Strong-coupling calculation of the lower critical field in high-Tcsuperconductors

Using the strong-coupling version of Usadel's dirty-limit theory, the lower critical field $H_{c1\mathrm{}}$ of high-$T_c$ superconductors is calculated. It is shown that the strong-coupling theory predicts a marked plateau in $H_{c1\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ at low temperatures, which is most pronounced for large electron-phonon coupling parameters $\lambda$. This effect can be used as a criterion for conventional electron-phonon superconductivity in high-$T_c$ superconductors. The plateau has not been observed in early measurements of $H_{c1\mathrm{}}$ in granular samples.