Group theory in solid state physics and photonics

382 Pages·2015·22.143 MB·English

by  Hergert, Wolfram

Wolfram Hergert and R. Matthias Geilhufe
Group Theory in
Solid State Physics
and Photonics
Problem Solving with Mathematica

WolframHergertand
R.MatthiasGeilhufe
GroupTheoryinSolidState
PhysicsandPhotonics

WolframHergertandR.MatthiasGeilhufe
Group Theory in Solid State Physics
and Photonics
ProblemSolvingwithMathematica
Authors AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.WolframHergert publisherdonotwarranttheinformation
MartinLutherUniversityHalle-Wittenberg containedinthesebooks,includingthisbook,
Von-Seckendorff-Platz1 tobefreeoferrors.Readersareadvisedtokeep
06120Halle inmindthatstatements,data,illustrations,
Germany proceduraldetailsorotheritemsmay
inadvertentlybeinaccurate.
Dr.R.MatthiasGeilhufe
LibraryofCongressCardNo.:
Nordita
appliedfor
Roslagstullsbacken23
10691Stockholm BritishLibraryCataloguing-in-PublicationData:
Sweden Acataloguerecordforthisbookisavailable
fromtheBritishLibrary.
Bibliographicinformationpublishedbythe
DeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhis
publicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableonthe
Internetathttp://dnb.d-nb.de.
©2018WILEY-VCHVerlagGmbH&Co.KGaA,
Boschstr.12,69469Weinheim,Germany
Allrightsreserved(includingthoseoftrans-
lationintootherlanguages).Nopartofthis
bookmaybereproducedinanyform–by
photoprinting,microfilm,oranyothermeans
–nortransmittedortranslatedintoamachine
languagewithoutwrittenpermissionfromthe
publishers.Registerednames,trademarks,etc.
usedinthisbook,evenwhennotspecifical-
lymarkedassuch,arenottobeconsidered
unprotectedbylaw.
PrintISBN 978-3-527-41133-7
ePDFISBN 978-3-527-41300-3
ePubISBN 978-3-527-41301-0
MobiISBN 978-3-527-41302-7
oBookISBN 978-3-527-69579-9
CoverDesign Formgeber,Mannheim,Germany
Typesetting le-texpublishingservicesGmbH,
Leipzig,Germany
Printedonacid-freepaper.
V
Contents
Preface XI
1 Introduction 1
1.1 SymmetriesinSolid-StatePhysicsandPhotonics 4
1.2 ABasicExample:SymmetriesofaSquare 6
PartOne BasicsofGroupTheory 9
2 SymmetryOperationsandTransformationsofFields 11
2.1 RotationsandTranslations 11
2.1.1 RotationMatrices 13
2.1.2 EulerAngles 16
2.1.3 Euler–RodriguesParametersandQuaternions 18
2.1.4 TranslationsandGeneralTransformations 23
2.2 TransformationofFields 25
2.2.1 TransformationofScalarFieldsandAngularMomentum 26
2.2.2 TransformationofVectorFieldsandTotalAngularMomentum 27
2.2.3 Spinors 28
3 BasicsAbstractGroupTheory 33
3.1 BasicDefinitions 33
3.1.1 IsomorphismandHomomorphism 38
3.2 StructureofGroups 39
3.2.1 Classes 40
3.2.2 CosetsandNormalDivisors 42
3.3 QuotientGroups 46
3.4 ProductGroups 48
VI Contents
4 DiscreteSymmetryGroupsinSolid-StatePhysicsandPhotonics 51
4.1 PointGroups 52
4.1.1 NotationofSymmetryElements 52
4.1.2 ClassificationofPointGroups 56
4.2 SpaceGroups 59
4.2.1 Lattices,TranslationGroup 59
4.2.2 SymmorphicandNonsymmorphicSpaceGroups 62
4.2.3 SiteSymmetry,WyckoffPositions,andWigner–SeitzCell 65
4.3 ColorGroupsandMagneticGroups 69
4.3.1 MagneticPointGroups 69
4.3.2 MagneticLattices 72
4.3.3 MagneticSpaceGroups 73
4.4 NoncrystallographicGroups,Buckyballs,andNanotubes 75
4.4.1 StructureandGroupTheoryofNanotubes 75
4.4.2 BuckminsterfullereneC 79
60
5 RepresentationTheory 83
5.1 DefinitionofMatrixRepresentations 84
5.2 ReducibleandIrreducibleRepresentations 88
5.2.1 TheOrthogonalityTheoremforIrreducibleRepresentations 90
5.3 CharactersandCharacterTables 94
5.3.1 TheOrthogonalityTheoremforCharacters 96
5.3.2 CharacterTables 98
5.3.3 NotationsofIrreducibleRepresentations 98
5.3.4 DecompositionofReducibleRepresentations 102
5.4 ProjectionOperatorsandBasisFunctionsofRepresentations 105
5.5 DirectProductRepresentations 112
5.6 Wigner–EckartTheorem 120
5.7 InducedRepresentations 123
6 SymmetryandRepresentationTheoryink-Space 133
6.1 TheCyclicBorn–vonKármánBoundaryCondition
andtheBlochWave 133
6.2 TheReciprocalLattice 136
6.3 TheBrillouinZoneandtheGroupoftheWaveVectork 137
6.4 IrreducibleRepresentationsofSymmorphicSpaceGroups 142
6.5 IrreducibleRepresentationsofNonsymmorphicSpaceGroups 143
Contents VII
PartTwo ApplicationsinElectronicStructureTheory 149
7 SolutionoftheSchrödingerEquation 151
7.1 TheSchrödingerEquation 151
7.2 TheGroupoftheSchrödingerEquation 153
7.3 DegeneracyofEnergyStates 154
7.4 Time-IndependentPerturbationTheory 157
7.4.1 GeneralFormalism 159
7.4.2 CrystalFieldExpansion 160
7.4.3 CrystalFieldOperators 164
7.5 TransitionProbabilitiesandSelectionRules 169
8 GeneralizationtoIncludetheSpin 177
8.1 ThePauliEquation 177
8.2 HomomorphismbetweenSU(2)andSO(3) 178
8.3 TransformationoftheSpin–OrbitCouplingOperator 180
8.4 TheGroupofthePauliEquationandDoubleGroups 183
8.5 IrreducibleRepresentationsofDoubleGroups 186
8.6 SplittingofDegeneraciesbySpin–OrbitCoupling 189
8.7 Time-ReversalSymmetry 193
8.7.1 TheRealityofRepresentations 193
8.7.2 Spin-IndependentTheory 194
8.7.3 Spin-DependentTheory 196
9 ElectronicStructureCalculations 197
9.1 SolutionoftheSchrödingerEquationforaCrystal 197
9.2 SymmetryPropertiesofEnergyBands 198
9.2.1 DegeneracyandSymmetryofEnergyBands 200
9.2.2 CompatibilityRelationsandCrossingofBands 201
9.3 Symmetry-AdaptedFunctions 203
9.3.1 Symmetry-AdaptedPlaneWaves 203
9.3.2 LocalizedOrbitals 205
9.4 ConstructionofTight-BindingHamiltonians 210
9.4.1 HamiltoniansinTwo-CenterForm 212
9.4.2 HamiltoniansinThree-CenterForm 216
9.4.3 InclusionofSpin–OrbitInteraction 224
9.4.4 Tight-BindingHamiltoniansfromabinitioCalculations 225
9.5 HamiltoniansBasedonPlaneWaves 227
9.6 ElectronicEnergyBandsandIrreducibleRepresentations 230
9.7 ExamplesandApplications 236
9.7.1 CalculationofFermiSurfaces 236
9.7.2 ElectronicStructureofCarbonNanotubes 238
9.7.3 Tight-bindingReal-SpaceCalculations 240
9.7.4 Spin–OrbitCouplinginSemiconductors 245
9.7.5 Tight-BindingModelsforOxides 247
VIII Contents
PartThree ApplicationsinPhotonics 251
10 SolutionofMaxwell’sEquations 253
10.1 Maxwell’sEquationsandtheMasterEquationforPhotonic
Crystals 254
10.1.1 TheMasterEquation 254
10.1.2 One-andTwo-DimensionalProblems 256
10.2 GroupoftheMasterEquation 257
10.3 MasterEquationasanEigenvalueProblem 259
10.4 ModelsofthePermittivity 260
10.4.1 ReducedStructureFactors 264
10.4.2 ConvergenceofthePlaneWaveExpansion 266
11 Two-DimensionalPhotonicCrystals 269
11.1 PhotonicBandStructureandSymmetrizedPlaneWaves 270
11.1.1 EmptyLatticeBandStructureandSymmetrizedPlaneWaves 270
11.1.2 PhotonicBandStructures:AFirstExample 273
11.2 GroupTheoreticalClassificationofPhotonicBandStructures 276
11.3 SupercellsandSymmetryofDefectModes 279
11.4 UncoupledBands 283
12 Three-DimensionalPhotonicCrystals 287
12.1 EmptyLatticeBandsandCompatibilityRelations 287
12.2 Anexample:DielectricSpheresinAir 291
12.3 Symmetry-AdaptedVectorSphericalWaves 293
PartFour OtherApplications 299
13 GroupTheoryofVibrationalProblems 301
13.1 VibrationsofMolecules 301
13.1.1 Permutation,Displacement,andVectorRepresentation 302
13.1.2 VibrationalModesofMolecules 305
13.1.3 InfraredandRamanActivity 307
13.2 LatticeVibrations 310
13.2.1 DirectCalculationoftheDynamicalMatrix 312
13.2.2 DynamicalMatrixfromTight-BindingModels 314
13.2.3 AnalysisofZoneCenterModes 315
14 LandauTheoryofPhaseTransitionsoftheSecondKind 319
14.1 IntroductiontoLandau’sTheoryofPhaseTransitions 320
14.2 BasicsoftheGroupTheoreticalFormulation 324
14.3 ExampleswithGTPackCommands 326
14.3.1 InvariantPolynomials 326
14.3.2 LandauandLifshitzCriterion 327