# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion
Both surfaces exhibit Gaussian distributions, allowing these peaks to be fitted with Gaussian peak functions to determine the positions of their centers, representing the most probable height of the given area. The distance between the centers of these peaks corresponds to the desired geometric thickness \(\delta\) of the metal layer on the substrate. Figure 5: The difference between the immersion depth of the probe in the case of two spheres, separated by smaller (**a**) and larger (**b**) distances between spheres compared to the size of the probe. The green dash-line indicates the upper and lower limits of the AFM topology that will be obtained in both cases. Figure 6: A three-dimensional image of the transition boundary between the film and the substrate. This technique enabled the accurate estimation of the geometric thickness of the film, contingent on identifying the transition limit. The digital processing of the AFM image in the area where the boundary between the film surface and the substrate is visible allows us to determine the spatial thickness of the film layer. Figure 8 illustrates the relationship between the geometric thickness of silver films, as determined from a series of AFM experiments, and their mass thickness. In cases of small mass thicknesses, the difference between \(d_{geom}\) and \(d_{m}\) is significant. This discrepancy arises because, at the initial stages of deposition, separate metal clusters form, resulting in large \(d_{geom}\) values. As the film thickness further increases, the difference between them should decrease, and so the dependence Figure 8 has a nonlinear character. With both the geometric thickness \(d_{geom}\) and the mass thickness \(d_{m}\) known, determining the filling factor \(p\) is straightforward. The packing factor, representing the position of the film's volume utilized by the metal, is computed as a ration of the mass thickness \(d_{m}\) to the geometric thickness \(d_{geom}\) obtained from AFM images. Figure 9 displays the relationship between the filling factor of the metallic phase \(p\) and the mass thickness. Figure 2:0 The dependence of the geometric thickness of the film \(d_{geom}\) on the mass thickness \(d_{m}\) for Ag films (symbols) and its fitting (red line). Figure 2:1 A histogram of the height distribution of image points (Figure 6) for the film–substrate transition boundary (green columns) and the approximation of each of the maxima of the distribution by Gaussian functions (blue and red solid lines). As expected, the filling factor increases nonlinearly with the increase in mass thickness and asymptotically approaches a value of one for bulk samples. A different approach was utilized to determine RMS and another surface parameter--the correlation length. The two-dimensional autocorrelation function (ACF) serves as a valuable tool for characterizing the statistical properties of a surface. For stationary surfaces, the following relationship was calculated [55]:  
Figure 2:2  
where \(S\) is the scanning area and \(z\)(\(x\),\(y\)) is the AFM topology height. The following equation can be applied to numerically calculate the correlation function [46]:  
Figure 2:3  
where the discrete Fourier transforms, both in reverse and direct, are denoted by \(F^{-1}\) and \(F\), respectively. The correlation function for the Gaussian distribution is expressed as follows [55]:  
Figure 2:4  
where the parameter \(\delta\) corresponds to the surface roughness, and \(\sigma\) is called the correlation length. Figures 10 and 11 depict the 2D autocorrelation functions of the surface topography for the Ag and Au films studied, respectively. The intersection of these ACFs with the \(y\) = 0 plane is also presented. The calculated ACFs reveal an anisotropic character in the cross-sections. While isotropy in the \(xy\)-plane implies a symmetrical correlation function, the observed anisotropy is likely caused by the elongated shape and, to some extent, the preferential orientation of the metallic nanoparticles.