Benchmarking van der Waals functionals with noncontact RPA calculations on graphene-Ag(111)

We have benchmarked long range behavior of seven different van der Waals functionals comparing them with our ACF-RPA correlation calculations for graphene on a Ag(111) system. Correlation given by the second version of van der Waals density functional vdW-DF2 agrees remarkably well with our random phase approximation (RPA) calculation in the long range region. In the intermediate and shorter range regions combining vdW-DF2 correlation with proper exchange functional becomes important. We compared the results of the van der Waals functionals in this region to the previous RPA calculations and to some extent to experimental observations, and calculated that the combined vdW-DF2(C09x) or rev-vdW-DF2 functionals show satisfactory behavior.

Figure 1

Geometry of the supercell used for our vdW DFT calculations. Upper part: $2\times 2$ graphene lattice matched to $\sqrt{3}\times \sqrt{3}$ Ag(111) lattice. Lower part: Geometry of the supercell in perpendicular direction.

Figure 2

Geometry of the jellium (orange)-graphene (blue) system. Jellium slab corresponds to metal surface shown as gray atoms.

Figure 3

Binding energies per carbon atom of graphene on Ag(111) for various vdW functionals and noncontact RPA correlation energy. Black circles represent data taken from Fig. 2 of Ref. [33].

Figure 4

Binding energies per carbon atom of graphene on Cu(111) and Al(111) for two vdW functionals and noncontact RPA correlation energy. Black circles represent data taken from Fig. 2 of Ref. [33].

Figure 5

Band structure of graphene on Ag(111) at the equilibrium position, calculated with the vdW-DF2(C09x) functional. Red (value 1) corresponds to graphene bands and blue (value 0) corresponds to silver bands.