The impurity effects on the transition temperature Tc and the isotope effect are examined in multiband superconductors with magnetic and nonmagnetic impurities, where the effect of Coulomb repulsion is considered. It is shown that an internal transition of the superconducting state is induced by impurity doping, and that the transition is accompanied by a jump of the isotope-effect coefficient α. In particular, the transition is illustrated in a system with two electron bands. In some special cases, extended Abrikosov and Gor'kov (AG) equations for Tc and the expressions of the isotope-effect coefficient α are obtained. Possible relevance of the present mechanism to the experimental results of Sr2RuO4 is discussed.KEYWORDS: Abrikosov-Gor'kov equation, superconducting transition temperature, isotope effect, multiband superconductor, impurity effect, weakly screened electron-phonon interactionThe isotope-effect coefficient provides important information on the mechanism of superconductivity. For example, the BCS theory based on the electron-phonon interactions is supported in nontransition-metal superconductors, such as Hg and Zn, by their isotope-effect coefficient α, which is nearly equal to 0.5. In transition metals and some compounds, large deviations from α = 0.5 have been observed, but the deviations can be explained in the context of electron-phonon interactions by taking into account strong Coulomb repulsion, anharmonicity of lattice vibrations, and van Hove singularity.
1)On the other hand, in exotic superconductors, such as high-T c cuprates, organic superconductors, Sr 2 RuO 4 , and heavy fermion superconductors, nonphonon mechanisms of superconductivity, such as spin-fluctuation exchange interactions, have been examined as possible mechanisms. In such nonphonon mechanisms, the isotope-effect coefficient must deviate from 0.5 markedly, or vanish completely. The presence of the isotope shift (α = 0) suggests the presence of a contribution to the pairing interaction from electron-phonon interactions.Therefore, it may be difficult to derive information on the pairing mechanism solely from a single value of α. Therefore, it is useful to systematically examine correlations between α and other quantities, such as impurity concentration, transition temperature, hole concentration (for high-T c cuprates), and pressure.We examine the impurity effect on the isotope effect in this study. First, we briefly review it in single-band systems. In the presence of impurities, T c is reduced except for nonmagnetic isotropic impurity scattering and isotropic pairing. The ratio T c /T c0 satisfies the Abrikosov and Gor'kov (AG) equation
2)ln T c0