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

Abstract: 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 $\mu^{\star}\in \langle 0.1, 0.3\rangle$.