Dependence of the quality factor of micromachined silicon beam resonators on pressure and geometry

An experimental study of damping and frequency of vibrating small cantilever beams in their lowest eigenstate is presented. The cantilever beams are fabricated from monocrystalline silicon by means of micromachining methods. Their size is a few millimeters in length, a few 100 µm in width, and a few 10 µm in thickness. Damping and resonance frequency are studied as a function of the ambient pressure p (1–105 Pa) and the geometry of the beam. The purpose of this research was to obtain design rules for sensors employing vibrating beams. The analysis of the experimental results in terms of a semiqualitative model reveals that one can distinguish three mechanisms for the pressure dependence of the damping: viscous, molecular, and intrinsic. For viscous damping a turbulent boundary layer dominates the damping at high pressures (105 Pa), while at smaller pressure laminar flow dominates. In the latter region, this leads to a plateau for the quality factor Q and in the former to Q p. The pressure pc at which the transition from laminar flow dominated damping to turbulent flow dominated damping occurs depends on the geometry of the beams. pc is independent on the length and decreases with both, the width and the thickness of the beams.