  We report ab initio calculations of the spin splitting of the uppermost
valence band (UVB) and the lowermost conduction band (LCB) in bulk and
atomically thin GaS, GaSe, GaTe, and InSe. These layered monochalcogenides
appear in four major polytypes depending on the stacking order, except for the
monoclinic GaTe. Bulk and few-layer $\epsilon$- and $\gamma$-type, and
odd-number $\beta$-type GaS, GaSe, and InSe crystals are noncentrosymmetric.
The spin splittings of the UVB and the LCB near the $\Gamma$-point in the
Brillouin zone are finite, but still smaller than those in a zinc-blende
semiconductor such as GaAs. On the other hand, the spin splitting is zero in
centrosymmetric bulk and even-number few-layer $\beta$-type GaS, GaSe, and
InSe, owing to the constraint of spatial inversion symmetry. By contrast, GaTe
exhibits zero spin splitting because it is centrosymmetric down to a single
layer. In these monochalcogenide semiconductors, the separation of the
non-degenerate conduction and valence bands from adjacent bands results in the
suppression of Elliot-Yafet spin relaxation mechanism. Therefore, the electron-
and hole-spin relaxation times in these systems with zero or minimal spin
splittings are expected to exceed those in GaAs when the D'yakonov-Perel' spin
relaxation mechanism is also suppressed.
