On the lifetime of the parallel vibrational motion of adsorbates on metallic surfaces

We point out that the relationship between the adatom-induced surface resistivity, rho(s), and the lifetime of the parallel vibrational motion of adatoms, tau, recently derived by Persson can be recast to read as F-f/J(f) = n(e)el(f) rho(s)/n(a), where F-f, J(f), n(e), I-f, and n(a) are the damping force on the adatom, the apparent current density in the metal seen from the adatom, the electron density in the metal, the him thickness, and the 2-dimensional density of adatoms, respectively. This has essentially the same form as the relationship between rho(s) and the indirect (''electron-wind'') force F-w on the adatom located in a current density J(w), i.e., F-w/J(w) = n(e)el(f) rho(s)/n(a), which has been known for a long time in the theory of electromigration. Using a first-principles linear response approach, we show that both the equations hold true if the substrate is translationally invariant in the plane (fiat jellium model). We derive a simple expression of the surface resistivity, rho(s), which can be evaluated from the scattering amplitudes of one-electron wavefunctions at the Fermi energy alone.