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An effective spin–orbital Hamiltonian for Sr2FeWO6
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 132–133
An effective spin–orbital Hamiltonian for Sr2FeWO6 N.B. Perkinsa,b,*, S. Di Matteoa,c, G. Jackelid a
Laboratori Nazionali di Frascati, INFN, Casella Postale 13, Frascati I-00044, Roma, Italy b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia c INFM UdR Roma III, via della Vasca Navale 84, Roma 00146, Italy d Institut Laue Langevin, B.P. 156, Grenoble F-38042, France
Abstract We have developed a superexchange theory for the insulating double-perovskite compound Sr2 FeWO6 : The theory has been implemented in an effective spin–orbital Hamiltonian formulated in terms of spin (S ¼ 2) and orbital pseudospin (t ¼ 1) degrees of freedom of the iron ion only, while W-sites have been integrated out by means of a fourth-order perturbative expansion. We have shown that for realistic values of the model parameters the ground state is antiferromagnetic, as experimentally observed. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.b; 75.30.Et; 75.50.Ee Keywords: Double perovskite; Effective spin–orbital Hamiltonian
Sr2 FeWO6 is an antiferromagnetic insulator which belongs to the family of double perovskites, and has very low Ne! el temperature TN ¼ 16–37 K Ref. [1]. Its electronic and magnetic properties differ significantly from those of the other members of the family which are mainly half-metallic ferrimagnets [2]. A possible reason why Sr2 FeWO6 shows different behavior has been suggested by band structure analysis [3]. In this picture, the anti-bonding W(5d) states are pushed higher in energy by the stronger hybridization with oxygen porbitals. 5d-electron prefer to move away from W-site and to stay on the Fe 3d-level even at the cost of paying an extra Coulomb energy. As a result, in Sr2 FeWO6 iron is in a Fe2þ ð3d6 Þ valence state, and the large Fe–W charge-transfer gap, in the presence of strong Coulomb interaction on Fe sites, causes the insulating behavior. There are several experimental evidences [1] suggesting that Fe2þ ion is in the high spin configuration t42g e2g with S ¼ 2: As four electrons occupy the threefold degenerate t2g -levels, the ground state of Fe2þ has a threefold *Corresponding author. Laboratori Nazionali di Frascati, INFN, Casella Postale 13, Frascati I-00044, Rome, Italy. Tel.: +39-06-9403-2882; fax: +39-06-9403-2582. E-mail address: [email protected] (N.B. Perkins).
orbital degeneracy. This degeneracy can be described by pseudospin t ¼ 1: The spin S ¼ 2 and pseudospin t ¼ 1 are thus the building blocks of an effective theory for insulating Sr2 FeWO6 : In order to study the ground state and give some quantitative estimates of superexchange constants, we derive an effective spin–orbital model from the Hubbard Hamiltonian. We assume that oxygen degrees of freedom are integrated out and consider an effective hybridization between correlated Fe-3d electrons and uncorrelated 5d electrons of W. The general form of the Hamiltonian turns out to be [4] X Þ ðAÞ Heff ¼ ½Si Sj þ 6OðF ð1Þ ij þ ½Si Sj 4Oij ; ij
where ij denote the summation over both nearest neighbor (nn) and next nearest neighbor (nnn) Fe-ions of the FCC lattice. The terms OijðF ðAÞÞ are the orbital, energy-dependent, contributions to the effective Hamiltonian for FM or AFM bonds ij; respectively, which are explicitly given elsewhere [4]. The two spin-dependent terms are projectors over Si þ Sj ¼ 4 (FM coupling) and Si þ Sj ¼ 0 (AFM coupling), respectively. We factorize spin and orbital degrees of freedom and perform a
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.069
ARTICLE IN PRESS N.B. Perkins et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 132–133
mean-field type analysis. It is justified because there are two different energy scales: spins are coupled by virtual processes that involve both eg and t2g electron transfers, while only t2g electrons generate the coupling between the orbital degrees of freedom. As the eg -hopping matrix elements are much larger than those of t2g electrons, it is reasonable to average out spin degrees of freedom, first, and then for each type of magnetic ordering find an orbital configuration, that minimizes the total energy. Our analysis suggests that the ground state is mainly determined by eg superexchange, which is antiferromagnetic and overwhelms the superexchange due to the t2g hopping. Therefore the ground state is antiferromagnetic and it turns out to be of AFM-II type, consisting of ferromagnetic f1 1 1g planes coupled antiferromagnetically. Given this magnetic structure, orbitals order in such a way to further reduce the ground state energy. There are two possible types of orbitally ordered ground states, depending on the considered parameters [4]. To understand the orbital structure it is convenient to divide the FCC lattice of Fe ions, in four simple cubic sublattices. Each sublattice is antiferromagnetic and the moments are coupled only by nnn exchange J2 ; while the coupling between sublattices is due to the nn exchange J1 . If we consider only spin degrees of freedom, then the ground state energy in the AFM-II phase depends only on J2 ; because there is an equal number of nn ferro- and antiferro-bonds. Therefore the contributions from J1 to the ground state energy are canceled out. However, in the present case, due to the orbital part of the effective Hamiltonian (1), inter-sublattice couplings also contribute to the ground state energy. The antiferromagnetic superexchange for a given sublattice is maximized when t2g orbitals are ordered ferromagnetically and all equally populated. Switching on the inter-sublattice coupling does not affect the ferromagnetic-type orbital order within each sublattices. However, the type of occupied
133
orbital state changes, in order to minimize the contribution of nn energy to the ground state. We have evaluated J1 and J2 averaged over the bond directions. They are mediated by O and W diamagnetic ions, along the path Fe–O–W–O–Fe, with 90 and 180
angles between Fe and W ions, and scale as the fourth power of the transfer integrals. Fixing the ratio of eg and t2g transfer integrals to be teg =tt2g ¼ 3; tt2g ¼ 0:25 eV; interorbital Coulomb repulsion U2 ¼ 5 eV and varying charge-transfer gap D in the range 2–7 eV we obtain the following range of J2 ¼ 0:03–0:14 meV: Our analysis also suggests that J1 is antiferromagnetic as well, but around half of J2 : The transition temperature TN can be directly linked to the nnn exchange energy J2 by the formula [4] TN ¼ 2SðS þ 1ÞzJ2 =3; where z ¼ 6 is a number of next-nearest neighbors. For S ¼ 2 and the experimental values of TN C16–37 K one obtains J2 C0:06–0:15 meV: From these estimates we can conclude that the superexchange theory gives the correct order of magnitude of the experimental transition temperature. To summarize, we have developed a superexchange theory for insulating double-perovskite compounds such as Sr2 FeWO6 : We have shown that for realistic values of the model parameters the ground state is antiferromagnetic, as experimentally observed.
References [1] H. Kawanaka, I. Hase, S. Toyama, Y. Nishihara, J. Phys. Soc. Jpn. 68 (1999) 2890. [2] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, Y. Tokura, Nature (London) 395 (1998) 677. [3] Z. Fang, K. Terakura, J. Kanamori, Phys. Rev. B 63 (2001) 180407(R). [4] S. Di Matteo, G. Jackeli, N.B. Perkins, Phys. Rev. B 67 (2003) 184427.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 132–133
An effective spin–orbital Hamiltonian for Sr2FeWO6 N.B. Perkinsa,b,*, S. Di Matteoa,c, G. Jackelid a
Laboratori Nazionali di Frascati, INFN, Casella Postale 13, Frascati I-00044, Roma, Italy b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia c INFM UdR Roma III, via della Vasca Navale 84, Roma 00146, Italy d Institut Laue Langevin, B.P. 156, Grenoble F-38042, France
Abstract We have developed a superexchange theory for the insulating double-perovskite compound Sr2 FeWO6 : The theory has been implemented in an effective spin–orbital Hamiltonian formulated in terms of spin (S ¼ 2) and orbital pseudospin (t ¼ 1) degrees of freedom of the iron ion only, while W-sites have been integrated out by means of a fourth-order perturbative expansion. We have shown that for realistic values of the model parameters the ground state is antiferromagnetic, as experimentally observed. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.b; 75.30.Et; 75.50.Ee Keywords: Double perovskite; Effective spin–orbital Hamiltonian
Sr2 FeWO6 is an antiferromagnetic insulator which belongs to the family of double perovskites, and has very low Ne! el temperature TN ¼ 16–37 K Ref. [1]. Its electronic and magnetic properties differ significantly from those of the other members of the family which are mainly half-metallic ferrimagnets [2]. A possible reason why Sr2 FeWO6 shows different behavior has been suggested by band structure analysis [3]. In this picture, the anti-bonding W(5d) states are pushed higher in energy by the stronger hybridization with oxygen porbitals. 5d-electron prefer to move away from W-site and to stay on the Fe 3d-level even at the cost of paying an extra Coulomb energy. As a result, in Sr2 FeWO6 iron is in a Fe2þ ð3d6 Þ valence state, and the large Fe–W charge-transfer gap, in the presence of strong Coulomb interaction on Fe sites, causes the insulating behavior. There are several experimental evidences [1] suggesting that Fe2þ ion is in the high spin configuration t42g e2g with S ¼ 2: As four electrons occupy the threefold degenerate t2g -levels, the ground state of Fe2þ has a threefold *Corresponding author. Laboratori Nazionali di Frascati, INFN, Casella Postale 13, Frascati I-00044, Rome, Italy. Tel.: +39-06-9403-2882; fax: +39-06-9403-2582. E-mail address: [email protected] (N.B. Perkins).
orbital degeneracy. This degeneracy can be described by pseudospin t ¼ 1: The spin S ¼ 2 and pseudospin t ¼ 1 are thus the building blocks of an effective theory for insulating Sr2 FeWO6 : In order to study the ground state and give some quantitative estimates of superexchange constants, we derive an effective spin–orbital model from the Hubbard Hamiltonian. We assume that oxygen degrees of freedom are integrated out and consider an effective hybridization between correlated Fe-3d electrons and uncorrelated 5d electrons of W. The general form of the Hamiltonian turns out to be [4] X Þ ðAÞ Heff ¼ ½Si Sj þ 6OðF ð1Þ ij þ ½Si Sj 4Oij ; ij
where ij denote the summation over both nearest neighbor (nn) and next nearest neighbor (nnn) Fe-ions of the FCC lattice. The terms OijðF ðAÞÞ are the orbital, energy-dependent, contributions to the effective Hamiltonian for FM or AFM bonds ij; respectively, which are explicitly given elsewhere [4]. The two spin-dependent terms are projectors over Si þ Sj ¼ 4 (FM coupling) and Si þ Sj ¼ 0 (AFM coupling), respectively. We factorize spin and orbital degrees of freedom and perform a
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.069
ARTICLE IN PRESS N.B. Perkins et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 132–133
mean-field type analysis. It is justified because there are two different energy scales: spins are coupled by virtual processes that involve both eg and t2g electron transfers, while only t2g electrons generate the coupling between the orbital degrees of freedom. As the eg -hopping matrix elements are much larger than those of t2g electrons, it is reasonable to average out spin degrees of freedom, first, and then for each type of magnetic ordering find an orbital configuration, that minimizes the total energy. Our analysis suggests that the ground state is mainly determined by eg superexchange, which is antiferromagnetic and overwhelms the superexchange due to the t2g hopping. Therefore the ground state is antiferromagnetic and it turns out to be of AFM-II type, consisting of ferromagnetic f1 1 1g planes coupled antiferromagnetically. Given this magnetic structure, orbitals order in such a way to further reduce the ground state energy. There are two possible types of orbitally ordered ground states, depending on the considered parameters [4]. To understand the orbital structure it is convenient to divide the FCC lattice of Fe ions, in four simple cubic sublattices. Each sublattice is antiferromagnetic and the moments are coupled only by nnn exchange J2 ; while the coupling between sublattices is due to the nn exchange J1 . If we consider only spin degrees of freedom, then the ground state energy in the AFM-II phase depends only on J2 ; because there is an equal number of nn ferro- and antiferro-bonds. Therefore the contributions from J1 to the ground state energy are canceled out. However, in the present case, due to the orbital part of the effective Hamiltonian (1), inter-sublattice couplings also contribute to the ground state energy. The antiferromagnetic superexchange for a given sublattice is maximized when t2g orbitals are ordered ferromagnetically and all equally populated. Switching on the inter-sublattice coupling does not affect the ferromagnetic-type orbital order within each sublattices. However, the type of occupied
133
orbital state changes, in order to minimize the contribution of nn energy to the ground state. We have evaluated J1 and J2 averaged over the bond directions. They are mediated by O and W diamagnetic ions, along the path Fe–O–W–O–Fe, with 90 and 180
angles between Fe and W ions, and scale as the fourth power of the transfer integrals. Fixing the ratio of eg and t2g transfer integrals to be teg =tt2g ¼ 3; tt2g ¼ 0:25 eV; interorbital Coulomb repulsion U2 ¼ 5 eV and varying charge-transfer gap D in the range 2–7 eV we obtain the following range of J2 ¼ 0:03–0:14 meV: Our analysis also suggests that J1 is antiferromagnetic as well, but around half of J2 : The transition temperature TN can be directly linked to the nnn exchange energy J2 by the formula [4] TN ¼ 2SðS þ 1ÞzJ2 =3; where z ¼ 6 is a number of next-nearest neighbors. For S ¼ 2 and the experimental values of TN C16–37 K one obtains J2 C0:06–0:15 meV: From these estimates we can conclude that the superexchange theory gives the correct order of magnitude of the experimental transition temperature. To summarize, we have developed a superexchange theory for insulating double-perovskite compounds such as Sr2 FeWO6 : We have shown that for realistic values of the model parameters the ground state is antiferromagnetic, as experimentally observed.
References [1] H. Kawanaka, I. Hase, S. Toyama, Y. Nishihara, J. Phys. Soc. Jpn. 68 (1999) 2890. [2] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, Y. Tokura, Nature (London) 395 (1998) 677. [3] Z. Fang, K. Terakura, J. Kanamori, Phys. Rev. B 63 (2001) 180407(R). [4] S. Di Matteo, G. Jackeli, N.B. Perkins, Phys. Rev. B 67 (2003) 184427.