Calculus of Variations and Geometric Measure Theory

E. Davoli - P. Piovano - U. Stefanelli

Wulff shape emergence in graphene

created by davoli on 22 Jan 2016
modified on 04 Sep 2020


Published Paper

Inserted: 22 jan 2016
Last Updated: 4 sep 2020

Journal: Math. Models Methods Appl. Sci.
Year: 2016


Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number $n$ of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as $n\to+\infty$. Precisely, ground states deviate from such hexagonal Wulff shape by at most $K n^{3/4} + o(n^{3/4})$ atoms, where both the constant $K$ and the rate $n^{3/4}$ are sharp.