Polarization-induced renormalization of molecular levels at metallic and semiconducting surfaces

On the basis of first-principles $G_0{W}_0$ calculations we systematically study how the electronic levels of a benzene molecule are renormalized by substrate polarization when physisorbed on different metallic and semiconducting surfaces. The polarization-induced reduction in the energy gap between occupied and unoccupied molecular levels is found to scale with the substrate density of states at the Fermi level (for metals) and substrate band gap (for semiconductors). These conclusions are further supported by self-consistent GW calculations on simple lattice models. By expressing the electron self-energy in terms of the substrate’s joint density of states we relate the level shift to the surface electronic structure, thus providing a microscopic explanation of the trends in the GW and $G_0{W}_0$ calculations. While image charge effects are not captured by semilocal and hybrid exchange-correlation functionals, we find that error cancellations lead to remarkably good agreement between the $G_0{W}_0$ and Kohn-Sham energies for the occupied orbitals of the adsorbed molecule.

Figure 1

(Color online) (a) Supercell used to represent benzene physisorbed on NaCl(001). (b) Reduction in a molecule’s energy gap when it approaches a polarizable surface. (c) Calculated LDA, PBE0, and $G_0{W}_0$ HOMO-LUMO gap of a benzene molecule lying flat at $z=4.5\text{Å}$ above different surfaces. Note that BaO(111) is metallic due to surface states in the BaO band gap.

Figure 2

(Color online) The lattice models representing a metal-molecule and semiconductor-molecule interface, respectively. We consider the weak coupling limit where no hybridization between the molecule and surface states occur. Thus the only interaction between the solid and molecule is via the nonlocal Coulomb interaction $U$.

Figure 3

(Color online) Calculated $G_0{W}_0$ energy gap of benzene on NaCl, ${\text{TiO}}_2$, and Ti surfaces (circles) as a function of the distance to the surface, and the best fit to the classical model Eq. (8) (full lines).

Figure 4

(Color online) (a) Feynman diagrams representing dynamic polarization of the substrate induced by an electron propagating in the molecule. (b) and (c): generic shapes of the imaginary and real parts of the self-energy of Eq. (15) for an occupied molecular level $|a⟩$ interacting with a metallic and semiconducting substrate assuming $V_{k{k}^{\prime },aa}$ to be energy independent.

Figure 5

(Color online) HOMO and LUMO positions obtained from the simple lattice models for a metallic substrate (upper panel) and semiconducting substrate (lower panel). In all plots we vary one parameter while keeping the remaining parameters fixed.[34]

Figure 6

(Color online) Calculated HOMO-LUMO gap of benzene at $z=4.5\text{Å}$ (same numbers as in Fig. 1) plotted as function of the LDA substrate band gap for the semiconductors (a) and the total DOS per volume evaluated at the Fermi level for the metals (b). Dashed lines have been added to guide the eye. (c) Average electron density in a plane lying $z=3.5\text{Å}$ above the clean surfaces.

Figure 7

LDA, PBE0, and $G_0{W}_0$ energies for the HOMO and LUMO levels of benzene at $z=4.5\text{Å}$ above the surfaces. The very good agreement between LDA and $G_0{W}_0$ energies for the HOMO level at the metallic surfaces is due to error cancellation in the LDA approximation.