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

• Published in: Journal of physics. Condensed matter Vol. 35; no. 13; pp. 135302 - 135309
• Main Authors: Li (李睿), Rui, Qi (齐新雨), Xin-Yu
• Format: Journal Article
• Language: English
• Published: England IOP Publishing 05.04.2023
• Subjects: hole ‘spin’; nanowire; subband dispersions; ‘spin’-orbit coupling; hole ‘spin’; subband dispersions; ‘spin’-orbit coupling; nanowire
• ISSN: 0953-8984; 1361-648X; 1361-648X
• DOI: 10.1088/1361-648X/acb8f5

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.