Microscopic mechanisms of spin-dependent electric polarization in3doxides

We address a systematic microscopic theory of spin-dependent electric polarization in $3d$ oxides starting with a generic three-site two-hole cluster. A perturbation scheme realistic for $3d$ oxides is applied which implies the quenching of orbital moments by low-symmetry crystal field, strong intra-atomic correlations, the $dp$-transfer effects, and rather small spin-orbital coupling. An effective spin operator of the electric-dipole moment is deduced incorporating both nonrelativistic $\ensuremath{\propto}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{s}}}_{1}\ensuremath{\cdot}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{s}}}_{2})$ and relativistic $\ensuremath{\propto}[{\mathbf{s}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathbf{s}}_{2}]$ terms. The nonrelativistic electronic polarization mechanism related to the effects of the redistribution of the local on-site charge density due to $pd$ covalency and exchange coupling is believed to govern the multiferroic behavior in $3d$ oxides. The relativistic exchange-dipole moment is mainly stems from the nonrelativistic one due to the perturbation effect of Dzyaloshinsky-Moriya coupling and is estimated to be a weak contributor to the electric polarization observed in the most of $3d$ multiferroics.