## 1 Introduction  
One of the special classes of materials that has attracted, in recent years, the attention of researchers and industry in general are multiferroics, because they show simultaneous coexistence of ferromagnetic and ferroelectric properties in the same phase. Thus, multiferroic materials have been of great interest in several areas, like spintronics, memory storage devices, sensors, electronic components, among others [1, 2, 3, 4, 5, 6, 7]. Moreover, the perovskite oxides of RMnO\({}_{3}\) type (R is rare earth element) are a novel class of materials exhibiting a set of remarkable interesting properties from the physics point of view, such as superconductivity,ferroelectrics, ferromagnetism, multiferroics and colossal magnetoresistance [8; 9]. The simplicity of the perovskite structure turns it into the most common crystalline structure among the studied multiferroic materials. This is due to its fairly simple organization and chemical constitution, with the general formula ABX\({}_{3}\), where A and B are cations and X is an anion, usually oxygen, but others, like fluorine, can be used.  
Among the magnetoelectric materials, GdMnO\({}_{3}\) has been studied as this material exhibits magnetic-induced ferroelectricity [10], although other study proposals are being carried out. As an example, [PERSON] et al. [11] carried out a study of the magnetic properties, and magnetic memory effect in nanocrystalline single-phase powders of GdMnO\({}_{3}\) synthesized by the sol-gel method. [PERSON] et al. [12] prepared GdMnO\({}_{3}\) ceramics by solid-state reaction method and analyzed the structure, defects, dielectric, and magnetic properties of those samples sintered at different temperatures. At room temperature, GdMnO\({}_{3}\) shows an orthorhombic distorted perovskite structure with Pbnm symmetry [13; 14], and it has attracted considerable interest in recent years due to its applications in multifunctional devices, and underlying physics, where its antiferromagnetic and ferroelectric properties are strongly connected. As can be noted, many recent studies focusing on the structural, ferroelectric, and magnetic properties of GdMnO\({}_{3}\) have been reported in the literature [11; 15; 16; 17; 18]. However, it is intriguing and challenging to note that there are rare reports involving the morphological study of its surfaces through the atomic force microscopy (AFM) technique, which is an important tool to evaluate their physical properties that are crucial for advanced technological applications. As mentioned before, the AFM technique is a peculiar technique, due to it being both sensible and accurate and recently, morphological studies have been done through topographical maps, providing several parameters, such as stereometric [19; 20; 21; 22; 23], multifractal [24; 25; 26; 27], and power spectrum density (PSD) [28; 21; 29]. All these parameters allow us to characterize surfaces at micro- or nanoscale, showing how the spatial patterns of the film surface, such as roughness, shape of peaks, peak density, heterogeneity of topographic texture distribution among others, are modified as the sintering temperature is varied.  
In this work, three GdMnO\({}_{3}\) films were synthesized from a specific precursor solution, and deposited by spin coating onto Pt(111)/TiO\({}_{2}\)/SiO\({}_{2}\)/Si substrates, and thereafter sintered at different temperatures. Morphological analysis, and nanoscale pattern results, along with the combined stereometric and advanced fractal parameters are discussed in detail, which has not yet been reported for the as-processed GdMnO\({}_{3}\) films. Much of the presented results were obtained using MountainsMap commercial software. It is worth stressing that the results extracted from fractal data, were carried out by self-developed analytical methods, which are not provided by commercial software.