Semi-empirical calculation for design of flexural mode silicon mechanical resonator with variable cross section

• A semi-empirical calculation for design of flexural mode silicon mechanical resonators with a variable cross section is proposed.
• The mode function of the resonator is semi-empirically formulated using some FEA results.
• The resonant frequency of the resonator is derived by applying the mode function to the Rayleigh-Ritz method.
• The resonant frequencies of the resonators are calculated within a mean absolute percentage error of 3% from the FEA results.

This paper newly proposes a semi-empirical calculation for design of electrostatically driven flexural mode silicon (Si) mechanical resonators with a variable cross section which is applicable to a feasibility study of their development without finite element analysis (FEA). In the design for Si mechanical resonators with a variable cross section, a mode function describing a shape of vibration is simply assumed to be a static deflection function, and a resonant frequency is derived by applying the Rayleigh-Ritz method. The static deflection function, which reflects an effect of stress concentration generated by a variation of cross section, is semi-empirically formulated using some FEA results. This research adopts a clamped–clamped beam Si mechanical resonator which a wide rectangular bar located at the center is supported by same two narrow rectangular bars clamped to anchors. The resonant frequencies of the Si mechanical resonators are calculated within a mean absolute percentage error of 3% from the FEA results. In order to confirm the applicability of the semi-empirical formula, twenty different sizes of clamped–clamped beam Si mechanical resonators having the first-mode resonant frequency of 5–20 MHz are designed and fabricated from a silicon-on-insulator (SOI) wafer, and their vibration characteristics in a vacuum environment are measured by a network analyzer. From the comparison with the measured resonant frequency of each Si mechanical resonator, the mean and maximum absolute percentage errors of the semi-empirical calculation are 28.6 and 51.3%, respectively, which are not negligible errors for the feasibility study. The fabricated Si mechanical resonators are different from an ideal clamped–clamped beam due to undercuts of the insulator layer under the anchors. The resonant frequency of each Si mechanical resonator is analyzed by the FEA model taking into account the effect of the undercuts. The mean and maximum absolute percentage errors of the FEA results from the measured ones are reduced to 9.61 and 19.9%, respectively. These results show that the large errors of the semi-empirical calculation are mainly caused by the undercuts, i.e., flexural stiffness degradation at both clamped ends. Under the condition that the FEA results obtained from FEA models precisely imitating actual geometry and boundary condition are used in the semi-empirical calculation, this is adequately applicable to the feasibility study for developing the Si mechanical resonator with a variable cross section.