative discussion, we consider only the Hubbard subband which crosses the Fermi energy and expand the corresponding GF G(q = k − Q − p, iωn ), in the expression (7.44) of the self-energy over p as well, G(q, iωn ) =

1 iωn − εk−Q + p · vk−Q − Σ(k − Q − p, iωn )

,

(7.46)

where vqα = ∂εq /∂qα . This yields the equation for the self-energy (7.44): Σ(k, iωn ) =

g 2 (k) 1 G(k − Q − p, iωn ), 2 + p2 N κ p

(7.47)

where the interaction is approximated by its value at q = k − Q (see (7.39)), g 2 (k) = ωs χ0 t2 (k − Q) = 2C(ξ) t2 (k − Q). In the lowest order approximation for the self-energy (7.47) for κ → 0 the GF in (7.43) can be written as ω − εk−Q , (7.48) [ω − E1 k ][ω − E2 k ] = (1/2)[εk + εk−Q )] ± [(1/2)(εk − εk−Q )]2 + Δ2AF (k),

G(k, ω) = E1,2 k

7.2 Electron Excitations in the Normal State

411

where the AF spectrum is determined by Δ2AF (k) =

gap in2 the 2electronic 2 2 g (k)(1/N ) p 1/(κ + p ) = 2 t (k − Q) S2i (see the normalization condition (7.39)). At the AF BZ, cos kx + cos ky = 0 , the gap ΔAF (k) ∼ |t(k − Q)| = 4|t | (cos kx )2 is deﬁned by the n.n.n. hopping integral t and shows maxima at (π, 0) points and nodes at (π/2, π/2) points, similar to the d-wave superconducting gap. To derive the electronic spectrum with the pseudogap instead of the AF gap in (7.48), we should take into account the momentum dependence of the GF (7.46) and integrate over p in (7.47). For a qualitative discussion, we may further simplify the static spin susceptibility model (7.45) by writing it as a product of one-dimensional components: [(κ/(κ2 + p2x )][κ/(κ2 + p2y )] as suggested by Schmalian et al. [1121] and Sadovskii [1080]. Then integration in the complex plane over the (px , py ) momenta can be done independently that results in the equation for the self-energy (7.47) Σ(k, iωn ) = g 2 (k)

1 , iωn − εk−Q + ivκ − Σ(k − Q, iωn )

(7.49)

y x | + |vk−Q |. This equation can be solved by iteration but even where v = |vk−Q in the lowest approximation for the self-energy we obtain a pseudogap-like behavior of the spectral function A(k, ω) = −(1/π) ImG(k, ω) :

A(k, ω) =

1 κ v g 2 (k) . π [(ω − εk )(ω − εk−Q ) − g 2 (k)]2 + [κ v(ω − εk )]2

(7.50)

The spectral function reveals a pseudogap structure in the vicinity of the X(π, 0) points and an incipient AF gap in the nodal direction. The system of equations for the GF (7.46) and the self-energy (7.47) is similar to those one derived in the TPSC approach by Vilk et al. [1315] and in the DMFT+Σk -model by Sadovskii et al. [1081]. In our approach, the interaction is determined by the k-dependent hopping parameter, g 2 (k) ∝ t2 (k − Q), while in the TPSC and the Σk -model the coupling constant is determined by the eﬀective Coulomb scattering, g 2 = U 2 (ni↑ ni↓ /n2 )S2i /3 which is k-independent. However, the absolute values of these vertices are close

to each other: the averaged over the BZ value |t(k)|2 k ∼ 2t is comparable with the coupling constant Δ ≤ 2t used by Sadovskii et al. [1081]. In the spinfermion model, the self-energy is also determined by spin-ﬂuctuations with the coupling constant ﬁtted from ARPES experiments, g ∼ 0.7 eV∼ 2t, of the same order [316]. As follows from (7.48), in the limit κ = 1/ξ → 0, the AF gap ΔAF (k) in the QP spectra emerges in the subband located at the Fermi energy. Thus, in our approach the pseudogap is a precursor of the AF long-range order, similar to TPSC theory and the generalized DMFT model [670, 1081]. To support this conclusion, we consider the last model in more detail. In this model, the single-electron GF is written as G(k, iωn ) =

1 , iωn − ε(k) − Σ(iωn ) − Σk (iωn )

(7.51)

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7 Theoretical Models of High-Tc Superconductivity

where Σ(iωn ) is the self-energy calculated within the DMFT, while Σk (iωn ) takes into account static short-range ﬂuctuations, induced by AF or other types of order parameters (charge density wave, superconducting order parameter). The self-energy Σk (iωn ) = Σ1 (k, iωn ) was calculated by using the recurrence relations: Σn (k, iωn ) =

Δ2 s(n) iωn − εn (k) + invn κ − Σ(iωn ) − Σn+1 (k, iωn )

(7.52)

y x where εn (k) = ε(k + Q), vn = |vk+Q | + |vk+Q | for odd n while εn (k) = y x ε(k), vn = |vk |+|vk | for even n. The pseudogap amplitude Δ2 and the inverse correlation length κ = 1/ξ are phenomenological parameters determined from experiments. The combinatorial factor s(n) depends on the type of the order parameter ﬂuctuations (for details, see [669, 1081]). Figure 7.7 shows the spectral function A(k, ω) = −(1/π)ImG(k, ω) for the GF (7.51) along the symmetry directions of the BZ, Γ (00) → X(π, 0) → M (π, π) → Γ (00) (left panel). The computations were performed for the AF-type ﬂuctuations (SDW) for the model parameters: the Hubbard energy U = 4t, Δ = 2t, ξ = 10, the ﬁlling factor n = 0.8 (hole doping), at temperature T = 0.088t. Two cases for the bare dispersion ε(k) = −4t γ(k) − 4t γ (k) were considered: t /t = −0.4 (left column) and t /t = 0 (right column). In both cases, a double-peak pseudogap structure is observed close to the X(π, 0) point at the Fermi energy (FE), ω = 0, while a precursor AF gap is found in the middle of the M → Γ nodal direction. We can see also a very ﬂat dispersion of

Fig. 7.7. Spectral function A(k, ω) along the symmetry directions (left panel) and the intensity plots of A(k, ω) close to the FS, ω = 0, at various pseudogap amplitude Δ (right panel) (after Kuchinskii et al. [669])

7.2 Electron Excitations in the Normal State

413

the lower Hubbard band close to the FE in the Γ → X and Γ → M directions (more pronounced for t /t = −0.4) with the spectral intensity transferred to the upper Hubbard band close to the M point. The FS shown as intensity plots of the spectral function A(k, ω = 0) is displayed in Fig. 7.7 (right panel) for the same model parameters for diﬀerent coupling parameters, 0.2t ≤ Δ ≤ 2t in panels (a)–(d). The “destruction” of the FS with transition from the large FS to the arc-type is clearly observed under the increase of the interaction Δ. The “bare” FS is shown by dashed lines, while the FS deﬁned as a solution of the conventional equation, ω−ε(k)−Σ(ω)−Σk(ω) = 0, is represented by solid lines. The latter deﬁnition of the FS is obviously inadequate in comparison with the intensity plots, which are measured in ARPES experiments. Now we discuss the numerical results of the self-consistent solution of the system of equations for the GFs (7.43) and the self-energy (7.44). The spectrum of the single-electron excitations is determined by the spectral function A(el) (k, ω) = A(h) (k, −ω) where the spectral function for holes reads A(h) (k, ω) = −(1/π) Im akσ | a†kσ ω+i0+ = [Q1 + P (k)]A1 (k, ω) + [Q2 − P (k)]A2 (k, ω),

(7.53)

with the hole annihilation akσ and creation a†kσ operators being deﬁned in terms of the Hubbard operators as akσ = Xi0σ + σXiσ¯ 2 , a†kσ = Xiσ0 + σXi2¯σ . Therefore, all the four components of the GF Gαβ (k, ω) (7.41) matrix give contributions to (7.53). In the normal state, the spectral function for two subbands is determined by the GFs (7.43): (7.54) A1(2) (k, ω) = −(1/π) ImG1(2) (k, ω). √ The parameter P (k) = (n − 1)b(k) − 2 Q1 Q2 W (k)/Λ(k) takes into account hybridization eﬀects both from diagonal and oﬀ-diagonal GFs (see Sect. A.3). An extensive study of the dispersion of the single-particle excitations, the spectral functions, and the Fermi surface in the normal state have been reported by Plakida et al. [1017] in the limit of strong correlations for U = 8t. Below we present several numerical results under the intermediate value of the Coulomb repulsion, Ueﬀ = 4t. The bare dispersion in the Hubbard model (7.8) is speciﬁed by the hoping parameters t = −0.13 t, t = 0.16 t found by Korshunov et al. [645] in the LDA calculations for the La2 CuO4 system. The spectral functions (7.53) and dispersion curves given by maxima of the spectral functions are shown in Figs. 7.8 and 7.9 along the symmetry directions Γ (0, 0) → M (π, π) → X(π, 0) → Γ (0, 0) for the hole doping δ = 0.1 and 0.3, respectively. At low hole doping, δ = 0.1, the spectral function demonstrates weak QP peaks at crossing the Fermi energy (FE) and a dispersive behavior far away from the FE as shown in Fig. 7.8 (left panel). The dispersion reveals a rather ﬂat hole-doped lower Hubbard band (LHB) at the FE (ω = 0) (right panel), in particular close to the X(π, 0) point of the Brillouin zone (BZ) in agreement with the quantum Monte Carlo calculations for the Hubbard model

414

7 Theoretical Models of High-Tc Superconductivity 12

Γ

Energy

8 4 0 -4

Γ

-8

Γ

M

X

Γ

Fig. 7.8. Electron spectral functions A(k, ω) (left panel) and dispersion curves (right panel) in units of t along the symmetry directions for δ = 0.1 [1017]

Γ

Energy

8 4 0 -4

Γ

-8

Γ

M

X

Γ

Fig. 7.9. Electron spectral functions A(k, ω) (left panel) and dispersion curves (right panel) in units of t along the symmetry directions for δ = 0.3 [1017]

(see [190] and references therein). A remarkable weight transfer from the LHB to the upper Hubbard band (UHB) is observed close to the M (π, π) point of the BZ. With doping, the dispersion and the intensity of the QP peaks at the FE substantially increase as demonstrated in Fig. 7.9, though a ﬂat band in X(π, 0) → Γ (0, 0) direction is still observed in agreement with ARPES measurements in the overdoped La1.78 Sr0.22 CuO4 [1390]. To study the inﬂuence of the AF spin-correlations on the spectra, we have calculated also the spectral functions at high temperatures [1017]. Computations at T = 0.3t under the neglect of the spin correlation functions in the MFA single-particle excitation spectra in (A.48), (A.49) and the assumption of a small AF correlation length (ξ = 1.0) in the spin-susceptibility (7.38) have revealed a strong increase of the dispersion and the intensity of the QP peaks at the FE similar to the overdoped region, δ = 0.3, which proves a strong inﬂuence of the AF spin-correlations on the spectra. In general, the

7.2 Electron Excitations in the Normal State

415

spectral functions in the Hubbard model are comparable to those in the t–J model (compare Figs. 7.4 and 7.8). Similar results for the spectral functions and the dispersion curves were obtained for the Hubbard model within the operator projection method by Onoda et al. [922], in particular, ﬂat bands close to the X(π, 0) points of the BZ and the weight transfer from the LHB to the UHB at the M (π, π) point of the BZ. It is interesting to compare our results with those obtained in the generalized DMFT [1081] discussed above. In fact, the spectral function shown in Fig. 7.7 for t = −0.4 demonstrates similar ﬂat QP bands in Γ (0, 0) → X(π, 0) and Γ (0, 0) → M (π, π) directions, as in our Figs. 7.8 and 7.9, a strong intensity transfer from the LHB to the UHB close to the M (π, π) point of the BZ and a splitting of the band close to the X(π, 0) point. Analogous temperature and doping behavior of the spectral functions and the pseudogap revealed in both theories support the spin-ﬂuctuation scenario of the pseudogap formation. A similar behavior was reported in the cluster perturbation theory [1134, 1265]. A noticeable doping dependence of the FS was found also. In Fig. 7.10a, the FS determined by a solution of the equation: ε2 (kF ) + Re Σ(kF , 0) = 0 is plotted for various hole concentrations. We see a large pocket at small doping δ = 0.05 which opens with doping at δ = 0.1 and changes with temperature. At the overdoping for δ = 0.3, the FS transforms to the electron-like as experimentally observed in the overdoped La1.78 Sr0.22 CuO4 [1390]. A remarkable feature of these results is that the part of the FS close to the Γ (0, 0) point in the nodal direction does not shift much with doping (or temperature) being pinned to a large FS as observed in ARPES experiments (see Fig. 5.17). The FS deﬁned by the maxima of the spectral function Ael (k, ω = 0) in the (kx , ky )-plane for δ = 0.05 is shown in Fig. 7.10b. It reveals an arc-type shape with maximum intensity located close to the nodal direction at the large FS in agreement with ARPES data (see Fig. 5.21). Thus, the theory explains the “destruction” of the FS due to the arc formation at low doping by strong AF correlations at large ξ, similar to the results of the generalized DMFT shown in Fig. 7.7 (right panel). The doping dependence of the FS was also reported in studies of the electron momentum distribution function N(el) (k)) deﬁned similar to (7.37). At low doping, N(el) (k) reveals hole-pockets close to (±π/2, ±π/2) points of the BZ which transform to a large FS with increasing doping. Similar to the t–J model, Fig. 7.6, in the underdoped case at δ = 0.1 the drop of the electron occupation numbers at crossing of the FS is rather small, while in the overdoped case when the AF spin correlations are suppressed, this change is substantially increased (for detail see [1017]). Thus, the arc-shape of the FS and a small change of the electron momentum distribution function at the FS at low doping further point to a large contribution of the spin correlations in the renormalization of the QP spectra. The wave vector and doping dependence of the intensity of spin-ﬂuctuation scattering can be inferred from the study of the self-energy (7.44). Calculations

416

7 Theoretical Models of High-Tc Superconductivity

Fig. 7.10. (a) Doping dependence of the conventional FS for δ = 0.05 (full line), δ = 0.1 (dotted line at T = 0.03t and dashed line at T = 0.3t), and δ = 0.3 (dotdashed line). (b) Intensity plot of A(k, ω = 0) close to the Fermi energy at δ = 0.05 [1017]

of the real and imaginary parts of the self-energy Σ(k, ω) demonstrated a strong dependence of the self-energy on the wave vector and the hole concen˜ ω)/∂ω)ω=0 trations. With doping, the coupling constant λk = −(∂ ReΣ(k, substantially decreases. At large binding energies (larger than the spinﬂuctuation energy ∼ωs ), the self-energy renormalization vanishes and the electron dispersion shows bending, the so-called “kink” in the electron dispersion as observed in ARPES experiments (see Sect. 5.2.2). The variation of the dispersion was observed in the intensity plots of the spectral function A(k, ω) below the Fermi level ω ≤ 0, which demonstrated a much stronger renormalization in the antinodal direction (X → M ) than in the nodal (Γ → M ) one (for details, see [1017]). In conclusion, this theory based on the numerical solution of the strongcoupling equations for the GFs (7.43) and the self-energy (7.44) has been able to explain, at least qualitatively, major ARPES experimental results. These include the appearance of the arc-type shape of the FS and the pseudogap at low doping, the strong renormalization eﬀects of the QP dispersion close to the Fermi energy (the “kinks”), the anomalous behavior of the momentum distribution function caused by strong correlations. At high temperatures or large hole concentrations (ξ ∼ 1), the AF ﬂuctuations become weak and the crossover to the Fermi-liquid-like behavior can be noticed. Similar results concerning the ﬂat bands, the pseudogap and the holepocket FS with the spectral function revealing arcs at low doping were obtained by Avella et al. [86, 87] within the equation of motion method for composite operators. Contrary to our phenomenological model for the dynamical spin susceptibility (7.38), in their approach the spectra of the collective excitations (spin and charge) has been calculated self-consistently in

7.2 Electron Excitations in the Normal State

417

the two-pole approximation. It should be mentioned, however, that even in the underdoped case the actual spin and charge susceptibility show a broad incoherent spectrum of ﬂuctuations instead of sharp modes given by the two-pole approximation. A self-consistent calculation of the self-energy of the singleelectron GF and the polarization operator of the dynamical (spin or charge) susceptibility is required for the development of a fully microscopic theory. To obtain quantitative results for the electron spectrum in cuprates, one should take into account also the real many-orbital band structure (see e.g., Fig. 7.1) beyond the simple one-band Hubbard model (7.3) or eﬀective twosubband model (7.8). Such calculations have been carried out within the generalized tight-binding (GTB) method by Ovchinnikov and coworkers (for review, see [936]). In the GTB method, a multiband p–d model is studied within the cell-cluster type perturbation theory. At ﬁrst, an exact diagonalization of the multiorbital Hamiltonian for CuO4 or CuO6 clusters is performed like in the simple two-orbital model in (7.7). Then the cell eigenvalues are used for the construction of the Hubbard operators of the model. The intercell hopping and residual interactions are considered as perturbations by the diagram technique for Hubbard operators. A hybrid LDA + GTB scheme was also developed for the calculation of the GTB model parameters from the LDA method [645, 646]. Within this approach, correct values of the gaps and the band dispersion were obtained in agreement with ARPES data. However, the method does not make allowance for the self-energy eﬀects, in particular inelastic scattering on spin and charge ﬂuctuations which are important as discussed above. In this respect, the LDA + DMFT method provides much better results by combining the “ﬁrst principle” LDA calculations with treating the strong Coulomb correlations within the DMFT. Unfortunately, this method is rather involved and demands complicated numerical calculations. As an example, we show in Fig. 7.11 the results of the LDA + DMFT calculations for the three p–d band model of La2−δ Srδ CuO4 at the doping δ = 0.24 by Weber et al. [1345]. The intensity plot for the spectral function A(k, ω) and the dispersion curves are shown in the panel (a) along the principal directions Γ (0, 0) → X(π, 0) → M (π, π) → Γ (0, 0) in units of electron volt. The ﬂat band crossing the Fermi energy (FE) is ascribed to coherent QP excitations of the Zhang–Rice (ZR) singlet band, while the dispersing part below the FE at about 1 eV is related to incoherent ZR excitations caused by the self-energy eﬀects in the DMFT. The abrupt change in the dispersion from the ﬂat ZR band to the dispersing part below the FE shown by the white dashed lines is associated by the authors with the so-called “waterfall” feature observed by ARPES experiments in cuprates (see e.g., [523, 1297]). In the panel (b), the local spectral functions A(ω) of the dx2 −y 2 band (dashed line) and of the pσ band (full line) are shown. The upper Hubbard band (U.H.B.) and the lower Hubbard band (L.H.B.) are marked along with the coherent ZR singlet QP peak (QU.P) separated by a dip from the incoherent part of the ZR singlet band (Z.R.S.). The parent (undoped) compound is found to be an insulator

418

7 Theoretical Models of High-Tc Superconductivity

Fig. 7.11. LDA + DMFT calculations (a) of the spectral function A(k, ω) and (b) of the density of states A(ω) for the hole doped La2−δ Srδ CuO4 compound at δ = 0.24. The local spectral functions of the dx2 −y2 band and of the pσ band are shown by the dashed and full lines, respectively (after Weber et al. [1345])

with a charge transfer gap of ∼1.8 eV where the ZR singlet band is located just below the FE. In the case of electron doping, a sharp QP peak develops close to the UHB. Although the DMFT is unable to reveal the low-energy phenomena, like the pseudogap and formation of the Fermi arcs, the theory in combination with LDA provides a realistic description of the band structure allowing for the strong electron correlations. A more accurate description of the ZR singlet dispersion was obtained within the cluster exact diagonalization based LDA + DMFT calculations by Yin et al. [1389]. Our results based on the self-consistent solution of the Dyson equation show, similar to the LDA + DMFT calculations, the ﬂat ZR singlet band at the FE with the spectral weight transfer to the one-hole subband close to the M (π, π) point and a “waterfall” feature below the FE as shown in Fig. 7.8. However, contrary to the LDA+DMFT data, the dispersion close to the X(π, 0) point and in the nodal direction Γ (0, 0) → M (π, π) reveal energy gaps as in the generalized DMFT (Fig. 7.7). Most likely, this diﬀerence is due to a momentum dependence of the self-energy (7.46) which is neglected in the DMFT, besides diﬀerence in the models and doping levels which may result in diﬀerent energy scales. Conclusion In conclusion, the following main results concerning the single-particle electron spectra should be singled out. 1. Studies of the hole motion in an AF background evidence the important role of the electron coupling with the spin-ﬂuctuations, which results in the appearance of a narrow QP spin-polaron band and a broad incoherent

7.2 Electron Excitations in the Normal State

419

multispin string excitation band. For a quantitative description of the spectra, the electron–phonon coupling and the lattice polaron eﬀects should be taken into account as well. 2. In the underdoped and optimally doped regions, the electron scattering by the spin ﬂuctuations plays an important role in the electron spectrum renormalization, as shown by studies of the t–J and the Hubbard models. In particular, the normal state pseudogap, the arc-type Fermi surface, the non-Fermi-liquid self-energy for electrons can be explained by their strong coupling with the AF spin ﬂuctuations as proved by the self-consistent solution of the Dyson equation. These phenomena are also revealed in numerical studies on ﬁnite clusters and by various cluster dynamic meanﬁeld methods. In the limit of weak correlations, the anomalous electronic normal state spectrum can be described also within a phenomenological spin-fermion model. 3. In the overdoped region, the spin correlations are suppressed and a Fermilike electron spectrum is revealed. However, strong electron correlations are still playing some role in preserving the physics of a doped Mott insulator. 7.2.2 Spin Dynamics In the previous sections, it was shown that the electron scattering by antiferromagnetic (AF) spin-ﬂuctuations in cuprates is of considerable importance in the understanding of the anomalous normal state properties and in elucidating the superconducting pairing mechanism (see [237, 1105, 1106]). This asks for the development of a theory of the spin dynamics. As discussed in Sects. 3.2.4 and 3.2.2, in the undoped insulating phase of the single-layer LCO and double-layer YBCO compounds, the quasi-two-dimensional Heisenberg model for localized spins (see (3.16) and (3.33), respectively) gives a reasonable description of the spin-wave spectrum, while in the metallic phase, it is much more diﬃcult to establish a theory of the spin-ﬂuctuation spectrum over broad ranges of doping and temperature. In particular, the interpretation of the spin gap and the resonance mode observed in superconducting cuprates (see Sect. 3.2.3) pose a challenging problem to the theoretical description. Similar to the theory of the single-particle electron excitations, there are two diﬀerent theoretical approaches in the theory of the spin dynamics. In the overdoped region, collective spin excitations are considered within the Fermi liquid of itinerant electrons. A phenomenological approach within the spin-fermion model [2,316,863] was used in the description of the spin dynamics in the normal and superconducting states. In early theoretical studies, a random phase approximation (RPA) in the weak correlation limit, U t, for the Hubbard model (7.3) was applied to calculate the dynamical spin susceptibility (see e.g., [1354, 1355]). A more accurate ﬂuctuation-exchange approximation (FLEX) was derived to study self-consistently the singleelectron GFs and the spin and charge dynamical susceptibility (for references,

420

7 Theoretical Models of High-Tc Superconductivity

see [311,787,861,862]). We consider this approach in more detail in Sect. 7.3.2 in connection with the spin-ﬂuctuation pairing. However, starting from the Fermi-liquid picture of itinerant electrons, the theory fails to describe the underdoped regime close to the insulating (and AF) state where well-deﬁned spin-wave-like excitations are observed. In this region, the strong AF correlations are of crucial importance and models of strongly correlated electrons like the t–J model (7.16) should be used. Here, studies of ﬁnite clusters by numerical methods were important in elucidating the static and dynamic spin correlations though they have limited energy and momentum resolutions (see the reviews by Dagotto [261], Jakliˇc et al. [550], Maekawa et al. [769]). In particular, a substantially diﬀerent behavior was found for the charge (electron density) and spin susceptibility at low doping in the t–J model [302]. Whereas the density response was of a bosonlike character with a characteristic energy scale of the order of the hopping integral t, the spin susceptibility was characterized by energy scale of the order of the exchange energy J. Among the analytical methods, a slave-particle approach was extensively used in studies of spin dynamics and single-particle excitations (see e.g., [369, 1232], and reviews by Ogata et al. [912], by Izyumov [543], and by Lee et al. [700]). The main diﬃculty in the slave-particle method is treatment of the local constraint which has to be imposed to reduce the enlarged Hilbert space for the fermion-boson slave particles to the physical one. In Sect. 7.3.1, this problem will be discussed in more detail. Contrary to the slave-particle representation, the Hubbard operator technique enables one to take rigorously into account the projected character of the electron operators in the t–J model. In particular, using a special diagram technique for the Hubbard operators (for a review see [539]) the dynamical spin susceptibility (DSS) (3.9) was calculated in the t–J model by Izyumov et al. [540]. Since the Hubbard operators are governed by a more complicated algebra of commutation relations (see (A.38)) than the Fermi or Bose operators, there arise, besides the vertex caused by the dynamical interaction J additional vertices due to a “kinematical” interaction. This term was introduced by Dyson [298] in the theory of spin waves in the Heisenberg model. As a result, three new loop diagrams appear in addition to the conventional electron–hole one in the RPA. The summation of all loop diagrams yields the DSS in a generalized random-phase approxi−1 mation: χ(k) = χ0 (k) {[1 − Λ(k)][1 − Q(k)] + χ0 (k)[J(k) + Φ(k)]} , where k = (k, iωn ) is a four-momentum, including the Matsubara frequency iωn . Here Λ(k), Q(k), Φ(k) are the additional loop diagrams. The “bare” susceptibility χ0 (k) besides the conventional hole-particle loop diagram Π(k) contains a localized contribution χloc ∝ 1/T at zero frequency ωn = 0, which is nonzero only for the electron concentration n > nc . Thus, it follows from this theory that the system behaves as an itinerant magnet for n < nc , while for n > nc ∼ 2/3 a contribution of localized moments with the Curie–Weiss susceptibility n/T arises at low hole concentration δc < 1 − nc ∼ 1/3. The

7.2 Electron Excitations in the Normal State

421

diagram technique for the Hubbard operators was used also by Onufrieva et al. [926, 927] to describe spin-ﬂuctuation spectra observed in INS experiments. The problem with application of the diagram technique in the t–J model is an absence of a small parameter which can justify a selection of a certain set of diagrams. In this case, a more general equation of motion method based on the Mori projection technique for the memory function seems to be more appropriate to treat the t–J model with strong coupling. This method has used by several groups in studies of spin and charge excitations (see e.g., [547, 548, 1128,1152–1154,1316,1317,1319]). It was also extensively used in the Hubbard model (for a review see [782]). Below we present several examples of studies of spin-excitation spectra both in the weak and in the strong correlation limits. One of the most interesting problem, in spin dynamics in cuprates is an explanation of the resonance mode (RM) phenomenon in superconducting state (see Sect. 3.2.3). Two basic approaches in the theory of the RM can be singled out. In the ﬁrst one, the RM is considered as a particle-hole bound state, usually referred to as a spin-1 exciton (for a review, see [316]). The state is formed below the continuum of particle-hole excitations which is gapped at a threshold energy Ωth ≤ 2Δ(q∗ ) determined by the superconducting d-wave gap 2Δ(q∗ ) at a particular wave vector q∗ on the Fermi surface (FS). In another approach, it is assumed that the collective spin excitations preexist in the normal state but they are overdamped, while in the superconducting state, due to a strong suppression of the spin excitation damping a sharp peak, the RM, appears. Below we discuss both scenarios. The spectrum of spin excitations measured in the inelastic magnetic neutron scattering is determined by the dynamic spin susceptibility (see (3.9), − (3.10) in Chap. 3) χ(q, ω) = −Sq+ |S−q ω . In the exciton model of the RM, the DSS is usually calculated within the random phase approximation (RPA) for the itinerant electrons χ(q, ω) =

χ0 (q, ω) , 1 − gq χ0 (q, ω)

(7.55)

where χ0 (q, ω) is the “bare” irreducible susceptibility given by a single loop of an electron–hole pair excitation (see (3.42)). The fermionic four-point vertex gq in the Hubbard model (7.3) is deﬁned by the q-independent Coulomb energy gq = U , while for the t–J model (7.16) it is given by the exchange interaction gq = −2J (cos qx + cos qy ) and sometimes a combination of the functions is used to ﬁt the experimental data. The irreducible susceptibility in the superconducting state within the BCStype theory can be written as 1 εk εk+q + Δk Δk+q fk+q − fk χ0 (q, ω) = − 1+ 2 Ek Ek+q ω − (Ek+q − Ek ) k εk εk+q + Δk Δk+q (Ek+q + Ek )(1−fk+q −fk ) , (7.56) + 1− Ek Ek+q ω 2 − (Ek+q + Ek )2

422

7 Theoretical Models of High-Tc Superconductivity

where εk is the electron dispersion in the normal state, Δk is the superconducting (SC) gap function, and fk = [exp(Ek /T ) + 1]−1 is the Fermi

function for Bogoliubov quasiparticles (QPs) with the dispersion Ek = ε2k + |Δk |2 . Here, the last contribution comes from the creation or the annihilation of a pair of QPs in the SC state. It should be emphasized that the q-dependence of this term is quite diﬀerent for an isotropic gap Δk = Δ0 and for a gap of d-wave symmetry, e.g., Δk = Δ0 (cos kx − cos ky )/2. In particular for QPs at the Fermi surface εkF = 0, EkF = |ΔkF | for the AF wave vector Q = (π, π) the last contribution for the isotropic gap vanishes, while for the d-wave symmetry gap Δk+Q = −Δk and the last term shows an enhancement of the bare DSS χ0 (Q, ω) in the SC state. For a model with nearest neighbor hopping only, εk = −εk+Q , the imaginary part of the bare susceptibility Im χ0 (Q, ω) =

1 (1 − 2 fk ) [δ(ω − 2Ek ) − δ(ω + 2Ek )], 2

(7.57)

k

exhibits a jump at a continuum threshold Ωth (q) = |ΔkF | + |Δ(k+q)F | under the creation of two QPs at the FS. The jump of Im χ0 (Q, ω) leads to a logarithmic singularity in Re χ0 (Q, ω). This results in the appearance of the RM at a frequency Ωres (Q) determined by the equations 1 − gQ Re χ(Q, Ωres ) = 0,

Im χ0 (Q, Ωres ) = 0.

(7.58)

In this condition, the RM appears below the continuum threshold Ωth (q) and represents a spin-triplet exciton bound state. The bound state extends over a region near (π, π) determined by the q-dependence of the continuum threshold Ωth (q) for the d-wave SC gap Δ(k+q)F , which closes toward the nodal direction at the FS for some wave vector q0 . This results in the decrease of Ωth (q) away from Q and the downward dispersion of the RM. Calculations of the RPA susceptibility (7.55) for the Hubbard model within the FLEX approximation have shown that in the underdoped regime, the frequency of the resonance mode decreases with the decrease of the doping rate, while its intensity is increasing in agreement with the experiment [786]. In the underdoped region Ωres ∼ ωsf , at the optimal doping Ωres ∼ 2Δ0 − ωsf and in the overdoped region Ωres ∼ 2Δ0 . Here, ωsf is a characteristic frequency of the spin ﬂuctuations in the normal state where the spin response is overdamped: Im χ0 (q, ω) ∝ (ω/ωsf )/[1 + (ω/ωsf )2 ]. In a later study by Eremin et al. [312], both the downward and upward dispersion for the RM as a spin exciton were found. The obtained results are in accord with experimental data reported by Pailh`es et al. [944, 946] including the observation of the “silent” band at Q0 (see Sect. 3.2.3). The spinexciton theory of the RM was generalized for bilayer cuprate superconductors [313]. Magnetic resonance in the electron-doped cuprate superconductors was discussed by Ismer et al. [534] and Kr¨ uger et al. [665]. The interpretation of the resonance mode as a spin exciton was conﬁrmed in a number of studies (for references see [316, 700]).

7.2 Electron Excitations in the Normal State

423

However, in the underdoped region and close to the AF state the DSS RPAbased theories for the description of the spin-wave-like excitations of localized spins observed in cuprates become inadequate. In this region, a phenomenological spin-fermion model can be used where collective spin excitations and their interaction with low-energy fermionic quasiparticles are considered (see e.g., [2, 236, 863]). The occurrence of the RM in the model is also viewed as a spin-1 collective mode caused by the disappearance of the damping in the d-wave superconductor. In the model, the DSS close to the AF wave vector Q = (π, π) is given by χ(q, ω) =

1+

ξ 2 (q

−

Q)2

χ0 ξ 2 , − (ω/Δsw )2 − Π(q, ω)

(7.59)

where Δsw = csw /ξ and ξ is the AF correlation length. The spectrum of the spin-wave collective mode ω 2 (q) = Δ2sw + c2sw (q − Q)2 shows an upward dispersion with a gap Δsw = ω(Q) which closes at ξ → ∞ in the AF state. The polarization operator Π(q, ω) describes the spin-wave decay into a particlehole pair. The damping of the spin-wave given by ImΠ(q, ω) ω γq ξ 2 ≡ ω/ωs in the normal state at low frequencies is large and the DSS (7.59) shows an incoherent response: Im χ(q, ω) ∝ (ω/ωs )/[(1 + ξ 2 (q − Q)2 ]2 + (ω/ωs )2 ]. In the superconducting state, ImΠ(q, ω) vanishes below the continuum threshold Ωth (q) as χ0 (q, ω) (7.57) in the RPA. If the gap satisﬁes Δsw < Ωth (Q), the occurrence of the RM is possible below the threshold with a downward dispersion related to the q-dependence of the d-wave SC gap Δ(k+q)F like in the RPA scenario above. The frequency and intensity of the RM Ωq decrease when q deviates from Q and show a rapid drop close to the wave vector Q0 0.8(π, π) where the RM vanishes in agreement with neutron data [236]. In the limit of strong electron correlations, the t–J model was used to develop a microscopic theory of spin dynamics. Using the equation of motion method for the memory function, both the static and the dynamic spin properties were studied in a broad region of temperatures and doping, including the undoped region of spin waves. The theory is based on the Mori-type projection technique for the memory function, which enables to derive a general representation for the DSS (see e.g., [1128, 1316]): − χ(q, ω) = −Sq+ |S−q ω =

mq , ωq2 + ω Σ(q, ω) − ω 2

(7.60)

− − where mq = [iS˙ q+ , S−q ] = [[Sq+ , H], S−q ] . The static spin susceptibil2 ity χ(q, 0) = χq = mq /ωq is determined by the spin-excitation spectrum ωq calculated in a generalized mean-ﬁeld approximation (GMFA) within the mode-coupling approximation (MCA) allowing for vertex corrections (see e.g., [1365]). The self-energy is given by the “proper” part of the Kubo–Mori relax− ))ω , where the irreducible operators ation function Σ(q, ω) = (1/mq )((S¨q+ |S¨−q ± ± ¨ Sq = −[[Sq , H], H] are determined by the commutation of the spin-density operator Sq± with the t–J model Hamiltonian H written in the Hubbard

424

7 Theoretical Models of High-Tc Superconductivity

operator representation (see (A.23)). The spin-excitation spectrum is determined by the spectral function given by imaginary part of the DSS (7.60): χ (q, ω) = Im χ(q, ω + i0+ ). In the underdoped region, the interaction is weak and the spectrum of the spin-wave-like excitations ωq is determined mostly by the exchange interaction J. In the overdoped region, a strong interaction between spin excitations and holes results in the incoherent response. The static magnetic properties were determined by solving a self-consistent system of equations for the spectrum ωq and the spin correlation functions

− CR = (1/N ) q Cq exp(iqR), Cq = Sq+ S−q = (mq /2ωq ) coth(ωq /2T ) [1317]. At zero temperature, a long-range AF order characterized in the spin

rotation-invariant theory by the staggered magnetization m2 = (3/2N ) R CR exp(iQR) = (3/2) C = 0 (or ωQ = 0) was found for hole concentrations δ < δc where the critical hole concentration δc = 0.025, 0.05 for J/t = 0.2, 0.4, respectively. For δ > δc at T = 0 or nonzero temperatures, a ﬁnite AF 2 correlation length ξ 2 = 8J 2 α1 |C1 | /ωQ (where α1 2.3) is obtained which shows temperature and doping dependences similar to those observed in the INS (see Fig. 3.4). The uniform static susceptibility χ = (1/2) limq→0 mq /ωq2 shows temperature and doping dependences in qualitative agreement with ED studies and experiments on LSCO. − In the calculation of the self-energy Σ(q, ω) = (1/mq )((S¨q+ |S¨−q ))ω , the MCA for the many-particle time-dependent correlation functions was used similar to (A.54). There are four

terms in the force operator, Fi = −S¨i+ = + [[Si , (Ht + HJ )], (Ht + HJ )] ≡ α Fi,α , α = tt, tJ, Jt, JJ corresponding to commutation with the hopping Ht and the exchange HJ terms in the t–J model. In comparison with previous studies based on the memory function method (e.g., [1025, 1128, 1154]), all 16 contributions to the self-energy Σα,β (q, ω) were taken into account from various combinations of the forces Fi,α and their temperature and doping dependence were thoroughly analyzed [1317]. It was found that the largest contributions come from the diagonal terms, ΣJJ,JJ ≡ ΣJ and Σtt,tt ≡ Σt . Depending on the damping of spin excitations, determined by the imaginary part of the self-energy, Γq = −(1/2)Σ (q, ωq ) , a smooth transition from the spin-wave-like excitations for (Γq /ωq ) 1 to the overdamped incoherent spin excitations occurs for (Γq /ωq ) ≥ 1. At low temperatures, (T J), the spin–spin scattering ΣJ is weak and for low doping, δ < 0.05, when the spin-hole scattering Σt is also weak, well-deﬁned spin-wave excitations are observed similar to the Heisenberg model at δ = 0 (see [783]). However, at higher doping Σt becomes large and already at moderate hole concentrations it exceeds by far the spin–spin scattering ΣJ . Figure 7.12a shows the spectrum of the spin excitations ωq and the damping ΓJ,q = −(1/2)ΣJ (q, ωq ) (dotted curve) and Γt,q = −(1/2)Σt(q, ωq ) (dashed curve) calculated at hole doping δ = 0.1 for J = 0.3t. The damping increases rapidly with doping and temperature and in the overdoped range the spectrum of the spin excitations reveals a broad distribution in energy even at the AF wave vector Q = (π, π). The calculated spectral function χ (q, ω) for spin-excitations are in good agreement

7.2 Electron Excitations in the Normal State

425

Fig. 7.12. (a) Spin-excitation spectrum ωq along symmetry directions (solid line) and damping Γt,q (dashed line) and ΓJ,q (dotted line) in units of t. (b) Im χ(q, ω) for various wave vectors (qx , qy ) = (n, m) π/5 in comparison with ED data (symbols, Prelovˇsek et al. [1025]) [1317]

with the numerical results obtained for the t–J model by an exact diagonalization (ED) method by Prelovˇsek et al. [1025] as demonstrated in Fig. 7.12b for various wave vectors at T = 0.15t and δ = 0.1. The theory

displays also a scaling behavior for the local susceptibility χL (ω) = (1/N ) q χ (q, ω). The scaling function f (ω/T ) = χL (ω, T )/χL (ω, T = 0) shows a good agreement with the data of the neutron-scattering experiments on La1.96 Sr0.04 CuO4 [595] (see Fig. 3.5). The DSS in the superconducting state and the RM within the memory function approach were studied by several groups (see [1027,1128,1129,1152– 1154]). An estimation of the damping Γq (ω) = −(1/2)Σ (q, ω) in the DSS (7.60) within the particle-hole bubble approximation by Sega et al. [1128,1129] revealed the results similar to the spin-exciton scenario within the RPA for the DSS (7.55), (7.56). A self-consistent solution of the system of equations for the single-electron GFs and the DSS within the Mori projection technique by Sherman et al. [1152–1154] has shown that the RM can be explained without evoking the spin-exciton model but by considering the doping and wave vector dependence of the spin-excitation damping near the AF wave vector. Here, we present the theory of the RM based on the extension of the microscopic theory of the DSS [1317] discussed above to the superconducting state (Vladimirov et al. 2010). It was found that the RM at low temperatures is caused by a strong suppression of the spin-excitation damping where the spin gap plays the major role instead of the superconducting gap considered in the spin-exciton scenario. To explain this result, let us consider the spin-excitation damping given by the spin-hole scattering Σt (q, ω). At a hole doping δ ≥ 0.05, where the superconducting state appears, this is the largest contribution to the self-energy. In the superconducting state, it can be written as

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7 Theoretical Models of High-Tc Superconductivity

π(2t)4 (eβω − 1) 1 m(q) ω N2 ∞ × dω1 dω2 dω3 N (ω2 )[1 − n(ω1 )]n(ω3 )Bq2 (ω2 )

Σt (q, ω) = −

q1 ,q2

−∞

N × (Λ2q1 ,q2 ,q3 + Λ2q3 ,q2 ,q1 ) AN q1 (ω1 ) Aq3 (ω3 ) − 2Λq1,q2 ,q3 Λq3 ,q2 ,q1 ASq1 σ (ω1 ) ASq3 σ (ω3 ) × δ(ω + ω1 − ω2 − ω3 ),

(7.61)

where q3 = q−q1 −q2 and n(ω) = (eβω +1)−1 and N (ω) = (eβω −1)−1 are the Fermi- and Bose-functions, respectively, and Λq1 q2 q3 = 4(γq3 +q2 − γq1 ) γq3 + γq2 − γq1 +q3 , where γq = (1/2) (cos qx + cos qy ). Here, the electronic spectral 0σ σ0 functions are deﬁned by the normal GF, AN q (ω) = −(1/π)ImXq |Xq ω , S 0σ 0¯ σ and the anomalous (pair) GF, Aqσ (ω) = −(1/π)ImXq |X−q ω (see Sect. A.4). In the superconducting state, the spectrum of electron excitations are deﬁned by the Bogoliubov QPs, ω = ±Eq as in (7.56). The spectral density of spin excitations Bq (ω) = (1/π) χ (q, ω) is approximated by the renormalized spin energy, ω = ± ωq where ω q is the pole of the DSS (7.60). Numerical computations of the self-energy (7.61) reveal quite a diﬀerent behavior of the spin-excitation damping Γ (q, ω) = −(1/2) Σt (q, ω) in the superconducting state in comparison with the particle-hole bubble approximation used in the RPA. Instead of a jump at a continuum threshold Ωth (q), as in (7.57), the damping shows a smooth increase with the energy of excitations ω and a deep minimum at the AF wave vector Q, which results in the appearance of the RM at Ωres ω Q . This diﬀerence is explained by occurrence of an additional spin excitation in the decay process in (7.61) besides the particle-hole pair contribution considered in (7.57). In this case, the decay process of the spin excitation with the energy ω(Q) at the wave vector Q at low temperatures is governed by a more general energy-conservation law, ω(Q) = Eq3 + Eq1 + ω q2 , than in the particle-hole bubble approximation, ω(Q) = EQ+q + Eq . Since the largest contribution from the spectral density Bq2 (ω2 ) in (7.61) comes from q2 Q, the energy-momentum conservation law strongly reduces the phase space for the decay, where Eq3 + Eq1 0 if we choose ω(Q) ω Q . In fact, the occurrence of an additional spin excitation in the scattering process with the ﬁnite energy ω Q Ωres plays a role similar to the superconducting gap 2Δk in the excitation of the particle-hole pair in (7.57). In the low doping region where the damping is extremely small, the RM is found even above Tc since a superconducting gap in the particle-hole continuum is much smaller than the spin gap and does not play an essential role. In the overdoped region, at hole concentration δ ∼ 0.2 and high Tc , the spinexcitation damping becomes large and opening of the superconducting gap enhances the intensity of the RM below Tc . This behavior is demonstrated in Fig. 7.13 where the spectral functions χ (Q, ω) are depicted. In the underdoped region, δ = 0.09, the RM Ωr does not change with temperature and

7.2 Electron Excitations in the Normal State

427

Fig. 7.13. Temperature dependence of the spectral function χ (Q, ω) for δ = 0.09 (a) and for δ = 0.2 (b) [1319]

is seen even in the normal phase at T = 1.4Tc in a good agreement with experiment ([1190], see Fig. 3.20). In the overdoped region, δ = 0.2, the RM Ωr is only slightly softens at T = 0.4Tc but becomes overdamped at Tc in accordance with experiment [170]. Due to the important role of gapped spin excitations in the damping of the RM, its energy Ωr does not critically depend on the superconducting gap energy 2Δk (T ) and hence, on the temperature and peculiarities of the electronic spectrum in cuprates, contrary to the theories based on the calculation of the damping within the particle-hole bubble approximation (see e.g., [312, 927, 1128]). Studies of the DSS in the Hubbard model in the limit of strong correlations is more involved and a two-pole approximation for spin and charge collective excitations is often used. In this approximation, the static properties of the model, namely, the spin and charge correlation functions, the static susceptibility and AF correlations show a good agreement with the numerical results for ﬁnite clusters [85, 782]. A diagram technique was used by Sherman et al. [1155] to calculate the magnetic susceptibility of the two-dimensional Hubbard model. Resume Summarizing the discussion in this section, we can point out that in the overdoped region the spin-excitation spectrum found within the RPA theory (7.55) qualitatively agrees with those obtained in the strong correlation limit for the t–J model in the memory function method (7.60). However, in the underdoped region, the RPA approach fails to reproduce the spin-wave-like excitations, while the memory function theory for the dynamical spin susceptibility (7.60) provides a reliable approach to describe spin dynamics in a broad region of temperatures and doping. The RM at the AF wave vector and the spin gap observed in the INS studies at low temperatures in cuprates can be explained by a strong suppression of the spin-excitation damping. In the spin-exciton scenario, an opening of a superconducting d-wave gap plays

428

7 Theoretical Models of High-Tc Superconductivity

the major role in the appearance of the RM. In a more general consideration, a weak spin-excitation damping is explained by an involvement of a gapped spin excitation in the decay process that results in the temperature independent RM. The role of AF spin ﬂuctuations in the superconducting pairing is discussed in Sect. 7.3.2.

7.3 Magnetic Mechanism of Superconductivity The experimental studies of the electronic properties of the cuprate superconductors considered in Chap. 5 have shown that the electron interaction with the spin-ﬂuctuations play an essential role in their anomalous normal state properties. Therefore, it is natural to assume that the same interaction may be responsible for superconductivity in cuprates materials. A number of various “magnetic” mechanisms of superconductivity has been proposed some of which are discussed in this section. They are based on the “doped Mott insulator” approach where only the electronic, spin or charge, degrees of freedom are taken into account. 7.3.1 Unconventional Ground State Anderson was the ﬁrst to note the possibility of an unconventional ground state and related to it a new mechanism of superconductivity in copperoxide compounds [61, 65]. In the Heisenberg model, the antiferromagnetic (AF) exchange interaction J in the three-dimensional lattice leads to the N´eel ground state with a long-range AF order. For a lower dimension, strong quantum ﬂuctuations for spin 1/2 can destroy the AF order and new phases can emerge. The hole doping further weakens the spin interaction and leads to frustration of the spin ordering. Anderson has suggested that in a twodimensional lattice the N´eel ground state is destroyed under hole doping and a disordered state in the form of a spin liquid of resonating valence bonds (RVB) occurs (for reviews, see [700, 702, 706, 912]). The RVB state is a quantum state of spin singlets b†ij for a pair of electrons on the lattice sites (i, j) coupled by the exchange interaction † 1 HJ = −J bij bij , b†ij = √ c†i↑ c†j↓ − c†i↓ c†j↑ (7.62) 2 i=j This very interaction is responsible for the creation of bound RVB states + = 0. The breaking of a singlet bond in (7.62) leads to the appearance of bij two Fermi excitations with spin 1/2 without charge instead of S = 1 magnons in the AF state. These neutral fermions were called “spinons” and they were suggested to form a Fermi surface (FS) of extended s-wave symmetry [111]. Later on a gapped d-wave FS was proposed by Kotliar et al. [652]. Upon doping, a new type of excitation, “holon”, which is a spinless hole with a

7.3 Magnetic Mechanism of Superconductivity

429

positive charge, i.e., a charged Bose quasiparticle, arises. Thus, the separation of the spin and charge degrees of freedom has been conjectured in analogy with the Luttinger liquid behavior in the one-dimensional systems [62] (see Sect. 7.1.1). The spin-singlet bound state formation leads to a pseudogap in the electronic spectrum and a concomitant reduction in the uniform magnetic susceptibility. At the same time, the transport properties are determined by the doped holes as vacancies in the spin liquid. This explains why the transport coeﬃcients are proportional to the hole concentration (see [68]). Slave-Particle Approach To treat the RVB state, various types of slave-boson (-fermion) representation for spin and charge degrees of freedom were considered (see e.g., [331,368,912, 1014, 1075, 1209], and references therein). In the simplest version of the slaveboson theory, the projected electron operators cˆ†iσ in the t–J model (7.16) are † and boson (holon) bi operators: replaced by a product of fermion (spinon) fiσ † + cˆiσ = fiσ bi . However, to reduce the enlarged Hilbert space required for the spinon–holon representation (four states per site) to the physical one of the projected electronic state (three states per site) one has to introduce a local constraint at each lattice site: † q i = b+ fiσ fiσ = 1. (7.63) i bi + σ

Using this representation, the Hamiltonian of the t–J model (7.16) can be written in the form † J + (bj bi χij + H.c.) − χij χij H = −t 2 i=j i=j † † −μ fiσ fiσ − λi (qi − 1), χij = Σσ fiσ fjσ , (7.64) iσ

i

where χij is the spinon operator for a bond (i, j) of the nearest neighbors. The Lagrange multiplier λi takes into account the local constraint (7.63). In the mean-ﬁeld approximation (MFA), the local constraint is substituted by a global one, q = qi = 1 and two-order parameters are introduced: † χ ˜ij = Σσ fiσ fjσ ,

† † † † Δij = fi↑ fj↓ − fi↓ fj↑ .

(7.65)

˜ several phases In the RVB state with a uniform order parameter χ ˜ij = χ can occur depending on the temperature T and the hole concentration x. The singlet d-wave pairing for fermions with the order parameter Δij of the d-wave (or s-wave) symmetry may appear below a temperature T ∗ ∝ J(1 − x/xc ). A bose-condensation bi = b0 ∝ x leads to the superconducting phase transition † † f−k↓ . Thus, in this for electrons with the order parameter ˆ c†k↑ cˆ†−k↓ = b20 fk↑

430

7 Theoretical Models of High-Tc Superconductivity

approximation both the pseudogap phase below T ∗ and the superconductivity with Tc ∝ x in the region T < T ∗ are observed. A variety of nonuniform RVB phases were also found with close ground state energies. In particular, a staggered ﬂux phase characterized by the order parameter χ ˜ij = χ0 exp{i(−1)ix +jy Φ0 } reveals a nonzero ﬂow of the ﬂux ﬁeld in going around a closed contour on the two-dimensional lattice [20, 1352]. An alternative description of the ﬂux phases is possible with the help of particles with fractional statistics, the so-called anyons, in the two-dimensional lattice. Under the permutation of two anyons, their wave function gets the supplementary phase factor exp(iΘ). In the case of Fermi and Bose particles, Θ = π and Θ = 0, respectively. The angle Θ is arbitrary for anyons, and these particles are called semions for Θ = ±π/2. One of the main properties of the ﬂux phases is the breaking of the time (T ) and spatial (P ) parity. The chiral spin liquid [1352], which can be characterized by the order parameter aijk = Si [Sj × Sk ] for noncoplanar spins Sl at the sites i, j, k of a two-dimensional lattice, is a simple example of a ground state with T - and P -broken symmetry. The superconductivity in a system of anyons can be related to a transition to a T - and P -broken state below Tc . In this respect, the model of intracell orbital currents which ﬂow between the Cu–O and O– O bonds suggested by Varma [1307] should be mentioned. Another model of time-reversal symmetry broken state is the d-density wave state (DDW), which is characterized by staggered orbital current order [220]. The coexistence of the DDW and superconductivity was found by Eremin et al. [310] for the extended t–J model in a certain range of hole doping which could be related to a pseudogap formation. Experimental investigations, however, have not produced reliable results concerning the time-reversal symmetry breaking or the occurrence of spontaneous moments due to orbital currents in cuprates (see e.g., [163] and references therein). Because in the MFA the local constraint (7.63) is taken into account in average, the enlargement of the Hilbert space in the slave-boson (-fermion) representation results in controversial representations of the basic physical parameters. For instance, in the slave-fermion hard-core (CP1 ) boson representation considered by Feng et al. [331], the charge and spin degrees of freedom are represented as products of a spinless fermion h†i for the charge degree of freedom (holon) and a hard-core boson biσ for the spin degree of freedom (spinon): cˆ†iσ = hi b†iσ , cˆiσ = h†i biσ . This representation has some advantage over other slave-particle representations since the constraint of no double occupancy can be fulﬁlled without introducing the Lagrange multiplier. The hard-core bosons b†iσ anticommute on at a same lattice site prohibiting double occupancy: † † cˆiσ cˆiσ = hi h†i biσ biσ = 1 − h†i hi ≤ 1, (7.66)

† σ biσ biσ

σ

σ

since = 1. However, the spin-charge separation imposed by this representation results in extra degrees of freedom: a spin 1/2 is assigned to any

7.3 Magnetic Mechanism of Superconductivity

431

lattice site – including an empty site, while for the projected electron operators we have only 3 states: an empty state and a ﬁlled state with spin ±1/2. For a rigorous treatment of this problem, one should introduce a projection operator to exclude the unphysical states. Otherwise, the commutation relations for the original projected electron operators and their slave-boson representation give diﬀerent results. For instance, the double counting of the empty sites occupied by bosons results in controversial equations for the average number of electrons entailing errors of the order of the hole concentration δ . This violates also the sum rule for the single-electron Green function (for details, see [1014]). To treat the constraint in a systematic way, a large-N expansion was proposed [652, 1075] with N/2 being a number of states (orbitals) at a lattice site. In those approaches, the constraint (7.63) is relaxed to a

local + much weaker one, qi = b+ i bi + σ fiσ fiσ = N/2. Using the 1/N expansion, the d-wave superconducting instability induced by the superexchange interaction was found in the generalized t–J model close to half ﬁlling [412]. The Baym–Kadanoﬀ variational technique for the Green functions in terms of the Hubbard operators within the 1/N expansion was also applied to study the superconducting pairing in the t–J model. It was observed that in the lowest order of 1/N there is a strong compensation of the diﬀerent contributions to the pairing interaction and at inﬁnite U (J → 0) the superconducting Tc is extremely small [408]. At a ﬁnite exchange interaction J, the d-wave superconducting instability mediated by the exchange and spin- and chargeﬂuctuations was obtained below Tc 0.01t [1417]. It was also proved that the results in the Hubbard operator technique diﬀer from those in the slave-boson representation even in the same order of 1/N expansion due to the diﬀerent Hilbert spaces used in the two approaches. Actually, in the limit of large N similar to the slave-boson method, the kinematic interaction is suppressed. This results, in particular, in suppression of the spin-ﬂuctuation contribution, which appears in the order 1/N 2 in comparison with the charge-ﬂuctuation one in the order 1/N [410]. Therefore, in spite of being rigorous in the limit N → ∞, this approach is diﬃcult to extrapolate to real spin systems with N = 2 where the spin-ﬂuctuation pairing contribution plays the major role. To go beyond the MFA in the slave-particle method, one should consider the ﬂuctuations of the RVB order parameters and the Lagrange factor λi in (7.64). As proposed by Baskaran et al. [112], this can be performed by the introduction of gauge ﬁelds. A time component of the gauge ﬁeld was connected with the ﬂuctuations of λi and spatial components were related to the phase of the RVB order parameters which were treated ﬁrst at a Gaussian level [524, 880]). Within the gauge ﬁeld approach, the investigation of twodimensional systems within the model (7.64) is reduced to the study of the gauge ﬁelds in the space with dimensionality (2 + 1). An application of the gauge theory to the high-Tc superconductivity problem is discussed in detail by Lee et al. [700].

432

7 Theoretical Models of High-Tc Superconductivity

Variational Approach To overcome the complications induced by the unconventional commutation relations of the original projected electron operators, i.e., the Hubbard operators, a variational approach has been developed for the t–J model within the RVB ground state (see [68, 706, 912, 1420], and references therein). The basic idea in the variational theory is to use conventional electron Fermi operators cˆiσ → ciσ in the original model (7.16) but with a renormalized hopping parameter t˜ = gt t and a superexchange AF interaction J˜ = gJ J. By using the Gutzviller variational wave function, these parameters were estimated as follows: gt = 2δ/(1 + δ) and gJ = 4/(1 + δ)2 where δ is the hole concentration. A self-consistent system of equations for two parameters, the Fock exchange self-energy χij = c†iσ cjσ and the gap function Δij = c†i↑ c†j↓ , was solved for the t–J model written in terms of the conventional Fermi operators but with ˜ The gap function was related to the pseudogap the eﬀective interactions t˜, J. characterized by the doping dependence Δij ∝ J(1 − δ/δc ) with δc ∼ 0.3. To calculate the superconducting order parameter, the projected character of c†i↑ cˆ†j↓ ∼ gt Δij ∝ δΔij the BCS state should be taken into account: ΔSC ij = ˆ which vanishes at δ → 0. In fact, the renormalized mean-ﬁeld approximation gives results similar to the slave-particle approach (cf. 7.65). In a more accurate approach, an eﬀective Hubbard model within the variational Monte Carlo method was considered by Paramekanti et al. [960, 961]. To exclude transitions to doubly occupied states, a unitary transformation for the Hubbard Hamiltonian was performed: H = exp(−iS) H exp(iS). The factor S was determined perturbatively in (t/U ) in the second order of t2 /U . In this way, three-site hopping terms were also taken into account contrary to the original t–J model (7.16) where only two-site hopping is considered. In the calculations, a trial variational wave function in the form of the Gutzwiller projected BCS wave function was used: (1 − ni↑ ni↓ ), |Ψ0 = exp(−iS) P|ΦBCS , P = |ΦBCS =

k

i

N/2 ϕ(k)c†k↑ c†−k↓

|0,

(7.67)

where the Gutzwiller projector P excludes the doubly occupied states. Extensive numerical studies have been performed to evaluate the doping dependence of various physical quantities at T = 0. In particular, it was found that in the underdoped region the pairing strength is strong but the phase stiﬀness is very weak leading to a pairing pseudogap above Tc . The superconducting d-wave pairing was observed for doping 0 < δ < δc = 0.35 with a transition to a Fermi liquid metal at δ > δc . The Drude weight appeared to be proportional to the spectral weight of nodal quasiparticles which vanished with underdoping, δ → 0, while the Fermi surface remained large, including 1 + δ holes. By using a fully projected Gutzwiller wave function in (7.67) a spin-liquid

7.3 Magnetic Mechanism of Superconductivity

433

Mott insulator state was found at δ → 0 with a ﬁnite eﬀective mass m∗ and the nodal Fermi velocity. The obtained results were found in a good agreement with experimental data. The variational approach based on the trial wave function (7.67) was used by many authors to calculate various physical properties of a strongly correlated d-wave superconductor (for reviews, see [301, 912]). A partially projected Gutzwiller wave function with P = i (1−α ni↑ ni↓ ), with α as a variational parameter was considered by Laughlin [694]. The solution reveals a BCS-like superconducting state with a very thin superﬂuid density in strongly correlated system, a “gossamer” superconductivity, as coined by Laughlin (see also [1423]). In this model, the superconducting state exists even at zero doping, δ = 0, contrary to the Mott insulating state found in fully projected Gutzwiller wave function in (7.67). The well-known limitation of the Raleigh–Ritz variational methods, however, comes from the introduction of a particular variational wave function which may be far away from the real ground state of the complicated system under consideration. Inhomogeneous Ground State As was discussed in Sect. 3.2.4, the observation of incommensurate spin and charge density waves, or stripes, in LBCO cuprates suggests the possibility of an inhomogeneous ground state. It was argued that electronic states in cuprates cannot be described as a gas of quasiparticles within the Fermi-liquid theory but rather as a collective quantum state in the form of ﬂuctuating dynamical stripes [205, 308, 1400]. It is then precisely this inhomogeneity that induces the pairing mechanism in HTSC (see [616] and references therein). In this approach, the pairing results from strong correlations in low dimensional systems as cuprates contrary to the BCS pairing of well-deﬁned and essentially free quasiparticles. However, the static stripes are detrimental to superconductivity as observed in the LBCO or LSCO one-layer systems, where the stripes have been clearly evidenced. In the YBCO compounds, the dynamical stripes can be suggested in the underdoped phase only (see Sect. 3.2.2) and it is unlikely that they could explain the superconductivity in the overdoped region. The importance of the ﬂuctuations of the superconducting order parameter was pointed out originally by Emery et al. [309]. It was argued that the two-component superconducting order parameter Ψ = Δ0 exp[iΘ] is determined by the maximum gap Δ0 and its phase Θ (see Sect. 4.1). Therefore, two characteristic temperatures can be introduced: the temperature of the pair formation Δ0 estimated by the mean-ﬁeld superconducting temperature (0) Tc and the temperature of the phase coherence TΘ below which the bosecondensation of pairs occurs. The latter temperature is proportional to the superﬂuid density TΘ ∝ ns /m ∝ λ−2 (0) which can be estimated from the magnetic penetration depth λ(0) at zero temperature. Whereas in conventional superconductors with high superﬂuid density ns the phase coherence

434

7 Theoretical Models of High-Tc Superconductivity

temperature TΘ is enormous and much higher than Tc , in the cuprate superconductors (and organic materials) with low concentrations of charge carriers TΘ can be close or even smaller than Tc [309]. Therefore, the observed Tc in the underdoped region is restricted by TΘ ∝ ns , while in the overdoped region it is determined by the energy scale related to the pair strength Δ0 . The pseu(0) dogap temperature T ∗ in the underdoped region in this scenario is set by Tc which may be high enough for a strong pairing interaction. An extensive theoretical review of precursor phase ﬂuctuations as a mechanism for pseudogap formation in low-dimensional systems is given by Loktev et al. [739]. Resume In summarizing the discussion of unconventional ground states, in general the proposed models are essentially based on strong electron correlations and low dimensionality. However, a number of unusual properties found for the one-dimensional Hubbard model or the t–J model have not been rigorously proved within the two-dimensional (2D) theory. Moreover, the real cuprate superconductors are three-dimensional (3D) systems, in particular the YBCO compounds with a modest 3D anisotropy. Occurrence of two phase transitions, to the pseudogap and the superconducting phases, suggested in the RVB theory, in the DDW phase and other “hidden” quantum phase transitions are not clearly detected in experiments: the pseudogap phase is a crossover without distinct anomaly in physical properties. Therefore, physical properties, predicted in the unconventional ground states, require further experimental veriﬁcation. 7.3.2 Spin-Fluctuation Pairing As was shown in the previous sections, the antiferromagnetic (AF) spin ﬂuctuations are important in explaining many of the anomalous properties of the cuprate superconductors in the normal phase. It is thus natural to suppose that these boson excitations are also responsible for the superconducting pairing in the cuprates. Earlier the magnetic pairing mechanism was proposed for systems with heavy fermions [259, 844, 1103, 1104] where the d-wave pairing was suggested. Later on, it was proposed that this mechanism could be responsible for the high-temperature superconductivity as well [133]. It was shown that, under the exchange of AF paramagnons, an attraction appears in the d-channel and acts most eﬀectively near the AF instability. The spinﬂuctuation pairing mechanism was considered by several groups within a phenomenological approach (for reviews, see [237, 544, 861, 862, 1105]). Interesting historical remarks concerning interrelation between superconductivity and magnetism were given by Norman [903], who also discussed distinctions between the spin-ﬂuctuation pairing and the singlet formation in the RVB theory.

7.3 Magnetic Mechanism of Superconductivity

435

To elucidate why the spin ﬂuctuation mechanism for the d-wave symmetry pairing is more favorable, we consider a simple BCS-type equation for the superconducting gap Δk (for detail, see [1105]): Δ(k) =

Δ(k ) E(k ) 1 V (k, k ) tanh , N q 2E(k ) 2T

(7.68)

where E(k) = ε2k + |Δ(k)|2 . In the case of the electron–phonon pairing, 2 V (k, k ) = ge−ph (k, k ) χph (k − k ) is determined by the electron–phonon matrix element ge−ph (k, k ) and the phonon susceptibility χph (k − k ) > 0. The largest contribution in this case arises from the isotropic component g0 of the matrix

element decomposition over the Legendre polynomials: ge−ph (k, k ) = l gl Pl (θ) where θ = (k · k )/kF2 (see e.g., [5]). To get a solution for the d-wave gap Δ(k) ∝ (cos kx − cos ky ), the interaction V (k, k ) should be anisotropic which can occur only for particular phonon modes. However, the contribution from the isotropic component g0 results in a suppression of the quasiparticle weight Z and a weak coupling in other channels with l = 0 (see Sect. 7.4.1). In the case of the spin-ﬂuctuation pairing, 2 V (k, k ) = −gsf (k, k ) χsf (k − k ), where the antiferromagnetic (AF) spin susceptibility χsf (q) > 0 has a maximum at the AF wave vector q = Q, while 2 2 gsf (k, k ) ∼ gsf (k − k = Q). The negative sign of the interaction V (k, k ) is due to the spin-ﬂip of the electrons in the Cooper pair at spin-ﬂuctuation scattering. Therefore, only the sign-dependent solution for the gap Δ(k) as, e.g., for the d-wave symmetry, can be obtained. In this case, the pairing occurs on diﬀerent lattice sites and therefore the strong on-site Coulomb repulsion as well as an isotropic interactions induced by phonons are canceled out that also favors the spin-ﬂuctuation pairing. Phenomenological Approach For the antiferromagnetic (AF) ground state, the transverse ﬂuctuations of the AF order parameter ensure an eﬀective attraction which is favorable for a magnetic pairing mechanism. To describe the pairing in the AF state, the “spin bag” concept was developed by Schrieﬀer et al. [1125]. This approach was based on the assumption of the local depression by a hole of the AF gap in the electron spectrum. As a result, a magnetic polaron – a spin bag, which moves together with the cloud of spin deformation, appears. In this case, two holes, i.e., polarons, attract each other due to the overlap of their regions of deformation of the AF order. An extended analysis has shown that the longitudinal spin ﬂuctuations lead to a singlet pairing with the maximum contribution coming from the d-wave channel. Although in the copper-oxide compounds, the superconductivity arises for hole concentrations nh ≥ 0.05, where the long-range AF order is already destroyed, due to the small superconducting correlation length ξ0 , the theory can be applied in the region ξN > ξ0 (where ξN is the AF correlation length). In the paramagnetic phase, the contribution of the spin ﬂuctuations near the AF wave vector Q = (π, π)

436

7 Theoretical Models of High-Tc Superconductivity

leads to the appearance of a pseudogap in the electron spectrum near halfﬁlling [582, 583]. The additional exchange of the AF spin ﬂuctuations for two quasiparticles (QPs), the “spin bags”, decreases their energy, similar to the case of the long-range AF order, and results in their mutual attraction. The interaction between QPs in a two-dimensional square lattice under the exchange of AF paramagnons was most extensively studied by Pines et al. (see [237,988,989]). They have considered the model of a nearly antiferromagnetic Fermi liquid (NAFL), proposed by Millis, Monien and Pines (MMP) (see → → σ2 Sect. 3.3.2). The interaction between two electrons (holes) with spins − σ 1, − mediated by an exchange of spin ﬂuctuations was written as: − → σ1 ·− σ 2. Vsf (q, ω) = g 2 χ(q, ω) →

(7.69)

The dynamical spin susceptibility described by the MMP model is given by χMMP (q, ω) =

αξ 2 , 1 + ξ 2 (Q − q)2 − iω/ωsf

(7.70)

where ξ(T ) is the AF correlation length and ¯hωsf ∝ 1/ξ 2 is the typical energy of AF ﬂuctuation (see (3.53)). The spin susceptibility (7.70) at large values of ξ(T ) can reach high values at the AF wave vector q = Q = (π, π). The authors solved numerically the strong-coupling Eliashberg-type equations in the (q, ω)-space and obtained superconductivity in the d-wave channel with Tc ∼ 90 K for a comparatively modest value of the coupling constant g [849, 852]. Studies of the temperature dependence of the superconducting gap Δ(T ) have shown that near Tc the gap grows rapidly when the temperature decreases and reaches a maximum value 2Δ(0)/kTc = 6 − 8, which agrees with experiments. In these calculations, the high Tc and the d-wave pairing are conditioned by a strongly anisotropic interaction due to the AF spin ﬂuctuations. Strong enhancement of the spin susceptibility (7.70) near the AF wave vector Q is capable of bringing about high Tc in spite of strong pairbraking eﬀects due to spin scattering. Studies of the normal state properties of the cuprate superconductors have supported the NAFL model [852]. Therefore, the authors have concluded that the mechanism for high-Tc is electronic and magnetic in origin. An important question of the vertex corrections in the spin-ﬂuctuation Elishberg theory was studied by Monthoux [853]. It was shown that at a large AF correlation length the renormalization is strong but for a short AF correlation length in doped cuprates the renormalization is weak and even leads to a stronger pairing potential in the d-wave channel in comparison with a one-loop approximation in the Eliashberg theory. Whereas in the NAFL theory the phenomenological spin ﬂuctuation susceptibility (7.70) was used as an input to the model, in the spin-fermion model the interaction of the low-energy fermionic quasiparticles with their own collective spin excitations was considered self-consistently (for a review, see [237]). The model contains only a small number of parameters which can be ﬁxed from experimental data. It has been shown that the low-lying spin excitations described by the spin susceptibility (7.59) are strongly overdamped

7.3 Magnetic Mechanism of Superconductivity

437

due to their decays into particle-hole pairs for a Fermi surface with hot spots. These overdamped low-lying spin excitations appear to be more eﬃcient for spin-mediated pairing in comparison with the magnon-like spin waves with high energy close to the AF transition. As pointed out by Schrieﬀer [1126], in the latter case a near cancelation occurs between the enhancement of the spin susceptibility and the strong reduction in the fermion–magnon interaction at the AF wave vector Q which greatly reduces the d-wave pairing interaction. For the overdamped collective spin ﬂuctuations, the spin-fermion vertex does not vanish at the magnetic transition and therefore the enhancement of the spin susceptibility (7.59) near Q results in the increase of the pairing interaction. In the spin-fermion model, the feedback from quasiparticle pairing to the form of the spin susceptibility and the resulting pairing interaction (7.69) was also studied. However, several critical remarks concerning the NAFL and spin-fermion model should be mentioned. It emphasized by Anderson [66], in the conventional metals only ferromagnetic correlations between electrons are observed, while the antiferromagnetic coupling results from the superexchange as described by the t–J model (7.16). But in the latter case, the existence of the projection operator on the hopping kinetic terms should be taken into account. This strongly complicates the theoretical treatment of the spinfermion model since a simple diagram technique for fermions and bosons becomes inapplicable as discussed in the previous section. A proper approach for the spin-ﬂuctuation pairing in the case of the model with strong electron correlations, the t–J and Hubbard models will be considered in Sect. 7.3.3. An important results supporting the spin-ﬂuctuation pairing mechanism in cuprates were found by [268]. By using ARPES and inelastic magnetic neutron scattering experiments done on the same twin-free Y Ba2 Cu3 O6.6 crystal, a direct evaluation of the eﬀective spin-ﬂuctuation pairing interaction (7.69) was performed. A numerical solution of the strong-coupling equations for the superconducting transition temperature for this interaction results in a quite high Tc ∼ 150 K. Weak Correlation Limit In the weak correlation limit, U t, a microscopic theory based on the perturbation approach for the Hubbard model (7.3) can be developed. In a number of studies, a self-consistent system of equations in the ﬂuctuationexchange approximation (FLEX) [134] for the single-electron Green function and the spin and charge susceptibility were numerically solved (see [266, 267, 708, 851, 861, 862, 959], and references therein). The FLEX approximation was also applied for the three-band p–d model with Coulomb repulsion U on copper sites [1219, 1220] and for electron doped cuprates [785]. Within this approach, a qualitative description of the copper-oxide materials was obtained at moderate and large doping regimes under intermediate values of the Coulomb repulsion, U ≤ W/2 = 4t. However, starting from the Fermiliquid picture, the theory fails to describe the underdoped regime with strong correlations and large values of the Coulomb repulsion, U ≥ W , close to the

438

7 Theoretical Models of High-Tc Superconductivity

insulating (and AF) state. A comparison of the FLEX results for the Hubbard model with the quantum Monte Carlo calculations have shown similar results for the single-particle density of states for U ≤ 4t but a qualitative diﬀerence for the larger value U = 8t [190]. In the framework of FLEX, a self-consistent system of equations for the diagonal G(p, ωn ) and the oﬀ-diagonal F (p, ωn ) single-electron Green functions was written as (see e.g., [851]): iωn Z(p, ωn ) + X(p, ωn ) , [iωn Z(p, ωn )]2 − X 2 (p, ωn ) − φ2 (p, ωn ) φ(p, ωn ) F (p, ωn ) = , [iωn Z(p, ωn )]2 − X 2 (p, ωn ) − φ2 (p, ωn )

G(p, ωn ) =

(7.71)

where X(p, ωn ) = p − μ + ξ(p, ωn ). Here, we used the standard notation for the renormalization parameter Z(p, ωn ), the energy shift ξ(p, ωn ), and the gap parameter φ(p, ωn ) in the Eliashberg-type equations for the Nambu frequencies ωn = πT (2n + 1) and the 2D wave vector p = (px , py ) (see Appendix (A.32)). The interaction is mediated by the spin Vs (q, ωn ) and the charge Vc (q, ωn ) ﬂuctuations calculated in the random phase approximation (RPA): Vs (q, ωn ) = (3/2) U 2 χs0 [1 − U χs0 ]−1 − (1/2)U 2 χs0 , Vc (q, ωn ) = (1/2) U 2 χc0 [1 + U χc0 ]−1 − (1/2)U 2 χc0 .

(7.72)

The irreducible spin χs0 (q, ωn ) and charge χc0 (q, ωn ) susceptibility in the RPA formulas are deﬁned self-consistently in a one-loop approximation by using the Green functions (7.71): χs,c 0 (q, ωn ) = −T

[G(k + q, ωn + ωm )G(k, ωm )

k,m

±F (k + q, ωn + ωm )F (k, ωm )] ,

(7.73)

where + (−) stands for χso (χc0 ). Numerical solutions of the self-consistent set of equations (7.71)–(7.73) for intermediate values of the Coulomb repulsion, U ≤ 4t, for the full momentum- and frequency dependent renormalization parameter Z(p, ω), the energy shift ξ(p, ω), and the gap function Δ(p, ω) = φ(p, ω)/Z(p, ω) have been obtained. The gap function exhibited the dx2 −y 2 symmetry. In the vicinity of AF instability near half-ﬁlling, the superconducting temperature Tc reaches values of the order of 0.02t 60 K. Quasiparticle and spin ﬂuctuation spectra in the normal and superconducting states, calculated also on the real frequency axis by Dahm et al. [267], reveal a strong dependence of the spectra on the temperature and a strong feedback eﬀect for the eﬀective interactions arising from the spin and charge susceptibilities as the superconducting gap opens. These direct numerical solutions of the strong coupling Eliashberg equations for the Hubbard model in the weak correlation

7.3 Magnetic Mechanism of Superconductivity

439

limit unambiguously conﬁrm the possibility of the d-wave pairing mediated by the spin ﬂuctuations. However, this model predicts superconductivity only in a narrow range of U , very close to the AF instability. In a more general treatment, a self-consistent account of all three types of instabilities driven by the spin or charge density waves or the transition into the superconducting state is required. In a number of investigations, the two-dimensional generalized t − t Hubbard model (7.3), allowing for the nextnearest-neighbor (n.n.n.) hopping t , was studied within the renormalization group technique (RG). As pointed out ﬁrst by Dzyaloshinskii [299], several instabilities driven by spin or charge density waves or superconducting transition compete with each other close to half-ﬁlling at the Van Hove singularity. The evolution from an AF instability due to nesting eﬀects at small t /t < 0 to a d-wave superconducting pairing at moderate t /t ∼ −0.3 and further to a ferromagnetic ordering at large values of |t /t| was observed in a one-loop approximation in the weak correlation limit U ∼ 3t [481]. The phase diagram as a function of the Coulomb interaction U/t and the doping was investigated in the one-loop parquet approximation by Irkhin et al. [527]. Studies of a more general extended U –V –J Hubbard model under the doping near the Van Hove singularity revealed a complex competition between various phases depending on the model parameters [591]. The RG technique was also applied to the study of the Fermi surface instability and breakdown of the Landau– Fermi liquid in the two-dimensional t–t Hubbard model close to half-ﬁlling (see e.g., [482,706], and references therein). However, these results obtained in the weak correlation limit U ≤ W/2, do not lead to an adequate description of the strong-coupling phases, revealing only a complicated competition between diﬀerent types of instabilities. Resume Thus, in the framework of the Landau–Fermi liquid model (in the weak correlation limit) the electronic interaction mediated by AF spin ﬂuctuations can lead to a superconducting pairing. The magnetic pairing mechanism is most eﬀectively manifested itself in the d-channel where the local Coulomb repulsion is suppressed. In spite of the pair-breaking eﬀects due to spin scattering, which considerably decrease Tc , the strong enhancement of the spin susceptibility near the AF instability results in the d-wave pairing. An exceptionally strong suppression of Tc in all cuprates by nonmagnetic Zn impurities which are also detrimental to the local magnetic order and in this way suppress the eﬀective spin-ﬂuctuation pairing, provides a “smoking gun” for the magnetic mechanism [989]. 7.3.3 Models with Strong Correlations Finite Cluster Calculations To deal with the strong correlation limit in the Hubbard or t–J models, various numerical methods for ﬁnite clusters, the quantum Monte Carlo (MC), the

440

7 Theoretical Models of High-Tc Superconductivity

exact diagonalization (ED), the density renormalization-group (DMRG), have been developed (for reviews, see [190, 261, 550, 1106]). Extensive MC numerical simulations have been carried out for the two-dimensional (2D) Hubbard model (7.3) and DMRG calculations for the 2D two-leg Hubbard model. These studies have revealed strong AF correlations and the tendency to the formation of the dx2 −y 2 pairing correlations for small clusters. The strong pairing amplitude in the particle–particle channel has been explained by an unusually ﬂat dispersion near the (0, π) and (π, 0) points of the Brillouin zone which results in high density of states in these regions and a strong AF scattering at q = (π, π) momentum transfer [190]. In the DMRG calculations for the two-leg Hubbard ladder, a strong pairing correlation at low temperature was also found which competed with the stripe formation. Usually in the quantum MC computations the pair-correlation function is studied, 1 Pα = Δα Δ†α , Δα = gα (l)c†m+l↓ c†m↑ . (7.74) N m,l

Here, N is the number of cells in the cluster, gα (l) = ±1, determines the pairing symmetry (s- or d-type) in the summation over the lattice sites l, nearest to the site m. To exclude the pair correlations from a free particle propagation, the irreducible pair-correlation vertex is studied: P = Pα − P¯α ,

1 P¯α = gα (l)gα (l )G↑mm G↓m +l N mm

ll

m+l ,

(7.75)

where Gσm m = c†m σ cmσ . Superconducting correlations are indicated when the value of the irreducible pair-correlation vertex P is positive. Generally, the ﬁnite cluster calculations due to known limitations (e.g., the sign problem) give only restricted information and show controversial results depending on the employed version of the numerical technique [190]. For instance, by applying the constrained-path MC method to the 2D Hubbard model, the dx2 −y2 pairing correlations were detected for small size lattices and a weak interaction, while with the increase of the lattice sizes or of the interaction, they vanished [1422]. MC studies of the t–t –U Hubbard model by Huang et al. [497] have revealed that the long range d-wave pairing correlations are much smaller than the local ones and the t hopping had a weak inﬂuence on them. Similar results were obtained within the pre-projected Gaussianbasis MC method by Aimi et al. [23], who observed a weak dx2 −y 2 pairing correlation at large distances. The correlation decayed as r−3 with the distance r and was an order of magnitude smaller than found in the dynamical cluster theories (see below). Therefore, the authors concluded that the 2D Hubbard model could not account for the high superconducting temperature in the cuprates. Controversial results were obtained also for the t–J model. In particular, no long-range order was found for the 2D t–J model in relation to cuprates

7.3 Magnetic Mechanism of Superconductivity

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parameter range J ≤ 0.5t within the power-Lanczos method by Shih et al. [1156]. Contrary to this conclusion, clear indications of superconducting pairing were found in the 2D t–J by Sorella et al. [1184] by applying various numerical techniques, quantum MC, DMRG, and Lanczos. However, it should be pointed out that in strictly 2D systems the phase ﬂuctuations of the order parameter destroy the long-range order at ﬁnite temperature, according to the Mermin–Wagner theorem [824], and only the Berezinskii–Kosterlitz–Thouless [126, 649] superconducting instability may occur (for discussion of phase ﬂuctuations in 2D superconductors see e.g., [739]). Generally, a very delicate balance is observed between diﬀerent phases, like spin or charge density waves, superconducting and stripe phases [1106]. Therefore, to prove the existence of the superconducting pairing in the strong correlation limit, a more general analytical theory is necessary. Dynamical Cluster Theory Among the analytical methods, the dynamical mean ﬁeld theory (DMFT) was eﬀectively used in treating strong electron correlations in studies of electronic spectra (see Sect. 7.2.1). However, the single-site DMFT is unable to describe wave vector dependent phenomena like AF correlations and the dwave superconductivity. To overcome this limitation of the DMFT, various types of the dynamical cluster theories have been developed . In the original dynamical cluster approximation (DCA), a cluster was considered in the reciprocal space (see e.g., [773] and references therein). The cellular DMFT (CDMFT) considers a cluster embedded in a self-consistent medium in the real space (see e.g., [1186] and references therein). A variational cluster approach (VCA) based on a self-energy-functional approach is a more general cluster method which incorporates the cluster-perturbation theory and C-DMFT in certain limits (see [269, 1021, 1265] and references therein). In particular, within the VCA the superconductivity and two-gap energy scales were found in the t–t –U Hubbard model by Aichhorn et al. [22] contrary to MC studies by Huang et al. [497]. It was shown that in the underdoped region the gap near the nodes decreases with lowering of hole concentration, while the gap near the antinodes increases. Two gaps merges into one gap in the overdoped region at the superconducting transition at Tc . The authors concluded that this complicated behavior can be explained within the spin-mediated pairing mechanism. A plaquette DMFT using a 2 × 2 cluster has been extensively used in studies of both the normal and, superconducting properties of the Hubbard model and the t–J model [437]. In this approach, an exact solution for the cluster is found which aﬀords to calculate the self-energy for diﬀerent lattice sites and to study the physical properties at several wave vectors, like k = (0, 0), (π, 0), (π, π). Therefore, within the C-DMFT the diﬀerentiation between the coherent nodal QPs and antinodal incoherent pseudogap states at the FS is possible. Studies of the anomalous self-energy for the nearest

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7 Theoretical Models of High-Tc Superconductivity

neighbors have shown a nonzero d-wave pairing order parameter and ﬁnite Tc ∼ 0.01t at optimal doping. The frequency dependence of the anomalous self-energy revealed unconventional behavior and complicated frequency dependence in comparison with the strong-coupling Eliashberg theory. A ﬁnite static value of the self-energy in the limit of inﬁnite frequency was found for the t–J model but was not observed in the Hubbard model. A systematic study of the pair correlations in the Hubbard model within the DCA and QMC simulations as a cluster solver has shown a substantial dependence of the d-wave superconducting instability on the size and geometry of small clusters [774]. However, for large enough clusters a ﬁnite superconducting Tc ≈ 0.023t at 10% doping was found in the Hubbard model (7.3) with U = 4t. Calculations of the momentum and frequency dependence of the particle–particle pairing interaction Γ pp have shown that the dominant contribution comes from the magnetic exchange channel, while the charge channel appears to be irrelevant to the d-wave pairing [775]. In a subsequent publication by Maier et al. [776], spectral properties of the superconducting gap in the t–J model and the Hubbard model within the DCA and exact diagonalization calculations were reported. It was shown that the main contribution to the d-wave pairing is due to the retarded frequency-dependent spin-ﬂuctuations which can be considered as a spin-ﬂuctuation “glue” for the pairing. A high-energy contribution due to the AF exchange interaction was also found which, in contradistinction to the spin ﬂuctuations gives an instantaneous contribution to the pairing. A similar two-component electron–boson coupling function was inferred from studies of optical spectra by van Heumen et al. [1303] (see Sect. 5.3.2). Although the results of numerical simulations for ﬁnite clusters and those found within various versions of the cluster studies are to some extent controversial, one can draw the general conclusion that the short-range AF spin correlations could be responsible for the dx2 −y 2 -wave pairing mechanism in the doped t–J and the Hubbard models. Diagram Technique A rigorous method to treat the unconventional commutation relations for the projected electron operators in the t–J model (7.16) is based on the Hubbard operator technique since in this representation the local constraint (7.63) is rigorously implemented by the Hubbard operator algebra. A superconducting pairing due to the kinematic interaction in the Hubbard model in the limit of strong electron correlations was ﬁrst found by Zaitsev and Ivanov [1404]. By applying a diagram technique for Hubbard operators [1403], they studied the two-particle vertex equation in the lowest order approximation and found the s-wave pairing. Systematic investigation of the t–J model within the diagram technique was performed by Izymov et al. [539]. In the framework of this approach, spin ﬂuctuations and superconducting pairing in the t–J model in the limit of small J were studied [541, 542]. The ﬁrst order diagrams for the

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443

self-energy reproduced the results of the MFA found by Plakida et al. [1000] and Yushankhai et al. [1396]. In calculations of the second-order diagrams only the exchange interaction J was taken into account, while more important contributions due to the kinematic interaction of the order t2 were disregarded. As a result, estimations in the weak coupling limit for the Eliashberg equation for a three-dimensional model near AFM instability resulted in quite a low superconducting Tc . Projection Operator Method The projection operator method in the equation of motion for the thermodynamic Green functions (GFs), similar to the Mori projection technique, was used in several studies of the Hubbard and the t–J model (see Appendix A.3, A.4). The method appears to be much simpler than the diagram technique, while it accurately reproduces results of the latter within the second order of an eﬀective interaction for the self-energy. We will discuss applications of this method to the t–J and the Hubbard models in the next section in detail, while here point out only the main publications concerning the method. An equation for the superconducting gap was ﬁrst derived for the Hubbard model [152] by applying a decoupling procedure in the equation of motion method for the GFs. The s-wave pairing due to the kinematic interaction proposed by Zaitsev et al. [1404] was obtained in MFA. However, as was shown later by Plakida et al. [1000] and Yushankhai et al. [1396], the s-wave pairing in the limit of strong correlations violates the exact requirement of no double occupancy of a lattice site in one Hubbard subband, σ ¯0 Xiσ0 Xiσ¯ 0 = k Xkσ0 X−k = 0, and should be disregarded. The BCS-type theory for the t–J model was formulated within the formally exact projection operator method for the GFs in terms of the Hubbard operators and a high Tc 0.1t for J 0.4 was found [1000]. It was proved that the d-wave superconducting pairing mediated by the exchange interaction J leads to the lowering of the ground state energy in the superconducting state [1396] (see Appendix (A.73)). A theory of the single-particle electron spectrum and superconducting pairing beyond MFA with the self-energy calculated in the noncrossing approximation was formulated for the t–J model by Plakida et al. [1012]. The normal state spectrum was considered in the previous Sect. 7.2.1, while the superconductivity will be discussed in the next section. The spin-ﬂuctuation mechanism of superconductivity in the t–J model was considered also by Prelovˇsek et al. [1026] within the projection operator method for the GFs in terms of projected electron operators and spin operators. The projection operator method for the Hubbard model was applied by several groups to study the superconducting pairing in the MFA (see [16, 88, 122, 1185]). We discuss these publications and a theory of the exchange and spin-ﬂuctuation pairing beyond the MFA developed by Plakida et al. [1009, 1015] in the section below.

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7 Theoretical Models of High-Tc Superconductivity

Spin-Polaron Model As discussed in Sect. 7.2.1, for a weakly doped CuO2 plane the spin-polaron model (7.26), (7.27) describes a hole motion dressed by AF spin ﬂuctuations as a quasiparticle propagation in a narrow band ∼J. So it is quite natural to expect that the same spin ﬂuctuations induce superconducting pairing of the spin polarons. This problem was treated originally in the framework of the standard BCS theory assuming a rigid band model for the quasiparticles with a numerically evaluated spectrum and an eﬀective pairing interaction in the atomic limit [263] or mediated by the AF magnon exchange [123, 1368]. Within this weak coupling approach, the d-wave pairing of spin-polarons was found with high superconducting transition temperature. However, since the pairing spin-ﬂuctuation energy J is of the same order as the QP band width, the weak coupling approximation is inadequate to treat the problem. Also, the rigid band approximation for QPs fails to describe a strong doping dependence of the QP spectrum as demonstrated in Fig. 7.3 (right panel). A consistent solution of the strong coupling spin–polaron model (7.26), (7.27) at ﬁnite temperatures and hole concentrations for normal and superconducting states was given by Plakida et al. [1009]. In Sect. 7.2.1., a selfconsistent system of equations for the normal and anomalous Green function (GF) (7.29) and the corresponding self-energy (7.30) was derived and the normal state QP spectrum was calculated. To determine the superconducting temperature Tc , one should consider a linearized equation for the anomalous GF Ghf (k, iωn ) in (7.29), which describes the singlet pairing of two holes with opposite spins in the diﬀerent AF sublattices. The linear equation for the superconducting gap function ϕ(k, iωn ) = Σhf (k, iωn ) is determined by the anomalous self-energy in (7.31) as λ12 (k, k − q | iωn − iωm ) Ghh (q, iωm ) ϕ(k, iωn ) = −T q,m

×Ghh (−q, −iωm ) ϕ(q, iωm ).

(7.76)

where the normal GF reads Ghh (k, iωn ) = {iωn + k − μ − Σhh (k, iωn )}−1 .

(7.77)

Taking into account the solution of the normal GF (7.77) discussed in Sect. 7.2.1, the gap equation (7.76) was solved by direct diagonalization in the (kx , ky , ωn )-space. By examining the temperature dependence of the highest eigenvalue in the (7.76) at diﬀerent hole concentrations, the superconducting temperature Tc was found. At this temperature, the normal state becomes unstable due to the singlet pairing of QPs. In Fig. 7.14 (left panel) the superconducting Tc dependence on the hole concentration δ is shown for J = 0.4t. The maximum value of Tc 0.013 t 60 K at δ 0.15 (for t = +0.1t) is related to the Fermi level crossing of the highest density of states when the FS exhibits a topological transformation from the 4-pocket-like shape to a large

7.3 Magnetic Mechanism of Superconductivity

445

Fig. 7.14. Superconducting Tc dependence (in units of t) on the hole concentration δ for t = −0.1t (solid line at right), t = 0 (dashed line), and t = +0.1t (solid line at left) (left panel) and momentum dependence of the gap function Δ(k, ω = 0) (right panel) in the spin-polaron model [1009]

FS. Superconducting Tc increased with J and saturated at Tc 0.025t for J 3. The nonmonotonic Tc dependence on δ with a maximum at optimal doping is in agreement with the experimental data and is diﬀerent from the monotonic increase of Tc with δ within the weak coupling limit of the BCS equation obtained by Belinicher et al. [123]. The d-symmetry superconducting gap was found in ﬁnite cluster calculations by Ohta et al. [915] of the Bogoliubov pair Green function for the t–J model at zero temperature. The gap reached the maximum value near half-ﬁlling, δ = 0.5, at small values of J. In Fig. 7.14 (right panel), the momentum dependence of the gap function Δ(k, 0) = φ(k, ω = 0)/Z(k, ω = 0) is shown in a quarter of the full Brillouin zone for δ = 0.25 and T /Tc ≈ 0.8. The gap function has the typical d-wave symmetry with two ridges resulting from sharp changes of the interaction function at the FS. The real and imaginary components of the gap function Δ(k, ω) revealed unconventional frequency dependence with a characteristic cutoﬀ energy of order J 0.4 for the pairing theory (for detail see [1009]). Since the value of the chemical potential μ is of the order of the exchange energy J we have here really a strong coupling limit for spin polaron pairing where all QPs contribute to the pairing state contrary to the weak coupling case in conventional superconductors. Although the two-sublattice representation in the spin-polaron model can be rigorously proved only for the long-range AF state, the model provides a reasonable description of the spin polarons in the spin liquid regime, provided the AF correlation length is suﬃciently large as compared to the Cooper pair

446

7 Theoretical Models of High-Tc Superconductivity

and the polaron radius. At moderate doping, this condition may be violated and the original t–J model should be considered as discussed in the next section. Superconducting Pairing in the t–J Model In this section, we consider the theory of the superconducting pairing in the t–J model developed by Plakida et al. [1012]. The normal state electron spectrum within the self-consistent Dyson equation for the t–J model was discussed in Sect. 7.2.1. By applying the projector operator method for the matrix Green function for Nambu operators, we have derived in Appendix A.4 equations for the anomalous part of the frequency matrix (A.66) which is the superconducting gap equation in the MFA and the anomalous self-energy Σσ12 (k, ω) in the noncrossing approximation (A.67). By taking into account both contributions we have arrived to the gap equation (A.69). To calculate the superconducting Tc and to deduce the symmetry of the gap, it is suﬃcient to study the gap equation (A.69) in the linear approximation for the anomalous GF G12 σ (q, ω). By using the imaginary frequency representation, the linearized gap equation can be written as [1012]: φσ (k, ωn ) =

T {J(k − q) − g 2 (q, k − q) χ(−) sc (k − q, iωn − iωm )} N q,m ¯ (q, −iωm )φσ (q, iωm ). ×Gσ11 (q, iωm )Gσ11

(7.78)

where the vertex g(q, k − q) = [ t(q) − (1/2) J(k − q) ] and the dynamical susceptibility is approximated by the model spin susceptibility (7.40) like in the normal state studies, χs (k − q, iων ) = χ(k − q) Fs (ων ). The normal GF is deﬁned by (7.35). The ﬁrst term in (7.78) determines the pairing induced by the AF superexchange interaction J. In the t–J model, it acts as an instantaneous interaction caused by virtual excitations above the Mott gap, which is much larger than the width of the Hubbard subbands. The second term arises from the interaction of the electrons in one Hubbard subband with the spin-charge ﬂuctuations deﬁned by the dynamical susceptibility. The energy scale of the spin ﬂuctuation spectrum is much smaller than the bandwidth, and this results in retardation eﬀects in the spin-ﬂuctuation pairing. A picture of the two energy scale pairing, the nonretarded one due to the exchange interaction J and the retarded spin-ﬂuctuation pairing has been conﬁrmed in exact diagonalization calculations for the t–J model by Maier et al. [776]. We emphasize that in this microscopic theory the pairing vertexes are not ﬁtting parameters since they are determined by the same parameters J(q) and t(q) of the original model. Therefore, the theory captures both the pairing caused by the strong correlations as in the RVB theory (determined by the exchange interaction J) and the pairing caused by the spin-ﬂuctuations as in the phenomenological spin-fermion model.

7.3 Magnetic Mechanism of Superconductivity

447

To analyze the weights of the two components in the pairing interaction, we consider the weak coupling approximation (WCA) in the gap equation (A.69). In this approximation, the kernel of the integral equation in (A.69) is approximated by its value near the Fermi surface for energies |ω, ω1 | μ as given by (A.33): ω1 1 K (−) (ω, ω1 |k, q − k) = −g 2 (q, k − q) χ(k − q) tanh , 2 2T where χ(k − q) is the static susceptibility. Taking the WCA for the GFs in (A.69) (i.e., neglecting the self-energy contributions), we obtain the BCS-like equation for the gap φ(k) =

E(q) 1 φ(q) tanh , {J(k − q) − λs (q, k − q)} N q 2E(q) 2T

(7.79)

where λs (q, k − q) = g 2 (q, k − q) χ(k − q)θ(ωs − |E(q)|) is the spinﬂuctuation pairing interaction conﬁned to an energy shell (±ωs ) close to the Fermi energy. The energy of the Bogoliubov quasiparticles E(q) = [E 2 (q) + φ2 (q)]1/2 is determined by the electronic normal state dispersion E(q) in MFA (A.65) and the gap function. Figure 7.15 (left panel) shows the superconducting Tc versus the hole concentration δ at the AF correlation length ξ = 1 for various contributions of the pairing interaction in (7.79) as explained in the caption. The exchange interaction J(k − q) alone results already in quite a high Tc . The spin-ﬂuctuation contribution λs (q, k − q) enhances Tc both due to the kinematic, t(q), and the exchange, J(k − q), interactions. For larger AF correlation lengths, the superconducting Tc is greatly enhanced, e.g., Tc 0.1t for ξ = 3 in WCA. Numerical solution of the full gap equation (7.78) by a direct diagonalization in the (kx , ky , ωn )-space has shown a substantial reduction in the

a

b 3

3

ky

2

ky

1

1 0

2

δ = 0.1

0

1

2

kx

3

0

δ = 0.2

0

1

2

3

kx

Fig. 7.15. Left panel: Superconducting Tc versus hole concentration δ in the weak coupling approximation for the full vertex (upper line), the vertex with t(q) = 0 (dashed line), and in the MFA for λs (q, k − q) = 0 (lower line). Right panel: the gap function φ(k, ω = 0) for two-hole concentrations δ = 0.1, 0.2 in the ﬁrst quadrant of the BZ [1012]

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7 Theoretical Models of High-Tc Superconductivity

superconducting Tc due to the strong coupling eﬀects in the normal GF (7.35) which suppress the quasiparticle weight in comparison with the WCA (7.79) (for detail, see [1012]). Figure 7.15 (right panel) shows the contour plots in the quarter of the BZ, (0 ≤ kx , ky ≤ π), of the static gap function φ(k) = φ(k, ω = 0) found in this solution for two-hole concentrations. At a small doping, δ = 0.1, it has a more complicated k-dependence than at larger hole concentration. δ ≥ 0.2 where only one positive and one negative maximum survive. In all cases, the gap φ(k) has a more complicated momentum dependence and cannot be ﬁtted by a simple function φ(k) ∝ (cos kx − cos ky ) usually used for the d-wave gap. However, in all cases the gap function shows the B1g symmetry: φ(kx , ky ) = −φ(ky , kx ), which violates the 4-fold symmetry of the FS in the normal state. The frequency dependence of the gap function φ(k, ω) is also unconventional (for detail, see [1012]). The maximum value of Tc (δ) found both in the WCA and in the solution of the full equation (7.78) occurs close to the hole concentration δ 0.33 and does not change much with ξ. At this concentration, the FS is close to the (X, Y ) = (±π, 0), (0, ±π) points of the BZ where the electronic spectrum is rather ﬂat which provides a high density of states. In the Hubbard model, a proper allowance for the weight transfer between the Hubbard subbands results in the maximum of Tc (δ) at optimal doping δ 0.13 (see Fig. 7.18). Since the pairing interaction is proportional to the spin susceptibility χs (k − q) with the maximal contribution at k − q = Q = (π, π) , the strongest pairing occurs for electrons at the FS in the vicinity of the (X, Y ) points of the BZ coupled by the AF wave vector Q. This scenario of emerging high-Tc due to the ﬂat dispersion near the (X, Y ) points of the BZ and a strong AF scattering at Q = (π, π) momentum transfer were also suggested in numerical simulations in small clusters [190]. It should be noted that close to the AF BZ the nearest neighbor hopping gives no contribution to the vertex t(q) since t(q)n.n. = 4t(cos qx +cos qy ) = 0. Therefore, for a large Fermi surface close to the AF BZ Tc should depend on the next nearest neighbor hopping parameter t as was found by Prelovˇsek et al. [1026]. The characteristic dependence of Tc on t was found in the spin-polaron model at the optimal doping value δ 0.16 (see Fig. 7.14). As discussed by Barabanov et al. [102], a large Fermi surface at low doping in spinsinglet state can be found if one properly takes into account the spin-polaron formation due to the short range AF order. The spin-polaron represents a spin-hole bound state described by the product of hole and on

spin operators nearest neighbor sites written in terms of the HOs as j,σ Xiσσ Xjσ 0 . The spin-polaron operator should be treated as a new dynamical variables in the equation of motion for the GFs. To conclude this section, we make several general remarks. The ﬁrst one concerns the robust short range pairing found in the ﬁnite cluster calculations. It may be explained by the short-range of the exchange interaction acting between the nearest neighbors, which gives a large contribution in the MFA. The spin-ﬂuctuation pairing is strong at large AF correlation lengths ξ

7.3 Magnetic Mechanism of Superconductivity

449

when it shows a sharp maximum at the large AF wave vector and therefore is short ranged also. Another remark concerns the role of the three-site terms in the extended t–J model which are disregarded in the two-site approximation: 21 12 2 HJ ∝ t12 ij tjn ∼ δn=i [tij ] (see (7.14)). In a number of studies it was shown that the three-site terms may be important in the AF exchange interaction pairing (see e.g., [1295]). However, to obtain quantitative results for Tc , one should take into account the spin-ﬂuctuation term as well in the gap equation (7.78) or (7.79) since it gives a comparable contribution as shown in Fig. 7.15. The important role of the three-site terms in the exchange interaction pairing was found also in studies of spin-polarons within the Kondo model Hamiltonian by Val’kov et al. [1296]. In the model, a system of localized spin-1/2 at the copper sites coupled by the s–d AF exchange interaction to itinerant holes at oxygen sites was considered. Within the operator perturbation theory, an eﬀective Hamiltonian for the s–d spin-polarons was constructed. The superconducting pairing of the spin-polarons induced by an eﬀective exchange interaction, similar to the t–J model, was treated in the BCS-like mean-ﬁeld approximation. It was shown that the three-site terms in the eﬀective exchange strongly enhanced Tc calculated in the two-site approximation which was found to be quite low. Hubbard Model As mentioned in the previous section, there are two contributions to the magnetic pairing mechanism in the t–J model: caused by the AF exchange interaction J and by the spin-ﬂuctuations determined by the dynamical spin susceptibility (see (7.78)). However, in the t–J model, the time-dependent interband transitions are eliminated with the help of the canonical transformation (7.13), leading to an instantaneous exchange interaction in a single Hubbard subband. This transformation is quite similar in the reduction of the electron–phonon model with retarded interaction to the reduced BCS model with instantaneous interaction in a restricted region of electron energies close to the Fermi level. Consequently, to conﬁrm the results obtained in the framework of the t–J model, it is important to ﬁnd out whether the retardation eﬀects in interband transitions can occur in the initial Hubbard model (7.3). The Dyson equation for the 4×4 matrix GF in terms of the four-component ˆ iσ , X ˆ † = (X 2σ X σ¯ 0 X σ¯ 2 X 0σ ) in the Hubbard model Nambu operator X i i i i iσ was considered in Sect. A.3, as originally derived by Plakida [1010]. The normal state electronic properties within the Dyson equation were discussed in Sect. 7.2.1. Below we present results for the superconducting state, in particular, superconducting Tc dependence on the hole concentrations and the momentum dependence of the superconducting gap φ(k) [1015]. The comparison of the results obtained for the t–J model with those for the Hubbard model will provide further insight into the origin of the magnetic mechanism of the pairing in the cuprates.

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7 Theoretical Models of High-Tc Superconductivity

Let us ﬁrst examine the MFA for the anomalous GF and the gap equation given by (A.50). For deﬁniteness, we consider a hole doped case with the gap 02 function Δ22 ijσ = −σtij Xi Nj /Q2 , where Q2 = n/2 and σ = ±1. As pointed out in Sect. A.3.2, the anomalous correlation function in the gap function can be written as Xi02 Nj = Xi0↓ Xi↓2 Nj = ci↓ ci↑ Nj , where aiσ are the conventional Fermi electron operators (see (7.11)). This shows that the MFA pairing in the Hubbard model occurs at a single site but in diﬀerent Hubbard subbands. This pairing does not require high energy excitations of the order of U since in the case of hole doping the lower Hubbard subband is ﬁlled, and the energies of the conﬁgurations Xi02 Nj and Ni Xj02 at hole pair hopping in the upper subband are equal. Similar results in the MFA for the original Hubbard model were reported in other publications (see [88, 122, 1185]). To calculate the anomalous correlation function ci↓ ci↑ Nj , Beenen et al. [122] and Stanescu et al. [1185] have used the Roth procedure based on a decoupling of the operators at a single lattice site in the time-dependent correlation function: ai↓ (t)|ai↑ (t )Nj (t ). However, the result of the Hubbard operator decoupling following this scheme is not unique (as it has been actually noticed in those calculations) and hence it is unreliable. To escape uncontrollable decoupling, Avella et al. [88] have used kinematic restrictions imposed on the correlation functions for the Hubbard operators which, however, have not produced a unique solution for the superconducting equations either. As shown in Sect. A.3.2, it is possible to calculate the correlation function Xi02 Nj from the equation of motion without any decoupling ansatz for the pair GF Lij (t − t ) = Xi02 (t) | Nj (t ) and to obtain expression (A.52) for the gap function in the two-site approximation: 02 σ2 σ ¯2 Δ22 ijσ = −σ tij Xi Nj /Q2 = Jij Xi Xj /Q2 ,

(7.80)

This equation corresponds to the conventional gap equation in the MFA for the t–J model in (7.79) with the exchange interaction Jij = 4 (tij )2 /U . We may therefore conclude that the anomalous terms in the frequency matrix (A.46) are just the conventional anomalous pairs of electrons in one Hubbard subband the coupling of which in the MFA is mediated by the exchange interaction already studied in the t–J model. As discussed in Sect. A.3.2, the retardation eﬀects associated with the interband transitions in the gap function (7.80) can be neglected due to a much higher excitation energy of the transition ∼ U in comparison with the characteristic excitation energies of the order of |tij | in one subband. Rigorous transforms of the spectral theorem representation of the matrix element Xi02 Nj based on the use of the equation of motion of the GF Xi02 (t)|Nj (t ) yield, besides the two-site contribution (7.80), three-site contributions as well [16]. Their approximate decoupling by use of the general rule of separating the fermion and boson ﬁeld contributions in the resulting matrix elements leads to two-site-like contributions which renormalize the expression (7.80) by a factor (1 − δ), where δ is the doping rate [17].

7.3 Magnetic Mechanism of Superconductivity

451

By taking into account the anomalous self-energy operator (A.60), the gap equation (A.58) for the upper Hubbard subband can be written as 22 (7.81) ϕ2,σ (k, ω) = Δ22 σ (k) + Φσ (k, ω)/Q2 +∞ 1 = dω1 J(k − q) + K (−) (ω, ω1 |q, k − q) N q −∞ 1 22 ImFσ (q, ω1 ) . × − πQ2

We neglect the anomalous GF Fσ11 (q, ω1 ) in the full equation (A.60) since for hole doping the lower Hubbard subband lies much below the Fermi level, E1 −U , and the contribution of this GF is negligible. To calculate the superconducting Tc and to deduce the symmetry of the gap, it is suﬃcient to study the gap equation (7.81) in the linear approximation for the anomalous GF Fσ22 (q, ω1 ) deﬁned in (A.56). By using the imaginary frequency representation, the linearized gap equation can be written as: ϕ2,σ (k, iωn ) =

T {J(k − q) − |t(q)|2 χ(−) sc (k − q, iωn − iωm )} N q m 22 ×G22 N (q, −iωm ) GN (q, iωm ) ϕ2,σ (q, iωm ).

(7.82)

For the spin-charge susceptibility, the model χs (k − q, iων ) (7.40) can be used. In the strong-coupling limit within the Eliashberg-type theory, the full normalstate GFs (7.42), (7.43) self-consistently calculated taking into consideration the self-energy (7.44) should be used. To calculate Tc and to ﬁnd the gap function, one should ﬁnd out the eigenvalue and the eigenfunction of the linear equation (7.82) in the (k, ωn )-space. The particular wave vector dependence of the interaction in (7.82) as a product of the (q)- and (k − q)-dependent functions aﬀords to use the fast-Fourier transformation which helps to simplify the calculations. This program has been realized for the single-band t–J model discussed in the previous section. However, the complicated two-subband form of GFs (7.42), (7.43) in the Hubbard model presents a complicated problem for numerical calculations which has not yet been solved. Therefore, as a ﬁrst step, a weak-coupling approximation (WCA) for the gap function (7.81) can be considered. In the WCA, the kernel in the integral equation (7.81) is approximated by its value near the Fermi surface for energies |ω, ω1 | μ as given in (A.33): K(ω, ω1 |q, k − q) = −|t(q)|2 χ(k − q)

1 ω1 tanh . 2 2T

In the WCA, the self-energy contribution to the normal-state GF (7.43) is neglected. This results in the following equation for the gap function at the Fermi energy ϕσ (k) = ϕ2,σ (k, ω = 0):

452

7 Theoretical Models of High-Tc Superconductivity

ϕσ (k) =

ε2 (q) 1 ϕσ (q) tanh [J(k − q) − |t(q)|2 χ(k − q)] . N q 2ε2 (q) 2Tc

(7.83)

In this equation, several contributions from the lower Hubbard band in the full GF Fσ22 (q, ω1 ) were omitted since they give negligible contributions due to the large energy ε1 (q) ∼ U in their denominators. This suggests that for estimation of Tc one can take into consideration only the contribution from a single Hubbard subband at the FS, similar to the gap equation (7.79) for the t–J model. The investigation of the superconductivity in the Hubbard model and, in particular, of the gap equations (7.81) or (7.82) suggests the following general conclusion concerning the mechanism of the pairing in the Hubbard model. As follows from the gap equation, there are two channels of pairing both of which being induced by the kinematic interaction. The ﬁrst one is mediated by the inter-subband hopping and is determined by the AF exchange interaction J(k − q) which is usually considered in the RVB-type theories (see e.g., [68]). There are no retardation eﬀects for the exchange pairing because of the large hopping energy U t that results in pairing of all the electrons in the hole subband as sketched in Fig. 7.16. The second contribution comes from the spin-charge ﬂuctuation pairing ∝ χsc (k − q, ω) induced by intrasubband hopping which is only possible in a range of characteristic energies ±ωs,ch near the FS, similar to the BCS theory, as sketched in Fig. 7.17. The spin-ﬂuctuation interaction is repulsive and can produce pairing only for a sign-varying gap at the FS, as the d-wave gap. This type of pairing is usually considered in spin-fermion models (see [237]). To estimate both contributions, we consider the WCA gap equation (7.83) for the conventional d-wave gap ϕ(k) = ϕ0 (cos kx − cos ky )/2 ≡ ϕ0 η(k). By integrating both sides of (7.83) multiplied by η(k) over k, we can derive an equation for Tc in the conventional BCS form: −˜ W μ

1= −˜ μ

d [J Nd ( ) + θ(ωs − | |) λs N ( )], tanh 2 2Tc

(7.84)

where the d-wave projected DOS Nd ( ) = (1/N ) q η 2 (q)δ( − ε2 (q)) and μ ˜ is the Fermi energy measured from the bottom of the upper (conduc . For the spin-ﬂuctuation term, tion) band of the (renormalized) width W we have taken into account that the static spin susceptibility shows a strong e2 e1

m t12 i

W 0

j

Fig. 7.16. AF exchange pairing mediated by the interband hopping t12 ij

7.3 Magnetic Mechanism of Superconductivity t 22

e2 e1

m i

j

453

W + ws – ws 0

Fig. 7.17. Spin-ﬂuctuation pairing mediated by the intraband hopping t22 ij

peak at the AF wave

vector k − q = Q = (π, π) that enables us to obtain the estimate (1/N ) k η(k)χs (k − q) −η(q)/ωs and to introduce the dwave projected coupling constant averaged over the Fermi surface: λs N ( ) =

(1/N ) q η 2 (q) (|t(q)|2 /ωs ) δ( − ε2 (q)). This projection suppresses the contribution from the nearest neighbor hopping tnn (q) = 2t (cos qx + cos qy ) such that it is the next-nearest neighbor hopping t which mostly determines the coupling constant, λs ∼ ((t )2 /ωs ). Therefore, the momentum dependence of the spin-ﬂuctuation coupling constant t(q) in (7.83) is essential to the determination of the superconducting Tc . The integration over the energy in the ﬁrst term in (7.84) extends over all the energies in the upper subband. This results in pairing of all the electrons in the subband and Tc is proportional to the Fermi energy, −μ Tcex μ ˜(W ˜) exp(−1/Vex ), (7.85) where Vex J Nd (˜ μ). Here, the chemical potential μ ˜ is measured from the bottom of the subband and therefore, both for the empty band, μ ˜ = 0, and , the exchange pairing vanishes: Tcex = 0. In the for the ﬁlled band, μ ˜ =W second term in (7.84) the integration is restricted to the region of energy ±ωs near the FS and Tcsf ωs exp(−1/Vsf ) with Vsf λs N (˜ μ). By taking into account both the contributions the following estimation of Tc can be derived: Tc = ωs exp(−1/Vsf ),

Vsf = Vsf +

Vex . 1 − Vex ln(˜ μ/ωs )

(7.86)

In the conventional superconductors, the nonretarded Coulomb repulsion decreases the pairing, while in (7.86) the nonretarded exchange attraction ˜/ωs and the Vex enhances the pairing, in particular, due to the large ratio μ sign minus in the denominator of the enhanced coupling Vex , contrary to the sign plus in the reduced Coulomb repulsion in the conventional superconductors: VC∗ = VC /[1 + VC ln(μ/ωph )], where ωph is the phonon energy. Therefore, even for originally weak coupling, Vsf Vex 0.2 (e.g., by assuming ωs J 0.13 eV, λs 0.2 eV, N (˜ μ) = 1 − 1.5 eV−1 ), and for a narrow bandwidth W 8t˜ = 1.6 eV (as suggested by Hashimoto et al. [434] for /2 Bi2 Sr2−x Lax CuO6+δ ) we obtain Tcex 60 K for a half-ﬁlled band μ ˜=W sf and Tc 10 K, while according to (7.86) Tc 200 K for Vsf 0.5. Results of a direct numerical solution of the gap equation (7.83) are shown in Fig. 7.18 for the superconducting transition temperature Tc (δ) and for the

454

7 Theoretical Models of High-Tc Superconductivity 0.14 (III) 0.12 0.10

Tc

1

0.08

(II)

0.06

Ky

0.04

(I)

0.8

0.1

0.6

+

0.4

0.02

0.2

0

0 0

0.05 0.1 0.15 0.2

n-1

– 0

0.2 0.4 0.6 0.8

1

Kx

Fig. 7.18. Left panel: superconducting Tc (δ) (in units of teﬀ 0.2 eV) as a function of hole doping δ = n − 1 for a spin-ﬂuctuation interaction (I), for the AF exchange interaction (II), and for the both contributions (III). Right panel: momentum dependence of the gap function ϕ(k) in the ﬁrst quarter of the BZ at the optimum doping (δ = 0.13). The Fermi surface is shown by circles. The (+/−) indicate the gap signs while ϕ(kx = ky ) = 0 [1015]

momentum dependence of the gap ϕ(k) [1015]. The following parameters are used: ξ = 3, J = 0.4t, ωs = 0.15 eV and t 0.2 eV, U = Δpd = 3 eV. The maximum Tc ∼ 280 K (upper curve) is achieved at the optimal doping δopt 0.12 for large Fermi surface shown by dots in the right panel of Fig. 7.18. The numerical results are in qualitative agreement with the crude estimates given above: the superconducting Tc (δ) induced by the AF exchange interaction (II) is higher than those due to the spin-ﬂuctuation interaction (I). An essential reduction of Tcmax in comparison with WCA was observed in the solution of the strong-coupling gap equation (7.82). The gap shows the d-wave symmetry, but a more complicated momentum dependence than the simple model form ϕ(k) ∝ (cos kx − cos ky ) similar to the t–J model. Further evidence in favor of a magnetic pairing mechanism can be drawn from the experimentally observed Tcmax increase with pressure in cuprates as discussed in Sect. 2.6. While in the electron–phonon superconductors Tc decreases under pressure, in cuprates Tc increases with the compression of the in-plane lattice constant a. In particular, as shown in Fig. 2.20 in mercury superconductors dTc /da −1.35 × 103 K/˚ A [737], and for Hg1201 compound d ln Tc /d ln a −50. From (7.85) we get an estimate: (d ln Tc /d ln a) (d ln Tc /d ln J) (d ln J/d ln a) −(14/Vex ) −47, which is quite close to the experimentally observed value [1013]. Here, we used Vex = JN (δ) 0.3 and took into account that for the exchange interaction an estimate J(a) ∝ t4pd where tpd ∝ 1/a7/2 for the p–d hybridization can be used. The same approximate formula (7.85) for Tcex provides at least a partial explanation, of the weak oxygen isotope eﬀect (OIE) on the substitution 18 O oxygen for 16 O in optimally doped cuprates, as we discussed in Sect. 6.3. By

7.4 Electron–Phonon Superconducting Pairing

455

using the experimentally observed isotope shift for the N´eel temperature in La2 CuO4 [1424]: αN = −(d ln TN /d ln M ) (d ln J/d ln M ) 0.05 we obtain from (7.80) αc = −(d ln Tc /d ln M ) = −(d ln Tc /d ln J) (d ln J/d ln M ) (1/Vex ) αN 0.1 − 0.5 for Vex 0.5 − 0.1 which is close to experimental data for the overdoped cuprates with strong coupling and underdoped cuprates with weak coupling [1013]. Resume The various theoretical approaches considered in this section, going from the phenomenological spin-ﬂuctuation model (7.69) to the microscopic theories: the Hubbard model (7.3) in the weak correlation limit, as well as in the strong correlation limit, and the t–J model (7.16), unambiguously demonstrate the possibility of the magnetic d-wave pairing with high-Tc . In particular, the study of the gap equation (7.81) in the Hubbard model in the strong correlation limit provides a valuable insight in the magnetic pairing mechanism. The models show that there are essentially two channels of superconducting pairing. The ﬁrst one is the AF exchange pairing caused by lowering of the electronic kinetic energy produced by the intersubband hopping in a lattice with short-range AF order. The retardation eﬀects in this pairing are negligible which results in the coupling of all the charge carriers in the conduction subband and in a Tc proportional to the Fermi energy. This pairing mechanism was originally proposed in the RVB theory by Anderson [61]. The second channel is the conventional spin-ﬂuctuation pairing due to hopping in one Hubbard subband, which is usually considered in the spin-fermion models. It is essential to stress that both the pairing channels in the Hubbard model are induced by the kinematic interaction emerging from a projected character of the electron operators in strongly correlated systems. These mechanisms of superconducting pairing are absent in the fermionic models (for a discussion, see [66]) and are generic for cuprates revealing strong Coulomb correlations. At the same time, the pseudogap emerging in the vicinity of the antinodal region of the BZ can also be explained by the short-range AF correlations in the underdoped region. Therefore, both the high-temperature superconducting pairing and the anomalous pseudogap state can be accommodated within a single model allowing for strong electron correlations, e.g., the Hubbard model (7.3) or the t–J model (7.16).

7.4 Electron–Phonon Superconducting Pairing The electron–phonon pairing mechanism, well established in conventional superconductors, was advocated by a number of researchers as an origin of the high temperature superconductivity from the very beginning of the HTSC discovery by Bednorz and M¨ uller [118]. This mechanism is usually considered within a Fermi-liquid model, which may be realized in the overdoped part of

456

7 Theoretical Models of High-Tc Superconductivity

the phase diagram for the cuprate superconductors. In the underdoped part of the phase diagram, the strong electron correlations play an essential role and should be properly taken into account in studies of the electron–phonon interaction. The electron–phonon pairing mechanism has been discussed in detail in several comprehensive reviews (see e.g., [673, 674, 778]). The eﬀect of the electron–phonon interaction (EPI) on the normal state electronic properties has been already considered in previous sections. In particular, in Sect. 5.2.2 the electron dispersion renormalization (the kink phenomenon in ARPES) caused by the EPI was considered (see the reviews by Campuzano et al. [196], Cuk et al. [258], Zhou et al. [1436]). The EPI eﬀects on the transport properties were discussed in Sect. 5.4.4. The EPI in the phonon spectrum renormalization was covered in Chap. 6: in neutron scattering experiments – in Sect. 6.1.2, in the Raman scattering – in Sect. 6.2. Theoretical calculations of the phonon spectrum and the EPI in the cuprates are presented in Sect. 6.4. Therefore, in this section we give only general considerations on the electron–phonon pairing mechanism and, in particular, on the polaron and bipolaron superconductivity (for recent reviews, see [43, 44]). 7.4.1 Anisotropic Electron–Phonon Interaction In the strong-coupling Eliashberg theory, a self-consistent solution for the matrix single-particle Green function and the self-energy is required to calculate the superconducting Tc (see Sect. A.2.2). To obtain a high Tc , it is necessary to assume the existence of a strong electron–phonon (EP) coupling λ as discussed in Sect. 1.1 (see (1.1)). In conventional superconductors, an average over the Fermi surface (FS) Eliashberg function (6.3) or the coupling constant (6.4) can be used in these kind of calculations. In the LDA calculations, the data for the EP coupling constant averaged over the Brillouin zone are scattered, λe−ph = 0.3 − 2 (see Sect. 6.4). Subsequent theoretical studies have shown that the electron–phonon coupling with speciﬁc phonon modes may strongly vary over the Fermi surface and therefore an estimate of the average coupling constant according to (6.3) is inadequate in the cuprates for Tc calculations. In particular, the EP coupling λk for the B1g bond-buckling phonon changes from a weak coupling in the nodal direction, λN ∼ 0.2 to a much stronger coupling in the antinodal direction, λAN ∼ 2, while for the breathing mode the largest coupling constant is observed in the nodal direction [279]. It was found that the electron–phonon matrix element |gj (p, m; p , n)|2 (see (6.2)) depends not only on the phonon scattered momentum q = p − p but also on the electronic momentum p at the Fermi surface. Therefore, the anisotropy of the electronic band structure, in particular, the vicinity of the momentum p to the van Hove singularity play essential roles in the anisotropy of the electron–phonon coupling. In realistic calculations of the EPI one should take into account the longrange Coulomb interaction which is poorly screened in the cuprates, especially

7.4 Electron–Phonon Superconducting Pairing

457

for the out-of plane phonon modes, and the strong local electron correlations as in the Hubbard model (7.3). These special features of the electronic structure result in an unconventional electron–phonon superconductivity in the cuprates (see e.g., [1304]). A general strong coupling EP theory allowing for these eﬀects within the Hubbard operator method has been developed by Kuli´c [673, 674]. Two distinctive characteristics of the EPI in cuprates were emphasized: a large contribution to the EPI from the variation of the Madelung energy at lattice vibrations (see (6.12)) and the strong anisotropy of electron scattering in the charge channel. In this case, the momentum-dependent Eliashberg function should be considered α2 F (p, q, ω) = Nsc (0) |geﬀ (p, q, j)|2 γc2 (p, q)δ(ω − ωqj ), (7.87) j

where Nsc (0) is the density of the electronic states renormalized by the doping dependent strong electron correlations. The eﬀective EPI matrix element geﬀ (p, q, j) = g(p, q, j)/εe (q) is renormalized by the long-range Coulomb interaction determined by the dielectric function of the electrons εe (q). The charge vertex γc2 (p, q) strongly suppresses the low-energy Coulomb scattering processes with large momentum transfer, q ∼ pF which results in the pronounced forward scattering peak in the EPI. Therefore, a large EP coupling constant λe−ph at small momentum transfer which may result in a strong superconducting pairing does not contradict the weak electron–phonon scattering observed in the transport experiments related to the electronic momentum relaxation. Another problem with the electron–phonon pairing mechanism is the d-wave symmetry of the gap which can appear only for a speciﬁc symmetry of the EPI. A large contribution of the d-wave symmetry to the EPI was found for the B1g bond-buckling phonon mode as mentioned above. However, an EPI linear in the atomic displacements arises only due to a buckling of the CuO2 plane, which breaks the mirror plane symmetry in YBCO, while in the tetragonal phase of LSCO the linear EPI is absent. It was suggested that the lower Tc in LSCO in comparison with YBCO could be explained just by this diﬀerence in the EPI for the B1g mode. However, this conjecture contradicts the observation of the highest Tc in the mercury compounds with the smallest buckling of the planes among the copper-oxide materials (see Sect. 2.5). It was shown also that the substantial in-plane versus out-ofplane sound speed anisotropy in cuprates could result in the d-wave pairing in the underdoped region where the anisotropy is large [45]. Therefore, various phonon modes should be taken into account in a realistic calculation of Tc . The strong Coulomb renormalization of the EPI, in particular the occurrence of ae forward scattering peak, may induce d-wave pairing by electronic coupling to all phonons [674]. In this respect, a spin-dependent electron–electron interaction can enhance the singlet d-wave pairing in comparison with the s-wave pairing usually observed in the electron–phonon pairing as discussed by Abrahams et al. [5].

458

7 Theoretical Models of High-Tc Superconductivity

In particular, the spin-ﬂuctuation scattering with a large momentum transfer brings about the d-wave pairing induced by the EPI with a small momentum transfer. As shown by Lichtenstein et al. [718], the spin-ﬂuctuation pairing can even be increased by a strongly momentum dependent electron–boson interaction. At the same time, an isotropic EPI is detrimental to the spin-ﬂuctuation pairing due to the reduction in the quasiparticle weight Z. Generally, the momentum dependence of the EPI determines whether the magnetic pairing induced by the strong correlations is enhanced or suppressed by the EPI. For instance, studies of the p–d model have shown that the d-wave pairing within the eﬀective t–J model is increased by an EPI of the same symmetry with the exchange interaction J(q) [1006]. Similar results were obtained by Shneider et al. [1159] within the generalized t–J model. It has been found that the EPI with the buckling mode increases Tc , while the EPI with the breathing mode suppresses it. The eﬀect of Holstein phonons on the properties of the 2D Hubbard model was studied by Macridin et al. [766] within the DCA and QMC methods (see Sect. 7.3.3). It has been found that EPI results in the formation of polarons, which enhance the nonlocal AF spin correlations. At the same time, the AF correlations enhance the EP coupling (as discussed in Sect. 6.4), but suppresses the d-wave pairing due to the reduction in the quasiparticle weight Z and the DOS N (0). We should mention other complications in the formulation of the strong coupling theory of the EP pairing in the cuprates. Quite a low Fermi energy EF in the cuprates which in some cases is comparable with the high-frequency oxygen phonon modes, EF ∼ ωph , may violate the Migdal adiabatic approximation. In that case, nonadiabatic eﬀects in the EPI and higher order vertex corrections in the parameter λωph /EF are to be taken into account (see e.g., [166, 198, 986]). The coupling of speciﬁc phonon modes with charge ﬂuctuations, e.g., with the low energy plasmon mode mentioned in Sect. 6.4 (see [322]), may give strong renormalization to the EPI as well. As discussed in Sects. 5.2.2 and 6.4, an important role in the underdoped cuprates is played by the hole coupling to the antiferromagnetic ﬂuctuations leading to the formation of magnetic polarons [837, 1063]. Lattice polarons will be considered below in relation with the bipolaron superconductivity. In certain cases, the anharmonic lattice eﬀects, especially in oxygen vibrations, can give a strong EP coupling constant. In particular, a strong EPI can be achieved within the double-well anharmonic model proposed by Vujiˇci´c et al. [1324] due to a large amplitude of atomic displacements, as, e.g., in the soft tilting mode in LSCO [998]. Within the anharmonic model a qualitative description of the anomalous oxygen isotope eﬀect and other peculiarities in the cuprate superconductors are possible (see [894, 1002] and references therein).

7.4 Electron–Phonon Superconducting Pairing

459

Resume In summary, realistic studies of the electron–phonon pairing in the cuprates require a consistent solution of several challenging problems: calculations of anisotropic electron–phonon matrix elements for a multi-mode phonon spectrum, consideration of strong electron correlations and the long-range Coulomb interaction, and, ﬁnally, solution of the strong-coupling anisotropic Eliashberg equation for a momentum dependent superconducting gap. This program has not yet been realized to prove unambiguously the crucial role of the EP coupling in the high-temperature superconductivity in cuprates. For a quantitative calculation of Tc , the spin-ﬂuctuation contribution in no case can be ignored. 7.4.2 Van Hove Singularity Scenario In early studies the important role of the van Hove singularity (vHs) at the (π, 0)-type points of the Brillouin zone in the electronic spectrum of the cuprates was emphasized by several researches (for a review see [352]). At the vHs in a two-dimensional (2D) lattice, the density of electronic states (DOS) diverges logarithmically, N (ε) N0 ln | 2D/|ε − εvHs | where D is the 2D electronic bandwidth and N0 is a normalized DOS. The topology of the vHs is a saddle-point one with an electron-like and a hole-like dispersion. This peculiarity of the electronic spectrum can result in various anomalous electronic properties in the normal state of the cuprates and have an inﬂuence on the superconducting pairing. A comprehensive review of the vHs scenario for the high-Tc superconductivity has been presented by Markiewicz [791] where a detailed discussion of various aspects of the vHs problem is given. Some consequences of a much stronger extended saddle point singularity are discussed by Abrikosov [11]. To show the inﬂuence of the vHs on the superconducting pairing, we consider the weak coupling BCS equation for Tc h ¯ωph

1=V −¯ hωph

dε ε , N (ε) tanh 2ε 2kB Tc

(7.88)

where ωph is a cutoﬀ phonon frequency and V is the coupling constant. If the DOS N (ε) is a smooth function, it can be replaced by the DOS at the FS N (0) and (7.88) leads to a conventional BCS formula for kB Tc ∝ ¯hωph exp(−1/λ) with λ = V N (0). However, if the FS lies close to the vHs, |EF − εvHs | ωph , the integration of the logarithmically divergent DOS function in (7.88) results in a more complicated formula for Tc (see e.g., [791]):

2 2D 1 ˜ − 1, (7.89) = + ln2 kB Tc = eD exp(−1/ λ), ˜ λ0 ¯hωph λ

460

7 Theoretical Models of High-Tc Superconductivity

where λ0 = N0 V . In the weak coupling limit,ln2 (2D/¯hωph ) 2/λ0 , the for mula (7.89) can be written as kB Tc = eD exp − 2/λ0 . An enhancement of Tc due to the vHs results from the square root dependence of the exponent in ˜ and a much larger electronic energy D in the prefactor in compar(7.89) on λ ison with the phonon energy h ¯ ωph in the BCS formula. However, in the weak coupling Tc is small, while in the strong coupling limit the enhancement caused by the vHs decreases. For instance, for λ0 = 2 the vHs enhancement is only e/2 = 1.65 [791]. In the case of an extended saddle-point singularity, the DOS shows a much stronger divergence, ∝ (ε − ε0 )−1/2 which results in a stronger enhancement of Tc (see [11]). It should be noted, however, that Tc (7.89) near vHs is calculated in the weak coupling approximation which ignores the quasiparticle weight renormalization near vHs. A fully self-consistent solution within the Eliashberg strong coupling theory has revealed a strong suppression of Tc for doping when the chemical potential crosses the extended saddle-point singularity (see e.g., [966]). However, a smoothing of the vHs by weak threedimensional eﬀects, impurities, etc., does not change essentially Tc (7.89) due to the ﬁnite region of integration in the derivation of the formula. The inﬂuence of the vHs on the isotope eﬀect in the cuprates was discussed in Sect. 6.3 (see also discussion by Markiewicz [791]). It should be mentioned that the vHs in a 2D lattice, in view of the Coulomb correlations, favors the development of spin and charge ﬂuctuations near half-ﬁlling. As the renormalization group technique studies show, several instabilities driven by spin (SDW) or charge density waves (CDW) or superconducting transition compete with each other close to half-ﬁlling at the vHs (see Sect. 7.3.1). Close to the CDW instability an enhancement of the DOS and an increase of Tc can occur as in the case of the vHs anomaly. We note that this mechanism for the enhancement of Tc was ﬁrst proposed by Kopaev (see e.g., [398] and references therein). Other aspects of the vHs in physical properties of the cuprates, like the dynamic vHs–Jahn–Teller eﬀect, the pseudogap and striped phases, are discussed in detail by Markiewicz [791]. 7.4.3 Polaron and Bipolaron Superconductivity In early studies of high-temperature superconductivity, much attention had been paid to models with local pairing due to the nonretarded on-site attraction (the negative-U Hubbard model) or intersite attraction, Vij < 0. The electron attraction in the models has been explained by strong polaron eﬀects which are well known in the ionic crystals. When an electron is embedded into the lattice it induces a local distortion of the lattice and for a suﬃciently strong electron–phonon (EP) coupling a formation of the polaron – a bound state of the electron dressed by a cloud of phonons occurs (for a review see e.g., [29]). The lattice deformation produced by polarons results in their mutual attraction. Provided that the lowering of the crystal energy overcomes the Coulomb repulsion of two polarons, the small bipolaron as a bound state of two polarons arises.

7.4 Electron–Phonon Superconducting Pairing

461

Originally, the bipolaronic superconductivity in narrow band crystals has been proposed by Alexandrov and Runninger [26]. After the discovery of the high-temperature superconductivity in cuprates, the concept of the local pairing in contrast to the BCS Cooper pairing in the momentum space has been advocated by many authors. This approach has been supported by the observation of a short coherence length in cuprates in comparison with conventional superconductors. The properties of the models with polaronic and bipolaronic superconductivity are discussed in detail by Micnas et al. [827] and by Alexandrov and Mott [29]. Brieﬂy, two limits can be considered. In the case of weak coupling, the polaron superconductivity is observed where a nonretarded pairing of the BCS-type takes place. An increase of the DOS due to the polaronic band narrowing enhances the superconducting Tc ∼ W exp(−1/λ) where λ =| U | /W and W = zt is the half of the bandwidth for polarons with hoping integral t and the number of the nearest-neighbors z. In the strong coupling limit, | U | W , a crossover to the bipolaron superconductivity occurs where the Bose–Einshtein condensation (BEC) of preformed local pairs takes place with Tc ∼ t2 / | U |. The Tc peaks at some intermediate coupling constant. Generally, a crossover from the BCS limit for pairs with a large coherence length to the BEC limit for local pairs takes place with increasing EP coupling (for references see e.g., [739]). However, several problems arise in the application of these models to the cuprate superconductors. In the strong coupling limit, the localization of the bipolarons at a large coupling constant takes place which destroys the superconductivity. It is diﬃcult to describe the d-wave pairing observed in cuprates by local pairs which have no internal structure. The bipolarons, as Bose quasiparticles, have no Fermi surface which has been unambiguously established in cuprate superconductors (see Sect. 5.2.2). To resolve these and some other problems of the bipolaron superconductivity, the “Fr¨olich–Coulomb” model (FCM) for bipolarons has been developed by Alexandrov [30] and Alexandrov et al. [39] which deals with the strong long-range Coulomb and the strong long-range EP interaction (see also [40]). We have already discussed several results of the theory concerning the anomalous upper critical magnetic ﬁeld in Sect. 4.3.2, the isotope eﬀect in Sect. 6.3. Various physical properties of the cuprates in the normal and superconducting states have been described within the bipolaron theory in detail by Alexandrov [43,44]. So below we present only basic results for the bipolaronic superconductivity within the FCM. In deriving the FCM, the occurence of a large electron–phonon Fr¨olich interaction in cuprates was emphasized. A theoretical estimate of the interaction can be obtained the calculation of the polaron energy level shift from −1 −1 3 3 Ep = (ε−1 d − ε ) q/(2π) ) (4πe2 /q 2 ) where ε∞ and ε0 are the high ∞ 0 frequency and static dielectric constants, respectively. By taking the experimental values ε∞ = 5 and ε0 = 30 for the La2 CuO4 , one obtains Ep = 0.65 eV [38]. The resulting strong polaron attraction of the order of 2Ep leads to the

462

7 Theoretical Models of High-Tc Superconductivity

formation of bipolarons. The inter-site bipolarons considered in the FCM are relatively light in comparison with the on-site bipolarons. The FCM Hamiltonian in the direct lattice space can be written as T (n − n ) c†n cn + Vc (n − n )c†n cn c†n cn H=− n=n

−ω

n,m

n=n

Ve,ph (m − n)c†n cn (b†m + bm ) + ω

m

(b†m bm + 1/2), (7.90)

where c†n , cn and b†m , bm are electron and phonon operators, respectively. For simplicity, only one optic phonon mode ω without dispersion is considered here and the following notations are introduced: T (n − n ) is the electron hopping integral in a rigid lattice, Vc (n − n ) is the Coulomb repulsion, and Ve,ph (m − n) = g(m − n)(e · um−n ) is the Fr¨olich EP interaction, where g(m − n) is the dimensionless EP coupling function, um−n = (m − n)/|m − n| is the unit vector in the direction from the lattice site m to n and e is the polarization vector of the phonon mode. It is

assumed that the EP coupling is strong, λ = Ep /D > 1 where Ep = ω n [Vep (n)]2 is the polaron level shift and D = zT (a) is close to the half-bandwidth in a rigid lattice (a is the lattice constant). In the strong coupling limit, the Migdal–Eliashberg theory cannot be applied and the “1/λ” multi-polaron expansion can be used which treats the kinetic energy as a perturbation in the model (7.90) [40]. Depending on the ratio of the intersite Coulomb repulsion Vc and the polaronic level shift Ep various phases can emerge [39]. When the Coulomb repulsion is large a polaronic Fermi liquid is the ground state of the model. At the intermediate Coulomb interaction, a bipolaronic superconducting phase arises, while at the weak Coulomb repulsion a charge-segregated insulator may appear. A speciﬁc energy dispersion of the mobile intersite bipolarons with four minima at k = 0 close to the (π, 0) points in the Brillouin zone explains the d-wave symmetry of the Bose–Einshtein condensate [35]. A schematic representation of the low-energy polaron–bipolaron electronic structure is shown in Fig. 7.19 (left panel). The bipolaronic bandwidth denoted by 2t and the polaronic bandwidth denoted by 2W are determined by the renormalized hopping integral t(n − n ). This can be estimated at low temperatures, T ω, as t(n − n ) = T (n − n ) exp[−g 2 (n − n )], g 2 (n − n ) = (Ep /ω) − [Ve,ph (m − n) Ve,ph (m − n )].

(7.91)

m

The eﬀective polaronic mass, m∗ = m exp(g 2 ) , is determined by the exponential factor g 2 = g 2 (a) which in general can be represented as g 2 = γ Ep /ω. The coeﬃcient γ, as follows from (7.91), depends on the interaction Ve,ph . To have light polarons and bipolarons with a small mass renormalization,

7.4 Electron–Phonon Superconducting Pairing

463

Fig. 7.19. The electronic structure (left panel) and the phase diagram (right panel) of the polaron–bipolaron model (after Alexandrov [43])

m∗ /m < 10, models of the intersite bipolarons were proposed. Their mass renormalization due to the long range Fr¨ olich EP interaction is smaller than those of the Holstein (on-site) polarons, γFr ∼ 0.5, while γH = 1 (see [43]). Since the perturbation theory in the physically interesting regime λ = 1 − 2 becomes unreliable, various numerical techniques have been used to calculate the polaron parameters and to conﬁrm the existence of light polarons and bipolarons (see e.g., [422, 423, 838, 840], and references therein). A temperature–doping phase diagram derived for the bipolaron model is shown in Fig. 7.19 (right panel). The superconducting phase transition occurs at the temperature of the Bose–Einstein condensation of bipolarons in the region between the underdoped xu and the overdoped xo charge concentrations. This temperature for small radius bipolarons can be estimated by the formula (1.5) for an ideal 3D Bose gas as discussed in Sect. 1.1: TBEC ∝ n2/3 /m∗ . A parameter-free ﬁtting for TBEC 1.64 (eRH /λ4ab λ2c )1/3 (in terms of the Hall number RH and the magnetic penetration depth λab,c ) for various cuprate superconductors has given results close to the experimentally observed Tc [36]. With doping, the EP interaction decreases due to screening eﬀects and the bipolaron binding energy Δ also decreases. This binding energy determines the pseudogap temperature T ∗ in the phase diagram. In the overdoped region, it is assumed that the chemical potential enters the bipolaron band and a mixture of bipolaron and polaron charge carriers appears. It is argued that the Pauli principle and the energy conservation prevent the bipolaron decay into a pair of fermions. In this region, the pseudogap disappears and a large FS should evolve as seen in ARPES experiments. By ﬁtting the normal-state spin susceptibility to the polaron-bipolaron theory, M¨ uller et al. [872] have suggested that at optimal doping a coexistence of approximately 60% of bipolarons and 40% Fermi-liquid-type charge carriers is realized.

464

7 Theoretical Models of High-Tc Superconductivity

An unconventional dynamic bipolaron formation has been suggested by Mihailovic et al. [831]. By taking into account the observation of strong anomalies of the LO phonon modes in the direction of the Cu–O bonds in the LSCO and YBCO crystals (see Sect. Ch06:sect6.1.1), the authors developed a theory of nonlocal Jahn–Teller (JT) pairing interaction caused by coupling between the degenerate planar oxygen px , py states and phonons with k0 ∼ π/2a (see also [573]). It is assumed that at a pseudogap temperature T ∗ of the order of the pairing energy gap EJT ∼ 32 meV preformed JT-type pairs appear. They attain a coherence below the BEC (or percolation temperature) which determines the superconducting transition temperature Tc [832]. A crossover from dynamical pairs and in the underdoped phase to a mixture of boson and fermions in the stripes in the overdoped region is proposed. Contrary to mobile light bipolarons and thermally excited Fermi particles, polarons, a model with coexisting localized bound electronic pairs (hard-core bosons) and itinerant electrons, possibly in diﬀerent energy bands, has been proposed by Ranninger et al. [1042] and independently by Ionov [526]. The boson–fermion model (BFM) is described by the Hamiltonian (see e.g., [1043]): † † HBF = (D − μ) ci,σ ci,σ − t ci,σ cj,σ i,σ

+(ΔB − 2μ)

i

i=j ,σ

b†i bi

+v

i,σ

(b†i ci↓ ci↑ + bi c†i↑ c†i↓ ),

(7.92)

where t is the hopping integral for electrons (D = zt), ΔB is the atomic level of the local pairs (bipolarons), and v describes the bipolaron – electron pair exchange interaction. The common chemical potential μ for electrons and is determined

bosons

† by the total number of charge carriers: † N = c c + 2 i,σ i,σ i,σ i bi bi . There are two coupled superconducting order parameters in the model, the BCS Fermi pairing parameter φ =

(1/N ) k c†k↑ c−k↓ and the bipolaron BEC density ρ = (1/N ) i b†i + bi . The commutation relations for the bosons which represent pairs of coupled electrons are described by those for the Pauli (or hard-core boson) operators: (b†i bj − bj b†i ) = δi,j (2b†i bi − 1). The solution of the system of equations for the two-order parameters and the number of charge carriers N in the MFA suggested the following general picture of the superconducting transition [1043]. For a low concentration of charge carriers n, the bosonic level being above the Fermi level EF is empty and the itinerant electrons show a weak BCS-type superconductivity induced by the exchange interaction. With increasing n, the chemical potential crosses the bosonic level at some concentration nc1 beyond which the bosonic occupation number nB = b†i bi increases linearly with n. The Fermi occupation

numbers nF = σ c†i,σ ci,σ changes only slowly which results in the pinning of the Fermi energy. At some nc2 , the bosonic level is completely ﬁlled, nB = 1, and extra charge carriers go into the system of the itinerant electrons. In the interval nc1 < n < nc2 both the Fermi and the Bose superconducting order

7.5 Charge Fluctuation Models

465

parameters are nonzero which may give a high Tc with a maximum at some intermediate occupation number when nF ∼ nB . Various thermodynamic and transport properties, single-particle and collective spectra of excitations, a pseudogap have been studied within the BFM (see e.g., [289, 290, 1044, 1045] and references therein). In the model (7.92), the exchange interaction v is local which results in the s-wave pairing order parameters φ and ρ. To obtain d-wave pairing, a momentum dependent interaction should be considered as, e.g., in the model proposed by Geshkenbein et al. [379]. In the model the bosons originate from fermions paired into the state with d-wave symmetry in the corners of the FS close to the (π, 0)-type points in the pseudogap region. The fermions, weakly interacting with bosons, are on the remaining part of the FS. The superconducting phase transition in the model reveals a more mean-ﬁeld-like character than the conventional Bose condensation with a wide ﬂuctuation region near Tc . However, studies of the BFM beyond the MFA by Alexandrov [32, 34, 37] have shown the important role of the boson energy renormalization E0 = ΔB −2μ caused by the boson self-energy Σb (q, ω). It has been pointed out that due to the energy conservation the Cooper pairing of the itinerant electrons via bosonic exchange involves only the zero-energy bosonic propagator. It has been shown that the corresponding bosonic self-energy Σb (q = 0, ω = 0) is divergent in the same manner as in the BCS equation for the order parameter. This results in an equation for Tc which has only a zero solution, Tc = 0 [37]. Therefore, Alexandrov has concluded that a superconducting transition can occur only due to the condensation of the bosons (in a 3D system), while boson interaction with itinerant electrons gives merely a damping of the longwave bosonic excitations further suppressing the condensation temperature (see also [43]). Thus, these studies question the results obtained in the BFM within the MFA. Resume In summary, the important role of the polaron eﬀects in the polaron–bipolaron theory of superconductivity should be emphasized. The Fr¨ olich–Coulomb model for mobile small bipolarons with high Tc deﬁned by the 3D BEC provides a plausible explanation for many normal and superconducting anomalous properties of cuprate superconductors. At the same time, a pure bipolaronic character of charge carriers is diﬃcult to reconcile with the observation of the large Fermi surface and a model of overlap bands of polarons and mobile bipolarons seems to be more feasible.

7.5 Charge Fluctuation Models As follows from the simple BCS formula for the superconducting transition temperature (1.1), a large enhancement of Tc can be obtained if an intermediate boson mediating the electron pairing will have a large energy, e.g.,

466

7 Theoretical Models of High-Tc Superconductivity

of the order of the electron energies, ¯hω ∼ 1 eV. Then, even in the case of weak coupling, λ 1, transition temperatures around the room temperature are possible. This exciton scenario, in which the coupling is due to the exchange of electronic excitations, was ﬁrst proposed by Little [728] for quasi-one-dimensional organic superconductors and by Ginzburg [390,391] for layered systems (for reviews, see [392, 729]). To realize these models, we must assume the existence of two groups of electrons: one of them is connected with the conductivity band where the superconducting pairing occurs due to the exchange of excitons living in the second group of almost localized electrons. In the cuprate superconductors, due to the multiple-band structure of the electron spectrum, it is tempting to suggest such a separation of the electronic states which has been used in various exciton models. In particular, a pairing induced by a screened Coulomb interaction within a plasmon model or by charge-transfer excitations in multiband models has been proposed. Several models where pairing occurs due to a Coulomb repulsion in multiband models have also been suggested. Below we consider some of these models. 7.5.1 Plasmon Model Let us consider a general representation for the eﬀective interaction of two electrons in a crystal. In the framework of the dielectric function formalism the interaction in a quasi–isotropic system can be written as (see e.g., [392]) Veﬀ (q, ω) =

4πe2 1 , q 2 ε(q, ω)

(7.93)

¯ ω = εp − εp are, respectively, the momentum and the where q = p − p and h energy transfer between two scattering electrons and ε(q, ω) is the dynamical dielectric function of the crystal. In a more general approach which takes into account the discrete crystal structure, the interaction (7.93) and the dielectric function depend not only the momentum transfer q but on the electron momenta p, p and the reciprocal lattice vectors G and therefore are matrices, e.g., εp+G,p +G (ω). For simplicity, we ignore here the matrix structure of these functions. The electron attraction and a possible pair formation may occur if the inverse dielectric function in (7.93) is negative in some region of wave vectors and frequencies. The inverse dielectric function is related to the density response function χ(q, ω) = ρq |ρ†q ω by the equation ε−1 (q, ω) = 1 + v(q) χ(q, ω) where v(q) is the Coulomb interaction. Therefore, the inverse dielectric function satisﬁes the dispersion relation (see (A.6), (A.7)) as any causal response function: Re ε

−1

2 (q, ω) = 1 + P π

∞

z dz Im ε−1 (q, z), − ω2

z2 0

(7.94)

7.5 Charge Fluctuation Models

467

where the symmetry relation Im χ(q, −ω) = −Im χ(q, ω) was used. It should be pointed out that the dielectric function ε(q, ω) satisﬁes the Kramers– Kronig relations only for q = 0 as discussed in Sect. 5.3.1 for the dynamical conductivity (5.24) related to ε(0, ω) (for detail, see [287]). Quite a general condition can be inferred from (7.94) for the static response function by taking into account that the spectral intensity for charge ﬂuctuations is a nonnegative function, S(q, ω) = −(1/π) Im ε−1 (q, ω) > 0 at ω > 0. This function is directly measured in the transmission electron energyloss spectroscopy (see Sect. 5.2.1). Therefore, from (7.94) we can derive the following inequalities for the static dielectric function [286]: ε−1 (q, 0) < 1 :

ε(q, 0) > 1,

or ε−1 (q, 0) < 0.

(7.95)

The requirement of a positive charge compressibility of the whole crystal leads to the stability condition limq→0 ε(q, 0) > 0 , or even to a more strict one, limq→0 ε(q, 0) > 1 resulting from (7.95). However, negative values of the ε−1 (q, 0) in some region of the wave vectors q = 0 given by the last inequality does not contradict the general condition in (7.95). Thus, the eﬀective interaction (7.93) may be attractive at ﬁnite wave vectors but must be repulsive in the static limit at long wavelengths. This shows that the long-range part of the Coulomb interaction in (7.93) is not favorable to the conventional superconducting pairing. As shown by Dolgov et al. [287], a negative static dielectric function at ﬁnite values of momentum q caused by local ﬁeld eﬀects turns out to occur in a wide class of electronic systems and this has important implications to the high-temperature superconductivity [288]. In this general discussion, a conventional s-wave pairing mediated by electron attraction is implicitly assumed. However, the d-wave pairing can be mediated by a repulsive interaction as discussed in Sect. 7.3. Let us consider a general representation for the dielectric function by taking into account both the electronic, αe (q, ω), and the ionic, αi (q, ω), polarizations: ε(q, ω) = 1 + αe (q, ω) + αi (q, ω). (7.96) Then the eﬀective interaction (7.93) takes the form Veﬀ (q, ω) =

v(q) α ˜ i (q, ω) v(q) − , εe (q, ω) εe (q, ω) 1 + α ˜ i (q, ω)

(7.97)

−1 where the ﬁrst term proportional to ε−1 determines e (q, ω) = [1 + αe (q, ω)] the contribution to the interaction from electronic excitations (excitons) and the second determines the contribution from phonons with a lattice polarization α ˜ i = αi /εe . In the BCS theory, the low-frequency limit in (7.97) should be considered when the ﬁrst term gives a static Coulomb repulsion and the second term induces the attraction at frequencies ω < ωph where ωph is a characteristic phonon frequency as discussed in Sect. 7.4.1. In the region of high-frequencies, an additional contribution to the phonon pairing can arise

468

7 Theoretical Models of High-Tc Superconductivity

from long wavelength plasmon oscillations. The latter can be represented by a simple model in the conventional random phase approximation (RPA): εe (q, ω) = 1 −

ωp2 . ω 2 − ω 2 (q)

(7.98)

The plasma frequency is determined by the equation: εe (q, Ωq ) = 0 which gives Ωq = [ωp2 + ω 2 (q)]1/2 with the plasma frequency ωp (see (5.26)) and the dispersion ω(q) = ωp (q/q0 ) where q0 is the Thomas–Fermi wave vector (for details, see [287, 730]). In this model, the attraction occurs in the frequency region ω(q) < ω < Ωq where ε−1 e (q, ω) < 0. Although a superconducting pairing can be obtained within the RPA for the simple plasmon model (7.98), the transition temperature appears to be quite small, Tc /EF ∼ 10−4, or even vanishes for a large plasma frequency ωp if one takes into account strong vertex corrections in this case (for details see [732]). Several more complicated plasmon models have been proposed for hightemperature superconductivity (see for example, [135,663,964,966,967,1210], and references therein). It was pointed out that in layered crystals a weakly damped quasi-acoustic plasmon mode appears due to a quasi-two-dimensional electronic spectrum and a layered structure. The plasmon spectrum Ωp (q , qz ) depends on the in-plane q and perpendicular to the plane qz wave vectors and in the long-wave limit for q → 0 decreases from Ωp (q , qz = 0) ≈ ωp to Ωp (q , qz = π/d) ≈ s q [663]. Here, ωp2 = 4πnl e2 /ε∞ m∗ d is the conventional optic plasmon mode for electrons of density nl in one layer and the distance d between the layers along the z-axis. The acoustic mode at qz = (π/d) is caused by the out-of-phase electron ﬂuctuations in the neighboring layers. A weak electron hopping between the layers tz results in a gap in the acoustic spectrum proportional to tz . The acoustic plasmons in a wide range of the wave vectors q reveal weakly damped excitations determined by the spectral density Spl (q, ω) = −(1/π) Im ε−1 (q , qz , ω). These excitations give an additional contribution to the phonon attraction leading to a noticeable enhancement (∼ 20%) of the phonon mediated Tc in the s-wave pairing channel (see e.g., [135]). However, it should be pointed out that the EELS study of the plasmon dispersion in Bi-2212 [908] has not revealed an acoustic A (see Sect. 5.3.1). plasmon mode for the in-plane wave vectors q ≥ 0.05 ˚ Extensive studies of the strong coupling Eliashberg-type equations for the normal and anomalous Green functions by taking into account the screened Coulomb interaction and the retarded electron–plasmon coupling have been performed by Pashitskii et al. (for references, see [964,966,967]). The eﬀective electron interaction is deﬁned by the static Coulomb repulsion Vc (q , qz ) and the attraction mediated by the plasmon excitations: ∞ Spl (q , qz , ω ) dω Veﬀ (q , qz , ω) = Vc (q , qz ) 1 + . (7.99) ω − ω + i0+ ∞ Here, the spectrum of plasmon excitations is determined by the spectral density Spl ((q , qz ), ω) = −(1/π)Im [1/ε(q , qz , ω + i0+ )] which can be directly

7.5 Charge Fluctuation Models

469

measured in the EELS experiment (see (5.13)). A linearized gap equation within the Eliashberg strong coupling theory was numerically solved for an averaged over the wave vector qz eﬀective interaction (7.99) and the superconducting Tc was calculated. A high transition temperature Tc ∼ 150 K was found for the s-wave pairing, higher than for the d-wave one, Tcs > Tcd . To obtain Tcd > Tcs , the exchange-correlation eﬀects should be taken into account determined by crossing diagrams of the ladder type [965]. It has been also shown that many-body-eﬀects given by the Coulomb vertexes in the electron polarization operator further enhance Tcd together with the charge vertex γc (p, q) in the electron–phonon interaction (7.87). At the same time, a fully self-consistent solution for the self-energy and the Green functions has shown a strong reduction and even vanishing of Tc with doping when the chemical potential crosses the extended saddle-point singularity in the electron spectrum due to the quasiparticle weight Z(ω) suppression (see [966] and the references therein). In conclusion, the authors have suggested that all the pairing channels, the electron–phonon, the electron–plasmon coupling, and the local Hubbard repulsion should be taken into account to describe the d-wave high-temperature superconductivity in the cuprates. 7.5.2 Exciton Models The charge transfer character of the insulating state of the cuprate superconductors described by the p–d model (7.4) (see Fig. 7.2) could result in collective electron excitations (excitons) related to a charge transfer from copper to oxygen sites, Cu2+ O2− → Cu1+ O1− , as proposed originally by Varma et al. [1305]. For a suﬃciently strong Coulomb repulsion Upd of the holes on the p- and d-orbital, Upd ∼ tpd in (7.4), the excitonic band energy may occur in the mid-infrared region, ω0 < 1 eV, below the particle-hole continuum. It was suggested that this collective charge-transfer excitations could mediate the high-temperature superconducting transition. Studies of the collective spin and charge excitations in the three-band model have shown the possibility of the superconducting pairing of dx2 −y 2 symmetry due to spin ﬂuctuations, while the charge transfer excitations between copper and oxygen ions are more favorable for the extended s or dxy symmetry pairing [731]. In a more detailed elaboration of this model, a circulating current phase below a certain temperature Tcc (x) depending on the hole concentration δ has been proposed [1307]. The phase is characterized by the intracell orbital currents between the Cu–O and O–O bonds in each unit cell ﬂowing in such a manner that the translational symmetry of the lattice is preserved while the time-reversal invariance and fourfold rotational symmetry are broken (see Sect. 7.3.1). A quantum critical point (QCP) was assumed at the concentration δ0 where Tcc (δ0 ) = 0 and various physical properties close to the QCP were considered. In particular, a superconducting pairing of the d-wave or the extended s-wave symmetry mediated by the current ﬂuctuations has been suggested. The current ﬂuctuation model can be used as the microscopic basis

470

7 Theoretical Models of High-Tc Superconductivity

for the phenomenological marginal Fermi-liquid (MFL) model discussed in Sect. 7.2.1. The superconductivity in the general MFL model was considered within the Eliashberg strong coupling theory by Kuroda et al. [679] who found the s-wave pairing induced by the q-independent spectrum of the collective excitations (7.22). A general two-band model was proposed by Kresin et al. [664] to describe an induced superconductivity by charge transfer between an intrinsically superconducting and a metallic normal subsystems. The model was used to explain the peculiar properties of the YBCO-type compounds. Charge transfer excitations within a system of CuO4 clusters were considered by Moskvin et al. [867]. It was assumed that the CuO4 cluster is unstable with respect to the disproportionation reaction: 2 CuO6− → [CuO5− 4 4 ]JT + 7− [CuO4 ]JT which results in a system of polar (hole-h or electron-e) pseudo Jahn–Teller (JT) centers in the lattice. It was suggested that a doped hole in a system of CuO4 clusters can occupy the nearly degenerate molecular terms 1 A1g (Zhang–Rice singlet) or the singlet 1 E1u (triplet 3 E1u ) state given by the conﬁgurations (b1g )2 or b1g eu , respectively. The polar oxygen eu -centers couple with the active local displacements of the corresponding symmetry resulting in the JT eﬀect. In the model, a doped hole motion can be considered as a local boson hopping in a lattice of singlet-triplet hole-centers. The transfer integral for bosons tBB ∝ Keh depends on the overlap integral for local oscillatory states for hole and boson Keh = χe |χh which determines a complicated spectrum of excitons. Related to the charge transfer excitations, a two-site negative-U center model has been proposed by Mitsen et al. [842]. It was assumed that upon doping the CuO2 plane by a hole or an electron, e.g., by Sr in the La2 CuO4 crystal or by Ce in the Nd2 CuO4 crystal, the dopant charges are localized near the impurity ions. In the case of La2 CuO4 , a doped hole occupies four nearest neighbor oxygen sites around the impurity ion lowering the 3d10 (Cu1+ ) energy levels of the nearest copper sites by Δd ∼ 1.8 eV due to the unscreened Coulomb attraction between the doped hole and the 3d10 electron states. If two copper sites with the reduced 3d10 energy√levels occur close to each other (e.g., at a distance of either l = 3a or l = a 5), two holes from the nearest neighbor oxygens tunnel to these copper levels creating a bound state of two holes on oxygen 2p5 sites and two electrons on 3d10 sites – a negative U -center. The energy position of the two-site negative-U center is suggested to be close to the O2p6 electronic band and a strong mixing between the localized states and band states could occur. The model therefore presumes that, at low doping level, the localized holes just create “active” negative U -centers, while the conductivity and superconductivity are determined by holes generated by these U -centers. In this scenario, both transitions from the insulator to the metallic state and further to the superconducting state is of the percolation orogon. The superconductivity occurs due to the exchange between localized singlet hole pairs – bosons, and holes in the oxygen conduction band in a similar way as in the considered above boson–fermion model. This scenario could

7.5 Charge Fluctuation Models

471

be justiﬁed for LMCO or NMCO compounds where the dopant ions, residing in the layer nearest to the CuO2 plane, can localize the charge carriers in the plane. However, in other cuprates, in particular in the mercury compounds, the dopant ions are far away from the CuO2 plane and produce a very weak perturbation on the electronic structure of the plane, as discussed in Sect. 5.1.2 (see [304]). A diﬀerent behavior of the chemical potential with doping in the La2−x Mx CuO4 and Bi-2212 crystals also points to a distinct mechanism of doping in these compounds as discussed in Sect. 5.2.2 (see [434]). In early studies an exciton d–d pairing mechanism was proposed by Weber [1343] and Gaididei et al. [374]. The pairing is mediated by the Jahn–Teller (JT) orbital excitations between the dx = d(x2 − y 2 ) and dz = d(3z 2 − r2 ) orbitals of the 3d9 Cu states where the JT excitation energy is of the order of EJT ∼ 0.5 eV. The Coulomb repulsion Vpd between the p-hole on the pσ (x, y) x orbitals and the dx -orbital, Vpd , is much larger than the Coulomb interacz x z tion with the dz -orbital, Vpd , where ΔVxz = Vpd − Vpd 0.5 eV. Therefore, an excitation of the d-system from the dx state to dz by p-hole hopping leads to a decrease of the Coulomb energy of the CuO4 -cluster, which is energetically favorable. Superconducting pairing of the p-holes mediated by exchange of these excitons could result in a high transition temperature Tc EJT exp(−1/λex ) where the coupling constant is λex N (0) (ΔVxz )2 /EJT . However, the theoretical description of the model and the experimental veriﬁcation of the quadruple d–d excitations appeared to be very complicated and this hindered realistic calculations of the transition temperature (see e.g., [1344]). In a number of studies the charge density ﬂuctuations close to a charge instability were invoked to explain the high-temperature superconductivity in the cuprates. We have already discussed charge density waves (CDW) and spin-density waves (SDW) related to stripe formation in the LBCO and LSCO crystals (see Figs. 2.5 and 3.8). A general discussion of the competition between the Cooper pairing, CDW and SDW is given by Gabovich et al. [373]. The competition between various phases studied within the renormalization group technique was considered also at the end of Sect. 7.3.2. Here we consider a superconducting pairing scenario induced by charge ﬂuctuations close to a quantum critical point related to the charge instability (for a review see [209, 210]). It has been suggested that the charge ﬂuctuations in the critical region result in a non-Fermi liquid behavior in the normal phase and can mediate a superconducting transition. In the proximity to the CDW instability, the charge density response function χch (q, ω) = ρq |ρ†q ω shows a critical behavior similar to the antiferromagnetic (AF) QCP described by the dynamical spin susceptibility (7.59). Whereas the AF QCP occurs at the AF wave vector Q = (π, π), the charge instability can occur at q = (0, 0) in the case of the phase separation instability or at some incommensurate wave vector qc in the case of the CDW instability. The gap in the charge ﬂuctuation spectrum (speciﬁed by the eﬀective mass m ) closes at the QCP as m ∝ (δ − δc )β where δc is the

472

7 Theoretical Models of High-Tc Superconductivity

critical doping. To study the superconducting pairing, a static model for the eﬀective electron interaction has been used by Perali et al. [976]: Veﬀ (qx , qy ) = U −

V , κ2 + qx2 + qy2

κ2 ∝ (δ − δc )β .

(7.100)

The parameters of the model U = 0.2 eV, V = 0.3 eV were chosen to ﬁt the estimates obtained for a Hubbard–Holstein model near phase-separation transition. By solving a conventional BCS equation (see (A.35)) with the eﬀective interaction (7.100), a superconducting gap at zero temperature and the superconducting Tc were calculated. It was found that the repulsion part U in the interaction is important for the occurrence of the d-wave symmetry of the gap. The doping dependence of Tc is determined by the temperature and doping dependence of κ2 in (7.100). Similar results have been obtained for the incommensurate CDW instability scenario when the wave vector dependence in the interaction (7.100) is given by (qα − qc,α )2 instead of (qα )2 . A related model of high temperature superconductivity mediated by critical charge ﬂuctuations at small wavenumbers have been proposed by Imada [519]. It was argued that close to the Mott transition in the region of the marginal quantum criticality, density ﬂuctuations are generated which are responsible for the superconducting pairing. The linearized Eliashberg equation was solved in the mode-coupling approximation for fermions and charge ﬂuctuations and a high transition temperature of the order of 100 K was found. It has been shown that the dx2 −y 2 -wave symmetry pairing wins over the extended s-wave symmetry because the eﬀective interaction is repulsive almost everywhere inside the Brillouin zone except for the region of small wave vectors. It was emphasized that the charge-ﬂuctuation mechanism induced by the marginal quantum Mott criticality may be even more eﬃcient in attaining high values of Tc than the spin ﬂuctuation pairing mechanism considered in Sect. 7.3.2. 7.5.3 Coulomb Repulsion Pairing In the end, we consider an electronic pairing mechanisms induced by the Coulomb repulsion. Contrary to the previous sections, where the pairing mechanisms were mediated by retarded electron interaction with spin- or charge-ﬂuctuations caused by Coulomb correlations, here a superconductivity is assumed to originate in the static Coulomb repulsion. This mechanism may occur in systems with a peculiar electronic structure. The simplest example is the two-band model of a superconductor proposed by Moskalenko [864] and later, independently, by Suhl et al. [1198] (for a review, see [947]). Within this model, the interband Coulomb scattering may induce the superconducting state if the order parameter Δα (k) has diﬀerent signs in the two bands α = 1, 2. The system of BCS-type equations in the two-band model reads

7.5 Charge Fluctuation Models

Δα (k) = −

α ,k

Vαα (k, k )

Eα (k ) Δα (k ) tanh , Eα (k ) 2T

473

(7.101)

where Eα (k) = [ε2α (k) + |Δα (k)|2 ]1/2 is the spectrum of the quasiparticles in the superconducting state with the normal state spectrum εα (k). The scattering potential Vαα (k, k ) describes the interband interaction at α = α , and the intraband interaction at α = α . The analysis shows that the equations have a nonzero solution for gaps of diﬀerent signs, Δ1 /Δ2 < 0, for repulsive interactions Vαα (k, k ) > 0, if the interband scattering is stronger than the intraband one: V11 V22 − V12 V21 < 0. In a simple model of two equivalent bands and equal interactions, V11 = V22 , V12 = V21 , one gets for the transition temperature an estimate Tc EF exp[−1/λ] where the eﬀective coupling constant λ ∼ N (0)[V12 − V11 ]. Even for a weak coupling, λ ∼ 0.2 a high transition temperature can be achieved for a large Fermi energy EF ∼ 1 eV. The two-band model can be used to explain the peculiar properties of the two-layer cuprate compounds concerning their doping and temperature dependence. However, the observation of the high-temperature superconductivity in the single-layer cuprates (e.g., Tc = 96 K in Hg-1201) shows that the pairing mechanism should be related to a single CuO2 plane, presumably described by a single p–d hybridized band crossing the FS. It is interesting to point out that the d-wave spin-ﬂuctuation pairing considered in Sect. 7.3.2. is another example of a pairing mediated by a repulsive interaction. In that case, the gap has diﬀerent signs in the regions of the Brillouin zone coupled by the AF wave vector Q = (π, π) which can be considered as two “bands”: Δ(k) = −Δ(k + Q) (see (7.68)). Generally, given an interaction V (r) the average of which over the whole space points to its repulsive character, V (r)dr > 0, the superconducting pairing, nevertheless, can occur if V (r) has negative values in a ﬁnite region of the space [626]. This happens, for example, for the screened Coulomb interaction in a degenerate electron gas where Friedel oscillations occur due to a sharp Fermi surface: V (r) ∝ r−3 cos(2kF )r. This brings about an attraction for some orbital moments l > 0 which can lead to pairing with nonzero l. However, the transition temperature in this case turns out to be extremely low, Tc EF exp[−(2l)4 ] ∼ EF 10−7 [626]. An unconventional Coulomb repulsion pairing was proposed by Kopaev et al. for Cooper singlet pairs c†k+ ↑ c†k− ↓ with a large momentum k± = K/2± k where K ∼ Q = (π, π) (for a review, see [125]). It was assumed that the pairing can occur in a ﬁnite domain Ξ in the two-dimensional Brillouin zone (BZ) deﬁned by a kinematic constraint for kinetic energy of a pair 2ξ(k) = ε(k+ ) + ε(k− ) − 2μ = 0. In the conventional pairing with a zero Cooper pair momentum K = 0, the kinematic constraint is fulﬁlled over the whole Fermi surface (FS) ε(k) = μ and the pairing occurs for electrons with large momenta, |k| = kF . The pairing with a large momentum K of the pair takes place at a small relative electron momentum, |k| K, when conditions for a

474

7 Theoretical Models of High-Tc Superconductivity

nesting ε(kα ) = ε(kα + Q) or a “mirror nesting”, ε(k+ ) = ε(k− ), are fulﬁlled in a ﬁnite domain Ξ in the BZ. A model Hamiltonian for two groups of electrons with momenta (k+ , k− ) was considered and the BCS equation similar to (7.101) for the superconduct

† † ing gap Δ(k) = k V (k − k ) ck+ ↑ ck− ↓ was derived. It was shown that the quasiparticle spectrum in the superconducting state has two branches, E± (k) = E(k) ± (ε(k+ ) − ε(k− ))/2, where E(k) = [ξ 2 (k) + |Δ(k)|2 ]1/2 . Therefore, the gap in the spectrum occurs under the mirror nesting condition ε(k+ ) = ε(k− ), while outside this region there is a gapless spectrum. The two-ﬂuid behavior emerging in the model can explain normal state features observed in some physical properties of the cuprates. The analysis of the Cooper pair instability in the model revealed that the pairing could occur for small scattering momenta of the screened Coulomb potential, V (k − k ) V0 (1 − (k − k )2 /κ2 ) where κ is related to the inverse screening radius. This potential shows an oscillatory behavior in the real space which enables the appearance of bound Cooper pairs with large momentum if the kinematic constraint is realized over a ﬁnite domain Ξ inside the BZ. At low doping, the Fermi surface (FS) is determined by four hole pockets lying along the AF BZ boundary. There are two distinct parts in the FS contour: one having a high spectral weight is related to the main AF BZ, while another with a lower spectral weight belongs to the shadow AF BZ. The mirror nesting condition is realized along the Fermi contour for electrons on the main and shadow parts of the FS contour, while the usual nesting condition is fulﬁlled for electrons in diﬀerent pockets. Upon doping, the nesting conditions and the Cooper pairing vary resulting in the superconducting temperature dependence on doping. Long-living quasi-stationary states found in the solution of the pairing equation were related to the pseudogap phase in the cuprates. Various physical properties of the cuprate superconductors have been explained within the Coulomb repulsion model for pairing with large momentum. Hole Superconductivity An original model of a “hole superconductivity” with a purely electronic pairing mechanism was proposed by Hirsh [463] (see also [462, 470, 794] and references therein). Taking into account the strong polarization of the oxygen ions, an additional correlated hopping term was introduced in the Hubbard model (7.3) that results in the kinetic energy dependence on the hole occupation numbers [tij + Δtij (ni,−σ + nj,−σ )] (c†iσ cjσ + H.c.), (7.102) Ht = − ij

where c†iσ is the creation operator of a hole of spin σ at site i and Y = Δtij /tij > 0 determines the increase of the hopping amplitude of a hole in the presence of other holes, ni,−σ = 1. The model (7.102) predicts lowering

7.5 Charge Fluctuation Models

475

of the hole eﬀective mass with increasing number of holes n as shows the renormalization of the hopping parameter (tij )eﬀ = tij (1 + Y n) in the meanﬁeld approximation. The correlated hopping bring about an eﬀective Coulomb interaction V (k, k ) = U + 2Y [ε(k) + ε(k )] where U is the one-site Coulomb repulsion in the Hubbard model and ε(k) = −2 (cos kx + cos ky ) is the bare hole dispersion with the bandwidth D = 2zt for the nearest neighbor hopping t and z = 4 for a square lattice. The superconducting transition was studied within the BCS equation similar to (7.101) with the eﬀective interaction V (k, k ) [463]. A solution was found for the gap function Δ(k) = Δ0 [c − ε(k)/(D/2)] the s-wave symmetry of which was imposed by the interaction V (k, k ). The gap is nonzero at the FS, ε(kF ) = μ, and changes sign for c < 1. Similar to the general model (7.101), in (7.102) a mutual compensation of the negative sign of the repulsion and the order parameter Δ(k) occurs: in the repulsion region where V (k, k ) > 0, the gap becomes negative ensuring a positive contribution to (7.101) for proper values of the constants Y, c. Superconductivity was found for a suﬃciently large value of Y > [(1 + u)(1 + w)]1/2 − 1 where u = U/D and w = zV /D, V is the Coulomb repulsion between the nearest neighbors. The pairing occurs in a ﬁnite region of hole concentrations n: 0 < n < nc ∼ 0.2 with supercon ducting temperature Tc ∝ D n(2 − n) exp(−a/b) where the parameters a, b depend on n. It has a maximum Tc ∼ 100 K at n ∼ 0.05. Various physical properties were calculated and their dependence on the model parameters was studied. A more general three-dimensional model was considered by Marsiglio et al. [794]. In the weak coupling limit, the results were close to the BCS model, while in the strong coupling limit deviation from the universal BCS behavior was observed. Exact diagonalization study for ﬁnite clusters has conﬁrmed the analytical results obtained within the BCS approximation for the Hubbard model with the correlated hopping (7.102) [725]. It is to be observed that a similar correlated hopping amplitude appears in the kinetic energy term in the t–J model (7.16) for the projected electron operators, cˆ†iσ = c†iσ (1 − ni,−σ ). Contrary to (7.102), the kinematic interaction imposed by the projection suppresses the hopping amplitude of the electrons at the already occupied sites, Y = −1. However, this interaction also results in the s-wave superconducting pairing with a k-independent gap Δ and a high

transition temperature Tc ∝ D n(1 − n) exp(−a /b ) with a maximum at a low hole concentration δ = (1 − n) ∼ 0.15 as was found originally by Zaitsev et al. [1404]. In subsequent publications, a general “dynamic Hubbard model” was proposed by Hirsch [468, 470]. In the model, the lowering of the Coulomb on-site repulsion energy with increasing number of electrons in the atom is taken into account by a coupling between a local boson displacement qi of an orbital in the atom i and electrons ni : Hi =

p2i 1 + K qi2 + [U + α qi ] ni↑ ni↓ , 2M 2

(7.103)

476

7 Theoretical Models of High-Tc Superconductivity

where K is the elastic constant of a harmonic oscillator characterized by the frequency ω0 = K/M . This interaction results in the renormalization of the Hubbard repulsion Ueﬀ = U − α2 /2K for a nonzero displacement qi = −α/K of a doubly occupied orbital: ni↑ = ni↓ = 1. For a singly occupied orbital qi = 0. The model (7.103) was studied at ﬁnite boson frequencies ω0 by using a generalized Lang–Firsov transformation usually considered in a model with linear coupling between bosons and fermions [796]. It was found that the “dressed” hopping amplitude depends not only on the boson operators a†i , ai (qi = a†i + ai ) as in the conventional Lang-Firsov transformation but also on † the fermion occupation numbers: Tiσ = exp[g (ai − a†i )(1 − ni,−σ )] where g =

α/2Kω0 . In the limit of an inﬁnite frequency ω0 related to the intra-atomic electron excitation energy, the dynamic Hubbard model leads to the model with correlated hopping amplitudes (7.102). The solution of the Eliashbergtype equations of the model at ﬁnite frequencies ω0 revealed an enhancement of the superconducting Tc and the increase of the region of hole concentrations over which superconducting pairing develops. The general conclusion from the dynamic Hubbard model study was that “the essential physics of the high-Tc phenomenon is undressing” [467]. It was argued that in the nearly ﬁlled bands the strong electron correlations heavily dress the quasiparticles in the normal state. The correlations can be suppressed by the hole doping or with the transition to the paired superconducting state. In these cases the eﬀective quasiparticle mass decreases which can be called as “undressing” of the quasiparticles. It was pointed out that the asymmetry between the undressed electrons at the bottom of the band and the heavily dressed holes, i.e., electrons at the top of the band, appears not only due to many-body eﬀects but is already caused by the electron–ion interaction [471]. It has been claimed that the related lowering of the kinetic energy in the superconducting state, giving rise to a higher mobility of the quasiparticles, is just the “glue” which mediates the pairing. As discussed in Sect. 5.3.3, the observation of the optical high-frequency spectral weight transfer at the superconducting transition supports this statement [469]. Although the undressing mechanism of superconducting pairing is universal for any hole-type superconductors, the s-wave symmetry of the pairing predicted in the model (7.102) precludes its direct application to the cuprate superconductors where the d-wave symmetry of the gap has been ﬁrmly established, at least in the hole-doped materials. Additional pairing mechanisms, such as spin-ﬂuctuation (Sect. 7.3.2) or highly anisotropic electron– phonon interaction (Sect. 7.4.1) should be invoked to reconcile the undressing mechanism with the d-wave pairing detected in the cuprate superconductors. Resume Summarizing the discussions of the purely electronic mechanisms of pairing mediated by the Coulomb interaction, we can point out that both the dynamic

7.6 Conclusion

477

charge ﬂuctuations and the static Coulomb repulsion can lead to a superconducting pairing provided that the electronic band structure displays speciﬁc features. In particular, a quasi-two-dimensionality of the electronic spectrum with nesting properties or with a strong van Hove singularity close to the Fermi surface are favorable to the occurrence of speciﬁc plasma excitations or charge instabilities in the system. To ensure the d-wave symmetry of the pairing, special conditions for the Coulomb interaction should be fulﬁlled or additional pairing mechanisms should be taken into account.

7.6 Conclusion To conclude the overview on theoretical models for the high-temperature superconductors, the following general results are worth mentioning. 1. The band structure calculations within the density functional method have failed to describe the charge-transfer gap in the undoped parent compounds. However, they are useful in the estimation of electronic parameters in eﬀective models allowing for strong correlations in the CuO2 plane as the Hubbard or t–J models (Sect. 7.1). 2. To describe the single-particle excitations in cuprates the many-body eﬀects must be taken into account, in particular, the strong electron correlations which can be accounted for by various methods (DMFT, DCA, etc.). The short-range antiferromagnetic (AF) correlations are the most natural and principal cause of the pseudogap state in the undoped region. The pseudogap is certainly detrimental to superconductivity and has a diﬀerent origin than the superconducting gap (Sect. 7.2.1). The strong electron correlations are also essential in the underdoped region for the spin dynamics which can be properly described within the t–J model (Sect. 7.2.2). 3. The experimental observation of extensive AF spin ﬂuctuations in a broad region of charge carrier density in cuprates suggests to consider the electron exchange by spin ﬂuctuations as the main “glue” for superconducting pairing, naturally explaining, in particular, the d-wave symmetry of the gap. However, the AF exchange interaction emerging from strong correlations provides a second channel for nonretarded pairing which is absent in the conventional metals and has been suggested in the RVB theory by Anderson [61]. Even for a weak coupling, the AF exchange interaction strongly enhances the superconducting temperature by pairing all electrons in the conduction band (Sect. 7.3). An importance of spin-ﬂuctuations in the superconducting pairing was also revealed in a new class of the iron-based superconductors with Tc up to 55 K (see e.g., [492]). 4. There is evidence of the strong coupling of the charge carriers with the charge ﬂuctuations in cuprates coming from both the lattice vibrations (phonons) and the itinerant electrons (Chap. 6). However, the d-wave symmetry of the pairing in the cuprates imposes strict restrictions on the

478

7 Theoretical Models of High-Tc Superconductivity

character of the electron– phonon interaction or the charge ﬂuctuations (Sects. 7.4 and 7.5). 5. As a whole, the “synergetic” pairing mechanism seems to be the most probable one in cuprates. This mechanism should include the spin-ﬂuctuation interaction, the electron–phonon interaction enhanced by charge ﬂuctuations and take into account various aspects of the strong Coulomb correlations in the CuO2 plane in a self-consistent way. The elaboration of the corresponding theory requires not only the performance of complicated numerical calculations, but also the solution of a number of important problems in the systems with strong electron correlations.

8 Applications

The conventional low-temperature superconductors (LTS) have found various technical applications. The most important one is the production of high-ﬁeld magnets in the range of 5–10 T, which are widely used both in the technical applications as in particle accelerators and in magnetic-resonance-imaging devices, and in research laboratories for production of high magnetic ﬁelds. In construction of these magnets, usually two materials are used: the Nb–Ti alloy with Tc = 9 K and Nb3 Sn with Tc = 18 K, which are cooled down to 4 K by liquid helium. The critical current in these materials is of the order of 105 A cm−2 in the magnetic ﬁelds up to 10 T. It is practically impossible to construct such high-ﬁeld magnets with room-temperature coils since it requires enormous electric power and complicated cooling equipment. Various high-quality superconducting electronic devices, such as superconducting quantum interference devices (SQUIDs), microwave devices, detectors, etc., have been also elaborated (for reviews, see [185, 892]). After discovery of the high-temperature superconductivity in the YBCO material by Wu et al. [1371] with Tc ∼ 92 K higher than the boiling temperature of liquid nitrogen, Tb,N2 = 77.4 K (see Sect. 1.2), wide-spread applications of the new cheap superconductors have been predicted. However, the cuprate high-temperature superconductors (HTS) have revealed inconvenient for their applications physical properties. First of all, due to a short coherence length, a giant ﬂux creep of the vortex lattice and its melting at temperatures Tm ∼ (1/2) Tc present a great problem for pinning of the vortex lattice (see Sect. 4.3.1, Fig. 4.9). The second problem concerns with weak links in superconductors, which diminish the critical “transport” current density Jct by several orders of magnitude in comparison with the local “intra-grain” current density Jc . Contrary to the conventional superconductors with the isotropic s-wave gap, the cuprate superconductors with the quasi-two-dimensional structure and the d-wave symmetry of the gap are especially prone to create weak intergrain links (see, e.g., [452]). A special treatment is necessary to align the c-axis of the superconducting ﬁlaments and the phases of the d-wave superconducting gaps in diﬀerent grains in the plane to suppress weak

480

8 Applications

links. Mechanical properties of the HTS materials, which are usually brittle ceramics, are also very inconvenient for fabricating wires. All these anomalous properties of the cuprate HTS have delayed their wide-spread technical applications. In recent years, a substantial progress have been achieved in overcoming these problems and implementation of superconducting technology in such diverse ﬁelds as energy transportation systems, energy production, new computer architectures, biomagnetism, microwave devices, etc. (see, e.g., [185]). Various applications of HTS are reported in Proceedings of numerous conferences devoted to this problem (see, e.g., International Symposium on Superconductivity [535], International Conferences on HTS and Materials and Mechanisms of Superconductivity (M2 -HTSC) [754–761], European Conferences on Applied Superconductivity [319]). Present state of art of HTS applications can be found in the Internet sites of various companies involved in implementation of HTS devices, for example, “American Superconductor Company” (http://www.amsc.com), SuperPower Inc., “Sumitomo Electric” (http://www.sumitomo.com), Southwire (http://www.southwire.com), “Long Island Power Authority” (LIPA) (http://www.lipower.org), etc. Below, we brieﬂy consider only several examples of successful applications of cuprate superconductors.

8.1 Electric Power Applications 8.1.1 Superconducting Tapes and Cables A possibility of electric energy transportation without loss is the most attractive property of superconductors. A high transition temperature of the cuprate HTS has enabled their practical applications by development of HTS power cables, which are able to carry up to ten times greater current and power capacity per cross section than conventional copper cables (see, e.g., [507, 815]). In the ﬁrst generation (1G) of the HTS wires, the Bi-2212 material was used in the “powder in tube” method: a silver tube is ﬁlled under high pressure with Bi-2212 powder. Due to a mica- or graphite-like structure, the Bi-2212 material forms long ﬁlaments of the order of 10 µm thick and 200 µm long, which being inserted in a silver tube produce a wire with good mechanical properties. The Bi-based wires were discovered in Japan and presently long cables up to 1,500 m are produced by Sumitomo Electric Company (http://www.sumitomo.com). The HTS cables have signiﬁcantly lower impedance than conventional cables, which is important for eﬃcient transportation of AC electric power within the grid. Typically, 1G wires have self ﬁeld critical currents ∼180 A at 77 K (Jct ∼ 104 A cm−2 ), which has enabled to use them in various applications. However, the 1G wires have several disadvantages: their critical current density Jc strongly reduced under magnetic ﬁeld and their production

8.1 Electric Power Applications

481

Fig. 8.1. Schematic geometry for the 2G coated-conductor (after [507])

demands a large amount of expensive silver presenting an obstacle to their wide commercial application. Therefore, a development of a new second generation (2G) of wires fabricated on the basis of YBCO compounds has accelerated superconducting cable commercial applications. The 2G wires are manufactured in the form of the tapes by deposition of about 1 µm thick YBCO epitaxially grown layers on the textured nickel-alloy substrate with several buﬀer layers in between as shown in Fig. 8.1. The tape is coated by the stabilizing layers made of copper, brass, or stainless steel (for details, see [780]). The YBCO 2G tapes exhibits critical current densities of the order of 250 A per cm width or Jct ∼ 3 × 106 A cm−2 (77 K, self ﬁeld), much higher than 1G Bi-based wires. Two processes used in the manufacture of the tapes: ion beam assisted deposition (IBAD) and rolling assisted biaxially textured substrates (RABiTS) can be carried out in a reel-to-reel conﬁguration, which greatly accelerates the production. Critical current densities up to ∼700 A per cm width at 77 K were reported by SuperPower Inc., in short, ∼1 m tapes produced by real-to-real continuous process [1133]. The critical current properties in the Bi-2223 silver-sheathed tapes and the YBCO coated conductors and a possibility of further improvement of their properties are discussed, for example, by Matsushita et al. [810]. These HTS coated conductor tapes are used in fabricating power cables. An example of the single-phase power cable produced by American Superconductor Company (AMSC) (http://www.amsc.com) is shown in Fig. 8.2. A higher than in YBCO tapes critical current has been obtained in the GdBCO coated HTS conductor [1157]. By using a pulsed laser deposit technique for fabricating the GdBCO tapes, critical current densities ∼300 A per cm width was achieved at 77 K (self ﬁeld) and Jct ∼ 40 A per cm width under the parallel magnetic ﬁeld 3 T. To reduce HTS cable costs, three-phase power cables were developed. Figure 8.3 shows the three-phase low impedance HTS 13.4 kV cable developed by AMSC, where three 2G wires are concentrically placed around the common central core, surrounded by a copper shield. The Triax HTS cable was developed jointly by Southwire and nkt cables companies. This compact construction has several advantages over the single-phase cable: in the threephase cable, only about one-half of the quantity of HTS wire is needed and the cables cold surface area is reduced, which lower the cost associated with cryogenic cooling equipment. A single HTS Triax cable operating at 13 kV carries 3,000 A, which is equivalent of 18 conventional underground cables.

482

8 Applications

Cold dielectric design:

Outer Protective Covering

Inner Cryostat Wall Liquid Nitrogen Coolant Copper Shield Wire HTS Shield Tape High Voltage Dielectric HTS Tape Copper Core

Thermal “Superinsulation”

American Superconductor™

Outer Cryostat Wall

Fig. 8.2. (Color online) Single-phase HTS cable demonstrated to 138 kV (from http: \www.amsc.com)

Fig. 8.3. (Color online) Three-phase low impedance HTS cable “Triax” (from http: \www.amsc.com)

The usage of the compact HTS cables is especially advantageous in adding electric power to densely populated areas since these systems release no heat or electromagnetic ﬁelds and due to small sizes no new duct construction is needed. As reported by Southwire company, the 200-m Triax HTS system has been operated successfully since its installation in August 2006 at the Bixby substation in the Columbus, Ohio suburb of Groveport, USA. Another successful HTS cable project has been realized by the US Long Island Power Authority (LIPA) in April 2008. In the LIPA project, three single-phase HTS power cables operating at 138 kV can transmit up to 574 MW of electricity over 600 m. There are several research and development projects concerning the 1G wires and 2G HTS cables applications in electric power transmission in USA, Japan, Germany, China, Russia, and other countries. It is believed that the HTS cables will be extensively used in electric transportation systems in near future.

8.1 Electric Power Applications

483

8.1.2 Fault Current Limiters To protect the transmission or distribution electric systems from outages caused by fault currents a special device – a fault current limiter (FCL) is installed in the transmission grid. A fault current occurs in the event of a short circuit caused by lightning, accidental contact between the lines or the ground, etc. In this case, the power current ﬂowing through a local network can rise enormously damaging electrical equipment. Conventional line reactors widely used as FCLs have high AC losses and can produce voltage drop in the grid in the case of a fault current. The HTS technology oﬀers a much better solution to the fault current problem and represents one of the most successful application of the cuprate superconductors. In the HTS FCL, a basic property of a superconductor is used, which is a transition from the zero resistance superconducting state to the normal resistive state when the electric current exceeds the material’s critical current. The HTS FCL represents a coated conductor consisted of layers of HTS material within layers of resistive materials. Under normal operating conditions, the current in the cable ﬂows through the HTS layers in the FCL. In the case of a fault, the current exceeds the HTS material’s critical current and the HTS layers become normal. In that case, the current is automatically shunted within a millisecond to ﬂow through the higher resistance layers, eﬀectively quenching the fault current amplitude. The very rapidly operated HTS FCLs greatly reduce damage to electrical equipment caused by system faults. They are fail-safe since they require no external sensing of the current to initiate the transition. Several types of FCLs were developed: a single-phase stand alone FCLs of high power or HTS rods and cylinders connected in parallel with conventional copper cables to reduce heating of the FCL and decreasing the recovery time (for details, see, e.g., [507, 780]). 8.1.3 Superconducting Rotating Machines HTS technology is promising for application in superconducting devices working on the principle of levitation: electric motors, generators, pumps for liquid-gas transfer, magnetic bearings, ﬂywheels, magnetically levitated train (MAGLEV), etc. In particular, a successful application of HTS wires has been found in construction of electric motors and generators. A Japanese group in collaboration with Sumitomo Electric Company has fabricated the world’s ﬁrst practical-level HTS 12.5 kW-power motor by using the 1G Bi-based wires. The motor was cooled by low-cost, easy-to-handle liquid nitrogen. The HTS motors generate high magnetic ﬁeld without using an iron core and, therefore, they are much smaller and lighter than conventional motors, for example, the 5-MW HTS motor is one-tenth in volume and one-ﬁfth in weight of a conventional motor (Sumitomo Electric). They produce little noise and there is no magnetic ﬂux leakage from the HTS motor.

484

8 Applications

Small size, high eﬃciency, and low energy consumption of the HTS motor is very attractive in construction of new ships. For example, a successful completion of full-power testing of the world’s ﬁrst 36.5-MW HTS ship propulsion motor at the US Navy’s Integrated Power System Land-Based Test Site in Philadelphia was reported by the AMSC and NOC (January 2009). It is expected that HTS motors will also ﬁnd various applications in industry, for example, as railcar motors, in steel rolling mills. Superconducting generators are another promising application of HTS technology. It is expected that HTS machines will be more compact and light, about half the size and weight of a conventional generator. They have several other beneﬁts: a low synchronous reactance, operative at very small load angles, and low losses (one-half to two-thirds the losses of a conventional generator). A power consumption for the cryogenic cooling system is only few percent of the total losses in the HTS generator (AMSC). The compact HTS generators can ﬁnd various technical applications, for example, for wind power systems. HTS superconducting synchronous condenser is a good example of superconducting rotating machine. The synchronous condenser controls voltage by generating or absorbing reactive power dynamically providing reliable ﬂow of electricity through AC power networks. The HTS synchronous condenser has several advantages over a conventional machine: low loss, low synchronous reactance enables high output, instantaneous response, high reliability, light weight, smaller size, etc. (see http://www.amsc.com). The HTS materials are ideal for nearly friction-free bearing mechanism, which enables to use them for ﬂywheel energy storage. High-speed ﬂywheels with HTS bearing have extremely low rotational losses, about two orders of magnitude lower than conventional high-speed ﬂywheels. Including the energy losses on the cooling system, etc., a construction of ﬂywheel energy storage systems with about 90% diurnal storage eﬃciency may be possible [507]. Another possibility of energy storage is the superconductive magnetic energy storage (SMES) device in which the DC magnetic ﬁeld is created by an inductor wound with superconductor wire. At present, the LTS SMESs are used commercially, whereas SMESs with HTS coils are under experimental testing (see, e.g., [1135]). Several other research and developments projects concerning the HTS technology applications should be mentioned. In particular, the MAGLEV project promises several remarkable beneﬁts in comparison with conventional train: no limit on traveling speed (business speed 500 km h−1 ), no exhaust gas, and quieter than automobiles. The train is ease of maintenance: resistant to sand, earthquakes, and snow. The HTS wires have been successfully used in transformers in which the magnetic ﬁeld is restricted by the iron core and, therefore, is quite low. This enables to use the low-cost liquid nitrogen cooling system. The HTS magnets are more eﬃcient than the LTS ones since they can generate higher magnetic ﬁelds, have greater thermal stability, and lower

8.2 Electronic Applications

485

operating cost. However, since the critical current in HTS wires is strongly suppressed by magnetic ﬁeld at elevated temperatures (∼70 K), the HTS magnet can operates only at low temperatures. HTS magnets are commercially available today. A large-grain HTS can be used for fabricating a permanent magnet of several tesla. However, a poor mechanical stability and low thermal conductivity of a bulk superconductor present obstacles to practical applications of such magnets. By a special treatment of a bulk YBCO sample, it was possible to fabricate a permanent magnet of 17 T trapped magnetic ﬁeld at 29 K [1253]. To conclude, HTS technology has already found practical applications in building up advanced components for the modern electric transportation systems. These include 1G and 2G HTS wires and cables, FCLs, synchronous condensers, transformers, motors, and generators. It is believed that in near future with increasing production and lowering the cost of HTS devices they will be available for broad commercial applications in industry.

8.2 Electronic Applications In general, electronic applications are based on two remarkable properties of superconductors: several orders of magnitude lower level of the microwave absorption in comparison with normal conductors and the Josephson eﬀect. As shown in Sect. 5.3.3, the microwave surface resistance Rs (T ) ∝ ω 2 σ1 (ω, T ) λ3 (T ) (see (4.41)) of the YBCO HTS ﬁlms reaches extremely small values due to strong suppression of the conductivity σ1 (ω, T ) at temperatures below 77 K (see Fig. 5.36). Interference eﬀects in a Josephson junction (JJ) and SQUIDs are discussed in many text-books (see, e.g., [185]). JJs was considered in Sect. 5.5.3 in relation with the phase-sensitive experiments and below important equations concerning the Josephson eﬀect will be given in more detail. A substantial progress in preparation of high-quality thin ﬁlms and multilayers have resulted in development of various HTS-based microwave devices and low-noise SQUIDs. There are several comprehensive reviews on electronic applications of HTS concerning passive microwave devices [687], highfrequency applications of HTS thin ﬁlms [619], and high-temperature (HT) SQUIDs [625]. Therefore, below we consider only several examples of important HTS electronic applications. 8.2.1 Josephson Junctions The supercurrent Is of Cooper pairs tunneling through a barrier between two superconductors, left (L) and right (R), is determined by the equation for the dc Josephson eﬀect Is = Ic sin(Δϕ),

Δϕ = ϕL − ϕR + 2 π φ/φ0 .

(8.1)

486

8 Applications

Here, Ic is the Josephson critical current and the phase diﬀerence Δϕ depends on the left ϕL and right ϕR phases of superconducting order parameters and the magnetic ﬂux φ = nφ0 threading the JJ, φ0 = h/2e = 2.067 (mV ps). In a closed loop with a Josephson weak link, the phase coherence must be maintained. This results in the phase shift at the junction ϕL − ϕR = 2π n for conventional s-wave superconductors, while for d-wave superconductors a π-shift is possible (see Sect. 5.5.3). To describe the current–voltage I–V characteristic of the JJ the resistively and capacitively shunted junction (RCSJ) model is usually employed. In the model, it is assumed that the JJ is shunted by a resistance R and a capacitance C (internal or external). The I–V characteristic is determined by the Stewart–McCumber parameter βc = 2πIc R2 C/φ0 . In the limit βc 1, the I–V dependence is nonhysteretic and is given by V (I) = R Is2 − Ic2 . If the supercurrent Is is higher than the critical current Ic a voltage V appears across the JJ and the ac Josephson eﬀect is observed. In this case, the phase Δϕ varies with the time (8.2) dΔϕ/dt = 2πV /φ0 giving rise to a generation of the ac current with frequency f0 = V /φ0 = V (2e/h) = 483 MHz V (meV). Figure 8.4 shows the current–voltage characteristic of the high-temperature (HT) JJ at T = 77 K with the resistivity R = 1.2 Ω in the case of the dc Josephson eﬀect (a) and the radio-frequency (rf) Josephson eﬀect (b) in the 93.37 GHz irradiation ﬁeld. Without the rf ﬁeld, a critical current jump Ic 0.2 mA close to the bias voltage V = 0 is observed, while in the case of the rf ﬁeld vertical Shapiro steps take place with ΔI1 = Ic at the ﬁrst step with voltage resolution 2 nV (after [619]). The JJs are used in the dc 0.8 0.6

Current (mA)

0.4

(a)

0.2 (b)

0.0 –0.2 –0.4 –0.6 –0.8 –0.4 –0.3 –0.2 –0.1 0.0

0.1

0.2

0.3

0.4

Voltage (mV)

Fig. 8.4. Current–voltage characteristic of the resistively shunted high-temperature JJ without microwave field (a) and with irradiation field (b) (after [619])

8.2 Electronic Applications

487

SQUID and in the rf SQUID in the external microwave ﬁeld (see Sect. 8.2.4). For details and references concerning the Josephson eﬀect see, for example, Barone et al. [107], Koelle et al. [625], Buckel et al. [185]. The HTS materials reveal these remarkable properties at temperatures near 77 K attainable with liquid nitrogen or low power cryocoolers, which make them attractive for practical operation. In fabrication of the low-temperature (LT) JJs, a well-established multilayer technology for the planar contact geometry with Nb as a superconductor ﬁlm is used. Both the superconductor– insulator–superconductor (SIS) and superconductor–normal metal–superconductor (SNS) multilayers are fabricated. To prepare HT JJs with controllable and reproducible parameters, an advanced thin ﬁlm and multilayer technology has been developed. In the most of HT JJs, the epitaxial YBCO thin ﬁlms with the c-axis orientation are used. In this case, due to a small coherence length in the c-direction, the in-plane contact geometry has been developed. Several types of the HT JJs have been fabricated. Generally, two classes of JJs are considered: junctions with intrinsic interfaces and junctions with extrinsic interfaces. In the ﬁrst class, the interface between two superconducting grains is used to fabricate the grain-boundary junctions (GBJs). The bicrystal GBJs are produced by the epitaxial growth of the HT ﬁlm on a bicrystal substrate, such as SrTiO3 , LaAlO3 , MgO, with a ﬁxed misorientation angle Θ (see Fig. 8.5a). An easy single layer process

a

b

bicrystal GBJ [001]

GB

[001]

[010] [100]

θ

step-edge GBJ [001]

GBs

[001]

[010] [100]

[001] α

c

d

step-edge SNS Au, Ag

barrier

[001]

[001]

[001]

[001]

S

ramp-edge

S N

S

S

Fig. 8.5. Various types of HT Josephson junctions (reprinted with permission by c 1999) APS from Koelle et al. [625],

488

8 Applications

and low level of 1/f noise make this type of the JJs most reliable and successful for SQUID applications though it is diﬃcult to use them in devices with a complicated circuit structure. In the step-edge GBJ, an epitaxially grown YBCO ﬁlm on a substrate with a steep step is used (see Fig. 8.5b). The two grain boundaries with diﬀerent orientations at the step play the role of the interface. The step-edge GBJs have an advantage over the bicrystal technique since they can be used in fabricating of a complex circuits though they show a greater spread of junction parameters and poor reproducibility. The junctions with extrinsic interface consist of a thin interlayer of insulating or normal metal between two superconductors. The step-edge junction with a noble metal interlayer is shown in Fig. 8.5c and the rump-edge junction with an epitaxial grown interlayer is shown in Fig. 8.5d. To control the properties of the interface and the parameters of the step-edge junctions, an advanced fabrication technology is required, which hinders usage of these types of the JJs. A more detail description and references concerning properties and fabrication of the JJs is given by Koelle et al. [625]. 8.2.2 Passive Microwave Devices Superconducting Cavity Resonators and Filters Extremely small values of the microwave surface resistance Rs (T ) enables to fabricate microwave resonators with a large unload quality factor Q0 = 2π (energy stored / energy lost per cycle) ∝ 1/Rs . High-quality microwave resonators for conventional conductors, as copper, should be of large sizes since the quality factor increases with the volume of cavity. Large values of Q0 ∼ 105 for superconductors permits to fabricate high-quality HTS resonators of small sizes, which have found applications in microwave communication systems. An important element in the communication system is the bandpass ﬁlters. By using HTS resonators, small size, low energy loss, and high selectivity of the ﬁlter can be achieved. Various types of HTS resonators and ﬁlters (e.g., HTS planar resonators, multipole HTS planar ﬁlters) were fabricated and used in base stations in mobile communication systems. Usage of HTS planar ﬁlters as satellite transponders has advantage in comparison to conventional technology due to their small weight, which reduces satellite launch costs (for details, see [619, 687]). Another possible application is HTS planar antennae, which can be made of small sizes due to high Q-values of electrically small (in comparison to the free space wavelength) resonators. HTS thin ﬁlms planar transmission lines are used in fabricating compact delay lines, which have various applications, for example, in modern radar systems. A delay time up to 40 ns are typical to HTS-based delay lines [687].

8.2 Electronic Applications

489

NMR and MRI Receiver Coils The nuclear magnetic resonance (NMR) and the magnetic resonance imaging (MRI) techniques are widely used in chemical analysis and imaging in modern medicine. To reach high sensitivity in these techniques, high-quality pickup coils are needed to measure rf signals of the magnetic resonance of particular molecules. Presently, in commercially available high-sensitive NMR systems LT pickup coils are used, since HTS coils exhibit high level of noise (see Sect. 8.2.4). There are several demonstrations of YBCO thin ﬁlm planar coils with acceptable noise properties, which show certain advantage over the LT coils (see [619]). Tuneable Microwave Devices In communication systems, tuneable bandpass ﬁlters are applied to improve the receiver sensitivity. For fabrication, a tunable resonator or phase shifter HTS devices can be used. A commonly employed technique is based on a thin ﬁlm of an YBCO structure deposited on a SrTiO3 ﬁlm. The SrTiO3 crystal is an incipient ferroelectric that shows a high permittivity at low temperatures. In the YBCO and SrTiO3 multilayer, the dielectric permittivity can be tuned by applying external electric ﬁeld. With such a ferroelectric varactor, a tunability of about 10% can be achieved. Other possibility of the resonator frequency and phase tuning is to use a ferrite-HTS device, which is operated by an external magnetic ﬁeld (for details, see [619]). 8.2.3 Active Microwave Devices Infrared Detection A direct detection of infrared (IR) radiation is based on the breaking of the Cooper pairs by absorption of IR photons. The pair breaking then can be measured from changes of certain superconducting characteristics. More eﬃcient for detection frequencies above 1 THz are the hot-electron bolometers (HEBs), which require ∼100 times smaller local oscillator power level to drive the device in comparison with the Schottky-diode mixer. In the IR detection by a HEB, a nonlinearity of the resistivity close to the superconducting phase transition is employed. It is convenient to use the HEB at temperatures much lower than the transition temperature but to operate the device below but very close to the critical current by applying an external voltage. An absorption of electromagnetic waves generates “hot” electrons resulting in heating of the HEB, which changes its I–V characteristic. The HEBs using YBCO ﬁlms could be attractive since they can work at temperatures accessible with low-power cryocoolers.

490

8 Applications

Microwave Mixers and Generators The relation (8.2) between a frequency f0 of oscillations and an applied voltage V in the ac Josephson eﬀect provides a possibility to detect and analyze microwave spectra or to generate electromagnetic waves in the terahertz region. In particular, in microwave mixers the nonlinearity of the I–V characteristic of the JJ in the overdamped regime, βc 1, is employed. The nonlinearity appears at a Shapiro step determined by the voltage V = fLO φ0 , where fLO is the frequency of a local oscillator. The HTS GBJs can be used as such microwave mixers with an upper frequency limit of about 750 GHz. In the HTS mixers, a much lower level of power is required for the local oscillator in comparison with the cooled Schottky mixers though the noise level for the latter is much lower than achieved in HTS mixers. By analyzing the height of the Shapiro steps emerging in the JJ under the incident radiation of frequency f at the biased voltage V , a spectroscopic analysis of the electromagnetic radiation can be performed within the Hilbert transform spectroscopy. The HT JJ array under a microwave irradiation can be used also in creation of highprecision voltmeters with a relative precision of about 10−8 (for details, see [619]). There are attempts to use a synchronized array of JJs biased at a voltage V as generators or detectors in the region of several hundred gigahertz. However, the low output power and a large linewidth of the radiation preclude to fabricate practical devices. Much more promising results have been achieved by using a stack of intrinsic JJ created by conducting layers of CuO2 planes in HTS crystals (see Sect. 5.5.2). In the resistive state, all JJs can generate coherently as sketched in Fig. 8.6 [621]. In particular, Ozyuzer et al. [941] have demonstrated that more than 500 junctions of the 1 µm thick Bi-2212 ﬁlm in the resonance cavity can produce a continuous coherent radiation power up to 0.5 µW at frequencies up to 0.85 THz. The radiation was observed up to 50 K. The synchronization of the JJs was induced by a Josephson resonance plasma mode in the cavity as in a conventional laser since the intensity of the radiation was proportional to N 2 of the number N of the junctions. Contrary to other experiments, the emission was obtained without an application of a magnetic ﬁeld. This discovery has shown that useful micrometer-sized sources of terahertz radiation and detectors made on the basis of the cuprate HTS like the Bi-2212 crystal can be developed. Digital Circuits It has been suggested to use circuits of a large number of JJs working on the principle of the “rapid single ﬂux quantum” (RSFQ) logic in a new generation of ultrafast digital technology (see [723]). RSFQ circuits has great advantages over the conventional metal-oxide transistor circuits because they have two or three orders of magnitude lower power dissipation per gate and can allow to clock frequencies up to more than 100 GHz. A RSFQ circuit operates by external current pulses that switch oﬀ an overdamped JJ biased with a current just below Ic in a superconducting ring. This results in the phase change

8.2 Electronic Applications

491

Current

1.5 nm

Sr

Bi

Ca Cu 0

Fig. 8.6. Coherent radiation induced by the intrinsic Josepson junctions in Bi-2212 (after [621])

across the JJ by 2π which generates one quantum ﬂux φ0 in the superconducting ring and concurrently a short voltage pulse. Under certain conditions for parameters of the JJ, the emerging current pulses in the superconducting ring enable to operate the circuit state. The switching time between the states with and without a quantum ﬂux in the ring is in the order of picoseconds, which allow to use clock frequencies in the 100 GHz region. A complicated RSFQ circuits of about 1,000 integrated JJs for niobium superconductors working at liquid helium temperature have been fabricated, while the HTS RSFQ circuits of similar sizes are diﬃcult to realize due to some problems with multilayer technology for HTS (for details, see [619]). Recently, a π-phase shift Josephson structure has been proposed to use in the ﬁeld of Josephson electronics and in particular in RSFQ circuits (see [453] and references therein). It is believed that by improving technology and optimization of circuit parameters ultra high-speed HTS digital devices will be developed. 8.2.4 Superconducting Quantum Interference Devices The LT SQUIDs based on Nb technology cooled by liquid 4 He have been extensively used in a broad range of applications in which a high-precision magnetic ﬁeld measurement is needed (see, e.g., [107, 185]). Therefore, the development of HT SQUID technology with a low operating cost at the liquid nitrogen temperature is very attractive. A fabrication of HT SQUIDs with the eﬃciency comparable to LT devices demands the development of the appropriate technology for manufacturing high-quality thin-ﬁlms and JJs

492

8 Applications

with reproducible parameters. For HTS, these requirements present a major challenge since an atomic disorder in complicated HTS structures leads to a strong variation of the JJ parameters due to a short and highly anisotropic superconducting coherence length. Other important problem for HT SQUIDs is a much higher 1/f noise in comparison with LT SQUIDs. Two diﬀerent sources of 1/f noise can be noted: critical current ﬂuctuations in the JJ associated with a Nyquist noise with a spectral density SI (f ) = 4kB T /R and a noise generated by motion of trapped ﬂux vortices, which are weakly pinned in HTS (see Sect. 4.3.1). The 1/f noise reduces the critical currents and lowers the SQUID resolution parameters (for details, see [625]). Despite these problems, impressive progress has been made in the HT SQUID fabrication with reasonable parameters by using various types of the JJ shown in Fig. 8.5. This allows to use them in a number of practical applications, in particular, in those ﬁelds in which nonsuperconducting alternatives fail (for references see, e.g., [1131]). Two primary types, dc and rf, HT SQUIDs have been developed by using mainly the YBCO thin ﬁlm technology. The dc SQUID consists of two JJ connected in parallel on a superconducting loop. For a constant bias current IB > 2Ic , the voltage V across the SQUID oscillates with a period φ0 with increasing the external magnetic ﬂux φ. To measure a small ﬂux variation threading the loop, one chooses the bias current that gives the maximum of the ﬂux-to-voltage transfer coeﬃcient Vφ = |∂V /∂φ|. By the change of the output voltage the ﬂux variation is evaluated: δV = Vφ δφ. The resolution of the ﬂux variation measurement is determined by the intrinsic noise of the SQUID, which is typically of a few φ0 Hz−1/2 for a white noise. The corresponding white noise for a magnetic ﬁeld achieved in the best magnetometers is below 10 fT Hz−1/2 . Whereas the sensitivity of HT SQUID magnetometers is suﬃcient for many practical applications, their usage in an unshielded environment with a background of a high level magnetic noise is problematic. In this case, a gradiometer should be used to suppress a distant noise with small gradients against the signals close to the detector. To measure the ﬁrst derivative axial gradient ∂Bz /∂z, two pickup loops are used mounted on a common axis with a small distance of 5–10 mm between them. A three-loop gradiometer enables to measure the second derivative ∂ 2 Bz /∂z 2 of the magnetic ﬁeld. An oﬀdiagonal gradient like ∂Bz /∂x can be measured by planar magnetometers. Various types of HT SQUID gradiometers, suitable for practical applications, have been developed with a low spectral density of ﬂux noise, for example, as in the HT dc-SQUID gradiometer with a resolution about 3 pT cm−1 Hz−1/2 fabricated by Seidel et al. [1132]. The rf SQUID has a single JJ in a superconducting loop inductively coupled to an LC-resonant circuit (tank) that is driven by the rf current. The dependence of total ﬂux φT in the SQUID on the applied ﬂux φ is determined by the expression: φT = φ − LIc sin(2πφT /φ0 ). So, in the nonhysteretic mode when βL = 2πLIc /φ0 < 1 the total ﬂux (and the amplitude of the rf voltage)

8.2 Electronic Applications

493

is a periodic function of the applied ﬂux φ. In this nondissipative mode, the SQUID is a parametric inductance that modulates the resonant frequency of the tank circuit with the ﬂux variation. In the opposite case, βL > 1, the hysteretic mode is realized when the SQUID transition between quantum states occurs periodically in φ with energy dissipation. Both modes are used in HT SQUIDs. SQUID Applications Presently, there are several successful demonstrations of HT SQUID applications in the ﬁelds of biomagnetism, nondestructive evaluation (NDE), geophysics, and scanning SQUID microscopes. Below, we brieﬂy discuss these applications, while a more detail description and respective references can be found in the comprehensive review by Koelle et al. [625] and in the monograph by Buckel et al. [185]. The LT SQUIDs have found an important applications in multichannel systems for magnetoencephalography (MEG) and magnetocardiology (MCG) – measurements of magnetic signals from the human brain and from the human heart, respectively. A magnetic-ﬁeld resolution of a few fT Hz−1/2 is required for the MEG measurements, while a lower resolution of several tens of picotesla can be suﬃcient for the MCG measurements. Since the HT SQUID magnetometers have a white noise about 10 fT Hz−1/2 , their application in MEG is problematic but more favorable in the MCG. As reported in several publications, single-channel, multilayer magnetometers have suﬃcient resolution in a magnetically shielded room to obtain high-quality results in MCG. For measurements in unshielded environment, magnetic gradiometers have been used. The HT SQUIDs have been applied for the NDE of structural defects in various materials. In particular, HT SQUIDs were used instead of induction coils in eddy-current imaging of subsurface damage in metallic structures as aircraft or reinforced rods in concrete structures. There are several reports of successful usage of HT SQUIDs, both dc and rf, in eddy-current NDE, which are very promising in future applications. Another practical application of HT SQUIDs is in geophysics, in magnetotellurics, which is used in surveying for oil and gas or locating subsurface fault lines. By using high-sensitive SQUID magnetometers, a measurement of the spatial variation of magnetic ﬁelds at the earth’s surface is performed, which helps to determine the spatial variation of the resistivity of the ground. The use of HT SQUID magnetometers is very attractive also for cross-borehole sounding instead of conventional coils. As was demonstrated by several groups, the three-axis HT magnetometers cooled by liquid nitrogen have proved to be convenient for operation in the ﬁeld conditions. Scanning LT SQUID microscopy has been developed to image static magnetic ﬁelds with high spatial resolution. For example, the LT Nb-based SQUID with a 4-µm-diameter pickup loop has been used in the detection of a half-ﬂux

494

8 Applications

quantum φ0 /2 in the thin-ﬁlm YBCO tricrystal ring sample by Tsuei et al. [1276] (see Sect. 5.5.3). In subsequent studies, HT SQUID microscopes have been developed in which the sample may be at 77 K or even at room temperature. A two-dimensional scan is performed by moving the sample with the help of stepper motors. The SQUID microscope can be also operated at high frequency up to several gigahertz. By applying a sinusoidal magnetic ﬁeld produced by a drive coil, the eddy currents are measured in the sample by detecting the ﬂux modulation in the SQUID. The scanning SQUID microscopes have a spatial resolution of about 5 µm with cold samples and about 30 µm with room temperature samples. A novel and very perspective application of the HT SQUID microscope for biology is tracing the motion of bacteria containing magnetite particles or an antibody labeled by a tiny magnetic particles.

8.3 Conclusion To summarize the discussion of the high-temperature cuprate superconductor applications in the ﬁelds of electric power and electronic devices, the following general results can be formulated: 1. HTS technology has already found practical applications in building up advanced components for the modern electric transportation systems. These include 1G and 2G HTS wires and cables, FCLs, synchronous condensers, transformers, motors, and generators. 2. It is believed that in near future with increasing production and lowering the cost of HTS cables they will be available for broad commercial applications in industry. 3. The achievements in manufacturing of high-quality HTS ﬁlms and the development of reproducible schemes for fabricating JJs and multilayer integrated circuits have enabled to propose a number of successful electronic applications similar to low-temperature devices. Among them are various passive and active microwave devices, in particular, superconducting cavity resonators and ﬁlters, infrared detectors and generators in the terahertz region, etc. The development of the HT SQUIDs with a suﬃciently low level of magnetic ﬁeld noise have made them competitive with respect to similar low-temperature devices, in particular in the magnetocardiology, the NDE, and geophysics. 4. The commercial applications of HTS electronic devices is presently at a low level, with only few examples of their implementation. A future development of the market of HTS electronic devices depends on further system optimization, cost reduction, and reliability including cryocooling systems.

A Thermodynamic Green Functions in Superconductivity Theory

The thermodynamic retarded and advanced double-time Green functions (GFs) introduced by Bogoliubov and Tyablikov [151] were found to be an eﬃcient tool in investigation of quasiparticle spectra in the theory of many-body systems and in the superconductivity theory, in particular [1438, 1440]. Originally, Gor’kov [399] used the temperature causal GFs for calculation of the energy spectrum of superconductors in the Bardeen–Cooper–Schrieﬀer (BCS) model [104]. To take the Cooper-pair condensate into account, he introduced a pair-correlation GF that does not conserve the number of particles. Using the temperature diagram technique and taking into account the creation and annihilation of Cooper pairs in the mean-ﬁeld approximation (MFA), he derived a system of two equations for the single-electron and pair GFs. Solution of these equations led to the BCS theory equations. This system of equations was subsequently called Gor’kov equations. Of much interest was the investigation of the more realistic electron– phonon model, which required going beyond the mean ﬁeld for taking the eﬀects of retardation of the electron–phonon interaction into account in calculating the self-energy operator. This problem was solved simultaneously and independently by Eliashberg [306], based on the method of causal GFs, and Zubarev [1439], using the method of retarded double-time GFs. In his work, Zubarev considered the ﬁrst-order perturbation theory for the GFs in the equation of motion method. In Eliashbergs theory, the diagrams for temperature GFs were summed, which allowed considering the Migdal strongcoupling approximation. Subsequently, the Eliashberg (or Migdal–Eliashberg) superconductivity theory has become rather widespread because it allows investigating realistic strong-coupled electron–phonon systems. As was shown later by Vujiˇci´c et al. [1325], Eliashberg-type equations can be also easily obtained in the method of the double-time GFs by diﬀerentiating them with respect to the two times. In this method, an exact Dyson equation for a matrix GF with the self-energy operator given by the multiparticle GF can be derived. Calculation of the self-energy operator in the noncrossing approximation (NCA) leads to the system of Eliashberg equations. A deﬁnite advantage of the

496

A Green Functions

method of the double-time GFs is the possibility to deduce the equations for the electron–boson interaction of an arbitrary form because the explicit form of the bosonic Hamiltonian is not used in the derivation. This method turned out to be especially useful in considering strongly correlated electron systems, in which well-deﬁned QP excitations are absent and the corresponding zerothorder GFs that are necessary in the diagram technique are ill-deﬁned. In this case, a Mori-type projection technique [859], suitable for investigating strongly correlated systems, can be used. This approach in the double-time GFs was considered by Plakida [997], Kuzemsky [680], and in a most general form by Tserkovnikov [1271,1272]. A certain advantage of the retarded GF method also is that a problem of the analytical continuation of imaginary frequency GFs, used in the temperature diagram technique, to real frequencies is avoided. In this Appendix, we give a brief introduction to the theory of retarded and advanced double-time GFs and then formulate a general strong-coupling superconductivity theory based on the method of the equation of motion for GFs [1018]. We consider diﬀerent types of the interaction leading to pairing: the electron–phonon interaction, scattering on spin ﬂuctuations, and the kinematic interaction caused by strong Coulomb correlations in the Hubbard model. By diﬀerentiating the GFs with respect to the two times, we derive an exact Dyson equation, which then can be solved in the NCA for the self-energy operator. This approach is used in studies presented in Sects. 7.2.1 and 7.3.3.

A.1 Thermodynamic Green Functions A.1.1 Green Function Definition We introduce the thermodynamic retarded (r), advanced (a), and causal (c) double-time GFs as deﬁned by Zubarev [1438]: GrAB (t − t ) = A(t)|B(t )r = −iΘ(t − t )A(t)B(t ) − ηB(t )A(t),

GaAB (t − t ) = A(t)|B(t )a = iΘ(t − t)A(t)B(t ) − ηB(t )A(t), (A.1) GcAB (t − t ) = A(t)|B(t )c = −iTˆA(t)B(t ), where the time-ordered product of operators is deﬁned as TˆA(t)B(t ) = Θ(t − t )A(t)B(t ) + ηΘ(t − t)B(t )A(t). Here, the step-function Θ(x) = 1, for x > 0 and Θ(x) = 0 for x < 0, A(t) = exp(iHt)A exp(−iHt) is a time-dependent operator in the Heisenberg representation and A = (1/Z)T r{exp(−βH)A}, Z = T r{exp(−βH)} is the statistical average (we set β = 1/T, kB = 1, ¯ h = 1). The parameter η = +1 is taken for commutator GFs and η = −1 for anticommutator GFs: [A, B]η = AB − ηBA. In this notation, equations of motion for all types of GFs are the same and can be written as d i A(t)|B(t ) = δ(t − t )[A(0), B(0)]η + [A(t), H]|B(t ) dt = δ(t − t )[A(0), B(0)]η − A(t)|[B(t ), H], (A.2)

A.1 Thermodynamic Green Functions

497

where we use the relation dΘ(t)/dt = δ(t) and take into account in the last line that the GFs (A.1) depend on the time diﬀerence (t − t ) only. By sequential diﬀerentiating GFs over time t or t , a chain of equations can be derived. To obtain a closed system of equations, an approximation should be used for higher order GFs, usually called as “decoupling” of GFs [1438]. We emphasize here that any type of decoupling of GFs (A.1) corresponds to a certain set of diagrams in the temperature diagram technique for causal GFs (for the temperature diagram technique see, e.g., [9]). This enables to evaluate which set of diagrams are neglected in the decoupling and, therefore, to estimate the accuracy of adopted approximation. However, there is no way of telling which decoupling should be done to take into account a given set of diagrams. A.1.2 Spectral Representation Let us introduce Fourier transformation for the GFs (A.1) and the respective time correlation functions: +∞ 1 GAB (ω)e−iω(t−t ) dω, (A.3) GAB (t − t ) = 2π −∞ +∞ 1 JBA (ω)e−iω(t−t ) dω, (A.4) B(t )A(t) = 2π −∞ where Fourier transformation for the correlation function A(t)B(t ) is similar to (A.4) with the spectral function JAB (ω) = e−βω JBA (−ω). The spectral representation for the retarded GF in (A.1), in view of the integral represen +∞ tation for the step-function Θ(t) = (i/2π) −∞ e−ixt dx/(x + i) and (A.3), (A.4) can be written as GrAB (E)

1 = 2π

+∞ −∞

eβω − η JBA (ω) dω, E − ω + i

→ 0+ .

(A.5)

The spectral representation for the advanced GF in (A.1) is similar to (A.5) but with a replacement +i → −i. The retarded GF GrAB (E) is an analytic function in the upper half-plane of the complex variable E, ImE > 0, while the advanced GF GaAB (E) is an analytic function in the lower half-plane of the complex variable E, ImE < 0 and, therefore, a single analytic function can be introduced, GAB (E) = GrAB (E) for ImE > 0 and GAB (E) = GaAB (E) for ImE < 0 [151]. The analytical GF GAB (E) obeys the dispersion relation Re Gr,a AB (ω) = ±

1 P π

+∞

−∞

dz Im Gr,a AB (z), z−ω

(A.6)

where P denotes the principal value of the integral. At the same time, the causal GF GcAB (E) in (A.1) has a more complicated than (A.5) spectral representation, which shows that it cannot be analytically continued

498

A Green Functions

into the complex E plane at nonzero temperature, T > 0 (see [9]). In this respect, GcAB (E) is less convenient for application than the analytical function GAB (E). The retarded commutator (η = +1) GFs GrAB (ω) = A|Brω have a simple physical meaning. According to the linear response theory of Kubo [667], they are directly related to the complex admittance or the generalized susceptibility χAB (ω) that describes the inﬂuence of an external perturbation determined by the operator B on the average value of dynamical variable A: χAB (ω) = −A|Brω, η=+1 .

(A.7)

From the spectral representation (A.5), we obtain the relation between the spectral density of the correlation function and GF: JBA (ω) = i [GAB (ω + i) − GAB (ω − i)] (eβω − η)−1 = −2Im GAB (ω + i)(eβω − η)−1 ,

(A.8)

where the last relation holds for a real function JBA (ω) for operators A, B of the same parity with respect to time inversion. This relation enables to calculate time correlation functions (A.4) from respective GFs. A.1.3 Sum Rules and Symmetry Relations The GFs (A.1) satisfy sum rules and certain symmetry relations that are important in applications. For the Fourier transform of the retarded GF in (A.3), we readily derive the integral relation, the sum rule: 1 +∞ r G (ω)dω = −i[A, B]η . (A.9) π −∞ AB From the equation of motion (A.2) written for the Fourier transform of the retarded GF, the following sum rule can be obtained: 1 +∞ {ωGrAB (ω) − [A, B]η dω = −i[[A, H], B]η . (A.10) π −∞ Subsequent diﬀerentiation +∞ of the GFs over time t yields sum rules for higher momenta of the GFs, −∞ {ω n GrAB (ω)}dω, similar to (A.10). Based on these sum rules, a continued fraction representation for GFs can be derived. Several useful symmetry relations for the retarded and advanced GFs follow from their deﬁnition (A.1). In particular, we have A(t)|B(t )r = ηB(t )|A(t)a , which yields the symmetry relation for the Fourier transform of the GFs: A|Brω+i = ηB|Aa−ω−i . (A.11) For the Hermitian conjugated GFs, we obtain the relation: [A|Brω ]† = ηA† |B † r−ω .

(A.12)

In view of the relation (A.7), similar sum rules and symmetry relations hold for the generalized susceptibility χAB (ω).

A.2 Eliashberg Equations for Fermion–Boson Models

499

A.2 Eliashberg Equations for Fermion–Boson Models A.2.1 Dyson Equation Let us consider a general model for electron interaction with phonons and spin ﬂuctuations: ε(p)a†p ap + W (p, p )a†p ap , (A.13) H= p

p,p

where p = (p, σ) denotes the momentum p and the spin σ = +(↑), −(↓) of an electron with the energy ε(p) = (p) − μ referenced to the chemical potential μ. The interaction matrix element consists of two contributions: α α Sp−p ˆσ,σ (A.14) W (p, p ) = δσ,σ Vph (p − p )ρp−p + Vsf (p − p ) τ . α

The ﬁrst and the second terms describe the respective scattering of electrons on charge ﬂuctuations of the lattice ρq (phonons) and on spin ﬂuctuations α Sqα ; the τˆσ,σ are the Pauli matrices. The scalar product of spin operators in (A.14) can be conveniently written in the standard form of the s–d model: Hsf = p,p Vsf (q){Sqz (a†p↑ ap ↑ − a†p↓ ap ↓ ) + Sq+ a†p↓ ap ↑ + Sq− a†p↑ ap ↓ } where q = p − p and Sq± = Sqx ± iSqx . In discussing the singlet superconducting pairing in the framework of model (A.13), we consider the matrix double-time anticommutator GF (A.1) † Gp,σ (t − t ) = Ψp,σ (t)|Ψp,σ (t ) =

for the Nambu operators: ap,σ Ψp,σ = , a†−p¯σ

+∞ −∞

dω Gp,σ (ω)e−iω(t−t ) , 2π

† Ψpσ = a†p,σ a−p¯σ ,

(A.15)

(A.16)

where σ ¯ = −σ. We represent the Fourier transform of matrix GF (A.15) as 11 Gp (ω) G12 Gpσ (ω) Fpσ (ω) p (ω) Gpσ (ω) ≡ ≡ , (A.17) † 22 Fpσ (ω) −G−p¯σ (−ω) G21 p (ω) Gp (ω) † where we introduced the normal Gp (ω) ≡ G11 p (ω) = ap,σ |ap,σ ω and 12 anomalous Fpσ (ω) ≡ Gp (ω) = ap,σ |a−p¯σ ω components of the GF. Here, we use the symmetry relations (A.11) and (A.12). The equation of motion (A.2) for the GF (A.15) in Fourier representation can be written as (0) † W (p, p )ˆ τ3 Ψp ,σ | Ψp,σ ω , (A.18) Gp,σ (ω) = G(0) p,σ (ω) + Gp,σ (ω) p

500

A Green Functions

where the zero-order GF is introduced as τ0 − ε(p)ˆ τ3 )−1 . G(0) pσ (ω) = (ωˆ

(A.19)

Here and hereafter, we use the standard matrix form of GF (A.17) involving the Pauli matrices, where τˆ0 is the unit matrix, τˆ3 = τˆz , and τˆ1 = τˆx . Furthermore, unlike in Zubarev’s method [1439] of repeatedly diﬀerentiating with respect to time t, we diﬀerentiate the multiparticle GF W (p, p )(t)Ψp ,σ (t) | † † Ψp,σ (t ) ≡ A(p, p )(t) | Ψp,σ (t ) in the right-hand side of (A.18) with respect to the second time t . This yields the second equation of the chain † † A(p, p ) | Ψp,σ ω = A(p, p ) | Ψp ,σ τˆ3 W † (p, p )ω G(0) p,σ (ω), (A.20) p

where we assume that a long-range magnetic order is absent in the system and the average is therefore zero, W (p, p ) = 0. Introducing the scattering matrix W (p, p )ˆ τ3 Ψp ,σ | Ψp† ,σ τˆ3 W † (p, p )ω , (A.21) Tp,σ (ω) = p ,p

and the self-energy operator Σp,σ (ω) as the proper part of scattering matrix deﬁned by the equation Tp,σ (ω) = Σp,σ (ω) + Σp,σ (ω)G(0) p,σ (ω)Tp,σ (ω),

(A.22)

we can solve system of equations (A.18), (A.20) and write the GF (A.17) in the form of the matrix Dyson equation

−1 −1 Gp,σ (ω) = G(0) − Σp,σ (ω) . p,σ (ω)

(A.23)

In contrast to the standard diagram technique, where the self-energy operator in the Dyson equation is written as the full vertex and the product of full GFs for fermions and bosons, the self-energy operator in our method is deﬁned by zeroth-order vertices and the full multiparticle fermion–boson GF:

11 12 (ω) Σp,σ (ω) Σp,σ Σp,σ (ω) = 21 22 Σp,σ (ω) Σp,σ (ω) = W (p, p )ˆ τ3 Ψp ,σ | Ψp† ,σ τˆ3 W † (p, p )proper , (A.24) ω p ,p

21 ∗ 11 22 12 (ω) = −Σ−p¯ according to the where Σp,σ σ (−ω) and Σp,σ (ω) = Σpσ (ω) symmetry relations (A.11) and (A.12). The multiparticle GFs in (A.24) describe ﬁnite life-time eﬀects caused by inelastic electron scattering on charge and spin ﬂuctuations.

A.2 Eliashberg Equations for Fermion–Boson Models

501

A.2.2 Noncrossing Approximation To obtain a closed system of equations for GF (A.23) and self-energy operator (A.24), we must evaluate the multiparticle GF in (A.24). For this, we consider the NCA, also known as the self-consistent Born approximation (SCBA) or the mode-coupling approximation. In this approximation, Fermi-like excitations described by operators Ψp,σ and Bose-like excitations described by operators ρq and Sqα in multiparticle GF (A.24) are considered to propagate independently, and their correlation functions, therefore, factor into a product of the corresponding functions: τ3 Ψp ,σ (t) | Ψp† ,σ τˆ3 W † (p, p ) W (p, p )(t)ˆ τ3 Ψp ,σ (t) | Ψp† ,σ τˆ3 . (A.25) = W (p, p )(t) | W † (p, p )ˆ Correlation functions in (A.25) are evaluated in terms of the respective GFs by using the spectral representations (A.4) and (A.8). In this approximation, we obtain the representation for self-energy operator (A.24) 1 +∞ 1 11(12) 11(12) (±) dzK (ω, z|p − p ) − ImGp σ (z) . (A.26) Σp,σ (ω) = N −∞ π p

The kernel of the integral equations for the self-energy has the same form as in the Eliashberg theory: +∞ z Ω λ(±) (q, Ω) 1 tanh . (A.27) K (±) (ω, z|q) = + coth dΩ ω−z−Ω 2 2T 2T −∞ The electron–electron coupling due to charge (phonon) or spin ﬂuctuations for the normal and anomalous components of the self-energy operator is given in accordance with (A.25) by the functions 3 1 λ(±) (q, ω) = |Vsf (q)|2 Imχsf (q, ω) ± |Vph (q)|2 Imχph (q, ω), π π

(A.28)

where we introduced the dynamic charge and spin susceptibilities χph (q, ω) = z −ρq |ρ†q ω+i0+ and χsf (q, ω) = −Sqz |S−q ω+i0+ as deﬁned by (A.7). It is assumed that the spin susceptibility is isotropic in the paramagnetic phase, and therefore χsf (q, ω) = χzz (q, ω) = (1/2)χ± (q, ω). Hence, we obtain a system of self-consistent equations for the normal and anomalous components of the self-energy operator (A.26) and the corresponding GFs that according to (A.23) can be written as G11 p,σ (ω) =

11 ω + ε(p) + Σ−p,¯ σ (−ω) , Dp (ω)

G12 p,σ (ω) =

12 Σp,σ (ω) , Dp (ω)

(A.29)

11 11 12 2 where Dp (ω) = [ω + ε(p) + Σ−p,¯ σ (−ω)][ω − ε(p) − Σp,σ (ω)] − |Σp,σ (ω)| . We emphasize that the GFs in the self-energy operator (A.26) are determined by

502

A Green Functions

full fermionic GFs (A.29) and bosonic GFs, which corresponds to calculating the self-energy operator from the ﬁrst skeleton diagram. It is remarkable that this does not require an explicit form of the bosonic Hamiltonian, which is only necessary in calculating the bosonic GF. This system of equations is equivalent to the Eliashberg equations for the electron–phonon system or their generalizations for the interaction of electrons with spin ﬂuctuations (see, e.g., [541], where equations for the GFs are derived in the case of both singlet and triplet pairing in the s–d model). To take only the one-phonon contribution into account in interaction (A.28), it suﬃces to consider the one-phonon approximation for the dynamic structure factor for lattice oscillations, S(q, ω) = ρq |ρ†q ω . The eﬀects of the Coulomb interaction of electrons can also be considered in the method of double-time GFs as described by Vujiˇci´c et al. [1325]. A standard form of Eliashberg equations is obtained by writing down the matrix of self-energy operator (A.24) as Σp,σ (ω) = ω[1 − Zp (ω)]ˆ τ0 + ξp (ω)ˆ τ3 + Φpσ (ω)ˆ τ1 ,

(A.30)

12 (ω) and the Eliashberg funcwhere the superconducting gap Φpσ (ω) = Σp,σ tions are determined by odd and even components of the normal self-energy 11 operator Σpσ (ω) ≡ Σp,σ (ω) with respect to the frequency ω:

1 [Σpσ (ω) − Σ−p,¯σ (−ω)] , 2 1 ξp (ω) = [Σpσ (ω) + Σ−p,¯σ (−ω)] . 2

ω[1 − Zp (ω)] =

(A.31)

Then, a formal solution of matrix Dyson equation (A.23) for the GF can be written in the conventional form Gp,σ (ω) =

ωZp (ω)ˆ τ0 + (ε(p) + ξp (ω))ˆ τ3 + Φpσ (ω)ˆ τ1 . 2 2 (ωZp (ω)) − (ε(p) + ξp (ω)) − | Φpσ (ω) |2

(A.32)

A system of integral equations for the Eliashberg functions Zp,σ (ω), ξp,σ (ω) and the gap Φp,σ (ω) that follows from (A.26), (A.30), and (A.32) is equivalent to representations (A.26) and (A.29). Various methods for solving this system of equations are detailed in several reviews (see, e.g., [202]). To obtain a BCS-type equation for the energy-independent gap, we should consider a weak-coupling approximation (WCA) for the integral equation for the superconducting gap (A.26). In this approximation, the kernel of integral equation (A.27) is approximated by its value near the Fermi surface for energies |ω, ω1 | μ as ω 1 1 (A.33) K (±) (ω, ω1 |q) − tanh λ(±) (q), 2 2T where the coupling is determined by the static susceptibility λ(±) (q) = 3|Vsf (q)|2 χsf (q) ± |Vph (q)|2 χph (q),

(A.34)

A.3 Superconductivity in the Hubbard Model

503

z for the spin and charge ﬂuctuations, χsf (q) = −ReSqz |S−q ω=0 > 0 and † χph (q) = −Reρq |ρq ω=0 > 0. In the WCA, we neglect the renormalization of the electron spectrum: Zp = 1 and ξp = 0. The equation for the gap on the Fermi surface Φpσ = Φpσ (0) has the form of the BCS equation:

Φp,σ =

1 N

|Vph (q)|2 χph (q) p =p−q

Φ Ep p ,σ , tanh −3|Vsf (q)|2 χsf (q) 2Ep 2T

(A.35)

where Ep = ε(p)2 + | Φpσ |2 is the Bogoliubov quasiparticle spectrum. We integrate over the momentum p in a range of energies near the Fermi surface for phonon ﬂuctuations |ε(p) − ε(p )| ≤ Ωph and for spin ﬂuctuations |ε(p) − ε(p )| ≤ Ωsf , where Ωph and Ωsf are the maximum energies of phonons and spin ﬂuctuations. We note that the pairing caused by spin ﬂuctuations in (A.35) is repulsive (because of a spin ﬂip in the scattering process) and can, therefore, be realized only for a variable-sign gap, for example, in the d-wave singlet channel. In the next section, we consider this pairing for systems with strong Coulomb correlations.

A.3 Superconductivity in the Hubbard Model Let us derive a self-consistent system of equations in the superconductivity theory developed within the retarded GF method for the Hubbard model Xiσσ + E2 Xi22 H = E1 i,σ

+

i

tij Xiσ0 Xj0σ + Xi2σ Xjσ2 + σ(Xi2¯σ Xj0σ + H.c.) . (A.36)

i=j,σ

Here, we use notation as in the eﬀective Hubbard model (7.8) for holes but for simplicity set the hopping integrals tij equal for all transitions by introducing the k representation of the hopping integral as t(k) = 4tγ(k) + 4t γ (k),

(A.37)

with γ(k) = (1/2)(cos kx + cos ky ), γ (k) = cos kx cos ky . We denote the singlesite repulsion energy by U and introduce E1 = −μ and E2 = U − 2μ as the energy levels for the one-hole and two-hole states. The Hubbard operators (HOs) Xiαβ = |iαiβ| (for notation, see Sect. 7.1.2) satisfy the multiplication rules Xiαβ Xiγδ = δβγ Xiαδ and the commutation relations (A.38) Xiαβ , Xjγδ = δij δβγ Xiαδ ± δδα Xiγβ . ±

504

A Green Functions

The upper sign in relations (A.38) pertains to Fermi-type operators like Xi0σ , which change a number of particles, and the lower sign pertains to Bose-type operators, for example, the particle number operator Ni = σ Xiσσ + 2 Xi22 or spin operators Siα deﬁned in (7.17). It is important to emphasize that in contrast to the fermion–boson model in (A.13), Hubbard model (A.36) does not involve a dynamical coupling of electrons (holes) to ﬂuctuations of spins or charges. Its role is played by the “kinematic interaction” caused by the non-fermionic commutation relations (A.38) for the HOs, as was already noted by Hubbard [498]. The term “kinematic interaction” was introduced originally by Dyson [298] in developing a general theory of spin–wave interactions in the Heisenberg model to distinguish those induced by the non-bosonic commutation relations for spin operators from the dynamical exchange interaction. For example, the equation of motion for the HO Xiσ2 has the form 22 σ 2 21 0¯ σ til Biσσ − σBiσσ idXiσ2 / dt = [Xiσ2 , H] = (E1 + U )Xiσ2 + Xl Xl −

l=i,σ

til Xi02 Xlσ0 + σXl2¯σ ,

l=i αβ where Biσσ are Bose-like operators related to the particle number operator Ni and spin operators Siα : 22 22 Biσσ + Xiσσ )δσ σ + Xiσ¯σ δσ σ¯ = (Ni /2 + Siz )δσ σ + Siσ δσ σ¯ , = (Xi

21 z σ Biσσ = (Ni /2 + Si )δσ σ − Si δσ σ ¯.

(A.39)

A.3.1 Dyson Equation To consider the superconducting pairing in Hubbard model (A.36), we introˆ† ˆ iσ and its conjugate operator X duce the four-component Nambu operator X iσ and deﬁne the double-time anticommutator GF as a 4 × 4 matrix similarly to (A.15) and (A.17): ˆ iσ | X ˆ † ω = Gijσ (ω) = X jσ

ˆ Fˆijσ (ω) Gijσ (ω) , † ˆ ji¯σ (−ω) Fˆijσ (ω) − G

(A.40)

ˆ † = (X 2σ X σ¯ 0 X σ¯ 2 X 0σ ). Because of the two-band nature of the where X i i i i iσ ˆ ijσ and anomalous Fˆijσ components of the GF model (A.36), the normal G are 2 × 2 matrices. To calculate the GF (A.40), we use the equation of motion method, as in Sect. A.2.1. Diﬀerentiating the GF with respect to time t, the Fourier representation of it leads to the equation ˆ iσ , H] | X ˆ † ω , ωGijσ (ω) = δij Q + [X jσ

(A.41)

A.3 Superconductivity in the Hubbard Model

505

ˆ iσ , X ˆ † } = τˆ0 × Q; ˆ Q ˆ = In state, the matrix Q = {X iσ the paramagnetic Q2 0 , where Q2 = Xi22 + Xiσσ = n/2 and Q1 = Xi00 + Xiσ¯ σ¯ = 1 − Q2 0 Q1 depends only on the occupation number n = Ni of holes (see (7.9)). In the Hubbard model, there is no well-deﬁned QP excitations speciﬁed by zeroth-order kinetic energy as ε(p) in the fermion–boson model (A.13). It is convenient to choose the mean-ﬁeld contribution to the energy of QPs in equations of motion (A.41) as the zeroth-order QP energy. To identify this contribution, we use the Mori-type projection method [859]. For this, we write ˆ iσ , H] in (A.41) as a sum of the linear part, proportional the operator Zˆiσ = [X ˆ iσ , and the irreducible part Zˆ (ir) orthogonal to X ˆ iσ : to the original operator X iσ ˆ iσ , H] = ˆ lσ + Zˆ (ir) . Eilσ X (A.42) Zˆiσ = [X iσ l (ir) ˆ † ˆ (ir) ˆ † ˆ † ˆ (ir) The orthogonality conditions {Zˆiσ , X jσ } = Ziσ Xjσ + Xjσ Ziσ = 0 determine the linear part, the frequency matrix:

ˆ iσ , H], X ˆ † }Q−1 . Eijσ = {[X jσ

(A.43)

The frequency matrix deﬁnes the QP spectrum in the generalized MFA and the corresponding zeroth-order GF. ˆ † (t ) in (A.41) with Diﬀerentiating the multiparticle GF Zˆiσ (t) | X jσ respect to the second time t and using the same projection procedure as in (A.42) leads to the Dyson equation for GF (A.40), as described in Sect. 2.1. In the (q, ω)-representation, the Dyson equation becomes −1 Gσ (q, ω) = ω˜ τ0 − Eσ (q) − Σσ (q, ω) Q,

(A.44)

where τ˜0 is the 4 × 4 unit matrix. The self-energy operator Σσ (q, ω) is deﬁned by the proper part of the scattering matrix similarly to the fermion–boson model in Sect. A.2.1: (ir) (ir)† (prop) −1 Σσ (q, ω) = Zˆqσ | Zˆqσ ω Q .

(A.45)

Dyson equations (A.44)–(A.45) give an exact representation for GF (A.40). To obtain a closed system of equations, we must evaluate the multiparticle GF in self-energy operator (A.45); this describes the processes of inelastic scattering of electrons (holes) on charge and spin ﬂuctuations due to kinematic interaction. A.3.2 Mean-Field Approximation In contrast to the fermion–boson model in (A.13), the superconducting pairing in the Hubbard model already occurs in the MFA and is caused by the

506

A Green Functions

kinetic exchange interaction [61]. It is therefore reasonable to consider the MFA electronic spectrum separately. Using commutation relations for HOs (A.38), we evaluate frequency matrix (A.43): Eijσ =

εˆijσ Δˆ∗

jiσ

Δˆijσ − εˆji¯σ

,

or

Eσ (k) =

εˆσ (k) ˆ∗ (k) Δ σ

Δˆσ (k) . − εˆσ¯ (k)

(A.46)

The matrix εˆσ (k) determines the QP spectrum in two Hubbard subbands in the normal phase (see [781, 1007, 1010]): ε1,2 (k) = (1/2)[ω2 (k) + ω1 (k)] ∓ (1/2)Λ(k), Λ(k) = {[ω2 (k) − ω1 (k)]2 + 4W (k)2 }1/2 ,

(A.47)

where the one-band excitation spectrum ω1,2 (k) and the hybridization parameter W (k) are ω1 (k) = 4tα1 γ(k) + 4t β1 γ (k) − μ, ω2 (k) = 4tα2 γ(k) + 4t β2 γ (k) + U − μ, W (k) = 4tα12 γ(k) + 4t β12 γ (k).

(A.48)

Here, t and t are hopping integrals for the nearest-neighbor and the next-tonearest neighbor lattice sites (see (7.5)). Because of the kinematic interaction, the spectrum is renormalized: α1(2) = Q1(2) [1 + C1 /Q21(2) ], β1(2) = Q1(2) [1 + √ √ C2 /Q21(2) ], α12 = Q1 Q2 [1 − C1 /Q1 Q2 ], β12 = Q1 Q2 [1 − C2 /Q1 Q2 ], which is determined by the static spin correlation functions C1 = Si Si±ax /ay ,

C2 = Si Si±ax ±ay .

(A.49)

We note that in the case of strong AF spin correlations, the hopping integral for the nearest neighbors is considerably suppressed. In particular, in the case of the N´eel AF order C1 −1/4 and at half-ﬁlling Q1 = Q2 = 1/2 the parameter α1(2) = 0. In this case, hopping to the nearest-neighbor sites does not occur, and the quasiparticle spectrum is deﬁned by the hopping integral t for the next-to-nearest neighbors. Whereas taking into consideration spin correlation functions (A.49), we neglect charge correlations in (A.48) by using the approximation Ni Nj = Ni Nj . For a moderate Coulomb energy U ≤ 8t, charge ﬂuctuations may be important reducing the strong AF spin renormalization mentioned above. ˆijσ of matrix (A.46), which We evaluate the anomalous component Δ determines the superconducting gap. In what follows, we consider only the singlet d-type pairing, which is determined by the anomalous averages at noncoincident sites, (i = j). The diagonal matrix components have the forms 02 Δ22 ijσ = −σtij Xi Nj /Q2 ,

02 Δ11 ijσ = σtij Nj Xi /Q1 .

(A.50)

A.3 Superconductivity in the Hubbard Model

507

Expressing the Fermi operators in terms of the HOs as aiσ = Xi0σ + σXiσ¯ 2 (see 7.11), we can write the anomalous averages in (A.50) as ai↓ ai↑ Nj |=Xi0↓ Xi↓2 Nj = Xi02 Nj , because the other products of the HOs do not contribute, in accordance with their multiplication rules Xiαγ Xiλβ = δγ,λXiαβ . This representation of the anomalous averages in terms of Fermi operators shows that the pairing occurs at a single site but in diﬀerent Hubbard subbands. The anomalous averages Xi02 Nj can be calculated directly by using the equation for the pair GF Lij (t − t ) = Xi02 (t) | Nj (t ) without any decoupling approximation [1015]. For deﬁniteness, we consider the hole-doped case n = 1 + δ > 1 in the Hubbard model (A.36) when the Fermi level lies in the upper subband, μ ≈ U , and the one-site excitation energies E2 − E1 = U − μ ≈ 0, E1 ≈ −U . From the equation for the Fourier component of pair GF Lij (ω), we can derive the following expression for the correlation function with i = j: +∞ dω 02 σ tim Xi Nj = −∞ 1 − exp(−ω/T ) m,σ 1 1 0¯ σ 0σ σ 2 σ ¯2 , × − Im Xi Xm − Xi Xm |Nj ω π ω − E2 where we neglect a small contribution given by the hopping parameter |tij | U in the pair excitation energy E2 ≈ −U . The contribution from the lower subband, which is proportional to 1 0σ |Nj ω δm,j Xi0¯σ Xj0σ δ(ω − 2E1 ), − ImXi0¯σ Xm π makes an exponentially small contribution of the order of exp(−2U/T ) 1. The contribution from the pole of the function [1/(ω − E2 )] is also exponentially small (of the order of exp(−U/T ) 1) and can be neglected. As a result, we obtain the following estimate for the anomalous correlation function: +∞ dω 1 Xi02 Nj = − Re ω − E2 −∞ 1 − exp(−ω/T ) 1 σ ¯2 σtim − ImXiσ 2 Xm |Nj ω (A.51) × π m,σ

−

1 U

m=i,σ

σ ¯ 2 σ tim Xiσ 2 Xm Nj −

4tij σ Xiσ2 Xjσ¯ 2 , U

σ ¯ 2 σ ¯ 2 N j = 2 Xm and the where in the last equation we have used the relation Xm two-site approximation, m = j, typically employed in a derivation of the t–J model (see (7.15)). Here, we evaluated the integral over ω in (A.51) disregarding the retardation eﬀects, that is, omitting the frequency dependence in the denominator [1/(ω − E2 )] 1/U since the excitation energy |E2 | = U is much

508

A Green Functions

higher than the characteristic excitation energies of the order of |tij | deter σ ¯2 |Nj ω . Thus, direct evaluation of mined by the poles of the GF Xiσ 2 Xm the anomalous correlation function shows that retardation eﬀects associated with an interband transition can be neglected, as for the exchange interaction J in the original t–J model. In this approximation, for the superconducting gap in (A.50) we obtain the expression 02 σ2 σ ¯2 Δ22 ijσ = −σ tij Xi Nj /Q2 = Jij Xi Xj /Q2 ,

(A.52)

corresponding to the conventional gap equation in the t–J model (see (A.66)) with the exchange interaction Jij = 4 (tij )2 /U . A similar expression can be obtained in the case of electron doping for the gap in the lower Hubbard 0¯ σ 0σ subband: Δ11 ijσ = Jij Xi Xj /Q1 . We thus conclude that anomalous averages (A.50) in the Hubbard model correspond to the conventional anomalous averages in one of the Hubbard subbands (depending on the position of the chemical potential) in the t–J model.

A.3.3 Self-Energy Operator Self-energy operator (A.46) can be conveniently written in the same form as GF (A.40): ˆ Mijσ (ω) Φˆijσ (ω) (A.53) Σijσ (ω) = Q−1 , ˆ ji¯σ (−ω) Φˆ†ijσ (ω) − M ˆ and Φˆ denote the respective normal and anomalous where the matrices M components of the self-energy operator. We calculate self-energy operator (A.53) in the NCA, as in Sect. 2.1. This approximation corresponds to factored fermion–boson time correlation functions at noncoincident lattice sites (1 = 1 , 2 = 2 ) in accordance with the equation B1 (t)X1 (t)B2 (t )X2 (t ) = X1 (t)X2 (t )B1 (t)B2 (t ).

(A.54)

Using spectral representations (A.4) and (A.8) for the time correlation functions in the right-hand side of the equation, we obtain a closed system of equations for GF (A.40) and self-energy operator (A.53), similar to the system of equations for the fermion–boson model in Sect. 2.2. The system of equations for the (4 × 4) matrix GF (A.40) and the selfˆ σ (k, ω) energy (A.53) can be reduced to a system of equations for the normal G ˆ and anomalous Fσ (k, ω) (2 × 2) matrix components. Using representations for the frequency matrix (A.46) and the self-energy (A.53), we derive for these components the following system of matrix equations [1019]: −1 ˆ (A.55) ˆ N (k, −ω) ϕˆ∗σ (k, ω) ˆ N (k, ω)−1 + ϕˆσ (k, ω)G ˆ σ (k, ω) = G Q, G ˆ N (k, −ω)ϕˆ∗ (k, ω)G ˆ σ (k, ω). Fˆσ∗ (k, ω) = −G σ

(A.56)

A.3 Superconductivity in the Hubbard Model

509

In (A.55), we introduced the normal state matrix GF and the matrix superconducting gap function: −1 ˆ (k, ω)/Q ˆ ˆ N (k, ω) = ωˆ τ0 − εˆ(k) − M , G ˆ ϕˆσ (k, ω) = Δˆσ (k) + Φˆσ (k, ω)/Q.

(A.57) (A.58)

The properties of the system under hole doping are determined by the GF and the self-energy for the two-hole subband. The normal Mσ22 (k, ω) and anomalous Φ22 σ (k, ω) diagonal components of the self-energy operator in this case are given by 1 +∞ dω1 K (+) (ω, ω1 |q, k − q) Mσ22 (k, ω) = N q −∞ 22 1 11 × − Im Gσ (q, ω1 ) + Gσ (q, ω1 ) , (A.59) π 1 +∞ dω1 K (−) (ω, ω1 |q, k − q) Φ22 σ (k, ω) = N q −∞ 22 1 11 × − Im Fσ (q, ω1 ) − Fσ (q, ω1 ) . (A.60) π The kernels of integral equations for these components are given by functions similar to (A.27) for the fermion–boson model, but with another coupling function, dependent on two momenta, 1 2 (±) Im χ λ(±) (q, k − q, ω) = |t(q)| (k − q, ω) . (A.61) sc sc π The interaction vertex is here given by the Fourier components of the hopping integral t(q) = 4t γ(q) + 4t γ (q). The spectral density of bosonic excitations is determined by the corresponding dynamic susceptibilities (A.7) of spin and charge ﬂuctuations: χ(±) sc (q, ω) = 3χs (q, ω) ± χc (q, ω),

(A.62)

z where χs (q, ω) = −Sqz |S−q ω , and χc (q, ω) = −(1/4)δNq |δN−q ω . These functions occur because of the correlation functions B1 (t)B2 (t ) of Bosetype operators (A.39) involved in factorization (A.54). The electron–phonon coupling can be easily taken into account in the Hubbard model similar to the model (A.13) by adding to the coupling function (A.61) the phonon contribution as in (A.28). The renormalized QP spectrum in the two-hole subband in the normal state is determined by the equation: ε˜2 (k) ε2 (k) + ReMσ22 (k, 22 ω = ε˜2 (k))/Q2 , while the gap function ϕ2,σ (k, ω) = Δ22 σ (k) + Φσ (k, ω)/Q2 . We thus see that the system of self-consistent equations for the GFs (A.55), (A.56) and the self-energy components (A.59), (A.60) is formally similar to

510

A Green Functions

the system of equations for the fermion–boson model considered in Sect. A.2.2. But the mechanisms of the electron coupling to bosons are diﬀerent in these models: in the fermion–boson model, the coupling has a dynamical nature and is determined by the corresponding coupling constants, but in the Hubbard model, the electron scattering is due to the kinematic interaction, which is determined by the hopping integral and is, therefore, not an independent parameter.

A.4 Superconductivity in the t–J Model Below, we brieﬂy present the results of the GF calculations [1012] for the planar t–t –J model (7.16) written in terms of HOs H= −

i=j,σ

tij Xiσ0 Xj0σ +

J σ¯σ σ¯ σ Xiσσ . Xi Xj − Xiσσ Xjσ¯ σ¯ − μ 4 i,σ

(A.63)

i=j,σ

We consider the lower Hubbard subband for electrons with the hopping parameters tij , which determines the bare electron dispersion: (k) = −t(k) = −4t γ(k) − 4t γ (k), where t and t are the hopping parameters for the nearest neighbor (n.n.) and the next n.n. sites, respectively (see (7.5) and Table 7.2). J is the exchange interaction for the n.n. sites. The chemical potential μ is determined by the average electron occupation number n = σ Xiσσ . To study the superconducting pairing within the model (A.63), we intro† duce the two-component GF for the Nambu operators: Ψiσ and Ψiσ = σ0 0¯ σ (Xi Xi ): 11 Gijσ (ω) G12 ijσ (ω) + . (A.64) Gij,σ (ω) = Ψiσ |Ψjσ ω = 22 G21 ijσ (ω) Gijσ (ω) Applying the projection technique as described in the previous section, we derive the Dyson equation similar to (A.44). The QP electronic spectrum in the MFA is determined within the Mori projection technique by the frequency † matrix Eij = {[Ψiσ , H], Ψiσ }/Q, where Q = 1 − n/2. The normal component of the matrix in the k-representation similar to (A.48) reads E(k) = −4tαγ(k) − 4t βγ (k) − J(k)

1 γ(q)N (q) − μ ˜, 2N q

(A.65)

where the factors α = Q [1+C1 /Q2 ] and β = Q [1+C2 /Q2 ] take into the renormalization caused by the short-range AF spin correlation functions C1 and C2 (A.49) as discussed in Sect. A.3.2. The shift of the chemical potential caused by interaction is given by δμ = μ ˜ − μ = (n − 2C1 /Q) J − (1/N ) q t(q)N (q). The anomalous part of the frequency matrix determines the superconducting gap in the MFA mediated by the AF exchange interaction 1 0¯ σ J(k − q)X−q Xq0σ . (A.66) Δσ (k) = NQ q

A.4 Superconductivity in the t–J Model

511

Here, we neglect the kinematic interaction determined by the hopping parameter t(q) since it gives no contribution to the d-wave pairing. The normal (Σ 11 ) and anomalous (Σ 12 ) components of the self-energy calculated in the NCA (A.54) similar to the Hubbard model are given by 1 +∞ Σσ11(12) (k, ω) = dω1 K (±) (ω, ω1 |q, k − q)A11(12) (q, ω1 ), (A.67) σ N q −∞ 11(12)

11(12)

(q, ω) = −(1/π Q) ImGσ (q, ω + i). where the spectral functions Aσ The kernel of the integral equation K (±) (ω, ω1 |q, k − q) is deﬁned by (A.27) similar to the fermion–boson model with the coupling function λ(±) (q, k − q, ω) = g 2 (q, k − q)(1/π) Im χ(±) sc (k − q, ω).

(A.68)

Here, the electron coupling is determined by the vertex g(q, k − q) = [ t(q) − (1/2) J(k − q) ] and by the spin-charge dynamical susceptibility (A.62) as in the Hubbard model. As we see, the equation for the self-energy (A.67) is similar to (A.59) and (A.60) derived for the Hubbard model if we disregard in the latter the small contribution from the second subband outside the Fermi level, which should be small for large Hubbard subband splitting. By taking into account the contribution to the gap in MFA (A.66) for the superconducting gap, we obtain the equation: φσ (k, ω) = Δσ (k) + Σσ12 (k, ω) 1 +∞ dω1 J(k − q) + K (−) (ω, ω1 |k, q − k) = N q −∞ φσ (q, ω1 ) 1 × − Im . π [ω1 − Eq − Σ(q, ω1 )][ω1 + Eq + Σ(q, −ω1 )] − |φσ (q, ω1 )|2 (A.69) To calculate the superconducting Tc , it is suﬃcient to consider a linearized gap equation (A.69) by neglecting the gap function in the denominator of the anomalous GF. We close the section by a derivation of an exact representation for an average energy of the t–J model E = Ht−J in terms of the single-electron GF following Yushankhai et al. [1396]. Whereas in the Hubbard model (A.36) an average energy can be easily calculated in terms of the matrix single-electron ˆ ijσ (ω) in (A.40), in the t–J model one should calculate many-particle GFs G GF to evaluate the exchange energy. For this, we consider equation of motion for the correlation function i

d X σ0 (t) Xj0σ = [Xiσ0 (t), Ht−J ] Xj0σ , dt i

(A.70)

where the Hamiltonian Ht−J ≡ Ht + HJ − μN is given by (A.63). By using the commutation relation for the HOs (A.38), the commutator [Xiσ0 , Ht−J ] is

512

A Green Functions

easily calculated. Then, by setting j = i in (A.70) and summing the obtained expression over (i, σ) the following equation can be derived: [Xiσ0 , Ht−J ] Xi0σ = −Ht − 2HJ + μN . (A.71) i,σ

Therefore, for the average energy of the t–J model we obtain the expression E = Ht−J = (1/2){Ht − μN − [Xiσ0 , Ht−J ] Xi0σ }. (A.72) i,σ

Using the Fourier transformation (A.4) for the correlation function in (A.70) and the spectral representation (A.8), we derive from (A.72) the following exact formula for the average energy of the t–J model 1 1 +∞ dω 0σ σ0 (ω − t(q) − μ) − Im Xq | Xq ω . (A.73) E= 2 q,σ −∞ eβω + 1 π Many-body eﬀects in this expression are taken into account through the selfenergy operator of the exact one-particle GF. The average kinetic Ht and the exchange HJ energies can be evaluated separately by using the derived formulas. Resume In conclusion, we have demonstrated that the strong-coupling Migdal– Eliasberg-type superconductivity theory can be easily formulated within the method of the thermodynamic GFs for a general fermion–boson model (A.13) with electron–phonon and electron-spin-ﬂuctuation interactions, or within the models for systems with strong-electron correlations: the Hubbard model (A.36) or the t–J model (A.63). Though it is diﬃcult within the proposed approach to consider higher-order corrections beyond the NCA, a solution of the self-consistent equations for the GFs and the self-energy provides a ﬁrst meaningful step in studies of superconductivity in strongly correlated electronic systems.

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Index

ac Josephson eﬀect, 486 ACAR – angular correlation of annihilation radiation, 231 Acoustic and optic spin excitations in YBCO, 91 Active microwave devices, 489 Andreev (Andreev-Saint-James) reﬂections, 347 Anisotropic electron–phonon interaction, 456 Antiferromagnetic (AF) structure of cupric oxycloride Sr2 CuO2 Cl2 (SCOC), 56 Antiferromagnetism in REBa2 Cu3 O6+x (REBCO), 98 ARPES angle-resolved photoemission spectroscopy, 214 antinodal kink, 240 bilayer band splitting, 220 conclusion, 250 Fermi surface arcs, 225 kink in electron dispersion, 236 metal-insulator transition (MIT), 224 nodal kink, 236 peak-dip-hump (PDH) structure, 221 shadow bands, 219 shadow Fermi surface, 223 superconducting gap, pseudogap, 245 umklapp bands, 219 Average energy in t–J model, 511

Bardeen–Cooper–Schrieﬀer (BCS) pairing theory of superconductivity, 2 Bogoliubov QP spectral function, 217 Bose–Einstein condensation (BEC), 137 Bi2212, 140 scenario, 461 theory of superconductivity, 4 Boson-fermion model (BFM), 464 Cell-cluster perturbation method, 388 Charge-density-wave excitation model, 471 Charge-transfer excitation models, 469 Charge-transfer insulator, 181 Condensation energy, 128 doping dependence, 169 Conventional and unconventional pairing, 328 Correlation length, 123 Coulomb repulsion pairing, 472 Critical magnetic ﬁelds, 122 Crystal structure, 13 BaBiO3 , 15 Bi-22(n-1)n compounds, 41 Hg-12(n-1)n compounds, 42 La2−x Mx CuO4 , 17 Nd(Pr)2−x Cex CuO4 compounds, 32 Ru-1212 (Ru-1222) compounds, 39 Tl-22(n-1)n compounds, 41 YBaCuO compounds, 33 YBCO-124, -247 compounds, 38

566

Index

dc Josephson eﬀect, 485 de Haas–van Alphen (dHvA) eﬀect, 315 d-density wave (DDW) state, 430 Density functional theory, 378 Diagonal incommensurate modulation, 75 Diagram technique for Hubbard operators, 442 Dielectric function, 252, 466 Discovery of high-temperature superconductivity, 4 Dispersion relations for GFs, 497 Drude-Lorentz model, 253 Dynamic Hubbard model, 475 Dynamical cluster theory, 408, 441 Dynamical conductivity, 252 Dynamical mean ﬁeld theory (DMFT), 408 Dynamical spin susceptibility (DSS) memory function method, 423 RPA, 421 spin-fermion model, 423 Dyson equation in Hubbard model, 505 Dzyaloshinsky-Moriya (DM) antisymmetric exchange interaction, 59 Electric power applications, 480 Electron energy-loss spectroscopy (EELS), 200 O1s and Cu2p spectra, 205 Electron paramagnetic resonance (EPR), 101 Electron spectral density in t-J model, 405 Electron spectral function – Hubbard model, 413 Electron–phonon interaction, 371 pairing, 455 Electronic applications, 485 Electronic entropy, 128 Electronic pairing – resume, 476 Electronic Raman scattering (RS), 290 resume, 301 Electronic structure of PrBCO, 210 Eliashberg equations for fermion–boson model, 502 Energy distribution curve (EDC), 215

Energy level crystal ﬁeld splitting, 178 Ettingshausen eﬀect, 303 Exciton models, 466 Extended Drude model, 253 Fano eﬀect, 359 Fault current limiter (FCL), 483 Fermi surface – Hubbard model, 415 Ferrel-Tinkham-Glover sum rule, 284 Finite cluster numerical studies, 440 First generation (1G) of HTS wires, 480 Fluctuation eﬀects in cuprate superconductors, 136 Fluctuation-exchange approximation (FLEX), 437 Fluctuations 2D-XY model, 137 3D-XY model, 137 3D-XY model scaling, 141 magnetic ﬁeld eﬀects, 138 magnetization scaling, 140 speciﬁc heat, 136 Flux creep, 144 Flux phases, 430 f -sum rule, 253 Gapless superconductivity in cuprates, theory, 195 Generalized susceptibility, 498 Generalized tight-binding (GTB) method, 417 Ginzburg – Landau parameter, 123 Ginzburg – Landau theory, 121 Gutzwiller projected BCS wave function, 432 Half-integer ﬂux quantum eﬀect, 346 Hall eﬀect, 302 in cuprates, 310 resume, 317 Hebel–Slichter peak, 103 Heisenberg model for bilayer YBCO, 85 Hidden Fermi liquid, 396 High-energy electron spectroscopy – Resume, 213 High-energy photoelectron spectroscopy, 202 High-pressure eﬀects on superconducting Tc , 45

Index High-temperature superconductivity (HTSC) – a general problem, 1 Hole superconductivity, 474 Hot and cold spot model, 325 HTSC cuprate superconductors generic properties, 6 table of representative classes, 6 Hubbard model, 383 Hyperﬁne interaction, 101 Impurity altervalent substitution, 183 Impurity isovalent substitution, 184 Impurity scattering resonance states, 197 Impurity substitution eﬀects, 183 eﬀects-resume, 198 Li, 189 Pr, 185 resonance peak, 192 spin-ﬂuctuation spectrum, 191 STM-STS spectra, 192 Zn, 187 induced magnetic moments, 188 and Ni, 189 Inelastic magnetic neutron scattering, 54 Infrared (IR) phonon studies, 362 Inhomogeneity in cuprate superconductors, 337 Inhomogeneous ground state, 433 Interlayer tunneling, 335 Intrinsic Josephson junction (IJJ), 335 Ioﬀe-Regel criterion, 306 Irreversibility line, 143, 153 Bi-2212, 154 YBCO, 154 Isotope eﬀect on Tc , 365 the penetration depth, 366 JJ – π-junction, 343 Josephson junction (JJ), 342 Josephson vortices, 155 Kinematic interaction, 504 Knight shift, 101 in YBCO, 103

567

Korringa relation, 102 Kramers-Kronig relation, 252 Ladder copper-oxide compounds, 29 Lattice dynamics, theory, 371 Lattice polaron, 460 LDA + DMFT method, 417 Lower critical ﬁeld – YBCO, 166 Lower critical magnetic ﬁeld measurements, 165 Lower magnetic critical ﬁeld, 124 Lowest Landau level model, 140 Luttinger liquid, 394 Magnetic ﬂux quantum, 124 Magnetic mechanism of superconductivity, 428 resume, 455 Magnetic neutron scattering, theory, 52 Magnetic penetration depth, 123 Bi-2212, 170 c-axis, 168 in cuprates, 168 linear T -dependence, 168 measurements, 167 microwave measurements, 171 μSR measurements, 173 optic measurements, 172 Magnetic phase diagram of La2 CuO4 , 55 YBCO, 82 Magnetic susceptibility of YBCO, 83 Marginal Fermi liquid (MFL), 395 Mean-ﬁeld approximation in Hubbard model, 506 Melting line, 147 Bi-2212, 151 pressure eﬀects, 150 small-angle neutron scattering, 150 Memory function, 254 Metal–insulator (MI) transition, 305 Microwave spectroscopy, 272 resume, 279 Mid-infrared absorption, 257 Millis, Monien and Pines (MMP) model, 110 Momentum distribution curve (MDC), 215

568

Index

Momentum distribution function in t-J model, 406 Mott–Hubbard insulator, 181 Nagaoka theorem, 384 Nearly antiferromagnetic Fermi liquid (NAFL) model, 436 Negative-U center model, 470 Hubbard model, 460 Nernst eﬀect, 163, 303 Nodal metal, 262 Noncrossing approximation (NCA), 501 Normal state optical spectra – Resume, 271 Normal-state pseudogap, 133 Bi2212, 134 YBCO, 134 Nuclear magnetic resonance (NMR), 101 Nuclear quadrupole resonance, 103 One-dimensional Hubbard model, 394 Optical conductivity, electron-doped cuprates, 269 Optical electron spectroscopy, 251 Optical spectral weight at SC transition, 282 Optical spectral weight transfer – Resume, 290 Orbital (d–d) excitation model, 471 p-d model Hamiltonian, 384 Pair-breaking peak in RS, 296 Pairing with large momentum of Cooper pair, 473 Pancake vortices, 144, 146 Parallel incommensurate modulation, 73 Paramagnetic Meissner eﬀect (Wohlleben eﬀect), 343 Parameters of t-J model, 392 Parameters of eﬀective Hubbard Hamiltonians, 385 Passive microwave devices, 488 Peltier coeﬃcient, 303 PES – photoemission spectroscopy, 199 Phase diagram of La2−x Mx CuO4 , 19

Nd(Pr)2−x Cex CuO4 compounds, 32 Phase ﬂuctuations of SC order parameter, 433 Phase sensitive experiment, 342 resume, 347 tricristal geometry, 345 Phonon neutron scattering studies, 350 Phonon optic studies, 359 Pinning energy, 144 Pinning force, 143 Plasmon dispersion, 256 Plasmon model, 468 Polar model of metals, 382 Polaron–bipolaron model – Resume, 465 phase diagram, 463 superconductivity, 461 Polaronic mass, 462 Projection operator method, 443 Pseudogap disorder eﬀects, 135 DMFT + Σk -model, 411 Hubbard model, 410 RS, 298 Quantum critical point (QCP), 396 Quantum nonlinear 2D sigma model (QNLSM), 64 Quantum phase transition in QNLSM, 64 Quantum spin-liquid, 68 Quasiparticle (QP) spectral function, 216 Raman scattering (RS) phonon studies, 359 Renormalization group (RG) technique, 439 Resistive upper critical ﬁeld in Tl-2201, 161 Resistivity, 305 saturation, 306 Resonance magnetic mode, 92 Resonance mode – theory, 421 Resonating valence bond (RVB) model, 428 state, 396 Ring exchange interaction, 63 Rutgers formula, 158

Index Second (II) type superconductors, 123 Second generation (2G) of HTS wires, 481 Self-energy operator for fermion–boson model, 501 in Hubbard model, 509 in t–J model, 511 Shubnikov–de Haas (SdH) eﬀect, 315 Single-particle electron spectra – Conclusion, 418 Single-particle spectral function, 214 Sommerfeld constant, 126 Speciﬁc heat, 126 LSCO, 130 magnetic ﬁeld dependence, 127 superconducting state, 127 YBCO, 132 Spectral representation for GFs, 497 Spin dynamics of bilayer YBCO, 85 of LCO, 68 resume, 427 theory, 419 Spin gap, 88 in LSCO, 79 in YBCO, 87 Spin pseudogap, 113 Spin structure in YBCO, 80 Spin structure of La2 CuO4 , 55 Spin–lattice relaxation rate, 101 YBCO, 108 Spin-ﬂuctuation pairing, 434 phenomenological approach, 435 resume, 439 Spin-ﬂuctuation spectrum in LSCO, 78 Spin-polaron model, 398 quasiparticle spectrum, 399 Spin-wave spectrum in monolayer cuprates, 61 Spinon-holon representation, 429 SQUID phase sensitive experiments, 344 Staging, 75 STM – scanning tunneling microscopy, 329 Stripes – incommensurate spin (SDW) and charge (CDW) density waves, 72

569

Structural phase transition in La2−x Mx CuO4 , 19 STS checkerboard-pattern modulation, 338 diﬀerential conductance spectrum, 330 Fourier-transform STS method, 338 local density of states (LDOS) , 330 quasiparticle dispersion, 338 quasiparticle interference, 338 scanning tunneling spectroscopy, 329 superconducting gap and pseudogap, 335 Sum rules for GFs, 498 Superconducting Tc (n) in Hg-12(n-1)n compounds, 43 Superconducting gap symmetry, 327 Superconducting glass, 143 Superconducting pairing Hubbard model, 449 kinematic interaction, 442 spin-polaron model, 444 two channel model, 452 Superconducting quantum interference device (SQUID), 491 Superconducting rotating machines, 483 Superconductivity in Hubbard model, 503 induced phonon renormalization, 356 in t–J model, 510 Superﬂuid density, 167 scaling relations, 173 Symmetry relations for GFs, 498 t-J model, 390 Theory of coupled magnetic and structural phase transitions, 65 Theory of structural phase transition in La2−x Mx CuO4 , 25 Thermodynamic Green Functions (GFs), 496 Thermoelectric eﬀects, 302 Thermoelectric power (TEP), 303, 321 Thermomagnetic eﬀects, 303 Transferred hyperﬁne interaction, 106 Transport properties, 301 resume, 326 theoretical models, 323

570

Index

Tunneling experiments – Resume, 341 Two-band superconductor, 472 Two-dimensional (2D) anisotropic Heisenberg model, 59 Two-dimensional (2D) quantum Heisenberg model (2DQHM), 63 Unconventional ground state – Resume, 434 Universal formula for superconducting Tc (p) in cuprates, 9 Universal thermal conductivity, 317 Upper critical ﬁeld in Bi-2212, 160 in cuprates, 159 measurements, 156 Upper critical magnetic ﬁeld, 124 Variational approach, 432 Vortex Bragg glass, 146 Vortex dynamics, 154

Vortex glass, 146 Vortex liquid, 145 Vortex matter, 142 Bi-2212 phase diagram, 152 disorder eﬀects, 146 phase diagram, 144 Vortex-lattice melting, 145 ﬁrst order transition, 147 Vortex-liquid state, 164 Weak-coupling approximation (WCA), 502 Wiedemann-Franz law, 303 Wilson ratio, 128, 135 X-ray absorption spectra (Y-Ca)BCO, 208 (Y-Pr)BCO, 210 Zhang–Rice (ZR) singlet, 385

High-Temperature Cuprate Superconductors: Experiment, Theory, and Applications (Springer Series in Solid-State Sciences, 166)