Characteristics of the Eliashberg formalism on the example of high-pressure superconducting state in phosphor

TL;DR

This study uses the Eliashberg formalism to analyze the high-pressure superconducting state in phosphor, revealing that low critical temperatures are due to weak electron-phonon coupling and strong Coulomb interactions, with thermodynamic ratios close to BCS predictions.

Contribution

It provides a detailed analysis of high-pressure superconductivity in phosphor using the Eliashberg formalism, emphasizing the roles of electron-phonon coupling and Coulomb interactions.

Findings

Critical temperature is limited by Coulomb interactions.

Weak coupling and retardation effects keep ratios close to BCS values.

Superconducting properties vary with pressure in phosphor.

Abstract

The work describes the properties of the high-pressure superconducting state in phosphor: $p\in\{20, 30, 40, 70\}$ GPa. The calculations were performed in the framework of the Eliashberg formalism, which is the natural generalization of the BCS theory. The exceptional attention was paid to the accurate presentation of the used analysis scheme. With respect to the superconducting state in phosphor it was shown that: (i) the observed not-high values of the critical temperature ($\left[T_{C}\right]_{p=30{\rm GPa}}^{\rm max}=8.45$ K) result not only from the low values of the electron - phonon coupling constant, but also from the very strong depairing Coulomb interactions, (ii) the inconsiderable strong - coupling and retardation effects force the dimensionless ratios $R_{\Delta}$, $R_{C}$, and $R_{H}$ - related to the critical temperature, the order parameter, the specific heat and the thermodynamic critical field - to take the values close to the BCS predictions.