Zero- and finite-temperature mean field study of magnetic field induced electric polarization in Ba${}_2$CoGe${}_2$O${}_7$: Effect of the antiferroelectric coupling

We investigate the spin induced polarization in the multiferroic compound Ba${}_2$CoGe${}_2$O${}_7$ using variational and finite-temperature mean field approaches, with the aim to reproduce the peculiar behavior of the induced polarization in a magnetic field observed experimentally in Murakawa et al. [Phys. Rev. Lett. 105, 137202 (2010)]. The compound is usually described by a spin-3/2 Heisenberg model extended with easy-plane anisotropy and Dzyaloshinskii-Moriya (DM) interaction. By applying a magnetic field parallel to the $\left[110\right]$ axis, three phases can be distinguished in this model: (i) At high magnetic field, we find a partially magnetized phase with spins parallel to the fields and uniform polarization. (ii) Below a critical field, the ground state is a twofold-degenerate canted antiferromagnet, where the degeneracy can be lifted by a finite DM interaction. (iii) At zero field, a U$\left(1\right)$ symmetry-breaking phase takes place, exhibiting a Goldstone mode. We find that extending the Hamiltonian with an antiferroelectric term results in the appearance of a canted ferrimagnetic phase for $h\lesssim 1$ T. This phase is characterized by a finite staggered polarization, as well as by a magnetization closing a finite angle with the applied field leading to torque anomalies.

Figure 1

A schematic figure of Ba${}_2$CoGe${}_2$O${}_7$. The Co${}^{2+}$ ions are located at the center of the tetrahedra and are represented by magenta circles. The unit cell contains two Co${}^{2+}$ ions that we denote by $A$ and $B$ (note that the neighboring tetrahedra are oriented differently). (a) The point group of this material can be constructed from the groups ${\mathcal{S}}_4$ and ${\mathcal{C}}_{2v}$ acting on the Co sites and in the middle of four sites, respectively. The mirror planes are labeled by their normal vector. (b) The symmetry-allowed DM vectors, with respect to the order of involved sites (black arrows), are indicated by purple arrows. At the $\Gamma$ point, the in-plane components of the DM vector cancel each other, thus, in the case of two-sublattice ordering, it is sufficient to take only the out-of-plane components into account.

Figure 2

Relation between the cases $h\left|\right|\left[110\right]$ and $h\left|\right|\left[100\right]$. In our notation, $P_j^z$ takes maximum value for a spin aligned parallel to the lower edge (dashed line) and minimum value along the upper (solid) edge. (a) Schematic spin configuration for $h\left|\right|\left[110\right]$ and $D_z<0$. The spin induced polarization is the same on the two sublattices because the spins are canted by the same angle from the same (lower) edge of the tetrahedra. (b) Spin configuration for $h\left|\right|\left[100\right]$ and $D_z<0$ after rotating the basis by the angle $\pi /4$. Note that due to the rotation of the tetrahedral environment, the induced polarization changes. The spins of the two sublattices are now canted from the different edges, consequently, the induced polarization on them have different signs.

Figure 3

(a), (b) Schematic figure of ground states in canted antiferromagnetic phase when $h\left|\right|\left[110\right]$. For $D_z=0$, (a) and (b) are degenerate. A finite DM coupling lifts the twofold degeneracy, and the ground-state configuration for $D_z>0$ is shown in (a) while (b) is selected when $D_z<0$. (c) The polarization $P_z$, the magnetization $m_{\left[110\right]}$, and the staggered magnetization $m_{\left[\overline{1}10\right]}^{\mathrm{st}}$ is shown. Black line indicates the twofold degeneracy of the $D_z=0$ case below the partially magnetized phase. The colored lines correspond to the $D_z>0$ and $D_z<0$ cases in accordance with the coloring of spin states in (a) and (b).

Figure 4

(a), (b) The canted antiferromagnetic ground state for $h\left|\right|\left[100\right]$. Similarly to the case of $h\left|\right|\left[110\right]$, (a) and (b) are degenerate when $D_z=0$. Whereas a finite $D_z$ lifts this degeneracy, and $D_z>0$ selects the canted state (a) while $D_z<0$ prefers the configuration of (b). (c) When $h\left|\right|\left[100\right]$, the uniform polarization vanishes ($P_z=P_z^A+P_z^B=0$) and the staggered polarization $P_z^{\mathrm{st}}=P_z^B-P_z^A$ takes a finite value. The coloring of the cases $D_z>0$ and $D_z<0$ follows that of the spin states used in (a) and (b).

Figure 5

Ground-state spin configurations when the antiferroelectric is present in the Hamiltonian. (a), (b) The fourfold-degenerate ground state in zero field for different signs of $D_z$. A spin configuration, with smaller canting angle, shown in (a) would correspond to the case $D_z=0$, too; (c) when $h\left|\right|\left[110\right],$ a canted ferrimagnetic phase emerges below $h_{c2}\approx 1$ T due to the polarization term. (d) For $h\left|\right|\left[100\right]$, since $K_z>0$, the canted ferrimagnetic phase is missing, and at finite field the ground state is the canted antiferromagnet similar to the case $K_z=0$. A ferromagnetic polarization term would have the opposite effect.

Figure 6

Behavior of magnetization (a) and polarization (b) in external field along $\left[110\right]$ at various temperature values. The results for 0 K and 2 K are indiscernible.

Figure 7

The canted ferrimagnetic phase for $h⪅1$ is the consequence of the antiferroelectric term (11). The figures show the total polarization (a), the magnetic susceptibility in the $\left[110\right]$ direction (b), the staggered polarization (c), and the $m\left[\overline{1}10\right]$ magnetization (d). Note that curves of 0 K and 2 K are indistinguishable.

Figure 8

Staggered polarization and magnetization results at finite temperature for the magnetic field parallel to the $\left[100\right]$ direction. In this setting, the canted ferrimagnetic phase is missing, and the lower polarization curve is selected when $K_z>0$. Similarly to the $h\left|\right|\left[110\right]$ case, the staggered polarization depends strongly on the temperature, however, the magnetization hardly changes ($J=J_z=1.885$ K, $\Lambda =15.08$ K, $D_z=-0.02$ K, $K_z=0.01$ K, and $g=2.2$). The 0 K and 2 K curves cannot be distinguished.