Crystallization for a Brenner-like potential

TL;DR

This paper demonstrates that a Stillinger-Weber type potential, inspired by the Brenner potential, leads to a honeycomb lattice structure as the energy minimizer in two dimensions, confirming the natural formation of graphene-like structures.

Contribution

It introduces a Stillinger-Weber type potential with Brenner-like features and proves the honeycomb lattice as the ground state configuration in the thermodynamic limit.

Findings

Ground state energy per particle matches that of a honeycomb lattice.

Minimizers under periodic boundary conditions are translated honeycomb lattices.

Thermodynamic limit confirms the stability of the honeycomb structure.

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 degrees 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.