Clamping Instability and van der Waals Forces in Carbon Nanotube

Apr 23, 2014 - We investigate the role of weak clamping forces, typically assumed to be infinite, in carbon nanotube mechanical resonators. Due to the...
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Clamping Instability and van der Waals Forces in Carbon Nanotube Mechanical Resonators Mehmet Aykol,§ Bingya Hou,*,§ Rohan Dhall,§ Shun-Wen Chang,† William Branham,§ Jing Qiu,‡ and Stephen B. Cronin§ §

Ming Hsieh Department of Electrical Engineering, †Department of Physics and Astronomy and ‡Department of Materials Science, University of Southern California, Los Angeles, California 90089, United States S Supporting Information *

ABSTRACT: We investigate the role of weak clamping forces, typically assumed to be infinite, in carbon nanotube mechanical resonators. Due to these forces, we observe a hysteretic clamping and unclamping of the nanotube device that results in a discrete drop in the mechanical resonance frequency on the order of 5−20 MHz, when the temperature is cycled between 340 and 375 K. This instability in the resonant frequency results from the nanotube unpinning from the electrode/trench sidewall where it is bound weakly by van der Waals forces. Interestingly, this unpinning does not affect the Q-factor of the resonance, since the clamping is still governed by van der Waals forces above and below the unpinning. For a 1 μm device, the drop observed in resonance frequency corresponds to a change in nanotube length of approximately 50−65 nm. On the basis of these findings, we introduce a new model, which includes a finite tension around zero gate voltage due to van der Waals forces and shows better agreement with the experimental data than the perfect clamping model. From the gate dependence of the mechanical resonance frequency, we extract the van der Waals clamping force to be 1.8 pN. The mechanical resonance frequency exhibits a striking temperature dependence below 200 K attributed to a temperature-dependent slack arising from the competition between the van der Waals force and the thermal fluctuations in the suspended nanotube. KEYWORDS: MEMS, resonator, mechanical, nanotube, clamping, suspended

T

strong coupling between resonance modes, which leads to spectral fluctuations that can account for the experimentally observed quality factors Q ∼ 100 at 300 K.13 Eichler et al. reported that symmetry breaking leads to spectral broadening of mechanical resonances due to the nonlinearity of a single mode.14 Several other loss mechanisms have been discussed in the literature, however, perfect clamping conditions are assumed in these studies.12 It is, therefore, important to understand the mechanisms limiting the clamping of carbon nanotube based mechanical resonators. Here, the role of weak clamping forces is studied by measuring the mechanical resonance of individual single-walled carbon nanotubes over a wide temperature range from 4−450 K. The suspended carbon nanotubes used in this study are grown on top of metal electrodes and are clamped down by weak van der Waals forces. In order to understand several phenomena that arise from this weak clamping, we develop a model, which includes the effects of van der Waals bonding. This model is used to fit our experimentally observed data, and is compared with a model

he small mass density (μ = 5 ag/μm) and high stiffness (Young’s modulus, E = 1 TPa) of carbon nanotubes provide a nearly ideal system for high frequency mechanical resonators.1,2 Since their first demonstration by Sazanova et al. in 2004,2 the frequency and quality factors of these devices have steadily risen.3−7 These unique devices have enabled interesting new phenomena to be studied. For example, Wang et al. were able to resolve subtle differences in the structural phases of argon atoms adsorbed on the nanotube surface.8 Also, optical phonon emission by quasi-ballistic electrons was observed through abrupt changes in the mechanical resonance frequency at high bias voltages.9 These devices have demonstrated a minimum mass resolution of 1.7 × 10−24 g,10,11 which is several orders of magnitude below other electromechanical systems. The loss (Q−1) in carbon nanotube resonators is attributed to the scattering of phonons at the contact of the carbon nanotube and the metal electrodes, which are minimized at low temperatures when there are almost no phonons populated. While the quality factor of carbon nanotube based nanomechanical oscillators should be around 105 at temperatures below 100 mK and decrease to 103 at room temperature,12 the typical Q-factors reported in the literature remain below the expected values. Recently, Barnard et al. found that thermal fluctuations induce © 2014 American Chemical Society

Received: January 8, 2014 Revised: April 5, 2014 Published: April 23, 2014 2426

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Figure 1. (a) Scanning electron microscope image of a carbon nanotube (CNT) mechanical resonator and (b) close-up of the carbon nanotube crossing the trench. (c) Schematic diagram of the device structure. (d) Mixing current plotted as a function of carrier frequency measured in FM mode.

assuming perfect clamping. It should be noted that the “weak clamping” discussed here does not entail slipping of the nanotube relative to the leads. In the fabrication of carbon nanotube mechanical resonators, a predefined trench, W = 1−2 μm, is patterned in an oxidized ptype silicon (Si) substrate with platinum (Pt) source, drain, and gate electrodes, as shown in Figure 1.15 During the fabrication process, isotropic etching of the oxidized silicon creates an undercut, as illustrated schematically in Figure 1c, which leaves an overhanging structure of width Woh that is equal to the trench depth. Nanotubes are grown on top of the Pt source and drain electrodes by chemical vapor deposition using argon bubbled through ethanol in a quartz tube furnace at 825 °C. This fabrication scheme differs from most previous studies which deposit metal electrodes on top of the carbon nanotube after growth, providing more rigid clamping. The electromechanical resonances of these devices were measured in a high vacuum (10−8 Torr) cryogenic chamber between 4 and 450 K. The resistivity of an on-chip platinum resistive temperature detector was continuously monitored to obtain the temperature of the nanotube during the measurements. The mechanical resonance frequencies were detected with a single source frequency modulation (FM) technique.16 A frequency modulated signal, Vsd(t) = Ac cos(2πfct + fΔ/f FM cos(2π f FMt)), is applied to the source electrode. Typical values for the signal amplitude, frequency deviation, and modulation frequency are Ac = 8 mV, fΔ = 50 kHz, and f FM = 654 Hz, respectively. The current through the nanotube is measured using a lock-in amplifier with the reference frequency set to f FM, while sweeping the carrier frequency fc. Figure 2d shows a

Figure 2. (a) Color plot of mixing current, INT, for a 1 μm long carbon nanotube (Sample 1) measured at Vg = −4 V. (b) Frequency of the fundamental mechanical resonance mode of Sample 2 observed during a high temperature cycle. (c) Schematic diagram showing the carbon nanotube (CNT) taut at Vg = 0 due to van der Waals forces at room temperature. When the temperature of the substrate is increased to 425 K, (d) the carbon nanotube delaminates from the platinum (Pt) electrode/trench sidewall by thermal excitation. This causes an elongation of the suspended length of the nanotube (Δl).

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and 2. Here, Γel arises from the electrostatic force between the nanotube and gate electrode and is given by

mechanical resonance peak detected using the frequency modulation mixing technique, which is consistent with previous reports by Gouttenoire et al.16 We confirm that the resonances are, in fact, mechanical in nature by tuning the resonance frequency with a DC gate voltage, Vg, which increases the tension on the carbon nanotube given by eq 2. We also performed measurements in air (760 Torr) in order to identify the electrical resonances that are simply due to the passive elements in the circuit. Since gas molecules dampen the mechanical oscillations of the carbon nanotube at small driving potentials, the resonances observed in gas can only originate from the passive elements of the circuit. Figure 2 shows the mechanical resonance frequency measured between 300 and 450 K for two different suspended nanotube samples. A sudden and discrete drop in the resonance frequency (on the order of 5−20 MHz) is observed around 340 K for both samples, as shown in Figure 2, parts a and b. The hysteresis of this behavior can be seen in Figure 2b when the temperature is cycled between 300 and 450 K. This drop in the mechanical resonance frequency was observed in a total of six devices and results from a sudden elongation in the suspended length of the nanotube, Δl, as depicted in Figure 2, parts c and d. Since the nanotubes are grown at 825 °C, the nanotube acquires slack (s) when the sample is cooled to room temperature. At ambient and low temperatures, this slack is taken up by van der Waals forces, which causes the nanotube to stick to the electrode/trench sidewall, as illustrated in Figure 2c. We can estimate the change in the suspended length of the nanotube corresponding to this drop in mechanical resonance frequency using eqs 1−3 described below. On the basis of this model, the 20 MHz drop shown in Figure 2a (second mode) corresponds to a change in the suspended nanotube length of Δl = 65 nm for sample 1, which has a 1 μm nominal suspended length. The 5 MHz drop shown in Figure 2b (lowest mode) corresponds to a change in the suspended nanotube length of Δl = 50 nm for sample 2, which also has a 1 μm nominal suspended length. Interestingly, the nanotube delaminating from the electrode/ trench sidewall does not affect the Q-factor of the resonance, as shown in Figure S1 of the Supporting Information, since the clamping conditions have not changed.13,17 That is, the clamping at the ends of the nanotube is limited by van der Waals forces for both configurations depicted in Figure 2, parts c and d. It is possible that the Q factor changes with the clamping, but that the sharpness of the resonance is limited by fluctuation broadening, as suggested by Barnard et al.13 This sudden drop in frequency at higher temperatures was observed in six of the seven samples tested, all in the temperature range 340−375 K. For repeated heating cycles, the critical temperature remains the same, as shown in Figure S2 of the Supporting Information. However, the transition becomes sharper after repeated cycles. Previous modeling of suspended nanotube resonators assumes perfect clamping and predicts a mechanical resonance frequency given by ftotal = =

Γel =

2 ⎛ ⎛ β ⎞2 ⎞2 Γel ⎞ 1 n EI ⎟ ⎜ ⎟ +⎜ ⎜ ⎟ μ ⎠ μ ⎟⎠ ⎝ 2π ⎝ L ⎠

dz

Vg 2

1 96s

(2)

where Cg, z, Vg, and s are the capacitance of the nanotube, distance between the gate electrode and the nanotube, gate voltage, and the slack of the nanotube, respectively. In the perfect clamping model, the tension Γ goes to zero at zero gate voltage (Γ → 0 at Vg = 0). The van der Waals clamping of the suspended carbon nanotube resonator depicted in Figure 2c differs significantly from the perfect clamping model described above. In the model presented here, there is always a finite tension on the nanotube due to the van der Waals bonding, and thus, Γ → ΓvdW at Vg = 0, where ΓvdW is the van der Waals force. Including this extra tension term, the equation for the gate voltage dependence of the mechanical resonance frequency becomes f (Vg ) =

⎛ 1 ⎜ ⎝ 2L

2 ⎛ 9.8075 Γel + ΓvdW ⎞ ⎟ +⎜ 2 μ ⎝ L ⎠

2 EI ⎞ ⎟ μ ⎠

(3)

This deviates from the hyperbolic function described above, and thus, this van der Waals clamping predicts a mechanical resonance frequency with a different gate voltage dependence, which can be evaluated experimentally, as described below. Figure 3 shows the gate voltage dependence of the mechanical resonance frequency for a suspended carbon nanotube taken at

Figure 3. Gate dependence of the mechanical resonance frequency of a suspended carbon nanotube device, plotted together with the perfect clamping (solid blue line) and van der Waals bonding (solid black line) models.

room temperature. The data is fit using eq 3 above with ΓvdW = 0 (perfect clamping model) and ΓvdW = 1.8 pN. The other parameters in the fit are d = 1 nm, E = 1 TPa, I = 2.325 × 10−15 kg/m, and L = 1.0−2.2 μm (depending on sample). The two models deviate significantly around zero gate voltage, as expected, when the van der Waals (vdWs) force/tension becomes dominant. In particular, the presence of a vdWs force makes the bottom of the hyperbola more “flat”. The fact that the vdWs model fits the data better in this gate voltage range validates that this physical mechanism is indeed playing a significant role in the mechanical resonance of the device. Figure 4a shows the temperature dependence of the mechanical resonance frequency below room temperature for two suspended nanotube devices. For both devices, the resonance frequency exhibits a minimum at approximately 180

ftension 2 + fbending 2

⎛ n ⎜ ⎝ 2L

dCg

(1)

where L, μ, E, and I are the suspended length, linear mass density, Young’s modulus, and the area moment of inertia of the nanotube, respectively.2,13,18 βn = 4.75, 7.85, and 11 for n = 0, 1, 2428

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expansion of the CNT and/or underlying substrate. In eq 2, however, the slack, s, is temperature dependent and is given by18 L ( T ) − W (T ) W (T ) =1− L (T ) L (T ) Wo + ΔW (T ) =1− Lo + ΔW (T ) + ΔLCNT(T )

s=

(5)

where L, W, Lo, Wo, and ΔLCNT are the nanotube suspended length, trench width, suspended length of the nanotube at 300 K, trench width at 300 K, and the change of the nanotube’s suspended length due to the thermal expansion/contraction. Here, the thermal expansion/contraction of the nanotube contributes to the slack, and is given by T

ΔLCNT(T ) = L

K and increases rapidly at lower temperatures, nearly doubling in mechanical resonance frequency at 4 K. This striking increase in frequency is not consistent with the perfect clamping model, as described below. Assuming perfect clamping, we can calculate the temperature dependence of the mechanical resonance frequency using eq 1. Here, the temperature dependence of the trench width, ΔW, can be calculated from the literature values of the thermal expansion coefficients of silicon and Pt, as T

(6)

where αCNT is the temperature-dependent thermal expansion coefficient of the suspended nanotube, as reported by Kahaly et al.22 From eqs 1, 4, 5, and 6, we calculate the temperature dependence of the mechanical resonance frequency of a 1 μm wide suspended nanotube device, as shown in Figure S3 of the Supporting Information. This model predicts a very small temperature dependence of the mechanical resonance frequency, varying by only 0.2% over the whole temperature range. This is 3 orders of magnitude smaller than the variation we observe experimentally, further indicating that the perfect clamping model is not valid in this system. Instead, we expect there to be temperature-dependent clamping, due to the van der Waals bonding. For perfectly clamped carbon nanotube resonators, a linearly increasing frequency with temperature was observed (Δf ∼ T) by Barnard et al., and is attributed to a change in length of the nanotube caused by thermal fluctuations in each eigenmodel.13 This model also predicts the temperature dependence of the Q factor quite accurately. For the weakly clamped nanotubes measured in our study, we observe the same temperature dependence (f increasing with T) above 200 K. However, below 200 K, the opposite behavior is observed, with the frequency nearly doubling between 200 and 4 K, as shown in Figure 4a. This anomalous behavior can be explained by considering the temperature-dependent adhesion brought about by the competition between the van der Waals force and the thermal fluctuations in the suspended nanotube, which leads to a temperature-dependent slack, s(T), as illustrated in Figure 4b. We can estimate the change in slack over this temperature range by solving eq 3 above for the slack. A derivation of the slack dependence on mechanical resonance frequency is given in the Supporting Information. The temperature dependence of the slack for these two data sets are shown in Figure 4c, which exhibit the same general trend, monotonically increasing with temperature. We believe that this change in slack is responsible for the temperature dependence of the mechanical resonance frequency observed in our work. The temperature-dependent slack model described above is based on the competition between the van der Waals force and the thermal fluctuations in the suspended nanotube. A rigorous theoretical model of this competition is beyond the scope of this paper, however, one could imagine that, in thermodynamic equilibrium, there is a trade-off between the binding enthalpy of the nanotube to the trench sidewall ΔH and the entropy of the suspended carbon nanotube ΔS. Such entropic considerations are known to be important in the nanomechanics of DNA.23 In

Figure 4. (a) Temperature dependence of the mechanical resonance frequency of sample 3 (solid red triangles) at Vg = −5 V and sample 4 (empty black squares) at Vg = −6.1 V. (b) Schematic diagram and (c) calculated temperature dependence of slack s at lower temperatures.

ΔW (T ) = (Wo + 2Woh)

∫300 αCNT(t ) dt

T

∫300 αSi(t ) dt − 2Woh ∫300 αPt(t ) dt (4)

where Wo, Woh, αSi, and αPt are the width of the trench at 300 K, width of the overhanging platinum electrodes at 300 K (illustrated in Figure 1c), and the temperature-dependent thermal expansion coefficients of silicon and platinum,19−21 respectively. We also assume that the nanotube is always slack in this calculation, since (1) the nanotube is grown at high temperatures (∼825 °C) and (2) the thermal expansion coefficient of the carbon nanotube is negative and small (∼10−7) compared to that of the trench (∼10−6). As a result, the tension on the nanotube is simply determined by eq 2, and we do not have to consider the tension increasing due to the thermal 2429

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(7) Peng, H.; Chang, C.; Aloni, S.; Yuzvinsky, T.; Zettl, A. Microwave electromechanical resonator consisting of clamped carbon nanotubes in an abacus arrangement. Phys. Rev. B 2007, 76, 035405. (8) Wang, Z. H.; Wei, J.; Morse, P.; Dash, J. G.; Vilches, O. E.; Cobden, D. H. Phase Transitions of Adsorbed Atoms on the Surface of a Carbon Nanotube. Science 2010, 327, 552−555. (9) Aykol, M.; Branham, W.; Liu, Z. W.; Amer, M. R.; Hsu, I. K.; Dhall, R.; Chang, S. W.; Cronin, S. B. Electromechanical resonance behavior of suspended single-walled carbon nanotubes under high bias voltages. J. Micromech. Microeng. 2011, 21, 085008. (10) Chaste, J.; Eichler, A.; Moser, J.; Ceballos, G.; Rurali, R.; Bachtold, A. A nanomechanical mass sensor with yoctogram resolution. Nat. Nano 2012, 7, 301−304. (11) Chiu, H. Y.; Hung, P.; Postma, H. W. C.; Bockrath, M. AtomicScale Mass Sensing Using Carbon Nanotube Resonators. Nano Lett. 2008, 8, 4342−4346. (12) Jiang, H.; Yu, M. F.; Liu, B.; Huang, Y. Intrinsic Energy Loss Mechanisms in a Cantilevered Carbon Nanotube Beam Oscillator. Phys. Rev. Lett. 2004, 93, 185501. (13) Barnard, A. W.; Sazonova, V.; van der Zande, A. M.; McEuen, P. L. Fluctuation broadening in carbon nanotube resonators. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 19093−19096. (14) Eichler, A.; Moser, J.; Dykman, M. I.; Bachtold, A. Symmetry breaking in a mechanical resonator made from a carbon nanotube. Nat. Commun. 2013, DOI: DOI:10.1038/ncomms3843. (15) Bushmaker, A. W.; Deshpande, V. V.; Bockrath, M. W.; Cronin, S. B. Direct observation of mode selective electron-phonon coupling in suspended carbon nanotubes. Nano Lett. 2007, 7, 3618−3622. (16) Gouttenoire, V.; Barois, T.; Perisanu, S.; Leclercq, J.-L.; Purcell, S. T.; Vincent, P.; Ayari, A. Digital and FM Demodulation of a Doubly Clamped Single-Walled Carbon-Nanotube Oscillator: Towards a Nanotube Cell Phone. Small 2010, 6, 1060−1065. (17) Eichler, A.; Moser, J.; Chaste, J.; Zdrojek, M.; Wilson Rae, I.; Bachtold, A. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene. Nat. Nano 2011, 6, 339−342. (18) Sazonova, V.; Yaish, Y.; Ü stünel, H.; Roundy, D.; Arias, T. A.; McEuen, P. L. A tunable carbon nanotube electromechanical oscillator. Nature 2004, 431, 284−287. (19) Kirby, R. Platinuma thermal expansion reference material. Int. J. Thermophys. 1991, 12, 679−685. (20) Lyon, K.; Salinger, G.; Swenson, C.; White, G. Linear thermal expansion measurements on silicon from 6 to 340 K. J. Appl. Phys. 1977, 48, 865−868. (21) Okada, Y.; Tokumaru, Y. Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. J. Appl. Phys. 1984, 56, 314−320. (22) Kahaly, M. U.; Waghmare, U. V. Vibrational Properties of SingleWall Carbon Nanotubes: A First-Principles Study. J. Nanosci. Nanotechnol. 2007, 7, 1787−1792. (23) Mossa, A.; Manosas, M.; Forns, N.; Huguet, J. M.; Ritort, F. Dynamic force spectroscopy of DNA hairpins: I. Force kinetics and free energy landscapes. Journal of Statistical Mechanics: Theory and Experiment 2009, 2009, P02060.

this case, the stable binding length of the nanotube would correspond to the minimum Gibbs free energy of the system, ΔG = ΔH − TΔS. As the temperature increases, we expect to see an entropically driven delamination of the nanotube from the trench sidewall, which effectively increases the slack in the nanotube. In conclusion, we report weak clamping behavior in suspended carbon nanotube nanomechanical resonators. Above room temperature (340−375 K), a discrete drop in the mechanical resonance frequency on the order of 5−20 MHz is observed due to weak van der Waals binding forces between the nanotube and its underlying electrodes/trench sidewall. This peeling event, however, does not change the quality factor of the mechanical resonance. The instability in the resonance frequency results from the nanotube delaminating from the electrode sidewall by thermal excitation, which corresponds to an additional length of approximately 50−65 nm for a 1 μm device. On the basis of a revised model, we estimate the out-of-plane van der Waals force between the carbon nanotube and the Pt surface to be ΓvdW = 1.8 pN. The strong temperature dependence of the mechanical resonance frequency observed at low temperatures (4−200 K) is attributed to a temperature-dependent slack arising from the competition between the van der Waals force and thermal fluctuations in the suspended nanotube.

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ASSOCIATED CONTENT

S Supporting Information *

Figures showing the temperature dependence of the mechanical resonance frequency and quality factor, and derivation of the temperature dependence of the slack. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*(B.H.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Department of Energy (DOE) Award No. DE-FG02-07ER46376. A portion of this work was carried out in the University of California Santa Barbara (UCSB) nanofabrication facility, part of the NSF funded National Nanotechnology Infrastructure Network (NNIN) network.

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REFERENCES

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