Self-consistent tight-binding model of B and N doping in graphene

Boron and nitrogen substitutional impurities in graphene are analyzed using a self-consistent tight-binding approach. An analytical result for the impurity Green's function is derived taking broken electron-hole symmetry into account and validated by comparison to numerical diagonalization. The impurity potential depends sensitively on the impurity occupancy, leading to a self-consistency requirement. We solve this problem using the impurity Green's function and determine the self-consistent local density of states at the impurity site and, thereby, identify acceptor and donor energy resonances.

Figure 1

Schematic illustration of the impurity model assuming identical hopping and overlap integrals between all nearest neighbors.

Figure 2

Green's function of the orthogonal (upper panel) and nonorthogonal $s$ = 0.15 (lower panel) models. In the lower panel, the circle indicates the pole contribution to the impurity occupancy.

Figure 3

Energy dependence of the local density of states (DOS) computed by numerical diagonalization (solid lines) and analytically (dashed lines). Both perturbed (Δ = −5 eV) and unperturbed (Δ = 0) cases are shown.

Figure 4

Occupancy of an impurity versus potential Δ (blue curve). The straight lines are derived from the occupancy-dependent on-site potentials of N (red) and B (green) atoms, respectively. Both isolated-atom (full) and effective (dashed) Hubbard parameters are considered. The intersections correspond to self-consistent solutions.

Figure 5

Local density of states (DOS) for N- (top) and B-impurities (bottom). Results for different values of the overlap $s$ are shown. The insets display the $s$ dependence of donor and acceptor resonances, both relative to the Fermi level.