Boundary Condition on the Order Parameter at a Tunneling Barrier in a Pure Superconductor (T=Tc)

Abstract

We consider the nucleation of superconductivity at a $\delta$-function plane barrier in a free-electron super-conductor. We calculate the exact kernel of Gorkov's linear integral equation for the order parameter $\Delta$, and compare it with the kernel derived by de Gennes for a similar model on the basis of a correlation-function argument which has not been rigorously justified as yet. The two kernels are not the same, particularly near the barrier, but the difference oscillates with wavelength $\frac{\pi }{k_F}$ and is negligible for the purpose of calculating asymptotic boundary conditions on the order parameter. We thus confirm for our model the boundary condition predicted by de Gennes, ${\psi }_{+}={\psi }_{-}+{\left(\frac{〈1-t〉}{〈t〉}\right)\frac{{\xi }_{0}d\psi }{\mathrm{dx}}|}_0$.