Superconductors with Plane Boundaries

The Gor'kov equations are solved approximately for various geometries to obtain information about the pair wave function ${\Delta }^{*}\mathrm{}\left(x\right)\mathrm{}$ in the vicinity of plane boundaries. The approximation method consists of assuming a model ${{\Delta }_m}^{*}\mathrm{}\left(x\right)\mathrm{}$ taken constant within the superconductor, and assuming it to be close to the correct ${\Delta }^{*}\mathrm{}\left(x\right)\mathrm{}$. The equations are then solved and a new ${\Delta }^{*}\mathrm{}\left(x\right)\mathrm{}$ is calculated. ${{\Delta }_m}^{*}$ is then chosen in a self-consistent manner. The problems considered are the finite and semi-infinite superconducting slabs, and semi-infinite superconducting and normal metals in contact. The effects of the boundary conditions are discussed. The calculations are performed both at zero temperature and near the critical temperature.