Molecular graphene under the eye of scattering theory

The recent experimental observations of designer Dirac fermions and topological phases in molecular graphene are addressed theoretically. Using scattering theory, we calculate the electronic structure of finite lattices of scattering centers dual to the honeycomb lattice. In good agreement with experimental observations, we obtain a V-shaped electron density of states around the Fermi energy. By varying the lattice parameter we simulate electron and hole doping of the structure, and by adding and removing scattering centers we simulate, respectively, vacancy and impurity defects. Specifically, for the vacancy defect we verify the emergence of a sharp resonance near the Fermi energy for increasing strength of the scattering potential.

Figure 1

(a) Topograph of a lattice consisting of hexagonal scatterers at an energy of $-10$ meV. The triangular and hexagonal lattices are added as guides to the eye. (b)–(d) Local DOS as a function of (b) the number of scattering points in each scattering center [seven points (hexagon), four points (triangle), and one point], (c) the diameter of the (hexagonal) scattering centers, and (d) lattice size: $5\times 6$, $9\times 10$, $13\times 14$, and $17\times 18$ (hexagonal) scattering centers. In (a), (b), and (d) we used a diameter of $4.8$ Å for the hexagonal scattering centers and lattice parameter $a=19.2$ Å. For all plots we have used ${\varepsilon }_{\mathbf{k}}-{\varepsilon }_F=E_0+{\hslash }^2{k}^2/2m^{*}$, with $E_0\simeq -0.45$ eV for Cu(111), and $m^{*}=0.38m$, where $m$ is the free-electron mass.

Figure 2

Electron DOS for (a) varying lattice parameter $a$, going from hole-doped (dotted line) via nearly neutral (solid line) to electron-doped (dashed line) molecular graphene, and (b) Kekulé texturing of the scattering centers. The inset shows the geometry of the Kekulé texturing. Other parameters are as in Fig. 1.

Figure 3

Molecular graphene under a pseudomagnetic field corresponding to 60 T ($q={10}^{-3}$ Å). (a) Spectral density and (b) electron DOS in the $A$ (solid line) and $B$ (dashed line) sublattices.

Figure 4

Effect of a single defect in molecular graphene. (a) Electron DOS for a varying number $\{7,13,25,37\}$ of scattering points within the vacancy site. The thin line shows the unperturbed electron DOS for reference. (b) Perturbed divided by unperturbed electron DOS. The inset shows the geometry of the modified lattice including the vacancy site. The spatial spectral density for the cases of (c) 7 and (d) 13 scattering points at the vacancy site. (e) Spatial spectral density for a single impurity in the plaquette position.