The higher superconducting transition temperature T_c and the functional derivative of T_c with α²F(ω) for electron-phonon superconductors

This work presents an analysis of the functional derivative of the superconducting transition temperature T_c with respect to the electron-phonon coupling function α²F(ω) [δT_c/δα²F(ω)] and α²F(ω) spectrum of H₃S (Im ^¯3m), in the pressure range where the high-T_c was measured (155-225 GPa). The calculations are done in the framework of the Migdal-Eliashberg theory. We find for this electron-phonon superconductor, a correlation between the maximums of δT_c/δα²F(ω) and α²F(ω) with its higher T_c. We corroborate this behavior in other electron-phonon superconductors by analyzing data available in the literature, which suggests its validity in this type of superconductors. The correlation observed could be considered as a theoretical tool that in an electron-phonon superconductor, allows describing qualitatively the proximity to its highest T_c, and determining the optimal physical conditions (pressure, alloying or doping concentration) that lead to the superconductor reaching its highest T_c possible.