Quasiparticle spectra and excitons of organic molecules deposited on substrates: $G_0{W}_0$-BSE approach applied to benzene on graphene and metallic substrates

We present an alternative methodology for calculating the quasiparticle energy, energy loss, and optical spectra of a molecule deposited on graphene or a metallic substrate. To test the accuracy of the method it is first applied to the isolated benzene (${\text{C}}_6{\text{H}}_6$) molecule. The quasiparticle energy levels and especially the energies of the benzene excitons (triplet, singlet, optically active and inactive) are in very good agreement with available experimental results. It is shown that the vicinity of the various substrates [pristine/doped graphene or (jellium) metal surface] reduces the quasiparticle highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO-LUMO) gap by an amount that slightly depends on the substrate type. This is consistent with the simple image theory predictions. It is even shown that the substrate does not change the energy of the excitons in the isolated molecule. We prove (in terms of simple image theory) that energies of the excitons are indeed influenced by two mechanisms which cancel each other. We demonstrate that the benzene singlet optically active ($E_{1u}$) exciton couples to real electronic excitations in the substrate. This causes it substantial decay, such as $\Gamma \approx 174$ meV for pristine graphene and $\Gamma \approx 362$ meV for metal surfaces as the substrate. However, we find that doping graphene does not influence the $E_{1u}$ exciton decay rate.

Figure 1

Bethe-Salpeter equation for general kernel $\Xi$.

Figure 2

Bethe-Salpeter equation in time-dependent screened Hartree-Fock (TDSHF) approximation.[16]

Figure 3

$GW$ approximation for ${\Sigma }_{\mathit{XC}}$.

Figure 4

Propagator of the dynamically screened Coulomb propagator obtained from the polarizability $L$.

Figure 5

Bare Coulomb interaction matrix element.

Figure 6

Bethe-Salpeter equation.

Figure 7

BSE kernel in TDSHFA. The first term represents the BSE-Hartree kernel, the second term represents the bare BSE-Fock kernel, and the third term represents the induced BSE-Fock kernel.

Figure 8

Feynman diagrams for (a) the optical-absorption process and (b) the energy-loss process. The squares represent current vertices, and dots show charge vertices.

Figure 9

Schematic representation of (a) an optical-absorption experiment and (b) a dipole energy-loss experiment.

Figure 10

Ground-state geometry of a benzene molecule deposited on graphene. Benzene carbon and hydrogen atoms are depicted as black and white spheres, respectively, while carbon atoms of the graphene substrate are depicted as gray spheres.

Figure 11

Ground-state electronic density of benzene horizontally deposited on graphene. The position of the graphene image plane is denoted by $z_{im}$, and the equilibrium benzene-graphene separation is denoted by $z_0$. The density is plotted in the $xz$ plane across the dashed line denoted in Fig. 10.

Figure 12

Screening of the bare Coulomb interaction by the polarization of the substrate.

Figure 13

Molecular orbitals of the $\pi$-${\pi }^{*}$ complex ($1a_{2u}$,$1e_{1g}$, $1e_{1u}$, and $1b_{2g}$) involved most dominantly in the formation of benzene excitons.

Figure 14

Optical-absorption and energy-loss spectra of benzene in vacuum (black solid line), on pristine graphene (red dashed line), and in the vicinity of a Ag (jellium) surface (blue dash-dotted line). For separation between the molecular plane and graphene plane or jellium edge we take the equilibrium distance $z_0\approx 6.0a_0$. Black dotted lines are Lorentzian fits to the $E_{1u}$ optical adsorption spectra.