Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

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

A Publisher Correction to this article was published on 20 July 2023

This article has been updated


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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

Data availability

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

Change history


  1. Jahn, H. A. & Teller, E. Stability of polyatomic molecules in degenerate electronic states I—orbital degeneracy. Proc. R. Soc. Lond. 161, 220–235 (1937).

    ADS  CAS  MATH  Google Scholar 

  2. Goodenough, J. B. Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO3. Phys. Rev. 100, 564–573 (1955).

    Article  ADS  CAS  Google Scholar 

  3. Gehring, G. A. & Gehring, K. A. Co-operative Jahn–Teller effects. Rep. Prog. Phys. 38, 1–89 (1975).

    Article  ADS  CAS  Google Scholar 

  4. Goodenough, J. B. Jahn–Teller phenomena in solids. Annu. Rev. Mater. Sci. 28, 1–27 (1998).

    Article  ADS  CAS  Google Scholar 

  5. Goodenough, J. B. Jahn–Teller distortions induced by tetrahedral-site Fe2+ ions. J. Phys. Chem. Solids 25, 151–160 (1964).

    Article  ADS  CAS  Google Scholar 

  6. Siegrist, T., Zahurak, S. M., Murphy, D. W. & Roth, R. S. The parent structure of the layered high-temperature superconductors. Nature 334, 231–232 (1988).

    Article  ADS  CAS  Google Scholar 

  7. Smith, M. G., Manthiram, A., Zhou, J., Goodenough, J. B. & Markert, J. T. Electron-doped superconductivity at 40 K in the infinite-layer compound Sr1–yNdyCuO2. Nature 351, 549–551 (1991).

    Article  ADS  CAS  Google Scholar 

  8. Crespin, M., Levitz, P. & Gatineau, L. Reduced forms of LaNiO3 perovskite. Part 1.—Evidence for new phases: La2Ni2O5 and LaNiO2. J. Chem. Soc. Faraday Trans. 2 79, 1181–1194 (1983).

    Article  CAS  Google Scholar 

  9. Hayward, M. A., Green, M. A., Rosseinsky, M. J. & Sloan, J. Sodium hydride as a powerful reducing agent for topotactic oxide deintercalation: synthesis and characterization of the nickel(I) oxide LaNiO2. J. Am. Chem. Soc. 121, 8843–8854 (1999).

    Article  CAS  Google Scholar 

  10. Tsujimoto, Y. et al. Infinite-layer iron oxide with a square-planar coordination. Nature 450, 1062–1065 (2007).

    Article  ADS  CAS  PubMed  Google Scholar 

  11. Kawakami, T. et al. Spin transition in a four-coordinate iron oxide. Nat. Chem. 1, 371–376 (2009).

    Article  CAS  PubMed  Google Scholar 

  12. Dixon, E., Hadermann, J., Ramos, S., Goodwin, A. L. & Hayward, M. A. Mn(I) in an extended oxide: the synthesis and characterization of La1–xCaxMnO2+δ (0.6 ≤ x ≤ 1). J. Am. Chem. Soc. 133, 18397–18405 (2011).

    Article  CAS  PubMed  Google Scholar 

  13. Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).

    Article  CAS  Google Scholar 

  14. von Helmolt, R., Wecker, J., Holzapfel, B., Schultz, L. & Samwer, K. Giant negative magnetoresistance in perovskitelike La2/3Ba1/3MnOx ferromagnetic films. Phys. Rev. Lett. 71, 2331–2333 (1993).

    Article  ADS  Google Scholar 

  15. Raveau, B., Hervieu, M., Maignan, A. & Martin, C. The route to CMR manganites: what about charge ordering and phase separation? J. Mater. Chem. 11, 29–36 (2001).

    Article  CAS  Google Scholar 

  16. Han, J. E., Gunnarsson, O. & Crespi, V. H. Strong superconductivity with local Jahn–Teller phonons in C60 solids. Phys. Rev. Lett. 90, 167006 (2003).

    Article  ADS  CAS  PubMed  Google Scholar 

  17. Millis, A. J., Littlewood, P. B. & Shraiman, B. I. Double exchange alone does not explain the resistivity of La1–xSrxMnO3. Phys. Rev. Lett. 74, 5144–5147 (1995).

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Millis, A. J., Shraiman, B. I. & Mueller, R. Dynamic Jahn–Teller effect and colossal magnetoresistance in La1–xSrxMnO3. Phys. Rev. Lett. 77, 175–178 (1996).

    Article  ADS  CAS  PubMed  Google Scholar 

  19. Röder, H., Zang, J. & Bishop, A. R. Lattice effects in the colossal-magnetoresistance manganites. Phys. Rev. Lett. 76, 1356–1359 (1996).

    Article  ADS  PubMed  Google Scholar 

  20. Guzmán-Verri, G. G., Brierley, R. T. & Littlewood, P. B. Cooperative elastic fluctuations provide tuning of the metal–insulator transition. Nature 576, 429–432 (2019).

    Article  ADS  PubMed  Google Scholar 

  21. Keller, H., Bussmann-Holder, A. & Müller, K. A. Jahn–Teller physics and high-Tc superconductivity. Mater. Today 11, 38–46 (2008).

    Article  CAS  Google Scholar 

  22. Wurzenberger, X., Piotrowski, H. & Klüfers, P. A stable molecular entity derived from rare iron(II) minerals: the square-planar high-spin-d6 FeIIO4 chromophore. Angew. Chem. Int. Ed. 50, 4974–4978 (2011).

    Article  CAS  Google Scholar 

  23. Li, D. et al. Superconductivity in an infinite-layer nickelate. Nature 572, 624–627 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Zeng, S. et al. Phase diagram and superconducting dome of infinite-layer Nd1–xSrxNiO2 thin films. Phys. Rev. Lett. 125, 147003 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  25. Boullay, P. et al. Structure determination of a brownmillerite Ca2Co2O5 thin film by precession electron diffraction. Phys. Rev. B 79, 184108 (2009).

    Article  ADS  Google Scholar 

  26. Xiang, L. et al. Exceptional oxygen evolution reactivities on CaCoO3 and SrCoO3. Sci. Adv. 5, eaav6262 (2022).

    Google Scholar 

  27. Wang, Z. L. & Yin, J. S. Cobalt valence and crystal structure of La0.5Sr0.5CoO2.25. Phil. Mag. B 77, 49–65 (1998).

    Article  ADS  CAS  Google Scholar 

  28. Li, H.-B. et al. Dehydration of electrochemically protonated oxide: SrCoO2 with square spin tubes. J. Am. Chem. Soc. 143, 17517–17525 (2021).

    Article  CAS  PubMed  Google Scholar 

  29. Hayward, M. A. et al. The hydride anion in an extended transition metal oxide array: LaSrCoO3H0.7. Science 295, 1882–1884 (2002).

    Article  ADS  CAS  PubMed  Google Scholar 

  30. Lu, N. et al. Electric-field control of tri-state phase transformation with a selective dual-ion switch. Nature 546, 124–128 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  31. Osada, M. et al. Nickelate superconductivity without rare-earth magnetism: (La,Sr)NiO2. Adv. Mater. 33, 2104083 (2021).

    Article  CAS  Google Scholar 

  32. Williams, J. H. The molecular electric quadrupole moment and solid-state architecture. Acc. Chem. Res. 26, 593–598 (1993).

    Article  CAS  Google Scholar 

  33. Radaelli, P. G. et al. Formation of isomorphic Ir3+ and Ir4+ octamers and spin dimerization in the spinel CuIr2S4. Nature 416, 155–158 (2002).

    Article  ADS  CAS  PubMed  Google Scholar 

  34. Khomskii, D. I. & Streltsov, S. V. Orbital effects in solids: basics, recent progress, and opportunities. Chem. Rev. 121, 2992–3030 (2021).

    Article  CAS  PubMed  Google Scholar 

  35. Chowdhury, S. et al. Negative charge-transfer energy in SrCoO2.5 thin films: an interplay between O-2p hole density, charge-transfer energy, charge disproportionation, and ferromagnetic ordering. ACS Appl. Electron. Mater. 2, 3859–3870 (2020).

    Article  CAS  Google Scholar 

  36. Potze, R. H., Sawatzky, G. A. & Abbate, M. Possibility for an intermediate-spin ground state in the charge-transfer material SrCoO3. Phys. Rev. B 51, 11501–11506 (1995).

    Article  ADS  CAS  Google Scholar 

  37. Cui, B. et al. Direct imaging of structural changes induced by ionic liquid gating leading to engineered three-dimensional meso-structures. Nat. Commun. 9, 3055 (2018).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  38. Hidaka, M., Inoue, K., Yamada, I. & Walker, P. J. X-ray diffraction study of the crystal structures of K2CuF4 and K2CuxZn1−xF4. Physica B+C 121, 343–350 (1983).

    Article  CAS  Google Scholar 

  39. Aguado, F., Rodríguez, F., Valiente, R., Señas, A. & Goncharenko, I. Three-dimensional magnetic ordering in the Rb2CuCl4layer perovskite—structural correlations. J. Phys. Condens. Matter 16, 1927–1938 (2004).

    Article  ADS  CAS  Google Scholar 

  40. Cammarata, A. & Rondinelli, J. M. Ferroelectricity from coupled cooperative Jahn–Teller distortions and octahedral rotations in ordered Ruddlesden–Popper manganates. Phys. Rev. B 92, 14102 (2015).

    Article  ADS  Google Scholar 

  41. Savitzky, B. H. et al. Image registration of low signal-to-noise cryo-STEM data. Ultramicroscopy 191, 56–65 (2018).

    Article  CAS  PubMed  Google Scholar 

  42. Campbell, B. J., Stokes, H. T., Tanner, D. E. & Hatch, D. M. ISODISPLACE: a web-based tool for exploring structural distortions. J. Appl. Crystallogr. 39, 607–614 (2006).

    Article  CAS  Google Scholar 

  43. Stokes, H. T., Hatch, D. M. & Wells, J. D. Group-theoretical methods for obtaining distortions in crystals: applications to vibrational modes and phase transitions. Phys. Rev. B 43, 11010–11018 (1991).

  44. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  ADS  CAS  PubMed  Google Scholar 

  45. Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys. Rev. B 57, 1505–1509 (1998).

    Article  ADS  CAS  Google Scholar 

  46. Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).

    Article  ADS  CAS  Google Scholar 

  47. Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).

    Article  ADS  PubMed  Google Scholar 

  48. Tassel, C. et al. CaFeO2: a new type of layered structure with iron in a distorted square planar coordination. J. Am. Chem. Soc. 131, 221–229 (2009).

    Article  CAS  PubMed  Google Scholar 

  49. Goodge, B. H. et al. Doping evolution of the Mott–Hubbard landscape in infinite-layer nickelates. Proc. Natl Acad. Sci. USA 118, e2007683118 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Momma, K. & Izumi, F. VESTA: a three-dimensional visualization system for electronic and structural analysis. J. Appl. Crystallogr. 41, 653–658 (2008).

    Article  CAS  Google Scholar 

Download references


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.

Author information

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.

Corresponding authors

Correspondence to Woo Jin Kim or Harold Y. Hwang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Eric Hoglund and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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)

Supplementary information

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing