Description of the thermodynamic properties of $\rm{BiH_{5}}$ and $\rm{BiH_{6}}$ superconductors beyond the mean-field approximation

Abstract: The ab initio calculations suggest (Y. Ma et al.), that the high-pressure ($p=200$~GPa) superconducting state in $\rm{BiH_{5}}$ and $\rm{BiH_{6}}$ compounds characterizes with a high value of the critical temperature ($T_{C}\sim 100$~K). Due to the large value of the electron-phonon coupling constant ($\lambda\sim 1.2$), the thermodynamic parameters of the superconducting phase in $\rm{BiH_{5}}$ and $\rm{BiH_{6}}$ have been determined beyond the mean-field approximation - in the framework of the Eliashberg equations formalism. We have calculated the dependence of the order parameter ($\Delta$) and the wave function renormalization factor on the temperature. Then we have estimated the free energy difference between the superconducting state and the normal state, the thermodynamic critical field ($H_{C}$) and the specific heat of the superconducting state ($C^{S}$) and the normal state ($C^{N}$). The values of the dimensionless ratios $R_{\Delta}=2\Delta\left(0\right)/k_{B}T_{C}$, $R_{C}=\Delta C\left(T_{C}\right)/C^{N}\left(T_{C}\right)$ and $R_{H}=T_{C}C^{N}\left(T_{C}\right)/H^{2}_{C}\left(0\right)$ are equal $R_{\Delta_{{\rm BiH_{5}}}}=4.17$ and $R_{\Delta_{{\rm BiH_{6}}}}=4.20$, $R_{C_{{\rm BiH_{5}}}}=2.54$ and $R_{C_{{\rm BiH_{6}}}}=2.58$, $R_{H_{{\rm BiH_{5}}}}=0.146$ and $R_{H_{{\rm BiH_{6}}}}=0.146$ respectively.