Abstract
Finding stimuli capable of driving an imbalance of spinpolarised electrons within a solid is the central challenge in the development of spintronic devices. However, without the aid of magnetism, routes towards this goal are highly constrained with only a few suitable pairings of compounds and driving mechanisms found to date. Here, through spin and angleresolved photoemission along with density functional theory, we establish how the pderived bulk valence bands of semiconducting 1THfSe_{2} possess a local, groundstate spin texture spatially confined within each Sesublayer due to strong sublayerlocalised electric dipoles orientated along the caxis. This hidden spinpolarisation manifests in a ‘coupled spinorbital texture’ with inequivalent contributions from the constituent porbitals. While the overall spinorbital texture for each Se sublayer is in strict adherence to timereversal symmetry (TRS), spinorbital mixing terms with net polarisations at timereversal invariant momenta are locally maintained. These apparent TRSbreaking contributions dominate, and can be selectively tuned between with a choice of linear light polarisation, facilitating the observation of pronounced spinpolarisations at the Brillouin zone centre for all k_{z}. We discuss the implications for the generation of spinpolarised populations from 1Tstructured transition metal dichalcogenides using a fixed energy, linearly polarised light source.
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Introduction
Spintronic devices aim to exploit the spin quantum number of an electron rather than the charge^{1}. Their operational principles are underpinned by a reversible external stimulus to which electrons of opposing spin species respond oppositely^{2}. In nonmagnetic systems the task of finding such stimuli is not straightforward due to the presence of timereversal symmetry (TRS : E(k, ↑) = E(−k, ↓) where E, ±k, and ↑↓ denote the electron energy, momentum and spin respectively), enforcing a netzero spin polarisation for all electronic bands across a material. One therefore must enforce an imbalance in spin species across the material by applying a magnetic field^{3,4}, or by selectively coupling to electrons at ±k unevenly, in addition to the selective coupling to a single spin species^{5,6,7}. The latter, allelectronic pathway permits easier integration with traditional components, and is thus more desirable^{5,8}, but only a few known compounds host the necessary spin textures within their electronic structure along with an inbuilt mechanism to couple from opposite k vectors reliably.
Of these compounds, several belong to the transition metal dichalocgenide (TMD) material family. TMDs consist of van der Waals separated layers of XMX formula units (X ∈ {S, Se, Te}), with the transition metal atom (M) positioned at the centre of each MX_{6} octahedron^{9}. The TMDs are renowned for their array of oftenoverlapping superconducting^{10,11,12,13}, charge density wave^{14,15,16,17,18} and topological^{13,19,20,21,22,23} phases, but the modern interest in this series stemmed from the similarity between their twodimensional honeycomb lattice structures to that of graphene. Indeed, the electronic structures of monolayer 1HMoS_{2} and 1HWSe_{2} can be derived from that of graphene by breaking AB sublattice symmetry and increasing the atomic spinorbit coupling strength^{24}. The resulting pair of d_{xy} and \({d}_{{x}^{2}{y}^{2}}\)derived spinsplit valence bands at the K and K’ points of the bulk Brillouin zone (BZ) couple differently to circularly polarised light, enabling the generation of spinpolarised currents in the conduction band and prompting the design of socalled valleytronic devices exploiting the spinvalley coupling^{6,7,25,26}.
In their bulk forms, MoS_{2} and WSe_{2} adopt an inversion symmetric (IS : E(k, ↑) = E(−k, ↑)) 2H structure. While the combination of IS and TRS enforce that the bulk band structure is entirely spindegenerate, a socalled ‘hidden spin polarisation’ is still directly observable at the K and K’ points with a surface sensitive probe of the electronic structure due to a strong localisation of the d_{xy} and \({d}_{{x}^{2}{y}^{2}}\) wavefunctions to within a single 1Hsublayer^{27,28,29}, with similar phenomena observable in other centrosymmetric compounds^{30,31,32}. However, the transition to the 2HTMD structure shifts the valence band maximum from the K point to the timereversal invariant momentum (TRIM), Γ, complicating the generation of clean spincurrents in the bulk compounds^{33,34,35}.
Here, we search for an analogous hidden spin mechanism in 1TTMDs. Through spin and angleresolved photoemission (spinARPES), we show that time and inversionsymmetric 1THfSe_{2} exhibits a hidden spinpolarisation in the \(\overline{{{\Gamma }}}\)centred Se pderived valence bands originating from the asymmetry of a single 1T unit cell along the caxis. Remarkably, though, we show how this spinpolarisation persists through the \(\overline{{{\Gamma }}}\) point of the surface Brillouin zone and retains the same sign for ±k vectors. Our density functional theory (DFT) calculations show how this effective net spinpolarisation is of entirely nonmagnetic origin, instead deriving from local spinorbital mixing within the porbital manifold, aided by a selective coupling of linearly polarised light to subsets of porbitals, and therefore to a partial ground state spintexture.
Results and discussion
Hidden spin mechanism in 1Tstructured TMDs
Figure 1 compares the hidden spin mechanism in 2H and 1T structured bulk TMDs using 2HWSe_{2} and 1THfSe_{2} as examples. In the 2H structure, the triangularly coordinated X atoms eitherside of the MX_{6} octahedron are orientated identically within a single sublayer. This causes a charge imbalance orientated parallel to the layers, breaking inversion symmetry on a local scale. Figure 1b schematizes the resulting layerresolved electronic structure^{27}. The Rashba effect produces a spinpolarisation in the directions perpendicular to any applied field^{36}, here producing an outofplane Zeemanlike splitting in the W d_{xy} and \({d}_{{x}^{2}{y}^{2}}\)derived bands at the K points, reversing sign at the K’ points in accordance with TRS. The spinpolarisation is exactly opposite in the second sublayer, and therefore nonzero spin polarisations are uncovered only for surface sensitive probes of electronic structure^{27,28}.
For the 1Tstructure, the chalcogen sublayers are 180^{∘} rotated from oneanother either side of the transition metal plane. Charge is thus equally distributed across a single XMX layer, and the 1T structure is inversion symmetric down to a single monolayer. A net dipole now exists outofplane, however, switching sign about the M plane. While the spatial scales for this local inversion symmetry are very small, previous studies have predicted that hidden spin physics can arise in 1TTMDs^{37}, with a local inplane Rashba polarisation verified in the Se p_{x,y}derived bands of monolayer 1TPtSe_{2} by spinARPES^{38}, inline with that schematised in Fig. 1d for the top and bottom sublayers. For a bulk system, the presence of hidden spin polarisation is yet to be confirmed. For the present case, Hf is amongst the least electronegative of all transition metals, producing a particularly strong electric dipole across a HfSe bond. The degree of spinsplitting from a hidden spin mechanism should be particularly pronounced in this compound.
Overview of the electronic structure of 1THfSe_{2}
Figure 2 overviews the electronic structure of 1THfSe_{2} as seen by ARPES, which probes only the occupied part of the band structure below the Fermi level, E_{F}. The bulk and surface BZs are displayed in Fig. 2a with high symmetry points indicated. The degree of atomic orbital overlap in 1Tstructured TMDs is, in general, low, with the bonding (B) and antibonding (AB) chalcogen derived states energetically separated from the e_{g} and t_{2g} manifolds. For the group IV TMDs to which HfSe_{2} belongs, the transition metal is in the d^{0} configuration, with the Fermi level falling between the unoccupied t_{2g} manifold, and the ABSe pderived states^{9,39}. HfSe_{2} is thus an indirect gap semiconductor (E_{G} ≈ 1.1 eV^{40}), with the Sederived valence band maximum (VBM) and Hfderived conduction band minimum (CBM) located at the Γ and M points of the bulk BZ, respectively.
Figure 2b displays a band dispersion along the k_{z} axis (along the AΓA line), obtained by varying the photon energy of the incident light source (see Methods). There are three dispersive bands visible within the energy range shown, all of which are of Se p character^{39}. At E − E_{F} ≈ −1.2 and −1.6 eV, a pair of lessdispersive bands are visible. We assign these to be the pair of spinorbit split Se p_{x,y}derived bands. Their limited dispersion along k_{z} originates from the disparity in the hopping strengths of p_{x} and p_{y} orbitals in the xy plane compared to that across the van der Waals gap. It is this quasi twodimensionality that is thought to enable a hidden spin polarisation of these bands, with their wavefunctions being sufficiently localised to one half of an XMX unit. Also visible in Fig. 2b is a strongly dispersing band spanning from the valence band maximum at E − E_{F} ≈ −1.1 eV down to ≈ −3.8 eV. This band crosses through the p_{x,y}derived states in the vicinity of the VBM. We attribute this band to be of Se p_{z} derivation, with enhanced hopping along the caxis deriving from the extended spatial extent of p_{z} orbitals into the van der Waals gap. The remaining twodimensional band visible at E − E_{F} ≈ −4.1 eV is not of relevance here.
As a point of interest, we note that this crossing pattern between threedimensional chalcogen p_{z}derived bands and quasi twodimensional chalcogen p_{x,y} bands along this rotationally C_{3v}symmetric ΓA line is known to generically generate highly tunable socalled ‘topological ladders’ across the TMD family^{20,41,42}. In short, due to the influence of trigonal crystal field on the chalcogen porbital manifold, the crossing between each p_{z}derived band with each pair of p_{x,y}derived bands produces a single symmetryprotected bulk crossing point (or socalled bulk Dirac point), and a hybridisation gap (HG) which is often topologically nontrivial^{43}. In Fig. 2c, we enhance the region of the band dispersion where the crossing pattern occurs and overlay orbitallyprojected DFT calculations. Each band is labelled with the corresponding space group representation. A HG between the p_{z}derived band (labelled R_{4}) and the lower energy p_{x,y} band (\({R}_{4^{\prime} }\)) is clear, although there is little evidence of a twodimensional state occupying the gap. This suggests a topologically trivial classification for the HG. It follows that a crossing between the p_{z}derived R_{4} and the topmost p_{x,y}derived state (R_{5,6}) would generate bulk Dirac points symmetrically located about Γ. However, due to a suppression in the photoemission matrix element of the R_{5,6} band in the vicinity of the Γ point, we cannot rule out that it avoids R_{4} entirely. In other words, the presence of bulk Dirac points in this compound is dependent on whether or not the valence band top is of p_{z} or p_{x,y} character. Our DFT calculations, which account for electron correlation effects (see Methods), predict that this crossing point does occur, with a 20 meV gap separating the p_{z}derived VBM and the p_{x,y}derived band below at the Γ point. The valence band top of HfSe_{2} can therefore be described as a pair of closelyspaced typeII bulk Dirac cones of the same origin as those widely studied in the (semi)metallic group X TMDs, recently found to have applications in THz optoelectronics^{44}.
In Figure 2d–g, we turn to the inplane band dispersions of these Se pderived states along the \(\overline{{{{{{{{\rm{K}}}}}}}}}\)\(\overline{{{\Gamma }}}\)\(\overline{{{{{{{{\rm{K}}}}}}}}}\) direction of the surface BZ for various photon energies corresponding to k_{z} planes between A and Γ. The significant inplane dispersion of the relatively sharp p_{x,y}derived bands is clear, sitting against a background of diffuse spectral weight forming umbrellalike dispersions down to approximately E − E_{F} = −3.8eV in each case. While a chosen photon energy will selectively excite a specific k_{z} plane, due to the surface sensitivity of our photoemission experiments, the envelope of k_{z} integration can span over several BZs^{45,46}. The severity of this effect correlates to the extent of the band dispersion along k_{z}. The k_{z}broadened spectral weight in Fig. 2d–g is therefore predominantly of p_{z} character. In Supplementary Fig. S1 (See Supplementary Information), we further explore the symmetry of the pair of p_{x,y}derived bands as they disperse inplane as function of photon energy. While these bands are very twodimensional relative to the p_{z}derived band, there is a periodic evolution in how these bands disperse away from the \(\overline{{{\Gamma }}}\) point as a function of k_{z}, providing further confirmation that these are indeed bulk states.
Apparent timereversal symmetry breaking in HfSe_{2}
Next, we turn to the spinpolarisation of the bands discussed in the previous section. There are several possible sources of spinpolarisation here. The 1Tstructured TMDs are known to have several surface states both topologically trivial and nontrivial with complex band dispersions due to the influence of the dense k_{z}projected bulk band manifold^{41}. Any such states are permitted to be spinpolarised due to the inversion asymmetry offered by the surface potential step. These should be Rashba split with a predominantly inplane momentumlocked chiral spin texture. We note that there is little evidence of surface states in the data presented in Fig. 2, and so this origin is unlikely. The other source is a spinpolarisation of bulk bands caused by the hiddenspin mechanism discussed in Fig. 1. Although unobserved to date, this too should produce a predominantly inplane chiral spin texture, and is expected to be most prominent in the p_{x,y}derived valence bands which are more likely to have sufficiently localised wavefunctions, as evidenced by their relative twodimensionality (Fig. 2). Finally, we note that the superposition of incoming and reflected wavefunctions from the surface potential step in elemental metals has been demonstrated to result in observable Rashbatype spin polarisations of bulk bands when probed by ARPES, with an analogous mechanism possibly applicable to other systems^{47,48}. We stress that measured spinpolarisations deriving from any of these origins should be strictly timereversal symmetric.
In Fig. 3a, spinresolved energy distribution curves (spinEDCs) are shown on each side of the timereversal invariant momentum \(\overline{{{\Gamma }}}\) along the \(\overline{{{{{{{{\rm{M}}}}}}}}}\)\(\overline{{{\Gamma }}}\)\(\overline{{{{{{{{\rm{M}}}}}}}}}\) (k_{y}) direction, alongside a spinintegrated band dispersion taken with 29 eV photons corresponding to a k_{z} plane where the p_{z} band is at the shallow binding energy turning point (Fig. 2b). Accounting only for the exponential attenuation of the emitted photoelectrons, for this photon energy we estimate that 45% and 64% of the photoemission intensity will originate from the top sublayer of the unit cell, and from sublayers equivalent to the top sublayer, respectively^{49}. This is sufficiently sensitive to probe any hidden spin polarisation, even when assuming purely destructive interference between the two structure types. We note, however, that the interference pattern is likely far more complex^{27}. There are five bands labelled in the central ARPES image. The states B1 and B2 together form the valence band top of HfSe_{2} for this photon energy. B1 is the p_{z}derived band, originating from the shallowenergy turning point along k_{z}, which is less dispersive inplane. B2 is the shallower binding energy p_{x,y}derived state from previous discussions, forming a narrower parabola in the vicinity of the VBM. B4, also very dispersive in plane, is the second p_{x,y}derived state from previous discussions. The high binding energy apex of the p_{z}character, k_{z}broadened spectral weight (originating from the turning point of the p_{z}derived band in k_{z} near the A point) is labelled as B5. A final band, B3, visible only at higher ∣k_{y}∣ values, appears to form a pair with either B1 or B2, separating only away from \(\overline{{{\Gamma }}}\). This description is consistent with a splitting of both p_{x,y}derived states shown in earlier DFT studies^{39}, but we note that there are other possible origins: A localised turning point or plateau in the k_{z} dispersion of B1 or B2 could give the impression of a separate, distinct band in k_{z}broadened spectra. Although less likely, we also note that a singlebranch state that disperses exactly through the k_{z}projected typeII Dirac nodes was observed by ARPES in both PdTe_{2} and PtSe_{2}, and was not replicated by the accompanying band structure calculations^{20,41}. Although not well understood, one may expect a similar state here given the common origin of the Dirac nodes.
Turning attention to the spinEDCs, it becomes clear that all five of these states are spinpolarised, with signal in each of the measured inplane chiral (x) and outofplane (z) channels. B1, B2 and B4 have an opposite, predominantly inplane, spinpolarisation to B3. The diffuse p_{z} weight from the higher binding energy turning point in k_{z}, B5, carries a pronounced polarisation, opposite to that of the shallower energy turning point, B1. This suggests that the bulk p_{z} band carries a spinpolarisation switching from Γ to A. This final observation is particularly surprising, suggesting that p_{z} orbitals can also carry a hidden spintexture despite their higher caxis delocalisation relative to the p_{x,y} orbitals. It is likely, therefore, that proximity to the more twodimensional states energetically positioned near the extrema of the p_{z}band k_{z} dispersion^{39} produces a hybridised state of mixed character, with a wavefunction sufficiently localised to carry a hiddenspin.
A second set of spinEDCs is displayed for a k_{y} point on the other side of \(\overline{{{\Gamma }}}\). Astonishingly, the direction of the spinpolarisation of all states remains unchanged when transitioning from −k_{y} to +k_{y}, strongly suggesting a TRSbreaking spin texture for each state. To further investigate the apparent TRS breaking, Fig. 3b–d show spinEDCs taken at the \(\overline{{{\Gamma }}}\) point (k_{x} = k_{y} = 0) for a series of photon energies covering approximately one ΓA line in k_{z}. As both A and Γ are TRIM points, all bands along the entire AΓA line should be entirely spin degenerate in the absence of magnetism. Figure 3b, c show measured spinpolarisations in the z and x channels respectively as a function of photon energy, with the spinintegrated ARPES image overlaid as an intensity filter (note the twodimensional colour bar). The spinEDCs in Fig. 3d are select EDCs from these same datasets along with higher energy EDCs taken with hν = 29 and 39 eV. Despite conventional wisdom stipulating that there should be strictly no spinpolarisation at \(\overline{{{\Gamma }}}\) for any choice of k_{z}, these data unambiguously demonstrate that the bands discussed previously retain a strong nonzero spinpolarisation both at the \(\overline{{{\Gamma }}}\) point and for all k_{z}. We believe this is the first such observation of \(\overline{{{\Gamma }}}\) point spinpolarisation in a nonmagnetic system. We note that the persistence of spin polarisations that remain largely unchanged for a large range of incident photon energies precludes a final state origin^{50}.
While these observations are seemingly at odds with the fundamental symmetries possessed by this compound, it is possible to reconcile the experimentallyobtained spin texture with a hiddenspin mechanism that is entirely respectful of TRS. The solution has two distinct parts, requiring both the consideration of individual porbital contributions to the overall spin texture, and the role played by orbitalselective photoemission matrix elements.
Local spinorbital magnetisation in HfSe_{2} subcells
In attempt to uncover the origin of the apparent TRSbreaking shown in the previous section, in Fig. 4a, we display the spinresolved valence electronic structure for HfSe_{2}, as determined by DFT (See Methods) for HfSe_{2} along the MΓM direction. The chiral component of spin, parallel to k_{y} here, is projected onto the top (1) and bottom (2) Se lattice sites, verifying the presence of a hidden spin mechanism in this compound.
The electronic structure of the valence bands itself arises from the interplay of the \(C_{3v}\) crystal field and spinorbit interaction (SOI). The former acts on both B and AB groups individually, splitting each into two submanifolds \(\{p_x,p_y\}\) and \(p_z\). Mixed by SOI, they further reduce to three doubly degenerate \(\bigJ,\pm {m}_{j}\big\rangle\) branches with J = 3/2 and 1/2, and 1/2 ≤ m_{j} ≤ J. Of highest relevance here is the AB group, composed of the three shallowestbinding energy bands in Fig. 4a. The Se sublayerlocalised chiral spin textures demonstrate the symmetryimposed interplay between the underlying lattice with the spin and orbital degrees of freedom of the occupying electrons. The two sites have opposite chiralities to respect the overall TRS and global IS of the system, consistent with the simplified Rashbamodel discussed in Fig. 1d.
Inline with previous predictions^{37}, these calculations show, therefore, that a sufficiently sensitive probe of the electronic structure would indeed measure a nonzero spinpolarisation of the nearE_{F} Se pderived bands, exactly switching from Se1 to Se2, but clearly do not predict symmetric spinpolarisations of any state at and around \(\overline{{{\Gamma }}}\) like those observed experimentally in Fig. 3. The behaviour of the p_{z}derived state is well reproduced, however, exhibiting a finite spinpolarisation despite the expectation of a more delocalised wavefunction along the caxis relative to the p_{x,y}derived bands. This is further explored in Fig. 4b, where the k_{z}projected band dispersion shows how a hidden spinpolarisation of all bands is present for all k_{z}, with a reversing polarisation of the p_{z}derived between the turning points of its k_{z}dispersion, and with an enhanced polarisation magnitude relative to the intermediate k_{z} planes. This behaviour highlights the significant role played by spinorbital mixing within the Se porbital manifold.
Now let us consider the distinct individual porbital contributions to the hidden spinpolarisation shown in Figs. 3 and 4. This distinction is significant for two reasons. Firstly, the driving force behind the hidden spin mechanism in the 1T class of TMDs is the caxis aligned net electric fields within each C_{3v}symmetric HfSe_{6} octahedron. The orthogonal Se p_{x,y,z} orbitals within each Se sublayer experience the trigonal field disparately, and so this asymmetry should be reflected by the individual porbital contributions to the spin texture of the locally Rashbasplit pderived bands. Inequivalent orbital contributions to a global spin texture, or ‘coupled spinorbital textures’, have been previously discussed in the context of the ‘giant’ Rashbasplit semiconductor BiTeI and the topological insulator Bi_{2}Se_{3}^{51,52,53,54}. Secondly, while the total porbital contribution to the \(\overline{{{\Gamma }}}\) point would still be expected to sum to zero in accordance with TRS, orbitals are not excited equally in the photoexcitation process. Indeed, if one considers any single band in Fig. 4 without its energydegenerate partner, a spinorbital magnetisation is uncovered, with the p_{x} and p_{y} contributions to the spin polarisation at the \(\overline{{{\Gamma }}}\) point finite and exactly opposite. An orbitalselective probe of electronic structure, applicable to the present experiment as explained in the next section, could then selectively probe the spintexture partially and thus detect a net spin polarisation at \(\overline{{{\Gamma }}}\). However, the energy degenerate partner behaves exactly oppositely, negating this effect.
Instead, motivated by the substantial contribution of the p_{z}derived band in both Figs. 3 and 4, we further explore the role of spinorbital mixing terms between constituent porbitals. To facilitate this, we form a 12band tightbinding model from the \(\bigJ,{m}_{j}\big\rangle\) states, \({\psi }_{v}={\sum }_{J,{m}_{j}}{\alpha }_{J,{m}_{J}}\bigJ,{m}_{j}\big\rangle\). This framework, detailed in full in Supplementary Note 2 (See Supplementary Information), enables the computation of individual Se porbital contributions to the three component i = {x, y, z} spinpolarisation localised at each Se site, a, using S_{i,a} = 〈ψ_{v,a}∣σ_{i}∣ψ_{v,a}〉, where σ_{i} are Pauli spin matrices.
Restricting the problem to the vicinity of a single Se layer, we find four overlap integrals \(\langle J,{m}_{J} {\sigma }_{i} J^{\prime} ,{m}_{J^{\prime} }\rangle\) with nonvanishing values at the Γ point. These terms obey \({{\Delta }}J=JJ^{\prime} =0\) or 1 and \({{\Delta }}{m}_{J}={m}_{J}{m}_{J^{\prime} }=2\). All the other overlap integrals vanish at the Γ point due to TRS (see Supplementary Note 2 in Supplementary Information for a full derivation). The resulting spin polarisation from these four terms undergoes a sign reversal when switching between Se sites, ensuring full compliance to TRS and IS. A careful inspection of the above ΔJ and Δm_{J} constraints reveals that only the spinorbital terms mixing {p_{x}, p_{y}} and p_{z} submanifolds can contribute to such a nonvanishing spin polarisation at a timereversal invariant momentum. This finding further signifies the critical role of p_{z} orbitals in the electronic and spin structures of the low energy bands in HfSe_{2}. It should be noted that all overlap integrals can hold finite contributions away from the Γ point, altogether producing the chiral spin textures shown in Fig. 4.
Figure 5a, b show the spinprojected band structure stemming from these ΔJ = 0 and 1 terms, denoted as \({S}_{y}^{1}\) and \({S}_{y}^{2}\), respectively, for the chiral (σ_{y}) component at Se1. These exhibit symmetric spinpolarisations about k = 0 as for the experimentally determined spin texture. This suggests that orbital mixing of inplane and outofplane orbitals is the driving force behind the the experimentallyobserved symmetric spinpolarisations. It is noteworthy that these two inequivalent contributors are not opposite, with their sum and k_{z}projection shown in Fig. 5c, d. This demonstrates that a small net spinpolarised contribution to the BZ centre detectable even to a probe of electronic structure blind to orbital type. The two nonzero contributions to the spintexture can be tuned between, however, and therefore enhanced, with a linearly polarised light source. This provides a convenient way to further verify this description, and explore routes towards the tunability of this effect for applications.
Disentangling local spinorbital textures at \(\overline{{{\Gamma }}}\)
To experimentally confirm that the hidden spinorbital textures, driven by the spinorbit mixing between p_{z} and inplane p_{x,y} orbitals, are indeed the source of the observed spinpolarisation, we can tune the orbital selectivity of the experimental probe and test that the measured spin polarisation responds accordingly. The photoemission matrix element for a transition between an initial (i) and final (f) is given as \( {M}_{f,i}^{k}{ }^{2}\propto  \langle {\phi }_{f}^{k} \overrightarrow{A}\cdot \overrightarrow{p} {\phi }_{i}^{k}\rangle { }^{2}\), where \({\phi }_{i,f}^{k}\) are initial and final state wavefunctions for an electronic band at k, and \(\overrightarrow{A}\) and \(\overrightarrow{p}\) are the vector potential and the momentum operator, respectively^{45}. This integral is required to be of overall even parity to be nonzero. Assuming an even parity final state, this integral is nonzero for initial state p_{x} and p_{z} orbitals with a ppolarised light source. Switching to spolarised light would probe p_{y} orbitals in isolation. Therefore for either choice of linear light polarisation, the ground state spinorbital texture is only partially probed, aiding the observation of apparent timereversal symmetry breaking spin textures.
In Fig. 6, we continuously rotate our sample in the xy plane while keeping a fixed ppolarised light source. A rotation of π/2 switches the parity of the photoemisison matrix elements for p_{x} and p_{y} orbitals, while leaving that for p_{z} orbitals unchanged. By performing a series of spinEDCs at \(\overline{{{\Gamma }}}\) as a function of the inplane sample azimuthal angle θ with a spindetector capable of probing the spinpolarisation in all threedirections, it is, therefore, possible to evaluate the spinpolarisation contribution from p_{x}, p_{y} and p_{z} orbitals in full, here allowing for the tuning the relative weight of the contributions from \({S}_{y}^{1}\) and \({S}_{y}^{2}\) overlap integrals due to their inequivalent inplane orbital compositions.
Figure 6 provides an overview of the \(\overline{{{\Gamma }}}\) point spin polarisation as a function of θ for 17 eV photons, where the spin polarisation is particularly clear, as also seen in Fig. 3d. Figure 6a shows the valence band structure along a \(\overline{{{{{{{{\rm{K}}}}}}}}}\overline{{{\Gamma }}}\overline{{{{{{{{\rm{M}}}}}}}}}\) path. There are two bands visible at \(\overline{{{\Gamma }}}\) with energies E − E_{F} = −1.3 and −1.9 eV. By crossreferencing to the spectra in Fig. 3, it can be determined that, at \(\overline{{{\Gamma }}}\), the top band labelled v_{1} is an amalgamation of up to three (doublydegenerate) bands, previously labelled as B1, B2 and B3. The lower band, labelled v_{2} is likely predominantly made up of band B4 discussed previously. Figure 6b show a series of threecomponent spinEDCs measured at 30 degree increments in θ. For clarity, we fix the k_{x} and k_{y} plane to be parallel and perpendicular to the initial \(\overline{{{{{{{{\rm{M}}}}}}}}}\overline{{{\Gamma }}}\overline{{{{{{{{\rm{M}}}}}}}}}\) orientation at θ = 0, respectively. The spin components S_{x}, S_{y} and S_{z} are the measured spin signals in this fixed laboratory reference frame.
A subset of the resulting series of spinEDCs are shown in Fig. 6b–f. There is clear periodicity to the measured spin polarisation in each channel. The \(S_x\) projection for \(\nu_1\) is dominant for all angles, and the sign of the \(S_x\) component is independent of \(\theta\) for both \(\nu_1\) and \(\nu_2\). This is not the case for either the S_{y} or S_{z} channels, with the net polarisation within v_{1} switching sign periodically in both channels. For v_{2}, the S_{z} spin polarisation magnitude is strongly θdependent, and has a reversing signal in S_{y}. These qualitative observations already show that photoemission matrix elements strongly influence the measured spinpolarisation at \(\overline{{{\Gamma }}}\), independently suggesting a spinorbital locking of the measured spinsignal^{55}.
To better visualise the changes imposed by the sample rotation, we extract the spinpolarisation from the peaks corresponding to v_{1} and v_{2} as a function of θ (see Methods). The result of this analysis is displayed in Fig. 6g. The overlaid solid lines are periodic functions that satisfy a simplified model that assumes that TRS is intact (see Supplementary Note 3 in Supplementary Information). Firstly, it assumes that the p_{x}, p_{y} and p_{z} contributions to the \(\overline{{{\Gamma }}}\)point spin polarisation exactly cancel out, and therefore that the observation of nonzero spin signals is only possible with an incomplete probing of the full set of porbitals, although as shown in the previous section, there is a route towards finite spinpolarisation even with an unpolarised light source due to a finite \({S}_{y}^{1}+{S}_{y}^{2}\) sum. The second constraint is more robust, however, restricting the periodicity of the measured spinpolarisation functions to be strictly of π/n, where \(n\in {\mathbb{Z}}\). Satisfying this second constraint entirely removes the possibility of spin vectors of a magnetic origin, as the photoemission orbitalselective matrix elements in 0 and π rotated geometries are equivalent, and so otherwise equivalent experiments should yield equivalent results. A real (inplane) magnetic spin vector would always switch sign with a π rotation.
All functions do satisify this constraint for both the v_{1} and v_{2} peaks, with the S_{x} and S_{y} signals sharing the same periodicity. For the higher (lower) binding energy state, v_{2} (v_{1}), the spinpolarisations in the x and y channels are well fit to a pair of functions that are π/2 (π) periodic. The measured S_{z} polarisation is π periodic for both v_{1} and v_{2}. We note that the measured spinpolarisation functions only satisfy the simplified TRSconserving model if there is a p_{z} contribution to the spinpolarisation. Due to the effective doublecounting of the p_{z} contribution as the sample is rotated, the spinpolarisation functions in the S_{x} and S_{y} channels do not have to be exactly opposite, instead summing to a periodic function with limits defined by the p_{z} contribution to the inplane channels. This observation is entirely consistent with the crucial role played by p_{z} orbitals uncovered in the previous section. The measured spinpolarisation at \(\overline{{{\Gamma }}}\) is thus consistent with a timereversal symmetric spinorbital texture driven by the mixing of p_{x,y} and p_{z} orbitals in the ground state.
Taken altogether, we conclude that the presence of a nonzero spinpolarisation at the BZ centre of HfSe_{2} is a direct consequence of a coupled hidden spinorbital texture. The adheration of the measured spinpolarisations to a model in which timereversal symmetry is not broken again confirms that the hidden coupled spinorbital texture observed here is a ground state property of bulk HfSe_{2}. Note that an exactly equivalent conclusion would be drawn when using s − polarised light, but with measured spin polarisation enitrely composed of the missing half of the p_{x,y} contribution.
Application for transient spincurrent generation
To finalise, we briefly discuss potential for this type of Γcentred orbitalselectivity in the context of applications in spinpolarised current generation. As demonstrated in Figs. 5 and 6 and discussed in the previous section, the sign of the spinpolarised signal from the valence band maximum of HfSe_{2} is dependent on the subset of the porbital contributions probed, thus opening opportunities to populate the unoccupied conduction bands with electrons belonging to a chosen spinspecies, with linearly polarised, fixed energy light sources. While it is important to ensure that the mean free path of photoexcited elections remains sufficiently small to be sensitive to the hiddenspin derived spinpolarisation, there is some scope to tune final states with the choice of photon energy.
These spinselective excitations are different to those possible in the monolayer 1HTMDs, where the transient spin polarisation of the conduction band minimum following excitation is dependent both on the helicity and the energy of a circularlypolarised light source^{7,25}. Moreover, any rotational domains in the 1H and 2HTMDs could randomize the spin signal from the oppositely polarised K and K’ points, requiring highquality single crystals with a known orientation. For HfSe_{2}, we demonstrated in Fig. 6 that changing the orientation of the crystal does not switch the sign of the dominant spin channel, greatly reducing any hindrance from the presence of multiple rotational domains. As a result, polycrystalline samples may also yield similar net polarisations to those seen here when using linear polarised light sources, potentially reducing barriers to commercialisation.
We expect that the new physics observed here is general to the other semiconducting 1T structured TMD without a significant transition metal dderived valence band contribution, although likely less pronounced than the present case due to a reduced dipolar field strength within the ITunit. For the (semi)metallic systems (e.g., (Ir, Ni, Pt, Pd)(Se,Te)_{2}) it is likely that screening effects suppress the effective crystal field splitting within chalcogen sites key to the phenomena observed here.
In conclusion, we have shown that the \(\overline{{{\Gamma }}}\)centred, selenium porbital derived valence bands of 1THfSe_{2} exhibit hidden ground state spinpolarisation owing to the strong outofplane dipole present within each half SeHfSe sublayer of the unit cell. This hiddenspin polarisation manifests as a coupled spinorbital texture wherein p_{x}, p_{y} and p_{z} orbitals contribute differently to the Rashbasplit states, and therefore a net spinpolarisation at the \(\overline{{{\Gamma }}}\) point can be measured and enhanced when selectively probing a subset of these orbitals, despite the presence of timereversal symmetry. These findings offer a new route towards the generation of spinpolarised carriers from polycrystalline 1Tstructured transition metal dichalcogenides by using a fixed energy, linearly polarised light source, revealing possibilities to exploit effective net spin polarisations in nonmagnetic systems more widely.
Methods
Experimental
ARPES measurements were carried out using ppolarised synchrotron light of energies between 17 and 64 eV at the U125PGM beamline of BESSYII in HelmholtzZentrum Berlin. Photoelectrons were detected with a Scienta R4000 analyzer at the SpinARPES endstation, and the base pressure of the setup was better than ~1 × 10^{−10} mbar. Highquality HfSe_{2} single crystals were cleaved in situ using a standard top post method at temperatures of 40–50 K. The data presented are from three samples, all of which showed qualitatively the same behaviour, including in the absolute directions of spinpolarisation. The spin data in Figs. 3a–d and 6 originate from samples 1, 2 and 3, respectively. No sample was magnetised.
To determine the HfSe_{2 }k_{z} dispersion from photonenergydependent ARPES, we employed a free electron final state model
where θ is the inplane emission angle and V_{0} is the inner potential. We find best agreement to the periodicity of the dataset when taking an inner potential of 10.5 eV for a caxis lattice constant of 6.160 Å.
Spin resolution was achieved using a Motttype spin polarimeter operated at 25 kV and capable of detecting all three components of the spin polarisation. Spinresolved energy distribution curves (spinEDCs) were determined according to
where \(i=\{x,y,z\},\,{I}_{i}^{{{{{{{{\rm{tot}}}}}}}}}=({I}_{i}^{+}+{I}_{i}^{})\) and \({I}_{i}^{\pm }\) is the measured intensity for the oppositely deflected electrons for a given channel, corrected by a relative efficiency calibration. The final spin polarisation is defined as follows
where S is the Sherman function.
For the quantitative spinpolarisation magnitudes in Fig. 6, we fit the two peaks corresponding to v_{1} and v_{2} to Lorentzian functions with a Shirley background and a Gaussian broadening to account for the finite experimental resolution. For each pair of EDCs, the peak widths and positions are held constant with only the intensity allowed to vary between the spin up and down channels. The spinpolarisation is then calculated from the relative areas of the Lorentzian peaks. The net polarisations from the two collections of bands, v_{1} and v_{2}, are therefore extracted.
Theoretical
The electronic structure of HfSe_{2} was calculated within DFT using the Perdew–Burke–Ernzerhof exchangecorrelation functional^{56}, modified by BeckeJohnson potential^{57} as implemented in the WIEN2K programme^{58}. Relativistic effects, including spinorbit coupling, were fully included. The muffintin radius of each atom, R_{MT}, was chosen such that its product with the maximum modulus of reciprocal vectors \({K}_{\max }\) become \({R}_{{{{{{{{\rm{MT}}}}}}}}}{K}_{\max }=7.0\). The Brillouin zone was sampled by a 15 × 15 × 10kmesh. To calculate the spinprojected band structures, we downfolded the DFT Hamiltonian into a 22band tightbinding model using maximally localised Wannier functions^{59} with Hf5d and Se4p as the projection centres.
Data availability
Data underpinning the figures that support this work are available from the authors upon reasonable request.
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Acknowledgements
The authors thank Lewis Bawden for useful discussions. O.J.C. and J.S.B. acknowledge financial support from the Impuls und Vernetzungsfonds der HelmholtzGemeinschaft under grant No. HRSF0067. O.D. and M.S.B. gratefully acknowledge the Research Infrastructures at the University of Manchester for allocations on the CSF3 high performance computing facilities.
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O.J.C. performed the experiment, analysed the experimental data, developed the model describing the periodicity of the measured spinpolarisation functions and wrote the manuscript with input from all coauthors. J.S.B. aided with data collection and interpretation, and is responsible for the U125PGM beamline and its spinARPES endstation. Following the initial Reviewer comments, O.D. and M.S.B. performed density functional theory calculations and developed the tightbinding model discussed in the Supplementary Information. O.J.C. was responsible for the overall project planning and direction.
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Clark, O.J., Dowinton, O., Bahramy, M.S. et al. Hidden spinorbital texture at the \(\overline{{{\Gamma }}}\)located valence band maximum of a transition metal dichalcogenide semiconductor. Nat Commun 13, 4147 (2022). https://doi.org/10.1038/s41467022315392
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DOI: https://doi.org/10.1038/s41467022315392
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