Theory of carrier density in multigated doped graphene sheets with quantum correction

The quantum capacitance model is applied to obtain an exact solution for the space-resolved carrier density in a multigated doped graphene sheet at zero temperature, with quantum correction arising from the finite electron capacity of the graphene itself taken into account. The exact solution is demonstrated to be equivalent to the self-consistent Poisson-Dirac iteration method by showing an illustrative example, where multiple gates with irregular shapes and a nonuniform dopant concentration are considered. The solution therefore provides a fast and accurate way to compute spatially varying carrier density, on-site electric potential energy, as well as quantum capacitance for bulk graphene, allowing for any kind of gating geometry with any number of gates and any types of intrinsic doping.

Figure 1

(a) Schematic of a graphene sheet subject to $N$ metalic gates. (b) Equivalent circuit plot of (a) with quantum capacitance of graphene $Q_C$ taken into account.

Figure 2

(a) Side view of a graphene sheet (with a hyperbolic-tangent-shaped intrinsic doping $n_0$) and a backgate (with $V_{\text{bg}}=-20$ V) sandwiching a SiO${}_2$ with two embedded local gates (with $V_{\text{lg1}}=-1.8$ V and $V_{\text{lg2}}=1.5$ V); the color shading shows the electric potential $u(x,z)$ obtained by the self-consistent Poisson-Dirac method. (b) The electric potential at the graphene layer $V_G\left(x\right)$ obtained by the Poisson-Dirac method and the quantum capacitance model.

Figure 3

Carrier density profiles of intrinsic doping $n_0\left(x\right)$, classical capacitance model $n_{\text{CC}}\left(x\right)$, Poisson-Dirac method $n_{\text{PD}}\left(x\right)$, quantum capacitance model $n_{\text{QC}}\left(x\right)$, and the difference $n_{\text{PD}}\left(x\right)-n_{\text{QC}}\left(x\right)$, with identical parameters used in Fig. 2.