  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.
