  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.
