## 3 Results
### Fractal Characterization
In addition, a probable greater number of defects in relation to other systems studied previously was registered due to a higher value of lattice microstrain. Naturally, the structural arrangement affected the surface morphology because for films sintered at 650 and 750 \({}^{\circ}\)C the surface topography was smooth, while the film sintered at 850 \({}^{\circ}\)C had a rougher surface. Samples sintered at 850 \({}^{\circ}\)C presented the highest roughness, negative asymmetry, and flattened height distribution. Several other stereometric parameters such as functional, feature and hybrid demonstrated that the film sintered at 850 \({}^{\circ}\)C had the best topographical properties. Qualitative rendering revealed interesting aspects of the surface microtexture of the samples and confirmed that deeper valleys were assigned to be of the sample sintered at 850 \({}^{\circ}\)C. The smoother texture distribution of the film sintered at 850 \({}^{\circ}\)C indicates that this film may have more homogeneous microtexture. The stereometric results were confirmed by fractal and advanced fractal analysis because there was a correlation between these parameters. The results showed that the roughness distribution affected the Hurst coefficient and spatial complexity, where the film sintered at 850 \({}^{\circ}\)C reached the biggest values, as well as the emergence of topographic uniformity. Surface percolation and the lacunarity exponent of GdMnO\({}_{3}\) sintered at 850 \({}^{\circ}\)C were lower values, providing a surface with less surface percolation and more homogeneous microtexture, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Unit** & **GdMnO650** & **GdMnO750** & **GdMnO850** \\ \hline FD & (-) & 2.242 \(\pm\) 0.034 & 2.207 \(\pm\) 0.007 & 2.281 \(\pm\) 0.025 \\ \hline HC & (-) & 0.169 \(\pm\) 0.172 & 0.357 \(\pm\) 0.023 & 0.813 \(\pm\) 0.038 \\ \hline \(\mid\beta\mid\) & (-) & 4.02 \(\times\) 10\({}^{-7}\) \(\pm\) 1.58 \(\times\) 10\({}^{-7}\) & 4.09 \(\times\) 10\({}^{-7}\) \(\pm\) 1.82 \(\times\) 10\({}^{-7}\) & 8.45 \(\times\) 10\({}^{-8}\) \(\pm\) 3.15 \(\times\) 10\({}^{-8}\) \\ \hline FS & (-) & 0.536 \(\pm\) 0.022 & 0.500 \(\pm\) 0.024 & 0.483 \(\pm\) 0.009 \\ \hline E & (-) & 0.962 \(\pm\) 0.019 & 0.967 \(\pm\) 0.017 & 0.996 \(\pm\) 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fractal and parameters advanced fractals of GdMnO\({}_{3}\) thin films. Our results indicate that the process of deposition of GdMnO\({}_{3}\) thin films and the subsequent sintering at 850 \({}^{\circ}\)C produces films with interesting topographic properties that can be used to improve the process of fabrication of thin films based on systems involving rare earths, known as excellent multiferroic materials for technological application. The following are available online at [[https://www.mdpi.com/article/10.3390/app11093886/s1](https://www.mdpi.com/article/10.3390/app11093886/s1)]([https://www.mdpi.com/article/10.3390/app11093886/s1](https://www.mdpi.com/article/10.3390/app11093886/s1)), Figure S1: SEM images of (a,b) GdMnO650, (c,d) GdMnO750, and (e,f) GdMnO850, Figure S2. 2D AFM topographic maps of (a,b) GdMnO650, (c,d) GdMnO750, and (e,f) GdMnO850., Figure S3. Graphical study of volume parameters (left) and Sk parameters (right) based upon the Abbott curve calculated for the samples: (a) GdMnO650, (b) GdMnO750, and (c) GdMnO850. Conceptualization, methodology, resources, J.P.d.C. and H.D.d.F.F. ; data curation, original draft preparation, Y.R.-B. ; writing--review and editing, supervision, R.S.M. and R.M.P.B.O. ; funding acquisition, reviewing, S.T. ; formal analysis, investigation and project administration, J.A.M. ; validation, reviewing, H.D.d.F.F. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to large amount of data.