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

Abstract

Abstract 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 e f = ℏ 2 k z 2 / ( 2 m h ∗ ) + α σ x k z + g h ∗ μ B B σ 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 α ≡ α ( 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.