# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion  
The pre-percolation topology of Ag thin films formed on substrates (Figure 1) was studied by atomic force microscopy. Figure 1d indicates the topology of the glass substrate without the applied film. From the comparison of the topologies of the surface of glass and films, it can be seen that the height difference of the glass surface is more than ten times smaller than that corresponding to films, at 3.5 nm versus 35-40 nm. This is because, at such magnifications, the glass has a slowly changing relief, and the inclusions that may be in the glass are huge compared to the size of the silver particles that make up the film. Therefore, for this experiment, the influence of substrate topology is negligible. The mass thickness of the films (\(d_{m}\)) was varied to study its effect on surface morphology. It was found that ultrathin films consisted of aggregated metal nanoparticles, with their size and shape being dependent on \(d_{m}\). This conclusion aligns with studies conducted using similar methods to obtain metal films on glass substrates under similar conditions [45; 46; 47; 48; 49]. Notably, pre-percolation Au films exhibited qualitatively similar surface features to Ag films (Figure 2), although with smaller nanoparticle dimensions. Figure 1: AFM visuals of the Ag films on substrates with thicknesses (\(d_{m}\)) of 1.1 nm (**a**), 3.6 nm (**b**), and 5.7 nm (**c**), and of the glass substrate without film (**d**) at dimensions of 2.0 \(\times\) 2.0 \(\upmu\)m.  
Figure 2: AFM visuals of the Au films on substrates with thicknesses (\(d_{m}\)) of 1.0 nm (**a**), 2.0 nm (**b**), and 3.0 nm (**c**) at dimensions of 0.2 \(\times\) 0.2 \(\upmu\)m.  
It should be noted that the AFM image dimensions, shown in Figures 1 and 2, are different. This is due to the significant size difference in surface inhomogeneities between silver and gold films. These images are presented to illustrate the structural evolution with the increasing mass thickness of the films. Most parameters were calculated using image sets with different dimensions. To quantitatively analyze the surface roughness, the root mean square (RMS) deviation of the topography from a reference plane was calculated based on the AFM data. In the case of two dimensions, this parameter represents the minimum RMS deviation and characterizes the overall variation in the surface profile [50]:  
\[S_{q}=\sqrt{\frac{\sum_{i=1}^{N}(Z_{i}-Z_{c})^{2}}{N}}, \tag{1}\]  
where \(Z_{i}\) is the point height, \(Z_{c}\) is the average surface height, and \(N\) signifies the total number of points considered on the surface for calculation. Figure 3 illustrates how the surface roughness depends on the thickness of the films studied. Further analysis of AFM and the objectivity of the parameters that can be obtained with its help is required to understand the reasons for the dependence of RMS on the mass thickness of metal films. Atomic force microscopy is a useful instrument for studying surface topology at the nanoscale [51; 52; 53]. The positioning of the probe system relative to the sample enables a fairly accurate reproduction of the surface profile. However, in nanophysics, the objects under study have sizes of tens of nanometers. In this case, the limited size of the scanning probe is a significant drawback and introduces considerable errors in the measurement results [51; 52]. According to the principles of operation of a scanning microscope, the signal obtained when scanning a line is a convolution of the surface shapes and the scanning element. It is widely known that mathematically each point of an AFM image can be represented as a part of the convolution of the surface relief signals, \(f(x,y)\), and the probe shape, \(p(x,y)\)[54]:  
\[g(x,y)=\int\limits_{0}^{x}\int\limits_{0}^{y}f(\xi,\eta)p(x-\zeta,y-\eta)d\xi d\eta. \tag{2}\]  
Figure 3: The dependence of the surface roughness of Ag (**a**) and Au (**b**) films deposited on substrates (depicted on Figure 1d) on mass thickness. Circles—calculated values, red line—linear fitting.