Exchange interaction and stability diagram of coupled quantum dots in magnetic fields

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

ExchangeinteractionandstabilitydiagramofcoupledquantumarXiv:cond-mat/0607571v1 [cond-mat.mes-hall] 22 Jul 2006dotsinmagnetic eldsL.-X.Zhang,D.V.Melnikov,andJ.-P.LeburtonBeckmanInstituteforAdvancedScience&TechnologyandDepartmentofElectricalandComputerEngineering,UniversityofIllinoisatUrbana-Champaign,Urbana,Illinois61801(Dated:February6,2008)AbstractThechargestabilitydiagramfortwocoupledquantumdotscontaininguptotwoelectronsiscomputedinmagnetic elds.One-andtwo-particleSchr¨odingerequationsaresolvedbyexactdiagonalizationtoobtainthechemicalpotentialsandexchangeenergyinthesesystems.Byan-alyzingthechemicalpotentialsvariationwithexternalbiasesandmagnetic elds,itispossibletodistinguishbetweentheweakandstronginter-dotcouplings.Thevariationofthechemicalpotentialcurvaturesandthedouble-triplepointseparationsinthestabilitydiagramscon rmstheinter-dotcouplingdecreasewithincreasingmagnetic elds.Thecomputedexchangeenergiesarealsofoundtobesigni cantlysmallerthanthevaluesestimatedfromthestabilitydiagram.PACSnumbers:,73.21.-b

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)

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

1h(r)=2

cA)+V(r),(4)

C(r1,r2)=e2/ |r1 r2|(5)

HZ=gµB B·Si(6)

i

Here,m =0.067meistheelectrone ectivemass, =13.1isthedielectricconstant,g= 0.44istheg-factorinGaAs,µBistheBohrmagnetonandA=1

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)=

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

µS(2)= 18meVford=50nmandµ(1)=µS(2)= 16.5meVford=60nmasthereferencevaluesofthechemicalpotentialinthesource/drainoftheQDdevice.11InFig.3wecanrecognizefourregionscorrespondingtofourstablechargestateswithdi erentnumbersofelectronsineachdot[thenumbersintheparenthesesineachregiongivethenumberofelectronsinthe(left,right)QD]separatedbythechemicalpotentialcontoursandthemaindiagonalVL=VR.Attheturningpointoneachbranchalongthemaindiagonal,threestablechargestatescoincideintermsofthetotalenergyofthesystem.Thedistancebetweentheturningpointsistheso-calleddouble-triplepoint(DTP)separation(alsocalledtheanti-crossingseparation).6,8FromFig.3wealsoobservethattheDTPseparation VL= VR=5.00meVinthed=50nmcaseissigni cantlylargerthanthecorrespondingvalue VL= VR=2.93meVinthed=60nmcase.Furthermore,thecurvatureofthebranchesaroundtheDTPissmallerford=50nmthanford=60nm.Accordingtothe“classical”theory,6asmallerDTPseparation(orequivalentlyalargercurvatureofthechemicalpotentialcontourlines)indicatesaweakerinter-dotcouplingwhichisconsistentwithour ndings.

FromthedatainTableI,wenotethatford=50nm,thecurvature(magnitude)κ(2)oftheµ(2)curveissmallerthanthecurvature(magnitude)κ(1)forµ(1),whileinthed=60nmcaseκ(1)<κ(2).Thispeculiarbehaviorcanbeclari edbynotingthatbothκ(1)andκ(2)aredeterminedbythedi erencesbetweenthecorrespondingcurvaturesofthetotalenergywhosebehaviorinthevoltageplaneisdiscussedabove.Thisindicatesthatingeneral,allbeingequal,intheweak-couplingregimethecurvatureofthechemicalpotentialfortwoelectronsislargerthanthatoneforoneelectron,κ(2)>κ(1),whileinthestrongcouplingregime,theoppositerelationshipκ(2)<κ(1)holds.

Inthepresenceofthemagnetic eld,thecurvaturesofthechemicalpotentialcontoursalsoincreaseascanbeseeninFig.4(a)and(b)whereweagainplotthechemicalpotentialcon-toursforµ(1),µS(2)andµT(2)atconstantreferencevaluesofµ(1)=µS(2)=µT(2)= 18( 16.5)meVford=50(60)nmatB=0,3and6T.NotetheorderofthecontoursforµS(2)andµT(2)atdi erentmagnetic elds.Asthemagnetic eldincreases,thecontoursshiftfromthelowerleftcornertotheupperrightcornerbecausethesingle-particleeigenen-ergiesincrease.4Inadditiontothecurvatureincrease,theDTPseparationbecomessmalleratlargermagnetic eldforbothsingletandthelowesttripletstates[forthedetailedexpla-nationofthise ect,seethediscussiononFig.5(a)].Fromthechangesinthecurvatureand

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

DTPseparation,oneconcludesthatthemagnetic eldindeedinducesaquantummechani-caldecouplingbetweenthetwodotsandresultsinmagneticlocalizationofelectronsineachdot.BycomparingthechemicalpotentialcurvesinFig.4(a)withthecorrespondingonesinFig.4(b),weseethatinthelattercasethechemicalpotentialcontourshavemuchlargercurvaturesthanintheformercase(seeTableIfordetails)duetotheincreasedinter-dotdecouplingandforeachvalueofthemagnetic eldtheDTPseparationford=60nmismorethan60%smallerthanford=50nm.

FromTableI,itisalsoshownthatthecurvaturesκ(1)andκ(2)progressivelyincreaseasthemagnetic eldbecomeslarger.Thisisduetoenhancedlocalizationofelectronscausedbythemagnetic eld.Themagneticlocalizationintheweakcouplingcasebecameprevalentatlower eldsthaninthestrongcouplingsituation[seelowerinsetsofFig.5],whichismanifestedbyamorerapidincreaseinthecurvatureofchemicalpotentialcontours.Figures5(a)and(b)showtheextractedDTPseparationalongVL(orVR,VL=VR)axisasafunctionofmagnetic eldsford=50nmand60nminter-dotseparations,respectively.Ineachplotthedataareshownforthesingletandlowesttripletstates.NotethatatB=0theDTPseparationforthesingletstateissmallerthanthatforthelowesttripletstatebecausethesingletisthegroundstate,whileatlargerB elds,thelowesttripletstatebecomesthegroundstateandtheorderoftheDTPseparationsisreversed.Inboth(a)and(b),theDTPseparationforthelowesttripletstatedecreasesfasterwithB eldsthanthatforthesingletstate.ThisisbecausetheDTPseparationisproportionaltoµ(2) µ(1)=EG(2) 2EG(1)fora xedVL=VRonthemaindiagonalofthestabilitydiagram(seeFigs.1and3).Forthesingletstate,EG(2)doesnotchangewiththeB eldwhileEG(1)decreaseswiththeB eldduetotheZeemane ect,thereforetheZeemancontributiontoµ(2) µ(1)increaseswiththeB eld.Forthetripletstate,theZeemancontributionstoEG(2)and2EG(1)cancelout,andµ(2) µ(1)isnota ectedbytheB eld.ThedecreaseoftheDTPseparationinthemagnetic eldwasalsorecentlyobservedexperimentally.8Theupper(lower)insetineach gureshowsthecorrespondingexchangeenergyJasafunctionofthemagnetic eldcalculatedbyEq.(8)with(without)theZeemane ect.Inbothcases,theZeemane ectinducesalineardepenedenceofJonB.However,in(a)giventhestrongcouplingbetweenthedots,theorbitalcontributiontoJdominatesatlowB eldsbeforebeingovercomebytheZeemaninduceddecreaseathigher eld;in(b),JistotallydominatedbytheZeemancontribution,whichdecreaseslinearlywiththeB parisonoftheB

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-

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

andtwo-electronHamiltonian.Twointer-dotseparationsd=50and60nmcorrespondingtostrongandweakinter-dotcouplingwereconsidered.Wefoundthatintheweak-couplingregimethecurvatureofthechemicalpotentialµ(2)islargerthanthatoneofµ(1)whileinthestrong-couplingcasethesituationisreversed.Hence,byanalyzingthechemicalpotentialvariationscausedbyexternalbiasesandmagnetic elds,itispossibletodistinguishbetweenstrongandweakinter-dotcoupling,evenifthecurvaturesareofthesameorderinbothcases.Theevolutionofthestabilitydiagramsinmagnetic eldsconformstothegeneralideaofenhancedelectronlocalizationwithincreasing eldstrength.Theexchangeenergiesextractedfromthestabilitydiagramsshowedthatthesevaluesaresigni cantlyoverestimatedwhencomparedwithnumericallyexactdata.

ThisworkissupportedbyDARPAQUISTprogramthroughAROGrantDAAD19-01-1-0659.TheauthorsthanktheMaterialComputationalCenterattheUniversityofIllinoisthroughNSFGrantDMR99-76550.LXZthankstheComputerScienceandEngineeringFellowshipProgramattheUniversityofIllinoisforsupport.

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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

2180(2005).

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11T.Hatano,M.Stopa,andS.Tarucha,Science309,268(2005).D.V.MelnikovandJ.-P.Leburton,Phys.Rev.B73,085320(2006).L.-X.Zhang,D.V.Melnikov,andJ.-P.Leburton(unpublished).Thevaluesofthechemicalpotentialsatwhichthecontourplotsaredrawnarechosentofacilitatedatarepresentation.Fortheinvestigatedmagnetic elds,therelativechangeofthecurvatureofaspeci cchemicalpotentialandtheDTPseparationislessthan5%fordi erentreferencevalues.

12Thesameconsiderationisobviouslyvalidforthetripletstatesaswell,albeitwithalargerquasilinearregion.

13D.V.MelnikovandJ.-P.Leburton(unpublished).

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

FIGURES

V [meV]x [nm] µ [meV]-10 µ [meV]-15-20

-25

VL=VR [meV]

FIG.1:(Coloronline)(a)Thecon nementpotentialford=50nm(red)andd=60nm(blue)atVL=VR=25meV.Chemicalpotentialsµ(1)(solid)andµS(2)(dashed)vs.VL=VRfor(b)d=50nmand(c)d=60nm.InbothofthemthehorizontallineindicatesthevaluesofthechemicalpotentialatwhichthecontoursinFig.3aredrawn.

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

VL [meV]

VL [meV]L L FIG.2:(Coloronline)Surface(contour)plotsofthetotalenergiesforN=1(leftcolumn)andN=2,singlet(rightcolumn)atd=50nm(toprow)andd=60nm(bottomrow).Inallplotstheenergiesdecreasefromthelowerlefttotheupperrightcorner.

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

29

27

VR [meV]25

23

21VL [meV]FIG.3:(Coloronline)Contourplotsofthechemicalpotentialsµ(1)andµS(2)asfunctionsofVLandVRford=50nm(red)andd=60nm(blue).Theturningpointsonthecontourlinesareindicatedbysoliddotsandthedottedlineisaguidefortheeyesalongthemaindiagonal(VL=VR).Thenumbersontheleft(right)withinparenthesesgivetheelectronnumberintheleft(right)dot[the(0,0)regionislocatedatthelowerleftcornerbelowtheµ(1)branchford=50nm].

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

FIG.4:(Coloronline)(a)Contourplotsofthechemicalpotentialsµ(1)(singleelectronstate,lowerbranches,solidcurves),µS(2)(two-electronsingletstate,upperbranches,solidcurves),andµT(2)(thelowesttwo-electrontripletstate,upperbranches,dashedcurves)atdi erentmagnetic eldsford=50nm.(b)Sameas(a)butatd=60nm.Inthecaseofd=60nmandB=0,thecontourlinesforµS(2)andµT(2)areindistinguishableonthescaleofthe gure.

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

FIG.5:(Coloronline)Double-tripletpointseparationalongVL(orVR,VL=VR)axisasafunctionofthemagnetic eldBfor(a)d=50nmand(b)d=60nm.Thedataforsingletandthelowesttripletstatesarelabeledby” ”(bluecurve)and”+”(redcurve),respectively.Theupper(lower)insetineach gureshowstheexchangeenergyJasafunctionofthemagnetic eldwith(without)theZeemane ect.ThedataintheinsetsareobtainedatVL=VR=25meV.

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

TABLES

TABLEI:Comparisonofthemagnitudeofthesecondorderderivativeκ(1)andκ(2)ofthechemicalpotentialcurvesµ(1)andµ(2)neartheirturningpointsforvariousinter-dotseparationsandmagnetic elds.κS(2)andκT(2)denotethevaluesforthesingletandthelowesttripletstate,respectively.

d=50nm

B(T)κ(1)κS(2)[κT(2)]κ(1)d=60nmκS(2)[κT(2)]

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