Piscanec Francesco Mauri A C Ferrari 2 and J Robertson Institut de Min57577ralogie et de Physique des Milieux Condens57577s 4 Place Jussieu 75252 Paris cedex 05 France Cambridge University Engineering Department 9 JJ Thompson Avenue Cambridge CB3 0F ID: 31493
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Nkk,i,jg branch,correspondingtoanatomicmotionparallelandperpendicularto.FromDFTcalculationswe=45.60eV/ÅforbothLOandTOmodes.A.PhononfrequenciesWehavepreviouslyshownthatgraphitehastwoKohnanomaliesinthephonondispersionsfortheDuetotheanomaly,thedispersionnear-LOmodeisalmostlinear,withslope 3 a02 =2.46Åisthegraphitelatticespacing,=5.52ÅeVistheslopeoftheelectronbandsnearthecarbonatomicmass,andisthepulsationofthe=196.0meVnoticethatinRef.6theband=14.1eV,becausethemomentumwasexpressedinunitsof2.Equationshowsthatcanbedi-rectlymeasuredfromtheexperimental.Thephononshavebeenmeasuredbyseveralgroupswithcloseagreement.FromaquadraticttothemostrecentdataofRef.25Fig.1weget=133cmÅ.FromEq.=39eV/Å,ingoodagreementwithDFT.B.PhononlinewidthsWenowshowthatthefullwidthathalfmaximumofthepeak,FWHM,givesanotherindependentEPCmeasurement.ThepeakofgraphiteisduetotheWeuseEq.tocomputethewidth,,forthismode.Inthevicinityof,thebandsdispersionhasaconicshape,withvertexattheFermilevelandslopeForexample,thebandsat,forsmall,are.Thus,fortheelectronicstatesthatsatisfytheenergyconservation,Fig.2.TheEisadoublydegeneratemodeconsistinginananti-phasemovementofthetwoatomsofthegrapheneunitcell,alongadirectioninthegrapheneplane.Fora,theEPCofthemodeis:1+cosistheanglebetween.ForbothLOandTO 3 a02 1+cos 3a02 2 Weusedthesubstitution1/,where istheareaofthegrapheneunit-cell.Afactor2inisduetothecountingofbothFermisur-faces.Equationasanenergy.Throughoutthepaperwewillexpressascm,theconversionfactoris,whereislightvelocity.AccordingtoEq.,theEPCisdirectlyobtainedbymeasuringthephononlinewidth.Wethenmeasureforasingle-crystalgraphitesimilartothatofRef.25.ItsRamanspectrumdoesnotshowanyReichinRefs.15and27,thusweexcludeextrabroad-eningduetodisorder.ByttingthepeakwithaLorent-zianwegetFWHM=13cm.Temperaturede-pendentmeasurementsshownoincreaseofFWHMinthe2900Krange.OurRamanspectrometerresolution1.5cmthustheexperimentalintrinsiclinewidth131.5=11.5cm.Byneglectingtheanharmoniccon-,weestimate11.5cm.Then,fromEq.eV/Å.ThiscompareswellwithDFT,con-rmingthatissmall.Finally,neartheconservationoftheenergyandmo-mentuminEq.,implies:=0ifissatisedbythephonon,involvedinthepeak.Ontheotherhand,thedoubleresonantmodecloseto,givingthepeakat1615cm,doesnotsatisfyit.isindeedsharperthantheInsummary,wepresentedtwoindependentmeasurementsofthegraphiteEPC.TheseareconsistentwitheachotherandwithDFT.WenowconsiderSWNTs.III.NANOTUBESpeakofSWNTscanbetwithtwocomponents,.SemiconductingSWNTshavesharp FIG.1.Graphitephonondispersionofthehighestopticalbranch.Dotsareinelasticx-raymeasurementsfromRef.25.Thelineisaquadratict. FIG.2.Electronbandsaroundingrapheneandiname-tallictube.ShadedareaisthegrapheneBrillouinzone.Dashedarrows:decayprocessesforaetal.PHYSICALREVIEWB,155426 haus,Chem.Phys.Lett.,383G.D.Mahan,Phys.Rev.B,125409G.PenningtonandN.Goldsman,Phys.Rev.B,045426J.Jiang,R.Saito,A.Grüneis,S.G.Chou,G.G.Samsonidze,A.Jorio,G.Dresselhaus,andM.S.Dresselhaus,Phys.Rev.BP.B.Allen,Phys.Rev.B,2577;P.B.AllenandR.Silberglitt,,4733J.Maultzsch,S.Reich,C.Thomsen,H.Requardt,andP.Ordejón,Phys.Rev.Lett.,075501F.TuinstraandJ.Koenig,J.Chem.Phys.,1126V.ScardaciandA.C.FerrariA.C.FerrariandJ.Robertson,Phys.Rev.B,14095,075414P.H.Tan,C.Y.Hu,J.Dong,W.Shen,andB.Zhang,Phys.Rev.,214301M.Paillet,P.Poncharal,A.Zahab,J.L.Sauvajol,J.C.Meyer,andS.Roth,Appl.Phys.Lett.,237401S.M.Bose,S.Gayen,andS.N.Behera,cond-mat/0506004O.Dubay,G.Kresse,andH.Kuzmany,Phys.Rev.Lett.;O.DubayandG.Kresse,Phys.Rev.BK.-P.Bohnen,R.Heid,H.J.Liu,andC.T.Chan,Phys.Rev.Lett.,245501D.Connétable,G.-M.Rignanese,J.-C.Charlier,andX.Blase,Phys.Rev.Lett.,015503R.Barnett,E.Demler,andE.Kaxiras,Phys.Rev.B,035429V.ZólyomiandJ.Kürti,Phys.Rev.B,085403J.P.Perdew,K.Burke,andM.Ernzerhof,Phys.Rev.Lett.N.TroullierandJ.L.Martins,Phys.Rev.B,1993S.Baroni,S.deGironcoli,A.DalCorso,andP.Giannozzi,Rev.Mod.Phys.,515S.Reich,C.Thomsen,andP.Ordejon,Phys.Rev.B,195416S.Piscanecetal.E.g.,anharmonicity,inhomogeneousbroadeningandcouplingwithasubstrate.FWHMandFWHM15cmwithtemperatureinthe80600KrangeforbothmetallicandsemiconductingSWNTsRefs.11and27.ThusC.ThomsenandS.Reich,Phys.Rev.Lett.,5214J.Maultzsch,S.Reich,andC.Thomsen,Phys.Rev.BM.A.Pimentaetal.,inProceedingsofIWEPNM2003AIP,Melville,NY,2003U.Fano,Phys.Rev.,1866etal.PHYSICALREVIEWB,155426