Three-dimensional model of single-electron tunneling between a conductive probe and a localized electronic state in a dielectric

Motivated by recent measurements of force detected single-electron tunneling, we present a three-dimensional model for the tunneling rate between a metallic tip and a localized electronic state in a dielectric surface. The tip is assumed to be semi-infinite, with electron wave functions approximated by plane waves. A localized electron state in the dielectric sample is approximated by a spherical quantum well. The tunneling rate is obtained with the help of Bardeen's approach and is compared with the results for a one-dimensional square barrier model. A comparison with experimental data is also presented.