  Ga$_2$O$_3$ and (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys are promising materials
for solar-blind UV photodetectors and high-power transistors. Basic key
parameters in the device design, such as band gap variation with alloy
composition and band offset between Ga$_2$O$_3$ and
(Al$_x$Ga$_{1-x}$)$_2$O$_3$, are yet to be established. Using density
functional theory with the HSE hybrid functional, we compute formation
enthalpies, band gaps, and band edge positions of (Al$_x$Ga$_{1-x}$)$_2$O$_3$
alloys in the monoclinic ($\beta$) and corundum ($\alpha$) phases. We find the
formation enthlapies of (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys are significantly
lower than of (In$_x$Ga$_{1-x}$)$_2$O$_3$, and that (Al$_x$Ga$_{1-x}$)$_2$O$_3$
with $x$=0.5 can be considered as an ordered compound AlGaO$_3$ in the
monoclinic phase, with Al occupying the octahedral sites and Ga occupying the
tetrahedral sites. The direct band gaps of the alloys range from 4.69 to 7.03
eV for $\beta$-(Al$_x$Ga$_{1-x}$)$_2$O$_3$ and from 5.26 to 8.56 eV for
$\alpha$-(Al$_x$Ga$_{1-x}$)$_2$O$_3$. Most of the band offset of the
(Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloy arises from the discontinuity in the
conduction band. Our results are used to explain the available experimental
data, and consequences for designing modulation-doped field effect transistors
(MODFETs) based on (Al$_x$Ga$_{1-x}$)$_2$O$_3$/Ga$_2$O$_3$ are discussed.

Materials Science