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 (R; r), expansions of G^A, the self-energy part, 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 (R) 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 (R), are derived. For temperatures near T₂, correction terms to the coefficients of the GLG equations occur which are proportional to powers of ||^2. For temperatures near 0^, the function multiplying the term (+2ieA) ^2 behaves like exp (-||). The first-order correction to the term proportional to A^2 is found to be proportional to {₀}^2H^2, for T near T₂ and near 0^ (H=magnetic field strength, ₀=coherencelength). Our results are consistent with the formula of Nambu and Tuan for the reduction of the gap at 0^ in the London region.