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Spinel structured compounds, $AB_2O_4$, are special because of their exoticmultiferroic properties. In $ACr_2O_4$ ($A$=$Co$, $Mn$,$Fe$), a switchablepolarization has been observed experimentally due to a non-collinear magneticspin order. In this article, we demonstrated the microscopic origin behind suchmagnetic spin order, hysteresis, polarisation and the so-called magneticcompensation effect in $ACr_2O_4$ ($A$=$Co$, $Mn$,$Fe$, $Ni$) using Monte Carlosimulation. With a careful choice of the exchange interaction, we were able toexplain various experimental findings such as magnetization vs. temperature (T)behavior, conical stability, unique magnetic ordering and polarization in arepresentative compound $CoCr_2O_4$ which is the best known multiferroiccompound in the $AB_2O_4$ spinel family. We have also studied the effect of$Fe$-substitution in $CoCr_2O_4$, with an onset of few exotic phenomena such asmagnetic compensation and sign reversible exchange bias effect. These effectsare investigated using an effective interactions mimicking the effect ofsubstitution. Two other compounds in this family, $CoMn_2O_4$ and $CoFe_2O_4$,are also studied where no conical magnetic order and polarisation was observed,as hence provide a distinct contrast. Here all calculations are done using thepolarisation calculated by the spin-current model. This model has certainlimitation and it works quite good for low temperature and low magnetic field.But the model despite its limitation it can reproduce sign reversible exchangebias and magnetic compensation like phenomena quite well.

Exotic multiferroic properties of spinel structured $AB_2O_4$ compounds: A Monte Carlo Study