Anchor losses of non-slender MEMS resonators and the impact of mode shape interference at frequency crossing and avoided crossing

Abstract

This work investigates frequency crossing phenomena in non-slender MEMS resonators. Special focus is put on the influence of anchor losses and the related quality factors (Q-factors). To evaluate the anchor losses and resonance frequencies of higher-order-out-of-plane (HO-OOP) modes, a mathematical model of the resonator is developed and solved with the finite element method (FEM). The results show that non-slender geometries vibrating in OOP modes exhibit very high Q-factors up to 1e7, while in-plane (IP) mode shapes have low Q-factors below 10. These results motivated an extensive investigation into the OOP and IP vibrational modes, their Q-factors, and resonance frequencies as the resonator changes from a slender beam to a wider plate. The resonance frequencies of different mode shapes evolve differently when the width of non-slender resonators is increased, and at some widths, the resonance frequencies cross. Three phenomena are identified: simple crossing, avoided crossing, and "Q-drop" crossing. Simple crossing does not exhibit an interference between the crossing mode shapes, and their Q-factors are not changed. The avoided crossing describes the repulsion of the natural frequencies while increasing the width due to the strong coupling of the natural modes. "Q-drop" crossing does not affect the resonance frequencies but drastically reduces the quality factor of a single mode. A novel method for mathematically analyzing modal displacement patterns is introduced, employing the Fourier transformation (FT) to analyze these mode shape interference patterns that significantly affect the Q-factor. In avoided crossing, the vibrational modes and Q-factors are exchanged while the resonance frequencies are repelled from each other. In "Q-drop" crossing, the displacement patterns of the interfering mode shapes are only slightly affected. Avoided crossing occurs between OOP-OOP modes, as well as OOP-IP, while "Q-drop" crossing occurs only in OOP-IP resonance frequency crossing. The identified phenomena explain the Q-factor drop in previous experimental studies.