Solid-State Physics

James D. Patterson • Bernard C. Bailey

Solid-State PhysicsIntroduction to the Theory

Third Edition

123

James D. PattersonRapid City, SDUSA

Bernard C. BaileyCape Canaveral, FLUSA

Complete solutions to the exercises are accessible to qualified instructors at springer.com onthis book’s product page. Instructors may click on the link additional information andregister to obtain their restricted access.

ISBN 978-3-319-75321-8 ISBN 978-3-319-75322-5 (eBook)https://doi.org/10.1007/978-3-319-75322-5

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Preface

First, we want to say a bit about solid-state physics, condensed matter, and materialsscience. These three names have overlapping meanings, and as far as we under-stand, there is no universal agreement on what each term signifies. Let us state whatwe signify by these terms and why we have decided to use the term solid-statephysics in our title.

Within the American Physical Society (APS), the Division of Solid-StatePhysics was formed in 1947 and the Division of Condensed Matter Physics(DCMP) replaced it in 1978. An outgrowth from DCMP was the eventual formationof the Division of Materials Physics (DMP) in 1990. According to APS, theDivision of Condensed Matter Physics was formed “to recognize that disciplinescovered in the division included liquids (quantum fluids) as well as solids.” Also theAPS states, “Materials Physics applies fundamental condensed matter concepts tocomplex and multiphase media, including materials of technological interest.” Aninteresting paper gives some insight as to what has been considered interesting inthe world of materials science in the last fifty years. Johnathan Wood, “The top tenadvances in materials science,” Materials Today, 11, Number 1–2, pp. 40–45,2008.

What we mean by solid-state physics is essentially defined by chapter titles andheaders in our book (a large part of solid-state physics is the physics of crystallinematter). Some authors tend to think of condensed matter physics as containing thefundamental aspects of solid-state physics as well as adding liquids. Some mighteven go so far as to say condensed matter physics is “more pure” than materialsphysics. Material physicists we believe tend to have a more applied or technologicalslant to their field, and I suppose in that sense some might consider it “less pure.”

The names “Condensed Matter,” and “Materials,” are also influenced by fund-ing. If there are several funding opportunities available in the fundamental under-pinnings of a solid-state area, a physicist in that field might wish to be considered a

v

condensed matter physicist. Similarly, if funding is going to technological areasmore generously, the same physicist might want to be thought of as working inmaterials.

All three of the areas are overlapping. In any case, when one is discussingintroductory material, there seems to be little reason to split hairs, however fluidsare not normally part of our considerations, although we added a short appendix onthem.

In recent years, two very instructive books have appeared in this area.

1. Marvin L. Cohen and Steven G. Louie, Fundamentals of Condensed MatterPhysics, Cambridge University Press, Cambridge, UK, 2016. This book is at thegraduate level.

2. Steven H. Simon, The Oxford Solid State Basics, Oxford University Press,Oxford, UK, 2013. This book is at a modern undergraduate level.

The principle changes to this book from early editions are:

1. An (idiosyncratic) set of very brief mini-biographies of men and women whohave made a major mark in solid-state physics. The mini-biographies aregathered from a variety of references both on and off the Internet. Every efforthas been made for their accuracy we hope with success. We found the obituariesin Physics Today as particularly helpful sources. We would also like to feel thelist is representative if not complete. (Note: Whenever the pronoun “I” is used inthe mini-biographies, it refers to the first author of this book—JDP)

2. Several other brief discussions of mostly modern work presented in a condensedand often qualitative way. These include:Batteries, BEC-to-BCS evolution, BJT and JFET, Bose–Einstein Condensation,Polymers, Density Functional Theory, Dirac Fermions, Drude Model, EmergentProperties, Excitonic Condensates, Five Kinds of Insulators, Fluid Dynamics,Graphene, Heavy Fermions, High Tc Superconductor, Hubbard and t-J Models,Invisibility Cloaks, Iron Pnictide Superconductors, Light-Emitting Diodes,Majorana Fermions, Moore’s Law, N-V Centers, Nanomagnetism, NanometerStructures, Negative Index of Refraction, (Carbon) Onions, Optical Lattices,Phononics, Photonics, Plasmonics, Quantum Computing, QuantumEntanglement, Quantum Information, Quantum Phase Transitions, QuantumSpin Liquids, Semimetals, Skyrmions, Solar Cells, Spin Hall Effect, Spintronics,Strong Correlations, Time Crystals, Topological Insulators, Topological Phases,Weyl Fermions.

3. A discussion of the recent Nobel Prize-winning work (and related matters) inTopological Phases and Topological Insulators.

4. A different set of solved problems.5. Some additional material on magnetism.

vi Preface

In addition to the acknowledgements in the prefaces of previous editions, wewould like to thank Prof. Marvin Cohen of the University of California/Berkeley,for suggesting some names of female physicists to include in our mini-biographies,and we continue to appreciate the aid of Dr. Claus Ascheron and the Staff ofSpringer.

Rapid City, South Dakota J. D. PattersonCape Canaveral, Florida B. C. BaileyJune 2017

Preface vii

Preface to the Second Edition

It is one thing to read science. It is another and far more important activity to do it.Ideally, this means doing research. Before that is practical however, we must “getup to speed.” This usually involves attending lectures, doing laboratory experi-ments, reading the material, and working problems. Without solving problems, thematerial in a physics course usually does not sink in and we make little progress.Solving problems can also, depending on the problems, mimic the activity ofresearch. It has been our experience that you never really get anywhere in physicsunless you solve problems on paper and in the lab.

The problems in our book cover a wide range of difficulty. Some involve fillingin only a few steps or doing a simple calculation. Others are more involved, and afew are essentially open-ended. Thus, the major change in this second edition is theinclusion of a selection of solutions in an appendix to show you what we expectedyou to get out of the problems. All problems should help you to think more aboutthe material. Solutions not found in the text are available to instructors throughSpringer.

In addition, certain corrections to the text have been made. Also very briefintroductions have been added to several modern topics such as plasmonics,photonics, phononics, graphene, negative index of refraction, nanomagnetism,quantum computing, Bose–Einstein condensation, optical lattices.

We have also added some other materials in an expanded set of appendices.First, we have included a brief summary of solid-state physics as garnered from thebody of the text. This summary should, if needed, help you get focused on asolution. We have also included another kind of summary we call “folk theorems.”We have used these to help remember the essence of the physics without themathematics. A list of handy mathematical results has also been added.

As a reminder that physics is an ongoing process, in an appendix we have listedthose Nobel Prizes in physics and chemistry that relate to condensed matter physics.

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In addition to those people we thanked in the preface to the first edition, wewould like to thank again Dr. Claus Ascheron and the Staff at Springer for addi-tional suggestions to improve the usability of this second edition.

Boa Viagem, as they say in Brazil!

Rapid City, South Dakota J. D. PattersonCape Canaveral, Florida B. C. BaileyJuly 2010

x Preface to the Second Edition

Preface to the First Edition

Learning solid-state physics requires a certain degree of maturity, since it involvestying together diverse concepts from many areas of physics. The objective is tounderstand, in a basic way, how solid materials behave. To do this, one needs botha good physical and mathematical background. One definition of solid-state physicsis that it is the study of the physical (e.g., the electrical, dielectric, magnetic, elastic,and thermal) properties of solids in terms of basic physical laws. In one sense,solid-state physics is more like chemistry than some other branches of physicsbecause it focuses on common properties of large classes of materials. It is typicalthat solid-state physics emphasizes how physical properties link to the electronicstructure. In this book, we will emphasize crystalline solids (which are periodic 3Darrays of atoms).

We have retained the term solid-state physics, even though condensed matterphysics is more commonly used. Condensed matter physics includes liquids andnon-crystalline solids such as glass, about which we have little to say. We have alsoincluded only a little material concerning soft condensed matter (which includespolymers, membranes, and liquid crystals—it also includes wood and gelatins).

Modern solid-state physics came of age in the late 1930s and early 1940s (seeSeitz [82]) and had its most extensive expansion with the development of thetransistor, integrated circuits, and microelectronics. Most of microelectronics,however, is limited to the properties of inhomogeneously doped semiconductors.Solid-state physics includes many other areas of course; among the largest of theseare ferromagnetic materials and superconductors. Just a little less than half of allworking physicists are engaged in condensed matter work, including solid state.

One earlier version of this book was first published 30 years ago (J. D. Patterson,Introduction to the Theory of Solid State Physics, Addison-Wesley PublishingCompany, Reading, Massachusetts, 1971, copyright reassigned to JDP 13December, 1977), and bringing out a new modernized and expanded version hasbeen a prodigious task. Sticking to the original idea of presenting basics has meantthat the early parts are relatively unchanged (although they contain new andreworked material), dealing as they do with structure (Chap. 1), phonons (2),electrons (3), and interactions (4). Of course, the scope of solid-state physics has

xi

greatly expanded during the past 30 years. Consequently, separate chapters are nowdevoted to metals and the Fermi surface (5), semiconductors (6), magnetism (7,expanded and reorganized), superconductors (8), dielectrics and ferroelectrics (9),optical properties (10), defects (11), and a final chapter (12) that includes surfacesand brief mention of modern topics (nanostructures, the quantum Hall effect, carbonnanotubes, amorphous materials, and soft condensed matter). The reference list hasbeen brought up to date, and several relevant topics are further discussed in theappendices. The table of contents can be consulted for a full list of what is nowincluded.

The fact that one of us (JDP) has taught solid-state physics over the courseof these 30 years has helped define the scope of this book, which is intended as atextbook. Like golf, teaching is a humbling experience. One finds not only that thestudents do not understand as much as one hopes, but one constantly discoverslimits to his own understanding. We hope this book will help students to begin alifelong learning experience, for only in that way they can gain a deep under-standing of solid-state physics.

Discoveries continue in solid-state physics. Some of the more obvious onesduring the last 30 years are: quasicrystals, the quantum Hall effect (both integer andfractional—where one must finally confront new aspects of electron–electroninteractions), high-temperature superconductivity, and heavy fermions. We haveincluded these, at least to some extent, as well as several others. New experimentaltechniques, such as scanning probe microscopy, LEED, and EXAFS, among othershave revolutionized the study of solids. Since this is an introductory book onsolid-state theory, we have only included brief summaries of these techniques. Newways of growing crystals and new “designer” materials on the nanophysics scale(superlattices, quantum dots, etc.) have also kept solid-state physics vibrant, and wehave introduced these topics. There have also been numerous areas in whichapplications have played a driving role. These include semiconductor technology,spin-polarized tunneling, and giant magnetoresistance (GMR). We have at leastbriefly discussed these as well as other topics.

Greatly increased computing power has allowed many ab initio methods ofcalculations to become practical. Most of these require specialized discussionsbeyond the scope of this book. However, we continue to discuss pseudopotentialsand have added a section on density functional techniques.

Problems are given at the end of each chapter (many new problems have beenadded). Occasionally, they are quite long and have different approximate solutions.This may be frustrating, but it appears to be necessary to work problems insolid-state physics in order to gain a physical feeling for the subject. In this respect,solid-state physics is no different from many other branches of physics.

We should discuss what level of students for which this book is intended. Onecould perhaps more appropriately ask what degree of maturity of the students isassumed? Obviously, some introduction to quantum mechanics, solid-state physics,thermodynamics, statistical mechanics, mathematical physics, as well as basicmechanics and electrodynamics is necessary. In our experience, this is most

xii Preface to the First Edition

commonly encountered in graduate students, although certain mature undergradu-ates will be able to handle much of the material in this book.

Although it is well to briefly mention a wide variety of topics, so that studentswill not be “blind sided” later, and we have done this in places, in general it is betterto understand one topic relatively completely than to scan over several. We cautionprofessors to be realistic as to what their students can really grasp. If the studentshave a good start, they have their whole careers to fill in the details.

The method of presentation of the topics draws heavily on many other solid-statebooks listed in the bibliography. Acknowledgment due the authors of these books ismade here. The selection of topics was also influenced by discussion with col-leagues and former teachers, some of whom are mentioned later.

We think that solid-state physics abundantly proves that more is different, as hasbeen attributed to P. W. Anderson. There really are emergent properties at higherlevels of complexity. Seeking them, including applications, is what keepssolid-state physics alive.

In this day and age, no one book can hope to cover all of solid-state physics. Wewould like to particularly single out the following books for reference and or furtherstudy. Terms in brackets refer to references listed in the Bibliography.

1. Kittel—7th edition—remains unsurpassed for what it does [23, 1996]. AlsoKittel’s book on advanced solid-state physics [60, 1963] is very good.

2. Ashcroft and Mermin, Solid State Physics—has some of the best explanations ofmany topics I have found anywhere [21, 1976].

3. Jones and March—a comprehensive two-volume work [22, 1973].4. J. M. Ziman—many extremely clear physical explanation [25, 1972], see also

Ziman’s classic Electrons and Phonons [99, 1960].5. O. Madelung, Introduction to Solid-State Theory—Complete with a very

transparent and physical presentation [4.25].6. M. P. Marder, Condensed Matter Physics—A modern presentation, including

modern density functional methods with references [3.29].7. P. Phillips, Advanced Solid State Physics—A modern Frontiers in Physics book,

bearing the imprimatur of David Pines [A.20].8. Dalven—a good start on applied solid-state physics [32, 1990].9. Also Oxford University Press has recently put out a “Master Series in

Condensed Matter Physics.” There are six books which we recommend.

a) Martin T. Dove, Structure and Dynamics—An atomic view of Materials[2.14].

b) John Singleton, Band Theory and Electronic Properties of Solids [3.46].c) Mark Fox, Optical Properties of Solids [10.12].d) Stephen Blundell, Magnetism in Condensed Matter [7.9].e) James F. Annett, Superconductivity, Superfluids, and Condensates [8.3].f) Richard A. L. Jones, Soft Condensed Matter [12.30].

Preface to the First Edition xiii

A word about notation is in order. We have mostly used SI units (althoughGaussian is occasionally used when convenient); thus E is the electric field, D is theelectric displacement vector, P is the polarization vector,H is the magnetic field, B isthe magnetic induction, and M is the magnetization. Note that the above quantitiesare in boldface. The boldface notation is used to indicate a vector. The magnitude ofa vector V is denoted by V. In the SI system, l is the permeability (l also representsother quantities). l0 is the permeability of free space, e is the permittivity, and e0 isthe permittivity of free space. In this notation, l0 should not be confused with lB,which is the Bohr magneton ½¼ ej j�h=2m, where e = magnitude of electronic charge(i.e., e means +|e| unless otherwise noted), �h = Planck’s constant divided by 2p, andm = electronic mass]. We generally prefer to write

RAd3r or

RAdr instead ofR

A dx dy dz , but they all mean the same thing. Both hijHjji and i Hj jjð Þ are used forthe matrix elements of an operator H. Both mean

Rw�Hwds where the integral over

s means to integrate over whatever space is appropriate (e.g., it could mean anintegral over real space and a sum over spin space). By

Pa summation is indicated

and byQ

a product. The Kronecker delta dij is 1 when i = j and zero when i 6¼ j. Wehave not used covariant and contravariant spaces; thus, dij and d

ji , for example, mean

the same thing. We have labeled sections by A for advanced, B for basic, and EE formaterial that might be especially interesting for electrical engineers, and similarlyMS for materials science, and MET for metallurgy. Also by [number], we refer to areference at the end of the book.

There are too many colleagues to thank, to include a complete list. JDP wishes tospecifically thank several. A beautifully prepared solid-state course by ProfessorW. R Wright at the University of Kansas gave him his first exposure to a logicalpresentation of solid-state physics, while also at Kansas, Dr. R. J. Friauf was veryhelpful in introducing JDP to the solid-state. Discussions with Dr. R. D. Redin,Dr. R. G. Morris, Dr. D. C. Hopkins, Dr. J. Weyland, Dr. R. C. Weger, and otherswho were at the South Dakota School of Mines and Technology were alwaysuseful. Sabbaticals were spent at Notre Dame and the University of Nebraska,where working with Dr. G. L. Jones (Notre Dame) and D. J. Sellmyer (Nebraska)deepened JDP’s understanding. At the Florida Institute of Technology,Drs. J. Burns, and J. Mantovani have read parts of this book, and discussions withDr. R. Raffaelle and Dr. J. Blatt were useful. Over the course of JDP’s career, avariety of summer jobs were held that bore on solid-state physics; these includedpositions at Hughes Semiconductor Laboratory, North American Science Center,Argonne National Laboratory, Ames Laboratory of Iowa State University,the Federal University of Pernambuco in Recife, Brazil, Sandia NationalLaboratory, and the Marshal Space Flight Center. Dr. P. Richards of Sandia andDr. S. L. Lehoczky of Marshall were particularly helpful to JDP. Brief, but verypithy conversations of JDP with Dr. M. L. Cohen of the University of California,Berkeley, over the years, have also been uncommonly useful.

xiv Preface to the First Edition

Dr. B. C. Bailey would like particularly to thank Drs. J. Burns and J. Blatt for themany years of academic preparation, mentorship, and care they provided at FloridaInstitute of Technology. Special thanks to Dr. J. D. Patterson who, while PhysicsDepartment Head at Florida Institute of Technology, made a conscious decision totake on a coauthor for this extraordinary project.

All mistakes, misconceptions, and failures to communicate ideas are our own.No doubt some sign errors, misprints, incorrect shading of meanings, and perhapsmore serious errors have crept in, but hopefully their frequency decreases with theirgravity.

Most of the figures, for the first version of this book, were prepared in prelim-inary form by Mr. R. F. Thomas. However, for this book, the figures are either newor reworked by the coauthor (BCB).

We gratefully acknowledge the cooperation and kind support of Dr. C. Ascheron,Ms. E. Sauer, and Ms. A. Duhm of Springer. Finally, and most importantly, JDPwould like to note that without the constant encouragement and patience of his wifeMarluce, this book would never have been completed.

Rapid City, South Dakota J. D. PattersonCape Canaveral, Florida B. C. BaileyOctober 2005

Preface to the First Edition xv

Contents

1 Crystal Binding and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Classification of Solids by Binding Forces (B) . . . . . . . . . . . . 3

1.1.1 Molecular Crystals and the van der WaalsForces (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Ionic Crystals and Born–Mayer Theory (B) . . . . . . . 71.1.3 Metals and Wigner–Seitz Theory (B) . . . . . . . . . . . . 111.1.4 Valence Crystals and Heitler–London Theory (B) . . . 121.1.5 Comment on Hydrogen-Bonded Crystals (B) . . . . . . 13

1.2 Group Theory and Crystallography . . . . . . . . . . . . . . . . . . . . 141.2.1 Definition and Simple Properties of Groups (AB) . . . 151.2.2 Examples of Solid-State Symmetry

Properties (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 Theorem: No Five-Fold Symmetry (B) . . . . . . . . . . . 231.2.4 Some Crystal Structure Terms and Nonderived

Facts (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.5 List of Crystal Systems and Bravais Lattices (B) . . . 271.2.6 Schoenflies and International Notation for Point

Groups (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.7 Some Typical Crystal Structures (B) . . . . . . . . . . . . 321.2.8 Miller Indices (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 341.2.9 Bragg and von Laue Diffraction (AB) . . . . . . . . . . . 34

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2 Lattice Vibrations and Thermal Properties . . . . . . . . . . . . . . . . . . . 472.1 The Born–Oppenheimer Approximation (A) . . . . . . . . . . . . . . 482.2 One-Dimensional Lattices (B) . . . . . . . . . . . . . . . . . . . . . . . . 57

2.2.1 Classical Two-Atom Lattice with PeriodicBoundary Conditions (B) . . . . . . . . . . . . . . . . . . . . 58

2.2.2 Classical, Large, Perfect Monatomic Lattice,and Introduction to Brillouin Zones (B) . . . . . . . . . . 61

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2.2.3 Specific Heat of Linear Lattice (B) . . . . . . . . . . . . . 722.2.4 Classical Diatomic Lattices: Optic and Acoustic

Modes (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.2.5 Classical Lattice with Defects (B) . . . . . . . . . . . . . . 812.2.6 Quantum-Mechanical Linear Lattice (B) . . . . . . . . . . 87

2.3 Three-Dimensional Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 962.3.1 Direct and Reciprocal Lattices and Pertinent

Relations (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.3.2 Quantum-Mechanical Treatment and Classical

Calculation of the Dispersion Relation (B) . . . . . . . . 982.3.3 The Debye Theory of Specific Heat (B) . . . . . . . . . . 1052.3.4 Anharmonic Terms in the Potential/The

Gruneisen Parameter (A) . . . . . . . . . . . . . . . . . . . . . 1122.3.5 Wave Propagation in an Elastic Crystalline

Continuum (MET, MS) . . . . . . . . . . . . . . . . . . . . . . 116Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3 Electrons in Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.1 Reduction to One-Electron Problem . . . . . . . . . . . . . . . . . . . . 129

3.1.1 The Variational Principle (B) . . . . . . . . . . . . . . . . . . 1293.1.2 The Hartree Approximation (B) . . . . . . . . . . . . . . . . 1313.1.3 The Hartree–Fock Approximation (A) . . . . . . . . . . . 1353.1.4 Coulomb Correlations and the Many-Electron

Problem (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.1.5 Density Functional Approximation (A) . . . . . . . . . . . 155

3.2 One-Electron Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.2.1 The Kronig–Penney Model (B) . . . . . . . . . . . . . . . . 1683.2.2 The Free-Electron or Quasifree-Electron

Approximation (B) . . . . . . . . . . . . . . . . . . . . . . . . . 1783.2.3 The Problem of One Electron in a Three-

Dimensional Periodic Potential . . . . . . . . . . . . . . . . 1963.2.4 Effect of Lattice Defects on Electronic States

in Crystals (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4 The Interaction of Electrons and Lattice Vibrations . . . . . . . . . . . . 2394.1 Particles and Interactions of Solid-State Physics (B) . . . . . . . . 2394.2 The Phonon–Phonon Interaction (B) . . . . . . . . . . . . . . . . . . . . 246

4.2.1 Anharmonic Terms in the Hamiltonian (B) . . . . . . . . 2464.2.2 Normal and Umklapp Processes (B) . . . . . . . . . . . . . 2484.2.3 Comment on Thermal Conductivity (B) . . . . . . . . . . 2504.2.4 Phononics (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

xviii Contents

4.3 The Electron–Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . 2534.3.1 Form of the Hamiltonian (B) . . . . . . . . . . . . . . . . . . 2534.3.2 Rigid-Ion Approximation (B) . . . . . . . . . . . . . . . . . 2584.3.3 The Polaron as a Prototype Quasiparticle (A) . . . . . . 261

4.4 Brief Comments on Electron–Electron Interactions (B) . . . . . . 2724.5 The Boltzmann Equation and Electrical Conductivity . . . . . . . 276

4.5.1 Derivation of the Boltzmann DifferentialEquation (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

4.5.2 Motivation for Solving the Boltzmann DifferentialEquation (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

4.5.3 Scattering Processes and Q Details (B) . . . . . . . . . . 2794.5.4 The Relaxation-Time Approximate Solution

of the Boltzmann Equation for Metals (B) . . . . . . . . 2844.6 Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

4.6.1 The Electrical Conductivity (B) . . . . . . . . . . . . . . . . 2874.6.2 The Peltier Coefficient (B) . . . . . . . . . . . . . . . . . . . . 2874.6.3 The Thermal Conductivity (B) . . . . . . . . . . . . . . . . . 2874.6.4 The Thermoelectric Power (B) . . . . . . . . . . . . . . . . . 2884.6.5 Kelvin’s Theorem (B) . . . . . . . . . . . . . . . . . . . . . . . 2894.6.6 Transport and Material Properties in Composites

(MET, MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

5 Metals, Alloys, and the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . 3015.1 Fermi Surface (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

5.1.1 Empty Lattice (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 3045.1.2 Exercises (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

5.2 The Fermi Surface in Real Metals (B) . . . . . . . . . . . . . . . . . . 3095.2.1 The Alkali Metals (B) . . . . . . . . . . . . . . . . . . . . . . . 3095.2.2 Hydrogen Metal (B) . . . . . . . . . . . . . . . . . . . . . . . . 3095.2.3 The Alkaline Earth Metals (B) . . . . . . . . . . . . . . . . . 3105.2.4 The Noble Metals (B) . . . . . . . . . . . . . . . . . . . . . . . 310

5.3 Experiments Related to the Fermi Surface (B) . . . . . . . . . . . . 3125.4 The de Haas–van Alphen Effect (B) . . . . . . . . . . . . . . . . . . . . 3125.5 Eutectics (MS, ME) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3165.6 Peierls Instability of Linear Metals (B) . . . . . . . . . . . . . . . . . . 317

5.6.1 Relation to Charge Density Waves (A) . . . . . . . . . . 3215.6.2 Spin Density Waves (A) . . . . . . . . . . . . . . . . . . . . . 322

5.7 Heavy Fermion Systems (A) . . . . . . . . . . . . . . . . . . . . . . . . . 3225.8 Electromigration (EE, MS) . . . . . . . . . . . . . . . . . . . . . . . . . . 323

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5.9 White Dwarfs and Chandrasekhar’s Limit (A) . . . . . . . . . . . . 3255.9.1 Gravitational Self-Energy (A) . . . . . . . . . . . . . . . . . 3265.9.2 Idealized Model of a White Dwarf (A) . . . . . . . . . . 327

5.10 Some Famous Metals and Alloys (B, MET) . . . . . . . . . . . . . . 330Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

6 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3336.1 Electron Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

6.1.1 Calculation of Electron and HoleConcentration (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 336

6.1.2 Equation of Motion of Electrons in EnergyBands (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

6.1.3 Concept of Hole Conduction (B) . . . . . . . . . . . . . . . 3456.1.4 Conductivity and Mobility in

Semiconductors (B) . . . . . . . . . . . . . . . . . . . . . . . . 3486.1.5 Drift of Carriers in Electric and Magnetic Fields:

The Hall Effect (B) . . . . . . . . . . . . . . . . . . . . . . . . . 3506.1.6 Cyclotron Resonance (A) . . . . . . . . . . . . . . . . . . . . 352

6.2 Examples of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 3606.2.1 Models of Band Structure for Si, Ge and II-VI

and III-V Materials (A) . . . . . . . . . . . . . . . . . . . . . . 3606.2.2 Comments About GaN (A) . . . . . . . . . . . . . . . . . . . 366

6.3 Semiconductor Device Physics . . . . . . . . . . . . . . . . . . . . . . . . 3676.3.1 Crystal Growth of Semiconductors

(EE, MET, MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3676.3.2 Gunn Effect (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . 3686.3.3 pn Junctions (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . 3706.3.4 Depletion Width, Varactors and Graded

Junctions (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3746.3.5 Metal Semiconductor Junctions—the Schottky

Barrier (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3766.3.6 Semiconductor Surface States

and Passivation (EE) . . . . . . . . . . . . . . . . . . . . . . . . 3786.3.7 Surfaces Under Bias Voltage (EE) . . . . . . . . . . . . . . 3806.3.8 Inhomogeneous Semiconductors not

in Equilibrium (EE) . . . . . . . . . . . . . . . . . . . . . . . . 3806.3.9 Solar Cells (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3886.3.10 Batteries (B, EE, MS) . . . . . . . . . . . . . . . . . . . . . . . 3946.3.11 Transistors (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3966.3.12 Charge-Coupled Devices (CCD) (EE) . . . . . . . . . . . 402

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

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7 Magnetism, Magnons, and Magnetic Resonance . . . . . . . . . . . . . . . 4057.1 Types of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

7.1.1 Diamagnetism of the Core Electrons (B) . . . . . . . . . 4067.1.2 Paramagnetism of Valence Electrons (B) . . . . . . . . . 4077.1.3 Ordered Magnetic Systems (B) . . . . . . . . . . . . . . . . 413

7.2 Origin and Consequences of Magnetic Order . . . . . . . . . . . . . 4277.2.1 Heisenberg Hamiltonian . . . . . . . . . . . . . . . . . . . . . 4277.2.2 Magnetic Anisotropy and Magnetostatic

Interactions (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4477.2.3 Spin Waves and Magnons (B) . . . . . . . . . . . . . . . . . 4527.2.4 Band Ferromagnetism (B) . . . . . . . . . . . . . . . . . . . . 4717.2.5 Magnetic Phase Transitions (A) . . . . . . . . . . . . . . . . 482

7.3 Magnetic Domains and Magnetic Materials (B) . . . . . . . . . . . 4977.3.1 Origin of Domains and General Comments (B) . . . . 4977.3.2 Magnetic Materials (EE, MS) . . . . . . . . . . . . . . . . . 5077.3.3 Nanomagnetism (EE, MS) . . . . . . . . . . . . . . . . . . . . 510

7.4 Magnetic Resonance and Crystal Field Theory . . . . . . . . . . . . 5117.4.1 Simple Ideas About Magnetic Resonance (B) . . . . . . 5117.4.2 A Classical Picture of Resonance (B) . . . . . . . . . . . . 5127.4.3 The Bloch Equations and Magnetic

Resonance (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5167.4.4 Crystal Field Theory and Related Topics (B) . . . . . . 530

7.5 Brief Mention of Other Topics . . . . . . . . . . . . . . . . . . . . . . . . 5437.5.1 Spintronics or Magnetoelectronics (EE) . . . . . . . . . . 5437.5.2 The Kondo Effect (A) . . . . . . . . . . . . . . . . . . . . . . . 5477.5.3 Spin Glass (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5497.5.4 Quantum Spin Liquids—A New State

of Matter (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.5.5 Solitons (A, EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

8 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5558.1 Introduction and Some Experiments (B) . . . . . . . . . . . . . . . . . 555

8.1.1 Ultrasonic Attenuation (B) . . . . . . . . . . . . . . . . . . . . 5598.1.2 Electron Tunneling (B) . . . . . . . . . . . . . . . . . . . . . . 5608.1.3 Infrared Absorption (B) . . . . . . . . . . . . . . . . . . . . . . 5608.1.4 Flux Quantization (B) . . . . . . . . . . . . . . . . . . . . . . . 5608.1.5 Nuclear Spin Relaxation (B) . . . . . . . . . . . . . . . . . . 5608.1.6 Thermal Conductivity (B) . . . . . . . . . . . . . . . . . . . . 561

8.2 The London and Ginzburg–Landau Equations (B) . . . . . . . . . . 5618.2.1 The Coherence Length (B) . . . . . . . . . . . . . . . . . . . 5648.2.2 Flux Quantization and Fluxoids (B) . . . . . . . . . . . . . 5688.2.3 Order of Magnitude for Coherence Length (B) . . . . . 570

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8.3 Tunneling (B, EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3.1 Single-Particle or Giaever Tunneling . . . . . . . . . . . . 5718.3.2 Josephson Junction Tunneling . . . . . . . . . . . . . . . . . 573

8.4 SQUID: Superconducting Quantum Interference (EE) . . . . . . . 5788.4.1 Questions and Answers (B) . . . . . . . . . . . . . . . . . . . 581

8.5 The Theory of Superconductivity (A) . . . . . . . . . . . . . . . . . . . 5818.5.1 Assumed Second Quantized Hamiltonian for

Electrons and Phonons in Interaction (A) . . . . . . . . . 5818.5.2 Elimination of Phonon Variables and Separation

of Electron–Electron Attraction Term Due toVirtual Exchange of Phonons (A) . . . . . . . . . . . . . . 585

8.5.3 Cooper Pairs and the BCS Hamiltonian (A) . . . . . . . 5888.5.4 Remarks on the Nambu Formalism and Strong

Coupling Superconductivity (A) . . . . . . . . . . . . . . . 6018.6 Magnesium Diboride (EE, MS, MET) . . . . . . . . . . . . . . . . . . 6038.7 Heavy-Electron Superconductors (EE, MS, MET) . . . . . . . . . . 6038.8 High-Temperature Superconductors (EE, MS, MET) . . . . . . . . 6038.9 Summary Comments on Superconductivity (B) . . . . . . . . . . . . 607Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

9 Dielectrics and Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6139.1 The Four Types of Dielectric Behavior (B) . . . . . . . . . . . . . . . 6139.2 Electronic Polarization and the Dielectric Constant (B) . . . . . . 6159.3 Ferroelectric Crystals (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

9.3.1 Thermodynamics of Ferroelectricity by LandauTheory (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

9.3.2 Further Comment on the Ferroelectric Transition(B, ME) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

9.3.3 One-Dimensional Model of the Soft Modelof Ferroelectric Transitions (A) . . . . . . . . . . . . . . . . 627

9.3.4 Multiferroics (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6309.4 Dielectric Screening and Plasma Oscillations (B) . . . . . . . . . . 631

9.4.1 Helicons (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6339.4.2 Alfvén Waves (EE) . . . . . . . . . . . . . . . . . . . . . . . . . 6359.4.3 Plasmonics (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

9.5 Free-Electron Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6379.5.1 Introduction (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6379.5.2 The Thomas–Fermi and Debye–Huckel

Methods (A, EE) . . . . . . . . . . . . . . . . . . . . . . . . . . 6379.5.3 The Lindhard Theory of Screening (A) . . . . . . . . . . 641

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

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10 Optical Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64910.1 Introduction (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64910.2 Macroscopic Properties (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 650

10.2.1 Kronig–Kramers Relations (A) . . . . . . . . . . . . . . . . 65410.3 Absorption of Electromagnetic Radiation—General (B) . . . . . . 65710.4 Direct and Indirect Absorption Coefficients (B) . . . . . . . . . . . . 65810.5 Oscillator Strengths and Sum Rules (A) . . . . . . . . . . . . . . . . . 66610.6 Critical Points and Joint Density of States (A) . . . . . . . . . . . . 66710.7 Exciton Absorption (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66810.8 Imperfections (B, MS, MET) . . . . . . . . . . . . . . . . . . . . . . . . . 67010.9 Optical Properties of Metals (B, EE, MS) . . . . . . . . . . . . . . . . 67010.10 Lattice Absorption, Restrahlen, and Polaritons (B) . . . . . . . . . 677

10.10.1 General Results (A) . . . . . . . . . . . . . . . . . . . . . . . . 67710.10.2 Summary of the Properties of ɛ(q, x) (B) . . . . . . . . . 68510.10.3 Summary of Absorption Processes: General

Equations (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.11 Optical Emission, Optical Scattering

and Photoemission (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.11.1 Emission (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68610.11.2 Einstein A and B Coefficients (B, EE, MS) . . . . . . . . 68810.11.3 Raman and Brillouin Scattering (B, MS) . . . . . . . . . 69310.11.4 Optical Lattices (A, B) . . . . . . . . . . . . . . . . . . . . . . 69510.11.5 Photonics (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69610.11.6 Negative Index of Refraction (EE) . . . . . . . . . . . . . . 69710.11.7 Metamaterials and Invisibility Cloaks

(A, EE, MS, MET) . . . . . . . . . . . . . . . . . . . . . . . . . 69910.12 Magneto-Optic Effects: The Faraday Effect (B, EE, MS) . . . . . 700Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

11 Defects in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70511.1 Summary About Important Defects (B) . . . . . . . . . . . . . . . . . 70511.2 Shallow and Deep Impurity Levels in

Semiconductors (EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70811.3 Effective Mass Theory, Shallow Defects,

and Superlattices (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70911.3.1 Envelope Functions (A) . . . . . . . . . . . . . . . . . . . . . 70911.3.2 First Approximation (A) . . . . . . . . . . . . . . . . . . . . . 71011.3.3 Second Approximation (A) . . . . . . . . . . . . . . . . . . . 711

11.4 Color Centers (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71411.5 Diffusion (MET, MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71711.6 Edge and Screw Dislocation (MET, MS) . . . . . . . . . . . . . . . . 71711.7 Thermionic Emission (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

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11.8 Cold-Field Emission (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72311.9 Microgravity (MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

12 Current Topics in Solid Condensed–Matter Physics . . . . . . . . . . . . 72912.1 Surface Reconstruction (MET, MS) . . . . . . . . . . . . . . . . . . . . 73012.2 Some Surface Characterization Techniques

(MET, MS, EE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73112.3 Molecular Beam Epitaxy (MET, MS) . . . . . . . . . . . . . . . . . . . 73312.4 Heterostructures and Quantum Wells . . . . . . . . . . . . . . . . . . . 73512.5 Quantum Structures and Single-Electron Devices (EE) . . . . . . 735

12.5.1 Coulomb Blockade (EE) . . . . . . . . . . . . . . . . . . . . . 73612.5.2 Tunneling and the Landauer Equation (EE) . . . . . . . 739

12.6 Superlattices, Bloch Oscillators, Stark–Wannier Ladders . . . . . 74212.6.1 Applications of Superlattices and Related

Nanostructures (EE) . . . . . . . . . . . . . . . . . . . . . . . . 74412.7 Classical and Quantum Hall Effect (A) . . . . . . . . . . . . . . . . . . 747

12.7.1 Classical Hall Effect—CHE (A) . . . . . . . . . . . . . . . . 74712.7.2 The Quantum Mechanics of Electrons in a

Magnetic Field: The Landau Gauge (A) . . . . . . . . . . 75012.7.3 Quantum Hall Effect: General Comments (A) . . . . . . 75212.7.4 Majorana Fermions and Topological Insulators

(Introduction) (A) . . . . . . . . . . . . . . . . . . . . . . . . . . 75712.7.5 Topological Insulators (A, MS) . . . . . . . . . . . . . . . . 75912.7.6 Phases of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 77612.7.7 Topological Phases and Topological Insulators

(A, MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77612.7.8 Quantum Computing (A, EE) . . . . . . . . . . . . . . . . . 77612.7.9 Five Kinds of Insulators (A) . . . . . . . . . . . . . . . . . . 78312.7.10 Semimetals (A, B, EE, MS) . . . . . . . . . . . . . . . . . . 784

12.8 Carbon—Nanotubes and Fullerene Nanotechnology (EE) . . . . 78412.9 Graphene and Silly Putty (A, EE, MS) . . . . . . . . . . . . . . . . . . 78812.10 Novel Newer Transistors (EE) . . . . . . . . . . . . . . . . . . . . . . . . 78812.11 Amorphous Semiconductors and the Mobility Edge (EE) . . . . 789

12.11.1 Hopping Conductivity (EE) . . . . . . . . . . . . . . . . . . . 79012.11.2 Anderson and Mott Localization and Related

Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79112.12 Amorphous Magnets (MET, MS) . . . . . . . . . . . . . . . . . . . . . . 79212.13 Anticrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79312.14 Magnetic Skyrmions (A, EE) . . . . . . . . . . . . . . . . . . . . . . . . . 793

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12.15 Soft Condensed Matter (MET, MS) . . . . . . . . . . . . . . . . . . . . 79412.15.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . 79412.15.2 Liquid Crystals (MET, MS) . . . . . . . . . . . . . . . . . . . 79512.15.3 Polymers and Rubbers (MET, MS) . . . . . . . . . . . . . 796

12.16 Bose–Einstein Condensation (A) . . . . . . . . . . . . . . . . . . . . . . 79912.16.1 Bose–Einstein Condensation for an Ideal Bose

Gas (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80112.16.2 Excitonic Condensates (A) . . . . . . . . . . . . . . . . . . . 803

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915

Index of Mini-Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

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