Microscopic mechanisms of spin-dependent electric polarization in 3d oxides

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 and relativistic terms. The nonrelativistic electronic polarization mechanism related with 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 behaviour 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.