We fabricate 2D NEMS resonators by first lithographically patterning substrates with surface microtrenches and contact electrodes, and then transferring the 2D MoS2 onto the surface microtrenches using a dry-transfer process based on polydimethylsiloxane (PDMS) stamp, which is widely used in 2D device transfer, especially for suspended devices.12,22,23,32,31,34,37,40 The DC gate voltage *V*GS and radio-frequency (RF) driving voltage *v*RF are applied to the gate electrode through a bias tee to capacitively drive the membrane, with the contact electrode grounded. The suspended MoS2 membrane is pulled down towards the gate electrode due to electrostatic force induced by *V*GS, which leads to the tension and resonance frequency tuning. The resonances are measured using a custom-built optical interferometry setup (Supporting Information section S1).31,41,42 All resonance measurements are performed in vacuum at room temperature.

To measure the *Q* factors at different diameters while minimizing the effects from the MoS2 material’s variation, we fabricate an array of 2D MoS2 NEMS resonators on a substrate with 8×8 circular microtrenches with various diameters (Figure 1a). The fully-clamped resonator structure avoids the undesirable effects of free edges on energy dissipation, and can enhance the *Q* factor.36,37 We measure resonances from 34 resonators in the array, and name each device as Device “*m*–*n*”, where *m* is the row number and *n* is the column number. The membrane surface is first characterized using AFM, which can show the surface quality of the suspended membranes clearly (Figure 1a). Comparisons show that resonators with larger diameters (>3 μm) typically contain more wrinkles (Figure 1b–1e) on the suspended membranes than those with smaller diameters (2–3 μm) (Figure 1f–1g). We then perform Raman mapping to measure the uniformity of the MoS2 material (Figure 1n–1q), where for the MoS2 with the same thickness, uniform suspended MoS2 membranes generally show slightly higher Raman peak intensity than a wrinkled membrane. Resonance spectrum for each resonator is recorded and fitted to the solution for equation of motion to extract the *Q* factor (Figure S4), with the *Q* factors summarized in Figure 1r. We find that the *Q* factors can vary by several times for different devices with the same diameter and thickness, *i.e.*, from 49 (Device 1–1) to 179 (Device 5–1). Furthermore, the *Q* factors of thicker resonators that are in the plate regime typically show slightly higher *Q* factors than thin resonators that are in the membrane regime. We then perform in-depth resonance measurements for two representative resonators (Devices 1–1 and 5–1). Mode mapping measurements are performed by fixing the laser and scanning the stage that holds the vacuum chamber with the resonator inside, which demonstrate that Device 1–1 with more wrinkles on the surface (Figure 1b) also shows spurious modes (Figure 1i–1j) and a nonuniform mode shape (Figure 1h). In contrast, Device 5–1 with a more uniform surface and less wrinkles (Figure 1d) shows clear fundamental flexural mode shape (Figure 1k), and a resonance mode with regular frequency tuning characteristics (Figure 1l). By gradually increasing *v*RF, Device 5–1 shows linear to Duffing nonlinear resonances with hardening (Figure 1m). The gate-tunable resonance frequency model can well fit the frequency tuning characteristics of Device 5–1 but not that for Device 1–1 due to the spurious modes (Supporting Information Section S4.1–S4.2), using:31

where *R* is the radius,* t* is the thickness,* E*Y is the Young’s modulus,* ν* is the Poisson’s ratio, *ε*r is the gate-tunable total strain, *g* is the initial vacuum gap, *ϵ*0 is the vacuum permittivity, and *ρ* is the mass density.

The lower *Q* factors for devices with more wrinkles and residues on the surface suggest the importance of surface-induced damping mechanisms in these 2D NEMS resonators. From the summarized *Q* *vs.* diameter relationship in Figure 1r, we find that the *Q* factor generally decreases with a larger diameter in the range of 2 μm to 5 μm diameter. If the anchor loss or the thermoelastic damping dominates, then according to previous models,32,43 *Q* factor should increase with the resonator diameter. Therefore, the results in Figure 1r suggest that as the resonator size increases, there is a higher chance to contain surface residues and wrinkles, which induce more energy loss.

To further validate the size dependence of *Q* for thinner resonators, and achieve a high *Q* factor in MoS2 NEMS resonators, we fabricate another single isolated MoS2 NEMS resonator with the diameter of 2 μm, which has bilayer thickness and is named “Device S1”. The clean, flat, and smooth surface in the suspended region is confirmed by the SEM images (Figure 2a–2b). Besides, the bilayer thickness and high quality of the MoS2 material are confirmed by the high PL intensity and the Raman peak separation of 22.1 cm-1 (Figure 2c–2d).41,44 The *V*GS tuning of resonance frequency* *shows clear trend that can be well fitted by the frequency tuning model (Figure 2e), and the resonance measurements by increasing *v*RF show clear transition from undriven thermomechanical resonance to nonlinear driven resonances (Figure 2f). Furthermore, the *Q* factor decreases with a larger *V*GS and *v*RF (Figure 2g–2j), which can be well fitted with the strain-modulated thermoelastic dissipation model (Supporting Information Section S4.3):

where *δ* is a fitting parameter representing the loss angle, *x*0 is the vibration amplitude that is proportional to |*V*GS×*v*RF|. From fitting to the gate tuning of *Q *factor, we can extract *δ* for each resonator. By optimizing the diameter, minimizing surface nonidealities, and decreasing the driving strength, we achieve a high *Q* factor up to 3,105±207 at *v*RF = 1 mV* *and* **V*GS = 1 V (Figure 2g). Such *Q* factor is comparable to some piezoelectric MEMS resonators measured under similar conditions.45,46,47

To more comprehensively study *Q vs.* diameter relationship, we further scale down the resonators and measure a single isolated resonator “Device S2” with 1-μm diameter (Figure 3a) and monolayer thickness (Figure 3d–3e). The flat and smooth surface is confirmed by the SEM images (Figure 3b–3c), showing no observable residue or wrinkle. From the measured resonances (Figure 3f–3g), we extract *Q* at each *V*GS and *v*RF, and obtain *Q* up to 1,051±77 (Figure 3h–3k). By comparing the results from Figures 2–3, we find that when the resonator surfaces are clean, the resonators with larger diameters show larger *Q* factors.

To further investigate the relationship between the *Q* factor and diameter, we measure 2D MoS2 NEMS resonators with larger diameters. For a resonator with 6-μm diameter (“Device S3”, Figure 4a) and bilayer thickness (Figure 4d–4e), the zoom-in SEM images clearly show the wrinkle (Figure 4b) and residue (Figure 4c) close to the clamping ends. From resonance measurements, we find that the frequency tuning characteristics before and after *V*GS = 10 V show different increasing trends (Figure 4f). The mode shapes and frequency tuning characteristics of resonators with surface residues or wrinkles are simulated using finite element method (FEM) (Figures S18–S20), where abnormal mode shapes for the fundamental flexural mode are observed. For the resonator with residue, we further demonstrate that the mode shape and the position of the maximum vibration amplitude change when the gate voltage increases (Figure S18). The irregular mode shape at each *V*GS determines the dynamic energy in the device, which could result in both non-ideal frequency tuning characteristics and larger dissipation. The extracted *Q* at each *V*GS and *v*RF show that *Q* factors are low, and *Q* decreases with a larger *V*GS and *v*RF (Figure 4h–4k). Fitting to the *Q* *vs.* *V*GS relationship also shows a turning point of different tendencies before and after *V*GS = 10 V, similar to the frequency tuning characteristics. The low *Q* factor and non-ideal characteristics for gate tuning of *Q* suggest correlations with the dynamic-energy-dependent loss angle *δ* (Supporting Information Section S4 and Table S1), where resonators with larger diameters and lower *Q* factors generally show higher *δ*. To confirm such effect, we further measure the frequency tuning characteristics of another device with the same 6-μm diameter named “Device S8” (Figure S7), which also shows relatively low *Q* factors and non-ideal resonance peak shapes.

For another MoS2 NEMS resonator with an even larger diameter of 8 μm named “Device S4” and with monolayer thickness (Figure 4o–4p), the SEM images also show that the suspended membrane contains several wrinkles (Figure 4l–4n). The frequency tuning characteristics (Figure 4q) and Duffing nonlinearity (Figure 4r) are also measured, but due to the multiple spurious modes, the frequency tuning trend is not clear and cannot be fitted consistently using the model. Multiple resonance peaks show similar resonance frequencies instead of well-separated resonances as predicted by the classical model (Figure 4s–4t),48 which could also be attributed to the surface nonidealities. The nonideal frequency tuning characteristics and mode shapes due to wrinkles are also simulated by FEM (Supporting Information Figure S19–S20). Summary of fitting to all the resonances at various *V*GS and *v*RF also show low *Q* factors, and decreasing *Q* with larger *V*GS and *v*RF, which cannot be fitted well after *V*GS = 15 V (Figure 4u and 4v). To further confirm the consistency of diameter and surface nonideality effects on device properties, we demonstrate measurement results from 13 additional single isolated MoS2 drumhead NEMS resonators with various diameters (Device S6–S18), which consistently show that larger membranes typically contain more surface nonidealities on the suspended membranes and thus show lower *Q* factors (Figures S5–S17).

We further investigate “Device S5” which contains an array of five microtrenches with the same diameter of 3 μm and uniform thickness of 10 nm (Figure 5a, 5g), but with a large bubble on the surface formed during device fabrication (Figure 5b, 5e). Despite the nominally same material and geometry, the five resonators in “Device S5” show different surface nonidealities on the suspended membrane: resonators “b” and “c” are influenced by the bubble (Figure 5e); resonator “a” shows a residue on the suspended membrane (Figure 5c); resonator “e” shows a small wrinkle near the clamping edge (Figure 5d); and resonator “d” has a very flat and clean suspended material (Figure 5g). The PL intensities for resonators “b” and “c” shows much smaller intensity than other resonators (Figure 5l–5o), further confirming the effects of the bubble. We measure the resonance tuning characteristics of all the five devices by varying *V*GS. Both resonators “e” and “d” show single clear resonances with high intensity, so that the frequency tuning characteristics can be well fitted with our model (Figure 5h–5i). However, resonators “c” and “a” show multiple spurious modes in the resonances (Figure 5j–5k). The extracted *Q* factors for the flat resonator “d” without surface nonidealities is the highest in Device S5 (Figure 5p–5s), which shows that surface bubbles can also lead to larger dissipation. The resonator “e” with a small wrinkle shows lower *Q* factor than that of resonator “d”, but a much higher *Q* factor than resonator “a” and “c”. Such control experiments further confirm the importance of surface nonidealities on resonance characteristics and damping properties of 2D NEMS resonators.

The measurement results from all 52 MoS2 NEMS resonators consistently show that devices with large diameters (3–8 μm) generally have irregular resonance frequency tuning characteristics and many spurious modes especially at large *V*GS, due to the higher chance to include surface nonidealities. In contrast, the devices with smaller diameters (1–2 μm) show clear resonance modes, with the frequency and *Q* factor tuning characteristics well fitted by the models. We summarize the measured *Q* factor *vs.* diameter relationship in Figure 5t, where the *Q* factor for each device is the largest *Q* factor at varying gate voltages. Because the dissipation increases with a larger drive (|*V*GS×*v*RF|), the largest *Q* factor is usually obtained near the first measurable resonance at small driving amplitude*.* The *Q*factor first increases and then decreases with a larger diameter, reaching a maximum value at 2 μm and minimum value at 8 μm. We model the overall effect from multiple damping mechanisms as:

where *Q*TED-1 is thermoelastic dissipation, *Q*Anchor-1 is anchor loss, *Q*SN-1 is the loss induced by surface nonidealities, and *Q*Other-1 is other diameter-independent damping mechanisms. Surface-induced energy dissipation has been found to limit or decrease the *Q* factor as the device size increases.49,50 For thermoelastic dissipation and anchor loss, a larger diameter can lead to a larger *Q* factor. When the diameter is 1–2 μm, thermoelastic dissipation and anchor loss dominates, and the *Q* factor increase with diameter, reaching up to ~3,105 at the diameter of 2 μm. When the diameter further increases, damping induced by surface nonidealities become more important, and the *Q* factors dramatically decrease down to ~10 at the diameter of 8 μm. Therefore, to achieve high *Q* factors in 2D NEMS resonators, a clean, smooth, and high-quality suspended membrane is the key for minimizing the surface-induced energy loss. We expect that when better fabrication techniques for these fully-clamped circular drumhead NEMS resonators can be developed, the *Q* factor should further increase for resonators with larger diameters.