Introduction

Two-dimensional (2D) superconductivity, which may occur in heterogeneous interfaces and atomic-thin layers, has attracted growing attention because of its fundamental interest and practical application1. Over the past decade, the advent of nanofabrication methods, such as molecular beam epitaxy and mechanical exfoliation2, has enabled the exploration of ultrathin and highly crystalline 2D superconductors3,4,5,6, and intriguing phenomena have been unveiled, such as quantum Griffiths singularity7,8, anomalous/bosonic metallic state9 and enhanced upper critical field6,10. Some 2D superconductors can achieve much higher superconducting transition temperature Tc than their bulk counterparts, such as the FeSe/SrTiO3 interface11,12 and surface-hydrogenated monolayer MgB213. Especially, monolayer Td-MoTe2 was discovered to have a 60-times Tc enhancement (~7.6 K) over the bulk phase (~0.1 K) by electrical gating14.

In general, practical application of 2D superconductors has to include the effect of substrate, which may suppress the superconductivity15,16 in particular the high Tc predicted, such as 24.7 K for borophene17. In this regard, van der Waals (vdW) layered materials are obviously advantageous. Representatively, the transition metal dichalcogenides (TMDs) have provided a fertile land for 2D superconductivity. For example, the coexistence of superconductivity and charge density wave phases in 2H-NbSe218,19, two types of Ising pairing in gated 2H-MoS2 and 1Td-PdSe210,20,21, and metastable superconductivity of IrTe222. 2M-WS2 presents the highest Tc = 8.8 K among intrinsic TMDs23. Beyond TMDs (atomic layers N = 3), recent computational studies have predicted a high Tc ~ 21 K in monolayer W2N3 (N = 5)24,25.

More interestingly, when superconductivity meets with nontrivial band topology26, their interplay can lead to more exotic topological superconducting (TSC) state, e.g., manifesting in Majorana fermionic mode, which offers quantum computing with high fidelity of fault tolerance27. In addition to interfacing superconducting and topological states in heterostructures by superconducting proximity effect28,29 or Berezenskii-Kosterlitz-Thouless phase transition30, the coexistence of superconducting and topological states offers a promising platform for realizing Majorana fermions without complex interface conditions25. Efforts have been made by searching either superconducting state in a topological material31,32,33,34,35 or conversely topological state in a superconductor36,37. Excitingly, a recent experiment has evidenced the Majorana bound states in superconducting 2M-WS238. Besides TSC, the additional electronic and phononic states at surface or edge can play a significant role of reinforcing electron-phonon coupling (EPC) and superconductivity by interacting with bulk phonons or electrons39,40. For instance, the nontrivial topological states on Be (0001) surface give rise to an anomalously large surface EPC41. Given these aspects, it is fundamentally interesting to explore 2D superconductors with electronic and phononic topology.

In this paper, we report the investigation of superconductivity and topological aspects of vdW-monolayered MX (M = Zr, Mo; X = F, Cl) based on first-principles calculations. 2D MX have strong EPC (λ > 1) and remarkable Tc (5.9–12.4 K) due to the softening of acoustic phonon modes. Anisotropic Migdal-Eliashberg theory unveils that monolayer ZrCl is a single-gap superconductor with Tc ~ 12.4 K and superconducting gap ∆0 = 2.14 meV, higher than the predicted 1.5 meV of 2M-WS242. Monolayer MoCl displays a distinct two-gap superconductivity. And the larger gap ∆0 = 1.96 meV has a ratio 2∆0/kBTc = 4.69 > 3.53, exhibiting a strong coupling beyond the Bardeen-Cooper-Schrieffer (BCS)43 superconductors. In addition, 2D MX are encoded with the Z2 band topology, and monolayer MoF, MoCl are TSC candidates. We also identify the Dirac phonons of monolayer ZrCl and MoCl with quantized ±π Berry phases at Brillouin zone boundaries. Corresponding zigzag edge states display w-shape dispersions, indicating a potential edge-enhanced EPC in the one-dimensional zigzag ribbons of ZrCl and MoCl. Our results demonstrate the intriguing superconducting and topological properties in 2D MX, deserving further experimental verifications.

Results

Atomic and electronic structures

As shown in Fig. 1a, our investigated vdW-monolayered transition-metal monohalides MX (M = Zr, Mo; X = F, Cl) have a 1:1 X-M-M-X sandwich structure with P\(\bar 3\)m1 symmetry (No. 164), where top and bottom sub-monolayers constitute two sublattices (pseudospins) in Bernal stacking. Bulk ZrCl has been synthesized in 1970s by reacting gas phase ZrCl4 with abundant electropolished Zr44,45. Similar synthetic method should be applicable to bulk MoCl, MoF and ZrF. For example, one may obtain MoF by reacting Mo with MoF6 gas46. The calculated binding energy is 17.1 meV Å−2 for ZrCl, 26.2 meV Å−2 for MoCl, 16.0 meV Å−2 for ZrF and 28.4 meV Å−2 for MoF, respectively, including the vdW-D3 correction47. The results of binding energy are close to known vdW-monolayers like 2H-MoS2 (23.7 meV Å−2)48, indicating our proposed monolayer MX can be mechanically exfoliated from the bulk.

Fig. 1: Atomic and electronic structures.
figure 1

a Crystal structure of monolayer transition-metal monohalides MX (M = Zr, Mo; X = F, Cl) with two (top and bottom) M-X sub-monolayers. Orbital-weighted band structures with spin-orbit coupling (SOC) and projected density of states (DOS) of b monolayer ZrCl and c monolayer MoCl. Here, + (−) denote parities. The two bands with crossing the Fermi level are labeled as α and β. The gray dashed lines exhibit the curved Fermi level, and the black dashed circles show SOC gaps at Dirac-type crossings (i.e., crossing points without SOC).

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Band structures in Fig. 1b, c, Supplementary Figs. 2a and 3a show that 2D MX are metallic with two spin-degenerate bands (α for hole-like and β for electron-like) crossing the Fermi level. Here, we focus on monolayer ZrCl, MoCl since monolayer ZrF, MoF exhibit similar electronic structures, respectively. The Fermi surface (FS) of monolayer ZrCl consists of one hole pocket FS1 and two electron pockets FS2, FS3 centered at Γ point (Fig. 3a), while that of monolayer MoCl has one flower-shaped hole pocket FS2 near Brillouin zone boundaries and one electron pocket FS1 at Γ point (Fig. 3d). Notably, the FS2 of monolayer MoCl is slightly different that the in-plane \(d_{x^2 - y^2,xy}\) orbitals dominate near K point and out-of-plane \(d_{z^2}\) orbital dominates near M point. The density of states (DOS) of monolayer ZrCl and MoCl are both about peaking at the Fermi level, with N(EF) equals to 2.78 states eV-1 and 1.71 states eV-1, respectively.

Superconductivity in 2D MX

We first show the phonon dispersions of 2D MX in Fig. 2a, c, Supplementary Figs. 2c and 3c. To investigate the EPC, we have calculated the Eliashberg function a2F(ω) and accumulated EPC coefficient λ(ω) in Fig. 2b, d, Supplementary Figs. 2d and 3d. As listed in Table 1, 2D MX all have a strong integral EPC coefficient λ > 1. Monolayer ZrCl exhibits the highest McMillan-Allen-Dynes49 Tc = 12.0 K (μ* = 0.1). Notably, the low-frequency phonons (< 150 cm−1) of 2D MX contribute mostly to the EPC. In particular, the mode-1 of monolayer ZrCl and mode-2, mode-3 of monolayer MoCl are acoustic soft modes hosting considerable weights of λqv (Fig. 2a, c). These soft modes prefer to form attractive electron-phonon interaction and contributes to the a2F(ω) peaks (Fig. 2b, d). We have also estimated their significant contributions to the enhancement of total EPC λ. It shows that mode-1 contributes 23% of λ of monolayer ZrCl, mode-2 and mode-3 together contribute 36% of λ of monolayer MoCl (Supplementary Note 2). We will discuss the origin of these soft modes in the section below.

Fig. 2: Phonon properties and EPC.
figure 2

a Phonon dispersion of monolayer ZrCl and b its Eliashberg function a2F(ω), phonon DOS F(ω) and accumulated EPC coefficient λ(ω). c Phonon dispersion of monolayer MoCl and d its a2F(ω), F(ω) and λ(ω). In a and c, the red weighted dots display momentum-mode resolved λqv by magnitude. The Dirac points (DPs) and acoustic soft modes at point 1, 2 and 3 are also marked. e Schematic of the vibration mode of monolayer ZrCl at point 1 with frequency 75.8 cm−1, where Zr atoms in the top (bottom) sub-monolayer vibrate circularly left-handed (right-handed), as indicated by the arrows. Schematics of the vibration modes of monolayer MoCl f at point 2 with frequency 37.1 cm−1 and g at point 3 (along Γ–K path) with frequency 77.1 cm−1.

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Table 1 Structural, superconducting and topological properties of 2D MX (M = Zr, Mo; X = F, Cl).
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Moreover, we have analyzed the EPC-favorable vibration modes in 2D MX (Fig. 2e–g and Supplementary Fig. 2e, f). Interestingly for monolayer ZrCl, the soft mode-1 at K point is mainly from Zr atoms that vibrate circularly but of opposite chirality between top and bottom sub-monolayers (Fig. 2e). The previous study50 reveals that chiral-phonon modes can be induced by breaking the inversion symmetry of monolayer ZrCl. Based on this, we have explored the Janus structure of monolayer Zr2FCl (P3m1 symmetry, No.156) in Supplementary Fig. 4. We find there are two chiral-phonon-related a2F(ω) peaks in monolayer Zr2FCl, and the predicted Tc is further enhanced to 13.2 K. Particularly in Supplementary Fig. 4c, one finds the lowest phonon branch seems to develop a roton-like minimum at the K valley. It may originate from the phonon-chirality-induced attractive intervalley dipole-dipole interaction, reminiscent of the roton mode in a BEC experiment51. Roton is a kind of quasiparticle first introduced to explain the spectrum of superfluid 4He. In a recent study52, it is also found the chiral modes can cause a roton-like minimum in the acoustic phonon branch. We believe this finding may help better understanding the relation between chiral phonon and superconductivity: when the chiral modes appear in the acoustic phonon branches, they may accelerate the so-called roton-like phonon softening, which in turn enhances EPC and superconductivity.

We have also employed Migdal-Eliashberg theory53 to obtain the finite-temperature superconducting properties. The estimated Tc of monolayer ZrCl and MoCl by anisotropic Migdal-Eliashberg theory are close to those from McMillan-Allen-Dynes formula (Table 1), while the isotropic Migdal-Eliashberg approximation leads to a ~20% overestimation of Tc (Supplementary Fig. 6). In Fig. 3b, monolayer ZrCl is found to have an averaged superconducting gap vanishing at Tc = 12.4 K and three FS sheets jointly contributing to a single-gap distribution with median EPC strength53 \(\lambda _{n{{{\mathbf{k}}}}}^{el}\) ~ 1.35. Such single-gap nature is also captured by the normalized quasiparticle DOS in Fig. 3c. The superconducting gap of monolayer ZrCl at zero temperature limit is ∆0 = 2.14 meV, yielding a ratio 2∆0/kBTc = 4.00 > 3.53 of BCS value. The one-gap α-model fitting54 of the averaged superconducting gap with ∆(T)/∆0 = (1 − (T/Tc)p)0.5 gives p = 3.4. In addition, monolayer MoCl exhibits a distinct two-gap superconductivity with Tc = 9.7 K and two peaks in the normalized quasiparticle DOS (Fig. 3e, f). The multigap superconductivity was first discovered in MgB2, where the two superconducting gaps are respectively from σ and π FS sheets55. Based on FS analysis, we find the spin textures of FS2 of monolayer MoCl are spirally flipped due to the aforementioned in-plane and out-of-plane orbital compositions, quite different from that of FS1 (Fig. 3d). We attribute the two-gap superconductivity to the well-separated FS1 and FS2 of monolayer MoCl. Since the cross pairings between bands with different orbital components are energetically disfavored, it further leads to the two-component \(\lambda _{n{{{\mathbf{k}}}}}^{el}\) distribution in the inset of Fig. 3e. Notably, the large superconducting gap ∆β0 = 1.96 meV displays a significant Cooper pairing with the ratio 2∆β0/kBTc = 4.69 > 3.53. While the small gap ∆α0 = 1.37 meV has the ratio 2∆α0/kBTc = 3.28 (close to 3.53). The two-gap α-model fitting ∆α,β(T)/∆(α,β)0 = (1 − (T/Tc)p)0.5 gives p = 2.8. Given the two-gap feature, several intriguing superconducting phenomena, such as anisotropy of the upper critical field56, are anticipated in monolayer MoCl.

Fig. 3: Superconducting properties.
figure 3

a Fermi surface (FS) with in-plane spin textures of monolayer ZrCl. The arrows show the in-plane spin orientations and the opposite spins at k and −k points are anti-parallel, indicating a spin-singlet s-wave pairing. b Temperature-dependent superconducting gap distribution of monolayer ZrCl based on anisotropic Migdal-Eliashberg theory. The red dashed line in b represents α-model fitting using the average gap values (black square dots) and the inset is the density distribution of EPC strength \(\lambda _{n{{{\mathbf{k}}}}}^{el}\). c Normalized superconducting quasiparticle DOS of monolayer ZrCl at 4 K and 11.5 K. d–f FS with in-plane spin texture, temperature-dependent superconducting gap distribution, and normalized superconducting quasiparticle DOS of monolayer MoCl, respectively.

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EPC mechanisms

Here we explain the origins of acoustic soft modes in monolayer ZrCl and MoCl. Phonon softening can originate from mechanisms of either FS nesting57 or incipient/latent lattice instability under fluctuations58. The latter mechanism happens when the Fermi level is near electronic singularities, and the fluctuations cause prominent phonon softening through electron-phonon interaction59,60. For monolayer ZrCl, we first rule out the FS nesting effect as the origin of soft mode-1 in Fig. 2a because there is no direct \(\chi ^{\prime\prime} \left( {{{\mathbf{q}}}} \right)\) nesting peak at K point (Fig. 4a). Instead, we notice in Fig. 1b that the DOS peak is very close to Fermi level (0.03 eV lower), suggesting the possibility of the second mechanism. So, following the approach in ref. 60, we have constructed a 3 × 3 supercell to examine the possible phonon softening at K point (Supplementary Fig. 9a). We find the frozen A1g mode related lattice fluctuations can indeed lift the electronic degeneracies and consequently reduce electronic densities close to the Fermi level, as shown in Supplementary Fig. 9b–e. It indicates the electronic states couple strongly to phonons at K point in the way analogous to the dynamical Jahn-Teller effect60, manifesting strong electron-phonon interaction. This is consistent with the mechanism of latent lattice instability58 for enhancing EPC through softening of mode-1 at K point. For monolayer MoCl, the nesting vector along Γ–K path is marked red in Fig. 4b. Corresponding FS nesting directly leads to the phonon softening of mode-3 in Fig. 2c. Also, we find several small and broad nesting peaks around M point, indicating that the soft mode-2 in Fig. 2c arises possibly from phonon renormalization caused by Kohn anomaly61.

Fig. 4: Unveiling the EPC mechanisms.
figure 4

FS nesting function \(\chi ^{\prime\prime} \left( {{{\mathbf{q}}}} \right)\) of a monolayer ZrCl and b monolayer MoCl. Notice the peaks at Γ point are irrelevant to the nesting. Electronic EPC strength λk of c monolayer ZrCl and d monolayer MoCl near Fermi level (within EF ± 0.2 eV) at 6 K.

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We then study the FS resolved EPC of monolayer ZrCl and MoCl, by inspecting the electronic EPC strength λk62 in Fig. 4c, d. We have considered a FS spread of 200 meV under 6 K, so that all phonon contributions are taken into account. For monolayer ZrCl, the three FS sheets overlap due to the energy spread (Fig. 4c). The hot spots form a closed hexagon, and the maximal λk = 1.66 appear long Γ–K directions. For monolayer MoCl, the λk hot spots appear along Γ-M directions of FS1, with a maximal value of 1.86 (Fig. 4d). One can clearly see the phonons are inclined to create strong electron-phonon interaction with FS1. The significant difference between EPC strengths of FS1 and FS2 eventually leads to the two-gap superconductivity as observed in Fig. 3e.

Electronic topology

Besides superconductivity, we have investigated the band topology in 2D MX. Strictly speaking, topological insulator state is defined with a global gap for the calculation of Z2 topological invariants. However, one may loosely calculate an effective Z2 using a “curved” Fermi level to identify a topological metal63. Accordingly, we have calculated the effective Z2 of 2D MX based on symmetry-indicator thoery64. Consequently, 2D MX all have an effective Z2 = 1 up to both α band and the band following below by assuming a “curved” Fermi level (see Fig. 1b, c). As shown in Fig. 1b, c, Supplementary Figs. 2a and 3a, the band topology of 2D MX originates from Dirac-type crossings under Fermi level, at which spin-orbit coupling open the nontrivial gaps. The electronic edge spectra of 2D MX along both zigzag and armchair edges are explored in Fig. 5 and Supplementary Fig. 5. We find the zigzag termination is advantageous for observing edge states. In Fig. 5a–d of the zigzag edge spectra, there exist several edge states around the Fermi level. However, only monolayer MoCl and MoF exhibit topological edge states (TESs) with distinct bulk-edge correspondence from the topology between the α and β bands. For monolayer ZrCl and ZrF, the edge states near Fermi level come from bands below α band, which are trivial for realizing TSC. All these indicate monolayer MoCl and MoF are TSC candidates with the coexistence of nontrivial TESs and spin-singlet s-wave superconductivity. One promising approach for triggering (helical) TSC states in monolayer MoCl or MoF is to induce Rashba-type band splitting by applying an electrical field65 and adjust the chemical potential by electrical gating33.

Fig. 5: Electronic and phononic topology in 2D MX.
figure 5

The zigzag electronic spectra of a ZrCl, b MoCl, c ZrF and d MoF. Here, the nontrivial topological edge states (TESs) in between α and β bands are labeled. The phononic spectra near DPs of e ZrCl along zigzag edge and f MoCl along zigzag edge. Here, the bulk projected DPs are marked with dots. The golden ones exhibit nontrivial Berry phases of ±π, while the black ones are projected from two DPs, having trivial Berry phases of ±2π. Notice the marked TESs have w-shape dispersions and connect to the nontrivial projected DPs.

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Phononic topology

The nontrivial phononic topology of monolayer ZrCl and MoCl can be identified in Fig. 2a, c. The phonon dispersions show that monolayer ZrCl possesses two crossing points at the K and K′ valleys, and monolayer MoCl possesses six crossing points along M-K paths. We have calculated the Berry phase of each crossing point by \(\gamma = \mathop {\sum }\limits_{n \in {{{\mathrm{occ}}}}}\oint_{L}A_{n}\left( {{{\mathbf{q}}}} \right) \cdot d{{{\mathbf{q}}}}\)66, where \(A_n\left( {{{\mathbf{q}}}} \right) = i{\langle{u_n\left( {{{\mathbf{q}}}} \right)|\nabla _{{{\mathbf{q}}}}|u_n\left( {{{\mathbf{q}}}} \right)\rangle}}\) is the Berry connection of the nth phonon branch. We find the crossing points of monolayer ZrCl at 176.8 cm−1 and the crossing points of monolayer MoCl at 183.2 cm−1 have Berry phases of ±π, evidencing they are Dirac phonons. The detailed Berry phase distributions of these Dirac phonons are available in Supplementary Fig. 11. Here we briefly explain the symmetry mechanism. These monolayers are in a P\(\bar 3\)m1 symmetry, whose symmetry operations follow D3d point group. All the symmetry-allowed band crossings, i.e., 2D Dirac points (DPs), are discussed in Supplementary Note 9. Importantly, these DPs are protected by the joint \({{{\mathcal{P}}}}{{{\mathcal{T}}}}\) symmetry (\({{{\mathcal{P}}}}\): inversion, \({{{\mathcal{T}}}}\): time-reversal), and are pair-related by \({{{\mathcal{T}}}}\): \({{{\mathbf{q}}}} \leftrightarrow - {{{\mathbf{q}}}}\). For monolayer ZrCl, the DP at K (K′) valley is guaranteed by a 2D irreducible representation E, which is C3z-symmetric. For monolayer MoCl, the six DPs (three pairs) along M-K paths are symmetry-allowed accidental crossings, with opposite eigenvalues of in-plane two-fold rotations (see irreducible representations marked in Fig. 2c).

We have also investigated the phononic edge spectra with both zigzag and armchair edge terminations in Supplementary Fig. 11. Interestingly, we discover one w-shape TES for both monolayer ZrCl and MoCl along zigzag edges (Fig. 5e, f). For MoCl in Fig. 5f, the six bulk DPs are projected into two groups: (1) two golden marked projected DPs from one bulk DP, which are topologically nontrivial; (2) two black marked projected DPs from two bulk DPs, which are trivial with total Berry phases of +2π and −2π. This point is explained in Supplementary Fig. 11d. Here, the w-shape TES only connects to the nontrivial projected DPs. If the two trivial projected DPs are connected, there will be two edge states from the same start point to the same end point, which are not stable. Very recently, the concept of topological-phonon-mediated superconductivity has been proposed67. It is a kind of boundary superconductivity that the electrons are mediated by the nontrivial surface/edge phonon modes, promising for a strong EPC68. In this sense, the one-dimensional zigzag ribbons of ZrCl or MoCl are potential for further realizing an edge-enhanced EPC or superconductivity than the 2D bulk, for the following reasons: (1) The TES contributed DOS are much larger than the bulk; (2) The TES has a w-shape (i.e., two-dip like) dispersion, which is favorable for the electron-phonon interaction.

Discussion

In summary, we have predicted the phonon-mediated superconductivity and topological aspects in vdW-monolayered MX (M = Zr, Mo; X = F, Cl). We highlight that the 2D MX family host remarkable Tc, arising from the soft-mode-enhanced EPC. Particularly, monolayer MoCl displays a striking two-gap superconductivity because of disparate pairing strengths at each FS sheet. We have also discussed about the interesting Janus monolayer Zr2FCl, which has the inversion breaking with respect to monolayer ZrCl. The discovered Tc-enhancement may facilitate the study of chiral-phonon-related superconductivity. Moreover, the electronic and phononic topology of 2D MX have been inspected. We demonstrate that monolayer MoF and MoCl are TSC candidates, and there exist Dirac phonons at Brillouin zone boundaries of monolayer ZrCl and MoCl. Our findings enrich the 2D superconducting and topological states in a single material platform.

Methods

Electronic and phononic properties

We performed first-principles calculations via QUANTUM ESPRESSO package69 with relativistic norm-conserving ONCV pseudopotentials70. The Perdew-Bruke-Ernzerhof exchange-correlation functional71 of generalized gradient approximation was adopted. We employed a plane-wave cutoff energy of 120 Ry and a force tolerance of 1.0 × 10−4 Ry Å-1 under atomic relaxations. An k-mesh of 24 × 24 × 1 and a Marzari-Vanderbilt smearing of 0.01 Ry were adopted for the monolayer structures. Electronic nesting function \(\chi ^{\prime\prime} \left( {{{\mathbf{q}}}} \right)\) was calculated using a dense 200 × 200 × 1 k-mesh, where \(\chi ^{\prime\prime} \left( {{{\mathbf{q}}}} \right)\) is obtained by72

$$\chi ^{\prime\prime} \left( {{{\mathbf{q}}}} \right) = \mathop {\sum }\limits_{{{\mathbf{k}}}} \delta (\varepsilon _{{{\mathbf{k}}}} - E_{{{\mathrm{F}}}})\delta (\varepsilon _{{{{\mathbf{k}}}} + {{{\mathbf{q}}}}} - E_{{{\mathrm{F}}}}).$$
(1)

The dynamical matrices and EPC matrices were computed by density functional perturbation theory with spin-orbit coupling. A q-mesh of 6 × 6 × 1 was chosen for the monolayers. We constructed maximally localized Wannier functions73 from d orbitals of Zr (Mo) atoms and p orbitals of Cl (F) atoms using Wannier90 code74. We generated the phononic tight-binding Hamiltonians by phonopyTB code75. The electronic and phononic edge spectra were calculated by the iterative Green’s function technique76, as implemented in WannierTools package75.

Superconducting properties

Under convergence tests (Supplementary Note 1), we employed an EPC matrix interpolation to the denser 72 × 72 × 1 k-mesh and 72 × 72 × 1 q-mesh via the EPW code77. Then the Eliashberg spectral function a2F(ω) was calculated by78

$$a^2F\left( \omega \right) = \frac{1}{{2\pi N(E_{{{\mathrm{F}}}})}}\mathop {\sum }\limits_{{{{\mathbf{q}}}}v} \frac{{\gamma _{{{{\mathbf{q}}}}v}}}{{\omega _{{{{\mathbf{q}}}}v}}}\delta (\omega - \omega _{{{{\mathbf{q}}}}v}),$$
(2)

where \(\gamma _{{{{\mathbf{q}}}}v}\) is the phonon linewidth and \(\omega _{{{{\mathbf{q}}}}v}\) is the phonon eigen frequency. The EPC constant λ was obtained from the integral of a2F(ω) by

$$\lambda = 2\mathop {\int }\nolimits_0^\infty \frac{{\alpha ^2F\left( \omega \right)}}{\omega }d\omega .$$
(3)

The critical superconducting temperature Tc was estimated using McMillan-Allen-Dynes formula49:

$$T_{{{\mathrm{c}}}} = \frac{{\omega _{log}}}{{1.2}}{{{\mathrm{exp}}}}\left( - \frac{{1.04\left( {1 + \lambda } \right)}}{{\lambda - \mu ^ \ast (1 + 0.62\lambda )}}\right),$$
(4)

where ωlog is a logarithmic average of the phonon frequency and μ* is the effective Coulomb repulsion constant, which typically values between 0.05 and 0.2. We adopted μ* = 0.1 for predicting Tc, as widely used in other 2D materials17,24,79. We also calculated the superconducting properties of monolayer ZrCl and MoCl by directly solving anisotropic (isotropic) Migdal-Eliashberg equations53 provided by EPW code. The electronic states between EF − 0.8 eV and EF + 0.8 eV were taken into account and the Matsubara frequency cutoff was set as 0.2 eV. The delta smearing was 25 meV for electrons and 0.05 meV for EPC sum-over.