Coulomb effects on the transport properties of qua

arXiv:cond-mat/0004110v1[cond-mat.mes-hall]7Apr2000CoulombeffectsonthetransportpropertiesofquantumdotsinstrongmagneticfieldV.Moldoveanu,A.Aldea,A.Manolescu,M.Nit˜aNationalInstituteofMaterialsPhysics,POBoxMG7,Bucharest-Magurele,RomaniaAbstractWeinvestigatethetransportpropertiesofquantumdotsplacedinstrongmagneticfieldusingaquantum-mechanicalapproachbasedonthe2Dtight-bindingHamiltonianwithdirectCoulombinteractionandtheLandauer-B¨uttiker(LB)formalism.TheelectronictransmittanceandtheHallresis-tanceshowCoulomboscillationsandalsoprovemultipleadditionprocesses.Weidentifythisfeatureasthe’bunching’ofelectronsobservedinrecentexper-imentsandgiveanelementaryexplanationintermsofspectralcharacteristicsofthedot.Thespatialdistributionoftheaddedelectronsmaydistinguishbe-tweenedgeandbulkstatesandithasspecificfeaturesforbunchedelectrons.Thedependenceofthechargingenergyonthenumberofelectronsisdis-cussedbothforstrongandvanishingmagneticfield.ThecrossoverfromthetunnelingtoquantumHallregimeisanalyzedintermsofdot-leadcoupling.PACS:73.23Hk,73.50-h,73.40HmTypesetusingREVTEX1I.INTRODUCTIONThebasicphenomenainquantumdots(QD)areusuallydescribedbythe’orthodoxtheory’[1,2]whichexplainstheCoulombblockadeeffects(chargequantizationandtheoscillationsoftheelectricalresistance)intermsofthecapacitivepropertiesoftheisolated(orveryweaklycoupled)dot.Generallyspeaking,thecorrespondingcapacitanceshoulddependonthedimensionandthedotshape,numberofelectronsaccommodatedinsideandtheelectron-electroninteraction(EEI).Nevertheless,theorthodoxtheoryconsidersthedotcapacitanceasbeingaconstant,independentofthenumberofelectronsandthesizequantizationeffects.ThisideacanbeacceptedforlargemetallicQDswhenthechargingenergyduetotheCoulombinteractionislargerthanthelevelspacingoftheone-electronenergyspectrum.HowevermanyspecificpropertiesofQDsremainbeyondthismodel.Suchaspectsoccurespeciallyforsmallsemiconductorquantumdots,whenthelevelspacingisrelativelylarge,thedistributionoftheenergylevelsdependsvisiblyonthedotshapeandthenumberofelectronsissmallertheninthemetalliccase.Insmalldotstheinterplaybetweenthequantumaspectsandthechargingeffectsisimportant.Thisiswhyamoreadvanceddescriptionpretendstopayattentiontotheone-particleenergyspectrumandtoconsideramorerealisticHamiltonian.InspiteofthegeneralacceptancethatthechargingenergyshoulddependonthenumberofelectronsinsidedotN,factwhichisprovedexperimentallybytheirregularCoulombblockadeoscillations[3],thiseffecthasbeensimulatednumericallyonlyrecently[4,5].Thesituationbecomesevenmorecomplicatedwhenthecouplingtotheleadsistakenintoaccountcarefully,eventuallyinthepresenceofamagneticfield.Infact,thiscouplingbetweendotandleads(orusingotherwords:thedegreeofpinching/constrinctionatthecontacts)decidesthedegreeofquantizationofthechargeinQDandaffectstheelectronictransmittancethroughthedotandtheHallresistance.Generally,intheliterature,thelead-dot(LD)couplingistakenintoonsiderationassuminganelectronictransitionrateΓbetweenleadsanddot.Thisphenomenologicalwayofthinkingisreplacedherebytheexplicitcalculationofthetransmittancepropertiesinthequantum-mechanicalLandauer-B¨uttikerformalism.ThisformalismworksaslongastheEEIispresentinthedotbutisneglectedintheleads[6].InthisframeworkweshowthattheLDcouplingplaysanevenmorepregnantroleinthepresenceofastrongmagneticfield,inwhichcaseaninterestingcrossoverfromCoulomboscillationstotheQuantumHallregimecanbenoticedforthetransverseresistancewithincreasingcoupling.Thestrongmagneticfieldperpendicularonthetwo-dimensionaldotgivesrisetoedge-statesevenifthedotissmall.Then,forlargeLDcoupling,QHE-typeeffectsappear,although,usually,onlythefirstplateauisvisibleinsmalldots(andnotalwaysveryclean).Suchaspectswereevidentiatedforthefirsttimein[7]bytheuseofatight-binding(TB)modelintheabsenceoftheelectron-electroninteraction.Thephilosophyofourapproachforcalculatingthetransportpropertiesofquantumdotsisthefollowing:theQDiscoupledweaklytofoursemi-infiniteleadssupportingmanychannels.ThenumberofdegreeoffreedomoftheterminalsisinfinitewhiletheQDhasonlyafinitenumberofdegreeoffreedom,sothattheFermilevelEFofthewholesystemisimposedbyterminals.Atagivenmagneticfield,whenagatepotentialVgisapplied2andvaried,thefixedEFscansthewholeenergyspectrum.Withoutmagneticfield,aspectsconcerningCoulombeffectsandtheimportanceofsizequantizationhavebeendiscussedin[5]intheHartreeapproximation:thecalculatedadditionspectrumshowsthedependenceofthechargingenergyonthenumberofparticles,theelectronictransmittancepeaks—whichcorrespondtothecharge-degeneracypoints—aredistributedirregularlykeepingtrackofthesizequantization.ThewidthofthetransmittancepeaksdependsnotonlyontheLDcouplingbutalsoonthestrengthoftheEEI,showingthecotunnelingeffectatresonances.Theaimofthispaperistodiscussthetransportpropertiesoftheopendotsintheregimeofastrongmagneticfield.Wecalculatetheadditionspectrum,thetransmittancematrixandtheHallresistance.Weshowthat,whenthecontactsarepinched,thequantumplateausdisappearandquantumoscillationsofRHareinstalledevenintheabsenceofEEI.NextweshowthecharacteristicsoftheRHoscillationsinthepresenceofthelong-