Origin of multiferroic spiral spin order in the $R{\text{MnO}}_3$ perovskites

The origin of the spiral spin order in perovskite multiferroic manganites $R{\text{MnO}}_3$ ($R=\text{Tb}$ or Dy) is here investigated using a two $e_g$-orbital double-exchange model. Our main result is that the experimentally observed spiral phase can be stabilized by introducing a relatively weak next-nearest-neighbor superexchange coupling ($\sim 10\%$ of the nearest-neighbor superexchange). Moreover, the Jahn-Teller lattice distortion is also shown to be essential to obtain a realistic spiral period. Supporting our conclusions, the generic phase diagram of undoped perovskite manganites is obtained using Monte Carlo simulations, showing phase transitions from the $A$-type antiferromagnet, to the spiral phase, and finally to the $E$-type antiferromagnet, with decreasing size of the $R$ ions. These results are qualitatively explained by the enhanced relative intensity of the superexchanges.

Figure 1

(Color online) (a) Sketch of the crystal structure ($a\text{−}b$ plane) of $R{\text{MnO}}_3$. Two types of coordinate axes ($a\text{−}b$ and $x\text{−}y$) are shown. (b) Illustration of the two kinds of distortions discussed in the text: ${\text{GdFeO}}_3$ type (oxygen moves perpendicular to the Mn–O–Mn bond) and Jahn-Teller type (oxygen moves along Mn–O–Mn bond). (c) Two Jahn-Teller distortion modes: $Q_2$ and $Q_3$. (d) Sketch of the spin patterns (in the $a\text{−}b$ plane) corresponding to seven ($S4$ and $S6$ belong to the same group) candidate phases considered here in the zero-temperature variational method. $A$, $G$, $CE$, $C$, and $E$ are standard notations for well-known AFM phases in manganites. Here, the ferromagnetic chains in the $C/CE/E$-AFM phases are within the $a\text{−}b$ plane. $D$ denotes the $\uparrow \uparrow \downarrow \downarrow$ dimer phase that is considered in our studies although it has not been observed in real manganites (Ref. 23). In the spiral phases SL, the length period is denoted by $L$ (along $x/y$). The $L=4$ case has the same energy as the $E$-AFM phase in the classical spin model context.

Figure 2

(Color online) (a) Zero-temperature phase diagram of the two-orbital DE model for $R{\text{MnO}}_3$. The possible path for the $A\text{−}S\text{−}E$ phase transition is indicated by the arrow. The phase diagram is independent of $J_{2a}$, as long as $J_{2a}<J_{2b}$. (b) $J_{\text{AF}}$, $K_{\text{A}}$, and $K_{\text{E}}$, as a function of the Mn–O–Mn angle, normalized to their values at $\varphi =180°$. (c) Wave vector number $q$ for (a). Here, $q$ is defined along the $x/y$ direction and equals half of the $q_{\text{Mn}}$ used in experimental papers.

Figure 3

(Color online) (a) Zero-temperature phase diagram of the two-orbital DE model with JT distortions $\left(\lambda \left|Q_2\right|=1.5\right)$ for $R{\text{MnO}}_3$. The possible values of $J_{\text{AF}}\text{−}{J}_{2b}$ for realistic $qs$ in ${\text{TbMnO}}_3$ and ${\text{DyMnO}}_3$ are shown with dashed lines. The four asterisks represent the set of couplings (first and second for $A$, third for $S$, and fourth for $E$) that are studied via the MC techniques. (b) Same as (a) except for a finite $J_{2a}=0.5J_{2b}$. The four asterisks in (a) remain within the $A\text{−}S\text{−}E$ region here. (c) Wave vector number $q$ for (a).

Figure 4

(Color online) (a) Spin structure factor of the four MC simulations at $T=0.004$, corresponding to the four sets of couplings described in the text. All characteristic peaks are prominent, suggesting stable spin orders. (b) Temperature-dependent polarization $P$ and $C\left(\frac{L}{2},\frac{L}{2}\right)$ corresponding to the spiral phase with $q=1/6$. All values are normalized to their saturation values. (c) Sketch of the finite-temperature phase diagram. The $T_N$’s for the $A$ and $E$ phases are determined from their $C\left(\frac{L}{2},\frac{L}{2}\right)$-temperature curves. The lines connecting $T_N$ $\left(T_c\right)$ are just to guide the eyes.