Superconductivity in Nontransition Metals. I

Abstract

The theoretical calculation of superconducting-state parameters $〈{V}_{\mathrm{ph}}〉$, $〈{V}_{\mathrm{C}}〉$, and ${Z}_{0}$ [$〈{V}_{\mathrm{ph}}〉$ estimates the phonon-mediated attractive electron-electron interaction in the metal; $〈{V}_{\mathrm{C}}〉$, the Coulomb pseudo-potential, measures the effectiveness of the Coulomb repulsion in inhibiting superconductivity; and ${Z}_{0}$, the quasiparticle mass (or energy) renormalization parameter, takes into account the many-body renormalization effects due to electron-phonon and Coulomb interactions] is undertaken for both nonsuperconductors and simple-metal (nontransition-metal) superconductors. The two questions faced are: why non-superconductors (alkali, alkaline-earth, and noble metals) are nonsuperconductors (this question arises from the startling prediction by Morel and Anderson that all metals should be superconductors); and whether better agreement can be achieved between theoretically estimated superconducting-state parameters and empirical parameters---better, e.g., than those of Pines and of Morel and Anderson. Our calculation is differentiated from that of Morel and Anderson first in using electron-phonon and Coulomb-interaction matrix elements between Bloch electron states, secondly in using for the electron-phonon matrix element the form suggested by Harrison for the orthogonalized-plane-wave form factor, and lastly in including the renormalization effects due to electron-phonon and Coulomb interactions. The renormalization due to Coulomb interaction is included by taking the values from the calculations by Rice. The Bloch electron states are calculated in a model in which the ionic core is replaced by a three-dimensional constant repulsive potential, and the wave function is determined in the spirit of the Wigner-Seitz approximation and first-order perturbation theory. The theoretically estimated parameters $〈{V}_{\mathrm{ph}}〉$, $〈{V}_{\mathrm{C}}〉$, and ${Z}_{0}$ are compared with the empirically estimated parameters. These are also compared with the parameters estimated by Garland. A good comparison---better than that of Morel and Anderson---is found between theoretical and empirical parameters. It is pointed out that if renormalization effects are included in the calculation by Morel and Anderson, then their apparently good comparison with empirical parameters does not remain so good. In the case of nonsuperconductors, we find that the phonon-mediated attractive interaction fails to dominate over the Coulomb repulsion, contrary to the conclusions of Morel and Anderson, and the effective interaction strength $\frac{(〈{V}_{\mathrm{ph}}〉\ensuremath{-}〈{V}_{\mathrm{C}}〉)}{{Z}_{0}}$, which is analogous to the BCS parameter ${N}_{0}V$, is repulsive or very slightly attractive. The nonsuperconducting status is thus restored to alkali and alkaline-earth metals.