Influence of the Rashba effect on the Josephson current through a superconductor/Luttinger liquid/superconductor tunnel junction

The Josephson current through a one-dimensional quantum wire with Rashba spin-orbit and electron-electron interactions is calculated. We show that the interplay of Rashba and Zeeman interactions gives rise to an anomalous phase shift in the current-phase relation for the supercurrent. The electron dispersion asymmetry induced by the Rashba interaction in a Luttinger-liquid wire plays a significant role when the electron-electron interaction is not strong and for poorly transmitting junctions. It is shown that for a weak or moderate electron-electron interaction the spectrum of plasmonic modes confined to the normal part of the junction becomes quasi-random in the presence of dispersion asymmetry. The resonance effects which are significant for transport properties of weakly interacting electrons in symmetric junctions survive in the presence of a strong Rashba interaction only for special boundary conditions at normal metal/superconductor interfaces.

Figure 1

Schematic energy spectrum of 1D spin-$1∕2$ electrons with dispersion asymmetry. The subbands “1” and “2” are characterized by their Fermi velocities $v_{1F}\ne v_{2F}$. In the case of weak Rashba interaction, the spin projections for a given momentum have opposite directions in the two subbands (solid arrows). For strong Rashba interaction the spin projections are parallel for all particles that move in the same direction irrespective of subband, but are different for right- and left-moving particles (dashed arrows).

Figure 2

$S∕\mathrm{LL}∕S$ junction of length $L$ formed by an effectively infinite Luttinger liquid coupled to bulk superconductors by side electrodes.

Figure 3

A Luttinger-liquid wire of length $L$ coupled to bulk superconductors via tunnel barriers with transparencies $D_{l\left(r\right)}$.