The thermodynamic critical field and specific heat of superconducting state in phosphorene under strain

In this work we present the thermodynamic properties of the superconducting state in phosphorene. In particular, we have examined the electron doped ($n_{D}=1.3\times 10^{14} \rm{cm^{-2}}$) and biaxially strained (4 %) monolayer of black phosphorous, which exhibits best thermodynamic stability and highest superconducting critical temperature ($T_{c}$) among all monolayer phosphorene structures. Due to the confirmed electron-phonon pairing mechanism and relatively high electron-phonon coupling constant in the studied material, we carried out the calculations in the framework of the Eliashberg formalism for a wide range of the Coulomb pseudopotential $\mu^{\star}\in\langle 0.1, 0.3\rangle$. We have determined the thermodynamic critical field ($H_{c}$), and the specific heat difference ($\Delta C$) between superconducting ($C^{S}$) and normal state ($C^{N}$) as the functions of the temperature. In addition, we have calculated the dimensionless parameters $R_{C}=\Delta C(T_{c})/C^{N}(T_{c})$ and $R_{H}=T_{c}C^{N}(T_{c})/H^{2}_{c}(0)$, and also found their significant deviation from the expectations of the BCS theory. In particular, $R_{C} \simeq \langle 2.724, 1.899\rangle$ and $R_{H} \simeq \langle 0.133, 0.155\rangle$ for $\mu^{\star}\in \langle 0.1, 0.3\rangle$.