Realization of a two-dimensional Weyl semimetal and topological Fermi strings

Weyl semimetals in three dimensions host spin-split massless quasiparticles with chiral nodal points and Fermi-arc surface states, leading to phenomena such as the chiral anomaly, unusual optical conductivity, and nonlocal transport. Extending the Weyl concept to two dimensions yields a distinct topological state—a 2D Weyl semimetal—featuring a spin-polarized analog of graphene with nontrivial winding numbers around Weyl nodes that enforce topologically protected edge states known as Fermi strings. Despite extensive theoretical interest predicting unique properties (e.g., parity anomaly in (2+1)D, charge fractionalization, spin-valley Hall effects, giant Berry curvature dipole, and topological quantum criticality), an intrinsic 2D Weyl semimetal had not been experimentally realized. This study aims to realize and characterize an intrinsic 2D Weyl semimetal in epitaxial bismuthene on SnS(Se) substrates, addressing both bulk Weyl fermion states and their edge counterparts and establishing the bulk-boundary correspondence in 2D.

Prior work established 3D Weyl semimetals with chiral anomaly signatures, exotic transport, and Fermi arcs (e.g., TaAs family). Theoretical studies in 2D predicted Weyl-like states and associated phenomena including parity anomaly, valley/spin Hall responses, Berry curvature dipole-driven nonlinear Hall effects, and topological phase transitions near critical points. Various proposals involved symmetry engineering, Zeeman fields, thickness control, or material-specific designs (e.g., silicene, iron-based 2D materials, Cd3As2 thin films, Bi–Sb films). However, experimental realization of an intrinsic 2D Weyl semimetal with direct observation of Weyl nodes and Fermi string edge states remained elusive. Bismuthene has been studied in orthorhombic (phosphorene-like) and honeycomb phases, with symmetry-enforced Dirac fermions or large-gap QSH behavior on certain substrates, motivating exploration of substrate-induced inversion breaking to drive a Weyl state in monolayer bismuthene.

Samples: Monolayer bismuthene (phosphorene-like α-phase) was grown by MBE on cleaved SnS and SnSe single crystals (Br-doped n-type). Base pressure <2×10^-10 mbar. High-purity Bi was evaporated from a Knudsen cell at 0.3 Å/min. Substrates were held at 50 °C to obtain smooth monolayers. STM/STS: In-situ Aarhus-150 STM for room-temperature topography and lattice parameters (bias 1.5 V, current 0.01 nA; atom-resolved: 5 mV, 0.15 nA). dI/dV spectra, maps, and QPI at 4.6 K with an Omicron LT-Nanoprobe; tunneling current 200 pA. Samples were transferred via UHV suitcase (<1×10^-8 mbar). ARPES and spin-ARPES: Measurements in a lab-based setup coupled to MBE using a Scienta DA30L analyzer (base pressure <5×10^-10 mbar, temperature 7–8 K). Linearly polarized 11 eV laser light (polarization perpendicular to sample and detector slits). Electronic dispersion: pass energy 2 eV, 0.3 mm slit, energy resolution ~2.5 meV, momentum resolution ~0.01 Å^-1. Spin resolution via dual VLEED detectors measuring Sx, Sy, Sz; for spin: pass energy 10 eV, 1×2 mm aperture, energy resolution ~50 meV, momentum resolution ~0.033 Å^-1. First-principles and modeling: DFT (VASP) with PBE exchange-correlation, SOC included self-consistently, 11×11×1 Monkhorst-Pack mesh, >20 Å vacuum, atomic relaxation to forces <0.01 eV/Å. Lattice of SnS(Se) adjusted to match epitaxial Bi. Wannierization (VASP2WANNIER90) using Bi p, Sn p, Se p orbitals to build tight-binding Hamiltonians without MLWF localization. Edge states computed for semi-infinite slabs using Green’s function techniques. Photoemission matrix-element effects simulated via non-trivial structure factors and unfolding procedures. QPI calculated with Green’s functions including spin-dependent scattering probability (SSP). Effective low-energy k·p models were built to capture Dirac/Weyl behavior and the impact of inversion breaking and Rashba terms. Structural characterization: STM topography established high-quality monolayer bismuthene with apparent height ~8.0 Å and lattice constants a,b ≈ (4.5, 4.8 Å) for Bi; (4.1, 4.5 Å) for SnS; (4.3, 4.6 Å) for SnSe.

- Theory: Free-standing bismuthene exhibits a gapped Dirac cone between Γ and X1 from Bi pz orbitals with SOC gap Egap ≈ 0.11 eV (Δsoc ≈ 55 meV). Fermi velocities: vx ≈ 3.17×10^5 m/s, vy ≈ 4.23×10^5 m/s; tilt parameter A ≈ 0.19×10^5 m/s. Substrate (SnSe/SnS) breaks inversion symmetry inducing spin splitting via in-plane dipole (Δpip) and Rashba coupling (λR), which closes the SOC gap and yields a gapless, spin-polarized 2D Weyl cone at EF. - Substrate-state separation: SnSe-derived bands lie ~0.9 eV below or ~0.4 eV above EF, ensuring the near-EF states are confined to the Bi monolayer, confirming the intrinsically 2D nature of the Weyl fermions. - ARPES: Bismuthene/SnSe shows two electron pockets along X1–Γ–X1; linear Weyl cones with no observable gap at the node (confirmed by second-derivative spectra and EDC maps). The Weyl node lies ~0.1 eV below EF (electron-doped). Bismuthene/SnS shows EF aligned with the Weyl node (charge-neutral 2D Weyl semimetal). Calculations including photoemission matrix elements reproduce intensity asymmetries. - Spin texture (spin-ARPES): Along cut1, Sy polarization matches theory with opposite spin polarizations in the two valleys (time-reversal partners), Sx vanishes due to glide-mirror symmetry. Along cut2, both linear branches show same-sign Sy but opposite-sign Sx, consistent with calculated canted spin texture. Out-of-plane Sz polarization is present but reduced due to rotated domains. - Bulk-boundary correspondence: Edge-state calculations for semi-infinite bismuthene/SnSe reveal Fermi string edge bands emanating from bulk Weyl nodes; in an sz-conserving model, two chiral edge bands (one per spin sector) connect nodes to band edges with opposite connections at top/bottom edges, consistent with charge conservation. In realistic strong SOC, hybridization occurs but node-edge connectivity remains topologically protected. - STS/LDOS: dI/dV on SnSe surface shows a large gap (~1.3 eV), while at bismuthene edges the LDOS is strongly enhanced in a narrow energy window near the Weyl-node energy (bias ~ -12 and +17 meV), consistent with Fermi string edge states, while interior Bi shows lower LDOS at those energies. - QPI: High-resolution dI/dV maps near the node energy show plane-wave-like interference with a dominant wavevector. Fourier transform exhibits a central oval (intravalley) and two satellites (intervalley) with Δq ≈ 0.42 Å^-1, matching the ARPES-measured node separation; SSP calculations reproduce the pattern.

The study directly addresses whether an intrinsic 2D Weyl semimetal can be realized in a crystalline monolayer. ARPES demonstrates gapless, spin-polarized linear band crossings at EF attributable to inversion-symmetry breaking by SnS(Se), transforming gapped Dirac bands of free-standing bismuthene into Weyl cones. Spin-ARPES confirms the predicted canted spin texture distinguishing the Weyl state from graphene-like Dirac states. Edge calculations and STS reveal Fermi string edge bands and enhanced edge LDOS near the Weyl-node energy, establishing the 2D bulk-boundary correspondence analogous to Fermi arcs in 3D Weyl semimetals. The robustness of gapless cones on both SnS and SnSe, despite differing doping levels and surface potentials, demonstrates stability against weak perturbations. These findings verify a long-sought topological phase in two dimensions and provide a platform to explore predicted 2D Weyl phenomena such as parity anomaly manifestations and nonlinear Hall effects driven by Berry curvature dipoles.

This work realizes an intrinsic 2D Weyl semimetal in monolayer bismuthene epitaxially grown on SnS(Se). The combination of ARPES and spin-ARPES reveals spin-polarized Weyl cones and nodes near EF (at EF on SnS and ~0.1 eV below EF on SnSe). STS and QPI measurements, together with edge-state calculations, identify Fermi string edge states and their characteristic scattering vectors matching the Weyl-node separation, confirming 2D bulk-boundary correspondence. The results establish a solid-state platform for Weyl fermions in 2D and open avenues to study reduced-dimensional topological quantum phenomena. Future research could pursue transport signatures (e.g., parity anomaly-related responses, nonlinear Hall effects from Berry curvature dipoles), tunability via gating/strain/substrate engineering, domain control to enhance spin-resolved measurements, and device integration for spintronic and optoelectronic applications.

- Rotated bismuthene domains in MBE-grown films reduce the measurable out-of-plane (Sz) spin polarization and complicate spin-texture analysis. - The realized 2D Weyl state corresponds to a critical point between distinct gapped phases and relies on substrate-induced inversion breaking; maintaining optimal interface conditions and doping may be necessary for reproducibility across substrates and samples. - Strong SOC in real materials means Sz is not a conserved quantum number, allowing hybridization of spin sectors and additional edge bands within bulk gaps, which may complicate isolation of pure Fermi string contributions. - The study focuses on spectroscopic evidence; direct transport demonstrations of predicted 2D Weyl responses were not included.