Superconductivity in pure $\omega \text{−}\mathrm{Hf}$ under high pressures $(40-60\mathrm{G}\mathrm{Pa})$ was investigated using density functional theory (DFT) combined with the Migdal-Eliashberg theory. The results indicate that interstitial anionic electrons (IAE) in pure $\omega \text{−}\mathrm{Hf}$ exhibit both kagome and hexagonal electride structures, with interstitial electrons distributed across two distinct crystal planes. The results also show that spin-orbit coupling significantly modifies the conduction electronic band-structure, inducing a Van-Hove singularity at the Fermi level, which enhances the superconducting critical temperature $\left(T_c\right)$ from approximately $3.0\mathrm{K}$ to $5.3\mathrm{K}$ at $60\mathrm{G}\mathrm{Pa}$. In addition, interstitial electrons in the valence band at the Fermi level also contribute to superconductivity. These $T_c$ values agree well with previously reported experimental results. Furthermore, we demonstrate that crystal planes hosting anionic interstitial electrons exhibit anisotropy and low work function of $3.01\mathrm{e}\mathrm{V}$ along the [001] direction, consistent with the characteristics of electride materials.

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• BCS theory
• Band structure methods
• Density functional approximations
• Density functional theory
• Eliashberg theory
• First-principles calculations
• Methods in superconductivity
• Quantum chemistry methods
• Superconductivity
• Superconductors
• Wannier function methods