Effects of time reversal symmetry on phonons in sapphire substrate for ZnO and GaN

Vibrational states in a crystal are classified according to the irreducible representations (irreps) of the corresponding factor group G0k/T. The wave vector k runs over the entire Brillouin zone (BZ). For trigonal BZs, the factor groups are determined by the symmetry points ΓΓ, F, L, T, and the symmetry lines ΛΛ, ΣΣ, Y. When the irreps are complex, the time reversal symmetry has to be taken into account. Using the Frobenuis–Schur criterion adapted to space groups with real and complex irreps, we have investigated high symmetry points and lines of the phonons in trigonal crystals: Cr2O3,Fe2O3,Ti2O3,V 2O3,FeCO3,CaCO3,CdCO3,MgCO3,MnCO3,NaCO3 and ZnCO3, with the common space group D3d6(R3̄c). We have found several phonons which are influenced by the time reversal symmetry. Therefore, an extra degeneracy of phonons arises. The theoretical results are also compared with available experimental data.