High-temperature superconductivity in the Ca-Sc-H system

The discovery of high critical temperature ${T}_{c}$ in compressed ${\mathrm{H}}_{3}\mathrm{S}$ has been followed by the prediction and experimental confirmation of even higher superconducting temperatures with ${T}_{c}$ approaching room temperature in ${\mathrm{LaH}}_{10}$ and ${\mathrm{CaH}}_{6}$. These works established the mechanism of the electron-phonon interaction and the dominant role of hydrogen in these materials. In the present work we focus on ${\mathrm{CaH}}_{6}$ and we follow the classic McMillan paper, which writes the electron-phonon coupling parameter $\ensuremath{\lambda}$ as a ratio of an electronic contribution $\ensuremath{\eta}$ over a force constant $k=M\ensuremath{\langle}{\ensuremath{\omega}}^{2}\ensuremath{\rangle}$ which contains the phonon contribution. First the numerator of McMillan's expression, the Hopfield-McMillan parameter $\ensuremath{\eta}$, is computed using the theory of Gaspari and Gyorffy (GG), and the force constants in the denominator are obtained from the paper of Quan et al. The resulting $\ensuremath{\lambda}$ is used in the Allen-Dynes equation to calculate ${T}_{c}$. We present an analysis of the different terms of the GG equation and conclude that the $sp$ channel of hydrogen has the most important contribution to obtain high values of ${T}_{c}$, as in the other hydrogenated materials. In addition, we further separate the three terms of the GG expression and assess the role of the phase shifts term versus the partial densities of states and free scatterers. We compare these quantities in ${\mathrm{CaH}}_{6}$ to those in ${\mathrm{SH}}_{3}$ and ${\mathrm{LaH}}_{10}$ and, in contrast to what is expected for high values of $\ensuremath{\lambda}$, we conclude that in ${\mathrm{CaH}}_{6}$ the coupling parameter $\ensuremath{\lambda}$ is the better indicator of high ${T}_{c}$ than $\ensuremath{\eta}$. However, we have found that in the strong coupling limit of large $\ensuremath{\lambda}$ the high values of ${T}_{c}$ are strongly dependent on the parameter $\ensuremath{\eta}$ and make ${T}_{c}$ a decreasing function of $\ensuremath{\eta}$. Unfortunately most of the researchers of these materials have ignored the importance of the parameter $\ensuremath{\eta}$ partly because the computational packages they are using do not separately compute $\ensuremath{\eta}$. Our view is that the parameters $\ensuremath{\eta}, \ensuremath{\kappa}$, and $\ensuremath{\lambda}$ must be examined on an equal footing as we study how to achieve high ${T}_{c}$ at low pressures. Finally, we present the following findings: (a) using the virtual crystal approximation and an extension of the Allen-Dynes finding that at the strong coupling limit ${T}_{c}$ depends on $\ensuremath{\eta}$, we predict that the alloy ${\mathrm{Ca}}_{1\ensuremath{-}x}{\mathrm{Sc}}_{x}{\mathrm{H}}_{6}$ can reach higher ${T}_{c}$ than ${\mathrm{CaH}}_{6}$. (b) If the higher hydrogen content material ${\mathrm{CaH}}_{10}$ can be made in the $Fm\overline{3}m$ structure it would have significantly higher ${T}_{c}$ than ${\mathrm{CaH}}_{6}$.