Domain Wall Roughness in Stripe Phase ${\mathrm{BiFeO}}_3$ Thin Films

Using the model system of ferroelectric domain walls, we explore the effects of long-range dipolar interactions and periodic ordering on the behavior of pinned elastic interfaces. In piezoresponse force microscopy studies of the characteristic roughening of intrinsic 71° stripe domains in ${\mathrm{BiFeO}}_3$ thin films, we find unexpectedly high values of the roughness exponent $\zeta =0.74\pm 0.10$, significantly different from those obtained for artificially written domain walls in this and other ferroelectric materials. The large value of the exponent suggests that a random field-dominated pinning, combined with stronger disorder and strain effects due to the step-bunching morphology of the samples, could be the dominant source of pinning in the system.

Figure 1

Topography (a), vertical (b), and lateral (c),(d) PFM phase images of an artificial rectangular domain in the as-grown type I film. Blue arrows indicate cantilever orientation. (e) Renormalized $n=2–8$ displacement-displacement correlation functions for the full left domain wall show fanning. (f) The correlation functions collapse when only the boxed segment is considered. (g) Roughness function of the boxed segment, with roughness exponent $\zeta$ extracted by fitting (solid red line).

Figure 2

Simultaneously obtained (a) topography and (b) lateral phase images of step bunching and intrinsic 71° stripe domains. Blue arrow indicates cantilever orientation. The inset in (b) indicates the in-plane polarization orientation. (c) Average roughness $\overline{B(r)}$ for domain walls $>6\mu \mathrm{m}$, with saturation of relative displacements at $\sim 100\mathrm{nm}$ (dashed red line).

Figure 3

(a) Average structure factor $S(q)$ versus the corrected wave vector $\tilde{q}$ for 92 monoaffine intrinsic 71° domain walls, with roughness exponent fitting (solid red line). Inset: Histogram of the corresponding individual roughness exponents $\zeta$, with mean $\overline{\zeta }$ indicated by the (solid red) line. (b) Roughness exponent $\zeta$ as a function of stitching length, with saturation value (dashed orange line) determined by fitting (solid red line).

Figure 4

(a) Schematic diagram of different possible pinning centers in a ${\mathrm{BiFeO}}_3$ film with step bunches (larger steps) and terraces with unit cell steps (smaller steps). 71° domain walls are drawn as solid (black) lines on top of the film and dashed (black) lines into the film. The small (dark blue) spots and the larger (dark red) spots indicate weak pinning centers and threading dislocations, respectively. The color gradient indicates the strain gradient introduced by the substrate’s vicinality, whereas the blue (dark) and the yellow (bright) areas indicate the strained and relaxed ${\mathrm{BiFeO}}_3$ regions. (b) Topography image of the type II ${\mathrm{BiFeO}}_3$ film overlaid with the edges of the step bunches (dark lines). Areas of higher step density and terraces are marked by the red and blue bar below the figure. (c) Overlay of the step-bunch edges (dark lines) with the domain walls (red lines) extracted from the lateral phase image of the same region as (b).