# Influence of f Electrons on the Electronic Band Structure of Rare-Earth Nickelates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}. Contrary to this, superconductivity in LaNiO

_{2}is still under debate. This indicates the crucial role played by the f electrons on the electronic structure and the pairing mechanism of infinite-layer nickelates. Here, we discuss the role of the electron correlations in the f electron states and their influence on the electronic structure. We show that the lattice parameters are in good agreement with the experimental values, independent of the chosen parameters within the DFT+U approach. Increasing Coulomb interaction U tends to shift the f states away from the Fermi level. Surprisingly, independently of the position of f states with respect to the Fermi energy, these states play an important role in the electronic band structure, which can be reflected in the modification of the NdNiO

_{2}effective models.

## 1. Introduction

_{2}at 9–15 K, in thin-film samples grown on SrTiO

_{3}[1,2,3]. Contrary to the thin layers, the bulk NdNiO

_{2}does not exhibit superconductivity [4]. Superconductivity was also reported in hole doped PrNiO

_{2}[5,6], while its occurrence in LaNiO

_{2}is still under debate [1,7].

_{2}were performed based on density functional theory (DFT) [11,12,13,14,15,16,17,18,19], DFT including correlation on mean field level (DFT+U) [19,20,21,22,23], or combination of DFT and dynamical mean-field theory (DFT+DMFT) [18,23,24,25,26,27,28,29,30,31] approaches.

_{2}[32] and magnetic and charge instabilities in NdNiO

_{2}[33,34,35]. The observed charge density wave (CDW), with the same in-plane wavevector (1/3, 0) for Nd $5d$ and Ni $3d$ orbitals, disappears when superconductivity emerges in doped NdNiO

_{2}[34]. Similarly, for LaNiO

_{2}, the CDW is characterized by incommensurate wavevector, and under doping the charge order diminishes and its wavevector shifts towards the commensurate order [36]. Such results suggest the existence of charge order and its potential interplay with antiferromagnetic fluctuations and superconductivity in infinite-layer nickelates.

_{3}) [46]. These values cover the expected range, while ${U}_{eff}=10$ eV, used for the description of NdNiO

_{2}[14], we consider too large. The range of used parameters raises the question of not only the role of the f electron in the electronic properties but also the realistic parameters describing correlation in rare-earth nickelates.

## 2. Calculation Details

**k**–point grid. For the A-AFM unit cell, built as a $1\times 1\times 2$ supercell and containing two formula units, a $10\times 10\times 4$

**k**-grid was used. Similarly, the C-AFM unit cell is related to the $\sqrt{2}\times \sqrt{2}\times 1$ supercell containing two formula units—here the $7\times 7\times 8$

**k**-grid was used. Finally, the G-AFM unit cell (related to the $\sqrt{2}\times \sqrt{2}\times 2$ supercell and containing four formula units) was optimized with the $7\times 7\times 4$

**k**-grid. In all of the cases, the

**k**-grid in the Monkhorst–Pack scheme [56] was used. As the convergence condition of an optimization loop, we took the energy differences of ${10}^{-5}$ eV and ${10}^{-7}$ eV for ionic and electronic degrees of freedom, respectively.

## 3. Results and Discussion

#### 3.1. Crystal Structure

_{2}crystallizes in the P4/mmm symmetry (space group No. 123) [57]. The atoms are located in the high symmetry Wyckoff positions: Ni $1a$(0, 0, 0), O $1d$(0, 1/2, 0), and Nd $2f$(1/2, 1/2, 1/2). Experimentally, the average values of lattice constants are $a=3.920$ Å and $c=3.275$ Å (the lattice constants obtained by minimization of magnetic structures for increasing values of ${U}_{Nd}$ are collected in Table 1) [57,58,59].

#### 3.2. Magnetic Ground State

_{2}[20], and are mostly twice as large as those observed in LaNiO

_{2}[61] (for similar Hubbard-like parameters assumed on Ni ions).

_{2}. However, experimental results suggest the realization of strong anitferromagnetic fluctuations, which can be reflected in the magnetic ground state found within the DFT+U manner.

#### 3.3. Electronic Density of States

#### 3.4. The Changes in the Electronic Bands due to f Electron States

_{2}system. Recently, many models have been developed, from one band models based on Ni ${d}_{{x}^{2}-{y}^{2}}$ orbitals [29,61] to the multiband models, such as: (i) Two orbital models (e.g., based on Ni ${d}_{{x}^{2}-{y}^{2}}$ and ${d}_{{z}^{2}}$ orbitals [17]; Nd ${d}_{{z}^{2}}$ and Ni ${d}_{xy}$ orbitals [53], Ni ${d}_{{x}^{2}-{y}^{2}}$ and extended s-like state [10]); (ii) Three orbital models (containing Ni ${d}_{xy}$ and ${d}_{{z}^{2}}$ orbitals with additional interstitial s orbital [23]; Ni ${d}_{{x}^{2}-{y}^{2}}$ and Nd ${d}_{{z}^{2}}$ orbitals with additional interstitial orbital mixing s and ${d}_{xy}$ orbitals centered on Nd ions [11]; effective Nd ${d}_{{z}^{2}}$, Nd ${d}_{{x}^{2}-{y}^{2}}$ and Ni ${d}_{xy}$ orbitals [20]; Ni ${d}_{{z}^{2}}$ and ${d}_{{x}^{2}-{y}^{2}}$ orbitals, and a self-doped s-like orbital [27]); or (iii) A full 13-orbitals tight binding model in the Wannier orbital basis [62]. Nevertheless, the role of f orbitals should be introduced indirectly to the model, by the effective modification of tight binding hopping parameters.

## 4. Summary and Conclusions

_{2}possesses the AFM ground state order. The calculations predict the C-AFM ground state order which we treat here as a prediction for future experimental studies. In the absence of Coulomb interactions at Nd atoms, the f states are located in the close vicinity of the Fermi level. The introduction of Coulomb interaction ${U}_{Nd}$ at Nd atoms, within the DFT+U manner, generates the shift of the f states, and then the energies of these states are far away from the Fermi level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Phases of NdNiO

_{2}with magnetic order discussed in the paper: (

**a**) ferromagnetic (FM); (

**b**) C-type antiferromagnetic (C-AFM); (

**c**) A-type antiferromagnetic (A-AFM); and (

**d**) G-type antiferromagnetic (G-AFM). The magnetic states follow from the orientation of Ni-spins (at gray balls).

**Figure 2.**Comparison of the electronic density of states (DOS) of NdNiO

_{2}for different magnetic configurations and model parameters. We take constant values of ${U}_{Ni}=5.0$ eV and ${J}_{Ni}=0.5$ eV; the Coulomb parameter at Nd ions ${U}_{Nd}$ increases as marked, while Hund’s exchange is constant and locally stabilizes high-spin states, ${J}_{Nd}=0.7$ eV. The total DOS is shown by the black line, Ni $5d$ states by the red line, while the multiplet structure of the Nd $4f$ electrons is indicated by gray-shaded maxima.

**Figure 3.**Electronic band structure (

**a**) and density of states (panels (

**b**,

**c**)) of NdNiO

_{2}with C-AFM spin order obtained for different treatments of the Nd f electrons. The band structures (

**a**) for the f electrons treated as core or as valence states are presented by red and blue lines, respectively. Related density of states, for the f electrons treated as core or as valence states are presented in panels (

**b**) and (

**c**), respectively. Results obtained within DFT+U (GGA PBE method) for the Coulomb interactions ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=8.0$ eV, ${J}_{Nd}=0.7$ eV.

**Table 1.**Comparison of the predicted NdNiO

_{2}system parameters. Bolded energies denote the magnetic ground states for given sets of interaction parameters.

Phase | Lattice Constant (Å) | E (eV/f.u.) | ${\mathit{\mu}}_{\mathit{Nd}}$ (${\mathit{\mu}}_{\mathit{B}}$) | ${\mathit{\mu}}_{\mathit{Ni}}$ (${\mathit{\mu}}_{\mathit{B}}$) | |
---|---|---|---|---|---|

a | c | ||||

Experimental values | |||||

Ref. [57] | 3.919 | 3.307 | - | - | - |

Ref. [58] | 3.914 | 3.239 | - | - | - |

Ref. [59] | 3.928 | 3.279 | - | - | - |

DFT (GGA PBE) | |||||

NM | 3.909 | 3.314 | −28.464 | - | - |

FM | 3.896 | 3.277 | −31.462 | 3.155 | 0.184 |

A-AFM | 3.894 | 3.312 | −31.397 | 3.123 | 0.081 |

C-AFM | 3.900 | 3.267 | −31.448 | 3.123 | 0.404 |

G-AFM | 3.903 | 3.283 | −31.424 | 3.083 | 0.406 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.925 | 3.249 | −29.618 | 3.129 | 0.854 |

A-AFM | 3.926 | 3.273 | −29.568 | 3.089 | 0.892 |

C-AFM | 3.952 | 3.216 | −29.594 | 3.076 | 1.007 |

G-AFM | 3.871 | 3.303 | −29.170 | 3.112 | 0.000 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=2.0$ eV, ${J}_{Nd}=0.7$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.937 | 3.292 | −29.287 | 3.125 | 0.924 |

A-AFM | 3.929 | 3.307 | −28.948 | 3.138 | 0.867 |

C-AFM | 3.941 | 3.263 | −29.303 | 3.046 | 0.962 |

G-AFM | 3.879 | 3.333 | −28.629 | 3.162 | 0.000 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=4.0$ eV, ${J}_{Nd}=0.7$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.940 | 3.284 | −28.712 | 3.061 | 0.893 |

A-AFM | 3.940 | 3.290 | −28.714 | 3.056 | 0.938 |

C-AFM | 3.950 | 3.263 | −28.742 | 3.031 | 0.940 |

G-AFM | 3.949 | 3.264 | −28.740 | 3.004 | 0.945 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=6.0$ eV, ${J}_{Nd}=0.7$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.943 | 3.286 | −28.483 | 3.016 | 0.905 |

A-AFM | 3.944 | 3.287 | −28.483 | 3.006 | 0.936 |

C-AFM | 3.955 | 3.274 | −28.517 | 3.017 | 0.941 |

G-AFM | 3.954 | 3.275 | −28.515 | 2.997 | 0.948 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=8.0$ eV, ${J}_{Nd}=0.7$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.950 | 3.294 | −28.321 | 3.014 | 0.910 |

A-AFM | 3.948 | 3.297 | −28.319 | 3.004 | 0.933 |

C-AFM | 3.958 | 3.285 | −28.351 | 3.013 | 0.944 |

G-AFM | 3.960 | 3.286 | −28.349 | 2.996 | 0.951 |

DFT+U (GGA PBE, ${U}_{Ni}=5.0$ eV, ${J}_{Ni}=0.5$ eV, ${U}_{Nd}=9.0$ eV, ${J}_{Nd}=0.7$ eV) | |||||

NM | 3.891 | 3.306 | −26.203 | - | - |

FM | 3.951 | 3.299 | −28.256 | 3.016 | 0.911 |

A-AFM | 3.951 | 3.301 | −28.255 | 3.006 | 0.933 |

C-AFM | 3.959 | 3.288 | −28.285 | 3.014 | 0.946 |

G-AFM | 3.962 | 3.289 | −28.283 | 2.998 | 0.953 |

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## Share and Cite

**MDPI and ACS Style**

Ptok, A.; Basak, S.; Piekarz, P.; Oleś, A.M. Influence of *f* Electrons on the Electronic Band Structure of Rare-Earth Nickelates. *Condens. Matter* **2023**, *8*, 19.
https://doi.org/10.3390/condmat8010019

**AMA Style**

Ptok A, Basak S, Piekarz P, Oleś AM. Influence of *f* Electrons on the Electronic Band Structure of Rare-Earth Nickelates. *Condensed Matter*. 2023; 8(1):19.
https://doi.org/10.3390/condmat8010019

**Chicago/Turabian Style**

Ptok, Andrzej, Surajit Basak, Przemysław Piekarz, and Andrzej M. Oleś. 2023. "Influence of *f* Electrons on the Electronic Band Structure of Rare-Earth Nickelates" *Condensed Matter* 8, no. 1: 19.
https://doi.org/10.3390/condmat8010019