# In-Situ Nanoindentation Surface Topography of Lead-Free Piezoelectric Thin Films
## 3 Results
### Relationship Between Mechanical Properties by Nanoindentation and Roughness Parameters  
The surface topography of the four surfaces under investigation reflects the different conditions of the surface production processes. It is thus appropriate to analyze the potential relationships between the mechanical properties (hardness and elastic modulus) of the material and the surface topography [32]. For this, we will examine the hypothesis of a linear relationship (Equation (3)) between a mechanical property of the surface (hardness or elastic modulus) and a morphological property of the surface (roughness parameter obtained with a given filter), i.e., the shape. \[m_{i}=a_{i,j,p,q}\ f_{j}^{s_{p},f_{q}}+b_{i,j,p,q}, \tag{3}\]  
where the variables are defined as follows:  
* \(m_{i}\), \(i\in\{1..I\}\) is a mechanical property of the surface (\(m_{1}\) is the hardness in GPa and \(m_{2}\) is the reduced elastic modulus in GPa);
* \(r_{j}^{s_{p},f_{q}}\) is the \(r_{j}\) roughness parameter computed with a \(s_{p}\) filter at a cut-off length \(f_{q}\) with \(j\in\{1..I\}\), \(p\in\{1..P\}\), \(q\in\{1..Q\}\);
* \(a_{i,j,p,q}\) et \(b_{i,j,p,q}\) are two constants. To assess the quality of the proposed model, the coefficients \(a_{i,j,p,q}\) and \(b_{i,j,p,q}\) are calculated using the least square method, and the correlation coefficient noted \(R^{2}\Big{(}m_{i},r_{j}^{s_{p},f_{q}}\Big{)}\) is calculated. By performing all possible permutations of the indices \(\{j,p,q\}\) for a mechanical property \(m_{i}\) and by solving Equation (3) for each element containing \(J\times P\times Q\), we will obtain the values of \({R_{i}}^{2}\Big{(}m_{i},r_{j}^{s_{p},f_{q}}\Big{)}\) for the \(m_{i}\) mechanical property. Figure 9 represents the values of \(R^{2}\Big{(}H,r_{j}^{s_{p},f_{q}}\Big{)}\) and \(R^{2}\Big{(}E,r_{j}^{s_{p},f_{q}}\Big{)}\) as a function of the order of classification, which is related to the tested roughness parameters. The first position in the 'Order of classification' axis corresponds to the roughness parameters exhibiting the best correlation (highest R\({}^{2}\)) with the mechanical parameters, specifically hardness (blue curve, Figure 9) and Young's modulus (red curve, Figure 9). Figure 8: Spc versus Vvc, at the whole scale (no filter); \(1.9\times 10^{-31}\%\) misclassified. In order to have a robust quantification of experimental error, ensuring that observed differences in mechanical properties are statistically significant and not merely due to experimental variation. The mean hardness and modulus values are calculated using a bootstrapping method over the 100 nanoindentation measurements for each surface. The roughness parameters that exhibited the strongest correlation were Smr2 with a low-pass filtering at 500 nm for the hardness (\(R^{2(H,J^{\text{\tiny{low-pass, 500 nm}}})}=0.82\)) and Vvc with a low-pass filtering at 2 nm for the reduced elastic modulus (\(R^{2(E,J^{\text{\tiny{low-pass, 2 nm}}})}=0.84\)). Figure 10 shows the correlation between these two parameters at their relevant spatial scales. Each data point in Figure 10 represents the bootstrap mean of two considered parameters, showing the variance across surfaces. Although the clusters appear discrete due to the limited number of measurements, Gaussian smoothing densities provide a continuous classification within clusters, supporting the linear relationship observed. Figure 10: Relation between mechanical properties obtained by nanoindentation and roughness parameters. The best relation (R\({}^{2}\) = 0.82) between the hardness and the roughness parameter Smr2 is obtained for a low-pass filtering at 500 nm (**a**). The best relation (R\({}^{2}\) = 0.84) between the reduced elastic modulus and the roughness parameter Vvc at the whole scale (equivalent to low-pass/2 nm) (**b**). Figure 9: R\({}^{2}\) values versus classification order of the roughness parameters according to the mechanical properties.