Energy Dissipation Pathways in Few-Layer MoS2 Nanoelectromechanical Systems

Free standing, atomically thin transition metal dichalcogenides are a new class of ultralightweight nanoelectromechanical systems with potentially game-changing electro- and opto-mechanical properties, however, the energy dissipation pathways that fundamentally limit the performance of these systems is still poorly understood. Here, we identify the dominant energy dissipation pathways in few-layer MoS2 nanoelectromechanical systems. The low temperature quality factors and resonant frequencies are shown to significantly decrease upon heating to 293 K, and we find the temperature dependence of the energy dissipation can be explained when accounting for both intrinsic and extrinsic damping sources. A transition in the dominant dissipation pathways occurs at T ~ 110 K with relatively larger contributions from phonon-phonon and electrostatic interactions for T > 110 K and larger contributions from clamping losses for T < 110 K. We further demonstrate a room temperature thermomechanical-noise-limited force sensitivity of ~8 fN/Hz1/2 that, despite multiple dissipation pathways, remains effectively constant over the course of more than four years. Our results provide insight into the mechanisms limiting the performance of nanoelectromechanical systems derived from few-layer materials, which is vital to the development of next-generation force and mass sensors.


2
(on the order of several days minimum) to assist in the removal of physisorbed water on the surface of the resonator and to ensure minimal energy dissipation due to gas friction.
An estimate of the dissipation due to gas friction S1 at room temperature and under vacuum can be made using the expression !"# !! = ! !"! !"# 1 + 8 where P ~ 1.0x10 -6 Torr is the gas pressure, ρ ~ 5,060 kg/m 3 is the MoS 2 resonator density, v avg ~ 500 m/s is the mean speed of thermal-molecular motion based upon the equipartition theorem, t and d are the resonator thickness and diameter, respectively, µ ~ 0.028966 kg/mol is the mean molecular weight of the gas, and R = 8.314 J/Kmol is the universal gas constant. We assume the same heat capacities at constant volume and constant pressure. The result is !"# !! ~ 9x10 -14 at T = 293 K, which is 9 orders of magnitude lower than the lowest measured Q -1 values reported in the main text. 4

Cubic fit to f(T):
The cubit fit to the data shown in Fig.3d of the main text captures the functional form of the fundamental frequency temperature dependence, and is used in the subsequent data analysis of Q -1 (T). We find that f(T) is well fitted by the following cubic equation: The fit to this functional form for f(T), shown in Fig. 3d of the main text, had an R 2 value of ~0.99975.

Akheiser regime discussion:
Dissipation due to phonon-phonon interactions is modelled using Akheiser theory (see page 9 of the main text) across the full range of temperatures (4.4 K -293 K). The Akheiser theory is applicable in our resonators since 1/τ ph >> ω where τ ph is the phonon relaxation time (~ 2x10 -12 s) and ω the resonator angular frequency (~1x10 8 s -1 ). Temperature dependent Raman measurements on few-layer MoS 2 samples have shown a slight decrease in the full-width at halfmaximum (FHWM) of the A 1g mode with decreasing temperature from 523 K to 83 K S2 ; the temperature-dependent linewidth of the Raman peak can be used to estimate phonon relaxation times using the Heisenberg uncertainty relation ~ ∆ = ℏ !! where ℏ is Planck's constant divided by 2π. The data shown in Fig. 4 of Reference S2 yields τ ph ~ 1.6 ps at 300 K and τ ph ~ 1.97 ps at 83 K, which demonstrates that τ ph changes by a factor of only ~ 1.2 between 5 room temperature and 83 K. The temperature-dependent Raman data from Reference S2 also show an apparent saturation in the FWHM for T < 200 K. These results demonstrate that the order of magnitude of τ ph (on the order of 1.0x10 -12 ) remains the same from room temperature down to temperatures below 100 K, which ensures that 1/τ ph >> ω and that the Akheiser regime remains applicable in few-layer MoS 2 devices at low temperatures.

Temperature-dependent data analysis:
The data fittings shown in Fig. 4 of the main text were carried out by using the expression where ! represents a constant coefficient and free-fitting parameter and  , specifically C v , C p , γ, ρ, v, E and α have all been experimentally determined for MoS 2 and we use these experimentally determined quantities within the data fittings (E is the Young's modulus found for our MoS 2 resonator, which is reported on page 7 of our manuscript). Second, the fitting parameters R and S contain additional information such as mode-participation factors S3 and information related to device geometry and 6 structure S4 . Prior studies S3,S4 have accounted for such factors and information by incorporating them into a constant of proportionality term to each dissipation pathway, which is the convention that we adopt for our data analysis. Third, the scattering times τ ph ~ 2 ps and τ s ~ 1 µs (the scattering times associated with !! !! and !"#$ !! , respectively) are separated by six orders of magnitude and it is this large difference between the scattering times that helps to separate the relative contributions of !! !! and !"#$ !! to the overall measured dissipation, which is why we refrain from using the scattering times as free fitting parameters in the data fittings.
The following Table S1 summarizes the results of the data fittings, and the dissipation contributions from each pathway (right four columns of Table S1) can be compared to the overall measured dissipation, !! , presented in Fig. 4 of the manuscript ( !! ~ 10 -2 at T = 293 K and !! ~ 10 -4 at T = 4.4 K):