Josephson $\varphi$-junctions based on structures with complex normal/ferromagnet bilayer

TL;DR

This paper proposes a method to realize Josephson $$-junctions with a nontrivial phase difference using structures with normal metal and ferromagnet layers, supported by numerical analysis and fabrication feasibility.

Contribution

It introduces a new design for $$-junctions based on N-F bilayer structures and provides quantitative criteria through numerical solutions of Usadel equations.

Findings

$$-junctions can be achieved with current technology.

The N-layer induces a negative second harmonic in the current-phase relation.

The F-layer's oscillatory coupling results in a small first harmonic.

Abstract

We demonstrate that Josephson devices with nontrivial phase difference $% 0<\varphi_g <\pi $ in the ground state can be realized in structures composed from longitudinally oriented normal metal (N) and ferromagnet (F) films in the weak link region. Oscillatory coupling across F-layer makes the first harmonic in the current-phase relation relatively small, while coupling across N-layer provides negative sign of the second harmonic. To derive quantitative criteria for a $\varphi$-junction, we have solved two-dimensional boundary-value problem in the frame of Usadel equations for overlap and ramp geometries of S-NF-S structures. Our numerical estimates show that $\varphi $-junctions can be fabricated using up-to-date technology.