Spin properties of single electron states in coupl

arXiv:cond-mat/0506610v1[cond-mat.other]23Jun2005SpinpropertiesofsingleelectronstatesincoupledquantumdotsPeterStano1,2andJaroslavFabian1,21InstituteofPhysics,Karl-FranzensUniversity,Universit¨atsplatz5,8010Graz,Austria2InstituteforTheoreticalPhysics,UniversityofRegensburg,93040Regensburg,GermanySpinpropertiesofsingleelectronstatesinlaterallycoupledquantumdotsinthepresenceofaperpendicularmagneticfieldarestudiedbyexactnumericaldiagonalization.Dresselhaus(linearandcubic)andBychkov-Rashbaspin-orbitcouplingsareincludedinarealisticmodelofconfineddotsbasedonGaAs.Grouptheoreticalclassificationofquantumstateswithandwithoutspinorbitcouplingisprovided.Spin-orbiteffectsontheg-factorareratherweak.Itisshownthatthefrequencyofcoherentoscillations(tunnelingamplitude)incoupleddotsislargelyunaffectedbyspin-orbiteffectsduetosymmetryrequirements.TheleadingcontributionstothefrequencyinvolvesthecubictermoftheDresselhauscoupling.Spin-orbitcouplinginthepresenceofmagneticfieldleadstoaspin-dependenttunnelingamplitude,andthustothepossibilityofspintochargeconversion,namelyspatialseparationofspinbycoherentoscillationsinauniformmagneticfield.ItisalsoshownthatspinhotspotsexistincoupledGaAsdotsalreadyatmoderatemagneticfields,andthatspinhotspotsatzeromagneticfieldareduetothecubicDresselhaustermonly.PACSnumbers:71.70.Ej,73.21.LaI.INTRODUCTIONThepossibilityoftuningspin-orbitcoupling1,2,3inlow-dimensionalsemiconductorelectronicstructureshasstirredgreatinterestinspinpropertiesoflateralsemicon-ductorelectronsystemsinthepresenceofDresselhaus4andBychkov-Rashba5,6spin-orbitcouplings.Thefor-merappearsinlow-dimensionalsystemslackinginver-sionsymmetryinthebulk(suchaszinc-blendesemicon-ductors),thelatterinlow-dimensionalstructureswithasymmetricconfiningpotentials.Theprincipalquestioniswhatspinandchargepropertiesandtowhatdegreecanbeaffectedandmanipulatedbythistuning.Suchquestionsareoffundamentalimportanceforspintronics.7Electronspinsincoupledquantumdotsystemshavebeenproposedtoperformuniversalgatingofquantumcomputers.8Thespinactsasaqubitandexchangecouplingprovidesthephysicalrealizationoftwo-qubitgates.9,10Anotherapplicationofacontrolledcouplingbetweendotsisspinentanglementdistillationinwhichsingletandtripletstatesgetspatiallyseparatedduringadiabaticpassagethroughtrappedstates.11Understand-ingofspin-orbitpropertiesofcoupleddotsisthusofgreatinteresttoquantuminformationprocessing.Spin-orbitcouplingprovidesawayfororbitalde-greesoffreedomtoinfluencespinstates.Asare-sultthespindynamicsisaffected,makingspinqubitoperationsmorecomplex(itwasshown,though,thattwoqubitoperationscanbeperformedreliablyeveninthepresenceofspin-orbitinteractionwhichleadstoanisotropicexchange12,13).Furthermore,spin-orbitcouplingleadstospindecoherenceandrelaxationduetophonons,14,15,16,17,18,19limitingtheoperationtime.Theimpressiveexperimentalprogressincoherentos-cillationsincoupleddotsystems20,21,22,23,aswellinspindephasingandspinmanipulationinsingle24,25anddoubledots26,providesadditionalstrongimpetusforinvestigatingspinstatesindoubledots.Spin-orbitpropertiesofsingledotshavebeenalreadyextensivelyinvestigated.27,28,29,30,31,32,33,34,35,36,37,38Inthispaperweinvestigatetheroleofspin-orbitcou-pling,representedbytheDresselhaus(bothlinearandcu-bic)andBychkov-Rashbaterms,inspinandchargeprop-ertiesoftwolaterallycoupledquantumdotsbasedonGaAsmaterialsparameters.Weperformnumericallyex-actcalculationsoftheenergyspectrumusingthemethodoffinitedifferences.Wefirststudythegeneralstructureoftheenergyspectrumandthespincharacterofthestatesofthedoubledotsystem.Weconstructthegrouptheoreticalcorrelationdiagramforthesingleanddoubledotstatesandindicatethepossibletransitionsduetospin-orbitcoupling.ThisgrouptheoreticalclassificationisusedincombinationwithL¨owdinperturbationtheorytoexplainanalyticallyournumericalresults.Inparticu-lar,weshowthatwhileallowedbysymmetry,thespecificformsofthelinearspin-orbitinteractionsdonotleadtospinhotspotsintheabsenceofmagneticfield(spinhotspotsarenominallydegeneratestatesliftedbyspin-orbitcoupling39).OnlythecubicDresselhaustermgivesspinhotspots.Ifidentifiedexperimentally,thestrengthofthecubictermcanbedetected.Wenextfocusontwoimportantmeasurableparam-eters:electronicg-factorandtunnelingamplitude.Insingledotsthevariationoftheeffectiveg-factorwiththestrengthofthespin-orbitinteractionhasbeeninvesti-gatedearlier.29Theeffectisnotlarge,amountingtoafractionofapercent.Similarbehaviorisfoundfordou-bledots.InourcaseofGaAsthecontributiontotheg-factorfromspin-orbitcouplingistypicallyabout1%,duetothelinearDresselhausterm.Moreexcitingistheprospectofinfluencingcoherenttunnelingoscillationsbetweenthedotsbymodulatingthespin-orbitcouplingstrength.Twoeffectscanap-pear:(i)thetunnelingamplitudeorfrequencycanbemodulatedbyspin-orbitcouplingand,(ii)thetunnel-ingamplitudecanbespindependent.Weshowhowa2naiveapplicationofperturbationtheoryleadstoamis-leadingresultthat(i)ispresentinthesecondorderinlinearspin-orbitcouplingstrengths,givingrisetoanef-fectivetunnelingHamiltonianinvolvingspin-fliptunnel-ingatzeromagneticfield.Bothnumericalcalcula