Introduction

In the past years, 2D materials have attracted considerable attention not only because they provide the possibility to easily control their properties by external methods1, but also because of the plenty of salient physical properties brought about by their reduced dimensionality and electron confinement2,3. Among various properties, Dirac semimetallicity and superconductivity are two active topics4,5,6. In general, one material cannot possess Dirac properties around its Fermi level and superconductivity simultaneously, since the Fermi surface (FS) of a Dirac semimetal usually consists of discrete points or nodal lines, which conflicts with the requirement of a high carrier density near the Fermi level to give rise to a high critical temperature (Tc). For instance, the intrinsic graphene (the first proposed 2D Dirac semimetal) is not superconducting and becomes a superconductor that needs the aid of doping and external strain7. The coexistence of Dirac properties and superconductivity possibly lures many exotic properties8,9, therefore, it is highly interesting to search for this kind of 2D materials harboring these two properties.

The band crossing points in Dirac semimetals are typically more robust in light-element (e.g. boron and carbon) materials owing to their intrinsic negligible spin-orbital coupling strengths, as exemplified by dozens of Dirac semimetals have been proposed in boron10,11,12 and carbon materials13,14,15. On the other hand, according to the conventional Bardeen-Cooper-Schrieffer (BCS) theory16, it is believed that lightweight metals have a better chance to induce high Tc, because the Debye temperatures within these metals are usually high enough to induce strong phonon-mediated superconducting pairing. Thus, it is natural to prospect a 2D material with both Dirac and superconducting properties has a higher opportunity to be found in light-element materials. Scanning the periodic table, it seems that boron is a desirable candidate that can combine lightweight and metallicity in a 2D material to simultaneously trigger Dirac and superconducting properties. In fact, this assumption has been partially confirmed in the 2D boron materials (i.e. borophenes). For example, the Dirac fermions17,18 and superconductivity19,20 have been experimentally or theoretically verified in the synthesized borophene β12 and χ3, the backbone of honeycomb borophene in the well-known superconductor MgB2 is also mainly responsible for the appearance of its Dirac nodal line21 and high Tc22,23.

2D borophenes have wide potential applications24,25,26,27,28,29. However, the bare surface makes the borophenes susceptible and easy to be oxidized at ambient conditions30. Chemical functionalization, such as hydrogenation, usually serves as an effective knob to remedy this problem and while be regarded as a desirable approach to modulate their different properties31. It is worthwhile to point out that hydrogenated borophenes (also known as borophanes) are also suitable for spawning Dirac fermions and superconductors since hydrogen and boron are both light elements. Hydrogenated borophenes have been theoretically found to be highly stable and possess ideal Dirac cones32,33,34,35 or Dirac nodal loops36. A honeycomb borophene hydride (named as h-B2H2 here) was successfully achieved by exfoliation and complete ion exchange between protons and magnesium cations in MgB237, and a hydrogenated borophene based on β12 (named as β12-B5H2 here) was also experimentally realized on the substrate38. The h-B2H2 is a normal metal with poor superconductivity39 and the existence of free-standing β12-B5H2 is still questionable (we will discuss it later). It is natural to wonder whether there exists a stable hydrogenated borophene with superb superconductivity, or clean Dirac elements or even both.

The answer is affirmative, in this work, based on synthesized borophene β12 [see Fig. 1a, b] and hydrogenated β12 borophene β12-B5H2 [see Fig. 1c, d], we propose a hydrogenated β12 borophene named as β12-B5H3 [see Fig. 1e, f] can fulfill these expectations. First, our first-principles calculations of mechanical properties and phonon spectrum suggest β12-B5H3 exhibits good stability, even better than its brother β12-B5H2. Second, β12-B5H3 is a Dirac semimetal with three clean Dirac cones (two type-I Dirac cones and one type-II Dirac cone) near the Fermi level at its equilibrium and can be tailored into a Dirac semimetal with a single type-I Dirac cone under external strain beyond 3.8% along the a direction. To our knowledge, such a feature has not been found before. In the end, based on the anisotropic Migdal-Eliashberg (ME) equations, we find that the Tc of β12-B5H3 is as high as 32.4 K and can be boosted to 42 K at 5.8% external strain along the b direction. Therefore, β12-B5H3 is a Dirac semimetal with triple Dirac cones and while with superconducting features, which make it a platform to study the exotic physics brought about by either Dirac points or superconductivity or both of them.

Fig. 1: Crystal structure.
figure 1

(a) Top view and (b) side view of β12. c Top view and (d) side view of β12-B5H2. e Top view and (f) side view of β12-B5H3. Green and pink balls represent boron and hydrogen atoms, respectively. The dashed purple rectangles indicate their unit cells.

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Results

Atomic structure and mechanical properties

Due to the nature of electron-deficient in boron, the 2D borophenes demonstrate various polymorphs with different hole densities referring to the triangular lattice40. The hole density of β12 [see Fig. 1a] is 1/6 making the whole electrons a little surplus in terms of the electron counting rule41, which partially explains why it would be quickly oxidized at ambient conditions. In a chemical sense, the excess electrons can be neutralized by chemical passivation, such as hydrogenation, and thus improve stability. The recent experiment has confirmed that the oxidation rate can be reduced by more than two orders of magnitude after hydrogenating β12 borophene to a well-ordered borophane on the substrate[i.e. β12-B5H2, see Fig. 1c, d]38. According to our first-principles calculations, the free-standing β12-B5H2 is not stable since there are sizable soft modes in its phonon spectrum [see Supplementary Fig. 1a–d and more discussion about their stabilities can be found in Supplementary Note 1]. However, this phonon instability can be eliminated by adding an enantiomer of the bridge hydrogen in β12-B5H2 to form a hydrogenated borophene called as β12-B5H3 here [see Fig. 1e, f], which its phonon spectrum shows no any soft modes throughout the entire Brillouin Zone (BZ) [see Supplementary Fig. 1e, f]. The optimized lattice constants of β12-B5H3 are a = 2.90 Å and b = 5.16 Å, varying not too much in comparison with those of its brother β12-B5H2 and mother β12 (see Table 1). Three different kinds of boron atoms are marked as B1, B2 and B3, respectively [see Fig. 1e]. In experiments, β12-B5H3 can be prepared by hydrogenating β12 borophene or β12-B5H2 via in situ and three-step thermal-decomposition process as conducted in the previous work31 once the free-standing β12 borophene or β12-B5H2 is produced.

Table 1 The structural and mechanical properties.
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It has been reported that β12 has good mechanical properties42, we are also curious about its performance in β12-B5H3. The calculated independent elastic constants of β12-B5H3 are C11 = 153.5 (N/m),C22 = 181.9 (N/m), C12 = 25.4 (N/m) and C44 = 70.3 (N/m), which obviously satisfy the Born-Huang mechanical stable criteria: \({C}_{11}{C}_{22}-{C}_{12}^{2} > 0\) and C44 >0. The small difference of Young’s moduli and Poisson ratios of β12-B5H3 along a (Ya = 149.9 N/m, νa = 0.17) and b (Yb = 177.7 N/m, νb = 0.14) directions indicate its mechanical anisotropy is not prominent [see Table 1, more details can be found in Supplementary Note 2 and Supplementary Fig. 3]. Although the elastic constants of β12-B5H3 are little smaller than those of β12 but better than those of synthesized h-B2H2 (see Table 1). We have also calculated the strain-stress curves along the a and b directions [see Supplementary Fig. 4], which shows that β12-B5H3 is within linear elastic regime up to at least 6% along the two directions, and its elastic strain limits are high(tensile strain limits are even beyond 20% along the two directions). The excellent mechanical properties of β12-B5H3 would facilitate the application of external strain for tuning its various properties.

Strain-tunable Dirac semimetal phase transition

The most fascinating feature of β12-B5H3 lies in its electronic properties. The electronic band structure for β12-B5H3 is shown in Fig. 2a. Notably, it can be observed that two bands cross each other three times (two times along Γ–X and one time along Γ–Y) to form three Dirac points near the Fermi level as indicated with green, red and blue circles in Fig. 2a. Their corresponding positions in BZ are sketched in Fig. 2g. Depending on the type of band dispersion, a Dirac point can be classified as type-I and type-II43: a type-I Dirac point is formed by the crossing between an electron-like band and a hole-like band; a type-II Dirac point is formed by the crossing between two electron-like bands or hole-like bands. Here, after examining the dispersions around these three Dirac points, we find that two of them are type-I Dirac points [see Fig. 2c, e] and the other one is a type-II Dirac point [see Fig. 2d], which can also be inferred from the three-dimensional (3D) band structure [see Fig. 2f]. Another highlight worth being pointed out is that there are only two Dirac bands in a broad energy range around the Fermi level, which should be beneficial for experimentally detecting these Dirac fermions. We further check the band structure by the more accurate hybrid functional method [see Supplementary Fig. 5], which confirms that the key band features are still maintained. Therefore, the following discussion would be based on the PBE results.

Fig. 2: The electronic properties of β12-B5H3.
figure 2

a The band structure and (b) the projected density of states of β12-B5H3. The band dispersions are coded with orbital characters. The color gradient from blue to red stands for a varying contribution from px to pz orbital of boron. The position of a van Hove singularity (vHS) is indicated with an arrow. c–e The magnified band structures correspond to the regions circled with different colors in (a). f The 3D band structure of β12-B5H3, the colored dashed circles indicate the positions of the Dirac points marked in (a). g The sketched positions of Dirac points in the Brillouin Zone, the dots with different colors correspond to the different Dirac points marked with the same colors in (a). h The Fermi surface of β12-B5H3 upon being projected with px and pz orbitals of boron.

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The orbital-resolved band structure denotes the two Dirac bands are mainly contributed by px and pz orbitals of boron, which are further consolidated by the projected density of states [PDOS, see Fig. 2b.] The two Dirac bands form two elliptical electron Fermi sheets along the a direction and two triangular-like hole Fermi sheets along the b direction [see Fig. 2h]. These Fermi sheets also mainly originated from px and pz orbitals of boron [see Fig. 2h] and demonstrate strong anisotropy.

The space group of β12-B5H3 is Pmm2 (NO. 25), the little group along Γ–X/Y is Cs containing an identical symmetry (E) and a mirror symmetry (σh). According to our calculations, the two Dirac bands have opposite mirror eigenvalues and thus belong to different irreducible representations [see Fig. 2c–e], which means the two Dirac bands cannot hybridize and the crossing between them cannot be gapped.

In the original β12 borophene, its FS consists of px, py and pz orbitals of boron20 and external strains cannot change its strong metallicity44,45. In contrast, the Fermi sheets of β12-B5H3 are contributed by px and pz orbitals of boron and three Dirac points are formed around the Fermi level. We are curious about how the electronic properties of β12-B5H3 are affected by the external strains. The appearance of the three Dirac points is attributed to the two times band inversions between the valence band and conduction band. We denote the magnitude of the band inversion that occurred at Γ is Δ [see Fig. 3a]. In Fig. 3d, we plot the Δ as a function of tensile strain along the a direction. One observes that there exists a critical strain of 3.8%, during which the band order is switched [see Fig. 3b]. Furthermore, beyond the critical into a single Dirac point phase [see Fig. 3c]. Since the mirror symmetry is preserved during the tensile strain along the a direction, the single Dirac point in Fig. 3c is still protected by the mirror symmetry. Thus, a quantum phase transition from Dirac semimetal with triple Dirac points into a Dirac semimetal with a single Dirac point is realized in β12-B5H3 by applying tensile strain along the a direction.

Fig. 3: The electronic properties of β12-B5H3 under uniaxial tensile strain along a direction.
figure 3

Electronic band structures of β12-B5H3 on the path X–Γ–Y under the applied tensile strain of (a) 0%, (b) 3.8%, and (c) 6%. The band inversion gap at Γ is defined as Δ as marked in (a). The insets in (a–c) indicate the positions of Dirac points in the Brillouin Zone. d The Δ as a function of tensile strain along a direction. TDPs and SDP are defined as triple Dirac points and single Dirac point, respectively.

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It is widely accepted that the type-I Dirac fermion in graphene is responsible for its various exotic properties, such as high carrier mobility, Klein tunneling and some quantum behaviors46. Type-II Dirac fermion however can give rise to many properties, such as a direction-dependent chiral anomaly47, an antichiral effect of the chiral Landau level48 and quantum oscillations due to momentum-space Klein tunneling49. The concurrence of type-I and type-II Dirac fermions in the β12-B5H3 provides an opportunity to observe how these properties coexist or whether they will coherent with each other. On the other side, the strain-tunable Dirac phase transition from triple Dirac fermions to a single Dirac fermion offers a platform to study how these exotic properties evolve in the course of this phase transition.

Isotropic and anisotropic superconducting properties

Generally speaking, a Dirac semimetal with a single Dirac point sitting rightly at the Fermi level such as in graphene cannot possess superconductivity. As for the six Dirac points in the whole BZ of β12-B5H3, they deviate slightly from the Fermi level would not influence their detection in experiments, but expands the six Fermi points to four Fermi loops [see Fig. 2g, h] and thus is an advantage for spawning superconductivity. Moreover, it can be observed that the pz-induced Dirac band of the type-II Dirac point is rather dispersionless and forms a van Hove singularity (vHS), leading to a quite large peak in the PDOS near the Fermi level [see Fig. 2b], which may trigger some exotic physics, such as superconductivity. On the other side, we have justified above that lightweight metal, such as the β12-B5H3 composed of boron and hydrogen, has a great potential to spawn good superconductivity. Therefore, it is highly interesting to investigate the superconducting properties in the β12-B5H3.

We proceed to investigate the vibrational properties and the electron-phonon coupling(EPC) in β12-B5H3 based on the density-functional perturbation theory50 as implemented in Quantum ESPRESSO51. The dynamic stability of the β12-B5H3 can be inferred from the phonon band structure, on which the mode-(ν) and momentum-(q) dependent EPC λqν are projected [see Fig. 4a]. We find that the phonon modes with low frequencies are key to achieving a high EPC in β12-B5H3, where about 75% of the total EPC is induced by the phonons with energy lower than 80 meV [see Fig. 4b]. Within this lower energy range, two modes [A1 and B2 modes as indicated in Fig. 4a] have large EPC strengths. A1 phonon mode is an out-of-plane shear mode involving mainly the B2 and B3 atoms, the contribution from the B3 and H atoms is relatively small [see Fig. 4d]. B2 phonon mode is also an out-of-plane shear mode but contributed mostly by the B3 and H atoms instead [see Fig. 4e]. Note that in these two modes, H atoms are involved, which is corroborated by the phonon density of states shown in Fig. 4c, implying that H atoms play an important role in the appearance of superconductivity in β12-B5H3. These two modes result in two large peaks in the Eliashberg spectral function α2F(ω) [see Fig. 4b].

Fig. 4: The phonon and superconducting properties of β12-B5H3.
figure 4

a Phonon dispersions of β12-B5H3, where the area of the orange dots is proportional to the magnitude of the mode-(ν) and momentum-(q) dependent electron-phonon coupling (EPC) λqν. b The calculated Eliashberg spectral function α2F(ω) and the cumulative EPC strength λ, where ω means phonon frequency. c The phonon density of states for H, B1, B2 and B3 atoms. Eigenvectors of two modes at (d) A1 and (e) B2 phonon frequencies as labeled in (a). The red arrows and their lengths indicate the directions and relative amplitudes of these two phonon modes, respectively.

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The Eliashberg spectral function α2F(ω) is a central parameter, through which we can obtain the EPC constant λ and the logarithmic average frequency ωlog with the following equations:

$$\lambda =\mathop{\sum}\limits_{q\nu }{\lambda }_{q\nu }=2\int\,\frac{{\alpha }^{2}F(\omega )}{\omega }{{{{{\rm{d}}}}}}\omega$$
(1)
$${\omega }_{log}=\exp \left[\frac{2}{\lambda }\int\,\frac{{{{{{\rm{d}}}}}}\omega }{\omega }{\alpha }^{2}F(\omega )\log\omega \right]$$
(2)

The calculated λ = 0.77 and ωlog = 44.27 meV for β12-B5H3, of which the value of λ (0.77) is comparable to that of the well-known phonon-mediated superconductor MgB2 (λ = 0.748 in the ref. 52), implying β12-B5H3 is also a potential phonon-meditated superconductor with good superconductivity. We can determine the Tc using the McMillian–Allen–Dynes (MAD) formula53,54 as follows:

$${{{{{T}}}}}_{{{{{{{{\rm{c}}}}}}}}}=\frac{{{{{{{{{\rm{\omega }}}}}}}}}_{\log }}{1.20}\,\exp \left[\frac{-1.04\,(1+\lambda )}{\lambda -{\mu }^{* }(1+0.62\lambda )}\right]$$
(3)

where μ* is the effective screened Coulomb repulsion constant.For β12-B5H3, the calculated Tc = 21.96 K (setting μ*= 0.1), which is a little larger than that of β12 (18.7 K)19.

The MAD formula works reasonably well for conventional bulk metals and for weakly anisotropic superconductors such as in bulk lead55. However, for layered systems, systems of reduced dimensionality, and those with complex multisheet Fermi surfaces, proper treatment of the anisotropic electron-phonon interaction is required, which has been demonstrated in the well-known MgB252. As mentioned above, the Fermi surface of β12-B5H3 is rather anisotropic, therefore, it is believed that the superconducting properties should be more accurate in the formalism of the ME approximation. For β12-B5H3, the variation of momentum-dependent EPC parameter λk and the superconducting gap Δk at 10 K are shown in Fig. 5a, b. Both quantities display similar anisotropy, with their maximum (λmax = 1.12 meV, Δmax = 5.73 meV) along the Γ–X direction and over double larger than their minimum (λmin = 0.51 meV, Δmin = 2.80 meV) along the Γ–Y direction, signifying that the anisotropy of the superconducting gap at the FS is strong and the necessity of predicting Tc by the anisotropic ME formula. In addition, compared with the orbital-resolved FS plotted in Fig. 2h, one can see that the relatively strong EPC along the Γ–X direction is mainly contributed by the electronic states on the red section of FS [i.e. the pz-induced bands, see Fig. 2h and Fig. 5a, b]. Figure 5c shows the evolution of the superconducting gap as a function of temperature by solving the ME equations in both the isotropic and the fully anisotropic approximations. The Tc computed within the isotropic approximation is 27.3 K, which is larger than the value (21.96 K) obtained from the MAD formula but smaller than the fully anisotropic result (32.4 K). Within the same anisotropic approximation, the Tc(32.4 K) of β12-B5H3 is almost identical to the value (33.0 K) of β1256.

Fig. 5: The anisotropic superconducting properties of β12-B5H3.
figure 5

a Momentum-dependent strength of electron-phonon coupling λk across the full Brillouin Zone and (b) momentum-dependent superconducting band gap Δk at 10 K projected onto the Fermi surface. c Variation of the superconducting gap Δk with temperature, calculated by solving the Migdal-Eliashberg equations in the isotropic approximation (yellow dots and dashed line interpolation) and with the fully anisotropic solution, where the gray shadowed regions indicate the magnitude distribution of the Δk and the blue dots connected with dashed line represents the average value of the entire anisotropic Δk.

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A Dirac semimetal phase transition has been observed in β12-B5H3 under external strain along the the a direction, we expect the external strain would have also a great effect on its superconducting properties. We calculate the effects of a tensile strain along the b direction up to 5.8% on the EPC λ, the logarithmic average frequency ωlog and the Tcs estimated within three different approximations [i.e. MAD, isotropic ME and anisotropic ME, see Fig. 6a, b]. One observes that λ decreases at first and then go arise to 1.42 at 5.8%. The ωlog keeps diminishing and drops down to 16.66 meV at 5.8%. Their combined result is the Tc initially decreases and then slowly increases [see Fig. 6a]. Note that the values of Tc overall becomes smaller, although not too much, after subjecting from tensile strain along the b direction in the MAD and isotropic ME approximations, but the Tc calculated with the more accurate anisotropic ME method is boosted to 42 K at 5.8% tensile strain, which is a record in the hydrogenated borophenes. It is revealed that the vibrational modes of H atoms are pushed towards a high-frequency range during the tensile strain along the b direction increases and their contributions on the strain-meditated Tc is weakening (more detailed discussion can be found in Supplementary Note 3). Interestingly, we notice that the U-shape of Tc versus tensile strain relationship also occurred in the original β12 borophene45. The trends of Tc going with the strains are analogous to that of the values of λ, implying the variation of strain-meditated Tc roots in the change of the strength of EPC. The great enhancement of λ and Tc at 5.8% can be ascribed to the Kohn anomaly induced by the huge softened phonon modes near the X point [see Fig. 6c]. Based on our results and discussion, we have partially confirmed that there is a higher chance of finding superior superconductors in the light element materials, such as in the boron hydrides discussed here. The effect of strain along the a direction on the superconducting properties of β12-B5H3 can be found in Supplementary Note 4, which indicates the Tc decreases rapidly during the tensile strain along the a direction increases. This denotes that despite the tensile strain along the a direction can trigger the Dirac semimetal phase transition but will deteriorate the superconductivity.

Fig. 6: The superconducting properties of β12-B5H3 under tensile straion along b direction.
figure 6

a Evolution of superconducting transition temperature (Tc) as a function of tensile strain along the b direction at three different approximations [e.g. the McMillian-Allen-Dynes(MAD), isotropic Migdal-Eliashberg (iso) and anisotropic Migdal-Eliashberg (aniso)]. b Evolution of electron-phonon coupling (EPC) λ and the logarithmic average frequency ωlog. c Phonon band structure, isotropic Eliashberg spectral function α2F and EPC λ under 5.8% tensile strain along the b direction. Data in the equilibrium (equil.) case are shown by the blue shadow and line for comparison.

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Discussion

Although Dirac fermions are not uncommon in borophanes, for instance, Dirac points in δ6-borophanes32,34 and ladder polyborane35, Dirac nodal loops in (5-7)-α-borophane and (5-6-7)-γ-borophane36; but, β12-B5H3 is the first case with ideal triple Dirac fermions. On the other hand, the vanishing electron density of states in these Dirac semimetallic borophanes will deteriorate the superconducting properties, thus none of the borophanes with Dirac fermions has been reported as having superconducting features. The experimentally realized h-B2H2 is a metal, but its Tc is only 11 mK39. It seems that it is challenging to possess Dirac fermions and superconductivity together in one borophane. In our work, the high Tc (32.4 K) and clean triple Dirac fermions in β12-B5H3 have proven this unreachable goal can be reached. The last but not least, β12-B5H3 is not just a fantasied structure, its mother (β12) and brother (β12-B5H2) have been synthesized, we have sufficient confidence that it can be realized in more consideration of the good stability it otherwise has.

A few 2D borides have been claimed to harbor Dirac elements and superconductivity concurrently, such as the triple Dirac cones in 2D AlB6 with a Tc of 4.7 K57, the Dirac nodal lines in bilayer TiB4 with a Tc of 0.82 K58 and the Dirac nodal loop in h-B2O59 with a Tc of 10.3 K60. But, it should be pointed out that either the Dirac states in these borides are intermixed with many other trivial electronic states near the Fermi level, not like the clean Dirac states in the β12-B5H3, or their superconducting critical temperatures are much lower than the value in β12-B5H3. The ideal Dirac properties and high Tc endow the β12-B5H3 a unique position among the 2D boron compounds.

In addition, we would like to mention that there are some other physical attributes worth being exploited in β12-B5H3. For instance, hydrogenated borophenes have been reported to possess high thermal conductivities61,62, which would be also expected in the β12-B5H3; furthermore, as confirmed that the β12 has a strong potential to exhibit plasmons in the visible and near-infrared regime range of frequencies with no need of doping63, how does it is affected in the β12-B5H3 is also an interesting question.

conclusion

In conclusion, a hydrogenated borophene named β12-B5H3 has been proposed that exhibits a bunch of properties. We determine its good stability. Its good elastic properties demonstrate that the material can be readily tailored by external strain. Its electronic band structure involves three clean Dirac points (two type-I Dirac points and one type-II Dirac point) near the Fermi level. A quantum phase transition in β12-B5H3 from a Dirac semimetal with triple Dirac points to a Dirac semimetal with a single Dirac point that can be tunable by external strain along the a direction. Another aspect is that β12-B5H3 is also a 2D phonon-mediated superconductor with a high Tc of 32.4 K by solving the fully anisotropic ME equations. We also reveal that the Tc may be slightly suppressed under the medium tensile strain along the b direction but eventually be enhanced to 42 K at 5.8% tensile strain along the b direction. The proposed 2D β12-B5H3 thus offers a platform not only for the investigation of quantum phase transition in Dirac semimetal but also for the study of the superconductivity and the potential rich physics brought about by their interplay.

Methods

Structural and electronic properties

First-principles calculations were carried out within the Vienna ab initio Simulation Package (VASP)64,65 based on density functional theory (DFT). The exchange-correlation functional of Perdew-Burke-Ernzerhof (PBE)66 along with the projector-augmented wave (PAW)67 pseudopotentials were employed for the self-consistent total energy calculations and geometry optimization. The kinetic energy cutoffs were chosen to be 500 eV. Atomic positions were relaxed until the energy difference was smaller than 10−5 eV and the maximum Hellmann-Feynman forces imposed on any atoms were below 10−2 eV/Å. The Brillouin Zone (BZ) was sampled with a 12 × 8 × 1 Monkhorst-Pack k-point mesh. The vacuum thick was set to 25 Å.

Phononic and superconducting properties

The phonon and superconducting properties were calculated in the Quantum-ESPRESSO (QE) package51. The PBE exchange-correlation functional and PAW pseudopotential with a 60 Ry cutoff energy were adopted to model the electron-ion interactions. The structural and electronic properties were recalculated by QE, and consistent results with those calculated by VASP were obtained. The vibrational properties and phonon perturbation potentials were calculated on a 12 × 8 × 1 mesh of q-points within the framework of density-functional perturbation theory (DFPT)50, combing with Methfessel-Paxton smearing scheme of width 0.02 Ry and 24 × 16 × 1k-point mesh. Once the phonon perturbation potentials were obtained in QE, then we shift our calculations to the Electron-phonon Wannier (EPW) 5.4 code68 to solve the ME equation both in the isotropic and anisotropic approximations to obtain the superconducting gap and its temperature evolution. Fine electron (240 × 160 × 1) and phonon (120 × 80 × 1) grids were used to interpolate the EPC constant through the maximally localized Wannier functions69 as implemented in the EPW code, where the results of electronic band structures calculated with QE and EPW can be found in Supplementary Fig. 6. In all cases, a 0.8 eV cutoff for the Matsubara frequency was chosen; the Dirac delta functions for electrons and phonons were smeared out with the widths of 15 meV and 0.3 meV, respectively; a typical value of 0.1 was used for the screened Coulomb parameter μ*.