Role of interface transparency and spin-dependent scattering in diffusive ferromagnet/superconductor heterostructures

We present a numerical study of the density of states in a ferromagnet/superconductor junction and the Josephson current in a superconductor/ferromagnet/superconductor junction in the diffusive limit by solving the Usadel equation with Nazarov’s boundary conditions. Our calculations are valid for an arbitrary interface transparency and an arbitrary spin-dependent scattering rate, which allows us to explore the entire proximity-effect regime. We first investigate how the proximity-induced anomalous Green’s function affects the density of states in the ferromagnet for three magnitudes of the exchange field $h$ compared to the superconducting gap $\Delta$: (i) $h\lesssim \Delta$, (ii) $h\gtrsim \Delta$, and (iii) $h⪢\Delta$. In each case, we consider the effect of the barrier transparency and allow for various concentrations of magnetic impurities. We clarify features that may be expected in the various parameter regimes accessible for the ferromagnetic film, with regard to thickness and exchange field. In particular, we address how the zero-energy peak and minigap observed in experiments may be understood in terms of the interplay between the singlet and the triplet anomalous Green’s functions and their dependence on the concentration of magnetic impurities. Our results should serve as a useful tool for the quantitative analysis of experimental data. We also investigate the role of the barrier transparency and spin-flip scattering in a superconductor/ferromagnet/superconductor junction. We suggest that such diffusive Josephson junctions with large residual values of the supercurrent at the $0\text{−}\pi$ transition, where the first harmonic term in the current vanishes, may be used as efficient supercurrent-switching devices. We numerically solve for the Josephson current in such a junction to clarify to what extent this idea may be realized in an experimental setup. It is also found that uniaxial spin-flip scattering has a very different effect on the $0\text{−}\pi$ transition points depending on whether one considers the width or the temperature dependence of the current. Our theory takes into account vital elements that are necessary to obtain quantitative predictions of the supercurrent in such junctions.

Figure 1

(Color online) The ferromagnet/superconductor heterostructures investigated in this paper. In (a), a ferromagnet/superconductor bilayer setup is shown. We model the barrier region at the normal/ferromagnet interface to be very strong, mimicking a free edge boundary condition or insulating region. Our proposed experimental setup is thus equivalent to the ones employed in the experimental works in Refs. 20, 23. In (b), we consider a superconductor/ferromagnet/superconductor junction. The superconductors are treated as reservoirs; thus, they are unaffected by the proximity effect. We will consider a ferromagnetic layer thickness ranging from $d/\xi =0.1$ to $d/\xi =1.0$, where $\xi$ is the superconducting coherence length.

Figure 2

(Color online) The local density of states at $x=0$ (see Fig. 1 for $h/\Delta =0.5$) For each thickness of the ferromagnetic layer and barrier transparency, we study the role of different types of spin-dependent scattering.

Figure 3

(Color online) The local density of states at $x=0$ (see Fig. 1 for $h/\Delta =1.5$) For each thickness of the ferromagnetic layer and barrier transparency, we study the role of different types of spin-dependent scattering.

Figure 4

(Color online) Local density of states at $x=0$ (see Fig. 1) for $h/\Delta =15$ (upper row) and $h/\Delta =50$ (lower row). Here, we have fixed $d/\xi =0.1$ and $\tau =0.5$ and study the role of different types of spin-dependent scattering.

Figure 5

(Color online) Plot of the DOS of a ferromagnet/superconductor junction with increasing exchange field $h$, using $d/\xi =1.0$ and $\tau =0.5$. Here, we have set the spin-dependent scattering to zero.

Figure 6

(Color online) Plot of the DOS of a ferromagnet/superconductor junction for a fixed exchange field $h/\Delta =0.3$ which gives a ZEP in the absence of spin-dependent scattering. We fix $d/\xi =1.0$ and $\tau =0.5$.

Figure 7

(Color online) Plot of the singlet $\left(\text{Im}\left\{f_s\right\}\right)$ and the triplet $\left(\text{Re}\left\{f_t\right\}\right)$ anomalous Green’s functions induced in the ferromagnet for $\epsilon =0$ right at the N/F interface $\left(x=0\right)$. We have used $\tau =0.5$ and $d/\xi =1.0$.

Figure 8

(Color online) Plot of the singlet $\left(\text{Im}\left\{f_s\right\}\right)$ and the triplet $\left(\text{Re}\left\{f_t\right\}\right)$ anomalous Green’s functions induced in the ferromagnet for $\epsilon =0$ right at the N/F interface $\left(x=0\right)$. We have used $\tau =0.5$ and $d/\xi =1.0$.

Figure 9

(Color online) Plot of the DOS at $x=0$ for a normal metal/superconductor junction using $d/\xi =1.0$ and $\tau =0.5$. As seen, spin-flip scattering closes the minigap, whether it is uniaxial or isotropic. However, the minigap shows a strong resilience against increasing spin-orbit scattering.

Figure 10

(Color online) Plot of the junction-width dependence of the $I_c{R}_N$ product for an S/F/S Josephson junction in the intermediate $\left(Z=0.5\right)$ and low $\left(Z=5.0\right)$ barrier transparency regimes. The temperature is fixed at $T/T_c=0.1$. For each case, the effect of increasing uniaxial spin-flip scattering is seen to severely reduce the residual value of the critical current at the $0\text{−}\pi$ transition points.

Figure 11

(Color online) Plot of the temperature dependence of the $I_c{R}_N$ product for an S/F/S Josephson junction in the intermediate $\left(Z=0.5\right)$ and low $\left(Z=5.0\right)$ barrier transparency regimes for different values of the normalized Thouless energy $\mathcal{E}$.

Figure 12

(Color online) (a) Plot of the temperature-dependence of the $I_c{R}_N$ product for weakly increasing spin-flip scattering. (b) Plot of the current-phase relationship for the $I_c{R}_N$ product for an S/F/S Josephson junction, illustrating the strong deviation from the sinusoidal dependence and the $0\text{−}\pi$ transition.