## 3 Results
### Advanced Stereometric Evaluation  
Several other stereometric parameters associated with the surface of GdMnO\({}_{3}\) thin films are shown in Table 3. According to [PERSON] [34; 35], these parameters, which are obtained from carrying out Fourier analysis to the height autocorrelation function, provide information about the spatial patterns of the material's microtexture. In this regard, almost all functional parameters showed a statistically significant differences (\(p<0.05\)). The microtexture of GdMnO850 has a different spatial configuration, where a greater inverse areal material ratio (Smc) was observed for GdMnO850 (\(\sim\)4.7 nm), while GdMnO650 and GdMnO750 exhibited lower values (Table 3). This behavior was also noted for peak extreme height (Sxp), clearly indicating that the topographic structures of GdMnO850 have a different rough profile regarding GdMnO650 and GdMnO750. As a result of differences in the surface roughness of the films, other differences were also observed regarding the portion of the material that makes up the surface microtexture (thickness) and the volume of the material that fills the representative peaks and valleys of the topographic texture. The graphical representation associated with those parameters can be found in Figure S3, whose full definition can be consulted in [35; 36]. The results show that GdMnO850 has the highest (\(\sim\)10 nm) core surface thickness (Sk), while GdMnO650 and GdMnO750 have lower and close values (Table 3). A persistent pattern was displayed for reduced valley depth (Svk), revealing that there are different contours between rough peaks due to the formation of different channels along with the surface microtexture of GdMnO850 in relation to GdMnO650 and GdMnO750. This generated different dale void volume (Vvv), core void volume (Vvc), and core material volume (Vmc) for GdMnO850. Nevertheless, the rough peak shape seems not to have been affected by the sintering \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Unit** & **GdMnO650** & **GdMnO750** & **GdMnO850** \\ \hline \hline  & \multicolumn{4}{c}{Functional} \\ \hline Smc & (nm) & \(2.07\pm 0.42\) & \(2.01\pm 0.27\) & \(4.67\pm 0.42\) \\ \hline Sxp & (nm) & \(2.72\pm 0.12\) & \(3.29\pm 0.51\) & \(7.09\pm 0.36\) \\ \hline Sk & (nm) & \(3.58\pm 0.69\) & \(3.88\pm 0.31\) & \(10.05\pm 90.01\) \\ \hline Spk * & (nm) & \(2.17\pm 0.51\) & \(2.11\pm 0.59\) & \(2.64\pm 0.30\) \\ \hline Svk & (nm) & \(1.33\pm 0.21\) & \(1.75\pm 0.47\) & \(2.93\pm 0.25\) \\ \hline Smr1 & (\%) & \(14.02\pm 1.55\) & \(10.47\pm 1.08\) & \(7.64\pm 0.52\) \\ \hline Smr2 * & (\%) & \(90.95\pm 2.01\) & \(89.20\pm 2.07\) & \(90.28\pm 0.38\) \\ \hline Vmp * & (\(\upmu\text{m}^{3}/\upmu\text{m}^{2}\)) & \(1.04\times 10^{-4}\pm 2.15\times 10^{-5}\) & \(1.03\times 10^{-4}\pm 2.79\times 10^{-5}\) & \(1.41\times 10^{-4}\pm 1.63\times 10^{-5}\) \\ \hline Vmc & (\(\upmu\text{m}^{3}/\upmu\text{m}^{2}\)) & \(1.31\times 10^{-3}\pm 2.38\times 10^{-4}\) & \(1.38\times 10^{-3}\pm 1.53\times 10^{-4}\) & \(3.52\times 10^{-3}\pm 3.45\times 10^{-4}\) \\ \hline Vvc & (\(\upmu\text{m}^{3}/\upmu\text{m}^{2}\)) & \(2.02\times 10^{-3}\pm 4.44\times 10^{-4}\) & \(1.92\times 10^{-3}\pm 2.60\times 10^{-4}\) & \(4.43\times 10^{-3}\pm 4.24\times 10^{-4}\) \\ \hline Vvv & (\(\upmu\text{m}^{3}/\upmu\text{m}^{2}\)) & \(1.54\times 10^{-4}\pm 8.26\times 10^{-6}\) & \(1.96\times 10^{-4}\pm 4.23\times 10^{-5}\) & \(3.80\times 10^{-4}\pm 1.50\times 10^{-5}\) \\ \hline Spd * & (\(1/\upmu\text{m}^{2}\)) & \(31.00\pm 17.93\) & \(10.00\pm 0.43\) & \(20.44\pm 10.62\) \\ \hline Spc * & (\(1/\upmu\text{m}^{2}\)) & \(11.86\pm 2.56\) & \(8.00\pm 1.00\) & \(10.98\pm 2.70\) \\ \hline \hline \multicolumn{4}{c}{Hybrid} \\ \hline Sdq & (-) & \(0.04\pm 0.00\) & \(0.03\pm 0.01\) & \(0.085\pm 0.002\) \\ \hline Sdr & (\%) & \(0.07\pm 0.01\) & \(0.05\pm 0.01\) & \(0.35\pm 0.01\) \\ \hline \hline \end{tabular}  
* Samples without significant difference, ANOVA one-way and Tukey test (\(p>0.05\)).  
\end{table}