Starting from the Mayadas-Shatzkes model for the conductivity in thin polycrystalline metal films, an exact expression for the dependence of the temperature coefficient of resistivity (t.c.r.) on film thickness is derived. In order to allow a quick comparison of experimental data with the model, approximate equations are also derived for the dependence on thickness of both resistivity and its temperature coefficient. These approximations are valid in the range 0.2≦ko≦5 (where kois the ratio between film thickness d and the electron mean free path lo). Numerical tables are also given (i) for the values of the exact function of the t.c.r. and (ii) to show the agreement of the approximate equations with the exact functions. © 1973.
Mola, E. E., & Heras, J. M. (1973). Exact and approximate equations for the thickness dependence of resistivity and its temperature coefficient in thin polycrystalline metal films. Thin Solid Films, 18(1), 137–144. https://doi.org/10.1016/0040-6090(73)90231-9