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Relaxation and Domain Wall Structure of Bilayer Moiré Systems

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Abstract

Moiré patterns result from setting a 2D material such as graphene on another 2D material with a small twist angle or from the lattice mismatch of 2D heterostructures. We present a continuum model for the elastic energy of these bilayer moiré structures that includes an intralayer elastic energy and an interlayer misfit energy that is minimized at two stackings (disregistries). We show by theory and computation that the displacement field that minimizes the global elastic energy subject to a global boundary constraint gives large alternating regions of one of the two energy-minimizing stackings separated by domain walls.

We derive a model for the domain wall structure from the continuum bilayer energy and give a rigorous asymptotic estimate for the structure. We also give an improved estimate for the \(L^{2}\)-norm of the gradient on the moiré unit cell for twisted bilayers that scales at most inversely linearly with the twist angle, a result which is consistent with the formation of one-dimensional domain walls with a fixed width around triangular domains at very small twist angles.

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Notes

  1. Note that in [1] the GSFE profile across the domain wall is approximated by the sine-Gordon potential \(\Phi (\Delta u) = \frac{V_{\text{SP}}}{2} (1 - \cos (2\pi \Delta u))\) for \(0 \leqslant \Delta u \leqslant 1\) allowing for an explicit solution, and a different normalization was used with \(k = \Phi [0] - \Phi _{\min}\) the saddle point energy, resulting in a slightly different expression for the thickness of the domain wall.

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Funding

PC’s research was supported in part by National Science Foundation Award DMS-189220 and Simons Collaboration Grants for Mathematicians No. 966604. DC’s research was supported in part by National Science Foundation Award DMS-1906129. RE’s and PK’s research was supported in part by National Science Foundation Award DMREF Award No. 1922165. ML’s research was supported in part by National Science Foundation Award DMREF Award No. 1922165 and Simons Targeted Grant Award No. 896630.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24060, USA

    Paul Cazeaux

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55408, USA

    Drake Clark & Mitchell Luskin

  3. Department of Physics, Harvard University, Cambridge, MA, 02138, USA

    Rebecca Engelke & Philip Kim

Authors
  1. Paul Cazeaux
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  2. Drake Clark
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  3. Rebecca Engelke
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  4. Philip Kim
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  5. Mitchell Luskin
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Contributions

P.C. and M.L. wrote the main manuscript text, D.C. and P.C. did the numerical simulations, and R.E. and P.K. did the experiments.

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Correspondence to Mitchell Luskin.

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Cazeaux, P., Clark, D., Engelke, R. et al. Relaxation and Domain Wall Structure of Bilayer Moiré Systems. J Elast 154, 443–466 (2023). https://doi.org/10.1007/s10659-023-10013-0

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  • DOI: https://doi.org/10.1007/s10659-023-10013-0

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