Effects of the Heavy- and Light-Hole Mixing

ÂA.Brum:ExcitonDynamicsinQWsM.CarolinadeO.AguiarandJose

phys.stat.sol.(a)178,89(2000)

Subjectclassification:71.35.Ày;73.20.Dx;S7.12;S7.1589

EffectsoftheHeavy-andLight-HoleMixingintheExcitonDynamicsinSemiconductorQuantumWells

ÂA.BrumM.CarolinadeO.AguiarandJose

DFESCM,InstitutodeFõÂsica,UniversidadeEstadualdeCampinas,CP6165,CEP13083-970,Campinas(SP),Brazil

(ReceivedAugust30,1999)

Wepresentthecalculationofthelowtemperatureexcitonbimolecularformationrateinquantumwellsthroughacousticphononscatteringtakingintoaccounttheheavy-andlight-holemixingintheholeandexcitonstates.Weobserveadecreaseintheexcitonformationratecomparedtotheparabolicapproximation.Forsymmetricquantumwells,thehole-subbandmixingallowsafiniteprobabilityofscatteringinvolvingthechangeofparity.Thisopensachannelforspin-fliptransi-tionsduringtheexcitonformationprocess.

IntroductionInnonresonantphotoluminescenceexperiments,opticallycreatedelec-tron±holepairsrapidlyreachaquasi-thermalequilibriumdistributionnearthesubbandedges.Theythendecayintoanexcitonstate.Atlowcarrierdensitiesthisisdominatedbyphonon-assistedscatteringanditisthenabimolecularexcitonformationprocess.Theexcitonsinthegroundstateareformedwithafinitecenter-of-mass(CM)wavevectorandhavetorelaxtoopticallyactivestatesbeforeemissiontakesplace.Thestudyoftheexcitondynamicsisparticularlyimportantinsemiconductorquantumwells(QWs)wheretheexcitonstatesdominatetheopticalemission.Time-resolvedopticalspectroscopyhasprovidedimportantinformationontheexci-tonformationprocessinsemiconductorQWs.Kusanoetal.[1]andDamenetal.[2]obtainedthefirstresultsontheexcitonformationtime.Morerecently,Kumaretal.[3]measuredabimolecularexcitonformationrateCequalto0X5cm2/sfora80#eGaAsQW.Fromthetheoreticalviewpoint,theexcitonformationprocesshasbeenstudiedfol-lowingdifferentapproaches.Guliaetal.[4]solvedasetofkineticequationsusinganensembleMonteCarlotechnique.Theyobtainedatimedependentexcitonformationrate.Piermarocchietal.[5]calculatedtheexcitondynamicsintermsofrateequationsforthecarrierpopulations.Inthisdescription,theexcitonformationrateistimeinde-pendent.Morerecently,Kiraetal.[6]studiedtheexcitonicluminescenceintheregimeofveryshorttimeusingthequantumtheoryoftheinteractingphotonelectron±holesystem.TheseworkshaveconsideredasimpleparabolicdescriptionfortheholeandexcitonstatesintheQW.Theactualholedispersionshowsacomplexpatternoncetheheavy-andlight-holecouplingistakenintoaccount[7].TriquesandBrum[8]showedthattheexcitonCMdispersionisalsostronglynon-parabolicasaconsequenceofthiscoupling.Whenweconsiderthecomplexityofthehole-subbandmixingforGaAsQWsthespinisnotagoodquantumnumber.ForsymmetricQWs,theholeandtheexcitonHamilto-7physica(a)178/1

90ÂA.BrumM.CarolinadeO.AguiarandJose

niansmaybeseparatedbyaunitarytransformationintotwouncoupledblocks,whichhavedegenerateeigenstateslabeledbythequantumparitynumberS.Theheavy-andlight-holemixingallowstheformationofexcitonswithholesshowingadifferentparitythanthatintheinitialstate(parity-flipprocess).ToaccountfortheseeffectsonehastoconsidertheLuttingerHamiltoniantode-scribethehole-subbandstatesaswellastheexcitonstates.Theexciton±phononinter-actionisthendescribedbytheBir-PikusHamiltonian[9].Thisconsiderablycomplicatestheexcitondynamicswhichinvolvestheparticipationoftheexcitedexcitonstates.Inthispaper,wefocusontheeffectsofheavy-andlight-holemixingintheformationofexcitonsintheirgroundstatethroughlongitudinal-acoustic(LA)phononscattering.ModelWefollowheretheapproachbasedontherateequationsforthecarrierpopu-lationsdevelopedbyPiermarocchietal.[5].Inourcase,theexcitonformationprocessischaracterizedbytwobimolecularformationrates,onefortheparity-conservingpro-cess,C44,andonefortheparity-flipprocess,C45,where4and5standfortheexcitonandholeparities,SexcandSh.Therateequationfortheelectrondensityis

dne??ÀC44nenhÀC45nenhYwheretheexcitonformationratesaregivenby??C44??5????dEfF44??5????Ef??g??Ef??

and??42p??4??5??nF44??5????Ef????jhn??q????n1YY4Yn??q??ij2fjHexcÀphjYiehn????YÀkeYkhYq

Âfe??ke??fh??kh??d??EfÀEi??nEph??Y

ne??1??fe??ke??YkeYsenh??1??fh??kh??YkhYSh??1????2????3????4??

keandkharetheelectronandholewavevectors,respectively,seistheelectronspinandSisthesamplearea.fe??ke??andfh??kh??are,respectively,theelectronandholedis-tributionsinthequasi-thermal-equilibriumsituationcharacterizedbyacommoncarriertemperatureTc.HnexcÀphistheexciton±phononinteractionHamiltonian,where

hn??????À??referstotheemission(absorption)process.YSiandEiarethewavefunctionandtheenergyoftheinitialstate,respectively,whichisformedbyanelectron±hole

excpair,wheretheCoulombinteractionisneglected.YSandEfarethewavefunctionfandtheenergyofthefinalstate,respectively.g??Ef??isthedensityofstatesoftheexci-tongroundstate.n??q??istheBose-EinsteindistributionofphononswithwavevectorqandenergyEphatthelatticetemperatureTL.TheholeandtheexcitonCMstatesarecalculatedintheframeofthe4Â4Luttin-gerHamiltonian[7,8].TheLA-phononsaredescribedbytheirbulkdispersionintheDebyeapproximation.Theholepartoftheexciton±LA-phononinteractionHamilto-nianisgivenbytheBir-PikusHamiltonian[9].Weareinterestedhereintheeffectsofthehole-subbandcomplexityintheelectronicstatesandwethereforesimplifytheBir-PikusHamiltonianbyconsideringonlyitsdiagonalpart.Weassumethattheinitialelectronandholedensitiesarelowandfe??ke??andfh??kh??areBoltzmanndistributions.ThemultipleintegralsinEqs.(2)and(3)aresolvedusingtheMonteCarlomethod.

ExcitonDynamicsinSemiconductorQuantumWells

91HnexcÀphTheparity-conservingchannelinvolvesboththeelectronandtheholepartsofwhileonlytheholepartofHnexcÀphispresentintheparity-flipchannel.IntheparabolicapproximationthespinisconservedandC45iszero.Themodificationsintroducedintheexcitonformationratebytakingintoaccounttheheavy-andlight-holemixingaretwofold:(i)TheholeandtheexcitonCMin-planedispersionsarenon-parabolic.TheiraveragemassesareheavierthanintheparabolicapproximationandstronglydependentontheQWwidth.(ii)Thereisanotherchannelforexcitonformationinwhichtheparityisnotconserved.

ResultsandDiscussionsWeconsideraGaAs±Ga0X7Al0X3AsQWatthelatticetem-peratureTL??10Kandcarrierquasi-temperatureTc??30K.Figures1aandbshowFasafunctionoftheexcitonCMkineticenergy,Ek??EfÀEfjkCM??0,intheparabolic(dashedlines)andnon-parabolic(fulllines)dispersionsforQWwidthsof100and150#e,respectively.Thecontributionsforthetwochannels,F44(dash-dottedlines)andF45(dottedlines),inthenon-parabolicdispersionarealsoshown.Atthelatticetem-peratureweconsidered,theLA-phononemissionlargelydominatesovertheLA-pho-nonabsorption(notshownhere).Theresultsshowthatmostoftheexcitonsareformedwithanexcessofkineticenergyoftheorderoftheexcitongroundstatebind-

Fig.1.FasafunctionofEkforaGaAs/Ga0X3Al0X7AsQWatTc??30KandTL??10Kinthenon-paraboliccase(solidlines)andtheparabolicapproximation(dashedlines)fora)L??100#e#andb)L??150e.Theparity-con-serving,F44(dash-dottedlines),andtheparity-flip,F45(dottedlines),contributionsarealsoshown

7*

92ÂA.

BrumM.CarolinadeO.AguiarandJose

Fig.2.CasafunctionoftheQWwidthLforthesameparametersasinFig.1inthenon-paraboliccase(solidline)andtheparabolicapproximation(dashedline).C44(dash-dottedline)andC45(dottedline)contributionsarealsoshown

ingenergy.Thisisaconsequenceofthelowenergyphononsthateffectivelyparticipateintheprocess.ThehighenergytailofFisduetothefinitecarriertemperature.FortheQWwidthL??150#etheformationofexcitonsathigherenergythantheexcitonbind-ingenergyisparticularlyfavored.ForthisvalueofL,theholesthermallyoccupytheanticrossingregionofthedispersionwhichhasahigherdensityofstates,favoringtheformationofmoreenergeticexcitons.Figure2showsthetotalbimolecularexcitonformationrate,C??C44??C45,asafunc-tionofQWwidthinthenon-paraboliccase(fullline)andintheparabolicapproxima-tion(dashedline)forthesameparametersasinFig.1.Thetwocontributions,C44(dash-dottedline)andC45(dottedline),arealsoshown.Theinsetgivesthedetailsofthenon-paraboliccaseatwideQWs.TheexcitonformationrateincreaseswiththedegreeofconfinementofthecarrierwavefunctionintheQW,reachingitsmaximumforL%20to30#einbothcases.Asaconsequenceofthehole±subbandcoupling,Cdecreasestolowervaluesascomparedtotheparabolicapproximation.ThiseffectispresentalreadyatnarrowQWsbecomingimportantastheQWwidthincreases.Mostoftheholestateswhicharethermallyoccupiedpresentapredominantlyheavy-holecharacter.Theexcitons,however,areformedwithafiniteCMwavevectorandalwayspresentamixedcharacter.Theresultisadecreaseinthescatteringmatrixelementwhichbecomesmoresignificantasthecouplingincreases.Thecontributionoftheparity-flipchannel,C45,isweak.Itincreaseswiththecou-plingatlargeQWs.Sinceonlytheholeflipsitsspin,thischannelplaysarolefortheformationofdarkexcitons.Inconclusion,wecalculatedthebimolecularexcitonformationrateassistedbyLA-phononscatteringinGaAsQWsincludingthehole±subbandmixingandexcitonCMdispersions.Themainresultsarethereductionoftheformationrateandtheopeningofaparity-flipchannelfortheexcitonformation.Ourresultssuggestthattheseeffectshavetobeconsideredforacompleteunderstandingoftheexcitondynamics.

AcknowledgementsWeacknowledgethefinancialsupportfromCNPq,FAEP-UNI-CAMPandFAPESP.

ExcitonDynamicsinSemiconductorQuantumWells93

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