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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

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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

concentration, Z isthetotalnumberofelectronsperatom,and e and me arethecharge andmassoftheelectron,respectively.The f -sumrulesprovideaconsistencycheckand enablevariousconstantstobeinterrelated.

1.3RefractiveIndexandDispersion

Thereareseveralpopularmodelsdescribingthespectraldependenceofrefractiveindex n inamaterial.Mostofthesearedescribedinthefollowingtext,althoughsome,such astheinfraredrefractiveindex,iscoveredinthediscussiononReststrahlenabsorption inPartII,sinceitiscloselyrelatedtothecouplingoftheEMwavetolatticevibrations. ThemostpopulardispersionrelationinopticalmaterialsisprobablytheSellmeierrelationship,sinceonecansumanynumberofresonance-typetermstogetaswidearange ofwavelengthdependenceaspossible.However,itsmaindrawbackisthatitdoesnot accuratelyrepresenttherefractiveindexwhenthereisacontributionarisingfromfree carriersinnarrowbandgapordopedsemiconductors.

Therearemanyhandbooks,books,andwebsitesthatnowprovideempiricalequations fortherefractiveindexofawiderangeofsolids,forexampleasinreferences[1–19,44].

1.3.1CauchyDispersionRelation

IntheCauchyrelationship,thedispersionrelationshipbetweentherefractiveindex(n) andthewavelengthoflight(��)iscommonlystatedinthefollowingform:

where A, B,and C arematerial-dependentspecificconstants.Equation(1.13)isknown as Cauchy’sformula;itistypicallyusedinthevisiblespectrumregionforvariousoptical glasses,anditappliesto normaldispersion,when n decreaseswithincreasing �� [45,46]. Thethirdtermissometimesdroppedforasimplerrepresentationof n versus �� behavior. Theoriginalexpressionwasaseriesintermsofthewavelength, ��,orfrequency, ��,or photonenergy ℏ�� oflightas:

or

where ℏ�� isthephotonenergy; ℏ��th = hc/��th istheopticalexcitationthreshold(e.g. bandgapenergy);and a0 , a2 ,… and n0 , n2 ,… areconstants.Ithasbeenfoundthata Cauchyrelationinthefollowingform[47]:

canbeusedsatisfactorilyoverawiderangeofphotonenergies.ThedispersionparametersofEq.(1.15)arelistedinTable1.1forafewselectedmaterialsoverspecificphoton energyranges.

Cauchy’sdispersionrelationsgiveninEqs.(1.13)–(1.14)wereoriginallycalledthe elasticethertheoryoftherefractiveindex.Ithasbeenwidelyusedformanymaterials,

Table1.1 Cauchy’sdispersionparametersofEq.(1.15)forGe,Si,andDiamondfrom[43].

although,inrecentyears,manyresearchershavepreferredtousetheSellmeierequation, describedinthefollowingtext.

1.3.2SellmeierEquation

TheSellmeierequation[48]isanempiricalrelationbetweentherefractiveindex n of asubstanceandwavelength �� oflightintheformofaseriesofsingledipoleoscillator terms,eachofwhichhastheusual ��2 /(��2 ��i 2 )dependenceasin

where A1 , A2 , A3 and ��1 , ��2 ,and ��3 areconstants,called Sellmeiercoefficients,which aredeterminedbyfittingthisexpressiontotheexperimentaldata.TheactualSellmeier formulaismorecomplicated.Ithasmoretermsofsimilarform,suchas Ai ��2 /(��2 – ��i 2 ), where i = 4,5,...,butthesecangenerallybeneglectedinrepresenting n vs. �� behavior overtypicalwavelengthsofinterestandbyensuringthatthethreetermsincludedin Eq.(1.16)correspondtothemostimportantorrelevanttermsinthesummation[49].

TheSellmeiercoefficientsforsomematerials,includingpureSilica(SiO2 )and86.5mol% SiO2 –13.5mol%GeO2 ,aregiveninTable1.2asexamples.Aquantitativeanalysisofthe applicationoftheSellmeierdispersionrelationtoarangeofmaterials,fromglassesto semiconductors,hasbeendiscussedbyTatian[49].

Therearetwomethodsfordeterminingtherefractiveindexofsilica–germania glass(SiO2 )1-x (GeO2 )x .Thefirstisasimple,butapproximate,linearinterpolationof therefractiveindexbetweenknowncompositions,forexample, n(x) n(0.135) = (x 0.235)[n(0.135) n(0)]/0.135,where n(x)isfor(SiO2 )1 x (GeO2 )x ; n(0.135)isfor 86.5mol%SiO2 –13.5mol%GeO2 ;and n(0)isforSiO2 .Thesecondisaninterpolation forcoefficients Ai and ��i betweenSiO2 andGeO2 as[50]: n2 1 = {A1 (S )+ X [A1 (G

where S and G inparenthesesrefertosilicaandgermania,respectively.ThetheoreticalbasisoftheSellmeierequationliesinrepresentingthesolidasasumof N lossless (frictionless)Lorentzoscillatorssuchthateachhastheusualformof ��2 /(��2 – ��i 2 )with different ��i andeachhasadifferentstrength,orweightingfactor; Ai , i = 1to N [51,52]. Suchdispersionrelationshipsareessentialindesigningphotonicdevicessuchaswaveguides.(Notethatalthough Ai weighsdifferentLorentzcontributions,theydonotsumto 1sincetheyincludeotherparametersbesidestheoscillatorstrength f i .)Therefractive indicesofmostopticalglasseshavebeenextensivelymodeledbytheSellmeierequation.

Table1.2 Sellmeiercoefficientsofafewmaterials,where ��1 , ��2 , ��3 arein μm.

Material

SiO2 (fusedsilica)0.6967490.4082180.8908150.06906600.1156629.900559

86.5%SiO2 –13.5%

GeO2 0.7110400.4518850.7040480.06427000.1294089.425478

GeO2 0.806866420.718158480.854168310.0689726060.1539660511.841931

Bariumfluoride0 33560 5067623 82610 0577890 10968146 38642

Sapphire1 0237981 0582645 2807920 06144820 11070017 92656

Diamond0.33064.33560.1750.106

Quartz, n o 1.354000.0100.99940.09261210.7009.8500

Quartz, n e 1 381000 01000 99920 09350511 3109 5280

KTP, n o 1 25400 01000 09920 096466 97775 9848

KTP, n e 1.130000.00010.99990.093517.671012.170

Source:Fromvarioussources.

VariousopticalglassmanufacturerssuchasSchottGlassnormallyprovidetheSellmeier coefficientsfortheirglasses[53].Theopticaldispersionrelationsforglasseshavebeen discussedbyanumberofauthors[7,25,54].

ThereareotherSellmeier–Cauchy-likedispersionrelationshipsthatinherentlytake accountofvariouscontributionstotheopticalproperties,suchastheelectronicand ionicpolarizationandtheinteractionofphotonswithfreeelectrons.Forexample,for manysemiconductorsandioniccrystals,twousefuldispersionrelationsare,

and

where A, B, C , D, E ,and ��o areconstantsparticulartoagivenmaterial.Eq.(1.18)is equivalenttotheSellmeierequation.Eq.(1.19)isknownasthe Herzbergerdispersion relation [52].Table1.3providesafewexamples.BothCauchyandSellmeierequations arestrictlyapplicableinwavelengthregionswherethematerialistransparent,thatis, theextinctioncoefficientisrelativelysmall.Therefractiveindexdispersionrelations

Table1.3 ParametersofEq.(1.19)forsomeselectedmaterials.

Source:SidatafromD.F.EdwardsandE.Ochoa, Appl.Optics 19,4130(1980),othersfromW.L.Wolfe,The HandbookofOptics,W.G.DriscollandW.Vaughan,McGraw-Hill,NewYork,1978.

1FundamentalOpticalPropertiesofMaterialsI

forawiderangeofsemiconductorshavebeencompiledbyMadelungin[14].There aremanyapplication-basedarticlesintheliteraturethatprovideempiricaldispersion relationsforavarietyofmaterials;arecentexampleonfarinfraredsubstrates(Ge,Si, ZnSe,ZnS,ZnTe)isgiveninreference[55].Therearebothwebsitesandvariousjournal articlesintheliteraturethatgivetherefractiveindexofnumerousmaterialsasafunction ofwavelength.

1.3.3RefractiveIndexofSemiconductors

1.3.3.1RefractiveIndexofCrystallineSemiconductors

Aparticularinterestinthecaseofsemiconductorsisin n and K forphotonenergiesgreaterthanthebandgap E g foroptoelectronicsapplications.Duetovarious featuresandsingularitiesinthe E -k diagramsofcrystallinesemiconductors,the opticalconstants n and K for ℏ��> E g arenotreadilyexpressibleinsimpleterms. Variousauthors,forexample,ForouhiandBloomer[42,56]andChenetal.[57],have nonethelessprovidedusefulandtractableexpressionsformodeling n and K inthis regime.Inparticular,Forouhi–Bloomer(FB)equationsexpress n and K intermsofthe photonenergy ℏ�� inaconsistentwaythatobeytheKramers–Kronigrelations[42], thatis

q ∑ i=1 Ai (ℏ�� Eg )2 (ℏ��)2 Bi (ℏ��)+ Ci and n = n(∞)+

i=1 Boi (ℏ��)+ Coi (ℏ��)2 Bi (ℏ��)+ Ci , (1.20) where(ℏ��)isthephotonenergy; q isanintegerthatrepresentsthenumberofterms neededtosuitablymodelexperimental n, K ; E g isthebandgapand Ai , Bi , C i , Boi , C oi are constants; Boi and C oi dependon Ai , Bi , C i ,and E g —onlythelatterfourareindependent parameters;and Boi = (Ai /Qi )[ (1/2)Bi 2 + E g Bi – E g 2 + C i ], C oi = (Ai /Qi )[(1/2)(E g 2 + C i ) Bi 2E g C i ],and Qi = (1/2)(4C i Bi 2 )1/2 .ForouhiandBloomerprovideatableof FBcoefficients, Ai , Bi , C i ,and E g forfourtermsinthesummationinEq.(1.20)[42]for anumberofsemiconductors;anexamplethatshowsanexcellentagreementbetween theFBdispersionrelationandtheexperimentaldataisshowninFigure1.3.Table1.4 providestheFBcoefficientsforafewselectedsemiconductors.

Otherusefultheoreticalorsomewhatsemiempiricaldispersionrelationshipshave alsobeenproposed,forexample,byAfromowitz[58],Adachi[59–63],Campiand Papuzza[64],andothers[65].Thesemodelshavebeenappliedtovarioussemiconductorsandtheiralloyswithrelativesuccessovercertainphotonenergyranges.Oneofthe usefulandstraightforwardapproachestomodelingthedispersionhasbeenbasedon writingthecomplexrelativepermittivity ��r (ℏ��)asafinitesumofanumberofdamped harmonicoscillators(theso-called harmonicoscillatorapproximation),andfittingthis expressiontotheexperimentaldataasinreferences[66,67],eventhoughmanyterms maybeneededandthecurvefitprocesshastobecarefullychosentoensureareliable representationofthedata.Oneofbestmodelsconsideredsofar,however,hasinvolved parametricmodeling[68–70],inwhichnotonlyasumofharmonicoscillatorsareused butalsoGaussianbroadenedpolynomialstorepresentthedispersionofthecomplex relativepermittivity,andhence n and K .