Simple microscopic model for magnetoelectric coupling in type-II antiferromagnetic multiferroics

We present a simple two-dimensional model in which the lattice degrees of freedom mediate the interactions between magnetic moments and electric dipoles. This model reproduces basic features, such as a sudden electric polarization switch-off when a magnetic field is applied and the ubiquitous dimerized distortion patterns and magnetic $\uparrow \uparrow \downarrow \downarrow$ ordering, observed in several multiferroic materials of different composition. The list includes E-type manganites, $R\mathrm{Mn}{\mathrm{O}}_3$, nickelates such as $\mathrm{Y}\mathrm{Ni}{\mathrm{O}}_3$, and other materials under strain, such as $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$. In spite of its simplicity, the model presented here captures the essence of the origin of multiferroicity in a large class of type-II multiferroics.

Figure 1

Interaction terms and phases. Aside from the paramagnetic phase, three ordered states are relevant to this study. Magnetic and elastic degrees of freedom are taken on an equal footing in our model, but the names chosen refer to spin configurations. (a) Ordered Ising state (ferromagnetic in the example pictured). For relatively small values of the magnetoelastic coupling $\alpha$, this is the preferred spin configuration. Distortions average zero in this state. (b) Checkerboard phase and (c) stripe phase. If the magnetoelastic coupling constant $\alpha$ is bigger than ${\alpha }_c$, it is favorable to deform the lattice. These deformations imply a dipolar electric moment ${\mathbf{p}}_i={\gamma }_i\delta {\mathbf{r}}_i$, represented here by pairs of colored circles. (d) Dipolar interactions, with an energy scale given by $\lambda$, do not affect the FM or the paramagnet, but change the energy balance between the ST and CB phases. (e) As shown here for an example with $\alpha =1.5{\alpha }_c$, and with ${\gamma }_i=1,2$ for odd and even sites, respectively, the ST becomes the ground state of the system above ${\lambda }_c=2.45$.

Figure 2

Effect of the magnetic field on charge and magnetic degrees of freedom (case of homogeneous charge distribution ${\gamma }_i=\gamma$). Susceptibility peaks and the first peak in $C_V$ coincide with the abrupt fall in $P$ and the rise in $M$. The behavior is similar for both $\lambda$ (i.e., for both ground states). In both cases, $\alpha =1.5{\alpha }_c$. The rise in $M$ is smooth and goes through an intermediate step marked by peaks in the specific heat.

Figure 3

$B-\lambda$ phase diagram. For low fields the ground state of the system transitions from a checkerboard (CB), below ${\lambda }_c$, to a stripe phase (ST). As the field is increased, both the CB and ST states transition into a ferromagnet (FM) going through an intermediate state (IM) where the system is mostly ordered ferromagnetically, with some remnants of the low-field phases in the form of aligned clusters that oppose the magnetic field direction. The snapshot here represents one of the many possible spin and structural configurations taken by the system.

Figure 4

Effect of the magnetic field on charge and magnetic degrees of freedom (nonhomogeneous charge distribution ${\gamma }_i$). In this case (sketched in the inset), where there are two sublattices with different charges, the system has a nonzero homogeneous polarization $P$ at low fields. As in the case with ${\gamma }_i=\gamma$, the transition towards the high-field state is also marked by an intermediate phase.

Figure 5

Signatures in scattering experiments. Structure factors in the spin channel (left panels) and diffuse charge channel, associated with the displaced charges (right panels). The stripe phase (upper panels) shows ${\mathrm{C}}_2$ symmetry in both channels, while the checkerboard phase (lower panels) retains the ${\mathrm{C}}_4$ symmetry of the lattice.

Figure 6

Unit cell and labels used for the calculation of ${\mathrm{\Phi }}_{\mathrm{CB}}$.