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Magneto-Optical Properties of Narrow-Gap Semiconductor Heterostructures

Permanent Link: http://ufdc.ufl.edu/UFE0042109/00001

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Title: Magneto-Optical Properties of Narrow-Gap Semiconductor Heterostructures
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Pan, Xingyuan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Next generation of semiconductor device will not only based on the charge transport properties of the carrier, but also their spin degree of freedom. In order to understand or predict how those devices work one need to understand the spin-dependent electronic structures of both bulk and low-dimensional semiconductors. We have theoretically studied the spin-dependent Landau levels for electrons or holes in bulk GaAs system and AlInSb/InSb multiple quantum wells system. We use the envelope function approximation for the electronic and magneto-optical properties of AlInSb/InSb superlattices. Our model includes the conduction electrons, heavy holes, light holes and the split-off holes for a total of 8 bands when spin is taken into account. It is a generalization of the Pidgeon-Brown model to include the wave vector dependence of the electronic states, as well as quantization of wave vector due to multiple quantum well superlattice effects. In addition, we take strain effects into account by assuming pseudomorphic growth conditions. For bulk GaAs system, we calculated the spin-dependent absorption coefficients which can be directly compared with the optically pumped NMR experiment. We show that the optically pumped NMR is a complimentary tool to traditional magneto optical absorption measurement, in the sense that optically pumped NMR is more sensitive to the light hole transitions which are very hard to resolve in the traditional magneto absorption measurement. For the AlInSb/InSb multiple quantum well system, we calculated both the magneto absorption spectra and the cyclotron resonance spectra. We compare both spectra to experimental results and achieve a good agreement. This agreement assures us that our understanding of the valence band structure of the narrow gap InSb materials are correct.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Xingyuan Pan.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Stanton, Christopher J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042109:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042109/00001

Material Information

Title: Magneto-Optical Properties of Narrow-Gap Semiconductor Heterostructures
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Pan, Xingyuan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Next generation of semiconductor device will not only based on the charge transport properties of the carrier, but also their spin degree of freedom. In order to understand or predict how those devices work one need to understand the spin-dependent electronic structures of both bulk and low-dimensional semiconductors. We have theoretically studied the spin-dependent Landau levels for electrons or holes in bulk GaAs system and AlInSb/InSb multiple quantum wells system. We use the envelope function approximation for the electronic and magneto-optical properties of AlInSb/InSb superlattices. Our model includes the conduction electrons, heavy holes, light holes and the split-off holes for a total of 8 bands when spin is taken into account. It is a generalization of the Pidgeon-Brown model to include the wave vector dependence of the electronic states, as well as quantization of wave vector due to multiple quantum well superlattice effects. In addition, we take strain effects into account by assuming pseudomorphic growth conditions. For bulk GaAs system, we calculated the spin-dependent absorption coefficients which can be directly compared with the optically pumped NMR experiment. We show that the optically pumped NMR is a complimentary tool to traditional magneto optical absorption measurement, in the sense that optically pumped NMR is more sensitive to the light hole transitions which are very hard to resolve in the traditional magneto absorption measurement. For the AlInSb/InSb multiple quantum well system, we calculated both the magneto absorption spectra and the cyclotron resonance spectra. We compare both spectra to experimental results and achieve a good agreement. This agreement assures us that our understanding of the valence band structure of the narrow gap InSb materials are correct.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Xingyuan Pan.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Stanton, Christopher J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042109:00001


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MAGNETO-OPTICALPROPERTIESOFNARROW-GAPSEMICONDUCTOR HETEROSTRUCTURES By XINGYUANPAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010XingyuanPan 2

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Idedicatethistomywife,ShuyangGu,andmyparents. 3

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ACKNOWLEDGMENTS Iwouldliketothankmythesisadvisor,Dr.ChristopherStan tonforhisconsistent supportingandadvisingduringmygraduatestudy.Hisinsig htincondensedmatter physicsalwayshelpmealotwheneverIwasintroubleduringm yresearch.Ialso learnedalotonhowtobeascientistfromhisprofessionalat titudetowardsscientic research.Icannotforgetthosenumeroushourswespenttoge therinhisofce, discussingallkindsofphysicsproblemingeneral,andmygr aduateresearchon semiconductorphysicsinparticular. IalsoliketothankmyPh.D.supervisorycommitteemembers, Dr.DavidReitze, Dr.PradeepKumar,Dr.Hai-PingChengandDr.CammyAbernath y,forspendingtheir valuabletimeonreviewingmyPh.D.proposalformyqualifyi ngexam,andonreviewing mymanuscriptforthisdissertation. TherearealsoseveralcolleaguesIwanttothank.Dr.GarySa ndershelped mealotduringmygraduateresearch,especiallywhenImetso medifcultiesinthe codingusingFortranprogramminglanguage.Healwaystried hisbesttohelpme locatethebug.IalsoliketothankDavidHansenandBrentNel sonfortheirsupportin computerrelatedproblems.Iwouldliketothankourcollabo ratorsDr.SophiaHayes inSt.Louis,andDr.MikeSantosandDr.RyanDoezemainOklah oma,forproviding theexperimentaldata.IwanttothankMs.KristinNicholafo rherhelpduringmystay here,andIalsoliketothankallmyfriendsinGainesville.L astIwanttothankmywife, ShuyangGu,andmyparents.Theirlovearealwaysthebestsup porttome. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2THEORY ....................................... 14 2.1 k p Method .................................. 14 2.2Spin-OrbitInteraction ............................. 16 2.3L ¨ owdin'sPerturbationTheory ......................... 18 3BANDSTRUCTUREOFIII-VSEMICONDUCTORMATERIALS ......... 23 3.1UnperturbedProblem—BasisFunctions ................... 23 3.2 k p PerturbationWithoutCouplingtoDistantBands ............ 27 3.3CouplingtoDistantBands ........................... 33 4LANDAULEVELSOFGALLIUMARSENIDE ................... 37 4.1TheEnvelopeFunctionApproximation .................... 37 4.2ExplicitFormoftheEFAHamiltonian ..................... 38 4.2.1LandauHamiltonian .......................... 41 4.2.2ZeemanHamiltonian .......................... 42 4.3EnergyandEnvelopeFunctions ....................... 45 4.4NumericalCalculationsoftheGaAsLandauLevels ............. 54 4.5Magneto-OpticalAbsorption .......................... 56 4.6Spin-PolarizedAbsorptionandOpticallyPumpedNMR ........... 59 5MAGNETO-PROPERTIESOFINDIUMANTIMONIDEQUANTUMEWELLS .. 86 5.1ExperimentalDetails .............................. 86 5.2ExtensionoftheTheoreticalModel ...................... 86 5.2.1NarrowEnergyGap .......................... 87 5.2.2StrainEffect ............................... 87 5.2.3QuantumConnementEffect ..................... 89 5.3Magneto-OpticalAbsorption .......................... 91 5.4ResultsandDiscussion ............................ 93 5

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6CONCLUSIONSANDFUTUREDIRECTIONS .................. 104 6.1Conclusions ................................... 104 6.2FutureDirections ................................ 106 6.2.1ExcitonAbsorption ........................... 106 6.2.2BeyondtheAxialApproximation .................... 107 6.2.3CarrierDynamics ............................ 107 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 112 6

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LISTOFTABLES Table page 1-1Electroneffectivemassand g -factorofGaAs,InAsandInSb .......... 13 4-1Percentageprobabilityofthezonecenterwavefunction sat4.7T ........ 62 4-2Percentageprobabilityofthezonecenterwavefunction sat7.0T ........ 64 4-3Selectionrules .................................... 66 7

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LISTOFFIGURES Figure page 4-1EnergybandstructureforGaAsat B =4.7T .................... 66 4-2EnergybandstructureforGaAsat B =7.0T .................... 67 4-3Valencebandstructurefor4.7T ........................... 68 4-4Valencebandstructurefor7.0T ........................... 69 4-5WavefunctioncomponentsvsLandaulevelquantumnumber at4.7T ...... 70 4-6WavefunctioncomponentsvsLandaulevelquantumnumber at7.0T ...... 71 4-7SpindependentabsorptionforB=4.7T( + polarization) ............. 72 4-8SpindependentabsorptionforB=4.7T( polarization) ............. 73 4-9SpindependentabsorptionforB=7.0T( + polarization) ............. 74 4-10SpindependentabsorptionforB=7.0T( polarization) ............. 75 4-11ActivecomponentsforabsorptionatB=4.7T( + polarization) .......... 76 4-12ActivecomponentsforabsorptionatB=4.7T( polarization) .......... 77 4-13ActivecomponentsforabsorptionatB=7.0T( + polarization) .......... 78 4-14ActivecomponentsforabsorptionatB=7.0T( polarization) .......... 79 4-15Calculatedabsorptioncomparedwithexperiments ................ 80 4-16OPNMRsignalintensityasafunctionofphotonenergyfo r excitation .... 81 4-17OPNMRsignalintensityasafunctionofphotonenergyfo r + excitation .... 82 4-18First6manifoldsofLandaulevelsat4.7T ..................... 83 4-19Spin-upandspin-downabsorptionfor lightat4.7T .............. 84 4-20ComparisonbetweenOPNMRspectrawithcalculatedspin polarization .... 85 5-1Samplestructureandexperimentalsetup ..................... 98 5-2BanddiagramforanInSb/AlInSbmultiplequantumwell ............. 98 5-3Absorptionspectrawaterfallplot .......................... 99 5-4MQWspectraforB=6T ............................... 100 5-5MQWfandiagram .................................. 101 8

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5-6MQWeigenfunctionsatB=6T ............................ 102 5-7Absorptionwithdifferentpolarization ........................ 103 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MAGNETO-OPTICALPROPERTIESOFNARROW-GAPSEMICONDUCTOR HETEROSTRUCTURES By XingyuanPan December2010 Chair:ChristopherStantonMajor:Physics Nextgenerationofsemiconductordevicewillnotonlybased onthechargetransport propertiesofthecarrier,butalsotheirspindegreeoffree dom.Inordertounderstand orpredicthowthosedevicesworkoneneedtounderstandthes pin-dependent electronicstructuresofbothbulkandlow-dimensionalsem iconductors.Wehave theoreticallystudiedthespin-dependentLandaulevelsfo relectronsorholesin bulkGaAssystemandAlInSb/InSbmultiplequantumwellssys tem.Weusethe envelopefunctionapproximationfortheelectronicandmag neto-opticalproperties ofAlInSb/InSbsuperlattices.Ourmodelincludesthecondu ctionelectrons,heavy holes,lightholesandthesplit-offholesforatotalof8ban dswhenspinistakeninto account.ItisageneralizationofthePidgeon-Brownmodelt oincludethewavevector dependenceoftheelectronicstates,aswellasquantizatio nofwavevectordueto multiplequantumwellsuperlatticeeffects.Inaddition,w etakestraineffectsintoaccount byassumingpseudomorphicgrowthconditions.ForbulkGaAs system,wecalculated thespin-dependentabsorptioncoefcientswhichcanbedir ectlycomparedwiththe opticallypumpedNMRexperiment.Weshowthattheoptically pumpedNMRisa complimentarytooltotraditionalmagnetoopticalabsorpt ionmeasurement,inthesense thatopticallypumpedNMRismoresensitivetothelighthole transitionswhicharevery hardtoresolveinthetraditionalmagnetoabsorptionmeasu rement.FortheAlInSb/InSb multiplequantumwellsystem,wecalculatedboththemagnet oabsorptionspectraand 10

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thecyclotronresonancespectra.Wecomparebothspectrato experimentalresultsand achieveagoodagreement.Thisagreementassuresusthatour understandingofthe valencebandstructureofthenarrowgapInSbmaterialsarec orrect. 11

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CHAPTER1 INTRODUCTION Understandingtheelectronic,transportandopticalprope rtiesofrealmaterials enableustoutilizetheuniquepropertiesofeachmaterialt omakedevicesapplications. Manyofthesepropertiesarecloselyrelatedtotheelectron icstructures.Nowadays certainexperimentaltoolscanbeusedtoprobetheelectron icstructures,forexample, thedeHaas-vanAlpheneffectisaveryusefultooltoprobeth eFermisurfaceformetals. Anotherexampleisusingthemagneto-opticalabsorptionto probetheLandaulevelsfor semiconductors,orusingcyclotronresonancemeasurement toprobetheeffectivemass tensor.Fromalltheseavailableexperimentaltoolswefoun dthatmagneticeldisvery importantinstudyingtheelectronicstructures.Theoreti callywealsowanttomodelthe electronicstructuresofmaterialsinthemagneticeld.Be causethematerialsweare interestedinaredirect-gapsemiconductors,andweareonl yinterestedinaverysmall rangeoftheBrillouinzonearoundthezonecenter,the k p methodismostsuitable hereforourpurpose. Thefactthatthesolidcanbecategorizedintometals,insul atorsandsemiconductors isbasedonitselectronicstructure.Metalsdonothaveaban dgap,insulatorshavea largebandgapandsemiconductorsusuallyhaveabandgaples sthan4electronvolts. Manytypesofelectronicdevicesaredesignedbasedonthese miconductorsmaterials anditisfairtosaysemiconductormaterialsarethebasisof themodernelectronic industry.Atthistimethemostimportantandmostfamoussem iconductorissilicon,not onlybecauseofitspropertiesbutalsobecauseitsmanufact ureissomaturethatthe costislowcomparedwithothermaterials.However,III-Vse miconductorsalsofound themselvesimportantdeviceapplications.Widebandgapse miconductorssuchasGaN ndtheirapplicationsinlight-emittingdiodes(LEDs),la sersanddetectorsinthevisible andultravioletrange.Attheotherend,narrowbandgapIIIVsemiconductorshavesome uniquepropertiesthatcanbeusedinanumberofelectronicd evices[ 1 – 3 ]. 12

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InSbistheIII-Vbinarysemiconductorwiththesmallestban dgap[ 4 ],smallest electroneffectivemassandlargest g -factor.Table 1-1 showstheelectroneffectivemass m and g -factorforthreeimportantIII-Vmaterials.Ithaspotenti alapplicationssuchasa optoelectronicdeviceintheinfraredrangeduetothenarro wgap[ 5 ],andfasttransistors duetothesmalleffectivemass[ 6 ].Italsohaspotentialapplicationsintheareaofspin electronics,sinceitslarge g factorcanbeusedtocontrolitsspindegreeoffreedom throughitsorbitalenvironment. Table1-1.Electroneffectivemassand g -factorofGaAs,InAsandInSb Electron m Electron g -factor GaAs0.067 m 0 -0.5 InAs0.023 m 0 -15 InSb0.014 m 0 -51 TheoreticallyitishardtocalculatethebandstructureofI nSbcomparedwithother III-Vsemiconductors[ 7 ],becauseofthestrongmixingbetweenconductionbandsand valencebandsthatcomesfromthenarrowbandgap.Beforewea ttackthiscomplicated problem,webeginwithasimplerproblemofcalculatingthee lectronicstructuresofbulk GaAsinthemagneticeld.Thetheoreticalmodelisbasedonr eference[ 8 ].Itissimpler thantheInSbproblembecauseitisnotaquantumconnedsyst em,andalsothereis noneedtoconsiderstraineffect.Howeverwewillstillkeep theeightbandmodeltofully accountforthecouplingbetweenconductionbandsandvalen cebands.Thismodelcan beextendedlatertodealwiththeInSbquantumwellsystem.B ecauseoftheverylarge g factorInSbhas,magneto-absorptioncanbeusedtodetermin ethespinsplittingsin theconductionband[ 9 10 ].Inordertogetadeeperunderstandingofthevalenceband structure,especiallythedependenceofthevalancebandso nstrainandconnement, weperformedtheoreticalstudiesofmagneto-opticaltrans itionsacrossthebandgapin strainedInSbquantumwellswithAlInSbbarriers,andcompa redtoexperimentalstudies bySantosgroupattheUniversityofOklahoma. 13

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CHAPTER2 THEORY 2.1 k p Method Theelectronenergyandwavefunctioninaperiodicpotentia listhestartingpointof thetheoreticalmodel.Thisisgivenbythewell-knowSchr ¨ odingerequationforaBloch electron H k ( r )= E k ( r )(2–1) where k isthewavevectorandtheHamiltonian H isgivenby H = p 2 2 m 0 + V 0 ( r ),(2–2) here p isthemomentumoperator, m 0 istheelectronmassand V 0 ( r )istheperiodic potential.RightnowtheHamiltonianisthesimpleformofas ingleBlochelectron,and IwilladdmoretermstothissimpleHamiltonianasIgotomore complicatedsituations suchasspin-orbitinteraction,connementpotentialandm agneticeldetc. Manydifferentmethodsexisttosolveequation( 2–1 )whichcanbefoundin standardtextbooksuchas[ 11 ].Iaminterestedinthemagneto-opticalpropertiesof theIII-Vsemiconductorsandforthiskindofsystem,thepro pertiesaredetermined bythelowestlyingexcitedstatesnearthebandedge.The k p methodisthemost suitablemethodheretosolvingforelectronenergyandwave functions.Thismethod isaperturbativemethod,assumingthesolutiontoequation ( 2–1 )foracertain k = k 0 valuearealreadyknownandexpandthesolutionsofother k intermsofthelinear combinationofthose k = k 0 solutions.Whenchoosing k 0 = 0 ,i.e.,thecenterofthe Brillouinzone,thesolutionsareofhighsymmetryandserve asanaturalbasis. FromBlochtheoremthewavefunction k ( r )inequation( 2–1 )canbechosento havetheform k ( r )=e i k r u k ( r ),(2–3) 14

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where u k ( r )hasthesameperiodicityasthecrystallattice.Substitut eequation( 2–2 ) andequation( 2–3 )intoequation( 2–1 ),weobtainandifferentialequationfor u k ( r ),the periodicpartofthewavefunction: p 2 2 m 0 + V 0 ( r )+ ~ m 0 k p u k ( r )= E ( k ) ~ 2 k 2 2 m 0 u k ( r ).(2–4) Thethirdtermintheaboveequationisproportionalto k p ,hencethename“ k p ” method. Tosolveequation( 2–4 )foranarbitrary k ,weassumethatthesolutionsfora particularpoint k 0 areknown,anddenotethemas f u n 0 k 0 ( r ) g ,where n 0 labelsthe differentsolutionsforthesame k 0 point.Thissetofsolutions f u n 0 k 0 ( r ) g provideabasis forsolutionsforother k point.Inpractice,wechoose k 0 = 0 ,i.e.,thecenterofthe Brillouinzone.Now u k ( r )inequation( 2–4 )canbeexpressedasalinearcombinationof thebasisfunctions f u n 0 0 ( r ) g : u k ( r )= X n 0 c n 0 ( k ) u n 0 0 ( r ).(2–5) Here u n 0 0 ( r )satisfythefollowingtwoequations p 2 2 m 0 + V 0 ( r ) u n 0 0 ( r )= E n 0 ( 0 ) u n 0 0 ( r )(2–6) and Z unitcell u n 0 ( r ) u n 0 0 ( r ) d 3 r = nn 0 .(2–7) Substituteequation( 2–5 )intoequation( 2–4 )andmultiplyfromtheleftby u n 0 ( r ),and thenintegratingovertheunitcell,byusingequation( 2–6 )and( 2–7 ),weobtain X n 0 E n 0 ( 0 )+ ~ 2 k 2 2 m 0 nn 0 + ~ m 0 k P nn 0 c n 0 ( k )= E ( k ) c n ( k ),(2–8) where P nn 0 = Z unitcell u n 0 ( r ) p u n 0 0 ( r ) d 3 r .(2–9) 15

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Equation( 2–8 )canbediagonalizedtoobtaintheenergyandwavefunctions forarbitrary k .Sonowwehaveaworkingroutinetosolvetheelectronicstat esneartheband edge.Howeverequation( 2–8 )willnotgiveusreasonableresultssincewehavenot consideredthespin-orbitinteractionyet.Thisisthesubj ectofthenextsection. 2.2Spin-OrbitInteraction Aspromised,therstcorrectiontotheHamiltonianinequat ion( 2–2 )isaterm comingfromthespin-orbitinteraction: H SO = ~ 4 m 2 0 c 2 p ( r V 0 ).(2–10) where c isthespeedoflightand isthePaulispinmatrixthatactingonthespinor states.Theoriginofthistermcanbetracedbacktothenon-r elativisticapproximation totheDiracequationwhichcanbefoundonanystandardquant ummechanictextbook suchas[ 12 ].TheimportanceofthistermisdiscussedbyElliott[ 13 ],Dresselhaus[ 14 15 ] andParmenter[ 16 ].WhenthistermisincludedintheoriginalHamiltonian,we havethe Schr ¨ odingerequation: p 2 2 m 0 + V 0 ( r )+ ~ 4 m 2 0 c 2 p ( r V 0 ) e i k r u k ( r )= E ( k )e i k r u k ( r ),(2–11) herewealreadywritethewavefunctionintermsoftheBlochf unction, k ( r )=e i k r u k ( r ). Notethat u k ( r )isacompactnotationofthetwo-componentcolumnspinorfu nction u k ( r )= 264 u (1) k ( r ) u (2) k ( r ) 375 = u (1) k ( r ) j"i + u (2) k ( r ) j#i .(2–12) NowevaluatingtheHamiltonianoperatoractingontheplane waveparte i k r ,wegota equationforthetwocomponentspinorpartofthewavefuncti on p 2 2 m 0 + V 0 ( r )+ ~ 2 k 2 2 m 0 + ~ m 0 k + ~ 4 m 2 0 c 2 p ( r V 0 ) u k ( r )= E ( k ) u k ( r )(2–13) 16

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where = p + ~ 4 m 0 c 2 r V 0 ,(2–14) Thisistheequationwewanttosolveinthepresenceofspin-o rbitinteraction. Likethecasewhenthereisnospin-orbitinteraction,thege neralsolutionsfor equation( 2–13 )canstillbeexpandedintermsofthezonecentersolutions. Wecanstill writethemas u k ( r )= X n 0 c n 0 ( k ) u n 0 0 ( r ).(2–15) Here u n 0 0 ( r )satisfythefollowingtwoequations p 2 2 m 0 + V 0 ( r ) u n 0 0 ( r )= E n 0 ( 0 ) u n 0 0 ( r )(2–16) and Z unitcell u y n 0 ( r ) u n 0 0 ( r ) d 3 r = nn 0 .(2–17) Substitutethisexpansionof u k ( r )intoequation( 2–13 )andmultiplyfromtheleftby u y n 0 ( r ),andthenintegrateoveraunitcell,byusingequation( 2–16 )and( 2–17 ),wegot thematrixeigenvalueequationfortheexpansioncoefcien t c n 0 ( k ) X n 0 E n 0 ( 0 ) nn 0 + ~ m 0 k nn 0 + nn 0 c n 0 ( k )= E 0 ( k ) c n ( k ),(2–18) where E 0 ( k )= E ( k ) ~ 2 k 2 2 m 0 (2–19) nn 0 = Z cell u y n 0 ( r ) u n 0 0 ( r ) d 3 r (2–20) nn 0 = ~ 4 m 2 0 c 2 Z cell u y n 0 ( r )[ p r V 0 ] u n 0 0 ( r ) d 3 r .(2–21) Notethat nn 0 hastwocontributions nn 0 = Z cell u y n 0 ( r ) p u n 0 0 ( r ) d 3 r + ~ 4 m 0 c 2 Z cell u y n 0 ( r )[ r V 0 ] u n 0 0 ( r ) d 3 r (2–22) 17

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thesecondcontributionisverysmallcomparedwiththe nn 0 term[ 7 ],thuscanbe neglected.Inthisapproximationthematrixelement nn 0 isgivenby nn 0 = p nn 0 = Z cell u y n 0 ( r ) p u n 0 0 ( r ) d 3 r .(2–23) Nowwecancalculatingtheelectronenergyandwavefunction sbysolvingthe eigenvalueproblemgivenbyequation( 2–18 ),butnotethatthematrixarisingfrom equation( 2–18 )isinnitedimensional,makingitinpracticalunsolvable .Evenifwe knowhowtosolveaninnitedimensionaleigenvalueproblem ,westillneedtoknow theinnitenumberofthebandedgeeigenstates.Thisisalso adauntingtask.The goodnewsisthatpeopleareusuallyonlyinterestedinafewa djacentbands,either neglectingremotebandscompletelylikethe8-bandKanemod el[ 7 ],ortreatingthemas aperturbation[ 17 18 ].Asystematicapproachtoreducethedimensionalitytodot he perturbationproblemisthesocalled“L ¨ owdin'sperturbationtheory”,andIwillintroduce thismethodinthenextsection. 2.3L ¨ owdin'sPerturbationTheory The k p Hamiltonianincludingthespin-orbitinteractiongivesus aneigenvalue problemtosolve.Thiseigenvalueproblem,givenbyequatio n( 2–18 ),involves diagonalizingainnitedimensionalmatrix,whichmakeiti mpossiblewithoutfurther simplications.OneofsuchsimplicationswasgivenbyL ¨ owdin[ 19 ].Theadvantageof L ¨ owdin'sperturbationtheoryovertraditionalperturbatio ntheoriessuchasRayleigh-Schr ¨ odinger perturbationsisthatwedon'tneedtodifferentiatethenon -degenerateanddegenerate cases,i.e.,wecantreatbothcasestogetherinasystematic way.Thisadvantage makeitwellsuitabletoproblemssuchasthevalencebandspi nstructuresofIII-V semiconductors.InthissectionIwillnotgiveadetailedde rivationofthetheoryitself, ratherIwouldsummarizethekeyresultsofthetheoryinafor mthatisreadilyapplicable toourcurrentproblems,theproblemofsolvingthe k p eigenvalueandeigenfunctions. 18

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AthroughtreatmentoftheL ¨ owdin'sperturbationtheorycanbefoundintheoriginal paperofL ¨ owdin[ 19 ],orthebookbyRolandWinkler[ 20 ]. Theproblemwewanttosolveisthetime-independentSchr ¨ odingerequation H j i = E j i ,(2–24) wheretheHamiltonian H canbedividedintotwoparts:amajorcontribution H 0 ,anda smallperturbationterm H 0 H = H 0 + H 0 .(2–25) Likeotherperturbationmethods,weassumethatwealreadyk nowthesetofsolutions fj n ig tothe H 0 partoftheproblem H 0 j n i = E n j n i .(2–26) Ourtaskistondthesolution j i toequation( 2–24 )intheformofalinearcombination ofthesetoffunctions fj n ig j i = X n c n j n i .(2–27) L ¨ owdinassumedthatthesetofunperturbedeigenstates fj n ig naturallyfallsinto twocategories,whichwecallset A andset B .Eacheigenstateof j n i belongstooneof them.Theenergyofthosestatesthatinset A aredifferentfromthoseinset B .Inother words,wemayormaynothavedegeneratestateswithinset A orset B butwearesure thatastatefromset A willbeneverdegeneratewithastatefromset B .Indoingthe linearcombinationtogettheeigenstatesofthetotalHamil tonian,weareonlyinterested inthosestatescomingfromset A ,treatingcontributionfromset B asaperturbation. Lateronwewillseethatinourproblemofsolvingthe k p Hamiltonian,weareonly interestedinthelowestconductionbandsandhighestvalen cebands(thesebands formset A ),treatingthedistantbandsasaperturbation(thedistant bandsformset B ). Thismethodwillreducethedimensionalityoftheproblemfr ominnitytothenumberof statesinset A 19

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Weusethefollowingnotations: m m 0 etcarequantumnumberslabelingstates fromset A ,and l l 0 etcarequantumnumberslabelingstatesfromset B .Wewanttond anapproximatesolutiontoequation( 2–24 )thatisvaliduptothedesiredorderofthe perturbatingHamiltonian H 0 : j i = X m 2 A c m j m i ,(2–28) notethathere j m i belongstosetAonly.Ifthematrixelementscouplingset A andset B vanish,i.e., h m j H j l i =0forevery m 2 A and l 2 B ,wealreadyhaveset A andset B decoupledandourinterestedeigenstatesareexactlyinthe formofequation( 2–28 ).If therearenon-zeromatrixelements h m j H j l i ,wewanttondaunitarytransformation generatedbytheanti-Hermitianoperator S e H =e S H e S ,(2–29) sothatthetransformedHamiltoniandoesnothavecouplingb etweenset A andset B h m j e H j l i =0.(2–30) Sincethisisaperturbativetheory,whatwereallymeanis h m j e H j l i vanishesuptothe desiredorderof H 0 .Thegeneratoroftheunitarytransformation S canbeexpressedin theasymptoticform S = S (1) + S (2) + S (3) +...(2–31) where S (1) isthe1st-orderinnitesimal, S (2) isthe2nd-orderinnitesimal,andsoon. Aftersomealgebra[ 20 ]wecanshowthattherst3termsofthisanti-Hermitianoper ator S aregivenby S (1) ml = H 0 ml E m E l (2–32a) S (2) ml = 1 E m E l X m 0 H 0 mm 0 H 0 m 0 l E m 0 E l X l 0 H 0 ml 0 H 0 l 0 l E m E l 0 # (2–32b) S (3) ml = 1 E m E l 20

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" X m 0 m 00 H 0 mm 00 H 0 m 00 m 0 H 0 m 0 l ( E m 00 E l )( E m 0 E l ) X l 0 l 00 H 0 ml 0 H 0 l 0 l 00 H 0 l 00 l ( E m E l 00 )( E m E l 0 ) + X l 0 m 0 H 0 mm 0 H 0 m 0 l 0 H 0 l 0 l ( E m 0 E l )( E m 0 E l 0 ) + X l 0 m 0 H 0 mm 0 H 0 m 0 l 0 H 0 l 0 l ( E m E l 0 )( E m 0 E l 0 ) + 1 3 X l 0 m 0 H 0 ml 0 H 0 l 0 m 0 H 0 m 0 l ( E m 0 E l 0 )( E m 0 E l ) + 1 3 X l 0 m 0 H 0 ml 0 H 0 l 0 m 0 H 0 m 0 l ( E m E l 0 )( E m 0 E l 0 ) + 2 3 X l 0 m 0 H 0 ml 0 H 0 l 0 m 0 H 0 m 0 l ( E m E l 0 )( E m 0 E l ) # ,(2–32c) where H 0 ml = h m j H 0 j l i .Notethatforevery j wehave S ( j ) lm = S ( j ) ml ,and S ( j ) mm 0 = S ( j ) ll 0 =0so thatwecanactuallyobtaineveryelementsof S fromequation( 2–32 ). Finallysubstitutingequation( 2–32 )intoequation( 2–29 )wegettheasymptoticform ofthetransformedHamiltonian e H = H (0) + H (1) + H (2) + H (3) +...,(2–33) where H (0) mm 0 = H 0 mm 0 (2–34a) H (1) mm 0 = H 0 mm 0 (2–34b) H (2) mm 0 = 1 2 X l H 0 ml H 0 lm 0 1 E m E l + 1 E m 0 E l # (2–34c) H (3) mm 0 = 1 2 X lm 00 H 0 ml H 0 lm 00 H 0 m 00 m 0 ( E m 0 E l )( E m 00 E l ) + H 0 mm 00 H 0 m 00 l H 0 lm 0 ( E m E l )( E m 00 E l ) # + 1 2 X ll 0 H 0 ml H 0 ll 0 H 0 l 0 m 0 1 ( E m E l )( E m E l 0 ) + 1 ( E m 0 E l )( E m 0 E l 0 ) # .(2–34d) Notethatforevery j wehave H ( j ) ml = H ( j ) lm =0,but H ( j ) ll 0 neednotbezero.Sinceweareonly interestedinset A ,wedonotneedtheexpressionfor H ( j ) ll 0 .Nowwehavereducedour original( m + l )dimensionaleigenvalueproblem( 2–24 )toan m dimensionalone.The rst4termsofthis m dimensionalHamiltonianisgivenbyequation( 2–34 ). 21

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Goingbacktoouroriginal k p eigenvalueproblem,ifweareonlyinterestedin lowestconductionbandsandthehighestvalencebands,then thesebandsarewhatwe calledset A andallotherdistantbandsmakeuptheset B .Dependingonthenumber ofbandsweareinterestedin(thenumberofbandsinset A ),wemayhavethe6-band Luttingermodel,8-bandKanemodel,14-bandextendedKanem odeletc.Theworkin thisthesisisdoneusingthe8-bandKanemodel.Inthenextch apter,wewillusethe k p methodandtheL ¨ owdin'sperturbationtheorytogethertostudythebandstru cture ofbulkIII-Vsemiconductormaterialsintheabsenceofaext ernalmagneticeld.Only afterweunderstandthisproblemweareabletotacklemoreco mplicatedproblems suchastheLandaulevelsinamagneticeld,quantumconnem entpotentialcoming fromthesemiconductorsuperlattices,andmagneto-optica lpropertiesofsemiconductor materials,whetheritisbulkorlow-dimensional. 22

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CHAPTER3 BANDSTRUCTUREOFIII-VSEMICONDUCTORMATERIALS 3.1UnperturbedProblem—BasisFunctions Wehavederivedaworkingmethodtodiagonalizethe k p Hamiltonian( 2–18 ), namely,theLowdin'sperturbationtheory.Likeanyotherqu antummechanicsperturbation theory,therststepistosolvetheunperturbedproblem.Si ncewearemostly concernedaboutaverysmallvolumeof k -spacenearthecenteroftheBrillouin zone, k isverysmallandinequation( 2–18 )wecantreatthetermlinearin k asa perturbation.Wedon'tneedtotreatthe ~ 2 k 2 2 m 0 termasaperturbationbecausethispartof theHamiltonianisalreadydiagonal.Theunperturbedeigen valueproblemis X n 0 [ E n 0 ( 0 ) nn 0 + nn 0 ] c n 0 ( k )= E 0 ( k ) c n ( k ),(3–1) where E 0 ( k )= E ( k ) ~ 2 k 2 2 m 0 .(3–2) Recallthat c n 0 ( k )istheexpansioncoefcientofthegeneralsolution u k ( r ): u k ( r )= X n 0 c n 0 ( k ) u n 0 0 ( r ),(3–3) where u n 0 0 ( r )isthebandedgebasisfunctionwithoutspin-orbitinterac tion.Although wedonotknowthefunctionformofthebandedgeeigenstates, weknowthatthey transformedaccordingtotheirreduciblerepresentationo fthepointgroupofthe crystal.ForIII-VsemiconductorssuchasGaAsandInSb,the pointgroupisthe tetrahedralgroup( T d group).FollowingthenotationofKoster[ 21 22 ],theve irreduciblerepresentationsof T d are i for i =1,2,3,4,5.Thelowestconduction bandbelongstothe 1 representationanditseigenstatetransformsastheatomic s -orbital(orbitalangularmomentum l =0),thuswecanlabelthisconductionbandedge functionas j S i .Similarly,thehighestvalencebandbelongstothe 5 representation andithasthreedegeneratedeigenstatesthattransformast heatomic p -orbital(orbital 23

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angularmomentum l =1),whichcanbelabeledas j X i j Y i and j Z i .Nowwehavefour bandedgestatesnearthefundamentalgap: j S i j X i j Y i and j Z i .Includingspindegree offreedombutnotspin-orbitinteraction,wehave8bandedg estates: j 0 0 i = fj S "i j X "i j Y "i j Z "i j S #i j X #i j Y #i j Z #ig (3–4) Whenspin-orbitinteractionisconsideredasinequation( 3–1 ),thebandedge eigenstatesareclassiedaccordingtothedoublegroupfor the T d group.The irreduciblerepresentationofthedoublegroupcanbeobtai nedfromtakingthe directproductofthecorrespondingsinglegrouprepresent ationwiththeirreducible representationofthespinorwhichwecall 6 .Grouptheorytellusthat 1 n 6 = 6 (3–5a) 5 n 6 = 7 + 8 (3–5b) Sotheoriginalconductionbandof 1 representationbecomes 6 andthevalenceband of 5 representationbecomes 7 and 8 .Thiswillsplitthe6-folddegeneratevalence bandat k =0intotwosets:the4-folddegenerate 8 bandandthe2-folddegenerate 7 band.Asexpectedtheconductionbandremainsthesameasift hereisnospin-orbit interactionbecausetheconductionbandis“ s like”andtheorbitangularmomentum l =0. Althoughthefunctionsgiveninequation( 3–4 )canserveasabasiswhenspin-orbit interactionisincluded,theyarenottheeigenstatesforto talangularmomentum J = L + S andthematrix 0 0 inequation( 3–1 )isnotdiagonal.Itisbettertochooseanewset ofbasisfunctionsthatcandiagonalizethespin-orbitinte ractionHamiltonian.Sucha setofbasisfunctionscanbeobtainedbydoingtheproblemof theadditionofangular momenta.Thefoureigenstatesoftheorbitangularmomentum j l m l i canbedenedas 24

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(similartothesphericalharmonics) j 0,0 i = j iS i (3–6a) j 1,1 i = j X + iY i = p 2(3–6b) j 1,0 i = j Z i (3–6c) j 1, 1 i = j X iY i = p 2,(3–6d) andthetwospineigenstates j s m s i canbewrittenas 1 2 ,+ 1 2 = j"i (3–7a) 1 2 1 2 = j#i .(3–7b) Forthe l =0conductionbands,thetotalangularmomentum j =0+1 = 2,andforthe l =1 valencebands,thetotalangularmomentum j =1 1 = 2.Usingthestandardtechnique ofadditionofangularmomentumwecanobtainthebasisfunct ions j j m j i fromequation ( 3–6 )and( 3–7 ), 1 2 ,+ 1 2 c = j iS "i (3–8a) 1 2 1 2 c = j iS #i (3–8b) 3 2 ,+ 3 2 v = 1 p 2 j ( X + iY ) "i (3–8c) 3 2 ,+ 1 2 v = 1 p 6 j ( X + iY ) # 2 Z "i (3–8d) 3 2 1 2 v = 1 p 6 j ( X iY ) +2 Z #i (3–8e) 3 2 3 2 v = 1 p 2 j ( X iY ) #i (3–8f) 1 2 ,+ 1 2 v = 1 p 3 j ( X + iY ) # + Z "i (3–8g) 1 2 1 2 v = 1 p 3 j ( X iY ) Z #i .(3–8h) 25

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Thephaseconventioninequation( 3–8 )ischosentobeagreewiththeClebsch-Gordan coefcients.Wehaveaslightlydifferentphaseconvention andre-orderthese8statesto formabasiswhichweuseinourcalculations.Ourbasisfunct ionsfromnowonreadas thefollowing: j 1 i = j CB "i = 1 2 ,+ 1 2 c = j S "i (3–9a) j 2 i = j HH "i = 3 2 ,+ 3 2 v = 1 p 2 j ( X + iY ) "i (3–9b) j 3 i = j LH #i = 3 2 1 2 v = 1 p 6 j ( X iY ) +2 Z #i (3–9c) j 4 i = j SO #i = 1 2 1 2 v = i p 3 j ( X iY ) + Z #i (3–9d) j 5 i = j CB #i = 1 2 1 2 c = j S #i (3–9e) j 6 i = j HH #i = 3 2 3 2 v = i p 2 j ( X iY ) #i (3–9f) j 7 i = j LH "i = 3 2 ,+ 1 2 v = i p 6 j ( X + iY ) # 2 Z "i (3–9g) j 8 i = j SO "i = 1 2 ,+ 1 2 v = 1 p 3 j ( X + iY ) # + Z "i .(3–9h) Wecandirectlyverifythat,usingthebasisfunctionsin( 3–9 ),theunperturbed Hamiltonianisdiagonalizedanditsmatrixforminequation ( 3–1 )is H (0) = 264 A 0 0 A 375 (3–10) where A = 266666664 E s 000 0 E p + = 300 00 E p + = 30 000 E p 2 = 3. 377777775 (3–11) 26

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Here E s and E p arethebandedgeenergywithoutspin-orbitinteraction E s = h S j p 2 2 m 0 + V 0 ( r ) j S i (3–12a) E p = h Z j p 2 2 m 0 + V 0 ( r ) j Z i ,(3–12b) andthespin-orbitsplit-offenergy isdenedas = 3 i ~ 4 m 2 0 c 2 X @ V 0 @ x p y @ V 0 @ y p x Y .(3–13) Choosingtheenergyreferencepointof V 0 ,wecanlet E p = = 3anddene E g = E s ,so thatthe4 4matrix A inequation( 3–10 )canbewrittenas A = 266666664 E g 000 00000000000 377777775 .(3–14) Wehavealreadysolvedtheeigenvalueproblemfortheunpert urbedHamiltonian, theeigenstatesofwhicharegivenbyequation( 3–9 ).Therst4statesaredegenerate withthelast4states,respectively,andtheenergyforthem are: E g (forconduction band),0(forheavyandlightholebands)and (forspin-orbitband).Thisconcludes ourrststepinL ¨ owdinperturbations.Inthefollowingsectionswewillgoto thenextstep inL ¨ owdinperturbations. 3.2 k p PerturbationWithoutCouplingtoDistantBands IntheprevioussectionwediagonalizedthezerothorderHam iltonianandobtained thebandedgeenergyandbasisstatesinthepresenceofspinorbitinteractionbut withoutthe k p couplingbetweendifferentbands.InthenextlevelofL ¨ owdin perturbationtheorytheperturbatingHamiltonianisgiven byequation( 2–34b ).In thecontextofoureigenvalueproblemgivenbyequation( 2–18 ),theperturbating Hamiltonianisjusttheterm ~ m 0 k nn 0 or ~ m 0 k p nn 0 .Notethatatthislevelofperturbation, 27

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the k p termonlycouplesthosestatescomingfrom“set A ”,i.e., n and n 0 arechosen fromtheeightbasisstatesinequation( 3–9 ).Couplingtodistantbandswillbetreatedin thehigherorderperturbation. Grouptheoryshowsthattheonlynon-zeromomentummatrixel ementswithinour basisstatesin( 3–4 )areoftheform h S j p z j Z i ,andalltheotheronessuchas h S j p x j Y i and h X j p z j Z i vanish.Weonlyhavethreenon-zeromomentummatrixelement sandthey areallequal,sowecandenetheKane'sparameter V as V = i ~ m 0 h S j p x j X i = i ~ m 0 h S j p y j Y i = i ~ m 0 h S j p z j Z i .(3–15) Itisalsoconvenienttodenethe“plus”and“minus”wavevec tor k = k x ik y .(3–16) NowourrstorderperturbationHamiltonianinthematrixfo rmis H (1) = 2666666666666666666664 0 i p 2 Vk + i p 6 Vk 1 p 3 Vk 00 q 2 3 Vk z i p 3 Vk z i p 2 Vk 0000000 i p 6 Vk + 000 i q 2 3 Vk z 000 1 p 3 Vk + 000 1 p 3 Vk z 000 00 i q 2 3 Vk z 1 p 3 Vk z 0 1 p 2 Vk 1 p 6 Vk + i p 3 Vk + 0000 1 p 2 Vk + 000 q 2 3 Vk z 000 1 p 6 Vk 000 i p 3 Vk z 000 i p 3 Vk 000 3777777777777777777775 (3–17) 28

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Wecanadd H (0) and H (1) togethertoobtainourHamiltonian H : H = H (0) + H (1) .The explicitformofthematrix H isgivenby H = 2666666666666666666664 E g i p 2 Vk + i p 6 Vk 1 p 3 Vk 00 q 2 3 Vk z i p 3 Vk z i p 2 Vk 0000000 i p 6 Vk + 000 i q 2 3 Vk z 000 1 p 3 Vk + 00 1 p 3 Vk z 000 00 i q 2 3 Vk z 1 p 3 Vk z E g 1 p 2 Vk 1 p 6 Vk + i p 3 Vk + 0000 1 p 2 Vk + 000 q 2 3 Vk z 000 1 p 6 Vk 000 i p 3 Vk z 000 i p 3 Vk 00 3777777777777777777775 (3–18) Foranywavevector k thatisintherstBrillouin(butnotneedtobesmall),wecan diagonalizethematrix H inequation( 3–18 )toobtaintheelectronenergyandtheir wavefunctionintermsofthelinearcombinationoftheeight basisstates.Numerically diagonalizinga8 8matrixcanbedoneveryeasilybutitishardtodoitanalytic ally. Howeverifthewavevector k isrestrictednearthebandedge,i.e., k issmallcompared withthesizeoftheBrillouinzone,thenwecandiagonalize H inequation( 3–18 )using thetwo-stepprocedure:[ 7 ]rstconsiderthespecialcasewhen k isintheˆ z direction, k = k ˆ z anddiagonalize H inthisspecialcase;thenusingtheappropriateunitary transformationtorotatethebasisfunctiontohandlethege neralcasewhen k isina generaldirection. 29

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Inthespecialcasewhen k = k ˆ z ,wehave k =0and k z isreplacedbythe magnitude k ,theHamiltonian H becomes H = 2666666666666666666664 E g 00000 q 2 3 Vk i p 3 Vk 000000000000 i q 2 3 Vk 000 000 1 p 3 Vk 000 00 i q 2 3 Vk 1 p 3 VkE g 000 00000000 q 2 3 Vk 0000000 i p 3 Vk 000000 3777777777777777777775 .(3–19) Wecanimmediatelyseethattheheavyholestates j HH "i and j HH #i decouplefrom therestofthestates.Sowehavethedoublydegenerateenerg y E 0 =0.Therestofthe statesalsodecoupleintotwodifferentblockswhichwewill callthe“upper”blockandthe “lower”block.Theupperblockcorrespondingtostates j CB "i j LH "i and j SO "i is 266664 E g q 2 3 Vk i p 3 Vk q 2 3 Vk 00 i p 3 Vk 0 377775 ,(3–20) andthelowerblockcorrespondingthestates j CB #i j LH #i and j SO #i is 266664 E g i q 2 3 Vk 1 p 3 Vk i q 2 3 Vk 00 1 p 3 Vk 0 377775 .(3–21) Wehavesimplifytheoriginal8 8dimensionalproblemintotwo3 3onesandtwo 1-dimensionalones.Althoughtheupperandlowerblockmatr ixaredifferent,theylead tothesamesecularequation E 0 ( E 0 E g )( E 0 + ) V 2 k 2 E 0 + 2 3 =0.(3–22) 30

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Thisisacubicequationwhichwecansolvebyusingthestanda rdcubicformula,butitis tootedious.Sinceweknowthat k issmall,thethreesolutionstoequation( 3–22 )willbe closeto E 0 = E g E 0 =0and E 0 = .Assumeoneofthethreerootsis E 0 = E g + ,we canreplaceinequation( 3–22 )( E 0 E g )with andeveryother E 0 with E g .Thiswillgive us = V 2 k 2 ( E g +2 = 3) E g ( E g + ) ,(3–23) sotherstrootis E 0 = E g + V 2 k 2 ( E g +2 = 3) E g ( E g + ) .(3–24) Similartricksgiveustheothertworootsfor E 0 E 0 = 2 V 2 k 2 3 E g ,(3–25) E 0 = V 2 k 2 3( E g + ) .(3–26) Becausetheactualenergy E ( k )isgivenby E ( k )= E 0 ( k )+ ~ 2 k 2 = 2 m 0 ,wecan summarizeourresultsatthispointthatwehavefourdiffere ntenergybands,eachis two-folddegenerate,andtheyaregivenby(intheorderofco nductionband,heavyhole band,lightholebandandspin-orbitsplit-offband) E cb ( k )= E g + ~ 2 k 2 2 m 0 + V 2 k 2 ( E g +2 = 3) E g ( E g + ) (3–27a) E hh ( k )= ~ 2 k 2 2 m 0 (3–27b) E lh ( k )= ~ 2 k 2 2 m 0 2 V 2 k 2 3 E g (3–27c) E so ( k )= + ~ 2 k 2 2 m 0 V 2 k 2 3( E g + ) .(3–27d) Ifthewavevector k isinageneraldirectionspeciedbytwopolarangles and k = k sin cos ˆ x + k sin sin ˆ y + k cos ˆ z ,(3–28) 31

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wecanrotatetoanewbasissothatinthenewbasisthewavevec torisintheˆ z 0 direction, k = k ˆ z 0 .Thebandedgefunctionsarealsorotatedtothenewprimedst ates 266664 X 0 Y 0 Z 0 377775 = 266664 cos cos cos sin sin sin cos 0 sin cos sin sin cos 377775 266664 XY Z 377775 (3–29a) 264 0 # 0 375 = 264 e i = 2 cos = 2 e i = 2 sin = 2 e i = 2 sin = 2 e i = 2 cos = 2 375 264 "# 375 .(3–29b) Nextwedenethe8primedangularmomentumbasisstatessimi lartothoseinequation ( 3–9 ),forexample 3 2 ,+ 3 2 0v = 1 p 2 j ( X 0 + iY 0 ) 0 i .(3–30) Wecandirectlycalculatethematrixelementof H (0) and H (1) withrespecttotheprimed basis,afteraverytediousalgebrawewillseethatboth H (0) and H (1) stilltakethesame formasinequations( 3–10 )and( 3–17 ).Thereasonforthiscanbeunderstandfrom twopointofview:mathematically,thetransformationmatr ixconnectingtheoldandnew basisisunitary,physically,boththespin-orbitinteract ionandthe k p couplingterm areisotropic[ 23 ].Since H (0) and H (1) staythesameinthenewprimedbasis,theenergy dispersionrelationinequation( 3–27 )stillholdsforthegeneral k direction. Equation( 3–27 )giveusfourisotropic,parabolicbandsneartheBrillouin zone center,eachofwhichisdoublydegenerate.Wecandeneasca lareffectivemassfor eachband.Notethatinthecurrentmodeltheheavyholedispe rsionisstillthesame asafreeelectron,whichiscertainlywrong.Thisisbecause wehavenotincludethe couplingbetweenour8basisstatesfromset A andremotebandsinset B .Once couplingwithremotebandsisconsidered,wewillgetabette rapproximationforthe effectivemass,andwewillalsondthatthebandsareanisot ropic[ 24 ]. 32

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3.3CouplingtoDistantBands Inordertogetthemoreaccuratebanddispersionrelationne arthezonecenter, andtoaccountfortheanisotropicpropertiesofthebands,w eneedtoconsiderthe couplingeffectsgeneratedbythe ~ m 0 k p termbetweenour8basisstatesinthe set A andtheremotebandsfromset B .Thusweneedtogotothenextorderinthe L ¨ owdin'sperturbationprocedure.Thenextordercorrection totheHamiltonianisgiven byequation( 2–34c ),whichIwillrewriteithere: H (2) mm 0 = 1 2 X l H 0 ml H 0 lm 0 1 E m E l + 1 E m 0 E l # .(3–31) TheperturbatingHamiltonianisstill H 0 = ~ m 0 k p .Notethat H (2) mm 0 dependsontheenergy E m and E m 0 ,whicharetheeigenenergiesofthebandedgebasisstatesin set A .This energydependencewillmakethenumberofindependentparam etersduetocoupling tothedistantbandsmuchlargethaniftheirisnoenergydepe ndence.Tosimplifyour problem,wemaketheapproximationthatalltheenergies E m and E m 0 inequation( 3–31 ) canbereplacedbyanaverageenergy E ,since E l aretheenergyofthedistantbandso thattheenergydifferencebetween E m and E m 0 issmallcomparedwiththedifference between E m (or E m 0 )and E l .Nowourperturbationis H (2) mm 0 = X l H 0 ml H 0 lm 0 E E l .(3–32) Grouptheorycanhelpustoreducethenumberoftheindepende ntparameters arisefromthecouplingduetotheperturbation H (2) mm 0 .Itturnsoutthat,withourchoiceof thebasisstatesinequation( 3–9 ),weneedfouradditionalcouplingparameters, A 0 = ~ 2 2 m 0 + ~ 2 m 2 0 X p x X p x X E E (3–33a) B 0 = ~ 2 2 m 0 + ~ 2 m 2 0 X p y X p y X E E (3–33b) C 0 = ~ 2 m 2 0 X p x X p y Y + p y X p x Y E E (3–33c) 33

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F 0 = 1 m 0 X p x S p x S E E .(3–33d) Parameters A 0 B 0 and C 0 describethecouplingbetweenthevalencebandcomponents j X i j Y i ,and j Z i ,withtheremotebandslabeledby .Theyaresimilartotheparameters A B and C denedbyLuttingerandKohnintheiroriginalpaper[ 17 ].Theonly differenceisthatourparametersdonothavethecontributi onthatcomesfromthe couplingbetweenthelowestconductionbandwiththehighes tvalencebands,because ourmodelis8-dimensionalwhichalreadyincludethiscontr ibution.Wehavetodene onemoreparameter F 0 becauseweneedtoconsiderthecouplingbetweenlowest conductionbandwithremotebands.Nextwedenetherenorma lizedLuttinger parameter ~ 2 2 m 0 r 1 = 1 3 ( A 0 +2 B 0 )(3–34a) ~ 2 2 m 0 r 2 = 1 6 ( A 0 B 0 )(3–34b) ~ 2 2 m 0 r 3 = 1 6 C 0 (3–34c) r 4 =1+2 F 0 .(3–34d) OurrenormalizedLuttingerparametersarerelatedtotheor iginalLuttingerparameters r L 1 r L 2 and r L 3 throughthefollowingequations r 1 = r L 1 E p = 3 E g (3–35a) r 2 = r L 2 E p = 6 E g (3–35b) r 3 = r L 3 E p = 6 E g ,(3–35c) where E P = 2 m 0 V 2 ~ 2 (3–36) isameasureofthecouplingstrengthbetweenconductionand valencebands. 34

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Nowwecancalculatethematrixelementofthesecondorderpe rturbating Hamiltonian H (2) mm 0 H (2) mm 0 + ~ 2 k 2 2 m 0 mm 0 = 2666666666666666666664 A 0000000 0 P Q Mi p 2 M 00 L i p 2 L 0 M P + Qi p 2 Q 0 L 0 i q 3 2 L 0 i p 2 M i p 2 Q P 0 i p 2 Li q 3 2 L 0 0000 A 000 00 L i p 2 L 0 P Q M i p 2 M 0 L 0 i q 3 2 L 0 M P + Qi p 2 Q 0 i p 2 L i q 3 2 L 00 i p 2 M i p 2 Q P 3777777777777777777775 (3–37) where A = r 4 ~ 2 k 2 2 m 0 (3–38a) P = r 1 ~ 2 k 2 2 m 0 (3–38b) Q = r 2 ~ 2 2 m 0 ( k 2 x + k 2 y 2 k 2 z )(3–38c) L = i p 3 r 3 ~ 2 m 0 k k z (3–38d) M = p 3 ~ 2 2 m 0 [ r 2 ( k 2 x k 2 y ) 2 i r 3 k x k y ].(3–38e) 35

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Byadding H (0) H (1) H (2) andthediagonalterm ~ 2 k 2 2 m 0 togetherwecangetthetotal Hamiltonian H H = 2666666666666666666664 E g + A i p 2 Vk + i p 6 Vk 1 p 3 Vk 00 q 2 3 Vk z i p 3 Vk z i p 2 Vk P Q Mi p 2 M 00 L i p 2 L i p 6 Vk + M P + Qi p 2 Q i q 2 3 Vk z L 0 i q 3 2 L 1 p 3 Vk + i p 2 M i p 2 Q P 1 p 3 Vk z i p 2 Li q 3 2 L 0 00 i q 2 3 Vk z 1 p 3 Vk z E g + A 1 p 2 Vk 1 p 6 Vk + i p 3 Vk + 00 L i p 2 L 1 p 2 Vk + P Q M i p 2 M q 2 3 Vk z L 0 i q 3 2 L 1 p 6 Vk M P + Qi p 2 Q i p 3 Vk z i p 2 L i q 3 2 L 0 i p 3 Vk i p 2 M i p 2 Q P 3777777777777777777775 (3–39) HereIwanttoemphasizethatthisHamiltonianinequation( 3–39 )havealreadytaken intoaccountthecrystalperiodicpotential,thespin-orbi tinteractions,the k p coupling effectswiththeremotebands,andnallythediagonalterm ~ 2 k 2 2 m 0 .Sowecangetthe energyoftheelectronbydiagonalizingthis H .ThisisamuchcomplicatedHamiltonian thantheoneinequation( 3–18 )anditisimpossibletosolveanalyticallywithoutfurther simplications.Itispossibletosolvethiseigenvaluepro blemsanalyticallyforsome specialcasesforexampleif k z vanishes.InthiscasetheHamiltonianinequation( 3–39 ) hasablockdiagonalform,decouplingintotwo4 4problems.Howeverwewillnotgo furtherinthisdirectionsincewecansolvethefull8 8problemsnumericallyanyway. Therealproblemweareinterestediniswhenthereisanexter nalmagneticeld,whatis thebandstructurelooklike. 36

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CHAPTER4 LANDAULEVELSOFGALLIUMARSENIDE 4.1TheEnvelopeFunctionApproximation WhenthereisanexternalmagneticeldappliedtothebulkII I-Vsemiconductor samplesuchasGaAs,theSchr ¨ odingerequationdoesnottaketheformofequation ( 2–11 )anymore.InsteadweneedtogobacktotheDiracequationfor theelectron intheperiodicpotentialinthepresenceoftheexternalel d.Thenon-relativistic approximationoftheDiracequationforthelargecomponent spinorwavefunction becomes ( i ~ r + e c A ) 2 2 m 0 + V 0 ( r )+ ~ 4 m 2 0 c 2 ( i ~ r + e c A ) ( r V 0 )+ B B ( r )= E ( r )(4–1) where A isthevectorpotentialand B istheBohrmagneton B = e ~ = 2 m 0 c .Wehave alreadyassumedthescalarpotentialiszerosinceweareonl yinterestedinthecaseof astaticmagneticeld.Themagneticeldisgivenby B = r A ( r ). WeknowthatwavefunctionfortheBlochelectroncanbeexpre ssedasaplain wavetimesafunctionwiththesameperiodicityasthecrysta llattice.Similarly,the solutiontoequation( 4–1 )canbewrittenas ( r )= X n 0 n 0 ( r ) u n 0 0 ( r ),(4–2) wherethe u n 0 0 ( r )arestillthebandedgespinorfunctionsincludingboththe set A andset B .Thespinorfunctionsinset A arestillgivenbyequation( 3–9 ).Thefunctions n 0 ( r )are calledenvelopefunctions.Substituteequation( 4–2 )intoequation( 4–1 ),multiplyfrom theleftby u y n 0 ( r ),andthenintegrateoveraunitcell,weobtain X n 0 n E n 0 ( 0 )+ ( i ~ r + e A = c ) 2 2 m 0 nn 0 + 1 m 0 ( i ~ r + e A = c ) nn 0 + nn 0 + B h n j B j n 0 i o n 0 ( r )= E n ( r ),(4–3) 37

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whichisasystemofcoupleddifferentialequations.Wecans tillusingtheL ¨ owdin's perturbationmethodtoreducetheinnitenumbersofequati onstoatractableset,for examplebyusingthebasisgivenbyequation( 3–9 )wecanobtaina8dimensional eigenvalueproblem.TheHamiltonianinequation( 4–3 )iscalledEFA(Envelope FunctionApproximation)Hamiltonian.Theeigenvectoroft heEFAHamiltonianis thesetofenvelopefunctions.Theelectronicwavefunction isobtainedbyusingequation ( 4–2 )onceweknowthoseenvelopefunctions. 4.2ExplicitFormoftheEFAHamiltonian WehavealreadyderivedtheexplicitformoftheHamiltonian whenthereisno magneticeld.Inthepresenceofanexternalmagneticeld, wewanttoderivea similarmatrixfromwhichwecanobtaintheeigenenergyandt heenvelopefunctions. Comparingequation( 2–18 )and( 4–3 )wecanseethatthemagneticeldeffectisbuilt intothesetofcoupleddifferentialequationsthroughrepl acingthewavevector ~ k with theoperator i ~ r + e A = c ,andaddingthe B B termexplicitly.Thissimilaritycan helpustoderivetheEFAmatrixinasimilarwaytothecasewhe nthereisnomagnetic eld.InsteadofderivingtheEFAmatrixfromequation( 4–3 )directlyfromthebeginning, wecanusethematrixforminequation( 3–39 )asastartingpoint.Replacingthewave vector ~ k inequation( 3–39 )withtheoperator i ~ r + e A = c ,andaddingthematrix element B h n j B j n 0 i ,wearriveatthematrixformoftheEFAHamiltonian.However thereisasubtlepointthatwemustpayspecialattentiontod uringthisprocess.Inwhat followsIwillexplainthissubtlepointinalittlebitdetai l. Intheabsenceofthemagneticeld,thedifferentcomponent ofthewavevector commutewitheachother([ k x k y ]=0etc.).Whenthereisamagneticeldaround,the differentcomponentsofthemechanicmomentum ~ k (= i ~ r + e A = c )vectordonot commute.Wecandirectlycalculatethecommutatorusingthe denitionofthemechanic momentumoperatortoget [ k i k j ]= e i ~ c ijk B k ,(4–4) 38

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orintermsofthevectorproduct k k = e i ~ c B .(4–5) Inouroriginalderivationofthematrix H (2) mm 0 inequation( 3–37 ),wewilltypically encounterthetermthatcontainingtheproductofthetwocom ponentsofthewave vector H (2) mm 0 = X l H 0 ml H 0 lm 0 E E l = ~ 2 m 2 0 X X l k k p ml p lm 0 E E l = X D mm 0 k k (4–6) wherethedenitionof D mm 0 isgivenby D mm 0 = ~ 2 m 2 0 X l p ml p lm 0 E E l (4–7) Ifthetwocomponentsofthewavevectorcommute,wecanconst ructthecoefcientsto besymmetricaboutinterchangingthe and indices, H (2) mm 0 = X D ( S ) mm 0 k k (4–8a) D ( S ) mm 0 = 1 2 D mm 0 + 1 2 D mm 0 .(4–8b) Inthepresenceofamagneticeld,thetwocomponentsofthe k operatordonot commute, k k 6 = k k ,andwecanwrite k k as k k = 1 2 f k k g + 1 2 [ k k ](4–9) wherethecommutatoris[ k k ]= k k k k andtheanticommutatoris f k k g = k k + k k .Nowthematrixelements H (2) mm 0 canbewrittenas H (2) mm 0 = 1 2 X D mm 0 f k k g + 1 2 X D mm 0 [ k k ].(4–10) 39

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Againwecandenethesymmetricandantisymmetricpartofth e D mm 0 coefcientstobe D ( S ) mm 0 = 1 2 D mm 0 + 1 2 D mm 0 (4–11a) D ( A ) mm 0 = 1 2 D mm 0 1 2 D mm 0 ,(4–11b) andwrite H (2) mm 0 as H (2) mm 0 = 1 2 X D ( S ) mm 0 f k k g + 1 2 X D ( A ) mm 0 [ k k ].(4–12) Wecanseethatthematrixelements H (2) mm 0 hastwopartofcontributions:asymmetric partgivenbytherstterminequation( 4–12 )andanantisymmetricpartgivenby thesecondterminequation( 4–12 ).Thesymmetrictermrequireustodenefour independentcouplingparameters A 0 B 0 C 0 ,and F 0 aswedidinequation( 3–33 ). Wewillseelaterthattheantisymmetrictermrequireustode neonemorecoupling parameter. Ifwesimplyreplacethe ~ k vectorwiththeoperator i ~ r + e A = c inthematrix giveninequation( 3–37 ),andinterpretanyproductofthewavevector k k asthe symmetrizedproduct 1 2 f k k g ,wewillobtainthesymmetricpart H ( S ) ofthe H (2) matrix. This H ( S ) matrix,togetherwiththe H (0) H (1) andthediagonalterm ~ 2 k 2 2 m 0 mm 0 giveusthe socalledLandauHamiltonian H L .Thesecondterminequation( 4–12 )willgiveusthe antisymmetricpart H ( A ) ofthe H (2) matrix,whichdidnotshowupwhentherewasno magneticeldapplied.This H ( A ) matrix,togetherwiththe B B termwillgiveusthe so-calledZeemanHamiltonian H Z .Sowehave H (2) mm 0 = H ( S ) mm 0 + H ( A ) mm 0 (4–13a) H mm 0 = H (0) mm 0 + H (1) mm 0 + ~ 2 k 2 2 m 0 mm 0 + H ( S ) mm 0 | {z } H L + H ( A ) mm 0 + B h m j B j m 0 i | {z } H Z .(4–13b) WewilldiscusstheLandauandZeemanHamiltonianonebyone. 40

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4.2.1LandauHamiltonian TheLandaupartoftheHamiltonianisgivenbyfourterms: H L = H (0) + H (1) + H ( S ) + ~ 2 k 2 2 m 0 I 8 8 .(4–14) WecanobtaintheLandauHamiltonian H L fromequation( 3–39 ),byreplacingthe k wavevectorwiththeoperator ˆ k = 1 ~ ( i ~ r + e A = c ),(4–15) andtreatingthewavevectorproduct k k asthesymmetrizedproduct 1 2 f k k g .Note thatintheoriginalmatrixinequation( 3–39 ),the H (0) and H (1) donotcontainatermlike k k ,andthediagonaltermisalreadysymmetrizedintheformof k k ,soweonlyneed tobecarefulaboutthe H (2) contribution. Nowwecanrewritethematrixinequation( 3–39 ),usingtheoperators ˆ k x ˆ k y and ˆ k z insteadofthewavevectorcomponents k x k y and k z ,toobtaintheLandaupartofthe Hamiltonian.TheexplicitformofLandauHamiltonianmatri xisthus H L = 2666666666666666666664 E g + A i p 2 V ˆ k + i p 6 V ˆ k 1 p 3 V ˆ k 00 q 2 3 V ˆ k z i p 3 V ˆ k z i p 2 V ˆ k P Q Mi p 2 M 00 L i p 2 L i p 6 V ˆ k + M y P + Qi p 2 Q i q 2 3 V ˆ k z L 0 i q 3 2 L y 1 p 3 V ˆ k + i p 2 M y i p 2 Q P 1 p 3 V ˆ k z i p 2 Li q 3 2 L y 0 00 i q 2 3 V ˆ k z 1 p 3 V ˆ k z E g + A 1 p 2 V ˆ k 1 p 6 V ˆ k + i p 3 V ˆ k + 00 L y i p 2 L y 1 p 2 V ˆ k + P Q M y i p 2 M y q 2 3 V ˆ k z L y 0 i q 3 2 L 1 p 6 V ˆ k M P + Qi p 2 Q i p 3 V ˆ k z i p 2 L y i q 3 2 L 0 i p 3 V ˆ k i p 2 M i p 2 Q P 3777777777777777777775 (4–16) wheretheoperators A P Q L and M aregivenby A = r 4 ~ 2 ˆ k 2 2 m 0 (4–17a) 41

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P = r 1 ~ 2 ˆ k 2 2 m 0 (4–17b) Q = r 2 ~ 2 2 m 0 ( ˆ k 2 x + ˆ k 2 y 2 ˆ k 2 z )(4–17c) L = i p 3 r 3 ~ 2 2 m 0 ( ˆ k ˆ k z + ˆ k z ˆ k )(4–17d) M = p 3 ~ 2 2 m 0 [ r 2 ( ˆ k 2 x ˆ k 2 y ) i r 3 ( ˆ k x ˆ k y + ˆ k y ˆ k x )].(4–17e) 4.2.2ZeemanHamiltonian Tobespecicwewillassumethemagneticeldisintheˆ z directionfromnowon, andwechoosetheLandaugauge A = By ˆ x (4–18) sothatthemagneticeldisgivenby B = r A = B ˆ z .(4–19) Thethreeoperators ˆ k x ˆ k y and ˆ k z aregivenby ˆ k x = i r x y 2 (4–20a) ˆ k y = i r y (4–20b) ˆ k z = i r z (4–20c) where isthemagneticlengthdenedby 2 = ~ c eB .(4–21) Thenon-zerocommutatorbetweendifferentcomponentsofth e k operatorisofthe form [ ˆ k x ˆ k y ]= eB i ~ c = 1 i 2 ,(4–22) andboth ˆ k x and ˆ k y operatorsstillcommutewith ˆ k z 42

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NextwewanttoderivetheexplictformoftheZeemanHamilton ian.TheZeeman Hamiltonianhastwocontributions: H Z mm 0 = H ( A ) mm 0 + B h m j B j m 0 i ,(4–23) webeginwiththerstone: ThedenitionoftheantisymmetricHamiltonianisgivenbyt hesecondtermof equation( 4–12 ), H ( A ) mm 0 = 1 2 X D ( A ) mm 0 [ ˆ k ˆ k ].(4–24) Whenthemagneticeldisintheˆ z direction,theonlynon-zerocommutatoris[ ˆ k x ˆ k y ]= [ ˆ k y ˆ k x ],andboth ˆ k x and ˆ k y stillcommutewith ˆ k z ,sowehavetwotermssurvivefrom thesummation: H ( A ) mm 0 = 1 2 D ( A ) xymm 0 [ ˆ k x ˆ k y ]+ 1 2 D ( A ) yxmm 0 [ ˆ k y ˆ k x ] = D ( A ) xymm 0 [ ˆ k x ˆ k y ] = 1 2 ( D xy mm 0 D yx mm 0 )[ ˆ k x ˆ k y ]. (4–25) Usingthedenitionofthe D mm 0 coefcientsinequation( 4–7 )andthecommutation relationgiveninequation( 4–22 ),wecanwrite H ( A ) mm 0 as H ( A ) mm 0 = 1 2 eB i ~ c ~ 2 m 2 0 X p x m p y m 0 p y m p x m 0 E E .(4–26) Wecanimmediatelyseethatweneedanothercouplingconstan t K 0 besidesthe alreadydenedconstants A 0 B 0 C 0 and F 0 inequation( 3–33 ).Thecouplingconstant K 0 isdenedas K 0 = ~ 2 m 2 0 X p x X p y Y p y X p x Y E E .(4–27) WealsodenetherenormalizedLuttingerparameter throughtherelation ~ 2 m 0 (3 +1)= K 0 (4–28) 43

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where isrelatedtotheoriginalLuttingerparameter L through = L E p = 6 E g .For theLuttingerparameter L ,weusetheapproximation[ 14 15 25 ] L = r L 3 + 2 3 r L 2 1 3 r L 1 2 3 .(4–29) Withthesedenitionswecancalculatetheexplictformofth e H ( A ) matrixfromequation ( 4–26 ),theresultis H ( A ) = B B 2666666666666666666664 000000000 3 1000000 00 + 1 3 i p 2( + 1 3 )0000 00 i p 2( + 1 3 )2 + 2 3 0000 00000000000003 +100 000000 1 3 i p 2( + 1 3 ) 000000 i p 2( + 1 3 ) 2 2 3 3777777777777777777775 (4–30) Nextwecalculatethematrixelementof B h m j B j m 0 i .Notingthatthemagnetic eldisintheˆ z directionwehave B = z B .Theresultofthe B h m j B j m 0 i matrix takestheform B h m j B j m 0 i = B B 2666666666666666666664 100000000100000000 1 3 2 p 2 3 i 0000 00 2 p 2 3 i 1 3 0000 0000 1000 00000 100 000000 1 3 2 p 2 3 i 000000 2 p 2 3 i 1 3 3777777777777777777775 .(4–31) 44

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Nowwecanaddequation( 4–30 )and( 4–31 )togethertogettheexplictformofthe ZeemanHamiltonian H Z =2 B B 2666666666666666666664 1 2 0000000 0 3 2 000000 00 1 2 i q 1 2 ( +1)0000 00 i q 1 2 ( +1) + 1 2 0000 0000 1 2 000 00000 3 2 00 000000 1 2 i q 1 2 ( +1) 000000 i q 1 2 ( +1) 1 2 3777777777777777777775 (4–32) NowwehavetheexplicitmatrixforboththeLandauandZeeman partofthe Hamiltonian,wecanaddthemtogethertogetthetotalEFAHam iltonianforsemiconductors inanexternalmagneticeld. 4.3EnergyandEnvelopeFunctions Accordingtotheenvelopefunctionapproximation,oureige nvalueproblemforthe Blochelectroninanexternalmagneticeldcanbephrasedas 8 X m 0 =1 H EFA mm 0 m 0 ( r )= E m ( r ),(4–33) where m and m 0 areintegersfrom1to8,andtheEFAHamiltonianisthesumoft he LandauHamiltonianandtheZeemanHamiltonian: H EFA = H L + H Z .(4–34) Theelectronwavefunction ( r )= 8 X m =1 m ( r ) u m 0 ( r ),(4–35) where m ( r )istheenvelopefunctionand u m 0 ( r )isthebasisfunctiongivenbyequation ( 3–9 ).Notethatequation( 4–33 )isasystemofcoupleddifferentialequationssince 45

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theelementsoftheEFAHamiltoniancontainsthedifferenti aloperators ˆ k x ˆ k y and ˆ k z ThereforetheHamiltonian H EFA isnotreadilytobediagonalized.Whatwewantisan algebraicmatrixeigenvalueproblem,thematrixelementso fwhicharejustnumbers insteadofoperators.Toachievethispurposewecanproceed asfollows: Firstweseparatevariablesinourenvelopefunctions m ( r ) m ( r )= e i ( k x x + k z z ) f m ( y ).(4–36) Thisseparationofvariableispossiblebecauseofourparti cularchoiceofthevector potentialinequation( 4–18 ):sincethevectorpotentialisonlydependon y butnot x and z ,the k x and k z aregoodquantumnumbers.Wethensubstituteequation( 4–36 )into equation( 4–33 )toget 8 X m 0 =1 H EFA mm 0 e i ( k x x + k z z ) f m 0 ( y )= Ee i ( k x x + k z z ) f m ( y ).(4–37) NotethattheEFAHamiltonianisthesumofLandauHamiltonia nandZeeman Hamiltonian,sothelefthandsideofequation( 4–37 )canbewrittenas 8 X m 0 =1 H EFA mm 0 e i ( k x x + k z z ) f m 0 ( y )= 8 X m 0 =1 H L mm 0 e i ( k x x + k z z ) f m 0 ( y )+ 8 X m 0 =1 H Z mm 0 e i ( k x x + k z z ) f m 0 ( y )(4–38) ThenextstepistoevaluatetheeffectofLandauHamiltonian actingontheenvelope functions.Sinceweknoweveryelements, H L mm 0 ,ofLandauHamiltonian H EFA mm 0 ,wecan evaluatetheoperatoractingontheplanewavepartoftheenv elopefunctionsoneby oneforevery m and m 0 ,theresultis H L mm 0 e i ( k x x + k z z ) f m 0 ( y )= e i ( k x x + k z z ) G L mm 0 f m 0 ( y ),(4–39) 46

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wherethe G L mm 0 matrixisgivenby G L = 2666666666666666666664 E g + A i p 2 V ˆ k + i p 6 V ˆ k 1 p 3 V ˆ k 00 q 2 3 Vk z i p 3 Vk z i p 2 V ˆ k P Q Mi p 2 M 00 L i p 2 L i p 6 V ˆ k + M y P + Qi p 2 Q i q 2 3 Vk z L 0 i q 3 2 L y 1 p 3 V ˆ k + i p 2 M y i p 2 Q P 1 p 3 Vk z i p 2 Li q 3 2 L y 0 00 i q 2 3 Vk z 1 p 3 Vk z E g + A 1 p 2 V ˆ k 1 p 6 V ˆ k + i p 3 V ˆ k + 00 L y i p 2 L y 1 p 2 V ˆ k + P Q M y i p 2 M y q 2 3 Vk z L y 0 i q 3 2 L 1 p 6 V ˆ k M P + Qi p 2 Q i p 3 Vk z i p 2 L y i q 3 2 L 0 i p 3 V ˆ k i p 2 M i p 2 Q P 3777777777777777777775 (4–40) wheretheredenedoperators ˆ k = k x y 2 i ˆ k y andtheoperators A P Q L and M are givenby A = r 4 ~ 2 2 m 0 [( k x y 2 ) 2 + ˆ k 2 y + k 2 z ] (4–41a) P = r 1 ~ 2 2 m 0 [( k x y 2 ) 2 + ˆ k 2 y + k 2 z ] (4–41b) Q = r 2 ~ 2 2 m 0 [( k x y 2 ) 2 + ˆ k 2 y 2 k 2 z ] (4–41c) L = i p 3 r 3 ~ 2 m 0 k z ( k x y 2 i ˆ k y ) (4–41d) M = p 3 ~ 2 2 m 0 n r 2 ( k x y 2 ) 2 ˆ k 2 y i r 3 ( k x y 2 ) ˆ k y + ˆ k y ( k x y 2 ) o .(4–41e) Thisnewmatrix G L onlycontainsoperator ˆ k y andthewavevectorquantumnumber k x and k z butnotoperator ˆ k x and ˆ k z ,sincewehavealreadyevaluatedtheeffectsof operators ˆ k x and ˆ k z actingontheplanewavepartoftheenvelopefunctions. Tofurthersimplytheexpressionofthe G L matrix,itisconvenienttodenethe operator a = 1 p 2 ( 2 k x y ) i p 2 ˆ k y (4–42) 47

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anditsHermitianconjugate a y = 1 p 2 ( 2 k x y )+ i p 2 ˆ k y .(4–43) Itcanbedirectlyveriedthatthecommutationrelation [ a a y ]=1,(4–44) sothatwecanthinkofoperators a and a y asannihilationandcreationoperatorsand denethenumberoperator N = a y a .(4–45) Usingtheannihilationandcreationoperators a and a y ,thematrix G L canbewrittenas G L = 2666666666666666666664 E g + Ai V a y i q 1 3 V a q 2 3 V a 00 q 2 3 Vk z i q 1 3 Vk z i V a P Q Mi p 2 M 00 L i q 1 2 L i q 1 3 V a y M y P + Qi p 2 Q i q 2 3 Vk z L 0 i q 3 2 L y q 2 3 V a y i p 2 M y i p 2 Q P q 1 3 Vk z i q 1 2 Li q 3 2 L y 0 00 i q 2 3 Vk z q 1 3 Vk z E g + A V a q 1 3 V a y i q 2 3 V a y 00 L y i q 1 2 L y V a y P Q M y i p 2 M y q 2 3 Vk z L y 0 i q 3 2 L q 1 3 V a M P + Qi p 2 Q i q 1 3 Vk z i q 1 2 L y i q 3 2 L 0 i q 2 3 V a i p 2 M i p 2 Q P 3777777777777777777775 (4–46) andtheoperators A P Q L and M aregivenby A = ~ 2 m 0 r 4 2 2 N +1 2 + k 2 z (4–47a) P = ~ 2 m 0 r 1 2 2 N +1 2 + k 2 z (4–47b) Q = ~ 2 m 0 r 2 2 2 N +1 2 2 k 2 z (4–47c) L = ~ 2 m 0 r 3 i p 6 k z a (4–47d) 48

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M = p 3 ~ 2 2 m 0 2 ( r 2 + r 3 ) a 2 +( r 2 r 3 ) a y 2 .(4–47e) Nowthatweknowtheelementsofthe G L matrixwecangobacktoequation( 4–37 ) toobtain 8 X m 0 =1 e i ( k x x + k z z ) G L mm 0 f m 0 ( y )+ 8 X m 0 =1 H Z mm 0 e i ( k x x + k z z ) f m 0 ( y )= Ee i ( k x x + k z z ) f m ( y ).(4–48) UsingthefactthattheZeemanmatrixelements H Z mm 0 arejustnumberswecancancel outthecommonexponentialfactortoget 8 X m 0 =1 ( G L mm 0 + H Z mm 0 ) f m 0 ( y )= Ef m ( y ),(4–49) ormorecompactly 8 X m 0 =1 G mm 0 f m 0 ( y )= Ef m ( y ),(4–50) where G = G L + H Z .(4–51) Equation( 4–50 )isamuchbiggerprogressovertheoriginalEFAeigenvalue probleminequation( 4–33 ),sincewehavealreadyseparatedvariablesandweonly needtofocusontheundeterminedfunction f m ( y ).Thematrix G containsonlythe y and ˆ k y operators,oraswealreadysimplied,the a and a y operators.Howeverthisisstillnot thenalformwewantbecauseitisstillnotanalgebraicmatr ixeigenvalueequations duetotheexistenceofthe a and a y operators.Thelaststepwetaketosimplifythe problemisusingthepropertiesoftheharmonicoscillatore igenfunctionstoconvert equation( 4–50 )intoanalgebraicmatrixeigenvalueequation.Fromourde nitionof theannihilationandcreationoperatorsinequation( 4–42 )and( 4–43 ),wecanwritethe eigenfunctionsofthenumberoperatoras n ( y 2 k x ),witheigenvalue n : N n ( y 2 k x )= n n ( y 2 k x ),(4–52) 49

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where n isnonnegativeintegers.Wedon'tneedtheexplictformofth isharmonic oscillatoreigenfunctionsintermoftheHermitepolynomia ls,instead,wewillusethe followingpropertiesof n : a n = p n n 1 (4–53a) a y n = p n +1 n +1 .(4–53b) Nextweexpandthe f m ( y )functionsintermsoftheseharmonicoscillatoreigenfunc tions f 1 ( y )= 1 X n =1 f n 1 n 1 f 2 ( y )= 1 X n =2 f n 2 n 2 (4–54a) f 3 ( y )= 1 X n =0 f n 3 n f 4 ( y )= 1 X n =0 f n 4 n (4–54b) f 5 ( y )= 1 X n =0 f n 5 n f 6 ( y )= 1 X n = 1 f n 6 n +1 (4–54c) f 7 ( y )= 1 X n =1 f n 7 n 1 f 8 ( y )= 1 X n =1 f n 8 n 1 (4–54d) where f n 1 f n 2 etc.areexpansioncoefcientsindependentof y .Notethathereweexpand theeightcomponentsoftheenvelopefunctioninaslightlyd ifferentway.Thisturnstobe abetterchoicethanifweexpandalleightcomponentsinthes ameformas P 1n =0 f n m n Thereasonforthiswillbeclearshortly. Whenweusethealreadyderivedoperatormatrix G actingonthe f m ( y )givenby equation( 4–54 ),foreachlineoftheoperation,wewillgetanequationofth eform X n G m 1 f n 1 n 1 + G m 2 f n 2 n 2 + G m 3 f n 3 n + G m 4 f n 4 n + G m 5 f n 5 n + G m 6 f n 6 n +1 + G m 7 f n 7 n 1 + G m 8 f n 8 n 1 = E X n f n m n 0 ( n m ) (4–55) 50

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where n 0 isafunctionofthenumber n and m : n 0 ( n m )= 8>>>>>>><>>>>>>>: n 1,if m =1,7,8 n 2,if m =2 n ,if m =3,4,5 n +1,if m =6 (4–56) Theequation( 4–55 )canalsobewrittenmorecompactlyas X n X m 0 G mm 0 f n m 0 n 0 ( n m 0 ) = E X n f n m n 0 ( n m ) .(4–57) Ifweneglectthetermproportionalto a y 2 inthedenitionofthe M operatorinequation ( 4–47 ),wecandirectlyveritythatthematrixelement G L mm 0 and H Z mm 0 actingonthe n 0 ( n m 0 ) canbeconvertedinto G L mm 0 n 0 ( n m 0 ) = J n mm 0 n 0 ( n m ) (4–58a) H Z mm 0 n 0 ( n m 0 ) = H Z mm 0 n 0 ( n m ) (4–58b) where J n mm 0 arejustnumberswithoutanyoperatorsandthequantumnumbe r n 0 areall thesamefordifferent m 0 .Theexplicitformofthe J n matrixis J n = 264 J n a J n c J n c y J n b 375 ,(4–59) 51

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wherethesubmatricesaregivenbyJ n a = 2666666666666666666664 E g + r 4 ~ 2 k 2 z 2 m 0 + ~ 2 r 4 m 0 2 ( n 1 2 ) i V p n 1 i V q n 3 V q 2 3 n i V p n 1 ~ 2 k 2 z m 0 ( r 2 r 1 2 ) ~ 2 m 0 r 1 + r 2 2 2 (2 n 3) ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 3 n ( n 1) i ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 6 n ( n 1) i V q n 3 ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 3 n ( n 1) ~ 2 k 2 z m 0 ( r 1 2 + r 2 ) ~ 2 m 0 r 1 r 2 2 2 (2 n +1) i p 2 ~ 2 m 0 [ r 2 k 2 z + r 2 2 ( n + 1 2 )] V q 2 3 n i ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 6 n ( n 1) i p 2 ~ 2 m 0 [ r 2 k 2 z + r 2 2 ( n + 1 2 )] r 1 ~ 2 k 2 z 2 m 0 ~ 2 m 0 r 1 2 ( n + 1 2 ) 3777777777777777777775 (4–60a) J n b = 2666666666666666666664 E g + r 4 ~ 2 k 2 z 2 m 0 + ~ 2 r 4 m 0 2 ( n + 1 2 ) V p n +1 V q n 3 i V q 2 3 n V p n +1 ~ 2 k 2 z m 0 ( r 2 r 1 2 ) ~ 2 m 0 r 1 + r 2 2 2 (2 n +3) ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 3 n ( n +1) i ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 6 n ( n +1) V q n 3 ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 3 n ( n +1) ~ 2 k 2 z m 0 ( r 1 2 + r 2 ) ~ 2 m 0 r 1 r 2 2 2 (2 n 1) i p 2 ~ 2 m 0 [ r 2 k 2 z + r 2 2 ( n 1 2 )] i V q 2 3 n i ~ 2 m 0 ( r 2 + r 3 ) 2 2 p 6 n ( n +1) i p 2 ~ 2 m 0 [ r 2 k 2 z + r 2 2 ( n 1 2 )] r 1 ~ 2 k 2 z 2 m 0 ~ 2 m 0 r 1 2 ( n 1 2 ) 3777777777777777777775 (4–60b) J n c = k z 266666664 00 V q 3 2 iV q 1 3 00 i ~ 2 m 0 r 3 p 6( n 1) ~ 2 m 0 r 3 p 3( n 1) iV q 2 3 i ~ 2 m 0 r 3 p 6( n +1)0 3 ~ 2 m 0 r 3 p n V q 1 3 ~ 2 m 0 r 3 p 3( n +1) 3 ~ 2 m 0 r 3 p n 0 377777775 (4–60c) 52

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Thisallowustorewriteequation( 4–55 )as X n X m 0 ( J n mm 0 + H Z mm 0 ) f n m 0 # n 0 ( n m ) = E X n f n m n 0 ( n m ) ,(4–61) Atthispointitisclearthatwhywechoosetheparticularexp ansionformfor f m ( y )given inequation( 4–54 ),sincethisformwillgiveusaquantumnumber n 0 independentof m 0 sothatwecanfactor n 0 ( n m ) outofthesummationover m 0 intheleftsideoftheabove equation.Denematrix R n as R n = J n + H Z (4–62) andbyusingtheorthogonalityoftheharmonicoscillatorei genstatesfordifferent quantumnumber,wearrived X m 0 R n mm 0 f n m 0 = Ef n m .(4–63) Diagonalizingtheabovealgebraicmatrixeigenvalueequat ionwillgiveustheeigenenergy whichdependonthequantumnumber n andwecallthisquantumnumbermanifold quantumnumber.Thesolutionsfordifferentmanifoldquant umnumberdonotmixed witheachother,sothatourinitiallyexpansionofthe f m ( y )intermsofthesummation overdifferentmanifoldquantumnumber n breaksapartformanifolds.Thismeansthat our f m ( y )functionsnallytakethefollowingform: f 1 ( y )= f n 1 n 1 f 2 ( y )= f n 2 n 2 (4–64a) f 3 ( y )= f n 3 n f 4 ( y )= f n 4 n (4–64b) f 5 ( y )= f n 5 n f 6 ( y )= f n 6 n +1 (4–64c) f 7 ( y )= f n 7 n 1 f 8 ( y )= f n 8 n 1 ,(4–64d) ormorecompactly f m ( y )= f n m n 0 ( n m ) .(4–65) Nowwecansummarizethetechniqueweusetocalculatetheele ctronicstructures whenthereisanexternalmagneticeldappliedtotheIII-Vs emiconductors.Whenthe 53

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magneticeldisintheˆ z direction,theelectronwavefunctionisgivenby ( r )= 8 X m =1 m ( r ) u m 0 ( r ),(4–66) where u m 0 ( r )isthebasisfunctiongiveninequation( 3–9 ),and m ( r )istheenvelope functiongivenby m ( r )= e i ( k x x + k z z ) f n m n 0 ( n m ) ( y 2 k x ).(4–67) Inequation( 4–67 ), k x and k z arewavevectorquantumnumbers, n ismanifoldquantum number,and n 0 = n 0 ( n m )isLandauquantumnumber.Thecoefcients f n m andthe electronenergycanbeobtainedfromthematrixeigenvaluee quation X m 0 R n mm 0 f n m 0 = Ef n m .(4–68) Thematrix R n isthesumofthe J n matrixandthe H Z matrix,andtheexplicitformofthe matrix J n and H Z aregivenbyequation( 4–60 )and( 4–32 ),respectively.Theenergy E willdependon n on k z butnot k x 4.4NumericalCalculationsoftheGaAsLandauLevels WeperformcalculationsofLandaulevelsforbulkGaAsattwo differentmagnetic eldstrength,4.7Tand7.0T,atatemperatureof6K.Fig. 4-1 andFig. 4-2 show conductionandvalencebanddiagramsforthetwoeldstreng th,justtogiveusanover allideaofwhattheenergybandslooklike.Wecanseeheretha tasthemagneticeld increases,theenergyspacingbetweendifferentlevelsals oincreases,asexpected.The conductionbandshaveasimplestructurebutthevalenceban dsarealotcomplicated andhardtoseeclearlyinFig. 4-1 andFig. 4-2 .Fig. 4-3 andFig. 4-4 giveusacloser lookatthevalencebandsnearthezonecenter,attheeldstr engthof4.7Tand7.0T, respectively. Besidestheenergylevelsshownhere,wealsoneedtoinvesti gatetheeigenstates ofthewavefunctions,i.e.,thebreakdownoftheeigenstate intermsofthelinear combinationoftheeightbasisstatesgiveninequation( 3–9 ),inordertoreally 54

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understandthebandstructuresofGaAsinthemagneticeld. Byusingthetheoretical modeldevelopedintheprevioussectionandnumericallysol vingtheeigenvalueproblem givenbyequation( 4–68 ),wecanobtaineacheigenstatesmanifoldbymanifold.We listourresultsforeigenstatesinTable 4-1 andTable 4-2 forthecasesof B =4.7Tand B =7.0T,respectively.TheinterpretationofTable 4-1 andTable 4-2 isthefollowing. Eachline(excepttherst)ofthetabledescribesaparticul areigenstatewhiletherst linegivesthetitleofeachcolumn.Therstcolumnassignse acheigenstateauniqueID numberwhichcanbeusedtorefertodifferenteigenstates.T hesecondcolumnisthe Pidgeon-Brownmanifoldnumber n [ 25 ].Thethirdcolumnlabelsdifferenteigenstates withineachmanifold,intheorderofincreasingenergies.T heresteightcolumnsgiveus thepercentageprobabilityoftheeigenstate.Forexamplet hesixthstateinTable 4-1 is thelowestenergystateinthe n =1manifold,andthisstateismixedwith0.2% j CB 0 "i and99.8% j SO 1 #i .ThesmallnumberontheshoulderistheLandauquantumnumbe r n 0 .NotethattheLandauquantumnumber n 0 isdifferentfromthePidgeon-Brown manifoldnumberaswecanseefromequation( 4–56 ).Table 4-1 andTable 4-2 canbe usedtoexplainthefeaturesfoundinthemagneto-opticalab sorptionspectralaterinthis dissertation. SimpleobservationofTable 4-1 andTable 4-2 cantellusalotinformation.Atthis pointIwanttopointoutseveralfacts.Therstfactisthat, althoughourgeneraltheory developedinprevioussectionsmakeusexpecttheeigenstat esismadeupwithlinear combinationoftheeightcomponents,thezonecenter,i.e., k =0statesaremadeup withlinearcombinationofeithertherstfour,orthesecon dfourcomponents.Therst fourcomponentsnevermixwiththesecondfourcomponentsat k =0.Onecanalso seethisfactdirectlyfromthematrixformof J n and H Z ,asboth J n and H Z haveablock diagonalformwhen k z =0.Wewillrefertherstfourcomponentstheuppersetandth e secondfourcomponentsthelowersetfromnowon. 55

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Thesecondfactworthnotingisthatforbotheldstrengthsu nderinvestigation, theconductionbandcomponentsarealmost100%purestatesb ythemselves.See,for example,theconductionbandspin-upstates j CB "i withID=11,19,27,35,onecan tellthesestateshavealarge( > 96%)percentageof j CB "i ,althoughthispercentage decreasewithincreasingLandaulevels.Thiscanbeeasilys eenfromFig. 4-5 (a)and Fig. 4-6 (a),wherethefourbluedatapointsareallquitecloseto100 %andslowly decreasingoverLandaulevelquantumenumbers.Similarsta tementscanbemade fortheconductionbandspin-downstates j CB #i withID=5,12,20,28,36,asshown inFig. 4-5 (b)andFig. 4-6 (b).OnecanalsotellfromTable 4-1 andTable 4-2 thatthe spin-orbitsplit-offbands j SO "i withID=7,14,22,30,and j SO #i withID=2,6,13,21,29 arealsoalmost100%purestates( > 99%),evenpurerthattheconductionbandstates. TheseareillustratedinFig. 4-5 (c,d)andFig. 4-6 (c,d).Thesespin-orbitsplit-offbands arelowlyingstateswhoseenergiesarebelowtheinterested range. Giventhefactthattheconductionbandsandspin-orbitspli t-offbandsarenearly purestates,andtheuppersetstatesnevermixwiththelower setstates,theonly mixingstatesthatweleftareheavy-holespin-upstates j HH "i mixedwithlight-hole spin-downstates j LH #i (thosewithID=4,9,16,18,24,26,32,34),andheavy-hole spin-downstates j HH #i mixedwithlight-holespin-upstates j LH "i (thosewith ID=1,3,8,10,15,17,23,25,31,33).Fig. 4-5 (e,h)andFig. 4-6 (e,h)illustratethemixing between j HH "i componentsand j LH #i components.Wecanseethatinonesetof statesthe j HH "i componentsgetsmallerwhilethe j LH #i componentsgetbigger;in anothersetofstatesthe j LH #i componentsgetsmallerwhilethe j HH "i statesget bigger.Similarobservationscanbemadeformixingbetween the j HH #i components andthe j LH "i components,whichcanbeseeninFig. 4-5 (f,g)andFig. 4-6 (f,g). 4.5Magneto-OpticalAbsorption Oncewehaveknowntheelectronicstructurewecancomputeth etransition probabilitiesundertheperturbationoftheradiationeld usingFermi'sgoldenrule,and 56

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thencomputetheopticalabsorptionspectraoftheGaAsunde rtheexternalmagnetic eld.Inthissectionwediscusshowwecalculatethespin-de pendentmagneto-optical propertiesandcomparewiththemagneto-absorptionexperi ments. Wecalculatethemagneto-opticalabsorptioncoefcientat thephotonenergy ~ from[ 26 ] ( ~ )= ~ ~ cn r 2 ( ~ ),(4–69) where 2 ( ~ )istheimaginarypartofthedielectricfunctionand n r istheindexof refraction.Theimaginarypartofthedielectricfunctioni sfoundusingFermi'sgolden rule.Theresultis 2 ( ~ )= e 2 2 ( ~ ) 2 X n ; n 0 0 Z 1 1 d k z j ˆ e P n 0 0 n ( k z ) j 2 [ f n ( k z ) f n 0 0 ( k z )] ( E n n 0 0 ( k z ) ~ ), (4–70) where E n n 0 0 ( k z )= E n 0 0 ( k z ) E n ( k z )isthetransitionenergy.Thefunction f n ( k z )in Eq.( 4–70 )istheprobabilitythatthestate( n k z ),withenergy E n ( k z ),isoccupied.Itis givenbytheFermidistribution f n = 1 1+exp[( E n ( k z ) E f ) = kT ] .(4–71) TheFermienergy E f inEq.( 4–71 )dependsontemperatureanddoping.If N D isthedonorconcentrationand N A theacceptorconcentration,thenthenetdonor concentration N C = N D N A canbeeitherpositiveornegativedependingonwhether thesampleis n or p type.ForaxedtemperatureandFermilevel,thenetdonor concentrationis N C = 1 (2 ) 2 2 X n Z 1 1 d k z [ f n ( k z ) v n ],(4–72) where v n =1ifthesubband( n )isavalencebandandvanishesif( n )isa conductionband.Giventhenetdonorconcentrationandthet emperature,theFermi energycanbefoundfromEq.( 4–72 )usingarootndingroutine. 57

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WhendoingthesummationinEq.( 4–70 ),weseparatethecontributionsintotwo casesaccordingtothenalstatesoftheopticaltransition s 2 ( ~ )= 2 ( ~ )+ 2 # ( ~ )(4–73) whereboth 2 ( ~ )and 2 # ( ~ )aregivenbythesamerighthandsideofEq.( 4–70 )with theexceptionthat 2 ( ~ )includesthosetermsifthenalstate( n 0 0 k z )isspin-upand 2 # ( ~ )includesthosetermsifthenalstate( n 0 0 k z )isspin-down.Wecandothis separationbecauseweknowthat,withineachmanifold,thee nergyeigenvaluescanbe putinordersothatthehighestenergyeigenvaluealwayscor respondstothespin-down conductionelectron j CB #i andthenexthighestenergyeigenvaluecorrespondstothe spin-upconductionelectron j CB "i ,notingthattheeffectivegfactorisnegative.The onlyexceptionstothisruleare n = 1and n =0manifolds.For n = 1manifold,thereis noconductionbandstatesatall,theonlyeigenstateinthis manifoldbeingaheavyhole state,andfor n =0manifold,thisisnoconductionbandspin-upstate. Accordingly,wecangetthespin-resolvedabsorptioncoef cients ( ~ )and # ( ~ ) fromEq.( 4–69 )andthetotalabsorptioncoefcientisjustthesumofthesp in-upand spin-downpartoftheabsorption ( ~ )= ( ~ )+ # ( ~ ).(4–74) Inthefollowingweshowourcalculatedspin-resolvedabsor ptionspectraatthe magneticeldof4.7Tand7.0Tforboth +and polarizations.InFig. 4-7 through Fig. 4-10 weseparatethespin-upabsorption (red)fromthespin-downabsorption # (blue),andeachtransitionislabeledwithanarrowandtwon umbers.Theseare thesamenumbersusedinTable 4-1 andTable 4-2 tolabeltheenergyeigenstate.For exampleinFig. 4-7 therstmajorpeakresultsfromatransitionfromtheID=1st ateto theID=5stateinTable 4-1 58

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WhenwelabelingthesetransitionswithapairofstateIDnum bers,notethateach stateisnotapurestatebutamixturewithcomponentscoming fromtheeightbasis statesgivenbyEq.( 3–9 ).Eachcomponentdoesnotcontributeequallytothetransit ion. Togetabetterunderstandingoftheoriginofthetransition ,i.e.,toknowwhetherthe initialstateofaparticulartransitionisaheavy-holesta teorlight-holestateandthespin orientationofthenalconductionelectron,weneedtoiden tifytheactivecomponents thatcontributetoeachtransition.Thistaskissimplieda ccordingtotheselectionrule summarizedinTable 4-3 .Fromtheseselectionrulesonecanimmediatelytellthat,f or +polarization,thespin-upelectionssolelycomefromthel ight-holecomponentofthe initialstateandthespin-downelectionssolelycomefromt heheavy-holecomponent. Thingsareexactlyoppositeforthe polarization.Nowwecanlabelthosetransitions withtheexactactivecomponentsofthewavefunctioninstea dofthenumericalID number,asshowninFig. 4-11 throughFig. 4-14 .Notethatforeverytransition,the Landaulevelquantumnumberisconserved: N =0,however,thePidgeon-Brown manifoldnumber n isnotconserved.For +polarization n =+1andfor polarization n = 1. Ourcalculationsarecomparedwithexperimentalabsorptio nspectraforamagnetic eldof B =7.5 T forboth +and polarizationasshowninFig. 4-15 ,wherethe experimentaldataaretakenfrom[ 27 ].Wecanseefromthiscomparisonthatour theoreticalmodelarewellandsufcienttoexplainthemajo randminorfeatures observedintheexperiments.Notethatthecalculatedcurve sareshiftedasawhole totheleftby0.0025eVtomatchtheobservedspectra. 4.6Spin-PolarizedAbsorptionandOpticallyPumpedNMR Inthissectionwecompareourcalculatedspin-dependentab sorptionwiththe opticallypumpedNMR(OPNMR)experiments.OPNMRmeasureme ntsinvolvetwo components:rstopticallypumpthesemiconductorsystemw ithcircularlypolarized photonsthenusetheNMRdetectionofthenuclearspin.Durin gtheopticallypumping, 59

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theangularmomentumofthephotoncanbetransferedtotheco nductionband electronsandtheresultingconductionbandelectronswill bespinpolarizedwitha majorityspinandaminorityspin.Thesespinpolarizedelec tronscaninteractwiththe nuclearspinsothattheOPNMRspectraisverysensitivetoth espinpolarizationof theconductionbandelectrons[ 28 – 32 ].Notethatthemagneto-absorptionexperiment, whichisatraditiontooltoprobetheelectronicstructures ofasemiconductorsystem, measuresthetotalexcitedelectrons(thesumofthespin-up andspin-downelectron populations)comingfromvalencebandstoconductionbands ,whereastheOPNMR measurementisonlysensitivetothespinpolarization(the differencebetweenspin-up andspin-downelectronpopulations)oftheexcitedelectro ns.Inthisdissertation theoscillatoryfeaturesintheOPNMRsignalofGaAsareattr ibutedtotheLandau leveltransitions.WewillshowthattheOPNMRsignalsaredo minatedbytheweaker light-holetransitionswhereasthemagneto-absorptionsa redominatedbythestronger heavy-holetransitions.ThesepropertiesoftheOPNMRmeas urementmakeitavery usefultooltoprobethespin-dependentelectronicstructu resofasemiconductorsystem. Theexperimentwasperformedbyourcollaborators,Dr.Soph iaHayes'sgroupin St.Louis.TheOPNMRspectraof 69 Gaspinsinbulksemi-insulatingGaAspolarized byanarrow-bandlaserweremeasuredintwodifferentextern aleldsof4.7Tand7.0T. WendthattheOPNMRsignalintensityoscillateasafunctio noftheabove-gapphoton energy,asshowninFig. 4-16 andFig. 4-17 .TheseoscillatoryfeaturesoftheOPNMR signalcanbewellexplainedusingourelectronicstructure calculation[ 33 ].Resultsof ourcalculatedenergylevelsareshowninFig. 4-18 .InFig. 4-19 weplotthetheoretical calculationsofthetotalmagneto-absorptionof light(upperblackline),spin-up absorption (bluedashedline),andspin-downabsorption # (reddotted-dashedline) inamagneticeldof4.7T.Thetheoreticalcalculationsfor allcurveswereshiftedin energyby6meVtoaccountfortheshiftduetoCoulombinterac tions(i.e.,theexciton bindingenergy)whichwewerenotincludedinourcalculatio ns. 60

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For excitationinFig. 4-19 ,thetotalabsorption(upperblackline)isdominated byopticaltransitionsfromtheheavy-holespin-upLandaul evels(solidblacklines, Fig. 4-18 )toconduction-bandspin-upLandaulevels.Howevertherea realsooptical transitionsfromthelight-holespin-upLandaulevels(sol idredlines,Fig. 4-18 )tothe conduction-bandspin-downLandaulevelsfor excitation.Theselight-holetransitions aredifculttoseeinabsorptionspectrasincethelight-ho letransitionsareweaker(by afactorof3)thantheheavy-holetransitionsandaresepara tedbyonlyafewmeVfrom thedominantheavy-holetransitions. InFig. 4-20 weplotacombinationof 69 GaOPNMRexperimentaldata(black symbols)for excitation.SuperimposedontotheexperimentalOPNMRdata are thecalculatedelectronspinpolarizations(redsolidline s).Aplotoftheelectronspin polarizationshowswhetherpeaksintheabsorptioncamefro mtransitionsfromheavyorlight-holeLandaulevels.Whenwelookattheconductionbandspinpolarization for excitation,weseethatthefeatureswhichariseintheelect ronspinpolarization aredominatedbythetransitionsfromthelight-holeLandau levels.Theselight-hole spin-uptoconduction-bandspin-downtransitionsarevery weakandbarelyvisiblein theplotofthetotalmagneto-absorptionfor excitation;howeverthesetransitions arewellresolvedintheOPNMRdataasafunctionofphotonene rgy.Wendthat theconduction-bandspinpolarizationisparticularlysen sitivetoregionsofphoton energywherethetotalspin-polarizedmagneto-absorption ( + # )andthedifferential magneto-absorption( # )aredifferentfromoneanother,whichoccursprincipallya t thepeaksofthelight-holetransitions. 61

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Table4-1.Percentageprobabilityofthezonecenterwavefu nctionsat4.7T ID n j CB "ij HH "ij LH #ij SO #ij CB #ij HH #ij LH "ij SO "i 1-1100000100HH 0 # 00 201000100SO 0 # 0000 30200000.3CB 0 # 99.7HH 1 # 00 40300100LH 0 # 00000 504000099.7CB 0 # 0.3HH 1 # 00 6110.2CB 0 0099.8SO 1 # 0000 71200000.2CB 1 # 0099.8SO 0 81300000.8CB 1 # 90.3HH 2 # 8.9LH 0 0 9140.1CB 0 099.9LH 1 # 00000 1015000009HH 2 # 91LH 0 0 111699.7CB 0 00.1LH 1 # 0.1SO 1 # 0000 1217000099.1CB 1 # 0.7HH 2 # 0.1LH 0 0.1SO 0 13210.3CB 1 0099.7SO 2 # 0000 142200000.3CB 2 # 0099.6SO 1 152300001.2CB 2 # 84.7HH 3 # 14.1LH 1 0 16240.5CB 1 29.8HH 0 69.6LH 2 # 00000 17250000014.3HH 3 # 85.7LH 1 0 1826069.8HH 0 30.1LH 2 # 00000 192799.1CB 1 0.3HH 0 0.2LH 2 # 0.3SO 2 # 0000 2028000098.5CB 2 # 1.0HH 3 # 0.2LH 1 0.3SO 1 21310.5CB 2 0099.5SO 3 # 0000 223200000.5CB 3 # 0.1HH 4 # 099.4SO 2 233300001.6CB 3 # 81.2HH 4 # 17.1LH 2 0.1SO 2 24340.9CB 2 46.3HH 1 52.7LH 3 # 0.1SO 3 # 0000 25350000017.4HH 4 # 82.6LH 2 0 2636053.0HH 1 46.9LH 3 # 00000 273798.6CB 2 0.7HH 1 0.3LH 3 # 0.4SO 3 # 0000 2838000098CB 3 # 1.3HH 4 # 0.3LH 2 0.4SO 2 62

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Table 4-1 .Continued ID n j CB "ij HH "ij LH #ij SO #ij CB #ij HH #ij LH "ij SO "i 29410.6CB 3 0.1HH 2 099.3SO 4 # 0000 304200000.7CB 4 # 0.1HH 5 # 099.2SO 3 314300001.9CB 4 # 78.8HH 5 # 19.1LH 3 0.2SO 3 32441.3CB 3 53.4HH 2 45.1LH 4 # 0.1SO 4 # 0000 33450000019.5HH 5 # 80.5LH 3 0 3446045.5HH 2 54.4LH 4 # 00000 354798CB 3 1.0HH 2 0.4LH 4 # 0.6SO 4 # 0000 3648000097.4CB 4 # 1.6HH 5 # 0.4LH 3 0.6SO 3 63

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Table4-2.Percentageprobabilityofthezonecenterwavefu nctionsat7.0T ID n j CB "ij HH "ij LH #ij SO #ij CB #ij HH #ij LH "ij SO "i 1-1100000100HH 0 # 00 201000100SO 0 # 0000 30200000.5CB 0 # 99.5HH 1 # 00 40300100LH 0 # 00000 504000099.5CB 0 # 0.5HH 1 # 00 6110.2CB 0 0099.8SO 1 # 0000 71200000.2CB 1 # 0099.7SO 0 81300001.1CB 1 # 90.0HH 2 # 8.8LH 0 0 9140.2CB 0 099.8LH 1 # 00000 1015000009HH 2 # 91LH 0 0 111699.6CB 0 00.2LH 1 # 0.2SO 1 # 0000 1217000098.6CB 1 # 1.0HH 2 # 0.2LH 0 0.2SO 0 13210.5CB 1 0099.5SO 2 # 0000 142200000.5CB 2 # 0.1HH 3 # 099.4SO 1 152300001.7CB 2 # 84.2HH 3 # 13.9LH 1 0.2SO 1 16240.7CB 1 29.5HH 0 69.7LH 2 # 00000 17250000014.3HH 3 # 85.7LH 1 0 18260.1CB 1 70.0HH 0 29.9LH 2 # 00000 192798.7CB 1 0.5HH 0 0.3LH 2 # 0.4SO 2 # 0000 2028000097.8CB 2 # 1.4HH 3 # 0.3LH 1 0.4SO 1 21310.7CB 2 0.1HH 1 099.2SO 3 # 0000 223200000.8CB 3 # 0.2HH 4 # 099.1SO 2 233300002.2CB 3 # 80.5HH 4 # 17.0LH 2 0.3SO 2 24341.3CB 2 45.7HH 1 52.9LH 3 # 0.1SO 3 # 0000 25350000017.5HH 4 # 82.5LH 2 0 e26360.1CB 2 53.3HH 1 46.6LH 3 # 00000 273797.9CB 2 1.0HH 1 0.5LH 3 # 0.6SO 3 # 0000 2838000097.0CB 3 # 1.9HH 4 # 0.5LH 2 0.6SO 2 64

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Table 4-2 .Continued ID n j CB "ij HH "ij LH #ij SO #ij CB #ij HH #ij LH "ij SO "i 29411.0CB 3 0.1HH 2 098.8SO 4 # 0000 304200001.0CB 4 # 0.3HH 5 # 098.6SO 3 314300002.7CB 4 # 77.9HH 5 # 18.9LH 3 0.5SO 3 32441.9CB 3 52.6HH 2 45.3LH 4 # 0.3SO 4 # 0000 33450000019.6HH 5 # 80.4LH 3 0 3446045.9HH 2 54.1LH 4 # 00000 354797.1CB 3 1.4HH 2 0.6LH 4 # 0.8SO 4 # 0000 3648000096.3CB 4 # 2.3HH 5 # 0.6LH 3 0.8SO 3 65

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Table4-3.Selectionrules +polarization j HH #i)j S #i Leftcircularpolarization j LH #i)j S "i polarization j HH "i)j S "i Rightcircularpolarization j LH "i)j S #i 1.5 1.52 1.54 1.56 1.58 1.6 Energy (eV) (a) 4.7T -0.02 -0.015 -0.01 -0.005 0 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Energy (eV)Wave vector (1/A) (b) 4.7T Figure4-1.EnergybandstructureforGaAsat B =4.7T 66

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1.5 1.52 1.54 1.56 1.58 1.6 Energy (eV) (a) 7.0T -0.02 -0.015 -0.01 -0.005 0 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Energy (eV)Wave vector (1/A) (b) 7.0T Figure4-2.EnergybandstructureforGaAsat B =7.0T 67

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-0.01 -0.008 -0.006 -0.004 -0.002 0 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Energy (eV)Wave vector (1/A) 4.7T Figure4-3.Valencebandstructurefor4.7T 68

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-0.01 -0.008 -0.006 -0.004 -0.002 0 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Energy (eV)Wave vector (1/A) 7.0T Figure4-4.Valencebandstructurefor7.0T 69

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0.8 0.85 0.9 0.95 1 0 1 2 3 (a) 0 1 2 3 4 (b) 0 1 2 3 (c) 0 1 2 3 4 (d) 0 0.2 0.4 0.6 0.8 1 0 1 2 (e) 0 1 2 3 4 5 (f) 0 1 2 3 (g) 0 1 2 3 4 (h) Figure4-5.WavefunctioncomponentsvsLandaulevelquantu mnumberat B =4.7T. Thesubgures(a)through(h)correspondtheeightcomponen tsintheorder of j CB "i j CB #i j SO "i j SO #i j HH "i j HH #i j LH "i j LH #i .The horizontalaxislabelsLandaulevelquantumnumberandthev erticalaxis givesthepercentageofthatparticularcomponents.Noteth atverticalaxisof thesubgure(a)through(d)havedifferentscalesfromthos efromthe subgure(e)through(h). 70

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0.8 0.85 0.9 0.95 1 0 1 2 3 (a) 0 1 2 3 4 (b) 0 1 2 3 (c) 0 1 2 3 4 (d) 0 0.2 0.4 0.6 0.8 1 0 1 2 (e) 0 1 2 3 4 5 (f) 0 1 2 3 (g) 0 1 2 3 4 (h) Figure4-6.WavefunctioncomponentsvsLandaulevelquantu mnumberat B =7.0T. Thesubgures(a)through(h)correspondtheeightcomponen tsintheorder of j CB "i j CB #i j SO "i j SO #i j HH "i j HH #i j LH "i j LH #i .The horizontalaxislabelsLandaulevelquantumnumberandthev erticalaxis givesthepercentageofthatparticularcomponents.Noteth atverticalaxisof thesubgure(a)through(d)havedifferentscalesfromthos efromthe subgure(e)through(h). 71

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0 2000 4000 6000 8000 10000 12000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 4.7T Left 1 5 3 12 8 20 15 28 23 36 4 11 9 1910 20 18 27 16 27 17 28 26 35 25 36 24 35 up down Figure4-7.SpindependentabsorptionforB=4.7T( + polarization) 72

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 4.7T Right 18 11 26 19 34 27 24 19 32 27 10 5 16 11 17 12 8 5 25 20 15 12 33 28 23 20 up down Figure4-8.SpindependentabsorptionforB=4.7T( polarization) 73

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0 2000 4000 6000 8000 10000 12000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 7.0T Left 1 5 3 12 8 20 4 11 9 1910 20 18 27 16 27 17 28 26 35 25 36 up down Figure4-9.SpindependentabsorptionforB=7.0T( + polarization) 74

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0 2000 4000 6000 8000 10000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 7.0T Right 18 11 26 19 34 27 24 19 32 27 10 5 16 11 17 12 8 5 25 20 15 12 33 28 up down Figure4-10.SpindependentabsorptionforB=7.0T( polarization) 75

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0 2000 4000 6000 8000 10000 12000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 4.7T Left h0 c0 h1 c1 h2 c2 h3 c3 h4 c4 l0 c0 l1 c1h2 c2 l2 c2 l2 c2 h3 c3 l3 c3 h4 c4 l3 c3 up down Figure4-11.ActivecomponentsforabsorptionatB=4.7T( + polarization) 76

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 4.7T Right h0 c0 h1 c1 h2 c2 h1 c1 h2 c2 l0 c0 h0 c0 l1 c1 l0 c0 l2 c2 l1 c1 l3 c3 l2 c2 up down Figure4-12.ActivecomponentsforabsorptionatB=4.7T( polarization) 77

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0 2000 4000 6000 8000 10000 12000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 7.0T Left h0 c0 h1 c1 h2 c2 l0 c0 l1 c1h2 c2 l2 c2 l2 c2 h3 c3 l3 c3 h4 c4 up down Figure4-13.ActivecomponentsforabsorptionatB=7.0T( + polarization) 78

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0 2000 4000 6000 8000 10000 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 Absorption (a.u.)Photon Energy (eV) 7.0T Right h0 c0 h1 c1 h2 c2 h1 c1 h2 c2 l0 c0 h0 c0 l1 c1 l0 c0 l2 c2 l1 c1 l3 c3 up down Figure4-14.ActivecomponentsforabsorptionatB=7.0T( polarization) 79

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Absorption (a.u.)(a)Left 7.45T Theory Experiment 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 Absorption (a.u.)Photon Energy (eV) (b)Right 7.45T 7.45T Theory Experiment Figure4-15.Calculatedabsorptioncomparedwithexperime nts 80

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1.51 1.52 1.53 1.54 1.55 1.56 1000000 1500000 2000000 2500000 3000000 3500000 Figure4-16.OPNMRsignalintensityasafunctionofphotone nergyfor excitation 81

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1.51 1.52 1.53 1.54 1.55 1.56 -6000000 -5500000 -5000000 -4500000 -4000000 -3500000 -3000000 -2500000 Figure4-17.OPNMRsignalintensityasafunctionofphotone nergyfor + excitation 82

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-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 -0.015 -0.010 -0.005 1.52 1.53 1.54 1.55 Energy (eV) Figure4-18.Calculatedspin-splitvalence-bandandcondu ction-bandLandaulevelsin GaAsat4.7T.Black(thick)linescorrespondtoheavyholes, red(thin)lines tolightholes,andbluelinestoconductionbandlevels.Sol idlinesarefor spin-upanddashedlinesareforspin-downstates.Theseass ignmentsare onlyapproximateduetobandmixing.OnlythelowestfewLand aulevelsof eachtypeareshown.Spin-upandspin-downstatesforthecon duction bandarenearlydegenerateandarenotresolvedinthisgure 83

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0 2 4 6 8 10 Figure4-19.Theoreticalcalculationsofabsorptionof lightbybulkGaAsat4.7T.Blue dashedlineshowstheabsorptionthatproducesspin-upelec trons( primarilyfromheavy-holetransitions).Reddotted-dashl ineshowsthe absorptionthatproducesspin-downelectrons( # ,primarilyfromlight-hole transitions).Blacksolidlinesshowstotalabsorption, + # 84

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-0.5 0.0 0.5 1 2 3 4 1.51 1.52 1.53 1.54 1.55 1.56 -0.5 0.0 0.5 Photon Energy (eV) -6 -5 -4 -3 Figure4-20.Depictionofthe 69 GaOPNMRsignalintensityasafunctionofphoton energyfor and + polarizedlightat4.7T.Theexperimentaldata(black symbols)arecomparedwiththecalculatedelectronpolariz ation(solidred line),( # ) = ( + # ). 85

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CHAPTER5 MAGNETO-PROPERTIESOFINDIUMANTIMONIDEQUANTUMEWELLS 5.1ExperimentalDetails ExperimentwascarriedoutbySantosandco-workersattheUn iversityof Oklahoma.TheInSb/AlInSbheterostructureasshowninFig. 5-1 wasgrownby molecularbeamepitaxyonan[001]GaAssubstrates.Thestru cturecontains40 InSbwellsthatare15nmthickandseparatedbyIn 0.9 Al 0.1 Sbbarrierlayersthatare50nm thick.A0.5 m-thickInAlSbbufferlayerwithagradedAlcompositionwas deposited betweenthemultiple-quantum-well(MQW)layersandthesub strateinordertoreduce thedensityofdislocationsthatresultfromthe 14%latticemismatchbetweenthe substrateandtheMQWlayers.A3 mIn 0.9 Al 0.1 Sblayer,whichisalmostcompletely relaxed,wasgrownjustpriortotheMQWlayers.TheInSbwell sarecompressively strainedtothelatticeconstantofthe3 mIn 0.9 Al 0.1 Sblayer.AFourierTransformInfrared spectrometerwasusedtonomitorthetransmissionthrought heMQWstructureasa functionofphotonfrequency,whichisalsoshowninFig. 5-1 .Thestructurewaswedged at4 toreduceunwantedFabry-Perotinterference.Intheirprev iousexcitonstudies withoutamagneticeld,theydeducedthebandoffsetsforIn Sb/AlInSb[ 34 ]andthe strainparametersforInSb[ 35 ].Inthecurrentstudy,aperpendicularmagneticeldof 0 B 7.5Twasappliedinfarinfraredtransmissionmeasurements atatemperature of4.2K.WeobservedrichspectraoftransitionsbetweenLan daulevelsofholeand electronsubbands. 5.2ExtensionoftheTheoreticalModel Wewanttolookatthemagneto-opticalpropertiesofInSbqua ntumwells.TheInSb quantumwellssystemhavethreemajordifferencesformtheb ulkGaAssystemwhich wealreadystudiedindetailinthepreviouschapter.Theset hreedifferencesbetween thetwophysicalsystemmakeitmuchmorecomplicatedtodeal withtheInSbquantum wellthanbulkGaAs.InthissectionIwilloutlinethesethre edifferencesandextend 86

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ourcurrenttheoreticalmodeltoaformthatcanbeusedtocal culatetheelectronicand opticalpropertiesoftheInSBquantumwells.5.2.1NarrowEnergyGap Therstthingwewanttonoticeisthat,unlikeGaAs,theInSb isanarrow-gap material.SimpleKanemodeltellsusthatnarrowbandgapwil lleadtostrongcoupling betweentheconductionbandsandthevalencebandssothatth eelectronicwave functionwillhavecomponentsmixedfromtheconductionban dandvalencebands.In ourprevioustreatmentofGaAssystem,theconductionbandw avefunctionsarenearly 100%,forexample,seeTable 4-1 andTable 4-2 ,andwecantaketheapproximationthat thevalencebandsareonlymixedwiththesix-dimensionalsu bspace.Howeverthiswill notbetrueforInSbsinceithasstrongcouplingbetweenthec onductionbandsandthe valencebands.Thegoodnewsforusisthatevenwecantakethe approximationtotreat conductionbandsandvalencebandsseperatellyforGaAs,we didnotdothatandwe stillkeepalleightbandsinourtheoreticalmodelintroduc edinpreviouschapters.Inthis concernweweretreatingGaAsasanarrowgapmaterialjustli keInSb,sowhenwedo needtotreatInSbwedonotneedtomodifyourprevioustheore cticalmodel. 5.2.2StrainEffect InourMQWstructure,theInSblayersarecompressivelystra inedtothelattice constantofthe3 mIn 0.9 Al 0.1 Sblayer.ThiswillgiveusonemoreterminourEFA Hamiltonian,sothatinsteadofusingequation( 4–34 ),wehavenow H EFA = H L + H Z + H S .(5–1) Thestraincontributiontotheenvelopefunctionapproxima tionis H S = 264 S a S c S y c S b 375 (5–2) 87

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wherethesubmatrices S a S b and S c aregivenby S a = 266666664 A 000 0 P Q M i p 2 M 0 M P + Q i p 2 Q 0 i p 2 M i p 2 Q P 377777775 ,(5–3a) S b = 266666664 A 000 0 P Q M i p 2 M 0 M P + Q i p 2 Q 0 i p 2 M i p 2 Q P 377777775 ,(5–3b) S c = 266666664 000000 L i q 1 2 L 0 L 0 i q 3 2 L 0 i q 1 2 L i q 3 2 L 0 377777775 .(5–3c) Intermsofthestraintensorcomponents ij ,thequantities A P Q L and M in equation( 5–3 )aregivenby A = a c ( xx + yy + zz ),(5–4a) P = a v ( xx + yy + zz ),(5–4b) Q = b 2 ( xx + yy 2 zz ),(5–4c) L = id ( xz i yz ),(5–4d) M = p 3 2 b ( xx yy )+ i 2 p 3 3 d xy .(5–4e) Inequation( 5–4 ), a c a v b and d aredeformationpotentials.Valuesofthedeformation potentialsforawiderangeofIII-Vsemiconductoralloysar etabulatedinRef.[ 4 ]. WeassumethatstrainintheAlInSb/InSbMQWispseudomorphi cwhichmeans thatthelatticeconstantintheInSbandAlInSblayersareeq ualtothelatticeconstantin 88

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theIn 0.9 Al 0.1 Sbsubstrate.Inourpseudomorphicstrainapproximation,t henon-vanishing straintensorcomponentsintheInSblayeraregivenby xx = yy = a 0 (AlInSb) a 0 (InSb) a 0 (InSb) (5–5a) zz = 2 c 12 c 11 xx (5–5b) where a 0 (InSb)istheunstrainedlatticeconstantinbulkInSb, a 0 (AlInSb)isthelattice constantoftheAlInSbsubstrate,and c 11 and c 12 areelasticstiffnessconstants tabulatedinRef.[ 4 ].SimilarexpressionsholdfortheAlInSblayers.Fromequa tion( 5–5 ) weseethat L and M inequation( 5–4 )vanishinthepseudomorphicstrainapproximation. Thisimplies S c =0and S a = S b inthestrainHamiltonian, H S .Thus H S isblockdiagonal. 5.2.3QuantumConnementEffect WeconsideranInSb/InAlSbmultiplequantumwell(MQW)grow nonathick (strain-relaxed)In 0.9 Al 0.1 SbbufferlayeronaGaAs(001)substrate.Thebanddiagram fortheMQWisshownschematicallyinFig. 5-2 .NotethebandgapofInSblieswithin thebandgapofAlInSb.ThebandgapmismatchbetweenInSband AlInSbis E g = E g (AlInSb) E g (InSb)=0.446eV 0.240eV=0.206eV,usingthebandgapvalues forInSbandAlInSbfromexperimentaldatafromDr.Santosgr oupattheUniversityof Oklahoma.Theconductionbandoffset, Q c ,whichisdenedastheratioofthedepth oftheconduction-bandsquarewelltothebandgapdifferenc eofthewell(InSb)andthe barrier(AlInSb),isassumedtobe0.62[ 34 ]andthevalencebandoffset Q v isassumed tobe0.38.Thustheconductionbandbarrierheightis E c = Q c E g =0.128eVandthe valencebandbarrierheightis E v = Q v E g =0.078eV.Withthesebandoffsetsboth theelectronsandtheholesareconnedintheInSblayers. Totakeintoaccountthequantumconnedstructureswetreat theMQWsampleas innitesuperlatticeswithwidebarrierregions.Stayinth esameLandaugaugegivenby 89

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equation( 4–18 ),theenvelopefunctionoftheEFAHamiltoniancanbewritte nas F n = e i ( k x x + k z z ) p A 2666666666666666666664 U n ,1, k z ( z ) n 1 U n ,2, k z ( z ) n 2 U n ,3, k z ( z ) n U n ,4, k z ( z ) n U n ,5, k z ( z ) n U n ,6, k z ( z ) n +1 U n ,7, k z ( z ) n 1 U n ,8, k z ( z ) n 1 3777777777777777777775 .(5–6) Inequation( 5–6 ), n isthemanifoldquantumnumberassociatedwiththeHamilton ian matrix, labelstheeigenvectors, A = L x L y isthecrosssectionalareaofthesamplein the xy plane, n ( )areharmonicoscillatoreigenfunctionsevaluatedat = y 2 k x and U n m k z ( z )areeightcomplexenvelopefunctions( m =1...8)forthe -theigenstate. FromBloch'stheorem,theenvelopefunctions, U n m k z ( z )havetheperiodicityofthe superlatticeandthewavevector k z ,isdenedwithintheminizone, = L k z = L where L isthesuperlatticeperiod.Theenvelopefunctionsarenorm alizedoverthe superlatticeunitcell,i.e. X m Z L = 2 L = 2 dz L U n m k z ( z ) U n m k z ( z )=1.(5–7) Notethatthewavefunctionsthemselveswillbegivenbythee nvelopefunctionsin equation( 5–6 )witheachcomponentmultipliedbythecorresponding k z =0Blochbasis statesgiveninequation( 3–9 ). Weuseanitedifferenceschemetoobtaintheenergiesandwa vefunctionsinthe superlattice.Wedividethesuperlatticeunitcellintoane venlyspacedgridofpoints, z i ,where i =1... N .Substituting F n fromequation( 5–6 )intotheEFAequationwith 90

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Hamiltoniangivenbyequation( 5–1 ),weobtainamatrixeigenvalueequation H n F n = E n ( k z ) F n ,(5–8) thatcanbesolvedforeachallowedvalueofthemanifoldquan tumnumber n andwave vector k z toobtaineigenvaluesandeigenvectors.Sincetheharmonic oscillatorfunctions n 0 ( ),areonlydenedfor n 0 0,itisnecessarytodeleterowsandcolumnsof H n for which n 0 < 0.Itfollowsfromequation( 5–6 )that F n isdenedfor n 1.Theresulting eigenvaluesaretheLandaulevels,denoted E n ( k z ),where n labelsthemanifold quantumnumberand labelstheeigenenergiesbelongingtothesamemanifoldin ascendingorder.Thecorrespondingeigenvectors, F n ,arethecellperiodicfunctions, U n m k z ( z i ),evaluatedatthegridpoints, z i .InnitedifferencingtheEFASchr ¨ odinger equation,weallowallthematerialparameterstovarywithp osition.Toensurethat H n is Hermitian,wemaketheoperatorreplacements B ( z ) @ 2 @ z 2 @ @ z B ( z ) @ @ z (5–9a) B ( z ) @ @ z 1 2 B ( z ) @ @ z + @ @ z B ( z ) (5–9b) whendifferencingderivativeswithrespectto z .Thecellperiodicboundaryconditionson U n m k z ( z i )aresatisedbyletting U n m k z ( z N + i )= U n m k z ( z i )inthedifferenceformulas forthederivatives. 5.3Magneto-OpticalAbsorption Wecanstilluseequation( 4–70 )tocalculatetheopticalpropertiesforour InSb/AlInSbMQWstructure.Sincetheenvelopefunctionsan dvectorpotentialsare slowlyvaryingoveraunitcell,thedominantcontributions totheopticalmatrixelements aregivenby P n 0 0 n ( k z )= X m m 0 Z dz L U n m k z ( z ) U n 0 m 0 k z 0 ( z ) h N ( n m ) j N ( n 0 m 0 ) ih m j P j m 0 i (5–10) 91

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where N ( n m ) areorthonormalizedharmonicoscillatorwavefunctions.T heirquantum numbers N ( n m )dependexplicitlyon n and m asdenedinequation( 5–6 ).In equation( 5–10 )wehaveneglectedatermthatdependsonthemomentummatrix element, h N ( n m ) j P j N ( n 0 m 0 ) i betweentheoscillatorstates.Owingtostrongbandmixing inthenarrowgapmaterials,thistermismuchsmallerthanth emomentummatrix elementsbetweentheBlochbasisfunctions,evenforintrab andopticalabsorptionsuch asforcyclotronresonance,henceweneglectit. Themomentummatrixelements h m j P x j m 0 i h m j P y j m 0 i and h m j P z j m 0 i are themomentummatrixelementsbetweentheBlochbasisfuncti ons j m i denedin equation( 3–9 ).For P x wehavetheexplicitrepresentation P x = 264 P a x 0 0 P b x 375 (5–11a) P a x = m 0 ~ 266666664 0 iV q 1 2 iV q 1 6 V q 1 3 iV q 1 2 000 iV q 1 6 000 V q 1 3 000 377777775 (5–11b) P b x = m 0 ~ 266666664 0 V q 1 2 V q 1 6 iV q 1 3 V q 1 2 000 V q 1 6 000 iV q 1 3 000 377777775 .(5–11c) Likewise,for P y wehave P y = 264 P a y 0 0 P b y 375 (5–12a) 92

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P a y = m 0 ~ 266666664 0 V q 1 2 V q 1 6 iV q 1 3 V q 1 2 000 V q 1 6 000 iV q 1 3 000 377777775 (5–12b) P b y = m 0 ~ 266666664 0 iV q 1 2 iV q 1 6 V q 1 3 iV q 1 2 000 iV q 1 6 000 V q 1 3 000 377777775 .(5–12c) andfor P z wehave P z = 264 0 P c z iP c z 0 375 (5–13a) P c z = m 0 ~ 266666664 00 V q 2 3 iV q 1 3 0000 iV q 2 3 000 V q 1 3 000 377777775 .(5–13b) Inperformingtheintegralinequation( 4–70 )theDiracdeltafunction ( x )inFermi's goldenruleisreplacedbytheLorentzianlineshapefunctio n r ( x )withfullwidthathalf maximum(FWHM)of r 5.4ResultsandDiscussion Inthissectionwediscussourresultsforourcalculationof theelectronicstructure andmagnetoopticalpropertiesofInSb/AlInSbMQW,andcomp arewithexperimental studies. InFig. 5-3 ,weshowthefullevolutionoftheabsorptionspectraupto8T ina waterfalldisplay.Fig. 5-3 (a)showstheexperimentaldata.Thespectraaredominated bytheabsorptionpeaksthat,withincreasingeld,evolvef romthezero-eldH 1 -C 1 transition(at0.295eV)andtoalesserdegreebythosefromt heH 2 -C 2 transition(at 93

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0.385eV).AlthoughtheL 1 -C 1 transitionisclearlyseenatzeroeldat0.350eV,no clearlyassociatedevolvingmagnetic-levelstructureiso bviousinthegure.Because theconduction-bandmassismuchlessthantheheavy-holema ss,thechangeofthe transitionenergieswithBisdominatedbytheLandau-level structureoftheconduction bandwheretheLandaulevelspacingismuchlargerthanthesp acinginthevalence band.Thustoextractthedetailsoftheholeband-structure willrequireadetailed comparisontotheory. Fig. 5-3 (b)showsthetheoreticalcalculationoftheabsorptionspe ctrumof theInSb/AlInSbMQWstructure.Thegureincludestheeffec tsofstrainatthe pseudomorphicinterface.InFig. 5-3 (c),weplotthetheoreticalcalculationofthe absorptionspectrumwithoutincludingtheeffectsofstrai n.Ascanbeseenby comparingFig. 5-3 (b)andFig. 5-3 (c)toFig. 5-3 (a),theinclusionofstrainhasadramatic effectonthemagneto-absorptionspectra.Weseethatstrai nisessentialtocalculatethe correctspectrum.ThiscanbeseenagaininFig. 5-4 ,whereweplottheexperimental spectrum(a)at6Tandcompareittothetheoreticalcalculat ionwithstrain(b)and withoutstrain(c).Clearlythecalculationswithstrainmo reaccuratelyreproducethe experimentaldata. FromFig. 5-3 itisclearthatthemainabsorptionfeaturesduetotheH 1 -C 1 and H 2 -C 2 transitionsaredominantinboththeoryandexperiment.Ina ddition,onecan seethatthe1stand2ndH 1 -C 1 Landauleveltransitionshavebeenspin-splitbutthe 0thLandauleveltransitiondoesnotshowspin-splitting.A nothercleardifferenceis theanti-crossing-likestructureat0.35eVnear6Tintheex perimentalplotthatisnot reproducedinthetheoryplot. Nextwewanttoexamineourresultsindetailsintermsofther elationbetween electronicstructureandthemagnetospectra.Fig. 5-5 ,showsthemagneticeld dependenceofsomeoftheconductionandvalencebandsforth esquarewell. Fig. 5-5 (a)showsthelowestLandaulevelsfortherstandsecondcon ductionsubbands 94

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andFig. 5-5 (b)showsthelowestLandaulevelsfortherstthreeheavyho lesubbands andtherstlightholesubband.ThebandsinFig. 5-5 arecolorcodedtoindicatethe Pidgeon-Brownmanifoldindex( N = 1,0,1,...)with N = 1black; N =0red; N =1 green; N =2blue; N =3magenta;and N =4yellow.Thenumberslabelingthebands inFig. 5-5 arethebandnumbersgiveninFig. 5-6 andallowthecomponentsofthe Landaulevelsanddominantcontributionstothewavefuncti onsat6Ttobedetermined. ThebandsarelabeledinFig. 5-6 accordingtothedominantwavefunctioncomponent, i.e.bandnumber10is85.3%heavyholeupandlabeled(1sthea vyholesubband,0th Landaulevel,spinup( m j =+3 = 2)).SolidlinesinFig. 5-5 indicate(primarily)spin-up bandswhiledottedlinesindicate(primarily)spin-downba nds. ThebanddiagraminFig. 5-5 aidsusinidentifyingthemajoropticaltransitions. Wecalculatetheabsorptionspectraforboth + and circularlypolarizedlightaswell asforlinearlypolarizedlight.Theopticaldipoleselecti onrulesareasfollows.Inthe axialapproximationthePidgeon-BrownManifoldindexchan gesby+1fortransitions involving + polarizedlightand-1fortransitionswith polarizedlight.Notethatthe Pidgeon-BrownManifoldindextakesonvalues 1,0,1,2,...andisnotthesameasthe LandauLevelindex(whichtakesonthevalue0,1,2,...).Eac hPidgeonBrownmanifold hasuptoeightBlochstatesandtheLandauLevelindexdepend sontheBlochstate.i.e. forthe N = 1Pidgeon-Brownmanifold,onlytheHeavyHolespindownBloc hstateis inthatmanifoldandtheLandaulevelindex n isrelatedtothePidgeonBrownManifold numberby n = N +1.WithinagivenPidgeonBrownmanifold, m j (fortheBlochstate) plus n (theLandaulevel)isaconstant. ThebanddiagraminFig. 5-5 togetherwithFig. 5-6 andtheselectionrulesaidus inidentifyingtransitionpeaksinthemagneto-absorption spectra.Someofthedominant transitionpeaksareshowninFig. 5-7 .For + polarization,(Fig. 5-7 (b))thedominant transitionsarebetweentherstheavyholedownandrstcon ductionbanddown subbands.Weseetransitionsfromthe0th,1st,2ndand3rdLa ndaulevels.Wealsosee 95

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weakertransitionsbetweentherstlightholedownandrst conductionbandspinup subbands(0th,1stand2ndLandaulevels).Thesetransition saremuchweakersince theiroscillatorstrengthisdownbyafactorof3comparedto theheavyholetransitions andtheyareincloseproximitytotheheavyholetransitions For polarization(Fig. 5-7 (c)),wealsoseethedominanttransitionsbetweenthe rstheavyholeuptotherstconductionbandupsubbands(fo rthe0th,1st,2ndand 3rdLandaulevels)togetherwiththedominanttransitionsb etweenthesecondheavy holeupsubband(0thLandaulevel)tothesecondconductionb andupsubband. Inaddition,wealsoseeweakertransitionsfromtherstlig htholeuptotherst conductionbanddownsubbands(0th,1stand2ndLandaulevel s),butnowtheyare furtherseparatedfromtheheavyholetransitions. The + and polarizationspectrainFig. 5-7 (b)and(c)canbeaddeduptogive thelinearlypolarizedspectruminFig. 5-7 (a)whichcanbecomparedtoexperiment. Lookingatthespectrum(andalsoFig. 5-3 andFig. 5-4 ),weseethatthe1stand 2ndLandaulevelsarespinsplit,butthe0thLandaulevelisn ot.Thisissomewhat surprisingsinceifwelookatthesplittingoftheLandaulev elsintheconductionbands inFig. 5-5 (a),weseethatthe0thLandaulevelintherstsubbandhasth elargest splitting.Thereasonforthiscanbeseenbyexaminingtheva lencebandLandaulevels inFig. 5-5 (b).The0thheavyholeLandaulevelsforthe1stholesubband (bands9, 10)spin-splitinexactlythesamemanneranddirectionasth e0thLandaulevelsfor theconductionband.Whilethe1st(bands11,12)and2nd(ban ds13,14)heavyhole Landaulevelsinitiallysplitinthesamedirectionastheco nductionbands,theycrossat eldsofabout4.5Tforthe1stlevelandabout3Tforthe2ndle vel.Thisisprecisely whereweseethe1stand2ndLandaulevelspin-splitinbothth eexperimentaland theoreticaldata(seeFig. 5-3 (a)andFig. 5-3 (b)).Whilethe0thheavyholeLandaulevels donotcross,theirsplittingissmallerthantheconduction band0thLandaulevelsplitting. 96

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Ourcalculationspredictthatthesplittingofthe0thLanda ulevelshouldbeobservable formagneticeldsgreaterthan10T. Ifwecarefullyexaminethesplittingofthe1stLandaulevel intheexperimental spectrum(Fig. 5-3 (a)),itappearsthatthreelevelsanticrossnearthepointw herethe splittingoccurs(5Tand0.34eV).Thisisnotreproducedint hetheoreticalcalculations (Fig. 5-3 (b)).IfwelookatthecalculatedvalencebandlevelsinFig. 5-5 (b).Thesplitting occurswhenthetwospin-splitheavyholesubbands(band11, 12)ofthe1stLandau levelcross.Whentheycross,theyalsocrosswiththe0thLan daulevelsofthesecond subband(bands17,18).Inthetheoreticalcalculations,we maketheaxialapproximation andassumethatthevalencebandsarecylindricallysymmetr icaboutthedirectionofthe magneticeld(ormoretechnicallythattheLuttingerparam eters r 2 r 3 areequal)andas aresult,theselevelsdonotmix.Intheaxialapproximation ,onlylevelswhichbelongto thesamePidgeon-Brownmanifoldcanmix,butifwegobeyondt heaxialapproximation, thenlevelsindifferentPidgeon-Brownmanifoldscanmix(a ndthecalculationsbecome muchmoredifcult).Thisshowsthattheexperimentalmeasu rementsaresensitive enoughtoshowdeviationsfromtheaxialapproximation. InthecalculatedspectrashowninFig. 5-7 (andalsointheexperimentalspectrum) wecanalsoseeminortransitions.Someofthesearelabeledi nFig. 5-7 (b,c)andthe assignmentofsomeoftheseminortransitionsmightseemstr ange.Forinstance,there isatransitionlabeled H 1 2 #! C 0 1 inFig. 5-7 (b)justtotherightedgeofthedominant H 0 1 #! C 0 1 # transitionthatisalsoseenintheexperimentaldatainFig. 5-3 (a).From Fig. 5-6 ,weseethatalthoughbandnumber19is85.4%heavyholedown( andhence labeled H 1 2 # ),itisthemixinginofthelightholedownstatethatisrespo nsibleforthe observedminortransition. 97

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Figure5-1.Samplestructureandexperimentalsetup Figure5-2.BanddiagramforanInSb/AlInSbmultiplequantu mwell 98

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(a) Expt.2 4 6 8 (b) Magnetic Field B (T) Theory with Strain 2 4 6 8 Theory without Strain (c) 0.30 0.35 0.40 Energy (eV)2 4 6 8 Figure5-3.Absorptionspectrawaterfallplotfor(a)exper imentand(b)theoryfora strainedinterfaceand(c)theoryforanunstrainedinterfa ce. 99

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0.280.300.320.340.360.380.400.420.44 (a)Experiment Absorption (a.u.)(b)Theory with Strain Energy (eV) (c)Theory without Strain Figure5-4.Experimentalandcalculatedabsorptionspectr afortheMQWforB=6T.(a) Theexperimentalmeasuredabsorption.(b)Theoreticallyc alculated absorptionincludingtheeffectsofstrain.(c)Theoretica llycalculated spectrumnotincludingtheeffectsofstrainattheinterfac e. 100

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012345678 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 012345678 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 E (eV)B (T) 1 2 3 4 5 6 7 8 9 10 11 12 13 / 14 15 / 16 17/ 18 / 19 / 20 21 /22/ 23 / 24 25 26 27 (a) Conduction Bands1CB 2CB 1HH2HH 3HH 1LH B (T) (b) Valence Bands Figure5-5.Calculatedconductionband(a)andvalenceband (b)Landaulevelsforthe MQWstructure.ThecolorsofthebandsindicatethePidgeion -Brown manifoldindex( N = 1,0,1,...)with N = 1black; N =0red; N =1green; N =2blue; N =3magenta;and N =4yellow.Thenumbersrefertotheband numbergiveninFig. 5-6 andallowonetodeterminethecomponentsofthe Landaulevelsanddominantcontributionsat6T. 101

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n r n |CB>|HH>|LH>|CB>|HH>|LH> #" $ # # #" % & $ & % & # "# % % # $ # % % # # & $ $ % &" ## # % ( % # & ( % & ( % % # $" # ( % % $ # ( % & & # $" # ## & ( % # # $ & # & ( $ $" #$ ( % ( # # $ & & ( % ) && # & ( % # & $ & ( % # # & # #" ( ) ## ( ) # $ $ $ "# ( ) % # & &" && ( ) % #" $ & % #$ & $ &#" % % # # #" $ $ % + ./0. 1 ,/ 2 3 4 5/ 2 60 7 8 5/ 1 ,/ 2 3 Figure5-6.Eigenfunctionsofthelowestlyingbandsforthe MQWforB=6T.The numbersgivethefractionofagivencomponentinthatband.( Thespin-split holecontributionswerenegligible.)Bandsarecolorcoded accordingtothe Pidgeon-Brownmanifoldindex( N = 1,0,1,...)with N = 1black; N =0 red; N =1green; N =2blue; N =3magenta;and N =4yellow.Theband numbercorrespondstothenumbershowninFig. 5-5 .Thebandsare labeledaccordingtothedominantcomponent,i.e.,bandnum ber10is 85.3%heavyholeupandlabeled H 0 1 (1stheavyholesubband,0thLandau level,spinup m j =+3 = 2). 102

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linear 0.280.300.320.340.360.380.400.420.44 (a) Absorption (a.u.)(b) Energy (eV) (c) 00 11 HC 11 11 HC 22 11 HC 33 11 HC 00 11 HC 11 11 HC 22 11 HC 33 11 HC 00 11 LC 11 11 LC 22 11 LC 00 11 LC 11 11 LC 22 11 LC 3 00 1 HC 22 00 HC s ss s s ss s + 1 2 0 1 HC 003 1 HC 2 1 0 1 HC 1 2 0 1 HC Figure5-7.Calculatedmagneto-absorptionspectrafor(a) linearpolarizedlight,(b) + circularlypolarizedlightand(c) circularlypolarizedlight.Peaksare labeledbythedominanttransitions.For + light,withintheaxial approximation,transitionsoccuronlybetweenstateswhos ePidgeon-Brown manifoldindexchangeby+1.For light,thePidgeon-Brownmanifold indexchangesby-1foratransition. 103

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CHAPTER6 CONCLUSIONSANDFUTUREDIRECTIONS 6.1Conclusions Inthissectionwesummarizethewholedissertationandemph asizethekey stepsinourdevelopmentofthetheoreticalmodel,andtheim portantresultsfromour calculations.WewanttostudytheInSbmaterialbecauseith asthenarrowestgap oftheIII-Vcompoundsemiconductors.Thisleadstoaverysm allconductionband effectivemassandalargeeffectiveg-factorwhichcanlead toimportantapplicationsin digitalnano-electronicsandspintronicsaswellaspossib leusesininfra-reddetectors. TheconductionbandsofInSbarerelativelysimpletounders tandbutthedetailsofthe valencebandstructuresareneededinordertomakeuseofthi smaterial,forexample, intheapplicationofp-channeleldeffecttransistors(FE T). Wehavemeasuredtheexperimentalmagneto-absorptioninaI nSb/InAlSbmultiple quantumwellstructuresandcomparedthemtodetailedcalcu lations.Ourtheoretical modelstartsfromthe8-band k p methodwithspin-orbitinteractionconsidered explicitly.Weusethequasi-degenerateperturbationtheo ry(L ¨ owdin'spartitioning)to treattheeffectsofcouplingwithremotebands.Theexterna lmagneticeldisbuiltinto ourmodelonthebasisofPidgeon-Brownmodel,andgeneraliz edtoincludethewave vectordependence. Becausethisisan8-bandmodel(notLuttinger's6-bandmode l),whichtreat thecouplingofconductionbandsandvalencebandsexplicit ly,itissuitedforboth wide-gapandnarrow-gapsemiconductors.Beforeapplyingt hismodelinthenarrow-gap InSb/AlInSbmultiplequantumwellstructures,weusethism odeltocalculatethe spin-dependentelectronicstructuresofbulkGaAssystemi ntheexternalmagnetic eld.Theopticalpropertiesthencanbeobtainedfromtheel ectronicstructuresby usingFermi'sGoldenrule.WefoundthatforbulkGaAssystem ,theconductionbands arealmostdegenerateattheeldof7T,butthevalencebands arespin-splitting.Our 104

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calculatedopticalabsorptionspredictthedominanttrans itions(comingfromtheheavy hole),secondarytransitions(comingfromthelighthole)a ndalsoveryweaktransitions whichareduetobandmixing.Weproposetwoexperimentaltoo lstoprobethedetails ofthevalencebandstructures:i)magneto-opticalabsorpt ion,whichissensitiveto heavy-holetransitions;ii)opticallypumpedNMR(OPNMR)m easurement,whichis sensitivetothelight-holetransitions.ByusingmagnetoabsorptionandOPNMR togetherwecanprobethevalencebandsplittingeasily. Nextweextendourtheoreticalmodelsothatitcanbeapplied totheInSb/AlInSb multiplequantumwellsystem.Wetreatthequantumconneme nteffectsviaanite differencemethod.BecausethelatticeconstantofInSbisb iggerthatthatofAlInSb, therewillbeacompressivestrainintheInSblayers.Weinco rporatethepseudomorphic straininourmodelanditturnsourthethisstraineffectwil ldeeplyaffecttheelectronic structures.Wefoundthatwithintheaxialapproximationth einnitedimensional eigenvalueproblemsdecouplesintodifferentPidgeon-Bro wnmanifolds.Theoptical propertiescanbestillcalculatedbyusingFermi'sgoldenr ule. JustlikethecaseofbulkGaAs,ourdetailedcalculationhel pustoidentifythe dominant(bright)transitions,secondarytransitions,an dveryweak(dark)transitions duetobandmixing,inthemagneto-absorptionspectrum.Bec auseofInSb'slarge g-factor,spinsplittingoftheLandaulevelscanbeseenint hespectrumatarelatively lowmagneticeld.However,the0thLandaulevelsplittingi snotseenforeldsupto8 Tsince0thheavyholeLandaulevelssplitinsamedirectiona sconductionband.We predictthatspin-splittingofthe0thlevelinthesquarewe llcanbeseenforeldsabove 10T.Wehavealsoidentiedseveralminor(dark)transition s.Thesetransitionsoccur becauseofband-mixingwhichmixesinsmallcomponentsofth e8-Blochfunctions intolevelsdominatedbyaBlochfunctionwhichwouldnotnor mallyleadtoanoptical transition.Furthermore,wehaveseenthattheexperimenta ldataisdetailedenough 105

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thatitleadstomixingofstatesbetweenPidgeon-Brownmani foldswhicharisesfrom non-axialsymmetrictermsinthepotential. Insummarize,wehaveveryaccuratelymodelthespin-depend entelectronic structuresofbothbulkandquantumconnedsemiconductors ystems.Ourmodel canbeappliednotonlytowide-bandmaterialbutalsonarrow -gapmaterialwhich hasstrongcouplingbetweenconductionbandsandvalenceba nds.Weexpectthat detailedunderstandingoftheInSbvalencebandstructures canhelpustomakedevice applicationssuchasopticaldetectorsandCMOS,aswellasa pplicationinspintronics. 6.2FutureDirections Althoughourcurrenttheoreticalmodelcanbesuccessfully appliedtopredict semiconductorelectronicstructures,itcanstillbeimpro vedfromseveralaspects.Inthis sectionIproposesomeimprovementsonthemodelwecanpossi blymakebeyondits currentform.6.2.1ExcitonAbsorption Inourpreviouscalculationofopticalpropertieswehaveco mpletelyneglectthe Coulombinteractionbetweentheexcitedelectroninthecon ductionbandandthehole leftbehindatthevalenceband.Thisapproximationmaybego odenoughforthose transitionwithphotonenergymuchbiggerthanthefundamen talbandgap,becausein thiscasetheelectronandtheholehavesufcientkineticen ergy.Howeverwhenthe excitationenergyisclosetothebandgap,weneedtoconside rtheattractionsbetween theelectronandholepair.Thisattractioncanleadtoaseto fdiscretehydrogen-like levelswhichcanbeobservedexperimentally.ForInSb,duet othesmalleffective mass,theexcitonbindingenergyis0.6meV.ForGaAstheexci tonbindingenergy is4meV.Weexpectthattheexcitonabsorptioneffectismore importantforGaAs system,especiallynearthebandgap.Infactwesawthediscr epancybetweenthe calculatedabsorptionandtheobservedabsorptionnearthe bandgap.Weexpectthat byconsideringexcitoneffectinthefutureworkthisdiscre pancycanberemoved. 106

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6.2.2BeyondtheAxialApproximation Inordertoconverttheinnitedimensionaleigenvalueprob lemintoatractableform, wetaketheLuttingerparameters r 2 = r 3 andneglectthetermproportionalto a y 2 inthe denitionofthe M operatorinequation( 4–47 ),andthisiscalledtheaxialapproximation. DifferentPidgeon-Brownmanifoldscanonlybeseparated(b lock-diagonalized)within thisapproximation.Decoupledmanifoldsmeansthatonlyst ateswithinthesame manifoldscanmixwitheachother,i.e.theinteractionbetw eenstatesinthesame manifoldwillmakethemanti-crosseachotherwhentheenerg ylevelsmeetatacertain magneticeld. Wesuccessfullyobservedlevelsanti-crossingsinboththe magneto-optical absorptionmeasurementandcyclotronresonancemeasureme nts.Partofthese anti-crossingscanbeunderstandwithincurrentmodel,int ermsofthemixingstates belongingtothesamemanifold,howeverwealsofoundanti-c rossingsthatcannotbe explainedwithincurrentmodel,sincethosemixingstatesb elongtodifferentmanifolds. Inordertohaveinteractionsbetweendifferentmanifoldsw ehavetoabandontheaxial approximationanddealwiththemuchmorecomplicatedprobl em.Onepossibleway ofdoingitisstillstartingwiththeaxialapproximationan donlyafterwardwetreatthe couplingbetweenmanifoldsasaperturbation.Onecanhopet hatnumericalresults canbeobtainedinthisway.Anotherpossibilityisstarting fromthegeneralprinciples ofgrouptheory[ 20 36 37 ]andseparatetheHamiltonianintotermsaccordingtothe symmetryhierarchy,i.e.,thefullHamiltoniancanbesepar atedintoaxialterm,cubic termandtetrahedralterm.Thiswaycanhelpusphysicallyun derstandwhentwolevels crosseachotherandwhentheyanti-cross,butitisdifcult togetanynumericalresults. 6.2.3CarrierDynamics Wehavesuccessfullycalculatedtheelectronicstructures inthemagneticeld andthenextstepwecangoistomodelthecarrierdynamics[ 38 – 40 ].Thegeneration andrelaxationofcarriers,andthetimedependentcarrierd istributionfunctioncan 107

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bemodeledandcomparedwithexperimentssothatinformatio naboutscattering mechanismsandcouplingconstantscanbeobtained. 108

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[21] G.F.Koster, SpaceGroupsandTheirApplications (AcademicPress,NewYork, 1957). [22] G.F.Koster,J.O.Dimmock,R.G.Wheeler,andH.Statz, PropertiesoftheThirtyTwoPointGroups (MITPress,Cambridge,1964). [23] E.O.Kane, The k p Method (AcademicPress,NewYork,1966). [24] G.Bastard, WaveMechanicsAppliedtoSemiconductorHeterostructures (Les EditionsdePhysique,lesUlis,1988). [25] C.K.PidgeonandR.N.Brown,Phys.Rev., 146 ,575(1966). [26] F.BassaniandG.P.Parravicini, ElectronicStatesandOpticalTransitionsinSolids (Pergamon,NewYork,1975). [27] D.V.Vasilenko,N.V.Luk'yanova,andR.P.Seisyan,Semicon ductors, 33 ,15 (1999). [28] D.Paget,G.Lampel,B.Sapoval,andV.I.Safarov,Phys.Rev. B, 15 ,5780(1977). [29] S.E.Barrett,R.Tycko,L.N.Pfeiffer,andK.W.West,Phys.R ev.Lett., 72 ,1368 (1994). [30] R.Tycko,S.E.Barrett,G.Dabbagh,L.N.Pfeiffer,andK.W.W est,science, 268 1460(1995). [31] C.R.Bowers,SolidStateNuclearMagneticResonance, 11 ,11(1998),ISSN 0926-2040. [32] S.E.Hayes,S.Mui,andK.Ramaswamy,TheJournalofChemical Physics, 128 052203(2008). [33] K.Ramaswamy,S.Mui,S.A.Crooker,X.Pan,G.D.Sanders,C.J .Stanton,and S.E.Hayes,Phys.Rev.B, 82 ,085209(2010). [34] N.Dai,G.A.Khodaparast,F.Brown,R.E.Doezema,S.J.Chung ,andM.B. Santos,AppliedPhysicsLetters, 76 ,3905(2000). [35] T.Kasturiarachchi,F.Brown,N.Dai,G.A.Khodaparast,R.E .Doezema,S.J. Chung,andM.B.Santos,AppliedPhysicsLetters, 88 ,171901(2006). [36] K.SuzukiandJ.C.Hensel,Phys.Rev.B, 9 ,4184(1974). [37] H.R.Trebin,U.R ¨ ossler,andR.Ranvaud,Phys.Rev.B, 20 ,686(1979). [38] G.Khodaparast,M.Bhowmick,M.Frazier,R.Kini,K.Nontapo t,T.Mishima, M.Santos,andB.Wessels,Proc.ofSPIE, 7608 ,760800(2010). [39] A.V.Kuznetsov,C.S.Kim,andC.J.Stanton,JournalofAppli edPhysics, 80 ,5899 (1996). 110

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BIOGRAPHICALSKETCH XingyuanPanwasbornin1981inQinhuangdao,Hebeiprovince ,China.He spent18yearsthereuntil1999whenhenishedhishighschoo lstudyinNo.1Middle SchoolinQinhuangdaoandwenttotheUniversityofSciencea ndTechnologyofChina (USTC)atHefei,Anhuiprovince,China.Hismajorisphysics andhegotanBachelorof SciencedegreeinJuly2004.Hethenwenttograduateschooli nAugust2004,inthe DepartmentofPhysicsattheUniversityofFlorida.Hisgrad uateworkwasontheoretical condensedmatterphysics,ormorespecically,thespindep endentphenomenain semiconductorphysics,underthesupervisionofDr.Christ opherStanton.XingyuanPan wasmarriedtoShuyangGu,whoisXingyuan'shighschoolclas smateontheChristmas dayoftheyear2007. 112