## Better performance under stress

Engineering stress or strain into materials can improve their performance. Adding mechanical stress to silicon chips, for instance, produces transistors with enhanced electron mobility. Ghadimi *et al.* explore the possibility of enhancing the vibrational properties of a micromechanical oscillator by engineering stress within the structure (see the Perspective by Eichler). By careful design of the micromechanical oscillator, and by building in associated stresses, exceptional vibrational properties can be produced. Such enhanced oscillators could be used as exquisite force sensors.

## Abstract

Extreme stresses can be produced in nanoscale structures; this feature has been used to realize enhanced materials properties, such as the high mobility of silicon in modern transistors. We show how nanoscale stress can be used to realize exceptionally low mechanical dissipation when combined with “soft-clamping”—a form of phononic engineering. Specifically, using a nonuniform phononic crystal pattern, we colocalize the strain and flexural motion of a free-standing silicon nitride nanobeam. Ringdown measurements at room temperature reveal string-like vibrational modes with quality (*Q*) factors as high as 800 million and *Q* × frequency exceeding 10^{15} hertz. These results illustrate a promising route for engineering ultracoherent nanomechanical devices.

Elastic strain engineering uses stress to realize unusual material properties (*1*). For instance, stress can enhance the electron mobility of a semiconductor, enabling more efficient solar cells (*2*) and smaller, faster transistors (*3*). In mechanical engineering, the pursuit of resonators with low dissipation (*4*) has led to studies of a complementary strain engineering technique known as dissipation dilution, whereby the stiffness of a stressed material is effectively increased without added loss (*5*–*8*). Unlike most bulk mechanical properties, dissipation dilution can improve with reduced device dimensions, implying that smaller-mass resonators can have higher quality factors *Q*. This unusual scaling is responsible for the anomalously high *Q* of Si_{3}N_{4} nanomechanical resonators (*8*–*11*) and has led to the emergence of “quantum-coherent” resonators with thermal decoherence times *ℏQ*/*k*_{B}*T* longer than one vibrational period (where *ℏ*, *k*_{B}, and *T* are the reduced Planck constant, Boltzmann constant, and bath temperature, respectively).

Whereas elastic strain engineering commonly relies on extreme inhomogeneous stresses produced by nanoscale deformation (*12*) [e.g., by lithographic patterning (*13*, *14*) or nano-indentation (*15*)], nearly all studies of dissipation dilution have focused on materials under weak, uniform stress produced during material synthesis. The main challenge in bridging these two approaches is to identify strategies to colocalize stress and mechanical motion at the nanoscale. Our strategy, based on phononic crystal patterning, is conceptually simple and entirely material-independent (Fig. 1): By weakly corrugating a prestressed nanobeam, we create a band gap for localizing its flexural modes around a central defect. By tapering the beam, we colocalize these modes with a region of enhanced stress. Reduced motion near the supports [“soft-clamping” (*8*)] results in higher dissipation dilution, while enhanced stress increases both dilution and mode frequency. We implemented this approach on tapered beams with extremely high aspect ratios (as long as 7 mm and as thin as 20 nm) made of 1.1 GPa–prestressed Si_{3}N_{4}, and achieved local stresses as high as 3.8 GPa.

To illustrate the basic features of our approach, we first consider a model for dissipation dilution of a nonuniform beam of length *L*, thickness *h*, and variable width *w*(*x*). Following an anelastic approach successfully applied to uniform nanobeams (*6*, *7*) and nanomembranes (*16*), we partition the potential energy of the beam *U* into two components: a dissipative component due to bending, , and a conservative component due to elongation, , where *u*(*x*) is the vibrational mode shape,* I*(*x*) = (1/12)*w*(*x*)*h*^{3} is the geometric moment of inertia, *E*_{0} is the Young’s modulus, T = *hw*(*x*)σ(*x*) is the tension, and σ(*x*) is the axial stress of the beam, respectively. The *Q* enhancement due to stress (the dissipation “dilution factor”) is given by the participation ratio of the lossy potential (*5*–*7*):(1)where *Q*_{0} is the intrinsic (undiluted) quality factor. For the familiar case of a uniform beam with a string-like mode shape described by *u*(*x*) ∝ sin(π*nx*/*L*), where *n* is the mode number, Eq. 1 implies that(2)where ρ is the material density and is the mode frequency. Nearly all stressed nanomechanical resonators studied to date have operated far below this limit. The main reason for this discrepancy is clamping loss; for example, in the case of a doubly clamped beam, boundary conditions *u*′(*x*_{0}) = *u*(*x*_{0}) = 0 require that the vibrational mode shape exhibit extra curvature [*u*′′(*x*)] near the supports (*x*_{0} = 0, *L*), resulting in a reduced dilution factor of the form(3)(*7*, *17*), where in units of axial strain ε = σ/*E*.

The uniform beam model (Eq. 3) gives several rules of thumb for maximizing the *Q* (or *Q* × *f* product) of a stressed nanomechanical resonator; namely, *Q* is typically highest for the fundamental mode (*n* = 1) and can be increased by increasing the aspect ratio (*L*/*h*) or stress. By contrast, *Q* × *f* is typically larger for high-order modes. Both strategies have been explored for a wide variety of beam- and membrane-like geometries (*17*, *18*). A third approach, recently demonstrated with a membrane (*8*), is to use periodic micropatterning [a phononic crystal (PnC)] to localize the mode shape away from the supports. By this soft-clamping approach, the leading term in Eq. 3 can be suppressed, giving access to the performance of an ideal clamp-free resonator (Eq. 2).

Complementary to soft-clamping, our approach consists of colocalizing the mode shape with a region of geometrically enhanced stress, making use of the tension balance relation σ(*x*) = *T*/[*w*(*x*)*h*] [similar to microbridge structures (*13*, *14*)]. Inhomogeneous stress has been exploited before to increase the *Q* × *f* product of a nanomechanical resonator (*19*); however, performance was in this case limited by rigid clamping. Combining geometrically enhanced stress with soft-clamping can lead to improved performance: For example, Eq. 2 suggests that the *Q* (for a fixed *f*) of a typical 1 GPa–prestressed Si_{3}N_{4} nanobeam can be enhanced by a factor of 50 before the stress in the thinnest part of the beam reaches the yield strength of Si_{3}N_{4} (σ_{yield} ≈ 6 GPa). This material limit, described by Eq. 2 with σ = σ_{yield} and illustrated by the hatched region in Fig. 1, can be shown to apply to an arbitrary beam profile *w*(*x*) (*20*). In gaining access to it, the main caveat of our approach is the small area in which the stress is enhanced, which implies that high-order flexural modes must be used to achieve sufficient colocalization.

Devices were patterned on 20-nm-thick films of high-stress Si_{3}N_{4} (*E*_{0} ≈ 250 GPa, σ_{0} ≈ 1.1 GPa) grown by low-pressure chemical vapor deposition on a Si wafer. A multistep release process (*20*) was used to suspend beams as long as 7 mm, enabling aspect ratios as high as 3.5 × 10^{5} and dilution factors in excess of (2λ)^{–1} ≈ 3 × 10^{4}. PnCs were realized by corrugating beams with a simple step-like unit cell (length *L*_{c}, minor width *w*_{min}, major width *w*_{max} ≈ 2*w*_{min}) (Fig. 2A). A uniform defect of length *L*_{d} was patterned at the center of each beam to define the position of localized modes. Colocalization of stress with these modes is achieved by adiabatically tapering the width of successive unit cells toward the defect according to a Gaussian envelope function (*20*).

Localized modes of PnC nanobeams (“1D phononic crystals”) have already been widely studied, as their ultralow mass and sparse mode spectrum make them highly promising for sensing applications. However, in contrast to 2D (membrane-like) resonators (*8*), ultrahigh *Q* in 1D PnCs has not been reported to date because of a focus on unstrained materials (*21*) and/or highly confined (high-curvature) modes (*18*) limited by radiation loss. With this discrepancy in mind, we first embarked on a study of uniform (untapered) PnC nanobeams, focusing on localized modes of our high–aspect ratio devices.

An experiment demonstrating soft-clamped 1D nanomechanical resonators is shown in Fig. 3. We studied 2.6-mm-long devices with unit cells of length *L*_{c} = 100 μm and width *w*_{min(max)} = 0.5 ± 0.1 μm. To characterize these devices, we carried out thermal noise and ringdown measurements in vacuo (<10^{–6} mbar) using a lensed-fiber interferometer (*20*). As a consequence of their simple geometry, mode frequencies (inferred from thermal noise spectra; Fig. 3E) were found to agree well with a numerical solution to the 1D Euler-Bernoulli equation (*20*). Particularly striking is the sparse mode spectrum inside the band gap, visualized by compiling spectra of beams with different defect lengths (Fig. 3F). A single defect mode appears to move in and out of the band gap as the defect length is varied. This mode is expected to be localized and therefore to have a reduced effective mass *m*. Comparing the area under thermal noise peaks and estimating the physical beam mass to be *m*_{0} = 100 pg, we infer that indeed *m* ≈ 5 pg << *m*_{0} (*20*). This value is in good agreement with the mode profile obtained from the Euler-Bernoulli equation (Fig. 3G) and is smaller than that of an equivalent 2D localized mode by roughly two orders of magnitude.

In accordance with Eq. 3, we also observed a marked increase in the *Q* of localized modes. To visualize this enhancement, we compiled measurements of *Q* versus mode frequency for 40 beams of different defect length (Fig. 3G). Outside the band gap, we find that *Q*(*f*) is consistent with that of a uniform beam, asymptoting at low mode order (*n* < ~20) to *Q* ≈ 2 × 10^{7}, implying *Q*_{0} ≈ 2λ*Q* ≈ 1500. Inside the band gap (*n* ≈ 26), *Q* approaches that of an idealized clamp-free beam [*Q* ≈ *Q*_{0}/(π*n*λ)^{2} ≈ 10^{8}]. The transition between these two regimes agrees well with a full model (gray dots in Fig. 3G) based on Eq. 1. In Fig. 4A we highlight the 19-s ringdown of a 2.46-MHz defect mode, corresponding to *Q* = 1.5 × 10^{8} and *Q* × *f* = 3.7 × 10^{14} Hz.

Having established near-ideal soft-clamping of uniform nanobeams, we next studied the performance of strain-engineered (tapered) nanobeams. A set of 4- and 7-mm-long tapered PnC nanobeams was fabricated with the length of the taper varied so as to tune the stress at the center of the beam σ(*x*_{c}) from 2 to 4 GPa (Fig. 2, D and E). We note that for our tapering strategy, the width of the beam center *w*(*x*_{c}) is fixed, so that the stress is tuned by changing the equilibrium tension T (*20*). Moreover, for each taper length the soft-clamped mode is engineered to be well localized inside the thin taper region by tuning the pitch of unit cells. Measurements of band gap frequency *f*_{bg} versus length of the central unit cell length *L*_{c,0} (parameterizing the taper length) corroborate enhanced stress through correspondence with the theoretical scaling (Fig. 2E).

The *Q* factors of uniform and tapered PnC nanobeams are compared in Fig. 4. Blue circles correspond to the measurements in Fig. 3G; red circles are compiled for localized modes of 4-mm-long tapered beams with various peak stresses, corresponding to *f*_{bg} = 1 to 6 MHz. According to a full model (*20*), *Q*(*f*_{bg}) should in principle trace out a line of constant *Q* × *f* ≈ 10^{15} Hz, exceeding the clamp-free limit of a uniform beam (*Q* × *f* ∝ 1/*f*) for sufficiently high frequency. We observe this behavior with an unexplained ~30% reduction, with *Q* factors exceeding the clamp-free model by a factor of up to 3 and reaching absolute values high as 3 × 10^{8}. Although theoretically this *Q* should be accessible by soft-clamping alone at lower frequency, our strain-engineering strategy gives access to higher *Q* × *f*, reaching a value as high as 8.1 × 10^{14} Hz for the 3.2-MHz mode of a 4-mm-long device. Higher *Q* and *Q* × *f* factors were achieved using longer beams (red squares in Fig. 4C). In Fig. 4A we highlight the 190-s ringdown of a 7-mm-long device excited in its 1.33-MHz defect mode, corresponding to *Q* = 8.0 × 10^{8} and *Q* × *f* = 1.1 × 10^{15} Hz. We note that at this low damping rate (*f/Q* ~ 1 mHz), photothermal effects become important. Stroboscopic ringdowns (Fig. 4, A and B) confirm that photothermal damping contributes less than 5% uncertainty (*20*).

Realization of *Q* × *f* ~ 10^{15} in a mechanical oscillator with *m* on the order of picograms has numerous intriguing implications. First, such an oscillator is an exquisite force sensor. For example, localized modes of the beam outlined in Fig. 3 are limited by thermal noise to a sensitivity of at *f* ~ 2.5 MHz and *T* = 300 K. This value is on par with a typical atomic force microscope cantilever operating at a frequency and absolute temperature two orders of magnitude lower (*22*), creating new opportunities for applications such as high-speed force microscopy (*23*). Of practical importance is that the reported devices also exhibit an exceptionally strong thermal displacement of , accessible by rudimentary detection techniques such as deflectometry. Indeed, their zero-point motion is orders of magnitude larger than the sensitivity of modern microcavity-based optical interferometers (*24*), offering possibilities in the field of quantum measurement and control (*25*). A fascinating prospect is to use measurement-based feedback to cool such an oscillator to its ground state from room temperature (*26*). A basic requirement is that the oscillator undergo a single oscillation in the thermal decoherence time *ℏQ*/*k*_{B}*T*. The devices reported are exceptional in this respect, capable of performing (2π*Q* × *f*)/(*k*_{B}*T*/*ℏ*) > 100 coherent oscillations at room temperature.

Looking forward, the performance of our devices seems far from exhausted. First, the dilution factors we have achieved are still an order of magnitude below the limit set by the yield stress of Si_{3}N_{4}. Our results may thus benefit from more aggressive strain engineering. [For example, Si microbridges have been fabricated with local stresses as high as 7.6 GPa (*14*).] We also emphasize that higher aspect ratios offer a direct route to higher *Q*. The aspect ratios of our longest beams (*L*/*h* = 3.5 × 10^{5}) appear to be anomalously high for a suspended thin film, including 2D materials (*27*); however, Si_{3}N_{4} membranes with centimeter-scale dimensions have recently been reported (*28*), hinting at a trend toward more extreme devices. Finally, we note that the source of intrinsic loss in our devices is unknown, although it is likely due to surface imperfections (*17*). To test this hypothesis, we compiled defect *Q*s for beams with thicknesses *h* = 20, 50, and 100 nm (Fig. 4D). The inferred thickness dependence of the intrinsic *Q*, *Q*_{0} ≈ 6900 · *h*/100 nm, is indeed a signature of surface loss and agrees well in absolute terms with a recent meta-study on Si_{3}N_{4} nanomechanical resonators (*17*). Remarkably, the *Q* ∝ *Q*_{0}/*h*^{2} scaling of soft-clamped resonators (*8*) preserves the advantage of thinner devices even in the presence of surface loss. It therefore seems appealing to apply our approach to epitaxially strained crystalline thin films (*29*), which can have *Q*_{0} values two orders of magnitude larger than amorphous films at temperatures below 10 K (*30*).

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank H. Schütz and E. Mansouri for valuable contributions during the initial phase of the experiment.

**Funding:**Supported by the EU Horizon 2020 Research and Innovation Program under grant agreement 732894 (FET Proactive HOT) and the SNF Cavity Quantum Optomechanics project (grant 163387); MSCA ETN-OMT grant 722923 (M.J.B.); ERC AdG QuREM grant 320966 (T.J.K.); and DARPA seedling grant HR0011181003. All samples were fabricated at the Center for MicroNanoTechnology (CMi) at EPFL.

**Author contributions:**Device design and simulation was led by A.H.G. and S.A.F. with early support from R.S.; devices were fabricated by A.H.G. and M.J.B.; S.A.F. developed the semi-analytical model; all authors contributed to measurements and/or development of the experimental apparatus; data analysis was led by N.J.E., S.A.F., and D.J.W. with support from A.H.G. and M.J.B.; the manuscript was initially drafted by D.J.W. and S.A.F. with support from N.J.E.; D.J.W., S.A.F., N.J.E., A.H.G., M.J.B., and T.J.K. all participated in editing of the final manuscript and supporting information; and D.J.W. and T.J.K. supervised the project.

**Competing interests:**None declared.

**Data and materials availability:**Data and data analysis code are available through Zonedo at doi:10.5281/zenodo.1202322. All other data needed to evaluate the conclusions in the paper are present in the paper or the supplementary materials.