The effect of time reversal symmetry on vibrational modes in ZnO and related compounds: GaN, CdS, BeO, ZnS, CdSe

Vibrational states of a crystal are classified according to the irreducible representations (irrps) of the corresponding factor groups Gk/T. The wave vector k runs over the entire first Brillouin Zone (BZ). For a hexagonal BZ the factor groups are determined by high symmetry points Γ=GM,M,K,L,H, and lines R,Q,S,Δ=LD, Σ=SM, Λ=LD,T,U, and P. The generators of irrps and the characters of the corresponding factor groups GkΓ/T,…,GkH/T and GkR/T,…,GkU/T have been tabulated (CDML 1979) [A.P. Cracknell, B.L. Davies, S.C. Miller, W.F. Love, Kronecker Product Tables, vol. 4, IFI/Plenum Press, New York, Washington, London, 1979]. When the irrps are complex, the time reversal symmetry must be taken into account. Using the Wigner criterion adapted to space groups on real and complex irrps, we have investigated high symmetry points and lines of hexagonal crystals with the common space group C6v4−P63/mc. We have found that the A1−6,L1−6,H1,2,S1,2,Δ1−6,U1−4,P1,2,3, and F1,2 irrps are complex. Therefore, an extra degeneracy of phonons arises. For example, a phonon with momentum ħkA1 (twofold generate) will be classified according to the A1⊕A1∗ representation. Experimental dispersion curves obtained by the neutron scattering technique for ZnO and related materials confirm the existence of time reversal symmetry in wurtzite crystals.