A general analytical method for finding the quantum capacitance of graphene

The quantum capacitance of graphene has been modeled in numerous articles using approximate analytical expressions and numerical methods. However, no article shows how to analytically find the quantum capacitance of graphene for the full temperature range and for any Fermi level. This, of course, would require a complete analytical evaluation of Fermi–Dirac-type integrals valid for any temperature. This article will illustrate a method for finding the quantum capacitance of monolayer graphene in the presence of electron–hole puddles for any Fermi level and for any temperature. The method employed is easily generalized to find the quantum capacitance of bilayer graphene as well as any other material when the density of states is a known function of energy.