Critical current of a Josephson junction containing a conical magnet

We calculate the critical current of a superconductor/ferromagnetic/superconductor (S/FM/S) Josephson junction in which the FM layer has a conical magnetic structure composed of an in-plane rotating antiferromagnetic phase and an out-of-plane ferromagnetic component. In view of the realistic electronic properties and magnetic structures that can be formed when conical magnets such as Ho are grown with a polycrystalline structure in thin-film form by methods such as direct current sputtering and evaporation, we have modeled this situation in the dirty limit with a large magnetic coherence length ($ξ_f$). This means that the electron mean free path is much smaller than the normalized spiral length $λ/2π$ which in turn is much smaller than $ξ_f$ (with $λ$ as the length a complete spiral makes along the growth direction of the FM). In this physically reasonable limit we have employed the linearized Usadel equations: we find that the triplet correlations are short ranged and manifested in the critical current as a rapid oscillation on the scale of $λ/2π$. These rapid oscillations in the critical current are superimposed on a slower oscillation which is related to the singlet correlations. Both oscillations decay on the scale of $ξ_f$. We derive an analytical solution and also describe a computational method for obtaining the critical current as a function of the conical magnetic layer thickness.