Exchange interaction and stability diagram of coupled quantu(7)

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

elddependencesoftheDTPseparationandexchangeenergyintheabsenceoftheZeemane ectinFig.5showsthatthelattersaturatesatmuchlowervaluesofthemagnetic eldthantheformer.ThisisbecausetheDTPseparationisdeterminedbytheCoulombinteractionbetweenelectronswhichdecreasesastheelectronsbecomelocalizedbythemagnetic eldinindividualdots(withintheHeitler-Londonapproximation,thisdecreaseisproportionaltoB 2,Ref.13),whiletheexchangeenergyinabsenceoftheZeemane ectapproacheszeromuchfasterthantheCoulombinteractionsinceitisproportionaltotheoverlapbetweentheindividualelectronwavefunctionsthatdecaysexponentiallyfastinstrongmagnetic elds.2,13

Itisalsointerestingtocomparetheexactvaluesoftheexchangeenergy(seetheinsetsinFig.5)withthoseextractedfromthestabilitydiagramsinmagnetic eldsusingtheHubbardmodel.2,8Accordingtothismodel,Jest=4t2/(Vintra Vinter)where2tisthetunnel(symmetric-asymetric)splitting,VintraandVinteraretheintra-dotandinter-dotCoulombinteractions.FromthedatashowninFig.5,weestimatethevalueoftheinter-dotCoulombinteractionVinter≈3.4(2.0)meVford=50(60)nm,whichisgivenbytheDTPseparation(forthelowesttripletstate)inthelimitoflargemagnetic elds.Thesenumbersareingoodagreementwiththecorrespondingexpectationvalues C(r1,r2) oftheCoulombinteractionmatrix(3.5and2.2meV,respectively)obtainedfromdirectcalculations,therebycon rmingelectronlocalizationandQDsdecoupling.Sinceatzeromagnetic eld,theDTPseparationisequalto2t+Vinter,weobtain2t50(60)≈1.6(0.7)meVwhichisconsistentwiththeenergydi erencesbetweenthetwolowestsingle-particlelevelsof1.9(0.4)meV.AsVintra≈8meVisgivenbytheelectronadditionenergyinoneQDwhichisthedistancebetweenthe”corners”ofthelinearregionwheresingleelectronre-localizationoccursfromonedottotheotherintheN=2energydiagram(seeFig.2),theestimatedvaluesoftheexchangeenergybecomeJest50(60)50(60)≈0.6(0.08)meV.Thesenumbersareofthesameorderasthenumericallyexactvaluesof0.24(0.012)meV,buttheybothsigni cantlyoverestimatethecomputeddata,andtherefore,canonlybeusedasageneralguidelinetogaugethemagnitudeoftheexchangecouplingindoubleQDs.Theoverestimationisduetothedi erencebetweenCoulombenergiesinthesingletandtripletstatesthatlowerstheexchangeenergy,2butwhichisnottakenintoaccountinthesimpleHubbardmodel.

Insummary,wecomputedthestabilitydiagramformodeldoubleQDsystemspopulatedwithuptotwoelectronsinmagnetic eldsusingnumericallyexactdiagonalizationoftheone-