Spin Josephson effect in ferromagnet/ferromagnet tunnel junctions

TL;DR

This paper predicts a Josephson-like spin current between ferromagnets, based on phase relations of magnetization components, drawing an analogy to superconducting Josephson effects.

Contribution

It introduces a theoretical framework for a spin Josephson effect in ferromagnet junctions, highlighting the phase-conjugate relationship of spin components.

Findings

Derives a spin tunnel current proportional to sin of phase difference

Establishes the conjugate relationship between spin phase and z-component

Discusses potential experimental implications

Abstract

We consider the tunnel spin current between two ferromagnetic metals from a perspective similar to the one used in superconductor/superconductor tunnel junctions. We use fundamental arguments to derive a Josephson-like spin tunnel current $I_J^{\rm spin}\propto\sin(\theta_1-\theta_2)$. Here the phases are associated with the planar contribution to the magnetization, $<c_\uparrow^\dagger c_\downarrow>\sim e^{i\theta}$. The crucial step in our analysis is the fact that the $z$-component of the spin is canonically conjugate to the phase of the planar contribution: $[\theta,S^z]=i$. This is analogous to the commutation relation $[\phi,N]=i$ in superconductors, where $\phi$ is the phase associated to the superconducting order parameter and $N$ is the Cooper pair number operator. We briefly discuss the experimental consequences of our theoretical analysis.