Elastic coupling between layers in two-dimensional materials

Journal name:
Nature Materials
Year published:
Published online


Two-dimensional materials, such as graphene and MoS2, are films of a few atomic layers in thickness with strong in-plane bonds and weak interactions between the layers. The in-plane elasticity has been widely studied in bending experiments where a suspended film is deformed substantially; however, little is known about the films elastic modulus perpendicular to the planes, as the measurement of the out-of-plane elasticity of supported 2D films requires indentation depths smaller than the films interlayer distance. Here, we report on sub-ångström-resolution indentation measurements of the perpendicular-to-the-plane elasticity of 2D materials. Our indentation data, combined with semi-analytical models and density functional theory, are then used to study the perpendicular elasticity of few-layer-thick graphene and graphene oxide films. We find that the perpendicular Youngs modulus of graphene oxide films reaches a maximum when one complete water layer is intercalated between the graphitic planes. This non-destructive methodology can map interlayer coupling and intercalation in 2D films.

At a glance


  1. Modulated nanoindentation experiments.
    Figure 1: Modulated nanoindentation experiments.

    a, Schematic diagram of the experimental set-up, where a spherical AFM tip vibrates while indenting a few-layer-thick film of graphene or GO. b, Experimentally measured indentation curves for single-crystal SiC, 10-layer-thick EG, and 10-layer-thick EGO. All three curves were obtained with the same AFM tip, R = 114 nm.

  2. Experimental, SAM-simulated and Hertz indentation curves.
    Figure 2: Experimental, SAM-simulated and Hertz indentation curves.

    a, Experimentally measured indentation curves in HOPG (filled circles), semi-analytical model simulations of indentation in graphite (open circles), and Hertzian fitting (continuum line) of the indentation curves on HOPG. The indenting tip radius was 100 nm. b, Contact-pressure distribution profiles for Hertz contacts and SAM simulations of indentation in graphite. Note that for bulk graphite and for a graphite film 50 nm thick, the SAM simulations and the contact distribution profiles almost overlap. c, Experimental indentation curves on 10-layer-thick EG, 1-layer-thick EG, buffer-layer EG, and SiC. d, Statistical analysis of exponent number b in the fitting function Fz = Czindentb. For EGO and GO, the RH is indicated. baverage is 1.40.

  3. DFT and experimental results for conventional GO films.
    Figure 3: DFT and experimental results for conventional GO films.

    a, The DFT-calculated Fz versus displacement curves for different water contents in graphene fully oxidized with hydroxyl groups. b, Experimental Fz versus indentation depth curves at different RHs in conventional GO. All of the curves were obtained with the same AFM tip. c,d, Experimental and DFT results of E of GO as a function of water content and RH, respectively. The insets are schematic diagrams of the corresponding atomistic structures showing how water molecules fill the interlayer spacing. Each experimental point of E is an average value of more than 30 different measurements.


  1. Park, J. Y., Kwon, S. & Kim, J. H. Nanomechanical and charge transport properties of two-dimensional atomic sheets. Adv. Mater. Interfaces 1, 130089 (2014).
  2. Berger, C. et al. Electronic confinement and coherence in patterned epitaxial graphene. Science 312, 11911196 (2006).
  3. Zhu, Y. et al. Microwave assisted exfoliation and reduction of graphite oxide for ultracapacitors. Carbon 48, 21182122 (2010).
  4. Lin, Y-M. et al. Operation of graphene transistors at gigahertz frequencies. Nano Lett. 9, 422426 (2008).
  5. Wang, X., Zhi, L. & Müllen, K. Transparent, conductive graphene electrodes for dye-sensitized solar cells. Nano Lett. 8, 323327 (2008).
  6. Yu, W. J. et al. Vertically stacked multi-heterostructures of layered materials for logic transistors and complementary inverters. Nature Mater. 12, 246252 (2013).
  7. Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nature Mater. 11, 409416 (2012).
  8. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666669 (2004).
  9. De Heer, W. A. et al. Epitaxial graphene. Solid State Commun. 143, 92100 (2007).
  10. Lee, C., Wei, X., Kysar, J. W. & Hone, J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385388 (2008).
  11. Baringhaus, J. et al. Exceptional ballistic transport in epitaxial graphene nanoribbons. Nature 506, 349354 (2014).
  12. Bolotin, K. I. et al. Ultrahigh electron mobility in suspended graphene. Solid State Commun. 146, 351355 (2008).
  13. Balandin, A. A. et al. Superior thermal conductivity of single-layer graphene. Nano Lett. 8, 902907 (2008).
  14. Chiu, H. C., Kim, S., Klinke, C. & Riedo, E. Morphology dependence of radial elasticity in multiwalled boron nitride nanotubes. Appl. Phys. Lett. 101, 103109 (2012).
  15. Loh, K. P., Bao, Q., Eda, G. & Chhowalla, M. Graphene oxide as a chemically tunable platform for optical applications. Nature Chem. 2, 10151024 (2010).
  16. Eda, G., Fanchini, G. & Chhowalla, M. Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material. Nature Nanotech. 3, 270274 (2008).
  17. Wei, Z. et al. Nanoscale tunable reduction of graphene oxide for graphene electronics. Science 328, 13731376 (2010).
  18. Ci, L. et al. Atomic layers of hybridized boron nitride and graphene domains. Nature Mater. 9, 430435 (2010).
  19. Ramakrishna Matte, H. S. S. et al. MoS2 and WS2 analogues of graphene. Angew. Chem. 122, 41534156 (2010).
  20. Castellanos-Gomez, A. et al. Mechanical properties of freely suspended semiconducting graphene-like layers based on MoS2. Nanoscale Res. Lett. 7, 14 (2012).
  21. Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V. & Kis, A. Single-layer MoS2 transistors. Nature Nanotech. 6, 147150 (2011).
  22. Kelly, B. T. Physics of Graphite (Applied Science, 1981).
  23. Kwon, S. et al. Probing nanoscale conductance of monolayer graphene under pressure. Appl. Phys. Lett. 99, 013110 (2011).
  24. Lee, C. et al. Frictional characteristics of atomically thin sheets. Science 328, 7680 (2010).
  25. Lee, C. et al. Elastic and frictional properties of graphene. Phys. Status Solidi B 246, 25622567 (2009).
  26. Stewart, J. A. & Spearot, D. E. Atomistic simulations of nanoindentation on the basal plane of crystalline molybdenum disulfide (MoS2). Modelling Simul. Mater. Sci. Eng. 21, 045003 (2013).
  27. Suk, J. W., Piner, R. D., An, J. & Ruoff, R. S. Mechanical properties of monolayer graphene oxide. ACS Nano 4, 65576564 (2010).
  28. Hajgató, B. et al. Out-of-plane shear and out-of plane Youngs modulus of double-layer graphene. Chem. Phys. Lett. 564, 3740 (2013).
  29. Nakamura, N., Ogi, H. & Hirao, M. Resonance ultrasound spectroscopy with laser-Doppler interferometry for studying elastic properties of thin films. Ultrasonics 42, 491494 (2004).
  30. Chiritescu, C. et al. Ultralow thermal conductivity in disordered, layered WSe2 crystals. Science 315, 351353 (2007).
  31. Riedl, C., Coletti, C. & Starke, U. Structural and electronic properties of epitaxial graphene on SiC(0 0 0 1): A review of growth, characterization, transfer doping and hydrogen intercalation. J. Phys. D 43, 374009 (2010).
  32. Lucas, M. et al. Hindered rolling and friction anisotropy in supported carbon nanotubes. Nature Mater. 8, 876881 (2009).
  33. Tan, P. H. et al. The shear mode of multilayer graphene. Nature Mater. 11, 294300 (2012).
  34. Lucas, M., Mai, W., Yang, R., Wang, Z. L. & Riedo, E. Aspect ratio dependence of the elastic properties of ZnO nanobelts. Nano Lett. 7, 13141317 (2007).
  35. Palaci, I. et al. Radial elasticity of multiwalled carbon nanotubes. Phys. Rev. Lett. 94, 175502 (2005).
  36. Kim, S. et al. Room-temperature metastability of multilayer graphene oxide films. Nature Mater. 11, 544549 (2012).
  37. Hummers, W. S. Jr & Offeman, R. E. Preparation of graphitic oxide. J. Am. Chem. Soc. 80, 13391339 (1958).
  38. Lantz, M., OShea, S., Welland, M. & Johnson, K. Atomic-force-microscope study of contact area and friction on NbSe2. Phys. Rev. B 55, 1077610785 (1997).
  39. Carpick, R. W., Ogletree, D. & Salmeron, M. Lateral stiffness: A new nanomechanical measurement for the determination of shear strengths with friction force microscopy. Appl. Phys. Lett. 70, 15481550 (1997).
  40. Grierson, D., Flater, E. & Carpick, R. Accounting for the JKR–DMT transition in adhesion and friction measurements with atomic force microscopy. J. Adhes. Sci. Technol. 19, 291311 (2005).
  41. Derjaguin, B. V., Muller, V. M. & Toporov, Y. P. Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci. 53, 314326 (1975).
  42. Johnson, K. L. Contact Mechanics (Cambridge Univ. Press, 1987).
  43. Turner, J. Contact on a transversely isotropic half-space, or between two transversely isotropic bodies. Int. J. Solids Struct. 16, 409419 (1980).
  44. Bagault, C., Nelias, D., Baietto, M-C. & Ovaert, T. C. Contact analyses for anisotropic half-space coated with an anisotropic layer: Effect of the anisotropy on the pressure distribution and contact area. Int. J. Solids Struct. 50, 743754 (2013).
  45. Bagault, C., Nelias, D. & Baietto, M-C. Contact analyses for anisotropic half space: Effect of the anisotropy on the pressure distribution and contact area. J. Tribol. 134, 031401 (2012).
  46. Jacq, C., Nelias, D., Lormand, G. & Girodin, D. Development of a three-dimensional semi-analytical elastic–plastic contact code. J. Tribol. 124, 653667 (2002).
  47. Palacio, I. et al. Atomic structure of epitaxial graphene sidewall nanoribbons: Flat graphene, miniribbons, and the confinement gap. Nano Lett. 15, 182189 (2014).
  48. Zhou, S. & Bongiorno, A. Origin of the chemical and kinetic stability of graphene oxide. Sci. Rep. 3, 2484 (2013).
  49. Zhou, S. et al. Film structure of epitaxial graphene oxide on SiC: Insight on the relationship between interlayer spacing, water content, and intralayer structure. Adv. Mater. Interface 1, 1300106 (2014).
  50. Giannozzi, P. et al. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
  51. Carroll, K. M. et al. Parallelization of thermochemical nanolithography. Nanoscale 6, 12991304 (2014).
  52. Troullier, N. & Martins, J. L. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B 43, 19932006 (1991).
  53. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 38653868 (1996).
  54. Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 17871799 (2006).

Download references

Author information

  1. These authors contributed equally to this work.

    • Yang Gao &
    • Suenne Kim


  1. School of Physics, Georgia Institute of Technology, 837 State Street Atlanta, Georgia 30332-0430, USA

    • Yang Gao,
    • Si Zhou,
    • Claire Berger,
    • Walt de Heer,
    • Angelo Bongiorno &
    • Elisa Riedo
  2. Advanced Science Research Center and City College New York, City University of New York, 85 St Nicholas Terrace New York, New York 10031, USA

    • Yang Gao &
    • Elisa Riedo
  3. Department of Applied Physics, Hanyang University, Ansan 426-791, South Korea

    • Suenne Kim
  4. Department of Physics, National Taiwan Normal University, 88, Sec.4, Ting-Chou Road Taipei 116, Taiwan

    • Hsiang-Chih Chiu
  5. Université de Lyon, CNRS, INSA-Lyon, LaMCoS UMR5259, Villeurbanne F69621, France

    • Daniel Nélias
  6. Institut Néel, Université Grenoble Alpes-CNRS, BP 166 38042 Grenoble, France

    • Claire Berger
  7. King Abdulaziz University, Department of Physics, Jeddah 21589, Saudi Arabia

    • Walt de Heer
  8. L-NESS, Department of Physics, Politecnico di Milano, Via Anzani 42 22100 Como, Italy

    • Laura Polloni &
    • Roman Sordan
  9. Department of Chemistry, College of Staten Island, City University of New York, New York, New York 10314, USA

    • Angelo Bongiorno


Y.G., S.K. and H-C.C. performed nanomechanics experiments and data analysis. S.Z. carried out DFT calculations. D.N. performed the SAM calculations. C.B., and W.d.H. synthesized the EG and EGO samples. L.P. and R.S. synthesized the GO samples. A.B. conceived and designed the theory and analysed the data. E.R. conceived and designed the experiments and analysed the data. All authors contributed to write the article.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (9,409 KB)

    Supplementary Information

Additional data