Charge transfer energies of benzene physisorbed on a graphene sheet from constrained density functional theory

Constrained density functional theory (CDFT) is used to evaluate the energy level alignment of a benzene molecule as it approaches a graphene sheet. Within CDFT the problem is conveniently mapped onto evaluating total energy differences between different charge-separated states, and it does not consist in determining a quasiparticle spectrum. We demonstrate that the simple local density approximation provides a good description of the level alignment along the entire binding curve, with excellent agreement to experiments at an infinite separation and to $GW$ calculations close to the bonding distance. The method also allows us to explore the effects due to the presence of graphene structural defects and of multiple molecules. In general, all our results can be reproduced by a classical image charge model taking into account the finite dielectric constant of graphene.

Figure 1

Top view of the (a) H, (b) S, (c) ${\mathrm{H}}_{\mathrm{sw}-\mathrm{T}}$, and (d) ${\mathrm{H}}_{\mathrm{sw}-\mathrm{A}}$ configurations. Red spheres are carbon atoms belonging to benzene ring, while yellow spheres are those of the graphene sheet. In the graphene sheet with a SW defect, the carbon atoms in all rings affected by the deformation are marked in gray.

Figure 2

Charge transfer energy gap $E_{\mathrm{CT}}^{+}-E_{\mathrm{CT}}^{-}$ for different unit cell sizes of graphene sheet. The results are presented for two different molecule-to-graphene distances: 3.4 and 6.8 Å.

Figure 3

Renormalization of the charge transfer gap as a function of the distance between the molecule and the graphene sheet (a). The dashed line denotes the difference between the IP and the EA calculated with the $\mathrm{\Delta }\text{SCF}$ method. (b) and (c) The excess charge in different parts of the system after the transfer of an electron from the molecule to the sheet for $d=3.4\text{Å}$. (d) and (e) The same plot for $d=6.8\text{Å}$. In both cases, red and blue denote positive and negative net charge, respectively. In practice the charge distribution over the graphene plane corresponds to the charge distribution of the image charge.

Figure 4

DOS of pristine graphene (black line), DOS of a graphene sheet with one SW defect (orange line), and PDOS for the atoms adjacent to the SW defect (blue shade). The latter has been multiplied by 3 for better visibility. These calculations are performed with a supercell of 200 atoms containing one SW defect.

Figure 5

Image charge formation for various configurations in which more than one benzene molecule is present. (a) and (b) The side and top view of the isovalue of $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$ after transferring one electron from the molecule to the sheet for the ${\mathrm{H}}_{\mathrm{M}1}$ case. (c) and (d) and (e) and (f) Plots of the same quantity for the cases of ${\mathrm{H}}_{\mathrm{M}2}$ and ${\mathrm{H}}_{\mathrm{L}}$, respectively.

Figure 6

$-E_{\mathrm{CT}}^{+}$ (circles) and $-E_{\mathrm{CT}}^{-}$ (squares) calculated for different molecule-to-substrate distances. The CDFT results are seen to agree well with the classically calculated curve given in red. The horizontal lines mark the same quantities for an isolated molecule (gas-phase quantities). The continuous blue line shows the position of the classically calculated level curve for adsorption on a perfect metal $\epsilon =\infty$.