BINDING ENERGY OF EXCITON IN QUANTUM DOTS

CommunicationsinPhysics,Vol.14,No.2(2004),pp.95–99

BINDINGENERGYOFEXCITONINQUANTUMDOTS

WITHTHECENTRAL-CELLCORRECTION

DEPENDINGONTHEDOTSIZES

TOTHITHAOANDNGUYENAIVIET

InstituteofPhysics&Electronics,VAST

Abstract.Thebindingenergyofexcitoninquantumdotswithaparaboliccon nementpotentialwascalculatedbyvariationalmethodsbeyondtheKohn-Luttingere ectivemasstheory,whenthecentral-cellcorrectionwastakenintoaccount.Wehaveassumedthatashortrangepotentialwithtwoparametersforstrengthandrangeforexciton,representingthecenter-celle ectalsodependsondotsize.Ourresultisingoodagreementwithexperiment.

I.INTRODUCTION

Semiconductornanoparticles-quantumdotshavebeenfabricatedandextensivelyinvestigatedinbothexperimentalandtheoreticalsides.Thequantumdotsaresuchsmallstructureasquasizero-dimensionalwithstrongcon nementinalldirections.Theelectronsandholesinquantumdotsarefullyquantizedinadiscretespectrumofenergylevels.Thestrongopticale ciencyobservedinquantumdotsmakesthempromisingcandidatesforoptolectronicandnanodevices[1,2].

Onewellknownthattheexcitonplayanimportantroleindeterminingtheopticalpropertiesofsystem.Thestudyofexcitonstatesinquantumdotsisarelevantaspecttowhichmanytheoreticalworkshavebeendevoted[3,4,5,6,7,8,9,10].

Thee ectivemasstheoryprovidesasimpletheoreticalmodeltocalculatebindingenergiesofexcitonsinquantumdots[3,4,5].Accordingtothismodel,theexcitonproblemisahydrogenatomembeddedinadiecletricmediumwithamassrenormalization(e ectivemass)fortheelectron-holepair.Whilethebindingenergyofexcitoninbigquantumdotsaredescribedquiteaccurately,therearelargedeviationforthebindingofsmallquantumdotsfromthosepredictedbythise ectivemasstheory.ThisCoulombicpotential(e2/ 0)assumedinthebreakdownofconceptofthestaticdielectricconstantandacorrectioncanbee ectivebyusingproperscreeningfunction (r)approaching 0asr→∞.

Recentlysomeauthors[11,12]havestudiedthecentral-cellcorrectionsfordonorsinsemiconductorsasanalogyproblemofexcitons.Beyondthee ectivemasstheory,insteadoftheCoulombicpotentialtheyhaveassumedashortrangepotentialwithtwoparametersforthestrengthandtherangefordonors,representingthecentral-celle ects.Inthiswork,usingthatideaweinvestigatetheproblemofexcitonsinsmallquantumdots.

Assumingthesametwoparametersmodelpotentiallikeinthecentral-cellcorrec-tionsproblem,wewillshowthatthebindingenergiesofexcitonsincreasewithreducingofthedotradiusR.

96TOTHITHAOANDNGUYENAIVIET

II.MODELHAMILTONIAN

Letusconsidertheconductionbandandvalencebandaresphericalwithe ective

massesm eandmhfortheelectronandhole.TheHamiltonianforanexcitoninspherical

quantumdotsisde nedas

2 2ppeHx=Eg++h +Vconf(r e,r h)+Uc(r e r h), 2me2mh(1)

inwhichEgisthebandgap,Vconf(r e,r h)isthecon nementpotentialfortheelectronandhole,whichcanbecalculatedapproximatelybyuseasimpleone-parameterparabolicpotential[4,5]2ωc2 2Vconf(r e,r h)=(2)(m ere+mhrh).2

TheparameterωcwillbechosenlaterbychangingtheradiusofquantumdotinEq.(8).

Thelasttermofequation(1)Uc(r e r h)istheparametere ectivepotentialrepre-sentingthedeviationfromthee ectivemasstheory,whichis

e2r2V0Uc( r)= [1+4/3exp( 2)], rλR(3)

where isthestaticdielectricconstant,Risdotradius,V0andλareparametersrepre-sentingthestrengthandtherangeofthepotentialinthecentral-cellregion,resp.

ThecrucialpropertyoftheparabolicpotentialisthattheHamiltonian(1)issepa-rableuponintroducingthecentral-of-massandrelativetransformationsforthecoordinateandmomentum[5,6].TheHamiltonianofcentral-of-massmotionisidenti edtobethewellknownoscillatoronewiththelowestenergylevelERO=32 ωc.TheHamiltonianoftherelativemotionequals

p2

Hr=(4)+Vc(r)+Uc(r),2µ

whereµ=m r=r e r h,emh/(me+mh)isthereducedmassoftheelectron-holepair,

andVc(r)iscon nementpotentialfortherelativemotion

122Vc(r)=µωcr.2

ChoosingtheexcitonRydbergenergyER=µe4/2 2 2astheunitofenergyandax=astheunitoflength,theHamiltonianHrindimensionlessformcanbewrittenas

γ2222H= +ξ (1+ve αξ),9ξ

whereξisthedimensionlessvariable;v,α,γarethedimensionlessparameters:

ra23 ωcV0ξ=,α=x,γ=,v=.axλ22ERR4/3(5) 2µe2(6)(7)

BINDINGENERGYOFEXCITONINQUANTUMDOTS...97

WewillpresentourresultsintermoftheunitofenergyER.

II.1.Bindingenergyofexcitonwithoutthecorrection

Whentheinteractionbetweentheelectronandholeisomitted,wehaveasimplerelationbetweentheparameterωcanddotsizeRas

π21ωc=.3µR2(8)

IncaseofV0=0,thepotentialUcin(3)http://wendang.chazidian.comingatrialfunction

ψ0(ξ)=N0e1 4aξ2

0,(9)

wherea0isthevariationparameterandN0isthenormalizationconstant.From(6)and

(9)weobtain

22π312<H>0=2 +a20γ.a034a0 (10)

Thevariationalparametera0canbeobtainedbydemanding <H>0/ a0=0.Thedotsizesarechangedbyparameterγ(see(7)and(8)).Minimizing<H>0withrespecttoa0,theminimumexcitonenergywasfound.OurresultispresentedinFig.1.Thisresultissimilartotheoneofotherauthors[1,2,3].

Fig.1.Thedashedlineisthesphericalcon nementpotentialmodelresult[11],the

solidlineisourresultandthedotsareexperimentresult[1].

98TOTHITHAOANDNGUYENAIVIET

Forcomparison,wealsoplottedthesevaluesobtainedbyusingthesphericalcon- nementpotentialmodel[1,2]

Eex 1.786e2 2π211 =+ 0.248ER,2R2memh R(11)

wheremeandmharee ectivemassesofelectronandhole,Risradiusofquantumdot, isdielectricconstantandERisbulkexcitonbindingenergy.

II.2.Bindingenergyofexcitonwiththecorrection

Inthissection,westudythecontributionofthebeyonde http://wendang.chazidian.comingatrailwavefunctionasinequation(9)

ψ0(ξ)=Ne 4aξ,12(12)

whereaandNisthedimensionlessvariationalparameterandnormalizationconstant,resp.TheexpectationvalueofHamiltonian(6)becomes

12262232<H>=γα+9(1+2aα)+4a(aγ 12[ 24a(1+v)+8aα)],(13)2(a2+2a4α)ππ

herev,αandγandhavethesamemeaningasin(7).

Similarly,thedotsizecanbechangedbychangingγ.Inotherwords,thecenter-celle ectcanbedescribedbychangingthedotsizeR.

Fig.2.Thedashedlineandthesolidlineisthesphericalcon nementpotentialmodel

result[11]andourresult,respectively.Thedotsareexperimentalresult[1].

Theminimumexcitonenergy,E Eg,wasfoundbyminimizingtheexpectationvaluesuccessivelywithrespecttoa.IntheunitofER,theenergyofexcitonversusthe

BINDINGENERGYOFEXCITONINQUANTUMDOTS...99

parameterV0andλispresentedinFig.2.Notethatthecorrectionenhancesthebindingenergyofexciton.

Insummary,westudyapresentsimplemodelforexcitons,whicharecon nedinaquantumdot,whenthedeviationsfrome ectivemasstheorywastakenintoaccountbyusingamodi edCoulombpotentiallikein[11].ThebindingenergyofexcitoncanbecalculatedbyusingEb=γ <H>min,where<H>minisexpressedinunitsoftheRydbergenergyER.ItispresentedinFig.3.Thecentral-cellcorrectionenhancesthebindingenergyofexcitonconsiderably.

Fig.3.The[11],the

solidlineisourresult.

ACKNOWLEDGMENT

ThisworkissupportedbytheNationalprogramonBasicsResearch4.1.1601:“Theoryoflowdimensionalsystemsandnano-structures”.

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Received17January2003