Ferromagnetic kinetic exchange interaction in magnetic insulators

The superexchange theory predicts dominant antiferromagnetic kinetic interaction when the orbitals accommodating magnetic electrons are covalently bonded through diamagnetic bridging atoms or groups. Here we show that explicit consideration of magnetic and (leading) bridging orbitals, together with the electron transfer between the former, reveals a strong ferromagnetic kinetic exchange contribution. First-principles calculations show that it is comparable in strength with antiferromagnetic superexchange in a number of magnetic materials with diamagnetic metal bridges. In particular, it is responsible for a very large ferromagnetic coupling ( − 10 meV) between the iron ions in a Fe 3 + -Co 3 + -Fe 3 + complex. Furthermore, we ﬁnd that the ferromagnetic exchange interaction turns into antiferromagnetic by substituting the diamagnetic bridge with magnetic one. The phenomenology is observed in two series of materials, supporting the signiﬁcance

Introduction.-Anderson'ssuperexchange theory [1] plays a central role in the description of exchange interactions in correlated magnetic insulators.It provides in particular an explanation of phenomenological Goodenough-Kanamori rules [2][3][4].This theory identifies the orbitals at which reside the unpaired (magnetic) electrons -the Anderson's magnetic orbitals (AMO) -via a minimization of electron repulsion on magnetic sites.For non-negligible electron transfer (b) between these magnetic orbitals, the theory predicts strong kinetic antiferromagnetic interaction between localized spins, J = 4b 2 /U , where U is the electron repulsion on magnetic sites.When b is suppressed e.g., on symmetry reasons [4,5], weaker ferromagnetic interactions of non-kinetic origin, such as, potential exchange [1,6], the Goodenough's mechanism [2,3] and the spin-polarization (the RKKY mechanism) [7][8][9] become dominant.
A different extension of the theory was proposed by Geertsma [41], Larson et al [42], and Zaanen and Sawatzky [43] through explicit consideration of the orbitals of bridging diamagnetic atoms/groups along with the orbitals accommodating the magnetic electrons.Such an extension allowed for a concomitant description of high-energy excitations and exchange interaction in charge-transfer insulators [44].Another reason for this extension was the claim that the Anderson's theory would break down when the ligand-to-metal electron transfer energy becomes lower than the metal-to-metal electron transfer energy [43].However, a detailed analysis has shown that the predictions of this extended model for the low-lying states are basically the same as of the Anderson's model when only metal-ligand electron transfer is taken into account [45].The situation changes crucially when the metal-to-metal electron transfer is added to the model.In this case a strong ferromagnetic contribution of kinetic origin can arise [46][47][48].Despite the fact that this mechanism has been mentioned on different occasions [46,[49][50][51], its relevance to existing materials has not been clarified.
In this work, we elucidate the conditions for strong ferromagnetic kinetic exchange interaction.Combining model description with first-principles calculations, we prove the importance of this exchange mechanism in ferromagnetic metal compounds and its dominant contribution in cases of very strong ferromagnetic coupling between distant metal sites.We show that also in materials not exhibiting (strong) ferromagnetism, kinetic ferromagnetic contribution is crucial for the annihilation of the antiferromagnetic superexchange.
Basic three-site model.-Ina first step, we derive the AMOs as minimizing the electron repulsion between magnetic electrons in a spin-restricted broken-symmetry band (molecular) orbital picture [1,6].Then we identify the common ligand orbitals in the composition of neighbor AMOs and approximate them by Wannier transformation of a group of suitable band (molecular) orbitals.The resulting localized bridging orbitals (LBO) mainly reside at the diamagnetic atom/group bridging the neighbor paramagnetic sites.Extracting these orbitals from the AMOs via an orthogonal transformation, we end up with localized magnetic orbitals (LMO), which are more localized on the paramagnetic sites than the corresponding AMOs but now strongly overlap with neighbor LBOs.The exchange interaction is derived from a many-body treatment of electrons in LMOs of two chosen paramagnetic sites and LBOs of the bridging diamagnetic atom/group.We first consider the simplest model involving only two LMOs and one LBO, Here, 1, 2 and d indicate the paramagnetic and the diamagnetic sites, respectively, t M D/DM and t M M are the corresponding electron transfer parameters, ∆ is the gap between the diamagnetic and paramagnetic orbital levels, and U M and U D are on-site Coulomb repulsion parameters within LMO and LBO, respectively.For symmetric magnetic sites considered below, the following relations hold: The model (1) always reduces to two unpaired particles localized at the LMOs, which are electrons when the LBO on the diamagnetic site is empty and holes when this is doubly occupied.In the latter case, all one-electron parameters in Eq. ( 1) change the sign except for ∆ which becomes −∆ + 2U D , remaining always positive in magnetic insulators.For t M M = 0, the Hamiltonian (1) reduces to the earlier considered 3-orbital model [41][42][43].We stress, however, that this limit is often unrealistic because the LMOs and the LBO are not atomic orbitals but instead have "tails" which extend on neighbor sites, in analogy with AMOs [1,6].
Ferromagnetic kinetic exchange interaction.-Thecalculated spectrum is shown in Fig. 1(a).One can see that the system exhibits strong ferromagnetism for relatively large values of t M M , further enhanced for small ∆ [Fig.1(b)].We emphasize that it arises without Hund's rule coupling and potential exchange interaction, which are not included in Eq. (1).To unravel the mechanism of ferromagnetism, we consider , and obtain in the fourth order of perturbation theory the expression for the exchange parameter, The first and second terms are always antiferromagnetic, and the fourth term is ferromagnetic.The third term becomes ferromagnetic for t M D t DM t M M > 0 and is antiferromagnetic otherwise.According to the order of the perturbation, the first and the third terms are dominant, and the nature of J is mainly determined by their competition.The generalization of four kinetic contributions in Eq. ( 2), further denoted as K1-K4, to a more realistic model including several LBOs on the diamagnetic site, is given in Eqs.(S6)-(S9) [52], respectively.The new feature is that K2 can be now both ferromagnetic and antiferromagnetic albeit giving a weaker ferromagnetic contribution than K3.The ferromagnetic contribution K3 originates from cyclic electron transfer processes avoiding double occupation of any of three orbitals [Fig.1(c)].It can be called ferromagnetic kinetic exchange interaction.Note that the contribution of this mechanism to the energy of the ferromagnetic state, −2t M D t DM t M M /∆ 2 (the factor 2 is due to a cyclic processes, similar to Fig. 1(c) but in anticlockwise sense), is opposite to the case of antiferromagnetic state, because of the sign change in the latter.It should be noted that this contribution is not fully captured by the Anderson's approach [1].Indeed, a finite t M M merely modifies the effective transfer parameter b between the AMOs, i.e., the antiferromagnetic kinetic exchange.The ferromagnetism in this approach can only arise indirectly, via the enhancement of the potential exchange contribution.However, it is much underestimated compared to the exact treatment (Fig. S1 [52]).
Condition for strong ferromagnetism.-Thenecessary condition for a dominant ferromagnetic kinetic contribution is the right sign and a large value of t M M .The existence of non-negligible t M M is expected for LMOs extending on neighbor paramagnetic sites.This occurs when the relevant bands (molecular orbitals) involves several atomic orbitals centered on different atoms in the unit cell (molecule).Then, the corresponding Wannier orbitals will not be completely localized, leading to non-negligible overlap between neighbor LMOs.In an opposite situation, when the common bridging orbitals in the composition of neighbor AMOs are contained in the same number of relevant bands (molecular orbitals), the Wannier transformation of the latter will result in LMOs almost coinciding with atomic orbitals and LBOs well localized on the bridging diamagnetic groups.An example are superconducting cuprates, in which the lowenergy states are described by a three-orbital model for the CuO 2 plane [53], involving almost net atomic 3d x 2 −y 2 orbital on Cu and 2p x (2p y ) orbitals on O.The latter lead to small t M M and negligible kinetic ferromagnetic exchange contribution (t M D t DM t M M > 0), which is in accord with a very large antiferromagnetic exchange interaction in cuprates [54].
According to Eq. ( 2), the t M M of a right sign not only gives rise to a ferromagnetic kinetic contribution but concomitantly reduces the antiferromagnetic one.However, the largest ferromagnetic J is not achieved at a t M M quenching K1 but at a larger value, t M M ≈ (t M D t DM /∆)(1 + U M /2∆).The expression in Eq. ( 2) then becomes Counter-intuitively, the ferromagnetic coupling increases with U D and U M .Besides, it rises very fast with diminishing ∆, a feature also confirmed by non-perturbative treatment [Fig.1(b)].Small ∆ (strong metal-ligand hybridization) is expected in late transition metal compounds, which are thus primary candidates for the observation of strong kinetic ferromagnetism.
A similar treatment shows that adding one electron/hole to the empty/doubly occupied LBO turns the initially dominant kinetic ferromagnetic interaction into antiferromagnetic one of comparable strength.This suggests that the kinetic ferromagnetic mechanism does not exist for next-nearest neighbor exchange pairs.Such behavior was observed in a series of trinuclear isostructural complexes with various electronic populations of the central metal ion [55] and between Cu ions in La In order to achieve a realistic description of exchange contributions, the results of first-principles calculations were mapped into an extended three-sites model (S1-S5) which, contrary to the basic model in Eq. ( 1), includes all relevant LBOs on the diamagnetic bridging site and the bielectronic interactions between the LMOs and LBOs.Electronic band structure calculations for all materials were performed on their experimental structure [55,58,61] with revised Perdew-Burke-Ernzerhof (PBE) functional [62] and optimized normconserving Vanderbilt pseudo-potentials [63].Using the Kohn-Sham orbitals, maximally localized Wannier functions [64] and one-particle interaction parameters, t and ∆, were derived.Screened intra-(U M/D ) and intersite Coulomb, and potential exchange parameters were calculated within constrained random phase approximation [65].Note that for quantitative description of the exchange interaction in these materials, the intersite Coulomb and potential exchange interactions ignored in Eq. ( 1) were also taken into account.Quantum ESPRESSO [66,67] and RESPACK [68][69][70][71][72] were used for electronic structure calculations, and VESTA [73] for plotting the orbitals.To match the experimental J, in molecular complexes ∆ was treated as a parameter.The obtained parameters of the extended three-site model for the four compounds are listed in Table S1 [52].J was derived by exact diagonalization of the corresponding Hamiltonian, Eqs.(S1)-(S5), and the kinetic contributions to the exchange parameters were calculated using the corresponding expressions, Eqs.(S7)-(S10).Due to the perturbative character of the latter, their sum (together with the contribution from potential exchange interaction between LMOs) deviates from the exact value of J (cf Table I).In the above DFT-based derivations, the obtained values of U M and ∆ are the less reliable.Accordingly, Figs.2(c), and 3 show J diagrams for the three molecular complexes as functions of U M and ∆ at fixed values of other parameters (Table S1).
1. Fe-Co-Fe complex.-Thiscomplex has a C 3 symmetry with respect to the Fe-Co-Fe axis (z), hence, the 3d orbitals on each metal site split into one totally symmetric a (d z 2 type) and two sets of doubly degenerate e levels.The calculated spin density shows that the a or-  bitals are the magnetic ones.Due to symmetry, only the a orbitals on Fe and Co sites are relevant to kinetic exchange interaction, while the Goodenough's mechanism is ruled out.Since the a orbital of the Co site is doubly occupied, the magnetism is described in the hole picture.
The calculated LMO on one Fe 3+ site and LBO on Co 3+ site [Figs.2 (a), (b)] are strongly hybridized with the 3p orbitals of the sulfur atoms between the metal ions, which makes t M M non-negligible.With the theoretical value of U M = 2.86 eV, experimental J is reproduced with ∆ = 0.60 eV which matches the estimation 1.05 eV from absorption spectra in solution [55].Table I shows that the ferromagnetic kinetic exchange (K3) is clearly dominant due to a relatively large value of t M M .The contributions K1 and K2 are similar in magnitude because of an efficient cancellation of t M D t DM /∆ by t M M in the former.Thus the observed very large ferromagnetic coupling (−10 meV) in this complex [55] is entirely due to the ferromagnetic kinetic exchange mechanism.
2. Cu-Cr-Cu and Cu-Mo-Cu complexes.-Thespin density analysis shows that the 3d x 2 −y 2 orbitals of Cu ions are the magnetic ones and the 3/4d zx orbital of Cr/Mo is the bridging one, where the Cu-Cu axis corresponds to the x axis and z is the out of plane axis (Fig. S2).In both complexes, due to a partial cancellation of t M M and t M D t DM /∆, the effective transfer parameter between the two LMOs, t M M − t M D t DM /∆, is reduced and hence the K1 contribution becomes small.The t M M in Cu-Cr-Cu is larger than in Cu-Mo-Cu, the same for the K3 contribution.Consequently, the former compound is ferromagnetic and the later antiferromagnetic.
3. Quasi 1D Cu chain.-Theorigin of ferromagnetism in La 4 Ba 2 Cu 2 O 10 was debated in the past [50,74].In this system, the magnetic orbitals are of 3d zx type of Cu [Fig.4(a)] and the bridging orbitals are the empty orbitals of La, Ba and O, where the plane of the Cu chain is taken zx.With first-principles parameters of Table S1, we obtained J = −0.65 meV close to the experimental value (−0.4 meV [60]).Remarkably, the contribution K2 is now ferromagnetic and of similar magnitude as K3.K2 ¡ 0 became possible due to numerous loop terms (involving different LBOs) in Eq. (S8) which can be of either sign because both t M D = t DM and t M D = −t DM relations hold for different LMOs.On the same reason both ferro-and antiferromagnetic contributions for dif-ferent LBOs are present in Eq. (S9) reducing the total K3 contribution [Fig.4(d)].Among the latter, the contributions via the 5d zx and the 4f z(x 2 −y 2 ) of in plane La ions [Figs.4(b),(c)] are dominant.Thus two kinetic ferromagnetic exchange mechanisms, K2 and K3, make together a dominant contribution rendering the resulting exchange interaction ferromagnetic.
Conclusions.-We have investigated the ferromagnetic kinetic exchange interaction between localized spins.This mechanism shows up at a higher level of treatment compared to Anderson's theory, through the separation and explicit consideration of relevant diamagnetic orbitals bridging the magnetic ones.The crucial point is that despite a stronger localization compared to AMOs, the LMOs and LBOs arising in the present treatment are by far not atomic like.This opens two paths for delocalization of magnetic electrons, via the LBOs and through-space.When the latter is sufficiently strong, the interference between the two kinetic paths can result in a ferromagnetic contribution which overcomes the conventional antiferromagnetic superexchange.The conditions for achieving strong ferromagnetism via this mechanism have been elucidated.In particular, it is favored by reduced orbital gap between magnetic and bridging orbitals, pointing to materials with strong metal-ligand covalency.
We have investigated the relevance of ferromagnetic kinetic exchange mechanism in several compounds by firstprinciples calculations.It was found that this exchange contribution is of comparable magnitude with the antiferromagnetic kinetic exchange.The calculations show that in the Fe-Co-Fe complex the observed very large ferromagnetic coupling is entirely due to a strong ferromagnetic kinetic contribution.The obtained results call for the reconsideration of the origin of ferromagnetism and week antiferromagnetism in insulating magnetic materials and complexes.
Z.H. and D.L. were supported by the China Scholarship Council.A.M. acknowledges funding provided by the Magnus Ehrnrooth Foundation.V.V. was postdoctoral fellow of the Research Foundation -Flanders (FWO).N.I. was partly supported the GOA program of KU Leuven, and the scientific research grant R-143-000-A80-114 of the National University of Singapore.The computational resources were provided by the VSC (Flemish Supercomputer Center).

FIG. 1 .FIG. 2 .
FIG. 1.(a) Energy levels diagram of the three-site model (1) for tMD = tDM , tMD/UM = ∆/UM = 0.2 and UD/UM = 1.The solid red and dashed blue lines indicate triplet and singlet states, respectively.(b) Exchange parameter diagram (other parameters than indicated on the axes are the same as in (a)).(c) Third-order process responsible for ferromagnetic kinetic exchange contribution.

a
First principles J PE is scaled down following Ref.[74].b ∆ was chosen to reproduce the experimental J.

TABLE I .
J and its kinetic (Kn) and potential exchange (PE) contributions (meV).