NANO LETTERS
Bending-Mode Vibration of a Suspended Nanotube Resonator
2006 Vol. 6, No. 12 2904-2908
Benoit Witkamp, Menno Poot, and Herre S. J. van der Zant* KaVli Institute of Nanoscience, Delft UniVersity of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Received September 19, 2006; Revised Manuscript Received November 8, 2006
ABSTRACT We have used a suspended carbon nanotube as a frequency mixer to detect its own mechanical motion. A single gate-dependent resonance is observed, which we attribute to the fundamental bending mode vibration of the suspended carbon nanotubes. A continuum model is used to fit the gate dependence of the resonance frequency, from which we obtain values for the fundamental frequency, the residual and gateinduced tension in the nanotube. This analysis shows that the nanotubes in our devices have no slack and that, by applying a gate voltage, the nanotube can be tuned from a regime without strain to a regime where it behaves as a vibrating string under tension.
In nano-electromechanical systems (NEMS), mechanical motion of a nanoscale object induces changes in the object’s electrical properties and vice versa.1 Compared to microelectromechanical systems, NEMS promise improvements in terms of speed and power consumption. NEMS are also interesting from a fundamental point of view; they can be used to study the fundamental limit of mechanical motion.2 This regime should be reached with high-frequency resonators (>1 GHz) at low temperatures ( Vg*) regime. The extracted values for T0L2/EI are larger than the value (T0L2/EI ) -4π2) where Euler-buckling occurs, indicating that the tubes in our devices do not exhibit slack, as confirmed by SEM images (see Figure 1d). This has to be contrasted to the work by Sazanova et al.,9 where slack is present, which makes the tubes acting as hanging chains. When slack is present, the nanotube can oscillate with several modes,9,18 making the identification of the observed modes difficult. The consistency of the model can further be checked by estimating the oscillation amplitude. From the fitted gate Nano Lett., Vol. 6, No. 12, 2006
Figure 4. Resonance frequency ν0 extracted from Figure 3a (device A) and from a second device, shown in Figure 1d (device B). At some gate voltages, the resonance could not be resolved, due to a low signal-to-noise ratio. The continuum model described in the text fits the gate dependence of the resonance frequency well (solid lines).
dependence we obtain the tension needed to calculate the amplitude. With the quality factors from the fits shown in parts b and c of Figure 3, the amplitudes for device A at Vg ) -7.5 V and Vg ) 2.9 V are uj ) 7.9 and 4.0 nm, respectively. These values are close to the values obtained from the peak height. Thus, the model, together with the fit parameters obtained from the data can be used to estimate uj and T at any gate voltage. For example, at the highest gate voltage (Vg ) 8 V), the nanotube is under a tension of T ) 0.14 nN due to a dc displacement of 8 nm. In addition, the model can explain why higher modes were not detected. The mechanical contribution to the current of the n ) 2 mode is almost 30 times smaller than for the fundamental mode. For the data shown in Figure 3, this would result in a resonance of 5 pA, which is of the same order as the background fluctuations. For higher mode numbers this contribution is even smaller; only the fundamental mode can therefore be resolved in our measurements. In conclusion, we have measured a single gate-dependent resonance due to mechanical motion of suspended nanotubes. By fitting the gate dependence to a continuum model, we have identified the resonance as the fundamental flexural bending mode of the nanotube. With the model, we extracted the fundamental frequency, the gate-induced and residual tension in the nanotube, which are in agreement with their predicted values. The good agreement between experiment and model is a starting point for a further study of the flexural bending mode in suspended carbon nanotubes that includes the nonlinear and quantum regime of operation and of damping mechanisms such as the one associated with the coupling to the clamping points. Acknowledgment. We thank Raymond Schouten for discussions and technical support. Financial support is obtained from the Dutch organization for Fundamental Research on Matter (FOM), by the ‘‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (VICI-grant), and from EC FP6 funding (Contract No. FP6-2004-IST-003673). This publication reflects the views of the authors and not 2907
necessarily those of the EC. The Community is not liable for any use that may be made of the information contained herein. References (1) Cleland, A. N. Foundations of Nanomechanics; Springer-Verlag: Berlin, 2003. (2) Knobel, R. G.; Cleland, A. N. Nature 2003, 424, 291. (3) Krishnan, A.; Dujardin, E.; Ebbesen, T. W.; Yianilos, P. N.; Treacy, M. M. J. Phys. ReV. B 1998, 58, (14013). (4) Reulet, B.; Kasumov, A. Yu.; Kociak, M.; Deblock, R.; Khodos, I. I.; Gorbatov, Yu. B.; Volkov, V. T.; Journet, C.; Bouchiat, H. Phys. ReV. Lett. 2000, 85, 2829. (5) LeRoy, B. J.; Lemay, S. G.; Kong, J.; Dekker, C. Nature 2004, 432, 371. (6) Sapmaz, S.; Jarillo-Herrero, P.; Blanter, Ya. M.; Dekker, C.; van der Zant, H. S. J. Phys. ReV. Lett. 2006, 96, 026801. (7) Babic´, B.; Furer, J.; Sahoo, S.; Farhangfar, S.; Scho¨nenberger, C. Nano Lett. 2003, 3, 1577. (8) Meyer, J. C.; Paillet, M.; Roth, S. Science 2005, 309, 1539. (9) Sazanova, V.; Yaish, Y; U ¨ stu¨nel, H.; Roundy, D.; Arias, T. A.; McEuen, P. L. Nature 2003, 431, 284. (10) Peng, H. B.; Chang, C. W.; Aloni, S.; Yuzvinsky, T. D.; Zettl, A. Phys. ReV. Lett. 2006, 97, 087203.
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(11) Babic´, B.; Iqbal, M.; Scho¨nenberger, C. Nanotechnology 2003, 14, 327. (12) Kong, J.; Soh, H. T.; Cassell, A. M.; Quate, C. F.; Dai, H. Nature 1998, 395, 878. (13) Walters D. A.; et al. Appl. Phys. Lett. 1999, 74, 3803. Nygård, J.; Cobden, D. H. Appl. Phys. Lett. 2001, 79, 4216. (14) Sapmaz, S.; Blanter, Ya. M.; Gurevich, L.; van der Zant, H. S. J. Phys. ReV. B 2003, 67, 235414. (15) The phase difference φ between the gate and source electrode depends on the frequency of the rf signals, on the (gate dependent) conductance of the nanotube, and on its derivative. Note also that the lock-in detected phase does not contain information about φ. (16) Postma, H. W. Ch; Kozinsky, I; Husain, A.; Roukes, M. L. Appl. Phys. Lett. 2005, 86, 223105. (17) Lefe`vre, R.; Goffman, M. F.; Derycke, V.; Miko, C.; Forro´, L.; Bourgoin, J. P.; Hesto, P. Phys. ReV. Lett. 2005, 95, 185504. (18) U ¨ stu¨nel, H; Roundy, D; Arias, T. A. Nano Lett. 2005, 5, 523. (19) We neglect corrections in the tension due to the ac motion of the tube. This means that the model is only valid in the linear regime. (20) The minimum of the resonance frequency of device B was located at Vg ) -1.7 V. This shift may be explained by a nonzero dc potential on the nanotube due to work function differences between the tube and the metallic electrodes. More research is required to confirm this.
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Nano Lett., Vol. 6, No. 12, 2006
