Reduced crystal symmetry as the origin of the ferroelectric polarization within the incommensurate magnetic phase of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$

The precise crystal symmetry, and hence the emergence of the electric polarization, still remains an open question in the multiferroic $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ ($R=$ rare earth, Bi, Y). While previous diffraction studies have indicated the centrosymmetric space group $Pbam$, an atomic displacement allowing for electric polarization would require a noncentrosymmetric crystal symmetry. Our single crystal neutron diffraction experiments on ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ provide direct evidence of a reduced crystallographic symmetry with the polar space group $P{12}_{1}1$ already above the multiferroic phase transition, indicating that a symmetric ${\mathbf{S}}_{\mathbf{i}}·{\mathbf{S}}_{\mathbf{j}}$ spin coupling, i.e. magnetostriction is the dominating mechanism in the commensurate magnetic phase. Furthermore, the commensurate magnetic reflections are in accordance with a quartile step spin spiral along the $c$ axis. Therefore, the antisymmetric ${\mathbf{S}}_{\mathbf{i}}\times {\mathbf{S}}_{\mathbf{j}}$ exchange via the inverse Dzyaloshinskii-Moriya interaction contributes as well and becomes the leading term in the low-temperature incommensurate spin-spiral magnetic phase. These findings provide important information for the understanding of the complex interplay between magnetic and structural order throughout the multiferroic $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ series.

Figure 1

Crystal structure of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ at room temperature (crystallographic data from Ref. [20]). (a) $ab$ plane. The key elements are edge-sharing ${\mathrm{Mn}}^{3+}{\mathrm{O}}_5-\mathrm{M}{\mathrm{n}}^{4+}{\mathrm{O}}_6-\mathrm{M}{\mathrm{n}}^{3+}{\mathrm{O}}_5$ pyramid-octahedron-pyramid units which are linked to zigzag chains along the $a$ direction (see light green lines). The red circle indicates the circular arrangement of five ${\mathrm{Mn}}^{4+}\text{−}{\mathrm{Mn}}^{3+}\text{−}{\mathrm{Mn}}^{3+}\text{−}{\mathrm{Mn}}^{4+}\text{−}{\mathrm{Mn}}^{3+}$ polyhedra, which are the origin of the spin frustration in ${\mathrm{TbMn}}_2{\mathrm{O}}_5$. (b) $ac$ plane. Edge-sharing ${\mathrm{Mn}}^{4+}{\mathrm{O}}_6$ octahedral chains along the $c$ direction are linked by two ${\mathrm{Mn}}^{3+}{\mathrm{O}}_5$ pyramids. The dark green lines in both panels indicate the magnetic exchange interactions $J_1\text{–}{J}_5$ in the $ab$ and $ac$ planes.

Figure 2

Reciprocal space maps of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ in the ($h$ 0 $l$) plane at various temperatures, i.e., at 4 K in the LT-ICM, at 28 K in the CM, at 40 K in the HT-ICM, and at 50 K in the PM phase. In the $Pbam$ space group nuclear reflections appear at the position with $h=2n$ and $l=n$ where $n$ is an integer (see white right angles). Additional nuclear Bragg peaks are observed at ($h$ 0 $l$) positions with $h=$ odd integer and $l=$ integer, and at ($h$ 0 $l$) positions with $h=$ integer and $l=$ half-integer indices. The magnetic Bragg reflections at $(\frac{1}{2}-{\delta }_x,0,\frac{1}{4}+{\delta }_z)$ are indicated by red right angles.

Figure 3

(a,b) RSM of the $hk$ plane of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ measured in the LT-ICM phase at 5.5 K and in the PM phase at 50 K. The nuclear Bragg reflections are highlighted by white right angles. At low-temperature out of plane magnetic Bragg peaks at ($h\pm 0.51$, $k$, ±0.31) are also visible due to the vertical detector range of $\mu =\pm 7.8^{\circ }$. (c,d) RSM of the $kl$ plane at 4.0 K (LT-ICM phase) and at 50 K (PM phase). Additional nuclear Bragg peaks appear at both temperatures at half-integer $l$ indices and are labeled underneath the reflection. In the LT-ICM phase at 4 K additional magnetic Bragg peaks appear. They can be attributed to out of plane reflections at $h=\pm 0.52$ and are indicated by red right angles.

Figure 4

Line cuts through the RSMs of the $kl$ plane along the $l$ direction, (a) along (0 4 $l$), and (b) along (0 5 $l$). The out of plane magnetic Bragg reflections at $(\pm 0.52,k,\frac{1}{4}+{\delta }_z)$ are indicated by vertical dashed lines. For the (0 5 $l$) direction, weak structural reflections are observed at $l$ half-integer and $l$ integer positions (see black arrows). They are present at all temperatures, also above the magnetic phase transition, and their intensity does not change with temperature.

Figure 5

Experimental and calculated integrated intensities of the fundamental nuclear Bragg reflections. The least-squares fitting method was applied to model the data taken at 28 K, i.e., in the CM phase of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$. The results for the parent space group $Pbam$ are shown in (a), for the polar space group $Pb{2}_{1}m$ in (b) and the designated polar space group $P{12}_{1}1$ in (c).

Figure 6

Temperature dependence of the $q_h$, $q_k$, and $q_l$ peak positions and peak intensities of selected magnetic Bragg reflections determined from the neutron diffraction measurements with the crystal being aligned in the $ab$, $ac$, and $bc$ planes. Note that the $q_l$ position for the measurement in the $ab$ plane, the $q_k$ position in the $ac$ plane, and $q_h$ in the $bc$ plane were difficult to determine precisely since they appear out of the scattering plane. The different magnetic phase transitions are indicated by vertical dashed lines.