Monolayer Kagome metals AV₃Sb₅

The kagome lattice is a two-dimensional network of corner-sharing triangles that hosts distinctive electronic features such as a flat band, Dirac points, and saddle-point van Hove singularities (VHSs). Vanadium-based kagome metals AV₃Sb₅ (A = K, Rb, Cs) exhibit correlated electronic states including charge density waves (CDWs) and superconductivity, together with unconventional phenomena like giant anomalous Hall effects and potential Majorana modes. Prior studies largely focused on three-dimensional (3D) layered crystals and often assumed decoupled 2D kagome layers, but dimensionality can be crucial in layered systems. While thin films of AV₃Sb₅ have been exfoliated, the role of dimensionality in this family remains unclear. This work addresses whether a true monolayer of AV₃Sb₅ exists, how its symmetry differs from bulk, and how such differences impact VHSs and the resulting competing electronic orders.

Experiments on bulk AV₃Sb₅ revealed cascades of correlated states linked to CDWs and superconductivity, alongside anomalous Hall effects without long-range magnetism and possible edge supercurrents. Theoretical analyses frequently employ effective 2D kagome models to interpret these phases, suggesting VHS-driven instabilities in conjunction with electron correlations. Dimensionality has been shown to matter in many layered systems. Initial exfoliation studies produced thin flakes of AV₃Sb₅, hinting at tunable behaviors with thickness; experiments report enhanced CDW tendencies and suppressed superconductivity in thinner films. Theoretical works have proposed various CDW patterns (star-of-David, inverse star-of-David, and time-reversal symmetry breaking CDWs) and unconventional superconducting pairings in AV₃Sb₅, often tied to VHS physics and sublattice interference effects in kagome lattices.

The study combines first-principles density-functional theory (DFT), tight-binding (TB) modeling, electronic susceptibility calculations, phonon calculations, and zero-temperature mean-field theory (MFT) to investigate monolayer AV₃Sb₅.

• DFT: VASP with PAW method; PBE GGA exchange-correlation; D3 van der Waals correction; plane-wave cutoff 300 eV; structure optimization force criterion 0.01 eV/Å; monolayer simulated with ~20 Å vacuum. k-point meshes: 10×17 for √3×√3 structure, 46×69 for DOS. Phonons via finite difference; pristine monolayer with 4×4 supercell and 9×9 k-points (Phonopy); ISD-1 phase with 8×8 supercell and 6×6 k-points using a nondiagonal supercell method. Exfoliation energies via Jung–Park–Ihm method. Onsite (U) and nearest-neighbor (V) interactions estimated by constrained RPA with weighting in VASP.
• Tight-binding model: Constructed for √3×1 unit cell with D₂h symmetry. Six-component spinor (two sites per sublattice A/B/C). Hamiltonian includes nearest (t), next-nearest (t₂) hoppings, onsite energies, and symmetry-lowering terms (δε, δt₅) that reduce symmetry from √3×1 D₆h to 3×1 D₂h. Parameters chosen to reproduce DFT bands and VHS irreducible representations for A = K (and validated for Rb, Cs). TRSB CDW phases included via a CDW magnitude φ.
• Electronic susceptibility: Real part of bare susceptibility χ(q) computed in the constant matrix approximation using TB bands with 200×348 k-points and β = 1000. Sublattice-resolved susceptibility analyzed to assess sublattice interference.
• Mean-field theory: Zero-temperature MFT on an extended Hubbard model with onsite U and nearest-neighbor V density–density interactions. Considered six 2×2 CDW configurations (ISD-1/2, SD-1/2, TRSB-1/2) and nine spin-singlet superconducting channels classified by irreducible representations. CDW order parameters modulate hoppings with Q-vectors connecting M points; relative phases determine patterns. Ground-state energies evaluated on 80×80 k-mesh for Φ ∈ [−0.08, 0.08] to map phase diagrams in U–V for different chemical potentials (μ).
• Berry curvature and anomalous Hall conductivity (AHC): Berry curvature Ωₙ(k) calculated from TB bands; AHC σ_xy(E) obtained by integrating Ωₙ(k) over the BZ using 500×500 k-mesh.
• Strain and doping tuning: DFT band calculations under uniform biaxial strain (±2%) and alkali atom doping A_{1+x}V₃Sb₅ to evaluate chemical potential shifts; CRPA used to assess strain dependence of U and V.

• Monolayer structure and symmetry: Due to stoichiometry and alkali atom arrangement, the monolayer adopts a √3×1 translational symmetry and D₂h point group, distinct from the bulk 1×1, D₆h symmetry. Rectangular alkali sublayers in the monolayer halve neighboring alkali coordination and break rotational symmetry, preventing a simple dimensional crossover from bulk.
• Rearranged VHSs: Symmetry lowering and BZ folding hybridize bulk M-point VHSs, annihilating a VHS at M and creating four off-M type-II VHS points P_i (i=1–4), while retaining a type-I VHS at a time-reversal-invariant momentum (I). DFT/TB for KV₃Sb₅ show DOS divergences at E = −6 meV (type-II, off-M at P_i ≈ M+(±0.054, ±0.021) Å⁻¹) and E = +9 meV (type-I, at I). The −6 meV peak is enhanced due to the quartet of type-II VHS points and mixed B/C sublattice flavor, whereas the +9 meV VHS is pure A-sublattice.
• Electronic susceptibility and instabilities: χ(q) exhibits strong, filling-dependent features. Near type-II VHS filling (μ ≈ −6 meV), prominent incommensurate nesting vectors q₁, q₂ connect the P_i points, suggesting possible IC-CDWs at T = 0. With increased μ (including neutral and type-I VHS fillings), peaks merge and χ(q) is strongly enhanced at M (commensurate 2×2 vectors), indicating robust 2×2 CDW tendencies over a wide μ range.
• Phonon softening: DFT phonons of the pristine monolayer show soft modes at M and I, linked to V-atom displacements and 2×2 3Q CDWs. The softening at M diminishes with increasing electronic temperature (smearing σ), implicating electronic (VHS-driven) contributions alongside electron–phonon coupling. The 2×2 instability is leading among various softened modes.
• Doublet CDWs: Due to √3×1 symmetry, each CDW forms a Z₂ doublet (e.g., SD-1/SD-2, ISD-1/ISD-2), captured by a two-component order parameter in Landau theory with a term splitting the doublet.
• Mean-field phase diagrams (T → 0): For μ₁ = −6 meV (near type-II VHS), dominant CDWs are SD-2 and ISD-2, with sizable ISD-1 regions for −0.6 eV < U < 0.4 eV and −20 meV < V < 40 meV. Conventional s-wave SC (A₁g) appears and competes with CDWs at negative U/positive V. For μ₂ = +35 meV, TRSB-2 CDW emerges broadly in U–V space, and an unconventional d-wave SC (B₁g) appears, competing with ISD-1 for positive U/negative V. SD-1 appears near the type-I VHS (μ₃ ≈ +9 meV). Thus, modest μ tuning (−6 to +40 meV) toggles among ISD-1/2, SD-1/2, TRSB-2 and SC states.
• Topological CDWs and transport: TRSB-1 and TRSB-2 gaps carry nontrivial Chern numbers. For TRSB-1, the lowest unoccupied and highest occupied bands have Chern numbers 2 and −1; for TRSB-2, −3 and 2, respectively, arising from Berry curvature sign changes near M₁, K₁, K₂. The monolayer’s asymmetry yields distinct anomalous Hall conductivity σ_xy(E) structures for TRSB-1 vs TRSB-2, including sign changes near E ≈ 0, providing experimental signatures.
• Tunability: Biaxial strain ±2% shifts μ across approximately −100 to +50 meV, and modifies U, V (CRPA), enabling access to different phases. Light alkali doping A_{1+x}V₃Sb₅ or Sb→Sn substitution can also tune μ; 2D geometry allows further control (e.g., ionic gating).
• Stability and exfoliation: Cohesive energy ≈ 3.8 eV/atom (monolayer) comparable to bulk ≈ 3.9 eV/atom, indicating thermodynamic stability. Energy landscape vs V-atom distortion shows local minima for CDWs with barriers ~4 meV/atom. ISD-1 phonon spectrum is free of imaginary frequencies (dynamic stability). Exfoliation energies are 42, 45, and 45 meV/Ų for A = K, Rb, Cs, respectively, comparable to known 2D materials, and thin layers have been experimentally exfoliated.

The monolayer’s stoichiometry-enforced symmetry lowering (to √3×1, D₂h) fundamentally alters VHS topology and enhances CDW propensities relative to bulk. The rearranged VHSs (addition of type-II VHSs) diversify electronic instabilities and create fertile competition among multiple CDW doublets and superconducting channels, tunable by small chemical potential shifts achievable via strain or doping. Strong χ(q) peaks at M over broad fillings rationalize the robustness of 2×2 CDW instabilities even when perfect VHS nesting is weakened, suggesting insights transferable to bulk AV₃Sb₅. At nonzero temperatures, monolayers are subject to strong fluctuations (Mermin–Wagner), suppressing long-range IC-CDW, so commensurate 2×2 orders are more likely; at T = 0, IC-CDW phases tied to type-II VHSs may emerge in restricted interaction/μ windows. TRSB CDWs with nontrivial Chern numbers lead to detectable anomalous Hall signatures distinct from bulk, offering transport probes of symmetry-broken phases. Overall, monolayer AV₃Sb₅ provides a controllable platform to study competing orders and topological responses in 2D kagome metals.

This work predicts that AV₃Sb₅ can exist as a thermodynamically and dynamically stable monolayer with fundamentally reduced symmetry relative to the bulk, leading to rearranged VHSs (including type-II), enhanced DOS near −6 meV, and a rich set of competing phases: Z₂-doublet CDWs (ISD/SD), TRSB CDWs with nontrivial Chern numbers, and s- and d-wave superconductivity. Electronic instabilities inferred from χ(q) and phonon softening favor commensurate 2×2 CDWs over a wide filling range, while mean-field phase diagrams highlight strong competition with superconductivity at T → 0. The chemical potential can be tuned via modest biaxial strain, alkali doping, substitution, or gating, enabling designer access to different orders; anomalous Hall responses provide experimental diagnostics of TRSB phases. Future research should explore zero-temperature IC-CDW regimes, fluctuation-driven phenomena (e.g., Kosterlitz–Thouless transitions), and detailed mechanisms of CDW formation, leveraging the monolayer as a model system to complement bulk studies.

• Theoretical framework relies on zero-temperature mean-field approximations; thermal fluctuations are significant in 2D and can suppress long-range orders at finite temperatures, particularly incommensurate CDWs.
• The number and precise positions of type-II VHS points depend on microscopic details; conclusions may be sensitive to model parameters and symmetry-breaking terms.
• DFT phonon spectra include electron–phonon coupling, but the relative roles of electronic versus phononic mechanisms in CDW formation are not uniquely resolved.
• Interactions (U, V) are estimated via CRPA and may vary with strain/substrate; substrate and long-range interactions are not explicitly modeled.
• SC phases are mapped at T → 0 and within selected pairing channels; additional channels/fluctuation effects could alter phase boundaries.
• IC-CDW phases at finite temperature are constrained by 2D fluctuation theorems and were mainly discussed at T = 0; substrate or long-range effects could modify this.