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 condensed-matter 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 computing1,2. 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 inversion-symmetry breaking to itinerant surface electrons than is typically achieved. The key element is a pronounced asymmetry of surface hopping energies—that is, a kinetic-energy-coupled inversion-symmetry breaking, the energy scale of which is a substantial fraction of the bandwidth. Using spin- and angle-resolved photoemission spectroscopy, we demonstrate that such a strong inversion-symmetry breaking, when combined with spin–orbit interactions, can mediate Rashba-like3,4 spin splittings that are much larger than would typically be expected. The energy scale of the inversion-symmetry breaking that we achieve is so large that the spin splitting in the CoO2- and RhO2-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 Rashba-like3,4 spin splittings. The core structural building blocks that facilitate the bandwidth-scaled inversion-symmetry breaking are common to numerous materials. Our findings therefore provide opportunities for creating spin-textured states and suggest routes to interfacial control of inversion-symmetry breaking in designer heterostructures of oxides and other material classes.
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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 Max-Planck Society and the International Max-Planck 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 tight-binding models, and H.R., D.K. and M.W.H. performed the first-principles calculations. M.H. and T.K.K. maintained the ARPES end station and J.F. and I.V. the spin-ARPES 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 co-authors, and were responsible for overall project planning and direction.
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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 (EF ± 5 meV; sum of measurements with circularly left (CL)- and circularly right (CR)-polarized light), as a function of incident photon energy in PtCoO2. 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 CoO2 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 in-plane momentum dependence indicating the same chirality of OAM for the two spin-split 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 PtCoO2, PdCoO2 (e) and PdRhO2 (f) surface states also show strong circular dichroism (hν = 90 eV) of the same sign on the two spin-split branches.
Extended Data Figure 2 OAM of CoO2 surface states in PtCoO2.
a, Fermi surface of the surface states calculated without SOC, with arrows indicating the expected value of the in-plane OAM (see Methods) in the direction normal to the momentum and the colouring indicating a small out-of-plane 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 no-SOC 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 pz PDOS for layers above (O1, pink) and below (O2, purple) the Co layer. The pz 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 pz 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 bulk-like. 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 CoO2 surface layer is very clearly demonstrated. Surface relaxation has only a minor effect on this scenario. d, Close-up of the band structure of PtCoO2 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 tight-binding model (see Methods) reveals the key elements for maximal Rashba-like 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) in-plane (c, e) and out-of-plane (d, f) OAM and chiral clockwise and anticlockwise in-plane (g) and out-of-plane (h) SAM are shown by colouring (see legends). If the two oxygens have the same on-site energy (no asymmetry, EO1 = EO2), and neglecting SOC (a, b), then the electronic structure closely resembles that of a Kagome model, which has previously been used to describe the CoO2 layer of NaxCoO2 (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 inversion-symmetric environment. Introducing asymmetry as a result of a difference in the on-site 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 in-plane momentum) along the 
and 
directions, and crosses over to the OAM having a large out-of-plane component close to 
where any in-plane component must vanish owing to symmetry. For such an in-plane and out-of-plane OAM to develop there must be an out-of-plane and in-plane ISB, respectively. The fact that the asymmetric hopping occurs via the layers above and below Co naturally gives rise to the out-of-plane ISB. The opposite orientation of nearest-neighbour Co–O bonds to the oxygen layers above and below the transition-metal plane (Fig. 4b) additionally provides the in-plane 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 pre-existing 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 spin-resolved ARPES (Fig. 3c) and CD–ARPES (Extended Data Fig. 1 and Fig. 2d) data, which show that the two spin-split branches of the CoO2-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 single-layer tight-binding model for edge-sharing (TM)O2 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 = (EO1 − EO2)/(EO1 + EO2). Two regimes are clearly observed. When the ISB energy scale EISB is smaller than the SOC, spin splitting is proportional to, and limited by, EISB (weak ISB limit). Once EISB becomes dominant, the spin splitting saturates to a value that is limited by the spin–orbit energy scale, which is now weaker than EISB (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 in-plane (b) and out-of-plane (c) OAM, and the chiral in-plane (d) and out-of-plane (e) spin. We find that the sign of the OAM switches between the two spin-split 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 first-principles calculations for the surface states of PtCoO2 (Extended Data Fig. 2b).
Extended Data Figure 6 Comparison of surface on-site 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 on-site energy shifts on the edge-sharing transition-metal oxide layer found in delafossites (a–c) with that on its corner-sharing counterpart found in 〈001〉 perovskites (d–f). In both cases the breaking of covalent bonds at the surface can lead to an on-site 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 tight-binding model parameters for the two structures (as quoted in Extended Data Table 3), and in particular the same on-site energy shift, to calculate the bandstructure for a single transition-metal oxide layer. It is clear that, despite the same on-site energy shift at the surface, there is negligible effect on the band structure of the corner-sharing layer (e, f). This is because the dominant hopping path between transition-metal 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 effect29,30. In contrast, for the delafossites, hopping between transition-metal atoms is via either the surface or subsurface oxygen layers (a), and so the effect of a pure on-site 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 tight-binding 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 Rashba-like spin splitting via kinetic-energy-coupled inversion-symmetry breaking. Nature 549, 492–496 (2017). https://doi.org/10.1038/nature23898
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