  Cation site disorder provides a degree of freedom in the growth of ternary
nitrides for tuning the technologically relevant properties of a material
system. For example, the band gap of ZnGeN$_2$ changes when the ordering of the
structure deviates from that of its ground state. By combining the perspectives
of carrier localization and defect states, we analyze the impact of different
degrees of disordering on electronic properties in ZnGeN$_2$, addressing a gap
in current studies which focus on dilute or fully disordered systems. The
present study demonstrates changes in the density of states and localization of
carriers in ZnGeN$_2$ calculated using band gap-corrected density functional
theory and hybrid calculations on partially disordered supercells generated
using the Monte Carlo method. We use localization and density of states to
discuss the ill-defined nature of a band gap in a disordered material,
comparing multiple definitions of the energy gap in the context of theory and
experiment. Decreasing the order parameter results in a large reduction of the
band gap in disordered cases. The reduction in band gap is due in part to
isolated, localized states that form above the valence band continuum and are
associated with nitrogen coordinated by more zinc than germanium. The
prevalence of defect states in all but the perfectly ordered structure creates
challenges for incorporating disordered ZnGeN$_2$ into optical devices, but the
localization associated with these defects provides insight into mechanisms of
electron/hole recombination in the material.
