Electrical Conduction in Graphene and Nanotubes

308 Pages·2013·2.53 MB·English

by  Shigeji Fujita

ShigejiFujitaandAkiraSuzuki
ElectricalConductioninGraphene
andNanotubes
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ISBN:978-1-84821-200-8 ISBN:978-3-527-40386-8
Shigeji Fujita and Akira Suzuki
Electrical Conduction in Graphene
and Nanotubes
TheAuthors AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.Dr.ShigejiFujita publisherdonotwarranttheinformation
UniversityofBuffalo containedinthesebooks,includingthisbook,to
SUNY,Dept.ofPhysics befreeoferrors.Readersareadvisedtokeepin
329FronczakHall mindthatstatements,data,illustrations,
Buffalo,NY14260 proceduraldetailsorotheritemsmay
USA inadvertentlybeinaccurate.
LibraryofCongressCardNo.:
Prof.Dr.AkiraSuzuki
appliedfor
TokyoUniversityofScience
Dept.ofPhysics BritishLibraryCataloguing-in-PublicationData:
Shinjuku-ku Acataloguerecordforthisbookisavailable
162-8601Tokyo fromtheBritishLibrary.
Japan
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publicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableonthe
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V
Contents
Preface XI
PhysicalConstants,Units,MathematicalSignsandSymbols XV
1 Introduction 1
1.1 CarbonNanotubes 1
1.2 TheoreticalBackground 4
1.2.1 MetalsandConductionElectrons 4
1.2.2 QuantumMechanics 4
1.2.3 HeisenbergUncertaintyPrinciple 4
1.2.4 BosonsandFermions 5
1.2.5 FermiandBoseDistributionFunctions 5
1.2.6 CompositeParticles 6
1.2.7 QuasifreeElectronModel 6
1.2.8 “Electrons”and“Holes” 7
1.2.9 TheGateFieldEffect 7
1.3 BookLayout 8
1.4 SuggestionsforReaders 9
1.4.1 SecondQuantization 9
1.4.2 SemiclassicalTheoryofElectronDynamics 9
1.4.3 FermiSurface 9
References 10
2 KineticTheoryandtheBoltzmannEquation 11
2.1 DiffusionandThermalConduction 11
2.2 CollisionRate:MeanFreePath 12
2.3 ElectricalConductivityandMatthiessen’sRule 15
2.4 TheHallEffect:“Electrons”and“Holes” 17
2.5 TheBoltzmannEquation 19
2.6 TheCurrentRelaxationRate 21
References 25
3 BlochElectronDynamics 27
3.1 BlochTheoreminOneDimension 27
3.2 TheKronig–PenneyModel 30
VI Contents
3.3 BlochTheoreminThreeDimensions 33
3.4 FermiLiquidModel 36
3.5 TheFermiSurface 37
3.6 HeatCapacityandDensityofStates 40
3.7 TheDensityofStateintheMomentumSpace 42
3.8 EquationsofMotionforaBlochElectron 46
References 51
4 PhononsandElectron–PhononInteraction 53
4.1 PhononsandLatticeDynamics 53
4.2 VanHoveSingularities 57
4.2.1 ParticlesonaStretchedString(CoupledHarmonicOscillators) 57
4.2.2 Low-FrequencyPhonons 59
4.2.3 Discussion 61
4.3 Electron–PhononInteraction 65
4.4 Phonon-ExchangeAttraction 71
References 75
5 ElectricalConductivityofMultiwalledNanotubes 77
5.1 Introduction 77
5.2 Graphene 78
5.3 LatticeStabilityandReflectionSymmetry 81
5.4 Single-WallNanotubes 84
5.5 MultiwalledNanotubes 85
5.6 SummaryandDiscussion 87
References 89
6 SemiconductingSWNTs 91
6.1 Introduction 91
6.2 Single-WallNanotubes 93
6.3 SummaryandDiscussion 98
References 98
7 Superconductivity 99
7.1 BasicPropertiesofaSuperconductor 99
7.1.1 ZeroResistance 99
7.1.2 MeissnerEffect 100
7.1.3 RingSupercurrentandFluxQuantization 101
7.1.4 JosephsonEffects 102
7.1.5 EnergyGap 104
7.1.6 SharpPhaseChange 104
7.2 OccurrenceofaSuperconductor 105
7.2.1 ElementalSuperconductors 105
7.2.2 CompoundSuperconductors 106
7.2.3 High-T Superconductors 107
c
7.3 TheoreticalSurvey 107
7.3.1 TheCauseofSuperconductivity 107
Contents VII
7.3.2 TheBardeen–Cooper–SchriefferTheory 108
7.3.3 QuantumStatisticalTheory 110
7.4 QuantumStatisticalTheoryofSuperconductivity 111
7.4.1 TheGeneralizedBCSHamiltonian 111
7.5 TheCooperPairProblem 114
7.6 MovingPairons 116
7.7 TheBCSGroundState 119
7.7.1 TheReducedGeneralizedBCSHamiltonian 119
7.7.2 TheGroundState 121
7.8 Remarks 126
7.8.1 TheNatureoftheReducedHamiltonian 126
7.8.2 BindingEnergyperPairon 127
7.8.3 TheEnergyGap 127
7.8.4 TheEnergyGapEquation 128
7.8.5 NeutralSupercondensate 130
7.8.6 CooperPairs(Pairons) 130
7.8.7 FormationofaSupercondensateandOccurrenceofSuperconductors 130
7.8.8 BlurredFermiSurface 131
7.9 Bose–EinsteinCondensationin2D 133
7.10 Discussion 137
References 139
8 Metallic(orSuperconducting)SWNTs 141
8.1 Introduction 141
8.2 Graphene 147
8.3 TheFullHamiltonian 149
8.4 MovingPairons 151
8.5 TheBose–EinsteinCondensationofPairons 153
8.6 SuperconductivityinMetallicSWNTs 157
8.7 High-FieldTransportinMetallicSWNTs 159
8.8 Zero-BiasAnomaly 161
8.9 TemperatureBehaviorandCurrentSaturation 162
8.10 Summary 162
References 164
9 MagneticSusceptibility 165
9.1 MagnetogyricRatio 165
9.2 PauliParamagnetism 167
9.3 TheLandauStatesandLevels 170
9.4 LandauDiamagnetism 171
References 176
10 MagneticOscillations 177
10.1 Onsager’sFormula 177
10.2 StatisticalMechanicalCalculations:3D 181
10.3 StatisticalMechanicalCalculations:2D 184
VIII Contents
10.4 AnisotropicMagnetoresistanceinCopper 189
10.4.1 Introduction 189
10.4.2 Theory 192
10.4.3 Discussion 194
10.5 Shubnikov–deHaasOscillations 196
References 201
11 QuantumHallEffect 203
11.1 ExperimentalFacts 203
11.2 TheoreticalDevelopments 206
11.3 TheoryoftheQuantumHallEffect 208
11.3.1 Introduction 208
11.3.2 TheModel 210
11.3.3 TheIntegerQHE 212
11.3.4 TheFractionalQHE 217
11.4 Discussion 218
References 219
12 QuantumHallEffectinGraphene 221
12.1 Introduction 221
References 227
13 SeebeckCoefficientinMultiwalledCarbonNanotubes 229
13.1 Introduction 229
13.2 ClassicalTheoryoftheSeebeckCoefficientinaMetal 232
13.3 QuantumTheoryoftheSeebeckCoefficientinaMetal 235
13.4 SimpleApplications 239
13.5 GrapheneandCarbonNanotubes 240
13.6 ConductioninMultiwalledCarbonNanotubes 242
13.7 SeebeckCoefficientinMultiwalledCarbonNanotubes 243
References 246
14 Miscellaneous 247
14.1 Metal–InsulatorTransitioninVanadiumDioxide 247
14.1.1 Introduction 247
14.2 ConductionElectronsinGraphite 249
14.3 CoronetFermiSurfaceinBeryllium 250
14.4 MagneticOscillationsinBismuth 251
References 251
Appendix 253
A.1 SecondQuantization 253
A.1.1 BosonCreationandAnnihilationOperators 253
A.1.2 Observables 256
A.1.3 FermionCreationandAnnihilationOperators 257
A.1.4 HeisenbergEquationofMotion 259
A.2 EigenvalueProblemandEquation-of-MotionMethod 261