c-Axis Penetration Depth in the Cuprates: Additional Evidence for Incoherent Hopping

Measurements of the $c$-axis penetration depth $\lambda_c$ in the cuprates reveal a low-temperature $T$ dependence which is inconsistent with simple models of coupling between the CuO$_2$ layers. In this paper, we examine whether a model based on {\it incoherent} hopping between the layers can account for this low-$T$ behavior. We compute $\lambda_c$ directly from linear response theory and compare our results with recent experimental measurements on $\rm YBa_2u_3O_{7-\delta}$ as a function of temperature and doping. We find that the data can be reproduced within this model providing the inter-layer scattering is anisotropic and the pairing is $d$-wave. In addition, our calculations demonstrate that $1/\lambda_c^2$ is proportional to the $c$-axis critical current, which seems to be a generic feature in weakly coupled layered superconductors.