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

The incoherent regime of multiple Andreev reflections ~MAR! is studied in long diffusive SNS junctions at applied voltages larger than the Thouless energy. 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 electrons and holes through ‘‘tunnel’’ and ‘‘Andreev’’ resistors. These resistors in diffusive junctions play roles analogous to the normal and Andreev reflection coefficients in Octavio-Tinkham-Blonder-Klapwijk theory for ballistic junctions. The theory is applied to the subharmonic gap structure ~SGS!; simple analytical results are obtained for the distribution function and current spectral density for the limiting cases of resistive and transparent NS interfaces. In the general case, the exact solution is found in terms of chain fractions, and the current is calculated numerically. SGS shows qualitatively different behavior for even and odd subharmonic numbers n52D/eV, and the maximum slopes of the differential resistance correspond to the gap subharmonics, eV52D/n. The influence of inelastic scattering on the subgap anomalies of the differential resistance is analyzed.