Properties of transducers and substrates for high frequency resonators and sensors

Properties of transducers and substrates for bulk acoustic wave resonators and sensors are described. These resonators utilize one-dimensional thickness vibrations of structures consisting of a low-loss substrate crystal surmounted by a thin active piezoelectric film that drives the composite in resonant modes to achieve gigahertz frequencies. The structures considered include oblique orientations of the substrate, leading to generation of coupled elastic modes in the composite. A modified Christoffel-Bechmann (CB) formalism is presented to calculate acoustic wave speeds and displacements in the piezoelectric film transducer and the substrate. The CB method also yields the piezoelectric coupling coefficients of arbitrarily oriented piezofilms, for electric fields impressed either along the thickness or laterally. The calculations apply generally to transducer and substrate crystals of any symmetry class. The piezoelectric portion is then made specific for films of class 6 mm (wurtzite structure) with arbitrary orientation on the substrate, and the substrate calculations are specified for class 3 ¯ m materials, but apply also to any substrate with 3 m or 32 symmetry. Zinc oxide and sapphire are used in an example of the acoustic resonator structure.