  In the present study we have demonstrated epitaxial stabilization of the
metastable magnetically-hard $\varepsilon$-Fe$_2$O$_3$ phase on top of a thin
MgO(111) buffer layer grown onto the GaN (0001) surface. The primary purpose to
introduce a 4\,nm-thick buffer layer of MgO in between Fe$_2$O$_3$ and GaN was
to stop thermal migration of Ga into the iron oxide layer. Though such
migration and successive formation of the orthorhombic GaFeO$_3$ was supposed
earlier to be a potential trigger of the nucleation of the isostructural
$\varepsilon$-Fe$_2$O$_3$, the present work demonstrates that the growth of
single crystalline uniform films of epsilon ferrite by pulsed laser deposition
is possible even on the MgO capped GaN. The structural properties of the 60\,nm
thick Fe$_2$O$_3$ layer on MgO / GaN were probed by electron and x-ray
diffraction, both suggesting that the growth of $\varepsilon$-Fe$_2$O$_3$ is
preceded by formation of a thin layer of $\gamma$-Fe$_2$O$_3$. The presence of
the magnetically hard epsilon ferrite was independently confirmed by
temperature dependent magnetometry measurements. The depth-resolved x-ray and
polarized neutron reflectometry reveal that the 10\,nm iron oxide layer at the
interface has a lower density and a higher magnetization than the main volume
of the $\varepsilon$-Fe$_2$O$_3$ film. The density and magnetic moment depth
profiles derived from fitting the reflectometry data are in a good agreement
with the presence of the magnetically degraded $\gamma$-Fe$_2$O$_3$ transition
layer between MgO and $\varepsilon$-Fe$_2$O$_3$. The natural occurrence of the
interface between magnetoelectric $\varepsilon$- and spin caloritronic
$\gamma$- iron oxide phases can enable further opportunities to design novel
all-oxide-on-semiconductor devices.
