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  • Open Access

Perimeter Modes of Nanomechanical Resonators Exhibit Quality Factors Exceeding 109 at Room Temperature

Mohammad J. Bereyhi, Amirali Arabmoheghi, Alberto Beccari, Sergey A. Fedorov, Guanhao Huang, Tobias J. Kippenberg, and Nils J. Engelsen
Phys. Rev. X 12, 021036 – Published 12 May 2022
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  • INTRODUCTION
  • SOFT-CLAMPED PERIMETER MODES
  • EXPERIMENTAL CHARACTERIZATION OF…
  • POLYGON RESONATORS WITH A WIDTH…
  • THERMAL-NOISE-LIMITED FORCE SENSING
  • PHONONIC DIMERS AND ARRAYS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Systems with low mechanical dissipation are extensively used in precision measurements such as gravitational wave detection, atomic force microscopy, and quantum control of mechanical oscillators via optomechanics and electromechanics. The mechanical quality factor (Q) of these systems determines the thermomechanical force noise and the thermal decoherence rate of mechanical quantum states. While the dissipation rate is typically set by the bulk acoustic properties of the material, by exploiting dissipation dilution, mechanical Q can be engineered through geometry and increased by many orders of magnitude Recently, soft clamping in combination with strain engineering has enabled room temperature quality factors approaching 109 in millimeter-scale resonators. Here we demonstrate a new approach to soft clamping which exploits vibrations in the perimeter of polygon-shaped resonators tethered at their vertices. In contrast to previous approaches, which rely on cascaded elements to achieve soft clamping, perimeter modes are soft clamped due to symmetry and the boundary conditions at the polygon vertices. Perimeter modes reach Q’s of 3.6×109—a record at room temperature—while spanning only two acoustic wavelengths. We demonstrate thermal-noise-limited force sensitivity of 1.3  aN/Hz for a 226 kHz perimeter mode with quality factor of 1.5×109 at room temperature. The small size of our devices makes them well suited for near-field integration with microcavities for quantum optomechanical experiments. Moreover, their compactness allows the realization of phononic lattices. We demonstrate a one-dimensional Su-Schrieffer-Heeger chain of high-Q perimeter modes coupled via nearest-neighbour interaction and characterize the localized edge modes.

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    • Received 11 August 2021
    • Accepted 17 March 2022
    • Corrected 21 December 2022

    DOI:https://doi.org/10.1103/PhysRevX.12.021036

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Optomechanics
    1. Physical Systems
    Micromechanical & nanomechanical oscillators
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Corrections

    21 December 2022

    Correction: The previously published Fig. 3 contained incorrect data in panel (c) and has been replaced. Corresponding changes to the caption have been made. The last sentence in the penultimate paragraph of Sec. III has been removed.

    Authors & Affiliations

    Mohammad J. Bereyhi*, Amirali Arabmoheghi*, Alberto Beccari, Sergey A. Fedorov, Guanhao Huang, Tobias J. Kippenberg†, and Nils J. Engelsen‡

    • Institute of Physics (IPHYS), Swiss Federal Institute of Technology Lausanne (EPFL), 1015 Lausanne, Switzerland

    • *These authors contributed equally to this work.
    • †tobias.kippenberg@epfl.ch
    • ‡nils.engelsen@epfl.ch

    Popular Summary

    Nanomechanical oscillators are among the most sensitive force and acceleration sensors and show promise as a quantum technology. However, mechanical loss fundamentally limits their performance by admitting thermal force noise from the environment and decreasing the lifetime of their quantum states. Here, we implement a new, compact mechanical-resonator design that reduces mechanical dissipation below that of state-of-the-art devices and demonstrates proof-of-principle force sensing.

    In mechanical resonators subjected to tensile stress, dissipation can be engineered by geometry. In particular, structures with higher aspect ratios have lower loss. However, such structures are challenging to fabricate and difficult to integrate in on-chip optomechanical systems, where their motion can be read out and controlled by light.

    In this work, we show that mechanical resonators shaped as polygons support mechanical modes around their perimeter that have extremely low loss. An added advantage of the design is that the aspect ratio of the resonator is much lower than the previous state-of-the-art devices. Stringlike devices such as ours typically have sub-optical-wavelength dimensions, which makes it hard to measure their motion optically. We show that we can widen a part of the resonator to allow high-fidelity optical measurement while maintaining low mechanical loss.

    The polygon resonator design can be used for force sensing, such as in magnetic resonance force microscopy, and to explore the limits of quantum measurements in integrated optomechanical systems.

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    Vol. 12, Iss. 2 — April - June 2022

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    • Figure 1
      Figure 1

      Comparison of state-of-the-art strained mechanical resonator designs. (a) Schematics of different mechanical resonators with their spatial extent illustrated by double-headed arrows (L). From left to right: polygon resonator, binary tree nanobeam [13], phononic crystal nanobeam [15], strain engineered nanobeam [15], and phononic crystal membrane [11]. (b) Simulated displacement profile of perimeter modes (plotted between two adjacent clamps) and phononic crystal soft-clamped modes with the same frequency. (c) Measured room temperature Q for polygons (blue circles) with different aspect ratios (L/h) and the best reported values for other designs of tensioned resonators [11, 13, 15]. For each data point, the corresponding resonance frequency is written on the plot. The green shaded area is accessible for the fundamental mode of a tensioned nanobeam with 20 nm thickness (h). The solid black line separating the gray and blue shaded areas shows the clampless limit for a 20-nm-thick beam [Eq. (2)]. The dashed arrow compares a polygon (Q=120×106) and a phononic crystal soft-clamped beam (Q=150×106) with the same thickness (h=20  nm) and frequency (f=2.5  MHz), showing 26 times lower aspect ratio for the polygon resonator, given the particular number of unit cells (n=26) chosen for the phononic crystal soft-clamped beam in Ref. [15]. (d) Size comparison between a 200−μm-long polygon resonator with Q=155×106 at 1.6 MHz and a 2×2  mm2 phononic crystal membrane with Q=74×106 at 1.46 MHz frequency [22], illustrating the reduced size.

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    • Figure 2
      Figure 2

      Ultrahigh-Q perimeter modes. (a) FEM simulation of the fundamental perimeter mode with an inset showing the generation of torsional deformation of the tethers. (b) False-colored scanning electron micrographs of polygon Si3N4 strained resonator devices with l0=700  μm, rl=0.6, w0=200  nm, and 20 nm thickness (scale bars correspond to 500  μm). Top right of (b) shows an enlarged junction of a stress-preserving square resonator (scale bar corresponds to 1.5  μm). (c) The effect of tether length (rl) on Q for samples with l0=250  μm, w0=300  nm, and 20 nm thickness. Blue, orange, and green data points represent square, hexagon, and octagon resonators, respectively. (d) The effect of polygon width (w0) on the Q for samples with l0=100  μm, rl=0.6, and 20 nm thickness. Open circles (joined by a dashed line) are FEM simulations and orange line is the Q computed from Eq. (3).

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    • Figure 3
      Figure 3

      Characterization of fundamental and higher-order perimeter modes. (a) A survey of all the measured Q’s of fundamental perimeter modes for square-, hexagon-, and octagon-shaped Si3N4 resonators (blue, orange, and green circles, respectively), with rl∈[0.2,0.4,0.6,0.8] and w0=200 and 300 nm. The solid black line shows the clampless limit for a 20 nm Si3N4 film assuming intrinsic quality factor of Qint=2500. FEM simulation Q’s (open circles) are plotted for optimal geometrical parameters (rl=0.6 and w0=200  nm). The dot-dashed line interpolates the FEM simulation Q’s and shows the λ2 scaling. The dashed line is the fundamental mode Q of a uniform nanobeam. (b) High-order perimeter mode Q’s (blue dots) of a square with l0=700  μm, rl=0.2, and w0=200  nm are shown with FEM predictions (open circles). Open squares show the calculated Q’s of a uniform beam with the same thickness and fundamental mode frequency. Displacement profiles of the first two perimeter modes are shown as insets. (c) Gated ringdown measurement (1 s on, 5 min off) of a square with l0=700  μm, rl=0.2, and w0=200  nm with a fundamental perimeter mode Q of 1.6×109 at 350 kHz. Orange line: exponential fit to the data shown by blue circles. Faded blue data are excluded from the fit as the laser is blocked at these times.

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    • Figure 4
      Figure 4

      Polygon resonators with width corrugations and interferometric readout. (a) Dark-field microscope image of a square-shaped polygon resonator with width corrugations (inset). White scale bars: main image, 100  μm; inset, 10  μm. Device parameters: l0=700  μm, rl=0.35, and w0 increased from 700 nm to 2.5  μm in the width corrugation. (b) The width profile of the pad shown in the inset of (a). (c) Perimeter mode thermomechanical sideband power at different positions on the side segment of a 1 mm square-shaped polygon resonator with width corrugations similar to the ones shown in (a) and (b). Blue circles: measured sideband power. Orange curve: expectation from the mode shape of the perimeter mode along a side segment (obtained from a FEM simulation). (d) Experimental setup used for interferometric position measurement and feedback control of the resonators. UHV: ultrahigh vacuum chamber with pressure below 10−8  mbar. PZT: piezoelectric actuator. Feedback: lock-in amplifier and digital phase-locked loop (PLL) employed to synthesize a feedback signal for the piezoelectric actuator. (e) Working principle of displacement calibration using nonlinear transduction in an interferometer. The gray line shows the transduction curve of the interferometer. The blue and red lines correspond to linear and nonlinear transduction regimes, respectively. (f) PSD of the homodyne signal SV(ω) (left) and corresponding calibrated displacement PSD Sx(ω) (right) for the driven motion of the perimeter mode. Boxes mark sidebands at the higher-order harmonics of the perimeter mode that are created due to the interferometer nonlinearity. (g) Fit of the first five Bessel functions to the five sidebands shown in (f). Gray dashed line corresponds to the fit value for β0 and boxes correspond to the normalized sideband amplitude An/V0qn(ϕ0).

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    • Figure 5
      Figure 5

      Feedback cooling and thermal-noise-limited force sensing. (a) Displacement fluctuation PSD of the feedback cooled perimeter mode. Blue line: experimental data. Orange line: Lorentzian fit to the data which gives a total linewidth of Γtot=2π×1.0  Hz. Red line: expected Brownian motion PSD. Dashed gray line: the imprecision noise background in the displacement measurement. (b) Force noise PSD. Blue and orange lines are the experimental data and the Lorentzian fit shown in (a), divided by the effective susceptibility [Eq. (7)]. Red line: the expected thermal force PSD. Dashed gray line: contribution from imprecision.

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    • Figure 6
      Figure 6

      Phononic dimers and arrays of polygon resonators. (a) FEM simulations of the in-phase and out-of-phase modes of a phononic dimer. The insets show the deformation of the coupler segment for the two cases. (b),(c) Optical micrographs of a fabricated dimer and a chain, both composed of square-shaped resonators with side lengths of 100  μm. The scale bars correspond to 100  μm. (d) The frequencies of the in-phase (square) and out-of-phase (circle markers) modes from data (filled markers) and FEM simulation (empty markers) for different coupling parameters. The inset indicates the coupling parameters of a phononic dimer. (e) Frequency splitting of the in-phase and the out-of-phase modes from data (filled markers) and FEM simulation (empty markers) for different coupling parameters. (f) FEM simulation of the edge mode profiles of a phononic array made of six square-shaped resonators. The slight asymmetry is due to the disorder in the mesh of the simulation. (g) Q of the hybridized modes from data (filled markers) and FEM simulation (empty markers) for different coupling parameters. The dashed line corresponds to quality factor of a single square-shaped resonator. (h) rms values of the Brownian motion of each site measured for the two split modes, showing localization of the mode at the edges of the chain.

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    • Figure 7
      Figure 7

      Theory model and FEM simulation comparison. We compare simulation and analytical predictions for a polygon resonator with N=4, l0=700  μm, and uniform stress (rw=2). (a) Dilution factor (DQ) for different perimeter modes (w0=200  nm, rl=0.4). (b) DQ of the first perimeter mode as the support length is varied (fixed w0=200  nm). The inset shows the mode displacement for r=1.9, with the color encoding the torsional energy density. (c) DQ of the first perimeter mode as the width of all the segments is varied (fixed rl=0.4).

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    • Figure 8
      Figure 8

      Strain engineered polygon resonators. (a) Simulated Q enhancement of polygon resonators via strain engineering (Qstress) compared to the stress-preserving design (Qnormal). (b) Schematic of the polygon width tapering for strain engineering (i) and stress-preserved design (ii) for a square-shaped polygon.

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    • Figure 9
      Figure 9

      Dependence of Q and SFth on the corrugation width. Square-shaped polygon resonators with corrugated widths are simulated. Q and SFth are shown in (a) and (b) for w0=0.2  μm (blue) and w0=0.7  μm (orange). The diamond marker corresponds to the device used in the force sensitivity experiment.

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    • Figure 10
      Figure 10

      Fabrication process flow. (1) LPCVD thin film deposition of Si3N4 on Si (red on gray). (2) ebeam lithography on Si3N4 wafer using FOX16 (blue). (3) Dry etching of Si3N4. (4) Recess ebeam (green corresponds to the second layer of FOX16). (5) DRIE etching of Si substrate. (6) KOH undercut and CPD.

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    • Figure 11
      Figure 11

      ebeam lithography stitching errors. An example of a device with (a) and without (b) stitching errors after the ebeam lithography process. The area of stitching is marked with the white dashed square for comparison. Scale bars correspond to 500 nm.

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    • Figure 12
      Figure 12

      Mode spectrum of polygon resonators. (a) Blue line: broadband thermomechanical noise spectrum, obtained from the device used in Sec. 5. Some of the identified modes are marked with their corresponding mode profile obtained from a FEM simulation. Orange line: signal with only local oscillator light. Gray line: signal with no light incident on detector. (b) Narrowband displacement spectrum around the perimeter mode of the same device; obtained while applying moderate dissipative feedback. The out-of-plane and in-plane perimeter modes are marked.

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    • Figure 13
      Figure 13

      Effect of chip mounting on the quality factor. Blue dots show the average measured Q when the chip is freely mounted on the sample holder and orange dots correspond to the average measured Q when fixing the chip tightly using metallic clamps pressing on the frame. Error bars show the standard deviation of four measurements for each sample.

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    • Figure 14
      Figure 14

      Frequency stability of perimeter modes. The measurement result of Allan deviation of a 1 mm square-shaped sample with w0=200  nm and rl=0.2 is plotted in blue dots. The dash-dotted line shows the calculated detection limit based on the measurement SNR, and the dashed line is the fit to the linear frequency drift in the long time limit. The fit yields a drift rate D∼10−8  s−1, which varies among different samples. The theoretical thermal mechanical limit is plotted in black line, which is many orders of magnitude below the measured Allan deviation, due to the exceptionally high quality factor. The scaling is reversed from the usual τ−1/2 to τ1/2 because we are in the limit where the mechanics lifetime is longer than the measurement time τ<1/Γm.

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    • Figure 15
      Figure 15

      Measurement of laser antidamping for different duty cycles. Ringdown measurement of a polygon resonator with varying duty cycles of the measurement strobes. The laser off time is swept while the on time is fixed to 1 s. Q’s are calculated from exponential fits to the data and are shown in the inset with marker color corresponding to the data traces.

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    • Figure 16
      Figure 16

      Frequency splitting of dimers. (a) FEM simulation of mode splitting for varying coupler length (lc) and different coupler widths (wc). (b) Compilation of all the simulated values in (a) plotted versus wc2/lc.

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    • Figure 17
      Figure 17

      Mode profiles of the collective perimeter modes. rms values of the Brownian motion of each site measured for all the collective perimeter modes of both topological (a) and trivial (b) chains of six square-shaped resonators.

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    • Figure 18
      Figure 18

      Disorder in phononic dimers and arrays. (a),(b) Spectra of the interferometric measurement when the laser beam is focused on sides of the two resonators in a phononic dimer. The dimer’s dimensions are similar to the ones shown in Fig. 6 with wc=2.1  μm and lc=11.1  μm corresponding to nominal frequency splitting of 1.1 kHz. (c) Asymmetry ratio [Eq. (m4)] obtained from the experimental spectra for devices with different coupling parameters, used for the measurement in Fig. 6. The dashed line is the model Eq. (m4) with δ set as its average value. (d) Histogram of the disorder parameter δ extracted from the data shown in (c) using Eq. (m5).

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