Correlative imaging of ferroelectric domain walls

The wealth of properties in functional materials at the nanoscale has attracted tremendous interest over the last decades, spurring the development of ever more precise and ingenious characterization techniques. In ferroelectrics, for instance, scanning probe microscopy based techniques have been used in conjunction with advanced optical methods to probe the structure and properties of nanoscale domain walls, revealing complex behaviours such as chirality, electronic conduction or localised modulation of mechanical response. However, due to the different nature of the characterization methods, only limited and indirect correlation has been achieved between them, even when the same spatial areas were probed. Here, we propose a fast and unbiased analysis method for heterogeneous spatial data sets, enabling quantitative correlative multi-technique studies of functional materials. The method, based on a combination of data stacking, distortion correction, and machine learning, enables a precise mesoscale analysis. When applied to a data set containing scanning probe microscopy piezoresponse and second harmonic generation polarimetry measurements, our workflow reveals behaviours that could not be seen by usual manual analysis, and the origin of which is only explainable by using the quantitative correlation between the two data sets.


Supplementary Discussion 1: Background subtraction
Ferroelectric thin films can exhibit additional SHG contributions due to symmetry breaking at surfaces and interfaces. Here, the studied system consists of a ferroelectric film (PZT 50 nm) grown on a thin buffer electrode (35 nm thick SrTiO 3 ) deposited on the surface of a single crystal (SrTiO 3 (001)). Given the elongated shape of the photon beam voxel and its micron size, each of these surfaces and interfaces contributes to the overall SHG response. More importantly, surface SHG has a specific symmetry (i.e., an anisotropy that is different from that of the domain walls) and its order of magnitude can be comparable to that of the domain walls in reflection geometry. Eliminating this background signal can thus be indispensable. However, a simple arithmetic background subtraction can represent a delicate task in the case of anisotropic surface SHG response. The left panel of Supplementary Figure 2 shows the background signal measured at the centre of the triangular shaped domains (internal BKG) and the surrounding region (external BKG). While the inner background displays a nearly isotropic response, the outer background shows elongated anisotropy plots. In the artisanal method, the local SHG at the domain wall regions (displayed in Fig. 2(c)) is derived by subtracting the averaged background signal 1/2 · (internal + externalBKG)) from the intensity collected at the domain wall regions.
The subtraction of the background is also needed in the clustering methods. To do this, the mean polar response is subtracted before clustering, and thus we observe in this case background signatures that are more defined than in the artisanal methods (see Supplementary Figure 2). Furthermore, two distinct external background components showing the same anisotropy but different intensities are revealed in the clustering method. Note that zero-second harmonic emission is expected at c-domains due to the tetragonal structure of PbZrTiO 3 in the geometry used for this study (see Methods). The non-zero SHG response observed in these regions can be explained either by surface SHG, as discussed above, or by a misalignment of the local polarization due to local strain or defects (see discussion in the main body text).

Supplementary Note 1: K-means clustering
The K-means clustering of the SHG data set is shown for n = 2 to n = 10 in Supplementary Figure 3. Each clustering is shown with the corresponding cluster map as the leftmost image, and two rows of polar plots. Within each polar plot are two curves, corresponding to the two polarizer angle configurations. The top row of the polar plots shows mean-subtracted and the bottom row shows complete SHG polar plots derived from the corresponding cluster of a given color.

Supplementary Note 2: Silhouette analysis
Silhouette analysis has been performed for K-means clustering with n = 2 to n = 39. The silhouette plot is shown in Supplementary Figure 4(a), demonstrating a sharp drop in silhouette score at n = 9. This can be seen in the corresponding silhouette analysis shown for n = 2 to n = 10 in Supplementary Figure 4(b-j), with the drop at n = 9 giving the most well defined clusters (with a significant amount of data points' individual scores situated above the average silhouette score).

Supplementary Note 3: SHG+SPM Correlated K-means analysis
As the SPM and SHG data sets have been corrected and aligned, it is possible to include the SPM data set within the Kmeans analysis alongside the SHG. Supplementary Figures 5 and 6 show this for n = 2 to n = 12 and and n = 13 to n = 14 respectively, with the two data sets given equal weight in the clustering. As can already be seen in Supplementary Figure  5(a), the initial clustering appears to be primarily influenced by the polarization orientation through the piezoresponse force microscopy phase signal. As the number of clusters is increased, however, additional features appear around domain walls, refining the structure more and more until a qualitatively similar distribution of distinct domain wall and background regions appears in Supplementary Figure 6(a) at n = 13. As the SPM data are a collection of 1D data points, the mean value of the topographic height, piezoresponse phase and piezoresponse amplitude is extracted for each cluster and written above each set of corresponding polar plots. For instance, one can confirm the qualitative observation in the main text regarding the segregation of the up-polarized domain background signal into two distinct clusters due to the topographic morphology. Indeed, the average height of the two up-background clusters as shown in Supplementary Figure 6(a) and (b) differs significantly.