Hole subband dispersions and strong `spin'-orbit coupling in a cylindrical Ge nanowire

Quasi-one-dimensional hole gas is achievable in a semiconductor Ge nanowire. The lowest two subband dispersions of the hole gas are just two shifted parabolic curves with an anticrossing at $k_z=0$. This peculiar low-energy subband structure manifests the existence of a strong spin' (pseudo spin)-orbit coupling. Based on the Luttinger-Kohn Hamiltonian in the axial approximation, we show two sets of combined dispersions that not only isolated from each other but also with strong spin’-orbit coupling are obtainable in the presence of strong magnetic field. Realistic calculations are performed for three representative nanowire growth directions [001], [111], and [110]. These results are further confirmed via constructing the low-energy effective Hamiltonian of the hole gas. We also calculate the external electric field induced spin splitting for comparison with the magnetic field induced spin splitting.