Observation of anomalous Hall resonance of massive Dirac fermions in topological kagome-lattice magnet

Topological materials hosting Dirac and Weyl quasiparticles exhibit transport and optical phenomena governed by Berry phase effects. Two-dimensional Dirac systems acquire a mass gap when time-reversal symmetry is broken, producing massive Dirac fermions and enabling quantized anomalous Hall (QAH) conductance e^2/h when the Fermi level lies in the gap. Corresponding magneto-optical (MO) effects at photon energies below the gap show quantized Faraday and Kerr rotations, observed in magnetic topological insulators. These quantized optical responses originate from interband transitions across the massive Dirac node; resonant enhancement is expected at photon energies exceeding the gap with a universal spectral shape described by a few band parameters, but direct observation of this resonance has been lacking. Kagome-lattice magnets generically host massive Dirac bands at K and K' due to spin–orbit coupling and their coupling to magnetic order, offering intrinsic, dopant-free platforms for robust AHE and potential high-temperature QAH physics. This study investigates whether kagome magnets exhibit the predicted resonant MO response associated with massive Dirac fermions and quantifies its contribution to anomalous Hall electrodynamics.

Prior works established Berry-phase-driven transport in topological materials and the anomalous Hall effect. Quantized MO Faraday and Kerr rotations have been reported in magnetically doped topological insulators at terahertz frequencies, reflecting universal electrodynamics of massive Dirac fermions in the QAH regime. Theoretical analyses predict universal MO resonance associated with interband transitions across the massive Dirac node, parameterized by the mass gap, chemical potential, and broadening. Kagome-lattice magnets (e.g., RMn6Sn6 family) are known to host massive Dirac bands at K/K' and display large AHE due to strong Berry curvature. STM studies in TbMn6Sn6 observed quasi-2D Dirac electrons, Landau levels, and side-surface modes within the mass gap, while transport and optical studies reported metallic conductivity with Drude response and relatively flat interband σxx(ω). However, the direct observation of the predicted MO resonance linked to the massive Dirac interband transition had not been experimentally demonstrated.

- Materials and growth: Single crystals of TbMn6Sn6 were grown by Sn-flux with starting ratio Tb:Mn:Sn = 1:6:30. The sealed quartz ampoule was heated to 900 °C and slowly cooled to 500 °C over six days; excess flux was removed by centrifugation. Plate-like crystals with (0001) kagome-plane surfaces were confirmed by Laue diffraction. - Transport: Magnetoresistivity and Hall resistivity were measured using a PPMS. - Optical reflectivity and σxx(ω): Reflectivity R(ω) was measured from 0.01–1 eV (FTIR) and 1–4 eV (monochromator). Kramers–Kronig analysis with ω^-2 high-energy extrapolation (and alternative core-level-including extrapolation checked) yielded σxx(ω). A Drude response below ~0.1 eV and flat interband continuum above were observed. - Magneto-optical Kerr effect (MOKE): Kerr rotation θK(ω) and ellipticity ηK(ω) were measured from 0.09–1 eV (FT spectrometer with PEM) and 1–2 eV (monochromator with PEM). Single-domain conditions were prepared: below 150 K, ±1 T field applied then removed; above 150 K, ±0.5 T applied during measurement. Spectra were antisymmetrized for ±M to extract MOKE. - Optical Hall conductivity σxy(ω): Calculated from measured σxx(ω), complex dielectric εxx(ω), θK(ω), and ηK(ω) using σxy(ω) = −σxx(ω) εxx(ω)^{1/2} [θK(ω) + i ηK(ω)]. Temperature-dependent σxy(ω) was obtained from 8–300 K. - First-principles DFT: Quantum ESPRESSO with fully relativistic PAW pseudopotentials (PBE-GGA), cutoffs 50/500 Ry (wavefunction/charge), 7×7×5 k-mesh. Ferromagnetic Mn spins; Tb 4f treated as core. An exchange-splitting penalty term adjusted magnetization to match STM-reported Dirac energy. Band structures and JDOS were computed; massive Dirac bands highlighted. - Wannierization and Kubo-Greenwood optics: Wannier90 basis (Tb/Mn d, Sn s/p; 59 orbitals per spin, 118 total) from −15 to +20 eV, frozen window −15 to 3 eV. Optical Hall conductivity via Kubo–Greenwood on a 175×175×175 k-grid with smearing η = 50 meV. Orbital-resolved σxy(ω) restricting initial/final states to Mn 3dxy contributions was computed to isolate Dirac-band transitions. - Analytical model: Two-dimensional massive Dirac Hamiltonian H(k) = −μσ0 + kxσx + kyσy + mσz. Optical Hall conductance for a single 2D Dirac cone derived via Kubo formula: σxy^MD(ω) = [e^2 m/(2ħ)] (ħω + iγ)^{-1} (ħω + iγ + 2μ)^{-1}. Extension to bulk TbMn6Sn6 accounted for valley degeneracy and two kagome layers per unit cell (factor 4) and c-axis lattice parameter, plus a constant background from higher-lying resonances. Finite-temperature effects included via Fermi-Dirac occupations without changing band parameters (m, μ, γ). - Parameter extraction and comparison: Fitting analytical σxy^MD(ω) + const to experimental σxy(ω) at 8 K yielded 2m, μ, and γ; the same parameters were used to simulate temperature evolution and to evaluate the Dirac-band contribution to DC AHC σxy^MD(0,T). Sensitivity to Dirac node position was assessed by modifying exchange splitting in DFT to shift the node and recalculating σxy(ω).

- A pronounced magneto-optical Hall resonance is observed in TbMn6Sn6 around 0.4 eV in Im σxy(ω), with a corresponding dispersive feature in Re σxy(ω), persisting from 8 K to 300 K. - The resonance onset aligns with the lowest interband transition within the massive Dirac bands at ~0.24 eV (twice the Dirac-node offset from EF), consistent between experiment and DFT. - Peak magnitude of Im σxy(ω) at 8 K is ~160 Ω^−1 cm^−1, comparable to the measured DC anomalous Hall conductivity (AHC) ~200 Ω^−1 cm^−1, indicating the resonance dominates the AHC spectral weight. - DFT reproduces key σxy(ω) features (main 0.4 eV resonance and low-energy rise <0.15 eV). Orbital-resolved calculations restricting transitions to Mn 3dxy (Dirac bands) still yield a strong 0.4 eV resonance despite their small JDOS fraction, demonstrating the Dirac-band origin. - Analytical massive-Dirac model quantitatively fits the 8 K spectra with parameters: mass gap 2m = 33 meV, chemical potential μ = 113 meV, and broadening γ = 22 meV, in agreement with STM (2m ≈ 34 meV, μ ≈ 130 meV). - Temperature evolution of σxy(ω) (broadening and red-shift of the resonance; enhanced low-energy spectral weight due to Fermi-edge smearing) is captured by the finite-temperature model using the same parameters. - The extracted Dirac-band contribution to DC AHC, σxy^MD(0,T), accounts for the dominant portion of the total AHC and slightly increases with temperature; the massive Dirac bands yield approximately 15–20% of the quantized Hall conductance e^2/h per kagome layer, remaining robust even at room temperature. - σxx(ω) remains a flat continuum above the Drude peak and is dominated by trivial-band interband transitions, underscoring that σxy(ω) selectively probes topological bands with strong Berry curvature.

The work directly observes and quantifies the predicted universal magneto-optical resonance of massive Dirac fermions in a kagome-lattice magnet, addressing the open experimental question of its existence and spectral form. The alignment of the resonance onset with twice the Dirac-node offset and the successful analytical fit using only the mass gap, chemical potential, and broadening substantiate the universality of the response. The dominance of the resonance’s spectral weight in the DC AHC confirms that Berry-curvature hot spots from the massive Dirac bands control the anomalous Hall electrodynamics in TbMn6Sn6. Differences between σxy(ω) and σxx(ω) highlight that transverse optical responses are more sensitive to topological band features than longitudinal conductivity, which is governed by JDOS and includes trivial bands. The persistence and sizeable fraction (15–20% per kagome layer) of the quantized conductance up to room temperature indicate robustness of the Dirac-related electrodynamics against thermal effects, linking the observations to the universal low-frequency QAH electrodynamics (quantized MO rotations) while noting deviations from the fully quantized limit due to finite μ and coexisting trivial bands.

Infrared magneto-optical spectroscopy of TbMn6Sn6 reveals a robust anomalous Hall resonance originating predominantly from two-dimensional massive Dirac fermions in the kagome lattice. Combining experiment, DFT, and an analytical model captures the resonance’s universal spectral characteristics with a few parameters (2m, μ, γ), quantitatively connecting it to a large, robust DC AHC dominated by Dirac bands. The observed MO resonance and AHC are consequences of intense Berry curvature from the massive Dirac bands and relate to universal QAH electrodynamics, though not fully quantized due to finite Fermi-level offset and trivial bands. These insights provide a general framework for interpreting σxy(ω) in massive-Dirac kagome systems and point to opportunities for terahertz/infrared MO devices and exploration of high-temperature QAH phenomena in intrinsic magnets. Future work could tune μ toward the Dirac gap (e.g., via gating or chemical substitution) to approach quantized optical responses, reduce trivial-band contributions, and probe kz dispersion effects and layer-resolved contributions.

- The material deviates from the quantized electrodynamics limit because the Fermi level lies a finite energy from the Dirac node and trivial bands coexist. - Discrepancies between experiment and DFT σxy(ω) appear (e.g., sign change in Re σxy(ω) and low-energy increase), likely due to contributions from many trivial bands not captured by simplified models. - Finite kz dispersion and bulk three-dimensionality can broaden the resonance; this is treated phenomenologically via a broadening parameter γ. - Optical measurements cover 0.09–1 eV; higher-energy resonances enter as an additive constant background, and low-energy (<0.09 eV) features are inferred via KK relations. - Above 150 K, weak hysteresis necessitated an applied field during MOKE, which could subtly influence domain configurations. - The analytical model assumes idealized 2D massive Dirac cones and may not capture all multi-band and interaction effects.