Quantum Theory of the Solid State. Volume 2 PDF
Preview Quantum Theory of the Solid State. Volume 2
Quantum Theory of the Solid State Part B JOSEPH CALLA WAY Department of Physics and Astronomy Louisiana State University Baton Rouge, Louisiana ACADEMIC PRESS New York and London 1974 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1974, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Callaway, Joseph. Quantum theory of the solid state. Includes bibliographies. 1. Solids. 2. Quantum theory. I. Title. QC176.C32 530.4Ί 72-12200 ISBN 0-12-155202-0 (v. 2) PRINTED IN THE UNITED STATES OF AMERICA To Mary Preface The purpose of this book is to describe the concepts and methods, and to introduce some of the central problems of the quantum theory of solids. It should be suitable as a textbook for students who have completed a one- year course in quantum mechanics and have some familiarity with the experimental facts of solid state physics. It should also be useful as a refer ence work. I have attempted a moderately comprehensive coverage: The physics of solids is, in fact, a rather diverse subject. A book with these aims must develop both principles and mathematical techniques; in addition, it should assist the reader in making his way through the more specialized periodical literature. To this end, fairly lengthy bibliographies have been included at the end of each chapter, although these are not intended to be complete in any area. If these refer ences are used in conjunction with the Science Citation Index, it should be possible to follow many specific subjects to the frontier of present research. This work is intended to be a single intellectual unit, although for reasons of convenience it has been divided into two parts. Part A contains much of the formalism required for the theoretical study of solids; Part B is oriented toward more specific problems. Thus, Part A includes phenomenological treatments of lattice vibrations and magnetic order, a discussion of sym metry groups, and a description of the properties of one-electron wave functions and the principal techniques for calculating energy levels. In Part B the machinery developed previously is applied to impurities, dis ordered systems, the effects of external fields, and transport phenomena (including superconductivity). The book concludes with an introduction to many-body theory, including some applications. The specific selection of topics is obviously a personal one, and some areas of considerable importance, such as mechanical properties, surfaces, electron diffraction, and amorphous materials, have not been included. Experimental results are used occasionally and illustratively. There is no ix X PREFACE detailed confrontation of specific approximations with experimental data. MKS, cgs, and atomic units have been used rather interchangeably. Some problems have been included. I am indebted to my colleague, Dr. John Kimball, and to several stu dents (W. Y. Ching, M. Eswaran, G. S. Grest, W. Y. Hsia, M. Singh, and C. S. Wang) for critical readings of portions of the manuscript. Contents of Part A 1. Lattice Dynamics 2. Phenomenological Theories of Magnetic Order 3. Symmetry and Its Consequences 4. Energy Bands Appendix A. Summation Relations Appendix B. Quantization of the Free Electromagnetic Field Appendix C. Character Tables and Compatibility Tables Appendix D. Second Quantization for a System of Fermions XI Note to the Reader In this volume superscript "plus" (+) is used to indicate a Hermitian adjoint instead of the superscript "dagger" (f) used in Part A. xiii CHAPTER 5 Impurities and Alloys 5.1 REPRESENTATION THEORY Throughout this chapter and the subsequent one we are concerned with solutions of a Schrödinger equation of the form (Ho + U)V = ih ΘΨ/dt (5.1.1) where H is a Hamiltonian which contains a periodic potential and U 0 represents an external potential. We will consider three different situations: (1) (this chapter) point impurities and alloys; (2) (Chapter 6) uniform electric and magnetic fields in crystals; and (3) (also Chapter 6) the response of electrons to electromagnetic radiation. The method of treat ment of these diverse problems has a unifying feature, the expansion of the wave function of an electron in a complete set of functions originally defined in the perfect crystal. 5.1.1 The Crystal Momentum Representation One rather obvious procedure is to expand quantities in terms of the Bloch functions ^ (k, r) which are eigenfunctions of the perfect crystal n Hamiltonian #o*n(k, r) = #n(k)* (k, r) (5.1.2) n where k is the wave vector and n is the band index. This procedure is known as the crystal momentum representation (CMR) (Adams, 1952, 1953) since it is based on states of definite k, the "crystal momentum." It was shown in Section 4.1.1, Part A, that the Bloch functions are orthonormal, yV(k,r)iMq,r) d*r = δ δ(ί - q), (5.1.3) η1 and in this equation and everywhere else, unless explicitly specified other- 371 372 5. IMPURITIES AND ALLOYS wise, the integral on r includes all space. They were also shown to be complete: Σ /Vn*(k, r)*»(k, r') fflk = δ(Γ - r'). (5.1.4) The integral includes a single Brillouin zone. Since the Bloch functions are complete, any one-electron function may be expanded in Bloch functions. Let Ψ(Γ, t) be a general wave function. We write *(r, 0 = Σ I *»(k, 0*»(k, r) d%. (5.1.5) n J The expansion coefficients <£ (k, t) describe the wave function in the crystal n momentum representation. They satisfy an effective Schrödinger equation which can be obtained by substituting (5.1.5) into (5.1.1), multiplying by ψι*(<1> r) i and integrating Σ if *l*(q, Γ) (C^n(k) + C/]0n(k, 0 - ih C^ (k, *)/**}*»(k, r) <Pfc (Pr n n JJ = C^i(q) - ih d/«>j(q, ί) + Σ /d3fc(iq I U | nk)« (k, i) = 0. n (5.1.6) We have introduced the abbreviation (lq\U\ nk) = f *,*(q, r) t/(r)* (k, r) d3r. (5.1.7) n Note that if we are concerned with an energy eigenstate, we have (HQ + U)V = E* (5.1.8) which implies ih a</>n(k, t)/dt = Εφ(ΐ£, t), (5.1.9) η so that the time-independent form of (5.1.6) is [J0,(q) - J5]0i(q) + Σ f **<ïq I ^ I nk>0 (k) = 0. (5.1.10) n n ' It is convenient in some of the following to use Dirac notations and work directly with the states | nk). The Bloch wave function ^ (k, r) may be n interpreted as a transformation coefficient *»(k,r) = (r|nk). (5.1.11) 5.1 Representation Theory 373 5.1.2 The Effective Mass Representation Use of the crystal momentum representation may require knowledge of crystal wave functions throughout the Brillouin zone. Although it is by no means impossible with modern computing equipment to obtain wave functions at a representative sample of general points in the zone, it is sometimes desirable to expand in terms of functions simply related to wave functions at a single point of a band. Such a representation was introduced by Luttinger and Kohn (1955) ; we shall call it the effective mass representa tion. The basis functions of the Luttinger-Kohn representation are denoted by Xn(k, r). They are defined in terms of the Bloch functions at some con veniently chosen reference point in the Brillouin zone k by 0 Xn(k, r) = exp[i(k - k ) -p]^»(ko, r) = exp(ikT)w»(ko, r). (5.1.12) 0 These functions were introduced in Section 4.1.2, Part A, in connection with k · p perturbation theory. We will find them particularly convenient in problems involving a small number of particles-electrons (or holes) in a nearly empty (or nearly full) band. They may be shown to obey the same orthogonality and completeness relations as the Bloch functions. If ko is not invariant under all the operations of the crystal point group, there will be other points (those in its star) which are equivalent to it. In such cases, it will usually be desirable to perform parallel expansions about all such points. However, for the present we will ignore this problem and consider only a single point k . 0 Consider the expansion of an arbitrary wave function in these functions: *(*, 0 = Σ / ^n(k, i)Xn(k, r) d3fc. (5.1.13) n J The A (k,t) are the expansion coefficients. The representation of the n Hamiltonian on the basis of the χ is somewhat more complicated than in η the case of the ψ since the x's are not eigenfunctions of H . The matrix η 0 elements of H are determined in the following manner: 0 [nk | Ho | Zq] = / Xn*(k, r)ffoX*(q, r) dh = / exp(i(q - k) -r)u *(ko, r) n X |jE,(ko) + (Ä(q - ko)Im) p + (Ä2/2m) (q2 - k 2)] 0 Xuz(ko,r) dh. (5.1.14)
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