Comparison of a Numerical Method and the WKB Approximation in the Determination of Transmission Coefficients for Thin Insulating Films

Abstract

A direct numerical solution of the Schrödinger equation for the case of electron tunneling through thin-film metal—insulator—metal sandwiches is described. Curves of transmission coefficient vs energy are obtained for an image-force barrier model by this method and are then compared, for applied fields ranging from 103 to 109 V/m, to analogous curves obtained by application of the WKB approximation. For a constant applied field, the WKB treatment predicts transmission coefficients which are smoothly varying functions of energy and monotonically decreasing functions of insulator thickness at all energies. On the other hand, the corresponding numerically computed quantities show definite periodic oscillations in energy and also thickness ``resonances.'' The dependence of these oscillations on energy and thickness is shown to be the result of the partial reflection and interference of electron waves as the electron beam penetrates the barrier region representing the insulator film. The numerically computed transmission curves indicate that these reflection effects are significant at very low energies and at energies approaching the barrier maximum and that the resulting interference becomes significant at high energies and fields where the electron wavelength becomes comparable to the dimensions of the barrier. At the low energies, the major portion of the reflection is shown to originate from the regions adjacent to the metal—insulator interfaces; at the high energies, the reflection is attributed mostly to regions contiguous to the first metal—insulator interface and the top of the potential barrier. In all cases, however, the conditions for validity of the WKB treatment are seen to rule out these effects. Finally, the numerical results confirm the expected breakdown of the WKB connection formulas at those energies where a major portion of the barrier region is reflecting.