Rashba effects in bulk wurtzite materials

The spin-splitting energies in strained bulk wurtzite AlN are studied using the linear combination of atomic orbital method. It is found that strain and crystal field induce not only a Rashba linear-$k (\alpha _{wz} )$ but also two Rashba cubic$-k$ terms ($\gamma _R$ and $\lambda _R )$ in the two-band $k\cdot p,$ Hamiltonian $H_{SO} (\vec {k})=(\alpha _{wz} -\gamma _R k_{//}^2 +\lambda _R k_z^2 )(\sigma _x k_y -\sigma _y k_x)+H_{SO}^0 $ where $H_{SO}^0 =(-\gamma _0 k_{//}^2 +\lambda _0 k_z^2 )(\sigma _x k_y -\sigma _y k_x )$ generates a cone-shaped minimum-spin-splitting (MSS) surface and ${\lambda _0} \mathord{\left/ {\vphantom {{\lambda _0} {\gamma _0}}} \right. \kern-\nulldelimiterspace} {\gamma _0}\approx 4$. As tensilely biaxial strain increases, the shape of the MSS surface changes from a hexagonal hyperboloid of two sheets in unstrained AlN to a hexagonal cone, and eventually becomes a hyperboloid of one sheet.