石墨烯电子局域态

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TransitionbetweenElectronLocalizationandAntilocalizationinGraphene

F.V.Tikhonenko,A.A.Kozikov,A.K.Savchenko,andR.V.Gorbachev

SchoolofPhysics,UniversityofExeter,EX44QLExeter,UnitedKingdom

(Received16July2009;published23November2009)

Weshowthatquantuminterferenceingraphenecanresultinantilocalizationofchargecarriers—an

increaoftheconductance,whichisdetectedbyanegativemagnetoconductance.Wedemonstratethat

dependingonexperimentalconditionsonecanobrveeitherweaklocalizationorantilocalizationof

carriersingraphene.Atransitionfromlocalizationtoantilocalizationoccurswhenthecarrierdensityis

decreadandthetemperatureisincread.Weshowthatquantuminterferenceingraphenecansurviveat

hightemperatures,uptoT$200K,duetoweakelectron-phononscattering.

DOI:10.1103/PhysRevLett.103.226801PACSnumbers:73.23.Àb,,

Quantuminterferenceingraphene,amonolayerofcar-,butalsoonelastic

bonatoms,isverydifferentfromthatinconventionaltwo-

dimensional(2D)systemsduetochiralityofchargecar-

riersandanadditionalquantumnumber,thepudospin

[1,2].ThecarriersingraphenehaveaBerryphaof󰀁:the

additionalphathattheelectronwavefunctionwillac-

quireifanelectroncompletesaclodtrajectory.This

shouldresultinweakantilocalization(WAL)ofcharge

carrierscomparedwiththeconventionallocalization

(WL)in2Dsystems[3,4].

Figure1(a)showshowanelectronscatteredbyimpuri-

tiescaninterfereonaclodtrajectorywhenitistreatedas

awave.Twoelectronwavespropagateinoppositedirec-

tionsaroundthetrajectoryandinterfereatthepointof

intercept.Asthetwopathsareidentical,thephaofthe

twowavesisthesameandtheinterferenceconstructive.

Thisincreastheprobabilityforelectronstoscatterback

anddecreastheelectricalconductancecomparedwithits

classicalDrudevalue.Inexperiment,thequantumcorrec-

tiontotheconductanceisusuallydetectedbyapplyinga

magneticfieldperpendiculartothe2Dsystem.Itaddsa

phadifferencetothetwowavesanddestroystheinter-

ference.Thisresultsinanincreaoftheconductance—

positivemagnetoconductance(MC),Á󰀂ðBÞ¼󰀂ðBÞÀ

󰀂ðB¼0Þ>0.Ingraphene,however,theBerrypha󰀁

addsaphadifferencetothetwointerferingtrajectories,

sothattheymeetinantiphaanddestructiveinterference

occurs.Thisshouldresultinanincreaoftheconductance

duetoquantuminterferenceandanegativemagnetocon-

ductance[1].Thisantilocalizationeffectisverydifferent

fromthatobrvedbeforein2Dsystemswithstrongspin-

orbitscattering[3,5–8]wherethetwowavesmeetinanti-

phabecauofspinflipsinscatteringbyimpurities—in

graphenespin-orbitinteractionisknowntobeweakdueto

thelowmassofcarbonatoms[9].

Sofartheexperimentalstudiesofquantuminterference

ingraphene-badsystemshavenotrevealednegative

magnetoconductance[10–14],althoughtheobrved

weaklocalizationhasbeenentobeunusualinthatit

dependsnotonlyoninelasticscatteringofelectrons,char-

0031-9007=09=103(22)=226801(4)226801-1Ó2009TheAmericanPhysicalSociety

acterizedbythedephasingtime󰀃

󰀄

scatteringcaudbyimpuritiesandimperfectionsinthe

crystalstructure.Weakantilocalizationwasexpectedtobe

enonlyinsampleswithoutdefects,i.e.,intheabnceof

intervalleyscatteringandchiralitybreakingscatteringby

atomicallysharpdefects.Inthisworkweprentclear

evidenceforWALingraphenebyobrvingnegative

MCatlowmagneticfields.Weestablishtheexperimental

conditionsforitsobrvationandshowthatinmechani-

callyexfoliatedgrapheneoneisabletodetectnegativeMC

evenintheprenceofsuchscattering.Wedemonstrate

thatinthesamesampleatransitionbetweenlocalization

andantilocalizationcanbeenbychangingtemperature

andelectrondensity.

Thetheory[1]ofquantuminterferencehasbeenshown

tobeapplicabletothedescriptionoftheMCinmechani-

callyexfoliatedgraphene[12]andagraphiticsystemon

thesurfaceofSiC[10]:

FIG.1(coloronline).(a)Thetrajectoriesofanelectronscat-

teredbyimpuritiesthatgiveritoaquantumcorrectiontothe

conductance.(b)Adiagramofthescatteringtimesrelatedto

quantuminterferenceingraphene.Thesolidcurveparatesthe

regionsofelectronlocalizationandantilocalization.Pointsare

experimentalvaluesfoundfromtheanalysisofthemagneto-

conductanceusingEq.(1),forthreeregionsofelectrondensity.

PRL103,226801(2009)

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󰀁󰀂󰀁󰀁󰀂

À112À

e󰀃󰀃

BB

Á󰀂ðBÞ¼

FÀF

À1À11À

󰀁h

󰀃󰀃þ2󰀃

󰀄󰀄

i

󰀂󰀂

󰀁

1À

󰀃

B

:À2F(1)

À11À1À

󰀃þ󰀃þ󰀃

󰀄

Ã

i

HereFðzÞ¼lnzþð0:5þz

c

À1

Þ,ðxÞisthedigamma

c

1À

function,󰀃

B

¼4eDB=@andDisthediffusioncoeffi-

cient.(Thetheoryassumesthatthemomentumrelaxation

1À

rate󰀃isthehighestinthesystemandcomesfrom

p

chargedimpurities,anddoesnotaffecttheelectroninter-

ference.)Thequantumcorrectiondependsnotonlyonthe

dephasingtime󰀃

󰀄iÃ

butonelasticscatteringtimes󰀃and󰀃.

Graphene’sbandstructurehastwovalleys,andquantum

interferenceofelectronsinonevalleycanbesuppresdby

scatteringonthedefectswiththesizeofthelatticespacing,

aswellasdislocationsandripples[1,2,15].Suchdefects

breakthechirality,whiledislocationsandripplesproduce

aneffectiverandommagneticfieldwhichdestroysthe

interference.Thecombinedeffectofthisintravalleyscat-

teringischaracterizedbythetime󰀃

Ã

.Intervalleyscattering

bysharpdefects(suchastheedgesofthesample)thatare

abletoscatterelectronsbetweenthetwovalleysischar-

acterizedbythetime󰀃

iÃ

.Whilesmall󰀃suppressinter-

ferencewithinavalley,smallenough󰀃

i

restoresitby

mixingthetwovalleys,whichhaveoppositechirality.

NegativeMCcorrespondingtoantilocalizationisdeter-

minedby(negative)condandthirdtermsinEq.(1).In

theabnceofintra-andintervalleyscatteringinadefect-

freegraphenelayer,󰀃

i;Ã

!1,Á󰀂ðBÞistotallycontrolled

bythethirdterm.Intheoppositecaofstrongintra-and

intervalleyscattering(small󰀃and󰀃),bothnegativeterms

Ãi

aresuppresdandthefirst(positive)termdominates,

whichcorrespondstoelectronlocalization.Itisthissitu-

ationthatwasrealizedintheexperiments[12]onmechani-

callyexfoliatedgraphene,wherethenegativeterms

comingfromthechiralityofelectronswereclearlyvisible

butnotlargeenoughtochangethesignoftheMCinlow

fields.

CarefulanalysisofEq.(1)demonstrates,however,that

antilocalizationcanstillbedetectedinsuchsamples.

UsingthefactthatthefunctionFðzÞforz(1(atsmall

magneticfields)canbereprentedbyasimplequadratic

dependenceFðzÞ¼z

2

=24,wesimplifyEq.(1)forsmall

fieldsas

󰀁󰀂

󰀁

e1

2

4eDB󰀃

󰀄

2

Á󰀂ðBÞ¼

1À

24󰀁h@

ð1þ2󰀃=󰀃Þ

󰀄i

2

󰀂

2

À

:(2)

ð1þ󰀃=󰀃þ󰀃=󰀃Þ

󰀄i󰀄Ã

2

tionsfortheobrvationofnegativeMCaresmallratios

󰀃

󰀄Ã󰀄i

=󰀃=󰀃

and󰀃.Thiscanberealizedbyincreasingthe

temperature(whichdecreas󰀃

󰀄

)andbyloweringthe

carrierdensity(whichincreas󰀃

i

[12]).

Thestudiedsampleisproducedbymechanicalex-

foliationofgraphiteanddepositedonanoxidizedSiwafer

[16].Usingelectron-beamlithographyasix-terminalHall

barisformedfromtheflakewithawidthof2󰀅manda

lengthbetweenthepotentialprobes(Au=Cr)of22:5󰀅m.

Thesamplewasannealedinvacuumatatemperatureof

140

󰀅

Cfortwohoursbeforecoolingdowninacryostat.

Figure2(a)showsaschematicofthemeasurementcircuit.

TheintshowsthequantumHalleffectwhereplateaux

athalf-integerfillingfactorsareevidencethatthesample

ismonolayergraphene[17].Themobilityofelectrons

outsidetheDirac(electro-neutrality)regionis

$12000cm

2À1À1

Vs

.TheresistanceRasafunctionof

thegatevoltageV

g

showsapeakattheDiracpointwhere

thedensityofchargecarriersiszero.Thebarsindicate

threeregionsofthegatevoltage(ofsizeÁV

g

¼1V)

wheretheMCismeasured.Toaverageouttheeffectof

universalconductancefluctuations[18],theresistanceat

eachmagneticfieldisaveragedovereachoftheregions.

Themeasurementsarethenrepeatedatdifferenttem-

peraturesintherangeT¼5–200K.

Figures2(b)–2(d)showtheevolutionoftheMCwith

changingcarrierdensityatthreetemperatures.Onecane

FIG.2(coloronline).(a)Resistivityasafunctionofthecarrier

density,withthethreeregionswherethemagnetoconductanceis

studiedindicatedbybars.Ints:adiagramofthesampleandthe

resultsofthequantumHalleffectmeasurement.(b),(c),(d)Evo-

lutionofthemagnetoconductancewithdecreasingelectron

density,atthreetemperatures.SolidcurvesarefitstoEq.(1).

Theexpressioninthebracketsdeterminesthesignofthe

MC.Figure1(b)showsadiagramwithacurveÁ󰀂¼0

thatparatestheregionsofpositiveandnegativeMC

(localizationandantilocalization).Thefavorablecondi-

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thatwithdecreasingdensity(movingfromregionIIItobecomestoosmalltosatisfytheconditionofantilocaliza-

regionI)theMCatT¼14and27Kchangesitssignfromtion,Fig.1(b).

positivetonegative.Indeed,theratioofthecharacteristicOurexperimentsareperformedatmuchhighertempera-

timesfoundfromtheanalysisoftheMCcurvesusingturesthanthepreviousstudiesofweaklocalizationin

Eq.(1)isentoentertheregionofantilocalizationinexfoliatedgraphene[12,13](T<20K).InFig.3one

Fig.1(b).(AsimilartransitionbetweenWLandWALwascanethatthetemperaturedependentMCexistsattem-

obrvedin2Dsystemswherethestrengthofspin-orbitperaturesT$200K,whileinconventional2Dsystems

scatteringwascontrolledbychangingcarrierdensitythequantumcorrectionusuallydisappearsatmuchlower

[7,8].)AtT¼5K,however,wherethevalueof󰀃temperatures,duetointensiveelectron-phononscattering

󰀄

is

larger,onecannotachievethetransitiontoantilocalization,

Fig.1(b).

Figure3showstheevolutionoftheMCwithincreasing

temperatureinthethreeregionsofcarrierdensity.Itfol-

lowsfromEq.(1)thatatlowtemperaturesthewidthofthe

dipinsmallBismainlycontrolledby󰀃

󰀄

,whilethe

bendingofthecurveatlargerBisdeterminedby󰀃and

i

󰀃

Ã

.TheanalysisoftheMCcurvesshowsthat,asexpected,

elastictimes󰀃and󰀃areesntiallytemperatureinde-

iÃ

pendentbutinelastictime󰀃

󰀄

stronglydecreaswithin-

creasingT.OnecaneinFig.3thatwithincreasingTthe

widthofthedipintheMCincreas(duetoadecreaof

󰀃

󰀄

),sothatthedependencebecomesflatatT¼27K.

Withfurtherincreaofthetemperaturethequantum

correctionstartsbeingenagain,butnowasapeakin

theMC.ItswidthcontinuestoincreawithincreasingT,

untilatT$200Kthedependencebecomesflatagain

whenantilocalizationdisappearsduetorapiddephasing

oftheelectrontrajectories.Notethatthetransitionfrom

WLtoWALiseninregionsIandII,butnotinthehigh-

densityregionIII.Inthisregiontheintervalleytime󰀃

i

[19].Ingraphene,however,electron-phononscatteringis

expectedtobeweak[20,21],andthusitisinterestingto

examinewhatthesourceofdephasingisathighT.

Inearlierstudiesofelectronlocalization[12]thetem-

1À

peraturedependenceofthedephasingrate󰀃

󰀄

wasfound

tofollowthelineartemperaturedependencecaudby

electron-electronscatteringinthe‘‘diffusive’’regime[22]:

󰀂󰀁

2E󰀃

Fp

kT

B

À1

󰀃¼󰀆;(3)ln

󰀄

2E

Fp

󰀃@

where󰀆isacoefficientoftheorderofunity.Thisregime

correspondstotheconditionk

Bp

T󰀃=@<1,whichmeans

thattwointeractingelectronsexperiencemanycollisions

withimpuritiesduringtheinteractiontime@=k

B

T.Figure4

showsthedephasingrate,obtainedfromtheanalysisofthe

MCusingEq.(1),inthethreeregionsofcarrierdensity.

Theexpectedtransportphononscatteringrateingraphene

isgivenbythefollowingrelation[20,21]:

1À

󰀃

e

-

ph

1ED

Fa

2

¼kT;(4)

322

B

@4V󰀇V

F

m

ph

whereD

am

isthedeformationpotentialconstant,󰀇isthe

densityofgraphene,Visthespeedofsound,andVis

phF

theFermivelocity.Assumingthatthisscatteringrateis

clotothedephasingrateduetoelectron-phononscatter-

ing,󰀃

󰀄eph

$󰀃

-

[23],weplotitinFig.4usingtheparame-

ters󰀇,V,

mph

¼7:6Â10kgm¼2Â10ms

À7À24À1

V

Fa

¼10ms%18eVtakenfromtheanalysis

6À1

,andD

FIG.3(coloronline).(a),(b),(c)Evolutionofthemagneto-FIG.4.Temperaturedependenceofthedephasingrateforthe

conductancewithincreasingtemperatureinthethreestudiedthreeregions.Solidcurvesarefitstotheelectron-electron

regions,showingatransitionfrompositivetonegativelow-fieldscatteringratesfoundasasumofEq.(3)and(5):(a)󰀆¼1:5,

magnetoconductance.Bottompanelsshowtheresultsathigh

temperatures.SolidcurvesarefitstoEq.(1).areelectron-phononratescalculatedusingEq.(4).

󰀈¼0;(b)󰀆¼1:5,󰀈¼2:5;(c)󰀆¼0,󰀈¼2:5.Dottedlines

226801-3

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ofthetemperaturedependenceoftheclassicalconduc-

tancein[24].Onecanethattheelectron-phononrate

istoolowtoexplaintheexperimentalvalues.Inaddition,

1À

theexperimental󰀃

󰀄

ðTÞinregionsIIandIIIhasstronger

thanlineartemperaturedependence.Thelatterexcludes

notonlyelectron-phononbutalsoelectron-electronscat-

teringinthediffusiveregime.Thealternativeinelastic

scatteringmechanism,whichgivesaparabolictemperature

dependence,iselectron-electronscatteringintheballistic

regime,k

Bp

T󰀃=@>1,whenelectroninteractionismedi-

atedbyonlyafewimpurities[25]:

1À

󰀃¼󰀈ln

󰀄

󰀂

󰀁

󰀁ðkTÞ2E

BF

2

:(5)

4

@EkT

FB

Thisdependence,withacoefficient󰀈oftheorderofunity,

hasbeenobrvedinexperimentsonhigh-mobility2D

systemsatlowtemperatures[26].InregionIwherethe

transitiontemperaturek

B0p

T󰀃=@$1betweenthetwore-

gimesishigh,T

0

$80K,thedephasingratecanbesat-

isfactorilyexplainedbythediffusiveelectron-electron

interaction,Eq.(3).InregionsIIandIIIwithT

0

$60K

and40K,respectively,thedephasingratecanberepre-

ntedasasumofthetworates,Eq.(3)and(5).Therefore,

ourresultsshowthatitiselectron-electronscattering

whichisthemainsourceofhigh-temperaturedephasing

ingraphene,duetoweakelectron-phononscattering.

Insummary,weshowthatelectronantilocalizationin

graphene,aconquenceoftheBerrypha󰀁,canbe

realizedexperimentally.Itcanbeenundertheconditions

whenthedephasingtimeismadesmallenoughcompared

withtheelasticinter-andintravalleyscatteringtimes.Our

experimentsshowthatquantuminterferenceingraphene

canexistatextremelyhightemperaturesduetosuppresd

electron-phononscattering.

WearegratefultoD.W.Horll,I.Gornyi,andE.

McCannforreadingthemanuscriptandvaluablecom-

ments,andacknowledgesupportfromtheEPSRCgrant

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