Spin-orbit coupling effects in zinc-blende InSb and wurtzite InAs nanowires: Realistic calculations with multiband $\vec{k} \cdot \vec{p}$ method

A systematic numerical investigation of spin-orbit fields in the conduction bands of III-V semiconductor nanowires is performed. Zinc-blende InSb nanowires are considered along [001], [011], and [111] directions, while wurtzite InAs nanowires are studied along [0001] and [10$\overline{1}$0] or [11$\overline{2}$0] directions. Realistic multiband $\vec{k} \cdot \vec{p}\,$ Hamiltonians are solved by using plane-wave expansions of real-space parameters. In all cases the linear and cubic spin-orbit coupling parameters are extracted for nanowire widths from 30 to 100 nm. Typical spin-orbit energies are on the $\mu$eV scale, except for InAs wurtzite nanowires grown along [10$\overline{1}$0] or [11$\overline{2}$0], in which the spin-orbit energy is about meV, largely independent of the wire diameter. Significant spin-orbit coupling is obtained by applying a transverse electric field, causing the Rashba effect. For an electric field of about 4 mV/nm the obtained spin-orbit energies are about 1 meV for both materials in all investigated growth directions. The most favorable system, in which the spin-orbit effects are maximal, are InAs WZ nanowires grown along [1010] or [11$\overline{2}$0], since here spin-orbit energies are giant (meV) already in the absence of electric field. The least favorable are InAs WZ nanowires grown along [0001], since here even the electric field does not increase the spin-orbit energies beyond 0.1 meV. The presented results should be useful for investigations of optical orientation, spin transport, weak localization, and superconducting proximity effects in semiconductor nanowires.