Asymmetric Cation stoichiometry in Spinels : Site occupancy in Co$_{2}$ZnO$_{4}$ and Rh$_{2}$ZnO$_{4}$*

Cations A and B in A$_{2}$BX$_{4}$ spinels normally appear in precise 2:1 Daltonian ratio only at low temperature. At finite temperature, they become either A-rich or B-rich, which control dopability of the compound. We survey the experimentally observed stoichiometry asymmetries and describe the first principles framework for calculating these. The results of the calculations compare well with the phase boundary determined from XRD and the site occupancy measured by anomalous-XRD on Co$_{2}$ZnO$_{4}$ and Rh$_{2}$ZnO$_{4}$ samples grown in thermodynamic equilibrium. Good comparison between theory and experiment allows us predict the co-existence line in composition range form first principle for other spinels, which in turn can be extended to predict the nature of electrical conductivity of a compound, while designing the material with the desired properties via principle of inverse design.