Spin Josephson effect in ferromagnet/ferromagnet tunnel junctions

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 $I_J^{\mathrm{spin}}\propto \sin ({\theta }_1-{\theta }_2)$. Here the phases are associated with the planar contribution to the magnetization, $<c_{\uparrow }^{†}{c}_{\downarrow }>\sim e^{i\theta }$. 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 $[\varphi ,N]=i$ in superconductors, where $\varphi$ 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.

LaTex, seven pages, no figures; version to appear in Europhys. Lett.; in order to make room for a more extended microscopic analysis, the phenomenological discussion contained in v2 was removed

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