Neither motion of the interface, nor hysteresis in the currents are observed, suggesting that changes in the domain wall structure smaller than the PFM resolution do not play a dominant role. Between -1.5 V and -2.7 V, currents are still limited at the domain wall with no visible changes in the polarisation configurations but at this higher bias range, a hysteresis in the forwards and backwards voltage ramps is observed. At higher bias, the currents increase significantly and are visible throughout the domain structure indicative of irreversible polarisation switching. Temperature dependent I-V measurement show a thermally activated conduction with the onset bias for conduction decreasing as the temperature increases.

The conduction mechanism inferred from all these measurements suggests the dominant process to be Poole-Frenkel hopping of carriers between trap states, assisted by tunnelling from the tip to the film surface. The density of trap states is not high enough for conduction to occur within the domains.

However, accumulation of charged defects such as oxygen vacancies at the domain walls [60] would allow for conduction to occur. Charged defects could be further attracted to the walls by local steps in the domain wall observed by transmission electron microscopy [28], which would necessitate screening.

Subsequent density functional theory calculations of domain wall conduction in PbTiO3 support these conclusions [61].

This scenario is consistent with the variations in the magnitude of the domain wall currents observed in the c-AFM scans shown in figure6.1. The domain wall currents appear to be locally varying in intensity, while current hotspots, some of which are highlighted with red circles, are visible at the same location throughout the measurement series.

In the present work, the alternating PFM and c-AFM scans are used in order to extract the domain wall positions and assess their local geometric distortions in order to establish whether the magnitude of such distortions can be directly correlated with the magnitude of the observed c-AFM currents.

**6.3** **Preprocessing steps**

In order to assess such a correlation, the positions of the domain wall in the successive PFM and c-AFM scans need to be matched and distortions due to AFM scanner artefacts need to be removed. To do this, the distortion correction algorithm presented in section 4.3.4 is applied to the PFM/c-AFM scan series, with the first PFM scan used as a reference. Following this correction, the domain wall positions in the PFM scans are extracted by binarising the phase signal in the PFM scan prior to the c-AFM measurement, directly giving the domain wall positions as the contours of the binarised contrast image. This results in known domain wall positions in the PFM scans that match the position of the corresponding domain wall currents in

the c-AFM scans.

**Figure 6.1:**Domain wall currents measured in c-AFM at (a)-0.625 V, (b) -1.0 V
and (c) -1.375 V. The current magnitude fluctuates along the domain walls. Areas
with higher currents highlighted with red circles in (a) can be seen to remain more
highly conductive throughout the measurement series. Panel (c) shows the labelling
of the domain walls as well as the polarisation orientations. The film is monodomain
polarised up and the written domains are polarised down.

**6.3.1** **Domain wall curvature**

To assess the magnitude of the domain wall distortions, two criteria are used based on the local domain wall curvature and local deviation from the average domain wall position.

To calculate the local curvature, segments of domain wall of fixed length
*l** _{dw}*are fitted with a circle and the fitted radius is assigned to the domain wall
position at the middle of the segment. The segment is then shifted by one
pixel and the process is repeated.

*l*

*is therefore an important parameter. As illustrated in figure6.2for domain walls 1 and 2 showing different roughness, the segment length affects the minimal size of the deviations in domain wall position that result in a significant curvature. If the interface length is too*

_{dw}**6.3 Preprocessing steps**

short, small-scale variations in the domain wall position due to instrument noise are fitted as circles with high curvatures. If the segment is too long, deviations that can be significant are ignored. In this work, interface lengths of 150 nm are used for the fitting. The choice of segment length is for now based on visual inspection and on knowledge of the typical noise in the domain wall positions due to noise in the PFM or imperfect corrections of the relative distortions in consecutive AFM measurements.

**Figure 6.2:** Configurations of domain walls (a) 1 and (b) 2, where the colour
code corresponds to the local domain wall radius. Low radii corresponding to high
curvatures appear in yellow. The domain wall radii are shown for fitting interface
lengths *l**dw* of 100, 150 and 200 nm. As this length increases, short regions yield
larger fit radii, providing a means of filtering fluctuations due to noise in the domain
wall positions. The domain wall positions are shown over three pixels in width to
improve visibility.

This procedure yields a distribution of fitted radii shown in figure6.3(a)
that shows two peaks at∼100 nm and 10^{9}nm. The latter peak is attributed
mostly to fitting artefacts due to changes in curvature directions within
the segment. The corresponding points in the interfaces are shown in figure
6.3(b) and are excluded from the analysis of correlations between domain
wall distortions and currents.

**6.3.2** **Domain wall displacement**

Another metric used to assess the distortion is the distance from the mean domain wall position illustrated in figure6.4, motivated by the description of elastic interfaces in terms of their displacement functions. As can be seen by comparing figures 6.2and6.4, the domain wall displacement typically gives smaller fluctuations of the interface distortion than the domain wall curvature.

**Figure 6.3:**(a) Histogram of the domain wall radii fitted with a sliding interface
length of 100 nm. The histogram shows two peaks, at∼100 and∼10^{9} nm. The
latter peak is attributed to issues in fitting regions where the curvature changes
direction. (b) Map of assigned peak in the domain wall radii histogram. Black and
orange colours are assigned to the low and high radii peaks respectively.

**Figure 6.4:**Domain wall displacements from the average position, for all four walls
studied. For wall 1, the average was calculated over a section excluding the bottom
half of the image, where large overhangs are observed.

**6.3 Preprocessing steps**

**6.3.3** **Topographical curvature**

Another effect that can influence the local domain wall currents is the local
tip-sample contact area. The tip radius is typically of ∼30−50 nm. In
principle, the better the radius of curvature of the sample matches that
of the tip, the higher the contact area, and thus intuitively the measured
currents should be larger. As can be seen in figure6.5, this would correspond
in principle to higher currents at locally convex surface curvature, and lower
currents at locally concave surface curvature for a material with otherwise
uniform conductive properties. To this effect, the local surface curvature was
also extracted. The procedure is similar to that described in section6.3.1,
extended to two spatial dimensions. A sphere is fitted on a square window
of the sample topography of width*w**z* and the extracted radius is assigned
to the pixel at the centre of the surface. The window is shifted by one pixel
and the process is repeated.

**Figure 6.5:** Illustration of the varying contact surface due to the topographical
radius of curvature. Convex and concave sections are assigned positive and negative
radii respectively.

To distinguish between convex and concave curvatures, which lead to different contact areas for the same radius, a positive or negative sign is given to the radius of curvature if the centre of the circle lies above or below the sample surface, respectively, as shown in figure6.5.

A fitting window width of *w** _{z}* = 40 nm, corresponding to the typical
radius of an AFM tip, is used to extract the curvature of the topography in
figure6.6(a). This process yields the surface radius of curvature map shown
in figure6.6(b). As can be seen from the topographical radii map and its
corresponding distribution histogram shown in figure6.7, the fitted radii are
mostly much larger than the tip radius, peaking at∼800 nm.

**Figure 6.6:**(a) AFM topography of the area shown in figure6.1and (b)
correspond-ing map of topographical radii of curvature.

**Figure 6.7:**Distribution of fitted topographical radii of curvature, showing a peak
at∼ ±800 nm.