Boundary Conditions and Quantum Effects in Thin Superconducting Films

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Abstract
Earlier calculations of quantum effects in thin superconducting films were based on the assumption that the wave functions of the "Cooper pairs" vanish at the surface of the film. The experimental confirmation of recent calculations of surface superconductivity indicate that the condition of no net electron flow normal to the surface is a valid approximation. This condition suggests that the gradient of the single-electron functions should also vanish at the surface. The present calculation follows closely the work of Thompson and Blatt except that we replace the boundary condition that the wave function vanish at the surface by the one that its gradient vanish at the surface. The main results are: The chemical potential decreases with decreasing film thickness and the superconducting energy gap increases with decreasing film thickness. These results differ from Thompson and Blatt's work in that the chemical potential now has the opposite thickness dependence and the energy-gap parameter increases more rapidly with decreasing film thickness.