Green's function approach to the lifetimes of image potential resonances at metal surfaces

Abstract

We present a theoretical study of the image potential resonances (IPRs) at metal surfaces. We develop the Green's functions approach allowing us to calculate binding energies ${E}_{n}$ and lifetimes ${\ensuremath{\tau}}_{n}$ of IPRs with high quantum numbers $n$ (up to 10 in this work). A systematic study is performed at the $\overline{\ensuremath{\Gamma}}$ point for the close-packed metal surfaces: Cu(111), Ag(111), Au(111), Al(001), Al(111), Be(0001), Mg(0001), Na(110), Li(110), and also at the $\overline{\mathrm{Y}}$ point on Cu(110). The calculated lifetimes of IPRs on close-packed surfaces demonstrate the scaling law ${\ensuremath{\tau}}_{n}\ensuremath{\propto}{n}^{3}$. Our results are in agreement with available experimental data. We show that at the $\overline{\mathrm{Y}}$ point on Cu(110) each quantum number $n$ corresponds to a pair of IPRs ${n}^{+}$ and ${n}^{\ensuremath{-}}$, where the energy difference ${E}_{n+}\ensuremath{-}{E}_{n\ensuremath{-}}$ is proportional to ${n}^{\ensuremath{-}3}$. The lifetimes ${\ensuremath{\tau}}_{n+}$ and ${\ensuremath{\tau}}_{n\ensuremath{-}}$ differ significantly, however, they both obey the scaling law ${\ensuremath{\tau}}_{n\ifmmode\pm\else\textpm\fi{}}\ensuremath{\propto}{n}^{3}$. Since the electrons trapped in the long-lived IPRs are strongly localized on the vacuum side, we argue that the inelastic electron-electron and electron-phonon scattering have a small contribution to the decay rate of these IPRs. The latter is dominated by the resonant electron transfer into the bulk.