Solvable Pair Potential for the Bogoliubov-de Gennes Equations of Space-Dependent Superconductivity

The Bogoliubov-de Gennes equations for the quasiparticle states of space-dependent superconductivity are solved exactly for a two-parameter pair potential representing a normal-metal-superconductor interface. For the simplest case of a superconductor filling the half-space $z>\mathrm{}0\mathrm{}$, $\Delta (z,T)=\left[{\Delta }_0\mathrm{}\right(T)+{\Delta }_1\mathrm{}(T\left)\mathrm{}{e}^{-\frac{z}{\xi }}\right]$, where ${\Delta }_0$ is the bulk value of the gap and ${\Delta }_1$ and $\xi$ are variational parameters. For $T\sim T_c$, ${\Delta }_1$ and $\xi$ can be determined by minimizing the Ginzburg-Landau free-energy functional. In addition, the effect of self-consistency on the low-lying bound states of a superconductor-normal-superconductor junction is discussed.