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Geometric frustration of Jahn–Teller order in the infinite-layer lattice


The Jahn–Teller effect, in which electronic configurations with energetically degenerate orbitals induce lattice distortions to lift this degeneracy, has a key role in many symmetry-lowering crystal deformations1. Lattices of Jahn–Teller ions can induce a cooperative distortion, as exemplified by LaMnO3 (refs. 2,3). Although many examples occur in octahedrally4 or tetrahedrally5 coordinated transition metal oxides due to their high orbital degeneracy, this effect has yet to be manifested for square-planar anion coordination, as found in infinite-layer copper6,7, nickel8,9, iron10,11 and manganese oxides12. Here we synthesize single-crystal CaCoO2 thin films by topotactic reduction of the brownmillerite CaCoO2.5 phase. We observe a markedly distorted infinite-layer structure, with ångström-scale displacements of the cations from their high-symmetry positions. This can be understood to originate from the Jahn–Teller degeneracy of the dxz and dyz orbitals in the d7 electronic configuration along with substantial ligand–transition metal mixing. A complex pattern of distortions arises in a \(2\sqrt{2}\times 2\sqrt{2}\times 1\) tetragonal supercell, reflecting the competition between an ordered Jahn–Teller effect on the CoO2 sublattice and the geometric frustration of the associated displacements of the Ca sublattice, which are strongly coupled in the absence of apical oxygen. As a result of this competition, the CaCoO2 structure forms an extended two-in–two-out type of Co distortion following ‘ice rules’13.

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Fig. 1: JT distortion in 3D and 2D oxide lattices.
Fig. 2: Synthesis and large-scale cation displacements in thin-film CaCoO2.
Fig. 3: GIXRD and refinement of the oxygen positions.
Fig. 4: Extended structure of CaCoO2, ice rules, quadrupolar ordering and electronic structure.

Data availability

The data presented in the figures and other findings of this study are available from the corresponding authors upon reasonable request.


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We thank W.-S. Lee for discussions. The work at SLAC and Stanford was supported by the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (contract number DE-AC02-76SF00515) and the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative (grant number GBMF9072, synthesis equipment and initial development). Electron microscopy at Cornell was support by the Department of Defense Air Force Office of Scientific Research (number FA 9550-16-1-0305) and the Packard Foundation, and made use of the Cornell Center for Materials Research Shared Facilities which are supported through the NSF MRSEC programme (DMR-1719875), with the Thermo Fisher Helios G4 UX focused ion beam also supported by NSF (DMR-1539918). The Thermo Fisher Spectra 300 X-CFEG was acquired with support from PARADIM, an NSF MIP (DMR-2039380), and Cornell University. M.A.S. acknowledges additional support from the NSF GRFP under award number DGE-1650441. The 3A beamline at PLS-II is supported in part by MSIT. B.-G.C. is currently affiliated to Korea Research Institute of Standards and Science (KRISS). D.J. acknowledges funding by the Alexander-von-Humboldt foundation via a Feodor Lynen postdoctoral fellowship. Raman spectroscopy measurement was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822. TOF-SIMS characterization was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility, and using instrumentation within ORNL’s Materials Characterization Core provided by UT-Battelle, LLC under contract number DE-AC05-00OR22725. The computational work for this project was performed on the Sherlock cluster in the Stanford Research Computing Center.

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Authors and Affiliations



W.J.K. and H.Y.H. conceived and designed the experiments. M.A.S., B.H.G. and L.F.K. performed the STEM and EELS measurements and analysis. C.J. performed the DFT calculations. C.J., B.M. and T.P.D. performed the cluster calculations. D.J. performed Raman spectroscopy measurements. W.J.K. grew the samples, which were characterized by W.J.K., K.L., D.J. and M.O. W.J.K. and B.-G.C. performed and analysed the synchrotron GIXRD measurements. A.V.I. performed TOF-SIMS measurements. W.J.K., T.P.D. and H.Y.H. wrote the manuscript, with input from all authors.

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Correspondence to Woo Jin Kim or Harold Y. Hwang.

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Extended data figures and tables

Extended Data Fig. 1 Structural characterizations for CaCoO2.5 and CaCoO2.

a, Atomic-resolution HAADF-STEM image along the [100]t zone-axis projection of CaCoO2.5 showing alternate stacking of tetrahedral and octahedral layers. b, X-ray diffraction reciprocal space map of CaCoO2 around the (−103) SrTiO3 diffraction peak, indicating that the film is relaxed from the substrate. c, Empirical relationship between perovskite and infinite-layer lattice parameters. c-axis lattice parameters for various transition metal oxide compounds are plotted for the perovskite phase and the infinite-layer phase after topotactic reduction. The dashed line is a linear fit for all the data points in the plot. Note that the CaFeO2 has relatively large cinfinite layer/cperovskite associate with out-of-plane displacement of both FeO4 and Ca layers48.

Extended Data Fig. 2 EELS measurements of CaCoO2.

a, Co-L3,2 edge; the blue (red) solid line indicates EEL spectra for CaCoO2.5 (CaCoO2). b, Ca-L3,2 edge EELS shows that there are no substantial changes in the spectra before (CaCoO2.5, blue) and after reduction (CaCoO2, red). c, A plot of the intensity ratio I(L3)/I(L2) of the Co-L3,2 edge for different Co compounds with different oxidation states. Note that the dashed line indicates a polynomial fit curve for four different compounds from ref. 27 (CoCO3, CoSO4, Co3O4, and CoSi4). I(L3)/I(L2) of the CaCoO2.5 and CaCoO2 films are depicted with blue and red circles, respectively. d, O K-edges EELS data. Spatially averaged O K-edge spectra of CaCoO2 (CaCoO2.5) in red (blue). The partially transparent, solid lines indicate the raw, background-subtracted data, and the dashed lines indicate the Gaussian filtered spectra. Upon reduction of the CaCoO2.5 films to CaCoO2, we observe a suppression of the distinct pre-peak at ~ 529 eV in the region of the O K-edge associated with hybridization between O 2p and transition metal d orbitals consistent with a nominal electronic transition from 3d6 to 3d7. This is similar to the pre-peak suppression observed upon reduction from perovskite to infinite-layer phase in the related nickelates49. We further see the emergence of a shoulder in the CaCoO2 spectrum at ~ 530 eV, which is similar to a feature attributed to ligand hole states in doped infinite-layer nickelates49. This feature is also consistent with published spectra acquired from SrCoO3-δ, which has negative charge transferred state37. An O K-edge spectrum of the SrTiO3 substrate is included in black for comparison.

Extended Data Fig. 3 Time-of-flight secondary-ion mass spectrometry (TOF-SIMS) and ABF-STEM measurements of CaCoO2.

a, Depth profiles of H+ and other ions from both CaCoO2.5 and CaCoO2 thin film on SrTiO3 substrate (with ~ 2 nm SrTiO3 capping layer) were measured with secondary-ion mass spectrometry. The Co ion signals from both CaCoO2.5 and CaCoO2 thin films were employed as a marker for the interface position. TOF-SIMS measurements show that the H+ concentration for CaCoO2 is similar to the background level of the as-synthesized CaCoO2.5 thin film. b, ABF-STEM image along the [100]t zone-axis projection with overlaid Co, Ca, and O atoms. c, Intensity line profiles for the blue and the orange dashed lines in b. The intensities of the line profiles are from inverted image b. The blue (orange) solid line indicates the line profile for the Co column (Ca and O column). The peak positions are the relative distances noted at the bottom of the image b. d, Atomic distances between Ca and Ca (black triangles), Co and Co (green diamonds), Ca and O (red squares), and Ca and Co (blue circles) layers are plotted. Note that the atomic layer numbers in d correspond to those in b. Error bars are taken as the full-width at half-maximum of the intensity peaks in c.

Extended Data Fig. 4 Powder XRD simulation and c-lattice parameter determination.

Lattice structure models for a, 2\(\sqrt{2}\)at \(\times \) 2\(\sqrt{2}\)at \(\times \) ct and b, 2\(\sqrt{2}\)at \(\times \) 2\(\sqrt{2}\)at \(\times \) 2ct. The second structure model is lattice doubled from the first model by stacking a half-unit-cell shifted layer along the in-plane direction. Powder XRD simulation results for both c, 2\(\sqrt{2}\)at \(\times \) 2\(\sqrt{2}\)at \(\times \) ct and d, 2\(\sqrt{2}\)at \(\times \) 2\(\sqrt{2}\)at \(\times \) 2ct models. Note that the XRD simulation for the 2\(\sqrt{2}\)at \(\times \) 2\(\sqrt{2}\)at \(\times \) 2ct model has a distinct half-order peak along the c-lattice direction. We first found e, the CaCoO2 (103)t XRD peak as a reference peak. Based on this reference peak position, we perform θ–2θ scans along the expected CaCoO2 (0.75, 0.25, 0.5) position. f, No XRD peak was observed at the expected CaCoO2 (0.75, 0.25, 0.5) peak position, indicating that CaCoO2 does not have a c-axis doubling of the simple tetragonal unit cell.

Extended Data Fig. 5 DFT calculations for CaCoO2.

a, Plan-view of the relaxed crystal structure for CaCoO2 from DFT + U calculations with U = 2 eV, U = 3 eV, U = 4 eV, U = 5 eV, and U = 6 eV. b, Calculated band dispersion of CaCoO2 (DFT + U for U = 5 eV). Green highlights dxz (and dyz) projections. The inset shows high-symmetry points in the tetragonal Brillouin zone. c, Resistivity versus temperature of CaCoO2 thin film. The inset shows that the resistivity is well fitted with an Arrhenius plot with an estimated (transport) gap of 0.337 + 0.001 eV. The spin-dependent partial density of states (PDOS) of d, Co(2) and e, Co(3) d orbitals from DFT + U (U = 5 eV). The spin-dependent PDOS of Co(2) shows the degeneracy lifting of the dxz/yx-orbitals.

Extended Data Fig. 6 Total energy calculation for CaCoO2 with \(\sqrt{2}\times \sqrt{2}\times 1\) and \(2\sqrt{2}\times 2\sqrt{2}\times 1\) supercell.

a, DFT+U (U = 5 eV) calculations for the total energy under purely Q2-JT-distortions in the \(\sqrt{2}\times \sqrt{2}\times 1\) supercell. b, \(2\sqrt{2}\times 2\sqrt{2}\times 1\) supercell with different distortion amplitudes. Approaching #10, the structure is approaches the experimentally refined structure. c, Normalized total energy for the structures depicted in b. Three different first-principle calculations are used for c (Methods).

Extended Data Table 1 Atomic coordinates for the initial (before GIXRD refinement) and refined structure of CaCoO2
Extended Data Table 2 Simulated and experimental CaCoO2 (HK0)t GIXRD peak positions and intensities
Extended Data Table 3 Structure symmetry and atomic coordinates for the refined structure of CaCoO2
Extended Data Table 4 Parameters used for multiplet calculations (in eV)

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Kim, W.J., Smeaton, M.A., Jia, C. et al. Geometric frustration of Jahn–Teller order in the infinite-layer lattice. Nature 615, 237–243 (2023).

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