Microscopic theory of spin-polarization coupling in multiferroic transition metal oxides

A systematic microscopic theory of magnetically induced ferroelectricity and lattice modulation is presented for various electron configurations of Mott-insulating transition metal oxides. The origin of polarization is classified as the spin-orbit interaction effective (i) within the magnetic $t_{2g}$ orbitals, (ii) between the $t_{2g}$ and $e_g$ orbitals, and (iii) within the ligand ion’s $p$ orbitals. Predictions for x-ray and neutron scattering experiments are proposed to clarify the microscopic mechanism of the spin-polarization coupling in different materials. Semiquantitative agreements with the multiferroic $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ are obtained.