Searching strong ‘spin’-orbit coupled one-dimensional hole gas in strong magnetic fields

Abstract

Abstract We show that a strong ‘spin’-orbit coupled one-dimensional hole gas is achievable via applying a strong magnetic field to the original two-fold degenerate (spin degeneracy) hole gas confined in a cylindrical Ge nanowire. Both strong longitudinal and strong transverse magnetic fields are feasible to achieve this goal. Based on quasi-degenerate perturbation calculations, we show the induced low-energy subband dispersion of the hole gas can be written as E = ℏ 2 k z 2 / ( 2 m h * ) + α σ z k z + g h * μ B B σ x / 2 , a form exactly the same as that of the electron gas in the conduction band. Here the Pauli matrices σ z,x represent a pseudo spin (or ‘spin’), because the real spin degree of freedom has been split off from the subband dispersions by the strong magnetic field. Also, for a moderate nanowire radius R = 10 nm, the induced effective hole mass m h * ( 0.065 ∼ 0.08 m e ) and the ‘spin’-orbit coupling α (0.35 ∼ 0.8 eV Å) have a small magnetic field dependence in the studied magnetic field interval 1 < B < 15 T, while the effective g-factor g h * of the hole ‘spin’ only has a small magnetic field dependence in the large field region.