# In-Situ Nanoindentation Surface Topography of Lead-Free Piezoelectric Thin Films
## 3 Results  
### Topography Analysis to Distinguish Between Surfaces  
A discriminant analysis is used to determine which roughness parameter better distinguishes surfaces. In this analysis, each surface is assigned to one of four classes based on its roughness parameter: BNTA, BNTB, BNTC, and BNTD. The discriminant analysis consists of calculating the percentage of misclassified surfaces according to roughness parameters (e.g., Sa in Figure 2a), that is, the percentage of surfaces that do not correspond with the assigned classes [16]. To enhance the finesse of the analysis, we assess the probability density function of the parameters using a multidimensional Gaussian curve subsequent to the classification. The percentage of misclassified surfaces is then calculated by integration. This approach allows for the proposal of a probabilistic estimation of misclassified surfaces. A global indicator of the percentage of misclassified areas is obtained by summation of the misclassified areas per category. The principal interest of this kind of statistical analysis is to obtain a classification of different roughness parameters to better distinguish the studied surfaces, based on the misclassification percentage.  
As a first step, the discriminant analysis described above was performed, selecting the parameter Sa (arithmetic mean height), as it is a standard measure in the domain of surface topography. This analysis makes it possible to determine if this parameter permits distinguishing the four surfaces. Figure 2a shows the histogram of the Sa mean values obtained through the bootstrap oversampling technique (1000 bootstraps) in accordance with a method developed by [PERSON] et al. [29]. The surface A presents a higher roughness (2.5 nm) than the surface C (1.7 nm), and the surfaces B and D are observed to have an equivalent level of roughness (2.0 nm). Regarding the Sa parameter, 25% of the surfaces were misclassified.  
Figure 1: Initial 3D topography of the piezoelectric surfaces: 450 nm-LNO/Si (**a**), 350 nm-LNO/Si (**b**), 220 nm-LNO/SiO\({}_{2}\)/Si (**c**), and 350 nm-LNO/SiO\({}_{2}\)/Si (**d**). For each figure, the histogram corresponds to the distribution of the amplitudes of the surface points.  
The roughness calculated here corresponds to the roughness of the bottom electrode (LNO) and the piezoelectric film (BNT). LNO films are deposited by sputtering at 450 \({}^{\circ}\)C followed by annealing at 700 \({}^{\circ}\)C to optimize electrical conductivity [30]. After this annealing treatment, a growth of the grains is observed that is not always homogeneous. The Sa values take into account these LNO grain size effects, which obviously affect the roughness of the heterostructure.  
The same analysis was conducted using the fractal dimension, which yielded a misclassification rate of only 2%. The fractal dimensions of the surfaces are, respectively, 2.35, 2.41, 2.46, and 2.53 for C, D, A, and B (Figure 2b). These results suggest that the studied surfaces exhibit fractal characteristics, as evidenced by values exceeding 2. Additionally, the fractal dimensions indicate that the mechanisms underlying the formation of these surfaces are different and occur at multiple scales. It is noteworthy that surface B represents a perfect Brownian surface, indicated by its fractal dimension being closest to 2.5.