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Spin wave theory of ferrimagnetic double perovskites
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 501–502
Spin wave theory of ferrimagnetic double perovskites G. Jackeli* Institut Laue Langevin 6, rue Jules Horawitz, Theory Group, B. P. 156, Grenoble F-38042, France
Abstract We present a theoretical study of magnetic properties of metallic double perovskite ferrimagnets such as Sr2 FeMoO6 and Sr2 FeReO6 : The analysis is based on the Kondo-type Hamiltonian in which charge carriers are constrained to be antiparallel to Fe local moments with spin S: The spectrum of spin wave excitations is derived based on the model Hamiltonian within the 1=S expansion. The ground state phase diagram as a function of carrier density is also discussed. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Lp; 75.30.Ds Keywords: Double perovskites; Spin wave theory
The recent discovery of the room temperature magnetoresistance in double perovskite compounds [1] has generated intense interest in these materials. The metallic ferrimagnetic (FiM) compounds such as Sr2 FeMoO6 and Sr2 FeReO6 are of particular recent interest, because both have high magnetic transition temperature (Tc ¼ 420; and 400 K; respectively) and their electronic structure has been suggested to be halfmetallic. In the double perovskite structure Fe and Mo/Re ions form two face-centered cubic sublattices and each pair of nearest-neighbor lattice sites is occupied by two different ions. In the ionic picture, Fe is in the 3+ valence state and has five d-electrons, which are strongly coupled by Hund’s rule in high spin state S ¼ 52 and are localized. The Mo is in 5+ valence state and has one t2g electron in 4d-shell. (The Re-based systems have one more charge carrier per formula unit.) The charge carriers (electrons form non-magnetic ions Mo/Re) and local moments of Fe ions are the building blocks of low energy theory for these compounds. Moreover, charge carriers are constrained to be antiparallel to the local moments due to the Pauli principle. As at Fe ion spin up states (with respect to the direction of the Fe moment) are already all occupied only an electron with spin down *Tel.: + 33-476-20-73-72; fax: + 33-476-88-24-16. E-mail address: [email protected] (G. Jackeli).
can hop to iron site. This picture can be described within the Kondo-type Hamiltonian in which core and itinerant spins are coupled by the infinite local antiferromagnetic (AFM) exchange J-N: This coupling is invoked to project out unphysical states and fulfill the constraint imposed by the exclusion principle.PThe model Hamiltonian can be written as H ¼ a Ha ; where a ¼ xy; xy; xz refers to three degenerate t2g ðdxy ; dyz ; dyz Þ orbital states. In the cubic lattice the t2g -transfer matrix is diagonal in the orbital space and is non-zero only in the corresponding plane. Each term Ha corresponds to a given orbital state and acts in the corresponding plane. All three terms have the same form given by X w X X H ¼ t ½dis d%js þ h:c: þ Dni Dn% j /ijSs
þ
X
J½Si si Ani ;
i
j
ð1Þ
i
where orbital index is omitted for simplicity. The first term describes an electron hopping between nearestneighbor Fe and Mo/Re ions, labeled by i and j; respectively. The operators dis ðni Þ and d%js ðn% j Þ correspond to Fe and Mo/Re sublattices, respectively, 2D ¼ EðFe d6 ; B0 d0 Þ EðFe d5 ; B0 d1 Þ is a charge transfer gap ðB0 ¼ Mo=ReÞ; and Si (si ) stands for the localized (itinerant) spin. The last constant term, where A ¼ ðS þ 1Þ=2; is chosen so that the exchange energy vanishes for a state with total spin Stot ¼ 2:
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1016
ARTICLE IN PRESS 502
G. Jackeli / Journal of Magnetism and Magnetic Materials 272–276 (2004) 501–502
Considering the ferromagnetic arrangement of local moments, one can introduce Fermion operators that generate eigenstates of the total spin at site i; i.e. operators with spin quantization axis directed parallel to the local moment. In these bases exchange term is diagonal and can be treated rigorously. The deviation of the local moments from the averaged value can be treated by introducing magnon operators and the controlled 1=S expansion of the model Hamiltonian can be developed (see Ref. [2]). As a result, one finds two branches of the spin excitations, similarly to the case of localized ferrimagnets. They describe gapless Goldstone mode and gapped optical mode. In addition to these modes, there exist Stoner continuum in the metallic systems. The dispersion law of the gapless branch at the quasi-classical level is Heisenberg-like oq ¼ 16SJ1 ½1 g1q þ 16SJ2 ½1 g2q ; t2 X g% 1ð2Þ J1ð2Þ ¼ nk ; 16S2 N k Ek
ð2Þ
where J1 and J2 are nearest and next-nearest neighbor exchange couplings, respectively, and g1ð2Þq is nearest (next-nearest) neighbor harmonic, nk is a Fermi distribution function and g% 1 ¼ 2 cos kx cos ky ; g% 2 ¼ cos 2kx þ cos 2ky : At a fixed carrier density n the spin wave stiffness ½D ¼ 16SðJ1 þ J2 Þ scales with the carrier bandwidth: DBt and DBt2 =D for tbD and t5D; respectively. As a function of density spin stiffness exhibits strong nonmonotonic behavior. It shows a maximum at some optimal filling nopt (nopt C1 for D ¼ 0) and then drops to
zero at some critical density nc (nc C1:9 for D ¼ 0). The highest value of spin stiffness is thus realized when the energy level of Fe d6 configuration is at resonance with B0 d-level and there is one charge carrier per unit cell. This agrees with the density dependence of Tc obtained in Ref. [3]. At n > nc the spin wave stiffness changes sign indicating the instability of FiM ordering. For n ¼ nc low energy spin wave spectrum becomes flat and vanishes identically in f1 1 1g direction. No special soft mode is singled out at this order. We have analyzed the quantum corrections to the linear theory in order to identify the symmetry of new magnetic order. The letter is found to correspond to a layered antiferromagnetic arrangement of local moments (so called AFM-II ordering). This state consists of ½1 1 1 FM planes, which are coupled antiferromagnetically. Transition from FiM to AFM-II ordering is a first order. There is finite window of carrier density when the inhomogeneous (mixed) phase has the lower energy than that of one of the homogeneous phases. By analyzing ground state thermodynamic potential we found that mixed phase is favored for nc1 ononc2 ; where nc1 C1:5 and nc2 C2 for D ¼ 0:
References [1] K.-I. Kobayashi, et al., Nature 395 (1998) 677. [2] G. Jackeli, Phys. Rev. B 68 (2003) 092401. [3] A. Chattopadhyay, A.J. Millis, Phys. Rev. B 64 (2001) 024424.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 501–502
Spin wave theory of ferrimagnetic double perovskites G. Jackeli* Institut Laue Langevin 6, rue Jules Horawitz, Theory Group, B. P. 156, Grenoble F-38042, France
Abstract We present a theoretical study of magnetic properties of metallic double perovskite ferrimagnets such as Sr2 FeMoO6 and Sr2 FeReO6 : The analysis is based on the Kondo-type Hamiltonian in which charge carriers are constrained to be antiparallel to Fe local moments with spin S: The spectrum of spin wave excitations is derived based on the model Hamiltonian within the 1=S expansion. The ground state phase diagram as a function of carrier density is also discussed. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Lp; 75.30.Ds Keywords: Double perovskites; Spin wave theory
The recent discovery of the room temperature magnetoresistance in double perovskite compounds [1] has generated intense interest in these materials. The metallic ferrimagnetic (FiM) compounds such as Sr2 FeMoO6 and Sr2 FeReO6 are of particular recent interest, because both have high magnetic transition temperature (Tc ¼ 420; and 400 K; respectively) and their electronic structure has been suggested to be halfmetallic. In the double perovskite structure Fe and Mo/Re ions form two face-centered cubic sublattices and each pair of nearest-neighbor lattice sites is occupied by two different ions. In the ionic picture, Fe is in the 3+ valence state and has five d-electrons, which are strongly coupled by Hund’s rule in high spin state S ¼ 52 and are localized. The Mo is in 5+ valence state and has one t2g electron in 4d-shell. (The Re-based systems have one more charge carrier per formula unit.) The charge carriers (electrons form non-magnetic ions Mo/Re) and local moments of Fe ions are the building blocks of low energy theory for these compounds. Moreover, charge carriers are constrained to be antiparallel to the local moments due to the Pauli principle. As at Fe ion spin up states (with respect to the direction of the Fe moment) are already all occupied only an electron with spin down *Tel.: + 33-476-20-73-72; fax: + 33-476-88-24-16. E-mail address: [email protected] (G. Jackeli).
can hop to iron site. This picture can be described within the Kondo-type Hamiltonian in which core and itinerant spins are coupled by the infinite local antiferromagnetic (AFM) exchange J-N: This coupling is invoked to project out unphysical states and fulfill the constraint imposed by the exclusion principle.PThe model Hamiltonian can be written as H ¼ a Ha ; where a ¼ xy; xy; xz refers to three degenerate t2g ðdxy ; dyz ; dyz Þ orbital states. In the cubic lattice the t2g -transfer matrix is diagonal in the orbital space and is non-zero only in the corresponding plane. Each term Ha corresponds to a given orbital state and acts in the corresponding plane. All three terms have the same form given by X w X X H ¼ t ½dis d%js þ h:c: þ Dni Dn% j /ijSs
þ
X
J½Si si Ani ;
i
j
ð1Þ
i
where orbital index is omitted for simplicity. The first term describes an electron hopping between nearestneighbor Fe and Mo/Re ions, labeled by i and j; respectively. The operators dis ðni Þ and d%js ðn% j Þ correspond to Fe and Mo/Re sublattices, respectively, 2D ¼ EðFe d6 ; B0 d0 Þ EðFe d5 ; B0 d1 Þ is a charge transfer gap ðB0 ¼ Mo=ReÞ; and Si (si ) stands for the localized (itinerant) spin. The last constant term, where A ¼ ðS þ 1Þ=2; is chosen so that the exchange energy vanishes for a state with total spin Stot ¼ 2:
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1016
ARTICLE IN PRESS 502
G. Jackeli / Journal of Magnetism and Magnetic Materials 272–276 (2004) 501–502
Considering the ferromagnetic arrangement of local moments, one can introduce Fermion operators that generate eigenstates of the total spin at site i; i.e. operators with spin quantization axis directed parallel to the local moment. In these bases exchange term is diagonal and can be treated rigorously. The deviation of the local moments from the averaged value can be treated by introducing magnon operators and the controlled 1=S expansion of the model Hamiltonian can be developed (see Ref. [2]). As a result, one finds two branches of the spin excitations, similarly to the case of localized ferrimagnets. They describe gapless Goldstone mode and gapped optical mode. In addition to these modes, there exist Stoner continuum in the metallic systems. The dispersion law of the gapless branch at the quasi-classical level is Heisenberg-like oq ¼ 16SJ1 ½1 g1q þ 16SJ2 ½1 g2q ; t2 X g% 1ð2Þ J1ð2Þ ¼ nk ; 16S2 N k Ek
ð2Þ
where J1 and J2 are nearest and next-nearest neighbor exchange couplings, respectively, and g1ð2Þq is nearest (next-nearest) neighbor harmonic, nk is a Fermi distribution function and g% 1 ¼ 2 cos kx cos ky ; g% 2 ¼ cos 2kx þ cos 2ky : At a fixed carrier density n the spin wave stiffness ½D ¼ 16SðJ1 þ J2 Þ scales with the carrier bandwidth: DBt and DBt2 =D for tbD and t5D; respectively. As a function of density spin stiffness exhibits strong nonmonotonic behavior. It shows a maximum at some optimal filling nopt (nopt C1 for D ¼ 0) and then drops to
zero at some critical density nc (nc C1:9 for D ¼ 0). The highest value of spin stiffness is thus realized when the energy level of Fe d6 configuration is at resonance with B0 d-level and there is one charge carrier per unit cell. This agrees with the density dependence of Tc obtained in Ref. [3]. At n > nc the spin wave stiffness changes sign indicating the instability of FiM ordering. For n ¼ nc low energy spin wave spectrum becomes flat and vanishes identically in f1 1 1g direction. No special soft mode is singled out at this order. We have analyzed the quantum corrections to the linear theory in order to identify the symmetry of new magnetic order. The letter is found to correspond to a layered antiferromagnetic arrangement of local moments (so called AFM-II ordering). This state consists of ½1 1 1 FM planes, which are coupled antiferromagnetically. Transition from FiM to AFM-II ordering is a first order. There is finite window of carrier density when the inhomogeneous (mixed) phase has the lower energy than that of one of the homogeneous phases. By analyzing ground state thermodynamic potential we found that mixed phase is favored for nc1 ononc2 ; where nc1 C1:5 and nc2 C2 for D ¼ 0:
References [1] K.-I. Kobayashi, et al., Nature 395 (1998) 677. [2] G. Jackeli, Phys. Rev. B 68 (2003) 092401. [3] A. Chattopadhyay, A.J. Millis, Phys. Rev. B 64 (2001) 024424.