Vertex correction to the Eliashberg equation for the superconducting critical temperature

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer

When considering a Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron-phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio λωD/εF ≈ 0.46, where λ is the electron-phonon coupling constant, ωD is the Debye frequency, and εF represents the Fermi energy. Due to nonadiabatic effects, the phonon-induced superconducting state in Li-hBN is characterized by much lower values of the critical temperature (T LOVC C ∈ {19.1, 15.5, 11.8} K, for μ* ∈ {0.1, 0.14, 0.2}, respectively) than would result from calculations not taking this effect into account (T ME C∈ {31.9, 26.9, 21} K). From the technological point of view, the low value of T C limits the possible applications of Li-hBN. The calculations were carried out under the classic Migdal-Eliashberg formalism (ME) and the Eliashberg theory with lowest-order vertex corrections (LOVC). We show that the vertex corrections of higher order (λ3) lower the value of T LOVC C by a few percent.