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Strained crystalline nanomechanical resonators with quality factors above 10 billion


In strained mechanical resonators, the concurrence of tensile stress and geometric nonlinearity dramatically reduces dissipation. This phenomenon, called dissipation dilution, is employed in mirror suspensions of gravitational-wave interferometers and at the nanoscale, where soft clamping and strain engineering have allowed extremely high quality factors. However, these techniques have so far been applied only to amorphous materials, specifically Si3N4. Crystalline materials exhibit substantially lower intrinsic damping at cryogenic temperatures. Applying dissipation dilution engineering to strained crystalline materials could, therefore, enable extremely low loss nanomechanical resonators, as they combine low internal friction, high intrinsic strain and high yield strength. This potential has not yet been fully exploited. Here we demonstrate that single-crystal strained silicon—a material developed for high-mobility transistors—can be used to realize mechanical resonators with ultralow dissipation. We fabricate strained silicon nanostrings with high aspect ratios supporting megahertz mechanical modes with quality factors exceeding 1010 at 7 K, a tenfold improvement over values reported in Si3N4. We estimate a thermal-noise-limited force sensitivity of (5 ± 2) × 10–20 N Hz–1/2 at 7 K—approaching that of carbon nanotubes—and a heating rate of only 60 quanta per second. The low mass and high quality factors of our nanomechanical resonators make them particularly promising for quantum sensing and transduction.

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Fig. 1: Dissipation in strained crystalline mechanical resonators.
Fig. 2: Fabrication of sSi nanostrings.
Fig. 3: Bandgap engineering and soft clamping of nanostrings.
Fig. 4: Temperature dependence of dissipation in an sSi nanostring.

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Source data are provided with this paper. Data supporting the figures and lithographic masks used for microfabrication are available on Zenodo (


  1. Braginskiĭ, V. B., Mitrofanov, V. P. & Panov, V. I. Systems with Small Dissipation (Univ. Chicago Press, 1985).

  2. Cady, W. G. The piezo-electric resonator. Proc. Inst. Radio Eng. 10, 83–114 (1922).

    Google Scholar 

  3. Braginsky, V. B., Gorodetsky, M. L. & Vyatchanin, S. P. Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae. Phys. Lett. A 264, 1–10 (1999).

    Article  ADS  Google Scholar 

  4. Hirose, E. et al. Sapphire mirror for the KAGRA gravitational wave detector. Phys. Rev. D 89, 062003 (2014).

    Article  ADS  Google Scholar 

  5. Cole, G. D., Zhang, W., Martin, M. J., Ye, J. & Aspelmeyer, M. Tenfold reduction of Brownian noise in high-reflectivity optical coatings. Nat. Photon. 7, 644–650 (2013).

    Article  ADS  Google Scholar 

  6. Kessler, T. et al. A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity. Nat. Photon. 6, 687–692 (2012).

    Article  ADS  Google Scholar 

  7. Chae, J., Kulah, H. & Najafi, K. A monolithic three-axis micro-g micromachined silicon capacitive accelerometer. J. Microelectromech. Syst. 14, 235–242 (2005).

    Google Scholar 

  8. Rugar, D., Budakian, R., Mamin, H. J. & Chui, B. W. Single spin detection by magnetic resonance force microscopy. Nature 430, 329–332 (2004).

    Article  ADS  Google Scholar 

  9. Degen, C. L., Poggio, M., Mamin, H. J., Rettner, C. T. & Rugar, D. Nanoscale magnetic resonance imaging. Proc. Natl Acad. Sci. USA 106, 1313–1317 (2009).

    Article  ADS  Google Scholar 

  10. Safavi-Naeini, A. H. et al. Squeezed light from a silicon micromechanical resonator. Nature 500, 185–189 (2013).

    Article  ADS  Google Scholar 

  11. Cripe, J. et al. Measurement of quantum back action in the audio band at room temperature. Nature 568, 364–367 (2019).

  12. Phillips, W. A. Two-level states in glasses. Rep. Prog. Phys. 50, 1657–1708 (1987).

    Article  ADS  Google Scholar 

  13. Pohl, R. O., Liu, X. & Thompson, E. Low-temperature thermal conductivity and acoustic attenuation in amorphous solids. Rev. Mod. Phys. 74, 991–1013 (2002).

    Article  ADS  Google Scholar 

  14. Grabovskij, G. J., Peichl, T., Lisenfeld, J., Weiss, G. & Ustinov, A. V. Strain tuning of individual atomic tunneling systems detected by a superconducting qubit. Science 338, 232–234 (2012).

    Article  ADS  Google Scholar 

  15. Bagdasarov, S., Braginsky, V. B. & Mitrofanov, V. P. Mechanical dissipation in single crystal sapphire. Kristallografiya 19, 883 (1974).

    Google Scholar 

  16. Galliou, S. et al. Extremely low loss phonon-trapping cryogenic acoustic cavities for future physical experiments. Sci. Rep. 3, 2132 (2013).

    Article  Google Scholar 

  17. McGuigan, D. F. et al. Measurements of the mechanical Q of single-crystal silicon at low temperatures. J. Low Temp. Phys. 30, 621–629 (1978).

    Article  ADS  Google Scholar 

  18. Unterreithmeier, Q. P., Faust, T. & Kotthaus, J. P. Damping of nanomechanical resonators. Phys. Rev. Lett. 105, 027205 (2010).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  20. Fedorov, S. A. et al. Generalized dissipation dilution in strained mechanical resonators. Phys. Rev. B 99, 054107 (2019).

    Article  ADS  Google Scholar 

  21. Nowick, A. S. Anelastic Relaxation In Crystalline Solids (Academic Press, 1972).

  22. Tsaturyan, Y., Barg, A., Polzik, E. S. & Schliesser, A. Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution. Nat. Nanotechnol. 12, 776–783 (2017).

    Article  Google Scholar 

  23. Reetz, C. et al. Analysis of membrane phononic crystals with wide band gaps and low-mass defects. Phys. Rev. Appl. 12, 044027 (2019).

    Article  ADS  Google Scholar 

  24. Bereyhi, M. J. et al. Clamp-tapering increases the quality factor of stressed nanobeams. Nano Lett. 19, 2329–2333 (2019).

  25. Beccari, A. et al. Hierarchical tensile structures with ultralow mechanical dissipation. Preprint at (2021).

  26. Ghadimi, A. H. et al. Elastic strain engineering for ultralow mechanical dissipation. Science 360, 764–768 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  27. Bereyhi, M. J. et al. Nanomechanical resonators with ultra-high-Q perimeter modes. Preprint at (2021).

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

    Article  ADS  Google Scholar 

  29. Rossi, M., Mason, D., Chen, J., Tsaturyan, Y. & Schliesser, A. Measurement-based quantum control of mechanical motion. Nature 563, 53–58 (2018).

    Article  ADS  Google Scholar 

  30. Tao, Y. et al. Permanent reduction of dissipation in nanomechanical Si resonators by chemical surface protection. Nanotechnology 26, 465501 (2015).

    Article  Google Scholar 

  31. Liu, J. et al. High-Q optomechanical GaAs nanomembranes. Appl. Phys. Lett. 99, 243102 (2011).

    Article  ADS  Google Scholar 

  32. Romero, E. et al. Engineering the dissipation of crystalline micromechanical resonators. Phys. Rev. Appl. 13, 044007 (2020).

    Article  ADS  Google Scholar 

  33. Kermany, A. R. et al. Microresonators with Q-factors over a million from highly stressed epitaxial silicon carbide on silicon. Appl. Phys. Lett. 104, 081901 (2014).

    Article  ADS  Google Scholar 

  34. Cole, G. D. et al. Tensile-strained InxGa1–xP membranes for cavity optomechanics. Appl. Phys. Lett. 104, 201908 (2014).

    Article  ADS  Google Scholar 

  35. Bückle, M. et al. Stress control of tensile-strained In1–xGaxP nanomechanical string resonators. Appl. Phys. Lett. 113, 201903 (2018).

    Article  ADS  Google Scholar 

  36. Chu, M., Sun, Y., Aghoram, U. & Thompson, S. E. Strain: a solution for higher carrier mobility in nanoscale MOSFETs. Annu. Rev. Mater. Res. 39, 203–229 (2009).

    Article  ADS  Google Scholar 

  37. Jacobsen, R. S. et al. Strained silicon as a new electro-optic material. Nature 441, 199–202 (2006).

    Article  ADS  Google Scholar 

  38. MacCabe, G. S. et al. Nano-acoustic resonator with ultralong phonon lifetime. Science 370, 840–843 (2020).

    Article  ADS  Google Scholar 

  39. Ghyselen, B. et al. Engineering strained silicon on insulator wafers with the Smart CutTM technology. Solid State Electron. 48, 1285–1296 (2004).

    Article  ADS  Google Scholar 

  40. Hÿtch, M., Houdellier, F., Hüe, F. & Snoeck, E. Nanoscale holographic interferometry for strain measurements in electronic devices. Nature 453, 1086–1089 (2008).

    Article  ADS  Google Scholar 

  41. De Wolf, I., Maes, H. E. & Jones, S. K. Stress measurements in silicon devices through Raman spectroscopy: bridging the gap between theory and experiment. J. Appl. Phys. 79, 7148–7156 (1996).

    Article  ADS  Google Scholar 

  42. Süess, M. J. et al. Power-dependent Raman analysis of highly strained Si nanobridges. Nano Lett. 14, 1249–1254 (2014).

    Article  ADS  Google Scholar 

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

  44. Honig, R. E. & Hook, H. O. Vapor pressure data for some common gases. RCA Rev. 21, 360–368 (1960).

    Google Scholar 

  45. Cannelli, G. et al. Reorientation of the B-H complex in silicon by anelastic relaxation experiments. Phys. Rev. B 44, 11486–11489 (1991).

    Article  ADS  Google Scholar 

  46. Gysin, U. et al. Temperature dependence of the force sensitivity of silicon cantilevers. Phys. Rev. B 69, 045403 (2004).

    Article  ADS  Google Scholar 

  47. Sementilli, L., Romero, E. & Bowen, W. P. Nanomechanical dissipation and strain engineering. Adv. Funct. Mater. 32, 2105247 (2022).

    Article  Google Scholar 

  48. Carney, D. et al. Mechanical quantum sensing in the search for dark matter. Quantum Sci. Technol. 6, 024002 (2021).

    Article  ADS  Google Scholar 

  49. Boureau, V., Reboh, S., Benoit, D., Hÿtch, M. & Claverie, A. Strain evolution of SiGe-on-insulator obtained by the Ge-condensation technique. APL Mater. 7, 041120 (2019).

    Article  ADS  Google Scholar 

  50. Hopcroft, M. A., Nix, W. D. & Kenny, T. W. What is the Young’s modulus of silicon? J. Microelectromech. Syst. 19, 229–238 (2010).

    Article  Google Scholar 

  51. Flinn, P. A., Gardner, D. S. & Nix, W. D. Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history. IEEE Trans. Electron Devices 34, 689–699 (1987).

    Article  ADS  Google Scholar 

  52. Chen, W. W., Sun, X. H., Wang, S. D., Lee, S. T. & Teo, B. K. Etching behavior of silicon nanowires with HF and NH4F and surface characterization by attenuated total reflection Fourier transform infrared spectroscopy: similarities and differences between one-dimensional and two-dimensional silicon surfaces. J. Phys. Chem. B 109, 10871–10879 (2005).

    Article  Google Scholar 

  53. Norte, R. A. Nanofabrication for On-Chip Optical Levitation, Atom-Trapping, and Superconducting Quantum Circuits. PhD thesis, California Institute of Technology (2015).

  54. Borselli, M., Johnson, T. J. & Painter, O. Measuring the role of surface chemistry in silicon microphotonics. Appl. Phys. Lett. 88, 131114 (2006).

    Article  ADS  Google Scholar 

  55. Anastassakis, E., Pinczuk, A., Burstein, E., Pollak, F. H. & Cardona, M. Effect of static uniaxial stress on the Raman spectrum of silicon. Solid State Commun. 8, 133–138 (1970).

    Article  ADS  Google Scholar 

  56. Ossikovski, R., Nguyen, Q., Picardi, G., Schreiber, J. & Morin, P. Theory and experiment of large numerical aperture objective Raman microscopy: application to the stress-tensor determination in strained cubic materials. J. Raman Spectrosc. 39, 661–672 (2008).

    Article  ADS  Google Scholar 

  57. Hart, T. R., Aggarwal, R. L. & Lax, B. Temperature dependence of Raman scattering in silicon. Phys. Rev. B 1, 638–642 (1970).

    Article  ADS  Google Scholar 

  58. Anastassakis, E., Cantarero, A. & Cardona, M. Piezo-Raman measurements and anharmonic parameters in silicon and diamond. Phys. Rev. B 41, 7529–7535 (1990).

    Article  ADS  Google Scholar 

  59. Catalini, L., Tsaturyan, Y. & Schliesser, A. Soft-clamped phononic dimers for mechanical sensing and transduction. Phys. Rev. Appl. 14, 014041 (2020).

    Article  ADS  Google Scholar 

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We thank D. Wilson and A. Ghadimi for their contributions in the early stages of the project, and R. Groth for assistance with sample characterization. This work was supported by funding from the Swiss National Science Foundation under grant agreement no. 182103, the EU H2020 research and innovation programme under grant agreement no. 732894 (HOT) and the European Research Council grant no. 835329 (ExCOM-cCEO). N.J.E. acknowledges support from the Swiss National Science Foundation under grant no. 185870 (Ambizione). This work was further supported by the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO) contract no. HR00111810003. All the samples were fabricated at the Center of MicroNanoTechnology (CMi) at EPFL.

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Authors and Affiliations



A.B. fabricated the devices with assistance from M.J.B. The devices were characterized by A.B., D.A.V. and N.J.E. The characterization setup was constructed by S.A.F., N.J.E. and A.B. TEM was performed by V.B. Data were analysed by A.B., D.A.V. and N.J.E. The manuscript was written by A.B. and N.J.E. with assistance from the other authors. The work was supervised by N.J.E. and T.J.K.

Corresponding authors

Correspondence to A. Beccari, N. J. Engelsen or T. J. Kippenberg.

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Nature Physics thanks Christopher Baker and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Fabrication of sSi nanostrings.

a, Suspended nanostrings without a stress-compensation layer are subject to a compressive load well beyond the critical value for out-of-plane buckling. Green - SiO2, blue - patterned sSi + FOX16 mask. b, The encapsulation layers (dark and light green) compensate compressive stress; the suspended beam is tensioned and flat. c, Determination of the intrinsic stress in PECVD SixNy through wafer bow. The wafer height profile is measured by scanning a laser beam and recording the angular deflection of the beam reflected off the surface. The magnitude and sign of the profile curvature permit to reconstruct the biaxial stress. Red circles - wafer profile before nitride deposition, orange circles - wafer bow after nitride deposition, with the initial profile subtracted.

Extended Data Fig. 2 Color-coded strain map in sSi, measured with dark-field electron holography.

a, Portion of Fig. 2f magnified around the sSi layer. b, Distribution of reconstructed strain values in the sSi layer, centered on the average strain 〈ϵ〉 = 0.85%.

Extended Data Fig. 3 Raman scans with variable laser power.

Micro-Raman scans along a PnC beam unit cell are conducted, with variable laser power. A linear dependence of the Raman scattering wavevector on the power is observed; the dotted line indicates the extrapolation to zero optical power.

Extended Data Fig. 4 Schematic of the measurement setup.

ECDL: external cavity diode laser. EOM: electro-optical modulator. WDM: fiber optic wavelength division multiplexer. PZT: piezoelectric transducer. ESA: spectrum analyzer.

Extended Data Fig. 5 Comparison of ringdowns with continuous and gated acquisition.

The probe laser, with an impinging power  1 mW, induces, in the continuous ringdown (green line), additional optical damping, leading to a faster amplitude decay rate than the one set by the intrinsic dissipation of the nanostring. For the gated ringdown (blue dots), the measurement duty cycle (ratio of gate duration over the time interval between successive gates) is 10%, and the gate repetition rate is 1/20 s. In both ringdowns, nonlinear damping is observed at the highest amplitudes59, and exponential fits (dashed lines) are performed only within the linear decay regions. The data were acquired at T ≈ 10 K for the 6.0 mm sample presented in the main text.

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Supplementary Information

Supplementary Sections 1–5 and Figs. 1–4.

Source data

Source Data Fig. 3

Plain text data for Fig. 3.

Source Data Fig. 4

Plain text data for Fig. 4.

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Beccari, A., Visani, D.A., Fedorov, S.A. et al. Strained crystalline nanomechanical resonators with quality factors above 10 billion. Nat. Phys. 18, 436–441 (2022).

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