Effect of strong scattering on the low-temperature penetration depth of a d-wave superconductor

For a pure superconductor in a d-wave-like state at temperatures T well below the critical temperature ${\mathit{T}}_{\mathit{c}}$, the deviation Δλ of the penetration depth from its zero-temperature value λ(0) is proportional to T. When the concentration ${\mathit{n}}_{\mathit{i}}$ of strongly scattering impurities is nonzero, Δλ∝${\mathit{T}}^{\mathit{n}}$, where n=2 for T<${\mathit{T}}^{\mathrm{*}}$≪${\mathit{T}}_{\mathit{c}}$ and n=1 for ${\mathit{T}}^{\mathrm{*}}$<T≪${\mathit{T}}_{\mathit{c}}$. The crossover temperature ${\mathit{T}}^{\mathrm{*}}$ and the increase in λ(0) scale as √${\mathit{n}}_{\mathit{i}}$ up to logarithmic corrections when resonant scattering is dominant. We argue that this case is relevant to recent measurements on ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ and present specific results for a model pairing state with ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ symmetry.