Subharmonic energy-gap structure in superconducting constrictions

A Boltzmann-equation approach for the calculation of the $I\ensuremath{-}V$ characteristics of superconducting constrictions is presented. This technique allows for the inclusion of normal scattering as well as Andreev reflection processes in the constriction. The computed $I\ensuremath{-}V$ characteristics exhibit subharmonic gap structure which varies strongly with scattering strength and temperature. For even small scattering strengths, the structure is found to persist to $T=0$, and its temperature dependence agrees qualitatively with experimental observations. In the limit of zero scattering, the technique is shown to be equivalent to the trajectory technique of Klapwijk, Blonder, and Tinkham.