Relaxor Ferroelectrics: Back to the Single-Soft-Mode Picture

The fluctuations of electric polarization in a disordered ferroelectric substance, relaxor crystal ${\mathrm{PbMg}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_3$ (PMN), were studied using a nonlinear inelastic light-scattering technique, hyper-Raman scattering, within a $5–100{\mathrm{cm}}^{-1}$ spectral interval and in a broad temperature range from 20 to 900 K. The split ferroelectric mode reveals a local anisotropy of up to about 400 K. Spectral anomalies observed at higher temperatures are explained as due to avoided crossing of the single primary polar soft mode with a temperature-independent, nonpolar spectral feature near $\text{45\hspace{0.17em}\hspace{0.17em}}{\mathrm{cm}}^{-1}$, known from Raman scattering. The temperature changes of the vibrational modes involved in the measured fluctuation spectra of PMN were captured in a simple model that accounts for the temperature dependence of the dielectric permittivity as well. The observed slowing down of the relaxational dynamics directly correlates with the huge increase of the dielectric permittivity.

Figure 1

Structure of complex perovskites. Left: $B^{\prime }$ and $B^{\prime \prime }$ atoms are distributed randomly so that on average, the crystal has a cubic $Pm\overline{3}m$ structure. Some experiments indicate the presence of $Fm\overline{3}m$ (checkerboard-ordered) clusters (bold square). Middle: Eigenvectors of the low-frequency vibrations emphasizing the in-phase motion ($F_{1u}$ mode) and the out-of-phase ($F_{2g}$ mode in the $Fm\overline{3}m$-symmetry cluster) motion of the lead atoms. Right: Characteristic temperatures delimiting regions of normal paraelectric ($p$), dipole gas ($g$), dipole liquid ($l$), and dipole ice ($s$) relaxor states, respectively.

Figure 2

Hyper-Raman spectra and their fit: (a) Temperature dependence of the HRS intensity, ${\mathrm{I}}_{\mathrm{HRS}}$. (b) Same spectra normalized by the Bose factor emphasizing an almost $T$-independent behavior of ${\omega }_2$. (c) Zoom of the spectra at 30 K showing three modes at low temperature. (d) Fit of the HRS spectra of PMN from 30 K to 900 K. (e) High resolution spectra (in log scale) in the temperature domain where the soft mode (${\omega }_1$) is overdamped. (f) Spectrum of PMT at 750 K.

Figure 3

Mode splittings and mode couplings: Temperature dependence of the modes ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$, observed by HRS. The limit at $\omega =0$ of the two gray regions defines the crossovers from the vibrational to the relaxational regime of the polar ${\omega }_1$ vibration. The splitting of the latter into $A_1$ and $E$ vibrations is the likely signature of the growth of a polar anisotropy on cooling below $\sim 400\mathrm{K}$. The full and dashed lines associated with $\{{\omega }_1,{\omega }_2\}$ and $\{{\omega }_{\mathrm{soft}},{\omega }_{\mathrm{hard}}\}$, respectively, result from a mode-coupling model (see text). The same model successfully applies to PMT (inset).

Figure 4

Soft polar dynamics and dielectric response. (a) The slowing down of the polarization dynamics in PMN is highlighted by (i) the lowering of the soft mode relaxational frequency (circles, ${\omega }_{\mathrm{rel}}$) down to $\sim 0.7{\mathrm{cm}}^{-1}$, (ii) the divergence of its response at $\omega =0$ [diamonds, $I_{F_{1u}}(\omega =0)$], and (iii) the rise of the total intensity at $\omega =0$ ($I_{\mathrm{SHG}}$ in the inset). For comparison, the cross symbols ($+$) in and near the shaded region stand for the neutron spin-echo data [25]. (b) Temperature dependence of the dielectric response at THz frequencies as extracted directly from the HRS data (arbitrary units). The maxima of $\epsilon (T)$ at 34 GHz and 74 GHz (circle) are reproduced from dielectric measurements of Ref. [41]. (c) Inverse permittivity of PMN at 1 kHz, obtained from dielectric measurement (line) and recalculated from the HRS data with (squares) and without (circles) mode coupling. Dashed lines in panels (a) and (c) are guides to the eyes emphasizing a linear behavior.