Abstract
Graphene and related twodimensional (2D) materials associate remarkable mechanical, electronic, optical and phononic properties. As such, 2D materials are promising for hybrid systems that couple their elementary excitations (excitons, phonons) to their macroscopic mechanical modes. These builtin systems may yield enhanced strainmediated coupling compared to bulkier architectures, e.g., comprising a single quantum emitter coupled to a nanomechanical resonator. Here, using microRaman spectroscopy on pristine monolayer graphene drums, we demonstrate that the macroscopic flexural vibrations of graphene induce dynamical optical phonon softening. This softening is an unambiguous fingerprint of dynamicallyinduced tensile strain that reaches values up to ≈4 × 10^{−4} under strong nonlinear driving. Such nonlinearly enhanced strain exceeds the values predicted for harmonic vibrations with the same root mean square (RMS) amplitude by more than one order of magnitude. Our work holds promise for dynamical strain engineering and dynamical strainmediated control of lightmatter interactions in 2D materials and related heterostructures.
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Introduction
Since the first demonstration of mechanical resonators made from suspended graphene layers^{1}, considerable progress has been made to conceive nanomechanical systems based on 2D materials^{2,3} with wellcharacterised performances^{4,5,6,7,8}, for applications in mass and force sensing^{9} but also for studies of heat transport^{10,11}, nonlinear mode coupling^{12,13,14} and optomechanical interactions^{5,15,16}. These efforts triggered the study of 2D resonators beyond graphene, made for instance from transition metal dichalcogenide layers^{8,11,17,18} and van der Waals heterostructures^{19,20,21}. In suspended atomically thin membranes, a moderate outofplane stress gives rise to large and swiftly tunable strains, in excess of 1%^{22,23}, opening numerous possibilities for strainengineering^{24}. These assets also position 2D materials as promising systems to achieve enhanced strainmediated coupling^{25,26,27,28} of macroscopic flexural vibrations to quasiparticles (excitons, phonons) and/or degrees of freedom (spin, valley). Such developments require sensitive probes of dynamical strain. Among the approaches employed to characterise strain in 2D materials, microRaman scattering spectroscopy^{29} stands out as a local, contactless and minimally invasive technique that has been extensively exploited in the static regime to quantitatively convert the frequency softening or hardening of the Raman active modes into an amount of tensile or compressive strain, respectively^{30,31,32,33}. Recently, the interplay between electrostaticallyinduced strain and doping has been probed in the static regime in suspended graphene monolayers^{34}. Dynamicallyinduced strain has been investigated using Raman spectroscopy in bulkier micro electromechanical systems^{35,36}, including mesoscopic graphite cantilevers^{37} but remains unexplored in resonators made from 2D materials.
In this article, using microRaman scattering spectroscopy in resonators made from pristine suspended graphene monolayers, we demonstrate efficient strainmediated coupling between "builtin” quantum degrees of freedom (here the Ramanactive optical phonons of graphene) of the 2D resonator, and its macroscopic flexural vibrations. The dynamicallyinduced strain is quantitatively determined from the frequency of the Ramanactive modes and is found to attain anomalously large values, exceeding the levels of strain expected under harmonic vibrations by more than one order of magnitude. Our work introduces resonators made from graphene and related 2D materials as promising systems for hybrid optoelectromechanics^{38} and dynamical strainmediated control of lightmatter interactions.
Results
Measurement scheme
As illustrated in Fig. 1a, the system we have developed for probing dynamical strain in the 2D limit is a graphene monolayer, mechanically exfoliated and transferred as is onto a prepatterned Si/SiO_{2} substrate. The resulting graphene drum is capacitively driven using a timedependent gate bias \({V}_{{\rm{g}}}(t)={V}_{{\rm{dc}}}+{V}_{{\rm{ac}}}\cos \Omega t\), with V_{ac} ≪ V_{dc} and Ω/2π the drive frequency. The DC component of the resulting force (\(\propto {V}_{{\rm{g}}}^{2}\), see “Methods” section) enables to control the electrostatic pressure applied to the graphene membrane (and hence its static deflection ξ, see Fig. 1a), whereas the AC bias leads to a harmonic driving force \((\propto {V}_{{\rm{dc}}}{V}_{{\rm{ac}}}\cos \Omega t)\). A single laser beam is used to interferometrically measure the frequencydependent mechanical susceptibility at the drive frequency, akin to ref. ^{1} and, at the same time, to record the microRaman scattering response of the atomically thin membrane. We have chosen electrostatic rather than photothermal actuation^{39} to attain large RMS amplitudes while at the same time avoiding heating and photothermal backaction effects^{10,11}, possibly leading to additional damping^{8}, selfoscillations^{10}, mechanical instabilities and sample damage. All measurements were performed at room temperature under high vacuum (see “Methods” section and Supplementary Notes 1 to 8).
Raman spectroscopy in strained graphene
The Raman spectrum of graphene displays two main features: the G mode and the 2D mode, arising from one zonecentre (that is, zero momentum) phonon and from a pair of nearzone edge phonons with opposite momenta, respectively (see Fig. 1a and Supplementary Note 1)^{29}. Both features are uniquely sensitive to external perturbations. Quantitative methods have been developed to unambiguously separate the share of strain, doping, and possibly heating effects that affect the frequency, full width at half maximum (FWHM) and integrated intensity of a Raman feature^{31,34,40,41,42} (hereafter denoted ω_{i}, Γ_{i}, I_{i}, respectively, here with i = G, 2D). Biaxial strain is expected around the centre of circular graphene drums^{22} and the large Grüneisen parameters of graphene (γ_{G} = 1.8 and γ_{2D} = 2.4, with \({\gamma }_{i}=\frac{1}{2{\omega }_{i}}\frac{\partial {\omega }_{i}}{\partial \varepsilon }\) and ε the level of biaxial strain)^{31,32} allow detection of strain levels down to a few 10^{−5}. The characteristic slope \(\frac{\partial {\omega }_{{\rm{2D}}}}{\partial {\omega }_{{\rm{G}}}}\approx 2.2\) in graphene under biaxial strain is much larger than in the case of electron or hole doping, where the corresponding slope is significantly smaller than 1^{41,42}. This difference allows a clear disambiguation between strain and doping (see “Methods” section for details).
Mechanical and Raman characterisation
Fig. 1b presents the main characteristics of a circular graphene drum (device 1) in the linear response regime. A Lorentzian mechanical resonance is observed at Ω_{0}/2π ≈ 33.8 MHz for V_{dc} = −6 V (Fig. 1b and Supplementary Notes 5 and 6). The mechanical mode profile shows radial symmetry (inset in Fig. 1b) as expected for the fundamental flexural resonance of a circular drum^{6}. The mechanical resonance frequency is widely gatetunable: it increases by ~70% as \(\left{V}_{{\rm{dc}}}\right\) is ramped up to 10 V and displays a symmetric, "Ushaped” behaviour with respect to a nearzero DC bias \({V}_{{\rm{dc}}}^{0}=0.75\,{\rm{V}}\), at which graphene only undergoes a builtin tension. These two features are characteristic of a low builtin tension^{4,8,10,43} that we estimate to be \({T}_{0}=\left(4\pm 0.4\right)\times 1{0}^{2}\,{{\rm{Nm}}}^{1}\), corresponding to a builtin static strain \({\varepsilon }_{{\rm{s}}}^{0}={T}_{0}\left(1\nu \right)/{E}_{{\rm{1LG}}}\approx \left(1.0\pm 0.1\right)\times 1{0}^{4}\), where E_{1LG} = 340 Nm^{−1}, ν = 0.16 are the Young modulus and Poisson ratio of pristine monolayer graphene^{44} (Supplementary Note 6). The quality factor Q is high, in excess of 1500 near charge neutrality. As \(\left{V}_{{\rm{dc}}}\right\) increases, Q drops down to ~200 due to electrostatic damping^{8}.
Fig. 1d shows that the Raman response of suspended graphene is tunable by application of a DC gate bias, as extensively discussed in ref. ^{34}. Once V_{dc} is large enough to overcome \({\varepsilon }_{{\rm{s}}}^{0}\), the membrane starts to bend downwards and the downshifts of the Gmode and 2Dmode features measured at the centre of the drum are chiefly due to biaxial strain (∂ω_{2D}/∂ω_{G} ≈ 2.2, see inset in Fig. 1d) with negligible contribution from electrostatic doping^{34} (see “Methods” section for details). At V_{dc} = −9 V, the 4 ± 0.5 cm^{−1} 2Dmode downshift relative to its value near \({V}_{{\rm{dc}}}^{0}\) yields a gateinduced static strain ε_{s} = 3 ± 0.3 × 10^{−4} that agrees qualitatively well with the value ε_{s} = 2 ± 0.2 × 10^{−4} estimated from the gateinduced upshift of Ω_{0} (Fig. 1c and Supplementary Note 6). This agreement justifies our assumption that the Young’s modulus of our drum is close to that of pristine graphene (see also Supplementary Note 5 for details on the drum effective mass).
Noteworthy, optical interference effects cause a large gatedependent modulation of I_{G} and I_{2D} (refs. ^{31,34} and see normalisation factors in Fig. 1d). Both straininduced Raman shifts and Raman scattering intensity changes are exploited to consistently estimate that ξ increases from about 30 nm to 70 nm when V_{dc} is varied from −5 V to −9 V (Supplementary Notes 2, 3 and 4).
Nonlinear mechanical response
We are now examining how the dynamicallyinduced strain can be readout by means of Raman spectroscopy. First, to obtain a larger sensitivity towards static strain (Supplementary Note 3), we apply a sufficiently high V_{dc} to reach a sizeable ξ. V_{ac} is then ramped up to yield large RMS amplitudes. After calibration of our setup (Supplementary Note 5), we estimate that resonant RMS amplitudes \({z}_{{\rm{rms}}}^{0}\) up to ~10 nm are attained in device 1 (Figs. 2 and 3). In this regime, graphene is a strongly nonlinear mechanical system that can be described to lowest order by a Duffinglike equation^{5,7,45}:
where z is the mechanical displacement at the membrane centre relative to the equilibrium position ξ, Ω_{0}/2π is the resonance frequency in the linear regime, Q is the quality factor and Ω_{0}/Q is the linear damping rate. The effective mass \(\widetilde{m}\) and effective applied electrostatic force \({\widetilde{F}}_{{\rm{el}}}\) account for the mode profile of the fundamental resonance in a rigidly clamped circular drum^{6,46} (see “Methods” section and Supplementary Note 6). The linear spring constant is \(\widetilde{m}\ {\Omega }_{0}^{2}\). Mechanical nonlinearities are considered using an effective thirdorder term \({\widetilde{\alpha }}_{3}\) that changes sign at large enough ξ, leading to a transition from nonlinear hardening to nonlinear softening^{5}. Such a behaviour is indeed revealed in our experiments, as shown in Figs. 2a and 3a, where nonlinear softening and nonlinear hardening are observed at V_{dc} = −8 V and V_{dc} = −6 V, respectively. At V_{dc} = −7 V, we observe a V_{ac}dependent softeningtohardening transition (Supplementary Notes 6 and 7).
Dynamical optical phonon softening
Fig. 2c–e shows the frequencies, linewidths and integrated intensities of the Raman features measured at V_{dc} = −8 V (where ξ ≈ 60 nm), with V_{ac} increasing from 0 to 150 mV and applied at a drive frequency that tracks the V_{ac}dependent nonlinear softening of the mechanical resonance frequency \({\widetilde{\Omega }}_{0}/2\pi\), that is the socalled backbone curve in Fig. 2a, f (Supplementary Note 6). Both Gmode and 2Dmode features downshift as the drum is nonlinearly driven. This phonon softening is accompanied by spectral broadening by up to ~10−15% (Fig. 2d) that increases with z_{rms}. The correlation plot between ω_{2D} and ω_{G} reveals a linear slope near 2 (see also Supplementary Note 1), which is a characteristic signature of tensile strain^{31,41} that gets as high as ≈2.5 × 10^{−4} for z_{rms} ≈ 9 nm.
In Fig. 3a, we compare, on device 1, the frequencydependence of z_{rms} to that of ω_{G,2D} and I_{2D}, for upward and downward sweeps under V_{dc} = −6 V and V_{ac} = 100 mV. As in Fig. 2c, sizeable Gmode and 2Dmode softenings are observed near the mechanical resonance (Fig. 3a–c) and assigned to tensile strain (see correlation plot in Fig. 3c). Remarkably, the hysteretic behaviour of the mechanical susceptibility, associated with nonlinear hardening at V_{dc} = −6 V, is wellimprinted onto the frequencydependence of ω_{G,2D} and I_{2D}. Looking further at Fig. 3a, we notice that while z_{rms} fully saturates at drive frequencies above 33.5 MHz and ultimately starts to decrease near the jumpdown frequency, the tensile strain keeps increasing linearly up to ≈2.5 × 10^{−4} as Ω/2π is raised from 33.2 MHz up to 34.5 MHz.
Equilibrium position shift
As our graphene drums are nonlinearly driven, including beyond the Duffing regime (Fig. 3a and Supplementary Notes 6 and 7), the large strains revealed in Figs. 2 and 3 could in part arise from an equilibrium position shift Δξ_{eq} due to symmetry breaking nonlinearities^{45,47} (inset in Fig. 2e). This effect can be quantitatively assessed through analysis of I_{G,2D}. As shown in Fig. 2e both I_{2D} and I_{G} decrease by about ~20% as V_{ac} increases up to 150 mV. These variations are assigned to optical interference effects (refs. ^{31,34}); in our experimental geometry they indicate an equilibrium position upshift Δξ_{eq} by up to ≈12 nm (Fig. 2e and Supplementary Note 4), that leads to a reduction of the static tensile strain Δε_{s} ≈ 1 × 10^{−4}, in stark contrast with the enhanced tensile strain unambiguously revealed in Fig. 2c. Similarly, the ≈10% drop in I_{2D} near the jumpdown frequency at 34.5 MHz indicates an equilibrium position upshift Δξ_{eq} ≈ 4 nm that is qualitatively similar to the results in Fig. 2e. The larger Δξ_{eq} measured at V_{dc} = −8 V is consistent with our observation of nonlinear mechanical resonance softening (Fig. 2a) due to an increased contribution from symmetry breaking nonlinearities at large ξ (refs. ^{5,45,47} and Supplementary Note 6). From these measurements, we conclude that the dynamical softening of ω_{G} and ω_{2D} is not due to an equilibrium position shift.
Evidence for dynamicallyinduced strain
We therefore conclude that the tensile strain measured in device 1 is dynamicallyinduced (hereafter denoted ε_{d}) and arises from the timeaveraged resonant vibrations of the graphene drum. Starting from a reference recorded at V_{dc} = −8 V and V_{ac} = 0 mV, ε_{d} recorded under resonant driving at V_{ac} = 150 mV (where z_{rms} ≈ 9 nm) is as high as the static strain ε_{s} induced when ramping V_{dc} from 0 V to −8 V (where ξ ≈ 60 nm). Along these lines, the small yet observable broadenings ΔΓ_{G,2D} of the Raman features (Fig. 2d) can be assigned to timeaveraged Raman frequency shifts due to dynamical strain^{48}. We have consistently observed dynamicallyenhanced strain in three graphene drums with similar designs, denoted device 1, 2, 3. Complementary results are reported in Supplementary Note 9 for device 1 and in Supplementary Notes 10 and 11 for devices 2 and 3, respectively. In device 3, we have measured ε_{d} ≈ 4 × 10^{−4} for z_{rms} ≈ 14 nm.
Spatiallyresolved dynamicallyinduced strain
Interestingly, our diffractionlimited Raman readout enables local mapping of ε_{d}. Fig. 4 compares ω_{2D} and I_{2D} recorded across the diameter of a graphene drum (device 2, similar to device 1) under V_{dc} = −6 V with and without resonant driving. Very similar results are observed when performing a linescan along the perpendicular direction (Supplementary Note 9). In the undriven case, we find a nearly flat ω_{2D} profile, which is consistent with the difficulty in resolving lowlevels of static strain below 1 × 10^{−4}. In contrast, finite ε_{d} (Fig. 4b) and equilibrium position upshift (Fig. 4c) are observed at the centre of the drum, as in Figs. 2 and 3. We find that ε_{d} and the equilibrium position upshift decrease as they are probed away from the centre of the drum and the spatial profile of ε_{d} resembles the static tensile strain profile measured on bulged graphene blisters, where strain is biaxial at the centre of the drum and radial at the edges^{49}.
Dynamicallyenhanced strain
It is instructive to compare the measured ε_{d} to \({\varepsilon }_{{\rm{d}}}^{{\rm{h}}}=2/3{\left({z}_{{\rm{rms}}}/a\right)}^{2}\), with a the drum radius, the timeaveraged dynamicallyinduced strain estimated for an harmonic oscillation with RMS amplitude z_{rms} (Supplementary Note 7). For the largest z_{rms} ≈ 9 nm attained in device 1, \({\varepsilon }_{{\rm{d}}}^{{\rm{h}}}\approx 6\times 1{0}^{6}\), i.e., about 40 times smaller than the measured ε_{d} (Figs. 2f and 3d). Under strong nonlinear driving, we expect sizeable Fourier components of the mechanical amplitude at harmonics of the drive frequency, which could in part be responsible for the large discrepancy between ε_{d} and \({\varepsilon }_{{\rm{d}}}^{{\rm{h}}}\). Harmonics are indeed observed experimentally in the displacement power spectrum of our drums (Supplementary Note 10, device 2) but display amplitudes significantly smaller than the linear component at the drive frequency. In addition, we do not observe any measurable fingerprint of internal resonances^{12,13,14} in the displacement power spectrum.
To get further insights into the unexpectedly large ε_{d} deduced from the Gmode and 2Dmode downshifts we plot ε_{d} as a function of the corresponding z_{rms} at the centre of the drum (Figs. 2f and 3d). This plot is directly compared to the backbone curves that connect the resonant z_{rms} to the nonlinear relative resonance frequency shift \(\delta =\left{\widetilde{\Omega }}_{0}{\Omega }_{0}\right/{\Omega }_{0}\), where \({\widetilde{\Omega }}_{0}\) is considered equal to the measured jumpdown frequency (Figs. 2a, 3a and Supplementary Notes 6 and 7). Remarkably, ε_{d} grows proportionally to δ, both in the case of nonlinear softening and hardening, including when z_{rms} fully saturates (Fig. 3). This proportionality is expected from elasticity theory with a third order geometrical nonlinearity^{50} and we experimentally show here that it still holds when symmetry breaking and higherorder nonlinearities come into play (Supplementary Note 7).
Discussion
The large values of \({\varepsilon }_{{\rm{d}}}\gg {\varepsilon }_{{\rm{d}}}^{{\rm{h}}}\) reported in Figs. 2–4 cannot be understood as a simple geometrical effect arising from the timeaveraged harmonic oscillations of mode profile that remains smooth over the whole drum area. Instead, the enhancement of ε_{d} could arise from socalled localisation of harmonics, a phenomenon recently observed in larger and thicker (~500 μm wide, ~500 nm thick) SiN membranes^{51} showing RMS displacement saturation similar to Fig. 3a. As the resonator enters the saturation regime, nonlinearities (either intrinsic^{44}, geometrical^{50,52} or electrostaticallyinduced^{5,7,53}) may lead to internal energy transfer towards harmonics of the driven mode (Supplementary Fig. 17) and, crucially, to the emergence of ringshaped patterns over length scales significantly smaller than the size of the membrane^{51}. The large displacement gradients associated with these profiles thus enhance ε_{d} (Supplementary Note 7). The mode profiles get increasingly complex as the driving force increases, explaining the rise of ε_{d} even when z_{rms} reaches a saturation plateau. Considering our study, with \({\varepsilon }_{{\rm{d}}} \sim 40\ {\varepsilon }_{{\rm{d}}}^{{\rm{h}}}\), we may roughly estimate that large mode profile gradients develop on a scale of \(a/\sqrt{40}\approx 500\,{\rm{nm}}\) that is smaller than our spatial resolution (see “Methods” section). Finally, the fact that ΔΓ_{G,2D} (Fig. 2d and Supplementary Fig. 16) is smaller than the associated Δω_{G,2D} (Figs. 2c and 3a) suggests that the oscillations of ε_{d}(t) are rectified under strong nonlinear driving, an effect that further increases the discrepancy between the timeaveraged ε_{d} we measure and \({\varepsilon }_{{\rm{d}}}^{{\rm{h}}}\).
Combining multimode optomechanical tomography and hyperspectral Raman mapping on larger graphene drums (effectively leading to a higher spatial resolution) would allow us to test whether localisation of harmonics occurs in graphene and to possibly correlate this phenomenon to the dynamicallyinduced strain field. More generally, unravelling the origin of the anomalously large ε_{d} may require microscopic models that may go beyond elasticity theory^{54} and explicitly take into account the ultimate thinness and atomic structure of graphene^{55,56}.
Concluding, we have unveiled efficient coupling between intrinsic microscopic degrees of freedom (here optical phonons) and macroscopic nonlinear mechanical vibrations in monolayer graphene resonators. Room temperature resonant mechanical vibrations with ≈10 nm RMS amplitude induce unexpectedly large timeaveraged tensile strains up to ≈ 4 × 10^{−4}. Realistic improvements of our setup, including phaseresolved Raman measurements^{35,36} could permit to probe dynamical strain in finer detail, including in the linear regime, where the effective coupling strength^{28} could be extracted. For this purpose, larger resonant displacements may be achieved at cryogenic temperatures. In addition, graphene drums, as a prototypical nonlinear mechanical systems, can be engineered to favour mode coupling and frequency mixing, which in return can be readout through distinct modifications of their spatiallyresolved Raman scattering response.
Our approach can be directly applied to a variety of 2D materials and related van der Waals heterostructures. In fewlayer systems, rigid layer shear and breathing Ramanactive modes^{29,33} could be used as invaluable probes of inplane and outofplane dynamical strain, respectively. Strainmediated coupling could also be employed to manipulate the rich excitonic manifolds in transition metal dichalcogenides^{57}, as well as the single photon emitters they can host^{58,59}. More broadly, light absorption and emission could be controlled electromechanically in nanoresonators made from customdesigned van der Waals heterostructures^{60}. Going one step further, with the emergence of 2D materials featuring robust magnetic order and topological phases^{61}, that can be probed using optical spectroscopy, we foresee new possibilities to explore and harness phase transitions using nanomechanical resonators based on 2D materials^{62,63}.
Methods
Device fabrication
Monolayer graphene flakes were deposited onto prepatterned 285 nmSiO_{2}/Si substrates, using a thermally assisted mechanical exfoliation scheme as in ref. ^{64}. The pattern is created by optical lithography followed by reactive ion etching and consists of hole arrays (5 and 6 μm in diameter and 250 ± 5 nm in depth) connected by ~1 μmwide venting channels. Ti(3 nm)/Au(47 nm) contacts are evaporated using a transmission electron microscopy grid as a shadow mask^{34} to avoid any contamination with resists and solvents. Our dry transfer method minimises rippling and crumpling effects^{65}, resulting in graphene drums with intrinsic mechanical properties (see ref. ^{31} and Supplementary Note 5 for details). We could routinely obtain pristine monolayer graphene resonators with quality factors in excess of 1500 at room temperature in high vacuum.
Optomechanical measurements
Electrically connected graphene drums are mounted into a vacuum chamber (5 × 10^{−5} mbar). The drums are capacitively driven using the Si wafer as a backgate and a timedependent gate bias \({V}_{{\rm{g}}}(t)={V}_{{\rm{dc}}}+{V}_{{\rm{ac}}}\cos \Omega t\) is applied as indicated in the main text. The applied force is given by \({\epsilon }_{0}\pi {a}^{2}\frac{{V}_{{\rm{g}}}^{2}\left(t\right)}{2{d}^{2}\left(\xi \right)}\), where a is the drum radius, ϵ_{0} the vacuum dielectric constant, \(d\left(\xi \right)=({d}_{{\rm{vac}}}\xi )+{d}_{{{\rm{SiO}}}_{2}}/{\epsilon }_{{{\rm{SiO}}}_{2}}\) the effective distance between graphene and the Si substrate, with ξ the static displacement, d_{vac} the grapheneSiO_{2} distance in the absence of any gate bias, \({d}_{{{\rm{SiO}}}_{2}}\) the thickness of the residual SiO_{2} layer. This force contains a static component proportional to \({V}_{{\rm{dc}}}^{2}\), which sets the value of ξ and a harmonic driving force proportional to \({V}_{{\rm{dc}}}{V}_{{\rm{ac}}}\cos \left(\Omega t\right)\). Note that since \({V}_{{\rm{ac}}}\ll \left{V}_{{\rm{dc}}}\right\), we can safely neglect the force \(\propto {V}_{{\rm{ac}}}^{2}\left(1+\cos \left(2\ \Omega t\right)\right)\) throughout our analysis.
A 632.8 nm HeNe continuous wave laser with a power of ~0.5 mW is focused onto a ~1.2 μmdiameter spot and is used both for optomechanical and Raman measurements. Unless otherwise stated, (e.g., insets in Figs. 1b and 4), measurements are performed at the centre of the drum. The beam reflected from the Si/SiO_{2}/vacuum/graphene layered system is detected using an avalanche photodiode. In the driven regime, the mechanical amplitude at Ω/2π is readout using a lockin amplifier. Mechanical mode mapping is implemented using a piezo scanner and a phaselocked loop. For amplitude calibration, the thermal noise spectrum is derived from the noise power spectral density of the laser beam reflected by the sample, recorded using a spectrum analyser. Importantly, displacement calibration is performed assuming that the effective mass of our circular drums is \(\widetilde{m}=0.27\ {m}_{0}\) (ref. ^{46}), with m_{0} the pristine mass of the graphene drum. As discussed in details in Supplementary Note 5, this assumption is validated by two other displacement calibration methods performed on a same drum. These calibrations are completely independent of \(\widetilde{m}\). We therefore conclude that to experimental accuracy, our graphene drums are pristine and do not show measurable fingerprints of contamination by molecular adsorbates^{66}, as expected for a resistfree fabrication process.
MicroRaman spectroscopy
The Raman scattered light is filtered using a combination of a dichroic mirror and a notch filter. Raman spectra are recorded using a 500 mm monochromator equipped with 300 and 900 grooves/mm gratings, coupled to a cooled CCD array. In addition to electrostaticallyinduced strain, electrostaticallyinduced doping might in principle alter the Raman features of suspended graphene^{34}. Pristine suspended graphene, as used here, is wellknown to have minimal unintentional doping (≲10^{11 }cm^{−2}) and charge inhomogeneity^{66,67}. Considering our experimental geometry, we estimate a gateinduced doping level near 3 × 10^{11 }cm^{−2} at the largest \(\left{V}_{{\rm{dc}}}\right=10\,{\rm{V}}\) applied here. Such doping levels are too small to induce any sizeable shift of the Gmode and 2Dmode features ^{34,40,66}. In the dynamical regime, the RMS modulation of the doping level induced by the application of V_{ac} is typically two orders of magnitude smaller than the static doping level and can safely be neglected. Similarly, the reduction of the gate capacitance induced by equilibrium position upshifts discussed in Figs. 2e and 3a–iv does not induce measurable fingerprints of reduced doping on graphene.
Let us note that since the lifetime of optical phonons in graphene (~1ps)^{68} is more than three orders of magnitude shorter than the mechanical oscillation period, Raman scattering processes provide an instantaneous measurement of ε_{d}. However, since our Raman measurements are performed under continuous wave laser illumination, we are dealing with timeaveraged dynamical shifts and broadenings of the Gmode and 2Dmode features. Raman Gmode and 2Dmode spectra are fit using one Lorentzian and two modified Lorentzian functions, as in refs. ^{31,67}, respectively (Supplementary Note 1). As indicated in the main text, Grüneisen parameters of γ_{G} = 1.8 and γ_{2D} = 2.4 are used to estimate ε_{s} and ε_{d}. These values have been measured in circular suspended graphene blisters under biaxial strain^{31}. Considering a number of similar studies^{31,32,34,49,69}, we conservatively estimate that the values of ε_{s} and ε_{d} are determined with a systematic error lower than 20%. Such systematic errors have no impact whatsoever on our demonstration of dynamicallyenhanced strain. Finally, the Raman frequencies and the associated ε_{s} and ε_{d} are determined with fitting uncertainties represented by the errorbars in the figures.
Data availability
The datasets generated during and/or analysed during this study are available from the corresponding authors (X.Z. and S.B.) on reasonable request.
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Acknowledgements
We thank T. Chen, A. Gloppe and G. Weick for fruitful discussions. We thank the StNano clean room staff (R. Bernard and S. Siegwald), M. Romeo, F. Chevrier, A. Boulard and the IPCMS workshop for technical support. This work has benefitted from support provided by the University of Strasbourg Institute for Advanced Study (USIAS) for a Fellowship, within the French national programme “Investment for the future” (IdExUnistra). We acknowledge financial support from the Agence Nationale de Recherche (ANR) under grants H2DH ANR15CE240016, 2DPOEM ANR18ERC10009, as well as the Labex NIE project ANR11LABX0058NIE.
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The project was originally proposed by S.B and P.V (GOLEM project, supported by USIAS). K.M. and D.M. built the experimental setup, with help from X.Z. X.Z. fabricated the samples, with help from H.M., D.M. and K.M. X.Z. carried out measurements, with help from K.M. and L.C. X.Z. and S.B. analysed the data with input from K.M., L.C. and P.V. X.Z. and S.B. wrote the manuscript with input from L.C. and P.V. S.B. supervised the project.
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Zhang, X., Makles, K., Colombier, L. et al. Dynamicallyenhanced strain in atomically thin resonators. Nat Commun 11, 5526 (2020). https://doi.org/10.1038/s41467020192613
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DOI: https://doi.org/10.1038/s41467020192613
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