Theoretical study on the possibility of high $T_c$ s$\pm$-wave superconductivity in the heavily hole-doped infinite layer nickelates

This paper theoretically proposes that heavily hole-doped infinite-layer nickelates La$_{1-x}$Sr$_x$NiO$_2$ can exhibit high-$T_c$ $s\pm$-wave superconductivity when tetragonal symmetry is preserved in thin films, a state driven by enhanced interactions between the $d_{x^2-y^2}$ band and lower-lying $3d$ bands due to the absence of apical oxygens.

The Gist

Imagine you are trying to build a super-fast highway for electricity, where cars (electrons) can zip along without any traffic jams or friction. This is what superconductivity is. For decades, scientists have been trying to build these highways in materials called "nickelates," which are cousins to the famous copper-based superconductors (cuprates).

This paper is a theoretical blueprint. The authors are saying, "We think we found a way to build an even better, faster highway in these nickel materials, but we need to change the rules of the road."

Here is the breakdown of their idea using simple analogies:

1. The Problem: The "Traffic Jam" at the Wrong Time

In the standard nickel materials (like the ones discovered recently), scientists usually think the electrons behave like a specific type of traffic pattern (called d-wave). This works okay, but the authors suspect there's a better pattern hidden inside the material if we change the ingredients.

Usually, to get this better pattern, scientists thought they needed to add a tiny bit of "hydrogen" (like a secret ingredient) to the mix. But there's a debate: some say the hydrogen is there, others say it's not. The authors wanted to find a way to get this better pattern without relying on that controversial hydrogen.

2. The Solution: Heavy "Substitution" and a "Stretchy Floor"

The authors propose a new recipe:

• Heavy Doping: Instead of just adding a little bit of a different element, they suggest replacing a huge chunk of the original material (Lanthanum) with a different element (Strontium). Imagine replacing half the bricks in a wall with a different kind of brick.
• The "Stretchy Floor" (Substrates): When you mix these bricks, the wall naturally wants to crumple or change shape. To stop this, the authors suggest growing these materials as very thin films on top of a "floor" (a substrate) that forces them to stay flat and square.

The Analogy: Think of the nickel material as a piece of clay. If you mix in too much of a new ingredient, the clay wants to shrink and crack. But if you press that clay onto a rigid, flat cookie cutter (the substrate) that holds it in a perfect square shape, the clay stays stable even with the heavy mixing.

3. The "Magic" Result: The $s_{\pm}$-Wave

When they ran their calculations with this "heavy mixing" and "flat floor" setup, they found something exciting.

• The Old Way: The electrons were dancing in a pattern where the "sign" of their movement was the same everywhere (like everyone waving their hands up).
• The New Way ($s_{\pm}$-wave): The electrons start dancing in a pattern where the "sign" flips. Imagine a checkerboard: on white squares, everyone waves their hands up; on black squares, everyone waves their hands down.

The authors found that when the material is heavily mixed (specifically, when the electron configuration gets close to a specific state called $d^8$), this "up-down" checkerboard dance becomes the most efficient way to move. This specific dance pattern is predicted to allow for higher temperatures (High $T_c$) where superconductivity works.

4. Why It Works: The "Missing Ceiling"

In these nickel materials, there is no "apical oxygen" (an atom that usually sits above the nickel like a ceiling). Because this "ceiling" is missing, the energy levels of the electrons shift.

The authors explain that this shift creates a situation where the "dance floor" for the electrons is perfectly set up for that checkerboard ($s_{\pm}$) pattern. It's like removing a wall in a room, which suddenly allows a group of people to form a perfect circle dance that was impossible before.

5. The Safety Check: Will It Fall Apart?

Before you can build a house, you need to make sure the foundation won't collapse. The authors ran "phonon calculations" (which check if the atoms vibrate in a way that breaks the structure).

The Result: They found that even with a huge amount of mixing (up to 100% replacement), the material remains stable and doesn't crumble, as long as it is held in that flat, square shape by the substrate. This validates their idea that this heavy mixing is physically possible.

Summary

The paper claims that if you take nickel superconductors, replace half of their atoms with Strontium, and force them to stay flat on a specific type of substrate, you can create a stable material where electrons dance in a special "checkerboard" pattern. This pattern is theoretically predicted to allow superconductivity to happen at much higher temperatures than currently seen, without needing any mysterious hydrogen ingredients.

Important Note: This is a theoretical study. The authors have done the math and the computer simulations. They have not yet built this specific material in a lab to prove it works, but their calculations suggest it is a very promising path to explore.

Technical Summary

1. Problem Statement

The discovery of superconductivity in infinite-layer nickelates ($RE_{1-x}AE_xNiO_2$) has sparked intense interest, often compared to cuprates. While most theoretical studies assume a $d^9$ electron configuration (similar to cuprates) leading to $d$-wave pairing, previous work by the authors suggested that an electron configuration closer to $d^8$ could yield even higher critical temperatures ($T_c$) via $s_{\pm}$-wave pairing.

In this scenario, $s_{\pm}$-wave symmetry implies a sign reversal of the superconducting gap function between the $d_{x^2-y^2}$ orbital and other $3d$ orbitals. Previously, achieving this $d^8$-like state was theorized to require the presence of residual hydrogen (which modifies the electronic structure). However, the existence and role of residual hydrogen are experimentally controversial.

The Core Question: Can the $d^8$-like configuration and subsequent high-$T_c$ $s_{\pm}$-wave superconductivity be realized in infinite-layer nickelates without relying on residual hydrogen? The authors propose achieving this by heavily hole-doping the material (substituting La with Sr or Ba) while maintaining the tetragonal crystal structure via epitaxial strain.

2. Methodology

The study employs a multi-step computational approach combining first-principles calculations with many-body perturbation theory:

• First-Principles Calculations (DFT):

  • Used the QUANTUM ESPRESSO code to calculate band structures for $La_{1-x}Sr_xNiO_2$ and $La_{1-x}Ba_xNiO_2$.
  • Strain Engineering: The in-plane lattice constants were fixed to match three different substrates: SrTiO$_3$ (STO), LSAT, and LaAlO$_3$ (LAO). This simulates thin-film growth where the substrate enforces the $P4/mmm$ symmetry even at high doping levels ($x$), preventing the structural transition to orthorhombic phases found in bulk Sr/Ba nickelates.
  • Phonon Calculations: Performed using Density Functional Perturbation Theory (DFPT) to verify the dynamic stability of the tetragonal structure across the entire doping range ($0 \le x \le 1$).

• Model Construction:

  • Extracted Maximally Localized Wannier Functions (MLWFs) using the RESPACK code to construct a five-orbital model (all Ni $3d$ orbitals).
  • Virtual Crystal Approximation (VCA): Used to account for band variance and energy level shifts ($\Delta E$) as the Sr/Ba content ($x$) increases, moving beyond the rigid band approximation.

• Interaction Parameters:

  • Calculated electron-electron interaction parameters (Coulomb repulsion $U$, Hund's coupling $J$, etc.) using the Constrained Random Phase Approximation (cRPA) for each specific $x$ and substrate combination.

• Many-Body Analysis:

  • Applied the Fluctuation Exchange (FLEX) approximation to solve the linearized Eliashberg equation.
  • The eigenvalue ($\lambda$) of the Eliashberg equation serves as the measure of superconducting instability (where $\lambda \to 1$ at $T_c$).
  • Calculations were performed at a fixed temperature of $T = 0.01$ eV.

3. Key Contributions

1. Alternative Path to $d^8$ Configuration: The paper demonstrates that heavy hole doping combined with epitaxial strain can achieve the necessary electronic configuration for $s_{\pm}$-wave pairing without relying on the controversial residual hydrogen mechanism.
2. Validation of Structural Stability: Through phonon calculations, the authors prove that the tetragonal $P4/mmm$ symmetry (essential for the theoretical model) remains dynamically stable even with significant Sr substitution ($x$ up to 1.0) when grown on specific substrates.
3. Quantitative Interaction Analysis: Unlike previous studies that assumed rigid bands and fixed interactions, this work uses VCA and cRPA to dynamically update band structures and interaction parameters as doping increases, providing a more realistic quantitative prediction.
4. Substrate Dependence: The study explicitly links the magnitude of the energy level offset ($\Delta E$) between orbitals to the lattice constant of the substrate, showing that smaller lattice constants (stronger strain) favor the high-$T_c$ regime.

4. Key Results

• Structural Stability: Phonon dispersion calculations for $x=0.5$ (and others in supplemental material) show no imaginary frequencies, confirming that the tetragonal structure is stable for the entire doping range on STO, LSAT, and LAO substrates.
• Electronic Structure Evolution:
  • As $x$ increases, the Fermi level moves down, reducing the occupancy of the $d_{x^2-y^2}$ band.
  • Crucially, the energy level offset ($\Delta E$) between the $d_{x^2-y^2}$ orbital and the other $3d$ orbitals (specifically $d_{3z^2-r^2}$ and $t_{2g}$ bands) increases significantly with shorter lattice constants (e.g., LAO).
  • This creates a regime where the $d_{x^2-y^2}$ band is nearly empty while the other bands are "incipient" (lying just below the Fermi level), mimicking the $d^8$-like physics.
• Superconducting Instability ($\lambda$):
  • Symmetry Transition: For low doping ($x < 0.3$), the dominant pairing symmetry is $d$-wave. For heavy doping ($x > 0.3$), it transitions to $s_{\pm}$-wave.
  • Peak $T_c$: The eigenvalue $\lambda$ (proxy for $T_c$) is maximized in the range $x \approx 0.5 - 0.7$.
  • Substrate Effect: Substrates with smaller lattice constants (like LaAlO$_3$) yield larger $\Delta E$ and higher maximum $\lambda$ values. This is because the large $\Delta E$ enhances the inter-orbital scattering favorable for $s_{\pm}$-wave pairing.
  • Comparison: The calculated $\lambda$ values for the optimal heavily doped regime are comparable to or exceed those calculated for cuprates within the same formalism.
• Gap Function: The gap function shows a sign reversal between the $d_{x^2-y^2}$ orbital and the other $3d$ orbitals, confirming the $s_{\pm}$-wave nature.

5. Significance

• Theoretical Guidance for Experiments: This work provides a concrete roadmap for experimentalists: to realize high-$T_c$ $s_{\pm}$-wave superconductivity in nickelates, one should grow heavily hole-doped ($x \approx 0.5-0.7$) infinite-layer nickelate thin films on substrates with small lattice constants (like LaAlO$_3$).
• Mechanism Clarification: It strengthens the theoretical case for $s_{\pm}$-wave pairing in nickelates driven by orbital physics (incipient bands and large $\Delta E$) rather than solely by hydrogen doping or simple cuprate-like physics.
• Robustness: The finding that the tetragonal phase is stable under heavy doping suggests that the "nickel age" of superconductivity may extend beyond the currently observed low-doping regimes, potentially unlocking materials with $T_c$ exceeding current records.

In summary, the paper theoretically establishes that heavily hole-doped, strained infinite-layer nickelates are a viable platform for high-$T_c$ $s_{\pm}$-wave superconductivity, driven by the specific interplay of orbital energy offsets and electron correlations in a $d^8$-like configuration.