Multiscaling Analysis of Ferroelectric Domain Wall Roughness

Using multiscaling analysis, we compare the characteristic roughening of ferroelectric domain walls in $\mathrm{Pb}({\mathrm{Zr}}_{0.2}{\mathrm{Ti}}_{0.8}){\mathrm{O}}_3$ thin films with numerical simulations of weakly pinned one-dimensional interfaces. Although at length scales up to $L_{\mathrm{MA}}\ge 5\mu \mathrm{m}$ the ferroelectric domain walls behave similarly to the numerical interfaces, showing a simple monoaffine scaling (with a well-defined roughness exponent $\zeta$), we demonstrate more complex scaling at higher length scales, making the walls globally multiaffine (varying $\zeta$ at different observation length scales). The dominant contributions to this multiaffine scaling appear to be very localized variations in the disorder potential, possibly related to dislocation defects present in the substrate.

Figure 1

(a) PDF of the relative displacements for different length scales taken over ${10}^4$ numerical disorder configurations in Gaussian units, compared with the Gaussian function (solid line). $\sigma$ is the standard deviation. (b) Collapse of the Gaussian-normalized displacement-displacement correlation functions for orders 2–8.

Figure 2

(a) Disorder-averaged roughness function with a roughness exponent ${\zeta }_{\mathrm{avg}}=0.66$, and (b) roughness exponents of all individual numerical interfaces, with mean value $\overline{\zeta }=0.64$, both in excellent agreement with theoretical ${\zeta }_{RB}^{1D}=2/3$. (c) ${\zeta }_{\mathrm{avg}}$ and $\overline{\zeta }$ (shifted by 0.1 for visual clarity) for small numbers of interfaces, with the values for ${10}^4$ configurations indicated by long-dashed lines, and $\pm \sigma$ by the short-dashed lines. (d) Converging FWHM of the $\zeta$ distribution for small numbers of interfaces.

Figure 3

$1.2\times 4.8\mu {\mathrm{m}}^2$ PFM phase map showing two domain walls, with light and dark contrast corresponding to up and down polarization, respectively. For the full domain walls, the normalized correlation functions show either collapse (b) or fanning (c) between different orders. Collapse is recovered (d) when considering only the lower part of the right wall (dashed box).

Figure 4

(a) Roughness exponents obtained on a set of 43 different monoaffine domain walls with $\overline{\zeta }=0.57\pm 0.05$. (b) Disorder-averaged roughness function for 22 domain walls, with similar roughness exponents ${\zeta }_{\mathrm{avg}}=0.57$.