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First-principles investigations into the thermodynamics of cation disorder and it's impact on electronic structure and magnetic properties of spinel $Co\left(Cr_{1-x}Mn_{x} \right)_{2}O_{4}$

Recent experiments on Mn doped multiferroic $CoCr_{2}O_{4}$ indicate that a possible distribution of Mn atoms among tetrahedrally and octahedrally coordinated sites in the spinel lattice give rise to different variations in the structural parameters and saturation magnetisations in different concentration regimes of Mn atoms substituting the Cr. A composition dependent magnetic compensation behaviour points to the role conversions of the magnetic constituents. In this work, we have investigated the thermodynamics of cation disorder in $Co\left(Cr_{1-x}Mn_{x}\right)_{2}O_{4}$ system and it's consequences on the structural, electronic and magnetic properties, using results from first-principles electronic structure calculations. We have computed the variations in the cation-disorder as a function of Mn concentration and the temperature and found that at the annealing temperature of the experiment many of the systems exhibit cation disorder. Our results support the interpretations of the experimental results regarding the qualitative variations in the sub-lattice occupancies and the associated magnetisation behaviour, with composition. We have analysed the variations in structural, magnetic and electronic properties of this system with variations in the compositions and the degree of cation disorder from the variations in their electronic structures and by using the ideas from crystal field theory. Our study provides a complete microscopic picture of the effects that are responsible for composition dependent behavioural differences of the properties of this system. This work lays down a general framework, based upon results from first-principles calculations, to understand and analyse the substitutional magnetic spinel oxides $A\left(B_{1-x}C_{x} \right)_{2}O_{4}$ in presence of cation disorder.