Nonlinear transport theory for hybrid normal-superconducting devices
  We report a theory for analyzing nonlinear DC transport properties of mesoscopic or nanoscopic normal-superconducting (N-S) systems. Special attention was paid such that our theory satisfies gauge invariance. At the linear transport regime and the sub-gap region where the familiar scattering matrix theory has been developed, we provide confirmation that our theory and the scattering matrix theory are equivalent. At the nonlinear regime, however, our theory allows the investigation of a number of important problems: for N-S hybrid systems we have derived the general nonlinear current-voltage characteristics in terms of the scattering Green's function, the second order nonlinear conductance at the weakly nonlinear regime, and nonequilibrium charge pile-up in the device which defines the electrochemical capacitance coefficients. 
Jian Wang, Yadong Wei, Hong Guo, Qing-feng Sun, and Tsung-han Lin