Condensed Matter Physics

This textbook is a reader-friendly introduction to the theory underlying the many

fascinating properties of solids. Assuming only an elementary knowledge of

quantum mechanics, it describes the methods by which one can perform calculations

and make predictions of some of the many complex phenomena that occur in solids

and quantum liquids. The emphasis is on reaching important results by direct and

intuitive methods, and avoiding unnecessary mathematical complexity. The authors

lead the reader from an introduction to quasiparticles and collective excitations

through to the more advanced concepts of skyrmions and composite fermions.

The topics covered include electrons, phonons, and their interactions, density

functional theory, superconductivity, transport theory, mesoscopic physics, the

Kondo e¬ect and heavy fermions, and the quantum Hall e¬ect.

Designed as a self-contained text that starts at an elementary level and proceeds to

more advanced topics, this book is aimed primarily at advanced undergraduate and

graduate students in physics, materials science, and electrical engineering. Problem

sets are included at the end of each chapter, with solutions available to lecturers on

the internet. The coverage of some of the most recent developments in condensed

matter physics will also appeal to experienced scientists in industry and academia

working on the electrical properties of materials.

˜˜. . . recommended for reading because of the clarity and simplicity of presentation™™

Praise in American Scientist for Philip Taylor™s A Quantum Approach to the Solid

State (1970), on which this new book is based.

p h i l i p t a y l o r received his Ph.D. in theoretical physics from the University of

Cambridge in 1963 and subsequently moved to the United States, where he joined

Case Western Reserve University in Cleveland, Ohio. Aside from periods spent as a

visiting professor in various institutions worldwide, he has remained at CWRU, and

in 1988 was named the Perkins Professor of Physics. Professor Taylor has published

over 200 research papers on the theoretical physics of condensed matter and is the

author of A Quantum Approach to the Solid State (1970).

o l l e h e i n o n e n received his doctorate from Case Western Reserve University in

1985 and spent the following two years working with Walter Kohn at the University

of California, Santa Barbara. He returned to CWRU in 1987 and in 1989 joined the

faculty of the University of Central Florida, where he became an Associate Professor

in 1994. Since 1998 he has worked as a Sta¬ Engineer with Seagate Technology. Dr

Heinonen is also the co-author of Many-Particle Theory (1991) and the editor of

Composite Fermions (1998).

This Page Intentionally Left Blank

A Quantum Approach to

Condensed Matter Physics

PHILIP L. TAYLOR

Case Western Reserve University, Cleveland

OLLE HEINONEN

Seagate Technology, Seattle

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)

FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 IRP

40 West 20th Street, New York, NY 10011-4211, USA

477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© Cambridge University Press 2002

This edition © Cambridge University Press (Virtual Publishing) 2003

First published in printed format 2002

A catalogue record for the original printed book is available

from the British Library and from the Library of Congress

Original ISBN 0 521 77103 X hardback

Original ISBN 0 521 77827 1 paperback

ISBN 0 511 01446 5 virtual (netLibrary Edition)

Preface

The aim of this book is to make the quantum theory of condensed matter

accessible. To this end we have tried to produce a text that does not demand

extensive prior knowledge of either condensed matter physics or quantum

mechanics. Our hope is that both students and professional scientists will ¬nd

it a user-friendly guide to some of the beautiful but subtle concepts that form

the underpinning of the theory of the condensed state of matter.

The barriers to understanding these concepts are high, and so we do not try

to vault them in a single leap. Instead we take a gentler path on which to

reach our goal. We ¬rst introduce some of the topics from a semiclassical

viewpoint before turning to the quantum-mechanical methods. When we

encounter a new and unfamiliar problem to solve, we look for analogies

with systems already studied. Often we are able to draw from our storehouse

of techniques a familiar tool with which to cultivate the new terrain. We deal

with BCS superconductivity in Chapter 7, for example, by adapting the

canonical transformation that we used in studying liquid helium in

Chapter 3. To ¬nd the energy of neutral collective excitations in the frac-

tional quantum Hall e¬ect in Chapter 10, we call on the approach used for

the electron gas in the random phase approximation in Chapter 2. In study-

ing heavy fermions in Chapter 11, we use the same technique that we found

successful in treating the electron“phonon interaction in Chapter 6.

Experienced readers may recognize parts of this book. It is, in fact, an

enlarged and updated version of an earlier text, A Quantum Approach to

the Solid State. We have tried to preserve the tone of the previous book by

emphasizing the overall structure of the subject rather than its details. We

avoid the use of many of the formal methods of quantum ¬eld theory, and

substitute a liberal amount of intuition in our e¬ort to reach the goal of

physical understanding with minimal mathematical complexity. For this we

pay the penalty of losing some of the rigor that more complete analytical

ix

Contents

Preface ix

Chapter 1

Semiclassical introduction 1

1.1 Elementary excitations 1

1.2 Phonons 4

1.3 Solitons 7

1.4 Magnons 10

1.5 Plasmons 12

1.6 Electron quasiparticles 15

1.7 The electron“phonon interaction 17

1.8 The quantum Hall e¬ect 19

Problems 22

Chapter 2

Second quantization and the electron gas 26

2.1 A single electron 26

2.2 Occupation numbers 31

2.3 Second quantization for fermions 34

2.4 The electron gas and the Hartree“Fock approximation 42

2.5 Perturbation theory 50

2.6 The density operator 56

2.7 The random phase approximation and screening 60

2.8 Spin waves in the electron gas 71

Problems 75

v

vi Contents

Chapter 3

Boson systems 78

3.1 Second quantization for bosons 78

3.2 The harmonic oscillator 80

3.3 Quantum statistics at ¬nite temperatures 82

3.4 Bogoliubov™s theory of helium 88

3.5 Phonons in one dimension 93

3.6 Phonons in three dimensions 99

3.7 Acoustic and optical modes 102

3.8 Densities of states and the Debye model 104

3.9 Phonon interactions 107

3.10 Magnetic moments and spin 111

3.11 Magnons 117

Problems 122

Chapter 4

One-electron theory 125

4.1 Bloch electrons 125

4.2 Metals, insulators, and semiconductors 132

4.3 Nearly free electrons 135

4.4 Core states and the pseudopotential 143

4.5 Exact calculations, relativistic e¬ects, and the structure factor 150

4.6 Dynamics of Bloch electrons 160

4.7 Scattering by impurities 170

4.8 Quasicrystals and glasses 174

Problems 179

Chapter 5

Density functional theory 182

5.1 The Hohenberg“Kohn theorem 182

5.2 The Kohn“Sham formulation 187

5.3 The local density approximation 191

5.4 Electronic structure calculations 195

5.5 The Generalized Gradient Approximation 198

vii

Contents

5.6 More acronyms: TDDFT, CDFT, and EDFT 200

Problems 207

Chapter 6

Electron“phonon interactions 210

6.1 The Frohlich Hamiltonian 210

¨

6.2 Phonon frequencies and the Kohn anomaly 213

6.3 The Peierls transition 216

6.4 Polarons and mass enhancement 219

6.5 The attractive interaction between electrons 222

6.6 The Nakajima Hamiltonian 226

Problems 230

Chapter 7

Superconductivity 232

7.1 The superconducting state 232

7.2 The BCS Hamiltonian 235

7.3 The Bogoliubov“Valatin transformation 237

7.4 The ground-state wave function and the energy gap 243

7.5 The transition temperature 247

7.6 Ultrasonic attenuation 252

7.7 The Meissner e¬ect 254

7.8 Tunneling experiments 258

7.9 Flux quantization and the Josephson e¬ect 265

7.10 The Ginzburg“Landau equations 271

7.11 High-temperature superconductivity 278

Problems 282

Chapter 8

Semiclassical theory of conductivity in metals 285

8.1 The Boltzmann equation 285

8.2 Calculating the conductivity of metals 288

8.3 E¬ects in magnetic ¬elds 295

8.4 Inelastic scattering and the temperature dependence of resistivity 299

8.5 Thermal conductivity in metals 304

viii Contents

8.6 Thermoelectric e¬ects 308

Problems 313

Chapter 9

Mesoscopic physics 315

9.1 Conductance quantization in quantum point contacts 315

9.2 Multi-terminal devices: the Landauer“Buttiker formalism 324

¨

9.3 Noise in two-terminal systems 329

9.4 Weak localization 332

9.5 Coulomb blockade 336

Problems 339

Chapter 10

The quantum Hall e¬ect 342

10.1 Quantized resistance and dissipationless transport 342

10.2 Two-dimensional electron gas and the integer quantum Hall e¬ect 344

10.3 Edge states 353

10.4 The fractional quantum Hall e¬ect 357

10.5 Quasiparticle excitations from the Laughlin state 361

10.6 Collective excitations above the Laughlin state 367

10.7 Spins 370

10.8 Composite fermions 376

Problems 380

Chapter 11

The Kondo e¬ect and heavy fermions 383

11.1 Metals and magnetic impurities 383

11.2 The resistance minimum and the Kondo e¬ect 385

11.3 Low-temperature limit of the Kondo problem 391

11.4 Heavy fermions 397

Problems 403

Bibliography 405

Index 411

x Preface

treatments can yield. The methods used to demonstrate results are typically

simple and direct. They are expedient substitutes for the more thorough

approaches to be found in some of the bulkier and more specialized texts

cited in the Bibliography.

Some of the problems at the ends of the chapters are su¬ciently challenging

that it took the authors a longer time to solve them than it did to create them.

Instructors using the text may therefore ¬nd it a time-saver to see our versions

of the solutions. These are available by sending to solutions@cambridge.org

an e-mail containing plausible evidence that the correspondent is in fact a

busy instructor rather than a corner-cutting student pressed for time on a

homework assignment.

The earlier version of this text owed much to Harold Hosack and Philip

Nielsen for suggested improvements. The new version pro¬ts greatly from the

comments of Harsh Mathur, Michael D. Johnson, Sankar Das Sarma, and

Allan MacDonald. Any mistakes that remain are, of course, ours alone. We

were probably not paying enough attention when our colleagues pointed

them out to us.

Philip Taylor Cleveland, Ohio

Olle Heinonen Minneapolis, Minnesota

Chapter 1

Semiclassical introduction

1.1 Elementary excitations

The most fundamental question that one might be expected to answer is

˜˜why are there solids?™™ That is, if we were given a large number of atoms

of copper, why should they form themselves into the regular array that we

know as a crystal of metallic copper? Why should they not form an irregular

structure like glass, or a super¬‚uid liquid like helium?

We are ill-equipped to answer these questions in any other than a quali-

tative way, for they demand the solution of the many-body problem in one of

its most di¬cult forms. We should have to consider the interactions between

large numbers of identical copper nuclei “ identical, that is, if we were for-

tunate enough to have an isotopically pure specimen “ and even larger num-

bers of electrons. We should be able to omit neither the spins of the electrons

nor the electric quadrupole moments of the nuclei. Provided we treated the

problem with the methods of relativistic quantum mechanics, we could hope

that the solution we obtained would be a good picture of the physical reality,

and that we should then be able to predict all the properties of copper.

But, of course, such a task is impossible. Methods have not yet been

developed that can ¬nd even the lowest-lying exact energy level of such a

complex system. The best that we can do at present is to guess at the form the

states will take, and then to try and calculate their energy. Thus, for instance,

we might suppose that the copper atoms would either form a face-centered or

body-centered cubic crystal. We should then estimate the relative energies of

these two arrangements, taking into account all the interactions we could. If

we found that the face-centered cubic structure had the lower energy we

might be encouraged to go on and calculate the change in energy due to

various small displacements of the atoms. But even though we found that

all the small displacements that we tried only increased the energy of the

1

2 Semiclassical introduction

system, that would still be no guarantee that we had found the lowest energy

state. Fortunately we have tools, such as X-ray di¬raction, with which we can

satisfy ourselves that copper does indeed form a face-centered cubic crystal,

so that calculations such as this do no more than test our assumptions and our

mathematics. Accordingly, the philosophy of the quantum theory of con-

densed matter is often to accept the crystal structure as one of the given

quantities of any problem. We then consider the wavefunctions of electrons

in this structure, and the dynamics of the atoms as they undergo small dis-

placements from it.

Unfortunately, we cannot always take this attitude towards the electronic

structure of the crystal. Because we have fewer direct ways of investigating

the electron wavefunction than we had for locating the nuclei, we must some-

times spend time questioning whether we have developed the most useful

picture of the system. Before 1957, for example, people were unsuccessful

in accounting for the properties of superconductors because they were start-

ing from a ground state that was qualitatively di¬erent from what it is now

thought to be. Occasionally, however, a new technique is introduced by

means of which the symmetry of electronic states can be probed. An example

is shown on the cover of this book. There the e¬ect on the electronic structure

of an impurity atom at the surface of a high-temperature superconductor is

shown. The clover-leaf symmetry of the superconducting state is clearly seen

in the scanning-tunneling-microscope image.

The interest of the experimentalist, however, is generally not directed

towards the energy of the ground state of a substance, but more towards

its response to the various stimuli that may be applied. One may measure its

speci¬c heat, for example, or its absorption of sound or microwaves. Such

experiments generally involve raising the crystal from one of its low-lying

states to an excited state of higher energy. It is thus the task of the theorist

not only to make a reasonable guess at the ground state, but also to estimate

the energies of excited states that are connected to the ground state in a

simple way. Because the ground state may be of little further interest once

its form has been postulated, it is convenient to forget about it altogether and

to regard the process of raising the system to a higher state as one of creating

something where nothing was before. The simplest such processes are known

as the creation of elementary excitations of the system.

The usefulness of the concept of elementary excitations arises from a

simple property that most many-body systems have in common. Suppose

that there are two excited states, and that these have energies above the

ground state of E 1 and E 2 , respectively. Then it is frequently the case that

there will also be one particular excited state whose energy, E 3 , is not far

3

1.1 Elementary excitations

removed from °E 1 þ E 2 Þ. We should then say that in the state of energy E 3 all

the excitations that were present in the other two states are now present

together. The di¬erence ÁE between E 3 and °E 1 þ E 2 Þ would be ascribed to

an interaction between them (Fig. 1.1.1). If the states of energy E 1 and E 2

could not themselves be considered as collections of other excitations of

lower energy then we say that these states represent elementary excitations

of the system. As long as the interaction energy remains small we can with

reasonable accuracy consider most of the excited states of a solid as collec-

tions of elementary excitations. This is clearly a very useful simpli¬cation of

our original picture in which we just had a spectrum of energy levels which

had no particular relationship to one another.

At this point it is useful to consider a division of the possible types of

elementary excitations into two classes, known as quasiparticle excitations

and collective excitations. The distinction between these is best illustrated

by some simple examples. We know that if we have a gas of noninteracting

particles, we can raise the energy of one of these particles without a¬ecting

the others at all. Thus if the gas were originally in its ground state we could

describe this process as creating an elementary excitation. If we were now to

raise the energy of another particle, the energies of the excitations would

clearly add up to give the energy of the doubly excited system above its

ground state. We should call these particle excitations. If now we include

some interactions between the particles of the gas, we should expect these

particle excitations to decay, since now the excited particle would scatter o¬

the unexcited ones, and its energy and momentum would gradually be lost.

However, if the particles obeyed the Pauli Exclusion Principle, and the energy

of the excitation was very low, there would be very few empty states into

which the particle could be scattered. We should expect the excitation to

have a su¬ciently long lifetime for the description in terms of particles to

Figure 1.1.1. When two elementary excitations of energies E 1 and E 2 are present

together the combined excitation has an energy E 3 that is close to E 1 þ E 2 .

4 Semiclassical introduction

be a useful one. The energies of such excitations will di¬er from those for

noninteracting particles because of the interactions. It is excitations such as

these that we call quasiparticles.

A simple example of the other class of excitation is that of a sound wave in

a solid. Because the interatomic forces in a solid are so strong, there is little

pro¬t in considering the motion of an atom in a crystal in terms of particle

motion. Any momentum we might give to one atom is so quickly transmitted

to its neighbors that after a very short time it would be di¬cult to tell which

atom we had initially displaced. But we do know that a sound wave in the

solid will exist for a much longer time before it is attenuated, and is therefore

a much more useful picture of an excitation in the material. Since a

sound wave is speci¬ed by giving the coordinates not of just one atom but

of every atom in the solid, we call this a collective motion. The amplitude of

such motion is quantized, a quantum unit of traveling sound wave being

known as a phonon. A phonon is thus an example of a collective excitation

in a solid.

We shall now consider semiclassically a few of the more important excita-

tions that may occur in a solid. We shall postpone the more satisfying

quantum-mechanical derivations until a later chapter. By that time the

familiarity with the concepts that a semiclassical treatment gives may reduce

somewhat the opacity of the quantum-mechanical procedures.

1.2 Phonons

The simplest example of collective motion that we can consider is that of a

linear chain of equal masses connected by springs, as illustrated in Fig. 1.2.1.

The vibrational modes of this system provide some insight into the atomic

motion of a crystal lattice.

If the masses M are connected by springs of force constant K, and we call the

displacement of the nth mass from its equilibrium position yn , the equations

Figure 1.2.1. This chain of equal masses and springs supports collective motion in

the form of traveling waves.

5

1.2 Phonons

of motion of the system are

d 2 yn

¼ K½°ynþ1 À yn Þ À °yn À ynÀ1 Þ

M

dt2

¼ K°ynþ1 À 2yn þ ynÀ1 Þ: °1:2:1Þ

These equations are easily solved for any boundary conditions if we remem-

ber the recursion formula for cylindrical Bessel functions,

1

dJn

¼ À ½Jnþ1 °tÞ À JnÀ1 °tÞ;

2

dt

from which

d 2 Jn 1

¼ ½Jnþ2 °tÞ À 2Jn °tÞ þ JnÀ2 °tÞ:

dt2 4

The problem we considered in Section 1.1 was to ¬nd the motion of the

masses if we displaced just one of them °n ¼ 0, say) and then released it.

The appropriate solution is then

yn °tÞ ¼ J2n °!m tÞ

where !2 ¼ 4K=M. This sort of behavior is illustrated in Fig. 1.2.2. The

m

displacement of the zeroth mass, being given by J0 °!m tÞ, is seen to exhibit

oscillations which decay rapidly. After just a few oscillations y0 °tÞ behaves as

tÀ1=2 cos °!m tÞ. This shows that particle-like behavior, in which velocities are

constant, has no relation to the motion of a component of such a system.

Figure 1.2.2. These Bessel functions are solutions of the equations of motion of the

chain of masses and springs.

6 Semiclassical introduction

And this is quite apart from the fact that in a crystal whose atoms are

vibrating we are not fortunate enough to know the boundary conditions of

the problem. This direct approach is thus not very useful.

We ¬nd it more convenient to look for the normal modes of vibration of

the system. We make the assumption that we can write

yn / ei°!tþknaÞ ; °1:2:2Þ

where ! is some function of the wavenumber k, and a is the spacing between

masses. This satis¬es the equations of motion if

À!2 M ¼ K°eika þ eÀika À 2Þ;

that is, if

À1 Á

! ¼ Æ!m sin ka :

2

The solution (1.2.2) represents traveling waves of frequency ! and wave-

number (de¬ned for our purposes by 2=, where is the wavelength)

equal to k. The group velocity v is given by d!=dk, the gradient of the

curve shown in Fig. 1.2.3. We note that as ! approaches its maximum

value, !m , the group velocity falls to zero. This explains why the Bessel

function solution decayed to an oscillation of frequency !m after a short

time, if we realize that the original equation for yn °tÞ can be considered as

a superposition of waves of all wavenumbers. The waves of low frequency,

having a large group velocity, travel quickly away from the zeroth site, leav-

ing only the highest-frequency oscillations, whose group velocity is zero.

Figure 1.2.3. The dispersion curve for the chain of masses and springs.

7

1.3 Solitons

It is formally straightforward enough to ¬nd the normal modes of vibra-

tion for systems more complicated than our linear chain of masses. The

extension to three dimensions leads us to consider the polarization of the

lattice waves, that is, the angle between k, which is now a vector, and the

direction of displacement of the atoms. We can also introduce forces between

atoms other than nearest neighbors. This makes the algebra of ¬nding !°kÞ

more involved, but there are no di¬culties of principle. Introduction of two

or more di¬erent kinds of atom having di¬erent masses splits the graph of

!°kÞ into two or more branches, but as long as the restoring forces are all

proportional to the displacement, then solutions like Eq. (1.2.2) can be

found.

A phonon is the quantum-mechanical analog of the lattice wave described

by Eq. (1.2.2). A single phonon of angular frequency ! carries energy 0!. A

classical lattice wave of large amplitude corresponds to the quantum situa-

tion in which there are many phonons present in one mode. We shall see later

that a collection of phonons bears some similarity to a gas of particles. When

two particles collide we know that the total momentum is conserved in the

collision. If we allow two phonons to interact we shall ¬nd that the total

wavenumber is conserved in a certain sense. For this reason phonons are

sometimes called quasiparticles, although we shall avoid this terminology

here, keeping the distinction between collective and particle-like behavior.

1.3 Solitons

The chain of masses connected by Hookean springs that we considered in the

previous section was a particularly easy problem to solve because the equa-

tions of motion (1.2.1) were linear in the displacements yn . A real solid, on

the other hand, consists of atoms or ions having hard, mutually repulsive

cores. The equations of motion will now contain nonlinear (i.e., anharmonic)

terms. How do these a¬ect the type of excitation we may ¬nd?

If the amplitudes of the phonons are small then the e¬ects of the anhar-

monic terms will be weak, and the problem can be treated as a system of

interacting phonons. If the atomic displacements are large, on the other

hand, then there arises a whole new family of elementary excitations

known as solitary waves or solitons. In these excitations a localized wave

of compression can travel through a solid, displacing the atoms momentarily

but then leaving them as stationary as they were before the wave arrived.

The term soliton suggests by its word ending that it is a purely quantum-

mechanical concept, but this is not the case. Solitary waves in classical

systems had been observed as long ago as 1834, but it was only when their

8 Semiclassical introduction

interactions were studied that it was found that in some cases two solitary

waves could collide and then emerge from their collision with their shapes

unchanged. This particle-like behavior led to the new terminology, which is

now widely applied to solitary waves of all kinds.

We can begin to understand the relation between phonons in a harmonic

solid and solitary waves in an anharmonic solid with the aid of an exactly

soluble model due to Toda. We start by considering the simplest possible

model that can support a soliton, namely a one-dimensional array of hard

rods, as illustrated in Fig. 1.3.1. If we strike the rod at the left-hand end of the

array, it will move and strike its neighbor, which in turn will strike another

block. A solitary wave of compression will travel the length of the array

leaving all but the ¬nal block at rest. The speed of this soliton will be deter-

mined entirely by the strength of the initial impact, and can take on any

positive value. The wave is always localized to a single rod, in complete

contrast to a sound wave in a harmonic solid, which is always completely

delocalized.

Toda™s achievement was to ¬nd a model that interpolated between these

two systems. He suggested a chain in which the potential energy of inter-

action between adjacent masses was of the form

a Àbr

V°rÞ ¼ ar þ e: °1:3:1Þ

b

In the limit where b ! 0 but where the product ab is equal to a ¬nite constant

c we regain the harmonic potential,

a1

V°rÞ ¼ þ cr2 :

b2

In the opposite limit, where b ! 1 but ab ¼ c, we ¬nd the hard-rod poten-

tial for which V ! 1 if r 0 and V ! 0 if r > 0.

Figure 1.3.1. Through a series of elastic collisions, a solitary wave of compression

propagates from left to right.

9

1.3 Solitons

We construct a chain of equilibrium spacing d by having the potential

Æn V°Rn À RnÀ1 À dÞ act between masses located at Rn and RnÀ1 . In the nota-

tion where the displacement from equilibrium is yn ¼ Rn À nd, the equations

of motion are

d 2 yn Àb°ynþ1 Àyn Þ Àb°yn ÀynÀ1 Þ

¼ Àa e Àe :

M

dt2

If we now put yn À ynÀ1 rn then we have

d 2 rn Àbrnþ1 Àbrn ÀbrnÀ1

M 2 ¼ a Àe þ 2e Àe :

dt

One simple solution of this set of equations is the traveling wave for which

eÀbrn À 1 ¼ sinh2 sech2 °n Æ tÞ °1:3:2Þ

p¬¬¬¬¬¬¬¬¬¬¬¬¬

with ¼ ab=M sinh , and a number that determines both the amplitude

of the wave and its spatial extent. Because the function sech2 °n Æ tÞ

becomes small unless its argument is small, we see that the width of the

solitary wave is around d=. The speed v of the wave is d=, which on

substitution of the expression for becomes

r¬¬¬¬¬

ab sinh

v¼d :

M

For large-amplitude solitons the hard-rod feature of the potential dominates,

and this speed becomes very large. For small-amplitude waves,¬¬¬¬¬¬¬¬¬¬¬¬¬ other

p on the

hand, sinh = ! 1, and we recover the speed of sound, d ab=M , of the

harmonic chain.

The example of the Toda chain illustrates a number of points. It shows

how the inclusion of nonlinearities may completely alter the qualitative

nature of the elementary excitations of a system. The complete solution of

the classical problem involves Jacobian elliptic functions, which shows

how complicated even the simplest nonlinear model system can be.

Finally, it also presents a formidable challenge to obtain solutions of the

quantum-mechanical version of this model for a chain of more than a few

particles.

10 Semiclassical introduction

1.4 Magnons

In a ferromagnet at a temperature below the Curie point, the magnetic

moments associated with each lattice site l are lined up so that they all point

in more or less the same direction. We call this the z-direction. In a simple model

of the mechanism that leads to ferromagnetism, the torque acting on one of

these moments is determined by the orientation of its nearest neighbors. Then

the moment is subjected to an e¬ective magnetic ¬eld, Hl , given by

X

Hl ¼ A kl 0

l0

where A is a constant, kl 0 is the moment at the site l 0 , and the sum proceeds

only over nearest neighbors. The torque acting on the moment at l is kl ‚ Hl

and this must be equal to the rate of change of angular momentum. Since the

magnetic moment of an atom is proportional to its angular momentum we have

X

dkl

/ k l ‚ Hl ¼ A kl ‚ kl 0 : °1:4:1Þ

dt l0

As in the problem of the chain of masses and springs we look for a wave-like

solution of these equations which will represent collective behavior. With the

assumption that deviations of the kl from the z-direction are small we write

kl ¼ kz þ k? ei°!tþk Á lÞ ;

where kz points in the z-direction and where we have used the useful trick of

writing the components in the xÀy plane as a complex number, kx þ iky . That

is, if k? is in the x-direction, then ik? is in the y-direction. On substitution in

(1.4.1) we have, neglecting terms in k2 , ?

X ik Á l 00

i!k? / kz ‚ k? °e À 1Þ:

l 00

Here the l 00 are the vectors joining the site l to its nearest neighbors. In a

crystal with inversion symmetry the summation simpli¬es to

X

sin2 ° 1 k Á l 00 Þ:

À2 2

l 00

This equation tells us that k? rotates in the xÀy plane with frequency

X

sin2 ° 1 k Á l 00 Þ;

! / jkz j °1:4:2Þ

2

l 00

11

1.4 Magnons

the phase di¬erence between atoms separated by a distance r being just k Á r.

This sort of situation is shown in Fig. 1.4.1, which indicates the direction in

which k points as one moves in the direction of k along a line of atoms.

Because the magnetic moment involved is usually due to the spin of the

electron, these waves are known as spin waves. The quantum unit of such a

wave is known as a magnon.

The most important di¬erence to note between phonons and magnons

concerns the behavior of !°kÞ for small k (Fig. 1.4.2). For phonons we

found that the group velocity, d!=dk, tended to a constant as k tended to

zero, this constant being of course the velocity of sound. For magnons,

Figure 1.4.1. The k-vector of this spin wave points to the left.

Figure 1.4.2. The dispersion curve for magnons is parabolic in shape for small wave

numbers.

12 Semiclassical introduction

however, the group velocity tends to zero as k becomes small. This is of great

importance in discussing the heat capacity and conductivity of solids at low

temperatures.

In our simpli¬ed model we had to make some approximations in order to

derive Eq. (1.4.2). This means that the spin waves we have postulated would

eventually decay, even though our assumption about the e¬ective ¬eld had

been correct. In quantum-mechanical language we say that a crystal with two

magnon excitations present is not an exact eigenstate of the system, and that

magnon interactions are present even in our very simple model. This is not to

say that a lattice containing phonons is an exact eigenstate of any physical

system, for, of course, there are many factors we left out of consideration that

limit the lifetime of such excitations in real crystals. Nevertheless, the fact

that, in contrast to the phonon system, we cannot devise any useful model of

a ferromagnet that can be solved exactly indicates how very di¬cult a pro-

blem magnetism is.

1.5 Plasmons

The model that we used to derive the classical analog of phonons was a

system in which there were forces between nearest neighbors only. If we

had included second and third nearest neighbors we might have found that

the dispersion curve (the graph of ! against k) had a few extra maxima or

minima, but ! would still be proportional to k for small values of k. That is,

the velocity of sound would still be well de¬ned. However, if we wanted to

consider a three-dimensional crystal in which the atoms carried an electric

charge e we would ¬nd some di¬culties (Problem 1.3). Although the

Coulomb force of electrostatic repulsion decays as rÀ2 , the number of neigh-

bors at a distance of around r from an atom increases as r2 , and the equation

for ! has to be treated very carefully. The result one ¬nds for longitudinally

polarized waves is that as k tends to zero ! now tends to a constant value p ,

known as the ion plasma frequency and given by

40 e2

2 ¼ °1:5:1Þ

p

M

where e is the charge and M the mass of the particles, and 0 the number of

particles per unit volume of the crystal. We thus conclude that a Coulomb

lattice does not support longitudinal sound waves in the usual sense, since p

is no longer proportional to the wavenumber k.

13

1.5 Plasmons

This raises an interesting question about the collective excitations in

metals. We think of a metal as composed of a lattice of positively charged

ions embedded in a sea of nearly free conduction electrons. The ions interact

by means of their mutual Coulomb repulsion, and so we might expect that

the lattice would oscillate at the ion plasma frequency, p . Of course, we

know from everyday experience that metals do carry longitudinal sound

waves having a well de¬ned velocity, and so the e¬ective interaction between

ions must be short-range in nature. It is clear then that the conduction

electrons must play some part in this.

This leads to the concept of screening. We must suppose that in a sound

wave in a metal the local variations in charge density due to the motion of the

positively charged ions are cancelled out, or screened, by the motion of the

conduction electrons. This in¬‚ux of negative charge reduces the restoring

force on the ions, and so the frequency of the oscillation is drastically

reduced. That is to say, the ions and the electrons move in phase, and we

should be able to calculate the velocity of sound by considering the motion of

electrically neutral atoms interacting through short-range forces.

But if there is a mode of motion of the metallic lattice in which the elec-

trons and ions move in phase, there should also be a mode in which they

move out of phase. This is in fact the case, and it is these modes that are the

true plasma oscillations of the system, since they do give rise to variations in

charge density in the crystal. Their frequency, as we shall now show, is given

for long wavelengths by Eq. (1.5.1), where now the ionic mass, M, is replaced

by the electron mass, m. (In fact m should really be interpreted as the reduced

mass of the electron in the center-of-mass coordinate system of an electron

and an ion; however, since the mass of the ion is so many times greater than

that of the electron this re¬nement is not necessary.)

We shall look for plasma oscillations by supposing that the density of

electrons varies in a wave-like way, so that

°rÞ ¼ 0 þ q cos qx: °1:5:2Þ

This density must be considered as an average over a distance that is large

compared with the distance between an electron and its near neighbors, but

small compared with qÀ1 . When the electrons are considered as point parti-

cles the density is really a set of delta-functions, but we take a local average of

these to obtain °rÞ. The electrostatic potential °rÞ will then be of the same

form,

°rÞ ¼ 0 þ q cos qx; °1:5:3Þ

14 Semiclassical introduction

and will be related to the density of electrons °rÞ and of ions ion °rÞ by

Poisson™s equation,

r2 °rÞ ¼ À4e½°rÞ þ ion °rÞ: °1:5:4Þ

If we take ion to be equal to 0 we have on substitution of (1.5.2) and (1.5.3)

in (1.5.4)

q2 q ¼ 4eq :

The potential energy density is then

1 4e q

22

1

e ¼ cos2 qx:

2 q2

2

The average kinetic energy density, 1 mv2 , is also altered by the presence of

2

the plasma wave. The amplitude of the oscillation is q =0 q and so an elec-

_

tron moving with the plasma su¬ers a velocity change of °q =0 qÞ sin qx with

_

q the time derivative of q . We must also take into account the heating of the

plasma caused by adiabatic compression; since the fractional increase in

density is °q =0 Þ cos qx this e¬ect will add to the velocity an amount of

the order of °v0 q =0 Þ cos qx. If we substitute these expressions into the

classical Hamiltonian and take the spatial average we ¬nd an expression of

the form

!

_

q 2 v0 q 2

1 4e q 1

22

"

H¼ þ m0 þ

0 q 0

4 q2 4

!

40 e2

m

_q

¼ 2 þ 2 þ 2 v2 q2

0

q

40 q2 m

with a constant of order unity. This is the Hamiltonian for a classical

oscillator of frequency

!q ¼ °!2 þ 2 v2 q2 Þ1=2 ;

0

p

where !p is the electron plasma frequency, °40 e2 =mÞ1=2 .

The important point to note about this approximate result is that !p is a

very high frequency for electrons in metals, of the order of 1016 Hz, which

corresponds to a quantum energy 0!p of several electron volts. Quanta of

such oscillations are known as plasmons, and cannot be created thermally,

15

1.6 Electron quasiparticles

since most metals melt at a thermal energy of the order of 0.1 eV. Thus the

plasma oscillations represent degrees of freedom of the electron gas that are

˜˜frozen out.™™ This accounts for the paradoxical result that the interaction

between electrons is so strong that it may sometimes be ignored. One may

contrast this situation with that of an atom in the solid considered in Section

1.2. There it was found that any attempt to give momentum to a single atom

just resulted in the creation of a large number of collective excitations of low

energy. An electron in the electron gas, on the other hand, retains its particle-

like behavior much longer as it may not have the energy necessary to create a

single plasmon.

1.6 Electron quasiparticles

Most of the phenomena we have considered so far have been collective

motions. Our method of solving the equations of motion was to de¬ne a

collective coordinate, yk , which was a sum over the whole lattice of some

factor times the particle coordinates, yl . If we had had an in¬nite number of

particles, then the coordinates of any one particle would only have played an

in¬nitesimal role in the description of the motion. We now turn to the con-

sideration of excitations in which the motion of one particle plays a ¬nite

role.

In Section 1.1 we have already brie¬‚y considered the problem of an assem-

bly of particles that obey the Pauli Exclusion Principle. A gas of electrons is

an example of such a system. As long as the electrons do not interact then the

problem of classifying the energy levels is trivial. The momentum p of each of

the electrons is separately conserved, and each has an energy E ¼ p2 =2m. The

spin of each electron may point either up or down, and no two electrons may

have the same momentum p and spin s. If there are N electrons, the ground

state of the whole system is that in which the N individual electron states of

lowest energy are occupied and all others are empty. If the most energetic

electron has momentum pF , then all states for which jpj < jpF j will be occu-

pied. The spherical surface in momentum space de¬ned by jpj ¼ pF is known

as the Fermi surface (Fig. 1.6.1). The total energy of the system is then

X p2

ET ¼ ;

2m

s;jpj<pF

the sum being over states contained within the Fermi surface. We can write

this another way by de¬ning an occupation number, np;s , which is zero when

16 Semiclassical introduction

Figure 1.6.1. The Fermi sphere in momentum space contains all electron states with

energy less than p2 =2m.

F

the state with momentum p and spin s is empty and equal to 1 when it is

occupied. Then

X p2

ET ¼ n:

2m p;s

all s;p

The usefulness of the concept of a quasiparticle rests on the fact that one may

still discuss the occupancy of a state even when there are interactions between

the particles. Although in the presence of interactions np;s will no longer have

to take on one of the two values 0 or 1, we can attach a meaning to it. We

might, for instance, suppose that in with the electrons there is a positron at

rest, and that it annihilates with one of the electrons. The total momentum of

the gamma rays that would be emitted by the annihilating particles would be

equal to their total momentum before annihilation. We could now ask what

the probability is that this momentum be equal to p. Since for the noninter-

P

acting system this probability is proportional to s np;s , this provides an

interpretation for np;s in the interacting system.

In the noninteracting system we had a clear view of what constituted a

particle excitation. The form of np;s di¬ered from that of the ground state in

that one value of p less than pF was unoccupied, and one greater than pF was

occupied (Fig. 1.6.2). We then consider the excited system as composed of

the ground state plus an excitation comprising a particle and a ˜˜hole,™™ the

particle“hole pair having a well de¬ned energy above that of the ground

state. If we introduce interactions between the particles, and in particular if

17

1.7 The electron“phonon interaction

Figure 1.6.2. The excited state (b) is formed from the ground state (a) by the creation

of a particle“hole pair.

we introduce the troublesome Coulomb interaction, it is hard to see whether

the concept of a particle“hole excitation survives. It is, in fact, not only hard

to see but also hard to calculate. One approach is to consider the e¬ect of

switching on the interactions between particles when the noninteracting sys-

tem contains a particle“hole pair of energy E. If the lifetime of the excita-

tion is large compared with 0=E, then it will still be useful to retain a similar

picture of the excitations. Since now the interactions will have modi¬ed their

energies, we refer to ˜˜quasielectrons™™ and ˜˜quasiholes.™™

1.7 The electron“phonon interaction

In Section 1.5 we discussed sound waves in a metal, and came to the con-

clusion that in these excitations the ions and electrons moved in phase. The

long-range potential of the positively charged ions was thus screened, and the

phonon frequency reduced from the ion plasma frequency p to some much

smaller value. The way in which this occurs is illustrated in Fig. 1.7.1. We

¬rst imagine a vibration existing in the unscreened lattice of ions. We then

18 Semiclassical introduction

Figure 1.7.1. The deep potential due to the displacement of the ions by a phonon is

screened by the ¬‚ow of electrons.

suppose that the electron gas ¬‚ows into the regions of compression and

restores the electrical neutrality of the system on a macroscopic scale.

There is, however, a di¬erence between the motion of the ions and the

electrons in that we assume the ions to be localized entities, while the

electrons are described by wavefunctions that, in this case, will be small

distortions of plane waves. When we increase the local density of electrons

we must provide extra kinetic energy to take account of the Exclusion

Principle. We might take the intuitive step of introducing the concept of a

Fermi energy that is a function of position. We could then argue that the

local kinetic energy density of the electron gas should be roughly equal to

that of a uniform gas of free electrons, which happens to be 3 E F 0 . The

5

sound wave in a metal is thus seen in this model as an interchange of kinetic

energy between the ions and the electrons. We can calculate an order of

magnitude for the velocity of sound by writing the classical Hamiltonian

for the system in a similar approach to that of Section 1.5. The kinetic

_

energy of the ions will be M°q Þ2 =20 q2 when a wave of wavenumber q

passes through a lattice of ions of mass M and average number density 0 .

The total kinetic energy of the electrons is only changed to second order in

q =0 , and so contributes an energy density of the order of E F 0 °q =0 Þ2 .

Then

!

E F q2

M

_

H™ °q Þ2 þ 2 ;

q

2

20 q M

where is a constant of order unity. The frequency of the oscillator that this

19

1.8 The quantum Hall effect

Figure 1.7.2. In a careful calculation the kinetic energy of the electrons is found to

prevent a complete screening of the potential due to the displaced ions. The residual

potential is shown as a dashed line.

Hamiltonian describes is

r¬¬¬¬¬¬¬¬

¬

E F

!¼ q

M

which shows that the velocity of sound, vs , can be written as

r¬¬¬¬¬

m

vs $ vF

M

where vF is the velocity of an electron with energy E F and m=M is the mass

ratio of electron to ion.

In a more careful treatment one would argue that the electron gas would

not completely screen the electric ¬eld of the ions. Instead the electrons

would ¬‚ow until the sum of the electric potential energy and the kinetic

energy of the electrons (the dotted line in Fig. 1.7.1) became uniform.

There would then be a residual electric ¬eld (the dashed line in Fig. 1.7.2)

tending to restore the ions to their equilibrium positions. It is the action of

this residual electric ¬eld on the electrons that gives rise to the electron“

phonon interaction which we shall study in Chapter 6.

1.8 The quantum Hall effect

We close this chapter with a ¬rst glimpse of a truly remarkable phenom-

enon. In the quantum Hall e¬ect a current of electrons ¬‚owing along a sur-

face gives rise to an electric ¬eld that is so precisely determined that it has

20 Semiclassical introduction

become the basis for the legal standard of electrical resistance. This result is

reproducible to better than one part in 108 , even when one changes the

material from which an experimental sample is made or alters the nature

of the surface in which the current ¬‚ows.

In the elementary theory of the Hall e¬ect one argues that when electrons

travel down a wire with average drift velocity vd at right angles to an applied

magnetic ¬eld H0 then they experience an average Lorentz force e°vd =cÞ ‚ H0

(Fig. 1.8.1). In order for the electrons to be unde¬‚ected in their motion this

force must be counterbalanced by the Hall ¬eld EH , which arises from accu-

mulations of charge on the surface of the wire. In the absence of applied

electric ¬elds we can then write

vd

Eþ ‚ H ¼ 0; °1:8:1Þ

c

and since the current density jy is given by 0 evd , we have

Ex ¼ H jy ; °1:8:2Þ

with the Hall resistivity H equal to H=0 ec. The product 0 e can be inter-

preted as giving the density of charge carriers in a metal and also the sign of

their e¬ective charge (which may be positive or negative as a result of the

e¬ects of the lattice potential, as discussed in Chapter 4).

A special situation arises if the electrons are con¬ned to a two-dimensional

surface held perpendicular to the magnetic ¬eld. A semiclassical electron in

the center of the sample will then travel in a circular orbit with the cyclotron

frequency !c ¼ eH=mc. The x-component of this circular motion is reminis-

cent of a harmonic oscillator, and so it is no surprise to ¬nd that its energy

Figure 1.8.1. The Hall ¬eld EH cancels the e¬ect of the Lorentz force due to the

applied magnetic ¬eld H0 .

21

1.8 The quantum Hall effect

levels are quantized, with E ¼ °n þ 1Þ0!c . These are known as Landau levels.

2

Because its motion is circular, it does not contribute to any net current

¬‚owing through the sample, and so the question arises as to the origin of

any such current.

The answer lies with the electrons at the edge of the sample. They cannot

complete their little circles, as they keep bumping into a wall, bouncing o¬ it,

and then curving around to bump into it again (Fig. 1.8.2). In this way they

can make their way down the length of the sample, and carry an electric

current, the current of electrons at the top of the ¬gure being to the right and

the current at the bottom being to the left. To have a net current to the right,

we must have more electrons at the top of the ¬gure than at the bottom. The

Fermi level must thus be higher there, and this translates into a higher

electrical potential, and thus a Hall-e¬ect voltage.

Suppose now that we gradually increase the density 0 of electrons in the

sample while keeping the Hall voltage constant. The number of circular

orbits and edge states will increase proportionately, and the Hall resistance

will decrease smoothly as 1=0 . This simple picture, however, is spoiled if

there are impurities in the system. Then there will exist bound impurity states

whose energies will lie between the Landau levels. Because these states carry

no current, the Hall resistance will stop decreasing, and will remain constant

until enough electrons have been added to raise the Fermi energy to lie in the

next-highest Landau level.

This existence of plateaus in the Hall resistance as a function of number of

electrons is known as the integral quantum Hall e¬ect. In very pure samples

plateaus can also be found when simple fractions (like 1/3 or 2/5) of the states

in a Landau level are occupied. This occurs for a di¬erent reason, and is

Figure 1.8.2. Only the ˜˜skipping orbits™™ at the edges can carry a current along the

sample.

22 Semiclassical introduction

known as the fractional quantum Hall e¬ect. The detailed origin of both these

e¬ects will be explored in Chapter 10.

Problems

1.1 The energy levels E ln of a diatomic molecule are characterized by an

angular momentum quantum number l and a vibrational one n. Assume

a Hamiltonian of the form

p2 p2 1

1

þ 2 þ K°d À d0 Þ2

H¼

2m 2m 4

where d is the interatomic separation and Kmd0 ) 02 . Calculate

4

E 10 À E 00 and E 01 À E 00 (the energies of the two kinds of elementary

excitation), and also the di¬erence between the sum of these quantities

and E 11 À E 00 . This di¬erence represents the energy of interaction

between the two excitations.

¨ssbauer E¬ect Suppose that an atom of 57 Fe emits a

-ray of

1.2 The Mo

frequency !0 in the x-direction while it is moving in the same direction

with velocity v. Then by the Doppler e¬ect a stationary observer will see

radiation of frequency approximately equal to !0 °1 þ v=cÞ. The spec-

trum of radiation emitted by a hot gas of iron atoms will thus be

broadened by the thermal motion. Now suppose the iron atom to be

bound in a solid, so that the x-component of its position is given by

X

x¼ aq sin °!q t þ q Þ;

q

where the phonon frequencies, !q , are much smaller than !0 , and the

phases, q , are random. Derive an expression for the Fourier spectrum

of the radiation intensity seen by a stationary observer, taking into

account the frequency modulation caused by the motion of the atom.

[The fact that when the aq are small a large proportion of the radia-

tion has the unperturbed frequency, !0 , is the basis of the Mo ¨ssbauer

e¬ect. The emitted

-ray may be resonantly absorbed by another iron

atom in a process that is the converse of that described above.]

1.3 Phonons in a Coulomb Lattice If a particle at l carrying charge Ze

is displaced a distance yl , the change in electric ¬eld experienced at

23

Problems

distances r from l that are large compared with yl is the ¬eld due to an

electric dipole of moment kl ¼ Zeyl , and is given by

El °rÞ ¼ jrjÀ5 ½3°kl Á rÞr À r2 kl :

If the lattice is vibrating in a single longitudinal mode of wavenumber q

then one may evaluate its frequency !q by calculating the total ¬eld

P

l El . For vanishingly small q the sum over lattice sites may be

replaced by an integral [why?]. Evaluate !°q ! 0Þ by (i) performing

the integration over spherical polar coordinates ; , and r, in that

order, and (ii) performing the integral in cylindrical polar coordinates

; r, and z, restricting the integration to points for which r < R, where R

is some large distance. State in physical terms why these two results

di¬er.

1.4 Phonon Interactions The velocity of sound in a solid depends, among

other things, on the density. Since a sound wave is itself a density

¬‚uctuation we expect two sound waves to interact. In the situation

shown in Fig. P1.1 a phonon of angular frequency ! 0 and wavenumber

q 0 is incident upon a region of an otherwise homogeneous solid contain-

ing a line of density ¬‚uctuations due to another phonon of wavenumber

q. By treating this as a moving di¬raction grating obtain an expression

for the wavenumber q 00 and frequency ! 00 of the di¬racted wave.

Figure P1.1. When two phonons are present simultaneously one of them may form

an e¬ective di¬raction grating to scatter the other.

24 Semiclassical introduction

1.5 In a long chain, atoms of mass M interact through nearest-neighbor

P

forces, and the potential energy is V ¼ n g°yn À ynÀ1 Þ4 , where g is a

constant and yn is the displacement from equilibrium of the nth atom. A

solitary wave travels down this chain with speed v. How does v vary

with the amplitude of this wave?

1.6 Assume that the density of allowed states in momentum space for an

electron is uniform, and that the only e¬ect of an applied magnetic

¬eld H is to add to the energy of a particular momentum state the

amount ÆB H, according to whether the electron spin is up or down

(the only two possibilities). Derive an expression in terms of B and the

Fermi energy E F for the magnetic ¬eld H that must be applied to

increase the kinetic energy of an electron gas at 0 K by 5 ‚ 10À8 of its

original value.

1.7 In a certain model of ferromagnetism the energy of a free-electron gas

has added to it an interaction term

4=3 4=3

E int ¼ K°N" þ N# Þ;

where N" and N# are the total numbers of up- and down-spin

electrons, respectively. By investigating the total energy of this system

as a function of N" À N# for constant N°¼ N" þ N# Þ decide for what

range of K the magnetized state (in which N" 6¼ N# Þ will be (a) stable;

(b) metastable; (c) unstable. Express your results for K in terms of N

and E F , the Fermi energy in the absence of interactions.

1.8 In a classical antiferromagnet there are two oppositely magnetized sub-

lattices, each of which is subject to a ¬eld

X

Hl ¼ ÀA kl 0 ;

0

l

the sum proceeding over sites l 0 that are nearest neighbors to l. Find

the form of the spin-wave spectrum in a simple cubic crystal,

and describe pictorially the motion of the spins at low and high

frequencies.

1.9 In problem 1.6, you were asked to ¬nd the magnetic ¬eld that would

increase the kinetic energy by a fraction 5 ‚ 10À8 . Now redo this

25

Problems

problem for the case where the total energy is decreased by a fraction

5 ‚ 10À8 .

1.10 Calculate the energy of the soliton described by Eq. (1.3.2) in an in¬nite

Toda chain. [Alternatively, as an easier problem just estimate this

energy in the limits of small and large .]

Chapter 2

Second quantization and the electron gas

2.1 A single electron

We have taken a brief look from a semiclassical point of view at some of the

kinds of behavior exhibited by many-particle systems, and have then used

intuition to guess at how quantum mechanics might modify the properties we

found. It is now time to adopt a more formal approach to these problems,

and to see whether we can derive the previous results by solving the

Schrodinger equation for the quantum-mechanical problem.

¨

For a single electron we have the time-independent Schrodinger equation

¨

H °rÞ ¼ E °rÞ; °2:1:1Þ

where

p2

H¼ þ V°rÞ

2m

and p is interpreted as the operator Ài0r. This equation has physically mean-

ingful solutions for an in¬nite number of energies E ° ¼ 1; 2; 3; . . .Þ. The

eigenfunctions u °rÞ, for which

Hu °rÞ ¼ E u °rÞ;

form a complete set, meaning that any other function we are likely to need

can be expanded in terms of them. The u °rÞ for di¬erent are orthogonal,

meaning that

°

u*°rÞu 0 °rÞ dr ¼ 0; ° 6¼ 0 Þ; °2:1:2Þ

where dr is an abbreviation for dx dy dz; u* is the complex conjugate of u,

26

27

2.1 A single electron

and the integration is over all space. If the wavefunctions u °rÞ are normal-

ized then the integral is equal to unity for ¼ 0 .

It is convenient to adopt what is known as the Dirac notation to describe

integrals of this kind. Because wavefunctions like u °rÞ have to be continuous,

we can think of the integral in Eq. (2.1.2) as being equal to the limit of a sum

like

X

Á u*°ri Þu 0 °ri Þ; °2:1:3Þ

i

where we have divided all space into a large number of cells centered on the

points ri , and each of which encloses a volume equal to the vanishingly small

quantity Á. If we look on u 0 °rÞ as being the column vector (i.e., the vertical

array of numbers)

0 1

u 0 °r1 Þ

B u 0 °r Þ C

B 2C

B C

BÁC

@ A

Á

Á

and u* °rÞ as the row vector

°u*°r1 Þ; u*°r2 Þ; . . .Þ;

then the sum in expression (2.1.3) is just the matrix product of u*°rÞ and

u 0 °rÞ. We adopt the notation of writing the row vector u* °rÞ as hj and of

0

writing the column vector u 0 °rÞ as j i. Then we write the integral of

Eq. (2.1.2) as hj 0 i. For normalized wavefunctions Eq. (2.1.2) then becomes

hj 0 i ¼ 0 °2:1:4Þ

where 0 is the Kronecker delta symbol, which is unity when ¼ 0 and

zero otherwise.

An example of such a set of functions u °rÞ are the plane waves that are a

solution of (2.1.1) when the potential V°rÞ ¼ 0. Then if °rÞ / eik Á r ,

02 2 02 k2

H ¼À r ¼ :

2m 2m

We can avoid the di¬culty of normalizing such plane waves (which extend

over all space) by only considering the wavefunction within a cubical box of

28 Second quantization and the electron gas

volume ¼ L3 , having corners at the points °ÆL=2; ÆL=2; ÆL=2Þ. Then we

can take

u °rÞ ¼ À1=2 eik Á r :

We then impose periodic boundary conditions, by stipulating that the form of

the wavefunction over any side of the box must be identical to its form over

the opposite side. That is,

L L

u À ; y; z ¼ u þ ; y; z ; etc:

2 2

This means that k can no longer be any vector, but is restricted to a discrete

set of values such that

eik Á °L;0;0Þ ¼ eik Á °0;L;0Þ ¼ eik Á °0;0;LÞ ¼ 1:

Hence

2mx 2my 2mz

k ¼ ; ;

L L L

where mx ; my , and mz are integers. Equation (2.1.4) is then obeyed. These

allowed values of k form a simple cubic lattice in k-space, the density of

allowed points being =°2Þ3 , which is independent of k. Summations over

can then be interpreted as summations over allowed values of k.

We can expand a function °rÞ in terms of the u °rÞ by writing

X

°rÞ ¼ C u °rÞ °2:1:5Þ

and forming the integral

°

u* 0 °rÞ°rÞ dr;

an integral that we would write in the notation of Eq. (2.1.4) as h 0 ji. On

substituting from Eq. (2.1.5) we have

X X

h 0 ji ¼ C h 0 ji ¼ C 0 ¼ C 0 ;

29

2.1 A single electron

so that

X

°rÞ ¼ u °rÞhji;

which in the Dirac notation becomes

X

ji ¼ jihji:

This allows us to consider the ¬rst part of the right-hand side of this equation

as an operator that is identically equal to one;

X

jihj 1: °2:1:6Þ

The combination jihj is the product of a column vector with a row vector;

it is not a number but is an operator. When it acts on a wavefunction ji it

gives the combination jihji, which is just the number hji multiplying the

wavefunction ji.

This notation can be extended by treating an integral like

°

u*°rÞV°rÞu 0 °rÞ dr °2:1:7Þ

in the same way that we handled Eq. (2.1.2). If we replace this integral by the

sum

X

Á u*°ri ÞV°ri Þu 0 °ri Þ

i

we ¬nd that we are considering V°rÞ as the diagonal matrix

0 1

V°r1 Þ ÁÁÁ

0 0

B C

V°r2 Þ 0 Á Á Á A:

@ 0

ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ

The matrix product that gives expression (2.1.7) would then be written as

hjVj 0 i:

30 Second quantization and the electron gas

We then make use of the identity (2.1.6) to write

X X

j 0 ih 0 j V j 00 ih 00 j

V°rÞ ¼

0 00

X

h 0 jVj 00 ij 0 ih 00 j:

¼ °2:1:8Þ

0 ; 00

The operator j 0 ih 00 j gives zero when it operates on any of the states ji

except that for which ¼ 00 , and then it gives the state j 0 i. We can thus

interpret j 0 ih 00 j as removing an electron from the state described by the

wavefunction u 00 °rÞ and putting it into the state described by u 0 °rÞ. In more

dramatic language the operator annihilates an electron in the state j 00 i and

creates one in the state j 0 i. We can write this symbolically another way by

introducing the rather di¬cult concept of the wavefunction j0i that denotes an

empty box! That is, we de¬ne a column vector j0i that is normalized, so that

h0j0i ¼ 1; °2:1:9Þ

but which is orthogonal to all the one-particle wavefunctions u °rÞ, so that

hj0i ¼ 0

for all . We call it the vacuum state and are careful not to confuse it with the