Magnetic and electronic structures of antiferromagnetic topological material candidate EuMg₂Bi₂

Since the discovery of Dirac fermions in graphene, interest in topological materials has accelerated along with the complexity of newly uncovered exotic electronic structures. Topologically protected electronic surface states in Dirac materials are distinctive from the bulk states and can be characterized by the linear energy-momentum dispersion near the Fermi level and massless chiral excitations of their electrons.¹ Recently, the focus has shifted from topological insulators (TIs) to topological semimetals (TSMs). Driving these materials even further, we can incorporate additional quantum phases such as superconductivity, magnetism, and charge density waves into the host material possessing the topological electronic surface states to explore the new physics that arises.^2,3 With the addition of magnetism in the TSM condensed matter, scientists have realized materials that can host nontrivial phenomena such as axion electrodynamics, the quantum anomalous Hall effect (QAHE), and Majorana fermions.^4–6 

Magnetic TSMs have the potential to be used in a range of applications including spintronics and optical devices as heterogeneous catalysts.^7,8 One such avenue being photonics and nonlinear optics applications where recent advances have shown that 3D Dirac semimetals can generate extreme THz harmonics, such as in Cd₃As₂.^9–11 TMSs have also been considered for electronic devices, including Na₃Bi that was discovered to be a high-efficiency electrical contact material for 2D semiconductors.¹² Emerging TSM-based electronic devices have included ultrafast broadband photodetectors and spin topological field effect transistors.¹³ However, the stability, cost, and ease of fabrication are crucial factors that must be taken into account if magnetic TSMs are to become technologically relevant today. Topological materials such as (Bi,Sb)₂Te_3,¹⁴ which have been magnetically doped as thin films, often produce sharp inhomogeneities; therefore, intrinsic magnetic topological materials provide a more reliable platform. In order to discover new intrinsic magnetic TSMs, attention must be directed to the current chemical factors that have proven to drive band inversion. This includes chemical bonding, spin–orbit coupling (SOC), nuclear and magnetic structure and symmetry, etc.^15,16 In the TSM Na₃Bi,^17,18 the formation of the Dirac nodes is based on the energetically low 6s Bi orbitals and a SOC. A similar dependency is also observed in the low-lying energy 5s Cd orbitals of the tetragonal Dirac semimetal Cd₃As₂.¹⁹ A chemical bonding motif that has proven to produce an even greater band inversion energy than in Na₃Bi and Cd₃As₂ is the 4⁴ square net.²⁰ The epitome of the square net motif is exhibited in the TSMs found in the PbFCl structural family, such as ZrSiS,²¹ and in AMnBi₂ (A = alkaline-earth-metal).²² Likewise, the ABX honeycomb structure (ZrBeSi-type, space group P63/mmc; No. 194) has been reported to vary in its topological properties as a direct result of an interplay between the atomic orbitals and ionic sizes present in the local structures (i.e., X[B₃A₆]).²³ 

After the band inversion has been considered, the next action is to incorporate the magnetic component, and an excellent platform for this is through rare earth elements, which can promote the coupling between topological quasiparticles and magnetism. Within the XMnBi₂ (X = rare earth elements) family, which acquires a Bi square net motif, various topological candidates have been discovered including EuMnBi₂;^24–26 a TSM candidate shown to exhibit phenomena such as the half-integer Quantum Hall effect and magnetopiezoelectric properties. Additionally, antiferromagnetic (AFM) YbMnBi₂^27,28 has been suspected to host Weyl fermions, the formation of which is believed to be dependent on the ∼10° spin canting of the Mn moments. Such an observation, resultantly, reveals the influence that magnetism has on controlling the electronic surface states. Subsequently, the layered triangular lattice compound EuCd₂As₂²⁹ has been identified as an AFM Dirac semimetal candidate. The triangular layers of ferromagnetic (FM) Eu moments provide a rich playing field for diverse magnetic behavior and interactions.

Mg₃Bi₂³⁰ was recently discovered by Chang et al. as a potential type-II nodal-line semimetal. The strong SOC from Bi in Mg₃Bi₂ combined with the weak interlayer interactions and a partial confinement of the electronic states has promoted the formation of topological states. Subsequent investigations into this structure type may potentially reveal a new motif for magnetic topological semimetals. Considering the potential interplay between the topological electronic states and magnetism with the replacement of one equivalent of Mg in Mg₃Bi₂ with one equivalent of magnetic Eu, we found interest in investigating EuMg₂Bi₂.^8,31,32 The previously reported semimetal EuMg₂Bi₂, isostructural to EuCd₂As₂ sharing the space group P-3m1 (no. 164), was found to be a topological candidate with an AFM behavior, T_N∼ 8 K. Angle-resolved photoemission spectroscopy (ARPES) data have been measured by our group and confirms the linear band observed around the Fermi level on the surface of EuMg2Bi2. Crystals of EuMg₂Bi₂ can be grown easily using a conventional solid-state metal flux reaction demonstrating a more efficient pathway for achieving the TSM technology. This paper reports our recent results studied by neutron scattering, ARPES, and DFT calculations. Note, a parallel neutron scattering work was also reported on the EuMg₂Bi₂ system by Pakhira et al.³³ 

The nominal composition EuMg₄Bi_∼12 yielded single crystals of EuMg₂Bi₂ via a metal flux reaction. Reactants included elemental europium pieces (Alfa Aesar, sublimed dendritic, 99.9%), magnesium turnings (∼4 mesh, 99.98%), and bismuth pieces (99.99%). Reactants were placed in an alumina crucible and subsequently sealed in an evacuated silica tube (<10⁻⁵ Torr) having a sample size of approximately 500 mg of stoichiometric Eu and Mg with a ∼5 g Bi metal flux. The sample was heated to 900 °C at a rate of 180 °C/h for 5 h, then cooled to 800 °C at a rate of 100 °C/h, and finally, slowly cooled to 650 °C at a rate of 3 °C/h where the samples were then centrifuged to remove the excess Bi flux. Hexagonal shaped single crystals resulted from the synthesis, and were found to be slightly oxidizing in air and moisture after one week.

A Riguku MiniFlex 600 powder x-ray diffractometer with Cu Kα radiation (λ = 1.5406 Å, Ge monochromator) and powder x-ray diffraction were employed to examine the phase identity and purity of EuMg₂Bi₂. A step of 0.005° and a scan speed of 1.250°/min over a Bragg angle (2θ) range of 5°–90° was analyzed. A full-profile LeBail refinement to determine phase identification and refinement of lattice parameters was performed using FullProf Suite.^34,35

The room-temperature crystal structure of EuMg₂Bi₂ was determined from a Bruker Apex II single crystal x-ray diffractometer equipped with Mo radiation (λ_Kα= 0.71073 Å) using single crystals from the nominal composition EuMg₄Bi_∼12. The sample was measured with an ω of 0.5° per scan and an exposure time of 10 s per frame. Subsequently, the crystal structure was solved using a SHELXTL package with the direct methods and full-matrix least squares on the F² model.^36,37 Chemical stoichiometry was analyzed using an FEI quanta 3D Field Emission Gun (FEG) Focused Ion Beam (FIB)/Scanning Electron Microscope (SEM) and Energy-Dispersive Spectroscopy (EDS). With a 20 kV accelerating voltage, various areas from a EuMg₂Bi₂ crystal were selected for the spectrum collection.

A quantum design physical property measurement system (PPMS) including a temperature range of 1.85–300 K and applied fields up to 9 T was employed for the temperature and field-dependent magnetization, resistivity, and magnetoresistivity measurements. Field-dependent and temperature-dependent magnetization measurements were performed perpendicular and parallel to the c axis. Normal electronic transport measurements were performed with a four-probe method using platinum wires in the basal plane of a single crystal.

The magnetic structure of EuMg₂Bi₂ was determined using single crystal neutron diffraction on the Dimensional Extreme Magnetic Neutron Diffractometer (DEMAND, HB-3A) at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory (ORNL).³⁸ A small single crystal of EuMg₂Bi₂ (approximately 1.2 × 1.2 × 0.2 mm³) was selected for the measurement as a result of the large neutron absorption coefficient of Eu. The sample was measured at 4 and 10 K using neutrons with the short wavelength of 1.008 Å from the bent Si-331 monochromator to reduce the heavy neutron absorption. PLATON software was utilized to apply the absorption corrections while FullProf refinement Suite was used to perform the nuclear and magnetic structure refinements.^34,39,40

Bulk band structure calculations were calculated using the generalized gradient approximation (GGA) and correlation parameter (U) with and without spin–orbit coupling (SOC) and were performed with VASP.⁴¹ The experimental structural parameters from both x-ray and neutron diffractions were used here. Spin–orbit coupling (SOC) was included self-consistently to examine the SOC effects on the electronic and magnetic structures. Spin polarization on Eu atoms using GGA + U (U = 6 eV) was set to investigate the magnetic models.⁴² Different U parameters were tested to match with the experimental ARPES measurements on Eu atoms. The Monkhorst–Pack k-point mesh in the Brillouin zone was set as 11 × 11 × 7 for the k-space integrations.⁴³ The converged energy was set up as 0.1 meV per atom.

Photoemission (ARPES) maps were obtained at the Electron Spectro-Microscopy (ESM) beamline of the National Synchrotron Light Source (NSLS-II) using a two-dimensional momentum space mapping analyzer Scienta DA30 set at angular and energy resolutions of <0.2° and 15 meV. The samples were glued to a cryostat, cooled to 10 K in ultrahigh vacuum, and cleaved prior to measurements, which resulted in flat, shiny (001) surfaces. Photon energies were in the range 60–100 eV.

Single crystal x-ray diffraction confirmed that the crystal structure of EuMg₂Bi₂ crystallizes into the tetragonal CaAl₂Si₂ structure type with space group P-3m1 (No. 164). Table I lists the lattice parameters of EuMg₂Bi₂, while Table II gives the atomic coordinates and the equivalent isotropic displacement parameters. Atomic site disorders for Eu and Bi have been refined but not observed. The structure consists of alternating Eu²⁺ and MgBi₄ edge-sharing tetrahedron layers. In Fig. 1(b), an image of the alternating layers can be observed, additionally, Fig. 1(a) displays the top view along the c axis showing the double stacked puckered Mg₃Bi₃ honeycombs. Three of the bond lengths found in the MgBi₄ tetrahedron share the same length of 2.914(3) Å, while the remaining Mg–Bi bond length is slightly longer at 2.965(7) Å, which is likely considering the bulky adjacent Mg²⁺ atoms. The bond angles in the MgBi₄ tetrahedron are all approximately equivalent at ∼109°, making this a relatively ideal tetrahedron environment. The tetragonal structure of EuMg₂Bi₂ is analogous to the well-established topological semimetal Mg₂Bi₃, except Eu²⁺ now replaces the Mg²⁺ layer in Mg₃Bi₂.³⁰ The chemical formula as determined by SEM-EDS and was found to be Eu_1.4(2)Mg_1.9(4)Bi_2.0(2), which further verifies the chemical composition and stoichiometry of the EuMg₂Bi₂ phase.

Multiple techniques have been used to understand the magnetic and electronic properties of EuMg₂Bi₂. Field-dependent magnetization curves up to 9 T were measured from 2 to 100 K with applied fields perpendicular and parallel to the c axis to verify with previously reported measurements, as seen in Fig. 2(a).³² Based on the apparent magnetic saturation (MH-curve) at 2 K, the saturated magnetic moment, μ_sat, is approximately 7.7 μ_B/Eu²⁺, which is in good agreement with the theoretical 7.94 μ_B per Eu²⁺ ion. A slight magnetic anisotropy is observed and is most prominent at 2 K, where the magnetic moment saturates at ∼3 T for fields perpendicular to the c axis and ∼3.4 T for fields parallel to the c axis. To further verify the magnetic behavior of EuMg₂Bi₂ temperature-dependent magnetic susceptibility, measurements were performed from 1.8 to 300 K at applied fields from 0.05 to 0.3 T both perpendicular and parallel to the c axis. As shown in Fig. 2(b), the sharp peaks at lower temperatures confirm the Néel temperatures of EuMg₂Bi₂, T_N//C∼ 7.0 K and T_N┴C ∼ 6.7 K. The magnetic susceptibility was fit using the modified Curie–Weiss formula, $χ=χ0+CT−θ,$ where θ is the paramagnetic Curie temperature and C is the Curie constant. The resulting Curie–Weiss temperatures and effective magnetic moments were determined to be θ_CW∼ 7.4(9) K and μ_eff∼ 7.8(4) μ_B/Eu²⁺ for fields perpendicular to the c axis and θ_CW∼ 5.5(6) K and μ_eff∼ 7.9(3) μ_B/Eu²⁺ for fields parallel to the c axis. The observed positive θ_CW temperature indicates that the magnetism of EuMg₂Bi₂ is dominated by ferromagnetic interactions.

The four-probe temperature and field-dependent resistivity measurements were performed in the ab-plane of a single crystal of EuMg₂Bi₂. As shown in Fig. 3(a), the temperature-dependent resistivity increases in resistivity from ∼1.82 × 10⁻⁵ Ωm to ∼2 × 10⁻⁵ Ωm over a temperature range of 1.8–300 K, in the absence of an applied magnetic field. This result confirms the semi-metallic behavior of EuMg₂Bi₂.⁸ Additionally, the temperature-dependent resistivity plot reaches a minimum at ∼20 K, then the resistivity increases slightly and finally proceeds to decrease. This jump at ∼8.6 K can be related to the AFM ordering temperature T_N//C∼ 7 K and T_N┴C ∼ 6.7 K. The observed minimum in the resistivity has been attributed to the Anderson localization, an interference phenomenon caused by a high enough disorder in the lattice and resulting in a localized electron wavefunction.³² In the case of EuMg₂Bi₂, the disorder has been potentially linked to the low carrier-density metal. A domelike feature is observed in the field-dependent resistivity and magnetoresistivity (MR) of EuMg₂Bi₂ when an external magnetic field is applied parallel to the ab-plane, reminiscent of the proposed magnetic topological candidates EuBiTe₃⁴⁴ and EuIn₂As₂.⁴⁵ At 2 K, the MR appears to saturate at ∼3 T consistent with the field-dependent magnetization plot at 2 K. The evolution of the MR with temperature can be seen in Fig. 3(c), where it transitions from largely negative ∼≤20 K to slightly positive ∼≥50 K. In theory,⁴⁶ the behavior of the MR for an AFM metal will either positively vary with H² when the applied magnetic field is along the easy axis or be negligible when the applied magnetic field is perpendicular to the easy axis. However, as can be seen in Fig. 3(c), this is clearly inconsistent with the MR for EuMg₂Bi₂. For this circumstance, the MR behavior may be explained by the crossover between weak localization (WL) and weak antilocalization (WAL) in terms of the competition between the AFM and FM orders, which has been speculated for previously reported materials in this family such as EuIn₂As₂.^44,45 Taking into account the positive Curie–Weiss temperature, which indicates the presence of a FM interaction, and the aligned in plane Eu magnetic moments determined from neutron diffraction, this kind of behavior is likely. Additionally, the high spin state of Eu²⁺ is verified in EuMg₂Bi₂ by the effective moment of μ_eff∼ 7.8(4) μ_B/Eu²⁺ and μ_eff∼ 7.9(3) μ_B/Eu²⁺ for fields perpendicular and parallel to the c axis from the temperature-dependent magnetization measurements, which suggests highly localized spins.⁴⁵ Therefore, at lower fields (≤1 T), the positive MR can originate from localized carriers that result from the long-range AFM order. While at higher fields, magnetic polarons (MP), or FM clusters, can form resulting from the large exchange interaction between localized spins and conduction electrons, giving rise to more itinerant carriers and causing the MR to decrease.^47,48 This can be observed at temperatures much higher than T_N. However, further investigations should be performed in order to fully understand the magnetoresistivity behavior in EuMg₂Bi₂. An extensive discussion on the electronic transport properties of EuMg₂Bi₂ has been provided by Pakhira et al..³² 

Based on the neutron single crystal diffraction experiment measured on the HB-3A Dimensional Extreme Magnetic Neutron Diffractometer (DEMAND), it was found that a set of magnetic peaks at 4 K can be reasonably indexed by the k vector (0, 0, ½), as shown in Fig. 4. Due to the heavy absorbance of Eu in EuMg₂Bi₂, we used the shortest available wavelength to reduce the absorption effect. The absorption correction was carefully taken by using PLATON software.^39,40 At 4 K, the spins align ferromagnetically within the ab plane, which can potentially explain the observed positive θ_CW determined from the bulk measurements; however, the antiferromagnetic spin alignment between the Eu layers [see Fig. 4(c)] is consistent with the antiferromagnetic transition observed in the bulk magnetic susceptibility data. Note, the magnetic structure is also consistent with the recently reported magnetic structure for EuMg₂Bi₂ by Pakhira et al.³³ This magnetic structure is similar to EuSn₂P₂,¹⁵ another potential magnetic topological Eu based 122 material discovered by our group. The resulting refined magnetic moment at 4 K was 6.610(37) μ_B/Eu, which is lower than the expected theoretical total magnetic moment of 7.94 μ_B per Eu²⁺ ion. Additionally, the refined magnetic moment reported here is slightly higher than the recently reported magnetic moment of 5.3(5) μ_B/Eu.³³ The temperature dependence of the magnetic scattering at (0 0 2.5) may follow the empirical power law I = A(T_M −T/T_M)^2β+ B, where T_M is the magnetic phase transition critical temperature, A is the proportionality constant, β is the order parameter critical exponent, and B is the background. The best fit was performed in the temperature range ∼5.2–8 K, shown as the red line in Fig. 4(a), which revealed a magnetic transition temperature of T_M∼ 6.72(7) K. As a result of the limited neutron scattering data, an accurate interpretation of the power law fitting could not be attained while allowing for all free variables, thus the β value was fixed. Considering the most desirable fitting was achieved with a fixed β value of 0.23, this system can be described by the 2D Heisenberg model, which is consistent with the weak interlayer antiferromagnetic coupling suggested by the well matched positive CW temperature and T_N (i.e., FM is the dominant interaction and the AFM interaction is negligible). From the power law, the calculated magnetic moment at T = 0 K was determined to be 6.79(4) μ_B/Eu, indicating that the magnetic moment continues to order past 4 K. This may help us to explain the tendency of the ordering parameter to saturate at lower temperatures, as shown in Fig. 4(a).

To understand the magnetic and electronic properties of EuMg₂Bi₂, electronic structure calculations were performed using the generalized gradient approximation (GGA) plus a correlation parameter U of 6 for both with and without spin–orbit coupling (SOC), as shown in Fig. 5. A small continuous bandgap of ∼0.4 eV is seen throughout the Brillouin zone without SOC, which has been previously attributed to the large hole mobility in EuMg₂Bi₂ and the strong hybridization between Bi and Eu atoms.^49-51 The Eu 4f electrons are roughly localized around −2 without SOC and in between −2 and −3 with SOC. The Mg and Bi states are hybridized and located near the Fermi level. The calculated magnetic moment on the Eu atoms was determined to be 7 μ_B, which is consistent with the bulk magnetic measurements. Considering the GGA + U with SOC, the small bandgap disappears, and Fermi surface states appear around the $Γ$ points.

The Z2 topological invariant was calculated by the Wilson loop method in Fig. 6. The Wilson band is an open curve traversing the entire Brillouin zone in the time reversal invariant plane kz = 0. The result shows that the Z2 invariant equals 1, indicating that EuMg₂Bi₂ is a good topological insulator in the presence of SOC.

The core level photoemission spectrum presented in Fig. 7(a) exhibits two characteristic high-intensity peaks of Bi 5d orbitals, and sharp peaks of Mg 2p and Eu 4f. This confirms the chemical composition of our samples, and the cleaved surface purity. ARPES maps taken at 10 K from the (001) surface, which is the surface at which EuMg₂Bi₂ naturally cleaves, show several hole-like bands encircling the Γ point of the surface Brillouin zone. The outer band, consistently the sharpest and most intense at all photon energies, exhibits hexagonal constant energy contours, consistent with the threefold symmetry of the material. Inside of the contour, at least two other distinct bands are present at the binding energies probed; their shape and visibility change with photon energies, which signifies different cuts of the bulk band structure at different perpendicular momenta. The outer band, as can be seen from our spectra along M–Γ–M at photon energies of 62 and 96 eV, Fig. 7(c), shows no measurable perpendicular momentum dependence. Namely, as these photon energies represent approximately a 0.95 1/Å change of kz, which is 2.4 times π/c, the spectra are guaranteed to come from different parts of the Brillouin zone; yet, the measured bands under the same pair of dashed lines added in Fig. 3(c) do not change position. Unsurprisingly, this band is similar to that observed in structurally related type-II nodal-line semimetal Mg₃Bi₂.²⁵ All bands appear highly linear, dispersing at 2.5 × 10⁵ m/s. Our results differ from a recent unpublished study by Kabir et al.²⁹ Inasmuch as our core level spectra show all the expected peaks, our surface Brillouin zone is hexagonal and not square, and the only bands crossing the Fermi level are those around Γ.

After refinement of the neutron scattering data including an applied absorption correction, the A-type magnetic structure of EuMg₂Bi₂ was revealed. Similar to other materials in the Eu based 122 family, the Eu²⁺ moments in EuMg₂Bi₂ are ferromagnetically ordered in the plane and antiferromagnetically stacked along the c axis. The bulk magnetic susceptibility measurements indicated a dominant FM interaction from the positive Curie–Weiss temperatures of θ_CW∼ 7.4(9) K and θ_CW∼ 5.5(6) K for fields perpendicular and parallel to the c axis, respectively. The reduced ordering moment of 6.61(4) μ_B/Eu is an indication of a secondary interlayer antiferromagnetic interaction. This is also consistent with the 2D Heisenberg model that well fits the magnetic ordering parameter. Likewise, the magnetoresistivity measurements exhibit a similar dependence on the competition between the AFM and FM interactions. The resulting ARPES measurements reveal the outer linearly dispersed Dirac surface states, similar to that observed in the type-II nodal-line semimetal Mg₃Bi₂,³⁰ and are consistent with our experimental prediction.

See the supplementary material for powder x-ray diffraction data and single crystal XRD information.

The work at Rutgers was supported by Beckman Young Investigator (to Weiwei Xie) and No. NSF-DMR-2053287. M.M. was supported by DOE Office of Science Graduate Student Research (SCGSR) Fellowship program. H.C. was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Early Career Research Program Award No. KC0402010 under Contract No. DE-AC05-00OR22725. This research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by ORNL. T.-R.C. was supported by the Young Scholar Fellowship Program from the Ministry of Science and Technology (MOST) in Taiwan under a MOST Grant for the Columbus Program No. MOST109-2636-M-006-002, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences, Taiwan. This work was supported partially by the MOST, Taiwan, Grant No. MOST107-2627-E-006-001. This research was supported, in part, by Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at National Cheng Kung University (NCKU).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.