ITINERANT ORDERING IN CRYSTALS

Transcription

1 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:i Trim: MM TS: Integra, India Models of ITINERANT ORDERING IN CRYSTALS

2 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:ii Trim: MM TS: Integra, India

3 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:iii Trim: MM TS: Integra, India Models of ITINERANT ORDERING IN CRYSTALS An Introduction By JERZY MIZIA and GRZEGORZ GÓRSKI Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

4 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:iv Trim: MM TS: Integra, India Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 84 Theobald s Road, London WC1X 8RR, UK First edition 2007 Copyright 2007 Elsevier BV. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) ; fax (+44) (0) ; permissions@elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Number: 2007XXXXX ISBN: For information on all Elsevier publications visit our web site at books.elsevier.com AU1 Printed and bound in [name of country] Working together to grow libraries in developing countries

5 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:v Trim: MM TS: Integra, India CONTENTS Preface xi Part One. Introduction to Theory of Solids 1 1. Periodic Structures Fundamental Types of Lattices Diffraction of Waves by a Crystal and the Reciprocal Lattice Reciprocal lattice vectors Brillouin Zones 8 References Various Statistics 11 Appendix 2A: Fermi Dirac and Bose Einstein Distribution Functions 14 References Paramagnetism and Weiss Ferromagnetism Paramagnetism Weiss Ferromagnetism 25 References Electron States The Nearly Free Electron Model General result for Use of DOS for evaluating lattice sums in momentum space Heat capacity of the free electron gas: an introduction The Tight-Binding Method Cohesion energy Bloch Theorem 43 Appendix 4A: Nearly Free Electrons, Two-Plane Waves Model 44 References 48 Part Two. Models of Itinerant Ordering in Crystals The Hubbard Model Simple Hubbard Model Extended Hubbard Model 54 References 58 v

6 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:vi Trim: MM TS: Integra, India vi Contents 6. Different Approximations for Hubbard Model Chain Equation for Green Functions Hartree Fock Approximation Hubbard I Approximation Atomic limit Finite bandwidth limit Extended Hubbard III Approximation Coherent Potential Approximation Relation between CPA, Hubbard III and extended Hubbard III approximations Different applications of the CPA Spectral Density Approach Modified Alloy Analogy Dynamical Mean-Field Theory Hubbard Model Extended by Inter-site Interactions Modified Hartree Fock approximation Coherent potential approximation for the extended Hubbard model 94 Appendix 6A: Equation of Motion for the Green Functions 95 Appendix 6B: Hubbard Solution for the Scattering and Resonance Broadening Effects 97 6B.1 The scattering effect 97 6B.2 The resonance broadening effect 100 Appendix 6C: Modified Hartree Fock Approximation for the Inter-site Interactions 106 References Itinerant Ferromagnetism Periodic Table Ferromagnetic Elements Ferromagnetic elements Introduction to Stoner Model Static magnetic susceptibility Stoner Model for Ferromagnetism Stoner Model for Rectangular and Parabolic Band Rectangular band Parabolic nearly free electron band Modified Stoner Model Modified Stoner model for a semi-elliptic band Beyond Hartree Fock Model General formalism Enhancement of magnetic susceptibility Critical values of interactions Numerical results The Critical Point Exponents Spin Waves in Ferromagnetism Energy of spin-wave excitations 161

7 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:vii Trim: MM TS: Integra, India Contents Dynamic susceptibility of ferromagnets Curie temperature 164 References 165 vii 8. Itinerant Antiferromagnetism Phenomenological Introduction Simple Model of Itinerant Antiferromagnetism Free Energy and the Magnetic Susceptibility Antiferromagnetism Induced by On-site and Inter-site Correlations Free Energy and the Magnetic Susceptibility Including Correlation Effects Longitudinal and transversal susceptibility Onset of Antiferromagnetism The case of zero Coulomb correlation: U = The case of the strong correlation: U>>D Numerical Results for Magnetization and Néel s Temperature Spin-Density Waves 193 Appendix 8A: Antiferromagnetism in the Presence of On-site and Inter-site Coulomb Correlation 199 References Alloys, Disordered Systems Introduction Order Disorder Transformation and Bragg Williams Approximation Bragg Williams approximation Relation to the Band Model Transition Metal alloys Different Types of Disorder in Bragg Williams Approximation 219 References Itinerant Superconductivity Phenomenological Introduction and Historical Background Physical Properties of the High-Temperature Superconductors General properties Crystal structure of the HTS Symmetry of the energy gap Dependence of the critical temperature on concentration Phase diagrams of the ordering Classic (BCS) Model for Superconductivity Electron Electron Interaction as a Source of Superconductivity Introduction Single-band model Model Hamiltonian Moments method for the superconductivity equation 245

8 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:viii Trim: MM TS: Integra, India viii Contents Analysis of the solution: critical temperature dependence on concentration Effect of internal pressure on superconductivity Symmetry of the energy gap Three-band model Introduction The Model Hamiltonian Hamiltonian diagonalization and the pairing interaction Results 266 Appendix 10A: Transformation of Superconductivity Hamiltonian to Momentum Space 269 Appendix 10B: Green Function Equations for Singlet Superconductivity 274 Appendix 10C: Effective Pairing Potential in the Single-Band Model 277 Appendix 10D: Bogoliubov Transformation 280 References The Coexistence Between Magnetic Ordering and Itinerant Electron Superconductivity Experimental Evidence of Coexistence Between Magnetic Ordering and Superconductivity Coexistence of Ferromagnetism and High-Temperature Superconductivity Model Hamiltonian Green function solutions General equations Ferromagnetism coexisting with singlet superconductivity Ferromagnetism coexisting with triplet opposite spins pairing superconductivity Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity Comparison with experimental results (for UGe 2 ZrZn 2, URhGe) Coexistence of Antiferromagnetism and High-Temperature Superconductivity Model Hamiltonian Formalism of the model Numerical examples Superconducting Gap in Stripe States 315 Appendix 11A: Coexistence of Ferromagnetism and Singlet Superconductivity 318 Appendix 11B: Coexistence of Antiferromagnetism and Singlet Superconductivity 321 References 323 Subject Index 325

9 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:ix Trim: MM TS: Integra, India PREFACE Over the last few years there has been extensive new research in the field of superconductivity (SC) interacting with antiferromagnetism (AF), stimulated by the experimental discoveries of high-temperature SC in cuprates (YBa 2 Cu 3 O 7, La 2 x(ba,sr) x CuO 4. Similarly there has been equally extensive research in the field of SC interacting with ferromagnetism (F), following the discovery of new superconducting materials, e.g. UGe 2, ZrZn 2, URhGe, showing weak ferromagnetic properties and SC under high pressure. In the more established field of itinerant F also, there have been slow but steady developments recently. There are similarities between all these processes, which have resulted in their close existence under experimental conditions. This should be reflected by the similarities in their formalism. The aim of this book is to present a unified itinerant model of these phenomena. Such a model will be an introduction to the field of itinerant F, itinerant AF and electronic SC (i.e. driven by electron electron interactions). Prompted by experimental evidence, this book also includes the areas of interaction between F and AF on one side and electronic SC on the other. Since we undertook this rather ambitious task, we apologize to the reader if in some places we did not rise to this difficult project. The book came to fruition during the preparation of our students for their M.Sc. dissertations in this field during a two-semester course. These students had rather weak preparation in elementary quantum mechanics and statistical physics. Therefore we included general introductory chapters on solid-state physics (Chapters 1 4) and the Hubbard model (Chapter 5), which will allow this book to be understood with minimal prerequisites; also the mathematical techniques are explained thoroughly. We hope that due to its tutorial nature, the book or parts of it will have the potential to become a textbook for teaching the course on a more introductory level. This book is primarily intended for undergraduate as well as graduate students and young scientists working in the field. The reader will be taken gradually, step by step from the rudiments of solidstate physics to the basics of the many-body theory and on to the understanding of the different approximations. We have considered (see Chapter 6) the Hartree Fock (H F) and the modified H F approximations, the classic CPA, the modified CPA which includes the inter-site correlation, the spectral density approximation (SDA) and the dynamical mean field theory (DMFT). These approximations are used to evaluate, quantitatively, the minimum energy of a system and the values of critical interactions for F, AF and SC types of ordering. ix

10 Elsevier UK Job code: MIOC Prelims-I :28p.m. Page:x Trim: MM TS: Integra, India x Preface This book also covers the close relation between the BCS (Bardeen, Cooper and Schrieffer) formalism of SC and the itinerant model for AF (Chapter 8). The comparison will start from the simple basic models inter-relating quantities describing the AF and SC. Next the Green function formalism for SC in the presence of AF or F is developed (Chapter 11) and the conclusions are put forward for the simultaneous appearance of SC and AF or F in the H F approximation and beyond the H F approximation to include the strong on-site Coulomb correlation. Due to the rapid development in this field, a concerted effort has been made to include and relate the results of the most recent research in the areas of electronic SC, itinerant F and AF. The authors realize that these matters are not covered wholly satisfactorily since there is a great deal of new literature published every month in this field. Rzeszów December, 2006 Jerzy Mizia Grzegorz Górski

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