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Molecular dynamics theory of complex magnetic structures in itinerant magnets
Journal of Magnetism and Magnetic Materials 177-181 (1998) 573 574
~
|ollrllld o|
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magnetism and magnetic
, J R materials
ELSEVIER
Molecular dynamics theory of complex magnetic structures in itinerant magnets Y. Kakehashi*, S. Akbar, N. K i m u r a Department o f Physics, Hokkaido Institute o f Technology, Maeda, Teine-ku, Sapporo 006, Japan
Abstract
A molecular dynamics method which determines the magnetic structures of itinerant magnets with a few hundred atoms in a unit cell has been developed. It is demonstrated with use of 108 and 256 atoms in a unit cell that the FCC transition metals and "¢-FeMn alloys show various complex noncollinear antiferromagnetic structures. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Itinerant magnetism; Magnetic structure; Competing interactions
The itinerant magnets with competing magnetic interactions such as Cr, Mn, Fe, and their alloys show complex magnetic structures, which prevent us from theoretical understanding of their peculiar magnetism as well as their magnetic structures. The past theories on the stability of magnetic structure for itinerant electron systems are based on a simple energy comparison among possible magnetic structures, minimization of energy with respect to the order parameter found in the experiments, or the susceptibility analysis, all of which are limited to a small subspace of magnetic structures, and never ensure the global minimum of the systems. We propose in this presentation a molecular dynamics method (MD) which automatically searches the global minimum in the free energy and, therefore, the magnetic structure of itinerant electron systems with a few hundred atoms in a unit cell. We start from the tight-binding model with intraatomic Coulomb and exchange interactions, and adopt the functional integral technique [1]. The local magnetic moment (LM) (mi) on site i at temperature T is then expressed by a semi-classical form [2] as
(mi)
=
((1
+
(4T/Ji~2))~i).
(1)
Here Jg is the effective exchange energy parameter, ¢i is the exchange field variable conjugate to the LM m~. Angle brackets stand for a classical average.
*Corresponding author. Tel.: +81-11-681-2161; fax: +8111-681-3622; e-mail: [email protected].
Assuming ergodicity of the system, we can calculate the thermal average of LM given by Eq. (1) as a time average of exchange fields {¢i}, whose equations of motion [3] are given by
~, = p~/~i,
(2)
Pia = ~ Ji((mi:)o -- ~i~) -- 2T~i~/¢ 2 -- (~ "Pia,
(3)
(~ = ( ~, (p2~/#i) -- N T ) Q . i
(4)
Here #i and Q are effective masses for the LM on site i and the thermal variable ~, respectively, pl is the fictitious momentum conjugate to the exchange field ¢i, and N is the number of atoms in the M D unit cell with periodic boundary condition. The first and second terms in Eq. (3) denote a 'magnetic force'. (m~)o is the thermal average of LM on site i, which is calculated from the Green function at each time step with use of the recursion method and the effective medium. The method similar to our M D approach has recently been proposed by Antropov et al. [4]. Our method has the following merits in comparison with their theory called spin dynamics method. (1) Amplitude fluctuations at finite temperatures are fully taken into account. (2) Site-dependent magnitudes of local moments are allowed. (3) The second-order phase transition is described by use of energy-dependent efl'ective medium. (4) The self-interactions and double counting in magnetic forces are removed. We have performed the calculations for various systems as a test of our approach. Using the Slater-Koster
0304-8853/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 88 5 3 ( 9 7 ) 0 0 4 5 7 - 5
574
Y. Kakehashi et aL / Journal of Magnetism and Magnetic Materials 177-181 (1998) 573-574
hopping parameters by Pettifor [5], the band width W = 0.45 Ry, J = 0.065 Ry, and the d electron number nd = 7.0, we verified that the M D calculations with use of the selfconsistent effective medium lead to the secondorder phase transition at the Curie temperature Tc = 950 K for a-Fe (N = 128). We applied the M D method to the determination of the magnetic structure of F C C transition metals in the range of nd from 6.0 to 7.0, where complex magnetic structures are expected due to competing interactions. Adopting N = 108 atoms, W = 0.443 Ry, J = 0.060 Ry, and T = 50 K, we obtained the first-kind antiferromagnetic (AF) structure with L M s [(mi)[ = 2.5#B for na = 6.2. It explains the magnetic structure of 7-Mn. Calculated magnetic structures show a modulation when nd is decreased. We found for na = 6.0 a modulated A F structure characterized by a stacking of the antiferromagnetic planes A and B with L M s I(mi)l = 2.9 #B along (1, 0, 0) direction as AB90A9oBA9oB , where the subscript 90 denotes ~/2 rotation of LMs on the AF plane. When nd is increased on the other hand, we obtain a helical structure with I(mi>l = 2.0~B and the wave vector Q = (0, ½, 1)2~/a for nd = 6.4. Here a is the lattice constant. F o r nd = 6.6, we found the helical structure with the same Q, but amplitude modulation from 0.75 to 1.30/~B, which originates in the magnetic energy gain caused by breaking a frustrated magnetic structure. In the case of 7-Fe, we performed the M D calculations at 25 K with use of N = 108, nd = 7.0, W = 0.39 Ry, and J = 0.065 Ry. After 8000 steps, we obtained a helical structure with Q = (0, ~, z3)2rt/a and I(mi)] = 0.75 PB as shown in Fig. 1. The result is consistent with the experiments, though the wave vector deviates from the value Q:(O,O.11,1)27t/a obtained for cubic y - F e l - ~ C o x (x ~ 0.02) precipitates in Cu [6]. We performed the M D calculations extending the size to N = 256 to examine the N dependence. The result shows the noncollinear A F structure with a unit cell consisting of 2 × 2 × 2 F C C lattices as seen from Fig. 2. The LMs have 16 nonequivalent directions and their magnitudes are distributed from 0.8 to 1.6 #B- The results are therefore sensitive to the size N and the interaction range. The rigid-band calculations mentioned above do not explain the concentration dependence of the average L M I(mi)l in 3,-FeMn alloys. We therefore performed the M D calculations for 3,-FeMn alloys taking into account the random configuration of Fe and M n atoms in a unit cell with N = 108. The results explain the peculiar minimum in average L M at 50 at% Fe [7]. It is characterized by a strong frustration of LMs, since there exist very slow and large spin fluctuations in the time development of simulations. In summary, we have presented the M D method to determine the complex magnetic structures of itinerantelectron systems. The theory seems to predict reasonable magnetic structures as well as the peculiar concentration
Fig. 1. The magnetic structure of 7-Fe obtained from the MD with N = 108 atoms in a unit cell at 25 K.
Fig. 2. The same as in Fig. 1, but for N = 256 atoms.
dependence of average LMs in 7 - F e M n alloys. In case of ~,-Fe, we found two possible structures: a helical structure and a complex noncollinear magnetic structure which cannot be described by a few Q vectors. M o r e detailed calculations with larger size N are desired for 7-Fe to reach the theoretical conclusion.
References [1] Y. Kakehashi, Phys. Rev. B 34 (1986) 3243. I-2] S. Akbar, Y. Kakehashi, N. Kimura, J. Appl. Phys. 81 (1997) 3862. [3] S. Nos6, J. Chem. Phys. 81 (1984) 511. [4] V.P. Antropov, M.I. Katsnelson, B.N. Harmon, Phys. Rev. B 54 (1996) 1019. 1-5] R. Haydock, M.J. Kelly, Surf. Sci. 38 (1973) 139. [6] Y. Tsunoda, J. Phys.: Condens. Matter 1 (1989) 10427. [7] Y. Endoh, Y. Ishikawa, J. Phys. Soc. Japan 30 (1971) 1614.
~
|ollrllld o|
;ii
magnetism and magnetic
, J R materials
ELSEVIER
Molecular dynamics theory of complex magnetic structures in itinerant magnets Y. Kakehashi*, S. Akbar, N. K i m u r a Department o f Physics, Hokkaido Institute o f Technology, Maeda, Teine-ku, Sapporo 006, Japan
Abstract
A molecular dynamics method which determines the magnetic structures of itinerant magnets with a few hundred atoms in a unit cell has been developed. It is demonstrated with use of 108 and 256 atoms in a unit cell that the FCC transition metals and "¢-FeMn alloys show various complex noncollinear antiferromagnetic structures. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Itinerant magnetism; Magnetic structure; Competing interactions
The itinerant magnets with competing magnetic interactions such as Cr, Mn, Fe, and their alloys show complex magnetic structures, which prevent us from theoretical understanding of their peculiar magnetism as well as their magnetic structures. The past theories on the stability of magnetic structure for itinerant electron systems are based on a simple energy comparison among possible magnetic structures, minimization of energy with respect to the order parameter found in the experiments, or the susceptibility analysis, all of which are limited to a small subspace of magnetic structures, and never ensure the global minimum of the systems. We propose in this presentation a molecular dynamics method (MD) which automatically searches the global minimum in the free energy and, therefore, the magnetic structure of itinerant electron systems with a few hundred atoms in a unit cell. We start from the tight-binding model with intraatomic Coulomb and exchange interactions, and adopt the functional integral technique [1]. The local magnetic moment (LM) (mi) on site i at temperature T is then expressed by a semi-classical form [2] as
(mi)
=
((1
+
(4T/Ji~2))~i).
(1)
Here Jg is the effective exchange energy parameter, ¢i is the exchange field variable conjugate to the LM m~. Angle brackets stand for a classical average.
*Corresponding author. Tel.: +81-11-681-2161; fax: +8111-681-3622; e-mail: [email protected].
Assuming ergodicity of the system, we can calculate the thermal average of LM given by Eq. (1) as a time average of exchange fields {¢i}, whose equations of motion [3] are given by
~, = p~/~i,
(2)
Pia = ~ Ji((mi:)o -- ~i~) -- 2T~i~/¢ 2 -- (~ "Pia,
(3)
(~ = ( ~, (p2~/#i) -- N T ) Q . i
(4)
Here #i and Q are effective masses for the LM on site i and the thermal variable ~, respectively, pl is the fictitious momentum conjugate to the exchange field ¢i, and N is the number of atoms in the M D unit cell with periodic boundary condition. The first and second terms in Eq. (3) denote a 'magnetic force'. (m~)o is the thermal average of LM on site i, which is calculated from the Green function at each time step with use of the recursion method and the effective medium. The method similar to our M D approach has recently been proposed by Antropov et al. [4]. Our method has the following merits in comparison with their theory called spin dynamics method. (1) Amplitude fluctuations at finite temperatures are fully taken into account. (2) Site-dependent magnitudes of local moments are allowed. (3) The second-order phase transition is described by use of energy-dependent efl'ective medium. (4) The self-interactions and double counting in magnetic forces are removed. We have performed the calculations for various systems as a test of our approach. Using the Slater-Koster
0304-8853/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 88 5 3 ( 9 7 ) 0 0 4 5 7 - 5
574
Y. Kakehashi et aL / Journal of Magnetism and Magnetic Materials 177-181 (1998) 573-574
hopping parameters by Pettifor [5], the band width W = 0.45 Ry, J = 0.065 Ry, and the d electron number nd = 7.0, we verified that the M D calculations with use of the selfconsistent effective medium lead to the secondorder phase transition at the Curie temperature Tc = 950 K for a-Fe (N = 128). We applied the M D method to the determination of the magnetic structure of F C C transition metals in the range of nd from 6.0 to 7.0, where complex magnetic structures are expected due to competing interactions. Adopting N = 108 atoms, W = 0.443 Ry, J = 0.060 Ry, and T = 50 K, we obtained the first-kind antiferromagnetic (AF) structure with L M s [(mi)[ = 2.5#B for na = 6.2. It explains the magnetic structure of 7-Mn. Calculated magnetic structures show a modulation when nd is decreased. We found for na = 6.0 a modulated A F structure characterized by a stacking of the antiferromagnetic planes A and B with L M s I(mi)l = 2.9 #B along (1, 0, 0) direction as AB90A9oBA9oB , where the subscript 90 denotes ~/2 rotation of LMs on the AF plane. When nd is increased on the other hand, we obtain a helical structure with I(mi>l = 2.0~B and the wave vector Q = (0, ½, 1)2~/a for nd = 6.4. Here a is the lattice constant. F o r nd = 6.6, we found the helical structure with the same Q, but amplitude modulation from 0.75 to 1.30/~B, which originates in the magnetic energy gain caused by breaking a frustrated magnetic structure. In the case of 7-Fe, we performed the M D calculations at 25 K with use of N = 108, nd = 7.0, W = 0.39 Ry, and J = 0.065 Ry. After 8000 steps, we obtained a helical structure with Q = (0, ~, z3)2rt/a and I(mi)] = 0.75 PB as shown in Fig. 1. The result is consistent with the experiments, though the wave vector deviates from the value Q:(O,O.11,1)27t/a obtained for cubic y - F e l - ~ C o x (x ~ 0.02) precipitates in Cu [6]. We performed the M D calculations extending the size to N = 256 to examine the N dependence. The result shows the noncollinear A F structure with a unit cell consisting of 2 × 2 × 2 F C C lattices as seen from Fig. 2. The LMs have 16 nonequivalent directions and their magnitudes are distributed from 0.8 to 1.6 #B- The results are therefore sensitive to the size N and the interaction range. The rigid-band calculations mentioned above do not explain the concentration dependence of the average L M I(mi)l in 3,-FeMn alloys. We therefore performed the M D calculations for 3,-FeMn alloys taking into account the random configuration of Fe and M n atoms in a unit cell with N = 108. The results explain the peculiar minimum in average L M at 50 at% Fe [7]. It is characterized by a strong frustration of LMs, since there exist very slow and large spin fluctuations in the time development of simulations. In summary, we have presented the M D method to determine the complex magnetic structures of itinerantelectron systems. The theory seems to predict reasonable magnetic structures as well as the peculiar concentration
Fig. 1. The magnetic structure of 7-Fe obtained from the MD with N = 108 atoms in a unit cell at 25 K.
Fig. 2. The same as in Fig. 1, but for N = 256 atoms.
dependence of average LMs in 7 - F e M n alloys. In case of ~,-Fe, we found two possible structures: a helical structure and a complex noncollinear magnetic structure which cannot be described by a few Q vectors. M o r e detailed calculations with larger size N are desired for 7-Fe to reach the theoretical conclusion.
References [1] Y. Kakehashi, Phys. Rev. B 34 (1986) 3243. I-2] S. Akbar, Y. Kakehashi, N. Kimura, J. Appl. Phys. 81 (1997) 3862. [3] S. Nos6, J. Chem. Phys. 81 (1984) 511. [4] V.P. Antropov, M.I. Katsnelson, B.N. Harmon, Phys. Rev. B 54 (1996) 1019. 1-5] R. Haydock, M.J. Kelly, Surf. Sci. 38 (1973) 139. [6] Y. Tsunoda, J. Phys.: Condens. Matter 1 (1989) 10427. [7] Y. Endoh, Y. Ishikawa, J. Phys. Soc. Japan 30 (1971) 1614.