## 1 Introduction  
Thin-Film Bulk Acoustic-wave Resonators (FBARs) represent one of the most promising technologies concerning other microelectromechanical resonators [1, 2]. An FBAR consists of a stack formed of a piezoelectric layer sandwiched between two electrodes on a suspended cavity (back-trench and air-gap) or an acoustic Bragg grating (Surface-Mounted Resonators, SMR) [3, 4]. The fabrication of the first FBAR was reported in 1980 [5], and nowadays, with the maturity of their fabrication processes [6], this technology has gained remarkable interest. FBARs present unique features: they can easily reach frequencies between 1 and 20 GHz [7] and are characterized by the highest Q-factor and the smallest footprint reported in the literature [8] and a low insertion loss combined with high selectivity [9]. In addition, their fabrication process allows large-scale production as it is fully compatible with silicon-integrated circuit technology. All these features make them the world's smallest and best-performing resonator technology [10]. The commonly used simulation methods for the design of FBARs, such as the Equivalent Circuit Analysis (ECA) [11] and the Finite Element Method (FEM) [12; 13; 14], always require knowledge of the specifications of the materials composing the stack. These specifications can differ from the experimental ones, as they depend on the equipment technology and status, the deposition recipes, environmental conditions, and so on. Furthermore, each deposition process introduces a degree of uncertainty on the thickness of each layer due to the fabrication tolerances. The latter becomes a crucial challenge to overcome as even small variations in the top electrode thickness cause resonant frequency drift because of the mass-loading effect. Moreover, the design procedure loses validity if the piezoelectric thickness changes. All these drawbacks, if included in the commonly used models, increase the computational time [12]. In Reference [15], the FEM simulations were combined with a deep learning approach based on artificial neural networks; however, thousands of simulated data samples are necessary, making the method unsuitable for fast prototyping. By contrast, in Reference [16], this problem was treated using FEM simulations whose parameters were refined with a closed feedback loop to minimize the error with measurements; however, this approach requires a very high computational cost as it exploits 3D FEM simulations. In addition to the design problems, using a back trench to withstand the structure causes resonator fragility. The use of SMR structures prevents the release of the piezoelectric layer but lowers the performance compared to other solutions [8]. The trade-off between the two approaches is represented by air-gap types, where the structure is suspended on a thin membrane fabricated by employing a sacrificial process [6; 17; 18]. The presence of a membrane support layer ensures higher robustness, but also, in this case, the fragility of the suspended structure is an open issue in the case of FBARs working with very thin films [17]. Moreover, removing the sacrificial layer always requires aggressive chemicals that can damage the other materials, complicating their integration and increasing times and costs. In this scenario, the fabrication of the first air-gap FBAR based on a highly soluble organic sacrificial layer made of Lift-Off Resist (LOR) and suspended on a polymeric flexible membrane is presented. Unlike other air-gap structures reported in the literature, with the innovative proposed fabrication protocol, the removal of the sacrificial material becomes very straightforward as it does not require any aggressive chemicals. In addition, in contrast to the state of the art, the presence of the flexible membrane support layer enhances the robustness of the structure by avoiding the formation of cracks and the collapse of the stack.