High-temperature superconductivity in electrides dominated by hybridized $\mathit{p}$-orbital-like electride states

High-pressure electrides have opened a promising path to high-temperature superconductors and attracted considerable attention. However, the origins of superconductivity from discrepant sources remain puzzling. In this study, we propose a different type of $p$-orbital-like electride state to shed light on the causality that induces high $T_c$. Taking our predicted $R-3\mathrm{m}$ phase in ${\mathrm{Li}}_6\mathrm{P}$ as a representative, with a high $T_c$ of 41.36 K, our first-principles studies unveil that the $p$-orbital-like electride states play a dominant role in $T_c$ by softening the acoustic phonon and forming itinerantly hybridized $p$-orbital-like (IHP) electride-states-dependent phonon-coupled bands, which are corroborated by hole doping. Compared to the nonitinerant $s$-orbital-like electride states with low $T_c$, the IHP electride states exhibit greater freedom of orbital multiplicity and hence a higher propensity to form Cooper pairs, promoting electron-phonon coupling (EPC), and demonstrating the derivation of differential $T_c\mathrm{s}$. Of particular note, the IHP electride states originate from the atypical nature of concurrent oxidization states, characterized by electrons donated from electronegative phosphorus and electropositive lithium. Our finding provides crucial insights into the role of electride states in EPC, elucidates the origin of superconductivity, and identifies the characteristics of high-$T_c$ electrides, with profound implications for exploring this class of multifunctional superconductors.

Figure 1

(a) The $R-3\mathrm{m}$ phase at 200 GPa in the hexagonal system and corresponding (b) 3D isosurface plot of electron localization function (ELF) with the partial charge density $0.80e/{\AA }^3$, and (c) in the trigonal system and corresponding (d) 2D map of $\left(110\right)$ plane.

Figure 2

Calculated superconducting parameters as functions of pressure, including $T_c$, λ, ${\omega }_{log}$, and $N\left({\varepsilon }_F\right)$ at ${\varepsilon }_F$.

Figure 3

(a) Phonon dispersion, projected phonon DOS, and ${\alpha }^{2}F\left(\omega \right)$ with electron-phonon integral $\lambda \left(\omega \right)$ at 200 GPa. The solid blue dot on the phonon spectra represents the contribution of the phonon linewidth in proportion to the spherical scale. (b) Nesting function ${\xi }_{\mathbf{q}}$ along typical q paths and the illustration of FS. (c) Integral DOS values of different atoms up to ${\varepsilon }_F$ on a per f.u. basis. (d) Band structures and projected DOS (PDOS) of $R-3m$ phase at 200 GPa.

Figure 4

The hole-doping modulated electronic properties and superconductivity at 200 GPa. (a) Integral DOS values of different elements up to ${\varepsilon }_F$ (set to zero) without hole doping (the solid line) and at doping level ($n_h$) of $1e$ (dashed line). (b) The calculated $T_c, \lambda \left(\omega \right), {\omega }_{log}$, and $N\left({\varepsilon }_F\right)$ with different hole doping concentrations. (c) Phonon dispersion, projected phonon DOS, and ${\alpha }^{2}F\left(\omega \right)$ along with integral $\lambda \left(\omega \right)$ at $n_h=1e$. (d) The nesting function ${\xi }_{\mathbf{q}}$ with different doping concentrations. (e) PDOS at $n_h=1e$. (f) Contribution percentages of electride-states-$p$ and Li-$p$ to $N\left({\varepsilon }_F\right)$ and the corresponding 3D FS at different doping levels.