Introduction

The frequency of a musical instrument, such as a guitar string, is tunable over more than one octave, i.e., a ratio of two between the maximum and minimum frequency. It defines as \(f_0 = \sqrt {\frac{k}{m}}\), in which k is the spring constant and m the resonator mass. To integrate successfully in devices or sensors, the resonator frequency must be tuned to the modes of the other components. When dimensions are reduced to nanometer sizes, such precise tuning is difficult to achieve in-situ. There is thus a demand for simple and robust techniques that can tune mechanical resonances at the nanoscale, and nano-resonators that can be tuned over a large frequency window. Among the different nano-resonators, carbon nanotube and two-dimensional (2D) materials have demonstrated the best performances for mechanical frequency tuning, including transition metal dichalcogenides (TMDs), 2D heterostructures1,2,3,4,5,6,7,8,9,10,11,12, graphene13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39, and nanotubes40,41,42,43,44,45,46. We review more than 50 articles with different methods of modulating the frequency of vibration modes using a simple voltage. Some hold record for mass sensing47 or force sensing13,48, which makes them very desirable for applications or device, but none has yet achieved an efficient frequency adjustment over a large window, i.e., more than an octave.

When combined with 2D nanomechanics, thermal transport turns quite exciting, as materials with a large panel of thermal conductivities49,50 and extreme anisotropy51, are brought together with techniques to characterize their intrinsic properties52 and thermal measurements with a temperature sensitivity better than standard Raman spectroscopy techniques12. It has also simple device applications, as shown by a single layer graphene membrane used fast bolometer (heat sensor) with a precision of the order of pW.Hz−0.5, comparable to the best current detectors33. It is worth noting that 2D materials have also radically altered our understanding of thermal transport at low dimensionality, proving that conventional models established for bulk crystals no longer apply at the nanoscale53,54,55,56,57.

We design here a thermally-tuned nanomechanical resonator, which sits at the interface between heat transport in 2D materials and nanomechanics, with a focus on the electrothermal actuation of mechanical vibrations and their frequency tuning.

Results

Experimental setup

We construct a nano-opto-electro-mechanical52 platform operating under vacuum (10−5–10−6 mbar) and at room temperature, to combine the electrical excitation and optical measurements of a suspended MoS2 membrane. The sample consists of a pre-structured SiO2 substrate with cavity and electrodes, covered by a few layers of MoS2 deposited by 2D transfer technique58,59,60. Figure 1a shows a schematic of the sample; the MoS2 is contacted laterally with gold electrodes in order to apply a transverse DC source drain voltage (Vds); the silicon (100) wafer is used as the back gate in a transistor configuration.

Fig. 1: Opto electromechanical setup.
figure 1

a Schematic of the suspended MoS2 membrane in a back-gate transistor configuration, contacted laterally by two gold electrodes. b Optical microscopy image of the sample. Dotted lines are guide to the eye to locate the cavity and the MoS2 crystal, scale bar is 2 µm. c Atomic force microscopy image of the suspended flake used for thermal measurements, scale bars is 2 µm. d Electrical characteristic Ids(Vds) of the sample (e) Mechanical response of the assembled membrane near 35.6 MHz, as measured by our optical technique. The quality factor at room temperature is Q = 1040.

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In the optical image, Fig. 1b, we see that the central cavity is fully covered by the MoS2 monocrystal, so that the 2D material is suspended over the circular hole and forms a resonant drum. To confirm the flatness and the surface quality of our suspended membrane, we proceed to atomic force microscopy characterization in Fig. 1c. The typical vertical deflection of the membrane at rest position is below 1 nm. We excited electrically the drum movement and measure the vibration with a laser reflectometry (see methods). A typical resonance is shown in Fig. 1e near f = 35 MHz.

From the literature, we know that the 2D material MoS2 is a semiconductor with thickness-dependent properties, the band gap is indirect 1.3 eV in the bulk and changes to a direct band gap of 1.8 eV at the monolayer61,62. Our red laser (633 nm, 1.96 eV), used for opto-mechanical monitoring, performs an above bandgap laser excitation in the material and the bulk of the charge carriers in our multilayer MoS2 is thus photo-generated. Experimentally, the transverse electrical conductivity of the membrane strongly depends on the impinging laser power.

By measuring the lateral current Ids against the corresponding voltage Vds at a given laser power, we obtain The Joule heating of the membrane.

Here, our optomechanical measurements are typically carried out at 1,5.104 W.cm-2 incident optical density (30 µW, spot size ~500 nm).

A highly tunable nano guitar

Mechanical resonators made of TMDs or 2D heterostructures have been shown to exhibit significant frequency tunability when varying electrical bias; generally only the gate voltage is considered1,2,3,4,5,6,7,8,9,10,11,12. In Fig. 2a, we plot the mechanical frequency f0 as a function of the bias voltage for our MoS2 resonator. The drum exhibits a huge tunability of the vibration frequency with respect to drain-source voltage Vds, with a maximum frequency shift Δf = fmax-f-min of 24 MHz, constructed from the maximum (fmax) and minimum (fmin) experimentally accessible. It is a tunability over 1.2 octave of the minimum frequency (10 MHz). Here fmax is obtained at Vds = 0 V while fmin used Vds = 6 V under a DC gate bias \(\overline {V_{{{\mathrm{g}}}}}\) = 1 V.

Fig. 2: Joule heating induced a tuning of the vibrations.
figure 2

a Variation of the frequency with respect to Vds (Vg = 1 V). It shows a huge frequency tunability from 34 to 10 MHz for a Vds variation of 7 V. A strong softening of the frequency is observed. b Values of the frequency tuning efficiency TfV and TfA for our work (index = 1) and other reference on TMDs and 2D heterostructures. c The experimental mode frequency and all different contributions fm,fz,fS,fB. d Reflectivity (R) and vertical deflection variation (Δz) of the sample with respect to Vds (Vg = 1 V). The measurement is done at the middle of membrane. We arbitrarily fix Δz = 0 nm at Vds = 0 V (red and blue for back and forth measurement). e The corresponding damping of the resonator Γ = f0/Q.

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The softening is related to the Joule heating of the membrane. The 2D drum can be heated locally, in the suspended part, for an applied electric power of a few mW. This has been confirmed by finite element analyses (see Supplementary Fig. 11) and, later on in the article, by Raman spectroscopy. To characterize the maximum of the frequency tuning efficiency (TE) for each device, we define a figure of merit. The observed frequency tunability via source drain voltage \(Tf_V = 1/f_0 \cdot \left( {\partial f_0/\partial V_{{{{\mathrm{ds}}}}}} \right)_{\max }\) is huge in our experiment (123%.V−1), for the maximum slope of f0(Vds) function with the applied bias. It is almost an order of magnitude greater than previous works see Fig. 2b-top1,2,3,4,5,6,7,8,9,10,11,12. on semiconducting (Fig. 2b, red-orange) and conducting (Fig. 2b, blue green) 2D materials. Only a single experimental report based on a graphene membrane obtained comparable electromechanical softening13,63. There, the reported frequency shift was mainly obtained by an extreme capacitive force, created by a very short distance between the membrane and the metallic backgate (33–85 nm), close to the pull-in event. It remains difficult to achieve experimentally. High TfV was obtained for suspended carbon nanotubes in extreme conditions40,41, this is due to the electron–phonon coupling inherent in low temperature measurements and high-quality samples with maximum tunability of only 1 MHz. Electrothermal actuation with high TfV of 30%.V−1 has already been reported with graphene membranes14,25. This confirms the effectiveness of such a frequency adjustment method.

The absolute frequency tunability TfA, defined as \(Tf_A = 2\left( {f_{\max } - f_{\min }} \right)/\left( {f_{\max } + f_{\min }} \right)\) draws a similar picture, Fig. 2b-bottom. It defines the amplitude of the frequency range that is tunable, closely related to the dynamic range \(DR = f_{\max }/f_{\min }\). We reached a value of 54%, it is among the best values reported so far. For all the devices studied, our work is the only one to have both a TfV and a TfA greater than 100%. To conclude, electrothermal actuation is a very simple and efficient way to adjust the frequency of a nanoresonator with a simple voltage.

Discussion

In our case, the situation is different and we review in the following the contribution of each possible softening sources. For a clamped circular membrane, the frequency of the fundamental mode f0 decomposes as follows.

$$f_0 = \frac{1}{{2\pi }}\sqrt {f_{{{\mathrm{m}}}}^2 + f_{{{\mathrm{B}}}}^2 + f_{{{\mathrm{z}}}}^2\left( z \right) - f_{{{\mathrm{S}}}}^2\left( {V_g} \right)}$$
(1)

Each of the four major contributions is defined as52

$$f_m^2 = \frac{{2.405^2}}{{R^2}}\sqrt {\frac{{\gamma _0}}{{\rho t}}}$$
(2)
$$f_B^2\sim t^3$$
(3)
$$f_z^2\left( z \right) = 2.405^2.12/3.Et/\,\left( {1 - \nu ^2} \right).\,z^2/R^4\rho$$
(4)
$$f_{{{\mathrm{S}}}}^2\left( {V_{{{\mathrm{g}}}}} \right) = 1.23{\,\it{\epsilon }}_0V_{{{\mathrm{g}}}}^2/d_0^3\rho$$
(5)

The first term \(f_{{{\mathrm{m}}}}^2\) is strictly mechanical and detailed in Eq. (2). It links the in-plane mechanical tension (γ0) to the geometrical considerations of the membrane (R, t its diameter and thickness, respectively) and to the material density (ρ). We notice that fm is temperature dependant since the material dilatation will vary γ0 as γ0(T). T is the membrane temperature profile convoluted by the mode shape. In our experiment, the membrane temperature T is mostly controlled by lateral electrothermal Joule heating between the drain and source contacts, with corresponding power Pds = Ids.Vds.

The second term fB is detailed in Eq. (3) relates to the bending stiffness and is proportional to the cube of the membrane thickness t3. As our membrane is near atomically thin, t = 5.8 nm for five layers, fB is neglected here, as in our previous work52.

The third term, fz(z) is detailed Eq. (4). It links the relative vertical displacement of the membrane center (z), measured against it rest position (z0 = 0 at Vds = 0 V in Fig. 2a), with the mechanical parameter Young Modulus (E) and the Poisson ratio (ν) of the 2D material, respectively. Here the vertical displacement includes the deflection due to the static capacitive force FC applied on the membrane and the photothermal forces or radiation pressure. We have \(F_{{{\mathrm{C}}}} = dC/dz\,.\,\left( {\bar V_{{{\mathrm{g}}}} - V_{{{{\mathrm{ds}}}}}/2} \right)^2\) in a diffusive and ohmic regime, with C the local capacitance.

The last term fS is the contribution related to the capacitive softening and detailed in Eq. (4). It links the vacuum permittivity \(\left( {{\it{\epsilon }}_0} \right)\) with the gate voltage (Vg) and the vertical distance (d0) between the membrane rest position (z0) to the bottom surface the cavity (i.e., “gate distance”). This last term can be estimated from the geometry of the device.

As fz only depends on the (known) gate voltage Vg and the distance d0, we can quantify its magnitude by an independent determination of d0 during the actual frequency measurements. To this end, we investigate in Fig. 2c, the variation of the laser reflectance (R) as a function the drain source voltage (Vds) at a fixed gain voltage (\(\overline {V_{{{\mathrm{g}}}}}\)=1 V). To exclude alternative softening source, we use a constant laser power (30 µW) and the membrane is illuminated continuously for 30 min prior to the data acquisition to avoid any potential thermal drift induced by the laser. Modeling of 2D membrane and the underneath SiO2 surface as a simple Fabry-Perot cavity, we obtain an expression of the absolute reflectance which depend on the actual length of the cavity (d = d0 + z) and thus yields an in situ value of the membrane apex position (z), see Supplementary Fig. 12. We measure a maximum deflection z = −6 nm when a polarized lateral bias Vds = 6 V is applied to the sample. Replacing this parameter in Eq. (4) we can evaluate the magnitude of fz for this particular drain source voltage. Repeating this procedure for all individual Vds, we deconvolve in Fig. 2d the contribution of fz to the observed experimental variation. From the shape and the numerical variation, we see that the capacitive softening fz, is not the source of the observed strong mechanical softening with lateral bias. Other experiments varying the gate voltage (see Supplementary Fig. 8) confirms the previous conclusion.

We can proceed similarly fS and fB, with similar conclusions, which leaves the fm as the only possible source of the electromechanical tuning in our sample. Based on this hypothesis, we fit the experimental frequency shift (Fig. 2d, black dots) to obtain the experimental variation of fm as a function of the drain source voltage (Fig. 2d, blue curve). As fm only depends on Vds via the thermal softening of the membrane in-plane tension γ0, we now have an experimental variation γ0(Vds, Ids), which we can investigate against to the proper electrothermal heating model of our membrane by Joule effect (see in Supplementary Information). This strongly supports our hypothesis that Joule heating is the cause of membrane softening. This is confirmed also by Raman measurements, described later on.

To investigate experimentally the temperature profile of our membrane in-situ, during the optomechanical measurements, we turn to non-contact Raman spectroscopy. In Fig. 3a, the variations of two main Raman mode A1g and E12g of the MoS2 membrane are shown for different Vds voltages. Without thermal heating (i.e at room temperature), the reference position of the A1g peak is 408 cm−1 and the E12g peak is 385 cm−1. The two main Raman peaks A1g and E12g are shifted with the electrical power applied to the sample by ΔA1g and \({\Delta}E_{2{{{\mathrm{g}}}}}^1\). The ratio of the two shift peaks being almost equal to 1, this is the signature of a temperature variation of the suspended membrane64,65. Knowing the variation \(dA_{1{{{\mathrm{g}}}}}/dT\) and \(dE_{2{{{\mathrm{g}}}}}^1/dT\) are respectively −0.016 cm−1 K−1 and −0.013 cm−1 K−1, and −0.013 cm−1 K−1 and −0.015 cm−1 K−1 for a multilayer 64, we compute the membrane temperature as follows;\(T = 24^\circ C + {\Delta}E_{2{{{\mathrm{g}}}}}^1\,.\,dE_{2{{{\mathrm{g}}}}}^1/dT.\) Proceeding similarly for all Vds voltages, we combine in Fig. 3b the variation of Raman derived temperature and the experimental DC Joule heating (P = Ids Vds). The temperature of the membrane rises to 300 °C at P = 18 mW. We also observe a strong linear correlation, which we model (red line) as the direct heating of the membrane in vacuum. There are additional peaks in Fig. 3a. A fortiori, these extra peaks come from the unsuspended and thermalized part since the position does not change with heating. We assume here that the focalization is not optimized or that there is an artifact of the optical measurement. We have verified that a broader Gaussian laser distribution does not affect the simulated temperature too much (less than 10 °C).

Fig. 3: Thermal transport measurement.
figure 3

a Raman spectroscopy of MoS2 membrane at different joule heating power (P). Blue line: no bias applied, the membrane is not heated. Orange and red: we applied different source drain bias and achieved a heating power of P = 8 mW and P = 19 mW. b Temperature, obtained from Raman, of the 2D material in function of the applied power. The red line is a linear fit of the curve and the gray area the incertitude of the slope. In inset, the spatial distribution of temperature measured by Raman during Joule Heating of 1 mW along a MoS2 membrane center at 3.5 µm along the axe x (c) Resonant frequency of MoS2 membrane at different Joule heating power (P). Blue line: no bias applied, the membrane is not heated. from P = 0 mW to P = 2.7 mW (Vg = 1 V). d Temperature of the membrane in function of the applied power extracted from the mechanical resonance of the membrane. In inset, the extracted thermal expansion coefficient α in function of the temperature.

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To this end, we consider a Comsol simulation of the Joule heating effect along the sample for applied power, see in see Supplementary Fig. 11 (temperature coupling and boundary electromagnetic heating modules). Our simulation needs to define a charge density profile along the device. In order to be accurate with our system. We consider that the current comes mainly from the photodoping and from the active traps which locally increase the doping and the charge density. The MoS2 is suspended and under vacuum so no heat dissipation is possible in the out of plane direction and electrical energy is well converted into heat. We deduce the variation of the temperature compared to the calorific power of the membrane \(dT/dP = 17.5\,K.mW^{ - 1}\). The resulting thermal conductivity k of the 2D material is k = 50 W m−1 K−1 which corresponds to our previous report measured on similar MoS2 and of typical values of thermal conductivity in suspended MoS264.

We heat our sample with Joule heating and this results in strong mechanical softening with Vds. There is an analogy between Raman spectroscopy measurements and the mechanical vibrations of our resonator regarding heat dependence and the temperature sensitivity12,33. Having independently established the temperature variations of our MoS2 membrane T(Vds,Ids) using Raman (Fig. 3b) and its mechanical softening \(\gamma _0\left( {V_{{{{\mathrm{ds}}}}},\,I_{{{{\mathrm{ds}}}}}} \right)\) in Fig. 3c, we finally correlate in Fig. 3d those two quantities. At first order, we see that the frequency of the fundamental mode of the membrane is well described by a linear variation \(\gamma _0\left( T \right) = \gamma _0 + \gamma _{{{\mathrm{T}}}}\left( T \right)\) where γT is the additional tension due to the thermal dilatation at temperature T and γ0 = 0.7 N m−1 the built-in tension at room temperature (T0 = 20 °C). It is expressed as \(\gamma _{{{\mathrm{T}}}} = - Et{\int}_{T_0}^T {\alpha \left( {T^\prime } \right)dT^\prime }\), α the thermal expansion coefficient (TEC). In first approximation, we neglect here any contribution from the underlying SiO2 substrate and only consider the thermal expansion of the MoS2 membrane. Using the Eq. (1), we can determine the total tension of the membrane at any heating power. Combining the Eq. (1) with our simulation of thermal conductivity, see in SI, we are able to extract the TEC for our material at each heating power, in the inset of Fig. 3d. We obtain a TEC almost constant, as expected and around the value α = 5 × 10-6 K−1. It is in accordance to the value α = 4.9 × 10-6 K−1 obtained experimentally for bulk MoS266,67,68,69 and by theory 70,71,72,73 and around 7 × 10–6 K−1 for monolayer MoS2. It corroborates our model of thermal heating. The slope for the variation of the mechanical vibration frequency is approximately \(df_0/dP = 5.26MHz.mW^{ - 1}\). In order to extract the temperature sensitivity of our device, we fix the drive signal frequency at mid-amplitude of the mechanical resonance, that is to say when the response slope \(\partial V_{{{{\mathrm{out}}}}}/\partial f\) is maximum, and we measure the voltage output Vout for a long time. In Fig. 4a, we measured Allan’s deviation of the frequency as a function of the integration time. We measured the output Voltage noise \(\delta V_{{{{\mathrm{out}}}}}\)when the drive frequency is set at half the maximum amplitude of the resonance f0. The noise in temperature is \(\delta T = \delta V_{{{{\mathrm{out}}}}}\,.\,\left( {dT/df_0} \right).\left( {df_0/dV_{{{{\mathrm{out}}}}}} \right)\). By converting the frequency noise, we get a temperature noise of 20 mK Hz−0.5. By way of comparison, the temperature resolution with the Raman measurements is of the order of 20 K for a few seconds of integration per signal.

Fig. 4: Long living photothermal softening of the 2D membrane.
figure 4

a The Allan deviation of the mechanical resonance in function of the integration time, measured over a period of hundreds of seconds47. In insert, a schematic to explain the measurement of frequency and temperature noise at mid amplitude along the mechanical response. b Spatial cartography of the two first mechanical modes of the drum at two different values of Vds. The higher mode (1.1) is observed at high Vds in (a). The field direction is indicated with the black dot arrow. Scale bar: 1 µm. c Measurement of the mechanical signal response and the damping rate Γ in function of time for different switching of the incoming laser power, from 300nW to 30 µW. For minimum power, at 300 nW, the response signal has been artificially enhanced for clarity. After some time without any light, the initial frequency is recovered. d Slow current variation over long time, Ids(t), when the laser is switch on and the laser is switch off.

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To confirm that our mechanical model is adequate to our system, we cartography the vertical displacement relative to two mechanical modes for different Vds in Fig. 4b. The modes of a circular drum are Bessel functions of different orders, noted by their (m,n) index. The observed modes are the fundamental mode (m = 0, n = 1) which has a circular shape and the mode (m = 1, n = 1)74,75,76. At Vds = 0 V, the modes show the expected circular symmetry. When a lateral electrical field Vds = −5 V is applied, the both modes are modified. We demonstrate the mode to be sensitive to the electric field in addition to the electrothermal softening since the temperature profile has a circular symmetry for a circular membrane, at less from the model.

Unlike electric heating, another strategy to address heat transport is to use local optical heating of 2D materials as with optomechanical studies12,77, Raman spectroscopy50,78 or pump probe experiments56,79. We have proceeded to measurement of the mechanical signal in function of time for different laser power, from 300 nW to 30 µW, in Fig. 4c. With a power of 30 µW, there is a strong softening of the vibration of 1.5 MHz and the damping rate increase from 50 kHz to 90 kHz. This photoinduced effect is highly similar to Joule heating experiment, in Figs. 2a and 2e, when a voltage around 3 V is applied to the resonator, the frequency decrease by almost 2 MHz and the damping increase from 60 kHz to 90 kHz. This similitude strongly suggests a photothermal heating of the membrane. The corresponding temperature increase is around 10 °C, which is also fully reasonable considering a laser power of few tens of µW on a suspended 2D material.

We also measured the slow time response of the mechanical frequency, with a timescale above 500 s, in Fig. 4c and in the Supplementary Fig. 20. We previously reported such persistent mechanical softening effect (PPMS) of the mechanical resonance which has been related to long term persistent photocurrent (PPC) as it exists in our samples (Fig. 4d)52,80,81. The PPC, as well as PPMS, results from charge traps located at the sulfur vacancies (SV) along the sample. The origin of the long-time scale seems to lie in the large spatial separation of SV traps and their random distribution and diffusion. It creates random potential energy for charge carriers. This phenomenon, and especially the timescale, is strongly temperature dependant80,81. On contrary to PPC which are related to a strong photodoping of the 2D material, the mechanical counterpart PPMS is almost the same for different current Ids or Vds and not directly related to the charge flow (see Supplementary Fig. 19). In addition, as mention previously, we are able to quantify the doping effect with the mechanical capacitive terms \(f_{{{\mathrm{z}}}}^2\left( {V_{{{\mathrm{g}}}}} \right)\) or \(f_{{{\mathrm{S}}}}^2\left( {V_{{{\mathrm{g}}}}} \right)\) . But the reflectance measurement has shown no out of plane deflection of the membrane, even few nanometer, during the long-time measurements. It means it is impossible to relate our slow mechanical softening measurement to any doping effect. Saying this, and since we suggest a photothermal heating for this very slow mechanical softening, our own interpretation consists to say that the phonons distribution and the local thermal conductivity is affected as well by the random potential energy and the diffusion of the SV traps. It can explain the long-time term reduction of the vibration frequency but the absence of a direct relation with the doping and PPC.

To finish, we observe a damping increase by a factor 3 for high polarization, the temperature increases of 70 °C is not enough to explain this additional dissipation. Beyond the taking into account of the stationary local expansion of the material, which affect the mechanical frequency and the damping of our resonator, the heat propagation along the 2D material is at the origin of a response delay between the force and the movement. A light is absorbed by the 2D material and the material is mechanically deformed due to heating, after a certain delay time τ. In backaction, this deformation modifies the leakage of the photon population in the cavity. It defines the light-vibration coupling and the appearance of photothermal forces in micro and nanoresonators12,15,82,83. It creates a drift \({\Delta}f = - f_0/2k\,.\,\partial F/\partial z\,.\,1/\left( {1 + \left( {2\pi f_0\tau } \right)^2} \right)\) and an additional damping \({\Delta}{\Gamma} = 4\pi f_0\tau \left| {{\Delta}f} \right|\). This type of photo-induced force is proportional to the light reflection intensity at the z position and appearing when a native deflection is applied to the membrane12. Deduced from the reflective measurements, the out-of-plane deflection z induced a term \(f_{{{\mathrm{z}}}}\left( z \right)\) which is almost of the order of f0 at high VDS. This gives a time delay \(s\tau \sim {\Delta}{\Gamma}/4\pi f_0^2\sim 3.10^{ - 11}s\) for our measurements.

To conclude, we performed mechanical measurements on suspended MoS2 membranes by combining optical reflection and electrical polarization. We heated the sample with an electrical current. The demonstration of the efficiency of Joule heating to tune mechanical resonance has been reported for our devices. We have compared the frequency tunability of vibrations by simple electric voltages, and the attainable frequency range, with many previous works and determined that the electrothermal method is very efficient and simpler than traditional capacitive methods or other solutions. Beyond that, the electrothermal tuning makes it possible to measure the thermal properties of the membrane such as thermal conductivity or thermal expansion of the material.

Methods

Experimental set-up

We electrically drive the movement of the membrane using the AC gate bias (\(\widetilde {V_{{{\mathrm{g}}}}}\))8,52 added to the DC back gate voltage (\(\overline {V_{{{\mathrm{g}}}}}\)) so that the total back gate voltage is \(V_{{{\mathrm{g}}}} = \overline {V_{{{\mathrm{g}}}}} + \widetilde {V_{{{\mathrm{g}}}}}\)52. Optically, the MoS2 membrane and the bottom Si surface of the cavity form a Fabry Perot interferometer8,84, which converts the vertical displacement of the MoS2 surface into modulations of the reflected intensity of the 633 nm laser (1.96 eV). This combination of electrical AC drive and optical detection allows one to scan the mechanical spectrum. The mechanical modulation of the laser reflection is the strongest when driving the membrane close to one of its resonant mechanical frequencies, see Fig. 1e near f = 35 MHz, which we characterize by its quality factor, Q = 1040 under vacuum (5 × 10–6 mbar), at room temperature.

Ids(Vds) curve are measured with Yokogawa sources and Femto current amplifier. We used a Long working distance objective ×100 (Mitutoyo) for the reflectance and optomechanical measurements. A photodetector transduces the intensity of the reflected laser spot into voltage, which is recorded by a fast lock-in amplifier (Zurich instrument). Raman and PL spectroscopy was performed within the same vacuum chamber with a Horiba HR evolution system.

Sample fabrication and 2D transfer

Large sheets of MoS2 are obtained by direct vapor deposition on silicon substrates. The transfer method consists, in a first step, in reducing the interaction between the 2D material and the substrate. For this, the substrate is covered with PMMA resin and immersed in liquid nitrogen, the difference in thermal expansion between 300 K and 77 K mechanically decoupling the different parts60. The protective PMMA is then removed in acetone. The 2D sheets are then pick-up on PDMS/PPC bubbles using an HQ graphene transfer station. In order to increase the interaction forces between the bubble and the 2D, during the extraction of the 2D sheets, the capillary force of a drop of water is used during the transfer at the interface PPC/substrate59. The sheets are then deposited on the pre-made structures over the etched holes in the SiO2.

Supplementary material

In addition to the main manuscript, we provide all the details of Fig. 2b on the frequency tunability of 2D and 1D nanoresonators in the literature. We present the experimental setup, additional characterizations of the sample, and additional measurements of the Joule heating experiment. The Comsol simulation of the 2D heating is also detailed.

Data avaibility

The data that support the findings of this study are available from the corresponding author upon reasonable request.