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.
Categories: cond-mat.mtrl-sci