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 $\Delta =\left|\Delta \right|\exp i\varphi$. However, the theory is still based on the assumption that the integral equation for the Green's function can be solved in powers of ${\mathrm{v}}_s=\frac{(2e\mathrm{A}-\nabla \varphi )}{2m}$ and successively higher space derivatives of $\left|\Delta \right|$ and ${\mathrm{v}}_s$. 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 $\left|\Delta \right|$ is close to the BCS gap ${\Delta }_{\mathrm{BCS}}\mathrm{}\left(T\right)\mathrm{}$ 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) $\left|\Delta \right|$ and ${\mathrm{v}}_{\mathrm{s}}$ vary slowly over distances $\xi$; and (2) ${\lambda }_{\mathrm{s}}=\frac{h}{2m{v}_{\mathrm{s}}}>\xi$. In the "dirty" limit ($l\ll {\xi }_0$) the length $\xi$ ranges from about ${\left(l{\xi }_0\right)}^{\frac{1}{2}}$ to $\left(\frac{T_c}{T}\right){\left(l{\xi }_0\right)}^{\frac{1}{2}}$ as $\left|\Delta \right|$ varies from ${\Delta }_{\mathrm{BCS}}\mathrm{}\left(T\right)\mathrm{}$ to zero.