Abstract
Spatial, momentum and energy separation of electronic spins in condensed-matter systems guides the development of new devices in which spin-polarized current is generated and manipulated1,2,3. Recent attention on a set of previously overlooked symmetry operations in magnetic materials4 leads to the emergence of a new type of spin splitting, enabling giant and momentum-dependent spin polarization of energy bands on selected antiferromagnets5,6,7,8,9,10. Despite the ever-growing theoretical predictions, the direct spectroscopic proof of such spin splitting is still lacking. Here we provide solid spectroscopic and computational evidence for the existence of such materials. In the noncoplanar antiferromagnet manganese ditelluride (MnTe2), the in-plane components of spin are found to be antisymmetric about the high-symmetry planes of the Brillouin zone, comprising a plaid-like spin texture in the antiferromagnetic (AFM) ground state. Such an unconventional spin pattern, further found to diminish at the high-temperature paramagnetic state, originates from the intrinsic AFM order instead of spin–orbit coupling (SOC). Our finding demonstrates a new type of quadratic spin texture induced by time-reversal breaking, placing AFM spintronics on a firm basis and paving the way for studying exotic quantum phenomena in related materials.
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Data availability
The data that support the findings of this study are available from the corresponding authors on request. Correspondence and requests of ARPES and DFT data are addressed to C.L. and those of the NV magnetometry data are addressed to J.W.
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Acknowledgements
We thank K. Miyamoto, K. Kuroda, T. Okuda, M. Zhang and L. Deng for the help in SARPES measurements, R. Peng for the help in scanning NV magnetometry measurements and J. Li for discussions. Work at SUSTech was supported by the National Key R&D Program of China (nos. 2022YFA1403700 and 2020YFA0308900), the National Natural Science Foundation of China (NSFC) (nos. 12074161 and 12274194), the Key-Area Research and Development Program of Guangdong Province (2019B010931001), the Guangdong Provincial Key Laboratory for Computational Science and Material Design (no. 2019B030301001), the Guangdong Innovative and Entrepreneurial Research Team Program (no. 2016ZT06D348) and Shenzhen Science and Technology Program (grant no. RCJC20221008092722009). The DFT calculations were performed at the Center for Computational Science and Engineering at SUSTech. Work at SIMIT was supported by the NSFC (nos. U1632266, 11927807 and U2032207) and the National Key R&D Program of China (no. 2022YFB3608000). Work at Westlake University was supported by the NSFC (no. 12274353), National Key R&D Program of China (no. 2022YFA1402200) and the Westlake Instrumentation and Service Center for Physical Sciences. D.S. acknowledges support from the NSFC (no. U2032208). C.L. acknowledges support from the Highlight Project (no. PHYS-HL-2020-1) of the College of Science, SUSTech.
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C.L. and Q.L. conceived and designed the research project. Y.-P.Z. grew and characterized the single crystals. S.Q., H.Z. and W.L. designed and built the image-type SARPES setup. Y.-P.Z., X.-R.L., H.Z., G.Q., C.H., Z.J., X.-M.M., Y.-J.H., M.-Y.Z., W.L., M.Z., J.D., S.M., K.T., M.A., Z.L., M.Y., D.S., Y.H., R.-H.H., S.Q. and C.L. performed the ARPES measurements. X.C., Y.L., P.L., J.L. and Q.L. performed the theoretical analysis and DFT calculations. S.J., M.L. and J.W. performed the scanning NV magnetometry measurements. Y.-P.Z., X.C., Q.L. and C.L. wrote the paper, with the help from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Structural, magnetic and compositional characterization of MnTe2 single crystals.
a, Single-crystal X-ray diffraction results of MnTe2, showing a (001) exposed plane. Inset, a MnTe2 single crystal against a millimetre grid. b, FC and ZFC temperature dependence of magnetization with H∥(001). The magnetic transition temperature is TN = 87 K. The downturn of the ZFC M–T curve at TN signals the AFM behaviour. Inset, magnetic-field dependence of magnetization at T = 2 K. c, A typical scanning electron microscopy image of the MnTe2 single crystal and corresponding energy-dispersive X-ray spectroscopy elemental maps, indicating a homogeneous distribution of the Mn and Te elements. d, Atomic ratio of Mn and Te at the seven regions indexed in c. A ratio close to 1:2 is found in all regions, indicating a high quality of the crystals.
Extended Data Fig. 2 Electronic structure along the kx–kz plane.
a, ARPES kx–kz map at binding energy EB = 0.25 eV. A clear kz dispersion is seen, consistent with the bulk nature of the observed bands. The inner potential is determined to be V0 = 10 eV from the data; high-symmetry points are marked accordingly. The red curve shows the kx–kz position of an ARPES measurement using 74-eV incident light, corresponding to kz = 10π/c (Γ5) at the in-plane Brillouin zone centre. The cyan curves mark the ARPES measurement positions in Fig. 3. hν = 21.2, 28, 66 and 82 eV correspond to kz = 5.8 (−0.2), 6.5 (0.5), 9.5 (−0.5) and 10.5 (0.5)π/c, respectively. b, kx–kz dispersion at binding energies EB = 0.25 and 0.65 eV, with corresponding CECs calculated by DFT. The experimental spectral intensities show rough agreement with the calculated results (indicated by red arrows). c, E–kz dispersion along kx = 0. The band at EB = 0.5–1.0 eV is seen to experience repetitive dispersion within our measurement range, which matches qualitatively with that of a theoretical bulk band in the dispersing period, bandwidth and energy locations of band tops.
Extended Data Fig. 3 DFT-calculated spin-resolved energy bands with and without SOC.
a, E–k dispersion with SOC. A high level of spin polarization also exists in the O–A–B–C plane when SOC is turned on. A comparison between this result and Fig. 1j proves that the existence of SOC is not a necessary condition for the AFM-induced spin splitting. Even though Te is a heavy element that has strong SOC, the relativistic spin–orbit interaction gives only mild, secondary effects on the spin splitting of the MnTe2 bulk bands. b,c, DFT-derived spin-resolved E–k dispersion along Cuts 1–4 with and without SOC, respectively. All calculated results show a plaid-like antisymmetric spin texture.
Extended Data Fig. 4 Electronic structure at different photon energies.
a–c, CECs at binding energy EB = 0.35 eV measured with 54-eV, 92-eV and 100-eV photons. The CEC at 92 eV show a rectangular feature close to the zone centre (in good agreement with the DFT result shown in the inset), whereas the other two maps show a cross shape in the first zone. This kz dispersive behaviour indicates the bulk nature of the bands. d–f, ARPES E–k dispersion taken with 76-eV, 92-eV and 102-eV photons. Purple curves are the bulk bands calculated by DFT. The qualitative match between the DFT bands and the ARPES data probably indicates that the ARPES intensity comes mostly from bulk states of the system.
Extended Data Fig. 5 Raw spin-resolved E–k images along Cuts 1–4 and corresponding polarization curves.
a, Spin-integrated E–k images along Cuts 1–4 obtained by the spin-imaging ARPES system, shown in Fig. 2e–j. b, Corresponding raw spin-resolved E–k images. The difference between these images and those in Fig. 2 is that the data here are not multiplied by the spin-integrated intensity of the bands. In other words, these are the ‘raw’ spin-polarized data obtained with the image-type spin detector. c, Spin polarization curves of Cuts 1–4 at all measured E–k areas. From these images and polarization curves, the same conclusions as those in the main text can be drawn: the x (y) component of the spin-polarization vector is antisymmetric about ky = 0 (kx = 0). Furthermore, we see that non-vanishing spin polarization also exists in E–k areas that have ‘no bands’ (low spin-integrated ARPES intensity), which possibly comes from spin signals of vague electronic states or from bands at other kz values (kz broadening effect).
Extended Data Fig. 6 Further evidence for the Dresselhaus-like in-plane spin texture.
a, DFT-calculated spin-resolved CEC at kz = −0.2π/c and binding energy EB = 0.45 eV. In-plane spins point mainly along ±Sx+y and ±Sx−y, resembling the Dresselhaus configuration. The direction of +Sx+y and +Sx−y are defined along kx + ky and kx − ky, respectively, shown in the bottom-left inset. The top-right inset depicts the schematic spin texture centred at point B. b, Spin-integrated ARPES band dispersion along Cut 5. Cuts 5 and 6 are defined in a; the bands along Cut 6 (not shown) seem the same. The observed bands exhibit two hole pockets and further intensity at k = 0 and EB = 0.4 eV. c,d, Sx+y-resolved and Sx−y-resolved raw E–k cuts along Cuts 5 and 6. Blue/red circles highlight the regions with +/− polarizations. The SARPES results (i) vaguely corroborate with the DFT-calculated results (ii), which becomes more obvious by overlaying the calculated spin-resolved energy bands on the SARPES images (iii). iv, Polarization curves (top) and spin-integrated intensity curves (bottom) at k = ±0.18 Å−1, integrated within the dashed rectangles in iii. Each point on a curve is an integrated intensity over a (E, k) range of (50 meV, 0.08 Å−1). These spin-resolved EDCs reveal three important features. First, the Sx+y polarizations at kx+y = ±0.18 Å−1 have opposite polarizations at the same binding energy. Second, the polarization at low binding energy (about 0.25 eV) and high binding energy (about 0.55 eV) is opposite. The corresponding Sx−y polarization of Cut 6 also exhibits these two features. Third, comparing the polarization curves at −0.18 Å−1 in Cuts 5 and 6, we find that their polarizations are opposite at the same binding energy. Extracted from the polarization curves, the energy scale of the spin splitting is about 274 ± 40 meV, which is in good agreement with the calculated 297 meV and is comparable with the well-known giant bulk Rashba effect38. v, Polarization curves versus momentum. We can trace the splitting along k and find that the +Sx+y (−Sx−y) spin-polarized peaks of Cut 5 (Cut 6) move downward in energy when k moves away from O. In summary, our SARPES data along Cuts 5 and 6 reveal a Dresselhaus-like in-plane spin texture and ruled out a Rashba-like spin texture.
Extended Data Fig. 7 Temperature dependence of the ARPES band structures.
ARPES band dispersion along Cuts i–xii and corresponding second-derivative analysis along the energy direction at T = 30 K and 150 K, respectively. The positions of Cuts i–xii are marked in the schematic of the first Brillouin zone. Apart from the data shown in Fig. 4, here the band structure is also seen to undergo a non-rigid shift and several energy-band changes across the Néel temperature (TN = 87 K).
Extended Data Fig. 8 Temperature-dependent electronic structure along Cut 3.
ARPES band dispersion below and above TN = 87 K and corresponding second-derivative analysis along the energy direction. The band structures are seen to undergo a structural modification associated with the AFM to paramagnetic transition.
Extended Data Fig. 9 Temperature evolution of the spin splitting and the spin-polarization signal.
a,b, Temperature dependence of the band splitting: ARPES band dispersion along Cut 5 at 30 K and 150 K (i); corresponding second derivatives to the raw data (ii); and spin-polarized DFT bulk bands for the low-T AFM state (iii). The abrupt change of bands between the two temperatures is evident for the vanishing of spin splitting above TN. c,d, Sx-resolved E–k images along Cut 1 and corresponding raw SARPES data. Data are taken at 30 K (c) and 60 K (d). Data in c and d are measured consecutively on the same sample with the same experimental settings and integration time. Because the bands at 30 K and 60 K, as well as the associated matrix elements, are not expected to change because they belong to the same magnetic phase, the decrease in spin polarization here is probably intrinsic, related to the origin of spin splitting.
Extended Data Fig. 10 Evidence for a large magnetic domain size.
a,b, Evidence from cryogenic scanning NV magnetometry. a, A typical micrograph of the MnTe2 sample. Four different areas, corresponding to b, are measured within the dashed rectangle. b, Stray magnetic fields along the NV centre axis (about 55° to the surface normal) measured at 2 K. The sample is scanned using a pulsed ODMR scheme (see Methods). The spatial resolution is set to about 30 nm. An external magnetic field of 0.6 mT is applied along the NV centre axis to lift the degeneracy of the NV ± 1 spin states. No feature was detected within the measured length scales over a larger scan area (50 μm), indicating that the stray-field variations of the sample fall below the sensitivity threshold of the NV probe (\(2\,{{\upmu }}{\rm{T}}/\sqrt{{\rm{H}}{\rm{z}}}\)). Because the NV spin would detect non-vanishing magnetic fields contributed by domain walls, the measured feature in MnTe2 suggests a uniform and large magnetic domain with no net magnetic-moment contributions58,59. c,d, Evidence from Sx-resolved electronic structures before and after rotating the sample by 90°. i, Spin-integrated E–k dispersion along Cuts 1 and 7 (defined in the insets). ii, Corresponding DFT-calculated Sx-resolved E–k dispersion. iii, Spectral intensity and Sx polarization at (kx, ky) = (0, 0.5π/a) (large black dot in the insets) measured along Cuts 1 and 7. These curves are integrated within the dashed rectangles in i. The definition of Sx and error bars are the same as in Fig. 2. The Sx-polarization curves have the same sign at similar binding energies before and after rotating the sample by 90°, which is reproduced by DFT calculations (blue/red arrows). This is indicative of a magnetic domain size comparable with the size of the incident beam spot.
Extended Data Fig. 11 DFT-calculated electronic structures from semi-infinite slab models.
a, Calculation model with two different surfaces marked as ‘top’ and ‘bottom’. Panels b–d/e–g show the calculated results on the top/bottom surface. b, Spin-integrated and spin-resolved CECs at binding energies EB = 0.25, 0.45 and 0.65 eV, respectively. The spin-integrated CECs contain the spectral weights of both the bulk and the surface states. The spin-resolved CECs show the spectral weights of surface states only. At the three binding energies, there is almost no spin-polarized surface states at the positions of Cuts 1–4. c, Spin-integrated E–k dispersions along Cuts 1–4. These images contain the spectral weights of both the bulk and the surface states. d, Corresponding spin-integrated E–k dispersion of the surface states only. The dashed rectangles in first panel of c and d mark the ARPES-measured E–k region along Cuts 1–4 in Fig. 2. Clearly, there is almost no surface state in the areas we have measured. e–g, Results of the bottom surface, yielding the same conclusion. The data in this figure are evidences for the bulk nature of the spin-polarized bands in Fig. 2.
Extended Data Fig. 12 Repeatability of the SARPES data.
Sx-resolved ARPES maps on Cut 1 (Fig. 2e) are measured with four different samples. The experimental results are clearly reproducible.
Extended Data Fig. 13 EDCs at k = 0, band dispersions and Sx-polarization curves before and after an offset along the energy direction.
This process eliminates the slight hole-doping effect caused by the slow but gradual ageing of the samples in the vacuum chamber. a, EDCs of Cuts 1–4 before the offset. The energy bands are found to shift rigidly upward along the energy direction with increasing loop number (measurement time). b, EDCs of Cuts 1–4 after the offset. The bands are offsetting downward referring to the first loop (L1), which represents the pristine energy positions of the bands. The EDCs are taken at k = 0 Å−1, integrated within the rectangles in the first panel of c and d. c,d, Band dispersions of Cuts 1–4 before and after the offset, respectively. e, Sx-polarization curves before and after the offset. The integrated positions are the same as Fig. 2e–h. One can find that the offsetting procedure introduced no qualitative difference to the spin polarization, except for a slight change of polarization magnitude near the Fermi level. The main conclusion of the paper is valid regardless of using the data before or after the offset.
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Zhu, YP., Chen, X., Liu, XR. et al. Observation of plaid-like spin splitting in a noncoplanar antiferromagnet. Nature 626, 523–528 (2024). https://doi.org/10.1038/s41586-024-07023-w
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DOI: https://doi.org/10.1038/s41586-024-07023-w
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