A 2022 scanning-tunneling experiment (O'Mahony et al., PNAS 119, e2207449119) combined single-electron and electron-pair (Josephson) STM in Bi2Sr2CaCu2O8+δ to report that the local pair density tracks the local charge-transfer energy as superexchange theory predicts, and an influential essay (Singh, arXiv:2211.00400) declared the 40-year-old pairing-mechanism problem thereby settled in favor of charge-transfer superexchange in the three-band Emery model. The status of three components of the mechanism problem is contested: the causal attribution of the observed pair-density/charge-transfer-energy correlations, the mechanism's generality across cuprate families and dopings, and the quantitative closure of TcT_c, the superconducting dome, and the proportionality A∝TcA\propto T_c.

Background

Cuprate superconductors, discovered in 1986, are layered copper oxides whose CuO2 planes, when hole-doped away from an antiferromagnetic insulating parent state, superconduct at critical temperatures TcT_c up to well above 100 K, with TcT_c following a dome-shaped dependence on hole doping pp. The parent compounds are charge-transfer insulators: in the classification of the three-band Emery model (V. J. Emery, Phys. Rev. Lett. 58, 2794 (1987)), the relevant orbitals are Cu 3dx2−y23d_{x^2-y^2} and O 2px,2py2p_x,2p_y, with a charge-transfer energy EE (often written ϵpd\epsilon_{pd} or Δpd\Delta_{pd}) separating the oxygen and copper levels. Virtual hopping tt across this gap generates an antiferromagnetic 'superexchange' interaction, which in the strong-coupling analysis of the 2022 PNAS experiment scales as J≈4t4/E3J\approx 4t^4/E^3. P. W. Anderson's resonating-valence-bond (RVB) proposal (Science 235, 1196 (1987)) — a quantum-melted state of spin singlets — and the subsequent one-band tt-JJ reduction identify this superexchange as the candidate glue that binds doped holes into Cooper pairs, in contrast to the phonon-mediated glue of BCS superconductivity. Independent of any microscopic proof, the maximum TcT_c of hole-doped cuprate families had earlier been found empirically to scale with the computed charge-transfer energy across compounds (C. Weber, C. Yee, K. Haule, G. Kotliar, EPL 100, 37001 (2012)), and cluster dynamical mean-field theory (c-DMFT) studies of the three-band Hubbard model (N. Kowalski et al., PNAS 118, e2106476118 (2021)) found superconductivity controlled by the charge-transfer gap, while density-matrix-renormalization-group (DMRG, a controlled one-dimensional-chain numerics) studies found pairing correlations in the same model (P. Mai et al., Phys. Rev. B 103, 144514 (2021)).

The pivotal 2022 experiment visualized both ingredients of this proposal simultaneously in Bi2Sr2CaCu2O8+δ (Bi-2212) at optimal doping: superconducting-insulating STM (SISTM, conventional tunneling with a normally conducting tip at elevated voltage) mapped the local charge-transfer energy E(r)E(\mathbf{r}), while superconducting-Josephson STM (SJTM, tunneling with a superconducting tip, whose Josephson critical current is proportional to the local electron-pair density nPn_P) mapped nP(r)n_P(\mathbf{r}). Bi-2212 has a natural crystal 'supermodulation' — a wavy periodic (~26 Å) displacement of the lattice, including the apical oxygen height δ(r)\delta(\mathbf{r}) above the planar Cu — which modulates EE. Eliminating the common variable, the experiment found dnP/dE≈−0.81±0.17dn_P/dE\approx -0.81\pm 0.17 eV−1^{-1} (pair density decreasing as the charge-transfer energy grows), compared with a theoretical d∣⟨c↑c↓⟩∣/dE≈−0.46±0.05d|\langle c_\uparrow c_\downarrow\rangle|/dE\approx -0.46\pm 0.05 eV−1^{-1} from three-band strong-correlation calculations. The authors concluded that, taken at face value, these data indicate charge-transfer superexchange as the pairing mechanism. Navinder Singh's essay (arXiv:2211.00400) then asked whether the cuprate problem is thereby 'solved', answering yes for the pairing mechanism while listing unsettled items: the linear-in-TT resistivity of the strange-metal regime, the non-Drude 1/τ∝ω1/\tau\propto\omega scattering, the descent of the overdoped dome, and the drop of the effective carrier count at the pseudogap critical doping p∗p^*. Understanding what is actually settled requires knowing that the causal chain from apical-oxygen displacement to pair-density modulation passes through several simultaneously varying quantities — the charge-transfer energy, hopping amplitudes, and the effective hole doping of the CuO2 plane — and that the SJTM critical current between an s-wave tip and a d-wave sample has strong, theory-dependent variations within the unit cell.

Problem Statement

Determine whether charge-transfer superexchange, in the three-band (Emery-model) description of the CuO2 plane, is the microscopic electron-pairing mechanism of cuprate high-temperature superconductivity — i.e., settle which components of the mechanism problem the 2022 Bi-2212 STM/Josephson-STM verification actually closed. Concretely, three jointly defining sub-questions structure the determination: (i) causal attribution — is the observed covariation of local pair density with local charge-transfer energy (equivalently, with apical-oxygen displacement) caused by superexchange-controlled pairing, or by competing effects such as apical-oxygen-induced redistribution of the effective CuO2 hole doping or Kondo-type sd exchange, given that the measured slope dnP/dEdn_P/dE must be compared through the site-dependent Josephson-tunneling kernel and against controlled three-band calculations with quantified uncertainties; (ii) generality — does the mechanism hold across cuprate families (Bi-, Y-, Hg-based, and ideally electron-doped compounds) and across the doping dome, not only near optimal doping in one material; and (iii) quantitative closure — can a microscopic three-band (or rigorously justified reduced) Hamiltonian with experimentally determined parameters quantitatively reproduce the pairing scale: the magnitude and doping dependence of TcT_c, the dome shape including its overdoped descent and the relation to the pseudogap critical doping p∗p^*, and the empirical proportionality between the linear-resistivity coefficient AA (defined by the strange-metal normal-state resistivity ρ(T)=ρ0+AT\rho(T)=\rho_0+AT) and TcT_c? An answer adjudicates each of these three sub-questions as closed or open: the verdict may be that the mechanism is established, that it is refuted (by showing the data admit a quantitatively superior non-superexchange account, or contradict superexchange predictions), or that it is not established or only partially closed. No independent derivation of TcT_c, the superconducting dome, or the proportionality A∝TcA\propto T_c is required to complete the task.

References

  1. Shane M. O'Mahony, Wangping Ren, Weijiong Chen, Yi Xue Chong, Xiaolong Liu, H. Eisaki, S. Uchida, M. H. Hamidian, J. C. Séamus Davis, 'On the electron pairing mechanism of copper-oxide high temperature superconductivity', Proc. Natl. Acad. Sci. U.S.A. 119, e2207449119 (2022). DOI: 10.1073/pnas.2207449119, https://arxiv.org/abs/2108.03655
  2. Navinder Singh, 'Is the problem of Cuprate high-Tc superconductivity a solved problem?', arXiv:2211.00400 (2022), https://arxiv.org/abs/2211.00400
  3. Navinder Singh Bathinda, '40 years of cuprate high-Tc superconductors: a perspective on theories', arXiv:2606.29249 (2026), https://arxiv.org/abs/2606.29249
  4. Samuel Vadnais, Rémi Duchesne, Kristjan Haule, A.-M. S. Tremblay, David Sénéchal, Benjamin Bacq-Labreuil, 'The role of the apical oxygen in cuprate high-temperature superconductors', arXiv:2601.16017 (2026), https://arxiv.org/abs/2601.16017
  5. Albert M. Varonov, Todor M. Mishonov, 'On the theory of supermodulation of the superconducting order parameter created by structural supermodulation of apex distance in optimally doped Bi2_2Sr2_2CaCu2_2O8+x_{8+x}', arXiv:2506.15726 (2025), https://arxiv.org/abs/2506.15726
  6. Peayush Choubey, P. J. Hirschfeld, 'Theory of Josephson scanning microscopy with ss-wave tip on unconventional superconducting surface: Application to Bi2_2Sr2_2CaCu2_2O8+δ_{8+\delta}', Phys. Rev. B 110, 184519 (2024). DOI: 10.1103/PhysRevB.110.184519, preprint arXiv:2406.09878, https://arxiv.org/abs/2406.09878
  7. Felix A. Buot, Roland E. S. Otadoy, Unofre Pili, 'Entanglement and confinement: A new pairing mechanism in high-T_C cuprates', arXiv:2410.02300 (2024), https://arxiv.org/abs/2410.02300
  8. Patrick Navez, 'Screened superexchange mechanism for superconductivity applied to cuprates', arXiv:2507.08750 (2025), https://arxiv.org/abs/2507.08750
  9. Nicolas Kowalski, Sidhartha Shankar Dash, Patrick Sémon, David Sénéchal, André-Marie Tremblay, 'Oxygen hole content, charge-transfer gap, covalency, and cuprate superconductivity', Proc. Natl. Acad. Sci. U.S.A. 118, e2106476118 (2021). DOI: 10.1073/pnas.2106476118, https://arxiv.org/abs/2104.07087
  10. Peizhi Mai, Giovanni Balduzzi, Steven Johnston, Thomas A. Maier, 'Pairing correlations in the cuprates: a numerical study of the three-band Hubbard model', Phys. Rev. B 103, 144514 (2021). DOI: 10.1103/PhysRevB.103.144514, https://arxiv.org/abs/2101.10965
  11. Shengtao Jiang, Douglas J. Scalapino, Steven R. White, 'Density-matrix-renormalization-group-based downfolding of the three-band Hubbard model: the importance of density-assisted hopping', Phys. Rev. B 108, L161111 (2023). DOI: 10.1103/PhysRevB.108.L161111, https://arxiv.org/abs/2303.00756
  12. V. J. Emery, 'Theory of high-Tc superconductivity in oxides', Phys. Rev. Lett. 58, 2794 (1987). DOI: 10.1103/PhysRevLett.58.2794, https://doi.org/10.1103/PhysRevLett.58.2794
  13. C. Weber, C. Yee, K. Haule, G. Kotliar, 'Scaling of the transition temperature of hole-doped cuprate superconductors with the charge-transfer energy', EPL 100, 37001 (2012). DOI: 10.1209/0295-5075/100/37001, https://doi.org/10.1209/0295-5075/100/37001
  14. P. W. Anderson, 'The Resonating Valence Bond State in La2CuO4 and Superconductivity', Science 235, 1196 (1987). DOI: 10.1126/science.235.4793.1196, https://doi.org/10.1126/science.235.4793.1196