Supporting Information
Imaging mechanical vibrations in suspended graphene sheets
D. Garcia-Sanchez, A. M. van der Zande, A. San Paulo, B. Lassagne, P. L. McEuen and A. Bachtold
A - Sample fabrication上海中学生英语报>few和afew的区别
股份支付Suspended graphene sheets are fabricated by mechanical exfoliation. A freshly cleaved piece of Kish graphite (Toshiba Ceramics) is rubbed onto a degenerately doped silicon wafer with 290 nm SiO2 grown by plasma enhanced chemical vapor deposition. Before depositing the graphene, the wafer is patterned with trenches using photolithography and plasma etching. The trenches are millimeter long, 0.5 – 10 µm wide, and 250 nm deep. Electrodes defined by photolithography between the trenches are deposited using electron-beam evaporation and consist of 5 nm Cr and 35 nm Au.
B- Description of the FEM model
To model the shape of the eigenmodes, we have developed a simulation bad on finite element methods (FEM) using ANSYS. The first step of the simulation is to account for the buckling of the suspended sheet by finding the adequate boundary conditions at the clamping edges. To do this, we h
old one clamping edge of the suspended region fixed, and impo an in plane displacement to the other clamping edge. Specifically, the displacement of this edge consists of a translation and a rotation within the undeformed resonator surface. Since the resulting out of plane displacement can be large, calculations are carried out taking into account geometric non-linear deformations1. To ensure that the buckling goes in the desired direction, we apply an out of plane perturbative force, which is then cleared at the end of the calculations. We make the assumption that the mechanical properties of the resonator are isotropic with 1TPa for the Young’s modulus and 0.17 for the Poisson ratio2. The exact value of the Poisson ratio has little effect on the output of the calculations. In the cond step of the simulation, a modal analysis is performed to determine the resonance frequency and the eigenmode shape of the deformed resonator. Here, the modal analysis is carried out in the linear regime becau the amplitude of the vibration is small. To check that this simulation is free of errors, it has been successfully compared to analytical predictions for a beam under tension3. The simulation also reproduces recent calculations on nanotubes with slack4. See ctions C and D.
C- Comparison between the FEM model and analytical expressions for resonators under tension . The resonance frequency for a beam under weak uniform tension (T <<EI/l 2 ) is 5
wtEI T wt EI l f ρπρπ1228.024
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We take the length l = 500 nm, the width w = 20 nm, the thickness t = 3.5 Å, the density ρ = 2200 kg/m 3, and the Young’s Modulus E = 1TPa. The bending moment of a rigid beam is I = wt 3/12.
For high tension (T>>EI/l 2) the frequency can be expresd as 3
[221)2/4(2121
ξπξρ+++=wt T l f ] (2)
with .
23212/Tl Etw =ξFigure S1 shows the resonance frequency as a function of tension using the FEM model and the above expressions. There is a good agreement between the theory and the FEM model.
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10
100
f (M H z )T(N)
Figure S1 Resonance frequency as a function of tension for the first eigenmode of a graphene resonator under tension.
D- Comparison between the FEM model and previous simulations on buckled SWNTs .
Previous work 4 has reported numerical studies on SWNT resonators that are buckled (slack). We compare this work to the FEM model that we have developed. For this, we u the same geometry and the same physical
characteristics as in reference 4. The resonator is a doubly clamped rod with l =1.75um, d =2nm, E =2.18TPa, ρ=2992kg/m3 and a slack of 0,3%. The slack is defined as the ratio of the excess length of the tube to the distance between the clamping points. Figure S2 shows the resonance frequency for different eigenmodes obtained with the two simulation techniques. A good agreement is obtained.
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400
f (M H z )
Eigenmode number
Figure S2 Resonance frequency for different eigenmodes of a SWNT resonator with 0.3% of slack.
Conclusions of ctions C and D. The FEM model shows a good agreement with established analytical expressions for beam resonators under tension and previous numerical calculations on buckled SWNTs. Note that the FEM model that we have developed can go beyond the cas. The model can be applied to resonators with any arbitrary geometry and any arbitrary stresd state.
E- SFM detection of mechanical vibrations.
语法填空
We have developed a technique bad on scanning force microscopy (SFM) to detect the mechanic
al vibrations of nanotube and graphene resonators 6. The resonator motion is electrostatically actuated with an oscillating voltage applied on a gate electrode. The frequency f RF of the driving voltage V RF is t at (or clo to) the resonance frequency of the resonator. In addition, V RF is modulated at f mod , )2cos())2cos(1(mod t f t f RF ππ−. While the SFM cantilever cannot follow the rapid oscillations at f RF , it can detect the modulation envelope.
The topography imaging is obtained in tapping mode using the cond eigenmode of the SFM cantilever. The vibrations are detected with the first eigenmode of the SFM cantilever. Figure S3 shows that the signal of the vibrations is significantly enhanced when f mod is matching the resonance frequency f tip of the first eigenmode of the SFM cantilever. As a result, measurements are carried out with f mod = f tip .
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2
A m p l i t u d e (a .u .)f mod (kHz)
Figure S3 Detected respon of the vibration of a nanotube driven at resonance as a function of f mod . The frequency of the first eigenmode of the SFM cantilever is 58 kHz. Nanotube resonance frequency f RF is 153 MHz. Measurements are taken at the nanotube position where the vibration amplitude is maximum.
Graphene resonators show a lorentzian respon to the rf drive frequency. Figure S4 a shows the frequency respon of a graphene resonator measured with the SFM technique. For comparison, Fig. S4 b shows the frequency respon of the same resonator measured using optical interferometry 7. The resonance frequencies are very similar for both techniques. However, the quality factor measured with the SFM technique is much lower due to energy dissipation to air, as the SFM technique is operated at atmospheric conditions, while the optical interferometry is performed in vacuum. Note that the low Q is not attributed to the disturbance of the SFM tip 6. Indeed, we have noticed no change in the quality factor as the amplitude t point of the SFM cantile
ver is reduced by 3%–5% from the limit of cantilever retraction, which corresponds to the enhancement of the resonator-tip interaction.
A m p . (a .u .)
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f (MHz)0
Figure S4 a Resonance peak of the fundamental mode of the graphene resonator shown in Figure 2 of the paper. The measurement is carried out using the SFM technique in air. The resonance is found at ~31 MHz with the quality factor Q = 5. Measurements are taken at the position where the vibration amplitude is maximum. b Resonance peak measured optically with a pressure of < 10-6 torr. The resonance is found at 32 MHz with Q = 64.
As shown in Eq. 1 of the paper, the radio frequency force F RF on the suspended sheet is a linear function of the offt voltage V DC and the radio frequency voltage V RF . Figure S5 shows vibration amplitude as a function of V DC for an edge eigenmode of a graphene resonator. We find that the vibration amplitude is a linear function of the DC voltage and thus of the force. By operating in this regime, we ensure that the resonators are operating in the linear respon regime, and the exotic edge eigenmodes are not a result of non-linear effects.
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