Robust Dirac point in honeycomb-structure nanoribbons with zigzag edges

The zigzag edge graphene nanoribbon, which is an antiferromagnetic insulator, is found from the density-functional theory calculation to display a robust Dirac point after N and B doping at the zigzag edge. More interestingly, we found that the robust Dirac point is a common feature of the honeycomb-structure nanoribbon with appropriate edge sites. The zigzag edge honeycomb nanoribbon is therefore expected to provide a very useful platform for material design and development.

Figure 1

(Color online) (a) The schematic illustration of atoms at the left-hand side edge sites with higher on-site energy $\Delta /2$; and the atoms at the right-hand side edge sites with lower on-site energy $-\Delta /2$. The dashed rectangle indicates the unit cell. (b) The schematic picture for the band structure of the topmost valence band and the lowest conduction band. Solid line: the band dispersion for ZGNR. Dashed line: expected band structure for the ZGNR with the on-site energies upshifted and downshifted at the left- and right-side edge sites.

Figure 2

(Color online) (a) The structure model of graphene nanoribbon with zigzag edge. The dangling bonds at the edge are all terminated by hydrogen atoms. The top views of the (b) ${\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_{m-2}{\left(\text{B-N}\right)}_1$, (c) $\text{C}{\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_{m-3}{\left(\text{B-N}\right)}_1\text{C}$, and (d) ${\left(\text{CBNC}\right)}_{m/2}$ nanoribbons with the nanoribbon width of $m=10$. The dashed rectangles indicate the unit cells.

Figure 3

The band structures of (a) ${\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_8{\left(\text{B-N}\right)}_1$ and (b) $\text{C}{\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_7{\left(\text{B-N}\right)}_1\text{C}$ nanoribbons. Fermi level is set at 0 eV. The coordinates of high-symmetry points are $\Gamma =0$ and $\text{X}=\pi /a_0$. $a_0$ is the lattice parameter of the unit cell in Fig. 2. The energy levels cross the Fermi level at $\stackrel{⃑}{k}=0.85\left(\pi /a_0\right)$ and $0.7\left(\pi /a_0\right)$ for ${\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_8{\left(\text{B-N}\right)}_1$ and $\text{C}{\left(\text{B-N}\right)}_1{\left(\text{C-C}\right)}_7{\left(\text{B-N}\right)}_1\text{C}$ nanoribbons, respectively.

Figure 4

(Color online) (a) The band structure of ${\left({\text{BC}}_2\text{N}\right)}_5$ nanoribbon. Fermi level is set at 0 eV. The energy levels cross the Fermi level at $\stackrel{⃑}{k}=0.65\left(\pi /a_0\right)$ for ${\left({\text{BC}}_2\text{N}\right)}_5$ nanoribbon. (b) The charge distributions of HOMO and LUMO at Dirac point in the real space.