Elementary Excitations In Solids

David Pines

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780367091705
• eBook ISBN: 9780429980480
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This text continues to fill the need to communicate the present view of a solid as a system of interacting particles which, under suitable circumstances, behaves like a collection of nearly independent elementary excitations. In addition to introducing basic concepts, the author frequently refers to experimental data. Usually, both the basic theory and the applications discussed deal with the behavior of '`'simple' metals, rather than the '`'complicated' metals, such as the transition metals and the rare earths. Problems have been included for most of the chapters.

Chapter 1

INTRODUCTORY SURVEY

1-1 GENERAL CONSIDERATIONS

Whenever we deal with solids we are dealing with a many-body problem. Thus we ask what happens when we bring together some 10²³ atoms/per cubic centimeter to make a crystal. This has certain important consequences. For instance, it means that we cannot expect exact solutions–that instead we must be continually developing approximate models to fit the situation at hand. Thus in making a theory it is usually essential that we be aware of the experimental work on the phenomenon under consideration and vice versa. Many of the important present-day developments arise out of such a close collaboration between the theoretical and the experimental physicist.

It is this use of approximate models which lends solid-state physics much of its fascination. Indeed, we may regard it as a marvelous proving ground for quantum mechanics and the ingenuity of the theoretical and experimental physicist. For unlike the nuclear or elementary particle physicist we know what our particles are, and what are the forces between them, but we must use all our intelligence and insight to understand the consequences of this interaction. Thanks to the work of many people, particularly during the last decade, it is now possible to view much of solid-state physics in terms of certain elementary excitations which interact only weakly with one another.

The use of an elementary excitation to describe the complicated interrelated motion of many particles has turned out to be an extraordinarily useful device in contemporary physics, and it is this view of a solid which we wish to adopt in this book.

Under what circumstances is it useful to regard a solid as a collection of essentially independent elementary excitations ? First of all, it is necessary that the excitations possess a well-defined energy. Let us suppose that the excitations are labeled by their momenta, which will be the case for a translationally invariant system. We shall see that the energy of a given excitation of momentum p will be of the form

${\tilde{\xi }}_{\mathrm{p}}={\xi }_{\mathrm{p}}-\mathrm{i}{\gamma }_{\mathrm{p}}$

(1-1)

where γ_p, the imaginary part of the energy, is inversely proportional to the lifetime of the excitation. In order that the excitation be well defined, it must be long-lived. This means that one must have

${\gamma }_{\mathrm{p}}\ll {\xi }_{\mathrm{p}}$

(1-2)

the real part of the energy of the excitation.

One may well ask how it is possible that in a system, which, like a solid, is composed of strongly interacting particles, it is possible to find elementary excitations which satisfy the requirement (1-2). To answer this question let us consider the ways in which an excitation may decay. There are essentially two: (1) scattering against another excitation, and (2) scattering against the “ground-state particles.”

The first mode of decay is negligible if one confines one’s attention to temperatures sufficiently low that only a comparatively small number of excitations are present. The second mode of decay is less easily inhibited; it turns out for the various systems of interest there exist coherence factors which limit the phase space available for the decay of an excitation of low momentum or long wavelength. (An obvious example is the limit placed by the Pauli principle on the scattering of an electron in the immediate vicinity of Fermi surface.)

The requirement, (1-2), usually limits one to comparatively low temperatures and often, as well, to phenomena that involve comparatively low frequencies and long wavelengths. Where it is not satisfied, it may still be useful to describe a given physical process in terms of the excitations involved, but it becomes essential to take into account the fact that the excitations possess a finite lifetime.

Where (1-2) is satisfied, in thermal equilibrium one may characterize the excitation by a distribution function,

${\mathrm{f}}_{\mathrm{p}}\left(\mathrm{T}\right)=\frac{1}{{\mathrm{e}}_{\mathrm{p}}^{\beta \xi }\pm 1}$

(1-3)

where β = 1/κT; the plus sign applies if the excitation obeys Fermi-Dirac statistics, the minus sign for Bose-Einstein statistics. f_p(T) gives the probability of finding an excitation of momentum p, energy ξ_p, at the temperature T; from a knowledge of all the f_p(T), one can determine in straightforward fashion the various thermodynamic properties of the system.

These remarks are quite general; to see what they mean, it is necessary that one consider some specific examples. But before doing that let us specify clearly the basic model we shall take to describe a solid throughout this book.

1-2 BASIC HAMILTONIAN

The basic Hamiltonian which describes our model of the solid is of the form

$\mathrm{H}={\mathrm{H}}_{\mathrm{ion}}+{\mathrm{H}}_{\mathrm{electron}}+{\mathrm{H}}_{\mathrm{electron}-\mathrm{ion}}$

(1-4)

where

${\text{H}}_{\text{ion}}={\displaystyle \sum }_{\text{i}}\frac{{\text{P}}_{\text{i}}^2}{2\text{M}}+1/2{\displaystyle \sum }_{\text{i}\ne \text{j}}\text{V}\left({\text{R}}_{\text{i}}-{\text{R}}_{\text{j}}\right)$

(1-5)

${\mathrm{H}}_{\mathrm{electron}}={\displaystyle \sum _{\mathrm{i}}\frac{{\mathrm{P}}_{\mathrm{i}}^2}{2\mathrm{m}}}+1/2{\displaystyle \sum _{\mathrm{i}\ne \text{j}}\frac{{\mathrm{e}}^2}{\left|{\mathrm{r}}_{\mathrm{i}}-{\mathrm{r}}_{\mathrm{j}}\right|}}$

(1-6)

$H_{\mathrm{electron}-\mathrm{ion}}={\displaystyle \sum _{\mathrm{i},\mathrm{j}}\mathrm{v}\left({\mathrm{r}}_{\mathrm{i}}-{\mathbf{R}}_{\mathrm{j}}\right)}$

(1-7)

H_ion describes a collection of ions (of a single species) which interact through a potential V(R_i − R_j) which depends only on the distance between the ions. By ion we mean a nucleus plus the closed-shell, or core, electrons, that is, those electrons which are essentially unchanged when the atoms are brought together to make a solid. H_electron describes the valence electrons (the electrons outside the last closed shell), which are assumed to interact via a Coulomb interaction. Finally, H_electron-ion describes the interaction between the electrons and the ions, which is again assumed to be represented by a suitably chosen potential.

In adopting (1-4) as our basic Hamiltonian, we have already made a number of approximations in our treatment of a solid. Thus, in general the interaction between ions is not well-represented by a potential, V(R), when the coupling between the closed-shell electrons on different ions begins to play an important role. Again, in using a potential to represent electron-ion interaction, we have neglected the fact that the ions possess a structure (the core electrons); again, where the Pauli principle plays an important role in the interaction between the valence electrons and the core electrons, that interaction may no longer be represented by a simple potential. It is desirable to consider the validity of these approximations in detail, but such a study lies beyond the scope of this book; we shall therefore simply regard them as valid for the problems we study here. (It may be added that compared to the approximations which of necessity we shall have to make later, the present approximations look very good indeed.)

In general one studies only selected parts of the Hamiltonian, (1-4). Thus, for example, the band theory of solids is based upon the model Hamiltonian,

${\mathrm{H}}_{\mathrm{B}}={\displaystyle \sum _{\mathrm{i}}\frac{{\mathrm{p}}_{\mathrm{i}}^2}{2\mathrm{m}}+{\displaystyle \sum _{\mathrm{i},\mathrm{j}}\mathrm{v}\left({\mathrm{r}}_{\mathrm{i}}-{\mathbf{R}}_{\mathrm{jo}}\right)+{\mathrm{V}}_{\mathrm{H}}\left({\mathrm{r}}_{\mathrm{i}}\right)}}$

(1-8)

where the R_jo represents the fixed equilibrium positions of the ions and the potential V_H represents the (periodic) Hartree potential of the electrons, One studies the motion of a single electron in the periodic field of the ions and the Hartree potential, and takes the Pauli principle into account in the assignment of one-electron states. In so doing one neglects aspects other than the Hartree. potential of the interaction between the electrons. On the other hand, where one is primarily interested in ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Foreword
• Preface
• 1. Introductory Survey
• 2. Phonons
• 3. Electrons and Plasmons
• 4. Electrons, Plasmons, and Photons in Solids
• 5. Electron-Phonon Interaction in Metals
• Appendixes