Torsional Resonators Based on Inorganic Nanotubes - Nano Letters

Letter pubs.acs.org/NanoLett

Torsional Resonators Based on Inorganic Nanotubes Yiftach Divon,† Roi Levi,† Jonathan Garel,† Dmitri Golberg,∥ Reshef Tenne,† Assaf Ya’akobovitz,§ and Ernesto Joselevich*,† †

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan § Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 8410501, Israel ∥

S Supporting Information *

ABSTRACT: We study for the first time the resonant torsional behaviors of inorganic nanotubes, specifically tungsten disulfide (WS2) and boron nitride (BN) nanotubes, and compare them to that of carbon nanotubes. We have found WS2 nanotubes to have the highest quality factor (Q) and torsional resonance frequency, followed by BN nanotubes and carbon nanotubes. Dynamic and static torsional spring constants of the various nanotubes were found to be different, especially in the case of WS2, possibly due to a velocitydependent intershell friction. These results indicate that inorganic nanotubes are promising building blocks for highQ nanoelectromechanical systems (NEMS). KEYWORDS: Nanotube, nanoelectromechanical systems (NEMS), torsion, oscillator, nanomechanics, inorganic nanotubes norganic nanotubes, first reported in 1992,1 are increasingly attracting interest as the rolled-up version of noncarbon 2D materials, and potential building blocks for nanotechnology.2 What is their potential for nanoelectromechanical systems (NEMS)? Carbon nanotubes (CNTs) have long been regarded as attractive building blocks for NEMS owing to their outstanding mechanical and electrical properties, as well as their unique electromechanical coupling.3 In particular, torsional electromechanical systems could be used as the basis for gyroscopes for navigation of ultraminiaturized unmanned aerial vehicles (UAVs),4 and for various chemical and biological sensors.5 Extensive work has been done with respect to CNTbased torsional devices: fabrication,6 characterization of torsional7 and electromechanical properties in single-walled CNTs8 (SWCNTs) and multiwalled CNTs9,10 (MWCNTs), and creation of MWCNT and SWCNT torsional resonators.11,12 One of the most critical factors determining the sensitivity of resonant NEMS is their quality factor−a dimensionless parameter corresponding to the ratio between the stored and dissipated energy per cycle. Namely, the higher the quality factor, the less energy gets dissipated during one oscillation cycle. Internal friction, interlayer coupling, crystallographic structure, and chemical composition can play a critical role in determining the torsional behavior of nanotubes and specifically their quality factor (Q). WS2 nanotubes (WS2 NTs) are a promising material owing to their significant electromechanical response,13 stick−slip torsional behavior,14 and high current-carrying capacity.15 Boron nitride NTs (BNNTs), with their ultrahigh torsional stiffness and torsional strength,16 and

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© XXXX American Chemical Society

their carbon-doped version, BCNNTs, which have shown a significant electromechanical response,17 seem very promising as well. Thus, these properties and the aspects influencing the quality factor have motivated us to examine inorganic nanotubes (INTs) as potential building blocks for torsional devices. Here we demonstrate the first torsional resonators based on inorganic nanotubes and study the effect of the NT material on the torsional resonator properties, in ambient conditions and in vacuum. INTs exhibit higher torsional resonance frequencies and quality factors, extending the available material toolbox for torsional NEMS devices. This work further demonstrates that INTs are promising building blocks for NEMS in general and torsional NEMS in particular. The torsional resonators used in this work are quite similar to devices previously used in our group to study the torsional properties of CNTs,9,10 BNNTs,16 and WS2 NTs,13,14 except for an intentional broken symmetry that enables their electrostatic actuation. A torsional resonator (Figure 1) consists of a suspended nanotube (MWCNT, BNNT or WS2 NT) clamped between metallic pads at its ends, with a suspended pedal attached to its top. The pedal is off-centered with respect to the nanotube, so that each end of the pedal stands at a different distance from the nanotube (Figures 1b,c). The resonators were fabricated using electron-beam lithography, Received: July 20, 2016 Revised: December 16, 2016

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DOI: 10.1021/acs.nanolett.6b03012 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. Resonance spectrum measurement of nanotube-based torsional resonators (schematic setup). (a) Schematic structure of multiwalled CNTs, BNNTs, and WS2 NTs. (b) Scanning electron microscope (SEM) image and (c) atomic force microscope (AFM) images of a nanotubebased torsional resonator. (d) Torsional resonator actuated by applying a DC and AC voltage between the substrate and the offset pedal, which is attached to the nanotube. The amplitude is detected by a laser Doppler vibrometer (LDV) and outputted to the network analyzer.

followed by wet etching and critical point drying (see Supporting Information for details). In order to measure the oscillatory behavior of the torsional resonators, a DC bias voltage and a smaller AC drive voltage were applied between the substrate and the pedal using a network analyzer. The frequency of the AC component was swept from 0.1 to 24 MHz (the upper limit of our detection system). The alternating voltage between the substrate and the pedal combined with the offset of the center of the pedal with respect to the nanotube created an oscillatory net torque on the pedal, thus periodically twisting the nanotube. The amplitude of the pedal was detected using a laser Doppler vibrometer (LDV), and presented as a function of the drive AC voltage frequency, in order to capture the resonant response of each nanotube-based resonator (Figure 1d). Figure 2 shows representative resonance spectra of CNT-, BNNT- and WS2 NT-based torsional resonators under atmospheric pressure. The resonance frequency and quality factor were extracted for each peak in the spectrum by fitting the results to a classical driven damped oscillator given in eq 1, where θmax is the amplitude of the pedal, κ is the torsional spring constant, τ0 is the maximal electrostatic torque on the pedal, ν is the driving frequency, ν0 is the natural resonance frequency and Q is the quality factor. A first-order of approximation of ν0 is given in eq 2, where I is the pedal mass moment of inertia. A more rigorous description may be found in ref 18. Here, increasing the drive amplitude linearly increases the peak maximum while the symmetrical shape of the peaks is retained (see Supporting Information and Figure 2 and Figure S2). This is typical of linear oscillators,11 and allows for a linear approximation of the resonant behavior.18 The results for all the CNT-, BNNT- and WS2 NT-based resonators are summarized in Tables S1, S2 and S3 (Supporting Information), respectively. A total of 25 devices were measured: 11 of CNTs, 9 of BNNTs, and 5 of WS2 NTs. While CNT-based resonators exhibited 2−5 peaks in the measured frequency range, WS2

NTs exhibited 1−2 peaks, and BNNTs displayed only one distinct peak. τ0v0 2 /k

θmax(v) =

ν0 =

1 2π

(v 2 − v0 2)2 +

κ I

v0 2 2 Q2

v

(1)

(2)

In order to assign the different peaks to their corresponding oscillation mode, and in particular, to identify the torsional mode, we conducted a finite element analysis (FEA) using COMSOL MULTIPHYSICS. The numerical convergence of our FEA simulations was verified through refining of the mesh. We performed a variety of simulations comprising the wide range of parameters that can exist in our systems, namely: (i) NT diameters between 5.8 and 88.4 nm; (ii) Young’s modulus between 170 GPa (for WS2 NTs19) and 0.8−1.2 TPa (for CNTs20 and BNNTs21); (iii) Poisson’s ratios between 0 and 0.3;22,23 (iv) densities between 1380 kg/m3 (for BNNTs24) and 7730 kg/m3 (bulk density of WS225); and (v) extents of intershell coupling ranging for the extreme case of a hollow cylinder (only the outermost shell carrying the load) to the other extreme case of a solid rod (all the shells coupled). In all these simulations, the lowest-frequency normal (eigen) mode was always the torsion, followed by significantly higher frequencies related to the other modes (in-plane rotation, inplane bending and out-of-plane bending). Following this detailed analysis, we can safely assign the first peak (i.e., lowest-frequency) of all our measured spectra to the torsional mode of the nanotube-based resonators. Figure 3 shows an example for such an analysis for the resonator whose resonance spectrum is depicted in Figure 2a. Comparing the FEA simulations to the experiments, it can be seen that the torsional mode is consistent with a hollow cylinder case, while the inB


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contains a small component of torsional motion, each mode having a different contribution from all the walls (solid rod case) or only from the outer wall (hollow cylinder case). In principle, our experimental setup is not designed to actuate nor detect in-plane motion (referred to in Figure 3 as “in plane bending” and “in plane rotation”), so these modes are ideally not expected to appear in the spectra. Nonetheless, due to misalignment of the resonator with respect to the laser beam, and the offset of the pedal with respect to the nanotube, x−y−z cross-talk and parasitic actuation is possible to a certain degree. This could explain the appearance of the in-plane bending mode and the absence of the in-plane rotation mode in the spectrum. Based on these considerations, we focus on the lowest-frequency peak, which was assigned to the torsional mode of the nanotube-based resonators. FEA simulations of BNNT- and WS2 NT-based torsional resonators are qualitatively similar to those of CNTs (Figure 3), and thus summarized in the Supporting Information (Figures S3 and S4, respectively). Comparing the measured resonance frequency of BNNT-based resonators to the simulation suggests that the resonant torsional motion of BNNTs is an intermediate case between the solid rod and hollow cylinder cases, i.e. there is some degree of intershell coupling during the torsional motion.The FEA simulation for WS2 NT-based resonators seemingly points out to a discrepancy: the measured torsional resonance frequency appears to be higher than the extreme case of solid rod, as if the number of shells twisting together was larger than the number of existing shells in the nanotube. This discrepancy may be related to the fact that the Young’s modulus used for the simulation was the most widely accepted value (170 GPa), while various larger values have been reported for the Young’s modulus of WS2 NTs (up to 615 GPa).19 Therefore, these results suggest that the resonant torsional motion of WS2 NTs is described by the solid rod case with the largest degree of intershell coupling. Furthermore, this is consistent with previous research on the small angle torsion of WS2 NTs.14 Figure 4a shows a comparison of the torsional resonance spectrum for a typical torsional device from each material under atmospheric pressure and in an estimated vacuum of 2· 10−2 mBar. Comparing the torsional resonance spectra of all the measured torsional resonators, WS2 NTs exhibit the highest average torsional resonance frequency (19.7 ± 4.1 MHz), followed by BNNTs (5.21 ± 1.57 MHz) and CNTs (1.26 ± 0.43 MHz). The same trend applies for the average quality factors as well: 86 ± 30 for WS2 NTs, 28 ± 4 for BNNTs, and 15 ± 9 for CNTs. Dynamic κ, namely the torsional spring constants extracted from the resonance spectra measured in air using eq 2, are plotted as a function of nanotube diameter d in Figure 4b (for our devices, the effective κ takes into consideration the two segments of suspended nanotube, which are simultaneously twisted in opposite directions). WS2 NTs exhibit the highest dynamic κ, followed by BNNTs and CNTs. This trend is consistent with the expected strong dependence of κ on the diameter of the nanotube (∼d4 assuming a solid rod case, and ∼d3 assuming a hollow cylinder case16). The power law of κ in the diameter can thus provide a measure of the intershell coupling within the available range of nanotube diameters: it should be closer to 4 if the shells are more coupled and closer to 3 if only the outermost shell carries the torsional load. We can see that the BNNTs exhibit a power law of ∼d3.6, which suggests a more significant intershell coupling than CNTs (∼d2.2). The power law of BNNTs is

Figure 2. Representative resonance spectra of the different nanotubebased torsional resonators: (a) CNT (device no. 2); (b) BNNT (device no. 5); and (c) WS2 NT (device no. 1). Insets show the fittings to eq 1 of the various resonance peaks, measured at a higher resolution. In part b, there is one distinct peak at 5.84 MHz and smaller features at higher frequencies that appeared regardless of the location of the LDV laser spot, which are hence most likely an artifact of the measurement due to the relatively low signal-to-noise ratio for some of the measurements (see relative amplitude signal scales).

plane and out-of-plane bending modes are consistent with a solid rod case. This result is consistent with the torsional behavior of MWCNTs, which is known to involve only the outer shell of the nanotube,7,9,10 and the intuitive assumption that the bending motion will have to involve all the shells. Discrepancies between measured peak positions and calculated resonance frequencies can be explained by the simplicity of the model, which does not take into account the complexity of the inner structure of the nanotubes and its anisotropy, as well as defects and imperfections appearing during the fabrication process. Also, the normal mode designated as “torsion” does in fact contain a small component of bending motion,18 and likewise, the normal mode designated as “out-of-plane” in fact C


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Figure 3. Comparison between the measured and simulated resonance frequencies of the different normal modes for a carbon nanotube-based torsional resonator (device no. 2, Figure 2a). The two extreme intershell coupling cases of solid rod and hollow cylinder were examined. The underlined simulated frequencies are the ones closer to the measured ones.

the pure drag-forces. A common approach to this estimation is to substitute the oscillating object by a superposition of spheres with the damping coefficient of each sphere as given by eq 4, where μ is the air viscosity, r is the radius of the sphere (in our case, this is roughly the pedal width), ρ is the air density, and v is the oscillation frequency.29 Inputting the frequency at resonance in eq 5 into eq 4 shows that we can expect b to go as ∼κ0−0.25, and using eq 3 we expect Q to go as ∼κ0.25−0.5. As seen in Figure 4c, Q goes as ∼κ0.30±0.07 which is consistent with this prediction.

consistent with an intermediate case between the two extreme cases suggested by the FEA simulations. WS2 NTs could not be fitted to any such power law, probably due to the high variance in the intershell coupling between the individual nanotubes constituting the resonators and the large variance in their effective Young’s moduli mentioned earlier. Figure 4c shows that the measured quality factors under atmospheric pressure increase with the dynamical torsional spring constant. This relationship can be attributed to the dominant effect of air losses rather than different intrinsic properties. When air losses are the dominant energy dissipation mechanism, as in ambient conditions,26 then Q is given by eq 3, where κ is the dynamic torsional spring constant, I is the pedal mass moment of inertia, and b is the damping coefficient due to air friction. The mass moment of inertia depends mainly on the geometry and density of the pedal, since the nanotube material and diameter have a negligible influence, and thus I should be quite similar for all resonators (except for small differences in the offset of the pedal position with respect to the nanotube due to nanofabrication inaccuracies). A calculated squeeze number of 0.04 and 0.15 (see Supporting Information) for CNT- and BNNT-based resonators, respectively, indicates that for these resonators, the damping coefficient b of the system is expected to be mainly contributed by pure drag-force damping (drag caused by a moving object in a fluid far away from other surfaces), as opposed to squeeze-film damping (increased damping caused by squeezing of the gas confined between two nearby surfaces).27 Due to their higher resonance frequency, the squeeze number of torsional resonators based on WS2 NT is higher than that of those based on CNT and BNNT (0.55), indicating a higher contribution of squeeze-film damping. However, the squeeze-film damping coefficient for torsional resonators at high frequencies is of the form of a converging series and thus does not have a simple power law dependence on the resonance frequency.28 Nonetheless, we may estimate

Q=

κI / b

(3)

bair − drag = 3πμr +

vres = v0 1 −

1 2Q 2

3 2 πr (2ρμν)1/2 4

(4)

(5)

On the basis of these considerations, the quality factor of the resonator is significantly affected by the pedal interaction with the air and does not directly reflect the intrinsic mechanical properties of the NTs. In order to observe the intrinsic behavior of the nanotube, i.e. the internal friction which is induced by the nanotube material and structure, the air damping has to be reduced down to the point where it is negligible compared to the internal friction of the NT. The air pressure range in which the intrinsic behavior is dominant can be referred to as the intrinsic region.30 Measurements of the torsional devices frequency response were thus conducted in a vacuum of 2 × 10−2 mBar (the low viscous regime) and are summarized in Table 1. As expected due to reduction of the interaction of the pedal with the air molecules, the vacuum caused an increase in the quality factor of all nanotubes. Comparing the average ratios of the quality factors in vacuum with those in air, it appears that all the quality factors have D


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than those of CNTs. In addition, we can roughly estimate the expected intrinsic Q for BNNTs and WS2 NTs. This is done by comparing the different components of the quality factor Q that we measured in air for MWCNT torsional devices with the Q that was measured in sufficient vacuum for similar devices performed in ref 11 (see Supporting Information for calculation). This estimate indicates that CNTs should have an average quality factor of 98 in vacuum, whereas the quality factors of BNNTs and WS2 should be higher (200 and 216 respectively), once again indicating that the intrinsic Q values of the INTs are higher than those of CNTs. Comparing the static and dynamic properties of the NTs may yield additional insight on the interactions between adjacent layers for different materials. Thus, we complemented the calculated dynamic κ with the measured static torsional spring constant (static κ) of the various nanotubes using the well-established method7,9,13,16 of pressing an atomic force microscope (AFM) tip against the pedal in various positions along the pedal, and measuring the force while twisting the nanotube (Figure 5a). The linear stiffness K of the system was calculated for each position across the pedal (See Supporting Information for details). The static κ was extracted by fitting the plot of K as a function of the tip position (Figure 5b) to eq 6, where x and a are the positions of the tip and the nanotube with respect to an arbitrary origin, respectively, κ is the static torsional spring constant and KB is the bending spring constant.16 In Table 2, we compare the torsional spring constants that were extracted from the resonance spectrum measurements (dynamic κ, Figure 4b) to the static ones that were extracted from the AFM measurements. All measured devices exhibit a higher dynamic κ than the static κ, with a significant increase in the case of BNNT and WS2 NT based devices. ⎤−1 ⎡ (x − a)2 −1 K=⎢ + KB ⎥ ⎦ ⎣ 2κ

(6)

The resonance spectrum measurements, from which the dynamic κ is extracted, involve twisting the nanotube at an average speed that is 6−7 orders of magnitude higher than in the static AFM-pressing measurement. Zhang et al.32 had found during pullout experiments in double-wall carbon nanotubes (DWCNTs) that the intershell friction between the outer- and inner-shell increases linearly with increasing pullout velocity. Although the measured pullout velocities were axial rather than torsional, and they were significantly smaller than in our dynamic experiments, Zhang’s findings are consistent with our results, since the higher the intershell friction, the higher the coupling between shells should be, and thus more shells share the load and contribute to the overall torsioni.e., the dynamic κ should be higher than the static one. The increased dynamic κ with respect to the static one does not seem to stem from squeeze-film effect,27 because, as seen in Table 1, there is no apparent difference between the resonance frequencies and thus dynamic torsional spring constants in air and in vacuum. To the best of our knowledge, the only comparison between static and dynamic κ of a CNT-based torsional device, based on a single MWCNT device measurement, had found the dynamic κ to be slightly smaller than the static one.11 The discrepancy between that and our results is not yet understood. BNNTs show an increase in the dynamic κ with respect to the static one. Similar to the CNT case, a velocity-dependent intershell friction mechanism might explain the higher dynamic

Figure 4. Comparison of torsional resonance characteristics of CNTs, BNNTs and WS 2 NTs: (a) Torsional resonance peaks of representative CNT-, BNNT-, and WS2 NT-based resonators in air and vacuum (CNT device no. 2, BNNT device no. 1 and WS2 NT no. 1, respectively). Values in parentheses represent the values measured in an estimated vacuum of 2 × 10−2 mBar. (b) Dynamic torsional spring constant of the torsional resonators measured in air as a function of NT diameter. (c) Quality factors of all resonators in air as a function of their torsional spring constant.

increased approximately by the same factor (2.4 ± 0.6 for CNT devices,31 1.7 ± 0.3 for BNNTs, and 2.3 ± 0.2 for WS2 NTs). This suggests that, although the quality factor that was measured was closer to the intrinsic Q, the vacuum level that was reached did not completely eliminate air damping. If the intrinsic region would have been reached, we would have expected to see the quality factor of each material change by a different factor under vacuum, since the intrinsic Q of each material should be independent of the Q in air. Nonetheless, the intrinsic component of the Q measured in vacuum was more significant when compared to the measurement in air. This suggests that the intrinsic Q values of the INTs are higher E


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Table 1. Comparison between Torsional Resonance Frequencies and Quality Factors at Atmospheric Pressure and under Vacuum

κ. The fact that the dynamic/static ratio for BNNTs is higher than the dynamic/static ratio for CNTs could be explained by the different chemical composition and structure of the two types of nanotubes, and by the difference in diameters between the two types of nanotubes (since intershell friction is contact area dependent, the larger the diameter the larger the contact area). An additional factor that could lead to the higher dynamic κ for BNNTs is related to their facets. It has been shown that BNNTs of large diameters (>27 nm) are faceted but undergo unfaceting when twisted using an AFM.16 It is possible that the time it takes for the BNNT to undergo unfaceting is longer than the time of oscillation, so the BNNT stays faceted through the whole oscillation. If this were the case, then the dynamic κ would be larger than the static one due to the intershell coupling of the faceted BNNT compared to the unfaceted one. Compared to CNTs and BNNTs, WS2 NTs exhibit the highest dynamic/static ratio. WS2 NTs are known to exhibit torsional stick−slip behavior.14 This behavior, in which energy is dissipated due to irreversible jumps between neighboring equilibrium positions, is known to be responsible for velocity dependent friction on the atomic scale.33 As described earlier for CNTs, the high torsional velocity during dynamic measurements might cause increased intershell friction which may lead to higher coupling between shells. This could imply that more shells are involved in the torsional movement thus increasing κ. It has been suggested before13 that during the “stick regime” the different shells of the WS2 NT are not necessarily locked or unlocked in an all-or-nothing situation. There is a possibility that the dynamic actuation causes the shells to have an increased degree of locking compared to the static AFM-pressing measurements. The high dynamic/static ratio might be explained by the difference in the mechanical and structural properties of WS2 NTs with respect to CNTs and BNNTs. Although these are all different materials with different mechanical properties and dynamic behaviors, the difference in diameters also needs to be considered, since

Figure 5. Measurement of the static torsional spring constant of CNTs, BNNTs and WS2 NTs resonators. (a) Schematic of the AFM cantilever and the pedal during a force−distance measurement. (b) Linear stiffness of a CNT resonator (device no. 5) is plotted as a function of the position of the cantilever across the pedal.

Table 2. Comparison between Dynamic and Static Torsional Behavior of CNTs, BNNTs, and WS2 NTsa

a

The AFM force−distance measurement was conducted prior to the dynamic measurements, as opposed to the rest of the devices which were measured by AFM after the dynamic measurements. F


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static torsional spring constant measurement procedure (4) resonance peaks of all measured CNT-based torsional devices; (5) resonance peaks of all measured BNNT-based torsional devices and a representative FEA simulation of a BNNT-based torsional device; (6) resonance peaks of all measured WS2 NT-based torsional devices and a representative FEA simulation of a WS2 NT-based torsional device; (7) estimation of intrinsic Q under vacuum; (8) calculation of squeeze number (PDF)

AUTHOR INFORMATION

Corresponding Author

*(E.J.) E-mail: [email protected]. ORCID

Yiftach Divon: 0000-0002-0365-1314 Reshef Tenne: 0000-0003-4071-0325 Ernesto Joselevich: 0000-0002-9919-0734 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS E.J. acknowledges support from the Office of Naval Research (ONR) Grant No. N62909-15-1-2022, the Israeli Ministry of Defense Grant No. 4440411487 for the CNT and BNNT parts, and the Israel Science Foundation Grant No. 1493/10. E.J. and R.T. acknowledge support from the FTA action “Inorganic nanotube-polymer composites” No. 711543 of the Israel National Nano Initiative for the WS2 NT part, the Irving and Cherna Moskowitz Center for Nano and Bio-Nano Imaging Grant No. 7208214, and the Perlman Family Foundation. R.T. acknowledges support from the Israel Science Foundation Grant No. 265/12, the Kimmel Center for Nanoscale Science Grant No. 43535000350000, the German-Israeli Foundation (GIF) Grant No. 712053, the Waltcher Foundation Grant No. 720821, the EU Project ITN- “MoWSeS” Grant No. 317451, the G. M. J. Schmidt Minerva Center for Supramolecular Chemistry Grant No. 434000340000, and the Irving and Azelle Waltcher Foundations in honor of Prof. M. Levy. E.J. holds the Drake Family Professorial Chair of Nanotechnology.

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ABBREVIATIONS CNT, carbon nanotube; BNNT, boron nitride nanotube; WS2 NT, tungsten disulfide nanotube

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REFERENCES

(1) Tenne, R.; Margulis, L.; Genut, M.; Hodes, G. Nature 1992, 360, 444−446. (2) Tenne, R. Nat. Nanotechnol. 2006, 1, 103−111. (3) Hierold, C.; Jungen, A.; Stampfer, C.; Helbling, T. Sens. Actuators, A 2007, 136, 51−61. (4) Joselevich, E. ChemPhysChem 2006, 7, 1405−1407. (5) Hall, A.; Paulson, S.; Cui, T.; Lu, J.; Qin, L.; Washburn, S. Rep. Prog. Phys. 2012, 75, 116501. (6) Williams, P.; Papadakis, S.; Patel, A.; Falvo, M.; Washburn, S.; Superfine, R. Appl. Phys. Lett. 2003, 82, 805−807. (7) Williams, P.; Papadakis, S.; Patel, A.; Falvo, M.; Washburn, S.; Superfine, R. Phys. Rev. Lett. 2002, 89, 255502.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b03012. (1) Nanofabrication methods and sample mounting; (2) experimental procedure for resonance spectrum measurements and resonance spectra of different nanotube-based torsional resonators at different drive amplitudes; (3) G


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Torsional Resonators Based on Inorganic Nanotubes - Nano Letters

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