Thermodynamics of the Zr-O system from first-principles calculations

Abstract

We investigate the electronic and thermodynamic properties of Zr and its oxides from first principles to elucidate phase stability in the Zr-O system. Hexagonally close-packed Zr is unusual in its ability to dissolve very high concentrations of oxygen over its interstitial octahedral sites, forming a variety of ordered suboxides that undergo both first-order and second-order phase transitions upon heating. We perform a first-principles, statistical-mechanical analysis of finite temperature phase stability of ZrO${}_{x}$ using a cluster expansion Hamiltonian and Monte Carlo calculations. This analysis predicts the existence of 0-K ground-state oxygen orderings at composition ZrO${}_{1/6}$, ZrO${}_{2/9}$, ZrO${}_{1/3}$, ZrO${}_{4/9}$, and ZrO${}_{1/2}$ along with evidence of an infinite sequence of ground-state suboxide orderings at intermediate oxygen concentrations consisting of different stackings of empty, $\frac{1}{3}$-filled and $\frac{2}{3}$-filled two-dimensional oxygen layers. We also predict the stability of a previously uncharacterized Zr-monoxide phase, which we label ${\ensuremath{\delta}}^{\ensuremath{'}}$-ZrO due to its crystallographic relation to $\ensuremath{\delta}$-TiO. The ${\ensuremath{\delta}}^{\ensuremath{'}}$-ZrO structure is equivalent to the high-pressure $\ensuremath{\omega}$-Zr phase but has interstitial oxygen ordering. Finally, as part of the technical implementation of our statistical mechanical study, we introduce a new algorithm to parametrize the coefficients of a cluster expansion Hamiltonian and apply a $k$-space analysis to rigorously track order-disorder phenomena at finite temperature.