Role of Normal Layers in Penetration Depth Determinations of the Pairing State in High-TcSuperconductors

We calculate the penetration depth $\lambda \left(T\right)$ for a model of the high- $T_c$ superconductors based upon proximity coupling between one superconducting $\left(S\right)$ and one normal $\left(N\right)$ layer per unit cell. The linear, low- $T$ region of the ${\lambda }_{\mathrm{ab}}\left(T\right)$ data of Bonn et al. on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ is fitted quantitatively for both $s$-wave and $d_{x^2{-y}^2}$ order parameters. However, these fitted models give very different predictions for ${\lambda }_c\left(T\right)$, which may serve as a new test for the order parameter, or at least for the role of $N$ layers.