Exchange interaction and stability diagram of coupled quantu(6)

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