Ferromagnetic kinetic exchange interaction in magnetic insulators
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Ferromagnetic kinetic exchange interaction in magnetic insulators © Authors, 2020 Published version Huang, Zhishuo; Liu, Dan; Mansikkamäki, Akseli; Vieru, Veacheslav; Iwahara, Naoya; Chibotaru, Liviu F. Huang, Z., Liu, D., Mansikkamäki, A., Vieru, V., Iwahara, N., & Chibotaru, L. F. (2020). Ferromagnetic kinetic exchange interaction in magnetic insulators. Physical Review Research, 2(3), Article 033430. https://doi.org/10.1103/PhysRevResearch.2.033430 2020
PHYSICAL REVIEW RESEARCH 2, 033430 (2020) Ferromagnetic kinetic exchange interaction in magnetic insulators Zhishuo Huang ,1Dan Liu ,2,1Akseli Mansikkamäki ,3,4Veacheslav Vieru,5,1 Naoya Iwahara ,6,1,*and Liviu F. Chibotaru 1,† 1Theory of Nanomaterials Group, KU Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium 2Institute of Flexible Electronics, Northwestern Polytechnical University, 127 West Youyi Road, Xi’an, 710072 Shaanxi, China 3NMR Research Unit, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland 4Department of Chemistry, Nanoscience Centre, University of Jyväskylä, FI-40014 University of Jyväskylä, Finland 5Maastricht Science Programme, Faculty of Science and Engineering, Maastricht University, Paul-Henri Spaaklaan 1, 6229 EN Maastricht, The Netherlands 6Department of Chemistry, National University of Singapore, Block S8 Level 3, 3 Science Drive 3, 117543 Singapore (Received 31 December 2019; accepted 9 July 2020; published 16 September 2020) 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 Fe3+-Co3+-Fe3+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 of the ferromagnetic kinetic exchange mechanism. DOI: 10.1103/PhysRevResearch.2.033430 I. INTRODUCTION Anderson’s superexchange 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 [2–4]. This theory identifies the orbitals at which reside the unpaired (magnetic) electrons—the Anderson magnetic orbitals (AMOs)—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= 4b2/U, where Uis the electron repulsion on magnetic sites. When bis suppressed, e.g., for symmetry reasons [4,5], weaker ferromagnetic interactions of nonkinetic origin, such as potential exchange [1,6], Goodenough’s mechanism [2,3], and spin polarization (the RKKY mechanism) [7–9], become dominant. Various developments of this theory have been proposed in the last decades [6,10–16]. Moreover, the AMOs have been used in the analysis of exchange interactions derived from first-principles calculations [5,17–19]. The physics of Anderson’s model lies in the basis of the derivation of exchange *naoya.iw[email protected] †li[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. parameters through spin-unrestricted broken-symmetry density functional theory (DFT) widely employed nowadays [20–24]. The superexchange theory [1,6] has been extended to treat exchange interactions between orbitally degenerate sites [25–29], in the presence of spin-orbit coupling on the metal ions [30–37], and beyond the second-order perturbation theory after b, leading to biquadratic [6,30,38,39] and ring [40–42] exchange interactions. A different extension of the theory was proposed by Geertsma [43], Larson et al. [44], and Zaanen and Sawatzky [45] through explicit consideration of the orbitals of bridging diamagnetic atoms or 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 [46]. Another reason for this extension was the claim that Anderson’s theory would break down when the ligand-to-metal electron transfer energy becomes lower than the metal-to-metal electron transfer energy [45]. However, a detailed analysis has shown that the predictions of this extended model for the low-lying states are basically the same as of Anderson’s model when only metal-ligand electron transfer is taken into account [47]. 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 [48–50]. Despite the fact that this mechanism has been mentioned on different occasions [48,51–55], 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 2643-1564/2020/2(3)/033430(12) 033430-1 Published by the American Physical Society
ZHISHUO HUANG et al. PHYSICAL REVIEW RESEARCH 2, 033430 (2020) FIG. 1. Basic three-site model for the system consisting of two paramagnetic (1,2) and a bridging diamagnetic (d) sites. The parameters correspond to Eq. (1). 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, the kinetic ferromagnetic contribution is crucial for the annihilation of the antiferromagnetic superexchange. II. BASIC THREE-SITE MODEL A. Model Hamiltonian In a first step, we derive the AMOs as minimizing the electron repulsion between magnetic electrons in a spinrestricted 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 (LBOs) mainly reside at the diamagnetic atom or group bridging the neighbor paramagnetic sites. Extracting these orbitals from the AMOs via an orthogonal transformation, we end up with localized magnetic orbitals (LMOs), 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 or group. We first consider the simplest model involving only two LMOs and one LBO (Fig. 1), ˆ H= σ=↑,↓ [ˆndσ+tMM(ˆa† 1σˆa2σ+ˆa† 2σˆa1σ) +tMD(ˆa† 1σˆadσ+ˆa† dσˆa1σ)+tDM(ˆa† 2σˆadσ+ˆa† dσˆa2σ)] +UM(ˆn1↑ˆn1↓+ˆn2↑ˆn2↓)+UDˆnd↑ˆnd↓.(1) Here, 1,2 and dindicate the paramagnetic and the diamagnetic sites, respectively, σ(=↑,↓) is electron spin projection, ˆa† iσ (ˆaiσ) is the electron creation (annihilation) operator in the localized orbital on the sites i(=1,2,d) with σ,ˆniσ=ˆa† iσˆaiσ, tMD/DM and tMM are the corresponding electron transfer parameters, >0 is the gap between the diamagnetic and paramagnetic orbital levels, and UMand UDare on-site Coulomb repulsion parameters within the LMO and LBO, respectively. For symmetric magnetic sites considered below, the following relations hold: tMD =tDM or tMD =−tDM. 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 −+ 2UD[50], remaining always positive in magnetic insulators. For tMM =0, the Hamiltonian (1) reduces to the earlier considered 3-orbital model [43–45]. 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]. B. Ferromagnetic kinetic exchange interaction The calculated energy spectrum of model (1)isshown in Fig. 2(a). One can see that the system exhibits strong ferromagnetism for relatively large values of tMM, further enhanced for small [Fig. 2(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 the ferromagnetism, we consider |tMD|,|tMM| UM,UD,||, and obtain in the fourth order of perturbation theory the expression for the exchange parameter Jfor the spin-1/2 Heisenberg model, ˆ Hex =Jˆ s1·ˆ s2.(2) The ground energies for the ferroand antiferromagnetic states are calculated as EF=−2t2 MD +4t4 MD 3−2tMDtDMtMM 2,(3) EAF =−2t2 MD +4t4 MD 3−4 UMtMM −tMDtDM 2 −8t2 MDt2 DM 2(UD+2)+2tMDtDMtMM 2+16t4 MM U3 M ,(4) respectively [56]. The energy gap between them, EF−EAF, corresponds to the exchange parameter J(2): J=4 UMtMM −tMDtDM 2 +8t2 MDt2 DM 2(UD+2) −4tMDtDMtMM 2−16t4 MM U3 M .(5) The first and second terms in Eq. (5), K1 and K2, are always antiferromagnetic (>0), and the fourth term (K4) is ferromagnetic (<0). The third term (K3) becomes ferromagnetic for tMDtDMtMM >0 and is antiferromagnetic otherwise. According to the order of the perturbation, the first and the third terms are dominant, and the nature of Jis mainly determined by their competition. The ferromagnetic contribution K3 originates from cyclic electron transfer processes avoiding double occupation of any of three orbitals [Fig. 2(c)]. It can be called the ferromagnetic kinetic exchange interaction. Note that the contribution of this mechanism to the energy of the ferromagnetic state, −2tMDtDMtMM/2[the factor 2 is due to a cyclic process, 033430-2
FERROMAGNETIC KINETIC EXCHANGE INTERACTION IN … PHYSICAL REVIEW RESEARCH 2, 033430 (2020) FIG. 2. (a) Energy level diagram of the three-site model (1)fortMD =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)]. Jis for Eq. (2). (c) Third-order process responsible for ferromagnetic kinetic exchange contribution. similar to Fig. 2(c) but in anticlockwise sense], is opposite to the case of antiferromagnetic state, because of the sign change in the latter [see the third and fifth terms in Eqs. (3) and (4), respectively]. C. Ferromagnetism within Anderson’s approach It should be noted that the ferromagnetic kinetic exchange contribution (K3) is not fully captured by Anderson’s approach [1]. Projecting the basic three-site model, Eq. (1), on the space of two AMOs, we obtain a tight-binding model (see for derivation Appendix B): ˆ H=EHF +ˆ HA+ σ=↑,↓ b(ˆ N1,−σ+ˆ N2,−σ) ×(ˆ A† 1σˆ A2σ+ˆ A† 2σˆ A1σ)+U[ˆ N1↑ˆ N2↓+ˆ N2↑ˆ N1↓ +(ˆ A† 1↑ˆ A2↑+ˆ A† 2↑ˆ A1↑)( ˆ A† 1↓ˆ A2↓+ˆ A† 2↓ˆ A1↓)],(6) ˆ HA= σ τb(ˆ A† 1σˆ A2σ+ˆ A† 2σˆ A1σ)+ i=1,2 Uˆ Ni↑ˆ Ni↓,(7) where EHF is the restricted open-shell Hartree-Fock energy for the ferromagnetic state, ˆ Aiσ(ˆ A† iσ) is the electron annihilation (creation) operator in the AMO centered at site i, ˆ Niσ=ˆ A† iσˆ Aiσ,τ=tMD/tDM,bis the effective electron transfer parameter between the two AMOs (B13), Uis the energy of electron promotion between AMOs (B17), Uis the intersite potential exchange interaction parameter (B18), and b is the Coulomb repulsion assisted transfer parameter (B19). Although the last term in Eq. (6) is different from the standard potential exchange interaction (17), we use the name because the resulting spin-dependent shift of energy levels resembles it. The second Hamiltonian (7) is regarded as the original Anderson tight-binding model (or Hubbard model) [1]. As is well known, the exchange parameter Jfrom the model (7) gives the antiferromagnetic contribution. The extended model (6) retains all the interaction terms appearing in the approach based on the AMOs, and hence describes the magnetic properties more accurately (see for detailed analysis of Eq. (1) without next-nearest-neighbor transfer, tMM =0, Ref. [47]). The calculated Jwithin this approach (dashed line) is shown in Fig. 3in comparison with the exact treatment (solid line). Indeed, a finite tMM merely modifies the effective transfer parameter bbetween the AMOs, i.e., the antiferromagnetic kinetic exchange: For tMM >0, the ferromagnetic kinetic exchange contribution [K3 in Eq. (5)] is only partly recovered within the extended Anderson theory through mainly the potential exchange contribution as shown below. However, it is much underestimated compared to the exact treatment (Fig. 3). For further insight, the exchange interaction parameter is calculated within perturbation theory in the case of |tMD|,|tMM|U. The energy level for the ferromagnetic state is EHF, and its expression becomes the same as Eq. (3) (see for calculations Appendix B). This coincidence is a consequence that the ferromagnetic ground state (with maximum spin projection) is described exactly within single Slater determinant. On the other hand, the antiferromagnetic ground state energy FIG. 3. Exchange parameters calculated by exact diagonalization (solid) and within Anderson’s model (dashed). The red, green, blue, and purple lines indicate /UM=2,1.5,1,0.75, respectively. tMD/UM=tDM/UM=0.2andUD/UM=1 are used in all calculations. 033430-3
ZHISHUO HUANG et al. PHYSICAL REVIEW RESEARCH 2, 033430 (2020) (B23) differs much from Eq. (4). The leading terms of the exchange parameter Jare J=−2U+4(b+b)2 U−U−16(b+b)4 (U−U)3.(8) The second term of Eq. (8) gives the strong antiferromagnetic contribution. The ferromagnetic contribution arises from the potential exchange term (the first term) and a contribution of K4 type (the third term). D. Condition for strong ferromagnetism The necessary condition for a dominant ferromagnetic kinetic contribution is the right sign and a large value of tMM. The existence of non-negligible tMM is expected for LMOs extending on neighbor paramagnetic sites. This occurs when the relevant bands (molecular orbitals) involve 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 is a family of superconducting cuprates, in which the low-energy states are described by a three-orbital model for the CuO2 plane [57], involving the almost net atomic 3dx2−y2orbital on Cu and 2px(2py) orbitals on O. The latter lead to small tMM and negligible kinetic ferromagnetic exchange contribution (tMDtDMtMM >0), which is in accord with a very large antiferromagnetic exchange interaction in cuprates [58]. On the other hand, the sign of tMM is unambiguously determined by the type of the localized orbitals (e.g., d,f), which is not influenced by the hybridization of the Wannier orbitals. According to Eq. (5), the tMM of a right sign not only gives rise to a ferromagnetic kinetic contribution but concomitantly reduces the antiferromagnetic one. However, the largest ferromagnetic Jis not achieved at a tMM quenching K1 but at a larger value, tMM ≈(tMDtDM/)(1 +UM/2). The expression in Eq. (5) then becomes Jmax ferro ≈−4(tMDtDM/)2 UD UD+2+UM 4.(9) Counterintuitively, the ferromagnetic coupling increases linearly with UM. Besides, it rises very fast with diminishing , a feature also confirmed by nonperturbative treatment [Fig. 2(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. E. Switching of ferromagnetic kinetic mechanism A similar treatment shows that adding one electron or hole to the empty or doubly occupied LBO turns the initially dominant ferromagnetic kinetic interaction into an antiferromagnetic one of comparable strength (see Fig. 4). The Heisenberg model for the electron or hole doped system is FIG. 4. J2diagram. The parameters are the same as in Fig. 2(b). The dashed lines outline the ferromagnetic domain in the case of diamagnetic bridge [Fig. 2(b)]. written as [59] ˆ H=J1(ˆ s1·ˆ sd+ˆ s1·ˆ sd)+J2ˆ s1·ˆ s2.(10) Within the perturbation theory (|tMM|,|tMD/DM| UM,UD,UM−), the exchange parameters J1and J2are calculated as J1=2t2 MD UM−+τtMM +2t2 MD UD+−τtMM ,(11) J2=4t2 MM UM+2tMDtDMtMM (UM−)2+4tMDtDMtMM UM(UM−) −2tMDtDMtMM (UD+)2−4tMDtDMtMM UM(UD+).(12) J1is antiferromagnetic because is smaller than UM:otherwise the added electron or hole occupies the paramagnetic sites rather than the diamagnetic site. The first term of J2 is antiferromagnetic and the remaining terms become both antiferromagnetic and ferromagnetic depending on the sign of tMDtDMtMM. When tMDtDMtMM >0, the last two terms of J2(12) become ferromagnetic as K3 (5), while the ferromagnetism is quenched due to the following reasons. First, contrary to the undoped system (3), the ferromagnetic state is not stabilized by hybridization because the cyclic electron transfer processes [Fig. 2(c)] are forbidden by Pauli’s exclusion principle. Second, the denominators of the ferromagnetic terms of J2tend to be large in comparison with the K3 term, resulting in weaker ferromagnetic contribution than the K3. Finally, these contributions are canceled by the antiferromagnetic contributions (the second and the third terms as well as the first term of J2). In particular, the second and the third terms of J2are large because of the partial cancellation of UMand , which gradually increases as UM−decreases (/UMincreases) as seen in our numerical analysis (Fig. 4). Therefore, the kinetic ferromagnetic mechanism for nextnearest-neighbor exchange pairs is quenched by adding an electron or hole to the LBO. On the other hand, in the case of tMDtDMtMM <0, the second and the third terms in J2(12) are ferromagnetic. 033430-4
FERROMAGNETIC KINETIC EXCHANGE INTERACTION IN … PHYSICAL REVIEW RESEARCH 2, 033430 (2020) These ferromagnetic contributions become comparable to the strongest antiferromagnetic contribution (the first term) when tMM is smaller than tMDtDM/(UM−). Indeed, our numerical calculations show that J2becomes very weak antiferromagnetic for −tMDtDM/(UM−)tMM <0 (Fig. 4). III. RELEVANCE OF THE FERROMAGNETIC KINETIC EXCHANGE MECHANISM TO MAGNETIC MATERIALS A. First-principles-based approach The ferromagnetic kinetic exchange mechanism is further investigated in several magnetic materials with diamagnetic metal bridges. Examples considered include complexes Fe3+-Co3+-Fe3+[60] and Cu2+-Cr6+-Cu2+and Cu2+-Mo6+-Cu2+[61], and a quasi-one-dimensional Cu chain in La4Ba2Cu2O10 [62,63]. In these systems, Fe3+(d5) and Cu2+(d9) ions are the magnetic ions with s=1/2, while Co3+(d6), Cr6+(d0), Mo6+(d0), and La3+belong to diamagnetic bridges. Despite the large distance between paramagnetic centers, they all (except Cu-Mo-Cu) display ferromagnetic exchange interaction. In order to achieve a realistic description of exchange contributions, the results of first-principles calculations were mapped into an extended three-site model which, contrary to the basic model in Eq. (1), includes all relevant LBOs on the diamagnetic bridging site and the Coulomb and potential exchange interactions between the LMOs and LBOs: ˆ H=ˆ H0+ˆ Ht+ˆ HCoul +ˆ HPE,(13) ˆ H0= d σ=↑↓ dˆndσ,(14) ˆ Ht= i=1,2 d σ=↑↓ (tMd ˆa† iσˆadσ+tdM ˆa† dσˆaiσ) + σ=↑↓ tMM(ˆa† 1σˆa2σ+ˆa† 2σˆa1σ) + d<d σ=↑↓ tdd(ˆa† dσˆadσ+ˆa† dσˆadσ),(15) ˆ HCoul = i=1,2 UMˆni↑ˆni↓+ d Udˆnd↑ˆnd↓+VMM ˆn1ˆn2 + i=1,2 d VMd ˆniˆnd+ d<d Vddˆndˆnd,(16) ˆ HPE = i=1,2 d σσ=↑,↓ JMd ˆa† iσˆa† dσˆaiσˆadσ + σσ=↑,↓ JMM ˆa† 1σˆa† 2σˆa1σˆa2σ.(17) Here dindicates the LBO on the bridging diamagnetic site, dis the energy gap between the LBO dand the LMO levels, tid are the electron transfer parameter between the corresponding orbitals, VMd the intersite Coulomb repulsion, Vddis the Coulomb repulsion between different LBOs, JMM and JMd the potential exchange parameters, and ˆnj=σ=↑,↓ˆnjσ(j= 1,2,d). As in the case of the basic three-site model, tMd and tdM fulfill either tMd =tdM or tMd =−tdM. The energy eigenstates of the Hamiltonian are derived in two ways: direct numerical diagonalization and perturbation theory. In the former case, the Hamiltonian matrix for the three center complexes or fragment is built using all electron configurations constructed with LMOs and LBOs as the basis and DFT parameters, and then numerically diagonalized. Within the fourth-order perturbation theory, the contributions to the Heisenberg exchange parameter are calculated as J=JK1 +JK2 +JK3 +JK4 +JPE,(18) JK1 =4 UM−VMM tMM − d tMdtdM d−VMM +VMd 2 ,(19) JK2 = dd 2tMdtdMtMdtdM d+d−VMM +Vdd ×1 d−VMM +VMd +1 d−VMM +VMd2 + d<d 4tddtMM(tMdtdM+tMdtdM) (d−VMM +VMd )(d−VMM +VMd) ×2 UM−VMM + d<d 4tddtMM(tMdtdM+tMdtdM) (d−VMM +VMd )(d−VMM +VMd) ×1 d−VMM +VMd +1 d−VMM +VMd,(20) JK3 =− d 4tMdtdMtMM (d−VMM +VMd )2,(21) JK4 =− 16t4 MM (UM−VMM )3,(22) JPE =−2JMM.(23) In Eq. (20), Vdd stands for Ud. The kinetic contributions (19)–(22) for the extended model correspond to the four terms in J(5) of the basic model (1), and are further denoted as K1–K4. In the last term of Eq. (20), there are many cross terms involving pairs of LBOs, tMdtdMtMdtdM. Since tMdtdM can be both positive (tMd =tdM) and negative (tMd =−tdM), this term becomes equally positive (antiferromagnetic) and negative (ferromagnetic). Electronic band structure calculations for all materials were performed on their experimental structure [60,61,65] with the revised Perdew-Burke-Ernzerhof (PBE) functional [66] and optimized norm-conserving Vanderbilt pseudopotentials [67]. Using the Kohn-Sham orbitals, maximally localized Wannier functions [68] and one-particle interaction parameters, tand , were derived. Screened intra- (UM/D) and intersite Coulomb and potential exchange parameters were calculated within the constrained random phase approximation [69]. Quantum ESPRESSO [70,71] and RESPACK [72–77] were used for electronic structure calculations, and pseudopotentials were taken from PSEUDO DOJO [78], and VESTA [79] for plotting the orbitals. See for details Appendix C. The obtained parameters of the extended three-site model for the four compounds are listed in Table I. The exchange 033430-5
ZHISHUO HUANG et al. PHYSICAL REVIEW RESEARCH 2, 033430 (2020) TABLE I. Microscopic parameters of the extended three-site model (eV). Fe-Co-FeaCu-Cr-Cu Cu-Mo-Cu La4Ba2Cu2O10 tMD 0.290 −0.499 −0.554 0.748 tMM 0.193 0.084 0.040 0.013 1.048b3.357 4.774 6.787 c0.595 3.246 3.685 UM2.912 4.848 4.482 3.178 UD2.859 3.786 2.789 1.563 VMD 1.672 2.463 2.109 0.681 VMM 1.347 1.474 1.380 0.441 JMDd0.0106 0.0084 0.0091 JMMd0.0025 0.0013 0.0005 8.4×10−5 atand are given in hole picture. bDerived from absorption spectrum in solution. cThe value allowing us to reproduce the experimental J. dScaled down following Ref. [64]. parameter Jof the spin-1/2 Heisenberg model (2) was determined to reproduce the energy gap between the ground highand low-spin term energies obtained by numerical diagonalization of the corresponding Hamiltonian, Eqs. (13)–(17). The kinetic contributions to the exchange parameters were calculated using the corresponding expressions, Eqs. (19)– (22). 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 II). B. Fe-Co-Fe complex The 3dorbitals of each metal ion split into eg(ein C3 group) and t2g[a(dz2) and e] orbitals because of the strong octahedral-like C3ligand field. In both Fe and Co, the t2g orbitals have much lower energy than the egorbitals and are filled by 5 and 6 electrons, respectively. The half-filled a orbital on the Fe site is the LMO, which is consistent with the calculated spin density. Due to the C3symmetry, only the a orbitals on Fe and Co sites are relevant to kinetic exchange interaction, while Goodenough’s mechanism is ruled out. Below, we use the hole picture [50]. The calculated LMOs and LBO [Figs. 5(a) and 5(b)]are strongly hybridized with the 3porbitals of the sulfur atoms between the metal ions, which makes tMM non-negligible. Figure 5(c) shows the Jdiagram as a function of parameters UM and (the less reliable among the DFT-extracted parameters) at DFT-calculated values of other parameters (Table I). TABLE II. Jand its kinetic (K1–K4) and potential exchange (PE) contributions (meV). System JK1 K2 K3 K4 PEa Fe-Co-Fe −10.4b27.1 23.7 −75.1 −6.2 −5.1 Cu-Cr-Cu −3.62b0.75 3.14 −4.67 −0.02 −2.58 Cu-Mo-Cu 1.26b1.14 4.45 −2.56 0.00 −0.99 Cu-chain −0.65 0.49 0.02 −0.27 0.00 −0.17 aFirst-principles JPE is scaled down following Ref. [64]. bwas chosen to reproduce the experimental J. FIG. 5. LMO on one Fe (a) and LBO on Co (b) sites in the FeCo-Fe complex (only core ligand atoms are shown). The LMO on the other Fe has the same phase as (a). Brown, blue, yellow, and gray balls stand for Fe, Co, S, and N, respectively. (c) Exchange parameter diagram. The solid line corresponds to J=0 and the dashed line to the experimental J. The Jdiagram shows the presence of ferromagnetism for a wide range of the parameters. To elucidate the realistic contributions to J,=0.60 eV was taken to reproduce its experimental value with the theoretical value of UM=2.86 eV. The value of matches the estimation 1.05 eV from absorption spectra in solution [60]. Table II shows that the ferromagnetic kinetic exchange (K3) is clearly dominant due to a relatively large value of tMM (Table I). The contributions K1 and K2 are similar in magnitude because of an efficient cancellation of tMDtDM/ by tMM in the former. Thus the observed very large ferromagnetic coupling (−10 meV) in this complex [60] is confirmed to be entirely due to the ferromagnetic kinetic exchange mechanism. C. Cu-Cr-Cu and Cu-Mo-Cu complexes In tetragonal and tetrahedral environments, 3dx2−y2and 3/4dzx orbitals become LMO and LBO on Cu and Cr/Mo, respectively [Figs. 6(a) and 6(b)], which agrees with the calculated spin density. Jdiagrams show that Cu-Cr-Cu and FIG. 6. LMO on Cu site (a) and LBO on Cr site (b) in the CuCr-Cu complex, exchange parameter diagrams for Cu-Cr-Cu (c) and Cu-Mo-Cu (d) complexes. The phase of the LMO at the other Cu site is opposite to (a). The blue, green, red, light gray, dark brown, and white balls are Cu, Cr, O, N, C, and H, respectively. The Cu-Cu axis corresponds to the xaxis and zis the out-of-plane axis. The meaning of the lines in (c) and (d) is the same as in Fig. 5. 033430-6
FERROMAGNETIC KINETIC EXCHANGE INTERACTION IN … PHYSICAL REVIEW RESEARCH 2, 033430 (2020) (a) (d) (b) (c) FIG. 7. LMO on Cu (a) and two LBO on the bridging La ions giving the strongest K3 contribution [(b) and (c)]. The phase of the LMO at the other Cu site is the same as (a). The blue, green, and red balls correspond to Cu, La, and O, respectively. (d) The contributions of individual LBOs to K3. The first two contributions correspond to LBO in (b) and (c), respectively. The red line indicates the total K3 contribution from all LBOs. Cu-Mo-Cu complexes become ferroand antiferromagnetic, respectively, for realistic and UM[Figs. 6(c) and 6(d)]. In both complexes, due to a partial cancellation of tMM and tMDtDM/, the effective transfer parameter between the two LMOs, tMM −tMDtDM/, is reduced and hence the K1 contribution becomes small. The tMM in Cu-Cr-Cu is larger than in Cu-Mo-Cu, and the same for the K3 contribution. Consequently, the former compound is ferromagnetic and the latter antiferromagnetic. D. Quasi-1D Cu chain The origin of ferromagnetism in La4Ba2Cu2O10 was debated in the past [52,64]. In this system, the magnetic orbitals are of 3dzx type (b2g) of Cu due to the tetragonal-like D2h ligand field [Fig. 7(a)] and the bridging orbitals are the empty orbitals of La, Ba, and O, where the caxis is taken as the z axis and the plane of the Cu chain zx. Because of the symmetry, Goodenough’s mechanism is irrelevant: the irreducible representations of the other 3dorbitals differ from b2g, and therefore the electron transfers between the 3dzx orbital and the other types of 3dorbitals are forbidden. With first-principles parameters (the parameters related to the 5dzx LBO are shown in Table I), we obtained J= −0.65 meV close to the experimental value (−0.4meV[63]). Remarkably, the first contribution in K2 (20), which is like theK2inEq.(5), is now ferromagnetic (−0.26) and of similar magnitude as K3 [the second (similar to K1) and the third (resembling K3) contributions are 0.18 and 0.09, respectively]. The first term of K2 <0 became possible due to numerous loop terms involving two different LBOs [the third term in Eq. (20)] which can be negative when tMD =tDM for one LBO and tMD =−tDM for the other. For the same reason both ferroand antiferromagnetic contributions for different LBOs are present in Eq. (21) reducing the total K3 contribution [Fig. 7(d)]. Among the latter, the contributions via the 5dzx and the 4 fz(x2−y2)of in-plane La ions [Figs. 7(b) and 7(c)] are dominant. Thus two kinetic ferromagnetic exchange mechanisms, K2 and K3, make together a dominant contribution rendering the resulting exchange interaction ferromagnetic. The potential exchange interaction between Anderson’s magnetic orbitals was attributed to the origin of ferromagnetism based on Anderson’s original approach [64] (see for further discussion Appendix D), while the present analysis shows that the ferromagnetic kinetic contribution which is missing in Anderson’s model (7) is more important. E. Fingerprint of ferromagnetic kinetic mechanism There is further evidence of the dominant contribution of the ferromagnetic kinetic exchange mechanism in the Fe complex and the Cu chain. As described in Sec. II E,this mechanism is quenched when replacing the bridging metal ion with a magnetic one. Such behavior was observed in a series of trinuclear isostructural Fe complexes with various electronic populations of the central metal ion [60] and between Cu ions in La4Ba2Cu2O10 under the substitution of the diamagnetic La3+by the paramagnetic Nd3+[80,81]. IV. 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 the reduced orbital gap between magnetic and bridging orbitals, pointing to materials with strong metalligand covalency. We have investigated the relevance of the 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-CoFe complex the observed very large ferromagnetic coupling is entirely due to a strong ferromagnetic kinetic contribution. We also discovered a fingerprint of the ferromagnetic kinetic mechanism: a switching of the ferromagnetism to antiferromagnetism by substituting the diamagnetic bridging site with magnetic one. The phenomenology is observed in two series of systems, supporting the importance of the mechanism in magnetic materials. The obtained results call for the reconsideration of the origin of ferromagnetism and weak antiferromagnetism in insulating magnetic materials and complexes. 033430-7
ZHISHUO HUANG et al. PHYSICAL REVIEW RESEARCH 2, 033430 (2020) ACKNOWLEDGMENTS Z.H. and D.L. were supported by the China Scholarship Council. A.M. acknowledges funding provided by the Magnus Ehrnrooth Foundation. V.V. received support as a postdoctoral fellow of the Research Foundation, Flanders (FWO). N.I. was partly supported the GOA program of KU Leuven and Scientific Research Grant No. R-143-000-A80-114 of the National University of Singapore. The computational resources were provided by the VSC (Flemish Supercomputer Center). APPENDIX A: A DERIVATION OF J We show an alternative derivation of the exchange parameter (5). We introduce ferromagnetic (upper) and antiferromagnetic (lower) configurations, |± ij= 1 √2(ˆa† i↑ˆa† j↓±ˆa† i↓ˆa† j↑)|0,(A1) where i,j=1,2,d, and |0is the vacuum state. The lowest energy configurations |± 12slightly hybridize with |± ex= 1 √2(τ|± 1d+|± d2), due to the electron transfer interaction between the magnetic and diamagnetic sites, where τ= tDM/tMD. Taking the electron transfer interaction as the perturbation and the rest in Eq. (1) as the unperturbed Hamiltonian, the ground states are calculated within second-order perturbation theory as |±=|± 12− √2tMD ∓τtMM |± ex+···,(A2) where ∓τtMM is the energy of |± ex, and the terms which are not directly relevant to the K3 term (5) are not explicitly written. The energies with respect to |±(A2) are calculated as E±=E± 0−2t2 MD ∓τtMM (A3) ≈E± 0−2t2 MD ±2tMDtDMtMM 2.(A4) The last term of Eq. (A4) corresponds to the K3 contribution, and E± 0contains all the other contributions. The ferromagnetic contribution arises by the spindependent covalency between paramagnetic centers. |± exis transformed as |± ex=1 2[(τˆa† 1↑∓ˆa† 2↑)ˆa† d↓±(τˆa† 1↓∓ˆa† 2↓)ˆa† d↑]|0.(A5) Note that the molecular orbital states, (τˆa† 1σ∓ˆa† 2σ)/√2, depend on the total spin (triplet or singlet), and consequently, their orbital energy levels (∓τtMM) too. A mathematical discussion on the relation between the symmetry of the wave function and the ferromagnetic ground state is found in Ref. [82]. APPENDIX B: ANDERSON’S MODEL The tight-binding model (7) is derived as follows [1]: (1) calculation of the molecular orbitals for the high-spin state within restricted open-shell Hartree-Fock calculations; (2) transformation of the magnetic molecular orbitals (a symmetric and an antisymmetric half-filled molecular orbital in the present case) into the localized orbitals (AMOs). The other molecular orbitals are ignored. (3) Transformation of the Hamiltonian in the space of the AMOs. From the two LMOs and one LBO in the basic three-site model, two symmetric (S,S) and one antisymmetric (A) molecular orbitals are constructed as ˆc† Sσ=cos θ √2(ˆa† 1σ+τˆa† 2σ)+sin θˆa† dσ,(B1) ˆc† Sσ=−sin θ √2(ˆa† 1σ+τˆa† 2σ)+cos θˆa† dσ,(B2) ˆc† Aσ=1 √2(ˆa† 1σ−τˆa† 2σ),(B3) where τ=tDM/tMD. We assume cos θ>|sin θ|and the orbital energy for the state Sis lower than that for the state S. The angle θis determined so that the high-spin-state energy becomes the minimum within the restricted open-shell Hartree-Fock method. Anderson’s magnetic orbitals are defined by using partially filled orbitals: ˆ A† 1σ=1 √2(ˆc† Sσ+ˆc† Aσ),(B4) ˆ A† 2σ=1 √2(ˆc† Sσ−ˆc† Aσ).(B5) In Anderson’s approach, the other orbital ˆcSσis omitted. Thus, within this approximation, the atomic orbitals are expressed as ˆa† 1σ=cos θ+1 2 ˆ A† 1σ+cos θ−1 2 ˆ A† 2σ,(B6) ˆa† 2σ=τcos θ−1 2 ˆ A† 1σ+τcos θ+1 2 ˆ A† 2σ,(B7) ˆa† dσ=sin θ √2(ˆ A† 1σ+ˆ A† 2σ).(B8) Substituting Eqs. (B6)–(B8) into the single-electron part ˆ H1of Eq. (1), ˆ H1=EHF + σ τb(ˆ A† 1σˆ A2σ+ˆ A† 2σˆ A1σ),(B9) where EHF is defined by EHF =−τtMM 2(1 −cos 2θ)+√2tMD sin 2θ =−τtMM 2−Rcos(2θ+α),(B10) Rand αare R=−τtMM 22 +√2tMD2 ,(B11) cos α=−τtMM 2R,sin α=√2tMD R,(B12) respectively, and bis b=tMM +τEHF 2.(B13) 033430-8