Physics of Graphene

356 Pages·2014·12.277 MB·English

by  Andrea F. Young, Yuanbo Zhang, Philip Kim (auth.), Hideo Aoki, Mildred S. Dresselhaus (eds.)

NanoScience and Technology
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BharatBhushan,Columbus,Ohio,USA
DieterBimberg,Berlin,Germany
KlausvonKlitzing,Stuttgart,Germany
HiroyukiSakaki,Tokyo,Japan
RolandWiesendanger,Hamburg,Germany
Forfurthervolumes:
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The series NanoScience and Technology is focused on the fascinating nano-world,
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Hideo Aoki (cid:2) Mildred S. Dresselhaus
Editors
Physics of Graphene
Editors
HideoAoki MildredS.Dresselhaus
FacultyofScience,DepartmentofPhysics PhysicsandElectricalEng.Department
UniversityofTokyo MassachusettsInstituteofTechnology
Tokyo,Japan Cambridge,MA,USA
ISSN1434-4904 ISSN2197-7127(electronic)
NanoScienceandTechnology
ISBN978-3-319-02632-9 ISBN978-3-319-02633-6(eBook)
DOI10.1007/978-3-319-02633-6
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Preface
Fascination with graphene has been growing very rapidly in recent years and the
physicsofgrapheneisnowbecomingoneofthemostinterestingaswellasthemost
fast-moving topics in condensed-matter physics. Needless to say, the Nobel prize
in physics awarded to Andre Geim and Konstantin Novoselov in 2010 has given
a tremendous impetus to this topic. Several years have passed since Andre Geim
andhisgroupputforwardtheirmethodforthefabricationofgraphene,asaresult
of which an anomalous quantum Hall effect was observed. This, however, is only
oneofthehallmarksoftheunusualpropertiesofthegraphenesystem.Thehorizon
of the physics of graphene is ever becoming wider, where physical concepts go
handinhandwithadvancesinexperimentaltechniques.Thuswearenowexpanding
our interests to not only transport but also optical and other properties for layered
systemsthatincludemultilayersaswellasmonolayergraphene.
Thus it should be very timely to publish a book that overviews the physics of
graphene,whichisexactlywhywehaveeditedthepresentvolume,wheregeneral
and fundamental aspects in the physics of graphene are overviewed by outstand-
ingauthorities.Thebookcomprisesfiveexperimentalandfivetheoreticalchapters.
Abirds’eyeviewofthechaptercontentsisgiveninthefigureoverleaf.
We have endeavored to have reasonable levels of accessibility to students as
well, with some heuristic introductions in each chapter. Given the unusually rapid
progressinthefieldofgraphene,wehavefounditimpossibletocoverallthefron-
tiers, while parts of the chapters do extend to advanced levels. For instance, we
havenotcoveredmuchongrapheneapplications,whichwouldrequireanothervol-
ume.Wehopethestate-of-the-artarticlespresentedhereongraphenephysics,which
startfromthe“masslessDiracparticle”andproceedwithfurtheruniqueaspectsof
graphene,willbenefitawideaudience,andencouragethemtogoevenfurtherand
toexplorenewavenuesinthisfascinatingtopic.
Tokyo,Japan HideoAoki
Cambridge,MA,USA MildredS.Dresselhaus
v
vi Preface
Fig.1 Planofthebook
Contents
PartI Experimental
1 ExperimentalManifestationofBerryPhaseinGraphene. . . . . . 3
AndreaF.Young,YuanboZhang,andPhilipKim
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 PseudospinChiralityinGraphene . . . . . . . . . . . . . . . . . 5
1.3 BerryPhaseinMagneto-Oscillations . . . . . . . . . . . . . . . 8
1.4 PseudospinandKleinTunnelinginGraphene. . . . . . . . . . . 16
1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 ProbingDirac Fermions in Grapheneby ScanningTunneling
MicroscopyandSpectroscopy . . . . . . . . . . . . . . . . . . . . . 29
AdinaLuican-MayerandEvaY.Andrei
2.1 ScanningTunnelingMicroscopyandSpectroscopy . . . . . . . . 29
2.2 FromDisorderedGraphenetoIdealGraphene . . . . . . . . . . 31
2.2.1 SurfaceTopographyofGraphene . . . . . . . . . . . . . 33
2.2.2 TunnelingSpectroscopyofGraphene . . . . . . . . . . . 35
2.2.3 DopingandElectronHolePuddles . . . . . . . . . . . . 36
2.2.4 LandauLevels . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5 MeasuringSmallGrapheneDeviceswithScanning
Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.6 GrapheneEdges . . . . . . . . . . . . . . . . . . . . . . 49
2.2.7 StrainandElectronicProperties . . . . . . . . . . . . . . 51
2.2.8 BilayerGraphene . . . . . . . . . . . . . . . . . . . . . 51
2.3 ElectronicPropertiesofTwistedGrapheneLayers . . . . . . . . 52
2.3.1 VanHoveSingularities . . . . . . . . . . . . . . . . . . 52
2.3.2 RenormalizationoftheFermiVelocity . . . . . . . . . . 55
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vii
viii Contents
3 ElectronandPhononTransportinGraphene
inandoutoftheBulk . . . . . . . . . . . . . . . . . . . . . . . . . 65
Jean-PaulIssi,PauloT.Araujo,andMildredS.Dresselhaus
3.1 GeneralIntroduction . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.1 Graphenes . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.3 InelasticScatteringofLight . . . . . . . . . . . . . . . . 69
3.1.4 GeneralReferencesandHistoricalBackground . . . . . . 70
3.1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1.6 TopicsAddressed . . . . . . . . . . . . . . . . . . . . . 71
3.2 ElectricalConductivity . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.2 ElectronicStructure . . . . . . . . . . . . . . . . . . . . 73
3.2.3 ChargeCarrierDensitiesandScattering. . . . . . . . . . 76
3.2.4 QuantumEffects . . . . . . . . . . . . . . . . . . . . . . 84
3.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 ThermalConductivityofGrapheneinandoutoftheBulk . . . . 88
3.3.1 PreliminaryRemarks . . . . . . . . . . . . . . . . . . . 88
3.3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.3 ComparingtheThermalConductivityofGraphene
inandouttheBulk. . . . . . . . . . . . . . . . . . . . . 90
3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4 InelasticScatteringofLight—RamanScattering . . . . . . . . . 101
3.4.1 ABriefOverviewofInelasticScatteringofLight . . . . 101
3.4.2 TheG-BandMode . . . . . . . . . . . . . . . . . . . . . 103
(cid:2)
3.4.3 TheG-Band(or2D)Mode . . . . . . . . . . . . . . . . 104
3.4.4 TheDisorder-InducedD-BandMode . . . . . . . . . . . 105
3.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4 OpticalMagneto-SpectroscopyofGraphene-BasedSystems . . . . 113
C.Faugeras,M.Orlita,andM.Potemski
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Magneto-SpectroscopyofGraphene. . . . . . . . . . . . . . . . 115
4.2.1 ClassicalCyclotronResonanceofDiracFermions . . . . 115
4.2.2 Magneto-OpticalResponseofGraphene:
QuantumRegime . . . . . . . . . . . . . . . . . . . . . 117
4.2.3 LandauLevelFanChartsandFermiVelocity . . . . . . . 120
4.2.4 BeyondSimpleBandModels . . . . . . . . . . . . . . . 121
4.2.5 Scattering/Disorder . . . . . . . . . . . . . . . . . . . . 121
4.2.6 Electron-ElectronInteraction . . . . . . . . . . . . . . . 122
4.2.7 EffectsofElectron-PhononInteraction . . . . . . . . . . 123
4.3 Magneto-SpectroscopyofBilayerGraphene . . . . . . . . . . . 124
4.4 Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Contents ix
4.4.1 SimplifiedModelsfortheBandStructure . . . . . . . . . 126
4.4.2 FullSlonczewski-Weiss-McClureModel . . . . . . . . . 128
4.4.3 BandStructureClosetotheNeutralityPoint:Proximity
toLifshitzTransition . . . . . . . . . . . . . . . . . . . 129
4.4.4 ScatteringEfficiency . . . . . . . . . . . . . . . . . . . . 131
4.4.5 Electron-PhononCoupling . . . . . . . . . . . . . . . . 132
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 GrapheneConstrictions . . . . . . . . . . . . . . . . . . . . . . . . 141
S.Dröscher,F.Molitor,T.Ihn,andK.Ensslin
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 GrapheneElectronics . . . . . . . . . . . . . . . . . . . 141
5.1.2 GrapheneNanostructures . . . . . . . . . . . . . . . . . 142
5.2 ConstrictionsinConventionalSemiconductors . . . . . . . . . . 143
5.3 ConductanceinGrapheneConstrictions. . . . . . . . . . . . . . 144
5.3.1 NanoribbonswithIdealEdges . . . . . . . . . . . . . . . 144
5.3.2 ExtensiontoDisorderedEdges . . . . . . . . . . . . . . 146
5.4 ExperimentalObservationsandMicroscopicPictures . . . . . . 146
5.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4.2 DependenceofTransportontheChargeCarrier
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.3 DependenceofTransportontheAppliedVoltageBias . . 148
5.4.4 MicroscopicPictures . . . . . . . . . . . . . . . . . . . 151
5.4.5 GeometryDependence. . . . . . . . . . . . . . . . . . . 152
5.5 FurtherExperimentsforMoreDetailedUnderstanding . . . . . . 153
5.5.1 TemperatureDependence . . . . . . . . . . . . . . . . . 153
5.5.2 MagneticFieldDependence . . . . . . . . . . . . . . . . 156
5.5.3 Side-GateInfluence . . . . . . . . . . . . . . . . . . . . 158
5.5.4 ThermalCycling . . . . . . . . . . . . . . . . . . . . . . 160
5.5.5 TunnelingCouplinginaDoubleQuantumDot . . . . . . 161
5.6 RecentAdvancesandOutlook . . . . . . . . . . . . . . . . . . . 164
5.6.1 Bottom-UpGrowthofNanoribbons . . . . . . . . . . . . 164
5.6.2 QuantizedConductanceinSuspendedNanoribbons . . . 165
5.6.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
PartII Theoretical
6 ElectronicPropertiesofMonolayerandMultilayerGraphene . . . 173
MikitoKoshinoandTsuneyaAndo
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 ElectronicStructureofGraphene . . . . . . . . . . . . . . . . . 174
6.2.1 EffectiveHamiltonian . . . . . . . . . . . . . . . . . . . 174
6.2.2 LandauLevels . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.3 BandGapinGraphene. . . . . . . . . . . . . . . . . . . 179