## 3 Results
### Surface Morphology Analysis  
In order to guarantee the formation of films with adequate thickness, at each deposition cycle, layers of 20 nm were deposited, which at the end of the ninth cycle provided films with a thickness of \(\sim\)180 nm. After sintering there was no change in thickness as a function of temperature increase. Scanning electron microscopy plan-view images of the GdMnO\({}_{3}\) thin films are shown in Figure S1 (Supplementary Material), revealing the evolution of the surface as a function of the increasing sintering temperature. In this figure, we present images with a larger viewing area (Figure S1a,c,e) and others with a greater magnification of the surface of thin films (Figure S1b,d,f). The samples annealed at temperatures of 650 \({}^{\circ}\)C (Figure S1a,b) and 750 \({}^{\circ}\)C (Figure S1c,d) exhibit smooth surfaces without visible grain outlines, which is characteristic of an amorphous or incipient crystalline phase. However, GdMnO\({}_{3}\) films annealed at 850 \({}^{\circ}\)C (Figure S1e,f) exhibit grains and a well-defined structure. Furthermore, 3D nanoscale morphology and a 2D representation of the relative frequency of topographic maps of GdMnO\({}_{3}\) thin films are shown in Figure 2. GdMnO650 and GdMnO750 exhibited smoother surfaces (Figure 2a,c). The topography of GdMnO850 (Figure 2e) is more irregular and has a distribution of sharper peaks, which can be attributed to the growth of GdMnO\({}_{3}\) grains due to the increase in temperature that promoted the emergence of larger structures. [PERSON] et al. [31] reported the evolution of GdMnO\({}_{3}\) film morphology through scanning electron microscopy measurements, whose results were similar to the AFM images shown here. [PERSON] et al. [15] reported the fabrication of GdMnO\({}_{3}\) multiferroic thin films on SrTiO\({}_{3}\) (110) substrate by pulsed laser deposition (PLD) technique and their AFM images showed the granular nature of the film [51]. A complementary 2D observation can be found in Figure S2. These qualitative observations can be confirmed by the increase in the roughness parameters that are shown in Table 1, where all parameters showed significant difference (\(p<0.05\)). GdMnO650 and GdMnO750 presented similar roughness values, which were observed for mean roughness (Sa) and standard quantitative statistical parameters of image amplitude (Sq), while GdMnO850 has the highest roughness (\(\sim\)3.6 nm), suggesting greater spatial complexity (greater topographic irregularity). Similarly, the maximum peak height (Sp), maximum pit height (Sv), and maximum height (Sz) also exhibited similar behavior because GdMnO850 presented the highest values (Table 2), revealing that the organized and nanostructured growth of GdMnO\({}_{3}\) crystals in GdMnO850 changes the topography of the film, because of the increase of the sintering temperature. Notably, the increase of roughness may be associated with the high microstrain value of the GdMnO\({}_{3}\) lattice, when compared to other values reported in previous studies, for example [11; 31], as mentioned in Section 3.1. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Unit** & **GdMnO650** & **GdMnO750** & **GdMnO850** \\ \hline \multicolumn{5}{c}{Height} \\ \hline Sq & (nm) & 1.58 \(\pm\) 0.25 & 1.65 \(\pm\) 0.22 & 3.63 \(\pm\) 0.30 \\ \hline Ssk & (-) & 0.44 \(\pm\) 0.21 & 0.15 \(\pm\) 0.27 & \(-\)0.08 \(\pm\) 0.08 \\ \hline Sku & (-) & 3.61 \(\pm\) 0.42 & 3.48 \(\pm\) 0.23 & 2.63 \(\pm\) 0.04 \\ \hline Sp & (nm) & 7.80 \(\pm\) 2.39 & 6.58 \(\pm\) 1.42 & 17.12 \(\pm\) 5.49 \\ \hline Sv & (nm) & 4.49 \(\pm\) 0.25 & 5.31 \(\pm\) 0.81 & 12.11 \(\pm\) 0.51 \\ \hline Sz & (nm) & 12.28 \(\pm\) 2.34 & 11.88 \(\pm\) 1.59 & 29.22 \(\pm\) 5.59 \\ \hline Sa & (nm) & 1.23 \(\pm\) 0.21 & 1.28 \(\pm\) 0.15 & 2.97 \(\pm\) 0.26 \\ \hline \end{tabular}
\end{table} 
Table 2: Height surface parameters of GdMnO\({}_{3}\) thin films, according to ISO 25178-2:2012.