  Using first-principles calculations combined with a semi-empirical van der
Waals dispersion correction, we have investigated structural parameters, mixing
enthalpies, and band gaps of buckled and planar few-layer In$_x$Ga$_{1-x}$N
alloys. We predict that the free-standing buckled phases are less stable than
the planar ones. However, with hydrogen passivation, the buckled
In$_x$Ga$_{1-x}$N alloys become more favorable. Their band gaps can be tuned
from 6 eV to 1 eV with preservation of direct band gap and well-defined Bloch
character, making them promising candidate materials for future light-emitting
applications. Unlike their bulk counterparts, the phase separation could be
suppressed in these two-dimensional systems due to reduced geometrical
constraints. In contrast, the disordered planar thin films undergo severe
lattice distortion, nearly losing the Bloch character for valence bands;
whereas the ordered planar ones maintain the Bloch character yet with the
highest mixing enthalpies.
