  We describe and demonstrate a method by which the nonlinear piezoelectric
properties of a piezoelectric material may be measured by detecting the force
that it applies on a suspended micromechanical resonator at one of its
mechanical resonance frequencies. Resonators are used in countless
applications; this method could provide a means for better-characterizing
material behaviors within real MEMS devices. Further, special devices can be
designed to probe this nonlinear behavior at specific frequencies with enhanced
signal sizes. The resonators used for this experiment are actuated using a
1-$\mu$m-thick layer of aluminum nitride. When driven at large amplitudes, the
piezoelectric layer generates harmonics, which are measurable in the response
of the resonator. In this experiment, we measured the second-order
piezoelectric coefficient of aluminum nitride to be
$-(23.1\pm14.1)\times10^{-22}\ \mathrm{m/V^2}$.
