  Charge carrier concentrations in acceptor-doped proton-conducting perovskites
are to a large extent determined by the hydration and oxidation of oxygen
vacancies, which introduce protons and holes, respectively. First-principles
modeling of these reactions involves calculation of formation energies of
charged defects, which requires an accurate description of the band gap and the
position of the band edges. Since density-functional theory (DFT) with local
and semi-local exchange-correlation functionals (LDA and GGA) systematically
fails to predict these quantities this can have serious implications on the
modeling of defect reactions. In this study we investigate how the description
of band gap and band edge positions affects the hydration and oxidation in
acceptor-doped BaZrO$_3$. First-principles calculations are performed in
combination with thermodynamic modeling in order to obtain equilibrium charge
carrier concentrations at different temperatures and partial pressures. Three
different methods have been considered: DFT with both semi-local (PBE) and
hybrid (PBE0) exchange-correlation functionals, and many-body perturbation
theory within the $G_0W_0$-approximation. All three methods yield similar
results for the hydration reaction, which are consistent with experimental
findings. For the oxidation reaction, on the other hand, there is a qualitative
difference. PBE predicts the reaction to be exothermic while the two others
predict an endothermic behavior. Results from thermodynamic modeling are
compared with available experimental data, such as enthalpies, concentrations
and conductivities, and only the results obtained with PBE0 and $G_0W_0$, with
an endothermic oxidation behavior, give a satisfactory agreement with
experiments.
