Size, Shape, and Impurity Effects on Transition Temperatures of Nanostructured Superconductors

We have investigated system size, shape, and nonmagnetic impurity effects on superconducting transition temperatures T c 's for nanoscaled superconducting plates using the Gor'kov equations with the finite-element method. We have considered a weak coupling superconductor to which the BCS theory is applicable. We have found that T c shows a sawtooth behavior as a function of the size and T c increases with decreasing superconductor size. Spatial distributions of superconducting order parameters abruptly change at drops in T c during the sawtooth behavior. The drops in T c and changes of distributions of order parameters are caused by retardation of the attractive interaction between electrons in the BCS theory. In addition, we have found that T c 's for rectangular plates are higher than those for square plates. Furthermore, we have found that impurity effects decrease T c 's.