Table of Contents

Cover

Preface

1 Introduction

1.1 Symmetries in Solid-State Physics and Photonics
1.2 A Basic Example: Symmetries of a Square

Part One: Basics of Group Theory

2 Symmetry Operations and Transformations of Fields
1. 2.1 Rotations and Translations
2. 2.2 Transformation of Fields
3 Basics Abstract Group Theory
1. 3.1 Basic Definitions
2. 3.2 Structure of Groups
3. 3.3 Quotient Groups
4. 3.4 Product Groups
4 Discrete Symmetry Groups in Solid-State Physics and Photonics
1. 4.1 Point Groups
2. 4.2 Space Groups
3. 4.3 Color Groups and Magnetic Groups
4. 4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes
5 Representation Theory
1. 5.1 Definition of Matrix Representations
2. 5.2 Reducible and Irreducible Representations
3. 5.3 Characters and Character Tables
4. 5.4 Projection Operators and Basis Functions of Representations
5. 5.5 Direct Product Representations
6. 5.6 WIGNER–ECKART Theorem
7. 5.7 Induced Representations
6 Symmetry and Representation Theory in k-Space
1. 6.1 The Cyclic BORNVON KÁRMÁN Boundary Condition and the BLOCH Wave
2. 6.2 The Reciprocal Lattice
3. 6.3 The BRILLOUIN Zone and the Group of the Wave Vector k
4. 6.4 Irreducible Representations of Symmorphic Space Groups
5. 6.5 Irreducible Representations of Nonsymmorphic Space Groups

Part Two: Applications in Electronic Structure Theory

7 Solution of the SCHRÖDINGER Equation
1. 7.1 The SCHRÖDINGER Equation
2. 7.2 The Group of the SCHRÖDINGER Equation
3. 7.3 Degeneracy of Energy States
4. 7.4 Time-Independent Perturbation Theory
5. 7.5 Transition Probabilities and Selection Rules
8 Generalization to Include the Spin
1. 8.1 The PAULI Equation
2. 8.2 Homomorphism between SU(2) and SO(3)
3. 8.3 Transformation of the Spin–Orbit Coupling Operator
4. 8.4 The Group of the PAULI Equation and Double Groups
5. 8.5 Irreducible Representations of Double Groups
6. 8.6 Splitting of Degeneracies by Spin–Orbit Coupling
7. 8.7 Time-Reversal Symmetry
9 Electronic Structure Calculations
1. 9.1 Solution of the Schrödinger Equation for a Crystal
2. 9.2 Symmetry Properties of Energy Bands
3. 9.3 Symmetry-Adapted Functions
4. 9.4 Construction of Tight-Binding Hamiltonians
5. 9.5 Hamiltonians Based on Plane Waves
6. 9.6 Electronic Energy Bands and Irreducible Representations
7. 9.7 Examples and Applications

Part Three: Applications in Photonics

10 Solution of Maxwell’s Equations
1. 10.1 MAXWELL’S Equations and the Master Equation for Photonic Crystals
2. 10.2 Group of the Master Equation
3. 10.3 Master Equation as an Eigenvalue Problem
4. 10.4 Models of the Permittivity
11 Two-Dimensional Photonic Crystals
1. 11.1 Photonic Band Structure and Symmetrized Plane Waves
2. 11.2 Group Theoretical Classification of Photonic Band Structures
3. 11.3 Supercells and Symmetry of Defect Modes
4. 11.4 Uncoupled Bands
12 Three-Dimensional Photonic Crystals
1. 12.1 Empty Lattice Bands and Compatibility Relations
2. 12.2 An example: Dielectric Spheres in Air
3. 12.3 Symmetry-Adapted Vector Spherical Waves

Part Four: Other Applications

13 Group Theory of Vibrational Problems
1. 13.1 Vibrations of Molecules
2. 13.2 Lattice Vibrations
14 Landau Theory of Phase Transitions of the Second Kind
1. 14.1 Introduction to Landau’s Theory of Phase Transitions
2. 14.2 Basics of the Group Theoretical Formulation
3. 14.3 Examples with GTPack Commands

Appendix A: Spherical Harmonics

A.1 Complex Spherical Harmonics
A.2 Tesseral Harmonics

Appendix B: Remarks on Databases

B.1 Electronic Structure Databases
B.2 Molecular Databases
B.3 Database of Structures

Appendix C: Use of MPB together with GTPack

C.1 Calculation of Band Structure and Density of States
C.2 Calculation of Eigenmodes
C.3 Comparison of Calculations with MPB and Mathematica

Appendix D: Technical Remarks on GTPack

D.1 Structure of GTPack
D.2 Installation of GTPack

References

Index

End User License Agreement

List of Tables

1 Introduction

Table 1.1 Conservation laws and symmetry in classical mechanics.

2 Symmetry Operations and Transformations of Fields

Table 2.1 The different cases of general rotations. The set of eigenvectors of type A contains one real and two mutually complex conjugate eigenvectors, while set B contains real eigenvectors only (complex conjugation ist denoted by ^⋆).

4 Discrete Symmetry Groups in Solid-State Physics and Photonics

Table 4.1 SCHÖNFLIES (SFL) notation and HERMANN–MAUGUIN (HM) notation of point group elements. A correspondence of the labeling of rotations is also given in Table 4.2.
Table 4.2 Comparison of proper and improper rotations in SCHÖNFLIES (SFL), HERMANN–MAUGUIN notation (HM), and the notation of GTPack (z-axis used as the reference).
Table 4.3 Properties of regular polyhedra.
Table 4.4 The crystal systems.
Table 4.5 Magnetic BRAVAIS lattices. Columns 2 and 3 show the notation and basis vectors of the BRAVAIS lattice for a colored structure. Column 4 gives the notation of the magnetic lattice and column 5 the translation combined with the color inversion operator [61].

5 Representation Theory

Table 5.1 Commands for the generation of matrix representations in GTPack and a classification of which kind of matrices are used. If the representation is faithful in general it is denoted with +, and otherwise with –. In the last column a reference to a more detailed description within the book is given.
Table 5.2 Comparison of BOUCKAERT notation and MULLIKEN notation for the irreducible representations of the point group O_h.

7 Solution of the Schrödinger Equation

Table 7.1 List of BST operator equivalents for a cubic crystal field according to Ref. [112]. The abbreviation X = J(J + 1) is used.
Table 7.2 List of the prefactors θ_l for rare-earth elements (cf. Ref. [113]).

9 Electronic Structure Calculations

Table 9.1 Notation of the irreducible representations at Γ and X of the BRILLOUIN zone of the fcc structure corresponding to BETHE, BOUCKAERT, SMOLUCHOWSKI and WIGNER (BSW), and MULLIKEN. The point groups at Γ and X are O_h and D_4h, respectively.
Table 9.2 Information from the character tables of C_4v and O_h to evaluate 9.10. Characteristic elements of the classes are: .

12 Three-Dimensional Photonic Crystals

Table 12.1 Transformation of vector operators under proper and improper rotations.
Table 12.2 Correspondence between spherical harmonics of angular momentum l and irreducible representations of the point group O_h for M- and N-fields.

13 Group Theory of Vibrational Problems

Table 13.1 Zone center modes for cubic structures (B – BOUCKAERT notation, M – MULLIKEN notation).

14 Landau Theory of Phase Transitions of the Second Kind

Table 14.1 Characters of symmetrized powers of a representation.

List of Illustrations

1 Introduction

Figure 1.1 Symmetry in nature and architecture. (a) Table 91 from HAECKEL’s ‘Art forms in Nature’ [4]; (b) Chrysler Building in New York City [5] (© JORGE ROYAN, www.royan.com.ar, CC BY-SA 3.0).
Figure 1.2 Symmetry in solid-state physics and photonics. (a) Atomically resolved STM image of two monolayers of MgO on Ag(001) (from [17], Figure 1) (With permission, Copyright © 2017 American Physical Society.)(b) SEM image of a width-modulated stripe (a) of macroporous silicon on a silicon substrate. The increasing magnification in (b)– (d) reveals a waveguide structure prepared by a missing row of pores. (from [18]). (With permission, Copyright © 1999 Wiley-VCH GmbH.)
Figure 1.3 Square with coordinate system and reflection lines. The vertices are numbered only to explain the effect of symmetry operations.
Figure 1.4 Symmetry of a square: Square colored in different ways.

2 Symmetry Operations and Transformations of Fields

Figure 2.1 Illustration of active and passive rotation. (a) Active rotation; (b) passive rotation.
Figure 2.2 Change from the passive definition to the active definition of rotations by means of GTReinstallAxes.
Figure 2.3 Result of the rotation by means of the matrices (2.6) on the vector r₁.
Figure 2.4 With the help of GTEulerAnglesQ it can be checked if the input is a list of EULER angles, {{α, β, γ},det }.
Figure 2.5 Obtaining EULER angles from a matrix or a symbol using GTGetEulerAngles. The symbol C_3z denotes a counterclockwise rotation about the angle 2/3π about the z-axis.
Figure 2.6 The usage of GTQuaternionQ and GTQMultiplication in GTPack.
Figure 2.7 The application of GTQInverse.
Figure 2.8 The calculation of the conjugate quaternion, the absolute value of a quaternion, and the polar angle of a quaternion by means of GTQConjugate, GTQAbs and GTQPolar, respectively.
Figure 2.9 The symmetry transformation IC_2y applied to an equilateral triangle in the xy plane (Numbers are introduced to illustrate the impact of the operations.) (a) Triangle at the beginning; (b) two-fold rotation (rotation by an angle π) about the y-axis; (c) application of inversion I.
Figure 2.10 Calculation of a rotation matrix in spin-space with the help of GTSU2Matrix.
Figure 2.11 SU(2) matrices can be obtained from EULER angles, quaternions, or symbols of symmetry elements with the help of GTGetSU2Matrix. The symbol C_3z denotes a counterclockwise rotation by about the z-axis.

3 Basics Abstract Group Theory

Figure 3.1 Installation of the point group C_4v. To verify that C_4v forms a group GTGroupQ is applied. In a second step, the group conditions are checked manually.¹⁾
Figure 3.2 Multiplication table of the non-Abelian point group C_4v calculated using GTMultTable.
Figure 3.3 Calculating the generators of the group C_4v using GTGenerators. The group was installed in Figure 3.1.
Figure 3.4 Installation of KLEIN’s four group from the multiplication table.
Figure 3.5 The application of GTCenter, GTClasses, and GTClassMultTable for calculating the center, the classes, and the class multiplication table of the point group C_4v.
Figure 3.6 Calculating the normalizer of C_s with respect to the covering group D_3h.
Figure 3.7 Calculating the left and right cosets of the subgroups C_s and C_2v with respect to C_4v by applying GTLeftCosets and GTRightCosets.
Figure 3.8 Determining the invariant subgroups of C_4v using GTInvSubGroups.
Figure 3.9 Calculate the multiplication table of the quotient group C_4v/C₄ using GTQuotientGroup.
Figure 3.10 Application of GTProductGroup to represent D_3h as the semidirect product .

4 Discrete Symmetry Groups in Solid-State Physics and Photonics

Figure 4.1 Illustration of high symmetry points and axes for a cube and for a hexagon. The labels are used for the denotation of symmetry elements in GTPack. (a) High symmetry points of a cube; (b) high symmetry points of a hexagon.
Figure 4.2 Installation of new symmetry elements using GTInstallAxis.
Figure 4.3 Illustration of symmetry elements within the point groups C_3v (a) and D_3h (b).
Figure 4.4 Twofold and threefold rotation axes of the point group T_d, illustrated using GTShowSymmetryElements.
Figure 4.5 Subgroup hierarchy of the 32 point groups (C_3i = S₆). Dashed and dotted lines are used to increase the discriminability.
Figure 4.6 Relationships between the ten 2-dimensional point groups. The relationship between the ten groups is illustrated in HERMANN–MAUGUIN notation (left) and SCHÖNFLIES notation (right).
Figure 4.7 Relationships between the groups D_3h, C_3h, C₃, C_3v and D₃.
Figure 4.8 Examples for a symmorphic and a nonsymmorphic space group. (a) The one-atomic square lattice is an example for a symmorphic space group. (b) Nonsymmorphic chain of atoms. A combination of reflection σ and nonprimitive translation τ leaves the chain invariant.
Figure 4.9 Receiving information about the space group by using GTSpaceGroups.⁴⁾
Figure 4.10 Construction of the WIGNER–SEITZ cell for a rectangular lattice. The procedure works as follows: Take one lattice site and connect it to all neighboring lattice sites. Within a 2-dimensional (3-dimensional) lattice, go to the middle of each connecting line and construct an orthogonal line (plane). Take all crossing points of two lines (three planes) and form the convex hull.
Figure 4.11 Reading the information of a cif-file and transforming them to GTPack standard form using GTImportCIF.
Figure 4.12 Construction of a 2-dimensional WIGNER–SEITZ cell using GTVoronoiCell.
Figure 4.13 Calculating the invariant subgroups of C_4v using GTInvSubGroups.
Figure 4.14 Coloring schemes corresponding to SHUBNIKOV groups of the third kind for a square.
Figure 4.15 Installation of magnetic point groups in GTPack.
Figure 4.16 Construction of magnetic lattices [61]. (a) Primitive lattice; (b) colored lattice; (c) magnetic structure of NiO [62].
Figure 4.17 Structural parameters for an (n, m)=(9, 0) zigzag nanotube.
Figure 4.18 Construction of carbon nanotubes by cutting a rectangular region out of a graphene layer. The procedure is demonstrated for (3,3) armchair and (3,0) zigzag nanotubes. Three unit cells of the tubes are cut out.
Figure 4.19 Structure of nanotubes. (a) One unit cell, (c) three unit cells of a (5,5) armchair nanotube; (b) one unit cell, (d) three unit cells of a (5,0) zigzag nanotube.
Figure 4.20 Implementation and properties of the groups D_n.
Figure 4.21 C₆₀ molecule and check of some symmetry properties.
Figure 4.22 Installation of the groups I and I_h and symmetry properties of C₆₀.

5 Representation Theory

Figure 5.1 Installation of the regular representation of the point group C₄v using GTRegular-Representation.
Figure 5.2 Installation of irreducible representations of the point group O using GTGetIrep.
Figure 5.3 Calculation of the CLEBSCH–GORDAN sum of two irreducible representations of the point group O using GTClebschGordanSum. The irreducible representations irep1 and irep3 were calculated in Example 21.
Figure 5.4 Verifying the orthogonality theorem for the point group C_3v.
Figure 5.5 Calculation of the characters of the regular representation of the point group C_4v.
Figure 5.6 The character table of the point group D_2h calculated using GTCharacterTable.
Figure 5.7 The character table of the point group O_h calculated using GTCharacterTable. For the denotation of the irreducible representations the MULLIKEN notation is used [90].
Figure 5.8 Decomposition of the regular representation of the point group C_4v using GTIrep.
Figure 5.9 Application of the projection operator and the character projection operator using GTProjectionOperator and GTCharProjectionOperator.
Figure 5.10 Application of the character projection operator of the irreducible representation T₁ of the point group O to the Cartesian tesseral harmonics with l = 2.
Figure 5.11 Application of the transformation operator of the element C_4z to the basis functions of the irreducible representation T₂ of the point group O (Functions are defined in Figure 5.10).
Figure 5.12 Calculation of the direct product representation E ⊗ E for the group C_3v using GTDirectProductRep.
Figure 5.13 Calculation of the character system of the direct product representation Γ^E⊗E of the point group C_3v.
Figure 5.14 Evaluation of CLEBSCH–GORDAN coefficients for the direct product representation E ⊗ E of the point group C_3v.
Figure 5.15 Block diagonalization of the direct product representation E ⊗ E of the point group C_3v. The representation matrices ΓE and the CLEBSCH–GORDAN coefficients cg1, cg2 and cg3 were calculated in Figure 5.14.
Figure 5.16 Character table of C₄.
Figure 5.17 The point group O_h can be written as the semidirect product .
Figure 5.18 The character tables of C_i, T_d and O_h.

6 Symmetry and Representation Theory in k-Space

Figure 6.1 Calculation of the reciprocal basis for a monoclinic lattice using GTReciprocalBasis.
Figure 6.2 BRILLOUIN zone for the simple cubic lattice and the cesium chloride structure.
Figure 6.3 Calculation of the group of the wave vector k as well as the star of k for the high symmetry points Γ and M and the high symmetry line Σ for the cesium chloride lattice.
Figure 6.4 Construction of the BRILLOUIN zone for the body-centered cubic lattice using GTVoronoiCell. The information about high symmetry points are obtained using GTBZPath.
Figure 6.5 Calculation of the character table of the monoclinic space group P2₁∕c (#14) at the Γ and Y point using GTCharacterTableOfK.

7 Solution of the Schrödinger Equation

Figure 7.1 Local symmetry groups of certain molecules and atomic clusters [107, 108]. (a) 1,3,5-Trithiane, (CH₂S)₃, C_3v; (b) borazine, B₃N₃H₆, D_3h; (c) nickel tetracarbonyl, Ni(CO)₄, T_d; (d) magnesium vacancy in a magnesium oxide crystal, O_h.
Figure 7.2 Splitting of d-states in a crystal field. (a) Octahedral crystal field with O_h symmetry. (b) Uniaxial strain reduces symmetry to D_4h. (c) Splitting of d-states within the crystal field.
Figure 7.3 Application of GTAngularMomentumChars, GTIrep, and GTCrystalFieldSplitting to estimate the splitting of d-states in crystal fields.
Figure 7.4 Illustration of an octahedral, a cubic, and a tetrahedral crystal field.
Figure 7.5 Splitting of the electronic d-state in an octahedral, a cubic and a tetrahedral field.
Figure 7.6 Calculation of the crystal field splitting of the electronic d-state in an octahedral, a cubic, and a tetrahedral field using GTPack (cf. Figure 7.5).
Figure 7.7 Example of illustrating and retrieving radial expectation values ⟨rl⟩ from a database in GTPack.
Figure 7.8 Calculation of matrix elements of STEVENS operator equivalents.
Figure 7.9 A time-dependent perturbation with duration T, starting at t₀ =–T∕2.
Figure 7.10 Verify the transformation behavior of the perturbations A_0,x x, A_0,y y, and A_0,z z.
Figure 7.11 Possible transitions of an initial state transforming like the first row of the irreducible representation E of D₄ into a final state transforming like another irreducible representation of D₄ for the perturbations A_0,x x, A_0,y y, and A_0,z z.
Figure 7.12 Verify the transformation behavior of the perturbations A_0,x x, A_0,y y, and A_0,z z.

8 Generalization to Include the Spin

Figure 8.1 Isomorphism between an abstract point group , the faithful representation , and the group of the SCHRÖDINGER equation .
Figure 8.2 Installation of the double group of O and O_h using GTGroupFromGenerators.
Figure 8.3 Installation of the double group of O and O_h using GTInstallGroup.
Figure 8.4 The character table of the double group .
Figure 8.5 Application of GTSpinCharacters and GTIrep to estimate the splitting of the T₂ state due to spin–orbit coupling.
Figure 8.6 Application of GTSOCSplitting to calculate the influence of spin–orbit coupling to states with cubic symmetry.
Figure 8.7 Application of GTReality to the irreducible representations of .
Figure 8.8 Specifying the option GOReality within the command GTCharacterTable prints the information about the reality of the irreducible representations.

9 Electronic Structure Calculations

Figure 9.1 BRILLOUIN zone of the fcc structure and band structure of Cu along lines of high symmetry in the BRILLOUIN zone. The band structure is taken from [121] (BSW notation used). (With permission, Copyright © 1963 American Physical Society.)
Figure 9.2 Compatibility of irreducible representations occurring at the symmetry point Γ and the symmetry line Δ in the BRILLOUIN zone of the fcc structure.
Figure 9.3 Construction of symmetrized plane waves at H in the body-centered structure. O_h is in this case.
Figure 9.4 Molecular graph of benzene. Carbon atoms numbered for discussion of symmetry operations.
Figure 9.5 Construction of a faithful representation and decomposition into irreducible representations of D_6h.
Figure 9.6 Part of the character table of D_6h corresponding to the irreducible representations occurring in the decomposition given in (9.25). The classes are characterized by elements selected in Figure 9.5.
Figure 9.7 Construction of symmetry-adapted linear combinations (SALC’s) for the irreducible representations A_2u and B_2g of the point group D_6h. The lists a2u, b2g contain the characters of the irreducible representations.
Figure 9.8 Schematic view of the different two-center integrals up to l = 2. Real spherical harmonics are used. Positive lobes are depicted dark and negative lobes are depicted light.
Figure 9.9 Analytic decomposition of three-center integrals E_i,j (R) in two-center form, for the interaction of a p_z orbital with orbitals s, p_y, p_z, p_x(m = 0, –1, 0, 1).
Figure 9.10 Construction of the tight-binding Hamiltonian for graphene.
Figure 9.11 Calculation of π bands of graphene.
Figure 9.12 Calculation of the complete band structure of graphene. Parameters are taken from MIN et al. [129].
Figure 9.13 Number of independent parameters for the first three shells in a three-center tight-binding Hamiltonian for a simple-cubic structure and a spd-basis. (a) First shell; (b) second shell; (c) third shell.
Figure 9.14 Construction of a three-center Hamiltonian, simple cubic structure, first neighbor shell.
Figure 9.15 Construction of a three-center Hamiltonian, simple cubic structure second and third neighbor shells.
Figure 9.16 Calculation of the band structure of GaAs with form factors from COHEN and BERGSTRESSER [142].
Figure 9.17 Construction of a tight-binding Hamiltonian in two-center approximation for fcc structures with spd-basis. Copper represents the prototype structure.
Figure 9.18 Calculation of band structure and density of states of Au without spin–orbit coupling.
Figure 9.19 Symmetry analysis of the band structure of Au.
Figure 9.20 Tight-binding Hamiltonian of Au including spin–orbit interaction.
Figure 9.21 Symmetry analysis for the relativistic band structure of Au.
Figure 9.22 Relativistic band structure of Au.
Figure 9.23 FERMI surface of Cu (E_F = 0.5688 Ry). Additionally, the BRILLOUIN zone and the standard path for band structure calculations are shown.²⁰⁾
Figure 9.24 Cuts through the FERMI surface of Cu (cf. Figure 9.23) The orientation of the cut plane is defined by its normal vector n. The vector s is a shift of the plane with respect to the Γ point. The vectors n, s are given in units 2π/a. The BRILLOUIN zone boundary is also indicated (dashed lines). (a) n = (0, 0, 1), s = (0, 0, 0); (b) n =(1, 1, 0), s =(0, 0, 0); (c) n =(1, 1, 1), s =(.5, .5, .5).
Figure 9.25 Calculation of band structure and density of states for a (5,5) armchair carbon nanotube (π-band approximation,
Figure 9.26 Band structures for armchair and zigzag carbon nanotubes. (a) (5,5) Armchair; (b) (9,0) zigzag; (c) (10,0) zigzag.
Figure 9.27 Construction of a k-space Hamiltonian for the zinc blende structure.
Figure 9.28 Band structure and density of states for GaAs, calculated by means of the tight-binding model according to VOGL et al. [156]. (a) Band structure of GaAs. (b) Density of states of GaAs (916 k points in the irreducible part of the BRILLOUIN zone).
Figure 9.29 Nearest neighbor interaction of p-electrons of atoms A and B. DM, DN, and DL are the direction cosines.
Figure 9.30 Construction of a real-space tight-binding Hamiltonian for GaAs.
Figure 9.31 Density of states of a real-space tight-binding calculation of GaAs (dashed line) in comparison with the corresponding k-space calculation (solid line). (a) Comparison of DOS (real-space cluster of 74 atoms); (b) comparison of DOS (real-space cluster of 2146 atoms).
Figure 9.32 Weights of eigenstates for a real-space tight-binding calculation of GaAs at cluster sites of a 74 atom cluster. (a) State 334 (E₃₃₄ = 1.0026 eV); (b) state 302 (E₃₀₂ =–1.9984eV).
Figure 9.33 Band structure of GaAs with spds basis and spin–orbit coupling (parameters taken from [158]) (a) Energy levels at Γ point (adapted from [159]); (b) band structure in the entire BRILLOUIN zone.
Figure 9.34 Influence of spin–orbit coupling on degenerated eigenstates at the Γ point within the BRILLOUIN zone for GaAs. Because of spin–orbit coupling, the threefold degenerate state belonging to Γ₁₅ is split into the twofold degenerated state Γ₈ and Γ₇. The plot shows the energy dispersion along L—Γ—X in the neighborhood of Γ. (a) Splitting of into Γ₇ and Γ₈; (b) splitting of into Γ₇ and Γ₈.
Figure 9.35 A rearrangement of the basis allows one to work with a Hamiltonian of smaller size (Hamiltonian at Γ point). The eigenvalues of the 3 × 3 p-matrix reveal the spin–orbit splitting of Γ₁₅.
Figure 9.36 Crystal structure of SrTiO₃. The light gray atoms represent Ti, the central atom is Sr, and the dark small atoms represent oxygen.
Figure 9.37 Construction of a k-space model Hamiltonian for the perovskite structure.
Figure 9.38 Model band structure of the perovskite SrTiO₃. The following parameters (in eV) are used (see [172], p. 83): E_e = –5.5, E_t = – 6.5, E_∥ = –10.5, E_⊥ = –10.0, (pdσ) = –2.0, (pdπ)= 1.0, (ppσ) = –0.15, (ppπ) = 0.05.

10 Solution of Maxwell’s Equations

Figure 10.1 Definition of the transverse magnetic (TM) and the transverse electric (TE) modes.
Figure 10.2 Distribution of ε in a square unit cell. The permittivity is constant within each of the N = 5 patches. Local coordinate systems are used for each individual patch.
Figure 10.3 A prismatic rod in general position in the unit cell. It is shifted with respect to the unit cell frame and rotated with respect to its standard orientation in the x′–y′ coordinate system.
Figure 10.4 Definition of ε(r) by means of a pixel map or an ensemble of objects in the unit cell.
Figure 10.5 Comparison of a FOURIER expansion for a smoothed and a nonsmoothed profile of the permittivity. (a) FOURIER coefficients of a slab with filling factor γ = 0.8 and a permittivity of ε_s = 12 of the slab and a background value ε_b = 1. The full width at half-maximum of the Gaussian is σ = 0.01. (b) Left: Number of plane waves N_G = 201 for the stepped ε(r)-profile (gray) but N_G = 101 for the smoothed profile (black solid line). Right: Comparison of the stepped profile with the smoothed profile (γ = 0.8, ε_s = 12, ε_b = 1, σ = 0.01).

11 Two-Dimensional Photonic Crystals

Figure 11.1 Empty lattice photonic band structure for a square lattice.
Figure 11.2 Determination of the irreducible representations at the high symmetry points Γ, X and M of the empty lattice band structure for the square lattice shown in Figure 11.1. The example shows the application of GTPwEmptyLatticeIrep.
Figure 11.3 Photonic band structures of circular rods of alumina (ε = 8.9, R = 0.2a) in a square lattice calculated using MPB. TE – dashed line, TM modes – solid line.
Figure 11.4 Calculation of photonic band structures for dielectric rods in air using the analytic form of κ(G) (solid line) and a definition of ε(r) by means of a pixel map (dashed line).
Figure 11.5 Comparison of the photonic band structures for dielectric rods in air calculated by MPB (solid line) and GTPack (dashed line) for different numbers of reciprocal lattice vectors.
Figure 11.6 Determination of the symmetry of the second and third band at the M point of the photonic band structure of dielectric alumina rods shown in Figure 11.3.
Figure 11.7 Symmetry analysis of a photonic band structure using GTPhSymmetryBands. The band structure refers to the example of dielectric alumina rods in air presented in Section 11.1.2.
Figure 11.8. Rods (ε = 8.9) with quadratic and rectangular cross section in air in a square lattice. The filling factor is γ = 0.08.
Figure 11.9 Photonic bands and density of states of the ideal structure (square lattice with rods in air, TM-polarization, ε = 9.0, r = 0.378a). (a) Band structure; (b) density of states.
Figure 11.10 Band structure of the perfect structure calculated using a 2 × 2 supercell. (a) Band structure of the 2 × 2 supercell. (b) Quarter of the BRILLOUIN zone of the primitive cell and the supercell.
Figure 11.11 Folding of the first band of Figure 11.10a into the smaller BZ of the 2 × 2 super-cell. (a) Band calculated with respect to the folding procedure. (b) Band folded into the smaller BZ of the supercell.
Figure 11.12 Eigenfrequencies of the localized defect modes in the three gaps of the TM band structure as a function of the permittivity ε_d of the defect rod. Numbers in brackets behind the names of the irreducible representations denote the multiplicity. The picture is taken from [183]. (With permission, Copyright © 1997, American Physical Society.)
Figure 11.13 Determination of the symmetry of the defect modes for ε_d = 15 in the first and second gap using GTPhMPBFields and GTPhSymmetryField. The result agrees with the symmetry classification given in Figure 11.12.
Figure 11.14 Basis functions of the irreducible representations of C_4v in terms of external plane waves.
Figure 11.15 Calculating the compatibility relation between C_4v and C_s ⊂ C_4v using GTCom-patibility.
Figure 11.16 Uncoupled bands in the square lattice (a) Band structure along Γ–X (dashed line: uncoupled bands); (b) full DOS along Γ–X (dashed line) and DOS without uncoupled bands (solid lines).

12 Three-Dimensional Photonic Crystals

Figure 12.1 Symmetrized plane waves at the X and Γ point of the BRILLOUIN zone for a simple cubic lattice. Only such irreducible representations are selected where the projection gives nonzero results.
Figure 12.2 Irreducible representations at the Γ and X point of the BRILLOUIN zone of the simple cubic lattice (empty lattice test).
Figure 12.3 Irreducible representations at the L and W point of the BRILLOUIN zone of the face-centered cubic lattice (empty lattice test).
Figure 12.4 Compatibility relations between the irreducible representations along Γ–Δ–X for the simple cubic and face-centered cubic structure.
Figure 12.5 Band structure and density of states for dielectric spheres in air in a simple cubic lattice (ε = 13, r = 0.3a), dotted line: GTPack using the matrix inversion method for κ(G, G′), solid line: MPB calculation. (a) Photonic band structure; (b) density of states.
Figure 12.6 Photonic bands for dielectric spheres in air in a simple cubic lattice (ε = 13, r = 0.3a) with symmetry analysis. (a) Output of the symmetry analysis by means of GTPhSymmetryBands; (b) plot after using GTBandsPlotImprove and/or interactive drawing tools.
Figure 12.7 Tesseral spherical harmonics as basis functions of irreducible representations of the point group C_4v (corresponds to N-waves).
Figure 12.8 Symmetry-adapted vector spherical waves for point group O_h. ohctN is the character table of the point group, ohctM is the character table corrected to include the sign change for improper rotations. Spherical harmonics that contribute to a certain irreducible representation are marked by “x”.

13 Group Theory of Vibrational Problems

Figure 13.1 Structure and symmetry of the ammonia molecule NH₃ (from the Mathematica database with GTMolChemicalData). (a) Structure of the ammonia molecule; (b) C_3v symmetry of the ammonia molecule (top view).
Figure 13.2 Permutation representation and displacement representation for the ammonia molecule NH₃.
Figure 13.3 Decomposition of the displacement representation of the ammonia molecule NH₃.
Figure 13.4 Structure of the planar XeF₄ molecule with D_4h symmetry.
Figure 13.5 Infrared and RAMAN selection rules for Xenontetrafluoride XeF₄ investigated using GTVibSpectroscopy.
Figure 13.6 Construction of the dynamical matrix and calculation of the phonon spectrum for a simple cubic lattice within GTPack.
Figure 13.7 Phonon band structure and density of states for crystalline argon in an fcc structure calculated using GTVibDynamicalMatrix, GTBandStructure and GTDensityOfStates (k₁ = 9.0, k₂ = 0.3, m = 1). (a) Phonon band structure. Dots show experimental values according to [207]. (b) Phonon density of states.
Figure 13.8 Calculation of the phonon band structure for a fcc lattice within the central force model from a tight-binding p-Hamiltonian.
Figure 13.9 Construction of zone center modes for BaTiO₃.

14 Landau Theory of Phase Transitions of the Second Kind

Figure 14.1 Body-centered tetragonal structure with symmetry elements. The A-atoms are located at the corners of the unit cell, while the B atom is located at the inversion center. (Plots to illustrate the reflection planes and rotation axes of D_4h can be obtained using GTShowSymmetryElements). (a) Reflection planes; (b) rotation axes.
Figure 14.2 Application of GTSymmetryElementQ to show that the point group of the body-centered tetragonal lattice is given by D_4h.
Figure 14.3 Symmetry groups of the low-symmetric phase in dependence on the shift of atom B.
Figure 14.4 Invariance properties of the quadratic terms within the series expansion of the GIBBS free energy.
Figure 14.5 Invariance properties of the quartic terms within the series expansion of the GIBBS free energy.
Figure 14.6 Construction invariant polynomials of C_3v up to degree p = 4.
Figure 14.7 Characters of symmetrized and antisymmetrized powers of a representation (c3vct contains the charcter table of the point group C_3v).
Figure 14.8 Example for the investigation of the LIFSHITZ criterion.

Appendix A: Spherical Harmonics

Figure A.1 Spherical coordinate system.
Figure A.2 Tesseral harmonics plotted from top l = 0 to bottom l = 4 and from left m = –l to right m = l using GTTesseralHarmonicY and SphericalPlot3D.

Appendix C: Use of MPB together with GTPack

Figure C.1 Density of states for dielectric rods in air (r = 0.2a, ε = 8.9). Bands are calculated by means of MPB, DOS is calculated with GTPack. TE modes – dashed line, TM modes – solid line.

Appendix D: Technical Remarks on GTPack

Figure D.1 Illustration of the subpackage structure of GTPack, its interaction with external databases, and interface to external ab initio software.

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Preface

Symmetry principles are present in almost all branches of physics. In solid-state physics, for example, we have to take into account the symmetry of crystals, clusters, or more recently detected structures like fullerenes, carbon nanotubes, or quasicrystals. The development of high-energy physics and the standard model of elementary particles would have been unimaginable without using symmetry arguments. Group theory is the mathematical approach used to describe symmetry. Therefore, it has become an important tool for physicists in the past century.

In some cases, understanding the basic concepts of group theory can become a bit tiring. One reason is that exercises connected to the definitions and special structures of groups as well as applications are either trivial or become quickly tedious, even if the concrete calculations are mostly elementary. This occurs, especially, when a textbook does not offer additional help and special tools to assist the reader in becoming familiar with the content. Therefore, we chose a different approach for the present book. Our intention was not to write another comprehensive text about group theory in solid-state physics, but a more applied one based on the Mathematica package GTPack. Therefore, the book is more a handbook on a computational approach to group theory, explaining all basic concepts and the solution of symmetry-related problems in solid-state physics by means of GTPack commands. With the length of the manuscript in mind, we have, at some points, omitted longer and rather technical proofs. However, the interested reader is referred to more rigorous textbooks in those cases and we provide specific references. The examples and tasks in this book are supposed to encourage the reader to work actively with GTPack.

GTPack itself provides more than 200 additional modules to the standard Mathematica language. The content ranges from basic group theory and representation theory to more applied methods like crystal field theory and tight-binding and plane-wave approaches to symmetry-based studies in the fields of solid-state physics and photonics. GTPack is freely available online via GTPack.org. The package is designed to be easily accessible by providing a complete Mathematica style documentation, an optional input validation, and an error strategy. Therefore, we believe that also advanced users of group theory concepts will benefit from the book and the Mathematica package. We provide a compact reference material and a programming environment that will help to solve actual research problems in an efficient way.

In general, computer algebra systems (CAS) allow for a symbolic manipulation of algebraic expressions. Modern systems combine this basic property with numerical algorithms and visualization tools. Furthermore, they provide a programming language for the implementation of individual algorithms. In principle, one has to distinguish between general purpose systems like, e.g., Mathematica and Maple, and systems developed for special purposes. Although the second class of systems usually has a limited range of applications, it aims for much better computational performance. The GAP system (Groups, Algorithms, and Programming) is one of these specialized systems and has a focus on group theory. Extensions like the system Cryst, which was built on top of GAP, are specialized in terms of computations with crystallographic groups.

Nevertheless, for this book we decided to use Mathematica, as Mathematica is well established and often included in the teaching of various Physics departments worldwide. At the Department of Physics of the Martin Luther University Halle-Wittenberg, for example, specialized Mathematica seminars are provided to accompany the theoretical physics lectures. In these courses, GTPack has been used actively for several years.

During the development of GTPack, two paradigms were followed. First, in the usual Mathematica style, the names of commands should be intuitive, i.e., from the name itself it should become clear what the command is supposed to be applied for. This also implies that the nomenclature corresponds to the language physicists usually use in solid-state physics. Second, the commands should be intuitive in their application. Unintentional misuse should not result in longer error messages and endless loop calculations but in an abort with a precise description of the error itself. To distinguish GTPack commands from the standard Mathematica language, all commands have a prefix GT and all options a prefix GO. Analogously to Mathematica itself, commands ending with Q result in logical values, i.e., either TRUE or FALSE. For example, the new command GTGroupQ[list] checks if a list of elements forms a group.

The combination of group theory in physics and Mathematica is not new in its own sense. For example, the books of EL-BATANOUNY and WOOTEN [1] and MCCLAIN [2] also follow this concept. These books provide many code examples of group theoretical algorithms and additional material as a CD or on the Internet. However, in contrast to these books, we do not concentrate on the presentation of algorithms within the text, but provide well-established algorithms within the GTPack modules. This maintains the focus on the application and solution of real physics problems. References for the implemented algorithms are provided whenever appropriate.

In addition to applications in solid-state physics we also discuss photonics, a field that has undergone rapid development over the last 20 years. Here, instead of discussing the symmetry properties of the Schrödinger, Pauli, or Dirac equations, Maxwell’s equations are in the focus of consideration. Analogously to the periodic crystal lattice in solids, periodically structured dielectrics are discussed. GTPack can be applied in a similar manner to both fields.

The book itself is structured as follows. After a short introduction, the basic aspects of group theory are discussed in Part One. Part Two covers the application of group theory to electronic structure theory, whereas Part Three is devoted to its application to photonics. Finally, in Part Four two additional applications are discussed to demonstrate that GTPack will be helpful also for problems other than electronic structure and photonics.

GTPack has a long history in terms of its development. In this context, we would like to thank Diemo Ködderitzsch, Markus Däne, Christian Matyssek, and Stefan Thomas for their individual contributions to the package. We would especially like to acknowledge the careful work of Sebastian Schenk, who contributed significantly to the implementation of the documentation system. Furthermore, we would like to thank Kalevi Kokko, Turku University Finland, who provided a silent work place for us on several occasions. At his department, we had the opportunity to concentrate on both the book and the package and many parts were completed in this context. This was a big help. We acknowledge general interest and support from Martin Hoffmann and Arthur Ernst. Also we would like to thank Wiley-VCH, especially Waltraud Wüst, Martin Preuss and Stefanie Volk.

Lastly, we would like to thank our families for their patience and support during this long-term project.

Stockholm and Halle (Saale), October 2017

R. Matthias Geilhufe, Wolfram Hergert

Group Theory in Solid State Physics and Photonics

Problem Solving with Mathematica

Preface

Part One
Basics of Group Theory