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Motivated by the recent theoretical and experimental progress in the heavyfermion system UCoGe, we study ferromagnetic chiral superconductors in thepresence of magnetic domains. Within mean field approximations, it is shownthat chiral superconducting domains are naturally induced by the ferromagneticdomains. The domain wall current flows in the opposite direction to the naivelyexpected one as in $^3$He-A phase due to contributions from "unpairedelectrons". Consequently, the domain wall current flows in the same directionwith that of surface currents when the magnetic domain wall lies parallel tothe sample surface, and therefore they contribute to the net current along thewhole sample. We find that, due to the non-cancellation between the domain wallcurrent and surface current, a Fulde-Ferrell-like superconducting state can bestabilized in an anisotropic sample for all the temperatures below thesuperconducting transition temperature.

Fulde-Ferrell state in ferromagnetic chiral superconductor with magnetic domain wall