Variational theory of superconductivity and application to the low-density electron gas

Abstract

The method of effective-potential expansion is employed to describe superconductivity in strongly correlated electron systems. We have obtained a gap equation in which both the pairing potential Vp(k,k’) and the single-particle energy ɛ̃k are expanded in terms of the effective potential. When the ring approximation or the random-phase approximation is used to evaluate each term in the expansion, Vp(k,k’) and ɛ̃k have the form VpKMK(k,k’)/zkzk’, and ɛk*/zk, respectively, where VpKMK is the pairing potential derived by Kirzhnits, Maksimov, and Khomskii (KMK), zk is the renormalization factor, and ɛk* tends to the bare single-particle energy in the weak-coupling limit. Thus our theory may be viewed as an extension of the KMK theory to the strong-coupling region. The present general theory is applied to the electron gas in order to investigate whether the Coulomb interaction alone can cause superconductivity. We have improved on the two-body approximation to describe the normal state of the electron gas and have determined the effective potential variationally. With the use of the effective potential thus obtained, we have evaluated all terms up to second order for both Vp(k,k’) and ɛ̃k. The pairing potential includes the plasmon-mediated attractive and the paramagnon-mediated repulsive parts as well as many complicated vertex corrections. We have solved the gap equation numerically and obtained the result that superconductivity appears at rather low carrier densities (i.e., rs>3.9). The highest transition temperature Tc in the range 30–60(m*/ε02) K is obtained at rs≊7, where m* is the band mass in units of the free-electron mass and ε0 is the dielectric constant. Our results might be useful to help explain the mechanism of superconductivity in the high-Tc oxide superconductors.

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