Abstract
Engineering and enhancing the breaking of inversion symmetry in solids—that is, allowing electrons to differentiate between ‘up’ and ‘down’—is a key goal in condensedmatter physics and materials science because it can be used to stabilize states that are of fundamental interest and also have potential practical applications. Examples include improved ferroelectrics for memory devices and materials that host Majorana zero modes for quantum computing^{1,2}. Although inversion symmetry is naturally broken in several crystalline environments, such as at surfaces and interfaces, maximizing the influence of this effect on the electronic states of interest remains a challenge. Here we present a mechanism for realizing a much larger coupling of inversionsymmetry breaking to itinerant surface electrons than is typically achieved. The key element is a pronounced asymmetry of surface hopping energies—that is, a kineticenergycoupled inversionsymmetry breaking, the energy scale of which is a substantial fraction of the bandwidth. Using spin and angleresolved photoemission spectroscopy, we demonstrate that such a strong inversionsymmetry breaking, when combined with spin–orbit interactions, can mediate Rashbalike^{3,4} spin splittings that are much larger than would typically be expected. The energy scale of the inversionsymmetry breaking that we achieve is so large that the spin splitting in the CoO_{2} and RhO_{2}derived surface states of delafossite oxides becomes controlled by the full atomic spin–orbit coupling of the 3d and 4d transition metals, resulting in some of the largest known Rashbalike^{3,4} spin splittings. The core structural building blocks that facilitate the bandwidthscaled inversionsymmetry breaking are common to numerous materials. Our findings therefore provide opportunities for creating spintextured states and suggest routes to interfacial control of inversionsymmetry breaking in designer heterostructures of oxides and other material classes.
Main
The natural breaking of inversion symmetry at surfaces and interfaces provides a way of stabilizing electronic structures that are distinct from those of the bulk^{4,5}. A notable example is found in materials that also have strong spin–orbit interactions, in which inversionsymmetry breaking (ISB) underpins Rashba^{3,4} spin splitting of surface or interfacelocalized twodimensional electron gases^{6,7,8,9,10,11}, which are generically characterized by a locking of the quasiparticle spin perpendicular to its momentum. Such effects are central to various applications that have been proposed in spinbased electronics and provide routes to stabilizing new physical regimes^{4,12,13,14,15}. Conventional wisdom about the way in which to maximize the Rashba effect has been to work with heavy elements with large atomic spin–orbit coupling (SOC). However, the energetic spin splittings that are obtained are usually only a small fraction of the atomic spin–orbit energy scale. This is because the key physics is not exclusively that of SOC, but rather an interplay between the spin–orbit (E_{SOC}) and inversionbreaking (E_{ISB}) energy scales.
Evidencing the importance of this interplay, the size of the spin splitting can often be controlled by tuning the strength of the ISB potential, for example, via applied electrostatic gate voltages^{7}. This sensitivity to E_{ISB} indicates a regime in which the achievable spin splitting is limited by E_{ISB}, typically to only a modest percentage of the atomic SOC (Fig. 1a)^{16,17,18}. If, however, an ISB energy scale can be achieved that is much larger than that of the SOC, then the spin–orbit Hamiltonian will act as a perturbation to a Hamiltonian dominated by the ISB. A fourfolddegenerate level will be split by ISB into two states of opposite orbital angular momentum (OAM). The weaker spin–orbit interaction cannot mix these states and instead further spinsplits them into states of spin that are parallel and antiparallel to the preexisting OAM, with a splitting that must therefore take the full atomic SOC value (Fig. 1b)^{16,17,18}. This limit is clearly desirable for maximizing the achievable spin splitting for a given strength of SOC, but is typically realized only when the absolute value of SOC, and consequently also the total spin splitting, is small.
Here we demonstrate a method for using the intrinsic energetics of hopping to realize large ISB energy scales at the surfaces of transitionmetalbased delafossite oxides (Pd,Pt)(Co,Rh)O_{2}. We show, from spin and angleresolved photoemission (ARPES), that this large energy scale leads to spinsplit Fermi surfaces with one of the largest momentum separations known, even for the unlikely environment of a 3delectronbased CoO_{2} metal, where atomic SOC is not strong. Crucially, we demonstrate that E_{ISB} scales with the bandwidth and so will grow concomitant with the SOC strength upon moving to systems of heavier elements. We show that this scaling enables the full atomic SOC to be retained in spinsplitting the states of a 4d RhO_{2} surface layer, leading to an unprecedented spin splitting for an oxide compound of approximately 150 meV, and providing new opportunities for tuning between giant spin splittings and strongly interacting Rashbalike states in oxides and other compounds.
We first consider the delafossite oxide^{19,20,21} PtCoO_{2}. Pt^{1+} cations sit on a triangular lattice, leading to extremely high bulk conductivity^{22}, and are separated by layers of edgesharing and trigonally distorted cobalt oxide octahedra (Fig. 2a). In the bulk, six 3d electrons fill the Co t_{2g} manifold, which makes the CoO_{2} block insulating. At the surface, however, the ‘missing’ Pt atoms from above the topmost oxygen layer cause the CoO_{2} block to have a formal valence of only 0.5^{−}, effectively holedoping the manifold of Coderived states^{23}. Consistent with this simple ionic picture, our ARPES data not only show sharp and rapidly dispersing spectral features that are representative of the bulk band structure^{22}, but also a pair of twodimensional hole bands that cross the Fermi level where none are expected from the calculated bulk electronic structure at any value of the outofplane momentum, k_{z} (Fig. 2b and Extended Data Fig. 1a).
The inner and outer bands form almost circular and hexagonally warped holelike Fermi surfaces located at the Brillouin zone centre (Fig. 3a). Their Luttinger count is approximately 0.4 holes per unit cell, very close to the 0.5 holes that would be expected from simple electron counting. This therefore seems like a remarkably clean example of an ‘electronic reconstruction’, a phenomenon that has been discussed extensively for polar transitionmetal oxide interfaces^{5}. The bands that we observe host heavy quasiparticles with masses of up to around 15m_{e}, where m_{e} is the freeelectron mass (see Extended Data Table 1 for a more detailed description of the quasiparticle masses and their anisotropy) and have clear spectroscopic signatures of electron–phonon and electron–electron interactions (Fig. 2b, c). This is in stark contrast to the almost freeelectron masses of the bulk Ptderived bands that cross the Fermi energy E_{F} (ref. 22), and suggests that these states are instead predominantly derived from much more local Co 3d orbitals. Consistent with this, we note that similar dispersions have previously been observed^{24} in the sister compound PdCoO_{2}. We therefore identify these bands as surface states of the CoO_{2}terminated surface of PtCoO_{2}, with very little intermixing of Pt; this is further confirmed by our density functional theory (DFT) calculations, described below.
Our ARPES measurements performed using circularly left and circularly rightpolarized light reveal a strong circular dichroism (CD) of these surface states (Fig. 2d and Extended Data Fig. 1). CD–ARPES is known to be sensitive to OAM structures in solids^{25}. Whereas OAM would naturally be expected to be quenched for the t_{2g} manifold, the observation of a pronounced circular dichroism across the entire bandwidth of the surface states here suggests that these instead host a large OAM. Moreover, from spinresolved ARPES measurements (Fig. 3c; see Methods) we find that the two surface states are strongly spinpolarized, with a large inplane spin component that points perpendicular to their momentum. This finding indicates chiral spin textures of the form that would be expected for a Rashbalike splitting^{3}. Whereas the spin chirality reverses sign between the inner and outer Fermi surface sheets, the circular dichroism of each spinsplit branch is of the same sign (Fig. 2d and Extended Data Fig. 1). This suggests that both branches carry the same sign of OAM, with spin splitting resulting from the parallel and antiparallel alignment of the spin to this OAM, in close agreement with the situation shown in Fig. 1b. Critically, this spin and OAM texture should lead to a spin splitting that is of the order of the full strength of atomic SOC.
Our DFT supercell calculations fully support the above analysis. They confirm that the surface states host dominantly chiral spin angular momentum (orbital angular momentum) textures, which are of the opposite (same) sign for the two surface Fermi surfaces (Fig. 3b, d; see also Extended Data Fig. 2). Moreover, the calculations reveal that these states host a spin splitting that remains large over the entire bandwidth of the states, except near timereversalinvariant momenta ( and ), at which timereversal symmetry dictates that it must vanish. Away from these points, the spin splitting reaches values as high as 60 meV at the average Fermi wavevector k_{F} along the direction. This is comparable to the atomic SOC strength of Co (ref. 26). A surface projection of our calculations (Fig. 3e) indicates that the wavefunctions of the surface states are almost entirely located in the topmost CoO_{2} block; they therefore exhibit a spin splitting that is indeed of the maximum strength possible for the relevant Co atomic orbitals.
Spin–orbit interactions are generally neglected for such 3dorbital systems. In contrast, Figs 1, 2, 3 demonstrate that these interactions can lead to a major restructuring of the electronic structure if a sufficiently strong ISB can be realized to unlock their full strength, opening up the potential for investigating the interplay between spin–orbit interactions and strong electronic correlations, the latter being more naturally associated with local 3d orbitals. For the CoO_{2}terminated surface of PtCoO_{2}, achieving the maximum possible energetic spin splitting as well as high quasiparticle masses leads to momentum spin splittings at the Fermi level (the relevant quantity for several technological applications) as high as Δk_{F} = 0.13 ± 0.01 Å^{−1} (see also Extended Data Table 1). This value is among the highest known for any Rashbalike system^{27}. It is approximately ten times larger than that of the enhanced Rashbalike splitting that is thought to occur at isolated momentum points where different t_{2g} bands intersect in the 3dorbital system of SrTiO_{3}based twodimensional electron gases^{10,28,29,30}, and is comparable in magnitude to the momentum splitting in the socalled giant Rashba semiconductor BiTeI (refs 27, 31), which has both very strong SOC and large internal electrostatic potential gradients. We note that our experimentally observed Fermi surfaces (Fig. 3a) are well reproduced by our DFT calculations, which consider an ideal bulktruncated supercell. This rules out subtle modulations in surface structure, such as those known to markedly enhance Rashba spin splitting in various surface alloy systems^{9,32}, as the origin of the large effects observed here. Instead, the agreement between theory and experiment identifies the spin splitting that we observe as an intrinsic property of the bulklike CoO_{2} layer, which is unlocked when it is placed in an environment in which inversion symmetry is broken by the presence of the surface.
We now show that the surprisingly large spin splitting occurs as a natural consequence of a strong asymmetry in effective Co–Co hopping paths through oxygen atoms located above and below the topmost Co layer. The structure of a single CoO_{2} layer is shown in Fig. 4a, b. This structure can be viewed as a trilayer unit, with a central triangular net of Co atoms, above and below which are two triangular oxygen sublattices that have opposite orientation with respect to Co. Given the local nature of 3d orbitals, an important effective Co–Co hopping will be via oxygen, either through the oxygen layer above (O1) or below (O2) the transitionmetal plane. In bulk, these effective hoppings must be equivalent owing to the inversion symmetry of the crystal structure. However, this requirement is lifted at the surface. To demonstrate this explicitly, we show in Fig. 4c the projection of the electronic structure from our DFT supercell calculations onto O1 and O2 of the surface CoO_{2} layer. It is clear that there is a much higher admixture of O1 than O2 in the CoO_{2}derived surface states that intersect the Fermi level identified above: we estimate that the O1 orbital contribution is approximately 2.4 times higher than that of O2 when averaged around the surface Fermi surfaces. This indicates that Co–O–Co hopping occurs dominantly via the oxygen network above, rather than below, the transitionmetal plane in the surface CoO_{2} layer.
The microscopic origin of the hopping being predominantly via O1 is evident from the layerprojected oxygen partial density of states (PDOS) shown in Fig. 4d. Although the PDOS of O2 is similar to those of subsequent layers of our supercell calculation, that of O1 is strongly shifted in energy, with the main centroid of spectral weight located at a binding energy that is approximately 4 eV lower than that of O2 (see also Extended Data Fig. 3a, b). In the bulk^{22}, and in the bulklike environment of O2, the bonding Pt–O combinations shift the predominantly oxygenderived levels to higher binding energy. An absence of this shift as a result of the missing Pt above O1 therefore causes the difference in onsite energy of the two oxygen atoms. In a tightbinding picture, Co–O–Co hopping via On (n = 1, 2) is described by the effective transfer integral, where t_{dp} is the Co 3d–O 2p hopping matrix element (Fig. 4a) and E_{Co} and E_{On} are the onsite energies of Co and On (Fig. 4e). The decrease in binding energy of the oxygenderived PDOS on site 1 leads to a strongly asymmetric hopping with t_{1} > t_{2} and thus directly to a large E_{ISB}.
To validate that this asymmetry drives the spin splitting observed here, we develop a minimal tightbinding model for a single CoO_{2} layer (see Methods and Extended Data Fig. 4). This model yields an estimate of the relative ISB that is introduced by the asymmetric hopping of α_{ISB} = (t_{1} − t_{2})/(t_{1} + t_{2}) > 40%. The asymmetry enters between two hopping paths, which are nominally equivalent in bulk and which define the entire bandwidth; the energy scale of the ISB is therefore large: E_{ISB} = α_{ISB}t ≈ 150 meV, where t is the average effective Co–O–Co hopping integral. This energy scale is approximately twice the atomic SOC and so places the CoO_{2} surface within the ‘strong ISB’ limit discussed above (Fig. 1b; see also Extended Data Fig. 5), explaining why the spin splitting is so large for the Coderived states that are observed here.
The critical feature of our analysis is that ISB need not be a weak perturbation to the Hamiltonian of the surface conduction electrons, but can enter through the kinetic part of that Hamiltonian directly. This differs substantially from previous treatments, which typically consider a dipole term arising from the surface electric field directly as the dominant symmetry breaking^{16}, allowing only a weak perturbation to a Hamiltonian governed by the kinetic energy^{30}. We illustrate the importance of structure in achieving this kineticenergycoupled symmetry breaking with a simple analysis, shown in Extended Data Fig. 6, of the difference between a twodimensional layer of edgesharing transitionmetal octahedra and the cornersharing octahedra of, for example, many cuprates and other transitionmetal oxides^{29,30}. The edgesharing geometry of the delafossites yields an ISB that is not only strong, but is also scaled by the conduction electron bandwidth. This result suggests strategies for maximizing spinsplitting in solids that are counter to conventional wisdom: instead of just increasing the SOC, which would normally lead to the achievable spin splitting becoming limited by E_{ISB} (Fig. 1a and Extended Data Fig. 5), the ‘bandwidthscaled’ E_{ISB} will simultaneously grow with increased orbital overlap of heavyelement systems, enabling the maximum possible spin splittings to be retained even in systems with much stronger SOC.
To verify the potential of this new approach, it is desirable to perform two further experiments. The first is to compare the surface states of PtCoO_{2} with those of PdCoO_{2} in order to rule out an influence of the strong SOC of the noblemetal cation in mediating our observed spin splittings. The second is to change the transition metal from Co to Rh, to test whether the spin splitting really scales with the factor of approximately 2.5 increase of transitionmetal SOC. The former is easy to do, because highquality PdCoO_{2} crystals are straightforward to synthesize^{20}. However, PdRhO_{2}, although known to be metallic in polycrystalline form^{19}, has not to our knowledge previously been synthesized in singlecrystal form. We did so for this project.
As shown in Fig. 5a, c and Extended Data Table 1, the surfacestate spin splitting of PdCoO_{2} is in good quantitative agreement with that of PtCoO_{2}. This confirms that the spin splitting that we observe is a property of the CoO_{2} block, with only minimal influence from the Asite (Pd,Pt) cation. We can clearly resolve the spinsplit bands across their bandwidth, extracting a spin splitting of 60 meV at the point for both compounds (Fig. 5b, d). As shown in our tightbinding model (Extended Data Fig. 4a, b), a fourfolddegenerate band crossing would be expected at this momentum point in the absence of ISB and SOC. It is therefore an ideal location to probe the energy scale of the spin splitting.
Our measurements of PdRhO_{2} (Fig. 5e) reveal qualitatively similar surface states. However, the momentum splitting at the Fermi level is higher, reaching Δk_{F} = 0.16 ± 0.01 Å^{−1} along , despite its lower quasiparticle masses (see also Extended Data Table 1). This is a result of a strongly enhanced energetic splitting, which we find to be 150 meV at the point (Fig. 5f). To the best of our knowledge, this is by far the largest spin splitting to be observed in an oxide so far. More importantly, it is of the order of the full atomic SOC strength of Rh^{3+} (ref. 26). The bandwidth scaling of E_{ISB} allows PdRhO_{2} to remain in a limit in which E_{ISB} > E_{SOC} even as the latter is markedly increased. This limit is further evidenced by our CD–ARPES measurements (Extended Data Fig. 1e, f), which demonstrate that both of the spinsplit branches retain the same sign of OAM, as in (Pt,Pd)CoO_{2}.
This key result—that the symmetry breaking that enters into the kinetic part of the Hamiltonian directly can lead to a bandwidthscaled E_{ISB}—is potentially realizable in any system in which hopping between the frontier orbitals is mediated by atoms above and below their plane, including the large and varied family of delafossites^{21} as well as, for example, (111)oriented surfaces and interfaces of perovskites. Immediate quantitative insight into the levels of ISB that are expected could in principle be obtained from DFT calculations, providing a reliable predictive capability of the ‘materials by design’ approach that is currently generating large interest from specialists in firstprinciples electronic structure calculations. Moreover, suitable mismatched interfaces could be fabricated by highly controlled thinfilm and multilayer techniques such as molecularbeam epitaxy, providing ways to access new physical regimes.
Methods
ARPES
Singlecrystal samples of the delafossite oxides^{19,20,21,22,33,34,35} PtCoO_{2}, PdCoO_{2} and PdRhO_{2} were grown by flux and vapour transport techniques in sealed quartz tubes. They were cleaved in situ at the measurement temperature of 6–10 K. (Pt,Pd) and (Co,Rh)O_{2}terminated surfaces would both be expected for a typical cleaved surface. Asiteterminated surface states have been reported previously^{36} for the sister compound PdCrO_{2}, and for some cleaves we observe states that we tentatively assign as derived from Pt/Pd terminations. However, in agreement with previous studies^{24}, our ARPES spectra often show spectroscopic signatures arising from only the bulk and the CoO_{2}terminated surface, which is the case for all of the data included here. ARPES measurements were performing using the I05 beamline of Diamond Light Source, UK. Measurements were performed using s, p and circularly polarized synchrotron light from 55 eV to 110 eV and using a Scienta R4000 hemispherical electron analyser. Spinresolved ARPES measurements were performed using ppolarized synchrotron light at the APE beamline of the Elettra synchrotron^{37}, using a Scienta DA30 hemispherical analyser equipped with a very lowenergy electron diffraction (VLEED)based spin polarimeter probing the spin polarization perpendicular to the analyser slit, 〈S_{y}〉. The finite spindetection efficiency was corrected using a Sherman function (S = 0.3), determined by comparison with the known spin polarization of the Rashbasplit surface states measured on the Au(111) surface. The spin polarization along momentum distribution curves at the Fermi level is extracted as
where I^{+} (I^{−}) is the VLEED channeltron intensity measured along momentum distribution curves for the target magnetization polarized in the positive (negative) direction, defined with respect to the DFT calculation.
Density functional theory
Relativistic DFT electronic structure calculations were performed using the fullpotential FPLO code^{38,39}, version fplo14.0047 (http://www.fplo.de). The exchange correlation potential was treated within the general gradient approximation (GGA), using the Perdew–Burke–Ernzerhof^{40} parameterization. SOC was treated nonperturbatively by solving the fourcomponent Kohn–Sham–Dirac equation^{41}. The density of states was calculated by applying the tetrahedron method. For all calculations, the appropriate experimental crystal structures were used. The bulk electronic structure calculations were carried out on a wellconverged mesh of 27,000 k points (30 × 30 × 30 mesh, 2,496 points in the irreducible wedge of the Brillouin zone). The strong Coulomb repulsion in the Co 3d shell was taken into account in a meanfield way, applying the GGA + U approximation in the atomiclimit flavour, with U = 4 eV (ref. 22).
The surface electronic structure was calculated using a symmetric slab containing nine CoO_{2} layers with Co in the centre, and separated by a vacuum gap of 15 Å along the z direction. The influence of including SOC in the calculations, which leads to spinsplitting of the surface states, is shown explicitly in Extended Data Fig. 3d. The layerprojected Co PDOS and the layerresolved charge accumulation are shown in Extended Data Fig. 3c. A dense k mesh of 20 × 20 × 4 points was used for the calculations. For this slab thickness the calculations are well converged with respect to the electronic states of the three central layers. However, owing to the slight offstoichiometry of the slab, the partial charges of these layers are slightly modified from their true bulk values. We therefore set the Fermi level in the calculations to match the experimental crossing vectors of the surface states. However, this causes a small upward shift of the holelike bands near , causing them to intersect the Fermi level in the slab calculation (Fig. 3e), contrary to the case for the true bulk electronic structure (Fig. 2b). We neglect these pockets in plotting the calculated Fermi surface (Fig. 3b). Supercell calculations were also performed for fully relaxed crystal structures, yielding qualitatively the same results as in the calculations performed for the ideal truncated bulk crystals used for the results presented here. The OAM shown in Fig. 3d and Extended Data Fig. 2 were calculated by downfolding the DFT supercell calculation (neglecting spin–orbit interactions) onto Wannier functions. SOC was added at the tightbinding level for these Wannier orbitals, enabling us to calculate the OAM of the surface states with and without SOC, as shown in Extended Data Fig. 2.
Tight binding models
To gain further understanding of the mechanism by which the difference in the onsite energy between the surface and subsurface oxygen stabilizes a giant ISB, ultimately leading to spin splitting of the full atomic SOC strength, we developed a minimal tightbinding model that describes the surface CoO_{2} layer (see Fig. 4a, b). This model is broadly justified by the DFT supercell calculations, which indicate that the surface states are almost entirely localized in the topmost CoO_{2} layer, and is described below.
A tightbinding Hamiltonian was constructed using the Slater–Koster parameterization of the energy integrals^{42}, in the cubic (xy, yz, 3z^{2} − r^{2}, xz, x^{2} − y^{2}, p1_{y}, p1_{z}, p1_{x}, p2_{y}, p2_{z}, p2_{x}) basis, where p1 and p2 refer to the orbitals on the two distinct oxygens, and the axis is taken to be normal to the crystal surface. For illustrative purposes and maximum simplicity, in Extended Data Fig. 4 we retain only Co–O–Co hopping between Co t_{2g} and O p_{z} orbitals, which are dominant at the Fermi level, and neglect the small trigonal crystal field splitting of the t_{2g} levels. Better agreement with our DFT calculations can be obtained by including both inplane O orbitals as well as direct Co d–d hopping (Extended Data Fig. 7).
Once the spin degree of freedom is taken into account the Hamiltonian is represented by a 22 × 22 matrix. It is a sum of three terms: the kinetic term H_{K}, the spin–orbit term H_{SO} and the crystal field term H_{CF}. The fact that the octahedral and crystallographic axes do not coincide (Fig. 4a) necessitates a coordinate transformation between cubic and trigonal orbital basis sets. Here we discuss all of the contributions to the total Hamiltonian, as well as the free parameters of the model.
The kinetic part of the Hamiltonian is constructed in the basis of cubic harmonics, with the axis normal to the crystal surface. This choice of coordinate system makes it easy to evaluate directional cosines between neighbouring atoms: l = sinθcosϕ, m = sinθsinϕ and n = cosθ, where θ and ϕ are the usual polar angles. The experimental crystal structure of PtCoO_{2} (ref. 22), with the hexagonal lattice constants of a = 2.82 Å and c = 17.8 Å, and the Pt–O distance z = 0.114c, was used to determine the relative positions of the atoms. If the Co atom is placed at the origin, (0, 0, 0), then the positions of two distinct oxygen atoms are given by and . Extended Data Table 2 lists all of the hopping paths considered in the extended model, along with the angles between the nearestneighbour atoms in the geometry outlined above, and the Slater–Koster parameters needed to describe the hopping^{42}.
The Co crystal field is diagonal in the trigonal basis, (u_{+}, u_{−}, x_{0}, x_{1}, x_{2}), where u_{+} and u_{−} are the two orbitals, x_{0} is a_{1g}, and x_{1} and x_{2} are of symmetry. The trigonal basis is related to the cubic one via the basis transformation^{43} (u_{+}, u_{−}, x_{0}, x_{1}, x_{2}) = B_{c→t}(xy, yz, 3z^{2} − r^{2}, xz, x^{2} − y^{2}), where
The coordinate transformation between the bases is then given by , and the crystal field Hamiltonian in the cubic basis can be found using , where is the diagonal crystal field Hamiltonian in the trigonal basis. Allowing for both the octahedral (C_{o}) and trigonal (C_{t}) crystal field, is
In the maximally simplified model, called model I (Extended Data Fig. 4), the orbitals are completely neglected, which is achieved technically by applying a very large octahedral crystal field. An overall onsite energy E_{Co} is added to all of the Co states. The onsite energy of the inplane and outofplane 2p orbitals of the first (second) oxygen are labelled and ( and ), respectively. In the simplified model I, which considers only the orbitals, large onsite energies are applied.
The spin–orbit Hamiltonian is H_{SO} = ξ L · S, where ξ is the atomic SOC constant of the Co 3d orbitals, taken to be 70 meV (ref. 26). The Hamiltonian in the cubic basis (xy↓, yz↓, 3z^{2} − r^{2}↓, xz↓, x^{2} − y^{2}↓, xy↑, yz↑, 3z^{2} − r^{2}↑, xz↑, x^{2} − y^{2}↑), where ↑ (↓) denotes spinup (spindown), is
The two sets of parameters used to calculate the tightbinding band structure are shown in Extended Data Table 3. Model I is maximally simplified, retaining only Co–O nearestneighbour hopping between Co t_{2g} and O p_{z} orbitals as discussed above, to isolate the key ingredients necessary for the large ISB energy scale (Extended Data Fig. 4). Co–O hopping is parameterized using two Slater–Koster parameters, V_{dpσ} and V_{dpπ}, using the empirical relation (ref. 44). The onsite energies of the oxygen p_{z} orbitals are estimated from the density of states shown in Extended Data Fig. 3, and chosen to be E_{O2} = −7 eV (E_{O1} = −3.2 eV) for oxygen below (above) Co. Guided by the bandwidth and filling of the CoO_{2}derived surface states from our DFT supercell calculations, the total bandwidth is required to be about 500 meV, and the Fermi level to be about 100 meV below the top of the band. This sets the onsite energy of the Co orbitals to be −600 meV, and the hopping parameter V_{dpσ} to 1.2 eV, a value similar to those obtained from fits to the DFT band structure of the related compound Na_{x}CoO_{2} (ref. 45). The differences between the oxygen and Co onsite energies, which are relevant for determining effective hopping parameters, are Δ_{O1} = −2.6 eV and Δ_{O2} = −6.4 eV. For the calculation with no asymmetry (Extended Data Fig. 4), the onsite energy difference was required to satisfy 2/Δ_{O} = 1/Δ_{O1} + 1/Δ_{O2}, yielding the oxygen onsite energy of E_{O} = −4.3 eV.
A slightly more advanced model, called model II, additionally incorporates direct Co–Co and O–O hopping, as well as the full Co 3d and O 2p orbital manifolds. Because the inplane oxygen orbitals do not directly participate in the Pt–O bonding, their onsite energy is allowed to be different to that of the p_{z} orbitals. In particular, the DFT PDOS suggests the difference between and is about 2 eV smaller than that between and . Allowing these different hopping paths, plausible for the real material, increases the degree of orbital mixing and therefore spin splitting, across k space. The calculated electronic structure for this model is shown in Extended Data Fig. 7.
Data availability
The data that underpin the findings of this study are available at http://dx.doi.org/10.17630/83be07cb5ea647d38136d75392da4a3d.
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Acknowledgements
We thank N. Nandi and B. Schmidt for discussions. We acknowledge support from the European Research Council (through the QUESTDO project), the Engineering and Physical Sciences Research Council, UK (grant no. EP/I031014/1), the Royal Society, the MaxPlanck Society and the International MaxPlanck Partnership for Measurement and Observation at the Quantum Limit. V.S., L.B., O.J.C. and J.M.R. acknowledge EPSRC for PhD studentship support through grant numbers EP/L015110/1, EP/G03673X/1, EP/K503162/1 and EP/L505079/1. D.K. acknowledges funding by the DFG within FOR 1346. We thank Diamond Light Source and Elettra synchrotrons for access to Beamlines I05 (proposal numbers SI12469, SI14927 and SI18267) and APE (proposal no. 20150019), respectively, that contributed to the results presented here. Additional supporting measurements were performed at the CASIOPEE beamline of SOLEIL, and we are grateful to I. Marković and P. Le Fèvre for their assistance.
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V.S., F.M., L.B., O.J.C., J.M.R. and P.D.C.K. measured the experimental data, and V.S. performed the data analysis. P.K and S.K. grew and characterized the samples. V.S. developed the tightbinding models, and H.R., D.K. and M.W.H. performed the firstprinciples calculations. M.H. and T.K.K. maintained the ARPES end station and J.F. and I.V. the spinARPES end station, and all provided experimental support. V.S., P.D.C.K. and A.P.M. wrote the manuscript with input and discussion from coauthors, and were responsible for overall project planning and direction.
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Correspondence to A. P. Mackenzie or P. D. C. King.
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Extended data figures and tables
Extended Data Figure 1 Photon energy and polarization dependent ARPES.
a, The intensity at the Fermi level (E_{F} ± 5 meV; sum of measurements with circularly left (CL) and circularly right (CR)polarized light), as a function of incident photon energy in PtCoO_{2}. The full purple lines correspond to the peak positions of fits to momentum distribution curves at the Fermi level. The Fermi crossing vectors do not depend on the photon energy, indicating that the states attributed to the CoO_{2} surface layer are indeed two dimensional. b–d, A strong circular dichroism of these states is evident over an extended photon energy range (b), and with an inplane momentum dependence indicating the same chirality of OAM for the two spinsplit bands (d; hν = 110 eV)^{25,46}. The grey lines in d represent the Fermi momenta extracted from the sum of the measurements in CL and CR polarizations (c) by fitting momentum distribution curves radially around the Fermi surface. e, f, Similar to PtCoO_{2}, PdCoO_{2} (e) and PdRhO_{2} (f) surface states also show strong circular dichroism (hν = 90 eV) of the same sign on the two spinsplit branches.
Extended Data Figure 2 OAM of CoO_{2} surface states in PtCoO_{2}.
a, Fermi surface of the surface states calculated without SOC, with arrows indicating the expected value of the inplane OAM (see Methods) in the direction normal to the momentum and the colouring indicating a small outofplane OAM canting. The OAM develops as a result of the ISM even in the absence of SOC. b, Once SOC is included, the Fermi surface is split into two, both retaining the same OAM direction as in the noSOC case. c, The two Fermi surfaces with the same OAM polarization carry opposite spin polarization, confirming that the surfaces states seen here are in the strong ISB limit (compare with Fig. 1).
Extended Data Figure 3 DFT supercell calculations.
a, The oxygen p_{z} PDOS for layers above (O1, pink) and below (O2, purple) the Co layer. The p_{z} PDOS of O1 is much larger than that of O2 close to the Fermi level (see also Fig. 4c, d). b, Except for this added weight, the p_{z} PDOS of O1, shifted by −3.8 eV in binding energy, well approximates that of O2. c, Co PDOS near the Fermi level for different Co layers. The surface state between about −0.45 eV and 0.1 eV has very little contribution from Co atoms below the first layer. The PDOS of the subsurface Co atoms is almost bulklike. This is also reflected by the charging of the surface shown in the inset. The plot shows the additionally accumulated charge versus depth below the surface, referenced to the constituent bulk charges. Only the surface O layer and the topmost Co layer deviate substantially from the bulk. In particular, the pronounced difference (asymmetry) between the two O layers of the CoO_{2} surface layer is very clearly demonstrated. Surface relaxation has only a minor effect on this scenario. d, Closeup of the band structure of PtCoO_{2} around the Fermi level. For the narrow Co–O surface band (between about −0.45 eV and 0.1 eV), the SOC (red lines) leads to a spin splitting of this band, with only small changes to the dispersion.
Extended Data Figure 4 Development of OAM and SAM.
a–h, The band structure obtained from a minimal tightbinding model (see Methods) reveals the key elements for maximal Rashbalike spin splitting. The calculations are shown without SOC or ISB (a, b), with ISB but no SOC (c, d), and with ISB and SOC (e–h). The chiral clockwise (CW) and anticlockwise (CCW) inplane (c, e) and outofplane (d, f) OAM and chiral clockwise and anticlockwise inplane (g) and outofplane (h) SAM are shown by colouring (see legends). If the two oxygens have the same onsite energy (no asymmetry, E_{O1} = E_{O2}), and neglecting SOC (a, b), then the electronic structure closely resembles that of a Kagome model, which has previously been used to describe the CoO_{2} layer of Na_{x}CoO_{2} (ref. 47): the lowest band is flat, and the other two bands cross at the Brillouin zone corner and along the line, where hybridization is forbidden by symmetry. OAM is quenched in this inversionsymmetric environment. Introducing asymmetry as a result of a difference in the onsite energy of O1 and O2 (c, d) allows orbital mixing, and hybridization gaps open where there are crossings in the absence of the symmetry breaking. The orbital mixing enables these bands to develop a large (magnitude approaching ħ) OAM even in the absence of SOC. This is largely chiral (OAM perpendicular to inplane momentum) along the and directions, and crosses over to the OAM having a large outofplane component close to where any inplane component must vanish owing to symmetry. For such an inplane and outofplane OAM to develop there must be an outofplane and inplane ISB, respectively. The fact that the asymmetric hopping occurs via the layers above and below Co naturally gives rise to the outofplane ISB. The opposite orientation of nearestneighbour Co–O bonds to the oxygen layers above and below the transitionmetal plane (Fig. 4b) additionally provides the inplane ISB. Together, this enables the OAM to remain large across a greater portion of the Brillouin zone, rather than being suppressed in the broad vicinity of the point. Crucially, the hybridization gaps between states of opposite OAM opened by such ISB are as large as 140 meV for the realistic parameters used here, about twice the size of atomic SOC. This difference in energy scales means that the spin–orbit interaction introduces an additional splitting between the states of spin that are parallel and antiparallel to the preexisting OAM, which is itself not greatly altered (e–h). The energetic splitting assumes the full value of the atomic SOC, validating the simple schematic shown in Fig. 1b. This picture is consistent with our spinresolved ARPES (Fig. 3c) and CD–ARPES (Extended Data Fig. 1 and Fig. 2d) data, which show that the two spinsplit branches of the CoO_{2}derived surface states host the same sign of OAM.
Extended Data Figure 5 Crossover from weak to strong ISB.
a, Spin splitting at the point, calculated using our minimal singlelayer tightbinding model for edgesharing (TM)O_{2} octahedra (Methods and Extended Data Tables 2 and 3), keeping the SOC strength fixed and varying the asymmetry of the upper and lower oxygen atoms, defined as a = (E_{O1} − E_{O2})/(E_{O1} + E_{O2}). Two regimes are clearly observed. When the ISB energy scale E_{ISB} is smaller than the SOC, spin splitting is proportional to, and limited by, E_{ISB} (weak ISB limit). Once E_{ISB} becomes dominant, the spin splitting saturates to a value that is limited by the spin–orbit energy scale, which is now weaker than E_{ISB} (strong ISB limit). b–e, To demonstrate that these two regimes correspond to the two limits illustrated in Fig. 1, we plot the band structure for a = 0.1, coloured by the chiral inplane (b) and outofplane (c) OAM, and the chiral inplane (d) and outofplane (e) spin. We find that the sign of the OAM switches between the two spinsplit bands, in contrast to what is found in the strong ISB limit of the same model (a = 0.4, Extended Data Fig. 4e–h) and in the firstprinciples calculations for the surface states of PtCoO_{2} (Extended Data Fig. 2b).
Extended Data Figure 6 Comparison of surface onsite energy shifts for different structural configurations.
a–f, To demonstrate the importance of the structural building blocks for the strength of ISB experienced by the relevant electrons, we compare the influence of surface onsite energy shifts on the edgesharing transitionmetal oxide layer found in delafossites (a–c) with that on its cornersharing counterpart found in 〈001〉 perovskites (d–f). In both cases the breaking of covalent bonds at the surface can lead to an onsite energy shift of the surface oxygen (O1) with respect to subsurface oxygens. However, the influence on the spin splitting is very different. We use the same tightbinding model parameters for the two structures (as quoted in Extended Data Table 3), and in particular the same onsite energy shift, to calculate the bandstructure for a single transitionmetal oxide layer. It is clear that, despite the same onsite energy shift at the surface, there is negligible effect on the band structure of the cornersharing layer (e, f). This is because the dominant hopping path between transitionmetal atoms is via the planar oxygen atoms (a), and so the relevant electrons do not feel this surface symmetry breaking strongly. Other mechanisms, such as surface distortions, are required to obtain a larger effect^{29,30}. In contrast, for the delafossites, hopping between transitionmetal atoms is via either the surface or subsurface oxygen layers (a), and so the effect of a pure onsite energy shift of the surface layer is already sufficient to drive a large OAM in the undistorted structure (b, c), as discussed in the main text.
Extended Data Figure 7 Tight binding model II.
The band structure calculated using the tightbinding model II (see Extended Data Table 3). Additional hopping paths allowed in this model increase the orbital mixing, and thus the spin splitting, across k space.
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Sunko, V., Rosner, H., Kushwaha, P. et al. Maximal Rashbalike spin splitting via kineticenergycoupled inversionsymmetry breaking. Nature 549, 492–496 (2017) doi:10.1038/nature23898
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