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Phase diagram of EuxSr1-xS
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534 www.elsevier.com/locate/jmmm
Phase diagram of Eux Sr1xS S. Ohigashi, S. Niidera, F. Matsubara Department of Applied Physics, Tohoku University, Sendai, Miyagi 980-8579, Japan Available online 7 November 2006
Abstract We have theoretically studied the spin ordering of Eux Sr1x S using a Monte Carlo method. We found the spin glass phase occurs for xo0:50 in addition to the ferromagnetic phase for x40:50. An obtained phase diagram is in agreement with the experimental one fairly well. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 75.10.Hk Keywords: Eux Sr1x S; Spin glass; Monte Carlo
1. Introduction Eux Sr1x S is a well known spin glass (SG) material in which the ferromagnetic (FM) phase appears for x40:50 and the SG phase for xo0:50. In addition, for x\0:50 the reentrant spin glass (RSG) transition is observed [1]. Theoretically the magnetic properties of this material was investigated only for the FM side [2]. Recently, it was found that a dilute FM with next nearest neighbor antiferromagnetic interactions can exhibit the SG phase transition [3]. Here, we reexamine the spin ordering of Eux Sr1x S in a wide range of x. 2. Model In Eux Sr1x S, magnetic atoms, Eu (S ¼ 72Þ, are located on the fcc lattice; the magnitudes of the FM nearest neighbor exchange interactions and that of the antiferromagnetic next nearest neighbor exchange interactions are J 1 ’ 0:48 ½K and J 2 ’ 0:24 ½K, respectively [4]. Then we consider the model on the fcc lattice of L3 described by Hamiltonian, H¼
n:n X
J~ 1 xi xj S i S j
hi;ji
n:n:n X
J~ 2 xk xl S k S l ,
(1)
where S i is the Heisenberg spin of jS i j ¼ 1; J~ 1 ¼ SðS þ 1ÞJ 1 and J~ 2 ¼ SðS þ 1ÞJ 2 ; xi ¼ 1 or 0 when the lattice site i is occupied, respectively, by a magnetic or a nonmagnetic atom. The average number of xð hxi iÞ is the concentration of the magnetic atoms. 3. Ferromagnetic phase transition We investigated, in addition to the magnetization M, the Binder parameter gL , 1 ½hM 4 i 53 gL ¼ (2) 2 ½hM 2 i2 to examine whether or not the FM phase transition occurs. We found that the FM phase appears for xX0:50. Fig. 1 shows gL ’s for x ¼ 0:50 for various L. We can see that gL ’s for different L intersect at T c =J~ 1 ¼ 0:53 0:02. This intersection is a usual one that is found in the FM phase transition. We do not see any other intersection of gL at lower temperatures. This fact suggests that the FM order does not vanish at lower temperatures. 4. Spin glass phase transition
hk;li
Corresponding author.
E-mail address: [email protected] (S. Ohigashi). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.450
Next we investigated the SG transition. Here, we considered the correlation length, xL , for different sizes L. If the SG transition occurs at T sg , data for the
ARTICLE IN PRESS S. Ohigashi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534
1
1533
0.6 L= 8
x = 0.52
0.9
10
0.5
x = 0.50
12 14 ξ ⊥L/L
gL
0.8 0.7
0.4
0.3
0.6 L = 12 0.5
16
0.2
20 0.4 0.1
0.2
0.3
0.4
0.5
0.1
0.6
0.14
0.18 T/J˜
T/J˜1
0.22
0.26
1
Fig. 3. The SG correlation length of the spin transverse component x? L divided by L for x ¼ 0:52.
Fig. 1. Binder parameter gL for x ¼ 0:50.
0.7 0.6
L=8
x = 0.30
10
EuxSr1-xS
12
0.5
15
ξL/L
14 0.4
Exp.
0.3
MC
0.2 T [K]
10
0.1
PM 0.04
0.08
0.12
0.16
0.2
0.24
FM
T/J˜1 5
Fig. 2. The SG correlation length xL divided by L for x ¼ 0:30.
dimensionless ratio xL =L will intersect at T sg [5]. The correlation length xL is calculated as follows. We consider thePSG component of the spin, i.e., S~ i ð S i mÞ with mð i xi S i =ðxNÞÞ being the FM component. We perform the MC simulation of a two replica system with fS i g and fT i g. The SG order parameter, generalized to wave vector k, qmn ðkÞ, is given as 1 X ~ m ~ n ikRi qmn ðkÞ ¼ , (3) S T e xN i i i where m; n ¼ x; y; z. Using qmn ðkÞ, the wave vector dependent SG susceptibility wSG ðkÞ is determined as X wSG ðkÞ ¼ xN ½hjqmn ðkÞj2 i. (4) m;n
The SG correlation length is then obtained from 1=2 1 wSG ð0Þ xL ¼ 1 , 2 sinðkmin =2Þ wSG ðkmin Þ
(5)
SG
Mixed
0 0
0.2
0.4
0.6
0.8
1
x Fig. 4. The comparison between the theoretical phase diagram and the experimental one of Eux Sr1x S. Mixed means coexistence of the FM and transverse SG phase.
where kmin ¼ ð2p=L; 0; 0Þ. Note that, in the FM phase (ma0 for L ! 1), the FM component will interfere with the development of the correlation length of the SG component S~ i . Then for xX0:50 we calculated x? L using the ? transverse spin component, S~ i ð ðS~ i mÞ mÞ. We found that for xo0:50, xL =L for different L intersect. Fig. 2 shows the temperature dependence of xL =L for x ¼ 0:30. From these facts, we speculate that the SG transition occurs for xo0:50. On the other hand, for
ARTICLE IN PRESS 1534
S. Ohigashi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534
xX0:50, xL =L for different L do not intersect but x? L =L for different L intersect. Fig. 3 shows the temperature dependence of x? L =L for x ¼ 0:52. These facts imply that the FM order and the transverse SG order coexists at lower temperatures, i.e., a mixed phase appears for xX0:50.
compatible with experimental observations. Some problems remain such as (1) T sg is remarkably lower than experimental one, and (2) for xX0:50, the FM order does not vanish.
5. Summary References Fig. 4 is a theoretical phase diagram obtained in this study. Properties of the phase diagram are as follows: (1) for xX0:50, the FM phase appears, (2) for xo0:50, the SG phase appears, and (3) for x\0:50, the mixed phase appears at low temperature. These results are almost
[1] [2] [3] [4] [5]
H. Maletta, W. Felsch, Z. Phys. B 37 (1980) 55. K. Binder, W. Kinzel, D. Stauffer, Z. Phys. B 36 (1979) 161. S. Niidera, S. Abiko, F. Matsubara, Phys. Rev. B 72 (2005) 214402. L. Passell, O.W. Dietrich, J. Als-Nielsen, Phys. Rev. B 14 (1976) 4897. L.W. Lee, A.P. Young, Phys. Rev. Lett. 90 (2003) 227203.
Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534 www.elsevier.com/locate/jmmm
Phase diagram of Eux Sr1xS S. Ohigashi, S. Niidera, F. Matsubara Department of Applied Physics, Tohoku University, Sendai, Miyagi 980-8579, Japan Available online 7 November 2006
Abstract We have theoretically studied the spin ordering of Eux Sr1x S using a Monte Carlo method. We found the spin glass phase occurs for xo0:50 in addition to the ferromagnetic phase for x40:50. An obtained phase diagram is in agreement with the experimental one fairly well. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 75.10.Hk Keywords: Eux Sr1x S; Spin glass; Monte Carlo
1. Introduction Eux Sr1x S is a well known spin glass (SG) material in which the ferromagnetic (FM) phase appears for x40:50 and the SG phase for xo0:50. In addition, for x\0:50 the reentrant spin glass (RSG) transition is observed [1]. Theoretically the magnetic properties of this material was investigated only for the FM side [2]. Recently, it was found that a dilute FM with next nearest neighbor antiferromagnetic interactions can exhibit the SG phase transition [3]. Here, we reexamine the spin ordering of Eux Sr1x S in a wide range of x. 2. Model In Eux Sr1x S, magnetic atoms, Eu (S ¼ 72Þ, are located on the fcc lattice; the magnitudes of the FM nearest neighbor exchange interactions and that of the antiferromagnetic next nearest neighbor exchange interactions are J 1 ’ 0:48 ½K and J 2 ’ 0:24 ½K, respectively [4]. Then we consider the model on the fcc lattice of L3 described by Hamiltonian, H¼
n:n X
J~ 1 xi xj S i S j
hi;ji
n:n:n X
J~ 2 xk xl S k S l ,
(1)
where S i is the Heisenberg spin of jS i j ¼ 1; J~ 1 ¼ SðS þ 1ÞJ 1 and J~ 2 ¼ SðS þ 1ÞJ 2 ; xi ¼ 1 or 0 when the lattice site i is occupied, respectively, by a magnetic or a nonmagnetic atom. The average number of xð hxi iÞ is the concentration of the magnetic atoms. 3. Ferromagnetic phase transition We investigated, in addition to the magnetization M, the Binder parameter gL , 1 ½hM 4 i 53 gL ¼ (2) 2 ½hM 2 i2 to examine whether or not the FM phase transition occurs. We found that the FM phase appears for xX0:50. Fig. 1 shows gL ’s for x ¼ 0:50 for various L. We can see that gL ’s for different L intersect at T c =J~ 1 ¼ 0:53 0:02. This intersection is a usual one that is found in the FM phase transition. We do not see any other intersection of gL at lower temperatures. This fact suggests that the FM order does not vanish at lower temperatures. 4. Spin glass phase transition
hk;li
Corresponding author.
E-mail address: [email protected] (S. Ohigashi). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.450
Next we investigated the SG transition. Here, we considered the correlation length, xL , for different sizes L. If the SG transition occurs at T sg , data for the
ARTICLE IN PRESS S. Ohigashi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534
1
1533
0.6 L= 8
x = 0.52
0.9
10
0.5
x = 0.50
12 14 ξ ⊥L/L
gL
0.8 0.7
0.4
0.3
0.6 L = 12 0.5
16
0.2
20 0.4 0.1
0.2
0.3
0.4
0.5
0.1
0.6
0.14
0.18 T/J˜
T/J˜1
0.22
0.26
1
Fig. 3. The SG correlation length of the spin transverse component x? L divided by L for x ¼ 0:52.
Fig. 1. Binder parameter gL for x ¼ 0:50.
0.7 0.6
L=8
x = 0.30
10
EuxSr1-xS
12
0.5
15
ξL/L
14 0.4
Exp.
0.3
MC
0.2 T [K]
10
0.1
PM 0.04
0.08
0.12
0.16
0.2
0.24
FM
T/J˜1 5
Fig. 2. The SG correlation length xL divided by L for x ¼ 0:30.
dimensionless ratio xL =L will intersect at T sg [5]. The correlation length xL is calculated as follows. We consider thePSG component of the spin, i.e., S~ i ð S i mÞ with mð i xi S i =ðxNÞÞ being the FM component. We perform the MC simulation of a two replica system with fS i g and fT i g. The SG order parameter, generalized to wave vector k, qmn ðkÞ, is given as 1 X ~ m ~ n ikRi qmn ðkÞ ¼ , (3) S T e xN i i i where m; n ¼ x; y; z. Using qmn ðkÞ, the wave vector dependent SG susceptibility wSG ðkÞ is determined as X wSG ðkÞ ¼ xN ½hjqmn ðkÞj2 i. (4) m;n
The SG correlation length is then obtained from 1=2 1 wSG ð0Þ xL ¼ 1 , 2 sinðkmin =2Þ wSG ðkmin Þ
(5)
SG
Mixed
0 0
0.2
0.4
0.6
0.8
1
x Fig. 4. The comparison between the theoretical phase diagram and the experimental one of Eux Sr1x S. Mixed means coexistence of the FM and transverse SG phase.
where kmin ¼ ð2p=L; 0; 0Þ. Note that, in the FM phase (ma0 for L ! 1), the FM component will interfere with the development of the correlation length of the SG component S~ i . Then for xX0:50 we calculated x? L using the ? transverse spin component, S~ i ð ðS~ i mÞ mÞ. We found that for xo0:50, xL =L for different L intersect. Fig. 2 shows the temperature dependence of xL =L for x ¼ 0:30. From these facts, we speculate that the SG transition occurs for xo0:50. On the other hand, for
ARTICLE IN PRESS 1534
S. Ohigashi et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1532–1534
xX0:50, xL =L for different L do not intersect but x? L =L for different L intersect. Fig. 3 shows the temperature dependence of x? L =L for x ¼ 0:52. These facts imply that the FM order and the transverse SG order coexists at lower temperatures, i.e., a mixed phase appears for xX0:50.
compatible with experimental observations. Some problems remain such as (1) T sg is remarkably lower than experimental one, and (2) for xX0:50, the FM order does not vanish.
5. Summary References Fig. 4 is a theoretical phase diagram obtained in this study. Properties of the phase diagram are as follows: (1) for xX0:50, the FM phase appears, (2) for xo0:50, the SG phase appears, and (3) for x\0:50, the mixed phase appears at low temperature. These results are almost
[1] [2] [3] [4] [5]
H. Maletta, W. Felsch, Z. Phys. B 37 (1980) 55. K. Binder, W. Kinzel, D. Stauffer, Z. Phys. B 36 (1979) 161. S. Niidera, S. Abiko, F. Matsubara, Phys. Rev. B 72 (2005) 214402. L. Passell, O.W. Dietrich, J. Als-Nielsen, Phys. Rev. B 14 (1976) 4897. L.W. Lee, A.P. Young, Phys. Rev. Lett. 90 (2003) 227203.