Effects of extrinsic and intrinsic perturbations on the electronic structure of graphene: Retaining an effective primitive cell band structure by band unfolding

We use a band unfolding technique to recover an effective primitive cell picture of the band structure of graphene under the influence of different types of perturbations. This involves intrinsic perturbations, such as structural defects, and external ones, comprising nitrogen substitutions and the inclusion of graphene in adsorbed systems. In such cases, the band unfolding provides a reliable and efficient tool for quantitatively analyzing the effect of doping and defects on the electronic structure of graphene. We envision that this approach will become a standard method in the computational analysis of graphene's electronic structure in related systems.

Figure 1

Band structure for a $4\times 4$ SC of graphene, before (blue lines) and after ($\times$ marks) unfolding, along the high-symmetry directions of graphene's PCBZ.

Figure 2

SW defect in graphene: (a) and (b) EBS, and (c) equilibrium geometry. In (b), the $k$ axis is perpendicular to the $K\text{−}\Gamma$ direction, and the point $K$ is at $k=0$. The color scale represents $\delta N$ (see main text).

Figure 3

V${}_2$(555-777) defect in graphene: (a) and (b) EBS, and (c) equilibrium geometry. In (b), the $k$ axis is perpendicular to the $K\text{−}\Gamma$ direction, and the point $K$ is at $k=0$. The color scale represents $\delta N$ (see main text).

Figure 4

N${}_2^{AA}$ substitutional defect in graphene: (a) and (b) EBS, and (c) equilibrium geometry. In (b), the $k$ axis is perpendicular to the $K\text{−}\Gamma$ direction, and the point $K$ is at $k=0$. The color scale represents $\delta N$ (see main text).

Figure 5

F4-TCNQ@graphene: (a) and (b) EBS, and (c) equilibrium geometry. In (b), the $k$ axis is perpendicular to the $K\text{−}\Gamma$ direction, and the point $K$ is at $k=0$. The color scale represents $\delta N$ (see main text).

Figure 6

Graphene@Cu(111): (a) and (b) EBS, and (c) employed SC. In (b), the $k$ axis is perpendicular to the $K\text{−}\Gamma$ direction, and the point $K$ is at $k=0$. The color scale represents $\delta N$ (see main text).