In this paper we demonstrate a method to extract the electric field dependence of first order piezoelectric parameters. We fit the Mason model to measured frequency dependent electrical impedance data of aluminum-nitride bulk acoustic wave resonators for various DC biasing. The impedance was fitted accurately at the mechanical resonances and in the quasi-static regime well below the resonance frequency. This revealed the dependence of the piezoelectric stiffness, charge constant, and permittivity on the electric field. From this the four second order piezoelectric parameters were calculated.
van Hemert, T., Reimann, K., & Hueting, R. J. E. (2012). Extraction of second order piezoelectric parameters in bulk acoustic wave resonators. Applied physics letters, 100(23), 232901:1-232901:4. https://doi.org/10.1063/1.4725503