Shallow Valence Band of Rutile GeO$_2$ and P-type Doping

GeO$_2$ has an $\alpha$-quartz-type crystal structure with a very wide fundamental band gap of 6.6 eV and is a good insulator. Here we find that the stable rutile-GeO$_2$ polymorph with a 4.6 eV band gap has a surprisingly low $\sim$6.8 eV ionization potential, as predicted from the band alignment using first-principles calculations. Because of the short O$-$O distances in the rutile structure containing cations of small effective ionic radii such as Ge$^{4+}$, the antibonding interaction between O 2p orbitals raises the valence band maximum energy level to an extent that hole doping appears feasible. Experimentally, we report the flux growth of $1.5 \times 1.0 \times 0.8$ mm$^3$ large rutile GeO$_2$ single crystals and confirm the thermal stability for temperatures up to $1021 \pm 10~^\circ$C. X-ray fluorescence spectroscopy shows the inclusion of unintentional Mo impurities from the Li$_2$O$-$MoO$_3$ flux, as well as the solubility of Ga in the r-GeO$_2$ lattice as a prospective acceptor dopant. The resistance of the Ga- and Mo-codoped r-GeO$_2$ single crystals is very high at room temperature, but it decreases by 2-3 orders of magnitude upon heating to 300 $^\circ$C, which is attributed to thermally-activated p-type conduction.