Intended Consequences Hemant Taneja
https://ebookmass.com/product/intended-consequences-hemant-taneja/
ebookmass.com
OpticalPropertiesofMaterialsandTheirApplications
WileySeriesinMaterialsforElectronicand OptoelectronicApplications
www.wiley.com/go/meoa
SeriesEditors
ProfessorArthurWilloughby, UniversityofSouthampton,Southampton,UK
DrPeterCapper, Ex-LeonardoMWLtd,Southampton,UK
ProfessorSofaKasap, UniversityofSaskatchewan,Saskatoon,Canada
PublishedTitles
BulkCrystalGrowthofElectronic,OpticalandOptoelectronicMaterials,EditedbyP.Capper PropertiesofGroup-IV,III—VandII—VISemiconductors,S.Adachi ChargeTransportinDisorderedSolidswithApplicationsinElectronics,EditedbyS.Baranovski OpticalPropertiesofCondensedMatterandApplications,EditedbyJ.Singh ThinFilmSolarCells:Fabrication,Characterization,andApplications,EditedbyJ.PoortmansandV. Arkhipov
DielectricFilmsforAdvancedMicroelectronics,EditedbyM.R.Baklanov,M.Green,andK.Maex LiquidPhaseEpitaxyofElectronic,OpticalandOptoelectronicMaterials,EditedbyP.CapperandM.Mauk MolecularElectronics:FromPrinciplestoPractice,M.Petty LuminescentMaterialsandApplications,A.Kitai CVDDiamondforElectronicDevicesandSensors,EditedbyR.S.Sussmann PropertiesofSemiconductorAlloys:Group-IV,III—VandII—VISemiconductors,S.Adachi MercuryCadmiumTelluride,EditedbyP.CapperandJ.Garland ZincOxideMaterialsforElectronicandOptoelectronicDeviceApplications,EditedbyC.Litton,D.C. Reynolds,andT.C.Collins
Lead-FreeSolders:MaterialsReliabilityforElectronics,EditedbyK.N.Subramunian SiliconPhotonics:FundamentalsandDevices,M.JamalDeenandP.K.Basu NanostructuredandSubwavelengthWaveguides:FundamentalsandApplications,M.Skorobogatiy PhotovoltaicMaterials:FromCrystallineSilicontoThird-GenerationApproaches,EditedbyG.Conibeer andA.Willoughby GlancingAngleDepositionofThinFilms:EngineeringtheNanoscale,MatthewM.Hawkeye,MichaelT. Taschuk,andMichaelJ.Brett
PhysicalPropertiesofHigh-TemperatureSuperconductors,R.Wesche SpintronicsforNextGenerationInnovativeDevices,EditedbyKatsuakiSatoandEijiSaitoh InorganicGlassesforPhotonics:Fundamentals,EngineeringandApplications,AnimeshJha AmorphousSemiconductors:Structural,OpticalandElectronicProperties,KazuoMorigaki,SandorKugler, andKoichiShimakawa
MicrowaveMaterialsandApplications,Twovolumeset,EditedbyMailadilT.Sebastian,RickUbic,andHeli Jantunen
MolecularBeamEpitaxy:MaterialsandApplicationsforElectronicsandOptoelectronics,EditedbyHajime AsahiandYoshijiKorikoshi
MetalorganicVaporPhaseEpitaxy(MOVPE):Growth,MaterialsProperties,andApplications,Editedby StuartIrvineandPeterCapper
OpticalPropertiesofMaterialsandTheir Applications
Editedby JaiSingh
CollegeofEngineering,ITandEnvironment
CharlesDarwinUniversity,Darwin,Australia
SecondEdition
Thiseditionfirstpublished2020
©2020JohnWiley&SonsLtd
EditionHistory
JohnWiley&SonsInc.(1e,2006)
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,in anyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermittedby law.Adviceonhowtoobtainpermissiontoreusematerialfromthistitleisavailableathttp://www.wiley.com/go/ permissions.
TherightofJaiSinghtobeidentifiedastheauthoroftheeditorialmaterialinthisworkhasbeenassertedin accordancewithlaw.
RegisteredOffices
JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA
JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UK
EditorialOffice
TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UK
Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationaboutWileyproductsvisitusat www.wiley.com.
Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand.Somecontentthat appearsinstandardprintversionsofthisbookmaynotbeavailableinotherformats.
LimitofLiability/DisclaimerofWarranty
Inviewofongoingresearch,equipmentmodifications,changesingovernmentalregulations,andtheconstantflow ofinformationrelatingtotheuseofexperimentalreagents,equipment,anddevices,thereaderisurgedtoreview andevaluatetheinformationprovidedinthepackageinsertorinstructionsforeachchemical,pieceofequipment, reagent,ordevicefor,amongotherthings,anychangesintheinstructionsorindicationofusageandforadded warningsandprecautions.Whilethepublisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,they makenorepresentationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthiswork andspecificallydisclaimallwarranties,includingwithoutlimitationanyimpliedwarrantiesofmerchantabilityor fitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysalesrepresentatives,writtensales materialsorpromotionalstatementsforthiswork.Thefactthatanorganization,website,orproductisreferredto inthisworkasacitationand/orpotentialsourceoffurtherinformationdoesnotmeanthatthepublisherand authorsendorsetheinformationorservicestheorganization,website,orproductmayprovideorrecommendations itmaymake.Thisworkissoldwiththeunderstandingthatthepublisherisnotengagedinrenderingprofessional services.Theadviceandstrategiescontainedhereinmaynotbesuitableforyoursituation.Youshouldconsultwith aspecialistwhereappropriate.Further,readersshouldbeawarethatwebsiteslistedinthisworkmayhavechanged ordisappearedbetweenwhenthisworkwaswrittenandwhenitisread.Neitherthepublishernorauthorsshallbe liableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedtospecial,incidental, consequential,orotherdamages.
LibraryofCongressCataloging-in-PublicationData
Names:Singh,Jai,editor.
Title:Opticalpropertiesofmaterialsandtheirapplications/editedby JaiSingh(CollegeofEngineering,IT,andEnvironment,CharlesDarwin University,Darwin,Australia)
Othertitles:Opticalpropertiesofcondensedmatterandapplications.| Opticalpropertiesofcondensedmatterandapplications.
Description:Secondedition.|Hoboken,NJ:JohnWiley&Sons,2020.| Series:Wileyseriesinmaterialsforelectronicandoptoelectronic applications|Previousedition:Opticalpropertiesofcondensedmatter andapplications,2006.|Includesbibliographicalreferencesandindex. Identifiers:LCCN2019023895(print)|LCCN2019023896(ebook)|ISBN 9781119506317(cloth)|ISBN9781119506065(adobepdf)|ISBN 9781119506058(epub)
Subjects:LCSH:Condensedmatter–Opticalproperties.|Materials–Optical properties.|Electrooptics–Materials.
Classification:LCCQC173.458.O66O682020(print)|LCCQC173.458.O66 (ebook)|DDC530.4/12–dc23
LCrecordavailableathttps://lccn.loc.gov/2019023895
LCebookrecordavailableathttps://lccn.loc.gov/2019023896
CoverDesign:Wiley
CoverImages:©mitchFOTO/Shutterstock Setin10/12ptWarnockProbySPiGlobal,Chennai,India
10987654321
Contents
ListofContributors xv
SeriesPreface xvii
Preface xix
1FundamentalOpticalPropertiesofMaterialsI1
S.O.Kasap,W.C.Tan,JaiSingh,andAsimK.Ray
1.1Introduction1
1.2OpticalConstants n and K 2
1.2.1RefractiveIndexandExtinctionCoefficient2
1.2.2 n and K ,andKramers–KronigRelations5
1.3RefractiveIndexandDispersion7
1.3.1CauchyDispersionRelation7
1.3.2SellmeierEquation8
1.3.3RefractiveIndexofSemiconductors10
1.3.3.1RefractiveIndexofCrystallineSemiconductors10
1.3.3.2BandgapandTemperatureDependence11
1.3.4RefractiveIndexofGlasses11
1.3.5Wemple–DiDomenicoDispersionRelation14
1.3.6GroupIndex15
1.4TheSwanepoelTechnique:Measurementof n and �� forThinFilms onSubstrates16
1.4.1UniformThicknessFilms16
1.4.2ThinFilmswithNon-uniformThickness22
1.5TransmittanceandReflectanceofaPartiallyTransparentPlate25
1.6OpticalPropertiesandDiffuseReflection:Schuster–Kubelka–Munk Theory27
1.7Conclusions31 Acknowledgments31 References32
2FundamentalOpticalPropertiesofMaterialsII37
S.O.Kasap,K.Koughia,JaiSingh,HarryE.Ruda,andAsimK.Ray 2.1Introduction37
2.2LatticeorReststrahlenAbsorptionandInfraredReflection40 2.3FreeCarrierAbsorption(FCA)42
2.4Band-to-BandorFundamentalAbsorption(CrystallineSolids)45
2.5ImpurityAbsorptionandRare-EarthIons48
2.6EffectofExternalFields54
2.6.1Electro-OpticEffects54
2.6.2Electro-AbsorptionandFranz–KeldyshEffect55
2.6.3FaradayEffect56
2.7EffectiveMediumApproximations58 2.8Conclusions61 Acknowledgments61 References62
3OpticalPropertiesofDisorderedCondensedMatter67 KoichiShimakawa,JaiSingh,andS.K.O’Leary
3.1Introduction67
3.2FundamentalOpticalAbsorption(Experimental)69
3.2.1AmorphousChalcogenides69
3.2.2HydrogenatedNano-CrystallineSilicon(nc-Si:H)72
3.3AbsorptionCoefficient(Theory)74
3.4CompositionalVariationoftheOpticalBandgap79
3.4.1InAmorphousChalcogenides79 3.5Conclusions80 References80
4OpticalPropertiesofGlasses83 AndrewEdgar
4.1Introduction83
4.2TheRefractiveIndex84 4.3GlassInterfaces86 4.4Dispersion88
4.5SensitivityoftheRefractiveIndex90
4.5.1TemperatureDependence90
4.5.2StressDependence91
4.5.3MagneticFieldDependence—TheFaradayEffect92
4.5.4ChemicalPerturbations—MolarRefractivity94 4.6GlassColor95
4.6.1ColorationbyColloidalMetalsandSemiconductors95
4.6.2OpticalAbsorptioninRare-Earth-DopedGlass96
4.6.3Absorptionby3dMetalIons99
4.7FluorescenceinRare-Earth-DopedGlass102
4.8GlassesforFiberOptics104
4.9RefractiveIndexEngineering106
4.10GlassandGlass–FiberLasersandAmplifiers109
4.11ValenceChangeGlasses111
4.12TransparentGlassCeramics114
4.12.1Introduction114
4.12.2TheoreticalBasisforTransparency116
4.12.3Rare-Earth-DopedTransparentGlassCeramicsforActive Photonics120
4.12.4FerroelectricTransparentGlassCeramics121
4.12.5TransparentGlassCeramicsforX-rayStoragePhosphors121
4.13Conclusions124 References124
5ConceptofExcitons129
JaiSingh,HarryE.Ruda,M.R.Narayan,andD.Ompong
5.1Introduction129
5.2ExcitonsinCrystallineSolids130
5.2.1ExcitonicAbsorptioninCrystallineSolids133
5.3ExcitonsinAmorphousSemiconductors135
5.3.1ExcitonicAbsorptioninAmorphousSolids137
5.4ExcitonsinOrganicSemiconductors139
5.4.1PhotoexcitationandFormationofExcitons140
5.4.1.1PhotoexcitationofSingletExcitonsDueto Exciton–PhotonInteraction141
5.4.1.2ExcitationofTripletExcitons142
5.4.2ExcitonUp-Conversion147
5.4.3ExcitonDissociation148
5.4.3.1ConversionfromFrenkeltoCTExcitons151
5.4.3.2DissociationofCTExcitons152
5.5Conclusions153 References154
6Photoluminescence157
TakeshiAoki
6.1Introduction157
6.2FundamentalAspectsofPhotoluminescence(PL)inMaterials158
6.2.1IntrinsicPhotoluminescence159
6.2.2ExtrinsicPhotoluminescence160
6.2.3Up-ConversionPhotoluminescence(UCPL)162
6.2.4OtherRelatedOpticalTransitions163
6.3ExperimentalAspects164
6.3.1StaticPLSpectroscopy164
6.3.2PhotoluminescenceExcitationSpectroscopy(PLE)and PhotoluminescenceAbsorptionSpectroscopy(PLAS)167
6.3.3TimeResolvedSpectroscopy(TRS)168
6.3.4Time-CorrelatedSinglePhotonCounting(TCSPC)171
6.3.5Frequency-ResolvedSpectroscopy(FRS)172
6.3.6QuadratureFrequencyResolvedSpectroscopy(QFRS)173
6.4PhotoluminescenceLifetimeSpectroscopyofAmorphous SemiconductorsbyQFRSTechnique175
6.4.1Overview175
6.4.2Dual-PhaseDoubleLock-in(DPDL)QFRSTechnique176
6.4.3ExploringBroadPLLifetimeDistributionina-Si:Hby WidebandQFRS178
6.4.3.1EffectsofExcitationIntensity,Excitation,and EmissionEnergies179
6.4.3.2TemperatureDependence184
6.4.3.3EffectofElectricandMagneticFields185
6.4.4ResidualPLDecayofa-Si:H189
6.5QFRSonUp-ConversionPhotoluminescence(UCPL)ofRE-Doped Materials192
6.6Conclusions197 Acknowledgments198 References198
7Photoluminescence,PhotoinducedChanges,andElectroluminescencein NoncrystallineSemiconductors203 JaiSingh 7.1Introduction203
7.2Photoluminescence205
7.2.1RadiativeRecombinationOperatorandTransitionMatrix Element206
7.2.2RatesofSpontaneousEmission211
7.2.2.1AtNonthermalEquilibrium212
7.2.2.2AtThermalEquilibrium214
7.2.2.3Determining E 0 215
7.2.3ResultsofSpontaneousEmissionandRadiativeLifetime216
7.2.4TemperatureDependenceofPL222
7.2.5ExcitonicConcept223
7.3PhotoinducedChangesinAmorphousChalcogenides225
7.3.1EffectofPhoto-ExcitationandPhononInteraction226
7.3.2ExcitationofaSingleElectron–HolePair228
7.3.3PairingofLikeExcitedChargeCarriers229
7.4RadiativeRecombinationofExcitonsinOrganicSemiconductors232
7.4.1RateofFluorescence233
7.4.2RateofPhosphorescence233
7.4.3OrganicLightEmittingDiodes(OLEDs)234
7.4.3.1Second-andThird-GenerationOLEDs:TADF235
7.5Conclusions236 Acknowledgments236 References237
8PhotoinducedBondBreakingandVolumeChangeinChalcogenideGlasses241 SandorKugler,RozáliaLukács,andKoichiShimakawa
8.1Introduction241
8.2Atomic-ScaleComputerSimulationsofPhotoinducedVolume Changes243
8.3EffectofIllumination244
8.4KineticsofVolumeChange245
8.4.1a-Se245
8.4.2a-As2 Se3 246
8.5AdditionalRemarks248 8.6Conclusions249 References249
9PropertiesandApplicationsofPhotonicCrystals251 HarryE.RudaandNaomiMatsuura
9.1Introduction251
9.2PCOverview252
9.2.1IntroductiontoPCs252
9.2.2NanoengineeringofPCArchitectures253
9.2.3MaterialsSelectionforPCs255
9.3TunablePCs255
9.3.1TuningPCResponsebyChangingtheRefractiveIndexof ConstituentMaterials256
9.3.1.1PCRefractiveIndexTuningUsingLight256
9.3.1.2PCRefractiveIndexTuningUsinganApplied ElectricField256
9.3.1.3RefractiveIndexTuningofInfiltratedPCs257
9.3.1.4PCRefractiveIndexTuningbyAlteringthe ConcentrationofFreeCarriers(UsingElectric FieldorTemperature)inSemiconductor-BasedPCs257
9.3.2TuningPCResponsebyAlteringthePhysicalStructure ofthePC258
9.3.2.1TuningPCResponseUsingTemperature258
9.3.2.2TuningPCResponseUsingMagnetism258
9.3.2.3TuningPCResponseUsingStrain258
9.3.2.4TuningPCResponseUsingPiezoelectricEffects259
9.3.2.5TuningPCResponseUsingMEMSActuation260
9.4SelectedApplicationsofPC260
9.4.1WaveguideDevices261
9.4.2DispersiveDevices262
9.4.3Add/DropMultiplexingDevices262
9.4.4ApplicationsofPCsforLight-EmittingDiodes(LEDs)and Lasers263
9.5Conclusions265 Acknowledgments265 References265
10NonlinearOpticalPropertiesofPhotonicGlasses269 KeijiTanaka
10.1Introduction269 10.2PhotonicGlass271
10.3NonlinearAbsorptionandRefractivity272
10.3.1Fundamentals272
10.3.2Two-PhotonAbsorption275
x Contents
10.3.3NonlinearRefractivity278 10.4NonlinearExcitation-InducedStructuralChanges280 10.4.1Fundamentals280 10.4.2Oxides281
10.4.3Chalcogenides283 10.5Conclusions285
10.AAddendum:PerspectivesonOpticalDevices286 References288
11OpticalPropertiesofOrganicSemiconductors295
TakashiKobayashiandHiroyoshiNaito 11.1Introduction295
11.2MolecularStructureof π-ConjugatedPolymers296 11.3TheoreticalModels298
11.4AbsorptionSpectrum300 11.5Photoluminescence304 11.6Non-EmissiveExcitedStates306
11.7Electron–ElectronInteraction309 11.8InterchainInteraction314 11.9Conclusions320 References321
12OrganicSemiconductorsandApplications323 FurongZhu
12.1Introduction323
12.1.1DeviceArchitectureandOperationPrinciple324
12.1.2TechnicalChallengesandProcessIntegration325
12.2AnodeModificationforEnhancedOLEDPerformance327
12.2.1Low-TemperatureHigh-PerformanceITO327
12.2.1.1ExperimentalMethods328
12.2.1.2MorphologicalProperties329
12.2.1.3ElectricalProperties331
12.2.1.4OpticalProperties333
12.2.1.5CompositionalAnalysis336
12.2.2AnodeModification339
12.2.3ElectroluminescencePerformanceofOLEDs340
12.3FlexibleOLEDs345
12.3.1FlexibleOLEDsonUltrathinGlassSubstrate346
12.3.2FlexibleTop-EmittingOLEDsonPlasticFoils347
12.3.2.1Top-EmittingOLEDs348
12.3.2.2FlexibleTOLEDsonPlasticFoils350
12.4Solution-ProcessableHigh-PerformingOLEDs353
12.4.1PerformanceofOLEDswithaHybridMoO3 -PEDOT:PSS HoleInjectionLayer(HIL)353
12.4.2MorphologicalPropertiesoftheMoO3 -PEDOT:PSSHIL361
12.4.3SurfaceElectronicPropertiesofMoO3 -PEDOT:PSSHIL363 12.5Conclusions368 References369
13TransparentWhiteOLEDs373 ChoiWingHongandFurongZhu
13.1Introduction—ProgressinTransparentWOLEDs373 13.2PerformanceofWOLEDs374
13.2.1OptimizationofDichromaticWOLEDs374
13.2.2 J -L-V CharacteristicsofWOLEDs377
13.2.3Electron-HoleCurrentBalanceinTransparentWOLEDs384
13.3EmissionBehaviorofTransparentWOLEDs386
13.3.1Visible-LightTransparencyofWOLEDs386
13.3.2 L-J CharacteristicsofTransparentWOLEDs390
13.3.3Angular-DependentColorStabilityofTransparentWOLEDs395 13.4Conclusions400 References400
14OpticalPropertiesofThinFilms403
V.-V.Truong,S.Tanemura,A.Haché,andL.Miao
14.1Introduction403 14.2OpticsofThinFilms404
14.2.1AnIsotropicFilmonaSubstrate404
14.2.2MatrixMethodsforMulti-LayeredStructures406
14.2.3AnisotropicFilms407
14.3Reflection-TransmissionPhotoellipsometryforDeterminationof OpticalConstants408
14.3.1PhotoellipsometryofaThickoraThinFilm408
14.3.2PhotoellipsometryforaStackofThickandThinFilms410
14.3.3RemarksontheReflection-TransmissionPhotoellipsometry Method412
14.4ApplicationofThinFilmstoEnergyManagementand Renewable-EnergyTechnologies412
14.4.1ElectrochromicThinFilms413
14.4.2PureandMetal-DopedVO2 ThermochromicThinFilms414
14.4.3Temperature-StabilizedV1-x Wx O2 SkyRadiatorFilms417 14.4.4OpticalFunctionalTiO2 ThinFilmforEnvironmentally FriendlyTechnologies420
14.5ApplicationofTunableThinFilmstoPhaseandPolarization Modulation424 14.6Conclusions430 References430
15OpticalCharacterizationofMaterialsbySpectroscopicEllipsometry435 J.Mistrík 15.1Introduction435
15.2NotionsofLightPolarization436
15.3MeasureableQuantities438 15.4Instrumentation441
15.5SingleInterface442 15.6SingleLayer448 15.7Multilayer454
15.8LinearGrating458
15.9Conclusions462 Acknowledgments463 References463
16ExcitonicProcessesinQuantumWells465 JaiSinghandI.-K.Oh
16.1Introduction465
16.2Exciton–PhononInteraction466
16.3ExcitonFormationinQWsAssistedbyPhonons467 16.4NonradiativeRelaxationofFreeExcitons474
16.4.1IntrabandProcesses475
16.4.2InterbandProcesses479
16.5Quasi-2DFree-ExcitonLinewidth485 16.6LocalizationofFreeExcitons491 16.7Conclusions499 References500
17OptoelectronicPropertiesandApplicationsofQuantumDots503 JørnM.Hvam
17.1Introduction503
17.2EpitaxialGrowthandStructureofQuantumDots504
17.2.1Self-AssembledQuantumDots504
17.2.2Site-ControlledGrowthonPatternedSubstrates505
17.2.3NaturalorInterfaceQuantumDots506
17.2.4QuantumDotsinNanowires507
17.3ExcitonsinQuantumDots508
17.3.1Quantum-DotBandgap509
17.3.2OpticalTransitions510
17.4OpticalProperties513
17.4.1RadiativeLifetime,OscillatorStrength,andInternal QuantumEfficiency514
17.4.2Linewidth,Coherence,andDephasing516
17.4.3TransientFour-WaveMixing517
17.5QuantumDotApplications520
17.5.1QuantumDotLasersandOpticalAmplifiers520
17.5.1.1GainDynamics522
17.5.1.2HomogeneousBroadeningandDephasing524
17.5.1.3Long-WavelengthLasers526
17.5.1.4NanoLasers527
17.5.2Single-PhotonEmitters527
17.5.2.1MicropillarsandNanowires530
17.5.2.2PhotonicCrystalWaveguide531 17.6Conclusions533 Acknowledgments534 References534
18Perovskites–RevisitingtheVenerableABX3 FamilywithOrganic FlexibilityandNewApplications537 JunweiXu,D.L.Carroll,K.Biswas,F.Moretti,S.Gridin,andR.T.Williams 18.1Introduction537
18.1.1Review537
18.1.2TheStructures538 18.1.2.1SimpleCubicFrameworks538 18.1.2.2TheMultiplicityofHybrids539 18.1.2.3StructuralVariation540
18.2HybridPerovskitesinPhotovoltaics544 18.2.1Review544 18.2.2ThePhenomenaCharacterizedas“DefectTolerance”548 18.3Light-EmittingDiodesUsingSolution-ProcessedLeadHalide Perovskites549
18.3.1Review549 18.3.2ConstructionandCharacterizationofLEDsUtilizing CsPbBr3 Nano-InclusionsinCs4 PbBr6 asthe ElectroluminescentMedium553 18.4IonizingRadiationDetectorsUsingLeadHalidePerovskite Materials:Basics,Progress,andProspects562 18.5Conclusions582 Acknowledgments583 References583
19OpticalPropertiesandSpinDynamicsofDilutedMagneticSemiconductor Nanostructures589
AkihiroMurayamaandYasuoOka 19.1Introduction589 19.2QuantumWells591 19.2.1SpinInjection591 19.2.2StudyofSpinDynamicsbyPump-ProbeSpectroscopy594 19.3FabricationofNanostructuresbyElectron-BeamLithography596 19.4Self-AssembledQuantumDots599 19.5HybridNanostructureswithFerromagneticMaterials604 19.6Conclusions607 Acknowledgments608 References609
20KineticsofthePersistentPhotoconductivityinCrystallineIII-V Semiconductors611
RubenJeronimoFreitasandKoichiShimakawa
20.1Introduction611
20.2AReviewofPPCinIII-VSemiconductors613
20.3KeyPhysicalTermsRelatedtoPPC615
20.3.1DispersiveReaction615
20.3.2SEFandPowerLaw616
20.3.3WaitingTimeDistribution617
20.4KineticsofPPCinIII-VSemiconductors617
20.5Conclusions623 Acknowledgments623
20.AOntheReactionRateUndertheUniformDistribution623 References625 Index 627
ListofContributors
TakeshiAoki JointResearchCenterofHigh-technology,DepartmentofElectronicsand InformationTechnology,TokyoPolytechnicUniversity,Atsugi,Japan
K.Biswas DepartmentofChemistryandPhysics,ArkansasStateUniversity,Jonesboro, USA
D.L.Carroll DepartmentofPhysicsandNanotechnologyCenter,WakeForestUniversity, Winston-Salem,NorthCarolina,USA
AndrewEdgar SchoolofChemicalandPhysicalSciences,VictoriaUniversityof Wellington,NewZealand
RubenJeronimoFreitas DepartmentofElectricalandElectronicEngineering,National UniversityofTimorLorosae,Díli,EastTimor
S.Gridin DepartmentofPhysicsandNanotechnologyCenter,WakeForestUniversity, Winston-Salem,NorthCarolina,USA
A.Haché Départementdephysiqueetd’astronomie,UniversitédeMoncton,New Brunswick,Canada
JørnM.Hvam DepartmentofPhotonicsEngineering,TechnicalUniversityofDenmark, KongensLyngby,Denmark
S.O.Kasap DepartmentofElectricalandComputerEngineering,Universityof Saskatchewan,Saskatoon,Canada
TakashiKobayashi DepartmentofPhysicsandElectronics,OsakaPrefectureUniversity, Sakai,Japan
K.Koughia DepartmentofElectricalandComputerEngineering,Universityof Saskatchewan,Saskatoon,Canada
SandorKugler DepartmentofTheoreticalPhysics,BudapestUniversityofTechnology andEconomics,Hungary
RozáliaLukács NorwegianUniversityofLifeSciences,Ås,Akershus,Norway
NaomiMatsuura CentreforNanotechnology,UniversityofToronto,Canada
L.Miao GuilinUniversityofElectronicTechnology,Guangxi,P.R.China
xvi ListofContributors
J.Mistrík CenterofMaterialsandNanotechnologies,FacultyofChemicalTechnology, UniversityofPardubice,CzechRepublic
F.Moretti LawrenceBerkeleyNationalLaboratory,Berkeley,California,USA
AkihiroMurayama GraduateSchoolofInformationScienceandTechnology,Hokkaido University,Sapporo,Japan
HiroyoshiNaito TheResearchInstituteforMolecularElectronicDevices,Osaka PrefectureUniversity,Sakai,Japan
M.R.Narayan CollegeofEngineering,InformationTechnologyandEnvironment, CharlesDarwinUniversity,Darwin,Australia
S.K.O’Leary SchoolofEngineering,TheUniversityofBritishColumbia,Kelowna, Canada
I.-K.Oh CollegeofEngineering,InformationTechnologyandEnvironment,Charles DarwinUniversity,Darwin,Australia
YasuoOka InstituteofMultidisciplinaryResearchforAdvancedMaterials,Tohoku University,Sendai,Miyagi,Japan
D.Ompong CollegeofEngineering,InformationTechnologyandEnvironment,Charles DarwinUniversity,Darwin,Australia
AsimK.Ray DepartmentofElectrical&ComputerEngineering,BrunelUniversity London,Uxbridge,UK
HarryE.Ruda CentreforNanotechnologyandElectronicandPhotonicMaterialsGroup, DepartmentofMaterialsScience,UniversityofToronto,Ontario,Canada
KoichiShimakawa DepartmentofElectricalandElectronicEngineering,GifuUniversity, Japan
JaiSingh CollegeofEngineering,InformationTechnologyandEnvironment,Charles DarwinUniversity,Darwin,Australia
W.C.Tan DepartmentofElectrical&ComputerEngineering,NationalUniversityof Singapore,KentRidge,Singapore
KeijiTanaka DepartmentofAppliedPhysics,GraduateSchoolofEngineering,Hokkaido University,Sapporo,Japan
S.Tanemura JapanFineCeramicsCentre,Mutsuno,Atsuta-ku,Nagoya,Japan
V.-V.Truong PhysicsDepartment,ConcordiaUniversity,Montreal,Quebec,Canada
R.T.Williams DepartmentofPhysicsandNanotechnologyCenter,WakeForest University,Winston-Salem,NorthCarolina,USA
ChoiWingHong,DepartmentofPhysics,HongKongBaptistUniversity,KowloonTong, China
JunweiXu DepartmentofPhysicsandNanotechnologyCenter,WakeForestUniversity, Winston-Salem,NorthCarolina,USA
FurongZhu DepartmentofPhysics,HongKongBaptistUniversity,KowloonTong,China
SeriesPreface
WileySeriesinMaterialsforElectronicandOptoelectronic Applications
Thisbookseriesisdevotedtotherapidlydevelopingclassofmaterialsusedforelectronicandoptoelectronicapplications.Itisdesignedtoprovidemuch-neededinformationonthefundamentalscientificprinciplesofthesematerials,togetherwithhow theseareemployedintechnologicalapplications.Thesebooksareaimedat(postgraduate)students,researchers,andtechnologistsengagedinresearch,development,andthe studyofmaterialsinelectronicsandphotonics,andatindustrialscientistsdeveloping newmaterials,devices,andcircuitsfortheelectronic,optoelectronic,andcommunicationsindustries.
Thedevelopmentofnewelectronicandoptoelectronicmaterialsdependsnotonly onmaterialsengineeringatapracticallevel,butalsoonaclearunderstandingofthe propertiesofmaterialsandthefundamentalsciencebehindtheseproperties.Itisthe propertiesofamaterialthateventuallydetermineitsusefulnessinanapplication.The seriesthereforealsoincludessuchtitlesaselectricalconductioninsolids,opticalproperties,thermalproperties,andsoon,allwithapplicationsandexamplesofmaterialsin electronicsandoptoelectronics.Thecharacterizationofmaterialsisalsocoveredwithin theseriesasmuchasitisimpossibletodevelopnewmaterialswithoutthepropercharacterizationoftheirstructureandproperties.Structure–propertyrelationshipshave alwaysbeenfundamentallyandintrinsicallyimportanttomaterialsscienceandengineering.
Materialsscienceiswellknownforbeingoneofthemostinterdisciplinarysciences. Itistheinterdisciplinaryaspectofmaterialssciencethathasledtomanyexcitingdiscoveries,newmaterials,andnewapplications.Itisnotunusualtofindscientistswith achemicalengineeringbackgroundworkingonmaterialsprojectswithapplicationsin electronics.Inselectingtitlesfortheseries,wehavetriedtomaintaintheinterdisciplinaryaspectofthefield,andhenceitsexcitementtoresearchersinthisfield.
ArthurWilloughby PeterCapper SafaKasap
Preface
Thesecondedition,beingpublishedmorethan10yearsafterthefirstedition,presents state-of-the-artdevelopmentsinalmostalltopicsrelatedtotheopticalpropertiesof materialsandtheirapplicationspresentedinthefirstedition.Sincethepublicationofthe firsteditionin2006,manyadvanceshavebeenmadeinfieldssuchastheopticalpropertiesofmaterials,electroluminescenceinorganiclight-emittingdevices,organicsolar cells,opto-electronicdevices,etc.Itishenceverytimelytoupdateallthechaptersin thefirsteditionbyaddingdevelopmentssince2006toproducethesecondedition.This secondeditioncontains15oftheoriginal16chapters,allofwhichhavebeenupdated, aswellas5brandnewchapters,contributedbyveryexperiencedandwell-knownscientistsandgroupsavailableondifferentaspectsoftheopticalpropertiesofmaterials.Thestudyofopticalpropertiesofmaterialshasnowbecomeaninterdisciplinary field,andscientistsofphysical,chemical,andbiologicalsciences;nanotechnologyengineers;andindustryresearchershavestronginterestsinthisfield.Thefieldoffersoneof thefastest-growingresearchplatformsinmaterialsciences.Thesecondeditioncovers manyexamplesandapplicationsinthefieldofelectronicandoptoelectronicproperties ofmaterials,andinphotonics.Mostchaptersarepresentedtoberelativelyindependentwithminimalcross-referencing,andchapterswithcomplementarycontentsare arrangedtogethertofacilitateareaderwithcross-referencing.
Bookswritteninthisfieldmostlyfollowoneofthetwopedagogies:chaptersareeither basedon(i)physicalprocesses,or(ii)thevariousclassesofmaterials.Thisbookcombinesthetwoapproachesbyfirstidentifyingtheprocessesthatshouldbedescribed indetail,andthenintroducingtherelevantclassesofmaterials.Manybooksalsomiss thedetailsofhowvariousopticalpropertiesaremeasured.Thisbookpresentsacomprehensivereviewofexperimentaltechniques,includingrecentadvancesinultrafast (femtosecond)spectroscopyofmaterials.Notmanybooksarecurrentlyavailablewith suchawidecoverageofthefieldwithclarityandlevelsofreadershipinasinglevolume asthisbook.
InChapters1and2byKasapetal.,thefundamentalopticalpropertiesofmaterials arereviewed,andassuchthesechaptersareexpectedtorefreshthereaderswiththe basicsbyprovidingusefulopticalrelations.InChapter3,Shimakawaetal.presentan up-to-datereviewoftheopticalpropertiesofdisorderedinorganicsolids,andChapter4 byEdgarpresentsanextensivediscussionontheopticalpropertiesofglasses.Chapter 5bySinghandco-workerspresentstheconceptofexcitonsininorganicandorganic semiconductors,bothcrystallineandnon-crystallinevariants.InChapter6,Aokihas presentedacomprehensivereviewoftheexperimentaladvancesinthetechniquesof
measuringphotoluminescencetogetherwithupdatesinluminescenceresultsinamorphoussemiconductors,andChapter7bySinghcomplementsthetheoreticaladvances inthefieldofphotoluminescenceandphotoinducedchangesinnon-crystallinesemiconductors.InChapter8byKugleretal.,recentadvancesinthesimulationofphotoinducedbondbreakingandvolumechangesinchalcogenideglassesarepresented.In Chapter9,RudaandMatsuurapresentacomprehensivereviewofthepropertiesand applicationsofphotoniccrystals.InChapter10,Tanakahaspresentedanup-to-date reviewofthenonlinearopticalpropertiesofphotonicglasses.
Chapter11byKobayashiandNaitodiscussesthefundamentalopticalpropertiesof organicsemiconductors.InChapter12,Zhuhaspresentedacomprehensivereview oftheapplicationsoforganicsemiconductors,inparticular,indevelopingorganic light-emittingdiodes(OLEDs).InChapter13,HongandZhuhavereviewedtherecent developmentsinthefabricationoftransparentwhitelight-emittingdiodes(WOLEDs). Thisisanewchapteraddedinthesecondedition.InChapter14,TruongandTanemura havepresentedanup-to-datereviewoftheopticalpropertiesofthinfilmsandtheir applications,andChapter15byMistrikdealswiththeopticalcharacterizationof materialsbyspectroscopicellipsometry.Thisisthesecondnewchapterinthesecond edition.InChapter16,SinghandOhhavediscussedtheexcitonicprocessesinquantum wells.InChapter17,thethirdnewchapterinthisedition,Hvamhaspresentedan up-to-datecomprehensivereviewoftheoptoelectronicpropertiesandapplications ofquantumdots.Chapter18byXuetal.presentsup-to-datedevelopmentsinthe applicationsofperovskites.Thisisthefourthnewchapterinthesecondedition. InChapter19,MurayamaandOkahavepresentedtheopticalpropertiesandspin dynamicsofdilutedmagneticsemiconductornanostructures.InthefinalChapter20, thefifthnewchapterinthisedition,FreitasandShimakawahavediscussedthekinetics ofthepersistentphotoconductivityinCrystallineIII–Vsemiconductors.Thus,the additionofthefivenewchaptersontransparentWOLELDs,ellipsometry,quantum dots,perovskites,andpersistentphotoconductivitywidensthescopeofthesecond editiontoanewlevel.Oneofthechaptersonthenegativeindexofrefractioninthe firsteditionhasnotbeenincludedinthesecondeditionattherequestoftheauthors.
Thereadershipofthebookisexpectedtobetheseniorundergraduateandpostgraduatestudents,andteachingandresearchprofessionalsinthefield.Inconclusion,I amverygratefultoallthecontributingauthorsofthesecondeditionfortheirutmost co-operationinmeetingthedeadlines,withoutwhichthisprojectwouldnothaveconcluded.IalsowouldliketoacknowledgethetechnicalsupportfromDrsStefanijaKlaric andLuisHerreraDiazinpreparingmychapters.Iwouldalsoliketothankmyfriend BethWoofforhersupportthroughoutthecourseofpreparationofthisvolume. JaiSingh
Darwin,Australia
FundamentalOpticalPropertiesofMaterialsI
S.O.Kasap 1 ,W.C.Tan 2 ,JaiSingh 3 ,andAsimK.Ray 4
1 DepartmentofElectricalandComputerEngineering,UniversityofSaskatchewan,57CampusDrive,Saskatoon,Canada
2 DepartmentofElectrical&ComputerEngineering,NationalUniversityofSingapore,KentRidge,Singapore
3 CollegeofEngineering,ITandEnvironment,Purple12,CharlesDarwinUniversity,EllengowanDrive,Darwin,Australia
4 DepartmentofElectrical&ComputerEngineering,BrunelUniversityLondon,KingstonLane,Uxbridge,UK
CHAPTERMENU
Introduction,1
OpticalConstants n and K ,2 RefractiveIndexandDispersion,7 TheSwanepoelTechnique:Measurementof n and �� forThinFilmsonSubstrates,16 TransmittanceandReflectanceofaPartiallyTransparentPlate,25 OpticalPropertiesandDiffuseReflection:Schuster–Kubelka–MunkTheory,27 Conclusions,31 References,32
1.1Introduction
Opticalpropertiesofamaterialchangeoraffectthecharacteristicsoflightpassing throughitbymodifyingitspropagationvectororintensity.Twoofthemostimportant opticalparametersaretherefractiveindex n andtheextinctioncoefficient K ,whichare genericallycalled opticalconstants,althoughsomeauthorsincludeotheropticalcoefficientswithinthisterminology.Thelatterisrelatedtotheattenuationorabsorptioncoefficient �� .InPartI,inthischapter,wepresentthecomplexrefractiveindex,thefrequency orwavelengthdependenceof n and K ,so-calleddispersionrelations,how n and K are inter-related,andhow n and K canbedeterminedbystudyingthetransmissionasa functionofwavelengththroughathinfilmofthematerial.Physicalinsightsinto n and K areprovidedinPartII(Chapter2).Inaddition,therehasbeenastrongresearchinterestincharacterizingtheopticalpropertiesofinhomogeneousmedia,suchasporous media,inwhichbothlightabsorptionandscatteringtakeplacesothatthereflectance isnotspecularbutdiffuse.Thelatterproblemisnowincludedinthissecondedition. Theopticalpropertiesofvariousmaterials,with n and K beingthemostimportant, areavailableintheliteratureinoneformoranother,eitherpublishedinjournals, books,andhandbooks,orpostedonwebsitesofvariousresearchers,organizations (e.g.NIST),orcompanies(e.g.SchottGlass).Nonetheless,thereaderisreferredtothe OpticalPropertiesofMaterialsandTheirApplications, SecondEdition.EditedbyJaiSingh. ©2020JohnWiley&SonsLtd.Published2020byJohnWiley&SonsLtd.
worksofGreenwayandHarbeke[1],Wolfe[2],Klocek[3],Palik[4,5],Ward[6], Efimov[7],PalikandGhosh[8],Nikogosyan[9],andWeaverandFrederikse[10] fortheopticalpropertiesofawiderangeofmaterials.Adachi’sbooksontheoptical constantsofsemiconductorsarehighlyrecommended[11–13],alongwithMadelung’s thirdeditionof Semiconductors:DataHandbook [14].Thereare,ofcourse,otherbooks andhandbooksthatalsocontainopticalconstantsinvariouschapters;see,forexample, references[15–20].Therearealsovariousbooksthatdescribeopticalproperties ofsolidsattheseniorundergraduateandintroductorygraduatelevels,suchasthose byTanner[21],JimenezandTomm[22],Stenzel[23],Fox[24],SimmonsandPotter [25],Toyozawa[26],Wooten[27],andAbeles[28],whicharehighlyrecommended.
Anumberofexperimentaltechniquesareavailableformeasuring n and K ,some ofwhichhavebeensummarizedbySimmonsandPotter[25].Forexample,ellipsometrymeasureschangesinthepolarizationoflightincidentonasampletosensitively characterizesurfacesandthinfilms(seeChapter23inthisvolume).Theinteraction ofincidentpolarizedlightwiththesamplecausesapolarizationchangeinthelight, whichmaythenbemeasuredbyanalyzingthelightreflectedfromthesample.Collins hasalsoprovidedanextensivein-depthreviewofellipsometryforopticalmeasurements[29].Oneofthemostpopularandconvenientopticalexperimentsinvolvesa monochromaticlightpassingthroughathinsample,andmeasuringthetransmitted intensityasafunctionofwavelength, T (��),usingasimplespectrophotometer.Forthin samplesonathicktransparentsubstrate,thetransmissionspectrumshowsoscillations in T (��)withthewavelengthduetointerferenceswithinthethinfilm.Swanepoel’stechniqueusesthe T (��)measurementtodetermine n and K ,asdescribedinSection1.4.
1.2OpticalConstants n and K
Oneofthemostimportantopticalconstantsofamaterialisitsrefractiveindex,whichin generaldependsonthewavelengthoftheelectromagnetic(EM)wave,througharelationshipcalled dispersion.InmaterialswhereanEMwavelosesitsenergyduringits propagation,therefractiveindexbecomescomplex.Therealpartisusuallytherefractiveindex, n,andtheimaginarypartiscalledthe extinctioncoefficient , K .Inthissection, therefractiveindexandextinctioncoefficientwillbepresentedindetail,alongwith somecommondispersionrelations.Amorepracticalandasemiquantitativeapproach istakenalongthelinesin[30]ratherthanafulldedicationtorigorandmathematical derivations.Moreanalyticalapproachescanbefoundinothertexts,suchas[25,26].
1.2.1RefractiveIndexandExtinctionCoefficient
Therefractiveindexofanopticalordielectricmedium, n,istheratioofthevelocity oflight c invacuumtoitsvelocity v inthemedium; n = c/v.UsingthisandMaxwell’s equations,oneobtainsthewell-knownMaxwell’sformulafortherefractiveindexofa substanceas n = √��r ��r ,where ��r isthestaticdielectricconstantorrelativepermittivityand �� r therelativemagneticpermeabilityofthemedium.As �� r = 1fornonmagnetic substances,onegets n = √��r ,whichisveryusefulinrelatingthedielectricpropertiesto opticalpropertiesofmaterialsatanyparticularfrequencyofinterest.As ��r dependson thewavelengthoflight,therefractiveindexalsodependsonthewavelengthoflight,and
thisdependenceiscalled dispersion.Inadditiontodispersion,anEMwavepropagating throughalossymediumexperiencesattenuation,whichmeansitlosesitsenergy,dueto variouslossmechanismssuchasthegenerationofphonons(latticewaves),photogeneration,freecarrierabsorption,scattering,etc.Insuchmaterials,therefractiveindex becomesacomplexfunctionofthefrequencyofthelightwave.Thecomplexrefractive indexinthischapterisdenotedby n* ,withrealpart n,andimaginarypart K ,calledthe extinctioncoefficient ,isrelatedtothecomplexrelativepermittivity, ��
where ��′ r and ��′′
are,respectively,therealandimaginarypartsof
Inexplicitterms, n and K canbeobtainedas
Somebooks(particularlyinelectricalengineering)use ��r = ��′ r i
and
iK insteadof ��
+
and n
= n + iK .Thepreferenceliesinwhatwasassumedforthe propagatingelectricfield,whetheritisrepresentedbyexpi(��t kx)orexpi(kx ��t ), where k isthepropagationconstant.Inalossymedium,theimaginarypartof n*must leadtoatravelingwavewhoseamplitudedecays.Noticethat,for ��′′
≪��
, n
��′ r and K = ��′′ r ∕2n—thatis,therefractiveindexisessentiallydeterminedbytherealpartof ��r and K isdeterminedbytheimaginarypartof ��r ,whichisknowntorepresentlossesin adielectricmedium.
Theextinctioncoefficient K representslossfromtheenergycarriedbythepropagatingEMwavebyconvenientlyincludingthislossastheimaginarypartinthecomplex refractiveindex.Theopticalattenuationcoefficient �� gaugestherateofthislossfrom thepropagatingEMwave.Intheabsenceofscattering,theattenuationwouldbedue toabsorptionwithinthemedium.ForanEMwavethatispropagatingalong x withan intensity I , �� isdefinedby
Wecanrelate �� and K quiteeasilybytakingaplanewavetravelingalong x forwhich theelectricfieldinthewavepropagatesas E = E o expi(kx ��t ),where E o isaconstant, �� istheangularfrequencyand k isthecomplexpropagationconstantinthemedium, relatedto n*byitsdefinition k = n*��/c = (n + iK )(��/c).Infreespace k = k o = ��/c = 2�� /��, where �� isthefreespacewavelength.Wecansubstitutefor n*andthenuse I isproportionalto|E |2 tofind I ∝ exp[ 2(��/c)Kx)]—thatis, I decaysexponentiallywiththe distancepropagated.Wecansubstitutefor I in(1.3)tofind
Theopticalconstants n and K canbedeterminedbymeasuringthereflectancefrom thesurfaceofamaterialasafunctionofpolarizationandtheangleofincidence.For normalincidence,thereflectioncoefficient, r ,isobtainedas
Thereflectance R isthendefinedby:
Noticethatwhenever K islarge,forexample,overarangeofwavelengths,theabsorptionisstrong,andthereflectanceisalmostunity.Thelightisthenreflected,andany lightinthemediumishighlyattenuated(typicalsamplecalculationsmaybefoundin [24,30]).
Opticalpropertiesofmaterialsaretypicallypresentedeitherbyshowingthefrequency dependences(dispersionrelations)of n and K or ��′ r and ��′′ r .Anintuitiveguidetoexplainingdispersionininsulatorsisbasedonasingleoscillatormodelinwhichtheelectric fieldinthelightinducesforceddipoleoscillationsinthematerial(displacestheelectron shellsinanatomtooscillateaboutthepositivenucleus)withasingleresonantfrequency ��o .Thefrequencydependencesof ��′ r and ��′′ r arethenobtainedas:
where N at isthenumberofatomsperunitvolume, ��o isthevacuumpermittivity,and �� ′ e and �� ′′ e are,respectively,therealandimaginarypartsoftheelectronicpolarizability, givenrespectivelyby:
where �� eo istheDCpolarizabilitycorrespondingto �� = 0and �� isthelosscoefficientthat characterizestheEMwavelosseswithinthematerialsystem.UsingEqs.(1.1)–(1.2)and (1.7)–(1.8),thefrequencydependenceof n and K canbestudied.Figure1.1ashowsthe dependenceof n and K onthenormalizedfrequency ��/��o forasimplesingleelectronic dipoleoscillatorofresonancefrequency ��o .
Figure1.1 Refractiveindex n andextinction coefficient K obtainedfromasingleelectronic dipoleoscillatormodel.(a) n and K versus normalizedfrequency,and(b)reflectance versusnormalizedfrequency.
1.2OpticalConstants n and K 5
ItisseenfromFigure1.1that n and K peakcloseto �� = ��o .Ifamaterialhasa ��′′ r ≫��′ r ,then ��r ≈ i��′′ r ,and n ≈ K ≈ √��′′ r ∕2isobtainedfromEq.(1.1b).Figure1.1bshows thedependenceofthereflectance R onthefrequency.Itisobservedthat R reachesits maximumvalueatafrequencyslightlyabove �� = ��o ,andthenremainshighuntil �� reachesnearly3��o ;thus,thereflectanceissubstantialwhileabsorptionisstrong.The normaldispersionregionisthefrequencyrangebelow ��o ,where n fallsasthefrequency decreases;thatis, n decreasesasthewavelength �� increases.Anomalousdispersion regionisthefrequencyrangeabove ��o where n decreasesas �� increases.Below ��o , K issmalland,if ��DC is ��r (0),theDCpermittivity,then
astheresonancewavelength,onegets:
Whileintuitivelyuseful,thedispersionrelationsinEq.(1.8)arefartoosimple.More rigorously,wehavetoconsiderthedipoleoscillatorquantummechanically,which meansaphotonexcitestheoscillatortoahigherenergylevel—see,forexample,Fox[24] orSimmonsandPotter[25].Theresultisthatwewouldhaveaseriesof ��2 /(��2 ��i 2 ) termswithvariousweightingfactors Ai thataddtounity,where ��i representdifferent resonancewavelengths.Theweightingfactors Ai involvequantummechanicalmatrix elements.
Figure1.2showsthecomplexrelativepermittivityandthecomplexrefractiveindex ofcrystallinesiliconintermsofphotonenergy h�� [31,32].Forphotonenergiesbelow thebandgapenergy(1.1eV),both ��′′ r and K arenegligibleand n iscloseto3.7.Both ��′′ r and K increaseandchangestronglyasthephotonenergybecomesgreaterthan3eV, farbeyondthebandgapenergy.Noticethatboth ��′′ r and K peakat h�� ≈ 3.5eV,which correspondstoadirectphotoexcitationprocesses,electronsexciteddirectlyfromthe valencebandtotheconductionband,asdiscussedinChapter2.
1.2.2 n and K ,andKramers–KronigRelations
Ifweknowthefrequencydependenceoftherealpart, ��′ r ,oftherelativepermittivityofa material,wecan,usingthe Kramers–Kronigrelations betweentherealandtheimaginary parts,determinethefrequencydependenceoftheimaginarypart ��′′ r ,andviceversa. Thetransformrequiresthatweknowthefrequencydependenceofeithertherealor imaginarypartoveraswidearangeoffrequenciesaspossible,ideallyfromzero(DC) toinfinity,andthatthematerialhaslinearbehavior,thatis,ithasarelativepermittivity thatisindependentoftheappliedfield.TheKramers–Kronigrelationsfortherelative permittivity ��r = ��′ r + i��′′ r aregivenby[33–35](seealsoAppendix1Cin[25]aswell as[27])
Figure1.2 (a)Complexrelativepermittivityofasiliconcrystalasafunctionofphotonenergyplotted intermsofreal(��′ r )andimaginary(��′′ r )parts.(b)Opticalpropertiesofasiliconcrystalvs.photon energyintermsofreal(n)andimaginary(K )partsofthecomplexrefractiveindex.Source:Adapted fromD.E.AspnesandA.A.Studna,1983[32]andH.R.PhilippandE.A.Taft,1960[31].
where ��′ istheintegrationvariable, P representstheCauchyprincipalvalueoftheintegral,andthesingularityat �� = ��′ isavoided.
Similarly,onecanrelatetherealandimaginarypartsofthepolarizability, ��
(��)and �� ′′ (��),andthoseofthecomplexrefractiveindex, n(��)and K (��),aswell.Foracomplex refractiveindexwrittenas n* = n(��) + iK (��),
Althoughitappears,intheory,thatoneneedstointegratethespectrumof n or K from DCtoinfinitefrequencies,thisisobviouslynotfeasible,andisunnecessary.Itshould benotedthattheexperimentalsetupusuallyhaslow-andhigh-frequencylimitations thattruncatetheprecedingintegrations.Moreover,inmanycases,weareinterestedin thespectrumof n and K inandaroundanabsorptionband.Thus,beforeandafterthe absorptionfrequencyrange, K wouldbenegligiblysmall,andwecanusethisabsorption frequencyrangeintheprecedingintegralsinEq.(1.12).Therearenumerousstudiesin theliteraturethatusetheprecedingKramers–Kronigrelationsinextractingthewavelengthdependenceof n fromthatof K ,andviceversa,especiallyaroundclearabsorption bands;afewselectedexamplescanbefoundin[36–40],andtherearemanyothersinthe literature.Therearealsoseveralusefulapproachesinwhichtheabsorptionspectrum, or K (��),isdescribedintermsofaparticularphysicalmodelwithaparticularexpression,andthecorrespondingrefractiveindex n(��)isderivedfromtheKramers–Kronig transformationforbothamorphousandcrystallinesolids—forexamples,see[41,42]. Itshouldbeemphasizedthattheopticalconstants n and K havetoobeywhatare called f-sumrules [43].Forexample,theintegrationof[n(��)–1]overallfrequencies mustbezero,andtheintegrationof ��K (��)overallfrequenciesgives(�� /2)��p 2 ,where ��p = ℏ(4�� NZe2 /me )1/2 isthefreeelectronplasmafrequencyinwhich N istheatomic
