# Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica

# Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica

ISBN: 978-3-527-41133-7

Jun 2018

377 pages

## Description

While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals.Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics.

The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.

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Preface VII

**1 Introduction 1**

1.1 Symmetries in Solid-State Physics and Photonics 4

1.2 A Basic Example: Symmetries of a Square 6

**Part One Basics of Group Theory 9**

**2 Symmetry Operations and Transformations of Fields 11**

2.1 Rotations and Translations 11

2.1.1 Rotation Matrices 13

2.1.2 Euler Angles 16

2.1.3 Euler–Rodrigues Parameters and Quaternions 18

2.1.4 Translations and General Transformations 23

2.2 Transformation of Fields 25

2.2.1 Transformation of Scalar Fields and Angular Momentum 26

2.2.2 Transformation of Vector Fields and Total Angular Momentum 27

2.2.3 Spinors 28

**3 Basics Abstract Group Theory 33**

3.1 Basic Definitions 33

3.1.1 Isomorphism and Homomorphism 38

3.2 Structure of Groups 39

3.2.1 Classes 40

3.2.2 Cosets and Normal Divisors 42

3.3 Quotient Groups 46

3.4 Product Groups 48

**4 Discrete Symmetry Groups in Solid-State Physics and Photonics 51**

4.1 Point Groups 52

4.1.1 Notation of Symmetry Elements 52

4.1.2 Classification of Point Groups 56

4.2 Space Groups 59

4.2.1 Lattices, Translation Group 59

4.2.2 Symmorphic and Nonsymmorphic Space Groups 62

4.2.3 Site Symmetry, Wyckoff Positions, and Wigner–Seitz Cell 65

4.3 Color Groups and Magnetic Groups 69

4.3.1 Magnetic Point Groups 69

4.3.2 Magnetic Lattices 72

4.3.3 Magnetic Space Groups 73

4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes 75

4.4.1 Structure and Group Theory of Nanotubes 75

4.4.2 Buckminsterfullerene C60 79

**5 Representation Theory 83**

5.1 Definition of Matrix Representations 84

5.2 Reducible and Irreducible Representations 88

5.2.1 The Orthogonality Theorem for Irreducible Representations 90

5.3 Characters and Character Tables 94

5.3.1 The Orthogonality Theorem for Characters 96

5.3.2 Character Tables 98

5.3.3 Notations of Irreducible Representations 98

5.3.4 Decomposition of Reducible Representations 102

5.4 Projection Operators and Basis Functions of Representations 105

5.5 Direct Product Representations 112

5.6 Wigner–Eckart Theorem 120

5.7 Induced Representations 123

**6 Symmetry and Representation Theory in k-Space 133**

6.1 The Cyclic Born–von Kármán Boundary Condition and the Bloch Wave 133

6.2 The Reciprocal Lattice 136

6.3 The Brillouin Zone and the Group of the Wave Vector k 137

6.4 Irreducible Representations of Symmorphic Space Groups 142

6.5 Irreducible Representations of Nonsymmorphic Space Groups 143

**Part Two Applications in Electronic Structure Theory 149**

**7 Solution of the SCHRÖDINGER Equation 151**

7.1 The Schrödinger Equation 151

7.2 The Group of the Schrödinger Equation 153

7.3 Degeneracy of Energy States 154

7.4 Time-Independent Perturbation Theory 157

7.4.1 General Formalism 159

7.4.2 Crystal Field Expansion 160

7.4.3 Crystal Field Operators 164

7.5 Transition Probabilities and Selection Rules 169

**8 Generalization to Include the Spin 177**

8.1 The Pauli Equation 177

8.2 Homomorphism between SU(2) and SO(3) 178

8.3 Transformation of the Spin–Orbit Coupling Operator 180

8.4 The Group of the Pauli Equation and Double Groups 183

8.5 Irreducible Representations of Double Groups 186

8.6 Splitting of Degeneracies by Spin–Orbit Coupling 189

8.7 Time-Reversal Symmetry 193

8.7.1 The Reality of Representations 193

8.7.2 Spin-Independent Theory 194

8.7.3 Spin-Dependent Theory 196

**9 Electronic Structure Calculations 197**

9.1 Solution of the Schrödinger Equation for a Crystal 197

9.2 Symmetry Properties of Energy Bands 198

9.2.1 Degeneracy and Symmetry of Energy Bands 200

9.2.2 Compatibility Relations and Crossing of Bands 201

9.3 Symmetry-Adapted Functions 203

9.3.1 Symmetry-Adapted Plane Waves 203

9.3.2 Localized Orbitals 205

9.4 Construction of Tight-Binding Hamiltonians 210

9.4.1 Hamiltonians in Two-Center Form 212

9.4.2 Hamiltonians in Three-Center Form 216

9.4.3 Inclusion of Spin–Orbit Interaction 224

9.4.4 Tight-Binding Hamiltonians from ab initio Calculations 225

9.5 Hamiltonians Based on Plane Waves 227

9.6 Electronic Energy Bands and Irreducible Representations 230

9.7 Examples and Applications 236

9.7.1 Calculation of Fermi Surfaces 236

9.7.2 Electronic Structure of Carbon Nanotubes 238

9.7.3 Tight-binding Real-Space Calculations 240

9.7.4 Spin–Orbit Coupling in Semiconductors 245

9.7.5 Tight-Binding Models for Oxides 247

**Part Three Applications in Photonics 251**

**10 Solution of MAXWELL’s Equations 253**

10.1 Maxwell’s Equations and the Master Equation for Photonic Crystals 254

10.1.1 The Master Equation 254

10.1.2 One- and Two-Dimensional Problems 256

10.2 Group of the Master Equation 257

10.3 Master Equation as an Eigenvalue Problem 259

10.4 Models of the Permittivity 260

10.4.1 Reduced Structure Factors 264

10.4.2 Convergence of the Plane Wave Expansion 266

**11 Two-Dimensional Photonic Crystals 269**

11.1 Photonic Band Structure and Symmetrized Plane Waves 270

11.1.1 Empty Lattice Band Structure and Symmetrized Plane Waves 270

11.1.2 Photonic Band Structures: A First Example 273

11.2 Group Theoretical Classification of Photonic Band Structures 276

11.3 Supercells and Symmetry of Defect Modes 279

11.4 Uncoupled Bands 283

**12 Three-Dimensional Photonic Crystals 287**

12.1 Empty Lattice Bands and Compatibility Relations 287

12.2 An example: Dielectric Spheres in Air 291

12.3 Symmetry-Adapted Vector Spherical Waves 293

**Part Four Other Applications 299**

**13 Group Theory of Vibrational Problems 301**

13.1 Vibrations of Molecules 301

13.1.1 Permutation, Displacement, and Vector Representation 302

13.1.2 Vibrational Modes of Molecules 305

13.1.3 Infrared and Raman Activity 307

13.2 Lattice Vibrations 310

13.2.1 Direct Calculation of the Dynamical Matrix 312

13.2.2 Dynamical Matrix from Tight-Binding Models 314

13.2.3 Analysis of Zone Center Modes 315

**14 Landau Theory of Phase Transitions of the Second Kind 319**

14.1 Introduction to Landau’s Theory of Phase Transitions 320

14.2 Basics of the Group Theoretical Formulation 324

14.3 Examples with GTPack Commands 326

14.3.1 Invariant Polynomials 326

14.3.2 Landau and Lifshitz Criterion 327

**Appendix A Spherical Harmonics 331**

A.1 Complex Spherical Harmonics 332

A.1.1 Definition of Complex Spherical Harmonics 332

A.1.2 Cartesian Spherical Harmonics 332

A.1.3 Transformation Behavior of Complex Spherical Harmonics 333

A.2 Tesseral Harmonics 334

A.2.1 Definition of Tesseral Harmonics 334

A.2.2 Cartesian Tesseral Harmonics 335

A.2.3 Transformation Behavior of Tesseral Harmonics 336

**Appendix B Remarks on Databases 337**

B.1 Electronic Structure Databases 337

B.1.1 Tight-Binding Calculations 337

B.1.2 Pseudopotential Calculations 338

B.1.3 Radial Integrals for Crystal Field Parameters 339

B.2 Molecular Databases 339

B.3 Database of Structures 339

**Appendix C Use of MPB together with GTPack 341**

C.1 Calculation of Band Structure and Density of States 341

C.2 Calculation of Eigenmodes 342

C.3 Comparison of Calculations with MPB and Mathematica 343

**Appendix D Technical Remarks on GTPack 345**

D.1 Structure of GTPack 345

D.2 Installation of GTPack 346

References 349

Index 359