Graphene Nanoelectromechanical Systems as Stochastic-Frequency

Temperature-Dependent Adhesion of Graphene Suspended on a Trench. Zoe Budrikis and Stefano Zapperi. Nano Letters 2016 16 (1), 387-391. Abstract | Full...
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Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators Tengfei Miao,† Sinchul Yeom,‡ Peng Wang,† Brian Standley,‡,§ and Marc Bockrath*,† †

Department of Physics and Astronomy, University of California, Riverside, California 92521, United States, and Department of Applied Physics, California Institute of Technology, Pasadena, California 91125, United States

‡

S Supporting Information *

ABSTRACT: We measure the quality factor Q of electrically driven few-layer graphene drumhead resonators, providing an experimental demonstration that Q ∼ 1/T, where T is the temperature. We develop a model that includes intermodal coupling and tensioned graphene resonators. Because the resonators are atomically thin, out-of-plane fluctuations are large. As a result, Q is mainly determined by stochastic frequency broadening rather than frictional damping, in analogy to nuclear magnetic resonance. This model is in good agreement with experiment. Additionally, at larger drives the resonance line width is enhanced by nonlinear damping, in qualitative agreement with recent theory of damping by radiation of inplane phonons. Parametric amplification produced by periodic thermal expansion from the ac drive voltage yields an anomalously large line width at the largest drives. Our results contribute toward a general framework for understanding the mechanisms of dissipation and spectral line broadening in atomically thin membrane resonators. KEYWORDS: Nanoelectromechanical systems, graphene, nonlinear dynamics, resonators, parametric oscillators

G

determine the line width in graphene12 or carbon nanotube13 resonators, and predict Q ∝ 1/T. However, ref 13 calculates the Q in SWNTs from tension fluctuations, but does not explicitly take into account the nonlinear mode coupling, and is not immediately applicable to two-dimensional (2D) membranes such as graphene. In ref 12 a tensioned membrane is not directly considered, which is most relevant to typical experimental conditions. Here we conceptually divide the graphene sheet modes into “fast” and “slow” modes (see, e.g., ref 14) interacting through nonlinear coupling, such as that observed previously between two individual modes in microfabricated Si beam resonators.15 The slow mode behavior is reminiscent to that found in nuclear magnetic resonance (NMR), in which Larmor frequency fluctuations from fluctuating environmental magnetic fields broaden the resonance as compared to isolated nuclei.16,17 The frequency-dependent oscillator susceptibility has an imaginary dissipative part with a characteristic width Γ ∝ T on the same scale as the resonance width, similar to a damped oscillator with a damping force Fd = −mΓż, where z is the oscillator coordinate and m its mass. Under steady-state conditions the slow mode motion modulates the sheet tension, transferring energy to the fast

raphene’s remarkable properties1 including mechanical robustness and high electrical conductivity makes it a highly promising material for nanoelectromechanical systems (NEMS).2,3 A key parameter of NEMS resonators is their quality factor Q, which is the ratio of the stored energy to the dissipation per oscillation cycle, and is inversely proportional to the resonance line width. The dissipation can be linear, in which Q is amplitude-independent, or nonlinear,4,5 in which Q is amplitude-dependent.6 Here, we measure both linear and nonlinear damping in monolayer and few-layer electrically driven graphene drum resonators. Addressing the linear damping regime first, we show that for sufficiently small drive voltage Vsd, Q follows a 1/T law in temperature T down to low temperatures, with Q ∼ 100 at room temperature. A canonical model for dissipation considers a primary oscillator coupled to a bath of environmental oscillators.7−9 However, membranes have a complex thermodynamic behavior that has been well-studied in the literature, including in graphene, for example in ref 10, that may exhibit new phenomena. Recently, Duffing forces in conjunction with fluctuations have been found that could account for Q in nanotube resonators.11 However, based on the measured Duffing forces in our samples, this phenomenon is too small to account for the observed Q. Other recent work has shown that thermal fluctuations in the fundamental mode’s effective spring constant and resonance frequency produced by coupling to thermally excited out of plane modes can potentially © 2014 American Chemical Society

Received: October 21, 2013 Revised: March 2, 2014 Published: April 17, 2014 2982

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Figure 1. Few-layer graphene resonator device image and nanomechanical resonance data. (a) Scanning electron image of a completed graphene drumhead resonator device. Devices are fabricated by exfoliation of graphene layers onto an array of holes etched into an oxidized Si wafer. To avoid collapsing the device when spinning resist on, a layer of PMMA is prepared on a layer of water-soluble poly(vinyl alcohol), lifted off in water then applied dry to the substrate and baked on. Areas are then exposed and developed, unwanted material is removed using a reactive ion etch. Finally, leads are attached using electron beam lithography and then the device is dried using a critical point dryer. Alternatively, to make monolayer graphene devices, CVD-grown monolayer graphene is transferred to a substrate with predefined holes and leads. (b) Color plot of signal from FM technique versus gate voltage Vg and drive frequency. The lowest frequency mode visible is taken to be the fundamental resonant mode. (c) Line trace along the dotted line in part b at Vg = 8 V showing the resonance signal.

Figure 2. Quality factor and resonance frequency of the lowest-frequency mode. (a) Resonant frequency f res vs Vg plotted as black circles. Solid line fit to the equation described in the text. (b) FM mixing signal vs frequency for different source-drain drive voltages Vsd taken at T = 57 K and Vg = 20 V. The curves have been offset horizontally to align the left minima and vertically for clarity. Each line trace shows a maximum at resonance surrounded by two minima. The resonance frequency width is determined by the spacing between the minima. Above Vsd ∼ 9 mV, the frequency width increases visibly because of the presence of nonlinear damping. Above Vsd ∼ 5 mV, the resonance frequency shifts upward from the presence of an anharmonic force on the oscillator. (c) Main panel: Log−log plot of measured quality factor vs temperature for different gate voltages. The straight line is a guide to the eye proportional to 1/T. Frequency values for Vg = 8 V are T = 16.0 K, f = 75.6 MHz; T = 29.5 K, f = 71.2 MHz; T = 35.0 K, f = 68.1 MHz; T = 57.0 K, f = 66.2 MHz; and T = 296.0 K, f = 71.4 MHz.

observed behavior. From the frequency width vs Vsd, we determine the thermal expansion coefficient to the 2D thermal conductivity ratio of ∼0.7 W−1 at T = 16 K, which is comparable to the room temperature value of ∼5 W−1, in agreement with expectation.19,20 We have studied 12 devices in total, but found observable resonances and obtained Q vs T data from 4 devices, 2 trilayer (denoted T1 and T2) and 2 monolayer, grown by chemical vapor deposition21 (CVD) on Cu foils22 (denoted M1 and M2). Figure 1a shows a scanning electron microscope (SEM) image of device T1, suspended over a hole in an oxidized highly doped Si wafer with attached source and drain electrodes. Except where otherwise noted, all data shown is from this device. The Si wafer acts as a back gate. A schematic diagram is shown in the Supporting Information, Figure S1. A frequencymodulated (FM) ac source-drain voltage Vsd at carrier frequency f with modulation frequency f L excites the vibration. This produces a current Imix at frequency f L proportional to the frequency derivative of the real part of the mechanical response,5,23,24 detected using a lock-in amplifier.

modes via parametric driving. This produces a viscous damping force ∝ T3. The fluctuation-driven broadening is expected to dominate the frictional broadening for typical experimental conditions, consistent with our observation of ∝ 1/T behavior of Q. We find that the measured Q varies with the gate voltage Vg, which we attribute to the changing tension as well as electrostatic forces. Furthermore, nonlinear damping in graphene and carbon nanotube NEMS has been recently reported,5 modeled by a cubic damping force term Fd = −ηz2ż, where η is the nonlinear damping parameter.5,6 Its origin, however, remains unknown. For sufficiently large Vsd, we observe nonlinear damping where the frequency width scales as Vsd2/3, consistent with previous work.5 The measured nonlinear damping is in qualitative agreement with recent theory of nonlinear damping by radiation of excited in-plane resonator phonons.18 Finally, for yet larger V sd, the line width increases unexpectedly rapidly, following an approximate Vsd2 power law. A model of parametric amplification by periodic thermal expansion from the drive voltage heating accounts for the 2983

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Figure 3. Quality factor temperature dependence and gate voltage tunability of the resonance frequency and quality factor at T = 57 K. (a) Main panel: Scaled quality factor Q̅ as described in the text vs T. The masses used for scaling are M1, 0.8 mm, M2, 5.2 mm, T1, 3.2 mm, and T2, 0.7 mm, where mm is the monolayer effective mass. The masses of T1, T2, and M1 were obtained by fitting to eq S9 to the measured f res vs Vg curves, while the mass of M2 was obtained using results for pressurized membranes under large load because only large Vg data was available, and obtaining an unambiguous mass using eq S9 was difficult. Blue and magenta correspond to trilayer devices while green and red correspond to single layer devices. Solid line: 1/T, plotted without free parameters. Inset: Q vs T for the same samples. (b) Q vs Vg for T1 plotted as solid circles normalized to the extrapolated value at Vg = 0. Solid curve plot of theoretical model without free parameters as described in the Supporting Information, section S5.

mechanisms can account for the magnitude, size,31 and temperature dependence of Q. Here, we consider a tensioned elastic membrane, with an isotropic strain c from the uniform tension T0 = 2c(λ + μ), where μ is the shear modulus and λ is Lamé’s first constant. The slow modes couple to the fast modes since out of plane bending fluctuations modulate the sheet tension. Fast mode thermal fluctuations stochastically shift the slow modes’ effective spring constant, broadening their resonances. As in NMR, the inverse correlation time ν of the frequency fluctuations is important. If ν ≫ δω, the root mean square of the fluctuations, the line width will be lowered by motional narrowing.16,17 We find that the static limit ν ≪ δω is likely relevant and the resonances should retain the full width δω. This yields for the fundamental mode with resonance frequency ω0 = 2πf 0 (Supporting Information, eq S34)

Figure 1b shows the current plotted as a colorscale versus gate voltage Vg and f. Bright features correspond to resonances with gate voltage tunable frequency. The black curve in Figure 1c shows a line trace along the dotted line in Figure 1b, showing a maximum at the resonance frequency f 0. The maximum is surrounded by two minima5,23 with frequency spacing δf, equal to the resonance frequency width; Q = f 0/δ f.5 Figure 2a shows the resonance frequency tunability, originating from both electrostatic force gradient softening as well as the increase in tension with Vg.25,26 A change in concavity of the f res vs Vg relationship is expected for sufficiently large gate voltages.2,25 However, as we do not observe such a change we use a small Vg approximation to fit the data over the entire Vg range. The solid curve is a fit to the function (Supporting Information, eq S9) f res = {[2πT0 − ε0πr02/(3d3) Vg2 + aVg4]/mef f}1/2/(2π), where ε0 is the vacuum permittivity, r0 is the resonator radius, and d is the distance between the resonator and the gate electrode, yielding the tension T0 at Vg = 0, the effective mass mef f, and a, which describes the tension increase with Vg. We obtain T0 = 6.9 × 10−2 N/m, meff = 2.5 × 10−18 kg, and a = 1.13 × 10−6 N/ V4·m. For small Vsd, the damping is linear and δf is nearly constant, but δf begins to visibly increase as Vsd increases above ∼9 mV, as shown in Figure 2b. This increase in δf is a signature of nonlinear damping.5 In the linear damping regime, Q follows a 1/T dependence over more than a decade of measured temperatures, denoted by the dashed line, down to T = 16 K as shown at various Vg in Figure 2c. In previous work, a larger exponent power law ∼1/T2 was observed in conjunction with a saturation to a lower exponent ∼1/T0.3 below T ∼ 100 K.2,24 However, variables such as membrane tension, which could in principle affect Q were not controlled or accounted for, nor were the effects of nonlinear damping. Recent work considers potential dissipation mechanisms, including thermoelastic losses and currents due to straininduced dynamic synthetic fields.18,27,28 Other mechanisms, for example clamping losses29 and surface state dissipation30 have also been considered. However, none of these proposed

Q=

ξT0 3/2r0 kBT

κ E

(1)

where ξ ≈ 6.84 is a numerical factor, E = N (λ + 4λμ + 4 μ ) ≈ N2 × 1.1 × 105 (N/m)2 with N the number of layers, κ ≈ 10−16 Nm is the bending modulus, and kB is the Boltzmann constant. Since the frequency does not change significantly as the temperature is varied, we assume that the tension is approximately constant25 for this sample. For some samples the frequency variation is more significant. Figure 3a inset shows the measured Q for all 4 samples. The dotted line shows a 1/T trend as a guide to the eye. Some samples follow it more closely, while others, for example sample T2, follow a somewhat different Q vs T scaling. To fit the data to eq 1 we write the tension T0 as T0 = meffω02/(2π). Then we rearrange eq 1 and define Q̅ = (QkB)/(ξω03r0mef f 3/2)(8π3E/κ)1/2. Note that Q̅ depends only on measured and known parameters. According to eq 1, Q̅ = 1/T. Plotting the measured Q̅ vs T along with the curve 1/T we find that the data follows the expected 1/T dependence for all four samples. No free parameters are used in the plot. Within the experimental sample-dependent variation of the Q vs T, the characteristic 1/T behavior and magnitude of the data is described by eq 1. This provides strong evidence that frequency broadening from nonlinear coupling to 2

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Figure 4. Nonlinear resonator dynamics. (a) Main panel: resonance frequency vs frequency width at T = 16 K and Vg = 18 V plotted as black circles. Solid curve is a straight line fit to the data. Inset: resonance frequency vs amplitude at T = 16 K and Vg = 18 V plotted as black circles. The amplitude is obtained from the FM line shape when it is still approximately Lorentzian (Supporting Information, eq S20). Solid curve is a fit to the theoretical expectation (Supporting Information, eq S58). (b) Frequency width δf vs force on the oscillator. Dashed blue curve is the expected trend based on a cubic nonlinear damping term; dashed red curve is the width in the linear damping regime as determined by extrapolating the data in the main panel of part (a) to the zero amplitude frequency determined by the fit to the part (a) inset; the solid black curve takes into account both nonlinear damping and a simultaneous parametric drive due to Joule heating by the source drain voltage. Data taken at T = 16 K and Vg = 18 V.

thermally excited modes determines the observed quality factor. The predicted Q increases for increasing size as ∝ r0, as observed,31 providing additional evidence for this picture. As as result, this mechanism is expected to dominate in micro- or nanoscale resonators. Q depends weakly on Vg as shown in Figure 3b. Previous work has reported a Vg-dependence of Q in carbon nanotube because of single-electron charging effects.32,33 However, as no Coulomb blockade is observed in our samples, we extend the above model to account for the electrostatic forces from an applied Vg, which change the frequencies of the fundamental and excited modes and the resulting intermodal perturbation of the fast modes on the slow modes. Figure 3b shows the measured Q/Q(Vg = 0) vs Vg, while the solid line shows the theoretically expected curve (Supporting Information, section S5) using the parameters obtained from Figure 2a, without performing a fit. The agreement is good considering the lack of free parameters in the model curve, providing further evidence that frequency broadening through nonlinear mode coupling determines Q in graphene resonators. A similar analysis for room temperature data is shown in Supporting Information, Figure S2, where both the measured Q modulation and frequency tuning with Vg are larger than at low temperature. More work will be required to fully understand the detailed changes of these curves with temperature. Nevertheless, the order of magnitude of the measured and predicted effects are similar, suggesting that the model captures the basic physics of the Q and f res tunability with Vg. Within this model, while the frequency broadening Γ does not correspond to a viscous damping force, in steady state under a sinusoidal drive it produces an average dissipation rate that is the same as if Γ originated from such a viscous force. This implies that the root-mean-square amplitude of the motion must become large enough for conventional friction to dissipate the input power from the drive (Supporting Information, eq S38). We calculate the energy transfer rate to the fast modes by assuming that the tension modulations during oscillation parametrically drive the fast modes in conjunction with a Langevin force. This produces a calculated frictional damping force proportional to velocity that is ∝ T3

and characteristically smaller than the frequency broadening given above for typical device parameters and measurement conditions (Supporting Information, eq S37). This is consistent with our observation of ∝ T behavior for the line width. We now turn to the nonlinear resonator dynamics. In addition to δf varying with Vsd due to nonlinear damping, the resonance frequency f res also shifts upward by Δf from f 0 with increasing Vsd, due to a nonlinear restoring force, consistent with previously reported work.5 This nonlinearity is represented by a cubic Duffing force term Fr = −αz3, with α a constant.5,6,34 Plotting Δf against f res reveals a straight line trend, as shown in Figure 4a. It can be shown that (Supporting Information eq S59) the slope of this line is related to α/η by α/η ≈ 4π (dΔf/dδf)f 0. Fitting the data gives α/η = 1.4 × 1010 s−1. We obtain α by plotting Δf versus amplitude (Figure 4a inset),5,6,34 and fitting to the theoretical expectation for α, which gives a frequency shift that is quadratic in the amplitude (Supporting Information, eq S58). From Figure 4a inset data with the trilayer mef f we obtain α = 2.4 × 1015 N/m3. This yields η ∼ 1.8 × 105 Ns/m3. To understand the origin of the Duffing nonlinearity, we use a variational method along with continuum elastic theory to determine α = 8π(λ + 2μ)/3r02 (Supporting Information, eq S10) based on the geometry leading to increasing tension with deflection. Putting in λ = 3 × 48 N/m, μ = 3 × 144 N/m,35 for a trilayer and r0 = 10−6 m we find α = 8.4 × 1015, in good agreement to the measured value. This suggests that the origin of the Duffing term is geometric in nature. A smaller value of α is found at a higher temperature T = 57 K (Supporting Information, Figure S3), which may indicate the development of a quadratic force that renormalizes α.6 To account for the nonlinear damping, previous work has proposed a viscoelastic model for damping in NEMS.4 However, the predicted nonlinear damping is much smaller than the typically observed magnitude.5 Another model for graphene resonators that considers in-plane and bending mode coupling and energy loss by in-plane phonon radiation into the substrate performs numerical calculations for a similar geometry to the one presented here, consisting of a doubly clamped graphene sheet that is 1 μm × 1 μm.18 A scaling 2985

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analysis shows that η ∼ T1σ3ω03, where T1 = λ + 2μ, and σ is the sheet mass density. A similar value to the one measured for our trilayer sample is obtained with a mass ∼3 trilayer masses, suggesting that this mechanism plays an important role in determining the nonlinear damping in graphene resonators. To investigate the resonator response at larger drives, in Figure 4b we plot δf versus the effective ac force on the sheet at the drive frequency f, given by Fac = (ε0πr02)/(4d2)VgVsd (Supporting Information eq S13), measured at T = 16 K. At small drive force, δf is approximately ∝ Vsd2/3 (shown by the blue dashed line) following the expected trend due to nonlinear damping.5 At larger biases, however, the measured frequency width becomes much larger, tending toward a Vsd2 power law. One possibility is that this originates from higher order terms such as a fifth order damping force term. However, detailed calculation shows such terms still lead to a smaller predicted linewidth power law than 2. We therefore consider the possibility that an unintentional parametric drive is being applied to the resonator. Such a parametric drive results when the resonance frequency is modulated by δf 0 by changing one of the parameters of the oscillator, such as its stiffness (see, e.g., ref 6 review and ref 36 for recent work on carbon nanotubes). The parametric drive is quantified by H = 2δ f 0/f 0 which is the dimensionless stiffness modulation,6 and can produce amplification or even self-oscillations.6 One possible source of parametric driving is electrostatic forces, however over the voltage range shown we estimate these forces are likely too small to produce the observed behavior. We then consider a parametric drive caused by the periodic thermal expansion of the graphene sheet due to Joule heating by Vsd. Then in terms of the ac force drive Fac given above, the power dissipated P0 and the temperature shift in the sheet ΔT, H ∝ ΔT ∝ P0 ∝ Vsd2 = bFac2 with b given by ⎛ 4d 2 ⎞2 |a |(λ + μ) ⎟⎟ T b ≈ ⎜⎜ ⎝ ε0AVg ⎠ 2πRκ2DT0

parametric driving is occurring. This result therefore constitutes the first low-temperature measurement of this ratio |aT|/κ2D. In sum, we measured Q for few-layer graphene drum resonators and found that Q ∝ 1/T due to frequency broadening from nonlinear coupling to thermally excited modes. We also measured η, and find qualitative agreement to a theory of nonlinear damping by in-plane phonon radiation. The resonator response is larger than expected at the largest drives, which we understand using a model of thermal expansion driven parametric amplification. From our data, we find a/κ2D ∼ 0.7 W−1, which constitutes the first low temperature measurement of this quantity. Our results indicate that to optimize Q in graphene membranes, a large area,31 and tension is desirable. Using several additional layers of graphene may also by beneficial since the increased bending modulus suppresses the fast mode amplitudes and their perturbation on the fundamental mode.

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Derivations of formulas used and additional figures showing a schematic diagram of device geometry, variation of frequency and Q with gate voltage, and nonlinear parameter analysis. This material is available free of charge via the Internet at http:// pubs.acs.org.

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3⎛ 3α ⎞ ⎜⎜1 + ⎟bFAC 2f + const 0 8⎝ 2πηf0 ⎟⎠

AUTHOR INFORMATION

Corresponding Author

*(M.B.) E-mail: [email protected]. Present Address §

Max Planck Institute for Plasma Physics, Wendelsteinstrasse 1, 17491, Greifswald, Germany Notes

The authors declare no competing financial interest.

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(2)

ACKNOWLEDGMENTS We thank Chun Ning Lau and Paul McEuen for helpful discussions. This work was supported by NSF-DMR-1106358, ONR/DMEA-H94003-10-2-1003 and the UCR CONSEPT Center. S.Y. and P.W. were supported by DOE ER 46940-DESC0010597. This material is based on research sponsored by the Defense Microelectronics Activity (DMEA) under agreement number H94003-10-2-1003. The United States Government is authorized to reproduce and distribute reprints for Government purposes, notwithstanding any copyright notation thereon.

where aT is the negative thermal expansion coefficient and the 2D thermal conductivity κ2D = κBτ with κB the bulk thermal conductivity, τ the layer thickness, A the resonator area, and R the electrical resistance. (Supporting Information, eq S57.) The nonlinear equations of motion are solved approximately using the method of harmonic balance.37 For large drive, δf is given by (Supporting Information, eq S56) δf =

ASSOCIATED CONTENT

S Supporting Information *

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(3)

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scaling as V sd for large drive, as observed. Further approximations yield a function describing the behavior from the linear to the nonlinear asymptotic regime (Supporting Information, section S6). The solid curve in Figure 4b shows a fit to this function with b and mef f as fit parameters, yielding b = 2.9 × 1017 N−2, and mef f = 5.7 × 10−18 kg (equivalent to ∼2.5 trilayer effective masses). From this, using the measured R of 1.7 kΩ we determine at T = 16 K, |aT|/κ2D ∼ 0.7 W−1. The expected room temperature value with aT = −10−5 K−1,20 and κ = 2000 W/m·K for a ∼1 nm thick trilayer19 gives ∼5 W−1. From elementary kinetic theory this ratio is unlikely to vary strongly with temperature since the phonon mean free path is expected to be impurity or surface limited below 300 K.19 The result is thus in reasonable agreement with expectations, providing strong evidence that thermal expansion driven

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