Theoretical investigation of isotope effect of the iron-based superconductors

► We derived the isotope effect exponent of phase-shifted s-wave two-band superconductor. ► Isotope effect exponent dependence on Tc is presented. ► The effect of antiferromagnetically-induced interaction and the phonon interaction on isotope exponent is considered. ► The implication of bands with different density of states is analyzed. A two-band BCS-like model in which the mediating interactions arise from both electron–phonon and antiferromagnetically induced interaction as well as ±s-wave gap symmetry has been used to investigate the isotope effect and transition temperature of iron-based superconductors within the Bogoliubov–Valatin approach. Our results show that electron–phonon interaction plays some role in the superconductors and experimentally determined values of the isotope effect exponent as well as transition temperatures of the compounds can be accounted for using the model.