Generalized Ginzburg-Landau Theory of Superconducting Alloys

The Ginzburg-Landau-Gorkov theory of superconducting alloys is generalized to arbitrary temperatures and magnitudes of the gap parameter Δ=|Δ|expiφ. However, the theory is still based on the assumption that the integral equation for the Green's function can be solved in powers of vₛ=(2eA-∇φ)2m and successively higher space derivatives of |Δ| and vₛ. The averaging over the positions of the impurities is carried out by means of the ladder-diagram technique developed by Abrikosov and Gorkov. The results are presented in terms of the free-energy functional. The form of this functional is found to be close to that proposed by Ginzburg-Landau if |Δ| is close to the BCS gap ΔBCS(T) and varies slowly in space. A comparison of the magnitudes of the fourth- and second-order terms in the free-energy functional shows that the local theory is valid if: (1) |Δ| and vₛ vary slowly over distances ξ; and (2) λₛ=h2mvₛ>ξ. In the "dirty" limit (lξ₀) the length ξ ranges from about (lξ₀)1/2 to (TcT)(lξ₀)1/2 as |Δ| varies from ΔBCS(T) to zero.