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# Covalent bonds against magnetism in transition metal compounds

Edited by Robert Joseph Cava, Princeton University, Princeton, NJ, and approved July 15, 2016 (received for review April 22, 2016)

## Significance

A mechanism by which orbital degrees of freedom can strongly affect magnetic properties of systems with correlated electrons is proposed. Using analytical treatment and numerical simulations, both in general and for some particular substances, we show that the formation of covalent bonds on part of the orbitals may strongly reduce magnetic moments and explain magnetic properties of different 4*d* and 5*d* transition metal oxides. In particular, orbital-selective effects may result in suppression of double exchange—one of the main mechanisms of ferromagnetism in transition metals and compounds, such as colossal magnetoresistance manganites or CrO_{2}—the materials used in many devices.

## Abstract

Magnetism in transition metal compounds is usually considered starting from a description of isolated ions, as exact as possible, and treating their (exchange) interaction at a later stage. We show that this standard approach may break down in many cases, especially in 4*d* and 5*d* compounds. We argue that there is an important intersite effect—an orbital-selective formation of covalent metal–metal bonds that leads to an “exclusion” of corresponding electrons from the magnetic subsystem, and thus strongly affects magnetic properties of the system. This effect is especially prominent for noninteger electron number, when it results in suppression of the famous double exchange, the main mechanism of ferromagnetism in transition metal compounds. We study this mechanism analytically and numerically and show that it explains magnetic properties of not only several 4*d*–5*d* materials, including Nb_{2}O_{2}F_{3} and Ba_{5}AlIr_{2}O_{11}, but can also be operative in 3*d* transition metal oxides, e.g., in CrO_{2} under pressure. We also discuss the role of spin–orbit coupling on the competition between covalency and magnetism. Our results demonstrate that strong intersite coupling may invalidate the standard single-site starting point for considering magnetism, and can lead to a qualitatively new behavior.

Transition metal (TM) compounds present one of the main playgrounds in a large field of magnetism (1⇓–3). Usually, when considering magnetic properties of these systems, one starts from the, as exact as possible, treatment of isolated TM ions or such ions in the surrounding of ligands, e.g., TMO_{6} octahedra. A typical situation for a moderately strong crystal field is the one in which *d* electrons obey the Hund’s rule, forming a state with the maximal spin. For a stronger crystal field low-spin states are also possible, but also in this case electrons in degenerate subshells, e.g., *d* levels (5), is then treated using this starting point with this total spin of isolated ions, taking into account the hopping between sites (leading in effect to magnetic interaction) as a weak perturbation, which does not break the magnetic state of an ion.

For heavier elements, such as *S* as dictated by the Hund’s rule, with the (effective) orbital moment *L*. In any case, it is usually assumed that the “building blocks” for further consideration of the magnetic interactions are such isolated TM ions with the corresponding quantum numbers.

However, especially when we go to heavier TM ions, such as *d* orbitals increases strongly, and with it the effective *d–d* hopping, *t* (2). One can anticipate a possibility for this hopping to become comparable with or even exceed the intra-atomic interactions, such as the Hund’s coupling, *d* electrons per ion, some electrons, namely those occupying the orbitals with the strongest overlap, form intersite covalent bonds, i.e., the singlet molecular orbitals (MO). In this case such electrons become essentially nonmagnetic and “drop out of the game,” so that only *d* electrons in orbitals without such a strong overlap may be treated as localized, contributing to localized moments and to eventual magnetic ordering. The result would be that the effective magnetic moment of a TM ion in such situation would become much smaller than the nominal moment corresponding to the formal valence of an ion. Below we demonstrate that this effect is indeed realized in many TM compounds, especially those of *d* levels.

Before presenting the main results, let us emphasize that the interplay between the formation of covalent bonds, spin–orbit coupling, and intra-atomic exchange interaction discussed in the present paper is very important for intensively studied nowadays _{3} (6, 7), Li_{2}RuO_{3} (8, 9), LiZn_{2}Mo_{3}O_{8} (10, 11), and Na_{2}IrO_{3} (12⇓–14). Competition of these interactions results in highly unusual physical properties in these systems: Kitaev spin liquid, valence–bond condensation, and different topological effects.

## Qualitative Considerations

We start by simple qualitative arguments, considering a two-site problem with, for example, three electrons per dimer, occupying two types of orbitals: orbital *c* (the corresponding creation and annihilation electron operators at the site *i* with the spin σ are *d*, with a very weak hopping *c* orbitals could be, for example, the strongly overlapping _{6} octahedra share edges, i.e., have two common oxygens, see Fig. 1*A*, or the *B*; the orthogonal orbitals (*d* orbitals (for a more detailed treatment of these situations see, e.g., refs. 15⇓–17).

In this case we can consider two different limiting states. In the state depicted in Fig. 2*A* we put two electrons into the localized *d* orbitals, and the remaining electron occupies the “itinerant” *c* orbital, hopping back and forth from site 1 to site 2, or occupying the bonding state

This DE state is definitely the most favorable state for a strong Hund’s interaction. Indeed, in all papers on double exchange (5, 18, 19) one makes such an approximation, most often by simply setting *d–d* hopping (direct, or via ligands), which is typically 0.1–0.3 eV. However, when we go to *d*-wave function, and the corresponding overlap and *d–d* hopping strongly increase, so that _{2}RuO_{3} (8) or Y_{5}Mo_{2}O_{12} (22). In this case we can form a different state (illustrated in Fig. 2*B*), redistributing electrons between *d* orbitals: we now put two electrons on the itinerant *c* orbitals, so that they form a singlet state in the bonding orbital of Fig. 2*B* (actually a molecular orbital state):*d* orbitals. In effect, the total spin of this state (we call it for short the MO state) is not 3/2, as for the DE state of Fig. 2*A*, but only **[1]** and **[4]**, we see that the MO state with suppressed double exchange and strongly reduced total spin is more favorable if

One can use a more general wave function of MO type, with two electrons in the *c* orbitals with the total *d* orbitals. For example, if we put *d* electron with spin **3**). That is, the electron hopping can be even more efficient in suppressing DE than it follows from the simple estimate in Eq. **5**.

## Model Consideration

In fact such a model with two orbitals per site and three electrons per dimer can be solved exactly (see the details in *Supporting Information*; here and below we used the Hubbard model within the Kanamori parameterization rather than a simplified expression for the interaction term given in Eq. **2**), including also the onsite Coulomb (Hubbard) repulsion *U*, besides the Hund’s rule exchange *d* and *c* orbitals (*U*, if *d* orbitals provide local moments, and electrons on *c* orbitals form spin singlet. For a finite *U* the simple estimates such as Eq. **5** do not hold anymore, and the critical value *c* electrons from real molecular orbital to the Heitler–London, or rather Coulson–Fisher-like, with an increased weight of the ionic terms (with respect to homopolar ones) (24). The transition from one state to another is discontinues, since they are characterized by different quantum numbers (

We can generalize this treatment onto larger systems, which we did for a dimerized chain with 2 orbitals and 1.5 electrons per site. The calculations have been performed using a cluster version of the dynamical mean-field theory (DMFT) (25). The cluster DMFT was shown to provide a very accurate description of the one-dimensional chain (25). The results of the cluster DMFT calculations for a dimerized chain at a fixed *U* are shown in Fig. 4. We see that indeed the MO-like state with *d*–5*d* materials) (20, 21) and the DE can be suppressed for small *U*. If *U* is small, then *U* we quickly arrive at the Heitler–London type of the wave function for *c* electrons with the energy gain

## Role of the Spin–Orbit Coupling

As the effects discussed in this paper are met mainly in _{2}), it is important to address the possible role of spin–orbit coupling (SOC), which is strong in these systems. It turns out that the effect of SOC is not universal and depends on a specific situation.

Consider first the same situation with three electrons per dimer and threefold degenerate *z* projection of the effective orbital moment *m* numerates orbitals). Then for weak SOC, *A*, and the state with the singlet MO and the total spin *B*. To gain at least some SO energy, we put localized electrons with spin **1**), because here for localized electrons only the “classical” part of the SOC contributes in lowest order, −*B* is**4**). Comparison of these expressions shows that in this case SOC stabilizes magnetic DE state: the condition for low-spin MO state is now, instead of Eq. **5**,

However, the situation is very different for strong SOC, *j* of a dimer 7/2 is the state shown in Fig. 5*C*, with localized *d* electrons on sites 1 and 2 in states

Similarly, an analog of the MO state is shown in Fig. 5*D*. In this state we put one electron on a localized state

Summing up, in the case of three electrons per dimer weak SOC acts in favor of the magnetic DE state, but strong SOC, instead, stabilizes the MO state with reduced moment. This is related to the fact that the energy of the bonding orbital (the lowest curve in Fig. S1) rapidly decreases with the increase of SOC, which makes the MO state (in which this orbital is occupied twice) more favorable.

The analytic treatment presented above is confirmed by the exact diagonalization results for the dimer with three orbitals per site. The resulting phase diagram is shown in Fig. 6. We see that it agrees with our analytical results for limiting cases of

One may expect that for the weak SOC the situation is similar also not for three electrons, but for three holes in ^{4+}(^{5+} (*d* electrons.

However, for example one can show that the situation is qualitatively different for five electrons per dimer: in this case the treatment similar to that for three electrons above, shows that both strong and weak SOC works in favor of a less-magnetic MO state, i.e., the corresponding curve separating MO and DE states in the phase diagram, similar to that of Fig. 6, would increase already at small ζ, without reentrant behavior. Thus, we see that, in contrast to the Hund’s coupling, which always competes with the electron hopping and tends to stabilize a “more magnetic” state, the SOC can work differently in different situations.

One more important factor, which can change the interplay of SOC with the Hund’s exchange and electron hopping, is a possible contribution of not only the direct *d–d* hopping, considered in our treatment, but also of the hopping via *p* orbitals of ligands, e.g., oxygens. To take into account all these effects, for real materials it is probably better to rely on ab initio calculations, the results of some of which are presented in the next section.

## Suppression of the Double Exchange and Magnetic Moment in Real Materials

We now turn to real materials and show that the physics described above (strong reduction of magnetic moment and eventual suppression of double exchange, due to formation of orbital-selective covalent bonds between TM) indeed works in real materials and allows us to explain the behavior of many _{2}O_{2}F_{3} (28) (see its structure in Fig. S2 and the results of our ab initio calculations in *Supporting Information*). Although, according to the chemical formula there has to be three *d* electrons per Nb dimer, i.e., *A*) is very large here,

According to the results of our model cluster DMFT calculations, see above, for *U* in Nb_{2}O_{2}F_{3} can be very different from what we used in the model calculations, so that comparison with the results (see Fig. S3) of the generalized gradient approximation (GGA) should be more appropriate. These results are summed up in *Supporting Information*, and indeed show that the value of the local magnetic moment in the high temperature phase is

We have discussed so far how covalent bonding competes with Hund’s rule coupling. However, as discussed above, for _{5}AlIr_{2}O_{11}, we show that this interaction may indeed take part in the competition between intra-atomic exchange and covalent bonding.

The crystal structure of Ba_{5}AlIr_{2}O_{11} consists of [Ir-Ir]^{9+} dimers (Fig. S4*A*) with nine electrons or three holes per dimer, so that this is exactly the example of a system, considered in the first part of our paper, where the DE and MO states could compete. Indeed, the effective magnetic moment in the Curie–Weiss law *d* levels, conspires with the covalency and finally helps to suppress the DE in this system, see *Supporting Information*. However, as stressed in the previous section, this is not a general conclusion: depending on the filling of the *d* shell and on a specific geometry, the spin–orbit interaction can act against the DE or support it.

Until now we have considered real materials in which the structure provides relatively well-separated dimers. As we saw, e.g., in our cluster DMFT calculations, the effect of suppression of the DE can survive even for solids, without well-defined dimers. In some of such systems singlet covalent bonds can form spontaneously, also acting against DE (cf. ref. 34).

One such example seems to be given by CrO_{2}—a classical DE system (35). In CrO_{2} one of the two 3*d* electrons is localized in the _{2}, having the same rutile structure, but one *d* electron per V, CrO_{2} does not dimerize at zero pressure.

Apparently, in normal conditions in this system the hopping via oxygens is more important, and in effect DE wins and provides ferromagnetism with large Curie temperature. However, under high pressure the situation can change: the direct *d–d* hopping can begin to dominate, the bonding–antibonding splitting even in this 3D system can become comparable with what we have in _{2} with pressure, with the formation of the dimerized monoclinic structure for P *d* electrons in the Cr–Cr dimer occupy this molecular orbital, and the remaining two electrons provide metallicity and eventual paramagnetism, similar to what is observed in MoO_{2} at normal conditions (37) or expected in MoCl_{4} at moderate pressure (38). We thus see that even some *d–d* overlap and hopping, making such systems more similar to the *d* ones.

It is also possible that some other factors, such as charge ordering (29), could modify the situation. The external perturbations, such as temperature or pressure, can drive a system from the DE to the MO state, so that the phase diagrams of systems with competing DE and MO states can be rather rich and their physical properties can be highly unusual.

## Conclusions

In conclusion, we demonstrated in this paper that the standard approach to describe the magnetism in solids with strongly correlated electrons, proceeding from the isolated ions in their ground state, and adding electron hopping between ions perturbatively, may break down in certain situations, especially for *d–d* hopping orbital-selective covalent bonds, or singlet molecular orbitals between transition metal ions may form in this case. This would lead to a strong suppression of the effective magnetic moment, and, for fractional occupation of *d* shells, can strongly reduce or completely suppress the well-known double exchange mechanism of ferromagnetism. Several examples of real material considered in our paper indeed demonstrate that this mechanism is very efficient in suppressing the double exchange in a number of

## Details of the Exact Diagonalization and Cluster DMFT

To study suppression of the DE we considered dimers and dimerized chain, described by the Hamiltonian, which reads as*U* and Hund’s rule exchange *i* numerates dimers and *j* all sites in the lattice. Here, *c* and *d* orbitals in the dimer, *j* for orbital *m* and spin σ.

In numerical calculations we used the Kanamori parameterization of

We used the exact diagonalization technique to treat a dimer and cluster DMFT (25) with Hirsch–Fye solver (40) to study dimerized chain.

## Details of the Ab Initio Calculations

All ab initio band structure calculations were performed within the GGA (41) and full-potential linearized augmented plane wave method using Wien2k code (42). The crystal structure of Nb_{2}O_{2}F_{3} were taken from ref. 28 and for Ba_{5}AlIr_{2}O_{11} from ref. 33. The parameter of the plane-wave expansion was chosen to be

Muffin-tin (MT) radii were chosen to be _{2}O_{2}F_{3} and _{5}AlIr_{2}O_{11}. We used following *k* meshes for the Brillouin-zone integration: 7 _{2}O_{2}F_{3} and 3 _{5}AlIr_{2}O_{11}.

## Nb_{2}O_{2}F_{3}

The crystal structure of Nb_{2}O_{2}F_{3} consists of Nb–Nb dimers, which are formed by two NbO_{3}F_{3} octahedra sharing their edges, see Fig. S2*A*. Nb ions are inside octahedra formed by O (red balls) and F (gray balls) ions. As a result the *A*) point to each other.

The density of states (DOS) plot obtained in the nonmagnetic GGA calculations for the high temperature phase is presented in Fig. S3. Analysis of the partial contributions to the total DOS shows that the bonding–antibonding splitting for the *B*. As a result the total spin and the double exchange turn out to be suppressed in this situation.

## Ba_{5}AlIr_{2}O_{11}

The main structural element of Ba_{5}AlIr_{2}O_{11} is isolated Ir–Ir dimers. Here, Ir ions are in the IrO_{6} octahedra, which share their faces, see Fig. S4*A*. Here, in contrast to Nb_{2}O_{2}F_{3}, the bonding–antibonding splitting is not large enough to stabilize the state with the lowest total spin. This is clearly seen in the ferromagnetic GGA calculations, where the ground state corresponds to configuration shown in Fig. S4*B* with the spin moment

Theory and experiment become consistent if one includes the spin–orbit coupling (SOC), which additionally splits antibonding Ir

## Details of Model Calculations with Spin–Orbit Coupling

To study effect of the spin–orbit coupling we considered isolated dimer with three electrons and three *m* numerates these orbitals. Corresponding level diagrams are given in Fig. S1.

In the calculations of the phase diagram, given in Fig. 6, we took into account the intra-atomic Hund’s exchange as

## Acknowledgments

This work was supported by the Russian Foundation of Basic Research via Grant 16-32-60070, Civil Research and Development Foundation via Program FSCX-14-61025-0, FASO (theme “Electron” 01201463326), Russian Ministry of Education and Science via act 11 Contract 02.A03.21.0006, by Köln University via the German Excellence Initiative, and by the Deutsche Forschungsgemeinschaft through SFB 1238.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: streltsov{at}imp.uran.ru.

Author contributions: S.V.S. and D.I.K. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1606367113/-/DCSupplemental.

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