Exact analytical solution to the electron density for monolayer and bilayer graphene

This article presents a mathematical method for analytically evaluating the electron density for both monolayer and bilayer graphene. Once an appropriate probability function is chosen for finding the electronic density of states, a complete analytical solution is found for the electronic density without recourse to any analytical approximations or asymptotic formulations. The form of the solution is in terms of a convergent series involving the complementary error function. The analytical solution developed in this paper is valid for all temperatures and for all Fermi energies greater than or equal to zero. The method is easily extended to include Fermi energies less than zero.