Microscopic investigation of the proximity effect in a finite geometry

The system of a finite superconductor $S_1$ in proximity with a weaker superconductor $S_2$ is considered. The order parameter is assumed to vary in space as ${\Delta }_{1}cos\mathrm{}\left[q\right(z-D\left)\right]$ ($0\le z\le D$) in $S_1$, and as ${\Delta }_{2}cosh\mathrm{}\left[K\right(z+L\left)\right]$ ($-L\le z\le 0$) in $S_2$. The Bogoliubov-deGennes equations were solved for the two regions: $z<\mathrm{}0\mathrm{}$. The transition temperature $T_{c\mathrm{vs}}$ of the system is evaluated for the case where $S_2$ is a thick normal layer. The dependence of $T_{c\mathrm{NS}}$ on the thickness $D$ is in agreement with experimental results.