Hole subband dispersions in a cylindrical Ge nanowire: exact results based on the axial Luttinger-Kohn Hamiltonian.

Based on the Luttinger-Kohn Hamiltonian in the axial approximation, the transcendental equations determining the hole subband dispersions in a cylindrical Ge nanowire are analytically derived. These equations are more general than that derived using the spherical approximation, and are suitable to study the growth direction dependence of the subband dispersions. The axial approximation almost gives rise to the accurate low-energy subband dispersions for high-symmetry nanowire growth directions [001] and [111]. The perturbation correction from the non-axial term is negligible for these two directions. The lowest two subband dispersions can be regarded as two shifted parabolic curves with an energy gap atkz=0for both growth directions [001] and [111]. At the position of the energy gap, the eigenstates for growth direction [111] are inverted in comparison with the normal eigenstates for growth direction [001]. A nanowire growth direction where the energy gap closes atkz=0is predicted to exist between directions [001] and [111].

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

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 asE=ℏ2kz2/(2mh*)+ασzkz+gh*µBBσx/2, a form exactly the same as that of the electron gas in the conduction band. Here the Pauli matricesσz,xrepresent 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 radiusR= 10 nm, the induced effective hole massmh*(0.065∼0.08me)and the 'spin'-orbit couplingα(0.35 ∼ 0.8 eV Å) have a small magnetic field dependence in the studied magnetic field interval 1

Low-energy subband wave-functions and effectiveg-factor of one-dimensional hole gas.

One-dimensional (1D) hole gas confined in a cylindrical Ge nanowire has potential applications in quantum information technologies. Here, we analytically study the low-energy properties of this 1D hole gas. The subbands of the hole gas are two-fold degenerate. The low-energy subband wave-functions are obtained exactly, and the degenerate pairs are related to each other via a combination of the time-reversal and the spin-rotation transformations. In evaluating the effectiveg-factor of these low-energy subbands, the orbital effects of the magnetic field are shown to contribute as strongly as the Zeeman term. Also, near the center of thekzspace, there is a sharp dip or a sharp peak in the effectiveg-factor. At the sitekz= 0, the longitudinalg-factorglis much less than the transverseg-factorgtfor the lowest subband, while away from the sitekz= 0,glcan be comparable togt.