Electrical-Resistivity Model for Polycrystalline Films: the Case of Arbitrary Reflection at External Surfaces

In this paper, the total resistivity of a thin metal film is calculated from a model in which three types of electron scattering mechanisms are simultaneously operative: an isotropic background scattering (due to the combined effects of phonons and point defects), scattering due to a distribution of planar potentials (grain boundaries), and scattering due to the external surfaces. The intrinsic or bulk resistivity is obtained by solving a Boltzmann equation in which both grain-boundary and background scattering are accounted for. The total resistivity is obtained by imposing boundary conditions due to the external surfaces (as in the Fuchs theory) on this Boltzmann equation. Interpretation of published data on grain-boundary scattering in bulk materials in terms of the calculated intrinsic resistivity, and of thin-film data in terms of the calculated total resistivity suggests that (i) the grain-boundary reflection coefficient in Al is \ensuremath{\approx} 0.15, while it is somewhat higher in Cu; (ii) the observed thickness dependence of the resistivity in thin films is due to grain-boundary scattering as well as to the Fuchs size effect; and (iii) the common observation that single-crystal films possess lower resistivities than polycrystalline films may be accounted for by grain-boundary effects rather than by differences in the nature of surface scattering.