Maximum reduced upper critical magnetic field in Eliashberg theory

Consideration of the functional derivative of the zero-temperature reduced upper critical magnetic field h/sub c2/(O) with electron-phonon spectral density α2F(omega) suggests that its value can be maximized, for a given area A under α2F(omega), with the choice of a delta-function spectral density. We show that h/sub c2/(O) is then independent of A and that it increases steadily as the Einstein frequency Omega/sub E/ of the delta function is lowered. We argue that there is a local maximum in h/sub c2/(O) at Omega/sub E/=O. The value at maximum is very sensitive to impurity content and increases sharply from a clean-limit value of 1.5 at T/sub c//Omega/sub E/ = 1.33 to more than 3.45 in the dirty limit