On-off switch and sign change for a nonlocal Josephson diode in spin-valve Andreev molecules

Andreev molecules consist of two coherently coupled Josephson junctions and permit nonlocal control over supercurrents. By making the barriers magnetic and thus creating a spin valve, we predict that a nonlocal Josephson diode effect occurs that is switchable via the magnetic configuration of the barriers. The diode effect is turned on, off, or changes its sign depending on whether the spin valve is in a parallel, normal, or antiparallel configuration. These results offer a way to exert complete control over a nonlocal Josephson diode effect via the spin degree of freedom.

Figure 1

(a) The spin-valve Andreev molecule consists of three 1D superconductors where the phase difference between the superconducting order parameters is fixed. By a relative rotation of the magnetic moments of the spin-active barriers separating the three superconducting regions, drastically different superconducting diode characteristics can be achieved in the molecule. (b) Experiment proposal for characterizing the spin-valve Andreev molecule. By pinching off the uppermost current loop, the phase difference between the middle and right superconductors can be fixed with a magnetic flux ${\mathrm{\Phi }}_R$.

Figure 2

Andreev bound states (ABSs) for the Andreev molecule with the length of the middle superconductor (a) significantly longer than, or (b) comparable to the superconducting coherence length ${\xi }_0$. For $l\gg {\xi }_0, E_{ABS}({\delta }_L,{\delta }_R)\to E_{ABS}\left({\delta }_L\right)$ for the ABSs localized at the left weak link as the nonlocal impact of ${\delta }_R$ diminishes. As $l$ becomes on the order of ${\xi }_0$, the ABSs from the two junctions hybridize into Andreev molecule orbitals. These states feature avoided crossings at ${\delta }_L=\pm {\delta }_R$ (dotted lines) related to the emergence of double-crossed Andreev reflection (dCAR) and double elastic cotunneling (dEC) processes.

Figure 3

Current through the left junction as well as the subgap Andreev bound states (ABSs) as a function of local phase ${\delta }_L$ for barrier strengths $\gamma =0.00, \gamma =0.25$, and $\gamma =0.75$ for the parallel (a), normal (b), and antiparallel (c) configurations. The nonlocal phase is set at ${\delta }_R=2.56$. As the strength of the magnetic barrier increases, the P ABS experiences a Zeeman-like spin splitting, leading to current discontinuities caused by band inversion as the gap between the ABS at the Fermi level closes at a critical value $\gamma \simeq 0.3$. In contrast, the AP configuration retains its spin degeneracy at the points ${\delta }_L=\pm {\delta }_R$ (black dotted lines) and the Fermi level gap remains open for all parameters investigated but with a reduced $I_{c+}$ due to spin-splitting induced mismatch in the phase gradient of the four bands. The N configuration ABS depicts both the reduction in $I_{c+}$ from AP and the band inversion and current discontinuities from N, behaving as an effective mix of the two extrema.

Figure 4

On-off switch as well as sign change in the nonlocal Josephson diode efficiency is shown for a spin-valve Andreev molecule with $l={\xi }_0$. (a) The diode efficiency dependence on the nonlocal phase ${\delta }_R$ and spin splitting of the barrier $\gamma$ shows that significant deviations arise between the three configurations for $\gamma >0.5$. (b) The ${\mathrm{JJ}}_{\text{L}}$ current-phase relation is shown for the parameter region denoted by red rings in (a). For specific combinations of ${\delta }_R$ and $\gamma$ shown in (c), $\eta >0.1$ for the P configuration, $-0.1<\eta <0.1$ for N, and $\eta <-0.1$ for the AP configuration. This enables an effective on-off switching as well as sign change of the diode efficiency by an appropriate rotation of the spin-valve configuration. (d) The diode efficiency is shown as a function of nonlocal phase ${\delta }_R$ along the dotted line in (a), for $\gamma \simeq 0.9$, highlighting the asymmetry between the three configurations.

Figure 5

Diode efficiencies for the parallel (P), normal (N), and antiparallel (AP) conditions for different choices of Fermi momentum, indicating the effect on the diode characteristics as the system deviates from the “off-resonance” condition $k_F=\pi /2\text{mod}2\pi$. The results indicate that the off-resonance condition is ideal for the switching effect due to the largest asymmetry between the P, N, and AP configurations. However, the effect can also persist when perturbed away from the $\pi /2$ condition.

Figure 6

Superconducting diode efficiency in the spin-valve Andreev molecule as a function of the middle superconductor length $l$. The maximum diode efficiency is denoted by the upper-left corner text. A significant increase in the diode efficiency is observed as $l$ decreases, with a maximum diode efficiency of $\eta =0.44$ observed for the parallel configuration at $l=0.30{\xi }_0$.