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 $\Delta \left(\mathrm{r}\right)$ 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 $\Delta \left(\mathrm{r}\right)$ 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 $\Delta$ is independent of position and H is weak, the equations reduce to those given by BCS for the dependence of $\Delta$ and the penetration depth on the temperature $T$. When $\Delta \ll T$, 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 $H$, $\Delta \left(\mathrm{r}\right)$ is slowly varying for $T\ll T_c$ only if $\Delta$ is near to its equilibrium, zero-field value.