A Study of Mode Dependent Energy Dissipation in 2D MEMS Resonators

Abstract

With the advent of micro and nano electromechanical systems (MEMS/NEMS), there has been rapid development in the design and fabrication of sensitive resonant sensors. Sensitivity of such devices depends on the resonant frequency and the quality factor (Q). The Q of these devices are dependent on process induced prestress in the structural geometry, interaction with the external environment, and the encapsulation method. For high frequency sensors operating in air and under encapsulation condition, the Q is dominated by structural and fluid-structure interaction losses. In this thesis, we set out to study the dominant energy dissipative mechanisms that are constituent of the experimentally observed loss (Q-factor) in two specific test geometries—uncapped and capped circular MEMS drumhead resonators. Considering the importance of various factors, we consider four important problems pertaining to the uncapped as well as capped resonators. In the first problem, the most important factors perhaps are the acoustic radiation losses emanating from the annular plate, and the effect of added mass effect on the natural frequencies of the annular plate. The second problem is to investigate the dominant contribution of squeeze film losses and acoustic radiation losses with respect to various natural frequencies of the annular plate. The third problem is to consider the effect of prestress on the natural frequencies of the annular plate and its associated fluid-structure interaction losses (quality factors due to squeeze film damping and acoustic radiation losses). The fourth problem is to study the dominant fluid-structure interaction losses and structural losses that are constituent of experimentally measured Q-factors of the encapsulated annular plate (conceptual representation of MEMS device under packaged conditions). In the first problem, we study the mode dependent acoustic radiation losses in an uncapped drumhead microresonator which is represented by a annular circular plate fixed at its outer edge, suspended over a fixed substrate. There are two main effects which are associated with such systems due to the fluid-structure interaction. First is the “added mass effect,” which reduces the effective resonance frequency of the structure. The second is the acoustic radiation loss from the top side of the resonator, that affects the quality factor of the vibrating structure. In deriving the analytical solution, we first obtain the exact mode shapes of the structure ignoring any effect of the surrounding fluid (air) on the mode shape. Subsequently, we use these mode shapes to study the effect of the surrounding fluid on the associated natural frequencies and the Q-factor. The effect of “added mass” on the frequencies of the structure is found to be negligible. However, the acoustic radiation losses found to be significant. Additionally, we found that the variation in Qac over the first few modes (< 40 MHz) is marked with a local maximum and a minimum. Beyond this range, Qac increases monotonically over the higher frequency modes. It is also found that such kind of variation can be described using different acoustics parameters. Finally, comparing the acoustics radiation loss based quality factor with the experimental results for the uncapped drumhead resonator, the acoustic damping dominates only at higher modes. Therefore, our second problem forms the basis of finding other fluid-related damping. In the second problem, we explore the fluid losses due to squeeze film damping in the uncapped drumhead micro resonator. In this case, the squeeze film loss is due to the flow of the fluid film between the bottom surface of the annular plate and the fixed substrate. Based on the literature survey, it is found that the squeeze film damping reduces with increase in the air-gap thickness and the operating frequencies respectively. However, the squeeze film effect can not be ignored at lower frequencies. In order to investigate the contribution of squeeze film damping in uncapped resonator, we determine squeeze-film damping based quality factor Qsq corresponding to different modes of the resonators using FEM based software, ANSYS. On comparing Qsq with the experiments, we found that Qsq matches well with the experiments corresponding to the lower modes. Therefore, it is found that Qsq dominates at low frequencies (< 20 MHz) and Qac plays significant role at high frequencies (> 40 MHz). Both types of damping should be considered while modeling the fluid damping in uncapped resonator. In the next study, we discuss the effects of prestress on the resonant frequencies and quality factor. In the third study, we discuss the applicability of thin-plate theory with prestress and membrane theory in computing the frequencies and quality factor due to acoustic and squeeze film losses in the uncapped drumhead resonator. In the first two studies, although the quality factor due to acoustic losses and the squeeze film captures the correct trend of the experimental results, there is a mismatch between the experimental and theoretical frequencies computed with added mass effect. In order to improve the computation of frequencies corresponding to measured modes, we first used membrane theory to predict the frequencies, and finally we quantify that there exists discrepancy between computed and the corresponding experimental frequencies with error of about 8–55%. Since, both the membrane as well as thin plate theory without prestress do not correctly model the frequencies, we used the thin plate theory with prestress. For a prestress level of 96 MPa, we found the match between the computed frequencies and the corresponding quality factors with the measured values. However, we also found that there exists strong dependence of prestress on the acoustic radiation loss, with decrease in the acoustic loss based quality factors with increase in the prestress level. In the subsequent problem, we focus on the computation of losses in capped drumhead resonator which leads to a design possibility of improving the quality factor by containing the acoustic radiation losses. In the fourth problem, we study the structural and fluid-structure interaction losses which are dominant constituent of net Q-factor observed in experiments due to encapsulation of uncapped drumhead resonator. Essentially, the geometry of the capped resonator constitutes upper and lower cavities subjected to fluid-structure interaction losses on both sides of the annular plate. The dominant fluid-structure interaction loss is found to be due to squeezing action acting simultaneously in the upper and lower cavities. However, as we go to the higher modes, squeeze film damping become very small and the damping due to structure related losses such as clamping and thermoelastic losses becomes significant. We found the thermoelastic damping to be the dominant source of structural damping at higher resonant modes, whereas, the clamping losses are found to be relatively smaller. Finally, on comparing the net quality factor with the experimental results, we observed that the squeeze film losses are dominant at lower frequencies, and thermoelastic losses dominate at the higher frequencies. However, there remains some discrepancy between theoretical and experimental Q-factors particularly over higher frequency range. Such discrepancy may be due to some unaccounted factors which may be explored to improve the modeling of damping in capped resonators. The emphasis of this work has been towards developing a comprehensive understanding of different dominant dissipative mechanisms, classified into the fluid-structure interaction and the structural losses, that are constituent of the Q-factor at various resonant modes of uncapped and capped drumhead resonators.