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# Electromechanical Resonators from Graphene Sheets

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Science  26 Jan 2007:
Vol. 315, Issue 5811, pp. 490-493
DOI: 10.1126/science.1136836

## Abstract

Nanoelectromechanical systems were fabricated from single- and multilayer graphene sheets by mechanically exfoliating thin sheets from graphite over trenches in silicon oxide. Vibrations with fundamental resonant frequencies in the megahertz range are actuated either optically or electrically and detected optically by interferometry. We demonstrate room-temperature charge sensitivities down to 8 × 10–4 electrons per root hertz. The thinnest resonator consists of a single suspended layer of atoms and represents the ultimate limit of two-dimensional nanoelectromechanical systems.

The miniaturization of electromechanical devices promises to be as revolutionary in the coming decades as the miniaturization of electronic devices was in the previous ones. Devices ranging from nanoscale resonators, switches, and valves have applications in tasks as diverse as information processing, molecular manipulation, and sensing. The prototypical nano-electromechanical system (NEMS) is a nanoscale resonator, a beam of material that vibrates in response to an applied external force (1, 2). The ultimate limit would be a resonator one atom thick, but this puts severe constraints on the material. As a single layer of atoms, it should be robust, stiff, and stable.

Graphite consists of stacked layers of graphene sheets separated by 0.3 nm and held together by weak van der Waals forces (3). It has extremely high strength, stiffness, and thermal conductivity along the basal plane. In addition, graphite can be exfoliated onto an insulating substrate, producing micron-sized graphene sheets with thicknesses down to a single atomic layer (48). Thus far, research on these thin graphene sheets has focused primarily on their electronic properties. We demonstrate a method of suspending single- and multilayer graphene sheets over trenches and show that such sheets can be mechanically actuated. This work also makes a detailed study of the mechanical properties of these graphene resonators, including resonance frequency, spring constant, built-in tension, and quality factor.

Suspended graphene sheets are fabricated with a peeling process similar to that reported previously (57). In our case, the graphene sheets are mechanically exfoliated over predefined trenches etched into a SiO2 surface (Fig. 1) (9). The result is a micron-scale doubly clamped beam or cantilever clamped to the SiO2 surface by van der Waals attraction. Some devices have prepatterned gold electrodes between the trenches to make electrical contact (Fig. 1, A and D).

A noncontact mode atomic force microscope (AFM) was used to quantitatively measure the thickness of the sheets on the substrate next to the trench, as shown in the inset in Fig. 1D. However, for sheets thinner than 2 to 3 nm, such measurements are unreliable (1012). For these we used spatially resolved Raman spectroscopy to determine the number of layers (Fig. 1C) (1012). The graphene sheet in Fig. 1B has an AFM-determined height of 0.9 nm. By comparison with previous results (1012), the shape of the Raman peak near 2700 cm–1 suggests the sheet is two layers thick over the area lying on the SiO2 substrate (Fig. 1C), whereas the section suspended over the trench is a single graphene layer.

All resonator measurements are performed at room temperature and a pressure of <10–6 torr unless otherwise indicated. The resonators are actuated by using either electrical (Fig. 1A) or optical modulation. In the case of electrical modulation, a time-varying radio frequency (rf) voltage δVg at frequency f is superimposed on top of a constant voltage and applied to the graphene sheet. The result is an electrostatic force between the suspended graphene sheet and the substrate $Math$(1) where Cg′ is the derivative of the gate capacitance with respect to the distance to the gate, and V dcg and δVg are, respectively, the dc and time-varying rf voltages applied to the gate (13). For optical actuation, the intensity of a diode laser focused on the sheet is modulated at frequency f, causing a periodic contraction/expansion of the layer that leads to motion. In both cases, the motion is detected by monitoring the reflected light intensity from a second laser with a fast photodiode (9).

Figure 2A shows the measured amplitude versus frequency for a 15-nm-thick sheet suspended over a 5-μm trench. Multiple resonances are observed, the most prominent one at the lowest frequency. We associate this dominant peak with the fundamental vibrational mode; its detected intensity is largest when the motion is in-phase across the entire suspended section. We will limit our discussion primarily to this fundamental mode. A fit to a Lorentzian yields a resonant frequency fo = 42 MHz andaquality factor Q = 210. Figure 2B shows similar results for the single-layer graphene resonator from Fig. 1B; f0 = 70.5 MHz and Q = 78. Figure 3 shows the results of measurements of 33 resonators with thicknesses varying from a single atomic layer to sheets 75 nm thick. The frequency f0 of the fundamental modes varies from 1 MHz to 170 MHz, with quality factor Q of 20 to 850.

For mechanical resonators under tension T, the fundamental resonance mode f0 is given by $Math$(2) $Math$ where E is the Young's modulus; ρ is the mass density; t, w, and L are the dimensions of the suspended graphene sheet; and the clamping coefficient, A, is 1.03 for doubly clamped beams and 0.162 for cantilevers (14). In the limit of small tension, Eq. 2 predicts that the resonance frequency f0 scales as t/L2. Figure 3A shows the resonant frequency of the fundamental mode for resonators with t > 7 nm as a function of t/L2 plotted as filled squares. Also plotted is the theoretical prediction, Eq. 2, in the limit of zero tension, for both cantilevers and beams, where we have used the known values for bulk graphite r = 2200 kg/m3 and E = 1.0 TPa (3). This is a valid comparison considering the extensive theoretical and experimental work that shows the basal plane of graphite to have a similar value for E as graphene and carbon nanotubes (3, 15). To account for possible errors in E, we plot dashed lines that correspond to values of E = 0.5 TPa and 2 TPa. The data follow the predictions reasonably accurately, indicating that thicker resonators are in the bending-dominated limit with a modulus E characteristic of the bulk material. This is among the highest modulus resonators to date, greater than 53 to 170 GPa in 12- to 300-nm-thick Si cantilevers and similar to single-walled carbon nanotubes and diamond NEMS (13, 16, 17). In contrast to ultrathin Si cantilevers, the graphene resonators show no degradation in Young's modulus with decreasing thickness (17).

The resonant frequencies versus t/L2 for the resonators with t < 7 nm are shown as open squares in Fig. 3A. The frequencies of these thinner resonators show more scatter, with the majority having resonant frequencies higher than predicted by bending alone. A likely explanation for this is that many of the resonators are under tension, which increases f0 (see supporting online text). The tension likely results from the fabrication process, where the friction between the graphite and the oxide surface during mechanical exfoliation stretches the graphene sheets across the trench.

The single-layer graphene resonator shown in Fig. 1B illustrates the importance of tension in the thinnest resonators. It has a fundamental frequency f0 = 70.5 MHz, much higher than the 5.4 MHz frequency expected for a tension-free beam with t = 0.3 nm, L = 1.1 μm, and w = 1.93 μm. From Eq. 2, this implies that the graphene resonator has a built-in tension of T = 13 nN. From the expression ΔL/L = T/(EA), this corresponds to a strain of 2.2 × 10–3%.

An important measure of any resonator is the normalized width of the resonance peak characterized by the quality factor Q=f0/Δf. A high Q is essential for most applications because it increases the sensitivity of the resonator to external perturbation. A plot of the Q versus the thickness for all the graphene resonators (Fig. 3B) shows that there is no clear dependence of Q on thickness. This contrasts with results on thicker NEMS resonators fabricated from silicon (18). The quality factors at room temperature are lower than diamond NEMS (2500 to 3000) of similar volume and significantly lower than high-stress Si3N4 nanostrings (200,000), yet similar to those reported in single-walled carbon nanotubes (50 to 100) (13, 16, 19). Preliminary studies on a 20-nm-thick resonator found a dramatic increase in Q with decreasing temperature (Q = 100 at 300 K to Q = 1800 at 50 K). This suggests that high Q operation of graphene resonators should be possible at low temperatures.

Even when a resonator is not being driven, it will still oscillate due to thermal excitation by a root mean square (RMS) amount xth =[kBTeff]1/2, where κeff = meff ω02 = 0.735Lwtρω02 is the effective spring constant of the mode (2). An example is shown in Fig. 4A, where a 5-nm-thick resonator with f0 = 35.8 MHz and κeff = 0.7 N/m has a room-temperature thermal RMS motion of xth = 76 pm. For resonators for which the thermal vibrations can be measured, we use this thermal RMS motion to scale the measured photodetector voltage with resonator displacement (see supporting online text). Figure 4B shows such a rescaled plot of the displacement amplitude versus rf drive voltage. The resonator is linear up to displacements of 3 nm, or on the order of its thickness, where nonlinearities associated with additional tension are known to set in (2). This nonlinearity is characterized as a deviation from a linear increase in amplitude with driving force and accompanied by a decrease in Q (Fig. 4B).

Two applications of nanomechanical resonators are ultralow mass detection (see supporting online text) and ultrasensitive force detection. The ultimate limit on the force sensitivity is set by the thermal fluctuations in the resonator: $Math$(3) For the resonator in Fig. 4A, this results in a force sensitivity of 0.9 f N/Hz½. From Eq. 1, this corresponds to a charge sensitivity of dQf = dFf d/Vgdc = 8 × 10–4 e/Hz½, where d is the distance between the graphene sheet and the gate electrodes. This is a high sensitivity demonstrated at room temperature; at low temperatures, with the onset of higher quality factors, it could rival those of rf single-electron transistor electrometers (1 × 105 e/Hz½) (20, 21). The high Young's modulus, extremely low mass, and large surface area make these resonators ideally suited for use as mass, force, and charge sensors (2228). The application of graphene NEMS extends beyond just mechanical resonators. This robust conducting membrane can act as a nanoscale supporting structure or atomically thin membrane separating two disparate environments.

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S3

References