Relaxation and dynamics of pre-displaced string resonators

Previously, we demonstrated that silicon nitride beams with a designed pre-displacement partially relax upon release, enabling geometric tuning of stress in micromechanical strings. To understand the statics and dynamics of such structures, an analytical model is developed. Expressions for the bending and tension energy are derived. The potential energies are functionals of the displacement profiles and a modified version of the Euler-Bernoulli equation with tension included is derived. After the projection of the energies and the partial differential equation onto the cosine shape of our devices, the mechanics is described by two variables only. This enables to visualize the energy as potential landscapes, whose shapes determine the resonance frequencies. The results are validated (and extended) with finite-element simulations. Experimentally, we measure the eigenmodes of a large number of devices optomechanically using on-chip MZIs, allowing a systematic study of geometric tuning of the frequencies and quality factors. A number of intriguing experimental results can be explained with the model. It is e.g. found that many of the observed static and dynamic effects are intimately related to buckling. More general applications of the geometric tuning method are also discussed.