Influence of ligand field and correlation on the electronic structure of NiO and CoO from DFT+DMFT calculations

This study employs charge self-consistent DFT+DMFT calculations to investigate how crystal structure, ligand fields, and varying correlation strengths (including oxygen 2p correlations) influence the electronic structure and spectral functions of paramagnetic NiO and CoO.

The Gist

Imagine a world made of tiny, dancing magnets and electrons. In certain materials called Transition Metal Oxides (like Nickel Oxide and Cobalt Oxide), these electrons don't just dance freely; they are "strongly correlated." This means they are so sensitive to each other's presence that if one moves, the others react instantly. It's like a crowded dance floor where everyone is holding hands; you can't move without pulling your neighbor with you.

This paper is a deep dive into understanding exactly how these electrons behave in Nickel Oxide (NiO) and Cobalt Oxide (CoO). The authors used a powerful computer simulation method called DFT+DMFT (a mix of two advanced physics theories) to create a "movie" of the electrons' energy levels, which they call spectral functions. Think of this as a map showing where the electrons like to hang out and how much energy they need to jump to a new spot.

Here is a breakdown of their findings using simple analogies:

1. The Two Main Ingredients: The "Crowd" and the "Room Shape"

The researchers looked at two main things that change how the electrons behave:

• The Crowd (Correlation Strength): How strongly the electrons push against each other. In the simulation, they adjusted a dial called $U$ to see what happens when the electrons get more "grumpy" and want to stay apart.
• The Room Shape (Ligand Field): The atoms in these materials are arranged in specific geometric shapes. The authors compared two shapes:
  • Rock-Salt (RS): Like a cube where the metal atom is surrounded by oxygen atoms on all sides (an octahedron). This is the natural, stable shape for these materials.
  • Zincblende (ZB): Like a pyramid shape (a tetrahedron). This is an unstable, "what-if" shape the authors tested to see how the geometry changes the electron dance.

2. The "Self-Interaction" Correction (SIC)

Standard computer models often make a mistake: they treat an electron as if it interacts with itself, which is physically impossible (like you trying to push yourself). The authors added a special fix called Self-Interaction Correction (SIC) specifically for the Oxygen atoms.

• The Analogy: Imagine a noisy party. The standard model thinks the Oxygen atoms are shouting at themselves, creating a lot of background noise that messes up the music. The SIC method turns down that specific noise.
• The Result: When they turned on this "noise-canceling" feature for the Oxygen, the simulation matched real-world experiments much better. It correctly predicted the size of the "gap" (the energy needed to make the material conduct electricity), landing right in the 5–6 eV range found in real labs.

3. Nickel vs. Cobalt: The "One-Step" Difference

Nickel and Cobalt are neighbors on the periodic table. The only difference is that Cobalt has one fewer electron in its "dance troupe" than Nickel.

• The Finding: Because Cobalt has one less electron, it has an extra "empty seat" (a hole) in its dance floor. The authors found that this extra empty seat makes the low-energy part of the spectrum (the music near the floor) much sharper and more distinct in Cobalt than in Nickel. It's like removing one dancer from a synchronized routine; the remaining dancers have to adjust, creating a clearer, more defined pattern.

4. The Shape Matters: Cubes vs. Pyramids

When they compared the stable Rock-Salt (cube) structure to the unstable Zincblende (pyramid) structure:

• The Energy Levels: The "rooms" (geometric shapes) change the energy levels of the electron orbits. In the cube, certain orbits are lower in energy; in the pyramid, the order flips.
• The Stability: The authors explain that the pyramid shape is less stable for these materials because it forces more electrons into "antibonding" states.
  • The Analogy: Think of "bonding" states as comfortable chairs where electrons are happy. "Antibonding" states are like sitting on a wobbly stool. The pyramid shape forces more electrons onto wobbly stools, making the whole structure shaky. This explains why Nickel Oxide naturally forms cubes, while the pyramid shape is rare and unstable.

5. The "Satellite" and the "Gap"

The paper looks at specific features in the electron energy map:

• The Satellite Peak: A high-energy "echo" caused by strong electron interactions. The authors found that their method could see this echo, but it was faint in some calculations.
• The Band Gap: The most important number. It's the energy wall that stops electricity from flowing (making the material an insulator).
  • The Result: Without the Oxygen correction (SIC), their computer models predicted a wall that was too low (around 2.5–3 eV). With the Oxygen correction, the wall jumped up to 5.1 eV, which matches what scientists measure in real experiments.

Summary

In short, this paper is a recipe for getting the perfect simulation of Nickel and Cobalt oxides. The authors discovered that to get the right answer, you can't just look at the metal atoms; you must also fix the way the computer treats the surrounding Oxygen atoms. When you do that, and you account for how the electrons push against each other, your digital model becomes a perfect mirror of the real physical world. This helps scientists understand why these materials act the way they do, which is crucial for designing better batteries, catalysts, and electronic devices.

Technical Summary

Technical Summary: Influence of Ligand Field and Correlation on the Electronic Structure of NiO and CoO

Problem Statement
Transition metal oxides (TMOs) like NiO and CoO exhibit complex electronic behaviors driven by the interplay of electron correlation, ligand fields, crystal structure, and hybridization. While these materials are promising for applications in thermoelectricity, optoelectronics, and catalysis, their electronic structure is difficult to model accurately. Standard Density Functional Theory (DFT) with Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA) often fails to reproduce experimental spectra, incorrectly predicting metallic ground states for these insulating compounds. Although DFT+U offers improvements, it lacks the dynamic treatment of charge and spin fluctuations necessary for paramagnetic high-temperature phases. The challenge lies in simultaneously capturing the strong correlations in transition metal (TM) $3d$ orbitals, the hybridization with oxygen $2p$ orbitals, and the influence of different crystal field environments (octahedral vs. tetrahedral) on spectral functions.

Methodology
The authors employ a charge self-consistent combination of Density Functional Theory and Dynamical Mean Field Theory (DFT+DMFT) to investigate paramagnetic NiO and CoO. The study utilizes the following specific methodological framework:

• Computational Setup: Calculations are performed using the mixed basis pseudopotential code (MBPP) for the DFT part and the continuous-time Quantum Monte Carlo (CTHYB) solver within the TRIQS package for the DMFT impurity problem.
• Correlated Subspace: The correlated subspace is defined using atomic-like $3d$ orbitals centered on the TM ions, projected onto a truncated Kohn-Sham basis.
• Interaction Parameters: The generalized multi-orbital Hubbard problem is solved using a rotationally invariant Slater Hamiltonian. The study varies the Hubbard $U$ parameter (6 eV and 9 eV for CoO; 10 eV for NiO) and Hund's coupling $J$ (approx. 1 eV).
• Self-Interaction Correction (SIC): To address correlation effects in the oxygen $2p$ orbitals, which are strongly hybridized with TM $3d$ states, the authors apply a self-interaction-correction pseudopotential scheme (DFT+sicDMFT). This involves adjusting oxygen pseudopotentials with specific weighting parameters ($w_l=1.0$ for $2s$ and $w_l=0.8$ for $2p$).
• Structures: The study compares two crystal structures: the stable rock-salt (RS) structure with octahedral ligand fields and the metastable zincblende (ZB) structure with tetrahedral ligand fields.
• Analysis: Real-frequency spectral functions are derived via analytical continuation (maximum entropy method) from imaginary Matsubara frequencies. The analysis focuses on total, element-projected (TM and O), and local orbital-resolved spectral functions.

Key Results

1. RS-CoO Spectral Functions:

   • The study identifies a low-energy peak below the Fermi level corresponding to a $d^7\underline{L}$ final state (where $\underline{L}$ is a ligand hole). Its position shifts from $-1.0$ eV to $-1.6$ eV as $U$ increases from 6 to 9 eV.
   • Applying O($2p$)-SIC shifts this peak to $-0.6$ eV, aligning better with specific UPS experimental data, though XPS data suggests a position near $-1.9$ eV.
   • The inclusion of O($2p$)-SIC significantly improves the description of the band gap. Without SIC, the gap between occupied and unoccupied peaks is underestimated. With $U=9$ eV and SIC, the calculated distance is 5.1 eV, which falls within the experimental range of 5–6 eV. Previous DFT+DMFT studies without SIC reported values as low as 2.5–3.8 eV.
   • The Lower Hubbard Band (satellite peak at $\approx -9.5$ eV in experiments) is not clearly resolved in the total spectral function without SIC but shows a slight bump with SIC, though shifted to $-7$ eV.

2. Structural and Compositional Effects (RS vs. ZB, NiO vs. CoO):

   • Crystal Field Theory Validation: The results confirm Crystal Field (CF) theory predictions. In RS structures (octahedral), the $t_{2g}$ states are lower in energy than $e_g$. In ZB structures (tetrahedral), the hierarchy reverses ($e$ lower than $t_2$).
   • Occupation: RS-NiO exhibits a $t_{2g}^6 e_g^2$ configuration, while RS-CoO shows $t_{2g}^5 e_g^2$. ZB structures show $e^4 t_2^4$ (NiO) and $e^4 t_2^3$ (CoO).
   • Stability: The ZB structures are found to be less stable due to the occupation of antibonding states. In ZB, the $t_2$ states are antibonding; CoO and NiO in ZB have 3 and 4 electrons in these states, respectively, compared to 2 electrons in the antibonding $e_g$ states of the stable RS phase.
   • Spectral Similarities: Despite structural differences, the local spectral functions of RS and ZB phases show qualitative similarities when comparing corresponding manifolds (e.g., RS-$t_{2g}$ resembles ZB-$e$).
   • CoO vs. NiO: CoO spectra exhibit sharper peaks and increased spectral weight at low energies compared to NiO. The authors attribute a sharp low-energy peak in CoO to $d^7\underline{Z}$ states (Zhang-Rice states), analogous to $d^8\underline{Z}$ in NiO, resulting from the additional hole in the valence configuration.

3. Impact of Correlation Settings:

   • SIC Effect: Applying O($2p$)-SIC consistently shifts unoccupied TM-projected bands (Upper Hubbard Bands) and O-projected bands to higher energies. It increases the energy difference ($\Delta E_{pd}$) between unoccupied TM and occupied O states, reducing charge fluctuations.
   • U Dependence: Increasing $U$ increases the distance between the centers of gravity of occupied and unoccupied TM-projected spectra, consistent with the Hubbard model.
   • Charge Transfer Energy ($\Delta$): The study distinguishes the effective charge-transfer energy $\Delta$ (derived from DFT single-particle levels) from the spectral gap. Calculated $\Delta$ values for RS-NiO (4.3 eV) and RS-CoO (4.9 eV) follow the experimental hierarchy (5.4 eV and 6.1 eV, respectively) but are slightly underestimated.

Significance and Claims
The paper claims that the DFT+sicDMFT approach provides a robust framework for describing the electronic structure of strongly correlated TMOs in their paramagnetic phases. The primary contribution is demonstrating that explicitly accounting for correlations in oxygen $2p$ orbitals via SIC is crucial for obtaining band gaps and spectral features that agree with experimental excitation spectra (specifically for RS-CoO). The study successfully decouples the effects of crystal structure (ligand field) and electron correlation strength, validating the application of Crystal Field Theory to the spectral features of these complex systems. The authors position these findings as descriptors that can support the research and optimization of TMOs for energy conversion devices, noting that characteristic features of electron excitation spectra correlate with catalytic activity. The work does not propose new experimental protocols but offers a refined computational methodology for interpreting existing experimental data and guiding the understanding of material functionality.