# In-Situ Nanoindentation Surface Topography of Lead-Free Piezoelectric Thin Films
## 3 Results
### Multiscale Analysis
#### 3.2.3 Multiscale Analysis: Two Parameters (Sa\({}^{81,f1}\), Sa\({}^{2,f2}\))  
It may also be pertinent to consider whether the surface topography would be more accurately described by two morphology parameters rather than one [16]. First, the Sa value is calculated at different filtering frequencies and filtering types (high-pass, low-pass, and band-pass). Then, the percentage of misclassified surfaces assigned to a pair of Sa  
Figure 5: Histograms of the Smr2 parameter for the four surfaces, for a low-pass filtering at 140 nm. For this filter and this scale, 0.1% misclassification is obtained.  
Figure 6: Best filtered (low-pass filter, 140 nm) surfaces described by only the Smr2 roughness parameter. For each figure, the histogram corresponds to the distribution of the amplitudes of the surface points: (**a**) BNTA; (**b**) BNTB; (**c**) BNTC; and (**d**) BNTD.  
roughness parameters noted (Sa\({}^{\text{{1,f,1}}}\), Sa\({}^{\text{{2,f2}}}\)) is calculated, where \(s\) represents the filtering type and \(f\) the size of the filtering cut-off length. By testing all two-by-two combinations of the different Sa, the relevance of these couples is classified by sorting them by increasing the order of misclassified surfaces. The optimal pair identified is (Sa\({}^{\text{No Filter, whole scale, Sa}}\) high-pass, 277 nm) with \(10^{-11}\%\) misclassified surfaces (Figure 7). The very low integral density probability values, such as \(10^{-11}\), serve as measures of discrimination distance rather than conventional probabilities. These values highlight the distinct separation between clusters, underscoring the discriminative power of the parameters. Therefore, the four studied surfaces can be described in a relevant way by a combination of the Sa parameter (no filter, full scale) and the Sa parameter at a finer scale (277 nm), including a nuance on the roughness amplitude at this spatial scale.  
The graph visualization (Sa\({}^{\text{No Filter, whole scale, Sa}}\) high-pass, 277 nm) is replete with information (Figure 7). As previously shown for the Sa parameter without filtering, the surfaces B and D have similar Sa values, which diverge from those observed for A and C. However, surfaces B and D display a more pronounced disparity for Sa\({}^{\text{high-pass, 277 nm}}\), with surface B exhibiting an approximate 0.2 nm higher value. This observation tends to support the fact that surface B has a higher fractal dimension (Figure 2b), approaching the Brownian surface (fractal dimension equal to 2.5). Moreover, the surfaces A, C, and D have values (Sa\({}^{\text{No Filter, whole scale, Sa}}\) high-pass, 277 nm) aligned on the same line of equation Sa\({}^{\text{No Filter}}=0.5\)\(\times\) Sa\({}^{\text{high pass, 277 nm}}\), showing a behavior of similar scale, different from the surface B.  
2.4 Multiscale Analysis: Two Parameters (Spc\({}^{\text{No Filter}}\), Vvc\({}^{\text{No Filter}}\)) and Best Couple of Parameters at Whole Scale  
The multiscale two-parameter analysis was conducted for additional parameter combinations without the application of any filter. The parameter pair (Spc\({}^{\text{No Filter}}\), Vvc\({}^{\text{No Filter}}\)) yielded the most relevant couple for discriminating surfaces, with a misclassified percentage of approximately \(10^{-31}\%\) (Figure 8). The surfaces C, D, and A are discriminated by the volume of roughness, with surface A exhibiting the greatest roughness, followed by surfaces D and C, respectively. The surface B is distinguished from the surface D by the curvature of the peaks, which are less curved in surface B than in surface D.  
Figure 7: Sa for a high-pass filter at 277 nm versus Sa at the whole scale (no filter); \(10^{-11}\%\) misclassified.