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Quality Factor

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Book cover Fundamentals of Nanomechanical Resonators

Abstract

The quality factor defines the rate with which a nanomechanical resonator dissipates energy. Low energy loss, i.e. a high quality factor, is desirable for most applications of nanomechanical resonators. In this chapter, the three main sources of energy loss in nanomechanical resonators are presented. Energy can be lost (1) to the surrounding medium, which can be a liquid or a gas, (2) through the clamping to the substrate via elastic waves, or (3) through dissipation mechanisms that are intrinsic to the resonator. Medium interaction losses can readily be circumvented by operation in vacuum, and clamping losses can be minimized by an optimized resonator design. This typically leaves intrinsic losses as the limiting mechanism defining the maximal obtainable quality factor. Intrinsic losses consist of material friction and fundamental loss mechanisms such as thermoelastic loss and phonon–phonon interaction loss. Generally, intrinsic losses can be reduced by decreasing the temperature. Damping dilution reduces the effect of intrinsic loss in resonators under tensile stress, resulting in quality factors up to several million even at room temperature.

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References

  1. B.C. Stipe, H.J. Mamin, T.D. Stowe, T.W. Kenny, D. Rugar, Noncontact friction and force fluctuations between closely spaced bodies. Phys. Rev. Lett. 87 (9), 096801 (2001)

    Google Scholar 

  2. A.N. Cleland, M.L. Roukes, External control of dissipation in a nanometer-scale radiofrequency mechanical resonator. Sens. Actuators A Phys. 72 (3), 256–261 (1999)

    Article  Google Scholar 

  3. K. Schwab, Spring constant and damping constant tuning of nanomechanical resonators using a single-electron transistor. Appl. Phys. Lett. 80 (7), 1276–1278 (2002)

    Google Scholar 

  4. J.E. Sader, Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 84 (1), 64 (1998)

    Google Scholar 

  5. C.A. Van Eysden, J.E. Sader, Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope: arbitrary mode order. J. Appl. Phys. 101 (4), 044908 (2007)

    Google Scholar 

  6. C.P. Green, J.E. Sader, Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 92 (10), 6262 (2002)

    Google Scholar 

  7. C.P. Green, J.E. Sader, Frequency response of cantilever beams immersed in viscous fluids near a solid surface with applications to the atomic force microscope. J. Appl. Phys. 98 (11), 114913 (2005)

    Google Scholar 

  8. M.K. Ghatkesar, T. Braun, V. Barwich, J.-P. Ramseyer, C. Gerber, M. Hegner, H.P. Lang, Resonating modes of vibrating microcantilevers in liquid. Appl. Phys. Lett. 92 (4), 043106 (2008)

    Google Scholar 

  9. J.W.M. Chon, P. Mulvaney, J.E. Sader, Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids. J. Appl. Phys. 87 (8), 3978 (2000)

    Google Scholar 

  10. C.A. Van Eysden, J.E. Sader, Resonant frequencies of a rectangular cantilever beam immersed in a fluid. J. Appl. Phys. 100 (11), 114916 (2006)

    Google Scholar 

  11. P. Enoksson, G. Stemme, E. Stemme, Fluid density sensor based on resonance vibration. Sens. Actuators A Phys. 47, 327–331 (1995)

    Google Scholar 

  12. T.P. Burg, M. Godin, S.M. Knudsen, W. Shen, G. Carlson, J.S. Foster, K. Babcock, S.R. Manalis, Weighing of biomolecules, single cells and single nanoparticles in fluid. Nature 446 (7139), 1066–1069 (2007)

    Article  Google Scholar 

  13. J. Lee, W. Shen, K. Payer, T.P Burg, S.R. Manalis, Toward attogram mass measurements in solution with suspended nanochannel resonators. Nano Lett. 10 (7), 2537–2542 (2010)

    Google Scholar 

  14. D. Sparks, R. Schneider, R. Smith, A. Chimbayo, M. Straayer, J. Cripe, N. Najafi, S. Anasari, Measurement of density and chemical concentration using a microfluidic chip. Lab Chip 3 (1), 19–21 (2003)

    Google Scholar 

  15. D. Westberg, O. Paul, G.I. Andersson, H. Baltes, A CMOS-compatible device for fluid density measurements fabricated by sacrificial aluminium etching. Sens. Actuators A Phys. 73 (3), 243–251 (1999)

    Article  Google Scholar 

  16. M.F. Khan, S. Schmid, P.E. Larsen, Z.J. Davis, W. Yan, E.H. Stenby, A. Boisen, Online measurement of mass density and viscosity of pL fluid samples with suspended microchannel resonator. Sens. Actuators B Chem. 185, 456–461 (2013)

    Google Scholar 

  17. T. Burg, J. Sader, S. Manalis, Nonmonotonic energy dissipation in microfluidic resonators. Phys. Rev. Lett. 102 (22), 1–4 (2009)

    Article  MATH  Google Scholar 

  18. J.E. Sader, T.P. Burg, S.R. Manalis, Energy Dissipation in Microfluidic Beam Resonators. J. Fluid Mech. 650, 215–250 (2010)

    Google Scholar 

  19. S. Schmid, C. Hierold, Damping mechanisms of single-clamped and prestressed double-clamped resonant polymer microbeams. J. Appl. Phys. 104 (9), 093516 (2008)

    Google Scholar 

  20. C.A. Van Eysden, J.E. Sader, Frequency response of cantilever beams immersed in compressible fluids with applications to the atomic force microscope. J. Appl. Phys. 106 (9), 094904 (2009)

    Google Scholar 

  21. M. Bao, Analysis and Design Principles of MEMS Devices (Amsterdam, The Netherlands, 2005)

    Google Scholar 

  22. S.S. Verbridge, R. Ilic, H.G. Craighead, J.M. Parpia, Size and frequency dependent gas damping of nanomechanical resonators. Appl. Phys. Lett. 93 (1), 13101 (2008)

    Google Scholar 

  23. S. Schmid, B. Malm, A. Boisen, Quality factor improvement of silicon nitride micro string resonators, in 24th International Conference on Micro Electro Mechanical Systems (MEMS) (New Jersey, USA, 2011), pp. 481–484

    Google Scholar 

  24. M. Bao, H. Yang, H. Yin, Y. Sun, Energy transfer model for squeeze-film air damping in low vacuum. J. Micromech. Microeng. 12, 341–346 (2002)

    Google Scholar 

  25. R.G. Christian, The theory of oscillating-vane vacuum gauges. Vacuum 16, 175–178 (1966)

    Article  Google Scholar 

  26. P. Li, R. Hu, On the air damping of flexible microbeam in free space at the free-molecule regime. Microfluid. Nanofluid. 3, 715–721 (2007)

    Google Scholar 

  27. D.M. Photiadis, J.A. Judge, Attachment losses of high Q oscillators. Appl. Phys. Lett. 85 (3), 482–484 (2004)

    Article  Google Scholar 

  28. M.C. Cross, R. Lifshitz, Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems. Phys. Rev. B 64 (8), 85324 (2001)

    Google Scholar 

  29. S. Schmid, S. Kühne, C. Hierold, Influence of air humidity on polymeric microresonators. J. Micromech. Microeng. 19 (6), 065018 (2009)

    Google Scholar 

  30. I. Wilson-Rae, Intrinsic dissipation in nanomechanical resonators due to phonon tunneling. Phys. Rev. B 77 (24), 245418 (2008)

    Google Scholar 

  31. I. Wilson-Rae, R.A. Barton, S.S. Verbridge, D.R. Southworth, B. Ilic, H.G. Craighead, J.M. Parpia, High-Q nanomechanics via destructive interference of elastic waves. Phys. Rev. Lett. 106 (4), 47205 (2011)

    Google Scholar 

  32. D.J. Wilson, C.A. Regal, S.B. Papp, H.J. Kimble, Cavity optomechanics with stoichiometric SiN films. Phys. Rev. Lett. 103 (20), 207204 (2009)

    Google Scholar 

  33. D.J. Wilson, Cavity optomechanics with high-stress silicon nitride films, Ph.D. thesis, California Institute of Technology, 2012

    Google Scholar 

  34. S. Chakram, Y.S. Patil, L. Chang, M. Vengalattore, Dissipation in ultrahigh quality factor SiN membrane resonators. Phys. Rev. Lett. 112 (12), 127201 (2014)

    Google Scholar 

  35. S. Schmid, K.D. Jensen, K.H. Nielsen, A. Boisen, Damping mechanisms in high-Q micro and nanomechanical string resonators. Phys. Rev. B 84 (16), 165307 (2011)

    Google Scholar 

  36. Y. Tsaturyan, A. Barg, A. Simonsen, L.G. Villanueva, S. Schmid, A. Schliesser, E.S. Polzik, Demonstration of suppressed phonon tunneling losses in phononic bandgap shielded membrane resonators for high-Q optomechanics. Opt. Express 22 (6), 6810 (2014)

    Google Scholar 

  37. P.-L. Yu, K. Cicak, N.S. Kampel, Y. Tsaturyan, T.P. Purdy, R.W. Simmonds, C.A. Regal, A phononic bandgap shield for high-Q membrane microresonators. Appl. Phys. Lett. 104 (2), 023510 (2014)

    Google Scholar 

  38. M.F. Ashby, F. Ashby, Overview No. 80: on the engineering properties of materials. Acta Metallur. 37 (5), 1273–1293 (1989)

    Google Scholar 

  39. G. Fantozzi, 1.1 phenomenology and definitions. Mater. Sci. Forum 366–368, 3–31 (2001)

    Google Scholar 

  40. I.M. Ward, J. Sweeney, An Introduction to the Mechanical Properties of Solid Polymers, 2nd edn. (Wiley, London, 2004)

    Google Scholar 

  41. J. Yang, T. Ono, M. Esashi, Energy dissipation in submicrometer thick single-crystal cantilevers. J. Microelectromech. Syst. 11 (6), 775–783 (2002)

    Google Scholar 

  42. P. Mohanty, D.A. Harrington, K.L. Ekinci, Y.T. Yang, M.J. Murphy, M.L. Roukes, Intrinsic dissipation in high-frequency micromechanical resonators. Phys. Rev. B 66 (8), 85416 (2002)

    Google Scholar 

  43. L.G. Villanueva, S. Schmid, Evidence of surface loss as ubiquitous limiting damping mechanism in SiN micro- and nanomechanical resonators. Phys. Rev. Lett. 113 (227201), 1–6 (2014)

    Google Scholar 

  44. K.Y. Yasumura, T.D. Stowe, E.M. Chow, T. Pfafman, T.W. Kenny, B.C. Stipe, D. Rugar, Quality factors in micron- and submicron-thick cantilevers. J. Microelectromech. Syst. 9 (1), 117–125 (2000)

    Article  Google Scholar 

  45. A.N. Cleland, Foundations of Nanomechanics (Springer, New York, 2003)

    Book  Google Scholar 

  46. T.V. Roszhart, The effect of thermoelastic internal friction on the Q of micromachined silicon resonators, in Solid-State Sensor and Actuator Workshop, 1990. 4th Technical Digest (New Jersey, USA, 1990), pp. 13–16

    Google Scholar 

  47. C. Zener. Internal friction in solids I. Theory of internal friction in reeds. Phys. Rev. 52, 230–235 (1937)

    Google Scholar 

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

    Article  Google Scholar 

  49. S. Prabhakar, S. Vengallatore, Thermoelastic damping in bilayered micromechanical beam resonators. J. Micromech. Microeng. 17 (3), 532–538 (2007)

    Article  Google Scholar 

  50. S. Joshi, S. Hung, S. Vengallatore, Design strategies for controlling damping in micromechanical and nanomechanical resonators. EPJ Tech. Instrum. 1 (1), 1–14 (2014)

    Article  Google Scholar 

  51. S.S. Verbridge, J.M. Parpia, R.B. Reichenbach, L.M. Bellan, H.G. Craighead, High quality factor resonance at room temperature with nanostrings under high tensile stress. J. Appl. Phys. 99, 124304 (2006)

    Article  Google Scholar 

  52. S.S. Verbridge, H.G. Craighead, J.M. Parpia, A megahertz nanomechanical resonator with room temperature quality factor over a million. Appl. Phys. Lett. 92 (1), 013112 (2008)

    Google Scholar 

  53. B.M. Zwickl, W.E. Shanks, A.M. Jayich, C. Yang, B. Jayich, J.D. Thompson, J.G.E. Harris, High quality mechanical and optical properties of commercial silicon nitride membranes. Appl. Phys. Lett. 92 (10), 103125 (2008)

    Google Scholar 

  54. G.I. Gonzfilez, P.R. Saulson, Brownian motion of a mass suspended by an anelastic wire. J. Acoust. Soc. Am. 96 (1), 207–212 (1994)

    Article  Google Scholar 

  55. G. Cagnoli, J. Hough, D. Debra, M.M. Fejer, E. Gustafson, S. Rowan, V. Mitrofanov, Damping dilution factor for a pendulum in an interferometric gravitational waves detector. Phys. Lett. A 272, 39–45 (2000)

    Google Scholar 

  56. P.-L. Yu, T. Purdy, C.A. Regal. Control of material damping in high-Q membrane microresonators. Phys. Rev. Lett. 108 (8), 083603 (2012)

    Google Scholar 

  57. Q.P. Unterreithmeier, T. Faust, J.P. Kotthaus, Damping of nanomechanical resonators. Phys. Rev. Lett. 105 (2), 27205 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Vienna University of Technology, Vienna, Austria

    Silvan Schmid

  2. École Polytechnique Fédérale de, Lausanne, Switzerland

    Luis Guillermo Villanueva

  3. California Institute of Technology, Pasadena, CA, USA

    Michael Lee Roukes

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  1. Silvan Schmid
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  2. Luis Guillermo Villanueva
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  3. Michael Lee Roukes
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Schmid, S., Villanueva, L.G., Roukes, M.L. (2016). Quality Factor. In: Fundamentals of Nanomechanical Resonators. Springer, Cham. https://doi.org/10.1007/978-3-319-28691-4_2

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  • DOI: https://doi.org/10.1007/978-3-319-28691-4_2

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