## 3 Results
### Fractal Characterization  
Fractal and advanced fractal tools are very useful modern approaches used for the characterization of thin films and others systems, for example [23; 56; 57; 58; 59; 24; 59]. Hurst coefficient (Hc) and fractal dimension (FD) are well-known parameters used to extract information about the spatial complexity of thin films, for example [19; 21; 29]. However, new advanced fractal parameters such as fractal lacunarity (FL), fractal succinolarity (FS), and surface entropy (E) were recently used by [PERSON] et al. [42], which expanded the analysis about surface spatial complexity. In this way, Figure 5 shows the power spectrum density  
Figure 4: Texture directions of surface microtexture of (**a**) GdMnO650, (**b**) GdMnO750, and (**c**) GdMnO850. and lacunarity distribution of the fractal region of representative experimental curves of each analyzed film. As can be observed in Figure 2a-c, the surfaces presented self-affine behavior, where well-adjusted fits were recorded. The Hurst coefficients are displayed in Table 5, where all fractal parameters exhibited a statistically significant difference (\(p<0.05\)). There was a larger difference between samples, as GdMnO650 and GdMnO750 exhibited lower values of Hc, while GdMnO850 showed the highest value (\(-0.8\)), revealing that it is dominated by lower spatial dominant frequencies (Hc \(>0.5\)) [21,23]. Figure 5: Surface lacunarity and power spectrum density (PSD) of (**a**,**d) GdMnO650, (**b**,**e) GdMnO750, and (**c**,**f) GdMnO850, respectively. The power spectrum density showed that the films have different microtexture, which was already observed in stereometric parameters. Despite this, it was not possible to obtain accurate data on spatial complexity, homogeneity of surface microtexture, surface percolation, and topographic uniformity. However, the computed fractal parameters revealed that GdMnO850 has the highest spatial complexity because it has greater fractal dimension (\(\sim\)2.281), which is notably due to their roughness distribution. The lacunarity distribution curves of the sample's distribution fractal region obey the power law exposed in Equation (1), proving that the distribution of lacunes decreases persistently as a function of the box size. The values of the lacunarity exponents (\(\mid\beta\mid\)) shown in Table 5 show that GdMnO850 has the most homogeneous surface microtexture, as it exhibited the lowest \(\mid\beta\mid\) value (\(\sim\)8.5 \(\times\) 10\({}^{-8}\)), suggesting that this material is less likely to fail because of its fabrication process. The surface percolation is less for GdMnO850 (FS \(\sim\) 0.48) than for GdMnO650 (\(\sim\)0.54) and GdMn750 (\(\sim\)0.50), revealing that although GdMnO850 is assigned to be rougher, it is less probable that lower bands can be penetrated by fluid through film upper bands, in addition to having low surface percolation (FS \(<\) 0.5) [41] (Table 5). Finally, GdMnO850 also exhibited the highest surface entropy (0.996), demonstrating that this surface has greater topographic uniformity than any other sample, although all surfaces have exhibited E \(\sim\) 1 [48]. Generally, failures in materials based on ceramic systems are derived from non-uniformity and corrosion by erosion, where the penetration of fluids are associated with the erosion process and therefore better physical properties of these materials are requested. As our surface percolation for GdMnO850 was lower, we can suggest that in relation to the other samples, failures due to these types of problems are less likely. In this paper, we have synthesized GdMnO\({}_{3}\) and deposited it onto Pt/TiO\({}_{2}\)/SiO\({}_{2}\)/Si substrates to obtain a correlation between fractal and stereometric parameters. A clear formation of the orthorhombic GdMnO\({}_{3}\) phase occurred for the film sintered at 850 \({}^{\circ}\)C, where a smaller grain size than for other systems previously reported was computed.