Introduction

Ising superconductivity in two-dimensional materials is a rapidly growing field of theoretical and experimental research1,2,3,4,5,6,7,8. The combination of broken-inversion symmetry and strong spin-orbit coupling present in single monolayers (MLs) of the two-dimensional transition metal dichalcogenides leads to Fermi surfaces where the spin of the electrons is perpendicular to the plane of the monolayer and the electron spin direction flips between time-reversal invariant points of the Brillouin zone. This has been experimentally confirmed by establishing, for example in NbSe2, that the superconducting critical field is significantly higher in-plane versus out-of-plane, and much larger than the Pauli limit1. While there have been extensive phenomenological descriptions of Ising superconductivity, there are several intriguing material-specific puzzles.

In NbSe2, which is the most widely studied Ising superconductor, the superconducting transition temperature, Tc, decreases from ~6 K to ~3−4 K, when it is reduced from bulk to a single monolayer1. Similar studies conducted on NbS2 provide an intriguing contrast. In 2H-NbS2 Tc is ~6 K, while superconductivity has not been observed in bulk 3R-NbS29,10. These two polytypes differ in the stacking of the individual monolayers, while within each ML Nb atoms are in a trigonal prismatic coordination with the chalcogen atom, similar to NbSe2. Reducing the thickness of NbS2 leads to a strong suppression in Tc11. Superconductivity has not been found in ML NbS2.

It was recently reported that when ML NbSe2 is alloyed with sulfur, S, Tc increases up to a S content of x = 0.412 in ML NbSxSe2−x alloys. For S content greater than ~0.4, Tc was then found to decrease monotonically12 exhibiting qualitatively similar behavior to the bulk alloys. A non-monotonic change in Tc was also found when silicon, Si, was deposited on the surface of the same NbSe2 samples, where it was assumed that Si was adsorbed on the monolayer. Up to a Si coverage of 0.05 Si atoms per NbSe2 formula unit, Tc increased. For larger concentrations of Si, Tc decreased and superconductivity was completely quenched at ~0.17 Si atoms per NbSe2 formula unit. These non-monotonic changes in Tc due to S and Si in NbSe2 were interpreted as disorder-induced enchancement of Tc, which possibly arises from the multifractality of the electronic wave functions13,14. Implicit in this assumption is that the effect of alloying (either with Si or S) on electronic and Coulomb interactions is sufficiently weak so as to not impact Tc directly. While this is an enticing consideration, there are several important questions and experimental puzzles that need to be addressed first, which we briefly outline.

The measurements where fractal superconductivity was observed report a Tc for ML NbSe2 that is ~2 K lower than the widely accepted Tc of ML NbSe2, ~3–4 K1,5. In fact, the peak Tc where fractal superconductivity is observed is ~3 K, which occurs for 0.2 ≤ x ≤ 0.5 due to alloying with S. We also note the experimental in-plane lattice constant is relatively unchanged for 0 ≤ x ≤ 0.2 (Supplementary information). If the Tc of NbSe2 reported by Zhao et al.12 occurred at the more widely accepted 3–4 K, this would not lead to a dome-shaped dependence of Tc on S and Si content, as illustrated in Fig. 1. Instead, Tc would decrease linearly with S content, as has been found when S is alloyed into bulk NbSe215,16.

Fig. 1: Experimental reports of the superconducting transition temperature versus alloy concentration.
figure 1

Data is for bulk and ML NbSxSe2−x alloys as a function of S content, x and Si content, y. The references associated with each marker is as follows: gray open squares16, gray circles15, teal and orange filled downward triangles and diamonds12, red triangles10 and blue filled square1. Open symbols correspond to measurements on bulk samples while filled symbols correspond to measurements on ML samples. The dotted lines correspond to a linear extrapolation of the ML data for x > 0.4. See the main text for the discussion on the extrapolation.

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Taken together, we arrive at three possible mechanisms that can lead to this non-monotonic dependence of Tc on S and Si content. The first is the role of fractal superconductivity, which was invoked in Refs. 12,17. While this exotic phenomenon may lead to a non-monotonic change in Tc18, the success of this model requires information on a plethora of material-dependent parameters that are often not accessible by experiment alone. A second possible mechanism is the role of the charge-density wave (CDW), which has been shown to lead to a pseudogapping of the Fermi surface in ML NbSe219,20, and thus to a reduction in Tc. However, recent studies have suggested the CDW transition temperature varies little when NbSe2 transitions from bulk to a single ML21,22, while Tc exhibits a large change1. This would imply that the coupling between the superconducting and CDW order parameters is weak, as has been found in studies on bulk NbSe223,24.

A third mechanism is the collective role of point defects4 and spin fluctuations6, both of which have been suggested as a source of pair breaking in ML NbSe2. Experimental studies on ML NbSe2 have found the selenium, Se, vacancy concentration can be large (equivalent to a bulk concentration of ~1021 cm−3), depending on the growth conditions25. Selenium vacancies, which are magnetic point defects in NbSe226, can act as a source of pair-breaking and decrease Tc. However, during the growth of NbSxSe2−x alloys, S, which is isovalent to Se, but more electronegative, can occupy the Se vacancies and lower the concentration of pair-breaking defects. This is analagous to the finding that oxygen can substitute for sulfur (both of which are isovalent) in sulfur-deficient ML TaS2, and lead to an increase in Tc compared to ML TaS227. Since Si and Se have approximately similar atomic radii the possibility for Si substitution for Se vacancy sites also exists.

Alloying will also lead to changes in the electronic structure, which may also affect the proximity of the material to magnetism or lead to changes in the density of states (DOS) at the Fermi level, and therefore Tc. There is a priori no means to determine how all of these properties change with alloying. Furthermore, if defects are indeed the source of the lower Tc in NbSe2, this raises questions on the purported relationship between the non-monotonic dependence of Tc and fractal superconductivity12.

In the present work we propose an alternative solution that reconciles these puzzles. Using first-principles density functional theory calculations (Sec. Methods) we show that this non-monotonic dependence of Tc on sulfur and silicon content can emerge from the interplay between defects and the effect of alloying on the electronic structure and spin-fluctuations. We show that S is completely miscible in NbSe2, across the entire alloy composition range. For finite concentrations of S in NbSxSe2−x we find a reduction of the density of states at the Fermi level and a weakening of magnetism, compared to the parent compounds, NbSe2 and NbS2. We also show there is a minimum energy pathway that would result in Si adatoms that are deposited on NbSe2 to be incorporated substitutionally on the Se site or as an interstitial. We conjecture a combination of these effects can lead to a non-monotonic dependence of Tc on S and Si content, without having to invoke the phenomenon of multifractality.

Results

Predictions for sulfur in NbSe2

We start by considering the properties of chalcogen vacancies in NbS2 and NbSe2 in the dilute limit. The formation energies of a S vacancy, VS, in NbS2 and a Se vacancy, VSe, in NbSe2 is listed in Table 1.

Table 1 Formation energy of chalcogen vacancies in NbSe2 and NbS2 under Nb-rich and Nb-poor conditions.
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The results show that the formation energy of VSe is lower than VS, even under Se-rich conditions that were used in the growth of the NbSe2 samples in the study by Zhao et al.12. This suggests that as-grown ML NbSe2 is likely to have a higher concentration of Se vacancies compared to S vacancies in NbS2. We also considered the possibility that S may substitute on the Nb site and calculated the formation energy of this defect, SNb, in ML NbSe2. In the dilute limit we find the formation energy of SNb to be larger than the formation energy of VSe. Hence, for the purposes of alloying beyond the dilute limit we only consider substitution of S on the Se site.

Next we check the stability of NbSxSe2−x alloys with respect to decomposing into their parent compounds, NbSe2 and NbS2. Figure 2 illustrates the lowest enthalpy structure for each composition. We find the T = 0 K formation enthalpy across the entire range of compositions is negative which suggests ordered NbSxSe2−x alloys are stable with respect to decomposition into the parent compounds.

Fig. 2: Formation enthalpy as a function of S content, x, in NbSxSe2−x.
figure 2

Calculations are performed using a (4 × 4 × 1) supercell of the monolayer structure.

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We now turn to the electronic and magnetic properties of the alloys. We first consider the parent compounds, NbSe2 and NbS2. In a single ML the Nb atoms are in a trigonal prismatic coordination with the chalcogen atoms. In ML NbSe2 the trigonal crystal field that acts on the 4d states of Nb4+ leads to one band that crosses the Fermi level, which generates Fermi contours at Γ, K and K\({}^{\prime}\)6. The combination of broken inversion symmetry in the monolayer and strong spin-orbit coupling leads to a spin-orbit splitting of the spin degenerate band along the M-K-Γ line of the Brillouin zone. Since the 4d states of Nb4+ in NbS2 are also in a trigonal prismatic coordination, albeit with a shorter Nb-S bond length compared to the Nb-Se bond length, the qualitative features of the band structure between the two materials are similar (Supplementary information).

ML NbSe2 exhibits strong spin fluctuations, which have been highlighted as a potential source of pair breaking6,28,29,30. First-principles calculations have shown that monolayer NbSe2 can host ferromagnetic spin fluctuations with a sizeable Stoner renormalization, and an antiferromagnetic spin spiral state with q vector (0.2, 0, 0)6,29. In ML NbSe2 we find the spin spiral state to be 1.7 meV/Nb atom lower in energy compared to the non-magnetic state. In NbS2, we find a spin spiral state at a q-vector of (0.2, 0, 0) is also stable (Supplementary information) and is is 1.9 meV/Nb atom lower in energy compared to the non-magnetic ground state. If spin fluctuations are sizeable in the alloy they can impact pairing interactions.

To study the effect of alloying on the spin spiral energies we use virtual crystal approximation (VCA) calculations (Sec. Methods) for S contents that correspond to x = 0.5, 1 and 1.5. Figure 3a illustrates the energy difference between the spin spiral state with respect to the non-magnetic state, ΔEspiral, in NbSxSe2−x When S is alloyed into NbSe2, the spin spiral state is less stable for intermediate values of S content than for either NbSe2 and NbS2. At x = 1 we find ΔEspiral decreases by a factor of 2.1 compared to NbS2 where the magnitude of ΔEspiral is the largest. The magnitude of the magnetic moment on the Nb atom is also suppressed by up to ≃ 25% in the spin-spiral state for the alloys with finite S content compared to the parent compounds, as illustrated in Fig. 3a.

Fig. 3: Electronic and magnetic properties of the NbSxSe2−x alloys.
figure 3

a Energy difference between the spin spiral state and the non magnetic state as a function of sulfur content, x, in NbSxSe2−x (blue filled squares, left vertical axis). Magnetic moment per Nb atom as a function of sulfur content in the spin spiral calculation with finite q (green circles, right vertical axis). b Collinear fixed-spin moment calculations of NbSe2 (gray squares), NbS2 (orange circles), NbS0.5Se1.5 (teal diamonds), NbSSe (red downward triangles), and NbS1.5Se0.5 (blue triangles) illustrate the change in energy per formula unit with respect to the non-magnetic state as a function of magnetic moment per Nb atom. The inset illustrates the coefficient a1 (see main text) normalized by the value of a1 in NbSe2. c Density of states at the Fermi level, EF, as a function of sulfur content, x. The magnitude of the DOS for NbSe2 and NbS2 correspond to the spin spiral state (Supplementary information).

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Next we consider whether ferromagnetic spin fluctuations, which are present in NbSe2, are also impacted due to alloying by using the VCA and collinear fixed-spin moment calculations for ML NbSe2, NbS2, NbSSe, NbS1.5Se0.5, and NbS0.5Se1.5. The quantity of interest is the ferromagnetic spin susceptibility, χ, which is defined as \(\chi ={a}_{1}^{-1}={\left(\frac{{\delta }^{2}E}{\delta {m}^{2}}\right)}^{-1}\) (see Methods). We find that  χ varies non-monotonically as a function of S content as illustrated in the inset of Fig. 3b, where it is large for NbSe2 and NbS2 and suppressed in the case of the alloys. Hence, it is reasonable to assume that the spin fluctuations for intermediate concentrations are supressed non-monotonically for all relevant wave vectors. In both cases these fluctuations are the weakest at roughly equal concentrations of S and Se.

The origin of the reduction in χ (and, probably, also ΔEspiral, given the relatively small spiral vector of (0.2, 0, 0)) can be understood by examining the density of states (DOS). The DOS at the Fermi level, N(EF) as a function of S content is illustrated in Fig. 3c. In NbSe2 and NbS2, N(EF) is suppressed from 2.8 states/eV/Nb atom in the nonmagnetic structure to 2.14 states/eV/Nb atom in NbSe2 and 2.18 states/eV/Nb atom in NbS2 in the spin spiral ground state (Supplementary information). We also find N(EF) is suppressed for the alloys at x = 0.5, 1, and 1.5, where in the non-magnetic state N(EF) is ~2.5 states/eV/Nb atom while in the spin spiral state it reduces to 1.9 states/eV/Nb atom. Hence, for the parent compounds and the alloys, our calculations indicate there is a gain in one-electron energy by transitioning to the spin spiral state. We also find that in the spin spiral state, the magnitude of N(EF) of the alloys decreases by 10% compared to N(EF) of the parent compounds. Such a small change in N(EF) as a function of S content is consistent with the fact that N(EF) is comprised almost entirely of Nb d-states in NbSe2 and NbS2.

Predictions for silicon in NbSe2

We now present our results for the properties of Si in monolayer NbSe2. The formation energies for Si incorporated substitutionally (on the Nb site, SiNb and on the Se site, SiSe), adsorbed at high-symmetry positions on top of the NbSe2 monolayer, Siads (where we consider the hollow site formed by the triangle of Se atoms, Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{hollow}}}}}}}}}\), and vertically above either a Se atom, Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Se}}}}}}}}}\), or Nb atom, Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\)) and Si incorporated interstitially, Sei, is summarized in Table 2.

Table 2 Formation energy of Si in NbSe2 under Nb-rich conditions.
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If Si is deposited on the NbSe2 surface we find that it is likely to initially adsorb on the Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\) site, not the Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Se}}}}}}}}}\) site (which was implicitly assumed to be the most stable configuration for Si in the study by Zhao et al.12). The likelihood of Si remaining adsorbed on the NbSe2 surface is determined in part by the migration barrier of Si adatoms. We calculated the minimum energy pathway for a Si adatom to migrate from the metastable site, Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Se}}}}}}}}}\), to the Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\) adsorption site and find it to be barrier-less as illustrated in Fig. 4a. Subsequent hops between Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\) sites occurs with a low migration barrier of 0.11 eV, which would render Si adatoms to be highly mobile even at low temperatures.

Fig. 4: Adsorption and migration properties of Si in NbSe2.
figure 4

a Theoretical minimum-energy pathway for Si adatoms on NbSe2 to migrate from the Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Se}}}}}}}}}\) metastable adsorption site to Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\). The migration barrier to hop between Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Nb}}}}}}}}}\) sites is 0.11 eV. The energies are reported with respect to the total energy of Si\({}_{{{{{{{{\rm{ads}}}}}}}}}^{{{{{{{{\rm{Se}}}}}}}}}\). Schematic top view and side view of (b) Si substituted on the Se site, SiSe and (c) Si incorporated interstitially within the same plane as the the Nb atoms, Sii.

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The low migration barrier of Si on the surface of NbSe2. makes it unlikely that Si adatoms will exist as isolated defects. Our formation energy calculations suggest that Si adatoms will likely be incorporated as SiSe or as Sii, which are the two lowest energy configurations for Si that is not adsorbed on NbSe2. If there is a large concentration of Se vacancies, as we suggest are present in the study by Zhao et al.12, Si that is deposited on NbSe2 will migrate and substitute for the Se sites when it encounters a VSe. The STM images of Si deposited on NbSe2 (cf. Fig. S9a in ref. 12) provides us a clue that supports this assertion. The STM images show bright spots associated with Si residing at Se sites. Our formation energy calculations in Table 2 show that it is unlikely for Si to adsorbed above Se. Hence, the experiments by Zhao et al.12 unambiguously shows that Si atoms indeed substitute for Se. When Si substitutes for Se as illustrated in Fig. 4b, Si is bonded to three nearest-neighbor Nb atoms with a Si-Nb bond length of 2.615 Å, which is 4.5% shorter than the equilibrium Nb-Se bond length.

For higher concentrations of Si, once all of the Se vacancy sites are occupied we expect Si to incorporate as an interstitial where it sits in the same plane as the Nb atoms. In this configuration, Sii displaces one of the Nb atoms that it is adjacent to and is six-fold coordinated by the Se atoms with a Si-Se bond length of ~2.60 Å, which is illustrated in Fig. 4c.

Discussion

Here we propose a framework to understand how this seemingly diverse set of results on Si and S in NbSe2 can be used to understand the non-monotonic change of Tc in NbSe2 observed by Zhao et al.12, without invoking multifractality. Since the formation energy of Se vacancies in NbSe2 is low, Se vacancies are likely to exist in considerable concentration in as-grown NbSe2, which is consistent with experimental observations. We have previously shown, using first-principles calculations, that VSe in NbSe2 is a magnetic point defect that results in a finite magnetization of ~0.6 μB26. This finite magnetization spans a length scale of up to 15 Å and is likely to have an easy axis along \(\hat{z}\). Hence, in the Ising superconductor NbSe2 they will be pair-breaking26 and suppress Tc, as in a regular s-wave superconductor31. This is consistent with measurements on ML NbSe2, where low values of Tc are found in samples where the residual resistivity ratio is low25. Our calculations show that the substitution of S on the Se site in NbSe2 is energetically favorable for all S compositions. Hence, during the growth of NbSxSe2−x we expect S to occupy the sites of missing Se atoms up to a critical S composition. This would lower the concentration of pair-breaking VSe defects and lead to an increase in Tc. This immediately explains why the putative multifractal behavior was observed in NbSe2 samples with suppressed Tc, compared to the samples in the literature with lower defect concentrations, which is correlated with higher Tc.

We also anticipate the presence of Se vacancies to play a key role when Si is deposited on NbSe212. The low migration barrier that we calculate for Si adatoms implies that the Si adatoms that are initially adsorbed vertically above the Nb site will migrate and occupy VSe or Sii sites. As VSe sites are occupied by Si, the pair-breaking effect of VSe will decrease and we expect Tc to increase as was indeed observed in by Zhao et al.12 until all of the VSe sites are occupied. In the experiments where fractal superconductivity was invoked12 we expect this to occur up to a Si coverage of ~0.05 Si atoms per NbSe2 formula unit, which is where the peak Tc in the NbSe2 samples with Si occurs (Fig. 1). Beyond this concentration, we expect the Si atoms to incorporate interstitially within the NbSe2 lattice as illustrated in Fig. 4c.

While the increase in Tc in the NbSe2 samples with S and Si can be understood by considering the role of VSe the subsequent reduction in Tc at higher concentrations of S and Si can have different origins. One possible consideration is the role of ionized impurity scattering. However, experimentally, there is no indication of ionized impurity scattering in the NbSxSe2−x alloys since Tc decreases monotonically for x ≥ 0.4 (Fig. 1) and does not exhibit any convex variations of Tc with x.

We also find that both NbSe2 and NbS2 host strong spin fluctuations at all wave vectors. First-principles calculations on monolayer NbSe2 suggest that spin fluctuations play a role in suppressing Tc when compared to the Tc obtained by electron-phonon coupling alone (S. Das, H. Paudyal, D. Agterberg, E.R. Margine and I. I. Mazin, Electron-phonon coupling and spin fluctuations in Ising superconductor NbSe2, Unpublished, 2022). However, for finite S compositions in NbSxSe2−x this tendency towards magnetism is weakened, which favors superconductivity. This reduction in the proximity to magnetism competes with a reduction in N(EF) of the alloys compared to NbSe2 and NbS2, which would decrease the electron-phonon coupling constant, λep, and weaken superconductivity. We can estimate the sign of the net effect using the general expression derived in ref. 32, under a simplifying assumption that the spin-fluctuations and phonons have comparable frequencies. Then \(d\log {T}_{c}/d\log {\lambda }_{ep}\propto {\lambda }_{ep}+2{\lambda }_{ep}{\lambda }_{sf},\) and \(-d\log {T}_{c}/d\log {\lambda }_{sf}\propto {\lambda }_{sf}+2{\lambda }_{sf}{\lambda }_{ep}.\) The DOS is reduced by ≈10% between the end composition and the midpoint. In contrast, the tendency to magnetism, as measured by the spin-spiral energy gain, decreases by a factor of 2, between the end composition and the midpoint. The latter is expected to be more important, at least for low concentrations of S. We reiterate, based on our considerations of the pair-breaking effect of selenium vacancies and the changes in the electronic structure due to alloying with S, the presence of VSe that are subsequently occupied by S, plays the primary role in leading to the non-monotonic change in Tc in the NbSxSe(2−x)12. We expect the changes in ΔEspiral, χ and N(EF) to have a secondary effect on Tc.

Silicon incorporation in NbSe2 leads to a more pronounced suppression of Tc compared to S. This suppression is unlikely to be due to ionized impurity scattering as evidenced by the linear change in Tc with Si content. We suggest Si incorporated interstitially leads to doping of NbSe2 that decreases Tc.

Taken together, these properties of S and Si in NbSe2 collectively imply that the non-monotonic dependence of Tc on S and Si content12 is not sufficient proof of fractal superconductivity, but likely has a rather prosaic origin: it is S or Si occupying a high concentration of Se vacancy sites (thus decreasing the concentration of pair-breaking defects). For the case of alloying with S this is also accompanied by a strong reduction of the tendency to magnetism as the S concentration increases from 0 to ~0.5. This effect overlaps with the general weaking of the electron-phonon matrix elements, as evidenced by the smaller coupling constant in NbS2 compared to NbSe2, despite their similar N(EF) (S. Das, H. Paudyal, D. Agterberg, E.R. Margine and I. I. Mazin, Electron-phonon coupling and spin fluctuations in Ising superconductor NbSe2, Unpublished, 2022)33,34. For the case of Si in NbSe2 we expect that when Si is incorporated interstitially once all of the VSe sites are occupied, Sii will act as a source of doping and decrease Tc.

We have discussed two possible mechanisms that can lead to an increase in Tc when S or Si is incorporated in NbSe2; (i) the presence of Se vacancies that are pair-breaking defects and (ii) a reduction in the tendency to magnetism when NbSe2 is alloyed with S. Of these two mechanisms we suggest the presence of Se vacancies is likely playing a larger role. If the concentration of Se vacancies were reduced their pair-breaking effect would be suppressed. This would lead to a higher value of Tc at x = 0 and the non-monotonicity that was observed would be less pronounced. Indeed, if we linearly extrapolate the Tc(x) data for x ≥ 0.4 to x = 0, in the NbSe2 samples alloyed with S we get a Tc of NbSe2 that ranges from 3.35 K to 4.2 K, as illustrated in Fig. 1. We find a similar Tc of ~4 K for NbSe2 if we linearly extrapolate Tc for y ≥ 0.05 for the NbSe2 samples with Si. Furthermore, this is consistent with optimizing growth conditions, which leads to a suppression in the concentration of selenium vacancies, which in turn leads to higher values of Tc in monolayer NbSe235.

Our results, when analyzed in the context of recent studies that have asserted the presence of fractal superconductivity when S and Si are incorporated in NbSe212,17, leads us to conclude that multifractality isn’t the only mechanism that can lead to non-monotonic changes in Tc. The following key factors emerge from our calculations: (1) the low formation energy of Se vacancies that are magnetic pair-breaking point defects, (2) the stability of NbSxSe2−x alloys across the entire composition range with respect to decomposition into the parent compounds, (3) the low migration barrier for Si adatoms on NbSe2 and the low formation energy for Si substitution on the Se site in NbSe2, (4) the reduction in the density of states at the Fermi level as a function of alloy content in NbSxSe2−x and (5) a reduction in the proximity to magnetism in NbSxSe2−x alloys compared to NbSe2 and NbS2.

These results suggest that as-grown NbSe2 hosts a large concentration of pair-breaking Se vacancies that upon alloying are occupied by sulfur or silicon atoms. This leads to an increase in Tc up to a critical composition where the concentration of sulfur or silicon is equal to the concentration of Se vacancies that are present during the growth. For the case of alloying with sulfur, Tc monotonically decreases once sulfur occupies all of the Se vacancy sites, reflecting a general weakening of the electron-phonon matrix elements toward NbS2. For high concentrations of silicon in NbSe2 we find silicon is also likely to incorporate interstitially, where it would act as a dopant and lead to a reduction in Tc. These two distinct regimes manifest in a non-monotonic change in Tc. Based on this scenario we also predict that if an experiment similar to that by Zhao et al.12 is performed on samples with a lower concentration of selenium vacancies and a higher initial Tc, the non-monotonic change in Tc will be less pronounced or even fully suppressed. Given that disorder-induced non-monotonic changes in Tc have been observed in other transition metal dichalcogenide alloys due to isovalent substitution36,37,38, we expect our findings to open new avenues for investigation in this broad class of materials.

Methods

Our calculations are based on density functional theory within the projector-augmented wave method39 as implemented in the VASP code40,41 using the generalized gradient approximation defined by the Perdew-Burke-Ernzerhof (PBE) functional42. We found it is essential that Nb 5s1, 4s2, 4p6, 4d4 electrons and Se 4s2, 4p4 electrons are treated as valence. All calculations use a plane-wave energy cutoff of 400 eV. We use a (18 × 18 × 1) Γ-centered k-point grid for the monolayer structure when performing structural optimization and calculating the electronic structure. The cell shape and atomic positions of each structure was optimized using a force convergence criteria of 5 meV/Å. All of the structures were optimized in the non-magnetic state. We verified that optimizing the structures in the spin spiral state leads to minor differences in the lattice parameters.

For the calculations of chalcogen vacancies we use a (10 × 10 × 1) supercell of ML NbSe2 and NbSe2. To simulate a chalcogen vacancy we remove a single chalcogen atom (S atom in NbS2 and Se atom in NbSe2), relax all of the atomic coordinates and determine the total energy. The formation energy, of for example, a Se vacancy, VSe in NbSe2 is defined as:

$${E}^{f}({V}_{{{{{{{{\rm{Se}}}}}}}}})={E}_{{{{{{{{\rm{tot}}}}}}}}}({V}_{{{{{{{{\rm{Se}}}}}}}}})-{E}_{{{{{{{{\rm{tot}}}}}}}}}({{{{{{{{\rm{NbSe}}}}}}}}}_{2})-{\mu }_{{{{{{{{\rm{Se}}}}}}}}}$$
(1)

where Ef(VSe) is the formation energy of the Se vacancy, Etot(VSe) is the total energy of the NbSe2 defect supercell with a Se vacancy, Etot(NbSe2) is the total energy of the pristine NbSe2 supercell, and μSe is the chemical potential of Se. For the calculation of Si in NbSe2 we use an orthorhombic supercell with 64 NbSe2 formula units. We use the formation enthalpy of SiSe2 as the limiting phase for our formation energy calculations of Si in NbSe2. All of the defect calculations were performed with a (3 × 3 × 1) k-point grid. The theoretical minimum energy pathway for adatom migration on NbSe2 was calculated using the using the nudged elastic-band method43.

For the calculations of the alloy properties with sulfur we consider two approaches; the virtual crystal approximation (VCA) and explicit supercell calculations using either a (4 × 1 × 1) and a (4 × 4 × 1) supercell that is constructed from the unit cell of the ML structure. For each alloy supercell we consider different arrangements of the S and Se atoms for compositions corresponding to x = 0.125, 0.25, 0.5, 1, and 1.5, and relax all of the atomic positions. The k-point grid for structural relaxation of each supercell is scaled with respect to the (18 × 18 × 1) Γ-centered k-point grid we use for calculations of the unit cell.

To determine the thermodynamics of alloy formation we calculated the formation enthalpy, ΔH(x), as a function of sulfur content, x, using the (4 × 4 × 1) supercell. ΔH(x) is defined as:

$${{\Delta }}H(x)=E(x)-xE({{{{{{{{\rm{NbS}}}}}}}}}_{2})-(1-{{{{{{{\rm{x}}}}}}}}){{{{{{{\rm{E}}}}}}}}({{{{{{{{\rm{NbSe}}}}}}}}}_{2})$$
(2)

where E(x) is the total energy of the alloy supercell with sulfur content, x, E(NbS2) is the total energy of the NbS2 supercell and E(NbSe2) is the total energy of the NbSe2 supercell.

We varied the lattice parameters for each alloy configuration linearly as a function of sulfur content in accordance with Vegard’s law and then relax all of the atomic coordinates. For a given sulfur content, x, the in-plane lattice constant, a(NbSxSe2−x) was varied as a(NbSxSe2−x) = \(x{a}_{{{{{{{{{\rm{NbS}}}}}}}}}_{{{{{{{{\rm{2}}}}}}}}}}\) + \((2-x){a}_{{{{{{{{{\rm{NbSe}}}}}}}}}_{{{{{{{{\rm{2}}}}}}}}}}\), where \({a}_{{{{{{{{{\rm{NbS}}}}}}}}}_{{{{{{{{\rm{2}}}}}}}}}}\) is the in-plane lattice parameter of bulk NbS2 and \({a}_{{{{{{{{{\rm{NbSe}}}}}}}}}_{{{{{{{{\rm{2}}}}}}}}}}\) is the in-plane lattice parameter of bulk NbSe2. We verified the accuracy of Vegard’s law for a subset of alloy structures by allowing the lattice parameters and atomic positions to relax. In all cases, the variation of the in-plane lattice parameters was linear (Supplementary information).

To calculate the spin spiral energies we used the generalized Bloch theorem formalism44 as implemented within VASP. We use a dense (36 × 36 × 1) Γ-centered k-point grid for the unit cell. We determine the energy difference between the spin spiral state with respect to the non-magnetic state, ΔEspiral, which is defined as ΔEspiral = E(q) − E(q = 0) where E(q) is the total energy of the unit cell with spin spiral wavevector q and E(q = 0) is the total energy of the non-magnetic unitcell.

To determine the ferromagnetic spin susceptibility, χ we used collinear fixed-spin moment (FSM) calculations (sometimes referred to as the constrained local moments approach). In our collinear FSM calculations we constrain the magnitude of the magnetic moment on the Nb atom. Performing these calculations allows us to determine the change in energy with respect to the non-magnetic ground state as a function of the total magnetization, m. We then fit our results to an expansion of the total energy as a function of m to the following expression, E(m) = a0 + a1m2 + a2m4 + a3m6 + a4m8, where E(m) is the total energy for a given magnetization m, to determine the ferromagnetic spin susceptibility, χ. The spin susceptibility, χ, obtained from FSM calculations is sensitive to the choice in energy convergence threshold, and the number of magnetization values used in the fit to expansion in the total energy as a function of magnetic moment. We use an energy convergence threshold of 10−8 eV, and up to 50 magnetization versus energy points between 0  μB and 0.6 μB for all of the FSM calculations.

The results on the formation enthalpy of the alloys are obtained using a (4 × 4 × 1) supercell with tetrahedron smearing and a (9 × 9 × 1) k-point grid. The spin spiral energies, fixed spin moment calculations, and the density of states of the alloys are obtained using VCA calculations for sulfur contents that correspond to x = 0.5, 1, and 1.5. The VCA calculations use the same k-point grid as the unit cell calculations. The in-plane lattice parameters for the x = 0.5, 1, and 1.5 VCA calculations are scaled linearly according to Vegard’s law. Furthermore, we also interpolate the vertical Nb-chalcogen bond length along the c-axis for each VCA alloy calculation.