Understanding cation ordering and oxygen vacancy site preference in Ba$_{3}$CaNb$_{2}$O$_{9}$ from first-principles*

We investigate the physical mechanism underlying the formation of the B-site cation ordering and the oxygen vacancy site selection in Ba$_{3}$CaNb$_{2}$O$_{9}$ using density functional theory calculations. We found that either cation site exchange or oxygen vacancy formation induces negligible lattice strain. This implies that the ionic radius plays an insignificant role in governing these two processes. Furthermore, the electrostatic interactions are found dominant in the ordering of mixed valence species on one or more sites, the ionic bond strength is identified as the dominant force in governing both the 1:2 B-site cation ordering along the \textless 111\textgreater direction and the oxygen vacancy site preference in Ba$_{3}$CaNb$_{2}$O$_{9}$. Specifically, the cation ordering can be rationalized by the increased mixing bonding energy of the Ca-O-Nb bonds over the Ca-O-Ca and Nb-O-Nb bonds, i.e., 1/2(Ca-O-Ca $+$ Nb-O-Nb) \textless Ca-O-Nb; while oxygen vacancy prefers a site to minimize the electrostatic energy and to break the weaker B-O-B bond.