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 GA(R; r), expansions of GA, the self-energy part Σ, 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 φ(R) are summed up. With the help of GA, the generalized Ginzburg-Landau-Gor'kov (GLG) equations, valid at all temperatures for slowly varying A(R) and φ(R), are derived. For temperatures near Tc, correction terms to the coefficients of the GLG equations occur which are proportional to powers of |βφ|². For temperatures near 0^∘{}K, the function multiplying the term (∇+2ieA)²φ behaves like exp(-|βφ|). The first-order correction to the term proportional to A² is found to be proportional to ξ₀²H², for T near Tc and near 0^∘{}K (H=magnetic field strength, ξ₀=coherencelength). Our results are consistent with the formula of Nambu and Tuan for the reduction of the gap at 0^∘{}K in the London region.