s- andp-wave pairings in the dilute electron gas: Superconductivity mediated by the Coulomb hole in the vicinity of the Wigner-crystal phase

The superconducting transition temperature Tc of both s- and p-wave pairings is calculated in the electron gas without phonons by the solution of the full Eliashberg equation in both frequency and momentum variables. The exchange and correlation effects are included in the form of the model proposed by Kukkonen and Overhauser with suitable local-field corrections. The ground state of the electron gas exhibits p-wave superconductivity for the electronic-density parameter rₛ around 4 or larger, but it is superseded by s-wave superconductivity for rₛ larger than about 8.5. The ratio of Tc to the Fermi energy increases monotonically with rₛ, but it is saturated to have the value of about 0.04 for rₛ>40. The physical origin of this superconductivity in the dilute electron gas is explained in terms of the pairing mediated by the Coulomb hole near the Wigner-crystal phase.