Anomalous Isotope Effect in BCS Superconductors with Two Boson Modes

The isotope effect in the superconducting transition temperature is anomalous if the isotope coefficient alpha < 0 or alpha > 1/2. In this work, we show that such anomalous behaviors can naturally arise within the Bardeen-Cooper-Schrieffer framework if both phonon and non-phonon modes coexist. Different from the case of the standard Eliashberg theory (with only phonon) in which alpha <= 1/2, the isotope coefficient can now take arbitrary values in the simultaneous presence of phonon and the other non-phonon mode. In particular, most strikingly, a pair-breaking phonon can give rise to large isotope coefficient alpha > 1/2 if the unconventional superconductivity is mediated by the lower frequency non-phonon boson mode. Based on our studies, implications on several families of superconductors are discussed.