Phonons in sapphire Al2O3 substrate for ZnO and GaN

By means of standard method of placing the vectors upon each ion in the basis (2-Al, 3-O) and acting by symmetry operators of sapphire space group onto the basis we obtain Lattice Mode Representation (LMR). The decomposition of the LMR onto irreducible representations (irrps) corresponding to the high symmetry point and lines results in symmetry allowed first order non-interacting modes originating from the entire first Brilloiun zone. First order Raman active modes are determined by symmetrized square of Kronecker products of sapphire vector representation. The selection rules for the second and third order Raman processes follow from the symmetrized Kronecker Products (square and cubes). Frequently the irrps are complex. In such cases the time reversal symmetry (TRS) must be taken into account. We have also investigated the effect of the TRS in sapphire and ZnO which leads to an extra degeneracy.