Clamping Instability and van der Waals Forces in Carbon NanotubeMechanical ResonatorsMehmet 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 clampingforces, typically assumed to be infinite, in carbon nanotubemechanical resonators. Due to these forces, we observe ahysteretic clamping and unclamping of the nanotube devicethat results in a discrete drop in the mechanical resonancefrequency on the order of 5−20 MHz, when the temperature iscycled between 340 and 375 K. This instability in the resonantfrequency results from the nanotube unpinning from theelectrode/trench sidewall where it is bound weakly by van derWaals forces. Interestingly, this unpinning does not affect theQ-factor of the resonance, since the clamping is still governedby van der Waals forces above and below the unpinning. For a1 μm device, the drop observed in resonance frequencycorresponds to a change in nanotube length of approximately 50−65 nm. On the basis of these findings, we introduce a newmodel, which includes a finite tension around zero gate voltage due to van der Waals forces and shows better agreement with theexperimental data than the perfect clamping model. From the gate dependence of the mechanical resonance frequency, weextract the van der Waals clamping force to be 1.8 pN. The mechanical resonance frequency exhibits a striking temperaturedependence below 200 K attributed to a temperature-dependent slack arising from the competition between the van der Waalsforce and the thermal fluctuations in the suspended nanotube.

KEYWORDS: MEMS, resonator, mechanical, nanotube, clamping, suspended

The 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 mechanicalresonators.1,2 Since their first demonstration by Sazanova et al. in2004,2 the frequency and quality factors of these devices havesteadily risen.3−7 These unique devices have enabled interestingnew phenomena to be studied. For example, Wang et al. wereable to resolve subtle differences in the structural phases of argonatoms adsorbed on the nanotube surface.8 Also, optical phononemission by quasi-ballistic electrons was observed throughabrupt changes in the mechanical resonance frequency at highbias voltages.9 These devices have demonstrated a minimummass resolution of 1.7 × 10−24 g,10,11 which is several orders ofmagnitude below other electromechanical systems. The loss(Q−1) in carbon nanotube resonators is attributed to thescattering of phonons at the contact of the carbon nanotube andthe metal electrodes, which are minimized at low temperatureswhen there are almost no phonons populated. While the qualityfactor of carbon nanotube based nanomechanical oscillatorsshould be around 105 at temperatures below 100 mK anddecrease to 103 at room temperature,12 the typical Q-factorsreported in the literature remain below the expected values.Recently, Barnard et al. found that thermal fluctuations induce

strong coupling between resonance modes, which leads tospectral fluctuations that can account for the experimentallyobserved quality factors Q ∼ 100 at 300 K.13 Eichler et al.reported that symmetry breaking leads to spectral broadening ofmechanical resonances due to the nonlinearity of a singlemode.14 Several other loss mechanisms have been discussed inthe literature, however, perfect clamping conditions are assumedin these studies.12 It is, therefore, important to understand themechanisms limiting the clamping of carbon nanotube basedmechanical resonators.Here, the role of weak clamping forces is studied by measuring

the mechanical resonance of individual single-walled carbonnanotubes over a wide temperature range from 4−450 K. Thesuspended carbon nanotubes used in this study are grown on topof metal electrodes and are clamped down by weak van derWaalsforces. In order to understand several phenomena that arise fromthis weak clamping, we develop a model, which includes theeffects of van der Waals bonding. This model is used to fit ourexperimentally observed data, and is compared with a model

Received: January 8, 2014Revised: April 5, 2014

Letter

pubs.acs.org/NanoLett

© XXXX American Chemical Society A dx.doi.org/10.1021/nl500096p | Nano Lett. XXXX, XXX, XXX−XXX

assuming perfect clamping. It should be noted that the “weakclamping” discussed here does not entail slipping of the nanotuberelative to the leads.In the fabrication of carbon nanotube mechanical resonators, a

predefined trench, W = 1−2 μm, is patterned in an oxidized p-type silicon (Si) substrate with platinum (Pt) source, drain, andgate electrodes, as shown in Figure 1.15 During the fabricationprocess, isotropic etching of the oxidized silicon creates anundercut, as illustrated schematically in Figure 1c, which leavesan overhanging structure of widthWoh that is equal to the trenchdepth. Nanotubes are grown on top of the Pt source and drainelectrodes by chemical vapor deposition using argon bubbledthrough ethanol in a quartz tube furnace at 825 °C. Thisfabrication scheme differs from most previous studies whichdeposit metal electrodes on top of the carbon nanotube aftergrowth, providing more rigid clamping. The electromechanicalresonances of these devices were measured in a high vacuum(10−8 Torr) cryogenic chamber between 4 and 450 K. Theresistivity of an on-chip platinum resistive temperature detectorwas continuously monitored to obtain the temperature of thenanotube during the measurements.The mechanical resonance frequencies were detected with a

single source frequency modulation (FM) technique.16 Afrequency modulated signal, Vsd(t) = Ac cos(2πfct + fΔ/f FMcos(2π f FMt)), is applied to the source electrode. Typical valuesfor the signal amplitude, frequency deviation, and modulationfrequency are Ac = 8 mV, fΔ = 50 kHz, and f FM = 654 Hz,respectively. The current through the nanotube is measuredusing a lock-in amplifier with the reference frequency set to f FM,while sweeping the carrier frequency fc. Figure 2d shows a

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

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

Nano Letters Letter

dx.doi.org/10.1021/nl500096p | Nano Lett. XXXX, XXX, XXX−XXXB

mechanical resonance peak detected using the frequencymodulation mixing technique, which is consistent with previousreports by Gouttenoire et al.16 We confirm that the resonancesare, in fact, mechanical in nature by tuning the resonancefrequency with a DC gate voltage, Vg, which increases the tensionon the carbon nanotube given by eq 2. We also performedmeasurements in air (760 Torr) in order to identify the electricalresonances that are simply due to the passive elements in thecircuit. Since gas molecules dampen the mechanical oscillationsof the carbon nanotube at small driving potentials, theresonances observed in gas can only originate from the passiveelements of the circuit.Figure 2 shows the mechanical resonance frequency measured

between 300 and 450 K for two different suspended nanotubesamples. A sudden and discrete drop in the resonance frequency(on the order of 5−20 MHz) is observed around 340 K for bothsamples, as shown in Figure 2, parts a and b. The hysteresis of thisbehavior can be seen in Figure 2b when the temperature is cycledbetween 300 and 450 K. This drop in the mechanical resonancefrequency was observed in a total of six devices and results from asudden elongation in the suspended length of the nanotube, Δl,as depicted in Figure 2, parts c and d. Since the nanotubes aregrown at 825 °C, the nanotube acquires slack (s) when thesample is cooled to room temperature. At ambient and lowtemperatures, this slack is taken up by van der Waals forces,which causes the nanotube to stick to the electrode/trenchsidewall, as illustrated in Figure 2c.We can estimate the change inthe suspended length of the nanotube corresponding to this dropin mechanical resonance frequency using eqs 1−3 describedbelow. On the basis of this model, the 20 MHz drop shown inFigure 2a (second mode) corresponds to a change in thesuspended nanotube length of Δl = 65 nm for sample 1, whichhas a 1 μmnominal suspended length. The 5MHz drop shown inFigure 2b (lowest mode) corresponds to a change in thesuspended nanotube length of Δl = 50 nm for sample 2, whichalso 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, asshown in Figure S1 of the Supporting Information, since theclamping conditions have not changed.13,17 That is, the clampingat the ends of the nanotube is limited by van der Waals forces forboth configurations depicted in Figure 2, parts c and d. It ispossible that theQ factor changes with the clamping, but that thesharpness of the resonance is limited by fluctuation broadening,as suggested by Barnard et al.13 This sudden drop in frequency athigher temperatures was observed in six of the seven samplestested, all in the temperature range 340−375 K. For repeatedheating cycles, the critical temperature remains the same, asshown in Figure S2 of the Supporting Information. However, thetransition becomes sharper after repeated cycles.Previous modeling of suspended nanotube resonators assumes

perfect clamping and predicts a mechanical resonance frequencygiven by

μ πβ

μ

= +

=Γ

+⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

f f f

nL L

EI2

12

n

total tension2

bending2

el2 2 2

(1)

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

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

Γ =C

zV

s

d

d196

ggel

2

(2)

where Cg, z, Vg, and s are the capacitance of the nanotube,distance between the gate electrode and the nanotube, gatevoltage, and the slack of the nanotube, respectively. In the perfectclamping 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 significantlyfrom the perfect clamping model described above. In the modelpresented here, there is always a finite tension on the nanotubedue to the van der Waals bonding, and thus, Γ→ ΓvdW at Vg = 0,where ΓvdW is the van der Waals force. Including this extratension term, the equation for the gate voltage dependence of themechanical resonance frequency becomes

μ μ=

Γ + Γ+

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟f V

L LEI

( )1

29.8075

gel vdW

2

2

2

(3)

This deviates from the hyperbolic function described above, andthus, this van der Waals clamping predicts a mechanicalresonance 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

room temperature. The data is fit using eq 3 above with ΓvdW = 0(perfect clamping model) and ΓvdW = 1.8 pN. The otherparameters 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 twomodels deviate significantly around zero gate voltage, asexpected, when the van der Waals (vdWs) force/tensionbecomes dominant. In particular, the presence of a vdWs forcemakes the bottom of the hyperbola more “flat”. The fact that thevdWs model fits the data better in this gate voltage rangevalidates that this physical mechanism is indeed playing asignificant role in the mechanical resonance of the device.Figure 4a shows the temperature dependence of the

mechanical resonance frequency below room temperature fortwo suspended nanotube devices. For both devices, theresonance frequency exhibits a minimum at approximately 180

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

Nano Letters Letter

dx.doi.org/10.1021/nl500096p | Nano Lett. XXXX, XXX, XXX−XXXC

K and increases rapidly at lower temperatures, nearly doubling inmechanical resonance frequency at 4 K. This striking increase infrequency is not consistent with the perfect clamping model, asdescribed 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, canbe calculated from the literature values of the thermal expansioncoefficients of silicon and Pt, as

∫ ∫α αΔ = + −W T W W t t W t t( ) ( 2 ) ( ) d 2 ( ) do oh

T

oh

T

300Si

300Pt

(4)

whereWo,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-dependentthermal expansion coefficients of silicon and platinum,19−21

respectively. We also assume that the nanotube is always slack inthis calculation, since (1) the nanotube is grown at hightemperatures (∼825 °C) and (2) the thermal expansioncoefficient 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, andwedo not have to consider the tension increasing due to the thermal

expansion of the CNT and/or underlying substrate. In eq 2,however, the slack, s, is temperature dependent and is given by18

= − = −

= −+ Δ

+ Δ + Δ

sL T W T

L TW TL T

W W TL W T L T

( ) ( )( )

1( )( )

1( )

( ) ( )o

o CNT (5)

where L, W, Lo, Wo, and ΔLCNT are the nanotube suspendedlength, trench width, suspended length of the nanotube at 300 K,trench width at 300 K, and the change of the nanotube’ssuspended length due to the thermal expansion/contraction.Here, the thermal expansion/contraction of the nanotubecontributes to the slack, and is given by

∫ αΔ =L T L t t( ) ( ) dT

CNT300

CNT (6)

where αCNT is the temperature-dependent thermal expansioncoefficient of the suspended nanotube, as reported by Kahaly etal.22 From eqs 1, 4, 5, and 6, we calculate the temperaturedependence of the mechanical resonance frequency of a 1 μmwide suspended nanotube device, as shown in Figure S3 of theSupporting Information. This model predicts a very smalltemperature dependence of the mechanical resonance frequency,varying by only 0.2% over the whole temperature range. This is 3orders of magnitude smaller than the variation we observeexperimentally, further indicating that the perfect clampingmodel is not valid in this system. Instead, we expect there to betemperature-dependent clamping, due to the van der Waalsbonding.For perfectly clamped carbon nanotube resonators, a linearly

increasing frequency with temperature was observed (Δf∼T) byBarnard et al., and is attributed to a change in length of thenanotube caused by thermal fluctuations in each eigenmodel.13

This model also predicts the temperature dependence of the Qfactor quite accurately. For the weakly clamped nanotubesmeasured in our study, we observe the same temperaturedependence ( f increasing with T) above 200 K. However, below200 K, the opposite behavior is observed, with the frequencynearly doubling between 200 and 4 K, as shown in Figure 4a. Thisanomalous behavior can be explained by considering thetemperature-dependent adhesion brought about by the com-petition between the van der Waals force and the thermalfluctuations in the suspended nanotube, which leads to atemperature-dependent slack, s(T), as illustrated in Figure 4b.We can estimate the change in slack over this temperature rangeby solving eq 3 above for the slack. A derivation of the slackdependence on mechanical resonance frequency is given in theSupporting Information. The temperature dependence of theslack for these two data sets are shown in Figure 4c, which exhibitthe same general trend, monotonically increasing with temper-ature. We believe that this change in slack is responsible for thetemperature dependence of the mechanical resonance frequencyobserved in our work.The temperature-dependent slack model described above is

based on the competition between the van der Waals force andthe thermal fluctuations in the suspended nanotube. A rigoroustheoretical model of this competition is beyond the scope of thispaper, however, one could imagine that, in thermodynamicequilibrium, there is a trade-off between the binding enthalpy ofthe nanotube to the trench sidewall ΔH and the entropy of thesuspended carbon nanotube ΔS. Such entropic considerationsare known to be important in the nanomechanics of DNA.23 In

Figure 4. (a) Temperature dependence of the mechanical resonancefrequency 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.

Nano Letters Letter

dx.doi.org/10.1021/nl500096p | Nano Lett. XXXX, XXX, XXX−XXXD

this case, the stable binding length of the nanotube wouldcorrespond to the minimumGibbs free energy of the system,ΔG= ΔH − TΔS. As the temperature increases, we expect to see anentropically driven delamination of the nanotube from the trenchsidewall, which effectively increases the slack in the nanotube.In conclusion, we report weak clamping behavior in suspended

carbon nanotube nanomechanical resonators. Above roomtemperature (340−375 K), a discrete drop in the mechanicalresonance frequency on the order of 5−20 MHz is observed dueto weak van der Waals binding forces between the nanotube andits underlying electrodes/trench sidewall. This peeling event,however, does not change the quality factor of the mechanicalresonance. The instability in the resonance frequency resultsfrom the nanotube delaminating from the electrode sidewall bythermal excitation, which corresponds to an additional length ofapproximately 50−65 nm for a 1 μm device. On the basis of arevised model, we estimate the out-of-plane van der Waals forcebetween the carbon nanotube and the Pt surface to be ΓvdW = 1.8pN. The strong temperature dependence of the mechanicalresonance frequency observed at low temperatures (4−200 K) isattributed to a temperature-dependent slack arising from thecompetition between the van der Waals force and thermalfluctuations in the suspended nanotube.

■ ASSOCIATED CONTENT*S Supporting InformationFigures showing the temperature dependence of the mechanicalresonance frequency and quality factor, and derivation of thetemperature dependence of the slack. This material is availablefree of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*(B.H.) E-mail: bhou@usc.edu.

NotesThe authors declare no competing financial interest.

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

■ REFERENCES(1) Bunch, J. S.; van der Zande, A. M.; Verbridge, S. S.; Frank, I. W.;Tanenbaum, D. M.; Parpia, J. M.; Craighead, H. G.; McEuen, P. L.Electromechanical Resonators from Graphene Sheets. Science 2007,315, 490−493.(2) Sazonova, V.; Yaish, Y.; Ustunel, H.; Roundy, D.; Arias, T. A.;McEuen, P. L. A tunable carbon nanotube electromechanical oscillator.Nature 2004, 431, 284−287.(3) Garcia-Sanchez, D.; San Paulo, A.; Esplandiu, M. J.; Perez-Murano,F.; Forro, L.; Aguasca, A.; Bachtold, A. Mechanical detection of carbonnanotube resonator vibrations. Phys. Rev. Lett. 2007, 99, 085501.(4) Huttel, A. K.; Steele, G. A.; Witkamp, B.; Poot, M.; Kouwenhoven,L. P.; van der Zant, H. S. Carbon nanotubes as ultrahigh quality factormechanical resonators. Nano Lett. 2009, 9, 2547−2552.(5) Island, J.; Tayari, V.; McRae, A.; Champagne, A. Few-HundredGHz Carbon Nanotube Nanoelectromechanical Systems (NEMS).Nano Lett. 2012, 12, 4564−4569.(6) Peng, H.; Chang, C.; Aloni, S.; Yuzvinsky, T.; Zettl, A. Ultrahighfrequency nanotube resonators. Phys. Rev. Lett. 2006, 97, 470.

(7) Peng, H.; Chang, C.; Aloni, S.; Yuzvinsky, T.; Zettl, A. Microwaveelectromechanical resonator consisting of clamped carbon nanotubes inan 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 CarbonNanotube. 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 ofsuspended 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. Nano2012, 7, 301−304.(11) Chiu, H. Y.; Hung, P.; Postma, H. W. C.; Bockrath, M. Atomic-Scale Mass Sensing Using Carbon Nanotube Resonators. Nano Lett.2008, 8, 4342−4346.(12) Jiang, H.; Yu, M. F.; Liu, B.; Huang, Y. Intrinsic Energy LossMechanisms in a Cantilevered Carbon Nanotube BeamOscillator. 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. Symmetrybreaking 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 insuspended 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 DoublyClamped Single-Walled Carbon-Nanotube Oscillator: Towards aNanotube 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 fromcarbon nanotubes and graphene. Nat. Nano 2011, 6, 339−342.(18) Sazonova, V.; Yaish, Y.; Ustunel, H.; Roundy, D.; Arias, T. A.;McEuen, P. L. A tunable carbon nanotube electromechanical oscillator.Nature 2004, 431, 284−287.(19) Kirby, R. Platinuma thermal expansion referencematerial. Int. J.Thermophys. 1991, 12, 679−685.(20) Lyon, K.; Salinger, G.; Swenson, C.; White, G. Linear thermalexpansion measurements on silicon from 6 to 340 K. J. Appl. Phys. 1977,48, 865−868.(21) Okada, Y.; Tokumaru, Y. Precise determination of latticeparameter and thermal expansion coefficient of silicon between 300and 1500 K. J. Appl. Phys. 1984, 56, 314−320.(22) Kahaly, M. U.; Waghmare, U. V. Vibrational Properties of Single-Wall Carbon Nanotubes: A First-Principles Study. J. Nanosci. Nano-technol. 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 freeenergy landscapes. Journal of Statistical Mechanics: Theory andExperiment 2009, 2009, P02060.

Nano Letters Letter

dx.doi.org/10.1021/nl500096p | Nano Lett. XXXX, XXX, XXX−XXXE