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 weakened 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 (${\xi }_f$). This means that the electron mean free path is much smaller than the normalized spiral length $\lambda /2\pi$ which in turn is much smaller than ${\xi }_f$ (with $\lambda$ 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 $\lambda /2\pi$. 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 ${\xi }_f$. 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.

Figure 1

(Color online) (a) An illustration of two singlet-type superconductors (S) sandwiching a ferromagnet with a conical magnetic structure (Ho). The two magnetic phases of Ho, (b) the in-plane antiferromagnetic spiral phase of Ho below its Néel temperature ${\Theta }_n$, and (c) the conical magnet phase of Ho below its Curie temperature ${\Theta }_c$; the antiferromagnetic component $I_r$ rotating in the $\left\{x,y\right\}$ plane; the ferromagnetic component $I_z$ pitched toward the $z$ axis; and the resultant magnetization vector $\mathbf{I}$ rotating on the surface of a cone. In the limit considered in this paper, , the ferromagnetic coherence length ${\xi }_f$ is much larger than $\lambda /2\pi$.

Figure 2

(Color online) Typical $I_c{R}_n$ dependence on the thickness $d$ of the FM layer ($\Delta =2.2\times {10}^{-22}\text{J}$, $Q=9\times {10}^8{\text{m}}^{-1}$, $D_f=6\times {10}^{-4}{\text{m}}^2{\text{s}}^{-1}$, $D_s=2.5\times {10}^{-4}{\text{m}}^2{\text{s}}^{-1}$, ${\sigma }_f=4\times {10}^6{\left(\Omega \text{m}\right)}^{-1}$, and ${\sigma }_s=6\times {10}^6{\left(\Omega \text{m}\right)}^{-1}$): (left) in the limit of ${ϵ}_z⪢{ϵ}_r^2$ $\left(I_r/k_B=I_z/k_B=100\text{K}\right)$ and (right) in the limit of ${ϵ}_z⪡{ϵ}_r^2$ $\left(I_r/k_B=130\text{K},I_z/k_B<4\text{K}\right)$ for two different values of the interfacial resistance $R$.

Figure 3

(Color online) Comparison of the analytical (solid curve) and computational (dashed curve) results for $I_c{R}_n$ in the function of the thickness $d$: (left) in the limit of ${ϵ}_z⪢{ϵ}_r^2$ and (right) in the limit of ${ϵ}_z⪡{ϵ}_r^2$. The parameters used are identical to those in Fig. 2 with $R=3.0\times {10}^{-15}\Omega {\text{m}}^2$.