  Rutile germanium dioxide (r-GeO$_2$) is a recently predicted
ultrawide-band-gap semiconductor with potential applications in high-power
electronic devices, for which the carrier mobility is an important material
parameter that controls the device efficiency. We apply first-principles
calculations based on density functional and density functional perturbation
theory to investigate carrier-phonon coupling in r-GeO$_2$ and predict its
phonon-limited electron and hole mobilities as a function of temperature and
crystallographic orientation. The calculated carrier mobilities at 300 K are
$\mu_{\text{elec},\perp \vec{c}}$=244 cm$^2$ V$^{-1}$ s$^{-1}$,
$\mu_{\text{elec},||\vec{c}}$=377 cm$^2$ V$^{-1}$ s$^{-1}$,
$\mu_{\text{hole},\perp \vec{c}}$=27 cm$^2$ V$^{-1}$ s$^{-1}$, and
$\mu_{\text{hole},||\vec{c}}$=29 cm$^2$ V$^{-1}$ s$^{-1}$. At room temperature,
carrier scattering is dominated by the low-frequency polar-optical phonon
modes. The predicted Baliga figure of merit of n-type r-GeO$_2$ surpasses
several incumbent semiconductors such as Si, SiC, GaN, and $\beta$-Ga$_2$O$_3$,
demonstrating its superior performance in high-power electronic devices.
