Josephson effect and spin-triplet pairing correlations in SF${}_1$F${}_2$S junctions

We study theoretically the Josephson effect and pairing correlations in planar SF${}_1$F${}_2$S junctions that consist of conventional superconductors (S) connected by two metallic monodomain ferromagnets (F${}_1$ and F${}_2$) with transparent interfaces. We obtain both spin-singlet and -triplet pair amplitudes and the Josephson current-phase relations for arbitrary orientation of the magnetizations using the self-consistent solutions of Eilenberger equations in the clean limit and for a moderate disorder in ferromagnets. We find that the long-range triplet correlations cannot prevail in symmetric junctions with equal ferromagnetic layers. Surprisingly, the long-range spin-triplet correlations give the dominant second harmonic in the Josephson current-phase relation of highly asymmetric SF${}_1$F${}_2$S junctions. The effect is robust against moderate disorder and variations in the layers thickness and exchange energy of ferromagnets.

Figure 1

Schematics of an SF${}_1$F${}_2$S heterojunction. The magnetization vectors lie in the $y-z$ plane at angles ${\alpha }_1$ and ${\alpha }_2$ with respect to the $z$ axis.

Figure 2

Spatial dependence of singlet and triplet pair amplitudes $f_s$, $f_{t0}^{<}$, and $f_{t1}^{<}$, normalized to the bulk singlet amplitude $f_{sb}$. We consider a symmetric SF${}_1$F${}_2$S junction ($d_1=d_2=50k_F^{-1}$) at low temperature, $T/T_c=0.1$, with the electron mean-free path in ferromagnets $l=200k_F^{-1}$ (thick curves) and in the clean limit $l\to \infty$ (thin curves). The phase difference is $\phi =0$. The magnetizations in ferromagnets are orthogonal, ${\alpha }_1=-\pi /4$, ${\alpha }_2=\pi /4$, and $h_1=h_2=0.1E_F$. Superconductor is characterized by ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$.

Figure 3

The same as in Fig. 2 for ten times thicker ferromagnetic films, $d_1=d_2=500k_F^{-1}$.

Figure 4

Comparison between normalized triplet amplitudes $2f_{t1}^{<}/f_{sb}$ (solid curve) and rotated $2{\tilde{f}}_{t1}^{<}/f_{sb}$ (dash-dotted curve). Parameters are the same as in Fig. 3 for $l\to \infty$.

Figure 5

The current-phase relation $I\left(\phi \right)$ for a symmetric SF${}_1$F${}_2$S junction with $d_1=d_2=500k_F^{-1}$, $h_1=h_2=0.1E_F$, $l=200k_F^{-1}$, $T/T_c=0.1$, and for three values of the relative angle between magnetizations: $\alpha =0$ (solid curve), $\pi /2$ (dashed curve), and $\pi$ (dash-dotted curve).

Figure 6

Spatial dependence of singlet and triplet pair amplitudes $f_s$, $f_{t0}^{<}$, and $f_{t1}^{<}$, normalized to the bulk singlet amplitude $f_{sb}$. We consider an asymmetric SF${}_1$F${}_2$S junction ($d_1=10k_F^{-1}$ and $d_2=990k_F^{-1}$) at low temperature, $T/T_c=0.1$, with the electron mean-free path in ferromagnets $l=200k_F^{-1}$ (thick curves) and in the clean limit $l\to \infty$ (thin curves). The phase difference is $\phi =0$. Magnetization in the thin layer is along $y$ axis, ${\alpha }_1=-\pi /2$, and along $z$ axis in the thick layer, ${\alpha }_2=0$. Here, $h_1=h_2=0.1E_F$ and ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$.

Figure 7

The current-phase relation $I\left(\phi \right)$ for an asymmetric SF${}_1$F${}_2$S junction with $d_1=10k_F^{-1}$ and $d_2=990k_F^{-1}$, $h_1=h_2=0.1E_F$, $l=200k_F^{-1}$, $T/T_c=0.1$, and for three values of the relative angle between magnetizations: $\alpha =0$ (solid curve), $\pi /2$ (dashed curve), and $\pi$ (dash-dotted curve).

Figure 8

The current-phase relation $I\left(\phi \right)$ for an asymmetric SF${}_1$F${}_2$S junction in the clean limit $l\to \infty$ with orthogonal magnetizations $\alpha =\pi /2$. For $d_1=10k_F^{-1}$, $d_2=990k_F^{-1}$, $h_1=h_2=0.1E_F$ three cases are shown: $T=0.1T_c$ (thick solid curve); $T=0.01T_c$ (dashed curve); $T=0.9T_c$ (dash-dotted curve, magnified $5\times {10}^2$ times).

Figure 9

The current-phase relation $I\left(\phi \right)$ for an asymmetric SF${}_1$F${}_2$S junction in the clean limit $l\to \infty$ with orthogonal magnetizations $\alpha =\pi /2$, and $T/T_c=0.1$. For $d_1=10k_F^{-1}$, $d_2=990k_F^{-1}$, three cases are shown: $h_1=h_2=0.1E_F$ (thick solid curve); $h_1=0.1E_F$, $h_2=0.2E_F$ (dashed curve); $h_1=0.05E_F$, $h_2=0.1E_F$ (dash-dotted curve). Influence of F layer thickness is shown for $d_1=20k_F^{-1}$, $d_2=980k_F^{-1}$, and $h_1=h_2=0.1E_F$ (thin solid curve).