Super-Exchange Theory for Polyvalent Anion Magnets

The Goodenough-Kanamori-Anderson (GKA) rules have been widely applied for explaining the magnetic properties induced by super-exchange interaction. As conclusions of the super-exchange theory, they reveal the antiferromagnetic (ferromagnetic) ordering along with bond angle of 180 degrees (90 degrees) in the cation-anion-cation interaction path, in which the theory sets a pre-condition that the electronic states of cations in all paths are identical. We observed that the GKA rules are in fact not universal and even invalid to materials containing anions with different valence states, for example, the layered CrOCl crystal (with two valence states of anions: O2- and Cl-). In this study, we propose an extended super-exchange theory (ESET) related to superposed electronic states of cation in a specific path. ESET is capable of predicting not only the sign and relative magnitude of magnetic exchange constants in different cation-anion-cation paths, but also the magnetic ground state. Through our proposed theory, we conclude that the magnetic ordering along with bond angle of 90 degrees in Cr-Cl-Cr path is moderately antiferromagnetic and of 180 degrees in Cr-O-Cr path is strongly ferromagnetic, which are opposite to the contents of GKA rules. Moreover, we clarify that monolayer CrOCl has antiferromagnetic ordering rather than ferromagnetic as reported recently. The reliability of ESET is verified via first-principles calculation and previous experimental report as well, and its universality is also demonstrated. Thus, our theory is powerful to predict the magnetic properties, which makes it possible to design new high Curie temperature two-dimensional semiconducting ferromagnets with polyvalent anion materials.

A new theory is badly needed.
In the present work, we take recently proposed 2D magnetic CrOCl [18] as an example to theoretically analyze the SEI of polyvalent anion materials (valence state -2 for O 2-, -1 for Cl -), which we call extended super-exchange theory (ESET). ESET firstly gives the valid electronic state of cation in specific super-exchange path, namely Cr-Cl-Cr. Then, we discuss how these cations interacting by using orbital symmetry relations between d and p orbitals acquired from quantum mechanics [19][20] . The results of ESET show that a 180º bond angle in the interaction path Cr-O-Cr favors strongly FM configuration and 90º bond angle in the interaction path Cr-Cl-Cr is moderately AFM ordering. Through our analytical results, the sign and relative magnitude of magnetic exchange constants in Heisenberg Spin Hamiltonian model [21] are given. In addition, the overall magnetic ground states of monolayer and bulk CrOCl are predicted, both revealed to be AFM. Our theory is then confirmed by both first-principles calculation and previous bulk experiment [22] . Besides CrOCl, ESET is also applied to monolayer FeOCl with a different electronic state (3d 5 for Fe 3+ [23] , see Supporting Information) to validate its universality. Since strongly ferromagnetic SEI does exist according to our theory, a new route for designing high Curie temperature semiconducting ferromagnets is opened. Thus, ESET is significant as more polyvalent anion materials are being studied, e.g. magnetic Mxenes [24][25] , many easily exfoliate 2D magnetic metals and semiconductors [26] .

Calculation Details
Our calculations were carried out using the projector-augmented-wave (PAW) method [27] as implemented in Vienna ab initio simulation package (VASP) [28] . For bulk CrOCl, van der Waals (vdW) correction was considered by DFT-D3 method [29][30] . We used 2 × 2 × 1 supercell containing 20 atoms and plane-wave cutoff energy of 500 eV for all calculations (monolayer and bulk). A Γcentered Monkhorst-Pack k-point mesh of 5 × 6 × 5 and 5 × 6 × 1 were employed for bulk and monolayer. A vacuum space of 20 Å along the z-direction was set for monolayer. The convergence criteria for the energy difference in electronic self-consistent loop were 10 −5 eV and residual forces on ions were less than 0.01 eV/Å. To count the electron correlation effects of Cr 3d orbital and obtain accurate electronic properties, we employed the screened hybrid HSE06 [31][32] functional that usually performs much better than the DFT+U methods [33][34][35] . Further details about optimized lattice constants and total energies of layered CrOCl can be found in Supporting Information.

Electronic Configurations of Cation
The electronic states of cations, are of great importance in SEI [5] . In order to find that, we firstly define four states where | ⟩ is the real state that we take account in SEI. It has three components.
which gives | ⟩ = −3| ⟩. The | ⟩ + | ⟩ means the electronic state owning to total ionic bond (e.g. | ⟩ + | ⟩ = | 3 ⟩ for Cr 3+ ). However, ionic character is not alone, as covalent character always exists in real materials. In ESET, | ⟩ for anions The difference between ESET and super-exchange theory is that ESET considers anions not in super-exchange path. For path Cr-Cl-Cr, the influence of O could be removed, based on the assumption that the polarized covalent bond between O and Cr are already formed, and vice versa. The population of electrons that anions lose due to covalent character, are assumed to be the half it receives due to fully ionic character. Eq. (2) is In super-exchange theory, | ⟩ in super-exchange path equals to zero. For one-valent state materials, | ⟩ is then zero by Eq. (4), resulting in | ⟩ = | ⟩ + | ⟩ which can be directly deduced from chemical formula as one always did.
However, for polyvalent anion materials, one specific path only related to one | ⟩ state anion. There has non-zero | ⟩ for anions not in super-exchange paths, which makes | ⟩ for cations non-zero. So, Cr cannot take as | 3 ⟩ state in CrOCl material.
In path Cr-O-Cr, | ⟩ = 0, | ⟩ = − 1 2 | ⟩. According to Eq. (1) and (4), the electronic state of Cr atom is which means the state | ⟩ in path Cr-O-Cr can regard as the superposition of states In path Cr-Cl-Cr, | ⟩ = −| ⟩, | ⟩ = 0, so It should be noticed that the electronic state of cation in path Cr-O-Cr is a superposed state, described as the linear combination of states | 3 ⟩ and | 4 ⟩ for convenience of the following analysis. The interaction is always via super-exchange, not double-exchange [36] .

Orbital Symmetry Relations in Distorted Octahedral Crystal Field
For super-exchange paths, the sign of exchange integral determines the magnetic ordering [6] . Here, we do not use the exchange integral method, but the basic principle of quantum mechanics [19][20] , as the latter one is much simple and intuitive. In quantum mechanics, d orbitals could be divided into five ( , , , 2 , 2 − 2 ) and p orbitals into three ( , , ) according to different magnetic quantum number.
Among them, symmetry relations exist [13] between d and p orbitals, such as is orthogonal to , and 2 is orthogonal to . To avoid the cumbersome fifteen symmetry relations in total, we inherit the notation 2 and from crystal field theory [37] of octahedral structure, denoting d orbitals in CrOCl as 2 ′ (contains , and orbitals) and ′ (contains 2 and 2 − 2 orbitals). If we choose z as the main axis, the pz orbital that parallel to it is regarded as , the other two orbitals that perpendicular to it (px and py) are called . In this simplification, the fifteen orthogonality relations can be expressed as The wavefunction has no overlap between orbitals ′ and , 2 ′ and , indicating of forbidden electrons transition. Besides, the correlation of orbitals ′ and is much stronger than 2 ′ and .

Four Possible Super-Exchange Paths in CrOCl
Now, we determine the magnetic ordering along different paths. Apart from orbital symmetry relations discussed above, we also need Pauli exclusion principle (PEP) and Hund's rule (HR) to achieve magnetic ordering.
The superposed electronic state of Cr can be regarded as the linear combination of two states, we denote them as cation1 and cation2. Path cation1-anion-cation2 can align to two bond angles, 90º or 180º . If the angle is 180º , cation1 and cation2 share the same main axis, which means orbital for anion in cation1-anion bond is also for cation2-anion bond and so does . For Cr-Cl-Cr in 90º (Fig. 2a, path P1 in Table 1), two d 4 states interact through two -bonds with mediate strength. Due to PEP, only in the condition that the spins of two cations are antiparallel, electron hoping between them is not forbidden, namely the interaction is AFM.
For Cr-O-Cr in 90º (Fig. 2b, path P2), the interaction spreads through twobonds while first -favors AFM according to PEP and the -bond favors FM due to HR that requires the maximum total magnetic moments. Since d 3 state has unfilled ′ orbital, HR takes advantage. In conclusion, this case is FM with extremely weak strength, owning to the competition between two -bonds.
For Cr-Cl-Cr in 180º (Fig. 2c, path P3), the interaction conducted through one -bond and one -bond. Both favor AFM (PEP). So, this case shall be AFM and extremely strong due to the strong character of -bond.
For Cr-O-Cr in 180º angle (Fig. 2d, path P4), the interaction happened through one -bond that favors FM according to HR and one -bond that is AFM thanks to PEP. As the interaction through -is weak, this case -bond would dominate which results in strong FM.  180º

FM Strong
The above predictions are automatically suitable for monolayer whose magnetic interactions are only through in-layer magnetic moments. However, for bulk CrOCl, from those evidence that (a) the exfoliation energy (Fig. S2) of monolayer is smaller than MnPSe3 [39] which possesses the lowest exfoliation energy as we know; (b) bulk is AA stacking; (c) experiment confirmed the magnetic moment orientation of bulk is in out-plane direction [22] (along z-axis), we can ignore the magnetic interaction between layers. So, predictions are suitable not only to monolayer, but also to bulk.
So far, ESET has fully predicted the magnetic orderings and strengths in different super-exchange paths. In real materials, magnetic exchange constants are directly related with those paths, which gives a effective way to verify our theory.

Magnetic Exchange Constants
In monolayer and bulk CrOCl, the SEI of cations has three different kinds that corresponding to first, second and third nearest neighbor magnetic exchange constants J1, J2 and J3, as shown in Fig. 3a. To clearly see their super-exchange paths, we extracted them from the CrOCl material (Fig. 4). For exchange constants J1, two Cr atoms interacted through two P2 paths. Since P2 are weak FM based on above analysis (as summarized in Table 1), J1 shall also be weak FM. J2 is through P1 and P2 paths. As P2 AFM character is stronger than P1 FM, J2 is mediate AFM. For J3, that one path P4 makes it FM with strong strength. The results based on ESET are summarized in Table   2.
In order to get magnetic constants through first-principles calculation, we express the total energy of one magnetic atom for four configurations (Fig. 3) by Heisenberg
On the one hand, that both J1 and J3 are positive indicates their FM interaction while J2 is AFM owning to its negative sign. On the other, the order of interaction strength is |J3| > |J2| > |J1|. No matter in sign or magnitude, the calculation results are highly accordance with our ESET prediction results, which proves our theory.

Supporting Information
Calculation details and procedures for structure optimization, total energy, exfoliation energy and phonon-related properties; Magnetic properties of layered FeOCl analyzed by ESET.
Additional figures for PDOS picture of monolayer CrOCl with AFM-stripy-x magnetic configuration, exfoliation energy of monolayer CrOCl and other common 2D materials, phonon pictures; the scheme of four possible super-exchange paths in FeOCl.
Additional tables for crystal constants and total energies of layered CrOCl; ESET prediction and first-principles calculation results of magnetic exchange constants for layered FeOCl.