Properties of strong-coupled superconductors

Abstract

By use of Eliashberg equations the system of electrons and phonons with the Einstein spectrum ${\ensuremath{\alpha}}^{2}$(\ensuremath{\omega})F(\ensuremath{\omega})=(\ensuremath{\lambda}\ensuremath{\Omega}/2)\ensuremath{\delta}(\ensuremath{\omega}-\ensuremath{\Omega}) is studied. The orbital and paramagnetic upper critical magnetic fields are obtained in the case of strong electron-phonon coupling \ensuremath{\lambda} (including \ensuremath{\lambda}\ensuremath{\gg}1). For \ensuremath{\lambda}\ensuremath{\gg}1, superconducting parameters at very low temperatures differ remarkably from those near ${T}_{c}$; the crossover takes place at a temperature corresponding to the phonon frequency \ensuremath{\Omega}. For \ensuremath{\lambda}\ensuremath{\gtrsim}4, at temperatures below about \ensuremath{\Omega}/3 the superconducting correlation length decreases, giving the positive curvature in the temperature dependence of the upper critical field. The coefficients of the Ginzburg-Landau functional are calculated. The absolute value of the specific-heat jump grows with \ensuremath{\lambda} while its relative value drops. For \ensuremath{\lambda}\ensuremath{\gtrsim}4, the resistivity above ${T}_{c}$ increases linearly with temperature.