The unbalanced phonon-induced superconducting state on a square lattice beyond the static boundary

Abstract

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron-phonon interaction for values of the unbalance parameter (γ = λD/λND) less than γC = 0.42. Symbols λD and λND denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter (∆ k (iωn)) and the wave function renormalising factor (Z k (iωn)) depend explicitly on the Matsubara frequency (ωn) and the wave vector (k). The value of γC in the static boundary (∆ k (iωn) → ∆ k (iωn=1)), equal to (0.93), is significantly greater than the obtained limit value of 0.42. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron-phonon interaction function explicitly dependent on the momentum transfer between electron states.Physical properties of the phonon-induced superconducting state on a square lattice were determined in the present work within the Eliashberg formalism. The initial results, obtained in the limit of momentum approximation (which neglects the self-consistency with respect to Matsubara frequency), were already presented by us elsewhere [1]. There we showed that the balanced superconducting state on a square lattice cannot arise in the case of the constant value of the electron-phonon coupling function (g q ∼ g). By the balanced superconducting state we mean such a state, for which the coupling constant λ D in the diagonal channel of the self-energy matrix (M k (iω n )) is equal to the coupling constant in its non-diagonal channel λ N D . Please notice that the diagonal elements of the self-energy determine, inter alia, the influence of the electron-phonon interaction on the value of the effective electron mass. The off-diagonal elements of the M k (iω n ) are directly related to the order parameter of the phase transition between the superconducting and the normal state.At present, we know many physical mechanisms which can lead to the unbalanced phonon-induced superconducting state. Particular attention should be paid to the spin-fluctuation (paramagnon) scattering, which contributes to the effective coupling constant in the diagonal channel in the opposite way than it does in the antidiagonal channel (s-wave symmetry) [2,3,4,5,6,7]. The unbalance of the superconducting state can also be generated by the non-conventional terms of the electron-phonon interaction, e.g. the electron-electron-phonon [8] or, similarly, the hole-hole-phonon interaction [9]. Additionally, they induce the asymmetry of the electron density of states, the pseudo-gap in the electron density of states, or the anomalous dependence of the energy gap on doping [10]. These effects can be of particular importance for the correct description of the superconducting state in cuprates [11,12,13], especially since they are o...

Nonadiabatic superconductivity in Li-intercalated hexagonal boron nitride bilayer

“…We are currently investigating this issue extensively. Preliminary results for ME formalism can be found in the papers [86,87].…”