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Crystallization for a Brenner-like Potential

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Abstract

Graphene is a carbon molecule with the structure of a honeycomb lattice. We show how this structure can arise in two dimensions as the minimizer of an interaction energy with two-body and three-body terms. In the engineering literature, the Brenner potential is commonly used to describe the interactions between carbon atoms. We consider a potential of Stillinger–Weber type that incorporates certain characteristics of the Brenner potential: the preferred bond angles are 180\({^\circ}\) and all interactions have finite range. We show that the thermodynamic limit of the ground state energy per particle is the same as that of a honeycomb lattice. We also prove that, subject to periodic boundary conditions, the minimizers are translated versions of the honeycomb lattice.

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Correspondence to Brittan Farmer.

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Communicated by F. Toninelli

P. Smereka: Deceased.

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Farmer, B., Esedoḡlu, S. & Smereka, P. Crystallization for a Brenner-like Potential. Commun. Math. Phys. 349, 1029–1061 (2017). https://doi.org/10.1007/s00220-016-2732-6

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  • DOI: https://doi.org/10.1007/s00220-016-2732-6

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