Mechanical effects in quantum dots in magnetic and electric fields

InstituteofPhysics,Wroc󰀬lawUniversityofTechnology,Wybrze˙zeWyspia´nskiego27,50-370Wroc󰀬law,Poland

b

InstituteofMathematics,UniversityofOpole,Oleska48,Opole,Poland

c

InstituteofPhysics,OdessaUniversity,Odessa,Ukraine

d

InstituteofMathematics,Wroc󰀬lawUniversityofTechnology,Wybrze˙zeWyspia´nskiego27,50-370Wroc󰀬law,Poland

Themechanicaleffectsinfinitetwo-dimensionalelectronsystems(quantumdotsordroplets)ina

strongperpendicularmagneticfieldarestudied.Itisshownthat,duetoasymmetryofthecyclotrondynamics,anadditionalin-planeelectricfieldcausesagroundstatetransitionaccompaniedbyachangeintheaveragetotalangularmomentumofthesystem,unlessthelateralconfiningpotentialisexactlyparabolic.Aprecisemechanicalexperimentisproposedinwhichamacroscopicangularmomentumofadensematrixofquantumdotscouldbemeasuredandusedtodetectandestimateanharmonicityoftheconfinement.73.21.La,75.80.+q

a

LucjanJacaka,JurijKrasnyjbc,DorotaJacakd,andArkadiuszW´ojsa

arXiv:cond-mat/0101133v1 [cond-mat.mes-hall] 10 Jan 2001I.INTRODUCTION

Themanufacturingofsemiconductorquantumdotswithacontrollednumberofconfinedelectronsallowstheex-perimentalstudyoffinitequantumsystemsintheareanotaccessibleinordinaryatomicphysics(cf.[1]forreview).

Particularlyinterestingarethemagneticeffectswhich󰀄inquantumdotsareoftheordersofmagnitudegreaterthaninatoms,asaconsequenceofthemagneticlengthlB=

ˆint=H

iN󰀂

e2

2m

󰀆epri+

×raretheexternal

magneticanditsvectorpotentialinthesymmetricgauge,E=(Ex,Ey,0)istheexternalelectricfield,ǫ0isthedielectricconstant,andµBandgareBohrmagnetonandthegyromagneticfactor.WeincludedthePaulitermeventhoughforsomematerialstheZeemansplittingisverysmallcomparedtothetypicalorbitalexcitationenergyofafewmeV(e.g.,forthebulkGaAsatsmallBtheZeemansplittingisonlyabout0.03meV/T).Thistermhasnoinfluenceonourresults.

Ifthelateralconfiningpotentialisa2Dharmonicwell,

V(ri)=

1

2B

2m

󰀆epri+

2

2mω0(ri+r0)2−

1

2mω0

.(7)

andtheeigenequation[cf.Eq.(4)]

Nowweformallychangethecoordinate,ri+r0=xi(notethatpr=px)toobtain󰀈21122ˆAxi−r0+mω0r0+gµBσiBHi=

c2

ˆiψ(xi−r0)=εψ(xi−r0).H

(8)

(9)

Wecannowusethelinearityofthevectorpotential,

Axi−r0=Axi−

1

2m

󰀆

pxi+

e

2

22

mω0xi−

1

2¯hc

(B×r0)·xiψ′(xi).2

󰀊

(13)

ˆ′isdifferentfromHˆo,the(axiallysymmetric)single-particleHamiltonianHˆiofEq.(3)intheFinally,wenotethatHii

absenceofanelectricfield,onlybyaconstant,

ˆi′=Hˆio−1H

2

22

mω0r0=εo−∆ε.

(16)

ThepairofEqs.(13)and(16)showthattheeffectofanelectricfieldEonthesingle-particlespectrumofaparabolic

dotconsistsmerelyofarigiddisplacementofthewavefunctionsbyr0∝E,aphasefactor,andaconstantshiftofenergiesby∆ε∝E2.Inparticular,despitebreakingoftherotationalsymmetrybytheelectricfield,thewavefunctionsremainrotationallysymmetric(althoughtheaxisofsymmetryisdisplacedfromthecenteroftheconfiningpotential

ˆintofthemany-bodyV).Similarconclusionsremainvalidforthemany-electronsystemsincetheinteractiontermH

ˆistranslationallyinvariant,HamiltonianH

N󰀂

e2

ǫ0|xi−xj|

.(17)

iTherefore,asaconsequenceofthegaugeinvarianceandtheharmonicformofconfinementV,thewavefunctionsΨλ

andenergiesEλoftheinteractingmany-electronsystemare:

󰀁ie

Ψλ(r1,r2,...,rN)=exp

2

′

whereΨ′λandEλdescribethesystematE=0,

22

Nmω0r0,

(19)

′′ˆoΨ′Hλ(r1,r2,...,rN)=EλΨλ(r1,r2,...,rN).

(20)

Letusstressthattheabove-discussedsymmetryofamulti-electronplanarquantumdotwithparabolicconfinement

isindependentofthesymmetryresponsiblefortheKohntheoremandconnectedwiththeseparationofthecenterofmassandrelativedynamics.

III.EFFECTSOFANIN-PLANEELECTRICFIELDINPARABOLICQUANTUMDOTS

Thetransformationpresentedaboveallowsthecalculationofvariouseffectscausedbyanin-planeelectricfieldinthepresenceofamagneticfieldorientedperpendicularlytothequantumdot.Forexample,usingEqs.(18)–(20)onecancalculatethechangeoftheaverageenergy󰀈E󰀉atatemperatureTbyapplyingthecanonicalGibbsdistribution,

󰀈E󰀉=

1

Z󰀂

λ

Eλe−βEλ=󰀈E′󰀉−

Ne2E2

Z′

󰀂

λ

′−βEλEλe

′

(22)

3

describesthedotatE=0.Here,Z=Tre−βHisthestatisticalfunctionandβ=(kBT)−1.Thepolarizabilityαis

α=−

1∂E

=Ne2

ˆ

2mω0

.(24)

Notethat∆ǫdependsonneitherTnorB.Onecanalsocalculatetheexpectationvalueofthegeneralizedmomentum,

󰀃󰀁N󰀅󰀂

priΨλ(r1,r2,...,rN)dr1dr2...drN󰀈p󰀉λ=Ψ∗λ(r1,r2,...,rN)

=󰀅

Ψ′∗λ(x1,x2,...,xN)

B×r0=

󰀁

i=1N󰀂

pxiΨ′λ(x1,x2,...,xN)dx1dx2...dxN+

Ne

i=12

󰀃

Ne

2c

p+mrim󰀆

󰀆

ee

pxi+

2c

󰀋󰀌=L′λ−

Ne3

r0×(B×r0)

(27)4B,2cm2ω0

󰀋󰀌󰀋󰀌

and,forthechangeofthethermodynamicaverage,∆L′=∆L′λ.Thekinetic(gaugeinvariant)angularmomen-tumM=r×mvdoesnotchangeintheelectricfield,

󰀁N󰀅󰀂e∗

ri×(pri+󰀈M󰀉λ=Ψλ(r1,r2,...,rN)

i=1

c

󰀋′󰀌

Axi)Ψ′(x,x,...,x)dxdx...dx=Mλ12N12Nλ

󰀃

(28)

Notealsothattheelectricfielddoesnotaffectmagnetization,

∂Eλ

∂B

.

(29)

4

IV.ANHARMONICEFFECTS

Thesimpledependenceofmany-electronwavefunctionsandenergiesontheelectricfieldgivenbyEqs.(13)and(16)dependedcriticallyontheharmonicformoftheconfinementV.Thissimpledependenceresultedintheinsensitivityofanumberofmeasurablequantitiestotheelectricfield,amongthemthekineticangularmomentumM.Inrealisticquantumdots,whoseconfinementisnotexactlyharmonic,theelectricfieldcausesmorethanarigiddisplacementandaphasechangeofthesingle-andmany-particlewavefunctions.Theharmoniccaseistheonlycaseinwhichthecombinationoftherotationallysymmetricconfiningpotentialandthepotentialoftheuniformin-planeelectricfieldremainsrotationallysymmetric.Inallothercases,theshapeofthejointsingle-particlepotentialdependsonE,andsodothesingle-andmany-particlechargedensityprofilesandanumberof(inprinciple)measurablequantities,suchasM.Oneexampleisthatoflargerdotswhoseconfinementcanbeusuallywellapproximatedbyarotationallysymmetrichard-wallpotential.Insuchsystems,anelectricfieldcreatesanasymmetricpotentialminimumwithinadot,andtheelectronsareconfinedtoasmallerarea.Anotherexampleisthatofveryshallowdotsinwhichastrongelectricfieldcanevencauseunbindingofelectrons.Alltheseeffectsarewell-knowninthecontextof1Dconfinementofelectronsinquantumwellsandheterojunctions.

Ifindeedexperimentallymeasurable,thedependenceofMonEcouldbeamechanicalprobeoftheanharmonicityoftheconfinement.ThemajordifficultycouldbeasmallmagnitudeofthechangesofMexpectedforquantumdots,whichhowevercanbeincreasedbymanyordersofmagnitudeinadense(3D,i.e.multi-layer)matrixofdots.ThespecificvaluesofthetotalMofndotsinaunitareadependonmanyfactorssuchasn,N,B,orV,butinthesimplestcaseofaverylargeB,inwhichso-calledfractionalquantumHalldropletsforminlargerquantumdots,MisoftheorderofnN(N−1)¯h.ToestimatetheorderofmagnitudeofthechangeofMonecanusetheexpression(27)forthechangeofL.UsingthefollowingparametersforadensematrixofGaAsdots:m=0.067,n=1010cm−2,hω0=3.3meV,N=10,E=109V/cm,B=10T,andS=1cm2,weobtain∆󰀈L󰀉=10−5gcm2/s.Notethat¯

−4

thefactorω0inEq.(27)stronglyfavorsshallowerconfinement,typicalfordotsdefinedelectrostatically[5,11]bymeansofapatternofelectrodesgrownovera2Dheterostructure.Whetherdetectionangularmomentumassmallasestimatedaboveispossibleornot,wefindtheideaofthemicroscopicmotionofagreatnumberofelectronscausingamacroscopicrotationofasamplequiteintriguing.Themostsensitivemeasurementwouldprobablyinvolvetheresonancebetweenthevibrationsofadotmatrixsuspendedintheformofatorsionpendulumandtheoscillationsofanelectricfield.

V.CONCLUSION

Wehavestudiedtheeffectofthein-planeelectricfieldEonthewavefunctionsandenergiesofmany-electronsystemsconfinedinquasi-two-dimensionalquantumdotsinaperpendicularmagneticfield.Wehaveshownthatforthespecialcaseoftheharmoniclateralconfiningpotentialtheeffectoftheelectricfieldisameredisplacementofthemany-electronwavefunctioninthedirectionofthefieldandachangeofphase.Inconsequence,anumberofmeasurablequantities,suchasthekineticangularmomentumMremainunchangedintheelectricfield.SincethelackofdependenceofMonEisauniquepropertyoftheharmonicconfinement,thechangeofangularmomentumunderthevariationofEisameasureoftheactualanharmonicityofthisconfinement.Anexperimentinwhichtherotationofadensequantumdotmatrixunderoscillationofanelectricfieldoccursisproposed.

ACKNOWLEDGMENT

ThisworkwassupportedbyKBNProjectNo:2PO3B05518.

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