We report on the characterization of torsional oscillators which use
multi-walled carbon nanotubes as the spring elements. Through
atomic-force-microscope force-distance measurements we are able to apply
torsional strains to the nanotubes and measure their torsional spring constants
and effective shear moduli. We find that the effective shear moduli cover a
broad range, with the largest values near the theoretically predicted value.
The data also suggest that the nanotubes are stiffened by repeated flexing.Comment: 4 page

A. M. Patel

J. Nygård

M. Buongiorno Nardelli

M. R. Falvo

P. A. Williams

R. Superfine

S. J. Papadakis

S. Washburn

T. W. Ebbesen

W. C. Young

English

arXiv

We report on the characterization of torsional oscillators which use multiwalled carbon nanotubes as the spring elements. Through atomic-force-microscope force-distance measurements we are able to apply torsional strains to the nanotubes and measure their torsional spring constants, and estimate their effective shear moduli. The data show that the nanotubes are stiffened by repeated flexing. We speculate that changes in the intershell mechanical coupling are responsible for the stiffening

Williams, P. A.

Papadakis, S. J.

Patel, A. M.

Falvo, M. R.

Washburn, S.

Superfine, R.

Carolina Digital Repository

arXiv:cond-mat/0207037v1  [cond-mat.mtrl-sci]  1 Jul 2002Torsional response and stiffening of individual multi-walled carbon nanotubesP. A. Williams,1 S. J. Papadakis,1 A. M. Patel,2 M. R. Falvo,2 S. Washburn,1, 2 and R. Superfine11Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 275992Curriculum in Applied and Materials Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599(Dated: October 30, 2018)We report on the characterization of torsional oscillators which use multi-walled carbon nanotubesas the spring elements. Through atomic-force-microscope force-distance measurements we are ableto apply torsional strains to the nanotubes and measure their torsional spring constants and effectiveshear moduli. We find that the effective shear moduli cover a broad range, with the largest valuesnear the theoretically predicted value. The data also suggest that the nanotubes are stiffened byrepeated flexing.The motivating vision of the future for nano-electro-mechanical systems (NEMS) [1, 2, 3, 4] is the emergenceof high frequency, high sensitivity devices which are com-mercially useful and also will serve as nanolabs for funda-mental investigations into physics of the mesoscale. Thegoal of high mechanical resonance frequency along withhigh sensitivity (high quality factor Q) requires that de-vices are small (low mass), stiff (high elastic modulus)and structurally defect free (low phonon scattering). Car-bon nanotubes (CNTs) may provide several advantagesover etched bulk semiconductors in this regard. Theyare naturally nanometer-scale objects, so little process-ing of the CNT itself is required. Furthermore, theirsurfaces are atomically ordered, and they have relativelylow chemical reactivity, so they may not suffer from someof the surface-to-volume ratio issues that limit the Q ofsemiconductor-based devices [5, 6].Because of these factors as well as their unique me-chanical and electrical properties [7, 8], CNTs may beideal candidates for use in NEMS. Nevertheless, therehave been only a handful of reports of CNTs used in anelectromechanical setting [9, 10]. We report here on theincorporation of CNTs into nano-electro-mechanical de-vices which allows the direct measurement of the CNTs’torsional properties.Torsional or ”paddle” oscillators are commonly studiedMEMS structures [3, 11, 12]. We fabricate paddle oscil-lators with multi-walled carbon nanotubes (MWNTs) asthe spring elements (Fig. 1). Fabrication is described indetail elsewhere [13]. Briefly, MWNTs are dispersed ontosilicon wafers which have 400-500 nm of oxide. Electron-beam lithography is used to pattern large metal pads overthe two ends of each MWNT to pin them down, and astrip of metal over the center of each MWNT to form thepaddle. The metal is thermally evaporated, 15 nm of Crfollowed by 100 nm of Au. The oxide is etched such thatthe paddles are completely undercut but the larger padspinning the MWNT ends are not [14]. The samples arethen critical-point-dried.Force measurements are performed in a hybrid atomicforce microscope (AFM)/scanning electron microscope(SEM) system; a Thermomicroscopes Observer AFMFIG. 1: a) Single-paddle torsional oscillator with MWNTspring-element. b) A double-paddle oscillator. c) One endof one of the paddles of the device in panel b has been pushedto the substrate. The deflection in the other paddle indicatesthat the MWNT is twisting uniformly along its length. Thescale bar in all three images corresponds to 1 µmmounted inside a Hitachi S-4700 cold-cathode SEM. Us-ing coarse-translation motors, we position the AFM tipabove a paddle such that the entire paddle is within therange of the AFM scan tube. Vertical force vs. dis-tance curves are taken at positions along the length ofthe paddle. Figure 2 shows an example of a force vs.distance curve in progress, and typical curves resultingfrom such measurements. In all of our analyses, we usethe data taken while the tip is being extended towardsthe sample. The data taken during tip retraction aresimilar. The AFM cantilevers are Si Nanosensor can-tilevers with nominal resonance frequencies of 70-80 kHzand force constants Kc = 1− 3 N/m.With the AFM we can apply forces to the paddlesand measure their displacements. We use these data todeduce the torsional properties of the MWNT spring-elements. Before performing a quantitative analysis, wecheck the assumption that the MWNTs twist uniformlyalong their lengths, rather than being strained in a lo-calized region. We fabricated a device with two paddlessuspended from one MWNT (Fig. 1b). When one of thepaddles was tilted with the AFM tip such that its endwas pinned to the substrate, the other deflected by abouthalf as much (Fig. 1c). The suspended portions of theMWNT are all of similar length, so this suggests that theMWNT is uniformly strained.To measure force quantitatively with an AFM, we must2FIG. 2: An example of a force-distance measurement on a paddle. a) The AFM tip above the paddle, before the measurementis started. b) The device during the measurement. The AFM tip has deflected the left end of the paddle downward by 300 nm.The right side of the paddle is raised, and the vertical deflection of the nanotube negligible. c) A schematic of the cantileverand paddle during measurement. d) Data from a force-distance curve on the bare substrate and from three consecutive tracesat different positions on device C. The slopes of the three curves appear, with the same symbols, as points in Fig. 3.characterize the AFM cantilever. We measure its dimen-sions with the SEM and its resonance frequency with theAFM. These values are used to calculate the spring con-stant (Kc) of the cantilever. Also, vertical force-distancetraces, which result in linear plots of photodiode signalvs. piezo-distance-traveled (Zp), are taken on the sub-strate. Under the assumption that the substrate is effec-tively an infinitely hard surface, the slopes of these traces(Ssub) give us the relationship between the detector sig-nal and the cantilever deflection (Fig. 2d) [15].To characterize an oscillator, vertical force-distancecurves are taken on the paddle itself, which yield traceswith slopes Spad (Fig. 2d). The force applied to the pad-dle is F = KcZpSpadSsuband the vertical displacement of thepoint of contact with the AFM tip is ∆Z = Zp(1−SpadSsub).The torsional compliance of the MWNT can be describedby a torsional spring constant (κ), which relates the ap-plied torque (T = F (x − x0), where x − x0 is the leverarm from the axis of the MWNT) to the angular deflec-tion (θ) of the paddle. The vertical compliance can becharacterized by a vertical spring constant (Kz), whichrelates F to the vertical displacement (∆h) of the MWNTpivot-point [16]:T = κθ, F = Kz∆h. (1)∆h and θ combine to yield the displacement of the pointof contact with the AFM tip, ∆Z = ∆h + (x − x0)θ.Combining these equations, we obtain:Spad =Ssub1 + KcKz+ Kcκ(x− x0)2. (2)On each paddle, we measure the slopes from a series offorce-distance curves at various positions x. Equation 2is fit to the resulting points to yield Kz and κ.The panels in Fig. 3, from left to right, depict threesets of data consecutively measured on device C. It issurprising, therefore, that κ changes from one panel tothe next. A rise in κ is seen in all of the devices wemeasure, and the percent change in κ from one pass to thenext is roughly correlated with the number of deflections-0.6 -0.3 0.0 0.30.00-0.01-0.02-0.03-0.04Pass 1κ (Nm)   =   4.1 x 10-14 5.6 x 10-14 7.3 x 10-14S pad (nA/nm)X-position (µm)-0.6 -0.3 0.0 0.3Pass 2   -0.6 -0.3 0.0 0.3Pass 3  FIG. 3: Vertical force-distance data from paddle C, shownwith fits of Eqn. 2. The curves are shifted on the X-axissuch that x0 = 0. The panels from left to right show threeconsecutive sets of data taken on one paddle.performed in the previous pass. κ values for some of ourmeasurements are summarized in Table I.We also positioned the AFM tip near the end of paddleD and repeatedly performed vertical force-distance mea-surements (Fig. 4). After 330 force curves, the tip wasmoved away from the device and then returned to within50 nm of the original position. In spite of the break in thedata, the trend is clear; the effective torsional stiffness ofthe paddle increases until about 400 deflections are per-formed, and then it saturates, showing a net change instiffness of about an order of magnitude.In principle, the MWNT could be accruing amorphouscarbon deposited during SEM imaging of the device, butthis is unlikely to explain its change in stiffness. In Fig.3, we imaged the device during pass 1, but we turnedoff the electron beam for the next two passes, and thestiffness still increased. For the data in Fig. 4, the elec-tron beam was off for most of the measurement, but thestiffness increases smoothly. After the measurements onthe paddles were completed, we used the AFM to imagethe edges of the regions the SEM had been scanning, andsaw no topographical step. The AFM has a vertical res-olution on the order of 1 nm, so if a significant thickness30 100 200 3000.000-0.005-0.010-0.015-0.020a S pad (nA/nm)Number of deflections400 500b FIG. 4: Repeated force curves taken on paddle D, showing afactor of ∼ 10 increase in stiffness. (a) The AFM tip was keptin one place for curves 1 to 330, after which its position on thepaddle was changed by about 50 nm, (b) where it remainedfor subsequent measurements.TABLE I: Summary of MWNT torsional spring constants (κ),outer radii (rout), and effective (Ge) and shell (Gs) shear mod-uli (See the text).Device/pass κ rout Ge Gs(10−14 Nm) (nm) (GPa) (GPa)A/1 15 15 600B/1 2.4 16 60 830B/2 4.5 120B/3 4.6 120B/4 10 280B/5 22 590B/6 46 1200C/1 2.5 18 15 210C/2 3.4 20C/3 4.4 26Da/i 1.4 16 30 430Da/f 17 400aA full pass was not made on paddle D. (x − x0) was estimatedfrom SEM images, and used in Eqn. 2 to calculate κ. The κ, Ge,and Gs given are (i) for the first few deflections and (f) for thesaturation (See the text and Fig. 4.)of carbon had been deposited during the measurement,it would have appeared as a topographical feature in theAFM image. To show that the stiffening is not an instru-mental artifact, both Ssub and the cantilever resonancefrequency were measured before and after each experi-mental run, and they showed no significant change. Also,we repeatedly deflected one AFM cantilever with another∼ 600 times and saw no change in stiffness. This impliesthat the repeated twisting of the nanotube is making itstiffer! The shear strain applied to the MWNT was typ-ically ∼ 0.5%. The applied tensile strain was very small.Most existing studies focus on the tensile propertiesof CNTs; we look to these results to gain some insight.Experiments have shown that MWNTs can withstandtensile strains of between 10% and 20% before breaking[17, 18], although details of the structural behavior beforethe actual break are not known. Simulations of single-walled nanotubes under tension predict that Stone-Walesdefects can form at tensile strains of ∼ 5% and defect mo-tion may occur at tensile strains as low as ∼ 3% [19]. Weare repeatedly applying shear strains of about ∼ 0.5%to the MWNTs, and few times in each pass approaching∼ 1%. There are two structural changes that could stiffenthe MWNTs. The individual shells of the MWNTs couldbe stiffened or the mechanical coupling between differentshells could be increased. It is unlikely that changes inthe connection between the nanotube and either the pin-ning metal or the paddle would result in an increase indevice stiffness. Mechanical flexing of those joints wouldbe expected to weaken them, which would result in adecrease in measured stiffness.From the κ, and the paddle dimensions, we can calcu-late the devices’ expected torsional resonant frequencies,which are in the 1-10 MHz range. In principle, we shouldalso be able to extract Kz from our fits to Eqn. 2. Formost of the devices, however, including device C shownin Fig. 3, Spad at (x − x0) = 0 is within experimentalerror of Ssub. This means that there is negligible verti-cal deflection of the paddle, and that Kc/Kz ∼ 0. Wecannot accurately measure Kz using a cantilever withKc << Kz. Nevertheless, given our experimental uncer-tainty in Kc, we can estimate a lower bound for Kz. Wereturn to this point below in our discussion of moduli.In order to make comparisons between MWNTs ofdifferent dimensions, we use the continuum mechanicsmodel for the shear (G) and Young’s (E) moduli. Weuse beam bending equations [20] combined with Eqs. 1:κ =π(r4out − r4in)G2l, Kz =48πE(r4out − r4in)l3, (3)where rout and rin are the outer and inner radii and lare the lengths of the suspended MWNT sections. Sincethe moduli depend on r4out−r4in, taking rin to be 0 yieldsmoduli only ∼ 6% smaller than taking rin/rout to be∼ 0.5. Based on TEM observations, rin of our arc-grownMWNTs are typically much less than half of rout. Wetherefore approximate the MWNTs as solid cylinders andcalculate effective shear moduli Ge, which are summa-rized in Table I.Uncertainty in rout contributes the bulk of the un-certainty in Ge. AFM rout measurements of the metalsurface under which the MWNTs were buried, and ofMWNTs pushed down to the substrate with the AFMtip, varied by more than a factor of two for a givenMWNT. The rout reported in Table I were arrived atby averaging the full-width at half-maximum of the SEMimage brightness along lines perpendicular to the sus-pended MWNTs. Applying this technique to multipleSEM images of the same MWNT, taken under differentimaging conditions (i.e. different image brightness, sec-ondary electron detector, etc.), yielded a variation in themeasured rout of about 20%, corresponding to an uncer-tainty in Ge of about a factor of 2. It is also possible thatthere is a systematic error in the estimation of rout fromSEM images, which can exaggerate nanotube diameters.4As mentioned, the MWNTs increased in stiffness as theexperiment progressed. Since the cause of this increase isstill unknown, we initially discuss data from the first passon each paddle, where the MWNT has been least affectedby the experiment. The Ge calculated from the first passon each paddle range over about an order of magnitude.We hypothesize that these differences are due to differ-ences in the mechanical coupling between the shells ofthe MWNTs in the various devices. Cumings and Zettlhave shown, by pulling inner shells of a MWNT out fromthe outer ones, that it is possible to have a very low in-tershell mechanical resistance [21]. Yu’s et al. study ofthe tensile failure modes of MWNTs shows that the in-dividual shells are not equally strained in all MWNTs,since some break at widely separated places along thetube’s length [17]. Our wide range of Ge suggests thatin some MWNTs, the stress is evenly distributed amongmany shells, while in others the stress is concentratedonly in the outer shell or few shells while the inner onesslide easily. Since our MWNTs are clamped by evapo-rating metal onto their outer shells, it is conceivable thatthe inner shells may be sliding both where the MWNTspass through the paddle and where they pass throughthe metal pinning the MWNT ends. Observations haveshown that some MWNTs have circular cross-sections,while others have polygonal cross-sections [22]. Suchpolygonal MWNTs could have greater inter-shell cou-pling than circular MWNTs under rotational strain. Thiswould explain the large range of effective Ge calculatedfrom the solid rod model where the entire diameter ofthe MWNT is assumed to share the stress. Other con-tributors to the large range of measured Ge could includedefects in the MWNTs or imperfect pinning of the nan-otubes within the metal.In Ref. 23, using a model which assumes that all shellsare strained, Lu predicted G = 541 GPa for a ten-wall 7.8nm-diameter CNT. For comparison, in diamond G = 576GPa and in graphite the basal plane G = 440 GPa. Ourlargest measured Ge are consistent with the theoreticalvalue. The experimental values which fall below this pre-diction can be explained by sliding of the inner shellswhen only the outer shell or few shells are clamped by theevaporated metal. Indeed, assuming that only the outershell carries the load, and using the inter-shell spacingas the effective thickness of that shell, we can calculatea shell shear modulus Gs. Some of these results are alsoin Table I. For devices B, C, and D, which initially havesmall Ge, Gs is near the theoretical value. The stiffen-ing of the MWNTs with repeated twisting suggests thatthe inter-shell coupling can be modified. Both devices Band D are stiffened by more than on order of magnitude,resulting in Ge consistent with the theoretical values. Itis noteworthy that device D shows a saturation at thisvalue. These results are consistent with the idea that,initially, only the outer shell is strained, and that re-peated deflections increase the inter-shell coupling untilall shells are strained. A more accurate rout measurementis required to confirm this conclusively. For Young’s mod-ulus, if we calculate lower bounds using the lower boundvalues of the Kz, we find E is typically greater than afew hundred GPa, which is consistent with previous ex-perimental and theoretical results [17, 23].In conclusion, we have fabricated and characterizedCNT paddle oscillators. We repeatedly apply torsionalstrains to the MWNT spring-elements and measure theirtorsional spring constants. We find that the MWNTs be-come stiffer with repeated deflection. The effective shearmoduli of the nanotubes calculated by approximating theMWNTs as solid cylinders vary by nearly two orders ofmagnitude, from near the theoretically predicted valueto significantly less than it. The results suggest that theinter-shell mechanical coupling varies widely from oneMWNT to the next and can be modified by applyingstrain.We thank the ONR and the NSF for funding for thisproject.[1] H. G. Craighead, Science 290, 1532 (2000).[2] A. Erbe, C. Weiss, W. Zwerger, and R. H. Blick, Phys.Rev. Lett. 87, 096106 (2001).[3] A. N. Cleland and M. L. Roukes, Nature 392, 160 (1998).[4] Y. T. Yang et al., Appl. Phys. Lett. 78, 162 (2001).[5] R. Lifshitz and M. L. Roukes, Phys. Rev. B 61, 5600(2000).[6] D. W. Carr et al., Appl. Phys. Lett. 75, 920 (1999).[7] H. W. C. Postma et al., Science 293, 76 (2001).[8] A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker,Science 294, 1317 (2001).[9] P. Poncharal, Z. L. Wang, D. Ugarte, and W. A. de Heer,Science 283, 1513 (1999).[10] P. Kim and C. M. Lieber, Science 286, 2148 (1999).[11] S. Evoy et al., Jour. Appl. Phys. 86, 6072 (1999).[12] D. W. Carr et al., Appl. Phys. Lett. 77, 1545 (2000).[13] P. A. Williams et al., in press.[14] J. Nyg̊ard and D. H. Cobden, Appl. Phys. Lett. 79, 4216(2001).[15] R. Lüthi et al., Surf. Sci 338, 247 (1995).[16] Bending of the metal paddle itself is negligible. The cal-culated spring constant due to the bending of the paddleis one to two orders of magnitude larger than the effectivespring constants of the CNT.[17] M.-F. Yu et al., Science 287, 637 (2000).[18] M. R. Falvo et al., Nature 389, 582 (1997).[19] M. Buongiorno Nardelli, B. I. Yakobson, and J. Bernholc,Phys. Rev. Lett. 81, 4656 (1998).[20] W. C. Young, Roark’s Formulas for Stress and Strain,sixth ed. (McGraw-Hill, New York, 1989).[21] J. Cumings and A. Zettl, Science 289, 602 (2000).[22] M. Liu and J. M. Cowley, Ultramicroscopy 53, 333(1994).[23] J. P. Lu, Phys. Rev. Lett. 79, 1297 (1997).

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