Determination of the nonlinear physical constants in a piezoelectric AlN film

Abstract

Recent models vary widely as to the mechanism in the piezoelectric AlN film that gives rise to the measured 2 nd and 3 rd 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 2F 1 -F 0 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.

Tailoring strain in microelectronic devices

“…We solved for all four δ parameters, as a result our measured δ 1 minus δ 3 should be comparable with the δ 1 reported earlier, i.e. δ 1 − δ 3 [this work] = δ 1 [36]. This is the case when a negative piezoelectric charge constant is assumed as was discussed earlier in section 3.6.…”

“…Earlier work [36] showed that small resonance frequency shifts are mainly caused by the electric field dependence of the stiffness c E 33 equation (3.6a). This changes the impedance and results in distortion of the signal.…”

“…It has been widely agreed upon that the nonlinearities stem from the nonlinear response of the piezoelectric layer [36]. For example a stress dependence of the stiffness, strain and electric field dependence of the dielectric displacement or strain dependence of the piezoelectric constant all have been modelled to accommodate the measured non-linearity.…”

“…For example a stress dependence of the stiffness, strain and electric field dependence of the dielectric displacement or strain dependence of the piezoelectric constant all have been modelled to accommodate the measured non-linearity. A more complete set, incorporating the second and third order piezoelectric response to accommodate the non-linearity was proposed by [36,37]. The second order equations are…”

“…These parameters can be estimated by measuring the second order harmonic and third order intermodulation distortion [36].…”