Quantitative analysis of nonadiabatic effects in dense H$_3$S and PH$_3$ superconductors

The comparison study of high pressure superconducting state of recently synthesized H$_3$S and PH$_3$ compounds are conducted within the framework of the strong-coupling theory. By generalization of the standard Eliashberg equations to include the lowest-order vertex correction, we have investigated the influence of the nonadiabatic effects on the Coulomb pseudopotential, electron effective mass, energy gap function and on the $2\Delta(0)/T_C$ ratio. We found that, for a fixed value of critical temperature ($178$ K for H$_3$S and $81$ K for PH$_3$), the nonadiabatic corrections reduce the Coulomb pseudopotential for H$_3$S from $0.204$ to $0.185$ and for PH$_3$ from $0.088$ to $0.083$, however, the electron effective mass and ratio $2\Delta(0)/T_C$ remain unaffected. Independently of the assumed method of analysis, the thermodynamic parameters of superconducting H$_3$S and PH$_3$ strongly deviate from the prediction of BCS theory due to the strong-coupling and retardation effects.