Ginzburg-Landau Equations and the Josephson Effect

A self-consistent model is developed for the Josephson effect based on the Ginzburg-Landau phenomenological equations. In this approach, the gap behaves like a weak superconductor, with the maximum Josephson current as its critical current. An expression is derived for the maximum Josephson current, for the case of two identical superconductors and no external magnetic field, in terms of an effective barrier height and width, the weak-field penetration depth, and the bulk critical field. The expression is first derived for a one-dimensional configuration, and then shown to remain valid for certain thin-film junctions. Using 1 eV for the barrier height and 10 Å for the barrier width, the maximum Josephson current turns out to be 0.38 A/${\mathrm{cm}}^2$ for pure-tin junctions. For the same sample a penetration depth of 2.06 mm is obtained at the center of the gap for the maximum current.