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

This study utilizes first-principles calculations to demonstrate that hydrostatic pressure and biaxial compressive strain both suppress octahedral tilts in the hybrid bilayer-trilayer nickelate La$_7$Ni$_5$O$_{17}$ to stabilize a tetragonal $C2/c$ structure, yet they induce distinct electronic behaviors by causing the trilayer $d_{z^2}$ band to cross the Fermi level under pressure while keeping it below the Fermi level under strain.

The Gist

Imagine a microscopic world made of tiny, rigid building blocks. In this paper, scientists are studying a specific, complex Lego structure made of Nickel and Oxygen atoms, called La7Ni5O17. This structure is a "hybrid," meaning it's built by stacking two different types of layers: some are double-decker (bilayer) and some are triple-decker (trilayer).

The researchers wanted to know: What happens to this Lego tower if we squeeze it? They tested two ways of squeezing it:

1. Pressure: Squeezing it from all sides (like deep underwater).
2. Strain: Stretching or compressing it from the sides (like stretching a rubber band).

Here is the story of what they found, explained simply:

1. The Wobbly Tower (Ambient Pressure)

At normal room conditions, this Lego tower isn't perfectly straight. It's a bit wobbly and tilted. The scientists discovered that the "floors" of the tower (the oxygen octahedra) are tilted at an angle, making the whole structure crooked.

• The Metaphor: Imagine a stack of books that has been slightly knocked over. The books aren't standing straight up; they are leaning against each other. This leaning is unstable.
• The Fix: The scientists used math to figure out exactly how to straighten the books. They found that if you let the structure relax into its natural, stable state, it settles into a lower, slightly more complex shape (called C2/c) where the "floors" are tilted but locked in place.

2. The Great Straightening (Hydrostatic Pressure)

Next, they simulated putting this tower under huge pressure (like being at the bottom of the ocean, but much deeper—about 30,000 times atmospheric pressure!).

• What happened: The pressure forced the wobbly, tilted floors to stand up straight. The "books" stopped leaning and aligned perfectly.
• The Result: The structure became a perfect, straight tower (called tetragonal).
• The Electronic Surprise: In the world of atoms, "straightening" changes how electricity flows. When the tower straightened, a new "highway" for electrons opened up. Specifically, a path made of a specific type of electron orbital (called dz2) from the triple-decker section suddenly appeared right at the "traffic light" (the Fermi level).
• Why it matters: In superconductors (materials that conduct electricity with zero resistance), having these specific electron highways is usually the key to making the magic happen. The scientists think this new highway might be the secret sauce for superconductivity in this material.

3. The Rubber Band Effect (Compressive Strain)

Then, they tried a different approach: Strain. Instead of squeezing from all sides, they squeezed the material from the sides, like compressing a spring or stretching a rubber band, while keeping the height the same.

• What happened: The material straightened up, just like it did with pressure. The "floors" became less tilted.
• The Twist: However, the electronic result was different! Even though the tower looked straight, the "highway" for the triple-decker electrons disappeared. It got pushed down below the traffic light.
• The Metaphor: Imagine you have a highway (pressure) where a new lane opens up. Now, imagine you take a different route (strain) that looks just as straight, but that new lane is closed off.
• Why it matters: This is a crucial discovery. It shows that looking straight isn't enough. You need the specific electronic conditions that come from pressure, not just strain, to get that special electron highway. This explains why some materials become superconductors under pressure but not under strain, even if they look the same.

The Big Picture

The scientists are trying to find the "Holy Grail" of superconductors: materials that conduct electricity perfectly without needing to be cooled down to near absolute zero.

• The Goal: They are testing this new hybrid material (La7Ni5O17) to see if it can superconduct.
• The Clue: They found that under high pressure, the material gets the right "electronic ingredients" (that extra electron highway) that other superconductors have.
• The Warning: If you try to make this material in a thin film (using strain instead of pressure), you might get the straight shape, but you lose the special electron highway.

In a nutshell:
Think of this material as a complex machine. The scientists found that squeezing it hard from all sides fixes the machine's gears and turns on a special power switch (superconductivity). However, squeezing it from the sides fixes the shape of the machine but leaves the power switch off. This helps them understand exactly what ingredients are needed to build the next generation of super-powerful, energy-saving electronics.

Technical Summary

1. Problem Statement

The discovery of high-temperature superconductivity ($T_c \sim 80$ K) in pressurized bilayer Ruddlesden-Popper (RP) nickelate $La_3Ni_2O_7$ ($n=2$) and subsequent findings in trilayer $La_4Ni_3O_{10}$ ($n=3$) has sparked intense interest in the RP family. Recent studies have identified hybrid RP phases (e.g., alternating single-layer/trilayer or bilayer/trilayer blocks) that also exhibit superconductivity under pressure.

However, the specific hybrid bilayer-trilayer phase $La_7Ni_5O_{17}$ (2323 configuration) remains theoretically scrutinized but experimentally unconfirmed at ambient conditions. Key open questions include:

• What is the true ground-state crystal structure of $La_7Ni_5O_{17}$ at ambient pressure (does it possess octahedral tilts)?
• How do hydrostatic pressure and biaxial compressive strain differently affect its structural stability and electronic structure?
• Specifically, how do these external perturbations influence the $d_{z^2}$ orbital bands derived from the trilayer block, which are hypothesized to be crucial for superconductivity?

2. Methodology

The authors employed first-principles Density Functional Theory (DFT) calculations using the Quantum ESPRESSO package.

• Functional & Potentials: Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) with ultrasoft pseudopotentials.
• Structural Analysis:
  • Investigated the dynamical stability of the high-symmetry $P4/mmm$ phase at ambient pressure via phonon dispersion calculations (using ALAMODE).
  • Identified unstable phonon modes and used group theory (irreducible representations) to construct distorted structures.
  • Relaxed structures under varying hydrostatic pressures (up to 30 GPa) and biaxial compressive strains (up to -2%).
• Electronic Analysis: Calculated band structures, Fermi surfaces, and Wannier-derived hopping parameters to analyze orbital character ($d_{x^2-y^2}$ and $d_{z^2}$) and inter-layer coupling.
• Validation: Performed DFT+$U$ calculations (with $U=4.7$ eV) to test the robustness of electronic features against electron correlations.

3. Key Contributions

1. Identification of the Ambient Pressure Ground State: The study establishes that the high-symmetry $P4/mmm$ structure is dynamically unstable at ambient pressure. By analyzing unstable phonon modes, the authors identified a dynamically stable $C2/c$ structure containing octahedral tilts as the true ground state.
2. Comparative Tuning Mechanisms: The paper provides a systematic comparison between hydrostatic pressure and biaxial compressive strain, revealing that while both suppress octahedral tilts, they induce distinct electronic modifications, particularly regarding the trilayer $d_{z^2}$ bands.
3. Electronic Structure of Hybrid Phases: It elucidates how the electronic structure of the hybrid 2323 phase is a superposition of conventional bilayer and trilayer physics, with specific sensitivity to the position of the trilayer bonding $d_{z^2}$ band relative to the Fermi level ($E_F$).

4. Detailed Results

A. Structural Stability and Evolution

• Ambient Pressure: The $P4/mmm$ phase exhibits imaginary phonon frequencies at the R-point. The unstable modes correspond to $A_{1u}$ (in-plane distortions) and $E_u$ (octahedral tilting). Relaxing these distortions yields a $C2/c$ structure with Ni-O-Ni bond angles between 160° and 170°.
• Under Hydrostatic Pressure:
  • Octahedral tilts are suppressed as pressure increases.
  • At ~20 GPa, the structure becomes tetragonal ($a=b$), and cross-plane Ni-O-Ni angles straighten to 180°.
  • At ~30 GPa, the structure is fully tetragonal ($P4/mmm$), and phonon calculations confirm dynamical stability (no imaginary frequencies).
• Under Biaxial Compressive Strain (-2%):
  • The out-of-plane (cross-plane) Ni-O-Ni angles also straighten, mimicking the pressure effect.
  • However, in-plane tilts persist even at -2% strain, unlike the pressure case where they are fully suppressed. The in-plane lattice constants at -2% strain roughly match those at ~15 GPa.

B. Electronic Structure Evolution

• General Features: The low-energy physics is dominated by Ni-$e_g$ orbitals ($d_{x^2-y^2}$ and $d_{z^2}$), highly hybridized with O-$p$ states. The system can be viewed as a superposition of bilayer and trilayer blocks.
• Effect of Pressure (30 GPa):
  • The bonding $d_{z^2}$ band from the trilayer block crosses the Fermi level, creating a small hole pocket at the zone corner (M point).
  • The bonding $d_{z^2}$ band from the bilayer block also crosses $E_F$, forming a corner pocket.
  • This emergence of the trilayer-derived pocket is considered a critical factor for superconductivity in this phase.
• Effect of Strain (-2%):
  • While the bilayer bonding $d_{z^2}$ band still crosses $E_F$, the trilayer bonding $d_{z^2}$ band is pushed below the Fermi level.
  • Consequently, the "extra" corner pocket from the trilayer block disappears under strain.
  • The electronic structure under strain resembles the structure at ~15 GPa (where the trilayer pocket is just emerging or absent) rather than 30 GPa.
• Hopping Parameters:
  • Pressure significantly enhances inter-layer hopping ($t_z^\perp$) due to the straightening of bonds.
  • Strain results in lower $t_z^\perp$ values (due to c-axis elongation) and planar hoppings similar to the 15 GPa case.
• Robustness (DFT+U): The trilayer $d_{z^2}$ pocket at 30 GPa is fragile; introducing a Hubbard $U$ (4.7 eV) or reducing pressure shifts this band below $E_F$, eliminating the pocket.

5. Significance and Implications

• Structural Insight: The work corrects the assumption that hybrid RP nickelates might be tetragonal at ambient pressure, confirming that octahedral tilts are intrinsic to the ground state of $La_7Ni_5O_{17}$.
• Superconductivity Mechanism: The study highlights a crucial divergence between pressure and strain tuning. Since superconductivity has been observed in strained bilayer $La_3Ni_2O_7$ films, the absence of the trilayer-derived $d_{z^2}$ pocket in the strained 2323 phase raises questions about the necessity of this specific pocket for superconductivity in hybrid systems.
• Future Directions: The authors note that if $La_7Ni_5O_{17}$ is synthesized (in bulk or thin film), the role of the trilayer corner pocket in superconductivity must be re-evaluated. The results suggest that while pressure creates a unique electronic environment favorable for high-$T_c$ (potentially via $s\pm$ pairing), strain may suppress specific electronic features (the trilayer pocket) while maintaining others (bilayer pocket and straightened angles).

Note: The paper includes a "Note added" acknowledging that experimental evidence for superconductivity in $La_7Ni_5O_{17}$-2323 was demonstrated by other researchers (Ref. [44]) during the completion of this work, validating the material's relevance.