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

Abstract

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 ${\ensuremath{\omega}}_{E}$, but assume that the energy gap $\ensuremath{\Delta}$ does not have any isotope effect. Nevertheless, we find an isotope effect for ${n}_{S}/{m}^{*}$ which is significant if $\ensuremath{\Delta}$ is sufficiently large that it becomes comparable to ${\ensuremath{\omega}}_{E}$, 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.