Magnetic Structure of CoO

The paper reports theoretical evidence that the multi-spin-axis magnetic structure proposed in 1964 by van Laar actually does exist in CoO. This tetragonal spin arrangement produces both the strong tetragonal and the weaker monoclinic distortion experimentally observed in this material. The monoclinic distortion is proposed to be a"monoclinic-like"distortion of the array of the oxygen atoms, comparable with the rhombohedral-like distortion of the oxygen atoms recently proposed to be present in NiO and MnO. The monoclinic-like distortion has no influence on the tetragonal magnetic structure which is generated by a special nonadiabatic atomic-like motion of the electrons near the Fermi level. It is argued that it is this special atomic-like motion which qualifies CoO to be a Mott insulator.


Introduction
Cobalt monoxide is antiferromagnetic with the Néel temperature T N = 289 K. Just as the other isomorphic transition-metal monoxides MnO, FeO, and NiO, it is a Mott insulator in both, the paramagnetic and the antiferromagnetic phase. While above T N , all the transition-metal monoxides possess the fcc structure Fm3m = Γ f c O 5 h (225) (in parentheses the international number), CoO occupies a special position in the magnetic phase: the magnetic structures of MnO, FeO and NiO are known to be monoclinic base-centered [1][2][3][4], but the magnetic structure of CoO is not fully understood. Two different models of antiferromagnetic CoO are discussed in the literature: first, the non-collinear multi-spin-axis magnetic structure with tetragonal symmetry proposed in 1964 by van Laar [5] and, second, a collinear monoclinic structure similar to that of the other monoxides. The models can hardly be distinguished by neutron diffraction data or by reverse Monte Carlo refinements of these data [6]. They are considered as alternative structures or as structures coexisting in antiferromagnetic CoO [7]. The non-collinear structure is suggested by the marked tetragonal distortion of CoO accompanying the antiferromagnetic state, and a collinear structure could be associated with the additional small monoclinic deformation unambiguously detected in the antiferromagnetic phase of CoO [8].
The present paper reports theoretical evidence that the tetragonal multi-spin-axis structure does exist in CoO and creates the tetragonal distortion of the crystal. The additional small monoclinic deformation is not connected with the magnetic order but is evidently a "monoclinic-like" deformation of the array of the oxygen atoms, comparable to the rhombohedral-like distortion proposed to exist in antiferromagnetic NiO [9] and MnO [10].
In the following Section 2 the magnetic group of the multi-spin-axis structure is determined. Just as in NiO and MnO, a problem emerges in this context: a system invariant under the already reported [5,6] type IV Shubnikov magnetic group I c 4 1 /acd given in Equation (1) cannot possess antiferromagnetic eigenstates. As shown in Section 3, for this reason the crystal undergoes a marked tetragonal distortion reducing the symmetry initially defined by the group I c 4 1 /acd. As demonstrated in Section 4, this tetragonal distortion evidently produces a monoclinic-like deformation of the array of the oxygen atoms in the deformed crystal. In Section 5 the nonadiabatic Heisenberg model (NHM) shall be applied to the electronic ground state of CoO. In Section 5.1, we will show that the electrons of paramagnetic CoO may occupy an atomic-like state allowing the system to be a Mott insulator. Having in Section 2 determined the active magnetic group of antiferromagnetic CoO, we may apply the NHM to the antiferromagnetic state of CoO, too. In Section 5.2, we will determine the magnetic band related to the magnetic group of the antiferromagnetic state. The electrons may perform an atomic-like motion in the nonadiabatic system stabilizing the tetragonal non-collinear magnetic structure. In addition, this atomic-like motion allows the electron system to be Mott insulating.

Magnetic group of the antiferromagnetic state
The multi-spin-axis structure of CoO [5] is indicated in Figure 1 (a) by red arrows. It is invariant under the space group I4 1 /acd (142) [5] and under the type IV Shubnikov magnetic group [6]: where K denotes the anti-unitary operator of time-inversion. The unitary subgroup I4 1 /acd has the tetragonal body-centered Bravais lattice Γ v q and contains (besides the pure translations) sixteen elements which are expressible as products of the three generating elements We write the symmetry operations {R|αβγ} in the Seitz notation: R is a point group operation written with respect to the x, y, and z axes in Figure 1 and αT 1 + βT 2 + γT 3 the subsequent translation [11], where the basis translations T i are also defined in Figure 1. We write the point group operations R as defined in Section 1.4 of [11], here R = E stands for the identity operation, R = C + 4z for the anti-clockwise rotation through 90 • about the z axis, R = I for the inversion, R = C 2a for the rotation through 180 • about the a axis as indicated in Figure 1, and R = σ da for the reflection IC 2a . Since the magnetic structure in Figure 1 (a) is invariant under the three generating elements (2) and under the anti-unitary element {K| 1 2 1 2 0}, it is invariant under the complete magnetic group (1).
In what follows, the magnetic group I c 4 1 /acd is referred to as M 142 because the unitary subgroup I4 1 /acd bears the international number 142,  [11], in the present case the one-dimensional representations follow Case (a) because they are real. According to Condition 1 in [9], the antiferromagnetic state cannot be an eigenstate of a system invariant under M 142 because I4 1 /acd does not possess at least one one-dimensional representation following Case (c) (as defined by Equation (7.3.47) of [11]).
These group-theoretical findings change drastically when we assume that the subgroup I4 1 /cd (110) of I4 1 /acd (142) is the unitary part of the magnetic group of antiferromagnetic CoO. I4 1 /cd contains one-half of the sixteen elements of I4 1 /acd (142) and may be defined by the generating elements see Table 3.7 of [11]. However, the origin used in the Tables of [11] for the space group I4 1 /cd (110) is different from the origin used in the present paper (and marked in Figure 1 by O). Thus, all the symmetry operations S BC used in [11] for the group I4 1 /cd (110) must be transformed by {E|0 1 4 3 4 } into the symmetry operations S paper used in this paper, Just as the group I4 1 /acd, the group I4 1 /cd has the tetragonal body-centered Bravais lattice Γ v q . From I4 1 /cd we may derive two magnetic groups, the type IV Shubnikov group (with a black and white Bravais lattice), and the type III Shubnikov group (with an ordinary Bravais lattice), both leaving invariant the magnetic structure. The one-dimensional representations at point Z in Table A2 now follow both requirements (i) and (ii) of Condition 1 of [9] for the magnetic group M 2 , but no one-dimensional representation of I4 1 /cd meets both requirements (i) and (ii) for M 1 .
Thus, the type III Shubnikov group M 2 is the magnetic group of antiferromagnetic CoO because it allows the system to possess antiferromagnetic eigenstates. In what follows, we refer M 2 to as

Tetragonal distortion of antiferromagnetic CoO
In the tetragonally distorted crystal the eight Co and eight O atoms in the unit cell are located at the positions Co(000) Co respectively, in the coordinate system defined by the basic translations T 1 , T 2 , and T 3 of Γ v q given in Figure 1. The positions (9) and (10) are the Wyckoff positions 16c and 16e (with x=0) of the space group I4 1 /acd (142) [5]. It should be noted that the unit cell of Γ v q does not contain 16 Co atoms and 16 O atoms as is often assumed (and is suggested by the notations 16c and 16e of the Wyckoff positions). In the present paper, the point in the center of the tetragonal prism is not (and must not be) an additional point within the unit cell, but is a lattice point connected by translation symmetry with the other lattice points. Thus, there are 8 Co and 8 O atoms in the unit cell.
The tetragonal magnetic structure produces initially a distortion invariant under the magnetic group M 142 in antiferromagnetic CoO. Since a system invariant under M 142 does not possess antiferromagnetic eigenstates, the crystal must be additionally distorted in such a way that the electronic Hamiltonian still commutes with the symmetry operations of M 110 , but does not commute with the symmetry operations of M 142 − M 110 , cf. Section 3 of [9]. The only distortion bringing the desired effect is a shift of the Co atoms in ±(T 1 + T 3 ) and ±(T 2 + T 3 ) direction from their positions (9), as indicated by the arrows in Figure 1 (b). In fact, the magnetic structure together with the shifts in Figure 1  This result may be understood by inspection of Figure 1, but also in terms of Wyckoff positions: the type III Shubnikov group M 110 may be written in the form [11] where is an ordinary (unitary) space group. Equation (11) Table 3.2 of [11]. Consequently, we receive the origin of G by shifting the origin of that means, by shifting the origin from a Co atom to an O atom, see Figure 1. Thus, the Wyckoff positions are interchanged: In the group G and, hence, in the magnetic group M 110 , the Co atoms lie on the position 16e of the group I4 1 /acd (142), and the O atoms on the position 16c. The position 16e contains a parameter (usually x) describing the shifts of the Co atoms indicated in Figure 1 (b), the O atoms on position 16c are fixed. This fact plays an essential role in the following Section 4.

Monoclinic-like distortion
As is the case with antiferromagnetic NiO and MnO, also antiferromagnetic CoO is clearly deformed by the magnetic structure but, additionally, by a slight distortion seemingly incompatible with the magnetic structure. In NiO [9] and MnO [10], this additional distortion is evidently produced by the oxygen atoms which form an rhombohedral-like array within the monoclinic magnetic group M 9 of antiferromagnetic NiO and MnO. This distortion was called "inner distortion" of the magnetic group M 9 because the symmetry of M 9 is not disturbed (and must not be disturbed) by the rhombohedral-like distortion. In CoO, on the other hand, the magnetic group M 110 is tetragonal and the additional distortion proved experimentally to be monoclinic [8]. This can be understood as follows: Figure 1 shows the distorted antiferromagnetic structure of CoO with the magnetic group M 110 . The arrows in Figure 1 (b) specify the shifts of the cobalt atoms from their positions in Equation (9). These shifts stabilize, on the one hand, the antiferromagnetic structure (Section 2) and produce, on the other hand, a strong tetragonal distortion of the crystal. Within this distortion, adjacent Co atoms are dislocated either in the same or in different directions and, consequently, their distance and, hence, their mutual attraction varies around its value in the paramagnetic state. Consequently, the crystal is tetragonally distorted in such a way that the basic translations T 1 , T 2 , and T 3 of Γ v q are no longer embedded in the cubic lattice since this lattice no longer exists.
The oxygen atoms, on the other hand, are not shifted from their positions in the tetragonal body-centered lattice, that means, from their position given in the list (10). Thus, their mutual distances are essentially the same as in the paramagnetic phase. Just as in NiO and MnO, they form a periodic array within the lattice of the Co atoms. We assume again that this array forms a Bravais lattice which does not contain eight O atoms but only one O atom in the unit cell. The array of the oxygen atoms is spanned by the vectors ρ 1 , ρ 2 , and ρ 3 in Figure 2. Though these vectors are symmetry operations in the paramagnetic lattice, they are no translation operators in the tetragonal distorted crystal. Nevertheless, they give an approximate picture of the array of the oxygen atoms. In antiferromagnetic NiO and MnO, the vectors ρ i define an rhombohedral-like array of the oxygen atoms because they form a trigonal basis within the monoclinic crystals of NiO and MnO, see Section 4 of [9]. In CoO, on the other hand, this basis is not if c = √ 2a (i.e., if the crystal is no longer cubic), see Figure 2. Thus, the vectors ρ i define a monoclinic base-centered array of the oxygen atoms, see Table 3.1 of [11]. As stated above, this array is not exactly monoclinic because the vectors ρ i are no translational symmetry operations in the distorted crystal. Therefore I call it a "monoclinic-like" distortion of antiferromagnetic CoO being an "inner distortion" of the magnetic group M 110 . This will say that the monoclinic-like distortion of the oxygen atoms is not connected with any change or modification of the tetragonal symmetry of the antiferromagnetic state which remains invariant under the symmetry operations of the magnetic group M 110 .

Application of the Nonadiabatic Heisenberg Model
The NHM defines in narrow, partly filled electronic energy bands a strongly correlated nonadiabatic atomic-like motion. The nonadiabatic localized states defining this atomic-like motion are represented by symmetry-adapted and optimally localized Wannier functions [12]. In the following Subsection 5.1 we will show that the band structure of paramagnetic CoO contains an "insulating band" [13] whose Bloch functions can be unitarily transformed into symmetry-adapted and optimally localized Wannier functions including all the electrons at the Fermi level. In the second Subsection 5.2 we will show that the band structure of antiferromagnetic CoO encloses two "magnetic bands" [9] related to the magnetic group M 110 of the antiferromagnetic phase. The Bloch functions of these two bands can be unitarily transformed into optimally localized Wannier functions symmetry-adapted to M 110 . They are even "magnetic super bands" [9] since all the electrons at the Fermi level belong to these two magnetic band.  Figure 3. Conventional band structure (Section 5 of [10]) of paramagnetic fcc CoO as calculated by the FHI-aims program [14,15] using the length a = 4.260 Å of the paramagnetic unit cell as given in [16]. The symmetry labels as defined in Table A1 of [9] are determined by the author (as described in Section 2 of [9]), the symmetry on the lines Σ and ∆ are defined by Table A6 (a). The notations of the points and lines of symmetry in the Brillouin zone for Γ f c follow Figure 3.14 of Ref. [11]. The insulating band is highlighted by the bold line. Figure 3 shows the band structure of paramagnetic CoO. The band indicated by the bold lines is characterized by Bloch functions with the symmetry:

Atomic-like Electrons in Paramagnetic CoO
of band 5 listed  [11]. If this d-like band is half-filled, it defines an atomic-like motion qualifying paramagnetic CoO to be a Mott insulator because it consists of all the branches crossing the Fermi level. In the paramagnetic system the atomic-like motion occurs solely between the Co atoms. Table A4 lists the only magnetic band related to the magnetic group M 110 of antiferromagnetic CoO. Figure 4 shows the band structure of paramagnetic CoO in Figure 3 as folded into the Brillouin zone Γ v q of the tetragonal body-centered magnetic structure. The band highlighted in Figure 3 by the bold line (the insulating band) becomes in Figure 4 twice the magnetic band in Table A4. The Bloch functions defining the sixteen branches of these two magnetic bands are highlighted in red. They can be unitarily transformed into optionally three types of optimally localized Wannier functions symmetry-adapted to M 110 : Either (i) the Wannier functions are centered twice at the eight Co atoms (that means two Wannier functions are at each Co atom), or (ii) they are centered twice at the eight O atoms; or (iii) they are centered at both the eight Co atoms and at the eight O atoms.

Atomic-like Electrons in Antiferromagnetic CoO
The atomic-like motion defined by the first and the second possibility (i) and (ii) clearly has a higher Coulomb energy than the atomic-like motion belonging to the third possibility (iii) because the Coulomb repulsion between two electrons occupying localized states at the same atom is greater than the repulsion of two electrons occupying localized states at different (adjacent) atoms. Thus, the last case (iii) defines the energetically most favorable nonadiabatic atomic-like motion which stabilizes the antiferromagnetic structure with the magnetic group M 110 [9,17]. Moreover, the two magnetic bands are magnetic super bands since they comprise all the branches crossing the Fermi level and, thus, qualify antiferromagnetic CoO to be a Mott insulator [9]. While in the paramagnetic system the atomic-like motion occurs solely between the Co atoms (Section 5.1), it comprises in the antiferromagnetic phase both, the Co and O atoms.
It should be noted that we can define in the band structure of antiferromagnetic CoO other narrow bands differing (slightly) from the bands highlighted in red, but coinciding also with the symmetry of the band in Table A4. They form likewise magnetic bands if they are half-filled. It is not easy to say which bands represent best the nonadiabatic atomic-like motion in antiferromagnetic CoO. In any case, the two highlighted bands are a first approach to describing the atomic-like motion because, in this case, the localized states in antiferromagnetic CoO most resemble the localized states in the paramagnetic phase. At this stage, we realize that two magnetic bands exist, but we do not know the exact combination of their branches.

Results
Under the assumption that the non-collinear multi-spin-axis magnetic structure as proposed by van Laar is realized in antiferromagnetic CoO, essential features of CoO can be understood:   Figure 4. The band structure of CoO given in Figure 3 folded into the Brillouin zone for the tetragonal body-centered lattice Γ v q of the magnetic group M 110 . As in Figure 3, the bands are calculated by the FHI-aims program [14,15]. At points Γ and Z, first the symmetry of the Bloch functions at the equivalent point in the fcc Brillouin zone is given. Then, after the colon, the symmetry of these Bloch functions in the tetragonal body-centered Brillouin zone is specified as given in Table A5. On the line Σ,F, the symmetry labels Σ 1 and Σ 2 are determined by Table A6 (b). The Bloch functions highlighted in red form two magnetic bands (i.e., twice the band in Table A4) consisting of sixteen branches assigned to the eight cobalt and eight oxygen atoms. The notations of the points and lines of symmetry follow Figure 3.10 (b) of Ref. [11]. centered at the eight O atoms. In each case, the Wannier functions are optimally localized and adapted to the symmetry of the magnetic state. Thus, the electrons of this band may perform a nonadiabatic atomic-like motion lowering their nonadiabatic condensation energy ∆E (as defined in Equation (2.20) in [12]) by activating an exchange mechanism producing a magnetic structure with the magnetic group M 110 (Section 5). • The two magnetic bands are magnetic super bands because they comprise all the branches crossing the Fermi level. Thus, they qualify antiferromagnetic CoO to be a Mott insulator (Section 5.2). The nonadiabatic atomic-like motion involves both, the Co and the O atoms.
Independently of the special magnetic symmetry it was shown in Section 5.1 that • there exists an insulating band [9] in the paramagnetic band structure of CoO. The electrons of this band may perform a nonadiabatic atomic-like motion that qualifies CoO to be a Mott insulator also in the paramagnetic phase. The related optimally localized Wannier functions are adapted to the cubic fcc symmetry of paramagnetic CoO, are centered on the Co atoms, are twofold degenerate and have d-like Γ + 3 symmetry. The atomic-like motion occurs exclusively between the Co atoms.

Discussion
The results of the present paper as listed in the preceding Section 6 provide strong evidence that the tetragonal non-collinear multi-spin-axis structure exists in CoO. The foundations of the presented theory are (i) the nonadiabatic Heisenberg model (NHM) published 20 years ago [12], and (ii) the group-theoretical theorem, stating that in a system invariant under time inversion, any magnetic eigenstate and its time-inverted state belong to a two-dimensional irreducible co-representation of the magnetic group [17][18][19][20].
Unfortunately, the three axioms of the NHM still are relatively unfamiliar though they are easy to recognize. Nevertheless, since being introduced, the NHM worked effectively in understanding superconductivity [20][21][22], magnetism and Mott insulation [9,10,13,19]. The group-theoretical theorem (ii) proved to be useful already in understanding distortions accompanying a magnetic state [9,10,17,23].
Funding: This publication was supported by the Open Access Publishing Fund of the University of Stuttgart.
Acknowledgments: I am much indebted to Guido Schmitz for his continuing support of my work.

Conflicts of Interest:
The author declares no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: NHM Nonadiabatic Heisenberg model E Identity operation I Inversion C + 4z anti-clockwise rotation through 90 • about the z axis C 2a Rotation through 180 • as indicated in Figure 1 σ da Reflection IC 2a K anti-unitary operator of time inversion

Appendix A Group-theoretical tables
This appendix provides Tables A1 -A6 along with notes to the tables.
Notes to Table A1: (i) The notations of the points of symmetry follow Figure 3.10 (b) of [11].
(ii) Only the two points of symmetry Γ and Z invariant under the complete space group are listed. (iii) The character tables are determined from Table 5.7 in [11].
(iv) K denotes the operator of time inversion. The entry (a) is determined by Equation (7.3.51) of [11] and indicates that the related co-representations of the magnetic group I4 1 /acd + {K|000}I4 1 /acd follow Case (a) as defined in Equation (7.3.45) of [11].
Notes to Table A2: (i) The notations of the points of symmetry follow Figure 3.10 (b) of [11].
(ii) The character tables are determined from   [13] showing only Band 5 with optimally localized Wannier functions of Γ + 3 symmetry centered at the Co atoms (Table (a)) and the O atoms (Table (b)).

(a)
Co(000) 2 ) Notes to Table A4 (i) The symmetry of the band related to the Co atoms in Table ( Table A2. (iii) The bands are determined following Theorem 5 of [20]. (iv) The point groups G 0Co and G 0O of the positions [20] of the Co and the O atoms contain only the identity operation: Thus, the Wannier functions at the atoms belong to the simple representation  Table A2 in [9]. • The entry "OK" indicates that the Wannier functions follow not only Theorem 5, but also Theorem 7 of [20]. Consequently, they may be chosen symmetry-adapted to the complete magnetic group M 110 , cf. note (xii) of Table A2 in [9]. Table A5. Compatibility relations between the Brillouin zone for the fcc space group Fm3m (225) of paramagnetic CoO and the Brillouin zone for the space group I4 1 /cd (110) of the antiferromagnetic structure in distorted CoO.
Notes to Table A5:  Table A4 of [9] and Table A2, respectively. (iv) The compatibility relations are determined as described in great detail in [24].