Superconductivity and topological aspects of two-dimensional transition-metal monohalides

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 application. Over the past decade, the advent of nanofabrication methods, such as molecular beam epitaxy and mechanical exfoliation, has enabled the exploration of ultrathin and highly crystalline 2D superconductors, and intriguing phenomena have been unveiled, such as quantum Griffiths singularity, anomalous/bosonic metallic state and enhanced upper critical field. Some 2D superconductors can achieve much higher superconducting transition temperature Tc than their bulk counterparts, such as the FeSe/SrTiO3 interface and surface-hydrogenated monolayer MgB2. Especially, monolayer Ta-MoTe2 was discovered to have a 60-times Tc enhancement (~7.6 K) over the bulk phase (~0.1 K) by electrical gating. In general, practical application of 2D superconductors has to include the effect of substrate, which may suppress the superconductivity. In this regard, van der Waals (vdW) layered materials are advantageous. Transition metal dichalcogenides (TMDs) have provided a fertile ground for 2D superconductivity, e.g., coexistence of superconductivity and charge density waves in 2H-NbSe2, Ising pairing in gated 2H-MoS2 and 1T-PdSe2, and metastable superconductivity of IrTe2. 2M-WS2 presents the highest Tc = 8.8 K among intrinsic TMDs. Beyond TMDs, computational studies predicted high Tc ~ 21 K in monolayer W2N3. More interestingly, when superconductivity meets non-trivial band topology, their interplay can lead to topological superconducting (TSC) states hosting Majorana modes, promising for fault-tolerant quantum computing. Beyond proximity-based heterostructures, the coexistence of superconducting and topological states in a single material offers a platform to realize Majorana fermions without complex interfaces. Prior efforts searched for superconductivity in topological materials and for topological states in superconductors; recent experiments evidenced Majorana bound states in superconducting 2M-WS2. Additionally, surface/edge electronic and phononic states can reinforce electron-phonon coupling (EPC) and superconductivity, e.g., topological states on Be(0001) yielding large surface EPC. Thus, it is compelling to explore 2D superconductors with both electronic and phononic topology. In this paper, we investigate superconductivity and topological aspects of vdW-monolayered MX (M=Zr, Mo; X=F, Cl) via first-principles calculations. 2D MX exhibit strong EPC (>1) and remarkable Tc (5.9–12.4 K) due to acoustic phonon softening. Anisotropic Migdal-Eliashberg theory shows monolayer ZrCl is a single-gap superconductor with Tc ~ 12.4 K and gap Δ0 = 2.14 meV, higher than predicted 1.5 meV of 2M-WS2. Monolayer MoCl displays two-gap superconductivity; the larger gap has 2Δ0/kBTc ~ 4.69, beyond weak-coupling BCS. 2D MX possess Z2 band topology; MoF and MoCl are TSC candidates. We identify Dirac phonons with quantized ±π Berry phases at Brillouin zone boundaries, and zigzag edge states show w-shape dispersions, indicating potential edge-enhanced EPC in one-dimensional zigzag ribbons of ZrCl and MoCl. Our results reveal intriguing superconducting and topological properties in 2D MX, deserving experimental verification.

The introduction surveys advances in 2D superconductivity enabled by epitaxial growth and exfoliation, highlighting phenomena such as quantum Griffiths singularity, bosonic metallic states, and enhanced upper critical fields. It notes systems with enhanced Tc relative to bulk, including FeSe/SrTiO3 interfaces and hydrogenated monolayer MgB2, and gating-induced Tc enhancement in monolayer Ta2MoTe2. Van der Waals materials, particularly TMDs (e.g., 2H-NbSe2, MoS2, PdSe2, IrTe2), are emphasized as platforms for intrinsic 2D superconductivity, with 2M-WS2 showing Tc = 8.8 K. Beyond TMDs, monolayer W2N3 was predicted to host high Tc (~21 K). The literature further connects superconductivity with nontrivial topology leading to potential topological superconductivity and Majorana modes, referencing both proximity-induced approaches and intrinsic coexistence in single materials. Prior experimental and theoretical efforts identified superconductivity in topological systems (e.g., CuxBi2Se3, gated monolayer topological insulators) and topological features in superconductors (e.g., MgB2 with Dirac-nodal-line fermions), with experimental evidence of Majorana bound states in 2M-WS2. The role of surface/edge states and topological phonons in enhancing EPC is also discussed, motivating the search for materials combining superconductivity with electronic/phononic topology.

First-principles DFT and DFPT calculations were performed using Quantum ESPRESSO with relativistic norm-conserving ONCV pseudopotentials and the PBE GGA functional. A plane-wave cutoff of 120 Ry and force tolerance of 1.0×10^−4 Ry Å^−1 were used. Brillouin-zone sampling employed a 24×24×1 k-mesh with Marzari-Vanderbilt smearing of 0.01 Ry for monolayers. Dynamical matrices and EPC matrices were computed via DFPT including spin-orbit coupling on a 6×6×1 q-mesh. Electronic nesting function χ″(q) was evaluated on a dense 200×200×1 k-mesh using χ″(q) = Σ δ(ε_k − ε_p) δ(ε_{k+q} − ε_p). Maximally localized Wannier functions (Wannier90) were constructed from metal d and halogen p orbitals to build tight-binding models; phononic tight-binding Hamiltonians were generated using phonopyTB. Electronic and phononic edge spectra were computed with iterative Green’s functions (WannierTools). Superconducting properties were obtained via EPC matrix interpolation (EPW) to 72×72×1 k- and q-meshes. The Eliashberg spectral function α^2F(ω) was computed as α^2F(ω) = (1/(2π N(E_F))) Σ_{αν} |γ_{αν}|^2 δ(ω − ω_{αν}), and the EPC constant λ from λ = 2 ∫_0^∞ dω α^2F(ω)/ω. Tc was estimated using the McMillan–Allen–Dynes formula with μ* = 0.1: Tc = (ω_log/1.2) exp{−1.04(1+λ)/[λ − μ*(1+0.62λ)]}. Anisotropic (and isotropic) Migdal–Eliashberg equations were solved using EPW, with electronic states within E_F ± 0.8 eV, Matsubara cutoff 0.2 eV, electron delta smearing 25 meV, and EPC sum-over smearing 0.05 meV. Topological analyses used symmetry-indicator theory for effective Z2 invariants with a curved Fermi level, and Berry phases of phonon band crossings were computed to identify Dirac phonons. Edge spectra along zigzag and armchair terminations were evaluated to locate topological edge states.

- Predicted vdW monolayer transition-metal monohalides MX (M=Zr, Mo; X=F, Cl) are 2D superconductors with strong EPC (λ > 1) and Tc ranging from 5.9 to 12.4 K. - Superconducting transition temperatures (μ* = 0.1): ZrCl Tc = 12.4 K (A-ME), 12.0 K (MAD); MoCl Tc = 9.7 K (A-ME), 10.1 K (MAD); ZrF Tc = 10.5 K (MAD); MoF Tc = 5.9 K (MAD). EPC constants: ZrCl λ = 1.37; MoCl λ = 1.06; ZrF λ = 1.03; MoF λ = 1.27. - Monolayer ZrCl is a single-gap s-wave superconductor with Δ0 = 2.14 meV and 2Δ0/kBTc = 4.00 (> BCS 3.53). α-model fit exponent p = 3.4. All three Fermi-surface sheets contribute to a single-gap distribution with median EPC λ ~ 1.35. - Monolayer MoCl exhibits two-gap superconductivity with Tc = 9.7 K: larger gap Δα = 1.96 meV (2Δα/kBTc = 4.69, strong-coupling), smaller gap Δβ = 1.37 meV (2Δβ/kBTc = 3.28). Two-gap α-model exponent p = 2.8. The two-gap feature is attributed to well-separated FS1 (electron pocket at Γ) and FS2 (flower-shaped hole pocket near BZ boundary) with distinct orbital characters and EPC strengths. - EPC and phonon softening mechanisms: Low-frequency phonons (<150 cm−1) dominate EPC. In ZrCl, an acoustic soft mode at K (mode-1) arises from latent lattice instability near an electronic DOS peak (0.03 eV below EF) and dynamical Jahn–Teller-like coupling; this mode contributes ~23% of total λ. In MoCl, phonon softening of mode-3 along Γ–K is linked to Fermi-surface nesting; mode-2 near M likely relates to Kohn anomaly; modes 2 and 3 together contribute ~36% of λ. - Spin textures indicate spin-singlet s-wave pairing (anti-parallel spins at k and −k on FS). FS-resolved EPC hot spots: ZrCl shows hexagonal hot spots with max Δk ≈ 1.66 along Γ–K; MoCl shows strong hot spots on FS1 along Γ–M with max Δk ≈ 1.86, consistent with two-gap superconductivity. - Electronic topology: Effective Z2 = 1 for all 2D MX (using a curved Fermi level). Zigzag edge spectra reveal that MoCl and MoF host topological electronic edge states (TESs) between α and β bands, coexisting with s-wave superconductivity, making them TSC candidates. ZrCl and ZrF edge states near EF originate from lower bands and are topologically trivial for TSC. - Phononic topology: ZrCl has two Dirac phonon points (DPs) at K/K′ with Berry phases ±π at 176.8 cm−1; MoCl has six DPs along M–K with Berry phases ±π at 183.2 cm−1. Zigzag phononic edge spectra show w-shaped topological edge states connecting nontrivial projected DPs, suggesting potential edge-enhanced EPC. - Janus Zr2FCl (inversion-broken derivative of ZrCl) shows enhanced Tc ≈ 13.2 K and chiral-phonon-related α^2F(ω) peaks; a roton-like minimum appears in the lowest phonon branch at K, hinting at chiral-phonon-mediated EPC enhancement.

The study addresses whether vdW monolayer transition-metal monohalides can host robust phonon-mediated superconductivity and simultaneously exhibit nontrivial electronic and phononic topology. First-principles and Migdal–Eliashberg calculations demonstrate that all considered MX monolayers have strong EPC and superconductivity with Tc up to 12.4 K. The gap structures—single-gap in ZrCl and two-gap in MoCl—are explained by their Fermi-surface geometries and orbital compositions, with FS-resolved EPC revealing band-selective coupling in MoCl that yields distinct pairing strengths on FS1 and FS2. The identified mechanisms of phonon softening—latent lattice instability and dynamical Jahn–Teller-like coupling in ZrCl, and nesting/Kohn anomalies in MoCl—clarify how low-frequency acoustic modes enhance EPC and Tc. The coexistence of superconductivity with effective Z2 nontrivial band topology and topological electronic edge states in MoCl and MoF positions these materials as promising intrinsic TSC candidates. The presence of Dirac phonons and associated w-shaped phononic edge states at zigzag terminations suggests an avenue for edge-enhanced EPC and possibly boundary superconductivity mediated by topological phonons. Collectively, these results substantiate the feasibility of realizing and tuning superconductivity intertwined with electronic and phononic topology in a single 2D material platform, offering prospects for Majorana physics and device applications via gating and electric-field control of Rashba splitting and chemical potential.

The work predicts and elucidates phonon-mediated superconductivity and topological characteristics in vdW monolayer MX (M=Zr, Mo; X=F, Cl). Monolayer ZrCl exhibits single-gap s-wave superconductivity with Tc ~ 12.4 K and strong-coupling gap ratio, while monolayer MoCl hosts two-gap superconductivity with a strongly coupled larger gap. The superconductivity is driven by enhanced EPC from acoustic soft modes whose origins are traced to latent lattice instability (ZrCl) and Fermi-surface nesting/Kohn anomalies (MoCl). All MX display effective Z2 = 1; MoCl and MoF possess topological electronic edge states near EF and are promising intrinsic TSC candidates. Phononic Dirac points with quantized Berry phases are identified in ZrCl and MoCl, and w-shaped phononic edge states suggest possible edge-enhanced EPC. Future directions include experimental synthesis and characterization of monolayer MX and Janus Zr2FCl, probing superconducting gaps and Tc, verifying electronic and phononic edge states, exploring electric-field-induced Rashba splitting and gating to access TSC phases, and investigating edge-mediated superconductivity driven by topological phonons in nanoribbon geometries.

- The results are computational predictions requiring experimental verification; the authors explicitly call for further experimental confirmation. - Tc estimates depend on the chosen Coulomb pseudopotential (μ* = 0.1); variations in μ* can change Tc. - Isotropic Migdal–Eliashberg approximations overestimate Tc by ~20% relative to anisotropic solutions, indicating sensitivity to anisotropy. - The electronic Z2 invariant is evaluated using an effective “curved” Fermi level appropriate for topological metals rather than a globally gapped insulator, which may limit direct topological classification at the true EF. - Substrate and environmental effects, strain, and disorder—relevant for 2D materials—are not explicitly included and may affect superconductivity and topology in practical devices. - Anharmonic effects and temperature-dependent phonon renormalization beyond DFPT are not treated and could modify soft-mode behavior and EPC in some regimes.