Circuit theory of multiple Andreev reflections in diffusive SNS junctions:The incoherent case

The incoherent regime of multiple Andreev reflections (MAR) is studied in l ong diffusive SNS junctions at applied voltages larger than the Thouless en ergy. Incoherent MAR are treated as a transport problem in energy space by means of a circuit theory for an equivalent electrical network. The current through NS interfaces is explained in terms of diffusion flows of electron s and holes through ''tunnel'' and ''Andreev'' resistors. These resistors i n diffusive junctions play roles analogous to the normal and Andreev reflec tion coefficients in Octavio-Tinkham-Blonder-Klapwijk theory for ballistic junctions. The theory is applied to the subharmonic gap structure (SGS); si mple analytical results are obtained for the distribution function and curr ent spectral density for the limiting cases of resistive and transparent NS interfaces. In the general case, the exact solution is found in terms of c hain fractions, and the current is calculated numerically. SGS shows qualit atively different behavior for even and odd subharmonic numbers n = 2 Delta /eV, and the maximum slopes of the differential resistance correspond to t he gap subharmonics, eV = 2 Delta /n. The influence of inelastic scattering on the subgap anomalies of the differential resistance is analyzed.