  Ultra-wide direct band gap semiconductors hold great promise for deep
ultraviolet opto-electronic applications. Here we evaluate the potential of
MgSiN$_2$-GaN alloys for this purpose. Although MgSiN$_2$ itself has an
indirect gap $\sim$0.4 eV below its direct gap of $\sim$6.5 eV, its different
sign lattice mismatch from GaN in two different basal plane directions could
avoid the tensile strain which limits Al$_x$Ga$_{1-x}$N on GaN for high $x$.
Two octet-rule preserving structures (with space groups $Pmn2_1$ and $P1n1$) of
a 50% alloy of MgSiN$_2$ and GaN are investigated and are both found to have
gaps larger than 4.75 eV using quasiparticle self-consistent (QS) $GW$
calculations. Both are nearly direct gap in the sense that the indirect gap is
less than 0.1 eV lower than the direct gap. Their mixing energies are positive
yet small, with values of 8 (31) meV/atom for $Pmn2_1$ ($P1n1$) indicating only
a small driving force toward phase separation.
