Magnetic Inversion Symmetry Breaking and Ferroelectricity in ${\mathrm{TbMnO}}_3$

${\mathrm{TbMnO}}_3$ is an orthorhombic insulator where incommensurate spin order for temperature $T_N<41\mathrm{K}$ is accompanied by ferroelectric order for $T<28\mathrm{K}$. To understand this, we establish the magnetic structure above and below the ferroelectric transition using neutron diffraction. In the paraelectric phase, the spin structure is incommensurate and longitudinally modulated. In the ferroelectric phase, however, there is a transverse incommensurate spiral. We show that the spiral breaks spatial inversion symmetry and can account for magnetoelectricity in ${\mathrm{TbMnO}}_3$.

Figure 1

Diffraction intensity along the $(0,k,1)$ direction for different temperatures (with a factor 10 offset between the sub-$T_N$ data sets), showing fundamental and higher-order Fourier components of the magnetic ordering. The $3t$- and $9t$-order peaks associated with the Tb order occur at $k=0.725$ and 0.175, respectively, and are difficult to identify unambiguously due to their proximity to strong peaks.

Figure 2

(a) $T$ dependence of the integrated intensity of the $(0,q,1)$ Bragg peak and the associated third-order peak at $(0,1-3q,1)$. (b) $T$ dependence of the wave vector $q$, corrected for the $T$ dependence of the $b$ lattice parameter, showing the absence of a lock-in transition at $T=28\mathrm{K}$.

Figure 3

Integrated intensities of magnetic Bragg peaks measured at $T=15\mathrm{K}$ and $T=35\mathrm{K}$ using TriCS are compared to various magnetic structures. Peaks are sorted by decreasing measured intensity. (a) The $T=35\mathrm{K}$ structure is described by a single irreducible representation, ${\Gamma }_3$, and the moments point along the $b$ axis. (b) The $T=15\mathrm{K}$ magnetic structure is a spiral described by a $y$ component of ${\Gamma }_3$ and an $x$ component of ${\Gamma }_2$ with a $\pi /2$ phase shift.

Figure 4

Schematic of the magnetic structure at (a) $T=35\mathrm{K}$ and (b) $T=15\mathrm{K}$, projected onto the $b\mathrm{\text{−}}c$ plane. Filled arrows indicate direction and magnitude of Mn moments. The longitudinally modulated phase (a) respects inversion symmetry along the $c$ axis, but the spiral phase (b) violates it, allowing an electric polarization (unfilled arrow). (c) Irreducible representation of the group $G_{\mathbf{k}}$ for the incommensurate magnetic structure with $\mathbf{k}=(0,q,0)$. (d) Phase diagram as a function of temperature and field applied along the $a$ axis. Solid circles indicate second order phase transitions. Open circles indicate the characteristic field for reduction of magnetic $(0,1-q,1)$ Bragg scattering from Tb moments by 50% from its zero-field intensity.

Figure 5

Field dependence of magnetic Bragg scattering from ${\mathrm{TbMnO}}_3$. (a) and (b) show data for the incommensurate peaks that occur for $T<41\mathrm{K}$. (a) The $(0,q,1)$ peak that is mostly sensitive to staggered magnetization on Mn sites and the $(0,1-q,1)$ peak that is sensitive to staggered magnetization on Tb sites. (c) and (d) show data for the incommensurate peaks that develop for $T<7\mathrm{K}$.