Revealing spin-orbit coupling in a cuprate Strong coupling between the spin and orbital degrees of freedom is crucial in generating the exotic band structure of topological insulators. The combination of spin-orbit coupling with electronic correlations could lead to exotic effects; however, these two types of interactions are rarely found to be strong in the same material. Gotlieb et al. used spin- and angle-resolved photoemission spectroscopy to map out the spin texture in the cuprate Bi2212. Surprisingly, they found signatures of spin-momentum locking, not unlike that seen in topological insulators. Thus, in addition to strong electronic correlations, this cuprate also has considerable spin-orbit coupling. Science, this issue p. 1271

Abstract Cuprate superconductors have long been thought of as having strong electronic correlations but negligible spin-orbit coupling. Using spin- and angle-resolved photoemission spectroscopy, we discovered that one of the most studied cuprate superconductors, Bi2212, has a nontrivial spin texture with a spin-momentum locking that circles the Brillouin zone center and a spin-layer locking that allows states of opposite spin to be localized in different parts of the unit cell. Our findings pose challenges for the vast majority of models of cuprates, such as the Hubbard model and its variants, where spin-orbit interaction has been mostly neglected, and open the intriguing question of how the high-temperature superconducting state emerges in the presence of this nontrivial spin texture.

Many of the exotic properties of quantum materials stem from the strength of spin-orbit coupling or electron-electron correlations. At one end of the spectrum are topological insulators, which have weak electron correlations but strong spin-orbit coupling (1, 2); at the other end are cuprate superconductors, where electron correlations are the dominant interaction. Although unusual forms of spin response in the cuprates have been reported previously (3, 4), the spin-orbit interaction has been mostly neglected or treated as a small perturbation to the Hubbard Hamiltonian and mean field theory in the context of the Dzyaloshinskii-Moriya interaction, leading to negligible changes to the electronic ground state of cuprates (5–9).

Recently, there has been an upsurge of interest in materials in which both spin-orbit coupling and strong correlations are important because of their potential to induce exotic quantum states (10–13). In the presence of superconductivity, for example, spin-orbit interaction can have fundamental consequences for the symmetry of the order parameter (14), driving unusual pairing mechanisms (11, 15), creating Ising pairs (16), and even realizing the conditions for the existence of previously unobserved particles (17–19).

Spin- and angle-resolved photoemission spectroscopy (SARPES) has been instrumental in studying the consequences of such interplay for the electronic structure of a variety of materials, from heavy fermions to iridates (20, 21), thanks to its ability to simultaneously probe the energy, momentum, and spin structure of quasiparticles. However, because of earlier predictions of negligible spin-orbit interaction in cuprates (6), the full spin character of quasiparticles has not been probed experimentally. Here, we report such a study, revealing unexpected consequences of the spin-orbit interaction for the electronic structure of cuprates.

Local inversion symmetry breaking We now present a possible explanation for the observed spin polarization and its momentum dependence and discuss possible implications for superconductivity. Perhaps the most studied spin texture is the Dresselhaus-Rashba effect (28, 29), which is manifested in noncentrosymmetric materials (i.e., materials lacking inversion symmetry) and gives rise to spin-dependent effects, inducing a momentum spin-splitting of the energy bands. Recently, it has been pointed out that even in centrosymmetric materials, a local electric field within the unit cell can lead to spin-split bands (30) whereas the net spin polarization remains zero as the electric field averages to zero within the unit cell. This local field can originate from specific structural characteristics that break local inversion symmetry centered on Cu atoms, such as layered structures or some types of lattice distortions that are present in the cuprates (31–36). In the case of a layered structure, the local field is perpendicular to the planes and the spin-split bands are spatially segregated in real space on top and bottom layers (30). In the case of a structural distortion, the spin-split bands are segregated within different parts of the unit cell. The model in (30) has been successfully applied to account for the nontrivial spin polarization observed in layered dichalcogenides (37, 38) and a BiS 2 -based superconductor (39), as well as to explain the nonzero nodal energy splitting between bonding and antibonding bands in a YBa 2 Cu 3 O 6+δ cuprate superconductor (9). We extend this model to the case of bilayer Bi2212 by using a tight-binding model in the presence of a local electric field, treated via Rashba-type spin-orbit coupling, as in (30); the details of the calculations are shown in (23). The field is induced by the local breaking of inversion symmetry in Bi2212. Although the crystallographic space group of Bi2212 is often regarded as centrosymmetric (40), the local environment of Cu is noncentrosymmetric: The Ca layer separating two Cu-O planes removes the inversion center from Cu. Each Cu-O layer is now subject to a different environment: One Cu-O layer has Bi-O ions above and Ca ions below, whereas this is reversed for the other layer in the unit cell, allowing for a nonzero electric field within the unit cell (see the schematic in Fig. 5A). Fig. 5 Spin structure within the unit cell. (A) Schematic view of the two-CuO 2 bilayer structure in Bi 2 Sr 2 CaCu 2 O 8+δ , where we omit layers of Bi-O and Sr-O which separate bilayers. Green atoms correspond to oxygen, yellow to copper, and red atoms in between are Ca. Arrows schematically depict the possible direction of the electric field, which leads to the spin-orbit coupling of the opposite sign on different layers. (B) Expected spin pattern of the antibonding band for two adjacent CuO 2 layers within the unit cell. Although one would expect both Rashba and Dresselhaus contributions to spin-orbit coupling [R2 and D2 according to the notations in (30)], it appears that the dominant components in our experiments come from the Rashba order. This is likely a consequence of the strong anisotropy between ab and c axes in Bi2212, making the Dresselhaus component subleading. Upon the addition of such spin-orbit coupling, the former bonding (antibonding) band loses its purely antisymmetric (symmetric) character under mirror symmetry. However, we retain this naming convention herein. Both bonding and antibonding bands remain doubly degenerate at any momentum in the Brillouin zone as the crystal retains unbroken inversion and time reversal symmetries. However, these bands acquire spin-momentum locking with opposite spin polarization on each individual Cu-O layer. The spin textures for the antibonding orbital in the two Cu-O layers that result from this model are shown in Fig. 5B. Photoemission measures the interference pattern of contributions from several near-surface layers (41) and in this case has different intensity from bonding and antibonding bands (42, 43). Therefore, a nonzero spin signal is expected, despite inversion symmetry and the lack of resolved band splitting. This spin texture stems from differences in photoemission matrix elements for different components of the wave function, as well as the surface sensitivity of the measurement and interference effects. We find that the spin polarization alternates as a function of photon energy, as discussed in (23), similarly to the change in the relative strength of photoemission intensity from bonding and antibonding bands (44). However, this could also be the result of a more complex dependence of the spin-orbit entanglement on photon energy, as shown extensively in other spin-orbit–coupled materials, such as topological insulators (41, 45), where the sign of spin polarization can change with photon energy and even be zero; more detailed studies and calculations are needed. By extending our tight-binding model to incorporate interference effects, we remove the perfect cancellation of spin polarizations between bonding and antibonding bands and get a spin texture that reverses sign across the Fermi surface (fig. S6). In addition, the interference effects can also explain the opposite direction of spin polarization between the original bands and their superstructure replicas shown in Fig. 3, as discussed in detail in (23). Although our model can reproduce qualitative aspects of the spin polarization observed in our experiment, it does not capture the magnitude and precise momentum dependence of the spin, which require more involved calculations. Reports in favor of a noncentrosymmetric space group for Bi2212 (31, 32, 46) might simply argue that it is the absence of any inversion center that allows for the reported nonzero spin texture, as in a standard Rashba system, rather than the creation of a local field. Such a scenario, however, would imply the presence of spin-split bands that have not yet been observed. Moreover, some of the structural distortions typical of cuprates, such as local Jahn-Teller distortions (32–34), modulations of the oxygens in the BiO slabs, and buckling of the CuO 2 planes (47), could break the local inversion symmetry and give rise to a nonzero electric field. The latter effects along with the presence of other atoms in a polar environment within the unit cell could also potentially contribute to the spin texture reported here and could be responsible for the nonzero spin polarization observed in single-layer Bi2201 (23, 25). Regardless of the origin of the observed spin-orbit interaction, it is clear that its effect on the symmetry of the Hamiltonian and on the ground state properties cannot be neglected. In the case of weak correlations, the interplay between spin-orbit coupling and superconductivity can affect spin susceptibility (48), alter the structure of the gap nodes, and allow for additional Amperean-like attraction channels coming from spin fluctuations (15, 49). In the case of strong correlations, spin-orbit coupling could enhance a charge density wave–type of order (50, 51), as observed in cuprates, and ultimately could affect the superconducting gap and the phase diagram (52). Our observation of spin-orbit coupling with a magnitude comparable to that of the interlayer tunneling and superconducting gap [see discussion in (23)] and the persistence of a nonzero spin polarization above T c (fig. S2) suggest that a complex correlation between superconductivity, spin-orbit coupling, and layer degrees of freedom might be at play in cuprates (52). As the effects of the coexistence of spin-orbit coupling, strong correlations, and superconductivity are still poorly understood, we hope that our results will stimulate further experimental and theoretical research exploring the physics in this emergent field.

Supplementary Materials www.sciencemag.org/content/362/6420/1271/suppl/DC1 Materials and Methods Supplementary Text Figs. S1 to S8 References (53–58) Data S1

http://www.sciencemag.org/about/science-licenses-journal-article-reuse This is an article distributed under the terms of the Science Journals Default License.

Acknowledgments: We thank D. H. Lee, C. Varma, E. Altman, and T. L. Miller for fruitful discussion. We thank K. Kurashima for sample preparation. Funding: This work was supported primarily by Berkeley Lab’s program on Ultrafast Materials Sciences, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-05CH11231. A.L. acknowledges partial support for this research from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4859. The theory component of this paper was supported by the Quantum Materials Program at Lawrence Berkeley National Laboratory, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-05CH11231. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract DE-AC02-05CH11231. M.S. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4307. W.Z. acknowledges support from the Ministry of Science and Technology of China (2016YFA0300501) and from NSF China (11674224). Author contributions: K.G. and A.L. were responsible for experimental design. K.G. and C.-Y.L carried out the experiments. Calculations were performed by M.S. and A.V. Samples were prepared by H.E. A.L. was responsible for experiment planning and infrastructure. All authors contributed to the interpretation and writing of the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All data are available in the manuscript or the supplementary materials.