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GraphITA pp 47-59 | Cite as

Elastic Properties and Electron–Phonon Coupling of Graphene/Metal Interfaces Probed by Phonon Dispersion

  • M. Alfano
  • C. Lamuta
  • G. Chiarello
  • A. PolitanoEmail author
Chapter
Part of the Carbon Nanostructures book series (CARBON)

Abstract

High-resolution electron energy loss spectroscopy is a suitable tool for investigating phonons in graphene, due to its exceptional energy resolution in both the energy and momentum domains. In this chapter, we show that the experimental phonon dispersion of graphene can be used to estimate elastic properties and electron–phonon coupling. Novel coupling mechanisms of Dirac cone electrons in graphene with out-of-plane optical phonons of the graphene lattice, activated only whenever the graphene sheet is supported by a solid substrate, are also discussed.

References

  1. 1.
    Aizawa, T., Souda, R., Ishizawa, Y., Hirano, H., Yamada, T., Tanaka, K.-I., Oshima, C.: Phonon dispersion in monolayer graphite formed on Ni(111) and Ni(001). Surf. Sci. 237, 194–202 (1990)CrossRefGoogle Scholar
  2. 2.
    AL Taleb, A., Yu, H.K., Anemone, G., Farías, D., Wodtke, A.M.: Helium diffraction and acoustic phonons of graphene grown on copper foil. Carbon 95, 731–737 (2015)Google Scholar
  3. 3.
    Allard, A., Wirtz, L.: Graphene on metallic substrates: suppression of the Kohn anomalies in the phonon dispersion. Nano Lett. 10, 4335–4340 (2010)CrossRefGoogle Scholar
  4. 4.
    Alzari, V., Nuvoli, D., Sanna, R., Scognamillo, S., Piccinini, M., Kenny, J.M., Malucelli, G., Mariani, A.: In situ production of high filler content graphene-based polymer nanocomposites by reactive processing. J. Mater. Chem. 21, 16544–16549 (2011)CrossRefGoogle Scholar
  5. 5.
    Arghavan, S., Singh, A.V.: Free vibration of single layer graphene sheets: lattice structure versus continuum plate theories. J. Nanotechnol. Eng. Med. 2, 031005–031006 (2011)CrossRefGoogle Scholar
  6. 6.
    Arroyo, M., Belytschko, T.: Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys. Rev. B 69, 115415 (2004)CrossRefGoogle Scholar
  7. 7.
    Aynajian, P., Keller, T., Boeri, L., Shapiro, S.M., Habicht, K., Keimer, B.: Energy gaps and kohn anomalies in elemental superconductors. Science 319, 1509–1512 (2008)CrossRefGoogle Scholar
  8. 8.
    Basko, D.M., Piscanec, S., Ferrari, A.C.: Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene. Phys. Rev. B 80, 165413 (2009)CrossRefGoogle Scholar
  9. 9.
    Batzill, M.: The surface science of graphene: Metal interfaces, CVD synthesis, nanoribbons, chemical modifications, and defects. Surf. Sci. Rep. 67, 83–115 (2012)CrossRefGoogle Scholar
  10. 10.
    Benedek, G., Hulpke, E., Steinhogl, W.: Probing the magnetic forces in fcc-Fe(001) films by means of surface phonon spectroscopy. Phys. Rev. Lett. 87, 027201 (2001)CrossRefGoogle Scholar
  11. 11.
    Bera, S., Arnold, A., Evers, F., Narayanan, R., Wölfle, P.: Elastic properties of graphene flakes: boundary effects and lattice vibrations. Phys. Rev. B 82, 195445 (2010)CrossRefGoogle Scholar
  12. 12.
    Blakslee, O.L., Proctor, D.G., Seldin, E.J., Spence, G.B., Weng, T.: Elastic constants of compression-annealed pyrolytic graphite. J. Appl. Phys. 41, 3373–3382 (1970)CrossRefGoogle Scholar
  13. 13.
    Bosak, A., Krisch, M., Mohr, M., Maultzsch, J., Thomsen, C.: Elasticity of single-crystalline graphite: inelastic x-ray scattering study. Phys. Rev. B 75, 153408 (2007)CrossRefGoogle Scholar
  14. 14.
    Cadelano, E., Palla, P.L., Giordano, S., Colombo, L.: Nonlinear elasticity of monolayer graphene. Phys. Rev. Lett. 102, 235502 (2009)CrossRefGoogle Scholar
  15. 15.
    Cadelano, E., Palla, P.L., Giordano, S., Colombo, L.: Elastic properties of hydrogenated graphene. Phys. Rev. B 82, 235414 (2010)CrossRefGoogle Scholar
  16. 16.
    Cao, A., Yuan, Y.: Atomistic study on the strength of symmetric tilt grain boundaries in graphene. Appl. Phys. Lett. 100, 211912 (2012)CrossRefGoogle Scholar
  17. 17.
    Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., GEIM, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)Google Scholar
  18. 18.
    Caudal, N., Saitta, A.M., Lazzeri, M., Mauri, F.: Kohn anomalies and nonadiabaticity in doped carbon nanotubes. Phys. Rev. B 75, 115423 (2007)CrossRefGoogle Scholar
  19. 19.
    Chis, V., Benedek, G.: Phonon-Induced surface charge density oscillations in quantum wells: a first-principles study of the (2 × 2)-K overlayer on Be(0001). J. Phys. Chem. A 115, 7242–7248 (2011)CrossRefGoogle Scholar
  20. 20.
    Clark, N., Oikonomou, A., Vijayaraghavan, A.: Ultrafast quantitative nanomechanical mapping of suspended graphene. Physica Status Solidi (b) 250, 2672–2677 (2013)Google Scholar
  21. 21.
    de Juan, F., Fertig, H.A.: Power-law Kohn anomaly in undoped graphene induced by Coulomb interactions. Phys. Rev. B 85, 085441 (2012)CrossRefGoogle Scholar
  22. 22.
    de Juan, F., Fertig, H.A.: Power law kohn anomalies and the excitonic transition in graphene. Solid State Commun. 152, 1460–1468 (2012)CrossRefGoogle Scholar
  23. 23.
    de Juan, F., Politano, A., Chiarello, G., Fertig, H.A.: Symmetries and selection rules in the measurement of the phonon spectrum of graphene and related materials. Carbon 85, 225–232 (2015)CrossRefGoogle Scholar
  24. 24.
    Dedkov, Y.S., Fonin, M., Laubschat, C.: A possible source of spin-polarized electrons: the inert graphene/Ni(111) system. Appl. Phys. Lett. 92, 052506 (2008)CrossRefGoogle Scholar
  25. 25.
    Elias, D.C., Gorbachev, R.V., Mayorov, A.S., Morozov, S.V., Zhukov, A.A., Blake, P., Ponomarenko, L.A., Grigorieva, I.V., Novoselov, K.S., Guinea, F., Geim, A.K.: Dirac cones reshaped by interaction effects in suspended graphene. Nat. Phys. 7, 701–704 (2011)CrossRefGoogle Scholar
  26. 26.
    Endlich, M., Molina-Sánchez, A., Wirtz, L., Kröger, J.: Screening of electron-phonon coupling in graphene on Ir(111). Phys. Rev. B 88, 205403 (2013)CrossRefGoogle Scholar
  27. 27.
    Eom, D., Prezzi, D., Rim, K.T., Zhou, H., Lefenfeld, M., Xiao, S., Nuckolls, C., Hybertsen, M.S., Heinz, T.F., Flynn, G.W.: Structure and electronic properties of graphene nanoislands on Co(0001). Nano Lett. 9, 2844–2848 (2009)CrossRefGoogle Scholar
  28. 28.
    Fair, K.M., Arnold, M.D., Ford, M.J.: Determination of the elastic properties of graphene by indentation and the validity of classical models of indentation. J. Phys.: Condens. Matter 26, 015307 (2014)Google Scholar
  29. 29.
    Gamo, Y., Nagashima, A., Wakabayashi, M., Terai, M., Oshima, C.: Atomic structure of monolayer graphite formed on Ni(111). Surf. Sci. 374, 61–64 (1997)CrossRefGoogle Scholar
  30. 30.
    Giovannetti, G., Khomyakov, P.A., Brocks, G., Karpan, V.M., van den Brink, J., Kelly, P.J.: Doping graphene with metal contacts. Phys. Rev. Lett. 101, 026803 (2008)CrossRefGoogle Scholar
  31. 31.
    Gui, G., Li, J., Zhong, J.: Band structure engineering of graphene by strain: first-principles calculations. Phys. Rev. B 78, 075435 (2008)CrossRefGoogle Scholar
  32. 32.
    Hwang, E.H., Das Sarma, S.: Screening, Kohn Anomaly, Friedel Oscillation, and RKKY interaction in bilayer graphene. Phys. Rev. Lett. 101, 156802 (2008)Google Scholar
  33. 33.
    Jhon, Y.I., Zhu, S.-E., Ahn, J.-H., Jhon, M.S.: The mechanical responses of tilted and non-tilted grain boundaries in graphene. Carbon 50, 3708–3716 (2012)CrossRefGoogle Scholar
  34. 34.
    Jiang, J.-W., Wang, J.-S., Li, B.: Young’s modulus of graphene: a molecular dynamics study. Phys. Rev. B 80, 113405 (2009)CrossRefGoogle Scholar
  35. 35.
    Kalosakas, G., Lathiotakis, N.N., Galiotis, C., Papagelis, K.: In-plane force fields and elastic properties of graphene. J. Appl. Phys. 113, 134307 (2013)CrossRefGoogle Scholar
  36. 36.
    Kam, K., Scarpa, F., Adhikari, S. & Chowdhury, R.: Graphene nanofilm as pressure and force sensor: a mechanical analysis. Physica Status Solidi (b) 250, 2085–2089 (2013)Google Scholar
  37. 37.
    Kilian, O., Allan, G., Wirtz, L.: Near Kohn anomalies in the phonon dispersion relations of lead chalcogenides. Phys. Rev. B 80, 245208 (2009)CrossRefGoogle Scholar
  38. 38.
    Kohn, W.: Image of the Fermi surface in the vibration spectrum of a metal. Phys. Rev. Lett. 2, 393–394 (1959)CrossRefGoogle Scholar
  39. 39.
    Kudin, K.N., Scuseria, G.E., Yakobson, B.I.: C2F, BN, and C nanoshell elasticity from ab initio computations. Phys. Rev. B 64, 235406 (2001)CrossRefGoogle Scholar
  40. 40.
    Kwon, S.Y., Ciobanu, C.V., Petrova, V., Shenoy, V.B., Bareño, J., Gambin, V., Petrov, I., Kodambaka, S.: Growth of semiconducting graphene on palladium. Nano Lett. 9, 3985–3990 (2009)CrossRefGoogle Scholar
  41. 41.
    Lazzeri, M., Mauri, F.: Nonadiabatic Kohn anomaly in a doped graphene monolayer. Phys. Rev. Lett. 97, 266407 (2006)CrossRefGoogle Scholar
  42. 42.
    Lazzeri, M., Piscanec, S., Mauri, F., Ferrari, A.C., Robertson, J.: Electron transport and hot phonons in Carbon nanotubes. Phys. Rev. Lett. 95, 236802 (2005)CrossRefGoogle Scholar
  43. 43.
    Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008)CrossRefGoogle Scholar
  44. 44.
    Lee, G.-H., Cooper, R.C., An, S.J., Lee, S., van der Zande, A., Petrone, N., Hammerberg, A.G., Lee, C., Crawford, B., Oliver, W., Kysar, J.W., Hone, J.: High-Strength chemical-vapor–deposited graphene and grain boundaries. Science 340, 1073–1076 (2013)CrossRefGoogle Scholar
  45. 45.
    Lin, Z., Zhigilei, L.V., Celli, V.: Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium. Phys. Rev. B 77, 075133 (2008)CrossRefGoogle Scholar
  46. 46.
    Liu, F., Ming, P., Li, J.: Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys. Rev. B 76, 064120 (2007)CrossRefGoogle Scholar
  47. 47.
    Lu, J.P.: Elastic properties of Carbon nanotubes and nanoropes. Phys. Rev. Lett. 79, 1297 (1997)CrossRefGoogle Scholar
  48. 48.
    Michel, K.H., Verberck, B:. Theory of the elastic constants of graphite and graphene. Physica Status Solidi (b) 245, 2177–2180 (2008)Google Scholar
  49. 49.
    Milošević, I., Kepčija, N., Dobardžić, E., Damnjanović, M., Mohr, M., Maultzsch, J., Thomsen, C.: Kohn anomaly in graphene. Mater. Sci. Eng B-Adv. Funct. Solid-State Mater. 176, 510–511 (2011)CrossRefGoogle Scholar
  50. 50.
    Milošević, I., Kepčija, N., Dobardžić, E., Mohr, M., Maultzsch, J., Thomsen, C., Damnjanović, M.: Symmetry based analysis of the Kohn anomaly and electron-phonon interaction in graphene and carbon nanotubes. Phys. Rev. B 81, 233410 (2010)CrossRefGoogle Scholar
  51. 51.
    Miniussi, E., Pozzo, M., Baraldi, A., Vesselli, E., Zhan, R.R., Comelli, G., Menteş, T.O., Niño, M.A., Locatelli, A., Lizzit, S., Alfè, D.: Thermal stability of corrugated epitaxial graphene grown on Re(0001). Phys. Rev. Lett. 106, 216101 (2011)CrossRefGoogle Scholar
  52. 52.
    Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V., Firsov, A.A.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005)CrossRefGoogle Scholar
  53. 53.
    Oshima, C., Aizawa, T., Souda, R., Ishizawa, Y., Sumiyoshi, Y.: Surface phonon dispersion curves of graphite (0001) over the entire energy region. Solid State Commun. 65, 1601–1604 (1988)CrossRefGoogle Scholar
  54. 54.
    Pan, Y., Zhang, H., Shi, D., Sun, J., Du, S., Liu, F., Gao, H.-J.: Highly ordered, millimeter-scale, continuous, single-crystalline graphene monolayer formed on Ru (0001). Adv. Mater. 21, 2777–2780 (2009)CrossRefGoogle Scholar
  55. 55.
    Park, S., An, J., Suk, J.W., Ruoff, R.S.: Graphene-based actuators. Small 6, 210–212 (2010)CrossRefGoogle Scholar
  56. 56.
    Peng, Q., Liang, C., Ji, W., De, S.: A theoretical analysis of the effect of the hydrogenation of graphene to graphane on its mechanical properties. Phys. Chem. Chem. Phys. 15, 2003–2011 (2013)CrossRefGoogle Scholar
  57. 57.
    Perebeinos, V., Tersoff, J.: Valence force model for phonons in graphene and carbon nanotubes. Phys. Rev. B 79, 241409 (2009)CrossRefGoogle Scholar
  58. 58.
    Piscanec, S., Lazzeri, M., Mauri, F., Ferrari, A.C., Robertson, J.: Kohn Anomalies and electron-phonon interactions in graphite. Phys. Rev. Lett. 93, 185503 (2004)CrossRefGoogle Scholar
  59. 59.
    Piscanec, S., Lazzeri, M., Robertson, J., Ferrari, A.C., Mauri, F.: Optical phonons in carbon nanotubes: Kohn anomalies, Peierls distortions, and dynamic effects. Phys. Rev. B 75, 035427 (2007)CrossRefGoogle Scholar
  60. 60.
    Pletikosić, I., Kralj, M., Pervan, P., Brako, R., Coraux, J., N’Diaye, A.T., Busse, C., Michely, T.: Dirac cones and minigaps for graphene on Ir(111). Phys. Rev. Lett. 102, 056808 (2009)CrossRefGoogle Scholar
  61. 61.
    Politano, A., Borca, B., Minniti, M., Hinarejos, J.J., Vázquez De Parga, A.L., Farías, D., Miranda, R.: Helium reflectivity and Debye temperature of graphene grown epitaxially on Ru(0001). Phys. Rev. B 84, 035450 (2011a)Google Scholar
  62. 62.
    Politano, A., Chiarello, G.: Probing Young’s modulus and Poisson’s ratio in graphene/metal interfaces and graphite: a comparative study. Nano Res. 8, 1847–1856 (2015)CrossRefGoogle Scholar
  63. 63.
    Politano, A., de Juan, F., Chiarello, G., Fertig, H.A.: Emergence of an out-of-plane optical phonon (ZO) Kohn anomaly in quasifreestanding epitaxial graphene. Phys. Rev. Lett. 115, 075504 (2015)CrossRefGoogle Scholar
  64. 64.
    Politano, A., Marino, A.R., Campi, D., Farías, D., Miranda, R., Chiarello, G.: Elastic properties of a macroscopic graphene sample from phonon dispersion measurements. Carbon 50, 4903–4910 (2012)CrossRefGoogle Scholar
  65. 65.
    Politano, A., Marino, A.R., Formoso, V., Chiarello, G.: Evidence of Kohn anomalies in quasi-freestanding graphene on Pt(111). Carbon 50, 734–736 (2012)CrossRefGoogle Scholar
  66. 66.
    Politano, A., Marino, A.R., Formoso, V., Farías, D., Miranda, R., Chiarello, G.: Evidence for acoustic-like plasmons on epitaxial graphene on Pt(111). Phys. Rev. B 84, 033401 (2011)CrossRefGoogle Scholar
  67. 67.
    Reddy, C.D., Ramasubramaniam, A., Shenoy, V.B., Zhang, Y.-W.: Edge elastic properties of defect-free single-layer graphene sheets. Appl. Phys. Lett. 94, 101904 (2009)CrossRefGoogle Scholar
  68. 68.
    Reddy, K.M., Gledhill, A.D., Chen, C.-H., Drexler, J.M., Padture, N.P.: High quality, transferrable graphene grown on single crystal Cu(111) thin films on basal-plane sapphire. Appl. Phys. Lett. 98, 113117 (2011)CrossRefGoogle Scholar
  69. 69.
    Reynaud, C., Sommer, F., Quet, C., el Bounia, N., Duc, T.M.: Quantitative determination of Young’s modulus on a biphase polymer system using atomic force microscopy. Surf. Interface Anal. 30, 185–189 (2000)CrossRefGoogle Scholar
  70. 70.
    Ruiz-Vargas, C.S., Zhuang, H.L., Huang, P.Y., van der Zande, A.M., Garg, S., McEuen, P.L., Muller, D.A., Hennig, R.G., Park, J.: Softened elastic response and unzipping in chemical vapor deposition graphene membranes. Nano Lett. 11, 2259–2263 (2011)CrossRefGoogle Scholar
  71. 71.
    Sánchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A., Ordejón, P.: Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B 59, 12678 (1999)CrossRefGoogle Scholar
  72. 72.
    Sasaki, K., Yamamoto, M., Murakami, S., Saito, R., Dresselhaus, M.S., Takai, K., Mori, T., Enoki, T., Wakabayashi, K.: Kohn anomalies in graphene nanoribbons. Phys. Rev. B 80, 155450 (2009)CrossRefGoogle Scholar
  73. 73.
    Scarpa, F., Adhikari, S., Srikantha Phani, A.: Effective elastic mechanical properties of single layer graphene sheets. Nanotechnology 20, 065709 (2009)Google Scholar
  74. 74.
    Scharfenberg, S., Rocklin, D.Z., Chialvo, C., Weaver, R.L., Goldbart, P.M., Mason, N.: Probing the mechanical properties of graphene using a corrugated elastic substrate. Appl. Phys. Lett. 98, 091908 (2011)CrossRefGoogle Scholar
  75. 75.
    Seldin, E.J., Nezbeda, C.W.: Elastic constants and electron-microscope observations of neutron-irradiated compression-annealed pyrolytic and single-crystal graphite. J. Appl. Phys. 41, 3389–3400 (1970)CrossRefGoogle Scholar
  76. 76.
    Shikin, A.M., Farías, D., Adamchuk, V.K., Rieder, K.H.: Surface phonon dispersion of a graphite monolayer adsorbed on Ni(111) and its modification caused by intercalation of Yb, La and Cu layers. Surf. Sci. 424, 155–167 (1999)CrossRefGoogle Scholar
  77. 77.
    Sicot, M., Bouvron, S., Zander, O., Rudiger, U., Dedkov, Y.S., Fonin, M.: Nucleation and growth of nickel nanoclusters on graphene Moiré on Rh(111). Appl. Phys. Lett. 96, 3–093115 (2010)CrossRefGoogle Scholar
  78. 78.
    Sutter, P., Sadowski, J.T., Sutter, E.: Graphene on Pt(111): growth and substrate interaction. Phys. Rev. B 80, 245411 (2009)CrossRefGoogle Scholar
  79. 79.
    Talwar, D.N., Vandevyver, M., Kunc, K., Zigone, M.: Lattice dynamics of zinc chalcogenides under compression: phonon dispersion, mode Grüneisen, and thermal expansion. Phys. Rev. B 24, 741–753 (1981)CrossRefGoogle Scholar
  80. 80.
    Tomasetti, E., Legras, R., Nysten, B.: Quantitative approach towards the measurement of polypropylene/(ethylene-propylene) copolymer blends surface elastic properties by AFM. Nanotechnology 9, 305 (1998)CrossRefGoogle Scholar
  81. 81.
    TSE, W.K., Hu, B.Y.K., Das Sarma, S.: Chirality-induced dynamic Kohn anomalies in graphene. Phys. Rev. Lett. 101, 066401 (2008)Google Scholar
  82. 82.
    Wagner, P., Ivanovskaya, V.V., Rayson, M.J., Briddon, P.R., Ewels, C.P.: Mechanical properties of nanosheets and nanotubes investigated using a new geometry independent volume definition. J. Phys.: Condens. Matter 25, 155302 (2013)Google Scholar
  83. 83.
    Wang, B., Günther, S., Wintterlin, J., Bocquet, M.-L.: Periodicity, work function and reactivity of graphene on Ru(0001) from first principles. New J. Phys. 12, 043041 (2010)CrossRefGoogle Scholar
  84. 84.
    Wang, C.G., Lan, L., Liu, Y.P., Tan, H.F.: Multiple component correlation model for elastic modulus of single layer graphene sheets. Physica E 56, 372–376 (2014)CrossRefGoogle Scholar
  85. 85.
    Wang, R., Wang, S., Wu, X., Liang, X.: First-principles calculations on third-order elastic constants and internal relaxation for monolayer graphene. Physica B 405, 3501–3506 (2010)CrossRefGoogle Scholar
  86. 86.
    Wang, Y., Yang, R., Shi, Z., Zhang, L., Shi, D., Wang, E., Zhang, G.: Super-elastic graphene ripples for flexible strain sensors. ACS Nano 5, 3645–3650 (2011)CrossRefGoogle Scholar
  87. 87.
    Wintterlin, J., Bocquet, M.L.: Graphene on metal surfaces. Surf. Sci. 603, 1841–1852 (2009)CrossRefGoogle Scholar
  88. 88.
    XIAO, J.R., Staniszewski, J., Gillespie Jr, J.W.: Fracture and progressive failure of defective graphene sheets and carbon nanotubes. Compos. Struct. 88, 602–609 (2009)Google Scholar
  89. 89.
    Zakharchenko, K.V., Katsnelson, M.I., Fasolino, A.: Finite temperature lattice properties of graphene beyond the quasiharmonic approximation. Phys. Rev. Lett. 102, 046808 (2009)CrossRefGoogle Scholar
  90. 90.
    Zhang, Y.Y., Wang, C.M., Cheng, Y., Xiang, Y.: Mechanical properties of bilayer graphene sheets coupled by sp3 bonding. Carbon 49, 4511–4517 (2011)CrossRefGoogle Scholar
  91. 91.
    Zhou, G., Duan, W., Gu, B.: First-principles study on morphology and mechanical properties of single-walled carbon nanotube. Chem. Phys. Lett. 333, 344–349 (2001)CrossRefGoogle Scholar
  92. 92.
    Zhou, J., Huang, R.: Internal lattice relaxation of single-layer graphene under in-plane deformation. J. Mech. Phys. Solids 56, 1609–1623 (2008)CrossRefGoogle Scholar
  93. 93.
    Zhou, L., Wang, Y., Cao, G.: Elastic properties of monolayer graphene with different chiralities. J. Phys.: Condens. Matter 25, 125302 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • M. Alfano
    • 1
  • C. Lamuta
    • 2
  • G. Chiarello
    • 1
  • A. Politano
    • 1
    • 3
    Email author
  1. 1.Department of PhysicsUniversity of CalabriaRendeItaly
  2. 2.Department of MechanicalEnergy and Management EngineeringUniversity of CalabriaRendeItaly
  3. 3.Fondazione Istituto Italiano di Tecnologia, Graphene LabsGenoaItaly

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