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By applying Berry-phase theory for the effective half-filled Hubbard model,we derive an analytical expression for the electronic polarization driven bythe relativistic spin-orbit (SO) coupling. The model itself is constructed inthe Wannier basis, using the input from the first-principles electronicstructure calculations in the local-density approximation, and then treated inthe spirit of the superexchange theory. The obtained polarization has thefollowing form: ${\bf P}_{ij} = \boldsymbol{\epsilon}_{ji} \boldsymbol{\calP}_{ij} \cdot [\boldsymbol{e}_i \times \boldsymbol{e}_j]$, where$\boldsymbol{\epsilon}_{ji}$ is the direction of the bond $\langle ij \rangle$,$\boldsymbol{e}_i$ and $\boldsymbol{e}_j$ are the directions of spins in thisbond, and $\boldsymbol{\cal P}_{ij}$ is the pseudovector containing all theinformation about the crystallographic symmetry of the considered system. Theexpression describes the ferroelectric activity in various magnets withnoncollinear but otherwise nonpolar magnetic structures, which would yield nopolarization without SO interaction, including the magnetoelectric (ME) effect,caused by the ferromagnetic canting of spins in the external magnetic field,and spin-spiral multiferroics. The abilities of this theory are demonstratedfor the the analysis of linear ME effect in Cr$_2$O$_3$ and BiFeO$_3$ andproperties multiferroic MnWO$_4$ and $\beta$-MnO$_2$. In all consideredexamples, the theory perfectly describes the symmetry properties of the inducedpolarization. However, in some cases, the values of this polarization areunderestimated, suggesting that other effects, besides the spin and electronicones, can also play an important role.

Superexchange theory of electronic polarization driven by relativistic spin-orbit interaction at the half-filling