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The higher superconducting transition temperature T$_c$ and the functional derivative of T$_c$ with $α^2F(ω)$ 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 $α^2F(ω)$ [$δT_c/δα^2$F($ω$)] and $α^2F(ω)$ spectrum of H$_3$S ($Im\bar3m$), 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 $δT_c/δα^2$F($ω$) and $α^2F(ω)$ 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.