Skip to main content
SpringerLink
Log in

Mechanics and Electromechanics of Two-Dimensional Atomic Membranes

  • Living reference work entry
  • First Online:
Handbook of Materials Modeling

  • 243 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Abanin DA, Pesin DA (2012) Interaction-induced topological insulator states in strained graphene. Phys Rev Lett 109:066802

    Article  ADS  Google Scholar 

  • Abedpour N, Asgari R, Guinea F, Strains and pseudomagnetic fields in circular graphene rings. Phys Rev B 84:(2011)115437

    Google Scholar 

  • 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 13:42–72

    Article  Google Scholar 

  • Alderson K, Alderson A, Anand S, Simkins V, Nazare S, Ravirala N (2012) Auxetic warp knit textile structures. Phys Status Solidi B 249(7):1322–1329

    Article  ADS  Google Scholar 

  • Bahamon DA, Qi Z, Park HS, Pereira VM, Campbell DK (2015) Conductance signatures of electron confinement induced by strained nanobubbles in graphene. Nanoscale 7:15300–15309

    Article  ADS  Google Scholar 

  • 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 95:235408

    Article  ADS  Google Scholar 

  • 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 4:562–566

    Article  ADS  Google Scholar 

  • Baughman RH, Galvao DS (1993) Crystalline networks with unusual predicted mechanical and thermal properties. Nature 365:735

    Article  ADS  Google Scholar 

  • Baughman RH, Shacklette JM, Zakhidov AA, Stafstrom S (1998) Negative poisson’s ratios as a common feature of cubic metals. Nature 392:362–365

    Article  ADS  Google Scholar 

  • 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):1191–1196

    Article  ADS  Google Scholar 

  • Bertoldi K, Reis PM, Willshaw S, Mullin T (2010) Negative poisson’s ratio behavior induced by an elastic instability. Adv Mater 22:361–366

    Article  Google Scholar 

  • 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):204–207

    Article  ADS  Google Scholar 

  • 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 14:783–802

    Article  ADS  Google Scholar 

  • Brum JA (1991) Electronic properties of quantum-dot superlattices. Phys Rev B 43:12082–12085

    Article  ADS  Google Scholar 

  • Büttiker M (1986) Four-terminal phase-coherent conductance. Phys Rev Lett 57:1761–1764

    Article  ADS  Google Scholar 

  • 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):1853–1869

    Article  ADS  Google Scholar 

  • Caroli C, Combescot R, Nozieres P, Saint-James D (1971) Direct calculation of the tunneling current. J Phys C Solid State Phys 4(8):916

    Article  ADS  Google Scholar 

  • Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (2009) The electronic properties of graphene. Rev Modern Phys 81:109–162

    Article  ADS  Google Scholar 

  • Chang T, Gao H (2003) Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solids 51:1059–1074

    Article  ADS  MATH  Google Scholar 

  • Chang T, Geng J, Guo X (2005) Chirality- and size-dependent elastic properties of single-walled carbon nanotubes. Appl Phys Lett 87(25):251929

    Article  ADS  Google Scholar 

  • Chen X, Xiong S, Wang G (1994) Tunneling in quantum-wire superlattices with random layer thicknesses. Phys Rev B 49:14736–14739

    Article  ADS  Google Scholar 

  • 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 

  • Clausen A, Wang F, Jensen JS, Sigmund O, Lewis JA (2015) Topology optimized architectures with programmable Poisson’s ratios over large deformations. Adv Mater 27:5523–5527

    Article  Google Scholar 

  • Datta S (1995) Electronic transport in mesoscopic systems. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Esaki L, Tsu R (1970) Superlattice and negative differential conductivity in semiconductors. IBM J Res Dev 14(1):61–65

    Article  Google Scholar 

  • Evans KE (1991) Auxetic polymers: a new range of materials. Endeavour 15(4):170–174

    Article  Google Scholar 

  • Evans KE, Alderson A (2000) Auxetic materials: functional materials and structures from lateral thinking! Adv Mater 12(9):617

    Article  Google Scholar 

  • 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):175301

    Article  ADS  Google Scholar 

  • Farjam M, Rafii-Tabar H (2009) Comment on “band structure engineering of graphene by strain: first-principles calculations”. Phys Rev B 80:167401

    Article  ADS  Google Scholar 

  • Fujita M, Wakabayashi K, Nakada K, Kusakabe K (1996) Peculiar localized state at zigzag graphite edge. J Phys Soc Jpn 65(7):1920–1923

    Article  ADS  Google Scholar 

  • Gallagher P, Todd K, Gordon DG (2010) Disorder-induced gap behavior in graphene nanoribbons. Phys Rev B 81:115409

    Article  ADS  Google Scholar 

  • 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):4107–4113

    Article  ADS  Google Scholar 

  • Geim AK, Novoselov KS (2007) The rise of graphene. Nat Mater 6:183–191

    Article  ADS  Google Scholar 

  • Golizadeh-Mojarad R, Datta S (2007) Nonequilibrium green’s function based models for dephasing in quantum transport. Phys Rev B 75:081301

    Article  ADS  Google Scholar 

  • Gong L, Kinloch IA, Young RJ, Riaz I, Jalil R, Novoselov KS (2010) Interfacial stress transfer in a graphene monolayer nanocomposite. Adv Matter 22:2694

    Article  Google Scholar 

  • González JW, Pacheco M, Rosales L, Orellana PA (2011) Transport properties of graphene quantum dots. Phys Rev B 83:155450

    Article  ADS  Google Scholar 

  • 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 27:1455–1459

    Article  Google Scholar 

  • Guinea F, Low T (2010) Band structure and gaps of triangular graphene superlattices. Philos Trans R Soc Math Phys Eng Sci 368(1932):5391–5402

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Guinea F, Horovitz B, Le Doussal P (2008) Gauge field induced by ripples in graphene. Phys Rev B 77:205421

    Article  ADS  Google Scholar 

  • 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):30–33

    Article  Google Scholar 

  • Guinea F, Geim AK, Katsnelson MI, Novoselov KS (2010b) Generating quantizing pseudomagnetic fields by bending graphene ribbons. Phys Rev B 81(3):035408

    Article  ADS  Google Scholar 

  • Han MY, Ozyilmaz B, Zhang Y, Kim P (2007) Energy band-gap engineering of graphene nanoribbons. Phys Rev Lett 98:206805

    Article  ADS  Google Scholar 

  • Han MY, Brant JC, Kim P (2010) Electron transport in disordered graphene nanoribbons. Phys Rev Lett 104:056801

    Article  ADS  Google Scholar 

  • 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 181–212

    Google Scholar 

  • Ho DT, Park S-D, Kwon S-Y, Park K, Kim SY (2014) Negative poisson’s ratios in metal nanoplates. Nat Commun 5:3255

    Article  Google Scholar 

  • Ho VH, Ho DT, Kwon S-Y, Kim SY (2016) Negative poisson’s ratio in cubic materials along principal directions. Phys Status Solidi B. https://doi.org/10.1002/pssb.201600017

  • Ho VH, Ho DT, Kwon S-Y, Kim SY (2016) Negative poisson’s ratio in periodic porous graphene structures. Phys Status Solidi B. https://doi.org/10.1002/pssb.201600061

  • Humphrey W, Dalke A, Schulten K (1996) VMD: visual molecular dynamics. J Mol Graph 14(1):33–38

    Article  Google Scholar 

  • Ihnatsenka S, Zozoulenko IV, Kirczenow G (2009) Band-gap engineering and ballistic transport in edge-corrugated graphene nanoribbons. Phys Rev B 80:155415

    Article  ADS  Google Scholar 

  • Jiang J-W, Park HS (2014) Negative poisson’s ratio in single-layer black phosphorus. Nat Commun 5:4727

    Article  Google Scholar 

  • Jiang J-W, Park HS (2016) Negative poisson’s ratio in single-layer graphene ribbons. Nano Lett 16:2657–2662

    Article  ADS  Google Scholar 

  • 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):235421

    Article  ADS  Google Scholar 

  • Jiang J-W, Kim SY, Park HS (2016) Auxetic nanomaterials: recent progress and future directions. Appl Phys Rev 3:041101

    Article  Google Scholar 

  • Jiang J-W, Chang T, Guo X, Park HS (2016) Intrinsic negative poisson’s ratio for single-layer graphene. Nano Lett 16:5286–5290

    Article  ADS  Google Scholar 

  • Jiang J-W, Park HS (2016) Negative poisson’s ratio in single-layer graphene ribbons. Nano Lett 16:2657–2662

    Article  ADS  Google Scholar 

  • Jiao L, Zhang L, Wang X, Diankov G, Dai H (2009) Narrow graphene nanoribbons from carbon nanotubes. Nature 458:877

    Article  ADS  Google Scholar 

  • Ji Z-L, Berggren K-F (1992) Quantum bound states in narrow ballistic channels with intersections. Phys Rev B 45:6652–6658

    Article  ADS  Google Scholar 

  • 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):2704–2708

    Article  ADS  Google Scholar 

  • Kane CL, Mele EJ (1997) Size, shape, and low energy electronic structure of carbon nanotubes. Phys Rev Lett 78:1932

    Article  ADS  Google Scholar 

  • Kim K-J, Blanter YM, Ahn K-H (2011) Interplay between real and pseudomagnetic field in graphene with strain. Phys Rev B 84(8):081401

    Article  ADS  Google Scholar 

  • Kitt AL, Pereira VM, Swan AK, Goldberg BB (2012) Lattice-corrected strain-induced vector potentials in graphene. Phys Rev B 85(11):115432

    Article  ADS  Google Scholar 

  • 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 

  • Kouwenhoven LP, Hekking FWJ, van Wees BJ, Harmans CJPM, Timmering CE, Foxon CT (1990) Transport through a finite one-dimensional crystal. Phys Rev Lett 65:361–364

    Article  ADS  Google Scholar 

  • Lakes RS (1987) Foam structures with a negative poisson’s ratio. Science 235:1038–1040

    Article  ADS  Google Scholar 

  • Lakes R (1993) Advances in negative poisson’s ratio materials. Adv Mater 5:293–296

    Article  Google Scholar 

  • Lammps (2018) http://lammps.sandia.gov/

  • Lee C, Wei X, Kysar JW, Hone J (2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321:385

    Article  ADS  Google Scholar 

  • Lethbridge ZA, Walton RI, Marmier AS, Smith CW, Evans KE (2010) Elastic anisotropy and extreme poisson’s ratios in single crystals. Acta Mater 58:6444–6451

    Article  Google Scholar 

  • Los JH, Fasolino A, Katsnelson MI (2006) Scaling behavior and strain dependence of in-plane elastic properties of graphene. Phys Rev Lett 116:015901

    Article  ADS  Google Scholar 

  • Milstein F, Huang K (1979) Existence of a negative poisson ratio in FCC crystals. Phys Rev B 19(4):2030

    Article  ADS  Google Scholar 

  • Mo Y, Turner KT, Szlufarska I (2009) Friction laws at the nanoscale. Nature 457:1116

    Article  ADS  Google Scholar 

  • 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):205433

    Article  ADS  Google Scholar 

  • 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):2301–2305

    Article  Google Scholar 

  • 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 3:483

    Article  Google Scholar 

  • 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 438:197–200

    Article  ADS  Google Scholar 

  • 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):1–23

    Google Scholar 

  • Pereira VM, Castro Neto AH (2009) Strain engineering of graphene’s electronic structure. Phys Rev Lett 103(4):4

    Article  Google Scholar 

  • Pereira VM, Castro Neto AH, Peres NMR (2009) Tight-binding approach to uniaxial strain in graphene. Phys Rev B 80:045401

    Article  ADS  Google Scholar 

  • Pereira VM, Castro Neto AH, Liang HY, Mahadevan L (2010) Geometry, mechanics, and electronics of singular structures and wrinkles in graphene. Phys Rev Lett 105:156603

    Article  ADS  Google Scholar 

  • Pereira VM, Ribeiro RM, Peres NMR, Castro Neto AH (2010) Optical properties of strained graphene. Eur Phys Lett 92:67001

    Article  ADS  Google Scholar 

  • Plimpton SJ (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19

    Article  ADS  MATH  Google Scholar 

  • Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19

    Article  ADS  MATH  Google Scholar 

  • 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):265702

    Article  ADS  Google Scholar 

  • Qi Z, Bahamon DA, Pereira VM, Park HS, Campbell DK, Castro Neto AH (2013) Resonant tunneling in graphene pseudomagnetic quantum dots. Nano Lett 13:2692

    Article  ADS  Google Scholar 

  • 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 90:125419

    Article  ADS  Google Scholar 

  • Qi Z, Campbell DK, Park HS (2014) Atomistic simulations of tension-induced large deformation and stretchability in graphene kirigami. Phys Rev B 90:245437

    Article  ADS  Google Scholar 

  • Ravirala N, Alderson A, Alderson KL (2007) Interlocking hexagons model for auxetic behaviour. J Mater Sci 42:7433–7445

    Article  ADS  Google Scholar 

  • Rothenburg L, Berlint AA, Bathurst RJ (1991) Microstructure of isotropic materials with negative poisson’s ratio. Nature 354:470

    Article  ADS  Google Scholar 

  • 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 328:213–216

    Article  ADS  Google Scholar 

  • Shen L, Li J (2004) Transversely isotropic elastic properties of single-walled carbon nanotubes. Phys Rev B 69:045414

    Article  ADS  Google Scholar 

  • Shenoy VB, Reddy CD, Ramasubramaniam A, Zhang YW (2008) Edge-stress-induced warping of graphene sheets and nanoribbons. Phys Rev Lett 101(24):245501

    Article  ADS  Google Scholar 

  • Sols F, Guinea F, Neto AHC (2007) Coulomb blockade in graphene nanoribbons. Phys Rev Lett 99:166803

    Article  ADS  Google Scholar 

  • Stampfer C, Güttinger J, Hellmüller S, Molitor F, Ensslin K, Ihn T (2009) Energy gaps in etched graphene nanoribbons. Phys Rev Lett 102:056403

    Article  ADS  Google Scholar 

  • Stuart SJ, Tutein AB, Harrison JA (2000) A reactive potential for hydrocarbons with intermolecular interactions. J Chem Phys 112(14):6472–6486

    Article  ADS  Google Scholar 

  • Stukowski A (2010) Visualization and analysis of atomistic simulation data with OVITO – the open visualization tool. Model Simul Mater Sci Eng 18:015012

    Article  ADS  Google Scholar 

  • Suzuura H, Ando T (2002) Phonons and electron-phonon scattering in carbon nanotubes. Phys Rev B 65:235412

    Article  ADS  Google Scholar 

  • Suzuura H, Ando T (2002) Phonons and electron-phonon scattering in carbon nanotubes. Phys Rev B 65:235412

    Article  ADS  Google Scholar 

  • Todd K, Chou H-T, Amasha S, Goldhaber-Gordon D (2008) Quantum dot behavior in graphene nanoconstrictions. Nano Lett 9:416

    Article  ADS  Google Scholar 

  • 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):3

    Article  Google Scholar 

  • Ulloa SE, Castao E, Kirczenow G (1990) Ballistic transport in a novel one-dimensional superlattice. Phys Rev B 41:12350–12353

    Article  ADS  Google Scholar 

  • Vozmediano MAH, Katsnelson MI, Guinea F (2010) Gauge fields in graphene. Phys Rep 496:109

    Article  ADS  MathSciNet  Google Scholar 

  • Wang ZF, Zhang Y, Liu F (2011) Formation of hydrogenated graphene nanoripples by strain engineering and directed surface self-assembly. Phys Rev B 83:041403

    Article  ADS  Google Scholar 

  • Wu Z, Zhang ZZ, Chang K, Peeters FM (2010) Quantum tunneling through graphene nanorings. Nanotechnology 21(18):185201

    Article  ADS  Google Scholar 

  • Yang HT (2011) Strain induced shift of dirac points and the pseudo-magnetic field in graphene. J Phys Condens Matter 23(50):505502

    Article  Google Scholar 

  • Yang W, Li Z-M, Shi W, Xie B-H, Yang M-B (2004) Review on auxetic materials. J Mater Sci 39:3269–3279

    Article  ADS  Google Scholar 

  • Yao YT, Alderson A, Alderson KL (2008) Can nanotubes display auxetic behaviour? Phys Status Solidi B 245(11):2373–2382

    Article  ADS  Google Scholar 

  • 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):1649–1656

    Article  ADS  Google Scholar 

  • Yue K, Gao W, Huang R, Liechti KM (2012) Analytical methods for the mechanics of graphene bubbles. J Appl Phys 112(8):083512

    Article  ADS  Google Scholar 

  • Zakharchenko KV, Katsnelson MI, Fasolino A (2009) Finite temperature lattice properties of graphene beyond the quasiharmonic approximation. Phys Rev Lett 102(4):046808

    Article  ADS  Google Scholar 

  • Zhang ZZ, Chang K, Chan KS (2008) Resonant tunneling through double-bended graphene nanoribbons. Appl Phys Lett 93(6):062106. https://doi.org/10.1063/1.2970957

    Article  ADS  Google Scholar 

  • Zhao H, Aluru NR (2010) Temperature and strain-rate dependent fracture strength of graphene. J Appl Phys 108(6):064321

    Article  ADS  Google Scholar 

  • Zhu S, Huang Y, Li T (2014) Extremely compliant and highly stretchable patterned graphene. Appl Phys Lett 104(17):173103

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Boston University, Boston, MA, USA

    Harold S. Park

Authors
  1. Harold S. Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harold S. Park .

Editor information

Editors and Affiliations

  1. Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Wanda Andreoni

  2. Department of Nuclear Science & Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Sidney Yip

Section Editor information

  1. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Ting Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Park, H.S. (2018). Mechanics and Electromechanics of Two-Dimensional Atomic Membranes. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-50257-1_44-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50257-1_44-1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50257-1

  • Online ISBN: 978-3-319-50257-1

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation