## 2 Materials and Methods  
In this work, we report the growth mechanism of the SAO sacrificial layer, the realization of STO, SDC, and GDC free-standing membranes onto a PDMS Gel-pack polymer (Gel-Pak, Hayward, CA, USA), and the subsequent transfer onto a silicon substrate. The film depositions were performed using an excimer laser charged with KrF, generating UV radiation (248 nm wavelength, pulses of 25 ns width; Coherent Corp., Saxonburg, PA, USA). The laser beam was focused to an energy density of 3 J/cm\({}^{2}\) on the slowly rotating target at an incidence angle of 45\({}^{\circ}\). The pulse repetition rate was fixed at 10 Hz. The substrate was glued by silver paint to the heater. The distance between the substrate and the target was fixed at about 5 cm. The base pressure in the deposition chamber was \(1\times 10^{-7}\) mbar. During the deposition, an oxygen partial pressure ranging between \(10^{-2}\) and \(10^{-1}\) mbar was maintained in the chamber. After deposition, the oxygen partial pressure was raised to 1 atm, and films were cooled to room temperature in about half an hour. The targets used were prepared by solid-state reaction starting from high-purity powders, which were successively calcinated in air. Powders were then re-grinded, pressed to form a disk, and partially reacted at 1450 \({}^{\circ}\)C in air.  
The surface quality of the deposited heterostructures was evaluated using the RHEED diagnostics. When the deposition process was conducted at oxygen pressures below \(10^{-3}\) mbar, RHEED provided real-time feedback on the crystallinity and surface morphology during the deposition. However, at oxygen pressures higher than \(10^{-3}\) mbar, RHEED analysis was performed only after cooling the sample to room temperature, with the oxygen pressure reduced to values below \(10^{-3}\) mbar. The RHEED was used at 15 kV and with a filament current of 1.5 A.  
To optimize the deposition conditions of the SAO layer, an epitaxial sacrificial layer of SAO was grown using PLD on an STO (100)-oriented substrate. Subsequently, an STO film was deposited onto the SAO layer using a multitarget carousel, which allows the fabrication of heterostructures.  
A bilayer structure consisting of SAO (sacrificial layer) and SDC was fabricated. The SAO layer was first deposited onto the STO substrate, followed by the growth of the SDC layer on top of the SAO. The PLD growth conditions for both materials are reported elsewhere [12]. The surface area of the SDC-buffered SAO on STO (100) measured 0.5 cm \(\times\) 0.5 cm. A PDMS Gelpack adhesive polymer was then applied to the SDC surface of the SDC/SAO/STO heterostructure.  
The SAO sacrificial layer was subsequently dissolved by immersing the PDMS/SDC/ SAO/STO structure in distilled water at room temperature. After a few hours, the SDC layer was successfully transferred to the PDMS. For further transfer to a Si wafer, the PDMS/SDC structure was placed directly onto the Si wafer's surface.  
Structural characterization of the films and membranes was performed using XRD, allowing us to analyze the crystallographic quality and identify any strain or defects induced by the substrate. Rocking curve analysis and 0-20 X-ray scans were carried out using a Rigaku diffractometer (Rigaku, Tokyo, Japan) in Bragg-Brentano configuration equipped with a Co anode as the X-ray source, operating at 30 kV and 30 mA.  
Furthermore, we studied the structural evolution, morphological characteristics, and surface integrity of the free-standing membranes through scanning electron microscopy (SEM), observing changes that occurred post-transfer and during dissolution of the sacrificial layer. The long-term stability of the SDC free-standing membrane was assessed during electrochemical impedance spectroscopy (EIS) measurements (see reference [12]). The EIS measurements were repeated daily, and the membranes demonstrated stability in air for two weeks.