Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction


This paper discussed the effect of environmental conditions (moisture and temperature) on the quality factors (Q-factor) of micro-electro-mechanical systems (MEMS) cantilever beam resonators in wide range of gas rarefaction (pressure (p), and accommodation coefficients (ACs)), and flexural mode of resonator. The modified molecular gas lubrication (MMGL) equation is applied for modeling the dominant squeeze film damping (SFD) problem on the quality factor of MEMS cantilever beam resonators to discuss the effect of environmental conditions. The external SFD and the internal structure damping (thermoelastic damping) and support loss) are accurately taken into account. Effective viscosity, which is ratio of dynamic viscosity and Poiseuille flow rate of moist air, is utilized to modify the MMGL equation to consider the environmental effects of moisture and temperature in gas rarefaction. In low pressures, mean free path changes more significantly with relative humidity and temperature than that of dynamic viscosity of moisture in gas rarefaction. Thus, effect of environmental conditions such as moisture and temperature must be discussed to improve Q-factors of MEMS cantilever beam resonators in wide range of gas rarefaction (p and ACs) and flexural modes of resonator. The results showed that Q-factor of SFD decreases significantly as moisture and temperature increase at higher gas rarefaction (lower p, and ACs), while Q-factor of SFD decreases and then increases slightly as moisture and temperature increase at lower gas rarefaction (higher p, and ACs). The total Q-factor is highly sensitive to the relative humidity and temperature in higher gas rarefaction (lower p and ACs) and lower flexural modes of resonator.

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  1. 1.

    Binnig, G., & Quate, C. F. (1986). Atomic force microscope. Physical Review Letters, 56(9), 930–934.

    Article  Google Scholar 

  2. 2.

    Thundat, T., Warmack, R. J., Chen, G. Y., & Allison, D. P. (1994). Thermal and ambient-induced deflections of scanning force microscope cantilevers. Applied Physics Letters, 64, 2894–2896.

    Article  Google Scholar 

  3. 3.

    Takahashi, H., Dung, N. M., Matsumoto, K., & Shimoyama, I. (2012). Differential pressure sensor using a piezoresistive cantilever. Journal of Micromechanics and Microengineering, 22, 055015–055021.

    Article  Google Scholar 

  4. 4.

    Chennippan, M., Bhaskaran, P. E., Adhulrasheed, I. S. K., Subramaniam, T., & Govindasamy, R. (2020). Vibration signals based bearing defects identification through online monitoring using LABVIEW. Journal Européen des Systèmes Automatisés, 53, 187–193.

    Article  Google Scholar 

  5. 5.

    Priyanka, E. B., Maheswari, C., Ponnibala, M., & Thangavel, S. (2019). SCADA based remote monitoring and control of pressure & flow in fluid transport system using IMCPID controller. Advances in Systems Science and Applications, 03, 140–162.

    Google Scholar 

  6. 6.

    Priyanka, E. B., Thangavel, S., & Pratheep, V. G. (2020). Enhanced digital synthesized phase locked loop with high frequency compensation and clock generation. Sensing and Imaging, 21–43.

  7. 7.

    Baller, M. K., Lang, H. P., Fritz, J., Gerber, Ch., Gimzewski, J. K., Drechsler, , et al. (2000). A cantilever array-based artificial nose. Ultramicroscopy, 82, 1–9.

    Article  Google Scholar 

  8. 8.

    Lang, H. P., Hegner, M., & Gerber, C. (2005). Cantilever array sensors. Materialstoday, 8(4), 30–36.

    Google Scholar 

  9. 9.

    Pratheep, V. G., Priyanka, E. B., & Prasad, P. H. (2019). Characterization and analysis of natural fibre-rice husk with wood plastic composites. IOP Conf. Series: Materials Science and Engineering, 561, 012066.

  10. 10.

    Gupta, A., Akin, D., & Bashir, R. (2004). Single virus particle mass detection using microresonators with nanoscale thickness. Applied Physics Letters, 84(11), 1976–1978.

    Article  Google Scholar 

  11. 11.

    Tamayo, J., Humphris, A. D. L., Malloy, A. M., & Miles, M. J. (2001). Chemical sensors and biosensors in liquid environment based on microcantilevers with amplified quality factor. Ultramicroscopy, 86, 167–173.

    Article  Google Scholar 

  12. 12.

    Cyril, V., Isabelle, D., Stephen, M. H., Fabien, J., & Andreas, H. (2008). Analysis of resonating microcantilevers operating in a viscous liquid environment. Sensors and Actuators A, 141, 43–51.

    Article  Google Scholar 

  13. 13.

    Singh, P., & Yadava, R. D. S. (2020). Stochastic resonance induced performance enhancement of MEMS cantilever biosensors. J. Phys. D: Appl. Phys. 53(46).

  14. 14.

    Fischeneder, M., Kucera, M., Hofbauer, F., Pfusterschmid, G., Schneider, M., & Schmid, U. (2018). Q-factor enhancement of piezoelectric MEMS resonators in liquids with active feedback. Sensor Actuat B-Chem., 260, 198–203.

    Article  Google Scholar 

  15. 15.

    Blake, N. J., & Raj, M. (2012). Biosensing using dynamic-mode cantilever sensors: A review. Biosensors & Bioelectronics, 32(1), 1–18.

    Article  Google Scholar 

  16. 16.

    Schneider, M., Pfusterschmied, G., Patocka, F., & Schmid, U. (2020). High performance piezoelectric AlN MEMS resonators for precise sensing in liquids. Elektrotechnik & Informationstechnik., 137(3), 121–127.

    Article  Google Scholar 

  17. 17.

    Amin, E., Habib, B. G., & Mousa, S. (2019). A Novel biosensor based on micromechanical resonator array for lab-on-a-chip applications. Sensing and Imaging, 20, 39.

    Article  Google Scholar 

  18. 18.

    Hosaka, H., Itao, K., & Kuroda, S. (1995). Damping characteristics of beam-shaped micro-oscillators. Sensors and Actuators A: Physical, 49(1–2), 87–95.

    Article  Google Scholar 

  19. 19.

    Zener, C. (1937). Internal friction in solids I theory of internal friction in reeds. Physical Review, 52(3), 230–235.

    MATH  Article  Google Scholar 

  20. 20.

    Zener, C. (1938). Internal friction in solids II general theory of thermoelastic internal friction. Physical Review, 53(1), 90–99.

    MATH  Article  Google Scholar 

  21. 21.

    Lifshitz, R., & Roukes, M. L. (2000). Thermoelastic damping in micro- and nanomechanical systems. Physical Review B, 61(8), 5600–5609.

    Article  Google Scholar 

  22. 22.

    Zhou, H., Li, P., & Zuo, W. (2016). Thermoelastic damping in microwedged cantilever resonator with rectangular cross-section. In: IEEE 2016 int. conf. on mechatronics and automation (ICMA), Harbin, China, 1590–1595.

  23. 23.

    Hao, Z., Erbil, A., & Ayazi, F. (2003). An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations. Sensors and Actuators A: Physical, 109(1–2), 156–164.

    Article  Google Scholar 

  24. 24.

    Jandak, M., Neuzil, T., Schneider, M., & Schmid, U. (2016). Investigation on different damping mechanisms on the Q factor of MEMS resonators. Procedia Engineering, 168, 929–932.

    Article  Google Scholar 

  25. 25.

    Yang, J., Ono, T., & Esashi, M. (2002). Energy dissipation in submicrometer thick single-crystal silicon cantilevers. Journal of Microelectromechanical Systems, 11(6), 775–783.

    Article  Google Scholar 

  26. 26.

    Kim, B., Hopcroft, M. A., Candler, R. N., Jha, C. M., Agarwal, M., Melamud, R., et al. (2008). Temperature dependence of quality factor in MEMS resonators. Journal of Microelectromechanical Systems, 17(3), 755–766.

    Article  Google Scholar 

  27. 27.

    Ghaffari, S., Ng, E. J., Ahn, C. H., Yang, Y., Wang, S., Hong, V., et al. (2015). Accurate modeling of quality factor behavior of complex silicon MEMS resonators. Journal of Microelectromechanical Systems, 24(2), 276–288.

    Article  Google Scholar 

  28. 28.

    Lee, J. W. (2011). Analysis of fuid-structure interaction for predicting resonant frequencies and quality factors of a microcantilever on a squeeze-film. Journal of Mechanical Science and Technology, 25(5), 3005–3013.

    Article  Google Scholar 

  29. 29.

    Bao, M., & Yang, H. (2007). Squeeze film air damping in MEMS. Sensors and Actuators A: Physical, 136(1), 3–27.

    MathSciNet  Article  Google Scholar 

  30. 30.

    Pandey, A. K., & Pratap, R. (2007). Effect of flexural modes on squeeze film damping in MEMS cantilever resonators. Journal of Micromechanics and Microengineering, 17(12), 2475–2484.

    Article  Google Scholar 

  31. 31.

    Kim, S. J., Dean, R., Jackson, R. L., & Flowers, G. T. (2011). An investigation of the damping effects of various gas environments on a vibratory MEMS device. Tribology International, 44(2), 125–133.

    Article  Google Scholar 

  32. 32.

    Burg, T. P., & Manalis, S. R. (2003). Suspended microchannel resonators for biomolecular detection. Applied Physics Letters, 83(2), 2698–2700.

    Article  Google Scholar 

  33. 33.

    Nguyen, C. C., & Li, W. L. (2016). Effect of gas rarefaction on the quality factors of micro-beam resonators. Microsystem Technologies, 23, 3185–3199.

    Article  Google Scholar 

  34. 34.

    Nguyen, C. C., & Li, W. L. (2016). Effects of surface roughness and gas rarefaction on the quality factor of micro-beam resonators. Microsystem Technologies, 23(8), 3489–3504.

    Article  Google Scholar 

  35. 35.

    Nguyen, C. C., & Li, W. L. (2017). Influences of temperature on the quality factors of micro-beam resonators in gas rarefaction. Sensors and Actuators A: Physical, 261, 151–165.

    Article  Google Scholar 

  36. 36.

    Nguyen, C. C., Ngo, V. K. T., Le, H. Q., & Li, W. L. (2018). Influences of relative humidity on the quality factors of MEMS cantilever resonators in gas rarefaction. Microsystem Technologies, 25, 2767–2782.

    Article  Google Scholar 

  37. 37.

    Hosseinian, E., Theillet, P. O., & Pierron, O. N. (2013). Temperature and humidity effects on the quality factor of a silicon lateral rotary micro-resonator in atmospheric air. Sensors and Actuators A: Physical, 189, 380–389.

    Article  Google Scholar 

  38. 38.

    Nieva, P. M., McGruer, N. E., & Adams, G. G. (2006). Design and characterization of a micromachined fabry-perot vibration sensor for high-temperature applications. Journal of Micromechanics and Microengineering, 16(12), 2618–2631.

    Article  Google Scholar 

  39. 39.

    Hosseinzadegan, H., Pierron, O. N., & Hosseinian, E. (2014). Accurate modeling of air shear damping of a silicon lateral rotary micro-resonator for MEMS environmental monitoring applications. Sensors and Actuators A: Physical, 216, 342–348.

    Article  Google Scholar 

  40. 40.

    Jan, M. T., Ahmad, F., Hamid, N. H. B., Khir, M. H. B. M., Shoaib, M., & Ashraf, K. (2016). Experimental investigation of moisture and temperature effects on resonance frequency and quality factor of CMOS-MEMS paddle resonator. Microelectronics Reliability, 63, 82–89.

    Article  Google Scholar 

  41. 41.

    Hwang, C. C., Fung, R. F., Yang, R. F., Weng, C. I., & Li, W. L. (1996). A new modified Reynolds equation for ultrathin film gas lubrication. IEEE Transactions on Magnetics, 32(2), 344–347.

    Article  Google Scholar 

  42. 42.

    Li, W. L. (1999). Analytical modelling of ultra-thin gas squeeze film. Nanotechnology, 10(4), 440–446.

    Article  Google Scholar 

  43. 43.

    Li, W. L. (2002). A database for couette flow rate considering the effects of non-symmetric molecular interactions. Journal of Tribology, 124(4), 869–873.

    Article  Google Scholar 

  44. 44.

    Li, W. L. (2003). A database for interpolation of Poiseuille flow rate for arbitrary Knudsen number lubrication problems. Journal of the Chinese Institute of Engineers, 26(4), 455–466.

    Article  Google Scholar 

  45. 45.

    Li, W. L. (2004). Modeling of head/disk interface—an average flow model. Tribology Letters, 17, 669–676.

    MathSciNet  Article  Google Scholar 

  46. 46.

    Li, W. L. (2008). Squeeze film effects on dynamic performance of MEMS μ-mirrors-consideration of gas rarefaction and surface roughness. Microsystem Technologies, 14(3), 315–324.

    Article  Google Scholar 

  47. 47.

    Hasan, M. H. (2018). Influence Of Environmental Conditions On The Response Of MEMS Resonators, Dissertation, University of Nebraska.

  48. 48.

    Morvay, Z. K., & Gvozdenac, D. D. (2008). Applied Industrial Energy and Environmental Management. in: Fundamentals for analysis and calculation of energy and environmental performance, (pp. 1–5), Wiley, Ltd.

  49. 49.

    Kreith, F., & Goswami, D. Y. (2005). The CRC HANDBOOK of Mechanical engineering. In: CRC Press LLC, (pp. 1385).

  50. 50.

    Tan, Z. (2014). Air pollution and greenhouse gases. Springer Science + Business Media, (pp. 33–34), Singapore.

  51. 51.

    ASHRAE. (2001). The 2001 ASHRAE Fundamentals Handbook. (pp. 6.2).

  52. 52.

    Saraireh M. (2012). Heat transfer and condesation of water vapour from humid air in compact heat exchangers. Doctor of Philosophy, Victoria University, (pp. 67), Footscray.

  53. 53.

    Leissa, A. W. (1969). Vibration of Plates, In: NASA, (pp. 1–6), Washington DC.

  54. 54.

    Nayfeh, A. H., & Younis, M. I. (2004). A new approach to the modeling and simulation of fexible microstructures under the effect of squeeze flm damping. Journal of Micromechanics and Microengineering, 14, 170–181.

    Article  Google Scholar 

  55. 55.

    COMSOL Multiphysics 5.5. (2021). Thermoelastic damping in a MEMS resonator, https://www.comsol.com/model/thermoelastic-damping-in-a-mems-resonator-1439. License Date to February 1, 2021.

  56. 56.

    Matthew, A. H., William, D. N., & Thomas, W. K. (2010). What is the young’s modulus of silicon? Journal of Microelectromechanical Systems, 19(2), 229–238.

    Article  Google Scholar 

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This research was supported by the Institute for Computational Science and Technology (ICST), Contract Number: 08/2019/HĐ-KHCNTT in October 24th, 2019 and series number: 082019-311.

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Appendix A

In Fig. 10, the Q factor of TED (QTED) is calculated as function of temperature (T) for various flexural modes of cantilever beam resonator. The result showed that QTED decreases as T increases for different modes of resonator. Also, QTED decreases more significantly as flexural modes of resonator increases because the TED increases with T and becomes dominantly in higher flexural mode of resonator. The calculated results of QTED from the present LR model [11] (Eq. 15 in [35]) showed good agreement with those obtained results from the Zener models [19, 20] (Eq. 14 in [35]), and those obtained results with COMSOL Multiphysics 5.5 [55] (Sect. "Quality Factors of MEMS Cantilever Beam Resonators" in [35]) in wide range of temperatures and resonator modes. Thus, the obtained results of QTED from the LR model, which are used in the present analysis, can be applied to calculate the total Q factor (QT) of MEMS cantilever beam resonators in wide range of temperature and flexural mode of resonator.

Fig. 10

The Q-factor of TED (QTED) versus temperature (T) for different flexural mode of resonator

Appendix B

In Table 2, \(Q_{\sup }\) is calculated by the model of Hao et al. [23] (Eq. 18 in [35]) for various flexural mode of MEMS cantilever beam resonator. The validation of this model has been proved by the assumption that the width of cantilever beam (\(w_{p}\)) is much less than the transverse elastic wavelength (\(\lambda_{T}\)) (\(\lambda_{T} /w_{p} > > 1\)). The result showed that \(Q_{\sup }\) decreases significantly as the mode of resonator increases because the support loss becomes a dominant source of energy loss on MEMS resonators in higher flexural mode of resonator. Thus, the results of \(Q_{\sup }\) can be used to calculate the total Q-factor (\(Q_{T}\)) of MEMS cantilever beam resonator in wide range of flexural mode of resonator conditions.

Table 2 The quality factor of support loss (\(Q_{\sup }\)) for various flexural mode of MEMS cantilever beam resonator

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Phan, M.T., Trinh, X.T., Le, Q.C. et al. Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction. Sens Imaging 22, 6 (2021). https://doi.org/10.1007/s11220-020-00329-9

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  • Quality factor
  • MEMS cantilever beam resonators
  • Temperature
  • Relative humidity
  • Gas rarefaction
  • Flexural mode of resonator