Ordinary Size Effects and Deviations from Matthiessen's Rule in the Resistance of Fine Wires

Contrary to previous statements in the literature, large deviations from Matthiessen's rule in fine wires are to be expected on the basis of a straight-forward solution of the ordinary transport equation, assuming the relaxation-time approximation and imposing the idealized condition of "diffuse" scattering of electrons at the boundaries. Using Chambers' path-integral method to evaluate the current density in a wire of arbitrary cross-sectional shape, the effects of boundary scattering on the resistivity in the regime d less than or similar to 0.1 lambda have been calculated for two model Fermi surface geometries. For the temperature-dependent part of the resistivity, Delta rho(d)(T) rho(d)(T) - rho(d)(0), two distinct types of behavior are found in the alternative cases: (1) for a spherical Fermi surface, Delta rho(d)(T) increases logarithmically with pal(0); (2) for a cylindrical Fermi surface, Delta rho(d)(T) increases essentially linearly with rho(d)(0). [In each case the qualitative dependence of rho(d)(0) on lambda/d is, for practical purposes, "linear." However, the correct value of the product rho(infinity)lambda in the cylindrical case is not simply given in the ordinary way by the slope of an empirical plot of rho(d)(0) vs. d(-1).] A comparison of theoretical results for the two simple models with the published data for indium and gallium shows that the actual temperature-dependent size effects are consistent, both qualitatively and, by a rough estimation, quantitatively, with the expected behavior.