Two-band description of the strong 'spin'-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire.

The low-energy effective Hamiltonian of the strong 'spin'-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire in the presence of a strong magnetic field is studied both numerically and analytically. Basing on the Luttinger-Kohn Hamiltonian in the spherical approximation, we show this strong 'spin'-orbit coupled one-dimensional hole gas can be accurately described by an effective two-band HamiltonianHef=ℏ2kz2/(2mh∗)+ασxkz+gh∗μBBσz/2, as long as the magnetic field is purely longitudinal or purely transverse. The explicit magnetic field dependent expressions of the 'spin'-orbit couplingα≡α(B)and the effectiveg-factorgh∗≡gh∗(B)are given. When the magnetic field is applied in an arbitrary direction, the two-band Hamiltonian description is still a good approximation.