Abstract
In noncentrosymmetric metals, spinorbit coupling induces momentumdependent spin polarization at the Fermi surfaces. This is exemplified by the valleycontrasting spin polarization in monolayer transition metal dichalcogenides with inplane inversion asymmetry. However, the valley configuration of massive Dirac fermions in transition metal dichalcogenides is fixed by the graphenelike structure, which limits the variety of spinvalley coupling. Here, we show that the layered polar metal BaMnX_{2} (X = Bi, Sb) hosts tunable spinvalleycoupled Dirac fermions, which originate from the distorted X square net with inplane lattice polarization. We found that BaMnBi_{2} has approximately onetenth the lattice distortion of BaMnSb_{2}, from which a different configuration of spinpolarized Dirac valleys is theoretically predicted. This was experimentally observed as a clear difference in the Shubnikovde Haas oscillation at high fields between the two materials. The chemically tunable spinvalley coupling in BaMnX_{2} makes it a promising material for various spinvalleytronic devices.
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Introduction
In crystals with broken inversion symmetry, the spin–orbit coupling (SOC) results in a spin and momentumdependent splitting of the energy bands. In 2D electron gases with structural inversion asymmetry, this leads to spinmomentum locking on the Fermi surface whose type is determined by the relative direction between the 2D plane and the internal electric field due to the asymmetric potential gradient. At surfaces and interfaces of heterostructures, for instance, the electric field is generated perpendicular to the 2D plane. This results in Rashbatype spin splitting and leads to a momentumdependent inplane helical spin polarization^{1,2,3,4}. More recently, giant Rashbatype spin splitting was also found in bulk materials, such as th`e layered polar semiconductors BiTeI^{5} and GeTe^{6}, in which novel transport^{7} and optical properties^{8} associated with spinmomentum locking have been experimentally observed.
In 2D systems with inplane inversion asymmetry, on the other hand, an outofplane Zeemanlike field is induced by the SOC, the sign of which switches depending on the position in momentum space. When the Fermi pockets (i.e., electronic valleys) are centered at lowsymmetry points, spinvalley coupling occurs and provides a platform for exploring the novel spin physics and spintronic applications associated with the valley degree of freedom. A promising system for this is the family of monolayer transition metal dichalcogenides (TMDCs) such as MoS_{2}, which has been attracting significant interest. The trigonal prismatic layer of a TMDC consists of a transition metal plane sandwiched between two chalcogen planes^{9}. The prismatic layer is regarded as a graphenelike material with different A and Bsite atoms that break the inversion symmetry. Therefore, the electron and hole valleys located at the K and K′ points are well described by massive Dirac fermions exhibiting valleycontrasting spin polarization perpendicular to the 2D plane^{10,11,12,13}. Such spinvalley coupling has indeed been demonstrated through its distinctive optical and transport properties, such as valleydependent circular dichroic photoluminescence^{14,15,16} and nonreciprocal charge transport^{17}.
To further explore these novel physical properties, it is necessary to control a variety of spinvalleycoupled states. For this, it is important to realize the states in the bulk form of the TMDCs as well as in the thin film form. In the former, chemical substitution can be used to tune the physical parameters determining the spinvalley coupling such as the magnitudes of the SOC and internal electric field. However, the spinvalley coupling is suppressed in the most common bulk phase of MoS_{2} (socalled 2H polytype) because the inversion symmetry is restored^{9}. Instead, another polytype (socalled 3R) with a noncentrosymmetric layer stacking needs to be stabilized by carefully optimizing the crystal growth^{18}. Furthermore, in principle, the valley positions in the TMDCs are fixed at the corners of the hexagonal Brillouin zone (the K and K′ points)^{12,13}, reflecting the graphenelike structure. Such structural restrictions may hamper the widerange control of the spinvalley coupling in TMDCrelated materials.
Recently, the layered Dirac material BaMnSb_{2} was found to have a polar structure with broken inplane inversion symmetry in the bulk form^{19,20}. BaMnSb_{2} belongs to the AMnX_{2} (A: alkaline earth, X: Bi, Sb) series of compounds^{21,22,23,24,25,26,27}, in which each compound consists of a A^{2+}Mn^{2+}X^{3−} blocking layer and a X^{−} Dirac fermion layer (see Fig. 1a). Since the former layer is an antiferromagnetic insulator with a Néel temperature around room temperature^{21,22,23,26,27,28,29,30}, the quasi2D Dirac fermion states are formed in the bulk form. For the layer stacking with a coincident arrangement of the Asites above and below the X^{−} layer, the X^{−} layer tends to form a square net, leading to a nonpolar tetragonal structure^{21,22,23,24,25}. However, in BaMnSb_{2}, the Sb^{−} layer is slightly distorted to an orthorhombic one with zigzag chains, leading to lattice polarization along the plane. Together with the SOC of Sb, this results in the formation of two Dirac valleys with valleycontrasting outofplane spin polarizations around the Y point, as was revealed by firstprinciples calculation and angleresolved photoemission spectroscopy^{19}.
Taking advantage of the bulk layered structure, it is anticipated that the spinvalleycoupled Dirac fermion states in BaMnSb_{2} can be widely tuned by chemical substitution. In particular, control of the SOC magnitude is most important as the primary determinant of the spinvalley coupling. For this purpose, the substitution of Sb with Bi in the Dirac fermion layer is a promising approach. However, it has been reported in the literature^{31} that BaMnBi_{2} has a nonpolar tetragonal structure (I4/mmm) on the basis of powder Xray diffraction. In contrast, by performing synchrotron Xray diffraction and optical measurements on BaMnBi_{2} single crystals, we find here that BaMnBi_{2} has a polar orthorhombic structure similar to that of BaMnSb_{2}. Reflecting the large difference in the orthorhombicity and SOC between the two materials, our firstprinciples calculation predicts that the configurations of the spinpolarized Dirac valleys are totally different between the two materials; BaMnBi_{2} has multiple valleys around the X and Y points. This is consistent with the complicated behavior of the Shubnikov–de Haas (SdH) oscillation observed at high fields in BaMnBi_{2}.
Results
Polar crystal structure
The crystal structure of BaMnBi_{2} (Fig. 1a) was determined by singlecrystal Xray structural analysis (for details see Methods, Supplementary Methods, and Supplementary Table 1). The obtained result shows that BaMnBi_{2} has a polar orthorhombic structure with the same space group as BaMnSb_{2} (Imm2), whereas the orthorhombicity (c − a)/a is estimated to be ~0.15%, which is only 1/10 that of BaMnSb_{2} (~1.3%)^{19}. Despite such a small orthorhombicity, twin domains are clearly observed with a polarized microscope at room temperature (>290 K), as shown in Fig. 1b. Therefore, the polar structure is stable irrespective of the antiferromagnetic order of Mn sublattice, which sets in at T_{N} = 290 K^{31}. Among the X = Bi compounds, all of which are known to have the Bi squarenet structure^{21,22}, BaMnBi_{2} is the only exception that has Bi zigzag chains accompanied by inplane lattice polarization along the caxis.
To verify the polar lattice structure directly, we measured the optical second harmonic generation (SHG), i.e., the frequency doubling of the probing light wave (Fig. 1c), in BaMnBi_{2} single crystals at room temperature. SHG is particularly useful for detecting polar states because, to leading order, SHG occurs only in noncentrosymmetric media. Figure 1d shows the rotational anisotropy of the SHG intensity from a nearly singledomain region on a cleaved surface. The rotational anisotropy was obtained by projecting the component of the SHG light oriented parallel to the polarization ϕ of the incident fundamental light while ϕ was rotated over 360^{∘}. A clear twofold rotation anisotropy with peaks at ϕ ~90^{∘} and 270^{∘} was observed despite the small orthorhombicity. The data is well fitted by assuming the mm2 symmetry with inplane lattice polarization along the caxis (solid curve in Fig. 1d, see Supplementary Methods, and Supplementary Fig. 1 for details), which is consistent with the results of the structural analysis.
Similar optical properties associated with polar lattice distortions have also been observed in BaMnSb_{2}^{20}. However, it should be noted that the orthorhombicity of BaMnBi_{2} is much smaller than that of BaMnSb_{2} (~1/10) while the SOC of Bi is stronger than that of Sb. Because the spinsplitting of the Dirac bands depends on the magnitudes of the polar lattice distortion and the SOC, the spinvalley coupling may significantly differ between these two materials. To demonstrate this, we calculated the band structure of BaMnBi_{2} based on the experimentally obtained crystal structure.
Electronic structure
The calculated band structure of BaMnBi_{2} is shown in Fig. 2a. Two sharp Diraclike bands accompanied by significant spin splitting are formed near the X and Y points (shaded regions in Fig. 2a). We show in Fig. 2b, the spinresolved Fermi surfaces at the Fermi energy ϵ_{F} = 0 eV. All the Fermi surfaces are 2D cylindrical, consistent with the large resistivity anisotropy ρ_{zz}/ρ_{xx} ~100 observed over the measured temperatures (see Supplementary Fig. 2), where ρ_{xx} and ρ_{zz} are the inplane and interlayer resistivities, respectively. Dirac bands form two and fourelectron valleys (named β and α) around the X and Y points, respectively. Each valley is spinpolarized because one of the spinsplit Dirac bands crosses ϵ_{F}. Because of the lattice polarization along the caxis, the spin polarization of s_{z} switches sign with respect to the ΓY line^{32}; upspin (red) and downspin (blue) occur k_{x} > 0 and k_{x} < 0, respectively. This k_{x}dependent s_{z} polarization (so called Zeemantype spin splitting) is qualitatively explained by the SOC Hamiltonian H_{SOC} ∝ σ ⋅ (k × E) = σ_{z}k_{x}E_{y}, where σ = (σ_{x}, σ_{y}, σ_{z}) is the vector of spin Pauli matrices and E = (0, E_{y}, 0) is the internal electric field. In Fig. 2d, f, we show the detailed band dispersion along the k_{y} = 0.8π/c and the k_{y} = 0 lines that are located at the centers of the α and β valleys, respectively. In both valleys, the magnitude of the spinsplitting (typically 200–300 meV as indicated by vertical arrows) is much larger than the energy gap at the Dirac points (~50 meV), resulting in spinpolarized nearlymassless Dirac bands. From the calculation, it is predicted that hole valleys also appear on the ΓM line in a similar fashion to the Bi squarenet analogs^{33,34,35}. However, in BaMnBi_{2}, the corresponding bands have a large band gap (~250 meV) due to the lattice distortion (Fig. 2e). Furthermore, in experiments, the Hall resistivity ρ_{yx} has a negative slope with respect to the field, indicating that electrontype carriers are dominant in BaMnBi_{2} (Fig. 3c). Therefore, it is plausible that the valence bands along the ΓM line do not cross ϵ_{F} in reality. Thus, the obtained Dirac valleys (α and β) for BaMnBi_{2} are totally different from those for BaMnSb_{2}; the latter hosts only two equivalent valleys near the Y point along each MY line^{19,20}. This suggests that the spinvalley coupling of the Dirac fermions in BaMnX_{2} is highly sensitive to the lattice distortion and the SOC.
Quantum oscillation reflecting spinpolarized Dirac fermions
To reveal this unique spinvalley coupling through the transport properties, we performed magnetoresistivity measurements with pulsed high magnetic fields of up to ~56 T. ρ_{zz} (Fig. 3a) and ρ_{xx} (Fig. 3b) exhibit positive magnetoresistance, on which clear SdH oscillations are superimposed at low temperatures (below ~70 K). To clarify the Landau level (LL) structure, we here analyze the SdH oscillation in the conductivity tensors. In the inset to Fig. 3d, we plot the inplane conductivity \({\sigma }_{{\mathrm{xx}}}={\rho }_{{{{\mathrm{xx}}}}}/({\rho }_{{\mathrm{xx}}}^{2}+{\rho }_{{\mathrm{yx}}}^{2})\) and interlayer conductivity σ_{zz} = 1/ρ_{zz} as a function of B_{F}/B, where B_{F} is the SdH frequency and B_{F}/B corresponds to the filling factor normalized by the spinvalley degeneracy factor^{19,36}. Importantly, the SdH oscillations of σ_{xx} and σ_{zz} resemble each other well, including the fine structures at high fields (B_{F}/B < 0.5). Therefore, both SdH oscillations directly reflect the density of states in the LLs. In the following, we use σ_{zz} for the detailed analysis owing to the higher S/N ratio and clearer oscillations even at low fields.
In Fig. 3d, we plot −d^{2}σ_{zz}/d(1/B)^{2} vs 1/B, where the oscillation nearly consists of a single frequency at 1/B ≥ 0.05 T^{−1}. From the Lifshitz–Kosevich equation, the oscillation component is given by^{37,38}
where γ is the Berry phase. Following this equation, we obtained the linear fan diagram (Fig. 3e) by indexing the integer (halfinteger) N to the peaks (dips). The gradient corresponds to the frequency B_{F1} ( = 12.9 T). The vanishing Nintercept ( = 0.01) indicates the nontrivial Berry phase γ = π consistent with the nearly massless Dirac bands (Fig. 2d, f). Note that at 1/B < 0.05 T^{−1} corresponding to the zeroth LL, additional structures (vertical arrows in Fig. 3d) appear, which cannot be explained by the extrapolation of the lowfield LLs (N > 1). We will discuss the origin of these complicated LLs in the quantum limit later.
The application of magnetic fields should lift the degeneracy of the equivalent valleys with opposite spin polarizations because of the Zeeman energy ϵ_{Z} = g^{*}μ_{B}B, where g^{*} is the effective g factor and μ_{B} is the Bohr magneton. However, no clear splitting is observed in the SdH oscillation for N ≥ 1 (Fig. 3d). In 2D systems, the ratio of ϵ_{Z} to the cyclotron energy \({\epsilon }_{{\mathrm{c}}}=e\hslash B\cos \theta /{m}_{{\mathrm{c}}}\) is given by \({\epsilon }_{{\mathrm{Z}}}/{\epsilon }_{{\mathrm{c}}}={g}^{* }({m}_{{\mathrm{c}}}/2{m}_{0})\cos \theta\), where m_{c} is the cyclotron mass, m_{0} is the bare electron mass, and θ is the tilt angle from the normal to the 2D plane. This shows that the spin splitting is enhanced by increasing θ, which leads to the variation of the amplitude and phase of the SdH oscillation with θ. However, this is not the case in BaMnBi_{2}. Figure 4 shows σ_{zz} plotted against \(1/B\cos \theta\). The oscillation period is almost constant for all values of θ up to 77^{∘} (as shown by vertical dotted lines), reflecting the cylindrical quasi2D Dirac valleys. In addition, the amplitude and phase of the SdH oscillation are almost unchanged upon θ without any enhancement of the splitting, indicating that ϵ_{Z}/ϵ_{c} independent of θ, i.e., \({\epsilon }_{{\mathrm{Z}}}\propto B\cos \theta\) (not ∝ B). This reflects that the spins of the Dirac bands are almost fixed along the s_{z} direction even under the tilted fields owing to the SOC, providing firm evidence for the spinvalley coupling. For BaMnBi_{2}, as shown in Fig. 2d, f, the magnitude of the SOCinduced spin splitting is theoretically estimated to be 200–300 meV at each valley. This corresponds to the effective Zeeman field of 1700–2500 T, which is much higher than the applied field. The similar tilt angle dependence has also been observed in other spinvalleycoupled systems, such as BaMnSb_{2}^{19} and monolayer MoS_{2}^{11}.
Discussion
In spite of the small orthorhombicity of (c − a)/a ~0.15% in BaMnBi_{2}, it is surprising that the Dirac bands around the X and Y points show the large spin splitting of 200–300 meV (Fig. 2d, f), which is comparable to that for BaMnSb_{2}^{19,20}. We estimate the magnitude of the inplane lattice polarization by calculating the offcenter displacement of the Bi^{−} layer relative to the Ba^{2+} layer. Note that the offcenter displacement of the [MnBi]^{−} layer relative to the Ba^{2+} layer is negligibly small compared to that of the Bi^{−} layer. The unit cell of the Bi^{−} layer together with the Ba^{2+} ion is shown in Fig. 2c. Two inequivalent Bi^{−} ions (Bi1 and Bi2) exist because of the zigzag chain structure. From the structural analysis, the center position of Bi1 (G_{Bi1}) is estimated to be G_{Bi1} = (0, −0.008(3)c) by taking the Ba^{2+} ion as the origin, while the center position of Bi2 G_{Bi2} = (0, 0.029(3)c). The total displacement of the Bi^{−} layer is given by G_{Bi} = (G_{Bi1} + G_{Bi2})/2 = (0, 0.011(4)c), and reaches ~1% of the caxis length. This value is roughly an order of magnitude larger than the orthorhombicity.
We also estimate the magnitude of the inplane lattice polarization in BaMnSb_{2}, which has a larger orthorhombicity of (c − a)/a ~1.3%^{19}. From the structural data in ref. ^{19}, it is estimated that G_{Sb1} = (0, −0.0082(2)c) and G_{Sb2} = (0, 0.0693(3)c). Note here that ∣G_{Sb1}∣ is similar to ∣G_{Bi1}∣, while ∣G_{Sb2}∣ is ~2.4 times as large as ∣G_{Bi2}∣, which suggests that the magnitude of the lattice polarization is determined by the displacement of Bi2 and Sb2. The resultant total displacement of the Sb^{−} layer is given by G_{Sb} = (G_{Sb1} + G_{Sb2})/2 = (0, 0.0305(3)c). Compared to BaMnBi_{2}, although the orthorhombicity differs by approximately an order of magnitude, the offcenter displacement differs only by approximately three times. Considering the stronger SOC in Bi^{−} than in Sb^{−}, it is natural that the spin splitting of the Dirac bands in BaMnBi_{2} is comparable to that in BaMnSb_{2}. Nevertheless, the valley configuration sensitively depends on each parameter of the lattice polarization and SOC and hence is totally different between the two compounds.
Finally, we discuss the anomaly in the SdH oscillation at 1/B < 0.05 T^{−1} (Fig. 3d). As denoted by triangles in Fig. 4, the position (\(1/B\cos \theta\) value) of the additional structure in the SdH oscillation is unchanged by θ, suggesting it is related to the cyclotron motion. Furthermore, because such an additional oscillation has not been reported in BaMnSb_{2}^{19,20}, it is characteristic of BaMnBi_{2}. Considering that BaMnBi_{2} has two inequivalent Dirac valleys (α and β in Fig. 2b), the additional oscillation may arise from another SdH oscillation superimposed on the main one (B_{F1} = 12.9 T, see Fig. 3e). By the analysis based on this assumption, we obtain the frequency of B_{F2} = 61(2) T from the oscillatory component at high fields (1/B < 0.05 T^{−1}, see Supplementary Note, Supplementary Figs. 4a–d and 5, and Supplementary Table 2). Importantly, the Fermi surfaces corresponding to B_{F1} and B_{F2} are semiquantitatively reproduced by the firstprinciples calculation taking account of a slight shift in ϵ_{F} (by ~30 meV, see Supplementary Fig. 4e). We also note that such multiple carriers in BaMnBi_{2} are also consistent with the nonlinear ρ_{yx} at low fields (Fig. 3c and Supplementary Fig. 3). For these reasons, it is likely that the anomaly at 1/B < 0.05 T^{−1} originates from the SdH oscillation of another Dirac valley. However, to clarify the precise origin, further investigations on the Fermi surface, such as photoemission spectroscopy, will be necessary as a future work.
In conclusion, we have revealed the crystal and electronic structure of the layered Dirac material BaMnBi_{2}. From the singlecrystal Xray diffraction and the SHG measurement, we found that the Bi^{−} squarenet is slightly distorted to form a zigzag chainlike structure with inplane lattice polarization. The firstprinciples calculation predicts that as a result of the broken inversion symmetry together with the SOC, the Dirac valleys show valleycontrasting spin polarization along the s_{z} direction. Such spinvalley coupling is indeed supported by the peculiar dependence of the SdH oscillation on the tilt angle of the field. Interestingly, it is also predicted that the valley configuration in BaMnBi_{2} (four and two valleys around the X and Y points, respectively) is totally different from that in the Sb analog BaMnSb_{2} (two valleys around the Y point). This difference originates from the differing magnitudes of the lattice polarization and the SOC; the experimentally obtained lattice distortion in BaMnBi_{2} is ~1/10 of that in BaMnSb_{2}, while the SOC in Bi is larger than that in Sb. Our results demonstrate that the spinvalley coupling of Dirac fermions in BaMnX_{2} is finely tunable by chemical substitution and is promising for a variety of novel applications, including electronic and optoelectronic devices utilizing the spin and valley degrees of freedom. Furthermore, the high sensitivity of BaMnX_{2} to lattice distortions may be taken advantage of to control spinvalleycoupled phenomena using mechanical stimuli, such as strain^{39}.
Methods
Crystal growth
Single crystals of BaMnBi_{2} were grown by the Bi selfflux method. 99.99% Ba, 99.9% Mn, and 99.999% Bi metal ingots were mixed in an alumina or carbon crucible at a ratio of Ba:Mn:Bi = 1: 1: 4 and sealed in an evacuated quartz tube. The ampoule was heated at 1000 ^{∘}C for 1 day and cooled down to 400 ^{∘}C at a 2 ^{∘}C/h cooling rate. After the ampoule was turned upside down, the flux and single crystals were separated by centrifugation. The synthesized crystals have a platelike shape with the typical size of 2 mm × 2 mm × 1 mm.
Single crystal Xray diffraction
A blockshaped pale black single crystal with the dimensions of 0.15 × 0.15 × 0.12 mm was mounted on a loop and set on a RIGAKU 1/4chi goniometer with PILATUS3 X CdTe 1M in SPring8 BL02B1. The diffraction data were collected using synchrotron radiation (λ = 0.4121 Å) monochromated by Si(311) at T = 100 K. The diffraction data from 7570 data points within 3.954^{∘} ≤ 2θ ≤ 59.298^{∘} were collected and merged to give 3236 unique reflections with the R_{int} of 0.049. The structure was solved by a dualspace method and refined on F^{2} by a leastsquares method using the programs SHELXS^{40} and SHELXL2018/3^{41}, respectively. The anisotropic atomic displacement parameters were applied for all atoms. The final R values from 3236 unique reflections (2θ_{max} = 59.298^{∘}) with I > 2σ(I) are 0.0659 and 0.1759 for R(F) and wR(F^{2}), respectively. The Flack parameter (χ = 0.39(2))^{42,43} from anomalous scattering suggests that two types of absolute structures ( + P and − P) are mixed at a ratio of 0.39:0.61. For further details, see the Supplementary information.
Optical measurements
Optical SHG is a nonlinear optical process in which two photons with a frequency of ω interact with the material and produce a photon with the frequency of 2ω. Under the electric dipole (ED) approximation, the SHG is given by \({P}_{i}(2\omega )={\varepsilon }_{0}{\sum }_{j,k}{\chi }_{ijk}^{(2)}{E}_{j}(\omega ){E}_{k}(\omega )\), where ε_{0}, E_{j}(ω), E_{k}(ω), and P_{i}(2ω) are the permittivity of vacuum, the j and kpolarized incident electric fields, and the induced ipolarized nonlinear polarization, respectively. The EDtype SHG is only allowed in noncentrosymmetric materials or surfaces/interfaces at which the inversion symmetry is broken. The nonlinear susceptibility \({\chi }_{ijk}^{(2)}\) is sensitive to the symmetry of the materials and its nonzero components were determined by the Neumann principle^{44,45}. A cleaved surface of a BaMnBi_{2} single crystal was irradiated by light pulses from a Ti:Sapphire laser with the wavelength of 800 nm, pulse width of 100 fs, and repetition rate of 80 MHz. The reflection geometry SHG measurements were at nearly normal incidence (~1^{∘}) with a typical laser power of 40 mW and laser beam spot size of ~250 μm. All the SHG measurements were performed in vacuum at room temperature.
Firstprinciples band calculations
We performed firstprinciples band structure calculations using the density functional theory with the generalized gradient approximation^{46} and the projector augmented wave method^{47} implemented in the Vienna ab initio simulation package^{48,49,50,51}. The planewave cutoff energy of 350 eV and a 12 × 12 × 12 kmesh were used with SOC included. The experimental crystal structure determined by this study was used. We assumed the Gtype antiferromagnetic order for the Mn atoms. After the firstprinciples calculation, we constructed the Wannier functions^{52,53} using the WANNIER90 software^{54}. We did not perform the maximal localization procedure to prevent the mixture of the different spin components. We took the Mnd and Bip orbitals as the Wannier basis. A 12 × 12 × 12 kmesh was used for the Wannier construction. By using the tightbinding model consisting of these Wannier functions, we obtained the band structure and the Fermi surface colored with the spin polarization, 〈s_{z}〉. The Fermi surface was depicted with a 140 × 140 × 140 kmesh using the FERMISURFER software^{55}.
Transport measurements
The temperature dependence of ρ_{xx} and ρ_{zz} was measured by a conventional fourprobe DC method using a Physical Property Measurement System (Quantum Design). The field dependence of ρ_{xx}, ρ_{yx}, and ρ_{zz} up to ~56 T was measured using the nondestructive midpulse magnet at the International MegaGauss Science Laboratory at the Institute for Solid State Physics, University of Tokyo. In this measurement, we used a fourprobe AC method with a typical current and frequency of 5 mA and 50 kHz, respectively. The typical sample sizes used in the inplane resistivity and interlayer resistivity measurements were 1 mm × 0.3 mm × 0.1 mm (S#3) and 0.8 mm × 0.3 mm × 0.2 mm (S#1, S#2), respectively. The transport properties are almost the same among the measured samples (see Supplementary Fig. 6 and Supplementary Table 3).
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We are grateful to M.Hagiwara and Y.Shimizu for fruitful discussions. We also thank H.Fujimura, and K.Nakagawa for their experimental support. This work was partly supported by the JST PRESTO (Grant No. JPMJPR16R2), the JSPS KAKENHI (Grant Nos. 19H01851, 19K21851, 19H05173, 21H00147, 17H04844, 21H04649, and 18H04226), and the IWATANI NAOJI Foundation. The synchrotron radiation experiments were performed at the BL02B1 of SPring8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2019B1402).
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H.S. conceived the project and planned the experiments. M.K. and H.S. worked on the single crystal growth. T.K., M.K. and H.S. measured the singlecrystal xray diffraction and T.K. analyzed the data. D.S. and M.M. performed the optical measurements. M.O. and K.K performed the firstprinciples calculations. M.K., H.S., R.K., A.M. and M.T. performed the highfield transport measurements. M.K., H.S., H.M. and N.H. measured the resistivity at low fields. M.K. and H.S. wrote the manuscript with the inputs from M.O., T.K., D.S., and M.M. All the authors discussed the results and commented on the manuscript.
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Kondo, M., Ochi, M., Kojima, T. et al. Tunable spinvalley coupling in layered polar Dirac metals. Commun Mater 2, 49 (2021). https://doi.org/10.1038/s4324602100152z
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DOI: https://doi.org/10.1038/s4324602100152z
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