Microscopic calculation of electromagnetic fields in refraction at a jellium-vacuum interface

TLDR

The spatial dependence of the vector potential for a long‑wavelength electromagnetic wave incident on a jellium‑vacuum interface is evaluated in the surface region. The authors propose using these vector‑potential results to compute refraction effects in surface photoemission and reflection spectroscopy. They derive a simple integral equation linking the vector potential to the jellium nonlocal conductivity tensor, solve it numerically within the random‑phase approximation using self‑consistent Lang–Kohn surface‑barrier potentials, and examine its sensitivity to the barrier shape. Graphical families of the vector potential versus frequency and versus bulk electron concentration are presented, illustrating its dependence on these parameters.

Abstract

For a long-wavelength electromagnetic wave of frequency $\ensuremath{\omega}$, incident on a jellium-vacuum interface, the spatial dependence of the vector potential, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\ensuremath{\omega})$, is evaluated in the surface region. A simple integral equation is derived which relates $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\ensuremath{\omega})$ to the jellium nonlocal conductivity tensor $\stackrel{\ensuremath{\leftrightarrow}}{\ensuremath{\sigma}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\ensuremath{\omega})$; numerical calculations based on this equation are reported, in which the random-phase approximation to $\stackrel{\ensuremath{\leftrightarrow}}{\ensuremath{\sigma}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\ensuremath{\omega})$ was used---the required single-electron wave functions were evaluated via the self-consistent surface-barrier potentials of Lang and Kohn. Families of graphs of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\ensuremath{\omega})$ are presented, for fixed bulk electron concentration as a function of frequency and for fixed $\frac{{\ensuremath{\omega}}_{p}}{\ensuremath{\omega}}$ (where ${\ensuremath{\omega}}_{p}$ is the plasma frequency) as a function of bulk electron concentration. The sensitivity of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\ensuremath{\omega})$ to the shape of the surface potential barrier, at fixed $\ensuremath{\omega}$ and bulk electron concentration, is also explored. The use of the results obtained for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}};\ensuremath{\omega})$ is proposed for the calculation of refraction effects in surface photoemission and in reflection spectroscopy.