The electrical conductivity of polycrystalline metallic films

We calculate the electrical conductivity of polycrystalline metallic films by means of a semi-numerical procedure that provides solutions of the Boltzmann transport equation, that are essentially exact, by summing over classical trajectories according to Chambers' method. Following Mayadas and Shatzkes (MS), grain boundaries are modeled as an array of parallel plane barriers situated perpendicularly to the direction of the current. Alternatively, according to Szczyrbowski and Schmalzbauer (SS), the model consists in a triple array of these barriers in mutual perpendicular directions. The effects of surface roughness are described by means of Fuchs' specularity parameters. Following SS, the scattering properties of grain boundaries are taken into account by means of another specularity parameter and a probability of coherent passage. The difference between the sum of these and one is the probability of diffuse scattering. When this formalism is compared with the approximate formula of Mayadas and Shatzkes (Phys. Rev. B 1, 103 (1986)) it is shown that the latter greatly overestimates the film resistivity over most values of the reflectivity of the grain boundaries. The dependence of the conductivity of thin films on the probability of coherent passage and grain diameters is examined. In accordance with MS we find that the effects of disorder in the distribution of grain diameters is quite small. Moreover, we find that it is not safe to neglect the effects of the scattering by the additional interfaces created by stacked grains. However, when compared with recent resitivity-thickness data, it is shown that all three formalisms can provide accurate fits to experiment. In addition, it is shown that, depending on the respective reflectivities and distance from a surface, some of these interfaces may increase or diminish considerably the conductivity of the sample. As an illustration of this effect, we show a tentative fit of resistivity data of gold films measured by Chen et al. (Appl. Phys. 60, 659 (2005)). Finally, we present a new version of Matthiessen's rule that describe, with high accuracy, the way in which the contributions from surface scattering and grain boundary combine to form the total resistivity of the sample. (C) 2016 Elsevier B.V. All rights reserved.