# Phase relations in superconductor-normal metal-superconductor tunnel junctions

## TL;DR

This paper investigates the complex phase relations and current behaviors in superconductor-normal metal-superconductor tunnel junctions, revealing double-valued dependencies and the critical role of phase incursion in their quantum properties.

## Contribution

It introduces a detailed analysis of phase relations in SINIS junctions, showing how phase incursion influences the current-phase relation and effective transmission.

## Key findings

- The phase difference $ heta_{1,2}$ passes through its domain twice, creating double-valued solutions.
- The overall phase difference $$ depends on two solutions that form a single-valued relation.
- Phase incursion $$ is crucial in the behavior of the current-phase relation.

## Abstract

The phase difference $\phi$, between the superconducting terminals in superconductor-normal metal-superconductor tunnel junctions (SINIS), incorporates the phase differences $\chi_{1,2}$ across thin interfaces of constituent $SIN$ junctions and the phase incursion $\varphi$ between the side faces of the central electrode of length $L$. It is demonstrated here that $\chi_{1,2}$ pass through over their proximity-reduced domain twice, there and back, while $\phi$ changes over the single period. Two corresponding solutions, that describe the double-valued order-parameter dependence on $\chi_{1,2}$, jointly form the single-valued dependence on $\phi$, operating in two adjoining regions of $\phi$. The phase incursion $\varphi$ plays a crucial role in creating such a behavior. The current-phase relation $j(\phi,L)$ is composed of the two solutions and, at a fixed small $L$, is characterized by the phase-dependent effective transmission coefficient.