  Ultra-wide-band-gap group-III nitrides are of interest for applications in
deep-ultraviolet optoelectronics and power electronics. Such devices must be
able to efficiently dissipate heat generated from their operation, making the
thermal conductivity of the constituent materials an important parameter for
high-power applications. We have investigated the phonon-limited thermal
conductivity of AlN, GaN, and Al$_x$Ga$_{1-x}$N using first-principles
calculations, with a focus on the effects of compositional and isotopic
disorder. Our Boltzmann-transport-equation calculations show that the maximum
thermal conductivity for AlN (GaN) is 348 W m$^{-1}$ K$^{-1}$ (235 W m$^{-1}$
K$^{-1}$) for with pure $^{14}$N, and 292 W m$^{-1}$ K$^{-1}$ for GaN with pure
$^{71}$Ga. Al$_x$Ga$_{1-x}$N alloys reach a minimum thermal conductivity at Al
mole fractions of x = 0.60 to 0.71 over the 100-1000K temperature range. Our
results provide understanding on the effects of isotope disorder on the thermal
conductivity of AlN and GaN. We also present an analytical model for the
evaluation of the thermal conductivity of Al$_x$Ga$_{1-x}$N alloys for
arbitrary composition and temperature, which can be applied for the thermal
design of AlGaN-based electronic and optoelectronic devices.
