Graphene edge from A to Z

We introduce an energy decomposition anzats, which leads to an analytical expression for the edge energy G(X) of arbitrary direction-cut angle X in two dimensional materials [1]. We further show that thermodynamic conditions at the edge simply add a ``chemical phase shift'' C, G(X) = cos(X + C), making the favorable shapes controllable, according to the Wulff construction. Direct atomistic computations and analysis for graphene, as well as 2D boron nitride (h-BN), and zinic oxide (ZnO) support the universal nature of the relationship.\\[4pt] [1] Y. Liu, A. Dobrinsky, and B.I. Yakobson, Phys. Rev. Lett., in press (Dec 10 2010 issue).