Heterophase fluctuations near $T_c$ in the relaxor ferroelectrics $(1-x)\mathrm{Pb}$(${\mathrm{Zn}}_{1/3}{\mathrm{Nb}}_{2/3}$)${\mathrm{O}}_3-x{\mathrm{PbTiO}}_3$ ($x=0.09$) studied by x-ray diffuse scattering and coherent x-ray scattering

The paraelectric (PE) to ferroelectric (FE) first-order phase transition of ($1-x$)Pb(${\mathrm{Zn}}_{1/3}{\mathrm{Nb}}_{2/3}$)${\mathrm{O}}_3\text{−}x{\mathrm{PbTiO}}_3$ ($x=0.09$) ($T_c^c=455$ K on cooling) has been studied by the complementary use of x-ray diffuse scattering (XDS) and coherent x-ray scattering (CXS). XDS was mainly used to investigate the FE regions, while CXS was mainly used to investigate the PE regions above $T_c^c$ on cooling. The diffuse scattering intensity due to the appearance of FE regions shows a maximum at $T_{max}=460$ K. The diffuse scattering is dynamic in nature and the softening trend changes to a hardening trend at $T_{max}$. This means that the FE instability is maximum at $T_{max}$ and therefore the FE regions are well stabilized below $T_{max}$. The spatial autocorrelation function obtained by CXS, corresponding to the texture of PE regions, starts to rapidly change at about $T_{max}$ and is most unstable at $T_c^c$. We conclude that a heterophase fluctuation occurs between $T_c^c$ and $T_{max}$ near the phase transition. The heterophase fluctuation can be expected to correlate to the low-frequency dielectric dispersion and contribute to the phase transition as a precursor phenomenon of the first-order phase transition.

Figure 1

Temperature dependencies of the (a) lattice constants and (b) dielectric permittivity of PZN-9%PT for various frequencies [13]. Solid and open circles represent cooling and heating, respectively.

Figure 2

A $10\times 10\times 0.25$ ${\mathrm{mm}}^3$ wafer of PZN-9%PT. Two rectangular electrodes were evaporated parallel to each other on the surface with a gap of 200 $\mu \mathrm{m}$. X rays are targeted at a location between the two electrodes, enabling simultaneous x-ray scattering and dielectric measurements.

Figure 3

Top view of optical alignment for x-ray scattering measurements at BL22XU of SPring-8. ${\mathrm{A}}_1$, upstream aperture; ${\mathrm{A}}_2$, receiving aperture; S, sample; H, sample holder; D, detector; K, scattering vector; and L, displacement sensor. ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are the distances from ${\mathrm{A}}_1$ to S and from S to D (or ${\mathrm{A}}_2$), respectively.

Figure 4

Temperature dependencies of (a) relative distance and (b) peak position (solid circles) and width (bars) of rocking curve.

Figure 5

Diffuse scattering distribution around the 200 Bragg point taken at (a) 460 K [13] and (b) 480 K. (c) Intensity difference between 460 and 480 K ($I_D^{460\text{K}}-I_D^{480\text{K}}$). An increase in the diffuse scattering intensity is clearly seen around area $A$, which means that the transverse-wave-like fluctuation is the dominant cause of the diffuse scattering near $T_c$.

Figure 6

Peak profile taken along ($2+\mathit{h},-\mathit{h},0$) ($-0.005\le h\le 0.05$) at 480 K shown on a logarithmic scale. The inset shows the results plotted on a linear scale.

Figure 7

Temperature dependence of the scattering intensity at $h=$ (a) 0, (b) 0.005 [13], (c) 0.01, (d) 0.02, (e) 0.03, and (f) 0.05.

Figure 8

${(1/I_D)}^{1/2}$ at all $h$ positions for selected temperatures of 480, 460, and 440 K.

Figure 9

Temperature dependence of $2\theta -\omega$ profiles across the 200 Bragg reflection.

Figure 10

Selected CXS patterns taken at (a) 480 K, (b) 462 K, and (c) 456 K. (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$) S-ACFs obtained from (a)–(c), respectively. Corresponding patterns from a perfect crystal (${\mathrm{KTO}}_3$) are also shown in (d) and (${\mathrm{d}}^{\prime }$).

Figure 11

One-dimensional patterns of S-ACFs obtained from Figs. 10(${\mathrm{a}}^{\prime }$)–10(${\mathrm{c}}^{\prime }$) along the black dotted vertical lines.

Figure 12

Temperature dependencies of $\langle \sigma \rangle$ and $\langle d\rangle$.

Figure 13

Contour plots of temperature vs S-ACF. The lower panel shows all the data between 480 and 455 K, while the upper panel shows a magnification of the region between 460 and 455 K. The temperature interval of the measurements was about 0.06 K.

Figure 14

Temperature dependence of $\Delta T$ as a function of reduced temperature obtained from Fig. 13.

Figure 15

Schematic drawing of the heterophase structure of a solid. The heterophase structure consists of many regions indexed as $i$, $i+1$, and so forth. The electron density of each region $\left[{\rho }_i(\mathbf{r},t)\right]$ can be separated into $\overline{{\rho }_i\left(\mathbf{r}\right)}+\Delta {\rho }_i(\mathbf{r},t)$. The coherent sum of $\overline{{\rho }_i\left(\mathbf{r}\right)}$ (overall density) contributes to the so-called speckle pattern of Bragg reflection.

Figure 16

(a) The average electron density $\overline{\rho \left(r\right)}$ can be assumed constant, $\overline{\rho \left(r\right)}={\rho }_0$, in the present CXS measurement. An x ray with a size of $b$ is irradiated. (b) S-ACF obtained from (a). (c) One-dimensional pattern of S-ACF obtained from Fig. 10(${\mathrm{d}}^{\prime }$). An ideal triangular graph can be seen.

Figure 17

Minimal model of PE phase of PZN-9%PT used for understanding of the S-ACF obtained from CXD. (a) Schematic diagram of PZN-9%PT. The PE regions are distributed in the x-ray-irradiated area and the interspaces among the PE regions are filled by newly appearing FE regions. ${\sigma }_i$ are the sizes of the PE regions, while $d_i$ are the distances between the PE regions. (b) Schematic drawing of S-ACF obtained from (a). The width of the central peak represents the average size of the PE regions $\langle \sigma \rangle$, while the distance between the center and the nearest peak represents the average distance between neighboring PE regions $\langle d\rangle$.

Figure 18

Meaning of the overall density. Not all PE regions contribute to the Bragg reflection at an angle.