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Research Papers

Interior and Edge Elastic Waves in Graphene

[+] Author and Article Information
Y. W. Zhang

e-mail: zhangyw@ihpc.a-star.edu.sg
Institute of High Performance Computing,
Singapore 138632

H. J. Gao

School of Engineering,
Brown University,
Providence, RI 02912

1Corresponding author.

Manuscript received October 4, 2012; final manuscript received November 12, 2012; accepted manuscript posted May 31, 2013; published online May 31, 2013. Editor: Yonggang Huang.

J. Appl. Mech 80(4), 040901 (May 31, 2013) (5 pages) Paper No: JAM-12-1464; doi: 10.1115/1.4024166 History: Received October 04, 2012; Revised November 12, 2012; Accepted April 08, 2013

Elastic waves propagating in graphene nanoribbons were studied using both continuum modeling and molecular dynamics simulations. The Mindlin's plate model was employed to model the propagation of interior waves of graphene, and a continuum beam model was proposed to model the propagation of edge waves in graphene. The molecular dynamics results demonstrated that the interior longitudinal and transverse wave speeds of graphene are about 18,450 m/s and 5640 m/s, respectively, in good agreement with the Mindlin's plate model. The molecular dynamics simulations also revealed the existence of elastic edge waves, which may be described by the proposed continuum beam model.

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References

Rayleigh, J. W. S., 1945, The Theory of Sound, Vol. 1 & 2, 2nd ed., Dover, New York.
Graff, K. F., 1975, Wave Motion in Elastic Solids, Dover, New York.
Wang, Q., 2005, “Wave Propagation in Carbon Nanotubes Via Nonlocal Continuum Mechanics,” J. Appl. Phys., 98, p. 124301. [CrossRef]
Wang, L., and Hu, H., 2005, “Flexural Wave Propagation in Single-Walled Carbon Nanotubes,” Phys. Rev. B, 71, p. 195412. [CrossRef]
Liu, P., Gao, H. J., and Zhang, Y. W., 2008, “Spontaneous Generation and Propagation of Transverse Coaxial Traveling Waves in Multiwalled Carbon Nanotubes,” Appl. Phys. Lett., 93, p. 013106. [CrossRef]
Hu, Y. G., Liew, K. M., Wang, Q., He, X. Q., and Yakobson, B. I., 2008, “Nonlocal Shell Model for Elastic Wave Propagation in Single- and Double-Walled Carbon Nanotubes,” J. Mech. Phys. Solids, 56, pp. 3475–3485. [CrossRef]
Arroyo, M., and Belytschko, T., 2004, “Finite Crystal Elasticity of Carbon Nanotubes Based on the Exponential Cauchy-Born Rule,” Phys. Rev. B, 69(11), p. 115415. [CrossRef]
Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I. V., and Firsov, A. A., 2004, “Electric Field Effect in Atomically Thin Carbon Films,” Science, 306(5696), pp. 666–669. [CrossRef] [PubMed]
Geim, A. K., and Novoselov, K. S., 2007, “The Rise of Graphene,” Nature Mater., 6(3), pp. 183–191. [CrossRef]
Thalmeier, P., Dora, B., and Ziegler, K., 2010, “Surface Acoustic Wave Propagation in Graphene,” Phys. Rev. B, 81, p. 041409. [CrossRef]
Liu, P., and Zhang, Y. W., 2009, “Temperature-Dependent Bending Rigidity of Graphene,” Appl. Phys. Lett., 94, p. 231912. [CrossRef]
Kim, S. Y., and Park, H. S., 2011, “On the Effective Plate Thickness of Monolayer Graphene From Flexural Wave Propagation,” J. Appl. Phys., 110(5), p. 054324. [CrossRef]
Cai, J. M., Ruffieux, P., Jaafar, R., Bieri, M., Braun, T., Blankenburg, S., Muoth, M., Seitsonen, A. P., Saleh, M., Feng, X. L., Mullen, K., and Fasel, R., 2010, “Atomically Precise Bottom-Up Fabrication of Graphene Nanoribbons,” Nature, 466(7305), pp. 470–473. [CrossRef] [PubMed]
Shenoy, V. B., Reddy, C. D., Ramasubramaniam, A., and Zhang, Y. W., 2008, “Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons,” Phys. Rev. Lett., 101(24), p. 245501. [CrossRef] [PubMed]
Son, Y. W., Cohen, M. L., and Louie, S. G., 2006, “Half-Metallic Graphene Nanoribbons,” Nature, 444(7117), pp. 347–349. [CrossRef] [PubMed]
Ritter, K. A., and Lyding, J. W., 2009, “The Influence of Edge Structure on the Electronic Properties of Graphene Quantum Dots and Nanoribbons,” Nature Mater., 8(3), pp. 235–242. [CrossRef]
Scarpa, F., Chowdhury, R., Kam, K., Adhikari, S., and Ruzzene, M., 2011, “Dynamics of Mechanical Waves in Periodic Grapheme Nanoribbon Assemblies,” Nanoscale Res. Lett., 6, p. 430. [CrossRef] [PubMed]
Shenoy, V. B., Reddy, C. D., and Zhang, Y. W., 2010, “Spontaneous Curving of Graphene Sheets With Reconstructed Edges,” ACS Nano, 4(8), pp. 4840–4844. [CrossRef] [PubMed]
Reddy, C. D., Ramasubramaniam, A., Shenoy, V. B., and Zhang, Y. W., 2009, “Edge Elastic Properties of Defect-Free Single-Layer Graphene Sheets,” Appl. Phys. Lett., 94(10), p. 101904. [CrossRef]
Konenkov, Y. K., 1960, “A Rayleigh-Type Flexural Wave,” Sov. Phys. Acoust., 6, pp. 122–123.
Kauffmann, C., 1998, “A New Bending Wave Solution for the Classical Plate Equation,” J. Acoust. Soc. Am., 104(4), pp. 2220–2222. [CrossRef]
Mindlin, R. D., 1951, “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” ASME J. Appl. Mech., 18(1), pp. 31–38.
Yakobson, B. I., Brabec, C. J., and Bernholc, J., 1996, “Nanomechanics of Carbon Tubes: Instabilities Beyond Linear Response,” Phys. Rev. Lett., 76(14), pp. 2511–2514. [CrossRef] [PubMed]
Timoshenko, S. P., 1921, “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars,” Philos. Mag., 41(245), pp. 744–746. [CrossRef]
Brenner, D. W., Shenderova, O. A., Harrison, J. A., Stuart, S. J., Ni, B., and Sinnott, S. B., 2002, “A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons,” J. Phys. Condens. Matter, 14(4), pp. 783– 802. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Transverse elastic waves traveling along graphene nanoribbons with zigzag edges and different widths at t = 12.18 ps. The right arrows indicate the front of a train of interior transverse waves while the left arrows indicate the front of a train of edge transverse waves. (b) The interior transverse wave speed VT-I versus the graphene nanoribbon width W. (c) The edge transverse wave speed VT-E versus the graphene nanoribbon width W. The left ends of the graphene nanoribbons are subjected to forced vibrations with frequency of 2.27 THz and amplitude of 0.05 nm. The vibration amplitude has been amplified 20 times in the plots.

Grahic Jump Location
Fig. 2

(a) Transverse wave traveling along graphene nanoribbons with armchair edges and different widths at t = 12.18 ps. The right arrows indicate the front of a train of interior transverse waves while the left arrows indicate the front of a train of edge transverse waves. (b) The interior transverse wave speed VT-I versus the graphene nanoribbon width W. (c) The edge transverse wave speed VT-E versus the graphene nanoribbon width W. The left ends of the graphene nanoribbons are subjected to forced vibrations with frequency of 2.27 THz and amplitude of 0.05 nm. The vibration amplitude has been amplified 20 times in the plots.

Grahic Jump Location
Fig. 3

Transverse waves traveling at different vibration frequencies at t = 12.18 ps. (a) The zigzag edged graphene nanoribbon with a width of 22.86 nm. (b) The armchair edged graphene nanoribbon with a width of 22.13 nm. In these simulations, the left ends of the graphene nanoribbons are subjected to forced vibrations with amplitude of 0.05 nm. The vibration amplitude has been amplified 20 times in these plots.

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