Superconductivity in pure hafnium kagome electride under high pressure

Superconductivity in pure $\ensuremath{\omega}\text{\ensuremath{-}}\mathrm{Hf}$ under high pressures $(40\ensuremath{-}60\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Pa})$ was investigated using density functional theory (DFT) combined with the Migdal-Eliashberg theory. The results indicate that interstitial anionic electrons (IAE) in pure $\ensuremath{\omega}\text{\ensuremath{-}}\mathrm{Hf}$ 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 $({T}_{c})$ from approximately $3.0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ to $5.3\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ at $60\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Pa}$. 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 $3.01\phantom{\rule{0.16em}{0ex}}\mathrm{e}\mathrm{V}$ along the [001] direction, consistent with the characteristics of electride materials.