Compressed hydrides as metallic hydrogen superconductors

The 2014–2015 prediction, discovery, and confirmation of record high-temperature superconductivity above 200 K in compressed ${\mathrm{H}}_3\mathrm{S}$, followed by the 2018 extension to superconductivity in the 250–260 K range in lanthanum hydride, mark a new era in the long-standing quest for room-temperature superconductivity: quest achieved, at the cost of supplying 1.5–2 Mbar pressure. Predictions of numerous high-temperature superconducting metal hydrides $X{\text{H}}_n$ ($X$=metal atom) have appeared but are providing limited understanding of why some transition temperatures $T_c$ are high while others are low. We make use of the small mass ratio $M_H/M_X$ to obtain an atomic decomposition of the coupling strength to reveal that although the $X$ atom provides coupling strength via ${\lambda }_X$ as commonly calculated, it is irrelevant for $T_c$ because the resulting lowering of frequency moments compensates (and sometimes overcompensates) for the increase in $\lambda$. It is important for analysis and for understanding that the $X$ atom contribution is neglected because $T_c$ depends more transparently on ${\lambda }_H.$ Five $X{\mathrm{H}}_n$ compounds, predicted in harmonic approximation to have $T_c$ in the 150–300 K range, are analyzed consistently for their relevant properties, revealing some aspects that confront conventional wisdom. A phonon frequency–critical temperature (${\omega }_2-T_c$) phase diagram is obtained that reveals a common structural phase instability boundary limiting $T_c$ at the low-pressure range of each compound. The hydrogen scattering matrix elements are obtained and found to differ strongly over the hydrides. A quantity directly proportional to $T_c$ in these hydrides is identified, indicating that (in common notation) $N_H\left(0\right)I_H^2/{\omega }_H={\eta }_H/{\omega }_H$ is the parameter combination to be maximized in hydrides.

Figure 1

Top: crystal structures of $n=3$ bcc ${\mathrm{SH}}_3, n=6$ bcc ${\mathrm{CaH}}_6$, and $n=10$ fcc ${\mathrm{LaH}}_{10}$. Bottom: corresponding band structures (in eV) and electronic densities of states (in states per eV unit cell). In each case several bands cross the Fermi level.

Figure 2

Interrelationships between the various materials characteristics for the H atoms in the hydrides we discuss. (a) Schematic ${\omega }_2-T_c$ phase diagram, with blue indicating the island of lattice instability. The blue arrow denotes the direction of increasing pressure $P$. (b) and (c) Plots of $\kappa$ and $\eta$, respectively, versus $\lambda =\frac{\eta }{\kappa }$. The increase of $\lambda$ correlates strongly with the decrease in $\kappa$ (frequencies). (d)–(g) Plots of $\lambda$. $\kappa =M{\omega }_2^2$ ($\mathrm{eV}/{\AA }^2$), $\eta =N\left(0\right)I^2$ ($\mathrm{eV}/{\AA }^2$), and $I^2$ (${\mathrm{eV}}^2{/\AA )}^2$, respectively, versus pressure. All panels show each of the five hydrides toward the lower end of their region of stability.

Figure 3

Views of the evolution under pressure of phonon coupling strength and frequencies for ${\mathrm{SH}}_3$. From the top: $F\left(\omega \right)$ for the three pressures indicated, the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$, the ratio ${\alpha }^{2}F\left(\omega \right)/\omega$ that determines $\lambda$ and frequency moments, and the coupling spectrum ${\alpha }^2\left(\omega \right)$. Note that in these zone-averaged functions there is no indication of the lattice instability that occurs just below 200 GPa.

Figure 4

Regions of unstable phonons. The indicated regions of the Brillouin zone indicate where phonons first become unstable, in harmonic approximation. For ${\mathrm{SH}}_3$ the instability regions are repeated outside the first zone for more clarity.

Figure 5

Pressure dependence of various superconducting quantities of cubic ${\mathrm{CaH}}_6$. $A$ denotes the area under ${\alpha }^{2}F$ (see the text). All quantities refer to the hydrogen contribution alone. As emphasized in the text, $\lambda$ and $T_c$ increase (rather strongly) at the low pressure end, before the instability. The bottom panel shows that the decrease of ${\omega }_2^2$ is responsible, even though from frequency moments no impending instability can be inferred. For these data, norm-conserving pseudopotentials were used.

Figure 6

Plot of area under ${\alpha }^2{\mathrm{F}}_H$ (the H contribution) versus $T_c$ for binary hydrides using H-derived quantities. The slope of 0.148 denotes the Leavens-Carbotte line for strongly coupled superconductors existing in 1974.