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

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 $E=\hbar^{2}k^{2}_{z}/(2m^{*}_{h})+\alpha\sigma^{z}k_{z}+g^{*}_{h}\mu_{B}B\sigma^{x}/2$, a form exactly the same as that of the electron gas in the conduction band. Here the Pauli matrices $\sigma^{z,x}$ 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 $m^{*}_{h}$ ($0.065\sim0.08~m_{e}$) and the `spin'-orbit coupling $\alpha$ ($0.35\sim0.8$ eV~\AA) 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.