  We investigate the electronic, structural and topological properties of the
SnTe and PbTe cubic nanowires using ab-initio calculations. Using standard and
linear-scale density functional theory, we go from the ultrathin limit up to
the nanowires thicknesses observed experimentally. Finite-size effects in the
ultra-thin limit produce an electric quadrupole and associated structural
distortions, these distortions increase the band gap but they get reduced with
the size of the nanowires and become less and less relevant. Ultrathin SnTe
cubic nanowires are trivial band gap insulators, we demonstrate that by
increasing the thickness there is an electronic transition to a spin-orbit
insulating phase due to trivial surface states in the regime of thin nanowires.
These trivial surface states with a spin-orbit gap of a few meV appear at the
same k-point of the topological surface states. Going to the limit of thick
nanowires, we should observe the transition to the topological crystalline
insulating phase with the presence of two massive surface Dirac fermions
hybridized with the persisting trivial surface states. Therefore, we have the
co-presence of massive Dirac surface states and trivial surface states close to
the Fermi level in the same region of the k-space. According to our estimation,
the cubic SnTe nanowires are trivial insulators below the critical thickness
tc1=10 nm, and they become spin-orbit insulators between tc1=10 nm and tc2=17
nm, while they transit to the topological phase above the critical thickness of
tc2=17 nm. These critical thickness values are in the range of the typical
experimental thicknesses, making the thickness a relevant parameter for the
synthesis of topological cubic nanowires. Pb(1-x)Sn(x)Te nanowires would have
both these critical thicknesses tc1 and tc2 at larger values depending on the
doping concentration.
