Design of etch holes to compensate spring width loss for reliable resonant frequencies

A pattern width loss during the fabrication of lateral silicon resonators degrades resonant frequency reliability since such a width loss causes the significant deviation of spring stiffness. Here we present a design guide for etch holes to obtain reliable resonant frequencies by controlling etch holes geometries. The new function of an etch hole is to generate the comparable amount of the width loss between springs and etch holes, in turn to minimize the effect of the spring width loss on resonant frequency shift and deviation. An analytic expression reveals that a compensation factor (CF), defined by the circumference (C_u) of a unit etch hole divided by its silicon area (A_u), is a key parameter for reliable frequencies. The protrusive etch holes were proposed and compared with square etch holes to demonstrate the frequency reliability according to CF values and etch hole shapes. The normalized resonant frequency shift and deviation of the protrusive etch hole (−13.0% ± 6.9%) were significantly improved compared to those of a square etch hole with a small CF value (−42.8% ± 14.8%). The proposed design guide based on the CF value and protrusive shapes can be used to achieve reliable resonant frequencies for high performance silicon resonators. (technical note)