LO Phonon-Induced Exciton Dephasing in Quantum Dots An Exact

It is widely believed that, due to its discrete nature, excitonic states in a quantum dot coupled to dispersionless LO phonons form everlasting mixed states (exciton polarons) showing no line broadening in the spectrum. This is indeed true if the model is

LOPhonon-InducedExcitonDephasinginQuantumDots:AnExactlySolvableModel

1

E.A.Muljarov1,2, andR.Zimmermann1

Institutf¨urPhysikderHumboldt-Universit¨atzuBerlin,Newtonstrasse15,D-12489Berlin,Germany2

GeneralPhysicsInstitute,RussianAcademyofSciences,Vavilova38,Moscow119991,Russia

(Dated:February5,2008)Itiswidelybelievedthat,duetoitsdiscretenature,excitonicstatesinaquantumdotcoupledtodispersionlessLOphononsformeverlastingmixedstates(excitonpolarons)showingnolinebroadeninginthespectrum.Thisisindeedtrueifthemodelisrestrictedtoalimitednumberofexcitonicstatesinaquantumdot.Weshow,however,thatextendingthemodeltoalargenumberofstatesresultsinLOphonon-inducedspectralbroadeningandcompletedecoherenceoftheopticalresponse.

PACSnumbers:78.67.Hc,71.38.-k,03.65.Yz,71.35.-y

arXiv:cond-mat/0605545v1 [cond-mat.other] 23 May 2006

Amongdi erentmechanismsoftheopticaldecoher-enceinsemiconductorquantumdots(QDs),interactionbetweenexcitonsandlatticevibrations(phonons)isthemostimportantoneatlowcarrierdensities,leadingtoatemperature-dependentlinewidthinopticalspectra.ThediscretenatureofexcitoniclevelsinaQDgaverisetotheso-calledphononbottleneckproblem[1]:Whenthephononenergydoesnot tthelevelseparation,realphonon-assistedtransitionsbetweendi erentexcitonicstatesarenotallowed.Ontheotherhand,themea-suredopticalpolarizationinQDsshowsapartialini-tialdecoherenceandatemperature-dependentexponen-tialdecayatlargertimesThebottleneckproblemispartlyremovedforacousticphononsastheyhaveanenergydispersionandthuscontributetocarrierrelax-ationMoreover,evenapartfromtheresonancebe-tweenthephononenergyandthelevelseparation,acous-ticphonons,duetotheirdispersion,areshowntoberesponsibleforpuredephasinginducedbyvirtualpro-cesses[4].

Longitudinaloptical(LO)phonons,inturn,haveanegligibledispersionwhichleadstoaquitedi erentbe-havior:ExcitonsinQDsandLOphononsarealwaysinastrongcouplingregimeinthesensethattheyformev-erlastingpolaronswithnospectralbroadeningThisalsomakesanyapproximateperturbativeapproachtotheexciton-LOphononprobleminappropriate.In-deed,forafewexcitoniclevelsinaQDcoupledtobulkLOphonons,aself-consistentsecondBornapproxima-tionfortheselfenergydevelopedin[10,11]resultsinalinebroadeningwhichisfullyarti cial.Thisartefactisrefutedbytheexactsolutionofthisproblem[6,9]showingthatthespectrumconsistsexclusivelyofdis-creteunbroadenedlines.Also,aGaussspectrallineshapewasfoundinthequadraticcouplingmodelbytruncatingthecumulantexpansioninsecondorderAgain,theexactsolutionofthismodelshowsnospectralbroaden-ing[13].

InthisLetterweshowthatpronouncedLOphonon-induceddephasingandspectralbroadeningdoneverthe-lessexistinQDs.Thisbroadeningiscalculatedmicro-

scopicallybyinclusionofin nitelymanyexcitonicstatesinaQDthathasneverbeendonebefore.Thusaquali-tativelynewsourceofthedephasinginQDsisfound.ThefullproblemofexcitonsinaQDlinearlycoupledtotheLOphonondisplacementcanbesolvedexactlyonlyforaverylimitednumberofexcitonicstates[9].Themajorobstacleareo -diagonaltermsintheexciton-phononinteractionwhichcoupledi erentexcitonicstatesinaQD.Inordertotakeintoaccountasmanyexcitonicstatesaswelike,wehavederivedmicroscopicallyanef-fectiveexciton-phononHamiltonianwhichhasonlylevel-diagonaltermsandthusallowsanexactsolutionforanarbitrarynumberofstates.Thise ectiveHamiltonian,however,preservesthemainfeaturesoftheoriginalprob-lem,sincetheo -diagonaltermsarealsointrinsicallyrep-resented:Bymeansofaunitarytransformationtheyaremappedintodiagonaltermsgivingrisetoanexciton-phononcouplingquadraticinthephonondisplacementoperators[4,Concentratingonthegroundexcitonstate|1 ,theef-fectiveHamiltoniantakestheform( =1):

H=ω0a qaq+(E1+VL+VQ)|1 1|,(1)

VL=

q

q

M11(q)(aq+a q),

(2)

VQ=

1

2(En E1)2 ω0

M1n(p)Mn1(q)

(4)

≡

n=1

Fn(p)Fn(q),

whereω0isthedispersionlessLOphononfrequency

andEnisthebaretransitionenergyofasingle-excitonstate|n inaQD.ThekernelQofthequadraticcou-plingtoLOphononshasthesameformastheef-fectivescatteringmatrixusedinRef.todescribethephononmodesboundtoaneutraldonor.Itisderivedfromthelevel-nondiagonalmatrixelements

It is widely believed that, due to its discrete nature, excitonic states in a quantum dot coupled to dispersionless LO phonons form everlasting mixed states (exciton polarons) showing no line broadening in the spectrum. This is indeed true if the model is

M1n(q)∝q 1 1|eiqre eiqrh|n ofthelinearexciton-phonon(Fr¨ohlich)interaction,http://www.77cn.com.cn-ingitsfactorizationproperty,QisexpressedinEq.(4)intermsoffunctionsFn(q).AquadraticcouplingmodelsimilartoEqs.(1–4)hasbeenalreadysuccessfullyusedforcalculationoftheexcitondephasinginInGaAsQDs,inducedbyacousticphonon-assistedvirtualtransi-tions[4].

Themethoddevelopedin[4]allowsusto ndtheexactsolutionoftheHamiltonian(1–4),usingthecumulantexpansion.ThelinearopticalpolarizationhastheformP(t)=θ(t)exp[ iE1t+KL(t)+KQ(t)+KM(t)],(5)wherethelinearandquadraticcumulantsare

KL(t)=[(2N+1)(cosω0t 1)+i(ω0t sinω0t)]

×|M11(q)/ω0|2,(6)

q

KQ(t)=

1

It is widely believed that, due to its discrete nature, excitonic states in a quantum dot coupled to dispersionless LO phonons form everlasting mixed states (exciton polarons) showing no line broadening in the spectrum. This is indeed true if the model is

holepair)statesaretakenintoaccount,wecalculatethelinearabsorptionspectrum,i.e.therealpartoftheFouriertransformofP(t).Itishopeless,however,to ndtheFouriertransformnumericallyifthefunctiondoesnotdecayatallordecaysveryweakly.Thatiswhyweusehereadi erentmethod:Wediagonalizeexactlytheex-citedstateHa …… 此处隐藏：8358字，全部文档内容请下载后查看。喜欢就下载吧 ……