Determination of the nonlinear physical constants in a piezoelectric AlN film

Recent models vary widely as to the mechanism in the piezoelectric AlN film that gives rise to the measured 2nd and 3rd order nonlinear behavior in BAW resonators. As an example one model suggests that a strain dependence of the bulk modulus in a piezoelectric AlN film is responsible for both the 2nd and 3rd order response of BAW and FBAR resonators [Collado]. We call this model the "bulk-bulk" model since the bulk modulus depends on the strain in each order respectively. We find that this "bulk-bulk" model is not capable of modeling our measured second harmonic (H2), and intermodulation distortion (IMD3) data simultaneously. When the 2nd order coefficient is chosen to match the measured 2nd harmonic response, it generates an IMD3 response through a process of remixing at a frequency 2F1-F0 which is larger than our measured data by ∼45 dBs when two +24 dBm tones are applied. It appears the authors did not fully incorporate their chosen non-linearity into their model. As a result the "bulk-bulk" non-linear model of the AlN film must be abandoned in favor of a new model — a "general" nonlinear Mason model, in which a complete set of 2nd and 3rd order nonlinear mechanisms can be evaluated to see which are consistent with the data. Such a model is described in this work and in a companion paper. Using this model we show that a strain dependent piezoelectric coefficient must be employed to model the H2 behavior without modeling an IMD3 response that is much larger than what is measured. To fit the IMD3 data a 3rd order strain dependent bulk modulus must also be incorporated. The resulting model is a "piezo-bulk" model for suggesting that the piezo coefficient and the bulk modulus have a 2nd and 3rd order dependence on strain, respectively. We also show that another recent model [Ueda] is not suitable for evaluating the underlying nonlinear physics because it violates conservation of energy and does not allow for remixing.