Impedance of DC-bias-controlled composite FBARs

A closed-form analytic formula is derived for composite resonators consisting of two layers of paraelectricphase ferroelectrics. It is used to analyze the properties of these resonators. Under an applied dc electric field, the layers exhibit an induced piezoelectric effect. The signs of the induced piezoelectric coefficients may be changed by changing the polarity of the dc voltages applied to the layers. In the case of a composite in which the layers have similar thicknesses, changing the sign of the piezoelectric coefficient causes a switching between the resonant frequencies of the first and second harmonics. In the case in which one of the layers is substantially thinner, changing the sign of the piezoelectric coefficient causes switching of the resonant frequency of the first harmonic between two values. In both cases, a continuous change in the dc bias voltage causes continuous changes in the resonant frequencies.