Group Theory in Solid State Physics and Photonics

Wolfram Hergert, extraordinary professor in Computational Physics, is member of the Theoretical Physics group at University Halle-Wittenberg, Germany. Main subjects of his work are solid state theory, electronic and magnetic structure of nanostructures and photonics. Prof. Hergert has experience in teaching group theory and in applying Mathematica to physical problems. He has published in renowned journals, like Nature and Physical Review Letters, and edited a books on Computational Materials Science and Mie Theory. He is also coauthor of a book on Quantum Theory.Matthias Geilhufe studied physics at the Martin Luther University Halle-Wittenberg (Germany) with specialization in theoretical and computational physics. From 2012-2015 he was employed as a PhD student at the Max Planck Institute of Microstructure Physics in Halle. In 2015 he obtained his PhD at the Martin Luther University Halle-Wittenberg. Currently, he is working at the Nordita Institute in Stockholm, Sweden. His work is based on the investigation of electronic and magnetic properties of complex materials. For his research, methods based on group theory or density functional theory are applied.

• Preface VII1 Introduction 11.1 Symmetries in Solid-State Physics and Photonics 41.2 A Basic Example: Symmetries of a Square 6Part One Basics of Group Theory 92 Symmetry Operations and Transformations of Fields 112.1 Rotations and Translations 112.1.1 Rotation Matrices 132.1.2 Euler Angles 162.1.3 Euler–Rodrigues Parameters and Quaternions 182.1.4 Translations and General Transformations 232.2 Transformation of Fields 252.2.1 Transformation of Scalar Fields and Angular Momentum 262.2.2 Transformation of Vector Fields and Total Angular Momentum 272.2.3 Spinors 283 Basics Abstract Group Theory 333.1 Basic Definitions 333.1.1 Isomorphism and Homomorphism 383.2 Structure of Groups 393.2.1 Classes 403.2.2 Cosets and Normal Divisors 423.3 Quotient Groups 463.4 Product Groups 484 Discrete Symmetry Groups in Solid-State Physics and Photonics 514.1 Point Groups 524.1.1 Notation of Symmetry Elements 524.1.2 Classification of Point Groups 564.2 Space Groups 594.2.1 Lattices, Translation Group 594.2.2 Symmorphic and Nonsymmorphic Space Groups 624.2.3 Site Symmetry, Wyckoff Positions, and Wigner–Seitz Cell 654.3 Color Groups and Magnetic Groups 694.3.1 Magnetic Point Groups 694.3.2 Magnetic Lattices 724.3.3 Magnetic Space Groups 734.4 Noncrystallographic Groups, Buckyballs, and Nanotubes 754.4.1 Structure and Group Theory of Nanotubes 754.4.2 Buckminsterfullerene C60 795 Representation Theory 835.1 Definition of Matrix Representations 845.2 Reducible and Irreducible Representations 885.2.1 The Orthogonality Theorem for Irreducible Representations 905.3 Characters and Character Tables 945.3.1 The Orthogonality Theorem for Characters 965.3.2 Character Tables 985.3.3 Notations of Irreducible Representations 985.3.4 Decomposition of Reducible Representations 1025.4 Projection Operators and Basis Functions of Representations 1055.5 Direct Product Representations 1125.6 Wigner–Eckart Theorem 1205.7 Induced Representations 1236 Symmetry and Representation Theory in k-Space 1336.1 The Cyclic Born–von Kármán Boundary Condition and the Bloch Wave 1336.2 The Reciprocal Lattice 1366.3 The Brillouin Zone and the Group of the Wave Vector k 1376.4 Irreducible Representations of Symmorphic Space Groups 1426.5 Irreducible Representations of Nonsymmorphic Space Groups 143Part Two Applications in Electronic Structure Theory 1497 Solution of the SCHRÖDINGER Equation 1517.1 The Schrödinger Equation 1517.2 The Group of the Schrödinger Equation 1537.3 Degeneracy of Energy States 1547.4 Time-Independent Perturbation Theory 1577.4.1 General Formalism 1597.4.2 Crystal Field Expansion 1607.4.3 Crystal Field Operators 1647.5 Transition Probabilities and Selection Rules 1698 Generalization to Include the Spin 1778.1 The Pauli Equation 1778.2 Homomorphism between SU(2) and SO(3) 1788.3 Transformation of the Spin–Orbit Coupling Operator 1808.4 The Group of the Pauli Equation and Double Groups 1838.5 Irreducible Representations of Double Groups 1868.6 Splitting of Degeneracies by Spin–Orbit Coupling 1898.7 Time-Reversal Symmetry 1938.7.1 The Reality of Representations 1938.7.2 Spin-Independent Theory 1948.7.3 Spin-Dependent Theory 1969 Electronic Structure Calculations 1979.1 Solution of the Schrödinger Equation for a Crystal 1979.2 Symmetry Properties of Energy Bands 1989.2.1 Degeneracy and Symmetry of Energy Bands 2009.2.2 Compatibility Relations and Crossing of Bands 2019.3 Symmetry-Adapted Functions 2039.3.1 Symmetry-Adapted Plane Waves 2039.3.2 Localized Orbitals 2059.4 Construction of Tight-Binding Hamiltonians 2109.4.1 Hamiltonians in Two-Center Form 2129.4.2 Hamiltonians in Three-Center Form 2169.4.3 Inclusion of Spin–Orbit Interaction 2249.4.4 Tight-Binding Hamiltonians from ab initio Calculations 2259.5 Hamiltonians Based on Plane Waves 2279.6 Electronic Energy Bands and Irreducible Representations 2309.7 Examples and Applications 2369.7.1 Calculation of Fermi Surfaces 2369.7.2 Electronic Structure of Carbon Nanotubes 2389.7.3 Tight-binding Real-Space Calculations 2409.7.4 Spin–Orbit Coupling in Semiconductors 2459.7.5 Tight-Binding Models for Oxides 247Part Three Applications in Photonics 25110 Solution of MAXWELL’s Equations 25310.1 Maxwell’s Equations and the Master Equation for Photonic Crystals 25410.1.1 The Master Equation 25410.1.2 One- and Two-Dimensional Problems 25610.2 Group of the Master Equation 25710.3 Master Equation as an Eigenvalue Problem 25910.4 Models of the Permittivity 26010.4.1 Reduced Structure Factors 26410.4.2 Convergence of the Plane Wave Expansion 26611 Two-Dimensional Photonic Crystals 26911.1 Photonic Band Structure and Symmetrized Plane Waves 27011.1.1 Empty Lattice Band Structure and Symmetrized Plane Waves 27011.1.2 Photonic Band Structures: A First Example 27311.2 Group Theoretical Classification of Photonic Band Structures 27611.3 Supercells and Symmetry of Defect Modes 27911.4 Uncoupled Bands 28312 Three-Dimensional Photonic Crystals 28712.1 Empty Lattice Bands and Compatibility Relations 28712.2 An example: Dielectric Spheres in Air 29112.3 Symmetry-Adapted Vector Spherical Waves 293Part Four Other Applications 29913 Group Theory of Vibrational Problems 30113.1 Vibrations of Molecules 30113.1.1 Permutation, Displacement, and Vector Representation 30213.1.2 Vibrational Modes of Molecules 30513.1.3 Infrared and Raman Activity 30713.2 Lattice Vibrations 31013.2.1 Direct Calculation of the Dynamical Matrix 31213.2.2 Dynamical Matrix from Tight-Binding Models 31413.2.3 Analysis of Zone Center Modes 31514 Landau Theory of Phase Transitions of the Second Kind 31914.1 Introduction to Landau’s Theory of Phase Transitions 32014.2 Basics of the Group Theoretical Formulation 32414.3 Examples with GTPack Commands 32614.3.1 Invariant Polynomials 32614.3.2 Landau and Lifshitz Criterion 327Appendix A Spherical Harmonics 331A.1 Complex Spherical Harmonics 332A.1.1 Definition of Complex Spherical Harmonics 332A.1.2 Cartesian Spherical Harmonics 332A.1.3 Transformation Behavior of Complex Spherical Harmonics 333A.2 Tesseral Harmonics 334A.2.1 Definition of Tesseral Harmonics 334A.2.2 Cartesian Tesseral Harmonics 335A.2.3 Transformation Behavior of Tesseral Harmonics 336Appendix B Remarks on Databases 337B.1 Electronic Structure Databases 337B.1.1 Tight-Binding Calculations 337B.1.2 Pseudopotential Calculations 338B.1.3 Radial Integrals for Crystal Field Parameters 339B.2 Molecular Databases 339B.3 Database of Structures 339Appendix C Use of MPB together with GTPack 341C.1 Calculation of Band Structure and Density of States 341C.2 Calculation of Eigenmodes 342C.3 Comparison of Calculations with MPB and Mathematica 343Appendix D Technical Remarks on GTPack 345D.1 Structure of GTPack 345D.2 Installation of GTPack 346References 349Index 359