Hubbard-$U$-corrected electron-phonon interactions in strongly correlated materials via the finite-displacement method

This paper presents a finite-displacement algorithm that integrates Hubbard $U$ corrections into electron-phonon calculations for strongly correlated materials, demonstrating that these corrections significantly alter phonon stability and electron-phonon coupling in LaNiO$_2$ and RuO$_2$ by modifying Fermi surface topology, thereby resolving discrepancies between theoretical predictions and experimental observations.

The Gist

Imagine a bustling city where two types of traffic jams are constantly happening. One is caused by cars bumping into each other (electrons repelling other electrons), and the other is caused by cars hitting potholes in the road (electrons bumping into the vibrating ground, or "phonons"). In the world of superconductors—materials that conduct electricity with zero resistance—scientists want to know how these two traffic jams interact. If they work together just right, the city can achieve a "super-highway" where traffic flows perfectly.

For years, scientists have used a standard map-making tool called DFT (Density Functional Theory) to study these materials. However, in "strongly correlated" materials (where the cars are very aggressive and constantly bumping into each other), this standard map is often inaccurate. To fix this, scientists added a "correction factor" called Hubbard U to the map.

The problem was that while scientists knew how to use this correction for the cars (electrons), they didn't know how to apply it to the potholes (phonons) or the crashes between them (electron-phonon coupling). They were correcting the map of the cars but ignoring the fact that the potholes themselves might change shape because of the aggressive driving.

The New Algorithm: A Complete Remodel
This paper introduces a new method (an algorithm) that applies the "Hubbard U" correction to everything: the cars, the potholes, and the crashes between them. Think of it as a construction crew that doesn't just fix the cars, but also re-paves the road and redesigns the traffic rules all at once, ensuring everything is consistent.

The researchers tested this new "full remodel" on two specific materials:

1. The Nickelate City (LaNiO₂)

• The Mystery: This material becomes a superconductor at very low temperatures. Recent studies using a different, very expensive method (called GW) suggested that the "crashes" between cars and potholes were huge—five times bigger than the standard map predicted. This implied the crashes were the main reason for the superconductivity.
• The Paper's Finding: When the authors used their new "full remodel" (DFT+U), they found the crashes were still small.
• The Analogy: Imagine the GW method said, "The cars are crashing into the potholes so hard that the whole road is shaking!" The new method says, "Actually, the cars are just driving normally."
• Why the difference? The two methods drew the city's layout (the Fermi surface) differently. The GW method drew a layout where cars were forced into a tight corner, causing massive crashes. The new method drew a layout where the cars had plenty of room to move, so the crashes remained minor. The authors conclude that for this material, the "crashes" alone are too weak to explain the superconductivity, suggesting something else is driving the phenomenon.

2. The Ruthenium Dioxide City (RuO₂)

• The Mystery: This material is a thin film grown on a specific substrate. Experiments show it becomes a superconductor, but only at a very low temperature (1.5 Kelvin). However, the standard map (plain DFT) predicted a disaster: it said the road was so unstable it would collapse (imaginary phonon modes) and that the crashes were so violent the city should be a superconductor at a much higher temperature (30 Kelvin).
• The Paper's Finding: When they applied the "full remodel" (adding Hubbard U to the road and the crashes), two things happened:
  1. The Road Stabilized: The "collapsing road" (imaginary modes) disappeared. The road became solid and stable, matching what we see in the real world.
  2. The Crashes Calmed Down: The violent crashes turned into gentle bumps. The total "crash energy" dropped significantly.
• The Result: This explains why the superconductivity is so weak (low temperature). The "correction" stiffened the road (phonon hardening), making it harder for the cars to crash into it. This perfectly matches the experimental reality.

The Big Takeaway

The paper argues that you cannot fix the "cars" (electrons) without also fixing the "roads" (phonons) and the "traffic rules" (coupling).

• If you only fix the cars (a "partial" correction), you might get the wrong answer. In the case of Ruthenium Dioxide, a partial fix would have predicted a super-strong superconductor that doesn't exist in reality.
• The authors show that for some materials (like Nickelates), the correction changes the layout slightly but not the outcome much. For others (like Ruthenium Dioxide), the correction is essential to stop the road from collapsing and to explain why the superconductivity is so weak.

In short, this paper provides a new, more consistent way to map out how electrons and vibrations interact in complex materials, showing that ignoring the "correlation" effects on the vibrations themselves leads to misleading predictions.

Technical Summary

Technical Summary: Hubbard-U-corrected electron-phonon interactions in strongly correlated materials via the finite-displacement method

Problem Statement
The interplay between electron-electron and electron-phonon interactions is central to understanding strongly correlated materials. While the Density Functional Theory plus Hubbard $U$ (DFT+U) method is standard for calculating electronic structures in these systems, its extension to electron-phonon coupling ($g$ matrices) and phonon spectra has been limited. Previous approaches often treat electron-electron and electron-phonon interactions separately, using plain DFT for phonons and empirical parameters for correlations. Although full GW corrections have been applied to electron-phonon calculations in materials like cuprates and nickelates, the computational cost remains prohibitively high for complex real materials. Furthermore, the specific impact of Hubbard $U$ corrections on phonon spectra and electron-phonon $g$ matrices has not been systematically investigated, despite the known importance of correlation effects in these systems.

Methodology
The authors implement an algorithm that integrates the DFT+U method with the finite-displacement method to calculate phonon spectra and electron-phonon $g$ matrices. This approach circumvents the convergence issues often encountered in Density Functional Perturbation Theory plus Hubbard $U$ (DFPT+U) calculations, particularly when large $U$ values are applied.

Key methodological features include:

• Finite-Displacement Approach: The method utilizes supercell DFT+U calculations to obtain phonon perturbation potentials in real space, which are then Fourier transformed to derive the electron-phonon $g$ matrix in momentum space.
• Basis Set: The calculations are performed using the OpenMX package with numeric atomic orbital (NAO) basis sets, chosen for their computational efficiency in large-scale supercell simulations compared to plane-wave bases.
• Fully U-Corrected Framework: Crucially, the Hubbard $U$ correction is applied consistently across three components: the electronic structure (band structure), the phonon spectrum (force constants), and the electron-phonon $g$ matrices. This distinguishes the work from "partially U-corrected" approaches where correlations are applied only to the electronic structure while phonons and coupling matrices remain at the plain DFT level.
• Materials Studied: The algorithm is applied to two prototypical correlated materials: infinite-layer nickelates (specifically 20% hole-doped LaNiO$_2$) and ruthenium dioxide (RuO$_2$), including strained RuO$_2$ thin films on TiO$_2$ substrates.

Key Results

• Infinite-Layer Nickelates (LaNiO$_2$):

  • Electronic Structure: Hubbard $U$ corrections on Ni-$d$ orbitals primarily push other Ni-$d$ bands away from the Fermi level but leave the primary Ni-$d_{x^2-y^2}$ band largely pinned. This results in a moderate modification of the Fermi surface, inducing a small zone-centered electron pocket but preserving the dominant Ni-$d_{x^2-y^2}$ sheet.
  • Phonons and Coupling: The phonon spectrum remains dynamically stable and largely insensitive to $U$ corrections. Consequently, the total electron-phonon coupling ($\lambda$) increases only slightly (from 0.18 at $U=0$ eV to 0.28 at $U=7$ eV).
  • Superconductivity: Even with full $U$ corrections, the estimated superconducting transition temperature ($T_c$) remains below 3 K, far below the experimental value of 10–30 K.
  • Comparison with GW: These results contrast with recent full GW studies that predict a five-fold increase in electron-phonon coupling. The authors attribute this discrepancy to the different Fermi surface topologies predicted by DFT+U (dominated by Ni-$d_{x^2-y^2}$) versus GW (which predicts a reconstruction including La-$d$ derived sheets).

• Ruthenium Dioxide (RuO$_2$):

  • Stability: Plain DFT calculations predict imaginary phonon modes (lattice instability) for the experimentally observed orthorhombic $Pnnm$ structure of strained RuO$_2$. The inclusion of Hubbard $U$ corrections on Ru-$d$ orbitals eliminates these imaginary modes, dynamically stabilizing the structure.
  • Phonon Hardening: Increasing $U$ induces significant phonon hardening, particularly for low-frequency modes.
  • Coupling Reduction: This phonon hardening substantially reduces the total electron-phonon coupling, decreasing $\lambda$ from 0.8 (at $U=1$ eV) to 0.2 (at $U=5$ eV).
  • Superconductivity: The reduction in coupling lowers the estimated $T_c$ from ~30 K (at low $U$) to a few Kelvin, aligning with the experimental observation of ~1.5 K. This resolves the discrepancy between the large coupling predicted by plain DFT and the low experimental $T_c$.
  • Partial vs. Full Correction: The "partially U-corrected" approach (using DFT phonons) fails for RuO$_2$, yielding unphysically large coupling due to the presence of imaginary modes. This highlights the necessity of applying correlation corrections to the phonon spectrum itself.

Significance and Claims
The paper claims to provide a fully self-consistent algorithm for calculating electron-phonon interactions in correlated materials that incorporates Hubbard $U$ corrections not only in the electronic structure but also in the phonon spectrum and electron-phonon coupling matrices.

The authors assert that their work highlights two distinct mechanisms by which correlation affects electron-phonon coupling:

1. Electronic Channel: Modifying the Fermi surface topology and scattering phase space (observed as a minor effect in LaNiO$_2$).
2. Phonon Channel: Directly renormalizing lattice dynamics, leading to phonon hardening or softening (observed as a dominant effect in RuO$_2$).

The study concludes that a reliable description of electron-phonon interactions in correlated materials requires correlation effects to be consistently included in all degrees of freedom (electronic and lattice). Neglecting these effects in the phonon sector can lead to qualitatively incorrect predictions, such as spurious lattice instabilities or gross overestimations of superconducting transition temperatures. The work offers a computationally efficient alternative to full GW methods for investigating these properties in complex real materials.