Emergent s+id Superconductivity from the Interplay between Electronic Correlations and Electron-Phonon Coupling in R₁₋ₓSrₓNiO₂

Summary

Combining first-principles calculations with fluctuation exchange-Migdal-Eliashberg theory, this study investigates the interplay between electron correlations and electron-phonon coupling in infinite-layer nickelate superconductors. The results show that spin fluctuations drive robust d-wave superconductivity in the Ni dx2-y2 orbital, while electron-phonon coupling induces s-wave pairing in interstitial orbitals, and their synergy gives rise to a mixed s+id superconducting state. The emergence of the s-wave component strongly depends on carrier density: a moderate electron-phonon coupling strength (λ=0.4) stabilizes the mixed state only at an electron density n=0.9, but not at n=0.8. In the thermodynamic limit, the critical coupling required to stabilize the s-wave component is about 0.6, but it can be reduced to 0.4 in finite-size systems. These results reveal that local oxygen defects, by modulating the local electron density, can form finite-size domains with distinct pairing symmetries, thereby providing a microscopic explanation for the spatially inhomogeneous superconducting gaps observed experimentally, and highlight the crucial influence of the cooperative effect of electron correlations and electron-phonon coupling on the pairing symmetry in nickelate superconductors.

Materials

• Nd₀.8Sr₀.2NiO₂

Methods

• DFT
• FLEX
• Migdal-Eliashberg theory
• Padé approximation

Keywords

• s+id superconductivity
• d wave pairing
• s wave pairing
• electron correlations
• electron phonon coupling
• spin fluctuations
• oxygen defects
• finite size domains

Highlights

• An intermediate electron-phonon coupling of λ=0.4 can stabilize the s+id state on finite-size lattices, consistent with physically estimated values for Nd₀.8Sr₀.2NiO₂.
• The s+id state emerges from the interplay between spin-fluctuation-driven d-wave pairing on the Ni orbital and phonon-driven s-wave pairing on the interstitial orbital.
• Our results provide a microscopic explanation for the spatially inhomogeneous superconducting gaps observed in scanning tunneling spectroscopy on Nd₀.8Sr₀.2NiO₂ thin films.

Conclusions

• Spin fluctuations drive robust d-wave superconductivity on the Ni dx2-y2 orbital, whereas electron-phonon coupling promotes s-wave superconductivity in the interstitial orbital, leading to an s+id superconducting state.
• The emergence of the s-wave component is strongly carrier-density dependent: an intermediate electron-phonon coupling stabilizes the s+id state at electron density n=0.9 but not at n=0.8.
• Local oxygen defects modulate local electron density and form finite-size domains with distinct pairing symmetries, explaining the spatially inhomogeneous superconducting gaps observed experimentally.

Main claims

• Spin fluctuations drive robust d-wave superconductivity on the Ni dx2-y2 orbital.
  • Evidence: FLEX-ME calculations show a large d-wave order parameter on orbital 1 across a wide doping range, persisting even without e-ph coupling (Fig. 2).
• Electron-phonon coupling induces s-wave pairing on the interstitial orbital.
  • Evidence: For non-zero lambda, an s-wave order parameter emerges on orbital 2, with magnitude increasing with lambda (Fig. 2 and 4).
• The interplay yields an s+id superconducting state at intermediate electron-phonon coupling and appropriate doping.
  • Evidence: Coexistence of d-wave and s-wave order parameters is seen at n=0.9, lambda=0.4 in the phase diagram and gap plots (Fig. 2, 4).
• Emergence of the s-wave component is carrier-density dependent: lambda=0.4 stabilizes s+id at n=0.9 but not at n=0.8.
  • Evidence: Phase diagrams in Fig. 2: at n=0.9, a finite s-wave T_c appears for lambda=0.4; at n=0.8, no s-wave order for lambda=0.4.
• Local oxygen defects create finite-size domains with varying electron density, giving rise to spatially distinct pairing symmetries and explaining scanning tunneling observations.
  • Evidence: The finite-size lattice results show that the critical lambda is reduced from 0.6 (thermodynamic limit) to ≈0.4; experimental spectra (Fig. 4e) show position-dependent gap features that can be matched by calculations with different defect densities.

Workflow

• phonon_properties_calculation — DFT reveals moderate electron-phonon coupling for an in-plane oxygen phonon mode, providing a lower bound for the true coupling strength in strongly correlated nickelates.
  • Materials: infinite-layer nickelate R_1-xSr_xNiO₂
  • Methods: density functional theory (DFT); Quantum ESPRESSO package; norm-conserving Vanderbilt pseudopotential
  • Observations: e-ph spectral function peak in 30–40 meV range; total dimensionless e-ph coupling lambda=0.3, oxygen mode contributes one-third (lambdaapprox0.1)
• model_construction — A minimal two-orbital model incorporating Coulomb repulsion and electron-phonon coupling captures the essential physics of superconducting nickelates.
  • Materials: two-orbital model with Ni dx2-y2 and interstitial s-like orbital
  • Methods: tight-binding Hamiltonian; on-site Hubbard U on Ni orbital; linear electron-phonon coupling to oxygen phonon mode
  • Observations: model captures both spin-fluctuation and phonon-mediated pairing channels
• FLEX-ME_simulations — The interplay of electronic correlations and electron-phonon coupling yields a mixed s+id superconducting state at moderate doping and coupling.
  • Methods: fluctuation-exchange (FLEX) + Migdal-Eliashberg (ME) formalism; self-consistent calculations in the superconducting state; projection onto d-wave and s-wave order parameters
  • Observations: d-wave order parameter from spin fluctuations, s-wave from e-ph coupling; coexistence regime for lambda=0.4 at n=0.9 but not at n=0.8
• phase_diagram_and_gap_analysis — Calculated spectral gaps and phase boundaries are consistent with experimental tunnelling data when finite-size domains and oxygen defects are considered.
  • Methods: temperature-dependent anomalous self-energy fitting; Padé analytic continuation of local Green's function; extrapolation to thermodynamic limit
  • Observations: d-wave gap ≈4 meV at lambda=0.4, decreasing with lambda; s-wave gap ≈1 meV at lambda=0.4, growing to ≈1.5 meV at lambda=0.5; thermodynamic limit critical lambda approx 0.6 for s+id
• interpretation — Local oxygen defects tune electron density, creating nanoscale finite-size domains with distinct pairing symmetries, explaining the spatially inhomogeneous superconducting gaps observed in scanning tunnelling experiments.