Gate-induced carrier density modulation in bulk graphene: theories and electrostatic simulation using MATLAB pdetool

This article aims at providing a self-contained introduction to theoretical modeling of gate-induced carrier density in graphene sheets. For this, relevant theories are introduced, namely, classical capacitance model (CCM), self-consistent Poisson-Dirac method (PDM), and quantum capacitance model (QCM). The usage of MATLAB pdetool is also briefly introduced, pointing out the least knowledge required for using this tool to solve the present electrostatic problem. Results based on the three approaches are compared, showing that the quantum correction, which is not considered by the CCM but by the other two, plays a role only when the metal gate is exceedingly close to the graphene sheet, and that the exactly solvable QCM works equally well as the self-consistent PDM. Practical examples corresponding to realistic experimental conditions for generating graphene pnp junctions and superlattices, as well as how a background potential linear in position can be achieved in graphene, are shown to illustrate the applicability of the introduced methods. Furthermore, by treating metal contacts in the same way, the last example shows that the PDM and the QCM are able to resolve the contact-induced doping and screening potential, well agreeing with the previous first-principles studies.