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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