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Theoretical Studies of the Electronic, Magneto-Optical, and Transport Properties of Diluted Magnetic Semiconductors

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IowemygratitudetoallthepeoplewhomadethisthesispossibleandbecauseofwhommygraduateexperiencehasbeenonethatIwillcherishforever.FirstandforemostIwouldliketothankmyadvisor,ProfessorChristopherJ.Stanton,forgivingmeaninvaluableopportunitytoworkonchallengingandextremelyinterestingprojectsoverthepastfouryears.HehasalwaysmadehimselfavailableforhelpandadviceandtherehasneverbeenanoccasionwhenIhaveknockedonhisdoorandhehasnotgivenmetime.Hisphysicsintuitionimpressedmealot.Hetaughtmehowtosolveaproblemstartingfromasimplemodel,andhowtodevelopit.Ithasbeenapleasuretoworkwithandlearnfromsuchanextraordinaryindividual.IwouldalsoliketothankProfessorDavidH.Reitze,ProfessorSelmanP.Hersheld,ProfessorDmitriiMaslovandProfessorCammyAbernathyforagreeingtoserveonmythesiscommitteeandforsparingtheirinvaluabletimereviewingthemanuscript.MycolleagueshavegivenmelotsofhelpinthecourseofmyPh.D.studies.GarySandershelpedmegreatlytodeveloptheprogramcode,andwealwayshadfruitfuldiscussions.ProfessorStanton'sformerpostdocFedirKyrychenkoalsogavemegoodadviceandsomeinsightfulideas.IwouldalsoliketothankRongliangLiuandHaidongZhang,whomademylifeheremoreinteresting.Iwanttothankourresearchcollaborators.Dr.Kono'sgroupfromRiceUniversityprovidedmostoftheexperimentaldata.CollaborationwithDr.Konowasawonderfulexperienceinthepastfouryears.IalsohadfruitfuldiscussionwithProf.MiuraandDr.MatsudafromUniversityofTokyo. iii

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iv

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page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... xiv CHAPTER 1INTRODUCTIONANDOVERVIEW ................... 1 1.1Spintronics ............................... 1 1.2TheII-VIDilutedMagneticSemiconductors ............ 4 1.2.1BasicPropertiesofII-VIDilutedMagneticSemiconductors 4 1.2.2ExchangeInteractionbetween3d5ElectronsandBandElec-trons ............................. 8 1.3TheIII-VDilutedMagneticSemiconductors ............ 13 1.3.1FerromagneticSemiconductor ................ 13 1.3.2EectiveMeanField ..................... 21 1.4OpenQuestions ............................ 23 1.4.1NatureofFerromagnetismandBandElectrons ....... 23 1.4.2DMSDevices .......................... 24 2ELECTRONICPROPERTIESOFDILUTEDMAGNETICSEMICON-DUCTORS ................................. 29 2.1FerromagneticSemiconductorBandStructure ........... 29 2.2ThekpMethod ........................... 30 2.2.1IntroductiontokpMethod ................. 30 2.2.2Kane'sModel ......................... 34 2.2.3CouplingwithDistantBands-LuttingerParameters .... 38 2.2.4EnvelopeFunction ....................... 42 2.3LandauLevels ............................. 43 2.3.1ElectronicStateinaMagneticField ............. 43 2.3.2GeneralizedPidgeon-BrownModel .............. 44 2.3.3WaveFunctionsandLandauLevels ............. 49 2.4ConductionBandg-factors ...................... 53 v

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........................ 56 3.1GeneralTheoryofCyclotronResonance ............... 56 3.1.1OpticalAbsorption ...................... 56 3.1.2CyclotronResonance ..................... 60 3.2UltrahighMagneticFieldTechniques ................ 63 3.3ElectronCyclotronResonance .................... 64 3.3.1ElectronCyclotronResonance ................ 64 3.3.2ElectronCyclotronMass ................... 72 3.4HoleCyclotronResonance ...................... 74 3.4.1HoleActiveCyclotronResonance .............. 74 3.4.2HoleDensityDependenceofHoleCyclotronResonance .. 83 3.4.3CyclotronResonanceinInMnAs/GaSbHeterostructures .. 83 3.4.4ElectronActiveHoleCyclotronResonance ......... 90 4MAGNETO-OPTICALKERREFFECT .................. 96 4.1RelationsofOpticalConstants .................... 96 4.2KerrRotationandFaradayRotation ................ 101 4.3Magneto-opticalKerrEectofBulkInMnAsandGaMnAs .... 104 4.4Magneto-opticalKerrEectofMultilayerStructures ....... 107 5HOLESPINRELAXATION ......................... 112 5.1SpinRelaxationMechanisms ..................... 113 5.2LatticeScatteringinIII-VSemiconductors ............. 115 5.2.1ScreeninginBulkSemiconductors .............. 117 5.2.2SpinRelaxationinBulkGaAs ................ 118 5.3SpinRelaxationinGaMnAs ..................... 122 5.3.1ExchangeScattering ...................... 122 5.3.2ImpurityScattering ...................... 124 6CONCLUSION ................................ 129 REFERENCES ................................... 132 BIOGRAPHICALSKETCH ............................ 138 vi

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Table page 1{1SomeimportantII-VIDMS ........................ 4 2{1SummaryofHamiltonianmatriceswithdierentn 50 2{2InAsbandparameters ........................... 51 3{1Parametersforsamplesusedine-activeCRexperiments ........ 67 3{2CharacteristicsoftwoInMnAs/GaSbheterostructuresamples ..... 85 5{1ParametersforGaAsphononscattering ................. 121 vii

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Figure page 1{1ThebandgapdependenceofHg1kMnkTeonMnconcentrationk. .. 5 1{2ThebandstructuresofHg1xMnxTewithdierentx. ......... 6 1{3Cd1xMnxTex-Tphasediagram. .................... 7 1{4Averagelocalspinasafunctionofmagneticeldat4temperaturesinparamagneticphase. ......................... 10 1{5Magnetic-elddependenceofHallresistivityHallandresistivityofGaMnAswithtemperatureasaparameter. ............. 14 1{6MncompositiondependenceofthemagnetictransitiontemperatureTc,asdeterminedfromtransportdata. ................ 16 1{7VariationoftheRKKYcouplingconstant,J,ofafreeelectrongasintheneighborhoodofapointmagneticmomentattheoriginr=0. 17 1{8CurietemperaturesfordierentDMSsystems.CalculatedbyDietlusingZener'smodel. .......................... 19 1{9SchematicdiagramoftwocasesofBMPs. ................ 20 1{10Averagelocalspinasafunctionofmagneticeldat4temperatures. 22 1{11Thephoto-inducedferromagnetisminInMnAs/GaSbheterostructure. 25 1{12Spinlightemittingdiode. ......................... 27 1{13GaMnAs-basedspindevice. ....................... 28 2{1ValencebandstructureofGaAsandferromagneticGa0:94Mn0:06Aswithnoexternalmagneticeld,calculatedbygeneralizedKane'smodel. .................................. 30 2{2BandstructureofatypicalIII-Vsemiconductornearthepoint. .. 35 2{3CalculatedLandaulevelsforInAs(left)andIn0:88Mn0:12As(right)asafunctionofmagneticeldat30K. ................. 52 2{4TheconductionandvalencebandLandaulevelsalongkzinamag-neticeldofB=20TatT=30K. ................. 53 viii

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................ 54 2{6g-factorsofferromagneticIn0:9Mn0:1As. ................. 55 3{1Quasi-classicalpicturesofe-activeandh-activephotonabsorption. .. 62 3{2Thecorepartofthedevicebasedonsingle-coilmethod. ........ 64 3{3Astandardcoilbeforeandafterashot. ................. 65 3{4WaveformsofthemagneticeldBandthecurrentIinatypicalshotinsingle-turncoildevice. ........................ 65 3{5WaveformsofthemagneticeldBandthecurrentIinatypicaluxcompressiondevice. ........................... 66 3{6ExperimentalelectronCRspectrafordierentMnconcentrationsxtakenat(a)30Kand(b)290K. ................... 68 3{7Zone-centerLandauconduction-subbandenergiesatT=30Kasfunctionsofmagneticeldinn-dopedIn1xMnxAsfor=0andx=12%. ................................ 69 3{8ElectronCRandthecorrespondingtransitions. ............ 70 3{9CalculatedelectronCRabsorptionasafunctionofmagneticeldat30Kand290K. ............................ 71 3{10Calculatedelectroncyclotronmassesforthelowest-lyingspin-upandspin-downLandautransitionsinn-typeIn1xMnxAswithphotonenergy0:117eVasafunctionofMnconcentrationforT=30KandT=290K. ............................. 73 3{11Holecyclotronabsorptionasafunctionofmagneticeldinp-typeInAsforh-activecircularlypolarizedlightwithphotonenergy0:117eV. 75 3{12CalculatedcyclotronabsorptiononlyfromtheH1;1H0;2andL0;3L1;4transitionsbroadenedwith40meV(a),andzonecenterLan-daulevelsresponsibleforthetransitions(b). ............. 76 3{13ExperimentalholeCRandcorrespondingtheoreticalsimulations. .. 77 3{14ObservedholeCRpeakpositionsforfoursampleswithdierentMnconcentrations. ............................. 78 3{15Thedependenceofcyclotronenergiesonseveralparameters. ..... 79 3{16HoleCRspectraofInAsusingdierentsetsofLuttingerparameters. 80 3{17CalculatedLandaulevelsandholeCRinmagneticeldsupto500T. 81 ix

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......... 82 3{19BandstructurenearthepointforInAscalculatedbyeight-bandmodelandfullzonethirty-bandmodel. ................ 82 3{20TheholedensitydependenceofholeCR. ................ 84 3{21CyclotronresonancespectrafortwoferromagneticInMnAs/GaSbsamples. ................................. 86 3{22TheoreticalCRspectrashowingtheshiftofpeakAwithtemperature. 87 3{23AveragelocalizedspinasafunctionoftemperatureatB=0,20,40,60and100Tesla. ............................ 88 3{24RelativechangeofCRenergy(withrespecttothatofhightempera-turelimit)asafunctionoftemperature. ............... 89 3{25BanddiagramofInMnAs/GaSbheterostructure. ............ 90 3{26SchematicdiagramofLandaulevelsandcyclotronresonancetransi-tionsinconductionandvalencebands. ................ 91 3{27ThevalencebandLandaulevelsande-activeholeCR. ......... 92 3{28ExperimentalandtheoreticalholeCRabsorption. ........... 93 3{29ValencebandstructureatT=30KandB=100TforIn1xMnxAsalloyshavingx=0%andx=5%. .................. 94 3{30Theprimarytransitioninthee-activeholeCRunderdierentMndoping. ................................. 95 4{1Diagramforlightreectionfromtheinterfacebetweenmedium1withrefractiveindexN1andmedium2withrefractiveindexN2. .... 100 4{2Schematicdiagramformagneticcirculardichroism. .......... 102 4{3DiagramsforKerrandFaradayrotation. ................ 103 4{4KerrrotationofInMnAs. ......................... 105 4{5ThebanddiagramforInAs. ....................... 106 4{6KerrrotationofGaMnAs. ........................ 107 4{7ThebanddiagramforGaAs. ....................... 108 4{8TheabsorptioncoecientsbothinInMnAsandGaSblayers(a)andthereectivityofInMnAs/GaSbheterostructure(b). ........ 109 x

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4{9ReectivityofIn 0 : 88 Mn 0 : 12 Asnm/GaSbnmheterostructure at T =5 : 5KmeasuredbyP.FumagalliandH.Munekata. ..... 110 heterostructureunderamagneticeldof3Tat T =5 : 5K. 111 5{1Light-inducedMOKE.Signaldecaysinlessthan2ps. ......... 113 5{2Light-inducedmagnetizationrotation. .................. 114 5{3Theheavyholespinrelaxationtimeasafunctionofwavevectora, andtemperatureatthepointb. .................. 123 5{4Spinrelaxationtimeforaheavyholeasafunctionof k along,0,1 direction. ................................ 126 5{5Spinrelaxationtimeofaheavyholeasafunctionofholedensityat directiona,0,1andb,1,1. .................. 127 xi 4{10MeasuredaandcalculatedbKerrrotationofInMnAsnm/ AlSbnm

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Spintronicshasrecentlybecomeoneofthekeyresearchareasinthemagnetic-recordingandsemiconductorindustries.Akeygoalofspintronicsistoutilizemagneticmaterialsinelectroniccomponentsandcircuits.Ahopeistousethespinsofsingleelectrons,ratherthantheircharge,forstoring,transmittingandprocessingquantuminformation.Thishasinvokedagreatdealofinterestinspineectsandmagnetisminsemiconductors.Inmywork,theelectronicandopticalpropertiesofdilutedmagneticsemiconductors(DMS),especially(In,Mn)Asanditsheterostructures,aretheoreticallystudiedandcharacterized.Theelectronicstructuresinultrahighmagneticeldsarecarefullystudiedusingamodiedeight-bandPidgeon-Brownmodel,andthemagneto-opticalphenomenaaresuccessfullymodeledandcalculatedwithintheapproximationofFermi'sgoldenrule.Wehavefoundthefollowingimportantresults:i)MagneticionsdopedinDMSplayacriticalroleinaectingthebandstructuresandspinstates.ThespdinteractionbetweentheitinerantcarriersandtheMndelectronsresultsinashiftofthecyclotronresonancepeakandaphasetransitionoftheIII-VDMS xii

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xiii

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Thereisawideclassofsemiconductingmaterialswhicharecharacterizedbytherandomsubstitutionofafractionoftheoriginalatomsbymagneticatoms.Thematerialsarecommonlyknownasdilutedmagneticsemiconductors(DMS)orsemi-magneticsemiconductors(SMSC). SincetheinitialdiscoveryofDMSinII-VIsemiconductorcompounds[ 1 ],morethantwodecadeshavepassed.TherecentdiscoveryofferromagneticDMSbasedonIII-Vsemiconductors[ 2 ]hasleadtoasurgeofinterestinDMSforpossiblespintronicsapplications.Manypapershavebeenpublishedinvestigatingtheirelectronic,magnetic,optical,thermal,statisticalandtransportproperties,inmanyjournals,andeveninpopularmagazines[ 3 ].ThisinterestnotonlycomesfromtheDMSthemselvesasgoodtheoreticalandexperimentalsubjects,butalsocanbebetterunderstoodfromabroaderviewfromtherelationofDMSresearchwithspintronics[ 4 ]. 1

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willeventuallybealimittohowmanytransistorstheycancramonachip.ButevenifMoore'sLawcouldcontinuetospawnever-tinierchips,smallelectronicde-vicesareplaguedbyabigproblem:energyloss,ordissipation,assignalspassfromonetransistortothenext.LineupallthetinywiresthatconnectthetransistorsinaPentiumchip,andthetotallengthwouldstretchalmostamile.Alotofusefulenergyislostasheataselectronstravelthatdistance.Spintronics,whichusesspinastheinformationcarriers,incontrastwithconventionalelectronics,consumeslessenergyandmaybecapableofhigherspeed. Spintronicsemergedonthestageinscienticeldin1988whenBaibichetal.discoveredgiantmagnetoresistance(GMR)[ 5 ],whichresultsfromtheelectron-spineectsinmagneticmaterialscomposedofultra-thinmultilayers,inwhichhugechangescouldtakeplaceintheirelectricalresistancewhenamagneticeldisapplied.GMRishundredsoftimesstrongerthanordinarymagnetoresistance.BasingonGMRmaterials,IBMproducedin1997newreadheadswhichareabletosensemuchsmallermagneticelds,allowingthestoragecapacityofaharddisktoincreasefromtheorderof1totensofgigabytes.AnothervaluableuseofGMRmaterialisintheoperationofthespinlter,orspinvalve,whichconsistsof2spinlayerswhichletthroughmoreelectronswhenthespinorientationsinthetwolayersarethesameandfewerwhenthespinsareoppositelyaligned.Theelectricalresistanceofthedevicecanthereforebechangeddramatically.Thisallowsinformationtobestoredas0'sand1's(magnetizationsofthelayersparallelorantiparallel)asinaconventionaltransistormemorydevice.Astraightforwardapplicationcouldbeinthemagneticrandomaccessmemory(MRAM)devicewhichisnon-volatile.Thesedeviceswouldbesmaller,faster,cheaper,uselesspowerandwouldbemuchmorerobustinextremeconditionssuchashightemperature,orhigh-levelradiationenvironments.

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Currently,besidescontinuingtoimprovetheexistingGMR-basedtechnology,peoplearenowfocusingonndingnovelwaysofbothgeneratingandutilizingspin-polarizedcurrents.Thisincludesinvestigationofspintransportinsemiconductorsandlookingforwaysinwhichsemiconductorscanfunctionasspinpolarizersandspinvalves.Wecancallthissemiconductorbasedspintronics,theimportanceofwhichliesinthefactthatitwouldbemucheasierforsemiconductor-baseddevicestobeintegratedwithtraditionalsemiconductortechnology,andthesemiconductorbasedspintronicdevicescouldinprincipleprovideamplication,incontrastwithexistingmetal-baseddevices,andcanserveasmulti-functionaldevices.Duetotheexcellentopticalcontrollabilityofsemiconductors,therealizationofopticalmanipulationofspinstatesisalsopossible. Althoughthereareclearmeritsforintroducingsemiconductorsintospintronicapplications,therearefundamentalproblemsinincorporatingmagnetismintosemiconductors.Forexample,semiconductorsaregenerallynonmagnetic.Itishardtogenerateandmanipulatespinsinthem.Peoplecanovercometheseproblemsbycontactingthesemiconductorswithother(spintronic)materials.However,thecontrolandtransportofspinsacrosstheinterfaceandinsidethesemiconductorisstilldicultandfarfromwell-understood.Fortunately,thereisanotherapproachtoinvestigatingspincontrolandtransportinall-semiconductordevices.ThisapproachhasbecomepossiblesincethediscoveryofDMS. ThemostcommonDMSstudiedintheearly1990swereII-VIcompounds(likeCdTe,ZnSe,CdSe,CdS,etc.),withtransitionmetalions(e.g.,Mn,FeorCo)substitutingfortheiroriginalcations.TherearealsomaterialsbasedonIV-VI(e.g.,PbS,SnTe)andmostimportantly,III-V(e.g.,GaAs,InSb)crystals.Mostcommonly,Mnionsareusedasmagneticdopants.

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1.2.1BasicPropertiesofII-VIDilutedMagneticSemiconductors 1 ],andhasbeengivenagreatdealofattentioneversince[ 6 ].ThemoststudiedII-VIDMSmaterialsarelistedinTable 1{1 Table1{1: SomeimportantII-VIDMS MaterialCrystalStructurexrange1 7 ]. 1{1 [ 8 ]. WiththedenitionofthebandgapasEg=E6E8,thebandstructuresofHg1xMnxTewithdierentxaregiveninFig. 1{2 [ 8 ].Withx0:075,Eg<0,andwithx>0:075,Eg>0.Withoutspin-orbitalcoupling,weshouldhaveasix-folddegeneratevalencebandatthepoint.Consideringspin-orbitalcoupling,thevalencebandsplitsintotwobands-7and8(split-oband),withanenergydierenceof. Theelectroneectivemass,i.e.,thebandcurvature,willalsochangewithx.Atsomexvalues,theeectivemassbecomessosmallthatthemobilityofelectronscanbeveryhigh.Forinstance,=106cm2=VsforHg1xMnxTewhenx=0:07at4:2K.

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Figure1{1: ThebandgapdependenceofHg1kMnkTeonMnconcentrationk.ReprintedwithpermissionfromBastardetal.Phys.Rev.B24:1961-1970,1981.Figure10,Page1967. 1{3 isthephasediagramofCd1xMnxTeobtainedfromspecicheatandmagneticsusceptibilitymeasure-ments[ 9 ].TheDMSsystemmaybeconsideredascontainingtwointeractingsubsystems.Therstoftheseisthesystemofdelocalizedconductionandvalencebandelectrons/holes.Thesecondistherandom,dilutedsystemoflocalizedmagneticmomentsassociatedwiththemagneticatoms.Thesetwosubsystemsinteractwitheachotherbythespinexchangeinteraction.Thefactthatboththestructureandtheelectronicpropertiesofthehostcrystalsarewellknownmeansthattheyareperfectforstudyingthebasicmechanismsofthemagneticinteractionscouplingthespinsofthebandcarriersandthelocalizedspinsofmagneticions.Thecouplingbetweenthe

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Figure1{2: ThebandstructuresofHg1xMnxTewithdierentx.isthespin-orbitalsplitting,HHindicatestheheavyholeband,andLHthelightholeband,respectively.ReprintedwithpermissionfromBastardetal.Phys.Rev.B24:1961-1970,1981.Figure1,Page1961. localizedmomentsresultsintheexistenceofdierentmagneticphasessuchasparamagnets,spinglassesandantiferromagnets.

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Figure1{3: Cd1xMnxTex-Tphasediagram.P:Paramagnet;A:Antiferromagnet;s:spin-glass,mixedcrystalwhenx>0:7.ReprintedwithpermissionfromGalazkaetal.Phys.Rev.B22:3344-3355,1980.Figure12,Page3352.

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10 ].Inthefollowing,IwillbrieyintroduceasimplequalitativetheoreticalapproachtoII-VIDMS. SupposethestateofMnionsinDMSmaterialisMn2+.Theelectronicstruc-tureofMn2+is1s22s22p63s23p63d5,inwhich3d5isahalf-lledshell.AccordingtoHund'srule,thespinoftheseve3d5electronswillbeparalleltoeachother,sothetotalspinisS=5=2.Theseveelectronsareinstatesinwhichtheorbitalangularmomentumquantumnumberl=0;1;2.Thusthetotalorbitalangularmomen-tumL=0.ThetotalangularmomentumforaMn2+ionthenisJ=S=5=2.TheLandeg-factoris 2J(J+1)=2:(1{1) AnalogoustotheexchangeinteractionintheHydrogenmolecule,theexchangeinteractionbetweena3d5electronandabandelectroncanbewrittenintheHeisenbergform whereisthespinofabandelectron/hole,Jistheexchangeconstant,andSisthetotalangularmomentumofall3d5electronsinaMn2+ion. Inthenon-interactingparamagneticphase,averysimpliedmodelwillbedescribedinthefollowing.SinceL=0forMn2+,themagneticmomentumfor

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Mn2+is=(ge=2m0)J=(ge=2m0)S.AssumingamagneticeldBalongthezdirection,theadditionalenergyinthiseldofaMn2+ionisB=gBmsB,wherems=5=2,3/2,1/2,-1/2,-3/2,-5/2.Assumingnon-interactivespins,andusingaclassicBoltzmandistributionfunctionegBmsB=kBT,theaveragemagneticmomentinthezdirectionisthen Thiscanbewrittenas whereBs(y)istheBrillouinfunction 2Scoth2S+1 2Sy1 2Scothy TheaveragespinofoneMn2+ionthenis TheantiparallelorientationofBandhSziisduetothedierenceinsignofthemagneticmomentandtheelectronspin.SinceBisdirectedalongthezaxis,theaverageMnspinsaturatesathSzi=5=2.The(paramagnetic)dependenceofhSzionmagneticeldandtemperatureisshowninFig. 1{4 FromEq. 1{2 ,theexchangeHamiltonianofonebandelectronwithspininteractingwiththe3d5electronsfromallMn2+ionsis, whereristhepositionvectorofthebandelectron,andRiisthepositionvectoroftheithMn2+ion,J(rRi)istheexchangecouplingcoecientofthebandelectron

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Figure1{4: Averagelocalspinasafunctionofmagneticeldat4temperaturesinparamagneticphase. withthe3d5electronsintheithMn2+ion.Siisthetotalangularmomentumofthe3d5electronsintheithMn2+ion. NextwewilluseavirtualcrystalapproximationtodealwithHamiltonian 1{7 .Duetothefactthethewavefunctionofabandelectronactuallyextendsoverthewholecrystal,itinteractswithalltheMn2+ionssimultaneously.Inthemeaneldframework,wecanreplacetheangularmomentumofeachMn2+ionbytheaveragevalue.Stillassumingamagneticeldalongzdirection,wehavehSi=hSzi,andSi=hSzimch2.mc=1=2hereindicatesthespinquantumnumberofthebandelectrons.ThehSziisgivenbyEq. 1{6 .Theexchange

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Hamiltonianthencanbewrittenas Becauseoftheextendednatureofthebandelectronstates,whichinteractwiththe3d5electronsinallMn2+ions,thepositionsoftheseMn2+ionsarenotimportant.WecandistributeapproximatelytheseMn2+ionsuniformlyatcationsites.ThisamountstoassumingwehaveanequivalentmagneticmomentofxhSziateachcationsite.So,Eq. 1{8 becomes HereRbecomesthepositionvectorofeachcationsite.InEq. 1{9 theexchangeHamiltoniannowhasthesameperiodicityasthecrystal. FromtheHamiltonian 1{9 ,theexchangeenergycanthenbeobtainedby Fortheelectronsattheconductionbandedge,thewavefunctionisck=uc0(k=0).J(rR)isthecouplingcoecientaswehavesaidabove,whichistheexchangeintegralbetweenthebandelectronsand3d5electrons.Duetothefactthatthe3d5electronsarestronglylocalized,wecanassumetheintegralisonlynonvanishinginaunitcellrangeforaspecicRinEq. 1{10 .ConsideringtheperiodicityofJ(rR),theEq. 1{10 canberewrittenas, whereNisthenumberofunitcellsinthecrystal. Forzinc-blendesemiconductors(mostII-VIandIII-Vsemiconductors),thestatesforconductionband-edge(k=0)electronsares-like,andthoseforvalence

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band-edgeholesarep-like.Sotheuseofmc=1=2isjustied.TheninamagneticeldB,theconductionbandenergyis, 2)h!c+mcgcBB+mcNxhSzi;(1{13) where!c=eB=mcisthecyclotronfrequency,andgcistheconductionbandg-factor.InEq. 1{13 ,thersttermistheLaudausplitting,thesecondtermistheZeemansplitting,andthethirdtermistheexchangesplitting,whichisuniquefortheDMS. Similarly,theenergystructureofthevalencebandcanalsobeobtained,ifwereplace!cby!v=eB=mv,gcbygv,mcbymv,andimportantly,by,where WecanintroduceaneectiveLandeg-factorintheconductionband whichindicatesthestrengthofthespinsplittingoftherstLandaulevelintheconductionband.Intheloweldapproximation,Eq. 1{6 becomeshSzi=gBS(S+1)B=3kBT,sointhislimit 3kBT:(1{16) Atlowtemperature,theeectiveg-factorcanreachverylargevalues.Theg-factordependsontemperaturethroughhSziinEq. 1{15 .Wewillhaveamoredetaileddiscussionofg-factorsinChapter2. Theabovediscussionisaverysimpliedqualitativemodel,andonlyappropri-ateforII-VIDMSinaparamagneticphase,wheretheMnconcentrationisnotso

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highthattheydon'thaveadirectexchangeinteraction.ThisdiscussioncanalsobeappliedtoparamagneticIII-VDMS,inwhichcommonlytheMnsolubilityareverylow.Asamatteroffact,althoughEq. 1{13 cangiveaqualitativediscriptionoftheconductionbandstructure,itdoesnotworkinrealcases.Chapter2givesaquantitativemodel. SincethediscoveryofferromagnetisminIII-VDMS,muchresearchnowfocusesonexploringferromagnetismmechanisms,lookingfornewmaterialsandobtaininghigherCurietemperatures.Recently,ferromagnetisminII-VIDMSwasalsoreportedbyseveralgroups[ 11 12 13 ]. 1.3.1FerromagneticSemiconductor 1{15 andEq. 1{16 ,wecanseethatatlowtempera-tures,theg-factorcanbeverylarge,butitisstronglytemperaturedependent.Aswementionedabove,theg-factoractuallyindicatesthespinsplitting.Toemployspinasasubjectinresearchanddevicedesign,alargespinsplittingisessen-tial.WhilemostII-VIDMSareparamagnetic,thespinsplittingbecomessmallathightemperatures,sotherealizationofroomtemperaturespintronicdevicesbecomesdicult.Theanswerforthisproblemisferromagneticsemiconductors.Wecanexpectalargespinsplittingevenathightemperaturesforferromagneticsemiconductors. TheleapfromII-VIDMStoIII-VDMSshouldhavebeenverynatural.ButunlikeII-VIsemiconductors,MnisnotverysolubleinIII-Vsemiconductors.Itcanbeincorporatedonlybynon-equilibriumgrowthtechniquesanditwasnotuntil1992thattherstIII-VDMS,InMnAswasgrownandinvestigated.Ferromagnetismwassoondiscoveredinthissystem[ 14 ].Higherferromagnetic

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Figure1{5: Magnetic-elddependenceofHallresistivityHallandresistivityofGaMnAswithtemperatureasaparameter.Mncompositionisx=0:053.TheinsetshowsthetemperaturedependenceofthespontaneousmagnetizationMsdeterminedfrommagnetotransportmeasurements;thesolidlineisfrommean-eldtheory.ReprintedwithpermissionfromMatsukuraetal.Phys.Rev.B57:R2037-R2040,1998.Figure1,PageR2037.

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transitiontemperatureswerealsoachievedinGaMnAs[ 15 ].ShowninFig. 1{5 isthemagnetic-elddependenceoftheHallresistivityandthenormalresistivityofGaMnAswithtemperatureasaparameter[ 16 ].Inthiscase,theferromagnetictransitiontemperatureisabout110K.ThediscoveryofferromagnetisminIII-VDMSledtoanexplosionofinterest[ 14 15 17 18 ].Manynewmatetialswereinvestigated,theoriesexplainingtheferromagnetismmechanismswerebroughtforward,andexperimentsaimedatincreasingtheCurietemperatureswerecarriedout. AlthoughInMnAswastherstMBEgrownIII-VDMS,itsCurietemperaturewasrelativelylowatabout7:5K.In1993,ahigherCurietemperatureof35KwasrealizedinaptypeInMnAs/GaSbheterostructure[ 17 ].Since1996,anumberofgroupsareworkingontheMBEgrowthofGaMnAsandrelatedheterostructures,inwhichthehighestCurietemperature(173K)hasbeenachievedrecentlyfor25nmthickGa1xMnxAslmswith8%nominalMndopingafterannealing[ 19 ].ThedependenceoftheCurietemperatureofGa1xMnxAsonMnconcentrationxisshowninFig. 1{6 [ 16 ].TheCurietemperaturereachesthehighestvaluewhenx=5:3%inthiscase. GaMnNandGaMnParealsocandidatesforhighCurietemperatureIII-VDMSmaterials.FerromagnetisminGaMnNiselusive.WhilesomegroupsfounditparamagneticwhendopedwithpercentlevelsofMn[ 20 ],somegroupshavereportedaferromagnetictransitiontemperatureabove900K[ 21 ].RoomtemperatureferromagnetismwasalsoreportedinGaMnP[ 22 23 ].BesidesIII-VDMS,MndopedIVsemiconductorslikeGeMn[ 24 25 ],SiMn[ 26 ],werealsoreportedferromagnetic. ThetheoryforferromagnetisminIII-VDMSisstillcontroversial,however,thereisconsensusthatitismediatedbytheitinerantholes.UnlikethecaseinII-VIDMSinwhichMnionshavethesamenumberofvalenceelectronsasthe

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Figure1{6: MncompositiondependenceofthemagnetictransitiontemperatureTc,asdeterminedfromtransportdata.ReprintedwithpermissionfromMatsukuraetal.Phys.Rev.B57:R2037-R2040,1998.Figure2,PageR2038. cations,MnionsinIII-VDMSarenotonlyprovidersofmagneticmoments,theyarealsoacceptors.DuetocompensatingdefectslikeAs-antisitesor/andMninterstitials[ 27 28 29 ],holeconcentrationsaregenerallymuchlowerthantheMnconcentration. Thetheoriesofcarrierinducedferromagnetismfallintofourcategories. 1. 30 ]andlaterextendedbyKasuya[ 31 ]andYosida[ 32 ]togivethetheorynowgenerallyknowastheRKKYinteraction.Ohnoetal.explainedtheferromagnetisminGaMnAsforMnconcentrationx=0:013usingtheRKKYmechanism[ 14 ].IntheinteractionHamiltonian, thecouplingcoecientJRKKYi;jassumestheform[ 33 ],

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wherekFistheradiusoftheconductionelectron/holeFermisurface,risthedistanceawayfromtheoriginwherealocalmomentisplaced.TheRKKYexchangecoecient,J,oscillatesfrompositivetonegativeastheseparationoftheionschangeswiththeperioddeterminedbytheFermiwavevectork1FandhasthedampedoscillatorynatureshowninFig. 1{7 .Therefore,dependingupontheseparationbetweenapairofionstheirmagneticcouplingcanbeferromagneticorantiferromagnetic.Amagneticioninducesaspinpolarizationintheconductionelectronsinitsneighborhood.Thisspinpolarizationintheitinerantelectronsisfeltbythemomentsofothermagneticionswithinrange,leadingtoanindirectcoupling. Figure1{7: VariationoftheRKKYcouplingconstant,J,ofafreeelectrongasintheneighborhoodofapointmagneticmomentattheoriginr=0. InthecaseofDMS,theaveragedistancebetweenthecarriersrc=4p 3isusuallymuchgreaterthanthatbetweenthespinsrS=4xN 3.AsimplecalculationshowthattherstzerooftheRKKYfunctionoccursatr1:17rc.Thismeansthatthecarrier-mediatedinteractionisferromagneticandeectivelylongrangeformostofthespins. TheRKKYinteractionasthemainmechanismfortheferromagnetisminIII-VDMSisquestionableinsomecasessuchasintheinsulatingphase(x<3%forGaMnAs),inwhichcarriersarenotitinerant.Whentheholedensityislow,andthereisnoFermisurface(Fermilevelinthegap),RKKYtheorycannotpredictferromagnetism.Theotherproblem,maybefatal,isthatintheRKKYapproximationtheexchangeenergyismuchsmallerthan

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theFermienergy,whichisnotcommonlythecaseinDMS.Asamatteroffact,thesetwoenergiesarecomparableinmostcases. 2. 34 ]tointerprettheferromagneticcouplingintransitionmetals.SimilartotheRKKYmodel,itdescribesanexchangeinteractionbetweencarriersandlocalizedspins.TheHamiltonianofZener'smodelinatransitionmetalis[ 34 ] 2S2dSsSc+1 2S2c;(1{19) whereSdandScarethemeanmagnetizationofthed-shellelectronandtheconductionelectron,respectively,and,,andarethreecouplingconstants.Themainassumptionhereisthattheexchangeconstantisalwayspositive,whichundercertaincircumstancesleadstoferromagneticcoupling.ComparingHamiltonian 1{17 and 1{19 ,wecanseethatinEq. 1{19 playsthesimilarroleofJinEq. 1{17 .OnebigdierenceisthatZener'smodelneglectstheitinerantcharacterandtheFriedeloscillationsoftheelectronspinpolarizationaroundthelocalizedspins. Dietl[ 35 ]appliedZener'smodeltoferromagneticsemiconductorsandpredictedtheCurietemperatureTCforseveralMndopedDMSsystems.TheresultsareshowninFig. 1{8 .Thisquiteaccuratelypredictsthe110KtransitiontemperatureinGaMnAs,butcertainlythisisstillaquitecoarsemodel.Evenso,thetrendshowninFig. 1{8 hasstimulatedtheenthusiasmofpeopleinvestigatingGaNbasedmaterialslookingforhighertransitiontemperatures. SomeoftheproblemsintheRKKYmodelremaininZener'smodel.Forinstance,Zener'smodelstillhaslimitedapplicationwhencarriersaremostlylocalizedbecauseitstillrequiresitinerantcarrierstomediatetheinteractionsbetweenlocalizedspins.Besides,whenthecarrierdensityishigherthantheMnconcentration,importantchangesintheholeresponsefunctionoccuratthelengthscaleofthemeandistancebetweenthelocalizedspins.Accordingly,thedescriptionofspinmagnetizationbythecontinuous-mediumapproximation,whichconstitutesthebasisoftheZenermodel,ceasestobevalid.Incontrast,theRKKYmodelisagoodstartingpointinthisregime. 3. 1{7 ,andthiseldcausesferromagneticcouplingoftheselocalspins.Thenetspinalignment

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Figure1{8: CurietemperaturesfordierentDMSsystems.CalculatedbyDietlusingZener'smodel. againcreatesaself-consistentexchangeeldforthecarriers.Inthisprocess,thecarrierspincreatesamagneticpotentialwellresultinginformationofa\spincloud",amagneticpolaron.DuetothelocalizedcharacterofthesemagneticpolaronsinDMS,theyarecalledboundmagneticpolarons(BMP). TherehavebeenextensivestudiesofBMPinII-VIDMS[ 6 ],inwhichBMPareaccountableformanyopticalandphasetransitionproperties.Recently,Bhattetal.[ 36 ]andDasSarmaetal.[ 37 ]generalizedBMPtheoryforIII-VDMS.TheystudiedthecouplingbetweentwoadjacentBMPs,andconcludedthattheexchangecouplingisferromagnetic.Therearetwodierentcases.Inonecasetwopolaronsoverlapandtheoverlapintegralaccountsfortheferromagneticcoupling.TheferromagnetictransitioncanberegardedasapercolationoccurringthroughthewholesystemwhenthetemperaturedropsbelowtheCurietemperature.Intheothercaseonedoesnotneedoverlappingpolarons,theireectonthemagneticmomentbeingtakenintoaccountthroughalocalmagneticeld.Ferromagneticcouplinghasbeenshowntoresultwhenthecarrierisallowedtohopbetweenthegroundstateofonemagneticatomandexcitedstatesoftheother.AdiagramofthesetwocasesareshowninFig. 1{9 TheBMPmodelquitenaturallyandsuccessfullyexplainsthemagnetismoftheDMSintheinsulatingphase.Withamuchhighercarrierdensity,most

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Figure1{9: SchematicdiagramoftwocasesofBMPs. carriersareconducting.Theyaremorelikefreebandcarriers.Insuchacase,theBMPmodelmaynotbeappropriate.Althoughsomepartofthecarriersarelocalizedandhaveexchangeinteractionwiththelocalizedspins,mostcarriershaveextendedwavefunctions,whichtendtointeractwiththeothercarriersandspinsinthewholeband.TheconditionfortheBMPmodeldoesnotexistanymore.Insuchacase,theRKKYmechanismshoulddominate. 4. 38 ]performedrstprincipleDFTcalculationswhichshowthatthemajorityofthecarrierscomesfromMndstates.Thehoppingofthecarriersbetweentheimpuritybandsandvalencebandscausestheferromagneticordering.Later,Inoueetal.[ 39 ]alsodiscussedasimilarmechanism.TheycalculatedtheelectronicstatesofIII-VDMSandfoundthatresonantstateswereformedatthetopofthedownspinvalencebandduetomagneticimpuritiesandtheresonantstatesgaverisetoastronglong-rangedferromagneticcouplingbetweenMnmoments.Theyproposedthatcouplingoftheresonantstates,inadditiontotheintra-atomicexchangeinteractionbetweentheresonantandnonbondingstateswastheoriginoftheferromagnetismofGaMnAs.Wecanclassifythiskindofmechanismcausedbythehoppingofcarriersbetweenimpuritystatesandvalencestatesasadoubleexchangemechanism.Double-exchange-likeinteractionsinGaMnAswerereportedbyHirakawaetal.[ 40 ]. InthefourmodelsofferromagnetisminIII-VDMS,therstthreearemean-eldbasedtheories,andthelastisbasedond-electrons.Thougheachofthemis

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capableofexplainingsomespecicaspectsofferromagnetism,noneofthemcanbeapplieduniversally. SupposeaHeisenberg-likeHamiltonian wherei,jspecifyatomicsites,say,ofthemagneticmomentsinthecrystal,andJi;jistheinteratomicexchangeinteractionconstant.Themoleculareld(eectivemeaneld)issimplygivenby wheregisthegfactor.UsingtheresultswegotinthediscussioninSection1.2.2,theaveragespinalongamagneticeldB(supposeitisdirectedalongz)willbe withBs,theBrillouinfunction,givenbyEq. 1{5 ,andwhere AftersubstitutionofEq. 1{22 toaboveequation,weget

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Figure1{10: Averagelocalspinasafunctionofmagneticeldat4temperatures.TheCurietemperatureis110K. Equation 1{22 canbesolvedbystandardrootndingprogramstondhSzi.ThesolutionforhSzi6=0existsevenwhenB=0duetotheinternalexchangeeld.WhenjhSzij1, 3(S+1)y:(1{26) WhenJ0>0,theconditionforhSzi6=0thenis ThisisconsistentwiththefactthatJ>0inHeisenbergHamiltonianleadstoferromagneticinteraction. Inarealisticcalculation,TCasameasurableparameteriseasytoobtain,hencewecanuseEq. 1{27 tondtheexchangeinteractionconstantJ0,andthusthespontaneousmagnetizationforT
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Thespontaneousmagnetizationhasfundamentaleectsoncarrierscatteringandspinscattering,andthusaectsthetransportpropertiesofbothcarriersandspins.WewilltalkaboutthisinChapter5. 41 42 43 ]performedrstprinciplecalculationsshowingthatcontrarytotheRKKYmodel,theholeinducedbyMnisnothost-like,whichunderminesthebasisofapplyingRKKYtheorytoDMS.TheensuingferromagnetismbytheholesinducedbyMnionsisthennotRKKY-like,but\hasacharacteristicdependenceonthelattice-orientationoftheMn-MninteractionsinthecrystalwhichisunexpectedbyRKKY".Theyclaimthatthedominantcontributiontostabilizingtheferromagneticstatewastheenergyloweringduetothepd

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hopping.Thenatureoftheferromagnetismtheniscloselyrelatedtothenatureofthebandelectrons.Photo-inducedferromagnetism[ 44 ]clearlyrevealstheroleofholesinmediatingtheferromagneticcoupling.Thereisnodoubtcarriersarecrucialinallthemechanismsaccountingfortheferromagnetism,butaretheyreallyhost-likeholes,ordotheyhavestrongdcomponentmixing?Howdotheybehaveintheprocessofmediatingtheferromagnetism?Onlyafterweknowtherightanswer,willthemanipulationofchargecarriersandalsothespinsbecomemorepredicable. 45 ].SinceII-VIDMSisparamagneticatroomtemperature,amagneticeldisneededtoobtainFaradayrotation.FerromagneticsemiconductorbasedonIII-VDMS,whichdoesnotneedanexternalmagneticeldtosustainthebigFaradayrotation,shouldhaveagoodpotentialforuseinopticalisolators. Photo-inducedferromagnetismhasbeendemonstratedbyKoshiharaetal.[ 44 ]andKonoetal.[ 46 ].InKoshihara'sexperiment,ferromagnetismisinducedbyphoto-generatedcarriersinInMnAs/GaSbheterostructures.TheeectisillustratedinFig. 1{11 .Duetothespecialbandalignmentofthisheterostructure,electronsandholesarespaciallyseparated,andholesaccumulateintheInMnAslayer.Thephoto-generatedholesthencauseatransitionoftheInMnAslayertoaferromagneticstate.Thisopensapossibilitytorealizeopticallycontrollablemagneto-opticaldevices.InKono'sexperiment,ultrafastdemagnetizationtakesplaceafteralaserpulseshinesonInMnAs/GaSbheterostructureandproduces

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ferromagnetism.Thetimescaleistypicallyofseveralps.Theyproposeanewandveryfastschemeformagneto-opticalrecording. Figure1{11: Thephoto-inducedferromagnetisminInMnAs/GaSbheterostructure.ReprintedwithpermissionfromKoshiharaetal.Phys.Rev.Lett.78:4617-4620,1997.Figure3,Page4619. Recently,Ohnoetal.[ 2 ]achievedcontrolofferromagnetismwithanelectriceld.Theyusedeld-eecttransistorstructurestovarytheholeconcentrationsinDMSlayersandthusturnthecarrier-inducedferromagnetismonandobyvaryingtheelectriceld.Rashbaetal.[ 47 ]alsoproposedtheelectronspinoperationbyelectricelds.Theyalsodiscussedthespininjectionintosemiconductors.The

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electriccontrolofferromagnetismorspinstatesmakespossibleaunicationofmagnetismandconventionalelectronics,andthushasaprofoundmeaning. Low-dimensionalstructuresusuallyhavedramaticallydierentpropertiesfrombulkmaterials.Muchlongerspincoherenttimeshavebeenreportedbyseveralgroupsinquantumdots[ 48 49 ],whichhavebeensuggestedforuseinquantumcomputerswherequantumdotscanbeusedasquantumbits,sincetheyoeratwo-levelsystemclosetotheidealcase.OneultimategoalofDMSspintronicsistoimplementquantumcomputing.Theuseofsemiconductorsinquantumcomputinghasvariousbenets.Theycanbeincorporatedintheconventionalsemiconductorindustry,andalso,low-dimensionalstructuresareveryeasytoconstruct,souniquelow-dimensionalpropertiescanbeemployed.Severalproposalshavebeenmadeforquantumcomputingusingquantumdots[ 50 51 52 ]. Spinmanipulationneedsinjection,transportanddetectionofspins.Themostdirectwayforspininjectionwouldseemtobeinjectionfromaclassicalferromagneticmetalinametal/semiconductorheterostructurebutthisraisesdicultproblemsrelatedtothedierenceinconductivityandspinrelaxationtimeinmetalsandsemiconductors[ 53 ].Althoughtheseproblemsarenowbetterunderstood,thishassloweddowntheprogressforspininjectionfrommetals.Ontheotherhand,thishasboostedtheresearchofconnectingDMSwithnonmagneticsemiconductorsforspininjection.Manyexperimentspursuinghigneciencyspininjectionhavebeencarriedout.ShowninFig. 1{12 isaspinlightemittingdiode[ 54 ],inwhichacurrentofspin-polarizedelectronsisinjectedfromthedilutedmagneticsemiconductorBexMnyZn1xySeintoaGaAs/GaAlAslight-emittingdiode.Circularlypolarizedlightisemittedfromtherecombinationofthespinpolarizedelectronswithnon-polarizedholes.Aninjectioneciencyof90%spinpolarizedcurrenthasbeendemonstrated.AsBexMnyZn1xySeisparamagnetic,thespinpolarizationisobtainedonlyinanappliedeldandatlowtemperature.

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Figure1{12: Spinlightemittingdiode. AferromagneticIII-VDMSbasedspininjectordoesnotneedanappliedeld.ShownintheleftpanelofFig. 1{13 isaGaMnAs-basedspininjectionanddetectionstructure[ 55 ],inwhichspin-polarizedholesareinjectedfromGaMnAstoaGaAsquantumwell.TheemitterandanalyzerarebothmadeoflayersofferromagneticsemiconductorGaMnAs.ThetemperaturedependenceofthespinlifetimeintheGaAsquantumwellfrommagnetoresistancemeasurementsisshownintherightpanel. Toobtaintheinformationwhichaspincarries,oneneedstodetectanelectronspinstate.Manymethodsfordoingthishavebeenbroughtforthandstructuresordeviceshavebeendesignedsuchasspinltersusingmagnetictunneljunctions[ 56 57 ],spinlters[ 58 ],andonedeviceinvolvingasingleelectrontransistortoreadoutthespatialdistributionofanelectronwavefunctiondependingonthespinstate[ 59 ]. ThedevelopmentofDMS-basedspintronicsisnowreceivingagreatattention,andmaybecomeakeyareainresearchandindustryinthefuture.Althoughenormouseorthasbeenmade,thereisstillalongwaytogoforDMStobeextensivelyusedinreallife.

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Figure1{13: GaMnAs-basedspindevice.Left:GaMnAs-basedspininjectorandanalyzerstructure.Right:TemperaturedependenceofspinlifeintheGaAsquan-tumwellinthestructureshowninleftpanel.

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TounderstandtheopticalandtransportpropertiesofDMSinthepresenceofanappliedmagneticeld,wehavetoknowtheelectronicbandstructureandtheelectronicwavefunctions.Foropticaltransitions,withtheknowledgeoftheinteractionHamiltonian,wemayuseFermi'sgoldenruletocalculatethetransitionrate.Inanexternalmagneticeld,oneenergylevelwillsplitintoaseriesofLandaulevels.OpticaltransitionscantakeplaceinsideoneseriesofLandaulevelsorbetweendierentseriesaccordingtothelightconguration.SotheknowledgeoftheparitiesoftheseLandaulevelsneedtobeinvestigated.Inthischapter,wewillusethekpmethodtostudythebandstructureofDMSmaterialsaroundthepoint.Specically,ageneralizedPidgeon-Brownmodel[ 60 ]willbeusedtostudytheLandaulevelstructures. TheextraenergygaininaferromagneticDMScanbetreatedinameaneldapproximation(seeSection1.3.2).Thelocalizedmagneticmomentslineupalongtheeectiveeld,soforeachmagneticion,ithasanonvanishingaverage 29

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spinalongtheelddirection.AccordingtothediscussioninSection1.2.2,anextraenergytermproportionaltotheexchangeconstantwillbeaddedtotheHamiltonian.Thistermisrelatedtospinquantumnumbers,thusdierentspinstateswillgaindierentenergies,leadingtospinsplittings.ShowninFig. 2{1 arecalculatedvalencebandstructuresforBext=0ofbulkGaAsandferromagneticGaMnAs,whichhasaCurietemperatureTC=55K.ThecalculationisactuallybasedonageneralizedKane'smodel[ 61 ],andtheeectiveeldisassumedtobedirectedinzdirection.Kane'smodelwasdevelopedfromkptheory,whichwewillintroduceinthefollowingsection. Figure2{1: ValencebandstructureofGaAsandferromagneticGa0:94Mn0:06Aswithnoexternalmagneticeld,calculatedbygeneralizedKane'smodel.Thespinsplittingofthebandsisshown. 2.2.1IntroductiontokpMethod 62 ]andSeitz[ 63 ].Itisaperturbationtheorybasedmethod,oftencalledeectivemasstheoryintheliterature,usefulforanalyzingthebandstructurenearaparticularpointk0,which

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isanextremumofthebandstructure.Inthecaseofthebandstructurenearthepointinadirectbandgapsemiconductor,k0=0. TheHamiltonianforanelectroninasemiconductorcanbewrittenas herep=ihristhemomentumoperator,m0referstothefreeelectronmass,andV(r)isthepotentialincludingtheeectivelatticeperiodicpotentialcausedbytheionsandcoreelectronsorthepotentialduetotheexchangeinteraction,impurities,etc.IfweconsiderV(r)tobeperiodic,i.e., whereRisanarbitrarylatticevector,thesolutionoftheSchrodingerequation satisesthecondition where andkisthewavevector.Equations 2{4 and 2{5 istheBlochtheorem,whichgivesthepropertiesofthewavefunctionofanelectroninaperiodicpotentialV(r). TheeigenvaluesforEq. 2{3 splitintoaseriesofbands[ 64 ].ConsidertheSchrodingerequationinthenthbandwithawavevectork, InsertingtheBlochfunctionEq. 2{4 intoEq. 2{6 ,weobtain m0kp+h2k2

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Inmostcases,spin-orbitcouplingmustalsobeconsideredandaddedintotheHamiltonian.Thespin-orbitinteractiontermis h Includingthespin-orbitinteraction,Eq. 2{7 becomes TheHamiltonianinEq. 2{9 canbedividedintotwoparts [H0+W(k)]unk=Enkunk;(2{10) where and OnlyW(k)dependsonwavevectork. IftheHamiltonianH0hasacompletesetoforthonormaleigenfunctionsatk=0,un0,i.e., thentheoreticallyanylatticeperiodicfunctioncanbeexpandedusingeigenfunc-tionsun0.Substitutingtheexpression

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intoEq. 2{9 ,andmultiplyingfromtheleftbyun0,andintegratingandusingtheorthonormalityofthebasisfunctions,weobtain Solvingthismatrixequationgivesusboththeexacteigenstatesandeigenenergies.Usually,peopleonlyconsidertheenergeticallyadjacentbandswhenstudyingthekexpansionofonespecicband.Itactuallybecomesverycomplicatedifonewantstopursueacceptablesolutionswhenkincreases.Onehastoincreasethenumberofthebasisstates,gotohigherorderperturbations,orboth. Whenkissmallandweneglectthenon-diagonaltermsinEq. 2{15 ,theeigenfunctionisunk=un0,andthecorrespondingeigenvalueisgivenbyEnk=En0+h2k2 where Inthecalculationshownabove,weusedthepropertyhun0jp+h thenthesecondordereigenenergiescanbewrittenas

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Equation 2{19 isoftenwrittenas where 1 istheinverseeectivemasstensor,and;=x;y;z.Theeectivemassgenerallyisnotisotropic,butwecanseeitisnotk-dependent,thisisbecauseatthislevelofapproximation,theeigenenergiesinthevicinityofthepointonlydependquadraticallyonk. 2{15 givesexactsolutionsforboththeeigenfunctionsandeigenenergies.Practically,itisnotfeasibletoincludeacompletesetofbasisstates,sousuallyonlystronglycoupledbandsareincludedinusualkpformalism,andtheinuenceoftheenergeticallydistantbandsistreatedperturbatively. InKane'smodel,electronicbandsaredividedintotwogroups.Intherstgroup,thereisastronginterbandcoupling.Usuallythenumberofbandsinthisgroupiseight,includingtwoconductionbands(oneforeachelectronspin)andsixvalencebands(twoheavyhole,twolightholeandtwosplit-oholebands).Thesecondgroupofbandsisonlyweaklyinteractingwiththerstgroup,sotheeectcanbetreatedbysecondorderperturbationtheory. ShowninFig. 2{2 isthebandstructureofatypicalIII-Vdirectbandgapsemiconductor.Duetocrystalsymmetry,theconductionbandbottombelongstothe6group,thevalencebandtopbelongstothe8group,andthesplit-obandbelongstothe7group.Thespatialpartofthewavefunctionsattheconductionbandedgeares-likeandthoseatthevalencebandtoparep-like.SymbolsofjSi,jXi,jYi,andjZiareusedtorepresenttheoneconductionbandedgeand

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Figure2{2: BandstructureofatypicalIII-Vsemiconductornearthepoint.Kane'smodelconsidersthedoublyspindegenerateconduction,heavyhole,lightholeandsplit-obands,andtreatsthedistantbandsperturbatively. threevalencebandedgeorbitalfunctions.Withspindegeneracyincluded,thetotalnumberofstatesiseight.TheseeightstatesjS"i,jS#i,jX"i,jX#i,etc,canserveasasetofbasisstatesintreatingtheseeightbands.Aunitarytransformationofthisbasissetisstillabasisset.Soinpractice,peopleusethefollowingexpressions,whicharetheeigenstatesofangularmomentumoperatorsJ

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andmJ,asthebasisstatesfortheeight-bandKane'smodel, 2;1 2i=jS"i=jS"i;u2=j3 2;3 2i=jHH"i=1 2;1 2i=jLH#i=1 2;1 2i=jSO#i=i 2;1 2i=jS#i=jS#i;u6=j3 2;3 2i=jHH#i=i 2;1 2i=jLH"i=i 2;1 2i=jSO#i=i Thissetofbasisstatesisaunitarytransformationofthebasiswhichwehavementionedabove,anditcanbeproventhattheyaretheeigenfunctionsoftheHamiltonian 2{11 .Becauseofspindegeneracyatk=0,theeigenenergiesforjSi,jHHi,jLHiandjSOiareEg,0,0,,respectively,withtheselectionofenergyzeroatthetopof8band,whereEgisthebandgap,and =3ih @xpy@V @ypxjYi;(2{23) isthesplit-obandenergy. Atthislevelofapproximation,thebandsarestillatbecausetheHamiltonian 3{10 isk-independent.IncludingW(k)inEq. 2{12 intotheHamiltonian,anddeningKane'sparameteras m0hSjzjZi;(2{24)

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weobtainamatrixexpressionfortheHamiltonianH=H0+W(k),i.e., 3Pk00q 3Pkz1 3Pkz000q 3Pk+00+h2k2 3Pkz1 3Pk+0000Pk+h2k2 3Pkz0001 3Pk00+h2k2 wherek+=kx+iky,k=kxiky,andkx,ky,kzarethecartesiancomponentsofk.TheHamiltonian 2{25 iseasytodiagonalizetondtheeigenenergiesandeigenstatesasfunctionsofk.Wehaveeighteigenenergies,butduetospindegeneracy,thereareonlyfourdierenteigenenergieslistedbelow.Fortheconductionband, Forthelightholeandsplit-obands, Fortheheavyholebandwehave Theeectivemassoftheheavyholebandisstillequaltothebareelectronmass,sincewehavenotincludedthedistantbandcouplingintheHamiltonian.The

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eectofthedistantbandcouplingwillmaketheheavyholebandcurvedownwardratherthanupward. 65 ],inwhichthebandsareclassiedasAandB.Inourcase,weselectthebasisstates 2{22 asclassAandlabelthemwithsubscriptnandalltheother(energeticallydistant)statesasclassBwhichwelabelwithsubscript. Supposeallstatesareorthonormal,theSchrodingerequationthentakestheform wherelandmrunoverallstates.RewritethisequationusingclassAandB,andweobtain (EHmm)am=AXn6=mHmnan+BX6=mHma(2{31) or wheretherstsumontherighthandsideisoverthestatesinclassAonly,whilethesecondsumisoverthestatesinclassB.WecaneliminatethosecoecientsinclassBbyaniterationprocedureandobtainthecoecientsinclassAonly, and AlittlealgebrashowsthatEq. 2{33 isequivalentto

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ThismeansthatwecanndtheeigenenergieswiththebasisinclassAbutstillincludetheremoteeectsfromclassBusingEq. 2{35 .TheeectfromclassBistreatedasaperturbationusingEq. 2{34 tosecondorder. TruncatingUAmntothesecondterm,andusingHamiltonianinEq. 2{9 ,itcanberewrittenas where and m0kju0i=Xahka wherea=x;y;zandam'pamform2Aand2B.Thus Applyingbasisset 2{22 ,wecandeneparametersA,B,CandFasfollow, Rewritingtheseparametersintermsof\Luttinger"parameters1,2,3and4denedas

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3(A+2B); 6(AB); wecanobtainthetheHamiltonianHmn=UmnincludingthedistantbandcouplingunderthebasissetlistedinEq. 2{22 as 3Vkzi 3VkzL0iq 2L+1 2L+000iq 3Vkz1 3VkzL+0iq 2L1 2L0i where

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and relatedtotheKane'sparameterPdenedinEq. 2{24 .Wecanseethatifk=0orkz=0,theHamiltonianisblockdiagonalized. Inpractice,oneimportantthingneedstobenotedthattheLuttingerparame-tersdenedinEq. 2{44 arenotthe\usualLuttinger"parameterswhicharebasedonasix-bandmodelsincethisisaneight-bandmodel,butinsteadarerelatedtotheusualLuttingerparametersL1,L2,andL3throughtherelations[ 66 ] Thistakesintoaccounttheadditionalcouplingofthevalencebandstotheconductionbandnotpresentinthesix-bandLuttingermodel.Wereferto1etc.astherenormalizedLuttingerparameters. TheHamiltonian 2{45 isbasedonaneight-bandKane'sHamiltonianincludingthecontributionsoftheremotebands.Withtheremotebandcoupling,theelectroneectivemassattheconductionbandminimumnowbecomes 1 InDMSmaterialswithoutmagneticelds,theHamiltonian 2{45 plustheexchangeinteractioncanbeusedtocalculatethebandstructurewhichwillbeappliedtothecalculationoftheopticalpropertiessuchasmagneto-opticalKerreect,whichistobestudiedinchapter4.

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Inamagneticeld,asingleenergylevelsplitsintoaseriesofLandaulevels.Opticaltransitionstakeplacebetweentwolevelsinoneseriesortwoindierentseries. Inlastsection,ifwewrite where theeigenequationisgivenby whereamisthesuperpositioncoecientsdenedas NowweconsideraspatialperturbationU(r)addedtotheHamiltonianHmn.Theeigenequationnowbecomes [H+U(r)](r)=E(r):(2{54)

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Ifwewritethesolutiontotheequationas LuttingerandKohn[ 67 ]haveshownthatweneedonlysolvethefollowingequa-tion, @xai@ @xb+U(r)mn#Fn(r)=EFm(r)(2{56) ThismeansthatweonlyneedtoreplacethewavevectorintheHamiltoniankabytheoperatorpa=h,andsolveanequationforF(r).ThefunctionF(r)iscalledtheeectivemassenvelopefunction. 2.3.1ElectronicStateinaMagneticField 1 2m(ihr+eA)2(x)=(x);(2{57) whereAisthevectorpotential,andeistheelectroncharge.Assumethemagneticeldisdirectedalongz.UsingLandau'sgauge, andassumingasolutionlike

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whereLx,Ly,andLzarelengthsforthebulkcrystalinthreedimensions.Aftersubstitutingintotheeectivemassequation,wehaveanequationfor(y), 1 2m(kxeBy Dening0h2k2z 2)h!c;(2{61) where!c=eB=misthecyclotronfrequency.Thusthetotalenergyis 2)h!c+h2k2z Thismeansthatinamagneticeld,themotionofanelectroninasemiconductornowhasquantizedenergiesinthexyplane,thoughitsmotioninthez-directionisstillcontinuous.TheoriginalstatesinonebandnowsplitintoaseriesofLandaulevelswhoseeigenfunctionsare TheelectronicenergiesinEq. 2{62 isonlyrelatedtonandkz.Theyaredegeneratefordierentkx.InEq. 2{63 thecenterofy0=hkx

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valencebandsisstrong,soitisnecessarytousetheeightbandmodeltocalculatetheLandaulevels.PidgeonandBrown[ 60 ]developedamodeltocalculatethemagneticelddependentLandaulevelsatk=0.Wewillgeneralizethismodeltoincludethewavevector(kz)dependenceoftheelectronicsstatesaswellasthesdandpdexchangeinteractionswithlocalizedMndelectrons. WewillstillutilizethebasissetdenedinEq. 2{22 .InthepresenceofauniformmagneticeldBorientedalongthezaxis,thewavevectorkintheeectivemassHamiltonianisreplacedbytheoperator hp+e cA;(2{64) wherep=ihristhemomentumoperator.Forthevectorpotential,westillusetheLandaugaugeasinEq. 2{58 ,thusB=rA=B^z. Nowweintroducetwooperators and whereisthemagneticlengthwhichisdenedas eB=s TheoperatorsdenedinEqs. 2{65 obeythecommutationrulesofcreationandannihilationoperators.Thestatestheycreateandannihilatearesimpleharmonicoscillatorfunctions,andaya=Naretheorderoftheharmonicfunctions.UsingthesetwooperatorstoeliminatekxandkyinHamiltonian 2{45 ,wearriveattheLandauHamiltonian

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withthesubmatricesLa,LbandLcgivenby aiq 3V ayq 3V ayiV ayPQMip 3V aMyP+Qip 3V aip ayq 3V aiq 3V aV aPQMyip 3V aMP+Qip 3V ayip 3Vkziq 3Vkz00Liq 2Liq 3VkzL0iq 2Lyq 3Vkziq 2Liq 3Ly0377777775(2{70) TheoperatorsA,P,Q,L,andMinEq. 2{67 noware !;(2{71d) and

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Theparameters1,2,3and4aredenedinEq. 2{48 and 2{44 .Usually,theLuttingerparameters2and3areapproximatelyequal(sphericalapproxima-tion),sowehaveneglectedaterminMproportionalto(23)(ay)2.ThistermwillcoupledierentLandaumanifoldsmakingitmorediculttodiagonalizetheHamiltonian.Theeectofthistermcanbeaccountedforlaterbyperturbationtheory. Foraparticlewithnon-zeroangularmomentum(thusanon-zeromagneticmoment)inamagneticeld,theenergyduetotheinteractionbetweenthemagneticmomentandthemagneticeldisB,whichiscalledZeemanenergywhichwediscussedinSection1.2.2.TheelectronsinIII-VDMSconductionorvalencebandspossessbothorbitalangularmomentaandspin,sothereisoneextraZeemantermproportionalto(K0LB+K1B),whereLandaretheorbitalangularmomentumandspinoperators,bothofwhichareinmatrixform.K0andK1arethemagneticelddependentcoecients.FollowingLuttinger[ 66 ],wedenetheparameteras where 3L21 3L12 3(2{73) istheLuttingerparameter,andweobtaintheZeemanHamiltonian wherethe44submatrixZaisgivenby 200003 200001 2iq 200iq 2377777775:(2{75)

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DuetoexistenceoftheMnimpurityions,theexchangeinteractionsbetweenthebandelectronsandlocalizedmomentsalsoneedstobeaccountedfor.Thistermisproportionalto(PIJ(rRI)SI).Underameaneldandvirtualcrystalapproximation(seeSection1.2.2),anddeningthetwoexchangeconstants hSjJjSi(2{76a) and hXjJjXi;(2{76b)wecanarriveatanexchangeHamiltonian wherexistheMnconcentration,N0isthenumberofcationsitesinthesample,andhSziistheaveragespinonaMnsitewhichisexactlytheonewederivedatSection1.2.2forparamagneticDMSorthatinSection1.3.2forferromagneticDMS.The44submatrixDais 200001 200001 6ip 300ip 31 2377777775:(2{78) Herewejusttreattheeectofmagneticionsasanadditionalinteraction.Wedon'tconsiderthepossibleeectofthesemagneticionsonthebandgap,etc.Thebandgapchangesasaresult. ThediscussionhereisverysimilartothatinSection1.2.2whereonlyaqualitativemodelisintroduced,buthereweusedarealisticbandstructure.Also

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similartothatdiscussion,thetotalHamiltonianherecanbewrittenas Wenotethatatkz=0,theeectivemassHamiltonianisalsoblockdiagonalliketheHamiltonian 2{45 2{58 ,translationalsymmetryinthexdirectionisbrokenwhiletranslationalsymmetryalongtheyandzdirectionsismaintained.ThuskyandkzaregoodquantumnumbersandtheenvelopeoftheeectivemassHamiltonian 2{79 canbewrittenas A2666666666666666666664a1;n;n1a2;n;n2a3;n;na4;n;na5;n;na6;n;n+1a7;n;n1a8;n;n13777777777777777777775(2{80) InEq. 2{80 ,nistheLandauquantumnumberassociatedwiththeHamilto-nianmatrix,labelstheeigenvectors,A=LxLyisthecrosssectionalareaofthesampleinthexyplane,n()areharmonicoscillatoreigenfunctionsevaluatedat=x2ky,andai;(kz)arecomplexexpansioncoecientsforthe-theigenstatewhichdependexplicitlyonnandkz.NotethatthewavefunctionsthemselveswillbegivenbytheenvelopefunctionsinEq. 2{80 witheachcomponentmultipliedbythecorrespondingkz=0BlochbasisstatesgiveninEq. 2{22

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SubstitutingFn;fromEq. 2{80 intotheeectivemassSchrodingerequationwithHgivenbyEq. 2{79 ,weobtainamatrixeigenvalueequation thatcanbesolvedforeachallowedvalueoftheLandauquantumnumber,n,toobtaintheLandaulevelsEn;(kz).Thecomponentsofthenormalizedeigenvectors,Fn;,aretheexpansioncoecients,ai. Sincetheharmonicoscillatorfunctions,n0(),areonlydenedforn00,itfollowsfromEq. 2{80 thatFn;isdenedforn1.TheenergylevelsaredenotedEn;(kz)wherenlabelstheLandaulevelandlabelstheeigenenergiesbelongingtothesameLandaulevelinascendingorder. Table2{1: SummaryofHamiltonianmatriceswithdierentn -1111(1;1)0444(0;),=141777(1;),=172888(n;),=18 Forn=1,wesetallcoecientsaitozeroexceptfora6inordertopreventharmonicoscillatoreigenfunctionsn0()withn0<0fromappearinginthewavefunction.Theeigenfunctioninthiscaseisapureheavyholespin-downstateandtheHamiltonianisnowa11matrixwhoseeigenvaluecorrespondstotheaheavyholespin-downLandaulevel.Pleasenotethatwhenwespeakaboutaheavy(light)holestate,itgenerallymeansthattheelectronicwavefunctioniscomposedmainlyoftheheavy(light)holeBlochbasisstatenearthek=0point. Forn=0,wemustseta1=a2=a7=a8=0andtheLandaulevelsandenvelopefunctionsarethenobtainedbydiagonalizinga44Hamiltonianmatrixobtainedbystrikingouttheappropriaterowsandcolumns.Forn=1,the

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Hamiltonianmatrixis77andforn2theHamiltonianmatrixis88.ThesummaryofHamiltonianmatricesfordierentnisgiveninTable 2{1 ThematrixHninEq. 2{81 isthesumofLandau,Zeeman,andexchangecontributions.TheexplicitformsfortheZeemanandexchangeHamiltonianmatricesaregiveninEq. 2{74 and 2{77 andareindependentofn. Table2{2: InAsbandparameters Energygap(eV)1Eg(T=30K)0.415Eg(T=77K)0.407Eg(T=290K)0.356Electroneectivemass(m0)me0.022Luttingerparameters1L120.0L28.5L39.2L7.53Spin-orbitsplitting(eV)10.39Mns-dandp-dexchangeenergies(eV)N0-0.5N01.0Opticalmatrixparameter(eV)1Ep21.5Refractiveindex2nr3.42 68 ]. 69 ]. 2{2 .ShowninFig. 2{3 aretheconductionbandLaudaulevelsforInAsandIn0:88Mn0:12Asasafunctionofmagneticeldatk=0foratemperatureof30K.Thedashedlinesrepresentspin-uplevels,andthesolidlinesrepresentthespin-downlevels.Thisillustratestheenergysplittingoftheconductionbandatthepoint.TherightpanelforInMnAsisonlydierentfromtheleftpanelfor

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Figure2{3: CalculatedLandaulevelsforInAs(left)andIn0:88Mn0:12As(right)asafunctionofmagneticeldat30K. InAsinthatithastheexchangecontributionsduetotheinteractionbetweenthebandelectronsandthelocalizedMnmoments.TheorderingoftheseLandaulevelscanbequalitativelyexplainedbythesimplemodelinEq. 1{13 wherewehaveananalyticalexpressionfortheLandaulevelenergy.Notethattheyarenotlinearfunctionsofthemagneticeld.Inthenextchapterwewillseethatthissimplemodelcannotpredictan(exchangeconstantdenedinEq. 2{76 )dependenceofthecyclotronenergy,whichistheenergydierencebetweentwoadjacentLandaulevelswiththesamespin.Theexchangeconstantdependenceisaconsequenceofkpmixingbetweenconductionandvalencebands. ThewavevectorkzdependenceofLandaulevelsinbothconductionbandandvalencebandsisshowninFig. 2{4 ,whereonlythevelowestorderLandaulevelsareshown.Becauseofthestrongstatemixing,thespinstatesinvalencebandsarenotindicated.ComparingtheleftandrightpanelsofFig. 2{3 andFig. 2{4 ,wecanseethatMndopingdrasticallychangestheelectronicstructure.Spinsplittingisgreatlyenhancedinbothconductionandvalencebands.Asamatteroffact,thespinstateorderingintheconductionbandisreversedwithMndoping.

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Figure2{4: TheconductionandvalencebandLandaulevelsalongkzinamag-neticeldofB=20TatT=30K.TheleftandrightguresareforInAsandIn0:88Mn0:12As,respectively. whereh!spinisthespin-splitting.Rothetal.[ 70 ]havecalculatedtheg-factorinsemiconductorsbasedonKane'smodel,andhaveshownthattheg-factorinthe

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conductionbandis 3Eg+2:(2{83) Usingthisequation,theg-factorforbulkInAsisabout15:1,whichisclosetotheexperimentalvalue15[ 71 ]. Figure2{5: Conductionbandg-factorsofIn1xMnxAsasfunctionsofmagneticeldwithdierentMncompositionx.Fortheleftgure,T=30Kandfortheright,T=290K.Noteathightemperatureswelosethespinsplitting. Duetotheexchangeinteraction,thespin-splittingisgreatlyenhanced.UsuallyinDMS,theexchangeenergyismuchbiggerthantheZeemanenergy,whichcanbeseenfromthesimpletheoryinEq. 1{15 forafewpercentofMndoping.Inthatcase,ifwetakex=0:1,N=0:5eV,andT=30K,thengeff256.Ifweonlyconsidertheexchangeinteraction,fromEq. 2{78 ,thespin-splittingintheconductionbandisexactlythatinEq. 1{13 .However,thisisnotcorrectbecausetherstconductionbandspin-downlevelcomesfromthen=0manifold,whiletherstconductionbandspin-uplevelcomesfromthen=1manifold.Dierentmanifoldnumbersresultindierentmatrixelements,whichwillcausedierentstatecoupling,andthusspin-splittingduetotheexchangeinteractionisnotwhatthesimplemodelpredicts.Theconductionbandg-factorsforInAsandInMnAsat30Kand290KareshowninFig. 2{5 .ThisclearlydemonstrateshowMndopingaectstheg-factors.At290K,theg-

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Figure2{6: factorsaredrasticallyreduced.Thisisbecauseathightemperatures,thermaluctuationsbecomesolargethatthealignmentofthemagneticspinsislessfavorable.However,ifferromagneticDMSareemployed,duetotheinternalexchangeeld,astrongalignmentcanbeexpectedevenathightemperatures.Nowwesupposeahigh-TCIn0:9Mn0:1AssysteminwhichaCurietemperatureof110Kisachieved.Theg-factorforthissystemisshowninFig. 2{6 .Evenatrelativelyhightemperature(stillbelowthetransitiontemperaturethough),bigg-factorsarestillobtained.Theg-factorreachesinnityatzeroeldwhentemperaturesarebelowTCbecausethereisstillspin-splittingeventhoughthereisnoexternaleld.

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Inchapter2,asystematicmethodofcalculatingtheelectronicstructureofDMSwasdevelopedanddescribedindetailandappliedtothenarrowgapInMnAs.IthasbeenseenthatthebandstructureofDMSdependsstronglyonMndopingwhichinducestheexchangeinteraction.Thebandstructurealsodependsonthestrengthoftheappliedmagneticeld,ascanbeseenfromFig. 2{3 and 2{5 .Apartfromthetheoreticalcalculation,opticalexperimentsarealwaysgoodwaystodetecttheelectronicpropertiesofsemiconductors.Amongthesemethods,cyclotronresonance(CR)isanextensivelyusedandapowerfuldiagnostictoolforstudyingtheinter-subbandopticalpropertiesandeectivemassesofcarriers.Cyclotronresonanceisahigh-frequencytransportexperimentwithallthecomplicationswhichcharacterizetransportmeasurements.Throughcyclotronresonance,onecangettheeectivemasses,whicharedeterminedbythepeakofaresonanceline,whilescatteringinformationisobtainedfromthelinebroadening.CyclotronresonanceoccurswhenelectronsabsorbphotonsandmakeatransitionbetweentwoadjacentLandaulevels.Fromcyclotronresonancemeasurementsonecaninferthemagneticelddependentbandstructureofthematerial.SincethebandstructureofaDMSissosensitivetomagneticelds,thisisausefulmeanstostudyandobtainbandinformationfromacomparisonbetweentheexperimentalresultsandtheoreticalcalculations. 3.1.1OpticalAbsorption 56

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energyuxoftheincidentlightisS,thenthephotonuxdensityisS=h!,andwehaveTdx=Sdx=h!,i.e. wherei;farethelabelsfortheinitialandnalstates.Thesummationrunsoverallstates.Forabsorptionbetweenstateiandf,thetransitionprobabilityfromFermi'sgoldenrule[ 72 ]is, andforemission whereEiandEfaretheenergiesoftheinitialandnalstates(hereweonlywantthenalexpressionforabsorption,soinemission,eventheelectronstransitfromstateftostatei,westillcallstateiistheinitialstate,andstatefthenalstate),respectively,andthefunctionensurestheconservationofenergyintheopticaltransition.H0istheelectronphotoninteractionHamiltonian.Essentially,inopticaltransitions,momentumshouldalsobeconserved.However,sincethephotonmomentump=h=ismuchsmallerthanthetypicalelectronmomentum,wegenerallyconsidertheopticaltransitiontobe\vertical",whichmeansanelectroncanonlytransittostateswiththesamek,i.e.,weeignorethephotonmomentum. Insemiconductorswhendealingwiththerealisticcaseofabsorption,weneedtotakeintoaccountthestateoccupationprobabilitybyelectrons,whichinthermal

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equilibriumisdescribedbyaFermi-Diracdistributionfunction 1+eEEF=kBT;(3{5) andsotherateofabsorptioninthewholecrystalcanbewrittenas andtheemissionrate DuetothehermitianpropertyofH0,jH0ifj=jH0fij.Thenetabsorptionrateperunitvolumethenis Whenasemiconductorisilluminatedbylight,theinteractionbetweenthephotonsandtheelectronsinthesemiconductorcanbedescribedbytheHamiltonian, 2m0(p+eA)2+V(r)(3{9) wherem0isthefreeelectronmass,eistheelectroncharge,Aisthevectorpotentialduetotheopticaleld,andV(r)isthecrystalperiodicpotential(inDMS,includingthevirtualcrystalexchangepotential).Thustheone-electronHamiltonianwithouttheopticaleldis andtheopticalperturbationtermsare m0Ap+e2A2

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OpticaleldsaregenerallyveryweakandusuallyonlythetermlinearinAisconsidered,i.e.,wetreattheelectron-photoninteractioninalinearresponseregimeandneglecttwo-photonabsorption.ThetransitionduetotheopticalperturbationinEq. 3{11 cantakeplaceeitheracrossthebandgaporinsideasingleband(conductionorvalenceband)dependingonthephotonenergy.Inthischapter,weonlyconsidercyclotronresonance,whichtakesplacebetweentheLandaulevelswithinconductionorvalencebands. Formonochromaticlightthevectorpotentialis whereKistheelectromagneticwavevector,!istheopticalangularfrequancy,pisthemomentumoperator,and^eistheunitpolarizationvectorinthedirectionoftheopticaleld,representingthelightconguration. TheenergyuxoftheopticaleldcanbeexpressedbythePoyntingvector,S=EH.UsingtherelationsE=@A=@t,H=rS=,and!=K=c=nr,theaveragedenergyuxthenis UsingthisrelationandEq. 3{8 ,theabsorptioncoecientthenis S=h! AccordingtoEq. 3{12 ,theinteractionHamiltoniancanbewrittenas sotheabsorptioncoecient 3{16 becomes

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Notethattheinteraction 3{15 isbasedonthedipoleapproximation.Sointhefollowingwhenwetalkaboutselectionrules,etc,theyareelectricdipoleselectionrules. Thescatteringbroadening(aswellasdisorder)canbeparameterizedbythelinewidththroughthereplacementofthefunctionbyaLorentzianfunction[ 72 ]as m0(3{18) wherem0isthefreeelectronmass(eectivemasswheninasemiconductor).Ifanelectromagneticwaveisappliedwiththesamefrequency,theelectronwillresonantlyabsorbthiselectromagneticwave. Quantummechanically,anelectroninamagneticeldwillhaveaquantizedmotion.ReferringtoEq. 2{62 ,theenergyoftheelectronsplitsintoaseriesofLandaulevels.Iftheenergyquantah!oftheappliedelectromagneticwaveareexactlythesameastheenergydierenceh!cbetweentwoadjacentLandaulevels,theelectronwillabsorbonephotontotransitfromthelowerLandauleveltothehigherone.Thisiscalledcyclotronresonance. Inthepresenceofamagneticeld,theHamiltonian 3{10 ,inDMSsystem,isreplacedbytheoneinEq. 2{79 .Wealreadyhavetheeigenstatesforthis

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Hamiltonian.Forconvenience,werewritethemhereas A2666666666666666666664a1;n;n1u1a2;n;n2u2a3;n;nu3a4;n;nu4a5;n;nu5a6;n;n+1u6a7;n;n1u7a8;n;n1u83777777777777777777775:(3{19) Theeigenfunctionabovecanbeconsideredasthelinearsuperpositionofeightbasisstates,eachofwhichiscomposedoftwoparts.nistheharmonicoscillatorenvelopefunction,whichisslowlyvaryingoverthelattice,andcanbeconsideredconstantoveraunitcelllengthscale.uiistheBlochpartofthewavefunction,whichvariesrapidlyoveraunitcellandhastheperiodicityofthelattice. NowletusinspectthepropertiesofthemomentummatrixelementinEq. 3{16 .Usingn;asthenewsetofquantumnumbers,andutilizingthespatialpropertiesofthewavefunctions,wecanfactorizetheintegralintotwopartsandwritethematrixelementas SincetheBlochfunctionsuiarequicklyvaryingfunctions,theirgradientsaremuchlargerthanthoseoftheenvelopesi.AsshowninRef.[ 73 ],thersttermontherighthandsidedominatesbothinnarrowgapandwidegapsemiconductors,sowehaveneglectedthesecondterminourcalculation.However,itiseasytocheckthatthesetwotermsobeythesameselectionrules.

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Wecanfactorize^epto^ep=^e+p+^ep++^ezpzwhere^e=(^xi^y)=p and Thismeansthatp+andpareraisingandloweringoperatorsfortheeigenstates.Forp+,anelectronwillabsorban^ephotontohaveann!n+1transition,whichusuallyhappensintheconductionbandforelectrons,sowecallthistransition\electron-active"(e-active).Forp,anelectronwillabsorban^e+photontohaveann!n1transition,whichusuallyhappensinthevalencebandsforholes,sowecallthistransition\hole-active"(h-active).Thequasi-classicalpictureforthetwotypesofabsorptionisshowninFig. 3{1 .Tocomplywithconservationof Figure3{1: Quasi-classicalpicturesofe-activeandh-activephotonabsorption. bothenergyandangularmomentum,inaquasi-classicalpicture,electronscanonlyabsorbphotonswithe-activepolarization,andholescanonlyabsorbphotonswithh-activepolarization.Inaquantummechanicaltreatment,wewillseethatthe

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truesituationismorecomplicatedthanthis.Inparticular,wendthate-activeabsorptioncanalsotakeplaceinp-typematerials. Whenthetemperatureisnotzero,EFinEq. 3{5 shouldbeunderstoodasthechemicalpotential,whichwestillcalltheFermienergy,anddependsontemperatureanddoping.IfNDisthedonorconcentrationandNAtheacceptorconcentration,thenthenetdonorconcentrationNC=NDNAcanbeeitherpositiveornegativedependingonwhetherthesampleisnorptype.ForaxedtemperatureandFermilevel,thenetdonorconcentrationis (2)22Xn;Z1dkz[fn;(kz)vn;];(3{23) wherevn;=1ifthesubband(n;)isavalencebandandvanishesif(n;)isaconductionband.Giventhenetdonorconcentrationandthetemperature,theFermienergycanbefoundfromEq. 3{23 usingarootndingroutine. 74 75 ].ThemegagaussexperimentshavebeendoneattheuniversityofTokyowherehighmagneticeldscanbegeneratedusingtwokindsofpulsedmagnets:thesingle-turncoiltechnique[ 76 77 ]andtheelectromagneticuxcompressionmethod[ 77 78 ].Thesingle-turncoilmethodcangenerate250Twithoutanysampledamageandthusmeasurementscanberepeatedonthesamesampleunderthesameexperimentalconditions.Theideabe-hindthismethodistoreleaseabigcurrentinaveryshortperiodoftime(severals)tothesingle-turncoiltogenerateanultrahighmagneticeld.Thecorepartofarealsingle-turncoildeviceisdemonstratedinFig. 3{2 [ 76 ].Althoughthesampleisintact,thecoilisdamagedaftereachshot.AstandardcoilisshowninFig. 3{3

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Figure3{2: Thecorepartofthedevicebasedonsingle-coilmethod.Thecoilisplacedintheclampingmechanismasseeninthegure.Thedomedsteelcylin-dersoneachsideofthecoilaresupportsforthesampleholderswhichprotecttheconnectiontothesample(e.g.,thinwires,heliumpipes)againstthelateralblast. beforeandafterashot.Dependingonthecoildimension,eachshotgeneratesapulsedmagneticeldupto250Tinseverals.ThetimedependenceofthepulsedmagneticeldandofthecurrentowingthroughthecoilisshowninFig. 3{4 [ 76 ]. Forhighereldexperimentsanelectromagneticuxcompressionmethodisused.Itusestheimplosivemethodtocompresstheelectromagneticuxsoastogenerateultrahighmagneticeldsupto600T.ThetimedependenceofthepulsedmagneticeldandcurrentisshowninFig. 3{5 [ 77 ].Thisisadestructivemethodandthesampleaswellasthemagnetisdestroyedineachshot. 3.3.1ElectronCyclotronResonance

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Figure3{3: Astandardcoilbeforeandafterashot. Figure3{4: WaveformsofthemagneticeldBandthecurrentIinatypicalshotinsingle-turncoildevice.

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Figure3{5: WaveformsofthemagneticeldBandthecurrentIinatypicaluxcompressiondevice. Intheconductionband,theLandausubbandsareusuallyalignedinsuchawaythatenergyascendswithquantumnumbern.Soforane-activetransition,bothangularmomentumandenergyforanelectron-photonsystemcanbeconserved. OurcollaboratorsKonoetal.[ 74 ]measuredtheelectronactivecyclotronresonanceinInMnAslmswithdierentMnconcentrations.Thelmsweregrownbylowtemperaturemolecularbeamepitaxyonsemi-insulatingGaAssubstratesat200C.Allthesampleswerentypeanddidnotshowferromagnetismfortemperaturesaslowas1:5K.TheelectrondensitiesandmobilitiesdeducedfromHallmeasurementsarelistedinTable 3{1 ,togetherwiththeelectroncyclotronmassesobtainedataphotonenergyof117meV(orawavelengthof10:6m). TypicalmeasuredCRspectraat30Kand290KareshownintheleftandrightpanelofFig. 3{6 ,respectively.Notethattocomparethetransmissionwithabsorptioncalculations,thetransmissionincreasesinthenegativeydirection.EachgureshowsspectraforallfoursampleslabeledbythecorrespondingMncompositionsfrom0to12%.Allthesamplesshowpronouncedabsorption

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Table3{1: Parametersforsamplesusedine-activeCRexperiments Mncontentx00.0250.0500.120 Density(4.2K)1:010171:010160:910161:01016Density(290K)1:010172:110171:810177:01016Mobility(4.2K)400013001200450Mobility(290K)4000400375450m=m0(30K)0.03420:03030:02740:0263m=m0(290K)0.03410:03340:03250:0272 peaks(ortransmissiondips)andtheresonanceelddecreaseswithincreasingx.Increasingxfrom0to12%resultsina25%decreaseincyclotronmass(seeTable 3{1 ).Athightemperatures[e.g.,Fig. 3{6 (b)]thex=0sampleclearlyshowsnonparabolicity-inducedCRspinsplittingwiththeweaker(stronger)peakoriginatingfromthelowestspin-down(spin-up)Landaulevel,whiletheotherthreesamplesdonotshowsuchsplitting.TheabsenceofsplittingintheMn-dopedsamplescanbeaccountedforbytheirlowmobilities(whichleadtosubstantialbroadening)andlargeeectivegfactorsinducedbytheMnions.Insampleswithlargex,onlythespin-downlevelissubstantiallythermallypopulated(seeFig. 2{5 ). UsingtheHamiltoniandescribedinSection2.3.2,thewavefunctionsinSection2.3.3,andthetechniquesforcalculatingFermienergy,theseverallowestLandaulevelsintheconductionbandattwoMnconcentrationsandtheFermienergyfortwoelectrondensities(11016=cm3and11018=cm3)arecalculated.TheconductionbandLandaulevelsandtheFermienergiesareshowninFig. 3{7 asafunctionofmagneticeldatT=30K.Fromthesegures,wecanseethatatresonance,thedensitiesandeldsaresuchthatonlythelowestLandaulevelforeachspintypeisoccupiedfortypicaldensitieslistedinTable 3{1 .Thus,alltheelectronswereinthelowestLandaulevelforagivenspinevenatroomtemperatureduetothelargeLandausplitting,precludinganydensity-dependentmassdue

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Figure3{6: ExperimentalelectronCRspectrafordierentMnconcentrationsxtakenat(a)30Kand(b)290K.Thewavelengthofthelaserwasxedat10:6mwithe-activecircularpolarizationwhilethemagneticeldBwasswept. tononparabolicity(expectedatzeroorlowmagneticelds)asthecauseoftheobservedtrend. ThecyclotronresonancetakesplacewhentheenergydierencebetweentwoLandaulevelswiththesamespinisidenticaltotheincidentphotonenergy.InFig. 3{8 ,wesimulatecyclotronresonanceexperimentsinn-typeInAsfore-activecircularlypolarizedlightwithphotonenergyh!=0:117eV.WeassumeatemperatureT=30Kandacarrierconcentrationn=1016=cm3.ThelowerpanelofFig. 3{8 showsthefourlowestzone-centerLandauconduction-subbandenergiesandtheFermienergyasfunctionsoftheappliedmagneticeld.Thetransitionattheresonanceenergyh!=0:117eVisaspin-upn=1transitionandisindicatedbytheverticalline.FromtheLandauleveldiagramtheresonancemagneticeldisfoundtobeB=34T.TheupperpanelofFig. 3{8 showstheresultingcyclotron

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Figure3{7: Zone-centerLandauconduction-subbandenergiesatT=30Kasfunc-tionsofmagneticeldinn-dopedIn1xMnxAsfor=0andx=12%.Solidlinesarespin-upanddashedlinesarespin-downlevels.TheFermienergiesareshownasdottedlinesforn=1016=cm3andn=1018=cm3. resonanceabsorptionassumingaFWHMlinewidthof4meV.Thereisonlyoneresonancelineinthecyclotronabsorptionbecauseonlytheground-stateLandaulevelisoccupiedatlowelectrondensities.Forhigherelectrondensities,moreLandaulevelsareoccupied.Forexample,ifbothspin-upandspin-downstatesoftherstLandaulevelareoccupied,oneobtainsmultipleresonancepeaks. Oursimulationoftheexperimentale-activecyclotronresonanceintheconductionbandshowninFig. 3{6 isshowninFig. 3{9 .Theleftandrightpaneldemonstratethecalculatedcyclotronresonanceabsorptioncoecientfore-active

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Figure3{8: ElectronCRandthecorrespondingtransitions.TheupperpanelshowstheresonancepeakandthelowerpanelshowsthelowestfourLandaulevelswithspin-upstatesindicatedbysolidlinesandspin-downstatesindicatedbydashedlines.Verticalsolidlineinthelowerpanelindicatesthetransitionaccountablefortheresonance. circularlypolarized10:6mlightintheFaradaycongurationasafunctionofmagneticeldat30Kand290K,respectively.Inthecalculation,thecurveswerebroadenedbasedonthemobilitiesofthesamples.ThebroadeningusedforT=30Kwas4meVfor0%,40meVfor2:5%,40meVfor5%,and80meVfor12%.ForT=290K,thebroadeningusedwas4meVfor0%,80meVfor2:5%,80meVfor5%,and80meVfor12%.AtT=30K,weseeashiftintheCRpeakasafunctionofdopinginagreementwithFig. 3{6 (a).ForT=290K,weseethepresenceoftwopeaksinthepureInAssample.Thesecondpeakoriginatesfromthethermalpopulationofthelowestspin-downLandaulevel.Thepeakdoes

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Figure3{9: CalculatedelectronCRabsorptionasafunctionofmagneticeldat30Kand290K.ThecurvesarecalculatedbasedongeneralizedPidgeon-BrownmodelandFermi'sgoldenruleforabsorption.TheyarebroadenedbasedonthemobilitiesreportedinTable 3{1 notshiftasmuchwithdopingasitdidatlowtemperature.ThisresultsfromthetemperaturedependenceoftheaverageMnspin.WebelievethattheBrillouinfunctionusedforcalculatingtheaverageMnspinbecomesinadequateatlargexand/orhightemperatureduetoitsneglectofMn-Mninteractionssuchaspairingandclustering. Thee-activeCRshowsashiftwithincreasingMnconcentration.FromthesimpletheoryinSection1.2.2,thecyclotronresonanceelddoesnotdependonxandbecausetheexchangeinteractionwillshiftalllevelsbythesameamount.Thisshiftcomesfromthecomplicatedconduction-valencebandmixing,anddependsonthevalueof()[ 79 ].Wecanqualitativelyexplainthisshiftusingthecyclotronmass,whichwillbediscussedinthefollowingsubsection.

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TheCRpeaksshowninFig. 3{9 arehighlyasymmetric.Thisisbecausewehavetakenintoaccountthenitekzeectinourcalculation,andtheenergydispersionalongkzshowshighnonparabolicity.Also,thecarrierllingeectduetotheFermienergysharpeningwillalsocontributetotheCRpeakasymmetry. ThisequationcanbederivedfromEq. 3{18 ifwesetmagneticeldBsothath!c=h!,whichisthecyclotronresonancecondition. Thecalculatedcyclotronmassesforthelowestspin-downandspin-uptran-sitionsareplottedinFig. 3{10 asafunctionofMnconcentrationxataphotonenergyofh!=0:117eV.Cyclotronmassesarecomputedforseveralsetsofandvalues.ThecyclotronmassesinFig. 3{10 (a)and(b)correspondtothecomputedcyclotronabsorptionspectrashowninFig. 3{9 (a)and(b),respectively.Inourmodel,theelectroncyclotronmassesdependontheLandausubbandenergiesandphotonenergiesandareindependentofelectronconcentration. Figure 3{10 clearlyshowsthatthecyclotronmassdependsonbothexchangeconstantsandx.Withincreasingx,spin-down(spin-up)cyclotronmassshowalmostalineardecrease(increase).Thecyclotronmassdoesnotdependononesingleexchangeconstant,itdependsonbothexchangeconstants.Investigationofthemassdependenceonthesetwoconstantsrevealsthemassshifthasacloserelationwiththeabsolutevalueof()[ 79 ].Thisshiftallowsusetomeasuretheexchangeinteraction. Thecalculatedcyclotronmasshastakenintoaccountalltheenergydepen-denceonnonparabolicityduetotheconduction-valencebandmixing,theexchange

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Figure3{10: Calculatedelectroncyclotronmassesforthelowest-lyingspin-upandspin-downLandautransitionsinn-typeIn1xMnxAswithphotonenergy0:117eVasafunctionofMnconcentrationforT=30KandT=290K.Electroncyclotronmassesareshownforthreesetsofandvalues.

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interactionconstantsand,andtheMncontentx.TheshiftoftheresonancepeakstolowereldswithincreasingMncontentxisnaturallyexplainedbythedecreaseofthespin-downcyclotronmass.Duetothesmallerdownwardslopeinthespin-downcyclotronmassat290Kascomparedto30K,theresonancepeakshiftat290Kisseentobelesspronouncedthanat30K. 3.4.1HoleActiveCyclotronResonance 2{4 ,theDMSvalencebandstructureismuchmorecom-plicatedthantheconductionbandstructure.Duetotheirenergeticproximity,heavyholeandlightholebandsarestronglymixedevennearthepoint.Thesplit-obandalsocontributesstronglytothevalenceband-edgewavefunctions.Inamagneticeld,theseholebandssplitintotheirownLandaulevels,butopticaltransitionscanhappenbetweenanytwolevelsifbothangularmomentumandenergyareconserved.Asintheconductionband,cyclotronresonancerequiresconduction-valencebandmixingtoproducestrongenoughoscillatorstrength.Interbandmixingacrossthebandgapissmallinwide-gapsemiconductors,soitismorediculttoobservecyclotronresonanceinthesesemiconductors.Asamatteroffact,nocyclotronresonancehasbeenreportedtodateinGaMnAs. InAsandInMnAsarenarrow-gapsemiconductors.Ourcollaborators[ 80 81 82 ]haveperformedcyclotronresonanceexperimentsonp-dopedInAsandInMnAsatultrahighmagneticeldsupto500T.Thetypicalh-activeCRabsorptionofInAsbelow150TisshowninFig. 3{11 ,inwhichtheincidentlightish-activecircularlypolarizedwithphotonenergy0:117eV.Twopeaksarepresentintheexperimentalobservation,onearound40T,andanotheraround125T.Atevenlowerelds,thereisabackgroundabsorption.Thetheoreticalsimulationusingaholedensityof11019=cm3andabroadeningfactorof40meVisalsodisplayedinFig. 3{11 forcomparison.

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Figure3{11: Holecyclotronabsorptionasafunctionofmagneticeldinp-typeInAsforh-activecircularlypolarizedlightwithphotonenergy0:117eV.Theup-percurveisexperimentallyobservedresultandtheloweroneisfromtheoreticalcalculation. Inourmodel,wearecapableofcalculatingtheabsorptionbetweenanytwoLandaulevels.Detailedcalculationrevealsthatthepeakatlowereldsisduetotheheavy-holetoheavy-holetransition,andthepeakathighereldsisfromthelight-holetolight-holetransition.WenowuseHn;tospecifytheheavyholelevel,andLn;tospecifythelightholelevel,where(n;)arethequantumnumbersdenedinEq. 3{19 .BecauseofstrongwavemixingHorLonlylabelsthezonecenter(k=0)characterofaLandaulevel.Usingtheselabels,weillustratethetwo-stateabsorptioninFig. 3{12 alongwiththeLandaulevelstructureasafunctionofmagneticeld. ItisseenfromFig. 3{12 thattheholesopticallyexcitedfromtheheavyholesubbandH1;1andlightholesubbandL0;3giverisetothetwostrongcyclotronabsorptionpeaksshowninFig. 3{11 .Thecyclotronabsorptionpeakaround40Tisduetoatransitionbetweenthespin-downgroundstateheavyholeLandaulevelH1;1,andheavyholeLandaulevelH0;2,whichnearthezonecenterisprimarily

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Figure3{12: CalculatedcyclotronabsorptiononlyfromtheH1;1H0;2andL0;3L1;4transitionsbroadenedwith40meV(a),andzonecenterLandaulevelsresponsibleforthetransitions(b). spin-down.Theotherabsorptionpeakaround140T,isaspin-downlightholetransitionbetweenL0;3andL1;4Landaulevels.ThebackgroundabsorptionatB<30TisduetotheabsorptionbetweenhigherLandaulevelswhichalsobecomeoccupiedbyholesatlowerelds. CyclotronresonanceabsorptionmeasurementsonIn1xMnxAswithx=2:5%havealsobeenperformed.TheyareshowninFig. 3{13 alongwithourtheoreticalsimulation.TheCRmeasurementsweremadeattemperaturesof17,46,and70Kinh-activecircularlypolarizedlightwithphotonenergyh!=0:224eV.Inoursimulation,theholedensityistakenas51018=cm3,andthecurvesare

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Figure3{13: ExperimentalholeCRandcorrespondingtheoreticalsimulations.ThelowtemperatureCRhasanabruptcutoatloweldsduetothefermilevelsharpeningeect. broadenedusingaFWHMlinewidthof120meV.ClearlytheabsorptionpeakisduetotheheavyholetransitionwhichwehaveseeninFig. 3{11 andFig. 3{12 .Duetothehigherphotonenergy,thispeakshiftsfromaround40Ttoaround85T.Theresonanceeldisinsensitivetotemperatureandthelineshapeisstronglyasymmetricwithabroadtailatlowelds.Thisbroadtailagaincomesfromthehigherordertransitionsresonantatlowelds.Weseethatinbothexperimentandtheoryatlowtemperatureandloweld,thereisasharpcutoof

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theabsorption.ThiscanbeattributedtothesharpnessoftheFermidistributionatlowtemperatures. Figure 3{14 showstheobservedCRpeaksasafunctionofmagneticeld.They-axisindicatesthephotonenergiesusedwhenobservingthecyclotronresonance.Thesolidcurvesshowthecalculatedresonancepositions.Thecurvelabeled`HH'(`LH')isjusttheresonanceenergybetweenLandaulevelsH1;1(L0;3)andH0;2(L1;4).Thetheoreticalcalculationshowsanoverallconsistencywiththeexperiments. Figure3{14: ObservedholeCRpeakpositionsforfoursampleswithdierentMnconcentrations.Thesolidcurvesaretheoreticalcalculations. Therearetwofactorsinourcalculationthataecttheresults.OneistheselectionofLuttingerparameters,theotheristhelimitationoftheeight-bandeectivemasstheoryitself.InFig. 3{11 ,thetheoreticallycomputedpeakathighereldsdoesnotttheexperimentalpeakexactly.Duetothefactthatthistransitiontakesplaceatthezonecenter,wherethekptheoryshouldbeveryaccurate,thisdeviationmaybetheresultofunoptimizedLuttingerparameters.

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TheempiricalparametersusedintheeectivemassHamiltoniancandrasticallychangethevalencebandstructureandtheresultingCRabsorptionspectra.Fig. 3{15 showsthedependenceoftheCRenergiesonseveralparameterssuchastheLuttingerparameters1,2,3,Kane'sparameterEpandtheeectiveelectronmassm.Thisgurerevealsthatthe`LH'transitionsareaectedmorebysmallvariationsintheseparametersthanthe`HH'transitions.Forinstance,a10%changein1willresultina0:025eVchangeatB=140TintheLHCRenergy,whichinturnwillresultinabouta50TCRpositionshiftintheresonanceeldwhenthephotonenergyis0:117eV.TheMndopingontheotherhandgenerallyenhancestheCRenergydependenceontheseparameters,whichcanbeseenfromcomparingthetwographsinFig. 3{15 Figure3{15: Thedependenceofcyclotronenergiesonseveralparameters.LeftpanelshowstheheavyholeCRenergydependence,andtherightpanelshowsthelightholeCRenergydependence. Figure 3{16 illustrateshowtheCRabsorptiondependsonthreeLuttingerparameterswhilekeepingalltheotherparametersunchanged.Itcanbeseenthat

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theCRspectraquitesensitivelydependsonthevaluesoftheLuttingerparameters,providinganeectivewaytomeasuretheseparametersthroughcomparisonwithexperiments. Figure3{16: HoleCRspectraofInAsusingdierentsetsofLuttingerparame-ters.LightholetransitionismoresignicantlyaectedbychangeoftheLuttingerparameters. InFig. 3{14 ,thereisonepeakaround450Tlabeledas`C'whenthelightenergyish!=0:117eV.Toaccountforthispeak,CRabsorptionspectraupto500Thavebeencomputed.Thek=0Landaulevelsasafunctionofmagneticeld,alongwiththeCRspectraareplottedinFig. 3{17 ,inwhichwecanseethatthispeakisduetothesuperpositionoftwotransitions:L1;5L2;5andH2;6H3;6.However,thecalculatedpeakpositionisaround360T,dierentfromtheexperiment.Therearetwopossiblereasonsforthisbigdeviation.Oneisthatatveryhighmagneticelds,theeight-bandPidgeon-Brownmodelmaybreakdown;theotheristhattransitionscontributingtothepeaktakeplaceawayfromthezonecenterwhereeight-bandkptheoryisnotadequatetodescribetheenergydispersion.ThebandstructurealongkzisplottedatFig. 3{18 ,whereweseethattheLandaulevelsH2;6andH3;6bothhavecamelbackstructures.Ataholedensityp=11019=cm3,thezonecenterpartofH2;6isnotoccupied.The

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Figure3{17: CalculatedLandaulevelsandholeCRinmagneticeldsupto500T.Theupperpanelshowsthek=0valencebandLandaulevelsasafunctionofmagneticeldandtheFermilevelforp=1019cm3(dashedline).TheholeCRabsorptioninp-typeInAsisshowninthelowerpanelforh-activelypolarizedlightwithh!=0:117eVatT=20Kandp=1019cm3.AFWHMlinewidthof4meVisassumed. lowestenergyforthisheavyholeLandaulevelresidesataboutkz=0:75(1=nm).Checkingthetransitionelementalongkz,itisalsofoundthatthistransitionindeedtakesplaceawayfromthezonecenter.DisplayedinFig. 3{19 isthecomparisonoftheeight-bandmodelversusafull-zonethirty-bandmodel.Atthezonecenter,theeight-bandmodeltswellwiththethirty-bandmodel.Notfarawayfromthezonecenter,abigdeviationoccurs.Wethinkthisdeviationoftheenergydispersionispossiblyresponsibleforthelargedeviationofthecalculatedresonancepeakposition.

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Figure3{18: k-dependentLandausubbandstructureatB=350T. Figure3{19: BandstructurenearthepointforInAscalculatedbyeight-bandmodelandfullzonethirty-bandmodel.

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3{16 ,thusCRspectradependstronglyoncarrierdensities.InFig. 3{12 theholedensityis11019cm3.Atsuchaholedensity,theFermienergyisbelowtheH1;1andL0;3statessothatwehavetwostrongtransitions.Iftheholedensityislower,theFermienergywillshiftupward,thusthesetwostateswillbecomelessoccupiedbyholes,andwecanexpectadecreaseintheCRstrength.However,thedecreaseinstrengthforthetworesonancepeaksisdierent.ShowninFig. 3{20 (a)aretheCRspectraforfourdierentholedensities.TheLandaulevelsalongwiththecorrespondingFermienergiesareplottedinFig. 3{20 (b).Resonanttransitionsat0:117eVareindicatedbyverticallines.WecanseethattheCRpeak2isalmostalwayspresent,becauseatlowmagneticelds,theheavyholestateH1;1isalmostalwaysoccupied.TheCRpeak1changesdramaticallywithholedensity,andnearlyvanishesatp=51018cm3.Therelativestrengthsoftheheavyandlight-holeCRpeaksissensitivetotheitinerantholedensityandcanbeusedtodeterminetheholedensity.BycomparingtheoreticalandexperimentalcurvesinFig. 3{20 (a),weseethattheitinerantholeconcentrationisaround21019cm3.FromFig. 3{20 (a),wecanruleoutp<1019cm3andn>41019cm3.Weestimatethatanerrorintheholedensityofaround25%shouldbeachievableatthesedensities.BecauseoftheexistenceinIII-VDMSoftheanomalousHalleect,whichcanoftenmakethedeterminationofcarrierdensitydicult,determiningcarrierdensitybycyclotronresonancecanserveasapossiblealternative. 83 ].ThesesamplesareferromagneticwithTCrangingfrom30to55KandwhosecharacteristicsaresummarizedinTable 3{2

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Figure3{20: TheholedensitydependenceofholeCR.(a)TheoreticalholeCRcurvesinInAsfrombottomtotopwithholedensitiesof51018,1019,21019and41019cm3;(b)LandaulevelsinvolvedinobservedCRalongwithFermilevelscorrespondingtotheoreticalcurvesin(a).

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Table3{2: CharacteristicsoftwoInMnAs/GaSbheterostructuresam-ples SampleNo.TC(K)MncontentxThickness(nm)Density(cm3) 1550.09251:110192300.1294:81019 3{21 (a)and(b),atvarioustemperaturesasafunctionofmagneticeld.Thelaserbeamwashole-activecircularlypolarized.IntheleftpanelofFig. 3{21 ,fromroomtemperaturedowntoslightlyaboveTC,abroadresonancefeature(labeled`A')isobservedwithalmostnochangeinintensity,position,andwidthwithdecreasingtemperature.ClosetoTC,quiteabruptanddramaticchangestakeplaceinthespectra.First,asignicantreductioninlinewidthandasuddenshifttoalowermagneticeldoccursimultaneously.Also,theresonancerapidlyincreasesinintensitywithdecreasingtemperature.Inaddition,asecondfeature(labeled`B')suddenlyappearsaround125T,whichalsorapidlygrowsinintensitywithdecreasingtemperatureandsaturates,similartofeatureA.Atlowtemperatures,bothfeaturesAandBdonotshowanyshiftinposition.Essentially,thesamebehaviorisseenintherightpanelinFig. 3{21 .Usingdierentwavelengthsoftheincidentlight,similarCRspectrumbehaviorhasalsobeenobserved. Forzinc-blendesemiconductors,theCRpeaksAandBareduetothetran-sitionsofH1;1!H0;2andL0;3!L1;4,respectively,whichwehavealreadypointedout.Weattributethetemperature-dependentpeakshifttotheincreaseinthecarrier-Mnionexchangeinteractionresultingfromtheincreaseofmagneticorderingatlowtemperatures.ThetheoreticallycalculatedresultsareshowninFig. 3{22 forbulkIn0:91Mn0:09As.TheCRspectrawasbroadenedusingaFWHMlinewidthof4meV.ThetheoreticalresultsclearlyshowashiftofpeakAtolower

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Figure3{21: CyclotronresonancespectrafortwoferromagneticInMnAs/GaSbsamples.Thetransmissionofhole-activecircularpolarized10:6mradiationisplottedvs.magneticeldatdierenttemperatures. eldswithdecreasingtemperature,althoughinbulkInAs,thetransitionoccursatabout40T,asopposedtotheheterostructurewheretheresonanceoccursat50T. TheCRpeakAonlyinvolvesthelowesttwoLandaumanifolds.AswasdiscussedinSection2.3.3,whenn=1,theHamiltonianis11,andwhenn=0,theHamiltonianfactorizesintotwo22matrices,soitiseasytoobtainanexactanalyticalexpressionforthetemperaturedependentcyclotronenergy.Withneglectofthesmalltermsarisingfromtheremotebandcontributions,thecyclotronenergy

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Figure3{22: TheoreticalCRspectrashowingtheshiftofpeakAwithtemperature. fortheH1;1!H0;2transitionis 4xhSzi()s Eg 4xhSzi()2+EpBB:(3{25) Intheeldrangeofinterest(40T),p 3{25 ,weobtainanexpressionoftheform 4xhSzi()(1)(3{26) where IfweassumethetemperaturedependenceofEgandEpissmall,itfollowsfromEq. 3{26 thattheCRpeakshiftshouldfollowthetemperaturedependenceofthemagnetizationhSzi,whichinameaneldtheoryframeworkisgivenby

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Figure3{23: AveragelocalizedspinasafunctionoftemperatureatB=0,20,40,60and100Tesla.TheCurietemperatureisassumedtobe55K. Eq. 1{22 .ThetemperaturedependenceofhSziisshowninFig. 3{23 atseveralmagneticelds.ThemagneticelddependenceofhSzihasalreadybeenshowninFig. 1{10 TherelativechangeoftheCRenergy,calculatedusingEq. 3{25 and 3{26 ,asafunctionoftemperatureispresentedinFig. 3{24 .Itshowsthatfromroomtemperatureto30Kthecyclotronenergyincreasesabout20%,whichcorrespondstoanapproximately20%decreaseintheresonantmagneticeld,approximatelytheresultobservedintheexperiment.Inaddition,wefoundthattheshiftisnonlinearintemperatureandthemainshiftoccursattemperatureswellaboveTC.Thesefeaturesarealsoconsistentwithexperiment. AlongwiththeCRpeakshift,experimentindicatesasignicantnarrowingofthelinewidth.Wespeculatethatthiseectmaybeassociatedwiththesuppressionoflocalizedspinuctuationsatlowtemperatures.AsimilareecthasbeenobservedinII-VIdilutemagneticsemiconductors(seeRef.[ 84 ]andreferencestherein).Spinuctuationsbecomeimportantwhenacarrierinthebandinteractssimultaneouslywithalimitednumberoflocalizedspins.Thistakesplace,for

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Figure3{24: RelativechangeofCRenergy(withrespecttothatofhightempera-turelimit)asafunctionoftemperature.VerticaldashedlineindicatesTC. example,inmagneticpolaronsandforelectronsindilutemagneticsemiconductorquantumdots.Thestrongin-planelocalizationbythemagneticeldmayalsoresultinareductionofthenumberofspinswhichacarrierinthebandfeels,thusincreasingtheroleofspinuctuations.However,itispossiblethattheCRpeaknarrowingistheresultoftheincreasedcarriermobilities.AlthoughtheInMnAslayerisheavilydopedandthustheholemobilityisverylow,holesintheGaSblayer,iftheyexist,willhavemuchhighermobilities.Sowecanspeculatethatnearthetransitiontemperature,thebandstructureofInMnAschangesinsuchawaythatafractionoftheholesmoveintotheGaSblayeror/andtheInMnAs/GaSbinterface,wheretheholeCRhasamuchnarrowerlinewidth.ShowninFig. 3{25 isthebanddiagramoftheInMnAs/GaSbheterostructure[ 46 ].TheinterfacestatesofInMnAs/GaSbareverycomplicated,andwehavenotcarriedoutcalculationsincorporatingtheminourCRsimulations.

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Figure3{25: BanddiagramofInMnAs/GaSbheterostructure. Thesituationinrealsemiconductors,however,diersfromthatinaclassicalfreeelectrongas.Matsudaetal.[ 82 ]haveexperimentallyobservede-activeCRinp-dopedInAsandInMnAs.Thetemperaturewasquitelow(12K)andtheholeconcentrationwashighenough(1019cm3)tosafelyeliminatethepossibilitythatthee-activeCRcomesfromthethermallyexcitedelectronsintheconductionband.Thepossibilityoftheexistenceofelectronsintheinterfaceorsurfaceinversionlayershasbeenalsoexcluded.Thus,theresultssuggestthate-activeCRcomesfromthevalencebandholes,incontradictionwiththesimplepictureofafreeholegas. Wendthate-activecyclotronresonanceinthevalencebandsisanintrinsicpropertyofcubicsemiconductorsandresultsfromthedegeneracyofthevalencebands.Aswediscussedbefore,heavyholeandlightholebandswillbothsplitinto

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aseriesofLandaulevels.ThiscomplexityallowsonetosatisfyconservationofangularmomentuminCRabsorptionforbothe+andepolarization,providedoneswitchesbandtype. Intheconductionband,increasingthemanifoldquantumnumberalwaysincreasestheenergy.Asaresult,onlytransitionswithincreasingnmaytakeplaceinabsorption,thatis,onlye-active(e)CRcanbeobservedintheconductionband. Figure3{26: SchematicdiagramofLandaulevelsandcyclotronresonancetransi-tionsinconductionandvalencebands.Bothh-activeande-activetransitionsareallowedinthevalencebandbecauseofthedegeneratevalencebandstructure.Onlye-activetransitionsareallowedintheconductionband. Thevalenceband,however,consistsoftwotypesofcarriers:heavyholes(J=3=2;Mj=3=2)andlightholes(J=3=2;Mj=1=2).EachofthemhastheirownLandauladderinthemagneticeld.Anincreaseofnalwaysdecreasestheenergyonlywithineachladder.Similartotheconductionbandcase,transitionswithinaladder(HH!HHorLH!LH)cantakeplaceonlyinh-active(e+)polarization.However,therelativepositionofthetwoladderscanbesuchthatinterladdertransitions(LH!HH)ine-activepolarizationareallowed.

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ThisprocessisschematicallyshowninFig. 3{26 .Notethatthisgureisextremelysimpliedandshouldbeusedonlyasaqualitativeexplanationoftheeect. Figure3{27: ThevalencebandLandaulevelsande-activeholeCR.(a)Thelow-estthreepairsofLandaulevelsinthee-activetransition;(b)TheseparateCRabsorptioncontributingtothee-activeCR. We'veexaminede-activeCRinp-typeInAsatT=12Kwithafreeholedensityof11019=cm3.Thecomputedk=0valencebandenergiesasafunctionofmagneticeld,thee-activeopticaltransitionsandthecorrespondingCRabsorptionspectraareshowninFig. 3{27 .Themostpronouncede-activetransitionstakeplacebetweentheHHstateH0;2andLHstateL1;5andbetweentheH1;3andL2;6states.Therearesomeotherlesspronouncedtransitions,whichcontributetotheabsorptionspectraatlowerelds.

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ThecalculatedandexperimentalCRareshowninFig. 3{28 forbothe-activeandh-activepolarizations.Thereisgoodagreementbetweentheoryandexperiment.Asdiscussedabove,theelectronactiveabsorptionisdeterminedbytheHH!LHtransitions.Themaincontributiontotheh-activeabsorption(leftpanelinFig. 3{28 )comesfromthetransitionswithintheheavyholeladder,whichwehavealreadydiscussedindetailinthelastsection. Thecalculationandobservationofe-activeCRinp-typeDMScanaidinunderstandingthevalencebandstructureofDMSsystems.Usingbothhande-activeCRonecanexplorethewholepictureofthevalencebands. Figure3{28: ExperimentalandtheoreticalholeCRabsorption.Solidlinesareex-perimentalholeCRspectraasafunctionofmagneticeldforh-activeande-activepolarizations.Correspondingtheoreticalcalculationsareshownindashedlines.

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Inthelastchapter,itwasseenthattheMndopinggreatlyaectsthevalencesubbandalignmentandthecyclotronresonance.Ontheonehand,dopingwithMnimpuritieswillgreatlyenhancethescatteringofcarriersthusincreasingthelinewidthandreducingthestrengthoftheCRspectra.Ontheotherhand,itwillalsoshifttheCRpeakpositionsduetothechangesinthevalencebandstructure. Figure3{29: ValencebandstructureatT=30KandB=100TforIn1xMnxAsalloyshavingx=0%andx=5%.Forx=0%,therstHHstate,H1;1,liesbelowthelightholestateL1;5.Forx=5%,theorderofthesetwostatesisreversed.TwopossibleCRtransitionsareshownusingupwardarrows,namelyanh-active()transitionbetweenH0;2andH1;1,andane-active(+)transitionfromH0;2toLH1;5.ThedashedlinesaretheFermienergiesforaholedensityof1019cm3. WeillustratehowMndopingaectstheopticaltransitionsinFig. 3{29 .Inthiscase,weassumeacarrierdensityof1019cm3,withoutandwithMndopingat30K,inamagneticledof100T.Theprimaryh-activeande-activetransitionsarebothindicatedinthesegures.Onlythreelevelsareinvolvedinbothh-ande-activetransitions.Theh-activetransitionisfromtheH1;1toH0;2stateaswementionedabove.Thee-activetransitionisfromtheL1;5toH0;2state.WeseethatwithoutMndoping,theL1;5statesitsonthetop,butwithdoping,H1;1stateisshiftedtothetopwhileL1;5isshiftedtoalowerposition.Thusbothh-activeande-activeabsorptionwillbeaected.

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LineshapesandpeakpositionsareverysensitivetoMndoping.Notethatabsorptiontakesplacenotonlyatthepoint,butalsoinregionsawayfromthezonecenter.EventhoughwebroadentheCRlineswiththesamelinewidth(4meV),thedopedsamplehasamuchbroaderlineshapeduetotheenergydispersionchangealongkzandtheenergypositionchangerelativetotheFermilevelbroughtaboutbythetheexchangeinteraction.ThisisshowninFig. 3{30 .Furthermore,theheightofthepeakoftheCRspectrumoftheMndopedsampleisreducedbyabout30timescomparedtotheundopedone.ThismaycomefromtheFermillingeect,sincetheL1;5statebecomelessoccupiedwhenthesampleisdopedwithMnionsatacarrierdensityof1019cm3. Figure3{30: Theprimarytransitioninthee-activeholeCRunderdierentMndoping.MndopingchangestheLandaulevelalignmentandthetransitionstrengthaswell.ItalsoshiftstheCRpeakpositiontoalowereld.

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Inmagneto-opticalexperiments,transmissionofthesampleisusuallymea-sured,sincetransmissionmeasuresabsorptioninsidethesample,andinmostcases,absorptionprobestheintrinsicelectronicandtransportproperties.However,whenasampleistoothick,directmeasurementoftransmissionisimpossible.Inthiscase,onecanmeasurethereectionofthesample,andusetherelationsbetweentheopticalconstantstoderivetheabsorptioncoecient,thusobtainingtheintrinsicpropertiesthroughquantitativeanalysis. Themagneto-opticalKerreect(MOKE)isrelatedtolightreection.Whenlinearlypolarizedlightisreectedbythesurfaceofaferromagneticsample,thepolarizationplanewillundergoarotation.Similarly,thereisalsotheFaradayeect,whichisrelatedtotherotationofthepolarizationplaneofthetransmittedlight.Magneto-opticaleectsmaybeobservedinnon-magneticmediasuchasglasswhenamagneticeldisapplied.However,theintrinsiceectsareusuallysmallinsuchcases.Inmagneticmedia(ferromagneticorferrimagnetic)theeectsaremuchlarger.Forcubiccrystals,whenthereisnoferromagnetism,noMOKEsignalwillbepresent,sowhenstudyingthedynamicalmagneticpropertiesofDMS,MOKEcanserveasapowerfultoolfordetectionandmeasurementofmagneticmomentsortimedependentmagneticmoments,withtimeresolvedoptics[ 46 ]. 96

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theotheropticalvariablesandtheabsorptioncoecient,especiallytherelationbetweenthereectionandabsorptioncoecients. Whenanelectromagneticwaveispropagatinginamediumwithamagneticrelativepermittivityr,andelectronicrelativepermittivityr,itsatisesMaxwellequations, @t; 0r: Whentherearenofreecharges,Eq. 4{4 becomes TakingthecurlofEq. 4{1 ,andusingtherelation Eq. 4{5 becomes Foraplanewavepropagatinginz-direction, and 0c2:(4{9) Ingeneral,thewavevectorcanbewrittenasacomplexnumber cN;(4{10)

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whereNisthecomplexrefractiveindexgiveby IfwewriteN=n+i,wherenandaretherealandimaginarypartofN,thenk=n! c+i! c.SubstitutingthisintoEq. 4{8 ,weobtain wherek0=k=Nisthewavevectorinvacuum.ThuswecanseethattheimaginarypartofN,,isrelatedtolightabsorption,i.e.theextinctioncoecient.ThelightintensityisproportionaltojEj2,sowecanwritetheintensityas cz:(4{13) Thustheabsorptioncoecientis c:(4{14) FromEq. 4{9 ,ifthereisnoenergyloss,then cp isreal.Butsincetherearelosses,wewrite cp wherethecomplexdielectricconstanterisdenedas 0!=1+i2:(4{17) where1and2aretherealandimaginarypartsofer. ComparingEq. 4{10 and 4{16 ,wehave

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Generally,risverycloseto1,insuchacase,therelationshipsbetweentherealandimaginarypartsofNand~rare and Thus,wecancalculatenandfrom1and2,andviceversa.Intheweaklyabsorbingcase,i.e.n,Eq.( 4{20 )canbesimpliedto FromEq. 4{14 ,therelationbetweentheabsorptioncoecientanddielectricconstantis Manymeasurementsofopticalpropertiesinsolidsinvolvenormallyincidentreectivity.Insidethesolid,thewavewillbeattenuated.Weassumeforthepresentdiscussionthatthesolidisthickenoughsothatreectionsfromthebacksurfacecanbeneglected. Considerthereectionofaplanewavemovinginthezdirection.Theinterfacebetweenahalf-innitemedium1withrefractiveindexN1andahalf-innitemedium2witharefractiveindexN2istakentobez=0.ThissituationisillustratedinFig. 4{1 .AssumingE==x,wehaveanincomingandreectedwavein

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Figure4{1: Diagramforlightreectionfromtheinterfacebetweenmedium1withrefractiveindexN1andmedium2withrefractiveindexN2. medium1 Inmedium2,thetransmittedwaveis ContinuityofelectriceldattheinterfacerequiresE0=E1+E2.WithEinthexdirection,thesecondrelationbetweenE0,E1andE2followsfromthecontinuityofthetangentialmagneticeldHyacrosstheinterface.FromEq. 4{1 ,wehave ThecontinuityconditiononHythusyieldsacontinuityrelationfor@Ex=@zsothatfromEq. 4{26 weobtain

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ThenormallyincidenctreectivityRis andthereectioncoecientrisgivenby whereN1,N2andrareallcomplexvariables.AccordingtoEq. 4{14 ,absorptionmeasurementscanbeusedtodeterminethereectioncoecient. Usually,therealandtheimaginarypartsoftheopticalconstantsarenotindependent.TheyarerelatedbytheKramers-Kronigrelation[ 85 ].Forexample,forN=n+i, orusingEq. 4{14 }Z10(!) where}Z10istheprincipalvalueoftheintegral.

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Figure4{2: Schematicdiagramformagneticcirculardichroism. four-folddegenerate.Withspontaneousmagnetizationwhichproducesaself-consistenteectivemagneticeld,theconductionandvalencebandstateswillsplit(seealso,Fig. 2{1 ).Thustheabsorptionfor+andpolarizationwillbedierent.AverysimpleschematicdiagramforthiseectisillustratedinFig. 4{2 ,inwhichweonlyshowthetwoheavyholevalencebands.Correspondingtothetwocircularpolarizations,wedenetwocomplexrefractiveindicesN+andN,whereN+=n++i+,andN=n+i.Therealpartsoftherefractiveindicesdonothaveastrongpolarizationdependence,sowesetn+=n=n. Attheinterface,thereectioncoecientfor+polarization,followingEq. 4{29 ,is andfor, Duetodierencebetween+and,r+andrhavedierentphasefactors.Thatmeansafterreection,+andlightwillhavedierentphases. Nowconsiderthecasewherelinearlypolarizedlightpropagatesnormaltothesurfaceofaferromagneticcrystalandisreectedbythesurface,asillustratedin

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Fig 4{3 (a).Linearlypolarizedlightcanbedecomposedinto+andcompo-nents.Fromthediscussionabove,thesetwocircularlypolarizedbeamswillhavedierentphasechangesafterreection,sothereectedlight,ifitisstilllinearlypolarized,willnotstayinthesamepolarizationplane.ThepolarizationplanewillberotatedandthisiscalledKerrrotation.Duetothedierencesinabsorptionofthetwocircularlypolarizedbeams,thereectedlightwillingeneralbeellipticallypolarized.Deningacomplexrotation K=K+iK(4{34) wehave K=i(r+r) Figure4{3: DiagramsforKerrandFaradayrotation.(a)Kerrrotation;(b)Fara-dayrotation. polarizedlight,andKistheellipticity,whichisdenedastheratiooftheminortothemajoraxesofanellipsoid.

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SimilartotheKerreect,afterbeingtransmittedthroughaferromagneticcrystal,linearlypolarizedlightwillalsohavearotationandingeneralbeellipti-callypolarized.ThisiscalledtheFaradayeect,whichisillustratedinFig. 4{3 (b).Itiseasytoshowthattherotationangleperunitlengthis andtheelliplicity Theellipticityisrelatedtothemagneticcirculardichroism(MCD),whichisdenedbythedierence(!)betweentheabsorptioncoecientoftherightandleftcircularlypolarizedlight.Fromtherelationsbetweentheopticalconstants,wehave (!)=+(!)(!)=4F(!) wherelstandsforthelighttransmittedlength. Notethattheincidentlightisalongthemagnetizationdirection,whichwedenedasthez-direction.Inthelongitudinalcasewherethemagnetizationvectorisintheplaneofthesurfaceandparalleltotheplaneofincidenceorinthetransversecasewherethemagnetizationvectorisintheplaneofthesurfaceandtransversetotheplaneofincidence,noKerrrotationisobservedatnormalincidence. 4{14 ,theextinction

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Figure4{4: KerrrotationofInMnAs.(a)AbsorptioncoecientofIn0:94Mn0:06Asasafunctionofphotonenergyfore-andh-activelightatT=30K;(b)ThecorrespondingKerrrotation. coecientsarecomputed.ThenfromEq. 4{35 ,weobtain SupposewehaveaIn1xMnxAssamplewithx=6%,aCurietemperatureTC=55K,atT=30K.Thecomputede-andh-activeabsorptioncoecientsareshowninFig. 4{4 (a).Itcanbeseenthatduetotheferromagnetism,thesamplehasdierentabsorptioncoecientsfore-andh-activepolarization.Thisgivesrisetoanon-zeroKerrrotation,whichisshowninFig. 4{4 (b).Therotationisaboutseveraltenthsofadegree. Actually,theeight-bandkptheoryisnotcapableofcalculatingthelightabsorptionforaverywiderangeofphotonenergies,becausenotonly-valley,butalso-Lvalley,andeven-Xvalleyabsorptionneedtobeconsidered.Thistaskrequiresafullzonebandstructure.TheL-valleyliesabout1:08eVabovethevalencebandedge,andtheX-valleyabout1:37eVabovethevalencebandedge.AschematicbanddiagramforInAsisshowninFig. 4{5 [ 86 ].Evenso,wecanstill

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Figure4{5: ThebanddiagramforInAs. getaqualitativepictureoftheKerrrotationforInMnAsforphotonenergiesbelow1eV. TheresultsforGa0:94Mn0:06AsareshowninFig. 4{6 (a)and(b),inwhichwesupposetheCurietemperatureisTC=110K,atT=30K.ComparingFig. 4{4 (b)andFig. 4{6 (b),weseethattheKerrrotationsareofthesameorder,aboutatenthofadegree. TheschematicdiagramfortheGaAsbandstructureisshowninFig. 4{7 [ 87 ].TheL-valleyliesabout1:71eVabovethevalencebandedge,andtheX-valleyabout1:90eVabovethevalencebandedge.Souseoftheeight-bandkptheoryinGaAsisevenworse.However,weexpectthetransitionfrom-Lvalleyisnotaseectiveas-valleytransitionbecausetheformerisanindirectprocess.

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Figure4{6: KerrrotationofGaMnAs.(a)AbsorptioncoecientofGa0:94Mn0:06Asasafunctionofphotonenergyfore-andh-activelightatT=30K;(b)ThecorrespondingKerrrotation. 2arg(r whererandr+arethetwocomplexreectioncoecientsforand+circularlypropagatinglightbeamsinthemedium,andarg(x)representsthephaseofthecomplexnumberx.ItiseasytoprovethatthisKisexactlytheKinEq. 4{35 AccordingtoRef.[ 88 ],foramultilayerstructure,thecoecientsrdependontheamplitudeofthereectioncoecientsri;i+1attheinterfacesofsuccessivelayersiandi+1.Ifweapproximateasinglequantumwellbyathree-layerstructure,inthecaseofnormalreection,rtakestheform where

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Figure4{7: ThebanddiagramforGaAs. Thereectioncoecientsri;i+1,canbeobtainedfromEq. 4{32 and 4{33 .i=(w=c)liNidenotesthedephazingoftheelectriceldradiationaftercrossingthelayeriofthicknessli. Attheinterfaceofi;i+1,thecomplexreectioncoecientbecomes whichcanbeeasilyobtainedfromthesameprocedurewedescribedinSection4.1.ThecomplexrefractiveindicesNareobtainedbycalculatingtheabsorptioncoecientsineachlayer. NowconsideraIn0:88Mn0:12As/GaSbheterostructurewithaInMnAslayerthicknessof9nminaparamagneticphase(thusr+=r)attemperatureT=5:5K.Tocomparewithexperiment,thereectivityofthisstructureiscalculatedandshowninFig. 4{8 (b).Alongwiththereectivity,theabsorptioncoecientsintheInMnAsandGaSblayerarealsoshowninFig. 4{8 .Inthe

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Figure4{8: TheabsorptioncoecientsbothinInMnAsandGaSblayers(a)andthereectivityofInMnAs/GaSbheterostructure(b). calculation,weassumeacarrierdensityofp=1019cm3intheInMnAslayer,andnocarriersintheGaSblayer. Theexperimentalresult[ 89 ]isillustratedinFig. 4{9 .Upto1:5eV,wecanseethatthecalculationsuccessfullyreproducestheoscillatingstructureofthereectivity,andthecalculatedmeanreectivityisveryclosethetheexperimentalone.Inourcalculation,thereareseveralenergieswherethereectivityisveryclosetozero,whileintheexperiments,thelowestreectivityisstillaround40%.Wesupposethisisbecauseinourcalculation,wehavenotconsideredinterfaceroughness,whichcansignicantlycontributetothereection. NowweconsideraferromagneticIn0:88Mn0:12As/AlSbheterostructurewitha9nmthickInMnAslayeranda136nmthickAlSblayergrownona400nmthickGaSblayer.ThisstructurehasaCurietemperatureof35K,andtheMOKEsignalhasbeenmeasuredatamagneticeldof3T.TheexperimentalresultisshowninFig. 4{10 (a).Followingthesameprocedure,theabsorptioncoecientsintheInMnAslayerarecalculatedincludingquantumconnementeectsusinganitedierencemethod[ 90 ].Thecalculationhasbeenperformedinthe0:31:5eVphotonenergyrange,inwhichtheAlSblayeristransparent.

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Figure4{9: ReectivityofIn0:88Mn0:12As(9nm)/GaSb(600nm)heterostructureatT=5:5KmeasuredbyP.FumagalliandH.Munekata.ReprintedwithpermissionfromP.FumagalliandH.Munekata.Phys.Rev.B53:15045-15053,1996.Figure3,Page15047. ThelightabsorptionintheGaSblayerisalsocalculated,andthelightreectionsfromtheInMnAs/AlSbandAlSb/GaSbinterfacearebothtakenintoaccount.TheKerrrotationisobtainedusingEq. 4{40 andthecalculatedresultisshowninFig. 4{10 (b).

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Figure4{10: Measured(a)andcalculated(b)KerrrotationofInMnAs(19nm)/AlSb(145nm)heterostructureunderamagneticeldof3TatT=5:5K.Panel(a)isreprintedwithpermissionfromP.FumagalliandH.Munekata.Phys.Rev.B53:15045-15053,1996.Figure8,Page15049.

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HolesinIII-VDMSplayanimportantroleinmediatingtheferromagnetismandparticipatinginmagneto-opticaltransportprocesses.UnlikeinII-VIDMSwhereMn2+ionsareisoelectronicwiththecations,MnionsinIII-VDMSareacceptors.DuetotheAsanti-sitedefectsandinterstitialMn,bothofwhichactasdoubledonors,theholeconcentrationisusuallymuchlowerthantheMncon-centration.However,themediationoftheexchangeinteractionbetweenlocalizedmagneticmomentsbyholesisthecornerstoneofmosttheoriesofferromagnetisminIII-VDMS(pleaserefertoSection1.3.1).Lightinducedferromagnetisminp-InMnAs/GaSbhasbeenobservedbyKoshiharaetal.[ 44 ],andKonoetal.[ 46 ],wherehole-electronpairshavebeenexcitedandtheholedensitygreatlyenhancedbytheincidentlight.Inthelatterexperiment,ultrafastlasershavebeenemployedandthetime-dependentMOKEsignalhasbeenmeasured.TheultrashortlaserpulsescreatealargedensityoftransientcarriersintheInMnAslayerandtheMOKEsignaldecayslessthan2psafterlaserpumping,asshowninFig. 5{1 .Recently,Mitsumorietal.[ 91 ]hasstudiedthephoto-inducedmagnetizationrotationinferromagneticp-GaMnAs.Theyfoundthatwhenshiningcircularlypolarizedlightnormaltothesamplesurface,whichisparalleltothemagnetiza-tion,andprobingwithlinearlypolarizedlight,anon-zeroKerrrotationisseenwhichhasadecaytimeof25ps.Aswediscussedinthelastchapter,thereisnolongitudinalKerrrotationatnormalincidence,sothisKerrrotationwasduetotherotationofthemagnetizationduetothelight-inducednon-equilibriumcarrierspins.Thislight-inducedrotationofmagnetizationisillustratedschematicallyinFig. 5{2 112

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Figure5{1: Light-inducedMOKE.Signaldecaysinlessthan2ps. Inthesepump-probeopticalexperiments,thebehaviorofthenon-equilibriumcarrierspinswhichinducetheexchangeinteractionisakeyfactorthatdeservestobestudied.Theelectronspinrelaxationintheconductionbandhasbeenthoroughlyinvestigatedbymanyauthors[ 4 92 ],however,theoreticalstudiesonholespins,especiallyonthenon-equilibriumholespins,aresparse.Inthischapter,wewillfocusontheholespinrelaxationinbulkIII-VDMS,anddiscussthemechanismswhichinducetherelaxation. 93 94 ],Dyakonov-Perel(DP)[ 95 ],Bir-Aronov-Pikus(BAP)[ 96 ],andhyperneinteractionprocesses.IntheEYmechanismelectronspinsrelaxbecausetheelectronwavefunctionsnormallyassociatedwithagivenspinhaveanadmixtureoftheopposite-spinstates,duetospin-orbitcouplinginducedbytheions,thusspin-ippingprocessesaccompanymomentumrelaxation.TheDPmechanismexplainsspindephasinginsolidswith

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Figure5{2: Light-inducedmagnetizationrotation.ReprintedwithpermissionfromMitsumorietal.Phys.Rev.B69:33203-33206,2004.Figure1,Page33203. inversionasymmetry,whichcausesspinsplitting.Spindephasingoccursbecauseelectronsfeelaneectivemagneticeld,duetothespin-splittingandspin-orbitinteractions,whichchangesinrandomdirectionseverytimetheelectronscatterstoadierentmomentumstate.TheBAPmechanismisimportantforp-dopedsemiconductors,inwhichtheelectron-holeexchangeinteractiongivesrisetouctuatinglocalmagneticeldswhichipelectronspins.Insemiconductorswithanuclearmagneticmoment,thereisalsoahyperneinteractionbetweentheelectronspinsandnuclearmomentswhichwillcausespin-ippingofelectronspins. Inthevalenceband,theLuttingerHamiltoniandescribingthe4heavyholeandlightholebandsis[ 97 ] 22k222(kJ)2(5{1) whereJx,JyandJzare44matricescorrespondingtoangularmomentumJ=3=2.Duetothespin-orbitinteraction,andthekJtermintheHamiltonian,thespinstatesinthevalencebandmixverystrongly.Thelightholeandsplit-oholebasisstatesarenotspineigenstates.Evenforapureheavystateatk=0,itbecomesstronglymixednotfarawayfromthepoint.Unlikeintheconductionband,wheretheoverlapintegralbetweentwoelectronspinstatesisverysmall

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(1)becauseoftheweakmixing,theoverlapintegralbetweentwostronglymixedvalencebandstatesisusuallybig.Thechancesarehighthataholescatteredfromoneelectronicstate(say,mainlyspin-up)toanother(say,mainlyspin-down)maytotallyreverseitsspin.Soweconcludethatforholespinrelaxationinthevalenceband,theEYmechanismismosteective. SupposeaholeinstatejkihasspinSk.Afteronescatteringevent,thisholetransitstoanotherstatejk0i,inwhichthespinisSk0.Inthisprocess,thespinchangeisS=Sk0Sk.Thespinrateofchangeis dt=ZW(k;k0)Sdk0=ZW(k;k0)(Sk0Sk)dk0(5{2) whereW(k;k0)isthescatteringratebetweenstatekandk0. Assumewecanuseonetimeconstanttodescribethisspinrelaxationprocess,thenwecanwritedownanequationlike dt=Sk whereisthespinrelaxationtime,whichstateshowlonganon-equilibriumholespinwilltaketocompletelyloseitspreviousorientation.Forcalculating,weneedtoknowthescatteringrateW(k;k0).

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andthelatterispolaropticalphononscattering.ThesetwomechanismsdominatephononscatteringinbulkIII-Vsemiconductors. UsingFermi'sgoldenrule,ageneral3-Dphononscatteringratecanbewrittenas[ 98 ] 21 2)(Ek0Ekh!q;b)dk0(5{4) whereVisthecrystalvolume,Nisthenumberofunitcellsinthecrystal,M0isthereducedmassoftheunitcell,h!q;bisenergyofaphononwithwavevectorqinmodeb,andn(!q;b)isthephonondensityforsuchamode.kqk0;Ktakesintoaccountmomentumconservationinascatteringevent,whereKisareciprocallatticevector.Fornormalprocesses,K=0,andprocesseswithK6=0areumklappprocesses.Usuallyweonlyconsiderlongwavelengthphonons,wherescatteringmainlytakesplaceneartheBrillouinzonecenter.Forthiskindofsituation,K=0.InEq. 5{4 ,theuppersignisforphononabsorptionandthelowersignforemission.C2q;bistheelectron-phononinteractioncouplingconstantfromtheinteractionHamiltonian,and istheoverlapintegralbetweeninitialandnalelectronicstates. InDMSmaterials,generallytheholedensityisratherhigh,soscreeningmustbetakenintoaccount.Whenconsideringopticalphononscattering,inwhichthevibrationfrequencyisveryhigh,theplasmamodemaybemixed,andthereforeadynamicscreeningmodelisdesirable.Buthere,wejustassumeastaticscreening,whichworkswellforacousticphononandimpurityscattering,andisagoodapproximationforopticalphononscattering.

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whereN(Ei)isthedensityofstatesatenergyEi,andf(Ei)istheFermi-Diracdistributionfunctionrepresentingtheoccupationprobability.Thesumrunsoverallenergylevels.WhenanelectricpotentialV(r)ispresent,itwillchangetheelectrondensityto Herewesupposetheperturbationpotentialissmallanddoesnotaectthedensityofstates.Becauseofthispotential,thereisaperturbationinthespacechargedensity ThechargedensityisrelatedtothepotentialbyPoisson'sequation, XiN(Ei)[f(EieV(r))f(Ei)](5{9) ConsiderV(r)tobeasmallperturbation,andexpandtheright-handsideoftheaboveequation.Theleadingtermgives DeningthereciprocalDebyescreeninglengthq0by

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thesolutionforEq. 5{10 is req0r(5{12) forsphericalsymmetry.ThevalueofAcanbeobtainedusingboundaryconditions.Forexample,forapointcharge,V(r)!0,r!1andV(r)!Ze2=4r,r!0,soA=Ze2=4.Equation 5{12 isknownastheYukawapotential. ThederivativeofFermi'sfunctionwithrespecttoEinEq. 5{10 is andthusEq. 5{11 becomes Wewillusethisequationtocalculatethereciprocalscreeninglengthlaterwhendealingwithphononandimpurityscattering. 99 ]measuredtheholespinrelaxationtimeintheGaAsvalencebandusingapump-probetechnique.Theygeneratedorientedholesusingan800nmlaserinheavyandlightholebands,andprobedusingalaserpulse(3m)withenergycorrespondingtothesplit-oholetoheavyholeorspilt-oholetolightholetransitions,thenmeasuredthecircularpolarizationchangeofthetransmittedlight.Withinanerrorof10%,theyobtainedaholespinrelaxationtimeof110fs.InpureintrinsicIII-Vsemiconductors,polaropticalphononandpiezoelectricscatteringareresponsibleforthisholespinrelaxation. PolaropticalphononscatteringdominatesinbothII-VIandIII-Vintrinsicsemiconductorswhentemperatureisnottoolow.Atverylowtemperature,duetothehighenergyoftheopticalphonons,theirdensityisverylow,too.Furthermore,theemissionofanopticalphononrequiresalargeenergydierence,whichisalsonotfavorableatlowtemperatures.

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Polarscatteringoccursinconnectionwiththecontrarymotionofthetwoatomsineachunitcellandonlytakesplaceinthelongitudinalopticalmode,asdescribedbyFrohlich[ 100 ]andCallen[ 101 ]. WecanwritethepolarinteractionHamiltonianas where(R)isthechargedensityoftheelectronsand(R)istheelectricpotentialassociatedwithpolarizationintheunitcellcenteredatR. Followingthediscussionin[ 98 ],ifwetakeintoaccountthescreeningeect,thentheHamiltonianis q2+q20(iQqeiqr+c:c:)(5{16) whereeistheeectivechargeontheatomsinaunitcellandV0isthevolumeofaunitcell.Qqarethenormalcoordinatesofthislongitudinalopticalmode.ThecouplingcoecientinEq. 5{4 isthegivenby andthepolaropticalphononscatteringrateis (q2+q20)2kqk0;0(n(!0)+1 21 2)(Ek0Ekh!0)dk0;(5{18) whereh!0istheopticalphononenergy,M0isthereducedmassinaunitcell,andq0isthereciprocalDebyescreeninglengthwederivedinthelastsection. The\!q"dispersioncurveforopticalphononsisatatthepoint,veryatinthewholeBrillouinzone,andperpendiculartotheBrillouinzoneboundary.Soinalong-waveapproximation,whichmeansscatteringneartheBrillouinzone

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center,weadopttheEinsteinapproximationandusetheLO-phononenergyatthepoint,h!0,forallqinEq. 5{18 TheeectivechargeinEq. 5{4 isrelatedtothedierencebetweenthepermittivitiesatlowandhighfrequencies,andisgivenby[ 102 103 104 ] SubstitutingEq. 5{19 intoEq. 5{18 ,weget (q2+q20)2kqk0;0(n(!0)+1 21 2)(Ek0Ekh!0)dk0:(5{20) Wecanseethatthepolarscatteringratedoesnotdependonthedetailsoftheunitcellsuchasthevolumeandreducedmass,etc. Theacousticphononenergyinthelongwavelengthlimitcanbeexpressedash!=hvsq,wherevsisthesoundvelocityinthecrystal.Withawavevectorq=107cm1,theacousticphononenergyisbelow1meV.Piezoelectricscatteringisanacousticphononeect,soforpiezoelectricscattering,thephonondensityinEq. 5{4 istheacousticphonondensity.Duetotheverylowenergy,thedensityisstillappreciableevenatlowtemperatures.Thepiezoelectriceectisduetotheacousticstrain,whichisincontrastwiththepolaropticaleectduetotheopticalpolarization. Theelectron-phononinteractionHamiltonianinthepiezoelectriccasecanbewrittenas whereD(R)istheelectricdisplacementvectorrelatedtotheelectriceldandacousticstrain,P(R)istheelectricpolarizationcausedbytheacousticvibration,andRisthepositionoftheunitcell.

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Followingthediscussionin[ 98 ],inaplanewaveapproximation,weobtain where,,arethedirectioncosineswithrespecttothecrystalaxisofthedirectionofpropagationofthewaves,a1,a2anda3arethecomponentsoftheunitpolarizationvector^a,ande14istheonlynonvanishingpiezoelectricconstantinzinc-blendecrystals.ThusthecouplingcoecientofEq. 5{4 is Foracousticphononscattering,wecanassumethattheratesforabsorptionandemissionarethesameduetothefactthatattemperaturesaboveseveralKelvin,n(!)1inthelongwavelengthlimit.Combiningthelongitudinalandtransversemodestogether,averagingoverallthedirectionaldependence,andusingtheequipartitionapproximation,wereachthefollowingexpression (q2+q20)2kqk0;0(Ek0Ekh!q)dk0(5{24) whereK2avisanaverageelectromechanicalcouplingcoecientrelatedtothesphericalelasticconstants.Asafurtherapproximation,wecanassumetheacousticphononscatteringisanelasticprocess,andtakethefunctionintheaboveequationtobeq-independent. Table5{1: ParametersforGaAsphononscattering LOphononwavelength()1285:0(cm1)K2av20:002521r10:6r12.4 105 ]. 98 ].

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Consideringbothpolaropticalphononandpiezoelectricscattering,wehavecalculatedtheheavyholespinrelaxationtimeinintrinsicGaAs.TheparametersusedarelistedinTable 5{1 Thecalculatedspinrelaxationtimeforaspin-upheavyholenearthepointasafunctionoftheelectronicwavevectoratT=300KisshowninFig. 5{3 (a).WecanseethatawayfromtheBrillouinzonecenter,thespinrelaxationtimedecreasesforaholedensityof1019=cm3,andincreasesforaholedensityof1018=cm3.ThisisbelievedtobeconnectedwithchangesintheavailabledensityofstatesatdierentFermienergies.InFig. 5{3 (b),thespinrelaxationtimeofaheavyholeatthepointisplottedasafunctionoftemperature.WecanseethatatT=300Kwithaholedensityof1018=cm3,whichwethinkshouldbeclosetotheexperimentalcase,theholespinrelaxationtimeisaround110fs,whichisveryclosetotheexperimentalvalue.ThisrevealsthatphononscatteringisindeeddominantinpureGaAs.

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Figure5{3: Theheavyholespinrelaxationtimeasafunctionofwavevector(a),andtemperatureatthepoint(b).

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Inameaneldapproximation,thisHamiltoniancanbewrittenas[ 106 ] where^istheorientationofthesubstitutionalMnlocalmoments,sisthecarrierspin,andJpdistheexchangeconstant.ThenthescatteringrateintheFermi'sgoldenruleapproximationis where Thus SinceS=Sxx+Syy+Szz,wehavehkjSjk0i=(Sxx)kk0+(Syy)kk0+(Szz)kk0.Thesquaredtermwillhave(Sxx)2kk0-liketermsinit.Intheabsenceofspontaneousmagnetization,(Sxx)2kk0=hS2xi(x)2kk0,andhS2xi=hS2yi=hS2zi.Forspontaneousmagnetization(ferromagneticcase),assumingthemagnetizationdirectionisalongz,hSzicanbefoundusingEq. 1{22 ,whichresultsfromtheself-consistenteectiveeldapproximation.FromtherelationS2=S2x+S2y+S2z,hS2xiandhS2xiarealsofound.ThehSxihSzi(x)kk0(z)kk0-likecrosstermsareallaveragedtozeroinrstorderapproximation.

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inIII-VDMSmaterialsmanyanti-sitedefectsandMninterstitials[ 27 28 29 ].TheybothserveasZ=2compensatingdefects.TheCoulombpotentialoftheseimpuritysitescancoupledierentelectronicstates. WewilluseBrooks-Herring'sapproach[ 107 ]inthefollowingtodealwiththeimpurityscatteringinGaMnAscrystals. WecanwritedowntheequationforascreenedCoulombpotentialas whereq0isthereciprocalDybyescreeninglength.UsingFermi'sgoldenrule,thescatteringrateduetothisscreenedCoulombcouplingisgivenby Inthisequation,ifweassumetheincidentcarrierscannotpenetrateveryclosetotheimpuritysite,wecanfactorizethematrixelementhk0jeV(r)jkiintotwoparts.OnepartistherapidlyvaryingBlochpart,theotheristheslowlyvaryingplanewaveparttimestheexponentiallydecayingCoulombpotential.Thuswehave wherezkistheeight-componentenvelopespinor.SoiftheimpuritydensityisNI,then (q2+q20)2jhz0kjzkij2(Ek0Ek)dk0:(5{33) InGaMnAs,Mnisanimpurity,itsconcentrationisx,anditsdensityNMnisproportionaltox.Supposetheholedensityisp,thenthedensityforthecompensatingdefectsis(NMnp)=2.Includingbothexchangeinteractionandimpurityscattering,thespinrelaxationtimeasafunctionofthewavevectoralongtheXdirectionisplottedinFig. 5{4 .Inthiscase,thesampleisferromagneticwithaCurietemperatureTC=55KatT=30K.Thehole

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Figure5{4: Spinrelaxationtimeforaheavyholeasafunctionofkalong(0,0,1)direction. densityisassumedtobep=1019=cm3.Fig. 5{4 revealsthatnormallyinDMS,impurityexchangephonon.Infact,theimpurityscatteringis1000timesstrongerthanphononscattering.Thisisnatural,becausefor6%Mndoping,theMnimpuritydensityitselfcanreach1021=cm3.TheotherpointwecanseeisthatthephononscatteringinGaMnAsisweakerthaninGaAs.Thisisbecausethevalencebandsplittingintheferromagneticphasemakessomestatesenergeticallyunavailableforscatteringduetoenergyconservation. ThespinrelaxationtimeattheholeFermisurfaceisillustrateinFig. 5{5 asafunctionofholedensity.Fig. 5{5 (a)showstheholespinrelaxationtimeintheXdirectionandFig. 5{5 (b)showstheholespinrelaxationtimeintheLdirection.HeretheFermisurfacecanbeconsideredasthatoftheunperturbedsystem.ThedierentbehavioralongdierentkdirectionisduetotheGaAsvalencebandanisotropy,whichisenhancedinGaMnAs.ThespinrelaxationtimeattheFermisurfacehelpsoneunderstandthepropertiesoftheholesmediating

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Figure5{5: Spinrelaxationtimeofaheavyholeasafunctionofholedensityatdirection(a)(0,0,1)and(b)(1,1,1).

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theferromagnetisminDMS.IntheRKKYmodel(oritslowdensitylimit,Zener'smodel),itistheholesattheFermisurfacethatmediatetheexchangeinteractionwhichresultsintoaferromagneticphasechange. Herewehaveonlytalkedaboutthespintransportpropertiesofasinglehole.However,thecollectivebehaviorofholesdeterminesthepropertiesoftheholesystem.InDMS,usuallythemagnetizationduetoholesthemselvesisnegligiblecomparedtothatduetothelocalizedmagneticmoments.Inthepump-probeexperimentswementionedinthebeginningofthischapter,itisthechangeofthemagnetizationduetothelocalizedmomentsthatgivesanobservableresult.Thechangeofthemagnetizationoflocalizedmoments,i.e.,therotationofthemagnetizationdirection,isinducedbythespinalignmentoftheitinerantholesthroughtheexchangeinteraction.Thechangeofthemagnetizationinreturnwillalsohaveafeedbacktotheseholes.Thusthehole-localizedmomentssystemisacomplicatedsystemandmustbetreatedinaself-consistentmanner.Thiscollectiveself-consistentsystemshouldhaveamuchlongerlifetimethanthatofasinglehole.

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Inthisthesis,thedevelopmentandcurrentresearchsituationofdilutedmagneticsemiconductors,includingbothII-VIandIII-VIsemiconductor-based,hasbeenintroducedanddiscussed.Incalculatingthebandstructure,aneight-bandkptheoryhasbeenemployedtogetherwiththespdexchangeinteraction.Intheabsenceofanexternalmagneticeld,ageneralizedKane'smodelisappropriateforcalculatingthebandstructure,whileinthecaseofanappliedmagneticeld,onebandwillsplitintoaseriesofLandaulevels.Inordertodealwiththis,wedevelopedageneralizedPidgeon-Brownmodelwhichincorporatestheexchangeinteractionandalsotakesintoaccountnitekzeects.Calculationshaveshownthatinadilutedmagneticsemiconductor,thebandstructureisverydierentfromthatinapuresemiconductor.Forexample,theg-factorsinInMnAscanbeabove100incontrastwithacomputedg-factorof15inInAs. Cyclotronresonanceinultrahighmagneticelds(upto500T)hasbeensimulatedandcomparedwithexperiments.ThemethodforcalculatingopticaltransitionshasbeenintroducedandFermi'sgoldenrulehasbeenutilized.WehavesuccessfullyreproducedthecyclotronabsorptioninbothconductionandvalencebandsinInMnAs.Wepointedoutthattheshiftofcyclotronresonancepeaksintheconductionbandhadadependenceontheexchangeconstants(),andthepeaklineshapedependedonthenonparabolicity.Theh-activeCRresonanceinvalencebandshasbeendecomposedintoheavy-holetoheavy-holeandlight-holetolight-holetransitionsinaeldupto150T.Theselectionrulesforopticaltransitionshavebeendiscussedinadipoleapproximationandwehavepointedoutthatduetothedegeneracyinthevalenceband,notonlyh-activecyclotron 129

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resonance,butalsoe-activecyclotronresonancecantakeplaceinsemiconductorvalencebands.Generallytheh-activetransitionstakeplacebetweenheavyholeorlightholeLandaulevelsthemselves,bute-activetransitionstakeplacebetweenheavyholeandlightholelevels.TheCRstrengthandlineshapestronglydependoncarrierdensity,whichprovideanalternativewaytomeasurethecarrierconcentration.WehavegivenananalyticalexpressionwhichexplainstheCRpeakshiftswithtemperatureinInMnAs/GaSbheterostructure.ThepronouncednarrowingmaybeduetothesuppressionofspinuctuationortransferoftheholestotheInMnAs/GaSbinterfaceatlowtemperatures. Wehavediscussedtherelationsbetweentheopticalconstants,andfromthecalculationofabsorptioncoecients,thereectioncoecientsandmagneto-opticalKerrrotationhasbeencalculatedinbulkInMnAsandInGaAsortheirheterostructures.Becauseofferromagnetism,thee-activeandh-activecross-bandabsorptionhavedierentdependenceonphotonenergies.Thismagneticcirculardichroismresultsinthepolarizationplanerotationoflinearlypolarizedlight.Wehavesimulatedthismagneticcirculardichroismandcomparedourresultstoexperiments. Duetotheimportanceofholesindilutedmagneticsemiconductorsystems,wehavecarriedoutcalculationsforholespinrelaxationtimesinbulkGaAsandGaMnAsvalencebands.InGaAs,phononscatteringdominatesandgivesaholespinrelaxationtimearound100fsatroomtemperature.InGaMnAs,duetothestrongexchangeinteractionandheavyimpuritydoping,exchangeandimpurityscatteringdominate.Wehavebrieyintroducedthetheoryofphononscattering,exchangescatteringandimpurityscattering,andshownincalculationsthatinMn-dopedDMSsystems,thephononscatteringisnolongerimportantandonlyimpurityscatteringdominates.Assumingtheexternaldisturbanceasmall

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perturbation,theholespinrelaxationtimeattheFermisurfaceisonlyafewfemtoseconds. Thereisstillmuchworktobedoneinthefuture.Themeaneldtheoryhasitsowndrawbacksintreatingferromagnetictransitions.Toobtainbetterresultswhencalculatingthebandstructureandopticalabsorptioninferromagneticsamples,abettertheoreticalframeworkishighlydesirable.OurcurrentmodelisnotadequatetocalculatetheCRabsorptioninaInMnAs/GaSbheterostructure,whichisatype-IIheterostructure(theconductionbandofInAsliesbelowthevalencebandsofGaSb).Amodelthatcanaccountfortheinterfacestatesneedstobedevelopedinthefuture.Atthecurrentlevel,wehaveonlycalculatedthestaticMOKEofDMS,whilethetime-resolvedMOKEisofmoreimportanceforstudyingthedynamicalpropertiesofDMS.Currentlywearetryingtodevelopamethodtostudytime-dependentmagneticphenomenainDMS.

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YongkeSunwasborninasmallvillageinZhumadian,HenanProvince,People'sRepublicofChina,onMarch6,1974.Hestayedtherefor15yearsuntilhenishedmiddleschoolstudies.Afterthat,hewent10milesawayfromhometostudyinahighschoolcalledYangzhuangHighSchool.In1992,hewasexemptedfromthenationalexamandadmittedtoPekingUniversityinBeijing,China.From1992to1997hestudiedinPekingUniversityandreceivedhisB.S.degreein1997.Hesubsequentlyparticipatedinthemaster'sprogramandobtainedtheM.S.degreein2000.Inthesameyear,hewasmarriedtohisbeautifulwife,YuanZhang,whowashisschoolmate.Inthefallof2000hecametotheUnitedStatesandbecameaGator.Inthesummerof2001,heenteredProf.Stanton'sgroupandhasbeenstudyingthepropertiesofdilutedmagneticsemiconductors,pursuingaPh.D.degree. 138


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THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETO-OPTICAL
AND TRANSPORT PROPERTIES OF DILUTED MAGNETIC
SEMICONDUCTORS














By

YONGKE SUN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005





































To my dear wife Yuan, and my parents.















ACKNOWLEDGMENTS

I owe my gratitude to all the people who made this thesis possible and because

of whom my graduate experience has been one that I will cherish forever.

First and foremost I would like to thank my advisor, Professor ('!Ch -I- .lh. r

J. Stanton, for giving me an invaluable opportunity to work on challenging and

extremely interesting projects over the past four years. He has ahv-- ~ made himself

available for help and advice and there has never been an occasion when I have

knocked on his door and he has not given me time. His physics intuition impressed

me a lot. He taught me how to solve a problem starting from a simple model, and

how to develop it. It has been a pleasure to work with and learn from such an

extraordinary individual.

I would also like to thank Professor David H. Reitze, Professor Selman P.

Hershfield, Professor Dmitrii Maslov and Professor Cammy Ab, i il,:,I for agreeing

to serve on my thesis committee and for sparing their invaluable time reviewing the

manuscript.

My colleagues have given me lots of help in the course of my Ph.D. studies.

Gary Sanders helped me greatly to develop the program code, and we ah--,i-i had

fruitful discussions. Professor Stanton's former postdoc Fedir Kyrychenko also gave

me good advice and some insightful ideas. I would also like to thank Rongliang Liu

and Haidong Zhang, who made my life here more interesting.

I want to thank our research collaborators. Dr. Kono's group from Rice

University provided most of the experimental data. Collaboration with Dr. Kono

was a wonderful experience in the past four years. I also had fruitful discussion

with Prof. Miura and Dr. Matsuda from University of Tokyo.










I would also like to acknowledge help and support from some of the staff

members, in particular, Darlene Latimer and Donna Balcom, who gave me much

indispensable assistance.

I owe my deepest thanks to my family. I thank my mother and father, and

my wife, Yuan, who have ah--lv- stood by me. I thank them for all their love and

support. Words cannot express the gratitude I owe them.

It is impossible to remember all, and I apologize to those I have inadvertently

left out.
















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iii

LIST OF TABLES ...................... ......... vii

LIST OF FIGURES ................... ......... viii

ABSTRACT ...................... ............ xiv

CHAPTER

1 INTRODUCTION AND OVERVIEW ................... 1

1.1 Spintronics ................... ........... 1
1.2 The II-VI Diluted Magnetic Semiconductors ............ 4
1.2.1 Basic Properties of II-VI Diluted Magnetic Semiconductors 4
1.2.2 Exchange Interaction between 3d5 Electrons and Band Elec-
trons ..................... ....... 8
1.3 The III-V Diluted Magnetic Semiconductors . . 13
1.3.1 Ferromagnetic Semiconductor ............... .. 13
1.3.2 Effective Mean Field ..... ........... .... 21
1.4 Open Questions . . ..... .......... 23
1.4.1 Nature of Ferromagnetism and Band Electrons ...... ..23
1.4.2 DMS Devices. .................. ... .. .. 24

2 ELECTRONIC PROPERTIES OF DILUTED MAGNETIC SEMICON-
DUCTORS ................... ..... ....... 29

2.1 Ferromagnetic Semiconductor Band Structure . ... 29
2.2 The k p Method. ..... ...... ........ 30
2.2.1 Introduction to k p Method ... . . 30
2.2.2 Kane's Model .... . . ...... 34
2.2.3 Coupling with Distant Bands-Luttinger Parameters . 38
2.2.4 Envelope Function .................. .. 42
2.3 Landau Levels .......... . . . ... 43
2.3.1 Electronic State in a Magnetic Field . . 43
2.3.2 Generalized Pidgeon-Brown Model . . ..... 44
2.3.3 Wave Functions and Landau Levels . . ..... 49
2.4 Conduction Band g-factors .................. ..... 53










3 CYCLOTRON RESONANCE ................... .... 56

3.1 General Theory of Cyclotron Resonance . . ..... 56
3.1.1 Optical Absorption .................. ..... 56
3.1.2 Cyclotron Resonance ................ .. .. 60
3.2 Ultrahigh Magnetic Field Techniques ............... .. 63
3.3 Electron Cyclotron Resonance ............ .. .. .. 64
3.3.1 Electron Cyclotron Resonance ............... .. 64
3.3.2 Electron Cyclotron Mass .................. .. 72
3.4 Hole Cyclotron Resonance .................. ..... 74
3.4.1 Hole Active Cyclotron Resonance . . ..... 74
3.4.2 Hole Density Dependence of Hole Cyclotron Resonance 83
3.4.3 Cyclotron Resonance in InMnAs/GaSb Heterostructures .83
3.4.4 Electron Active Hole Cyclotron Resonance . ... 90

4 MAGNETO-OPTICAL KERR EFFECT ....... . . 96

4.1 Relations of Optical Constants ............... .. 96
4.2 Kerr Rotation and Farid1,i Rotation . . . .... 101
4.3 Magneto-optical Kerr Effect of Bulk InMnAs and GaMnAs . 104
4.4 Magneto-optical Kerr Effect of Multilayer Structures ...... 107

5 HOLE SPIN RELAXATION. ................... .. .. 112

5.1 Spin Relaxation Mechanisms ............... .. .. .. 113
5.2 Lattice Scattering in III-V Semiconductors . . 115
5.2.1 Screening in Bulk Semiconductors . . ..... 117
5.2.2 Spin Relaxation in Bulk GaAs . . ..... 118
5.3 Spin Relaxation in GaMnAs .................. .. 122
5.3.1 Exchange Scattering ................. . 122
5.3.2 Impurity Scattering ............. .... . 124

6 CONCLUSION .................. ........... .. 129

REFERENCES .................. ................ .. 132

BIOGRAPHICAL SKETCH ............. . . .... 138















LIST OF TABLES
Table page

1-1 Some important II-VI DMS .................. ...... 4

2-1 Summary of Hamiltonian matrices with different n .......... ..50

2-2 InAs band parameters .................. ........ .. 51

3-1 Parameters for samples used in e-active CR experiments . ... 67

3-2 C'!i i i''teristics of two InMnAs/GaSb heterostructure samples . 85

5-1 Parameters for GaAs phonon scattering ................ .121















LIST OF FIGURES
Figure page

1-1 The band gap dependence of Hgl_, Mi:, Te on Mn concentration k. .. 5

1-2 The band structures of Hgi__,.i..Te with different x. . . 6

1-3 Cdl__?!. i,.Te x-T phase diagram. ............... . 7

1-4 Average local spin as a function of magnetic field at 4 temperatures
in paramagnetic phase. .................. ..... 10

1-5 Magnetic-field dependence of Hall resistivity pHall and resistivity p of
GaMnAs with temperature as a parameter. ........... ..14

1-6 Mn composition dependence of the magnetic transition temperature
T,, as determined from transport data. ............ 16

1-7 Variation of the RKKY coupling constant, J, of a free electron gas in
the neighborhood of a point magnetic moment at the origin r = 0. 17

1-8 Curie temperatures for different DMS systems. Calculated by Dietl
using Zener's model. .................. .... 19

1-9 Schematic diagram of two cases of BMPs. .............. ..20

1-10 Average local spin as a function of magnetic field at 4 temperatures.. 22

1-11 The photo-induced ferromagnetism in InMnAs/GaSb heterostructure. 25

1-12 Spin light emitting diode. .................. ..... 27

1-13 GaMnAs-based spin device. ............. .... 28

2-1 Valence band structure of GaAs and ferromagnetic Ga,,,,Mi_,,,,,,As
with no external magnetic field, calculated by generalized Kane's
model .. ......................... .... 30

2-2 Band structure of a typical III-V semiconductor near the F point. 35

2-3 Calculated Landau levels for InAs (left) and In,, ..--i,,, ,.As (right) as
a function of magnetic field at 30 K. ............. .. .. 52

2-4 The conduction and valence band Landau levels along kz in a mag-
netic field of B = 20 T at T = 30 K. ............. 53










2-5 Conduction band g-factors of Inl_--,i1,.As as functions of magnetic
field with different Mn composition x. .............. 54

2-6 g-factors of ferromagnetic In,,.,M\li,,,As. ............... 55

3-1 Quasi-classical pictures of e-active and h-active photon absorption. .. 62

3-2 The core part of the device based on single-coil method . .... 64

3-3 A standard coil before and after a shot. ................ 65

3-4 Waveforms of the magnetic field B and the current I in a typical shot
in single-turn coil device. ............... .... 65

3-5 Waveforms of the magnetic field B and the current I in a typical flux
compression device. ............... ...... 66

3-6 Experimental electron CR spectra for different Mn concentrations x
taken at (a) 30 K and (b) 290 K. ............... . 68

3-7 Zone-center Landau conduction-subband energies at T = 30 K as
functions of magnetic field in n-doped Inl_,7-1i..As for = 0 and
x = 12" .................. ............. .. 69

3-8 Electron CR and the corresponding transitions. . . 70

3-9 Calculated electron CR absorption as a function of magnetic field at
30 K and 290 K. ............... ....... 71

3-10 Calculated electron cyclotron masses for the lowest-lying spin-up and
spin-down Landau transitions in n-type Inl_-MnxAs with photon
energy 0.117 eV as a function of Mn concentration for T = 30 K
and T = 290 K. .................. .. ..... 73

3-11 Hole cyclotron absorption as a function of magnetic field in p-type
InAs for h-active circularly polarized light with photon energy 0.117 eV. 75

3-12 Calculated cyclotron absorption only from the H_1,1 H,2 and L0,3 -
L1,4 transitions broadened with 40 meV (a), and zone center Lan-
dau levels responsible for the transitions (b). ............ ..76

3-13 Experimental hole CR and corresponding theoretical simulations. 77

3-14 Observed hole CR peak positions for four samples with different Mn
concentrations. ............... .......... 78

3-15 The dependence of cyclotron energies on several parameters. . 79

3-16 Hole CR spectra of InAs using different sets of Luttinger parameters. 80

3-17 Calculated Landau levels and hole CR in magnetic fields up to 500 T. 81










3-18 k-dependent Landau subband structure at B = 350 T . .... 82

3-19 Band structure near the F point for InAs calculated by eight-band
model and full zone thirty-band model. ............ 82

3-20 The hole density dependence of hole CR. .............. 84

3-21 Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSb
samples ........ .......... ......... .... 86

3-22 Theoretical CR spectra showing the shift of peak A with temperature. 87

3-23 Average localized spin as a function of temperature at B = 0, 20, 40,
60 and 100 Tesla. ............... ...... 88

3-24 Relative change of CR energy (with respect to that of high tempera-
ture limit) as a function of temperature. ............. 89

3-25 Band diagram of InMnAs/GaSb heterostructure. . . 90

3-26 Schematic diagram of Landau levels and cyclotron resonance transi-
tions in conduction and valence bands. ............ 91

3-27 The valence band Landau levels and e-active hole CR. . ... 92

3-28 Experimental and theoretical hole CR absorption. ......... ..93

3-29 Valence band structure at T = 30 K and B = 100 T for Inl_-Mi,..As
alloys having x = and x = 5'. ................. 94

3-30 The primary transition in the e-active hole CR under different Mn
doping. .................. .............. ..95

4-1 Diagram for light reflection from the interface between medium 1 with
refractive index N1 and medium 2 with refractive index N2. .... ..100

4-2 Schematic diagram for magnetic circular dichroism. . ... 102

4-3 Diagrams for Kerr and Far,-idi rotation. .. . . .... 103

4-4 Kerr rotation of InMnAs. ................ .... 105

4-5 The band diagram for InAs. ............. ... 106

4-6 Kerr rotation of GaMnAs. .................. ..... 107

4-7 The band diagram for GaAs. .................. .... 108

4-8 The absorption coefficients both in InMnAs and GaSb lV. -i (a) and
the reflectivity of InMnAs/GaSb heterostructure(b). . ... 109









4-9 Reflectivity of In,, ..it,, ,,As(9 nm)/GaSb(600 nm) heterostructure
at T = 5.5 K measured by P. Fumagalli and H. Munekata. ..... ..110

4-10 Measured (a) and calculated (b) Kerr rotation of InMnAs(19 nm)/
AlSb(145 nm) heterostructure under a magnetic field of 3 T at T 5.5 K. 111

5-1 Light-induced MOKE. Signal decays in less than 2 ps. . ... 113

5-2 Light-induced magnetization rotation. ................ 114

5-3 The heavy hole spin relaxation time as a function of wave vector (a),
and temperature at the F point (b). ...... . . ...... 123

5-4 Spin relaxation time for a heavy hole as a function of k along (0,0,1)
direction. .................. ............ 126

5-5 Spin relaxation time of a heavy hole as a function of hole density at
direction (a) (0,0,1) and (b) (1,1,1). ................. 127















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETO-OPTICAL
AND TRANSPORT PROPERTIES OF DILUTED MAGNETIC
SEMICONDUCTORS

By

Yongke Sun

December 2005

CI, ,ir: Chi i-I.l !h. r J. Stanton
Major Department: Physics

Spintronics has recently become one of the key research areas in the magnetic-

recording and semiconductor industries. A key goal of spintronics is to utilize

magnetic materials in electronic components and circuits. A hope is to use the

spins of single electrons, rather than their charge, for storing, transmitting and

processing quantum information. This has invoked a great deal of interest in spin

effects and magnetism in semiconductors. In my work, the electronic and optical

properties of diluted magnetic semiconductors(DMS), especially (In,Mn)As and

its heterostructures, are theoretically studied and characterized. The electronic

structures in ultrahigh magnetic fields are carefully studied using a modified eight-

band Pidgeon-Brown model, and the magneto-optical phenomena are successfully

modeled and calculated within the approximation of Fermi's golden rule. We

have found the following important results: i) Magnetic ions doped in DMS

pli,- a critical role in affecting the band structures and spin states. The sp d

interaction between the itinerant carriers and the Mn d electrons results in a

shift of the cyclotron resonance peak and a phase transition of the III-V DMS









from paramagnetic to ferromagnetic; ii) g-factors of the electrons in DMS can be

enhanced to above 100 by large spin splitting due to strong sp d interaction. Also

the effective masses of DMS systems strongly depend on interaction parameters;

iii) Two strong cyclotron resonance peaks present in p-doped DMS arise from

the optical transitions of heavy-hole to heavy-hole and light-hole to light-hole

Landau levels, in lower and higher magnetic fields, respectively; iv) Electron-active

cyclotron resonance takes place in p-doped DMS samples. This is unusual since a

simple quasi-classical argument would sil-.-: -1 that one could not simultaneously

increase angular momentum and energy for this type of polarized light in a hole

system. This occurs because of the degeneracy in the valence bands; v) Due to

the magnetic circular dichroism, nonvanishing magneto-optical Kerr rotation up to

several tenths of a degree occurs in DMS systems. The Kerr rotation in muiltiliv. r

structures depends on quantum confinement and multi-reflections from the surfaces;

vi) Quantitative calculations show that in intrinsic bulk GaAs, the hole spin life

time is around 110 femtoseconds, which is due to phonon scattering. However, in

DMS, the p d exchange interaction and the high density of impurities give rise

to other spin flip scattering channels. The nonequilibrium spin life time is only

a few femtoseconds. These research results should be helpful for gaining more

understanding of the properties of DMS systems and should be useful in designing

novel devices based on DMS.















CHAPTER 1
INTRODUCTION AND OVERVIEW

There is a wide class of semiconducting materials which are characterized by

the random substitution of a fraction of the original atoms by magnetic atoms.

The materials are commonly known as diluted magnetic semiconductors (DMS) or

semi-magnetic semiconductors (SMSC).

Since the initial discovery of DMS in II-VI semiconductor compounds [1], more

than two decades have passed. The recent discovery of ferromagnetic DMS based

on III-V semiconductors [2] has lead to a surge of interest in DMS for possible

spintronics applications. Many papers have been published investigating their

electronic, magnetic, optical, thermal, statistical and transport properties, in many

journals, and even in popular magazines [3]. This interest not only comes from

the DMS themselves as good theoretical and experimental subjects, but also can

be better understood from a broader view from the relation of DMS research with

spintronics [4].

1.1 Spintronics

Spintronics, or spin electronics, refers to the study of the role 1p i, d by elec-

tron (and nuclear) spin in solid state physics, and possible devices that specifically

exploit spin properties instead of or in addition to the charge degrees of freedom.

Spin relaxation and spin transport in metals and semiconductors are of fundamen-

tal research interest not only for being basic solid state physics issues, but also for

the already demonstrated potential these phenomena have in electronic technology.

There is a famous Moore's Law in the conventional electronics industry, that -,i-

the number of transistors that fit on a computer chip will double every 18 months.

This may soon face some fundamental roadblocks. Most researchers think there









will eventually be a limit to how many transistors they can cram on a chip. But

even if Moore's Law could continue to spawn ever-tinier chips, small electronic de-

vices are plagued by a big problem: energy loss, or dissipation, as signals pass from

one transistor to the next. Line up all the tiny wires that connect the transistors in

a Pentium chip, and the total length would stretch almost a mile. A lot of useful

energy is lost as heat as electrons travel that distance. Spintronics, which uses spin

as the information carriers, in contrast with conventional electronics, consumes less

energy and may be capable of higher speed.

Spintronics emerged on the stage in scientific field in 1988 when Baibich et al.

discovered giant magnetoresistance (GMR) [5], which results from the electron-spin

effects in magnetic materials composed of ultra-thin rmuiltil iv- rs, in which huge

changes could take place in their electrical resistance when a magnetic field is

applied. GMR is hundreds of times stronger than ordinary magnetoresistance.

Basing on GMR materials, IBM produced in 1997 new read heads which are able

to sense much smaller magnetic fields, allowing the storage capacity of a hard

disk to increase from the order of 1 to tens of gigabytes. Another valuable use of

GMR material is in the operation of the spin filter, or spin valve, which consists

of 2 spin lV-. -i which let through more electrons when the spin orientations in

the two Iiv-r s are the same and fewer when the spins are oppositely aligned. The

electrical resistance of the device can therefore be changed dramatically. This

allows information to be stored as 0's and l's (magnetizations of the lv- -ir parallel

or antiparallel) as in a conventional transistor memory device. A straightforward

application could be in the magnetic random access memory (!I RAM) device which

is non-volatile. These devices would be smaller, faster, cheaper, use less power and

would be much more robust in extreme conditions such as high temperature, or

high-level radiation environments.










Currently, besides continuing to improve the existing GMR-based technology,

people are now focusing on finding novel v-i-, of both generating and utilizing spin-

polarized currents. This includes investigation of spin transport in semiconductors

and looking for v--,v- in which semiconductors can function as spin polarizers and

spin valves. We can call this semiconductor based spintronics, the importance of

which lies in the fact that it would be much easier for semiconductor-based devices

to be integrated with traditional semiconductor technology, and the semiconductor

based spintronic devices could in principle provide amplification, in contrast with

existing metal-based devices, and can serve as multi-functional devices. Due to

the excellent optical controllability of semiconductors, the realization of optical

manipulation of spin states is also possible.

Although there are clear merits for introducing semiconductors into spintronic

applications, there are fundamental problems in incorporating magnetism into

semiconductors. For example, semiconductors are generally nonmagnetic. It is hard

to generate and manipulate spins in them. People can overcome these problems

by contacting the semiconductors with other (spintronic) materials. However, the

control and transport of spins across the interface and inside the semiconductor is

still difficult and far from well-understood. Fortunately, there is another approach

to investigating spin control and transport in all-semiconductor devices. This

approach has become possible since the discovery of DMS.

The most common DMS studied in the early 1990s were II-VI compounds

(like CdTe, ZnSe, CdSe, CdS, etc.), with transition metal ions (e.g., Mn, Fe or Co)

substituting for their original cations. There are also materials based on IV-VI

(e.g., PbS, SnTe) and most importantly, III-V (e.g., GaAs, InSb) crystals. Most

commonly, Mn ions are used as magnetic dopants.










1.2 The II-VI Diluted Magnetic Semiconductors

1.2.1 Basic Properties of II-VI Diluted Magnetic Semiconductors

The first II-VI DMS was grown in 1979 [1], and has been given a great deal of

attention ever since [6]. The most studied II-VI DMS materials are listed in Table

1-1.

Table 1-1: Some important II-VI DMS

Material Crystal Structure x range1
Hg1ixM!i, Te Zinc-blende x < 0.30
Hg1i__-Mi, Se Zinc-blende x < 0.30
Cd_1 MnTe Zinc-blende x < 0.75
Cdl_M,?i. ,Se Wurtzite x < 0.50
Znl_?!Mi., Te Zinc-blende x < 0.75
1 x refers the range of x for which the crystals are usually studied. When
z's become relatively large, phases like MnTe or MnTe2 occur, and the
crystal qualities are poor [7].


The II-VI DMS have attracted so much attention since their discovery because

of the following important properties.

Unique electronic properties: The wide variety of both host < i- ,1-
and magnetic atoms provides materials which range from wide gap to zero
gap semiconductors, and reveal many different types of magnetic interaction.
Several of the properties of these materials may be tuned by changing the
concentration of the magnetic ions. The bandgap, E,, of Hgi__?- ,.Te can
even change from negative to positive. This property becomes favorable
as far as designing infrared devices is concerned. The dependence of E, of
Hgl_?j.Mi,Te on x is given in Fig. 1-1 [8].

With the definition of the band gap as E, E6r Ers, the band structures
of Hgi_ _-i,..Te with different x are given in Fig. 1-2 [8]. With x < 0.075,
Eg < 0, and with x > 0.075, E, > 0. Without spin-orbital coupling, we
should have a six-fold degenerate valence band at the F point. Considering
spin-orbital coupling, the valence band splits into two bands-F7 and Fs
(split-off band), with an energy difference of A.

The electron effective mass, i.e., the band curvature, will also change with x.
At some x values, the effective mass becomes so small that the mobility of
electrons can be very high. For instance, p 106 cm2/V s for HgIi_.li,.Te
when x 0.07 at 4.2 K.







5

300 -

*
200-



100



oo

0 0

-100
o0 0
00 x Hgl Mn1 Te

Ox T= 4.2K
0, M.O
-200 ox M0
x SdH
A FarlR
-3001

0 5 10 15

k (ot.%)

Figure 1-1: The band gap dependence of Hgi_, Mi,, Te on Mn concentration k.
Reprinted with permission from Bastard et al. Phys. Rev. B 24: 1961-1970, 1981.
Figure 10, Page 1967.

Broad phase behavior: With different Mn concentration x and temper-
ature T, each II-VI DMS presents a different (phase) property, but their x-T
phase diagrams are very similar. Shown in Fig. 1-3 is the phase diagram of
Cdl_-,i ,.Te obtained from specific heat and magnetic susceptibility measure-
ments [9]. The DMS system may be considered as containing two interacting
subsystems. The first of these is the system of delocalized conduction and
valence band electrons/holes. The second is the random, diluted system of
localized magnetic moments associated with the magnetic atoms. These
two subsystems interact with each other by the spin exchange interaction.
The fact that both the structure and the electronic properties of the host
< i -I Ji-; are well known means that they are perfect for studying the basic
mechanisms of the magnetic interactions coupling the spins of the band
carriers and the localized spins of magnetic ions. The coupling between the


















Sh.




L h.








0.02 0.04 OD06
Mn c


"positive gap"


I r
0.08 0.10
ontent (k)


Figure 1-2: The band structures of Hgl_, ,.iTe with different x. A is the spin-
orbital splitting, HH indicates the heavy hole band, and LH the light hole band,
respectively. Reprinted with permission from Bastard et al. Phys. Rev. B 24:
1961-1970, 1981. Figure 1, Page 1961.

localized moments results in the existence of different magnetic phases such as
paramagnets, spin glasses and antiferromagnets.

Important magnetic phenomena: As described above, if we don't
consider the spin exchange interaction between the band electrons and
localized magnetic moments, DMS materials are just the same as the other
semiconductors. When we consider the spin exchange interaction, however,
DMS materials present many important properties, such as very big Land6
g-factors, extremely large Zeeman splitting of the electronic bands, giant
Fa,-idiv rotation, and huge negative magnetoresistance. Therefore, to study
DMS, one has to first understand the spin exchange interaction between the
localized magnetic ions and band electrons.


8


0


1 1
0.12 0.14


h.h.7


,


























Vp mixed
P crystal
20- phases

IA
10 I S
S I
0I

0 0.2 0.4 0.6 0.8 1.0
X
Figure 1-3: Cdl_- i:,.Te x-T phase diagram. P: Paramagnet; A: Antiferromagnet;
s: spin-glass, mixed i< -I I1 when x > 0.7. Reprinted with permission from Galgzka
et al. Phys. Rev. B 22: 3344-3355, 1980. Figure 12, Page 3352.










1.2.2 Exchange Interaction between 3d5 Electrons and Band Electrons

Many features of DMS, such as the special electronic properties, unique

phase diagrams, and important magnetic and magneto-optical characteristics,

are induced by the exchange interaction between the localized d shell electrons of

the magnetic ions and the delocalized band states (of s or p origin). The s d,

p d exchange, and its consequences and origin have been pointed out from the

very beginning of the history of DMS and the Heisenberg form of the exchange

interaction Hamiltonian was successfully used for this interaction [10]. In the

following, I will briefly introduce a simple qualitative theoretical approach to II-VI

DMS.

Suppose the state of Mn ions in DMS material is Mn2+. The electronic struc-

ture of Mn2+ is ls22s22p63s23p63d5, in which 3d5 is a half-filled shell. According to

Hund's rule, the spin of these five 3d5 electrons will be parallel to each other, so the

total spin is S = 5/2. These five electrons are in states in which the orbital angular

momentum quantum number 1 = 0, 1, 2. Thus the total orbital angular momen-

tum L = 0. The total angular momentum for a Mn2+ ion then is J = S = 5/2. The

Land6 g-factor is

+ J(J + 1) + S(S +1) L(L +1)
g = 1 + =2. (1-1)
2J(J + 1)

Analogous to the exchange interaction in the Hydrogen molecule, the exchange

interaction between a 3d5 electron and a band electron can be written in the

Heisenberg form

H, = -J S, (1-2)

where a is the spin of a band electron/hole, J is the exchange constant, and S is

the total angular momentum of all 3d5 electrons in a Mn2+ ion.

In the non-interacting paramagnetic phase, a very simplified model will be

described in the following. Since L = 0 for Mn2+, the magnetic momentum for









Mn2+ is p = (-ge/2mo)J = (-ge/2mo)S. Assuming a magnetic field B along the

z direction, the additional energy in this field of a Mn2+ ion is -p B = gpBmsB,

where ms = 5/2, 3/2, 1/2, -1/2, -3/2, -5/2. Assuming non-interactive spins, and

using a classic Boltzman distribution function egpB"5B/kBT, the average magnetic

moment in the z direction is then

Y5/2 (-B-5 2 s)e-g"pgB/kn
Y5/2 egpS-mB/kBT
m s -5/2

This can be written as

(p) =-gpBSB,(y), (1-4)

where B,(y) is the Brillouin function

2S + t 2S 2S 2S

S = 5/2, y = gpBSB/kBT. (1-5)

The average spin of one Mn2+ ion then is


(S) -SB,(y). (1-6)

The antiparallel orientation of B and (S,) is due to the difference in sign of the

magnetic moment and the electron spin. Since B is directed along the z axis, the

average Mn spin saturates at (Sz) = -5/2. The paramagneticc) dependence of (S,)

on magnetic field and temperature is shown in Fig. 1-4.

From Eq. 1-2, the exchange Hamiltonian of one band electron with spin a

interacting with the 3d5 electrons from all Mn2+ ions is,

Hex = J(r Rj)Si a, (1-7)

where r is the position vector of the band electron, and Ri is the position vector of

the ith Mn2+ ion, J(r Ri) is the exchange coupling coefficient of the band electron










2.5


2.0 30K


A 1.5 K
N
C)
V
1.0- 150K


0.5 290K


0.0
0 20 40 60 80

B (T)

Figure 1-4: Average local spin as a function of magnetic field at 4 temperatures in
paramagnetic phase.

with the 3d5 electrons in the 1th Mn2+ ion. Si is the total angular momentum of
the 3d5 electrons in the 1th Mn2+ ion.
Next we will use a virtual < i I 1 approximation to deal with Hamiltonian
1-7. Due to the fact the the wave function of a band electron actually extends
over the whole i -- I 1 it interacts with all the Mn2+ ions simultaneously. In the
mean field framework, we can replace the angular momentum of each Mn2+ ion
by the average value. Still assuming a magnetic field along z direction, we have
(S) = (S,), and Si a = (Sz)mch2. m, = 1/2 here indicates the spin quantum
number of the band electrons. The (S,) is given by Eq. 1-6. The exchange









Hamiltonian then can be written as


H,, (S )m, J(r- Ri). (1-8)


Because of the extended nature of the band electron states, which interact

with the 3d5 electrons in all Mn2+ ions, the positions of these Mn2+ ions are not

important. We can distribute approximately these Mn2+ ions uniformly at cation

sites. This amounts to assuming we have an equivalent magnetic moment of x(S,)

at each cation site. So, Eq. 1-8 becomes


Hex = x(S,)mc J(r R). (1-9)
R
Here R becomes the position vector of each cation site. In Eq. 1-9 the exchange

Hamiltonian now has the same periodicity as the (i I I1

From the Hamiltonian 1-9, the exchange energy can then be obtained by


E = (. kL "'ck) x(S,)m Tc(ck J R) Ik). (1-10)
R

For the electrons at the conduction band edge, the wave function is Qck uco(k

0). J(r R) is the coupling coefficient as we have said above, which is the exchange

integral between the band electrons and 3d5 electrons. Due to the fact that the 3d5

electrons are strongly localized, we can assume the integral is only nonvanishing

in a unit cell range for a specific R in Eq. 1-10. Considering the periodicity of

J(r R), the Eq. 1-10 can be rewritten as,


E4 =- Nx(S,)mc ua J(r)ucodr = aNx(S,)mr, (1-11)

a J uoJ(r)ucodr. (1-12)


where N is the number of unit cells in the crystal.

For zinc-blende semiconductors (most II-VI and III-V semiconductors), the

states for conduction band-edge (k = 0) electrons are s-like, and those for valence









band-edge holes are p-like. So the use of m, = 1/2 is justified. Then in a

magnetic field B, the conduction band energy is,


Enc = (n + -)khW + nmc9cpBB + mcnaNx(Sz), (1-13)
2

where wc eB/mu is the cyclotron frequency, and gc is the conduction band

g-factor. In Eq. 1-13, the first term is the Laudau splitting, the second term is the

Zeeman splitting, and the third term is the exchange splitting, which is unique for

the DMS.

Similarly, the energy structure of the valence band can also be obtained, if we

replace w by wv = eB/m*, gc by g, me by m,, and importantly, a by 3, where


3 uoJ(r)uodr. (1-14)

a and 3 are called exchange constants for s d and p d exchange interactions

between band electrons and localized Mn2+ ions.

We can introduce an effective Land6 g-factor in the conduction band

SaNx (Sz)
geff = 9cB+ (1-15)
pBB

which indicates the strength of the spin splitting of the first Landau level in

the conduction band. In the low field approximation, Eq. 1-6 becomes (Sz) =

-gpBS(S + 1)B/3kBT, so in this limit

ff aNxgS2(S +1)
geff = gc (1-16)
3kBT

At low temperature, the effective g-factor can reach very large values. The

g-factor depends on temperature through (S,) in Eq. 1-15. We will have a more

detailed discussion of g-factors in C'! lpter 2.

The above discussion is a very simplified qualitative model, and only appropri-

ate for II-VI DMS in a paramagnetic phase, where the Mn concentration is not so










high that they don't have a direct exchange interaction. This discussion can also

be applied to paramagnetic III-V DMS, in which commonly the Mn solubility are

very low. As a matter of fact, although Eq. 1-13 can give a qualitative description

of the conduction band structure, it does not work in real cases. C'! lpter 2 gives a

quantitative model.

Since the discovery of ferromagnetism in III-V DMS, much research now

focuses on exploring ferromagnetism mechanisms, looking for new materials and

obtaining higher Curie temperatures. Recently, ferromagnetism in II-VI DMS was

also reported by several groups [11, 12, 13].

1.3 The III-V Diluted Magnetic Semiconductors

1.3.1 Ferromagnetic Semiconductor

Although II-VI DMS combine both semiconducting and magnetic properties

and manifest spectacular properties, other characteristics such as ferromagnetism

are also desirable. From Eq. 1-15 and Eq. 1-16, we can see that at low tempera-

tures, the g-factor can be very large, but it is strongly temperature dependent. As

we mentioned above, the g-factor actually indicates the spin splitting. To employ

spin as a subject in research and device design, a large spin splitting is essen-

tial. While most II-VI DMS are paramagnetic, the spin splitting becomes small

at high temperatures, so the realization of room temperature spintronic devices

becomes difficult. The answer for this problem is ferromagnetic semiconductors.

We can expect a large spin splitting even at high temperatures for ferromagnetic

semiconductors.

The leap from II-VI DMS to III-V DMS should have been very natural. But

unlike II-VI semiconductors, Mn is not very soluble in III-V semiconductors.

It can be incorporated only by non-equilibrium growth techniques and it was

not until 1992 that the first III-V DMS, InMnAs was grown and investigated.

Ferromagnetism was soon discovered in this system [14]. Higher ferromagnetic








14










6.0
2K
IOK 112
25 K .125K
4.0 40K
55 K / 150K
E 70K .



2.0T
l ... ....85K /








'0 100 M







0.8 a-----

0 K


0.0


-8 -4 0 4
B (T)

Figure 1-5: Magnetic-field dependence of Hall resistivity pHall and resistivity p of
GaMnAs with temperature as a parameter. Mn composition is x = 0.053. The
inset shows the temperature dependence of the spontaneous magnetization 1.,
determined from magnetotransport measurements; the solid line is from mean-
field theory. Reprinted with permission from Matsukura et al. Phys. Rev. B 57:
R2037-R2040, 1998. Figure 1, Page R2037.
R2037-R2040, 1998. Figure 1, Page R2037.










transition temperatures were also achieved in GaMnAs [15]. Shown in Fig. 1-5 is

the magnetic-field dependence of the Hall resistivity and the normal resistivity of

GaMnAs with temperature as a parameter [16]. In this case, the ferromagnetic

transition temperature is about 110 K. The discovery of ferromagnetism in III-V

DMS led to an explosion of interest [14, 15, 17, 18]. Many new materials were

investigated, theories explaining the ferromagnetism mechanisms were brought

forward, and experiments aimed at increasing the Curie temperatures were carried

out.

Although InMnAs was the first MBE grown III-V DMS, its Curie temperature

was relatively low at about 7.5 K. In 1993, a higher Curie temperature of 35 K was

realized in a p type InMnAs/GaSb heterostructure [17]. Since 1996, a number of

groups are working on the MBE growth of GaMnAs and related heterostructures,

in which the highest Curie temperature (173 K) has been achieved recently for

25 nm thick Gal-) iM..As films with 's. nominal Mn doping after annealing [19].

The dependence of the Curie temperature of Gal_,Mi,.AAs on Mn concentration x

is shown in Fig. 1-6 [16]. The Curie temperature reaches the highest value when

x 5.;;' in this case.

GaMnN and GaMnP are also candidates for high Curie temperature III-

V DMS materials. Ferromagnetism in GaMnN is elusive. While some groups

found it paramagnetic when doped with percent levels of Mn [20], some groups

have reported a ferromagnetic transition temperature above 900 K [21]. Room

temperature ferromagnetism was also reported in GaMnP [22, 23]. Besides III-

V DMS, Mn doped IV semiconductors like GeMn [24, 25], SiMn [26], were also

reported ferromagnetic.

The theory for ferromagnetism in III-V DMS is still controversial, however,

there is consensus that it is mediated by the itinerant holes. Unlike the case in

II-VI DMS in which Mn ions have the same number of valence electrons as the









120

(C) /(

80 /



40-


0
0.00 0.04 0.08
X

Figure 1-6: Mn composition dependence of the magnetic transition temperature Tc,
as determined from transport data. Reprinted with permission from Matsukura et
al. Phys. Rev. B 57: R2037-R2040, 1998. Figure 2, Page R2038.

cations, Mn ions in III-V DMS are not only providers of magnetic moments, they

are also acceptors. Due to compensating defects like As-antisites or/and Mn
interstitials [27, 28, 29], hole concentrations are generally much lower than the Mn
concentration.

The theories of carrier induced ferromagnetism fall into four categories.

1. RKKY mechanism: Indirect exchange couples moments over relatively
large distances. It is the dominant exchange interaction in metals where there
is little or no direct overlap between neighboring magnetic impurities. It
therefore acts through an intermediary which in metals are the conduction
electrons (itinerant electrons) or holes. This type of exchange was first
proposed by Ruderman and Kittel [30] and later extended by Kasuya [31] and
Yosida [32] to give the theory now generally know as the RKKY interaction.
Ohno et al. explained the ferromagnetism in GaMnAs for Mn concentration
x = 0.013 using the RKKY mechanism [14]. In the interaction Hamiltonian,

H = JRKKYSi .Sj (1-17)

the coupling coefficient JyKKY assumes the form[33],


JRKKY(r) [sin(2kFr) 2kFrcos(2kFr)]/(2kFr)4


(1-18)









where kF is the radius of the conduction electron/hole Fermi surface, r is
the distance away from the origin where a local moment is placed. The
RKKY exchange coefficient, J, oscillates from positive to negative as the
separation of the ions changes with the period determined by the Fermi
wavevector kF1 and has the damped oscillatory nature shown in Fig. 1
7. Therefore, depending upon the separation between a pair of ions their
magnetic coupling can be ferromagnetic or antiferromagnetic. A magnetic ion
induces a spin polarization in the conduction electrons in its neighborhood.
This spin polarization in the itinerant electrons is felt by the moments of
other magnetic ions within range, leading to an indirect coupling.

JMC7 1








0 r







Figure 1-7: Variation of the RKKY coupling constant, J, of a free electron gas in
the neighborhood of a point magnetic moment at the origin r = 0.



In the case of DMS, the average distance between the carriers rc (4- -p 3

is usually much greater than that between the spins rs ) A
simple calculation show that the first zero of the RKKY function occurs at
r t 1.17rc. This means that the carrier-mediated interaction is ferromagnetic
and effectively long range for most of the spins.

The RKKY interaction as the main mechanism for the ferromagnetism in
III-V DMS is questionable in some cases such as in the insulating phase
(x < :;'. for GaMnAs), in which carriers are not itinerant. When the hole
density is low, and there is no Fermi surface (Fermi level in the gap), RKKY
theory cannot predict ferromagnetism. The other problem, maybe fatal, is
that in the RKKY approximation the exchange energy is much smaller than









the Fermi energy, which is not commonly the case in DMS. As a matter of
fact, these two energies are comparable in most cases.

2. Zener's model: Zener's model is a continuous-medium limit of the RKKY
model. Zener's model was first proposed by C. Zener in 1950 [34] to interpret
the ferromagnetic coupling in transition metals. Similar to the RKKY model,
it describes an exchange interaction between carriers and localized spins. The
Hamiltonian of Zener's model in a transition metal is [34]

1 1
Hs = aS, /3SSc + 7YS (119)
2 2
where Sd and Sc are the mean magnetization of the d-shell electron and
the conduction electron, respectively, and a, 3, and 7 are three coupling
constants. The main assumption here is that the exchange constant 3 is
alv-b-, positive, which under certain circumstances leads to ferromagnetic
coupling. Comparing Hamiltonian 1-17 and 1-19, we can see that 3 in
Eq. 1-19 p]1 il the similar role of J in Eq. 1-17. One big difference is that
Zener's model neglects the itinerant character and the Friedel oscillations of
the electron spin polarization around the localized spins.

Dietl [35] applied Zener's model to ferromagnetic semiconductors and
predicted the Curie temperature Tc for several Mn doped DMS systems.
The results are shown in Fig. 1-8. This quite accurately predicts the 110 K
transition temperature in GaMnAs, but certainly this is still a quite coarse
model. Even so, the trend shown in Fig. 1-8 has stimulated the enthusiasm
of people investigating GaN based materials looking for higher transition
temperatures.

Some of the problems in the RKKY model remain in Zener's model. For
instance, Zener's model still has limited application when carriers are mostly
localized because it still requires itinerant carriers to mediate the interactions
between localized spins. Besides, when the carrier density is higher than
the Mn concentration, important changes in the hole response function
occur at the length scale of the mean distance between the localized spins.
Accordingly, the description of spin magnetization by the continuous-medium
approximation, which constitutes the basis of the Zener model, ceases to be
valid. In contrast, the RKKY model is a good starting point in this regime.

3. Bound polaron model: Paramagnetic spins can be aligned to form ferro-
magnetic domains even in the absence of an external magnetic field under
certain conditions. In DMS, localized moments can also be aligned in the
vicinity of carriers to form what are known as ii, ,iw,.tic 1p .! iiii- The
carrier spin creates an effective exchange field for the magnetic ions due to
the exchange interaction which is similar in form to Eq. 1-7, and this field
causes ferromagnetic coupling of these local spins. The net spin alignment











Si
Ge
AlP
AlAs
GaN
GaP
GaAs
GaSb
InP
InAs
ZnO
ZnSe
ZnTe
0 100 200 300 400
Curie Temperature (K)

Figure 1-8: Curie temperatures for different DMS systems. Calculated by Dietl
using Zener's model.


again creates a self-consistent exchange field for the carriers. In this process,
the carrier spin creates a magnetic potential well resulting in formation of
a -I"p' cloud", a magnetic polaron. Due to the localized character of these
magnetic polarons in DMS, they are called bound magnetic polarons (BMP).

There have been extensive studies of BMP in II-VI DMS [6], in which BMP
are accountable for many optical and phase transition properties. Recently,
Bhatt et al. [36] and Das Sarma et al. [37] generalized BMP theory for III-V
DMS. They studied the coupling between two .,1i i,'ent BMPs, and concluded
that the exchange coupling is ferromagnetic. There are two different cases.
In one case two polarons overlap and the overlap integral accounts for the
ferromagnetic coupling. The ferromagnetic transition can be regarded as a
percolation occurring through the whole system when the temperature drops
below the Curie temperature. In the other case one does not need overlapping
polarons, their effect on the magnetic moment being taken into account
through a local magnetic field. Ferromagnetic coupling has been shown to
result when the carrier is allowed to hop between the ground state of one
magnetic atom and excited states of the other. A diagram of these two cases
are shown in Fig. 1-9.

The BMP model quite naturally and successfully explains the magnetism of
the DMS in the insulating phase. With a much higher carrier density, most














a- \ L 4 1 N




a. Two overlapping BMPs b. Two non-overlapping BMPs.
Electrons can hop between two
localized moments.


Figure 1-9: Schematic diagram of two cases of BMPs.

carriers are conducting. They are more like free band carriers. In such a case,
the BMP model may not be appropriate. Although some part of the carriers
are localized and have exchange interaction with the localized spins, most
carriers have extended wave functions, which tend to interact with the other
carriers and spins in the whole band. The condition for the BMP model does
not exist any more. In such a case, the RKKY mechanism should dominate.

4. Double exchange theory: Double exchange can be considered as charge
transfer exchange which leads to ferromagnetism in ferromagnetic perovskites
Such as LaMnO3. Akai et al. [38] performed first principle DFT calculations
which show that the i1i i ,i i ly of the carriers comes from Mn d states. The
hopping of the carriers between the impurity bands and valence bands causes
the ferromagnetic ordering. Later, Inoue et al. [39] also discussed a similar
mechanism. They calculated the electronic states of III-V DMS and found
that resonant states were formed at the top of the down spin valence band
due to magnetic impurities and the resonant states gave rise to a strong
long-ranged ferromagnetic coupling between Mn moments. They proposed
that coupling of the resonant states, in addition to the intra-atomic exchange
interaction between the resonant and nonbonding states was the origin of the
ferromagnetism of GaMnAs. We can classify this kind of mechanism caused
by the hopping of carriers between impurity states and valence states as a
double exchange mechanism. Double-exchange-like interactions in GaMnAs
were reported by Hirakawa et al. [40].

In the four models of ferromagnetism in III-V DMS, the first three are mean-

field based theories, and the last is based on d-electrons. Though each of them is









capable of explaining some specific aspects of ferromagnetism, none of them can be

applied universally.

1.3.2 Effective Mean Field

Each of the models we discussed above utilized one type of interaction, namely,

the interaction between two spins. In the following, we discuss how to solve this

kind of interaction inside a mean field framework.

Suppose a Heisenberg-like Hamiltonian


H =- J,,(Si Sj), (1 20)


where i, j specify atomic sites, -iw, of the magnetic moments in the crystal, and

Jj is the interatomic exchange interaction constant. The molecular field (effective
mean field) is simply given by


B,, -= j(S), (1-21)
91-B

where g is the g factor. Using the results we got in the discussion in Section 1.2.2,

the average spin along a magnetic field B (suppose it is directed along z) will be


(S,) -SB(y), (1-22)

with Bs, the Brillouin function, given by Eq. 1-5, and where


Y = [gpS(B + Be.)]/kBT. (1-23)

After substitution of Eq. 1-22 to above equation, we get


y = [gpSSB + JoS(S,)]/kBT, (1-24)



Jo Z= J,. (1-25)
J










2.5 ____r_


2.0 80K


A 1.5- T=110K 150K
V
1.0
290K
0.5


0.0 1 1
0 20 40 60 80

B (T)

Figure 1-10: Average local spin as a function of magnetic field at 4 temperatures.
The Curie temperature is 110K.

Equation 1-22 can be solved by standard root finding programs to find (S,).

The solution for (Sz) / 0 exists even when B = 0 due to the internal exchange

field. When I(S,) < 1,

B,(y) ( (S + )y. (1-26)
3
When Jo > 0, the condition for (Sz) / 0 then is

T < Tc JoS(S + 1)/3kB. (1-27)

This is consistent with the fact that J > 0 in Heisenberg Hamiltonian leads to

ferromagnetic interaction.

In a realistic calculation, Tc as a measurable parameter is easy to obtain,

hence we can use Eq. 1-27 to find the exchange interaction constant Jo, and thus

the spontaneous magnetization for T < Tc.









The spontaneous magnetization has fundamental effects on carrier scattering

and spin scattering, and thus affects the transport properties of both carriers and

spins. We will talk about this in C'! Ilpter 5.

1.4 Open Questions

Although the research of III-V DMS has been carried on for more than one

decade, and people have gained lots of understanding of their properties, there

are still a lot of open questions which deserve a deep and thorough investigation.

Among these outstanding problems, the nature and origin of the ferromagnetism,

the nature of the band electrons, and the possible device applications are most

fundamental and crucial.

1.4.1 Nature of Ferromagnetism and Band Electrons

As we discussed in the last section, people have proposed a variety of theories

to explain ferromagnetism in III-V DMS, each of which has its drawbacks. The

importance of the mechanism of ferromagnetism lies in the fact that it can predict

trends and lead people to search for suitable materials to achieve applications.

The first and widely publicized RKKY (Zener) model made predictions of above

room temperature ferromagnetism and prompted a worldwide search for materials

satisfying the conditions. The model asserts that localized spins in the III-V DMS

will introduce host-like-hole states that will interact via RKKY-type coupling with

the Mn local moments to produce the observed ferromagnetism. Recently, Zunger

et al. [41, 42, 43] performed first principle calculations showing that contrary to

the RKKY model, the hole induced by Mn is not host-like, which undermines

the basis of applying RKKY theory to DMS. The ensuing ferromagnetism by

the holes induced by Mn ions is then not RKKY-like, but !h 1 a characteristic

dependence on the lattice-orientation of the Mn-Mn interactions in the i i-1I I

which is unexpected by RKKY". They claim that the dominant contribution

to stabilizing the ferromagnetic state was the energy lowering due to the p d









hopping. The nature of the ferromagnetism then is closely related to the nature

of the band electrons. Photo-induced ferromagnetism [44] clearly reveals the role

of holes in mediating the ferromagnetic coupling. There is no doubt carriers are

crucial in all the mechanisms accounting for the ferromagnetism, but are they really

host-like holes, or do they have strong d component mixing? How do they behave

in the process of mediating the ferromagnetism? Only after we know the right

answer, will the manipulation of charge carriers and also the spins become more

predictable.

1.4.2 DMS Devices

The attraction of DMS mostly comes from their promising application

prospects. The special optical and magnetic properties can both be employ, ,1

designing novel devices. Semiconductor optical isolators based on II-VI DMS,

CdMnTe, which has a low absorption and large Faraday rotation for light with

0.98 pm wavelength, have been developed. This is the first commercial semiconduc-

tor spintronic device [45]. Since II-VI DMS is paramagnetic at room temperature, a

magnetic field is needed to obtain Far-d1vi rotation. Ferromagnetic semiconductor

based on III-V DMS, which does not need an external magnetic field to sustain the

big Faraday rotation, should have a good potential for use in optical isolators.

Photo-induced ferromagnetism has been demonstrated by Koshihara et al.

[44] and Kono et al. [46]. In Koshihara's experiment, ferromagnetism is induced

by photo-generated carriers in InMnAs/GaSb heterostructures. The effect is

illustrated in Fig. 1-11. Due to the special band alignment of this heterostructure,

electrons and holes are specially separated, and holes accumulate in the InMnAs

1 -v-r. The photo-generated holes then cause a transition of the InMnAs ?1,-r to

a ferromagnetic state. This opens a possibility to realize optically controllable

magneto-optical devices. In Kono's experiment, ultrafast demagnetization takes

place after a laser pulse shines on InMnAs/GaSb heterostructure and produces









ferromagnetism. The time scale is typically of several ps. They propose a new and
very fast scheme for magneto-optical recording.

8

(a) (-*) *i*"*"
S 4- *
O0

2 -/*
0
N
I -8 |
w
z -4 (

-8











H((T)
-2
-0.30 -0.15 0.0 0.15 0.30
H (

Figure 1-11: The photo-induced ferromagnetism in InMnAs/GaSb heterostructure.
Reprinted with permission from Koshihara et al. Phys. Rev. Lett. 78: 4617-4620,
1997. Figure 3, Page 4619.

Recently, Ohno et al. [2] achieved control of ferromagnetism with an electric
field. They used field-effect transistor structures to vary the hole concentrations in
DMS 1 ,i. -i and thus turn the carrier-induced ferromagnetism on and off by varying
the electric field. Rashba et al. [47] also proposed the electron spin operation by
electric fields. They also discussed the spin injection into semiconductors. The









electric control of ferromagnetism or spin states makes possible a unification of

magnetism and conventional electronics, and thus has a profound meaning.

Low-dimensional structures usually have dramatically different properties from

bulk materials. Much longer spin coherent times have been reported by several

groups in quantum dots [48, 49], which have been -,.-.- -I- 1. for use in quantum

computers where quantum dots can be used as quantum bits, since they offer a

two-level system close to the ideal case. One ultimate goal of DMS spintronics is to

implement quantum computing. The use of semiconductors in quantum computing

has various benefits. They can be incorporated in the conventional semiconductor

industry, and also, low-dimensional structures are very easy to construct, so unique

low-dimensional properties can be employ, -1 Several proposals have been made for

quantum computing using quantum dots [50, 51, 52].

Spin manipulation needs injection, transport and detection of spins. The

most direct way for spin injection would seem to be injection from a classical

ferromagnetic metal in a metal/semiconductor heterostructure but this raises

difficult problems related to the difference in conductivity and spin relaxation

time in metals and semiconductors [53]. Although these problems are now better

understood, this has slowed down the progress for spin injection from metals. On

the other hand, this has boosted the research of connecting DMS with nonmagnetic

semiconductors for spin injection. Many experiments pursuing hign efficiency spin

injection have been carried out. Shown in Fig. 1 12 is a spin light emitting diode

[54], in which a current of spin-polarized electrons is injected from the diluted
magnetic semiconductor BetMi Zi:1_,_ySe into a GaAs/GaAlAs light-emitting

diode. Circularly polarized light is emitted from the recombination of the spin

polarized electrons with non-polarized holes. An injection efficiency of 911' spin

polarized current has been demonstrated. As BeA Mi Zii,_,_ySe is paramagnetic,

the spin polarization is obtained only in an applied field and at low temperature.



















ai-GaAs
n-contact








dNM dSM
p-contact
100 nm 300 nm 100 nm 15 nm 500nm 300mm


Figure 1-12: Spin light emitting diode.


A ferromagnetic III-V DMS based spin injector does not need an applied

field. Shown in the left panel of Fig. 1-13 is a GaMnAs-based spin injection and

detection structure [55], in which spin-polarized holes are injected from GaMnAs

to a GaAs quantum well. The emitter and analyzer are both made of l-. 'ri of

ferromagnetic semiconductor GaMnAs. The temperature dependence of the spin

life time in the GaAs quantum well from magnetoresistance measurements is shown

in the right panel.

To obtain the information which a spin carries, one needs to detect an electron

spin state. Many methods for doing this have been brought forth and structures

or devices have been designed such as spin filters using magnetic tunnel junctions

[56, 57], spin filters [58], and one device involving a single electron transistor to

read out the spatial distribution of an electron wave function depending on the spin

state [59].

The development of DMS-based spintronics is now receiving a great attention,

and may become a key area in research and industry in the future. Although

enormous effort has been made, there is still a long way to go for DMS to be

extensively used in real life.




























1 .0
08 0 O08
Ga1,JMnAs o0 aMagelazatim
06 O
S 0 0.6
"W life
0o.4 t0 04

GaNnAs 02 o02

0 10 20 30 40 50 60 70
Teerature (K)

Figure 1-13: GaMnAs-based spin device. Left: GaMnAs-based spin injector and
analyzer structure. Right: Temperature dependence of spin life in the GaAs quan-
tum well in the structure shown in left panel.















CHAPTER 2
ELECTRONIC PROPERTIES OF DILUTED MAGNETIC SEMICONDUCTORS

To understand the optical and transport properties of DMS in the presence

of an applied magnetic field, we have to know the electronic band structure and

the electronic wave functions. For optical transitions, with the knowledge of the

interaction Hamiltonian, we may use Fermi's golden rule to calculate the transition

rate. In an external magnetic field, one energy level will split into a series of

Landau levels. Optical transitions can take place inside one series of Landau levels

or between different series according to the light configuration. So the knowledge of

the parities of these Landau levels need to be investigated. In this chapter, we will

use the k p method to study the band structure of DMS materials around the F

point. Specifically, a generalized Pidgeon-Brown model [60] will be used to study

the Landau level structures.

2.1 Ferromagnetic Semiconductor Band Structure

Ferromagnetic DMS's are different from normal semiconductors in that they

are doped with magnetic ions. These magnetic ions usually have indirect exchange

interaction resulting in an internal effective magnetic field. The electrons experienc-

ing this effective field will have an extra energy gain. For a paramagnetic DMS, or

for a ferromagnetic DMS with Tc much smaller than the typical temperature, when

there is no applied magnetic field, there is no internal exchange field, so it is just

like the host semiconductor. The special properties are present only in an applied

field.

The extra energy gain in a ferromagnetic DMS can be treated in a mean

field approximation (see Section 1.3.2). The localized magnetic moments line up

along the effective field, so for each magnetic ion, it has a nonvanishing average









spin along the field direction. According to the discussion in Section 1.2.2, an

extra energy term proportional to the exchange constant will be added to the

Hamiltonian. This term is related to spin quantum numbers, thus different spin

states will gain different energies, leading to spin splitting. Shown in Fig. 2-1 are

calculated valence band structures for B,,t = 0 of bulk GaAs and ferromagnetic

GaMnAs, which has a Curie temperature Tc = 55 K. The calculation is actually

based on a generalized Kane's model [61], and the effective field is assumed to be

directed in z direction. Kane's model was developed from k p theory, which we

will introduce in the following section.




0.0 -0.0



0 -0.2 -0.2
Pure GaAs, GaMnAs T=30 K
>, T=30 K E x=6% T=55 K

S-0.4 -0.4



-0.6 -0.6
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
(1,1,1) k (1/nm) (o,o,1) (1,1,1) k (1/nm) (0,0o,1)

Figure 2-1: Valence band structure of GaAs and ferromagnetic Ga,,t ., ,,, As
with no external magnetic field, calculated by generalized Kane's model. The spin
splitting of the bands is shown.


2.2 The k p Method

2.2.1 Introduction to k p Method

The k p method was introduced by Bardeen [62] and Seitz [63]. It is a

perturbation theory based method, often called effective mass theory in the

literature, useful for analyzing the band structure near a particular point ko, which









is an extremum of the band structure. In the case of the band structure near the F

point in a direct bandgap semiconductor, ko = 0.

The Hamiltonian for an electron in a semiconductor can be written as


2
H=
2mo


+ V(r),


(2-1)


here p = -ihV is the momentum operator, mo refers to the free electron mass, and

V(r) is the potential including the effective lattice periodic potential caused by the

ions and core electrons or the potential due to the exchange interaction, impurities,

etc. If we consider V(r) to be periodic, i.e.,


V(r) V(r + R),


where R is an arbitrary lattice vector, the solution of the Schr6dinger equation


H k(r) = E k(r)


satisfies the condition


k (r) = eikrUk(r)


where


Uk(r + R) = Uk(r),


and k is the wave vector. Equations 2-4 and 2-5 is the Bloch theorem, which gives

the properties of the wave function of an electron in a periodic potential V(r).

The eigenvalues for Eq. 2-3 split into a series of bands [64]. Consider the

Schr6dinger equation in the nth band with a wave vector k,


(2-6)


p + V(r)j Qnk(r) E- (k)Qnk(r).
2mo I

Inserting the Bloch function Eq. 2-4 into Eq. 2-6, we obtain

r p2 hk h2k2 ]
-+ k p + + V(r) Unk(r) E,(k)Unk(r).
2mo mo 2mo


(2-7)


(2-2)


(2-3)


(2-4)


(2-5)










In most cases, spin-orbit coupling must also be considered and added into the

Hamiltonian. The spin-orbit interaction term is


S x. ( X VV) p. (2-8)

Including the spin-orbit interaction, Eq. 2-7 becomes

P2 Vk V \2k2 V)
S+ -- p + 2 +x V) + -- p + ))p + V (r) k (r)
2mo mo ii. *,-,. 2mo

= E,(k)Uk (r).

(2-9)

The Hamiltonian in Eq. 2-9 can be divided into two parts


[Ho + W(k)]uk = EnkUnk, (2-10)


where
p2 h
Ho + a x VV) p + V(r) (2-11)
2m o 1',,. -

and
W k ) hk ( h $2) h2k2
W(k) p + h(x VV) + k (2 12)
mo 1 ,,:2 2mo
Only W(k) depends on wave vector k.

If the Hamiltonian Ho has a complete set of orthonormal eigenfunctions at

k = 0, uno, i.e.,

HoU.n = EnOUno, (2 13)

then theoretically any lattice periodic function can be expanded using eigenfunc-

tions Uno. Substituting the expression


Unk Z Cn(k)Umo (2-14)
m









into Eq. 2-9, and multiplying from the left by u*0, and integrating and using the

orthonormality of the basis functions, we obtain


z[ (Eo En+ 2 k nm U n0 (p+ 2 (x VV) U) mO) c$(k) 0.
n2mo mo 2m .o,,,:
(2-15)

Solving this matrix equation gives us both the exact eigenstates and eigenenergies.

Usually, people only consider the energetically .,i.i i:ent bands when studying the k

expansion of one specific band. It actually becomes very complicated if one wants

to pursue acceptable solutions when k increases. One has to increase the number of

the basis states, go to higher order perturbations, or both.

When k is small and we neglect the non-diagonal terms in Eq. 2-15, the

eigenfunction is Unk = UnO, and the corresponding eigenvalue is given by Ek =

Eno + This solution can be improved by second order perturbation theory, i.e.


hE Z i 2k2 + ((Uno H' umTo) (umo H' uno) (26)
Enk = En0 + -2 + o (2-16)
2mo Eno Emo
where

H' --. (p+ 2 (x VV). (2-17)

In the calculation shown above, we used the property (uo (p + 4mToc2( xVV)) u 0)

0, which holds for a cubic lattice periodic Hamiltonian due to the
If we write

7 p + 21,-( (x VV) (2 18)

then the second order eigenenergies can be written as

Sh2k2 h2 Inm-k2
Enk = Eno + 2-+ 2 nm-k (2-19)
2mo m Eo Enmo
rn7nn










Equation 2-19 is often written as


Ek= E0 + h ( kk,, (2-20)
a,i3 a 3

where
1 1 2 7a( _)3
S6 E + m (2-21)
m* nmo ai 7 m Eo Emo

is the inverse effective mass tensor, and a, 3 x, y, z. The effective mass generally

is not isotropic, but we can see it is not k-dependent, this is because at this level

of approximation, the eigenenergies in the vicinity of the F point only depend

quadratically on k.

2.2.2 Kane's Model

As we mentioned in the last section, expanding in a complete set of orthonor-

mal basis states in Eq. 2-15 gives exact solutions for both the eigenfunctions and

eigenenergies. Practically, it is not feasible to include a complete set of basis states,

so usually only strongly coupled bands are included in usual k p formalism, and

the influence of the energetically distant bands is treated perturbatively.

In Kane's model, electronic bands are divided into two groups. In the first

group, there is a strong interband coupling. Usually the number of bands in this

group is eight, including two conduction bands (one for each electron spin) and six

valence bands(two heavy hole, two light hole and two split-off hole bands). The

second group of bands is only weakly interacting with the first group, so the effect

can be treated by second order perturbation theory.

Shown in Fig. 2-2 is the band structure of a typical III-V direct band gap

semiconductor. Due to crystal symmetry, the conduction band bottom belongs to

the F6 group, the valence band top belongs to the Fs group, and the split-off band

belongs to the F7 group. The spatial part of the wave functions at the conduction

band edge are s-like and those at the valence band top are p-like. Symbols of

IS), IX), |Y), and |Z) are used to represent the one conduction band edge and










Distant bands


Distant bands



Figure 2-2: Band structure of a typical III-V semiconductor near the F point.
Kane's model considers the doubly spin degenerate conduction, heavy hole, light
hole and split-off bands, and treats the distant bands perturbatively.


three valence band edge orbital functions. With spin degeneracy included, the

total number of states is eight. These eight states IS T), IS 1), IX ), IX 1),

etc, can serve as a set of basis states in treating these eight bands. A unitary

transformation of this basis set is still a basis set. So in practice, people use the

following expressions, which are the eigenstates of angular momentum operators J









and mj, as the basis states for the eight-band Kane's model,


1
2
3
U2 -
2
3
2
1
4 = | ,
2
1
2
3
U6 = ,
2
3
U7 = | 2
2
1
8 2'


1
2
3
2-
1
)-


2
1
2

3
)-


2
1

1


ST)- IsT),

HH T) -1 (X + iY)
1



2
ILH T) --l(X iY) 1 -2Z T),




V6



ILH 1) = (X + iY) -2 ),


V3


(2-22)


This set of basis states is a unitary transformation of the basis which we have

mentioned above, and it can be proven that they are the eigenfunctions of the

Hamiltonian 2-11. Because of spin degeneracy at k = 0, the eigenenergies for IS),

|HH), |LH) and ISO) are Eg, 0, 0, -A, respectively, with the selection of energy

zero at the top of Fs band, where E, is the band gap, and


3ikh 0V
A= 2 (XI p
4m2C2 Ox


OV
Px|Y),
iiy


(2-23)


is the split-off band energy.

At this level of approximation, the bands are still flat because the Hamiltonian

3-10 is k-independent. Including W(k) in Eq. 2-12 into the Hamiltonian, and

defining Kane's parameter as

-ih
P= (SIr|Z), (2-24)
Tmo









we obtain a matrix expression for the Hamiltonian H = Ho + W(k), i.e.,

E, + Pk^ PkP PkI_ 0 0 Pk, Pk,
Pk_ hk 0 0 0 0 0 0
2mo
Pk 0 0 Pk 0 0 0

Pk+ 0 0 -A+ 2Pk, 0 0 0
0 0 /jPPk Pk + Pk_ Pk+ Pk+

Pk 0 20 0 PI k 0 0+
+ S2o 0
-- Pkz 0 0 0 Pk_ 0 2tko 0
Pk 0 0 0 Pk_ 0o o -A + h 2
(2-25)

where k+ = k + iky, k_ = kx iky, and k1, ky, kz are the cartesian components

of k. The Hamiltonian 2-25 is easy to diagonalize to find the eigenenergies

and eigenstates as functions of k. We have eight eigenenergies, but due to spin

degeneracy, there are only four different eigenenergies listed below. For the

conduction band,

S2k2 1 1 4P2 2P2
E= E,-+ --++ (2-26)
2m,' m mno 32E, 3h2(E, + A)

For the light hole and split-off bands,

h2k2 1 1 4P2
Emlh = + 42 (2-27)
2mih T mh mo 3h2E


h2k2 1 1 2P2
Eo -A -2k+ 2= (2-28)
2mso mso mo 3h2(E, + A)
For the heavy hole band we have

h2k2 1 1
Ehh 2 1 (2-29)
2mhh mhh mo

The effective mass of the heavy hole band is still equal to the bare electron mass,

since we have not included the distant band coupling in the Hamiltonian. The










effect of the distant band coupling will make the heavy hole band curve downward

rather than upward.

2.2.3 Coupling with Distant Bands-Luttinger Parameters

The coupling with distant bands can be parameterized by L6wdin's pertur-

bation method [65], in which the bands are classified as A and B. In our case, we

select the basis states 2-22 as class A and label them with subscript n and all the

other (energetically distant) states as class B which we label with subscript a.

Suppose all states are orthonormal, the Schr6dinger equation then takes the

form

Y(Hi Ezi)a, 0, (2-30)

where I and m run over all states. Rewrite this equation using class A and B, and

we obtain
A B
(E H,,m)am = Hmnna, + H,,a, (2-31)
n4m a4m
or
A B
amn= "E + "E-H a,, (2-32)
/:E i + E Hnm
n4m n4a
where the first sum on the right hand side is over the states in class A only, while

the second sum is over the states in class B. We can eliminate those coefficients in

class B by an iteration procedure and obtain the coefficients in class A only,

A UA HI-_ nn
am m H m na (2-33)


and

B B
Ur HT + H ,1m mH,-,lan + YH,- + (2-34)
z E Haa (E Haa)(E H)..
a#3

A little algebra shows that Eq. 2 33 is equivalent to

A
5(U[/ E6m,)a, = 0. (2-35)
n










This means that we can find the eigenenergies with the basis in class A but still

include the remote effects from class B using Eq. 2-35. The effect from class B is

treated as a perturbation using Eq. 2-34 to second order.

Truncating U,, to the second term, and using Hamiltonian in Eq. 2-9, it can

be rewritten as

B B H H
UTn Hmn + HmH- Hmn+ Y -a an (236)
a7m,n a4m,n

where

Hn = (umo H uno) = E,(0) + h2k 62 n (2-37)

and

H (oh = o k o)= hka, (2-38)
mo mo

where a = x, y, z and r, a_ pa for m E A and a E B. Thus

F h2k21 h 2 B kk rPa
n Ew(0) + 22mo 6 + 2+ ka P (2-39)
L OJ 7m,n a,b

Applying basis set 2-22, we can define parameters A, B, C and F as follow,

h2 h2 B p x
S2r- Xa aX
2mo m02 z E, Ea'

B= +
B 2mo + zm0 E, Ea'

h2 pB Y 7 p7
Co-
m E2- Ea
B
F -= 1 PaPs (2-40)
Tmo Ec Ea

Rewriting these parameters in terms of "L- il i ,, parameters 71, 72, 73 and 74

defined as












h2
O71

h2
2nnO 72
h2
<73
2mo


(A + 2B),
3( )

(A B),
6


74 1 + 2F,


we can obtain the the Hamiltonian Hmn

under the basis set listed in Eq. 2-22 as


Umn including the distant band coupling


Vk_ Vk_

-M iz2M

-P+Q i 2Q


i2MM+ -iv2Q

0 i /lVk,


P-A


0

0

-i Vk,

Vk,
v-3^-


SVk E, 2 7 2
73 1 E+2 0<74


0 VVkk


0

L

SVL

SVk_
72


Vk,


-L L

0 16L+

1/i L+ 0

- Vk+ Vk+


k2 k2 + k2 + k2,
K y ~
h2
P = o 7k2,
2mo

2T

L -i-h v373(kx -
mo
h2
M 3 -
2mo
h2

0 = 2
V o 2 '


1Vk_

'Vk_
73'-


- Q -M+ i2M

M -P+Q iz2Q

2M -i2Q -P A
(2-45)


- 2k2),

iky)kz,

- 2) 21- I-. ) ,
y~ Ij


(2-46)


(2-41)

(2-42)

(2-43)

(2-44)


Eh2 k2
vg + 2To 74


"Vk_
72


P-Q


V~Vk
-3 Vk,


0

-i L


P



i\


where










and

E = 2 P2 (2-47)

related to the Kane's parameter P defined in Eq. 2-24. We can see that if k = 0 or

k, = 0, the Hamiltonian is block diagonalized.

In practice, one important thing needs to be noted that the Luttinger parame-

ters defined in Eq. 2-44 are not the "usual L iIIli:, i parameters which are based

on a six-band model since this is an eight-band model, but instead are related to

the usual Luttinger parameters 7
L Ep

L Ep
72 -72 6E'
6 Eg
73 3 EL p (248)
3E,

This takes into account the additional coupling of the valence bands to the

conduction band not present in the six-band Luttinger model. We refer to 71

etc. as the renormalized Luttinger parameters.

The Hamiltonian 2-45 is based on an eight-band Kane's Hamiltonian including

the contributions of the remote bands. With the remote band coupling, the

electron effective mass at the conduction band minimum now becomes

S- 4 + + (2-49)
m, mo 3 E9 E9 +A

In DMS materials without magnetic fields, the Hamiltonian 2-45 plus the

exchange interaction can be used to calculate the band structure which will be

applied to the calculation of the optical properties such as magneto-optical Kerr

effect, which is to be studied in chapter 4.










In a magnetic field, a single energy level splits into a series of Landau levels.

Optical transitions take place between two levels in one series or two in different

series.

2.2.4 Envelope Function

In the treatment of Kane's model (or six-band Luttinger model), all the bands

in class A are considered as degenerate at the F point. Away from the F point

or/and taking the remote band coupling into account, the electronic wave functions

become linear superposition of the basis states.

In last section, if we write

2k2 2 B kkba b
Hrmn mn Em(O) + 2mo 6mn + hn 2 k T-man
2Tmo Mc Er- Ea
aam,n a,b (2-50)

= Em (0)6-mn + D jb kakb,






the eigenequation is given by


H a.(k) Em(0)6 + Dab kakb a.(k)= E(k)am(k) (2-52)
n 1 n 1 a,b

where am is the superposition coefficients defined as


fnk(r) a,(k)uno. (2 53)


Now we consider a spatial perturbation U(r) added to the Hamiltonian Hm.

The eigenequation now becomes


[H + U(r)]b(r) = E(r).


(2-54)










If we write the solution to the equation as
8
(r) Fm(r)umo(r), (2-55)
nm=l

Luttinger and Kohn [67] have shown that we need only solve the following equa-

tion,


SEm(0)6mn + a D + U(r)6+ m F,(r) = EFm(r)

(2-56)

This means that we only need to replace the wave vector in the Hamiltonian k" by

the operator pa/h, and solve an equation for F(r). The function F(r) is called the

effective mass envelope function.

2.3 Landau Levels

2.3.1 Electronic State in a Magnetic Field

Using the simple effective mass theory, the motion of an electron in semicon-

ductors is like that of free electrons. In the simplest case, we consider a parabolic

band, and assume the effective mass to be m. The wave equation under the

effective mass approximation is

1 (-ihV + eA)2(x) c~(x), (2-57)
2m

where A is the vector potential, and e is the electron charge. Assume the magnetic

field is directed along z. Using Landau's gauge,


A = -Byx. (2-58)


and assuming a solution like


(x, y, z) = i +kz) )Q(y), (2-59)
V&xyLL









where L,, Ly, and L, are lengths for the bulk crystal in three dimensions. After

substituting into the effective mass equation, we have an equation for ((y),


2m (k e )2 h2 2+ 2k2 O(y) = e(). (2-60)
k2 72
Defining c' e 2, the equation above is a simple harmonic oscillator equation

with

= (n + n :. (2-61)
2
where wc = eB/m is the cyclotron frequency. Thus the total energy is

1 h2k2
S(n + )h + (2-62)
2 2m

This means that in a magnetic field, the motion of an electron in a semiconductor

now has quantized energies in the x y plane, though its motion in the z-direction

is still continuous. The original states in one band now split into a series of Landau

levels whose eigenfunctions are

S( hk )i(kx+k (2-63)
VLLX eB

The electronic energies in Eq. 2-62 is only related to n and kz. They are

degenerate for different kx. In Eq. 2-63 the center of yo = can only be from 0

to Ly. Using the periodic boundary condition, the interval for kx is 27/Lx, thus the

interval for yo is h/eBLx, the corresponding number of values for yo is eBLxL,/h.

Therefore, for given n and k,, the degeneracy is eBLxL,/h.

2.3.2 Generalized Pidgeon-Brown Model

In a realistic calculation for electronic states in a magnetic field, the simple

qualitative theory is not adequate. From the discussion of the k p theory, we

know that the band structure is very complicated. So in this section, we will

use the k p based Hamiltonian to calculate Landau levels in DMS. For narrow-

gap semiconductors such as InAs, the coupling between the conduction and









valence bands is -I ri,.- so it is necessary to use the eight band model to calculate

the Landau levels. Pidgeon and Brown [60] developed a model to calculate the

magnetic field dependent Landau levels at k = 0. We will generalize this model to

include the wave vector (k,) dependence of the electronics states as well as the s d

and p d exchange interactions with localized Mn d electrons.

We will still utilize the basis set defined in Eq. 2-22. In the presence of a

uniform magnetic field B oriented along the z axis, the wave vector k in the

effective mass Hamiltonian is replaced by the operator


k (p + A), (2-64)

where p = -ihV is the momentum operator. For the vector potential, we still use

the Landau gauge as in Eq. 2-58, thus B = x A = Bz.

Now we introduce two operators


at A (k + ik) (2-65a)
v2

and

a (k iky) (2-65b)
2
where A is the magnetic length which is defined as

hc h2 1
rh h2 (2-66)
eB 2m pBB

The operators defined in Eqs. 2-65 obey the commutation rules of creation and

annihilation operators. The states they create and annihilate are simple harmonic

oscillator functions, and ata = N are the order of the harmonic functions. Using

these two operators to eliminate kx and ky in Hamiltonian 2-45, we arrive at the

Landau Hamiltonian

HL La L (2-67)
Lt Lb










with the submatrices La, Lb and Lc given by


-i at


v


Eg+A
a



V
Va



0

0

-i 3Vk,

- 1Vk,


iza

P-Q
-Mt


- at

P-Q


-M -P + Q i2Q

-iM -i2Q P -P- A

0 Vk, i Vk



L 0 6i Lt

- 2 L Lt 0


The operators A, P, Q, L, and M in Eq. 2-67 now are


A 2 Y4 (2N + 1
mo 2 A2


p 2 71
P
mo 2


h2 72
Qmo 2
mo 2


+ k2)
12) 1


2N +k
A 2 ) 1


2N +1
A 2


S h2
L = -73
mo


2k 2


iW6
\


M -- + )3(
mo 2


A 2


Mat

-M


VMat
i 2
iA/at


P +Q iz2Q


iz 2Mt -i 2Q


-P-A


I a

iV2Mt


1 a
-Mt


(2-68)








(2-69)


(2-70)


(2-71a)


and


(2-71b)



(2-71c)



(2-71d)


(2-71e)










The parameters 71, 72, 73 and 74 are defined in Eq. 2-48 and 2-44. Usually,

the Luttinger parameters 72 and 73 are approximately equal (spherical approxima-

tion), so we have neglected a term in M proportional to (72 73)(at)2. This term

will couple different Landau manifolds making it more difficult to diagonalize the

Hamiltonian. The effect of this term can be accounted for later by perturbation

theory.

For a particle with non-zero angular momentum (thus a non-zero magnetic

moment p) in a magnetic field, the energy due to the interaction between the

magnetic moment and the magnetic field is -p B, which is called Zeeman energy

which we discussed in Section 1.2.2. The electrons in III-V DMS conduction or

valence bands possess both orbital angular moment and spin, so there is one extra

Zeeman term proportional to (KoL B + Klra B), where L and a are the orbital

angular momentum and spin operators, both of which are in matrix form. K0

and K1 are the magnetic field dependent coefficients. Following Luttinger [66], we

define the parameter K as

KL = (2-72)
6E,

where
L L 2 ,L 1 L 2
S= 73 + 72 371 (2-73)

is the Luttinger K parameter, and we obtain the Zeeman Hamiltonian


h2 1 Za 0
Hz --- (2-74)
mo A2 0 -Z*


where the 4 x 4 submatrix Za is given by

1 0 0 0
2
0 0 0
Za 2 (2-75)

0 0 i
0 0 i,/ K i









Due to existence of the Mn impurity ions, the exchange interactions between

the band electrons and localized moments also needs to be accounted for. This

term is proportional to (, J(r RI)SI o a). Under a mean field and virtual i I I1

approximation (see Section 1.2.2), and defining the two exchange constants


a= (SJIS) (2-76a)

and

P 3 (X|J|X), (2-76b)

we can arrive at an exchange Hamiltonian


HM x No(S) D (2-77)
0 -D*

where x is the Mn concentration, No is the number of cation sites in the sample,

and (S,) is the average spin on a Mn site which is exactly the one we derived at

Section 1.2.2 for paramagnetic DMS or that in Section 1.3.2 for ferromagnetic

DMS. The 4 x 4 submatrix Da is

Si 0 0 0

0 >0 0 0
Da= 2 (2-78)
0 0 -$3 -(i2p

0 0 i -3

Here we just treat the effect of magnetic ions as an additional interaction. We

don't consider the possible effect of these magnetic ions on the band gap, etc. The

band gap changes as a result.

The discussion here is very similar to that in Section 1.2.2 where only a

qualitative model is introduced, but here we used a realistic band structure. Also










similar to that discussion, the total Hamiltonian here can be written as


H = HL + Hz + HMn. (2-79)


We note that at k, = 0, the effective mass Hamiltonian is also block diagonal like

the Hamiltonian 2-45.

2.3.3 Wave Functions and Landau Levels

With the choice of Gauge 2-58, translational symmetry in the x direction is

broken while translational symmetry along the y and z directions is maintained.

Thus ky and k, are good quantum numbers and the envelope of the effective mass

Hamiltonian 2-79 can be written as

al,n,v (n-1

a2,n,v .' -2

a3,n,v

Ci(kyy+kzz) a4,n,v n
,,' A = (2-80)


a6,n,,v n+1

a7,n,,v n-1

a8,n,v -1

In Eq. 2-80, n is the Landau quantum number associated with the Hamilto-

nian matrix, v labels the eigenvectors, A = LxLy is the cross sectional area of the

sample in the x y plane, ,(0) are harmonic oscillator eigenfunctions evaluated at

( = x A2ky, and ai,,(kz) are complex expansion coefficients for the v-th eigenstate

which depend explicitly on n and k,. Note that the wave functions themselves will

be given by the envelope functions in Eq. 2-80 with each component multiplied by

the corresponding k, = 0 Bloch basis states given in Eq. 2-22.









Substituting T,, from Eq. 2-80 into the effective mass Schrodinger equation

with H given by Eq. 2-79, we obtain a matrix eigenvalue equation


H. F.,, = E,, (k,) F,,,, (2-81)


that can be solved for each allowed value of the Landau quantum number, n, to

obtain the Landau levels E,,,(k,). The components of the normalized eigenvectors,

F,,,, are the expansion coefficients, ai.

Since the harmonic oscillator functions, n,'((), are only defined for n' > 0,

it follows from Eq. 2-80 that F,,, is defined for n > -1. The energy levels are

denoted E,,,(k,) where n labels the Landau level and v labels the eigenenergies

belonging to the same Landau level in ascending order.

Table 2-1: Summary of Hamiltonian matrices with different n

n Dimension of Hamiltonian Eigenenergy No. Label as
-1 1 x 1 1 (-1,1)
0 4x4 4 (0,v), v 1.- 4
1 7x7 7 (1,, v), 1---
>2 8x8 8 (n,v), v 1--8


For n = -1, we set all coefficients ai to zero except for a6 in order to prevent

harmonic oscillator eigenfunctions 0,, (0) with n' < 0 from appearing in the

wavefunction. The eigenfunction in this case is a pure heavy hole spin-down state

and the Hamiltonian is now a 1 x 1 matrix whose eigenvalue corresponds to the a

heavy hole spin-down Landau level. Please note that when we speak about a heavy

(light) hole state, it generally means that the electronic wave function is composed

mainly of the heavy (light) hole Bloch basis state near the k = 0 point.

For n = 0, we must set al = a2 = a7 = a8 = 0 and the Landau levels

and envelope functions are then obtained by diagonalizing a 4 x 4 Hamiltonian

matrix obtained by striking out the appropriate rows and columns. For n = 1, the










Hamiltonian matrix is 7 x 7 and for n > 2 the Hamiltonian matrix is 8 x 8. The

summary of Hamiltonian matrices for different n is given in Table 2-1.

The matrix Hn in Eq. 2-81 is the sum of Landau, Zeeman, and exchange

contributions. The explicit forms for the Zeeman and exchange Hamiltonian

matrices are given in Eq. 2-74 and 2-77 and are independent of n.

Table 2-2: InAs band parameters

Energy gap (eV)1
E, (T = 30 K) 0.415
Eg (T = 77 K) 0.407
E, (T = 290 K) 0.356
Electron effective mass (mo)
m 0.022
Luttinger parameters 1
71L 20.0
72 8.5
73 9.2
KL 7.53
Spin-orbit splitting (eV) 1
A 0.39
Mn s-d and p-d exchange energies (eV)
No a -0.5
No f 1.0
Optical matrix parameter (eV) 1
Ep 21.5
Refractive index 2
n, 3.42
1 Reference [68].
2 Reference [69].


Now we study the Landau level of InAs and InMnAs, which in the following

we assume paramagnetic. The parameters used in the calculation are listed in

Table 2-2. Shown in Fig. 2-3 are the conduction band Laudau levels for InAs

and In,,.. --i,,, As as a function of magnetic field at k = 0 for a temperature of

30 K. The dashed lines represent spin-up levels, and the solid lines represent the

spin-down levels. This illustrates the energy splitting of the conduction band at

the P point. The right panel for InMnAs is only different from the left panel for












0.9 0.9 -

0.6 o.6

0.5 0.5

0.4 0.4
0 20 40 60 80 0 20 40 60 80
B (Tes a) B (Tesca)


Figure 2-3: Calculated Landau levels for InAs (left) and In, ..Mi!,, ,As (right) as a
function of magnetic field at 30 K.


InAs in that it has the exchange contributions due to the interaction between the

band electrons and the localized Mn moments. The ordering of these Landau levels

can be qualitatively explained by the simple model in Eq. 1-13 where we have an

analytical expression for the Landau level energy. Note that they are not linear

functions of the magnetic field. In the next chapter we will see that this simple

model cannot predict an a (exchange constant defined in Eq. 2-76) dependence of

the cyclotron energy, which is the energy difference between two .,i1] ient Landau

levels with the same spin. The exchange constant dependence is a consequence of

k p mixing between conduction and valence bands.

The wave vector kz dependence of Landau levels in both conduction band and

valence bands is shown in Fig. 2-4, where only the five lowest order Landau levels

are shown. Because of the strong state mixing, the spin states in valence bands are

not indicated. Comparing the left and right panels of Fig. 2-3 and Fig. 2-4, we can

see that Mn doping drastically changes the electronic structure. Spin splitting is

greatly enhanced in both conduction and valence bands. As a matter of fact, the

spin state ordering in the conduction band is reversed with Mn doping.








53


0.8 0.8
S0.7 0.7

> 0.6 0.6



-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
Kz (1/nm) Kz (1/nm)














Figure 2-4: The conduction and valence band Landau levels along kz in a mag-
netic field of B = 20 T at T = 30 K. The left and right figures are for InAs and
0In, 2, As, respectively.
0.2 0.2

0.3 ,0.3
04 0 .2 0 0.2 0.4 0.4 0.2 0.0 0.2 0.4



Figure 2-4: The conduction and valence band Landau levels along k, in a mag-
netic field of B 20 T at T 30 K. The left and right figures are for InAs and
In,, .. 11_,,,, 1,A s, respectively.


2.4 Conduction Band g-factors

In practice, spin-splitting is represented the g-factor. For a free electron, the

g-factor is the ratio between the magnetic moment due to spin in units of pB and

the angular momentum in units of h. The g-factor for a free electron is 2 (if the

influence of the black body radiation in the universe is accounted for, it is 2.0023).

In the solid state, due to the spin-orbital interaction (and other interactions, for

example in DMS, the exchange interaction), the g-factor for an electron is not 2.

Usually the g-factor in the solid state is defined as


g o=pi, (2-82)
P/BB

where hw8spi, is the spin-splitting. Roth et al. [70] have calculated the g-factor in

semiconductors based on Kane's model, and have shown that the g-factor in the








54

conduction band is

2r1+ 1 <- j. (283)
m, 3E,+ 2A

Using this equation, the g-factor for bulk InAs is about -15.1, which is close to the

experimental value -15 [71].

150 I I 0' O
.......... 12%
0 290 K -......... 12%



-- .. .. -- 2 .% :

S50 20 40 60 0 20 40
.15 .
C -`- ~--- -------------" ---- 5
.. .. .. .. ... ...................................... 2 .5 %
-0%
50 I I I I I I I I I 25 F
0 20 40 60 0 20 40 60
B (Tesl) 8 (Tesl)

Figure 2-5: Conduction band g-factors of In-lMn1As as functions of magnetic
field with different Mn composition x. For the left figure, T = 30 K and for the
right, T = 290 K. Note at high temperatures we lose the spin splitting.


Due to the exchange interaction, the spin-splitting is greatly enhanced.

Usually in DMS, the exchange energy is much bigger than the Zeeman energy,

which can be seen from the simple theory in Eq. 1-15 for a few percent of Mn

doping. In that case, if we take x = 0.1, Na = -0.5 eV, and T = 30 K, then

gff ~ 256. If we only consider the exchange interaction, from Eq. 2-78, the

spin-splitting in the conduction band is exactly that in Eq. 1-13. However, this

is not correct because the first conduction band spin-down level comes from the

n = 0 manifold, while the first conduction band spin-up level comes from the n = 1

manifold. Different manifold numbers result in different matrix elements, which

will cause different state coupling, and thus spin-splitting due to the exchange

interaction is not what the simple model predicts. The conduction band g-

factors for InAs and InMnAs at 30 K and 290 K are shown in Fig. 2-5. This

clearly demonstrates how Mn doping affects the g-factors. At 290 K, the g-










200 .

S----30K
150- \\ In MnAs ---80K
\' To=110K .... 150K
0\\ c -- 290 K
o \
o 100 \\



50 -


0-

0 20 40 60 80

B (Tesla)

Figure 2-6: ,j-f Ii.. 1 of ferromagnetic In, ., i,,, As. Tc = 110 K.

factors are drastically reduced. This is because at high temperatures, thermal
fluctuations become so large that the alignment of the magnetic spins is less

favorable. However, if ferromagnetic DMS are employ, .1 due to the internal
exchange field, a strong alignment can be expected even at high temperatures. Now

we suppose a high-Tc In,, ..,Mi As system in which a Curie temperature of 110 K

is achieved. The g-factor for this system is shown in Fig. 2-6. Even at relatively
high temperature (still below the transition temperature though), big g-factors are

still obtained. The g-factor reaches infinity at zero field when temperatures are
below Tc because there is still spin-splitting even though there is no external field.















CHAPTER 3
CYCLOTRON RESONANCE

In chapter 2, a systematic method of calculating the electronic structure

of DMS was developed and described in detail and applied to the narrow gap

InMnAs. It has been seen that the band structure of DMS depends strongly

on Mn doping which induces the exchange interaction. The band structure also

depends on the strength of the applied magnetic field, as can be seen from Fig. 2-3

and 2-5. Apart from the theoretical calculation, optical experiments are ahliv-

good v--,V- to detect the electronic properties of semiconductors. Among these

methods, cyclotron resonance (CR) is an extensively used and a powerful diagnostic

tool for studying the inter-subband optical properties and effective masses of

carriers. Cyclotron resonance is a high-frequency transport experiment with all

the complications which characterize transport measurements. Through cyclotron

resonance, one can get the effective masses, which are determined by the peak of a

resonance line, while scattering information is obtained from the line broadening.

Cyclotron resonance occurs when electrons absorb photons and make a transition

between two .i,.] ient Landau levels. From cyclotron resonance measurements one

can infer the magnetic field dependent band structure of the material. Since the

band structure of a DMS is so sensitive to magnetic fields, this is a useful means to

study and obtain band information from a comparison between the experimental

results and theoretical calculations.

3.1 General Theory of Cyclotron Resonance

3.1.1 Optical Absorption

The absorption coefficient, a, can be determined by calculating the absorption

rate T of incident light with angular frequency w in a unit volume. Suppose the









energy flux of the incident light is S, then the photon flux density is S/hw, and we

have Tdx = Sadx/hw, i.e.

a(WU) = T() (3-1)
S

T(w) is the sum of the transition probabilities Wif under the illumination of light

with angular frequency w divided by the volume, namely


T = W (3-2)
i,f

where i, f are the labels for the initial and final states. The summation runs over

all states. For absorption between state i and f, the transition probability from

Fermi's golden rule [72] is,

2r
Wabs = Hf 26(Ef Ej hw), (3-3)

and for emission
27 1
Wems 2= H |26(E- Ef+ hw), (3-4)

where Ei and Ef are the energies of the initial and final states (here we only want

the final expression for absorption, so in emission, even the electrons transit from

state f to state i, we still call state i is the initial state, and state f the final state),

respectively, and the 6 function ensures the conservation of energy in the optical

transition. H' is the electron photon interaction Hamiltonian. Essentially, in

optical transitions, momentum should also be conserved. However, since the photon

momentum p = h/A is much smaller than the typical electron momentum, we

generally consider the optical transition to be v,-i I ical", which means an electron

can only transit to states with the same k, i.e., wee ignore the photon momentum.

In semiconductors when dealing with the realistic case of absorption, we need

to take into account the state occupation probability by electrons, which in thermal










equilibrium is described by a Fermi-Dirac distribution function

1
f = t (3-5)
1 + E-EFkBT' (5)

and so the rate of absorption in the whole
1 2 1 1
Tif= H h 26(Ef E, hw)fi( ff) (3-6)
iv

and the emission rate

1 2x
Tf = 2 IH, |2I(E, Ef + hw)ff(1 fi) (3-7)
i,f

Due to the hermitian property of H', IHi, = IH' I. The net absorption rate per

unit volume then is


T = Tf T E- hu)(f, ff). (3-8)
i,f

When a semiconductor is illuminated by light, the interaction between

the photons and the electrons in the semiconductor can be described by the

Hamiltonian,
1
H (p + eA)2 + V(r) (3-9)
2mo

where mo is the free electron mass, e is the electron charge, A is the vector

potential due to the optical field, and V(r) is the < i--il periodic potential (in

DMS, including the virtual crystal exchange potential). Thus the one-electron

Hamiltonian without the optical field is


Ho + V(r) (3-10)
2mo

and the optical perturbation terms are

e e2A2
H' A .p+-- (3-11)
mo 2mo









Optical fields are generally very weak and usually only the term linear in A is

considered, i.e., we treat the electron-photon interaction in a linear response regime

and neglect two-photon absorption. The transition due to the optical perturbation

in Eq. 3-11 can take place either across the band gap or inside a single band

(conduction or valence band) depending on the photon energy. In this chapter, we

only consider cyclotron resonance, which takes place between the Landau levels

within conduction or valence bands.

For monochromatic light the vector potential is


A = eA,,, -.(K r uj) eire-wt + e -iKr eit (3-12)
2 2

where K is the electromagnetic wave vector, w is the optical angular frequency, p

is the momentum operator, and e is the unit polarization vector in the direction of

the optical field, representing the light configuration.

The energy flux of the optical field can be expressed by the Poynting vector,

S E x H. Using the relations E = -OA/Ot, H V x S/p, and u/K c/n,, the

averaged energy flux then is
n, 2A2
S = rnA (313)
2/ic
Using this relation and Eq. 3-8, the absorption coefficient then is

hwT hw 1 2r
a() (nw2A/2pc) v IH I26 (EF E ,- hw)(f, f). (3 14)
i,f

According to Eq. 3-12, the interaction Hamiltonian can be written as

eAo eAo
H "f 0(f le pi) Pfi, (315)
2mo 2mo

so the absorption coefficient 3-16 becomes

a(w) pif 26 (Ef E hw)(f, ff). (3 16)
,icf










Note that the interaction 3-15 is based on the dipole approximation. So in the

following when we talk about selection rules, etc, they are electric dipole selection

rules.

The scattering broadening (as well as disorder) can be parameterized by the

linewidth F through the replacement of the 6 function by a Lorentzian function [72]

as

6(Ef E w) 2 (3-17)
(Ef E, h)2 + (F/2)2

3.1.2 Cyclotron Resonance

From a classical mechanical point of view, in the presence of a magnetic field,

an electron moves along the field direction in a spiral, whose projection in the

perpendicular plane is a circle. The angular frequency for this circular motion is

eB
c = e (3-18)
mo

where mo is the free electron mass (effective mass when in a semiconductor). If

an electromagnetic wave is applied with the same frequency, the electron will

resonantly absorb this electromagnetic wave.

Quantum mechanically, an electron in a magnetic field will have a quantized

motion. Referring to Eq. 2-62, the energy of the electron splits into a series of

Landau levels. If the energy quanta hk of the applied electromagnetic wave are

exactly the same as the energy difference hkc between two .il1i ient Landau levels,

the electron will absorb one photon to transit from the lower Landau level to the

higher one. This is called cyclotron resonance.

In the presence of a magnetic field, the Hamiltonian 3-10, in DMS system,

is replaced by the one in Eq. 2-79. We already have the eigenstates for this










Hamiltonian. For convenience, we rewrite them here as


al,n,v On-l l

a2,n,v ,' -2U2

a3,n,v QnU3

ei(kyy+kzz) a4,n,v OnU4
nv = -(319)
VIA a,, .

a6,n,v Qn+lU6

a7,n,v in-1U7

a8,n,v -18

The eigenfunction above can be considered as the linear superposition of eight

basis states, each of which is composed of two parts. '. is the harmonic oscillator

envelope function, which is slowly varying over the lattice, and can be considered

constant over a unit cell length scale. ui is the Bloch part of the wave function,

which varies rapidly over a unit cell and has the periodicity of the lattice.

Now let us inspect the properties of the momentum matrix element in Eq. 3

16. Using n, v as the new set of quantum numbers, and utilizing the spatial

properties of the wave functions, we can factorize the integral into two parts and

write the matrix element as


Pn"' ax ((upu)( |. + (uju,)( p| ')). (3-20)
i,i'

Since the Bloch functions ui are quickly varying functions, their gradients are much

larger than those of the envelopes 0i. As shown in Ref. [73], the first term on the

right hand side dominates both in narrow gap and wide gap semiconductors, so we

have neglected the second term in our calculation. However, it is easy to check that

these two terms obey the same selection rules.









We can factorize e-p to e-p = +ep_ + e_p+ + ep where e (x iy)/V2, and

p = (p, ipy)/ /2. In the Fa ,-d'v configuration (light incident along the magnetic
field B), the circularly polarized light can be represented by unit polarization
vector e. In this case, we only need to consider the matrix elements of p. It is

easy to check that

(n, v p+ n', v') oc 6,_1,n/ (3-21)

and

(n, vlp_ n', v') oc 6,i,/. (3-22)

This means that p+ and p_ are raising and lowering operators for the eigenstates.

For p+, an electron will absorb an e_ photon to have an n --- n+1 transition, which

usually happens in the conduction band for electrons, so we call this transition
"electron- I ., (e-active). For p_, an electron will absorb an e+ photon to have

an n n 1 transition, which usually happens in the valence bands for holes,
so we call this transition "hole- .. I ,ii (h-active). The quasi-classical picture for

the two types of absorption is shown in Fig. 3-1. To comply with conservation of


BB






electron orbit hole orbit

Figure 3-1: Quasi-classical pictures of e-active and h-active photon absorption.


both energy and angular momentum, in a quasi-classical picture, electrons can only

absorb photons with e-active polarization, and holes can only absorb photons with
h-active polarization. In a quantum mechanical treatment, we will see that the









true situation is more complicated than this. In particular, we find that e-active

absorption can also take place in p-type materials.

When the temperature is not zero, EF in Eq. 3-5 should be understood

as the chemical potential, which we still call the Fermi energy, and depends on

temperature and doping. If ND is the donor concentration andNA the acceptor

concentration, then the net donor concentration No = ND NA can be either

positive or negative depending on whether the sample is n or p type. For a fixed

temperature and Fermi level, the net donor concentration is

1 of
Nc = ( 2Y dk, [f,,(k,) (3-23)


where 6,, = 1 if the subband (n, v) is a valence band and vanishes if (n, v) is a

conduction band. Given the net donor concentration and the temperature, the

Fermi energy can be found from Eq. 3-23 using a root finding routine.

3.2 Ultrahigh Magnetic Field Techniques

Since the mobility of a ferromagnetic III-V DMS is generally low, using ultra-

high magnetic fields exceeding 100 T (megagauss field) is essential for the present

study in order to satisfy the CR condition wjr > 1, where uc is the cyclotron

frequency and 7 is the scattering time [74, 75]. The megagauss experiments have

been done at the university of Tokyo where high magnetic fields can be generated

using two kinds of pulsed magnets: the single-turn coil technique [76, 77] and the

electromagnetic flux compression method [77, 78]. The single-turn coil method

can generate 250 T without any sample damage and thus measurements can be

repeated on the same sample under the same experimental conditions. The idea be-

hind this method is to release a big current in a very short period of time (several

pis) to the single-turn coil to generate an ultrahigh magnetic field. The core part of

a real single-turn coil device is demonstrated in Fig. 3-2 [76]. Although the sample

is intact, the coil is damaged after each shot. A standard coil is shown in Fig. 3-3




























Figure 3-2: The core part of the device based on single-coil method. The coil is
placed in the clamping mechanism as seen in the figure. The domed steel cylin-
ders on each side of the coil are supports for the sample holders which protect the
connection to the sample(e.g., thin wires, helium pipes) against the lateral blast.

before and after a shot. Depending on the coil dimension, each shot generates a

pulsed magnetic field up to 250 T in several ps. The time dependence of the pulsed

magnetic field and of the current flowing through the coil is shown in Fig. 3-4 [76].


For higher field experiments an electromagnetic flux compression method is

used. It uses the implosive method to compress the electromagnetic flux so as to

generate ultrahigh magnetic fields up to 600 T. The time dependence of the pulsed

magnetic field and current is shown in Fig. 3-5 [77]. This is a destructive method

and the sample as well as the magnet is destroyed in each shot.

3.3 Electron Cyclotron Resonance

3.3.1 Electron Cyclotron Resonance

According to the discussion in Section 3.1.2, for e-active cyclotron resonance,

the light polarization vector is e_ = (x iy)/ V2, corresponding to momentum

operator p+ = (p, + ipy)//2. This operator will result in an n -- n + 1 transition.

























Figure 3-3: A standard coil before and after a shot.


300


200


100


0


0 5 10 15
Time(gs)


Figure 3-4: Waveforms of the
in single-turn coil device.


magnetic field B and the current I in a typical shot











-4

500 I
3
400

300- 2
OD 2 :

200

100

0 0

0 10 20 30 40 50 60
Time (gs)

Figure 3-5: Waveforms of the magnetic field B and the current I in a typical flux
compression device.


In the conduction band, the Landau subbands are usually aligned in such a way

that energy ascends with quantum number n. So for an e-active transition, both

angular momentum and energy for an electron-photon system can be conserved.

Our collaborators Kono et al. [74] measured the electron active cyclotron

resonance in InMnAs films with different Mn concentrations. The films were grown

by low temperature molecular beam epitaxy on semi-insulating GaAs substrates

at 200 oC. All the samples were n type and did not show ferromagnetism for

temperatures as low as 1.5 K. The electron densities and mobilities deduced from

Hall measurements are listed in Table 3-1, together with the electron cyclotron

masses obtained at a photon energy of 117 meV (or a wavelength of 10.6 pm).

Typical measured CR spectra at 30 K and 290 K are shown in the left and

right panel of Fig. 3-6, respectively. Note that to compare the transmission with

absorption calculations, the transmission increases in the negative y direction.

Each figure shows spectra for all four samples labeled by the corresponding

Mn compositions from 0 to 1"' All the samples show pronounced absorption









Table 3-1: Parameters for samples used in e-active CR experiments

Mn content x 0 0.025 0.050 0.120
Density (4.2 K) 1.0 x 1017 1.0 x 1016 0.9 x 1016 1.0 x 1016
Density (290 K) 1.0 x 1017 2.1 x 1017 1.8 x 1017 7.0 x 1016
Mobility (4.2 K) 4000 1300 1200 450
Mobility (290 K) 4000 400 375 450
m/mo (30 K) 0.0342 0.0303 0.0274 0.0263
m/mo (290 K) 0.0341 0.0334 0.0325 0.0272


peaks (or transmission dips) and the resonance field decreases with increasing

x. Increasing x from 0 to 1'"-. results in a '",. decrease in cyclotron mass (see

Table 3-1). At high temperatures [e.g., Fig. 3-6(b)] the x = 0 sample clearly

shows nonparabolicity-induced CR spin splitting with the weaker (stronger) peak

originating from the lowest spin-down (spin-up) Landau level, while the other three

samples do not show such splitting. The absence of splitting in the Mn-doped

samples can be accounted for by their low mobilities (which lead to substantial

broadening) and large effective g factors induced by the Mn ions. In samples

with large x, only the spin-down level is substantially thermally populated (see

Fig. 2-5).

Using the Hamiltonian described in Section 2.3.2, the wave functions in Section

2.3.3, and the techniques for calculating Fermi energy, the several lowest Landau

levels in the conduction band at two Mn concentrations and the Fermi energy

for two electron densities (1 x 1016/cm3 and 1 x 1018/cm3) are calculated. The

conduction band Landau levels and the Fermi energies are shown in Fig. 3-7 as

a function of magnetic field at T = 30 K. From these figures, we can see that at

resonance, the densities and fields are such that only the lowest Landau level for

each spin type is occupied for typical densities listed in Table 3-1. Thus, all the

electrons were in the lowest Landau level for a given spin even at room temperature

due to the large Landau splitting, precluding any density-dependent mass due










0.99 a) 12% 0.982- J) 12%

:0 11.00-

0.96 0.92
S0.98 5%-0.96- 5%


0.92- 0.88-
0.96- y2.5%- 0.94- %

1.00-
0.20 0.20 -
0.60- 0%- 0.60 2 0%
30 K 290 K
1.00 1_ 31_ 1.002
0 40 80 0 40 80
B (T) B (T)

Figure 3-6: Experimental electron CR spectra for different Mn concentrations x
taken at (a) 30 K and (b) 290 K. The wavelength of the laser was fixed at 10.6pm
with e-active circular polarization while the magnetic field B was swept.

to nonparabolicity (expected at zero or low magnetic fields) as the cause of the
observed trend.
The cyclotron resonance takes place when the energy difference between
two Landau levels with the same spin is identical to the incident photon energy.
In Fig. 3-8, we simulate cyclotron resonance experiments in n-type InAs for e-

active circularly polarized light with photon energy h = 0.117eV. We assume a
temperature T = 30 K and a carrier concentration n = 1016/ cm3. The lower panel
of Fig. 3-8 shows the four lowest zone-center Landau conduction-subband energies

and the Fermi energy as functions of the applied magnetic field. The transition at
the resonance energy hu = 0.117eV is a spin-up An = 1 transition and is indicated
by the vertical line. From the Landau level diagram the resonance magnetic field is
found to be B = 34 T. The upper panel of Fig. 3-8 shows the resulting cyclotron









0.8
(0) x = 0%
0.7
> T = 30 K

8 0.6
S10'8
Lii
0.5

0.4
0 20 40 60
B (Teslo)
0 .8 1' I I '
(b) x = 12% s :
0.7
> T = 30 K

o 0.6 -101'




0.4
0 20 40 60
B (Tesla)

Figure 3-7: Zone-center Landau conduction-subband energies at T =30 K as func-
tions of magnetic field in n-doped In i_ l,.,As for = 0 and x = 1'-. Solid lines
are spin-up and dashed lines are spin-down levels. The Fermi energies are shown as
dotted lines for n 1016/ cm3 and n 1018/ cm3.

resonance absorption assuming a FWHM linewidth of 4 meV. There is only one

resonance line in the cyclotron absorption because only the ground-state Landau

level is occupied at low electron densities. For higher electron densities, more

Landau levels are occupied. For example, if both spin-up and spin-down states of

the first Landau level are occupied, one obtains multiple resonance peaks.

Our simulation of the experimental e-active cyclotron resonance in the

conduction band shown in Fig. 3-6 is shown in Fig. 3-9. The left and right panel

demonstrate the calculated cyclotron resonance absorption coefficient for e-active









4 .. II "IIII.IIIII.I.IIII 111111111
1016 cm-3 x = 0%
3 30 K
E = e-active
2 0.117 eV
0 2




-----
r 1 -







t 0.6


0 .4 ... .. .. .. .. .. .... .
c 0.5


0 .4 _,, ,, ,1. ..,, ,, ,, ,, ,I ,, ,, ,
20 30 40 50 60
B (Teslo)

Figure 3-8: Electron CR and the corresponding transitions. The upper panel shows
the resonance peak and the lower panel shows the lowest four Landau levels with
spin-up states indicated by solid lines and spin-down states indicated by dashed
lines. Vertical solid line in the lower panel indicates the transition accountable for
the resonance.

circularly polarized 10.6pm light in the Faraday configuration as a function of

magnetic field at 30 K and 290 K, respectively. In the calculation, the curves

were broadened based on the mobilities of the samples. The broadening used for

T =30 K was 4 meV for ( 40 meV for 2.5' 40 meV for 5' and 80 meV for

12'. For T = 290 K, the broadening used was 4 meV for 0' 80 meV for 2.5' ,

80 meV for 5' and 80 meV for 12'"- At T = 30 K, we see a shift in the CR peak

as a function of doping in agreement with Fig. 3-6(a). For T = 290 K, we see

the presence of two peaks in the pure InAs sample. The second peak originates

from the thermal population of the lowest spin-down Landau level. The peak does











"30K 8 -290

6 12% 6 12%


= 5%
4- 4 5%

SI
2 .5
2- 25 2-
2.5%
xlOo 5 x5"
0% 0 0% 0o 0%

0 20 40 60 80 100 0 20 40 60 80 100
B(Tesla) B ( Tesla)

Figure 3-9: Calculated electron CR absorption as a function of magnetic field at
30 K and 290 K. The curves are calculated based on generalized Pidgeon-Brown
model and Fermi's golden rule for absorption. They are broadened based on the
mobilities reported in Table 3-1.

not shift as much with doping as it did at low temperature. This results from the
temperature dependence of the average Mn spin. We believe that the Brillouin
function used for calculating the average Mn spin becomes inadequate at large x
and/or high temperature due to its neglect of Mn-Mn interactions such as pairing
and clustering.
The e-active CR shows a shift with increasing Mn concentration. From the
simple theory in Section 1.2.2, the cyclotron resonance field does not depend on x
and a because the exchange interaction will shift all levels by the same amount.
This shift comes from the complicated conduction-valence band mixing, and
depends on the value of (a 3) [79]. We can qualitatively explain this shift using
the cyclotron mass, which will be discussed in the following subsection.









The CR peaks shown in Fig. 3-9 are highly ..i-mmetric. This is because

we have taken into account the finite kz effect in our calculation, and the energy

dispersion along kz shows high nonparabolicity. Also, the carrier filling effect due to

the Fermi energy sharpening will also contribute to the CR peak ..i-iin.i. I ry.

3.3.2 Electron Cyclotron Mass

The electron cyclotron mass mcR for a given cyclotron absorption transition is

related to the resonance field B* and photon energy he by the definition

mcR 2PBB*
(3-24)
mo hw

This equation can be derived from Eq. 3-18 if we set magnetic field B so that

hkw = hw, which is the cyclotron resonance condition.

The calculated cyclotron masses for the lowest spin-down and spin-up tran-

sitions are plotted in Fig. 3-10 as a function of Mn concentration x at a photon

energy of hw = 0.117 eV. Cyclotron masses are computed for several sets of a and

3 values. The cyclotron masses in Fig. 3-10(a) and (b) correspond to the computed

cyclotron absorption spectra shown in Fig. 3-9 (a) and (b), respectively. In our

model, the electron cyclotron masses depend on the Landau subband energies and

photon energies and are independent of electron concentration.

Figure 3-10 clearly shows that the cyclotron mass depends on both exchange

constants and x. With increasing x, spin-down (spin-up) cyclotron mass show

almost a linear decrease (increase). The cyclotron mass does not depend on one

single exchange constant, it depends on both exchange constants. Investigation

of the mass dependence on these two constants reveals the mass shift has a close

relation with the absolute value of (a f) [79]. This shift allows use to measure

the exchange interaction.

The calculated cyclotron mass has taken into account all the energy depen-

dence on nonparabolicity due to the conduction-valence band mixing, the exchange



















5.0



4.5



4.0


0 2 4 6 8
Mn concentration, x (%)


down
3 .6.. ....... .



3.4- ......

...... ....l U P. .


0 2 4 6 8
Mn concentration, x (%)


10 12


10 12


Figure 3-10: Calculated electron cyclotron masses for the lowest-lying spin-up and
spin-down Landau transitions in n-type In-, MnAs with photon energy 0.117 eV
as a function of Mn concentration for T = 30 K and T = 290 K. Electron cyclotron
masses are shown for three sets of a and 03 values.


I I I I I I I I I


-0.5, = 1.3 E = 0.117 eV
photon
-0.5, = 1.0
-0.3, = 1.0 T = 290 K


'

-.-.-a=
a=
a=










interaction constants a and 3, and the Mn content x. The shift of the resonance

peaks to lower fields with increasing Mn content x is naturally explained by the

decrease of the spin-down cyclotron mass. Due to the smaller downward slope in

the spin-down cyclotron mass at 290 K as compared to 30 K, the resonance peak

shift at 290 K is seen to be less pronounced than at 30 K.

3.4 Hole Cyclotron Resonance

3.4.1 Hole Active Cyclotron Resonance

As shown in Fig. 2-4, the DMS valence band structure is much more com-

plicated than the conduction band structure. Due to their energetic proximity,

heavy hole and light hole bands are strongly mixed even near the F point. The

split-off band also contributes strongly to the valence band-edge wave functions. In

a magnetic field, these hole bands split into their own Landau levels, but optical

transitions can happen between any two levels if both angular momentum and

energy are conserved. As in the conduction band, cyclotron resonance requires

conduction-valence band mixing to produce strong enough oscillator strength.

Interband mixing across the band gap is small in wide-gap semiconductors, so it is

more difficult to observe cyclotron resonance in these semiconductors. As a matter

of fact, no cyclotron resonance has been reported to date in GaMnAs.

InAs and InMnAs are narrow-gap semiconductors. Our collaborators [80, 81,

82] have performed cyclotron resonance experiments on p-doped InAs and InMnAs

at ultrahigh magnetic fields up to 500 T. The typical h-active CR absorption of

InAs below 150 T is shown in Fig. 3-11, in which the incident light is h-active

circularly polarized with photon energy 0.117 eV. Two peaks are present in the

experimental observation, one around 40 T, and another around 125 T. At even

lower fields, there is a background absorption. The theoretical simulation using a

hole density of 1 x 1019/cm3 and a broadening factor of 40 meV is also di-pl i' 1 in

Fig. 3-11 for comparison.













Expt. 27.5KI
4 L \X=M

u 3 h-active
b E0.117eV
S2 Theory 30K -




0 50 100 150
B Tesla)


Figure 3-11: Hole cyclotron absorption as a function of magnetic field in p-type
InAs for h-active circularly polarized light with photon energy 0.117 eV. The up-
per curve is experimentally observed result and the lower one is from theoretical
calculation.

In our model, we are capable of calculating the absorption between any two

Landau levels. Detailed calculation reveals that the peak at lower fields is due

to the heavy-hole to heavy-hole transition, and the peak at higher fields is from

the light-hole to light-hole transition. We now use H,,, to specify the heavy hole

level, and L,,,, to specify the light hole level, where (n, v) are the quantum numbers

defined in Eq. 3-19. Because of strong wave mixing H or L only labels the zone

center (k = 0) character of a Landau level. Using these labels, we illustrate

the two-state absorption in Fig. 3-12 along with the Landau level structure as a

function of magnetic field.

It is seen from Fig. 3-12 that the holes optically excited from the heavy hole

subband H_1,1 and light hole subband L0,3 give rise to the two strong cyclotron

absorption peaks shown in Fig. 3-11. The cyclotron absorption peak around 40 T

is due to a transition between the spin-down ground state heavy hole Landau level

H_1,1, and heavy hole Landau level Ho,2, which near the zone center is primarily










101
Total (a)

C- 6 I
c6

(0 4





0 .1 1 1I I I -

(b)
H_c Ho .


CM -0H
1.1 0- L L -

-0.2


-0.2 .. .... 14



0 50 100 150 200
B (Teslo)

Figure 3-12: Calculated cyclotron absorption only from the H-1,1 H,2 and
Lo,3 L1,4 transitions broadened with 40 meV (a), and zone center Landau levels
responsible for the transitions (b).


spin-down. The other absorption peak around 140 T, is a spin-down light hole

transition between L0,3 and L1,4 Landau levels. The background absorption at

B < 30 T is due to the absorption between higher Landau levels which also become

occupied by holes at lower fields.

Cyclotron resonance absorption measurements on Inl_-, li,.As with x = 2.5'.

have also been performed. They are shown in Fig. 3-13 along with our theoretical

simulation. The CR measurements were made at temperatures of 17, 46, and

70 K in h-active circularly polarized light with photon energy he = 0.224 eV.

In our simulation, the hole density is taken as 5 x 1018/ cm3, and the curves are










1.0 I
SExperiment
0.8-
x=2.5%
0.6 70 K

z 0.4 46 K

0.2 0.224 eV
17 K h-active
0.0

10
Theory
8 p=5x10M cm-3
x = 2.5%
_6 6
ad 70 K
t 4
46 K
2 0.224 eV
17 K h-active
0 I I .
-50 0 50 100 150
Magnetic Field (T)

Figure 3-13: Experimental hole CR and corresponding theoretical simulations.
The low temperature CR has an abrupt cutoff at low fields due to the fermi level
sharpening effect.

broadened using a FWHM linewidth of 120 meV. Clearly the absorption peak is

due to the heavy hole transition which we have seen in Fig. 3-11 and Fig. 3-12.

Due to the higher photon energy, this peak shifts from around 40 T to around

85 T. The resonance field is insensitive to temperature and the line shape is

strongly .,-viii i I ic with a broad tail at low fields. This broad tail again comes

from the higher order transitions resonant at low fields. We see that in both

experiment and theory at low temperature and low field, there is a sharp cutoff of










the absorption. This can be attributed to the sharpness of the Fermi distribution

at low temperatures.

Figure 3-14 shows the observed CR peaks as a function of magnetic field. The

y-axis indicates the photon energies used when observing the cyclotron resonance.

The solid curves show the calculated resonance positions. The curve labeled

'HH' ('LH') is just the resonance energy between Landau levels H_1,1 (Lo,3)and

Ho,2 (L1,4). The theoretical calculation shows an overall consistency with the

experiments.


0.4
o InAs
SHH / x = 0.0015
0.3 o x = 0.006
>. v x = 0.025

>, 0.2
2D LH

) 00
,c
0.1



0.0
0 100 200 300 400 500

Magnetic Field (T)

Figure 3-14: Observed hole CR peak positions for four samples with different Mn
concentrations. The solid curves are theoretical calculations.


There are two factors in our calculation that affect the results. One is the

selection of Luttinger parameters, the other is the limitation of the eight-band

effective mass theory itself. In Fig. 3-11, the theoretically computed peak at

higher fields does not fit the experimental peak exactly. Due to the fact that this

transition takes place at the zone center, where the k p theory should be very

accurate, this deviation may be the result of unoptimized Luttinger parameters.








79

The empirical parameters used in the effective mass Hamiltonian can drastically

change the valence band structure and the resulting CR absorption spectra. Fig. 3

15 shows the dependence of the CR energies on several parameters such as the

Luttinger parameters 71, 72, 73, Kane's parameter Ep and the effective electron

mass m*. This figure reveals that the 'LH' transitions are affected more by small

variations in these parameters than the 'HH' transitions. For instance, a 10''.

change in 71 will result in a ~ 0.025 eV change at B = 140 T in the LH CR energy,

which in turn will result in about a 50 T CR position shift in the resonance field

when the photon energy is 0.117 eV. The Mn doping on the other hand generally

enhances the CR energy dependence on these parameters, which can be seen from

comparing the two graphs in Fig. 3-15.


HH
0.2- T = 30 K LH









HH ~LH
0.2 T=30K 0.2 T=30K K





=5% x=5%
.. ...... ..... ..>.x
-. Y 3-.-----











0.0 ._0.0_
U E .-. P







-0.2 (a) -0.2 (b) ..
HH LH




0 5= 30 100 150 0 50 100 150
x=5% x=5% 71--.-

-0.0 (b)....2.(b)........,..,,....





Magnetic Field (T) Magnetic Field (T)

Figure 3-15: The dependence of cyclotron energies on several parameters. Left
panel shows the heavy hole CR energy dependence, and the right panel shows the
light hole CR energy dependence.


Figure 3-16 illustrates how the CR absorption depends on three Luttinger

parameters while keeping all the other parameters unchanged. It can be seen that










the CR spectra quite sensitively depends on the values of the Luttinger parameters,

providing an effective way to measure these parameters through comparison with

experiments.

10 T = 30K
10 =21, y2=8.5, y3--9.2 1x101cmK

8-

6




2 ,=20, y2=8.5, y3=9.2

0
0 50 100 150 200
Magnetic Field (T)

Figure 3-16: Hole CR spectra of InAs using different sets of Luttinger parame-
ters. Light hole transition is more significantly affected by change of the Luttinger
parameters.


In Fig. 3-14, there is one peak around 450 T labeled as 'C' when the light

energy is h = 0.117 eV. To account for this peak, CR absorption spectra up to

500 T have been computed. The k = 0 Landau levels as a function of magnetic

field, along with the CR spectra are plotted in Fig. 3-17, in which we can see

that this peak is due to the superposition of two transitions: L1,5 L2,5 and

H2,6 H3,6. However, the calculated peak position is around 360 T, different

from the experiment. There are two possible reasons for this big deviation. One

is that at very high magnetic fields, the eight-band Pidgeon-Brown model may

break down; the other is that transitions contributing to the peak take place away

from the zone center where eight-band k p theory is not adequate to describe the

energy dispersion. The band structure along k, is plotted at Fig. 3-18, where we

see that the Landau levels H2,6 and H3,6 both have camel back structures. At a

hole density p = 1 x 1019/ cm3, the zone center part of H2,6 is not occupied. The











0.1
(a) T=20 K p=1x1019 cm-
0.0 h-active ,3) E =0.117 eV
00 ^ (0,3) photon
(1, 5)
.? -0.1

0) -0.2
-0.2 (2,5) -
W (3,6
InAs
-0.3

10
(b)
HH LH
8 HH LH T=20 K
: p=1x109 cm-3
c 6 Ephoton=0.117eV
c h-active
5. InAs
0 c
|n 2-



0 100 200 300 400 500
Magnetic Field (T)

Figure 3-17: Calculated Landau levels and hole CR in magnetic fields up to 500
T. The upper panel shows the k = 0 valence band Landau levels as a function of
magnetic field and the Fermi level for p = 1019 cm-3 (dashed line). The hole CR
absorption in p-type InAs is shown in the lower panel for h-actively polarized light
with u = 0.117 eV at T 20 K and p 1019 cm-3. A FWHM linewidth of 4 meV
is assumed.


lowest energy for this heavy hole Landau level resides at about k, = 0.75(1/nm).

C('!. L:.ig the transition element along k, it is also found that this transition indeed

takes place away from the zone center. Di- ,' 't1 in Fig. 3-19 is the comparison of

the eight-band model versus a full-zone thirty-band model. At the zone center, the

eight-band model fits well with the thirty-band model. Not far away from the zone

center, a big deviation occurs. We think this deviation of the energy dispersion

is possibly responsible for the large deviation of the calculated resonance peak

position.


















-0.1



-0.2


-0.3

-1.0


-0.5 0.0 0.5


K (1/nm)


Figure 3-18: k-dependent Landau subband structure at B


-0.5


350 T.


L A K A X

Figure 3-19: Band structure near the F point for InAs calculated by eight-band
model and full zone thirty-band model.









3.4.2 Hole Density Dependence of Hole Cyclotron Resonance

Cyclotron resonance depends on Fermi energy through Eq. 3-16, thus CR

spectra depend strongly on carrier densities. In Fig. 3-12 the hole density is

1 x 1019 cm-3. At such a hole density, the Fermi energy is below the H-1,1 and

L0,3 states so that we have two strong transitions. If the hole density is lower, the

Fermi energy will shift upward, thus these two states will become less occupied by

holes, and we can expect a decrease in the CR strength. However, the decrease

in strength for the two resonance peaks is different. Shown in Fig. 3-20(a) are

the CR spectra for four different hole densities. The Landau levels along with the

corresponding Fermi energies are plotted in Fig. 3-20(b). Resonant transitions at

0.117 eV are indicated by vertical lines. We can see that the CR peak 2 is almost

ah--,i- present, because at low magnetic fields, the heavy hole state H-1,1 is almost

ah--,v- occupied. The CR peak 1 changes dramatically with hole density, and

nearly vanishes at p = 5 x 1018 cm-3. The relative strengths of the heavy and

light-hole CR peaks is sensitive to the itinerant hole density and can be used to

determine the hole density. By comparing theoretical and experimental curves in

Fig. 3-20(a), we see that the itinerant hole concentration is around 2 x 1019 cm-3

From Fig. 3-20(a), we can rule out p < 1019 cm-3 and n > 4 x 1019 cm-3. We

estimate that an error in the hole density of around 25'. should be achievable at

these densities. Because of the existence in III-V DMS of the anomalous Hall effect,

which can often make the determination of carrier density difficult, determining

carrier density by cyclotron resonance can serve as a possible alternative.

3.4.3 Cyclotron Resonance in InMnAs/GaSb Heterostructures

Hole CR in InMnAs/GaSb heterostructures has also been experimentally

studied by Kono et al. [83]. These samples are ferromagnetic with Tc ranging from

30 to 55 K and whose characteristics are summarized in Table 3-2.


















Expt. h-active (a)
4 27.5 K 0.117 eV

2' 2(\X=O%
Si 9 0%


S -'1 0 I \ *

1 ETheory

..3 0 K ".. .............. ................. ,


0.1
B (Tesl(b))
-1,1 0,2 L
0.0curves 03to top with hole densities of 5 x 9, 2 x


S-- -*..-- ..........
C-0.1 E
-0. w \ ~I Fermi



0 50 100 150 200
B (Tesla)

Figure 3 20: The hole density dependence of hole CR. (a) Theoretical hole CR
curves in InAs from bottom to top with hole densities of 5 x 1018s, 1019, 2 x 1019 and
4 x 1019 cm-3; (b) Landau levels involved in observed CR along with Fermi levels
corresponding to theoretical curves in (a).







85

Table 3-2: Characteristics of two InMnAs/GaSb heterostructure sam-
ples

Sample No. Tc(K) Mn content x Thickness (nm) Density(cm-3)
1 55 0.09 25 1.1 x 1019
2 30 0.12 9 4.8 x 1019


The experimentally observed CR transmission of a 10.6 pm laser beam

through these two samples (Tc = 55 and 30 K, respectively) are shown in Fig. 3

21(a) and (b), at various temperatures as a function of magnetic field. The laser

beam was hole-active circularly polarized. In the left panel of Fig. 3-21, from

room temperature down to slightly above Tc, a broad resonance feature (labeled

'A') is observed with almost no change in intensity, position, and width with

decreasing temperature. Close to Tc, quite abrupt and dramatic changes take place

in the spectra. First, a significant reduction in line width and a sudden shift to a

lower magnetic field occur simultaneously. Also, the resonance rapidly increases

in intensity with decreasing temperature. In addition, a second feature (labeled

'B') suddenly appears around 125 T, which also rapidly grows in intensity with

decreasing temperature and saturates, similar to feature A. At low temperatures,

both features A and B do not show any shift in position. Essentially, the same

behavior is seen in the right panel in Fig. 3-21. Using different wavelengths of the

incident light, similar CR spectrum behavior has also been observed.

For zinc-blende semiconductors, the CR peaks A and B are due to the tran-

sitions of H-1,1 Ho,2 and L0,3 L1,4, respectively, which we have already

pointed out. We attribute the temperature-dependent peak shift to the increase

in the carrier-Mn ion exchange interaction resulting from the increase of magnetic

ordering at low temperatures. The theoretically calculated results are shown in

Fig. 3-22 for bulk Ino.90 g!i,,.,As. The CR spectra was broadened using a FWHM

linewidth of 4 meV. The theoretical results clearly show a shift of peak A to lower










Sample 1 Sample 2

RT RT

152 K 140 K
107 K 107 K
85 K --" 80 K
68 K 60 K


,42 K 30 K

25 K ,21 K

13K
15K 13K

A B

0 50 100 150 0 50 100 150

Figure 3-21: Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSb
samples. The transmission of hole-active circular polarized 10.6 pm radiation is
plotted vs. magnetic field at different temperatures.

fields with decreasing temperature, although in bulk InAs, the transition occurs

at about 40 T, as opposed to the heterostructure where the resonance occurs at

S50 T.

The CR peak A only involves the lowest two Landau manifolds. As was

discussed in Section 2.3.3, when n = -1, the Hamiltonian is 1 x 1, and when n = 0,

the Hamiltonian factorizes into two 2 x 2 matrices, so it is easy to obtain an exact

analytical expression for the temperature dependent cyclotron energy. With neglect

of the small terms arising from the remote band contributions, the cyclotron energy










10 1

p=1xl019 cm-3 In1xMnxAs
8 Ephoton=0.117 eVx=9%
h-active TC=55 K

6- 6
=3 290 K


42 K
2 30 K- A=
'15 K
0 1 *
0 20 40 60

B (T)

Figure 3-22: Theoretical CR spectra showing the shift of peak A with temperature.

for the H_1,1 HO,2 transition is


ECR = + x(S)(a ) E/ x(S) a 4) 2 + EppBB. (3-25)

In the field range of interest (~ 40 T), IEp pBB is the same order as E,/2,
while the exchange interaction is much smaller even in the saturation limit.
Expanding the square root in Eq. 3-25, we obtain an expression of the form

ECR =E (- I) + x(S,)(a -3)(1 ) (3-26)

where
6 = E (3-27)
/E + 4EpPBB
If we assume the temperature dependence of E, and Ep is small, it follows
from Eq. 3-26 that the CR peak shift should follow the temperature dependence
of the magnetization (S,), which in a mean field theory framework is given by