Contents I. Introduction
Resonators consisting of thickness mode transducers positioned at the surface of a non-piezoelectric substrate have been proposed by Plessky et al. as a new type of acoustic resonators [1] . The transducers are arranged in a periodic array excited with alternate polarities ( Fig. 1(a) ). The coupling between collective modes of the transducers and surface modes of the substrate induces so-called “hybrid SAW/BAW” modes. At first, the inertia added by the transducers slows down substrate modes. This leads to a “pseudo-SAW” mode, represented in Fig. 1(b) , confined in the vicinity or within the array of transducers. This mode is naturally confined, as its wave velocity is lower than the bulk modes of the substrate and is thus expected to exhibit high quality factors ( Q ). As demonstrated in Fig. 1(b) , the “pseudo-SAW” mode is characterized by a rigid translation of the transducer where a Rayleigh wave propagates on the top surface of the substrate. On the other hand, the thickness resonance of each individual transducer, mechanically loaded by the acoustic impedance of the substrate, promotes a collective mode developing a large electromechanical coupling factor ( k 2 ). This mode is known as the “pseudo-BAW” mode, illustrated in Fig. 1(c) . It is characterized by a deformation of the substrate when the transducers extend/compress. The frequency at which this mode occurs is mostly defined by the thickness ( e ) of the transducers. The condition for this mode to be confined close to the surface is that its effective velocity ( V ), defined by the product of its frequency ( f ) and the wavelength (2 p ), remains smaller than the bulk wave velocities in the substrate (5800 m/s) for Si). Considering the influence of these two geometric parameters, the transducer thickness and pitch, indicates that high electromechanical coupling factors (theoretically, close to 6 % for AlN transducers [1] ) and high quality factors can only be achieved if the transducers exhibit sufficiently high thickness to pitch aspect ratios ( e / p ).
