On the quality factors of mems resonators

@ if- & A # i &X Dissertation for Doctor of Philosophy On die Quality’ Factors of MEMS Resonators Student: Chi-Cuong Nguyen Ẩ' 4s * Kí âì Advisor: Wang-Long Li National Cheng Kung University Department of Materials Science and Engineering Tainan, Taiwan, R.o.c May 2017 t 106 4- 05 24 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 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 &&&& iM ỹ*E Ỹ.M íL5ỉí?ẩttíit(MMGO ’ M ’ ili&4ĩ %^^(FEM)A44«Hfi.|5l5Ì4íR«r>^ĩ MMQL fĩK ’ ft ôtf^ế »1 in Eq (2.83) can be satisfied decreases significantly (anchor loss for various resonant mode conditions The increases) as the mode of resonator increases (higher vibrational frequency) Table 4.4 Q-fac tor versus with the anchor loss (i? riA) with different flexural modes of the resonator iutural frt q /„ CW mod* £h^p* fir 2nrf 3rd 4?ft 5th 6ih 7th Wi 9th 10íA —— — w - •* - * • • — * -*- *■ * -A * *•-■* *• < , K - » ^»1 Ổorr* »4 550631 1394107 879.17 314 160 52 42875 22107 0.1211 2081 43792.89 00193 000691 0.00353 0.173 0D64 274278.3 767380.4 0033 1502159 0.00213 0.00143 00102 7.75t-4 0020 0013 0009 0007 2479854 3698408 5155430 6847967 9723 65 20 46.77 352 05e-4 0006 8772506 27.48 4019 86e-4 0004 10925000 2207 2679 115896 13398 8708 6029 4689 In Fig 44, the ^.TO(the external SFD only) are plotted as functions of resonant frequency for various ambient pessure (orD0 =0.03,0.1,1, 10 and 100) Qru, (Table 4.3) and ổ,Rrt (Table 4) are also plotted as functions of resonant frequency, since they are not functions of gas rare faction parameters (ACs and Do) Qspj) increases as Dt decreases ox the mode increases, Qm and increase as the mode increases They are very close between mode to mode for the present micro-beam resonators Under the first mode the SFD IS totally dominant Thus, tike condition, we have from Eq (4.4), since the Q-total (£?r) can be only determined by the and the if Ó"* are negligible in calculating of the Qr From the plot, we can find cross points 59 nearby the resonant frequencies in range of mode (2nd, 3id, 4th, 5th, 6th mode) We can tune the operation condition ( £>□) to find the plot of the Qrrjj arid find the cross points at die resonant frequencies, then we have approximately = Tte ^'iree = sources of damping (SFD, TED and anchor loss) are equal contribution in calculating the total Q-factor in appropriate mode arid gas rarefaction conditions ĩhus, we can have a good opportunity to nnpoove the Q-factor by controlling of various gas rarefaction (£•,, ACs) in various modes icon inrrm 1OECD immm Frequency (Hz) Figure 4.4 The Q-factor are plotted with both the SFD, TED and anchor loss for various resonant frequencies The effect ofvibrational modes of the resonator is a manifestation of the contribution of the TED and the anchor loss on the Q-factors of the resonator in low ambient pressure conditions In Fig 4.5(a), the increases as the ambient pressure decreases or ACs (a, = a,) decreases and the mode increases In Fig 4.5(b), the mere ares as the mode increases in higher ambient pressure (or higher Dj), whereas the Qr decreases significantly 60 due to the increase of the TED and the anchor toss in higher modes and in lower ambient pressure (tower Do) The Qr of the 1st mode and the 2nd mode are slightly affected by TED and anchor loss We can conclude that the SFD dominates in 1st and 2nd modes, in higher ambient pressure and larger ACs (a,,a2) conditions, whereas the TED and the anchor toss are dominant in higher re sonant modes and tower ambie nt pre ssure conditions in the first two modes of the Thus, the total Q-factor IS primarily determined by the resonator, high ambient pressure conditions and by all contribution of the Qrz,, the Qny-, and the in higher modes of the micro-beam resonator Tins result IS significant because the Q-factor can go up at the 2nd mode and then down quickly as mode increases at tower ambient pressure conditions Then, we can easily optimize the Q-faclor of the resonator by changing the operating conditions from Fig 5(b) For example, we can choose the 2nd mode if we ward to operate the micro-beam resonator with ambient pressure 100 Pa, and we can choose the 4th mode if we want to operate the micro-beam resonator with ambient pressure 1000 Pa OOI Ũ.1 IO t 1tu ■> 1Í1III1I1 11- cq-0^-0 inmD V Frzce s» rcôe ãmoat > frcde lLULU Q ft 2- rrôlr niQtf tnm txu im 1Ũ 1[HD Ambient press ure(Pd) 61 lULXlXi □ 0D1 Itu 10 versus ambient pressure (or £>g) fox various Figure 4.5 (a) The Q-factorby the SFE> ( modes of the resonator with a =0.7 and £X=1.0,(b) The total Q-factor (£^.) by the coupled SFD, TED and anchor loss versus ambient pressure (or D#) for various modesofthe resonator with a =0.7 and a=1.0 4.2.4 weightiiig of SFD To understand the contributions of different damping sources on the Q-factor of a micro-beam resonator, we need to examine the percentage of dissipation mechanisms of damping, e.g the SFD ((&ro)_|), the TED ((QrujY") and the anchor loss ((£,J'1), for various modes of the resonator in various gas rarefaction regions (Do, O|, at) Weighting of SFD IS defined as the contribution of the external SFD ((ikro)’1) on the total damping, contributed by SFD, TED and anchor loss Tlbk 43 62 ) i.e 44 ” c“ obtain tlie total Q-factor by Qr = (Qw + X(1 - In Fig 4.6, the ỈỈỈỊmC/i) IS plotted as functions of ambient pressure or Dd for various flexural modes of the resonators The ĩWfjPĨ>(40 %) is an example of weighting of the SFEi are equal to 40% of Qr (íQsrũ)"' - 40% ) 1^ie results show that the ỈWíre(%) decreases as ambient pressure decreases, Do decreases or mode of the resonators increases Since flie SFD decreases and its contribution reduces with higher gas rarefaction (lower D(j) and lower modes of tile resonator As shown in Fig 4.6(a), die K4rao(%) is still high and slightly changed in the 1st mode and the 2nd mode of the resonator or m higher Dd regions, in which the SFD IS very dominant in lower mode and gas rarefied regions Whereas Hl Fig 46(b), the is reduced obviously as the ACs decrease to = 0.5) and then intersects with W?OT(40 %) due to the SFD reduces as the ACs decrease Thus, in higher regions of W?fZD(40 %), it’s clear to see that the SFD is more dominant with respect to the TED and anchor loss (higher #?„(%)), then Jf4tTO(%) slowly changed with ambient pressure and the mode of the micro-beam resonator While in tower regions of W4rro(40 %), the TED and anchor loss are dominant and WffTO(%) strongly reduced as ambient pressure decreases, the ACs decrease and the mode of the resonator increases 63 A m b ie n t p re s s u re (P a ) rao lữ O W‘w\ 'OJS JO Bww6ị»m (tó“MTaJSỈ» 6uịWBị»m Figure 4.6 (a) Weighting of SFD () plotted with various ambient pressure or Dg for different modes of the resonator and ACs (a, = a, = 0),(b) Weighting ofSFD plotted with various ambient pressure or D„ for different modes of the resonator and ACs () decreases as symmetric ACs «(«! = &,) decreases The decrease of with symmetric ACs afa, =(Xj) IS due to the momentum exchange of tile collision of gas molecules and the solid surfaces decreases as ACs departs from a = 1.0 to a = 0.1, thus results in lower gas flow restriction, higher gas rarefaction effect and lower SFD contribution Furthermore, the decrease of with symmetric ACs a(a, = Oj) is slightly in die 1st mode of the resonator because of then dominant SFD in the fundamental mode of the resonator, while ttie decrease of with the symmetric ACs a(a, = otị) enhances obviously in higher modes of resonator in which the effect of external SFD reduces and the effect of internal TED and anchor loss become dominant, the contribution of SFD ((Ổro)”1) can be obviously reduced on the total damping 65 Figure 4.7 weighting of s FD () are plotted with various symmetric ACs (a, = a,) for different modes of the resonator In Fig 4.8, the effects of asymmetric surface ACs (a„at) (different ACs on two squeeze solid surfaces (