2023年6月24日发(作者：)

Non-adiabaticKohn-anomalyinadopedgraphenemonolayerMicheleLazzeriandFrancescoMauriIMPMC,Universit´esParis6et7,CNRS,IPGP,140ruedeLourmel,75015Paris,France(Dated:February6,2008)arXiv:cond-mat/0611708v1

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28

Nov

2006Wecompute,fromfirst-principles,thefrequencyoftheE2g,Γphonon(RamanGband)ofgraphene,ationsaredoneusingi)theadiabaticBorn-Oppenheimerapproximationandii)time-dependentperturbationtheorytoexploredynamiceffapproachesprovideverydiff,theadiabaticphononfrequencyweaklydependsonthedoping,,weshowthatdopedgrmbers:,,,heentdemonstra-tionofafield-effecttransistor(FET)basedonafew-layersgraphenesheethasboostedtheinterestinthissystem[1,2,3].Inparticular,bytuningtheFETgate-voltageVgualpossibil-ityofbuildingaFETwithjustonegraphenemonola-basedexperiment,graphenecanbedopedupto31013cm−2electronconcentra-tion[1,2],corresponding,inamonolayer,toa0.2%ultingchemical-bondmodificationcouldinduceavariationofbond-lengthsandphonon-frequenciesofthesameorder,uldrealizethedreamoftuningthechemistry,withinanelectronicdevice,senceofKohnanomalies(KAs)[4,5]ingraphenecouldactasamagnifyingglass,leadingtoavariationoftheopticalphonon-frequenciesmuchlargerthanthe0.2%therhand,thephonon-frequencychangeinducedbyFET-dopingcouldprovideamuchmoreprecisede-terminationoftheKA,ifestasasuddenchangeinthephonondispersionforawavevectorq∼2kF,wherekFisaFermi-surfacewavevector[4].Th,inelasticx-ray,echniqueshaveafiniteresolution,inqandenergy,hene,ggestsanalternativewaytostudytheKA,thatistomeasurethephononfrequencyatafithisapproach,onecoulduseRamanscat-tering,whichhasamuc-proachisfeasibleforgraphene,whichhasaKAfortheRaman-activeE2gΓ-phonon[5](RamanG-band).Inthispaper,wecomputethevariationofphononfrequencyoftheRamanG-band(E2gmodeatΓ)inagraphenemonolayer,,thecalculationsaredoneusingafullyab-initioa,time-dependentperturbationtheory(TDPT)-initiocalculationsaredonewithindensityfunc-tionaltheory(DFT),usingthefunctionalofRef.[6],planewaves(30Rycutoff)andpseudopotentials[7].TheBrillouinzone(BZ)integrationisdoneonauni-form64×64×tronicsmearingof0.01RywiththeFermi-Diracdistributionisused[8].Thetwo-dimensionalgraphenecrystalissimulatedusingasuper-cellgeometrywithaninterlayerspacingof7.5˚A(ifnototherwisestated).Phononfrequenciesarecalcu-latedwithintheapproachofRef.[9],usingthePWSCFcode[10].TheFermi-energyshiftissimulatedbyconsid-eringanexcesseendenceoftheFermienergyǫFonthesurfaceelectron-concentrationσisdeterminedbyDFT(Fig1).Ingraphene,thegapiszeroonlyforthetwoequivalentKandK’BZ-pointsandtheelectronenergyǫcanbeapproximatedasǫπ∗/π(K+k)=±βkfortheπ∗andπbands,thisapproxi-mation,atT=0Ktemperatureσ=sign(ǫF)ǫ2FeV210.361013cm−2(1)whereβ=5.52eV·˚AfromDFT,sign(x)isthesignofxandǫF=0attheπrkthat,fromFig1,thetypicalelectron-concentrationobtainedinexperiments[1,2]correspondstoanimportantFermi-levelshift(∼0.5eV).Forsuchshift,thelinearizedbandsarestillagoodapproximation(Fig.1).Thedependenceofthegraphenelattice-spacingaonσ,a(σ),isobtainedbyminimizingF(σ,A)=[E(σ,A)−E(0,A0)]/AwithrespecttoA,whereE(σ,A)istheen-ergyofthegrapheneunit-cell,Aisunit-cellareaandA0=5.29˚A2istheequilibriumA[11]atzeroσ.E(σ,A)iscomputedbyDFTlettingtheinter-layerspacing,L,tendtoinfinityinordertoeliminatethespuriousinterac-Electrons in the surface-unit-cell)1.0-0.06

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y0.5ab-initio (DFT)linear bandsgren0

T = 300 Ke

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Electron concentration (1013 cm-2)3-01(3ab-initio (DFT)

)fit to ab-initio02(aPietronero-Strässler/1])00(a--1)σ(-2a[-3-10-505Electron concentration (1013 cm-210)FIG.1:anel:ǫFasafunctionofthesurfaceelectron-concentrationσfromDFTcalculationsandfromlinearizedbands(atT=300K).Lowerpanel:in-planelatticespacingaasafunctionofσ.ThefittingfunctionisEq.2andthedashedlineisfromRef.[13].tionbetweenthebackgroundandthechargedsheet[12].∆a(σ)=[a(σ)−a(0)]/a(0)wasdeterminedinRef.[13]hesamefunctionaldependenceasinRef.[13],ourDFTcalculationsarefittedby∆a(σ)=6.748·10−6|σ|3/2+1.64·10−4σ,(2)whereσisinunitsof1013cm−σ=31013cm−2,thelatticespacingvariationis∼0.05%,whichis,asex-pected,ofthesameorderofthevalence-chargevariation.

dynamic+expanded lattice)140adiabatic+expanded lattice-madiabatic+constant latticec(

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ycn0euqerF-20-40-10Electron concentration (10-50135 cm-2)10FIG.2:FrequencyoftheE2gΓphonon(RamanGband)asafunctionofσ:ationsaredoneusingstandardDFT(adia-batic)orTDPT(dynamic),keepingthelattice-spacingcon-stant(constantlattice)orvaryingitaccordingtoEq.2(ex-pandedlattice).mentsshouldbecomparedwiththecontinuousline.2ThefrequencyoftheE2gΓphononiscomputedbystaticperturbationtheoryoftheDFTenergy[9],elinearizedforcesactingontheatomsduetotheproachisbasedontheadiabaticBorn-Oppenheimerapproximation,whichisthestandardtextbookapproachforphononcalculationsandisalwaysused,toourknowledge,putedzero-dopingphononfrequencyisω0a/(2πc)=1554cm−1,quencyvariation∆ωwithσ-culationsaredonekeepingthelattice-spacingconstantata(0),lattercase,∆ωisfittedby∆ωk+q)m,kn|2[f(k+q)m−fkn]Nkk