Electric Polarization, Magnetoelectric Effect, and Orbital State of a Layered Iron Oxide with Frustrated Geometry

A layered iron oxide $R{\mathrm{Fe}}_2{\mathrm{O}}_4$ ($R$ denotes rare-earth-metal elements) is an exotic dielectric material with charge-order (CO) driven electric polarization and magnetoelectric effect caused by spin-charge coupling. In this paper, a theory of electronic structure and dielectric property in $R{\mathrm{Fe}}_2{\mathrm{O}}_4$ is presented. Charge frustration in paired-triangular lattices allows a charge imbalance without inversion symmetry. Spin frustration induces reinforcement of this polar CO by a magnetic ordering. We also analyze an orbital model for the Fe ion which does not show a conventional long-range order.

Figure 1

(a) A pair of the Fe-O triangular lattices. Thick arrows represent the Coulomb interactions $V_{\mathrm{abNN}}$, $V_{\mathrm{cNN}}$ and $V_{\mathrm{cNNN}}$. (b) A schematic view of ${\mathrm{CO}}_{1/3}$ accompanied by the electric polarization.

Figure 2

(a) A mean-field phase diagram at finite $T$ calculated in ${\mathcal{H}}_V$. (b) Charge correlation function $N(\mathbf{q})$ at $V_{\mathrm{cNNN}}/V_{\mathrm{abNN}}=0.6$. (c) $N(\mathbf{q})$ at $V_{\mathrm{cNNN}}/V_{\mathrm{abNN}}=0.5$ (bold lines) and 0.7 (broken lines). The insets show schematic views of the ${\mathrm{CO}}_{1/4}$ and ${\mathrm{CO}}_{1/2}$ structures. (d) Electric polarization $P$ for several $V_{\mathrm{cNNN}}/V_{\mathrm{abNN}}$ values. A value of $V_{\mathrm{cNN}}/V_{\mathrm{abNN}}$ is chosen to be 1.2 in (a)–(d).

Figure 3

Charge-correlation function $N(\mathbf{q})$ (a), electric polarization $P$ (b), and spin correlation function $S(\mathbf{q})$ (c) calculated in ${\mathcal{H}}_V+{\mathcal{H}}_J$. A schematic view of the charge and spin structure below the ferrimagnetic ordering temperature is shown in (d). Spin directions at the sites marked by broken circles are not uniquely determined.

Figure 4

(a) A schematic picture of the ${\mathrm{Fe}}^{2+}$ sublattice in the $2{\mathrm{Fe}}^{2+}\mathrm{\text{−}}{\mathrm{Fe}}^{3+}$ layer. Bold arrows and bold bars represent the orbital PS direction and the PS projection components ${\tau }_i^l$ on the NN bonds, respectively. (b) One of the ground state PS configurations. (c) Specific heat $C(T)$ for several cluster sizes. Broken line represents $⟨{cos}^{2}3\theta ⟩$ in $L=12$. (d) One of the PS configurations at finite temperature below $T_{\mathrm{O}}$.