Ab initiocalculations on the defect structure ofβ-Ga2O3

The intrinsic point defects of $\ensuremath{\beta}$-${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ are investigated using density functional theory. We have chosen two different exchange-correlation potentials: the generalized gradient approximation (GGA) and a hybrid potential (HSE06). Defect formation energies were determined taking into account finite-size effects. Schottky, anti-Frenkel, and Frenkel energies have been extracted for $T=0$ K. We calculate formation entropies for an oxygen and a gallium vacancy and determine the Gibbs energy of Schottky disorder. Furthermore, we investigate the defect concentrations as a function of the oxygen partial pressure. The obtained purely intrinsic defect concentrations for charged defects are very small and result in a $p{\mathrm{O}}_{2}$ dependence of the electron concentration of $[{\mathrm{e}}^{\ensuremath{'}}]\ensuremath{\sim}$ $p{\mathrm{O}}_{2}^{\ensuremath{-}1/6}$, whereas experimentally $[{\mathrm{e}}^{\ensuremath{'}}]\ensuremath{\sim}$ $p{\mathrm{O}}_{2}^{\ensuremath{-}1/4}$ is found. So we assume that, experimentally, a small unintentional donor doping is unavoidable. A small extrinsic donor concentration $[{\mathrm{D}}^{\ifmmode\cdot\else\textperiodcentered\fi{}}]$ $=$ 10${}^{18}$ cm${}^{\ensuremath{-}3}$ (10 ppm) changes the electron concentration to $[{\mathrm{e}}^{\ensuremath{'}}]\ensuremath{\sim}$ $p{\mathrm{O}}_{2}^{\ensuremath{-}1/4}$ and gives an activation energy of the conductivity $\ensuremath{\sigma}$ of 1.7 eV in good agreement to experimental values. So we propose as majority disorder $3[{\mathrm{V}}_{\mathrm{Ga}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}]$ $=$ $[{\mathrm{D}}^{\ifmmode\cdot\else\textperiodcentered\fi{}}]$ with electrons being minority defects.