Introduction

Topological semimetals have become a fruitful platform for the discovery of quasiparticles that behave as massless relativistic fermions predicted in high-energy particle physics1,2,3,4,5,6. A prominent example are Weyl fermions, which are realized at topologically protected crossing points between spin-polarized electronic bands in the bulk band structure of non-centrosymmetric or ferromagnetic semimetals1,2,7,8,9,10,11. Near a Weyl node the momentum-resolved two-band Hamiltonian takes the form H ∝ ±σ ⋅ k, giving rise to the linear energy–momentum dispersion relation of a massless quasiparticle4. The topological structure of the Weyl node, however, is not encoded in the energy spectrum, but rather manifests in the momentum-dependence of the eigenstates, i.e., the electronic wave functions. The pseudospin σ and the Berry curvature Ω wind around the Weyl node, forming a Berry flux monopole in three-dimensional (3D) momentum space12,13. The nontrivial winding of σ stabilizes the Weyl node by a topological invariant, a nonzero Chern number of C = ±1 (ref. 4).

Until today, angle-resolved photoelectron spectroscopy (ARPES) experiments have confirmed a number of materials as Weyl semimetals (WSM), based on a comparison of the measured bulk band structure to band calculations and the observation of surface Fermi arcs1,9,10,14,15. Manifestations of the nontrivial topology have also been found, accordingly, in magnetotransport experiments16, by scanning tunneling microscopy17, and via optically induced photocurrents18. The winding of the electronic wave functions in momentum space, however, which characterizes the immediate effect of a Berry flux monopole and thus the topology of the WSM, has so far remained elusive.

While the pseudospin σ for a Dirac-Hamiltonian universally shows nontrivial winding, the underlying microscopic degrees of freedom may vary from one system to another19. In two dimensions, examples include the sublattice degree of freedom for graphene and the spin-angular momentum (SAM) for the surface states in topological insulators (TI). Accordingly, the nontrivial Berry-phase properties have been addressed by quasiparticle interference STM imaging and dichroic ARPES in graphene20,21 and by spin-resolved ARPES in TI22. Likewise in 3D WSM, one may expect the relevant degrees of freedom to depend on the considered material system. While previous theories have suggested orbital-sensitive dichroic effects in momentum-resolved spectroscopies, as a probe of topological characteristics in topological semimetals23,24,25, such approaches have previously not reached application to experimental data.

In the present work, we find that the orbital-angular momentum (OAM) L plays a crucial role in the Weyl physics of the paradigmatic WSM TaAs and, like the pseudospin σ, displays a topologically nontrivial winding at the Weyl points. Our experiments are based on soft X-ray (SX) ARPES, which allows for the systematic measurement of bulk band dispersions by virtue of an increased probing depth and excitation into photoelectron final states, with well-defined momentum along kz perpendicular to the surface26, when compared to surface-sensitive ARPES experiments at VUV photon energies. SX-ARPES has been the key method to probe chiral-fermion dispersions in topological semimetals1,2,5,27,28,29, but its combination with spin resolution (SR) and circular dichroism (CD) is challenging and not widely explored30. On the other hand, at lower excitation energies SR is routinely used to detect the SAM of electronic states22 and CD has been introduced as a way to address the OAM31,32,33,34,35. We performed SX-ARPES measurements combined with SR and CD to probe the SAM and OAM in the bulk electronic structure of TaAs.

Results

Bulk band structure of TaAs

TaAs crystallizes in the non-centrosymmetric space group I41md, as shown in Fig. 1a. According to the results of first-principles calculations, its bulk band structure features 12 pairs of Weyl points in the Brillouin zone, which divide into two inequivalent sets, W1 and W2 (refs. 7,8). Here, we focus on the W2 points which are located in kx–ky planes at \({k}_{z}=\pm 0.59\ \frac{2\pi }{c}\), with the length c along z denoting the conventional unit cell. In agreement with earlier works1,2, our SX-ARPES data in Fig. 1 predominantly reflect the bulk bands of TaAs, allowing us to address the bulk band structure through variation of the photon energy. The experimental dispersions along the ΓΣ and ZS high-symmetry lines compare well with the calculated bands (Fig. 1c–d). We further performed hν-dependent measurements to trace the band dispersion along kz, for which we likewise achieve agreement with theory (Fig. 1e and Supplementary Note 1). Based on this, we determine a photon energy of approximately hν = 590 eV that allows us to reach a final-state momentum corresponding to the W2 Weyl points within experimental uncertainty (Fig. 1e, f).

Fig. 1: Bulk electronic structure of TaAs.
figure 1

a Bulk crystal structure of TaAs with space group I41md. b Experimental geometry of the angle-resolved photoemission spectroscopy (ARPES) experiment using circularly polarized soft X-ray radiation. c, d ARPES data sets along the ΓΣ and ZS high-symmetry directions of the bulk Brillouin zone. e ARPES data set along kz at kx = −0.53 Å−1, corresponding to the calculated kx position of the W2 Weyl nodes (cut Z in f). In c–e, the red lines represent the calculated band dispersion. f Bulk Brillouin zone structure in the kx–kz plane. W2 Weyl nodes at \({k}_{z}=\ 23.41\ \frac{2\pi }{c}\) are addressed at a photon energy of approximately hν = 590 eV (cf. Figs. 2 and 3). The star marks the (kx,kz) position of the data set in Fig. 2a.

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Orbital and spin-angular momentum of the bulk states

The formation of Weyl nodes in TaAs relies on the coexistence of inversion-symmetry breaking (ISB) and spin–orbit coupling (SOC)7,8, which induces a spin splitting into nondegenerate bands. The results of our first-principles calculations in Fig. 2a, b show this splitting for the bulk valence and conduction bands, which split up into branches v+ and v−, as well as c+ and c−, respectively. Note that the crossing points between the upper valence band v+ and the lower conduction band c− define the Weyl nodes in TaAs (see below). Going beyond the band dispersion, we explore the impact of ISB and SOC on the electronic wave functions. According to our calculations, the key consequence of ISB is the formation of a sizable OAM L in the Bloch wave functions (Fig. 2a and Supplementary Fig. 3). The spin-split branches in the valence and conduction band carry parallel OAM, while the SAM is antiparallel (Fig. 2a, b). This indicates a scenario for the bulk states in TaAs, where the energy scale associated with ISB dominates over the one of SOC32,33,36 (see also Supplementary Fig. 4). Moreover, our calculations show that v± and c± carry opposite OAM, indicating that the Weyl nodes are crossing points between bands of opposite OAM polarization.

Fig. 2: Orbital and spin-angular momentum of bulk electronic states in TaAs.
figure 2

a Calculated bulk band dispersion of TaAs projected on the component Lx of the orbital-angular momentum (OAM). The cut along ky is taken at \({k}_{z}=-0.59\frac{2\pi }{c}\) and kx = −1 Å−1. The spin–orbit split valence band v± and conduction band c± are indicated. The calculation is compared to a corresponding CD-ARPES data set in the bottom panel, obtained at hν = 590 eV (see indication in Fig. 1f). The red/blue color code represents the circular dichroism (CD) measured with circularly polarized X-rays incident in the xz-plane (cf. Fig. 1b). b Calculated band dispersion projected on the component Sx of the spin-angular momentum (SAM), showing opposite Sx for v±. c ARPES data set along ky taken at hν = 590 eV and kx = −1 Å−1. Dashed lines indicate the ky positions of the spin-resolved energy distribution curves (EDC) in e. d, e, Spin-resolved EDC and measured spin polarization at ±ky, as indicated in c. The spin quantization axis is along x. Data sets at +ky (−ky) were measured with left (right) circularly polarized light (see Supplementary Note 2 for details).

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To confirm the formation of OAM experimentally, we measured the CD signal, i.e., the difference in photoemission intensity for excitation with right and left circularly polarized light, which has been shown to be approximately proportional to the projection of L on the light propagation direction kp (refs. 31,32). The grazing light incidence in the xz-plane of our experimental geometry (Fig. 1b), thus implies that the measurements predominantly reflect the Lx component of the OAM. Note that the relation between CD and OAM, derived in refs. 31,32, relies on the free-electron final-state approximation, which can be expected to hold particularly well at the high excitation energies used in the present experiment. Indeed, a comparison of the measured CD to the Lx-projected band structure shows a remarkable agreement over wide regions in momentum space, as seen in Figs. 2a and 3a, b. The detailed match between experimental data and theory proves that the measured CD indeed closely reflects the momentum-resolved local OAM of the bulk states in TaAs. As expected theoretically31,32, the OAM of the initial state is thus the most important source of CD under the present experimental conditions as opposed to mere geometric or final-state effects. Particularly the latter can become relevant at lower excitation energies, where the final state is not well-approximated in the free-electron picture37,38.

While OAM is induced by ISB already in the absence of SOC, the presence of Weyl nodes requires a SOC-induced formation of spin polarization7,8. An experimental proof of spin-polarized bulk states in TaAs and related transition-metal monopnictides has remained elusive up to now. Our spin-resolved SX-ARPES measurements directly verify the predicted SAM of the bands v±. They confirm an opposite Sx for v+ and v− (Fig. 2b–e), while our CD-ARPES data prove a parallel alignment of Lx, in line with our calculations (Fig. 2a). Moreover, the spin-resolved data confirm an opposite sign of Sx at +ky and −ky. Overall, our experiments establish sizable OAM and SAM polarizations of the bands forming the Weyl nodes in TaAs.

Circular dichroism and OAM near the Weyl nodes

To explore the momentum-dependence of the OAM in more detail, we consider momentum distributions of the measured CD and the calculated Lx for the band v± in an equi-kz plane of the W2 nodes (Fig. 3a, b). Besides the good agreement of experiment and theory, it is evident from the data that the Weyl-node pair near kx = −0.5 Å−1, marked in Fig. 3b, is located at a distinctive position within the OAM texture. A quantitative comparison of the CD and the calculated OAM along kx paths through the two Weyl nodes of opposite chirality is shown in Fig. 3c. The experiment reveals sign changes of Lx close to the Weyl nodes and an opposite overall sign of Lx for the two nodes, consistent with the calculated OAM. Although the data in Fig. 3 reflects information integrated over both bands v+ and v−, these observations suggest a role of the OAM in the Weyl physics of TaAs.

Fig. 3: Momentum-space texture of circular dichroism and orbital-angular momentum.
figure 3

a, b Momentum distributions of measured circular dichroism (CD) and calculated Lx component of the orbital-angular momentum (OAM) at \({k}_{z}=-0.59\frac{2\pi }{c}\), corresponding to the W2 Weyl nodes. The data sets were obtained by integrating over an energy range from 0 eV to −1.2 eV, the width of the bands v± (cf. Fig. 2a). The calculated positions of the W2 Weyl nodes are indicated in b. The CD-ARPES data were acquired at hν = 590 eV. c CD signal normalized to the total photoemission intensity and Lx component of the OAM along kx cuts through Weyl nodes of opposite chirality. Sign changes are observed near the Weyl nodes.

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To examine the OAM and CD of the Weyl cone, we focus on the band dispersion along kx across a W2 Weyl node in Fig. 4a–e. For the band v+, which forms the lower part of the Weyl cone, the two branches on the left and on the right side of the Weyl node are selectively probed by the circular light polarization: using right circularly polarized light (Fig. 4d), there is a high photoemission intensity for the left branch of v+, while the right branch is suppressed. Vice versa, for left circularly polarized light, we find the left branch to be suppressed, while the right branch is more strongly excited (Fig. 4e). Accordingly, the band v+, shows an inversion of OAM across the Weyl node (Fig. 4b). This behavior is supported by a quantitative analysis of the intensities, shown in Fig. 4c. These observations indicate an opposite OAM for the upward and downward dispersing branches of v+ and thus an OAM sign change across the Weyl node, in agreement with our calculations. It is noteworthy that the magnitude of the CD in general does not reach 100%, as seen, e.g., from the analysis in Fig. 4c.

Fig. 4: Topological winding and orbital-angular momentum of Weyl-fermion states.
figure 4

a–e ARPES data sets and calculated Lx-projected band dispersion along a kx cut through a W2 Weyl node at ky = 0.037 Å−1 and \({k}_{z}=-0.59\frac{2\pi }{c}\). The measurements were taken with left IR (d) and right IL (e) circularly polarized light. Panel a shows the sum and b the difference (circular dichroism, CD) of IL and IR. Panel c shows the results of Gaussian peak fits to energy distribution curves (EDC) in the data sets IR (blue) and IL (red) for different kx (cf. Fig. S11). The dot size represents the extracted peak intensity. f Momentum distributions of CD, Lx, and the Ωx component of the Berry curvature (BC) in the vicinity of the W2 Weyl nodes. The data sets were obtained by integrating over an energy range from −0.16 eV to −0.24 eV. g OAM and Berry curvature calculated on a small sphere around a W2 Weyl node and the corresponding azimuthal equidistant projections. Both textures are characterized by a nontrivial Pontryagin index of S = 1. h EDC of IL and IR (upper panel) and the corresponding CD (lower panel) taken at kx = −0.45 Å−1. The three peaks in the EDC are assigned to the bands v+, v−, and c−.

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The fact that the Weyl node is located slightly below the Fermi level further allows us to estimate the relative OAM orientation of the lower and the upper part of the Weyl cone. In Fig. 4h, we consider energy distribution curves (EDC) at a wave vector kx close to the Weyl node. A three-peak structure is observed in the EDC and attributed to the bands v−, v+, and c−. The CD for the bands v+ and c− is opposite confirming the opposite OAM of these bands predicted by our calculations (Fig. 4b–e). Note that for left circularly polarized light the intensity of the band v+ drops toward the Fermi energy, which, in combination with finite linewidth broadening, gives rise to some deviations between the CD signal and calculated OAM in Fig. 4b (see also Supplementary Figs. 10–12 for a more detailed discussion). In particular, unlike for the band v+, our present data does not allow us to discern a CD reversal of the band c− across the Weyl point close to the Fermi level.

Our measurements and calculations in Fig. 4 directly image characteristic modulations of the Bloch wave functions near the W2 Weyl nodes. Further evidence for momentum-dependent changes also of the Ly component near the W2 nodes is presented in Supplementary Fig. 9. It is important to note that the observed sign reversal of Lx across the Weyl node is not enforced by any symmetry. Rather, it reflects a true band-structure effect that characterizes the Weyl-fermion wave functions, namely the crossing of two bands carrying opposite OAM (Fig. 4b–e). This situation is different from the surface states in TI22,31 or related examples39,40, where a sign reversal across the nodal point is strictly imposed by time-reversal or crystalline symmetries. From an experimental point of view, this makes the presently observed sign reversal of Lx more forceful, as it is a band-structure-specific and not a symmetry-imposed texture change.

Topological winding of OAM and Berry curvature

Our results reveal a close correspondence between measured CD and calculated OAM. In this work, the OAM is computed in the local limit by projecting the Bloch wave function onto atom-centered spherical harmonics, i.e., we consider the quantum-mechanical expectation value of the atomic L angular momentum operator. This approach offers a viewpoint complementary to the OAM given by the modern theory of polarization41. The fact that the CD rather corresponds to the local OAM31,32 could be related to the local nature of the photoemission process itself42.

The formation of OAM, which we observe here, provides direct evidence for the influence of ISB on the bulk Bloch states and, thus, suggests that these states also carry finite Berry curvature Ω, which likewise originates from ISB. In Fig. 4f, we consider momentum distributions of CD and Lx for the band v+, revealing a characteristic sign change along the contour around the Weyl nodes. The calculated momentum texture of the Ωx-component of the Berry curvature qualitatively resembles the characteristics of the OAM and the measured CD (Fig. 4f), suggesting that these quantities reflect the topologically nontrivial winding of the wave functions near the Weyl nodes21,34,35,43. Indeed, our calculations show explicitly that for TaAs the nontrivial topology of the Berry flux monopoles is encoded in the momentum-dependence of the OAM, while the SAM shows topologically trivial behavior. The topological nature of the field configuration is determined by the Pontryagin index calculated on a sphere surrounding the Weyl point

$$S=\frac{1}{4\pi }\int {\bf{n}}\cdot \left[\frac{\partial {\bf{n}}}{\partial x}\times \frac{\partial {\bf{n}}}{\partial y}\right]{\mathrm{d}}x{\mathrm{d}}y$$
(1)

where n is the unit vector of the field and the integral is taken over the 2D manifold covering the sphere. S is proportional to the Berry phase in momentum space accumulated by electrons encircling the Weyl point. In our calculations, the Pontryagin index of the vector fields of Berry curvature and OAM is S = 1 (Fig. 4g), while the SAM has S = 0 (Supplementary Fig. 8). Analogously to the Berry curvature, the Pontryagin index of the OAM is nonvanishing only if the sphere surrounds a Weyl node, and changes sign when the other Weyl node of the pair is considered. The Berry flux monopoles in TaAs thus derive from the orbital degrees of freedom of the electronic wave functions, and their topology manifests in a nontrivial texture winding of the OAM.

We indeed theoretically verified that in the WSM TaP44 and LaAlGe45 the Pontryagin index of the OAM vector field winds around the type-I Weyl nodes. Interestingly, LaAlGe hosts two type-I and one type-II Weyl dispersions46. Our analysis brings further evidence of a nontrivial winding of OAM only at type-I Weyl cones (see also Supplementary Note 5), raising intriguing questions about the underlying conditions of spectroscopic manifestations of Berry flux monopoles. Preliminary investigations suggest that non-symmorphic symmetries and a large atomic-angular momentum (d orbitals involved in the low-energy description) may play a decisive role. Irrespectively, our work shows that the OAM can constitute—though not universally—an experimentally accessible quantity that reflects the nontrivial band topology in a WSM.

Energy hierarchy of ISB and SOC

For two-dimensional systems, the formation of OAM due to ISB has been shown to be the microscopic origin of SOC-induced spin splittings36,47. The present experiments establish a first instance where OAM-carrying bands are observed in a 3D bulk system. Our observations of SAM and OAM in TaAs indicate that an OAM-based origin of the spin splitting also applies to bulk systems with ISB and, as such, underlies the WSM state at the microscopic level. Specifically, the relative alignments of OAM and SAM in the bands v±, determined from our CD-ARPES and spin-resolved data (cf. Fig. 2), imply an energy hierarchy of the bulk band structure in which ISB dominates over SOC32,33. In this case, the electronic states can be classified in terms of their OAM, as SOC does not significantly mix states of opposite OAM polarization. The reversal of Lx, that we observe at the Weyl node, thus indicates an orbital-symmetry inversion accompanying the crossing of valence and conduction bands in TaAs. This supports an earlier theory predicting that the Weyl-semimetal state in TaAs originates from an inversion between bands of disparate orbital symmetries7.

Discussion

Previous SX-ARPES experiments succeeded in measuring the bulk band dispersion in WSM and other topological semimetals1,2,5,6,28,29,48. The present experiment for TaAs goes beyond the dispersion and probes the spin–orbital-components of the wave functions, which encode the topological properties. Our measurements unveil a characteristic reversal in the OAM texture of the bulk Weyl-fermion states at momenta consistent with the termination points of the surface Fermi arcs in TaAs1,9,48. Together, this constitutes a remarkably explicit spectroscopic manifestation of bulk-boundary correspondence in a WSM. In this regard, our results push forward the use of circularly polarized light as a probe of Weyl-fermion chirality to the momentum-resolved domain18.

Our results open a pathway to probe SAM and OAM textures in the bulk band structures of recently discovered magnetic WSM with broken time-reversal symmetry10,11, and of other topological semimetals hosting chiral fermions of higher effective pseudospin and higher Chern numbers, and thus with topological properties distinct from Weyl fermions5,6,28,29,49. This may allow to address the topological features in these band structures. For example, DFT calculations indicate a topological winding of the SAM at the Weyl nodes in the magnetic WSM HgCr2Se4 (ref. 50). More broadly, our results establish a possibility of probing the topological electronic properties of a condensed matter system directly from a bulk perspective without resorting to the corresponding boundary modes.

Methods

Experimental details

We carried out SX angle-resolved photoemission spectroscopy (SX-ARPES) experiments at the ASPHERE III endstation at the Variable Polarization XUV Beamline P04 of the PETRA III storage ring at DESY (Hamburg, Germany). SX photons with a high degree of circular polarization impinge on the sample surface under an angle of incidence of α ≈ 17° (cf. Fig. 2b). TaAs single crystals were cleaved in situ with a top-post at a temperature <100 K and a pressure better than 5 × 10−9 mbar. ARPES data were were collected using a SCIENTA DA30-L analyzer at a sample temperature of ~50 K and in ultra-high vacuum <3 × 10−10 mbar. The angle and energy resolution of the ARPES measurements was ca. Δθ ≈ 0.1° and ΔE ≈ 50 meV. Spin-resolved ARPES was performed by use of a Mott detector (Scienta Omicron). The Au scattering target had an effective Sherman function of 0.1. The energy resolution was ca. ΔE ≈ 600 meV, while the angle resolution was Δθ ≈ 6° in kx direction. The CD-ARPES and spin-resolved ARPES data were obtained using the deflection mode of the spectrometer, so that the experimental geometry stays fixed during the data acquisition procedures.

The growth of single crystals was performed via chemical vapor transport reactions. Sealed silica ampoules with inner diameter of 14 mm and length of 10 cm were loaded with Ta foil and As chunks in a stoichiometric ratio so that the total mass of the reactants was 1.2 g. After adding 0.055 g I2 as transport agent, the ampoules were sealed airtight under a vacuum of ≈10−5 mTorr. Then they were loaded into a single zone tube furnace with a defined temperature gradient across the length of the tube. The angle of the crucibles against the horizontal was ≈20 °. The temperature was increased from room temperature up to 640 °C at 5 K h−1, where it was held constant for 24 h and subsequently heated to 1000 °C at a rate of 2.5 K h−1. Crystal growth proceeded over the following 3 weeks in a temperature gradient of 1000 °C: 950 °C (ref. 51). After cooling radiatively to room temperature, TaAs single crystals with a volume up to 2 mm3 were obtained. The crystal structure and stoichiometry were confirmed using single-crystal X-ray diffraction52 and energy dispersive X-ray spectroscopy methods (Zeiss 1540 XB Crossbeam Scanning Electron Microscope).

Theoretical details

We consider in our theoretical study the non-centrosymmetric primitive unit cell of TaAs (space group I41md) with a lattice constant of 6.355 Å. We employ state-of-the-art first-principles calculations based on the density functional theory as implemented in the Vienna ab initio simulation package (VASP)53, within the projector-augmented plane-wave (PAW) method54,55. The generalized gradient approximation as parametrized by the PBE-GGA functional for the exchange-correlation potential is used56 by expanding the Kohn–Sham wave functions into plane waves up to an energy cutoff of 400 eV. We sample the Brillouin zone on an 8 × 8 × 8 regular mesh by including SOC self-consistently57. For the calculation of the OAM, the Kohn–Sham wave functions were projected onto a Ta s, p, d-, and As s, p-type tesseral harmonics basis as implemented in the WANNIER90 suite58. The OAM expectation values were then obtained in the atom-centered approximation by rotating the tesseral harmonics basis into the eigenbasis of the OAM-operator, i.e., the spherical harmonics.