GENERALIZED LANDAUER FORMULAS FOR QUASI-PARTICLE TRANSPORT IN DISORDERED SUPERCONDUCTORS

When a quasi-particle current passes into a disordered superconductor, from ideal normal leads connected to external reservoirs, a finite electrical resistance R(S) arises from scattering processes within the superconductor. A new formula for R(S) is obtained, which reduces to the well-known Landauer formula in the absence of superconductivity. If R0, R(a) (T0, T(a)) are reflection (transmission) coefficients associated with normal and Andreev scattering respectively, one finds, in one dimension at zero temperature, R(S) = (h/2e2)(R0 + T(a) + delta)/(R(a) + T0 - delta) where delta is a small parameter arising from the absence of inversion symmetry. Generalizations of this result to finite temperatures and higher dimensions are also obtained.