Introduction

Transition metal carbides (TMCs) are a broad class of materials formed by early transition metals and carbon, which crystallise in a rock-salt (NaCl-type) structure1. They serve as high-performance materials for cutting-tools, coatings, heat and chemical shields, due to their exceptional resistance. These carbides are metallic conductors (ρ ~ 10−5Ωm)2 with high thermal conductivity3. When the transition metal (TM) belongs to groups V (V, Nb, Ta) and VI (Mo, W), TMCs also exhibit superconductivity, with critical temperatures (Tc) reaching up to 18 K in NbN-NbC-TiN alloys4,5,6,7,8,9, and high critical fields10.

So far, TMCs have found limited use in superconducting technologies due to their brittleness. However, there is a clear potential for applications as shock- and radiation-hard superconductors in extreme environments - such as cryogenic current leads on deep-space probes, kinetic-energy dampers, or compact fusion diagnostics - where the unparalleled hardness, erosion resistance, and thermal stability of TMCs could outweigh manufacturing challenges. Understanding in detail the relationships between crystal chemistry and superconducting properties is a crucial prerequisite to further optimize these materials and unlock their full technological potential.

To date, experimental data remains sparse and largely outdated, with most superconducting critical temperatures (Tc) reported in the 1960s and 1970s from samples synthesized at high temperatures. The data for TMCs with a finite Tc is summarized in Table 1, where we included, to the best of our knowledge, all experimental reports5,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27. The table highlights a high variability in reported Tc’s for elements in groups V and VI, which has been historically attributed to differences in carbon vacancy concentrations14,15,17,28,29.

Table 1 Summary of the thermodynamic and superconducting properties of various TMCs in the stoichiometric and vacant rock-salt structure
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The microscopic understanding of superconductivity in TMCs is still superficial and fails to explain most experimental observations. In particular, although Tc is known to depend strongly on carbon content, existing studies on the role of vacancies are limited to the thermodynamic and mechanical stability of nonstoichiometric phases30,31. All first-principles studies of superconductivity so far have been based on the ideal 1:1 stoichiometric rock-salt structure, implicitly assuming its validity across the entire composition range. Recent work on group-IV and group-V compounds has interpreted variations in Tc in terms of the effect of electron filling of the transition-metal d states on the electron-phonon interaction19,32,33, but failed to address the strong dependence of Tc on vacancy concentration highlighted in Table 1. TMCs containing group-VI elements were shown to be dynamically unstable in the 1:1 rock-salt structure, highlighting even further the limitations of describing these compounds in the stoichiometric phase34,35.

In this work, we go beyond the 1:1 stoichiometric limit and systematically investigate the role of carbon vacancies on superconductivity across the TMC series, combining ab initio electron-phonon superconductivity theory and crystal structure prediction. Using unbiased evolutionary searches at variable compositions, we construct the TM-C phase diagrams, explicitly addressing the effect of non-stoichiometry. This leads us to identify a family of low-energy nonstoichiometric phases that preserve the rock-salt structure type, providing a natural explanation for the superconducting properties observed experimentally across the whole TMC series considered.

The main results across the whole TMC series are summarized by color-coding in Table 1. The TMs are ordered by the groups of the periodic table, and colored in green when the stoichiometric phase is thermodynamically (meta)stable (within 50 meV/atom from the hull), and red otherwise. The table reports the main superconducting parameters for the stoichiometric (1:1) phase, and a carbon-vacant (6:5) phase which preserves the rock-salt geometry. We note that elements of group IV and V (Nb and Ta) are the only ones for which the stoichiometric phase is stable and here the calculated and measured values of Tc match. In groups III, VI and VII, the stoichiometric phase is instead unstable, and the calculated Tc’s do not match experiments, while the carbon-vacant phase is found in much better agreement.

Our results demonstrate that thermodynamic stability is a crucial element in the realistic modeling of real-world superconductors28,36.

Results and discussion

Thermodynamical properties

In Fig. 1, we show the ab-initio convex-hull diagrams calculated at ambient pressure for all TMCs in Table 1, using evolutionary algorithms for crystal structure prediction37,38 (See the “Methods” section and the Supplementary Information for further details regarding DFT calculations, as well as additional figures detailing the electronic and structural properties).

Fig. 1: Convex hulls for TMC’s at ambient pressure.
Fig. 1: Convex hulls for TMC’s at ambient pressure.
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The figures are ordered with increasing atomic number, and follow the disposition of the respective TMs in the periodic table. Thermodynamically stable and unstable structures that lie on/above the hull are indicated with blue circles and red squares, respectively. Metastable rock-salt structures with carbon vacancies are marked as green circles when above the convex hull.

The plots are arranged in order of increasing atomic numbers, from ScC to ReC. In these diagrams, each point indicates the formation energy of one specific structure, while the lines indicate the convex hull; points on the hull (blue circles) indicate structures thermodynamically stable against decomposition, whereas structures that lie above the hull are indicated as red squares. All computed convex hulls contain at least one stable composition; however, the depth varies systematically across the d-block: from about 1 eV/atom in group-IV compounds (TiC, ZrC, HfC) to around 0.1 eV in Mo-C, W-C and Re-C. As we will show, this trend mirrors the progressive filling of antibonding d-states, which are empty in group-IV TMCs, and the consequent weakening of the M-C bond39. The stoichiometric rock-salt phase is stable for elements of group IV and Ta (The 1:1 stoichiometry is stable in the MoC and WC system, but with a hexagonal structure, while the rock-salt geometry is high in energy, as shown in the diagram.). However, for elements in groups III, V, and VI, we also find a plethora of sub-stoichiometric, carbon-deficient phases—such as M2C, M3C2, and M6C5—with rock-salt-like geometries (highlighted as green circles in Fig. 1). In several cases, these phases are more thermodynamically stable than the ideal 1:1 compound, or at least weakly metastable. While defects and vacancies typically increase the enthalpy and reduce the stability of a compound, our results point out that in rock-salt TMC they do the opposite.

In many cases, their powder diffraction patterns are nearly indistinguishable from that of the ideal 1:1 NaCl-type structure (see Supplementary Fig. 4). This suggests that many TMC samples for which finite critical temperatures were reported in past experiments may have actually contained nonstoichiometric, vacancy-rich phases rather than the nominal 1:1 compound. Indeed, this hypothesis is strongly supported by the fact that one such structure—M6C5— is an excellent model for capturing the effect of carbon vacancies in several TMCs. Before discussing superconducting properties, we first analyze general trends in thermodynamic stability, based on the calculated convex hulls.

Group III (Sc, Y): stable compositions include Y2C and Y4C5; Sc2C, Sc4C3, and Sc3C4, which have been experimentally reported40,41,42,43,44,45,46,47,48. All these the thermodynamically stable structures are qualitatively different from the NaCl structure, as they contain C2, or C3 chains.

The stoichiometric rock-salt structure is over 300 meV/atom above the convex hull. However, as the concentration of carbon vacancies increases, the relative energy of the rock-salt-like phases decreases drastically, to the point that the vacant rock-salt structure of Sc2C (Y2C) lies just 7 meV/atom (18 meV/atom) above the convex hull. These carbon-deficient phases may explain experimental reports of superconductivity in Sc and Y compounds nominally described as rock-salt carbides.

Experimental reports of a rock-salt ScC phase always involve a rapid quenching from extreme temperatures11,12,46. As for YC, although its Tc is cited in ref. 8, we were unable to retrieve the original source (ref. 49) to determine the synthesis conditions.

Group IV (Ti, Zr, Hf): the stoichiometric rock-salt structure is always thermodynamically stable. Carbon-vacant phases are present (Ti6C5, Ti3C2, Zr6C5, Hf4C3, Hf3C2), and their crystal structures retain a clearly recognizable rock-salt geometry throughout the 2:1 to 1:1 composition range. These results are in excellent agreement with the experimental report of synthesis of rocksalt structures with a wide range of vacancy concentrations13,50. In addition, for Zr we find on the convex hull a layered hexagonal phase with 2:1 composition, as also predicted in ref. 51, which has not been experimentally synthesized.

Group V (V, Nb, Ta): the stoichiometric rock-salt phase becomes progressively more stable down the group: the formation enthalpy is 92 meV/atom for V, 28 meV/atom for Nb, and negative for Ta. For all three elements, we also find stable carbon-deficient rock-salt phases (V8C7, Nb6C5, Ta4C3) which, in the case of Nb and V, are more stable than the stoichiometric phase.

The presence of vacancies in group-V TMCs has been widely documented experimentally52,53,54,55,56. Some authors suggested the formation of vacancy superstructures57,58,59 for both vanadium (V6C560,61 and V8C752,53) and niobium (Nb6C517,54). These observations strongly support our thermodynamic analysis.

Group VI (Mo, W): the stoichiometric rock-salt phase has a very high formation energy of 288 meV/atom (420 meV/atom) for Mo (W), making it strongly unstable. The thermodynamically stable phase for the 1:1 composition is, in fact, a hexagonal phase for both Mo and W, as confirmed by experiments62. We also found a 1:2 hexagonal phase on that is on the convex hull for Mo and very close for W63,64,65,66.

Introducing vacancies in the rock-salt phase lattice lowers its relative formation energy dramatically, down to 26 meV/atom in Mo2C (green circles in Fig. 1). Indeed, the experimental synthesis of the rock-salt phase through high-temperature melting and extreme quenching was reported for both MoC and WC4,7,21. This resulted in an intermediate, metastable phase stabilized by a certain amount of carbon vacancies67.

Group VII (Re) In rhenium carbide the 1:1 rock-salt phase is 932 meV/atom above the hull, and even carbon vacancies are not enough to stabilize the lattice, with Re6C5 lying 688 meV/atom above the hull. Recent experiments also failed to reproduce a rock-salt phase68. We observe instead that ref. 24, which reports a rock-salt phase of ReC, reports a lattice parameter of 4.0 Å. This value is much closer to the calculated equilibrium lattice constant of Re (3.9) than that of ReC (4.4), suggesting that the samples of ref.24 may have contained mostly pure Re, and, possibly, a small concentration of C impurities.

In summary, our results indicate that rock-salt-like TMC phases can be synthesized for a wide range of transition metals, even in cases where the 1:1 rock-salt structure is thermodynamically or dynamically unstable. Carbon vacancies play a key role in stabilizing these structures, lowering their formation energies and making them accessible as metastable phases.

This is reflected in the synthesis conditions reported in the literature: rock-salt-like phases are typically obtained via rapid quenching from the high-temperature melt16,69. Non-equilibrium synthesis methods are consistent with their metastable nature and the presence of a high vacancy concentration. Note that C-vacancies may not be always easy to detect through X-Ray Diffraction analysis, especially if they do not form an ordered superstructure.

Electronic structure

Figure 2 shows the electronic band structures and projected densities of states (DOS) of the eleven stoichiometric rock-salt carbides, arranged in the same order as the corresponding metals appear in the periodic table. For each TMC the left panel shows the electronic bands, while the right panel displays the DOS decomposed into C-2s (dark red), C-2p (orange), and M-d (teal) contributions.

Fig. 2: Electronic structure of TMC’s at ambient pressure.
Fig. 2: Electronic structure of TMC’s at ambient pressure.
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Electronic band structures (left panels) and projected densities of states (right panels) for rock-salt MC (M = Sc-Re), ordered with increasing atomic number, and following the disposition of the respective TMs in the periodic table. Colour code: C-2s (orange), C-2p (dark red) and M-d (teal). The Fermi level is at E = 0. The DOS is in units of states/eV/spin/atom.

The band structures of all compounds comprise the same three groups of bands: (i) a nearly flat, purely C-2s band between −9 and −8 eV; (ii) a broad ~10-eV-wide manifold around the Fermi level, which consists of six bands; (iii) weakly dispersive, metal-dominated states above.

The DOS corresponding to the broad manifold around the Fermi level exhibits a characteristic symmetric double feature: a sharp van-Hove peak with C-2p character, which declines towards a pseudo-gap, followed by a rise into another peak, with M-d character. These six bands derive from the hybridization of the three \({d}_{{t}_{2g}}\) orbitals of the M atom with the three C-p orbitals, and are clearly split into three bonding and three antibonding states, separated by a pseudogap.

The hybrid C-p/M-d manifold behaves in a rigid-band fashion across the TM series from group III to VII. For ScC and YC the Fermi level sits on the shoulder of the C-2p peak, resulting in a moderate DOS at the Fermi level (listed in Table 1). In TiC, ZrC and HfC EF lines up almost exactly with the pseudogap. For these elements, the TM-C bond strength is maximum, because the tetravalent M atoms match the preferred carbon valence.

When, in group V (V, Nb, Ta) TMCs, an additional d electron is added, EF is pushed onto the metal-dominated DOS peak above the gap. Finally, in Mo, W and Re (group VI), the Fermi level moves closer to the van-Hove singularity, so the DOS is even larger. However, as we shall see, this causes a dynamical instability of the stoichiometric rock-salt phase.

Moving down a group (e.g., Ti → Zr → Hf) mainly reduces the bandwidth, since TM with a larger atomic radius form TMCs with larger lattice parameters. This effect is particularly relevant in group V, where the theoretical equilibrium lattice parameter of Nb and Ta is almost identical (4.47 and 4.48 Å, respectively), but is significantly larger than that of V (4.15 Å). As a result, the DOS of V exhibits significantly sharper features; in particular, the value at EF is 50% larger, resulting in a significantly larger Tc predicted for the stoichiometric 1:1 phase. (See Supplementary Table III for the list of calculated lattice parameters.

The thermodynamic stability indeed reflects quite closely the filling of the t2g-C states: in group IV, where all bonding states are full, the depth of the hull is largest; in groups V–VII, as antibonding states are gradually filled, the 1:1 structure becomes first gradually less stable, and finally unstable in Mo, W and Re.

Lattice dynamics and electron-phonon coupling

We computed the phonon and superconducting properties of the TMC series using Density Functional Perturbation Theory70,71 (See the “Methods” section for further details) in the stoichiometric rock-salt phase for all elements. The crystal structures, as well as further computational details are reported in the Supplementary Information.

Figure 3 presents the phonon dispersion along the same high-symmetry directions used for the electronic bands, together with the total and atom-projected phonon DOS F(ω) and the Eliashberg function α2F(ω). The panels for different TMCs are arranged in the same order as Fig. 1; the color code indicates the projection of the eigenmodes on carbon (orange) and transition-metal (teal) vibrations. The dashed curve in the rightmost panels indicates the frequency-dependent electron-phonon coupling constant \(\lambda (\omega )=2\mathop{\int}\nolimits_{0}^{\omega }{\alpha }^{2}F(\Omega )/\Omega \,{\rm{d}}\Omega\).

Fig. 3: Phonon and electron-phonon properties of TMC’s.
Fig. 3: Phonon and electron-phonon properties of TMC’s.
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Phonon dispersions, phonon DOS F(ω) and Eliashberg function α2F(ω) for rock-salt MC (M = Sc-Re). Orange (teal) shading indicates the projection onto carbon (transition metal) modes. The dashed line in the rightmost panel is the cumulative integral λ(ω). Grey panels mark dynamically unstable phases, for which we did not compute the electron-phonon properties.

All spectra exhibit a gap between modes with mainly TM character, below 30 meV, and modes with dominant carbon character, at higher frequencies; the extent of the spectrum depends on the nature of the TM mode, and range from 60 meV in ScC and YC to 85 meV in all other cases. C-based vibrations are softer in TMCs containing group III elements because the incomplete filling of the corresponding bonding bands weakens the TMt2g-C bonds. The dispersions of compounds containing group-IV and group-V elements are quite similar, although a pronounced softening is clearly seen along the W-X path. In TMCs containing groups VI and VII elements entire branches become imaginary: the stoichiometric MoC, WC and ReC structures are dynamically unstable. This is consistent both with their high formation energies and with the calculated electronic structure. In all three compounds, the Fermi level is very close to a van-Hove peak in the Density of States, which generally points to an instability towards a lower-symmetry structure with a lower DOS.

The variation in the relative shape of the phonon density of states - F(ω) - and electron-phonon spectral function - α2F(ω) - reveal marked differences in the nature of the electron-phonon coupling across the series. These differences reflect the same rigid-band trend observed in the electronic structure. In group III TMCs, where electronic states at the Fermi level have essentially C-p character, 75% of the total λ is concentrated in carbon vibrations. In group IV TMCs, where EF lies in a pseudo-gap of the electronic DOS, the electron-phonon coupling is essentially zero; in group V TMCs, where the electronic DOS at EF is dominated by TM \({d}_{{t}_{2g}}\) states, 75% of the total λ is due to TM vibrations.

Overall, the lattice dynamics and the nature of the electron-phonon coupling across the TMC series can be understood in terms of the filling of the metal t2g-carbon bonding/antibonding manifold:

  • in group III, partial filling of bonding states leads to soft phonons and strong coupling via carbon vibrations;

  • in group IV, full bonding-state filling results in stiff lattices and negligible coupling due to the presence of a pseudogap at EF;

  • in group V, progressive filling of antibonding states enhances coupling through metal vibrations;

  • in groups VI and VII, further filling of the antibonding states ultimately drives dynamical instabilities.

Superconductivity

Using the calculated α2F(ω) we computed the superconducting Tc using the McMillan formula72,73 (See Eq. (1)), with a a constant value of μ* of 0.15.

The results are summarized in columns 5–8 of Table 1. The Tc calculated for the 1:1 stoichiometric rock-salt phase is within 30% of the experimental value for only five (Ti, Zr, Nb, Hf, Ta), out of the eleven elements studied, highlighted in green. For the remaining six, the Tc is either largely overestimated (Sc, V, Y), or could not be computed (Mo, W, Re), as the structure is dynamically unstable.

These discrepancies can be understood by analyzing not only the Tc, but considering also the enthalpy relative to the hull (ΔH). In all cases for which the predicted Tc for the stoichiometric rock-salt phase does not agree with experiment, the formation enthalpy is higher than 50 meV/atom, which is usually considered a practical threshold for metastability; moreover, in Mo, W, and Re, the rock-salt structure is even dynamically unstable. In other words, the disagreement does not indicate a limitation of conventional Migdal–Eliashberg theory; rather, it reflects the fact that quantitative predictions of Tc become unreliable when based on an incorrect structural model, instead of the true thermodynamic ground-state.

In fact, our analysis raises a crucial point: if the structure if the measured Tc does not correspond to the stoichiometric phase—and the calculated values confirm it—then what structure is actually measured in the experiments?

In the previous section, we have shown that carbon-vacant rock-salt-like structures lie on or close to the convex hull across the whole TMC series. In particular, in groups III, V, and VII, they are much more stable than the 1:1 phase. The most plausible explanation is that experiments were performed on such carbon-vacant phases, but the resolution of the powder diffraction spectra was not sufficient to resolve the concentration of carbon vacancies. For reference, in Fig. 4b, we compare the simulated X-Ray powder diffraction pattern for stoichiometric and vacant MoC and NbC. The main structure of the peaks is similar in both cases. In Mo, the main peak at 36° splits more evidently, while in Nb the vacancy-induced distortion requires a resolution of the order of 0.01-0.02° to be detected.

Fig. 4: Comparison of the structures of stoichiometric (M1C1) and carbon-vacant (M6C5) TMCs.
Fig. 4: Comparison of the structures of stoichiometric (M1C1) and carbon-vacant (M6C5) TMCs.
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a The TM, C, and vacancies are indicated as large green, small orange, and small black spheres, respectively. b Simulated X-Ray powder diffraction pattern for stoichiometric (black line) and carbon-vacant (red line) MoC and NbC assuming a Cu K-α wavelength of 1.54059 Å. The pattern was convolved with a Gaussian width of 0.05 °.

Carbon-vacant phases

To test this hypothesis, we computed the superconducting Tc for a carbon-vacant rock-salt structure with M6C5 composition (M1C0.83), represented in a reduced monoclinic cell (Structural information is given in the Supplementary Information). A visual comparison of the two structures is shown in Fig. 4. This structure corresponds to the experimentally observed crystal structure of Nb6C574, and was also independently reproduced by our unbiased crystal structure prediction calculations, which systematically placed it on or close to the convex hull across several TMCs. We selected it as a general, representative example of carbon-deficient rock-salt-like phases. The specific choice was motivated by the relatively small and symmetric unit cell, which makes ep calculations affordable, and by the weak vacancy-vacancy interactions expected for this geometry30. Comparing the stoichiometric and carbon-vacant phases, we see that all elements except W and Re undergo an expansion in the octahedral directions around the vacancy, i.e., the M atoms neighboring the vacancy are pushed away from it. In W and Re, the opposite trend occurs. For further details on this aspect, we refer the reader to Supplementary Figs. 2 and 3.

The calculated Tc’s, reported in the 12th column of Table 1, are in significantly closer agreement with experiments for all elements that were not already described by the stoichiometric phase, except for rhenium. The comparison between the stoichiometric and carbon-vacant phase reveals two main effects: (i) in TMCs containing elements of group III and V, carbon vacancies suppress the Tc drastically. In group V, this suppression—combined with the presence of several weakly metastable vacancy-rich phases—naturally explains the broad spread of experimental Tc values; (ii) in group VI compounds, the carbon-vacant structure is dynamically stable, unlike the stoichiometric phase, and the calculated Tc’s – around 18 K for Mo6C5, and 5 for W6C5—are in good agreement with the reported experimental values. In contrast, Re6C5 is unstable, and no amount of vacancies can stabilize it. It is therefore likely that the Tc measured in ref. 24 originated from pure Re, or some other impurity in the sample.

For group V TMC, the experimental dependence on Tc on vacancy concentration has been investigated in detail. For VC, the nominal concentration is usually VC0.88, which is not superconducting. Superconductivity can be induced by carbon implantation14, reaching a maximum Tc of 3.2 K. In NbC, the dependence of Tc on C-vacancy concentration has been studied systematically. The rock-salt phase exhibits a maximum Tc of 11.1 K9,16, but superconductivity is strongly suppressed as the number of vacancies is increased15,17,75, vanishing completely for concentrations below x = 0.77. Tantalum behaves in a similar way, with a maximum Tc of 10.3 K9, and a sharp decrease with carbon vacancies15,17.

These observations naturally raise the question: why do carbon vacancies suppress superconductivity so strongly in group V TMCs?

The Tc of a superconductor can be written in terms of the McMillan formula72

$${T}_{{\rm{c}}}=\frac{{\omega }_{\log }}{1.20}\,\exp \left[-\,\frac{1.04\left(1+\lambda \right)}{\lambda -{\mu }^{* }\left(1+0.62\,\lambda \right)}\right]$$
(1)

Where λ, ωlog, and μ* are the ep coupling constant, the logarithmic average phonon frequency, and the Morel–Anderson pseudopotential, respectively. The ep coupling constant can be approximated by Hopfield’s expression

$$\lambda =\frac{N({E}_{F})\,\langle {g}^{2}\rangle }{M\langle {\omega }^{2}\rangle },$$
(2)

A decrease in the superconducting Tc is most commonly associated to a decrease in λ which, in turn, can originate either from a decrease of N(EF), 〈g2〉, or both.

The suppression of N(EF) is a well-established mechanism, which is often invoked to explain the suppression of Tc due to crystal imperfections, as some of us have recently shown in NbTi76.

However, this mechanism does not apply to TMCs, as N(EF) remains essentially unchanged with vacancy concentration in group III and V TMC, while Tc is strongly suppressed – see columns 5-6 and 11-12 of Table 1 (We refer the reader to Supplementary Fig. 1 for the full DOS’s).

Hence, the suppression of Tc must originate from a reduction of the ep matrix elements. To isolate this effect in Fig. 5, we compare the Eliashberg spectral functions of the stoichiometric and carbon-vacant phases, scaled by their respective N(EF) values. This scaling removes the trivial effect of the DOS in Hopfield’s formula—Eq. (2)—so that any remaining discrepancy reflects a genuine change in the electron-phonon interaction strength and spectral distribution.

Fig. 5: Electron-phonon properties of the stoichiometric and carbon-vacant phases.
Fig. 5: Electron-phonon properties of the stoichiometric and carbon-vacant phases.
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Comparison of the Eliashberg function (α2F(ω) for transition-metal carbides in the rock-salt stoichiometric phase (1:1, black, solid lines) and in the vacant M6C5 phase (6:5, red, dashed lines), rescaled by the electronic DOS at the Fermi energy, across the 3d, 4d, and 5d series.

Figure 5 shows that in the TMCs of group V the suppression of coupling due to the introduction of carbon vacancies is particularly strong, and is accompanied by a sizable renormalization of the spectrum towards higher frequencies.

In this work, we reassessed superconductivity in rock-salt transition-metal carbides by combining first-principles electron-phonon calculations with variable-composition evolutionary structure prediction. Despite decades of study and the potential relevance of these compounds for rugged-environment superconductivity applications, a consistent microscopic description is still lacking. Our electron-phonon calculations show that the commonly assumed 1:1 stoichiometric rock-salt structure fails to reproduce experimental trends in Tc and is often dynamically or thermodynamically unstable. However, theory and experiment can be reconciled by properly accounting for carbon vacancies. In fact, unbiased evolutionary structure prediction identifies a family of low-energy, rock-salt-like nonstoichiometric phases that are both thermodynamically stable and consistent with the observed superconducting properties. The effect of carbon vacancies on Tc is highly nontrivial and can only be captured by structurally accurate models.

Our results definitively establish thermodynamic stability as a key factor for the modeling of real-world superconductors. Most large-scale technological applications rely on low-temperature superconductors synthesized under strongly non-equilibrium conditions, where defects, disorder, and synthesis history play a central role—yet are typically neglected in state-of-the-art computational techniques. By integrating realistic structural thermodynamics into first-principles calculations, our approach sets a practical foundation for predictive modeling of superconductors under realistic synthesis conditions.

Methods

Computational details

Evolutionary crystal structure prediction calculations were performed with the Universal Structure Predictor: Evolutionary Xstallography (USPEX) code 37,38 using variable-composition sampling. The local relaxations were performed in a five-step procedure with progressively tighter constraints using VASP77,78,79. We employed projector-augmented wave pseudopotentials, and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, a plane-wave cut-off gradually raised from the recommended minimum up to 500 eV, Gaussian smearing down to 0.02 eV, and a k-point density down to 0.35 Å−1.

All electronic, vibrational and superconducting properties were computed with QUANTUM ESPRESSO 7.3.170,71, after re-optimizing the structures. Optimized norm-conserving Vanderbilt PBE pseudopotentials80 were adopted with a 100 Ry cut-off on the plane-wave expansion. A Methfessel–Paxton smearing of 0.02 Ry was used for charge-density integration, and a Gaussian smearing of 0.015 Ry for electron-phonon coupling integration.

Harmonic force constants were obtained within density-functional perturbation theory and Fourier-interpolated to obtain phonon DOS and dispersions.

Critical temperatures were obtained from the McMillan–Allen–Dynes formula using the calculated λ and \({\omega }_{\log }\) and a Coulomb pseudopotential μ* = 0.10−0.15. This choice is consistent with experimental determinations in classical superconductors and modern RPA-based ab-initio estimates for transition-metal systems81,82,83,84.

Further details are available in the Supplementary Information.