Ab initio thermodynamics of intrinsic oxygen vacancies in ceria

Nonstoichiometric ceria(CeO$_{2-\delta}$) is a candidate reaction medium to facilitate two step water splitting cycles and generate hydrogen. Improving upon its thermodynamic suitability through doping requires an understanding of its vacancy thermodynamics. Using density functional theory(DFT) calculations and a cluster expansion based Monte Carlo simulations, we have studied the high temperature thermodynamics of intrinsic oxygen vacancies in ceria. The DFT+$U$ approach was used to get the ground state energies of various vacancy configurations in ceria, which were subsequently fit to a cluster expansion Hamiltonian to efficiently model the configurational dependence of energy. The effect of lattice vibrations was incorporated through a temperature dependent cluster expansion. Lattice Monte Carlo simulations using the cluster expansion Hamiltonian were able to detect the miscibility gap in the phase diagram of ceria. The inclusion of vibrational and electronic entropy effects made the agreement with experiments quantitative. The deviation from an ideal solution model was quantified by calculating as a function of nonstoichiometry, a) the solid state entropy from Monte Carlo simulations and b) Warren-Cowley short range order parameters of various pair clusters.