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2D crystals

The Wulff construction goes low-symmetry

An apparent quirk of mathematics draws on a symmetry and resolves the issue of how to determine the equilibrium shape of crystals of two-dimensional materials with asymmetric terminations.

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Fig. 1: A simple example of a Wulff construction for an asymmetric crystal.

References

  1. Gibbs, J. W. On the Equilibrium of Heterogeneous Substances (Connecticut Academy of Arts and Sciences, 1875–1878).

  2. Curie, P. Bull. Soc. Min. Fr. 8, 145–150 (1885).

  3. Wulff, G. XXV Z. Krystallog. Min. 34, 449–530 (1901).

  4. Laue, M. Z. Kristallog. 105, 124–133 (1943).

  5. Marks, L. D. & Peng, L. J. Phys. Condens. Matter 28, 053001 (2016).

  6. Rahm, J. M. & Erhart, P. Nano Lett. 17, 5775–5781 (2017).

  7. Cahn, J. W. & Handwerker, C. A. Mater. Sci. Eng. A 162, 83–95 (1993).

  8. Wang, L., Shirodkar, S. N., Zhang, Z. & Yakobson, B. I. Nat. Comput. Sci. https://doi.org/10.1038/s43588-022-00347-5 (2022).

    Article  Google Scholar 

  9. Chhowalla, M. Nat. Chem. 5, 263–275 (2013).

  10. Karunadasa, H. I. Science 335, 698–702 (2012).

  11. Rossi, T. P. Phys. Rev. B 96, 155407 (2017).

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Acknowledgements

This work was funded by the Swedish Research Council (grant numbers 2018-06482, 2020-04935 and 2021-05072) and the Excellence Initiative Nano at Chalmers.

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Correspondence to Paul Erhart.

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Erhart, P., Rahm, J.M. The Wulff construction goes low-symmetry. Nat. Mater. (2022). https://doi.org/10.1038/s41563-022-01407-x

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  • DOI: https://doi.org/10.1038/s41563-022-01407-x

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