Exchange interaction and stability diagram of coupled quantu(2)

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

Coupledquantumdots(QDs)areofparticularimportanceforspin-basedquantumcom-putationbecauseuniversalquantumlogicalgates(suchasaControl-NOTgate)canberealizedviatheinteractionbetweentwoquantumbits(qubits),i.e.,thespinsoftwoelec-trons,eachtrappedinonequantumdot.1Insuchdevices,theinteractionbetweenthetwospinsisproportionaltotheexchangeenergyJ,whichisequivalenttothesplittingbetweenthelowestsingletandtriplettwo-electronstates.

WhileextensivetheoreticalworkfocusesonthedependenceofJonthesystempara-menterssuchastheinter-dotseparation,thetunnelingbarrierbetweentheQDs,andtheexternalmagnetic eld,2,3,4,5thechargestabilitydiagramofcoupledQDs6hasbeenstudiedtoalesserextent.Meanwhile,recentadvancesinexperimentaltechniqueshavemadeitpossibletostudycoupledQDsinthefew-electronregimewheneachQDcontainsonlyoneconductionelectron(see,e.g.,Refs.[6,7,8]).Inthiscasethestabilitydiagrambecomesapowerfultooltostudyinter-dotcouplingandelectronictransportthroughdoubleQDsys-tems.Analysisofthestabilitydiagramanditsevolutioninmagnetic eldsallowsonetoestimatethevaluesoftheexchangeenergyaswasdemonstratedrecentlyinthecaseofthetwolaterallycoupledverticalQDs.8

Ingeneral,inthestabilitydiagramtheboundariesbetweendistinctstablechargestates,i.e.,betweenthestateswith xednumberofelectronsN1andN2ineachofthecoupleddots,arerepresentedasfunctionsofthetwocontrollinggatebiases,oneforeachdot.6TheseequilibriumchargesaredeterminedfromtheconditionthatthechemicalpotentialoftheQDstructureµ(N1+N2)de nedas:6

µ(N1+N2)=EG(N1+N2) EG(N1+N2 1),(1)

whereEG(N)isthegroundstateenergyoftheN-electronstate,islessthanthatoftheleads(sourceanddrain).

Inthispaper,wenumericallycomputethestabilitydiagramincoupledQDswithN1+N2≤2electronsinexternalmagnetic elds,andinvestigateitspropertiesfordi erentinter-dotcouplingstrengths.TheHamiltonianforthecoupledsystemisgivenby

H(r1,r2)=Horb+HZ,(2)

Horb=h(r1)+h(r2)+C(r1,r2)

(3)