- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Doping evolution of the charge density wave and charge density fluctuations in La2−xSrxCuO4

Charles C. Tam1,2,*, Mengze Zhu1,†,‡, Maud C. Barthélemy1, Lauren J. Cane1, Oliver J. Lipscombe1, Stefano Agrestini2, Jaewon Choi2, Mirian Garcia-Fernandez2, Ke-Jin Zhou2,§ et al.

Stephen M. Hayden1,∥

  • *Contact author: ctam@ucsb.edu
  • †Contact author: zhumen@phys.ethz.ch
  • ‡Present address: Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland.
  • §Present address: National Synchrotron Radiation Laboratory and School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, China.
  • ∥Contact author: s.hayden@bristol.ac.uk
PDF

Phys. Rev. B 113, 174506 – Published 13 May, 2026

DOI: https://doi.org/10.1103/6hcs-2y4y

Abstract

Cuprate superconductors show various collective charge correlations that are intimately connected with their electronic properties. In particular, charge order in the form of an incommensurate charge density wave (CDW) order with an in-plane wave vector δCDW≈ 0.23–0.35 reciprocal lattice units appears to be universally present. In addition to CDW, dynamic charge density fluctuations (CDFs) are also present with wave vectors comparable to δCDW. CDFs are present up to ≈300K and have relatively short correlation lengths of ξ≈20Å. Here we use Cu-L3 and O-K resonant inelastic x-ray scattering (RIXS) to study the doping dependence of CDW and CDFs in La2−xSrxCuO4. We fit our data with (quasi)elastic peaks resulting from the CDW and up to four inelastic modes associated with oxygen phonons that can be strongly coupled to the CDFs. Our analysis allows us to separate the charge correlations into three components: the CDW with wave vector δ4a-CDW≈0.24 and two CDF components with δ4a-CDF≈0.24 and δ3a-CDF≈0.30. We find that for T≈Tc the CDW coexists with the CDFs for dopings near x=p≈1/8. The 4a-CDW disappears beyond x=0.16 and the 4a-CDF beyond x=0.19, leaving only a weak 3a-CDF at the highest doping studied, x=0.22. Our data suggest that low-energy charge fluctuations exist up to doping x=0.19=p★, where the pseudogap disappears; however, we find no evidence that they are associated with a quantum critical point.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (50)

  1. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
  2. S. M. Hayden and J. M. Tranquada, Charge correlations in cuprate superconductors, Annu. Rev. Condens. Matter Phys. 15, 215 (2024).
  3. P. Monceau, Electronic crystals: An experimental overview, Adv. Phys. 61, 325 (2012).
  4. G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Long-range incommensurate charge fluctuations in (Y, Nd)BaCu3O6δ, Science 337, 821 (2012).
  5. W. Tabis, B. Yu, I. Bialo, M. Bluschke, T. Kolodziej, A. Kozlowski, E. Blackburn, K. Sen, E. M. Forgan, M. v. Zimmermann, Y. Tang, E. Weschke, B. Vignolle, M. Hepting, H. Gretarsson, R. Sutarto, F. He, M. Le Tacon, N. Barišić, G. Yu, et al., Synchrotron x-ray scattering study of charge-density-wave order in HgBa2CuO4+δ, Phys. Rev. B 96, 134510 (2017).
  6. W. S. Lee, K.-J. Zhou, M. Hepting, J. Li, A. Nag, A. C. Walters, M. Garcia-Fernandez, H. C. Robarts, M. Hashimoto, H. Lu, B. Nosarzewski, D. Song, H. Eisaki, Z. X. Shen, B. Moritz, J. Zaanen, and T. P. Devereaux, Spectroscopic fingerprint of charge order melting driven by quantum fluctuations in a cuprate, Nat. Phys. 17, 53 (2021).
  7. J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan, and S. M. Hayden, Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67, Nat. Phys. 8, 871 (2012).
  8. T. P. Croft, C. Lester, M. S. Senn, A. Bombardi, and S. M. Hayden, Charge density wave fluctuations in La2−xSrxCuO4 and their competition with superconductivity, Phys. Rev. B 89, 224513 (2014).
  9. V. Thampy, M. P. M. Dean, N. B. Christensen, L. Steinke, Z. Islam, M. Oda, M. Ido, N. Momono, S. B. Wilkins, and J. P. Hill, Rotated stripe order and its competition with superconductivity in La1.88Sr0.12CuO4, Phys. Rev. B 90, 100510(R) (2014).
  10. C. C. Tam, M. Zhu, J. Ayres, K. Kummer, F. Yakhou-Harris, J. R. Cooper, A. Carrington, and S. M. Hayden, Charge density waves and Fermi surface reconstruction in the clean overdoped cuprate superconductor Tl2Ba2CuO6+δ, Nat. Commun. 13, 570 (2022).
  11. H. Y. Huang, A. Singh, C. Y. Mou, S. Johnston, A. F. Kemper, J. van den Brink, P. J. Chen, T. K. Lee, J. Okamoto, Y. Y. Chu, J. H. Li, S. Komiya, A. Komarek, A. C. Fujimori, C. T. Chen, and D. J. Huang, Quantum fluctuations of charge order induce phonon softening in a superconducting cuprate, Phys. Rev. X 11, 041038 (2021).
  12. R. Arpaia, S. Caprara, R. Fumagalli, G. De Vecchi, Y. Y. Peng, E. Andersson, D. Betto, G. M. De Luca, N. B. Brookes, F. Lombardi, M. Salluzzo, L. Braicovich, C. Di Castro, M. Grilli, and G. Ghiringhelli, Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-Tc superconductor, Science 365, 906 (2019).
  13. B. Yu, W. Tabis, I. Bialo, F. Yakhou, N. B. Brookes, Z. Anderson, Y. Tang, G. Yu, and M. Greven, Unusual dynamic charge correlations in simple-tetragonal HgBa2CuO4+δ, Phys. Rev. X 10, 021059 (2020).
  14. J.-J. Wen, H. Huang, S.-J. Lee, H. Jang, J. Knight, Y. S. Lee, M. Fujita, K. M. Suzuki, S. Asano, S. A. Kivelson, C.-C. Kao, and J.-S. Lee, Observation of two types of charge-density-wave orders in superconducting La2−xSrxCuO4, Nat. Commun. 10, 3269 (2019).
  15. H. Miao, G. Fabbris, R. J. Koch, D. G. Mazzone, C. S. Nelson, R. Acevedo-Esteves, G. D. Gu, Y. Li, T. Yilimaz, K. Kaznatcheev, E. Vescovo, M. Oda, T. Kurosawa, N. Momono, T. Assefa, I. K. Robinson, E. S. Bozin, J. M. Tranquada, P. D. Johnson, and M. P. M. Dean, Charge density waves in cuprate superconductors beyond the critical doping, npj Quantum Mater. 6, 31 (2021).
  16. K. von Arx, Q. Wang, S. Mustafi, D. G. Mazzone, M. Horio, D. J. Mukkattukavil, E. Pomjakushina, S. Pyon, T. Takayama, H. Takagi, T. Kurosawa, N. Momono, M. Oda, N. B. Brookes, D. Betto, W. Zhang, T. C. Asmara, Y. Tseng, T. Schmitt, Y. Sassa, et al., Fate of charge order in overdoped La-based cuprates, npj Quantum Mater. 8, 7 (2023).
  17. B. Vignolle, S. M. Hayden, D. F. McMorrow, H. M. Rønnow, B. Lake, C. D. Frost, and T. G. Perring, Two energy scales in the spin excitations of the high-temperature superconductor La2−xSrxCuO4, Nat. Phys. 3, 163 (2007).
  18. R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. Nohara, H. Takagi, C. Proust, and N. E. Hussey, Anomalous criticality in the electrical resistivity of La2−xSrxCuO4, Science 323, 603 (2009).
  19. K.-J. Zhou, A. Walters, M. Garcia-Fernandez, T. Rice, M. Hand, A. Nag, J. Li, S. Agrestini, P. Garland, H. Wang, S. Alcock, I. Nistea, B. Nutter, N. Rubies, G. Knap, M. Gaughran, F. Yuan, P. Chang, J. Emmins, and G. Howell, I21: An advanced high-resolution resonant inelastic x-ray scattering beamline at diamond light source, J. Synchrotron Radiat. 29, 563 (2022).
  20. J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of transition-metal compounds, Phys. Rev. Lett. 55, 418 (1985).
  21. F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B 37, 3759 (1988).
  22. C. T. Chen, F. Sette, Y. Ma, M. S. Hybertsen, E. B. Stechel, W. M. C. Foulkes, M. Schulter, S.-W. Cheong, A. S. Cooper, L. W. Rupp, B. Batlogg, Y. L. Soo, Z. H. Ming, A. Krol, and Y. H. Kao, Electronic states in La2−xSrxCuO4+δ probed by soft-x-ray absorption, Phys. Rev. Lett. 66, 104 (1991).
  23. L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, Resonant inelastic x-ray scattering studies of elementary excitations, Rev. Mod. Phys. 83, 705 (2011).
  24. T. P. Devereaux, A. M. Shvaika, K. Wu, K. Wohlfeld, C. J. Jia, Y. Wang, B. Moritz, L. Chaix, W.-S. Lee, Z.-X. Shen, G. Ghiringhelli, and L. Braicovich, Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic x-ray scattering, Phys. Rev. X 6, 041019 (2016).
  25. L. Chaix, G. Ghiringhelli, Y. Y. Peng, M. Hashimoto, B. Moritz, K. Kummer, N. B. Brookes, Y. He, S. Chen, S. Ishida, Y. Yoshida, H. Eisaki, M. Salluzzo, L. Braicovich, Z.-X. Shen, T. P. Devereaux, and W.-S. Lee, Dispersive charge density wave excitations in Bi2Sr2CaCu2O8+δ, Nat. Phys. 13, 952 (2017).
  26. T. Fukuda, J. Mizuki, K. Ikeuchi, K. Yamada, A. Q. R. Baron, and S. Tsutsui, Doping dependence of softening in the bond-stretching phonon mode of La2−xSrxCuO4(0≤x≤0.29), Phys. Rev. B 71, 060501(R) (2005).
  27. S. R. Park, T. Fukuda, A. Hamann, D. Lamago, L. Pintschovius, M. Fujita, K. Yamada, and D. Reznik, Evidence for a charge collective mode associated with superconductivity in copper oxides from neutron and x-ray scattering measurements of La2−xSrxCuO4, Phys. Rev. B 89, 020506(R) (2014).
  28. L. Pintschovius, N. Pyka, W. Reichardt, A. Rumiantsev, N. Mitrofanov, A. Ivanov, G. Collin, and P. Bourges, Lattice dynamical studies of HTSC materials, Phys. C (Amsterdam, Neth.) 185–189, 156 (1991).
  29. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6hcs-2y4y for fits of RIXS data to three-peak and five-peak models and E−H RIXS colormaps with the quasi-elastic component subtracted.
  30. J. Q. Lin, H. Miao, D. G. Mazzone, G. D. Gu, A. Nag, A. C. Walters, M. García-Fernández, A. Barbour, J. Pelliciari, I. Jarrige, M. Oda, K. Kurosawa, N. Momono, K.-J. Zhou, V. Bisogni, X. Liu, and M. P. M. Dean, Strongly correlated charge density wave in La2−xSrxCuO4 evidenced by doping-dependent phonon anomaly, Phys. Rev. Lett. 124, 207005 (2020).
  31. R. Arpaia, L. Martinelli, M. M. Sala, S. Caprara, A. Nag, N. B. Brookes, P. Camisa, Q. Li, Q. Gao, X. Zhou, M. Garcia-Fernandez, K.-J. Zhou, E. Schierle, T. Bauch, Y. Y. Peng, C. Di Castro, M. Grilli, F. Lombardi, L. Braicovich, and G. Ghiringhelli, Signature of quantum criticality in cuprates by charge density fluctuations, Nat. Commun. 14, 7198 (2023).
  32. C. Zou, J. Choi, Q. Li, S. Ye, C. Yin, M. Garcia-Fernandez, S. Agrestini, Q. Qiu, X. Cai, Q. Xiao, X. Zhou, K.-J. Zhou, Y. Wang, and Y. Peng, Evolution from a charge-ordered insulator to a high-temperature superconductor in Bi2Sr2(Ca,Dy)Cu2O8+δ, Nat. Commun. 15, 7739 (2024).
  33. O. Cyr-Choinière, R. Daou, F. Laliberté, C. Collignon, S. Badoux, D. LeBoeuf, J. Chang, B. J. Ramshaw, D. A. Bonn, W. N. Hardy, R. Liang, J.-Q. Yan, J.-G. Cheng, J.-S. Zhou, J. B. Goodenough, S. Pyon, T. Takayama, H. Takagi, N. Doiron-Leyraud, and L. Taillefer, Pseudogap temperature T* of cuprate superconductors from the Nernst effect, Phys. Rev. B 97, 064502 (2018).
  34. K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. Greven, M. A. Kastner, and Y. J. Kim, Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in La2−xSrxCuO4, Phys. Rev. B 57, 6165 (1998).
  35. E. Blackburn, J. Chang, A. H. Said, B. M. Leu, R. Liang, D. A. Bonn, W. N. Hardy, E. M. Forgan, and S. M. Hayden, Inelastic x-ray study of phonon broadening and charge-density wave formation in ortho-II-ordered YBa2Cu3O6.54, Phys. Rev. B 88, 054506 (2013).
  36. M. Le Tacon, A. Bosak, S. M. Souliou, G. Dellea, T. Loew, R. Heid, K.-P. Bohnen, G. Ghiringhelli, M. Krisch, and B. Keimer, Inelastic x-ray scattering in YBa2Cu3O6.6 reveals giant phonon anomalies and elastic central peak due to charge-density-wave formation, Nat. Phys. 10, 52 (2014).
  37. X. Hong, Y. Yan, L. Martinelli, I. Biało, K. von Arx, J. Choi, Y. Sassa, S. Pyon, T. Takayama, H. Takagi, Z. Li, M. Garcia-Fernandez, K.-J. Zhou, J. Chang, and Q. Wang, Observation of collective charge excitations in a cuprate superconductor, arXiv:2503.04334.
  38. K. Fujita, C. K. Kim, I. Lee, J. Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J. Lawler, E.-A. Kim, and J. C. Davis, Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking, Science 344, 612 (2014).
  39. M. H. Hamidian, S. D. Edkins, C. K. Kim, J. C. Davis, A. P. Mackenzie, H. Eisaki, S. Uchida, M. J. Lawler, E.-A. Kim, S. Sachdev, and K. Fujita, Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state, Nat. Phys. 12, 150 (2016).
  40. G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, and J. Kulda, Nearly singular magnetic fluctuations in the normal state of a high-Tc cuprate superconductor, Science 278, 1432 (1997).
  41. M. Zhu, D. J. Voneshen, S. Raymond, O. J. Lipscombe, C. C. Tam, and S. M. Hayden, Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor, Nat. Phys. 19, 99 (2023).
  42. J. Radaelli, O. J. Lipscombe, M. Zhu, J. R. Stewart, A. A. Patel, S. Sachdev, and S. M. Hayden, Critical spin fluctuations across the superconducting dome in La2−xSrxCuO4, Nat. Commun. (2026).
  43. J. Zaanen and O. Gunnarsson, Charged magnetic domain lines and the magnetism of high-Tc oxides, Phys. Rev. B 40, 7391 (1989).
  44. J. M. Tranquada, Cuprate superconductors as viewed through a striped lens, Adv. Phys. 69, 437 (2020).
  45. W. A. Atkinson, A. P. Kampf, and S. Bulut, Charge order in the pseudogap phase of cuprate superconductors, New J. Phys. 17, 013025 (2015).
  46. A. Thomson and S. Sachdev, Charge ordering in three-band models of the cuprates, Phys. Rev. B 91, 115142 (2015).
  47. P. Mai, B. Cohen-Stead, T. A. Maier, and S. Johnston, Fluctuating charge-density-wave correlations in the three-band Hubbard model, Proc. Natl. Acad. Sci. USA 121, e2408717121 (2024).
  48. B. Michon, C. Girod, S. Badoux, J. Kačmarčík, Q. Ma, M. Dragomir, H. A. Dabkowska, B. D. Gaulin, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Verret, N. Doiron-Leyraud, C. Marcenat, L. Taillefer, and T. Klein, Thermodynamic signatures of quantum criticality in cuprate superconductors, Nature (London) 567, 218 (2019).
  49. S. A. Hartnoll and A. P. Mackenzie, Colloquium: Planckian dissipation in metals, Rev. Mod. Phys. 94, 041002 (2022).
  50. C. Tam and S. Hayden, Data for Doping evolution of the charge density wave and charge density fluctuations in La2−xSrxCuO4, Zenodo (2026), doi:10.5281/zenodo.19711629.

Outline

Information

Phys. Rev. B 113, 174506– Published 13 May, 2026
  • Received 12 August 2025
  • Revised 17 February 2026
  • Accepted 22 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation