Microscopic toy model for magnetoelectric effect in polar ${\mathrm{Fe}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$

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 $\left(t\right)$ and octahedral $\left(o\right)$ 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 ${\mathrm{Fe}}^{2+}\text{−}{\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{1+}\text{−}{\mathrm{Fe}}^{3+}$ 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.

Figure 1

Fragments of the crystal structure of ${\mathrm{Fe}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$ with the notations of main exchange interactions: (a) alternation of honeycomb layers formed by ${\mathrm{FeO}}_4$ tetrahedra, ${\mathrm{FeO}}_6$ octahedra, and trimerized kagome layers of ${\mathrm{MoO}}_6$ octahedra in the unit cell of ${\mathrm{Fe}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$; (b) nearest-neighbor interactions in the honeycomb layers; (c) interlayer interactions between tetrahedral and octahedral Fe sites located in the first $\left(J_{\perp }^1\right)$ and second $\left(J_{\perp }^2\right)$ coordination spheres; (d), (e) interlayer interactions between tetrahedral and octahedral Fe sites, respectively. Fe, Mo, and O atoms are denoted by large, medium, and small spheres, respectively.

Figure 2

Left panel: Total and partial densities of states in the local density approximation. Right panel: Corresponding band structure calculated in the full LMTO basis (solid curved) and in the Wannier basis for the Fe $3d$ bands. The Fermi level is at zero energy (shown by dot-dashed line). Notations of the high-symmetry points of the Brillouin zone are taken from Ref. [23].

Figure 3

(a) The ${\mathrm{Mo}}_3{\mathrm{O}}_{13}$ cluster with the notations of Mo-O-Mo paths mediating the hybridization between $t_{2g}$ orbitals in each of the Mo-Mo bonds. (b) Schematic view of the hybridization in the ${\mathrm{Mo}}_3$ trimer: each Mo site donates one $t_{2g}$ orbital for the hybridization in each of the Mo-Mo bonds, resulting in the formation of bonding and antibonding molecular states. These orbitals are denoted as $t_1$ and $t_2$ and are shown by the color of the bond in which they operate. The third $t_{2g}$ orbital is nonbonding and denoted as $t_3$. (c) Schematic view of the bonding-nonbonding-antibonding splitting in the ${\mathrm{Mo}}_3$ trimer resulting in the nonmagnetic state, where six $4d$ electrons of ${\mathrm{Mo}}_3$ reside at the bonding molecular orbitals. The molecular levels are shown by the same color as forming them the atomic orbitals.

Figure 4

Atomic level splitting at the tetrahedral (left) and octahedral (right) Fe sites.

Figure 5

Antiferromagnetic (AFM) and ferrimagnetic (FRM) structure of ${\mathrm{Fe}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$. Fe and O atoms are denoted by large and small spheres, respectively. The directions of local magnetic moments at the tetrahedral and octahedral sites are shown by small (brown) and big (blue) arrows, respectively. The direction of net magnetization in each layer is shown by the thick (red) arrow next to this layer.

Figure 6

Partial densities of states as obtained in the mean-field Hartree-Fock calculations for the antiferromagnetic (AFM) and ferrimagnetic (FRM) charge-disproportionated $d_t^7{d}_o^5$ phases. The contributions of the $t$-Fe and $o$-Fe atoms are shown by red and blue colors, respectively. In the AFM case, the contributions of atoms located in the antiferromagnetically coupled adjacent layers are shown by solid and dashed lines. The Fermi level, defined as the midpoint of the band gap, is at zero energy.

Figure 7

Partial densities of states as obtained in the mean-field Hartree-Fock calculations for the antiferromagnetic (AFM) and ferrimagnetic (FRM) charge homogeneous $d_t^6{d}_o^6$ phases. The contributions of the $t$-Fe and $o$-Fe atoms are shown by red and blue colors, respectively. In the AFM case, the contributions of atoms located in the antiferromagnetically coupled adjacent layers are shown by solid and dashed lines. The Fermi level, defined as the midpoint of the band gap, is at zero energy.

Figure 8

Orbital ordering obtained in constrained Hartree-Fock calculations for the configuration $d_t^6{d}_o^6$ in the case of the antiferromagnetic (AFM) and ferrimagnetic (FRM) spin order. A single occupied orbital of minority spin is shown.

Figure 9

Distance dependence of exchange interactions around the tetrahedral and octahedral Fe sites as obtained in the Green's function method for the antiferromagnetic $d_t^5{d}_o^7$ and $d_t^6{d}_o^6$ solutions. The main exchange interactions are labeled and explained in Fig. 1.

Figure 10

Results of molecular-field theory for the spin model: temperature dependence of magnetization $M$ at the $t$-Fe and $o$-Fe sites, net magnetic moment in the honeycomb layer, $\mathrm{\Delta }M=\left(\right|M_o|-|M_t\left|\right)/2$, recalculated per one Fe site, and the total-energy difference $\mathrm{\Delta }E$ between ferrimagnetic and antiferromagnetic phases calculated using parameters for the $d_t^7{d}_o^5$ and $d_t^6{d}_o^6$ states.

Figure 11

Results of molecular-field theory for the spin model: temperature dependence of the spin-dependent part of the electric polarization $\left(P^z\right)$ in the antiferromagnetic phase and the polarization jump $\left(\mathrm{\Delta }{P}^z\right)$ caused by the antiferromagnetic-to-ferrimagnetic transition calculated using parameters for the $d_t^7{d}_o^5$ and $d_t^6{d}_o^6$ states.