Fractal Dynamics in a Single Crystal of a Relaxor Ferroelectric

We report the high-resolution and broadband light-scattering spectroscopy of a single crystal of a prototypical relaxor ferroelectric, $\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_3$. A self-similar broad central peak, whose intensity is expressed as $I(\omega )\propto {\omega }^{\alpha }$ has been observed, indicating the presence of a fractal in the crystal. A strong correspondence exists between the temperature dependence of the exponent $\alpha$ and that of the reported behaviors of polar nanoregions. The estimated fractal dimension ($d_f\approx 2.6$) at low temperatures clearly indicates a percolation transition of the polar nanoregions at around 240 K.

Figure 1

Self-similar quasielastic light scattering observed in PMN at 297 K. The frequency ranges of (a)–(d) are [$-1000$, 1000 GHz], [$-250$, 250 GHz], [$-60$, 60 GHz], and [$-15$, 15 GHz], respectively. The solid black line is a fit of the power-law function; $I\propto {\omega }^{\alpha }$, with $\alpha =-0.67$. The LA, w, S, and G denote the Brillouin scattering from the longitudinal acoustic phonon, the cryostat window, the surface acoustic wave, and the first order transmission from the interferometer (ghost line), respectively.

Figure 2

Temperature variation of the power-law light-scattering spectrum observed in PMN. The open circles that are connected by lines represent the spectra recorded by a tandem FP (Brillouin spectra), and the solid lines are the spectra recorded by a triple diffraction monochromator [8]. The temperatures of the Brillouin and Raman spectra (indicated as B- and R-, respectively, in the legend) are not exactly the same, and differ by as much as 10%. The Brillouin spectra above 500 K are not shown because of the poor signal-to-noise ratio. The dashed lines are used as visual guides.

Figure 3

Temperature variation of the reduced intensity, $\mathcal{I}(\nu )\equiv \nu I(\nu )/[n(\nu )+1]$, for selected temperatures. The spectra are vertically shifted only by the multiplication of constants for visual clarity. The solid straight lines represent the least-square fits of the power function, $\mathcal{I}=A{\nu }^{\mu }$.

Figure 4

Temperature dependence of the power exponent of the reduced intensity obtained in PMN. The dashed straight line is the fitted asymptotic value ($\approx 1.33$) obtained by constant-line fitting to the data points below 200 K. The inset shows the temperature dependence of the estimated fractal dimension, $d_f$, assuming constant values for $d_{\varphi }$ and $d_s$.