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14 Engineering MEMS Resonators with Low Thermoelastic Damping Amy E. Duwel, 1 Rob N. Candler, 2 Thomas W. Kenny, 2 Mathew Varghese 1 Copyright © 2006, IEEE. Published in IEEE JMEMS, Vol. 15, No. 6, 2006 Nomenclature Variable Physical Definition E Young’s modulus a Coefficient of thermal expansion To Nominal average temperature (300 K) r Density of solid Csp Specific heat capacity of a solid Cv Heat capacity of a solid, Cv = rCsp k Thermal conductivity of a solid wmech Mechanical resonance frequency tn Characteristic time constant for thermal mode n s Stress e Strain l, µ Elastic Lamé parameters T Temperature S Entropy [u v w] Components of displacement in x,y, and z directions, respectively = [u, v] 2D vector of mechanical displacements Mechanical mode amplitude U m abstract This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions (2D and 3D) for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices. m Mechanical eigenmode shape function wm Mechanical resonant frequency for eigenmode m An Thermal mode amplitude Tn Thermal eigenmode shape function wth Characteristic frequency of dominant thermal mode DW Energy lost from mechanical resonator system W Energy stored in mechanical resonator 1 <strong>Draper</strong> <strong>Laboratory</strong>, Cambridge, MA 2 Stanford University, Departments of Mechanical and Electrical Engineering, Stanford, CA
Introduction Micromechanical resonators are used in a wide variety of applications, including inertial sensing, chemical and biological sensing, acoustic sensing, and microwave transceivers. Despite the distinct design requirements for each of these applications, a ubiquitous resonator performance parameter emerges. This is the resonator’s Quality factor (Q), which describes the mechanical energy damping. In all applications, it is important to have design control over this parameter, and in most cases, it is invaluable to minimize the damping. Over the past decade, both experimental and theoretical studies [1]-[6],[9],[22] have highlighted the important role of thermoelastic damping (TED) in micromechanical resonators. However, the tools available to analyze and design around TED in typical micromechanical resonators are limited to analytical calculations that can be applied to relatively simple mechanical structures. These are based on the defining work done by Zener in References [7] and [8]. Zener developed general expressions for thermoelastic damping in vibrating structures, with the specific case study of a beam in its fundamental flexural mode. In Reference [8], Zener’s calculation was based on fundamental thermodynamic expressions for stored mechanical energy, work, and thermal energy that used coupled thermalmechanical constitutive relations for stress, strain, entropy, and temperature. However, in order to evaluate these energy expressions for a specific resonator, Zener proposed that the strain and temperature solutions from uncoupled dynamical equations could be sufficient. He found the eigensolutions of the mechanical equation, and, separately, the eigensolutions of the uncoupled thermal equation. By applying these to the coupled thermodynamic energies, Zener calculated the thermoelastic Q of an isotropic homogenous resonator to be: where the physical constants are listed in the Nomenclature, w mech is the mechanical resonance frequency, and t n is the characteristic time constant of a given thermal mode. This takes into account the fact that multiple thermal modes may add to the damping of a single mechanical resonance. The contribution of a given mode, n, is determined by its weighting function, f n . Zener explicitly calculated the weighting functions for a simple beam resonating in its fundamental flexural mode. In order to make the analysis tractable, he assumed that only thermal gradients across the beam width (dimension in the direction of the flexing) were significant. This left only a 1D thermal equation to solve. Zener found that a single thermal mode dominated, giving (1) (2) Few structures are amenable to the simplifications that led to expression (2) for Q. However, Zener’s expression (1) is quite general. In the section “Weakly Coupled Approach to TED Solutions,” we show how numerical solutions to the uncoupled mechanical and thermal dynamics of a resonator can be used to evaluate (1). This adds a great deal of power to Zener’s approach, since arbitrary geometries can be considered. We show how Zener’s weighting function approach offers an intuition into the details of the energy transfer. At the same time, our results highlight the limits of intuition in identifying the thermal modes of interest. For example, we find that the simplification Zener made in assuming only thermal gradients in one direction along the beam were significant does not capture the most important thermal mode, even for a simple beam. In addition, past efforts to estimate Q without explicitly calculating the weighting functions have been shown [9] to greatly overestimate the damping behavior of real systems. This “modified” interpretation of Zener’s method can be misleading. In this paper, we describe a method for using full numerical solutions to evaluate Q using Zener’s approach. We call this a “weakly coupled” approach. We also present our numerical method for solving the fully coupled thermoelastic dynamics equations to calculate Q for an arbitrary structure. Using numerical solutions in the weakly coupled approach offers powerful guidance in engineering around thermoelastic damping, while fully coupled solutions offer the ability to precisely evaluate and optimize the thermoelastic Q of a resonator. Numerical solution of the Fully Coupled teD equations The coupled equations governing thermoelastic vibrations in a solid are derived in Reference [19]. The following section, “Governing Equations in 3D,” outlines the basic principles of this derivation. “Governing Equations in 2D with Plane Stress Approximations” highlights modifications required for a 2D plane stress formulation. The full 2D and 3D equations are written explicitly so that they are accessible to the user community. We numerically solve the 2D and 3D dynamical equations using the finite-elements based package Comsol Multiphysics. [11] The Comsol implementation is described in References [12] and [13]. This analysis can be applied to the wide variety of microelectromechanical system (MEMS) resonator structures reported in the literature. It is a useful tool for determining whether TED limits performance or whether other damping mechanisms, such as anchor damping, [23] should be investigated instead. “Quality Factor Calculations for Typical MEMS Resonators” demonstrates the application of TED simulations to a few example MEMS resonator structures. Quality factors are calculated and compared with the analytical Eq. (1) as well as with experimental measurements reported in the literature. Engineering MEMS Resonators with Low Thermoelastic Damping 15
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