A first-principles thermodynamic model for the Ba$\unicode{x2013}$Zr$\unicode{x2013}$S system in equilibrium with sulfur vapour

The chalcogenide perovskite BaZrS$_3$ has strong visible light absorption and high chemical stability, is nontoxic, and is made from earth-abundant elements. As such, it is a promising candidate material for application in optoelectronic technologies. However the synthesis of BaZrS$_3$ thin-films for characterisation and device integration remains a challenge. Here we use density functional theory and lattice dynamics to calculate the vibrational properties of elemental, binary and ternary materials in the Ba-Zr-S system. This is used to build a thermodynamic model for the stability of BaZrS$_3$, BaS$_x$, and ZrS$_x$ in equilibrium with sulfur gas, across a range of temperatures and sulfur partial pressures. We highlight that reaction thermodynamics are highly sensitive to sulfur allotrope and the extent of allotrope mixing. We use our model to predict the synthesis conditions in which BaZrS$_3$ and the intermediate binary compounds can form. At a moderate temperature of 500 {\deg}C we find that BaS$_3$, associated with fast reaction kinetics, is stable at pressures above 3x10$^5$ Pa. We also find BaZrS$_3$ is stable against decomposition into sulfur-rich binaries up to at least 1x10$^7$ Pa. Our work provides insights into the chemistry of this promising material and suggests the experimental conditions required for the successful synthesis of BaZrS$_3$.