Complex-Structure-of-Triangular-Graphene--Electron

arXiv:1101.3612v1[cond-mat.mes-hall]19Jan2011ComplexStructureofTriangularGraphene:Electronic,MagneticandElectromechanicalPropertiesMotohikoEzawaDepartmentofAppliedPhysics,UniversityofTokyo,Hongo7-3-1,113-8656,JapanWehaveinvestigatedelectronicandmagneticpropertiesofgraphenenanodisks(nanosizetriangulargraphene)aswellaselectromechanicalpropertiesofgraphenenanojunctions.Nanodisksarenanomagnetsmadeofgraphene,whicharerobustagainstperturbationsuchasimpuritiesandlatticedefects,wheretheferro-magneticorderisassuredbyLieb’stheorem.Wecangenerateaspincurrentbyspinfilter,andmanipulateitbyaspinvalve,aspinswitchandotherspintronicdevicesmadeofgraphenenanodisks.Wehaveanalyzednan-odiskarrays,whichhavemulti-degenerateperfectflatbandsandareferromagnet.Byconnectingtwotriangulargraphenecorners,weproposeananomechanicalswitchandarotator,whichcandetectatinyanglerotationbymeasuringcurrentsbetweenthetwocorners.BymakinguseofthestraininducedPeierlstransitionofzigzagnanoribbons,wealsoproposeananomechanicalstretchsensor,inwhichtheconductancecanbeswitchoffbyananometerscalestretching.Keyword:graphene,nanoribbons,nanodisks,graphene-nanodiskarray,quasiferromagnet,spintronics,nanomechanics,electromechanicsI.INTRODUCTIONGraphene,whichisaone-layerthickhoneycombstructureofcarbon,isanamazingmaterial1.Electronsexhibithighmo-bilityandtravelmicrondistanceswithoutscatteringatroomtemperature.Itisaverythinandstrongmaterial,showingveryhighthermalconductivity.Grapheneisnowamaintopicofnanoscience.Muchattentionhasbeenfocusedongraphenenanorib-bons,whichisaone-dimensionalribbon-likederivativesofgraphene.Theyhavevariousbandstructuredependingontheedgeandwidth.Inparticular,zigzaggaphenenanoribbonsshowedgeferromagnetismduetoalmostflatlow-energybandattheFermilevel.Thereareaprofusionofpapersonthem,amongwhichwecitesomeofearlyworks2–4.Anotherba-sicelementofgraphenederivativesisagraphenenanodisk5.Itisananometer-scaledisk-likematerialwhichhasaclosededge.Itmaybeconsideredasagiantmoleculemadeofaromaticcompound.ItispossibletomanufacturethembyetchingagraphenesheetbyNinanoparticles6.Amongthem,trigonalzigzagnanodiskshaveanovelelectricpropertythatthereexisthalf-filledzero-energystatesinthenon-interactingregime,aswasrevealedfirstbythetight-bindingmodel5andthenbyfirst-principlecalculations7–9.Variousremark-ablepropertiesofnanodiskshavebeeninvestigatedexten-sivelyinaseriesofworks5,10–12.Nanodiskisalsoreferredtoasnanoisland7,nanoflake9,13,14,nanofragment15orgraphenequantumdot16,17.Nanoribbonsandnanodiskscorrespondtoquantumwiresandquantumdots,respectively.Theyarecandidatesoffu-turecarbon-basednanoelectronicsandspintronicsalternativetosilicondevices.Ananoribbon-nanodiskcomplexcaninprinciplebefabricated,embodyingvariousfunctions,onlybyetchingagraphenesheet.Furthermore,grapheneiscom-monmaterialandecological.Inthispaper,exploringelec-tronic,magneticandelectromechanicalpropertiesoftrigonalzigzaggraphenenanodisks[seeFig.1],weproposesomeap-plicationofnanodisk-nanoribboncomplextonanoelectronics,spintronicsandelectromechanicsdevicesFIG.1.(a)Geometricconfigurationofatrigonalzigzagnanodisk.ThenanodisksizeisdefinedbyN=Nben−1withNbenthenumberofbenzenesononesideofthetrigon.Here,N=5.TheA(B)sitesonthelatticeareindicatedbyreddots(whilecircles).Theelectrondensityisfoundtobelocalizedalongtheedges.(b)Thenanodisk-leadsystem.Ananodiskisconnectedtotherightandleftleadsbytunnelingcoupling.Itmayactasaspinfilter.II.ENERGYSPECTRUMWecalculatetheenergyspectrumofthenanodiskbasedonthenearest-neighbortight-bindingmodel,whichhasprovedtodescribeaccuratelytheelectronicstructureofgraphene,car-bonnanotubes,graphenenanoribbonsandothersp2carbonmaterials.TheHamiltonianisdefinedbyH0=Xiεic†ici+Xhi,jitijc†icj,(1)whereεiisthesiteenergy,tijisthetransferenergy,andc†iisthecreationoperatoroftheπelectronatthesitei.Thesummationistakenoverallnearestneighboringsiteshi,ji.Owingtotheirhomogeneousgeometricalconfiguration,wemaytakeconstantvaluesfortheseenergies,εi=εFandtij=t≈2.70eV.Thereexistsoneelectronperonecarbon,andtheband-fillingfactoris1/2.Then,thediagonaltermyieldsjustaconstant,εFNC,andcanbeneglectedintheHamiltonian,whereNCisthenumberofcarbonatoms.WedefinethesizeNofananodiskbyN=Nben−1,whereNbenisthenumberofbenzenesononesideofthetrigonasinFig.1(a).Itcanbeshown5thatthedeterminantassociatedwith2theHamiltonian(1)hassuchafactorasdet[εI−H(NC)]∝εN,(2)implyingN-folddegeneracyofthezero-energystates.ThegapenergyisaslargeasafeweVfornanodiskswithsmallN,whereitisagoodapproximationtoinvestigatetheelectron-electroninteractionphysicsonlyinthezero-energysector,byprojectingthesystemtothesubspacemadeofthosezero-energystates.AsweshallseeinSectionV,theapproximationremainstobegoodevenforthosewithlargeN.III.TRIGONALSYMMETRYThesymmetrygroupofatrigonalnanodiskisC3v,whichisgeneratedbythe2π/3rotationc3andthemirrorreflectionσv.Ithastherepresentation{A1,A2,E}.TheA1representationisinvariantundertherotationc3andthemirrorreflectionσv.TheA2representationisinvariantunderc3andantisymmetricunderσv.TheErepresentationacquires±2π/3phaseshiftunderthe2π/3rotation.TheA1andA2are1-dimensi