Tailored Etch Profiles for Wafer-Level Frequency Tuning of Axisymmetric Resonators

Abstract

Abstract-This paper reports a wafer-level technique for the systematic elimination of the modal frequency difference between a nominally degenerate pair of modes in an axisymmetric resonator design. A targeted etch process is developed in which masking resist and a conformal layer are ablated at specific sites on the resonator thereby exposing the underlying silicon and enabling site-specific mass removal by SiDRIE. A model of the perturbed resonator dynamics guides the selection of the ablation sites so that the subsequent timed etch reduces the modal frequency differences by a prescribed amount. This waferlevel process is demonstrated on seven resonators whose modal frequency differences are reduced below 100 mHz from initial splits as large as 15 Hz for a pair of modes with 13.5 kHz nominal frequencies.

Multimodal Tuning of an Axisymmetric Resonator

“…Although the phases of the Fourier components differ from one case to the other, the amplitude ratios are essentially constant and independent of the perturbation magnitude as shown in Ref. [8] where only the n ¼ 2 pair of modes is analyzed. An example of the mode shapes associated with the n ¼ 2, 3 pairs of modes is shown in Fig.…”

“…This essential resonator design is also employed in Refs. [7] and [8]; however, there are differences in the way mass perturbations are achieved and, consequently, there are small changes to the top surface of the resonator to support one particular method or another. In this paper, a 1 lm thick, 130 lm diameter, gold film is deposited on the top surface of all 96 "large" spokes.…”

“…This reference lacked the smaller diameter solder spheres so the ink was necessary for fine tuning. More recently, [8] demonstrated selective etching of the top surface of the large spokes, i.e., mass is removed.…”

“…Reducing the modal frequency difference within the n ¼ 2 pair of modes has been demonstrated in Refs. [7] and [8] for the resonator under consideration. In the case of mass loading, the general strategy is to deposit mass in a neighborhood of the antinodes of the higher frequency mode since this has a greater effect on modifying the kinetic energy of the higher frequency mode relative to the lower frequency mode.…”

“…In addition to the experiments with the same resonator design reported in Refs. [7] and [8], gold was ablated from the lip of a microscale hemispherical resonator in Ref. [9] and laser cutting of a silicon ring was reported in Ref.…”