Josephson current in a normal-metal nanowire coupled to superconductor/ferromagnet/superconductor junction

TL;DR

This paper investigates how the Josephson current in a superconductor/ferromagnet/superconductor nanowire junction is affected by the decay length of magnetization, revealing a transition from zero to non-zero energy states and a resulting shift to a second-harmonic current-phase relation.

Contribution

It demonstrates the impact of magnetization decay length on Andreev bound states and Josephson current, highlighting the emergence of a sin(2φ) relation linked to ABS crossing points.

Findings

Shorter decay length splits zero-energy states into non-zero energy states.

The current-phase relation can become almost sin(2φ) under certain conditions.

Odd-frequency spin-triplet pairing is dominant in the ferromagnetic region for long decay lengths.

Abstract

We consider superconducting nanowire proximity coupled to superconductor / ferromagnet / superconductor junction, where the magnetization penetrates into superconducting segment in nanowire decaying as $\sim\exp[-\frac{\mid n \mid}{\xi}]$ with site index $n$ and the decay length $\xi$. We tune chemical potential and spin-orbit coupling so that topological superconducting regime hosting Majorana fermion is realized for long $\xi$. We find that when $\xi$ becomes shorter, zero energy state at the interface between superconductor and ferromagnet splits into two away from zero energy. Accordingly, the behavior of Josephson current is drastically changed due to this "zero mode-non-zero mode crossover". By tuning the model parameters, we find an almost second-harmonic current-phase relation, $\sin2\varphi$, with phase difference $\varphi$. Based on the analysis of Andreev bound state (ABS), we clarify that current-phase relation is determined by coupling of the states within the energy gap. We find that the emergence of crossing points of ABS is a key ingredient to generate $\sin2\varphi$ dependence in current-phase relation. We further study both the energy and $\varphi$ dependence of pair amplitudes in the ferromagnetic region. For long $\xi$, odd-frequency spin-triplet $s$-wave component is dominant. The magnitude of the odd-frequency pair amplitude is enhanced at the energy level of ABS.