Surface plasmons on Pd(110): An ab initio calculation

The surface excitation spectra of the Pd(110) surface is analyzed in the framework of the time-dependent density functional theory. The ab initio electronic structure of this surface is included into the evaluation of the surface response function using a linear response approach. At small momentum transfer the extrapolated energy of the surface plasmon is close to electron-energy-loss-spectroscopy measurements and can be understood considering the bulk Pd dielectric function. However, upon momentum increase the evaluated dispersion of the surface plasmon presents significant discrepancies with the published experimental data. While the experimental surface plasmon dispersion initially presents a strong negative slope, our calculations indicate a weak positive dispersion at all finite momentum transfers where this collective excitation is observed. Interestingly, we do not observe any noticeable effect on the collective surface excitations induced by the large number of surface states in the valence band of Pd(110). These states do not produce any additional mode at energies below the surface plasmon.

Figure 1

Band structure of the 19-atomic layer Pd(110) film calculated along the lines of high symmetry in the surface Brillouin zone. The states with strong localization in the surface atomic layer are highlighted by blue dots.

Figure 2

ALDA surface loss function of Pd(110) calculated for the 19-atomic layer slab along the lines of high symmetry, $\overline{\mathrm{\Gamma }}\overline{X}$ and $\overline{\mathrm{\Gamma }}\overline{Y}$, in the surface Brillouin zone. The dashed lines show the classical dispersion relationship for surface modes in a thin film, where the surface plasmon SP splits into two hybrid surface modes ${\mathrm{SP}}^{-}$ and ${\mathrm{SP}}^{+}$ for small momentum transfers. This dispersion has been obtained for a film of the same thickness as in the ab initio calculations and using a Drude model to describe the permittivity of Pd, where the Drude surface plasmon frequency ${\omega }_{\mathrm{SP}}=7.35\mathrm{eV}$ is obtained from the extrapolation of the TDDFT results to $q_{\parallel }=0$. The experimental data points indicated by green dots represent the measured SP dispersion, according to Ref.  [43].

Figure 3

(a) Surface loss function of a 19-atomic layer Pd(110) slab as obtained with a classical electromagnetic calculation using the experimental bulk dielectric function of Pd [79]. (b) Surface loss function of Pd(110) calculated within the ab initio approach along the $\overline{\mathrm{\Gamma }}\overline{X}$ line (zoom in of Fig. 2). The dashed lines show the dispersion of the two hybrid surface modes ${\mathrm{SP}}^{-}$ and ${\mathrm{SP}}^{+}$ according to the Drude model as in Fig. 2. The blanked region in (b) as well as in Fig. 2 at $q_{\parallel }<0.017{\AA }^{-1}$ is inaccessible in our ab initio calculations.

Figure 4

The RPA and ALDA surface loss functions of Pd(110) calculated at $q_{\parallel }=0.370 {\AA }^{-1}$.

Figure 5

Distribution of the (a) real and (b) imaginary parts of the ALDA dynamical charge density, $n^{\mathrm{ind}}(z,{\mathbf{q}}_{\parallel },\omega )$, induced at the Pd(110) surface by an external potential $V^{\mathrm{ext}}(\mathbf{r},t)$. Results are plotted as a function of energy $\omega$ and coordinate $z$ for $q_{\parallel }=0.084 {\AA }^{-1}$ along the $\overline{\mathrm{\Gamma }}\overline{\mathrm{X}}$ symmetry direction. The origin of the $z$ coordinate is at the geometrical surface position shown by the horizontal dashed lines. The horizontal solid lines define the palladium atomic layer positions.

Figure 6

Distribution of the (a) real and (b) imaginary parts of the ALDA dynamical charge density, $n^{\mathrm{ind}}(z,{\mathbf{q}}_{\parallel },\omega )$, induced at the Pd(110) surface by an external potential $V^{\mathrm{ext}}(\mathbf{r},t)$. Results are plotted as a function of energy $\omega$ and coordinate $z$ for $q_{\parallel }=0.336 {\AA }^{-1}$ along the $\overline{\mathrm{\Gamma }}\overline{\mathrm{X}}$ symmetry direction. The origin of the $z$ coordinate is at the geometrical surface position shown by the horizontal dashed lines. The horizontal solid lines define the palladium atomic layer positions.

Figure 7

ALDA surface loss function of Pd(110) calculated for the indicated momentum transfers. The dashed line shows the surface loss function calculated at $q_{\parallel }=0$ classically employing the experimental bulk dielectric function of Pd reported in the textbook of Palik [79]. For the latter, the inset shows a zoom in the 2.5–5.5 eV energy interval.