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Bending-induced extension in two-dimensional crystals

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Abstract

We find by ab initio simulations that significant overall tensile strain can be induced by pure bending in a wide range of two-dimensional crystals perpendicular to the bending moment, just like an accordion being bent to open. This bending-induced tensile strain increases in a power law with bent curvature and can be over 20% in monolayered black phosphorus and transition metal dichalcogenides at a moderate curvature of \(2\,\hbox {nm}^{-1}\) but more than an order weaker in graphene and hexagon boron nitride. This accordion effect is found to be a quantum mechanical effect raised by the asymmetric response of chemical bonds and electron density to the bending curvature.

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Notes

  1. The energy and force convergence criteria are set at \(10^{-4}\) eV and 0.1 eV/nm, respectively. The kinetic energy cutoff for the plane wave basis set is adopted as 450 eV, which guarantees that the absolute energies converge to around 2  MeV. The reciprocal space for the unit cell of BP is meshed at \(14 \times 10 \times 1\) using Monkhorst-Pack meshes centered at the \(\varGamma \) point. For one-dimensional periodic cells, a vacuum space of 2  nm is included for all tubes and a dense k-point mesh (\(1 \times 1 \times 12\)) is used for Brillouin zone sampling. The lattice constants in all directions are fully optimized for all tubes.

  2. See supplemental material for the determination of the lengths of the outer and inner half-layers of bent BP monolayers, bending-induced asymmetrical partial charge distribution, the bending to tensile deformation in other transitional metal dichalcogenides, the influence of finite temperature on the bending deformation, and the bending rigidity of monolayered BP with fixed midplane length.

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Acknowledgments

This project was supported by the 973 program (Grants 2012CB937500, 2013CB932604), the National Natural Science Foundation of China (Grants 51535005, 51472117, 11021262, 11172303, 11132011), and the Fundamental Research Funds for the Central Universities (Grant NP2013309). We thank X.F. Liu for helpful discussions and W.H. Tang for help in drawing some of the geometrical figures.

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    Authors and Affiliations

    1. State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

      Douxing Pan & Tzu-Chiang Wang

    2. University of Chinese Academy of Sciences, Beijing, 100049, China

      Douxing Pan

    3. State Key Laboratory of Mechanics and Control for Mechanical Structures and Key Laboratory for Intelligent Nano Materials and Devices (MOE), Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

      Yao Li & Wanlin Guo

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    1. Douxing Pan
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    2. Yao Li
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    3. Tzu-Chiang Wang
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    4. Wanlin Guo
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    Correspondence to Tzu-Chiang Wang or Wanlin Guo.

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    Douxing Pan and Yao Li have contributed equally to this work.

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    Pan, D., Li, Y., Wang, TC. et al. Bending-induced extension in two-dimensional crystals. Acta Mech. Sin. 33, 71–76 (2017). https://doi.org/10.1007/s10409-016-0602-2

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