Temperature dependence of the functional derivative δTc/δ[α2F(ω)] and empirical estimation of Tc in the superconductors H3S and D3S

Abstract

The functional derivative of the superconducting critical temperature Tc with respect to the Eliashberg electron-phonon spectral function δTc/δα2F(ω) is commonly used as a diagnostic tool to identify the phonon-frequency regions that most effectively enhance Tc. In this work, the explicit effect of temperature on the calculation of δTc/δα2F(ω) is investigated within the isotropic Eliashberg formalism for the conventional high-Tc superconductors H3S and D3S over the pressure ranges in which their Tc have been measured. The results show that the functional derivative is strongly temperature dependent and, for every pressure in both systems, exhibits a robust local minimum in the ω−T landscape, characterized by (ωL,TL). This characteristic point corresponds to a thermal reference condition where Tc is least sensitive to redistributions of spectral weight in α2F(ω). The extracted ωL correlates approximately linearly with ωln, while the reduced ratio ωL/(kBTL) correlates with λ−1 and clusters around 7, placing (ωL,TL) close to the canonical optimal-frequency scale ωopt∼7–8kBTc. Motivated by these scalings, an empirical relation (mapping) is proposed that establishes a correlation between the experimental critical temperature Tc and characteristic quantities extracted from the temperature-dependent functional response (ωL,TL,λ). Over the explored pressure ranges, the resulting estimates reproduce the experimental Tc values with absolute errors of 0.1–4.9 K (mean absolute error of 2.56 K). While the mapping is empirical and has been validated only within the closely related H3S/D3S family and an isotropic treatment, the systematic behavior with pressure and isotopic substitution suggests that the identified local minimum is a robust feature of the functional derivative, motivating broader tests in other electron-phonon superconductors, including anisotropic and multiband systems.

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