  CoCr$_2$O$_4$ has attracted significant attention recently due to several
interesting properties such as magnetostriction, magnetoelectricity etc.. More
recent experiments on Fe substituted CoCr$_2$O$_4$ observed a variety of novel
phenomena such as the magnetic compensation accompanied by the occurrence of
exchange bias, which reverses its sign. Understanding of such phenomena may
lead to control the properties of these material in an efficient way to enhance
its potential for multifunctional applications. In this paper, we study the
microscopic understanding of Fe doping in modifying the structural and magnetic
properties of CoCr$_{2}$O$_{4}$ with varying composition and substitution of Fe
at different sublattices by first-principles density functional calculations.
We have analysed in detail the effect of Fe substitution on crystal field and
exchange splittings, magnetic moments and interatomic exchange parameters. It
is also observed that with increasing concentration of Fe impurity, the system
has a tendency towards forming an "Inverse Spinel" structure as observed in
experiments. Such tendencies are crucial to understand this system as it would
lead to modifications in the magnetic exchange interactions associated with
sites with different symmetry finally affecting the magnetic structure and the
multiferrocity in turn.
