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Magneto-Optical Properties of Narrow Gap Semiconductor Nanostructures

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Title:
Magneto-Optical Properties of Narrow Gap Semiconductor Nanostructures
Creator:
Saha, Dipta
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[Gainesville, Fla.]
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University of Florida
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english
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1 online resource (123 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
STANTON,CHRISTOPHER JAY
Committee Co-Chair:
DUFTY,JAMES W
Committee Members:
MUTTALIB,KHANDKER A
TANNER,DAVID B
BOWERS,CLIFFORD RUSSELL
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Charge carriers ( jstor )
Conduction bands ( jstor )
Electronic structure ( jstor )
Electrons ( jstor )
Landau levels ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnets ( jstor )
Semiconductors ( jstor )
Signals ( jstor )
Physics -- Dissertations, Academic -- UF
opnmr -- optical -- photonics -- polarization -- properties -- spin -- spintronics
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Physics thesis, Ph.D.

Notes

Abstract:
I have theoretically studied different experiments for a range of III-V nanostructures including: (1) time resolved differential transmission (TRDT) spectra in ferromagnetic InMnAs without magnetic field, (2) cyclotron resonance (CR) in ferromagnetic InMnAs and InMnSb, (3) optically pumped NMR (OPNMR) in strained AlGaAs/GaAs square multi quantum wells (MQW) and (4) magneto-absorption in strained AlInSb/InSb parabolic MQW. The calculations and modeling have helped explain interesting phenomena: 1) carrier relaxation and dynamics, 2) sign changes in the TRDT spectra, 3) higher Curie temperature of the MOVPE grown ferromagnetic samples, 4) the sensitivity and sign of the OPNMR signal, and 5) the effects of strain and confinement on the magneto-absorption. I have used an 8-band model modified to include sp-d coupling between electrons and holes and Mn impurities to calculate the band structure of the ferromagnetic InMnAs. To calculate the Landau subband structure for the other materials, I have used the 8-band modified Pidgeon Brown model. Fermi's golden rule is used to calculate the optical properties. The band structure calculations, along with the identification of the optical transitions, explain the carrier dynamics and relaxation mechanisms and the sign changes in the TRDT spectra as a function of probe wavelength. The calculations of CR spectra, valence Landau suband structure, and average z component of the spins explain the differences between the CR measurements that are observed in molecular beam epitaxy (MBE) and metal organic vapor phase epitaxy (MOVPE) samples, and the higher Curie temperature in the MOVPE structures. In the OPNMR experiments on AlGaAs/GaAs MQW, the sign change of the calculated electron spin polarization agrees with the sign change of the OPNMR signal. This sign change has not been observed in bulk GaAs. The strain effect and density of states in the strained MQW are responsible for this sign change. The calculated magneto-absorption with biaxial strain effects included more accurately reproduces the experimental results of AlInSb/InSb parabolic MQW. Comparing the results with that of the AlInSb/InSb square MQW indicates that the shape of the confinement affects the magneto-absorption significantly. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: STANTON,CHRISTOPHER JAY.
Local:
Co-adviser: DUFTY,JAMES W.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-05-31
Statement of Responsibility:
by Dipta Saha.

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University of Florida
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University of Florida
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Copyright by Dipta Saha. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
5/31/2016
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MAGNETO-OPTICALPROPERTIESOFNARROWGAPSEMICONDUCTORNANOSTRUCTURESByDIPTASAHAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014DiptaSaha 2

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Idedicatethistomymother. 3

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ACKNOWLEDGMENTS Iamgratefultomyadvisorandmentor,ProfessorChristopherJ.Stanton,forthemostimportantrolehehasplayedasaguideinmyPhD.Itwouldbeimpossibleformetoworkonmultipleresearchtopicswithouthisproperguidanceandencouragement.Ilearnedalotoftechniquesfromhim,whichhelpedmetoanalyzethepropertiesofthesemiconductormaterialsmoreefciently.Hisinsightfulremarksalwayshelpedmechoosethecorrectpathtosolveanyproblem.Healwaysencouragedmetotackleanyresearchproblemindependently,whichIthinkwasagoodtrainingformyfutureaccomplishments.Workingwithhimistrulyaniceexperience.IwouldliketothankallthemembersofmyPhDsupervisorycommittee,ProfessorsJamesDufty,K.A.Muttalib,DavidTannerandRussbowersforservingonmycommittee.IwouldliketoacknowledgeProfessorDavidTannerforhisinspiringteaching.MyspecialthankstoProfessorRussBowersforusefuldiscussions.IamgratefultoDr.GarySandersforhisinsightfuldiscussionsandsuggestions,whichhelpedmetosolvemyresearchproblemsmoreefciently.Iwouldliketoacknowledgeallmyresearchcollaborators,ProfessorsGitiKhodaparast,RussBowers,andMikeSantos,forprovidingtheirinvaluableexperimentaldata.IwouldliketothankDr.X.Pan,formerstudentofourgroup.Myspecialthankstograduatestudent,RyanWoods,forusefuldiscussions.IwouldliketothankallthestaffsoftheHPCcenter.IamindebtedtoPamMarlin,KristinNicholaandBillieHermansenfortheiradministrativesupportthroughoutmygraduateprogram.FinallyIwouldliketoexpressmygratitudetomyparentsfortheirsupportinmyupbringingandeducation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2PERTURBATIONTHEORY ............................. 19 2.1KPMethod ................................... 19 2.2Spin-OrbitInteraction ............................. 20 2.3SolvingSchrodingerEquationinInniteDimensionalHamiltonian ..... 21 2.4Lowdin'sPerturbationTheory ......................... 22 3EIGHTBANDMODELFORIII-VBULKSEMICONDUCTORMATERIALS ... 26 3.1BasisStates: .................................. 26 3.2SixBandLuttingerModelandEightBandModel .............. 28 3.3HamiltonianMatrixofBulkIII-VSemiconductorMaterials ......... 29 3.4BandStructureCalculation .......................... 32 4CARRIERDYANAMICSINNARROWGAPFERROMAGNETICSEMICONDUCTORS ................................ 33 4.1ExperimentalDetails .............................. 33 4.2TheoryandModeling ............................. 34 4.2.1EffectiveMassHamiltonian ...................... 34 4.2.2CouplingtoSpontaneousMagnetization ............... 37 4.3ResultsandDissussion: ............................ 38 4.3.1CalculationofElectronicStructure .................. 38 4.3.2ContributionstoTRDTSpectra .................... 39 5ELECTRONICSTATESANDCYCLOTRONRESONANCEINNARROWGAPFERROMAGNETICSEMICONDUCTORS ..................... 44 5.1ExperimentalDetails .............................. 45 5.1.1Samples ................................. 45 5.1.2CRMeasurements ........................... 45 5.2TheoryandModeling ............................. 46 5.2.1EffectiveMassHamiltonian ...................... 46 5.2.2EnergiesandWavefunctions ..................... 50 5

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5.2.3Landaulevels,CRSpectra,andFermiEnergies ........... 51 5.3AverageZComponentsofSpins ....................... 54 5.4Results ..................................... 54 5.5Discussion ................................... 67 6SPINPOLARIZATIONANDOPTICALLYPUMPEDNMRINSTRAINEDMULTIQUANTUMWELLSYSTEM ............................. 70 6.1ExperimentalDetails .............................. 70 6.2TheoryandModeling ............................. 72 6.2.1EnvelopeFunctionFramework .................... 72 6.2.2EffectiveMassHamiltonian ...................... 73 6.2.3QuantumConnementPotential .................... 78 6.2.4ElasticStrainandQuadrupoleSplitting ................ 79 6.2.5EnvelopeFunctionsandWavefunctions ............... 83 6.2.6LandauSubbandEnergiesandEigenvectors ............ 84 6.2.7Magneto-OpticalAbsorptionandFermiEnergy ........... 85 6.2.8SpinPolarization ............................ 87 6.3NumericalCalculationofLandauSubbandStructure,Magneto-AbsorptionandSpinPolarization ............................. 87 6.4ResultsandDiscussion ............................ 88 7MAGNETO-ABSORPTIONINNARROWGAPPARABOLICMULTIQUANTUMWELL ......................................... 97 7.1ExperimentalDetails .............................. 97 7.2TheoryandModeling ............................. 98 7.2.1EffectiveMassHamiltonian ...................... 98 7.2.2QuantumConnementPotential .................... 99 7.2.3PseudomorphicStrainEffects ..................... 100 7.2.4EnvelopeFunctionsandWavefunctions ............... 101 7.2.5LandauSubbandEnergiesandEigenvectors ............ 101 7.2.6Magneto-OpticalAbsorptionandFermiEnergy ........... 101 7.3SimulationofLandauSubbandStructureandMagneto-Absorption .... 102 7.4Results ..................................... 103 7.5Discussion ................................... 104 8CONCLUSIONS ................................... 112 9FUTUREDIRECTIONSOFRESEARCH ..................... 117 REFERENCES ....................................... 120 BIOGRAPHICALSKETCH ................................ 123 6

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LISTOFTABLES Table page 5-1Characteristicsofthesamplesstudiedinthiswork. ............... 45 5-2MeasuredCRmassandmobilityfordifferentexcitationwavelengths. ...... 45 6-1Subbandlabelsandwavefunctionprobabilitiesfor+polarization ....... 89 6-2Labelsoftransitionsfor+polarization ...................... 89 6-3Subbandlabelsandwavefunctionprobabilitiesfor)]TJ /F1 11.955 Tf 12.62 0 Td[(polarization ....... 93 6-4Labelsoftransitionsfor)]TJ /F1 11.955 Tf 12.62 0 Td[(polarization ...................... 93 7

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LISTOFFIGURES Figure page 1-1CharacterizationTechniques ............................ 13 1-2Experimentalmeasurementsfordifferentnanostructures ............ 15 3-1Bandstructurenearthebandedge. ........................ 27 4-1TRDTmeasurementsinInMnAsfordifferentprobewavelengthsinMIR .... 35 4-2TRDTmeasurementsinInMnAsfortheprobewavelength2m. ........ 36 4-3ElectronicstructureofInMnAsandtransitionsforthepumpwavelength .... 40 4-4ElectronicstructureofInMnAsandtransitionsfordifferentprobewavelengths 42 5-1ExperimentalandcalculatedCRspectraforInMnAsandInMnSblms. .... 56 5-2Zone-centerLandauvalancesubbandstructuresinInMnAsandInMnSblmsfordifferentMnconcentrations ........................... 57 5-3CalculatedmanifoldresolvedCRspectraandLandauvalencesubbandstructureinInMnSbat10.7m. ............................... 58 5-4ValencebandstructureinInMnAsandInMnSbfordifferentMnconcentrations ............................................. 59 5-5CRspectraandLandauleveltransitionsat16.9m. ............... 61 5-6CRspectraandLandauleveltransitionsat5.53m. ............... 62 5-7CalculatedLandaulevelsandvalencebandstructureforInMnAs ........ 63 5-8MeasuredandcalculatedCRspectrafortheMOVPEgrownInMnSb ...... 64 5-9CalculatedLandauLevelsandvalencebandstructureforInMnAswith5.6%Mncontent ...................................... 65 5-10MeasuredandcalculatedCRspectrafortheMBEgrownInMnSb ........ 66 5-11CalculatedLandaulevelsandaveragespinfortwodifferentInMnSbsamples 68 6-1Banddiagraminatype-Iheterostructuresemiconductormaterials ....... 71 6-2OPNMRandcalculatedspinpolarizationinAl0.1Ga0.9As/GaAsSquareWellfor+polarizationand4.94T. ............................ 90 6-3OPNMRandcalculatedspinpolarizationinAl0.1Ga0.9As/GaAsSquareWellfor)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(polarizationand4.94T. ............................ 91 6-4ZonecenterLandausubbandstructuresforthesquarewell ........... 92 8

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7-1Thealuminumconcentrationasafunctionofpositionfortheparabolicquantumwell. .......................................... 106 7-2Parabolicwellabsorptionspectra .......................... 107 7-3AbsorptionspectrafortheparabolicwellforB=6T. ............... 108 7-4CalculatedLandausubbandstructurefortheparabolicwell ........... 109 7-5Calculatedmagneto-absorptionspectrafordifferentpolarization. ........ 110 7-6EigenfunctionsofthelowestlyingbandsfortheparabolicwellforB=6T. ... 111 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMAGNETO-OPTICALPROPERTIESOFNARROWGAPSEMICONDUCTORNANOSTRUCTURESByDiptaSahaMay2014Chair:ChristopherJ.StantonMajor:PhysicsIhavetheoreticallystudieddifferentexperimentsforarangeofIII-Vnanostructuresincluding:(1)timeresolveddifferentialtransmission(TRDT)spectrainferromagneticInMnAswithoutmagneticeld,(2)cyclotronresonance(CR)inferromagneticInMnAsandInMnSb,(3)opticallypumpedNMR(OPNMR)instrainedAlGaAs/GaAssquaremultiquantumwells(MQW)and(4)magneto-absorptioninstrainedAlInSb/InSbparabolicMQW.Thecalculationsandmodelinghavehelpedexplaininterestingphenomena:1)carrierrelaxationanddynamics,2)signchangesintheTRDTspectra,3)higherCurietemperatureoftheMOVPEgrownferromagneticsamples,4)thesensitivityandsignoftheOPNMRsignal,and5)theeffectsofstrainandconnementonthemagneto-absorption.Ihaveusedan8-bandmodelmodiedtoincludesp-dcouplingbetweenelectronsandholesandMnimpuritiestocalculatethebandstructureoftheferromagneticInMnAs.TocalculatetheLandausubbandstructurefortheothermaterials,Ihaveusedthe8-bandmodiedPidgeonBrownmodel.Fermisgoldenruleisusedtocalculatetheopticalproperties.Thebandstructurecalculations,alongwiththeidenticationoftheopticaltransitions,explainthecarrierdynamicsandrelaxationmechanismsandthesignchangesintheTRDTspectraasafunctionofprobewavelength. 10

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ThecalculationsofCRspectra,valenceLandausubandstructure,andaveragezcomponentofthespinsexplainthedifferencesbetweentheCRmeasurementsthatareobservedinmolecularbeamepitaxy(MBE)andmetalorganicvaporphaseepitaxy(MOVPE)samples,andthehigherCurietemperatureintheMOVPEstructures.IntheOPNMRexperimentsonAlGaAs/GaAsMQW,thesignchangeofthecalculatedelectronspinpolarizationagreeswiththesignchangeoftheOPNMRsignal.ThissignchangehasnotbeenobservedinbulkGaAs.ThestraineffectanddensityofstatesinthestrainedMQWareresponsibleforthissignchange.Thecalculatedmagneto-absorptionwithbiaxialstraineffectsincludedmoreaccuratelyreproducestheexperimentalresultsofAlInSb/InSbparabolicMQW.ComparingtheresultswiththatoftheAlInSb/InSbsquareMQWindicatesthattheshapeoftheconnementaffectsthemagneto-absorptionsignicantly. 11

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CHAPTER1INTRODUCTIONThefundamentalgoalsofspintronicsareunderstandingofcarrierandspindynamics,thegenerationofcarrierspinpolarization,andthespintransportinsemiconductors.Understandingthecarrierdynamicsinanysemiconductormaterialcanprovideinsightabouttheelectronicstructureandcarrierrelaxationmechanisms.Measurementsliketimeresolveddifferentialtransmission(TRDT)isanimportanttooltoprobethecarrierdynamicsinIII-Vsemiconductors.Thegenerationofcarrierspinpolarizationcanleadtoavarietyofnoveldevicesrangingfromspintransistors[ 1 ]tospin-polarizedLEDs[ 2 ]andlasers.InIII-Vsemiconductors,themostgeneralwaytocreatespinpolarizedcarriersistohavespin-splitbands.Thistypicallyoccursinthepresenceofanexternaleldorinferromagneticsemiconductorswithoutanyexternalmagneticeld.Anaccuratemeasurementoftheband-splittinginthepresenceofanexternalmagneticeldisveryimportantbecauseitwillallowonetodeterminetheg-factorsandeffectivemassesinbothbulksemiconductorsandheterostructures.Thisinformationcanbeimportantformodelingconventionalaswellasspintronicdeviceapplications.TheusualmeasurementtoolsusedtostudytheIII-Vbulkorheterostructurematerialsinthepresenceofanexternalmagneticeldarecyclotronresonance(CR),magneto-absorption,andopticallypumpedNMR(OPNMR).MeasurementtoolslikeTRDT,CR,Magneto-Absorption,andOPNMRwhencombinedwiththeorycanprovideinformationaboutthebandmixinginthebandstructure,averagespin,quantitativevaluesofelectronspinpolarization,andthevalenceandconductionbandsthatareresponsibleforanyopticaltransition.ThatiswhyIaminterestedinthetheoreticalstudiesoftheopticalormagneto-opticalpropertiesinthebulkandheterostructureIII-Vsemiconductormaterials.Iaminterestedintheopticalproperties(Fig. 1-1 )oftheIII-Vsemiconductormaterials,whicharedirectbandgapmaterials. 12

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Figure1-1. Differentcharacterizationtechniquesfornarrowandmidgapsemiconductors Therefore,Iaminterestedinasmallregionnearthebandedge()]TJ /F1 11.955 Tf 10.09 0 Td[(pointork0)wheretheopticaltransitionsoccur.Andthatiswhythekpperturbationschemeistheheartofthemodeling.MytheoreticalstudiesonIII-Vnarrowgapsemiconductors(NGS),aregenerallymotivatedbysomeinterestingpropertiesofNGSlikenarrowbandgap,smalleffectivemass,highelectronmobility,largeeffectiveg-factor,strongspin-orbitinteraction,etc.Becauseofthenarrowbandgapthesematerialsactiveintheinfraredregion,whichmakesthempotentialcandidatesforinfrareddetectorsandThzlaserdevices.Thesenarrowgapmaterialscanbeusedtobuildfasttransistorsbecauseoftheirhighelectronmobilities.Theyarepotentialcandidatesforspintronicdevicesbecauseoftheirlargeeffectiveg-factorandstrongspin-orbitinteraction.Moreover,someIII-Vnarrowgapferromagneticsemiconductors(NGFS)arealsostudied.Becauseoftheirferromagneticnature,whichisactuallyaresultofthetransferofmagnetizationofMnionstothecarriersthroughthespinexchangecouplingbetweenthes/pstateofthecarriersand 13

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thedstateoftheMnions,theNGFSmaterialshavehighereffectivegfactorthanthatoftheothernarrowgapsemiconductors(NGS)withoutMndoping.ThisuniquepropertygivestheNGFSmaterialsanextraadvantageoverNGSmaterialsintermsofspintronicdeviceapplications.ThatiswhyI'minterestedinthetheoreticalstudiesofsomecharacterizationtechniqueslikeTRDT,CR,andMagneto-absorptionforawiderangeofNGFSandNGSnanostructures,whichgivetheinformationaboutthecarrierandspindynamics,electronicstructures,magneto-opticalproperties,spin-splitbands,etc.However,theconventionalcharacterizationtechniquelikeMagneto-absorptionisalmostinsensitivetotheweakerlighthole(LH)transitionsinunstrainedbulkGaAsduetotheirverysmalleffectiveg-factor.ButthecharacterizationtechniquelikeOPNMRisverysensitivetotheLHtransitions,whichattractedmetomodeltheOPNMRexperimentsinGaAsquantumwells(QW).NowIwouldliketodiscussabouttheinterestingfeaturesoftheexperimentsthatspecicallymotivatedmetopursuethetheoreticalstudiesonthefollowingprojects:CarrierDynamicsinNGFS:TRDTexperimentusuallyprovidessomeimportantinformationaboutthecarrierdynamicsinanysemiconductormaterials.InTRDTspectroscopy,rst,apumppulsewithshorterwavelengthisusedatsometimet=0togeneratephotoexcitedcarriers.Thenaprobepulsewithlongerwavelengthisusedatsometimedelayt=.Thenexperimentallyanegativedifferentialtransmissionsignalismeasuredasafunctionofpump-probedelay.TRDTspectroscopyshowstwointerestingfeatureofcarrierdynamicsinInMnAs.First,thechangeinthenatureofthecarrierdynamicprocess,whichisshowninFig. 1-2 AandFig. 1-2 B.InFig. 1-2 A,thesignoftheTRDTsignalispositivefortheprobepulseswithlongerwavelengths.Ontheotherhand,inFig. 1-2 B,thesignoftheTRDTsignalbecomesnegativefortheprobepulsewithrelativelyshorterwavelength.Second,theobservedslowercarrierrelaxationtime(10ps),whichisshowninFig. 1-2 B,whichismuchslowerthanthetypicalcarrierrelaxationtime(2ps).Thesetwointriguingfeaturesdraggedmetopursuethe 14

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Figure1-2. Experimentalmeasurementsfordifferentnanostructures.A)TRDTmeasurementsatlongerwavelength.B)TRDTmeasurementsatshorterwavelength.C)MeasuredCRspectraforMOVPEandMBEgrownsamples.D)OPNMRsignalforthebulkGaAs.E)OPNMRsignalfortheALGaAs/GaAsquantumwell.F)Magneto-absorptionspectrafortheInSb/AlInSbparabolicquantumwellatdifferentmagneticelds 15

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theoreticalstudiesthatIamgoingtopresentinthisdissertation.ElectronicStatesandCRinNGFS:CRmeasurementsshowtwosignicantdifferencesbetweenthemetal-organicvaporphaseepitaxy(MOVPE)andmolecularbeamepitaxy(MBE)grownp-dopedInMnAs/InMnSbsamples.First,thedifferenceintheCRpeakpositionsbetweentheMOVPEandMBEgrownsamples,whichisshowninFig. 1-2 C.Second,theMOVPEsampleshavesignicantlyhigherCurietemperaturethanthatoftheMBEsamples.ThesetwointerestingfeaturesmotivatemetopursuethetheoreticalstudiestounderstandthecharacteristicsofCRspectraindifferentsamplesandthehigherCurietemperatureintheMOVPEsamples.SpinPolarizationandOPNMRinStrainedMulti-QuantumWell(MQW)system:InOPNMR,eitherrightorleftcircularlypolarizedphotonsareusedtoexcitetheelectronstotheconductionbandssothatthedesiredpolarizationoftheconductionelectronscanbeobtained.Thentheseelectronswillcoupleweaklywithnucleibyhypernecoupling.Thenaftercross-relaxation,thenucliewillbepolarizedwhicheventuallyenhancestheintensityoftheNMRsignals.TheOPNMRsignalintensityisdirectlyproportionaltotheelectronspinpolarization.ThemostinterestingfeaturewhichmotivatedmetocalculatetheSpinpolarizationinstrainedMQWisthatthesignchangeintheOPNMRsignaloccursnearlightholetransitionsinquantumwells(Fig. 1-2 E),whichwasnotseenintheunstrainedbulkGaAs(Fig. 1-2 D).Magneto-absorptioninNarrowGapParabolicMultiQuantumWell:Thespinsplittingisobservedrelativelyatalowermagneticeldinthemagneto-absorptionspectra(Fig. 1-2 F)oftheInSb/AlInSbparabolicquantumwell,whichcanbeattributedtothelargeeffectiveg-factorinInSb.Inmydissertation,themagneto-absorptioniscalculatedincludingtheeffectsofpseudomorphicstrainandparabolicquantumconnementtoverifytheimportanceofstrainandconnementontheLandausubbandstructures.Therestofthechaptersarearrangedinthefollowingway: 16

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InChapter 2 ,Idiscusssomegeneraltheoreticalaspectsinsemiconductorslikekpmethod,spin-orbitinteraction.IalsoshowhowsolvingtheSchrodingerequationforaninnitedimensionalHamiltoniancanposeaseriousproblem.Asaremedy,IintroducetheLowdin'sperturbationscheme,whichisdiscussedindetailinthatchapter.InChapter 3 ,followingtheLowdin'sPerturbationScheme,IsolvefortheHamiltonianofthebulkIII-Vsemiconductorsexactlyfortheeightbandbasisstatesandtakingtheinteractionsoftheremotebandsasperturbation.Thentheenergyeigenvaluesgivemethebandstructure.ThusIgetthe8-bandkpmodelforthebulkIII-Vsemiconductorsintheabsenceofmagneticeld.InChapter 4 ,IpresentsomeexperimentalTRDTmeasurementsforInMnAsatdifferentprobewavelengthsintheabsenceofthemagneticeld.Narrowgapferromagneticsemiconductors(NGFS)suchasInMnAshavesignicantpotentialforapplicationsininfraredspinphotonicsandspintransportdevicesduetotheirsmallenergygapandmuchhigherelectronandholemobilities[ 3 ].Icalculatethebandstructure,usingthe8-bandkpmodelofChapter 3 ,whichismodiedtoincludethecouplingofelectronsandholestothemagneticMnimpurities.IinvestigatetherelaxationdynamicsofthecarrierandthesignchangephenomenaintheTRDTspectra,usingthecalculatedbandstructurealongwiththeidenticationoftheopticaltransitions.InChapter 5 ,IpresenttheCRmeasurementsandcorrespondingtheoreticalcalculationsinMBEandMOVPEgrownpdopedferromagneticInMnAsandInMnSbwithdifferentMncontentsatvariouscarrierconcentration.Iusethe8-bandPidgeonBrownmodel(whichistheextensionofthe8-Bandkpmodelwithmagneticeld),whichisgeneralizedtoincludethewavevectordependenceoftheelectronicstatesalongkzaswellasthes-dandp-dexchangeinteractionswiththelocalizedMnd-electronstocalculatetheLandausubandstructure.TheCurietemperatureistakenasaninputparameterandtheaverageMnspinistreatedinmeaneldtheory.IcalculateabsorptionusingtheFermi'sgoldenrule.Iinvestigatewhetherthedifferencesbetween 17

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theCRmeasurementsareseenformolecularbeamepitaxy(MBE)andmetalorganicvaporphaseepitaxy(MOVPE)samplesarisefromdifferencesinthecarrierdensitiesbasedontheCR,FermienergyandLandausubbandstructurecalculations.IalsocalculatetheaveragezcomponentofthespinsinthematerialstoexplorethecauseofhigherCurietemperatureforMOVPEgrownsamples.InChapter 6 ,IinvestigatethesensitivityofthemeasuredOPNMRsignalintensityinthestrainedAlGaAs/GaAssquareMQWtothecalculatedelectronspinpolarization.Iusethe8-bandPidgeon-Brownmodel,whichisgeneralizedtoincludethesquarewellconnementeffectandthestraineffecttocalculatethesubbandstructure.Thestaininthesampleiscalculatedfromtheexperimentalquadrupolespinsplitting.IuseFermi'sgoldenruletocalculatethemagneto-absorption.Theelectronspinpolarizationiscalculatedfromtheabsorptions.InChapter 7 ,Imodeltheexperimentalmagneto-absorptionspectraforthestrainedAlInSb/InSbparabolicMQWtoinvestigatehowtheconnementandstrainaffectthemagneto-absorptionortheLandausubbandstructure.InSbhasthenarrowestbandgapofIII-Vsemiconductors.Moreover,thelargespin-orbitinteractionandg-factorinInSbalsomakesitattractiveforspintronicapplications[ 4 6 ].Iusethe8-bandPidgeon-Brownmodel,whichisgeneralizedtoincludethesquarewellconnementeffectandthepseudomorphicstraineffecttocalculatethesubbandstructure.IuseFermi'sgoldenruletocalculatethemagneto-absorption.Icalculatethemagneto-absorptionspectrawithandwithoutstrainandcomparethemwiththeexperimentalmagneto-absorptionspectratoexploretheeffectofthestrain.IalsocomparetheresultswiththatofthesquareMQW.toexploretheeffectoftheshapeoftheconnement. 18

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CHAPTER2PERTURBATIONTHEORY 2.1KPMethodTheproblemofinteractingparticlesinacrystalisamanybodyproblem.Becausethetotalpotentialofthesystemincludestheone-electronpotentialsthatcomefromtheinteractionsoftheelectronswiththeionsandthepairpotentialsthatarisefromtheelectron-electroncoulombinteractions.Butintheindependentelectronapproximationthemanybodyproblembecomesasingleelectronproblem.Thenthetotalpotentialofthesystemcanbewrittenasanone-electronperiodicpotentialV(r)foraperfectcrystal.TheelectroniscalledBlochelectron.SotheSchrodingerequationfortheBlochelectronisgivenbyHn,k(r)=En(k)n,k(r) (2)wherekistheBlochwavevector,nisthebandindexandtheHamiltonianisgivenbyH=p2 2m0+V(r) (2)wherep,m0andV(r)arethemomentumoperator,electronmassandperiodicpotentialrespectively.Thewavefunctionn,k(r)inEq. 2 hasthefollowingformaccordingtotheBlochtheoremn,k(r)=eikrun,k(r) (2)wherenisthestateorbandindexoftheelectronandun,k(r)arecellularfunctionswhichhavethesameperiodicityasthelattice. 19

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NowsubstitutingEq. 2 andEq. 2 intoEq. 2 andwrittingitintermsofcellperiodicfunctionsun,k(r)Igetp2 2m0+V(r)+~2k2 2m0+~ m0kpun,k(r)=En(k)un,k(r) (2)Becauseofthisfourthtermthemethodiscalledthekpmethod. 2.2Spin-OrbitInteractionInsemiconductors,thespin-orbitinteractionisalsopresent.SothetotalHamiltoniancanbewrittenbyincludingthespin-orbitinteractioninEq. 2 .H=p2 2m0+V(r)+~ 4m20c2(rV)p (2)wherethethirdtermrepresentsthespin-orbitinteractionwhichcomesfromthenonrelativisticapproximationtotheDiracequation,cisthespeedoflightandisthePaulispinmatrix.NowtheSchodingerequationtakesthefollowingformintermsofcellperiodicfunctions:p2 2m0+V(r)+~2k2 2m0+~ m0k+~ 4m20c2(rV)pu0n,k(r)=En(k)u0n,k(r) (2)Nowtheu0n,k(r)isacompactnotationofthetwo-componentcolumnspinorfunctionofthefollowingformu0n,k(r)=0B@u(1)n,k(r)u(2)n,k(r)1CA=u(1)n,k(r)j"i+u(2)n,k(r)j#i (2)where=p+~ 4m0c2(rV) (2) 20

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2.3SolvingSchrodingerEquationinInniteDimensionalHamiltonianInthissection,IwouldliketosolvetheEq. 2 whichIgotfromtheexactformofHamiltonian,whichwasgivenbyEq. 2 .TosolvetheEq. 2 foranarbitraryvalueofk,itisassumedthatasetofsolutionsfu0m,k0(r)gisalreadyknownforagivenpointk0,wheremlabelsthedifferentsolutionsforthesamevalueofpointk0.Thisknownsetofsolutionswillactasabasissetofsolutionsforotherkpoints.Thepracticalchoiceofthepointk0forIII-Vdirectbandgapsemiconductorisk0=0,whichisthecenterofBrillouinzone.SothegeneralsolutionsforEq. 2 canbeexpressedasalinearcombinationofthezonecentersolutionsfu0m,0(r)g.u0n,k(r)=Xmcm(k)u0m,0(r) (2)Nowthefollowingtwoequationsaresatisedbyu0m,0(r)p2 2m0+V(r)u0m,0(r)=Em(0)u0m,0(r) (2)andZunitcellu0yn,0(r)u0m,0(r)d3r=nm (2)Nowsubstitutingu0n,k(r)intoEq. 2 andmultiplyingu0yn,0(r)totheleftoftheequation,andthenintegratingoverthevolumeofaunitcell,andnallyusingtheEq. 2 andEq. 2 ,thematrixeigenvalueequationisfoundfortheexpansioncoefcientsXmEm(0)+~2k2 2m0nm+~ m0knm+nmcm(k)=E(k)cn(k) (2)wherenm=Zunitcellu0yn,0(r)u0m,0(r)d3r (2)nm=~ 4m20c2Zunitcellu0yn,0(r)[(rV)p]u0m,0(r)d3r (2) 21

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FromEq. 2 itcanbeseenthatmnhastwocontributionsnm=Zunitcellu0yn,0(r)pu0m,0(r)d3r+~ 4m0c2Zunitcellu0yn,0(r)(rV)u0m,0(r)d3r (2)IntheEq. 2 ,thesecondtermofEq. 2 willeventuallygivethekdependentspin-orbitinteraction,whichissmallcomparedwiththetermnm.Becausethecrystalmomentumneark00issmallerthantheelectronmomentumintheatomicorbit[ 7 ].Sobyneglectingthesecondtermthematrixelementmnisgivenby,nmpnmZunitcellu0yn,0(r)pu0m,0(r)d3r (2)NowtheenergyandwavefunctionscanbecalculatedbysolvingthematrixeigenvalueEq. 2 .ButthematrixarisingfromtheEq. 2 isinnitedimensional,whichisnotpracticallysolvable.Becauseeventhoughitwasknownhowtosolveaninnitedimensionaleigenvalueproblem,theknowledgeoftheinnitenumberofbandedgeeigenstateswouldstillbeneeded,whichisquiteimpossible.SinceIaminterestednearthebandedgesofthedirectband-gapsemiconductors,onlyaadjacentbandsareimportant.Usuallytwoapproachesareverypopularinthiscase.1)Solvingthematrixeigenvalueequationonlyfor8bandsbyneglectingtheremotebandscompletely,oneofthemethodslikethisisthe8-bandKanemodel[ 8 ].Butthismodelgivestheincorrectdispersionrelationandeffectivemassforthecarrierintheheavyholeband.2)Solvingthematrixeigenvalueequationexactlyforasetofbandsandtreatingtheeffectoftheremotebandsasperturbation,oneoftheperturbationschemelikethisistheLowdin'sperturbationtheory.Thisperturbationmethodisaveryusefulmethodincaseofreducingthedimensionalityofanymatrixeigenvalueequation,andthismethodwillbediscussedinthenextsection. 2.4Lowdin'sPerturbationTheoryAsmentionedearlier,theeigenvalueproblem,givenbythematrixeigenvalueEq. 2 ,isimpossibletosolvewithoutsimplicationsasIneedtodiagonalizetheinnite 22

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dimensionalmatrix.OneofthepopularsimplicationtechniquesisLowdin'sperturbationtheory.Inthissection,onlythesummaryofthekeyresultsofthemethodwillbegiven.ThedetailedderivationcanbefoundinLowdin'spaper[ 9 ],orinthebookbyMortenWillatzen[ 10 ].Theband-edgebasissetfu0m,0(r)gcanbedividedintotwoclassesbyexpandingthefollowingfunctionu0n,k(r)=Xj2ACju0j,0(r)+Xl2BClu0l,0(r) (2)IfIaminterestedinthesetAthenthesetAwillbechosenasthenormalsetandtheinuenceofthestatesinsetBonthestatesinsetAwillbetreatedasaperturbation.AlthoughtheusualmethodgivesaseculardeterminantwiththedimensionDIM(A)+DIM(B)inthiscase,theLowdin'sperturbationmethodwilleventuallyprovideadeterminantofdimensionDIM(A).ThisisquiterightwhenonlythelowestsetofDIM(A)eigenvaluesisneeded.ButanewsetofstatesA(perturbedfromAbysetB)isneededtoobtainthesetofDIM(A)eigenvalues.Sointhiscase,thesecularequationisrequired,nottheseculardeterminant.NowfollowingthemethodsofLowdin'sperturbationtheory(thebookbyMortenWillatzen[ 10 ]fordetails),onlytheeigenequationneedstobesolvedXi,j2A)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(UAij)]TJ /F8 11.955 Tf 11.95 0 Td[(EijCj=0 (2)whereUAij=Hij+Xl2BH0ilH0lj E0)]TJ /F8 11.955 Tf 11.95 0 Td[(El (2)Herethesecondtermistheperturbationterm.E0istheaverageenergyofthestatesthatlieinsetA.ThersttermisgivenbyHij=Ej+~2k2 2m0ij+pcijkc+ij(i,j2A) (2) 23

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TheperturbationtermisgivenbyH0il=Xs~ka m0pail(i2Aandl2B) (2)IthusobtainUAij=Ej(0)+~2k2 2m0ij+pcijkc+ij+~2 m20Xl2BXa,bkakbpailpblj E0)]TJ /F8 11.955 Tf 11.96 0 Td[(El (2)LetUAijDij.IgetthefollowingmatrixDij=Ej(0)ij+ij+pcijkc+Xa,bDabijkakb (2)whereDabij=~2 2m0"ijab+Xl2Bpailpblj+pbilpalj m0(E0)]TJ /F8 11.955 Tf 11.95 0 Td[(El)# (2)Herepij,pilandpljarethemomentummatrixelementsbetweentwostatesthatcanbedenedbyEq. 2 .Thespin-orbitinteractionmatrixelementsijbetweentwostatescanbedenedbyEq. 2 .Eachoftheindicesa,bandcrepresentsthex,yandzco-ordinatesofthemomentummatrixelementsandwavevector.Notethattherst,secondandthirdtermsdependonlythestatesofthesetAandthefourthtermhasthedependencyonthestatesofthesetB.Thersttermisalreadydiagonalwhereastheotherstermsarenot.However,thesecondterm(spin-orbitinteractionHamiltonian)canbediagonalizedbychoosingaparticularsetofzonecenterbasisstatesinthesetA.IaminterestedinthelowestlyingconductionbandsandthehighestlyingvalencebandsasmyreseachworkisrelatedtotheopticaltransitionsintheIII-Vbulkandheterostructuresemiconductormaterials.SothebasicideaistochoosetheinterestedlowestlyingconductionbandsandhighestlyingvalencebandsandcallthemthesetA.ThentheotherremotebandswillberegardedasthesetB.Finally,togetthebandstructureofthematerialIhavetocalculatethematrixelementsDijusingEq. 2 withEq. 2 24

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Intheforthcomingchapters,IwillcalculatebandstructureofthebulkandheterostructureIII-Vsemiconductorswithandwithoutmagneticeldbyfollowingtheproceduresmentionedabove. 25

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CHAPTER3EIGHTBANDMODELFORIII-VBULKSEMICONDUCTORMATERIALS 3.1BasisStates:Intheabsenceofspin-orbitcoupling,theelectronicstatesareplikestatesnearthevalencebandedgewithorbitalangularmomentum,l=1andtheml=1,0,)]TJ /F4 11.955 Tf 9.3 0 Td[(1.However,inthepresenceofthespin-orbitcoupling,thetotalangularmomentum(j)ofthesestatescantaketwopossiblevalues:j=l+s=1+1=2=3=2andj=l)]TJ /F8 11.955 Tf 12.32 0 Td[(s=1)]TJ /F4 11.955 Tf 12.32 0 Td[(1=2=1=2.Nowthej=3=2andj=1=2bandedgestateswillsplitinenergybyanamount(Fig. 3-1 ).Forthej=3=2states,mj=3=2,1=2,)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2,)]TJ /F4 11.955 Tf 9.3 0 Td[(3=2andforthej=1=2states,mj=1=2,)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2.Moreover,fors-likeconductionbandedgestateswithl=0,jcantakethevalue:j=l+s=0+1=2=1=2.Nowforthej=1=2states,mj=1=2,)]TJ /F4 11.955 Tf 9.3 0 Td[(1=2.Nowinthejj,mjinotation,j1 2,+1 2i,j1 2,)]TJ /F5 7.97 Tf 10.5 4.7 Td[(1 2i,j3 2,+3 2i,j3 2,)]TJ /F5 7.97 Tf 10.49 4.7 Td[(3 2i,j3 2,1 2i,j3 2,)]TJ /F5 7.97 Tf 10.49 4.7 Td[(1 2i,j1 2,+1 2iandj1 2,)]TJ /F5 7.97 Tf 10.5 4.71 Td[(1 2irepresenttheconductionband(CB)spinup,conductionbandspindown,heavyhole(HH)spinup,heavyholespindown,lighthole(LH)spinup,lightholespindown,splitoff(SO)holespinupandsplitoffholespindownstatesatthebandedgerespectively.NotethattheseCB,HH,LH,andSObandedgestatesaredoubly(upanddown)degenerate(Fig. 3-1 ).Sonallytotal8bandedgestatesarefoundinthepresenceofthespin-orbitcoupling.SinceIaminterestedintheopticaltransitionsbetweentheconductionbandsandvalence,Ineedminimum8bandstofullltherequirements.AccordingtothePidgeon-Brownmodel[ 11 ],IconsidereightBlochbasisstatesforthesystemwhichcanbeseparatedintotwosets,anuppersetandalowerset,sothatthistwosetsdecoupleatthebandedge. 26

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Figure3-1. Bandstructurenearthebandedge. Theuppersetisgivenby, jCB"i=j1 2,+1 2i=jS"i (3) jHH"i=j3 2,+3 2i=1 p 2j(X+iY)"i (3) jLH#i=j3 2,)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 2i=1 p 6j(X)]TJ /F8 11.955 Tf 11.95 0 Td[(iY)"+2Z#i (3) jSO#i=j1 2,)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 2i=i p 3j)]TJ /F4 11.955 Tf 17.93 0 Td[((X)]TJ /F8 11.955 Tf 11.96 0 Td[(iY)"+Z#i (3) whichcorrespondtoconductionbandspinup,heavyholespinup,lightholespindown,andsplitoffholespindownrespectively.Thelowersetisgivenby, 27

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jCB#i=j1 2,)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 2i=jS#i (3) jHH#i=j3 2,)]TJ /F4 11.955 Tf 10.5 8.08 Td[(3 2i=i p 2j(X)]TJ /F8 11.955 Tf 11.95 0 Td[(iY)#i (3) jLH"i=j3 2,+1 2i=i p 6j(X+iY)#)]TJ /F4 11.955 Tf 18.6 0 Td[(2Z"i (3) jSO"i=j1 2,+1 2i=1 p 3j(X+iY)#+Z"i (3) whichcorrespondstoconductionbandspindown,heavyholespindown,lightholespinup,andsplitoffholespinup.HerejXi,jYiandjZistateshavethesimilaritytotheatomicporbitalwavefunctions.ThejSistatehasthesimilaritytotheatomicsorbitalwavefunction.Thebasisstatesjj,mjicanbeobtainedbyusingthestandardruleofadditionoftheorbitalangularmomentumandspinangularmomentum.ThedetailedderivationcanbefoundinthebookbyManuelCardona[ 12 ].HerejisthetotalangularmomentumoftheBlochbasisstates(1 2fortheconductionband,3 2fortheheavyholeandlightholebandsand1 2forthesplit-offband).Oneofthegoodreasonsofchoosingthissetofeightbasisstatesisthatnowthespin-orbitinteractionmatrixijinEq. 2 canbediagonalized. 3.2SixBandLuttingerModelandEightBandModelInsixbandLuttingermodel,sixvalencebandsareusedasbandedgebasisstatesandtheconductionbandsareregardedasremotebands.Theeffectsoftheremoteconductionbandsareincludedinthesystemasperturbation.However,thecouplingoftheconductionandvalencebandsarenottakenintoaccountexactlyduetothesixvalencebandedgebasisstates,whichisveryimportantforthenarrowbandgapsemiconductors.Butintheeightbandkpmodel,thecouplingofthetwoconductionbandsandsixvalencebandsarenottakenintoaccountexactlywhichprovidesthenonparabolicityoftheconductionbandstructure,whichisindispensableforthenarrowgapsemiconductors. 28

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3.3HamiltonianMatrixofBulkIII-VSemiconductorMaterialsNowfollowingtheframeworkofLowdin'sPerturbationTheoryIconsiderthesetofeightbasisstatesinthesetAandtheotherremotebandsinthesetB.ThenwiththechoiceofthebasisstatesandusingtheEq. 2 IcanwritethematrixelementsDijexplicitlybydeningthefollowingfourcouplingparameters.whicharisefromthecouplingbetweenthebasisstatesinthesetAandtheremotebandsinthesetB. A0=~2 2m0+~2 m20Xl2BpXXlpXlX E)]TJ /F8 11.955 Tf 11.95 0 Td[(El (3) B0=~2 2m0+~2 m20Xl2BpYXlpYlX E)]TJ /F8 11.955 Tf 11.95 0 Td[(El (3) C0=~2 m20Xl2BpXXlpYlY+pYXlpXlY E)]TJ /F8 11.955 Tf 11.96 0 Td[(El (3) F0=1 m0Xl2BpXSlpXlS E)]TJ /F8 11.955 Tf 11.95 0 Td[(El (3) wheretheparametersA0,B0andC0representthecouplingbetweenthecomponentsofvalencebandsandtheremotebandslabeledasl.TheyaresimilartotheparametersA,BandCdenedinthe6bandLuttinger-Kohnmodel[ 13 ].However,inthatmodel,onlysixvalencebandsareconsideredinsetAandtheotherremotebandsareinsetB.Sointhecouplingparameters,thecouplingbetweenthelowestlyingtwoconductionbands(whichareinsetB)andvalencebandsaretakenintoaccountasaremotebandcouplingcontribution.Butinthe8bandmodel,thecouplingparameters(A0,B0andC0))donothavethecontributionsfromthecouplingbetweenthelowestlyingtwoconductionbandsandthevalencebands.BecausethesetwoconductionbandslieinthesetA.Sotheyarenotremotebandsinthecase.However,IhavetodeneanewparameterF0becausethecouplingbetweenthetwolowestlyingconductionbandsandremotebandsisneededtobetakenintoaccount.NowtherenormalizedLuttingerparametersare 29

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denedasbelow. ~2 2m01=1 3(A0+2B0) (3) ~2 2m02=1 6(A0)]TJ /F8 11.955 Tf 11.95 0 Td[(B0) (3) ~2 2m03=1 6C0 (3) 4=1+2F0 (3) TherenormalizedLuttingerparametersarerelatedtotheoriginalLuttingerparametersL1,L2andL3throughthefollowingrelations[ 14 ] 1=L1)]TJ /F8 11.955 Tf 16.34 8.09 Td[(Ep 3Eg (3) 2=L2)]TJ /F8 11.955 Tf 16.34 8.09 Td[(Ep 6Eg (3) 3=L3)]TJ /F8 11.955 Tf 16.34 8.08 Td[(Ep 6Eg (3) whereEp=2m0V2 ~2 (3)isaopticalmatrixparameterwhichmeasuresthestrengthofthecouplingbetweenvalanceandconductionbands.Theterm'V'iscalledtheKanemomentummatrixelementwhichcanbedenedasV)]TJ /F8 11.955 Tf 9.3 0 Td[(i~ m0hSjpxjXi=)]TJ /F8 11.955 Tf 9.3 0 Td[(i~ m0hSjpyjYi=)]TJ /F8 11.955 Tf 9.3 0 Td[(i~ m0hSjpzjZi (3)whereeachofthematrixelementsrepresentsthemomentummatrixelementbetweentheslikeconductionbandandplikevalenceband.Fortheconvenience,theplusandminuswavevectorsaredenedas k+=kx+iky (3) k)]TJ /F4 11.955 Tf 17.05 1.79 Td[(=kx)]TJ /F8 11.955 Tf 11.96 0 Td[(iky (3) 30

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SothetotalHamiltonianMatrixforthebulkIII-VsemiconductorwithoutmagneticeldcanbewrittenasHB=264HaHcHycHb375 (3)withsubmatricesHa,HbandHcaregivenbyHa=266666664Eg+Ai p 2Vk+i p 6Vk)]TJ /F5 7.97 Tf 28.98 6.5 Td[(1 p 3Vk)]TJ /F2 11.955 Tf -193.62 -22.12 Td[()]TJ /F9 7.97 Tf 14.9 4.71 Td[(i p 2Vk)]TJ /F2 11.955 Tf 18.37 1.79 Td[()]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.95 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Mip 2M)]TJ /F9 7.97 Tf 14.9 4.71 Td[(i p 6Vk+)]TJ /F8 11.955 Tf 9.3 0 Td[(M)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Q1 p 3Vk+)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.96 0 Td[(377777775 (3) Hb=266666664Eg+A)]TJ /F9 7.97 Tf 14.9 4.7 Td[(i p 2Vk)]TJ /F2 11.955 Tf 17.05 1.8 Td[()]TJ /F9 7.97 Tf 14.9 4.7 Td[(i p 6Vk+i p 3Vk+)]TJ /F5 7.97 Tf 14.02 4.71 Td[(1 p 2Vk+)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.96 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Mip 2M)]TJ /F5 7.97 Tf 14.02 4.71 Td[(1 p 6Vk+)]TJ /F8 11.955 Tf 9.3 0 Td[(M)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Q)]TJ /F9 7.97 Tf 14.9 4.71 Td[(i p 3Vk)]TJ /F2 11.955 Tf 18.66 1.79 Td[()]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.96 0 Td[(377777775(3) Hc=26666666400q 2 3Vkziq 1 3Vkz00)]TJ /F8 11.955 Tf 9.3 0 Td[(L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 2 3VkzL0iq 3 2L)]TJ /F12 11.955 Tf 9.3 13.72 Td[(q 1 3Vkz)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2Liq 3 2L0377777775(3)whereEgistheenergygapoftheIII-Vbulksemiconductormaterials,andisthespin-orbitsplitoffenergy. 31

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Theparametersaredenedasfollowing A=4~2k2 2m0 (3) P=1~2k2 2m0 (3) Q=2~2 2m0(k2x+k2y)]TJ /F4 11.955 Tf 11.95 0 Td[(2k2z) (3) L=)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 33~2 m0k)]TJ /F8 11.955 Tf 7.08 1.8 Td[(kz (3) M=p 3~2 2m0[2(k2x)]TJ /F8 11.955 Tf 11.95 0 Td[(k2y))]TJ /F4 11.955 Tf 11.96 0 Td[(2i3kxky] (3) IntheEq. 3 ,theHamiltonianincludesthecrystalperiodicpotential,thespin-orbitinteractions,thecouplingeffectsbetweeneightbandsandremotebandsandthediagonalterms. 3.4BandStructureCalculationNowtheenergyeigenvaluesandthecomponentsofnormalizedeigenvectorscanbeobtainedbydiagonalizingthis88HamiltonianmatrixgivenbyEq. 3 numericallyforagivenvaluesofwavevectork.ThenplottingenergyeigenvaluesEfordifferentvaluesofwavevectorkthebandstructurediagramcanbeobtainedforthebulkmaterial. 32

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CHAPTER4CARRIERDYANAMICSINNARROWGAPFERROMAGNETICSEMICONDUCTORSInthischapter,IpresentsometheoreticalcalculationsofthebandstructureforInMnAswithoutmagneticeld,usingan8-bandkpmodel,whichincludesconductionandvalencebandmixingaswellasthecouplingofelectronsandholestothemagneticMnions.Thetheoreticalcalculationsofthebandstructurealongwiththeidenticationoftheopticaltransitionsallowmetoexplainsomeofthecarrierdynamicsandthesignchangesinthedifferentialtransmission[ 3 ]. 4.1ExperimentalDetailsExperimentalTimeResolvedDifferentialTransmission(TRDT)measurementswerecarriedoutbyProf.KhodaparastgroupatVirginiaTech,Prof.WesselsatNorthwesternUniversity,andDr.McGillatNHMFL.TheyappliedNondegenerateDifferentialTransmission(NDDT)schemetomeasuretheTRDTsignal.InNDDTscheme,thepumpandprobelaserpulsescomefromdifferentwavelengths.Inadegeneratepump-probescheme,whenthepump/probeexcitationsarefromthesamesource(samewavelenghts),theopticalexcitationofthecarriers,followedbyfastrelaxationinthebands,canresultinasaturationoftheband-to-bandabsorption[ 15 16 ]Inordertoavoidpossiblenonlineareffects,theyemployedNDDTschemewherethepumpandprobelaserpulsescomefromdifferentsources(differentwavelengths).IntheNDDTschemethepumppulsesweretunedabovethefundamentalgap.Inthisscheme,thephotoexcitedcarrierswerecreatedbyNIRpulsesxedat800nmabovetheInMnAsfundamentalgapandprobedbylaserpulsesrangingfrom1.3)]TJ /F1 11.955 Tf 12.62 0 Td[(3.8m.Thepumpuencewastunablefrom1)]TJ /F1 11.955 Tf 12.62 0 Td[(5mJcm)]TJ /F5 7.97 Tf 6.59 0 Td[(2correspondingtoa ReprintedwithpermissionfromM.Bhowmick,T.R.Merritt,G.A.Khodaparast,B.W.Wessels,S.A.McGill,D.Saha,X.Pan,G.D.Sanders,andC.J.Stanton,Phys.Rev.B85,125313(2012),Copyright(2012)bytheAmericanPhysicalSociety. 33

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photoexcitedcarrierdensityintherangeof51018)]TJ /F1 11.955 Tf 12.62 0 Td[(11019cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3,respectively.Thebandstructurecalculations,presentedinSection 4.3.1 ,showtheopticaltransitionsforapumpwavelengthof800nm.AsshowninFig. 4-1 ,forthepumpxedat800nmwithsampletemperature290K,andauenceof3.8mJcm)]TJ /F5 7.97 Tf 6.59 0 Td[(2,tuningtheprobewavelengthsinmidinfrared(MIR)resultedinseveraldifferencesinthetimeresolveddifferentialtransmission(TRDT)patterns,whereboththeamplitudeandrelaxationtimedemonstratestrongwavelengthdependence.ItshowsphotoinducedabsorptionorpositiveTRDTsignal.AsshowninFig. 4-2 ,forthepumpxedat800nmwithsampletemperature290K,theTRDTspectrawasmeasuredatprobewavelength2m.ItshowsphotoinducedbleachingornegativeTRDTsignal. 4.2TheoryandModeling 4.2.1EffectiveMassHamiltonianThemodelisbasedontheaneightbandkpmodel,whichincludestheconductionandvalencebandmixing.ThemodelissimilartothePidgeon-Brownmodelexceptthereisnoexternalmagneticeldinthiscasebutthekxandkydependenciesoftheelectronicstructureareallowed.FollowingthePidgeon-Brownmodel,Iseparatethe8k=0Blochbasisstatesintoanupperandlowersetswithnoexternalmagneticeld.TheBlochbasisstatesfortheuppersetarejS"i,jHH"i,jLH#iandjSO#i,whichcorrespondtoelectronspinup,heavyholespinup,lightholespindownandsplitoffhole,respectively.Similarly,theBlochbasisstatesforthelowersetarejS#i,jHH#i,jLH"iandjSO"icorrespondingtoelectronspinup,heavyholespinup,lightholespindownandsplit-offhole,respectively.Theexplicitexpressionsforthese8BlochbasisstatesaregiveninfromEq. 3 toEq. 3 .SincethematerialsareMn-dopedferromagneticsemiconductors,Ineedtoincludetheeffectsoftheemphsp-dcouplingoftheitinerantcarrierstotheMnionlocalizeddelectrons. 34

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Figure4-1. Two-colordifferentialtransmissionmeasurementsinInMnAsferromagneticlmat290KfordifferentprobewavelengthsinMIR.Therelaxationdynamicisdominatedbyphotoinducedabsorption.FromRef.[ 3 ]. 35

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Figure4-2. Two-colordifferentialtransmissionmeasurementsinInMnAsferromagneticlmat290Kfortheprobewavelength2m.Therelaxationdynamicisdominatedbyphotoinducedbleaching.FromRef.[ 3 ]. 36

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SothetotaleffectivemassHamiltonianisthesumoftheBulksemiconductorHamiltonian(HB),andthesp-dexchangeinteractionHamiltonian(Hex), H=HB+Hex(4)TheexplicitexpressionofHBisgivenbyEq. 3 .Theexchangeinteractionwillbediscussedinthefollowingsection. 4.2.2CouplingtoSpontaneousMagnetizationIntheMnexchangeHamiltonianofEq. 4 ,theeffectsofthespontaneousmagnetizationoftheMnions(MagneticImpurities)andthecouplingofthismagnetizationtotheconductionbandelectronsandvalancebandholesareincludedthroughthes-dandp-dexchangeinteractions,respectively,followingthemethodofKossut[ 17 ].TheresultingexchangeinteractionHamiltonianisgivenby, Hex=xN0hSzi264Da00)]TJ /F8 11.955 Tf 9.3 0 Td[(Da375(4)wherexistheconcentrationoftheMnions,N0isthenumberofcationsitesinthematerial,andhSziistheaveragespinontheMnsite.The44submatrixDaisgivenby Da=2666666641 200001 20000)]TJ /F5 7.97 Tf 10.49 4.71 Td[(1 6)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2 300ip 2 31 2377777775.(4)Thevaluesoftheexchangeconstantsforthes-dinteraction()andp-dinteraction()aretakentobe0.5eVand-1.0eVrespectively[ 18 ].Thespontaneousmagnetizationiscalculatedusingmean-eldtheory[ 18 21 ]dependsontheMnconcentration(x)andthez-componentoftheaverageMnspin,hSzi.ThevaluesofxarevariedinthecalculationsaccordingtotheactualMnfractioninthesamples. 37

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ThevalueofhSziisdeterminedbynumericallysolvingatranscendentalequation(whichdependsontemperature,T,andexperimentallymeasuredCurietemperature,Tc,ofthesample)usingarootndingroutine.hSzi=SBSgS kBTBB)]TJ /F4 11.955 Tf 13.15 8.09 Td[(3kBTChSzi gS(S+1) (4)wheretheeffectiveg-factor,g=2andspin,S=5 2forthe3d5electronsoftheMnion[ 22 ]TheBrillouinfunction,BS(x),canbedenedbythefollowingrelationBS(x)=2S+1 Scoth2S+1 Sx)]TJ /F4 11.955 Tf 17.17 8.08 Td[(1 2Scothx 2S (4)Inthesimulation,thelatticetemperatureTandtheCurietemperatureTcofthesamplesareusedasinputstothemodel.WiththeferromagneticorderingoftheMnions,thespontaneousmagnetizationanditssp-dcouplingtotheitinerantcarrierswillspin-splitthebandsevenintheabsenceofanexternalmagneticeld. 4.3ResultsandDissussion: 4.3.1CalculationofElectronicStructureTounderstandtheeffectsoftheferromagneticorderontheelectronicstructureandsubsequentlythecarrierrelaxationdynamics,theelectronicstructureforbulkInMnAshasbeencalculatedbydiagonalizingtotal88HamiltonianmatrixgivenbyEq. 4 .Thepreliminarycalculationshavefocusedonthebandstructure.Bycalculatingtheelectronicbandstructure,Icandeterminewherephotoexcitedcarriersaregeneratedbythepumppulseandwhichregionsoftheelectronicstructurearesampledbytheprobepulse.Later,Iwillfocusonthecarrierdynamics.Fig. 4-3 showsthecalculatedbandstructureforIn0.96Mn0.04Asat290KforTc=330K.ItcanbeseenthattheferromagneticorderoftheMnionscausesspinsplittingofthebands.TheredarrowsinFig. 4-3 ,showtheopticaltransitionsthatarepossibleforan 38

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800nmpumppulse.Itcanbeseenthattransitionsarepossiblefromtheheavy-hole,light-hole,andspin-orbitsplitvalencebands.Furthermore,thetransitionsfromtheheavyandlightholescreatephotoexcitedelectronsintheconductionbandthatareabovetheenergythresholdforscatteringintothesatelliteLvalleys.Theseelectronsrapidlyscatterintothesatellitevalleys(whichhavealargeeffectivemassandhenceagreaterdensityofstates)andtakealongtimetoreturntothe)]TJ /F1 11.955 Tf 10.09 0 Td[(valleyandthenrelaxtothebottomofthe)]TJ /F1 11.955 Tf 10.09 0 Td[(valley,similartothe620nmphotoexcitationinGaAs[ 23 25 ]. 4.3.2ContributionstoTRDTSpectraTheTRDTmeasureschangestotheabsorptionoftheprobepulsethatresultfromthepumppulse.Forthinsamples,T/T0)]TJ /F4 11.955 Tf 21.92 0 Td[(L,whereListhethicknessofthesampleandisthechangeintheabsorptioncoefcient.WhileadetailedunderstandingoftheabsorptionchangesrequiresusingthesemiconductorBlochequations[ 26 ],onecangaininsightintowhattypesofprocessesinuencetheTRDTsignalbylookingatasimplied,approximateexpressionfortheabsorptioncoefcient:(!,t)=1 !Zd!0Zdt0N(!0,t)]TJ /F8 11.955 Tf 11.95 0 Td[(t0)XtransitionsZdkjHkj2[Ec(k))]TJ /F8 11.955 Tf 11.95 0 Td[(Ev(k))]TJ /F3 11.955 Tf 11.96 0 Td[(!0][1)]TJ /F8 11.955 Tf 11.96 0 Td[(fec(k,t0))]TJ /F8 11.955 Tf 11.95 0 Td[(fec(k,t0)]. (4)Here,(!,t)istheabsorptionoftheprobepulsecenteredatfrequencyasafunctionofdelaytimetwithrespecttothepumppulse,N(!0,t)isthetransientphotonenergydensityoftheprobepulse,Hkk0istheopticalmatrixelementbetweenconductionandvalencebandstates,EcandEvarethecarrierenergiesintheconductionandvalencebands,respectively,andfec(k,t0)andfhv(k,t0)aretheelectronandholedistributionfunctionsintheconductionandvalencebands,respectively.Therearefourmaincontributionstothedifferentialtransmissionsignal)]TJ /F4 11.955 Tf 9.3 0 Td[(T/T0=(T0)]TJ /F1 11.955 Tf 12.62 0 Td[(T)/T0[ 27 ].Thesecanbeunderstoodbylookingatchangesintheabsorptioncoefcientoftheprobe,Eq. 4 above,andseeingwhatchangesasaresultofthe 39

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Figure4-3. ElectronicstructureofIn0.96Mn0.04AsatT=290KforB=0,alongthe[001]and[111]directions.Theenergybandsarespin-splitduetotheferromagnetism.Theallowedopticaltransitionsforapumpwavelengthof800nm(1.55eV)areshownbythe(red)arrows.Thedottedlineat1.08eVshowsthethresholdfor)]TJ /F1 11.955 Tf 10.1 0 Td[(valleyelectronstoscattertothesatelliteLvalley.FromRef.[ 3 ]. 40

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pumppulse.Thefourmaincontributionsarethefollowing.(1)Phase-spacelling:thiscomesfromtheprobepulsecreatingadditionalelectronsandholesthatblocktheabsorptionofadditionalcarriersfromtheprobepulsebythePauliexclusionprincipal.Onecannotcreateanadditionalelectron-holepairwiththeprobepulseifthepumppulsehasalreadycreatedonesincethePauliprincipleexcludestwoelectronsfrombeinginthesamestate.Asthephotoexcitedcarriersrelax,theabsorptionoftheprobepulseincreaseswithtime.PhasespacellinggivesanegativecontributiontoT/T0(providedthereisnocarrierinversionbeforethepumppulse).(2)Band-gaprenormalization:thisresultsfromthepumppulsephotoexcitingelectron-holepairs,whichthroughthemany-bodyinteractions,causetheelectronandholeenergies[inthedeltafunctioninEq. 4 ]tochange.Thiscausesthebandgaptoshrink.Band-gaprenormalizationgivesapositivecontributionforprobelaserenergiesbelowthepump-inducedphotoexcitedcarriersandanegativecontributionatenergiesabovethepump-inducedphotoexcitedcarriers.Thiseffecttendstobestrongjustbelowthebandedge.Withnopumppulse,probingbelowthebandedgewillnotleadtoabsorptionoftheprobepulse.However,withapumppulse,thebandgapnarrowsandtheprobepulsewillnowbeabsorbed.(3)Localeldeffects:photoexcitedelecton-holepairsinteractthroughtheCoulombinteraction.Thisleadstotheformationofexcitonsandchangestotheelectronandholewavefunctions.Evenstatesexcitedabovethebandgapareunboundexcitons.ThesechangestothewavefunctionswillchangetheopticalmatrixelementsinEq. 4 andleadtoaCoulombenhancementoftheabsorptionoftheprobepulse.Nowwithapumppulseon,additionalelectron-holepairsarecreatedthatscreentheCoulombinteractionandchangetheopticalmatrixelements.Thisgivesanegativecontributionatenergiesbelowthephotoexcitedcarriersandapositivecontributionatenergiesabovethephotoexcitedcarries.(4)Freecarrierabsorption:thisisduetotheintrabandabsorptionofthephotoexcitedcarriers(whichmustbeassisted 41

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Figure4-4. ElectronicstructureofIn0.96Mn0.04AsatT=290KforB=0.Theenergybandsarespin-splitduetotheferromagnetism.Theallowedopticaltransitionsareshownforaprobewavelengthof3.5m(blackarrows)and2.0m(redarrows).FromRef.[ 3 ]. 42

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byphononsorimpurities).Thiswillgiveapositivecontributionto)]TJ /F4 11.955 Tf 9.3 0 Td[(T/T0duetothephotoinducedabsorption.Fig. 4-4 showsthecalculatedbandsofIn0.96Mn0.04Asat290KforTc=330K.ThistimeIshowthetransitionsfromtheprobepulse.Thisallowsmetoseewhichregionsarebeingmonitoredbytheprobepulse.Transitionsareshownforprobewavelengthsof3.5m(blackarrows)and2m(redarrows).Itcanbeseenthatthe3.5mprobeprimarilyprobesthestatesatornearthebandedgewhilethe2.0mprobeisdeepintothebands.ThiscanexplainthesignchangeinthedifferentialtransmissionsignalshowninFig. 4-2 FromFig. 4-4 ,itcanbeseenthatthereasonforthechangeofthesignbetween3.5and2m.The2mprobesdeepintothebandswerethephasespacellingcontributionto)]TJ /F4 11.955 Tf 9.3 0 Td[(T/T0isdominant.However,the3.5mtransitionsprobeclosetoandslightlybelowthebandedge.Inthiscase,thebandgaprenormalizationtermwilldominateandgiveapositivesignalto)]TJ /F4 11.955 Tf 9.29 0 Td[(T/T0.Inaddition,freecarrierabsorptioncanalsocontributetotheobservedpositivesignalat3.5m. 43

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CHAPTER5ELECTRONICSTATESANDCYCLOTRONRESONANCEINNARROWGAPFERROMAGNETICSEMICONDUCTORSCyclotronResonance(CR)spectroscopyisanextremelypowerfultoolforthestudyofelectronicstatesinsemiconductors,and,inthischapterIpresentandcomparetheexperimentalandtheoreticalstudiesofthemagneto-opticalpropertiesofp-typeIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xMnxAsandIn1)]TJ /F9 7.97 Tf 6.58 0 Td[(xMnxSbferromagneticsemiconductorslmsinultrahighmagneticeldsorientedalong[001][ 28 ].InthepresenceofanexternalstaticmagneticeldalonganycrystaldirectionofanyIII-VbulksemiconductorthemotionoftheBlochelectronisdescribedasthequantizationofthecyclotronorbits,whicharecalledtheLandaulevels.Soifthemagneticeldisappliedalong^zdirectionthentheenergyoftheBlochelectronisquantizedintheplaneperpendiculartothemagneticeld.ButtheBlochelectronisnotconnedinthe^zdirection.Moreover,therewillbethesplittingofLandaulevelsduetotheZeemaneffect.Inthe8-bandPidgeon-Brownmodel[ 11 ],wheretheLandauHamiltonianandZeemanHamiltonianmatricesarecalculatedforthezonecenter,kz=0only.Sothe8-bandPidgeon-Brownmodelisextendedtoincludethewavevectordependenceoftheelectronicstatesalongkzaswellasthes-dandp-dexchangeinteractionswiththelocalizedMnd-electrons.TheCurietemperatureistakenasaninputparameterandtheaverageMnspinistreatedinmeaneldtheory[ 20 29 ].Inthemodel,thetotaleffectivemassHamiltonianisthesumoftheLandau,Zeeman,andsp-dexchangeinteraction.ThedifferencesbetweenthepositionoftheCRresonancesforMBEvsMOVPEsamplesarefound.ThecalculationsagreereasonablywellwiththeexperimentsandshowthatdifferencesbetweenCRintheMBEandMOVPEsamplesareprimarilyduetodifferentcarrierdensitiesinthetwotypesofsamples. ReprintedwithpermissionfromG.A.Khodaparast,Y.H.Matsuda,D.Saha,G.D.Sanders,C.J.Stanton,H.Saito,S.Takeyama,T.R.Merritt,C.Feeser,B.W.Wessels,etal.,Phys.Rev.B88,235204(2013),Copyright(2013)bytheAmericanPhysicalSociety. 44

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5.1ExperimentalDetails 5.1.1SamplesThesamples'characteristicsaresummarizedinTable 5-1 Table5-1. Characteristicsofthesamplesstudiedinthiswork. SampleGrowthDensityFilmThicknessxTcMethodcm)]TJ /F5 7.97 Tf 6.58 0 Td[(3nm%K InMnAsMOVPE1.3510186004330InMnAsMOVPE1.610182352330InMnAsMOVPE4.810189701330InMnSbMOVPE1.910185005.6590InMnSb(A)MBE3.01020230210InMnSb(B)MBE1.01020230210 ThemeasuredCRmassandmobilityforthesamplesarelistedinTable 5-2 Table5-2. MeasuredCRmassandmobilityfordifferentexcitationwavelengths.Thecyclotronmobility,CR,wasextractedfromthewidthoftheresonancepeaks. SampleGrowthmCRTempMnMethodmm0cm2=VsK% InMnAsMOVPE10.70.0364802952InMnAsMOVPE10.70.0376001212InMnSbMBE10.60.0575002952InMnSbMBE10.70.0517001212InMnAsMOVPE5.530.046750304InMnAsMOVPE5.530.045650402InMnAsMOVPE5.530.043400201InMnSb(A)MBE5.530.0364002952InMnSb(A)MBE5.530.033500652InMnSb(B)MBE5.530.0374002952InMnSbMOVPE5.530.0351000615.6 5.1.2CRMeasurementsCRmeasurementswereperformedusingCO2,H2O,andCOlasers,providinglaserradiationat10.6,10.7,and16.9and5.53m.Magneticeldsexceeding100TeslaweregeneratedbyasingleturncoiltechniqueTheexternalmagneticeldwasappliedalongthegrowthdirectionandmeasuredbyapick-upcoilaroundthesample,placedinsideacontinuousowheliumcryostatandseveralIRwavelengthswereused 45

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astheexcitationsourceandthetransmittedsignalthroughthesamplewascollectedusingafastliquid-nitrogen-cooledHgCdTedetectororaCu-dopedGe.Amulti-channeldigitizerplacedinashieldedroomrecordedthesignalsfromthedetectorandthepick-upcoil.Althoughthesingle-turncoildestroysineachshot,thesampleandpick-upcoilremainintact,makingitpossibletocarryoutdetailedtemperatureandwavelengthdependencemeasurementsonthesamesample.Sincethetransmissionsignalcanberecordedduringboththeupanddownsweeps,eachCRresonancewasobservedtwiceinasinglepulse.ThisfactallowedmetocheckthereproducibilityoftheobservedresonancepeaksandtomakesurethatthespectrawerefreefromanyslowheatingeffectsThepresentedtransmissionmeasurementsherearetherelativechangeinthetransmissionT(B)=T(B=0). 5.2TheoryandModelingAneffectivemasstheorywasdevelopedforCRinbulknarrowgapdilutemagneticsemiconductorssuchasInMnAs.DetailsaredescribedinRef.[ 20 ].Inthatpaper,thetheorywasappliedtothestudyofn-typeInMnAsalloysandcomparedtheresultswithexperimentalCRmeasurements.HereIwillhighlightthemainpointsconcerningthetheoreticalmodelandrefertheinterestedreadertothemoredetailedtreatmentsintheabovereference. 5.2.1EffectiveMassHamiltonianThemodelisbasedonthe8-bandkpPidgeon-Brown(PB)modelforanarrowgapsemiconductorinastaticmagneticeldparalleltothe(001)direction(takentobethezdirection,Bz).ThePidgeon-Brownmodelisgeneralizedtoincludethewavevector(kz)dependenceoftheelectronicstatesaswellasthesp-dcouplingoftheitinerantcarrierstotheMnionlocalizeddelectrons.IusethismodeltoanalyzetheexperimentalCRspectraofInMnAsandInMnSbferromagneticsemiconductorlms.FollowingthePidgeon-Brownmodel,the8Blochbasisstatesareseparatedintoanupperandlowersetof4Blochbasisstateswhichdecoupleatthezonecenter,i.e.kz=0.TheBloch 46

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basisstatesfortheuppersetarejS"i,jHH"i,jLH#iandjSO#i,whichcorrespondtoelectronspinup,heavyholespinup,lightholespindownandsplitoffhole,respectively.Similarly,theBlochbasisstatesforthelowersetarejS#i,jHH#i,jLH"iandjSO"icorrespondingtoelectronspinup,heavyholespinup,lightholespindownandsplit-offhole,respectively.Theexplicitexpressionsforthese8BlochbasisstatesaregiveninfromEq. 3 toEq. 3 .ThetotaleffectivemassHamiltonianforthisferromagneticbulkmaterialconsistsoftheLandau(HL),Zeeman(HZ),andsp-dexchange(Hex)contributions, H=HL+HZ+Hex(5)TheexplicitexpressionoftheExchangeinteractionHamiltonianisgivenbyEq. 4 .InthepresenceofauniformmagneticeldBinthezdirection,theLandauHamiltonianHLcanbegivenby[ 20 ],HL=264LaLcLycLb375 (5)withsubmatricesHa,HbandHcaregivenby La=266666664Eg+AiV ayiq 1 3V aq 2 3V a)]TJ /F8 11.955 Tf 9.29 0 Td[(iV a)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.96 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Mip 2M)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 3V ay)]TJ /F8 11.955 Tf 9.3 0 Td[(My)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Qq 2 3V ay)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2My)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.96 0 Td[(377777775(5) Lb=266666664Eg+A)]TJ /F9 7.97 Tf 10.5 4.71 Td[(V a)]TJ /F12 11.955 Tf 9.29 13.73 Td[(q 1 3V ayiq 2 3V ay)]TJ /F9 7.97 Tf 10.5 4.71 Td[(V ay)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.95 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Myip 2My)]TJ /F12 11.955 Tf 9.3 13.72 Td[(q 1 3V a)]TJ /F8 11.955 Tf 9.3 0 Td[(M)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 2 3V a)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.95 0 Td[(377777775(5) 47

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Lc=26666666400q 2 3Vkziq 1 3Vkz00)]TJ /F8 11.955 Tf 9.3 0 Td[(L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 2 3VkzL0iq 3 2Ly)]TJ /F12 11.955 Tf 9.29 13.72 Td[(q 1 3Vkz)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2Liq 3 2Ly0377777775(5)wheretheLandaugauge,~A=xB^y,ischosenandthewavevector,~k,isgivenby ~k=1 ~~p+e c~A(5)here~p=)]TJ /F8 11.955 Tf 9.3 0 Td[(i~~risthemomentumoperator.Thecreationanddestructionoperatorsaredenedby ay= p 2(kx+iky),(5)and a= p 2(kx)]TJ /F8 11.955 Tf 11.95 0 Td[(iky).(5)Themagneticlength,,isgivenby =r ~c eB=s ~2 2m01 BB(5)whereB=5.78910)]TJ /F5 7.97 Tf 6.59 0 Td[(5eV/TeslaistheBohrmagnetonandm0isthemassofafreeelectron.Egisthebandgapofbulkmaterial,andisthespin-orbitsplitting.TheKanemomentummatrixelementV=)]TJ /F8 11.955 Tf 9.3 0 Td[(im0 ~hSjpxjXiisdenedby[ 30 ] V=s ~2 m0Ep 2.(5)whereEpistheopticalmatrixparametersTheoperatorsA,P,Q,LandMaredenedasfollowing A=~2 m04 22N+1 2+k2z,(5) P=~2 m01 22N+1 2+k2z,(5) 48

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Q=~2 m02 22N+1 2)]TJ /F4 11.955 Tf 11.96 0 Td[(2k2z,(5) L=~2 m03 )]TJ /F8 11.955 Tf 9.3 0 Td[(ip 6kza !,(5)and M=~2 m02+3 2 p 3 2a2!.(5)ThesecondterminMproportionalto(2)]TJ /F3 11.955 Tf 12.65 0 Td[(3)(ay)2isneglectedfortworeasons:1)(2)]TJ /F3 11.955 Tf 12.65 0 Td[(3)issmalland2)thistermwillcoupletheLandaumanifoldswithdifferentPBmanifoldnumberswhichwilleventuallymakethediagonalizationoftheeffectivemassHamiltonianmoredifcult.ThenumberoperatorisdenedasN=aay.Theparameters1,2,and3arerelatedtotheusualLuttingerparametersL1,L2,andL3throughtherelationsgivenbyfromEq. 3 toEq. 3 Theparameter4isgivenby[ 31 ] 4=1 me)]TJ /F8 11.955 Tf 13.15 8.09 Td[(Ep 32 Eg+1 Eg+.(5)TheZeemanHamiltonianisgivenby[ 20 ] HZ=~2 m01 2264Za00)]TJ /F8 11.955 Tf 9.3 0 Td[(Za375(5)wherethe44submatrixZaisgivenby Za=2666666641 20000)]TJ /F5 7.97 Tf 10.49 4.71 Td[(3 200001 2)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2(+1)00iq 1 2(+1)+1 2377777775.(5)TherelationshipbetweentheparameterandthetheLuttingerparameter,L,canbedenedbytherelation[ 14 ] =L+Ep 6Eg.(5) 49

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Thefollowingapproximation[ 11 32 33 ]isusedfortheLuttingerparameter,L, L=L3+2 3L2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 3L3)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2 3.(5)TheexpressionfortheexchangeinteractionHamiltonian,Hex,isgivenbyEq. 4 .Thedetaileddescriptionofthesp-dcouplingtothespontaneousmagnetizationcanbefoundintheSection 4.2.2 .IusedthestandardLuttingerparameters[ 30 ]andtemperaturedependentenergygapsforInAsandInSb 5.2.2EnergiesandWavefunctionsTheenvelopefunctionsoftheeffectivemassHamiltonianEq. 5 aregivenby Fp,=ei(kyy+kzz) p A2666666666666666666664Cp,1,(kz)p)]TJ /F5 7.97 Tf 6.59 0 Td[(1Cp,2,(kz)p)]TJ /F5 7.97 Tf 6.59 0 Td[(2Cp,3,(kz)pCp,4,(kz)pCp,5,(kz)pCp,6,(kz)p+1Cp,7,(kz)p)]TJ /F5 7.97 Tf 6.59 0 Td[(1Cp,8,(kz)p)]TJ /F5 7.97 Tf 6.59 0 Td[(13777777777777777777775(5)ThedetailsaregiveninRef.[ 20 34 ].ThecompletewavefunctionscanbeobtainedbymultiplyingtheseenvelopefunctionsbythecorrespondingzonecenterBlochbasisstatesgiveninfromEq. 3 toEq. 3 .InEq. 5 ,pisthePidgeon-Brown(PB)manifoldindex,denotestheeigenstateswithinthepthmanifold,Aistheareaofthecrosssectioninthesample'splaneperpendiculartothemagneticelddirection.Theharmonicoscillatoreigenfunctionsaredenotedasi(),whicharedeterminedat=x)]TJ /F3 11.955 Tf 12.34 0 Td[(2ky,whereisthemagneticlength.ThecomplexexpansioncoefcientsaredenotedbyCp,m,(kz)foreachtheigenstate.ThefollowingsetofmatrixeigenvalueequationscanbeobtainedbysubstitutingtheenvelopefunctionsintotheeffectivemassSchrodingerequation.Thedetailsaregiven 50

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inRef.[ 20 34 ]. HpFp,=Ep,(kz)Fp,(5)Tocalculatetheelectronicenergyeigenvaluesandeffectivemassenvelopefunctions,theeffectivemassSchrodingerequationissolvednumericallyforeachallowedvalueofthePidgeon-Brown(PB)manifoldindex,p,andzcomponentofwavevector,kz.Asi()areonlydenedfori0,onedeletestherowsandcolumnsofHpforwhichi<0.FromEq. 5 itcanbeseenthatFp,isdenedforp)]TJ /F4 11.955 Tf 23.59 0 Td[(1.Thecalculatedenergyeigenvalues,Ep,(kz),aretheLandaulevels,whereprepresentsthePBmanifoldandrepresentstheenergyeigenvalues,whichbelongtothesamePBmanifoldintheincreasingorder.Inthismodel,thes-dparameter()=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5eVandp-dinteractionparameter=1.0eVwereused[ 18 ]. 5.2.3Landaulevels,CRSpectra,andFermiEnergiesInthissection,IdiscussthetheoreticalaspectsoftheLandau-levels,CRspectraandFermienergies.ThedetailsaregiveninRef.[ 20 34 ].TheLandaulevelsatthezonecenterasafunctionofmagneticeldareobtainedusingtheenergyeigenvaluesatwavevectorkz=0,inadditionIcalculatetheLandaulevelenergyasafunctionofwavevector,kz,atagivenmagneticeldinthez-direction,Bz.TheCRabsorptionspectraareobtainedfromthemagneto-opticalabsorptionduetotransitionsbetweendifferentLandaulevels.FromFermi'sgoldenrule,themagneto-opticalabsorptioncoefcientatagivenmagneticeldBzandphotonenergy~!isgivenby[ 35 ] (~!))=e3Bz (~!)(~c)2nrXp,;p0,0Z1dkzj^e~Pp0,0p,(kz)j2(fp,(kz))]TJ /F8 11.955 Tf 11.95 0 Td[(fp0,0(kz)))]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Ep,p0,0(kz))]TJ /F13 11.955 Tf 11.96 0 Td[(~!, (5) where^eistheunitpolarizationvectoroftheincidentlight,~Pp0,0p,(kz)istheopticalmatrixelement,Ep,p0,0(kz)=Ep0,0(kz))]TJ /F8 11.955 Tf 12.32 0 Td[(Ep,(kz)istheopticaltransitionenergy,andnristhe 51

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refractiveindex.TheFermidistributionfunctionfp,(kz)inEq. 5 isgivenby fp,(kz)=1 1+exp[(Ep,(kz))]TJ /F8 11.955 Tf 11.95 0 Td[(Ef)=kT].(5)TheFermienergyEfinEq. 5 dependsonboththetemperatureandthenetcarrierconcentrationandalsoshowsaslightdependenceontheexternalmagneticeld.Thesignofthenetcarrierconcentrationdependsonthetypeofthedopingofthesample.Foragivenvaluesofthemagneticeld,temperature,andFermilevel,thenetcarrierconcentrationisgivenby NC=eBz (2)2(~c)Xp,Z1dkz(fp,(kz))]TJ /F3 11.955 Tf 11.95 0 Td[(vp,)(5)wherevp,istheKronckerdeltafunction,whichgivesthevalueeither1or0dependingonwhetherthe(p,)isaLandauvalencesubbandorLandauconductionsubandrespectively.TheFermienergycanbecalculatedfromEq. 5 usingarootndingprogramforthegivenvaluesoftemperature,netcarrierconcentrationandmagneticeld.Theopticalmatrixelementscanbegivenbythefollowingequationwhereonlythemostprominentcontributionsaretakenintoaccount. ~Pp0,0p,(kz)=Xm,m0Cp,m,(kz)Cp0,m0,0(kz)hN(p,m)jN(p0,m0)ihmj~Pjm0i (5) HereCp,m,(kz)arethecomplexcoefcientsdenedbyEq. 5 .TheorthonormalizedharmonicoscillatorwavefunctionsaredenotedasN(p,m),wheretheLandauquantumnumbersN(p,m)explicitlydependontwootherquantumnumbers,pandm,whicharedenedinEq. 5 .ThemomentummatrixelementsbetweentheBlochbasisstatesaregivenbyhmjPxjm0i,hmjPyjm0i,andhmjPzjm0iTheexplicitrepresentationsofPx,Py,andPzaregivenbelow 52

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Px=264Pax00Pbx375(5) Pax=2666666640iVq 1 2iVq 1 6Vq 1 3)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 2000)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 6000Vq 1 3000377777775(5) Pbx=2666666640)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 2)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 6iVq 1 3)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 2000)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 6000)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 3000377777775(5) Py=264Pay00Pby375(5) Pay=2666666640)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 2Vq 1 6)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 3)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 2000Vq 1 6000iVq 1 3000377777775(5) Pby=2666666640iVq 1 2)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 6)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 3)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 1 2000iVq 1 6000)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 3000377777775(5) Pz=2640PcziPcz0375(5) 53

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Pcz=26666666400Vq 2 3iVq 1 30000)]TJ /F8 11.955 Tf 9.3 0 Td[(iVq 2 3000)]TJ /F8 11.955 Tf 9.3 0 Td[(Vq 1 3000377777775(5)whereVistheKanematrixelementdenedinEq. 5 .InthenumericalcalculationoftheintegralinEq. 5 ,IhaveusedtheLorentzianfunction,(x),insteadoftheDiracdeltafunction,(x).IntheLorentzianfunction,theparameterspeciesthehalfwidthathalfmaximum. 5.3AverageZComponentsofSpinsTheaveragezcomponentofthespinsisgivenbythefollowingrelation:hzi=Pp,R1dkz(fp,(kz))]TJ /F3 11.955 Tf 11.96 0 Td[(vp,)hFp,jzjFp,i Pp,R1dkz(fp,(kz))]TJ /F3 11.955 Tf 11.95 0 Td[(vp,) (5) 5.4ResultsFigure 5-1 AshowstheCRmeasurementsforMBEgrownInMnSb(sampleA)andMOVOPEgrownInMnAswith2%Mncontent.ForthesameMncontent,theMBEgrownInMnSbhasmoreholedensitycomparedtotheMOVPEgrownInMnAs.ThisfacthasbeenreectedinalargercyclotronmassintheInMnSb(A)thanthatinMOVPEgrownInMnAs.ThelargerholedensityandtheFermienergywillbeassociatedwiththeCRoriginatedfromLandauleveltransitionsathighermagneticeldsandtherefore;thecyclotronmasscanbeenhanced.TheCRoftheholesinInMnSbhasbeenobservedforthersttime,andinthecaseofFig. 5-1 A,theCRmassesare0.057m0atRTand0.051m0at121K.ThesearemuchsmallerthanthebandedgeHHmassinInSb(0.32m0).TheCRabsorptionspectra,showninFig. 5-1 B,areobtainedfromthemagneto-opticalabsorptionduetotransitionsbetweendifferentLandaulevels.FromFermi'sgoldenrule,themagneto-opticalabsorptioncoefcientsatagivenphoton 54

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energyandforamagneticeldperpendiculartothesamplecanbecalculated.Thedetailsofthecalculationsaredescribedinthetheorysection.InordertodemonstratetheeffectoftheMncontentonthebandstructure,asshowninFig. 5-2 ,theenergyspectrumasafunctionofmagneticeldforMncontent,rangingfromzeroto4%hasbeencalculated.Thebandstructureandmorespecically,thebandsatthetopofthevalencebandcanbemodiedstronglywithincreasingtheMn.Itisimportanttonotethatthetopmostband,indicatedbytheblueline,isapurelyspinstatefortheMncontentof2%from0-80Teslaandfrom0-120Teslafor4%Mn.Figure 5-3 AshowsthecalculatedTotalCRtransmission,aswellastheindividualtransmissionsduetothetransitionsbetweentwoparticularLandaulevelswithdifferentPidgeon-Brown(PB)manifoldindices(-1,0,1,2,3,4,..),forthemeasurementinFig. 5-1 A.ThesameconventionofnotationismaintainedintheFig. 5-7 B.ThevalencebandstructureforT=121KandB=47TforInMnSb(A)with2%Mncontent,ispresentedinFig. 5-3 Bdemonstratingtwopossibletransitions,andFig. 5-3 CshowsLandaulevelscalculationsfortheCRtransitionsinInMnSbarisingfrombothLHandHHtransitions.InFig. 5-3 BandFig. 5-3 C,eachcolorrepresentsaLandaulevelwithaspecicPBmanifoldindex(-1,0,1,2,..)andtheFermienergyisalsoshownasareddashedline.Ialsousethenotation(p,q)todenotetheLandaulevelswhichareresponsibleforthetransitions,wherepcorrespondstothePBmanifoldindexandqrepresentsthestatewithinthemanifold.IwillmaintainthesameconventionsofnotationsforalltheupcomingguressimilartoFig. 5-3 .AsshowninFig. 5-4 fromadifferentprospective,foraxedmagneticeldof47TeslawhereseveralCRtransitionshavebeenobserved,thebandmixingbetweendifferentmanifoldreduceswithincreasingtheMncontent.InordertoprobetheCRindifferentmagneticeldregions,theCRspectrainInMnAsandInMnSbat16.9mand5.53mweremeasured.Figure 5-5 Ashowsthemeasurementsat16.9mwheretheresonancefeatureofInMnAsislessclearthanthat 55

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Figure5-1. CRspectraforInMnAsandInMnSblms.A)ExperimentalCRspectraforInMnAsandInMnSblms.B)CalculatedCRspectraforInMnAsandInMnSbwheretheCRresonanceinInMnAsoriginatesfromasingletransition;whereas,inInMnSbitarisesfrommultipletransitions.InthecaseoftheInMnSb,abetterttoexperimentaldatawasgeneratedforaholedensityof3.01020cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3.FromRef.[ 28 ]. 56

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Figure5-2. Zone-centerLandauvalancesubbandenergies(atT=121KandTc=330K)asfunctionsofmagneticeldinIn1)]TJ /F9 7.97 Tf 6.58 0 Td[(xMnxAsandIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xMnxSbfordifferentMnconcentrations(x).FromRef.[ 28 ]. 57

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Figure5-3. CalculatedmanifoldresolvedCRspectraandLandauvalencesubbandstructureinInMnSbat10.7m.A)ManifoldresolvedCRspectraofInMnSbat10.7m.B)ZonecenterLandauvalencesubbandenergiesatT=121Kasfunctionsofmagneticeldinp-dopedInMnSb.TheFermienergyisshownfortheholeconcentration,p=11020cm)]TJ /F5 7.97 Tf 6.58 0 Td[(3.FromRef.[ 28 ]. 58

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Figure5-4. ValencebandstructureatB=47T(withT=121KandTc=330K)inIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xMnxAsandIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xMnxSbfordifferentMnconcentrations(x).FromRef.[ 28 ]. 59

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foundat10.6and10.7m.Thisfactcanbeduetothelowmobilityofthesampleandtheplasmaabsorptioneffect.ForInMnSb,theresonancepatternisdifferentcomparedtoatypicalsingleresonancepeak.AccordingtotheLandaulevelcalculations,theoverlapofthelightholeandheavyholeresonancescanresultinsuchanunusualCRpattern.Figure 5-6 AshowstheCRtracesforthethreeMOVPEgrownInMnAswithdifferentMncontentsat5.53m.ThesampleswiththelowestMncontenthaveanonsymmetricresonancelineshapewhichcanbeexpectedforsampleswithlowcareermobility.AstheMncontentincreases,thepositionoftheCRshiftstothehighermagneticeldsandthelinebroadeningdecreases,suggestionahigherholemobility(750cm2=Vs)inthesamplewith4%Mncontentcomparedtotheothertwostructures.TheareaunderthecurveoftheCRtracesisrelatedtheholeconcentrationswheretuningtheMncontentfrom2%to4%didn'tincreasetheconcentrationssignicantly.ThisfactcouldberelatedtothelackofvariationsintheTcwiththesmallvariationsintheMncontentsinthismaterialsystems.TheincreaseintheholeconcentrationasafunctionoftheMncontenthasbeenattributedtoincreasetheexchangeinteraction.LandaulevelcalculationspredictthepositionsoftheCRinallthreesamples,presentedinFig. 5-6 BforInMnAswith4%MnandinFig. 5-7 fortheothertwosamples.CRspectraweremeasuredforoneMOVPEgrownandtwoMBEgrownInMnSblmsusingthe5.53mexcitation.TheCRoftheMOVPEgrownstructureatthiswavelengthwasobservedonlyatlowtemperatures,similartothecasefortheInMnAslmshowninFig. 5-6 B.AsshowninFig. 5-8 B,fortheMOVPEgrownInMnSb,tworesonanceswereobservedandthebandstructurecalculationinFig. 5-8 Bshowsthepossibletransitionsforthestrongandweakresonance.Figure 5-9 ,theLandaulevelcalculations,relatetheoriginofthestrongresonanceandweakresonancetotheCRoriginatingfromtheHHandLHtransitionsrespectively. 60

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Figure5-5. CRspectraandLandauleveltransitionsat16.9m.A)CRspectrainInMnAsandInMnSbat16.9m.B)CalculatedCRspectraforInMnAsandInMnSbfromtheexperimentalresultsinFig. 5-5 A.C)TheLandauLevelcalculationatT=7Kinp-dopedInMnAsdemonstratesonlyonepossibletransition.TheFermienergyisshownfortheholeconcentration,p=21018cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3.FromRef.[ 28 ]. 61

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Figure5-6. CRspectraandLandauleveltransitionsat5.53m.A)CRspectrainInMnAsat5.53musingthehole-activecircularlypolarizationofradiation.B)CalculatedCRspectraforInMnAscorrespondingtotheexperimentalresultsinFig. 5-6 A.C)CalculatedLandauLevelsforInMnAswith4%Mncontentfor5.53m.FromRef.[ 28 ]. 62

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Figure5-7. LandaulevelsandvalencebandstructureforInMnAs.A)CalculatedLandauLevelsforInMnAswith2%Mncontent,demonstratingonepossibleCRtransitionfor5.53mexcitation.B)CalculatedvalencebandstructureforT=40KandB=47Tinp-dopedInMnAswiththeweakintersubbandtransitioncorrespondingtotheCRspectrumofInMnAswith1%Mncontent.TheFermienergyisshownfortheholeconcentrationofp=4.81018cm)]TJ /F5 7.97 Tf 6.58 0 Td[(3.FromRef.[ 28 ]. 63

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Figure5-8. MeasuredandcalculatedCRspectrafortheMOVPEgrownInMnSb.A)CRfortheMOVPEgrownInMnSb.B)Thecalculatedtransmissionindicatingthetwoobservedresonances,oneweakfeaturearound20Tandastrongresonancearound60T.FromRef.[ 28 ]. 64

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Figure5-9. CalculatedLandauLevelsandvalencebandstructureforInMnAswith5.6%Mncontent.A)LandauLevelsforInMnAswith5.6%Mncontent,demonstratingthestrongCRtransitionfor5.53mexcitation.B)Bandstructureofthep-dopedInMnSbforB=20TwiththeweakintersubbandtransitioncorrespondingtotheCRspectruminFig. 5-8 A.Thewavevectorkisinthedirectionofmagneticeld.TheFermienergyisshownfortheholeconcentration,p=1.91018cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3.FromRef.[ 28 ]. 65

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Figure5-10. CRspectrafortheMBEgrownInMnSb.A)CRwithelectronandholeactivepolarizationforMBEgrownInMnSb.B)CalculatedCRspectrafortheMBEgrownInMnSb(B)withtheelectron-active,demonstratingseveraltransitionsintheconductionband,andthehole-activepolarizationoflight.FromRef.[ 28 ]. Fig. 5-10 AshowsthemeasuredCRspectrafor+(e-active)and)]TJ /F1 11.955 Tf 12.63 0 Td[((h-active)circularpolarizationoflightatroomtemperature,withtheholedensityof1.01020cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3.Fig. 5-10 BshowsthecorrespondingcalculatedCRspectrafordifferentpolarization. 66

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ThedifferencesinthecarrierdensitybetweentheMBEgrwonandMOVPEgrownstructurescouldbeanimportantfactorinthedifferencesintheirCurietemperatures.ThepositionoftheFermilevelisalsoimportanttodenethemagnitudeoftheaveragespininferromagneticsemiconductors.Figure 5-11 Aand 5-11 BshowthepositionoftheFermilevelsfortheMBEgrownInMnSb(A)andMOVPEgrownInMnSb,respectively.TheholedensityofMOVPEgrownsampleisafactor100smallerthantheMBEstructurewherecouldsignicantlychangethez-componentoftheaveragespinasshowninFig. 5-11 C.ThischangeoccursbecauseinthePidgeon-Brownmodel,atlowdensities,onlythelowestenergystate[topmostlevel,showninblueinFig. 5-11 A]isoccupied.Thisstateisa100%heavyholedownstate,whichisapurespin-downstate.Athigherdensities,theFermilevelisinthemiddleofthevalencebandstatesandhenceseveralofthevalencebandstatesareoccupied[Fig. 5-11 B].Thehigherlevelvalencebandstatesaremixturesofheavyandlightholestatesandasaresult,spinisnotagoodquantumnumber.Inaddition,severalvalencebandstatesareoccupiedandtheaveragespinpolarizationgoesdown. 5.5DiscussionIntheexperimentalobservations,theCRoriginatesonlyfromthetransitionsinthevalenceband,suggestingtheabsenceofthemidgapimpuritystates.Therecanbetwoeffectsoftheimpuritystatesinteractingwiththevalencebandstates.Thersteffectcomesfromtheexchangeinteractionbetweentheimpurityspinsandthevalencebandspins.Thiseffectisincludedinthemodel.Thesecondeffectcanresultfromhybridizationoftheimpuritylevelswiththevalencebandstates.Idonotincludethishybridizationeffectinthemodel,sinceIhaveassumedthattheimpuritystatesliedeepwithinthevalencebands;hencethishybridizationisunimportantfortheCR.ThismightbedifferentforGaMnAs[ 36 ],wheretheimpuritystatesmightlieinthegaporneartheedgeofthevalencebandstates.IhaveassumedthattheimpuritybandstatesliedeepwithinthevalencebandandnotinthegaporneartheedgeforInMnAsand 67

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Figure5-11. CalculatedLandaulevelsandaveragespinfortwodifferentInMnSbsamples.CalculatedLandaulevelsinp-dopedIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xMnxSbforA)x=5.6%(withTc=590KandT=61K)andB)x=2%(withTc=10KandT=290K).TheFermienergiesareshownasreddottedlineforp=1.91018cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3andp=31020cm)]TJ /F5 7.97 Tf 6.59 0 Td[(3.C)Thecalculatedz-componentoftheaveragePaulimatrixcorrespondingtothesetwodifferentsamples.FromRef.[ 28 ]. 68

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InMnSbbecauseiftheydid,IshouldseeimpuritystateCRtransitions,whichwouldbedistinguishedfromthevalencebandCR.Experimentally,impuritybandCRisnotseeninInMnAs[ 29 ].Furthermore,theobservedvalencebandCRisfairlyaccuratelyreproducedbythemodelwhichindicatesthateithertheimpuritybandisdeepwithinthevalenceband,orifitisneartheedge,thehybridizationwiththevalencebandisweak.ThismaynotbethecaseinGaMnAs,butInotedthatCRhasnotbeenobservedinGaMnAsandthismightbeareasonforthat.WhileInddifferencesinthepeakpositionsoftheCRpeaksbetweentheMBEandMOVPEsamples,thesedifferencescanbeattributedtothedifferencesincarrierdensityandhencethepositionoftheFermilevelsinthesamples.ThechangeinFermilevelalsoplaysanimportantpartindeterminingtheaveragezcomponentofthespins.Forlowdoping,theFermilevelliesinthelowestenergy(heavy-holedown)valencebandwhichisapurespinstate.Athigherdoping,theFermilevelliesintothevalencebandsandseveralstatesareoccupiedandtheaveragespingoesdown.AsshowninFig. 5-11 CforInMnSbgrownbyMOVPEwith5.6%Mncontent,thelowercarrierdensityinthestructureresultsinamuchhigheraveragezcomponentofthespinscomparedtotheMBE-grownstructure.ThisfactcouldberesponsibleforthehigherCurietemperatureintheMOVPEstructureswhereinthiscase,thetopmostlevelisapurespinstatefrom0-80T. 69

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CHAPTER6SPINPOLARIZATIONANDOPTICALLYPUMPEDNMRINSTRAINEDMULTIQUANTUMWELLSYSTEMInthepresenceofanexternalstaticmagneticeldalonganycrystaldirectionofanyIII-VQuantumWell(QW)structurethemotionoftheBlochelectronisdescribedasthequantizationofthecyclotronorbits,whicharecalledtheLandaulevels.SoifthemagneticeldisappliedalongzdirectionthentheenergyoftheBlochelectronisquantizedintheplaneperpendiculartothemagneticeld.Notethatinaddition,theBlochelectronisconnedinthezdirection,whichisthegrowthdirectionoftheQW.Moreover,therewillbeastrainwhicharisesfromthestraingeneratedintheQWsystemduringthecoolingprocess.SotocalculatethebandstructureofthiskindofQWsystemIhavetoincludetheeffectsofquantumconnementandstrain.Inthischapter,Ipresentthecalculatedelectron(conductionband)spinpolarizationinthestrainedAl0.1Ga0.9As/GaAsmultiquantumwell(MQW)structureresultsforthebothhelicitiesofpolarizedlightandcomparethemwiththecorrespondingopticallypumpedNMR(OPNMR)experiments.InOPNMR,either+orleftcircularlypolarizedphotonsareusedtoexcitetheelectronstotheconductionbandssothatthedesiredpolarizationoftheconductionelectronscanbeobtained.Thentheseelectronswillcoupleweaklywithnucleibyhypernecoupling.Thenaftercross-relaxation,thenucliewillbepolarizedwhicheventuallyenhancestheintensityoftheNMRsignals[ 37 40 ]. 6.1ExperimentalDetailsTheexperimentwascarriedoutonanAl0.1Ga0.9As/GaAsmultiquantumwell(MQW)structurebythecollaborators,Prof.BowersandRyanWoodatUniversityofFlorida,Dr.Kuhns,Dr.McGill,andDr.ReyesatNHMFL.ThisMQWstructurewasepoxybondedtoaSisupport.Thissampleconsistsof21wellswithGaAswellwidth30nmandAl0.1Ga0.9Asbarrierwidth360nm.ThesampleisSi--dopedwith2Delectrondensity71010cm)]TJ /F5 7.97 Tf 6.59 0 Td[(2.AnNMRcoilwithdimensionsslightlylargerthanthesamplewasused.InsidetheNMRcoil,thesamplewasinthephysicalcontacttothesapphire 70

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blocktomitigatethelaserheating.ThesapphireblockwasinstalledintoaNMRprobeandtheNMRcoilwascomposedofanLCcircuitwhichcouldbetunedtothedesiredresonantfrequencyofthenucleusatlowtemperature.Theexperimentbeginswiththe Figure6-1. Banddiagraminatype-Iheterostructuresemiconductormaterials saturationofthenuclearspinpolarizationwhichisachievedbyatrainofresonantpulses[ 41 ].Thenthemagneticeldwasrampedataconstantrateuntilthetargetmagneticeldhadbeedreached.Whenthetargetmagneticeldwasachieved,thesamplewasopticallypumpedusingalaserlightsourcefor20-80sec.ThesamplewasthenrampeddowntoadetectionmagneticeldandtheopticallypumpedNMR(OPNMR)signalwasacquired.Atagivenmagneticeld,theOPNMRspectrawasacquiredbyvaryingthephotonenergyandpolarizationofthelightsource.Theexperimentwasrepeatedatmagneticelds3.9T,4.9T,and9.4Twithtemperaturearound6Kfordifferentpolarizationsofthelight.Thewavelength(orphotonenergy)tunableTi:SapphireLaserwasusedfortheopticalpumping. 71

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6.2TheoryandModeling 6.2.1EnvelopeFunctionFrameworkFig. 6-1 showsconductionandvalencebandprolesinatype-IheterostructurematerialwhichismadeoftwodifferenttypesofbulksemiconductorlayersAandB.Inatype-Iheterostructurematerial,thesamelayeractsaseitherawellorabarrierbothforconductionandvalencebandelectrons.NotethatthelayerAiswellforboththeconductionandvalencebandelectrons.Inthesameway,thelayerBisbarrierforboththeconductionandvalencebandelectrons.Someexamplesofthetype-IheterostructuresareGaAs/AlGaAs,InSb/AlInSb,etc.Theenvelopefunctionframeworkisbasedontwofollowingkeyassumptions[ 42 ]:1)Ineachlayeroftheheterostructure,thewavefunctioncanbeexpandedbythecellperiodicpartsofthezonecenterBlochfunctionsorBlochbasisstates (r)=Xf(A)(r)u(A)n,k0(r) (6) (r)=Xf(B)(r)u(B)n,k0(r) (6) whereun,k0(r)isthefastvaryingcellperiodicpartoftheBlochfunctionsatthebandedgek0andf(r)areslowlyvaryingenvelopefunctions.Thesummationoverrunsoverthenumberofblochbasisstatesinthesystem.2)Ineachlayeroftheheterostructure,thecellperiodicpartsoftheBlochfunctionsareassumedtobethesame.u(A)n,k0(r)u(B)n,k0(r) (6)Thusthewavefunctionofanyheterostructurecanbewrittenas, (r)=Xf(A,B)(r)un,0(r) (6)wherek0=0istakenasextremumpoint. 72

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SotheEq. 6 impliesthatthetheinterbandmatrixelementisthesameinbothlayers.Letthegrowthaxisbeinthezdirectionandtheplanez=z0betheinterface.Nowasun,0arelinearlyindependentand (r)iscontinuousattheinterfacez=z0therefore,f(A)(r?,z0)=f(B)(r?,z0) (6)wherer?isabi-dimensionalvector.Thef'sarefactorizedinto: f(A)(r?,z)=1 p Aexp(ik?r?)U(A)(z) (6) f(B)(r?,z)=1 p Aexp(ik?r?)U(B)(z) (6) orinshortf(A,B)(r?,z)=1 p Aexp(ik?r?)U(A,B)(z) (6)whereAisthecross-sectionalareaofthesampleandk?=(kx,ky)isabi-dimensionalwavevector.IftheheterostructuresareobtainedbygrowingasequenceoftwolayersAandBrepetitivelyalongthezdirectionthenthesuperlatticewavevectorkzfollowsthefollowingcondition)]TJ /F3 11.955 Tf 10.49 8.09 Td[( L
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wavevector(kz)dependenceoftheelectronicstates.ThedetailedcalculationsoftheHamiltonianmatriceswillbefoundintheC.K.Pidgeon,etalpaper[ 11 ].Ialsoincludethequantizationofkzwhichcomesfromthemultiplequantumwellsuperlatticeeffect.Inaddition,Itakethestraineffectintoaccountbycalculatingthestraintensorelementsfromarelationthatassociatestheexperimentalvalueofthequadruplesplittingwiththestrainalonganygivendirection.FollowingthePidgeon-Brownmodel,Iseparatethe8Blochbasisstatesintoanupperandlowersetof4Blochbasisstateswhichdecoupleatthezonecenter,i.e.kz=0.TheBlochbasisstatesfortheuppersetarejS"i,jHH"i,jLH#iandjSO#i,whichcorrespondtoelectronspinup,heavyholespinup,lightholespindownandsplitoffhole,respectively.Similarly,theBlochbasisstatesforthelowersetarejS#i,jHH#i,jLH"iandjSO"icorrespondingtoelectronspinup,heavyholespinup,lightholespindownandsplit-offhole,respectively.Theexplicitexpressionsforthese8BlochbasisstatesaregiveninfromEq. 3 toEq. 3 .TheeffectivemassHamiltonianforthisstrainedQWsystemconsistsoftheLandau,Zeemanandstraincontributions,i.e., H=HL+HZ+HS.(6)TheexplicitformofLandauandZeemanHamiltonianaregiveninRef.[ 20 ]InthepresenceofauniformmagneticeldBinthezaxis,theLandauHamiltonianHLcanbegivenby[ 20 ],HL=264LaLcLycLb375 (6)withsubmatricesHa,HbandHcaregivenby 74

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La=266666664Eg+AiV ayiq 1 3V aq 2 3V a)]TJ /F8 11.955 Tf 9.29 0 Td[(iV a)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.96 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Mip 2M)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 3V ay)]TJ /F8 11.955 Tf 9.3 0 Td[(My)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Qq 2 3V ay)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2My)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.96 0 Td[(377777775(6) Lb=266666664Eg+A)]TJ /F9 7.97 Tf 10.5 4.7 Td[(V a)]TJ /F12 11.955 Tf 9.29 13.72 Td[(q 1 3V ayiq 2 3V ay)]TJ /F9 7.97 Tf 10.5 4.71 Td[(V ay)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F8 11.955 Tf 11.95 0 Td[(Q)]TJ /F8 11.955 Tf 9.3 0 Td[(Myip 2My)]TJ /F12 11.955 Tf 9.3 13.73 Td[(q 1 3V a)]TJ /F8 11.955 Tf 9.3 0 Td[(M)]TJ /F8 11.955 Tf 9.3 0 Td[(P+Qip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 2 3V a)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M)]TJ /F8 11.955 Tf 9.29 0 Td[(ip 2Q)]TJ /F8 11.955 Tf 9.3 0 Td[(P)]TJ /F4 11.955 Tf 11.95 0 Td[(377777775(6) Lc=26666666400q 2 3V^kziq 1 3V^kz00)]TJ /F8 11.955 Tf 9.3 0 Td[(L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2L)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 2 3V^kzL0iq 3 2Ly)]TJ /F12 11.955 Tf 9.29 13.73 Td[(q 1 3V^kz)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2Liq 3 2Ly0377777775(6)wheretheLandaugauge,~A=xB^y,ischosenandthewavevector,~k,isgivenby ~k=1 ~~p+e c~A(6)here~p=)]TJ /F8 11.955 Tf 9.3 0 Td[(i~~risthemomentumoperator.Sothethez-componentofthewavevectorwillbereplacedbythefollowingoperator^kz=)]TJ /F8 11.955 Tf 9.3 0 Td[(i^z@ @z (6)Thecreationanddestructionoperatorsaredenedby ay= p 2(kx+iky),(6)and a= p 2(kx)]TJ /F8 11.955 Tf 11.95 0 Td[(iky).(6) 75

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Themagneticlength,,isgivenby =r ~c eB=s ~2 2m01 BB(6)whereB=5.78910)]TJ /F5 7.97 Tf 6.59 0 Td[(5eV/TeslaistheBohrmagnetonandm0isthemassofafreeelectron.Egisthebandgapofbulkmaterial,andisthespin-orbitsplitting.TheKanemomentummatrixelementV=)]TJ /F8 11.955 Tf 9.3 0 Td[(im0 ~hSjpxjXiisdenedby[ 30 ] V=s ~2 m0Ep 2.(6)whereEpistheopticalmatrixparametersTheoperatorsA,P,Q,LandMaredenedasfollowing A=~2 m04 22N+1 2+^k2z,(6) P=~2 m01 22N+1 2+^k2z,(6) Q=~2 m02 22N+1 2)]TJ /F4 11.955 Tf 11.96 0 Td[(2^k2z,(6) L=~2 m03 )]TJ /F8 11.955 Tf 9.3 0 Td[(ip 6^kza !,(6)and M=~2 m02+3 2 p 3 2a2!.(6)ThesecondterminMproportionalto(2)]TJ /F3 11.955 Tf 12.41 0 Td[(3)(ay)2isneglectedfortworeasons:1)(2)]TJ /F3 11.955 Tf 12.28 0 Td[(3)issmalland2)thistermwillcoupletheLandaumanifoldswithdifferentPBmanifoldnumberswhichwilleventuallymakethediagonalizationoftheeffectivemassHamiltonianmoredifcult. 76

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ThenumberoperatorisdenedasN=aay.Theparameters1,2,and3arerelatedtotheusualLuttingerparametersL1,L2,andL3throughtherelationsgivenbyfromEq. 3 toEq. 3 .Theparameter4isgivenbyEq. 5 .TheexplicitformofZeemanHamiltonian(HZ)isgiveninEq. 5 .ThestrainHamiltonianisgivenby[ 43 45 ] HS=264SaScSycSb375(6)thesubmatricesSa,SbandScaregivenbelow Sa=266666664A"0000)]TJ /F8 11.955 Tf 9.3 0 Td[(P")]TJ /F8 11.955 Tf 11.95 0 Td[(Q")]TJ /F8 11.955 Tf 9.3 0 Td[(M"ip 2M"0)]TJ /F8 11.955 Tf 9.3 0 Td[(M")]TJ /F8 11.955 Tf 9.3 0 Td[(P"+Q"ip 2Q"0)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M")]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2Q")]TJ /F8 11.955 Tf 9.3 0 Td[(P"377777775,(6) Sb=266666664A"0000)]TJ /F8 11.955 Tf 9.3 0 Td[(P")]TJ /F8 11.955 Tf 11.96 0 Td[(Q")]TJ /F8 11.955 Tf 9.3 0 Td[(M"ip 2M"0)]TJ /F8 11.955 Tf 9.3 0 Td[(M")]TJ /F8 11.955 Tf 9.3 0 Td[(P"+Q"ip 2Q"0)]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2M")]TJ /F8 11.955 Tf 9.3 0 Td[(ip 2Q")]TJ /F8 11.955 Tf 9.3 0 Td[(P"377777775,(6)and Sc=266666664000000)]TJ /F8 11.955 Tf 9.29 0 Td[(L")]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2L"0L"0iq 3 2L"0)]TJ /F8 11.955 Tf 9.3 0 Td[(iq 1 2L"iq 3 2L"0377777775.(6)ThequantitiesA",P",Q",L"andM"inEqs. 6 6 aregivenbythefollowingrelationsintermsofthestraintensorcomponents"i,j, A"=ac("xx+"yy+"zz),(6) 77

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P"=)]TJ /F8 11.955 Tf 9.3 0 Td[(av("xx+"yy+"zz),(6) Q"=)]TJ /F8 11.955 Tf 10.5 8.08 Td[(b 2("xx+"yy)]TJ /F4 11.955 Tf 11.96 0 Td[(2"zz),(6) L"=id("xz)]TJ /F8 11.955 Tf 11.96 0 Td[(i"yz),(6)and M"=)]TJ 10.49 18.04 Td[(p 3 2b("xx)]TJ /F3 11.955 Tf 11.95 0 Td[("yy)+i2p 3 3d"xy.(6)wherethevaluesofthedeformationpotentialsac,av,b,anddcanbefoundinRef.[ 30 ].Sinceinthesample,"xx="yyandthereisnoshearstraininthesystem;i.e."xy="yz="xz=0,thetermL"inandthetermM"vanish.Asaresult,Sa=SbandSc=0inEq. 6 6.2.3QuantumConnementPotentialIconsideranndopedAlGaAs/GaAstype-IsuperlatticeheterostructureonathickAlGaAsbufferlayeronaSisubstrate.Sinceitisatype-Isuperlattice,thebandgapoftheGaAswelllayercompletelylieswithinthebandgapoftheAlGaAsbarrierlayerTheAl-concentrationbothforthebarrierandbufferlayersis10%.Thebarrierlayeristhickenoughtopreventinteractionsamongthewells.ThebandgapmismatchbetweenthelayersAlGaAsandGaAsisgivenby,Eg=Eg(AlGaAs))]TJ /F8 11.955 Tf 11.95 0 Td[(Eg(GaAs) (6)TheconductionbandbarrierheightVCBisgivenbyVCB=QCBEg (6)whereQCBistheconductionbandoffset. 78

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ThevalencebandbarrierheightVVBisgivenbyVVB=QVBEg (6)whereQVBisthevalencebandoffsetwhichisinfactQCB=1)]TJ /F8 11.955 Tf 11.95 0 Td[(QVB.HeretheEgistemperaturedependentwhichiscalculatedfromVarshni'sempiricalformula[ 46 ]. Eg(T)=Eg(T=0K))]TJ /F3 11.955 Tf 17.87 8.09 Td[(T2 T+,(6)whereandarecalledVarshniparameters,whosevaluesfortheAlGaAsandGaAslayersarefoundinRef.[ 30 ].ThevaluesofVCBandVVBarecalculatedfortheconductionbandoffset,QCB=.6IhavetoaddtheseasconnementpotentialsVCBandVVBintheeffectivemassHamiltonianasafunctionofz,whichisthegrowthdirectionofQWsystem 6.2.4ElasticStrainandQuadrupoleSplittingTocalculatethestrainintheGaAsquantumwellfromthemeasuredquadrupolesplittingintheNMRsignal,Irstrelatethequadrupolesplitting(Qtothecomponentoftheelectriceldgradienttensoralongthelabframez0axis(thedirectionoftheappliedmagneticeld),Vz0z0.Thequadrupolesplitting(QisdenedasthedifferenceinthefrequenciesofthecentralandsatellitetransitionsintheNMRsignal.ForanucleusofspinI>1/2,thefrequencyofanysingletransitionisgivenby[ 47 ]mI!mI)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(0=3(2mI)]TJ /F4 11.955 Tf 11.96 0 Td[(1)eQ 4I(2I)]TJ /F4 11.955 Tf 11.95 0 Td[(1)hVz0z0 (6)wheremItakesthevalues)]TJ /F21 11.955 Tf 9.3 0 Td[(I,)]TJ /F21 11.955 Tf 9.3 0 Td[(I+1,,+I,0isthenuclearLarmorfrequency,Qisthequadrupolemomentofthenucleus,hisPlanck'sconstant,andVz0z0=@2V @z0@z0,whichisthesecondderivativesoftheelectricpotentialV.NowforI=3 2,mI=)]TJ /F5 7.97 Tf 10.5 4.71 Td[(3 2,)]TJ /F5 7.97 Tf 10.5 4.71 Td[(1 2,1 2and3 2. 79

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SousingEq. 6 andtheselectionrules,theallowedthreetransitionsaregivenby +3 2!+1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(0=eQ 2hVz0z0 (6) +1 2!)]TJ /F15 5.978 Tf 16.25 3.26 Td[(1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(0=0 (6) )]TJ /F15 5.978 Tf 7.79 3.25 Td[(1 2!)]TJ /F15 5.978 Tf 16.25 3.25 Td[(3 2)]TJ /F3 11.955 Tf 11.96 0 Td[(0=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(eQ 2hVz0z0 (6) wheretheEq. 6 andEq. 6 representthefrequenciesofthe1stand2ndsatellitetransitionsrespectively,andEq. 6 representsthefrequencyofthecentraltransitiononly.NowaccordingtothedenitionofthequadrupolesplittingQ=+3 2!+1 2)]TJ /F4 11.955 Tf 11.96 0 Td[(+1 2!)]TJ /F15 5.978 Tf 16.25 3.25 Td[(1 2=+1 2!)]TJ /F15 5.978 Tf 16.25 3.25 Td[(1 2)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F15 5.978 Tf 7.78 3.25 Td[(1 2!)]TJ /F15 5.978 Tf 16.25 3.25 Td[(3 2 (6)SotheEq. 6 takesthefollowingformVz0z0=2h eQQ (6)Inthecaseoftheuniaxialelectriceldgradienttensor,theelectriceldgradienttensorcomponentalongthez0axis(Vz0z0)ofthelaboratoryframe(appliedmagneticelddirection)canberelatedtotheelectriceldgradienttensorcomponentalongthezprincipalaxis(Vzz)usingthefollowingrelation[ 48 ]Vz0z0=Vzz 2(3cos2)]TJ /F4 11.955 Tf 11.96 0 Td[(1) (6)whereistheanglebetweentheprincipleaxiszandthemagneticelddirectionz0.Inthisexperiment,themagneticeldisappliedalongthedirectionoftheprincipleaxisz.Therefore,=0.i.e.,Vz0z0=Vzz 80

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FinallytheEq. 6 canbewrittenintermsoftheelectriceldgradienttensorcomponentalongthezprincipalaxisVzz=2h eQQ (6)Nowthecomponentsoftheeldgradienttensorintheprincipalaxisframe,Vij,canberelatedthecomponentsoftheelasticstraintensor,"kl,usingthecomponentsofafourthrankgradientelastictensor,S0ijkl[ 49 ]Vij=Xk,lS0ijkl"kl(i,j,k,l=x,y,z) (6)whereVij=@2V @xi@xj.xiandxjtakethevaluesx,yandzfori,j=1,2and3respectively.Andx,yandzaretheprincipalaxesoftheelectric-eld-gradienttensor.Nowthenonvanishingcomponentsofthegradientelastictensor,whichsatisfythefollowingrelationsduetothecrystalsymmetryoftheGaAs,canbewrittenintheVoightnotation: S011=S022=S033(6)and S012=S013=S023=S021=S031=S032(6)Inthissamplethereisnoshearingstrain.Therefore,intheabsenceoftheshearingstrainandbecauseoftherelationsEq. 6 andEq. 6 theEq. 6 reducesto 266664VxxVyyVzz377775=266664S011S012S012S012S011S012S012S012S011377775266664"xx"yy"zz377775(6) 81

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SofromEq. 6 with"xx="yy(sincethestrainsalongxandydirectionsareequaltoeachother)IgetVzz=2S012"xx+S011"zz (6)Iassumethatthechargedistributiongeneratingtheelectricpotential(V)isexternaltotheGa69nucleus[ 49 ].ThusapplyingtheLaplace'sEquationIgetr2V=0 (6))Vxx+Vyy+Vzz=0 (6)whereVxx=@2V @x@x,Vyy=@2V @y@y,andVzz=@2V @z@zrespectively.AndVistheelectricpotential.InordertosatisfytheLaplace'sequationtheelectriceldgradienttensorshouldbetraceless.NowusingtherelationEq. 6 intheEq. 6 itcanbeshownthat S011=)]TJ /F4 11.955 Tf 9.3 0 Td[(2S012(6)NowsubstitutingthetheEq. 6 intoEq. 6 IgetVzz=()]TJ /F3 11.955 Tf 9.29 0 Td[("xx+"zz)S011 (6)NowsubstitutingtheEq. 6 intoEq. 6 andrearranginggive)]TJ /F3 11.955 Tf 9.3 0 Td[("xx+"zz=2h eQS011Q (6)Nowintheabsenceoftheshearstrain,foranyisotropicmateriallikeGaAscrystal,thecomponentsofthestraintensor,"ij,canberelatedtothecomponentsofthestresstensor,klusingthecomponentsofthecompliancetensor,Sij,intermsofVoight 82

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notation,0BBBB@"xx"yy"zz1CCCCA=0BBBB@S11S12S12S12S11S12S12S12S111CCCCA0BBBB@xxyyzz1CCCCA (6)Inthissample,thereareequalstressesalongthexandydirections,i.e.xx=yy=.Andthereisnostressalongzdirection,i.e.zz=0.NowusingtheserelationsintheEq. 6 givethefollowings "xx="yy=(S11+S12) (6) "zz=2S12 (6) RearrangingEq. 6 andEq. 6 givestensilestraincomponents(xandydirection)"xx="yy=1 21+S11 S12"zz (6)FinallysubstitutingEq. 6 intoEq. 6 andrearranginggivesthevalueofthestrainalongthezdirectionoftheQWsample"zz=4h eQS011"1 1)]TJ /F25 7.97 Tf 13.15 5.18 Td[(S11 S12#Q (6)ThevaluesoftheparametersS11andS12aregiveninRef.[ 50 ].AndthevaluesoftheparametersS011andQaregiveninRef.[ 51 ]. 6.2.5EnvelopeFunctionsandWavefunctionsAlthoughtranslationalsymmetryisbrokeninthexdirectionwiththechoiceoftheLandaugauge~A=xB^y,itstillholdsintheothertwodirections. 83

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TheenvelopefunctionsoftheeffectivemassHamiltonianEq. 6 canbegivenby Fp,=ei(kyy+kzz) p A2666666666666666666664Up,1,kz,(z)p)]TJ /F5 7.97 Tf 6.59 0 Td[(1Up,2,kz,(z)p)]TJ /F5 7.97 Tf 6.59 0 Td[(2Up,3,kz,(z)pUp,4,kz,(z)pUp,5,kz,(z)pUp,6,kz,(z)p+1Up,7,kz,(z)p)]TJ /F5 7.97 Tf 6.59 0 Td[(1Up,8,kz,(z)p)]TJ /F5 7.97 Tf 6.59 0 Td[(13777777777777777777775(6)ThedetailsaregiveninRef.[ 20 34 ].ThecompletewavefunctionscanbeobtainedbymultiplyingtheseenvelopefunctionsbythecorrespondingzonecenterBlochbasisstatesgiveninfromEq. 3 toEq. 3 .InEq. 6 ,pisthePidgeon-Brown(PB)manifoldindex,denotestheeigenstateswithinthepthmanifold,Aistheareaofthecrosssectioninthesample'splaneperpendiculartothemagneticelddirection.Theharmonicoscillatoreigenfunctionsaredenotedasi(),whicharedeterminedat=x)]TJ /F3 11.955 Tf 12.95 0 Td[(2ky,whereisthemagneticlength.ThecomplexenvelopefunctionsaredenotedbyUp,m,kz,(z)foreachtheigenstate,whichhavetheperiodicityofthesuperlatticestructureandthezcomponentofwavevector,kz.Forthissuperlatticestructure,theminizoneisdenedbyjkzj=L,withasuperlatticeperiodL.ApplyingthenormalizationconditionIget, XmZ)]TJ /F9 7.97 Tf 6.59 0 Td[(L=2L=2dz LUp,m,kz,(z)Up,m,kz,(z)=1.(6) 6.2.6LandauSubbandEnergiesandEigenvectorsInthesuperlatticestructure,theenergyeigenvaluesandeigenfunctionsarecalculatedusingthenitedifferencemethod.Thesuperlatticeunitcellisdividedintoanequallyspacedgridofpoints,zi(i=1...N)toapplythenitedifferencemethod. 84

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ThefollowingsetofmatrixeigenvalueequationscanbeobtainedbysubstitutingtheenvelopefunctionsintotheeffectivemassSchrodingerequation.ThedetailsaregiveninRef.[ 20 34 ]. HpFp,=Ep,(kz)Fp,,(6)Tocalculatetheelectronicenergyeigenvaluesandeffectivemassenvelopefunctions,theeffectivemassSchrodingerequationissolvednumericallyforeachallowedvalueofthePidgeon-Brown(PB)manifoldindex,p,andzcomponentofwavevector,kz.Asi()areonlydenedfori0,onedeletestherowsandcolumnsofHpforwhichi<0.FromEq. 6 itcanbeseenthatFp,isdenedforp)]TJ /F4 11.955 Tf 22.45 0 Td[(1.Thecalculatedenergyeigenvalues,Ep,(kz),aretheLandaulevels,whereprepresentsthePBmanifoldandrepresentstheenergyeigenvalues,whichbelongtothesamePBmanifoldintheincreasingorder.Thecalculatedeigenstates,Fp,,areinfactcomplexfunctions,Up,m,kz,(zi),whicharedeterminedateachvalueofthegridpoints.Thecellperiodicityismaintainedbyapplyingtheboundarycondition,Up,m,kz,(zi)=Up,m,kz,(zN+i).Allthematerialparametersareallowedtovarywithposition. 6.2.7Magneto-OpticalAbsorptionandFermiEnergyInthissection,Idiscussthetheoreticalaspectsofthemagneto-absorptionandFermienergies.ThedetailsaregiveninRef.[ 20 34 ].FromFermi'sgoldenrule,themagneto-opticalabsorptioncoefcientatagivenmagneticeldBzandphotonenergy~!isgivenby[ 35 ] (~!))=e3Bz (~!)(~c)2nrXp,;p0,0Z1dkzj^e~Pp0,0p,(kz)j2(fp,(kz))]TJ /F8 11.955 Tf 11.95 0 Td[(fp0,0(kz)))]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Ep,p0,0(kz))]TJ /F13 11.955 Tf 11.96 0 Td[(~!, (6) where^eistheunitpolarizationvectoroftheincidentlight,~Pp0,0p,(kz)istheopticalmatrixelement,Ep,p0,0(kz)=Ep0,0(kz))]TJ /F8 11.955 Tf 12.32 0 Td[(Ep,(kz)istheopticaltransitionenergy,andnristhe 85

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refractiveindex.TheFermidistributionfunctionfp,(kz)inEq. 6 isgivenby fp,(kz)=1 1+exp[(Ep,(kz))]TJ /F8 11.955 Tf 11.95 0 Td[(Ef)=kT].(6)TheFermienergyEfinEq. 6 dependsonboththetemperatureandthenetcarrierconcentrationandalsoshowsaslightdependenceontheexternalmagneticeld.Thesignofthenetcarrierconcentrationdependsonthetypeofthedopingofthesample.Foragivenvaluesofthemagneticeld,temperature,andFermilevel,thenetcarrierconcentrationisgivenby NC=eBz (2)2(~c)Xp,Z1dkz(fp,(kz))]TJ /F3 11.955 Tf 11.95 0 Td[(vp,)(6)wherevp,istheKroneckerdeltafunction,whichgivesthevalueeither1or0dependingonwhetherthe(p,)isaLandauvalencesubbandorLandauconductionsubandrespectively.TheFermienergycanbecalculatedfromEq. 6 usingarootndingprogramforthegivenvaluesoftemperature,netcarrierconcentrationandmagneticeld.Theopticalmatrixelementscanbegivenbythefollowingequationwhereonlythemostprominentcontributionsaretakenintoaccount. ~Pp0,0p,(kz)=Xm,m0Zdz LUp,m,kz,(z)Up0,m0,kz,0(z)hN(p,m)jN(p0,m0)ihmj~Pjm0i (6) whereUp,m,kz,(z)arecomplexquantities,whichareinfacttheenvelopefunctionsinEq. 6 .TheorthonormalizedharmonicoscillatorwavefunctionsaredenotedasN(p,m),wheretheLandauquantumnumbersN(p,m)explicitlydependontwootherquantumnumbers,pandm,whicharedenedinEq. 6 .ThemomentummatrixelementsbetweentheBlochbasisstatesaregivenbyhmjPxjm0i,hmjPyjm0i,andhmjPzjm0i 86

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TheexplicitrepresentationsofPx,Py,andPzaregiveninEq. 5 ,Eq. 5 ,andEq. 5 respectively.InthenumericalcalculationoftheintegralinEq. 6 ,IhaveusedtheLorentzianfunction,(x),insteadoftheDiracdeltafunction,(x).IntheLorentzianfunction,theparameterspeciesthehalfwidthathalfmaximum. 6.2.8SpinPolarizationThespinpolarizationalongthezdirection(Sz)atanyphotonenergyforthegivenpolarizationofthelightiscalculatedfromthefollowingrelation:Sz(~!)=")]TJ /F3 11.955 Tf 11.96 0 Td[(# "+# (6)where"(~!)and#(~!)representtheamountofabsorptionthatcreatespin-upandspin-downelectronsrespectively.Theterm~!isthephotonenergywithangularfrequency!. 6.3NumericalCalculationofLandauSubbandStructure,Magneto-AbsorptionandSpinPolarizationInthismodel,Imaketheaxialapproximationsothatthevalencebandstructureinaplaneperpendiculartothemagneticeldiscylindricallysymmetric.ThissimpliestheproblembyallowingmetosolvefortheelectronicstructureineachPidgeon-Brown(PB)manifoldseparately.ThismeansthereisnocouplingbetweenmanifoldswithdifferentPBindexes.TheMulti-QuantumWell(MQW)structureistreatedasaninnitesuperlatticewithwidebarriers.Ireplacethez-componentofthewavevectorkzwithadifferentialoperator^kz=)]TJ /F8 11.955 Tf 9.3 0 Td[(irzalongtheconnementdirection(z)totakeintoaccountthesuperlatticequantumconnementeffects.ThenIdividethesuperlatticeunitcellwithGequallyspacedgridpointssothatthisrzoperatorcanbeapproximatedbynitedifferencesonthegrid.Inaddition,allthematerialparametersareallowedtovarywithposition(z).IuseanitedifferenceapproachoneachPBmanifold.Itallowsmetocalculateall 87

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theLandaulevelsforeachquantumconnedsubband.Butitincreasesthesizeofthematrixtobediagonalizedfrom8x8(sizeoftheeffectivemassHamiltonianMatrixwithoutgridpoints)to8G8G.Inthismodel,IchooseG=101.Sothenalsizeofthematrixis808808.Thenthis808808matrixisdiagonalizedusingasoftwareprogramtoobtainLandausubbandenergiesandcorrespondingwavefunctions.Usingthesevalues,Icomputethetotalmagneto-absorptioncoefcient((~!)asafunctionofphotonenergyforagivenmagneticeld(Bz)inthezdirectionfromtheEq. 6 .Ihavedevelopedaspecialalgorithminthesoftwareprogram,whichseparatesthistotalmagneto-absorptionintotwodifferentcomponentslike"(~!)and#(~!),where"(~!)and#(~!)representtheamountofmagneto-absorptionthatcreatespin-upandspin-downelectronsrespectivelyforanygivenpolarizationoflightwithmagneticeldBz.Theterm~!isthephotonenergywithangularfrequency!.Thensubstitutingthevaluesof"(~!)and#(~!)intoEq. 6 Icalculatethespinpolarization(Sz(~!)asafunctionofphotonenergyforagivenpolarizationoflight. 6.4ResultsandDiscussionFig. 6-2 AandFig. 6-3 AshowtheexperimentalOPNMRsignalintensitiesasthefunctionofphotonenergyatmagneticeld4.94Tfor+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizationsoflightrespectively.TheOPNMRspectraarisefromtheenhancednuclearspinpolarizationoftheGa69nucleiinthedeltadopedAl0.1Ga0.9As/GaAssquareQWduetotheopticalpumpingofthepolarizedlight.Fromthesegures,itisclearthattheintensityoftheOPNMRsignalsoscillatesasafunctionofphotonenergy.Moreover,theoscillatorypatternoftheexperimentalOPNMRspectradependsonthepolarizationofthelight.Fig. 6-2 BandFig. 6-3 Bshowthecalculatedelectron(orconductionband)spinpolarizationsfortheQWsampleasthefunctionofphotonenergyatmagneticeld4.94Tfor+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizationsoflightrespectively.NowcomparingthecalculatedelectronspinpolarizationresultswiththeexperimentalOPNMRsignalintensitiesitisevidentthatthephaseandthemagnitudeoftheOPNMRsignalissensitivetothephase 88

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andmagnitudeofthecalculatedelectronspinpolarizationforthebothhelicitiesofthelight. Table6-1. Subbandlabelsandwavefunctionprobabilitiesfor+polarization SubbandlabelWavefunctionprobaility LH01#92%LH#CB11"99%CB"HH)]TJ /F5 7.97 Tf 6.59 0 Td[(11#100%HH#CB01#99.6%CB#LH11#85%LH#+14%HH#CB21"99.3%CB"HH01#45%HH#+54%LH#CB11#99%CB#HH02#71%HH#+28%LH#CB11#99%CB#LH21#58.5%LH#+34.5%HH"CB31"99%CB"HH)]TJ /F5 7.97 Tf 6.59 0 Td[(12#100%HH#CB02#99%CB# Table6-2. Labelsoftransitionsfor+polarization TransitionLabel T1LH01#!CB11"T2HH)]TJ /F5 7.97 Tf 6.59 0 Td[(11#!CB01#T3LH11#!CB21"T4HH01#!CB11#T5HH02#!CB11#andLH21#!CB31"T6HH)]TJ /F5 7.97 Tf 6.59 0 Td[(12#!CB02# Fig. 6-2 CandFig. 6-3 CshowthecalculatedPBmanifoldresolvedtransitionsasthefunctionofphotonenergyfortheQWsampleatmagneticeld4.94Tfor+and)]TJ /F1 11.955 Tf -453.14 -28.25 Td[(polarizationsoflightrespectively.Ineachgure,eachcolorrepresentstheabsorptionarisesfromtheopticaltransitionsbetweenaparticularpairofPBmanifoldsrepresentingvalenceandconductionLandausubbandsrespectivelyandthedashedlinerepresentsthetotalabsorptionineachcase.Buttheopticaltransitionsdependontheselectionrulesforthepolarizationofthelight,Becauseoftheselectionrules,onlythep!p+1transitionsareallowedforthe+polarizationwhereasonlythep!p)]TJ /F4 11.955 Tf 12.01 0 Td[(1transitionsare 89

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Figure6-2. OPNMRandcalculatedspinpolarizationinAl0.1Ga0.9As/GaAsSquareWellfor+polarizationand4.94T.A)ExperimentalOPNMR,B)Theoreticalspinpolarization,andC)PidgeonBrown(PB)manifoldresolvedabsorptionasafunctionofphotonenergyinAl0.1Ga0.9As/GaAsSquareWellfor+polarizationand4.94T. 90

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Figure6-3. OPNMRandcalculatedspinpolarizationinAl0.1Ga0.9As/GaAsSquareWellfor)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizationand4.94T.A)ExperimentalOPNMR,B)Theoreticalspinpolarization,andC)PidgeonBrown(PB)manifoldresolvedabsorptionasafunctionofphotonenergyinAl0.1Ga0.9As/GaAsSquareWellfor)]TJ /F1 11.955 Tf -343.67 -18.79 Td[(polarizationand4.94T. 91

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Figure6-4. CalculatedzonecenterLandausubbandstructures.A)conductionLandausubbandsandB)valenceLandausubbandsasafunctionofmagneticeldforAl0.1Ga0.9As/GaAsSquareWell.ThecolorsofthesubbandsindicatethePBmanifoldindex(p=-1,0,1,..) allowedforthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarization,wherepisthePBmanifoldnumber.Asforexamples,forthe+polarizationinFig. 6-2 C,theblacklinerepresents0!1transitions,andforthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizationinFig. 6-3 C,thebluelinerepresents2!1transitions.Notethatallthetheoreticalcurvesareshiftedinphotonenergyby12.6meVinordertotakeaccountfortheshiftduetotheexcitonicCoulombmanybodyinteractionswhicharenotincludedinthecalculations(Becausethemodelisbasedonmeaneldapproximationorsingleelectronmodel)Fig. 6-4 AandFig. 6-4 BshowthecalculatedzonecenterconductionandvalenceLandausubandsforthestrainedAl0.1Ga0.9As/GaAssquareQWsampleasafunctionofmagneticeldrespectively.ThecolorsofthesubbandsindicatethePidgeon-Brownmanifoldindices(p=-1,0,1,2,3,4.)InFig. 6-4 B,notethatthelightholesubbandsareliftedupduetothestrainintheQW. 92

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Table6-3. Subbandlabelsandwavefunctionprobabilitiesfor)]TJ /F1 11.955 Tf 12.62 0 Td[(polarization SubbandlabelWavefunctionprobaility LH11"89%LH"+8%HH#CB01#99.6%CB#HH21"33%HH"+15%HH#+13%LH#+38%LH"CB11"99.7%CB"LH21"77%LH"+13.3%HH"CB11#99%CB#HH31"35%HH"+57%LH#CB21"99%CB"HH21"33%HH"+15%HH#+38%LH"+13%LH#CB12"99%CB"LH31"74%LH"+15%HH"CB21#98.8%CB"LH4153.5%LH#+36.7%HH"CB31"99%CB" Table6-4. Labelsoftransitionsfor)]TJ /F1 11.955 Tf 12.62 0 Td[(polarization TransitionLabel T1LH11"!CB01#T2HH21"!CB11"T3LH21"!CB11#T4HH31"!CB21"T5HH21"!CB12"andLH31"!CB21#T6LH41!CB31" Forthe+helicityofthepumpinglight,inFig. 6-2 B,therstsixextremaoftheelectron(orconductionband)polarizationarelabeledasT1,T2,T3,T4,T5,andT6.ThesignsofthespinpolarizationforthepeaksT1,T3arepositive,butforthepeakT5itisnegative.ThesignsforthetroughsT2,T4,andT6arepositive.Themagnitudesofthespinpolarizationofthepeaksarelowerthanthatofthetroughs.TherstsixextremaoftheOPNMRsignalintensityfor+polarizationoflightfollowthesamepatternoftheextremaofelectronspinpolarizationintermsofsignsandmagnitudes.Thetransitions,responsibleforthesesixextremaofthespinpolarization,areidentiedusingtheFig. 6-2 C,Fig. 6-4 A,Fig. 6-4 B,andTable 6-1 .TheidentieddominanttransitionsaregiveninTable 6-2 93

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Forthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(helicityofthepumpinglight,inFig. 6-3 B,therstsixextremaoftheelectron(orconductionband)polarizationarelabeledasT1,T2,T3,T4,T5,andT6.ThesignsofthespinpolarizationforthetroughsT1,T3arenegative,butforthepeakT5itispositive.ThesignsforthepeaksT2,T4,andT6arenegative.Themagnitudesofthespinpolarizationofthepeaksarelowerthanthatofthetroughs.TherstsixextremaoftheOPNMRsignalintensityfor)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(polarizationoflightfollowthesamepatternoftheextremaofelectronspinpolarizationintermsofsignsandmagnitudes.Thetransitions,responsibleforthesesixextremaofthespinpolarization,areidentiedusingtheFig. 6-3 C,Fig. 6-4 A,Fig. 6-4 B,andTable 6-3 .TheidentieddominanttransitionsaregiveninTable 6-4 .NotethatintheTable 6-2 andTable 6-4 ,theidentiedtransitionsarelabeledasVBps"=#!CBp0s0"=#.HereVBandCBarevalenceandconductionLandausubbandsrespectively.VBcanbeeitherHH(HeavyHole)orLH(LightHole).sands0arethesubbandindices.Andpandp0arethePBmanifoldindices.The"=#representsspinup/downstateofthesubband.ButtheonlyexceptionistheT6transition(Table 6-4 )forthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarization.Thereisnoarrowforthevalencesubband,LH41.AlthoughintheFig. 6-4 B,therstfewvalenceLandausubbandsarelabeledasLH1,theyarenotpurelightholes.FromTable 6-3 ,itcanbefoundthatthevalencesubband,LH41,isinfactthecombinationof53.5%lightholedownstateand36.7%heavyholeupstate.Forthe)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(polarizationduetotheselectionrule,notransitionsareallowedfromthelightholedownstatetoconductionbandupstate.Onlythetransitionsfromheavyholeupstatetoconductionbandupstate.Sothe36.7%heavyholeupstateoftheLH41subbandisresponsiblefortheT6transitionincaseofthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarization.Thustobetechnicallycorrectthearrowsignisnotassigned.Forthe+polarization,fromtheTable 6-2 itisfoundthattheT1andT3transitionscomefromthedominantlightholetransitions,sothespinpolarizationispositive.TheT5transitioncomesfromboththelighthole(LH)transitionandheavyhole(HH) 94

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transition.ButtheHHtransitionisdominantinthiscase,asaresultthespinpolarizationisnegative.TheT2,T4,andT6transitionscomefromthedominantHHtransitions,sothespinpolarizationisnegative.ThespinpolarizationshowsmoresensitivitytotheweakerLHtransitions,andsodotheexperimentalOPNMRsignal.Forthe)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarization,fromtheTable 6-4 itisfoundthattheT1andT3transitionscomefromthedominantlightholetransitions,sothespinpolarizationisnegative.TheT5transitioncomesfromboththelighthole(LH)transitionandheavyhole(HH)transition.ButtheHHtransitionisdominantinthiscase,asaresultthespinpolarizationispositive.TheT2,T4,andT6transitionscomefromthedominantHHtransitions,sothespinpolarizationispositive.ThespinpolarizationshowsmoresensitivitytotheweakerLHtransitions,andsodotheexperimentalOPNMRsignal.Therefore,forthebothpolarizationsofpumpinglight,theOPNMRsignalintensityismoresensitivetotheLHtransitions.Moreover,becauseofthestraininthesampletheLHLandausubbandsareliftedupandstayovertheHHLandausubbands.Asaresult,therstextremumpointofthecalculatedelectronspinpolarizationisalwayssensitivetotheweakerLHtransitionsforthebothhelicitiesofthepolarizedlightandsodotherstextremumpointoftheOPNMRsignalintensity.However,themostimportantfeatureisthattheOPNMRsignalintensitychangesitssignforbothpolarizationsofpumpinglight,whichwasnotseenincaseofunstrainedbulkGaAsmaterial[ 52 ].Thecalculatedspinpolarizationalsochangesitssignforthebothpolarizedlight.Incaseofquantumwell(QW)withmagneticeld,thedensityofstateisdeltafunctionwhicheventuallyprovidessharperHHandLHabsorptionpeaksthanthatofthebulkmaterialwithmagneticeldwherethedensityofstateisnotdeltafunctionanymore.Therefore,thechanceofoverlappingoftheHHandLHabsorptionpeaksismuchlowerinGaAsQWthanthatofthebulkGaAswherethetailsoftheHHabsorptionpeaksalwayssubduetheLHabsorptionpeaks[ 52 ]becauseofthedensityofthestateofthebulkmaterialinthepresenceofthemagneticeld.Moreover,inthe 95

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GaAsQWsystem,becauseofthestraininthesampletheLHLandausubbandsareliftedupandstayovertheHHLandausubbandswhicheventuallyfacilitatesthelargeseparationbetweentheHHandLHabsorptionpeaks.Asaresult,thestraineffect,alongwiththedensityofthestates,isresponsibleforthesignchangeoftheOPNMRsignalinthestrainedGaAsQW. 96

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CHAPTER7MAGNETO-ABSORPTIONINNARROWGAPPARABOLICMULTIQUANTUMWELLInthepresenceofanexternalstaticmagneticeldalonganycrystaldirectionofanyIII-VQuantumWell(QW)structurethemotionoftheBlochelectronisdescribedasthequantizationofthecyclotronorbits,whicharecalledtheLandaulevels.SoifthemagneticeldisappliedalongzdirectionthentheenergyoftheBlochelectronisquantizedintheplaneperpendiculartothemagneticeld.Notethatinaddition,theBlochelectronisconnedinthezdirection,whichisthegrowthdirectionoftheQW.Moreover,therewillbethepseudomorphicstrainwhicharisesfromthelatticemismatchbetweenthewelllayerandthebufferlayer.SotocalculatethebandstructureofthiskindofQWsystemwehavetoincludetheeffectsofquantumconnementandstrain.Inthischapter,Ipresentsometheoreticalmagneto-absorptioncalculationsforstrainedAlInSb/InSbparabolicmultiquantumwellsandcomparethemwiththecorrespondingexperimentalresultsforstrainedAlInSb/InSbparabolicmultiquantumwells. 7.1ExperimentalDetailsExperimentalmagneto-opticalmeasurementsonmultipleparabolicwellstructures(StructureS578)werecarriedoutbyProf.Santos'groupatUniversityofOklahoma.Theparabolicquantumwell(PQW)structurecontains25AlxIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xSbwells,eachwithathicknessof50nmwiththealuminumconcentration,x,variedquadraticallyfromx=0atawellscentertox=0.09atawellsedgetoproduceagradedparabolicpotential.ThequadraticgradingisaccomplisheddigitallybyfollowingMiller'srecipe[ 53 54 ]Theparabolicwellsareseparatedby50nmthickAl0.09In0.91Sbbarriersandtheyweregrownbymolecularbeamepitaxyonsemi-insulating(001)GaAssubstrates[ 55 ].Theyarenominallyundopedwithabackgrounddensityofionizedimpuritieslessthanabout1015cm)]TJ /F5 7.97 Tf 6.58 0 Td[(3.A2-3mAlxIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xSbbufferlayer,x=0.09forPQWstructure,wasdepositedpriortothegrowthofthequantumwellsinordertoisolatedislocationsduetothelargelattice 97

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mismatchbetweenthesubstrateandtheepilayer.Thebufferlayeristhickenoughforthelatticeconstantinthebufferlayertorelaxtoitsbulkvalue.ThereforetheAlxIn1)]TJ /F9 7.97 Tf 6.59 0 Td[(xSbbarrierlayersareunstrainedandthethinwelllayersarepseudomorphicallystrainedtothelatticeconstantofthebufferlayer.Transmissionmeasurementsnear4.2KweremadeusingaglobarsourceinaBrucker66V/SFourier-transformspectrometerconnectedbyagold-platedlight-pipesystemtoasamplechamberwithinasuperconducting8TmagnetandfollowedbyanInSbdetector.Thetransmission,T,ismonitoredasafunctionofphotonfrequency,inaconstantmagneticeld,B,appliedintheFaradaygeometrywheretheconstantmagneticeldandthePoyntingvectoroftheincidentlightareparalleltothequantumconnementdirection.on.ThebandoffsetforInSb/AlInSb[ 54 ]andthestrainparametersforInSb[ 56 ]werededucedinthepreviousexcitonicabsorptionstudieswithoutanyexternalmagneticeld 7.2TheoryandModeling 7.2.1EffectiveMassHamiltonianInthe8bandPidgeon-BrownModel[ 11 ],wheretheLandauHamiltonianandZeemanHamiltonianmatricesarecalculatedatthethezonecenter,kz=0.Inthissection,the8-bandPidgeon-BrownModelismodiedtoincludethez-componentwavevector(kz)dependenceoftheelectronicstates.ThedetailedcalculationsoftheHamiltonianmatriceswillbefoundintheC.K.Pidgeon,etalpaper[ 11 ].Ialsoincludethequantizationofkzwhichcomesfromthemultiplequantumwellsuperlatticeeffect.Inaddition,IalsotakethepseudomorphicstraineffectsintoaccountwhicharisefromthelatticemismatchbetweentheQWlayerandthebufferlayer.FollowingthePidgeon-Brownmodel,Iseparatethe8Blochbasisstatesintoanupperandlowersetof4Blochbasisstateswhichdecoupleatthezonecenter,i.e.kz=0.TheBlochbasisstatesfortheuppersetarejS"i,jHH"i,jLH#iandjSO#i,whichcorrespondtoelectronspinup,heavyholespinup,lightholespindownandsplitoffhole,respectively. 98

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Similarly,theBlochbasisstatesforthelowersetarejS#i,jHH#i,jLH"iandjSO"icorrespondingtoelectronspinup,heavyholespinup,lightholespindownandsplit-offhole,respectively.Theexplicitexpressionsforthese8BlochbasisstatesaregiveninfromEq. 3 toEq. 3 .ThetotaleffectivemassHamiltonianisisthesumoftheLandau(HL),Zeeman(HZ),andstraincontributions(HS)i.e. H=HL+HZ+HS(7)ExplicitexpressionsfortheLandau,ZeemanandStrainHamiltoniansaregivenbyEq. 6 ,Eq. 5 andEq. 6 respectively.IusedthestandardLuttingerparameters[ 30 ]andtemperaturedependentenergygapsintheLandauandZeemanHamiltonians. 7.2.2QuantumConnementPotentialIconsideranundopedAlInSb/InSbtype-IsuperlatticeheterostructureonathickAlInSbbufferlayeronaGaAssubstrate.Sinceitisatype-Isuperlattice,thebandgapoftheInSbwelllayercompletelylieswithinthebandgapoftheAlInSbbarrierlayerThebarrierlayeristhickenoughtopreventinteractionsamongthewells.ThebandgapmismatchbetweenthelayersAlInSbandInSbisgivenby,Eg=Eg(AlInSb))]TJ /F8 11.955 Tf 11.95 0 Td[(Eg(InSb) (7)TheconductionbandbarrierheightVCBisgivenbyVCB=QCBEg (7)whereQCBistheconductionbandoffset.ThevalencebandbarrierheightVVBisgivenbyVVB=QVBEg (7)whereQVBisthevalencebandoffsetwhichisinfactQCB=1)]TJ /F8 11.955 Tf 11.95 0 Td[(QVB. 99

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HeretheEgistemperaturedependentwhichiscalculatedfromVarshni'sempiricalformula[ 46 ]. Eg(T)=Eg(T=0K))]TJ /F3 11.955 Tf 17.87 8.09 Td[(T2 T+,(7)whereandarecalledVarshniparameters,whosevaluesfortheAlInSbandInSblayersarefoundinRef.[ 30 ].ThevaluesofVCBandVVBarecalculatedfortheconductionbandoffset,QCB=.62(providedbySantosgroup)IhavetoaddtheseasconnementpotentialsVCBandVVBintheeffectivemassHamiltonianasafunctionofz,whichisthegrowthdirectionofQWsystem. 7.2.3PseudomorphicStrainEffectsInthesuperlatticestructure,thestrainHamiltonian,HS,inEq. 7 dependsonposition.IassumethatthestrainintheInSb/AlInSbsuperlatticestructureispseudomorphicwhichmeansthatthelatticeconstantsintheInSbwelllayersandtheAl0.09In0.91SbbarrierlayersareequaltothelatticeconstantintheAl0.09In0.91Sbbuffer.InbulkInSbandAl0.09In0.91Sbattemperature4K,theunstrainedlatticeconstantsare6.473Aand6.4423Arespectively.(providedbyProf.Santosexperimentalgroup)Inthesuperlattice,becauseofthepseudomorphicstrainapproximationthelatticeconstantthroughoutthestructureisthesameasintheAl0.09In0.91Sbbuffer.Sotheonlynon-vanishingcomponentsofthestraintensorintheInSbwelllayeraregivenby "xx="yy=a0(AlInSb))]TJ /F8 11.955 Tf 11.95 0 Td[(a0(InSb) a0(InSb)(7)and "zz=)]TJ /F4 11.955 Tf 9.3 0 Td[(2c12 c11"xx(7)wherec12andc11areelasticstiffnessconstants,whichcanbefoundinRef.[ 30 ].ButtheAlInSblayersareunstrained.Sinceinthesample,"xx="yyandthereisnoshear 100

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straininthesystem;i.e."xy="yz="xz=0,thetermL"inEq. 6 vanish.Asaresult,Sa=SbandSc=0inEq. 6 7.2.4EnvelopeFunctionsandWavefunctionsAlthoughtranslationalsymmetryisbrokeninthexdirectionwiththechoiceoftheLandaugauge~A=xB^y,itstillholdsintheothertwodirections.TheenvelopefunctionoftheeffectivemassHamiltonian,Eq. 7 canbegivenbyEq. 6 .ThecompletewavefunctionscanbeobtainedbymultiplyingtheseenvelopefunctionsbythecorrespondingzonecenterBlochbasisstatesgiveninfromEq. 3 toEq. 3 .ThedetailscanbefoundintheSection 6.2.5 7.2.5LandauSubbandEnergiesandEigenvectorsSubstitutingtheenvelopefunctionFp,fromEq. 6 intotheeffectivemassSchrodingerequationwithHgivenbyEq. 7 ,Iobtainasetofmatrixeigenvalueequations HpFp,=Ep,(kz)Fp,(7)thatcanbesolvedforeachallowedvalueofthePidgeon-Brownmanifoldindex,p,andwavevector,kz,toobtaintheelectronicenergiesandeffectivemassenvelopefunctions.ThecalculatedenergyeigenvaluesaretheLandaulevels,denotedbyEp,(kz),wherethePBindex,plabelstheLandaulevelmanifoldandlabelstheeigenenergiesbelongingtothesamepthLandaulevelmanifoldintheascendingorder.TheFp,,arethecorrespondingeigenstates.ThedetaileddescriptioncanbefoundintheSection 6.2.6 7.2.6Magneto-OpticalAbsorptionandFermiEnergyFromFermi'sgoldenrule,themagneto-opticalabsorptioncoefcientatthephotonenergy~!foramagneticeldBziscalculatedusingtheEq. 6 [ 35 ].TheFermienergyEfinEq. 6 dependsonboththetemperatureandthenetcarrierconcentrationandalsoshowsaslightdependenceontheexternalmagneticeld.Thenetcarrierconcentrationcanbeeitherpositiveornegativedependingonwhether 101

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thedopingofthesampleisn-orp-type.TheFermienergycanbecalculatedfromEq. 6 usingarootndingprogramforthegivenvaluesoftemperature,netcarrierconcentrationandmagneticeld.InthenumericalcalculationoftheintegralinEq. 6 theDiracdeltafunction,(x),isreplacedbytheLorentzianlineshapefunction(x)withfullwidthathalfmaximum(FWHM)of.Thedetaileddescriptionsofthecalculationofthemagneto-opticalabsorptionandFermienergycanbefoundintheSection 6.2.7 7.3SimulationofLandauSubbandStructureandMagneto-AbsorptionInthemodel,Imaketheaxialapproximationsothatthevalencebandstructureinaplaneperpendiculartothemagneticeldiscylindricallysymmetric.ThissimpliestheproblembyallowingmetosolvefortheelectronicstructureineachPidgeon-Brown(PB)manifoldseparately.ThismeansthereisnocouplingbetweenmanifoldswithdifferentPBindexes.TheMulti-QuantumWell(MQW)structureistreatedasaninnitesuperlatticewithwidebarriers.Ireplacethez-componentofthewavevectorkzwithadifferentialoperator^kz=)]TJ /F8 11.955 Tf 9.3 0 Td[(irzalongtheconnementdirection(z)totakeintoaccountthesuperlatticequantumconnementeffects.ThenIdividethesuperlatticeunitcellwithGequallyspacedgridpointssothatthisrzoperatorcanbeapproximatedbynitedifferencesonthegrid.Inaddition,allthematerialparametersareallowedtovarywithposition(z).IuseanitedifferenceapproachoneachPBmanifold.ItallowsmetocalculatealltheLandaulevelsforeachquantumconnedsubband.Butitincreasesthesizeofthematrixtobediagonalizedfrom8x8(sizeoftheeffectivemassHamiltonianMatrixwithoutgridpoints)to8G8G.Inthemodel,IchooseG=101.Sothenalsizeofthematrixis808808.Thenthis808808matrixisdiagonalizedusingasoftwareprogramtoobtainLandausubbandenergiesandcorrespondingwavefunctions.Usingthesevalues, 102

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Icomputethetotalmagneto-absorptioncoefcient((~!)asafunctionofphotonenergyforagivenmagneticeld(Bz)inthezdirectionfromtheEq. 6 7.4ResultsIchosetostudyaparabolic-well(PW)samplebecauseweakertransitions,darkinthesquare-well(SW)samplesandforbiddeninaninnitebarriersquare-well,canbeseeninparabolicwellswithnearlythesamestrengthastransitionsthatarebrightinthesquare-wells[ 54 ].Intheparabolicmulti-quantumwell(MQW)structure,theparabolicwellsareobtainedbygrowingaseriesofnarrowInSbandAl0.09In0.91Sblayerswhosewidths(<25Angstromsinallcases)produceawidequantumwellwithaneffective50nmwideparabolicpotentialprole.Intheparabolicwellregion,43suchlayersareusedandthethicknessesofeachlayerareusingMiller'srecipe[ 53 ].The50nmthickAl0.09In0.91Sbbarrierbetweentheparabolicwellsiswideenoughthattheparabolicwellsareeffectivelyisolatedfromeachother.Inthesimulations,convergencetotheMQWlimitoccursifweassumetheparabolicwellsinthesuperlatticeareseparatedby25nmAl0.09In0.91Sbbarriers.Inthemodel,thealuminumproleinthesuperlatticeunitcellisdenedonanevenlyspacedmesh.Thealuminumconcentrationateachmeshpointisobtainedbyaveragingthealuminumconcentrationovermeshcellscenteredonthemeshpoints.Ifthemeshisneenough,theexactaluminumproleofthe43InSbandAl0.09In0.91Sblayersisrecovered.IstudiedconvergenceoftheresultsasafunctionofthenumberofmeshpointsperunitcellandIndthattheresultsconvergewithasuperlatticeunitcelldenedonanevenlyspacedgridof101points.Themeshcellaveragedaluminumconcentration,atthe101gridpointsusedinthesuperlatticeunitcellintheconvergedcalculations,isshowninFig. 7-1 Themagneto-absorptionresultsfortheparabolicsampleareshowninFig. 7-2 A.Figure 7-2 BshowsthetheoreticalcalculationincludingtheeffectsofbiaxialstrainatalltheMQWinterfaces,whileFigure 7-2 Cshowsthetheoreticalcalculations 103

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withouttheeffectsofstrain.Biaxialstrainagainhasastronginuenceontheoreticalmagneto-absorptionspectrumandisneededtoaccuratelyreproducetheexperimentalresults.InFig. 7-3 ,Icomparetheexperimentalmagneto-absorptionat6T(a)withthecalculatedabsorptionwithbiaxialstrain(b)andwithoutbiaxialstrain(c).Again,itisclearlyseenthatthetheoreticalmagneto-opticalabsorptionspectrumwithbiaxialstraineffectsincludedmoreaccuratelyreproducestheexperimentalresults.IhavealsocomparedtheresultswiththatofthesquareMQWstructure[ 34 ].IseethattheabsorptionspectrumintheparabolicMQWstructureshowsmorefeaturesthantheabsorptionspectruminthesquarewellMQWstructure.Thewidthoftheparabolicwellsandtherelaxationofthebrightvs.darkrestrictionleadtoaricherabsorptionspectrumascomparedtotheabsorptionspectrainthesquare-wellMQWsamples. 7.5DiscussionTheelectronicstructurefortheparabolicwellMQWsampleisshowninFig. 7-4 .AgainthebandsinFig. 7-4 arecolorcodedasbeforetoindicatethePidgeon-Brownmanifoldindex(p=-1,0,1,..)..ThenumberslabelingthebandsinFig. 7-4 correspondtothebandnumbersinFig. 7-6 ,whichalsoliststhecompositionoftheLandauLevelwavefunctionsat6T.TheconductionsubbandlabelsinFig. 7-6 gofrom1to12startingatthebottom,whilethevalencesubbandsgofrom13to39startingatthetop.SolidlinesinFig. 7-4 indicateprimarilyspin-upsubbandswhiledottedlinesindicateprimarilyspin-downsubbands.NotethatthePidgeon-BrownManifoldindextakesonvalues-1,0,1,2,..andisnotthesameastheLandauLevelindex(whichtakesonthevalue0,1,2,..).EachPidgeonBrownmanifoldhasuptoeightBlochstatesandtheLandauLevelindexdependsontheBlochstate.i.e.forthep=-1Pidgeon-Brownmanifold,onlytheHeavyHolespin-downBlochstateisinthatmanifoldandtheLandaulevelindexnisrelatedtothePidgeonBrownmanifoldnumberbyn=p+1.WithinagivenPidgeonBrownmanifold,mj(fortheBlochstate)plusn(theLandauLevel)isaconstant. 104

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IhavealsocomparedtheresultswiththatofthesquareMQWstructure[ 34 ].TheparabolicMQWbandsaresimilartothesquareMQWbands.However,theparabolicMQWsubbandsaremorecloselyspacedsincetheparabolicMQWstatesarelesswellconned.Intheparabolicwellthereare5heavyholesubbandsbelowtherstlightholesubbandasopposedto3heavyholesubbandsinthesquarewellcase.Inotethatthe1stspin-downLandaulevelforthe1stconductionband,C11#,anticrosseswiththe0thspin-upLandaulevelforthe2ndconductionband,C02".ThisanticrossingoccursbecausethesestatesbelongtothesamePidgeon-Brownmanifold.Ingeneral,Cnsb#willmixandanticrosswithCn)]TJ /F5 7.97 Tf 6.59 0 Td[(1sb+1"(wheresbisthesubbandindexandnistheLandaulevelindex)sincetheseLandaulevelsbelongtothesamePidgeon-Brownmanifold.FromFig. 7-4 A,IseethattheC21#andC12"Landaulevels(bluelines)anticrossasdotheC12#andC03"levels(greenlines).Thesesameanticrossingsarealsoseeninthesquarewellsample,butonehastogotohighermagneticeldstoobservethemsincethesquarewellconductionsubbandsaremorewidelyspacedduetostrongerconnement.Theheavy-holesubbandLandaulevelsfortheparabolicMQWaresimilartothoseforthesquareMQW.Thespin-upandspin-down0thLandaulevelsforthe1stheavyholesubbanddonotcross,butthe1stand2ndspinLandaulevelsdocross.However,thespin-splittingforthe0thheavyholeLandaulevelsisslightlysmallerthanforthesquarewell,andtheconductionbandspin-splittingisslightlylargerfortheparabolicwell.Asaresult,onecanseespin-splittingofthemagneto-absorptionintheheavy-holetoconductionband0thLandauleveltransitionoccurataeldofaround5Twhichisobservableintheexperimentaldata(Fig. 7-2 A)),butisonlypredictedtooccurinthesquarewellabove10T.ThereismorestructureintheparabolicMQWmagneto-absorptionspectrathaninthesquareMQWcasesincetheparabolicMQWsubbandlevelsaremorecloselyspacedasshowninFig. 7-5 .ThebanddiagraminFig. 7-4 ,alongwithFig. 7-6 and 105

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theselectionrulesfor+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(circularlypolarizedlight,allowsmetodeterminetheLandaulevelsinvolvedinthetransitionsobservedinthemagneto-absorptionspectra.Again,IseethebrighttransitionsfromtheheavyholetoconductionbandLandaulevelswiththesamespin,aswellastransitionsfromthelightholetoconductionbandLandaulevelsofoppositespin(whicharemuchweakerbecauseoftheirsmalleropticalmatrixelements).Inaddition,asinthesquarewellcase,Iseeminordarktransitionsthatoccurduetobandmixing,ascanbeseenfromFig. 7-6 Figure7-1. ThealuminumconcentrationasafunctionofpositionusedinthecalculationoftheLandauLevelsfortheparabolicquantumwell.101gridpointswereusedinthecalculationtoapproximatetheactualstructure 106

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Figure7-2. Parabolicwellabsorptionspectra.AbsorptionspectrafromsampleS578fromA)experimentandB)theoryforastrainedinterfaceandc)theoryforanunstrainedinterface 107

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Figure7-3. ExperimentalandcalculatedabsorptionspectrafortheparabolicwellforB=6T.A)Theexperimentalmeasuredabsorption.B)Theoreticallycalculatedabsorptionincludingtheeffectsofstrain.C)Theoreticallycalculatedspectrumnotincludingtheeffectsofstrainattheinterface. 108

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Figure7-4. CalculatedLandaulevels.A)ConductionbandandB)ValencebandLandaulevelsfortheparabolicwell.Thecolors(online)ofthebandsindicatethePidgeon-Brownmanifoldindex(p=-1,0,1,..)withp=-1grey;p=0red;p=1green;p=2blue;p=3magenta;andp=4yellow.ColorcodingofthebandsmatcheswiththeFig. 7-6 .TodeterminethecompositionoftheLandauLevelsat6TusingFig. 7-6 ,theconductionbandsgofrom1to12startingatthebottomandthevalencebandsgofrom13to39startingatthetop. 109

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Figure7-5. Calculatedmagneto-absorptionspectrafortheparabolicwell.ForA)linearpolarized,B)+circularlypolarizedlightandC))]TJ /F1 11.955 Tf 10.41 -4.34 Td[(circularlypolarizedlight.Peaksarelabeledbythedominanttransitions.For+light,withintheaxialapproximation,transitionsoccuronlybetweenstateswhosePidgeon-Brownmanifoldindexchangeby+1.For)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(light,thePidgeon-Brownmanifoldindexchangesby-1foratransition. 110

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Figure7-6. EigenfunctionsofthelowestlyingbandsfortheparabolicwellforB=6T.Thenumbersgivethefractionofagivencomponentinthatband.(Thespin-splitholecontributionswerenegligible).BandsarecolorcodedaccordingtothePidgeon-BrownManifoldindex(p=-1,0,1,..)withp=-1grey;p=0red;p=1green;p=2blue;p=3magenta;andp=4yellow.ThebandnumbercorrespondstothenumbersshowninFig. 7-5 .Thebandsarelabeledaccordingtothedominantcomponent,i.e.bandnumber14is83.1%heavyholespin-upat6TandlabeledH01"(1stheavyholesubband,0thLandaulevel,spin-up(+3/2)) 111

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CHAPTER8CONCLUSIONSInthiswork,IhaveconstructedmodelstoexplainsomeexperimentalphenomenainIII-Vsemiconductornanostructures.Inparticular,Ihaveconstructedmodelsforbothbulkandquantumwell(QW)materials.Incaseofbulkmaterials,IhavestudiedthenarrowgapferromagneticsemiconductorslikeInMnAsandInMnSbwithorwithoutmagneticeld.IhavetriedtoexplainsomephenomenaofcarrierdynamicsintheferromagneticInMnAsmaterialwithoutthepresenceoftheexternalmagneticeld.Ihavealsomodelledthecyclotronresonance(CR)spectrainthemolecularbeamepitaxy(MBE)andmetalorganicvaporphaseepitaxy(MOVPE)grownferromagneticInMnAsandInMnSbsamplestogureoutthedifferencesbetweentheMBEandMOVPEgrownsamples.IncaseofQWheterostructures,IhavestudiedtwodifferenttypesofmaterialslikeAlGaAs/GaAsmultisquarequantumwell(SQW)andAlInSb/InSbmultiparabolicquantumwell(PQW)inthepresenceofthemagneticeld.IhavecalculatedelectronspinpolarizationinthestrainedAlGaAs/GaAssquaremultiquantumwell(MQW)toshowthesensitivityoftheopticallypumpedNMR(OPNMR)signaltotheelctronspinpolarization.Ihavealsomodelledthemagneto-opticalabsorptionforthestrainedAlInSb/InSbparabolicmultiquantumwell(MQW)structuretounderstandhowtheconnementandpseudomorphicstrainaffectthemagneto-opticalabsorptionspectra.Experimentaltimeresolveddifferentialtransmission(TRDT)measurementswerecarriedoutintheMOVPEgrownferromagneticInMnAsat290Kintheabsenceofthemagneticeld.Themeasurementdemonstratedthatthecarrierrelaxationtimecanbetunedandlastlongerthanafewpicoseconds,whentheexcitedcarriersarecloseorabovetheL-valleythreshold.Inaddition,thechangeinthenatureofthedynamicalprocesses,switchedbetweenthephotoinducedabsorptionandbleachinginaferromagneticsemiconductor,wasalsoobserved.Toexplaintheseexperimentalphenomena,IhavecalculatedtheelectronicstructureforInMnAsusingan8-bandkp 112

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modelat290KthatincludesthenonparabolicityandcouplingoftheelectronsandholestotheMnimpurities,tomodeltheobserveddynamics.ThecalculatedbandstructurewiththebandssplittingduetotheMnimpuritiesshowsthat(1)the800nmpumppulsecreatesphotoexcitedcarriersfromalltheholebandsandsomeofthephotoexcitedelectronscanscattertothesatelliteLvalleysthatslowsdowntherelaxationoftheelectrons.(2)Thesignchangebetweenprobingwith3.5mand2.0mcanbeexplainedbywhatregionsofthebandsaresampledbytheprobepulses.At2.0m,theprobeisdeepintothebandswherethephase-space-llingtermdominates.At3.5m,theprobeisrightatandslightlybelowtheband-edgewhereband-gaprenormalizationdominates.IhavemodeledtheexperimentalCRspectrameasurementsinsomeMBEandMOVPEgrownp-typeInMnAsandInMnSbferromagneticsemiconductorlmswithdifferentMncontentsatvarioustemperaturesusingan8-bandPidgeon-Brownmodel,whichhasbeengeneralizedtoincludethewavevectordependenceoftheelectronicstatesalongkzaswellasthes-dandp-dexchangeinteractionswiththelocalizedMnd-electrons.TheCurietemperatureistakenasaninputparameterandtheaverageMnspinistreatedinmeaneldtheory.IhavecalculatedthezonecenterLandauvalencesubbandstructures,CRspectra,andFermilevelsforbothInMnAsandInMnSbwithdifferentMncontentsatvarioustemperatures.ThecalculationsindicatethatthedifferencesbetweentheCRmeasurementsareseenforMBEandMOVPEsamplescanbeattributedtothedifferencesincarrierdensityandhencethepositionoftheFermilevelsinthesamples.IalsocalculatedtheaveragezcomponentofthespinsfortheferromagneticInMnAswithdifferentcarrierdensity,whichconcludesthatthelowercarrierdensityinthestructureresultsinamuchhigheraveragezcomponentofthespinscomparedtothestructurewithhighercarrierdensity.ThisfactcouldberesponsibleforthehigherCurietemperatureintheMOVPEstructures,whichhavelowercarrierdensities.Thetheoreticalstudiesalongwithexperimentalobservationsmight 113

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playanimportantrolefordesigningnewspinbasedsemiconductordevicesandprovidenewinsightsinthebandstructureoftheferromagneticstructures.ExperimentalOPNMRspectraweremeasuredinthestrainedAlGaAs/GaAssquareMQWatvariousmagneticelds,fortwodifferentpolarizationofthepumpinglight.ToinvestigatethesensitivityofthemeasuredOPNMRsignalintensitytotheelectronspinpolarizationIhaveusedthe8-bandPidgeon-Brownmodel,whichhasbeengeneralizedtoincludethesquarewellconnementeffectandthestraineffect.Thestaininthesampleiscalculatedfromtheexperimentalquadrupolespinsplitting.IusedFermi'sgoldenruletocalculatethemagneto-absorption.Ihavecalculatedtheelectronspinpolarizationsat4.94T,for+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizedlightandcomparedthemwiththecorrespondingOPNMRsignalintensities.TheOPNMRspectraisfoundsensitivetothesignandmagnitudeoftheelectronspinpolarizationforbothhelicitiesofthepolarizedlight.ThecalculationsoftheLandausubbandstructurealongwithwavefunctionprobabilityshowthattheelectronspinpolarizationismoresensitivetotheweakerlighthole(LH)toconductionband(CB)transitions.Asaresult,theOPNMRsignalismoresensitivetotheweakerLHtransitions.ItisinterestingbecausefortheGaAs,theeffectivegfactoroftheconductionbandisverylow.Soitisverydifculttodetectthespinsplittingusingconventionaltoolslikemagneto-absorptionspectra.Therefore,IthinktheOPNMRspectracanbeusedaneffectivetooltoinvestigatethespinsplittingoftheconductionLandausubbandstructuresofheterostructuresemiconductors.AnotherinterestingthingIfoundthattheLHsubbandsareliftedupduetothestrainofthesample.Thisisthereasonwhytherstpeak/trough(dependingonthepolarizationoflight)oftheOPNMRspectracomesfromtheweakerLHtransitions.However,themostimportantfeatureisthattheOPNMRsignalintensitychangesitssignforbothpolarizationsofpumpinglight,whichwasnotseenincaseofunstrainedbulkGaAsmaterial[ 52 ].Iconcludethatthestraineffect,alongwiththedensityofstatesinthestrainedAlGaAs/GaAssquareMQW,isresponsibleforthissignchange.Ithinkthe 114

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variationofstrainandphotonenergy(opticalpumping)mightplayanimportantroletomanipulatethenuclearspinpolarizationinanyIII-Vsemiconductornanostructures,whichalsoimpliesitspotentialapplicationfornuclearspinbaseddevices.Ihavemodeledtheexperimentalmagneto-absorptionspectraforthestrainedAlInSb/InSbparabolicMQW.Usingthe8-bandPidgeon-Brownmodel,whichhasbeengeneralizedtoincludetheparabolicwellconnementeffectandthepseudomorphicstraineffectIhavecalculatedtheLandausubbandstructure.Ihavecalculatedthemagneto-absorptionspectrawithandwithoutbiaxialstrainandcomparedwiththeexperimentalmagneto-absorptionspectratoinvestigatetheimportanceofthestrain.Ifoundthatthetheoreticalmagneto-opticalabsorptionspectrawithbiaxialstraineffectsincludedmoreaccuratelyreproducestheexperimentalresults.IalsocomparedtheresultswiththatofsquareMQW[ 34 ]tounderstandhowtheshapeoftheconnementaffectsthemagneto-absorptionspectra.Oneinterestingfeatureisthatonecanseespin-splittingofthemagneto-absorptionintheheavy-holetoconductionband0thLandauleveltransitionoccurataeldofaround5TintheparabolicMQW,butforthesquareMQWthesamespinsplittingoccuredabove10T.ThecalculatedLandausubbandstructuresindicatethatthespin-splittingforthe0thheavyholeLandaulevelsisslightlysmallerthanforthesquarewell,andtheconductionbandspin-splittingisslightlylargerfortheparabolicwell.Asaresult,thespin-splittingintheparabolicMQWisobservedatalowermagneticeld.ThecalculatedLandausubbandstructures,thecalculatedwavefunctionprobabilities,andtheselectionrulesfortheopticaltransitionsfor+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(polarizedlightallowmetoidentifythetransitions.IhaveseenboththebrightanddarktransitionsintheparabolicMQW.Butthemostinterestingfeatureisthatweakertransitions,darkinthesquare-well(SW)samplesandforbiddeninaninnitebarriersquare-well,canbeseeninparabolicwellswithnearlythesamestrengthastransitionsthatarebrightinthesquarewells.ThusIcanconcludethattheshapeofthe 115

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connementandthepseudomorphicstraininanysemiconductorheterostructureaffectthemagneto-absorptionspectra. 116

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CHAPTER9FUTUREDIRECTIONSOFRESEARCHInthissection,Iwouldliketoproposesomefuturestudiesthatcanbeperformedtogetbetteragreementwiththeexperiments.Althoughthe8-bandkpmodelhasexplainedtheslowerrelaxationmechanismsinthebulknarrowgapferromagneticsemiconductors,ithassomelimitations.Themodelcangeneratethebandstructureaccuratelyaroundthe)]TJ /F1 11.955 Tf 10.1 0 Td[(point(orbandedge)oftherstBrillouinzone,whichmightbegoodenoughwhentheopticaltransitionsoccurnearthebandedge.However,amoreaccuratebandstructureisneededtounderstandtherelaxationmechanismsmorepreciselywhenthe)]TJ /F1 11.955 Tf 10.1 0 Td[(valleyelectronscatterstothesatelliteXvalleyduetothepumppulsewithhighphotonenergy.Becauseinthe[100]directionoftheBrillouinzone,especiallyforInAs,theconductionbandofthe8-bandkpmodelcrosses(aroundatahalfofthedistancebetweenthesetwovalleys)withotherconductionbandwhichisregardedasaremoteband.Asaresult,themodelwillbreakdown.Therefore,infuture,thecurrentmodelcanbemodiedtoincludemorethaneightstatesasbandedgestatessothatitcanexplainthe)]TJ /F2 11.955 Tf 12.71 0 Td[($Xintervalleyscatteringmoreaccuratelybygeneratingmoreaccuratebandstructure,whichisanimportantpartofcarrierrelaxationmechanisms.Anotherimportantlimitationofthecurrentmodelisthatitdoesn'tincludethemanybodyinteractionslikeelectron-electronscatteringandelectron-phononscatteringwhichplaykeyroletothecarrierdynamics.Futurestudiescanbepursuedtoincludethemanybodyinteractions.Moreover,theabsorptioncoefcientinthecurrentmodeliscalculatedfromthebandstructureandtheequilibriumFermi-Diracdistributionsforthecarriers.Sotheabsorptionisnottimedependent.Buttomodelcarrierdynamicsofanymaterialthetimedependencyshouldbeincludedintheabsorption.Itcanbedoneinfuturebyincludingthenon-equilibriumFermi-Diracdistributionofthecarrierswhichvarieswithpump-probedelay.Ialsowould 117

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liketoproposesomefuturecarrierdynamicsstudiesintheconned(2D)systemtounderstandimportanceofconnementinthecarrierdynamics.Thecurrentmodelsuccessfullygeneratedthecyclotronresonance(CR)spectrafornarrowgapsemiconductors,whichagreeswiththeexperimentsinmostcases.However,theopticalmatrixelementofthemodeldoesnotincludethetermwhichcalculatesthemomentummatrixelementbetweentheharmonicoscillatorstatesoftheLandaulevels.Becauseinthenarrowgapsemiconductors,thetermwiththemomentummatrixelementsbetweentheBlochbasisstatesdominatesoverthethemomentummatrixelementbetweentheharmonicoscillatorstatesduetostrongconductionandvalencebandmixing.However,incaseofwidegapsemiconductorlikeGaN,thebandmixingisnottoostrongtoneglectthetermintheCRstudiesoftheGaNmaterial.Therefore,thistermcanbeincludedinthemodelforthefutureCRstudiesofthethewidegapmaterials.Althoughthecurrentmodelalmostsuccessfullysimulatetheexperimentalmagneto-absorptionspectra,anaxialapproximationofthebandstructureisassumedwhichdecouplestheLandaulevelswithdifferentPidgeon-Brown(PB)manifoldnumberbuttheLandaulevelswiththesamePBmanifoldnumberscancouplewitheachother.ThisapproximationhelpsdiagonalizetheHamiltonianmatrixmoreeasily.Becauseofthisapproximation,theLandaulevelcalculationsshowonlytheanti-crossingoftheLandaulevelswiththesamePBnumber,buttheLandaulevelswiththedifferentPBmanifoldnumberscross.Asresult,someoftheobservedanti-crossingbehaviorsinmagneto-absorptionspectracanbeexplainedbythemodelwheretheLandaulevelshavethesamePBmanifoldnumber.Butinsomecasesthemodelfailstoexplaintheobservedanti-crossingbehaviorswhichiscomingfromtheanti-crossingoftheLandaulevelswithdifferentPBmanifoldnumbers.Therefore,themodelshouldbemodiedtoincludetheanti-crossingoftheLandaulevelswithdifferentPBmanifoldnumbersbyeliminatingtheaxialapproximation. 118

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Inthecurrentmodel,theexcitonabsorptioneffectisneglectedduringthecalculationofthespinpolarizationoftheAlGas/GaAsmulti-quantumwell.Itisincludedasashiftofphotonenergyinthespinpolarization.InthebulknarrowgapmaterialslikeInAsandInSb,theCoulombbindingenergiesarearound1.3meVand0.65meVrespectively.Butinthebulkmid-gapmaterialslikeGaAs,itisaround4.6meV.SofortheGaAs,theexcitonabsorptioneffectisveryimportant.Moreover,inthequantumwellthevalueoftheCoulombbindingenergyshouldincreaseduetotheconnementeffect.SincetheCoulombinteractionintherealsystemisamanybodyinteraction,anewtoolisneededinfuturetocalculatetheexcitonabsorptioneffectintheAlGaAs/GaAswell. 119

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BIOGRAPHICALSKETCH DiptaSahawasborninDhaka,Bangladesh.HeobtainedhisB.Sc.degreeinphysicsfromtheUniversityofDhaka.HeobtainedhisMSdegreeinphysicsfromtheUniversityofDhaka.ThenhejoinedinthePhDprograminphysicsattheUniversityofFlorida.HeworkedinsemiconductorphysicswithProf.ChristopherJ.Stanton.HereceivedhisPhDdegreeintheSpringof2014. 123