Introduction

In conventional superconductors, electron pairing arises from virtual phonon exchange1. In this case, the slowly moving ions mediate an effective attractive interaction that allows the electrons to overcome their mutual repulsion and form Cooper pairs. This mechanism favours s-wave pairing, where the total spin of the electron pair is zero, and the gap function is isotropic in momentum space. Kohn and Luttinger have proposed an alternative pairing mechanism originating from pure Coulomb repulsion2. It is mediated by electron-hole fluctuations of the metal, which result in long-range Friedel oscillations3 of the screened Coulomb interaction. For conventional superconductors, however, the effect is subleading compared to phonon-induced pairing2. When superconductivity emerges from screened repulsion, the free energy is minimised by an anisotropic gap function4, and the pairing symmetry is associated to irreducible representations of the crystal. If the symmetry of the Bravais lattice allows a two-dimensional representation, Coulomb repulsion can even mediate a superconducting chiral phase, with spontaneous time-reversal symmetry breaking5,6. Two-dimensional (2D) systems with a trigonal lattice are potential candidates for becoming chiral superconductors7,8,9,10. Evidence for chiral superconductivity has recently been reported for tin monolayers on a Si(111) surface11, rhombohedral multilayer graphene12, and a transition from a nematic to a chiral phase has been proposed for TaS213,14,15.

Pairing from repulsive interaction has been considered for a wide range of different material classes, including cuprates16 and iron-based superconductors17,18,19, heavy fermion superconductors4, and Sr2Ru0420. More recently, these ideas have been extended to few-layer materials with flat bands12,21 and superconducting transition metal dichalcogenide (TMD) monolayers22,23,24,25,26. The latter have attracted a lot of interest both for their potential application in novel two-dimensional electronics27, but also their rich phenomenology, which combines strong Ising spin-orbit coupling (SOC)28,29,30, superconductivity31,32,33, violations of the Pauli limit in strong in-plane magnetic fields22,34,35,36,37,38, and an incommensurate charge density wave35,39,40. The expected enhanced importance of unconventional pairing in 2D TMDs arises from their fragmented Fermi surface, cf. Fig. 1, which implies distinct intervalley and intravalley scattering processes, and whose amplitude can be large due to the weaker screening in lower dimensions41,42. Indeed, model calculations have shown that an effective attraction from competing repulsive interactions requires dominance of the short-range intervalley scattering over the long-range intravalley one22,24,43.

Fig. 1: Band structure, Fermi surfaces and orbital contributions in monolayer NbSe2.
Fig. 1: Band structure, Fermi surfaces and orbital contributions in monolayer NbSe2.
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a Tight-binding Bloch bands and Fermi surfaces (inset). The Fermi energy lies within the spin-split bands crossing the Fermi level; the spin degeneracy is removed by the Ising spin-orbit coupling, resulting in spin-resolved Fermi surfaces. b Sketch of the three atomic dl,m orbitals used in the calculation and their contribution to the composition of the bands crossing the Fermi level. The quantum numbers l, m denote the total angular momentum of the orbital and its azimuthal projection.

In this work, we microscopically investigate the possibility of unconventional superconductivity in monolayer 1H-NbSe2, a TMD whose low-energy physics is dominated by its Nb atoms. Starting from the full unscreened Coulomb interaction, we show that a generalised random phase approximation (RPA) on the trigonal lattice of Nb atoms yields a selectively screened interaction that favours short-range intervalley scattering of Cooper pairs. While bulk NbSe2 is commonly assumed to be a conventional phonon-mediated superconductor44,45, the mechanism for the monolayer limit is still actively discussed and ranges from phonon-mediated46,47 to more exotic pairings22,24,26,48. Our findings, based on a microscopic calculation of the screened interaction, reinforce the possibility of unconventional superconductivity in the monolayer case. The multi-orbital character of the band crossing the Fermi level and the umklapp processes play an important role in this context. While in homogenous systems the period of Friedel oscillations in the interaction potential is provided by the Fermi wavelength2,49, here also faster oscillations with alternating sign on neighbouring lattice sites are present. Upon solving the coupled linearised gap equations for the full screened potential, we find near Tc two degenerate p-like solutions in the \({E}^{{\prime} }\) irreducible representation of the lattice’s symmetry group D3h. By solving self-consistency equations, we predict the gapped p + ip-like chiral solution to provide the stable phase down to the lowest temperatures. The spectral function we find for this chiral gap is in very good agreement with signatures in low-temperature spectroscopy measurements.

Results

The starting point for our considerations is a tight-binding model for the Bloch band structure of monolayer NbSe2 obtained by using the three most relevant orbitals of Nb, \({d}_{{z}^{2}}={d}_{2,0}\) and \({d}_{2,\pm 2}={d}_{{x}^{2}-{y}^{2}}\pm i{d}_{xy}\)50, cf. Fig. 1, fitted to ab initio calculations. Here, the parametrization by He et al.51 is used, but similar results are obtained from the one reported by Kim & Son52. There are two metallic bands relevant for superconducting pairing at the Fermi level, with distinct spin, and which are split by Ising SOC50. As seen in the inset of Fig. 1a, multiple disjoint Fermi surfaces are present, which suggests multi-patch superconductivity similar to the one arising in the cuprates53 and iron pnictides16,18. While the planar d-orbitals mostly contribute to the band composition around the K and K’ valleys, d2,0 is dominant at the Γ valley, see Fig. 1b.

In crystalline systems, the screened interaction is in general a function of both positions of the two interacting charges. It can be expanded as41

$$V({{\bf{r}}},{{{\bf{r}}}}^{{\prime} })=\frac{1}{{{\mathcal{V}}}}{\sum} _{{{{\bf{q}}}}^{{\prime} },{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}{V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}({{{\bf{q}}}}^{{\prime} }){e}^{i({{{\bf{q}}}}^{{\prime} }+{{\bf{G}}})\cdot {{\boldsymbol{r}}}}{e}^{-i({{{\bf{q}}}}^{{\prime} }+{{{\bf{G}}}}^{{\prime} })\cdot {{{\boldsymbol{r}}}}^{{\prime} }},$$
(1)

where G, \({{{\bf{G}}}}^{{\prime} }\) are reciprocal lattice vectors, \({{{\bf{q}}}}^{{\prime} }\) is a crystal momentum not confined to lie in the plane, and \({{\mathcal{V}}}\) is the sample volume. An exact evaluation of the screened interaction is not possible. However, a closed form can be obtained in the random phase approximation (RPA), in which the polarisability is approximated by that of independent Bloch electrons41. With a focus on NbSe2, we view the orbitals as tightly confined to the plane of the material, cf. Fig. 1. This enables us to obtain a two-dimensional version of the RPA-screened interaction, which properly accounts for the long-range behaviour. The resulting two-dimensional screened interaction \({V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}({{\bf{q}}};\omega )\) (see Supplementary Eq. (5)) can be expressed in terms of the 2D unscreened one, \({V}_{0}^{2{{\rm{D}}}}({{\bf{q}}})=\frac{{e}^{2}}{2{\epsilon }_{0}q}\), as

$${V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}({{\bf{q}}};\omega )={\epsilon }_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{-1}({{\bf{q}}};\omega ){V}_{0}^{2{{\rm{D}}}}({{\bf{q}}}+{{{\bf{G}}}}^{{\prime} }),$$
(2)

with \({\epsilon }_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{-1}\) the inverse of the dielectric tensor

$${\epsilon }_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}({{\bf{q}}};\omega )={\delta }_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}-{V}_{0}^{2{{\rm{D}}}}({{\bf{q}}}+{{\bf{G}}}){P}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}({{\bf{q}}};\omega ).$$
(3)

The elements of the polarisability tensor in Eq. (3) are given by

$${P}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}({{\bf{q}}};\omega )={\sum} _{{{\bf{p}}},\sigma }{\chi }_{\sigma }({{\bf{p}}}+{{\bf{q}}},{{\bf{p}}};\omega ){{{\mathcal{F}}}}_{{{\bf{p}}},{{\bf{p}}}+{{\bf{q}}}}^{\sigma }({{\bf{G}}}){{{\mathcal{F}}}}_{{{\bf{p}}},{{\bf{p}}}+{{\bf{q}}}}^{\sigma*}({{{\bf{G}}}}^{{\prime} }),$$
(4)

with χσ accounting for electron-hole fluctuations, cf. Eq. (9), and the Bloch overlaps \({{\mathcal{F}}}\) defined as

$${{{\mathcal{F}}}}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} }}^{\sigma }({{\bf{G}}})={\int}_{{{{\mathcal{V}}}}_{p}}d{{\bf{r}}}{e}^{-i{{\bf{G}}}\cdot {{\bf{r}}}}{{{\bf{u}}}}_{\sigma,{{\bf{k}}}}^{{\dagger} }({{\bf{r}}}){{{\bf{u}}}}_{\sigma,{{{\bf{k}}}}^{{\prime} }}({{\bf{r}}}).$$
(5)

Here, uσ,k are Bloch spinors, i.e., the periodic part of the Bloch functions holding the information about their orbital composition, and \({{{\mathcal{V}}}}_{p}\) is the volume of the unit cell. We show in Fig. 2a the macroscopic static polarisability, the element P0,0(q; 0) of the polarisability tensor, properly accounting for the Bloch overlaps. In Fig. 2d the same quantity is shown in the constant-matrix-element (CME) approximation39,52, where such overlaps are set to unity. In both cases, a plateau is observed for small momentum transfers, q ≈ qΓ. However, while the magnitude ∣P0,0∣ of the full macroscopic polarisability is maximal there, it is minimal for the approximated one. As such, retaining the directionality of the Bloch overlaps in momentum space (see Supplementary Fig. 5) is crucial to correctly account for the long-range behaviour of the screened interaction. This qualitative difference is visualised in Fig. 2b, e, where we display the effective statically screened interaction potential \({V}^{{{\rm{RPA}}}}({{\bf{q}}}):={\sum }_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}{V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}({{\bf{q}}};0)\) calculated with and without the CME approximation for the polarisability, respectively (see Methods). The effective RPA interaction potential shown in Fig. 2b is strongly suppressed for q ≈ qΓ, and displays maxima at  ± qK. The strong suppression for low momentum exchanges q supports the scenario of dominant intervalley scattering discussed by Hörhold et al., Roldán et al.24,43. In CME approximation, the effective interaction potential is, in contrast, largest at the qΓ plateau, see Fig. 2e, while being in general smaller throughout the whole Brillouin zone. Hence, the impact of screening for large momentum electron-hole fluctuations is overestimated in the CME approximation. This is in line with earlier works that discuss the limits of the validity of this approximation for the calculation of susceptibilities in multi-orbital systems54,55. When turning to the real space, we find that \({V}^{{{\rm{RPA}}}}({{\bf{r}}},{{{\bf{r}}}}^{{\prime} })\), calculated according to Eq. (1), displays sizable Friedel oscillations. The interference pattern originates from the superposition of contributions from qK and  − qK regions and is a signature of the trigonal lattice. As seen in Fig. 2c, the potential as a function of r, with \({{{\bf{r}}}}^{{\prime} }={{\bf{R}}}\) fixed to a Nb site, exhibits an on-site repulsion at the origin and repulsive regions around the next-nearest-neighbour lattice sites, while it is strongly attractive around the nearest-neighbour sites. Figure 2f shows the real-space interaction potential in CME approximation. The screened potential there remains mostly repulsive due to weaker Friedel oscillations and displays only narrow regions of weak attraction.

Fig. 2: Macroscopic static polarisability and screened interaction in the reciprocal and in real space.
Fig. 2: Macroscopic static polarisability and screened interaction in the reciprocal and in real space.
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a The macroscopic static polarisability P0,0 displays a plateau for small momentum transfers q ≈ 0 (0 = : qΓ), where its magnitude ∣P0,0∣ attains its maximal value, and decreases for large momentum transfers q ≈ ± K (K = : qK). This behaviour reflects the strong momentum dependence of the Bloch overlaps. b The effective statically screened interaction is correspondingly strongly suppressed for small momentum transfers and exhibits maxima around  ± qK. c In real space the screened potential displays Friedel oscillations which bear signatures of the underlying trigonal lattice. The potential is strongly repulsive on-site, but attractive at the nearest-neighbour sites. d–f Same observables as in the top panels evaluated in constant-matrix-element approximation, i.e., setting the Bloch overlaps to 1. In this case, the screening at large momentum transfers is overestimated, and hence the screened interaction is largest around qΓ. In turn, the oscillations in real space are weak and the potential remains repulsive everywhere except in small regions of weak attraction.

The short-range part of the interaction in Fig. 2c is peaked at the origin and adjacent lattice sites. Interactions of this type are commonly approximated by the ‘top-down’ Hubbard-Kanamori model, which has previously been applied to discuss unconventional pairing in multi-orbital superconductors18,19,55. The Hubbard-Kanamori model has also been used for monolayer TMDs6,7,26,51,56,57,58,59, and predicts a parameter-dependent crossover of the symmetry of the induced superconducting order7,26. A realistic ‘bottom-up’ calculation, either of the Hubbard-Kanamori model parameters60 or the spatially resolved screened interaction as done here, has the advantage of enabling a determination of the interaction strength without free model parameters. These findings thus constitute a major result of the present work. As evident from Fig. 2c, and consistent with earlier works60, the weaker screening in two dimensions results in a significant off-site and long-range contribution to the interaction in 1H-NbSe2. Deriving the screened interaction first in momentum space and then obtaining its real-space representation provides a benchmark for the validity of such real-space models.

We next discuss the properties of the superconducting instability mediated by the screened potential. In general, condensates may be formed with finite center-of-mass momentum for Cooper pairs. However, the matrix elements of the screened interaction contributing to pairing are complex, with a strongly fluctuating phase resulting from the product of Bloch overlaps, cf. Eq. (10). In absence of external magnetic fields, this tends to favour pairing between time-reversed states, whose Bloch spinors obey the relation \({{{\bf{u}}}}_{\bar{\sigma },\bar{{{\bf{k}}}}}={{{\bf{u}}}}_{\sigma,{{\bf{k}}}}^{*}\), with \(-{{\bf{k}}}=:\bar{{{\bf{k}}}},-\sigma=:\bar{\sigma }\), resulting in a sum of real and positive contributions. We hence retain only the scattering between time-reversal partners (\({{\bf{k}}},\sigma ;\bar{{{\bf{k}}}},\bar{\sigma }\)), and consider in the following the pairing between opposite spin electrons mediated by interaction matrix elements \({V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }:=\left\langle {{{\bf{k}}}}^{{\prime} },\sigma ;{\bar{{{\bf{k}}}}}^{{\prime} },\bar{\sigma }\right\vert \hat{V}\left\vert {{\bf{k}}},\sigma ;\bar{{{\bf{k}}}},\bar{\sigma }\right\rangle\) involving Kramers pairs of Bloch states. Due to Ising-SOC, the spin singlet and z-triplet can mix if they lie in the same irreducible representation6. This invalidates the typical expansion in spherical harmonics (s/p/d/f-wave), which is why we refer in the following to s/p/d/f-like gaps. A finite in-plane magnetic field would necessitate the inclusion of pairing in the x and y-triplet channels37,56. Restricting to time-reversal pairs yields a low-energy Hamiltonian of the BCS form1. Performing a mean-field approximation leads to the familiar BCS gap equation

$${\Delta }_{{{\bf{k}}},\sigma }=-{\sum} _{{{{\bf{k}}}}^{{\prime} }}{V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }\Pi ({E}_{{{{\bf{k}}}}^{{\prime} },\sigma }){\Delta }_{{{{\bf{k}}}}^{{\prime} },\sigma },$$
(6)

where the pairing function \(\Pi (E)=\tanh (\beta E/2)/2E\) depends on the gap through the quasiparticle energies \({E}_{{{\bf{k}}},\sigma }=\sqrt{{\xi }_{{{\bf{k}}},\sigma }^{2}+| {\Delta }_{{{\bf{k}}},\sigma }{| }^{2}}\), and on the inverse temperature β = 1/kBT. To access observables near the phase transition, we linearise Eq. (6) by replacing the excitation energies Ek,σ in the pairing function by the single-particle energies ξk,σ and solving the resulting linear problem within a range of  ± Λ around the Fermi energy22,26, cf. Methods.

The leading instabilities with the highest Tc are shown in Fig. 3 for the parameter set by He et al.51 and a truncation to third nearest neighbours of the first Brillouin zone in the calculation of the matrix \({V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}\). Due to the symmetry of the gap function under σh, the solutions can only belong to the irreducible representations \({A}_{1}^{{\prime} },{A}_{2}^{{\prime} }\), and \({E}^{{\prime} }\) of the D3h symmetry group of freestanding 1H-NbSe2. We find as the dominant pairings two degenerate nematic p-like solutions of the \({E}^{{\prime} }\,z\)-triplet type (the singlet admixture is small), in line with the results obtained by Roy et al.26 for their model interaction. These solutions are followed by the SOC-introduced combinations of the \({A}_{1}^{{\prime} }\,s\)-like singlet and f-like z-triplet that were previously discussed as candidates for the superconducting phase of monolayer NbSe222,23,24,26,33, and which are the leading instabilities in multiple previous works that did not account for pairing in the \({E}^{{\prime} }\) irrep22,56,61. The critical temperatures of these solutions depend on the associated eigenvalues λ of the pairing matrix and the cutoff in energy. Note that a range of eigenvalues λ is reported in the literature for corresponding solutions of model Hamiltonians22,23,26,33. This is in line with these works using the interaction strength (and the cutoff Λ) as a fit parameter to match the critical temperature observed in experiments, e.g., Tc ≈ 2 K in epitaxially grown samples33 and in mechanically exfoliated ones31. Solving the linearised gap equation for a cutoff Λ = 100 meV, we find that the pairing strength required to yield Tc = 2 K is λ ≈ − 0.154. The difference between our result and this value is well within the uncertainty of our calculation due to the differences in the reported band structures for monolayer NbSe226,51,52, and to the charge transfer from the substrate which was not included in our calculation (see the discussion in the Methods and Supplementary Section III B). In order to account for these uncertainties, we keep Λ = 100 meV and instead choose to rescale the resulting interaction \(\hat{V}\to \gamma \hat{V}\), with γ ≈ 2.4 to fix Tc = 2 K. Furthermore, an additional phonon-mediated interaction1,47, e.g., by considering a momentum-independent interaction ∣Vph∣ ≲ VRPA(qΓ) would be sufficient to explain this discrepancy, even with unscaled \(\hat{V}\). The two leading solutions of the pristine system are degenerate nematic gaps, which remain quasi-degenerate for weak enough breaking of the symmetry of the pristine system, e.g., by a substrate. This allows for the emergence of a chiral gap at temperatures below Tc5.

Fig. 3: Leading instabilities as found from solving the linearised gap equations.
Fig. 3: Leading instabilities as found from solving the linearised gap equations.
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The solutions are labeled by their approximate symmetry, the irreducible representation they belong to and whether they are majority spin singlet (s) or z-triplet (z). Lower eigenvalue λ indicates larger Tc of the corresponding instability. The two solutions with highest critical temperature are the two degenerate \({E}^{{\prime} }\,z\)-triplets of the D3h group. The two following solutions are combinations of the \({A}_{1}^{{\prime} }\,z\)-triplet and singlet.

The extent to which a nodal nematic solution or the gapped chiral phase is the preferred ground state in our model cannot be determined at the level of the linearised gap equation used so far. Instead, we solve the self-consistency equation, Eq. (6), as a function of temperature. Retaining the full momentum dependence in the gap equation renders the problem highly multidimensional. However, the relevant physics is captured by expanding both the interaction and the order parameter in basis functions of the appropriate irreducible representations of the crystal symmetry group22,23,62, and solving a set of coupled equations for the expansion coefficients as a function of the temperature (see Methods). By benchmarking the pairing strength λ of the expanded solutions against the linearized calculation, this approach offers a controlled reduction of the computational cost of the calculation without significant sacrifices to the allowed momentum dependence of the order parameter. We tracked the evolution of the coefficients down to zero temperature (see Supplementary Section III A). The square root of the average square gaps (see Methods for the definition) is shown in Fig. 4a. The chiral nature of the stable solution is measured by Rch and remains close to unity until very close to Tc, where any small splitting between px and py (induced here by the projection on the basis functions) results in a chiral to nematic transition. The magnitude and phase of this chiral gap at T = 340 mK are displayed in Fig. 4b. The free energy shown in Fig. 4c for the different candidate phases confirms that the chiral gap is physically stable within the considered temperature range (see Methods). Hence, similarly to what was possibly observed in 4Hb-TaS214, our model predicts a chiral ground state also for monolayer NbSe2.

Fig. 4: Chiral ground state and comparison with spectroscopic measurements.
Fig. 4: Chiral ground state and comparison with spectroscopic measurements.
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a Square root of the average square amplitude of the first three gaps solving the self-consistent gap equation as a function of temperature. The thick curve is the solution with the lowest free energy; its colour encodes the ratio Rch of the contribution from a purely chiral mixture px + ipy of the two nematic solutions. The stars indicate the position at which the fits to the experimental curves in panel e were performed. b At T = 340 mK the gap equation predicts a fully gapped chiral phase of p + ip type. The phase of the order parameter winds clockwise around each contiguous Fermi surface. c Free energy per electronic state for the first three gap solutions, calculated in the range of energies  ± 2 meV. The absence of nodes makes the chiral solution the favoured solution at low T. d Differential conductance spectra, with curves offset for clarity, recorded by STM at 340 mK, at different positions along a NbSe2 monolayer grown on a bilayer graphene/SiC(0001) substrate. A clear hard gap is observed, which can be fitted using the chiral gap in (a). e One of the experimental dI/dV curves in (d), fitted with the gaps Δkσ obtained from the nematic and chiral solutions of the self-consistent gap equation (see the marker in a). A very good agreement is found for the chiral solution on the Γ Fermi surface; the nematic ones display a clear V-shape gap that departs from the experimental data. f Experimental traces for different temperatures and corresponding theoretical fits with the leading Δch shown in (b, c).

A natural question is thus if this finding is compatible with available spectroscopic data. To this end, we turn to our own low-temperature scanning tunnelling spectroscopy (STS) measurements carried out on monolayer NbSe2 on a bilayer graphene/SiC(0001) substrate (see ref. 33 for details of the setup). The measured observable is the tunnelling differential conductance (dI/dV), which is proportional to the spectral function of the sample (see Methods and Eq. (29)). The experimental dI/dV spectra recorded at different locations at 340 mK are shown in Fig. 4d and fitted to the tunnelling density of states (DOS) obtained from the chiral gap. As shown in Fig. 1, in NbSe2 the quasiparticle states near the K and \({K}^{{\prime} }\) valleys arise from the planar d2,±2 orbitals, while the ones at the Γ valley have d2,0 character and extend farther beyond the NbSe2 plane. This, combined with the sensitivity of STM to states with small in-plane momentum32,63, suggests a larger coupling to the states on the Γ Fermi surface compared to those near the K and \({K}^{{\prime} }\) valleys. The theoretical differential conductance shown in Fig. 4d is calculated using the coupling to the Γ and K Fermi pockets as free parameters (see Methods). Figure 4e shows the fit to a representative experimental dI/dV spectrum recorded at 340 mK, using both the chiral and the nematic solutions on the Γ Fermi pocket. The presence of nodal lines in the nematic phases is reflected in a V-like shape of the gap, which was not observed in the experiment. Such a shape was instead recently reported for spectroscopy experiments on monolayer 1H-TaS264. Finally, the temperature evolution of the differential conductance at one fixed tip location is shown in Fig. 4f. The agreement with the theoretical curves which use the temperature dependence of the gap as displayed in Fig. 4a is very good.

Discussion

While one can confidently state that the STS data are consistent with the theory presented in this work, and, in particular, with a chiral p-like \({E}^{{\prime} }\) gap at the lowest temperatures, it is clear that they alone are not conclusive evidence for this pairing symmetry. Additional insight is provided by nematic features observed in resistance experiments38,61 under application of in-plane magnetic fields. Their presence can be explained with a transition to a nematic order parameter38,65. Our theory intrinsically undergoes such a transition to nematicity close to Tc if one of the in-plane symmetries of D3h is broken, which can be realised, e.g., by the application of strain65. A fully gapped spectral function and even nematic behaviour for in-plane magnetic field are, however, also compatible with a dominant conventional s-wave pairing33,61. With both the Kohn-Luttinger mechanism discussed in this work and phonon-mediated pairing47 potentially active simultaneously, the effect of their coexistence on the resulting pairing remains yet to be explored. To gather further evidence on the characterisation of the dominant mechanism, we propose testing for signatures that more strongly differentiate between the chiral p-like ground state of our theory and the s-like pairing expected from phonon-mediated pairing. Such signatures can be revealed in quasiparticle interference patterns66, measurements of the gap at the K Fermi pockets using dynamic-STS measurements, or in the enhancement of the zero-bias conductance near the edges of superconducting domains11.

Methods

We consider the low-energy Hamiltonian for the metallic bands of monolayer NbSe2

$$\hat{H}={\sum} _{{{\bf{k}}},\sigma }{\xi }_{{{\bf{k}}},\sigma }{\hat{c}}_{{{\bf{k}}},\sigma }^{{\dagger} }{\hat{c}}_{{{\bf{k}}},\sigma }+\hat{V},$$
(7)

where the first term describes independent Bloch electrons in the bands crossing the Fermi level with energies ξk,σ, and \({\hat{c}}_{{{\bf{k}}},\sigma }^{({\dagger} )}\) are the creation and annihilation operators of Bloch electrons with crystal momentum k. The second term is the statically screened Coulomb interaction which, in the position representation (1 ≡ r, σ), is defined by a Dyson equation for its matrix elements \(V(1,2)=\left\langle (1,2)\right\vert \hat{V}\left\vert (1,2)\right\rangle\) as

$$V(1,2)={V}_{0}(1,2)+\int\,d(3,4){V}_{0}(1,3)P(3,4)V(4,2),$$
(8)

with V0(1, 2) the matrix elements of the unscreened interaction and P the polarisability of the interacting system. We consider a trigonal 2D lattice with periodic boundary conditions composed of N prismatic unit cells of area Ω and height L with sample volume \({{\mathcal{V}}}=N\Omega L\). The derivation of Eq. (2) from Eq. (8) can be found in the Supplementary Material.

We treat the screening at the RPA level, where the full polarisability in Eq. (8) is replaced with the non-interacting polarisability tensor given by Eq. (4) where

$${\chi }_{\sigma }({{\bf{p}}}+{{\bf{q}}},{{\bf{p}}};\omega )=\frac{1}{N\Omega }\frac{f({\xi }_{{{\bf{p}}}+{{\bf{q}}},\sigma })-f({\xi }_{{{\bf{p}}},\sigma })}{\omega+{\xi }_{{{\bf{p}}}+{{\bf{q}}},\sigma }-{\xi }_{{{\bf{p}}},\sigma }+i\eta },$$
(9)

f is the Fermi function evaluated from the chemical potential μ and the overlaps \({{\mathcal{F}}}\) are defined in Eq. (5). For momenta \({{\bf{k}}}/{{{\bf{k}}}}^{{\prime} }\) outside the first Brillouin zone, Eq. (5) is modified by decomposing \({{\bf{k}}}={{\mathcal{P}}}({{\bf{k}}})+{{\mathcal{Q}}}({{\bf{k}}})\) (see Supplementary Fig. 3), where \({{\mathcal{P}}}\) projects onto the reciprocal lattice, \({{\mathcal{Q}}}\) projects onto the associated crystal momentum and using \({{{\bf{u}}}}_{{{\bf{k}}},\sigma }({{\bf{r}}})={e}^{i{{\mathcal{P}}}({{\bf{k}}})\cdot r}{{{\bf{u}}}}_{{{\mathcal{Q}}}({{\bf{k}}}),\sigma }({{\bf{r}}})\). We express the interaction \(\hat{V}\) entering Eq. (7) in the Bloch basis. It holds for the statically screened Coulomb interaction between Bloch states (see the Supplementary Material for the full derivation),

$$ \left\langle {{\mathcal{Q}}}({{\bf{k}}}+{{\bf{q}}}),\sigma ;{{\mathcal{Q}}}({{{\bf{k}}}}^{{\prime} }-{{\bf{q}}}),{\sigma }^{{\prime} }\right\vert \hat{V}\left\vert {{\bf{k}}},\sigma ;{{{\bf{k}}}}^{{\prime} },{\sigma }^{{\prime} }\right\rangle=\\ \frac{1}{N\Omega }{\sum} _{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}{V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}({{\bf{q}}};{0}^{+}){{{\mathcal{F}}}}_{{{\bf{k}}}+{{\bf{q}}},{{\bf{k}}}}^{\sigma }(-{{\bf{G}}}){{{\mathcal{F}}}}_{{{{\bf{k}}}}^{{\prime} }-{{\bf{q}}},{{{\bf{k}}}}^{{\prime} }}^{{\sigma }^{{\prime} }}({{{\bf{G}}}}^{{\prime} }),$$
(10)

where \({{\bf{k}}},{{{\bf{k}}}}^{{\prime} }\) and q are in-plane momenta restricted to the first Brillouin zone. Noting that the dominant contribution to the matrix element is due to the first few nearest-neighbour sites (see Supplementary Section II D for a discussion) we introduce an effective interaction potential

$$ \left\langle {{\mathcal{Q}}}({{\bf{k}}}+{{\bf{q}}}),\sigma ;{{\mathcal{Q}}}({{{\bf{k}}}}^{{\prime} }-{{\bf{q}}}),{\sigma }^{{\prime} }\right\vert \hat{V}\left\vert {{\bf{k}}},\sigma ;{{{\bf{k}}}}^{{\prime} },{\sigma }^{{\prime} }\right\rangle=\\ \approx \frac{1}{N\Omega }\left({\sum} _{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}{V}_{{{\bf{G}}},{{{\bf{G}}}}^{{\prime} }}^{2{{\rm{D}}},{{\rm{RPA}}}}({{\bf{q}}})\right){{{\mathcal{F}}}}_{{{\bf{k}}}+{{\bf{q}}},{{\bf{k}}}}^{\sigma }({{\bf{0}}}){{{\mathcal{F}}}}_{{{{\bf{k}}}}^{{\prime} }-{{\bf{q}}},{{{\bf{k}}}}^{{\prime} }}^{{\sigma }^{{\prime} }}({{{\bf{0}}}}^{{\prime} })\\ =:\frac{1}{N\Omega }{V}^{{{\rm{RPA}}}}({{\bf{q}}}){{{\mathcal{F}}}}_{{{\bf{k}}}+{{\bf{q}}},{{\bf{k}}}}^{\sigma }({{\bf{0}}}){{{\mathcal{F}}}}_{{{{\bf{k}}}}^{{\prime} }-{{\bf{q}}},{{{\bf{k}}}}^{{\prime} }}^{{\sigma }^{{\prime} }}({{{\bf{0}}}}^{{\prime} }),$$
(11)

where we restrict \({{\bf{G}}},{{{\bf{G}}}}^{{\prime} }\) to the range where \({{\mathcal{F}}}({{\bf{G}}})\approx {{\mathcal{F}}}({{\bf{0}}})\) holds, i.e., up to third nearest neighbouring Brillouin zones. This effective interaction potential enables a discussion of the screened interaction on the lattice in terms of a simple q-dependent quantity shown in Fig. 2, but is not used for the calculation of the pairing, where we keep the dependence of the Bloch factors \({{\mathcal{F}}}\) on G.

To access the possible superconducting instabilities described by Eq. (6), we solve a linearised form of the gap equation. First, we project the gap equations on the Fermi surfaces by introducing a local density of states ρk,σ along the Fermi surfaces. We restrict the allowed momenta \({{{\bf{k}}}}^{{\prime} }\) in Eq. (6) to a range around the Fermi surfaces given by a cutoff  ± Λ in energy. Next, we convert the sum over the transverse component \({{{\bf{k}}}}^{{\prime} }-{{{\bf{k}}}}_{{{\rm{F}}}}\) into an integral in energy over this range [ − Λ, Λ]. Assuming that the gap and the interaction vary slowly in this thin shell around each Fermi surface, we can approximate \({\Delta }_{{{{\bf{k}}}}^{{\prime} },\sigma }\approx {\Delta }_{{{{\bf{k}}}}_{{{\rm{F}}}},\sigma }\) and \({V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }\approx {V}_{{{\bf{k}}},{{{\bf{k}}}}_{{{\rm{F}}}},\sigma }\). The resulting gap equation with \({{\bf{k}}},{{{\bf{k}}}}^{{\prime} }\) on the Fermi surfaces takes the form

$${\Delta }_{{{\bf{k}}},\sigma }=-{\sum} _{{{{\bf{k}}}}^{{\prime} }\in {{\rm{FS}}}}{V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }\int_{-\Lambda }^{\Lambda }d\xi {\rho }_{{{{\bf{k}}}}^{{\prime} },\sigma }(\xi )\Pi ({E}_{{{{\bf{k}}}}^{{\prime} }\sigma }){\Delta }_{{{{\bf{k}}}}^{{\prime} },\sigma },$$
(12)

with the pairing function \(\Pi (E)=\tanh (\beta E/2)/2E\). Close to Tc we have Π(E) ≈ Π(ξ) and Eq. (12) becomes a set of linear equations. Instead of starting from the statically screened interaction and applying the mean-field approximation, an alternative approach to the problem is provided by investigating the pair-scattering vertex where the problem also simplifies to the real and static part of the effective pairing interaction18,55. The problem of finding the shape of the order parameter along the Fermi surfaces is equivalent to determining the kernel of a matrix \({{\mathbb{M}}}_{\sigma }\), whose indices are the momenta kF on the Fermi surface, as

$${{\mathbb{M}}}_{\sigma }{{{\mathbf{\Delta }}}}_{\sigma }=0,$$
(13)

where, \(\forall {{\bf{k}}},{{{\bf{k}}}}^{{\prime} }\in {{\rm{FS}}}\),

$${({{\mathbb{M}}}_{\sigma })}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} }}={\delta }_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} }}+{V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }{\rho }_{{{{\bf{k}}}}^{{\prime} },\sigma }{\alpha }^{0}(T,\Lambda ),$$
(14)
$${\alpha }^{0}(T,\Lambda )=\int_{0}^{\Lambda }d\xi \frac{\tanh (\beta \xi /2)}{\xi }.$$
(15)

Here, we assumed a flat density of states ρk,↑(ξ) ≈ ρk,↑ within the energy range  ± Λ around the chemical potential. We rewrite \({{\mathbb{M}}}_{\sigma }\) by factoring out the energy integral α0 in the second summand as

$${{\mathbb{M}}}_{\sigma }={\mathbb{1}}+{\alpha }^{0}(T,\Lambda ){{\mathbb{U}}}_{\sigma },$$
(16)

where \({{\mathbb{U}}}_{\sigma }\) is the pairing matrix containing the interaction and local density of states along the Fermi surfaces. The possible superconducting pairings solving Eq. (13) are the eigenvectors of \({{\mathbb{U}}}_{\uparrow }\) with the corresponding eigenvalues λi fulfilling

$${\lambda }_{i}=-1/{\alpha }^{0}({T}_{{{\rm{c}}},i},\Lambda ),$$
(17)

and Tc,i the critical temperature of the respective instability. The cutoff Λ sets an energy scale against which to measure the gaps and Tc, similar to the Debye frequency in the usual BCS-theory1. In general, not all eigenvectors of \({{\mathbb{U}}}_{\sigma }\) found in this way correspond to physically stable situations67. Similarly, the requirement of vanishing Δk,σ close to Tc,i is only fulfilled if no other superconducting phase with higher Tc is present. As such, we expect only the solution with the highest Tc to be realised. Due to the monotonic nature of α0(T, Λ) as a function of both T and Λ, the solution with the highest critical temperature always corresponds to the lowest negative eigenvalue λ of the pairing matrix. As Λ is a model parameter, we classify and discuss the strength of the different instabilities by their eigenvalues λi.

To investigate how the gaps evolve at temperatures below Tc, we solve Eq. (12) without linearizing. Since we want to use the approximations \({\Delta }_{{{{\bf{k}}}}^{{\prime} },\sigma }\approx {\Delta }_{{{{\bf{k}}}}_{{{\rm{F}}}},\sigma }\) and \({V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\sigma }\approx {V}_{{{\bf{k}}},{{{\bf{k}}}}_{{{\rm{F}}}},\sigma }\), we fix the cutoff to the value of Λ = 0.1 eV which, together with the rescaling of the screened interaction to \({\hat{V}}_{\gamma }:=\gamma \hat{V}\) as discussed in the main text, yields Tc = 2 K. This enables us to fit to the experimental Tc while retaining the shape of the interaction as found from our calculation of the screened Coulomb interaction. Since the macroscopic polarisability P0,0 calculated from DFT (see Supplementary Section I B) is even lower at intermediate momentum transfers than our results from a tight-binding calculation, it is plausible that the screened interaction in absence of the substrate is even stronger than our \(\hat{V}\). This is to be expected, since the inclusion of the Se p-orbitals, which contribute most noticeably near the M points, would further diminish the overlaps between states on the Fermi surface, resulting in lower screening.

We expand the gap for one spin species in terms of a restricted set of basis functions, corresponding to the irreducible representations of the symmetry group D3h, as

$${\forall }_{{{\bf{k}}}\in \pi }:\quad {\Delta }_{{{\bf{k}}},\uparrow }={\sum} _{\mu }{f}_{\mu }^{\pi }({{\bf{k}}}){\Delta }_{\mu }^{\pi },$$
(18)

where μ labels basis functions and π ∈ {Γ, K} the Fermi surfaces22,43. We orthonormalise our basis functions with respect to the number Nπ of momenta on each Fermi surface π to satisfy

$${\left\langle {f}_{\mu }^{\pi },{f}_{{\mu }^{{\prime} }}^{\pi }\right\rangle }_{\pi }:=\frac{1}{{N}_{\pi }}{\sum} _{{{\bf{k}}}\in \pi }{f}_{{\mu }^{{\prime} }}^{\pi }({{\bf{k}}})\;{f}_{\mu }^{\pi }({{\bf{k}}})={\delta }_{\mu,{\mu }^{{\prime} }}.$$
(19)

Using this orthogonality, we reduce the dimensionality of the self-consistency problem drastically by projecting the gap equation into a form that couples only the expansion coefficients \({\Delta }_{\mu }^{\pi }\)22. It reads

$${\Delta }_{\mu }^{\pi }=-\frac{1}{{N}_{\pi }}{\sum} _{{\pi }^{{\prime} },{\mu }^{{\prime} }}{\sum} _{\begin{array}{c}{{\bf{k}}}\in \pi \\ {{{\bf{k}}}}^{{\prime} }\in {\pi }^{{\prime} }\end{array}}{f}_{\mu }^{\pi }({{\bf{k}}}){V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\uparrow }\,{\rho }_{{{{\bf{k}}}}^{{\prime} },\uparrow }\,{\alpha }_{{{{\bf{k}}}}^{{\prime} }}^{{\pi }^{{\prime} }}\,{f}_{{\mu }^{{\prime} }}^{{\pi }^{{\prime} }}({{{\bf{k}}}}^{{\prime} }){\Delta }_{{\mu }^{{\prime} }}^{{\pi }^{{\prime} }},$$
(20)

where the pairing strength \({\alpha }_{{{\bf{k}}}}^{\pi }\) is given by

$${\alpha }_{{{\bf{k}}}}^{\pi }=\int_{0}^{\Lambda }d\xi \,\frac{\tanh (\beta E(\xi,{\Delta }_{{{\bf{k}}}}^{\pi })/2)}{E(\xi,{\Delta }_{{{\bf{k}}}}^{\pi })}.$$
(21)

At Tc, the pairing strength \({\alpha }_{{{\bf{k}}}}^{\pi }\) becomes independent of k and π. In this case, the orthogonality Eq. (19) yields a linearised gap equation for the expansion coefficients

$${\Delta }_{\mu }^{\pi }=-{\alpha }^{0}{\sum} _{{\pi }^{{\prime} },{\mu }^{{\prime} }}{U}_{\mu,{\mu }^{{\prime} }}^{\pi,{\pi }^{{\prime} }}{\Delta }_{\mu }^{{\pi }^{{\prime} }},$$
(22)

where the pairing matrix \({\mathbb{U}}\) in this representation is given by

$${U}_{\mu,{\mu }^{{\prime} }}^{\pi,{\pi }^{{\prime} }}=\frac{1}{{N}_{\pi }}{\sum} _{{{\bf{k}}}\in \pi,{{{\bf{k}}}}^{{\prime} }\in {\pi }^{{\prime} }}{f}_{\mu }^{\pi }({{\bf{k}}}){V}_{{{\bf{k}}},{{{\bf{k}}}}^{{\prime} },\uparrow }\,{\rho }_{{{{\bf{k}}}}^{{\prime} },\uparrow }{f}_{{\mu }^{{\prime} }}^{{\pi }^{{\prime} }}({{{\bf{k}}}}^{{\prime} }).$$
(23)

To account for the mixing between different symmetries due to the k dependence of \({\alpha }_{{{\bf{k}}}}^{\pi }\) below Tc, we insert a unity in Eq. (20) to arrive at

$${\Delta }_{\mu }^{\pi }=-{\sum} _{{\pi }^{{\prime} },{\mu }^{{\prime} },{\mu }^{{\prime\prime} }}{U}_{\mu,{\mu }^{{\prime} }}^{\pi,{\pi }^{{\prime} }}{\Upsilon }_{{\mu }^{{\prime} },{\mu }^{{\prime\prime} }}^{{\pi }^{{\prime} }}{\Delta }_{{\mu }^{{\prime\prime} }}^{{\pi }^{{\prime} }},$$
(24)

where the mixing of different basis functions is described by

$${\Upsilon }_{\mu,{\mu }^{{\prime} }}^{\pi }=\frac{1}{{N}_{\pi }}{\sum} _{{{\bf{k}}}\in \pi }{f}_{\mu }^{\pi }({{\bf{k}}}){\alpha }_{{{\bf{k}}}}^{\pi }{f}_{{\mu }^{{\prime} }}^{\pi }({{\bf{k}}}).$$
(25)

For an incomplete set of basis functions this insertion of unity becomes approximate, but in our case still suffices to capture the weak mixing between basis functions induced by \({\alpha }_{{{\bf{k}}}}^{\pi }\). As such, we employ Eq. (24) to numerically determine the evolution of the solutions obtained from the linearised equations below their Tc,i. The basis functions we used and the temperature dependence curves for the respective expansion coefficients are shown in Supplementary Section III A.

We define a purely chiral solution as

$${\Delta }_{{{\rm{p}}}+{{\rm{ip}}}}=\frac{| | {\Delta }_{{{\rm{py}}}}| | }{| | {\Delta }_{{{\rm{px}}}}| | }{\Delta }_{{{\rm{px}}}}+i{\Delta }_{{{\rm{py}}}},$$
(26)

where \(| | \Delta | | :=\sqrt{{\sum }_{\pi=G,K}{\langle \Delta,\Delta \rangle }_{\pi }}\), with the scalar product defined in Eq. (19). The Δpx component is renormalised so that both nematic solutions enter with the same weight. The ratio of admixture Rch of Δp+ip in the chiral solution Δch in Fig. 4a is defined as

$${R}_{{{\rm{ch}}}}:=\frac{| {\sum}_{\pi=\Gamma,K}{\langle {\Delta }_{{{\rm{p}}}+{{\rm{ip}}}},{\Delta }_{{{\rm{ch}}}}\rangle }_{\pi }| }{| | {\Delta }_{{{\rm{ch}}}}| | \,| | {\Delta }_{{{\rm{p}}}+{{\rm{ip}}}}| | }.$$
(27)

The square root of mean squared Δ shown in Fig. 4a for each gap symmetry is \(\sqrt{\langle {\Delta }^{2}\rangle }\equiv | | \Delta | |\).

The free energy of a superconductor with gap Δkσ can be written as

$$F=\frac{1}{2}{\sum} _{{{\bf{k}}}\sigma }\left({\xi }_{{{\bf{k}}}\sigma }-{E}_{{{\bf{k}}}\sigma }+\Pi ({E}_{{{\bf{k}}}\sigma })| {\Delta }_{{{\bf{k}}}\sigma }{| }^{2}\right)\\ -{k}_{TB}T{\sum} _{{{\bf{k}}}\sigma }{{\rm{In}}} \left(1+{e}^{-{E}_{{{\bf{k}}}\sigma }/{k}_{TB}T}\right).$$
(28)

Note that this free energy is written assuming that the gap obeys the self-consistent equations (see Supplementary Section III). It cannot be used to find the physical gaps by minimization with respect to Δkσ, but does yield the correct result once a solution is found by solving the self-consistent gap equations67. We therefore use it to rank the solutions by their free energy gain within the energy window of  ± 2 meV where the contributions of different gap symmetries are distinguishable.

NbSe2 monolayers2 were grown on graphitised (bilayer graphene) on 6H-SiC(0001) by molecular beam epitaxy (MBE) at a base pressure of ~ 2 ⋅ 10−10 mbar in our home-made ultrahigh-vacuum (UHV) MBE system. SiC wafers with resistivities ρ ~120 Ωcm were used68.

Reflective high-energy electron diffraction (RHEED) was used to monitor the growth of the NbSe layer2. During the growth, the bilayer graphene/SiC substrate was kept at 580 °C. High-purity Nb (99.99%) and Se (99.999%) were evaporated using an electron beam evaporator and a standard Knudsen cell, respectively. The Nb:Se flux ratio was kept at 1:30, while evaporating Se led to a pressure of ~ 5 ⋅ 10−9 mbar (Se atmosphere). Samples were prepared using an evaporation time of \(30\,\min\) to obtain a coverage of  ~ 1 ML. To minimise the presence of atomic defects, the evaporation of Se was subsequently kept for an additional 5 minutes. Atomic force microscopy (AFM) under ambient conditions was routinely used to optimise the morphology of the NbSe2 layers. The samples used for AFM characterization were not further used for the scanning tunnelling microscope (STM). Lastly, to transfer the samples from our UHV-MBE chamber to our STM, they were capped with a  ~10 nm amorphous film of Se, which was subsequently removed by annealing at the STM under UHV conditions at 280 °C.

Scanning tunnelling microscopy/spectroscopy experiments were carried out in a commercial low-temperature, high magnetic field (11 T) UHV-STM from Unisoku (model USM1300) operated at T = 0.34 K unless otherwise stated. STS measurements (using Pt/Ir tips) were performed using the lock-in technique with typical ac modulations of 20–40 μV at 833 Hz. All data were analyzed and rendered using WSxM software69. To avoid tip artifacts in our STS measurements, the STM tips used for our experiments were previously calibrated on a Cu(111) surface. A tip was considered calibrated only when tunnelling spectroscopy performed on Cu(111) showed a sharp surface state onset at  −0.44 eV followed by a clean and monotonic decay of the differential conductance signal (i.e., dI/dV). In addition to this, we also inspected the differential conductance within  ±10 mV to avoid strong variations around the Fermi energy.

For spectroscopy in the tunnelling regime with a normal metal tip, the differential conductance is given by

$$G(V)={\sum}_{{{\bf{k}}},\sigma }{C}_{{{\bf{k}}}\sigma }\int_{-\infty }^{\infty }\frac{{{\rm{d}}}E}{2\pi }\,{A}_{{{\rm{s}}}}({{\bf{k}}},\sigma ;E)\left(-\frac{\partial f(E+eV)}{\partial E}\right),$$
(29)

where As is the spectral function of the superconductor. The coupling Ckσ accounts for the spectral function of the tip and the tunnelling overlap between the tip’s evanescent states and quasiparticle states from the superconductor with momentum k and spin σ. We do not expect the coupling to vary much on any given Fermi surface π, and as such we approximate Ck,σ ≈ Cπ. However, due to the different orbital composition and in-plane momenta of the quasiparticle states at the K, \({K}^{{\prime} }\) valleys and at the Γ surface, cf. Fig. 1b, the coupling constant will be in general different. In particular, since the d2,0 orbital extends farthest out of plane and the STM favours states with small in-plane momentum63, as has been observed in previous STM measurements32, we expect CΓ ≫ CK.

In the tunnelling regime, where the experimental data was taken, the lifetime of the quasiparticles is weakly affected by the coupling to the tip and the spectral function can be approximated with the local density of states of the quasiparticles, As(k, σ, E) = 2πDk,σ(E). The tunnelling to each of the Fermi surfaces represents a distinct transport channel, whose strength is given by the sum over the local density of states

$${D}_{\pi,\sigma }(E)={\sum}_{{{\bf{k}}}\in \pi }{D}_{{{\bf{k}}},\sigma }(E).$$
(30)

The latter are modeled by assuming a BCS form factor modifying the local density of states ρk,σ in the normal conducting state as

$${D}_{{{\bf{k}}},\sigma }(E)={\rho }_{{{\bf{k}}},\sigma }{{\rm{Re}}}\sqrt{\frac{{E}^{2}}{{E}^{2}-| {\Delta }_{{{\bf{k}}},\sigma }{| }^{2}}}.$$
(31)

In the absence of magnetic fields, we can assume that, due to time reversal symmetry, Dπ,σ = Dπ/2 and Cπ,σ = Cπ. As such, Eq. (29) predicts for the total differential conductance a superposition of contributions from the distinct transport channels with weights provided by the couplings Cπ as

$$G(V)\approx {\sum}_{\pi }{C}_{\pi }{G}_{\pi }(V),$$
(32)
$${G}_{\pi }(V)=\int_{-\infty }^{\infty }{{\rm{d}}}E\,{D}_{\pi }(E)\left(-\frac{\partial f(E+eV)}{\partial E}\right).$$
(33)

For low temperature, the form of the gap on each separate Fermi surface results in characteristic signatures in the differential conductance of the respective transport channel. We use as a fit function this superposition of the differential conductance of the individual transport channels. For the gaps on the Fermi surfaces Δk,σ, we use the form of the gaps as found from Eq. (24) at the experimental temperature \({T}_{\exp }\) indicated by stars in Fig. 4a. For single-band s-wave superconductors the ratio between the amplitude of the gap and the critical temperature is fixed as \(\Delta (T=0\,{{\rm{K}}})/({k}_{{{\rm{B}}}}{T}_{{{\rm{c}}}})=\pi {e}^{-{\gamma }_{{{\rm{e}}}}}\approx 1.76\), with Euler’s constant γe1. This ratio appears to be slightly violated in monolayer 1H-NbSe2 as the expected s-wave gap for the Tc ≈ 2 K is Δ(T = 0 K) ≈ 0.3 meV instead of the roughly 0.4−0.5 meV gap we find in the spectral function, in agreement with earlier works33. Such violations of the BCS-ratio are known to occur in multi-band or multi-patch superconductors and have been observed previously18,70. To account for this possibility, we allow for a fit of the amplitude of our solution by a common rescaling of all Δk,σ by a parameter A as Δk,σ → AΔk,σ. The latter also takes care of the uncertainty of the actual experimentally realised Tc and the rescaling factor γ of the interaction.

To qualitatively account for additional sources of broadening in the experiment, we fit with an effective temperature Teff in the calculation of the derivative in Eq. (33) which ranges between 1.3 and 1.4 times the recorded base temperature of the STM in Fig. 4d, e and is kept at 1.4 times the measurement temperature for Fig. 4f. To account for offsets in the calibration, we further allow for both a small constant offset G0 in the measured conductivity and V0 in the recorded voltage. The fit coefficients for the other traces shown in Fig. 4d, f only differ by small quantitative changes from the ones reported in Table 1 for Fig. 4e. For the fitting we use the trust region reflective algorithm as implemented in SciPy’s “curve_fit” routine. We consider a range of  ± 0.5 mV containing the main coherence peaks, but not the satellite features due to Leggett modes33, which are not accounted for in our theory. The resulting fit parameters are listed in Table 1. The best fits for the solutions associated with the \({E}^{{\prime} }\) irreducible representation are obtained by considering strongly selective coupling of the tip to the Γ pockets.

Table 1 Fit coefficients for comparison between STS experiment and theory in Fig. 4(e)
Full size table