Exchange interaction and stability diagram of coupled quantu

1、exchange interaction and stability diagram of coupled quantu 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

2、 exchange energy exchangeinteractionandstabilitydiagramofcoupledquantumarxiv:cond-mat/0607571v1 cond-mat.mes-hall 22 jul 2021dotsinmagnetic eldsl.-x.zhang,d.v.melnikov,andj.-p.leburtonbeckmaninstituteforadvancedsciencetechnologyanddepartmentofelectricalandcomputerengineering,universityofillinoisatur

3、bana-champaign,urbana,illinois61801(dated:february6,2021)abstractthechargestabilitydiagramfortwocoupledquantumdotscontaininguptotwoelectronsiscomputedinmagnetic elds.one-andtwo-particleschrodingerequationsaresolvedbyexactdiagonalizationtoobtainthechemicalpotentialsandexchangeenergyinthesesystems.bya

4、n-alyzingthechemicalpotentialsvariationwithexternalbiasesandmagnetic elds,itispossibletodistinguishbetweentheweakandstronginter-dotcouplings.thevariationofthechemicalpotentialcurvaturesandthedouble-triplepointseparationsinthestabilitydiagramscon rmstheinter-dotcouplingdecreasewithincreasingmagnetic

5、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

6、 are solved by exact diagonalization to obtain the chemical potentials and exchange energy coupledquantumdots(qds)areofparticularimportanceforspin-basedquantumcom-putationbecauseuniversalquantumlogicalgates(suchasacontrol-notgate)canberealizedviatheinteractionbetweentwoquantumbits(qubits),i.e.,thesp

7、insoftwoelec-trons,eachtrappedinonequantumdot.1insuchdevices,theinteractionbetweenthetwospinsisproportionaltotheexchangeenergyj,whichisequivalenttothesplittingbetweenthelowestsingletandtriplettwo-electronstates. whileextensivetheoreticalworkfocusesonthedependenceofjonthesystempara-menterssuchasthein

8、ter-dotseparation,thetunnelingbarrierbetweentheqds,andtheexternalmagnetic eld,2,3,4,5thechargestabilitydiagramofcoupledqds6hasbeenstudiedtoalesserextent.meanwhile,recentadvancesinexperimentaltechniqueshavemadeitpossibletostudycoupledqdsinthefew-electronregimewheneachqdcontainsonlyoneconductionelectr

9、on(see,e.g.,refs.6,7,8).inthiscasethestabilitydiagrambecomesapowerfultooltostudyinter-dotcouplingandelectronictransportthroughdoubleqdsys-tems.analysisofthestabilitydiagramanditsevolutioninmagnetic eldsallowsonetoestimatethevaluesoftheexchangeenergyaswasdemonstratedrecentlyinthecaseofthetwolaterally

10、coupledverticalqds.8 ingeneral,inthestabilitydiagramtheboundariesbetweendistinctstablechargestates,i.e.,betweenthestateswith xednumberofelectronsn1andn2ineachofthecoupleddots,arerepresentedasfunctionsofthetwocontrollinggatebiases,oneforeachdot.6theseequilibriumchargesaredeterminedfromtheconditiontha

11、tthechemicalpotentialoftheqdstructure(n1+n2)de nedas:6 (n1+n2)=eg(n1+n2) eg(n1+n2 1),(1) whereeg(n)isthegroundstateenergyofthen-electronstate,islessthanthatoftheleads(sourceanddrain). inthispaper,wenumericallycomputethestabilitydiagramincoupledqdswithn1+n22electronsinexternalmagnetic elds,andinvesti

12、gateitspropertiesfordi 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 schroedi

13、nger 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=gb bsi(6) i here,m =0.067meistheelectrone ectivemass, =13.1isthedielectricconstant,g= 0.44istheg-factoringaas,bisthebohrmagnetonanda=1 the charge s

14、tability 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 chemicalpotentialswithslightlylargerslopesthanshownwasfo

15、undforvariousmagnetic eldsaswell.10 swe rstanalyzethedependencesofthetotalenergieseg(1)andeg(2)foroneandtwo electronsonvlandvrshowninfig.2.weseethatintheone-electroncase(leftcolumn),thecurvatureofeg(1)intheregionwherevlandvrareneareachother(vlvr)islargerford=60nmthanford=50nmbecauseoftheweakercoupli

16、ngbetweentheqdsintheformercase.wealsonotethatforbothvaluesoftheinter-dotdistance,thecurvaturesoftheenergycontourplotsincreasealongthemaindiagonalsincethecouplingbetweenthedotsdecreasesasvl=vrgetslarger. however,whenthetwoelectronspopulatetheqdsystem,thesituationbecomesradicallydi erent:intheweakcoup

17、lingcase(d=60nm,bottomright),thetotalenergyofthetwoelectronsysteminthevlvrregionisalmostlinearlydependentonvl(vr),i.e.,thecurvatureisvanishinglysmall,whileford=50nm(topright)theenergycurvesclearlyexhibitanon-linearbehaviorwithnon-zerocurvatures.thelargeoverlapbetweentheelectronsinthestronginter-dotc

18、ouplingcase(d=50nm)isresponsibleforthesmoothnon-lineardependenceoftheenergyonvl(vr).however,intheweak-couplingcase(d=60nm),thetwoelectronsarewelllocalizedintheindividualqdsbycoulombrepulsionandthelargebarrierbetweenthedots,sothatthepotentialchangeinoneqdcausedbythevariationofvl(vr)doesnota ecttheele

19、ctronchargedistributionbutonlyactsasaconstantadditiontothetotalenergy.thisleadstoalineardependenceofthetotalenergyonvl(vr).12whenthedi erencebetweenvlandvrbecomessu cientlylargetoovercomethecoulombrepulsion,thetwoelectronsmoveintooneqd.thisisaccompaniedbyachangeintheslopeoftheenergycurveswhichbecome

20、eitherhorizontalorverticalasvl(orvr)nolongera ectsthetotalenergyandwhichcorrespondstothe(0,2)/(2,0)regionsonthestabilitydiagram(notshown).6weemphasizethattheobservedquasi-linearbehaviorofthetotalenergyeg(2)whenvlvrintheweakcouplingregime(d=60nm)isphysicallydi erentfromthesituationintwocoalesceddotsw

21、herebotheg(1)andeg(2)arealsostraightlinesperpendiculartothemaindiagonalinthevl vrplane.6thisisbecauseinthatcaseonedealswithasingleqdandchangingvl(vr)modi esthetotalenergyofthesystem. fig.3displaysthecontourplotsof(1)(lowerbranches)ands(2)(upperbranches)asfunctionsofvlandvratzeromagnetic eld.wechoose

22、constantvaluesof(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 s(2)= 18mevford=50nmand

23、(1)=s(2)= 16.5mevford=60nmasthereferencevaluesofthechemicalpotentialinthesource/drainoftheqddevice.11infig.3wecanrecognizefourregionscorrespondingtofourstablechargestateswithdi erentnumbersofelectronsineachdotthenumbersintheparenthesesineachregiongivethenumberofelectronsinthe(left,right)qdseparatedb

24、ythechemicalpotentialcontoursandthemaindiagonalvl=vr.attheturningpointoneachbranchalongthemaindiagonal,threestablechargestatescoincideintermsofthetotalenergyofthesystem.thedistancebetweentheturningpointsistheso-calleddouble-triplepoint(dtp)separation(alsocalledtheanti-crossingseparation).6,8fromfig.

25、3wealsoobservethatthedtpseparation vl= vr=5.00mevinthed=50nmcaseissigni cantlylargerthanthecorrespondingvalue vl= vr=2.93mevinthed=60nmcase.furthermore,thecurvatureofthebranchesaroundthedtpissmallerford=50nmthanford=60nm.accordingtothe“classical”theory,6asmallerdtpseparation(orequivalentlyalargercur

26、vatureofthechemicalpotentialcontourlines)indicatesaweakerinter-dotcouplingwhichisconsistentwithour ndings. fromthedataintablei,wenotethatford=50nm,thecurvature(magnitude)(2)ofthe(2)curveissmallerthanthecurvature(magnitude)(1)for(1),whileinthed=60nmcase(1)(2).thispeculiarbehaviorcanbeclari edbynoting

27、thatboth(1)and(2)aredeterminedbythedi erencesbetweenthecorrespondingcurvaturesofthetotalenergywhosebehaviorinthevoltageplaneisdiscussedabove.thisindicatesthatingeneral,allbeingequal,intheweak-couplingregimethecurvatureofthechemicalpotentialfortwoelectronsislargerthanthatoneforoneelectron,(2)(1),whil

28、einthestrongcouplingregime,theoppositerelationship(2)(1)holds. inthepresenceofthemagnetic eld,thecurvaturesofthechemicalpotentialcontoursalsoincreaseascanbeseeninfig.4(a)and(b)whereweagainplotthechemicalpotentialcon-toursfor(1),s(2)andt(2)atconstantreferencevaluesof(1)=s(2)=t(2)= 18( 16.5)mevford=50

29、(60)nmatb=0,3and6t.notetheorderofthecontoursfors(2)andt(2)atdi erentmagnetic elds.asthemagnetic eldincreases,thecontoursshiftfromthelowerleftcornertotheupperrightcornerbecausethesingle-particleeigenen-ergiesincrease.4inadditiontothecurvatureincrease,thedtpseparationbecomessmalleratlargermagnetic eld

30、forbothsingletandthelowesttripletstatesforthedetailedexpla-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 equation

31、s are solved by exact diagonalization to obtain the chemical potentials and exchange energy dtpseparation,oneconcludesthatthemagnetic eldindeedinducesaquantummechani-caldecouplingbetweenthetwodotsandresultsinmagneticlocalizationofelectronsineachdot.bycomparingthechemicalpotentialcurvesinfig.4(a)with

32、thecorrespondingonesinfig.4(b),weseethatinthelattercasethechemicalpotentialcontourshavemuchlargercurvaturesthanintheformercase(seetableifordetails)duetotheincreasedinter-dotdecouplingandforeachvalueofthemagnetic eldthedtpseparationford=60nmismorethan60%smallerthanford=50nm. fromtablei,itisalsoshownt

33、hatthecurvatures(1)and(2)progressivelyincreaseasthemagnetic eldbecomeslarger.thisisduetoenhancedlocalizationofelectronscausedbythemagnetic eld.themagneticlocalizationintheweakcouplingcasebecameprevalentatlower eldsthaninthestrongcouplingsituationseelowerinsetsoffig.5,whichismanifestedbyamorerapidinc

34、reaseinthecurvatureofchemicalpotentialcontours.figures5(a)and(b)showtheextracteddtpseparationalongvl(orvr,vl=vr)axisasafunctionofmagnetic eldsford=50nmand60nminter-dotseparations,respectively.ineachplotthedataareshownforthesingletandlowesttripletstates.notethatatb=0thedtpseparationforthesingletstate

35、issmallerthanthatforthelowesttripletstatebecausethesingletisthegroundstate,whileatlargerb elds,thelowesttripletstatebecomesthegroundstateandtheorderofthedtpseparationsisreversed.inboth(a)and(b),thedtpseparationforthelowesttripletstatedecreasesfasterwithb eldsthanthatforthesingletstate.thisisbecauset

36、hedtpseparationisproportionalto(2) (1)=eg(2) 2eg(1)fora xedvl=vronthemaindiagonalofthestabilitydiagram(seefigs.1and3).forthesingletstate,eg(2)doesnotchangewiththeb eldwhileeg(1)decreaseswiththeb eldduetothezeemane ect,thereforethezeemancontributionto(2) (1)increaseswiththeb eld.forthetripletstate,th

37、ezeemancontributionstoeg(2)and2eg(1)cancelout,and(2) (1)isnota ectedbytheb eld.thedecreaseofthedtpseparationinthemagnetic eldwasalsorecentlyobservedexperimentally.8theupper(lower)insetineach gureshowsthecorrespondingexchangeenergyjasafunctionofthemagnetic eldcalculatedbyeq.(8)with(without)thezeemane

38、 ect.inbothcases,thezeemane ectinducesalineardepenedenceofjonb.however,in(a)giventhestrongcouplingbetweenthedots,theorbitalcontributiontojdominatesatlowb eldsbeforebeingovercomebythezeemaninduceddecreaseathigher eld;in(b),jistotallydominatedbythezeemancontribution,whichdecreaseslinearlywiththeb pari

39、sonoftheb 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 elddependencesofthedtpseparationan

40、dexchangeenergyintheabsenceofthezeemane ectinfig.5showsthatthelattersaturatesatmuchlowervaluesofthemagnetic eldthantheformer.thisisbecausethedtpseparationisdeterminedbythecoulombinteractionbetweenelectronswhichdecreasesastheelectronsbecomelocalizedbythemagnetic eldinindividualdots(withintheheitler-l

41、ondonapproximation,thisdecreaseisproportionaltob 2,ref.13),whiletheexchangeenergyinabsenceofthezeemane ectapproacheszeromuchfasterthanthecoulombinteractionsinceitisproportionaltotheoverlapbetweentheindividualelectronwavefunctionsthatdecaysexponentiallyfastinstrongmagnetic elds.2,13 itisalsointeresti

42、ngtocomparetheexactvaluesoftheexchangeenergy(seetheinsetsinfig.5)withthoseextractedfromthestabilitydiagramsinmagnetic eldsusingthehubbardmodel.2,8accordingtothismodel,jest=4t2/(vintra vinter)where2tisthetunnel(symmetric-asymetric)splitting,vintraandvinteraretheintra-dotandinter-dotcoulombinteraction

43、s.fromthedatashowninfig.5,weestimatethevalueoftheinter-dotcoulombinteractionvinter3.4(2.0)mevford=50(60)nm,whichisgivenbythedtpseparation(forthelowesttripletstate)inthelimitoflargemagnetic elds.thesenumbersareingoodagreementwiththecorrespondingexpectationvalues c(r1,r2) ofthecoulombinteractionmatrix

44、(3.5and2.2mev,respectively)obtainedfromdirectcalculations,therebycon rmingelectronlocalizationandqdsdecoupling.sinceatzeromagnetic eld,thedtpseparationisequalto2t+vinter,weobtain2t50(60)1.6(0.7)mevwhichisconsistentwiththeenergydi erencesbetweenthetwolowestsingle-particlelevelsof1.9(0.4)mev.asvintra8

45、mevisgivenbytheelectronadditionenergyinoneqdwhichisthedistancebetweenthe”corners”ofthelinearregionwheresingleelectronre-localizationoccursfromonedottotheotherinthen=2energydiagram(seefig.2),theestimatedvaluesoftheexchangeenergybecomejest50(60)50(60)0.6(0.08)mev.thesenumbersareofthesameorderasthenume

46、ricallyexactvaluesof0.24(0.012)mev,buttheybothsigni cantlyoverestimatethecomputeddata,andtherefore,canonlybeusedasageneralguidelinetogaugethemagnitudeoftheexchangecouplingindoubleqds.theoverestimationisduetothedi erencebetweencoulombenergiesinthesingletandtripletstatesthatlowerstheexchangeenergy,2bu

47、twhichisnottakenintoaccountinthesimplehubbardmodel. insummary,wecomputedthestabilitydiagramformodeldoubleqdsystemspopulatedwithuptotwoelectronsinmagnetic eldsusingnumericallyexactdiagonalizationoftheone- the charge stability diagram for two coupled quantum dots containing up to two electrons is comp

48、uted 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.wefoundthatinthewea

49、k-couplingregimethecurvatureofthechemicalpotential(2)islargerthanthatoneof(1)whileinthestrong-couplingcasethesituationisreversed.hence,byanalyzingthechemicalpotentialvariationscausedbyexternalbiasesandmagnetic elds,itispossibletodistinguishbetweenstrongandweakinter-dotcoupling,evenifthecurvaturesare

50、ofthesameorderinbothcases.theevolutionofthestabilitydiagramsinmagnetic eldsconformstothegeneralideaofenhancedelectronlocalizationwithincreasing eldstrength.theexchangeenergiesextractedfromthestabilitydiagramsshowedthatthesevaluesaresigni cantlyoverestimatedwhencomparedwithnumericallyexactdata. thisw

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55、science309, 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(2021). 8 9 10 11t.hatano,m.

56、stopa,ands.tarucha,science309,268(2021).d.v.melnikovandj.-p.leburton,phys.rev.b73,085320(2021).l.-x.zhang,d.v.melnikov,andj.-p.leburton(unpublished).thevaluesofthechemicalpotentialsatwhichthecontourplotsaredrawnarechosentofacilitatedatarepresentation.fortheinvestigatedmagnetic elds,therelativechange

57、ofthecurvatureofaspeci cchemicalpotentialandthedtpseparationislessthan5%fordi erentreferencevalues. 12thesameconsiderationisobviouslyvalidforthetripletstatesaswell,albeitwithalargerquasilinearregion. 13d.v.melnikovandj.-p.leburton(unpublished). the charge stability diagram for two coupled quantum do

58、ts 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 mevx nm mev-10 mev-15-20 -25 vl=vr mev fig.1:(coloronline)(a)thecon nementpotentialford=50nm(red)andd=60nm(blue)atvl=vr=25mev.chemicalpotentials(1)(solid)ands(2)(dashed)vs.vl=vrfor(b)d=50nmand(c)d=60nm.inbothofthemthehorizontallineindicatesthevaluesofthechemicalpotentialatwhichthecontoursinfig.3aredrawn. the charge stability diagram for two