Introduction

Superconductivity energy gap opens when the Cooper pairs are condensed into a coherent ground state. The microscopic mechanism sustaining such pairing of electrons typically involves electron–phonon coupling as the glue, but in unconventional cases magnetic excitations and other delicately balanced factors may dominate.1,2,3 A consequence of the complexity is that discovering new superconducting materials largely remains a rather haphazard process.4,5,6 Our favored approach is to examine a group of uniquely structured compounds in which the crystallographic cavity space confines excess electrons rendering them an anionic nature. Conceptually, we describe these materials as electrides.1, 7,8,9,10,11,12 A canonical example is the completely reduced mayenite, [Ca12Al14O32]2+:2e – which exhibits remarkably rich chemistries.13,14,15,16 Of particular note is the observation of a superconducting transition (T c ~ 0.4 K) intimately related to the anionic electrons in this electride, thereby opening a new frontier in the field of superconductors.17

The range of electrides has been extended from cage structured mayenite to both layered and one-dimensional (1D) types. Compared to the strongly reactive layered electrides, such as [Ca2N]+:e – and [Y2C]1.8+:1.8e –, the air stable 1D analogues promise a more friendly system to work on.18,19,20,21 From a structural perspective, the topology of the cation arrays in our 1D apatite electride [La8Sr2Si6O24]4+:4e – approximates that of the Mn5Si3 structure which hosts another 1D electride Y5Si3.20, 21 The presence of anionic electrons is associated with the formation of new energy states near the Fermi level with weak electron–phonon interaction, which is indicative of a good source for superconductivity. We are, therefore, motivated to look for possible combined electride and superconducting states by further exploring the chemical variety of Mn5Si3-type A5B3 phases (A: rare earth or transition metals; B: Ga, Si, Ge, etc).

Crystallizing in hexagonal symmetry (space group P6 3 /mcm), the Mn5Si3 unit cell is characterized by three distinct crystallographic sites, Si, Mn(I) and Mn(II), see Fig. 1a, b. The Si atoms generate a network of twisted trigonal prisms so as to accommodate Mn(I) atoms in the center. Such slightly distorted prisms are linked by planes down the c-axis stacking direction forming a linear Mn(I) skewer structure. The in-plane edge sharing gives rise to a zeolitic framework to accommodate an octahedral array of remaining Mn(II) atoms. Like in many chalcogenides and halides of the 4d and 5d transition elements,22 all like octahedral units here are interconnected forming infinite [Mn3] channels along the [001] direction. In fact, such structures have been considered to be favorable to superconductivity as in A15-type superconducting phases.23 As an example, a recent study has claimed the first superconductor Zr5Sb3 with T c ~ 2.3 K within this structural domain.24

Fig. 1
figure 1

An illustration of the Mn5Si3-type crystal structure and x-ray diffraction patterns for Nb5Ir3. a Shaded polyhedral Si (the plum spheres) complexes are twisted Si6 trigonal prisms which accommodate Mn(I) atoms (the blue spheres) in the centre. b The 6g Mn(II) atoms (the green spheres) form chains of confacial trigonal antiprisms in which the channel space can be filled by guest atoms (the red spheres), while the 4d Mn(I) atoms construct another linear structure. c XRD patterns collected on a series of Nb5Ir3O x (0 ≤ x ≤ 1) phases quenched in liquid nitrogen unless specified. d A Rietveld refinement analysis of data for single-phased Nb5Ir3O

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In this work, the isostructural Nb5Ir3 attracts our attention in consideration of the superconducting nature of the Nb element and sporadic reports on the superconducting behavior of the binary Nb–Ir phases.25,26,27 In virtue of the experiments and theoretical calculations, we find that the phase diagram for Nb–Ir hosts a hexagonal phase with interesting interplay of superconductivity and electride state. The characteristic band structure can be engineered via introduction of guest atoms into the cavity sites for targeted properties, showing a rather intriguing base for the development of intrinsic, stable and functional electrides.

Results

Synthesis and band structures

Our consideration is to synthesize all-transition-metal variants for electron-enriched compounds. With Nb and Ir as the chemical pair, a solid state reaction of element powders in a molar ratio of 5:3 by heating at 1200 °C in an evacuated silica jacket yielded air stable black products. X-ray diffraction (XRD) analysis shows that the obtained samples crystallize as a mixture of tetragonal and hexagonal phases. As shown in Fig. 1c, the diffraction peaks of the XRD pattern for Nb5Ir3 can be indexed into a Mn5Si3-type unit cell (a = 7.858(4) Å, c = 5.101(3) Å, space group: P6 3 /mcm) and a second phase with a tetragonal structure (a = 9.841(3) Å, c = 5.069(3) Å, space group: P4 2 /m). The former hexagonal phase of Nb5Ir3 receives little attention in the literature. In contrast, the latter tetragonal symmetry belongs to the σ phase and has been extensively studied concerning superconductivity. Synthesizing monophasic Nb5Ir3 proved to be challenging, nevertheless, upon quenching in liquid nitrogen, the concentration of the hexagonal phase increases (Fig. 2c), indicating the possibility to stabilize the hexagonal structure at ambient conditions. A Rietveld refinement analysis suggests a ratio of 60.8:39.2 between the hexagonal and tetragonal phases, see Supplementary Fig. 1. Introducing stabilizing oxygen atoms into the octahedral voids indeed enables pure hexagonal phases and we shall discuss the details later.

Fig. 2
figure 2

Electronic structures of different materials. Calculated band structures and density of states (DOS) for a [Nb5Ir3]2+:2e − with corresponding electron density for the channel bands (ChBs) and electron localization function (ELF) map on the (1−100) plane parallel to the c-axis in c and d, respectively, and f Nb5Ir3O. b, e Band structures of one-dimensional (1D) electrides b [La8Sr2(SiO4)6]4+:4e – and e Y5Si3. The characteristic ChBs are circled in red for all electrides

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At first, we show the calculated band structures of Nb5Ir3, Nb5Ir3O, and relevant electride materials for insight into the electron states by using density functional theory (DFT). From Fig. 2a, Nb5Ir3 is found to be a multiband metal with the states at the Fermi level mainly consisting of Nb 4d and Ir 5d orbitals. More interestingly, in analogy to the known 1D apatite electride [La8Sr2(SiO4)6]4+:4e – where four anionic electrons occupy two channel bands (ChBs) as in Fig. 2b, Nb5Ir3 shows two similar ChBs just below the Fermi level (marked by red circles in Fig. 2a), which are associated with the two interstitial sites, i.e., at (0, 0, 0) and (0, 0, 1/2), in the channel space. These two ChBs are localized along the direction perpendicular to the channels (e.g., the A–H direction), but delocalized along the channel directions (e.g., the H–K direction), implying that the ChBs states are confined within the 1D channel space, similar to the 1D apatite electride [La8Sr2(SiO4)6]4+:4e –.20 A further examination of the ChBs-associated electron density (Fig. 2c) and electron localization function (Fig. 2d) confirms that the confinement of ChBs electrons in the channel with ionic bonding to the [Nb5Ir3]2+ framework. As a result, we are able to conclude that Nb5Ir3 is an intermetallic electride with a [Nb5Ir3]2+:2e – configuration. For a simple comparison, we revisited the recently reported isostructural Y5Si3 electride and confirmed the existence of two similar characteristic ChBs (i.e., the two bands red circled in Fig. 2e). On the other hand, it is found that the oxygen-containing Nb5Ir3O presents a very different band structure, as the ChBs are now far away from the Femi level, which may be evidence for the disappearing electride property. In the following text, both Nb5Ir3 and Nb5Ir3O are revealed to be superconducting with relative high T c. It is to say that Nb5Ir3 exhibits both electride and superconducting behaviors, whereas only superconductivity remains in Nb5Ir3O. The finding on the combination of electride and superconductivity is quite unusual and we set a record for the transition temperature for this category of superconductors.

Superconducting properties

Shown in Supplementary Fig. 2a is the temperature dependence of resistivity under zero magnetic field. The hexagonal Nb5Ir3 shows a metallic behavior on cooling. Resistivity drops abruptly at a critical temperature \(T_{c1}^{{\rm{onset}}}\) ~ 9.4 K, and down further until zero at \(T_{c2}^{{\rm{onset}}}\) ~ 2.8 K, suggestive of the two coexisting superconducting phases. Here, \(T_c^{{\rm{onset}}}\) is defined as the temperature at which the resistivity begins to decrease. The temperature dependence of susceptibility of quenched Nb5Ir3 at a magnetic field of 10 Oe is shown in Supplementary Fig. 2b, consistent with the resistivity data. Around \(T_{c1}^{{\rm{onset}}}\), the transition is sharp with a transition width of ~0.5 K in both resistivity and susceptibility. The pronounced hysteresis between magnetization curves on zero-field cooled and field cooled substantiates a type-II superconductivity. Thus, the two phases of Nb5Ir3 are both superconductors. Meanwhile, the low-temperature superconductivity is found to vanish gradually when the tetragonal phase diminishes from the XRD patterns; consistent with previous findings.27 This is illustrative of their competing relationship. Thus, the higher T c is from hexagonal Nb5Ir3 while tetragonal Nb5Ir3 shows the lower T c. In other words, hexagonal Nb5Ir3 is not only an intermetallic electride, but actually a superconductor with a transition temperature much higher than that of the first superconducting electride [Ca12Al14O32]2+:2e– (T c ~ 0.4 K).

Compared to the known Mn5Si3-type superconductors, Nb5Ir3 exhibits the highest T c. However, the presence of the competing tetragonal phase necessitates an optimized synthesis for a pure phase. In many cases, Mn5Si3-type materials have to bind interstitial atoms to stabilize the host structure.28,29,30 For the current composition, Horyn and co-worker31 have tried to incorporate interstitial oxygen atoms, showing that the resultant compound remains superconductive; the superconducting mechanism was however not discussed. Using Nb2O3 as the oxygen source, we prepared a series of Nb5Ir3O x samples with 0 ≤ x ≤ 1. As evidenced in Fig. 1c, with growing oxygen concentration, the tetragonal phase is increasingly suppressed, whereas the hexagonal structure becomes more dominant. At a nominal composition of Nb5Ir3O, a single-phased Mn5Si3-type structure is formed. A Rietveld refinement of the XRD profile, see Fig. 1d, reveals that the oxygen occupies the channel cavity sites (0, 0, 0) and (0, 0, 1/2), in consistence with previous studies on the isostructrual compounds. 28,29,30 The almost linear change of unit cell parameters as a function of oxygen content in Nb5Ir3O x further substantiates the insertion of oxygen into the vacant lattice sites (Supplementary Fig. 3). The Nb–O interatomic distance is 2.28 Å, indicative of the bond formation. The Nb1 atoms define a linear chain with a short Nb–Nb interatomic distance of c/2 ~ 2.55 Å compared with ~2.99 Å separations in face centred cubic Nb.32 This chain of very closely spaced transition metals is a notable feature of the present crystal structure. The Nb2 atoms are more loosely packed in the octahedral units with the Nb–Nb distance between 3.17 and 3.28 Å. In other words, the relatively weak Nb2–Nb2 bonding in Nb5Ir3 is converted to Nb–O bonding in Nb5Ir3O.

Subsequent characterizations of the superconducting property of the stabilized hexagonal Nb5Ir3O were carried out. Figure 3a shows the temperature dependence of resistivity and susceptibility. Resistivity decreases sharply to zero at an onset temperature ~10.5 K. Susceptibility as a function of temperature was measured at 10 Oe in Fig. 3b. A sharp drop indicating the superconductivity appears at around 10.5 K, which is in excellent agreement with the zero resistivity temperature. For the merit of discussion hereafter, we define the onset temperature as the superconducting temperature for both Nb5Ir3 and its derivative Nb5Ir3O. The nature of superconductivity is supported by the presence of a large anomalous jump in the specific heat, which also signals a homogeneous sample. As can be seen from Fig. 3c inset, as the magnetic field increases, the jump was gradually suppressed and shifted toward a lower temperature. Also, the superconducting transition is found to broaden with increasing magnetic field (Fig. 3c and inset), indicating strong magnetic pinning effect as revealed in many known type-II superconductors.

Fig. 3
figure 3

Evidence for bulk superconductivity in Nb5Ir3O. a Temperature-dependent resistivity at zero magnetic field. b Magnetic susceptibility curve under an applied field of H = 10 Oe. c The electronic contribution to specific heat fitted according to a two-gap model. Inset is the specific heat measurement with and without magnetic field

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To extract more information, the specific heat at the normal state C N(T) was firstly estimated from the data under 5 T. As in Supplementary Figure 4, C N(T)/T departs from a linear dependence on T 2, suggesting the existence of low-lying phonons. Accordingly, C N(T) was fitted based on the function C N(T) = γT + βT 3 + δT 5 where the first and latter two terms represent the contributions from electrons and phonons, respectively. The obtained parameters are γ = 33.89 mJ mol−1 K−2, β = 0.40963 mJ mol−1 K−4 and δ = 1.413 × 10–3 mJ/mol K6. The specific heat jump at T c, ΔC/γT c, is approximately 1.86, notably larger than the expected value of 1.43 for a typical BCS (Bardeen–Cooper–Schrieffer) superconductor with a weak coupling limit, indicative of a strong electron–phonon coupling that serves as the glue of Cooper pairs in this system.33 The Debye temperature is estimated to be θ D = 350 K, using the relationship β = (12π 4/5)N R/θ D 3, where N is the atom number in a formula unit and R is the gas constant. From the McMillan formula,34 T c = (θ D/1.45) exp{[−1.04(1 + λ ph )]/[λ ph  − µ *(1 + 0.62 λ ph )]}, together with the assumption of the Coulomb pseudopotential µ* = 0.15,35 the electron–phonon coupling constant was estimated to be λ ph  = 0.82, indicating a strongly coupled superconductor, similar to LaMo6S8 with λ ph  = 0.85.36 In Fig. 3c, the electronic contribution C el (T) to specific heat is extracted by subtracting the lattice contribution C N (T) from the total. We find that neither a simple BCS weak-coupling limit nor a single gap α model could reproduce the data in Fig. 3c. It implies that C el (T) below T c behaves like a strong coupling superconductivity with a finite energy gap. A quantitative analysis of C el (T) by employing the two-gap model leads to a perfect fitting (Fig. 3c), yielding α 1 = 1.14 and α 2 = 3.06, indicating that two superconducting energy gaps of an s-wave symmetry with the different size develop simultaneously at T c. The larger gap corresponding to the electron states strongly coupled to phonons is believed to be dominant in the superconductivity. The derived 2Δ 0 /k B T c of ~6.12 far exceeded the weak coupling BCS value 3.52. Such a strong coupling superconductivity is relatively rare, but comparable to Pb–Bi alloy, pyrochlore osmates and SrPt3P.37,38,39 Also, we have noticed evidence of strong coupling of electrons with low-lying phonons in resistivity as shown in Fig. 3a. Above 100 K, the resistivity becomes increasingly saturated because the electron mean free path becomes essentially close to the atomic lattice spacing. Such resistivity saturation has been observed in other strong coupling superconductors and it is usually seen as evidence for strong coupling between electrons and low-lying phonons.39 Thus, the above results indicate that Nb5Ir3O is a two-gapped s-wave, strongly coupled superconductor.

The Wilson ratio R w was also determined based on R w  = π 2 k 2 B χ s /3μ 2 B γ where k B is the Bohr magneton.40 The magnetic susceptibility in the normal state (Supplementary Fig. 5) can be fitted to the formula χ = χ 0  + C w /(T  −  θ), where χ 0 is a temperature-independent term, C w is the Curie-Weiss constant, and θ is the Weiss temperature. The fitting yields χ 0  = 4.09 × 10–4 emu mol−1, C w  = 1.04 × 10–3 emu K mol−1 and θ = −6.2 K. Assuming the orbital susceptibility is negligible, that is to use the obtained χ 0 as spin susceptibility, we acquire R w  = 0.81, a value that tends to be overestimated due to the non-negligible spin-orbit interaction, but is in full support of the strong electron–phonon coupling.

Assuming most Cooper pairs are broken at T c, we can plot the upper critical field H c2 (T) from the ρ data under various magnetic fields (Supplementary Fig. 6a). As shown in Supplementary Figure 6b, by using the Ginzburg–Landau (GL) equation H c2 (T) = H c2(0) × (1 − t 2)/(1 + t 2), where the reduced temperature t = T/T c, the experimental data were fitted generating a ground state value H c2(0) = 11.6 ± 0.6 T. According to the GL theory,30 the H cc2 value is related to the GL coherent length (ξ GL) in the form of H c2 = φ 0 /2πξ GL 2 (φ 0 being the magnetic quantum). From this formula, ξ GL = 5.33 nm can be obtained. The lower critical field H c1, extracted from the magnetic field (Supplementary Figure 6c) where the magnetization curve departs from the linear response, was plotted in Supplementary Figure 6d. Applying the empirical function H c1(T) = H c1[1 − α(T/T c)2] (α is a fitting parameter) gives rise to H c1(0) = 147 ± 5 Oe. The London penetration depth (λ) derived from the BCS formula H c2 /H c1 = 2κ 2lnκ with κ = λ/ξ is 201 nm, while the GL parameter (κ) is 37.87).

Properties under high pressure

To reveal more details with respect to the pairing mechanism, a study under pressure was further conducted. Combining resistivity and magnetization results in Supplementary Fig. 7 gives the pressure dependence of \(T_c^{\it{M}}\) and \(T_c^{{\it{zero}}}\) which are referred to as the temperatures where diamagnetism appears and resistivity becomes zero, respectively, see Fig. 4a. While both temperatures are closely related, it is noted \(T_c^{onset}\) decreases monotonically from 10.1 K at ambient pressure to 9.2 K at 8 GPa. A linear fitting in the range of 0.1 MPa < P < 13 GPa yields a negative differential pressure coefficient of −0.08 K GPa–1. An analysis of the normal-state resistivity (ρ) just above T c was performed through a fitting against the residual resistivity ρ 0 , coefficient A and the exponent n based on the function ρ = ρ 0  + AT n, see Supplementary Fig. 8. The estimated parameters were shown in Fig. 4b–d. It can be seen that both \(T_c^{{\rm{onset}}}\) and \(T_c^{{\rm{zero}}}\) show the same trend in the whole pressure range. At ambient pressure n is nearly 2.5, and it increases gradually up to 3 above 10 GPa. The derived n is much smaller than the typical value (n = 5) for an electron–phonon scattering dominated case, consistent with strong-coupling superconductivity. The ρ 0 is found to decrease under pressure due to the polycrystalline nature of the sample. The inelastic scattering term A can serve as a measure of N(E F). Hence, the smaller A under pressure (nearly one order of magnitude decrease) is related to the reduced N(E F) due to the renormalized wide bandwidth. The reduced T c under pressure is expected resulting from the decreasing density of states (DOS) at the Fermi level associated with the increased orbital hybridizations. But, it is unusual that the pressure coefficient is notably smaller than that of other traditional BCS superconductors such as metals, indicating some other interactions or correlations are in favor of the superconducting state. As one example, the introduction of Ir atoms or Ir–Nb bonds in an itinerant electronic superconductor may provide some local magnetic moments, and their interplay has been revealed to play a critical role in unconventional superconductivity. This implies a unique superconducting state in Nb5Ir3O. One may also consider the pressure effect is due to the depression of electride, i.e. the electron concentration at interstitial sites, which shifts electrons to d orbitals of Nb and Ir. This would serve to further increase the DOS at Fermi level which will in turn favor the superconductivity, although overall T c decreases.

Fig. 4
figure 4

Measured transport parameters under pressure. Pressure (P) dependence of superconducting transition temperature a, and resistivity fitting parameters, ρ 0 and n b–d

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Discussion

Now we consider the enhanced T c of Nb5Ir3O compared to the parent Nb5Ir3. From Supplementary Fig. 9, the major variable that affects the superconductivity seems to be the amount of O because T c is enhanced as a generally linear response to increasing oxygen content. This is in stark contrast to the case of Zr5Sb3 in which filling the same channel sites with guest atoms, e.g. C, O and Sb, would suppress the superconductivity.24 We may link such an increase to the crystal chemistry of Nb5Ir3O. Upon binding O, the principal dimensional change taking place is the contraction of all cells down the c-axis, though the cell is expanded along the (001) plane (Supplementary Fig. 3). This means the octahedral chains along the c-axis become increasingly distorted when Nb–O bonds are formed. As in a similar interstitial filled Sb5RE3Ti (RE: rare earth),41 the position of the Fermi level is quite sensitive to even minor changes that lead to small contractions in the structure. Ir is the most electronegative component among these metal elements and, therefore, the Zintl concept may be applicable by assuming that a full electron transfer takes place. Hence, the Nb atom donates its valence electrons to both oxygen and iridium atoms whereas the remaining electrons are utilized to form a single pair bond with a neighboring metal atom. In this sense, we may expect O 2p and Nb 4d will contribute extra states to the energy region that hosts the Fermi level, while the σ-overlap of Nb dz2 orbitals within the linear Nb skewers still plays a dominating role. The larger DOS at Fermi level or γ value in Nb5Ir3O should account for the higher T c. In support of this scenario, a smaller γ value (24.1 mJ mol−1 K−2) in Zr5Sb3 has been reported with a lower T c. On the other hand, as can be found in calculations, inserting O atoms can push the ChBs down to approximately 6 eV below Fermi level, yet the transition temperature increases. Another likely reason is the involvement of O phonon modes that may have a higher frequency and stronger coupling to electrons close to the Fermi level.

As for electrides, while the apatite structured [La8Sr2(SiO4)6]4+ :4e – has to be prepared by extracting two channel oxygens from La8Sr2(SiO4)6O2, Nb5Ir3 is an intrinsic one with open channel spaces. When the two channel sites are completely filled by oxygen, Nb5Ir3O is formed. Figure 2f shows the band structure and DOS for Nb5Ir3O. It can be observed that the loosely bound ChBs states (right below the Fermi level) for Nb5Ir3 are transferred to oxygen to form oxide anions. The resulting tightly bound O 2p 6 states are pushed down into a much deeper energy level. Interestingly, among the six O 2p bands, two of them (orange circled in Fig. 2f) retain the dispersion character of the ChBs in 1D electrides. Therefore, in general, the ChBs correspond to the channel sites directly and their energy levels rely on binding force of the anions. When the channel structure is open, the ChBs states are loosely bound and serve as anions themselves. To take one step further, electrides can be viewed to be intimately related to materials that are amenable to cation filling but undergo no structural degradations. This general rule may serve as the fundamental tenet for the electride development and define a direction towards more superconducting electrides.

To summarize, we have demonstrated an entirely new intermetallic electride with electrons detached from the lattice framework atoms but localized in the 1D channel space. This electride is also a superconductor below a critical transition temperature of T c = 9.4 K. The superconducting behavior is tunable by controlling the ChBs arising from the electrons that are responsible for the electride state. Filling the free spaces with interstitial oxygen not only stabilizes the host hexagonal lattice, but also helps to engineering the band topology. As a result, the superconducting transition temperature (T c) was increased up to 10.5 K in Nb5Ir3O. Such a T c enhancement is in distinct contrast to the case of Zr5Sb3 whose superconducting state is subject to suppression upon binding any guest species. Specific heat analysis indicates the appearance of low-lying phonons and two-gap s-wave superconductivity. Strong electron–phonon coupling is revealed to be the pairing glue with an anomalously large ratio between the superconducting gap Δ 0 and T c, 2Δ 0 /k B T c = 6.12. Under pressure, the decreasing DOS at the Fermi level associated with the increased orbital hybridizations leads to decreased T c. We also show a general rule for the formation of electrides which is closely related to the structural stability against the cation filling/extraction in the channel site, suggesting a new avenue towards the discovery of more electride materials.

Methods

Materials synthesis

Polycrystalline samples of Nb5Ir3 and Nb5Ir3O x were synthesized by the solid state reaction. Stoichiometric amounts of Nb, Ir and Nb2O3 precursor powders were mixed and pressed into pellets before sealed in a quartz tube under vacuum of around 10−3 mbar. The reaction was at 1200 °C for 72 h followed by a quenching process in either air or liquid nitrogen.

Characterizations

The purity of the precursors and the products was checked by XRD using a D8 ADVANCE diffractometer with Cu Kα radiation. Structural parameters were generated from Rietveld refinement using GSAS software with EXPGUI interface.42, 43 Magnetic susceptibilities of all samples were measured by a superconducting quantum interface device magnetometer (SQUID) in the temperature range of 2–300 K under applied external magnetic fields. The electrical resistively and the specific heat under the ambient pressure were measured above 1.8 K by using the commercial Physical Properties Measurement System (MPMS). The resistively under high pressure was measured by using a cubic anvil apparatus, which generated quasi-hydrostatic pressures up to 8 GPa. A preheated pyrophyllite cube was used as the gasket and the glycerol was used as the pressure transmitting medium. Each measurement was carried out at a fixed pressure on warming process.

DFT Calculations

DFT calculations were performed for Nb5Ir3 and Nb5Ir3O together with two reported 1D electrides [La8Sr2(SiO4)6]4+:4e– and Y5Si3 using the generalized gradient approximation (GGA) with the Perdew−Burke−Ernzerhof (PBE)44 functional and the projector augmented plane-wave method as implemented in the VASP code.45 The plane wave cutoff energy was set to 500.0 eV. A 6 × 6 × 10 Γ-centered k-mesh was used to sample the first Brillouin zone. Prior to the electronic structure calculations, the crystal structures were fully relaxed and considered to be converged when the total force on each atom was less than 0.01 eV/Å.

Data availability

The data that support the findings of this study are available in the article and Supplementary Information.