Upper critical field for a high-Tcelectron-phonon superconductor: Regime ofTcω1n∼1

We have calculated the upper critical field $H_{c2\mathrm{}}$ for an electron-phonon superconductor in the limit when the critical temperature $T_c$ is of the same order as a typical phonon energy ${\omega }_{1n}$. This extends well beyond the conventional range for which $\frac{T_c}{{\omega }_{1n}}\lesssim 0.25$. In this very-strong-coupling regime, the strong-coupling corrections to $H_{c2\mathrm{}}$ are found to be much less than one, in sharp contrast to the usual case for which they are invariably greater than one. On the other hand, the value of the reduced critical field $h_{c2\mathrm{}}$ at zero temperature can be larger than in Bardeen-Cooper-Schrieffer theory by more than a factor of 3. Also, as a function of reduced temperature ($t$), $h_{c2\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ displays an unusual positive curvature for $t$ in the middle of its range and $h_{c2\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$ acquires a very strong impurity dependence.