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 ${\tau }_n$ of IPRs with high quantum numbers $n$ (up to 10 in this work). A systematic study is performed at the $\overline{\Gamma }$ 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 ${\tau }_n\propto n^3$. 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 $n^{-}$, where the energy difference $E_{n+}-E_{n-}$ is proportional to $n^{-3}$. The lifetimes ${\tau }_{n+}$ and ${\tau }_{n-}$ differ significantly, however, they both obey the scaling law ${\tau }_{n\pm }\propto n^3$. 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.

• Received 5 April 2013

DOI:https://doi.org/10.1103/PhysRevB.88.035449