Multi-mode Sensitive Layout Synthesis of Microresonators

Microresonator layouts are synthesized such that their preferred mode of oscillation is well-separated from the higher order in-plane and out-of-plane modes. Building on our previous work, we have incorporated models for four out-of-plane modes. All these modes are modeled as springmass systems. The spring constants and the effective masses of these modes are analytically derived. Synthesis is accomplished by encoding a design quality metric as the design objective while simultaneously constraining the design to meet user specifications. These constraints require that the resonant frequency in the preferred direction is sufficiently lower than (and, hence, dominates over) the resonant frequency of other modes of vibration of the structure. The models are verified by comparison with 3D FEM simulations and also with experimental measurements on fabricated resonators. The usefulness of these models is illustrated by comparing the oscillation modes of layouts synthesized with and without these models. This exercise also shows that such mode-separation can be achieved only if the microresonators have a structural thickness larger than flexure width.