Proximity effect in ferromagnet/superconductor hybrids: From diffusive to ballistic motion

We present an analytical study of the proximity effect in ferromagnet/superconductor (F/S) heterostructures, allowing for an arbitrary magnetic exchange energy as well as arbitrary impurity and spin-flip scattering rates within a quasiclassical approach. While previous studies mainly have focused on the clean or dirty limits, our results grant access to the regime of intermediate impurity concentrations, thus allowing us to probe the crossover from the clean to the dirty limit. We find that in the crossover regime, all possible symmetry correlations of the proximity-induced anomalous Green’s function are induced in the ferromagnet. We also point out that the local density of states oscillates spatially not only for an F/S bilayer but also for a normal/superconductor (N/S) bilayer in the diffusive limit, a fact which appears to have gone unnoticed in the literature. Within the weak proximity-effect regime, we present compact analytical expressions valid for arbitrary exchange fields and impurity-scattering rates for (i) the local density of states in an F/S bilayer, (ii) the Josephson current in an S/F/S junction, and (iii) the critical temperature in an F/S/F multilayer. For all cases, we study in particular the crossover regime between diffusive and ballistic motion. Our results may be useful for analyzing experimental data in cases when the dirty limit is not fully reached, thus invalidating the use of the Usadel equation.

Figure 1

(Color online) Overview of the superconductor/ferromagnet heterostructure that we study in this paper. We take into account an arbitrary strength of the exchange field as well as an arbitrary rate of nonmagnetic and magnetic scatterings within a quasiclassical approach.

Figure 2

(Color online) Plot of the proximity-induced anomalous Green’s functions in the middle of the ferromagnetic region $\left(x/d=0.5\right)$ using $h/\Delta =15$ and $\epsilon /{\Delta }_0=0.5$.

Figure 3

(Color online) Setup for our study of the density of states.

Figure 4

(Color online) Plot of the (a) energy-resolved DOS at $x=d$ and (b) spatially resolved DOS at $\epsilon =0$ for several values of the impurity-scattering rate. Here, the exchange field is set to $h/{\Delta }_0=15$ and $d/\xi =0.5$.

Figure 5

(Color online) Plot of the (a) energy-resolved DOS at $x=d$ and (b) spatially resolved DOS at $\epsilon /{\Delta }_0=0.5$ for several values of the impurity-scattering rate. Here, the exchange field is set to zero, corresponding to a normal metal, and $d/\xi =5.0$.

Figure 6

(Color online) Plot of the spatially resolved DOS for a diffusive N/S and F/S junctions, respectively. In both cases, oscillations of the DOS are seen at finite energies. We have here fixed $d/\xi =3.0$ and $\tau =0.2$ using the notation of Ref. 41 (here, $\xi =\sqrt{D/{\Delta }_0}$ while $\tau$ denotes the barrier transparency).

Figure 7

(Color online) Setup for our study of the Josephson current.

Figure 8

(Color online) Plot of the critical current as a function of junction width $d$. We have used $T/T_c=0.2$.

Figure 9

(Color online) Plot of the oscillation length of the critical current as a function of the impurity-scattering strength $g_{\text{imp}}$. We have used $T/T_c=0.2$.

Figure 10

(Color online) Plot of the $d$ dependence and the $T$ dependence of the critical current for $g_{\text{imp}}/{\Delta }_0=15$, corresponding to $h{\tau }_{\text{imp}}=1$.

Figure 11

(Color online) Phase diagram in the $d\text{−}T$ plane for the $0\text{−}\pi$ transitions of the critical current for several values of the impurity concentration.

Figure 12

(Color online) Setup for our study of the critical temperature.

Figure 13

(Color online) Plot of the critical temperature in an F/S/F structure as a function of the ferromagnetic layer thickness $d_F$ for fixed $d_S/\xi =0.03$. Note the different scale for $d_F$ in the middle panel. The symbols denote the result for $x/d_S=0.50$ while the dashed lines correspond to $x/d_S=0.01$.