Berry-phase

TopologicalAspectsoftheSpinHallEffectYong-ShiWuDept.ofPhysics,UniversityofUtahCollaborators:Xiao-LiangQiandShou-ChengZhang(XXIIIInternationalConferenceonDifferentialGeometricMethodsinTheoreticalPhysicsNankaiInstituteofMathematics;August21,2005)Motivations•Electronscarrybothchargeandspin•ChargetransporthasbeenexploitedinElectricandElectronicEngineering:Numerousapplicationsinmoderntechnology•SpinTransportofElectronsTheory:Spin-orbitcouplingandspintransportExperiment:InduceandmanipulatespincurrentsSpintronicsandQuantumInformationprocessing•IntrinsicSpinHallEffect:Impurity-IndependentDissipation-lessCurrentkijksHijEJ•Keyadvantages:•Electricfieldmanipulation,ratherthanmagneticfield•Dissipation-lessresponse,sincebothspincurrentandelectricfieldareevenundertimereversal•IntrinsicSHEoftopologicalorigin,duetoBerry’sphaseinmomentumspace,similartotheQHE•VerydifferentfromOhmiccurrent:lkhewhereEJFcjcj22Electricfieldinducestransversespincurrentduetospin-orbitcouplingTheSpinHallEffectp-GaAsExyzFamilyofHallEffects•ClassicalHallEffectLorentzforcedeflectinglike-chargecarriers•QuantumHallEffectLorentzforcedeflectinglike-chargecarriers(Quantumregime:Landaulevels)•Anomalous(Charge)HallEffectSpin-orbitcouplingdeflectinglike-spincarriers(measuringmagnetizationinferromagneticmaterials)•SpinHallEffectSpin-orbitcouplingdeflectinglike-spincarriers(inducingandmanipulatingdissipation-lessspincurrentswithoutmagneticfieldsorferromagneticelements)TimeReversalSymmetryandDissipativeTransport•MicroscopiclawsinsolidstatephysicsareTinvariant•MostknowntransportprocessesbreakTinvarianceduetodissipativecouplingtotheenvironment•DampedharmonicoscillatorlkhewhereEJFjj22(onlystatesclosetotheFermienergycontribute!)kxxxm•Ohmicconductivityisdissipative:underT,electricfieldisevenchargecurrentisodd•ChargesupercurrentandHallcurrentarenon-dissipative:tAcEAJjjjSj1,BEJHH/1,underTvectorpotentialisodd,whilemagneticfieldisoddSpin-OrbitCoupling•Origin:``Relativistic’’effectinatomic,crystal,impurityorgateelectricfield=Momentum-dependentmagneticfieldStrengthtunableincertainsituations•TheoreticalIssues:ConsequencesofSOCinvarioussituations?InterplaybetweenSOCandotherinteractions?•Practicalchallenge:ExploitSOCtogenerate,manipulateandtransportspinsTheExtrinsicSpinHalleffect(duetoimpurityscatteringwithspin-orbitcoupling)D’yakonovandPerel’(1971)Hirsch(1999),Zhang(2000)•TheIntrinsicSpinHallEffectBerryphaseinmomentumspaceIndependentofimpuritiesimpurityscattering=spindependent(skew)Mottscatteringplusside-jumpeffectSpin-orbitcoupingup-spindown-spinimpurityCf.MottscatteringBerryPhase(VectorPotential)inMomentumSpacefromBandStructurexdkuuiknkknikAdiknkninicellunit*)((:periodicpartoftheBlochwf.)knuxkiknknexux)()()()(kAkBnkn:Magneticfieldinmomentumspace:BandindexnWave-PacketTrajectoryinRealSpace)(,)(kBEemkxEekAnomalousvelocity(perpendiculartoand)k//zE//HolespinS00ESpincurrent(spin//x,velocity//y)ChangandNiu(1995);P.Horvarthetal.(2000)IntrinsicHallconductivity(KuboFormula)Thouless,Kohmoto,Nightingale,denNijs(1982)Kohmoto(1985)knnznFxykBkEnhe,2)()()()(kAkBnkn:fieldstrength;:bandindexn(DegeneracypointMagneticmonopole)FieldTheoryApproach•Electronpropagatorinmomentumspace•Ishikawa’sformula(1986):•HallConductanceintermsofmomentumspacetopology),p(with)(0pppSF(p)]S(p)S(p)Str[pεdheπiσFλFνFμμνλxy11132224p-GaAsExyzCf.Ohm’slaw:Ej:oddundertimereversal=dissipativeresponse:evenundertimereversal=reactiveresponse(dissipationless)i:spindirectionj:currentdirectionk:electricfieldkijksijEjs•Nonzeroinnonmagneticmaterials.IntrinsicspinHalleffectinp-typesemiconductorsInp-typesemiconductors(Si,Ge,GaAs,…),spincurrentisinducedbytheexternalelectricfield.(Murakami,Nagaosa,Zhang,Science(2003))ValencebandofGaAsLuttingerHamiltonian(:spin-3/2matrix,describingtheP3/2band)S2222122521SkkmH2/300002/100002/100002/302/3002/30100102/3002/3002/3002/300002/3002/30zyxSSiiiiiiSSPSP3/2P1/2LuttingermodelExpressedintermsoftheDiracGammamatrices:SpinHallCurrent(GeneralizingTKNN)ijkLFHFkkijHLijkkijkjikkekGknknVEj26)()]()([4xJzyJ•Oftopologicalorigin(Berryphaseinmomentumspace)•Dissipation-less•AlloccupiedstatecontributeSpinAnalogoftheQuantumHallEffectAtRoomTemperature(Sinova,Culcer,Niu,Sinitsyn,Jungwirth,MacDonald,PRL(2003))8esRashbaHamiltonianmkikkikkmkkmkHxyxyz2)()(22222IntrinsicspinHalleffectfor2Dn-typesemiconductorsinheterostructureKuboformula:zSyxJJindependentofzySySJJz,212DheterostructurexyzEffectivemagneticfieldSHE:Spinprecessionby“k-dependentZeemanfield”Note:isnotsmallevenwhenthespinsplittingissmall.duetoaninterbandeffectS)ˆ(intkzBSpinHallinsulator•Motivation:TrulydissipationlesstransportGapfulbandinsulator(togetridofOhmiccurrents)•NonzerospinHalleffectinbandinsulators:-Murakami,Nagaosa,Zhang,PRL(2004)•TopologicalquantizationofspinHallconductance:-Qi,YSW,Zhang,cond-mat/05