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Elastic coupling between layers in two-dimensional materials


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 Young’s 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.

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Figure 1: Modulated nanoindentation experiments.
Figure 2: Experimental, SAM-simulated and Hertz indentation curves.
Figure 3: DFT and experimental results for conventional GO films.


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Y.G., S.K., H-C.C. and E.R., acknowledge the support of the Office of Basic Energy Sciences of the US Department of Energy (DE-FG02-06ER46293). S.Z. and A.B. acknowledge the support of the National Science Foundation (NSF) grant CMMI 1436375. S.Z., A.B., C.B. and W.d.H. acknowledge the support of the NSF grant DMR-0820382. C.B. acknowledges partial financial support from the European Flagship Graphene. A.B. acknowledges the support of the NSF grant CHE-0946869. R.S. acknowledges the support of the Italian Cariplo Foundation, project No. 2011-0373. We thank J-P. Turmaud for the EG on Si sample.

Author information




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.

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Correspondence to Angelo Bongiorno or Elisa Riedo.

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The authors declare no competing financial interests.

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Gao, Y., Kim, S., Zhou, S. et al. Elastic coupling between layers in two-dimensional materials. Nature Mater 14, 714–720 (2015).

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