Role of electron Coulomb interaction in superconductivity

The effect of the dynamically screened electron-electron interaction on the superconducting transition temperature is investigated by solving the Eliashberg equation in both frequency and momentum variables. The present calculations are performed within the random-phase-approximation (RPA)—screened free-electron model and for electronic density parameters $r_s<5\mathrm{}$. The full normal-state self-energy is included self-consistently. In parallel, the equation for the Coulomb pseudopotential ${\mu }^{*}$ is set up and solved using the imaginary-frequency representation. We find that in RPA plasmon exchange leads to a change in sign for ${\mu }^{*}$ at $r_s\sim 2.5$. For $r_s>2.5\mathrm{}$, ${\mu }^{*}<0\mathrm{}$ and the system becomes superconducting even without consideration of phonons. These results show that the effect of electron-electron attraction by plasmon exchange is overestimated in RPA, and that vertex corrections must be included. Similar conclusions have to be drawn for electron attraction through other high-frequency bosons such as excitons.