Table Of ContentQuantum 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.
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United Kingdom Edition published by
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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)
