Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications
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Solid State Physics
John J. Quinn · Kyung-Soo Yi
Solid State Physics
Principles and Modern Applications
123
Prof. John J. Quinn
University of Tennessee
Dept. Physics
Knoxville TN 37996
USA
jjquinn@utk.edu
Prof. Kyung-Soo Yi
Pusan National University
Dept. Physics
30 Jangjeon-Dong
Pusan 609-735
Korea, Republic of (South Korea)
ksyi@pusan.ac.kr
ISBN 978-3-540-92230-8 e-ISBN 978-3-540-92231-5
DOI 10.1007/978-3-540-92231-5
Springer Dordrecht Heidelberg London New York
c
Springer-Verlag Berlin Heidelberg 2009
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
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liable to prosecution under the German Copyright Law.
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Library of Congress Control Number: 2009929177
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The book is dedicated to Betsy Quinn and Young-Sook Yi. 1
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Preface 3
This textbook had its origin in several courses taught for two decades (1965– 4
1985) at Brown University by one of the authors (JJQ). The original assigned 5
text for the first semester course was the classic “Introduction to Solid State 6
Physics” by C. Kittel. Many topics not covered in that text were included in 7
subsequent semesters because of their importance in research during the 1960s 8
and 1970s. A number of the topics covered were first introduced in a course 9
on “Many Body Theory of Metals” given by JJQ as a Visiting Lecturer at 10
the University of Pennsylvania in 1961–1962, and later included in a course at 11
Purdue University when he was a Visiting Professor (1964–1965). A sojourn 12
into academic administration in 1984 removed JJQ from teaching for 8 years. 13
On returning to a full time teaching–research professorship at the University 14
of Tennessee, he again offered a 1 year graduate course in Solid State Physics. 15
The course was structured so that the first semester (roughly the first half 16
of the text) introduced all the essential concepts for students who wanted 17
exposure to solid state physics. The first semester could cover topics from the 18
first nine chapters. The second semester covered a selection of more advanced 19
topics for students intending to do research in this field. One of the co-authors 20
(KSY) took this course in Solid State Physics as a PhD student at Brown 21
University. He added to and improved the lecture while teaching the subject 22
at Pusan National University from 1984. The text is a true collaborative effort 23
of the co-authors. 24
The advanced topics in the second semester are covered briefly, but thor- 25
oughly enough to convey the basic physics of each topic. References point the 26
students who want more detail in the right direction. An entirely different 27
set of advanced topics could have been chosen in the place of those in the 28
text. The choice was made primarily because of the research interests of the 29
authors. 30
In addition to Kittel’s classic I ntroduction to Solid State Physics, 7th 31
edn. (Wiley, New York, 1995), other books that influenced the evolution of 32
this book are: Methods of Quantum Field Theory in Statistical Physics ed. by 33
A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinsky (Prentice-Hall, Englewood, 34
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VIII Preface
NJ, 1963); Solid State Physics ed. by N.W. Ashcroft, N.D. Mermin 35
(Saunder’s College, New York, 1975); Introduction to Solid State Theory ed. 36
by O. Madelung (Springer, Berlin, Heidelberg, New York, 1978); and Fun- 37
damentals of Semiconductors ed. by P.Y. Yu, M. Cardona (Springer, Berlin, 38
Heidelberg, New York, 1995). 39
Many graduate students at Brown, Tennessee, and Pusan have helped to 40
improve these lecture notes by pointing out sections that were difficult to 41
understand, and by catching errors in the text. Dr. Alex Tselis presented 42
the authors with his carefully written notes of the course at Brown when 43
he changed his field of study to medical science. We are grateful to all the 44
students and colleagues who have contributed to making the lecture notes 45
better. 46
Both the co-authors want to acknowledge the encouragement and support 47
of their families. The book is dedicated to them. 48
Knoxville and Pusan, John J. Quinn 49
August 2008 Kyung-Soo Yi 50
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Contents 52
Part I Basic Concepts in Solid-State Physics 53
1 Crystal Structures ......................................... 3 54
1.1 CrystalStructure and SymmetryGroups.................. 3 55
1.2 CommonCrystalStructures............................. 10 56
1.3 ReciprocalLattice ..................................... 15 57
1.4 Diffraction ofX-Rays................................... 16 58
1.4.1 BraggReflection............................... 17 59
1.4.2 LaueEquations................................ 17 60
1.4.3 EwaldConstruction ............................ 19 61
1.4.4 AtomicScatteringFactor ....................... 20 62
1.4.5 Geometric Structure Factor ..................... 22 63
1.4.6 ExperimentalTechniques ....................... 23 64
1.5 ClassificationofSolids.................................. 24 65
1.5.1 CrystalBinding................................ 24 66
1.6 BindingEnergyofIonicCrystals......................... 27 67
Problems ................................................... 34 68
2 Lattice Vibrations ......................................... 37 69
2.1 MonatomicLinearChain................................ 37 70
2.2 NormalModes......................................... 41 71
2.3 M¨ossbauerEffect ...................................... 44 72
2.4 OpticalModes......................................... 48 73
2.5 LatticeVibrations inThreeDimensions................... 50 74
2.5.1 NormalModes................................. 52 75
2.5.2 Quantization .................................. 53 76
2.6 HeatCapacityof Solids................................. 54 77
2.6.1 EinsteinModel ................................ 55 78
2.6.2 Modern Theory of the Specific Heat of Solids . . . . . . 57 79
2.6.3 DebyeModel .................................. 59 80
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2.6.4 Evaluation of Summations over Normal Modes 81
fortheDebye Model............................ 61 82
2.6.5 Estimate of Recoil Free Fraction in M¨ossbauer Effect 62 83
2.6.6 Lindemann Melting Formula .................... 63 84
2.6.7 Critical Points in the Phonon Spectrum . . . . . . . . . . . 65 85
2.7 Qualitative Description of Thermal Expansion . . . . . . . . . . . . . 67 86
2.8 AnharmonicEffects .................................... 69 87
2.9 Thermal Conductivity of an Insulator . . . . . . . . . . . . . . . . . . . . 71 88
2.10 Phonon Collision Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 89
2.11 PhononGas........................................... 73 90
Problems ................................................... 75 91
3 Free Electron Theory of Metals ............................ 79 92
3.1 DrudeModel.......................................... 79 93
3.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 94
3.3 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 95
3.4 Wiedemann–FranzLaw................................. 82 96
3.5 CriticismsofDrudeModel .............................. 82 97
3.6 LorentzTheory........................................ 82 98
3.6.1 Boltzmann Distribution Function ................ 83 99
3.6.2 Relaxation TimeApproximation ................. 83 100
3.6.3 SolutionofBoltzmannEquation ................. 83 101
3.6.4 Maxwell–BoltzmannDistribution ................ 84 102
3.7 SommerfeldTheoryofMetals ........................... 84 103
3.8 ReviewofElementaryStatistical Mechanics ............... 86 104
3.8.1 Fermi–DiracDistributionFunction ............... 87 105
3.8.2 Density ofStates............................... 88 106
3.8.3 ThermodynamicPotential....................... 89 107
3.8.4 Entropy ...................................... 89 108
3.9 FermiFunction IntegrationFormula...................... 91 109
3.10 Heat Capacityofa FermiGas ........................... 93 110
3.11 EquationofStateofaFermiGas ........................ 94 111
3.12 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 112
3.13 Electrical and Thermal Conductivities . . . . . . . . . . . . . . . . . . . . 95 113
3.13.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 97 114
3.13.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 97 115
3.14 CritiqueofSommerfeldModel........................... 99 116
3.15 Magnetoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 117
3.16 Hall EffectandMagnetoresistance .......................100 118
3.17 DielectricFunction.....................................101 119
Problems ...................................................104 120
4ElementsofBandTheory..................................109 121
4.1 EnergyBandFormation ................................109 122
4.2 TranslationOperator...................................110 123
4.3 Bloch’sTheorem.......................................111 124
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Contents XI
4.4 CalculationofEnergyBands ............................112 125
4.4.1 Tight-BindingMethod..........................112 126
4.4.2 Tight Binding in Second Quantization 127
Representation ................................115 128
4.5 PeriodicPotential......................................116 129
4.6 FreeElectronModel....................................118 130
4.7 NearlyFreeElectronModel .............................119 131
4.7.1 Degenerate PerturbationTheory.................120 132
4.8 Metals–Semimetals–Semiconductors–Insulators . . . . . . . . . . . . 121 133
Problems ...................................................124 134
5 Use of Elementary Group Theory 135
in Calculating Band Structure .............................129 136
5.1 BandRepresentationofEmptyLatticeStates .............129 137
5.2 ReviewofElementaryGroupTheory .....................129 138
5.2.1 Some Examplesof Simple Groups................130 139
5.2.2 GroupRepresentation ..........................131 140
5.2.3 Examples of Representations of the Group 4 mm . . . 132 141
5.2.4 Faithful Representation.........................134 142
5.2.5 Regular Representation.........................134 143
5.2.6 Reducible and Irreducible Representations . . . . . . . . 134 144
5.2.7 Important Theorems of Representation Theory 145
(without proof)................................135 146
5.2.8 Characterof aRepresentation ...................135 147
5.2.9 Orthogonality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 136 148
5.3 EmptyLattice Bands,Degeneraciesand IRs...............137 149
5.3.1 Group of the Wave Vector k .....................138 150
5.4 Use ofIrreducibleRepresentations .......................140 151
5.4.1 Determining the Linear Combinations 152
ofPlaneWavesBelongingto DifferentIRs ........142 153
5.4.2 Compatibility Relations . . . . . . . . . . . . . . . . . . . . . . . . . 144 154
5.5 Using the Irreducible Representations in Evaluating 155
EnergyBands .........................................146 156
5.6 EmptyLattice BandsforCubic Structure.................148 157
5.6.1 PointGroupofa CubicStructure ................148 158
5.6.2 FaceCentered CubicLattice ....................150 159
5.6.3 BodyCenteredCubicLattice....................153 160
5.7 Energy Bands of Common Semiconductors . . . . . . . . . . . . . . . . 155 161
Problems ...................................................158 162
6 More Band Theory and the Semiclassical Approximation . . 161 163
6.1 Orthogonalized Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 164
6.2 PseudopotentialMethod ................................162 165
6.3 k ·p MethodandEffectiveMassTheory ..................165 166