Continuum scaling laws often break down when materials approach atomic length scales, reflecting changes in their underlying physics and the opportunities to access unconventional properties. These continuum limits are evident in two-dimensional materials, where there is no consensus on their bending stiffnesses or how they scale with thickness. Through combined computational and electron microscopy experiments, we measure the bending stiffness of graphene, obtaining 1.2–1.7 eV for a monolayer. Moreover, we find that the bending stiffness of few-layer graphene decreases sharply as a function of bending angle, tuning by almost 400% for trilayer graphene. This softening results from shear, slip and the onset of superlubricity between the atomic layers and corresponds with a gradual change in scaling power from cubic to linear. Our results provide a unified model for bending in two-dimensional materials and show that their multilayers can be orders of magnitude softer than previously thought, among the most flexible electronic materials currently known.
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The data and findings of this study are available from the corresponding authors on reasonable request.
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This work was supported in full by NSF-MRSEC award no. DMR-1720633. The work was carried out in part in the Micro and Nano Technology Laboratory and the Materials Research Laboratory Central Facilities at the University of Illinois, where electron microscopy support was provided by J. Mabon, C. Chen and H. Zhou. The authors acknowledge the use of facilities and instrumentation supported by the NSF through the University of Illinois Materials Research Science and Engineering Center (DMR-1720633). Computational resources were provided by the Blue Waters sustained petascale computing facilities and the Illinois Campus Computing Cluster. The authors acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. The authors acknowledge helpful discussions with J. Krogstad, H. Johnson, S. Kim and B. Janicek. The authors also acknowledge B. Janicek for the design of Fig. 1a.
The authors declare no competing interests.
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Han, E., Yu, J., Annevelink, E. et al. Ultrasoft slip-mediated bending in few-layer graphene. Nat. Mater. 19, 305–309 (2020). https://doi.org/10.1038/s41563-019-0529-7
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