# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion
When using an atomic force microscope in contact or semi-contact scanning modes, the condition of the minimum distance of the probe from the examined surface will be fulfilled. Mathematically, this condition is written as follows [54]:  
\[p(x,y)-f(x,y)\geq 0. \tag{3}\]  
Considering these two factors, a simple algorithm for modeling the AFM scanning process was proposed for use if the shape of the probe \(p\)(\(x\),\(y\)) and the relief of the real surface \(f\)(\(x\),\(y\)) are known. This algorithm is carried out in one: (1) assume that the center of the probe tip is positioned over a given point; (2) the initial height value at this point on the new (modeled) surface is set to be the same as that on the real surface \(f\)(\(x\),\(y\)); (3) find the maximum difference \(p(x,y)-f(x,y)\) over the entire area of the probe with the center at the given point; (4) if the obtained maximum difference is positive, the height value at this point on the new surface \(g\)(\(x\),\(y\)) is increased by the obtained difference value (Figure 4). On the other hand, metal films in the initial stages of sputtering form islands, the shapes of which can be roughly considered spheres or hemispheres depending on the wetting level of the substrate surface by the metal, which, in turn, depends on various thermodynamic parameters. As the amount of metal increases, the islands formed increase in size, and the distance between them decreases, accordingly. Figure 5 is built by analogy with the approach used for the formation of Figure 4. It schematically shows the formation of an AFM image of two spheres in cases when the distance between them is greater (Figure 5a) and less (Figure 5b) than the size of the scanning element. Condition (3) in the case of Figure 4, a does not allow the probe to sink sufficiently to reach the surface of the substrate. From Figure 5, it can be seen that the height \(h_{1}\) is smaller than \(h\), and since the distance between the particles decreases with the increase in mass thickness, the roughness decreases. The influence of the probe on the AFM visuals of island films is more pronounced at the boundary between the 10 nm thick silver film and the glass substrate (Figure 6). To produce these images, several series of experiments were conducted using a sharp tool with a hardness lower than that of glass to scratch the samples' surfaces. By scanning with a gradual increase in scale, an area without debris and with a clear boundary was identified. Figure 4: Conceptual illustration of AFM scanning procedure. 1–\(f\)(\(x\)), the real shape of the surface; 2—\(p\)(\(x\)), the shape of the probe; 3—\(g\)(\(x\)), the resulting topology. To obtain the image shown in Figure 6, a typical technique for processing AFM data--polynomial background removal--was used. In this case, only the part of the image corresponding to the substrate surface was applied as the basis for calculating the surface parameters. It is known that such a film is not yet conductive and consists of islands; however, Figure 6 gives the impression that the film is continuous. This representation makes it evident that during the scanning of the film, the probe does not reach the surface of the substrate, but only slightly penetrates it. In this case, the size of the probe affects both the RMS values determined from the AFM images and the estimation of the metal particle sizes. In both regions depicted in Figure 6, heterogeneity and different surface roughness values can be observed. These areas correspond to the silver film and the glass substrate; direct thickness measurement is not feasible, however. To accurately estimate thickness under these conditions, a height histogram was utilized (Figure 7). In this histogram, the height \(h\) is plotted on the \(x\)-axis, and the point quantity in the AFM image at that height is plotted on the \(y\)-axis. The histogram exhibits two distinct peaks. The left peak is attributed to the substrate, whereas the right peak corresponds to the film surface.