Thermal quench effects on ferroelectric domain walls

Using piezoresponse force microscopy on epitaxial ferroelectric thin films, we have measured the evolution of domain wall roughening as a result of heat-quench cycles up to 735 ${}^{\circ }$C, with the effective roughness exponent $\zeta$ changing from 0.25 to 0.5. We discuss two possible mechanisms for the observed $\zeta$ increase: a quench from a thermal one-dimensional configuration and from a locally equilibrated pinned configuration with a crossover from a two- to one-dimensional regime. We find that the postquench spatial structure of the metastable states, qualitatively consistent with the existence of a growing dynamical length scale whose ultraslow evolution is primarily controlled by the defect configuration and heating process parameters, makes the second scenario more plausible. This interpretation suggests that pinning is relevant in a wide range of temperatures and, in particular, that purely thermal domain wall configurations might not be observable in this glassy system. We also demonstrate the crucial effects of oxygen vacancies in stabilizing domain structures.

Figure 1

PFM images of linear ferroelectric domains in a process-gas-cooled 50-nm PZT film (a) immediately after writing and [(b)–(g)] after thermal cycling to the indicated temperatures (${}^{\circ }$C). Similar measurements [(j)–(l)] in an O${}_2$-cooled 100-nm film and [(m)–(o)] an Ar-cooled 40-nm film, where higher oxygen vacancy densities appear to stabilize the domain walls. Vertical 1-$\mu$m scale bars are shown in all images. (h) Average $B\left(L\right)=\overline{\langle {[u(z+L)-u\left(z\right)]}^2\rangle }$ extracted after thermal cycling to the indicated temperatures (${}^{\circ }$C) for all the domain walls in the 50-nm film. (i) The power-law growth region of $B\left(L\right)$ for same measurement (identical color code for the temperatures). The extrapolated fit to the power-law growth region of $B\left(L\right)$ (first 20 points) with $\ln \left[B\right(L\left)\right]\propto 0.890\ln \left[L\right]$ is shown in dark gray (red), and the fit to the saturation region of $B\left(L\right)$ in light gray (aquamarine) for the 685${}^{\circ }$ data set. The intersection of these fits allows the saturation length scale $L^{☆}$ to be extracted. The dashed vertical line in (h) and (i) indicates the length scale $L$ equal to the film thickness.

Figure 2

Numerical simulation of an overdamped elastic string in a random medium undergoing a temperature quench $T_1\to T_2<T_1$ at $t=0$. (a) From bottom to top: Typical evolution of the elastic string each $10000$ steps. The bold line is the configuration at the quench. (b) Curves from top to bottom: Evolution of the disorder-averaged correlation function of relative displacements $B(L,t)$ after the quench (symbols). The roughness exponent ${\zeta }_{\mathrm{th}}=0.5$ describes the geometry of the string at large length scales (solid line).

Figure 3

Disorder dominated scenario as a function of length scale. (a) At ambient temperature before heating, the as-written, initially flat domain walls are only able to locally equilibrate at short length scales. [(b) and (c)] Heating facilitates thermally activated glassy relaxation and longer length scales become locally equilibrated as the heating temperature (or heating time) increases. (d) If fully in equilibrium (reference state), the system displays a two-dimensional to one-dimensional crossover at length scale $L_{\times }$, with no memory of the written state. This may be reflected by the effective roughness exponent ${\zeta }_{\mathrm{eff}}$ in the correlation function $B\left(L\right)\sim L^{2{\zeta }_{\mathrm{eff}}}$ of the quenched metastable states, as in (b) and (c), evolving from the two-dimensional value ${\zeta }_{1\mathrm{D}}$, when $L_{\mathrm{dyn}}\le L_{\times }$, toward the one-dimensional value ${\zeta }_{1\mathrm{D}}$ when $L_{\mathrm{dyn}}\gg L_{\times }$, especially if the crossover length scale is smeared out. In the 1D regime the “dynamical length” $L_{\mathrm{dyn}}(t;T)$ is expected[20] to grow as $L_{\mathrm{dyn}}(t;T)\sim {\left[(T/U_0)\log (t/t_0)\right]}^{1/\theta }$, with $\theta$ a positive exponent, $U_0$ a characteristic temperature, and $t_0$ a characteristic microscopic time. This prediction agrees qualitatively with the experimentally observed increase in $\zeta$ and $L^{☆}$ values with heating time and temperature.

Figure 4

PFM images of nanoscale ferroelectric domains written at ambient conditions in the 91-nm PZT film (${\mathrm{P}}_{\mathrm{UP}}$ as grown) with (a) 1-s, $+$12-V, (c) 10-s, $-$10-V, (e) 1-ms, $+$12-V pulses, and in a 52-nm PZT film (${\mathrm{P}}_{\mathrm{DOWN}}$ as grown) with (g) 100-ms, $+$12-V pulses. Domain evolution after thermal cycling [(f) and (h)] to 200 ${}^{\circ }$C and [(b), (d), and (i)] 585 ${}^{\circ }$C. Vertical bar $=250$ nm. All images are shown to the same scale. The average domain size as a function of the writing time at ambient conditions, and after thermal cycling up to the indicated temperatures (j) for the 91-nm film and (k) for the 52-nm film.