Boundary conditions for the order parameter and the proximity influenced internal phase differences in double superconducting junctions

This paper gives an overview of the unconventional dependence of internal phase differences on the external phase difference in superconductor-normal metal-superconductor (SINIS) and superconductor-superconductor-superconductor (SISIS) tunnel double junctions. The results are obtained within the Ginzburg-Landau (GL) approach that includes boundary conditions for the superconductor order parameter in the presence of a Josephson coupling through interfaces. The boundary conditions are derived within the GL theory and substantiated microscopically. The absence of the one-to-one correspondence between external $\phi$ and internal $\chi_{1,2}$ phase differences in the junctions is shown to occur in two qualitatively different ways, both of which result in the range of $\chi_{1,2}$ reduced and prevent the $4\pi$-periodic current-phase relation $j=j_c\sin\frac{\phi}{2}$. In SINIS junctions, the effect of the supercurrent-induced phase incursion $\varphi$ between the end faces of the central electrode of mesoscopic length $L$ can play a crucial role. In SISIS junctions, there occurs the regime of interchanging modes, which is modified as $L$ decreases. The proximity and pair breaking effects in the double junctions with closely spaced interfaces are addressed.