Generalized Landauer formulae 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.