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.

A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2-xNaxCuO2Cl2


The phase diagram of hole-doped copper oxides shows four different electronic phases existing at zero temperature. Familiar among these are the Mott insulator, high-transition-temperature superconductor and metallic phases. A fourth phase, of unknown identity, occurs at light doping along the zero-temperature bound of the ‘pseudogap’ regime1. This regime is rich in peculiar electronic phenomena1, prompting numerous proposals that it contains some form of hidden electronic order. Here we present low-temperature electronic structure imaging studies of a lightly hole-doped copper oxide: Ca2-xNaxCuO2Cl2. Tunnelling spectroscopy (at energies |E| > 100 meV) reveals electron extraction probabilities greatly exceeding those for injection, as anticipated for a doped Mott insulator. However, for |E| < 100 meV, the spectrum exhibits a V-shaped energy gap centred on E = 0. States within this gap undergo intense spatial modulations, with the spatial correlations of a four CuO2-unit-cell square ‘checkerboard’, independent of energy. Intricate atomic-scale electronic structure variations also exist within the checkerboard. These data are consistent with an unanticipated crystalline electronic state, possibly the hidden electronic order, existing in the zero-temperature pseudogap regime of Ca2-xNaxCuO2Cl2.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



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

Figure 1: Atomic-scale explorations of electronic states in Ca2-xNaxCuO2Cl2.
Figure 2: Energy dependence of the modulation wavevectors in the electronic crystal state.
Figure 3: Electronic structure imaging within a representative 4a0 × 4a0 ‘tile’.
Figure 4: Doping dependence of electronic structure images.


  1. Timusk, T. & Statt, B. The pseudogap in high-temperature superconductors: an experimental survey. Rep. Prog. Phys. 62, 61–122 (1999)

    Article  ADS  CAS  Google Scholar 

  2. Hoffman, J. E. et al. A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi2Sr2CaCu2O8+δ . Science 295, 466–469 (2002)

    Article  ADS  CAS  Google Scholar 

  3. Vershinin, M. et al. Local ordering in the pseudogap state of the high-Tc superconductor Bi2Sr2CaCu2O8+δ . Science 303, 1995–1998 (2004)

    Article  ADS  CAS  Google Scholar 

  4. McElroy, K. et al. Destruction of antinodal state coherence via ‘checkerboard’ charge ordering in strongly underdoped superconducting Bi2Sr2CaCu2O8+δ. Preprint at 〈〉 (2004).

  5. Howald, C., Fournier, P. & Kapitulnik, A. Inherent inhomogeneities in tunneling spectra of Bi2Sr2CaCu2O8-x crystals in the superconducting state. Phys. Rev. B 64, 100504 (2001)

    Article  ADS  Google Scholar 

  6. Cren, T., Roditchev, D., Sacks, W. & Klein, J. Nanometer scale mapping of the density of states in an inhomogeneous superconductor. Europhys. Lett. 54, 84–90 (2001)

    Article  ADS  CAS  Google Scholar 

  7. McElroy, K. et al. Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+δ . Nature 422, 592–596 (2003)

    Article  ADS  CAS  Google Scholar 

  8. Ronning, F. et al. Photoemission evidence for a remnant Fermi surface and a d-wave-like dispersion in insulating Ca2CuO2Cl2 . Science 282, 2067–2072 (1998)

    Article  ADS  CAS  Google Scholar 

  9. Ronning, F. et al. Evolution of a metal to insulator transition in Ca2-xNaxCuO2Cl2 as seen by angle-resolved photoemission. Phys. Rev. B 67, 165101 (2003)

    Article  ADS  Google Scholar 

  10. Hiroi, Z., Kobayashi, N. & Takano, M. Probable hole-doped superconductivity without apical oxygens in (Ca, Na)2CuO2Cl2 . Nature 371, 139–141 (1994)

    Article  ADS  CAS  Google Scholar 

  11. Kohsaka, Y. et al. Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several GPa. J. Am. Chem. Soc. 124, 12275–12278 (2002)

    Article  CAS  Google Scholar 

  12. Kohsaka, Y. et al. Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator Ca2-xNaxCuO2Cl2 . Phys. Rev. Lett. (2004) (in the press); preprint at 〈

  13. Wei, J. Y. T. et al. Quasiparticle tunneling spectra of the high-Tc mercury cuprates: Implications of the d-wave two-dimensional van Hove scenario. Phys. Rev. B 57, 3650–3662 (1998)

    Article  ADS  CAS  Google Scholar 

  14. Fu, H. C., Davis, J. C. & Lee, D.-H. On the charge ordering observed by recent STM experiments. Preprint at 〈〉 (2004)

  15. Varma, C. M. Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997)

    Article  ADS  CAS  Google Scholar 

  16. Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001)

    Article  ADS  Google Scholar 

  17. Lee, P. A. Pseudogap and competing states in underdoped cuprates. Preprint at 〈〉 (2003).

  18. Zaanen, J. & Gunnarsson, O. Charged magnetic domain lines and the magnetism of high-Tc oxides. Phys. Rev. B 40, 7391–7394 (1989)

    Article  ADS  CAS  Google Scholar 

  19. Machida, K. Magnetism in La2CuO4 based compounds. Physica C 158, 192–196 (1989)

    Article  ADS  CAS  Google Scholar 

  20. White, S. R. & Scalapino, D. J. Phase separation and stripe formation in the two-dimensional t-J model: A comparison of numerical results. Phys. Rev. B 61, 6320–6326 (2000)

    Article  ADS  CAS  Google Scholar 

  21. Emery, V. J., Kivelson, S. A. & Tranquada, J. M. Stripe phases in high-temperature superconductors. Proc. Natl Acad. Sci. USA 96, 8814–8817 (1999)

    Article  ADS  CAS  Google Scholar 

  22. Sachdev, S. Quantum criticality: Competing ground states in low dimensions. Science 288, 475–480 (2000)

    Article  ADS  CAS  Google Scholar 

  23. Bosch, M., van Saarloos, W. & Zaanen, J. Shifting Bragg peaks of cuprate stripes as possible indications for fractionally charged kinks. Phys. Rev. B 63, 092501 (2001)

    Article  ADS  Google Scholar 

  24. Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator. Nature 393, 550–553 (1998)

    Article  ADS  CAS  Google Scholar 

  25. Vojta, M. Superconducting charge-ordered states in cuprates. Phys. Rev. B 66, 104505 (2002)

    Article  ADS  Google Scholar 

  26. Podolsky, D., Demler, E., Damle, K. & Halperin, B. I. Translational symmetry breaking in the superconducting state of the cuprates: Analysis of the quasiparticle density of states. Phys. Rev. B 67, 094514 (2003)

    Article  ADS  Google Scholar 

  27. Chen, H.-D., Vafek, O., Yazdani, A. & Zhang, S.-C. Pair density wave in the pseudogap state of high temperature superconductors. Preprint at 〈〉 (2004).

  28. Chen, H.-D., Hu, J.-P., Capponi, S., Arrigoni, E. & Zhang, S.-C. Antiferromagnetism and hole pair checkerboard in the vortex state of high Tc superconductors. Phys. Rev. Lett. 89, 137004 (2002)

    Article  ADS  Google Scholar 

  29. Tešanović, Z. Charge modulation, spin response, and dual Hofstadter butterfly in high-Tc cuprates. Preprint at 〈〉 (2004).

  30. Anderson, P. W. A suggested 4 × 4 structure in underdoped cuprate superconductors: a Wigner supersolid. Preprint at 〈〉 (2004).

Download references


We acknowledge and thank P. Coleman, E. Demler, M. Franz, J. E. Hoffman, P. A. Lee, K. Machida, K. McElroy, D. Pines, S. Sachdev, T. Senthil, T. Timusk, M. Vojta and J. Zaanen for discussions and communications. This work was supported by the ONR, NSF, MEXT, JST and NEDO. C.L. acknowledges support from a NSERC Postdoctoral Fellowship and Y.K. from a JPSJ Research Fellowship for Young Scientists.

Author information

Authors and Affiliations


Corresponding authors

Correspondence to T. Hanaguri or J. C. Davis.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Supplementary information

Supplementary Figure 1

Doping dependence of the spectra from x = 0.08 to 0.12. (PDF 84 kb)

Supplementary Figure 2

Topography and associated Fourier transform showing the existence of 1/4 and 3/4 signal in topographic signal. (PDF 282 kb)

Supplementary Figure 3

Raw spectra and averages from the white and red points of Fig. 3b from the main text. (PDF 100 kb)

Supplementary Figure 4

Doping dependence of the low temperature resistivity from x = 0.06 to 0.10. (PDF 101 kb)

Supplementary Figure 5

Low energy spectra along a line cut showing heterogeneous features possibly related to superconductivity in x = 0.12 sample. (PDF 278 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hanaguri, T., Lupien, C., Kohsaka, Y. et al. A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2-xNaxCuO2Cl2. Nature 430, 1001–1005 (2004).

Download citation

  • Received:

  • Accepted:

  • 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