Superexchange interaction revisited: the role of the A-site cations in A CuO3 (A=Se, Te)

To investigate the effects of A-site cations on the magnetic interaction in highly distorted perovskite ACuO3 (A=Se, Te), we have set the microscopic Hamiltonian for the Cu–O–Cu–A cluster, taking into account the or hybridization between A- or A- and O-2p orbitals. We have found that (i) the superexchange interaction of the simple Cu–O–Cu triad without considering the A-site effect is inadequate to explain the ferromagnetism of SeCuO3, (ii) the hybridization weakens the superexchange interactions and (iii) the hybridization induces the ferromagnetic superexchange interaction along the c-axis. We have also verified that the hybridization affects the magnetic interaction through anisotropic charge transfer energy shift in the Cu–O–Cu interaction path.


Introduction
The magnetism in transition metal oxides is closely correlated with structural properties, such as bond geometry, local crystal symmetry and Jahn-Teller (JT) distortion. Indeed, the empirical Goodenough-Kanamori-Anderson (GKA) rule [1]- [3] and the Kugel and Khomskii model [4] have been successful in describing the magnetic interactions in transition metal oxides in relation to bond angle and orbital ordering. However, there are some cases that are not consistent with the prediction based on the above theoretical models. One typical example is the magnetic interaction in ACuO 3 (A = Se, Te), which has a highly distorted perovskite structure (Pbnm space group) due to small ionic radii of Se and Te [5]- [7].
Subramanian et al [7] reported that the magnetic ground state of SeCuO 3 changes when substituting Te for Se from the ferromagnetic (FM) to the antiferromagnetic (AFM) phase. They observed that the Cu-O-Cu bond angles increase from 122.3 • to 123.1 • along the c-axis (φ 1 ) and from 127.1 • to 130.5 • in the ab-plane (φ 2 ) (see figure 1). Because a structural tuning of φ 2 is much larger than φ 1 , they attributed the change in the magnetic state to the sign change of the magnetic interaction in the ab-plane by applying the GKA rule. They thus predicted that the magnetic ground state of TeCuO 3 would be the C-type AFM state, in which FM chains along the c-axis are coupled antiferromagnetically. However, subsequent electronic structure calculations [8,9], as well as microscopic model analysis [10], showed that, going from SeCuO 3 to TeCuO 3 , the magnetic interaction along the c-axis changes from FM to AFM, whereas the interaction in the ab-plane always remains FM. This implies that the A-type AFM state, in which FM layers in the ab-plane are coupled antiferromagnetically along the c-axis, is realized for TeCuO 3 rather than the C-type AFM state.
As shown in figure 1(a), 3 out of 12 surrounding oxygens around A in ACuO 3 are much closer to A than others, and so a structural unit of the AO 3 trigonal pyramid is formed. Such a structural character causes anisotropic bonding between A and three oxygens and induces nonbonded orbitals of A to appear along the empty corner of an AO 3 pyramid [5,11]. Moreover, the A-O distance is a bit smaller than the Cu-O distance, and accordingly strong hybridization between the orbitals of A and oxygen is expected to occur. For this reason, the hybridization effect between the orbitals of A and O is required to be taken into account to understand the physical properties of ACuO 3 . In fact, Iñiguez and Yildirim [8] showed that the magnetic ground state of TeCuO 3 is changed from the AFM to the FM phase by decreasing artificially the Te-O distance, and they suggested that thesp bonding between A-s and O-2p orbitals would play a role in switching the sign of the magnetic interaction in ACuO 3 .
Other ABO 3 (B are transition metal elements) perovskites also have structural details similar to ACuO 3 . Hence, the strong hybridization between A and O has been considered to investigate their electronic and magnetic properties [5], [11]- [14]. However, the microscopic treatment of this effect is far from complete.
In this study, we have performed a systematic study to explore the microscopic mechanism of the magnetic interaction in ACuO 3 , considering not only structural specifications, such as Cu-O-Cu bond angle and orbital ordering, but also the A-site orbital contribution. Firstly, we have investigated the electronic structures of A-site ions in SeCuO 3 to inspect their hybridization behavior with other orbitals. Secondly, we have set the microscopic model for a Cu-O-Cu-A cluster and evaluated the magnetic exchange parameters with respect to the hybridization strength between orbitals of A and oxygen. As a result, we have demonstrated that thesp Due to cooperative JT distortion, C-type orbital ordering is realized.
hybridization between the occupied A-s and the O-2p orbital plays a significant role in the magnetic interaction of the ACuO 3 system.

Electronic structure
To investigate the electronic properties of the A-site ion in ACuO 3 (A = Se, Te), we have performed the electronic structure calculation employing the full-potential augmented plane wave band method [15]. The generalized gradient approximation (GGA) [16] was used for the exchange-correlation potential. The density of states (DOS) and charge density difference (CDD) for the FM phase of SeCuO 3 are presented in figure 2. The CDD corresponds to the valence charge density of the crystalline solid after subtraction of the superposed atomic charge density of the valence electrons. According to the DOS, figure 2 shows that FM SeCuO 3 has the insulating ground state with an energy gap of ∼0.3 eV and the valence state of Cu ion is nearly 2+, consistent with existing band structure results [9,10]. Focusing on the Se band, the 4s band is fully occupied and shows a small band overlap with the oxygen 2p band. Its band center is about 9 eV lower than that of oxygen 2p. On the other hand, the unoccupied Se 4p band is about 7 eV higher than the oxygen 2p band center. Strong mixing with the oxygen 2p band causes a small peak near −6 eV below the Fermi energy. This feature corroborates the strong hybridization between orbitals of Se and neighboring oxygens.
The CDD plot in the inset of figure 2 shows clearly the hybridization bonding property in SeCuO 3 . An SeO 3 unit is known to produce the nonbonded Se lone-pair orbital, which is directed toward the apex of each trigonal pyramid [5,11]. The CDD plot reveals that charges around Se ions are accumulated (red) on one side, but depleted (blue) on the other side, whereas those around oxygen ions are all accumulated. The distribution of positive CDD around Se ions, which represents the formation of a nonbonded orbital, indicates that the lone-pair orbital is located toward the apex direction of trigonal pyramid. It thus results in an anisotropic pyramidalshape bonding network. For A-type AFM TeCuO 3 too, we have obtained a similar band structure and confirmed the anisotropic nature of a strong hybridization between Te and oxygens.

The microscopic model
As discussed in figure 2, there exists a strong hybridization between the Se band and the O-2p band. This implies that oxygen states can be modified by A-site ions and accordingly the magnetic interaction mediated by oxygen can be changed in highly distorted perovskite ACuO 3 . To understand the magnetic interaction in ACuO 3 , we have considered the Cu-O-Cu-A clusters depicted in figure 1(b) and evaluated superexchange parameters in the framework of our previous study [17]. We have assumed that the ACuO 3 system follows the ideal Pbnm structure to consider systematically the relation between the Cu-O-Cu bond angle and the local coordinates of two CuO 6 octahedra [18]. In this assumption, a Cu-O-Cu triad is described by the following Hamiltonian:  9 , the charge transfer is given by E(p 5 d 10 ) − E(p 6 d 9 ). According to Slater and Koster [19], the hopping integrals between O-2p and Cu-3d orbitals are expressed by two parameters t pdσ and t pdπ that follow the relation t pdπ = −0.46 t pdσ [20].
Coulomb interaction parameters of 3d and 2p orbitals, respectively. The hopping parameters t µ i p are evaluated as functions of the two p-d hopping parameters, t pdσ and t pdπ , according to Slater and Koster [19].
In order to investigate the effect of the hybridization between orbitals of A and oxygen ions, we have adopted the additional Hamiltonian where c † aσ is the creation operator of the A-site orbital with a σ spin. Possible A-site orbitals are occupied 4s and empty 4p orbitals in the case of SeCuO 3 . According to Slater and Koster [19], thesp andpp hybridizations are characterized by one (ts pσ ) and two (tp pσ and tp pπ ) parameters, respectively.
The physical parameters used in the superexchange calculation are provided in table 1. The charge transfer energy is defined by E(3d 10 2p 5 ) − E(3d 9 2p 6 ). s and p are the charge transfer energies of the occupied A-s and the empty A-p orbital, respectively, e.g. for A = Se, s = E(4s 1 2p 6 ) − E(4s 2 2p 5 ) and p = E(4p 1 2p 5 ) − E(4p 0 2p 6 ). The distortion angle θ , which represents the local JT distortion, and the occupied e g orbital state |θ are set in between 104 • and 110 • according to real ACuO 3 systems [7]. JT in table 1 corresponds to the JT splitting energy, which is defined by the energy difference between two e g orbitals, |θ and |θ + π . 10Dq is the crystal field splitting between t 2g and e g states of Cu.
Physical parameters such as 10Dq, JT , J d , , s and p can be estimated from the DOS results in figure 2, which are consistent with our choices in table 1. For U d , the typical value (U d ≈ 7-8 eV) in cuprates is employed [21]. For J d , considering that the GGA band calculation usually underestimates the Coulomb interaction parameters, we adopted the value of J d = 1.5 eV that is in between the band calculation result ∼1.1 eV and the atomic multiplet calculation result ∼1.6 eV. We have checked that the reduction of J d tends to suppress the ab-plane FM interaction, but overall magnetic behaviors are really robust in the range of J d ≈ 1.0-1.5 eV.
In insulating ACuO 3 systems, electrons are nearly localized and so only the hopping between adjacent ions would give weak orbital overlaps. Then the ground state of the Cu-O-Cu-A cluster is mainly contributed by a Cu(3d 9 The superexchange parameters along the c-axis (J c ) and (b) those in the ab-plane (J ab ) for the d 9 perovskite oxides with given parameters in table 1. There is no change in the magnetic ground state for the bond angle change from 180 • to 110 • : the A-type AFM state is always stable. Arrows in (a) and (b) represent experimental values of φ 1 and φ 2 for SeCuO 3 (black) and TeCuO 3 (purple) systems.
configurational state. For this reason, we have considered the restricted Hilbert space, in which states are overlapped with a Cu(3d 9 )-O(2p 6 )-Cu(3d 9 )-A(s 2 orp 0 ) configurational state through off-diagonal elements of H and H 2 . It is reminiscent of the fourth-order perturbation approximation in the hopping parameter t pd . Because the Hamiltonian has no spin-flip part, we have calculated the ground state energies of the FM (E ↑↑ ) and AFM (E ↑↓ ) cases separately. Assuming that the magnetic energy is given by the Heisenberg type, E = 2J S 1 · S 2 , the superexchange interaction parameters are calculated by the relation 4J S 2 = E ↑↑ − E ↑↓ .

Magnetic interaction
Since ACuO 3 exhibits the C-type orbital ordering, its magnetic interaction would be similar to that of the perovskite manganite. In the ab-plane where the antiferro-orbital ordering appears, the FM interaction (J ab ) is dominant, whereas, along the c-axis having the ferro-orbital ordering, the AFM interaction (J c ) prevails. As the Cu-O-Cu bond angles (φ 1 and φ 2 ) decrease with the distortion of GdFeO 3 type, the magnitudes of two exchange parameters J ab and J c are reduced. This behavior is well described in figure 3. The magnetic structure of TeCuO 3 , whose bond angles φ 1 and φ 2 are estimated to be 123.5 • and 130.5 • , respectively, should be A-type AFM, because J ab is negative and J c is positive at those φ 1 and φ 2 (see purple arrows in figure 3). This finding does not agree with the magnetic structure proposed by Subramanian et al [7], but agrees with the results of the electronic structure calculations [8,9] and the microscopic model calculation [10]. Note that both exchange parameters in figure 3 never change signs until φ 1 and φ 2 become as small as 110 • , and so this result does not explain the FM ground state of SeCuO 3 that has φ 1 = 122.3 • and φ 2 = 127.1 • (black arrows in figure 3). Of course, the FM ground state is possible when φ 1 is smaller than about 100 • [10]. But this value is far from φ 1 of SeCuO 3 . Furthermore, as shown in figure 3, such a tendency is robust with respect to the variation in the  SeCuO 3 (square in figure 4(a)) and 0.93 for TeCuO 3 (circle in figure 4(a)) [5,6], which are close to the values for the ideal Pbnm structure. Smaller d A induces stronger hybridization between the outermost occupied A-s or unoccupied A-p and O-2p orbitals, and its strength overwhelms the pd hybridization between Cu-3d and O-2p orbitals. In fact, according to the tight-binding calculation [8], thesp hybridization is five or six times larger than the pd hybridization. Therefore, we have investigated the effect of the hybridization between orbitals of A and O on exchange parameters J ab and J c . In the calculation, we have assumed tã pσ ∝ d −2 A (ã =s orp) and tp pσ = −0.25ts pσ , following Harrison's relation [20]. If the bond strength for d A = d 0 is tã pσ 0 , tã pσ is given by tã pσ = tã pσ 0 (d A /d 0 ) −2 . Figure 4(b) provides bond angle-dependent behaviors of tã pσ /t pdσ for some tã pσ 0 values. Figures 4(c) and (d) show the effect of thepp hybridization on J ab and J c . The magnitudes of both J ab and J c decrease monotonically with increasingpp. However, J c is still positive down to φ 1 = 110 • , and so no magnetic transition from the AFM to the FM state takes place, even with strongpp hybridization. This means that unoccupied A-p states can modify the strengths of two exchange parameters, but cannot change the magnetic ground state. Thus, to explain the FM state in SeCuO 3 , the additional effect should be invoked. Now we have considered the effect of thesp hybridization on J c and J ab . Even though the band mixing between A-s and O-2p orbitals looks rather small, strongsp hybridization is estimated in the tight-binding model [8]. Figures 5(a) and (b) show the modification of magnetic interactions J c and J ab with respect to the change insp hybridization. Most prominent in figure 5(a) is the realization of FM J c for small φ 1 . The change in the magnetic interactions is not notable for ts pσ 0 < 3.0 eV. However, with increasing ts pσ 0 further, the FM region appears and the critical angle, where the sign of J c is reversed, becomes close to φ 1 of SeCuO 3 . It is coincidental that the in-plane FM interaction J ab is much strengthened near the critical angle, as shown in figure 5(b). It is seen in figure 5(a) that, for ts pσ 0 = 4.5 eV, J c is changed from about 3.8 to −2.8 meV when φ 1 varies from 123.5 • to 120.5 • . Also, in figure 5(b), J ab is seen to be in between −2.0 and −1.6 meV when φ 2 varies from 130.2 • to 127.1 • . These values are consistent with calculated exchange parameters based on the band structures of ACuO 3 [8,9]. Besides ts pσ , the charge transfer energy s also affects J ab and J c . Because an electron in the occupieds state can hop into the empty O-2p states easily for small s , the FM region will be broadened with decreasing s . The inset in figure 5(b) presents the magnetic phase diagram for φ 1 = 122.33 • as functions of ts pσ and s . It is evident that the strongsp hybridization modifies largely the superexchange interaction, and proper values of ts pσ and s cause its sign to be reversed and stabilize the FM interaction.
When the charge transfer energy is similar to or smaller than the on-site Coulomb energy, the charge transferred states with oxygen 2p holes can play an important role in the magnetic interaction. We have investigated the relation between thesp orpp hybridization and oxygen 2p hole states. For this purpose, we introduce the projected 2p hole weight (PTHW) that refers to the weight of one or two O-2p hole states in the collapsed state obtained by projecting the ground state of a Cu-O-Cu-A cluster on the subspace where A-site configuration iss 2 orp 0 . Figure 6(a) presents the bond angle-dependent behavior of PTHW. PTHW is useful to examine the effective change of the O-2p hole weight in the Cu-O-Cu cluster due to the hybridization with A-site orbitals. Without the hybridization, the PTHW along the c-axis is nearly constant and the weight for the AFM case is always a little larger than that for the FM case. The difference becomes nearly zero with decreasing φ 1 . By contrast, with the hybridization, the PTHW shifts up or down depending on the A-site orbitals. For thepp hybridization, the PTHW decreases monotonically, and the difference between the FM and AFM cases becomes reduced similarly to the prior case of no hybridization. For thesp hybridization, however, the weights of both the FM and AFM cases increase rapidly near the critical angle, and their magnitudes are reversed. Note that, according to the second-order perturbation theory, the weight coefficient of charge transferred states is approximately proportional to −t pd / . This implies that thesp hybridization causes the charge transfer energy of 2p hole states to be effectively lowered, whereas thepp hybridization gives rise to an opposite effect 3 .

10
In the Cu-O1-Cu interaction path depicted in figure 1(a), the displacement vector between A and O1 is along the a-axis. Then the hybridization between the A-s and O-2p x orbitals, whose lobe is directed to the a-axis, will be much stronger than others. Combining this fact with the finding of effective shift of in figure 6(a), one expects that a system in which a p x orbital has a different energy from others by x would exhibit behaviors similar to figure 5(a). Figure 6(b) shows J c for various x . Indeed, positive x produces the FM interaction of J c . Here, positive x means the reduction in for p x orbital. Thus figure 6(b) indicates that the anisotropic reduction in the charge transfer energy can stabilize the FM interaction. Moreover, the inset in figure 6(a) shows that the PTHW for positive x also increases with decreasing bond angle and their magnitudes of FM and AFM cases are reversed near the critical bond angle. It is thus clear that the effect of thesp hybridization on charge transferred states is equivalent to that of positive x . Further, we have found that the geometry of the Cu-O-Cu-A cluster is crucial to stabilize the FM interaction. As an example, we have considered an artificial Cu-O-Cu-A cluster where the A-O displacement is perpendicular to the Cu-O-Cu plane. In this case, the magnetic interaction along the c-axis is found to be almost inert against thesp hybridization. This behavior is related to the type of oxygen 2p orbital which bonds with the A-s orbital. Because the O-2p orbital that performs the strongsp hybridization is perpendicular to the Cu-O-Cu plane, its bonding with Cu 3d orbitals is too weak to give a significant contribution to the superexchange interaction. In the same context, it is verified that, if the p y orbital is shifted by y instead of p x , the Cu-O1-Cu system does not exhibit the FM stabilization. Based on these findings, one can construct a new scenario on the role of thesp hybridization. When an A-site cation is located in a position where the A-O displacement is included in the Cu-O-Cu plane and bisects the Cu-O-Cu bond angle, the strongsp hybridization can give rise to a maximal effect in modifying the magnetic interaction and stabilizing the FM interaction.
In the case of J ab , the PTHW for the FM case is always larger than that for the AFM case, and their values are nearly constant (∼ 0.1). But when J ab becomes enhanced suddenly for largẽ sp ( figure 5(b)), the PTHW also increases abruptly. This indicates that the behavior of J ab with respect to thesp hybridization is also understandable in view of the effective reduction in the charge transfer energy.
In contrast to thesp hybridization, thepp hybridization just suppresses the PTHW. One might expect that an electron in the oxygen 2p orbitals can hop into unoccupied A-p orbitals easily and so the oxygen 2p hole weight grows up. In actuality, total 2p hole weight (TTHW) of the Cu-O-Cu-A cluster increases with including thepp hybridization and becomes two to three times larger than that of the nonhybridized case. However, the TTHW increment does not bring about the PTHW increment, because an unoccupied A-p orbital can increase the O-2p hole weight by overlapping not only with the charge transferred states but also with the nontransferred states, for which no charge transfer between Cu-3d and O-2p orbitals takes place. If the O-2p hole contribution of nontransferred states prevails, the effective shift of the charge transfer energy, which is determined by the energy difference between the charge transferred and nontransferred states, would be positive. As a consequence, the enhancement of effective charge transfer energy reduces the magnitudes of J ab and J c , as seen in figures 4(c) and (d).
As mentioned earlier, the effect of thesp bonding on ACuO 3 was suggested by Iñiguez and Yildirim [8]. In their analysis, however, the role ofsp hybridization was simply restricted to the reduction of the effective pd hopping. That is, its effect suppresses the magnitude but cannot change the sign of the superexchange interaction. Hence they assumed an additional but seemingly artificial FM direct exchange interaction that is induced by the intersite exchange Coulomb interaction between two facing Cu-d orbitals or between Cu-d and O-p orbitals, and concluded that the competition between the constant FM direct exchange and the varying superexchange due to thesp hybridization would give rise to the magnetic transition of ACuO 3 systems. For the intersite exchange interaction parameters, they adopted typical values of cuprates such as Ca 2 CuO 3 and Sr 2 CuO 3 , for which the direct exchange interaction (K ) was estimated to be 10-30 meV [22]. Note, however, that Cu-O distances along the c-axis of ACuO 3 (2.06-2.09 Å) are considerably longer than those of the above conventional cuprates (1.89-1.96 Å). Then the K value of ACuO 3 is expected to be much smaller than that of conventional cuprates, and the direct exchange effect in ACuO 3 would be much weaker. In contrast, in our analysis, thesp hybridization could bring about the anisotropic change of the charge transfer energy as well as the reduction of the pd hopping. Moreover, it not only suppresses the magnitude but also reverses the sign of the superexchange interaction. Thus the artificial intersite exchange interaction is not necessary to stabilize the FM interaction in ACuO 3 . Nevertheless, it is hard at the moment to determine which scenario is more relevant in real ACuO 3 systems. There is no concrete experimental or theoretical evidence to justify physical parameters such as p and K in ACuO 3 . More precise determination of physical parameters and more information on the relation between structural change and magnetism are required. Investigation of the pressure effect on magnetic properties of ACuO 3 is desirable to test the relevant mechanism.
The proposed mechanism of superexchange interaction exploiting the anisotropic shift of the charge transfer energy can be applied to any magnetic insulator of perovskite ABO 3 type, in which the charge transfer energy is relatively smaller than the on-site Coulomb repulsion and the hopping-mediating anions bond strongly with nonmagnetic cations. ABO 3 magnetic insulators with B = Co, Ni and Cu belong to the former kind of group, whereas those with nonmagnetic cations, such as A = Se and Te, belong to the latter. Thus typical candidates are ABO 3 systems with A = Se, Te and B = Mn, Co, Ni [5]. When the A cation is replaced from Se to Te, the bond angle change of these systems is only about 5 • . But the Neel temperature (T N ) is enhanced by about 20-50%. To understand this feature, we think that not only the change of bond angle and the pd hopping but also the charge transfer energy shift proposed in the present study should be taken into account properly, as in ACuO 3 .

Conclusion
We have investigated the effects of hybridization of A-site orbitals on the magnetic interactions in highly distorted ACuO 3 (A = Se, Te). Band mixing betweens andp orbitals of the A and oxygen 2p orbital is viewed with the help of the electronic structure calculation for ACuO 3 (A = Se, Te). Using the microscopic model that includes the effect ofsp orpp hybridization, we have found that the simple Cu-O-Cu superexchange interaction model that considers only the orbital ordering and the bond angle change is inadequate to explain the transition from the AFM to the FM state in ACuO 3 , as A is substituted from Te to Se. We have provided a new mechanism where thesp hybridization gives rise to an anisotropic change in the effective charge transfer energy, which stabilizes the FM interaction along the c-axis. On the other hand, thepp hybridization weakens both magnetic exchange parameters J ab and J c , but does not induce the transition from the AFM to the FM state for the bond angles relevant for ACuO 3 .