Dimensional quantization and zero-field spin splitting of holes in GaAs nanowires

Nanowires (NWs) of III-V semiconductor materials have been of interest to researchers for the last two decades. Knowledge of the subband spectrum of charge carriers in NWs and NW-based structures is very important for current applications. The electronic subband spectrum in NWs is currently known in detail, while for holes it is found with significant simplifications. One or more of the following crucial features are usually neglected: the real NW cross section shape, the crystal orientation of the NW, an accounting for the real anisotropic Hamiltonian of the bulk host material, and contributions that are due to the lack of an inversion center in the crystal lattice. Here we present a detailed calculation of hole subbands in GaAs NWs with the [111] orientation with a zinc blende crystal lattice, taking into account all the above four features. The spectrum of hole subbands based on the $4\ifmmode\times\else\texttimes\fi{}4$ Luttinger Hamiltonian is numerically calculated taking into account two main contributions arising from the lack of inversion symmetry (the ${T}_{d}$ point group) in the lattice of the host crystal. Accounting for these contributions leads to the appearance of spin splitting only for some subbands, in accordance with symmetry considerations. However, a significant rearrangement also occurs in the spectrum of nonsplit subbands. The hole densities are visualized, and it is shown that the contribution of terms with ${T}_{d}$ symmetry significantly changes the structure of the multicomponent wave function. Thus, taking into account the lack of an inversion center is essential for the spectrum of hole subbands and wave functions in GaAs NWs. This can be more pronounced for NWs of III-V materials constituted by heavy elements, such as InSb, where spin-orbit interaction is stronger. The effect of a transverse electric field leading to so-called Rashba spin splitting is considered as well.