Abstract
Since the early days of Dirac flux quantization, magnetic monopoles have been sought after as a potential corollary of quantized electric charge. As opposed to magnetic monopoles embedded into the theory of electromagnetism, Weyl semimetals (WSM) exhibit Berry flux monopoles in reciprocal parameter space. As a function of crystal momentum, such monopoles locate at the crossing point of spinpolarized bands forming the Weyl cone. Here, we report momentumresolved spectroscopic signatures of Berry flux monopoles in TaAs as a paradigmatic WSM. We carried out angleresolved photoelectron spectroscopy at bulksensitive soft Xray energies (SXARPES) combined with photoelectron spin detection and circular dichroism. The experiments reveal large spin and orbitalangularmomentum (SAM and OAM) polarizations of the Weylfermion states, resulting from the broken crystalline inversion symmetry in TaAs. Supported by firstprinciples calculations, our measurements image signatures of a topologically nontrivial winding of the OAM at the Weyl nodes and unveil a chiralitydependent SAM of the Weyl bands. Our results provide directly bulksensitive spectroscopic support for the nontrivial band topology in the WSM TaAs, promising to have profound implications for the study of quantumgeometric effects in solids.
Introduction
Topological semimetals have become a fruitful platform for the discovery of quasiparticles that behave as massless relativistic fermions predicted in highenergy particle physics^{1,2,3,4,5,6}. A prominent example are Weyl fermions, which are realized at topologically protected crossing points between spinpolarized electronic bands in the bulk band structure of noncentrosymmetric or ferromagnetic semimetals^{1,2,7,8,9,10,11}. Near a Weyl node the momentumresolved twoband Hamiltonian takes the form H ∝ ±σ ⋅ k, giving rise to the linear energy–momentum dispersion relation of a massless quasiparticle^{4}. The topological structure of the Weyl node, however, is not encoded in the energy spectrum, but rather manifests in the momentumdependence 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 threedimensional (3D) momentum space^{12,13}. The nontrivial winding of σ stabilizes the Weyl node by a topological invariant, a nonzero Chern number of C = ±1 (ref. ^{4}).
Until today, angleresolved 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 arcs^{1,9,10,14,15}. Manifestations of the nontrivial topology have also been found, accordingly, in magnetotransport experiments^{16}, by scanning tunneling microscopy^{17}, and via optically induced photocurrents^{18}. 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 DiracHamiltonian universally shows nontrivial winding, the underlying microscopic degrees of freedom may vary from one system to another^{19}. In two dimensions, examples include the sublattice degree of freedom for graphene and the spinangular momentum (SAM) for the surface states in topological insulators (TI). Accordingly, the nontrivial Berryphase properties have been addressed by quasiparticle interference STM imaging and dichroic ARPES in graphene^{20,21} and by spinresolved ARPES in TI^{22}. Likewise in 3D WSM, one may expect the relevant degrees of freedom to depend on the considered material system. While previous theories have suggested orbitalsensitive dichroic effects in momentumresolved spectroscopies, as a probe of topological characteristics in topological semimetals^{23,24,25}, such approaches have previously not reached application to experimental data.
In the present work, we find that the orbitalangular 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 Xray (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 welldefined momentum along k_{z} perpendicular to the surface^{26}, when compared to surfacesensitive ARPES experiments at VUV photon energies. SXARPES has been the key method to probe chiralfermion dispersions in topological semimetals^{1,2,5,27,28,29}, but its combination with spin resolution (SR) and circular dichroism (CD) is challenging and not widely explored^{30}. On the other hand, at lower excitation energies SR is routinely used to detect the SAM of electronic states^{22} and CD has been introduced as a way to address the OAM^{31,32,33,34,35}. We performed SXARPES 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 noncentrosymmetric space group I4_{1}md, as shown in Fig. 1a. According to the results of firstprinciples calculations, its bulk band structure features 12 pairs of Weyl points in the Brillouin zone, which divide into two inequivalent sets, W_{1} and W_{2} (refs. ^{7,8}). Here, we focus on the W_{2} points which are located in k_{x}–k_{y} 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 works^{1,2}, our SXARPES 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 highsymmetry lines compare well with the calculated bands (Fig. 1c–d). We further performed hνdependent measurements to trace the band dispersion along k_{z}, 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 finalstate momentum corresponding to the W_{2} Weyl points within experimental uncertainty (Fig. 1e, f).
Orbital and spinangular momentum of the bulk states
The formation of Weyl nodes in TaAs relies on the coexistence of inversionsymmetry breaking (ISB) and spin–orbit coupling (SOC)^{7,8}, which induces a spin splitting into nondegenerate bands. The results of our firstprinciples 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 spinsplit 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 SOC^{32,33,36} (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.
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 k_{p} (refs. ^{31,32}). The grazing light incidence in the xzplane of our experimental geometry (Fig. 1b), thus implies that the measurements predominantly reflect the L_{x} component of the OAM. Note that the relation between CD and OAM, derived in refs. ^{31,32}, relies on the freeelectron finalstate 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 L_{x}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 momentumresolved local OAM of the bulk states in TaAs. As expected theoretically^{31,32}, 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 finalstate effects. Particularly the latter can become relevant at lower excitation energies, where the final state is not wellapproximated in the freeelectron picture^{37,38}.
While OAM is induced by ISB already in the absence of SOC, the presence of Weyl nodes requires a SOCinduced formation of spin polarization^{7,8}. An experimental proof of spinpolarized bulk states in TaAs and related transitionmetal monopnictides has remained elusive up to now. Our spinresolved SXARPES measurements directly verify the predicted SAM of the bands v_{±}. They confirm an opposite S_{x} for v_{+} and v_{−} (Fig. 2b–e), while our CDARPES data prove a parallel alignment of L_{x}, in line with our calculations (Fig. 2a). Moreover, the spinresolved data confirm an opposite sign of S_{x} at +k_{y} and −k_{y}. 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 momentumdependence of the OAM in more detail, we consider momentum distributions of the measured CD and the calculated L_{x} for the band v_{±} in an equik_{z} plane of the W_{2} nodes (Fig. 3a, b). Besides the good agreement of experiment and theory, it is evident from the data that the Weylnode pair near k_{x} = −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 k_{x} paths through the two Weyl nodes of opposite chirality is shown in Fig. 3c. The experiment reveals sign changes of L_{x} close to the Weyl nodes and an opposite overall sign of L_{x} 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.
To examine the OAM and CD of the Weyl cone, we focus on the band dispersion along k_{x} across a W_{2} 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.
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 k_{x} close to the Weyl node. A threepeak 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 W_{2} Weyl nodes. Further evidence for momentumdependent changes also of the L_{y} component near the W_{2} nodes is presented in Supplementary Fig. 9. It is important to note that the observed sign reversal of L_{x} across the Weyl node is not enforced by any symmetry. Rather, it reflects a true bandstructure effect that characterizes the Weylfermion wave functions, namely the crossing of two bands carrying opposite OAM (Fig. 4b–e). This situation is different from the surface states in TI^{22,31} or related examples^{39,40}, where a sign reversal across the nodal point is strictly imposed by timereversal or crystalline symmetries. From an experimental point of view, this makes the presently observed sign reversal of L_{x} more forceful, as it is a bandstructurespecific and not a symmetryimposed 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 atomcentered spherical harmonics, i.e., we consider the quantummechanical expectation value of the atomic L angular momentum operator. This approach offers a viewpoint complementary to the OAM given by the modern theory of polarization^{41}. The fact that the CD rather corresponds to the local OAM^{31,32} could be related to the local nature of the photoemission process itself^{42}.
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 L_{x} for the band v_{+}, revealing a characteristic sign change along the contour around the Weyl nodes. The calculated momentum texture of the Ω_{x}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 nodes^{21,34,35,43}. Indeed, our calculations show explicitly that for TaAs the nontrivial topology of the Berry flux monopoles is encoded in the momentumdependence 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
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 TaP^{44} and LaAlGe^{45} the Pontryagin index of the OAM vector field winds around the typeI Weyl nodes. Interestingly, LaAlGe hosts two typeI and one typeII Weyl dispersions^{46}. Our analysis brings further evidence of a nontrivial winding of OAM only at typeI 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 nonsymmorphic symmetries and a large atomicangular momentum (d orbitals involved in the lowenergy 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 twodimensional systems, the formation of OAM due to ISB has been shown to be the microscopic origin of SOCinduced spin splittings^{36,47}. The present experiments establish a first instance where OAMcarrying bands are observed in a 3D bulk system. Our observations of SAM and OAM in TaAs indicate that an OAMbased 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 CDARPES and spinresolved data (cf. Fig. 2), imply an energy hierarchy of the bulk band structure in which ISB dominates over SOC^{32,33}. 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 L_{x}, that we observe at the Weyl node, thus indicates an orbitalsymmetry inversion accompanying the crossing of valence and conduction bands in TaAs. This supports an earlier theory predicting that the Weylsemimetal state in TaAs originates from an inversion between bands of disparate orbital symmetries^{7}.
Discussion
Previous SXARPES experiments succeeded in measuring the bulk band dispersion in WSM and other topological semimetals^{1,2,5,6,28,29,48}. The present experiment for TaAs goes beyond the dispersion and probes the spin–orbitalcomponents of the wave functions, which encode the topological properties. Our measurements unveil a characteristic reversal in the OAM texture of the bulk Weylfermion states at momenta consistent with the termination points of the surface Fermi arcs in TaAs^{1,9,48}. Together, this constitutes a remarkably explicit spectroscopic manifestation of bulkboundary correspondence in a WSM. In this regard, our results push forward the use of circularly polarized light as a probe of Weylfermion chirality to the momentumresolved domain^{18}.
Our results open a pathway to probe SAM and OAM textures in the bulk band structures of recently discovered magnetic WSM with broken timereversal symmetry^{10,11}, and of other topological semimetals hosting chiral fermions of higher effective pseudospin and higher Chern numbers, and thus with topological properties distinct from Weyl fermions^{5,6,28,29,49}. 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 HgCr_{2}Se_{4} (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 angleresolved photoemission spectroscopy (SXARPES) 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 toppost at a temperature <100 K and a pressure better than 5 × 10^{−9} mbar. ARPES data were were collected using a SCIENTA DA30L analyzer at a sample temperature of ~50 K and in ultrahigh vacuum <3 × 10^{−10} mbar. The angle and energy resolution of the ARPES measurements was ca. Δθ ≈ 0.1° and ΔE ≈ 50 meV. Spinresolved 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 k_{x} direction. The CDARPES and spinresolved 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 I_{2} 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 mm^{3} were obtained. The crystal structure and stoichiometry were confirmed using singlecrystal Xray diffraction^{52} and energy dispersive Xray spectroscopy methods (Zeiss 1540 XB Crossbeam Scanning Electron Microscope).
Theoretical details
We consider in our theoretical study the noncentrosymmetric primitive unit cell of TaAs (space group I4_{1}md) with a lattice constant of 6.355 Å. We employ stateoftheart firstprinciples calculations based on the density functional theory as implemented in the Vienna ab initio simulation package (VASP)^{53}, within the projectoraugmented planewave (PAW) method^{54,55}. The generalized gradient approximation as parametrized by the PBEGGA functional for the exchangecorrelation potential is used^{56} 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 selfconsistently^{57}. For the calculation of the OAM, the Kohn–Sham wave functions were projected onto a Ta s, p, d, and As s, ptype tesseral harmonics basis as implemented in the WANNIER90 suite^{58}. The OAM expectation values were then obtained in the atomcentered approximation by rotating the tesseral harmonics basis into the eigenbasis of the OAMoperator, i.e., the spherical harmonics.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through ProjectID 258499086SFB 1170 (projects A01 and C07), the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter –ct.qmat ProjectID 390858490EXC 2147, and RE1469/131. We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out at PETRA III and we would like to thank Kai Bagschik, Jens Viefhaus, Frank Scholz, Jörn Seltmann, and Florian Trinter for assistance in using beamline P04. Funding for the photoemission spectroscopy instrument at beamline P04 (Contracts 05KS7FK2, 05K10FK1, 05K12FK1, and 05K13FK1 with Kiel University; 05KS7WW1 and 05K10WW2 with Würzburg University) by the Federal Ministry of Education and Research (BMBF) is gratefully acknowledged. The research leading to these results has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie Grant Agreement No. 897276. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gausscentre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (www.lrz.de). J.N.N. and T.S. acknowledge support from the National Research Foundation, under Grant No. NSF DMR1606952. The crystal synthesis and characterization was carried out at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation, Division of Materials Research under Grant No. DMR1644779 and the state of Florida. This publication was supported by the Open Access Publication Fund of the University of Würzburg.
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M.Ü. and T.F. performed the experiments, with support from H.B., B.G., F.D., S.R., J.B., M.H., M.K., and K.R. M.Ü. and T.F. analyzed the experimental data. P.E. and D.D.S. performed the firstprinciples and model calculations. J.N.N., T.F., and T.S. synthesized and characterized the TaAs samples. All authors contributed to the interpretation and the discussion of the results. H.B. wrote the manuscript with contributions from M.Ü., T.F., P.E., R.T., D.D.S., G.S., and F.R. H.B. conceived and planned the project.
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Ünzelmann, M., Bentmann, H., Figgemeier, T. et al. Momentumspace signatures of Berry flux monopoles in the Weyl semimetal TaAs. Nat Commun 12, 3650 (2021). https://doi.org/10.1038/s41467021237273
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DOI: https://doi.org/10.1038/s41467021237273
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