Gap Equation and Current Density for a Superconductor in a Slowly Varying Static Magnetic Field

The energy gap equation and the current density expression for a superconductor in a slowly varying static magnetic field are derived on the basis of a generalization of Nambu's Green's function formalism to finite temperatures. In the integral equation for the quasiparticle Green's function $G^A(\mathrm{R}; \mathrm{r})$, expansions of $G^A$, the self-energy part $\Sigma$, and the vector potential A, about the center-of-mass coordinates R, are introduced. The integral equation is solved by iteration, and the contributions of all orders in the gap $\phi \left(\mathrm{R}\right)$ are summed up. With the help of $G^A$, the generalized Ginzburg-Landau-Gor'kov (GLG) equations, valid at all temperatures for slowly varying A(R) and $\phi \left(\mathrm{R}\right)$, are derived. For temperatures near $T_c$, correction terms to the coefficients of the GLG equations occur which are proportional to powers of ${\left|\beta \phi \right|}^2$. For temperatures near 0°K, the function multiplying the term ${(\nabla +2\mathrm{ie}\mathrm{A})}^2\phi$ behaves like $\exp \mathrm{}(-|\mathrm{}\beta \phi \mathrm{}\left|\right)\mathrm{}$. The first-order correction to the term proportional to $A^2$ is found to be proportional to ${{\xi }_0}^2{H}^2$, for $T$ near $T_c$ and near 0°K (H=magnetic field strength, ${\xi }_0=\mathrm{coherence}\mathrm{length}$). Our results are consistent with the formula of Nambu and Tuan for the reduction of the gap at 0°K in the London region.