Self-consistent dynamic image potential in tunneling

We discuss the self-consistent determination of the dynamic polarization or ‘‘image’’ potential in tunneling within linear-response theory. A generalized WKB expression for the tunneling exponent is derived for tunneling into or out of a conductor (e.g., in field emission), as well as between two semi-infinite polarizable electrodes separated by a potential barrier. Numerical results are presented for the tunneling exponent, the barrier traversal time, and effective thickness for simple models assuming coupling to dispersionless surface plasmons of frequency ${\omega }_s$. These results demonstrate a systematic reduction of the image-potential contribution to the tunneling exponent if the barrier traversal time becomes shorter than ${\omega }_s^{\mathrm{-}1}$. We estimate that, in tunneling from a metal tip to a metal surface, the static image potential overestimates the polarization contribution to the tunneling exponent by 20–30 % in a typical scanning-tunneling-microscope measurement.