On the solution of the modified Ginzburg-Landau type equation for one-dimensional superconductor in presence of a normal layer

We perform an analytical and numerical study of the crossover from the Josephson effect to the bulk superconducting flow for two identical one-dimensional superconductors, co-existing with a layer of normal material. A generalized Ginzburg-Landau (GL) model, proposed by S.J. Chapman, Q. Du and M.D. Gunzburger [1] was used in modeling the whole structure. When the thickness of the normal layer is very small, the introduction of three effective δ-function potentials of specified strength leads to an exact analytical solution of the modified stationary GL equation. The resulting current density-phase offset relation is analyzed numerically. We show that the critical Josephson current density jc corresponds to a bifurcation of the solutions of the nonlinear boundary value problem coupled with the modified GL-equation. The influence of the second term in the Fourier-decomposition of the supercurrent density-phase relation is also investigated. We derive also a simple analytical formula for the critical Josephson current.