Searching strong `spin'-orbit coupled one-dimensional hole gas in strong magnetic fields

Abstract

We show that a strong `spin'-orbit coupled one-dimensional (1D) 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 $E={\hslash }^2{k}_z^2/(2m_h^{\ast })+\hat{I}\pm \ddot{I}{ƒ}^z{k}_z+g_h^{\ast }\hat{I}{¼}_{B}B\ddot{I}{ƒ}^x/2$, a form exactly the same as that of the electron gas in the conduction band. Here the Pauli matrices $\ddot{I}{ƒ}^{z,x}$ 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 $m_h^{\ast }$ ($0.065\sim 0.08m_e$) and the `spin'-orbit coupling $\hat{I}\pm$ ($0.35\sim 0.8$ eV~à ) have a small magnetic field dependence in the studied magnetic field interval $1<B<15$ T, while the effective $g$-factor $g_h^{\ast }$ of the hole `spin' only has a small magnetic field dependence in the large field region.

14 pages, 5 figures