Material nonlinearity limits on a Lamé-mode single crystal bulk resonator

In this paper, the material nonlinearity in single crystal silicon (SCS) square-plate bulk-mode resonators is described as a strain-dependent shear modulus and incorporated into a lumped element model. The equivalent circuit of the fully differential setup is developed for the nonlinear oscillation cases. According to measurement results, the spring hardening effect in a bulk resonator is reported for the first time. Opposite nonlinear behaviors (spring softening and spring hardening) are detected in devices with same dimensions and driving mechanism but different crystalline orientations ([110] and [100]) indicating the material nonlinearity dominance. The experimental results are compared against the circuit simulation results in various conditions which validate the proposed model.