Several approaches for calculating support loss in micro-electro-mechanical system resonators undergoing in-plane vibration are reviewed. In each of them, the support is approximated as a semi-infinite domain. The first approach is analytical and models the support as a semi-infinite thin plate. This is compared with two different finite element approaches that introduce artificial boundaries to their finite domain. In order to absorb outgoing waves and model the infinite support, a perfectly matched layer method and the use of infinite elements are considered. Simple test cases are studied and the results for the support losses predicted by the different methods are compared. It is shown that each of the methods yields similar trends. Using the developed analytical model, a parametric study is performed on the support losses of a ring-based resonator. General strategies for improving the quality factor by reducing support losses are provided.
LifshitzR.RoukesM. L.Thermoelastic damping in micro- and nano-mechanical systems. Phys. Rev. B, 2000, 61(8), 5600–5609.
2.
YiY. B.Geometric effects on thermoelastic damping in MEMS resonators. J. Sound Vibr., 2008, 309, 588–599.
3.
YasumuraK. Y.StoweT. D.ChowE. M.PfafmanT.KennyT. W.StipeB. C.RugarD.Quality factors in micron- and submicron-thick cantilevers. J. Microelectromech. Syst., 2000, 9(1), 117–125.
4.
ChandorkarS. A.AgarwalM.MelamudR.CandlerR. N.GoodsonK. E.KennyT. W.Limits of quality factor in bulk-mode micromechanical resonators. In Proc. IEEE MEMS 2008, Tucson, USA, January 2008, pp. 74–77.
5.
TabrizianR.Rais-ZadehM.AyaziF.Effect of phonon interactions on limiting the f.Q product of micromechanical resonators. In Proc. Transducers 2009, Denver, USA, January 2009, pp. 2131–2134.
6.
YangJ.OnoT.EsashiM.Investigating surface stress: surface loss in ultrathin single-crystal silicon cantilevers. J. Vac. Sci. Technol., 2001, 19(2), 551–556.
7.
HenryJ. A.WangY.HinesM. A.Controlling energy dissipation and stability of micromechanical silicon resonators with self-assembled monolayers. Appl. Phys. Lett., 2004, 84(10), 1765–1767.
8.
LangdonR. M.Resonator sensors – a review. J. Phys. E: Sci. Instrum., 1985, 18(2), 103–115.
9.
HaueisM.DualJ.BuserR.A mechanical isolation of a bending resonator. Sens. Actuators A, 2006, 128(2), 257–264.
10.
JudgeJ. A.PhotiadisD. M.VignolaJ. F.HoustonB. H.JarzynskiJ.Attachment loss of micromechanical and nanomechanical resonators in the limits of thick and thin support structures. J. Appl. Phys., 2007, 101(1), 013521.
11.
ParkY. H.ParkK. C.High-fidelity modeling of MEMS resonators – Part I: Anchor loss mechanisms through substrate. J. Microelectromech. Syst., 2004, 13(2), 238–247.
12.
ParkY. H.ParkK. C.High-fidelity modeling of MEMS resonators – Part II: Coupled beam-substrate dynamics and validation. J. Microelectromech. Syst., 2004, 13(2), 248–257.
13.
HaoZ.ErbilA.AyaziF.An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations. Sens. Actuators A, 2003, 109, 156–164.
14.
MillerG. F.PurseyH.The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid. Proc. R. Soc. Lond., 1954, 223, 521–541.
15.
JimboY.ItaoK.Energy loss of a cantilever vibrator. J. Horolog. Inst. Jpn., 1968, 47, 1–15, (in Japanese).
16.
CrossM. C.LifshitzR.Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems. Phys. Rev. B, 2001, 64(8), 1–22.
17.
HaoZ.XuY.Vibration displacement on substrate due to time-harmonic stress sources from a micromechanical resonator. J. Sound Vibr., 2009, 322, 196–215.
18.
ChouvionB.Vibration transmission and support loss in MEMS sensors. PhD Thesis, Univ. Nottingham, UK, 2010.
19.
GraffK. F.Wave motion in elastic solids, 1975 (Ohio State University Press, Ohio, USA).
20.
ChouvionB.FoxC. H. J.McWilliamS.PopovA. A.In-plane free vibration analysis of combined ring-beam structural systems by wave propagation. J. Sound Vibr., 2010, 329(24), 5087–5104.
21.
ChouvionB.PopovA. A.McWilliamS.FoxC. H. J.Vibration modelling of complex waveguide structures. Comput. Struct., 2011, 89, 1253–1263.
22.
EngquistB.MajdaA.Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 1977, 31(139), 629–651.
23.
GivoliD.PatlashenkoI.KellerJ. B.Discrete Dirichlet-to-Neumann maps for unbounded domains. Comput. Method. Appl. M., 1998, 164, 173–185.
24.
AstleyR. J.Infinite elements for wave problems: a review of current formulations and an assessment of accuracy. Int. J. Numer. Methods Eng., 2000, 49(7), 951–976.
25.
ChengA. H. D.ChengD. T.Heritage and early history of the boundary element method. Eng. Anal. Bound. Elem., 2005, 29(3), 268–302.
26.
BérengerJ. P.A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys., 1994, 114, 185–200.
27.
KosloffR.KosloffD.Absorbing boundaries for wave propagation problems. J. Comp. Phys., 1986, 63(2), 363–376.
28.
BindelD. S.GovindjeeS.Elastic PMLs for resonator anchor loss simulation. Int. J. Numer. Meth. Eng., 2005, 64(6), 789–818.
LysmerJ.KuhlemeyerR. L.Finite dynamic model for infinite media. J. Eng. Mech. div.-ASCE, 1969, 95(4), 859–877.
33.
Dassault Systèmes, Abaqus Online documentation Abaqus Theory manual Version 6.7, 2007.
34.
GriggH. T. D.Towards a parametrically pumped xylophone microbar magnetometer: design optimisation of xylophone bar resonator. Proc. IMAPS DPC, 2011(In Press).
35.
WongS. J.Thermoelastic damping in MEMS ring resonators. PhD Thesis, Univ. Nottingham, UK, 2005.
36.
WuC. M.LundbergB.Reflection and transmission of the energy of harmonic elastic waves in a bent bar. J. Sound Vibr., 1996, 190(4), 645–659.
37.
HaoZ.AyaziF.Support loss in micromechanical disk resonators. In Proc. IEEE MEMS 2005, Miami, January 2005, pp. 137–141.
38.
DuwelA.CandlerR. N.KennyT. W.VargheseM.Engineering MEMS resonators with low thermoelastic damping. J. Microelectromech. Syst., 2006, 15(6), 1437–1445.
39.
HaoZ.PourkamaliS.AyaziF.VHF single-crystal silicon elliptic bulk-mode capacitive disk resonators—Part I: Design and modeling. J. Microelectromech. Syst., 2004, 13(6), 1043–1053.
40.
AbdelmoneumM. A.DemirciM. U.NguyenC. T.-C.Stemless wine-glass-mode disk micromechanical resonators. In Proc. IEEE MEMS 2003, Kyoto, Japan, 19–23 January 2003, pp. 698–701.
41.
FoxC.McWilliamS.EleyR.FellC.Development of multi-axis rate sensors based on vibrating silicon ring structures. In Proc. of NATO RTO Symposium on Inertial Sensor Technology, Istanbul, Turkey, 2002, pp. 13.1–13.12.
