Discovery of a magnetic Dirac system with large intrinsic non-linear Hall effect

Magnetic materials exhibiting topological Dirac fermions are attracting significant attention for their promising technological potential in spintronics. In these systems, the combined effect of the spin-orbit coupling and magnetic order enables the realization of novel topological phases with exotic transport properties, including the anomalous Hall effect and magneto-chiral phenomena. Herein, we report experimental signature of topological Dirac antiferromagnetism in TaCoTe2 via angle-resolved photoelectron spectroscopy (ARPES) and first-principles density functional theory (DFT) calculations. In particular, we find the existence of spin-orbit coupling-induced gaps at the Fermi level, consistent with the manifestation of a large intrinsic non-linear Hall conductivity. Remarkably, we find that the latter is extremely sensitive to the orientation of the N\'eel vector, suggesting TaCoTe2 a suitable candidate for the realization of non-volatile spintronic devices with an unprecedented level of intrinsic tunability.

LV is only sensitive to the in-plane orbitals, while LH is equally sensitive to in-plane as well as out-of-plane orbitals (given the 45 degrees of incidence angle). By using the combination of LH and LV, we conclude that the near E F electronic structure is given by an admixture of both in and out of plane orbitals, with very little variation in the spectral intensity. This allows us to better identify the spectral features across the BZ and to make a more reliable comparison with the DFT calculations.
To understand the experimental electronic structure, we collected energy vs momentum spectra by using ARPES. We first notice that the layers of bulk TaCoTe 2 are weakly bound together by van der Waals forces. Thus, we expect that the electronic structure is intrinsically two-dimensional, with small, if any, electronic dispersion along the z direction (perpendicular to the layers). The absence of k z dispersion in TaCoTe 2 is confirmed by the qualitatively   TaCoTe 2 in the (f) AFM z , (g) AFM y , and (h) NM configuration. Clearly, the theoretical results for the AFM z configuration are the closest to the measured ARPES spectrum. Note that the ARPES spectra have been shown as 2LH+LV to account for both light polarizations and also the projections of LH, which is 50% in plane and 50% out of plane. This ensures a better visibility for the spectral features which might be dependent on the light polarization. similar shapes of the spectra at various photon energies 25,28 ( Supplementary Fig. S1). This behaviour is fully consistent with a two-dimensional electronic structure. The only change observed experimentally as a function of photon energy is a variation of the spectral intensity across the BZ, likely connected to the photoelectron matrix elements, 26,29 as also suggested by the data collected for both LH and LV polarization ( Supplementary Fig. S2). Our results demonstrate that TaCoTe 2 behaves electronically as a two-dimensional system, thus, without loss of generality, we will describe the electronic properties of this compound with those of a monolayer. From DFT, the system's lowest energy configuration is obtained for magnetic moments on the Co sites arranged in a bi-collinear type AFM order in each layer (Fig. 1a). Such moments are along the z direction (AFM z ). The energy difference between this and the AFM y order is very small, i.e. 9 meV/ unit cell, 15 but their effect on the electronic structure is sizeable, with the realization of markedly different features identifiable in the spectra (as summarized in Fig. 2). Both AFM z and AFM y break the time-reversal, as well as the inversion symmetry of the system (T and P), while preserving the combined PT symmetry, important for the transport and optical properties of the system. [30][31][32] The main differences between AFM z and AFM y are visible in the nonsymmorphic symmetry of the system (See Supplementary information), which manifests directly in different energy vs momentum electronic structures.
To unveil the magnetic ordering of this compound, we compared non-magnetic (NM) DFT calculations along with those for the antiferromagnetic AFM z and AFM y orders, with the measured energy-momentum spectra ( The differences between AFM z and AFM y along X'-Γ-X are also measured and calculated: Along the M-Y-M path, as a consequence of the magnetism along y, the AFM y order shifts (about 25%) the maximum of the band at Y towards the M point, creating an extremely large asymmetry in the electronic dispersion, see Fig. 2g. This asymmetry, in contrast, is neither observed in the experiment nor predicted in the AFM z calculation (Fig. 2f). Fur-thermore, the opening of a gap of a Dirac-like dispersion at -0.3 eV at the X and X' points for the AFM y configuration (indicated as D 3 ) is absent in the AFM z order and not detected by ARPES, despite its size is expected to be much larger than the experimental resolutions (12 meV and 0.018Å −1 for energy and momentum, respectively). This allows us to conclude that the electronic structure of TaCoTe 2 is consistent with a magnetic order hosting magnetic moments aligned along the z direction, i.e., AFM z . This is also consistent with our magnetization measurements, which suggest the existence of an easy axis mainly along the out-of-plane direction ( Supplementary Fig. S3). Such a magnetization, shows also non-trivial magnetic signatures in the curves, i.e. saturating magnetization, which are in full agreement with what predicted in Ref. 15 In addition, as mentioned above, AFM z is the most stable configuration found for this system. We also notice that ARPES does not resolve the fine details that DFT calculations reveal for binding energies lower than −0.4 eV, regardless of the magnetic order. We believe that the large k z broadening of the samples, which indeed manifests as 'shadowing' of the electronic structure, combined to the strongly varying matrix elements (as shown in Supplementary information) might be the reasons behind the apparent discrepancies. It is still worth mentioning that although transport, ARPES and DFT results indicate that TaCoTe 2 realizes a possible AFMz order, neutron scattering experiments would be also beneficial to conclusively determine the precise magnetic ground state, but it is beyond the current scope of this work.
Together with the identification of the magnetic order in TaCoTe 2 , our data demonstrates that SOC plays an important role in opening energy gaps at the Dirac points. This can be seen in the AFM z electronic structure calculations of Fig. 2b, where D 1,2 develops energy gaps right at the Fermi energy, while no gap is observed in D 3 . Such gaps are candidate k-space loci to enable topologically non-trivial behaviour. [33][34][35][36] We also experimentally detect such gaps: we show a zoom around D 1 and D 3 (Fig. 3 a), the intensity curvature plot in Fig. 3 b, 37 and the extracted energy-distribution curved across the X' point and exactly across D 1 (Fig. 3 c,d). The energy profile at the X' point D 3 does not show any gap, but rather a single peak, consistently with the predicted antiferromagnetic behavior along the z direction (Fig. 3c). As for D 2 , the small SOC-induced gap at the Fermi level predicted by DFT (≈ 20 meV) is more challenging to be observed with state-of-the-art experimental apparatuses. However, by fitting the dispersion with Lorentzian curves (purple markers in Fig. 3a (See Supplementary Information for details), we estimate that the top of the band is located ∼ 25 ± 10 meV below the Fermi level. We stress that a proper quantification of this gap is difficult via ARPES, because it involves unoccupied states that are not accessible to ARPES. However, the value of the gap (∼ 25±10 meV) adduced from the experimental data is a lower limit compatible with the calculated gap value. We also show in the Supplementary information how the observed peak is lower in binding energy compared to the Fermi level edge, extracted from EDCs in a region without bands, i.e. at k x =−0.5Å −1 . We used DFT to compute the INHE (σ) for the monolayer TaCoTe 2 in the AFM y and AFM z magnetic configurations. The NM state preserves both the P and T symmetry, and, as a result, all the components of the INHE vanish identically in this case. In the AFM phases, both P and T symmetries are broken, but the combined PT symmetry results in non-vanishing INHE. Specifically, in the AFM z phase, the relevant symmetry isC 2y , which forces the σ xyy component to vanish, while the σ yxx is non-zero. In contrast, the AFM y structure hosts theM y symmetry, which results in a vanishing σ yxx and a non-zero σ xyy . Fig. 4a for the AFM z configuration, and in Fig. 4c for the AFM y configuration. In both cases, the INHE vanishes identically inside the bandgap, and develops peaks near the band edge and decays rapidly when the chemical potential is shifted away.

The INHE is shown in
It exhibits an opposite sign for the electron and hole doping for both AFM z and AFM y order. The INHE value is found to be of the order of 1 mA/V 2 , which is comparable to the recently reported values in metallic antiferromagnets. [38][39][40][41] Interestingly, the INHE is almost an order of magnitude larger for AFM z compared to AFM y . Finally, given the direct link between the INHE and Λ (band resolved contribution to the INHE as defined in Eq.4 of the Supplementary information), we present the distribution of the latter on the band structure in Figs. 4b and 4d. As expected, near the band edge, Λ n αβγ (k) has the maximum value, and it decays away from the band edge.
In conclusion, we demonstrate that TaCoTe 2 is an antiferromagnetic Dirac system, which hosts SOC-driven bandgaps at the Fermi level. The combination of SOC effects, magnetism, and time-reversal symmetry breaking is found to generate a non-vanishing INHE, which influences to the transport properties of the system. The INHE in TaCoTe 2 is highly sensitive to the direction of the Néel vector of the AFM order, opening a novel pathway for using this compound in dissipationless electronics and spintronics. Our study indicates that TaCoTe 2 would provide a promising new materials platform for exploring the interplay of Dirac fermiology, SOC, magnetism, and topology.

Acknowledgements
The experiments were performed at the NFFA APE-LE beamline on the Elettra synchrotron AFM z and AFM y differences The main differences between the AFM z and AFM y orders are expected due to the nonsymmorphic symmetries, which are characterized by the combination of P, a two-fold screw rotation symmetry along the y-axisC 2y = {C 2y | 1 2 1 2 0}, and a glide mirror symmetry perpendicular to the y-axisM y = {M y | 1 2 1 2 0}. In particular, the AFM z order preserves theC 2y , while the AFM y order hostsM y . This difference is not only important for transport properties, but it is responsible for giving rise to significantly different k-resolved electronic dispersion.  Figure 5: ARPES energy-momentum spectra collected wit linear horizontal light polarization and for different photon energies, as indicated in the panels. The main difference is a shift of the spectral weight in the spectra, but no significant change in the electronic structure can be observed. This confirms the lack of k z dispersion, thus a two dimensional character for the electronic structure of TaCoTe 2 .  Figure 6: ARPES energy-momentum spectra at selected photon energies (where the intensity was prominent) and for both LH and LV. The main differences can be brought back to a redistribution of the spectral weight, indicating a mixed orbital character for the electronic structure collected within the energy region presented.

Sample growth and characterization
TaCoTe 2 single crystals were grown by a chemical vapor transport method . The stoichiometric mixture of Ta, Co, and Te powders was sealed in a quartz tube with TeCl 4 being used as transport agent. Thin plate-like single crystals with metallic luster can be obtained via the chemical vapor growth with a temperature gradient of 900 • C -750 • C. The composition and structure of the crystals were checked by Energy-dispersive x-ray spectrometer and x-ray diffractometer respectively.

Methods and samples preparations
The samples were cleaved in ultrahigh vacuum at the base pressure of 1 × 10 −10 mbar. The ARPES measurements were performed at the NFFA APE-Low Energy beamline, at 77 K, by using a Scienta DA30 hemispherical analyzer with energy and momentum resolutions better than 12 meV and 0.02Å −1 , respectively.

ARPES Fitting Details
The electronic structure of Fig.3a of the main text has been fitted by using Lorentzian curves convoluted by a Gaussian, which accounts for the energy resolution of the instrument. For performing the fit, the ARPES spectra have been decomposed in energy distribution curves (EDCs) and each EDC has been fitted the way here described.

Details of the ab-initio calculations
The density functional theory based ab − initio calculations were performed using the generalized gradient approximation framework (GGA-PBE) as implemented in VASP code . [45][46][47][48] A k-grid of 12×12×8 was used for the BZ integration. The kinetic energy cutoff for the plane wave basis was set to 400 eV. The Wannier function-based tight binding modeling is done by considering the s and d orbitals of Ta, the p orbital of Te, and the d orbitals of Co . 49,50 Temperature Dependent Resistivity and Magnetization Data Figure 7: Temperature dependent magnetization data for TaCoTe 2 for the applied magnetic field along (a) B||c direction, and (b) B||ab direction. Clearly, the isothermal magnetization displays a typical moment polarization behavior for all measured temperatures up to 300 K, under both out-of-plane (B||c) and in-plane (B||ab) magnetic field orientations. Such behavior signatures the presence of magnetic order in the system. The out-of-plane magnetization is roughly twice as large as the in-plane one, suggesting that the easy axis is mainly along the out-of-plane direction. Figure 8: Temperature dependent resistivity of TaCoTe 2 . The resistivity does not strongly vary with the temperature. Overall, it displays an non-metallic transport with resistivity increases upon cooling, followed by a resistivity down turn around 50 K. A transition-like behavior characterized by resistivity upturn is observed around 380K, which is reproducible in multiple samples with slight variation in upturn temperature.

INHE and Berry curvature connection
In a simple semi-classical model, INHE can be understood from the presence of a finite Berry connection polarizability (BCP) in the system. The BCP for the the n'th band can be expressed as, where, A nm a = u n i∂ a u m is the interband Berry connection, and u n are the unperturbed Bloch states with band energies n . In the presence of an external electric field (E b ), the net generated Berry connection can be obtained as, The field-induced Berry curvature Ω E (k) = ∇ × A E (k) results in a non-linear Hall-like response. The most important feature of this response is that it is intrinsic, and unlike the Berry curvature dipole induced nonlinear Hall conductivity, INHE does not depend on the relaxation time (τ ). The final expression for the INHE is given by, where, f ( n ) is the Fermi Dirac distribution function. The momentum-resolved Λ n αβγ (we suppress the argument k for brevity) is expressed in terms of the interband Berry connection and the velocity operator (v n ) as,

Fermi level and observed gap
The observed peak in the electronic structure, which forms the top of the bands which develop a gap, is lower in binding energy compared to the Fermi level edge, extracted from EDCs in a region without bands, i.e. at k x =−0.5Å −1 . This is visible in the figure below. Intensity (arb. units) Figure 9: The red dashed line has been extracted from EDc around k x =−0.5Å −1 , where there are no bands in the ARPES data. This allows us to determine the leading edge of the Fermi level as a calibration. We observe that the peak of the electronic structure discussed in the main text, formed by the band which delevlops a gap, is shifted away from the edge. The latter also, despite the increase (as observed in the figure and caused by the presence of additional bands at lower binding energies) is still visible.