First-principles study on the high-$T_ ext{c}$ superconductivity of Mg-Ti-H ternary hydrides up to the liquid-nitrogen temperature range under high pressures

Ternary hydrides have emerged as the primary focus of the new wave of research into superconducting hydrides. In this work, Mg-Ti-H ternary hydrides are explored under high pressures up to 300 GPa using the prediction method of the particle swarm optimization algorithm combined with first-principles calculations. Two new structures, $P4/nmm$-MgTiH$_6$ and $Pmm2$-Mg$3$TiH$6$, are identified to be thermodynamically stable at both 200 GPa and 300 GPa. Thermodynamically stable structures of Mg$3$TiH${12}$ are also identified, whose space groups are $R3/m$ at 200 GPa and $Pm\bar{3}m$ at 300 GPa, respectively. Among these Mg-Ti-H structures, $P4/nmm$-MgTiH$6$ achieves a record-high $T\text{c}$ of 81.9 K at 170 GPa, exceeding the boiling point of liquid nitrogen. Such a high $T\text{c}$ is primarily attributed to strong electron-phonon coupling (EPC) driven by low-frequency acoustic phonon modes, with the EPC strength reaching a large value of 1.54. The $T\text{c}$ of $Pm\bar{3}m$-Mg$3$TiH${12}$ is predicted to be 40 K at 300 GPa. Furthermore, element substitution of Zr(Hf) for Ti achieves considerable enhancement of superconducting properties in our predicted hydrogen-rich and high-symmetric crystal structures, i.e., $P4/nmm$-MgTiH$_6$ and $Pm\bar{3}m$-Mg$3$TiH${12}$. The high pressure required for dynamical stability is lowered to 100 GPa in both $Pm\bar{3}m$-Mg$3$ZrH${12}$ and $Pm\bar{3}m$-Mg$3$HfH${12}$, and to 90 GPa and 120 GPa for $P4/nmm$-MgZrH$_6$ and $P4/nmm$-MgHfH$_6$, respectively. Particularly, the electronic structure near the Fermi level is significantly modified in the $P4/nmm$-MgHfH$6$ phase, and pronounced softening of low-frequency acoustic phonon modes occurs. As a result, the EPC strength is enhanced to 1.72, leading to a higher $T\text{c}$ of 86 K.

💡 Research Summary

The authors performed an extensive high‑pressure crystal‑structure search for the Mg‑Ti‑H ternary system using the CALYPSO particle‑swarm algorithm combined with density‑functional theory (PBE‑GGA). Calculations were carried out at 200 GPa and 300 GPa, covering compositions MgₓTi_yH_z with x = 1‑3, y = 1‑3 and z = 3‑8. Two previously unknown phases, P4/nmm‑MgTiH₆ and Pmm2‑Mg₃TiH₆, lie on the convex hull at both pressures, indicating thermodynamic stability throughout the 200‑300 GPa range. In addition, Mg₃TiH₁₂ adopts the R3̅m structure at 200 GPa and transforms to the high‑symmetry Pm̅3m structure at 300 GPa. Two metastable hydrogen‑rich phases, I4₁/amd‑MgTiH₈ and P4/nmm‑MgTiH₁₀, are also identified with formation energies below 70 meV/atom.

Electronic‑structure analysis shows that all six compounds are metallic. Bader charge analysis and electron‑localization‑function (ELF) maps reveal an essentially ionic interaction between the metal atoms (Mg, Ti) and hydrogen, with charge transfer from the metals to H. The density of states at the Fermi level, N(E_F), is particularly high for P4/nmm‑MgTiH₆ and Pm̅3m‑Mg₃TiH₁₂, which is favorable for strong electron‑phonon coupling (EPC).

Phonon calculations using density‑functional perturbation theory (Quantum ESPRESSO) confirm dynamical stability of the identified phases. The Eliashberg spectral function α²F(ω) and the integrated EPC constant λ were evaluated. For P4/nmm‑MgTiH₆, λ reaches 1.54, driven mainly by low‑frequency acoustic phonon modes (≈200–300 cm⁻¹). The logarithmic average phonon frequency ω_log is modest (~500 K), and the Allen‑Dynes‑modified McMillan formula predicts a superconducting critical temperature T_c = 81.9 K at 170 GPa (μ* = 0.10). This value exceeds the boiling point of liquid nitrogen (77 K) and represents the highest T_c reported for Mg‑Ti‑H compounds. In contrast, Pm̅3m‑Mg₃TiH₁₂ exhibits a larger ω_log (~900 K) but a smaller λ (≈0.78), resulting in a predicted T_c of 40 K at 300 GPa.

To explore chemical‑tuning routes, the authors substituted Ti with isovalent Zr and Hf while retaining the same crystal frameworks. The Zr/Hf‑substituted phases become dynamically stable at considerably lower pressures: P4/nmm‑MgZrH₆ at 90 GPa, P4/nmm‑MgHfH₆ at 120 GPa, and Pm̅3m‑Mg₃ZrH₁₂ / Mg₃HfH₁₂ at 100 GPa. Notably, P4/nmm‑MgHfH₆ shows a pronounced modification of the electronic band structure near the Fermi level, an enhanced N(E_F), and a further softening of the acoustic phonon branch. Consequently, λ increases to 1.72 and the calculated T_c rises to 86 K (μ* = 0.10), again surpassing liquid‑nitrogen temperature. The Zr‑substituted analogues display modest improvements, with T_c values around 45–48 K.

The study highlights three key insights: (i) high‑symmetry, hydrogen‑rich frameworks can host strong EPC, especially when low‑frequency acoustic modes are present; (ii) the choice of transition‑metal element strongly influences both the electronic density of states and the phonon spectrum, offering a lever to tune λ and ω_log; and (iii) element substitution can dramatically lower the pressure required for dynamical stability, bringing experimental synthesis within reach of current diamond‑anvil‑cell capabilities.

Overall, the work demonstrates that Mg‑Ti‑H ternary hydrides can achieve superconductivity above liquid‑nitrogen temperature under high pressure, and that strategic Zr/Hf substitution further enhances the superconducting performance while easing the experimental pressure constraints. This provides a clear pathway for future experimental verification and for the design of new high‑T_c hydride superconductors based on ternary systems.