Electronic theory for scanning tunneling microscopy spectra in bilayer nickelate thin films

Summary

This paper theoretically analyzes scanning tunneling microscopy spectra of superconducting bilayer nickelate films using a two-orbital bilayer model based on first-principles Wannier functions and the continuous Green's function method. The study finds that the multi-orbital character and the spatial anisotropy of Wannier functions render the local density of states highly sensitive to the tip position: as the tip height increases, the relative weights of coherence peaks from different bands change significantly, thereby enabling distance-dependent measurements to distinguish the orbital origins of the controversial γ-band and β-band coherence peaks. Furthermore, in impurity-containing systems, quasiparticle interference patterns can clearly resolve the symmetry of s-wave and d-wave superconducting order parameters. This work provides explicit theoretical guidance for experimentally identifying the band attribution of superconducting gaps and the pairing symmetry.

Materials

• La₂PrNi₂O₇

Methods

• Tight-binding model
• Green's function formalism
• Bogoliubov-de Gennes mean-field theory
• T-matrix impurity scattering
• Wannier function projection
• Quasiparticle interference analysis

Keywords

• scanning tunneling microscopy
• superconducting gap symmetry
• incipient gamma band
• quasiparticle interference
• orbital filtering effect
• multiorbital physics

Highlights

• The continuum Green's function approach captures tip-height-dependent orbital filtering, explaining the weak gamma-band shoulder observed in STM experiments.
• Star-shaped quasiparticle interference features from accidental nodes of the gamma-band uniquely identify d-wave pairing in the presence of mirror-symmetric impurities.
• HAEM antisymmetrization provides a direct probe of the sign change in the superconducting order parameter, confirming s+- or d-wave symmetry.
• Wannier-resolved real-space QPI patterns reveal atomic-scale features due to orbital selectivity, improving upon prior lattice-based models.

Conclusions

• Continuum Green's function modeling reveals that Wannier function spatial profiles produce a tip-height-dependent orbital filtering effect, suppressing gamma-band coherence peaks relative to beta-band peaks.
• Varying STM tip height can distinguish band origins of coherence peaks and probe the incipient nature of the gamma-band.
• Apical oxygen impurities lead to mirror-symmetric scattering that exposes clear qualitative differences between s+- and d-wave gap symmetries in quasiparticle interference patterns.
• Quasiparticle interference and HAEM analysis reliably detect sign-changing order parameters, providing experimental tools to determine superconducting gap symmetry in bilayer nickelates.

Main claims

• Continuum Green's function calculations reveal that tip‑height‑dependent LDOS changes allow band‑resolved identification of superconducting coherence peaks.
  • Evidence: Fig. 5 shows that for incipient γ‑band, the γ‑band coherence peak at ≈7 meV is strongly suppressed relative to the β‑band peak at ≈19 meV as tip height increases.,Text: 'the spectral weight of the γ‑band coherence peak at 7 meV decreases significantly faster than that of the β‑band coherence peak at 19 meV with increasing distance d.'
• Quasiparticle interference patterns with apical oxygen impurities exhibit distinct features for s± and d‑wave order parameters, enabling gap symmetry discrimination.
  • Evidence: Fig. 9(p) and surrounding text: 'the star‑shaped fringe pattern associated with the scattering vector q_aa_β appears exclusively in the d‑wave case'.,For apical oxygen impurity, different dominant QPI signals for s‑wave (β–β circular features) vs. d‑wave (star‑shaped feature).
• The HAEM antisymmetrized QPI signal confirms the sign‑changing nature of the superconducting gap, ruling out a simple s++ state.
  • Evidence: Fig. 10(b) shows large HAEM response for both s± and d‑wave; such response is characteristic of sign‑changing gaps and would be absent for s++.,Text: 'the HAEM method is an ideal tool to probe the phase of the superconducting gap and confirm the predicted sign changing behavior of the s± or d‑wave symmetry.'
• The theoretical model reproduces the experimentally observed weak γ‑band shoulder, supporting the incipient γ‑band scenario.
  • Evidence: Fig. 5(a,b) shows that at larger tip heights (d > 3 Å) the γ‑band coherence peak becomes very weak, matching STM experiments [11,29,59] where only a shoulder is seen.,Text: 'the γ‑band LDOS is more strongly suppressed with increasing height, which may provide a natural explanation for the weak shoulder observed in STM experiments.'

Workflow

• Model setup — A realistic two-orbital bilayer model captures the low-energy electronic structure and candidate superconducting states of bilayer nickelates.
  • Materials: two-orbital bilayer tight-binding model from DFT Wannier functions; maximally-localized Wannier orbitals for Ni-3d (3dx2-y2 and 3dz2)
  • Methods: slave-boson renormalization of hopping integrals; bonding-antibonding transformation; mean-field Bogoliubov-de Gennes Hamiltonian for s± and d-wave pairing
  • Observations: three bands (α, β, γ) crossing the Fermi level in normal state; orbital weight distribution on Fermi surfaces
• Continuum LDOS and STM spectra calculation — The continuum approach reveals a strong orbital filtering effect: tip‑height‑dependent LDOS allows band‑selective identification of coherence peaks and can distinguish an incipient γ‑band from a crossing one.
  • Materials: continuum Green's function formalism; Wannier functions with spatial extension above the surface
  • Methods: calculation of local density of states (LDOS) via continuum Green's function with Wannier basis; varying STM tip height d from 1.5 to 5.5 Å; comparison with lattice-projected DOS
  • Observations: in incipient case, γ‑band coherence peak strongly suppressed with increasing tip height while β‑band peak persists; in crossing case, tip‑height dependence is weak; only minor changes in relative spectral weights
• Impurity QPI analysis and predictions — QPI patterns with symmetry‑selective impurities and the HAEM signal provide experimental criteria to discriminate gap symmetries, detect the incipient γ‑band, and verify the sign‑changing character of the order parameter.
  • Materials: single impurity models (Ni site on each layer, apical oxygen between layers); T‑matrix formalism for impurity scattering
  • Methods: real‑space LDOS correction via T‑matrix; Fourier transform to obtain QPI patterns; HAEM antisymmetrization to detect sign change of order parameter
  • Observations: apical oxygen impurity yields star‑shaped QPI feature from accidental β‑band nodes only in d‑wave; Ni‑layer impurities lead to dominant interband scattering with similar patterns for s± and d‑wave; HAEM response large for both s± and d‑wave, confirming sign‑changing gap; absent for s++