Introduction

Topological Dirac semimetals (TDSs) materialize a new state of quantum matter which hosts exotic quantum phenomena such as high carrier mobility1,2, giant linear magnetoresistance1, and Fermi-arc-mediated surface transport3,4. Such physical properties are governed by the Dirac-cone energy band showing linear dispersions in all the directions in the three-dimensional (3D) momentum (k) space (kx, ky, and kz) characterized by the crossing of bulk valence band (VB) and conduction band (CB) at discrete points (Dirac points) in k space. Such peculiar electronic structure is formed as a result of the band inversion due to the strong spin–orbit coupling (SOC) and the protection of band degeneracy by the time-reversal and specific crystalline symmetries. The most intriguing aspect of TDS is that it can be switched to other topological states such as quantum spin Hall states, topological insulators (TIs), Weyl semimetals, and topological superconductors5,6,7,8,9,10,11,12. For instance, breaking the time-reversal symmetry by an external magnetic field leads to a Weyl semimetal state possessing a pair of spin-split massless Weyl cones in the bulk and novel Fermi arcs at the surface13,14. Another way with a potentially high degree of freedom in manipulating the topological state is chemical substitution which may enable, e.g. to control the Dirac-cone dispersion via tuning SOC or to trigger a topological superconducting state with Majorana fermions by carrier doping. Angle-resolved photoemission spectroscopy (ARPES) has played a pivotal role in clarifying TDSs by directly visualizing the bulk Dirac-cone band, as demonstrated in Na3Bi, Cd3As2, and α-Sn/InSb(111)5,7,15,16,17,18 which host Dirac-cone bands protected by the rotational symmetry. Despite many theoretical predictions for TDS candidates19,20,21,22,23,24,25,26,27,28,29, experimental investigations hitherto performed on TDSs are mainly focused on these prototypical TDSs, which, however, have inherent problems such as high reactivity in the atmosphere and/or difficulty of chemical substitution of elements in crystals. Thus, it is highly desirable to explore new TDS materials which overcome such difficulties.

Recently, it was proposed that a ternary compound BaMg2Bi2 is a 3D TDS30 from the automated construction of Wannier functions and high-throughput screening out of non-topological materials. In BaMg2Bi2 crystal, Ba atoms can be easily replaced with other elements, and its general chemical composition is expressed as XMg2Bi2 where X represents alkaline earth or rare earth metal (Ca, Sr, Ba, Yb, Eu, etc.)31,32,33,34,35. XMg2Bi2 crystalizes in the CaAl2Si2-type structure (space group of P\(\overline{3}\)m1, No. 16436 which has the C3 rotational symmetry with respect to the c-axis (Fig. 1a; the Brillouin zone (BZ) is shown in Fig. 1b). Our first-principles band-structure calculations including SOC shown in Fig. 1c reveal that BaMg2Bi2 possesses a Dirac cone close to the Fermi level (EF) along the ΓA line of bulk BZ (Fig. 1b) due to the bulk-band inversion and the protection by the C3 symmetry, consistent with the previous calculation30. The predicted simple band structure with no EF crossings of other bands with topologically trivial origin makes BaMg2Bi2 an excellent system to search for exotic properties associated with bulk Dirac fermions. Moreover, the XMg2Bi2 family has a high potential to realize different types of topological states. For example, controlling the SOC by replacing the X element and breaking the crystal symmetry e.g. by applying pressure may trigger the topological phase transition to other phases such as TI and ordinary insulator phases30,37. Despite such interesting proposals, the electronic states of XMg2Bi2 family have been scarcely investigated. It is thus urgently required to clarify its fundamental band structure.

Figure 1
figure 1

(a,b) Crystal structure and bulk hexagonal BZ of XMg2Bi2 (X = Sr and Ba), respectively. (c) Calculated band structure of BaMg2Bi2 along high-symmetry lines in bulk BZ obtained from the first-principles band-structure calculations, and its expanded view around the Dirac point (right panel). (d) Photograph of BaMg2Bi2 single crystal. (e) X-ray Laue backscattering image of BaMg2Bi2 obtained for the (0001) surface.

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In this work, we report high-resolution angle-resolved photoemission spectroscopy (ARPES) of XMg2Bi2 (X = Sr and Ba) single crystals. By utilizing energy-tunable soft-X-ray (SX) and vacuum ultraviolet (VUV) photons from synchrotron radiation, we experimentally established the electronic structure of BaMg2Bi2 over the 3D bulk BZ and found an evidence for the existence of bulk Dirac fermions characterized by the band touching of bulk VB and CB. In sharp contrast, we found a bulk-band gap in SrMg2Bi2 suggestive of its ordinary insulator nature. We discuss the obtained results in relation to the band-structure calculation, SOC, and topological phase transition.

Results and discussion

Samples and experimental

XMg2Bi2 single crystals with typical size of 2 × 2 × 1 mm3 (see Fig. 1d) were synthesized by the self-flux method (for details, see “Methods”). ARPES measurements were performed with micro-focused VUV synchrotron light at BL28 in Photon Factory and BL5U in UVSOR, and also with SX photons at BL2 in Photon Factory. Samples were cleaved in situ along the (0001) plane of the hexagonal crystal in an ultrahigh vacuum of 1 × 10–10 Torr. Prior to ARPES measurements, the crystal orientation was determined by the X-ray Laue backscattering measurements which signify clear six-fold symmetric diffraction spots consistent with the (0001) cleaved plane, as shown in Fig. 1e.

Valence-band structure of BaMg2Bi2

We at first discuss the overall electronic structure of BaMg2Bi2. Figure 2a displays the energy distribution curve in a wide energy range measured at the photon energy (hν) of 500 eV. One can recognize several core-level peaks which are assigned to the Ba (5s, 5p), Mg (2p), and Bi (5p, 5d) orbitals. The sharp spectral feature and the absence of core-level peaks from other elements confirm the clean sample surface. Besides the core levels, one can find a weak feature near EF originating from the VB composed mainly of the Bi 6p orbitals. To determine the 3D bulk electronic states, we performed ARPES measurements at the normal-emission setup with varying hν in the SX region (250–522 eV). To visualize the bulk band in 3D k space, it is useful to use bulk-sensitive SX photons, because the increase of photoelectron mean-free path compared to VUV photons reduces the intrinsic uncertainty of the out-of-plane wave vector kz through the Heisenberg's uncertainty principle and as a result allows the accurate 3D band mapping. Figure 2b shows the obtained VB dispersion along the wave vector perpendicular to the sample surface (kz). One can recognize some energy bands displaying a finite kz dispersion, e.g., at the binding energies EB’s of EF-1.5 eV, 2–3 eV, and 3.7–4.3 eV. The observed band dispersions well follow the periodicity of the bulk BZ; for example, the near-EF band appears to have the top and bottom of dispersion at Γ and A points, respectively, consistent with the bulk-band calculations including SOC (red curves). A good matching of periodic oscillation in the band dispersion between the experiment and calculation is seen in the entire EB region, showing their bulk-origin nature.

Figure 2
figure 2

(a) Energy distribution curve of BaMg2Bi2 in a wide binding-energy (EB) range measured at hν = 500 eV. (b) ARPES intensity of BaMg2Bi2 measured at T = 40 K along the ΓA cut by varying hν from 250 to 522 eV, together with the calculated band structure (red curve). The inner potential was estimated to be V0 = 15.0 eV from the periodicity of out-of-plane band dispersion. (c) ARPES intensity along the \(\overline{{\Gamma \text{M}}}\) cut measured at hν = 489 eV. Calculated band structure along the ΓM cut is also overlaid. (d) Plots of ARPES intensity as a function of two-dimensional wave vector (kx and ky) at representative EB slices (EB = 0, 0.2, 0.4, and 0.6 eV) measured at hν = 70 eV for BaMg2Bi2.

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Since the VB has the top of dispersion around the Γ point of bulk BZ, next we carried out ARPES measurements along the in-plane kx cut crossing the Γ point (ΓM cut), as shown in Fig. 2c. One can recognize several bands whose energy dispersion is reproduced semi-quantitatively by the band calculation along the ΓM cut, suggesting no discernible band-mass renormalization, indicative of a weak electron correlation in BaMg2Bi2. One can also see two hole bands which rapidly disperse toward EF with approaching the Γ point. The outer one appears to cross EF and forms a hole pocket centered at the Γ point. To visualize the topmost VB more clearly, we plot in Fig. 2d the ARPES intensity as a function of in-plane wave vector, kx and ky, at selected EB slices (EB = EF, 0.2, 0.4, 0.6 eV) at kz ~ 0 (in reduced BZ scheme; kz = 7 × 2π/c in extended BZ), measured with VUV photons (hν = 70 eV) that achieve higher energy and k resolution. At EB = EF, one can clearly see a circular Fermi surface centered at the Γ point. This Fermi surface follows the periodicity of hexagonal BZ (for details, see Supplementary Fig. S1). With increasing EB, the circular intensity pattern gradually expands and shows a stronger hexagonal deformation, signifying that the topmost VB forms a hole pocket, consistent with the positive sign of Hall coefficient in our sample (see Supplementary Fig. S2). Taking into account the sizable kz dispersion of the topmost VB (Fig. 2b), it is suggested that the hole pocket has the 3D character. By assuming an isotropic spherical shape, we have estimated the hole carrier concentration to be 7 × 1018 cm−3. This value is in rough agreement with the value estimated from the Hall coefficient (2.4 × 1018 cm−3 at T = 50 K; for details, see Supplementary Sect. 2 of Supplementary Information), suggesting that the ARPES data reflect intrinsic bulk properties.

Now that the overall VB dispersion is established, we present the near-EF band structure to examine the existence of bulk Dirac fermions. Figure 3a shows the ARPES intensity (bottom panel) and its expanded view in the vicinity of EF (top panel), measured almost along the ΓK (ky) cut (cut #1 in Fig. 3b) at hν = 70 eV. One can see linearly dispersive inner and outer hole bands, and the latter crosses EF. Since hole carriers are doped into the crystal due to the slight off-stoichiometry of chemical composition, we were unable to see the bulk CB on the bare surface. To search for a possible band touching of VB and CB predicted in the calculation, we deposited K atoms onto the surface of BaMg2Bi2 in ultrahigh vacuum. One can clearly see in Fig. 3c that the hole bands in K-deposited BaMg2Bi2 are shifted downward as a whole by ~ 0.3 eV with respect to those in pristine BaMg2Bi2 due to the electron doping to the surface. Intriguingly, VB in the negative ky region appears to almost continuously disperse across k = 0 and cross EF at positive ky, without losing its intensity at ky = 0, suggestive of the absence of band gap. Taking into account that the linearly dispersive VB at positive ky shows a weaker intensity due to the dipole matrix-element effect of photoelectron intensity, together with the expectation that the band dispersion is symmetric with respect to ky = 0, it is suggested that the VB and CB touch each other at the Γ point and forms the Dirac-cone-like dispersion. To corroborate the band touching, we show in Fig. 3d,e the momentum distribution curves (MDCs) corresponding to the ARPES intensity of Fig. 3a,c, respectively. One can see that, in K-deposited BaMg2Bi2 (Fig. 3e), the peak for the bulk VB is smoothly connected to that for the bulk CB without obvious discontinuity around the Γ point, supporting the absence of an energy gap. This is also visualized by the experimental band dispersion obtained by tracing the peak position of MDCs shown in Fig. 3f. The band touching only at a single k point in the kx-ky plane (i.e., the \(\overline{\Gamma }\) point of the surface Brillouin zone) is confirmed by the ARPES intensity measured along the off-\(\overline{\Gamma }\) cut (cut #2 in Fig. 3b) in Fig. 3g that signifies the apparent absence of CB due to an energy separation between the CB and the VB, supporting the cone-shaped band dispersion. To further support this conclusion, we have carried out ARPES measurements with different experimental conditions. Figure 3h,i show the representative ARPES intensity after the K deposition on the surface of BaMg2Bi2 obtained by another cleaving, measured at hν = 110 eV nearly along the ΓM cut (#3 and #4, respectively, in Fig. 3b) thus with different light polarization with respect to the sample surface. While the distribution of ARPES intensity is different from that of Fig. 3c (note that somewhat fuzzy character of the ARPES spectrum along the ΓM cut is attributed to the matrix-element effect of photoelectron intensity), these intensity patterns commonly show that CB is located rightly above the VB without a band-gap opening. As shown in Fig. 3j, the plot of MDCs in the EB = 0–0.35 eV region of Fig. 3i reveals two broad peaks at EB = EF (triangles) whose k separation becomes gradually smaller with increasing EB, and the two peaks eventually merge into a single peak at EB = 0.3 eV. This signifies the existence of V-shaped CB. We have experimentally determined the energy position of VB and CB from numerical fittings of MDCs in Fig. 3c,h,i and found that outer hole band appears to touch the CB at the Γ point within our experimental uncertainty, as shown in the two panels on the left side of Fig. 3k. The band dispersion determined by ARPES shows an overall agreement with the calculated band dispersions along the ΓM cut (kz = 0; dashed curves, slightly away from the Dirac point) and off-ΓM cut (kz = 0.04 Å−1; solid curves, passing through the Dirac point) as shown in the second panel from the right in Fig. 3k. As shown in the right panel of Fig. 3k, the calculated Dirac points exist not exactly at the Γ point but slightly (by 0.04 Å-1) away from the Γ point along the ΓA cut. By taking into account the finite kz broadening (0.1–0.2 Å−1, estimated from the photoelectron escape depth) which is wider than the k separation between adjacent Dirac points, the present ARPES data would reflect the band dispersion integrated over the kz region involving two Dirac points. Thus, although it is difficult to resolve each Dirac point due to the inevitable kz broadening effect, the overall agreement between the experimental and calculated band dispersions is consistent with the existence of Dirac points around the Γ point, and therefore supportive of the TDS nature of BaMg2Bi2. To support the 3D nature of the band touching, we show in Fig. 3m,n the ARPES intensity measured with hν = 80 and 90 eV which correspond to kz ~ 0.5π (between AH and ΓK cut) and π (AH cut), respectively (Fig. 3l). One can recognize that the CB seen at hν = 70 eV (along the ΓK cut; Fig. 3c) is absent for both hν = 80 and 90 eV because it moves upward and enters the unoccupied region, suggesting that the band touching occurs in a particular kz point (kz ~ 0) in the 3D Brillouin zone, indicative of the overall 3D cone-shaped nature of the bulk band dispersion. This also suggests that the influence of K-deposition-induced electron doping extends into the bulk region, so that the quantized two-dimensional CB, as observed in semiconductor surfaces38, is not formed in K-deposited BaMg2Bi2.

Figure 3
figure 3

(a) VB ARPES intensity (bottom panel) and expanded image in the vicinity of EF (top panel) for BaMg2Bi2 measured nearly along the ΓK cut at hν = 70 eV. (b) Brillouin zone of BaMg2Bi2 in the kx–ky plane together with k cuts where the ARPES data shown in (c) (cut #1), (g) (cut #2), (h) (cut #3), and (i) (cut #4) were obtained. (c) ARPES intensity near EF measured along cut #1 after K deposition onto the surface of BaMg2Bi2. (d,e) MDCs at several EB’s (EF-1.1 eV) corresponding to the ARPES intensity of (a) and (b), respectively. Dashed curves are a guide to the eyes to trace the band dispersion. (f) Experimental band dispersions along cut #1 for pristine and K-deposited BaMg2Bi2, obtained by tracing the peak position of MDCs in (d) and (e). Band dispersion for pristine sample is shifted downward by 0.3 eV with respect to that of K-deposited counterpart to align the energy position of the Dirac point. (g) ARPES intensity measured along cut #2 for K-deposited BaMg2Bi2. (h,i) ARPES intensity measured along cuts #3 and #4, respectively, for K-deposited BaMg2Bi2. (j) MDCs in the area enclosed by red rectangle in (i). (k) Experimental band dispersion (circles and dashed curves) obtained from the MDC analyses along cuts #1, #3, and #4, compared with the calculated band dispersion along the kx cut at kz = 0 (ΓM cut; red dashed curve) and 0.04 Å−1 (k cut crossing the Dirac point; red solid curve), and with that along the kz axis (ΓA cut; blue curves). Experimental and calculated band dispersions are aligned in energy with respect to the Dirac point. (l) Brillouin zone of BaMg2Bi2 in the ky–kz plane together with k cuts where ARPES measurements shown in panels (m) and (n) were performed. (m,n) ARPES intensity for K-deposited BaMg2Bi2 measured at hν = 80 eV and 90 eV, respectively.

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Switching of band inversion

Next we address an important question whether or not the electronic structure of isostructural SrMg2Bi2 is similar to that of BaMg2Bi2. Figure 4a displays the ARPES intensity along the ΓM cut for SrMg2Bi2 measured with SX photons (hν = 489 eV). One can recognize qualitatively similar VB dispersions to those of BaMg2Bi2 (Fig. 2c), for example, three prominent hole bands centered at the Γ point and relatively flat band at around 4–4.5 eV, are clearly observed, while there exist some quantitative differences such as a slight downward shift of the flat band in SrMg2Bi2 relative to BaMg2Bi2. The topmost hole band crosses EF in SrMg2Bi2, as better visualized in the high-resolution ARPES data in the vicinity of EF shown in Fig. 4b measured with VUV photons which probe the Γ point (hν = 80 eV). To examine the possible band touching of VB and CB, we deposited K atoms on the surface of SrMg2Bi2 as in the case of BaMg2Bi2. As shown in Fig. 4c, although the hole bands shift downward after K deposition, surprisingly, the spectral weight associated with the CB is absent and no Fermi-edge cut-off is observed, in stark contrast to the case of K-deposited BaMg2Bi2 (Fig. 3c,h,i). This contrast behavior is clearly seen in the EDCs at the Γ point (red curves in Fig. 4c,d). The absence of spectral weight above the VB top cannot be explained in terms of the existence of bulk CB with suppressed intensity, because the photoionization cross-section of Ba 6s and Sr 5s orbitals that dominantly contribute to the CB30 is almost equal to each other at hν = 70–80 eV39 (also see Supplementary Fig. S3 which shows the absence of the CB-derived intensity in the ARPES data measured along different k cuts or at different hν’s). It is thus concluded that an intrinsic band gap opens in SrMg2Bi2. This suggests the occurrence of topological phase transition from the TDS state to the insulating phase upon gradually replacing Ba with Sr.

Figure 4
figure 4

(a) VB ARPES intensity of SrMg2Bi2 measured at T = 40 K along the ΓM cut at hν = 489 eV. (b,c) ARPES intensity in the vicinity of EF of SrMg2Bi2 before and after K deposition, respectively, measured almost along the ΓK cut (cut #1 in Fig. 3b) at hν = 80 eV. (d) ARPES intensity for K-deposited BaMg2Bi2 (same as Fig. 3c). EDCs at the Γ point are shown in (c) and (d). (e) Schematic band diagram of XMg2Bi2 suggested from the ARPES results.

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Now we discuss the origin of observed drastic difference in the near-EF band structure between BaMg2Bi2 and SrMg2Bi2. According to the calculation30, the existence of bulk Dirac-cone band in BaMg2Bi2 is associated with the protection of band degeneracy along the ΓA line by C3 rotational symmetry of the crystal, besides the band inversion of bulk VB and CB with different parities. Taking into account these conditions, one can think of two possibilities to account for the transition to the bulk insulating phase upon replacement of Ba with Sr. One is a transition to the TI phase caused by the C3-symmtery breaking, and another is transformation into an ordinary insulator by dissolving the band inversion. The former possibility is ruled out because (i) SrMg2Bi2 has the same space group as BaMg2Bi2 so that the C3 symmetry is preserved, and (ii) no evidence for the topological Dirac-cone surface state was found within the bulk band gap. One may think that crystal imperfections such as disorders, defects, vacancies, and/or strain may break the C3 symmetry. To break the C3 symmetry globally and open a large Dirac gap throughout the crystal, a substantial amount of disorders, defects, and/or vacancies must be present in our SrMg2Bi2 crystals. Local symmetry breaking by a small amount of imperfections unlikely causes a clear gap opening in the ARPES spectrum because ARPES reflects information of photoelectron signal emitted from a wide area over few-tens of micrometers. While such substantial disorders, defects, and/or vacancies may lead to significant spectral broadening in SrMg2Bi2, our ARPES results indicate that the sharpness of ARPES spectrum is comparable between SrMg2Bi2 and BaMg2Bi2 (e.g., compare Fig. 4a and Supplementary Fig. S1b). Therefore, a clear gap opening observed only in SrMg2Bi2 is unlikely triggered by disorders, defects, and/or vacancies. Also, it is unlikely that accidentally introduced strain in the crystal could globally and reproducibly break the C3 symmetry (note that we have confirmed the reproducibility of the gap opening for different crystals). Therefore, the crystal imperfection would not be a main cause of the observed gap in SrMg2Bi2. Thus, the latter possibility, transformation into an ordinary insulator, is likely the case. This conclusion is reasonable because the replacement of Ba with lighter Sr reduces the strength of SOC and may dissolve the band inversion (Fig. 4e) (note that SrMg2Bi2 can be either a trivial insulator or a TDS with a small band inversion by our calculations, depending on the approximation used, likely because SrMg2Bi2 is in the vicinity of the topological phase transition; for details, see Supplementary Fig. S5).

Finally, we compare the TDS characteristics of BaMg2Bi2 with other well-known TDSs, Na3Bi and Cd3As25,7,15,16,17. These TDSs share several common features, for example, the Dirac-cone band is protected by the rotational (C3 or C4) symmetry of crystal and the band inversion is associated with the s orbital of CB and p orbital of VB. We found that the Fermi velocity of BaMg2Bi2 (4.2 eV Å) is comparable to that of Na3Bi (2.4–2.8 eV Å)5 and Cd3As2 (8.47–9.8 eV Å)16,17, implying a potential to achieve a high mobility in BaMg2Bi2. Also, the SOC of BaMg2Bi2 is as strong as that of Na3Bi and Cd3As2 because the Dirac-cone band is formed by the orbital of heavy elements, i.e. Bi 6p and Ba 6s. Regarding the robustness of crystal against the exposure to the air, BaMg2Bi2 and Cd3As2 have an advantage over Na3Bi, because Na3Bi is known to be very unstable in the atmosphere. It is noted that our XMg2Bi2 crystals were stable for at least 1 week in the atmosphere. The robustness is important for reliable measurements of physical properties as well as applications to practical devices. Most intriguing aspect of BaMg2Bi2 compared to other TDSs is that chemical substation can be carried out with ease; e.g. Ba atoms can be completely replaced with other elements in BaMg2Bi2 whereas only partial Cd or As can be replaced with other elements in Cd3As240,41 and even partial substitution is hard to be done in Na3Bi. These characteristics make the XMg2Bi2 system an excellent platform to investigate the topological phase transition starting from the TDS phase.

In conclusion, we have reported the ARPES results of XMg2Bi2 (X = Sr and Ba). Through the band-structure mapping in the 3D BZ by utilizing energy-tunable SX and VUV photons, we revealed that the electronic structure of BaMg2Bi2 is characterized by the touching of bulk VB and CB around the Γ point, supporting its TDS nature. On the other hand, a sizable bulk band gap observed in SrMg2Bi2 suggests its ordinary insulator nature. We concluded that the chemical substitution in XMg2Bi2 works as an effective means to control the topological phase. The present result paves a pathway toward understanding the interplay among SOC, crystal symmetry, and topological phase transition in TDSs.

Methods

Sample preparation

XMg2Bi2 single crystals were synthesized by the self-flux method. The purity of starting materials is > 99% for Sr and Ba, > 99.9% for Mg, and 99.999% for Bi, respectively. Those materials were loaded in an alumina crucible with the molar ratio of X:Mg:Bi = 1:4:6 and the heat treatment was performed in a sealed quartz tube. The growth process is essentially the same as the previous report32, including removal of the flux with a centrifuge at a high temperature of ~ 650 °C. A 2θ–θ X-ray diffraction scan confirmed that the growth facet is (0001) plane and the obtained c-axis lattice constant is consistent with that of the target phase.

ARPES experiments

VUV-ARPES measurements with micro-focused synchrotron light were performed with a DA30 electron analyzer at BL28 in Photon Factory and also with a MBS-A1 analyzer at BL5U in UVSOR. We used circularly/linearly polarized light of 36–200 eV. SX-ARPES measurements were performed with a SES2002 analyzer at BL2 in Photon Factory with 230–600 eV photons with horizontal linear polarization. The energy resolutions for VUV- and SX-ARPES measurements were set to be 10–30 meV and 150 meV, respectively. Samples were cleaved in situ along the (0001) plane of the hexagonal crystal in an ultrahigh vacuum of 1 × 10–10 Torr, and kept at T = 4.5–40 K during the measurements.

Calculations

First-principles band-structure calculations shown in Figs. 1, 2 and 3 were carried out by using the Quantum Espresso code package42 with generalized gradient approximation (GGA)43. Ultrasoft pseudopotential was used and the SOC was included in the calculations. The plane-wave cutoff energy and the k-point mesh were set to be 70 Ry and 8 × 8 × 8, respectively. We have also performed band-structure calculations by using a projector augmented wave method implemented in Vienna Ab initio Simulation Package (VASP) code44 with GGA, local-density approximation (LDA), and Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional, as shown in Supplementary Fig. S5.