Magnetic Topological Dirac Semimetal Transition Driven by SOC in EuMg$_2$Bi$_2$

J. M. Wang(王佳萌)$^{1,2}$, H. J. Qian(钱浩吉)$^{3}$, Q. Jiang(姜琦)$^{4}$, S. Qiao(乔山)$^{1,2,5\hyperlink{s}{*}}$, and M. Ye(叶茂)$^{6,1,2\hyperlink{s}{*}}$

doi:10.1088/0256-307X/41/1/017101

⇦Chinese Physics Letters, 2024, Vol. 41, No. 1, Article code 017101⇨ Magnetic Topological Dirac Semimetal Transition Driven by SOC in EuMg$_2$Bi$_2$ J. M. Wang (王佳萌)1,2, H. J. Qian (钱浩吉)3, Q. Jiang (姜琦)4, S. Qiao (乔山)1,2,5*, and M. Ye (叶茂)6,1,2* Affiliations 1State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China 3Research Center for Intelligent Chips and Devices, Zhejiang Lab, Hangzhou 311121, China 4Center for Transformative Science, ShanghaiTech University, Shanghai 201210, China 5School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 6Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China Received 12 October 2023; accepted manuscript online 29 November 2023; published online 7 January 2024 ^*Corresponding authors. Email: qiaoshan@mail.sim.ac.cn; yem@sari.ac.cn Citation Text: Wang J M, Qian H J, Jiang Q et al. 2024 Chin. Phys. Lett. 41 017101 Abstract Magnetic topological semimetals have been at the forefront of condensed matter physics due to their ability to exhibit exotic transport phenomena. Investigating the interplay between magnetic and topological orders in systems with broken time-reversal symmetry is crucial for realizing non-trivial quantum effects. We delve into the electronic structure of the rare-earth-based antiferromagnetic Dirac semimetal EuMg$_2$Bi$_2$ using first-principles calculations and angle-resolved photoemission spectroscopy. Our calculations reveal that the spin–orbit coupling (SOC) in EuMg$_2$Bi$_2$ prompts an insulator to topological semimetal transition, with the Dirac bands protected by crystal symmetries. The linearly dispersive states near the Fermi level, primarily originating from Bi 6$p$ orbitals, are observed on both the (001) and (100) surfaces, confirming that EuMg$_2$Bi$_2$ is a three-dimensional topological Dirac semimetal. This research offers pivotal insights into the interplay between magnetism, SOC and topological phase transitions in spintronics applications.

DOI:10.1088/0256-307X/41/1/017101 © 2024 Chinese Physics Society Article Text Topological Dirac semimetals represent a novel non-trivial quantum phase that has evolved from a topological insulator, which is protected by crystal rotational symmetry and possesses four-fold degenerate Dirac points near the Fermi surface, as exemplified by Na$_3$Bi and Cd$_3$As$_2$.[1-4] Furthermore, when a magnetic order coexists with a non-trivial topological order and breaks time-reversal symmetry, topological Dirac semimetals can transform into exotic topological states, showing novel quantum phenomena such as the quantum anomalous Hall effect,[5,6] axion insulators,[7-10] and chiral Majorana fermions.[11,12] Through angle-resolved photoemission spectroscopy (ARPES) measurements, Wang et al. revealed the intrinsic anomalous Hall effect in the magnetic Weyl semimetal Co$_3$Sn$_2$S$_2$,[13] as well as the existence of Fermi arcs and the linearly dispersive bulk states.[14] Moreover, Pierantozzi et al. observed the gapless surface states of the magnetic axion insulator EuSn$_2$P$_2$ protected by inversion symmetry and mirror symmetry perpendicular to the magnetic moments.[10] These findings indicate the importance of probing the interplay between magnetism and topological materials, which could pave the way for advances in topological spintronic devices. Consequently, given their intriguing properties, it is imperative to continue the exploration of new magnetic topological semimetal materials. Recently, Zhang et al. predicted a new type-II nodal line semimetal Mg$_3$Bi$_2$, which exhibits both strong spin–orbit coupling (SOC) and weak interlayer interactions.[15-18] By replacing Mg atoms with rare earth elements, various intermetallic compounds $A$Mg$_2$Bi$_2$ with excellent transport properties could be obtained, which may also possess diverse non-trivial topological properties.[19-24] The hybridization between rare earth 4$f$ and Bi 6$p$ electrons significantly enhances both carrier mobility and concentration.[19,25] At the same time, substituting different $A$ elements in $A$Mg$_2$Bi$_2$ provides a method to tune the SOC strength for regulating topological phase transitions.[23,24] However, inhomogeneous elemental doping and substitution complicates the regulation of their electronic properties. Although Marshall et al. observed the linear dispersion of antiferromagnetic EuMg$_2$Bi$_2$, the underlying mechanism and the origin of the observed linear dispersion remain unclear,[26] and a comprehensive characterization of the surface band structure still needs to be accomplished. Therefore, detailed experimental and theoretical investigations into the electronic structure of EuMg$_2$Bi$_2$ are essential for further exploration of the topological Dirac semimetal phase. In this study, we employ first-principles calculations and ARPES to investigate the electronic structure of EuMg$_2$Bi$_2$. By tuning the SOC strength with density functional theory (DFT), we unveil the transition from an insulator to a topological Dirac semimetal. Through photon energy-dependent measurements, our experimental results verify the existence of bulk linearly dispersive states in EuMg$_2$Bi$_2$ and reveal the electronic structure on both (001) and (100) surfaces, confirming EuMg$_2$Bi$_2$ to be a three-dimensional (3D) topological semimetal. Our research holds significant implications for exploring the physical mechanisms and potential device applications of magnetic topological quantum materials. Experiments and Methods. High-quality single crystals of EuMg$_2$Bi$_2$ were synthesized using the Bi self-flux method. Within an argon-filled glovebox, an Eu ingot (Alfa Aesar; 99.99%) and Bi and Mg powders (Alfa Aesar; 99.999%) were mixed in a molar ratio of Eu : Mg : Bi=1 : 4 : 6. The resulting mixture was transferred to a ceramic alumina crucible, then hermetically sealed in a quartz ampoule under high vacuum. Subsequently, the ampoule was rapidly heated from room temperature to 900 ℃, cooled to 850 ℃ within an hour and then slowly cooled down to 750 ℃ over a span of ten hours. After that, EuMg$_2$Bi$_2$ single crystals were quickly separated from the solvent by cooling to 650 ℃ for 24 hours.[19] The obtained crystals showed typical hexagonal edges and were about 0.5 mm in thickness. ARPES measurements were conducted at the BL03U beamline of the Shanghai Synchrotron Radiation Facility (SSRF) with a hemispherical electron-energy analyzer (Scienta-Omicron DA30) using synchrotron light sources. The energy resolution of the ARPES measurement was set to 20 meV. Samples were cleaved in situ along the (001) and (100) planes of the hexagonal crystal in an ultrahigh vacuum better than $1\times10^{-8}$ Pa and were kept at 15 K throughout the measurements. Electronic band structure calculations were performed in the framework of DFT using the Vienna ab initio simulation package (VASP)[27] within the generalized gradient approximation schemes (GGA) and the GGA plus Hubbard $U$ (GGA + $U$) scheme.[28] The on-site $U = 4.0$ eV was used for Eu 4$f$ orbitals. The location of the $f$ band was adjusted by modifying the magnitude of $U$ to achieve agreement with experimental data. The SOC effects were included in calculations. A 9$\times$9$\times$3 Monkhorst–Pack $k$-point mesh was used in the computations. We established an antiferromagnetic unit cell model with a magnetic moment of 6.7$\mu_{\scriptscriptstyle{\rm B}}$ for a single Eu atom. The surface state calculations were performed using the WANNIERTOOLS software package,[29] and using Eu $f$ and $d$ orbitals, Mg $s$ and $p$ orbitals, and Bi $p$ orbitals as projections to construct Wannier functions.\ucite{30,31

Fig. 1. Crystal and electronic structure of bulk single-crystal EuMg$_2$Bi$_2$. [(a), (b)] Side and top views of the EuMg$_2$Bi$_2$ crystal structure, respectively. The interlayer Eu atoms adopt an A-type antiferromagnetic arrangement. (c) Three-dimensional and projected Brillouin zones in (001) and (100) surfaces with highlighted high-symmetry points. (d) Calculated bulk band structure along the ${\varGamma}$–${K}$–${M}$–${\varGamma}$–${A}$–${L}$–${H}$–${A}$ high symmetry directions, with colors indicating orbital contributions: Bi (green), Mg (blue), and Eu (red). (e) Photograph of an EuMg$_2$Bi$_2$ single crystal. (f) Powder crystal x-ray diffraction patterns of single crystal EuMg$_2$Bi$_2$.

Results. EuMg$_2$Bi$_2$ exhibits a crystal structure that belongs to the trigonal space group ($P3/m1$, No. 164) with lattice parameters $a=b=4.764(1)$ Å, $c=8.20(2)$ Å, and $\alpha=\beta=90^{\circ}$, $\gamma=120^{\circ}$. As depicted in Figs. 1(a) and 1(b), Eu atoms form an interlayer antiferromagnetic structure. The grey arrows represent the magnetic moment directions of antiferromagnetically coupled Eu layers ($T_{\scriptscriptstyle{\rm N}}=6.7$ K). The atoms of Bi and Mg together form a three-fold symmetric structure, which is staggered within the $ab$ plane. In Fig. 1(c), the pink and blue planes represent the surface Brillouin zones (BZs) [(001) and (100)] for the top and side cleavage planes, respectively, projected from the bulk BZ. The grey dots mark the high symmetry points. The bulk band structure, as illustrated in Fig. 1(d), exhibits no band gap proximate to the Fermi level ($E_{\rm F}$), consistent with the semimetallic ground state. The red, green, and blue lines denote the orbital contributions of Eu $f$, Mg $s$, and Bi $p$, respectively. The Eu 4$f$ electrons are localized below the $E_{\rm F}$ at 2–3 eV, forming a typical flat band, and the Bi $p$ orbitals are dominant within the bands near the $E_{\rm F}$. Figures 1(e) and 1(f) depict the optical microscopy image and XRD pattern of the sample at room temperature, respectively. The sample surface is smooth with a typical size of 2 mm, exhibiting clear hexagonal edges, and the sharp XRD peaks further attest to the high quality of the samples.

Fig. 2. (a)–(d) Calculated band structure along the ${K}$–${M}$–${\varGamma}$–${A}$–${L}$–${H}$ directions with varying SOC strength. (e) Schematic band diagram of EuMg$_2$Bi$_2$ suggested from calculations.

The SOC effect cannot be neglected in heavy elements and plays a crucial role in the evolution of topological states, as exemplified by the large bulk gaps in the topological insulator Bi$_2$Se$_3$ family. As for EuMg$_2$Bi$_2$, the formation of its linear dispersion remains unclear. To further explore the band evolution, we conducted DFT calculations by artificially modifying the SOC strength ($\lambda$). Figures 2(a)–2(d) illustrate the band structure along the high symmetry ${K}$–${M}$–${\varGamma}$–${A}$–${L}$–${H}$ directions with varying $\lambda$. Here $\lambda=0$ indicates that SOC is turned off, and $\lambda = 0.25,\, 0.5$, and 1 represent 25%, 50%, and 100% SOC strengths, respectively. As shown in Fig. 2(a), for $\lambda=0$, the B2 (green) and B3 (blue) bands are degenerate along the ${\varGamma}$–${A}$ direction from the P1 to the P2 point, which is protected by the $C_3$ rotational symmetry.[32] An energy gap of approximately 1 eV exists between the valence band maximum and the conduction band minimum, indicating a regular insulator state. When considering the SOC effect, the degeneracy along the ${\varGamma}$–${A}$ direction is gradually broken with increasing $\lambda$ as shown in Fig. 2. In addition, the band gap size at the Fermi level decreases until Dirac points are formed by complete contact. Moreover, during the whole band evolution process, the bulk band changes dramatically. Notably, although the B1 (orange) band at $\lambda=0$ and the B2 (green) band at $\lambda=1$ have similar shapes, they are two distinct bands before and after turning on the SOC, and the former band splits into deeper energy levels with complete SOC strength. This could be easily misunderstood in previous studies[26] as a band gap opened by the SOC effects. Meanwhile, Dachi et al. experimentally observed an insulator to topological semimetal transition from SrMg$_2$Bi$_2$ to BaMg$_2$Bi$_2$ by replacing Sr with Ba.[24] It is explained that the heavy Ba elements have stronger SOC strength than Sr elements, aligning well with our theoretical results. Figure 2(e) schematically summarizes the topological phase transition in EuMg$_2$Bi$_2$ under the influence of SOC.

Fig. 3. Electronic structure of the EuMg$_2$Bi$_2$ (100) surface. [(a), (b)] Experimental and calculated band structures along the $\bar{\varGamma}$–$\bar{A}$ direction. (c) ARPES measured Fermi surface and constant energy contours, compared with those obtained from first-principles calculations.

To demonstrate the 3D nature of the Dirac linear dispersion along $k_z$ in EuMg$_2$Bi$_2$, we analyzed the electronic structure of the (100) surface as illustrated in Fig. 3. The constant energy contours broaden with increasing binding energy $E_{\rm b}$ and the pocket boundary inwardly curves exhibit two-fold symmetry. Our calculated constant energy contour plots are consistent with the experimental results. Figure 3(a) presents the fine band structure along the $\bar{\varGamma}$–$\bar{A}$ direction. This structure primarily reveals highly linearly dispersive bands along the $k_z$ direction in EuMg$_2$Bi$_2$, largely constituted by Bi $p_z$ and $p_x$ orbitals. These electronic configurations exhibit high electron mobility, presenting significant potential for applications in electronic transport and devices.

Fig. 4. Electronic structure of the EuMg$_2$Bi$_2$ (001) surface. (a) Band structure along the $k_z$ direction probed with varying photon energy. (b) Calculated bulk-only states along the $\bar{\varGamma}$–$\bar{M}$ direction. (c) Experimental spectra along the $\bar{\varGamma}$–$\bar{M}$ direction. (d) Theoretical surface states for Bi termination compared with the experimental band structure. The position of the Fermi level is aligned to the experimentally measured value. The momentum distribution curve (MDC) is shown at the top of (d) as a dark purple dotted line. (e) The $k_z$ dependence of the surface states in the $k_z\sim0$ and $k_z\sim\pi$ planes. (f) Experimental and calculated constant energy contours at $E_{\rm b} = 0.4$ eV. The MDC is displayed on the left as a light purple dotted line.

Furthermore, we performed photon-energy-dependent ARPES measurements on the (001) surface to acquire the band structure of EuMg$_2$Bi$_2$ throughout the 3D BZ. Figure 4(a) displays the electronic states at binding energy 0.8 eV along the $k_x$–$k_z$ plane measured with 70–140 eV photon energies, uncovering the high-symmetry points along the $k_z$ direction in the bulk BZ. Clear linear dispersions of the electronic states are observed, corresponding to the S1 and S2 surface states distinct from the bulk bands. Figure 4(b) depicts the calculated bulk only band structure, and the Eu-4$f$ flat band appears at $E_{\rm b}=0.8$ eV, agreeing with the experimental results in Fig. 4(c). Figure 4(c) displays the second derivative of the bulk electronic structure along the $\bar{M}$–$\bar{\varGamma}$–$\bar{M}$ high-symmetry direction with 80 eV photon energy. Several hole-like pockets surrounding the $\varGamma$ point of the surface BZ are visible, with the outermost band exhibiting higher intensity. The fine spectral structure near $E_{\rm F}$ shown in Fig. 4(d), where red and black dashed lines correspond to surface and bulk states, respectively, indicates that the S1 and S2 surface bands associated with the Bi-termination. Furthermore, the hole-like S1 and S2 linear bands consistently exist at continuous photon energies ($k_z$) in the $\bar{\varGamma}/\bar{A}$ plane as shown in Figs. 4(a) and 4(e), demonstrating the nature of the surface states, and $h\nu = 80$, 106 eV probe the $\varGamma$ point at the BZ center, whereas $h\nu = 93$, 119 eV near the A point. The hexagonal equal-energy contours in the theoretical and experimental spectra near the $\varGamma$ point show the three-fold symmetry of EuMg$_2$Bi$_2$, as shown in Fig. 4(f), and the MDC curve can clearly resolve the S1 and S2 bands, further confirming the existence of surface states. In summary, we have demonstrated a topological Dirac semimetal phase transition driven by SOC in EuMg$_2$Bi$_2$ through DFT calculations. By using combined ARPES experiments and theoretical calculations, we reveal linearly dispersive surface states on the (001) and (100) surfaces originating from the Bi 6$p$ orbitals, confirming that EuMg$_2$Bi$_2$ is a 3D antiferromagnetic topological Dirac semimetal. We propose that the topological phase may be further tuned by adjusting the SOC strength via chemical doping. Remarkably, our electronic structure study on EuMg$_2$Bi$_2$ offers a promising platform for understanding the interplay between magnetism and non-trivial topology. Acknowledgements. This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1604302), and the National Natural Science Foundation of China (Grant Nos. U1632266, 11927807, and U2032207). The ARPES experiments were performed with the approval of the Proposal Assessing Committee of SiP.ME$^2$ platform project (Proposal No. 11227902) supported by the National Science Foundation of China. The calculations were carried out at the HPC Platform of SPST, ShanghaiTech University, and we also extend our gratitude to Gang Li. We acknowledge Q. S. Wu for the development of the user-oriented WANNIERTOOLS package. References Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3BiLarge linear magnetoresistance in Dirac semimetal

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