Scattering-theoretic approach to elastic one-electron tunneling through localized barriers: Application to scanning tunneling microscopy

Abstract

A new formulation of elastic one-electron tunneling through three-dimensional (3D), nonseparable, spatially localized barriers is developed in terms of potential-scattering theory. To illustrate the principles of the method, a model metal-vacuum-metal junction is used, consisting of two parallel electrodes, one of which has a hemispherical protrusion. The electronic structure of each metal electrode is assumed to be free-electron-like, for simplicity. The bias and multiple-image tunneling barriers for this model are constructed on the basis of classical electrostatics and a simple quantum correction at the metal surfaces. Regarding the barrier as made of a planar, separable part plus a nonseparable, localized perturbation due to the spherical boss, the exact, unperturbed, one-electron Green’s function of the planar part is first obtained by numerical integration of the corresponding, effectively 1D Schrödinger equation. Then the localized boss potential is treated to all orders of perturbation by solving the Dyson equation for the full barrier Green’s function, using a real-space discretization of the integral equation on a finite grid.