First-principles study of edge states of H-terminated graphitic ribbons

First-principlesstudyofedgestatesofH-terminatedgraphiticribbons

YoshiyukiMiyamoto

FundamentalResearchLaboratories,NECCorporation,34Miyukigaoka,Tsukuba305-8501,Japan

KyokoNakada*andMitsutakaFujita

InstituteofMaterialsScience,UniversityofTsukuba,Tennodai,Tsukuba305-8573,Japan

͑Received28April1998;revisedmanuscriptreceived19August1998͒

Theexistenceoftheedgestatesofsingle-layeredH-terminatedgraphiticribbonspredictedbyNakadaetal.͓Phys.Rev.B,179͑1996͔͒hasbeenconfirmedbyfirst-principlescalculationswithintheframeworkofthelocal-densityapproximationandthepseudopotentials.Theedgestatesincasesofstackedribbonshavealsobeenexamined,andithasbeenfoundthattheexistenceoftheedgestatesisdependentonthestackingmanner.ThisphenomenoncanbeunderstoodfromtheelectronicstructuresofAA-andAB-stackedbulkgraphite.͓S0163-1829͑99͒14615-0͔

Breakageoftranslationalsymmetryofsolidstatesinacertaindirectiongenerateslocalizedstates.Forinstance,acleavedbulkcrystalgeneratessurfacestatesthathavetwo-dimensionalextensionsparalleltothecleavedsurface.Thesesurfacestatesoriginatefromthesurfacedanglingbondslo-catedattheFermilevel(EF).Whenallsurfacedanglingbondsareterminated,forinstance,byhydrogenatoms͑Hatoms͒,thesurfacestatesmoveawayfromEFandbecomeresonantstatesofbulkbands.Similarly,onecanimaginethatlocalizedstatesaregeneratedwhenagraphenesheetiscutintographiticribbons.Uponthecutting,agraphiticribbonlackstheoriginaltwo-dimensionalperiodicboundariesandthenewlygeneratedC-danglingbondsattheribbon’sedgecauselocalizedstatesatEF.WhenalldanglingbondsareterminatedbyHatoms,seeFig.1͑a͒,theirlevelsshouldmoveawayfromEF.͓Hereafter,theribbonslikeFig.1͑a͒arecalled‘‘zigzagribbons.’’͔

However,itwaspredictedthattheH-terminatedgraphiticribbonshavepeculiarlocalizedstatesattheribbons’edgeswithcorrespondingenergylevelsatEF.1Theselocalizedstatesoriginatefrom␲orbitalsofH-terminatedCatomsextendingnormaltothegraphiticsheet.Theselocalizedstateswerecallededgestates.Theedgestatesappearwhenthewavevectorskalongtheribbonsareintheregionof2␲/3Ͻkр␲inaunitofinverseofaperiod(2.46Å).Thisfactcanbeanalyticallyderivedwhenonlythenearest-neighborhoppingsof␲electronsaretakenintoaccount,1whichresultsinperfectcancelingofoff-sitehoppingsofthe␲orbitalsattheH-terminatedCsiteswithkas2␲/3Ͻkр␲.Sincethelocalizedfeatureoftheedgestatesisex-pectedtocauseremarkableelectron-electroncorrelation,someinterestingpropertiesofthezigzagribbonsareex-pected,e.g.,spinpolarization,realizationofone-dimensionalMott-Hubbard-typeinsulators,andsoon.

Thispredictionwasbasedonatight-bindingschemewithonlynearest-neighborhoppings,soitshouldbecheckedwhethertheedgestatesareseenwhenhoppingdistanceisincreased.Indeed,wehaveanexperienceofdisappearanceoftheedgestatewhenweartificiallyintroducethehoppingparameterforalongdistance.Therefore,aquestionhasarisenastowhethertheedgestatesareobtainedbyfully

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self-consistentcalculationsthatexpressoff-siteelectronhop-pingsforareasonablerange.Furthermore,theconsideredstructuresoftheribbonswererestrictedinthecaseofasinglelayer.Ontheotherhand,stackedlayersofribbonsarethoughttobemorerealisticinavailablesamples.Theexis-tenceoftheedgestatesshouldthereforebeexaminedforthestackedgeometryinwhichelectronhoppingsbetweenneigh-boringlayershavetobetakenintoaccount.

Inthispaper,weperformedthefirst-principlesband-structurecalculationforH-terminatedzigzagribbonswithintheframeworkofthelocal-densityapproximation͑LDA͒byusingpseudopotentialsandtheplane-wavebasissets.Wehavefoundthatthepredictededgestatesinsingle-layeredribbonsarewellreproducedinthepresentfirst-principlescalculations.Furthermore,wehaveexaminedtheedgestateswhenthezigzagribbonsarecondensedinamanneroftheAAstackingandtheABstackingofthebulkgraphite.TheedgestatescanexistonlyinthecaseoftheABstacking,inwhichhalfoftheCatomsofoneribbonarelocateddirectlyabovethecenterofeachhexagonontheneighboringrib-bons.ThisfactcanbeunderstoodfromtheenergybandstructureoftheAA-andAB-stackedbulkgraphite.Intherestofthispaper,detailsofthecomputationalconditionsandtheresultsareshown.

Wehaveperformedtheband-structurecalculationswithintheLDAinwhichtheexchangecorrelationenergyofmany-bodyelectronsaretreatedasafunctionalform2ofthechargedensity,whichwasfittedtothenumericalresultsofelectrongas.3Toexpresseffectsof1scoreelectronsofCatomsonvalenceelectrons,thenormconservingnonlocalpseudopo-tentialsaregeneratedbyusingaschemeofsoftcoreradius.4Onlythescomponentsareusedasthenonlocalpart.WhilefortheHatoms,thelocalpseudopotentialformedas1/r⌺2iCierf(r/rci)isusedwithrbeingadistancefrompro-ton.Thepresentparametersare1.0519andϪ0.0519forC1andC2,0.20702bohrand0.39993bohrforrc1andrc2,respectively.Acutoffenergyof40Ryisusedforplane-waveexpansionofvalencewavefunctions.Tomimicasituationofanisolatedzigzagribbon,weadoptedthree-dimensionalrepeatingcellsinwhichindividualribbons

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FIG.1.͑a͒Atomicgeometryofazigzagribbon(Nϭ2ϳ6)terminatedbyHatoms.Shadedcirclesdenotecarbonatomswhilesmallercirclesdenotehydrogenatoms.͑b͒Bandstructureofthecorrespondinggeometry(Nϭ2ϳ6).ThepositionsofEFaresetas0eVindicatedbydottedlines.ArrowsindicatethestateshavinglocalizedC–H␴characters.

areseparatedbyavacuumregion.Intheadoptedvacuumregion,theintervalsamongtheribbonsarekeptas8.5Åand6Åforedge-edgeandlayer-layerdistances,respec-tively.Theseintervalshavebeenfoundtobeenoughtopre-ventartificialinterribboninteractions.5Asforthek-pointsamplingforthemomentumspaceintegration,ninekpointsinthewedgeofthefirstBrillouinzone͑BZ͒areused.Ac-cordingtothecalculatedHellmann-Feynmanforces,theatomicgeometryisoptimizedtoachievethetotal-energyminimumwithin6arestrictionoftheplanarconfigurationoftheribbon.However,thegeometryoptimizationisfoundnottoconsiderablychangetheatomicstructurefromtheidealone,i.e.,aperfecthoneycombpatternwithC–Clengthof1.42ÅandHterminationswithC–Hlengthof1.01Å.IndeedtheC–Cbondlengthsareshortenedonlyby0.02ÅattheH-terminatedsite.Therefore,aqualitativefeatureofelectronicstructureislesssensitivetothegeometryoptimi-zation.

Figure1͑b͒showsbandstructuresofzigzagribbonsinFig.1͑a͒.Thewidthofeachribboncorrespondsto2ϳ6

arraysofzigzagC–Cchains.WecallthiswidthNϭ2ϳ6accordingtoRef.1.Aremarkablefeatureinthebandstruc-tureofaribbonwithNϭ6isthattwobandsmeetatthekvectorbeyondtwothirdsonthewayfrom⌫toXinthefirstBZ,where⌫ϭ0andXϭ␲/awithaϭ2.46Å͑ribbon’spe-riod͒.ThenthesebandsbecomenearlyflatuptokϭX,andthelocationofEFcorrespondstotheseflatbands.Whileinthethinnerribbons(Nϭ2ϳ4),flatbandsarenotseen.Thoseribbonsaretoonarrowtoseparatetheedgestatespenetratingfrombothedges.ThenearlyflatbandsseenintheribbonwithNϭ6suggestthattheelectrontransferoflongerrangesgivesnoremarkableeffectforwiderribbons.Theseresultsarequalitatively1

thesameasthatobtainedinthetight-bindingcalculations.Next,thewavefunction’scharacteroftheflatbandsinthecaseofNϭ6hasbeeninvestigated.WehavecalculatedtheoverlapbetweentheatomicvalencewavefunctionsofeachCsiteandtheBlochwavefunctionsoftheflatbands.TheBlochwavefunctionshavetheirmaximumamplitudeatthe␲orbitalsofH-terminatedCsites,andthesecondmaximumamplitudeislocatedatsecondneighborCsites.Again,thesefeaturesagreewellwiththeformertight-bindingcalculations.1Fromthesewavefunction’scharacters,theseflatbandsshowninFig.1͑b͒withNϭ6canbeattributedtotheedgestates.

Theexistenceoftheedgestatesinsingle-layeredgra-phiticribbonshasthusbeenconfirmedbyperformingthefirst-principlescalculations.OnecanconsiderthattheedgestatesmayhavetheFermiinstabilitysincethereshouldbeasharppeakofthedensityofstatesatEreticalF.However,atheo-͑SSH͒discrepanciesmodelinvestigation7basedontheSu-Schrieffer-Heegerdeniedthispossibility.Sincetherearenose-riousintheresultsforhydrocarbonsystemsbetweenSSHmodelandthefirst-principlescalculation,weexpectthattheFermiinstabilitywillalsobedeniedinLDAcalculations.

Inadditiontotheexistenceoftheedgestate,onemaybeinterestedintheinfluenceofhydrogen1sstatesontheelec-tronicstructure,whichcanbeexpectedtocausehybridiza-tionwithC–C␴orbital.Wehavefoundthattherearesev-eraleigenstateshavingC–H␴characters.ThestateswithconsiderableamplitudesattheC–HbondsaredenotedbyarrowsinFig.1͑b͒.Ontheotherhand,theC–H␴*͑anti-bonding͒statesarefoundfaraboveEF,whichareoutofthescaleinFig.1͑b͒.TheseC–H␴*stateshavenocontributionfortheconductingpropertiesoftheribbonssincetheenergylevelsareawayfromEFbymorethan5eV.TheyalsohavenoinfluenceontheedgestatesincetheorbitalsofC–H␴and␴*extendparalleltotheribbonwhilethoseoftheedgestates(␲orbitals͒extendnormaltotheribbon.

Ournextinterestiswhethertheedgestatescanbeseeninstackedlayersofthezigzagribbons.Frompracticalview-points,realizationofasinglelayerofthezigzagribbonisratherhardcomparedtothatofamultilayeredsample.Wehaveperformedband-structurecalculationsofthestackedribbonsbyassumingtheAA-andtheAB-stackinggeom-etries.IntheAAstacking,allCatomsofoneribbonarelocateddirectlyaboveallCatomsontheneighboringrib-bons,whileintheABstacking,halfofCatomsofonerib-bonarelocateddirectlyabovethecenterofeachhexagonof

9860BRIEFREPORTSPRB59FIG.2.͑a͒ThefirstBrillouinzoneofAB-stackedzigzagrib-bonswithNϭ6andcorrespondingdirectionsinrealspace.⌫-XandZ-Mlinesarealongtheribbon’saxiswhile⌫-ZandX-Mlinesarealongthestackingdirection.͑b͒BandstructuresofAA-and͑c͒AB-stackedribbonswithNϭ6.ThepositionsEFaresetas0eVshownasdottedlines.

theneighboringribbons.Theinterlayerdistanceisassumedtobe3.34Å,whichisthesameasthatofthegraphite.HerewemustcommentontheLDAapproachindescrib-ingtheinterlayerinteractionofgraphite.ItiswidelythoughtthatthegraphiteinterlayerinteractionisdominatedbythevanderWaalsinteraction.Ifthisisthecase,theLDAcal-culationfailstoexpressthegraphiteinterlayerinteractionand͑maybe͒theelectronicbanddispersionalongthestack-ingdirection.However,SchabelandMartins8showedthat

theapplicationoftheLDAandpseudopotentialsschemefortheAB-stackedgraphitegivesreasonablevaluesofinterlayerbindingenergy,theoptimumdistance,andthecompressibil-ity,eachofwhichagreeswellwiththeexperimentalvaluescited͑inRef.foundincludingtobethe8.consistentdispersionFurthermore,withalongthethethecalculatedphotoemissionstackingbanddirectionstructuredata,which͒wasarealsocitedinRef.8.Sincetheabsolutevalueofthein-terlayerbindingenergyistiny͑fewtensmeVperatom͒,therestillremainsanargumentforthenumericalaccuracyinenergetics.Ontheotherhand,therearenoseriousproblemsoftheLDAindescribingenergy-bandstructuresofstackedgraphiteasfarasweknow.WehaveconfirmedthatourpresentcomputationalconditionwellreproducedthebandstructuresshowninRef.8aswellasRef.9.WethereforebelievethatthebanddispersionofthestackedribbonsisalsowelldescribedwithintheLDAscheme.Ourpresentcutoffenergyisnotsohighasthepreviousone,8soinsteadoftheoptimizationoftheinterlayerdistancewehavejustadjustedtheidealinterlayerdistanceof3.34Å.

Thebanddispersionofstackedribbonsbecomestwodi-mensionalindirectionsalongtheribbonandalongthestack-ing.Figure2͑a͒showsthetwo-dimensionalBZandthere-lateddirectionsinrealspace.͑OnlythecaseoftheABstackingisshownhere.͒ThecalculatedbandstructuresforAA-andAB-stackedribbonsareshowninFigs.2͑b͒and2͑c͒,respectively.TheassumedwidthoftheribbonsisNϭ6inbothofcases.theABThestacking.

flatbandslocatedatEFareseenonlyinthecaseWeagaininvestigatedthewavefunction’scharacterofthesenearlyflatbandsseenintheAB-stackedribbonsandfoundthesimilaritytotheedgestateofasinglelayerofazigzagribbon.IntheAB-stackedribbons,theedgestatesareindividuallylocalizedontheedgeofeithereachAlayeroreachBlayer.TherearenoremarkableelectronhoppingsbetweentheedgestatesofAandBlayersdespiteextensionofthecorrespondingwavefunctionsnormaltoeachlayer.However,thisfactisnotsosurprisingwhenwecomparethepresentresultswiththatoftheAB-stackedgraphitebulk,8,9whichalsohasverysmallbanddispersionaroundEK-HlineoftheBZ͑inthedirectionofthestackingFalong͒.Asmentionedbefore,theABstackingallowshalfofCatomstobeneighboredtothecenterofeachhexagoninthestackingdirection.Theedgestateofasingleribbonisessentiallycharacterizedbyanonbondingorbital(␲electrons͒whoseamplitudeismainlydistributedattheH-terminatededgesitesandtheirsecondneighbors.RememberthatsuchCsitesarelocatedeitherabovethecenterofthehexagonintheneighboringribbonordirectlyabovethenodalsiteoftheneighboringribbon.Theinterlayerhoppingsofthe␲elec-tronsonthosesitesarethussuppressed,whichresultsintheverysmalldispersionnearEstacking,allCatomsofoneF.WhileinthecaseoftheAAlayerarelocateddirectlyaboveall␲Catomsontheneighboringlayers,whichallowsthepearance-electronofhoppingstheflatbandsintheisdirectioninhibited.

ofthestacking,soap-Fromthepresentresults,weconcludethattheedgestatesexistinbothsingleandAB-stackedH-terminatedgraphiteribbons.Thesestatesareexpectedtobedetectedbyscanningtunnelingmicroscopy͑STM͒.WewouldliketopointoutherethatthesubstratewithoutelectronlevelsnearEFshould

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bechoseninSTMmeasurementinordertopreventhybrid-izationbetweentheedgestatesandthesubstratestatesnearEF.Ifthehybridizationtakesplace,thelocalizednatureoftheedgestatesmightbedestroyed.*Presentaddress:CollegeofScienceandEngineering,Aoyama

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