Weyl nodal ring states and Landau quantization with very large magnetoresistance in square-net magnet EuGa4

Abstract

Magnetic topological semimetals allow for an effective control of the topological electronic states by tuning the spin configuration. Among them, Weyl nodal line semimetals are thought to have the greatest tunability, yet they are the least studied experimentally due to the scarcity of material candidates. Here, using a combination of angle-resolved photoemission spectroscopy and quantum oscillation measurements, together with density functional theory calculations, we identify the square-net compound EuGa₄ as a magnetic Weyl nodal ring semimetal, in which the line nodes form closed rings near the Fermi level. The Weyl nodal ring states show distinct Landau quantization with clear spin splitting upon application of a magnetic field. At 2 K in a field of 14 T, the transverse magnetoresistance of EuGa₄ exceeds 200,000%, which is more than two orders of magnitude larger than that of other known magnetic topological semimetals. Our theoretical model suggests that the non-saturating magnetoresistance up to 40 T arises as a consequence of the nodal ring state.

Fig. 1. EuGa₄ as a candidate to host Weyl NR states.

a The proposed mechanism to create Weyl NR states in square-net magnetic materials: one spinless NR evolves into four spinful NRs and eventually two symmetry-protected Weyl NRs. FM ferromagnetic, SOC spin-orbit coupling. The gray planes represent the mirror symmetry plane, which is parallel to the NR plane in the k space. With SOC, the two yellow NRs survive, while the two green NRs are gapped. The blue and red structures represent the energy surfaces above and below the energy of the NR, respectively. b Magnetic phase diagram (H−T) for EuGa₄. SP spin-polarized phase, AFM antiferromagnetic phase. Inset shows the top and side views of the EuGa₄ crystal structure. The empty circle symbols mark the magnetic phase boundary determined by magnetization measurements, see Supplementary Fig. 1 for the full M(H) data. c–e Band structures of EuGa₄ in the paramagnetic (PM) phase without SOC, SP phase without SOC, and SP phase with magnetic moment along c axis with SOC, respectively. c The nodes circled in red (blue) represent the ones residing on (off) the mirror invariant planes. The vertical dashed lines mark the high-symmetry k-points. d Blue and red indicate two sets of spin-split bands. e The bands that host protected crossings are colored. f 3D view of the Weyl NRs from DFT calculations. Three pairs of NRs are shown in green, cyan, and red/blue, respectively. Note that small parts of the red/blue NRs near S on the k_z = ± 2π/c planes extend outside of the BZ. Symmetry operations fold these extended segments back to the k_z = 0 plane of the BZ. g Energy (E) surface of the bands that form the red/blue NRs. Blue and red indicate that the energy is above and below that of the NRs, respectively. h Top view of the red/blue Weyl NRs, with the color indicating the energy. Inset in h shows the zoom-in NR pair from the top of the panel. The legend is shown on the top.

Fig. 2. Electronic structure of EuGa₄ in the PM phase.

a Three groups of FS pockets: α, β, and γ, based on DFT calculations. The cross-sectional cut of the β pocket at the k_z = 0 plane is illustrated with dashed red lines. b ARPES measured FS with hν = 118 eV and T = 25 K. Two high-symmetry k − paths (yellow lines) are indicated for band dispersion analysis. The white lines mark the BZ boundary. The dashed red lines are the k_z = 0 cross sections of the β pocket, the same as those shown in a. The dashed yellow lines delineate the boundary within which the effect of k_z-broadening is observed, as discussed in the text. c ARPES band dispersion along path 1 with hν = 120 eV. The solid lines are band structures from DFT calculations. Red and orange indicate the bands that form the NR1 and NR2, respectively, while the gray bands are irrelevant ones. Same applies to e. d Zoom-in view of the boxed region in c, with the MDC stacks shown on the right. e, ARPES band dispersion along the Z−Y path. The colorbar in c is shared for d and e as well.

Fig. 3. Fermi surface geometry of EuGa₄ in SP phase from quantum oscillations.

a A series of QO curves with θ ranging from 0^∘ to 90^∘. For the QO at 0^∘, an L–K fit is shown. b Two representative FFT spectra of the QOs at θ = 0^∘ and 45^∘. The QO frequencies associated with the α, β, and γ pockets are labeled accordingly. The two insets in c show the zoom-in views of the FFT spectra near α₁/α₂ and ${\beta }_1^{\prime }$/${\beta }_2^{\prime }$ frequencies at θ = 0^∘. c, d Angle dependent QO frequencies (circles) above and below 300 T, respectively. The QO frequencies shown in circles are measured using a lab magnetometer, while those shown in squares are determined by high-field measurements. The shadings in c act as a guide to the eyes. Inset in c illustrates the definition of rotation angle, θ. H indicates the applied magnetic field. The cyan and green lines in c are from theoretical predictions for the α and β pockets, respectively. The red, blue, and orange lines in d represent the theoretical prediction associated with the γ pockets of the same color as those illustrated in Fig. 2a. d Illustrates the extremal cyclotron orbits associated with the measured γ₄ frequency at θ = 0^∘. In Supplementary Fig. 7d, we illustrated all the extremal orbits with the γ pockets. e Illustration of the extremal orbits (black lines) of the α and β pockets, when θ = 0^∘ and 40^∘. f Band structure along Γ − Σ with the feedback from QO measurements. The energy of the gray-shaded bands is accurately described by theory based on the QO measurements, while that of the green-shaded band is underestimated by theory. The actual bands should have slightly higher energy, as indicated by the yellow shades. g Temperature dependent QO amplitude for the γ₄, γ₃, and γ₂ oscillations at θ = 0^∘ and their L–K fits (solid lines).

Fig. 4. Large, non-saturating MR in EuGa₄.

a Temperature-dependent resistivity at selected fields. Inset shows the same data in a logarithmic scale, highlighting the low-T behaviors. The field is applied along the c axis (H ∥ c), while the current is along the a axis (j ∥ a). RRR, residual resistivity ratio; T_N, Néel temperature. b, c The low- and high-field MR behaviors, respectively. The AFM-to-SP magnetic phase transition is indicated by the arrow. The H² fit is performed on the MR curve from 0 to 3.5 T in b, while a power function fit is performed above H_c up to 41.5 T in c, with 1.8 as the exponent. Inset in b shows the MR curve up to 14 T. d Isothermal magnetization curves (H ∥ c) at T = 2 K. Fully spin-polarized phase is reached above μ₀H_c = 7.4 T. Inset illustrates the SP state with the moments on Eu sublattices along the c axis. e Comparison of the measured MR in EuGa₄ (marked by *) with other known TSMs. The nonmagnetic compounds are colored gray, while the ferromagnetic and antiferromagnetic ones are colored green and blue, respectively. See Supplementary Table 1 for the field, temperature, and reference information on the compounds in this plot.