Singularities in core-level spectra near metallic surfaces

Calculations of the screening shift and dynamical singularity index of the structureless core hole placed near and inside the metallic slab are presented, where the screening mechanism is described using the quantum-mechanical response function calculated in the local density approximation. Core-hole screening shifts show oscillations which are compared with the experimental results. Spatial dependence of the static singularity index, besides experimentally observed enhancement at the surface, shows some new features (bulk oscillations) which were not yet experimentally observed. The dynamical singularity index is used to calculate complete core hole spectrum, including singular line shapes and bulk and surface plasmon losses, and the results explain very well some observed features in the experimental photoemission spectra.

Figure 1

(a) Hole-electron interaction. (b) Electron-electron interaction.

Figure 2

Cumulant expansion. (a) One of the neglected diagrams of higher order in $V_1$ (or $w$). (b) Cumulant sum containing all diagrams of second order in $V_1$. (c) Irreducible polarizability $\Pi$.

Figure 3

Response function $\chi ({\mathbf{r}}_1,{\mathbf{r}}_2,t_1,t_2)$.

Figure 4

Metallic slab electron density profile (thick line), positive background of width $a$ (thin line), and electron density edges (dashed lines).

Figure 5

Frequency dependence of the singularity index $\alpha (Z,\omega )$, for $a=100a_0$, $r_s=2$.

Figure 6

Spatial dependence of the singularity index $\alpha (Z,\omega )$, for $a=40a_0$, $r_s=2$. The picture in the upper right corner shows the $r_s=4$ case.

Figure 7

(a) Core-hole energy shift $\mathrm{\Delta }E\left(Z\right)$: $r_s=2$ (full line); $r_s=4$ (dashed line). The vertical dotted line represents the jellium edge. [(b) and (c)] Oscillations of the surface core-level shift (SCLS) with amplitudes 75 and $225\mathrm{meV}$ for $r_s=2$ and $4$, respectively. The dots show the approximate positions of the monolayers for $\left(100\right)$ orientated surfaces.

Figure 8

Full lines represent the core-hole spectra for various hole positions inside and outside the metallic slab ($a=50a_0$ and $r_s=2$); dashed lines are the appropriate DS line shapes. Core-hole decay constant is $\gamma =0.002{\omega }_S$.

Figure 9

$\gamma$ dependence of the strength of the no-loss peak. $Z=-10a_0$.

Figure 10

Core-hole spectra for various core-hole lifetimes $\gamma$ (thin lines); thick lines represent the corresponding DS line shapes. The core-hole position is $Z=-10a_0$, $a=50a_0$, and $r_s=2$.