London penetration depth at zero temperature and near superconducting transition

A simple relation is established between the zero-$T$ penetration depth $\lambda (0)$ and the slope of ${\lambda }^{-2}(T)$ near $T_c$, similar to Helfand-Werthamer’s relation for $H_{c2}(0)$ and the slope of $H_{c2}(T)$ at $T_c$ for the isotropic s-wave case with non-magnetic scattering.\cite{HW} When the scattering parameter $\rho =\mathrm{\hslash }v\mathrm{/}2\pi T_c\mathrm{\ell }$ ($v$ is the Fermi velocity and $\mathrm{\ell }$ is the mean-free path) varies from 1 to 10, the coefficient of proportionality between ${\lambda }^{-2}(0)$ and $T_c(d{\lambda }^{-2}\mathrm{/}dT{)}_{T_c}$ changes from 0.43 to 0.38. Combining this relation with the Rutgers thermodynamic identity, one can express $\lambda (0)$ in terms of the slope $(d{H}_{c2}\mathrm{/}dT{)}_{T_c}$ and the density of states.