High-temperature superconductivity in the Ti-H system at high pressures

Search for stable high-pressure compounds in the Ti-H system reveals the existence of titanium hydrides with new stoichiometries, including $Ibam\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_5$, $I4/m\text{−}{\mathrm{Ti}}_5{\mathrm{H}}_{13}$, $I\overline{4}\text{−}{\mathrm{Ti}}_5{\mathrm{H}}_{14}$, $Fddd\text{−}{\mathrm{TiH}}_4$, $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$, $P\overline{1}\text{−}{\mathrm{TiH}}_{12}$, and $C2/m\text{−}{\mathrm{TiH}}_{22}$. Our calculations predict $I4/mmm\to R\overline{3}m$ and $I4/mmm\to$ $Cmma$ transitions in TiH and ${\mathrm{TiH}}_2$, respectively. Phonons and the electron-phonon coupling in all searched titanium hydrides are analyzed at high pressure. It is found that $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$, rather than the hydrogen-richest $C2/m\text{−}{\mathrm{TiH}}_{22}$, exhibits the highest superconducting critical temperature $T_c$. The estimated $T_c$ of $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$ and $C2/m\text{−}{\mathrm{TiH}}_{22}$ are, respectively, 127.4–149.4 K (${\mu }^{*}=0.1$–0.15) at 350 GPa and 91.3–110.2 K at 250 GPa, as found by numerically solving the Eliashberg equations.

Figure 1

Convex hull diagrams for the Ti-H system at (a) 0, (b) 50, (c) 100, (d) 150, (e) 200, (f) 250, (g) 300, and (h) 350 GPa. The blue dashed convex hull at 0 GPa shows the experimental result [55]. Black lines with solid squares and red lines with open squares, respectively, represent the calculated results without and with ZPE. The structures indicated in gray are those that lose stability after considering ZPE.

Figure 2

Pressure-composition phase diagram of the Ti-H system.

Figure 3

Crystal structures of titanium hydrides. Large spheres represent titanium atoms and small ones represent hydrogen atoms.

Figure 4

Enthalpy as a function of pressure (without ZPE) for phases of (a) TiH, referenced to the $I4/mmm$-TiH phase, and (b) ${\mathrm{TiH}}_2$, referenced to the $I4/mmm\text{−}{\mathrm{TiH}}_2$ one.

Figure 5

(a) Polyhedral representation of the $Fddd\text{−}{\mathrm{TiH}}_4$ structure. Ti-centered hexagonal prisms are shown in purple. (b) The fundamental “sandwich” of ${\mathrm{TiH}}_4$. (c) Distorted H-graphene layer in $Fddd\text{−}{\mathrm{TiH}}_4$, and (d) electronic band structure and density of states (DOS) of $Fddd\text{−}{\mathrm{TiH}}_4$ at 300 GPa; DOS is in the unit of states/eV/formula and Fermi energy (red dashed line) is set to zero.

Figure 6

(a) Polyhedral representation of the $C2/m\text{−}{\mathrm{TiH}}_{22}$ structure. (b) Expanded view of a ${\mathrm{H}}_{20}$ cage encapsulating a Ti atom. (c) ELF isosurface ($\mathrm{ELF}=0.7$) for $C2/m\text{−}{\mathrm{TiH}}_{22}$ at 350 GPa. Green and pink atoms represent molecular hydrogen and hydrogen chains, respectively. (d) Electronic band structure and DOS for $C2/m\text{−}{\mathrm{TiH}}_{22}$ at 350 GPa; DOS is in the unit of states/eV/formula and Fermi energy (red dashed line) is set to zero.

Figure 7

Phonon dispersion curves, atom-projected phonon density of states, Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$, and the electron-phonon coupling (EPC) parameter $\lambda$ for (a) $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$ at 350 GPa, (b) $C2/m\text{−}{\mathrm{TiH}}_{22}$ at 250 GPa, (c) $C2/m\text{−}{\mathrm{TiH}}_{14}$ at 200 GPa, (d) $P\overline{1}\text{−}{\mathrm{TiH}}_{12}$ at 350 GPa, (e) $R\overline{3}m$-TiH at 200 GPa, and (f) $Fddd\text{−}{\mathrm{TiH}}_4$ at 350 GPa. The magnitude of the phonon linewidths is indicated by the size of the blue haze, which is proportional to the respective coupling strength.

Figure 8

Dependence of the superconducting order parameter on the number $m$ for select values of temperature and Coulomb pseudopotential for (a) $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$ at 350 GPa and (d) $C2/m\text{−}{\mathrm{TiH}}_{22}$ at 250 GPa. The wave-function renormalization factor $Z_m$ on the imaginary axis for select values of temperature and Coulomb pseudopotential for (b) $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$ and (e) $C2/m\text{−}{\mathrm{TiH}}_{22}$. The influence of temperature on the maximum value of the order parameter (${\mathrm{\Delta }}_{m=1}$) for selected ${\mu }^{*}$ of (c) $Immm\text{−}{\mathrm{Ti}}_2{\mathrm{H}}_{13}$ and (f) $C2/m\text{−}{\mathrm{TiH}}_{22}$. Solid circles correspond to the exact numerical solutions of the Eliashberg equations, the red and blue lines represent the results obtained using analytical formula [Eq. (10)]. Black lines are predicted by BCS model [Eq. (10), where $\mathrm{\Gamma }=3]$.