Scattering by magnetic and spin-orbit impurities and the Josephson current in superconductor-ferromagnet-superconductor junctions

We analyze the Josephson current in a junction consisting of two superconductors (S) and a ferromagnetic layer (F) for arbitrary impurity concentration. In addition to nonmagnetic impurities, we consider also magnetic ones and spin-orbit scattering. In the limit of weak proximity effect we solve the linearized Eilenberger equation and derive an analytical expression for the Josephson critical current valid in a broad range of parameters. This expression enables us to obtain not only known results in the dirty and clean limits but also in a intermediate region of the impurity concentration, which may be very important for comparison with experimental data.

Figure 1

(Color online) Dependence of the absolute value of the Josephson critical current $I_c$ on the thickness $2d$ of the ferromagnetic layer for different values of $h\tau$. Here $\mathrm{\Delta }\tau =0.1$ and $T\tau =0.05$. Only scattering by nonmagnetic impurities is considered.

Figure 2

(Color online) Dependence of the absolute value of the Josephson critical current $I_c$ on the thickness $2d$ of the ferromagnetic layer for different values of $h\tau$ and ${\lambda }_z$. Here $\mathrm{\Delta }{\tau }_t=0.1$, $T{\tau }_t=0.05$, and ${\lambda }_x=0$. Without loss of generality we have set ${\tau }_N^{-1}\to 0$.

Figure 3

(Color online) Dependence of the absolute value of the Josephson critical current $I_c$ on the thickness $2d$ of the ferromagnetic layer for $h\tau =1.2$ and different values of ${\lambda }_{\perp }$. Other parameters are taken as in Fig. 1.