Microscopic model and phase diagrams of the multiferroic perovskite manganites

Orthorhombically distorted perovskite manganites, $R\mathrm{Mn}{\mathrm{O}}_3$ with $R$ being a trivalent rare-earth ion, exhibit a variety of magnetic and electric phases including multiferroic (i.e., concurrently magnetic and ferroelectric) phases and fascinating magnetoelectric phenomena. We theoretically study the phase diagram of $R\mathrm{Mn}{\mathrm{O}}_3$ by constructing a microscopic spin model, which includes not only the superexchange interaction but also the single-ion anisotropy (SIA) and the Dzyaloshinsky-Moriya interaction (DMI). Analysis of this model using the Monte Carlo method reproduces the experimental phase diagrams as functions of the $R$-ion radius, which contain two different multiferroic states, i.e., the $ab$-plane spin cycloid with ferroelectric polarization $P\parallel a$ and the $bc$-plane spin cycloid with $P\parallel c$. The orthorhombic lattice distortion or the second-neighbor spin exchanges enhanced by this distortion exquisitely controls the keen competition between these two phases through tuning the SIA and DMI energies. This leads to a lattice-distortion-induced reorientation of $P$ from $a$ to $c$ in agreement with the experiments. We also discuss spin structures in the $A$-type antiferromagnetic state, those in the cycloidal spin states, origin and nature of the sinusoidal collinear spin state, and many other issues.

Figure 1

(Color online) (a) Perovskite structure with $\mathrm{Gd}\mathrm{Fe}{\mathrm{O}}_3$-type distortion. The unit cell contains four Mn ions, which are referred to as Mn A, Mn B, Mn C, and Mn D, respectively. (b) Experimentally obtained magnetoelectric phase diagram of $R\mathrm{Mn}{\mathrm{O}}_3$ in plane of temperature and ionic $R$-site radius reproduced from Refs. 3, 15. The region between Gd and Tb was studied using a solid solution system ${\mathrm{Gd}}_{1-x}{\mathrm{Tb}}_x\mathrm{Mn}{\mathrm{O}}_3$ (Ref. 15). Averaged $R$-site radii for the solid solutions are deduced by interpolation. Insets show spin configurations of the $A$-type and $E$-type antiferromagnetic [$\mathrm{AFM}\left(A\right)$ and $\mathrm{AFM}\left(E\right)$] states. Along the $c$ axis, spins stack antiferromagnetically. PE and FE denote paraelectric and ferroelectric phases, respectively. In the $\mathrm{AFM}\left(E\right)$ phase, possible ferroelectricity due to an exchange-striction mechanism was theoretically proposed (Refs. 21, 22, 23), and it was confirmed experimentally (Refs. 24, 25, 26). (c) Experimentally obtained magnetoelectric phase diagram of a solid solution system ${\mathrm{Eu}}_{1-x}{\mathrm{Y}}_x\mathrm{Mn}{\mathrm{O}}_3$ in plane of temperature and Y concentration $x$ reproduced from Ref. 18. Insets show spin configurations of the $ab$-cycloidal, $bc$-cycloidal, and sinusoidal collinear states. Spin configuration of the $\mathrm{AFM}\left(A\right)$ state with weak ferromagnetism (WFM) seen along the $a$ axis is also shown.

Figure 2

(Color online) (a) Relationship between the spin-helicity vector ${\mathit{h}}_s={\sum }_i{\mathit{S}}_i\times {\mathit{S}}_{i+1}$ and the spontaneous electric polarization $\mathit{P}$ in the $ab$-cycloidal spin structure expected from the spin-current model (Refs. 9, 11, 12), and (b) that in the $bc$-cycloidal spin structure.

Figure 3

(Color online) (a) Superexchange interactions in $R\mathrm{Mn}{\mathrm{O}}_3$ and tilted local coordinate axes ${\xi }_i$, ${\eta }_i$, and ${\zeta }_i$ attached to the $i\text{th}$ $\mathrm{Mn}{\mathrm{O}}_6$ octahedron. With regard to the superexchange interactions, we consider ferromagnetic exchanges $J_{ab}$ on the Mn-Mn bonds along the cubic $x$ and $y$ axes, antiferromagnetic exchanges $J_2$ on the in-plane diagonal Mn-Mn bonds along the orthorhombic $b$ axis, and antiferromagnetic exchanges $J_c$ on the Mn-Mn bonds along the $c$ axis. (b) Configuration of the occupied $e_g$ orbitals and superexchange interactions ($J_{ab}$ and $J_c$, solid curves) in the $ab$ plane. Exchange path for the superexchange $J_2$ via two oxygen $2p$ orbitals is indicated by a dashed line.

Figure 4

(Color online) (a) Opposite contributions to the nearest-neighbor ferromagnetic exchange $J_{ab}$ from the $t_{2g}$- and $e_g$-orbital sectors. (b) Cooperative contributions to the second-neighbor antiferromagnetic exchange $J_2$ from the $t_{2g}$- and $e_g$-orbital sectors.

Figure 5

(Color online) Dzyaloshinsky-Moriya vectors associated with different Mn-O-Mn bonds, which are expressed by five parameters, ${\alpha }_{ab}$, ${\beta }_{ab}$, ${\gamma }_{ab}$, ${\alpha }_c$, and ${\beta }_c$. The vectors on the in-plane Mn-O-Mn bonds direct nearly perpendicular to the plane consisting of the corresponding Mn, O, and Mn ions.

Figure 6

(Color online) Theoretically obtained magnetoelectric phase diagram in plane of temperature and $J_2$. Here, $\mathrm{AFM}\left(A\right)+\mathrm{WFM}$ denotes the $A$-type antiferromagnetic phase with weak ferromagnetism due to the spin canting, and PE and FE denote paraelectric and ferroelectric phases expected in the spin-current model (Refs. 9, 11, 12), respectively.

Figure 7

(Color online) Calculated spin correlation functions ${\widehat{S}}_{\alpha }\left(\mathit{k}\right)$ for $k_c=\pi$ (left panels) and spin-helicity correlation functions ${\widehat{H}}_{\alpha }^b\left(\mathit{k}\right)$ for $k_c=0$ (right panels) in the $k_x\text{−}{k}_y$ plane ($\alpha =a$, $b$, and $c$) for different magnetic states—at the points indicated by cross symbols in the inset; (a) the canted $\mathrm{AFM}\left(A\right)$ state [$\mathrm{AFM}\left(A\right)+\mathrm{WFM}$ state] at $J_2=0.44\mathrm{meV}$ and $k_{\mathrm{B}}T=0.5\mathrm{meV}$, (b) the sinusoidal collinear spin state at $J_2=0.80\mathrm{meV}$ and $k_{\mathrm{B}}T=3.5\mathrm{meV}$, (c) the $bc$-cycloidal spin state at $J_2=0.80\mathrm{meV}$ and $k_{\mathrm{B}}T=2.5\mathrm{meV}$, and (d) the $ab$-cycloidal spin state at $J_2=0.80\mathrm{meV}$ and $k_{\mathrm{B}}T=0.5\mathrm{meV}$. For the $\mathrm{AFM}\left(A\right)+\mathrm{WFM}$ state, spin correlation function ${\widehat{S}}_{\alpha }\left(\mathit{k}\right)$ for $k_c=0$ is also displayed, which indicates small ferromagnetic moments along the $c$ axis.

Figure 8

(Color online) Calculated temperature profiles of specific heat $C_s\left(T\right)$ and spin-helicity vector ${\mathit{h}}_s^b\left(T\right)$ for various values of $J_2$ (along the vertical dashed lines in inset): (a) $J_2=0.36\mathrm{meV}$, (b) $J_2=0.62\mathrm{meV}$, and (c) $J_2=0.80\mathrm{meV}$. Here, $h_{\gamma }\left(T\right)$ ($\gamma =a$, $b$, and $c$) denotes the γ component of ${\mathit{h}}_s^b\left(T\right)$. In the $ab$-cycloidal [$bc$-cycloidal] phase, the value of $h_c\left(T\right)$ $\left[h_a\left(T\right)\right]$ is much larger than the other two components.

Figure 9

(Color online) (a) [(b)] Left panel: spin structure in the $ab$-cycloidal ($bc$-cycloidal) state and arrangement of the $c$ $\left(a\right)$ components of DM vectors on the in-plane (out-of-plane) Mn-O-Mn bonds. The symbols ☉ and ⊗ express their signs, i.e., ☉ for the positive sign and ⊗ for the negative sign. Right panel: (upper panel) spin directions in the $ab$-cycloidal ($bc$-cycloidal) state with uniform rotation angles in the absence of DM interactions, and (lower panel) spin directions with modulated rotation angles in the presence of DM interactions.

Figure 10

(Color online) Theoretically obtained magnetoelectric phase diagrams for (a) ${\alpha }_c=0.24\mathrm{meV}$, (b) ${\alpha }_c=0.34\mathrm{meV}$, and (c) ${\alpha }_c=0.38\mathrm{meV}$. Here ${\alpha }_c$ expresses magnitude of the $a$ component of DM vector on the out-of-plane Mn-O-Mn bond. PE and FE denote paraelectric and ferroelectric phases expected in the spin-current model (Refs. 9, 11, 12), respectively.

Figure 11

Temperature dependence of $\sqrt{{\widehat{S}}_c\left(\mathit{q}\right)∕{\widehat{S}}_b\left(\mathit{q}\right)}$ in $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ measured in the spin-polarized neutron-scattering experiment from Ref. 10. Here $\mathit{q}$ is the propagation wave vector of the $bc$-cycloidal spin state.

Figure 12

(Color online) (a) Temperature dependence of ${\widehat{S}}_a\left(\mathit{q}\right)$, ${\widehat{S}}_b\left(\mathit{q}\right)$, and $\sqrt{{\widehat{S}}_a\left(\mathit{q}\right)∕{\widehat{S}}_b\left(\mathit{q}\right)}$ for the $ab$-cycloidal spin state with $J_2=0.80\mathrm{meV}$ and ${\alpha }_c=0.24\mathrm{meV}$. (b) Temperature dependence of ${\widehat{S}}_b\left(\mathit{q}\right)$, ${\widehat{S}}_c\left(\mathit{q}\right)$, and $\sqrt{{\widehat{S}}_c\left(\mathit{q}\right)∕{\widehat{S}}_b\left(\mathit{q}\right)}$ for the $bc$-cycloidal spin state with $J_2=0.80\mathrm{meV}$ and ${\alpha }_c=0.38\mathrm{meV}$. The data for finite temperatures obtained by the Monte Carlo simulations are smoothly extrapolated to the data at $T=0$ obtained by numerically solving the Landau-Lifshitz-Gilbert equation as indicated by dashed lines. Insets of (a) and (b) show the spin-correlation functions in the $k_x\text{−}{k}_y$ plane for $k_c=\pi$ at $T=0$. (c) Spin alignment (left panel) and spin directions (right panel) at $T=0$ for the $ab$-cycloidal spin state. Here the orthorhombic unit cell contains alternately stacked two different MnO planes with $z_i=0$ and 0.5. (d) Those for the $bc$-cycloidal spin state.

Figure 13

(Color online) (a) Local hard magnetization axes (solid arrows) and local easy magnetization axes (dashed arrows) at Mn A and Mn B sites due to the single-ion anisotropies, ${\mathcal{H}}_{\mathrm{sia}}={\mathcal{H}}_{\mathrm{sia}}^D+{\mathcal{H}}_{\mathrm{sia}}^E$. Owing to the tilting of $\mathrm{Mn}{\mathrm{O}}_6$ octahedra, these axes deviate from the cubic $x$, $y$ and $z$ axes. (b) Polar representation of the $S=2$ classical-vector spin. The angle ${\varphi }_i$ is defined with respect to the orthorhombic $a$ and $b$ axes. [(c) and (d)] ${\varphi }_i$ dependence of energies $E_{\mathrm{sia}}^E$, $E_{\mathrm{sia}}^D$, and $E_{\mathrm{cub}}$ at (c) Mn A and (d) Mn B for a spin sticking on the $ab$ plane. [(e) and (f)] ${\varphi }_i$ dependence of energies $E_{\mathrm{siaA}}^E$, $E_{\mathrm{siaB}}^E$, and $E_{\mathrm{siaA}}^E+E_{\mathrm{siaB}}^E$ in the (e) presence and (f) absence of the $\mathrm{Mn}{\mathrm{O}}_6$ tilting. Here $E_{\mathrm{siaA}}^E$ and $E_{\mathrm{siaB}}^E$ are the energies of ${\mathcal{H}}_{\mathrm{sia}}^E$ at Mn A and Mn B sites, respectively. For the $\mathrm{Mn}{\mathrm{O}}_6$ tilting, we use the structural parameters of $\mathrm{Eu}\mathrm{Mn}{\mathrm{O}}_3$ taken from Ref. 87.

Figure 14

(Color online) $T\text{−}E$ phase diagram obtained by the Monte Carlo analysis of Hamiltonian (2). Here the parameter $E$ is a variable which expresses the strength of single-ion anisotropy ${\mathcal{H}}_{\mathrm{sia}}^E$, and $J_2$ is fixed at 0.80 meV. Values of other model parameters in Eq. (2) are fixed at the values listed in Table (see Sec. 2). The temperature range over which the sinusoidal collinear spin phase emerges increases as $E$ is increased.

Figure 15

(Color online) $T\text{−}K$ phase diagram obtained by the Monte Carlo calculation for the case without $\mathrm{Gd}\mathrm{Fe}{\mathrm{O}}_3$-type distortion (see text). The term ${\mathcal{H}}_K$ given in Eq. (27) is added to Hamiltonian (2), which generates the easy-axis spin anisotropy along the $b$ axis. Here, the parameter $K$ is a variable which expresses the strength of easy-axis spin anisotropy. Values of the model parameters except for $K$ and $J_2=0.80\mathrm{meV}$ are fixed at the values listed in Table (see Sec. 2). The temperature range of the sinusoidal collinear spin phase increases as $K$ is increased.

Figure 16

(Color online) Third-neighbor spin exchange $J_3$ considered in Refs. 115, 116.

Figure 17

(Color online) Calculated superexchange parameters, $J_{ab}$ and $J_c$, plotted as functions of the ionic $R$-site radius. Structural data used in the calculation are taken from Refs. 84, 85, 86, 87, 88.

Figure 18

(Color online) Estimated values of $J_2$ in $R\mathrm{Mn}{\mathrm{O}}_3$ and several solid solutions plotted as functions of the ionic $R$-site radius. Inset describes the $J_1\text{−}{J}_2$ classical Heisenberg model on the anisotropic triangular lattice and the spiral spin order with uniform rotation angles of θ.

Figure 19

(Color online) Calculated single-ion anisotropy parameters, $D$ and $E$, plotted as functions of the ionic $R$-site radius. Structural data used in the calculation are taken from Refs. 84, 85, 86, 87, 88. Optical absorption data are taken from different references between (a) and (b), i.e., Ref. 89 for (a) and Ref. 90 for (b).