  We report on a theoretical study of the electronic structures of the
[111]-oriented, free-standing, zincblende InAs and InP nanowires with hexagonal
cross sections by means of an atomistic $sp^{3}s^{*} $, spin-orbit interaction
included, nearest-neighbor, tight-binding method. The band structures and the
band state wave functions of these nanowires are calculated and the symmetry
properties of the bands and band states are analyzed based on the $C_{3v}$
double point group. It is shown that all bands of these nanowires are doubly
degenerate at the $\Gamma$-point and some of these bands will split into
non-degenerate bands when the wave vector $k$ moves away from the
$\Gamma$-point as a manifestation of spin-splitting due to spin-orbit
interaction. It is also shown that the lower conduction bands of these
nanowires all show simple parabolic dispersion relations, while the top valence
bands show complex dispersion relations and band crossings. The band state wave
functions are presented by the spatial probability distributions and it is
found that all the band states show $2\pi/3$-rotation symmetric probability
distributions. The effects of quantum confinement on the band structures of the
[111]-oriented InAs and InP nanowires are also examined and an empirical
formula for the description of quantization energies of the lowest conduction
band and the highest valence band is presented. The formula can simply be used
to estimate the enhancement of the band gaps of the nanowires at different
sizes as a result of quantum confinement.
