Intrinsic point-defect equilibria in tetragonal ZrO2: Density functional theory analysis with finite-temperature effects

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We present a density functional theory (DFT) framework taking into account the finite temperature effects to quantitatively understand and predict charged defect equilibria in a metal oxide. Demonstration of this approach was performed on the technologically important tetragonal zirconium oxide, T-ZrO${}_{2}$. We showed that phonon free energy and electronic entropy at finite temperatures add a nonnegligible contribution to the free energy of formation of the defects. Defect equilibria were conveniently cast in Kr\oger--Vink diagrams to facilitate realistic comparison with experiments. Consistent with experiments, our DFT-based results indicate the predominance of free electrons at low oxygen partial pressure (${P}_{{\mathrm{O}}_{2}}\ensuremath{\le}{10}^{\ensuremath{-}6}$ atm) and low temperature ($T\ensuremath{\le}1500\phantom{\rule{0.28em}{0ex}}\mathrm{K}$). In the same regime of ${P}_{{\mathrm{O}}_{2}}$ but at higher temperatures, we discovered that the neutral oxygen vacancies (F-centers) predominate. The nature of the predominant defect at high oxygen partial pressure has been a long-standing controversy in the experimental literature. Our results revealed this range to be dominated by the doubly charged oxygen vacancies at low temperatures ($T\ensuremath{\le}1500$ K) and free electrons at high temperatures. T-ZrO${}_{2}$ was found to be hypostoichiometric over all ranges of $T$ and ${P}_{{\mathrm{O}}_{2}}$, mainly because of the doubly charged oxygen vacancies, which are responsible for inducing $n$-type conductivity via a self-doping effect. A range of 1.3 eV in the band gap of T-ZrO${}_{2}$ starting from the middle of the gap toward the conduction band is accessible to the chemical potential of electrons (Fermi level) by varying $T$ and ${P}_{{\mathrm{O}}_{2}}$ without extrinsic doping. The approach presented here can be used to determine the thermodynamic conditions that extremize certain desirable or undesirable defects to attain the optimal catalytic and electronic performance of oxides.