double-dotsystemconnectedtothreeterminals
arXiv:cond-mat/0406577v1 [cond-mat.mes-hall] 24 Jun 2004LevG.Mourokh
DepartmentofPhysicsandEngineeringPhysics,StevensInstituteofTechnology,Hoboken,NJ07030
AnatolyYu.Smirnov
D-WaveSystem,Inc.,320-1985W.Broadway,Vancouver,
BritishColumbia,CanadaV6J4Y3
(February2,2008)
Abstract
Weexaminetransportandmicrowavepropertiesoftwocoupledquantumdotstakeninparallelconnectiontothecommonleftleadandconnectedtoseparateleadsattheirrightside.Inaddition,theareabetweentheleftleadandthedouble-dotstructureisthreadedbyAharonov-Bohmmagneticflux.Wedeterminetheenergiesandpopulationsofdouble-dotlevelsonthemicroscopicbasistakingintoaccounttheinterdotCoulombinteractionandshowthatatlargelead-to-leadbiasthepopulationinversioncanbeachieved.ForthecaseofstrongCoulombrepulsion,thisinversionleadstolevelcrossingaccompaniedbytheregionofnegativedifferentialconductivityinthecurrent-voltagecharacteristics,whereasforweakerCoulombrepulsion,theresonantmicrowaveabsorptionbecomesnegativeathighlead-to-leadvoltage.
Recentachievementsinnanotechnologyhaveledtoanewgenerationofsemiconductordevicesbasedonquantumdotsthatcanbeviewedasartificialatomsandmultiplequan-tumdotsystemsplayingtheroleofartificialmolecules[1].Inthesedevicesthequantum
1
propertiesofcarriersareexploredandtheCoulombinteractionisofcrucialimportanceinsuchsmalldevices.Initially,inmoststudies,thedotsweretakentobeconnectedinseries,but,recently,double-dotstructuresinwhichthetwoconstituentdotsareplacedinaparallelarrangementbetweenleadshavebeenalsoanalyzedtheoretically[2,3]aswellasexperimentally[4].Suchatwo-dimensionalsystemcanbethreadedbyAharonov-Bohm(AB)magneticfluxtoexamineinterferenceeffects.Inparticular,itwasshown[3]thatforasymmetricdouble-dotsystemhavingmolecularbondingandantibondingstates,oneofthesestatescanbedisconnectedfromtheleadsatappropriatevaluesoftheABflux,Φ.ThesevaluesareΦ/Φ0=2nfortheantibondingstateandΦ/Φ0=2n+1forthebondingstate,respectively,whereΦ0=hc/eisthefluxquantumandn=0,1,2,....Theelectronspinentanglerbasedonthetriple-dotsystemwithseparateleadsconnectedtoeachofthedotswasproposedinRef.[5].
InthisworkweexamineasystemcombiningthepropertiesofstructuresanalyzedinRefs.[3]and[5],i.e.thedouble-dotconnectedtothesameleadintheircommonleftsideinparallelarrangement,whileintheirrightsidethetwodotareconnectedtotheseparateleads(Figure1).Inaddition,theleftsideofstructureisthreadedbyABfluxwhichcanbeusedtocontroltheconnectionofthedouble-dotsystemtotheleftlead.Wedeterminethedouble-dotlevelpopulationsandshowthat,incontrasttothecaseofsymmetricdouble-dotcouplingtothesameleadsbothattheleftandattherightsidesofthestructure[3],theyexhibitdependenceontheABmagneticflux.Accordingly,thepopulationoftheantibondingstatecanbelargerathighlead-to-leadbiasthanthepopulationofthebondingstate,andeithertheantibondingstatebecomesthegroundstateofthesystemorthepopulationinversionisachieved.Wealsoanalyzethelead-to-leadcurrentandshowthat,inthecaseofstronginterdotCoulombcoupling,levelcrossingisaccompaniedbytheregionwithnegativedifferentialconductivityonthecurrent-voltagecharacteristics.
Thesecond-quantizedHamiltonianofthedouble-dotelectronsincludingtheinterdotCoulombinteractionisgivenby
2
+++++
H0=E0(a+1a1+a2a2)−∆(a1a2+a2a1)+Ua1a2a2a1,
(1)
wherea+i,aiarethecreation/annihilationoperatorsforelectronsini-thdot(i=1,2),−∆isatunnelingconstantbetweenthedots,U=e2/2dε(εisthedielectricconstant).TheHamiltonianoftheleadshastheform
HLeads=
k
ELkc+LkcLk+
k
ER1kc+R1kcR1k+
k
ER2kc+R2kcR2k,
(2)
wherec+αk(t),cαk(t)arecreation/annihilationoperatorsofelectronswithmomentumkintheα-lead(α=L,R1,R2).TheeffectofABfluxonquantumtransportcanbetakenintoaccountusingthePeierlsgaugephasefactorsexp(±iφ)inatransfermatrixdescriptionoftunnelingbetweentheleftleadanddots,withφ=Bld/4Φ0isthephaseexperiencedbyanelectronduringthelead-double-dottunnelingprocess.Withthis,theHamiltonianfortunnelingbetweendotsandleadsiswrittenas
Htun=
k
iφ
Lkc+Lk(a1e
+a2e
−iφ
)+
k
R1kc+R1ka1
+
k
R2kc+R2ka2+h.c.
(3)
EmployingtheproceduresofRef.[3],wederiveandsolvetheequationsofmotionforGreen’sfunctionsofthedouble-dotelectrons,obtainingaself-consistentsetofequationsforthelevelpopulations,
NA,B=
andthelevelenergies
EA,B=E0±∆+UNB,A,
(5)
fL(EA,B)(1±cos2φ)
2(2±cos2φ)
,
(4)
wherefα(EA,B)arethedistributionfunctionsofelectronsintheleftandrightleadstakenattheenergiesofthedouble-dotlevelsas
fα(EA,B)=exp
EA,B−µ−(−1)ǫαeV/2
leads.ItshouldbenotedthatthemagneticfieldinducedphaseisnotcancelledinEq.(4)asitwasinthecaseofsymmetricalconnectionstothesingleleftandrightleads[3],wherethepopulationsweregivenbyNA,B=(fL(EA,B)+fL(EA,B))/2.Inthelattercasethelimitinghigh-biasvaluesforthepopulationsofbothlevelswere1/2withthepopulationoftheantibondingstateapproachingthisvaluefromaboveandthepopulationofthebondingstateapproachingthisvaluefrombelow.Forthecaseexaminedinthepresentpaper,thepopulationsaremagneticfluxdependentandtheirlimitingvaluesaredeterminedbytheABphase.Moreover,thepopulationoftheantibondingstatecanbelargerthanthepopulationofthebondingstateatappropriatevaluesoftheappliedmagneticfieldandlead-to-leadbias,whichcanleadtothepopulationinversionor,forstrongCoulombrepulsion,tothesituationwhentheantibondingstatebecomesthegroundstateofthesystem.Itshouldbeemphasizedthatthepopulationinversioncanbeachievedonlyforthestronglynonequilibriumsituation.Forexample,atzerobiasthedistributionfunctionsarethesameforallleads,theABphasedependenceiscanceled,andthelevelpopulationsaredeterminedbythisdistributiontakenattheenergyofthecorrespondinglevel.Consequently,thepopulationoftheantibondingstateisalwayslessthanthepopulationofthebondingstate.
Eqs.(4)and(5)weresolvedself-consistentlywiththevoltagedependenciesoflevelpopulationsandenergiesshowninFigure2(a)andFigure2(b),respectively,forφ=π/8,lowtemperatureT=0.2∆,largeCoulombenergyU=8∆,andseparationbetweentheequilibriumchemicalpotentialoftheleads,µ,andtheenergyofthesingledotgroundstate,E0,chosenas7∆.Inthiscase,inequilibriumthebondingstateisbelowtheFermilevelandtheantibondingstateisabovetheFermilevelwithinitialpopulationstobeNB=1andNA=0.Withvoltageincreasing,thechemicalpotentialoftheleftleadpassesthroughtheantibondingstate(modulothermalbroadening)resultinginitspopulation(Figure2(a)).Accordingly,theenergyofthebondingstateincreases(Figure2(b))andwithfurthervoltageincreasingthechemicalpotentialoftherightleadspassesthroughtheenergyofthisstateresultinginitsdepopulationandcorrespondingdecreaseoftheantibondingstateenergy.TheCoulombenergyischosentobesufficientlylarge,sothattheenergyoftheantibond-4
ingstatebecomeslessthantheenergyofthebondingstateand,moreover,lessthanthechemicalpotentialoftherightlead.Consequently,thislevelbecomesnonconductiveanditspopulationincreasesuptoone.Withfurthervoltageincreasingtheantibondingstateremainsthegroundstateofthesystemandbecomesconductiveagain.
Thesepopulationchangesmanifestthemselvesinthecurrent-voltagecharacteristicsofthedouble-dotsystem.Forsymmetriccouplingtotheleadsandlowtemperaturethecurrentthroughthestructureisgivenby
I=
eΓ
2+cos2φ2−cos2φ
fL(EA)−
fR1(EA)+fR2(EA)
fL(EB)−
fR1(EB)+fR2(EB)
fortheweakerCoulombenergy(Figure5(b))thesystemcanamplifytheexternalfieldforthevoltageshigherthansomethresholdvalue.ThisamplificationcanbecontrolledbythemagneticfieldascanbeseeninFigure6wheretheabsorbedenergyisplottedasafunctionoftheABphaseφ.
Inconclusion,wehaveexaminedthetransportandmicrowavepropertiesofthedouble-dotstructureconnectedinparalleltothesameterminalfromitsleftsideandhavingthetwodotsconnectedtotheseparateterminalsfromtheirrightside.Inaddition,theleftsideofthestructureisthreadedbyAharonov-Bohmmagneticflux.WehaveshownthatbothlevelpopulationsandelectriccurrentthroughthestructurearefunctionsofABfluxandatappropriatechoiceofparametersboththenegativedifferentialconductivityandthepopulationinversioncanbeachievedwithpossibleamplificationofexternalmicrowavefield.
6
FigureCaptions
Figure1.Schematicofthedouble-dotsystemwithconnectionstothreeterminals.Figure2.(a)Levelpopulationsand(b)energiesasfunctionsoftheappliedvoltagebiasforthecaseofstrongCoulombrepulsion.
Figure3.Current-voltagecharacteristicsofthestructure.
Figure4.(a)Levelpopulationsand(b)energiesasfunctionsoftheappliedvoltagebiasforthecaseofweakCoulombrepulsion.
Figure5.Absorbedresonantmicrowaveenergyasfunctionoftheappliedvoltagebiasfor(a)strongCoulombrepulsionand(b)weakCoulombrepulsion.
Figure6.AbsorbedresonantmicrowaveenergyasfunctionoftheAharonov-Bohmphase.
7
REFERENCES
[1]W.G.vanderWiel,S.DeFranceschi,J.M.Elzerman,T.Fujisawa,S.Tarusha,andL.P.Kouwenhoven,Rev.Mod.Phys.75,1(2003).
[2]D.LossandE.V.Sukhorukov,Phys.Rev.Lett.84,1035(2000).
[3]A.Yu.Smirnov,N.J.M.Horing,andL.G.Mourokh,Appl.Phys.Lett.77,2578(2000);L.G.Mourokh,N.J.M.Horing,andA.Yu.Smirnov,Phys.Rev.B66,085332(2002).[4]H.Qin,A.W.Holleitner,K.Eberl,andR.H.Blick,Phys.Rev.B64,241302(2001);A.W.Holleitner,C.R.Decker,H.Qin,K.Eberl,andR.H.Blick,Phys.Rev.Lett.87,256802(2001);A.W.Holleitner,R.H.Blick,A.K.H¨uttel,K.Eberl,andJ.P.Kotthaus,Science297,70(2002).
[5]D.SaragaandD.Loss,Phys.Rev.Lett.90,166803(2003).
8
Dot 1 Left Lead 2 1st Right Lead Dot 2 2nd Right Lead Aharonov-Bohm magnetic flux
L. G. Mourokh and A. Yu. Smirnov, Figure 1 of 6
(a)
1.00.90.8
NA NB
φ = π/8U = 8∆µ - E0 = 7∆
Populations0.70.60.50.40.30.20.10.0
0
2
4
6
8
10
12
14
Voltage bias (eV/∆)
(b)
98
µL
Energies ((EA,B-E0)/∆)76543210-1-20
2
4
6
8
10
EA EB
φ = π/8U = 8∆µ - E0 = 7∆
12
µR
14
Voltage bias (eV/∆)
L. G. Mourokh and A. Yu. Smirnov, Figure 2(a,b) of 6
0.10
Current (arb. units)0.08
0.06
φ = π/8U = 8∆µ - E0 = 7∆
0.04
0.02
0.00
0
2
4
6
8
10
12
14
Voltage bias (eV/∆)
L. G. Mourokh and A. Yu. Smirnov, Figure 3 of 6
(a)
1.00.90.8
NA NB
Populations0.70.60.50.40.30.20.10.0
0
2
4
6
8
10
φ = π/8U = 2∆µ - E0 = 2∆
Voltage bias (eV/∆)
(b)
3.02.5
EA EB
φ = π/8U = 2∆µ - E0 = 2∆
Energies ((EA,B-E0)/∆)2.01.51.00.50.0-0.5-1.0
0
2
4
6
8
10
Voltage bias (eV/∆)
L. G. Mourokh and A. Yu. Smirnov, Figure 4(a,b) of 6
(a)
10
Absorption (arb. units)8
6
φ = π/8U = 8∆µ - E0 = 7∆
4
2
0
02468101214
Voltage bias (eV/∆)
(b)
4
Absorption (arb. units)3
2
φ = π/8U = 2∆µ - E0 = 2∆
1
0
-10
2
4
6
8
10
Voltage bias (eV/∆)
L. G. Mourokh and A. Yu. Smirnov, Figure 5(a,b) of 6
2.5
Absorption (arb. units)2.0
eV = 8∆U = 2∆µ - E0 = 2∆
1.5
1.0
0.5
0.0
-0.5
0
1
2
3
4
φ/π
L. G. Mourokh and A. Yu. Smirnov, Figure 6 of 6
因篇幅问题不能全部显示，请点此查看更多更全内容