Possible existence of a Josephson effect in ferromagnets

When a current density $j_x$ crosses a 180° domain wall in a metallic ferromagnet, the spin s of each conduction electron exerts an s-d exchange torque on the localized wall spins. Hence, the wall moment of a Bloch wall is canted out of the wall plane by an angle ψ, given by $j_x$=(eC/ħ)sin(2ψ), where C is the maximum restoring torque at ψ=45°. This equation is the exact analog of the dc Josephson effect, and 2ψ is the analog of the superconducting phase difference φ across a junction. For ‖$j_x$‖>eC/ħ≃${10}^6$ A/${\mathrm{cm}}^2$, the s-d exchange torque overcomes the restoring torque, and the wall moment precesses with a frequency ω=d(2ψ)/dt. A dc voltage δV is expected to appear across the wall, satisfying the famous ac Josephson relation 2eδV=-ħω. This wall precession can be described as a translation of Bloch lines, and the Bloch lines are the exact analog of superconducting vortices. The electric current exerts a transverse force on Bloch lines.