- 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

Charge gap and charge redistribution among copper and oxygen orbitals in the normal state of the Emery model

G. L. Reaney1, N. Kowalski2, A.-M. S. Tremblay2, and G. Sordi1,*

  • *Contact author: giovanni.sordi@rhul.ac.uk
PDF

Phys. Rev. B 112, 125106 – Published 2 September, 2025

DOI: https://doi.org/10.1103/7bbg-m38l

Abstract

Unraveling the behavior of the electrons in the copper-oxygen planes of cuprate superconductors remains a challenge. Here we examine the electronic charge redistribution among planar copper and oxygen orbitals and the charge gap using the Emery model in the normal state, solved with cellular dynamical mean-field theory at finite temperature. We quantify the charge redistribution as a function of the onsite Coulomb repulsion on the copper orbitals, the bare copper-oxygen energy difference, and the hole or electron doping. We find that the position relative to the metal to insulator boundary of the Zaanen-Sawatzky-Allen diagram determines the charge redistribution among copper and oxygen orbitals. For a fixed bare Cu-O energy difference, an increase in the Cu electron repulsion leads to a transfer of the electronic charge from Cu to O orbitals. For a fixed charge gap size of the undoped state, as the system evolves from a charge-transfer to a Mott-Hubbard regime, the electronic charge is transferred from Cu to O orbitals. Our findings posit the Coulomb repulsion and the bare charge-transfer energy as key drivers of the microscopic process of charge redistribution in the CuO2 plane. They quantify the anticorrelation between the charge gap size and oxygen hole content. They show that for fixed band-structure parameters, the charge gap and the charge redistribution between Cu and O orbitals provide a way to understand observed trends in cuprates.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (45)

  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. E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys. 66, 763 (1994).
  3. M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).
  4. D. Rybicki, M. Jurkutat, S. Reichardt, C. Kapusta, and J. Haase, Perspective on the phase diagram of cuprate high-temperature superconductors, Nat. Commun. 7, 11413 (2016).
  5. M. Jurkutat, C. Kattinger, S. Tsankov, R. Reznicek, A. Erb, and J. Haase, How pressure enhances the critical temperature of superconductivity in YBa2Cu3O6+y, Proc. Natl. Acad. Sci. USA 120, e2215458120 (2023).
  6. N. Kowalski, S. S. Dash, P. Sémon, D. Sénéchal, and A.-M. Tremblay, Oxygen hole content, charge-transfer gap, covalency, and cuprate superconductivity, Proc. Natl. Acad. Sci. USA 118, e2106476118 (2021).
  7. S. M. O'Mahony, W. Ren, W. Chen, Y. X. Chong, X. Liu, H. Eisaki, S. Uchida, M. H. Hamidian, and J. C. S. Davis, On the electron pairing mechanism of copper-oxide high temperature superconductivity, Proc. Natl. Acad. Sci. USA 119, e2207449119 (2022).
  8. V. J. Emery, Theory of high-Tc superconductivity in oxides, Phys. Rev. Lett. 58, 2794 (1987).
  9. C. Varma, S. Schmitt-Rink, and E. Abrahams, Charge transfer excitations and superconductivity in ionic metals, Solid State Commun. 62, 681 (1987).
  10. J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of transition-metal compounds, Phys. Rev. Lett. 55, 418 (1985).
  11. T. A. Maier, M. Jarrell, T. Pruschke, and M. Hettler, Quantum cluster theories, Rev. Mod. Phys. 77, 1027 (2005).
  12. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006).
  13. A.-M. S. Tremblay, B. Kyung, and D. Sénéchal, Pseudogap and high-temperature superconductivity from weak to strong coupling. Towards a quantitative theory, Low Temp. Phys. 32, 424 (2006).
  14. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  15. O. Andersen, A. Liechtenstein, O. Jepsen, and F. Paulsen, LDA energy bands, low-energy hamiltonians, t′,t″,t⊥(k) and J⊥, J. Phys. Chem. Solids 56, 1573 (1995).
  16. E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time Monte Carlo methods for quantum impurity models, Rev. Mod. Phys. 83, 349 (2011).
  17. P. Werner, A. Comanac, L. de Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impurity models, Phys. Rev. Lett. 97, 076405 (2006).
  18. K. Haule, Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Phys. Rev. B 75, 155113 (2007).
  19. P. Sémon, C.-H. Yee, K. Haule, and A.-M. S. Tremblay, Lazy skip-lists: An algorithm for fast hybridization-expansion quantum Monte Carlo, Phys. Rev. B 90, 075149 (2014).
  20. L. Fratino, P. Sémon, G. Sordi, and A.-M. S. Tremblay, Pseudogap and superconductivity in two-dimensional doped charge-transfer insulators, Phys. Rev. B 93, 245147 (2016).
  21. N. Kowalski, Dopage, temperature critique et étude du modèle de Hubbard à trois bandes, Master's thesis, Université de Sherbrooke, Sherbrooke, QC, Canada, 2021.
  22. G. Sordi, G. L. Reaney, N. Kowalski, P. Sémon, and A.-M. S. Tremblay, Ambipolar doping of a charge-transfer insulator in the emery model, Phys. Rev. B 111, 045117 (2025).
  23. H. Park, K. Haule, and G. Kotliar, Cluster dynamical mean field theory of the Mott transition, Phys. Rev. Lett. 101, 186403 (2008).
  24. A. Go and A. J. Millis, Spatial correlations and the insulating phase of the high-Tc cuprates: Insights from a configuration-interaction-based solver for dynamical mean field theory, Phys. Rev. Lett. 114, 016402 (2015).
  25. D. Bergeron and A.-M. S. Tremblay, Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation, Phys. Rev. E 94, 023303 (2016).
  26. R. T. Scalettar, D. J. Scalapino, R. L. Sugar, and S. R. White, Antiferromagnetic, charge-transfer, and pairing correlations in the three-band Hubbard model, Phys. Rev. B 44, 770 (1991).
  27. E. Arrigoni, M. Aichhorn, M. Daghofer, and W. Hanke, Phase diagram and single-particle spectrum of CuO2 high- Tc layers: Variational cluster approach to the three-band Hubbard model, New J. Phys. 11, 055066 (2009).
  28. Z.-H. Cui, C. Sun, U. Ray, B.-X. Zheng, Q. Sun, and G. K.-L. Chan, Ground-state phase diagram of the three-band Hubbard model from density matrix embedding theory, Phys. Rev. Res. 2, 043259 (2020).
  29. P. W. Anderson, Antiferromagnetism. theory of superexchange interaction, Phys. Rev. 79, 350 (1950).
  30. P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  31. N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17, 1133 (1966).
  32. B. Kyung, S. S. Kancharla, D. Sénéchal, A.-M. S. Tremblay, M. Civelli, and G. Kotliar, Pseudogap induced by short-range spin correlations in a doped Mott insulator, Phys. Rev. B 73, 165114 (2006).
  33. M. Jurkutat, D. Rybicki, O. P. Sushkov, G. V. M. Williams, A. Erb, and J. Haase, Distribution of electrons and holes in cuprate superconductors as determined from O17 and Cu63 nuclear magnetic resonance, Phys. Rev. B 90, 140504(R) (2014).
  34. S. R. White and D. J. Scalapino, Doping asymmetry and striping in a three-orbital CuO2 Hubbard model, Phys. Rev. B 92, 205112 (2015).
  35. Y. F. Kung, C.-C. Chen, Y. Wang, E. W. Huang, E. A. Nowadnick, B. Moritz, R. T. Scalettar, S. Johnston, and T. P. Devereaux, Characterizing the three-orbital Hubbard model with determinant quantum Monte Carlo, Phys. Rev. B 93, 155166 (2016).
  36. B. Ponsioen, S. S. Chung, and P. Corboz, Superconducting stripes in the hole-doped three-band Hubbard model, Phys. Rev. B 108, 205154 (2023).
  37. 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).
  38. A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle-resolved photoemission studies of the cuprate superconductors, Rev. Mod. Phys. 75, 473 (2003).
  39. L.-B. St-Cyr and D. Sénéchal, Effect of the Coulomb repulsion and oxygen level on charge distribution and superconductivity in the Emery model for cuprates superconductors, SciPost Phys. Core 8, 043 (2025).
  40. C. Weber, K. Haule, and G. Kotliar, Strength of correlations in electron- and hole-doped cuprates, Nat. Phys. 6, 574 (2010).
  41. C. Weber, C. Yee, K. Haule, and G. Kotliar, Scaling of the transition temperature of hole-doped cuprate superconductors with the charge-transfer energy, Europhys. Lett. 100, 37001 (2012).
  42. C. Weber, K. Haule, and G. Kotliar, Apical oxygens and correlation strength in electron- and hole-doped copper oxides, Phys. Rev. B 82, 125107 (2010).
  43. Z.-H. Cui, H. Zhai, X. Zhang, and G. K.-L. Chan, Systematic electronic structure in the cuprate parent state from quantum many-body simulations, Science 377, 1192 (2022).
  44. B. Bacq-Labreuil, B. Lacasse, A.-M. S. Tremblay, D. Sénéchal, and K. Haule, Toward an ab initio theory of high-temperature superconductors: A study of multilayer cuprates, Phys. Rev. X 15, 021071 (2025).
  45. Z.-H. Cui, J. Yang, J. Tölle, H.-Z. Ye, S. Yuan, H. Zhai, G. Park, R. Kim, X. Zhang, L. Lin, T. C. Berkelbach, and G. K.-L. Chan, Ab initio quantum many-body description of superconducting trends in the cuprates, Nat. Commun. 16, 1845 (2025).

Outline

Information

Phys. Rev. B 112, 125106– Published 2 September, 2025
  • Received 16 May 2025
  • Revised 23 July 2025
  • Accepted 19 August 2025
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