Long-range interactions between substitutional nitrogen dopants in graphene: Electronic properties calculations

Being a true two-dimensional crystal, graphene has special properties. In particular, a pointlike defect in graphene may induce perturbations in the long range. This characteristic questions the validity of using a supercell geometry in an attempt to explore the properties of an isolated defect. Still, this approach is often used in ab initio electronic structure calculations, for instance. How does this approach converge with the size of the supercell is generally not tackled for the obvious reason of keeping the computational load to an affordable level. The present paper addresses the problem of substitutional nitrogen doping of graphene. DFT calculations have been performed for $9\times 9$ and $10\times 10$ supercells. Although these calculations correspond to N concentrations that differ by $\sim$10%, the local densities of states on and around the defects are found to depend significantly on the supercell size. Fitting the DFT results by a tight-binding Hamiltonian makes it possible to explore the effects of a random distribution of the substitutional N atoms, in the case of finite concentrations, and to approach the case of an isolated impurity when the concentration vanishes. The tight-binding Hamiltonian is used to calculate the STM image of graphene around an isolated N atom. STM images are also calculated for graphene doped with 0.5 at% concentration of nitrogen. The results are discussed in the light of recent experimental data and the conclusions of the calculations are extended to other point defects in graphene.

Figure 1

(a) DFT and (b) $\pi$ tight-binding local densities of states (LDOS) of graphene doped with nitrogen at a concentration of one impurity per $9\times 9$ supercell (162 atoms). The different curves represent the local DOS on the N atom, on the first-neighbor carbons (C1), and on a few distant carbons (bulk). (c) Labels assigned to the nitrogen dopant (N) and its first, second, third, and fourth neighbors (C1, C2, C3, and C4). (d) Tight-binding LDOS of the $9\times 9$ superstructure (same as in panel b) without energy broadening. (e) DFT and (f) tight-binding LDOS on and around a substitutional N atom in graphene with $10\times 10$ periodic distribution (200 atoms per cell). The curves, shifted vertically for clarity, correspond to the atomic sites N, C1, C2, and C3 shown in panel (c). For all LDOS curve plotted in this figure, the zero energy corresponds to the Fermi level.

Figure 2

DFT band structures of the $9\times 9$ (top) and $10\times 10$ (bottom) N-doped superstructures shown along the high-symmetry lines of their first Brillouin zone. The zero of energy coincides with the Fermi level.

Figure 3

Tight-binding local $\pi$ densities of states on and around an isolated impurity in graphene. The labels correspond to the defect site (N), the first, second, and third neighbors (C1, C2, and C3). The curves have been shifted vertically for clarity. The zero of energy is ${\varepsilon }_C$.

Figure 4

Configurational average of the local $\pi$ densities of states on and around substitutional N atoms in a graphene sheet containing 0.5 at$\%$ of nitrogen atoms randomly distributed on the two sublattices. The labels correspond to the nitrogen dopant (N), the first, second, and third neighbors (C1, C2, and C3). The curves have been shifted vertically for clarity. The zero of energy is ${\varepsilon }_C$.

Figure 5

Configurational average of the local $\pi$ densities of states on and around substitutional N atoms in a graphene sheet containing 0.5 at$\%$ of nitrogen atoms randomly distributed on the same sublattice. The labels correspond to the nitrogen dopant (N), the first, second, and third neighbors (C1, C2, and C3). The curves have been shifted vertically for clarity. The zero of energy is ${\varepsilon }_C$.

Figure 6

Local DOS on three N atoms randomly selected in graphene doped at concentration 0.5 at$\%$. The unpolarized case (left) refers to dopants located on the two sublattices, whereas the dopants all sit on the same sublattice in the polarized case (right). The zero of energy is ${\varepsilon }_C$.

Figure 7

Tight-binding STM constant-current image of graphene with an isolated N impurity located near the center, computed for a negatively polarized tip ($V_{\text{tip}}=-0.5$ V). The vertical scale is the tip height (Å) above sample. The image size is 2 nm along both sides.

Figure 8

Constant-current STM images computed for two configurations of graphene doped with 0.5 at$\%$ N. The distribution of the N atoms is random. The two nitrogen atoms visible in image (a) lie on the same sublattice, which is not the case in image (b). In both cases, the STM tip is polarized at $-$0.5 V with respect to the sample. Each image is a square of 4 nm edge.

Figure 9

DFT calculation of constant-current STM images of $10\times 10$ N-doped graphene for two levels of the tunnel current: (a) high and (b) low (see text). The tip bias polarization is $-$0.5 V, the vertical scale is the tip height (Å) above sample. Each image represents a square of 4 nm edge centered on a N dopant.

Figure 10

Real [$F^0\left(E\right)$] and imaginary [$n^0\left(E\right)$] parts of the Green function of the graphene $\pi$ tight-binding Hamiltonian along the real axis, in units of $|{\gamma }_0|$. The intersection of $F^0\left(E\right)$ with the reciprocal of the perturbed level $1/U$ (dashed line) in the vicinity of the Dirac point gives the energy $E_r$ at which the local DOS of the impurity has a resonance (open circle symbol). There is another intersection (filled symbol) giving rise to a localized state below the $\pi$-electron density of states.

Figure 11

Local densities of states deduced from the simple tight-binding model for an isolated nitrogen impurity in graphene. (a) On-site perturbation energy $U=-10$ eV on the N site only, (b) on-site perturbation on the N ($-$4 eV), and the three C1 sites ($-$2.57 eV).