Charge and Spin Transport in Disordered Graphene-Based Materials (2016)

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SpringerThesesRecognizingOutstandingPh.D.ResearchDinhVanTuanChargeandSpinTransportinDisorderedGraphene-BasedMaterials SpringerThesesRecognizingOutstandingPh.D.Research AimsandScopeTheseries“SpringerTheses”bringstogetheraselectionoftheverybestPh.D.thesesfromaroundtheworldandacrossthephysicalsciences.Nominatedandendorsedbytworecognizedspecialists,eachpublishedvolumehasbeenselectedforitsscientificexcellenceandthehighimpactofitscontentsforthepertinentfieldofresearch.Forgreateraccessibilitytonon-specialists,thepublishedversionsincludeanextendedintroduction,aswellasaforewordbythestudent’ssupervisorexplainingthespecialrelevanceoftheworkforthefield.Asawhole,theserieswillprovideavaluableresourcebothfornewcomerstotheresearchfieldsdescribed,andforotherscientistsseekingdetailedbackgroundinformationonspecialquestions.Finally,itprovidesanaccrediteddocumentationofthevaluablecontributionsmadebytoday’syoungergenerationofscientists.Thesesareacceptedintotheseriesbyinvitednominationonlyandmustfulfillallofthefollowingcriteria•TheymustbewritteningoodEnglish.•ThetopicshouldfallwithintheconfinesofChemistry,Physics,EarthSciences,EngineeringandrelatedinterdisciplinaryfieldssuchasMaterials,Nanoscience,ChemicalEngineering,ComplexSystemsandBiophysics.•Theworkreportedinthethesismustrepresentasignificantscientificadvance.•Ifthethesisincludespreviouslypublishedmaterial,permissiontoreproducethismustbegainedfromtherespectivecopyrightholder.•Theymusthavebeenexaminedandpassedduringthe12monthspriortonomination.•Eachthesisshouldincludeaforewordbythesupervisoroutliningthesignifi-canceofitscontent.•Thethesesshouldhaveaclearlydefinedstructureincludinganintroductionaccessibletoscientistsnotexpertinthatparticularfield.Moreinformationaboutthisseriesathttp://www.springer.com/series/8790 DinhVanTuanChargeandSpinTransportinDisorderedGraphene-BasedMaterialsDoctoralThesisacceptedbyAutonomousUniversityofBarcelona,Spain123 AuthorSupervisorDr.DinhVanTuanProf.StephanRocheCatalanInstituteofNanoscienceCatalanInstituteofNanoscienceandNanotechnologyandNanotechnologyBarcelonaBarcelonaSpainSpainandCaseWesternReserveUniversityCleveland,OHUSAISSN2190-5053ISSN2190-5061(electronic)SpringerThesesISBN978-3-319-25569-9ISBN978-3-319-25571-2(eBook)DOI10.1007/978-3-319-25571-2LibraryofCongressControlNumber:2015952026SpringerChamHeidelbergNewYorkDordrechtLondon©SpringerInternationalPublishingSwitzerland2016Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeenmade.Printedonacid-freepaperSpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www.springer.com) Iwouldliketodedicatethisthesistomylovingparents Supervisor’sForewordResearchintographenestartedin2005,followingthediscoverythatasinglemonolayerofgraphenecouldbeseparatedoutsimplyusingascotchtape-inducedmechanicalexfoliationofnatural(buthighquality)graphiterawmaterial.Themagnificentpropertiesofgraphenewerefirstrevealedbytheinspiringandfecundtheoreticalanalogybetweenhypotheticalpropertiesofrelativisticparticles(pre-dictedbutneverobserved,suchastheKleintunnelling)andthelow-energyexci-tationsingraphene,formallydescribedasmasslessDiracfermions(withanewquantumdegreeoffreedom,namelythepseudospin).Manyotherunconventionalandspectaculartransportpropertieshavebeenrevealedduringthepastdecade,including(tocitethemostsalientones)weakantilocalizationdrivenbythepseu-dospindegreeoffreedom,absenceofstronglocalizationandminimumconduc-tivity,thehalf-integerquantumHalleffect,quantumHallferromagnetism,andtheobservationoftheHofstadter’sbutterflyforgraphenedepositedonboronnitridesubstrate.Thesimplicity(andbeauty)ofthegraphenehoneycomblatticestructureisinsharpcontrasttotheendlessrichnessoffascinatingtransportphenomena,whichtodayarestillundergoingin-depthexploration.Ontheotherhand,thenatureofdisorderinsuchapeculiartwo-dimensionalmaterialismanifold,andalthoughchargetransportingrapheneisrobusttoawideclassofsurroundingimperfectionssuchasmechanicaldeformations,nearbychargedimpurities,orsurfaceadsorbed(light)atoms,thereexistsanotherlargecategoryofdefectswhichcausesmuchmoredamagetotheremarkablemasslessDiracfermiontransportphysics.Butbeyondasimpletransitiontoabadconductorupondisordering,“moredamaged”graphenecanalsoexhibitunprecedentedtransportfeaturesespeciallywhendefectscreatelow-energyimpurityresonanceswhich,giventheirrealspacelong-rangecharacter,usuallyprovokenewpercolationpathsforunconventionaltransport.Thisisparticularlymanifestinthehighmag-neticfieldregime,whereexotictransportfeaturesappearinthephasediagramofthequantumHalleffect,oruponsurfacesegregationofheavyad-atoms(suchasindiumorthallium),whichlocallystronglyenhancesspin–orbitinteraction,andbyproximityeffectsthatgeneratelocalenergybandgapsandfrontierchiralstates.vii viiiSupervisor’sForewordAdditionally,quantifyingtheimpactofdisorderonchargemobilitiesandgra-phenedevicecharacteristicsisessentialfordevelopingfuturegrapheneapplicationsinflexibleandtransparentelectronics,long-lifebatteries,orspintronics.Graphenespintronicsactuallystandsasaparticularlypromisingandchallengingtopic,asevidencedbytheeffortsof2007physicsNobellaureateAlbertFert,whoispar-ticularlyactiveinthefield,claimingthatgrapheneistheperfectmaterialfordevelopinglateralspintronicsandintegratingnoveltypesofnon-charge-basedinformationprocessingtechnologies,withhopefor“asecondspintronicrevolu-tion”afterthediscoveryofgiantmagnetoresistanceanditsmassiveimpactonstoragetechnologies.ThisbeliefissharedbyapoolofEuropeanresearchers,who,togetherwithProfessorFert,areengagedinthegraphenespintronicsworkpackageoftheGrapheneFlagshipEU-project,aimingtorealizethisvisionaryprediction(graphene-flagship.eu/).DinhVanTuanhasjoinedthisadventureoftheexplorationofthefundamentalsofchargeandspintransportindisorderedgraphenein2011,andhasfocusedhisPh.D.ondevelopingnewtheoreticalmethodologiestoscrutinizethetransportpropertiesinlarge-scalemodelsofdisorderedgraphene,payingparticularattentiontofundamentaldefectssuchasgrainboundariesinpolycrystallinegraphene,structuraldefects(oftenencounteredinreducedgrapheneoxides),orvarioustypesofadsorbedad-atoms(suchastransitionmetalatomsorheavyatomssuchasthallium).Hehassuccessfully,technicallyintegratedspin–orbitinteractionintheformalismoftime-dependentevolutionofwavepacketsinrealspace,whichhasprovidedessentialinformationaboutpeculiaritiesofthespindynamicsofmasslessDiracfermions.TheendeavoraccomplishedbyDinh,asevidencedbythishigh-qualitythesis,hasresultedinanoutstandingpieceofwork,whichhasbeenacknowledgedbymanypublicationsinhigh-impactjournals,startingwiththe2014publicationinNaturePhysicsofthediscoveryofanunprecedentedmechanismforspinrelaxation,uniquetographeneanddrivenbyspin/pseudospinentanglement(D.VanTuanetal.,NaturePhysics,10,857–863(2014)).Thistheoreticalworkshinesnewlightonexperimentalcontroversiesandhasallowedustorevisitourunderstandingofspintransportphenomenaingraphene-relatedmaterials,bydemonstratinghowweakspin–orbitinteractiondrivenbytransitionmetalad-atomscouldinduceacomplexspinandpseudospindynamicsattheoriginofuniquespindecoherenceandrelaxationeffects.ThefindingsnotonlyestablishamoresolidfoundationforspindynamicsofmasslessDiracfermions,butalsoopeninspiringavenuestowardsthemanipulationofthespindegreeoffreedom(byactingonpseudospin)andtherealizationofspin-basedinformationprocessingtechnologies.Amassivescientificimpactofthisworkisthusexpected,andtheadventurousspiritandhardworkofDinhhavebeenessentialinthisaccomplishment.Thisthesisalsopresentsseveralotherhigh-levelscientificresultssuchastheimpactofvarioustypesofdefectsonquantumtransportingraphene,andparticularlytheeffectofpolycrystallinemorphologyonchargemobility(withthediscoveryofanewtransportscalinglaw).Takenasawhole,it Supervisor’sForewordixprovidesguidanceandinspirationforcurrentandfutureexperimentalwork,withalikelycollateralimpactontheimprovementofgraphenedeviceengineeringandapplicationsofgraphenematerials.BarcelonaProf.StephanRocheApril2015 AbstractThisthesisisfocusedonmodelingandsimulationofchargeandspintransportintwo-dimensionalgraphene-basedmaterialsaswellastheimpactofgraphenepolycrystallinityontheperformanceofgraphenefield-effecttransistors.TheKubo–Greenwoodtransportapproachhasbeenusedasthekeymethodtocarryoutnumericalcalculationsforchargetransportproperties.Thestudycoversawiderangeofdisordersingraphene,fromvacanciestochemicaladsorbatesongrainboundariesofpolycrystallinegraphene,andtakesintoaccountimportantquantumeffectssuchasquantuminterferenceandspin–orbitcouplingeffects.Forspintransport,anewmethodbasedontherealspaceorderO(N)transportformalismisdevelopedtoexplorethemechanismofspinrelaxationingraphene.Anewspinrelaxationphenomenonrelatedtospin/pseudospinentanglementisunveiledandcouldbethemainmechanismatplaygoverningfastspinrelaxationinultra-cleangraphene.xi AcknowledgmentsFirstofall,IwouldliketoexpressmydeepgratitudetoProf.Dr.StephanRocheforhisassistanceasmentorofmythesis.Withouthiskindencouragement,support,constructiveguidance,andalsoproofreadingthemanuscript,Iwouldnothavebeenabletofinishthisthesis.IwouldliketothankDr.FrankOrtmann,Dr.DavidSoriano,Dr.AronCummings,Prof.SergioValenzuela,Prof.DavidJiménez,Dr.JaniKotakoski,Dr.JoseEduardoBarriosVargas,Dr.NicolasLeconte,Prof.PabloOrdejón,Dr.AlessandroCresti,Prof.JannikC.Meyer,Mr.ThibaudLouvet,Mr.PawełLenarczyk,Prof.YoungHeeLee,Dr.DinhLocDuong,Mr.VanLuanNguyen,Dr.FerneyChaves,Prof.M.F.Thorpe,andDr.AvishekKumarfortheirguidance,interestingdiscussions,suggestions,andcollaborativework.IamalsoverythankfultoDr.NicolasLecontefortakinghistimetoreadandcorrectthisthesis.IwillneverforgetthehospitalityofInstitutCatalàdeNanociènciaiNanotecnologia(ICN2),andforthatIwouldliketothankMrs.RosaJuanNebot,Mrs.AnabelRodrguezSandá,Mrs.InmaculadaCañoZafra,Mrs.SandraDomeneMegias,Mrs.EmmaNietoFumanal,Mrs.AnadelaOsaChaparro,andmydearcolleaguesintheTheoreticalandComputationalNanoscienceGroup.IacknowledgeProf.JordiPascualforacceptingtobemytutor,Prof.DavidJiménez,Prof.FranciscoPacoGuinea,Prof.Jean-ChristopheCharlier,Prof.NicolasLorente,Dr.RiccardoRurali,Dr.XavierCartoixàSoler,andDr.XavierWaintalforacceptingtobethejurymembersonmythesisdefense.Deepinmyheart,Iwouldliketothankmylovingparents,mysisters,andmywholefamilyfortheirlove,supportinallrespects,andcontinuousencouragement,whichismeaningfulnotonlytomyworkbutalsotomylife.Onthechallengingroadofmyscientificlife,Iamhappyandproudtohavemywife,NguyenThiThanhThuy.Sheisalwayswithmetosharehappinessaswellasdisappointment.Iwanttothankherforherunderstanding,support,andespeciallyforherpresentoflove,ourdaughterDinhQuynhChi.xiii Contents1Introduction.........................................1References..........................................32ElectronicandTransportPropertiesofGraphene.............52.1Introduction......................................52.2GrapheneandDiracFermions.........................62.2.1Graphene..................................62.2.2Low-EnergyDispersion........................92.3ElectronicandTransportPropertiesofDisorderedGraphene....122.4SpinTransportinGraphene...........................202.4.1Spin-OrbitCouplinginGraphene..................202.4.2SpinTransportinGraphene......................27References..........................................323TheRealSpaceOrderOðNÞTransportFormalism............353.1ElectricalTransportFormalism........................353.1.1ElectricalResistivityandConductivity..............353.1.2SemiclassicalApproach........................363.1.3TheKubo-GreenwoodFormula...................403.1.4ThreeTransportRegimes.......................443.1.5TheKuboFormalisminRealSpace................473.2SpinTransportFormalism............................493.2.1WavefunctionandRandomPhaseStatewithSpin......503.2.2SpinPolarization.............................513.2.3TechnicalDetails.............................53References..........................................544TransportinDisorderedGraphene........................554.1TransportPropertiesofGraphenewithVacancies............554.1.1Introduction................................554.1.2Zero-EnergyModesandTransportProperties.........57xv xviContents4.2ChargeTransportinPoly-G...........................654.2.1Introduction................................654.2.2StructureandMorphologyofGGBs................664.2.3MethodsofObservingGGBs....................694.2.4TransportPropertiesofIntrinsicPoly-GbySimulation...734.2.5MeasurementofElectricalTransportAcrossGGBs.....804.2.6ManipulationofGGBswithFunctionalGroups........874.2.7ChallengesandOpportunities....................934.3ImpactofGraphenePolycrystallinityonthePerformanceofGrapheneField-EffectTransistors.....................944.3.1Introduction................................944.3.2Poly-GEffectontheGateElectrostaticsandI-VCharacteristicsofGFETs.................954.4TransportPropertiesofAmorphousGraphene..............1014.4.1Introduction................................1014.4.2ModelsofAmorphousGraphene..................1024.4.3ElectronicProperties...........................104References..........................................1085SpinTransportinDisorderedGraphene....................1155.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism..............................1155.1.1Introduction................................1155.1.2SpinRelaxationinGold-DecoratedGraphene.........1175.1.3FurtherDiscussion............................1245.2QuantumSpinHallEffect............................1305.2.1Introduction................................1305.2.2AdatomClusteringEffectonQSHE................131References..........................................1376Conclusions.........................................141AppendixA:TimeEvolutionoftheWavePacket................143AppendixB:LanczosMethod..............................147CurriculumVitae.......................................151 Chapter1IntroductionGraphene,anatomicmonolayerofcarbonatomsarrangedintoahoneycomblattice,isafascinatinganduniquesystem.Itisanextreme2Dcondensedmattersystemwherethechargecarrierdynamicscanbedescribedasquasi-relativisticparticleswithzeroeffectivecarriermassandthetransportpropertiesaregovernedbytheDiracequation,wherebytheirmobilitieshaveunprecedentedlylargevalues.Manyoftheinterestingpropertiesingrapheneresultfromthesecharacteristicswhichareanalogoustothoseofrelativistic,masslessfermions.Duringthepasttenyearsafteritsdiscovery,graphenehasattractedagreatattention.Eversince,numerousuniqueelectrical,optical,andmechanicalpropertiesofgraphenehavebeendiscoveredsuchasopticaltransparence,highstrengthandstiffness,Kleintunneling,half-integerquantumHalleffect(QHE),weakantilocalization(WAL),etc.However,disordersareunavoidablefactorsthataffecttransportpropertiesofgrapheneanditiscrucialtostudytheirdetrimentaleffectstohaveacomprehensionofrealgraphenesamples.Moreover,inordertodeveloptechnologyandapplicationbasedongraphenetheintegrationofthematerialatwaferscaleismandatory.Thechemicalvapordeposition(CVD)growthtechniqueisthebestcandidateforachievingacombinationofhighstructuralqualityandwafer-scalegrowth.However,theresultingCVDgrapheneispolycrystallinegraphene(Poly-G)[14],formedbymanysingle-crystalgrainswithdifferentorientations[5].Inordertoaccommodatethelatticemismatchbetweenmisorientedgrains,thegraphenegrainboundaries(GGBs)inPoly-Garemadeupofavarietyofnon-hexagonalcarbonrings,whichcanactasasourceofscatteringduringchargetransport.ThepropertiesofPoly-Garethereforedictatedbytheirgrainsizeandbytheatomicstructureatthegrainboundaries(GBs).Effectsofstructuraldefectsontheelectronic,mechanicalandtransportpropertiesofgraphenehaverecentlybeenanalyzedtheoretically[6,7].Moreover,severaltheoreticalstudieshavereportedontheeffectofasingleGBonelectronic[8,9],magnetic[10],chemical[11],andmechanical[1214]propertiesofgraphene.However,veryfewstudies[11,14]havediscussedmorecomplexformsofGBs(notrestrictedtoinfinitelineararrangementsofdislocationcores),whichwouldbettercorrespondtotheexperimentallyobservedstructures[5,15,16].Furthermore,becauseofexperimentalchallengesonlyafewexperimentalworks[17]havesystematicallyinvestigatedtheimpactofGBson©SpringerInternationalPublishingSwitzerland20161D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2_1 21Introductionelectronictransport,mainlyconfirmingthereducedconductivityascomparedtosingle-crystallinesamples.VeryrecentelectricalmeasurementsonindividualGBsinCVD-graphenealsoreportedthatagoodinterdomainconnectivityisafundamentalgeometricalrequirementforimprovedtransportcapability[18].However,todatelittleisknownabouttheglobalcontributionofcomplexdistributionsofGBstomeasuredchargemobilities[19].Therefore,tounderstandthelarge-scaleelectricaltransportpropertiesofPoly-G,itisimportanttoperformadetailedexplorationoftheroleplayedbytheGBs.Inregardstothepotentialofgrapheneforspintronics,theextremelysmallintrinsicspin-orbitcoupling(SOC)ofgrapheneandthelackofhyperfineinteractionwiththemostabundantcarbonisotopehaveledtointenseresearchintopossibleapplicationsofthismaterialinspintronicdeviceswiththeanticipatedpossibilityoftransportingspininformationoververylongdistances[2022].However,thespinrelaxationtimesarestillfoundtobeordersofmagnitudesmallerthaninitiallypredicted[2327],whilethemajorphysicalprocessforspinequilibrationanditsdependenceonchargedensityanddisorderremainelusive.ExperimentshavebeenanalyzedintermsoftheconventionalElliot-Yafet(EY)andDyakonov-Perel(DP)processes,yieldingcontradictoryresults.Recently,amechanismbasedonresonantscatteringbylocalmagneticmomentshasalsobeenproposed[28]butcontainstoomanyfreeparametersanddoesnotsolvethecontroversialresultsreportedexperimentally[29].In2005,thequantumspinHall(QSH)statewaspredictedingraphenebyKaneandMele[30].TheKaneandMelemodelistwocopiesoftheHaldanemodel[31]suchthatthespinupelectronexhibitsachiralintegerQHEwhilethespindownelectronexhibitsananti-chiralintegerQHE.Thisnovelelectronicstateofmatterisgappedinthebulkandsupportsthetransportofspinandchargeingaplessedgestatesthatpropagateatthesampleboundaries.Theedgestatesareinsensitivetodisorder(whichdoesnotbreaktimereversalsymmetry)becausetheirdirectionalityiscorrelatedwithspin.However,thisbeautifulstateisunobservableingrapheneduetoweakSOCinintrinsicgraphene.Asolutionforthisproblemisendowinggraphenewithcertainheavyadatomssuchasthalliumorindium[32],buttodatetheclusteringeffectoftheseadatomsmaketheQSHstateseemtojeopardizeitsobservation.Thepurposeofthisthesisistoaddressaboveproblems.Thethesisisorganizedinto6chaptersand2appendices.Thecontentsaredevelopedasfollows:Thischaptergivesthepurposeofthisthesisandoverviewstheproblemsofinterest.Thecontentofeachchapterinthisthesisisalsomentionedinthisintroductorychapter.Chapter2presentstheelectronicandtransportpropertiesofcleangraphene.Inthischapterthelinearbandstructureofgrapheneisderived,andsomespecialcharac-teristicsofDiracfermionssuchaschirality,zeroeffectivemass,etc.arementioned.Thechapteralsocoverstheliteratureofelectronictransportandspintransportingraphene.Inthislaterpart,spin-orbitinteractionsarederivedandtheirmodificationsontheDiracbandstructurearereviewed.Thefinalpartofthischapterisdevotedtoadiscussiononthediscrepancyofexperimentalandtheoreticalresultsconcerningspinrelaxationingraphene.Twomechanismsforspinrelaxationingraphene,EYandDP,arealsoderived. 1Introduction3Chapter3brieflyoverviewstheKubo-Greenwoodtransportformalismwhichisextensivelyusedinthisthesis.Inthischapter,twodifferentapproachesarediscussednamely,thesemiclassicalandquantumapproaches,whichleadtotheEinsteinrela-tionforconductivity.TherealspacetransportmethodfortheKuboconductivitycalculationisalsointroduced.AnextensionofrealspaceorderO(N)transportfor-malismisdevelopedtostudyspintransportintherealisticsystem.Chapter4focusesontheelectronictransportpropertiesofdisorderedgraphene.Thetransportpropertiesarestudiedwithgraduallyincreasingdisorder,frompointdefectsingraphenewithvacanciestolinedefectsinPoly-Gandfinallytotheextremelydisorderedformofgraphene,amorphousmembranesofsp2graphene.Thestudiesaresystematicallyconcentratedondifferentaspectsofgrapheneinper-spectivesofapplications.Chapter5dealswiththegraphenespinrelaxationproblems.InthischapterwepointoutthelimitationsofEYandDPmechanismsforgraphene,andweproposeanewmechanismdrivenbytheentanglementbetweenspinandpseudospinquantumdegreeoffreedoms,whichgovernsthefastspinrelaxationclosetoDiracpointingraphene.Attheendofthischapter,weexplainthedifficultyofobservingtheQSHeffectingraphenewhendepositingheavyadatoms.ThenaturalclusteringtrendofsuchadatomsweakenstheSOCeffectwhichisacrucialfactoroftheformationoftopologicaledgestate.Thechapteralsoreportstheformationofarobustmetallicstatewhichisrelatedtotheenhancedpercolationofpropagatingstatesbetweenislands.Chapter6summarizesthethesisandsuggestssomeopeningdirectionsforthenearfuture.References1.X.S.Li,W.W.Cai,J.H.An,S.Kim,J.Nah,D.X.Yang,R.Piner,A.Velamakanni,I.Jung,E.Tutuc,S.K.Banerjee,L.Colombo,R.S.Ruoff,Science324,13121314(2009)2.A.Reina,X.Jia,J.Ho,D.Nezich,H.Son,V.Bulovic,M.S.Dresselhaus,J.Kong,NanoLett.9,30(2009)3.X.S.Li,C.W.Magnuson,A.Venugopal,J.H.An,J.W.Suk,B.Y.Han,M.Borysiak,W.W.Cai,A.Velamakanni,Y.W.Zhu,L.F.Fu,E.M.Vogel,E.Voelkl,L.Colombo,R.S.Ruoff,NanoLett.10,43284334(2010)4.S.Bae,H.Kim,Y.Lee,X.F.Xu,J.S.Park,Y.Zheng,J.Balakrishnan,T.Lei,H.R.Kim,Y.I.Song,Y.J.Kim,K.S.Kim,B.Ozyilmaz,J.H.Ahn,B.H.Hong,S.Iijima,Nat.Nanotechnol.5,574578(2010)5.P.Y.Huang,C.S.Ruiz-Vargas,A.M.vanderZande,W.S.Whitney,M.P.Levendorf,J.W.Kevek,S.Garg,J.S.Alden,C.J.Hustedt,Y.Zhu,J.Park,P.L.McEuen,D.A.Muller,Nature469,389(2011)6.A.V.Krasheninnikov,F.Banhart,Nat.Mater.6,723(2007)7.S.Roche,N.Leconte,F.Ortmann,A.Lherbier,D.Soriano,J.-C.Charlier,SolidStateCommun.152,14041410(2012)8.N.M.R.Peres,F.Guinea,A.H.C.Neto,Phys.Rev.B73,125411(2006)9.O.V.Yazyev,S.G.Louie,Nat.Mater.9,806(2010)10.J.Cervenka,M.I.Katsnelson,C.F.J.Flipse,Nat.Phys.5,840(2009) 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Chapter2ElectronicandTransportPropertiesofGraphene2.1IntroductionGraphenehasreceivedagreatattentionsinceitwasfirstisolatedbyNobelLaureatesKonstantinNovoselovandAndreK.Geimin2004.Thereasonforsuchexcitementisthatgrapheneisthefirsttruly2Dcrystaleverobservedinnatureandpossessesremarkableelectrical,chemicalandmechanicalproperties.Furthermore,electronsingrapheneshowaquasi-relativisticbehavior,andthesystemisthereforeanidealcandidateforthetestofquantumfield-theoreticalmodelsthathavebeendevelopedinhigh-energyphysics.Mostprominently,electronsingraphenemaybeviewedasmasslesschargedfermionsexistingin2Dspace,particlesthatoneusuallydoesnotencounterinourthree-dimensionalworld.Indeed,allmasslesselementarypar-ticles,suchasphotonsorneutrinos,happentobeelectricallyneutral.Grapheneisthereforeanexcitingbridgebetweencondensedmatterandhigh-energyphysics,andtheresearchonitselectronicpropertiesunitesscientistswithvariousthematicbackgrounds.Grapheneisalsoanattractivematerialforspintronicsduetothetheoreticalpos-sibilityoflongspinlifetimesarisingfromlowintrinsicSOCandweakhyperfineinteraction[1].However,Hanlespinprecessionmeasurementsandnon-localspinvalvegeometryhavereportedspinlifetimesthatareordersofmagnitudeshorterthanexpectedtheoretically[2–5].Severalstudieshaveinvestigatedspinrelaxationincludingtherolesofimpurityscattering[5]andgraphenethickness[6]andspe-cially,ferromagnetcontact-inducedspinrelaxationwaspredictedtoberesponsiblefortheshortspinlifetimesobservedinexperiments[7].However,theseexplanationshavenotgivenasatisfyinganswerforthediscrepancybetweentheoreticalresultsandexperimentaldata.Thishaspromptedtheoreticalstudiesoftheextrinsicsourcesofspinrelaxationsuchasimpurityscattering[8],ripples[1],andsubstrateeffects[9].Theproblemremainshoweverstillpuzzlingandunsolved.Inthischapter,wewillbrieflyreviewsometheoreticalandexperimentalresultsaboutfundamentalelectricandspintransportpropertiesofgraphene.Firstly,wewillderivegraphenebandstructureandmasslessDiracequationforgraphenein©SpringerInternationalPublishingSwitzerland20165D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2_2 62ElectronicandTransportPropertiesofGrapheneSect.2.2.Next,someexperimentalandtheoreticalstudiesabouttransportpropertiesofgraphenearediscussedinSect.2.3.Section2.4discussessomeaspectsofSOCingraphene,whichplaysanimportantroleforstudyingspinrelaxationinChap.5.2.2GrapheneandDiracFermionsThemostinterestingpropertyofgraphenemightbetheDirac-coneenergydispersion.Thisistheconsequenceofsp2hybridizationandgraphenesymmetry.Inthissection,Ibrieflyreviewitsstructure,thecommonlyusedtight-binding(TB)descriptionandthedeviationofthelinearenergydispersionofgraphene.2.2.1GrapheneGrapheneisasingleatomiclayerofgraphite,anallotropeofcarbonthatismadeupofverytightlybondedcarbonatomsorganisedintoahexagonallattice.Whatmakesgraphenesospecialisitssp2hybridizationandverythinatomicthickness(seeFig.2.1).Thesepropertiesarewhatenablegraphenetobreaksomanyrecordsintermsofstrength,electricity,heatconduction,etc.Carbonisacommonelementinthenature,withatomicnumber6,group14ontheperiodictable.Theelectronicconfigurationofcarbonis1s22s22p2whichshowsthatcarbonhas4electrons(2sand2p)initsoutershellwhichisavailableforformingchemicalbonds.Ingraphene,thesefourvalenceelectronsformsp2hybridizationinwhichthreeelectronsaredistributedintothreein-planeσbonds,whicharestronglycovalent,determiningtheenergeticstabilityandtheelasticpropertiesofgraphene.(a)(b)Fig.2.1ElectronicstructureofgrapheneaGraphenesampleandthesp2hybridizationingraphene.bEnergyrangeoforbitalsingraphene(Figureistakenfrom[10]) 2.2GrapheneandDiracFermions7Fig.2.2Real(a)andreciprocal(b)spaceofgraphenelattice(Figureistakenfrom[10])Theremainingelectroninthepzorbitals,whichisperpendiculartographeneplane,formstheπbondingraphene(SeeFig.2.1).Thecalculationfortheenergyrangesofσandπbands(SeeFig.2.1b)showsthatonlyelectronsintheπbondcontributetotheelectronicpropertiesofgraphenebecausetheσbandsarefarawayfromtheFermilevel.Becauseofthis,itissufficienttotreatgrapheneasacollectionofatomswithsinglepzorbitalspersite.Ingraphene,carbonatomsarelocatedattheverticesofahexagonallattice.GrapheneisabipartitelatticewhichconsistsoftwosublatticesAandBandbasisvectors(a1,a2)(SeeFig.2.2):√√3131a1=a,,a2=a,−,(2.1)2222√witha=3acc,whereacc=1.42Åisthecarbon-carbondistanceingraphene.ThesebasisvectorsbuildahexagonalBrillouinzonewithtwoinequivalentpointsKandK(K+andK−respectivelyinFig.2.2)atthecorners√√4π314π31K=,−,K=,,(2.2)3a223a22AsmentionedaboveandfromBloch’stheorem,wecanwritethewavefunctionintheformofpzorbitalswavefunctionatsublatticesA(ϕ(r−rA))andB(ϕ(r−rB))(k,r)=cA(k)φA(k,r)+cB(k)φB(k,r)(2.3)whereφA(k,r)=√1eik.Rjϕ(r−rA−Rj),(2.4)NRjφB(k,r)=√1eik.Rjϕ(r−rB−Rj),(2.5)NRj 82ElectronicandTransportPropertiesofGraphenewherekistheelectronwavevector,Nthenumberofunitcellsinthegraphenesheet,andRjisaBravaislatticepoint.UsingtheSchrödingerequation,H(k,r)=E(k,r),oneobtainsa2×2eigenvalueproblem,cA(k)HAA(k)HAB(k)cA(k)SAA(k)SAB(k)cA(k)H(k)==E(k).cB(k)HBA(k)HBB(k)cB(k)SBA(k)SBB(k)cB(k)(2.6)whereSαβ(k)=φα(k)|φβ(k)andthematrixelementsoftheHamiltonianaregivenby:HAA(k)=1eik.(Rj−Ri)ϕA,Ri|H|ϕA,Rj(2.7)NRi,RjHAB(k)=1eik.(Rj−Ri)ϕA,Ri|H|ϕB,Rj,(2.8)NRi,RjwithHAA=HBBandHAB=H∗,andintroducingthenotation:ϕA,Ri=ϕ(r−BArA−Ri)andϕB,Ri=ϕ(r−rB−Ri).Ifweneglecttheoverlaps=ϕA|ϕBbetweenneighboringpzorbitals,then,Sαβ(k)=δα,βandEq.(2.6)becomesHAA(k)HAB(k)cA(k)cA(k)=E(k).(2.9)HBA(k)HBB(k)cB(k)cB(k)Ifweconsideronlythefirst-nearest-neighborsinteractionsthenHAB(k)=ϕA,0|H|ϕB,0+e−ik.a1ϕA,0|H|ϕB,−a1+e−ik.a2ϕA,0|H|ϕB,−a2=−γ0α(k)(2.10)whereγ0standsforthetransferintegralbetweenfirstneighborsπorbitals(γ0=2.7eVinthisthesis)andα(k)isgivenby:α(k)=(1+e−ik.a1+e−ik.a2).(2.11)TakingHAA(k)=HBB(k)=0astheenergyreference,wecanwriteH(k)as:0−γ0α(k)H(k)=∗.(2.12)−γ0α(k)0DiagonalizingthisHamiltoniangivestheenergydispersionrelationsforπ∗(con-duction)band(+)andπ(valence)band(−):E±(k)=±γ0|α(k)|√=±γ03+2cos(k.a1)+2cos(k.a2)+2cos(k.(a2−a1))√3kxakya2kya=±γ01+4coscos4cos.(2.13)222 2.2GrapheneandDiracFermions9Fig.2.3Bandstructureofgraphene(a),thezoom-infigureatclosetoKandKpoints(b,c)andthedensityofstateofgraphene(d)(Figureistakenfrom[11])ThisbandstructureisplottedinFig.2.3withthesymmetrybetweentheconduc-tionbandandthevalencebandwhichtouchatthreeKandKpointswithzerodensityofstateatthisenergy(Fig.2.3d).Becauseofthis,grapheneiscalledgaplesssemiconductororsemi-metal.Inneutralgraphene,theFermilevellieexactlyatthesepoints.2.2.2Low-EnergyDispersionBecauseofthefactthattheycanonlyexperimentallytunetheFermilevelasmallrange(0.3eV)aboutthetouchingpoints,thiscorrespondstoasmallvariationaroundtheKandKpointsinmomentumspace.Therefore,itissufficienttoexpandtheenergydispersioninthevicinityofKandKpointsbyreplacingk→K(K)+k,whichletsuswriteEq.(2.12)intheformH=vF(ησxkx+σyky).(2.14) 102ElectronicandTransportPropertiesofGrapheneandEq.(2.13)becomesEs(k)=svF|k|,(2.15)√wherevF=3γ0a/2istheelectronicgroupvelocity,η=1(−1)forK(K)points,s=±1isthebandindex(+1forconductionbandand–1forvalenceband)andthePaulimatricesaredefinedasusual:010−i10σx=,σy=,σz=.(2.16)10i00−1Equation(2.14)isalmostthesametheDiracequationforthemasslessfermionsinquantumelectrodynamicsexceptforthefactthatthePaulimatriceshererepresentthesublatticedegreesoffreedominsteadofspinandthespeedoflightcisreplacedbygraphenevelocityvFc/300.Therefore,thesublatticedegreesoffreedomandthetouchingpointsarecalledpseudospinandDiracpoint,respectively.ThelinearenergydispersioninEq.(2.15)leadstothefactthattotaldensityofstatesisdirectlyproportionaltoenergyandcarrierdensityisproportionaltoenergysquared.Indeed,12πkdk2|E|ρ(E)=δ(E−E(k))=gsgvδ(E−E(k))=(2.17)L2(2π)2π2v2kFwhichisplottedinFig.2.3d,wheregs=2andgv=2accountforspinandvalleydegeneracies,respectively.Thecarrierdensityisgivenby1k2E2gF=n(E)=sgv=gsgv(2.18)L24ππ2v2|k|≤kFFTofindtheeigenstatesofDiracHamiltonian(2.14),itisusefultowritethisHamiltonianinthetermofmomentumdirectionθk0e−iηθkHη(k)=vFke+iηθk0(2.19)whereθk=arctan(ky/kx).ThisHamiltonianhastheeigenvaluesgivenbyEq.(2.15)andtheeigenfunctions11|η,s(k)=√iηθk.(2.20)2se 2.2GrapheneandDiracFermions11Next,wearegoingtofindeigenvaluesofthehelicityoperator(averyimportantfeatureofDiracparticle)whichhereisdefinedas:phˆ=ˆσ·.(2.21)|p|wherep=kistheelectronmomentumoperator.Inordertodothat,itisconvenienttoexchangethespinorcomponentsattheKpoint(forη=−1)[12],|K(k)=cA(k),|K(k)=cB(k)(2.22)cB(k)cA(k)i.e.,toinverttheroleofthetwosublattices.Inthiscase,theeffectivelow-energyHamiltonianinEq.(2.14)mayberepresentedasHη(k)=ηvF(σxkx+σyky)=vFτz⊗σk.(2.23)whereτarePaulimatricesrepresentingthevalleydegreeoffreedomscalledvalleypseudospin.UsingEqs.(2.23)and(2.21)Hη(k)=ηvFkhˆ(2.24)wefindthathelicityoperatorcommuteswiththeHamiltonian,theprojectionofthepseudospinisawell-definedconservedquantitywhichcanbeeitherpositiveornegative,correspondingtopseudospinandmomentumbeingparallelorantiparalleltoeachother.Thebandindexs,whichdescribesthevalenceandconductionbands,isthereforeentirelydeterminedbythechiralityandthevalleypseudospin,andonefindss=ηh(2.25)whichhelpusfindoutthatchiralitychangessignfromconductionbandtovalencebandandfromKtoKpoints.Thefactthatpseudospinisblockedwithmomentumhasastronginfluenceinmanyofthemostintriguingpropertiesofgraphene.Forexample,foranelectrontobackscatter(i.e.changingpto−p)itneedstoreverseitspseudospin(seeFig.2.3c).SobackscatteringisnotpossibleiftheHamiltonianisnotperturbedbyatermwhichflipsthepseudospin.Thismakeselectronsingrapheneinsensitivetolong-rangescatterers.Thischaracteristicmanifestsitselfinsomephe-nomenasuchasKleintunnelingorWAL[13,14].Kleintunneling[15]isaspectacu-larmanifestationoftheDiracfermionsphysicswhichdescribesthatwhentheDiracchargecrossesatunnelingbarrier,theincomingelectronispartiallyortotallytrans-mitteddependingontheincidentangleoftheincomingwavepacket.Especially,thebarrieralwaysremainsperfectlytransparentforanglesclosetonormalinci-denceregardlessoftheheightandwidthofthebarrier,standingasafeatureunique 122.ElectronicandTransportPropertiesinGraphenetomasslessDiracfermionsandbeingcompletelydifferentformtheusualchargewhosetransmissionprobabilitydecaysexponentiallywiththebarrierwidth.KleintunnelinghasbeenstudiedtheoreticallyanditshowsthatforlongrangepotentialswhichpreserveABsymmetryandprohibitsintervalleyscattering,thebackscatteringistotallysuppressed.Innextsection,wewilldiscussinmoredetailtheeffectofspecialbandstructureandpseudospin-momentumcouplingonthetransportpropertiesofgraphene.2.3ElectronicandTransportPropertiesofDisorderedGrapheneThedisorderingraphenesampleispracticallyaninevitablefactorinanyexperi-ment.Insomeways,artificialdisordersarealsotoolstoengineer,functionalizethematerials.Forinstance,puresemiconductorsarepoorconductorsandpoorinsula-tors.However,theirmagnificentpropertieshavebeenachievedbyfunctionalizationusingn−andp−typedopants,leadingtop−njunctions,transistors,junctionlasers,light-emittingdiodes,andanentiretechnologicalrevolution.Similarlytosemiconductors,inspiteofhavinguniquepropertiessuchassuperbmechanicalstrengthandcarriermobility,pristinegrapheneisnotusefulforpracticalapplicationsbecauseofitslowcarrierdensity,zerobandgap,andchemicalinertness.ThelackofelectronicgapinpristinegrapheneisanissuethathastobeovercometoachievehighIon/Ioffcurrentratioingraphene-basedfield-effectdevices.Therefore,itisimportanttostudythedisordereffectontheelectronicpropertiesofgraphenenotonlytoconqueritsdetrimentaleffectsbutalsouseartificialdefectstofunctionalizegraphenedevices.Thestudyoftransportpropertiesisattheheartofgrapheneresearch.Experimentsshowthattheconductivity(downtoafewKelvin)isalmostconstantclosetotheDiracpoint,σ∼2−5e2/h,andweaklydependentonthevalueofthechargemobility[15–18].Onthetheoreticalside,withintheself-consistentBornapproximation(SCBA)thesemiclassicalpartoftheconductivityduetoshortrangedisorderisfoundtobeσmin=4e2/πhwhichisknownasthequantumlimitedconductivityofgrapheneincleanlimit[19].However,transportpropertiesofgraphenearestronglydependentonthenatureofpossiblesourcesofdisorder.Therearemanykindsofdisordersingraphene,somearelong-rangedisorderssuchasCoulombinteractionsofchargedimpuritiesinthesub-strate,electron-holepuddle,longrangestraindeformations,distortionofgraphenestructure,etc.Otherformsarerelatedtothesp3defectssuchasepoxidedefects,theabsorptionofhydroxyl,hydrogen,fluorine,etc.ongraphene(SeeFig.2.4).Finallytopologicaldisorderswhichkeepthesp2hybridizationofgraphenebutchangethehexagonalstructure,involvestructuralpointdefectsandlinedefectsorGBs.Theelectronicpropertiesofgraphenearewelldescribedbytheπ−orbitaltight-bindingHamiltonianinwhichthedisorderintherealsamplecanbesimulatedbychangingtheon-siteenergiesδofπ−orbital.Oneofthesimplestdisordermodelin 2.3.ElectronicandTransportPropertiesinDisorderedGraphene13Fig.2.4Somekindsofsp3disorderingrapheneFig.2.5Mainframe:The5semiclassicalconductivityW=1.510BoltzmannforAndersondisorder.InsetW=2.0)hSCBA_thecomparisonof4W=2.52W=2.0/π2Kubo-Greenwoodapproach(e5σ2/πGwiththeBoltzmannand2/h)03self-consistentBorn=2e00-4-2024approximation(FigureisE(ε)0takenfrom[20])(Gsc2σ12/π0-4-2024Energy(eV)grapheneistheshort-rangescatteringpotential,namelytheAndersondisorder[20,21].Thiswhitenoiseuncorrelateddisorderisintroducedthroughrandommodula-tionsoftheonsiteenergiesδ∈[−W/2,W/2]γ0.Thisdisordercouldinprinciplemimicneutralimpuritiessuchasstructuraldefects,dislocationlines,oradatoms,althoughthelocalgeometryandchemicalreactivityofdefectsandimpuritiesactu-allydemandformoresophisticatedabinitiocalculationsifaimingatquantitativepredictions.Figure2.5showstheenergydependenceofsemiclassicalconductivityfromtheKuboformalismforsomevaluesofW(mainframe)andthecomparisonwithonefromSCBA(inset).Theresultsareingoodagreementatlowenergyandclosetothetheoreticallypredictedminimumconductivityσmin=4e2/πh.Forhigherenergies,theagreementwithSCBAislostduetohigherorderdeviations.Furthermore,theSCBAisnotsufficienttodescribesuchasystemwithallsymmetriesbroken. 142.ElectronicandTransportPropertiesinGrapheneTheroleofdisorderonWLandWALingraphenehasalsobeenintensivelyinvestigated.Fromageneralperspective,theconductanceofasystemcanbeviewedasthesum(PA→B)overallprobabilityamplitudesofpropagatingtrajectoriesstartingfromonelocationAandgoingtoanotheroneBinrealspace,ormoreexplicitly2e2G=PA→B(2.26)hPA→B=i|Ai|2+i=jAiA∗(2.27)jHereAi=|Ai|eiϕiisthepropagationamplitudealongthepathi.Thefirsttermdenotestheclassicalprobabilitycorrespondingtosemiclassicalconductivityσscwhilethesecondoneistheinterferencetermwhichgivesthequantumcorrectionδσ(L)ofthesemiclassicalresultσ(L)=σsc+δσ(L).Forthemajorityofthetrajectoriesthephasegains,Bϕ=−1pdl1(2.28)Aandtheinterferencetermvanishes.However,forsomespecialtrajectorieswithself-crossings,ifwechangethedirectionofpropagation,p→−p,dl→−dl,thephasegainsarethesame,i.e.AiA∗=|Ai|2,andquantuminterferencethuseventuallyjenhancetheprobabilityofreturntosomeorigin.Thiscontributionofquantuminter-ferencesgivesrisetotheincreaseofthequantumresistance,i.e.δσ(L)<0,knownaslocalization.Therearetwodifferentscallingbehaviorsoflocalization:theWLwith[10,20,22]2e2Lδσ(L)=−ln(2.29)πhleandstronglocalizationdescribedbyLσ(L)∼exp−(2.30)ξwhereL,le,andξdenotethesamplelength,meanfreepathandthelocalizationlength,respectively.However,wehaven’tconsideredthecontributionofadditionaldegreesoffreedomsuchasspinorpseudospiningraphene.Thedetailcalculations(formoredetails,seeRef.[10]andreferencestherein)showedthatthesecontributionscanleadtothesignreversalofquantumcorrectionofconductivity2e2Lδσ(L)=+ln(2.31)πhlewhichisthescallinglawforWAL. 2.3.ElectronicandTransportPropertiesinDisorderedGraphene15Fig.2.6Thecontributionfromintraandintervalleyscattering(Figureistakenfrom[23])Asmentionedabove,theDiracfermionsingrapheneareexpectedtoexhibittheWALbehaviorbutanothereffectshouldalsobeinvolvedtoconsiderthewholepicture,thatistrigonalwarpingwhichisrelatedtothemomentumcontributionfromhigherorderintoEq.(2.15).Thetrigonalwarpingispredictedtosuppressantilocalizationandtogetherwithintervalleyscattering,itrestorestheweaklocal-ization(WL)[13].ThecrossoversfromWALtoWLandtheeffectofdisor-dersonintra-andintervalleyscatteringwerestudiedinmanyRefs.[13,14,20,23]inwhichthelongrangedisorderissimulatedbychangingonsiteenergiesN22Vi=j=1jexp[−(ri−Rj)/(2ξ)]wherejarechosenatrandomwithin−W,−W.Thesecalculationsshowthatthestrengthoflocalpotentialprofile22controlthecontributionofintra-andintervalleyscatteringsontheconductivity.FollowingthetheoreticalstudyinRef.[23],theintravalleyscatteringdominatesatsmallvalueofW(W<γ0)andvalleymixingstrengthwascontinuouslyenhancedfromW=γ0toW=2γ0.TheintervalleyscatteringcontributionislargeenoughasW>2γ0(See.Fig.2.6).Asaconsequence,grapheneexhibitsthecrossoverfromWALtoWLasWincrease(SeeFig.2.7).Indeed,thepositivemagnetoconductanceforthecaseW=2γ0(toppanels)agreeswiththestrongcontributionofintervalleyscattering,sinceallgraphenesymmetrieshavebeenbroken.HoweverbydecreasingthedisorderstrengthfromW=2γ0toW=1.5γ0(bottompanels),WALisindeedrecoveredgiventhereductionofintervalleyprocesses.ChemicalabsorptioningrapheneisusuallyrelatedtooxidationorhydrogenationofgraphenewhicharestronglyinvasiveforelectronicandtransportpropertiesandsystematicallydrivegraphenetoastrongAndersoninsulator[25].Thetheoreticalandexperimentalstudiesshowthathighcoveragesp3formationswhichbreaklocalABsymmetrysuchasinhydrogenatedorfluorinatedgrapheneinduceenergybandgapinthehighdensitylimit.Especially,graphane,fullyhydrogenatedgraphene,ispredictedtobeastablesemiconductorwiththeenergygapaslargeas3.5eV[26],somerecentDFTcalculationsusingthescreenedhybridfunctionalofHeyd,Scuseria, 162.ElectronicandTransportPropertiesinGrapheneFig.2.7MagnetoconductanceforW=2γ0(toppanels)andW=1.5γ0(bottompanels),thedataisextractedfromtheoretical(leftpanels)andexperimental(rightpanels)study(Figureistakenfrom[20])andErnzerhof(HSE)evengavealargerenergygapupto4.5eVforgraphaneand5.1eVforfluorographene(fullyfluorinatedgraphene)(SeeFig.2.8).Thecaseoflowcoverageofhydrogenismoreinterestingwiththetransportpropertiesstronglydependingontheabsorbingposition.TheorypredictedthatgrapheneexhibitsWLforthecompensatedcase(hydrogenabsorbsequallyintwosublattices)whereasthequantuminterferencesandlocalizationsaresuppressedifhydrogendefectsarerestrictedtooneofthetwosublattices[20].Theanalogyoftransportpropertiesofchemicalabsorptionsandlong-rangepotentialshavealsobeenstudied.AsonecanseeinFig.2.9,somechemicalabsorptionsatbridgepositionsuchasepoxidedefectswhichpreservelocalABsymmetryinduceenergy-dependentelasticscatteringtime(τe(E))ressemblingthecaseoflongrangeimpuritieswithsmallonsitepotentialdepth,whereassomeadsorbatesatthetoppositionsuchashydrogenorfluorinedefectswhichbreaklocalsp2andABsymmetrygiverisetoelasticscatteringtimeressemblingthecaseofstronglongrangepotentials.Theseareduetothefactthattransporttimebehavioriscontroledbythecontributionofinter-andintravalleyscatteringswhicharemainlydeterminedbythebreakingofABsymmetry. 2.3.ElectronicandTransportPropertiesinDisorderedGraphene17Fig.2.8Theelectronicbandstructureandprojecteddensityofstatesinthevicinityofthebandgapforgraphane(a)andfluorographene(b)(Figureistakenfrom[24])Fig.2.9Elasticscatteringtime(τe)versusenergyforthreedifferentlong-rangepotentialstrengthsW.Leftinsetτeforvariousdensitiesofepoxidedefects.Rightinsetτeforvariousdensitiesofhydrogendefects(Figureistakenfrom[20])Inparticular,theformationofsp3hybridizationsormonovacanciesingraphenecangiverisetolocalsublatticeimbalancesandthusinducelocalmagneticmomentaccordingtoLieb’stheorem[27].Theexistenceofmagnetismingrapheneaswellasmagnetism-dependenttransportpropertieshavebeenstudiedinmanyRefs.[28–30].Especially,whenhalfofthehydrogeningraphanesheetisremoved,theresult-ingsemihydrogenatedgraphene(graphone)becomesaferromagneticsemiconductorwithasmallindirectgap[31].Structuralpointdefectsusuallyexistinvariousgeometricalformsingraphene.Theycanbeobtainedforinstancewhenirradiatinggraphenesamples.Inthiskindofgraphene,thedisorderiscreatedlocallyinthesamplebylocallychangingthehexago-nalstructuresuchasremovingacarbonatomfromthegraphenesheet(monovacancy) 182.ElectronicandTransportPropertiesinGrapheneFig.2.10Somestructuralpointdefects(toppanels)andtheirexperimentalTEMimages(bottompanels)(Figureistakenfrom[35])orrotatingapairofcarbon90◦ingrapheneplane(Stone-Walesdefects).Somestud-ies[32]showedthatmonovacanciesareverymobileandunstable,recombiningindi-ormultivacanciesorlocalstructureswithsomenonhexagonalringswhicharemorestable.Thetransportpropertiesofgrapheneundertheinfluenceofstructuralpointdefectssuchasvacancies,divacancies,Stone-Walesdefects,585divacancies(SeeFig.2.10),etc.havebeennowwidelystudied[33,34],revealinginterestingfeaturessuchaselectron-holetransportasymmetry[33,34]duetothepresenceofdefect-inducedresonances.Underelectronirradiation,graphenechangesfrompristineformtostructuraldefectsandfinallytoanewtwo-dimensionalamorphouscarbonlattice[36]whichiscomposedofsp2-hybridizedcarbonatoms,arrangedasarandomtilingoftheplanewithpolygonsincludingfour-memberedrings.Mostthe-oreticalstudies[37,38]foundoutthatthereisahugeincreaseofthedensityofstateatthechargeneutralitypointinthisamorphousgrapheneandthesestatesarelocal-ized,suggestingthattheamorphousgrapheneisanAndersoninsulator.However,usingastochasticquenchingmethod,Ref.[39]claimedthat“wepredictatransitiontometallicitywhenasufficientamountofdisorderisinducedingraphene...”.InChap.4,byusingKubo-Greenwoodcalculations,weshowthatthisconclusionismisleadingandsimilarresultshavealsobeenobtainedrecentlyinRef.[40].Althoughpossessingmanyexcellentelectrical,opticalandmechanicalproper-ties,perfectgraphene(single-crystalgraphene)isonlyfabricatedinsmallsizebyexfoliationmethod.Sofar,themostpromisingapproachforthemassproductionoflarge-areagrapheneisCVD,whichresultsinagraphenewithmanylinedefects(SeeFig.2.11)orPoly-G.ThispolycrystallinityarisesduetothenucleationofgrowthsitesatrandompositionsandorientationsduringtheCVDprocess.Inorderto 2.3.ElectronicandTransportPropertiesinDisorderedGraphene19Fig.2.11TwoclassesofelectrontransportthroughGBs(Figureistakenfrom[41])accommodatethelatticemismatchbetweenmisorientedgrains,theGBsinPoly-Garemadeupofavarietyofnon-hexagonalcarbonrings,whichcanactasasourceofscatteringduringchargetransport.Becauseofitspotentialforapplications,thetransportpropertiesofPoly-Garethesubjectofintenseresearch.SomecalculationsshowedthattheeffectofGBsonthecarriertransportdifferdependingontheGBgeometry(SeeFig.2.11)resultinginatunablemobility(tunabletransportgaps)[41]whichallowstocontrolchargecurrentswithouttheneedtointroducebulkbandgapsingraphene.Inso-calledclassIGBs(toppanelsofFig.2.11),includingallsym-metricalGBs,theprojectedperiodicitiesofthelatticeoneachsidematchinawaythatallowscarrierstocrossfreelyevenattheDiracpoint.InclassIIGBs(bottompanelsofFig.2.11),nosuchmomentum-conservingtransmissionispossible,exceptforcarrierswithmuchhigherenergy.Anothercalculationpointedoutthatsomelinedefectscanplaytheroleassemitransparent“valleyfilter”.Itwasfoundthatcarriersarrivingatthislinedefectwithahighangleofincidencearetransmittedwithavalleypolarizationnear100%[42].ManyexperimentalworkshavestudiedthetransportpropertiesofPoly-GandshowedthattheGBsgenerallydegradetheelectricalper-formanceofgraphene[43,44]andspecifically,theinterdomainconnectivityplaysanimportantroletocontroltheelectricalpropertiesofPoly-G,withtheelectricalconductancethatcanbeimprovedbyoneorderofmagnitudeforGBswithbetterinterdomainconnectivity[43].However,justafewtheoreticalworkshavestudied 202.ElectronicandTransportPropertiesinGraphenethecomplexstructuresofGBsandcorrespondingelectronictransport.InChap.4,byusingmoleculardynamics,wesimulatethePoly-Gwithvariablegrainsizes,andtunableinterdomainconnectivities,andreportonascalinglawfortransportproper-tiesofPoly-G,whichpointsoutthatthesemiclassicalconductivityandmeanfreepatharedirectlyproportionaltograinsizeandbotharestronglyaffectedbygrainconnectivity.However,aspointedoutinournextcalculation,theGBresistivityfornon-contaminatedPoly-Gisverylowcomparedtotheexperimentalresults[43,45,46].TheexplanationforthisproblemisthattheGBswhichcontainmanynonhexag-onalstructurehavegreaterchemicalreactivity[47]andareusuallyfunctionalizedbymanydifferenttypesofchemicaladsorbates.Thishasbeenconfirmedinsev-eralexperiments[48,49].ByusingthenumericalsimulationswereportontheroleplayedbychemicaladsorbatesonGBsinchargetransportinChap.4.2.4SpinTransportinGrapheneBesidemanyinterestingelectronicproperties,grapheneisalsoconsideredtobeapromisingcandidateforspintronicapplications.Thespinrelaxationtimeinintrinsicgrapheneisexpectedtobeverylongandthereforegraphenehashighpotentialasaspin-conserversystemwhichcantransmitspin-encodedinformationacrossadevicewithhighfidelity.Theunderlyingreasonforlongspinrelaxationtimeisthelowhyperfineinteractionsofthespinwiththecarbonnuclei(naturalcarbononlycontains1%13C)andtheweakSOCduetothelowatomicnumber[50].Thetheoreticalpredictionshowedthatthespinrelaxationtimeingrapheneisintheorderofmicroseconds.However,thereportedexperimentalspinrelaxationtimesremainseveralordersofmagnitudelowerthantheoriginaltheoreticalpredictions.BecausespinrelaxationbasedonthegrapheneintrinsicSOCcouldnotgiveaconvincingexplanation,otherextrinsicsourcesofspinrelaxationarebelievedtocomeintoplay.Proposalstoexplaintheunexpectedlyshortspinrelaxationtimesincludespindecoherenceduetointeractionswiththesubstrate,theextrinsicSOCinducedbyimpurities,adatoms,ripplesorcorrugations,etc.whichwillbereviewedbelow.Thepuzzlingcontroversyofthespinrelaxationmechanismwillbementionedinthenextsection.2.4.1Spin-OrbitCouplinginGrapheneInordertoderivetheSOCtermintheHamiltonian,itisnecessarytostartfromtherelativisticHamiltonian,theDiracequation:H|ψ=E|ψwith0cp.σmc20H=+2+V(2.32)cp.σ00−mc 2.4SpinTransportinGraphene21andwherethewavefunctionisatwo-componentsspinor:|ψ=(ψTA,ψB).FromtheDiracequationweobtaintwoequationsforspinorcomponents:cp.σψB=ψA(2.33)E−V+mc2c2p.σψ2p.σA=(E−V−mc)ψA(2.34)E−V+mc2Inthenonrelativisticlimit,thelowercomponentψBisverysmallcomparedtotheuppercomponentψA.Indeed,withtherelativisticenergyE=mc2+andVmc2,Eq.(2.33)driveustop.σψB=ψAψA(2.35)2mcandEq.(2.34)leadsustotheSchrodingerequation.1p2+VψA=ψA(2.36)2mInotherwords,inthefirstorderof(v/c),ψAisequivalenttotheSchrodingerwavefunctionψ.InordertoobtaintheanalogyofψAandψathigherorderof(v/c),weusethenormalizationcharacteristicofthewavefunction+ψ+=1(2.37)ψA+ψψBABTofirstorder,usingEq.(2.35),thisgivesp2ψ+1+ψA=1(2.38)A4m2c2p2Apparently,tohaveanormalizedwavefunction,weshoulduseψ=1+8m2c2ψA.c2SubstitutingthisintotheDiracequation,andusingtheexpansion2E−V+mc11−−V+···,weobtain,aftersomerearrangement,thePauliequation2m2mc2p2p42+V−−σ.p×∇V+∇2Vψ=ψ(2.39)2m8m3c24m2c28m2c2thefirstandthesecondtermsaretheusualtermsintheHamiltonian,thethirdtermissimplyarelativisticcorrectiontothekineticenergy.ThefourthtermistheSOCtermandthefinaltermgivetheenergyshiftduetothepotential.1Using(σ.A)(σ.B)=A.B+iσ.(A×B). 222.ElectronicandTransportPropertiesinGrapheneHereafter,IwillderivetheSOCterminthemoreintuitivewaywhichgivesthephysicalmeaningofSOCinteration.Supposeanelectronismovingwithvelocityvinanelectricfield−eE=−∇V.ThiselectricfieldmightbeinducedbythepotentialVoftheadatomsorthesubstrate.Inrelativistictheory,thismovingelectronfeelsamagneticfieldB=−v×Einitsrestframe.Theinterationbetweenthismagneticcfieldandtheelectronspinleadstothepotentialenergyterm:gsμBgsVμs=−μsB=−2ecσ.v×∇V=−4m2c2σ.p×∇V=−2m2c2σ.p×∇V(2.40)ThisresultsistwicetheSOCterminPauliequations.Actually,thiswasthemajorpuzzle,untilitwaspointedoutbyThomas[51]thatthisargumentoverlooksasecondrelativisticeffectthatislesswidelyknown,butisofthesameorderofmagnitude:electricfieldEcausesanadditionalaccelerationoftheelectronperpendiculartoitsinstantaneousvelocityv,leadingtoacurvedelectrontrajectory.Inessence,theelectronmovesinarotatingframeofreference,implyinganadditionalprecessionoftheelectron,calledtheThomasprecession.Asaresult,theelectron“sees”themagneticfieldatonlyone-halftheabovevaluev×EB=−(2.41)2cwhichleadstothefullSOCtermVSOC=−σ.p×∇V(2.42)4m2c2Nowlet’srewritetheSOCterminformoftheSOCforceFHSOC=α(F×p).s=−α(s×p).F(2.43)whereαisanundeterminedparameter.Hereweusesinsteadofσtorepresentthespindegreeoffreedomtoavoidanymisunderstandingwithpseudospiningraphene.Ifweconsiderintrinsicgraphene,theinversionsymmetrydictatestheelectricfield(force)inplaneandthisSOCiscalledintrinsicSOC.Becauseofstructure’smirrorsymmetrywithrespecttoanynearest-neighborbond(SeeFig.2.12a),thenearest-neighborintrinsicSOCiszero,whilethenextnearest-neighborintrinsicSOCisnonzero.Accordingtosymmetry,2iHI=iγ2F//×d.s=√VIs.(dˆkj×dˆik)(2.44)ij3whereγ2andVIareundeterminedparameters,anddˆijistheunitvectorfromatomjtwoitsnext-nearestneighborsi,andkisthecommonnearestneighborofiandj 2.4SpinTransportinGraphene23Fig.2.12SOCingraphene:aIntrinsicSOCforces.bRashbaSOCforceInthepresenceoftheoutofplaneelectricfield(SeeFig.2.12b)whichcanoriginatefromagatevoltageorchargedimpuritiesinthesubstrate,adatoms,etc.,thebandstructureofgraphenechanges.Thisexternalelectricfieldbreaksspatialinversionsymmetryandcausesanearest-neighborextrinsicSOC.ThisSOCisRashbaSOCandhastheformHR=iγ1s×dˆij.F⊥ez=iVRzˆ.(s×dˆij)(2.45)wherejisthenearest-neighborofiandγ1andVRareundeterminedparameters.Finally,wegettheTBHamiltonian:2i+++zˆ.(s×dˆH=−γ0ccj+√VIcs.(dˆkj×dˆik)cj+iVRcij)cjiii3ijijij(2.46)ByperformingFouriertransformations,weobtainthelowenergyeffectiveHami-toniamaroundtheDiracpointinthebasis{|A,|B}⊗{|↑,|↓}h(k)=h0(k)+hR(k)+hI(k)(2.47)whereh0(k)=vF(ησxkx+σyky)⊗1shR(k)=λRησx⊗sy−σy⊗sxhI(k)=λIησz⊗sz(2.48)withFermivelocityv33F=γ0,RashbaSOCλR=VRandintrinsicSOCλI=√2233VI[52]. 242.ElectronicandTransportPropertiesinGrapheneFig.2.13ElectronicbandstructureofgraphenewithSOC(Figureistakenfrom[53])TheremarkablethingaboutSOCingrapheneisthattheSOCtermsaremomentum-independent.Thespindirectlycoupleswithpseudospininsteadofmomentumasinconventionalmetalsorsemiconductors,theusualSOCterm(k×s)issmallandcanbedisregarded.DiagonalizingtheHamiltonianinEq.(2.47)givestheelectronicbandsclosetotheDiracpoint[53,54]:μν(k)=μλR+ν(vFk)2+(λR−λI)2(2.49)whereμandν=±1arebandindexes.Ifweconsiderintrinsicgraphene,theRashbaSOCisvanishinglysmall,theintrin-sicSOCopensagap=2λI(SeeFig.2.13a).WhenRashbaSOCisturnedonbyinversionsymmetrybreaking(effectfromthesubstrate,theelectricfield,thecorru-gations,etc.),thecompetitionofRashbaandintrinsicSOCleadstogapclosing.Thegapremainsfinite=2(λI−λR)for0<λR<λI(Fig.2.13b).ForλR>λIthegapclosesandtheelectronicstructureisthatofazerogapsemiconductorwithquadraticallydispersingbands(Fig.2.13d).TheeigenfunctionscorrespondingtotheeigenvaluesinEq.(2.49)areηη−iηϕμν−λI−i(1+η)ϕ−iϕλI−μνψμν(k)=χ−|ηe,1+μχ+|−iηe,ie/CμννvFkνvFk√withtanϕ=ky/kxandthenormalizationconstant[53]Cμν=21+2η2λI−μν.Theexpectationvalueofthespin[53,54],vFkvF(k×zˆ)vFksμν(k)==n(k)(2.50)(vFk)2+(λI−μλR)2(vFk)2+(λI−μλR)2 2.4SpinTransportinGraphene25wheren(k)=(sinϕ,−cosϕ,0)istheunitvectoralongthespindirection,calledspinvector.Theremarkablecharacteristicofspininspin-orbitcoupledgrapheneinEq.(2.50)isthatitispolarizedin-planeandperpendiculartoelectronmomentumk.Themag-nitudeofspinpolarizationsvanisheswhenk→0.TheChap.5willshowthatthesebehaviorsareduetothefactthatspinandpseudospinarestronglycoupledclosetotheDiracpointwherethecouplingbetweenpseudospinandmomentumiszerobecauseofthedestructiveinterferencebetweenthethreenearest-neighborhoppingpaths.Andthisleadstothespin-pseudospinentanglement,thecomponentofnewspinrelaxationmechanismthatplaysamajorroleinspinrelaxationattheDiracpointinultracleangraphene.InthecaseofhighenergyvFkλR+λI,thepseudospinismainlycontroledbymomentumviah0(k)andalignsinthesamedirectionwithmomentum(inplane).SpinisdictatedbypseudospinviahR(k),asaconsequence,spinpolarizationforacertainmomentuminEq.(2.50)saturatesto1.Bysuccessiveunitaryrotationofh(k)firstintotheeigenbasisofh0(k)andthenintothespinbasiswithrespecttothedirectionn(k)aneffectiveBychkov-Rashba-type2×2Hamiltoniancanbeobtainedforbothholesandelectrons[9],h˜(k)=ν(vFk−λI)−νλRn(k).s(2.51)TheanalogyofthesecondterminaboveequationandtheoriginalBychkov-RashbaHamiltonianinsemiconductorheterostructuresHk=(k).s/2showsthatSOCcouplingingrapheneeffectivelyactsontheelectronsspinasanin-planemagneticfieldofconstantamplitudebutperpendiculartok.Inthiseffectivefieldthespinprecesseswithafrequencyandaperiodof[9]2λRπ=,T=(2.52)λRTheseresultswillbeobtainedagaininChap.5withthenumericalcalculationsofthereal-spaceorderNmethodimplementedforspin.Furthermore,wewillpointoutthatthisresultisonlyvalidathighenergy.Atlowenergythespin-pseudospinentanglementcomesintoplayandcreatesamorecomplicatedpicture.ThemagnitudeofSOCinteractionsisalsoamatteroflargeconcern.Itisacrucialfactortodeterminenotonlyquantitativelyspinrelaxationbutalsothemechanismatplay.ThenumericalestimatesforintrinsicSOCλIingrapheneremainsrathercontroversial.Atthebeginning,KaneandMele[56]estimatedthevalueof100µeV.ThisoptimisticestimatewasdrasticallyreducedbyMinetal.[57]tothevalueof0.5µeVbyusingmicroscopicTBmodelandsecond-orderperturbationtheory.ThisvaluewaslaterconfirmedbyHuertas-Hernandoetal.[50]withTBmodelandYaoetal.[58]withfirst-principlescalculations.AdensityfunctionalcalculationofBoettgerandTrickey[59],usingaGaussian-typeorbitalfittingfunctionmethod-ology,gave2µeV.ThreeRefs.[50,57,58]gavethesamevalueofλI,butthesecal-culationsonlyinvolvedtheSOCinducedbythecouplingofthepzorbitals(forming 262.ElectronicandTransportPropertiesinGrapheneFig.2.14Twopossiblehoppingpathsthroughsandporbitals(toppanels)andthroughdorbital(bottompanels)leadtothefirstandthesecondterms,respectivelyinEq.(2.53)(Figureistakenfrom[55])theπbands)tothesorbitals(formingtheσband).However,aspointedoutinRef.[55],thecouplingofthepzorbitalstothedorbitals(SeeFig.2.14)dominatestheSOCatK(K).Duetoafiniteoverlapbetweentheneighboringpzanddxz,dyzorbitals,theintrinsicsplittingλIislinearlyproportionaltothespin-orbitsplittingofthedstates,ξd(orbitalshigherthandhaveasmalleroverlapandcontributeless).Incontrast,duetotheabsenceofthedirectoverlapbetweenthepzandσ-bandorbitals,theusuallyconsideredspin-orbitsplitting[50,57,58]inducedbytheσ−πmixingdependsonlyquadraticallyonthespin-orbitsplittingofthepzorbital,ξp,givinganegligiblecontribution.2(εp−εs)9V22pdπλI9V2ξp+2(ε2ξd(2.53)spσd−εp)whereεs,p,daretheenergiesofs,p,dorbitals,respectivelyandVspσandVpdπarehoppingparametersoftheporbitaltothesanddorbital,respectively(Fig.2.14).ThisTBcalculationgavethevalueofintrinsicSOCλI=12µeV[55]andwasconfirmedbythefirstprinciplecalculation[53].Thesecalculationsalsoshowedthat 2.4SpinTransportinGraphene27Fig.2.15ArepresentativehoppingpathisresponsibleforRashbaSOCinEq.(2.54)(Figureistakenfrom[55])theRashbaterm(zeroinabsenceofelectricfield)istunablewithanexternalelectricfieldEwhichisperpendiculartographeneplane(Fig.2.15)2eEzsp√eEzsp3VpdπλRξp+3ξd(2.54)3Vspσ(εd−εp)(εd−εp)wherezspandzpdaretheexpectationvaluess|zˆ|pzandpz|ˆz|dz2,respectively,oftheoperatorzˆ.AllthesecalculationspredictedthattheRashbaSOCisdirectlyproportionaltotheelectricfieldE,buttheestimatedvaluesvarybyaboutanorderofmagnitudefrom5µeVinRef.[53]to47µeVinRef.[50]andto67µeVinRef.[57],foratypicalelectricfieldofE=1V/nm.Furthermore,AstandGierz[60]usedtheTBmodelanddirectlyconsideredthenearest-neighborcontributionfromtheelectricfieldandobtainedλR=37.4µeV.Ingeneral,theintrinsicSOCofgrapheneisveryweak,intheorderofµeVandisunmeasurable.ThismakessomephenomenasuchasQSHeffectunobservableingraphene,thematerialinwhichitwasoriginallypredicted[56].AwaytoobserveQSHeffectingrapheneisendowingitwithheavyadatomswhichincreaseSOCingraphene.ThisproblemwillbementionedinChap.5.2.4.2SpinTransportinGrapheneThegrapheneSOCintheorderofµeVasmentionedaboveshouldleadtospinrelaxationtimesinthemicrosecondscale[9].However,theexperimentalresultsis 282.ElectronicandTransportPropertiesinGrapheneintheorderofnanoseconds,severalordersofmagnitudelowerthantheoriginaltheoreticalprediction.Inordertoclarifythelimitationsandmechanismsforspinrelaxationingraphenealotofefforthasbeendonebybothexperimentalistsandtheoreticians,butuptonowthistopicisstillunderdebate.ThefirstmeasurementofelectronspinrelaxationwasperformedbyTombrosetal.[2]usingthenon-localspinvalvemeasurementandHanlespinprecessionmethodtostudyspinrelaxationinmechanicalexfoliatedsingle-layergraphene(SLG)onSiO2substratewithmobilityofthedevicesabout2,000cm2V−1s−1.Theyextractedthespinrelaxationtimeoffewhundredsofpsandspinrelaxationlengthoffewµmatroomtemperature,similartowhatonemightexpectforconventionalmetalsorsemiconductors.Thisvaluehasbeenconfirmedbyseveralmeasurements[7,61].Thespintransportwasfoundtoberelativelyinsensitivetothetemperatureandweaklydependentonthedirectionofspininjectionandchargedensity.DuetothefastspinrelaxationwasattributedtotheextrinsicSOCinthesubstrateandthewaytogrowgraphene,spinmeasurementsinmanyotherkindsofgrapheneandsubstrateshavebeenreported.ThemeasurementofspinrelaxationonepitaxiallygrowngrapheneonSiC(0001)[62]isthefirstreportofspintransportingrapheneonadifferentsubstratethanSiO2.Thevalueofspinrelaxationτswasobtainedintheorderoffewnanoseconds,oneorderofmagnitudelargerthaninexfoliatedgrapheneonSiO2.HoweverthespindiffusioncoefficientDs≈4cm2/sisabout80timessmaller,yieldingto70%lowervalueforspinrelaxationlengthλs.ThelongerτsbutmuchsmallerDswaslaterexplainedbytheinfluenceoflocalizedstatesarisingfromthebufferlayerattheinterfacebetweenthegrapheneandtheSiCsurfacethatcoupletothespintransportchannel[63].ThemeasurementalsoreportedthatτsisweaklyinfluencedbythetemperaturewithreductionsofDsbymorethan40%andτsbyabout20%atroomtemperature.WiththeexpectationthatremovingtheunderneathsubstratehelpstoreducetheextrinsicSOCandleadstolongspinrelaxationtime,thespinmeasurementonsuspendedgraphenewasperformed[64].Althoughahighmobilityμ≈105cm2V−1s−1,anincreaseuptoanorderofmagnitudeinspindiffusioncoefficient(Ds=0.1m2/s)comparedtoSiO2supportedgrapheneandlongmeanfreepathintheorderofaµmwereobserved,indicatingthatmuchlessscatteringhappens,thespinrelaxationtimeremainsafewhundredsofpsandspinrelaxationlengthfewµm.OthergroupusedCVDmethodtogrowgrapheneoncopper(Cu)substrateandstudiedtheeffectofcorrugationonspinrelaxationtime[65].Theyobservedthesamespinrelaxationtimeasinexfoliatedgrapheneandshowedthatripplesingrapheneflakeshaveminoreffectsonspintransportparameters.Thenatureofspinrelaxationisactuallyafundamentaldebatedissue.TheDP[1,66,67]andtheEY(EY)[68,69]aretwomechanismsusuallydiscussedinthecontextofgraphene.TheEYmechanismisasuitablemechanismforspinrelax-ationinmetals.IntheEYmechanism,electronspinchangesitsdirectionduringthescatteringeventthankstotheSOCwhichproducesadmixturesofspinandelec-tronmomentuminthewavefunctions.Duetotheseadmixtures,scatteringchangeselectronmomentumandinducesspin-flipprobabilityatthesametimeandleadstoatypicalscalingbehaviorofspinrelaxationtimewithmomentumrelaxationtimeτsEY∼τp.Ontheotherhand,DPmechanismisanefficientmechanismformaterials 2.4SpinTransportinGraphene29withbrokeninversionsymmetry.Inthesekindsofmaterials,SOCinducesaneffectivemomentum-dependentmagneticfieldaboutwhichelectronspinprecessesbetweenscatteringevents.Thelongertimeelectrontravels,thelargerangleelectronspinprecessesandasaconsequence,themorespindephasingbetweenelectronsintheensembleisaccumulated.Therefore,spinrelaxationtimeisinverselyproportionaltoelasticscatteringtimeτsDP∼τp−1.W.HanandR.K.Kawakamiperformedsys-tematicstudiesofspinrelaxationinSLGandbilayergraphene(BLG)spinvalveswithtunnelingcontact[61].TheyfoundthatinSLG,thespinrelaxationtimevarieslinearlywithmomentumscatteringtimeτp,indicatingthedominanceofEYspinrelaxationwhereasinBLG,τsandτpexhibitaninversedependence,whichindicatesthedominanceofDPmechanism.However,Pietal.reportedasurprisingresultthatτsincreaseswithdecreasingτpinthesurfacechemicaldopingexperimentwithAuatomsongraphene[5],indicatingthattheDPmechanismisimportantthere.Thisexperimentledtotheconclusionthatchargedimpurityscatteringisnotthedominantmechanismforspinrelaxation,despiteitsimportanceformomentumscattering.Evenmorepuzzling,Zomeretal.[70]performedspintransportmeasurementsongraphenedepositedonboronnitridewithmobilitiesupto4.104cm2V−1s−1andshowedthatneitherEYnorDPmechanismsaloneallowforafullyconsistentdescriptionofspinrelaxation.Furthermore,electronspinisexpectedtorelaxefasterinBLGthaninSLGbecausetheSOCinBLGisoneorderofmagnitudelargerthantheoneinSLGduetothemixingofπandσbandsbyinterlayerhopping[71],buttheexperimentalresultsshowedanoppositebehavior[61,72].ThespinrelaxationtimeinBLGhasbeenreportedintheorderoffewnanosecondsandshowthedominanceofDPspinscattering[61,72].Nowwelookatbothmechanismsinmoredetail.DPmechanism:AsonecanseefromEq.(2.51),electronprecessesabouttheeffectivemagneticfieldinplaneB(k)∼(k)betweenscatteringevents.Randomscatteringinducesmotionalnarrowingofthisspinprecessioncausingspinrelaxation(SeeFig.2.16).Thespinrelaxationratesfortheα-thspincomponentfollowingtheDPmechanismare[9]1∗22DP=τ(k)−α(k)(2.55)τs,αwhereτ∗isthecorrelationtimeoftherandomspin-orbitfield.Ingraphenethisvaluecoincideswithmomentumrelaxationtimeτ∗=τp[9,73]andthesymbol2(k)=(2λ2···expressesanaverageovertheFermisurface.BecauseofR/),2(k)=0and2(k)=1(2λ2zx,y2R/),theDPrelationforspinrelaxationingrapheneis[1,9]22τDP=,andτDP=2τDP=(2.56)s,z2s,{x,y}s,z24λτp2λτpRR 302.ElectronicandTransportPropertiesinGrapheneFig.2.16DPspinrelaxationingraphene:aDiraccoinwhenSOCisincluded.bB(k)alongtheFermicircle.cChargedimpuritiesinsubstrateinduceelectricfieldingraphene.dIllustrationofthespinrelaxationinaspatiallyrandompotentialduetothechargedcarriers.eCalculatedspinrelaxationtimeτsasafunctionoftheFermienergyEf(Figureistakenfrom[9])Becausethespinrelaxationtimeisinverselyproportionaltothemomentumrelax-ationtime,theDPspinrelaxationlengthisindependentofmeanfreepath[1].1vFλs=Dτs=v2τpτs=√(2.57)2F22λRTheanalyticalestimatesandMonteCarlosimulations[9]withDPmechanismshowthatthecorrespondingspinrelaxationtimesarebetweenmicro-tomilliseconds(SeeFig.2.16)severalordersofmagnitudelargerthentheexperimentalresults.EYmechanism:Asmentionedabove,intrinsicSOCobtainedbyTBmodelanddensityfunctionalcalculationisintheorderoftensµeV[50,55,57,58],muchsmallerandcanbeneglectedincomparisontotheRashbaSOC.InthecaseofslowlyvariedRashbaSOCinducedbyelectricfieldorripples,theHamiltoniancanbewritteninformH=−ivFσ.+λR(σ×s)(2.58)BecauseoftheRashbaSOC,Blochstateswithwell-definedspinpolarizationarenolongereigenstatesoftheHamiltonian.TheBlocheigenstatesofaboveHamiltonianare[69]1k±eiθk=⊗|↑±ivFk⊗|↓eikr.(2.59)k,±k±eiθke2iθkvFkwhereθk=arctan(ky/kx)andtheenergyk±=±λR+(vFk)2+λ2isRobtainedfromEq.(2.49)withλI=0.WhenλR=0,eigenstatesinEq.(2.59) 2.4SpinTransportinGraphene31Fig.2.17SketchofscatteringbyapotentialU(r)inthechiralchannels(Figureistakenfrom[69])havetheirspinpointingalong(helicity+)oroppositeto(helicity−)thedirectionofmotion.ThisisnottruewhenλR=0butinthecaseofλR/F1,usingperturbationtheorywecanidentifyeachoftheseeigenstateswithchiralstates±[69].Let’sconsidertheBornapproximationofthescatteringproblemofelectroninthegrapheneunderthelocalscatteringpotentialU(r)whichisdiagonalinthesublatticeandspindegreesoffreedom.Thescatteringamplitudesf±0(θ)forchiralchannels±ofanincomingelectronwithpositivechiralityinthecaseofλR=0are(Fordetailderivation,seeRef.[69])f0(θ)=−(v−1k−iθ+F)8πUqe(1+cosθ)f0(θ)=−(v−1k−iθ−F)8πUqiesinθ(2.60)whereUqistheFouriertransformationofthescatteringpotentialevaluatedatthetransferredmomentumq=k−kandangleθ(seeFig.2.17)betweentheoutgoingmomentumkandincomingmomentumk.WhenRashbaSOCisturnedontheseamplitudesbecomeλR(θ)=−(v−21−iθf+F)8πk+(+(−2λR)cosθ)Uq+efλR(θ)=−(v−21−iθ−F)8πk−(+2λR)Uq−iesinθ(2.61)wherek±=(vF)−12∓2λRandq±=k±−k.Letusdefinetheprobabilityforaspin-flipprocessfromthechangesinthescat-teringinbothchiralchannelsduetothepresenceoftheSOC.λ0R0±1|f±(θ)||f±(θ)−f±(θ)|S(θ)=(2.62)±1|f±0(θ)|2 322.ElectronicandTransportPropertiesinGrapheneThisistheamountofspinrelaxedinthedirectiondefinedbyθ.ThetotalamountofspinrelaxationduringascatteringeventcanbedefinedastheaverageofthisquantityovertheFermisurface:1S=S(θ)=dθS(θ,=F)(2.63)2πλR(θ)−f0(θ)∼λItiseasytoseethatf±±R/FfromexpandingofEq.(2.61)inpowersofλR/F.ThisimpliesthatS(θ)∼λR/FwhichisindependentwiththescatteringpotentialsU(r).ThisresultwasobtainedinRef.[1]forthecaseofweakscatterers,andlaterinRef.[69]forthecasesofscatteringbyboundary,strongscatterersandclustersofimpuritieswhichcannotbetreatedinperturbationtheory.Assumingthisbehavior,theEYrelationforgraphenecanbeeasilyfound.Indeed,thechangeofspinorientationateachcollisionisS∼λR/F.Thetotalchange√ofspinorientationafterNcolcollisionsisoftheorderofNcolF/λR.Dephasing√occurswhenNcolF/λR∼1andofcourse,afteratimeτsEY=Ncolτp.HenceweobtaintheEYrelation2EY≈Fττs2p(2.64)λRThisistheEYrelationforgraphene.ItisworthtomentionthatthespinrelaxationtimeτsherenotonlyisproportionaltomomentumrelaxationtimeτpbutalsodependsonthecarrierdensitythroughFermienergyF.ThespinrelaxationlengthinEYmechanismisproportionaltomeanfreepathe1Fλs=Dτs=v2τpτs∼e√(2.65)2F2λRDespitethefactthatsomeexperimentshavereportedthatτs∼τp,indicatingthedominanceofEYmechanisminspinrelaxationingraphene,thediscrepancybetweentheoreticalcalculationsandexperimentaldataisstilllarge.Furthermore,thederivationsofbothEYandDPforgraphenearebasedonthestrongcouplingofmomentumandpseudospinwhichisunsuitableclosetotheDiracpoint.InChap.5,weproposeanewmechanismwhichistheheartofthisPhDthesistoexplainsthefastspinrelaxationingraphenebytheentanglementofspinandpseudospindegreesoffreedom.References1.D.Huertas-Hernando,F.Guinea,A.Brataas,Phys.Rev.Lett.103,146801(2009)2.N.Tombrosetal.,Nature(London)448,571(2007)3.N.Tombrosetal.,Phys.Rev.Lett.101,046601(2008)4.C.Jozsaetal.,Phys.Rev.Lett.100,236603(2008)5.K.Pietal.,Phys.Rev.Lett.104,187201(2010) 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Chapter3TheRealSpaceOrderO(N)TransportFormalismQuantumssimulationsareveryimportanttoolstostudytransportphenomenainthenanoscale.Therearetwonumericalapproachesforquantumtransportsimulationsatthepresent,oneisthewidelyusednon-equilibriumGreensfunction(NEGF)method,theotheristheKubo-Greenwoodmethod.WhileNEGFisusuallyusedforsmallsystemssuchascarbonnanotubes(CNT),graphenenanoribons(GNRs),duetothecubic-scalingtimeconsumption,thelinear-scalingKubo-Greenwoodquantumtransportsimulationmethodisaveryeffectivemethodtoinvestigatethetransportpropertiesofthelarge-scaledisordersystems.Inthischapter,thetheoreticalbackgroundofKubo-Greenwoodformalism,thereal-spaceKuboformulasforconductivityandtheEinsteinrelationsarederivedinthefirstsection.Attheendofthissectionthethreedifferentregimesoftransportisdiscussed.Inthesecondsection,anewformalismbasingontherealspaceorderO(N)isfirstlydevelopedtostudythespintransportinlargescale2Dsystem.ThismethodisappliedinChap.5tostudythespintransportindisorderedgraphene.3.1ElectricalTransportFormalism3.1.1ElectricalResistivityandConductivityWhenthereisaelectricfieldE(ω)insideamaterial,itwillcauseelectriccur-renttoflow.Electricalresistivityρ(ω)isameasureofhowstronglyamaterialopposestheflowofelectriccurrent.Alowresistivityindicatesamaterialthatreadilyallowsthemovementofelectriccharge.Theelectricalresistivityisdefinedastheratiooftheelectricfieldtothedensityofthecurrentj(ω),implyingE(ω)j(ω)=(3.1)ρ(ω)©SpringerInternationalPublishingSwitzerland201635D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2_3 363TheRealSpaceOrderO(N)TransportFormalismConductivityσ(ω)istheinverseofresistivity1σ(ω)=(3.2)ρ(ω)Therefore,wehavej(ω)=σ(ω)E(ω)(3.3)Weusuallymeasuretheresponseofsystemtotheelectricfieldalong1direction(ex.thexdirection).Inthiscase,theconductivityσ(morespecificallyσxx)inthisdirectionis:jx(ω)=σ(ω)Ex(ω)(3.4)σ(ω)istheconductivityinthegeneralcase.Forthedirectcurrent(DC),theDCconductivityσDCcanbeobtainbysettingω→0.3.1.2SemiclassicalApproachFirstly,letsusethesemiclassicalapproachtohavethegeneralpictureofthemotionofelectroninthesystemundertheinfluenceofelectricfield.InthissectionsomeformulassuchasDrudeconductivity,Einsteinrelation,andLandauerformulaarederivedwhichwillgiveabettervisionofthequantumapproachwhichwillbegiveninthenextsection.InthepresenceofanelectricfieldEintheplaneofthetwodimensionalelectrongas(2DEG)system,besidethermalmotion,electronmovesalongthedirectionofelectricforce.However,thisdriftmotiononlyremainsforashorttimebeforeitsdirectionisrandomizedbyscatteringondisorder.Anelectronacquiresadriftvelocityvdrift=−eEt/minthetimetsincethelastimpuritycollision.Theaverageoftisthescatteringtimeτp(ormomentumrelaxationtime),sotheaveragedriftvelocityvdriftisgivenby[1]eτpvdrift=−μeE,μe=(3.5)mwhereμeiselectronmobility.Ifthesheetdensityisnsthenthecurrentdensityisj=−ensvdrift=σE(3.6)TheresultisthefamiliarDrudeconductivity[2]whichcanbewritteninseveralequivalentforms:ensτpe2kFlσ=ensμe==gsgv(3.7)mh2 3.1ElectricalTransportFormalism37Inthelastequalitywehaveusedtheidentityns=gsgvk2/4πwhichistrueforFall2DEGsystemincludinggraphene[3]andhavedefinedthemeanfreepathe=vFτp.Thevalleydegeneracyfactorsaretypicallygv=2forgraphene(KandK)andSi100based2DEGsystem,whereasgv=1(6)for2DEGsysteminGaAs(Si111).Thespindegeneracyisalwaysgs=2,exceptathighmagneticfields.Itisobviousthatthecurrentinducedbytheappliedelectricfieldiscarriedbyallconductionelectrons,sinceeachelectronacquiresthesameaveragedriftvelocity.Nonetheless,todeterminetheconductivityitissufficienttoconsidertheresponseofelectronsneartheFermileveltotheelectricfield.ThereasonisthatthestatesthataremorethanafewtimesthethermalenergykBTbelowEFareallfilledsothat,inresponsetoaweakelectricfield,onlythedistributionofelectronsamongstatesatenergiesclosetoEFischangedfromtheequilibriumFermi-DiracdistributionE−EF−1f(E−EF)=1+ekBT(3.8)Inthermodynamicequilibriumatzerotemperature,whichischaracterizedbyaspatiallyconstantelectrochemicalpotentialμ,thesumofthedriftcurrentdensity−σE/eandthediffusioncurrentdensity−D∇ns(Disdiffusionconstant)vanishes−σE/e−D∇ns=0when∇μ=0(3.9)Theelectrochemicalpotentialμisthesumoftheelectrostaticpotentialenergy−eVandthechemicalpotentialatEF.SincedEF/dns=1/ρ(EF),onehas∇ns∇μ=−e∇V+∇EF=eE+(3.10)ρ(EF)ThecombinationofEqs.(3.9)and(3.10)yieldstheEinsteinrelationfortheconduc-tivityσσ=e2ρ(EF)D(3.11)ToverifythatEq.(3.11)isconsistentwiththeearlierexpression(3.7)fortheDrudeconductivity,onecanusetheresult(seebelow)forthe2Ddiffusionconstant:121D=vFτp=vFl(3.12)22incombinationwiththedensityofstates:ρ(E)=gsgvE/2π(vF)2forgrapheneandρ(E)=gsgvm/2π2for2DEGsystems. 383TheRealSpaceOrderO(N)TransportFormalismTheresultinEq.(3.12)canbeexplainedinthefollowingway[1].Considerthediffusioncurrentdensityjxinducedbyasmallconstantdensitygradient,n(x)=n0+cx.Wewritejx=limvx(t=0)n(x(t=−t))t→∞=limcvx(0)x(−t)t→∞t=lim−cdtvx(0)vx(−t)t→∞0wherethebrackets...denoteanisotropicangularaverageovertheFermisurface.Thetimeintervalt→∞,sothevelocityoftheelectronattime0isuncorrelatedwithitsvelocityattheearliertime−t.Thisallowsustoneglectatx(−t)thesmalldeviationsfromanisotropicvelocitydistributioninducedbythedensitygradient[whichcouldnothavebeenneglectedatx(0)].Sinceonlythetimedifferencemattersinthevelocitycorrelationfunction,onehasvx(0)vx(−t)=vx(t)vx(0).WethusobtainforthediffusionconstantD=−jx/cthefamiliarlinearresponseformula[4]∞D=dtvx(t)vx(0)(3.13)0Since,inthesemiclassicalrelaxationtimeapproximation,eachscatteringeventisassumedtodestroyallcorrelationsinthevelocity,andsinceafractione−t/τpoftheelectronshasnotbeenscatteredinatimet,onehas(in2D)theresultinEq.(3.12)∞∞2−t/τp12−t/τp12D=dtvx(0)e=vFedt=vFτp(3.14)0202Theconductanceratherthantheconductivityisusuallymeasuredinexperiments.Theconductivityσrelatesthelocalcurrentdensitytotheelectricfield,j=σE,whiletheconductanceGrelatesthetotalcurrenttothevoltagedrop,I=GV.Becausetheconductancefor2DEGsystemofwidthWandlengthLinthecurrentdirectionisWG=σ(3.15)LSotheconductanceisidenticaltotheconductivityinalargehomogeneousconduc-tor(squaredsample)andisusuallycalledsquaredconductance.Ifwedisregardtheeffectsofphasecoherence,Eq.(3.15)isonlycorrectwhenthesamplesizesaremuchlargerthanmeanfreepath(W,Le).Thisisthediffusivetransportregime,illus-tratedinFig.3.1a.Whenthedimensionsofthesamplearereducedbelowthemeanfreepath,oneenterstheballistictransportregime,showninFig.3.1c.Onecanfurtherdistinguishanintermediatequasi-ballisticregime,characterizedbyW1×10−4e/atom)inred.dLocalDOSforatomsA1,A2andA3markedinpanel(c).eLocalDOSforatomA4markedinpanel(c)ascomparedtotheaverageDOSforpristinegraphene(PG)andaverageLDOSforallatomsatGBsinthesamesample(GB)Tobetterunderstandthedeviationsfromthepristinegrapheneforthewell-connectedstructures,wenextidentifiedatomsresidingatGBsofthe“18nm”samplebysearchingforatomsforwhichthebondlengthofatleastonenearestneighbordif-fersfromthecarbonspacinginpristinegraphene(aCC=1.42Å)by0.03Åormore.Wethencalculatedthelocalchargedensitydeficiencyδi(orself-doping)foreachGBsite.InFig.4.13cwepresenttheatomicstructureoftheelectron-holedensityfluc-tuations(δivariationsgreaterthan10−4electronsperatom)formedatasmallareaaroundoneGB.Theseself-dopingeffectsstemfromlocalfluctuationsintheelectro-staticpotential.Experimentsonexfoliatedgraphenedepositedoversilicondioxide 4.2ChargeTransportinPoly-G77[122,123]haveshownsimilarpotentialinhomogeneities;however,thesewerespreadoveramuchlongerscale(∼30nm)andwereinducedbyproximityeffectsgeneratedbychargestrappedintheoxide.Inourcase,averagingoverallcarbonatomsbelong-ingtotheGBsofthe18nmsamplegaveδGB=0.008electronsperatom,whichcorrespondstoameancarrierdensityofn(E=0)6.1×1011cm−2.(δfluctuatesbetween−0.096and0.08electronspercarbonatom,or,respectively,6.1×1012and−7.3×1012cm−2.)Thelocalchargedensityfluctuationsoccuronalengthscaleonlyafewtimeslargerthanthelatticespacing,whichisverysmallcomparedtothatinsupportedexfoliatedgraphene,suggestingamuchstrongerlocalscatteringefficiency.Wepointoutthatourresultsshownostraightforwardcorrelationbetweentheself-dopingvalueandthelocaldefectedmorphologyofthelattice.Figure4.13dshowstheplotofthecorrespondinglocalDOS(LDOS)ofthreeselectedatomsattheboundary(A1,A2andA3).Allofthemshowincreasedcontri-butionsofmidgapstates[16,17],significantlyreducedvanHovesingularities,andamarkedlyenhancedelectron-holeasymmetry,owingtotheodd-memberedcarbonrings[32].Theyalsoexhibitstrongresonantpeaks,whicharecharacteristicofquasi-localizedelectronicstatesinthevicinityofdefects.ThelocalelectronicconfigurationalongtheGBalsostronglydiffersfromonesitetoanother,aneffectarisingfromaninterferenceeffectbetweencoherentwavefunctionsoftheconnectedadjacentgrains.Inclearcontrast,anatomonlyfourlatticevectorsawayfromthebound-ary(A4)showsaLDOSnearlyindistinguishablefromthatofthepristinegraphene(Fig.4.13e).ComparisontotheaverageLDOScalculatedforallatomsattheGBsrevealsthatthechangesintheDOSseeninthepolycrystallinesamples(Fig.4.13a)ariselocallyfromtheatomicconfigurationsoftheGBsitself.Next,wediscussthetransportpropertiesofthesamples.Figure4.14ashowsthetimedependencyofthediffusioncoefficientD(t)attheDiracpointforallsamples.Ontheonehand,thewell-connectedsamplesdisplayaveryslowtime-dependentdecayofD(t)afterthesaturationvalue,indicatingweakcontributionofquantuminterferences.Ontheotherhand,thepoorlyconnectedsample“br-18nm”exhibitsamuchfasterdecay,eventuallydrivingtheelectronicsystemtoastronglocaliza-tionregime(asobservedinsometransportmeasurements[102]).Wenextdeducedthemeanfreepathe(E)fromthemaximumvaluesofD(t)(Fig.4.14b).Genuineelectron-holeasymmetryisapparentine(E),butonlyforenergies|E|>3eV(farfromtheexperimentallyrelevantenergywindow).Atlowerenergiesaroundthechargeneutralitypoint(|E|<1eV),e(E)changes,albeitonlyweakly,forallsamples.Thesamplewithbrokenboundaries,“br-18nm”,showstheshorteste<5nmandtheweakestdependenceonenergy,exceptforapronounceddipatE=0.Interestingly,thecurvesforthetwowell-connectedsampleswithsimilardbutdifferentd-distributions(“18nm”and“avg-18nm”)areverysimilarandclearlydifferentfromsampleswitheithersmallerorlargergrains.However,thisdifference√canbyaccountedforbyaconstantfactor.Remarkably,itturnsoutthat2×13nme≈√18nmeand2×18nme≈e25.5nm(seethescaledvaluesinFig.4.14b),whichcorrespondexactlytothedifferencesintheaveragegrainsizesinthesesamples 784TransportinDisorderedGraphene(a)(b)8nm13nmavg-18nm18nm13nmx25.5nm25.5nm/)-1fs2(nm))(nm60.6(ps)(c)(d)//Vs)2(cm/cmFig.4.14aDiffusioncoefficient(D(t))forthesamplespresentedinFig.4.12.bMeanfreepathe(E)forequivalentstructureswithscalede(E)forsampleswithd≈13nmandd≈25.5nm,showingthescalinglaw.cSemi-classicalconductivity(σsc(E))forallsamplesandasscaledforthesamecasesasabove.dChargemobility(μ(E)=σsc(E)/en(E))asafunctionofthecarrierdensityEn(E)ineachofthesamples(n(E)=1/Sρ(E)dE,Sbeinganormalizationfactor)0√√(2×13≈18and2×18≈25.5).Moreover,thegrain-sizedistributiondoesnot18nmavg−18nmenterintothisscalingbehaviour(e≈e).Hence,wehaveidentifiedaremarkablysimplescalinglawthatlinkstheaveragegrainsizetotransportlengthscalesinPoly-Gwithrandomlyorientedgrains.Thecomputedsemi-classicalconductivityσsc(E)exhibitsenergy-dependentvari-ationssimilartoe(E),ascanbeseeninFig.4.14c.WealsopointoutthelineardependencyofewithchargedensityintheDiracpointvicinity.Again,thesamescalinglaw(presentedaboveforthemeanfreepath)applies:theratioofσscfortwosampleswithdifferentaveragegrainsizesmatchescloselywiththeratioofthedvaluesthemselves.OneadditionalinterestingfeatureseeninFig.4.14cisthattheconductivityremainsmuchhigherthantheminimumvalue4e2/πh(horizontalline),whichfixesthetheoreticallimitinthediffusiveregime,asderivedwithinthe 4.2ChargeTransportinPoly-G79Table4.1MobilitiesforallsamplesatselectedchargedensitiesMobilities(cm2/Vs)13nm18nmAvg-18nm25.5nmbr-18nmμ(n=2.5×1011cm−2)5.1×1037×1036.8×1031044×103μ(n=2.5×1012cm−2)510700685950360μ(n=2.5×1013cm−2)6910510415045self-consistentBornapproximationvalidforanytypeofdisorder[43].Thisindi-catesthatPoly-Gremainsagoodconductor,evenforthepoorlyconnectedstructure“br-18nm”.Localizationlengthofelectronstates(ξ(E))cannowbeestimatedusingthevaluesforeandσsc.Scalinganalysis(ξ(E)=e(E)exp(πhσsc(E)/2e2)[13])revealsthatξ1−10µmoveralargeenergywindowaroundthechargeneutralitypoint.Thiscontrastswiththevalues(ontheorderof10nm)obtainedforgraphenestructureswith∼1%structuraldefects,stronglybondedadatoms,orothertypesofshortrangeimpurities[32,33].Finally,wemoveontothechargecarriermobilityμ(n)(Fig.4.14d).Asexpected,thepoorlyconnectedsample“br-18nm”showsthelowestmobility(reducedbyafactorofaboutthreewhencomparedtothewell-connectedsampleswithsimilard).Wepointoutthatthecomputedvaluesarevaliddowntothechargeneutralitypoint(thatis,tothesmallestchargedensityn(E)),sinceweaccountedforthedisorder-inducedfiniteDOS,whichyieldsanon-zerochargedensity(andthusnosingularityat1/n(E)).Table4.1givesthemobilitiesatseveralchargedensitiesforallstudiedsamples.Itisworthobservingthatthescalinglawalsoroughlyappliestochargemobilitiesversusaveragegrainsize,sincethesuperimposedeffectofdensityofstateschangestheratioonlybyafewpercent(forinstance,atn=2.5×1012cm−2,μ18nm/μ13nm≈1.37).Ifweextrapolatethemobilityforwellconnectedgrainsaccordingtoourscalinglawtoagrainsizeof1µmandachargedensityofn=3×1011cm−2asinthebestsamplesofRef.[100],weobtain300,000cm2V−1,whichisabouttentimeshigherthanthemeasuredvalues.Thisdiscrepancysuggeststhatsubstrate-relateddisordereffects,aswellassupplementarydefectsintroducedduringthetransferprocess,shouldaccountforanevengreaterlimitationforchargemobilitiesthantheactualGBmorphology.TheexistenceofmoredisorderedGBsasreportedinRefs.[87,100],orsampleswithoverlappinggrains,asobservedinRef.[124],yieldtolowermobilityvalues,whichhasbeenpartlyillustratedherewiththestructuralmodel“br-18nm”.MoreworkishoweverneededtodesignproperatomisticstructuralmodelsthatwillcaptureessentialgeometricalfeaturesofthosemorefragmentedstructuresofPoly-G.Inconclusion,wehavecreatedPoly-Gsampleswithnon-restrictedGBstructuresandrealisticmisorientationanglesandringstatistics.ThesesamplesenabledustoconfirmthesimplerelationshipbetweentheaveragegrainsizeandchargetransportpropertiesofintrinsicPoly-G.ThisscalinglawwillbeexplainedmorebelowinSect.4.2.5.3.ThedisorderscatteringstrengthinPoly-Gwasfoundtodependon 804TransportinDisorderedGraphenetheatomicstructureofGBs(inducingquasi-boundstatesatresonantenergy)andwavefunctionmismatchbetweenthegrains,whichgeneratestronglyfluctuating,buthighlylocalizedelectron-holedensityfluctuationsalongtheinterfacesbetweengrains.Ourresultssignificantlyimprovethepresenttheoreticalunderstandingontheinfluenceofthedetailedmorphologyofpolycrystallinematerialstotheirmeasurableelectronicproperties.TheyofferthepossibilityforestimatingchargemobilitiesinsuspendedCVD-graphenesamplesbasedontheaveragegrainsizesandqualityoftheGBs.Furthermore,theyestablishquantitativefoundationsforestimatingtheintrinsiclimitsofchargetransportinPoly-G,whichisofprimeimportanceforgraphene-basedapplicationsinthefuture.4.2.5MeasurementofElectricalTransportAcrossGGBsVariousmeasurementshavebeenmadetounderstandtheelectricalpropertiesofGGBs.Thesemeasurementsfallintothreeprimaryapproaches.Thefirstapproachinvolveslocaltwo-pointmeasurements,whichareaccomplishedwithSTMandscan-ningtunnelingspectroscopy(STS)[87,125–129].Withthesemeasurements,itispossibletodeducethelocalelectronicdensityofstates,thelocalchargedensity,andthechargescatteringmechanismsassociatedwithGGBs,thuspermittingthespatially-dependentelectricalcharacterizationofGGBsattheatomicscale.Thesecondapproachinvolvesfour-probemeasurements,whichcanbeusedtoanalyzetheinfluenceofindividualGGBsatascaleofseveralmicrometers[100–102].Bysubtractingthecontributionofeachgraphenegrainfromaninter-grainresistancemeasurement,theresistivityofasingleGGBcanbeestimated.Incombinationwithmicroscopicorspectroscopictechniques,thisapproachallowsonetocorrelatetheresistivityofasingleGGBwithitsstructuralorchemicalproperties.Finally,theglobalimpactofGGBscanbestudiedbymeasuringthesheetresistanceofPoly-Gsamplesoverawiderangeofaveragegrainsizesanddistributions,whicharetunablebytheCVDgrowthconditions.Byemployingasimplescalinglaw(asdiscussedbelow),itisthenpossibletoextrapolatetheaverageGBresistivity[89,130].Takentogether,thesemeasurementtechniquesprovidetheelectricalcharacterizationofGGBsatvariouslengthscales,thushelpingtorevealacomprehensivepictureofchargetransportinPoly-G.Amoredetailedoverviewofthesemethodsisgivenbelow.4.2.5.1Two-ProbeMeasurementsTwo-probeSTMandSTStechniquescanbeusedtolocallystudytheelectricalpropertiesofGGBs[87,125,127–129].ByvaryingthevoltageandpositionoftheSTMtip,itispossibletodeterminethenatureoflocalizedstates,thechargedoping,andthelocalscatteringmechanismcorrespondingtoagivenmorphologyofthe 4.2ChargeTransportinPoly-G81Fig.4.15Two-probemeasurementofGGBs.aDifferentialtunnelingconductanceatvariouspointson(bluelines)andaround(redlines)aGGB.TheappearanceofdefectstatesisevidentontheGGBs.Reproducedwithpermission[129].Copyright2013,ElsevierPublishing.bSTMimageoftheGGBstudiedinpanel(a)wherethecoloreddotsindicatethepositionsofdI/dVmeasurements.cdI/dVmapacrossaGGB.dLocationofthedI/dVminimumasafunctionoftipposition,indicatingthepresenceofanelectrostaticbarrierattheGGB.Reproducedwithpermission[128].Copyright2013,ACSPublishingGGB.OneexampleofsuchanalysisisshowninFig.4.15a,b[129].Figure4.15ashowsthedifferentialtunnelingconductance,dI/dV,takenatvariouspointson(bluecurves)andnextto(redcurves)aGGBinCVD-growngraphene.ASTMprofileoftheGGBandthepointswherethemeasurementsweremadeisshowninFig.4.15b.TheseresultsindicatethepresenceofapeakinthetunnelingconductanceneartheDiracpointwhenevertheSTMtipliesontopoftheGGB.Meanwhile,thispeakdoesnotappearformeasurementsawayfromtheGGBs.Densityfunctionaltheory(DFT)calculationshaveattributedthispeaktothelocalizedstatesarisingfromtwo-coordinatedcarbonatomsintheGGBs[129].TheSTMmap(notshownhere)alsorevealsinterferencesuperstructuresduetoscatteringfromtheGGBs,indicatingthecontributionofsignificantinter-valleyscattering.Thissupportsthehypothesisaboutthepresenceoftwo-coordinatedatoms,sinceinter-valleyscatteringstemsfromatomic-scalelatticedefects[35].Figure4.15cshowsanothermapofdI/dVcurvesastheSTMtipisscannedacrossaGGB[22].SimilartoFig.4.15a,anenhancedlocaldensityofstatesisobservedatpositivevoltagewhenthetipislocatedovertheGGB.ThevoltageassociatedwiththeminimumofdI/dV,asshowninFig.4.15d,indicatesastrongnegativeshiftaroundthepositionoftheGGB,revealingn-typedopingoftheGGBcomparedtobulkp-typedopingofthegraphenegrains.ThisshiftindopingcorrespondstoanelectrostaticpotentialbarrierofafewtensofmeV.Finally,STMinterferencepatternsindicatethatsomeGGBsaredominatedbyinter-valleyscatteringwhileothersaredominatedbybackscattering.Thetypeofscatteringappearstodependon 824TransportinDisorderedGraphenethestructureoftheGGB,whereaGGBconsistingofacontinuouslineofdefectsshowsprimarilybackscatteringbehaviorandaperiodiclineofisolateddefectsisdominatedbyinter-valleyscattering.OtherSTMstudiesofGGBsrevealsimilarresultstothosementionedabove,withGGBsformingp−n−porp−p−pjunctionswiththebulk-likegraphenegrains,wherepρGB/lGforthelargestgrains).SSForexample,basedontheextractedvaluesofRGandρGB,theGGBsbegintoSdominatethesheetresistanceofthesesampleswhentheaveragegrainsizeislessthanlG=ρGB/RG≈10µm.ThisinformationcanserveasausefuldesignparameterSwhenconsideringlarge-scaleapplicationsofPoly-G.4.2.6ManipulationofGGBswithFunctionalGroups4.2.6.1ChemicalReactivityofGGBsInadditiontothegeneralelectricaltransportpropertiesofPoly-G,thechemicalprop-erties(reactivity,functionalization,etc.)ofGGBshavebeenextensivelydiscussed.Forexample,ithasbeenshowntheoreticallythatnon-hexagonalatomicarrange-mentsingraphene,suchastheStone-Walesdefect,yieldhigherchemicalreactivitythantheidealhexagonalstructure[136–140],andthisbehaviorhasbeenextendedtoGGBs.AschematicrepresentationisshowninFig.4.20a,whereoxygenatomspref-erentiallyattachtothenon-hexagonalsiteslocatedintheGGBs.SelectiveoxidationofGGBscanbedemonstratedbytransferringCVDgraphenetoamicasubstrateandheatingthesamplefor30minat500◦C.ThisprocessselectivelyburnsawaytheGGBs[116],givingaccesstothegrainmorphologywithinthesampleswithAFM.ArepresentativeAFMimageisgiveninFig.4.20b,wherethedarklinesindicatethelocationoftheremovedGGBs.ThisprocedurenotonlyprovidesasimplemeansofcharacterizingthegrainmorphologyinthesamplesbutalsohighlightstheenhancedchemicalreactivityoftheGGBs.UVtreatmentofPoly-Gonacoppersubstratecanalsorevealselectivefunction-alizationoftheGGBs[89].Underhumidenvironment,OandOHradicalsgeneratedbytheUVlightpreferentiallyattachtotheGGBs,makingthedefectsattheGGBsinert.Thisallowsnextincomingradicalstodiffusethroughlarge-poreheptagonsandhigher-orderdefectstoeventuallyoxidizeandexpandtheunderlyingcoppersubstrate,asexplainedabove.Thedegreeofvolumeexpansioncanbeengineeredbycontrollingoxidationtimes,andthemorphologicalchangesaroundGGBsareeasilyidentifiedbyAFMandopticalmicroscopy.ThedarklinesinFig.4.20crevealthegrainstructureofthePoly-G.ThegrainstructureisalsorevealedviaRamanmap-pingofthesample,asshowninFig.4.20d–g.Figure4.20doutlinestheformationofastrongD-bandassociatedwiththeGGBsafterUVtreatment.TheD-peakalsoformswithinthegraphenegrains,butitsmagnitudeismuchsmaller,highlightingthehigherchemicalreactivityoftheGGBs.RedshiftsoftheGand2D(G)bandsintheGGBsafterUVtreatmentareattributedtostraininducedbytheoxidizedcopperbelowtheGGBs.Figure4.20e–gshowthatafterUVtreatment,spatialmappingsoftheD,G,and2DpeakscorrelatewellwiththeopticalimageoftheGGBs.ItshouldbenotedthatRamanmappingshowsnoevidenceoftheGGBspriortoUVtreatment,indicatingthestronginfluencebytheoxidationoftheGGBs. 884TransportinDisorderedGrapheneFig.4.20ChemicalreactivityofGGBsbyexperiments.aRepresentationofselectivechemicalfunctionalizationofGGBs.bThelocationofGGBscanbeimagedwithAFMafterburningthemawayathightemperature,whichhighlightstheirselectiveoxidation.Reproducedwithpermission[116].Copyright2011,AIPPublishing.cAnopticalimageofPoly-Gindicatestheselectiveoxi-dationofanunderlyingcoppersubstratebelowtheGGBs.dRamanspectroscopyindicatesthestrongoxidationattheGGBsafterUVtreatment.e,fRamanmappingindicatesstrongoxidationoftheGGBs(D-band),aswellasstrainduetotheexpansionoftheoxidizedcoppersubstratebelowtheGGBs(GandGbandshifts).Reproducedwithpermission[89].Copyright2012,NaturePublishingGroup 4.2ChargeTransportinPoly-G89TheexperimentaldemonstrationsofthechemicalreactivityofGGBsreportedtodatesuggestthatPoly-Gmaybeagoodmaterialforthedevelopmentofchemicalsensors.Forexample,gassensorsbasedonpristine(single-grain)andPoly-Ghaveyieldedhighlydifferentresponsestotolueneand1,2-dichlorobenzene,withthePoly-Gsensorshowingaresponse50timesgreaterthanthatofpristinegraphene[75].ThisimprovementinthesensitivityofthesensorisattributedtotheincreasedreactivityoftheGGBsandtheenhancedimpactthatlinedefectshaveoverpointdefectsontransportfeaturesintwodimensions.ThishighlightsthecombinedrolethatchemistryandchargetransportplayintheelectricalpropertiesofPoly-G.4.2.6.2SelectiveFunctionalizationofGGBsAsdescribedabove,GGBsaremorechemicallyactivethanthegraphenebasalplane.However,selectivefunctionalizationofGGBswithanappropriatereactantisstillanon-goingareaofresearch.OurmainconcernisaselectivefunctionalizationofGGBs,althoughdefectsinsidegraincouldbefunctionalizedaswell.Thewholegraphenelayerstillretainsmetallicitywithslightlyincreasedsheetresistance.Thisisgoodcontrastwithheavilyfunctionalizedgrapheneoxidethatleadstoaninsula-tor.Ozoneisagoodcandidateforthispurposebecauseitisinertwiththegraphenebasalplane[141,142].Figures4.21and4.22showsmeasurementsoftheelectricalreponseofthegraphenebasalplaneandGGBstoozonegeneratedbyUVexposureunderanO2environment.Afour-probedevicewasfabricatedonthemergedregionoftwographenedomains(describedinFig.4.16),asshowninFig.4.21.SeriesofHallbargeometry(5×5µm2)wasfabricatedacrossthroughanexpectedGGBlineasshowninFig.4.21a.ThefinaldeviceisshowninFig.4.21b,caftergrapheneparterning,metaldepositionandlift-offprocessbye-beamlithography.Afterfab-ricationprocessesincludinggraphenetransferande-beamlithography,theGGBsandpartialgraphenebasalplaneareexpectedtobecontaminated.Therefore,thesamplewasheat-treatedatdifferentconditionsundervacuum(102Torr).Physicaladsorbatesweresimplyremovedat150◦Cfor1h,andthetransportcharacteristicsFig.4.21Opticalimageofthefour-probedeviceacrossaGGB.aE-beamlithographyresist(PMMA)locationatamergingregionincludingaGGB.b,cAfinaldevicewithHallbargeometryatmergingregionoftwographenedomains 904TransportinDisorderedGrapheneFig.4.22O2SelectivefunctionalizationofGGBsbyUVtreatmentunderenvironment.a,bEffectofannealingat250◦Cin3h.FunctionalgroupsareremovedfromaGGB.c,dEffectofUVtreat-mentunderO2environment.Theexclusivechangeoftheinter-grainresistanceindicatesselectivefunctionalizationattheGGB.TheUVtreatmentissaturatedafter1minofUVtreatmentofthegrainsandtheGGBweremeasured,asshowninFig.4.22a.Here,theblackandbluelinesrepresenttheintra-grainresistancesRLandRR,andtheredlineisthemergingregionresistanceRB.Asexpected,RBislargerthanRLandRR,duetotheextraresistancecontributedbytheGGB.Next,thesamplewasfurtherannealedat250◦Cfor3h.Figure4.22bshowsthattheresistanceofthegraphenebasalplanewasnotchanged,whiletheresistanceacrosstheGGBdecreasedsignificantly.ThisdecreaseinresistanceimpliesthatfunctionalgroupsattheGGBwereremoved,assupportedbythesimulationresultsinthenextsection.ThesamplewasthenexposedtoUVunderanO2environment(0.5Torr).TheresistanceacrosstheGGBincreased,whiletheresistanceofthegraphenebasalplanewasstillunchanged,asshowninFig.4.22c.ThisstronglysuggeststhattheGGBsareselectivelyfunctionalizedbyozonegeneratedbyUV.ThissystematicseriesofmeasurementsleadsustoconcludethattheGGBscanbeselectivelyfunctionalizedbyozone.ThisisakeysteptowardsfurtherbiochemicalmodificationofGGBs.WenoticethattheUVtreatmentissatu-ratedafter1minexposure.LongertimeUVexposuredoesntincreasetheresistanceatGGBs. 4.2ChargeTransportinPoly-G914.2.6.3EffectofFunctionalGroupsonElectricalTransportatGGBsbySimulationAsdiscussedabove,theresistanceattheGGBscanbemodifiedbychangingtheirfunctionalgroups.ProvingthisconceptwithacurrentmeasurementtechniqueisachallengebecausethechemicalreactionoccursonthenanometerscaleattheGGBs.Therefore,numericalsimulationisakeystrategytounderstandthisprocess.SeveraltheoreticalandnumericalapproacheshavebeenemployedtostudychargetransportacrossindividualGGBs[21,74,143–146].Here,anapproachwhichallowsthestudyoflarge-areaPoly-GwitharandomdistributionofGBorientationsandmorphologiesisoutlined.ThePoly-GsampleiscreatedusingmoleculardynamicssimulationsthatmimicthegrowthofCVDgraphene[86],anditselectricalpropertiesaredescribedwiththeTBformalism.Tostudytransport,thetimeevolutionofanelectronicwavepacketwithinthegraphenesampleistracked[147].TheconductivitycanthenbecalculatedwiththeKuboformulainEq.(3.41).Byassumingawavepacketthatinitiallycoverstheentiresample,onecangetaglobalpictureofthescatteringinducedbyGGBs.Oncetheconductivityisknown,thesheetresistanceisgivenbyRS=1/σ.Bydoingthissimulationforarangeofaveragegrainsizes,theGGBresistivitycanbeextractedusingthescalinglawdescribedinSect.4.2.5.3.Toincludetheeffectofchemicalfunctionalization,adsorbatesarerandomlyattachedtotheGBatomsatdifferentconcentrations(asillustratedinFig.4.20a).Tight-bindingparametersfordescribinghydrogen,hydroxyl,andepoxygroupshavebeentakenfromtheliterature[19,20,148].Figure4.23a,bshowsatypicalexampleofa5-7GGBfunctionalizedbyOandOHgroups,respectively.TheresistivityoftheGGBswithdifferentfunctionalgroupsatvariousconcentrationsisextracted,asshowninFig.4.23c,whereρGBisplottedasafunctionofadsorbatecoverage,definedasthenumberofadsorbatesrelativetothetotalnumberofGGBtomsinthesample.Forcoveragegreaterthan100%,theadsorbatesareallowedtofunctionalizethecarbonatomsnexttotheGGBs.Foralltypesofadsorbates,ρGBincreaseswithcoverage,regardlessoftheirtype.However,itisalsonotedthatρGBisstronglyadsorbate-dependent.Forexample,whilebothHandOHgroupsarechemisorbedtothetopsiteofasinglecarbonatom,HgroupshaveastrongereffectontransportthroughtheGGBsthanOHgroups,withρGBnearly4timeslargerat200%coverage.Thisdifferencecanbeascribedtotheelectronicstructureofeachtypeofadsorbate.Thesimulationsemployaresonantscatteringmodel,whereeachadsorbateischaracterizedbyanon-siteenergyεadsandacouplingtoasinglecarbonatomγads.Theneteffectofthismodelistointroduceanenergy-dependentscatteringpotential[20],Vads(E)=γ2/(E−εads).UsingadsparametersforHandOHtakenfromtheliterature[20,147],thisgivesVH(E=0)=−40γ0andVOH(E=0)=1.8γ0.SinceσDC,andhenceRSandρGB,arecalculatedattheDiracpoint,theHgroupspresentamuchstrongerscatteringpotentialthantheOHgroups.CalculationshavealsoshownthatHgroupsinducestronglylocalizedstatesneartheDiracpoint,whileOHadsorbatesresultinamoredispersiveimpuritybandlyinginthevalencebandofgraphene[20].Meanwhile,theOgroupchemisorbsinthebridgesitebyformingapairwithadjacentcarbonatomsinthe 924TransportinDisorderedGrapheneFig.4.23SimulationoftheeffectoffunctionalgroupsatGGBs.a,bSchematicofGGBsfunction-alizedbyHandOHgroups,respectively.cDependenceoftheresistivityofGGBsonfunctionalgroupswithvariousconcentrations.dSummaryofexperimentalandsimulatedresultsfortheresis-tivityofGGBsgraphenelattice(epoxide)[148].ThesimulationsclearlyshowthattheresistanceatGGBswithfunctionalgroupsismuchhigherthanthatofpureGGBs.Figure4.23dshowsasummaryofthevaluesofρGBderivedfrommeasurementscomparedtothesimulationresults[21,88,99,100,129,148].ThesolidsymbolsarefromtheelectricalmeasurementsdescribedearlierInthissection,andtheopensymbolsarethenumericalsimulations.Here,mostmeasurementsgiveρGBintherangeof1to10k .µm,exceptforonethatgivesvaluesonetotwoordersofmagnitudesmaller[149].Thisdifferencecouldbecausedbythemeasurementtechnique,whereρGBwasmeasuredwithfour-probeSTMunderultra-highvacuum,whiletheothergroupsfabricatedphysicalcontactsontheirsamples.Thisextrafabricationstepcouldleadtoadditionalcontamination,increasingρGB.Accordingly,thenumericalsimulationsshowthatitispossibletobridgethegapbetweenthevariousmeasurementsbysystematicallyincreasingtheamountofchemicalfunctionalizationoftheGGBs.ThesituationbecomesmorecomplicatedbyseveralotherparameterssuchasthestructureandresistivityoftheGGBs,asmentionedpreviously[21].Thisishighlightedbythemeasurementslabeled“smallgrain”and“largegrain”inFig.4.23d,wheregrowthconditionsyieldinglargegrainsamplesalsotendtoyieldpoorlyconnectedandhighlyresistiveGGBs[100].Nevertheless,theseresultshighlightthestrongimpactthatchemicalfunctionalizationcanhaveontheelectricalpropertiesofGGBs. 4.2ChargeTransportinPoly-G934.2.7ChallengesandOpportunitiesTheobservationandcharacterizationofGGBsatbothatomicandmacroscopicscaleismandatorytounderstandthetransportpropertiesandtherelatedunderlyingphysicsandchemistryofPoly-G.AsdescribedinthisChapter,TEMandSTM,combinedwiththeoryandsimulation,canprovideinformationattheatomicscale,withtherelatedtransportpropertiesrevealedwiththeassistanceofSTS.UV-treatmentandliquidcrystalcoating,combinedwithopticalmicroscopy,canprovideinformationonboththeGBdistributionatthemacroscaleandtheorientationofeachdomain,whilemacroscopictransportpropertiescanbederivedusingthescalinglaw.Withallthesepowerfulmethodsavailable,onecanenvisiontheirapplicationtotheengineeringofGBsduringgraphenesynthesis.Forinstance,idealmonocrystallinegraphenecouldbeobtainedbydesigningseamlessboundariesbetweencoalescinggraphenegrains.Withavailablelarge-areamonocrystallinegraphene,bilayergraphenewithcontrolledstackingordercanbeconstructedbyalignedtransfertechniques.Therelativeorientationofthelayerscanbeidentifiedbyeitherlow-energyelectrondiffractionorRamanspectroscopy.Thisopensanewresearchdirectionofbilayergraphenefordesigningverticaltunnelingdevicesandplanarswitchingdevices.AGBlineisa1Dstructureconsistingofaseriesofpentagonal,hexagonal,andheptagonalcarbonrings.ItispossibletoselectivelyfunctionalizeaswellasdepositdesignedmaterialsonlyattheGGBsduetotheirhigherchemicalreactivitycomparedtoidealbasalgraphene.ThisimpliesthatGGBscanbeagoodtemplateforthesynthesisof1Dmaterials.Atomiclayerdeposition,whoseprecursorisquiteinertwiththegraphenebasalplane,wouldbeagoodmethodforthesynthesisofsub-nanometer1Dmetalsandsemiconductors.AnotherresearchdirectiontoutilizeGBsistocontroltheirdensitytodesignsensorsfordetectinggasesandmoleculesunderdifferentenvironmentalconditions.Asrevealedbyournumericalsimulationsandourexperimentalmeasurements,thetransportpropertiesofGBscanbestronglyalteredwithchemicalmodificationsoftheGBs.Togetherwithhighlyconductivegraphene,electro-biochemicalsensingdeviceswithhighsensitivityandselectivitycouldbedesigned.Membranescienceisanotheropenresearcharea.Althoughtheidealhexago-nalgraphenelatticeimpedesthediffusionofgases,defectsitessuchasheptagons,octagons,vacancies,anddivacanciesallowselectivediffusionoflimitedgasesandmolecules,asmentionedabove.ThisprovidesnewopportunitiestoexploreultrafinemembraneperformanceviathecontrolledengineeringofGBsandpointdefects.Althoughmuchprogresshasbeenmadeinthevisualizationandelectricalcharac-terizationofGGBsfromatomicscaletomacroscale,issuesstillremain.ThestructureofGGBsisdeterminedbythedifferentorientationsbetweenmergingdomains,andtherelatedphysicalandchemicalpropertiesarepredictedtobestronglychirality-dependent.However,noelectricalmeasurementshaverevealedsucheffects.ThequestioniswhetherthisoriginatesfromadevicefabricationprocesswhichinevitablyfunctionalizesGGBs,orifthenativestructureofGGBsisdisordered,differentfromtheoreticalpredictions. 944TransportinDisorderedGrapheneGGBsalsopresentchallengesforthedevelopmentoflargescalegraphene-basedspintronicdevices[150],andforharvestingtheuniqueopticalpropertiesofgraphene.Forinstance,GGBsintroducenon-triviallocalsymmetrybreakingwhichcouldsig-nificantlyimpactspin/pseudospincouplingandspinrelaxationtimes,aswellastheformationandpropagationofplasmonicexcitations.Similarly,thepeculiarstructureofinterconnectedGGBscouldaffecttransportpropertiesinhighmagneticfields,suchastheQHE.Overall,controllingtheatomicstructureofGGBsbyCVDisabigchallengefromascientificpointofview,butwouldbeahugestepforwardintherealizationofnext-generationtechnologiesbasedonthismaterial.4.3ImpactofGraphenePolycrystallinityonthePerformanceofGrapheneField-EffectTransistors4.3.1IntroductionIntheefforttosuccessfullyrealizenext-generationtechnologiesbasedongraphenefield-effecttransistors(GFETs),theoryanddevicemodelingwillplayacrucialrole.Specifically,itisimportanttodevelopmodelsthatcanaccuratelydescribeboththeelectrostaticsandthecurrent-voltage(I-V)characteristicsofgraphene-basedelec-tronicdevices[151–154].Thiscapabilitywillenabledevicedesignoptimizationandperformanceprojections,willpermitbenchmarkingofgraphene-basedtechnologyagainstexistingones[53,155],andwillhelptoexplorethefeasibilityofanalog/RFcircuitsbasedongraphene[156–158].Ultimately,graphene-baseddevicescouldprovideneworimprovedfunctionalitywithrespecttoexistingtechnologies,suchasthosebasedonsiliconorIII-Vmaterials.TheCVDtechniqueforgrowingwafer-scalegrapheneonmetallicsubstrates[64,159–161]producesapolycrystallinepattern.Thisisbecausethegrowthofgrapheneissimultaneouslyinitiatedatdifferentnucleationsites,leadingtosampleswithrandomlydistributedgrainsofvaryinglatticeorientations[72].IthasrecentlybeenpredictedthattheelectronicpropertiesofPoly-Gdifferfromthoseofpristinegraphene,wherethemobilityscaleslinearlywiththeaveragegrainsize[21].Basedontheseresults,wereportonhowtheelectronicpropertiesofPoly-Gimpactthebehaviorofgraphene-baseddevices.Specifically,weconcentrateourstudyontheeffectthatPoly-GhasonthegateelectrostaticsandI-VcharacteristicsofGFETs.Wefindthatthesource-draincurrentandthetransconductanceareproportionaltotheaveragegrainsize,indicatingthatthesequantitiesarehamperedbythepresenceofGBsinthePoly-G.However,oursimulationsalsoshowthatcurrentsaturationisimprovedbythepresenceofGBs,andtheintrinsicgainisinsensitivetothegrainsize.TheseresultsindicatethatGBsplayacomplexroleinthebehaviorofgraphene-basedelectronics,andtheirimportancedependsontheapplicationofthedevice. 4.3ImpactofGraphenePolycrystallinityonthePerformance…954.3.2Poly-GEffectontheGateElectrostaticsandI-VCharacteristicsofGFETsThestartingpointofourstudyisthecharacterizationofalarge-areamodelofdis-orderedPoly-Gsamples,containinghundredsofthousandsatomsanddescribedbyvaryinggrainmisorientationangles,realisticcarbonringstatistics,andunrestrictedGBstructures,basedonthemethodreportedinRef.[118].Tocalculatetheelec-tronicandtransportproperties,weusedaTBHamiltonianandanefficientquantumtransportmethod[41,43],whichisparticularlywell-suitedforlargesamplesofdisorderedlow-dimensionalsystems.Thetransportcalculationswerebasedonareal-spaceorder-Nquantumwavepacketevolutionapproach,whichallowedustocomputetheKubo-Greenwoodconductivity(Eq.3.41).Withthisquantity,thechargecarriermobilitycanbeestimatedasμ(E)=σ(E)/q∗Qc(E),whereQcisthe2Dchargedensityinthegraphene.Itshouldbenotedthatweassumethecarriermobilityisnotlimitedbythesubstrate,thatis,wedonotconsideradditionalscatteringduetochargetrapsorsurfacephononsintheinsulatorthatcouldfurtherdegradethecar-riermobility[162].Thus,ourresultsrepresentanupperboundontheperformancemetricsoftheGFETsthatwearestudying.Inthiswork,wefocusonadual-gateGFETastheonedepictedinFig.4.24.Thistransistorisbasedonametal/oxide/Poly-G/oxide/semiconductorstructurewhereanexternalelectricfieldmodulatesthemobilecarrierdensityinthePoly-Glayer.TheelectrostaticsofthisdualgatestructurecanbeunderstoodwithanapplicationofGausslaw∗−V∗Qc=Ct(Vgsc)+Cb(Vbs−Vc)(4.7)Fig.4.24aSchematicofthedual-gateGFET,consistingofapoly-Gchannelontopofaninsulatorlayer,whichisgrownonaheavily-dopedSiwaferactingasthebackgate.Anartisticviewofthepatchworkofcoalescinggraphenegrainsofvaryinglatticeorientationsandsizeisshownin(b).Thesourceanddrainelectrodescontactthepoly-Gchannelfromthetopandareassumedtobeohmic.Thesourceisgroundedandconsideredthereferencepotentialinthedevice.Theelectrostaticmodulationofthecarrierconcentrationingrapheneisachievedviaatop-gatestackconsistingofthegatedielectricandthegatemetal 964TransportinDisorderedGraphenewhereQc=q(p−n)isthenetmobilechargedensityinthegraphenechannel,CtandCbarethegeometricaltopandbottomoxidecapacitances,andVg∗sandV∗arethebseffectivetopandbottomgate-sourcevoltages,respectively.Here,Vg∗s=Vgs−Vgs0andV∗=Vbs−Vbs0,whereVgs0andVbs0arequantitiesthatcomprisetheworkbsfunctiondifferencesbetweeneachgateandthegraphenechannel,chargedinterfacestatesatthegraphene/oxideinterfaces,andpossibledopingofthegraphene.Thegraphenechargedensitycanbedeterminednumericallyusingtheprocedure0∞Qc(Vc)=qDOSp−G(E)f(qVc−E)dE−qDOSp−G(E)f(E−qVc)dE(4.8)−∞0whereDOSp−G(E)hasbeencalculatedwiththeprocedureoutlinedinRef.[21].ThepotentialVcrepresentsthevoltagedropacrossthegraphenelayer,andisrelatedtothequantumcapacitanceCqofthePoly-GbyCq=−dQc/dVc.Whentheentirelengthofthetransistorisconsidered,theeffectivegatevoltagescanbewrittenasVg∗s=Vgs−Vgs0−V(x)andV∗=Vbs−Vbs0−V(x),whereV(x)(theso-calledbsquasi-Fermilevel)representsthepotentialalongthegraphenechannel.TheboundaryconditionsthatshouldbesatisfiedareV(0)=0atthesourceandV(L)=Vdsatthedrain.Tomodelthedraincurrent,weemployadrift-diffusionmodelwiththeformIds=−W|Qc(x)|v(x),whereWisthegatewidth,Qc(x)isthefreecarriersheetdensityinthechannelatpositionx,andv(x)isthecarrierdriftvelocity.ThelatterisrelatedtothetransverseelectricfieldEasv=μE,sonovelocitysaturationeffecthasbeenincludedinthismodel.Thelow-fieldcarriermobilityμ(Qc)isdensity-dependentandcalculatedviatheprocedureofRef.[21].AfterapplyingE=−dV(x)/dx,includingtheaboveexpressionforv,andintegratingtheresultingequationoverthedevicelength,thesource-draincurrentbecomesVWdsIds=μ|Qc|dV.(4.9)L0InordertocalculateIds,theintegralinEq.(4.9)issolvedusingVcastheintegrationvariableandsubsequentlyexpressingμandQcasfunctionsofVc,basedonthemappinggivenbyEq.(4.8).ThisgivesVWcddVIds=μ(Vc)|Qc(Vc)|dVc(4.10)LVcsdVcwhereVcisobtainedbyself-consistentlysolvingEqs.(4.7)and(4.8).ThechannelpotentialatthesourceisdeterminedasVcs=Vc(V=0)andthechannelpotentialatthedrainisdeterminedasVcd=Vc(V=Vds).Finally,Eq.(4.7)allowsustodVCqevaluatethederivativeappearinginEq.(4.10),namely,=−1+,whichdVcCt+CbshouldbedeterminednumericallyasafunctionoftheintegrationvariableVc. 4.3ImpactofGraphenePolycrystallinityonthePerformance…97Fig.4.25Quantumcapacitance(a)anddensityofstates(b)ofPoly-Gconsideringdifferentaveragegrainsizes.ThepristinegraphenecasehasalsobeenplottedforthesakeofcomparisonNext,weapplythemulti-scalemodeltotheGFETshowninFig.4.24.Itconsistsofadual-gatestructurewithL=10µmandW=5µm.Thetopandbottomgateinsulatorsarehafniumoxideandsiliconoxidewiththicknessesof4and300nm,respectively.Fortheactivechannel,weconsideredpoly-Gwithdifferentaveragegrainsizestogetherwiththesimplepristinegraphenecase,whichservesasaconve-nientreferenceforcomparison.Forthisstudy,wecreatedsampleswiththreedifferentaveragegrainsizes(averagediameterd≈13,18,and25.5nm)anduniformgrainsizedistributions.TheatomicstructureattheGBsconsistspredominantlyoffive-andseven-membercarbonringsandassumesmeanderingshapessimilartotheexperi-mentallyobservedones.Wealsocreatedonesamplewithd≈18nmand“broken”(poorlyconnected)boundaries(“br-18nm”).Thequantumcapacitance(Cq)ofeachsampleispresentedinFig.4.25a,whichreflectsthestructureoftheDOS,showninFig.4.25b.Anenhanceddensityofzero-energymodesaroundthechargeneutralitypoint(CNP)canbeobserved,whichariseslocallyfromtheatomicconfigurationsoftheGBs,givingrisetoafiniteCq.AzeroCqwouldcorrespondtoidealgateeffi-ciency,meaningthatthegatevoltagewouldhave100%controloverthepositionofthegrapheneFermilevel.AwayfromtheCNP,bothCqandtheDOSoftheanalyzedstructureslookverysimilar.Forthepoorlyconnectedsample“br-18nm”,apeakisobservedaroundtheCNPbecauseofahigherdensityofmidgapstates,resultinginanegativedifferentialCq.Figure4.26ashowsthetransfercharacteristicsoftheGFETunderconsidera-tionfordifferentgrainsizes.Thelow-fieldcarriermobilitywascalculatedfromtheKubo-Greenwoodconductivityasμ(E)=σ(E)/q∗Qc(E),andhasbeenplot-tedasafunctionofQcinFig.4.26b.Themobilitycorrespondingtoagrainsizeof1µmwasestimatedfromthemobilityat25.5nmwithasimplescalinglaw[21],μ1µm(Qc)=(1µm/25.5nm)µ25.5nm(Qc).TheresultingI-Vcharacteristicsexhibit 984TransportinDisorderedGrapheneFig.4.26Transfercharacteristics(a)andtransconductance(c)ofthegraphenefield-effecttran-sistorconsideringdifferentsamplesofPoly-Gastheactivechannel.bEstimatedlow-fieldcarriermobilityasafunctionofthecarrierdensityforeachofthesamplestheexpectedV-likeshapewithanON-OFFcurrentratiointherangeof2–4,andonecanseethatthesource-draincurrentisproportionaltotheaveragegrainsize.Thisisduetothescalingofthemobilitywithgrainsize,asshowninFig.4.26b.InFig.4.26c,weplotthetransconductanceoftheGFET,definedasgm=dIds/dVgs,whichisakeyparameterindeterminingthetransistorvoltagegainorthemaximumoperationfrequency.Itappearsthatsmallgrainsizesaredetrimentaltothisfactor.Thereasonbehindsuchadegradationisthecombinationoftwofactorsasthegrainsizeisreduced:(a)anincreaseinCqatlowcarrierdensities(Fig.4.25a),whichisrelatedwiththeincreaseintheDOSneartheCNP(Fig.4.25b)andleadstoreducedgateefficiency;and(b)thereductionofthelow-fieldcarriermobility(Fig.4.26b)becauseofscatteringduetothedisorderedatomicstructureoftheGBs.Figure4.26bindicatesthatthemobilityisproportionaltotheaveragegrainsizeofthePoly-G;ahigherdensityofGBsresultsinmorescatteringandalowermobility.ThescatteringeffectoftheGBshasbeenfurtherquantifiedinRef.[21],whichshowsthescalingoftheconductivityandthemeanfreepathofthePoly-Gfordifferentgrainsizes.Forexample,thesamplewith25.5-nmgrainshasameanfreepathof10nmneartheDiracpoint,comparedwith5nmforthesamplewith13-nmgrains.InFig.4.27a,weplottheGFEToutputcharacteristicsfordifferentgrainsizesandgatebiases.Theoutputcharacteristicexhibitsaninitiallinearregiondominatedbyholetransport(p-typechannel),followedbyaweaksaturationregion.Theonsetofsaturation(Vsd,sat)happenswhenthechannelbecomespinchedoffatthedrainside.AfurtherincreaseinVsddrivesthetransistortowardsthesecondlinearregion,characterizedbyachannelwithamixedp-andn-typebehavior.Interestingly,areductionofthegrainsizeimprovesthecurrentsaturation,whichcanbeseeninaplotoftheoutputconductance(Fig.4.27b),definedasgd=VVdIds/dVds.Here,theminimumofgdismuchflatterandbroaderforsmallergrainsizes.BothgmandgddeterminetheintrinsicgainAv=gm/gd,whichisakeyfigureofmeritinanalogorRFapplications.OursimulationsdemonstratethatAvisinsensitivetothegrainsize(Fig.4.28),becauseanincreaseingmisalmostexactlycompensatedbyasimilarincreaseingd.Thissuggeststhatpolycrystallinityisnotalimitingfactorinanalog/RF 4.3ImpactofGraphenePolycrystallinityonthePerformance…99Fig.4.27Outputcharacteristics(a)andoutputconductance(b)ofthegraphenefield-effecttran-sistorconsideringdifferentsamplesofPoly-GastheactivechannelFig.4.28Intrinsicgainasafunctionofthedrainvoltage.Thetransconductanceandoutputcon-ductancearealsoplottedatVgs=0.25Vdeviceswhoseperformancedependsontheintrinsicgain.However,thereareotherperformancemetrics,suchastheintrinsiccutoff(fT)andmaximumfrequencies(fmax),whichareseverelydegradedbythepresenceofGBs.Todemonstratethis,wehavecalculatedbothfTandfmaxforthedeviceunderconsideration,butassumingachannellengthof100nm.ThecutofffrequencyisgivenbyfT≈gm/2πCgs,where 1004TransportinDisorderedGrapheneFig.4.29Intrinsicmaximumandcutofffrequencyforthesimulatedtransistorassumingachannellengthof100nmCgsisthegate-to-sourcecapacitance.12GiventhatthegeometricalcapacitanceCtismuchsmallerthanthequantumcapacitanceCq,Cgs∼=Ct.Themaximumfrequency√isgivenbyfmax≈gm/(4πCgsgd(RS+RG),whereRSandRGarethesourceandgateresistances,respectively[155].Here,wehaveassumedstateoftheartvalues,suchas[163]RS∼100 .µmandRG∼6.AsshowninFig.4.29,fmaxandfTaredegradedbyoneandtwoordersofmagnitude,respectively,whentheaveragegrainsizedecreasesfrom1µmtonm.RealisticGFETsarelimitedinperformancebyinteractionwiththesubstrateandtopgates.Comparingwiththeextractedmobilityfromsomereportedstate-of-the-artdevices[164],ourcalculations,whichrepresentthelimitingcaseofuncoveredgraphene,overestimatethemobilityofthesedevicesby∼10×.Asaconsequence,gm,gd,andfTshouldbereducedbythatamountwhenconsideringsubstrateandtopgateeffects.Meanwhile,Avisexpectedtoremainconstantandfmaxisexpectedtobereducedby∼3×.Thementioned∼10×factorofmobilityreductioncouldbemade 4.3ImpactofGraphenePolycrystallinityonthePerformance…101significantlysmallerbyusinganappropriatesubstrate,suchasdiamond-likecarbon[162](DLC),whichhelpstominimizeinteractionwiththesubstrate.Inconclusion,wehavedevelopedadrift-diffusiontransportmodelfortheGFET,basedonadetaileddescriptionofelectronictransportinpoly-G.Thismodelallowsustodeterminehowagraphenesample’spolycrystallinityalterstheelectronictransportinGFETs,enablingthepredictionandoptimizationofvariousfiguresofmeritforthesedevices.WehavefoundthatthepresenceofGBsproducesaseveredegradationofboththemaximumfrequencyandthecutofffrequency,whiletheintrinsicgainremainsinsensitivetothepresenceofGBs.Overall,polycrystallinityispredictedtobeanundesirabletraitinGFETstargetinganalogorRFapplications.4.4TransportPropertiesofAmorphousGraphene4.4.1IntroductionThephysicsofdisorderedgrapheneisattheheartofmanyfascinatingpropertiessuchasKleintunneling,WALoranomalousQHE(seereviews[43,165]).Thepreciseunderstandingofindividualdefectsonelectronicandtransportpropertiesofgrapheneiscurrentlyofgreatinterest[166].Forinstance,graphenesamplesobtainedbylarge-scaleproductionmethodsdisplayahugequantityofstructuralimperfectionsanddefectswhichjeopardizetherobustnessoftheotherwiseexceptionallyhighchargemobilitiesoftheirpristinecounterparts[8].Indeed,thelatticemismatch-inducedstrainbetweengrapheneandtheunderlyingsubstrategeneratesPoly-GwithGBswhichstronglyimpactontransportproperties[143](SeeSect.4.2).However,despitethelargeamountofdisorder,suchgrapheneflakesusuallymaintainafiniteconduc-tivitydowntoverylowtemperatures(whendepositedontooxidesubstrates)owingtoelectron-holepuddles(chargeinohomogeneitiesfluctuations)-inducedpercolationeffectswhichlimitlocalizationphenomena[9].ThepredictedAndersonlocalizationintwo-dimensionaldisorderedgraphenehasbeenhardtomeasureinnonintention-allydamagedgraphene,incontrasttochemicallymodifiedgraphene[167,168].Inarecentexperiment,itwashoweverpossibletoscreenoutelectron-holespud-dlesusingsandwichedgrapheneinbetweentwoboron-nitridelayers,togetherwithanadditionalgraphenecontrollayer[12].Asaresultofpuddlesscreening,alargeincreaseoftheresistivitywasobtainedattheDiracpoint,evidencinganonsetoftheAndersonlocalizationregime.Beyondindividualdefectsandpolycrystallinity,ahigherlevelofdisordercanbeinducedongraphenetothepointofobtainingtwo-dimensionalamorphousnetworkscomposedofsp2hybridizedcarbonatoms.Suchnetworkscontainringsotherthanhexagonsinadisorderedarrangement.TheaverageringsizeissixaccordingtoEuler’stheorem,allowingsuchasystemtoexistasaflat2Dstructure.Experimen-tally,suchamorphoustwo-dimensionallatticeshavebeenobtainedinelectron-beamirradiationexperiments[86],anddirectlyvisualizedbyhighresolutionelectrontrans- 1024TransportinDisorderedGraphenemissionmicroscopy.Previously,indirectevidencefortheformationofanamorphousnetworkwasobtainedbyRamanspectroscopyinsamplessubjecttoelectron-beamirradiation[169],ozoneexposure[170]andionirradiation[171].Inallthesecases,anevolutionfrompolycrystallinetoamorphousstructureswasobserveduponincreaseofthedamagetreatment.In[171],furtherevidenceoftheformationofanamorphousnetworkwasobtainedthroughtransportmeasurements.TheseindicatethetransitionfromaWLregimeinthepolycrystallinesamplestovariablerangehoppingtrans-portinthestronglylocalizedregimeforamorphoussamples,asevidencedbythetemperaturedependenceoftheconductivity.Localizationlengthswereestimatedtobeoftherange0.1–10nmintheamorphoussamples,dependingonthedegreeofamorphization.Fromthetheoreticalside,modelsoftheamorphousnetworkhavebeenproposedusingstochasticquenchingmethods[172],andmoleculardynamics[46,173,174].Electronicstructurecalculationsshowthattheamorphizationyieldsalargeincreaseofthedensityofstatesatandintheenvironmentofthechargeneutralitypoint[172–174].Despitetheexpectedreductionoftheconductionprop-ertiesduetostronglocalizationeffects,Holmströmetal.[173]suggestthatdisordercouldenhancemetallicityinamorphizedsamples,incontrastwiththeexperimentalevidence.Here,weexplorethetransportpropertiesoftwo-dimensionalsp2latticeswithmas-siveamountoftopologicaldisorder,encodedinageometricalmixtureofhexagonswithpentagonandheptagonringswithagivenringstatistics.Thecalculationsaredoneusingtwoapproaches:aKuboformulationinwhichtheconductivityofbulk2Damorphousgraphenelatticeswasdetermined,andaLandauer-Büttikerformulationwheretheconductanceofribbonsofamorphousgraphenecontactedtosemi-infinitepristinegrapheneelectrodeswascalculated.Bothapproachesleadtosimilarfind-ings.Dependingontheratiobetweenoddversuseven-memberedrings,atransitionformagraphene-likeelectronicstructuretoatotallyamorphousandsmoothelec-tronicdistributionofstatesisobtained.Thestrongerthedeparturefromthepristinegraphene,themoreinsulatingisthecorrespondinglattice,whichtransformsintoastrongAndersoninsulatorwithelasticmeanfreepathsbelowonenanometerandveryshortlocalizationlengthalloverthewholeelectronicspectrum.Thosestructuresarethereforeinefficienttocarryanysizablecurrent,andarethereforeuselessforanypracticalelectronicapplicationssuchastouchscreensdisplaysorconductingelec-trodes,butinterestingforscrutinizinglocalizationphenomenainlowdimensionalmaterials.4.4.2ModelsofAmorphousGrapheneAmorphousmodelsofgraphenearepreparedusingtheWooten-Winer-Weaire(WWW)method[175,176],introducingStone-Walesdefects[177]intotheper-fecthoneycomblattice.Togeneratethestructures,periodicboundaryconditionsareimposedandtheentirenetworkwasrelaxedwiththeKeating-likepotential[172,178].PiecesoftwodifferentnetworksareshowninFig.4.30a,b.Thesamplescon- 4.4TransportPropertiesofAmorphousGraphene103Fig.4.30aandbshowdetailsofamorphousgraphenesamplesS1andS2,respectively,usedtocomputetheconductivitywiththeKuboapproach.cTotaldensityofstatesofthetwoamorphoussamples.Thepristinecrystallinegraphenecase(dashedlines)isalsoshownforcomparisonTable4.2ComparisonofsamplespecificationsS1S2Numberofatoms10032101640Percent.ofn-memberedrings24/52/2444/12/44(n=5/6/7)n2− n20.470.88RMSdeviationofbondangles11.02◦18.09◦RMSdeviationofbondlengths0.044Å0.060ÅFermienergy(γ0)0.030.05tain10032and101640atoms,respectively,allofthemwiththree-foldcoordinationasthehoneycomblattice,buttopologicallydistinct.Samples1and2arecharac-terizedbyanumberofparametersgiveninTable4.2.ForSample1,24%oftheelementaryringsarepentagons,52%hexagonsand24%heptagons,whilesample2hasalargershareofodd-memberedrings.Inbothsamples,thenumberofheptagonsisthesameasthatofpentagons,accordingtoEuler’stheorem,andthesesystemscanexistwithoutanoverallcurvatureasflat2Dstructureswithsomedistortionsofbondlengthsandangles,althoughmaypuckerundersomecircumstances.Wewillonlybeconcernedwiththeplanarstructureshere.ForthecalculationoftheLandauer-Bütikkerconductance,wesetupmodelsinwhichanamorphousribboniscontactedbytwopristinegrapheneelectrodesatadistanceL.Modelswithdifferentribbonlengthoftheamorphouscontactarebuilttostudythedependenceoftheconductanceonthedistancebetweenelectrodes.Themodelsareperiodicinthedirectionperpendiculartotheribbon,withaperiodicityofW=11.4nm,andhavethesameringstatisticsasthebulksample1describedabove. 1044TransportinDisorderedGraphene4.4.3ElectronicPropertiesTheelectronicandtransportpropertiesofthesedisorderedlatticesareinvestigatedusingπ-π∗orthogonalTBmodelwithnearestneighborshoppingγ0andzeroonsiteenergies.Novariationofthehoppingelementswithdisorderisincludedinthemodelasbond-lengthvariationdoesnotexceedafewpercent(cf.Table4.2);alldependenceondisorderstemsfromtheringstatisticswhichisthedominatingeffect.Figure4.30cshowsthedensityofstates(DOS)ofthetwodisorderedsamples,togetherwiththepristinecase(dashedline)forcomparison.Sample1,whichkeeps52%ofhexagonalrings,displaysseveralnoticeablefeatures,similartothosefoundinpreviousstudies[172,173].First,theDOSatthechargeneutralitypointisfoundtobeincreasedbyalargeamount.Additionally,theelectron-holesymmetryofthebandstructureisbrokenduetothepresenceofodd-memberedringsandtheresonantstatesthattheseinduce[32].TheholepartofthespectrumisstillreminiscentofthegrapheneDOS,withasmoothenedpeakatthevanHovesingularitywhileintheelectronpartasecondmaximumappearsclosetotheupperconductionbandedge.Byreducingfurthertheratioofevenversusodd-memberedrings(Sample2),thesecondmaximumdevelopstoastrongpeakataboutE=2.5γ0whilespectralweightatE=3γ0issuppressed.TheredistributionofDOSattheupperconductionbandedgeisasignatureofodd-memberedringsanditsstrengthwithincreasingnumberofsuchringsrelatesthestatisticaldistributionofringswiththeDOSfeatures.TransportMethodology.Toexplorequantumtransportinthesetopologicallydis-orderedgraphenebulksamples,weemployareal-spaceorder-NquantumwavepacketevolutionapproachinChap.3tocomputetheKubo-Greenwoodconductivity[179].Theconductanceofamorphousstripes(ribbons)contactedtographeneelectrodesiscomputedusingtheLandauer-Büttikerapproach[180]:2e2G(E)=GTrt†t(4.11)0T(E)=hwhereT(E)andt(E)arethetransmissionprobabilityandtransmissionmatrix,respectively,whichcanbecomputedfromtheGreen’sfunctionG(E)inthecon-tactregionandthebroadening(E)ofthestatesduetotheinteractionwiththeleftandrightelectrodes.Wecalculatetheconductanceoftheribbon,whichisinfiniteandperiodicinthedirectionparalleltotheinterfacebetweenthepristinegrapheneelectrodesandtheamorphousribbons.Despitetheverylargeperiodicityofourmod-els,weperformathoroughsamplingofthek-pointsinthatdirection[181,182],toobtaintheappropriateV-shapedconductanceofgrapheneinthethermodynamiclimit.GisgivenpersupercellofperiodicityW=11.4nm.Notethatconductivityandconductancearerelatedthoughσ=LG.WMeanFreePath,ConductivityandLocalizationEffects.Figure4.31showstimedependenceofthenormalizeddiffusioncoefficientD(t)/Dmaxfortwochosenener-gies,forthetwobulksamples.ForenergyE=−2γ0,itisfoundtoincreaseballisti-callyatshorttime,butthensaturatestypicallyafter0.1ps.Thissaturationallowsto 4.4TransportPropertiesofAmorphousGraphene1051S1,E=-2γ01000.8S2,E=-2γ080S160S2[nm]40ξnorm20D0.60-10123E[γ]00.4S1,E=0S2,E=00.200.20.40.60.81t[ps]Fig.4.31Normalizedtime-dependentdiffusioncoefficientsfortwoselectedenergiesforbothsamplesS1andS2.Insetlocalizationlengthsasafunctionofthecarrierenergyextractthecorrespondingmeanfreepathse(E).Localizationeffects,manifestedinadecayofthediffusioncoefficientwithtime,areapparentforthelinescorrespondingtothechargeneutralitypoint,butarelessclearforE=−2γ0.TheelasticmeanfreepathandthesemiclassicalconductivitiesareshowninFig.4.32,asobtainedfromthemaximumofthediffusioncoefficient.Astrikingfeatureistheverylowvalueofthemeanfreepathebelow0.5nmfortheenergywindowaroundtheFermilevel,inwhichtheDOSdepartsfromthatofthepristinegraphenestructure.Fornegativeenergies(holes)farfromthechargeneutralitypoint,aconsiderableincreaseofmorethanoneorderofmagnitudeinthemeanfreepathsisobserved.Theincreaseoccursforsmallerbindingenergiesforsample1thanforsample2,ingoodcorrelationwiththechangesobservedintheDOS(which,aroundthevanHovesingularity,deviatesfromthepristinegrapheneonemorestronglyforsample2).ThesemiclassicalconductivitiesshowaminimumvalueattheFermilevelclosetoσscmin=4e2/πh,inagreementwiththevaluesforgrapheneinthepresenceofdisorderinducedbyimpuritiesorscatterers[28,29].Wenote,however,thattheconductivityremainsnearlyconstantatthatvalueforanenergyrangeofseveraleVaroundtheFermilevel.Thisindicatesthattransportisstronglydegradedintheamorphousnetworkcomparedtopristinegraphene,inwhichtheconductivityincreasesrapidlyawayfromtheFermilevel.Thechargemobility,μ(E)=σsc(E)/en(E),withn(E)beingthecarrierdensity,isfoundtobeoftheorderof10cm2V−1s−1forn=1011–1012cm−2,whichisordersofmagnitudeslowerthanthoseusuallymeasuredingraphenesamples[11].Suchlowconductivityandmobilityvaluesshouldbemeasuredatroomtemperature,wherethesemiclassicalapproximationisexpectedtohold. 1064TransportinDisorderedGraphene4S13.56]0S23[G42/πscσ22.502-2-1012[nm]elE[γ]01.51S10.5S2rescaledDOS0-3-2-10123E[γ]0Fig.4.32Elasticmeanfreepathversusenergyforthetwosamples.DOSofsampleS1isalsoshownforcomparisoninrescaledunits.InsetsemiclassicalconductivityofcorrespondinglatticesTheveryshortmeanfreepathsobtainedindicateafurthersignificantcontributionofquantuminterferencesturningthesystemtoaweakandstronginsulatingsystemasthetemperaturedrops.Interferenceeffectsareevidencedbythetime-dependentdecayofthediffusioncoefficientD(t)/Dmax.Basedonthescalingtheoryoflocaliza-tion[183],anestimateofthelocalizationlengthofelectronicstatescanbeextractedfromthesemiclassicalparametersbyξ(E)=e(E)exp(πhσsc(E)/2e2).TheresultsareshowninFig.4.31(inset).Theamorphoussamplesareextremelypoorconduc-tors,withlocalizationlengthsaslowasξ∼5–10nmoveralargeenergywindowaroundthechargeneutralitypoint.Tofurtherconfirmthelocalizationlengthsestimatedusingscalingtheory,wecomputeexplicitlytheconductanceoftheamorphousgrapheneribbonscontactedwithpristinegrapheneelectrodes,asafunctionoftheribbonlengthL.Figure4.33showstheconductancecurvesfortworibbonsof1.6and8.6nm,respectively,com-paredtothatofagraphenecontactwiththesamelateralsizeinthesupercell(11.4nm).Itisclearthattheconductanceoftheamorphoussamplesisgreatlyreducedwithrespecttothatofgraphene,andthatthereductionismorepronouncedasthelengthoftheamorphousribbonbecomeslarger.Also,whiletheconductancefortheribbonwiththesmallestlengthisrelativelysmooth,itbecomesmorenoisyastheribbonbecomeslonger.Thisreflectsthetransitionfromadiffusivesystem,inwhichtheribbonislongerthanthemeanfreepath,butshorterthanthelocalizationlength,toastronglylocalizedoneinwhichthelocalizationlengthisshorterthantheribbonlength. 4.4TransportPropertiesofAmorphousGraphene107201,2]00,9L=1.6nm>[G0,6σL=8.6nm<0,3graphene2/h]010051015202530L(nm)G[2e0-0,6-0,300,30,6E[γ]0Fig.4.33Landauer-Büttikerconductance(forW=11.4nm)oftwoamorphousribbonscontactedtographeneelectrodeswithL=1.6and8.6nm,respectively.Theconductanceofapristinegraphenecontactwiththesamelateralsize(11.4nm)isshownforcomparison.TheinsetshowsthedependenceoftheconductivityontheribbonsizeL;symbols:calculatedpoints;linefittoσ(L)∼Le−L/ξWFromthevariationoftheLandauer-BüttikerconductancewithsizeL,wecanextractreliablevaluesofthelocalizationlengths,asintheAndersonregimetheconductanceshoulddecayasG(L)∼e−L/ξ.TheinsetinFig.4.33showsthevalueoftheconductivity,obtainedfromtheconductance,foreachsizefrom1.6to15.3nm,averagedoveranenergywindowof1.5γ0aroundtheFermienergy.Afitoftheresultstoσ(L)∼Le−L/ξyieldsavalueofξ=5.8nm.ThisvalueisconsistentWwiththatobtainedaboveusingscalingtheory,forenergiesclosetotheFermilevel,andconfirmsthat,intheseamorphousstructures,stronglocalizationeffectsshouldoccuratlowtemperaturesatdistancesoflessthan10nm.TheseestimatesareingoodagreementwiththeexperimentalresultsfromtransportmeasurementsbyZhouetal.[171],whichshowvaluesintherangebetween0.1and10nmforsamplesamorphizedbyionradiation.Inconclusion,wehaveshownthatamorphousgrapheneisastrongAndersoninsulator.Theincreaseofthedensityofstatesclosetothechargeneutralitypointisconcomitantwithmarkedquantuminterferenceswhichinhibitcurrentflowatlowtemperature.Veryshortmeanfreepathsandlocalizationlengthsarepredicted,inlinewithrecentexperimentalevidenceingrapheneunderheavyionirradiationdam-age[171]. 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Chapter5SpinTransportinDisorderedGrapheneCarbonhasaweakatomicSOC,sographeneisexpectedtohavelongspinrelaxationtimeandphasecoherencelengths.However,thespininjectionmeasurementsbasedonanon-localspinvalvegeometry[13]revealedsurprisinglyshortspinrelaxationtimesofonlyabout100200ps,beingonlyweaklydependentonthechargedensityandtemperature.Thelongestspinrelaxationtimehasbeenmeasureduptonowisalsointheorderofafewns[4].Therearemanyexplanationsforshortspinrelaxationtimesingraphene.SomeexplanationsarerelatedtoenhancedSOCinducedbyadatomsingraphenesheetorbythebreakinginversionsymmetryduetotheelectricfieldcreatedbysubstrate.AnotherpossibilitycouldbetheGaugefieldduetoripples[5]whichinducesaneffectivemagneticfieldB⊥perpendiculartothegraphenesheet.Thereisalsoanotherexplanationsayingthattheformationofsp3hybridizationenhanceslocalSOC[6]whichleadstofastspinrelaxation.However,thesetheoreticalresultscouldntgivesatisfyingexplanationsforexperimentaldatatodate.Inthissection,weperformsometheoreticalcalculationstoinvestigatethespinrelaxationinultracleangraphene,andweproposeanewmechanismforspinrelax-ationingraphenewhichisrelatedtothedisconnectionofspinandmomentumclosetotheDiracpoint.AttheendofthisChapter,someresultsoftheeffectofthesegre-gationofstrong-SOCadatomsongrapheneontheQHEareshown.5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism5.1.1IntroductionTheelectronicpropertiesofmonolayergraphenestronglydifferfromthoseoftwo-dimensionalmetalsandsemiconductorsinpartbecauseofinherentelectron-holebandstructuresymmetryandaparticulardensityofstateswhichvanishesattheDirac©SpringerInternationalPublishingSwitzerland2016115D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2_5 1165SpinTransportinDisorderedGraphenepoint[6].Additionally,thesublatticedegeneracyandhoneycombsymmetryleadtoeigenstatesthatholdanadditionalquantum(Berrys)phase,associatedwiththeso-calledpseudospinquantumdegreeoffreedom.AlloftheseelectronicfeaturesaremanifestedthroughtheKleintunnelingphenomenon[7],WAL[8]ortheanomalousQHE[9].Thepossibilityofusingthepseudospinasameanstotransportandstoreinformationhasalsobeentheoreticallyproposed[10,11].There,theroleofthepseudospinisequivalenttothatofthespininspintronics,suchasinthepseudospinanalogueofthegiantmagnetoresistanceinbilayergraphene[11].Eventhoughpseudospin-relatedeffectsdrivemostoftheuniquetransportsignaturesofgraphene,theroleofthepseudospinonthespinrelaxationmechanismhasnotbeenexplicitlyaddressedandquantified.Pseudospinandspindynamicsareusuallyperceivedasdecoupledfromeachother,withpseudospinlifetimesbeingmuchshorterandpseudospindynamicsmuchfasterthanthoseforspins.However,thispicturebreaksdowninthevicinityoftheDiracpoint,aregionthatisusuallyoutofreachofperturbativeapproachesandthatisparticularlyrelevantforexperi-ments,becauseFermienergiescanonlybeshiftedbyabout0.3eVviaelectrostaticgating.Moreover,inthepresenceofSOC,spincouplestoorbitalmotion,andthere-foretopseudospin[12],sothatspinandpseudospindynamicsshouldnotbetreatedindependently.Thereasonforoverlookingtheroleofthepseudospinonthespindynamicsisperhapsrootedinthefactthatthespintransportpropertiesappearremarkablysimilartothosefoundincommonmetalsandsemiconductors[13].Indeed,spinprecessionmeasurementsinnonlocaldevicesresultinexperimentalsignaturesthatwouldbeindistinguishablefromthoseobtainedinametalsuchasaluminium[14],orasemi-conductorsuchasGaAs[15],withextractedspinrelaxationtimesτsthatarealsotypicallyofthesameorderofmagnitude(afewnanosecondsorlower).Spinrelax-ationingraphenehasthereforebeeninterpretedusingtheconventionalexperimentalmanifestationsofeithertheEYorDPmechanism[1620].IntheEYscenario,thespinrelaxationtimeisdeterminedbythespinmixingofcarriersandtheSOCofthescatteringpotential,andthusitisusuallyassumedtobeproportionaltothemomentumrelaxationtimeasτs≈α·τp,withα1(forinstanceinalkalimetalsα∼104−106)[13].Incontrast,intheDPmechanismspinprecessesaboutaneffectivemagneticfieldwhoseorientationisfixedbythemomentumdirectiondur-ingfreepropagationofelectrons.Suchorientationchangesateachscatteringevent,whichresultsinadifferentscalingbehavioras1/τsDP∼2τp[13](withtheaver-agemagnitudeoftheintrinsicLarmorfrequencyoverthemomentumdistribution).Experimentalestimatesofτsandτparegenerallyobtainedinaphenomenologicalwaybyfittingtheexperimentalresistivitycurvestothetheoreticalformulaobtainedusingsemi-classicaltransportequations[1,17].However,thisphenomenologicalanalysisisnotwellconnectedwiththemicroscopicinterpretation.Firstofall,theweakSOCingraphenewouldsuggestτsinthemicrosecondrange[21,22],incleardisagreementwithexperimentaldata.Inaddition,theτsestimatedinhigh-mobilitygraphenewithlongmeanfreepathsremainsunsatisfactorilyinterpretedwithasinglerelaxationmechanism,sayEYorDP[18,23,24].Thesuppressionofτsinclean 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism117graphenehasbeententativelyassociatedtoanenhanced(intrinsicorextrinsic)SOCduetomechanicaldeformationssuchasripples,orunavoidableadatomsincorpo-ratedduringthedevicefabricationprocess,buttheultimateandmicroscopicnatureofspinrelaxationatplayremainscontroversialandelusive.Here,weunravelaspinrelaxationmechanismfornonmagneticsamplesthatfollowsfromanentanglementofspinandpseudospindegreesoffreedomdrivenbyrandomSOC,whichmakesituniquetographeneandismarkedlydifferentfromcon-ventionalprocesses.Weshowthatthemixingbetweenspinandpseudospin-relatedBerrysphasesresultsinunexpectedlyfastspindephasing,evenwhenapproach-ingtheballisticlimit,andleadstoincreasingspinrelaxationtimesawayfromtheDiracpoint,asobservedexperimentally.Thishithertounknownphenomenonpointstowardsrevisitingtheoriginofthesmallspinrelaxationtimesfoundingraphene,whereSOCcanbecausedbyadsorbedadatoms,ripplesoreventhesubstrate.Italsoopensnewperspectivesforspinmanipulationusingthepseudospindegreeoffreedom(orviceversa),atantalizingquestfortheemergenceofradicallynewinfor-mationstorageandprocessingtechnologies.5.1.2SpinRelaxationinGold-DecoratedGrapheneInthefollowing,weexplorespincharacteristicsingraphenebyinvestigatingtheeffectofweakperturbationinducedbylowdensitiesofad-atoms(downto1012cm−2),whichintroducearandomRashbafieldinrealspacebutvanishinglysmallintervalleyscattering,yieldinglongmeanfreepaths.Here,fortypicalelec-√trondensitieswithin[1010,1012]cm−2,theFermiwavelength(λF=2π/n,nthechargedensity)liesbetween20and200nmandthusexceedsthemeanseparationbetweenadatoms(∼10nm)wherespin-orbitscatteringoccurs,thereforequestioningtheuseofastandardsemiclassicaldescription.Tostudyspindynamics(andspinrelaxation),weuseanon-perturbativemethodbysolvingthefulltime-dependentevo-lutionofinitiallyspinpolarizedwavepackets,eitherthroughadirectdiagonalizationofacontinuummodel,orarealspacealgorithmforamicroscopicdisordermodel,definedinaTBbasis.Wedescribethesystemofagraphenemonolayerfunctionalizedwitharandomdistributionofadatoms.Theelectronicstructureofcleangrapheneiscapturedbytheusualπ-π*orthogonalTBmodel(withasinglepz-orbitalpercarbonsite,zeroonsiteenergiesandnearestneighborshoppingγ0).Thepresenceofnon-magneticadatomsrandomlyadsorbedatthehollowpositionsonthegraphenesheetintroducesadditionallocalSOCterms(Fig.5.1a,b),definedas[25].2ic+c+H=−γ0j+√VIcs·(dkj×dik)cjii3ijij∈R+iV++Rcz·(s×dij)cj−μcci(5.1)iiij∈Ri∈R 1185SpinTransportinDisorderedGrapheneFig.5.1SpinDynamicsindisorderedgraphene.aBall-and-stickmodelofarandomdistributionofadatomsontopofagraphenesample.bTopviewofthegoldadatomsittingonthecenterofanhexagon.c,dTime-dependentprojectedspinpolarizationSz(E,t)ofchargecarriers(symbols)initiallypreparedinanout-of-planepolarization(atDiracpoint(redcurves)andatE=150meV(bluecurves)).Analyticalfitsaregivenassolidlines(seetext).ParametersareVI=0.007γ0,VR=0.0165γ0,μ=0.1γ0,ρ=0.05%(c)andρ=8%(d)Thefirsttermisthenearestneighborhoppingtermwithγ0=2.7eV.ThesecondtermisacomplexnextnearestneighborhoppingtermwhichrepresentstheintrinsicSOCinducedbytheadatoms,withdkjanddiktheunitvectorsalongthetwobondsconnectingsecondneighbors,sisavectordefinedbythePaulimatrices(sx,sy,sz),andVItheintrinsicSOCstrength.ThethirdtermdescribestheRashbaSOCwhichexplicitlyviolatesz→−zsymmetry,withzbeingaunitvectornormaltothegrapheneplaneandVRtheRashbaSOCparameter.ThelasttermisthepotentialshiftμassociatedwiththecarbonatomsintherandomplaquettesRadjacenttoadatoms(Fig.5.1b).Suchshiftisduetoweakelectrostaticeffectsthatarisefromchargeredistributioninducedverylocallyaroundtheadatom[25].ARashbasplittinghasbeenobservedexperimentallyatthegraphene/nickelandgraphene/gold(Au)interfaceswithspinsplittingofupto100meV[26,27].Goldandnickelaswellasothermaterialsliketitanium,cobaltorchromium,areusuallypresentduringthefabricationofthenonlocalspinvalvesthatareusedtodetermineτsandlikelyleaveresiduesontheexposedgraphenesurface.Hereafter,weconsiderthecaseofAuadatomswhoseinfluenceonthetransportpropertiesofgraphenehasbeenstudiedexperimentally[28].TheTBparameterstodescribebothintrinsicandRashbaSOCsinducedbysuchadatomsareextractedfromab-initiocalculations[27].Basedonsuchparameters,weexplorehowthespinrelaxationtimesscaleasafunctionoftheadatomdensityandadatom-inducedlocalpotentialshift.Thespindynamicsingrapheneareinvestigatedbycomputingthetime-dependenceofthespinpolarizationdefinedby(SeeSect.3.2.2fortechnicaldetails)(t)|sδ(E−H)+δ(E−H)s|(t)S(E,t)=(5.2)2(t)|δ(E−H)|(t) 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism119andassumingthatspinsareinitiallyinjectedout-of-plane(zdirection),i.e.|(t=0)=|ψ↑.Thetimeevolutionofthewavepackets|(t)isobtainedbysolvingthetime-dependentSchrödingerequation.Wefocusontheexpectationvalueofthespinz-componentSz(E,t).Figure5.1showsthetypicalbehaviorofSz(E,t)fortwoselectedenergies(attheDiracpointandatE=150meV)andtwoadatomdensitiesρ=0.05%(about1012adatomspercm2)(c)andρ=8%(d).ThetimedependenceofSz(E,t)isverywelldescribedbycos(2πt/T)e−t/τs,introducingthespinpre-cessionperiodTandthespinrelaxationtimeτs,whichareextractedfromfittingthenumericalsimulations(solidlines).Thetimedependenceofthemodulusofthefullspinpolarizationvector|S|=|(sx,sy,sz)|alsoexhibitsanunambiguoussignatureofspinrelaxation(SeeSect.5.1.3).Figure5.2givesτsandTextractedfromthefitsofSz(E,t)forvaryingadatomdensity.OnefirstobservesthatthespinprecessionperiodisenergyindependentandispreciselyequaltoT=π/λ¯R(withλ¯R=3ρVRanaverageSOCstrength)evenforthelowestcoverage,whichFig.5.2Spinrelaxationtimesandtransportmechanisms.Spinrelaxationtimes(τs)forρ=0.05%(a)andρ=8%(b).Black(red)solidsymbolsindicateτsforμ=0.1γ0(μ=0.2γ0).TversusEisalsoshown(opensymbols).τp(dottedlinein(b))isshownoverawiderenergyrange(topx-axis)inordertostressthedivergencearoundE=0(μ=0.2γ0).Wecannotevaluateτpbelow100meV,sincethediffusiveregimeisnotestablishedwithinourcomputationalreach.Panels(c)and(d):TimedependentdiffusioncoefficientD(t)forρ=0.05%andρ=8%withμ=0.2γ0 1205SpinTransportinDisorderedGrapheneFig.5.3Spinrelaxationtimesdeducedfromthecontinuumandmicroscopicmodels.aSpinrelaxationtimes(τs)forvaryingρbetween0.05and8%extractedfromthemicroscopicmodel(withμ=0.1γ0).Insetτsvaluesusingthecontinuummodelforρ=1and8%(filledsymbols).Acomparisonwiththemicroscopicmodel(withμ=0)isalsogivenforρ=8%(opencircles).bScalingbehaviorofTandτsversus1/ρ.TheTvaluesobtainedwiththemicroscopic(resp.continuum)modelaregivenbyreddiamonds(resp.redsolidlines).τsvaluesforthemicroscopicmodel(bluesquares)andthecontinuummodel(blackcircles)areshownfortwoselectedenergiesE=150meV(solidsymbols)andE=0(opensymbols).Solidlinesarehereguidestotheeyeagreeswiththeestimatebasedonthecontinuummodel[21](SeeEq.(2.52)).Incontrast,thespinrelaxationtimedisplaysasignificantenergydependence.AV-shapeisobtainedforlowenergy,withτsbeingminimalattheDiracpointwithvaluesrangingfrom0.1to200pswhentuningtheadatomdensityfrom8to0.05%(asgiveninFig.5.3a,mainframe).Basedontheobservedscalingτs∼1/ρ(seeFig.5.3b),onecanfurtherextrapolatethespinrelaxationtimesforevensmallerdefectdensity,obtainingτs∼1−10nsforadsorbatedensitiesdecreasingfrom1011cm−2downto1010cm−2.TheobtainedV-shapedenergydependenceandtheabsolutevaluesofτsareremarkablysimilartothosereportedexperimentally[1,16,17,28].ThefasterrelaxationattheDiracpointisactuallyevidentinFig.5.1candd.ThereasonforthisbehaviouristhedecreaseofthecouplingbetweenthepseudospinandmomentumandtheincreasingdominanceoftheSOCinteraction,whichleadstospin-pseudospinentanglement.ThedetailsoftheentanglementarefurtherdescribedinEq.(5.3)belowandintheSect.5.1.3Asdiscussedabove,theusualapproachtodiscriminatebetweenconventionalEYandDPrelaxationmechanismsinmetalsandsemiconductorsistoscrutinizethescalingofτsversusτp.Suchproceduredoesnotnecessarilyapplyifthedominantprocessesthatleadtomomentumandscatteringrelaxationarenotthesame.Forinstance,inmonolayertransition-metaldichalcogenides,itwasdemonstratedthatthecarrierscatteringbyflexuralphononsleadstofastspinflipsbutnottomomentumscatteringand,therefore,thespintransportisdecoupledfromthecarriermobility.Inthefollowingdiscussion,weshowthatsimpleEYorDPscalingisalsonotsuitabletodescribeourfindings. 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism121Withinourmicroscopiccalculations,weanalyzethetime-dependenceofthediffusioncoefficientforvaryingenergiesandad-atomdensities(Fig.5.2c,d).Forthelowestimpuritydensity(0.05%,Fig.5.2c),regardlessoftheconsideredenergy,D(E,t)isseentoincreaseintimewithnosignofsaturationwithinourcompu-tationalcapability,indicatingaballistic-likeregimefortheconsideredtimescales.Onlyforthelargestad-atomdensity(8%)doesD(t)eventuallysaturateathighenoughenergies(above100meV,D(t)→Dmax),allowingfortheevaluationofthetransporttimeusingτp(E)=Dmax(E)/2v2(E)(seedashedlinesinFig.5.2b).AsharpincreaseofτpisseenwhenapproachingtheDiracpoint,whereτsreachesitsminimumvalue,withτsτp.Thisenergydependenceinτpisnotuniquetogoldad-atomsbuthasalsobeenobservedforothertypesofdisorderwithaweakintervalleyscatteringcontribution,suchasepoxidedefectsorlongrangescatterers[29].AsseeninFig.5.3b,τs∼1/ρ,whichdoesnotallowustodiscriminatebetweenEYandDPprocesses.However,theabsolutevaluesofτsandτp(withτsτp)areaclearmanifestationofthebreakdownofthetypicalscalingassociatedtobothmechanisms.EventheunconventionalDPregimedescribedinRef.[13]forthecaseofτp/T≥1where1/τs∼(withaneffectivewidthofthedistributionofprecessionfrequencies)cannotaccountfortheobservationthataweakvariationinthelocaldisorderaffectstheabsolutevaluesofτs(whileρisunchanged)asobservedinFig.5.2.Herelocaldisorderismonitoredbytheμparameter.(AlthoughμbelongstotheTBparameterizationoftheadatom,weuseittemporarilytoincreaselocaldis-order.)Infact,itsvaluecouldslightlychangewhenmodifyingthesubstratescreeningorinpresenceofamorestronglybondedadsorbantthanAu.Asaconsequenceoftheabovefindings,thespinrelaxationmechanismatplayisincompatiblewithboththeEYandtheDPmechanisms,afactwhichcouldshednewlightonthecurrentdebateonthemicroscopicnatureofspinrelaxationincleangraphene[18,23,24].Wenowfurtherstudytheoriginoftheτsminimumatlowenergyanditsuncon-ventionalscalingwithτp.Giventhatoursimulationswiththemicroscopicmodelgiveτsτp,wefurtherexplorethelow-energyspindynamicswithaneffectivecontinuummodel,inwhichthespin-orbitscatteringistreatedasahomogeneouspotential[21].WesolvetheDiracequationinthecontinuummodelbyusinga4×4effectiveHamiltoniantakingintoaccountthepseudospindegreeoffreedomh(k)=h0(k)+hR(k)+hI(k)(5.3)Whilethehoppingfromthreenearestneighborsh0(k)=vF(ζσxkx+σyky)⊗1sdominatesathighenergyandvanishesattheDiracpoint(ζ=±1forKandKvalleys,σarepseudospinPaulimatricesand1sisa2×2identitymatrix),theintrinsicSOChI(k)=λ¯Iζσz⊗szandtheRashbainteractionhR(k)=λ¯Rζσx⊗sy−σy⊗sxplayanextremelyimportantroleattheDiracpoint,wherethecouplingbetweenspinandpseudospinbecomespredominant,andgovernsthequantumdynamicsanddephasingofthewavepacketsasdescribedbelow.Withinthecontinuummodelspinrelaxationisachievedbyintroducinganad-hocenergybroadening.Weuseaninitiallyz-polarizedstateforinjectionandconsider 1225SpinTransportinDisorderedGrapheneonlytheKvalley.AcertaindensityofAuimpurities(inducinglocalSOCs)is√describedbytheeffectiveSOCsλ¯R=3ρVRandλ¯I=33ρVI.Notethatnoadditionallocal(static)scatteringpotentialisintroducedhere(μ=0).Bycomputingthespindynamicsofinitiallyspin-polarizedwavepackets,onealsoobtainsaspinrelaxationeffectdefinedbythetwotimescalesTandτs(SeeSect.5.1.3).Itisinstructivetocontrasttheresultsofthecontinuummodel(Fig.5.3a,inset)withthosefromthemicroscopicmodel(Fig.5.3a,mainframe).AlthoughthespinprecessionperiodTobtainedbybothmodelsisidentical(Fig.5.3b)andtheenergydependenceofτsissimilar,theabsolutevaluesofτsdiffersubstantially,especiallyinthehighenergyregime,whereτsisclearlyoverestimatedusingthecontinuummodel.Ofkeyimportance,suchdifferencebecomesincreasinglylargeupondecreas-ingthead-atomdensitybecauseτspresentsadifferentscalingwithdefectdensity(seeFig.5.3b).Thisclearlyevidencestheimportanceofdisorder,asintroducedbytherandomdistributionofimpurities,andillustratesthelimitsofaphenomenologicalapproachusingthecontinuummodelforquantitativecomparisonwithexperimentaldata.Notwithstanding,thequalitativeagreementbetweenbothmodels(particularlyforhighcoverage)andtheweakmomentumrelaxationeffectsobservedinthemicro-scopicmodel(asseeninthelongτp)suggestsomegeneralityintheunconventionalspinrelaxationobservedneartheDiracpoint.Tofurthersubstantiatetheoriginofthespinrelaxation,wescrutinizethespinandpseudospindynamicsofwavepacketsusingthecontinuummodel.Pseudospinisintrinsicallyrelatedtothegraphenesublatticedegeneracyand,aslongasvalleymixingisnegligible,pseudospinisalignedinthedirectionofthemomentumathighenergy(h0(k)dominatestheHamiltonian(5.3)).TheRashbaspin-orbittermhR(k)entanglesspinswiththelatticepseudospinσ,overridingthelockingrulebetweenpseudospinandmomentumsinceh0(k)becomesvanishinglysmallinthevicinityoftheDiracpoint(seeSect.5.1.3)[12,19].Figure5.4highlightsthespindynamicsatdifferentchosenenergiesE=0,E=−5meV(lowenergy)andE=130meV(highenergy),whicharerepre-sentativeoftheunderlyingphysics(notethatnorelaxationtakesplaceforfixedenergy,thustherequirementofthead-hocbroadening).Athighenergy,thespinprecessesquiteregularlyasseeninFig.5.4a,whichshowsanoscillatorypatternofSz(t)dominatedbyasingleperiodT=π/λR=0.19ps.ThespinprecessionoccursaboutaneffectivemagneticfieldBRdictatedbytheRashbainteractionandpointingtangentiallytotheFermicircle(asseenfromtheprecessionfrombluetopinkinrightpanelsfromt1tot4).Incontrast,thepseudospinσ(t)pointsapproxi-matelyinthesamedirectionofthemomentum(evolvingfromgreentoorange).ItsoscillatorypatternisdrivenbytheRashbaperiodTtogetherwithasuperimposedandmorerapidoscillation(describedintheSect.5.1.3).Thesituationatlowenergy(Fig.5.4b,c)ismarkedlydifferent.Weobserveahighlyunconventionalspinandpseudospinmotionwhichisanalyzedmorecloselyforthespinandpseudospinz-componentsattwolowenergies(attheDiracpointandatE=−5meV).Incontrasttothehigh-energycase,theamplitudeofthepseudospinoscillationisstronglyenhancedsincepseudospinisnolongerlockedwithmomentumbutstartstoprecessaboutaneffectivepseudo-magneticfield.The 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism123Fig.5.4Spinandpseudospindynamicsingraphenewithρ=8%ofadatomsac.Timedepen-denceofspin-polarizationSz(blue)andpseudospinpolarizationσz(green)inzprojectionforenergiesE=130meV(a),E=0(b),andE=−5meV(c).Notethatallquantitiesarenormal-izedtotheirmaximumvaluetobettercontrasttheminthesamescale.Rightpanelsshowthetimeevolutionforbothspin(frombluetopink)andpseudospin(fromgreentoorange).Thesnapshotsaretakenatdifferenttimesfromt1tot4samplingtheshadedregionsin(a)(c).dFouriertransformofSz(t)plottedoveroscillationperiod,andshowingnon-dispersivespectraathighenergy(betweenE=125,130and135meV).Low-energyspectra(forE=−5,0and5meV)changestronglywithenergy(dispersive)showingagradualreductionandblueshiftoftheoriginalRashbapeakatabout0.19psandtheappearanceofadditionalfeaturespseudo-magneticfielddependsstronglyonthespinorientation,thusyieldingcom-plextime-dependentdynamicsofspinandpseudospin(seerightpanelsofFig.5.4correspondingto5.4b,c).SuchaneffectderivesfromtheincreasedpseudospinpspsprecessionperiodT=π/E(aboutB),whichdecreasessignificantlyatlow00energy.Thereforeσicannolongerbereplacedbyitstimeaverageσi(incontrasttothehigh-energysituation,seeSect.5.1.3),whichinconsequencealsoholdsfortheRashbafieldBR.ThetimedependenceofBRwithvariabilityonatimescalesimilartotheRashbaperiodleadsthentostrongnon-lineardynamicsofspinand 1245SpinTransportinDisorderedGraphenepseudospinmotion.Asaresultofsuchcoupleddynamics,thespinprecessioncannotbedescribedbyasingleperiodTasbecomesevidentfromthecomplexFourierspectraofSz(t)inFig.5.4d.ThetimedependenceofBRincludesalsochangesofitsdirection,thusimpactingthepseudospinandliftingthepseudospin-momentumlock-ing.Bothoftheseeffectsfinallyproduceajointspin/pseudospinmotionprohibitingpsthede-couplingofdrivingforces(B,BR)thatwaspossibleathigherenergies.0Whilethecontinuummodelprovidesqualitativeinsightintothespin-pseudospincouplingandentanglementoftheircorrespondingwavefunctions,themicroscopicmodelenablesthequantificationofspinrelaxationtimesforagivenmicroscopicdisorder.Byscrutinizingthegeneralformofthespinpolarization(Eq.(5.2)),asimpleunderstandingofthespinrelaxationmechanismcanbedrawn.Inthemicroscopicmodel,thepropagationofaninitiallyspin-polarizedwavepacket|ψ↑(t=0),isdrivenbytheevolutionoperatore−iHt/|ψ↑(t=0),withHconsistingofthecleangraphenetermplustheSOCpotential,whichactsasalocal(andrandom)perturbationontheelectronspin.Thetime-dependenceofthetotalspinpolarizationresultsfromtheaccumulateddephasingalongscatteringtrajectoriesdevelopedundertheevolutionoperator.Asthedistributionofscatteringcentersisrandominspace,alldifferenttrajectoriesaccumulatedifferentphaseshiftsintheirwavefunctions(eachbeingtheresultoflocalspin/pseudospincouplinganddisorderpotential).Whenphaseshiftsforupanddowncomponentsaverageout,thespinpolarizationof|ψ↑(t=0)islost.5.1.3FurtherDiscussionLow-energyeffectiveHamiltonianandanalysisofelectronicstatesclosetotheDiracpointToillustratethatspinandpseudospinarefullyentangledforcertainstatesclosetotheDiracpoint,wecalculatethebandstructureandthemodulusofthespinpolarizationvector|S(k)|.Figure5.5showsthecomputedbandstructureobtainedbydiagonalizingtheKane-Mele-RashbaHamiltonian(Eq.(5.3))for8%goldadatomcoverage.TheRashbaterminducesacounter-propagatingspintextureinthekx,kyplanethattendstovanishclosetotheDiracpointas[12]:μvF(k×z)Sνμ(k)=(5.4)λ2+2v2k2RFWefurthercalculatethemodulusofthespinpolarizationvector|S|=|(sx,sy,sz)|fromtheeigenstatesofthefullHamiltonianinEq.(5.3)withbothintrinsicandRashbaSOCcA,↑cA,↓ikrk,±=⊗|↑±i⊗|↓e.(5.5)cB,↑cB,↓ 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism125Fig.5.5BandstructurecalculatedusingtheKane-Mele-Rashbamodelfor8%adatomconcentra-tion.TheinsetshowsthetypicalRashba-likespintexturefortheconductionbandsInpresenceoftheRashbaSOCterm,Blochstateswithwell-definedspinpolarizationarenolongereigenstatesofthecompleteHamiltonian[19].Theclearsignatureofspin-pseudospinentanglementisfoundatlowenergies(k→0),forwhichwegetthefollowingsolutionsI0i=⊗|↑±⊗|↓(5.6)k,±10II10=⊗|↑±⊗|↓.(5.7)k,±0iInbothcases,achangeinsublattice(pseudospin)indexentailsachangeinspinindex.Thismeansthatatlowenergyspinandpseudospinarecompletelylockedand|S|≈0.Thesituationisdifferentforhighenergies(|k|>0),whenpseudospin-momentumcouplingcomesintoplay,allcoefficientsbecomeequallyweighted(|cσ,s|≈0.5)andspinandpseudospinareunlockedleadingto|S|≈1.SuchenergydependenceisshowninmoredetailsinFig.5.6,wherethespinpolarization|S|ofthestatesinthetwofirstconductionbandsarecomputedbydiagonalizingtheeffectiveHamiltonian(Eq.(5.3))forad-atomconcentrationsρ=25%(1/4MLgoldcoverageasreportedbyMarchenkoetal.[27])andρ=8%(whichallowstomakeaconnectionwiththemicroscopicmodelresultsinFig.5.7).Thelowerconduction-bandstatesarecompletelyentangledclosetotheDiracpoint(redcurves),butbecomedisentangledatrelativelylowenergies25and100meVforrespectivelylowandhighad-atomdensities(seeverticaldashedlines).Interestingly,abovetheseenergies,theeigenstatesofthesecondconductionband(bluecurves)comeintoplaywithastrongerspin/pseudospinentanglement(|S| 1)evenforhighenergyvalues:E≈150meVforρ=8%andE≈300meVforρ=25%. 1265SpinTransportinDisorderedGrapheneFig.5.6Energydependenceofthespinpolarizationvector|S|forstatesinthetwoconductionbandsobtainedwiththe4-bandslow-energymodel.Theresultscorrespondtoadatomconcentration8%(1/4ML)(leftpannel)and25%(rightpannel).Inbothcases,closetotheDiracpoint,spinandpseudospinentanglementisveryhighgiventhesmallvaluesof|S| 1Fig.5.7Time-dependenceofthemodulusofthespinpolarizationvector|S(E,t)|inthemicro-scopicmodelwithrealisticdisorderandgoldad-atomconcentrations0.05and8%attwospecificenergies:DiracpointE=0andE=150meVEnergycrossoverofspin/pseudospindynamicsandeffectivemagnetic/pseudo-magneticfieldsFigure5.4exhibitsdifferentoscillatingperiodsforspinandpseudospin.Athighenergy,spinprecessionleadstooscillationsinS(t)withRashbaperiodTwhilethepseudospinoscillations(σz(t)inFig.5.4)aredrivenbyTtogetherwithamorerapidsuperimposedoscillation.Acrossovertocomplexlow-energydynamicsis 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism127observedwherespin-andpseudospinmotionaremorecloselyrelatedtooneanother.Toillustratetherelationbetweenspin,pseudospinandmomentum,weintroducethreedifferenteffectivepseudomagneticfields:Bps(k)=vF(ηkx,ky,0)0Bps(s)=λR(ηsy,−sx,0)(5.8)RBps(s)=λI(0,0,ηsz)IandtwoeffectivemagneticfieldswhichareextractedfromEq.(5.3):BR(σ)=λR(−σy,ησx,0)(5.9)BI(σ)=λI(0,0,ησz)whereσi=k|σi⊗1s|kandsi=k|si⊗1σ|karetheexpectationvaluesofthepseudospinandspinoperators,σiandsi(analogoustodecoupledsubsystems),andkaretheeigenstatesoftheKMRHamiltonian(seeEq.(5.5)).Fromtheformoftheeffectivemagneticfields,itisseenthatunlikethecaseofsemiconductorswhereSOCdirectlycouplesspinwithmomentum,ingraphenespincouplesdirectlywithpspseudospin(seeBRandB),andisrelatedtomomentumviathecouplingbetweenRpseudospinandmomentumh0(k),atermwhichvanishesattheDiracpoint.TheseeffectivemagneticfieldshelpillustratingtheenergycrossoverinFig.5.4.WhiletheoccurrenceofthesameRashbaprecessionforspinandpseudospinathighenergy(Fig.5.4a)isrelatedtotheanalogyoftheeffectivefields(BRforspinpsandBforpseudospin),thesuperimposedrapidoscillationinσzcanberationalizedRasfollows.Weobservethatathighenergythenearestneighborhoppingfromthreeneighbors,h0∝k,dictatesadditionalpseudospinprecessionaboutaradialin-planepsfieldB(∝k)=hvF(kx,ky,0)withsmallamplitudesforσz(t)andwithaperiod0psgivenbyT=π/E(0.016psforE=130meV).Fortheoveralldynamicsitis0psimportantthatthisrapidpseudospinprecessionaboutBdoesnotaffecttheslower0spindynamicsimposedbyhR.Indeedwecanreplaceσxbyitstimeaverageσxandσy→σyinBR.Asaresult,thereisonlyweakinterference(feedback)betweenspinandpseudospindynamicsandbothdegreesoffreedomcanbeunderstoodasbeingdrivenindependentlybytheirrespectiveeffectivefields.Incontrast,atlowenergy,theabovereplacementsarenolongerjustifiedandBRbecomestimedependentthroughthetimedependenceofσ(analogouslyforpsBands)resultingincomplexspin-pseudospindynamicswithnewcharacteristicRperiods.Momentumrelaxation,spinrelaxationandentanglementofstatesingold-decoratedgraphenesamplesFromtheanalysisofspindynamicsusingthemicroscopicandcontinuummodels(Figs.5.3and5.4),wehaveshownthatthespinrelaxationmechanismclosetotheDiracpointisinconsistentwithEYorDPscalinglaws.ForEY,thespinrelaxationtimeisproportionaltothemomentumrelaxationtimeasτsEY≈Ncollisions·τp,where 1285SpinTransportinDisorderedGrapheneNcollisions1denotesthenumberofscattering-off-impurityeventsbeforespinflipoccursandτpisthetransporttime.BydefinitionτsEYτpwhichisoppositetoourestimatesinthelowimpurityregime.FortheDPmechanism,thescalingbehaviorbetweenspinandmomentumrelax-ationtimesisinvertedτsDP∝1/τp.Theessentialcharacteristicofsuchmechanismishoweverthatifdisorderincreases(accompaniedbyadecayofτp)thenτsincreasesconsistently.Ourresultscannotbedescribedbysuchscalingsincebyincreasingdis-orderμ,bothτpandτsdecreasesimultaneously.Also,whenapproachingtheDiracpoint,τpseemstoincreasecontinuouslywhileτstendstosaturatetoaminimumfinitevalue.Thisissimilarlyseeninthetime-dependenceofthemodulusofthespinpolarizationvector|S(t)|inthemicroscopicmodelwithrealisticdisorder.Figure5.7shows|S(t)|for0.05and8%goldad-atomconcentrationsandarecomplementarytoFig.5.2ofthemainpaperinshowingthespinpolarizationlossaccumulatedintime[30].ThefactthatthetotalspinpolarizationdecreasesfasterwhenapproachingtheDiracpoint,wheremomentumrelaxationtime(τp)islarger,isafurtherconfirmationofourinterpretationthatspin-pseudospinentanglementdrivenbyRashba-typeSOCisattheheartofthespinrelaxationmechanismofgold-decoratedgrapheneatlowenergies.Influenceofchargepuddles.ItiswellknownthatchargepuddlesingraphenecanmaketheDiracpointenergyfluctuatingduetolocalchangesinthechemicalpotential[31]possiblyhinderingtheobservationofthediscussedspin-relaxationmechanismatlowenergy.AsrecentlyreportedbyXueetal.[32],thefluctuation(standarddeviation)oftheDiracpointenergyinsupportedgraphenesamplesdependsonthesubstrateandrangefromE≈56meVforSiO2toE≈5meVforhBN[32]inconsistencywithapreviouspaperreporting50meVforSiO2[33].ThedifferenceintheenergeticpositionoftheDiracpointisintimatelyrelatedtothesizeofthechargepuddlesinducedbythesubstratewhichreachanapproximatesizeof10nmforSiO2andaround100nmforhBN.Bycomparingtherelevantenergiesforthebandonsets(i.e.energieswhere|S| 1)inFig.5.6andtheFermienergyfluctuationsfromliterature,weexpectthatthespinrelaxationmechanismproposedinthismanuscript,basedonspin-pseudospinentanglement,shouldbeexperimentallyaccessible.Analogytospin-lessbilayergrapheneWeobservethattheHamiltonianinEq.(5.3)isverysimilartotheoneofspin-lessbilayergraphene(BLG)atlowenergiesandshowsaverysimilarbanddispersionaroundbothvalleys[3436],althoughthenatureofeigenstatesisquitedifferent[12].BelowwecomparetheHamiltonianmatricesofbothcases.TheKMRHamiltonianinonevalleyreads⎛⎞λIv(kx−iky)00(K)⎜⎜v(kx+iky)−λI−2iλR0⎟⎟H=(5.10)KMR⎝02iλR−λIv(kx−iky)⎠00v(kx+iky)λI 5.1SpinTransportinGraphene:PseudospinDrivenSpinRelaxationMechanism129andthespin-lessBLG-Hamiltonian,initsmostreducedversion[36],canbeexpressedas:⎛⎞−v(kx−iky)00(K)⎜⎜v(kx+iky)γ10⎟⎟HBLG=⎝0γ1v(kx−iky)⎠(5.11)00v(kx+iky)−whereγ1istheinterlayerhoppingwhichconnectsaB-siteinthetoplayerwithanA-siteinthebottomlayerinBernalstackedbilayergraphene.Thisinteractioninducesastaggeredpotential±withineachlayerdistinguishingcarbonatomsintoppositionandthoseathollow-sites.Interestingly,thisstaggeredpotentialchangessignatoppositelayerssimilarlytotheintrinsicSOCλIingraphene.ItishelpfultowritetheaboveHamiltonianintermsofPaulimatricesinordertocomparewithEq.(5.3):BLG(k)=vh0F(ησxkx+σyky)⊗1shBLG(k)=γγ1[σx⊗ξx]+σy⊗ξyhBLGσ(k)=z⊗ξz(5.12)wherethelayeroperatorξinEq.(5.12)playstheroleofthespinoperatorsinEq.(5.3),whiletheseconddegreeoffreedomisthepseudospinσinbothcases.Itisimportanttonotethat,whilethefirstandthirdtermsinEq.(5.12)resembletheonesinEq.(5.3),thesecondonehasadifferentstructureintermsofPaulimatriceswhencomparedtotheRashbaterm.However,italsoleadstoin-planeeffectivepseudomagneticandmagneticfieldsoftheform:Bps(ξ)=γ1(ξx,ξy,0)(5.13)γBγ(σ)=γ1(σx,σy,0).(5.14)Also,theeigenstatesoftheBLGHamiltonian,whilenolongercomplex,stillshowalayer-pseudospinentanglementatlow-energiesallowingfornewinterestingphenomenaregardinglayerrelaxationinBernalstackedbilayergrapheneBLG,I01=⊗|1±⊗|2(5.15)k,±10BLG,II10=⊗|1±⊗|2.(5.16)k,±01TheapparentsimilarityofbothHamiltoniansindicatesthepossibilitytoobservephysicaleffectssimilartothepresentlystudiedspinrelaxationandspin-pseudospinentanglementwhenconsideringlayer-polarizedcarriertransportingraphene.It 1305SpinTransportinDisorderedGraphenewouldbeinterestingtostudytheeffectoflayer-pseudospinentanglementinsuchasituation.Inconclusion,ourspintransportsimulationsingraphene,chemicallymodifiedbyarandomdistributionofad-atoms,haverevealedahithertounknownphenom-enonrelatedtotheentangleddynamicsofspinandpseudospin,whichisinducedbySOCandleadstofastspinrelaxationinaquasi-ballistictransportregime.Theentanglementbetweenspinandorbitaldegreesoffreedomhasbeendiscussedformodelsofballisticsemiconductingnanowires[30].Here,theenergy-dependenceofspin/pseudospinentanglementinducedbySOChasbeenshowntodirectlyimpacttheresultingspindynamicsandspinrelaxationtimes.Fasterspinrelaxationdevel-opswhenspin-pseudospinentanglementismaximizedattheDiracpoint,wherethemomentumscatteringtimebecomesincreasinglylargebecausedisorderpreservespseudospinsymmetry.Thisrelaxationmechanism,occurringincleangraphenewithlongmeanfreepaths,hasnoequivalentincondensedmatterandcannotbedescribedbyEYorDPscaling.Suchaphenomenonishererevealedforthespecificcaseofgoldadsorbates,butshouldalsobeatplayforothersourcesoflocalSOC(ripples,defects,etc.),thuscontributingtoadeepgeneralunderstandingofspintransportingraphene-basedmaterialsanddevices[1,1618,37],whilethespecificspinrelaxationtimedependsontheeffectivestrengthoftheSOCbeingdifferentfordifferentsources.Theeffectoflateralconfinementinstripeorribbongeometrydeservesfurtherinvestigationregardingitsinfluenceonspinrelaxation(whichwasobservedinsemiconductornanowires[38]),whilesomegeneralmechanismduetoflexuralphononsforspinrelaxationin2Dmembraneshasbeenproposed[39].Finally,thespin-pseudospinentanglementcouldopenthepathtocontrolthepseudospinbymodifyingthespinorviceversa.Forexample,spinscouldbemanip-ulatedbyinducingpseudomagneticfieldsbystraininggraphene.Suchpossibilitiescouldleadtothedevelopmentofnovelapproachesfornon-charge-basedinforma-tionprocessingandcomputing,resultinginanewgenerationofactive(CMOS-compatible)spintronicdevicestogetherwithnon-volatilelow-energyMRAMmem-ories[40].5.2QuantumSpinHallEffect5.2.1IntroductionIn2005,KaneandMelepredictedtheexistenceoftheQuantumSpinHallEffect(QSHE)ingrapheneduetointrinsicSOC[41,42].WithintheQSHE,thepresenceofSOC,whichcanbeunderstoodasamomentum-dependentmagneticfieldcouplingtothespinoftheelectron,resultsintheformationofchiral(anti-chiral)integerQHEforspinup(spindown)electronpopulation.TheobservationofQSHEhasbeenhow-everprohibitedincleangraphenebythevanishinglysmallintrinsicSOCintheorder 5.2QuantumSpinHallEffect131ofμeV[43],butfurtherrealizedinstrongSOCmaterials(suchasCdTe/HgTe/CdTequantumwellsorbismuthselenideandtelluridealloys),givingrisetothenewexcit-ingfieldoftopologicalinsulators[4447].Recentproposalstoinduceatopologicalphaseingrapheneincludefunctionalizationwithheavyadatoms[25,48],covalentfunctionalizationoftheedges[49],proximityeffectwithothertopologicalinsulators[5052],orintercalationandfunctionalizationwith5dtransitionmetals[53,54].Inparticular,theseminaltheoreticalstudy[25]byWeeksandco-workershasrevealedthatgrapheneendowedwithmodestcoverageofheavyadatoms(suchasindiumandthallium)couldexhibitasubstantialbandgapandQSHEfingerprints(detectableintransportorspectroscopicmeasurements).Forinstance,signatureofsuchatopo-logicalstatecouldbeseeninarobustquantizedtwo-terminalconductance(2e2/h),withanadatomdensitydependentconductanceplateauextendinginsidethebulkgapinducedbySOC[25,55,56].Todate,suchapredictionlacksexperimentalconfirmation,despitesomerecentresultsonindium-functionalizedgraphenehaveshownasurprisingreductionoftheDiracpointresistancewithincreasingindiumdensity[57].Ontheotherhand,itisknownthatadatomsdepositedongraphenewillinevitablysegregate,formingislandsratherthanahomogeneousdistribution[58].Suchaclusteringeffectmayseriouslyimpactonthetransportfeatures[5961].InthisLetter,weshowthattheclusteringofthalliumadatomsongraphenecouldsuppresstheformationofaquantumspin-Hallphase,whiletheresultingfunction-alizedstructureswouldexhibitunconventionalbulktransportcharacteristics,withabsenceoftransitiontoaninsulatingregimeandarobustDiracpointconductivitycloseto4e2/h.ThepresenceofadatomislandslocallyintroducingstrongSOCisactuallyfoundtopreventthedevelopmentofquantuminterferencesandlocalizationphenomenainducedbyadditionalstrongdisordersources.5.2.2AdatomClusteringEffectonQSHEModelandMethods.Whenathalliumatomisgraftedongraphene,itplacesinthemiddleofahexagonalplaquetteofcarbonatoms,abovethesurface,seeFig.5.8.Asshownin[25],thedegreesoffreedomcorrespondingtotheadatomcanbecon-venientlydecimatedandtheireffectincludedintoaneffectiveπ-π*orthogonalTBmodelwithSOC.InthepresenceofadatomsrandomlydistributedoverasetRofplaquettes,theHamiltonian[41,42]readsas2iHˆ=−γ0cicj+√λcis·(dkj×dik)cj3ijij∈R−μcci+Vicci,(5.17)iii∈Riwhereci=[ci↓,ci↑]isthecoupleofannihilationoperatorsforelectronswithspindownandspinupontheithcarbonatom,andcisthecorrespondingcoupleofi 1325SpinTransportinDisorderedGrapheneFig.5.8aBall-and-stickmodelofagraphenesubstratewithrandomlyadsorbedthalliumatoms(concentrationis15%).bSameas(a)butwithadatomsclusteredinislandswitharadiusdistributionvaryingupto3nm(histogramshownin(d)).cZoom-inofatypicalthalliumad-atoms-basedisland.Allthalliumatomsarepositionedinthehollowpositionandequallyconnectedtothe6carbonatomsformingthehexagonunderneath(following[25])creationoperators.ThefirstcontributioninEq.(5.17)isthenearestneighborhoppingTBterm,withcouplingenergyγ0=2.7eV.ThesecondcontributionisanextnearestneighborhoppingtermthatrepresentstheSOCinducedbytheadatoms,withdkjanddiktheunitvectorsalongthetwobondsconnectingsecondneighborsandsthespinPaulimatrices.TheSOCissettoλ=0.02γ0,asextractedfromab-initiosimulationsinRef.[25].Thethirdtermdescribesthepotentialenergyinducedbychargetransferbetweenadatomsandgraphene.Thelasttermrepresentsthelong-rangeinteractionN22ofgrapheneandimpuritiesinthesubstrateVi=j=1jexp[−(ri−Rj)/(2ξ)][62],whereξ=0.426nmistheeffectiverangeandthesumrunsoverNimpuritycenterswithrandompositionsRjandmagnitudeofthepotentialjrandomlychosenwithin[−,].TheHamiltoniandoesnotconsidertheeffectofafurtherstructurerelaxationinthecaseofclusteredadatoms.Thiswillnotalterourconclusions.Forthestudyofelectronictransportinthallium-functionalizedribbons,wecon-siderastandardtwo-terminalconfigurationwithhighlydopedcontacts.ThedopingismimickedbyanappropriatepotentialenergyVonsourceanddrain.Thesimula-tionsarebasedonthenonequilibriumGreensfunctionformalism[63].Inadditiontotheelectronicconductance,thisapproachprovidesuswiththespin-resolvedlocal 5.2QuantumSpinHallEffect133density-of-occupied-states.Thisquantityillustrateshowelectronsinjectedfromthesourcespatiallydistributeinthesystemdependingontheirspin.Morespecifically,thezero-temperaturedifferentialconductanceasafunctionoftheelectronenergyisobtainedbytheLandauer-BüttikerformulaG(E)=(e2/h)Tr[GR(E)(S)GA(E)(D)],(5.18)whereGR/AaretheretardedandadvancedGreensfunctionsand(S/D)aretherateoperatorsforthesourceanddraincontacts.Thelocaldensity-of-occupied-statesisobtainedas<(E)]ρiη(E)=m[Giη,iη/(2π)(5.19)where[G<]iη,iηisthediagonalelementofthelesserGreensfunctioncorrespondingtotheelectronwithspinη(↓,↑)oftheithcarbonatom,andmindicatestheimaginarypart.Wealsostudyquantumtransportintwo-dimensionalfunctionalizedgraphenebymeansoftheKuboapproach[29,64].ThescalingpropertiesoftheconductivitycanbefollowedthroughthedynamicsofelectronicwavepacketsusingEq.(3.41).Calculations,basedontheuseofChebyshevpolynomialexpansionandcontinuedfractions,areperformedonsystemscontainingmorethan3.5millioncarbonatoms,whichcorrespondstosizeslargerthan300×300nm2.Suchasizeguaranteesthatourresultsareweaklydependentonthespecificspatialdistributionofadatomsorclustersofadatoms.SuppressionofQSHEbyadatomclustering.WestartbyconsideringanarmchairribbonofwidthW=50nmfunctionalizedwithaconcentrationn=15%ofran-domlyscatteredthalliumadatomsoveralengthL=50nm.Asalreadyreportedintheliterature[25],thedifferentialconductance[continuouslineinFig.5.9a]clearlyshowsa2e2/hplateau,whichissignatureofquantumspin-Hallphase.NotethattheplateauiscenteredatE≈−120meVandhasanextensionofabout100meV.Theobservedchargeneutralitypointshiftisconsistentwiththeconcentrationofcarbonatoms∼3nthatundergoachargetransferdopingeffect,i.e.E≈−3nμ=−121.5meV.Thewidthoftheplateauapproximatelycorrespondstothetopologicalgap√inducedbythalliumfunctionalization,andisgivenby63λeff≈84.2meV,wheretheeffectiveSOCisλeff=nλ≈8.1meV[65].Acloserinspectionoftheconduc-tanceshowsthatactuallytheplateauregionisnotperfectlyflat,butvarieswithintherange[1.92,2.02]e2/h.Thisindicatesthattheseparationbetweenspinpolar-izedchiraledgechannelsisnotcomplete.AbetterquantizationmaybeachievedbyincreasingW,Lortheadatomconcentration.Figures5.9b,calsoshowthespinresolvedlocaldensity-of-occupied-statesρforelectronsinjectedfromtherightcon-tactatenergyE=−100meV,indicatedbyanarrowinFig.5.9a.Weobserveahighρforx>50nm,i.e.theregionofthesource(injectedelectronsareindi-catedbyarrows),andspin-polarizedchannelsalongtheupperedgeforspindown(b)andalongtheloweredgeforspinup(c).Thewidthofthepolarizededgechan-nelsinarmchairribbonsdoesnotdependontheenergybutonlyontheSOCas√aγ0/(23λeff)≈13.5nm(see[66]).Theseparationbetweentheright-to-leftand 1345SpinTransportinDisorderedGrapheneFig.5.9aDifferentialconductanceforanarmchairribbonofwidthW=50nmwithaconcen-trationn=15%ofrandomlyscatteredthalliumadatomsoverasectionwithlengthL=50nm.ThepotentialenergyonthecontactsissettoV=−2.5eV.Thepresenceoflong-rangedisorderwithupto1eVistakenintoaccount.bLocaldensity-of-occupied-statesforspindownelectronsinjectedfromtherightcontactfor=0atenergyE=−100meV,seethearrowin(a).cSameas(b)butforspinupelectronsleft-to-rightmovingchannels,whichisoppositefordifferentspinpolarizations,isattheoriginoftheQSHE.Totestitsrobustness,weconsiderthepresenceofaconcen-trationnLR=0.5%oflong-rangedisorderwithdifferentstrength.AsreportedinFig.5.9a,aplateau,thoughnarrower,isobservedupto=1eV.Thispictureisactuallystronglymodifiedwhenadatomssegregateandformislands.Figure5.10ashowstheevolutionofthedifferentialconductancewhenislandshavearadiusrvaryingfrom0(non-segregatedcase)to1nmandfinallytorandomvaluesbetween2and3nm.Theadatomconcentrationiskeptatn=15%.Whileasignatureoftheplateauremainsuptor=1,forlargerradiusthequantizationiscom-pletelylostdespitetheshortinterclusterdistance.ThisindicatesthatsegregationhasadetrimentaleffectontheformationofaQSHphaseingraphenebyheavyadatomfunctionalization.Consideringthatadatomclusteringisunavoidableatroomtemper-ature,ourfindingsprovideanexplanationforthemissingexperimentalobservation 5.2QuantumSpinHallEffect135Fig.5.10aDifferentialconductanceforanarmchairribbonofwidthW=50nmwithacon-centrationn=15%ofclusteredthalliumadatoms(inislandswithradiusrupto23nm)overasectionwithlengthL=50nm.ThepotentialenergyonthecontactsissettoV=−2.5eV.bLocaldensity-of-occupied-statesinthecaser∈[2,3]nm,forspindownelectronsinjectedfromtherightcontactatenergyE=−100meV,seethearrowin(a).cSameas(b)butforspinupelectronsoftheQSHEinsuchsystems.Adeeperinsightintotheeffectofsegregationisfurtherprovidedbythespin-resolveddensity-of-occupied-statesreportedinFigs.5.10b,cforthecaseofislandradiiintherange[2,3]nm.Theρdistributionisverysimi-larforspindownandspinupelectrons,thismeansthatmostofthespin-couplingrelatedeffectissuppressed.Moreover,theinjectedelectronslargelyspreadallovertheribbonandshowahigherconcentrationinsidetheislands.Toexplainthesefea-tures,wehavetoconsiderthatsegregationreducesthehomogeneouscoverageofadatomsandleaveslargeregionsofpristinegraphene.Asaconsequence,thetopo-logicalgapcannotdevelopintheseregions,whereelectronsflowthesamewayasinnon-topologicalsystems.Moreover,theclustersaretoosmallandtheSOCistooweaktoinduceatopologicalphaseinsidethem.Togetherwiththehighlynegativevalueofthechargeneutralitypointinsidetheislands(E=3µ=−810meV),thisdeterminestheconsiderablyhighelectrondensityobservedinthefigure.However, 1365SpinTransportinDisorderedGrapheneasshownbelow,clusteringofthalliumadatomsproducesaremarkablebulktransportfingerprintoftheSOCintwo-dimensionalgraphene.Robustmetallicstateandminimumconductivity.Weinvestigatetheintrinsicbulkconductivityofthallium-functionalizedgraphenebycomputingtheKubo-Greenwoodconductivity.Wefocusonlargethalliumdensity(about15%),withthalliumclusterssizedistributionshowninFig.5.8dandconsidersuperimposeddistributionoflong-rangeimpuritiestomimicadditionalsourcesofdisorder(suchaschargeddefectstrappedintheunderneathoxide,additionaldopants,structuraldefects...).InFig.5.11(mainframe),weshowtheKuboconductivityforvariousdensities(nLR=0.2−0.5%)oflong-rangeimpuritieswith=2.7eV.Astrik-ingfeatureistheenergy-dependentimpactofadditionaldisorderonthetransportfeatures.IndeedaplateauisformedneartheDiracpoint,wheretheconductivityreachesaminimumvalue,regardlessofthesuperimposeddisorderpotential.Differ-ently,amoreconventionalscalingbehaviorσ∼1/nLRisobtainedforhighenergies,followingasemiclassicalFermigoldenrule.Theminimumconductivityobtainedσmin∼4e2/hremindsthecaseofcleangraphenedepositedonoxidesubstratesandsensitivetoelectron-holepuddles[67].However,heretheroleofspin-orbitinterac-tioniscriticalforpreservingarobustmetallicstate.ThisisshowninFig.5.11(inset),wherethetime-dependenceofthediffusioncoefficientatenergy(E=−120meV)isreportedfornLR=0.5%,inpresenceofthethalliumislandswithandwith-outspin-orbitinteraction.TheabsenceofSOCirremediablyproducesaninsulatingstateasevidencedbythedecayofthediffusioncoefficient,whereasonceSOCisswitchedon,thediffusivityisfoundtosaturatetoitssemiclassicalvalues,showingnosignofquantuminterferencesandlocalization,inagreementwithapercolationscenarioforthecorrespondingelectronicstates.NotethatsuchamechanismisnotFig.5.11Kuboconductivityversusenergyforthalliumclusteringandadditionalvaryingdensity(nLR)oflong-rangeimpurities.InsetDiffusioncoefficientforwavepacketwithenergyE=−120meV,forthecasenLR=0.5%,with(solidblueline)andwithouttheSOCofthalliumadatomsactivated 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Chapter6ConclusionsInthisthesis,Ihavepresentedthechargetransportofdisorderedgrapheneaswellasexplainedthefastspinrelaxationingraphenewhichisoneofthemostinterestingtopicsingrapheneatthemoment.Theroleofdefect-inducedzero-energymodesonchargetransportingrapheneisinvestigatedusingKuboandLandauertransportcalculations.Bytuningthedensityofrandomdistributionsofmonovacancieseitherequallypopulatingthetwosublat-ticesorexclusivelylocatedonasinglesublattice,allconductionregimesarecoveredfromdirecttunnelingthroughevanescentmodestomesoscopictransportinbulkdis-orderedgraphene.Dependingonthetransportmeasurementgeometry,defectdensity,andbrokensublatticesymmetry,theDirac-pointconductivityiseitherexceptionallyrobustagainstdisorder(supermetallicstate)orsuppressedthroughagapopeningorbyalgebraiclocalizationofzero-energymodes,whereasweaklocalizationandtheAndersoninsulatingregimeareobtainedforhigherenergies.Thesefindingsclar-ifythecontributionofzero-energymodestotransportattheDiracpoint,hithertocontroversial.Wealsoreportednewinsightstothecurrentunderstandingofchargetransportinintrinsicpolycrystallinegeometries.WecreatedrealisticmodelsoflargeCVD-growngraphenesamplesandthencomputedthecorrespondingchargecarriermobilitiesasafunctionoftheaveragegrainsizeandthecoalescencequalitybetweenthegrains.Ourresultsrevealaremarkablysimplescalinglawforthemeanfreepathandconductivity,correlatedtoatomic-scalechargedensityfluctuationsalonggrainboundaries.Furthermore,weusednumericalsimulationsandtransportmeasurementstodemonstratethatelectricalpropertiesandchemicalmodificationofgraphenegrainboundariesarestronglycorrelated.Thisnotonlyprovidesguidelinesfortheimprove-mentofgraphenedevices,butalsoopensanewresearchareaofengineeringgraphenegrainboundariesforhighlysensitiveelectro-biochemicaldevices.Weinvestigatedthechargetransportpropertiesofplanaramorphousgraphenethatisfullytopologicallydisordered,intheformofsp2threefoldcoordinatednetworksconsistingofhexagonalringsbutalsoincludingmanypentagonsandheptagons©SpringerInternationalPublishingSwitzerland2016141D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2_6 1426Conclusionsdistributedinarandomfashion.UsingtheKubotransportmethodologyandtheLanczosmethod,thedensityofstates,meanfreepaths,andsemiclassicalconductiv-itiesofsuchamorphousgraphenemembranesarecomputed.Despitealargeincreaseinthedensityofstatesclosetothechargeneutralitypoint,allelectronicpropertiesaredramaticallydegraded,evidencinganAndersoninsulatingstatecausedbytopolog-icaldisorderalone.TheseresultsaresupportedbyLandauer-Büttikerconductancecalculations,whichshowalocalizationlengthasshortas5nm.WereportedonthetransitionfromaQuantumSpinHalleffectregimetoarobustmetallicstate,uponsegregationofthalliumadatomsadsorbedontoagraphenesurfaceandintroducinggiantenhancementofspin-orbitcoupling.OurtheoreticalmethodologycombinesefficientcalculationofboththeLandauer-Büttikerconduc-tanceandtheKubo-Greenwoodconductivity,givingaccesstobothedgeandbulktransportphysicsindisorderedthallium-functionalizedgraphenesystemsofrealisticsizes.Ourfindingsquantifythedetrimentaleffectsofadatomsclusteringinobserv-ingtheQSHE,butprovideadditionalbulksignatureofarobustmetallicstatewithminimumbulkconductivityofabout4e2/h,whichshouldbehelpfulforguidingfurtherexperiments.Finally,wehavedevelopedanewspin-transport-simulationmethodtoinvesti-gatethespintransportingraphene.Weshowedthatthepresenceofalowdensityofrandomlydistributedadatoms(inducinglocalRashbaspin-orbitcoupling)yieldsultrafastspinrelaxationtimesattheDiracpoint,togetherwithanunconventionalrelationbetweenthespinandmomentumrelaxationtimes.Ourquantumtransportsimulationsshowedthatcertaintypesofadatoms(suchasNickelorGoldimpu-rities)triggerstrongspindecoherenceattheDiracpoint,althoughthetransportregimeeventuallyreachestheballisticlimit.Thisphenomenonhithertounknownisanewtypeofspindephasingmechanism,drivenbyentanglementbetweenspinandpseudospindegreesoffreedom.Thosefindingsbringanunprecedentedinsightofspinrelaxationmechanismsingraphene,suggestingapossibleoriginofreportedlowspinrelaxationtimes,andclarificationofthecontroversialdescriptionofrelaxationmechanismsinvarioustypesofgraphenesamples. AppendixATimeEvolutionoftheWavePacketThisappendixpresentshowtocalculatetheevolutionofthewavepacketUˆ(t)|ϕRPand[Xˆ,Uˆ(t)]|ϕRPwhichareusedintheapplicationoftherealspacemethodtocalculatethetransportproperties.Inordertodothat,wedividethetimetintosmalltimestepsT=t/NandapproximateUˆ(T)withtheseriesoforthogonalChebyshevpolynomialsQn(Hˆ)∞−iHTˆUˆ(T)=e=cn(T)Qn(Hˆ)(A.1)n=0TheoriginalChebyshevpolynomialsTnwhichsatisfytherecurrentrelationsT0(x)=1(A.2)T1(x)=x(A.3)T22(x)=2x−1(A.4)...Tn+1(x)=2xTn(x)−Tn−1(x)(A.5)andactontheinterval[−1;1]arerescaledtotherescaledChebyshevpolynomialsQnwhichcoverthebandwidthofsystemHamiltonianE∈[a−2b:a+2b],withthebandcenterandbandwidthareaand4b,respectively.TheserescaledChebyshevpolynomialsQnsatisfy√E−aQn(E)=2Tn(∀n≥1)(A.6)2bQ0(E)=1(A.7)√E−aQ1(E)=2(A.8)2b©SpringerInternationalPublishingSwitzerland2016143D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2 144AppendixA:TimeEvolutionoftheWavePacket2√E−a√Q2(E)=22−2(A.9)2b...E−aQn+1(E)=2Qn(E)−Qn−1(E)(A.10)2bWithabovedeÞnition,wehavetheorthonormalrelationsforQn(E)Qn(E)Qm(E)pQ(E)dE=δmn(A.11)withrespecttotheweight1pQ(E)=(A.12)22πb1−E−a2bOncetheQnpolynomialsarewelldeÞned,onecancomputetherelatedcn(T)coef-Þcients−iETcn(T)=dEpQ(E)Qn(E)e(A.13)√E−a2TnE2b−iT=dEe(A.14)22πb1−E−a2b√12Tn(x)−i(2bx+a)T=dx√e(A.15)π−11−x2√a2bn−iT=2ieJn−T,n≥1(A.16)an−iT2bandtheÞrstcoefÞcientsc0(T)=ieJ0−TwithJn(x)istheBesselfunc-tionoftheÞrstkindandordernWecannowcalculate|ϕRP(T)|ϕRP(T)=Uˆ(T)|ϕRP(A.17)NN|ϕRP(T)cn(T)Qn(Hˆ)|ϕRP=cn(T)|αn(A.18)n=0n=0where|αn=Qn(Hˆ)|ϕRP.WiththedeÞnitionsintroducedinEqs.(A.7,A.8andA.9)andtherecurrencerelationEq.(A.10),weobtain AppendixA:TimeEvolutionoftheWavePacket145|α0=|ϕRP(A.19)Hˆ−a|α1=√|α0(A.20)2bHˆ−a√|α2=|α1−2|α0(A.21)bHˆ−a|αn+1=|αn−|αn−1(∀n≥2)(A.22)bFollowingthesamereasoningasfor|ϕRP(T),|ϕ(T)canbeevaluatedÞrstRPwritting(T)=[Xˆ,Uˆ(T)]|ϕ|ϕRPRP(A.23)NN(T)c|ϕRPn(T)[Xˆ,Qn(Hˆ)]|ϕRP=cn(T)|βn(A.24)n=0n=0with|βn=[Xˆ,Qn(Hˆ)]|ϕRP.UsingtheEqs.(A.10)and(A.19ÐA.22),weobtaintherecurrencerelationfor|βn|β0=0(A.25)Xˆ,Hˆ|β1=√|ϕRP(A.26)2bHˆ−a1|βn+1=|βn−|βn−1+[Xˆ,Hˆ]|αn(∀n≥1)(A.27)bbwhichcontain|αnandthecommutator[Xˆ,Hˆ]determinedbythehopingsandthedistancesbetweenneighbours⎛⎞0⎜..⎟⎜⎜.HijXij⎟⎟⎜..⎟[Xˆ,Hˆ]=⎜.⎟(A.28)⎜⎟⎜..⎟⎝HjiXji.⎠0whereXij=(Xi−Xj)isthedistancebetweenorbitals|ϕiand|ϕj. AppendixBLanczosMethodInthisappendixtheLanczosmethodisintroduced.InsteadofdiagonalizingtheHamiltoniantheLanczosmethodisausefulmethodtotransformtheHamiltonianintotridiagonalmatrixwhichismoreconvenienttocomputethedensityofstateorspinpolarization.Thegeneralideaofthismethodisbuildingfromtheinitialstate|ϕRPanewbasisinwhichtheHamiltonianistridiagonal.Herearethebasicsteps:TheÞrststepstartswiththeÞrstvectorinthenewbasis|ψ1=|ϕRPandbuildsthesecondone|ψ2whichisorthonormaltotheÞrstonea1=ψ1|Hˆ|ψ1(B.1)|ψ˜2=Hˆ|ψ1−a1|ψ1(B.2)b1=|ψ˜2=ψ˜2|ψ˜2(B.3)1|ψ2=|ψ˜2(B.4)b1Allotherrecursionsteps(∀n≥1)areidentical,webuildthe(n+1)thvectorwhichisorthonormaltothepreviousonesandgivenbyan=ψn|Hˆ|ψn(B.5)|ψ˜n+1=Hˆ|ψn−an|ψn−bn−1|ψn−1(B.6)bn=ψ˜n+1|ψ˜n+1(B.7)1|ψn+1=|ψ˜n+1(B.8)bnThecoefÞcientsanandbnarenamedrecursioncoefÞcientswhicharerespectivelythediagonalandoff-diagonalofthematrixrepresentationofHˆintheLanczosbasis˜ˆ(thatwewriteH).©SpringerInternationalPublishingSwitzerland2016147D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2 148AppendixB:LanczosMethod⎛⎞a1b1⎜b1a2b2⎟⎜⎟˜ˆ⎜⎜....⎟⎟H=⎜b2..⎟(B.9)⎜....⎟⎝..bN⎠bNaNWithsimplelinearalgebra,oneshowsthatϕRP|δ(E−Hˆ)|ϕRP=ψ1|δ(E−Hˆ)|ψ111=lim−mψ1||ψ1η→0πE+iη−Hˆwhile11ψ1||ψ1=˜ˆb2E+iη−HE+iη−a11−b2E+iη−a22−b2E+iη−a33−...(B.10)whichisreferredasacontinuedfractionG1withthedeÞnitionofGnas,1Gn=(B.11)b2E+iη−ann−b2n+1E+iη−an+1−b2n+2E+iη−an+2−...1G1=(B.12)E+iη−a1−b2G211Gn=(B.13)E+iη−an−bn2Gn+1SincewecomputeaÞnitenumberofrecursioncoefÞcients,thesubspaceofLanc-zosifofÞnitedimension(N),soitiscrucialtoterminatethecontinuedfractionbyan AppendixB:LanczosMethod149appropriatechoiceofthelast{an=N,bn=N}elements.Letusrewritethecontinuedfractionas1G1=b2E+iη−a11−b2E+iη−a22−b2E+iη−a33−...E+iη−aN−b2GN+1N(B.14)whereGN+1denotessuchtermination.ThesimplestcaseiswhenallthespectrumiscontainedinaÞnitebandwidth[a−2b;a+2b],athespectrumcenterand4bitsbandwidth.RecursioncoefÞcientsanandbnoscillatearoundtheiraveragevalueaetb,andthedampingisusuallyfastafterafewhundredsofrecursionsteps.TheterminationthensatisÞes11GN+1==(B.15)E+iη−a−b2GN+2E+iη−a−b2GN+1fromwhichapolynomialofseconddegreeisfound2)G2+(E+iη−a)G−(bN+1N+1−1=0(B.16)andstraightforwardlysolved=(E+iη−a)2−(2b)2(B.17)√(E+iη−a)∓i−GN+1=(B.18)2b2(E+iη−a)−i(2b)2−(E+iη−a)2GN+1=(B.19)2b2 CurriculumVitaeDinhVanTuanContactinformationPostdoctoralResearcherE-mail:tuan.dinh@icn.cat;dinhvantuan1984@gmail.comTheoreticalandComputationalNanoscienceGroup,CatalanInstituteofNanoscienceandNanotechnologyResearchInterestsQuantumCondensedMatterTheory:Chargeandspintransport,quantumHalleffect,spinHalleffect,topologicalelectronicphasesanddisorderedelectronicsystems.ProfessionalPreparation•Ph.D.,MaterialsScience,2011Ð2014ÐDepartmentofPhysics,AutonomousUniversityofBarcelona,SpainÐThesisTopic:ChargeandSpinTransportinDisorderedGraphene-BasedMate-rialsÐSupervisor:Prof.StephanRoche©SpringerInternationalPublishingSwitzerland2016151D.V.Tuan,ChargeandSpinTransportinDisorderedGraphene-BasedMaterials,SpringerTheses,DOI10.1007/978-3-319-25571-2 152CurriculumVitae•M.Sc.,TheoreticalandMathematicalPhysics,2008-2010ÐDepartmentofTheoreticalPhysics,HoChiMinhcityUniversityofScience,VietnamÐThesisTopic:TheGraphenePolarizabilityandApplicationsÐSupervisor:AssociateProf.NguyenQuocKhanhProfessionalAppointments9/2014PostdoctoralResearcher,CatalanInstituteofNanoscienceandNanotechnology9/2011Ð9/2014Ph.D.student,CatalanInstituteofNanoscienceandNano-technology9/2007Ð9/2011ResearchAssistant,HoChiMinhcityUniversityofSciencePublications1.DinhVanTuan,FrankOrtmann,DavidSoriano,SergioO.Valenzuela,andStephanRoche.Pseudospin-drivenspinrelaxationmechanismingraphene.NaturePhysics,10,857Ð863(2014)2.AlessandroCresti,DavidSoriano,DinhVanTuan,AronW.Cummings,andStephanRoche.MultipleQuantumPhasesinGraphenewithEnhancedSpin-OrbitCoupling:fromQuantumSpinHallRegimetoSpinHallEffectandRobustMetallicState.PhysicalReviewLetter,113,246603(2014)3.AronW.Cummings,DinhLocDuong,VanLuanNguyen,DinhVanTuan,JaniKotakoski,JoseEduardoBarriosVargas,YoungHeeLeeandStephanRoche.ChargeTransportinPolycrystallineGraphene:ChallengesandOpportunities.AdvancedMaterials,26,Issue30,5079Ð5094(2014)4.DavidJimnez,AronW.Cummings,FerneyChaves,DinhVanTuan,JaniKotakoski,andStephanRoche.Impactofgraphenepolycrystallinityontheper-formanceofgrapheneÞeld-effecttransistors.Appl.Phys.Lett.,104,043509(2014)5.DinhVanTuan,JaniKotakoski,ThibaudLouvet,FrankOrtmann,JannikC.Meyer,andStephanRoche.ScalingPropertiesofChargeTransportinPolycrys-tallineGraphene.NanoLetters,13(4),1730Ð1735(2013)6.AlessandroCresti,FrankOrtmann,ThibaudLouvet,DinhVanTuan,andStephanRoche.BrokenSymmetries,Zero-EnergyModes,andQuantumTrans-portinDisorderedGraphene.Phys.Rev.Lett.,110,196601(2013).7.AlessandroCresti,ThibaudLouvet,FrankOrtmann,DinhVanTuan,PawełLenarczyk,GeorgHuhsandStephanRoche.ImpactofVacanciesonDiffusiveandPseudodiffusiveElectronicTransportinGraphene.Crystals,3,289Ð305(2013).8.DinhVanTuanandNguyenQuocKhanh.Plasmonmodesofdouble-layergrapheneatÞnitetemperature.PhysicaE:Low-dimensionalSystemsandNanos-tructures,54,267(2013) CurriculumVitae1539.DinhVanTuan,AvishekKumar,StephanRoche,FrankOrtmann,M.F.Thorpe,andPabloOrdejon.Insulatingbehaviorofanamorphousgraphenemembrane.Phys.Rev.B,86,121408(RapidCommunications)(2012)10.DinhVanTuanandNguyenQuocKhanh.TemperatureeffectsonPlasmonmodesofdouble-layergraphene.CommunicationsinPhysics,22,45(2012)11.DavidSoriano,DinhVanTuan,SimonM.-M.Dubois,MartinGmitra,AronW.Cummings,DenisKochan,FrankOrtmann,Jean-ChristopheCharlier,JaroslavFabian,andStephanRoche.SpinTransportinHydrogenatedGraphene.acceptedforpublicationin2DMaterialsasaTopicalReview,(2015)HonorsandAwards•Awardforthebestgraduatestudent,2010•VietnameseMinistryofEducationaward,2005•SliverMedalattheNationalPhysicsOlympiadforthestudentsofnationaluniversities,2005•BronzeMedalattheNationalPhysicsOlympiad,2003