Magnetodielectric phenomena in a charge- and spin-frustrated system of layered iron oxide

Dielectric and magnetic phenomena in spin- and charge-frustrated system $R{\text{Fe}}_2{\text{O}}_4$ ($R$ is a rare-earth metal ion) are studied. An electronic model for charge, spin, and orbital degrees in a pair of triangular-lattice planes is derived. We analyze this model by utilizing the mean-field approximation and the Monte Carlo simulation in a finite size cluster. A threefold-type charge-ordered structure with charge imbalance between the planes is stabilized in finite temperatures. This polar charge order is reinforced by spin ordering of Fe ions. This magnetodielectric phenomenon is caused by spin frustration and charge-spin coupling in the exchange interaction. We show cross-correlation effect in magnetic- and electric-field responses. Oxygen-deficiency effect as an impurity effect in a frustrated charge-spin coupled system is also examined.

Figure 1

(a) A pair of triangular-lattice planes (W layer). (b) A ${\text{FeO}}_5$ cluster.

Figure 2

Three two-dimensional coordinates $\left({\alpha }_x,{\alpha }_y\right)$, $\left({\beta }_x,{\beta }_y\right)$, and $\left({\gamma }_x,{\gamma }_y\right)$ in a FeO triangular lattice. Filled and dotted circles represent Fe and O ions, respectively.

Figure 3

Intersite Coulomb interactions between Fe ions. The solid, broken, and dotted arrows represent interactions between the nearest neighbor $\left(V_{c\text{NN}}\right)$, the next nearest neighbor $\left(V_{ab\text{NN}}\right)$, and the third neighbor $\left(V_{c\text{NNN}}\right)$ Fe-Fe bonds, respectively.

Figure 4

The lowest-energy spin and orbital configurations (a) for ${\text{Fe}}^{2+}{\text{-Fe}}^{2+}$ bond, (b) for ${\text{Fe}}^{3+}{\text{-Fe}}^{2+}$, and (c) for ${\text{Fe}}^{3+}{\text{-Fe}}^{3+}$. The open, filled, and dotted circles represent ${\text{Fe}}^{2+}$, ${\text{Fe}}^{3+}$, and O ions, respectively. Spin and orbital configuration in (a) is energetically close to ferromagnetic spin alignment with $d_{x^2-y^2}$ and $d_{{\beta }_x{\beta }_y}$ orbitals.

Figure 5

Charge structures in a W layer: (a) ${\text{CO}}_{1/3}$, (b) ${\text{CO}}_{1/4}$, (c) ${\text{CO}}_{1/2}-\text{I}$, and (d) ${\text{CO}}_{1/2}-\text{II}$. The filled and open circles represent ${\text{Fe}}^{3+}$ and ${\text{Fe}}^{2+}$, respectively, and large and small circles are for Fe ions in the upper and lower planes, respectively. Lower panel in each figure is a side view from $\left[1\overline{1}0\right]$.

Figure 6

Mean-field phase diagram for charge order at zero temperature. Three charge structures, ${\text{CO}}_{1/4}$, ${\text{CO}}_{1/2}-\text{II}$, and ${\text{CO}}_{1/3}$, are degenerate on a line of $V_{c\text{NN}}=2V_{c\text{NNN}}$ and the two, ${\text{CO}}_{1/4}$ and ${\text{CO}}_{1/2}-\text{I}$, are degenerate on a line of $V_{c\text{NNN}}=0$. The shaded area corresponds to a region for $R{\text{Fe}}_2{\text{O}}_4$. Phase diagram in finite temperatures, shown in Fig. 7, is calculated on broken line.

Figure 7

Mean-field phase diagram for charge structure in finite temperatures. The Coulomb interaction $V_{c\text{NN}}/V_{ab\text{NN}}$ is chosen to be 1.2.

Figure 8

(Color online) Charge correlation functions $N\left(\mathbf{q}\right)$ at (a) $V_{c\text{NNN}}/V_{ab\text{NN}}=0.60$, (b) 0.58, and (c) 0.62 calculated in ${\mathcal{H}}_V$. The Coulomb interaction $V_{c\text{NN}}/V_{ab\text{NN}}$ is chosen to be 1.2.

Figure 9

(Color online) Electric polarization $P$ as a functions of $V_{c\text{NNN}}/V_{ab\text{NN}}$ calculated in ${\mathcal{H}}_V$. The Coulomb interaction $V_{c\text{NN}}/V_{ab\text{NN}}$ is chosen to be 1.2.

Figure 10

Two-sublattice structure in the ${\text{CO}}_{1/3}$ phase. The amplitude of the mean field at Fe sites surrounded by broken circles is $0.9V_{ab\text{NN}}$ and that at other sites is $2.1V_{ab\text{NN}}$.

Figure 11

(Color online) (a) Charge correlation functions $N\left(\mathbf{q}\right)$, (b) spin correlation functions $S\left(\mathbf{q}\right)$, and (c) electric polarization $P$ calculated in ${\mathcal{H}}_V+{\mathcal{H}}_J$. The dashed lines in (a) and (c) are results obtained in ${\mathcal{H}}_V$. The parameters are chosen to be $V_{c\text{NN}}/V_{ab\text{NN}}=1.2$ and $V_{c\text{NNN}}/V_{ab\text{NN}}=0.60$.

Figure 12

(Color online) (a) Charge correlation functions $N\left(\mathbf{q}\right)$, (b) spin correlation functions $S\left(\mathbf{q}\right)$, and (c) electric polarization $P$ calculated in ${\mathcal{H}}_V+{\mathcal{H}}_J$. The dashed lines in (a) and (c) are results obtained in ${\mathcal{H}}_V$. Parameters are chosen to be $V_{c\text{NN}}/V_{ab\text{NN}}=1.2$ and $V_{c\text{NNN}}/V_{ab\text{NN}}=0.59$.

Figure 13

Charge and spin structures in the polar ${\text{CO}}_{1/3}$ phase at $V_{c\text{NNN}}/V_{ab\text{NN}}=0.6$. The filled and open circles represent ${\text{Fe}}^{3+}$ and ${\text{Fe}}^{2+}$, respectively, and large and small circles are for Fe ions in the upper and lower planes, respectively. The arrows represent spin directions. At Fe sites surrounded by broken circles, spin directions are not uniquely determined. The spin structures in (a) and (b) are almost degenerate.

Figure 14

(a) Charge and spin structure at $V_{c\text{NNN}}/V_{ab\text{NN}}=0.59$ and that at 0.61 (b).

Figure 15

(Color online) Magnetic-field effect in (a) charge correlation function $N\left(\mathbf{q}\right)$ and (b) electric polarization $P$. The parameters are chosen to be $V_{c\text{NN}}/V_{ab\text{NN}}=1.2$ and $V_{c\text{NNN}}/V_{ab\text{NN}}=0.6$. The inset of (b) shows magnetic-field dependence of the electric polarization at $\tilde{T}=0.05$.

Figure 16

(Color online) (a) Magnetic-field effect in charge correlation function $N\left(\mathbf{q}\right)$ and (b) electric polarization $P$. Parameters are chosen to be $V_{c\text{NN}}/V_{ab\text{NN}}=1.2$ and $V_{c\text{NNN}}/V_{ab\text{NN}}=0.59$. The inset of (b) shows magnetic-field dependence of the electric polarization. The solid circles and squares are calculated at $\tilde{T}=0.01$ and 0.05, respectively.

Figure 17

(Color online) Electric-field effect of spin correlation function $S\left(\mathbf{q}\right)$ at ${\mathbf{q}}_{1/3}$. The parameters are chosen to be (a) $V_{c\text{NNN}}/V_{ab\text{NN}}=0.6$ and (b) 0.58 with $V_{c\text{NN}}/V_{ab\text{NN}}=1.2$. The insets show electric-field dependence of $S\left({\mathbf{q}}_{1/3}\right)$. Temperatures are $\tilde{T}=0.05$ in (a) and 0.03 in (b).

Figure 18

(Color online) Oxygen-deficiency effects of electric polarization $P$ at (a) $V_{c\text{NNN}}/V_{ab\text{NN}}=0.6$ and (b) 0.58. The bold, broken, and dotted lines are for result without deficiency, with charge-imbalance effect and with electrostatic potential, respectively.

Figure 19

Intermediate states of the exchange processes in a ${\text{Fe}}^{2+}{\text{-Fe}}^{2+}$ bond represented by hole picture. (a)–(c) are for the Hamiltonian ${\mathcal{H}}^{\left(22\right)-1}$, ${\mathcal{H}}^{\left(22\right)-2}$, and ${\mathcal{H}}^{\left(22\right)-3}$ in the $dd$ processes, respectively, and (d)–(f) are for ${\mathcal{H}}^{\left(22\right)-4}$, ${\mathcal{H}}^{\left(22\right)-5}$, and ${\mathcal{H}}^{\left(22\right)-6}$ in the $dpd$ ones, respectively. Long and short arrows represent spins with $S=3/2$ and 1/2, respectively.

Figure 20

Intermediate states of the exchange processes in a ${\text{Fe}}^{2+}{\text{-Fe}}^{3+}$ bond represented by hole picture. (a)–(c) are for the Hamiltonian ${\mathcal{H}}^{\left(23\right)-1}$, ${\mathcal{H}}^{\left(23\right)-2}$, and ${\mathcal{H}}^{\left(23\right)-3}$ in the $dd$ processes, respectively, and (d)–(f) are for ${\mathcal{H}}^{\left(23\right)-4}$, ${\mathcal{H}}^{\left(23\right)-5}$, and ${\mathcal{H}}^{\left(23\right)-6}$ in the $dpd$ ones, respectively. Long, medium, and short arrows represent spins with $S=2$, 3/2, and 1/2, respectively.

Figure 21

Intermediate states of the exchange processes in a ${\text{Fe}}^{3+}{\text{-Fe}}^{3+}$ bond represented by hole picture. (a) is for the Hamiltonian ${\mathcal{H}}^{\left(33\right)-1}$ in the $dd$ processes, respectively, and (b)–(d) are for ${\mathcal{H}}^{\left(33\right)-2}$, ${\mathcal{H}}^{\left(33\right)-3}$, and ${\mathcal{H}}^{\left(33\right)-4}$ in the $dpd$ ones, respectively. The long, medium, and short arrows represent spins with $S=2$, 3/2, and 1/2, respectively.

Figure 22

(a) Schematic pictures of the ${\text{CO}}_{1/3}$ structure and (b) one of ${\text{CO}}_A$. When, in ${\text{CO}}_{1/3}$, ${\text{Fe}}^{3+}$ in the upper plane and ${\text{Fe}}^{2+}$ in lower one on chains indicated by the broken lines are exchanged, the ${\text{CO}}_A$ structure in (b) is obtained.

Figure 23

One of the ${\text{CO}}_B$ structures.