Behavior of grain boundary resistivity in metals predicted by a two-dimensional model

Abstract

The behavior of a model for the specific grain boundary resistivity in metallic bamboo conductor lines is developed and compared to other theoretical treatments, and to experiment. The grain boundary is modeled as an array of scatterers on a plane. The scatterers are called “vacancy-ion” complexes, in which the vacancy represents the boundary free volume, and the ion is an atom adjacent to the vacancy. Three cases are investigated, that of noninterfering scatterers, a continuum of interfering scatterers, and discrete interfering scatterers. The approximations used lead to a specific grain boundary resistivity ${\text{∼10}}^{\text{−16}}\mathrm{ \Omega }{\mathrm{m}}^{\mathrm{2}}$ for aluminum, in agreement with experiment, for the first two cases. In the noninterfering case, the specific resistivity is independent of the grain boundary area. For the continuum interfering case it is found that the grain boundary resistivity is only weakly dependent on the grain boundary area, and that the grain boundary has a high probability of perfect reflection or transmission of incident electrons. The source of resistivity is from reflection of electrons. This behavior is independent of the exact interaction potential between the incident electrons and the defects which comprise the grain boundary free volume. The discrete interfering case produces specific resistivities several of orders of magnitude too large, and a strong dependence on the grain boundary area. A connection is established between the grain boundary resistivity and the electromigration wind force.