Advanced McMillan's equation and its application for the analysis of highly-compressed superconductors

A theory of electron-phonon mediated superconductivity requires the knowledge of full phonon spectrum to calculate superconducting transition temperature, T$_c$. However, there is no experimental technique which can measure phonon spectrum in highly-compressed near-room-temperature (NRT) superconductors to date. In this paper we propose to advance McMillan's approach (1968 Phys Rev 167 331) which utilizes the Debye temperature, T$_{\theta}$ (an integrated parameter of full phonon spectrum), that we deduced by the fit of experimentally measured temperature-dependent resistance data, R(T), to Bloch-Gruneisen equation for highly-compressed black phosphorous, boron, GeAs, SiH$_4$, H$_x$S, D$_y$S, LaH$_x$ and LaD$_y$. By utilizing relations between T$_c$, Debye temperature and electron-phonon coupling strength constant, $\lambda_{e-ph}$ it is possible to affirm/disprove the electron-phonon mechanism in given superconductors. We show that computed $\lambda_{e-ph}$ for highly-compressed black phosphorous, boron, GeAs, SiH$_4$ and for one sample of LaH$_{10}$ are in a good agreement with $\lambda_{e-ph}$ values deduced from experimental data. It is also found remarkable constancy of Debye temperature in H$_3$S at different ageing stages. We also show that if phonon spectra of two isotopic counterparts have similar shape then within electron-phonon phenomenology these materials should obey the relation of T$_{c,1}$/T$_{c,2}$=T$_{\theta,1}$/T$_{\theta,2}$ (where 1 and 2 designate isotopes). We report that these ratios for H$_3$S-D$_3$S predicted by electron-phonon phenomenology are largely different from ratios deduced from experiment. This alludes that NRT superconductivity in H3S-D3S system is originated from more than one mechanism, where the electron-phonon coupling lifts T$_c$ in H$_3$S vs D$_3$S, but primary origin for NRT background of in both H$_3$S and D$_3$S remains to be discovered.