Self-consistent model of edge doping in graphene

Dopants positioned near edges in nanostructured graphene behave differently from bulk dopants. Most notable, the amount of charge transferred to delocalized states (i.e., doping efficiency) depends on position as well as edge chirality. We apply a self-consistent tight-binding model to analyze this problem focusing on substitutional nitrogen and boron doping. Using a Green's-function technique, very large structures can be studied, and artificial interactions between dopants in periodically repeated simulations cells are avoided. We find pronounced signatures of edges in the local impurity density of states. Importantly, the doping efficiency is found to oscillate with sublattice position, in particular, for dopants near zigzag edges. Finally, to assess the effect of electron-electron interactions, we compute the self-energy corrected Green's function.

Figure 1

Unit cells of (a) armchair and (b) zigzag edged nanoribbons with numbering of edge sites indicated.

Figure 2

Green's function (negative of imaginary part) vs site for armchair (top panel) and zigzag (bottom panel) edges.

Figure 3

Local density of states for $n$-type impurities ($\Delta =-5\mathrm{eV})$ for various sites and edge types.

Figure 4

Impurity occupancy for different impurity potentials $\Delta$ and sites. The straight dashed lines denoted N and B are derived from the dependence of $\Delta$ on $n$ in the self-consistent tight-binding model of these atomic species. Acceptable solutions are given as intersections.

Figure 5

Self-consistent impurity occupancies for N and B dopants as a function of positions for armchair (AC) and zigzag (ZZ) edges.

Figure 6

Imaginary part of the conduction-band self-energy. The black curve indicates the band edge.

Figure 7

Negative imaginary part of the Green's function with and without electron self-energy effects.

Figure 8

Impurity occupancy with and without electron self-energy effects.