Microscopic toy model for magnetoelectric effect in polar Fe2Mo3O8

The kamiokite ${\mathrm{Fe}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$ is regarded as a promising material exhibiting a giant magnetoelectric (ME) effect at the relatively high temperature $T$. Here, we explore this phenomenon on the basis of first-principles electronic structure calculations. For this purpose, we construct a realistic model describing the behavior of magnetic Fe $3d$ electrons and further map it onto the isotropic spin model. Our analysis suggests two possible scenarios for ${\mathrm{Fe}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$. The first one is based on the homogeneous charge distribution of the ${\mathrm{Fe}}^{2+}$ ions among tetrahedral $(t)$ and octahedral $(o)$ sites, which tends to lower the crystallographic $P{6}_{3}mc$ symmetry through the formation of an orbitally ordered state. Nevertheless, the effect of the orbital ordering on interatomic exchange interactions does not seem to be strong, so that the magnetic properties can be described reasonably well by averaged interactions obeying the $P{6}_{3}mc$ symmetry. The second scenario, which is supported by obtained parameters of on-site Coulomb repulsion and respects the $P{6}_{3}mc$ symmetry, implies the charge disproportionation involving the somewhat exotic $1+$ ionization state of the $t$-Fe sites (and $3+$ state of the $o$-Fe sites). Somewhat surprisingly, these scenarios are practically indistinguishable from the viewpoint of exchange interactions, which are nearly identical in these two cases. However, the spin-dependent properties of the electric polarization are expected to be different due to the strong difference in the polarity of the ${\mathrm{Fe}}^{2+}\text{\ensuremath{-}}{\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{1+}\text{\ensuremath{-}}{\mathrm{Fe}}^{3+}$ bonds. Our analysis uncovers the basic aspects of the ME effect in ${\mathrm{Fe}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$. Nevertheless, the quantitative description should involve other ingredients, apparently related to the lattice and orbitals degrees of freedom.