# Structural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiO

## Abstract

**:**

## 1. Introduction

## 2. Group-Theoretical and Computational Methods Used in the Paper

## 3. Magnetic Group of the Antiferromagnetic State: First Stability Condition

**Condition**

**1.**

- (i)
- following Case (a) with respect to the magnetic group $S+K\left\{R\right|\mathit{t}\}S$ in Equation (5) and
- (ii)
- following Case (c) with respect to the magnetic group $S+KS$.

## 4. Rhombohedral-Like Distortion

- ${\mathit{T}}_{1}$ still passes through the plain $\left(\overline{1}10\right)$ and
- in relation of this plane, ${\mathit{T}}_{2}$ and ${\mathit{T}}_{3}$ stay symmetrical to one another.

- (i)
- ${\mathit{\rho}}_{1}$ passes through P and
- (ii)
- ${\mathit{\rho}}_{2}$ and ${\mathit{\rho}}_{3}$ are symmetrical to each other with respect to P.

## 5. Wannier Functions Symmetry Adapted to M${}_{9}$: Second Stability Condition

#### 5.1. Atomic-Like Motion

- (i)
- to be adapted only to the magnetic group M of the magnetic structure or
- (ii)
- to be spin dependent,

**Condition**

**2.**

#### 5.2. Magnetic Band of NiO

## 6. Mott Insulator: Third Condition of Stability

- (i)
- The magnetic band with the symmetry in Table A2a occurs twice in the active band. Since Band 1 in Table A2b has the same symmetry, we can unitarily transform the Bloch functions of two branches of the active band into optimally localized Wannier functions centered at the Ni atoms, and the Bloch functions of the two remaining branches into optimally localized Wannier functions centered at the O atoms. Thus, the electrons perform an atomic-like motion with localized states situated at both the Ni and the O atoms.
- (ii)
- All the electrons at the Fermi level take part in the atomic-like motion because the active band comprises all the branches crossing the Fermi level.

**Definition**

**1.**

**Condition**

**3.**

## 7. Results

- The rhombohedral-like deformation of antiferromagnetic NiO,
- the stability of the antiferromagnetic state, and
- the insulating ground state.

#### 7.1. The Rhombohedral-Like Deformation

#### 7.2. The Stability of the Antiferromagnetic State

#### 7.3. The Insulating Ground State

## 8. Discussion

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\Delta E$ | Nonadiabatic condensation energy as defined in Equation (2.20) of [4] |

NHM | Nonadiabatic Heisenberg model |

E | Identity operation |

I | Inversion |

${C}_{2b}$ | Rotation through $\pi $ as indicated in Figure 2 |

${\sigma}_{db}$ | Reflection $I{C}_{2b}$ |

K | Anti-unitary operator of time inversion |

## Appendix A. Wyckoff Positions

## Appendix B. Group-Theoretical Tables

**Table A1.**Character tables of the single valued irreducible representations of the cubic space group $Fm3m={\mathsf{\Gamma}}_{c}^{f}{O}_{h}^{5}$ (225) of paramagnetic NiO.

**Table A2.**Symmetry labels of the Bloch functions at the points of symmetry in the Brillouin zone for $Cc$ (9) of all the energy bands with symmetry adapted and optimally localized Wannier functions centered at the Ni (Table (a)) and O (Table (b)) atoms, respectively.

- (i)
- The space group $Cc$ is the unitary subgroup of the magnetic group ${M}_{9}=Cc+K\left\{{C}_{2b}\right|\mathbf{0}\}Cc$ leaving invariant both the experimentally observed [2,29,30,31,32] antiferromagnetic structure and the dislocations of the Ni atoms shown in Figure 2a. K still denotes the operator of time-inversion.
- (ii)
- Each band consists of two branches (Definition 2 of [9]) since there are two Ni and two O atoms in the unit cell.
- (iii)
- Band 1 of Ni forms the magnetic band responsible for the antiferromagnetic structure of NiO.
- (iv)
- Band 1 of Ni and Band 1 of O form together the magnetic super band responsible for the Mott insulator.
- (v)
- The notations of the points of symmetry in the Brillouin zone for ${\mathsf{\Gamma}}_{m}^{b}$ follow Figure 3.4 of [3].
- (vi)
- The symmetry notations of the Bloch functions are defined in Table A4.
- (vii)
- The bands are determined following Theorem 5 of [9].
- (viii)
- Table (a) is valid irrespective of whether or not the Ni atoms are dislocated as shown in Figure 2a.
- (ix)
- The Wannier functions at the Ni or O atoms listed in the upper row belong to the representation ${\mathit{d}}_{1}$ included below the atom.
- (x)
- Applying Theorem 5, we need the representation ${\mathit{d}}_{1}$ of the point groups ${G}_{0Ni}$ and ${G}_{0O}$ of the positions of the Ni and O atoms, respectively (Definition 12 of [9]). In NiO, both groups contain only the identity operation,$${G}_{0Ni}={G}_{0O}=\left\{\left\{E\right|\mathbf{0}\}\right\}.$$Thus, the Wannier functions belong to the simple representation defined by:
$\left\{E\right|\mathbf{0}\}$ ${\mathit{d}}_{1}$ 1 - (xi)
- Each row defines a band with Bloch functions that can be unitarily transformed into Wannier functions being:
- as well localized as possible (according to Definition 5 of [9]);
- centered at the Ni (Table (a)) or O (Table (b)) atoms; and
- symmetry adapted to $Cc$; that means (Definition 7 of [9]) that they satisfy Equation (15) of [9], reading in NiO as:$$\begin{array}{ccc}P\left(\left\{{\sigma}_{db}\right|\mathit{\tau}\}\right){w}_{N{i}_{1}}\left(\mathit{r}\right)\hfill & =& {w}_{N{i}_{2}}\left(\mathit{r}\right),\hfill \\ P\left(\left\{{\sigma}_{db}\right|\mathit{\tau}\}\right){w}_{N{i}_{2}}\left(\mathit{r}\right)\hfill & =& {w}_{N{i}_{1}}\left(\mathit{r}\right),\hfill \\ P\left(\left\{{\sigma}_{db}\right|\mathit{\tau}\}\right){w}_{{O}_{1}}\left(\mathit{r}\right)\hfill & =& {w}_{{O}_{2}}\left(\mathit{r}\right),\hfill \\ P\left(\left\{{\sigma}_{db}\right|\mathit{\tau}\}\right){w}_{{O}_{2}}\left(\mathit{r}\right)\hfill & =& {w}_{{O}_{1}}\left(\mathit{r}\right),\hfill \end{array}$$

- (xii)
- The entry “OK” indicates that the Wannier functions follow also Theorem 7 of [9] with $\mathit{N}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ in Table (a) and $\mathit{N}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ in Table (b). That means that the Wannier functions may even be chosen symmetry adapted to the magnetic group $M=Cc+K\left\{{C}_{2b}\right|\mathbf{0}\}Cc$. Thus, Equation (62) of [9] is valid, reading in NiO as:$$\begin{array}{ccc}KP\left(\left\{{C}_{2b}\right|\mathbf{0}\}\right){w}_{N{i}_{1}}\left(\mathit{r}\right)\hfill & =& {w}_{N{i}_{1}}\left(\mathit{r}\right)\hfill \\ KP\left(\left\{{C}_{2b}\right|\mathbf{0}\}\right){w}_{N{i}_{2}}\left(\mathit{r}\right)\hfill & =& {w}_{N{i}_{2}}\left(\mathit{r}\right)\hfill \\ KP\left(\left\{{C}_{2b}\right|\mathbf{0}\}\right){w}_{{O}_{1}}\left(\mathit{r}\right)\hfill & =& {w}_{{O}_{2}}\left(\mathit{r}\right)\hfill \\ KP\left(\left\{{C}_{2b}\right|\mathbf{0}\}\right){w}_{{O}_{2}}\left(\mathit{r}\right)\hfill & =& {w}_{{O}_{1}}\left(\mathit{r}\right)\hfill \end{array}$$
- (xiii)
- Within the NHM, Equations (A8) and (A9) have only one, but an important meaning: they ensure that the nonadiabatic Hamiltonian of the atomic-like electrons commutes with the symmetry operators of ${M}_{9}$ [4].

**Table A3.**Character tables of the single valued irreducible representations of the monoclinic base centered space group $C2/c={\mathsf{\Gamma}}_{m}^{b}{C}_{2h}^{6}$ (15).

**Table A4.**Character tables of the single valued irreducible representations of the monoclinic base centered space group $Cc={\mathsf{\Gamma}}_{m}^{b}{C}_{1h}^{4}$ (9).

- (i)
- The notations of the points of symmetry follow Figure 3.4 of [3].
- (ii)
- The character tables are determined from Table 5.7 in [3].
- (iii)
- K denotes the operator of time inversion. The entries (a) and (c) are determined by Equation (7.3.51) of [3]. They indicate whether the related co-representations of the magnetic groups $Cc+KCc$, $Cc+K\left\{E\right|\mathit{\tau}\}Cc$, and $Cc+K\left\{{C}_{2b}\right|\mathbf{0}\}Cc$ follow Case (a) or Case (c) as defined in Equations (7.3.45) and (7.3.47), respectively, of [3].

**Table A5.**Compatibility relations between the Brillouin zone for the fcc space group $Fm3m$ (225) of paramagnetic NiO and the Brillouin zone for the space group $Cc$ (9) of the antiferromagnetic structure in distorted NiO.

- (i)
- The Brillouin zone for $Cc$ (9) lies diagonally within the Brillouin zone for $Fm3m$ (225).
- (ii)
- The upper rows list the representations of the little groups of the points of symmetry in the Brillouin zone for $Fm3m$, and the lower rows list representations of the little groups of the related points of symmetry in the Brillouin zone for $Cc$.The representations in the same column are compatible in the following sense: Bloch functions that are basis functions of a representation ${\mathit{D}}_{i}$ in the upper row can be unitarily transformed into the basis functions of the representation given below ${\mathit{D}}_{i}$.
- (iii)
- The notations of the points of symmetry follow Figures 3.14 and 3.4, respectively, of [3].
- (iv)
- (v)
- Within the Brillouin zone for $Fm3m$, the primed points are equivalent to the unprimed point.
- (vi)
- The compatibility relations are determined by a C++ computer program in the way described in great detail in [33].

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**Figure 1.**Conventional band structure of paramagnetic fcc NiO as calculated by the FHI-aims program [7,8], using the structure parameters given in [1], with symmetry labels determined by the author. The notations of the points of symmetry in the Brillouin zone for ${\mathsf{\Gamma}}_{c}^{f}$ follow Figure 3.14 of [3], and the symmetry labels are defined in Table A1 in Appendix B. The “active” band highlighted by the bold line becomes a magnetic super band when folded into the Brillouin zone for the magnetic structure; see Figure 3.

**Figure 2.**Nickel (

**a**) and oxygen (

**b**) atoms in distorted antiferromagnetic NiO with the magnetic group ${M}_{9}$ in Equation (12) possessing the monoclinic base centered Bravais lattice ${\mathsf{\Gamma}}_{m}^{b}$. The Ni atoms represented by red circles bear a magnetic moment parallel or antiparallel to $\left[11\overline{2}\right]$ and the atoms represented by green circles the opposite moment. The magnetic structure is orientated as in [2]. The vectors ${\mathit{T}}_{i}$ are the basic translations of ${\mathsf{\Gamma}}_{m}^{b}$.

**Figure 3.**The band structure of NiO given in Figure 1 folded into the Brillouin zone for the monoclinic base centered Bravais lattice ${\mathsf{\Gamma}}_{m}^{b}$ of the magnetic group ${M}_{9}$. The band highlighted in Figure 1 by the bold lines is still highlighted by bold lines in the folded band structure. It now forms a magnetic “super” band consisting of four branches assigned to the two nickel and the two oxygen atoms. The symmetry labels are defined in Table A4 and are determined from Figure 1 by means of Table A5. The notations of the points of symmetry follow Figure 3.4 of [3]. The midpoint ${\mathsf{\Lambda}}_{\mathrm{M}}$ of the line $\overline{\mathsf{\Gamma}\mathrm{Z}}$ is equivalent to the points ${W}^{\prime}\left(\overline{\frac{1}{4}}\frac{1}{4}\frac{1}{2}\right)$ and ${\Sigma}^{\prime}\left(\frac{1}{4}\overline{\frac{1}{4}}0\right)$ in the Brillouin zone for the paramagnetic fcc lattice.

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Krüger, E.
Structural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiO. *Symmetry* **2020**, *12*, 56.
https://doi.org/10.3390/sym12010056

**AMA Style**

Krüger E.
Structural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiO. *Symmetry*. 2020; 12(1):56.
https://doi.org/10.3390/sym12010056

**Chicago/Turabian Style**

Krüger, Ekkehard.
2020. "Structural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiO" *Symmetry* 12, no. 1: 56.
https://doi.org/10.3390/sym12010056