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Graphene:

PropertiesandProductions

ZhuChen2011103077

Grapheneisawondermaterialwhichisanallotrope of carbon.Itisaflatcarbonmonolayertightlypackedintoatwo-dimensionalhoneycombcrystallattice.Graphemeisthethinnestmaterialintheuniverseandthestrongestevermeasured.Itschargecarriershaveextremelymobility,havezeroeffectivemass.Inaddition,ithassomespecialopticalproperties.Sincethephysicistsproducegraphenebyusingtensingtape,researchershavefoundseveralmethods.Inthisreview,Iwillattempttoanalyzesomeinterestingpropertiesofthegrapheneandseveralpreparationmethods,andintroducesomeapplicationsaswellastherecenttrends.

1.

Figure1:

schematicviewofthesp2hybridization.Theorbitalsformanglesof120O

Introduction

Grapheneisasinglecarbonlayerofthegraphiticstructure,andthesp2-bondedcarbonatomsthataredenselypackedwiththebondlengthofabout0.142nanometers.Thecrystallineformofgraphiteconsistsofmanygraphenesheetsstackedtogetherbytheinterplanarspacingof0.335nm.Inaddition,grapheneisthefundamentalstructuralelementofsomeothercarbonallotropesincludingcharcoal,carbonnanotubesandfullerenes.

Figure2:

Motherofallgraphiticforms.Grapheneisa2Dbuildingmaterialforcarbonmaterialsofallotherdimensionalities.Itcanbewrappedupinto0Dfullerene,rolledinto1Dnanotubeorstackedinto3Dgraphite.

In2004,theManchestergroupobtainedgraphenebyusingtensingtaperepeatedlysplitgraphitecrystalsintoincreasingthinnerpieces.In2010,theywereawardedtheNobelPrizeinPhysics“forgroundbreakingexperimentsregardingthe two-dimensional materialgraphene”.

Grapheneresearchhasdevelopedquicklywhiletheliteratureongraphenekeepsrapidlyincreasingovertheseveralyears.Somanyaspectsarehotanditisarealstruggletochoicewhichseveralpointstolearn.Asastudentofexperimentalgroup,I’malwaysinterestedinsomestructuralpropertiesandelectronicpropertieswhichcanshowthesomeobviouscharacteristic.Itseemstomethatthemostimportantthingistopreparegrapheneeffectivelyandeconomically.Inthispaper,Iwillasksomequestionsaboutgrapheneandthenattempttofindtheanswersfromothers’paper,inaddition,Iwillputforwardmyownunderstandsandsomeopinions.

2.Properties

2.1Atomicstructure

Firstly,asgrapheneisatwo-dimensionalstructure,differentfromothermaterialsasweknow,Iwanttolearnabouttheatomicstructureofitandthestabilityofit.

Generally,grapheneisaflatmonolayerofcarbonatomstightlypackedintoatwo-dimensionalhoneycomblattices,thestructurecanberegardasatriangularlatticewithabasisoftwoatomsperunitcell.

Figure3:

latticestructureofgraphene,madeoutoftwointerpenetratingtriangularlattices

However,LandauandPeierlsarguedthatstrictlytwo-dimensionalcrystalsdidnotexistbecauseofthethermodynamicallyunsteady.Thephysicsintheirtheoryisthatthethermalfluctuationsinlow-dimensionalcrystallatticesshouldleadtosuchdisplacementsofatomsanddestroythecrystalstructure.Thistheorywasstronglysupportedbyaseriesofexperimentalobservations.Indeed,thefilmsbecomeunstableatathicknessofdozensofatomiclayers.Forthisreason,atomicmonolayersareusuallygrownontopofsubstrateswithmatchingcrystallattices.

Since2004,researchersobtainedtwo-dimensionalcrystalsontopofnon-crystallinesubstrates,inliquidsuspensionandassuspendedmembranes,refutingabovetheory,andattractingmemuch.Iamwonderfulofthestableofthesetwo-dimensionalmaterials.Suspendedgraphenealsoshowed"rippling"oftheflatsheet,withamplitudeofaboutonenanometer.Theseripplesmaybeintrinsictographeneasaresultoftheinstabilityoftwo-dimensionalcrystals.Inmyopinions,thegentlecrumplinginthethirddimensionleadstoagaininelasticenergybutsuppressesthermalvibrations,whichaboveacertaintemperaturecanminimizethetotalfreeenergy.Inaddition,theirsmallsizeandstronginteratomicbondsensurethatthermalfluctuationscannotleadtothegenerationofdislocationsorothercrystaldefectsevenatelevatedtemperature.

Figure4:

suspendedgraphenesheetstendtoripple

2.2Electronicproperties

Grepheneisdifferentfrommostconventionalthree-dimensionalmaterials,anditissemimetal.Inordertounderstandtheelectronicproperties,Iusetight-bindingmodelforelectronsonthehoneycomblattice.Thephysicsinthetight-bindingmodelisthatthreeelectronspercarbonatomingrapheneareinvolvedintheformationofstrongcovalentσbonds,andoneelectronperatomyieldstheπbonds.Theπelectronshappentobethoseresponsiblefortheelectronicpropertiesatlowenergies,whereastheσelectronsformenergybandsfarawayfromtheFermienergy.Thus,Icandevotetoabriefdiscussionoftheenergybandsofπelectronswiththetight-bindingapproximation,whichwasoriginallycalculatedforthehoneycomblatticebyP.R.Wallacein1947.

Thetight-bindingHamiltonianforelectronsingrapheneconsideringthatelectronscanhoptobothnearest-neighboratomshastheform.ThesecondquantizationHamiltonianofgraphenehastheform

t=2.7eVisthehoppingenergyofelectrons.

Fouriertransformasfollows

ThesecondquantizationHamiltonianis

Whileweknowthat

Andthen

TheenergybandsderivedfromthisHamiltonianhavetheform

Theplussignappliestotheupperbandandtheminussignappliestothelowerband.Itisevidentthatthespectrumissymmetricaroundthezeroenergy.

Figure5:

Electronicdispersioninthehoneycomblatticewitht=2.7eV.AndtherightisthezoominthebandsclosetooneoftheDiracpoints.

Inordertodescribethelow-energyexcitations,electronicexcitationswithanenergyismuchsmallerthanthenbandwidth~|t|.AnditmayrestricttheexcitationstoquantumstatesinthevicinityoftheDiracpointsandexpandtheenergydispersionaroundK.Wediscomposedthewavevectorask=K+q=,where|q||K|,therefore|q|a1.

Wecangetthat:

Whilewecangetthat:

Thiswillderivetheeffectivelow-energyHamiltonian:

WherewehavedefinedtheFermivelocity

Withavalue

AndusedthePaulimatrices

and

DiscussingthephysicsaboutresultsoftheSchorderingfunction,duetothislineardispersionrelationatlowenergy,electronsandholesnearthesesixpointsbehavelikerelativisticparticlesdescribedbytheDiracequation.HencetheelectronsandholesarecalledDiracfermions.

Figure6:

Brillouinzoneofgraphene,andtheDiracconesarelocatedattheKandK’points.

Inaddition,theenergydispersionresemblestheseparticlesaredescribedbythemasslessDiracequation.AnimmediateconsequenceofthismasslessDirac-likedispersionisacyclotronmassthatdependsontheelectronicdensityasitssquareroot.Thecyclotronmassisdefined,withinthesemi-classicalapproximation,as

TheelectronicdensitynisrelatedtotheFermimomentumkFaskF2/π=n,whichleadsto

AsfarasIamconcerned,ifwecanobservethisrelationofdependenceonthecyclotronmassinexperiment,wecanprovetheexistenceofmasslessDiracquasiparticlesingraphene.

Diracfermionsingraphenewillleadtoaseriesofinterestingphysicalphenomenon.IfsubjectedtomagneticfieldstheanomalousintegerquantumHalleffectmeasuredexperimentally.Moreover,ingraphenetheseremarkableanomaliescanevenbemeasuredatroomtemperature.ThisanomalousbehaviorisadirectresultoftheemergentmasslessDiracelectronsingraphene.Thephysicsofthisunusualphenomenonisthatinamagneticfield,theirspectrumhasaLandaulevelwithenergypreciselyattheDiracpoint,likefractionalquantumHalleffect,thislevelishalf-filledleadingtothe1/2intheHallconductivity.

2.3Bilayergrapheme

Bilayergrapheneistwolayersofgraphene,andithasbeenshowntohavesomeinterestingproperties,suchasanomalousbehaviorofintegerquantumHalleffect,andalsoagapcanopenbetweentheconductionbandandvalenceband,whichattractmeverymuch.Inordertodiscusstheappearanceofthebandgap,Itriedtogetthebandstructureofbilayergraphenebyusingthetight-bindingmodel.Thetight-bindingmodeldevelopedforgraphitecanbeeasilyextendedtostackswithbilayersgraphene.Thebilayerstructure,withtheABstacking,isshowinginfig7:

Figure7:

LatticestructureofbilayergrapheneandtheBrillouinzone.

Thetight-bindingHamiltonianofthebilayergrapheneis

Wheream,I,σandbm,j,σannihilatesanelectronwithspinσ,andϒ0=tisthein-planehoppingenergyandϒ1=t’~0.4eVisthehoppingenergybetweenatomA1andA2,ϒ4~0.04eVisthehoppingenergybetweenatomA1andB2,andϒ3~0.3eVisthehoppingenergybetweenB1andB2.

TheHamiltoniancanberegardedas

And

Ignoringcomplicatedcomputing,wetalkaboutthephysicsoftheHamiltonianandthebandstructurequalitatively.

Firstly,weignoringϒ4forthetimebeing,thephysicsisthatthehoppingenergyϒ4leadstoak-dependentcouplingbetweenthesublattices,andthesameroleisplayedbytheinequivalencebetweensublatticeswithinalayer.Then

Wherek=kx+ikyisacomplexnumber,andVistheshiftinelectrochemicalpotentialbetweentwolayers.

IfV=0andϒ3,vFkϒ1,wecanwritetheeffectiveHamiltonianas

Andifϒ3=0,wecangettwoparabolicbands,E（k）~vF2k2/t’,andthespectrumissymmetric.Bythisapproximation,wecangetthemetallicbandofbilayergraphene,andmaycometotheconclusionthatϒ3affectsthespectrumatlowenergy.Ithinkthatthephysicsofthisshouldbecausethatϒ3willdescribeatrigonaldistortionofthebands,anditcanalso