Abstract
In nonmagnetic materials the combination of inversion symmetry breaking (ISB) and spinorbit coupling (SOC) determines the spin polarization of the band structure. However, a local spin polarization can also arise in centrosymmetric crystals containing ISB subunits. This is namely the case for the nodalline semimetal ZrSiTe where, by combining spin and angleresolved photoelectron spectroscopy with ab initio band structure calculations, we reveal a complex spin polarization. In the bulk, the valence and conduction bands exhibit opposite spin orientations in two spatially separated twodimensional ZrTe sectors within the unit cell, yielding no net polarization. We also observe spinpolarized surface states that are well separated in energy and momentum from the bulk bands. A layerbylayer analysis of the spin polarization allows us to unveil the complex evolution of the signal in the bulk states near the surface, thus bringing the intertwined nature of surface and bulk effects to the fore.
Introduction
In the absence of a magnetic field, the spin degeneracy of the band structure is preserved in the bulk of materials that exhibit inversion symmetry. It is therefore generally assumed that a finite spin polarization can only arise in noncentrosymmetric (NC) materials. A typical example is that of III–V semiconductors with the zincblend structure, where spin–orbit coupling (SOC) determines the momentumdependent spin splitting of the lighthole and heavyhole bands (j = 3/2), the Dresselhaus effect^{1}. If the NC crystal contains polar atomic sites^{2}, the Rashba effect^{3} adds an additional source of spin–orbit splitting, with a characteristic helical spin texture^{4}. The Rashba term is also responsible for the spin polarization observed at surfaces^{5,6} and heterostructures^{7} that exhibit a structural inversion asymmetry (SIA).
While breaking the inversion symmetry is necessary for the appearance of a net polarization, a recent theory^{8,9} predicted a so far overlooked source of spin polarization in centrosymmetric (CS) materials containing spatially separated structural subunits that break space inversion. In this case a local and momentumresolved spin polarization develops, even if the material remains globally nonpolarized. The conditions for the emergence of such “hidden” spin polarization are then determined by the local point group at the atomic sites, rather than by the space group of the crystal. One can distinguish two possible situations, illustrated in Fig. 1. If the NC site has no electric dipole moment, as for a tetrahedral site, the system exhibits the Dresselhaus effect and a local spin polarization. If the local dipole moment is nonzero, as for a pyramidal site, the Dresselhaus and the Rashba effects are simultaneously present. They were dubbed, respectively, D2 and R2 in refs. ^{8,9} to distinguish them from the usual—D1 and R1—cases of NC crystals. The resulting kresolved local spin polarization may play a role in the mechanism behind the large nonsaturating magnetoresistance observed in Dirac semimetals^{10,11,12}, as recently proposed for WTe_{2}^{13}. Both in R2 and D2, however, the contributions from all sites add to zero, which hampers the direct detection of the local polarization by spatially averaging probes. Spinresolved and angleresolved photoelectron spectroscopy (spARPES) is a laterally averaging probe, but its signal decays exponentially with the distance of the probed region from the surface, according to the photoelectron mean free path. The very short (λ ~ 5−10 Å) probing depth can then be exploited to gain local sensitivity. In layered materials where a kresolved local spin polarization appears, this alternates in successive planes parallel to the surface. As a result, the contribution from the two (even and odd) families of planes to the spARPES signal can be remarkably different. For a typical interplane distance d = 5 Å and a 10 Å lattice parameter, their ratio is (I_{odd}/I_{even}) ≃ 1.7−2.7 for λ = 5−10 Å. Final state effects and possible interference terms in the photoemission process can modify these values^{14,15}, but a net spin signal is still expected. Indeed, spARPES measurements have revealed spinpolarized bands in WSe_{2}^{16} and in the superconductors Bi2212^{17,18} and LaO_{0.55}F_{0.45}BiS_{2}^{19}. Theory and conventional ARPES experiments have reached similar conclusions for BaNiS_{2}^{20} and MoS_{2}^{21}. Those results have established the concept of hidden spin polarization in CS materials, but questions about the relative importance of the local site asymmetry vs. the surface SIA in determining the bulk polarization remain open.
Here we address the spin–orbit splitting of the electronic states and the spin texture in the nodalline semimetal ZrSiTe^{22,23}. The crystal structure of ZrSiTe is CS, but contains polar sites with a local electric dipole field (Fig. 2). Therefore, one expects the D2 and R2 effects to impose a spin polarization on its bands. Our spARPES data and fully relativistic ab initio bulk calculations confirm that the Dirac quasiparticles are spinpolarized. ZrSiTe also exhibits surface states that are well separated in energy and momentum from the bulk bands^{24}, and we find that these surface states are also spin polarized. Slab calculations and a layerbylayer analysis unveil the complex evolution of the spin signal of the bulk states near the surface. This observation suggests that the interference between wave functions forming the bulk continuum with their reflection from the surface barrier might influence the bulk spin polarization, as previously discussed for the cases of Bi^{25} and noble metals^{26,27}.
Results
Crystal and electronic structure
ZrSiTe is a layered material. It crystallizes in the tetragonal nonsymmorphic space group P4/nmm (no. 129), with lattice parameters a = 3.692 Å and c = 9.499 Å^{28}. The structure is obtained by stacking (Te−Zr−Si−Zr−Te) quintuple layers along the caxis (Fig. 2a). Each unit cell contains two ZrTe bilayers built from ZrTe_{4} square pyramids, indicated by I and III in Fig. 2b. Inversion symmetry is broken within the bilayers, and each pyramid carries a local electric dipole perpendicular to the layer. The two ZrTe sectors, ~5 Å apart, are separated by a planar Si squarelattice, which coincides with a glide plane for the structure. Owing to the large c/a ratio—the largest in the ZrSiX series (X = S, Se, Te)—and to the presence of the Si interlayer, the interaction between the ZrTe bilayers is weak. This is a favorable condition for the appearance of a sizeable local polarization, as discussed in ref. ^{9}. Adjacent quintuple layers are weakly coupled through a van der Waals gap, so that the crystals are easily mechanically exfoliated to expose a Teterminated (001) surface.
Compounds of the ZrSiX family have recently been in the limelight, following suggestions that they would realize the topological nodalline semimetal phase^{22,23,24,29,30,31}. Figure 2c, d shows a cartoon of the lowenergy band structure. In the absence of SOC the linearly dispersing valence band (VB) and conduction band (CB) would cross each other at the Fermi level (E_{F}) forming a nodal line, i.e. a line of Dirac points in momentum space^{32}. SOC lifts the degeneracy of the nodal line throughout the Brillouin zone (BZ) and opens a hybridization gap between VB and CB. Owing to the local breaking of inversion symmetry, and under the action of SOC, the bands can acquire a local hidden spin polarization, as we show in the next section.
ARPES data and band structure calculations of Fig. 3 illustrate the prominent features of the electronic structure of ZrSiTe. The ARPES measurements were performed with s (blue color scale) and p (brown color scale) linearly polarized light at a photon energy of 107 eV, corresponding to a perpendicular wave vector k_{z}~π/c. The complex experimental Fermi surface follows the periodicity of the square (001)projected surface BZ in Fig. 3a, b. It consists of two concentric diamondlike contours centered at \(\overline{{{\Gamma }}}\), with additional smaller features around the \(\overline{{\rm{X}}}\) points. A comparison with the calculated (001)projected spectral weight at the Fermi level in Fig. 3c assigns the former to the residual nodalline bulk states and the latter to surface states (labeled SS), in agreement with the literature^{22,24,29}. E vs. k ARPES intensity maps are shown in Fig. 3d for the \(\bar{{\rm{M}}}\bar{\Gamma }\bar{{\rm{M}}}\) and in Fig. 3e for the \(\bar{{\rm{X}}}\bar{\Gamma }\bar{{\rm{X}}}\) highsymmetry directions, with overlaid calculated bulk (red lines) and surface (green lines) bands. ARPES matrix elements are quite strong and parts of the band structure have appreciable intensity only with one polarization, as a consequence of the different orbital character of the bands. First principles^{33} and simple tight binding calculations^{34} assign the negative effective mass VB to predominantly Si p_{x} + p_{y} orbitals and the positive effective mass CB to Zr \({d}_{{x}^{2}{y}^{2}}\) orbitals. VB and CB intersect above E_{F} along \(\bar{{\rm{M}}}\bar{\Gamma }\bar{{\rm{M}}}\) and below E_{F} along \(\bar{{\rm{X}}}\bar{\Gamma }\bar{{\rm{X}}}\), which indicates that the nodal line disperses in energy. The surface state SS is visible in the projected bulk gap along \(\bar{{\rm{M}}}\bar{{\rm{X}}}\bar{{\rm{M}}}\) in Fig. 3f. Theory predicts a small energy spin splitting for this band, which is later confirmed in the next sections of the manuscript.
Bulk states: hidden spin polarization
We now address the spin polarization of the bulk bands that form the nodal line. Let us first consider an unrealistic limit where the valence and the CB were perfectly confined within the Si plane and, respectively, within the ZrTe layers. In this case one would expect a large local spin polarization in the CB from the polar Zr sites and no polarization at all in the VB from the CS Si sites. In reality the VB and CB hybridize, but retain a largely twodimensional character, which is confirmed by their weak dispersion along the c axis (see Supplementary Note 1) and by independent quantum oscillations experiments^{35}. Therefore, their overlap remains small. As a result, the local polarization of the CB decreases from the ideally confined situation^{36}, but at the same time the VB acquires a spin polarization. The partial delocalization of the hybrid bands satisfies both the criteria of wave function’s segregation and overlap with polar sites, the key ingredients for the occurrence of a hidden spin polarization^{9}.
The hidden spin polarization is confirmed by bulk calculations. To assess the spatially resolved spin structure of the VB and CB, we project the spin expectation value on the top and bottom (I and III) ZrTe sectors and on the intermediate (II) Si sector, with a methodology described in details in the Supplementary Discussion. For each sector we plot in Fig. 4a–c the x component of the spin polarization (P_{x}), and in Fig. 4d–f the y component (P_{y}), along a cut at k_{y} = 0.53 (π/a) and for k_{z} = π/c in the bulk BZ, indicated by the black dashed line of Fig. 3b. Additional results for different values of k_{z} are discussed in the Supplementary Discussion. In the ZrTe sector we observe a sizeable polarization, both above and below E_{F}. P_{x} and P_{y} have opposite sign in the top and bottom sectors, while the polarization is essentially zero in the intermediate Si plane. Furthermore, P_{x} is symmetric with respect to k_{x} = 0, while P_{y} is antisymmetric, consistent with the helical Rashba R2 texture. As expected by the presence of polar atomic sites, the D2 term alone could not reproduce the observed spin texture. A quantitative assessment of the relative strength of the two contributions would require measuring the spin texture over the whole BZ and a theoretical analysis, e.g. by a k ⋅ p model, as recently done to analyze the spin texture of the surface alloy Bi/PbAg_{2}^{37}, which is beyond the scope of the present work.
Bulk calculations alone cannot elucidate the role of the surface SIA in the formation of the spin polarization. To this end we have also performed a fully relativistic ab initio calculation on a 9unit cells slab. Before assessing the effect of the surface SIA, we compare the calculations of the projected spin polarization in the bulk and in the slab. The results are shown in Fig. 4g–l, for the same cut at k_{y} = 0.53 (π/a), and projected on the innermost unit cell of the slab. We can see a fairly good agreement between bulk and slab calculations. We ascribe the somewhat smaller values obtained from the slab calculations to a reduced segregation of the wave function in the different sectors (see Supplementary Discussion for further details).
We have measured by spARPES the spin polarization of the CB at three wave vectors connected by reflections about highsymmetry planes: A = (−0.13, 0.53)(π/a), B = (0.13, 0.53)(π/a), and C = (0.13,−0.53)(π/a). The light was ppolarized, and the photon energy hν = 48.5 eV was chosen near the minimum of the photoelectron mean free path to enhance the depth selectivity, as previously discussed. A sketch of the experimental setup is given in the Supplementary Note 2. We stress here that during the experiment the sample position was fixed, and different regions of the BZ were accessed by means of an electrostatic deflection lens.
The spinaveraged ARPES intensity map of Fig. 5a, also collected at k_{y} = 0.53 (π/a), is well reproduced by the calculated bands of Fig. 4. Figure 5b–d shows spARPES results for the x spin component, and Fig. 5e–g displays the corresponding spin polarization P_{x}, measured at A and B along the red lines of Fig. 5a and at C, respectively. The data show a clear spin signal within ~0.6 eV of E_{F}, corresponding to the CB. The polarization P_{x} extracted from the spectra has the same sign at A and B, but opposite sign at C. Finally, the polarization P_{y} is very weak and of opposite sign at A and B (see Supplementary Note 3), as predicted by the calculations in Fig. 4d–f.
These results are consistent with the theoretical predictions. A fully quantitative comparison would require to evaluate possible final state effects and interference effects from different unit cells^{14,15}, and is beyond the scope of the present paper. Nevertheless, the consistency between theory and experiment already provides sufficient evidence for the existence of a hidden spin polarization in the bulk states of a CS nodalline semimetal. This leaves open questions about the possible influence of the surface SIA on the bulk polarization. This point is addressed in the following.
Surface states: net spin polarization
We now focus on the SS that linearly disperses across E_{F} around the \(\overline{{\rm{X}}}\) point^{22,24,29}. Theory has predicted the existence of topologically protected SSs (TSS) in gapless nodalline semimetals^{32}, and indeed there is evidence for TSS in TlTaSe_{2}^{38}, PbTaSe_{2}^{39}, and ZrB_{2}^{40}. For the ZrSiX family, despite some suggestions^{24,41}, the topological nature of the SSs has not been established, and their origin is probably different. The nonsymmorphic space group requires the bulk states to be fourfold degenerate along the \(\overline{{\rm{X}}}\) \(\overline{{\rm{M}}}\) line at the BZ boundary, but the surface breaks translational symmetry and removes all nonsymmorphic symmetries. This lifts the degeneracy and leaves a SS state “floating” on top of the bulk bands^{24}. Under the influence of the SIA and SOC the Rashba effect can then split the SS and induce a spin polarization. This is confirmed by our results.
Figure 6a displays the band structure calculated for the slab geometry near the \(\overline{{\rm{X}}}\) point, with the “floating” SS in the gap between the VB and the CB, from which it is split off. The splitting of the SS predicted by theory is now resolved in the ARPES constantenergy map of Fig. 6b, which covers a portion of the experimental Fermi surface. The spinresolved calculation of Fig. 6c captures all features of the E vs. k intensity map of Fig. 6d measured along the dashed line near and parallel to \(\bar{{\rm{M}}}\bar{{\rm{X}}}\bar{{\rm{M}}}\), including the small splitting of the SS and the bottom of the CB at the \(\overline{{\rm{X}}}\) point. The calculated spin polarization P_{y}, in the direction perpendicular to \(\bar{{\rm{M}}}\bar{{\rm{X}}}\bar{{\rm{M}}}\), is encoded in the color of the symbols. It has odd parity with respect to \(\overline{{\rm{X}}}\), while the small P_{x} component is even (see Supplementary Note 4). This finding is in good agreement with a recent study of the spin polarization in the surface state of HfSiS^{42}, where it is shown that the Dresselhaus and the Rashba terms similarly contribute to form an unique unidirectional spin texture.
We have probed by spARPES the spin polarization of the SS at various k_{x} values, indicated by dotted lines (1–7) in Fig. 6d, symmetrically located with respect to the k_{x} = 0 plane. The experimental data are presented in Fig. 6e for the spin projection along k_{y}. The spinresolved spectra indicate the presence of two components with opposite spin, separated by ~50 meV, corresponding to the spinsplit partners of the SS, which cannot be fully resolved owing to the experimental broadening. The sign of P_{y} is opposite at the (1) and (7) extremes, where the inner subband has already crossed E_{F}. Between these extremes the two peaks overlap, but it is still possible to recognize that their relative position is reversed on opposite sides of k_{x} = 0, consistent with the calculation. By contrast, the spectra for the k_{x} spin projection are essentially identical on both sides, and the polarization P_{x} is small (see Fig. S4d). In summary, both theory and experiment confirm that the SS is split into spinpolarized bands due to the action of the surface SIA. In the following section we will discuss how this term can also influence the spin polarization of the bulk states.
Discussion
The results of the previous sections demonstrate that both the bulk and the surface states of ZrSiTe are spin polarized. The surface SIA is the origin of the polarization of the SS. Bulk calculations predict that the interaction with the polar ZrTe sites will induce a combined hidden R2 and D2 polarization in the bulk states. The wave functions of the bulk states however experience the presence of the surface, and one cannot exclude an influence of the surface SIA also on the bulk polarization. To address this question we performed a detailed layerbylayer analysis of the slab calculations. We select three representative states from SS and from the bulk valence band (BS). They are indicated by a circle and, respectively, a square and a triangle on the band dispersion of Fig. 7a, corresponding to the same path investigated in Figs. 4 and 5. For these states we then plot in Fig. 7b–d the calculated local probability density ∣Ψ(z)∣^{2} within each quintuple layer of the 9layers slab. We also encode in the color of the symbols the local polarization P_{x}(z).
For the SS, Fig. 7b, ∣Ψ_{SS}(z)∣^{2} decreases by three orders of magnitude within three layers of the surface. The spin polarization is already strongly reduced in the second layer, and disappears in the deeper layers. Therefore, we attribute the polarization to the rapid variation of the surface potential as described by the surface Rashba–Bychkov effect. The situation is more complex in the bulk continuum. Depending on the state considered, ∣Ψ_{BS}(z)∣^{2} is partially reduced (Fig. 7c) or approximately constant at the surface (Fig. 7d), but the surface SIA has nonetheless a sizable effect on the spin polarization. Even when the spin polarization in the innermost layers alternates between opposite ZrTe sectors, as expected for the hidden spin polarization, the surface layers behave differently. In Fig. 7c, the polarization over a surface region of ~20 Å shows no alternation and a net polarization is thus established. For the state displayed in Fig. 7d, no clear signature of a spin oscillation between the sectors is observed. This suggests that the influence of the surface on the spin polarization dominates over the bulk effect. A possible explanation for this observation is the interference between the incident Bloch wave and the wave reflected at the surface. It results in a spin density with energy and spatial dependence^{26}, as previously observed in Bi^{25} and in Au^{27}.
In comparing the values of the spin polarization of the surface and bulk states, one should be aware that final states effects may affect the ARPES measurement, therefore we refrain from drawing quantitative conclusions, which we leave for a future investigation. From the point of view of the calculations, we notice that, in the slab model, the value of the spin polarization for both the BS and the SS is small, and the latter is comparably larger than the former. This difference results from the asymmetry of the wave functions^{43,44}, but also reflects the relative intensities of the surface electric field and the local field at the bulk polar sites. This is explicitly assessed in Fig. 8, in order to compare their relative importance. Figure 8a displays the layer dependence of the electrons potential energy within the first two unit cells below the surface, and of the z component of the electric field (Fig. 8b). As expected from the structure, a local dipole is formed within each ZrTe sector with an average value of ∣E_{z}∣ = 0.7 V/Å as computed over the shaded area, a value that is large but two times smaller than the one predicted for BaNiS_{2}^{20}. In order to evaluate the electric field induced by the SIA, Fig. 8c shows the difference between E_{z} calculated in the first (black line) and second (red line) unit cell of the slab with respect to the result of the bulk calculations. We clearly see that the behavior in the second unit cell is already an excellent approximation of the bulk, whereas in the first unit cell the difference becomes sizeable only above the topmost Te atoms. This shows that the SIA is responsible for an electric field with a peak value of 3 V/Å. Although the electric field is confined in a shallow region of ~1 Å, it might still influence the bulk continuum by introducing a spindependent dephasing in the formation of the stationary states.
In summary, spARPES data and firstprinciples calculations of the electronic structure show that the layered nodalline semimetal ZrSiTe hosts spinpolarized bulk and surface states. Our bulk calculations confirm that CS materials can develop a local hidden spin polarization in the presence of polar sites. Our extensive analysis of the slab calculations illustrate the possible role of the surface SIA in contributing to a net spin polarization in the bulk state at the surface, which is accessible to spARPES.
Finally, the ZrTe sectors are an example of a highmobility 2D electron system with sizeable SOC, and can be considered an experimental realization of an intrinsic spin Hall semimetal^{45}. Their integration on a suitable insulating substrate, such as SrTiO_{3}^{33}, would thus potentially produce an efficient spinHall filter of interest for spintronic devices.
Methods
Sample growth and angleresolved photoemission spectroscopy measurements
Highquality single crystals of ZrSiTe grown by vapor transport were mechanically exfoliated in situ under ultrahigh vacuum conditions (base pressure in low 10^{−10} mbar). All the photoemission spectra have been taken using hemispherical VGSCIENTA electron analyzers operating in deflection mode at the 7.0.2 MAESTRO beamline, ALS, Berkeley (US), at the APE beamline, ELETTRA, Trieste (IT), and at the ASPHERE III endstation of beamline P04 at PETRA III. In the latter setup, the sample temperature was kept at ~30 K, and the bulk band periodicity was investigated with circularly polarized light in the energy range between 420 and 500 eV, with energy and angular resolution of 50–100 meV and 0.07 Å^{−1}, respectively. The other ARPES data have been acquired at 80 K, and we used 107 eV photon energy with variable polarization for the data taken at ALS and 48.5 eV with ppolarization at APE. The energy and momentum resolution was better than 20 meV and 0.01 Å^{−1} for the ARPES spectra.
Spin and angleresolved photoemission spectroscopy measurements
The spinresolved photoemission spectra were measured at APE using a VLEEDbased spin detector. Energy distribution curves (EDCs) of the spinpolarized photoelectron intensity (I) were measured after magnetizing the detectors along different directions and orientations, using coils. The spectra were renormalized to the same acquisition time and corrected for possible differences in the detectors’ efficiency (of the order of few %) for opposite magnetizations. For a specific wave vector, we have used the two detectors to acquire I_{i} along three orthogonal directions in the reference system of the electron analyzer. These quantities have been combined in order to obtain the electron spin components in the reference system of the crystal surface, and the spin polarization according to
where S = 0.3 is the value of the Sherman function previously determined by measuring the spin polarization of the Shockley surface states of Au(111). \({I}_{i}^{+}\) and \({I}_{i}^{}\) are the spinpolarized photoelectron intensities for the target magnetized (+) or (−) along the i axis. From P_{i} we retrieve the relative populations of spin up and down electrons in the material’s band structure \({S}_{i}^{\uparrow /\downarrow }\), according to \({S}_{i}^{\uparrow /\downarrow }=(1\pm {P}_{i})\frac{{I}_{i}^{+}+{I}_{i}^{}}{2}\). The uncertainty \(\delta {I}_{i}^{+/}\) associated to the measured photoelectron intensities \({I}_{i}^{+/}\) is estimated assuming a Gaussian statistics, and it is equal to \(\sqrt{{I}_{i}^{+/}}\). The uncertainty over the spin polarization P_{i} is \(\delta {P}_{i}={P}_{i}\cdot \sqrt{\frac{{(\delta {I}_{i}^{+})}^{2}+{(\delta {I}_{i}^{})}^{2}}{{({I}_{i}^{+}+{I}_{i}^{})}^{2}}+\frac{{(\delta {I}_{i}^{+})}^{2}+{(\delta {I}_{i}^{})}^{2}}{{({I}_{i}^{+}{I}_{i}^{})}^{2}}}\). Finally, the uncertainty over \({S}_{i}^{\uparrow /\downarrow }\) is \(\delta {S}_{i}^{\uparrow /\downarrow }={S}_{i}^{\uparrow /\downarrow }\cdot \sqrt{\frac{{(\delta {P}_{i})}^{2}}{{(1\pm {P}_{i})}^{2}}+\frac{{(\delta {I}_{i}^{+})}^{2}+{(\delta {I}_{i}^{})}^{2}}{{({I}_{i}^{+}+{I}_{i}^{})}^{2}}}\). These quantities are indicated by the error bars in the figures of this manuscript and of the Supplemental Information. The energy and momentum resolution was better then 100 meV and 0.05 Å^{−1} for the spinresolved ARPES spectra.
Ab initio calculations
The electronic structure of both bulk and slab models were obtained within density functional theory using the planewave pseudopotentials method as implemented in the Quantum Espresso package^{46}. We used fully relativistic optimized normconserving vanderbilt pseudopotentials (ONCVPSP)^{47} obtained from the sg15 database^{48}. The exchange and correlation were treated within the Perdew–Burke–Ernzerhof (PBE) approximation^{49}. The energy cutoff of the planewave was set to 80 Ry, and we used a 8 × 8 × 3 and a 12 × 12 × 1 Monkhorst–Pack grid for the bulk and slab calculations, respectively. In order to compute the spin expectation value projected on atomic sites, we projected the eigenstates onto orthogonalized atomic wavefunctions using the projwfc.x code of the Quantum Espresso package. Subsequently, in the case of noncollinear calculations, we rotated the basis set from the total angularmomentum basis set to the atomic and spin angular momentum^{50,51,52}. For the slab calculations we used a tetragonal unit cell where the ZrSiTe unit cells were arranged along the [001] direction leaving a 23 Å vacuum region between slabs. The weight of the spectral function in the (001) surface was computed with the surface Green’s function of a semiinfinite system using the method described by A. Umerski^{53}. To compute the bulk Green’s function we used the Hamiltonian in the basis of maximally localized Wannier functions obtained with the Wannier90 code^{54}. The maximally localized Wannier functions were obtained by projecting the Bloch bands into the d orbitals of Zr, p orbitals of Te and the s and p orbitals of Si.
Data availability
Source data are available for this paper in ref. ^{55}. Additional data used to support the findings of this work are available, upon reasonable request, from the corresponding authors.
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Acknowledgements
We acknowledge financial support by the Swiss National Science Foundation (SNSF), in particular M.F. acknowledges the SNFN support through grant P2ELP2_181877. D.G.M. and O.V.Y. acknowledge the support by the NCCR Marvel. All firstprinciples calculations were performed at the Swiss National Supercomputing Centre (CSCS) under the project s1008. This work has been partly performed in the framework of the nanoscience foundry and fine analysis (NFFAMIUR Italy Progetti Internazionali) facility. We thank 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. 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.
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G.G. and A.C. conceived the experiment. G.G., A.C., S.R., M.F., M.Z., performed the ARPES experiments with the assistance of M.K., K.R., C.J., A.B., E.R., I.V., J.F. G.G., A.C., S.R., M.F., M.Z., carried out the spinresolved ARPES experiment with the assistance of I.V. and J.F. D.G.M. and O.V.Y. performed the ab initio calculations. H.B. and A.M. grew the highquality single crystals. G.G. wrote the manuscript and coordinated the project under the supervision of A.C. and M.G. All authors discussed the data and actively contributed to the final manuscript.
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Gatti, G., GosálbezMartínez, D., Roth, S. et al. Hidden bulk and surface effects in the spin polarization of the nodalline semimetal ZrSiTe. Commun Phys 4, 54 (2021). https://doi.org/10.1038/s4200502100555x
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DOI: https://doi.org/10.1038/s4200502100555x
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