Josephson junction on one edge of a two dimensional topological insulator affected by magnetic impurity

TL;DR

This paper investigates the effects of magnetic impurities on the current-phase relations of a Josephson junction on a topological insulator edge, revealing robustness of certain features and impurity-induced smoothing of others.

Contribution

It provides a detailed analysis of how magnetic impurities influence the current-phase relations, including the robustness of the $2 ext{-} ext{pi}$ jump and impurity effects on the $4 ext{-} ext{pi}$ relation.

Findings

Sharp jumps at specific phase values are identified in the clean junction.

Impurity strength does not affect the $2 ext{-} ext{pi}$ jump significantly.

Asymmetrical impurity distribution increases the maximum current.

Abstract

Current-phase relation in a Josephson junction formed by putting two s-wave superconductors on the same edge of a two dimensional topological insulator is investigated. We consider the case that the junction length is finite and magnetic impurity exists. The similarity and difference with conventional Josephson junction is discussed. The current is calculated in the semiconductor picture. Both the $2\pi$- and $4\pi$-period current-phase relations ($I_{2\pi}(\phi), I_{4\pi}(\phi)$) are studied. There is a sharp jump at $\phi=\pi$ and $\phi=2\pi$ for $I_{2\pi}$ and $I_{4\pi}$ respectively in the clean junction. For $I_{2\pi}$, the sharp jump is robust against impurity strength and distribution. However for $I_{4\pi}$, the impurity makes the jump at $\phi=2\pi$ smooth. The critical (maximum) current of $I_{2\pi}$ is given and we find it will be increased by asymmetrical distribution of impurity.