Free energy formula for a strong coupling superconductor with energy dependent electronic density of states

Abstract

Generalized expressions for the free energy difference between normal and superconducting states are derived for the case of an energy dependent (ED) electronic density of states (EDOS) with normal and paramagnetic impurity scattering included. The derivation starts from the Eliashberg expression for the grand thermodynamic potential and ends in explicit formulas for the case of a symmetric Lorentzian model EDOS. Some numerical results are given for the critical magnetic field deviation function and the ratio 2Δ₀/k_BT_c of the Δ₀ to the critical temperature T_c. For a peak at the Fermi energy of half width less than the Debye energy (ω_D), the modified Eliashberg theory is needed to describe its effect on the thermodynamic properties, whereas for a half width greater than ω_D, the corrections to a flat approximation are small.