Exchange interaction and stability diagram of coupled quantu(4)

The charge stability diagram for two coupled quantum dots containing up to two electrons is computed in magnetic fields. One- and two-particle Schroedinger equations are solved by exact diagonalization to obtain the chemical potentials and exchange energy

chemicalpotentialswithslightlylargerslopesthanshownwasfoundforvariousmagnetic eldsaswell.10

SWe rstanalyzethedependencesofthetotalenergiesEG(1)andEG(2)foroneandtwo

electronsonVLandVRshowninFig.2.Weseethatintheone-electroncase(leftcolumn),thecurvatureofEG(1)intheregionwhereVLandVRareneareachother(VL～VR)islargerford=60nmthanford=50nmbecauseoftheweakercouplingbetweentheQDsintheformercase.Wealsonotethatforbothvaluesoftheinter-dotdistance,thecurvaturesoftheenergycontourplotsincreasealongthemaindiagonalsincethecouplingbetweenthedotsdecreasesasVL=VRgetslarger.

However,whenthetwoelectronspopulatetheQDsystem,thesituationbecomesradicallydi erent:intheweakcouplingcase(d=60nm,bottomright),thetotalenergyofthetwoelectronsystemintheVL～VRregionisalmostlinearlydependentonVL(VR),i.e.,thecurvatureisvanishinglysmall,whileford=50nm(topright)theenergycurvesclearlyexhibitanon-linearbehaviorwithnon-zerocurvatures.Thelargeoverlapbetweentheelectronsinthestronginter-dotcouplingcase(d=50nm)isresponsibleforthesmoothnon-lineardependenceoftheenergyonVL(VR).However,intheweak-couplingcase(d=60nm),thetwoelectronsarewelllocalizedintheindividualQDsbyCoulombrepulsionandthelargebarrierbetweenthedots,sothatthepotentialchangeinoneQDcausedbythevariationofVL(VR)doesnota ecttheelectronchargedistributionbutonlyactsasaconstantadditiontothetotalenergy.ThisleadstoalineardependenceofthetotalenergyonVL(VR).12Whenthedi erencebetweenVLandVRbecomessu cientlylargetoovercometheCoulombrepulsion,thetwoelectronsmoveintooneQD.ThisisaccompaniedbyachangeintheslopeoftheenergycurveswhichbecomeeitherhorizontalorverticalasVL(orVR)nolongera ectsthetotalenergyandwhichcorrespondstothe(0,2)/(2,0)regionsonthestabilitydiagram(notshown).6Weemphasizethattheobservedquasi-linearbehaviorofthetotalenergyEG(2)whenVL～VRintheweakcouplingregime(d=60nm)isphysicallydi erentfromthesituationintwocoalesceddotswherebothEG(1)andEG(2)arealsostraightlinesperpendiculartothemaindiagonalintheVL VRplane.6ThisisbecauseinthatcaseonedealswithasingleQDandchangingVL(VR)modi esthetotalenergyofthesystem.

Fig.3displaysthecontourplotsofµ(1)(lowerbranches)andµS(2)(upperbranches)asfunctionsofVLandVRatzeromagnetic eld.Wechooseconstantvaluesofµ(1)=