A Modern Course in the Quantum Theory of Solids PDF
Preview A Modern Course in the Quantum Theory of Solids
A MODERN COURSE I N QUANTUM THEORY THE OF SOLIDS 8556_9789814417143_tp.indd 1 7/8/12 12:13 PM July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk A MODERN COURSE QUANTUM THEORY IN THE OF SOLIDS Fuxiang Han Dalian University of Technology, China World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8556_9789814417143_tp.indd 2 7/8/12 12:13 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. A MODERN COURSE IN THE QUANTUM THEORY OF SOLIDS Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4417-14-3 Printed in Singapore. Julia - A Modern Course in the Quantum.pmd 1 8/6/2012, 6:22 PM July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han To my wife Pan Yanmei Without her constant support and encouragement, I could not have finished writing this book. v July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han Preface This book contains advanced subjects in solid state physics with emphasis on the theoretical exposition of various physical phenomena in solids with the use of quantum theory, hence entitled “A modern course on quantum theory of solids”. The use of the adjective “modern” in the title is to reflect the fact that some of the new developments in condensed matter physics have been included in the book. The new developments contained in the book are mainly in experimental methods (the inelastic neutron scattering and the photoemission spectroscopy), in magnetic properties of solids (the itinerant magnetism, the superexchange, the Hubbard model, and the giant and colossal magnetoresistance), and in optical properties of solids (the Raman scattering). Besides the new developments, the Green’s functionmethodusedinmany-bodyphysicsandthestrong-couplingtheory of superconductivity are also expounded in great details. Thisbookgrewoutofmylecturenotespreparedforthecourseonquan- tumtheoryofsolidsforgraduatestudents atDalianUniversityofTechnol- ogy. There exists somewhat overlappingin lattice dynamics and electronic energy band theory with elementary solid state physics. This overlapping serves the purpose of providing basic knowledge on motions of atoms (or ions) and electrons in solids and setting up the playground for studying their influences onvariousphysical propertiesof solids. Excitationsarising from vibrations of atoms (or ions), from electrons, and from interactions between electrons and vibrations of atoms (or ions) play a crucial role in shaping physical properties of many solids. During the writing of the book, special attention was paid to the clar- ity in the elucidation of physical concepts and to the completeness of the derivations of physical results. The relevant algebras are given as much as possible. However, due to space limitation, many tedious details must be vii July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han viii A Modern Course inQuantum Theory of Solids omitted. In such a case, the procedure leading to the final result is often furnished succinctly in words so that it takes only a little extra effort for the reader to supply the omitted algebras. To help the reader grasp the materials in the book, problems are pro- vided at the end of each chapter. Some of these problems can be solved straightforwardly using the principles elucidated in the main text; others may require extra efforts since they contain extensions of the materials covered in the main text. Working out these problems will definitely ac- celerate the learning of the relevant subjects. When the book is used as a textbook in a course, some of these problems can be used in homework as- signments. Detailedsolutionstoalltheproblemsinthebookarecontained in the book “Problems in solid state physics with solutions” published by World Scientific Publishing (ISBN-13 978-981-4365-02-4), in which many other problems in solid state physics can be also found. This book was written using LATEX. To achieve the best quality for illustrations, all the illustrations in the book were (re)produced by the author in the Encapsulated PostScript (EPS) format using the PostScript programminglanguage. Dataformanyfigureswereobtainedbytheauthor throughsolvingnumericallytherelevantproblemsduringthewritingofthe book. Forabetterunderstandingofasubjectthatinvolveslightnumerical work, the reader is encouraged to make attempts to translate the problem into computer codes in a computer programminglanguagesuchas Fortran or C or C++ and to obtain the relevant numerical data. The writing of this book has been supported by a grant for teaching reform from the Graduate School at Dalian University of Technology. The work of the editors of this book at World Scientific Publishing is greatly appreciated. Dalian, May 2012 Fuxiang Han July31,2012 11:38 WorldScientificBook-9inx6in Julia8556-QuantumTheoryofSolids han Contents Preface vii 1. Lattice Dynamics 1 1.1 Born–Oppenheimer Approximation . . . . . . . . . . . . . 4 1.2 Lattice Potential Energy and Harmonic Approximation . 6 1.3 Normal Modes of a Three-Dimensional Crystal with a Multi-Atom Basis. . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Equations of motion of atoms . . . . . . . . . . . 11 1.3.2 Allowed values of wave vector k . . . . . . . . . . 12 1.3.3 Allowed values of frequency ω . . . . . . . . . . . 13 1.3.4 Polarizationvectors . . . . . . . . . . . . . . . . . 17 1.3.5 Displacements of atoms . . . . . . . . . . . . . . . 17 1.3.6 Hamiltonian of a crystal with a multi-atom basis. 18 1.4 Classical Theory of the Lattice Specific Heat . . . . . . . 19 1.5 Quantization of Lattice Vibrations . . . . . . . . . . . . . 20 1.5.1 Statistics for phonons . . . . . . . . . . . . . . . . 22 1.6 Phonon Density of States . . . . . . . . . . . . . . . . . . 23 1.7 Lattice Specific Heat of Solids . . . . . . . . . . . . . . . . 24 1.7.1 General expression of the lattice specific heat . . . 25 1.7.2 High-temperature limit . . . . . . . . . . . . . . . 25 1.7.3 Low-temperature limit . . . . . . . . . . . . . . . 26 1.8 Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.8.1 High-temperature limit . . . . . . . . . . . . . . . 29 1.8.2 Low-temperature limit . . . . . . . . . . . . . . . 30 1.8.3 Debye temperature . . . . . . . . . . . . . . . . . 30 1.9 Einstein Model . . . . . . . . . . . . . . . . . . . . . . . . 31 1.10 Effect of Thermal Expansion on Phonon Frequencies . . . 33 ix
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