BOUNDARY-CONDITIONS FOR QUASI-CLASSICAL EQUATIONS IN THE THEORY OF SUPERCONDUCTIVITY

In this paper we derive effective boundary conditions connecting the q uasiclassical Green's function through tunnel barriers in superconduct ing-normal hybrid (S-N or S-S') structures in the dirty limit. Our wor k extends previous treatments confined to the small transparency limit . This is achieved by an expansion in the small parameter r(-1) = T/2( 1 - T) where T is the transparency of the barrier. We calculate the ne xt term in the r(-1) expansion for both the;normal and, the supercondu cting case. In both cases this involves the solution of an integral eq uation, which we obtain numerically. While in the normal case our trea tment only leads to a quantitative change in the barrier resistance Rb , in the superconductor case, qualitative different boundary condition s are derived. To illustrate the physical consequences of the modified boundary conditions, we calculate the Josephson current and show that the next term in the r(-1) expansion gives rise to the second harmoni c in the current-phase relation.