Upper Critical Field of Two-Band Superconductors Containing Nonmagnetic Impurities

By making use of a 4×4-matrix Green's function suitable for the two-band model of the superconducting phase of the transition metals, generalized 2×2-matrix Gor'kov equations are established. With these, Maki's theory of the upper critical field of one-band superconductors in the "dirty" limit is extended to the two-band model in the "dirty" limit and in the strong intraband phonon-coupling limit. It is found that in the temperature range between the $d$-band critical temperature and the $s$-band critical temperature, the upper critical field is uniformly decreased because of the interband impurity scattering. The general behavior of the upper critical field as a function of temperature is discussed.