# In-Situ Nanoindentation Surface Topography of Lead-Free Piezoelectric Thin Films
## 4 Conclusions  
The methodology proposed in this study is designed to identify meaningful roughness parameters that can capture complex topographic features that influence mechanical properties (hardness and reduced elastic modulus). This methodology is based on a structured discriminant analysis associated with a bootstrapping technique and the multiscale study of the surfaces.  
The study demonstrates that the surfaces of piezoelectric thin films exhibit fractal characteristics, with the fractal dimension providing better surface distinction compared to traditional roughness parameters. For instance, the Sa parameter misclassified 25% of surfaces, whereas the fractal dimension reduced misclassification to 2%. Surfaces with higher fractal dimensions, particularly those close to a Brownian surface (fractal dimension \(\approx\) 2.5), exhibited higher hardness, especially when large valleys dominated the topography. A multiscale roughness analysis using various filters (high-pass, low-pass, and band-pass) further refined the surface distinctions. Band-pass filtering at a 500 nm cut-off for the Sa parameter reduced misclassification to 0.7%, and low-pass filtering at 140 nm for the Smr2 parameter brought it down to just 0.1%.  
Additionally, the study identified a significant correlation between mechanical properties and surface roughness. The valley regime, represented by the Smr2 parameter, exhibited the strongest correlation with hardness (R\({}^{2}\) = 0.82), particularly with low-pass  
Figure 11: Reduced elastic modulus versus hardness.  
filtering at 500 nm, where larger valleys were linked to increased hardness. Similarly, the void volume of the roughness core (Vvc) exhibited a strong correlation with the reduced elastic modulus (R\({}^{2}\) = 0.84), indicating that surface voids contribute to increased stiffness. These results highlight the significance of specific roughness characteristics in influencing the mechanical behavior of piezoelectric films.  
Despite the limited number of samples, representing typical configurations of lead-free piezoelectric films used in MEMS and covering a significant range of surface properties, the results are robust, with strong correlations observed between roughness parameters (Smr2 and Vvc) and mechanical properties. These findings provide a solid foundation for future studies with more samples, which could further validate and expand the conclusions.  
Methodology, M.B.; software, M.B.; validation, M.B., J.L., and A.M.; resources, D.R.; writing--original draft preparation, M.B., J.L., and A.M.; writing--review and editing, M.B., J.L., and A.M.; supervision, M.B. and D.R. All authors have read and agreed to the published version of the manuscript.  
This work was carried out within the framework of the Junior Professor Chair MesuRufo, cofounded by ANR, UPHF, and CNRS.  
Not applicable.  
Data are contained within the article.  
Author [PERSON] was employed by the company U.R Concept. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.