Energy decay plays a central role in a wide range of phenomena1,2,3, such as optical emission, nuclear fission, and dissipation in quantum systems. Energy decay is usually described as a system leaking energy irreversibly into an environmental bath. Here, we report on energy decay measurements in nanomechanical systems based on multilayer graphene that cannot be explained by the paradigm of a system directly coupled to a bath. As the energy of a vibrational mode freely decays, the rate of energy decay changes abruptly to a lower value. This finding can be explained by a model where the measured mode hybridizes with other modes of the resonator at high energy. Below a threshold energy, modes are decoupled, resulting in comparatively low decay rates and giant quality factors exceeding 1 million. Our work opens up new possibilities to manipulate vibrational states4,5,6,7, engineer hybrid states with mechanical modes at completely different frequencies, and to study the collective motion of this highly tunable system.
Your institute does not have access to this article
Open Access articles citing this article.
Scientific Reports Open Access 21 June 2022
Nature Communications Open Access 29 April 2022
Nonlinear Dynamics Open Access 05 April 2021
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
Rayleigh, J. W. Some general theorems relating to vibrations. Proc. London Math. Soc. s1–4, 357–368 (1871).
Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931).
Caldeira, A. O. & Leggett, A. J. Path integral approach to quantum Brownian motion. Physica 121A, 587–616 (1983).
Faust, T., Rieger, J., Seitner, M. J., Kotthaus, J. P. & Weig, E. M. Coherent control of a classical nanomechanical two-level system. Nat. Phys. 9, 485–488 (2013).
Okamoto, H. et al. Coherent phonon manipulation in coupled mechanical resonators. Nat. Phys. 9, 480–484 (2013).
De Alba, R. et al. Tunable phonon–cavity coupling in graphene membranes. Nat. Nanotech. 11, 741–746 (2016).
Mathew, J. P., Patel, R. N., Borah, A., Vijay, R. & Deshmukh, M. M. Dynamical strong coupling and parametric amplification of mechanical modes of graphene drums. Nat. Nanotech. 11, 747–751 (2016).
Gao, J., Zmuidzinas, J., Mazin, B. A., LeDuc, H. G. & Day, P. K. Noise properties of superconducting coplanar waveguide microwave resonators. Appl. Phys. Lett. 90, 102507 (2007).
Eichler, A. et al. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene. Nat. Nanotech. 6, 339–342 (2011).
Zaitsev, S., Shtempluck, O., Buks, E. & Gottlieb, O. Nonlinear damping in a micromechanical oscillator. Nonlinear Dynamics 67, 859–883 (2012).
Imboden, M., Williams, O. A. & Mohanty, P. Observation of nonlinear dissipation in piezoresistive diamond nanomechanical resonators by heterodyne down-mixing. Nano Lett. 13, 4014–4019 (2013).
Mahboob, I. et al. Dispersive and dissipative coupling in a micromechanical resonator embedded with a nanomechanical resonator. Nano Lett. 15, 2312–2317 (2015).
Polunin, P. M., Yang, Y., Dykman, M. I., Kenny, T. W. & Shaw, S. W. Characterization of MEMS resonator nonlinearities using the ringdown response. J. Microelectromech. Sys. 25, 297–303 (2016).
Dykman, M. I. & Krivoglaz, M. A. Theory of nonlinear oscillator interacting with a medium. Sov. Phys. Rev. 5, 265–442 (1984).
Wilson-Rae, I. Intrinsic dissipation in nanomechanical resonators due to phonon tunneling. Phys. Rev. B 77, 245418 (2008).
Rieger, J., Isacsson, A., Seitner, M. J., Kotthaus, J. P. & Weig, E. M. Energy losses of nanomechanical resonators induced by atomic force microscopy-controlled mechanical impedance mismatching. Nat. Commun. 5, 3345 (2014).
Tao, Y., Boss, J. M., Moores, B. A. & Degen, C. L. Single-crystal diamond nanomechanical resonators with quality factors exceeding one million. Nat. Commun. 5, 3638 (2014).
Midtvedt, D., Croy, A., Isacsson, A., Qi, Z. & Park, H. Fermi-Pasta-Ulam physics with nanomechanical graphene resonators: intrinsic relaxation and thermalization from flexural mode coupling. Phys. Rev. Lett. 112, 145503 (2014).
Gil-Santos, E. et al. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat. Nanotech. 5, 641–645 (2010).
Hanay, M. S. et al. Single-protein nanomechanical mass spectrometry in real time. Nat. Nanotech. 7, 602–608 (2012).
Chaste, J. et al. A nanomechanical mass sensor with yoctogram resolution. Nat. Nanotech. 7, 301–304 (2012).
Moser, J., Eichler, A., Güttinger, J., Dykman, M. I. & Bachtold, A. Nanotube mechanical resonators with quality factors of up to 5 million. Nat. Nanotech. 9, 1007–1011 (2014).
Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nat. Commun. 3, 806 (2012).
Eichler, A., del Álamo Ruiz, M., Plaza, J. A. & Bachtold, A. Strong coupling between mechanical modes in a nanotube resonator. Phys. Rev. Lett. 109, 025503 (2012).
Chen, C. et al. Performance of monolayer graphene nanomechanical resonators with electrical readout. Nat. Nanotech. 4, 861–867 (2009).
Singh, V. et al. Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators. Nanotechnology 21, 165204 (2010).
Barton, R. A. et al. Photothermal self-oscillation and laser cooling of graphene optomechanical systems. Nano Lett. 12, 4681–4686 (2012).
Miao, T., Yeom, S., Wang, P., Standley, B. & Bockrath, M. Graphene nanoelectromechanical systems as stochastic-frequency oscillators. Nano Lett. 14, 2982–2987 (2014).
Weber, P., Güttinger, J., Tsioutsios, I., Chang, D. E. & Bachtold, A. Coupling graphene mechanical resonators to superconducting microwave cavities. Nano Lett. 14, 2854–2860 (2014).
Song, X., Oksanen, M., Li, J., Hakonen, P. J. & Sillanpää, M. A. Graphene optomechanics realized at microwave frequencies. Phys. Rev. Lett. 113, 027404 (2014).
Singh, V. et al. Optomechanical coupling between a graphene mechanical resonator and a superconducting microwave cavity. Nat. Nanotech. 9, 820–824 (2014).
Weber, P., Güttinger, J., Noury, A., Vergara-Cruz, J. & Bachtold, A. Force sensitivity of multilayer graphene optomechanical devices. Nat. Commun. 7, 12496 (2016).
Eichler, C., Salathe, Y., Mlynek, J., Schmidt, S. & Wallraff, A. Quantum-limited amplification and entanglement in coupled nonlinear resonators. Phys. Rev. Lett. 113, 110502 (2014).
Lifshitz, R. & Cross, M. Nonlinear dynamics of nanomechanical and micromechanical resonators. Rev. Nonlinear Dyn. Complex. 1, 1–48 (2008).
Seoánez, C., Guinea, F. & Castro Neto, A. H. Dissipation in graphene and nanotube resonators. Phys. Rev. B 76, 125427 (2007).
Eriksson, A. M., Midtvedt, D., Croy, A. & Isacsson, A. Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators. Nanotechnology 24, 395702 (2013).
Benyamini, A., Hamo, A., Kusminskiy, S. V., von Oppen, F. & Ilani, S. Real-space tailoring of the electron–phonon coupling in ultraclean nanotube mechanical resonators. Nat. Phys. 10, 151–156 (2014).
Shoshani, O., Shaw, S. W. & Dykman, M. I. Anomalous dissipation of nanomechanical modes going through nonlinear resonance. Preprint at https://arxiv.org/abs/1702.00769 (2017).
We thank M. Dykman, S. Shaw, D. Lopez, F. Guinea and N. Noury for discussions. We acknowledge G. Ceballos and the ICFO mechanical and electronic workshop for support. We acknowledge financial support by the ERC starting grant 279278 (CarbonNEMS), the EU Graphene Flagship (contract no. 604391), the Foundation Cellex, Severo Ochoa (SEV-2015-0522) and grant MAT2012-31338 of MINECO, the Fondo Europeo de Desarrollo Regional (FEDER), and the Generalitat through AGAUR. A.I. and A.M.E. acknowledge financial support through the Swedish Research Council and the Knut and Alice Wallenberg foundation.
The authors declare no competing financial interests.
About this article
Cite this article
Güttinger, J., Noury, A., Weber, P. et al. Energy-dependent path of dissipation in nanomechanical resonators. Nature Nanotech 12, 631–636 (2017). https://doi.org/10.1038/nnano.2017.86
Nature Communications (2022)
Scientific Reports (2022)
Nonlinear Dynamics (2022)
Microsystems & Nanoengineering (2021)
Nature Communications (2021)