Pressure and strain tuning of the alternating bilayer-trilayer Ruddlesden-Popper nickelate: crystal and electronic structure

We use first-principles calculations to investigate the crystal and electronic structure of the hybrid bilayer-trilayer Ruddlesden-Popper (RP) nickelate La$7$Ni$5$O${17}$ under hydrostatic pressure and biaxial compressive strain. By analyzing the irreducible representations of the dynamically unstable phonon modes in the high-symmetry $P4/mmm$ structure, we identify a dynamically stable lower-symmetry $C2/c$ structure containing octahedral tilts. The application of both pressure and compressive strain tends to suppress the octahedral tilts, effectively tetragonalizing the structure, in analogy with the conventional RPs. The electronic structure under hydrostatic pressure and strain has similarities, but it differs in the position of the $d{z^2}$ bonding band from the trilayer block. This band crosses the Fermi level at a pressure of 30 GPa, but it remains below it for any level of compressive strain. This strain-induced modification mirrors the electronic structure changes observed in the conventional bilayer nickelate.

💡 Research Summary

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In this work the authors employ density‑functional theory (DFT) calculations to explore how hydrostatic pressure and biaxial compressive strain affect the crystal and electronic structures of the hybrid bilayer‑trilayer Ruddlesden‑Popper (RP) nickelate La₇Ni₅O₁₇ (often denoted as the 2323 phase). The study is motivated by the discovery that conventional RP nickelates with n = 2 (La₃Ni₂O₇) and n = 3 (La₄Ni₃O₁₀) become superconducting under pressure, accompanied by a structural transition that suppresses octahedral tilts and straightens Ni–O–Ni bond angles. The authors ask whether a mixed‑layer system, which has not yet been synthesized experimentally, can display similar or even enhanced behavior, and how strain—known to induce superconductivity in thin‑film versions of the n = 2 compound—compares with pressure in this more complex material.

Methodology
All calculations are performed with the plane‑wave code Quantum ESPRESSO using the PBE‑GGA functional and ultrasoft pseudopotentials. A kinetic‑energy cutoff of 80 Ry (charge density 640 Ry) and Monkhorst‑Pack k‑point meshes of 12 × 12 × 2 (for the high‑symmetry P4/mmm cell) and 8 × 8 × 2 (for the low‑symmetry C2/c cell) are employed. Phonon spectra of the P4/mmm phase are obtained with the ALAMODE package on a 2 × 2 × 1 116‑atom supercell. Structural relaxations are converged to forces below 10⁻⁴ Ry/Bohr. Hydrostatic pressure is simulated by uniformly scaling the cell volume up to 30 GPa; biaxial strain is imposed by fixing the in‑plane lattice constants to values corresponding to –2 % strain (mimicking growth on SrLaAlO₄) while allowing the out‑of‑plane lattice constant to relax.

Structural stability at ambient conditions
The high‑symmetry tetragonal P4/mmm structure, which contains no octahedral tilts, exhibits five imaginary phonon modes at the Brillouin‑zone R point. One mode belongs to the A₁u irreducible representation (an in‑plane distortion of the NiO₆ octahedra) and the remaining four form two degenerate Eu pairs (tilting of the octahedra). By applying a combination of the A₁u and the two Eu distortions to a 2 × 2 × 2 supercell and fully relaxing the atomic positions, the system converges to a monoclinic C2/c structure. This lower‑symmetry phase contains characteristic perovskite‑like octahedral tilts with Ni–O–Ni bond angles ranging from 160° to 170°. The phonon dispersion of the C2/c phase shows no imaginary frequencies, confirming its dynamical stability.

Effect of hydrostatic pressure
Under increasing pressure the lattice parameters a and b become equal at ≈ 20 GPa, indicating a tetragonalization of the crystal. Simultaneously the apical Ni–O–Ni angles straighten to 180° at the same pressure, while the in‑plane angles reach 180° slightly later (≈ 25 GPa). All Ni–O bond lengths contract monotonically with pressure. Phonon calculations reveal that the soft modes harden with pressure, and at ≈ 30 GPa the P4/mmm structure itself becomes dynamically stable, consistent with the trend observed in conventional RP nickelates.

The electronic structure at 30 GPa (the pressure at which the tetragonal phase is fully realized) shows that the Ni average valence is 2.6+ (d⁷·⁴ configuration). The t₂g manifold is fully occupied, leaving 1.4 electrons to fill the e_g orbitals. Both d_{x²‑y²} and d_{z²} bands cross the Fermi level. Importantly, a bonding‑antibonding pair derived from the trilayer block’s d_{z²} orbitals forms a small hole pocket at the Brillouin‑zone corner (M point). This pocket, previously identified in random‑phase‑approximation (RPA) studies as crucial for an s± pairing channel, is present but rather fragile; modest on‑site Coulomb repulsion U or a reduction of pressure can push it below the Fermi level. The d_{z²} band from the bilayer block also crosses the Fermi level, generating a corner pocket that is more robust. Hopping parameters extracted via Wannier functions show a strong pressure‑induced enhancement of the interlayer d_{z²}–d_{z²} hopping (t⊥^z ≈ 0.6 eV) and the in‑plane d_{x²‑y²}–d_{x²‑y²} hopping (t∥^x ≈ 0.5 eV), as well as a moderate d_{x²‑y²}–d_{z²} hybridization (t_{xz} ≈ 0.2 eV). These values are comparable to those of the isolated n = 2 and n = 3 compounds, confirming that pressure restores the electronic characteristics of the constituent layers while also strengthening interlayer coupling.

Effect of biaxial compressive strain
Applying a –2 % biaxial compressive strain reproduces many structural trends seen under pressure: the apical Ni–O–Ni angle straightens toward 180°, and the in‑plane lattice constants become nearly equal, resembling the lattice dimensions attained at ≈ 15 GPa hydrostatic pressure. However, the planar Ni–O–Ni angles behave non‑monotonically; the inner trilayer layer tends to buckle slightly, and one of the two inequivalent bilayer angles also deviates from linearity. Consequently, the C2/c symmetry persists under strain, albeit with reduced tilt amplitudes.

Electronic‑structure calculations under strain reveal a crucial difference from the high‑pressure case: the trilayer‑derived d_{z²} bonding band remains below the Fermi level, and the only d_{z²} contribution crossing the Fermi surface originates from the bilayer block, producing a corner pocket similar to that of La₃Ni₂O₇. The absence of the trilayer d_{z²} pocket means that the electronic structure under strain more closely resembles that of the pure bilayer RP nickelate, which is known to become superconducting when grown on compressive substrates. The hopping parameters under strain are slightly reduced compared with the 30 GPa case but still show sizable interlayer d_{z²} coupling (t⊥^z ≈ 0.6 eV) and in‑plane d_{x²‑y²} hopping (t∥^x ≈ 0.5 eV). The strain therefore enhances bandwidths while suppressing the specific trilayer d_{z²} feature that pressure promotes.

Discussion and Outlook
The study demonstrates that La₇Ni₅O₁₇ can be driven into a dynamically stable, nearly tetragonal structure either by hydrostatic pressure or by biaxial compressive strain, but the two routes lead to distinct electronic reconstructions. Pressure restores the trilayer d_{z²} bonding band at the Fermi level, generating a small hole pocket that may be essential for the s± pairing mechanism proposed for RP nickelates. In contrast, compressive strain keeps this band below the Fermi level, yielding an electronic landscape akin to the bilayer compound where superconductivity has already been observed in thin films. These findings suggest that the presence or absence of the trilayer d_{z²} pocket could be a decisive factor for superconductivity in mixed‑layer RP nickelates.

The authors conclude that La₇Ni₅O₁₇ is a promising platform for tuning superconductivity via external parameters. Experimental synthesis of the bulk 2323 phase, followed by high‑pressure measurements or epitaxial growth on suitably mismatched substrates, would allow direct testing of the predicted structural transitions and electronic‑band evolution. Moreover, incorporating strong correlation effects (e.g., DFT + U or dynamical mean‑field theory) could refine the position of the fragile d_{z²} pocket and clarify its role in pairing. Ultimately, the work provides a roadmap for engineering higher‑Tc nickelate superconductors by exploiting the interplay between octahedral tilts, interlayer coupling, and orbital-selective band shifts induced by pressure or strain.