Cluster formation and non-metal-to-metal transition in a diamond-shaped graphene-like lattice

This work is a numerical simulation of the bond percolation in an array of junctions and bifurcations mimicking a section of a graphene slab. We calculate the size distribution of graphene clusters as a filler of a polymer aiming to obtain the percolation threshold. We obtained the sigmoidal distribution of graphene clusters as a function of concentration of persistent conducting bonds creating these clusters. The probability density of this distribution shows a universal complementary Fermi–Dirac behavior as a signature of a topological response. Using a tight-binding model for the transmission from the source to the drain, we obtain a smooth transition from an insulator to a conductor through a dirty metal as the concentration of conductive bonds increases for small arrays. As the array size increases, the simulation shows a sharp non-metal-to-metal transition from a pure polymer into a pristine suspended graphene layer.