  Rutile GeO$_2$ is a visible and near-ultraviolet-transparent oxide that has
not been explored for semiconducting applications in electronic and
optoelectronic devices. We investigate the electronic and optical properties of
rutile GeO$_2$ with first-principles calculations based on density functional
theory and many-body perturbation theory. Our band-structure calculations
indicate a dipole-forbidden direct band gap at $\Gamma$ with an energy of 4.44
eV and effective masses equal to $m^*_{e\perp}$ =0.43$m_0$ , $m^*_{e||}$
=0.23$m_0$ , $m^*_{h\perp}$ =1.28$m_0$ , and $m^*_{h||}$=1.74$m_0$ . In
contrast to the self-trapped hole polarons by lattice distortions in other
wide-band-gap oxides that reduce the hole mobility, holes in rutile GeO$_2$ are
delocalized due to their small effective mass. The first allowed optical
transitions at $\Gamma$ occur at 5.04 eV ($\vec{E} \perp \vec{c}$) and 6.65 eV
($\vec{E} ||\vec{c} $). We also evaluate the optical absorption coefficient and
refractive index along both crystallographic directions. Our estimates for the
exciton binding energies using the Bohr model are close to the reported
experimental value. The ultra-wide-band-gap and light carrier effective masses
of rutile GeO$_2$, coupled with its optical transparency in the visible and
near UV are promising for applications in UV-transparent conductors and
solar-blind photodetectors.
