Theory of a Local Superconductor in a Magnetic Field

The behavior of a superconductor in a static magnetic field H (r) is considered in the limit that the field and the gap function (r) vary slowly in space compared with the correlation distance of a superconducting electron pair (local limit). The superconductor is described by a pair of coupled differential equations for (r) and the induced vector potential A (r). The equations are analogous in form to those proposed phenomenologically by Ginzburg and Landau (GL), but contain additional nonlinear terms. When is independent of position and H is weak, the equations reduce to those given by BCS for the dependence of and the penetration depth on the temperature T. When, the equations reduce to those derived by Gor'kov, in confirmation of the GL theory. The region of validity of the derivation is examined, showing that while a local (London) electrodynamics can be correct for some materials over a wide range of T and H, (r) is slowly varying for TT₂ only if is near to its equilibrium, zero-field value.