Theory of spin-phonon coupling in multiferroic manganese perovskites $R$MnO${}_3$

Magnetoelectric phase diagrams of the rare-earth ($R$) Mn perovskites $R$MnO${}_3$ are theoretically studied by focusing on crucial roles of the symmetric magnetostriction or the Peierls-type spin-phonon coupling through extending our previous work [M. Mochizuki et al., Phys. Rev. Lett. 105, 037205 (2010)]. We first construct a microscopic classical Heisenberg model for $R$MnO${}_3$ including the frustrated spin exchanges, single-ion anisotropy, and Dzyaloshinskii-Moriya interaction. We also incorporate the lattice degree of freedom coupled to the Mn spins via the Peierls-type magnetostriction. By analyzing this model using the replica-exchange Monte Carlo technique, we reproduce the entire phase diagram of $R$MnO${}_3$ in the plane of temperature and magnitude of the orthorhombic lattice distortion. Surprisingly it is found that in the $ab$-plane spiral spin phase, the ($\mathit{S}·\mathit{S}$)-type magnetostriction plays an important role for the ferroelectric order with polarization $\mathit{P}$$\parallel$$\mathit{a}$ whose contribution is comparable to or larger than the contribution from the ($\mathit{S}\times \mathit{S}$)-type magnetostriction, whereas in the $bc$-plane spiral phase, the ferroelectric order with $\mathit{P}$$\parallel$$\mathit{c}$ is purely of ($\mathit{S}\times \mathit{S}$) origin. This explains much larger $\mathit{P}$ in the $ab$-plane spiral phase than the $bc$-plane spiral phase as observed experimentally and gives a clue how to enhance the magnetoelectric coupling in the spin-spiral-based multiferroics. We also predict a noncollinear deformation of the $E$-type spin structure resulting in the finite ($\mathit{S}\times \mathit{S}$) contribution to the ferroelectric order with $\mathit{P}$$\parallel$$\mathit{a}$, and a wide coexisting regime of the commensurate $E$ and incommensurate spiral states, which resolve several experimental puzzles.

Figure 1

(a) Relationship between the spiral-plane orientation and the spontaneous ferroelectric polarization ${\mathit{P}}_{\mathrm{AS}}$ in the $bc$-plane spiral spin structure predicted by the spin-current model.[26, 27, 28] (b) That in the $ab$-plane spiral spin structure.

Figure 2

(a) Experimentally obtained magnetoelectric phase diagram of $R$MnO${}_3$ and solid solutions Gd${}_{1-x}$Tb${}_x$MnO${}_3$ and (b) that of solid-solution systems Eu${}_{1-x}$Y${}_x$MnO${}_3$ and Y${}_{1-y}$Lu${}_y$MnO${}_3$ in the plane of temperature and (effective) ionic radius of the $R$ ion.[38]

Figure 3

(a) Spin-exchange interactions in $R$MnO${}_3$ and tilted local coordinate axes ${\xi }_i$, ${\eta }_i$, and ${\zeta }_i$ attached to the $i$th MnO${}_6$ octahedron. Here FM (AFM) denotes (anti-)ferromagnetic exchange interaction. For the spin-exchange interactions, we consider ferromagnetic exchange $J_{ab}$ on the Mn-Mn bonds along the pseudocubic $x$ and $y$ axes, (anti-)ferromagnetic exchange $J_a$ ($J_b$) on the in-plane diagonal Mn-Mn bonds along the $a$ ($b$) axis, and antiferromagnetic exchange $J_c$ along the $c$ axis. (b) Mn($i$)-O-Mn($j$) bond in the orthorhombic lattice and local vector ${\mathit{n}}_{i,j}$. The O ion is displaced from its cubic position (0) to the orthorhombic position (${\Delta }_o$) at higher temperatures. At low temperatures, a further shift ${\delta }_{i,j}$ along ${\mathit{n}}_{i,j}$ can be induced by the spin-lattice coupling in the presence of magnetic order. (c) ${\Delta }_o$ vs. $J_{ab}$ for several $R$MnO${}_3$ compounds calculated in Ref. 34, which gives $J_{ab}^{\prime }=\partial J_{ab}$/$\partial {\Delta }_o=2.5$. Here ${\Delta }_o$ is normalized by the MnO bond length. (d) Main exchange path for the next-neighbor ferromagnetic exchange $J_a$ along the $a$ axis [(blue) dotted line].

Figure 4

Dzyaloshinskii-Moriya vectors ${\mathit{d}}_{i,j}$ associated with the Mn($i$)-O-Mn($j$) bonds.

Figure 5

Theoretical phase diagram of $R$MnO${}_3$ in the plane of temperature and $J_b$. Here ICS denotes the incommensurate spiral phase. In the shaded area, incommensurate spin states can coexist with the $E$-type state (see text).

Figure 6

Temperature profiles of specific heat $C_s\left(T\right)$, spin helicity $h_{\gamma }\left(T\right)$, and polarizations due to the ($\mathit{S}·\mathit{S}$)-type magnetostriction ${\tilde{P}}_{\gamma }\left(T\right)$ for $J_b=2.4$ meV.

Figure 7

(a) Real-space spin configuration of the $E$-type phase for two kinds of $ab$ planes, $z_i=0$ and 0.5. (b) Spin alignment on the zigzag Mn-O chain along the $x$ axis.

Figure 8

(a) Pure collinear up-up-down-down spin structure with Mn spins parallel to the $b$ axis. (b) Because of the staggered $d_{3{\xi }^2-r^2}$/$d_{3{\eta }^2-r^2}$ orbitals, the local easy magnetization axes are alternately arranged on the zigzag Mn-O chain (upper figure), which can cause deviation of each Mn-spin direction from the $b$ axis toward the local easy axis, resulting in the cycloidal deformation (lower figure). (c) On the zigzag Mn-O chain, the $c$-axis components of the Dzyaloshinskii-Moriya vectors are arranged in a staggered way, with which the Mn spins cant and rotate in the $ab$ plane. Here $\odot$ ($\otimes$) denotes the positive (negative) $c$-axis component of the Dzyaloshinskii-Moriya vector. Note that both the single-ion anisotropy ${\mathcal{H}}_{\mathrm{sia}}^E$ with alternate easy magnetization axes and the Dzyaloshinskii-Moriya interaction ${\mathcal{H}}_{\mathrm{DM}}^{ab}$ give the equivalent spin canting or rotation as can been seen in lower figure of (b) and (c).

Figure 9

Temperature dependence of expectation values for ${\mathcal{H}}_{\mathrm{DM}}^{ab}$ and ${\mathcal{H}}_{\mathrm{sia}}^E$ for $J_b=1.4$ meV. At the transition to the $E$-type phase with lowering temperature, the value for ${\mathcal{H}}_{\mathrm{sia}}^E$ abruptly decreases with a large jump, while that for ${\mathcal{H}}_{\mathrm{DM}}$ slightly increases, indicating that the term $H_{\mathrm{sia}}^E$ is a source of the cycloidal deformation.

Figure 10

In the spirally deformed up-up-down-down structure, both ($\mathit{S}·\mathit{S}$)-type and ($\mathit{S}\times \mathit{S}$)-type mechanisms contribute to the ferroelectric $\mathit{P}$. Concerning the former mechanism, alternate large and small spin turn angles cause a uniform shift of the O ions to strengthen and weaken the ferromagnetic exchanges through decreasing and increasing the Mn-O-Mn bond angle, respectively. Shifts of the O ions due to the ($\mathit{S}·\mathit{S}$)-type magnetostriction are shown by small (gray) arrows. Large (red and blue) arrows represent the dominant $b$-axis components of the Mn spins.

Figure 11

(a) Spin exchanges of the classical Heisenberg model given by Eq. (25). In the limit of large antiferromagnetic $J_b$, three kinds of magnetic states are degenerate, i.e., (b) (pure) $E$-type, (c) stripe, and (d) 90${}^{\circ }$ spiral states. The nearest-neighbor bonds with energy gain (cost) of $-|J_{ab}|S^2$ ($+|J_{ab}|S^2$) are shown by thick (dotted) lines. The $E$-type state becomes energetically stabilized by bond alternation shown in (b), while the stripe state by that shown in (c). (e) Energy diagram of these three states (see text).

Figure 12

(a) Calculated correlation function ${\widehat{J}}_{\gamma {\gamma }^{\prime }}(\mathit{k},T)$ given by Eq. (26) for the $E$-type state at $J_b=2.4$ meV and $k_{B}T=0.5$ meV, which shows sharp peaks at $\mathit{k}=$($\pm \pi$, $\pm \pi$, 0). Magnitudes of these peaks are all $1.0\times {10}^{-4}$ meV${}^2$. (b) Modulations of the nearest-neighbor ferromagnetic exchanges. The ferromagnetic coupling is strengthened (weakened) by $\Delta J_{ab}\sim 0.01$ meV on the thick (thin) bonds.

Figure 13

(a) Temperature profiles of specific heat $C_{\mathrm{s}}\left(T\right)$, spin helicity $h_{\gamma }\left(T\right)$, and polarizations due to the ($\mathit{S}·\mathit{S}$)-type magnetostriction ${\tilde{P}}_{\gamma }\left(T\right)$ for $J_b=0.7$ meV. (b) Those for $J_b=1.2$ meV.

Figure 14

Alternation of the spin turn angles in the $ab$-plane spiral state due to the staggered Dzyaloshinskii-Moriya vectors is depicted in an exaggerated manner where $\odot$ ($\otimes$) denotes the positive (negative) $c$-axis component of the vector. Induced shifts of the O ions due to the ($\mathit{S}·\mathit{S}$)-type magnetostriction are shown by gray arrows.

Figure 15

Calculated $J_b$ dependence of the magnetic wave number $q_b$ and the $\gamma$-axis components of the spin helicity $h_{\gamma }$ ($\gamma =a$, $b$, $c$) at $T\to 0$.

Figure 16

$J_b$ dependence of polarizations at $T\to 0$, i.e., ($\mathit{S}·\mathit{S}$) contribution $P_S$, ($\mathit{S}\times \mathit{S}$) contribution $P_{\mathrm{AS}}$, and experimentally measured $P$ in Eu${}_{1-x}$Y${}_x$MnO${}_3$ and Y${}_{1-y}$Lu${}_y$MnO${}_3$.[38] The summation $P_S$+$P_{\mathrm{AS}}$ reproduces the experimental $P$ well.

Figure 17

Real-space spin configuration obtained in the Monte Carlo simulation starting with an incommensurate spiral spin configuration as the initial state at $J_b=2.4$ meV (see text). The spin $\gamma$-axis components ($\gamma =a$, $b$, $c$) are plotted for two kinds of $ab$ plane, i.e., $z_i=0$ and $z_i=0.5$.

Figure 18

(a) Calculated spin-correlation functions in the momentum space at $J_b=2.4$ meV for a metastable incommensurate state obtained in the Monte Carlo simulation starting with the incommensurate spiral state (see text). Their peaks are located at $q_b=\pm 0.458\pi$. (b) Those for the commensurate $E$-type state whose peaks are located at $q_b=\pm 0.5\pi$. Here ${\widehat{S}}_{\gamma }\left(\mathit{k}\right)$ denotes the correlation function for the spin $\gamma$-axis components given by Eq. (19).