# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion
This approach helped mitigate errors arising from unstable height variations in AFM images. After identifying the boundaries of particles in the image, parameters describing their shape and size were determined. The primary parameter is the region area \(S\) related to each particle. The number \(N\) of image points enclosed by the contour was counted, yielding \(S=N\Delta x\Delta y\), where \(\Delta x\) and \(\Delta y\) are the discretization steps along the \(x\) and \(y\) axes, respectively. From the area \(S\), the equivalent diameter \(d_{\alpha\text{f}}\) of the particle is derived. This is the diameter of a circle whose area matches the particle's area \(S\), i.e., \(d_{\alpha\text{f}}=\sqrt{4S/\pi}\). The height \(h\) of the particle is determined by finding the maximum \(z_{max}\), and minimum \(z_{min}\), height values in the area where the particle appears on the image: \(h=z_{max}-z_{min}\). The length \(L\) is different in comparison with the maximum distance among two points within the region, while width \(w\) is not necessarily the minimum distance between two boundary points on opposite sides and is only accurate for convex regions. The same approach was employed as in [59] to determine these parameters: a rectangle with the minimum area enclosing the region was described, where the width of this rectangle was taken as the width of the region, and the length of the rectangle was the length of the region. This method also allows us to estimate the direction of the metal particle with respect to the \(x\)-axis of the image, which corresponds to the orientation (angle \(\varphi\)) of the longer side of the rectangle. Figure 14a presents the dependencies of the mean (average) equivalent diameter (\(d_{\alpha\text{f}}\)) of films on the time of deposition at a rate of 0.0016 nm/s. Figure 14b shows the subjection of the same \(d_{\alpha\text{f}}\) parameter, but after it has affected the temperature of the substrate at a constant \(d_{m}\). It was determined that the increase in \(d_{\alpha\text{f}}\) with the rising deposition time \(t\) is nearly linear during the initial deposition stages. It is noticed that there is a linear relationship between the characteristic structure size and the substrate temperature \(T\). This trend can be attributed to the metal particle's enhanced thermal energy \(kT\) at higher temperatures. This increased energy facilitates surface mobility, enabling particles to coalesce and form biggest clusters. \[S_{q}=\sqrt{\frac{\sum_{i=1}^{N}(Z_{i}-Z_{c})^{2}}{N}}, \tag{1}\]0 Mean (average) particle diameter \(d_{\alpha\text{f}}\) for Ag films formed on the substrate at constant \(T=293\) K vs. deposition time (**a**) and at a constant thickness of \(d_{m}=6\) nm, and the substrate temperature (**b**). It was demonstrated that, at a low thickness, the metal particle can be modeled in the form of ellipsoids with the sizes of \(a=L\), \(\text{b}=w\), and \(\text{c}=2h\) (\(L\) is the length, \(w\) is the width of the area, \(h\) is the elevation above the watershed). The values obtained are displayed in \[S_{q}=\sqrt{\frac{\sum_{i=1}^{N}(Z_{i}-Z_{c})^{2}}{N}}, \tag{1}\]1 Anisotropy in the \(xy\) plane is evident, as indicated by the significant difference between the dimensions of the \(a\) and \(b\) axes and the uneven distribution of particles in the visible region. This can be seen better when the particle distribution is along a certain direction because anisotropy occurs when there is simultaneous particle elongation and non-uniform distribution in the same direction. Combining methods to identify structural components and determine parameters allows for the precise characterization of each particle's geometry individually. The method for defining the area surrounding each particle makes it possible to evaluate both its geometric parameters and its alignment with the \(x\) and \(y\) axes along the rectangle's longest side.