Josephson-like spin current in junctions composed of antiferromagnets and ferromagnets

We study Josephson-like junctions formed by materials with antiferromagnetic (AF) order parameters. As an antiferromagnet, we consider a two-band material in which a spin density wave (SDW) arises. This could be Fe-based pnictides in the temperature interval ${T_{\text{c}}\leq T\leq T_{N}}$, where $T_{c}$ and $T_{N}$ are the critical temperatures for the superconducting and antiferromagnetic transitions, respectively. The spin current $j_{\text{Sp}}$ in AF/F/AF junctions with a ballistic ferromagnetic layer and in tunnel AF/I/AF junctions is calculated. It depends on the angle between the magnetization vectors in the AF leads in the same way as the Josephson current depends on the phase difference of the superconducting order parameters in S/I/S tunnel junctions. It turns out that in AF/F/AF junctions, two components of the SDW order parameter are induced in the F\nobreakdash-layer. One of them oscillates in space with a short period ${\xi_{\text{F,b}} \sim \hbar v/\mathcal{H}}$ while the other decays monotonously from the interfaces over a long distance of the order ${\xi_{\text{N,b}}=\hbar v/2\pi T}$ (where $v$, $\mathcal{H}$ and $T$ are the Fermi velocity, the exchange energy and the temperature, respectively; the subindex $\text{b}$ denotes the ballistic case). This is a clear analogy with the case of Josephson S/F/S junctions with a nonhomogeneous magnetization where short- and long\nobreakdash-range condensate components are induced in the F\nobreakdash-layer. However, in contrast to the charge Josephson current in S/F/S junctions, the spin current in AF/F/AF junctions is not constant in space, but oscillates in the ballistic F\nobreakdash-layer. We also calculate the dependence of $j_{\text{Sp}}$ on the deviation from the ideal nesting in the AF/I/AF junctions.