# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 4 Conclusions  
Pre-percolation silver and gold films, formed by thermal evaporation at the initial deposition stages, consist of discrete metal particles. AFM effectively studies the topology of these structures, but significant errors occur as films approach percolation, where the distances between metal particles become smaller than the probe tip size. A decrease in surface roughness indicates this and is further supported by examining the film-substrate  
Figure 19: Dependencies of the Hausdorff fractal dimension (blue curve 1), and fractal dimension (red curve 2) on \(d_{m}\) for Ag (**a**) and Au (**b**) films.  
border. For much thinner films, such errors are negligible, making AFM-determined parameters reliable. The average particle sizes for silver films with a mass thickness of up to 2 nm range from 30 to 60 nm.  
Correlation analysis of film topologies provides further insights into structures. The increase in correlation length with mass thickness and the anisotropy of the central peak of the autocorrelation function indicate growth in the size of surface inhomogeneities (metal particles) and some heterogeneity in their shape and size.  
Detailed analysis of the size and shape of silver particles for films with a mass thickness from 1 to 2 nm, performed using image processing, indicates that mainly elliptical particles are formed. The average length and width of these particles increase with the growth of mass thickness, ranging from 43 to 56 nm and from 31 to 36 nm, respectively. Studies also reveal that particle size increases with the substrate temperature during metal deposition. This could be due to the faster crystallization of the metal at lower temperatures, leading to the creation of larger amounts of smaller particles.  
The different approaches used in this study to determine fractal dimensionality generally yield predictable results. The fractal dimensionality of the metallic phase asymptotically approaches 2 with increasing mass thickness. However, the peak in the fractal dimensionality of the boundary at \(d_{m}\)\(\sim\)3 nm can indicate a maximum in the boundary length between metal and voids, suggesting that there is a metal filling factor of around 0.5 at this thickness. Contrarily, based on the film-substrate boundary examination, the metal filling factor should be 0.5 at approximately \(d_{m}\)\(\sim\)7 nm. This discrepancy is likely due to errors introduced by the AFM probe.  
Conceptualization, I.B.; methodology, I.K.; software, O.K.; validation, I.K., M.K. and H.K.; investigation, O.K.; writing--original draft preparation, O.K. and H.K.; writing--review and editing, O.K., H.K., I.B., I.K. and A.I.P.; data curation, O.K., H.K., M.K. and I.K.; project administration, I.B. and A.I.P. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Education and Science of Ukraine (Project \"Multisensor robotic systems using nanomaterials to enhance defense capability and safety in emergency situations\"). In addition, the research of M.K. and A.I.P was partly supported by COST Action CA20129 \"Multiscale Irradiation and Chemistry Driven Processes and Related Technologies\" (MultChem) and by HORIZON 2020 RISE-RADON Project \"Irradiation driven nanofabrication: computational modelling versus experiment\".  
The data presented in this study are available on request from the corresponding author.  
H.K. and O.K. are grateful for the support from the Education and Science of Ukraine (Project \"Multisensor robotic systems using nanomaterials to enhance defense capability and safety in emergency situations\").  
The authors declare no conflicts of interest.