Solid State Physics: Principles and Modern Applications

537 Pages·2009·16.74 MB·English

by  John J. Quinn

Solid State Physics
·
John J. Quinn Kyung-Soo Yi
Solid State Physics
Principles and Modern Applications
123
Prof.JohnJ.Quinn Prof.Kyung-SooYi
UniversityofTennessee PusanNationalUniversity
Dept.Physics Dept.Physics
KnoxvilleTN37996 30Jangjeon-Dong
USA Pusan609-735
[email protected] Korea,Republicof(SouthKorea)
[email protected]
ISBN978-3-540-92230-8 e-ISBN978-3-540-92231-5
DOI10.1007/978-3-540-92231-5
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009929177
(cid:2)c Springer-VerlagBerlinHeidelberg2009
Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermitted onlyundertheprovisions oftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
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Printedonacid-freepaper
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The book is dedicated to Betsy Qurinn and Young-Sook Yi. 1
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Preface
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This textbook had its originin severalcou rses taught for two decades (1965– 4
1985)atBrownUniversitybyoneoftheauthors(JJQ).Theoriginalassigned 5
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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
subsequentsemestersbecauseoftheireimportanceinresearchduringthe1960s 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
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theUniversityofPennsylvaniain1961–1962,andlaterincludedinacourseat 11
Purdue University when he wasca Visiting Professor (1964–1965). A sojourn 12
into academicadministrationin1984removedJJQ fromteaching for 8years. 13
On returning to a full time teaching–researchprofessorship at the University 14
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ofTennessee,heagainoffereda1yeargraduatecourseinSolidStatePhysics. 15
The course was structured so that the first semester (roughly the first half 16
of the text) introduced arll the essential concepts for students who wanted 17
exposuretosolidstatephysics.Thefirstsemestercouldcovertopics fromthe 18
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firstnine chapters.The secondsemestercovereda selectionofmoreadvanced 19
topicsforstudentsintendingtodoresearchinthisfield.Oneoftheco-authors 20
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(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
atPusanNationalcUniversityfrom1984.Thetextisatruecollaborativeeffort 23
of the co-authors. 24
The advanced topics in the second semester are covered briefly, but thor- 25
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oughlyenoughtoconveythe basic physicsofeachtopic.Referencespointthe 26
students who want more detail in the right direction. An entirely different 27
set of advanUced 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 Introduction to Solid State Physics, 7th 31
edn. (Wiley, New York, 1995), other books that influenced the evolution of 32
this bookare: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
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damentals of Semiconductors ed. by P.Y. Yu, M. Cardona (Springer, Berlin, 38
Heidelberg, New York, 1995). 39
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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. Dor. 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
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students and colleagues who have contributed to making the lecture notes 45
better. 46
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Boththe co-authorswanttoacknowledgethe encouragementandsupport 47
of their families. The book is dedicated to them. 48

Knoxville and Pusan, d John J. Quinn 49
August 2008 Kyung-Soo Yi 50
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Contents
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Part I Basic Concepts in Solid-State Physics 53
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1 Crystal Structures ......................................... 3 54
1.1 Crystal Structure and Symmetry Groups.................. 3 55
1.2 Common Crystal Structueres............................. 10 56
1.3 Reciprocal Lattice ..................................... 15 57
1.4 Diffraction of X-Rays................................... 16 58
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1.4.1 Bragg Reflection ............................... 17 59
1.4.2 Laue Equatcions................................ 17 60
1.4.3 Ewald Construction ............................ 19 61
1.4.4 Atomic Secattering Factor ....................... 20 62
1.4.5 Geometric Structure Factor ..................... 22 63
1.4.6 Experimental Techniques ....................... 23 64
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1.5 Classification of Solids.................................. 24 65
1.5.1 Crystal Binding................................ 24 66
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1.6 Binding Energy of Ionic Crystals......................... 27 67
Problems ......o............................................. 34 68
2 Lattice Vibrations ......................................... 37 69
2.1 Monatcomic Linear Chain................................ 37 70
2.2 Normal Modes......................................... 41 71
2.3 Mo¨ssbauer Effect ...................................... 44 72
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2.4 Optical Modes......................................... 48 73
2.5 Lattice Vibrations in Three Dimensions................... 50 74
2U.5.1 Normal Modes................................. 52 75
2.5.2 Quantization .................................. 53 76
2.6 Heat Capacity of Solids................................. 54 77
2.6.1 Einstein Model ................................ 55 78
2.6.2 Modern Theory of the Specific Heat of Solids...... 57 79
2.6.3 Debye Model .................................. 59 80
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X Contents
2.6.4 Evaluation of Summations over Normal Modes 81
for the Debye Model............................ 61 82
2.6.5 Estimate of RecoilFree Fractionin Mo¨ssbauerEffect 62 83
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2.6.6 Lindemann Melting Formula .................... 63 84
2.6.7 Critical Points in the Phonon Spectrum........... 65 85
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2.7 Qualitative Description of Thermal Expansion............. 67 86
2.8 Anharmonic Effects .................................... 69 87
2.9 Thermal Conductivity of an Insulator .o................... 71 88
2.10 Phonon Collision Rate.................................. 72 89
2.11 Phonon Gas........................................... 73 90
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Problems ................................................... 75 91
3 Free Electron Theory of Metals....P........................ 79 92
3.1 Drude Model .......................................... 79 93
3.2 Electrical Conductivity ................................. 79 94
3.3 Thermal Conductivity ....... ........................... 80 95
3.4 Wiedemann–Franz Law....d............................. 82 96
3.5 Criticisms of Drude Model .............................. 82 97
3.6 Lorentz Theory........................................ 82 98
3.6.1 Boltzmann Distreibution Function ................ 83 99
3.6.2 Relaxation Time Approximation ................. 83 100
3.6.3 Solution of Btoltzmann Equation ................. 83 101
3.6.4 Maxwell–Boltzmann Distribution ................ 84 102
3.7 Sommerfeld Theorycof Metals ........................... 84 103
3.8 Review of Elementary Statistical Mechanics ............... 86 104
3.8.1 Fermi–Deirac Distribution Function ............... 87 105
3.8.2 Density of States............................... 88 106
3.8.3 Thermodynamic Potential....................... 89 107
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3.8.4 Entropy ...................................... 89 108
3.9 Fermi Functrion Integration Formula...................... 91 109
3.10 Heat Capacity of a Fermi Gas ........................... 93 110
3.11 Equationoof State of a Fermi Gas ........................ 94 111
3.12 Compressibility ........................................ 94 112
3.13 Electrical and Thermal Conductivities .................... 95 113
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3.13.1 Electrical Conductivity ......................... 97 114
3.13.2 Thermal Conductivity .......................... 97 115
3.14 Critnique of Sommerfeld Model ........................... 99 116
3.15 Magnetoconductivity ................................... 99 117
3.16 Hall Effect and Magnetoresistance .......................100 118
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3.17 Dielectric Function.....................................101 119
Problems ...................................................104 120
4 Elements of Band Theory..................................109 121
4.1 Energy Band Formation ................................109 122
4.2 Translation Operator ...................................110 123
4.3 Bloch’s Theorem.......................................111 124
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Contents XI
4.4 Calculation of Energy Bands ............................112 125
4.4.1 Tight-Binding Method..........................112 126
4.4.2 Tight Binding in Second Quantization 127
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Representation ................................115 128
4.5 Periodic Potential......................................116 129
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4.6 Free Electron Model....................................118 130
4.7 Nearly Free Electron Model .............................119 131
4.7.1 Degenerate Perturbation Theoroy .................120 132
4.8 Metals–Semimetals–Semiconductors–Insulators ............121 133
Problems ...................................................124 134
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5 Use of Elementary Group Theory 135
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in Calculating Band Structure .............................129 136
5.1 Band Representation of Empty Lattice States .............129 137
5.2 Review of Elementary Group Theory .....................129 138

5.2.1 Some Examples of Simple Groups................130 139
5.2.2 Group Representatidon ..........................131 140
5.2.3 Examples of Representations of the Group 4mm ...132 141
5.2.4 Faithful Representation.........................134 142
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5.2.5 Regular Representation.........................134 143
5.2.6 Reducible and Irreducible Representations ........134 144
5.2.7 Important Thteorems of Representation Theory 145
(without proof)................................135 146
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5.2.8 Character of a Representation ...................135 147
5.2.9 Orthogonality Theorem.........................136 148
5.3 Empty Lattice Baends, Degeneracies and IRs...............137 149
5.3.1 Group of the Wave Vector k.....................138 150
5.4 Use of Irreducirble Representations .......................140 151
5.4.1 Determining the Linear Combinations 152
of Prlane Waves Belonging to Different IRs ........142 153
5.4.2 Compatibility Relations.........................144 154
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5.5 Using the Irreducible Representations in Evaluating 155
Energy Bands .........................................146 156
5.6 Empty Lattice Bands for Cubic Structure .................148 157
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5.6.1 Point Group of a Cubic Structure ................148 158
5.6.2 Face Centered Cubic Lattice ....................150 159
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5.6.3 Body Centered Cubic Lattice....................153 160
5.7 Energy Bands of Common Semiconductors ................155 161
Problems ...................................................158 162
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6 More Band Theory and the Semiclassical Approximation ..161 163
6.1 Orthogonalized Plane Waves ............................161 164
6.2 Pseudopotential Method ................................162 165
6.3 k·p Method and Effective Mass Theory ..................165 166