Role of surface Cooper pair interactions on critical temperature of ultra thin film superconductors

The superconductivity mechanism of Pb thin film on a Si substrate in the weak interaction regime is investigated. A discrete Fermi surface is constructed depend on the film thickness and electron density and crystallographic orientation. We consider two types of Cooper pair interactions, Cooper pair interaction at thin film surfaces and Cooper pair in the thin film volume. We have chosen the surface Cooper pair interaction, of ultra thin films superconductor proportional to the inverse of thin film thickness, while the volume Cooper pair interaction has been considered as a constant. By these assumptions, we have found oscillation feature of critical temperature Tc and energy gap Δ in terms of thin film thickness similar to the experimental results for Pb/Si(111) thin film superconductor. However, by increasing number of Pb layers, the thin film Tc goes to bulk Tc. In contrast to the previous claimed constant value for the Δb(T=0)/kBTc in bulk, we have found oscillation of this parameter in terms of thin film thickness similar to the Tc oscillation.