We studied the ferroelectricity of magnetic oxides, in which its emergence coincides with the onset of a second incommensurate magnetic order. We solved for the wave function of $e_g$ electrons in the presence of magnetic orders. The coupling between the magnetic and electric orders is provided by the spin-orbit interaction. It was found that the net electric dipole moment of the system came from the bond between the transition metal and oxygen atoms. The anisotropy of $d$ orbitals also played an important role. Finally, the form of the coupling leads us to the conclusion that it is an improper ferroelectricity.

• Received 19 April 2007

DOI:https://doi.org/10.1103/PhysRevB.75.172106