Exchange interaction and stability diagram of coupled quantu(5)

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