Under compression, the increase in orbital energy at interstitial cavities is less pronounced than that of atomic orbitals,1 causing electrons to detach from atoms and localize in the cavities to form off-center lobes of electrons, also known as non-nuclear attractors (NNAs).2,3 These NNAs are the hallmark feature of high-pressure electrides and can be identified in electron-rich, -neutral, and -deficient systems.4–6 The alkali metal lithium (Li), with its strong electron-donating capacity, serves as an ideal candidate as a cation.7–10 To date, a multitude of high-pressure electrides involving the alkali metal Li has been identified.11–15 The magnitude and topology of NNAs depend on the stoichiometric ratios and chemical properties of the electron donors and acceptors, as well as external conditions, particularly pressure. These variabilities significantly expand the potential chemical space and the fascinating variety of physical properties of electrides.16,17 Even more intriguingly, unexpected superconductivity has been induced in pressure-stabilized Li-rich electrides such as CaLi₂, Li₃Ar, Li₅He₂, Li₅C, Li₅N, Li₆Al, Li₈Cs, Li₉Te, Li₁₀Se, and Li₁₁Sb₂.18–25 Not only that, but the critical temperatures T_c of these electrides have continuously rewritten the superconducting records, ranging from 10 K for Li₆C to 39.3 K for Li₆P,26,27 closing in on the McMillan limit (40 K), and reaching 73.1 K for the designed Li₈Au,28 and progressively approaching the temperature of liquid nitrogen (77.4 K).

The discovery of these pressure-induced superconducting electrides has sparked an intense wave of interest in novel Li-based superconductors. Unfortunately, however, the detailed influence of NNAs on superconductivity remains to be elucidated, and a consensus concerning their role in superconductivity has yet to be reached. Carbon (C) and phosphorus (P), as main group nonmetallic elements, follow the diagonal rule and exhibit similar chemical properties.29,30 When combined with Li, they have demonstrated the ability to form superconducting Li-based electrides. Although the NNAs in both Li₆C and Li₆P act as bridges linking anions to form electron-conducting channels, their contributions to superconductivity are strikingly distinct, resulting in a considerable disparity in T_c. Specifically, the dumbbell-like interconnected NNAs in Li₆P play a dominant role in the superconducting transition, with the Fermi surface nesting associated with bands of these NNAs inducing phonon softening and thereby enhancing electron–phonon coupling (EPC) and favoring a high T_c close to the McMillan limit.27 By contrast, the superconductivity of Li₆C is attributed to synergy of the interatomic coupling effect and phonon-coupled bands, which are predominantly governed by host sp-hybridized electrons rather than the cage-like NNAs.26 In addition to these two typical instances, other theoretical studies have also suggested that NNAs make a remarkable contribution to the superconductivity of electrides such as Li₆Al,31 Li₁₀Te,23 Li₈Au,28 and Li₈H₄.32 However, an alternative point of view holds that the contribution of NNAs to superconductivity is negligible, as observed in electrides like Li₅C20 and Li₅Si.33

Therefore, it is imperative to elucidate the underlying mechanism by which NNAs contribute to superconductivity, which is vital for understanding their properties as well as providing potentially valuable insights to aid in the further development of superconducting electrides. Given the current opposing views regarding the relationship between NNAs and superconductivity in Li₆C and Li₆P, further exploration of ternary compounds that combine them is warranted. Ternary compounds, compared with binary ones, can integrate the merits of different elements and provide a more abundant range of chemical compositions and stronger synergistic charge transfer.34,35 To date, a variety of ternary electrides have been successfully predicted.6,36 Consequently, the reaction between Li, C, and P under high pressure is expected to produce diverse NNA morphology, allowing for a more thorough study of the contribution of NNAs to superconductivity.

Here, we choose ternary Li–C–P as an ideal system, focusing on the influence of NNAs on superconductivity at high pressure and attempting to reconcile the conflicting views on the contribution of NNAs to superconductivity in Li₆P and Li₆C. A novel high-pressure electride Li₁₄CP, with an R3 space group, is predicted using a crystal structure search technique combined with first-principles calculations. Excitingly, R3-Li₁₄CP is found to be a potential superconductor, with a T_c of 10.6 K at 300 GPa, positioned between the T_c values of Li₆C and Li₆P. The cage-like NNAs bond covalently with Li, leading to the formation of satellite interstitial electrons (SIEs) near the Fermi level E_F. Our first-principles studies reveal that the SIEs play a leading role in determining T_c by softening the low-frequency phonon vibrations. Most importantly, the mechanism by which SIEs contribute to superconductivity revealed here is general and can be applied to the previously predicted Li₆P and Li₆C, effectively resolving the controversy regarding whether NNAs contribute to superconductivity.

High-pressure structural predictions of the ternary Li–C–P system were performed from first principles with the implementation of the Ab Initio Random Structure Searching (AIRSS) and Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) codes,37,38 whose effectiveness has been confirmed by successful applications to the discovery of numerous structures of solids, surfaces, and clusters.39–41 The energy stability of different ternary stoichiometries was evaluated by their enthalpy of dissociation into the most competing element and binary compounds.20,27,42–44 Structural relaxations during the predictions used ultrasoft pseudopotentials in the Cambridge Sequential Total Energy Package (CASTEP) code.45 High-accuracy structural relaxations and enthalpy calculations employed the all-electron projector augmented wave (PAW) method pseudopotentials in the Vienna Ab initio Simulation Package (VASP) code with a kinetic cutoff energy of 500 eV.46 The valence electrons of the pseudopotentials were 1s²2s¹ for Li, 2s²2p² for C, and 3s²3p³ for P. The exchange–correlation functional was given by the Perdew–Burke–Ernzerhof (PBE) parametrization within the generalized gradient approximation (GGA).47 A Monkhorst–Pack k-point grid was generated with a reciprocal space resolution of 2π × 0.03 Å⁻¹ to ensure that the enthalpy calculations converged to <1 meV/atom. Bader charge analysis in the quantum theory of atoms in molecules (QTAIM) theory was used to reveal the extent of charge transfer.48 Chemical bonding analysis was investigated by the crystal orbital Hamilton population (COHP) using the Local Orbital Basis Suite Toward Electronic-Structure Reconstruction (LOBSTER) package.49 Density functional perturbation theory was adopted for calculation of EPC through the Quantum ESPRESSO (QE) code.50 Ultrasoft pseudopotentials were used with kinetic energy cutoffs of 80 Ry and 800 Ry for wave functions and charge density, respectively.51 A 24 × 24 × 24 k-point grid was used for Brillouin zone (BZ) integration in the electronic calculations and a 4 × 4 × 4 q-point grid was used to compute the EPC matrix elements. The superconducting critical temperatures T_c of all stable structures were estimated from the McMillan–Allen–Dynes equation.52

Comprehensive structure predictions were performed for ternary Li_xC_yP_z (x = 1–16, y = 1–3, and z = 1–3) compounds utilizing the AIRSS and CALYPSO structure search methods under pressures ranging from 150 to 350 GPa, with the main focus on lithium-rich species. A detailed description of the structure prediction is provided in the “Calculational details” section of the supplementary material, and the high-pressure phase diagrams of the ternary Li–C–P system are constructed in Figs. 1(a) and S1 (supplementary material). Thermodynamic stability of a compound is quantified by its energy above the convex hull E_hull.53 In Fig. 1(a), the stars represent stable ternary compounds with E_hull = 0 that are stable against decomposition into pure elements, together with other energetically most favorable binary and ternary phases, while the gray squares represent other metastable and unstable compounds. Four stable compounds Li₃CP, Li₈C₂P, Li₁₆C₂P, and Li₁₄CP were predicted at 300 GPa (detailed structural information is presented in Table S1, supplementary material). There is a near-linear increase in Li content, resulting in an enhanced number of electron donors and a greater abundance of electron-confined spaces available for accommodating NNAs, thereby providing the conditions necessary for the formation of diverse NNAs. All predicted stable Li–C–P ternaries are metallic and dynamically stable on the basis of electronic energy band structures and phonon dispersion curves (Figs. S2 and S3, supplementary material), but only Li₁₄CP and Li₁₆C₂P exhibit NNA characteristics (Fig. S4, supplementary material). Subsequently, we concentrate on analyses of Li₁₄CP, owing to its high crystal symmetry and lithium concentration.

Li₁₄CP crystallizes in a rhombohedral primitive cell with an R3 space group at 300 GPa [Fig. 1(b)]. The thermodynamically stable pressure of Li₁₄CP can be ascertained from enthalpy difference calculations (Fig. S5, supplementary material), and it is found that the rhombohedral phase is the most stable phase against decomposition into Li₆P, Li₂C, and Li above 145 GPa. Within the primitive cell, the Li atoms occupy two different crystallographic Wyckoff sites 3b and 1a (Fig. S6, supplementary material). The 12 Li atoms at the 3b site (Li1, Li2, Li3, and Li4) are arranged in four equilateral triangles that are parallel to each other, with the body diagonal of the primitive cell as the central axis. Additional Li5 and Li6 atoms (1a site) are distributed along this central axis in conjunction with C and P atoms. Each C and P atom has 16-fold coordination, assembling the C–Li and P–Li polyhedrons that serve as the building block in the construction of the three-dimensional crystal structure [Fig. 1(b)]. In Li₁₄CP, the Li atom serves as the electron donor, with a portion of the electrons occupying the electron shells of C and P atoms, and the excess electrons are accommodated in the interstitial cavities formed by coplanar connected polyhedrons, resulting in the formation of the NNAs. The calculated electron localization function (ELF) reveals the localized distribution of electrons within Li₁₄CP [Fig. 1(b)], where the electron densities at the centers of both the double hexagonal pyramid (NNA1) and dodecahedron (NNA2) exceed the superposition of unbound atoms’ free electrons, thus providing evidence for the existence of two cage-like NNAs. It is worth mentioning that the NNAs in Li₁₄CP exhibit similar characteristics to those in Li₆C and Li₆P, with the cage-like NNAs interconnecting with each other across intermediate extranuclear electrons of C and P atoms, forming the electron-conducting channels. However, the distinction lies in the presence of the electron conduction channel between the two cage-like NNAs in Li₁₄CP.

The electronic band structure in Fig. 2(a) shows that three bands, labeled I, II, and III, cross E_F and form a broad density of states (DOS) peak around E_F, which is predominantly attributed to the Li 2p orbitals and the non-nuclear centered orbital of the NNAs, with minor contributions from the atomic orbitals of C and P [Fig. 2(b)]. Furthermore, a pronounced Van Hove singularity (VHS) is observed near E_F, corresponding to the flat band structure around the high-symmetry point T. Meanwhile, bands I, II, and III passing through E_F along the Γ–T, H2–L, and Γ–S0 directions exhibit steep slopes. These not only characterize the metallicity but also imply potential superconductivity in Li₁₄CP. Subsequent EPC calculations reveal that Li₁₄CP is a superconductor with a T_c of 10.6 K at 300 GPa. Given that the interaction between NNAs and lattice originates primarily from the interaction of NNAs with the valence electrons of atoms, the electron distribution near the Fermi surface is crucial in elucidating the role of NNAs in superconducting electrides. The band-decomposed electron density maps for bands I, II, and III crossing E_F on the (1$\bar{1}$0) plane in the rhombohedral primitive cell are displayed in Fig. 2(c). The conduction electrons occupying electronic states within band I are distributed in the interstices of the double hexagonal pyramid and diffuse toward adjacent Li atoms, emphasizing a more pronounced influence of NNA1 than NNA2 on the electrical conductivity of Li₁₄CP. Specifically, the diffused NNA1 surrounds the Li5 atom like a satellite, forming the striking satellite interstitial electrons (SIEs). Furthermore, the band-decomposition electron density in the energy window −1 eV < E − E_F < 0 eV is also calculated (Fig. S7, supplementary material), and the electron distribution thus obtained is consistent with the above. The presence of SIEs is attributed to the covalent interaction between NNA1 and Li5, as evidenced by the two-dimensional ELF of the total electron density on the (1$\bar{1}$0) plane [Fig. 2(c)]. Valuable bonding information can be gleaned from the crystal orbital Hamilton population (COHP), and the integrated COHP (ICOHP) can be used as a qualitative measure of covalent bonding.41 The calculated ICOHP value of the NNA1–Li5 pair is −0.6, and its absolute value exceeds that of the typical NNA–Mg pair in Mg₂Xe (−0.4), indicating strong chemical interactions with a large covalent component. Consequently, the striking SIEs occupying the electronic state of band I contribute significantly to the electrical conductivity, even superconductivity, of Li₁₄CP.

To investigate the influence of SIEs on the superconductivity in Li₁₄CP, we analyzed the electron density alongside the most significant variations in SIEs (N_SIEs) and the related EPC parameters at pressures of 200, 250, and 300 GPa (Table S2, supplementary material). As shown in Figs. 3(a) and 3(b), N_SIEs is proportional to T_c and the EPC parameter λ, both of which progressively increase under pressure (5.4 K and 0.5 at 200 GPa, 10.6 K and 0.68 at 300 GPa). Strong compression reduces the volume of the interstitial cavity within the double hexagonal pyramid, hindering the entry of electrons released from the Li atoms. Consequently, a portion of the electrons are localized between Li and NNA1, thereby enhancing the covalent interaction between them. The pink histograms in Fig. 3(b) illustrate the variation of the ICOHP value of the NNA1–Li5 pair. It is evident that all ICOHPs are negative, and their absolute values gradually increase with increasing pressure, which provides compelling evidence for the enhancement of the covalent interaction between NNA1 and Li5; that is more SIEs are formed under strong compression. As expected, N_SIEs grows by 28.6%, from 6.9 × 10⁻³ at 200 to 8.7 × 10⁻³ Å⁻³ at 300 GPa. Meanwhile, the electronic energy band I associated with the SIEs gradually flattens out near the high-symmetry point T (Fig. S8, supplementary material), promoting the formation of Cooper pairs and enhancing electron–phonon interactions.

There is obvious phonon softening in the low-frequency acoustic branches along the F–Γ, Γ–T, H2–L, and Γ–S0 directions within the phonon dispersion curves of Li₁₄CP at 300 GPa [Fig. 3(c)], and the q-resolved λ_q is especially large in these softened regions. At the same time, the Eliashberg spectral function α²F(ω) illustrated in Fig. 3(d) reveals that the low-frequency acoustic branches dominate the superconductivity so that modes below 8 THz make a contribution of 64.7% to the EPC parameter λ at 300 GPa. It is noteworthy that as pressure increases, the softening along the F–Γ, Γ–T, H2–L, and Γ–S0 directions progressively intensifies (Fig. S9, supplementary material), and the low-frequency vibrations contribute more significantly to the EPC, which is usually related to Fermi surface nesting (FSN). To elucidate this phenomenon, the nesting function ξ(q) of Li₁₄CP at 300 GPa is calculated. This is a geometrical property of the Fermi surface and is particularly large for wave vectors that connect parallel portions of the surface.54 As can be seen in Fig. 3(e), ξ(q) exhibits a notable feature of sharp peaks along the Γ–T, H2–L, and Γ–S0 directions, which, except for that at Γ (i.e., q = 0, where the peak represents the entire Fermi surface nesting into itself and has no actual physical meaning),55 signify the occurrence of FSN. Coincidently, the nesting direction corresponds not only to the phonon softening direction, but also to the region where the electronic energy band I forms the SIEs crossing E_F in Li₁₄CP [Fig. 2(a)]. These results indicate that the highly localized SIEs play a major role in the coupling with phonons, thus contributing to enhancing the superconductivity of Li₁₄CP.

NNAs, including the SIEs proposed in this study, are essentially anions, and so the Li₁₄CP framework is positively charged at high pressure. To further substantiate the leading role of the SIEs in the superconductivity of Li₁₄CP, a hypothetical charged system is constructed by removing the valence electrons, in a process commonly referred to as hole doping.56–58 The electrons around E_F are gradually neutralized by this operation (Table S3, supplementary material). The hole-doping-dependent EPC parameters at 300 GPa are calculated and presented in Fig. 4(a), from which it can be seen that both T_c and λ gradually decrease with increasing doping concentration and ultimately almost disappear as the doping level approaches 1e. By comparing the phonon dispersion curves, it is found that there is a clear disappearance of the softening in the low-frequency acoustic branches [Figs. 4(b) and S10, supplementary material], leading to a significant reduction in the percentage contribution of acoustic branches to λ [pink columns in Fig. 4(a)]. The disappearance of acoustic branches induced by hole doping is attributable to the vanishing of FSN, which can be corroborated by the changes in the nesting function ξ(q) in Fig. 4(c). Most importantly, a topological analysis of Li₁₄CP reveals a bond critical point (BCP) between NNA1 and Li5 [Fig. 4(d) insert]. The charge density distribution at the BCP provides valuable insights into the bonding types and properties, while the sign of the Laplacian of the electron density [∇²ρ(r)] indicates whether the density is locally concentrated (negative) or depleted (positive). The calculated ∇²ρ(r) has a negative value in the primitive structure, confirming the covalent interaction between NNA2 and Li5, which manifests as SIEs near E_F. Furthermore, as the doping level increases, ∇²ρ(r) gradually transforms from negative to positive, with its absolute value decreasing first and then increasing [Fig. 4(d)]. These results directly illustrate how the electron density of SIEs and the soft phonon modes caused by FSN lead to the strong EPC, thereby inducing the superconductivity of Li₁₄CP at high pressure.

A leading role of SIEs in superconductivity has been proposed in the present study. Here, based on our point of view, the superconducting mechanisms of the previously predicted Li₆P and Li₆C are reanalyzed. From the available information,26,27 it can be concluded that both the connected dumbbell-like NNAs in Li₆P and the cage-state NNAs in Li₆C contribute to their metallic state [Fig. 5(a)], with these contributions ranking just slightly below that of Li. Calculations of the decomposed electron densities of the energy bands crossing E_F in these two electrides provide further understanding of the mechanisms of superconductivity. As depicted in Fig. 5(b), there are obvious SIEs between two types of NNAs and their surrounding Li atoms, and the electron density of SIEs N_SIEs exhibits an opposite trend with increasing pressure. Nevertheless, it is exciting to note that the variations of N_SIEs in Li₆P and Li₆C are proportional to their respective critical temperatures T_c [Fig. 5(c)]. The larger value of N_SIEs enhances the electron–phonon interaction, directly facilitating the improvement in T_c. Overall, in the binary Li–C and Li–P systems, as well as the ternary Li–C–P system, the existence of SIEs is conducive to superconductivity, and regulating the electron density of the SIEs is expected to uncover novel superconducting electrides with high T_c. We have thus unified the mechanisms of superconductivity in these three electrides and have successfully revealed the relationship between NNAs and superconductivity.

We have presented a new mechanism by which NNAs contribute to superconductivity in the ternary superlithide Li₁₄CP. Under compression, the excess electrons accumulate in double hexagonal pyramid interstitial cavities and form cage-like NNAs. These loosely bound NNAs have a covalent interaction with surrounding Li, leading to the formation of unique SIEs near E_F. First-principles calculations show that the SIEs progressively increase in number and couple more strongly with phonons at higher pressure. Moreover, the Fermi surface nesting associated with SIEs induces phonon softening, further enhancing the electron–phonon coupling and enabling the superlithide Li₁₄CP to achieve a T_c of up to 10.6 K at 300 GPa. This mechanism by which SIEs contribute to superconductivity also applies to the predicted compounds Li₆P and Li₆C, effectively reconciling previously conflicting views. Our work presented here thus provides deep insights into the mechanism of superconductivity in high-pressure electrides.

See the supplementary material for computational details, supplementary figures, and supplementary tables.

Acknowledgment. This work was supported by the National Key R&D Program of China (Grant No. 2023YFA1406200), the National Natural Science Foundation of China (Grant Nos. 12374004 and 12174141), and the High Performance Computing Center of Jilin University, China.