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Cation-ordered A$'_{1/2}$A$''_{1/2}$B$_2$X$_4$ magnetic spinels as magnetoelectrics

We show that 1:1 ordering of A$'$ and A$''$ cations in A$'_{1/2}$A$''_{1/2}$B$_2$X$_4$ magnetic spinels results in appearance of magnetoelectric properties. Possible value of magnetically induced electric polarization is calculated using the recently proposed microscopic model, which takes into account spin-dependent electric dipole moments of magnetic ions located in noncentrosymmetric crystallographic positions. We build phenomenological models of magnetic phase transitions in cation-ordered spinels, which describe ferromagnetic and antiferromagnetic ordering patterns of B cation spins, and calculate the respective magnetoelectric responses. We find that magnetoelectric coefficients diverge at ferromagnetic or weak ferromagnetic phase transitions in ordered spinels.