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Strained crystalline nanomechanical resonators with quality factors above 10 billion

Abstract

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.

Data availability

Source data are provided with this paper. Data supporting the figures and lithographic masks used for microfabrication are available on Zenodo (https://doi.org/10.5281/zenodo.4606602).

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Acknowledgements

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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Contributions

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.

Supplementary information

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). https://doi.org/10.1038/s41567-021-01498-4

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