Dynamical mechanism of phase transitions in $A$-site ferroelectric relaxor $\left({\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}\right){\mathrm{TiO}}_3$

The dynamical phase transition mechanism of $\left({\mathrm{Na}}_{1/2}{\mathrm{Bi}}_{1/2}\right){\mathrm{TiO}}_3$ (NBT) was studied using inelastic neutron scattering. Softening was observed of multiple phonon modes in the phase transition sequence of NBT. As usual, the softening of the zone center transverse optical modes ${\Delta }_5$ and ${\Sigma }_3$ was observed in the (200) and (220) zones, showing the Ti vibration instabilities in ${\mathrm{TiO}}_6$ octahedra for both cubic-tetragonal (C-T) and tetragonal-rhombohedral (T-R) phase transitions. In these two phase transitions, however, ${\mathrm{Ti}}^{4+}$ has different preferential displacement directions. Surprisingly, the longitudinal optic mode also softens significantly toward the zone center in the range of the transition temperature, indicating the ${\mathrm{Na}}^{+}/{\mathrm{Bi}}^{3+}$ vibration instability against ${\mathrm{TiO}}_6$ octahedra during the T-R phase transition. Strong inelastic diffuse scattering shows up near $M$(1.5, 0.5, 0) and $R$(1.5, 1.5, 0.5) in the tetragonal and rhombohedral phases, respectively, indicating the condensations of the $M_3$ and $R_{25}$ optic modes for the corresponding transitions. This reveals the different rotation instabilities of ${\mathrm{TiO}}_6$ in the corresponding transition temperature range. Bottleneck or waterfall features were observed in the dispersion curves at certain temperatures but did not show close correlations to the formation of polar nanoregions. Additional instabilities could be the origin of the complexity of phase transitions and crystallographic structures in NBT.

Figure 1

Longitudinal (a), (c), (e) and transverse (b), (d), (f) phonon dispersion curves near the (200) zone at 950 K, 750 K, and 300 K, respectively. The black squares show the fitted peak centers from the constant-$E$ scans. The empty diamonds, squares, and circles indicate the peak centers from the constant-$Q$ scans. The horizontal error bars and vertical error bars correspond to the full width at half-maximum (FWHM) of the fitted peaks in constant-$E$ scans and constant-$Q$ scans, respectively. The dash-dot lines are guides for the eyes. The intensity scales of all these contour maps are linear for good contrast. The white area at the bottom of each map is cut off because the elastic scattering intensities are out of the range of the scale.

Figure 2

(a) Constant energy scans for Δ$E=8\mathrm{meV}$ at 300 K, 750 K, and 950 K. (b) Constant energy scans for $\Delta E=7,8$, and 9 meV at 950 K. The curves in (a) and (b) are fitted to double Gaussians. Extra signals were detected near the zone center in some results.

Figure 3

Energy scans at (1.875, 0, 0) (a) and at (2, 0.125, 0) (b) at 300 K, 750 K, and 950 K.

Figure 4

Contour map of inelastic neutron scattering intensity from NBT near the (220) zone at 700 K (a) and 300 K (b). The black squares show the fitted peak centers from the constant-$E$ scans. The horizontal error bars correspond to the FWHM of the fitted peaks in constant-$E$ scans. The dash-dot lines are guides for the eyes.

Figure 5

Constant-$E$ scans at 300 K and 700 K with energy transfer at 3 meV (a) and 5 meV (b). In (b), the curves are fitted with double Gaussian peaks. The hatched area indicates the difference between the fitted and measured intensities.

Figure 6

Temperature dependence of peak intensity at the $M$ point (1.5, 0.5, 0) and the $R$ point (1.5, 1.5, 0.5). The black squares are the diffuse intensity of the $R$ point and the red circles indicate the sum of the diffuse and Bragg peak intensities.