Main

The powerful combination of inversion symmetry [E(k, ↑) = E(− k, ↑)] with time-reversal symmetry [E(k, ↑) = E(− k, ↓)] ensures that electronic states of non-magnetic centrosymmetric materials must be doubly spin-degenerate. If inversion symmetry is broken, however, relativistic spin–orbit interactions can induce a momentum-dependent spin splitting via an effective magnetic field imposed by spatially-varying potentials. If the resulting spin polarizations can be controllably created and manipulated, they hold enormous promise to enable a range of new quantum technologies. These include routes towards electrical control of spin precession for spin-based electronics1,9, new ways to engineer topological states10,11 and possible hosts of Majorana fermions for use in quantum computation5. There are two generally accepted methods for stabilizing spin-polarized states without magnetism, both exploiting breaking of global inversion symmetry. Structural inversion asymmetry can be created in a centrosymmetric host by imposing an electrostatic potential gradient, for example within an asymmetric quantum well, leading to Rashba-split12 states localized at surfaces or interfaces13,14,15,16. Alternatively, a lack of global inversion symmetry in the unit cell can mediate spin splitting of the bulk electronic states, either through a Dresselhaus-type interaction17, or a recently discovered bulk form of the Rashba effect4,18.

Here, we present direct experimental evidence that 2H-WSe2, a material which retains bulk inversion symmetry, nonetheless exhibits a large spin polarization of its bulk electronic states. This layered compound is composed of stacked Se–W–Se planes (Fig. 1a), each of which contains an in-plane net dipole moment which is proposed to lead to a strong spin–valley coupling for an isolated monolayer19,20,21. The bulk unit cell contains two such monolayers, stacked in a staggered ‘AB’ configuration, restoring inversion symmetry and necessitating spin degeneracy of the bulk electronic states. Nevertheless, combining spin- and angle-resolved photoemission spectroscopy (ARPES) with electronic structure calculations, we observe a large layer- and momentum-dependent spin polarization of these bulk bands.

Figure 1: Bulk electronic structure of WSe2.
figure 1

a, Side and top views of the bulk crystal structure of WSe2. The unit cell contains two Se–W–Se units in which there is a net in-plane dipole pointing to the right and left, respectively. b, Corresponding bulk and surface Brillouin zone. c,d, ARPES measurements (hν = 125 eV, T = 30 K) of the electronic structure along the –– direction (c) and isoenergy contours throughout the surface Brillouin zone (d) reveal sharply-defined bands (for example, the upper valence bands at ) with significant in-plane dispersion, indicative of two-dimensional electronic states. We also observe broader ‘filled-in’ pockets of spectral weight characteristic of three-dimensional states, where the finite kz resolution of ARPES leads to broadening. e, We directly confirm this absence or presence, respectively, of significant kz dispersion from photon-energy-dependent ARPES measurements (Methods and Supplementary Fig. 1). Our measured electronic structure is in excellent agreement with that calculated from density functional theory (solid lines in c,e), confirming that we are probing the bulk electronic states of WSe2.

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We first summarize the bulk electronic structure of WSe2 (Fig. 1). The material is known to be a semiconductor, consistent with our experimental observations, where we find the Fermi level located within the bandgap. We find the band extrema of the valence bands at Γ and K to be almost degenerate22, but here can resolve that the valence band maximum is located at the bulk Γ point, with significant dispersion of these zone-centre states along the surface normal (kz) direction (Fig. 1c, e). Our measured band dispersions are in excellent agreement with those calculated from density functional theory (DFT; see also Supplementary Fig. 1), confirming that we are probing the bulk electronic states of WSe2. The broad total bandwidth of more than 4 eV of the cosine-like upper valence bands along Γ–A reflects the spatially extended nature of W 5d and Se 5p orbitals from which these states predominantly derive.

As well as these dispersive states we find a series of quasi-two-dimensional states, predominantly of planar d x 2 − y 2 , dxy and px/y orbital character. The small overlap of these orbitals along the z direction, combined with suppressed interlayer hopping due to spin–orbit coupling7, results in minimal dispersion along kz, while their extended nature in-plane ensures significant dispersion throughout the surface Brillouin zone (Fig. 1d). The lowest binding energy two-dimensional states form a pair of hole-like bands centred at the Brillouin zone corners, contributing concentric, approximately circular pockets near the band top. These become trigonally warped as they grow in size with increasing binding energy, eventually merging with the zone-centre bands to form bone-shaped pockets centred at . The large splitting of ∼0.5 eV of the top of these bands at signifies the strong atomic spin–orbit interaction in this compound, which is further reflected by our observation of hybridization gaps, for example between two- and three-dimensional states along the Γ–A line (Fig. 1e). Despite such strong spin–orbit coupling, we stress that all states remain spin-degenerate in our calculations, as expected from the bulk inversion symmetry of the crystal.

Intriguingly, however, our spin-resolved photoemission measurements reveal a strong spin polarization of the upper pair of valence band states at the point of the Brillouin zone (Fig. 2). The measured polarization is entirely out of the surface plane within experimental error, with up (down) orientation for the upper (lower) valence band, respectively. From fitting the measured energy distribution curves (EDCs, described in Methods), we estimate the magnitude of the spin polarization to exceed 90%, suggestive of an almost fully spin-polarized band. Moreover, the signs of all polarizations are reversed at the point, confirming that time-reversal symmetry remains unbroken, and thus there is no net magnetic moment. This indicates a non-magnetic origin of the observed spin polarization, seemingly at odds with the centrosymmetric nature of the bulk crystal structure (Fig. 1a). We attribute this to the local inversion asymmetry of individual WSe2 layers, leading to spin-polarized states whose texture is strongly modulated in both real and momentum space despite the global inversion symmetry of the unit cell. For these quasi-two-dimensional bands around , our calculations reveal that the electronic wavefunctions are almost completely localized on individual Se–W–Se layers of the bulk crystal. This is consistent with a spin–orbit-mediated suppression of interlayer hopping predicted at the point for bilayer WSe2, which was proposed to lead to a strong coupling of the real spin with the layer pseudospin7. Such spin–layer locking was subsequently attributed as the origin of characteristic circularly and linearly polarized photoluminescence from bilayer WSe2 (ref. 8).

Figure 2: Observation of spin-polarized bulk bands in an inversion-symmetric host.
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a,b, Energy distribution curves (in arbitrary units (a.u.)) from spin-resolved ARPES measurements (hν = 25 eV, T = 300 K) at the point measured by the out-of-plane (I⊥) (a) and in-plane (I‖) (b) detectors in the Mott scattering chamber (corresponding measurements correcting for the finite Mott-detector efficiencies are shown in Supplementary Fig. 2). c, The extracted polarizations show a strong out-of-plane spin polarization, opposite for the two valence band peaks. d–f, Plots as for a–c, but for the point, showing that the signs of all components are reversed. g, Our projection of the calculated bulk band structure onto the first layer of the unit cell reveals a strong spin polarization of electronic bands localized on this layer (for example, at ), whose sign is reversed in the second layer of the unit cell, as shown in h. i, Measured spin polarization at exhibiting a strong photon energy dependence. Error bars reflect an approximate estimate of the uncertainty in extracting the polarization from the experimental measurements, incorporating statistical errors in peak fitting, systematic errors and uncertainty in sample alignment. Our model calculations (solid line, see Supplementary Information) reveal how the photon energy-dependent modulations of the measured spin polarization result from interference between outgoing photoelectrons originating from the different layers of the crystal.

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As in the bilayer, with the electronic wavefunctions localized on a single Se–W–Se layer (half of the unit cell) of the bulk crystal around , the D6h symmetry of the crystal is effectively reduced to D3h, allowing a net dipole moment within the ab-plane (Fig. 1a). A recent theory has established the general grounds by which such a lack of inversion symmetry of the crystal site point group can lead to a macroscopic spin polarization, driven by the local nature of spin–orbit coupling6. Indeed, our calculated bulk wavefunctions projected onto either WSe2 layer of the unit cell are almost fully spin polarized for the topmost two valence bands at (Fig. 2g). The 180° rotation of neighbouring layers in AB-stacked WSe2, however, ensures that the sign of the spin polarization is opposite between adjacent layers (Fig. 2h). This leads to a strong spin–layer locking7,8, with an overall spin degeneracy of the bulk electronic structure as required for a centrosymmetric material. Photoemission, being extremely surface sensitive, can be expected to predominantly probe the top layer of this material. We thus attribute the strong measured spin polarization we observe here to be a direct observation of a layer-localized spin-polarization of bulk electronic states in WSe2.

This is further supported by our photon-energy-dependent measurements (Fig. 2i), which show how the measured photoelectron spin polarization at can be tuned nearly to zero. Our model calculations (see also Supplementary Fig. 3 and associated discussion) show how this arises as a result of the interference23,24 of spin-up and spin-down polarized photoelectrons emitted from different layers of the material. This strongly supports our conclusions of a huge momentum-dependent spin splitting of up to ∼0.5 eV for bulk states localized in a constituent layer of the unit cell in WSe2, with a spin orientation that is directly tied to the layer pseudospin. We note that the size of this observed spin splitting greatly exceeds spin–orbit-mediated splittings typically observed so far, even in surface Rashba systems with strong local in-plane field gradients14,25. This is because, here, the energy of the spin splitting at the band extrema is directly set by the atomic spin–orbit coupling strength7.

Figure 3 reveals how the underlying spin-polarized states evolve along the –– direction. We find a marked suppression of the out-of-plane spin polarization approximately half way along this line, with negligible polarization observed around the zone centre. This is reproduced by our ab initio calculations (Fig. 4a), and can be understood by considering the orbital character of the underlying states (Fig. 4e). Close to , the electronic states are derived mostly from dxy and d x 2 − y 2 orbitals. There is thus significant orbital overlap within the surface plane, which, together with the net in-plane dipole, favours strong out-of-plane spin polarization26. Around , however, the orbital character becomes dominantly d z 2 /pz-like, causing this component to be strongly suppressed, as found experimentally, while also driving the observed increase in dimensionality of the electronic states. Intriguingly, we also find a small in-plane spin component emerges along ––, which again switches sign either side of . This component would not naively be expected given the symmetry of the Se–W–Se layer, which has no net dipolar field along the out-of-plane direction, as we have confirmed by explicit slab calculations for an ideal bulk-like termination. The emergence of this component therefore reflects further complexity beyond that considered in our theoretical approach, such as small surface relaxations leading to a non-negligible contribution of the dipole out of the surface plane. We stress, however, that this has only a small effect and the predominant contributions to the strong out-of-plane spin polarizations observed here are intrinsic to the bulk electric structure.

Figure 3: Evolution of spin texture along ––.
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a, Dispersion measured by ARPES (hν = 125 eV, T = 30 K) along the –– direction. b–e, Energy distribution curves (EDCs; in arbitrary units (a.u.)) measured using spin-resolved photoemission (hν = 25 eV, T = 300 K) and corresponding extracted spin polarizations, measured at the locations labelled by the corresponding vertical lines in a. f, The out-of-plane (P⊥) and in-plane (P‖) spin polarizations determined from fitting these and additional EDCs (hν = 25 eV, T = 80 K [squares and vertical diamonds, respectively] and T = 300 K [circles and horizontal diamonds, respectively]) reveal some canting of the spin into the surface plane away from the points, and a total suppression of the measured spin polarization around the zone centre. Error bars reflect an approximate estimate of the uncertainty in extracting the polarization from the experimental measurements, incorporating statistical errors in peak fitting, systematic errors and uncertainty in sample alignment. The lines in f are provided as guides to the eye.

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Figure 4: Momentum-dependent suppression of layer-resolved spin polarization.
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a, Momentum dependence of the out-of-plane spin polarization of the top two valence bands throughout the Brillouin zone, calculated for the kz = 0 plane and projected onto the first Se–W–Se layer of the unit cell. White regions indicate suppression of the layer-resolved spin-polarization. b–d, At the point, our ARPES measurements (hν = 125 eV, T = 30 K) (b) and corresponding spin polarization determined from spin-ARPES (hν = 25 eV, T = 300 K) along the coloured momentum (c) and energy (d) distribution curves show how this occurs through the crossing of strongly spin-polarized bands. e, In contrast, towards the zone centre, our orbitally projected band structure calculations reveal how this is correlated with the emergence of significant out-of-plane orbital character of the electronic states.

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We also find a suppression of this out-of-plane spin polarization along the entire – direction. Unlike at the zone centre, however, this cannot be attributed to a change in orbital character: the electronic states close to are predominantly derived from planar orbitals, similar to around , and we accordingly find strong layer-resolved spin polarizations of the underlying bands close to (Fig. 4). Rather, the suppression of spin polarization along – is mediated by the degeneracy of two oppositely polarized bands within a single layer. At the point itself, this is a natural consequence of time-reversal symmetry, as is a time-reversal invariant momentum. Along the – line, such degeneracies are enforced by the combination of time-reversal with the rotational D3h symmetry of a single monolayer within the unit cell, ensuring that the out-of-plane component of the spin must have opposite sign in neighbouring sextants of the Brillouin zone.

Together, our calculations and experiment thus point to an extremely rich real- and momentum-space-dependent spin texture of bulk transition-metal dichalcogenides. They provide a direct demonstration of a pronounced coupling between the spin, valley and layer degrees of freedom, of key importance to widespread proposals to use these materials in exotic devices exploiting the valley pseudospin7,8,19,20,21,27,28,29. More generally, our experimental measurement of spin-polarized bulk electronic states in a centrosymmetric material opens a wealth of new opportunities for creating, probing and controlling spin and valley polarization in bulk solids via local inversion asymmetry.

Methods

ARPES.

ARPES measurements were performed at the I05 beamline of Diamond Light Source, UK, and spin-ARPES measurements at the I3 beamline of MAX-III synchrotron, Sweden30. Single-crystal samples of WSe2, grown by the chemical vapour transport method, were cleaved in situ and measured at temperatures ranging from 30 to 300 K. Measurements were performed using p-polarized synchrotron light from 20 to 130 eV (ARPES) and from 20 to 40 eV (spin-ARPES), and employing Scienta R4000 hemispherical electron analysers. For the spin-ARPES measurements, a mini-Mott detector scheme was used, permitting simultaneous detection of the out-of-plane and one in-plane (along the analyser slit direction) component of the photoelectron spin30. A Sherman function of S = 0.17 was used to generate the measured spin polarizations30

where Pi is the photoelectron spin polarization measured along the out-of-plane, i = ⊥, or in-plane, i = ‖, direction, and Ii± is the measured intensity on the individual detectors in the Mott scattering chamber, corrected by a relative detector efficiency calibration. To extract numerical values of the polarization, we fitted the measured EDCs to two Lorentzian peaks and a Shirley background, convolved with a Gaussian function to account for the instrumental resolution, with the corresponding Lorentzian peak areas used to derive the measured spin polarization. We also applied a geometrical correction to account for the finite angle between the sample and the electron spectrometer, and the corresponding influence of this on the spin polarization measured in the reference frame of the spectrometer. To determine the kz dispersion from photon-energy-dependent ARPES, we employed a free electron final state model

where θ is the in-plane emission angle and V0 is the inner potential. Our photon energy range covers more than six complete Brillouin zones along kz, and we find best agreement taking an inner potential of 13 eV and a c-axis lattice constant of 13.45 Å.

Calculations.

Electronic structure calculations were performed within the context of density functional theory (DFT) using the modified Becke–Johnson exchange potential and Perdew–Burke–Ernzerhof correlation functional as implemented in the WIEN2K programme31. Relativistic effects, including spin–orbit coupling, were fully included. The Brillouin zone was sampled by a 12 × 12 × 6 k-mesh. For the orbital and layer projection calculation, a tight binding Hamiltonian for the bulk band structure was constructed by downfolding the DFT results using maximally localized Wannier functions32,33,34, employing W 5d and 5s orbitals and Se 5p and 5s orbitals as a basis.