Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity

We study the Josephson current through a ballistic, normal, one-dimensional quantum channel in contact with two superconducting electrodes. A single point impurity having reflection coefficient R is placed in the normal conductor. The impurity couples the Andreev energy levels of forward and reverse moving electrons inside the junction, opening energy gaps in the quasiparticle level spectrum versus superconducting phase difference φ. These ‘‘Andreev’’ energy gaps suppress the Josephson current in much the same way as disorder suppresses the magnetic flux driven currents in a normal mesoscopic ring. Finite-temperature ‘‘energy averages’’ the contribution of Andreev levels above the Fermi energy with those below μ, further suppressing the Josephson current. The portion of the Josephson current carried by scattering states outside the superconducting gap is similarly suppressed by disorder and finite temperature.