Quantum size effects in transport properties of metallic films

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Abstract
The confinement of electrons in small dimensions can lead to a discretization of energy levels. The associated quantum size effects in turn lead to an out-of-plane conductivity that shows nonanalytic behavior in the approach to the classical limit. The principal size dependence of the conductivity is ∼1/d for a film of thickness d; however, there is also a correction term that has an essential singularity in the small parameter l/d, where l is the mean free path in a bulk sample. Surface roughness in the film is introduced by establishing two physical length scales. Variations in d on length scales shorter than l are treated quantum mechanically by a suitable coordinate transformation. On the other hand, large-scale fluctuations which might reflect the presence of grains are incorporated classically by segmenting the film into independent units of length l. Impurity effects have also been included and in fact crossover behavior in the conductivity is found from a surface-dominated to an impurity-dominated regime. The transport coefficients—conductivity and thermopower—are found to show oscillations as a function of d with a period of half the Fermi wavelength. DOI: http://dx.doi.org/10.1103/PhysRevB.38.12298 © 1988 The American Physical Society