Asymmetric current-phase relation due to spin-orbit interaction in semiconductor nanowire Josephson junction

We theoretically study the current-phase relation in semiconductor nanowire Josephson junction in the presence of spin-orbit interaction. In the nanowire, the impurity scattering with strong SO interaction is taken into account using the random matrix theory. In the absence of magnetic field, the Josephson current I and phase difference φ between the superconductors satisfy the relation of I(φ) = −I(−φ). In the presence of magnetic field along the nanowire, the interplay between the SO interaction and Zeeman effect breaks the current-phase relation of I(φ) =−I(−φ). In this case, we show that the critical current depends on the current direction, which qualitatively agrees with recent experimental findings.