Crossing 1

Polarisation pattern: 0^◦ images

At 0^◦, consistent with the scan covering all the investigated crossings of figure7.5, the lateral amplitude shows a vertical line of minimum amplitude highlighted in red in all images of figure 7.6 with values higher than the background on both sides of the line. The lateral phase shows a 180^◦contrast at the position of the red line.

This could suggest the presence of a domain wall separating in-plane polarisation vectors with non-zero components along the horizontal direction of the image, schematically indicated by the red arrows as a tail-to-tail domain wall. In principle, one could argue that a head-to-head domain wall is also possible. Indeed, when assessing in-plane polarisation configurations, the absolute orientations are usually not known and only the relative angles between the in-plane polarisation components of domains are known. In our images however, the region of lateral amplitude higher than the background seems to extend further to the left of the red line than to the right, with a gradual amplitude decrease.

This suggests that the twin domain plunges into the film to the left of the line and the gradual decrease of the amplitude is due to the decreasing

7.2 PFM measurements of twin domain crossings

Figure 7.5:Vector PFM image of the investigated crossings. Crossings 1 and 2 are discussed in detail. Crossings 3 and 4 show similar characteristics to crossings 1 and 2 respectively. (a, c) Vertical PFM amplitude image showing the ferroelastic domains with drop shaped features located slightly off-centre of some of the crossings. The corresponding phase image shows no significant contrast. (b,d) Lateral PFM image showing higher amplitudes at the ferroelastic domain positions with distortions and pinching at the crossings. The vertical lines of minimum amplitude in (b) correspond to the boundary where the phase shifts by 180^◦in the phase image.

Figure 7.6:Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 0^◦between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The red lines indicate the region of low lateral amplitude and high phase contrast, with the red arrows indicating the horizontal in-plane polarisation components inferred from the images.

tip electric field leading to a lower in-plane PFM signal. A similar effect is seen in the vertical amplitude, where the amplitude increases back to the background value over a longer distance on the left than on the right.

From this, it can be assumed due to electrostatic considerations that an in-plane polarisation component of the a-domain pointing to the left is more favourable. If the polarisation of the twin domain pointed to the right, tail-to-tail and head-to-head polarisations would appear at the left and right interfaces of the twin domain respectively, leading to electrostatically unfavourable charged domain walls.

Polarisation pattern: 90^◦ images

A similar analysis can be performed on images where the structure is rotated counter-clockwise by∼90^◦ with respect to the tip and displayed in figure 7.7.This time, the ferroelastic twin domain appears to be plunging into the thickness of the film on the right side. The amplitude increase to the left of the green line is faint however, possibly due the fact that the a-domain highlighted by the green line is not perfectly vertical on the image, leading to a smaller horizontal polarisation component. Tip wear can also contribute to lower signal as the images taken at∼90^◦and∼45^◦ are more noisy and the amplitude values are lower.²

2Blunting of the tip with scanning is a known effect, well illustrated in [174], which decreases the lateral resolution and blurs small features. This is probably a significant effect as large scans needed to be performed when the angle was changed in order to find the investigated crossings again. Because images were taken for each angle at all crossings before changing to the next angle, all images taken at∼45^◦and∼90^◦show lower resolution. The phase signal, which is the most sensitive, still shows a 180^◦contrast at the green line corresponding to the position of the a-domain.

7.2 PFM measurements of twin domain crossings

Figure 7.7:Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 90^◦between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The green lines indicate the region of low lateral amplitude and high phase contrast, while the arrows indicate the in-plane polarisation components, inferred in a similar way as in the previous caption.

Polarisation pattern: 45^◦ images

Figure 7.8:Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 45^◦between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The green and red lines indicate the regions of low lateral amplitude and llarge changes in phase contrast. The arrows of corresponding colour indicate the polarisation orientations inferred from the previous images. The phase contrast is self-consistent; all areas with an in-plane polarisation component pointing to the left and right have bright and dark phase contrasts respectively.

To check whether these measurements are consistent, scans were per-formed at an angle of ∼ 45^◦ with respect to the configuration shown in figure7.6. The in plane polarisation orientations inferred from figures7.6 and7.7 are illustrated with the red and green arrows, while the lines of corresponding colour highlight the line where the lateral PFM phase shifts by 180^◦ and where the lateral amplitude is at a minimum in7.8. The assigned polarisation components from images taken at 0^◦ and 90^◦ are consistent in that all regions where the horizontal component of the assigned polarisation directions in figure 7.8 points to the left exhibit a bright phase contrast,

while those pointing to the right consistently show a dark phase contrast.

The interfaces between the dark and bright phase contrasts show signifi-cant deviations from straight lines and the red and green lines seem to pinch off close to the centre of the crossing. The lateral amplitude at the centre of the pinch is higher, suggesting that the polarisation might rotate from the configurations shown in red arrows to the one shown in green.

Close to the centre of the pinch, in the region highlighted by a black circle, the vertical amplitude is lower than anywhere else in the crossing, suggesting that the out-of-plane polarisation component is overall at its lowest.

Polarisation pattern: interpretation

Based solely on the images shown above, the PFM data suggests a polarisa-tion pattern as summarised schematically in figure7.9. Figure7.9(a) shows a schematic cut of an a-domain through the film thickness showing the tail-to-tail polarisation component. The polarisation component at the back of the a-domain is illustrated with a non-zero vertical component as the vertical amplitude in figure7.6goes back to the background value within a shorter distance than the corresponding decay of the lateral amplitude. This picture is consistent with phase field simulations of ferroelastic domains in Pb(Zr0.2Ti0.8)O3 where the transverse flexoelectric coefficient is shown to control polarisation rotations at the edges of the twin domains, particularly at the film-substrate interface [172], as shown in figure7.1(c). This interpre-tation is also consistent with local tilting and lattice deformations in order to accommodate the out-of-plane lattice mismatch between the in-plane and out-of-plane polarised regions [175,176].

Figure7.9(b) shows the polarisation pattern interpretation at and around the crossing, as seen from above the film. The centre area is indicated by a black circle to highlight the fact the structure at the heart of the crossing is complex and not well understood. The exact polarisation configuration close to the centre is very difficult to determine through the PFM alone, as the acquired signals are an integral of the piezoresponse throughout the thickness of the film and the three-dimensional structure of the crossing itself is unknown. Although the electric field decays through the surface, the PFM measurements probe a significant thickness of the film. The depth up to which PFM signals are above the noise threshold can be roughly estimated from the width of the slow decrease of the lateral amplitude in figure7.6, which was used to infer the direction in which the twin domain buries into the film. The width of the region where the lateral amplitude slowly decreases and is significantly above noise level is of ∼50−70 nm, suggesting that the measurement integrates the piezoresponse through at least∼50−70 nm of the film thickness (since the twin domains usually go

7.2 PFM measurements of twin domain crossings

into the film at an angle of approximately 45^◦). This means that the signals observed at the crossings could be superpositions of different polarisation configurations occurring at different depths.

Figure 7.9:Schematic interpretation of the overall polarisation structure. (a) Side view of the polarisation pattern around a twin domain. (b) Top view of the crossing.

The centre area is marked with a black circle to indicate low certainty on the detailed polarisation structure.