Mechanical resonator device

Abstract

A mechanical resonator device. The resonator device includes a resonator element made of an elastic material under tensile stress and adapted for sustaining at least one oscillation mode; and a clamping structure supporting the resonator element. The clamping structure has a phononic density of states exhibiting a bandgap or quasi-bandgap such that elastic waves of at least one polarisation and/or frequency are not allowed to propagate through the clamping structure. The resonator element and the clamping structure are configured to match with a soft-clamping condition that elastic waves of polarisation and/or frequency corresponding to the at least one oscillation mode of the resonator penetrate evanescently into the clamping structure in a manner such as to minimize bending throughout the entire resonator device. Thereby, bending related loss may be minimized and the Q-factor of the mechanical resonator may be maximized.

Claims

1. Mechanical resonator device, the resonator device comprising a resonator element made of an elastic material under tensile stress and adapted for sustaining at least one oscillation mode; and a clamping structure supporting the resonator element; wherein the clamping structure has a phononic density of states exhibiting a bandgap or quasi-bandgap such that elastic waves of at least one polarisation and/or frequency are not allowed to propagate through the clamping structure; and wherein the resonator element and the clamping structure are configured in a manner such that elastic waves of polarisation and/or frequency corresponding to the at least one oscillation mode of the resonator element penetrate evanescently into the clamping structure so as to provide a soft-clamping of the resonator element; wherein configuration of the resonator element and the clamping structure includes integral minimization of bending related loss over tensile energy over the entire resonator device.

2. Resonator device according to claim 1, wherein an energy-normalized mode shape curvature integral for said oscillation mode of the resonator device is less than an energy-normalized mode shape curvature integral for a corresponding mode with the same frequency of a reference resonator directly suspended from fixed anchoring means on a substrate.

3. Resonator device according to claim 1, wherein the bandgap or quasi-bandgap is produced in the clamping structure by a periodic pattern with lattice constant a.

4. Resonator device according to claim 1, wherein the resonator element and the clamping structure are made of the same elastic material under tensile stress.

5. Resonator device according to claim 1, wherein the resonator element and the clamping structure are formed in a membrane.

6. Resonator device according to claim 1, wherein the at least one oscillation mode of the resonator element is an out-of-plane oscillation mode.

7. Resonator device according to claim 1, wherein the elastic material under tensile stress is one of silicon nitride, diamond, quartz, aluminium nitride, silicon carbide, gallium arsenide, indium gallium arsenide, aluminium gallium arsenide, aluminium, gold, graphene, polymer materials, or combinations thereof.

8. Resonator device according to claim 1, wherein the elastic material under tensile stress is one of dielectrics, metals, semiconductors, metal dichalcogenides, ceramics or piezoelectric materials, or combinations thereof.

9. Resonator device according to claim 1, wherein an initial stress in the elastic material under tensile stress is between 10 MPa and 50 GPa.

10. Resonator device according to claim 1, wherein the resonator device comprises at least one further resonator element supported by the clamping structure, wherein each further resonator element is made of an elastic material under tensile stress and adapted for sustaining at least one respective further oscillation mode; and wherein each of the further resonator elements is configured with respect to the clamping structure in a manner such that elastic waves of polarisation and frequency corresponding to the at least one further oscillation mode of the at least one further resonator element penetrate evanescently into the clamping structure so as to provide a soft-clamping of the further resonator element.

11. Resonator device according to claim 10, wherein the resonator element, the at least one further resonator element, and the clamping structure are made of the same elastic material under tensile stress.

12. Resonator device according to claim 1, wherein a decay length of evanescent elastic waves is in the range of 0.1 to 20 times the wavelength of the elastic waves in the clamp.

13. Method of providing a mechanical resonator device, the resonator device comprising a resonator element and a clamping structure supporting the resonator element, the method comprising the steps of: determining at least one oscillator mode for the resonator element, determining a phononic density of states for the clamping structure, the phononic density of states exhibiting a bandgap or quasi-bandgap such that elastic waves of at least one polarisation and/or frequency are not allowed to propagate through the clamping structure; and matching the resonator element and the clamping structure in a manner such that elastic waves of polarisation and/or frequency corresponding to the at least one oscillation mode of the resonator penetrate evanescently into the clamping structure so as to provide a soft-clamping of the resonator element; wherein matching the resonator element and the clamping structure includes integrally minimizing bending related loss over tensile energy over the entire resonator device.

14. Method according to claim 13, wherein an energy-normalized mode shape curvature integral for said oscillation mode of the resonator device is less than an energy-normalized mode shape curvature integral for a corresponding mode with the same frequency of a reference resonator directly suspended from fixed anchoring means on a substrate.

15. Method according to claim 13, wherein matching the resonator element and the clamping structure includes determining a quality factor Q of the resonator device according to the equation:
Q.sup.−1=ΔW/(2πW), wherein ΔW is the bending related loss per oscillation cycle of a given mode, and W is the total energy of the mode.

16. Method according to claim 15, wherein the given mode is an out-of-plane oscillation mode, wherein u(x,y) denotes out-of-plane displacement of the resonator device in a z-direction as a function of a lateral position denoted by lateral position coordinates x and y, and wherein the bending related loss per oscillation cycle ΔW is determined as: $\Delta W\approx \int \frac{\pi E_2}{1-{\nu }^2}{z^2\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)}^2\mathrm{dV},$ wherein E.sub.2 denotes a complex part of the Young's modulus, wherein v denotes a Poisson ratio, wherein dV denotes a volume element; and wherein the total energy W of the given mode is determined as the tensile energy: $W\approx \int \frac{\overline{\sigma }}{2}\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial y}\right)}^2\right)dV.,$ wherein denotes tensile pre-stress in the elastic material of the resonator device.

17. Sensor for the detection of mass and/or forces, the sensor comprising a resonator device, the resonator device comprising a resonator element made of an elastic material under tensile stress and adapted for sustaining at least one oscillation mode; and a clamping structure supporting the resonator element; wherein the clamping structure has a phononic density of states exhibiting a bandgap or quasi-bandgap such that elastic waves of at least one polarization and/or frequency are not allowed to propagate through the clamping structure; and wherein the resonator element and the clamping structure are configured in a manner such that elastic waves of polarization and/or frequency corresponding to the at least one oscillation mode of the resonator element penetrate evanescently into the clamping structure so as to provide a soft-clamping of the resonator element; the sensor further comprising a readout device adapted to sensing a displacement of the resonator element and provide a detection signal representative of the displacement, wherein the read-out device is configured for sensing an out-of-plane oscillation of the resonator element.

18. Sensor according to claim 17, wherein the read-out device uses an optical and/or electronic readout element for sensing displacement of the resonator element.

Description

BRIEF DESCRIPTION OF THE FIGURES

(1) In the following, an embodiment of the invention is discussed in detail by way of example with reference to the appended figures, which show in

(2) FIG. 1 Measured and simulated data related to device characterization: a) Micrograph of a silicon nitride membrane patterned with a phononic crystal structure (left) and measured out-of-plane displacement pattern of the fundamental localized mode (right), of a device with lattice constant a=160 μm; b) Simulation of the stress redistribution in a unit cell of the hexagonal honeycomb lattice (left) and the corresponding first Brillouin zone (right); c) Simulated band diagram of a unit cell (left) and measured Brownian motion in the central part of the device shown in (a). Localized modes A-E are color-coded (A-red, B-orange, C-yellow, D-green, E-blue), the peak around 1.5 MHz an injected for calibration of the displacement amplitude; d) Ring-down measurements of A (red/bottom) and E (blue/top) modes of two membrane resonators with a=346 μm;

(3) FIG. 2 Quality factor statistics: a) Measured Q-factors of A-E defect modes in membranes of h=35 nm thickness and different size. Black (grey) dashed lines is a Q∝f.sup.−2 (Q∝f.sup.4) guide to the eye; Symbols (A-red diamond, B-orange square, C-yellow triangle, D-green pentagon, E-blue circle) indicate different localized defect modes corresponding to those shown in FIG. 1c; b) Corresponding Q f-products; For reference, solid orange (I), red (II) and blue (III) lines indicate the quantum limit of crystalline silicon, quartz and diamond resonators, respectively [28]; Solid green line (IV) shows expected value for the fundamental mode of a square membranes under σ=1 GPa stress (4), and dashed grey line (V) indicates Q f=6×10.sup.12 Hz required for room-temperature quantum opto-mechanics and reached by trampoline resonators [21] at f≈0.2 MHz (not shown);

(4) FIG. 3 Scaling of quality factors: (a)-(e) Measured mode shapes of localised defect modes (top) with frequencies f.sub.A, f.sub.B, f.sub.C, f.sub.D, f.sub.E=1.4627 MHz, 1.5667 MHz, 1.5697 MHz, 1.6397 MHz, 1.6432 MHz (for a device with a=160 μm), as well as characteristic scalings with the membrane thickness h (middle row) and lattice constant a (bottom row), when either are varied; Dashed grey lines in the middle row indicate a Q/a.sup.2∝h.sup.−1 scaling, while the solid black lines take into account the additional losses due to the bulk following eq. (5); Dashed grey line in bottom row indicates Q×h∝a.sup.2 scaling; The semitransparent points correspond to the individual membranes, while the solid points (error bars) indicate their mean value (standard deviation);

(5) FIG. 4 Enhancing dissipation dilution: a) Simulated displacement field of the fundamental mode and zoom on the defect (inset); b) Absolute value of mode curvature and zoom on the defect (inset); c) Simulated displacement along a vertical line through the defect (points); the curve labelled [red] is an exponential function as a guide to the eye, while the curve labelled [grey] represents a simplistic model of an exponentially decaying sinusoid as further detailed below; d) Absolute value of mode curvature (blue line labelled soft) along the same section as c); Curvature is normalized to the square-root of the total stored energy in the resonator; Also shown, for comparison, is the normalized curvature of a square membrane with the same frequency (grey line labelled rigid); Inset is a zoom on the membrane clamp, revealing the exceedingly large curvature of a rigidly clamped membrane, and absent with soft clamping; e) Compilation of measured (transparent markers and error bars, indicating standard deviations, colour coded according to FIG. 1c (A-red, B-orange, C-yellow, D-green, E-blue) and simulated (hollow circles) quality factors, normalized to a.sup.2/h, consistent with the observed scaling with the corresponding quantities for h={35 nm; 66 nm; 121 nm};

(6) FIG. 5 Alternative structures: a) Micrograph of a trampoline-like resonator embedded in a phononic crystal membrane; b) Measured higher-order localised mode of a large defect;

(7) FIG. 6 Simulated displacement fields: (a-e) Mode shapes of the localised vibrational modes A E of the defect, showing excellent agreement with the measurements in FIG. 3;

(8) FIG. 7 Projections of the displacement fields: (a-e) Projections of the simulated displacement fields along the x- (orange, thin solid line) and y-directions (red, thick solid line) for modes.

DETAILED DESCRIPTION

(9) The following is a detailed discussion of an exemplary embodiment of the invention.

(10) “Ultra-coherent nanomechanical resonators via soft clamping and dissipation dilution”

(11) Abstract

(12) Their small mass and high coherence render nanomechanical resonators the ultimate force probe, with applications ranging from bio-sensing and magnetic resonance force microscopy, to quantum opto-mechanics. A notorious challenge in these experiments is thermomechanical noise related to dissipation to internal or external loss channels. Here, we introduce a novel approach to defining nanomechanical modes, which simultaneously provides strong spatial confinement, full isolation from the substrate, and dilution of the resonator material's intrinsic dissipation by five orders of magnitude. It is based on a phononic bandgap structure that localizes the mode, without imposing the boundary conditions of a rigid clamp. The reduced curvature in the highly tensioned resonator enables Q>10.sup.8 at 1 MHz, yielding the highest mechanical Qf-products (>10.sup.14 Hz) yet reported at room temperature. The corresponding coherence times approach those of optically trapped dielectric particles. Extrapolation to 4.2 Kelvin predicts ˜quanta/ms heating rates, similar to trapped ions.

(13) Introduction

(14) Nanomechanical resonators offer exquisite sensitivity in the measurement of mass and force. This has enabled dramatic progress in several frontier fields of contemporary physics, such as the detection of individual biomolecules [1], spins [2], and mechanical measurements of quantum vacuum fields [3]. Essentially, this capability originates from the combination of two features: first, a low mass in, so that small external perturbations induce relatively large changes in the motional dynamics. Second, high coherence, quantified by the quality factor Q, implying that random fluctuations masking the effect of the perturbation are small. In practice, a heuristic Q∝M.sup.1/3 rule, likely linked to surface losses [4], often forces a compromise, however.

(15) A notable exception to this rule has been reported recently, in the form of highly stressed silicon nitride string [5] and membrane [6] resonators, achieving Q˜10.sup.6 at MHz-resonance frequencies f, and nanogram (ng) effective masses m.sub.eff. By now it is understood [7, 8, 9] that the pre-stress “dilutes” the dissipation intrinsic to the material (or its surfaces), a feat known and applied also in the mirror suspensions of gravitational wave antennae [10]. The resulting exceptional coherence has enabled several landmark demonstrations of quantum effects with nanomechanical resonators [11, 12, 13, 14, 15] already at moderate cryogenic temperatures.

(16) Systematic investigations [16] of such silicon nitride resonators have identified an upper limit for the product Q.Math.f<6×10.sup.12 Hz≈k.sub.RT.sub.R/(2πℏ) for the low-mass fundamental modes, insufficient for quantum experiments at room temperature T.sub.R. Better Qf-products have been reported in high-order modes of large resonators, but come at the price of significantly increased mass and intractably dense mode spectrum [17, 18]. Consequently, the revived development of so-called trampoline resonators [19] has received much attention recently [20, 21]. In these devices, four thin, highly tensioned strings suspend a small, light (m.sub.eff˜ng) central pad. The fundamental oscillation mode of the pad can (marginally) achieve Q.Math.f≈6×10.sup.12 Hz, provided that radiation losses at the strings' clamping points are reduced through a mismatched, i.e. very thick, silicon substrate [21].

(17) In this work, we choose a different approach based on phononic engineering [22]. Our approach not only suppresses radiation to the substrate [23] strongly, it also enhances dissipation dilution dramatically. This is because it allows the mode to penetrate, evanescently, into the “soft” clamping region, which exhibits a phononic bandgap around the mode frequency. This strongly reduces the mode's curvature, whose large value close to a rigid clamp usually dominates dissipation if radiation loss is absent [10, 7, 8, 9].

(18) As a result, we obtain Qf-products exceeding 10.sup.14 Hz at MHz frequencies, combined with rig-masses—an ideal combination for quantum opto-mechanics experiments. Remarkably, to the best of our knowledge, this is the highest room-temperature Qf-product of any mechanical resonator fabricated to date. This includes silicon MEMS devices and bulk quartz resonators, which are fundamentally limited to Qf≤3×10.sup.13 Hz by Akhiezer damping, but also LIGO's mirror suspensions [24, 25, 26, 27, 28].

(19) Key Design Features

(20) FIG. 1 shows the key characteristics of the devices fabricated following this new approach. A thin (thickness h=35 . . . 240 nm) silicon nitride film is deposited on a standard silicon wafer with a homogenous in-plane stress of σ≈1.27 GPa. The film is subsequently patterned with a honeycomb lattice (lattice constant a) of air holes over a ˜(19×19.5)a.sup.2 square region, where a=87 . . . 346 μm in the batch studied here. Back-etching the silicon substrate releases membranes of a few mm side length (Methods). Crucially, the lattice is perturbed by a small number of removed and displaced holes. They form a defect of characteristic dimension ˜ a in the centre of the membrane, to which the mechanical modes of interest are confined.

(21) In contrast to earlier opto-mechanical devices featuring phononic bandgaps [29, 30, 31, 15], a full bandgap is not expected [32] here, due to the extreme ratio h/a≤10.sup.−3. A quasi-bandgap can nonetheless be opened [33, 34], whereby only in-plane modes with high phase velocity persist in the gap (FIG. 1c). Under high tensile stress, a honeycomb lattice achieves a relatively large bandgap—about 20% of the centre frequency 251 m/s.Math.α.sup.−1—with a hole radius r=0.26a. At the same time, the design allows straightforward definition via photolithography, given that the tether width is still above 5 μm even for the smallest a. Evidently, the phonon dispersion is altered dramatically by the in-plane (d. c.) stress, which relaxes to an anisotropic and inhomogeneous equilibrium distribution that must be simulated (FIG. 1b) or measured [35] beforehand.

(22) We characterise the membranes' out-of-plane displacements using a home-built laser interferometer, whose sampling spot can be raster-scanned over the membrane surface (Methods and [36]). FIG. 1b shows the displacement spectrum obtained when averaging the measurements obtained in a raster scan over a (1210 μm).sup.2-square inside the defect, while the (a=160 μm) membrane is only thermally excited. The spectral region outside ˜1.41 . . . 1.68 MHz is characterised by a plethora of unresolved peaks, which can be attributed to modes delocalised over then entire membrane. In stark contrast, within this spectral region, only a few individual mode peaks are observed, a direct evidence for the existence of a bandgap. Its spectral location furthermore agrees with simulations to within 2%. Extracting the amplitude of a few spectral bins around the first peak effectively tunes the interferometer to this mode, and allows mapping out its (r. m. s.) displacement pattern when raster-scanning the probe. FIG. 1a shows an image constructed in this way, zooming on the defect region. The pattern resembles a fundamental mode of the defect, with no azimuthal nodal lines, and its first radial node close to the first ring of holes defining the defect. Its eigenfrequency is f.sub.A≈235 mls.Math.a.sup.−1. Outside the defect, the displacement follows the hexagonal lattice symmetry, but decays quickly with increasing distance to the centre. This is expected due to the forbidden wave propagation in the phononic bandgap, and leads to a strong localisation of the mode to the defect.

(23) Ultrahigh Quality Factors

(24) To assess the mechanical quality of the mode, we subject the defect to a second “excitation” laser beam, whose amplitude is modulated with the resonance frequency of the mode. Instead of a spatial scan, we now continuously monitor the defect's motion at the mode frequency, by lock-in detection of the interferometer signal. When the excitation laser is abruptly turned off, we observe the ring-down of the mechanical mode (Methods). Under a sufficiently high vacuum (p≤10.sup.−6 mbar) but room temperature, it can last for several minutes at MHz frequencies. FIG. 1d shows an example of an “E”-mode with f=777 kHz and amplitude ringdown time 2r=(87.7±0.8)s. This corresponds to Q=2πfτ=(214±2)×10.sup.6 and Q.Math.f=(1.66±0.02)×10.sup.14 Hz.

(25) To corroborate and explain this result, we have embarked on a systematic study of more than 400 modes in devices of varying thickness and size (rescaling the entire pattern with a). FIG. 2 shows a compilation of quality factors and Qf-products measured in 5 different modes of ˜20 devices with varying size a=87 . . . 346 μm but fixed thickness h=35 nm. Clearly, the Qf-products exceed those of trampoline resonators by more than an order of magnitude, reaching deeply into the region of Q.Math.f>6×10.sup.12 Hz. It also consistently violates the “quantum” (Akhiezer) damping limit of crystalline silicon, quartz and diamond at room temperature, which fundamentally precludes mechanical resonators made from these materials from reaching beyond Qf˜3×10.sup.13 Hz [24, 25, 26, 28].

(26) Our data, in contrast, do not seem to be limited by Akhiezer damping. Indeed a crude estimate following [28] indicates Q.sub.Akhf˜(10.sup.15 Hz) for silicon nitride. Furthermore, since the relaxation times are much faster than the mechanical oscillation period, we would expect constant Qf, rather than the Q∝f.sup.−2 trend discernible in our data. Thermoelastic damping, another notorious dissipation mechanism in micro- and nanomechanical resonators [37], has previously been estimated [6, 18] to allow Q>10.sup.11 at ˜1 MHz in highly stressed SiN resonators, and is therefore disregarded.

(27) In absence of radiation loss [23]—an assumption we discuss below—stressed membrane resonators are usually limited by internal dissipation. Its microscopic nature is not known, but evidence is accumulating that it is caused by two-level systems [38, 39] located predominantly in a surface layer [16]. Their effect is well captured by a Zener model [7, 8, 9], in which the oscillating strain (ϵ(t) Re[{tilde over (ϵ)}.sub.0e.sup.i2πft]) and stress (t)=Re[.sub.0e.sup.i2πft]) fields acquire a phase lag, {tilde over (σ)}.sub.0=E{tilde over (ϵ)}.sub.0, from a complex-valued Young's modulus E=E.sub.1+iE.sub.2. Per oscillation cycle, mechanical work amounting to Δw=(t){tilde over ({dot over (ϵ)})}(t)dt=πE.sub.2{tilde over (ϵ)}.sub.0|.sup.2 is done in each dissipating volume element. Integrating up the contributions yields the loss per cycle ΔW=∫ΔwdV. The comparison with the mode's total energy W determines its quality factor via

(28) $\begin{array}{cc}{Q}^{-1}=\frac{\Delta W}{2\pi W}& \left(1\right)\end{array}$

(29) In highly stressed strings and membranes, W is dominated by the large pre-stress σ, counteracting the membrane's elongation. In contrast, for small amplitudes, the oscillating strain and thus per-cycle loss is dominated by pure bending. As a result, W and ΔW in eq. (1) depend on different parts of the strain tensor {tilde over (ϵ)}.sub.0={tilde over (ϵ)}.sub.0.sup.along+{tilde over (ϵ)}.sub.0.sup.bend associated with the mode's displacement profile. For pure out-of-plane displacement u(x, y) of a clamped membrane, this translates into the imperative to minimise bending-related loss

(30) $\begin{array}{cc}\Delta W\approx \int \frac{\pi E_2}{1-{\nu }^2}{z^2\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)}^2\mathrm{dV}& \left(2\right)\end{array}$

(31) over the tensile energy

(32) $\begin{array}{cc}W\approx \int \frac{\overline{\sigma }}{2}\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial y}\right)}^2\right)\mathrm{dV}.& \left(3\right)\end{array}$

(33) For the fundamental mode of a plain square membrane of size L, this analysis predicts
Q.sub.□.sup.−1=(2λ+2π.sup.2λ.sup.2)Q.sub.int.sup.−1≈2λQ.sub.int.sup.−1,  (4)

(34) in very good agreement with available data [9, 16]. Here, λ=√{square root over (E.sub.1/(12))}h/L quantifies the “dilution” of the intrinsic dissipation Q.sub.int.sup.−1≡E.sub.2/E.sub.1 by the large internal stress . That is, λ<<1, given the extreme aspect ratio h/L˜(10.sup.−4) and the Young's modulus E.sub.1=270 GPa and pre-stress σ=1.27 GPa. In an extension of this model [16], extra loss in a δh-thick surface layer E.sub.2(z)=E.sub.2.sup.Vol+E.sub.2.sup.Surfθ(|z|−(h/2−δh)) can be mapped on a thickness dependent
Q.sub.int.sup.−1(h)=Q.sub.int,Vol.sup.−1+(βh).sup.−1.  (5)

(35) β=E.sub.1/(6δhE.sub.2.sup.Surf). If the latter dominates, it yields a total scaling Q.sub.□.sup.−1∝h.sup.0/L.sup.1 with the geometry of the device. Our devices, however, follow a rather different scaling (FIG. 3), even though they are embedded in square membranes.

(36) In this context, it is important to understand the origin of the two terms in eq. (4): the first, dominating term is associated with bending in the clamping region, while the second arises from the sinusoidal mode shape in the centre of the membrane [9]. The former is necessary to match this sinusoidal shape with the boundary conditions u({right arrow over (r)}.sub.cl)=({right arrow over (n)}.sub.cl.Math.{right arrow over (∇)})u({right arrow over (r)}.sub.cl)=0, where {tilde over (r)}.sub.cl=(x.sub.cl, y.sub.cl) are points on, and {right arrow over (n)}.sub.cl the corresponding normal vectors to, the membrane boundary. It requires, in particular, that the membrane lie parallel to the substrate directly at the clamp, before it bends upwards supporting the sinusoidal shape in the centre. The extent, and integrated curvature of this clamping region is determined by its bending rigidity.

(37) The boundary conditions differ dramatically in our case,
u.sub.d({right arrow over (r)}.sub.cl)−u.sub.pc({right arrow over (r)}.sub.cl)=0
({right arrow over (n)}.sub.cl.Math.{right arrow over (∇)})(u.sub.d({right arrow over (r)}.sub.cl)−u.sub.pc({right arrow over (r)}.sub.cl))=0,  (6, 7)

(38) requiring only the matching of the defect mode u.sub.d with the mode in the patterned part u.sub.pc. If the phononic crystal clamp supports evanescent waves of complex wavenumber k.sub.pc, it stands to reason that this “soft” clamping can be matched to a sinusoidal mode of the defect, characterised by a wavenumber k.sub.d≈Re(k.sub.pc)>>|Im(k.sub.pc)| without requiring significant extra bending. This eliminates the first term in eq. (4), leaving only the dramatically reduced dissipation

(39) $\begin{array}{cc}{Q}^{-1}=\eta \frac{E}{\overline{\sigma }}\frac{h^2}{a^2}{Q}_{int}^{-1}\left(h\right),& \left(8\right)\end{array}$

(40) dominated by the sinusoidal curvature in the defect (and evanescent displacement fields) ∝k.sub.d.sup.2∝1/a.sup.2, whereby the numerical pre-factor η depends on the exact mode shape. In the surface damping (thin-membrane) limit, we have again Q.sub.int.sup.−1(h)≈(βh).sup.−1 and obtain the overall scaling Q∝a.sup.2/h. This is indeed the scaling we observe over a wide range of parameters, in all five defect modes, supporting our argumentation (FIG. 3). For the largest thickness, slightly better agreement is found assuming contributions from bulk loss (5), in agreement with expectations.

(41) Simulations

(42) Finite element simulations (Methods) further support the hypothesis of coherence enhancement by soft clamping. As in the simulations of the band diagrams of FIG. 1, all modelling was performed in a two-step process, solving first for the new d.c. stress distribution compliant with the boundary conditions introduced by patterning. Subsequently the eigenmodes and -frequencies were determined. Only five modes with substantial out-of-plane displacements are found in the bandgap. Their displacement patterns match the measured ones excellently, and the measured and simulated frequencies agree to within 2%.

(43) FIG. 4 shows the simulated displacement pattern of mode A, reproducing the measured mode pattern. It features strong localisation to the defect and a short, evanescent wave tail in the phononic crystal clamp. Already a simplistic model u(0, y)∝Re[exp(ik.sub.pc|x|)], with k.sub.pc, 2π(0.57+i0.085)/a reproduces a cross-sectional cut remarkably well (FIG. 4), supporting the scaling of curvature, and thus damping ∝1/a.sup.2. Note that more accurate modelling of the Bloch waves, their complex dispersion, and interaction with the defect [40, 33], is possible, but, in general, has to take bending rigidity into account to obtain the correct mode shape and curvature.

(44) With the full simulated displacements at hand, we are in a position to evaluate the bending energy (2) and the total stored energy (3) for a prediction of the quality factor (1). For computational efficiency, we use the maximum kinetic energy W.sub.kin.sup.max=(2πf).sup.2∫ρu(x,y).sup.2 dV/2=W, equivalent to the stored energy (3) (ρ=3200 kg/m.sup.3 is the density of SiN). A comparison of the normalised curvature |(∂.sub.x.sup.2+∂.sub.y.sup.2)u(x,y)|/√{square root over (W)} reveals the advantage of phononic crystal clamping over the fundamental mode of a square membrane: the latter exhibits a 2-order of magnitude larger curvature in the clamping region (FIG. 4). The somewhat larger integration domain of the phononic crystal membrane does not overcompensate this. Indeed, carrying out the integrals leads to quality factors in very good agreement with our measured values, much higher than the square membranes'. FIG. 4 shows the normalised quality factor Q×h/a.sup.2 for the five defect modes of more than 30 samples, assuming Q.sub.int (h=66 nm)=3750 .sub.[16]. Not only are the observed extreme quality factors consistent with simulations, the latter also confirm the trend for the highest Q's to occur in mode E, apparent (albeit not very significant) in the measurements.

(45) Not all the modes' measured features are in quantitative agreement with the simulations. Small (<2%) deviations in the resonance frequency are likely due to small disagreements between the simulated and fabricated devices' geometry and material parameters, and deemed unproblematic for the purpose of this study. It is remarkable, however, that mode D exhibits significantly lower measured quality factors than simulated. We attribute this to the insufficient suppression of the mode amplitude at the silicon frame, leading to residual radiation losses. Indeed we observe in simulations that mode D has the largest amplitude at the silicon frame, and the fact that mode D responds most sensitively to the clamping conditions of the sample (SI).

(46) Applications in Optomechanics and Sensing

(47) The ultrahigh quality factors enabled by soft clamping enable the creation of mechanical devices with unique advantages for experiments in quantum optomechanics, and mass and force sensing. In quantum opto-mechanics [3], the presence of a thermal reservoir (temperature T) has the often undesired effect that it leads to decoherence of a low-entropy mechanical quantum state: for example, a phonon from the environment excites the mechanical device out of the quantum ground state. This decoherence occurs at a rate

(48) $\begin{array}{cc}\gamma =\frac{k_{B}T}{Q}=1/\tau & \left(9\right)\end{array}$

(49) and sets the timescale τ over which quantum-coherent evolution of mechanical resonators can be observed. It is a basic experimental requirement that this time exceeds the oscillation period, so that coherent evolution can be tracked over a number of ˜2πfτ>1 cycles. At room temperature T=300 K, this translates to Q.Math.f>6×10.sup.12 Hz already discussed above. Our measured devices fulfil this condition with a significant margin.

(50) The more challenging requirement typically is to optically measure and/or prepare the mechanical quantum state within the time τ. Since the measurement rate is proportional to the inverse effective mass 1/m.sub.eff, the latter constitutes another important figure of merit. For a device with a=160 μm, we have measured effective masses m.sub.eff of {4.3, 4.7, 4.2, 9.8, 7.2}.Math.(1±0.11) ng for the five defect modes, which compare very favourably with m.sub.eff,□=4.9 ng of a square membrane with the same fundamental frequency f=1.46 MHz as mode A. Note that we have, in a recent experiment [15], realised optical measurements on similar square membranes at rates close to Γ.sub.meas=2π×100 kHz, which exceeds γ of the new resonators already at room temperature. In principle, it is thus possible to ground-state cool, or entangle the novel mechanical resonators at room temperature. The limits in force and mass sensitivity due to thermomechanical noise are also improved by the devices' enhanced coherence and low mass, given the Langevin force noise power spectral density

(51) $\begin{array}{cc}{S}_{FF}=2m_{eff}\frac{2\pi f}{Q}{k}_{B}T.& \left(10\right)\end{array}$

(52) The table below gives an overview of the figures of merit that ensue for the best device we have measured at room temperature. It shows key figures of merit of the E-mode in the best (a=320 μm) sample at room temperature, where all measurements were performed, and extrapolated to liquid helium temperature.

(53) TABLE-US-00001 TABLE 1 Temperature T 300 4.2 K Frequency f 777 kHz Effective mass m.sub.eff  16 ng Quality factor Q 214 535 10.sup.6 fQ.sub.-product f × Q 166 416 THz Decoherence rate γ/2π 33000 175 Hz Coherence time τ = 1/γ 5 910 μs # coherent oscillations 2πfτ 23 4400 1 Thermal force noise √{square root over (S.sub.FF)} 55 4.1 aN/√{square root over (Hz)}

(54) It also includes an extrapolation of these parameters to liquid helium temperatures. Here we assumed a 2.5-fold reduction of intrinsic dissipation (5) upon cooling, a factor consistently observed in SiN films [30, 38]. Note that the expected decoherence rates are about one order of magnitude slower than those of optically trapped dielectric particles [41], and reach those achieved with trapped ions [42]. It combines with the low effective mass to thermomechanical force noise at the aN/√{square root over (Hz)}-level, attractive for force sensing and -microscopy, such as magnetic resonance force microscopy (MRFM) of electron and nuclear spins [2, 43], as well as mass detection [44].

(55) Efficient optical and electronic readout techniques are readily available [45, 38], facilitating also applications beyond cavity optomechanics. Further, due to the relatively high mode frequencies, 1/f-type noise, and technical noise such as laser phase noise, is less relevant. On a different note, due to the relatively low density of holes, it can be expected that the heat conductivity (provided by unaffected high-frequency phonons) is higher than that of trampoline resonators, an advantage in particular in cryogenic environments, and a fundamental difference to dielectric particles trapped in ultrahigh vacuum. Finally, the sparse spectrum of well-defined defect modes provides an ideal platform for multimode quantum opto-mechanics [15], or may be harnessed for multimode sensing, e.g. for mass imaging [46].

(56) Outlook

(57) Clearly, the devices we have discussed above are just specific examples of soft clamping, and many other designs are possible. Engineering of defect shape and size will modify its mode spectrum, mass, and dilution properties, and it is evident that our design can be further optimised, depending on the application. For example, larger defects will exhibit a richer multi-mode structure, of interest for multimode optomechanics and mass moment imaging [15, 46]. Small, trampoline-like defects have a potential to further reduce mass, as desired for force sensing. To illustrate this point, FIG. 5 shows two other examples we have realised in our laboratory, and verified to possess a phononic bandgap enhancing dissipation dilution. Similarly, the phononic crystal clamp can be engineered for stronger confinement, optimised dilution, and/or directional transport. Exploiting these new opportunities, it will be interesting to apply our soft-clamping approach to truly one-dimensional resonators [34], and to explore networks and arrays of defects with ultra-coherent modes with defined couplings.

(58) In summary, we have introduced a novel type of mechanical resonator, which combines soft clamping and dissipation dilution. Its extremely weak coupling to any thermal reservoir can, on one hand, be harnessed to relax cooling requirements, and thus allow more complex experiments with long-coherence mechanical devices. On the other hand, if combined with cryogenic cooling, it enables ultraslow decoherence, which can be overwhelmed even by very weak coherent couplings to other physical degrees of freedom. A wide range of scientific and technical fields can thus benefit from this new development, including, but not limited to cavity opto-mechanics [3, 15], MRFM [2, 43], mass sensing and imaging [44, 46], and hybrid quantum systems [47, 48, 49].

(59) Methods

(60) A) Fabrication

(61) The membrane resonators are fabricated by depositing stoichiometric silicon nitride (Si.sub.3N.sub.4) via low-pressure chemical vapor deposition (LPCVD) onto a double-side polished 500 μm single-crystal silicon wafer. A 1.5 μm layer of positive photoresist (AZ MiR 701) is spin-coated on both sides of the wafer and patterns are transferred onto both sides of the wafer via UV illumination, corresponding to the phononic crystal patterns on one side and rectangular patterns on the other side of the wafer. The regions exposed to UV radiation are developed and the silicon nitride is etched in these regions using reactive ion etching. The photoresist is removed using acetone and oxygen plasma. In order to protect the phononic patterned side of the wafer during the potassium hydroxide (KOH) etch, we use a screw-tightened PEEK wafer holder, only allowing the KOH to attack the side with square patterns. Finally, after a 6-hour-etch the wafers are cleaned in a piranha solution, thus completing the fabrication process.

(62) B) Characterisation

(63) Optical measurements of the mechanical motion are performed with a Michelson interferometer at a wavelength of 1064 nm. We place a sample at the end of one interferometer arm and spatially overlap the reflected light with a strong local oscillator. The relative phase between the two beams is detected by a high-bandwidth (0-75 MHz) InGaAs balanced receiver and analysed with a spectrum analyser. In the local oscillator arm a mirror is mounted on a piezoelectric actuator that follows an electronic feedback from the slow monitoring outputs of the receiver, stabilizing the interferometer at the mid fringe position. Furthermore, the piezo generates a peak with a known voltage and frequency. By measuring the full fringe voltage, the power of this peak is converted into a displacement, which is then used to calibrate the spectrum. Using an incident probe power of ˜1 mW the interferometer enables shot noise limited sensitivity of 10 fm/√{square root over (Hz)},

(64) To image mechanical modes the probe beam is focused down to a spot diameter of 2 μm and raster-scanned over the sample surface by means of a motorized 3-axis translation stage with a spatial resolution of 1.25 μm. At each position we extract the amplitude of a few spectral bins around a mechanical peak and thereby construct a 2D map of displacement. The effective masses of mode A-E are extracted from the maximum of the displacement maps after subtracting a background (˜1 pm) and smoothing. Uncertainties in the mass are based on a 10% error of the above-mentioned displacement calibration. Quality factor measurements are performed by continuously monitoring the membrane motion at a fixed spot on the sample and optically exciting a given mechanical mode using a laser at a wavelength of 880 nm and incident power of 0.5-1 mW, which is amplitude modulated at the mode frequency using an acousto-optic modulator. We use a lock-in amplifier to analyse the driven motion and record mechanical ring-downs.

(65) For our systematic study of >400 mechanical modes, we place a 4-inch wafer each with 40 membranes in a high vacuum chamber at a pressure of a few 10.sup.−7 mbar and gently clamp down the wafer at its rim. By measuring mechanical damping as a function of vacuum pressure we verify that modes with Qf>10.sup.14 Hz are unaffected by viscous damping to within 10%.

(66) C) Simulations

(67) We use COMSOL Multiphysics to simulate the phononic patterned membrane resonators. The simulations are typically carried out in two steps. First, we perform a stationary study to calculate the stress redistribution due to perforation, assuming a homogeneous initial in-plane stress σ.sub.xx=σ.sub.yy. The redistributed stress is subsequently used in an eigenfrequency analysis, where we either calculate the eigenmodes of an infinite array for different wave vectors {right arrow over (k)} in the first Brillouin zone, or simply simulate the eigenmodes of actual devices.

(68) The mechanical quality factors are extracted by calculating the curvature of a given localized mode, which is obtained from an eigenfrequency simulation, as described above. In order to minimize numerical errors, the geometry is densely meshed. We ensure that increasing the number of mesh elements by a factor of 3 only results in 10% change in the integrated curvature.

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