Surface Plasmons in Thin Films

TLDR

Surface plasmon oscillations are revisited in the context of transition radiation and undamped behavior in multiple‑film systems, especially relevant to the ac Josephson effect. The study aims to determine the dispersion relations of surface plasma oscillations in normal metals for single and multiple films, including retardation effects, and to reinterpret the transition‑radiation peak observed in thin metal films. The authors analyze dispersion relations with retardation, propose a new interpretation of the transition‑radiation peak, and extend the model to superconducting metals by replacing the plasma frequency with a nearly constant value derived from the penetration depth. They find that a simple dielectric function accurately describes high‑frequency, undamped oscillations, identify two distinct mode types—low‑frequency linear modes and high‑frequency plasma‑frequency modes—and confirm that the same functional form, with a modified plasma frequency, applies to superconducting metals below the energy gap.

Abstract

The dispersion relations for surface plasma oscillations in normal metals are investigated for single- and multiple-film systems taking retardation effects into account. The simple dielectric function $\ensuremath{\epsilon}(\ensuremath{\omega})=1\ensuremath{-}\frac{{{\ensuremath{\omega}}_{p}}^{2}}{{\ensuremath{\omega}}^{2}}$ is found to be adequate for the high-frequency region in which oscillations remain undamped. Two types of possible modes of oscillation are found. One type corresponds to dispersion relations which behave linearly for not-so-high frequency, with a phase velocity always smaller than the velocity of light in the dielectric, but at least ten times larger than the Fermi velocity, while the other type consists of high-frequency modes ($\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{p}$). The role of these oscillations in the problem of transition radiation is reexamined. In the case of a thin metal film, a new interpretation is proposed for the peak observed in the transition radiation spectrum. Finally, the work is extended to superconducting metals where, in the frequency range $\ensuremath{\hbar}\ensuremath{\omega}<2\ensuremath{\Delta}$ ($2\ensuremath{\Delta}$ is the superconducting energy gap), we have justified the use of a dielectric function of the same functional form as given above but with ${{\ensuremath{\omega}}_{p}}^{2}$ replaced by an almost frequency-independent quantity ${{\ensuremath{\omega}}_{\mathrm{ps}}}^{2}$, where ${\ensuremath{\omega}}_{\mathrm{ps}}=\frac{c}{{\ensuremath{\lambda}}_{\mathrm{ps}}}$ and ${\ensuremath{\lambda}}_{\mathrm{ps}}$ is the actual penetration depth. In this frequency range, the oscillations are essentially undamped and play an important role in the electromagnetic properties of the multiple-film systems, and particularly when the systems exhibit the ac Josephson effect.