Mechanics and Electromechanics of Two-Dimensional Atomic Membranes

  • Harold S. ParkEmail author
Living reference work entry


Two-dimensional (2D) materials have, over the past decade, attracted significant amounts of research interest due to their exceptional and unique physical properties. Here, two areas of graphene mechanics are overviewed where computational techniques, both existing (classical molecular dynamics) and new (electromechanical coupling techniques), have enabled new insights. First, we discuss the prediction of and insights gained with regard to atomistic simulations of auxetic behavior in 2D materials. Second, new computational techniques are discussed that couple molecular dynamics, tight-binding, and quantum transport to examine how mechanical strain can impact, in interesting and often unanticipated fashion, the electronic properties of graphene kirigami.


  1. Abanin DA, Pesin DA (2012) Interaction-induced topological insulator states in strained graphene. Phys Rev Lett 109:066802ADSCrossRefGoogle Scholar
  2. Abedpour N, Asgari R, Guinea F, Strains and pseudomagnetic fields in circular graphene rings. Phys Rev B 84:(2011)115437Google Scholar
  3. Akinwande D, Brennan CJ, Bunch JS, Egberts P, Felts JR, Gao H, Huang R, Kim J-S, Li T, Li Y, Liechti KM, Lu N, Park HS, Reed EJ, Wang P, Yakobson BI, Zhang T, Zhang Y-W, Zhou Y, Zhu Y (2017) A review on mechanics and mechanical properties of 2D materials – graphene and beyond. Extreme Mech Lett 13:42–72CrossRefGoogle Scholar
  4. Alderson K, Alderson A, Anand S, Simkins V, Nazare S, Ravirala N (2012) Auxetic warp knit textile structures. Phys Status Solidi B 249(7):1322–1329ADSCrossRefGoogle Scholar
  5. Bahamon DA, Qi Z, Park HS, Pereira VM, Campbell DK (2015) Conductance signatures of electron confinement induced by strained nanobubbles in graphene. Nanoscale 7:15300–15309ADSCrossRefGoogle Scholar
  6. Bahamon DA, Qi Z, Park HS, Pereira VM, Campbell DK (2016) Graphene kirigami as a platform for stretchable and tunable quantum dot arrays. Phys Rev B 95:235408ADSCrossRefGoogle Scholar
  7. Bao W, Miao F, Chen Z, Zhang H, Jang W, Dames C, Lau CN (2009) Controlled ripple texturing of suspended graphene and ultrathin graphite membranes. Nat Nanotechnol 4:562–566ADSCrossRefGoogle Scholar
  8. Baughman RH, Galvao DS (1993) Crystalline networks with unusual predicted mechanical and thermal properties. Nature 365:735ADSCrossRefGoogle Scholar
  9. Baughman RH, Shacklette JM, Zakhidov AA, Stafstrom S (1998) Negative poisson’s ratios as a common feature of cubic metals. Nature 392:362–365ADSCrossRefGoogle Scholar
  10. Berger C, Song Z, Li X, Wu X, Brown N, Naud C, Mayou D, Li T, Hass J, Marchenkov AN, Conrad EH, First PN, de Heer WA (2006) Electronic confinement and coherence in patterned epitaxial graphene. Science 312(5777):1191–1196ADSCrossRefGoogle Scholar
  11. Bertoldi K, Reis PM, Willshaw S, Mullin T (2010) Negative poisson’s ratio behavior induced by an elastic instability. Adv Mater 22:361–366CrossRefGoogle Scholar
  12. Blees MK, Barnard AW, Rose PA, Roberts SP, McGill KL, Huang PY, Ruyack AR, Kevek JW, Kobrin B, Muller DA, McEuen PL (2015) Graphene kirigami. Nature 524(7564):204–207ADSCrossRefGoogle Scholar
  13. Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB (2002) A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J Phys Condens Matter 14:783–802ADSCrossRefGoogle Scholar
  14. Brum JA (1991) Electronic properties of quantum-dot superlattices. Phys Rev B 43:12082–12085ADSCrossRefGoogle Scholar
  15. Büttiker M (1986) Four-terminal phase-coherent conductance. Phys Rev Lett 57:1761–1764ADSCrossRefGoogle Scholar
  16. Capasso F, Mohammed K, Cho AY (1986) Resonant tunneling through double barriers, perpendicular quantum transport phenomena in superlattices, and their device applications. IEEE J Quantum Electron 22(9):1853–1869ADSCrossRefGoogle Scholar
  17. Caroli C, Combescot R, Nozieres P, Saint-James D (1971) Direct calculation of the tunneling current. J Phys C Solid State Phys 4(8):916ADSCrossRefGoogle Scholar
  18. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (2009) The electronic properties of graphene. Rev Modern Phys 81:109–162ADSCrossRefGoogle Scholar
  19. Chang T, Gao H (2003) Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solids 51:1059–1074ADSzbMATHCrossRefGoogle Scholar
  20. Chang T, Geng J, Guo X (2005) Chirality- and size-dependent elastic properties of single-walled carbon nanotubes. Appl Phys Lett 87(25):251929ADSCrossRefGoogle Scholar
  21. Chen X, Xiong S, Wang G (1994) Tunneling in quantum-wire superlattices with random layer thicknesses. Phys Rev B 49:14736–14739ADSCrossRefGoogle Scholar
  22. Choi S-M, Jhi S-H, Son Y-W (2010) Effects of strain on electronic properties of graphene. Phys Rev B 81:081407(R)+Google Scholar
  23. Clausen A, Wang F, Jensen JS, Sigmund O, Lewis JA (2015) Topology optimized architectures with programmable Poisson’s ratios over large deformations. Adv Mater 27:5523–5527CrossRefGoogle Scholar
  24. Datta S (1995) Electronic transport in mesoscopic systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. Esaki L, Tsu R (1970) Superlattice and negative differential conductivity in semiconductors. IBM J Res Dev 14(1):61–65CrossRefGoogle Scholar
  26. Evans KE (1991) Auxetic polymers: a new range of materials. Endeavour 15(4):170–174CrossRefGoogle Scholar
  27. Evans KE, Alderson A (2000) Auxetic materials: functional materials and structures from lateral thinking! Adv Mater 12(9):617CrossRefGoogle Scholar
  28. Faria D, Carrillo-Bastos R, Sandler N, Latgé A (2015) Fano resonances in hexagonal zigzag graphene rings under external magnetic flux. J Phys Condens Matter 27(17):175301ADSCrossRefGoogle Scholar
  29. Farjam M, Rafii-Tabar H (2009) Comment on “band structure engineering of graphene by strain: first-principles calculations”. Phys Rev B 80:167401ADSCrossRefGoogle Scholar
  30. Fujita M, Wakabayashi K, Nakada K, Kusakabe K (1996) Peculiar localized state at zigzag graphite edge. J Phys Soc Jpn 65(7):1920–1923ADSCrossRefGoogle Scholar
  31. Gallagher P, Todd K, Gordon DG (2010) Disorder-induced gap behavior in graphene nanoribbons. Phys Rev B 81:115409ADSCrossRefGoogle Scholar
  32. Garza HHP, Kievit EW, Schneider GF, Staufer U (2014) Controlled, reversible, and nondestructive generation of uniaxial extreme strains (> 10%) in graphene. Nano Lett 14(7):4107–4113ADSCrossRefGoogle Scholar
  33. Geim AK, Novoselov KS (2007) The rise of graphene. Nat Mater 6:183–191ADSCrossRefGoogle Scholar
  34. Golizadeh-Mojarad R, Datta S (2007) Nonequilibrium green’s function based models for dephasing in quantum transport. Phys Rev B 75:081301ADSCrossRefGoogle Scholar
  35. Gong L, Kinloch IA, Young RJ, Riaz I, Jalil R, Novoselov KS (2010) Interfacial stress transfer in a graphene monolayer nanocomposite. Adv Matter 22:2694CrossRefGoogle Scholar
  36. González JW, Pacheco M, Rosales L, Orellana PA (2011) Transport properties of graphene quantum dots. Phys Rev B 83:155450ADSCrossRefGoogle Scholar
  37. Grima JN, Winczewski S, Mizzi L, Grech MC, Cauchi R, Gatt R, Attard D, Wojciechowski KW, Rybicki J (2015) Tailoring graphene to achieve negative poisson’s ratio properties. Adv Mater 27:1455–1459CrossRefGoogle Scholar
  38. Guinea F, Low T (2010) Band structure and gaps of triangular graphene superlattices. Philos Trans R Soc Math Phys Eng Sci 368(1932):5391–5402ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. Guinea F, Horovitz B, Le Doussal P (2008) Gauge field induced by ripples in graphene. Phys Rev B 77:205421ADSCrossRefGoogle Scholar
  40. Guinea F, Katsnelson MI, Geim AK (2010a) Energy gaps and a zero-field quantum hall effect in graphene by strain engineering. Nat Phys 6(1):30–33CrossRefGoogle Scholar
  41. Guinea F, Geim AK, Katsnelson MI, Novoselov KS (2010b) Generating quantizing pseudomagnetic fields by bending graphene ribbons. Phys Rev B 81(3):035408ADSCrossRefGoogle Scholar
  42. Han MY, Ozyilmaz B, Zhang Y, Kim P (2007) Energy band-gap engineering of graphene nanoribbons. Phys Rev Lett 98:206805ADSCrossRefGoogle Scholar
  43. Han MY, Brant JC, Kim P (2010) Electron transport in disordered graphene nanoribbons. Phys Rev Lett 104:056801ADSCrossRefGoogle Scholar
  44. Haug H, Jauho A-P (2008) Transport in mesoscopic semiconductor structures. In: Quantum kinetics in transport and optics of semiconductors. Solid-state sciences, vol 123. Springer, Berlin/Heidelberg, pp 181–212Google Scholar
  45. Ho DT, Park S-D, Kwon S-Y, Park K, Kim SY (2014) Negative poisson’s ratios in metal nanoplates. Nat Commun 5:3255CrossRefGoogle Scholar
  46. Ho VH, Ho DT, Kwon S-Y, Kim SY (2016) Negative poisson’s ratio in cubic materials along principal directions. Phys Status Solidi B.
  47. Ho VH, Ho DT, Kwon S-Y, Kim SY (2016) Negative poisson’s ratio in periodic porous graphene structures. Phys Status Solidi B.
  48. Humphrey W, Dalke A, Schulten K (1996) VMD: visual molecular dynamics. J Mol Graph 14(1):33–38CrossRefGoogle Scholar
  49. Ihnatsenka S, Zozoulenko IV, Kirczenow G (2009) Band-gap engineering and ballistic transport in edge-corrugated graphene nanoribbons. Phys Rev B 80:155415ADSCrossRefGoogle Scholar
  50. Jiang J-W, Park HS (2014) Negative poisson’s ratio in single-layer black phosphorus. Nat Commun 5:4727CrossRefGoogle Scholar
  51. Jiang J-W, Park HS (2016) Negative poisson’s ratio in single-layer graphene ribbons. Nano Lett 16:2657–2662ADSCrossRefGoogle Scholar
  52. Jiang J-W, Tang H, Wang B-S, Su Z-B (2008) Raman and infrared properties and layer dependence of the phonon dispersions in multilayered graphene. Phys Rev B 77(23):235421ADSCrossRefGoogle Scholar
  53. Jiang J-W, Kim SY, Park HS (2016) Auxetic nanomaterials: recent progress and future directions. Appl Phys Rev 3:041101CrossRefGoogle Scholar
  54. Jiang J-W, Chang T, Guo X, Park HS (2016) Intrinsic negative poisson’s ratio for single-layer graphene. Nano Lett 16:5286–5290ADSCrossRefGoogle Scholar
  55. Jiang J-W, Park HS (2016) Negative poisson’s ratio in single-layer graphene ribbons. Nano Lett 16:2657–2662ADSCrossRefGoogle Scholar
  56. Jiao L, Zhang L, Wang X, Diankov G, Dai H (2009) Narrow graphene nanoribbons from carbon nanotubes. Nature 458:877ADSCrossRefGoogle Scholar
  57. Ji Z-L, Berggren K-F (1992) Quantum bound states in narrow ballistic channels with intersections. Phys Rev B 45:6652–6658ADSCrossRefGoogle Scholar
  58. Joe YS, Ikeler DS, Cosby RM, Satanin AM, Kim CS (2000) Characteristics of transmission resonance in a quantum-dot superlattice. J Appl Phys 88(5):2704–2708ADSCrossRefGoogle Scholar
  59. Kane CL, Mele EJ (1997) Size, shape, and low energy electronic structure of carbon nanotubes. Phys Rev Lett 78:1932ADSCrossRefGoogle Scholar
  60. Kim K-J, Blanter YM, Ahn K-H (2011) Interplay between real and pseudomagnetic field in graphene with strain. Phys Rev B 84(8):081401ADSCrossRefGoogle Scholar
  61. Kitt AL, Pereira VM, Swan AK, Goldberg BB (2012) Lattice-corrected strain-induced vector potentials in graphene. Phys Rev B 85(11):115432ADSCrossRefGoogle Scholar
  62. Kitt AL, Pereira VM, Swan AK, Goldberg BB (2013) Erratum: lattice-corrected strain-induced vector potentials in graphene. Phys Rev B 87:159909(E); Phys Rev B 85:115432 (2012)Google Scholar
  63. Kouwenhoven LP, Hekking FWJ, van Wees BJ, Harmans CJPM, Timmering CE, Foxon CT (1990) Transport through a finite one-dimensional crystal. Phys Rev Lett 65:361–364ADSCrossRefGoogle Scholar
  64. Lakes RS (1987) Foam structures with a negative poisson’s ratio. Science 235:1038–1040ADSCrossRefGoogle Scholar
  65. Lakes R (1993) Advances in negative poisson’s ratio materials. Adv Mater 5:293–296CrossRefGoogle Scholar
  66. Lee C, Wei X, Kysar JW, Hone J (2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321:385ADSCrossRefGoogle Scholar
  67. Lethbridge ZA, Walton RI, Marmier AS, Smith CW, Evans KE (2010) Elastic anisotropy and extreme poisson’s ratios in single crystals. Acta Mater 58:6444–6451CrossRefGoogle Scholar
  68. Los JH, Fasolino A, Katsnelson MI (2006) Scaling behavior and strain dependence of in-plane elastic properties of graphene. Phys Rev Lett 116:015901ADSCrossRefGoogle Scholar
  69. Milstein F, Huang K (1979) Existence of a negative poisson ratio in FCC crystals. Phys Rev B 19(4):2030ADSCrossRefGoogle Scholar
  70. Mo Y, Turner KT, Szlufarska I (2009) Friction laws at the nanoscale. Nature 457:1116ADSCrossRefGoogle Scholar
  71. Mohiuddin TMG, Lombardo A, Nair RR, Bonetti A, Savini G, Jalil R, Bonini N, Basko DM, Galiotis C, Marzari N, Novoselov KS, Geim AK, Ferrari AC (2009) Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, gruneisen parameters, and sample orientation. Phys Rev B 79(20):205433ADSCrossRefGoogle Scholar
  72. Ni ZH, Yu T, Lu YH, Wang YY, Feng YP, Shen ZX (2008) Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening. ACS Nano 2(11):2301–2305CrossRefGoogle Scholar
  73. Ni ZH, Yu T, Lu YH, Wang YY, Feng YP, Shen ZX (2009) Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening. ACS Nano 3:483CrossRefGoogle Scholar
  74. Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV, Firsov AA (2005) Two-dimensional gas of massless dirac fermions in graphene. Nature 438:197–200ADSCrossRefGoogle Scholar
  75. Pastawski HM, Medina E (2001) “Tight binding” methods in quantum transport through molecules and small devices from the coherent to the decoherent description. Rev Mex de Física 47(S1):1–23Google Scholar
  76. Pereira VM, Castro Neto AH (2009) Strain engineering of graphene’s electronic structure. Phys Rev Lett 103(4):4CrossRefGoogle Scholar
  77. Pereira VM, Castro Neto AH, Peres NMR (2009) Tight-binding approach to uniaxial strain in graphene. Phys Rev B 80:045401ADSCrossRefGoogle Scholar
  78. Pereira VM, Castro Neto AH, Liang HY, Mahadevan L (2010) Geometry, mechanics, and electronics of singular structures and wrinkles in graphene. Phys Rev Lett 105:156603ADSCrossRefGoogle Scholar
  79. Pereira VM, Ribeiro RM, Peres NMR, Castro Neto AH (2010) Optical properties of strained graphene. Eur Phys Lett 92:67001ADSCrossRefGoogle Scholar
  80. Plimpton SJ (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19ADSzbMATHCrossRefGoogle Scholar
  81. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19ADSzbMATHCrossRefGoogle Scholar
  82. Qi Z, Zhao F, Zhou X, Sun Z, Park HS, Wu H (2010) A molecular simulation analysis of producing monatomic carbon chains by stretching ultranarrow graphene nanoribbons. Nanotechnology 21(26):265702ADSCrossRefGoogle Scholar
  83. Qi Z, Bahamon DA, Pereira VM, Park HS, Campbell DK, Castro Neto AH (2013) Resonant tunneling in graphene pseudomagnetic quantum dots. Nano Lett 13:2692ADSCrossRefGoogle Scholar
  84. Qi Z, Kitt AL, Park HS, Pereira VM, Campbell DK, Castro Neto AH (2014) Pseudomagnetic fields in graphene nanobubbles of constrained geometry: a molecular dynamics study. Phys Rev B 90:125419ADSCrossRefGoogle Scholar
  85. Qi Z, Campbell DK, Park HS (2014) Atomistic simulations of tension-induced large deformation and stretchability in graphene kirigami. Phys Rev B 90:245437ADSCrossRefGoogle Scholar
  86. Ravirala N, Alderson A, Alderson KL (2007) Interlocking hexagons model for auxetic behaviour. J Mater Sci 42:7433–7445ADSCrossRefGoogle Scholar
  87. Rothenburg L, Berlint AA, Bathurst RJ (1991) Microstructure of isotropic materials with negative poisson’s ratio. Nature 354:470ADSCrossRefGoogle Scholar
  88. Seol JH, Jo I, Moore AL, Lindsay L, Aitken ZH, Pettes MT, Li X, Yao Z, Huang R, Broido D, Mingo N, Ruoff RS, Shi L (2010) Two-dimensional phonon transport in supported graphene. Science 328:213–216ADSCrossRefGoogle Scholar
  89. Shen L, Li J (2004) Transversely isotropic elastic properties of single-walled carbon nanotubes. Phys Rev B 69:045414ADSCrossRefGoogle Scholar
  90. Shenoy VB, Reddy CD, Ramasubramaniam A, Zhang YW (2008) Edge-stress-induced warping of graphene sheets and nanoribbons. Phys Rev Lett 101(24):245501ADSCrossRefGoogle Scholar
  91. Sols F, Guinea F, Neto AHC (2007) Coulomb blockade in graphene nanoribbons. Phys Rev Lett 99:166803ADSCrossRefGoogle Scholar
  92. Stampfer C, Güttinger J, Hellmüller S, Molitor F, Ensslin K, Ihn T (2009) Energy gaps in etched graphene nanoribbons. Phys Rev Lett 102:056403ADSCrossRefGoogle Scholar
  93. Stuart SJ, Tutein AB, Harrison JA (2000) A reactive potential for hydrocarbons with intermolecular interactions. J Chem Phys 112(14):6472–6486ADSCrossRefGoogle Scholar
  94. Stukowski A (2010) Visualization and analysis of atomistic simulation data with OVITO – the open visualization tool. Model Simul Mater Sci Eng 18:015012ADSCrossRefGoogle Scholar
  95. Suzuura H, Ando T (2002) Phonons and electron-phonon scattering in carbon nanotubes. Phys Rev B 65:235412ADSCrossRefGoogle Scholar
  96. Suzuura H, Ando T (2002) Phonons and electron-phonon scattering in carbon nanotubes. Phys Rev B 65:235412ADSCrossRefGoogle Scholar
  97. Todd K, Chou H-T, Amasha S, Goldhaber-Gordon D (2008) Quantum dot behavior in graphene nanoconstrictions. Nano Lett 9:416ADSCrossRefGoogle Scholar
  98. Tomori H, Kanda A, Goto H, Ootuka Y, Tsukagoshi K, Moriyama S, Watanabe E, Tsuya D (2011) Introducing nonuniform strain to graphene using dielectric nanopillars. Appl Phys Express 4(7):3CrossRefGoogle Scholar
  99. Ulloa SE, Castao E, Kirczenow G (1990) Ballistic transport in a novel one-dimensional superlattice. Phys Rev B 41:12350–12353ADSCrossRefGoogle Scholar
  100. Vozmediano MAH, Katsnelson MI, Guinea F (2010) Gauge fields in graphene. Phys Rep 496:109ADSMathSciNetCrossRefGoogle Scholar
  101. Wang ZF, Zhang Y, Liu F (2011) Formation of hydrogenated graphene nanoripples by strain engineering and directed surface self-assembly. Phys Rev B 83:041403ADSCrossRefGoogle Scholar
  102. Wu Z, Zhang ZZ, Chang K, Peeters FM (2010) Quantum tunneling through graphene nanorings. Nanotechnology 21(18):185201ADSCrossRefGoogle Scholar
  103. Yang HT (2011) Strain induced shift of dirac points and the pseudo-magnetic field in graphene. J Phys Condens Matter 23(50):505502CrossRefGoogle Scholar
  104. Yang W, Li Z-M, Shi W, Xie B-H, Yang M-B (2004) Review on auxetic materials. J Mater Sci 39:3269–3279ADSCrossRefGoogle Scholar
  105. Yao YT, Alderson A, Alderson KL (2008) Can nanotubes display auxetic behaviour? Phys Status Solidi B 245(11):2373–2382ADSCrossRefGoogle Scholar
  106. Yeh NC, Teague ML, Yeom S, Standley BL, Wu RTP, Boyd DA, Bockrath MW (2011) Strain-induced pseudo-magnetic fields and charging effects on CVD-grown graphene. Surf Sci 605(17–18):1649–1656ADSCrossRefGoogle Scholar
  107. Yue K, Gao W, Huang R, Liechti KM (2012) Analytical methods for the mechanics of graphene bubbles. J Appl Phys 112(8):083512ADSCrossRefGoogle Scholar
  108. Zakharchenko KV, Katsnelson MI, Fasolino A (2009) Finite temperature lattice properties of graphene beyond the quasiharmonic approximation. Phys Rev Lett 102(4):046808ADSCrossRefGoogle Scholar
  109. Zhang ZZ, Chang K, Chan KS (2008) Resonant tunneling through double-bended graphene nanoribbons. Appl Phys Lett 93(6):062106. ADSCrossRefGoogle Scholar
  110. Zhao H, Aluru NR (2010) Temperature and strain-rate dependent fracture strength of graphene. J Appl Phys 108(6):064321ADSCrossRefGoogle Scholar
  111. Zhu S, Huang Y, Li T (2014) Extremely compliant and highly stretchable patterned graphene. Appl Phys Lett 104(17):173103ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringBoston UniversityBostonUSA

Section editors and affiliations

  • Ting Zhu
    • 1
  1. 1.Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

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