Giant triplet proximity effect in $\pi$-biased Josephson junctions with spin-orbit coupling

In diffusive Josephson junctions the phase difference $\varphi$ between the superconductors strongly influences the spectroscopic features of the layer separating them. The observation of a uniform minigap and its phase modulation were only recently experimentally reported, demonstrating agreement with theoretical predictions up to now—a vanishing minigap at $\varphi =\pi$. Remarkably, we find that in the presence of intrinsic spin-orbit coupling a giant proximity effect due to spin-triplet Cooper pairs can develop at $\varphi =\pi$, in complete contrast to the suppressed proximity effect without spin-orbit coupling. We here report a combined numerical and analytical study of this effect, proving its presence solely based on symmetry arguments, which makes it independent of the specific parameters used in experiments. We show that the spectroscopic signature of the triplets is present throughout the entire ferromagnetic layer. Our finding offers a way to artificially create, control, and isolate spin-triplet superconductivity.

Figure 1

The proximity effect in the standard SNS Josephson junction (a) manifests in a minigap that closes with increasing phase difference $\varphi$ between the superconductors, closing entirely for $\varphi =\pi$. Conversely, the proximity effect in the SFS junction with intrinsic SO coupling in the junction direction (b) results in a giant peak in the density of states at zero energy for $\varphi =\pi$. The different behavior is due to the symmetry of the anomalous Green's function: for SNS, the singlet component is symmetric around the center of the junction at $\varphi =0$ and antisymmetric for $\varphi =\pi$ whereas the SFS case retains a symmetric triplet component at $\varphi =\pi$. For the examples used in the text we take the junction direction to lie in the $z$ direction and use bulk values for the superconductors.

Figure 2

The local density of states $D\left(\epsilon \right)$ at $z=L_F/2$ for phase difference $\varphi$ between the superconductors of an SFS junction with spin-orbit coupling aligned in the junction direction, with exchange field $\underline{h}=10\mathrm{\Delta }\widehat{y}$ and spin-orbit coupling $\alpha =0.4/L_F$. The giant triplet proximity effect at $\varphi =\pi$ manifests as a large peak in the $D\left(\epsilon \right)$ at $\epsilon =0$. A comparison with the standard SNS and SFS junctions without SO coupling is also provided.

Figure 3

The spatial dependence of the local density of states $D(\epsilon ,z)$ for a phase differences $\varphi =\pi$ between the superconductors of an SFS junction with spin-orbit coupling. Parameter values are the same as in Fig. 2. As seen, the giant proximity effect is virtually independent of the position from the interfaces, thus persisting throughout the entire system. Inset: Top view of the density of states surface plot.

Figure 4

The local density of states $D(\varepsilon =0)$ at $z=L_F/2$ for a $\pi$-biased SFS junction with spin-orbit coupling vector aligned in the junction direction (corresponding to a nanowire setup) as a function of magnetization exchange field $\underline{h}=h\mathrm{\Delta }\widehat{y}$ and strength of spin-orbit coupling $\alpha L_F$ aligned in the junction direction.

Figure 5

The local density of states $D\left(\epsilon \right)$ at $z=L_F/2$ for phase differences $\varphi$ between the superconductors of an SFS junction with spin-orbit coupling aligned in the junction direction, with exchange field $\underline{h}=3\mathrm{\Delta }\widehat{z}$ and spin-orbit coupling $\alpha =2/L_F$. At small phase differences $D\left(\epsilon \right)$ displays a minigap while retaining a zero-energy peak at $\varphi =\pi$.

Figure 6

Spatial variation of the singlet and triplet components throughout the ferromagnet of the SFS junctions considered in the main text.