A coherent nanomechanical oscillator driven by single-electron tunnelling

Article metrics

Abstract

A single-electron transistor embedded in a nanomechanical resonator represents an extreme limit of electron–phonon coupling. While it allows fast and sensitive electromechanical measurements, it also introduces back-action forces from electron tunnelling that randomly perturb the mechanical state. Despite the stochastic nature of this back-action, it has been predicted to create self-sustaining coherent mechanical oscillations under strong coupling conditions. Here, we verify this prediction using real-time measurements of a vibrating carbon nanotube transistor. This electromechanical oscillator has some similarities with a laser. The single-electron transistor pumped by an electrical bias acts as a gain medium and the resonator acts as a phonon cavity. Although the operating principle is unconventional because it does not involve stimulated emission, we confirm that the output is coherent. We demonstrate other analogues of laser behaviour, including injection locking, classical squeezing through anharmonicity and frequency narrowing through feedback.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Strongly coupled single-electron electromechanics.
Fig. 2: Mechanical resonance and oscillation.
Fig. 3: Coherence of the free-running oscillator.
Fig. 4: Tuning the coherence with a gate voltage.
Fig. 5: Injection locking of the nanomechanical oscillator.
Fig. 6: Stabilizing the oscillator with feedback.

Data availability

The data represented in Figs. 26 are available as source data in Supplementary Data 15. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. 1.

    Clerk, A. A. et al. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 82, 1155–1208 (2010).

  2. 2.

    Schoelkopf, R. J., Wahlgren, P., Kozhevnikov, A. A., Delsing, P. & Prober, D. E. The radio-frequency single-electron transistor (RF-SET): a fast and ultrasensitive electrometer. Science 280, 1238–1242 (1998).

  3. 3.

    LaHaye, M. D., Buu, O., Camarota, B. & Schwab, K. C. Approaching the quantum limit of a nanomechanical resonator. Science 304, 74–77 (2004).

  4. 4.

    Mozyrsky, D., Hastings, M. B. & Martin, I. Intermittent polaron dynamics: Born-Oppenheimer approximation out of equilibrium. Phys. Rev. B 73, 035104 (2006).

  5. 5.

    Steele, G. A. et al. Strong coupling between single-electron tunneling and nanomechanical motion. Science 325, 1103–1107 (2009).

  6. 6.

    Lassagne, B., Tarakanov, Y., Kinaret, J., Daniel, G. S. & Bachtold, A. Coupling mechanics to charge transport in carbon nanotube mechanical resonators. Science 325, 1107–1110 (2009).

  7. 7.

    Naik, A. et al. Cooling a nanomechanical resonator with quantum back-action. Nature 443, 193–196 (2006).

  8. 8.

    Vahala, K. et al. A phonon laser. Nat. Phys. 5, 682–686 (2009).

  9. 9.

    Grudinin, I. S., Lee, H., Painter, O. & Vahala, K. J. Phonon laser action in a tunable two-level system. Phys. Rev. Lett. 104, 083901 (2010).

  10. 10.

    Mahboob, I., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Phonon lasing in an electromechanical resonator. Phys. Rev. Lett. 110, 127202 (2013).

  11. 11.

    Sazonova, V. et al. A tunable carbon nanotube electromechanical oscillator. Nature 431, 284–287 (2004).

  12. 12.

    Wen, Y., Ares, N., Pei, T., Briggs, G. A. D. & Laird, E. A. Measuring carbon nanotube vibrations using a single-electron transistor as a fast linear amplifier. Appl. Phys. Lett. 113, 153101 (2018).

  13. 13.

    de Bonis, S. L. et al. Ultrasensitive displacement noise measurement of carbon nanotube mechanical resonators. Nano Lett. 18, 5324–5328 (2018).

  14. 14.

    Khivrich, I., Clerk, A. A. & Ilani, S. Nanomechanical pump–probe measurements of insulating electronic states in a carbon nanotube. Nat. Nanotechnol. 14, 161–167 (2019).

  15. 15.

    Armour, A. D., Blencowe, M. P. & Zhang, Y. Classical dynamics of a nanomechanical resonator coupled to a single-electron transistor. Phys. Rev. B 69, 125313 (2004).

  16. 16.

    Rodrigues, D. A., Imbers, J. & Armour, A. D. Quantum dynamics of a resonator driven by a superconducting single-electron transistor: a solid-state analogue of the micromaser. Phys. Rev. Lett. 98, 067204 (2007).

  17. 17.

    Bennett, S. D. & Clerk, A. A. Laser-like instabilities in quantum nano-electromechanical systems. Phys. Rev. B 74, 201301 (2006).

  18. 18.

    Usmani, O., Blanter, Y. M. & Nazarov, Y. V. Strong feedback and current noise in nanoelectromechanical systems. Phys. Rev. B 75, 195312 (2007).

  19. 19.

    Hüttel, A. K., Witkamp, B., Leijnse, M., Wegewijs, M. & van der Zant, H. S. J. Pumping of vibrational excitations in the Coulomb-blockade regime in a suspended carbon nanotube. Phys. Rev. Lett. 102, 225501 (2009).

  20. 20.

    Eichler, A., Chaste, J., Moser, J. & Bachtold, A. Parametric amplification and self-oscillation in a nanotube mechanical resonator. Nano Lett. 11, 2699–2703 (2011).

  21. 21.

    Tsioutsios, I., Tavernarakis, A., Osmond, J., Verlot, P. & Bachtold, A. Real-time measurement of nanotube resonator fluctuations in an electron microscope. Nano Lett. 17, 1748–1755 (2017).

  22. 22.

    Barnard, A. W., Zhang, M., Wiederhecker, G. S., Lipson, M. & McEuen, P. L. Real-time vibrations of a carbon nanotube. Nature 566, 89–93 (2019).

  23. 23.

    Wu, C. C., Liu, C. H. & Zhong, Z. One-step direct transfer of pristine single-walled carbon nanotubes for functional nanoelectronics. Nano Lett. 10, 1032–1036 (2010).

  24. 24.

    Schupp, F. J. et al. Radio-frequency reflectometry of a quantum dot using an ultra-low-noise SQUID amplifier. Preprint at https://arxiv.org/abs/1810.05767(2018).

  25. 25.

    Liu, Y.-Y. et al. Semiconductor double quantum dot micromaser. Science 347, 285–287 (2015).

  26. 26.

    Cassidy, M. C. et al. Demonstration of an ac Josephson junction laser. Science 355, 939–942 (2017).

  27. 27.

    Pistolesi, F., Blanter, Y. M. & Martin, I. Self-consistent theory of molecular switching. Phys. Rev. B 78, 085127 (2008).

  28. 28.

    Fox, M. Quantum Optics: An Introduction (Oxford University Press, 2006).

  29. 29.

    Fujisawa, T. et al. Spontaneous emission spectrum in double quantum dot devices. Science 282, 932–935 (1998).

  30. 30.

    Steeneken, P. G. et al. Piezoresistive heat engine and refrigerator. Nat. Phys. 7, 354–359 (2011).

  31. 31.

    Stover, H. L. & Steier, W. H. Locking of laser oscillators by light injection. Appl. Phys. Lett. 8, 91 (1966).

  32. 32.

    Liu, Y.-Y., Stehlik, J., Gullans, M. J., Taylor, J. M. & Petta, J. R. Injection locking of a semiconductor double-quantum-dot micromaser. Phys. Rev. A 92, 053802 (2015).

  33. 33.

    Knünz, S. et al. Injection locking of a trapped-ion phonon laser. Phys. Rev. Lett. 105, 013004 (2010).

  34. 34.

    Seitner, M. J., Abdi, M., Ridolfo, A., Hartmann, M. J. & Weig, E. M. Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes. Phys. Rev. Lett. 118, 254301 (2017).

  35. 35.

    Adler, R. A study of locking phenomena in oscillators. Proc. IRE 34, 351–357 (1946).

  36. 36.

    Huber, J. S. et al. Detecting squeezing from the fluctuation spectrum of a driven nanomechanical mode. Preprint at https://arxiv.org/abs/1903.07601v2 (2019).

  37. 37.

    Stambaugh, C. & Chan, H. B. Supernarrow spectral peaks near a kinetic phase transition in a driven nonlinear micromechanical oscillator. Phys. Rev. Lett. 97, 110602 (2006).

  38. 38.

    Schawlow, A. L. & Townes, C. H. Infrared and optical masers. Phys. Rev. 112, 1940–1949 (1958).

  39. 39.

    Wiseman, H. M. Light amplification without stimulated emission: beyond the standard quantum limit to the laser linewidth. Phys. Rev. A 60, 4083–4093 (1999).

  40. 40.

    Chaste, J. et al. A nanomechanical mass sensor with yoctogram resolution. Nat. Nanotechnol. 7, 301–304 (2012).

  41. 41.

    Stipe, B. C. et al. Electron spin relaxation near a micron-size ferromagnet. Phys. Rev. Lett. 87, 277602 (2001).

  42. 42.

    Maryam, W., Akimov, A. V., Campion, R. P. & Kent, A. J. Dynamics of a vertical cavity quantum cascade phonon laser structure. Nat. Commun. 4, 2184 (2013).

  43. 43.

    Wiseman, H. M. Defining the (atom) laser. Phys. Rev. A 56, 2068–2084 (1997).

  44. 44.

    Öttl, A., Ritter, S., Köhl, M. & Esslinger, T. Correlations and counting statistics of an atom laser. Phys. Rev. Lett. 95, 090404 (2005).

  45. 45.

    Brandes, T. & Lambert, N. Steering of a bosonic mode with a double quantum dot. Phys. Rev. B 67, 125323 (2003).

  46. 46.

    Ohm, C., Stampfer, C., Splettstoesser, J. & Wegewijs, M. Readout of carbon nanotube vibrations based on spin-phonon coupling. Appl. Phys. Lett. 100, 143103 (2012).

  47. 47.

    Pályi, A., Struck, P. R., Rudner, M. S., Flensberg, K. & Burkard, G. Spin-orbit-induced strong coupling of a single spin to a nanomechanical resonator. Phys. Rev. Lett. 108, 206811 (2012).

  48. 48.

    Ogata, K. Modern Control Engineering (Prentice Hall, 1970).

Download references

Acknowledgements

We acknowledge A. Bachtold, E. M. Gauger, Y. Pashkin, A. Romito and M. Woolley for discussions, and T. Orton for technical support. This work was supported by EPSRC (EP/N014995/1, EP/R029229/1), DSTL, Templeton World Charity Foundation, the Royal Academy of Engineering, the European Research Council (grant agreement 818751), and the EU H2020 European Microkelvin Platform (grant agreement 824109).

Author information

Y.W. fabricated the device following a recipe devised by T.P., and performed the experiment and analysis with contributions from N.A., F.J.S. and E.A.L. Y.W. and E.A.L. wrote the manuscript. All authors discussed the results and commented on the manuscript.

Correspondence to E. A. Laird.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–9, Methods, Discussion and References.

Supplementary Data 1

Source Data for Fig. 2.

Supplementary Data 2

Source Data for Fig. 3.

Supplementary Data 3

Source Data for Fig. 4.

Supplementary Data 4

Source Data for Fig. 5.

Supplementary Data 5

Source Data for Fig. 6.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wen, Y., Ares, N., Schupp, F.J. et al. A coherent nanomechanical oscillator driven by single-electron tunnelling. Nat. Phys. (2019) doi:10.1038/s41567-019-0683-5

Download citation

Further reading