Spin of valence-band holes in wurtzite semiconductors

Spin polarization properties of a ballistic hole in wurtzite semiconductors are analyzed using the concept of spin surface. It is shown that, in general, the spin surface in a three-dimensional spin space has the shape of spheroid whose minor/major axis and orientation depend on a relative strength of the spin-orbit and crystal-field interactions as well as on the hole wave vector direction and length. Analytical expressions for the spin surfaces of heavy-mass, light-mass, and split-off bands were found for two experimentally important cases, when the hole wave vector is parallel and perpendicular to the hexagonal axis. For an arbitrary direction of the wave vector the numerical results are presented which show that the spin surface remains the spheroid in this case as well. However, the surface transforms to a line or sphere at small and large wave vectors. The properties of the spin surfaces are illustrated for valence band parameters of GaN, where the crystal field dominates in the band-edge splitting.