Domain wall motion and precursor dynamics in ${\mathrm{PbZrO}}_3$

Single crystals of ${\mathrm{PbZrO}}_3$ have been studied by dynamic mechanical analysis measurements in the low-frequency range $f=0.02$–50 Hz. The complex Young's modulus exhibits a quite rich behavior and depends strongly on the direction of the applied dynamic force. In pseudocubic ${\left[100\right]}_c$ direction, we found intrinsic elastic behavior as expected from the Landau theory; at the antiferroelectric transition $T_c\approx 510$ K, a downwards cusp anomaly in $Y^{\prime }$ accompanied by a peak in $Y^{\prime \prime }$ points to a quadratic/linear order parameter/strain coupling in the Landau free energy. Both anomalies are increasing with decreasing frequency showing that the measurements are performed in the limit $\omega {\tau }_{\mathrm{th}}>1$. Frequency scans around $T_c$ show energy dissipation, which could result from interphase boundary motion and/or heat diffusion. Above $T_c$, we observe a pronounced precursor softening, quite similar as it was found in other perovskites, which can be perfectly fitted including isotropic order parameter fluctuations. The low-frequency elastic response in ${\left[110\right]}_c$ direction is different. Below $T_c$, we find in addition to the intrinsic anomaly a strong contribution from ferroelastic domains, which leads to an additional softening in $Y^{\prime }$. With decreasing temperatures this superelastic softening gradually disappears, due to an increasing relaxation time ${\tau }_{\mathrm{DW}}$ for domain wall motion, indicating glassy behavior of domain freezing in ${\mathrm{PbZrO}}_3$. In contrast to the ${\left[100\right]}_c$ direction, for forces along ${\left[110\right]}_c$, we found a pronounced precursor hardening, starting at about 60 K above $T_c$. Since this anomaly is of dynamic nature, starting at the same temperature as the observed birefringence and piezoelectric anomalies [Ko et al. Phys. Rev. B 87, 184110 (2013)], we conclude that it originates from slow dynamic polar clusters, which freeze at $T^{*}\approx 550\text{K}>T_c$.

Figure 1

Crystal structure of ${\mathrm{PbZrO}}_3$ in orthorhombic $Pbam$ phase. The pseudocubic unit cell is indicated by blue dotted lines in the upper right figure.

Figure 2

Models of domain walls in AFE orthorombic phase of ${\mathrm{PbZrO}}_3$. The ${90}^{\circ }$ domain boundary sited on ${\left(100\right)}_c$, corresponding to ${\left(120\right)}_o$, plane and ${60}^{\circ }$ domain boundary sited on ${\left(101\right)}_o$, corresponding to ${\left(122\right)}_c$ plane. The coordinate systems label the axis directions of the orthorhombic cell. (Note that oxygen atoms have been omitted in these images for simplicity.)

Figure 3

Sketch of the geometrical situation and elastic response for two different directions of applied force (red and blue arrows) during a parallel plate measurement of a ${\mathrm{PbZrO}}_3$ crystal containing two domain states and a domain wall (green line) oriented in ${\left[010\right]}_o$ direction, corresponding to ${\left[1\overline{2}0\right]}_c$. Top left image shows a polarized light microscopy image of the measured sample with domain walls. The black arrow indicates the direction of applied force during the measurement of $Y_{\left[110\right]c}$, corresponding to red arrows in sketch.

Figure 4

Temperature dependencies of real and imaginary parts of Young's modulus of ${\mathrm{PbZrO}}_3$ measured in ${\left[100\right]}_c$ direction at various frequencies. Insets show magnifications near the phase transition.

Figure 5

Temperature dependencies of real and imaginary parts of Young's modulus of ${\mathrm{PbZrO}}_3$ measured in ${\left[110\right]}_c$ direction at various frequencies. Red lines show Cole-Cole fits using Eq. (4).

Figure 6

Temperature dependence of the relaxation time ${\tau }_{\mathrm{DW}}$ of domain wall motion plotted in semilogarithmic scale. The blue line corresponds to a VF fit, the red dot-dashed line to an Arrhenius fit.

Figure 7

Temperature dependencies of real and imaginary parts of Young's modulus of ${\mathrm{PbZrO}}_3$ measured in ${\left[110\right]}_c$ direction at various frequencies. The plot region is changed with respect to Fig. 5 to focus on the high-temperature region.

Figure 8

Sketch of the different contributions to the overall elastic anomalies in ${\mathrm{PbZrO}}_3$.

Figure 9

Magnified view of the high-temperature region of $Y_{\left[100\right]c}^{\prime }$ to show the behavior above $T_c$ more clearly. The line is a fit using isotropic fluctuation corrections in mean-field approximation, yielding an exponent of $1/2$.

Figure 10

Frequency scans revealing $Y_{\left[100\right]c}^{\prime }$(f) and $Y_{\left[100\right]c}^{\prime \prime }$(f) in the vicinity of the antiferroelectric phase transition. The inset shows the Cole-Davidson behavior of the relaxation at 509 K, yielding an exponent $\beta =0.27$.

Figure 11

$Y_{\left[110\right]c}^{\prime \prime }$ in the precursor region above $T_c$ together with the temperature dependence of the relaxation time $\tau$ calculated from the positions of the peak maxima.