## 1 Introduction
Improving the manufacturing of complex oxide thin films and their realization as free-standing membranes could substantially impact the performance and the field of utilization of materials like samarium-doped ceria (SDC) and gadolinium-doped ceria (GDC). Indeed, non-stoichiometric doped ceria (CeO\({}_{2,\text{-}8}\)), with Ce\({}^{+4}\) replaced by trivalent cations such as Gd\({}^{+3}\) or Sm\({}^{+3}\), is an attractive redox-active material due to its high ionic and electronic conductivity and stability [14; 15; 16; 17; 18]. For this reason, doped ceria-based materials represent a very promising class of electro-catalysts for CO\({}_{2}\) reduction in solid oxide electrolysis cells (SOECs) due to their high stability and efficiency in converting CO\({}_{2}\) to CO with remarkable carbon tolerance [10; 11; 19; 20; 21]. However, fabricating high-quality thin-film membranes of SDC and GDC remains a significant challenge. This process necessitates using a sacrificial support layer that can be removed post-deposition without compromising the structural integrity of the film [9; 12]. In particular, reducing electrochemical components such as cathodes, electrolytes, and anodes to two-dimensional configurations with nanometer-scale thicknesses can enhance device performance by increasing energy density, lowering operating temperatures, and enabling integration into compact systems. To produce complex oxide thin films with excellent crystallographic quality, Pulsed Laser Deposition (PLD) is a widely used technique for the fabrication of thin films with precise control over their structural, chemical, and physical properties. The PLD process employs a high-energy pulsed laser to ablate material from a solid target, generating a plasma plume containing ions, atoms, and molecules ejected from the target. This material is deposited onto a substrate under precisely controlled conditions, such as temperature and ambient gas pressure, forming a thin film. A key advantage of PLD is its ability to maintain the stoichiometry of complex materials, making it especially effective for fabricating oxide thin films, superconductors, and multilayer heterostructures [9; 12; 13]. Additionally, it supports the growth of films with smooth surfaces and sharp interfaces, which are critical for applications in microelectronics and nanotechnology. Reflection High-Energy Electron Diffraction in the last decades has become an in situ diagnostic tool in PLD, providing real-time monitoring of film growth at the atomic scale. By directing a high-energy electron beam at a grazing incidence angle to the substrate surface and analyzing the resulting diffraction pattern, RHEED enables the precise assessment of surface morphology, crystallinity, and layer-by-layer growth dynamics during deposition. One of the key advantages of integrating RHEED into PLD systems is its ability to monitor the evolution of thin-film surfaces without interrupting the deposition process. This real-time feedback allows the optimization of growth parameters, such as laser fluence, substrate temperature, and ambient gas pressure, ensuring high-quality film deposition. Furthermore, RHEED facilitates the study of growth modes, distinguishing between layer-by-layer (2D), step-flow, or island (3D) growth mechanisms [22], which are crucial for tailoring film properties to specific applications. RHEED is particularly valuable for achieving epitaxial growth, where precise control of crystallographic alignment and interface sharpness is required. For instance, the oscillations observed in the RHEED intensity during layer-by-layer growth can be used to determine deposition rates with sub-monolayer precision, enabling the fabrication of complex heterostructures and super-lattices [2; 9; 22]. Beyond its role in film characterization, RHEED also aids in detecting deviations from ideal growth conditions, such as surface roughening, amorphization, or phase transitions.