- Email: [email protected]
Magnetic behavior of the KFeS2
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 521–522
Magnetic behavior of the KFeS2 A.M.C. Souza*, S.R. Oliveira Neto, C.A. Macedo Departamento de Fisica, Universidade Federal de Sergipe, 49100-000 Sao Cristovao, SE, Brazil
Abstract The magnetic properties of KFeS2 are dominated by (FeS4)n chains and by the strongly reduced magnetic moments that indicate the presence of partial itinerance of the Fe 3d electrons. Here, we present a numerical study of the susceptibility of the KFeS2, following the Hubbard model in order to consider the itinerant electrons. Our results are in good agreement with experimental data and strengthen the evidences that the fermionic dynamics represented by the Hubbard model expresses an important part of the properties of KFeS2. r 2003 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 75.10.Lp; 75.50.Ee; 75.40.Mg Keywords: Magnetic materials; Numerical analysis; Semiconductor materials; Simulation
The KFeS2 is crystallized in monoclinic space group with (FeS4)n tetrahedra chains separated by potassium atoms [1] where the first and second neighbor intrachain separation are much shorter than the smallest interchain separation. It follows that the magnetic properties of KFeS2 are dominated by the intrachain spin interactions [2]. The one-dimensional character of this compound (and other AFeS2 thioferrates family, A=Cs, Rb, Ti) was confirmed by high-temperature magnetic susceptibility measurements to be typically of the short-range antiferromagnetic chains [2–4]. The KFeS2 involves trivalent iron and ordered magnetic moment of 2.43 mB that conflicts with the ionic high-spin value of 5 mB of the Fe3+ ions (3d5). The Fe– Fe intrachain distance (0.270 nm) is only 8.9% larger than the Fe–Fe distance in metallic iron (0.248 nm). This gives rise to strong covalence effects and indicates the presence of partial itinerance of the Fe 3d electrons for the KFeS2 [4,5]. Therefore, the theoretical study of the magnetic properties of this compound (as well as those of the thioferrates family) must be made considering itinerant electron magnetic models. However, the theoretical studies of the magnetic properties of these
*Corresponding author. Tel.: +55-79-212-6634; fax: +5579-212-6807. E-mail address: [email protected] (A.M.C. Souza).
compounds are dominated by localized spin models, often in terms of the Heisenberg model [2–4]. Here, we apply the idea of considering the itinerant electrons of KFeS2, such that the Fe3+ ions (3d5) are thermally ionized to produce Fe3+ ions (3d4) and itinerant electrons (for instance, see Ref. [6] for similar case). We present a numerical study of the magnetic susceptibility of the KFeS2, following the Hubbard model in order to consider the itinerant electrons of KFeS2. The Hamiltonian of the Hubbard model [7] is written as X X H ¼ t cþ nim nik ; ð1Þ is cjs þ U oij>s
cþ is ðcis Þ
i
where is the creation (annihilation) operator for a conduction band electron with spin s at site i; nis cþ is cis is the number of electrons operator. The first term represents the hopping energy (t) of the conductionband electrons, and the following term is the intraatomic Coulomb repulsion (U) between conduction electrons. A chemical potential m is introduced to impose the conservation of the total number of electrons in the system. We will neglect the orbital degeneracy associated with the d electrons. However, we believe that despite this simplification, the model puts into evidence the effect of itinerant electrons in the thermodynamic properties of KFeS2. This is not evidenced in magnetic
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.221
ARTICLE IN PRESS A.M.C. Souza et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 521–522
522 3.0
χ
χ(10-4 emu/mol)
2.5
2.0
1.5
χ
1.0
//
0.5
0.0
0
200
400
600
800
T (K) Fig. 1. Susceptibility of KFeS2 versus temperature. Experimental data points (full circles) are taken from Ref. [2]. The solid curve is due to our theoretical result using the U ¼ 0:34 eV and t ¼ 0:085 eV.
localized models, but is important to describe a variety of experimental results in KFeS2 which are associated with the conduction band. We study the temperature dependence of the susceptibility of the one-dimensional half-filled-band Hubbard model. For this purpose, we used the method of smallcluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain [8]. Fig 1 shows the magnetic susceptibility data, taken from Ref. [2], for two directions of the applied field: parallel and perpendicular to the chain axis. The result is typical of an antiferromagnetic compound. Below TN ¼ 250 K, we have an antiferromagnetic ordered phase and the susceptibility splits into parallel (wJ ) and perpendicular (w> ) components. Above TN ; the susceptibility becomes isotropic and increases with temperature indicating a one-dimensional character [2]. The fitting using the one-dimensional Hubbard model obtained for U ¼ 0:34 eV and t ¼ 0:085 eV model are represented by a solid line in Fig. 1. It is very similar to the form of the susceptibility of the KFeS2 above TN : Unfortunately, it
is not possible to study the susceptibility below TN using the one-dimensional Hubbard model because in this case we have an antiferromagnetic ordered phase and the three-dimensional interaction nature must be considered. Here a possible treatment is to consider a mixing model, where the Hubbard model is used to the intrachain interactions and the Heisenberg model to the interchain interactions. Further investigation in this line will be given elsewhere. We remark that the spin 1/2 one-dimensional Heisenberg model also presents a good fitting for the susceptibility of KFeS2. However, the susceptibility maximum at a temperature above the intrachain exchange that fits the KFeS2 data is unusual for onedimensional spin chains [4]. This suggests that the Hubbard model is more appropriate for studying the KFeS2. This point of view is in agreement with the temperature dependence of the electrical resistivity of the KFeS2 that presents a semiconducting behavior [5]. We can observe that our fitting was obtained considering the relation of the Hubbard model parameters as being equal to U=t ¼ 4 with is exactly the value of the bandwidth of the noninteracting one-dimensional Hubbard model, and is closely related to a semiconducting behavior. In conclusion, our results show that the fermionic dynamics represented by the Hubbard model expresses an important part of the magnetic properties of KFeS2. We thank M.E. de Souza for helpful discussions, and CAPES (Brazilian agency) for the financial support.
References [1] [2] [3] [4] [5] [6]
A. Mauger, M. Escorne, et al., Phys. Rev. B 30 (1984) 5300. S.K. Tiwary, S. Vasudevan, Phys. Rev. B 56 (1997) 7812. D. Welz, M. Nishi, Phys. Rev. B 45 (1992) 9806. Z. Seidov, et al., Phys. Rev. B 65 (2001) 014433. S. Nishioka, et al., Synth. Met. 71 (1995) 1877. C.A. Macedo, A.M.C. Souza, IEEE Trans. Magn. 38 (2002) 2875. [7] J. Hubbard, Proc. R. Soc London Ser. A 276 (1963) 238. [8] C.A. Macedo, A.M.C. Souza, Phys. Rev. B 64 (2001) 184441.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 521–522
Magnetic behavior of the KFeS2 A.M.C. Souza*, S.R. Oliveira Neto, C.A. Macedo Departamento de Fisica, Universidade Federal de Sergipe, 49100-000 Sao Cristovao, SE, Brazil
Abstract The magnetic properties of KFeS2 are dominated by (FeS4)n chains and by the strongly reduced magnetic moments that indicate the presence of partial itinerance of the Fe 3d electrons. Here, we present a numerical study of the susceptibility of the KFeS2, following the Hubbard model in order to consider the itinerant electrons. Our results are in good agreement with experimental data and strengthen the evidences that the fermionic dynamics represented by the Hubbard model expresses an important part of the properties of KFeS2. r 2003 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 75.10.Lp; 75.50.Ee; 75.40.Mg Keywords: Magnetic materials; Numerical analysis; Semiconductor materials; Simulation
The KFeS2 is crystallized in monoclinic space group with (FeS4)n tetrahedra chains separated by potassium atoms [1] where the first and second neighbor intrachain separation are much shorter than the smallest interchain separation. It follows that the magnetic properties of KFeS2 are dominated by the intrachain spin interactions [2]. The one-dimensional character of this compound (and other AFeS2 thioferrates family, A=Cs, Rb, Ti) was confirmed by high-temperature magnetic susceptibility measurements to be typically of the short-range antiferromagnetic chains [2–4]. The KFeS2 involves trivalent iron and ordered magnetic moment of 2.43 mB that conflicts with the ionic high-spin value of 5 mB of the Fe3+ ions (3d5). The Fe– Fe intrachain distance (0.270 nm) is only 8.9% larger than the Fe–Fe distance in metallic iron (0.248 nm). This gives rise to strong covalence effects and indicates the presence of partial itinerance of the Fe 3d electrons for the KFeS2 [4,5]. Therefore, the theoretical study of the magnetic properties of this compound (as well as those of the thioferrates family) must be made considering itinerant electron magnetic models. However, the theoretical studies of the magnetic properties of these
*Corresponding author. Tel.: +55-79-212-6634; fax: +5579-212-6807. E-mail address: [email protected] (A.M.C. Souza).
compounds are dominated by localized spin models, often in terms of the Heisenberg model [2–4]. Here, we apply the idea of considering the itinerant electrons of KFeS2, such that the Fe3+ ions (3d5) are thermally ionized to produce Fe3+ ions (3d4) and itinerant electrons (for instance, see Ref. [6] for similar case). We present a numerical study of the magnetic susceptibility of the KFeS2, following the Hubbard model in order to consider the itinerant electrons of KFeS2. The Hamiltonian of the Hubbard model [7] is written as X X H ¼ t cþ nim nik ; ð1Þ is cjs þ U oij>s
cþ is ðcis Þ
i
where is the creation (annihilation) operator for a conduction band electron with spin s at site i; nis cþ is cis is the number of electrons operator. The first term represents the hopping energy (t) of the conductionband electrons, and the following term is the intraatomic Coulomb repulsion (U) between conduction electrons. A chemical potential m is introduced to impose the conservation of the total number of electrons in the system. We will neglect the orbital degeneracy associated with the d electrons. However, we believe that despite this simplification, the model puts into evidence the effect of itinerant electrons in the thermodynamic properties of KFeS2. This is not evidenced in magnetic
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.221
ARTICLE IN PRESS A.M.C. Souza et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 521–522
522 3.0
χ
χ(10-4 emu/mol)
2.5
2.0
1.5
χ
1.0
//
0.5
0.0
0
200
400
600
800
T (K) Fig. 1. Susceptibility of KFeS2 versus temperature. Experimental data points (full circles) are taken from Ref. [2]. The solid curve is due to our theoretical result using the U ¼ 0:34 eV and t ¼ 0:085 eV.
localized models, but is important to describe a variety of experimental results in KFeS2 which are associated with the conduction band. We study the temperature dependence of the susceptibility of the one-dimensional half-filled-band Hubbard model. For this purpose, we used the method of smallcluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain [8]. Fig 1 shows the magnetic susceptibility data, taken from Ref. [2], for two directions of the applied field: parallel and perpendicular to the chain axis. The result is typical of an antiferromagnetic compound. Below TN ¼ 250 K, we have an antiferromagnetic ordered phase and the susceptibility splits into parallel (wJ ) and perpendicular (w> ) components. Above TN ; the susceptibility becomes isotropic and increases with temperature indicating a one-dimensional character [2]. The fitting using the one-dimensional Hubbard model obtained for U ¼ 0:34 eV and t ¼ 0:085 eV model are represented by a solid line in Fig. 1. It is very similar to the form of the susceptibility of the KFeS2 above TN : Unfortunately, it
is not possible to study the susceptibility below TN using the one-dimensional Hubbard model because in this case we have an antiferromagnetic ordered phase and the three-dimensional interaction nature must be considered. Here a possible treatment is to consider a mixing model, where the Hubbard model is used to the intrachain interactions and the Heisenberg model to the interchain interactions. Further investigation in this line will be given elsewhere. We remark that the spin 1/2 one-dimensional Heisenberg model also presents a good fitting for the susceptibility of KFeS2. However, the susceptibility maximum at a temperature above the intrachain exchange that fits the KFeS2 data is unusual for onedimensional spin chains [4]. This suggests that the Hubbard model is more appropriate for studying the KFeS2. This point of view is in agreement with the temperature dependence of the electrical resistivity of the KFeS2 that presents a semiconducting behavior [5]. We can observe that our fitting was obtained considering the relation of the Hubbard model parameters as being equal to U=t ¼ 4 with is exactly the value of the bandwidth of the noninteracting one-dimensional Hubbard model, and is closely related to a semiconducting behavior. In conclusion, our results show that the fermionic dynamics represented by the Hubbard model expresses an important part of the magnetic properties of KFeS2. We thank M.E. de Souza for helpful discussions, and CAPES (Brazilian agency) for the financial support.
References [1] [2] [3] [4] [5] [6]
A. Mauger, M. Escorne, et al., Phys. Rev. B 30 (1984) 5300. S.K. Tiwary, S. Vasudevan, Phys. Rev. B 56 (1997) 7812. D. Welz, M. Nishi, Phys. Rev. B 45 (1992) 9806. Z. Seidov, et al., Phys. Rev. B 65 (2001) 014433. S. Nishioka, et al., Synth. Met. 71 (1995) 1877. C.A. Macedo, A.M.C. Souza, IEEE Trans. Magn. 38 (2002) 2875. [7] J. Hubbard, Proc. R. Soc London Ser. A 276 (1963) 238. [8] C.A. Macedo, A.M.C. Souza, Phys. Rev. B 64 (2001) 184441.