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Fundamentals of Nanomechanical Resonators pp 1–56Cite as

Resonance Frequency

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Abstract

Nanomechanical resonators are continuum mechanical structures, such as beams, strings, plates, or membranes. In this chapter the eigenmodes of such ideal lossless continuum mechanical structures are estimated by simple analytical models. Specific resonance modes of a damped continuum mechanical structure are best described by an effective lumped-element model. In this chapter, the eigenmodes of the most common continuum mechanical structures used as nanomechanical resonators are derived. Then linear, coupled, and nonlinear damped and driven resonators are discussed by means of lumped-element models.

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Notes

  1. 1.

    For convenience, the term “frequency” is subsequently used in place for the actual correct term “angular velocity.”

References

  1. E. Ventsel, T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications (Marcel Dekker, New York, 2001)

    Google Scholar 

  2. W.F. Stokey, Vibration of systems having distributed mass and elasticity, in Harris’ Shock and Vibration Handbook (McGraw-Hill Education, New York, 2002), pp. 7.1–7.50

    Google Scholar 

  3. W. Weaver, S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering (Wiley Interscience, New York, 1990)

    Google Scholar 

  4. S. Schmid, Electrostatically Actuated All-Polymer Microbeam Resonators - Characterization and Application. Scientific Reports on Micro and Nanosystems, vol. 6 (Der Andere Verlag, Tönning, 2009)

    Google Scholar 

  5. M. Li, E.B Myers, H.X. Tang, S.J. Aldridge, H.C. McCaig, J.J. Whiting, R.J. Simonson, N.S. Lewis, M.L. Roukes, Nanoelectromechanical resonator arrays for ultrafast, gas-phase chromatographic chemical analysis. Nano Lett. 10 (10), 3899–3903 (2010)

    Google Scholar 

  6. S. Schmid, M. Kurek, A. Boisen, Towards airborne nanoparticle mass spectrometry with nanomechanical string resonators. SPIE Def. Secur. Sens. 8725, 872525–872528 (2013)

    Google Scholar 

  7. A.M. Van Der Zande, R.A Barton, J.S. Alden, C.S. Ruiz-Vargas, W.S. Whitney, P.H.Q. Pham, J. Park, J.M. Parpia, H.G. Craighead, P.L. McEuen, Large-scale arrays of single-layer graphene resonators. Nano Lett. 10, 4869–4873 (2010)

    Google Scholar 

  8. E. Gil-Santos, D. Ramos, J. Martínez, M. Fernández-Regúlez, R. García, A. San Paulo, M. Calleja, J. Tamayo. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat. Nanotechnol. 5 (9), 641–645 (2010)

    Google Scholar 

  9. 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 

  10. A.N. Cleland, M.L. Roukes, Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653 (1996)

    Article  Google Scholar 

  11. H.B. Peng, C.W. Chang, S. Aloni, T.D. Yuzvinsky, A. Zettl, Ultrahigh frequency nanotube resonators. Phys. Rev. Lett. 97 (8), 2–5 (2006)

    Google Scholar 

  12. T. Bagci, A. Simonsen, S. Schmid, L.G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. Sørensen, K. Usami, A. Schliesser, E.S. Polzik. Optical detection of radio waves through a nanomechanical transducer. Nature 507 (7490), 81–85 (2014)

    Google Scholar 

  13. M.B. Sayir, S. Kaufmann, Ingenieurmechanik 3: Dynamik (Vieweg+Teubner, Wiesbaden, 2005)

    Book  Google Scholar 

  14. L. Aigouy, P. Lalanne, H. Liu, G. Julié, V. Mathet, M. Mortier, Near-field scattered by a single nanoslit in a metal film. Appl. Opt. 46 (36), 8573–8577 (2007)

    Article  Google Scholar 

  15. A. Boisen, S. Dohn, S.S. Keller, S. Schmid, M. Tenje, Cantilever-like micromechanical sensors. Rep. Prog. Phys. 74 (3), 036101 (2011)

    Google Scholar 

  16. J. Bochmann, A. Vainsencher, D.D. Awschalom, A.N. Cleland, Nanomechanical coupling between microwave and optical photons. Nat. Phys. 9 (9), 1–5 (2013)

    Google Scholar 

  17. A.D. O’Connell, M. Hofheinz, M. Ansmann, R.C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, Others, A.D. O’Connell, J. Wenner, J.M. Martinis, A.N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator. Nature 464 (7289), 697–703 (2010)

    Google Scholar 

  18. J.H. Hales, J. Teva, A. Boisen, Z.J. Davis, Longitudinal bulk acoustic mass sensor, in International Solid-State Sensors, Actuators and Microsystems Conference, 2009. TRANSDUCERS 2009 (IEEE, New York, 2009), pp. 311–314

    Google Scholar 

  19. J.D. Thompson, B.M. Zwickl, A.M. Jayich, F. Marquardt, S.M. Girvin, J.G.E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. Nature 452 (7183), 72–75 (2008)

    Google Scholar 

  20. J.D. Teufel, T. Donner, D. Li, J.W. Harlow, M.S. Allman, K. Cicak, A.J. Sirois, J.D. Whittaker, K.W. Lehnert, R.W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state. Nature 475 (7356), 359–363 (2011)

    Google Scholar 

  21. J. Lee, Z. Wang, K. He, J. Shan, P.X.-L. Feng, High frequency MoS2 nanomechanical resonators. ACS Nano 7 (7), 6086–6091 (2013)

    Article  Google Scholar 

  22. 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 

  23. S. Schmid, T. Bagci, E. Zeuthen, J.M. Taylor, P.K. Herring, M.C. Cassidy, C.M. Marcus, L. Guillermo Villanueva, B. Amato, A. Boisen, Y. Cheol Shin, J. Kong, A.S. Sørensen, K. Usami, E.S. Polzik, Single-layer graphene on silicon nitride micromembrane resonators. J. Appl. Phys. 115 (5), 054513 (2014)

    Google Scholar 

  24. S.P. Timoshenko, S. Woinowsky-Krieger, W. -Krieger, Theory of Plates and Shells, 2nd edn. (McGraw-Hill, New York, 1959)

    Google Scholar 

  25. S.J. Papadakis, A.R. Hall, P.A. Williams, L. Vicci, M.R. Falvo, R. Superfine, S. Washburn, Resonant oscillators with carbon-nanotube torsion springs. Phys. Rev. Lett. 93 (14), 1–4 (2004)

    Google Scholar 

  26. X.C. Zhang, E.B. Myers, J.E. Sader, M.L. Roukes, Nanomechanical torsional resonators for frequency-shift infrared thermal sensing. Nano Lett. 13 (4), 1528–1534 (2013)

    Article  Google Scholar 

  27. A.N. Cleland, M.L. Roukes, A nanometre-scale mechanical electrometer. Nature, 392, 160–162 (1998)

    Article  Google Scholar 

  28. M Bao, Analysis and Design Principles of MEMS Devices (Elsevier, Amsterdam, 2005)

    Google Scholar 

  29. E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, J. Tamayo, Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers. Nano Lett. 9 (12), 4122–4127 (2009)

    Article  Google Scholar 

  30. M. Spletzer, A. Raman, H. Sumali, J.P. Sullivan, Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays. Appl. Phys. Lett. 92 (11), 2006–2009 (2008)

    Article  Google Scholar 

  31. P. Thiruvenkatanathan, J. Yan, J. Woodhouse, A.A. Seshia, Enhancing parametric sensitivity in electrically coupled MEMS resonators. J. Microelectromech. Syst. 18 (5), 1077–1086 (2009)

    Article  Google Scholar 

  32. S. Pourkamali, F. Ayazi, Electrically coupled MEMS bandpass filters: Part I: with coupling element. Sensors Actuators A Phys. 122 (2), 307–316 (2005)

    Article  Google Scholar 

  33. V. Singh, S.J. Bosman, B.H. Schneider, Y.M. Blanter, a Castellanos-Gomez, G. a Steele, Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity., Nat. Nanotechnol., 9 (10), 820–824 (2014)

    Google Scholar 

  34. M. Aspelmeyer, T.J. Kippenberg, F. Marquard, Cavity optomechanics. Rev. Mod. Phys. 86 (4), 1391–1452 (2014)

    Article  Google Scholar 

  35. R. Lifshitz, M.C. Cross, Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators, vol. 1, book section 1 (Wiley-VCH, Weinheim, 2008)

    Google Scholar 

  36. A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. Pure and Applied Mathematics (Wiley, New York, 1979)

    Google Scholar 

  37. J.M. Gere, B.J. Goodno, Mechanics of Materials, 8th edn. (Cengage Learning, Stamford, CT, 2013)

    Google Scholar 

  38. V. Kaajakari, T. Mattila, A. Oja, H. Seppa. Nonlinear limits for single-crystal silicon microresonators. J. Microelectromech. Syst. 13 (5), 715–724 (2004)

    Google Scholar 

  39. M.H. Matheny, L.G. Villanueva, R.B. Karabalin, J.E. Sader, M.L. Roukes, Nonlinear mode-coupling in nanomechanical systems. Nano Lett. 13 (4), 1622–1626 (2013)

    Google Scholar 

  40. I. Kozinsky, H.W.C. Postma, I. Bargatin, M.L. Roukes, Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators. Appl. Phys. Lett. 88 (25), 253101 (2006)

    Google Scholar 

  41. N. Kacem, S. Hentz, D. Pinto, B. Reig, V. Nguyen, Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20 (27), 275501 (2009)

    Google Scholar 

  42. N. Kacem, J. Arcamone, F. Perez-Murano, S. Hentz, Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive NEMS gas/mass sensor applications. J. Micromech. Microeng. 20 (4), 45023 (2010)

    Google Scholar 

  43. L.G. Villanueva, R.B. Karabalin, M.H. Matheny, D. Chi, J.E. Sader, M.L. Roukes, Nonlinearity in nanomechanical cantilevers. Phys. Rev. B 87 (2), 24304 (2013)

    Google Scholar 

  44. M.R.M.C. Dasilva. Non-linear flexural flexural torsional extensional dynamics of beams 2. Response analysis. Int. J. Solids Struct. 24 (12), 1235–1242 (1988)

    Article  Google Scholar 

  45. M.R.M. Crespodasilva, C.C. Glynn, Out-of-plane vibrations of a beam including nonlinear inertia and nonlinear curvature effects. Int. J. Non Linear Mech. 13 (5–6), 261–271 (1978)

    Google Scholar 

  46. S. Perisanu, T. Barois, A. Ayari, P. Poncharal, M. Choueib, S.T. Purcell, P. Vincent, Beyond the linear and Duffing regimes in nanomechanics: circularly polarized mechanical resonances of nanocantilevers. Phys. Rev. B 81 (16), 165440 (2010)

    Google Scholar 

  47. A. San Paulo, R. Garcia, Tip-surface forces, amplitude, and energy dissipation in amplitude-modulation (tapping mode) force microscopy. Phys. Rev. B 64 (19), 193411 (2001)

    Google Scholar 

  48. J.F. Rhoads, S.W. Shaw, K.L. Turner, Nonlinear dynamics and its applications in micro- and nanoresonators. J. Dyn. Syst. Meas. Control Trans. ASME 132 (3), 34001 (2010)

    Google Scholar 

  49. S. Zaitsev, O. Shtempluck, E. Buks, O. Gottlieb, Nonlinear damping in a micromechanical oscillator. Nonlinear Dyn. 67 (1), 859–883 (2012)

    Article  MATH  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). Resonance Frequency. In: Fundamentals of Nanomechanical Resonators. Springer, Cham. https://doi.org/10.1007/978-3-319-28691-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-28691-4_1

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