Impact of electron-phonon interaction on metal-organic interface states

The interface state (IS) between a noble metal and an organic layer shifts to lower energy by $\sim 100$ meV when heated from zero to room temperature. Here, we show that this shift can only be explained including the energy renormalization caused by a quantum-mechanical electron-phonon interaction (EPI) between molecular vibrations and the IS. We investigate the EPI by a simplified model on the bases of ab initio data. By combining EPI with a classical thermal lifting effect of the molecular layer, we explain the recently published temperature-dependent experimental data for the NTCDA/Ag(111) system.

Figure 1

Classical lifting effect of the interface state at the NTCDA/Ag(111) system following the MD and DFT results in Ref. [23]. The NTCDA molecule as shown in (a) from a top or side perspective is bound to the Ag(111) surface with an average binding distance $d$ by an anharmonic potential $V\left(d\right)$ which is printed in (b). This anharmonicity yields a temperature dependency of the distance $d\left(T\right)$ as plotted in (c) and (d). (c) shows the distribution of the binding distance for temperatures of 73, 144, 205, 249, 302, and $365\mathrm{K}$. In a static and classical view, the distributions and average values of the binding distance in (c) and (d) can be transferred to the non-EPI-corrected interface state energies $E_{\mathrm{IS}}^0\left(d\right)$ in (e) and (f) from DFT or model calculations [7, 18, 23], thus leading to an implicit temperature dependence of the latter, i.e., $E_{\mathrm{IS}}^0\left[d\left(T\right)\right]$.

Figure 2

Energy correction of the NTCDA/Ag(111) interface state through a quantum-mechanical electron-phonon interaction. (a) shows those two eigenmodes of the NTCDA/Ag(111) system that have the strongest out-of-plane deflection, i.e., the largest effective displacement amplitudes ${\overline{\xi }}^{\left(\nu \right)}$ as defined in the main text. This out-of-plane deflection is illustrated in (b) from a side perspective for both modes. Large displacements ${\overline{\xi }}^{\left(\nu \right)}$ lead to temperature-dependent modulations of the average binding distance $d$ to the Ag(111) surface. As the IS onset energy (cf. Fig. 3) is directly proportional to such modulations, this induces a linear dependency of the Hamiltonian on these displacement amplitudes, i.e., ${\widehat{H}}_{\mathrm{ep}}\sim {\overline{\xi }}^{\left(\nu \right)}\overline{\lambda }$. According to the result of second-order perturbation theory in Eq. (2), each mode then contributes a negative energy correction term to the IS which is proportional to ${\left({\overline{\xi }}^{\left(\nu \right)}\overline{\lambda }\right)}^2$. The sum of these contributions is drawn in (c). In (d), the obtained EPI corrections are added to the non-EPI-corrected IS onset energies (open blue/red triangles) from Fig. 1, yielding the solid gray triangles, in comparison to the experimental IS energies from Ref. [7]. The dashed lines are linear fits to guide the eye.

Figure 3

(a) Energy and dispersion of the IS from DFT calculations projected with a method from Ref. [24] for different vertical adsorption heights. The first height (gray) corresponds to the experimental binding distance at room temperature of $d_{\mathrm{expt}}\left(300\mathrm{K}\right)=2.95\text{Å}$. The second height (black) is the predicted zero-temperature distance of $d\left(0\mathrm{K}\right)=d_{\mathrm{expt}}\left(300\mathrm{K}\right)-\mathrm{\Delta }{d}_{\mathrm{RT}}=2.89\text{Å}$ that is obtained by subtracting our predicted room-temperature shift $\mathrm{\Delta }{d}_{\mathrm{RT}}\approx 0.06\text{Å}$ from the experimental value. The energetic position of the curves is obtained from a sum of the calculated energy shift between interface and clean surface state and the measured energy onset of the surface state [25]. The full dispersion is depicted in Ref. [23]. (b) The onset energies at $\mathit{k}=\mathrm{\Gamma }$ are compared to the IS model from Refs. [7, 18] that we adapted in Ref. [23] to our ab initio data. Additionally, a linear fit of the DFT data is drawn, showing a slope of $\overline{\lambda }=\mathrm{\Delta }E/\mathrm{\Delta }d=-1.07\mathrm{eV}/\text{Å}$.