# Magnetic Bands Producing a Monoclinic Magnetic Structure in NiO, FeO, MnO, and a Tetragonal One in CoO

## Abstract

**:**

## 1. Introduction

## 2. Paramagnetic CoO, NiO, FeO, and MnO

- being roughly half-filled,
- comprising all the Bloch electrons at the Fermi level, and
- defining Bloch functions which can be unitarily transformed into Wannier functions that
- –
- are adapted to the fcc structure,
- –
- possess the two-dimensional ${\Gamma}_{3}^{+}$ symmetry,
- –
- are optimally localized, and
- –
- are situated on the Co atoms.

- An insulating band defining Wannier functions with the two-dimensional ${\Gamma}_{3}^{+}$ symmetry exists neither in paramagnetic FeO nor in MnO because in these materials the Bloch states with ${L}_{1}^{+}$, ${\Sigma}_{1}$, ${X}_{5}^{+}$, and ${W}_{3}$ symmetry lie above the Fermi energy. Thus, in FeO and in MnO more than two branches cross the Fermi level and, consequently, the optimally localized and symmetry-adapted Wannier functions have a five-dimensional ${\Gamma}_{3}^{+}+{\Gamma}_{5}^{+}$ symmetry [11] because we demand that the band comprises all the Bloch electrons at the Fermi level.
- NiO possesses an insulating band with optimally localized Wannier functions of the two-dimensional ${\Gamma}_{3}^{+}$ symmetry [10]. However, the Fermi energy in NiO is moved upward by roughly 0.5 eV (Figure 1). As a consequence, the highlighted band is in NiO not so precisely half-filled than in CoO.

## 3. Magnetic Bands

#### 3.1. CoO

#### 3.2. MnO

- possesses eight Mn atoms and eight O atoms in the unit cell and
- is invariant under the magnetic group ${M}_{110}$ in Equation (1).

#### 3.3. FeO

#### 3.4. NiO

**Band red:**- Defined by the bold lines in black and in red.
**Band green:**- Defined by the bold lines in black and in green.

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NHM | Nonadiabatic Heisenberg model |

I | Inversion |

K | Antiunitary operator of time inversion |

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**Figure 1.**Conventional band structure of paramagnetic fcc CoO as determined in [9] serving in this section as approximation of the band structures of paramagnetic fcc NiO, FeO, and MnO when the Fermi energies are adjusted. The colored horizontal lines denote the different Fermi energies, and the insulating band of CoO is highlighted by the bold lines. The symmetry labels are defined in Table A1 of [10].

**Figure 2.**The conventional band structure of paramagnetic MnO in Figure 2 of [11] folded into the Brillouin zone for the tetragonal body-centered lattice ${\Gamma}_{q}^{v}$ as given in Figure 3.10b of [12]. The bands are calculated by the FHI-aims program [13,14]. The symmetry labels are defined in Table A2 of [9] and the compatibility relations between the line $\Lambda $ and its endpoints $\Gamma $ and Z are given in Table A6b of [9].

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**MDPI and ACS Style**

Krüger, E.
Magnetic Bands Producing a Monoclinic Magnetic Structure in NiO, FeO, MnO, and a Tetragonal One in CoO. *Symmetry* **2022**, *14*, 1285.
https://doi.org/10.3390/sym14071285

**AMA Style**

Krüger E.
Magnetic Bands Producing a Monoclinic Magnetic Structure in NiO, FeO, MnO, and a Tetragonal One in CoO. *Symmetry*. 2022; 14(7):1285.
https://doi.org/10.3390/sym14071285

**Chicago/Turabian Style**

Krüger, Ekkehard.
2022. "Magnetic Bands Producing a Monoclinic Magnetic Structure in NiO, FeO, MnO, and a Tetragonal One in CoO" *Symmetry* 14, no. 7: 1285.
https://doi.org/10.3390/sym14071285