Unconventional Ferroelectric Transition in the Multiferroic Compound ${\mathrm{TbMnO}}_3$ Revealed by the Absence of an Anomaly in $c$-Polarized Phonon Dispersion

${\mathrm{TbMnO}}_3$ exhibits a spontaneous electric polarization along $\mathit{c}$ concomitantly with a spiral spin ordering modulated along $\mathit{b}$ below $T_C=28\mathrm{K}$. We have performed inelastic x-ray scattering measurements on a single crystal of ${\mathrm{TbMnO}}_3$ to clarify whether phonon anomalies related to the ferroelectricity exist. We measured transverse modes, especially the Mn-O-Mn bending mode polarized along $\mathit{c}$ and propagating along $\mathit{b}$, which we expect is most relevant to the ferroelectricity. However, no anomaly was found in the phonon dispersion below 50 meV across $T_C$. The present result suggests that the mechanism of ferroelectricity in ${\mathrm{TbMnO}}_3$ is different from that of a conventional displacive-type ferroelectric. The weak coupling between electric polarization and lattice in ${\mathrm{TbMnO}}_3$ strongly suggests that the ferroelectricity is mainly derived from the spiral spin ordering.

Figure 1

Schematic of the cycloidal spiral ordering of manganese spins, displacements of oxygen ions, and induced electric polarizations in ${\mathrm{TbMnO}}_3$. Circles indicated by “Mn” and “O” represent manganese ions and oxygen ions. Arrows on the manganese ions and those indicated by “P” represent manganese spins and electric polarizations. For oxygen ions, open and closed circles show their original and displaced positions, respectively. (a) The case when spins rotate counterclockwise. (b) The case when spins rotate clockwise.

Figure 2

(a), (b) IXS intensity map on the $q\mathrm{\text{−}}\hslash \omega$ plane measured near Bragg point (0,-2,7) along the ${\mathit{b}}^{*}$ direction at (a) $T=35\mathrm{K}$ and (b) $T=11\mathrm{K}$. (c) Calculated dispersion curve and IXS intensity of phonons propagating along the ${\mathit{b}}^{*}$ axis near (0,-2,7).

Figure 3

IXS spectra measured at $\mathit{Q}=(0,-2+q,7)$ at $T=11$, 27, 35, and 50 K. (a) Spectra near the $\Gamma$ point (at $q=0.05$), (b) near the wave vector of the magnetic structure (at $q=0.29$), and (c) near the wave vector of the lattice modulation (at $q=0.53$).

Figure 4

Temperature dependence of high-energy part of IXS spectra measured at $\mathit{Q}=(0,0,6)+\mathit{q}$ and $(0,0,7)+\mathit{q}$ with $\mathit{q}\parallel {\mathit{b}}^{*}$. The measured positions are (a) (0,0.19,6), (b) (0,0.31,6), (c) (0,0.44,6), (d) (0,0.06,7), (e) (0,0.19,7), and (f) (0,0.31,7). Open and closed circles show data measured at $T=11$ and at $T=35\mathrm{K}$, respectively.