Optimal alloying in hydrides: Reaching room-temperature superconductivity in ${\mathrm{LaH}}_{10}$

Doping represents one of the most promising avenues for optimizing superconductors, such as high-pressure conventional superconductors with record-breaking critical temperatures. In this work, we perform an extensive search for substitutional dopants in ${\mathrm{LaH}}_{10}$, looking for elements that enhance its electronic structure. In total, 70 elements were investigated as possible substitutions of La-sites at doping ratio of 12.5% under high pressure. To accelerate the screening of the ternary phases, our protocol to scan the chemical space is 1) to be constrained to highly symmetric patterns of hydrogen atoms, 2) focusing on phases with compact basis of lanthanum atoms (minimize enthalpy) and 3) to choose candidate dopants that preserve the van Hove singularity around the Fermi level. We found Ca as the best candidate dopants, which shift the van Hove singularity and increase the electronic DOS at the Fermi level. By using harmonic-level phonon calculations and performing first-principles calculation of $T_{\mathrm{c}}$, Ca-doped ${\mathrm{LaH}}_{10}$ shows $T_{\mathrm{c}}$ which is 15% higher than the one of ${\mathrm{LaH}}_{10}$. It provides a promising route to reach room-temperature superconductivity in pressurized hydrides by doping.

Figure 1

Steps implemented in our computational study to search for suitable doping substitutions in ${\mathrm{LaH}}_{10}$ under pressure.

Figure 2

Geometry relaxation results of ${\mathrm{LaH}}_{10}$ doped by different elements with doping ratio of 12.5% at 150 GPa pressure, and color code gives the behavior for different elements. (a) The periodic table with elements classfied to four types of relaxation results. Dashed-squares are elements for which DOS is plotted in Fig. 4. Elements coloured in dark grey were not considered in this study. (b) Typical elements with corresponding structural patterns after relaxations for four cases. At the doping ratio of 12.5%, most of the elements destabilized the symmetric pattern, and only 3 elements preserve the high symmetry.

Figure 3

Optimized volume for 12.5% doping ratio with different atoms at 150 GPa. Volume is optimized without breaking symmetry, thus the highly symmetric H-cages are preserved. Volume has been scaled to a single formula unit for comparison. Representative elements are marked as a guide, for which the DOS of these cases are plotted in Fig. 4. Red line represents the volume of the undoped case.

Figure 4

Electronic density of states near the Fermi level for different doping atoms in ${\mathrm{LaH}}_{10}$ (fully optimized geometry at 150 GPa). Solid lines represent interesting dopants and lighter-dashed colors represent less relevant cases. For comparison, the DOS of ${\mathrm{LaH}}_{10}$ (pure phase) is shown in black-line (a grey area) calculated in the supercell.

Figure 5

Comparison of DOS for ${\mathrm{LaH}}_{10}$ and the case with a single vacancy in the La-site, for three different concentrations (one La atom missing for every $2\times 2\times 2$ supercell (88 atoms); one La atom missing for every $3\times 3\times 3$ supercell (297 atoms); one La atom missing for every $4\times 4\times 4$ supercell (704 atoms)).

Figure 6

The phonon DOS of ${\mathrm{LaH}}_{10}$ (blue) and Ca-doped ${\mathrm{LaH}}_{10}$ (red) with doping ratio of 12.5% at 300 GPa.

Figure 7

The Eliashberg spectral functinon ${\alpha }^{2}F\left(\omega \right)$ at 300 GPa for (a) ${\mathrm{LaH}}_{10}$ and (b) Ca-doped ${\mathrm{LaH}}_{10}$ with doping ratio of 12.5%. (c) The Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ of Ca-doped ${\mathrm{LaH}}_{10}$ used in our practical calculation, in which the low frequency part $\omega <$ 15 meV (blue) is from the one of (a) ${\mathrm{LaH}}_{10}$, and the part of $\omega <$ 15 meV (red) is the same as the one of (b) Ca-doped ${\mathrm{LaH}}_{10}$.

Figure 8

Eigenvalue ${\tilde{\lambda }}_{\max }$ vs $T$ in the Migdal-Eliashberg calculation. Red and blue circles represent the results of Ca-doped ${\mathrm{LaH}}_{10}$ and ${\mathrm{LaH}}_{10}$ respectively at 300 GPa. Solid lines are fitting results using fitting formula ${\tilde{\lambda }}_{\max }=A\log T+B$ via parameter $A$ and $B$. Extrapolation of the fitting results shows $T_{\mathrm{c}}=203.2$ K for ${\mathrm{LaH}}_{10}$ and $T_{\mathrm{c}}^{\mathrm{doped}}=233.9$ K for the doped case ${\mathrm{Ca}}_x{\mathrm{La}}_{1-x}{\mathrm{H}}_{10}$.

Figure 9

Geometry relaxation results of ${\mathrm{LaH}}_{10}$ doped by different elements with doping ratio of 3.70% at 150 GPa pressure, and color code gives the behavior for different elements. (a) The periodic table with elements classfied to three types of relaxation results. Dashed-squares are elements for which DOS is plotted in Fig. 12. Elements coloured in dark grey were not considered in this study. (b) Typical elements with corresponding structural patterns after relaxations for three cases. At the doping ratio of 3.70%, only 2 elements preserve the high symmetry.

Figure 10

Geometry relaxation results of ${\mathrm{LaH}}_{10}$ doped by different elements with doping ratio of 1.56% at 150 GPa pressure. (a) The periodic table with elements which show results with fixed symmetry (highly symmetric H-cages). Dashed squares are elements for which DOS is plotted in Fig. 12. Elements colored in dark grey were not considered in this study. Especially, Elements colored in white were not considered in this case of 1.56% doping ratio case. (b) Typical elements with corresponding structural patterns after relaxations. At low doping ratio of 1.56%, many elements preserve the H-cages.

Figure 11

Optimized volume for (a) 3.70% and (b) 1.56% doping ratio with different atoms at 150 GPa. Volume is optimized without breaking symmetry, thus the highly symmetric H-cages are preserved. Volume has been scaled to a single formula unit for comparison. Representative elements are marked as a guide, for which the DOS of these cases are plotted in Fig. 12. Red line represents the volume of the undoped case.

Figure 12

Electronic density of states near the Fermi level for different doping atoms in LaH10 (fully optimized geometry at 150GPa) with (a) 3.70% and (b) 1.56% doping ratio. For comparison, the DOS of LaH10 (pure phase) is shown in black-line (a grey area) calculated in the supercell.

Figure 13

DOS calculated for different dopants and different ratios of (a) 12.5%, (b) 3.70% and (c) 1.56% considering the quantum effects. Here we have assumed the symmetry of the fcc phase is preserved. The best elements for tuning the electronic structure in ${\mathrm{LaH}}_{10}$ at 150 GPa are Al, Si, Ge, Ir, Mg, Ca, Sr, and Ba at 12.5% doping ratio. For 3.7% doping ratio the best elements are Mn, N, Br, Ru, and H. In the case of even lower doping ratio, the incorporation of dopants, though in favor of thermodynamical stability, results in downturns of the stability of the van Hove singularity. In solid-line (in a grey area), the DOS of ${\mathrm{LaH}}_{10}$ for a single formula unit is shown.