  The key obstacle toward realizing integrated gallium nitride (GaN)
electronics is its low hole mobility. Here, we explore the possibility of
improving the hole mobility of GaN via epitaxial matching to II-IV nitride
materials that have recently become available, namely ZnGeN$_2$ and MgSiN$_2$.
We perform state-of-the-art calculations of the hole mobility of GaN using the
ab initio Boltzmann transport equation. We show that effective uniaxial
compressive strain of GaN along the $[1\bar{1}00]$ by lattice matching to
ZnGeN$_2$ and MgSiN$_2$ results in the inversion of the heavy hole band and
split-off hole band, thereby lowering the effective hole mass in the
compression direction. We find that lattice matching to ZnGeN$_2$ and MgSiN$_2$
induces an increase of the room-temperature hole mobility by 50% and 260% as
compared to unstrained GaN, respectively. Examining the trends as a function of
strain, we find that the variation in mobility is highly nonlinear; lattice
matching to a hypothetical solid solution of
Zn$_{0.75}$Ge$_{0.75}$Mg$_{0.25}$Si$_{0.25}$N$_2$ would already increase the
hole mobility by 160%.
