Criteria for Real and Complex Irreducible Representation. Point Groups

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Lattice Mode Representationfor Al2O3

In this section we derive the LMR. Our aim is to obtain the total number of the first order non-interacting modes, their symmetry and degeneracy. To this end we introduce a basis consisting of displacments and (stretch and bending) bond length angles between atoms in the unit cell. Imposing symmetry operations onto the basis, we generate the matrices of LMR. Using the multipication table Table A.1 and generator matrices we obtain the characters of the LMR. The first explicit derivation of a LMR of a full set of reducible matrices has been performed for GaN, which belongs to space group C 4 6v /T [20].

Sapphire

The primitive cell of Al2O3 is rhombohedral, containing two formula units. Each unit contains three oxgygen and two aluminium atoms. Taking the two unit formula in the primitive cell we obtain ten atoms with three degrees of freedom each, which yields thirty phonon modes for sapphire. In here, we define the 27-basis vector consisting of three (3) displacements and twenty four (24) angles for the reducible LMR as shown in the Figure (see Appendix B-1, page 86). Acting on the basis vector by symmetry operators we obtain the LMR. It is sufficient to generate three 27 × 27-matrices using the three augmenters (generators) C + 3 (3), {C 00 21/τ} (7.1) and {σv1|τ} (19.1). The angle between x-axis and y-axis is 60◦ . The z-axis is perpendicular to both x- and y-axes. The operator {σv1|τ} (19.1) vertical mirror, is placed along the y-axis and perpendicular to the axis of the symmetry operator {C 00 21|τ} (7.1), whose axis is contained in the oxygen basal planes [29].

Spinors. Double Valued Spin Irreducible Representation

Owing to large energy gap of sapphire Eg = 9.3 eV the formation of excitons in sapphire is rather unlikely and consequently the excitonic transitions will not be discussed. An exciton (electron-hole bounded complex) symmetry results in KP of electron and hole symmetries. In ZnO we have three free main excitons: A. ΓCB 7 ⊗ Γ V B 9 , B. ΓCB 7 ⊗ Γ V B 8 and C. ΓCB 7 ⊗ Γ V B 9 . There are a number of bound excitons to neutral and ionized donors and acceptors. It is clear that due to the presence of TRS the electrons in the CB and holes in VB should be classified according to the joint irrps as follows: ΓCB 7 ⊕ (ΓCB 7 ) ∗ , Γ V B 7 ⊕ (ΓV B 7 ) ∗ , ΓV B 8 ⊕ (ΓV B 8 ) ∗ , ΓV B 9 ⊕ (ΓV B 9 ) ∗ and therefore the corresponding exciton symmetries are: A, (ΓCB 7 ⊕(ΓCB 7 ) ∗ )⊗(ΓV B 7 ⊕(ΓV B 7 ) ∗ ); B, (ΓCB 7 ⊕(ΓCB 7 ) ∗ )⊗(ΓV B 8 ⊕(ΓV B 8 ) ∗ ) and C, (ΓCB 7 ⊕ (ΓCB 7 ) ∗ ) ⊗ (ΓV B 9 ⊕ (ΓV B 9 ) ∗ ). In the following section we analyze the effect on TRS on exciton symmetries and interpret the available experimental data.

Excitonic States in ZnO

In this section we briefly provide the necessary group theoretical tools to enable the discussion of excitonic excitations in ZnO and related materials. The spinor representations for ZnO are Γ7, Γ8 and Γ9. Due to TRS the states of electrons in the conduction band and holes in the valence band are classified according to ΓCB 7 ⊕ (ΓCB 7 ) ∗ ,. . . ,ΓV B 9 ⊕ (ΓV B 9 ) ∗ rep. The irrp’s Γ7, Γ8 and Γ9 belong to case (c) and consequently an extra degeneracy is introduced. The wurzite exciton is made up of s-like state ΓCB 1 ⊗ D1/2 = ΓCB 7 in CB and three p-like hole (px, py, pz) orbitals which transform according to the vector rep (ΓV B 1 (z) ⊕ Γ V B 5 (x, y)) ⊗ D1/2 = ΓV B 1 (z) ⊗ D1/2 ⊕ Γ V B 5 (x, y) ⊗ D1/2 = ΓV B 7 ⊕ Γ V B 7 ⊕ Γ V B 9 20 for hole states in VB. The symmetry of excitons are: (ΓCB 7 ⊕ (ΓCB 7 ) ∗ ) ⊗ (ΓV B 9 ⊕ (ΓV B 9 ) ∗ ) and (ΓCB 7 ⊕ (ΓCB 7 ) ∗ ) ⊗ (ΓV B 7 ⊕ (ΓV B 7 ) ∗ ), abbreviated as 7 − 9 exciton and 7 − 7 exciton respectively. Decomposition of the 7−9 and 7−7 excitons onto irrps gives 4Γ1⊕4Γ2⊕4Γ5 and 4Γ5 ⊕ 4Γ6 respectively [18].

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Multiphonon Processes. Selection

Rules As mentioned the number of modes and thier symmetry (degeneracy) of primary noninteracting phonons follows from the LMR, listed in, section 2.3 (subsection 2.3.1). In this section we investigate the multiphonon processes in Al2O3 and ZnO. Higher processes arise from the mutual interaction between first order phonons. The phonon scattering processes are permitted via deformation potential together with Fr¨ohlich interaction [30]. The selection rules for multiphonon processes and the phonon replicas replicas measured by PL are obtained from KP’s governed by WVSRs. The frequencies of phonons can be measured by means of infrared absorption, X-ray, Raman, neutron scattering, and PL.

Electronic Band Structure

The classification of electronic states in the electronic band structure is according to the same irrps used for phonons states. The electronic band structure of sapphire consists of two well separated valence bands and a conduction band. Evaristov et al. considered the band structure theoretical using the self-consistent method with symmetry adapted functions of the sapphire group. They however did not use Time Reversal Symmetry. Nevertheless, they concluded that the bottom of the conduction band has Γ-symmetry. Their simulation yielded a flat band and a larger band gap. The valence band is flat along the Λ−direction, no experimental data were obtained and therefore no conclusive evidence for an indirect band gap was established [37]. They were however able to classify the states of the band structure. In order to gain a complete knowledge of the band structure the matrix representations must be available in order to ascertain whether or not complex conjugation create new representations.

Contents :

1 Introduction
- 1.1 Vibrational Properties of Al2O3 and ZnO
- 1.2 Opto-Electronic Properties of Al2O3 and ZnO
2 Symmetry Aspects
- 2.1 Space Groups: Irreducible Representations and their Characters
  - 2.1.1 Sapphire Al2O3 Structure
  - 2.1.2 Wurtzite: ZnO Structure
- 2.2 Time Reversal Symmetry
  - 2.2.1 Criteria for Real and Complex Irreducible Representation. Point Groups
  - 2.2.2 Criteria for Real and Complex Irreducible Representation. Space Groups
  - 2.2.3 Sapphire Al2O3. Irreducible Representations Under Time Reversal Symmetry
  - 2.2.4 Wurtzite ZnO. Irreducible Representation Under Time Reversal Symmetry
- 2.3 Lattice Mode Representationfor Al2O
  - 2.3.1 Sapphire
- 2.4 Connectivity Relations
- 2.5 Spinor Representation
  - 2.5.1 Spinors. Double Valued Spin Irreducible Representation
  - 2.5.2 Excitonic States in ZnO
3 Multiphonon Processes. Selection Rules
- 3.1 First Order Modes in Sapphire and ZnO
  - 3.1.1 First Order Raman Processes
- 3.2 Second Order Raman Processes
  - 3.2.1 Third Order Raman Processes
4 Electronic Band Structure
5 Experimental Results
- 5.1 Raman and Infrared Modes
- 5.2 Dispersion Curves
6 Conclusion
A Calculations and Tables
- A.1 Multiplication Table for Hexagonal and Trigonal Points Groups
- A.2 Vector Representation for Hexagonal and Trigonal Point/Space Groups
- A.3 Table of Spinor Representations SU(2)
- A.4 Character Tables for Hexagonal C 6v and D6 3d Space groups
- A.4.1 Irreducible Representation and Factor Groups
- A.5 Generators for Trigonal D6 3d and Wurtzite C4 6v Space groups
- A.6 Matrix Representations for D6 3d and C4 6v Space Groups
- A.7 Classification of Irreducible Representations. Reality Test
- A.8 Wave Vector Selection Rules
- A.8.1 Symmetrized Wave Vector Selection Rules
- A.9 Characters for Lattice Mode Representation
- A.10 Table for Connectiviy Relations
- A.11 Kronecker Products
- A.12 Spinors Calculations for Hexagonal and Trigonal Groups
B Figures and Diagrams
- B.1 Lattice Mode Representation
- B.2 Brillouin Zone for Sapphire and Wurtzite
- B.3 Dispersion Curves for Sapphire and Wurtzite
- B.4 Raman Spectra
- B.5 Electronic Band Gap of Sapphire and ZnO
- B.5.1 Discussion of line Γ − ∆ − A for Wurtzite
- B.5.2 Discussion of the Γ − Λ − Z Line in Sapphire
C Publications