Cross-over from BCS superconductivity to Bose condensation and High-Tc Superconductors

We consider the Eliashberg theory in the coupling region where some fundamental qualitative deviations from the conventional BCS-like behaviour begin to appear. These deviations are identified as the onset of a cross-over from BCS superconductivity to Bose condensation. We point out that the beginning of this cross-over occurs when the gap $\Delta_g$ becomes comparable to the boson energies $\Omega_{ph}$. This condition is equivalent to the condition of Ref. \cite{Strinati} $k_F\xi\approx 2\pi$ and traduces the physical constraint that the distance the paired electron covers during the absorbtion of the virtual boson, cannot be larger than the coherence length. The frontier region of couplings is of the order of $\lambda\approx 3$, and high-$T_c$ materials are concerned. A clear qualitative indication of the occurence of a cross-over regime should be a dip structure above the gap in the density of states of excitations. Comparing our results with tunneling and photoemission experiments we conclude that high-$T_c$ materials (cuprates and fullerides) are indeed at the beginning of a cross-over from BCS superconductivity to Bose condensation, even though the fermionic nature still prevails. Taking into account the analysis of Ref. \cite{Strinati}, we predict a dip structure in heavy fermion and organic superconductors. Non-adiabatic effects beyond Migdal's theory are considered and give insight on the robustness of Eliashberg theory in describing qualitatively this cross-over regime, although for the quantitative interpretation of the results the inclusion of non-adiabatic corrections can be important.Comment: 37 pages, latex , 16 figures available upon request to [email protected]

The boson mediators of high-Tc superconductivity: phonons versus composite bosons from the superconducting phenomenology

We address the question of whether boson mediators of high-$T_c$ superconductivity are composite (electronic) or independent phonons. For s-wave superconductivity we show from the available experiments that the hypothesis of composite bosons is rather unlikely. Our analysis points naturally towards phonon mediators. In addition we point out that the eventual presence of a peak in the temperature dependence of the microwave conductivity while the Hebel-Slichter peak is absent in the temperature dependence of the NMR relaxation rate, can be understood within a phonon mechanism if one takes into account the modulation of the electron-phonon coupling (predominance of forward scattering) induced by Coulomb correlation of the carriers.Comment: Accepted for publication in the Rapid Communications section of Physical Review B, 4 pages latex (revtex),4 figures available upon request to [email protected]

Spin susceptibility in small Fermi energy systems: effects of nonmagnetic impurities

In small Fermi energy metals, disorder can deeply modify superconducting state properties leading to a strong suppression of the critical temperature $T_c$. In this paper, we show that also normal state properties can be seriously influenced by disorder when the Fermi energy $E_{\rm F}$ is sufficiently small. We calculate the normal state spin susceptibility $\chi$ for a narrow band electron-phonon coupled metal as a function of the non-magnetic impurity scattering rate $\gamma_{\rm imp}$. We find that as soon as $\gamma_{\rm imp}$ is comparable to $E_{\rm F}$, $\chi$ is strongly reduced with respect to its value in the clean limit. The effects of the electron-phonon interaction including the nonadiabatic corrections are discussed. Our results strongly suggest that the recent finding on irradiated MgB$_2$ samples can be naturally explained in terms of small $E_{\rm F}$ values associated with the $\sigma$-bands of the boron plane, sustaining therefore the hypothesis that MgB$_2$ is a nonadiabatic metal.Comment: 7 pages, 6 eps figures, to appear on Eur. Phys. J.

The physical origin of the electron-phonon vertex correction

The electron-phonon vertex correction has a complex structure both in momentum and frequency. We explain this structure on the basis of physical considerations and we show how the vertex correction can be decomposed into two terms with different physical origins. In particular, the first term describes the lattice polarization induced by the electrons and it is essentially a single-electron process whereas the second term is governed by the particle-hole excitations due to the exchange part of the phonon-mediated electron-electron interaction. We show that by weakening the influence of the exchange interaction the vertex takes mostly positive values giving rise to an enhanced effective coupling in the scattering with phonons. This weakening of the exchange interaction can be obtained by lowering the density of the electrons, or by considering only long-ranged (small q) electron-phonon couplings. These findings permit to understand why in the High-Tc materials the small carrier density and the long ranged electron-phonon interaction may play a positive role in enhancing Tc.Comment: 11 pages, 5 postscript figure