Basic aspects of ferroelectricity induced by noncollinear alignment of spins

Basic principles of ferroelectric activity induced by the noncollinear spins are reviewed. There is a fundamental reason why the inversion symmetry can be broken by magnetic order. Such situation occurs when the magnetic order simultaneously involves ferromagnetic ($F$) and antiferromagnetic ($A$) patterns, transforming under the spatial inversion $\mathcal{I}$ and time reversal $\mathcal{T}$ as $\mathcal{I}F=F$ and $\mathcal{IT}A=A$. The incompatibility of these two conditions breaks the inversion symmetry, imposing a constraint on possible dependencies of polarization on directions of spins, which can include only antisymmetric coupling and single-ion anisotropy in the from $\vec{P} = \vec{\boldsymbol{\mathcal{P}}}_{12} [ \boldsymbol{e}_{1} \times \boldsymbol{e}_{2} ] + \boldsymbol{e}_{1} \vec{\mathbb{\Pi}} \boldsymbol{e}_{1} - \boldsymbol{e}_{2} \vec{\mathbb{\Pi}} \boldsymbol{e}_{2}$. $\vec{\boldsymbol{\mathcal{P}}}_{12}$ can be evaluated in the framework of superexchange theory, resulting in $\vec{\boldsymbol{\mathcal{P}}}_{12} \sim \vec{\boldsymbol{r}}_{12}^{\phantom{0}}$, where $\vec{\boldsymbol{r}}_{12}^{\phantom{0}}$ is the part of the position operator produced by the spin-orbit coupling. $\vec{\boldsymbol{r}}_{12}$ remains invariant under $\mathcal{I}$, explaining why noncollinear spins can induce $\vec{P}$ even in the centrosymmetric case. The properties of $\vec{\boldsymbol{r}}_{12}$ are rationalized from the viewpoint of symmetry of the Kramers states. The Katsura-Nagaosa-Balatsky rule $\vec{P} \propto \vec{\epsilon}_{21} \times [\boldsymbol{e}_{1} \times \boldsymbol{e}_{2}]$ ($\vec{\epsilon}_{21}$ being the bond direction) is justified only for relatively high symmetry. The single-ion anisotropy vanishes for the spin 1/2 or if magnetic ions are located in the inversion centers. The properties of known multiferroics are reconsidered from the viewpoint of these principles.