Thermoelastic Dissipation in Composite Silicon MEMS Resonators with Thin Film Silicon Dioxide Coating
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Thermoelastic Dissipation in Composite Silicon MEMS Resonators with Thin Film Silicon Dioxide Coating Shirin Ghaffari and Thomas W. Kenny Mechanical Engineering Department, Stanford University, Stanford, CA 94305, U.S.A. ABSTRACT We analyze thermoelastic dissipation in composite silicon MEMS resonators that exhibit multiple mechanical and thermal modes with complex dynamics. Silicon resonators that are coated with thin films of silicon dioxide can have near-zero temperature coefficients of frequency, making them attractive for use as precision time references. The quality factor of MEMS resonators can be dominated by thermoelastic dissipation (TED), which is triggered by the relaxation of mechanically induced temperature gradients. Recently, Chandorkar et al. (2009) have shown an expression of TED based on entropy generation as a weighted sum of the modal solutions of the three-dimensional heat transfer equation. This expression was obtained for weak coupling between mechanical and thermal dynamics. Applying this same technique to a fully coupled solution to the dynamics, we show that the TED contribution of the dominant thermal modes can be inhibited in the presence of a thin silicon dioxide film. Reduction of the contribution from the dominant thermal mode is shown with increasing oxide. We studied the effects of varying oxide film thickness and beam length. The quality factor was simulated for each unique case and compared to multimode energy dissipation. Our results suggest with some variability, thin film oxide coating affects the thermal relaxation of the composite resonator in the direction of lower TED and increased quality factor. INTRODUCTION An important characteristic of micromechanical resonators is the linear temperature coefficient of frequency (TCF) that needs to be minimized for frequency stability over temperature. The temperature dependence of the Young modulus is the main cause of the linear TCF in MEMS. One technique for temperature compensation of MEMS resonators is through the addition of thin films of other materials. Typically, the base resonator (single crystal or poly silicon or silicon nitride,) is coated with a layer of a material with opposing temperature coefficient of Young modulus (TCE). A convenient compensator for Si resonators is SiO2; SiO2 features a positive TCE and can be grown thermally on the Si base. Si/SiO2 composite resonators with zero TCF have been recently fabricated and characterized by many groups [1-5]. The performance of the resonator can also depend on how well it can store energy; any source of damping that takes away from total vibrational energy contributes to a low quality factor (Q). Thermoelastic dissipation (TED) is one such mechanism particularly pertaining to silicon MEMS resonators. During TED, periodic deformation of the solid results in a cyclic temperature gradient from warmer compressive regions to cooler tensile regions and energy is irreversibly dissipated by entropy generation through thermal relaxation. TED is maximized when the relaxation of the
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