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 ${\mathit{T}}_{\mathit{c}}$ 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p-wave superconductivity for the electronic-density parameter ${\mathit{r}}_{\mathit{s}}$ around 4 or larger, but it is superseded by s-wave superconductivity for ${\mathit{r}}_{\mathit{s}}$ larger than about 8.5. The ratio of ${\mathit{T}}_{\mathit{c}}$ to the Fermi energy increases monotonically with ${\mathit{r}}_{\mathit{s}}$, but it is saturated to have the value of about 0.04 for ${\mathit{r}}_{\mathit{s}}$>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.