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Model for the inverse isotope effect of FeAs-based superconductors in the $π$-phase-shifted pairing state

The isotope effects for Fe based superconductors are considered by including the phonon and magnetic fluctuations within the two band Eliashberg theory. We show that the recently observed inverse isotope effects of Fe, $α_{Fe} \approx -0.18 \pm 0.03 $,\cite{Shirage0903.3515} as well as the large positive isotope exponent ($α\approx 0.35$) can naturally arise for the magnetically induced sign revered s-wave pairing state within reasonable parameter range. Either experimental report can not be discarded from the present analysis based on the parameter values they require. The inverse and positive isotope effects mean, respectively, the interband and intraband dominant eletron-phonon interaction. We first make our points based on the analytic result from the square well potential model and present explicit numerical calculations of the two band Eliashberg theory.