Statistical Model for the Size Effect in Electrical Conduction

A statistical model for the reflection of scalar plane waves from a rough surface leads to a plane wave in the direction of specular reflection and to a contribution with a finite angular spread about that direction, depending on the tangential correlation of the surface asperities. Based upon on this result, a new semiclassical model, which satisfies the requirement of flux conservation, is proposed for the boundary condition for the distribution function of the size effect in the electrical conductivity. In the absence of correlation, the resultant expression replaces the constant specularity parameter p of Fuchs by the function exp−(4π(h/λ) cosθ0)2 with θ0 the angle of the electron wave vector with the surface normal. Correlation produces an additional forward component within the diffuse contribution. Numerical results of the size effect for zero correlation are compared to the Fuchs model as well as a more recent model, and show a different thickness dependence for thin samples. The effect of correlation is to add to the conductivity, as a result of the diffuse contribution whose velocity has a finite expectation value in the direction of the current.