Possible existence of a Josephson effect in ferromagnets

When a current density jₓ 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 jₓ=(eC/{})sin(2{ψ}), where C is the maximum restoring torque at {ψ}=45^∘{}. This equation is the exact analog of the dc Josephson effect, and 2{ψ} is the analog of the superconducting phase difference {φ} across a junction. For {}jₓ{}>eC/{}{}10⁶ A/cm², the s-d exchange torque overcomes the restoring torque, and the wall moment precesses with a frequency {ω}=d(2{ψ})/dt. A dc voltage {δ}V is expected to appear across the wall, satisfying the famous ac Josephson relation 2e{δ}V=-{}{ω}. 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.