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CdSe量子棒的线偏振光学性质 第27卷第12期 2006年12月 半导体 CHINESEJOURNALOFSEMICONDUCTORS Vo1.27No.12 Dec.,2006 Linear.PolarizationOpticalPropertyofCdSeQuantumRods ZhangXiuwenandXiaJianbai (InstituteofSemiconductors?ChineseAcademyofSciences.Beijing100083,China) Abstract:Thelinear-polarizationopticalpropertyofCdSequantumrodsisstudiedintheframeworkofeffective. massenvelopefunctiontheory.Theeffectsofshapeandmagneticfieldonthelinearpolarizationfactorsarcinves. tigated.ItisfoundthatCdSequantumsphereshavenegativepolarizationfactors(xy.polarizedemission)andquan. tumlongrodswithsmallradiushavepositivelinearpolarizationfactors(z-polarizedemission).Thez-directionis thedirectionofthecaxis.Quantumlongrodswithlargeradiushavenegativeli nearpolarizationfactors.dueto thehexagonalcrystalsymmetryandthecrystalfieldsplittingenergy.Thelinearpolarizationfactorsdecreaseand maychangefromapositivevaluetoanegativevalue;i.e.,thez-polarizedemissionsdecreaserelativetoxy.polar. izedemissionsasthemagneticfieldappliedalongthezdirectionincreases. Keywords:quantumrotis;linearpolarization;CdSe PACCl7360L;7865K CLCnumher:0469Documentcode:AArticleID:0253.4177(2006)12.2094.07 1IntrOductiOn ThemethodtosynthesizeCdSenanostruc. tureshasimproved.SingleCdSedotshavebeena. chievedt’~,andCdSequantumrodshavealsobeen synthesizedE引.whoseshapecanbecontrolledC.. Recentlymuchattentionhasbeenpaidtotheline— ar-polarizationopticalpropertyofthesenano- structuresE1,2,5--8].Thesenanostructureshavebe. comeamajorsubjectofattentionbecauseoftheir prospectiveapplicationsindevices.xy—polarized emissionsfromCdSesphereswereobservedand usedbyEmpedoclesetaf.E6]tomonitortherota. tionalmotionofquantumdotsinvarioushostma- trices.Atwo.dimensionaltransitiondipolewasin+ troducedtoinvestigatethexy..polarizedemis. sionsE..whichareactuallyduetothehexagona1. crystalsymmetryandthecrystalfieldsplittingen- ergy.z-polarizedemissionsfromCdSerodswere observedE.andaone.dimensionaltransitiondi. polewasintroducedtoinvestigateitUJ. Motivatedbytheexperimentalprogress,we studytheeffectsoftheshapeandmagneticfield onthelinear-polarizationopticalpropertyof wurtzitequantumrodsusingtheformermod. el/9,13-l5] 2Modelandcalculation Ifwetakethebasicfunctionsofthevalence. bandtopas ll,I)=(1/?2)(X+iY) I1,1)=Z andl1,一1)=(1/?2)(X—iY)(1) theeffective.massHamiltonian[.]ofaholeinthe zeroSOCcaseiswrittenas where —— 1H 2to— ho— o P1 S 丁 P1y1P.一 P3=),P.+ P3 丁 P1 y2P. T=‘7P+ 丁=‘7P+ (2) S=Ap.P+~/P S=一Ap.PI”一PI(3) P.andP’?arethesecondandfirst.ordertensors ofthemomentumoperator,respectively,p0 *ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No. 90301007)andtheSpecialFundsforNationalKeyBasicRe? searchPlanofChina(No.G001CB3095) tCorrespondingauthor.Email:zhxw99@semi.ac.cn Received24April2006,revisedmanuscriptreceived22June2006@2006Ch ineseInstituteofElectronics 第12期ZhangXiuweneta1.:Linear—PolarizationOpticalPropertyofCdSeQuan一 O 00 00 0 一 ? O 000 00 0一?20 00 000 00一 (4) Table1ParametersforCdSeintheactualcalculation mm71珐AMmcV?c/mcV 0.17560.17281+79850.71350.79701.44922.1660.37790.6532139.325 Here,wetakethebasisfunctionstobe Il,1)十,ll,0)十,ll,一1)十,ll,l>,ll,0) andl1,一1).Theenvelopefunctionsare = ??M;m+1/21.H af..一1.十Cf._『(kt.r)Y.一1(口,) b…tC2(kr)YIIm(口,) d..+1.十C.f(k,)Y.+(口,) af.,..Cf,f(足,)Y”(口,) bf.,+l,’C,』f(k,)Y”+1(口,) df_n.m+’Cf(k,)Y+(口,) (5) Misthezcomponentofthetotalangular momentum.TheeffectivemassHamiltonianofan electroniswrittenas H印=一? 1 = (去+1) = 1(1一击)? WetakethebasicfunctionsasS十andS, whereSistheBlochstateoftheconduction-band bottom.Theenvelopefunctionsare = (麓:::)? Forquantumrods,weintroduceacoordinate transformationt.] =x Y=Y Zez P:P一/e(9) whereeistheaspectratio;i?e.豢’Lithe length,andRisthetransverseradiusofthequan_ tumrods. Weassumethattheexternalmagneticfieldis appliedalongthezaxisofthecrystalstructure? Wechoosethesymmetricgauge,sothatthevector potentialiswrittenas A=(一丢B:y,Bz,o)(1O) Inthepresenceofanexternalmagneticfield, themomentumoperatorbecomesP+eA.Because thedifferentcomponentsofPdonotcommute, thentheP.ermsintheLuttingerHamiltonian arenotsymmetric.Luttinger[4]introducedthe symmetrizedproduct {PP口)=寺(PP.+p.P)(11) HedividedtheLuttingerHamiltonianinto twoparts,thesymmetricpartandtheantisym_ metricpart.Theantisymmetricpartissimply writtenas H…=KpBI?B(12) ThewholeHamiltoniansofelectronand hole,respectively,are H.=H.o+HJrL.+Hz..n_.(13) Hh=Hh.+H?+Hmm_h—H删m—Hzm?h (14) ThetermsH一 ., H_h,Hz?.,andHzc—n_h aregivenbyZhangela1.. Ifthewavepropagationisalongtheaxisof thecrystalstructure,thelinearpolarizationfactor iswrittenas P=(I:一I)/(I+I)(15) IandIaretheintensitiesofthezandy polarizedtransitions.Withtheopticaltransition betweenagivenelectronstateandagivenhole state,theyareproportionalto ,={?b.i..卅. ,={?(al,n,m~sPJl…,f.n.巩- e{.n…). dl,n,m,s…,)/} 口f.n.卅.,bf,n,卅.,d1.n,m,ande!.n.m.are inEq.(5)andEq.(8). 3Resultsanddiscussion (17) given Theeffectivemassparameters[.]forCdSe aregiveninTable1.Weusetheenergyunit,o ooo OOOO 2O96半导体第27卷 (簧)andthedimensionlessmagneticfield strengthun=. 3.1Structureandshapeeffects Theeffectofthecrystalfieldsplittingenergy polarizationopticalpropertyof ?onthelinear— CdSequantumrodsiSshowninFig.1.FromFig.1 (a),weseethatthelinearpolarizationfactorsof opticaItransitionsofCdSequantumspheresare negative.ThisiSduetothehexagonalcrystalsym. metryandthecrystalfieldsplittingenergy?, whichlifttheenergiesofthestatewithZBloch stateandmakethestatewithXYBlochstatesthe loweststateofthevalenceband,withtheresult 2 葛 .凹 急 山 ?/meV thatthelinearpolarizationfactorsarenegative.in agreementwiththeexperimentresults~?,. With bigger?,theenergiesofthestatewithZ—Bloch stateareliftedmore,andthelinearpolarization factorsbecomeevenmorenegative.Wealsosee thatwithbigger?,thelinearpolarizationfactors ofopticaltransitionsofCdSequantumlongrods withsmallradiusarelesspositiveandhavesma1. 1ersaturationvalues.FromFig.1(b),weseethat aseincreases,thenormalizedintensityof口tran. sitionsdecreases,andthenormalizedintensityof transitionsincreasesatfirst,thendecreases slowlyduetothedecreaseoftheoverlapofthe electronandholewavefunctionswheneissufficient. 1ylarge.FromFig.1(c),weseethatas?increases, j 暑 誊 呈 .坌 凹 急 P Fig.1(a)LinearpolarizationfactorsofopticaltransitionsofCdSequantumrod swithR2nmatT300Kas functionsofe;(b)Normalizedintensitiesofand:transitionsofCdSequantumr odswithR2nmand? 25meVatT300Kasfunctionsofe;(c)Linearpolarizationfactorsofopticaltran sitionsofCdSequantum sphereswithR=2nmatT=300Kasfunctionsof?;(d)LinearpolarizationfactorsofopticaltransitionsoflnAs quantumrodswithR=2nmatT=300Kasfunctionsofe 第12期 ZhangXiuweneta1.:Linear-PolarizationOpticalPropertyofCdSeQuantumR ods2097 thenegativelinearpolarizationfactordecreases. WithA=0,thelinearpolarizationfactoriSnot zero,butnegative.Thisisduetothehexagonal crystalsymmetryasweusetheeffective—masspa- rametersinTable1.Forexample,y1andy1arenot equal,whiletheyareequalinzinc-blendestruc- turesemiconductormateria1.Thelinearpolariza- tionfactorofazinc.blendequantumsphere shouldbezero.asshowninFig.1(d). Theeffectofthelateraldimensiononthe linear—polarizationopticalpropertyofCdSequan— tumrodsiSshowninFig.2.FromFig.2(a),in whicheisalwaysgreaterthan1.66,weseethat thelinearpolarizationfactorsdecreasefromposi- tivevaluestonegativevaluesasRincreases.The linearpolarizationfactorsofopticaltransitionsof 善 磊 山 暑 ‘磊 CdSequantumlongrodswithlargeradiusareneg— ative.ThisiSbecausewhenRissufficientlylarge, thequantumconfinementeffectthatmakesthe statewiththeZBlochstatetheloweststateofthe valencebandissoweakthatitcannotcompensate theeffectofthehexagonalcrystalsymmetryand thecrystalfieldsplittingenergy?,whichmakes thestatewithXyBlochstatestheloweststateof thevalence.band.Athighertemperatures,thelin- earpolarizationfactorsdecreasemoreslowly,Itis explicitlyshowninFig,2(c)thatCdSequantum longrodswithsmallradiushavepositivelinear polarizationfactors,andCdSequantumlongrods withlargeradiushavenegativelinearpolarization factors.Thenormalizedintensitiesareshownin Fig.2(b)andFig.2(d),respectively. Fig.2(a)LinearpolarizationfactorsofopticaltransitionsofCdSequantumrodswithL=20nmand?.=25meV asfunctionsofR;(b)NormalizedintensitiesofandztransitionsofCdSequantumrodswithL=20nmand?. =25meVatT=300KasfunctionsofR:(c)LinearpolarizationfactorsofopticaltransitionsofCdSequantum rodswithe=2andA=25meVasfunctionsofR;(d)NormalizedintensitiesofandtransitionsofCdSequan— turnrodswithe=2andA=25meVatT=300KasfunctionsofR 三?lu_【扫【sII?葚一 2098半导体第27卷 3.2Magneticfieldeffect Thelinearpolarizationfactorsofoptical transitionsofCdSequantumsphereswithR= 2nmand?.=25meVasfunctionsofbareshown inFig.3(a),andfittedinFig.3(b).FromFig.3 (a),weseethatthenegativelinearpolarization factordecreasesasbincreases.Thisismainlydue totheantisymmetricsplitting.Athighertempera? tures,thenegativelinearpolarizationfactorsare b(164.56T) Fig.3(a)Linearpolarizationfactorsofopticaltran- sitionsofCdSequantumsphereswithR=2nmandA. =25meVasfunctionsofb:(b)FunctionY= c—eme’d c+edx+e—dx bigger,anddecreasemoreslowlywithincreasing b.Wecanuseasimplemodelthattakesintoac— countonlytheantisymmetricsplittingtoexplain thisphenomenon.Inthesimplemodel,asbin— creases,theenergyofthe『1,1)stategoesdown, theenergyoftheJl,一1)stategoesup,andthe energyofthel1,0)statedoesnotchange.Consid. eringtheBoltzmanndistribution,thelinearpolar. izationfactorinthesimplemodelisafunctionof b,whoseformis Y=导筹(18)(18) whereYandxrepresentthelinearpolarization factorandthemagneticfieldstrengthb,respec- tively.WefitthethreecurvesinFig.3(a)atdif— ferenttemperatureswithdifferentparametersc anddinEq.(18)inFig.3(b).Theparameterd intheBoltzmannfactorisproportionaltothere ciprocalofthetemperature.Thesimplemodelfits verywel1.Thelinearpolarizationfactorsofopti. caltransitionsofCdSequantumrodswithR= 2nmande2and?.=25meVasfunctionsofb areshowninFig.4(a),andfittedinFig.4(b).We seethatthelinearpolarizationfactordecreasesas bincreases,thesameasFig.3(a).Athighertem— peratures,thenegativelinearpolarizationfactors havebiggervaluesatb=0,anddecreasemore slowlywithincreasingb.AtT=100Kthelinear polarizationfactorchangesfromapositivevalue toanegativevalueasbincreases.Wecallthe magneticfield,underwhichthelinearpolariza— tionfactoriszero,thecrossingmagneticfield,as showninFig.4(a).Atlowertemperatures,the linearpolarizationfactorhasbiggervalueatb= 0,anddecreasesmorequicklyasbincreases,so thecrossingmagneticfieldmaybesmaller.The linearpolarizationfactorofopticaltransitionsof CdSequantumrodswithR=3nmande=2andA =25meVatT=10Kasafunctionofbisshown inFig.5.Weseethatthelinearpolarizationfac— torchangesfromapositivevaluetoanegative valuearoundB=17.55Tasthemagneticfieldin- 4Conclusion Thelinear-polarizationopticalpropertyof CdSequantumrodsisstudiedindetail.Itisfound thatCdSequantumsphereshavenegativepolari- zationfactors(xy-polarizedemission).TheCdSe 0一葛高一0Q’I一1 第12期 ZhangXiuweneta1.:Linear—PolarizationOpticalPropertyofCdSeQuantumRods2099 havenegativelinearpolarizationfactors.Undera magneticfieldappliedalongthezaxisofthe crystalstructure.thelinearpolarizationfactors decrease.Thatistosay,thez-polarizedemission decreasesrelativetoxy.polarizedemissionasthe magneticfieldincreasesinthesphereandrodca— ses.ThelinearpolarizationfactorofCdSequan— rumrodswithsmallradiusmaychangefroma positivevaluetoanegativevalueasthemagnetic fieldincreases. 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