Rashba Splitting of Cooper Pairs

We investigate theoretically the properties of a weak link between two superconducting leads, which has the form of a nonsuperconducting nanowire with a strong Rashba spin-orbit coupling caused by an electric field. In the Coulomb-blockade regime of single-electron tunneling, we find that such a weak link acts as a “spin splitter” of the spin states of Cooper pairs tunneling through the link, to an extent that depends on the direction of the electric field. We show that the Josephson current is sensitive to interference between the resulting two transmission channels, one where the spins of both members of a Cooper pair are preserved and one where they are both flipped. As a result, the current is a periodic function of the strength of the spin-orbit interaction and of the bending angle of the nanowire (when mechanically bent); an identical effect appears due to strain-induced spin-orbit coupling. In contrast, no spin-orbit induced interference effect can influence the current through a single weak link connecting two normal metals.

Figure 1

Schematic illustrations of the lowest-order perturbation expansion steps for tunneling (in the Coulomb-blockade regime) through a straight nanowire weak link subjected to the Rashba spin-orbit (SO) interaction caused by an electric field along the $\widehat{\mathbf{z}}$ direction. In a semiclassical picture, the spin of each electron (the arrows) is rotated in the $XZ$ plane as it goes through the link. (a) Single-electron tunneling from one normal metal to another, via an intermediate (rotating) state (the dashed circle). When the electron enters the second normal metal, its spin has been rotated. (b) Sequential tunneling in four steps of a Cooper pair between two superconductors connected by the same weak link. Because the two electrons that form the Cooper pair are in time-reversed states, the SO interaction rotates their spins in opposite directions. (c) As they enter the second superconductor, the Cooper pairs are in a coherent mixture (the dash-dotted circle) of a spin-singlet and a spin-triplet state. Inside this superconductor, this state is then projected onto the singlet state (the full circle).

Figure 2

(a) Sketch and (b) simplified model of a device that would allow the effects predicted in the text to be studied.

Figure 3

The Josephson current $J(\phi )$ divided by its value without the SO interaction, $J_0(\phi )$, for the genuine Rashba configuration [Eqs. (7) and (8)] as a function of $k_{\text{so}}d/(2\pi )$. The largest amplitude is for the zero bending angle, $\theta =0$, decreasing gradually for $\theta =\pi /6$, $\pi /5$, $\pi /4$, $\pi /3$, $\pi /2.5$ [Fig. 2]. Relevant values of $k_{\text{so}}$ are estimated in the text.