# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion
This enables the determination of mean values, as well as distributions for the parameters obtained, given an adequate particle number in the collection of visuals for the same sample. Figure 15 displays the topography of the Ag film's surface at 1.1 nm, with a colored mask applied to the picture and dispensation by equivalent diameters and via the particle's direction. The angle \(\varphi\) amid the particle's lateral axis and the positive orientation of the \(x\)-axis specified the orientation of the particles. The distribution of diameters (Figure 15b) shows a Gaussian shape, with a peak in the region of the average diameter. A similar pattern is observed in different images of all samples. The distribution of particle orientations is non-uniform, showing a significant peak, while other orientations exhibit non-zero values. This observation confirms previous findings of a partially uniform orientation of nanoparticles, deduced from the elliptical \begin{table}
\begin{tabular}{c c c c} \hline \(d_{m}\) (\(\pm\)0.1), nm & \(a\) (\(\pm\)2), nm & \(b\) (\(\pm\)2), nm & \(c\) (\(\pm\)1), nm \\ \hline
1.0 & 40 & 30 & 31 \\
1.1 & 45 & 32 & 32 \\
1.2 & 47 & 33 & 32 \\
1.5 & 49 & 33 & 35 \\
1.8 & 53 & 34 & 34 \\ \hline \end{tabular}
\end{table} 
Table 1: The averaged values of the axes of metal particles for silver films of different mass thicknesses for different images of the same sample. Figure 15: Marked metal particles on the AFM image of the Ag film surface \(d_{m}\) = 1.1 nm (**a**), distributions of the number of particles from the equivalent diameter (**b**), and the orientation of longest side (**c**) in relation to the \(x\)-axis of the image. cross-section of the auto-correlative function. It is noteworthy this allocation is not typical of all films and can be related to the production technology used. The determination of the fractal dimensions of the studied films is crucial, as metal-dielectric composites are well established to possess a fractal structure. In this study, the fractal dimension, known as the Hausdorff dimension, was defined using the box-counting method. In a 2D context, the fractal dimension of metallic phase (\(D_{p}\)) can be differentiated from the fractal dimension of the boundaries between the metal and dielectric (\(D_{b}\)) using the box-counting approach. When determining \(D_{p}\), boxes cover the complete area corresponding to the particles  
The fractal dimension \(D_{p}\) of the metallic phase of both Ag and Au films demonstrates a positive correlation with rising mass thickness. Such a trend can be related to the growing metal content as the film thickness increases, leading to more extensive metal phase filling of the surface. In contrast, the fractal dimension of the metal-dielectric boundary \(D_{b}\) exhibits a distinct peak at \(d_{m}\approx 3\) nm in the case of Ag films. Such behavior can be elucidated by the elongation of the boundary, given that the metal concentration is amplified during the initial stages of deposition. When areas of metal and dielectric parts are proportionally balanced (with a metal filling \(p\approx 0.5\)), the boundary length reaches its maximum value. With further increases in mass thickness, the boundary length decreases. In contrast, there is no distinct peak in the \(D_{b}\) dependence in the case of Au films (Figure 17b), and the dependence on \(D_{b}\) by \(d_{m}\) has a decreasing character over the entire range. This suggests that the metal concentration in the studied films exceeds the critical level, which would maximize the boundary length. In the three-dimensional case, the film surface is described by a height function \(z(x,y)\). Then, the fractal dimension can be determined by covering the surface with three-dimensional boxes. However, the correlation dimension is more commonly considered for such analysis. The Wiener-Khinchin theorem [60] establishes a link between the autocorrelation function and the spectral power density. This connection allows us to infer the fractal nature of the surface topography from its power spectrum.