Impurity resistivity of an ideal metallic thin film

In the free-electron model of a metallic thin film with hard-wall boundary conditions, the discretization of the energy levels makes it impossible to treat both the Fermi energy and the electron density as independent of the thickness. One can assume the electron Fermi energy to remain at its bulk value independent of thickness. In this case, the electron density will decrease in a nonlinear way with decreasing film thickness. Consequently, the impurity conductivity will decrease. In the opposite extreme, one can assume the electron density to be a constant. In this case, the Fermi energy will increase with decreasing film thickness. Consequently, the impurity conductivity will increase. To investigate and compare these two models, one has to calculate the impurity resistivity of an ideal metallic thin film with the constant electron density. The calculation shows that the two models display quite different behavior for impurity resistivity as a function of film thickness. copyright 1997 The American Physical Society