The ferroelectric polarization of triangular-lattice antiferromagnets induced by helical spin-spiral order is not explained by any existing model of magnetic-order-driven ferroelectricity. We resolve this problem by developing a general theory for the ferroelectric polarization induced by spin-spiral order and then by evaluating the coefficients needed to specify the general theory on the basis of density functional calculations. Our theory correctly describes the ferroelectricity of triangular-lattice antiferromagnets driven by helical spin-spiral order and incorporates known models of magnetic-order-driven ferroelectricity as special cases.

By performing density functional calculations, we investigate the origin of the Skyrmion state and ferroelectricity in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. We find that the Dzyaloshinskii-Moriya interactions between the two different kinds of Cu ions are extremely strong and induce the helical ground state and the Skyrmion state in the absence and presence of a magnetic field, respectively. On the basis of the general model for the spin-order induced polarization, we propose that the ferroelectric polarization of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ in the collinear ferrimagnetic state arises from an unusual mechanism, i.e., the single-spin-site contribution due to the spin-orbit coupling.