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 Hamiltonian H-ef = h(2)k(2)z/(2m(h)(*))+ alpha sigma(x)k(z) + g(h)(*)mu B-B sigma(z)/2, as long as the magnetic field is purely longitudinal or purely transverse. The explicit magnetic field dependent expressions of the "spin'-orbit coupling alpha -alpha(B) and the effective g-factor g(h)(*) -g(h)(*)(B) are given. When the magnetic field is applied in an arbitrary direction, the two-band Hamiltonian description is still a good approximation.