Introduction

The recently established new class of altermagnetic materials is characterized by the presence of particular alternating opposite-spin sublattices. They defy conventional magnetic classifications by breaking time-reversal symmetry while simultaneously maintaining a zero net magnetization enforced by spin-lattice symmetry1,2,3,4. Precisely this intrinsic trait allows altermagnets to combine advantageous features of both ferromagnets and antiferromagnets5,6,7,8, including spin-transport properties and compatibility with a diverse range of materials9,10,11,12,13,14,15,16, from superconductors17,18,19,20,21 to insulators22,23,24. As the vanishing net magnetization in altermagnets avoids effects of long-range magnetic stray fields typical for ferromagnets, this combination opens up promising avenues for applications in magneto-optics25, spintronics12,26, and beyond.

Interestingly, symmetry analysis reveals that the electronic structure of altermagnets may in principle provide fertile ground for non-trivial electronic topology, due to the unique interplay between time-reversal and crystalline symmetries27,28,29,30,31. Despite extensive experimental and theoretical scrutiny of altermagnets22,23,24,32,33,34,35,36,37,38,39 and topological semimetals40,41,42,43,44,45,46,47 separately, the simultaneous experimental manifestation of their characteristic phenomena has remained elusive so far. Here we establish CrSb as a topological Weyl system with two distinct types of spin-carrying nodal structures. The giant spin splitting that we find from high-resolution angle-resolved photoemission spectroscopy (ARPES) and spin-ARPES measurements aligns well with density functional theory (DFT) calculations on CrSb, as do the bulk Weyl points (WPs) with linear dispersion and ensuing altermagnetic surface Fermi arcs (SFAs) that, as we will show, connect same-spin WPs at the surface.

Results

For both its altermagnetic and topological features, the symmetries of CrSb are crucial. Figure 1a shows the crystal and magnetic structure of CrSb, which forms a hexagonal structure with the space group P63/mmc (No. 194)48. The magnetic space group is \(P6^{{\prime} }_{3}/m^{\prime} m^{\prime} c\) (No. 194.268) and the spin space group is \({P}^{-1}{{6}_{3}/}^{-1}{m}^{1}{m}^{-1}{c}^{\infty m}1\) with a six-fold screw rotation symmetry connecting Cr-atoms of opposite magnetic moment in real space. The three-dimensional (3D) Fermi surface of CrSb from DFT calculations in Fig. 1c illustrates that sixfold rotation connecting opposite-spin sublattices in real space (Fig. 1a) also connects opposite-spin electronic states in momentum space (Fig. 1c), as is mandatory for altermagnets1,2. The calculated Fermi surface also illustrates the four mirror planes in the Brillouin zone (BZ) on which the altermagnetic bands are spin-degenerate in absence of spin-orbit coupling (SOC), as shown in Fig. 1b.

Fig. 1: Crystal structure, WPs and calculated electronic structure of CrSb.
figure 1

a The crystal structure of CrSb with the space group P63/mmc (no. 194). b The 3D BZ of the original unit cell of CrSb, and the corresponding two-dimensional BZ projected on the (001) plane (green lines). The red, green, blue, and orange planes are the four mirror planes in the BZ on which the altermagnetic bands are spin-degenerate in absence of SOC. c Calculated bulk Fermi surface in the first BZ. The colormap from blue to orange of Fermi surface sheet represents the expectation value of spin along the z direction. Illustration of the two types of WPs: opposite-spin (d) and same-spin (e). The yellow, red, and blue lines connecting WPs are surface Fermi arcs (SFAs). The purple lines represent spin-up and spin-down degeneracy. Top (f) and side (g) views of the distribution of same-spin WPs in the 3D BZ. Red dots represent nodes with chirality χ = +1, green dots χ = −1. The calculated surface state of (001) on Sb (h) and Cr (i) terminations along \({\overline{{\rm{M}}}}_{1}-\overline{\Gamma }-{\overline{\rm{M}}}_{2}-{\overline{\rm{K}}}_{2}-\overline{\Gamma }-{\overline{\rm{K}}}_{1}-{\overline{\rm{M}}}_{1}\). The WPs are marked by the red (χ = 1) and green (χ = −1) dots. Star, square and round dots represent the projected positions of opposite-spin WPs on the high symmetry line/plane (WP1), opposite-spin WPs on general momenta (WP2) and same-spin WPs (WP3), respectively.

Full size image

The altermagnetic spin-splitting allows for topological electronic structures far beyond antiferromagnets with completely spin-degenerate bands27,28,29,30,31. We identify two mechanisms for the generation of topological nodes in the electronic structure of CrSb. The first relies on the observation that spin-splitting of fully spin-polarized altermagnetic bands automatically leads to nodal lines when opposite-spin bands cross. The degenerate states that form the nodal lines have opposite spins. Such crossings in momentum space can be symmetry enforced, for instance at a mirror plane, or be accidental. The effect of SOC in CrSb is to lift this spin-degeneracy for most, but not all momenta on nodal lines. At the remaining Weyl nodes also the spin degeneracy remains, so that the points correspond to doublets with vanishing total spin-projection Sz = 0 as illustrated in Fig. 1d. The altermagnetic symmetries and resulting spin-degeneracies thus provide a very natural setting for such opposite-spin WPs governed by the SOC energy scale. Indeed from DFT we identify 10 groups of opposite-spin WPs within 1 eV from the Fermi energy, see “Appendix B” in supplementary material (SM) for details.

Interestingly, there is also a route that generates purely altermagnetic WPs, reflecting the bulk altermagnetic symmetries. They give rise to same-spin WPs with a total spin-projection of Sz = ±1 (in units of ℏ) and a spin-splitting (energy-splitting at the nodal momentum) governed by the exchange interaction energy scale1,2, as illustrated in Fig. 1e. These altermagnetic Weyl nodes thus carry apart from a topological charge also a magnetic quantum number Sz, and with it, a finite magnetic moment. It is precisely the altermagnetic crystal symmetry that relates same-spin WPs with different Sz. The weak SOC that formally breaks the altermagnetic symmetry in CrSb causes the expectation value ∣〈Sz〉∣ to slightly deviate from unity (see supplementary material). These WPs, and the topological surface states they imply, render the topological features of Weyl altermagnets fundamentally distinct from ferro- or antiferromagnetic Weyl semimetals.

For CrSb, where lattice inversion symmetry connects atoms with the same magnetization, the “Appendix B and M” in SM provides details of the topological same-spin nodal line without SOC and three groups of same-spin WPs that we identified. Figure 1f–g shows the distribution of same-spin WPs in momentum space, with their chiralities. Fermi-arcs connect the surface projections of these WPs of opposite chirality. As a consequence also their topological surface states are fully spin-polarized in the altermagnetic limit, with additional SOC affecting the details of their dispersion and spectral weight. The electronic structures of both Sb (Fig. 1h) and Cr (Fig. 1i) terminated surfaces show that the surface states are very extended in momentum space and carry substantial spectral weight. Detailed representations of purely surface-related states and the full 3D bulk band structures are provided in the “Appendix C and D” in SM.

To experimentally access the spin-splitting and topological electronic structure of CrSb we use ARPES, which is to a certain extent complicated by CrSb having a 3D electronic structure. In order to establish the correspondence between photon energy and out-of-plane momentum kz, we performed broad-range (40 to 120 eV) photon energy dependent ARPES measurements along the \(\overline{{\rm{M}}}-\overline{\Gamma }-\overline{\rm{M}}\) direction. Figure 2f shows the photon energy dependent ARPES spectral intensity map (kx-kz Fermi surface) at the Fermi level along the \(\overline{{\rm{M}}}-\overline{\Gamma }-\overline{\rm{M}}\) direction. From the periodic structure along the kz direction, the correspondence between the high symmetry points of the BZ along the kz direction and the photon energy is determined as shown in Fig. 2f. To observe the spin-split Fermi surface of CrSb experimentally, we determined the kz dependent Fermi surfaces for Sb (Fig. 2h) and Cr (Fig. 2i) terminations, at photon energies of 97, 107, and 122 eV (see SM “Appendix F” for intermediate values). Their features indicated in red, blue, and pink are well captured by the calculated bulk Fermi surface with corresponding ckz/π = 1, 0.6, and 0 (Fig. 2c–e). At 107 eV (kz = 0.6 π/c) the hexagram consisting of two intersecting equilateral triangles is clearly visible in both the measurements and calculations. The altermagnetic order in CrSb causes these two triangles to become mirror partners with opposite-spin polarization (Fig. 2d)—thus the observed hexagram directly evidences the altermagnetic spin-splitting of bands. By symmetry the spin-splitting vanishes without SOC for Fermi surfaces at the high symmetry planes ckz/π = 1 or 0, and indeed it is observed that in these cases the spin-split triangles merge into a petal shape (see SM “Appendix F”). Interestingly, the spectra show additional features apart from the bulk Fermi surface further out from the Γ point, for both the Sb and Cr termination (Fig. 2h, i). These features agree well with the surface features marked by colored lines and arrows in the calculated surface spectral functions in Fig. 2a–b) and the measurements in Fig. 2h, i. As we will detail later, comparison to the calculated surface spectral density (Fig. 2a, b) establishes that the root cause of these surface states at the Fermi energy is the topological SFAs that originate from the WPs (see SM “Appendix I, L and P”).

Fig. 2: Fermi surface of CrSb.
figure 2

Projected Fermi surfaces for Sb (a) and Cr (b) surface terminations with WPs overlaid. Red dots represent nodes with chirality χ = +1, green dots χ = −1; The yellow and blue arrows mark the surface Fermi arcs (SFAs). Calculated bulk Fermi surfaces at ckz/π = 1 (c), 0.6 (d), and 0 (e) planes, integrated over a  ±0.1 interval. The Fermi level in here shifts by −80 meV compared to the calculated value. f Photon energy dependent ARPES spectral intensity map at the Fermi level along \(\overline{{\rm{M}}}-\overline{\Gamma }-\overline{\rm{M}}\). g Corresponding calculated kz-kx intensity map with the Fermi level shifts by −80 meV compared to the calculated value. h Photon energy dependent Fermi surfaces measured with photon energies of 97 eV (i), 107 eV (ii), 122 eV (iii) on Sb terminated surfaces, with the extracted bulk Fermi surfaces from (c–e) overlaid. i The same measurements on Cr terminated uneven surfaces. The red and blue lines denote the spin-polarized bulk Fermi surfaces, while the non-spin-polarized bulk Fermi surfaces are marked by pink lines. The SFAs are indicated by green, orange, yellow, and cyan lines and arrows. Dashed lines and arrows indicate the position of SFAs where they are not easily discerned immediately in the data. The corresponding sample photos are shown in the bottom left corner of (h(i)) and (i(i)).

Full size image

First we establish that there is a very large momentum dependent spin-splitting of the bulk bands in CrSb1,33,37,38,39,49. Figure 3a–f show the photon energy dependent dispersions along the \(\overline{{\rm{M}}}-\overline{\Gamma }-\overline{\rm{M}}\) direction measured with photon energies of 102 eV (Fig. 3a, d), 107 eV (Fig. b, e) and 112 eV (Fig. 3c, f) for Sb (Fig. 3a–c) and Cr (Fig. 3d–f) terminated surfaces. The corresponding kz-dependent calculated spin-polarized band structures are shown alongside it (Fig. 3g–i) (see “Appendix D and L” in SM for details). Due to the altermagnetism in CrSb, the bands exhibit spin splitting, as shown in Fig. 3g–i. The calculated bands and their spin splittings are in very good agreement with the ARPES measurements (Fig. 3a–f). We determine the band splitting to be up to 1 eV (see “Appendix J” in the SM). In addition to the bulk bands, features that are weakly dependent on photon energy are observed in all the experimental band plots. The absence of a kz dependence is consistent with their surface state nature and their location at the Fermi energy is consistent with the Fermi arc features in Fig. 2h–i.

Fig. 3: Band dispersions of CrSb.
figure 3

The photon energy dependent ARPES spectra along \(\widetilde{\rm{M}}-\widetilde{\Gamma }-\widetilde{\rm{M}}\) measured with photon energies of 102 eV (a, d), 107 eV (b, e), and 112 eV (c, f) on Sb (a–c) and Cr (d–f) terminated surfaces, with corresponding DFT bulk calculations (pink) overlaid. The green and yellow lines indicate the surface states of the Sb and Cr terminated surfaces along \(\overline{\Gamma }-\overline{\rm{M}}\) directions, respectively, and are inserted as a guide to the eye. The calculated spin dichroism spectra with SOC along the \(\widetilde{\rm{M}}-\widetilde{\Gamma }-\widetilde{\rm{M}}\) direction at kz = 0.8 π/c (g), 0.6 π/c (h), and 0.4 π/c (i) planes. The spectra are plotted using a two-dimensional color scale, with saturation representing the spectral intensity ρ(E, k) and red/blue color corresponding to the dichroism \(({\rho }_{\uparrow }-{\rho }_{\downarrow })/({\rho }_{\uparrow }+{\rho }_{\downarrow })\) where spin is projected onto the z axis. Spin-polarized EDCs along the left (j) and right (k) cut lines indicated in (d). The spin-polarized EDCs were measured with a photon energy of 102 eV, while the red (blue) curve corresponds to spin-up (spin-down). The EDC intensities are already normalized in the non-spin polarized region. In order to obtain sufficient spin statistics, more than 30 h were accumulated in total. l The normalized spin-polarized intensity difference between spin-up and spin-down, with the red (blue) colors indicating spin-up (spin-down) polarization for the left (right) bands.

Full size image

To probe the spin content of the altermagnetic bands, we performed spin resolved ARPES measurements with a photon energy of 102 eV, as shown in Fig. 3j–l. Figure 3j, k are the spin-polarized energy distribution curves (EDCs) along the left and right cut lines in Fig. 3d. The spin-polarized intensity difference between spin-up and spin-down of left and right bands is shown in Fig. 3l. Due to the presence of a non-spin polarized background, the measured spin polarization observed in the EDCs is decreased. Likely of greater importance, is the fact that two altermagnetic domains are expected to be present. Their opposing spin polarizations cancels out and a beam spot larger than the domain size is expected to result in a small but non-zero measured spin polarization. While it is not an experimental evidence for highly spin polarized bands, the small but robustly detectable difference in energy-dependent spin polarization observed experimentally (Fig. 3l) is consistent with the theoretical prediction and indicates that the left and right energy bands (Fig. 3d) have different spin projections along the z direction.

Having identified and quantified the altermagnetic spin-splitting in CrSb, we focus on the resulting topological properties. Both ARPES measurements and DFT calculations show that CrSb is metallic with a complex Fermi surface. In an energy range from  −1 to 1 eV around the Fermi level, we identify 13 groups of WPs in the calculated band structure, as listed in Table S1 of the SM. All WPs in CrSb arise from time-reversal symmetry breaking and their relative positions are given by the symmetries in the magnetic space group \(P6^{\prime} /m^{\prime} m^{\prime} c\) (No. 194.268) reflecting their unique altermagnetic nature. Many projected WPs are along or near the \(\overline{\Gamma }-\overline{\rm{M}}\) direction (see Fig. S2 in SM)—this is the high-symmetry line whose little co-group allows non-zero chirality. The pair of WPs in the \(\overline{\Gamma }-\overline{\rm{M}}\) direction below the Fermi level (marked by round dots in Fig. 4a, b) are same-spin WPs. Long SFAs connect neighboring WPs of opposite chirality χ = ±1 and identical spin-projection across the BZ. This topological surface state happens to appear in a region without bulk states so that it may be clearly discerned in ARPES. Even if the presence of these SFAs is dictated by topology, the shape of the surface spectra for two different terminations differ due to the lack of inversion and mirror symmetry on the surface, as shown in Fig. 1h and i. Due to the large distance between the two WPs projected on the (0001) surface, the arc is even detectable along \(\overline{\Gamma }-\overline{\rm{K}}\) (see SM “Appendix L”).

Fig. 4: WPs and Fermi arcs in CrSb.
figure 4

a Calculated total spectral density along \(\overline{\rm{M}}_1-\overline{\Gamma }-\overline{\rm{M}}_2\) for a Sb terminated CrSb (0001) surface, b surface electronic structure as in (a) but with the bulk spectral weight subtracted. The WPs and their chirality are marked by red (χ = 1) and green (χ = −1) dots. Loops as they are defined in the surface BZ, overlaid with Fermi surface maps at a photon energy of 102 eV (c) and 107 eV (d) on Sb (c) and Cr (d) terminated surfaces. The photon energy dependent EDC second derivative spectra on a Sb terminated surface for photon energies of 77 eV (e), 82 eV (f), 85 eV (g), 93 eV (h) under LH polarization along \(\overline{\rm{M}}_1-\overline{\Gamma }-\overline{\rm{M}}_2\), with corresponding DFT calculations overlaid. The pink lines mark the bulk states. The green (yellow) and orange (cyan) lines in (c, d) indicate the surface Fermi arcs (SFAs) of the Sb (Cr) terminated surfaces along the \(\overline{\Gamma }-\overline{\rm{M}}\) and \(\overline{\Gamma }-\overline{\rm{K}}\) directions, respectively. The green, blue, and red lines and arrows in (b, e–h) mark the SFAs. Chiral charge analysis for the experimental (i, k) and calculated (j, l) spectral density along the loops around the surface projected WP momentum.

Full size image

To track these WPs and SFAs experimentally, we conducted photon energy-dependent ARPES measurements along \({\overline{{\rm{M}}}}_{1}-\overline{\Gamma }-{\overline{{\rm{M}}}}_{2}\) on Sb (Fig. 4e–h) and Cr terminated surfaces. The band features measured on Sb (marked by green lines and arrows in Fig. 4e–h) terminated surfaces exhibit negligible photon energy dependence, indicating that it corresponds to a surface state. Comparison with the corresponding calculated surface states, shows that their band features (marked by colored arrows in Fig. 4e–h) agree well with the SFAs (marked by corresponding colored arrows in Fig. 4b), supporting that the SFAs are intrinsic and robust.

In addition, we measured the band dispersion along \(\overline{\Gamma }-\overline{\rm{K}}\) on Sb and Cr terminated surfaces and along \(\overline{{\rm{M}}}-\overline{\rm{K}}\) on Sb terminations. SFAs are also observed along these directions (SM “Appendix L and P”). The WPs at −0.36 eV, which are indicated by round dots in Fig. 4a and b, have a kz location of  ±0.27 π/c according to the DFT calculations, corresponding to a photon energy of about 82 eV. The second derivative data at this photon energy (see Fig. 4f) indeed evidences a crossing corresponding to the calculated Weyl point energy and momentum (see SM “Appendix P” for details).

To resolve the topological character of the observed surface states we perform a chiral charge analysis50. To this end we define closed loops in the surface BZ around projected WP momenta for both the Sb and Cr terminations, see Fig. 4c, d. The chiral charge can be determined from the spectral weight as a function of the loop momentum, which we analyse for both the experimental data (Fig. 4i, k) and calculated surface spectral function (Fig. 4j, l), see SM “Appendix R” for details. Due to the bulk-boundary correspondence having a WP in the center of the loop implies the presence of an unequal number of left (negative velocity) and right movers (positive velocity) at the SFA energy. Indeed in all cases we find precisely one uncompensated band crossing the Fermi level (see Fig. 4i–k), consistent with the WP chirality, and confirming the topological character of the SFAs.

Thus, we have quantified experimentally the altermagnetic spin-spliting of bands in CrSb with high-resolution and spin-resolved ARPES measurements. The presence of WPs close to the Fermi level and the topological Fermi arc surface states that result from those, indicates that CrSb is a topological Weyl system. Whereas in general, Weyl nodes carry only a topological charge, corresponding to their chirality χ, in altermagnets they also carry spin-projection Sz. Besides opposite-spin nodes with zero spin-projection, altermagnetic CrSb also has same-spin nodes with spin-projection  ±ℏ and chirality-spin locking. For each set of n symmetry-related altermagnetic WPs one can define a locking number \(\zeta ={\sum }_{i = 1,n}{\chi }_{i}{S}_{i}^{z}\). For CrSb ζ = 0 by symmetry, but interestingly other magnetic symmetry groups allow sets of altermagnetic nodes with non-zero integer locking numbers. The distribution of same-spin nodes in 3D momentum space reflects all altermagnetic symmetries as do their spin-polarized topological surface states, the altermagnetic Fermi arcs. Spin-orbit coupling weakly breaks altermagnetic symmetries and consequently the quantization of the spin-projection. As these properties derive from the interplay of symmetry and topology, they are generic for Weyl altermagnets. These findings may well imply that the unique bulk altermagnetic spin-transport properties are promoted by topology to also become properties of the Fermi arcs at the surface, which can render same-spin Weyl altermagnets interesting interface materials for spintronics. These insights not only spotlight the distinctive altermagnetic attributes of CrSb but also its potential to induce novel physics and applications in the area of topological materials.

Note added: During preparation of this manuscript, two preprints with ARPES measurements on CrSb appeared, focusing on the band spin splittings37, and the topological properties of CrSb51, respectively.

Methods

Sample

The CrSb single crystals were grown by the chemical vapor transport (CVT) method. A stoichiometric ratio of chromium and antimony powders, together with iodine of 2.5 mg/ml as the transport agent, were mixed and sealed in an evacuated quartz ampoule. The ampoule was slowly heated and finally exposed to a temperature gradient of 925–900 ∘C where the CVT preceded for one week, then naturally cooled down to room temperature. CrSb crystals in size of 5 mm with regular shapes and shiny surfaces were obtained.

ARPES measurements

High-resolution ARPES measurements were performed at the Bloch beamline of MAX IV and at the I05 beamline of the Diamond synchrotron light source. The total energy resolution (analyzer and beamline) was set at 15~20 meV for the measurements. The angular resolution of the analyzer was  ~0.1∘. The beamline spot size on the sample was about 10 × 12 μm at the Bloch beamline of MAX IV and about 70 × 70 μm at the I05 beamline of the Diamond synchrotron. The samples were cleaved in situ and measured at about 18 K at the Bloch beamline of MAX IV and about 10 K at the I05 beamline of the Diamond synchrotron in ultrahigh vacuum with a base pressure better than 1.0 × 10−10 mbar. The spin-resolved ARPES (SARPES) measurements were performed at the Bloch B-branch beamline of MAX IV with photon energy of 102 eV and a beam spot of about 30 × 30 μm. The samples were cleaved in situ and measured at about 77 K at the Bloch B-batch beamline of MAX IV. The total energy resolution (analyzer and beamline) was set at  ~50 meV for the measurements. The angular resolution of the analyzer was  ~1.3 ∘.

DFT calculations

Full relativistic spin-polarized electronic structure and Fermi surface calculations are done using the full potential local orbital code FPLO52,53, on a k-mesh of 20 × 20 × 13 points. We use the generalized gradient approximation to the exchange and correlation potential by Perdew, Burke, and Ernzerhof (PBE). The surface state and WPs calculations are performed based on the DFT calculation from VASP54 employing the projector augmented wave method55 and LDA functional56. The Brillouin zone is sampled with a 10 × 10 × 8, with Gamma-centered k-point. The energy cutoff of the plane wave basis is set to 550 eV. The Hubbard term was introduced and set to be 0.8 eV in the d orbitals of Cr-atom in the DFT framework (DFT+U) in order to account for the electron-electron correlation. The Wannier based Hamiltonian is symmetrized based on the maximally localized Wannier functions generated by the WANNIER90 interface57. The projectors are d orbitals of Cr and p orbitals of Sb atoms with the well-fitted region from −2 to 2 eV. To locate the WPs and calculate the chirality, Wanniertools is implemented58.