Theory of Magnetic Switching of Ferroelectricity in Spiral Magnets

We propose a microscopic theory for magnetic switching of electric polarization ($\mathit{P}$) in the spin-spiral multiferroics by taking ${\mathrm{TbMnO}}_3$ and ${\mathrm{DyMnO}}_3$ as examples. We reproduce their phase diagrams under a magnetic field ${\mathit{H}}_{\mathrm{ex}}$ by Monte Carlo simulation of an accurate spin model and reveal that competition among the Dzyaloshinskii-Moriya interaction, spin anisotropy, and spin exchange is controlled by the applied ${\mathit{H}}_{\mathrm{ex}}$, resulting in magnetic transitions accompanied by reorientation or vanishing of $\mathit{P}$. We also discuss the relevance of the proposed mechanisms to many other multiferroics such as ${\mathrm{LiCu}}_2{\mathrm{O}}_2$, ${\mathrm{MnWO}}_4$, and ${\mathrm{Ni}}_3{\mathrm{V}}_2{\mathrm{O}}_4$.

Figure 1

(a) Crystal structure, spin exchanges, and local axes ${\xi }_i$, ${\eta }_i$, and ${\zeta }_i$ in $R{\mathrm{MnO}}_3$. Here FM (AFM) denotes (anti)ferromagnetic exchange. (b) $bc$-plane transverse spin spiral in ${\mathrm{TbMnO}}_3$ and ${\mathrm{DyMnO}}_3$, which induces ferroelectric polarization $\mathit{P}\parallel \mathit{c}$. (c) [(d)] Application of ${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{b}$ [${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{c}$] is expected to stabilize the longitudinal [transverse] spin spiral with magnetization $\mathit{M}\parallel {\mathit{H}}_{\mathrm{ex}}$ where $\mathit{P}=0$ [$\mathit{P}\parallel \mathit{a}$] is expected within the spin-current model. (e) [(f)] Experimental $T\mathrm{\text{−}}{H}_{\mathrm{ex}}$ phase diagram of ${\mathrm{DyMnO}}_3$ [${\mathrm{TbMnO}}_3$] for ${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{b}$ [${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{c}$] from Ref. 4, which shows reorientation of $\mathit{P}$ from $\mathit{P}\parallel \mathit{c}$ to $\mathit{P}\parallel \mathit{a}$ [disappearance of $\mathit{P}$] [4]. Here FE (PE) denotes ferroelectric (paraelectric) phase.

Figure 2

Theoretical $T\mathrm{\text{−}}{H}_{\mathrm{in}}$ phase diagrams of (a) ${\mathrm{TbMnO}}_3$ and (b) ${\mathrm{DyMnO}}_3$ for ${\mathit{H}}_{\mathrm{in}}\parallel \mathit{b}$. (c) $T$ profiles of specific heat $C(T)$ and spin chiralities ${\chi }_{\gamma }(T)$ ($\gamma =a,b,c$) for ${\mathrm{TbMnO}}_3$ at $H_{\mathrm{in}}^b=8\mathrm{T}$. (d) Spin structure in the $bc$-plane spiral state at $H_{\mathrm{in}}=0$, and arrangement of the $a$-axis components of DM vectors on the out-of-plane Mn-O-Mn bonds. The symbols ⊙ and $\otimes$ express their signs, i.e., positive and negative, respectively. In the inset, the arrows (dashed lines) show the spin directions in the presence (absence) of DM interaction.

Figure 3

(a) [(b)] Intensity map of internal magnetic field $H_{\mathrm{in}}^b$ [$H_{\mathrm{in}}^c$] for ${\mathrm{DyMnO}}_3$ [${\mathrm{TbMnO}}_3$] in plane of $T$ and external magnetic field $H_{\mathrm{ex}}^b$ [$H_{\mathrm{ex}}^c$] calculated from experimental magnetization data $m_b(T,H_{\mathrm{ex}}^b)$ [$m_c(T,H_{\mathrm{ex}}^c)$], which reproduces the experimental $T\mathrm{\text{−}}{H}_{\mathrm{ex}}$ diagram in Fig. 1e [1f]. (c) [(d)] Arrangement of the Dy [Tb] $f$ moments in ${\mathrm{DyMnO}}_3$ [${\mathrm{TbMnO}}_3$] under ${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{b}$ [${\mathit{H}}_{\mathrm{ex}}\parallel \mathit{c}$] where $\theta \sim 60°$ [$\theta \sim 30°$] [4].

Figure 4

Theoretical $T\mathrm{\text{−}}{H}_{\mathrm{in}}$ phase diagrams of (a) ${\mathrm{TbMnO}}_3$ and (b) ${\mathrm{DyMnO}}_3$ for ${\mathit{H}}_{\mathrm{in}}\parallel \mathit{c}$, and spin structures in (c) $\mathrm{PE}(P=0)$ phase in ${\mathrm{TbMnO}}_3$ and (d) $\mathrm{FE}(\mathit{P}\parallel \mathit{a})$ phase in ${\mathrm{DyMnO}}_3$. Dashed line (open circles) in (a) denote the crossover line (points) where $C(T)$ shows a broad maximum.