Motivated by the observation of superconductivity in ${\mathrm{SrTiO}}_{3}$ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\ensuremath{\mu}$ is comparable to or even smaller than the characteristic phonon frequency ${\ensuremath{\omega}}_{L}$. We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg approximation becomes invalid when $\ensuremath{\mu}\ensuremath{\le}{\ensuremath{\omega}}_{L}$, chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to ${T}_{c}$ in the absence of phase fluctuations, as a function of $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$. We find both analytically and numerically that ${T}_{c}$ increases as the ratio $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$ becomes smaller. In particular, in the dilute regime, $\ensuremath{\mu}\ensuremath{\rightarrow}0$, it holds that ${T}_{c}\ensuremath{\propto}{\ensuremath{\omega}}_{L}{(\frac{Ry}{{\ensuremath{\omega}}_{L}})}^{\ensuremath{\eta}}$, where Ry is the Rydberg constant and $\ensuremath{\eta}\ensuremath{\sim}0.2$.Motivated by the observation of superconductivity in ${\mathrm{SrTiO}}_{3}$ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\ensuremath{\mu}$ is comparable to or even smaller than the characteristic phonon frequency ${\ensuremath{\omega}}_{L}$. We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg approximation becomes invalid when $\ensuremath{\mu}\ensuremath{\le}{\ensuremath{\omega}}_{L}$, chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to ${T}_{c}$ in the absence of phase fluctuations, as a function of $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$. We find both analytically and numerically that ${T}_{c}$ increases as the ratio $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$ becomes smaller. In particular, in the dilute regime, $\ensuremath{\mu}\ensuremath{\rightarrow}0$, it holds that ${T}_{c}\ensuremath{\propto}{\ensuremath{\omega}}_{L}{(\frac{Ry}{{\ensuremath{\omega}}_{L}})}^{\ensuremath{\eta}}$, where Ry is the Rydberg constant and $\ensuremath{\eta}\ensuremath{\sim}0.2$.

arXiv:1901.00344Physical review. B./Physical review. BMar 29, 201933 citations

Electron-phonon superconductivity mechanisms in the Hubbard model

Chan Minh Tien

Theoretical and Mathematical PhysicsJun 1, 19920 citations

Superconductivity in Systems with Diluted Interactions

Grzegorz Litak A, Thereza Paiva, Raimundo Dos Santos +2 more

APSMar 1, 20020 citations

Motivated by the observation of superconductivity in

{\mathrm{SrTiO}}_{3}

and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\ensuremath{\mu}$ is comparable to or even smaller than the characteristic phonon frequency ${\ensuremath{\omega}}_{L}$. We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg approximation becomes invalid when $\ensuremath{\mu}\ensuremath{\le}{\ensuremath{\omega}}_{L}$, chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to

{T}_{c}

in the absence of phase fluctuations, as a function of $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$. We find both analytically and numerically that

{T}_{c}

increases as the ratio $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$ becomes smaller. In particular, in the dilute regime, $\ensuremath{\mu}\ensuremath{\rightarrow}0$, it holds that ${T}_{c}\ensuremath{\propto}{\ensuremath{\omega}}_{L}{(\frac{Ry}{{\ensuremath{\omega}}_{L}})}^{\ensuremath{\eta}}$, where Ry is the Rydberg constant and $\ensuremath{\eta}\ensuremath{\sim}0.2$.