@ if- & A # i &X Dissertation for Doctor of Philosophy On die Quality’ Factors of MEMS Resonators Ẩ' Student: Chi-Cuong Nguyen 4s * Kí. âì Advisor: Wang-Long Li National Cheng Kung University Department of Materials Science and Engineering Tainan, Taiwan, R. May 2017 t 106 4- 05 24 8 On the quality factors of MEMS Resonators c lii-C uong N guy on A dissertation submitted to rhe Department of Materials Science and Engineering in partial fulfillment of the requirement for the degree of Doctor of Philosophy Nat iorut 1 Cheng-Kung University" Department of Materials Science and Engineering Tainan, Taiwan. Republic of China Approved by May 2017 Abstract Controlling of the energy dissipation mechanisms and investigations of then physical effects play key roles on the performances of many microelectromechanical system (MEMS) resonators such as accelerators, micro gyroscopes, torsional minors, microphones used in many applications such as sensors, actuates, transductors, and energy harvesting.
Two important dynamic characteristics of the MEMS resonators are its resonant frequency, and the quality factor (Q-factor) of resonators. High Q-factor (low energy loss) IS one of the major requirements for MEMS resonators operating in high sensitivities, resolutions, and overall stability of sensing system. The external squeeze film damping (SFD) of IvIE-IvIS resonators IS a dominant factor to lower the Q-factor due to their large surface area to volume ratio and as the gas flow is trapped into an ultra-thin spacing in ambient gas environment conditions. To improve the Q-factor of micro-beam resonators, lower pressure is introduced in a thin gap spacing to reduce the SFD, then the effects of gas rarefaction and surface roughness become im portant and must be taken into account.
The numerical methods were based on so bring the modified molecular gas lubrication (MMjL) equation with the databases for Poiseuille flowrate and the pressure flow factors applied for modelling the external dominant SFD problem on micro-beam resonators with coupling effects of gas rarefaction and surface roughness in a wide range of inverse Knudsen numbers and accommodation coefficients (ACs) conditions. Then, die MM3L equation, die transverse vibration equation and their corresponding boundary conditions of the micro-beams are simultaneously solved in die eigenvalue problem by a finite element methods (FEM) to evaluate die resonant frequency and the Q-factor of die SFD of micro-beam resonators. The internal thermoelastic damping (TED) is obtainedby sohring the thermal equation and the transverse vibration equation numerically in the eigenvalue problems by FEM The anchor loss is analytically evaluated from the theoretical model in the literature for micro-beam resonators. Then, the combined external SFD, thermoelastic damping (TED) and the anchor loss are included on the total Q-factor of the micro-beam resonators.
In this dissertation, the quality factors of micro beam resonators and contributions of SFD on total Q-factor (weighting of SFD) are I analyzed and designed over a wide range of gas rarefaction conditions (inverse Knudsen number and accommodation coefficients (ACs)), surface roughness (film thickness ratio and Peklenik number), modes of the resonators The results shown that the external SFD is dominant to control the Q-factor of the micro-beam resonators in lower modes, high pressure and ACs conditions, while the internal TED and the anchor loss are dominant in higher modes and low pressure conditions. Thus, weighting of SFD decreases significantly in higher modes and/or higher gas rarefaction (lower pressures and ACs) regions. Furthermore, effects of surface roughness are diluted by the gas rarefaction effects. The Q-factors depends significantly on the effects of surface roughness (film thickness ratio and Peklemk number) in higher gas rarefaction conditions and higher modes of the resonators.
This research provides insights into decreasing of the SFD in wide range of gas rarefaction, surface roughness and resonator modes conditions, thus help in ữn proving the total Q-factor of such resonators operating in controlled conditions. Keywords Squeeze film damping (SFD); Thermoelastic damping (TED); Quality factor; Micro-beam resonator, Gas rarefaction; Accommodation coefficient; Surface roughness n MEMS KM iW ft# N A 4 &&&& iM ỹ<#.MEMS)i»<8 ft ft$«s ’ MZ ’ *|#x frit-gilts 'MEMS igfc3ftiMMl&«i$44fi£Ai& fcriiMS ftSgE^(Q0ị)’<ậộE-T (4ỔJkỶ $ K)Ẵ. u ỉSị # ^^«ẤJtfcft^«4tíl^ĩft MEMS i»<Sfti-ft-ft-^^ ’ MEMS i&#.3ft ^4fW*<aX(5FD)j?; fg-te QmWH’ 0^A<fckftẬ(&^ỉí^i. ><> »M8 ft Q E-T ’ UtM'ftM^toflAfcteft 5FD ’ *»&-j|lâ.*fcMWmWÍ*’ ’ &te*^Ẩ£^O*ftíUrtUd$ert ^a.fcf*^£*fcM£Wầ'&.fcíSM?ỈJtft*l'SF*.4 5FD MQft Pội seui lie 2*3 &#,$♦>*E Ỹ.M íL5ỉí?ẩttíit(MMGO ft ’ M.
’ ’ «tf^<^ítf<3ft SFDftWJ£4^Q®^ • «fiffl. MQ ■ftttte^flft • íCX&4,ft^4ftĩ!Wjk^#ia4.«H ’ ÍM > SFD *L»fia^ơED)fJíírlS-«Kftte^£Ỳ-Ổi*lA,*#iSftft Q E9^ -t.SftSie^^SFD^eQE-? (SFDft4é) ft -Ồ KfcEftíUÍ^&4C£ tnuđseh ^4^íè«i(AC)) Peklenik K) ' iSMaft^AT^T^fr^a^ ’ ’ ^ểf SFDà<Wíft A.Ị?ft ọ ERCU 4é<&t>4T ’ TED ^S^Kàí-*^ ’ C)£JO ’ 5FD ft#í ®ỹ^4ổ.^Ạđ&^^ft^« ’Q E Peklenik fc)ft^5 ’^ẰWTàkg® ftầ.15ft^QE^ ’ flMtq ĩ »JỄ«ra. Then supports, that have a great meaning for me to finish this dissertation for Doctor of Philosophy. Secondly, I would like to send a special appreciation to my advisor Professor Wang- Long Li (£ ai JfiO for his patient guidance, support and encouragement through the Ph D degree.
His advice was very important to keep me various valuable comments and suggestions throughout my research, and help me to complete this dissertation In my heart, lie not only kept me a lot of lessons and deepvision in scientific research, but also shares for me a lot social experiences, keeps me the way for how to work as a real scientist. Furthermore, I want to keep my thanks to Qie-Cb Chen -ftn 4)> Hsiang-Chin Jao (-ÍẾ & i£) and anybody in my lab, because their helps, supports, and encourage through the Ph D program at all the time. TV Table of Contents ABSTRACT.IV TABLE OF CONTENTS. V LIST OF FIGURES.
VII LIST OF TABLES.XI NOMENCLATURE •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••a XII CHAPTER 1.2 Literature Studies and Research Focus.1 Squeeze Film Damping (SFD) in MEMS Resonators.2 Thermoelastic Damping (TED) in MEMS Resonators.3 Anchor Loss in MEMS Resonators.4 Other Losses in MEMS Resonators.3 Scope of Study and Research Motivation. THE ORETICAL MODELS.1 The Modified Molecular Gas Lubrication (MMGL) Equation for SFD problem .21 2 11 The Navier-Stokes Equations.2 The Continuity Equations.3 The Reynolds Equation.4 The Modified Molecular Gas Lubrication (MMOL) Equation.5 The Poiseuille Flow Rate and Pressure Flow Factors.2 The Transverse Vibration Equation of Micro-beam Resonators.3 The Thermoelastic Damping of Micro-beam Resonators.4 The Anchor Loss of Micio-beam Resonators.1 Eigenvalue Problems Procedures.2 Derivations for Eigenvalue Problems.2 Quality Factors of Micro-beam Re senators Calculations.1 Quality Factors in the Eigenvalue Problems.2 Total Quality Factors of Micro-be am Calculations. EFFECTS OF GAS RAREFACTION ON THE QUALITY FACTORS OF MICRO-BEAM RESONATORS.2 Results and Discussion.1 Pbiseuille Flow Rate Correctors.2 Quality Fac tor of Mic ro-beam Resonator in the First Three Modes.3 Quality Factor of Micro-beam Resonator in Modes Higher Than Three. COUPLED EFFECTS OF SURFACE ROUGHNESS AND GAS RAREFACTION ON THE QUALITY FACTORS OF MICRO-BEAM RESONATORS.2 Results and Discussion.1 Pressure Flow Factors.2 Effects of Gap Film Thickness.3 Quality Fac tors, c SPU arid 83 5.4 Effects of Surface Accommodation Coefficients, ACs.86 5 2 5 Effects ofFilm Thickness Ratio.
87 5 2 6 Effects of Peklenik Number.7 Effects of Modes of Micro-Beam Resonators. CONCLUSIONS AND FUTURE WORK.2 Recommendations for Future Work.■••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••a 106 VI List of Figures Page Figure i 1 Simple vibrational structures: (a) micro-cantileverbeams [5], (b) microbridges [6], (c) thin membranes [7] 1 Figure 1.2 Controlled dynamic performance of micro-cantilever beam in high frequencies [11] 2 Figure 1.3 Different MEMS resonators from the simple structure to various degrees of structural complexities Pandey and Pratap [12] such as (a) a micro-cantilever be am [5], (b) a comb-drive folded beam [14|; (c) a proof mass gyroscope suspended bymulti pie beams, (d) a hexagonal membrane used in a capacitive micromachined ul trasound transduce r (cMUT) [15] 3 Figure 2.1 Schematic representation of dynamic vibration of the resonators underthe SFD (Nguyenand Li (2016) [32]) 22 Figure 2.2 Stresses on two or three surfaces ofa fluid element in a viscous fluid in Hamroc k [42] 23 Figure 2.3 Velocities and densities for mass flow balance through a flux volume eleme nt in two dimensions in Hamrock [42] 29 Figure 2.1 Qu versus with various combined asymmetric ACs (ff|,at) for vanous Da conditions 53 Figure 4.2 Convergence of the computed Q-factor (Qsroi due to the SFD contribution with the number ofelements N' (N, X jVjp X ỈỈN) 54 Figure 4.3 (a) Real part of complex eigenvalue (damping factor (6 )) versus with various ambient pressure for different modes of the resonator by the SFD, (b) Imaginary part of complex eigenvalue (resonant frequency(úJ„)) versus with various ambient pressure for different modes of the re senator by the SFD 58 Figure 4.4 The Q-factor is plotted with both the SFD, TED and anchor loss VII for various re sonant frequencies.5 (a) The Q-factorbythe SFD (versus ambient pressure (or Do) for various modes of the resona tor with a =0.0; (h) The total Q-factor(Pr) bythe coupled SFD, TED and anchor loss versus ambient pressure (or Do) for various modes of the resonator with a =0.7 and ex=10 62 Figure 4.6 (a) Weighting of SFD () plotted with various ambient pressure or Da for different modes of the re sona tor and ACs (a, = dij = 1.0); (b) Weighting of SFD () plotted with various ambient pressure or for different modes of the resonator and ACs (a, = (Xj - 0.7 Weighting of s FD () are plotted with various symme tnc AC’s (a, = (Xj) for different modes of the resonator Ổ6 Figure 4.8 Weighting of SFD () are plotted with various asymmetric ACs(a|;izj forthe 2nd mode of the resonator Ổ7 Figure 4.1 A schematic diagram of the two squeezed rough surfaces (Lietal.3 Pressure flow factor (versus Peklenik number (Y) with ch fie rent film thickness ratios Hg (=3,4, 6) for (a) various ACs (<x, = <Zj)atD0 = 10; (b) various Do at ACs(ơ, = <z2=0.4 Specific Pekleruk number (Yi) fo* (i** =1 ) versus Da for various ACs (<z( = Oj) 20 Figure 5 5 Imaginary parts (resonant frequency ứ>n = 2^fn), <b> real parts (damping factor, <5 ) of complex eigenvalue ( z ); (c) Resultant Q-factor of the SFD (Qgrt = CVB fQ 6)) ve rsus ambient pressure for various hf, (=4,6, 8,10 pm) with three kinds of Pekleruk number (Y =9,1,1/9) 83 Figure 5.7 Ratio of Qgro K.Qĩpd) versus film thickness ratios (Hg) for Peklenik number, Y (=9,1,1/9) and ACs (a, = Oj) conditions for (a) Dfl=0.l, 3rdmode,(d) D0=l, 1st mode, (e) j?0=l, 2nd mode, (f) I{|=1, 92 3rd mode, (g) Da=10,1st mode of the resonator Figure 5.TOOf* in the 2U mode versus Peklemk number (x) for various ACs (at = <Xj) 9Ố Figure 5.9 fôtio of (b) ratio of ổr versus Peklenik number (x) for different modes of the resonator 98 K Figure 5 10 Weighting of SFD (KỸím(%)) versus ambient pressure or Do for various modes of the resonator with different Peklemk number (r =9,1,1 /9) for (a) ACs ((X, = a, = i .2) X List of Tables Page Table 4.1 A rectangular micro-beam parameters and gas film operating parameters 52 Table 4.2 Comparison of the damping ratio (^ = l/2p) results and (%) Errors with Pande y and Pratap (2007) [5] 56 Table 4.3 Q-factorversus with the TED (Ôxd) with different flexural modes of micro-beam resonator 58 Table 4.4 Q-factor versus with the anchor loss () with different flexural modes of the resonator 59 Table 5.1 A rectangular micro-beam parametersand gas film operating parameters 75 Table 5.2 Resultant Q-factor of SFD for smooth cases, (Ổszo)12 calculated with diffe rent gas rare faction conditions of Do (=0.0) for various modes of resonator conditions Table 5.3 Resultant Q-fac tor of TED (Qrsa) are calculated for various resonant frequenc les (/„) of the resonator 85 Table 5.