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 ωₛ. These results demonstrate a systematic reduction of the image-potential contribution to the tunneling exponent if the barrier traversal time becomes shorter than ωₛ^-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.