Ginzburg-Landau equations and the upper critical field of a mixed d- and s-wave superconductor

Based on the Abrikosov-Gor'kov theory of a superconducting alloy, we have microscopically derived Ginzburg-Landau (GL) equations of a mixed d- and s-wave superconductor with impurities. Diagrammatic relations for the impurity-averaged product of Green's functions have been algebraically established for the superconductor. Since cuprate materials with high superconducting transition temperatures (T_c) are believed to have a dominant d-wave pairing symmetry, we investigated effects of impurities on the superconductor using GL equations we have obtained. It has been qualitatively shown that the temperature ( T) dependence of London penetration changes from T to T2 dependence as the density of impurities increases, which is consistent with experimental results.; We have developed a quantum mechanical method to calculate ( H_c2) of a superconductor. Using this method, we discussed the impurity effect on H_c2 of a d-wave superconductor in a dilute limit of impurities, and calculated H _c2 of a mixed d- and s-wave superconductor in the whole temperature region. We have shown that H _c2 linearly increases with decreasing T near T_c, and becomes saturated consistent with the thermodynamical requirement. Since mass anisotropy may occur in a orthorombic matrial such as YBa₂Cu₃O_7–δ (YBCO), and paramagnetic Zeeman interaction is important in a strong magnetic field, we have taken these effects into account. We have found that the mass anisotropy effect enhances H_c2 against Zeeman suppression. The theoretical curve is in good agreement with the experimental data for YBCO.