Confirming the trilinear form of the optical magnetoelectric effect in the polar honeycomb antiferromagnet Co${}_2$Mo${}_3$O${}_8$

Abstract

Magnetoelectric phenomena are intimately linked to relativistic effects and also require the material to break spatial inversion symmetry and time reversal invariance. Magnetoelectric coupling can substantially affect light-matter interaction and lead to non-reciprocal light propagation. Here, we confirm on a fully experimental basis, without invoking either symmetry-based or material-specific assumptions, that the optical magnetoelectric effect in materials with non-parallel magnetization ($\mathbf{M}$) and electric polarization ($\mathbf{P}$) generates a trilinear term in the refractive index, $\hat{I}´n\propto \mathbf{k}\cdot (\mathbf{P}\times \mathbf{M})$, where $\mathbf{k}$ is the propagation vector of light. Its sharp magnetoelectric resonances, that are simultaneously electric and magnetic dipole active excitations, make Co${}_2$Mo${}_3$O${}_8$ an ideal compound to demonstrate this fundamental relation via independent variation of $\mathbf{M}$, $\mathbf{P}$ and $\mathbf{k}$. Remarkably, the material shows almost perfect one-way transparency in moderate magnetic fields at some of the magnetoelectric resonances.