Electronic structure trends in La$_{2}R$Ni$_2$O$_7$ ($R=$ Pr, Nd, Sm) from first-principles

This first-principles study reveals that rare-earth doping (Pr, Nd, Sm) in La$_3$Ni$_2$O$_7$ induces chemical pressure that drives a tetragonal structural transition and enhances planar electronic hopping, providing microscopic insights into the emergence of superconductivity in these Ruddlesden-Popper nickelates.

The Gist

Imagine a microscopic city built from layers of atoms, where the "residents" are electrons zipping around. Scientists have recently discovered that in a specific type of this city called Ruddlesden-Popper nickelates (specifically a material named $La_3Ni_2O_7$), these electrons can suddenly stop acting like individual commuters and start moving in perfect unison, creating a phenomenon called superconductivity (electricity flowing with zero resistance). This happens when you squeeze the city very hard with high pressure.

Recently, researchers found that if you swap some of the city's original "landlords" (Lanthanum atoms) with slightly smaller landlords from the rare-earth family (like Praseodymium, Neodymium, or Samarium), the city can conduct electricity even better, reaching superconductivity at higher temperatures.

This paper is a computer simulation study that asks: What exactly happens inside the city when we swap these landlords?

Here is the breakdown of their findings using simple analogies:

1. The "Chemical Squeeze"

Think of the rare-earth atoms (Pr, Nd, Sm) as suitcases. A Praseodymium suitcase is big, while a Samarium suitcase is small. When you swap a big suitcase for a small one in a crowded room, the room naturally shrinks to fill the gap.

• The Finding: As the researchers swapped larger atoms for smaller ones, the entire crystal structure got tighter and smaller. They call this "chemical pressure." It's like squeezing the material without using an external machine; the atoms themselves are doing the squeezing.

2. The Great Shape-Shifting

At normal pressure, this atomic city has a slightly squashed, irregular shape (monoclinic). But when you squeeze it hard enough (either with a machine or by using smaller atoms), it snaps into a perfect, symmetrical square shape (tetragonal).

• The Finding: The computer showed that this shape-shifting happens at the exact same time the material becomes a superconductor. It's as if the city has to rearrange its furniture into a perfect grid before the electricity can flow without friction.

3. The "Flat Highway" for Electrons

Inside this atomic city, electrons travel on different "roads" (energy bands). One type of road is for electrons moving flat across the floor ($dx^2-y^2$), and another is for electrons moving up and down ($dz^2$).

• The Finding: When the city gets squeezed, a new, very flat "highway" appears right at the level where electrons live (the Fermi level). This flat road is made of the up-and-down electrons ($dz^2$). The researchers believe this flat highway is a crucial ingredient that allows the electrons to team up and become superconductors. This happens in all the doped versions, just like it does when you use a machine to squeeze the material.

4. The Surprising Twist: The Elevator vs. The Sidewalk

This is the most interesting part of the story. Usually, when you squeeze a building, you expect the elevators (vertical connections) to get faster because the floors are closer together.

• The Expectation: Squeezing the material should make the vertical electron "hops" stronger.
• The Reality: The researchers found the opposite. When they used smaller atoms (chemical pressure), the vertical connections actually got weaker.
• Why? Imagine the building has two types of elevators. One connects the main floors, and another connects the floors to the roof (the "rocksalt" layer). When the smaller atoms are used, they pull the roof down, shortening the distance to the roof. This actually breaks the connection between the main floors and the roof, making the vertical "elevator" slower.
• The Silver Lining: However, the sidewalks (horizontal connections) got much wider and faster. The electrons could zip across the floor much more easily.

The Bottom Line

The paper concludes that while the vertical connections get weaker when you swap in smaller atoms, the horizontal connections get stronger. The superconductivity in these materials seems to rely on a delicate dance between the vertical and horizontal electron paths.

The researchers suggest that the key to making these materials superconduct at even higher temperatures isn't just about squeezing them tighter, but about how the atoms rearrange to create that specific "flat highway" and how the horizontal and vertical electron paths mix together. It's a reminder that in this atomic world, making things smaller doesn't always make the vertical connections stronger; sometimes, it changes the whole game in unexpected ways.

Technical Summary

Technical Summary: Electronic Structure Trends in La$_2$RNi$_2$O$_7$ (R = Pr, Nd, Sm) from First-Principles

Problem Statement
The discovery of superconductivity with a critical temperature ($T_c$) of $\sim$80 K in bilayer Ruddlesden-Popper (RP) La$_3$Ni$_2$O$_7$ under high pressure has intensified interest in this material family. Recent experimental reports indicate that rare-earth (R) doping (specifically Pr, Nd, and Sm) in La$_3$Ni$_2$O$_7$ can further elevate $T_c$, reaching up to 96 K in Sm-doped samples at $\sim$22 GPa. However, the microscopic mechanisms driving this enhancement remain debated. While some theoretical approaches suggest $T_c$ increases with R substitution, others predict a decrease. A systematic understanding of how chemical pressure (via R doping) alters the crystal structure and electronic properties compared to the undoped counterpart is required to clarify the relationship between doping, structural transitions, and superconductivity.

Methodology
The authors employed first-principles density-functional theory (DFT) calculations using the QUANTUM-ESPRESSO package.

• Functional and Correlation: The study utilized the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) combined with the Hubbard $U$ correction (GGA+$U$) with an effective on-site Coulomb repulsion $U = 3.5$ eV to account for strong electronic correlations in Ni-3d states.
• Structural Models: The calculations focused on La$_2$RNi$_2$O$_7$ (R = La, Pr, Nd, Sm). Based on enthalpy comparisons and experimental X-ray diffraction (XRD) data, the rare-earth dopants were modeled as occupying the La sites within the rocksalt layers. Both ambient pressure (monoclinic $P2_1/m$) and high-pressure (tetragonal $I4/mmm$) phases were investigated.
• Computational Parameters: Plane-wave kinetic energy and charge density cutoffs were set to 80 Ry and 320 Ry, respectively. Brillouin zone sampling utilized Monkhorst-Pack k-point grids, enhanced to $30\times30\times2$ for Fermi surface topology refinement.
• Analysis Tools: Maximally localized Wannier functions (MLWFs) were constructed using Wannier90 to extract and analyze dominant hopping parameters ($t_{\perp}^z$, $t_{\parallel}^x$, $t_{\parallel}^{xz}$) associated with Ni-$d_{z^2}$ and Ni-$d_{x^2-y^2}$ orbitals.

Key Results

1. Structural Trends and Phase Transitions:

   • The volume of La$_2$RNi$_2$O$_7$ progressively decreases as the ionic radius of R decreases from Pr to Sm, consistent with chemical pressure effects.
   • A pressure-induced structural transition from monoclinic ($P2_1/m$) to tetragonal ($I4/mmm$) symmetry was observed for all compounds. The critical transition pressure increases with decreasing R size (9.5 GPa for La, 14 GPa for Pr, 17 GPa for Nd, and 20.5 GPa for Sm), qualitatively matching experimental observations.
   • This transition coincides with the emergence of superconductivity in all cases and is characterized by the straightening of the apical Ni-O-Ni bond angle to 180$^\circ$.

2. Electronic Structure Evolution:

   • Ambient Pressure: The electronic structure shows subtle modifications upon doping. The Ni-$d_{z^2}$ bonding band near the Fermi level at the $\Gamma$ point is pushed down in energy, and the bandwidth of the Ni-$d_{x^2-y^2}$ bands increases slightly.
   • High Pressure (21 GPa): A critical feature emerges in the tetragonal phase: flat Ni-$d_{z^2}$ bonding bands cross the Fermi level at the S point of the Brillouin zone, creating extra hole-like Fermi surface pockets ($\gamma_1, \gamma_2$). This metallization of the $d_{z^2}$ bands is identified as a key electronic signature of superconductivity in bilayer RP nickelates.
   • The Fermi surface topology remains largely similar between doped and undoped cases, though pocket sizes shift with R doping.

3. Hopping Parameters and Hybridization:

   • Contrary to the effects of hydrostatic pressure, where interlayer coupling ($t_{\perp}^z$) typically increases, chemical pressure (decreasing R size) leads to a decrease in the dominant out-of-plane hopping ($t_{\perp}^z$).
   • This reduction is attributed to the significant shortening of the apical Ni-O$_{rocksalt}$ bond length, which compresses the rocksalt slab while the Ni-O$_{middle}$ distance remains relatively stable.
   • Conversely, the dominant in-plane hopping ($t_{\parallel}^x$) increases as the R size decreases due to the compression of the in-plane lattice constant.
   • The hybridization between the out-of-plane $d_{z^2}$ and itinerant planar $d_{x^2-y^2}$ states increases as the Ni-O planar distance decreases.

Significance and Claims
The paper provides microscopic insights into the effects of rare-earth doping in RP nickelate superconductors. The authors claim that while the structural transition to tetragonal symmetry and the emergence of flat $d_{z^2}$ bands at the Fermi level are consistent across doped and undoped systems, the specific trends in hopping parameters differ significantly from hydrostatic pressure effects.

Specifically, the study highlights that:

• A reduction in the $c$-lattice constant via chemical pressure does not necessarily correlate with an increase in interlayer hopping ($t_{\perp}^z$); rather, $t_{\perp}^z$ decreases due to the compression of the apical Ni-O$_{rocksalt}$ bond.
• The increase in $T_c$ observed with chemical pressure (smaller R ions) occurs despite the reduction in $t_{\perp}^z$.
• Consequently, the results emphasize the importance of hybridization effects between the out-of-plane $d_{z^2}$ orbitals and the itinerant planar $d_{x^2-y^2}$ states, as well as the increase in in-plane hopping, as potential drivers for the enhanced superconductivity in these doped systems, rather than simple interlayer coupling enhancement.