Isotope effect on the superfluid density in conventional and high-temperature superconductors

We investigate the isotope effect on the London penetration depth of a superconductor which measures n S/m*, the ratio of superfluid density to effective mass. We use a simplified model of electrons weakly coupled to a single phonon frequency ω E, but assume that the energy gap Δ does not have any isotope effect. Nevertheless, we find an isotope effect for n S/m* which is significant if Δ is sufficiently large that it becomes comparable to ω E, a regime of interest to high-T c cuprate superconductors and possibly other families of unconventional superconductors with relatively high T c. Our model is too simple to describe the cuprates and it gives the wrong sign of the isotope effect when compared with experiment, but it is a proof of principle that the isotope effect exists for n S/m* in materials where the pairing gap and T c are not of phonon origin and have no isotope effect.