  We present a comprehensive computational study of the electronic, thermal,
and thermoelectric (TE) properties of gallium nitride nanowires (NWs) over a
wide range of thicknesses (3--9 nm), doping densities ($10^{18}$--$10^{20}$
cm$^{-3}$), and temperatures (300--1000 K). We calculate the low-field electron
mobility based on ensemble Monte Carlo transport simulation coupled with a
self-consistent solution of the Poisson and Schr\"odinger equations. We use the
relaxation-time approximation and a Poisson-Schro\"dinger solver to calculate
the electron Seebeck coefficient and thermal conductivity. Lattice thermal
conductivity is calculated using a phonon ensemble Monte Carlo simulation, with
a real-space rough surface described by a Gaussian autocorrelation function.
Throughout the temperature range, the Seebeck coefficient increases while the
lattice thermal conductivity decreases with decreasing wire cross section, both
boding well for TE applications of thin GaN NWs. However, at room temperature
these benefits are eventually overcome by the detrimental effect of surface
roughness scattering on the electron mobility in very thin NWs. The highest
room-temperature $ZT$ of 0.2 is achieved for 4-nm-thick NWs, while further
downscaling degrades it. In contrast, at 1000 K, the electron mobility varies
weakly with the NW thickness owing to the dominance of polar optical phonon
scattering and multiple subbands contributing to transport, so $ZT$ increases
with increasing confinement, reaching 0.8 for optimally doped 3-nm-thick NWs.
The $ZT$ of GaN NWs increases with increasing temperature beyond 1000 K, which
further emphasizes their suitability for high-temperature TE applications.
