Canted-spin-caused electric dipoles: A local symmetry theory

A pair of magnetic atoms with canted spins Sₐ,Sb can give rise to an electric dipole moment P. Several forms for the behavior of such a moment have appeared in the theoretical literature, some of which have been invoked to explain experimental results found in various multiferroic materials. The forms that require canting of the spins are ${P}₁{∝}R×{}({S}ₐ×{}{S}b),{P}₂{∝}{S}ₐ×{}{S}b$, and ${P}₃{∝}{S}ₐR·{}{S}ₐ{-}{S}bR·{}{S}b$, where $R$ is the relative position of the atoms and ${S}ₐ,{S}b$ are unit vectors. To unify and generalize these various forms, we consider $P$ as the most general quadratic function of the spin components that vanishes whenever ${S}ₐ$ and ${S}b$ are collinear, i.e., we consider the most general expressions that require spin canting. The study reveals new forms. We generalize to the vector $P$, Moriya's symmetry considerations regarding the (scalar) Dzyaloshinskii-Moriya energy $D·{}{S}ₐ×{}{S}b$ (which led to restrictions on $D$). This provides a rigorous symmetry argument that shows that ${P}₁$ is allowed no matter how high the symmetry of the atoms plus environment, and gives restrictions for all other contributions. The analysis leads to the suggestion of terms omitted in the existing microscopic models, suggests a new mechanism behind the ferroelectricity found in the ``proper screw structure'' of CuXO₂, X=Fe,Cr, and predicts an unusual antiferroelectric ordering in the antiferromagnetically and ferroelectrically ordered phase of RbFe(MoO₄)₂.