Accurate explicit formulae of the fundamental mode resonant frequencies for FBAR with thick electrodes

This paper presents the explicit formulae of the parallel and series resonant frequencies of the fundamental thickness mode for the FBAR structures with improved accuracy. A hypothesis is proposed that the input acoustic impedance becomes continuous under resonance, at the two opposite faces of a virtual cut plane at any position along with the acoustic transmission line, no matter at interfaces or not. On the other words, only correct resonant frequency can meet the input acoustic impedance continuity condition. When the frequency deviation is small, the input acoustic impedance can be approximated as a linear function of frequency around the resonant frequency. We deduce the frequency correction formulae to eliminate the input impedance difference. The proposed parallel/series resonant frequency formulae are composed of two parts. The first part is the resonant frequency with mass loading effect, and the second part is the correction item for the input acoustic impedance difference correction mentioned above. To check the accuracy of the proposed formulae, the simulations with different thickness and material combinations of the piezo-layer (ZnO and AlN) and the electrodes (Al, Au, Ag, Cu, Mo, W) are conducted. We calculate the electrical impedance of the multi-layered transmission line and get the maximum resistance/conductance frequencies as the parallel/series resonant frequencies for comparison. The results show, the errors of the explicit formulae we proposed can be lower than 0.3% under the conditions that the unit area mass ratio of electrodes to piezo-layer is less than 0.5 for heavy metal (Au, Ag, Cu, Mo, W), and thickness ratio is less than 0.25 for light metal (Al). These conditions are suitable for usual FBAR. Under the same conditions, the error of our formulae can be decreased to less than 5% of the formulae with mass loading effect only.