Superconductivity is described by a gap function as the order parameter. For conventional electron-phonon induced superconductivity, the gap function usually does not have momentum dependence. However, for unconventional superconductivity, the momentum-dependence of gap function is a key property for formulating underlying theories. The momentum-dependent gap function can be obtained by solving the linearized BCS gap equation. Constrained by the lattice symmetry, the gap functions can also be categorized according to their k-space symmetries such as s, d_x²_−y², d_xy or p_x+ip_y, etc. For cuprates, the gap function symmetry is dominantly d_x²_−y².

Recently, La₃Ni₂O₇ exhibits superconductivity near 80 K under around 14 GPa pressure [1]. Unlike cuprates that have a single-sheet Fermi surface, this material has a three-sheet Fermi surface (see Fig. 1a), which consists of the two e_g-orbitals of the two Ni atoms (d_xy and d_3z²_-r² orbitals) [2].  Many theoretical studies construct different models to study this superconductor, yet no consensus has been reached on its superconducting gap symmetry. Some show that the gap symmetry is s-wave (s_± with A_1g symmetry) [3-5], while others show that it is d-wave (d_x²_−y² with B_1g symmetry  or d_xy with B_2g symmetry) [6-8].

In our recent work “Sensitive dependence of pairing symmetry on Ni-e_g crystal field splitting in the nickelate superconductor La₃Ni₂O₇” published in Nature Communications, we show that the superconducting gap symmetry of La₃Ni₂O₇ sensitively depends on its low-energy electronic structure, in particular the crystal field splitting between two Ni-e_g orbitals. As the first-principles electronic structure is exactly reproduced by a downfolded model, the gap symmetry is d_xy. However, slightly increasing the Ni-e_g crystal field changes the gap symmetry from d_xy to s_± (see Fig. 1b). Such a transition is associated with the change in inverse Fermi velocity and susceptibility.  But the shape of Fermi surface remains almost unchanged, as the Ni-e_g crystal field splitting is tuned in a small range.

Our work demonstrates that based on a given tight-binding model, a prediction of La₃Ni₂O₇ gap symmetry could be biased and does not provide the complete picture, considering the sensitive dependence of the gap symmetry on its low-energy electronic structure and inevitable approximations/uncertainties introduced in the modelling. Instead, the trend that a smaller (larger) Ni-e_g crystal field splitting favors d_xy (s_±) pairing symmetry is robust. This also implies that La₃Ni₂O₇ may exhibit distinct pairing symmetries under experimental perturbations that can tune Ni-e_g crystal field splitting. The sensitive dependence of pairing symmetry on the low-energy electronic structure may be general to other multi-orbital superconductors, such as La₄Ni₃O₁₀. Thus, all the “fine details” of Fermi surface need to be considered for the study of gap symmetry of multi-orbital superconductors.

For further information, please read our published article: https://www.nature.com/articles/s41467-025-56206-0

Reference:

[1] Sun, H. et al. Signatures of superconductivity near 80 K in a nickelate under high pressure. Nature 621, 493 (2023).

[2] Luo, Z. et al. Bilayer two-orbital model of La₃Ni₂O₇ under pressure. Phys. Rev. Lett. 131, 126001 (2023).

[3] Liu, Y.-B. et al. s^±-wave pairing and the destructive role of apical-oxygen deficiencies in La₃Ni₂O₇ under pressure. Phys. Rev. Lett. 131, 236002 (2023).

[4] Zhang, Y. et al. Structural phase transition, s_±-wave pairing, and magnetic stripe order in bilayered superconductor La₃Ni₂O₇ under pressure. Nat. Commun. 15, 2470 (2024).

[5] Qu, X.-Z. et al. Bilayer t−J−J_⊥ model and magnetically mediated pairing in the pressurized nickelate La₃Ni₂O₇. Phys. Rev. Lett. 132,036502 (2024).

[6] Lechermann, F. et al. Electronic correlations and superconducting instability in La₃Ni₂O₇ under high pressure. Phys. Rev. B 108, L201121 (2023).

[7] Heier, G., Park, K. & Savrasov, S. Y. Competing d_xy and s_± pairing symmetries in superconducting La₃Ni₂O₇: LDA + FLEX calculations. Phys. Rev. B 109, 104508 (2024).

[8] Fan, Z. et al. Superconductivity in nickelate and cuprate superconductors with strong bilayer coupling. Phys. Rev. B 110, 024514 (2024).