Weyl metallic state induced by helical magnetic order

Weyl semimetals (WSMs) host topologically protected band crossings (Weyl nodes) where singly degenerate bands intersect, yielding massless-fermion-like carriers and exotic transport. Realizing a WSM requires breaking inversion symmetry (P) and/or time-reversal symmetry (T). Magnetic routes that break T offer tunability of topology via magnetic order or applied fields. Ferromagnetic WSMs are attractive but generate stray fields problematic for device miniaturization. Antiferromagnetic (AFM) WSMs are more stable but many collinear AFMs preserve combined P×T symmetry, maintaining double degeneracy and preventing Dirac points from splitting into Weyl pairs. Non-collinear AFMs that lack P×T, such as chiral 120° AFM in Mn3X (X=Sn, Ge), can host Weyl nodes, though their correlated, broadened bands complicate experiments. Non-collinear Eu-based centrosymmetric compounds (EuCo2P2, EuNi2As2, EuZnGe, EuCuSb) can strongly exchange-couple localized Eu 4f moments to itinerant electrons, potentially lifting degeneracies with sizable splittings (~0.1 eV), but large magnetic supercells challenge accurate DFT, especially for incommensurate structures. This work targets EuCuAs, a centrosymmetric EuTX (T = Cu, Ag, Au; X = P, As, Sb, Bi) material, to test whether a commensurate non-collinear AFM (helical) order can induce Weyl nodes while maintaining zero net magnetization.

Prior studies established WSMs through broken P or T, with magnetic topological materials offering control of band topology (refs. 5–10). Ferromagnetic WSMs show large anomalous transport but suffer from stray fields and scaling issues (refs. 7,11–13). AFM systems often retain P×T symmetry, hindering Weyl node formation unless non-collinear order breaks it. Mn3X (X=Sn, Ge) exhibits chiral non-collinear order supporting Weyl nodes (ref. 14) but with strong correlations and broadened bands (ref. 15). A range of Eu-based layered pnictides (EuCo2P2, EuNi2As2, EuZnGe, EuCuSb) display non-collinear magnetism (refs. 16–19) and strong exchange between Eu 4f and conduction states allowing large band splittings (ref. 20), yet accurate DFT for their incommensurate orders is challenging. EuTX (T=Cu,Ag,Au; X=P,As,Sb,Bi) materials (refs. 21–30) provide platforms to explore magnetism–topology interplay. Earlier assumptions of A-type AFM in EuCuAs existed, but direct microscopic probes (neutrons, resonant x-ray) are needed to resolve the spin structure and its topological consequences.

- Crystal growth and characterization: Single crystals of EuCuAs grown by self-flux (per ref. 24). Structure verified via laboratory single-crystal x-ray diffraction (6-circle diffractometer) and Laue diffraction, confirming hexagonal Ni2In-type structure (P63/mmc). - Magnetization: PPMS with VSM option. Temperature-dependent susceptibility from 2–50 K in B=0.1 T with fields parallel and perpendicular to c. Field-dependent magnetization at T≈2 K up to 5 T in both orientations. - Magnetotransport: Four-probe resistivity on PPMS with fields up to 5 T applied perpendicular to c, temperatures down to 2 K. Field- and temperature-dependent ρxx measured, including isotherms with B∥c. - Neutron diffraction (powder and single crystal): Powder neutron diffraction (PND) and inelastic neutron scattering on WISH and LET (ISIS). ~1.3 g powder in 3 mm V can, 2–20 K. Severe Eu absorption addressed via modeling but limited exact moment/thermal parameter accuracy. Single-crystal neutron diffraction (ND) on D9 (ILL) with hot neutrons (λ=0.84 Å) to reduce absorption; 229 reflections collected at 2 K (structural and magnetic with qm=(0,0,0.5)); attenuation corrections applied based on crystal shape/path lengths. Field-dependent ND up to 2.5 T with b-axis vertical to access h0l reflections. Spherical neutron polarimetry on D3 (ILL) with Cryopad, polarized neutron setup using Heusler polarizer and He spin filter; polarization matrices Pij measured at magnetic Bragg positions; field-cooling protocol (1 T ∥ b from 25→2 K) used to test domain imbalance. - Resonant elastic x-ray scattering (REXS): I16 (Diamond). Photon energy at Eu L3 edge. Zero-field in vertical geometry, σ→π′ channel to enhance magnetic scattering; field-dependent in horizontal geometry with π→σ′ channel. - Angle-resolved photoemission spectroscopy (ARPES): SIS-ULTRA beamline (SLS). kz mapping via photon energies 50–150 eV; main data at 74 eV. Samples cleaved at ~15 K under ~1e-8 Torr; measurements at 5–22 K with various light polarizations. Band alignment compared to DFT with +0.4 eV rigid shift. - Density functional theory (DFT): VASP v6.2.1 with PAW, PBE-GGA, SOC included; U=5.0 eV on Eu 4f; cutoff 480 eV; Monkhorst-Pack k-mesh 9×9×7. Electronic structures computed for commensurate period-4 helix (τ=0.5) and for DP-AFM for comparison; topological features identified (Weyl nodes, nodal lines). - Spin Hamiltonian and spin waves: Semiclassical mean-field model with Heisenberg exchanges J0 (in-plane), J1 and J2 (interlayer), single-ion easy-plane anisotropy D, Zeeman term. Analytical results for helix turn angle and fields B1 (metamagnetic) and Bsat. Parameters extracted using measured τ and fields; linear spin-wave theory (LSWT) used to simulate powder-averaged magnon spectra; best-fit J0 obtained by matching LET INS data (dominant in-plane dispersion).

- Magnetic order: Below TN = 14.5 K, Eu moments form a planar helical structure with spins confined to the ab plane, rotating along c with propagation vector q_m = (0,0,τ). τ varies by sample: τ = 0.591(1) (PND), 0.42(1) (REXS), and 0.50(1) (ND). Upper bound on out-of-plane tilt ≲ 3°. - Metamagnetic transition: For B⊥c at T ≈ 2 K, a kink at B1 ≈ 0.3 T in M(B) corresponds to a transition from the helix to a canted double-period AFM (DP-AFM) state. In REXS, an incommensurate magnetic peak near L=1.58 jumps to commensurate L=1.5 at ~0.1 T (higher temperature measurement). ND shows intensity transfer from half-integer L to integer L above ~0.3 T. Above ~0.8 T, half-integer peaks vanish; full polarization reached by ~1.5 T. - Magnetization and anisotropy: Saturation magnetizations: Msat = 7.0(1) μB per f.u., consistent with Eu2+ (S=7/2, g=2). Bsat ≈ 1.5 T for B⊥c; Bsat ≈ 2.5 T for B∥c. χ∥c > χ⊥c at low T indicates easy-plane anisotropy. - Transport: Resistivity ρxx shows a peak at TN in zero field due to critical spin fluctuations (onset ~50 K); peak suppressed by magnetic field; negative magnetoresistance for fields above Bsat. - Magnetic structure determination: SNP decisively favors the helical model (χ2=1.85) and rules out a single-domain DP-AFM after field cooling (χ2≈649; any domain imbalance <5%). Magnetic space group for τ=0.5 helix: C2 (#5.16 OG setting). - Electronic topology: DFT for the period-4 helix yields two quadratic Weyl nodes with Chern numbers C=±2 at the A point and additional nodal-line features. In contrast, DP-AFM preserves mz×T, maintaining double degeneracy along kz and no Weyl nodes at A. ARPES agrees with DFT upon +0.4 eV rigid upward shift, indicating slight hole doping; observed two concentric hole pockets at Γ and linear dispersions along K–Γ–K match calculations. - Spin Hamiltonian parameters: Extracted interlayer exchanges and anisotropy (meV): for τ=0.42, J1≈0.021, J2≈−0.021, D≈0.009; for τ=0.59, J0≈0.060(3), J1≈−0.0077, J2≈−0.0069, D≈0.011. Calculated B1 using these agrees with experiment (0.46 T for τ=0.42; ~0.29 T for τ=0.59 vs observed ~0.3 T). J0 dominates, indicating highly two-dimensional magnetism; c-axis correlation length ~100 Å. - Central result: Helical magnetic order with zero net magnetization induces a Weyl metallic state in EuCuAs, specifically generating C=±2 quadratic Weyl nodes at A.

The study directly addresses whether non-collinear, fully compensated helical antiferromagnetism can lift band degeneracies and induce Weyl fermions in a centrosymmetric material. Neutron and resonant x-ray diffraction establish a planar helical ground state in EuCuAs below TN, and SNP removes ambiguity with alternative DP-AFM structures in zero field. The helical order breaks both inversion and time-reversal symmetries, removing Kramers degeneracy across the Brillouin zone and enabling topological band crossings. DFT reveals C=±2 quadratic Weyl nodes at A only for the helix, with ARPES confirming the calculated band structure after a modest energy shift. The measured magnetization, metamagnetic transition, and spin-wave spectrum are consistent with a frustrated interlayer exchange model with strong easy-plane anisotropy and dominant in-plane coupling, explaining the emergence and field evolution of the helix and DP-AFM states. Compared to ferromagnetic WSMs, the helical state avoids stray fields while still achieving tunable topological features. In contrast to NdAlSi where Weyl physics stabilizes helimagnetism, here helimagnetism produces the Weyl state. Relative to Mn3X, EuCuAs exhibits weakly correlated conduction bands and decoupled local-moment magnetism, providing a cleaner platform to control topological band structure via magnetic ordering.

This work demonstrates that a commensurate planar helical antiferromagnetic order in EuCuAs induces a Weyl metallic state, generating a pair of quadratic Weyl nodes (C=±2) at the A point. Comprehensive neutron (powder, single-crystal, polarized), resonant x-ray diffraction, magnetization, transport, ARPES, and DFT collectively establish the helical structure below TN≈14.5 K, its field-driven transition to a canted DP-AFM phase near 0.3 T, and the resulting topological electronic structure. The findings broaden the class of magnetic WSMs to include fully compensated helices, offering prospects for device integration without stray fields. Future research should aim to: (i) identify or engineer materials where the helical-order-induced Weyl nodes lie closer to EF with reduced trivial bands; (ii) map the full nodal structure across the Brillouin zone; (iii) tune carrier density (e.g., via doping or gating) to align EF with Weyl nodes; and (iv) explore dynamical/topological responses (anomalous transport, optical, and nonlinear effects) under controlled magnetic textures.

- Sample variability: The helical propagation vector τ varies between samples (≈0.42, 0.50, 0.59), likely due to compositional differences, which may affect the precise topological features and field scales. - Neutron absorption: Strong Eu absorption complicates quantitative refinement from unpolarized neutron diffraction (moment sizes, thermal parameters), necessitating SNP and complementary probes. - Modeling approximations: Mean-field expressions for B1 and Bsat neglect helix distortion, possible fan phases near saturation, and in-plane anisotropy, introducing quantitative uncertainty. The simple J0–J1–J2–D model cannot by itself stabilize τ=0.5 without additional terms (e.g., in-plane sixfold anisotropy, biquadratic exchange, RKKY contributions). - Electronic structure: DFT band structures required a +0.4 eV rigid shift to match ARPES, indicating slight hole doping; Weyl nodes are at different energies and offset from EF, and trivial bands at EF complicate a clean Weyl response. The full Brillouin zone was not mapped, leaving nodal-line details incomplete.