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antiferromagnetic superlattice
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 305 (2006) 233–239 www.elsevier.com/locate/jmmm
Magnetization and magnetic susceptibility of the Ising ferromagnetic/ antiferromagnetic superlattice An Du, Yan Ma, Zeng-hui Wu College of Sciences, Northeastern University, Shenyang 110004, PR China Received 8 August 2005; received in revised form 5 January 2006 Available online 26 January 2006
Abstract A linear cluster mean-field approximation is used to study the magnetic properties of the Ising ferromagnetic/antiferromagnetic superlattice, which is composed of a spin-1/2 ferromagnetic monolayer and a spin-1 antiferromagnetic monolayer with a single-ion anisotropy alternatively. By using the transfer matrix method, we calculate the magnetization and the initial magnetic susceptibility as functions of temperature for different interlayer coupling, single-ion anisotropy. We summarize the changing behaviors of the spin structure in ferromagnetic and antiferromagnetic layers and the characteristics of the corresponding magnetic susceptibilities, give the transition temperature as a function of the interlayer exchange coupling for different single-ion anisotropy, and analyze the features of the magnetization and the magnetic susceptibility. r 2006 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 75.10.Hk; 75.40.Cx; 75.70.Cn Keywords: Ferromagnetic/antiferromagnetic superlattice; Ising model; Linear cluster mean-field approximation; Magnetization; Magnetic susceptibility
1. Introduction Since the discovery of the magnetic interlayer coupling and the giant magnetoresistance in Fe/Cr superlattice [1,2], magnetic multilayers are of central interest in thin film magnetism. Ferromagnetic(FM)/FM multilayer with FM interface exchange coupling and antiferromagnetic(AFM) interface exchange coupling such as Co/Cu/Ni [3] and Gd/ Fe [4,5], FM/AFM multilayer such as Fe/NiO [6], NiO/Co [7] and NiO/Co84Fe16 [8], AFM/AFM multilayer such as FeF2/CoF2 [9] and CoO/NiO [10], ferrimagnetic/ferrimagnetic superlattice such as Fe3O4/Mn3O4 [11] and ferrimagnetic/AFM superlattice such as Fe3O4/CoO [12], and so on, have been studied experimentally. The interface structure, spin orientations of the constituents, the type and the strength of the interface exchange coupling, and the magnetic properties are measured, and some new phenomena have been found. Corresponding author. Tel.: +86 02483681370; fax: +86 02483691370. E-mail address: [email protected] (A. Du).
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.007
Theoretically, for a multilayer or a superlattice that consists of two materials with different Curie temperatures and an FM interface coupling, phenomenological approaches have been carried out by using the Landau–Ginzburg theory [13,14] which predicted that the magnetic susceptibility should have a single maximum (singularity) if the films are thin, but have two maxima (one maximum at lower temperature, one singularity at higher temperature) if the films are thick, which are consistent with the experimental results [3]. Meanwhile the microscopic approaches also find a similar phenomena in the superlattice systems with AFM interface coupling [15,16]. Mean-field approach on AFM/AFM superlattice reveals that a number of magnetic transitions appear depending on the thicknesses of the films in the superlattice and on the interface exchange constant [17]. Using the same method, the dependence of Curie temperatures on the thickness and exchange constants of the superlattices ABAByAB and ABAyBA are calculated [18]. The temperature-dependent magnetization in an FM/FM bilayer with different Curie temperatures is also discussed [19], and the surface spin reorientation in thin Gd films on Fe in an applied magnetic
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field is analyzed [20]. The effective ‘‘Curie temperature’’ enhancement of Ni films in the Co/Cu/Ni trilayer system [3] is explained by using the Green function theory with a random phase approximation (RPA) [21]. In the FM/AFM multilayer system, besides the common phenomenon like that in the FM/FM multilayer system, a phenomenon called exchange bias (EB), which is a shift of the hysteresis loop along the magnetic field axis, can occur. Matsuyama et al. [6] estimated the Fe and Ni spin configurations phenomenologically using a simple model, and found that the NiO spins at the interface cannot form the easy-spin axis towards the Fe spin because of the exchange coupling. Using Heisenberg FM and the diluted Ising AFM model and Monte Carlo simulation, Nowak et al. [22] gave the hysteresis loops of domain state (DS) in FM/AFM bilayer, and indicated that the irreversible part of the DS magnetization at the AFM interface leads to EB. Using the same model but the mean-field-type calculations, Scholten et al. [23] calculated the coercive field and the EB field, and found the different origins of the enhancement of the coercivity around TN and the occurrence of EB. Using the Ising model composed of alternate chains of spin-1/2 and -1 lying on a plane forming a hexagonal lattice and Monte Carlo simulation and mean-field calculation, Godoy et al. [24] calculated the sublattice magnetizations and the specific heats, found the compensation point of the model and discussed the dependence of the compensation point on the parameters of the system. In fact, there is a frustration effect in the FM/AFM superlattice. The frustrating bond in the magnetic system has greater ability to resist the thermal fluctuation of the spin than the bond without frustration [25]. Thus, there are many interesting properties in the FM/AFM superlattice than in the FM/FM superlattice. In this paper, we will use the mixed spin-1/2 and spin-1 Ising model to describe the FM/AFM superlattice, the linear cluster mean-field approximation to study the properties of the FM/AFM superlattice, calculate the magnetizations, magnetic susceptibilities and the transition temperature and discuss the orientations of the magnetizations in FM and AFM layers. In Section 2 we present the model and the linear cluster mean-field approximation, and give the basic formulations to calculate the physical quantities of interest. In Section 3, the magnetization, initial magnetic susceptibility and the transition temperature are examined by changing the interlayer interaction strength and the value of the singleion anisotropy in the AFM layer. The concluding remarks are given in Section 4.
2. Model and formulation We consider an infinite simple cubic superlattice in which spin-1/2 layers of material A with FM exchange within the layer alternate with spin-1 layers of material B with AFM exchange within the layer; a yABABy structure is
formed. The Hamiltonian is given by Refs. [16,26] X X H¼ J mn mzm mzn J ij S zi Szj hmni
X hmii
hiji
J mi mzm S zi
X
Di ðSzi Þ2 ,
ð1Þ
i
where mzm ðmzn Þ ¼ 1=2 in layer A, Szi ðSzj Þ ¼ 1 and 0 in layer B, and the summations are carried out only over the nearest-neighbor pair of spins. Jmn and Jij are the FM exchange and AFM exchange between spins in the spin-1/2 layers and spin-1 layers, respectively, J mn ¼ J A 40 and J ij ¼ J B o0. J mi ¼ J AB 40 is the FM exchange between spins in material A and its nearest neighbors in material B. Di ¼ D is the single-ion anisotropy parameter in layer B. It is found that the FM exchange between FM layer A and AFM layer B introduces frustration into the system, so the ground state of the FM/AFM system is different from that of the FM/FM system. In the FM/FM superlattice, the spin structure of every chain perpendicular to the FM layers is þ þ þ þ þ þ . However, in the FM/AFM superlattice, the spin structure of a chain perpendicular to the FM and AFM layers is þ þ þ þ þ þ , but the spin structure of its nearestneighbor chains is þ þ þ ; thus, the expectation value of a spin on a chain is not equal to that on its nearest-neighbor chains. For convenience, the lattice on layer A should be divided into two sublattices as that on layer B, and the two sublattices with a square lattice structure in every layer penetrate each other. In order to calculate the properties of the FM/AFM system, we adopt a linear cluster mean-field approximation which is an extension of the original mean-field approximation, in which a spin chain and its four nearest-neighbor chains perpendicular to the FM and AFM layers are taken as an infinite cluster, and the spins on the border chains are in an effective field produced by the other spins outside the cluster. The effective field is proportional to the expectation value of the spin on the central chain. Because of the different spin structures on the central chain and on the border chains in the original lattice, the expanded BethePeierls approximation used in Ref. [16], in which the effective field parameters acting on the spins on the border chains are determined by the condition that the expectation value of a spin on the central chain is equal to that on the border chains, is no longer suitable to the present problem. A linear mean-field cluster approximation with only an infinite chain in the cluster is also used by Wiatrowski to study the FM superlattice [27]. Because the effective field produced by the spins on its nearestneighbor chains is proportional to the expectation value of the spin on the nearest-neighbor chains rather than that on the selected chain, Wiatrowski’s linear mean-field cluster approximation cannot be used in the present problem. Compared with the original mean-field approximation, a bigger cluster is considered in the present method, and it is expected that a more accurate result can
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k
235
transfer matrix method can be used, and the partition function of the system can be written as
k+1
2
Z ¼ Tr½expðbH L Þ ¼ TrðPN Þ
3
¼ TrðV1 VPV1 VP . . . V1 VPÞ l =1
N N ¼ TrðkN Þ ¼ lN 1 þ l2 þ þ ln ,
ð4Þ
where b ¼ 1=kB T, N is the length of the chains, and P is the transfer matrix. It can be expressed as P ¼ PAB PBA , and PAB is defined as
5 4 A
B
A
B
A
B
ðPAB Þmn ¼ hmk ðmÞj expðbH Ak;Bk ÞjSk ðnÞi,
Fig. 1. Simplified lattice: : sublattice A1 on layer A, : sublattice A2 on layer A, m: sublattice B1 on layer B, ,: sublattice B2 on layer B. The spin quantum numbers in layers A and B are 1/2 and 1, respectively. The orientations of spins on sublattices A1 and B1 are supposed to be parallel.
be obtained. The simplified lattice in the linear cluster mean-field approximation is given in Fig. 1. Hamiltonian (1) is simplified as X X H k;kþ1 ¼ ðH Ak;Bk þ H Bk;Akþ1 Þ, (2) HL ¼ k
k
JA X z z JB X z z m1k mlk S S 2 lðla1Þ 2 lðla1Þ 1k lk X DX z 2 J AB mzlk Szlk ðS lk Þ 2 l l ðz 1ÞJ A hmzA1 i X z mlk 2 lðla1Þ
H Ak;Bk ¼
ðz 1ÞJ B hSzB1 i X z Slk , 2 lðla1Þ
(6)
(7)
and the free energy of the system is given by F ¼ kB T ln Z ¼ NkB T ln limax .
(8)
The expectation value per spin on monolayers A and B on the lth chain is given by ð3aÞ
JB X z z JA X z S 1k S lk m mz 2 lðla1Þ 2 lðla1Þ 1kþ1 lkþ1 X DX z 2 J AB S zlk mzlkþ1 ðSlk Þ 2 l l ðz 1ÞJ B hSzB1 i X z Slk 2 lðla1Þ ðz 1ÞJ A hmzA1 i X z mlkþ1 , 2 lðla1Þ
VPV1 ¼ k,
Z ¼ Tr½expðbH L Þ lN imax
H Bk;Akþ1 ¼
where jmk ðmÞi ¼ jm1k ðmÞm2k ðmÞm3k ðmÞm4k ðmÞm5k ðmÞi and jS k ðnÞi ¼ jS1k ðnÞS 2k ðnÞS 3k ðnÞS4k ðnÞS 5k ðnÞi are two eigenstates of five spins in layers A and B, respectively, so m ¼ 32 and n ¼ 243 and PAB is a matrix with dimensions of 32 243. PBA is the transposed matrix of matrix PAB; thus P is a square matrix with dimensions of 32 32. Supposing the eigenvalues of matrix P are l1 ; l2 ; . . . ; l32 , they can be obtained from the following matrix equation:
where k is a diagonal matrix. Suppose limax is the biggest eigenvalue among the eigenvalues of l1 ; l2 ; . . . ; ln . The partition function of the system is given by
where
(5)
hmzlk i ¼
ðl0 Þi i 1 Tr½mzlk expðbH L Þ ¼ l max max , Z lmax
(9a)
hS zlk i ¼
ðS0 Þi i 1 Tr½S zlk expðbH L Þ ¼ l max max , Z lmax
(9b)
where
ð3bÞ
where l ¼ 1; 2; 3; 4; 5, z ¼ 4 is the number of the nearestneighbors in every layer, and the parameters ðz 1ÞJ A hS zA1 i and ðz 1ÞJ B hSzB1 i are the effective field produced by the spins outside the cluster, which act on the spins on the border chains in monolayers A and B, respectively, and the hS zA1 i and hS zB1 i are the expectation values of spins on the central chain in layers A and B, respectively. Hamiltonian (2) is in a block form, and every block consists of three neighboring layers of ABA (three neighboring layers of BAB can also be taken as a block). Any two blocks are commutation with each other; thus, the
l0l ¼ Vðll PAB PBA ÞV1 ,
(10a)
S0l ¼ VðPAB Sl PBA ÞV1
(10b)
and both matrix ll and matrix Sl are diagonal matrixes. The diagonal elements are defined as ðll Þmm ¼ hmk ðmÞjmlk jmk ðmÞi,
(11a)
ðSl Þnn ¼ hS k ðnÞjSlk jSk ðnÞi.
(11b)
From Eq. (9), we can get the thermodynamic mean values of the spins on the central chain and the border chain in layers A and B. Because Hamiltonian (3a) includes the thermodynamic mean values of the spins on the central chain in layers A and B, the thermodynamic mean values of the spins must be solved from Eqs. (9a) and (9b) simultaneously. We define the sublattice magnetizations of the spins on the central chain and on the border chains in layers A and B as follows: mA1 ¼ hmz1k i,
(12a)
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236
mB1 ¼ hSz1k i, *
mA2 ¼
wB ¼
+
4 1 X ¼ Sz 4 l¼1 lk
wS j
(15b)
j¼1
4 1 X mz , 4 l¼1 lk
*
mB2
5 X
(12b)
(12c)
and the total magnetic susceptibility is given by w ¼ wA þ wB .
+ (12d)
(16)
Using the condition that the magnetic susceptibility is divergent at the transition temperature, we can determine the transition temperature of the superlattice system.
and the total magnetization is defined as mT ¼ ðmA1 þ mA2 þ mB1 þ mB2 Þ.
(13)
In order to calculate the initial magnetic susceptibility, a small magnetic field h is supposed to beP added to the system; P thus, a new term gmB m mzm gmB i S zi will appear in Hamiltonians (1) and (2). Here the same g factor for the two kinds of spin is assumed. Based on the definition of the initial magnetic susceptibility, the magnetic susceptibility of each sublattice in the linear cluster mean-field approximation can be obtained. For spin pl (pl ¼ mzl , or Szl , l ¼ 1; 2; 3; 4; 5), the initial magnetic susceptibility is given by q X gmB hplk i qh k "* + 5 X X X 2 z ¼ ðgmB Þ b plk mjk X
plk
k
"* 2
þ ðgmB Þ b *
X k
plk
X
j¼1
k
X
+*
k
5 X
plk
j¼1
k
j¼1
+#
mzjk
E g1L ¼ Nð4zJ A m2 þ 6J AB mS þ 4zJ B S 2 5DS 2 Þ.
5 X X
5 X X
+
j¼2
k
+#
S zjk
E g2 ¼ N 0 ð12 zJ A m2 þ 12 zJ B S 2 2J AB mS DS 2 Þ.
j¼2
k
"*
1.0
5 X j¼1
w mj ,
mB1 mA1
m
0.5
+
0.0
mA2
JB/JA=-0.5
-0.5 -1.0 20
ð14Þ
k
where the correlation function can be calculated by using the transfer matrix formula [16]. Thus, by solving the equations simultaneously, the magnetic susceptibilities of every layer can be obtained, and the magnetic susceptibilities of layers A and B are given by wA ¼
(18a)
k
5 X X X w plk Szjk þ gmB bðz 1ÞJ B S1 N j¼2 k k * +* +# 5 X X X z plk S jk , k
(17b)
If layer A changes into the AFM ground state, but layer B is still in the AFM ground state, the ground state energy of the system is
S zjk
"* + 5 X X X w m1 z plk mjk þ gmB bðz 1ÞJ A N j¼2 k k * +* +# 5 X X X z plk mjk k
(17a)
where N0 is the total number of spins on layer A or B. In the linear cluster mean-field approximation shown in Fig. 1 and described by Eq. (2), this ground state energy is changed into
(15a)
χ[ N(gµB)2/JA]
j¼1
k
+*
Using Eqs. (12)–(17), we can calculate the sublattice magnetizations and the initial magnetic susceptibilities of the superlattice. The results for various parameters are given in Figs. 2–7. In these figures, we set J B ¼ J A =2. Before discussing the results, we firstly analyze the ground state of the system. If layer A is in the FM ground state and layer B in the AFM ground state, the ground state energy of the system is E g1 ¼ N 0 ð 12 zJ A m2 þ 12 zJ B S2 DS 2 Þ,
wp l ¼
*
3. Numerical results and discussion
15 10
D/JA=0.0 mB2 JAB/JA=0.2
χA
χB
JAB/JA=1.2
χB χA
JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 2. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 0:0.
ARTICLE IN PRESS A. Du et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 233–239 0.02
m
mA1
1.0
m
0.5
1.0 1.2 1.4
0.0
-0.02
A2
-0.04
kBT/JA
0.0
-0.5
-0.5 -1.0 20
χA
JAB/JA=0.2
15 χ[ N(gµB)2/JA]
D/JA=1.0
mB2
χB
JAB/JA=2.0
10
D/JA=0.0
20
χB
JAB/JA=1.2
JB/JA=-0.5
-1.0
JB/JA=-0.5
mA2
15
χ[ N(gµB)2/JA]
m
0.5
mA1
0.00
mT
mB1
1.0
χA
10
JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
5
5
0 0.00
0.25
0.50
0.75
0 0.00
0.25
0.50
0.75
1.00 kBT/JA
1.25
1.50
1.75
2.00
Fig. 3. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
mT
0.0
mA2
-0.5
χ[ N(gµB)2/JA]
10
D/JA= -1.0
mB2 JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
χA
χA
χB
χB
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
In the linear cluster mean-field approximation, it is changed into E g2L ¼ Nð4zJ A m2 10J AB mS þ 4zJ B S 2 5DS 2 Þ.
JB/JA=-0.5 D/JA=1.0
-1.0 20
Fig. 4. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
(18b)
Thus, the transition point satisfies the following condition: J AB ¼ J A .
2.00
0.0 -0.5
JB/JA=-0.5
5 0 0.00
1.75
0.5
mA1
0.5
15
1.50
1.0
mB1
-1.0 20
1.00 1.25 kBT/JA
Fig. 5. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 0:0.
χ[ N(gµB)2/JA]
1.0
m
237
(19)
In selecting parameters, we consider two cases: J AB oJ A and J AB 4J A . From Figs. 2 and 3 we find an interesting phenomenon for J AB ¼ 0:2J A . At zero temperature, layer A is in the FM state and layer B is in the AFM state. With increasing temperature, the sublattice magnetizations decrease. When
JAB/JA=0.2
15 10
JAB/JA=1.2 JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 6. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
temperature reaches a definite value, the spins in layer A reorientate and direct to the opposite direction abruptly, but the sublattice magnetizations of layer B do not change much. With further increase in temperature, the sublattice magnetizations decrease further and get to zero at the transition temperature. Another phenomenon that should be brought to attention is that the sublattice magnetization mA1 strides across the horizontal axis again to return to the original direction with increasing temperature, but the sublattice magnetization mA2 remains in the opposite state (see the inset of Fig. 3; if we increase the value of D or JB, this phenomenon will be more clear). Layer A changes into an AFM structure paralleling completely to
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1.0
mT
0.5 0.0 -0.5
JB/JA=-0.5
χ[ N(gµB)2/JA]
-1.0 20 15 10
D/JA= -1.0 JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 7. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
the AFM structure in layer B. The magnetic susceptibility of layer A exhibits a singularity to the point that the sublattice magnetizations in layer A change directions; the magnetic susceptibility of layer B has only a maximum. When J AB ¼ 1:2J A and 2.0JA, layer A is in the AFM state like that in layer B, and the sublattice magnetizations and the susceptibilities of layers A and B exhibit normal behaviors. In a magnetic system, the frustration bond is usually superior to resist the thermal fluctuation of spins to the ordinary bond without frustration [25] at a definite temperature. In the vicinity of the point that the magnetization in layer A reorientates, the sublattice magnetization of sublattice A2 exceeds the sublattice magnetization of sublattice A1 by flipping their orientations. As a result the spins in sublattice A1 become frustrated. With the increase in the temperature, the spins in sublattice A1 flip again and cancel out the frustration. In the Fe film on the (0 0 1) surface of AFM NiO, it is found that the Fe domains change to follow the NiO domains [6]. The maximum of the magnetic susceptibility of layer A between the flipping point and the transition point for J AB ¼ 0:2J A corresponds to the transition of pure monolayer A. If J AB ¼ 0, we find that the transition temperature ðkB T C =J A Þ of layer A is 0.86, and that of layer B is kB T N =J A ¼ 1:18. Although they are bigger than those obtained by using the expanded Bethe-Peierls approximation [28], they are more accurate than those obtained by the original mean-field approximation. Fig. 4 shows another kind of phenomenon. When the interlayer exchange coupling is weak, J AB =J A ¼ 0:2, with increasing temperature, the sublattice magnetizations of layer B decrease initially, and they decrease abruptly at a definite temperature, and afterwards the sublattice magnetization of B2 flips to the opposite direction, gets to the
same direction as that of B1, and the spin structure in layer B becomes completely parallel to that of layer A. Meanwhile, the sublattice magnetizations of layer A decrease monotonously. Magnetic susceptibility of layer B exhibits a maximum at the flipping point. There may be two reasons as to why this phenomenon occurs: the first is the negative single-ion anisotropy in layer B, the second is the interlayer exchange coupling between layers A and B. The negative single-ion anisotropy D tends to make the system lie in the S ¼ 0 state, so the first-order phase transition usually occurs in the integer S Ising system [16]. From Eq. (17b) we find that the critical value of D at which layer B is in the S ¼ 0 state is ð16J B þ 3J AB Þ=5; the present value of D=J A ¼ 1:0 does not get this critical value. However, the tendency of the first-order phase transition in layer B still exists. Through the interlayer exchange coupling, the stable FM layer A brings the unstable AFM layer B to a new state in which the orientation of the magnetization of sublattice B1 turns to the direction of FM layer A. The spin reorientation near the NiO interface with FM layer has been detected in Refs. [6–8]. When J AB =J A ¼ 1:2, layer A is in the AFM ground state. The critical value of the single-ion anisotropy at which the first-order phase transition occurs in layer B is J AB þ 16J B =5. The present value of D=J A ¼ 1:0 is far from the critical value, so the tendency of the first-order phase transition in layer B is very small, and the magnetic structure of layer B is stable. However, layer A is in a frustrated state. When the temperature gets to a definite value, the FM exchange within layer A is superior to the interlayer exchange coupling between layers A and B in the competition between the two exchange interactions, and layer A returns to an FM state. This is a process of resisting the thermal fluctuation, and the transition is smooth, so the susceptibilities have a normal behavior with only one singularity. The total magnetization and the total magnetic susceptibility are given in Figs. 5–7. From Figs. 5 and 6, it is clearly seen that a compensation point appears for J AB ¼ 0:2J A , and magnetic susceptibility exhibits two singularities and a maximum. During the interlayer exchange coupling, J AB ¼ 1:2J A , no compensation point appears. For J AB ¼ 2:0J A , the total magnetization is close to zero at a low temperature; however, when the temperature gets to a definite value, it appears and disappears again at the transition temperature. The magnetic susceptibility exhibits only a singularity for the two interlayer exchange couplings. For D ¼ 1:0J A in Fig. 7, at J AB ¼ 0:2J A , the behavior of the total magnetization is different from those in Figs. 5 and 6, and the magnetic susceptibility exhibits a maximum and a singularity. For J AB ¼ 1:2J A and 2.0JA, the behaviors of the total magnetization and the magnetic susceptibility are similar to those for the single-ion anisotropy equal to zero or greater than zero. Fig. 8 shows the dependence of the transition temperature on the interlayer exchange coupling between layers A
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reaches a definite value, the sublattice magnetizations in layer B decrease abruptly and get to the same direction as that of the magnetization in layer A, the corresponding susceptibility of layer B exhibits a maximum at the flipping point, and there is no compensation in the total magnetization. At J AB =J A ¼ 1:2, layer A is in an AFM state at ground state; when the temperature increases, it changes to the FM state at finite temperature. When J AB =J A ¼ 2:0, this transition does not appear.
1.7 1.6
D/JA= 1.0
kBTN/JA
1.5 1.4
0.0
1.3 1.2 1.1
-1.0
1.0
Acknowledgments
JB=-0.5JA
0.9 0.0
239
0.5
1.0 JAB/JA
1.5
2.0
Fig. 8. The transition temperature as a function of the interlayer coupling for a few values of the single-ion anisotropy parameters. J B =J A ¼ 0:5.
This work was supported by the Special Funds for the Major State Basic Research Projects of China under Grant no. G2000067104. References
and B. The transition temperature increases as the interlayer exchange coupling increases. 4. Concluding remarks The linear cluster mean-field approximation is applied to study the magnetic properties of the magnetic superlattice consisting of an FM monolayer A with spin-1/2 and an AFM monolayer B with spin-1 and single-ion anisotropy. The interlayer interaction between the A and B monolayers is FM, so the system is in a frustrated state. As the AFM intralayer exchange interaction in layer B is half of the FM intralayer exchange interaction in layer A, J B ¼ 0:5J A , we calculate the temperature dependence of layered (total) spontaneous magnetization and initial susceptibility and the transition temperature for different values of interlayer coupling, single-ion anisotropy on the B monolayers. When the single-ion anisotropy in layer B is equal or greater than zero, DX0, for weak interlayer exchange coupling, J AB =J A ¼ 0:2, it is found that the two sublattice magnetizations in layer A change their directions simultaneously, from an FM state to another FM state with opposite spontaneous magnetization direction at a definite temperature, the total spontaneous magnetization appears as a compensation point, and the corresponding magnetic susceptibilitiy exhibits a singularity at this flipping point besides the singularity at the transition temperature and a maximum corresponding to the layer A monolayer. When interlayer exchange coupling is big, J AB =J A ¼ 1:2 or 2.0, the spin structure in layer A is AFM at ground state, and the total magnetization for J AB =J A ¼ 2:0 has a compensation phenomenon and the magnetic susceptibility exhibits only one singularity at the transition temperature for the two cases. When the single-ion anisotropy in layer B is smaller than zero, D ¼ 1:0, for weak interlayer exchange coupling, J AB =J A ¼ 0:2, layer A is in an FM state and layer B is in an AFM state at ground state; however, as the temperature
[1] P. Gru¨nberg, R. Schreiber, Y. Pang, M.B. Brodsky, C.H. Sowers, Phys. Rev. Lett. 57 (1986) 2442. [2] M.N. Baibich, F.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472. [3] U. Bovensiepen, F. Wilhelm, P. Srivastava, P. Poulopoulos, M. Farle, A. Ney, K. Baberschke, Phys. Rev. Lett. 81 (1998) 2368. [4] M. Taborelli, R. Allenspach, G. Boffa, M. Landolt, Phys. Rev. Lett. 56 (1986) 2869. [5] D. Haskel, G. Srajer, J.C. Lang, J. Pollmann, C.S. Nelson, J.S. Jiang, S.D. Bader, Phys. Rev. Lett. 87 (2001) 207201. [6] H. Matsuyama, C. Haginoya, K. Koike, Phys. Rev. Lett. 85 (2000) 646. [7] H. Ohldag, A. Scholl, F. Nolting, S. Anders, F.U. Hillebrecht, J. Sto¨hr, Phys. Rev. Lett. 86 (2001) 2878. [8] W. Zhu, L. Seve, R. Sears, B. Sinkovic, S.S.P. Parkin, Phys. Rev. Lett. 86 (2001) 5389. [9] A.S. Carrico, R.E. Camley, Phys. Rev. B 45 (1992) 13117. [10] M.J. Carey, A.E. Berkowitz, J.A. Borchers, R.W. Erwin, Phys. Rev. B 47 (1993) 9952. [11] G. Chern, L. Horng, W.K. Shieh, T.C. Wu, Phys. Rev. B 63 (2001) 94421. [12] Y. Ijiri, J.A. Borchers, R.W. Erwin, S.-H. Lee, P.J. van der Zaag, R.M. Wolf, Phys. Rev. Lett. 80 (1998) 608. [13] N. El Aouad, B. Laaboudi, M. Kerouad, M. Saber, J. Phys.: Condens. Matter 13 (2001) 797. [14] R.W. Wang, D.L. Mills, Phys. Rev. B 46 (1992) 1168. [15] R.E. Camley, D. Li, Phys. Rev. Lett. 84 (2000) 4709. [16] A. Du, H.J. Liu, B. Wang, Physica A 342 (2004) 583. [17] A.S. Carrico, R.E. Camley, Phys. Rev. B 45 (1992) 13117. [18] X.-G. Wang, N.-N. Liu, S.-H. Pan, G.-Z. Yang, J. Magn. Magn. Mater. 212 (2000) 121. [19] J. Wu, G.S. Dong, X. Jin, Phys. Rev. B 70 (2004) 212406. [20] R.E. Camley, Phys. Rev. B 35 (1987) 3608. [21] P.J. Jensen, K.H. Bennemann, K. Baberschke, P. Poulopoulos, M. Farle, J. Appl. Phys. 87 (2000) 6692. [22] U. Nowak, K.D. Usadel, J. Keller, P. Milte´nyi, B. Beschoten, G. Gu¨ntherodt, Phys. Rev. B 66 (2002) 14430. [23] G. Scholten, K.D. Usadel, U. Nowak, Phys. Rev. B 71 (2005) 64413. [24] M. Godoy, V.S. Leite, W. Figueiredo, Phys. Rev. B 69 (2004) 54428. [25] G.-Z. Wei, W. Jiang, A. Du, Phys. Lett. A 224 (1996) 137. [26] E.F. Sarmento, J.C. Cressoni, R.J.V. dos Santos, J. Appl. Phys. 75 (1994) 5820. [27] G. Wiatrowski, Physica A 290 (2001) 107. [28] A. Du, H.J. Liu, Y.Q. Yu¨, Phys. Stat. Sol. (b) 241 (2004) 175.
Journal of Magnetism and Magnetic Materials 305 (2006) 233–239 www.elsevier.com/locate/jmmm
Magnetization and magnetic susceptibility of the Ising ferromagnetic/ antiferromagnetic superlattice An Du, Yan Ma, Zeng-hui Wu College of Sciences, Northeastern University, Shenyang 110004, PR China Received 8 August 2005; received in revised form 5 January 2006 Available online 26 January 2006
Abstract A linear cluster mean-field approximation is used to study the magnetic properties of the Ising ferromagnetic/antiferromagnetic superlattice, which is composed of a spin-1/2 ferromagnetic monolayer and a spin-1 antiferromagnetic monolayer with a single-ion anisotropy alternatively. By using the transfer matrix method, we calculate the magnetization and the initial magnetic susceptibility as functions of temperature for different interlayer coupling, single-ion anisotropy. We summarize the changing behaviors of the spin structure in ferromagnetic and antiferromagnetic layers and the characteristics of the corresponding magnetic susceptibilities, give the transition temperature as a function of the interlayer exchange coupling for different single-ion anisotropy, and analyze the features of the magnetization and the magnetic susceptibility. r 2006 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 75.10.Hk; 75.40.Cx; 75.70.Cn Keywords: Ferromagnetic/antiferromagnetic superlattice; Ising model; Linear cluster mean-field approximation; Magnetization; Magnetic susceptibility
1. Introduction Since the discovery of the magnetic interlayer coupling and the giant magnetoresistance in Fe/Cr superlattice [1,2], magnetic multilayers are of central interest in thin film magnetism. Ferromagnetic(FM)/FM multilayer with FM interface exchange coupling and antiferromagnetic(AFM) interface exchange coupling such as Co/Cu/Ni [3] and Gd/ Fe [4,5], FM/AFM multilayer such as Fe/NiO [6], NiO/Co [7] and NiO/Co84Fe16 [8], AFM/AFM multilayer such as FeF2/CoF2 [9] and CoO/NiO [10], ferrimagnetic/ferrimagnetic superlattice such as Fe3O4/Mn3O4 [11] and ferrimagnetic/AFM superlattice such as Fe3O4/CoO [12], and so on, have been studied experimentally. The interface structure, spin orientations of the constituents, the type and the strength of the interface exchange coupling, and the magnetic properties are measured, and some new phenomena have been found. Corresponding author. Tel.: +86 02483681370; fax: +86 02483691370. E-mail address: [email protected] (A. Du).
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.007
Theoretically, for a multilayer or a superlattice that consists of two materials with different Curie temperatures and an FM interface coupling, phenomenological approaches have been carried out by using the Landau–Ginzburg theory [13,14] which predicted that the magnetic susceptibility should have a single maximum (singularity) if the films are thin, but have two maxima (one maximum at lower temperature, one singularity at higher temperature) if the films are thick, which are consistent with the experimental results [3]. Meanwhile the microscopic approaches also find a similar phenomena in the superlattice systems with AFM interface coupling [15,16]. Mean-field approach on AFM/AFM superlattice reveals that a number of magnetic transitions appear depending on the thicknesses of the films in the superlattice and on the interface exchange constant [17]. Using the same method, the dependence of Curie temperatures on the thickness and exchange constants of the superlattices ABAByAB and ABAyBA are calculated [18]. The temperature-dependent magnetization in an FM/FM bilayer with different Curie temperatures is also discussed [19], and the surface spin reorientation in thin Gd films on Fe in an applied magnetic
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field is analyzed [20]. The effective ‘‘Curie temperature’’ enhancement of Ni films in the Co/Cu/Ni trilayer system [3] is explained by using the Green function theory with a random phase approximation (RPA) [21]. In the FM/AFM multilayer system, besides the common phenomenon like that in the FM/FM multilayer system, a phenomenon called exchange bias (EB), which is a shift of the hysteresis loop along the magnetic field axis, can occur. Matsuyama et al. [6] estimated the Fe and Ni spin configurations phenomenologically using a simple model, and found that the NiO spins at the interface cannot form the easy-spin axis towards the Fe spin because of the exchange coupling. Using Heisenberg FM and the diluted Ising AFM model and Monte Carlo simulation, Nowak et al. [22] gave the hysteresis loops of domain state (DS) in FM/AFM bilayer, and indicated that the irreversible part of the DS magnetization at the AFM interface leads to EB. Using the same model but the mean-field-type calculations, Scholten et al. [23] calculated the coercive field and the EB field, and found the different origins of the enhancement of the coercivity around TN and the occurrence of EB. Using the Ising model composed of alternate chains of spin-1/2 and -1 lying on a plane forming a hexagonal lattice and Monte Carlo simulation and mean-field calculation, Godoy et al. [24] calculated the sublattice magnetizations and the specific heats, found the compensation point of the model and discussed the dependence of the compensation point on the parameters of the system. In fact, there is a frustration effect in the FM/AFM superlattice. The frustrating bond in the magnetic system has greater ability to resist the thermal fluctuation of the spin than the bond without frustration [25]. Thus, there are many interesting properties in the FM/AFM superlattice than in the FM/FM superlattice. In this paper, we will use the mixed spin-1/2 and spin-1 Ising model to describe the FM/AFM superlattice, the linear cluster mean-field approximation to study the properties of the FM/AFM superlattice, calculate the magnetizations, magnetic susceptibilities and the transition temperature and discuss the orientations of the magnetizations in FM and AFM layers. In Section 2 we present the model and the linear cluster mean-field approximation, and give the basic formulations to calculate the physical quantities of interest. In Section 3, the magnetization, initial magnetic susceptibility and the transition temperature are examined by changing the interlayer interaction strength and the value of the singleion anisotropy in the AFM layer. The concluding remarks are given in Section 4.
2. Model and formulation We consider an infinite simple cubic superlattice in which spin-1/2 layers of material A with FM exchange within the layer alternate with spin-1 layers of material B with AFM exchange within the layer; a yABABy structure is
formed. The Hamiltonian is given by Refs. [16,26] X X H¼ J mn mzm mzn J ij S zi Szj hmni
X hmii
hiji
J mi mzm S zi
X
Di ðSzi Þ2 ,
ð1Þ
i
where mzm ðmzn Þ ¼ 1=2 in layer A, Szi ðSzj Þ ¼ 1 and 0 in layer B, and the summations are carried out only over the nearest-neighbor pair of spins. Jmn and Jij are the FM exchange and AFM exchange between spins in the spin-1/2 layers and spin-1 layers, respectively, J mn ¼ J A 40 and J ij ¼ J B o0. J mi ¼ J AB 40 is the FM exchange between spins in material A and its nearest neighbors in material B. Di ¼ D is the single-ion anisotropy parameter in layer B. It is found that the FM exchange between FM layer A and AFM layer B introduces frustration into the system, so the ground state of the FM/AFM system is different from that of the FM/FM system. In the FM/FM superlattice, the spin structure of every chain perpendicular to the FM layers is þ þ þ þ þ þ . However, in the FM/AFM superlattice, the spin structure of a chain perpendicular to the FM and AFM layers is þ þ þ þ þ þ , but the spin structure of its nearestneighbor chains is þ þ þ ; thus, the expectation value of a spin on a chain is not equal to that on its nearest-neighbor chains. For convenience, the lattice on layer A should be divided into two sublattices as that on layer B, and the two sublattices with a square lattice structure in every layer penetrate each other. In order to calculate the properties of the FM/AFM system, we adopt a linear cluster mean-field approximation which is an extension of the original mean-field approximation, in which a spin chain and its four nearest-neighbor chains perpendicular to the FM and AFM layers are taken as an infinite cluster, and the spins on the border chains are in an effective field produced by the other spins outside the cluster. The effective field is proportional to the expectation value of the spin on the central chain. Because of the different spin structures on the central chain and on the border chains in the original lattice, the expanded BethePeierls approximation used in Ref. [16], in which the effective field parameters acting on the spins on the border chains are determined by the condition that the expectation value of a spin on the central chain is equal to that on the border chains, is no longer suitable to the present problem. A linear mean-field cluster approximation with only an infinite chain in the cluster is also used by Wiatrowski to study the FM superlattice [27]. Because the effective field produced by the spins on its nearestneighbor chains is proportional to the expectation value of the spin on the nearest-neighbor chains rather than that on the selected chain, Wiatrowski’s linear mean-field cluster approximation cannot be used in the present problem. Compared with the original mean-field approximation, a bigger cluster is considered in the present method, and it is expected that a more accurate result can
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k
235
transfer matrix method can be used, and the partition function of the system can be written as
k+1
2
Z ¼ Tr½expðbH L Þ ¼ TrðPN Þ
3
¼ TrðV1 VPV1 VP . . . V1 VPÞ l =1
N N ¼ TrðkN Þ ¼ lN 1 þ l2 þ þ ln ,
ð4Þ
where b ¼ 1=kB T, N is the length of the chains, and P is the transfer matrix. It can be expressed as P ¼ PAB PBA , and PAB is defined as
5 4 A
B
A
B
A
B
ðPAB Þmn ¼ hmk ðmÞj expðbH Ak;Bk ÞjSk ðnÞi,
Fig. 1. Simplified lattice: : sublattice A1 on layer A, : sublattice A2 on layer A, m: sublattice B1 on layer B, ,: sublattice B2 on layer B. The spin quantum numbers in layers A and B are 1/2 and 1, respectively. The orientations of spins on sublattices A1 and B1 are supposed to be parallel.
be obtained. The simplified lattice in the linear cluster mean-field approximation is given in Fig. 1. Hamiltonian (1) is simplified as X X H k;kþ1 ¼ ðH Ak;Bk þ H Bk;Akþ1 Þ, (2) HL ¼ k
k
JA X z z JB X z z m1k mlk S S 2 lðla1Þ 2 lðla1Þ 1k lk X DX z 2 J AB mzlk Szlk ðS lk Þ 2 l l ðz 1ÞJ A hmzA1 i X z mlk 2 lðla1Þ
H Ak;Bk ¼
ðz 1ÞJ B hSzB1 i X z Slk , 2 lðla1Þ
(6)
(7)
and the free energy of the system is given by F ¼ kB T ln Z ¼ NkB T ln limax .
(8)
The expectation value per spin on monolayers A and B on the lth chain is given by ð3aÞ
JB X z z JA X z S 1k S lk m mz 2 lðla1Þ 2 lðla1Þ 1kþ1 lkþ1 X DX z 2 J AB S zlk mzlkþ1 ðSlk Þ 2 l l ðz 1ÞJ B hSzB1 i X z Slk 2 lðla1Þ ðz 1ÞJ A hmzA1 i X z mlkþ1 , 2 lðla1Þ
VPV1 ¼ k,
Z ¼ Tr½expðbH L Þ lN imax
H Bk;Akþ1 ¼
where jmk ðmÞi ¼ jm1k ðmÞm2k ðmÞm3k ðmÞm4k ðmÞm5k ðmÞi and jS k ðnÞi ¼ jS1k ðnÞS 2k ðnÞS 3k ðnÞS4k ðnÞS 5k ðnÞi are two eigenstates of five spins in layers A and B, respectively, so m ¼ 32 and n ¼ 243 and PAB is a matrix with dimensions of 32 243. PBA is the transposed matrix of matrix PAB; thus P is a square matrix with dimensions of 32 32. Supposing the eigenvalues of matrix P are l1 ; l2 ; . . . ; l32 , they can be obtained from the following matrix equation:
where k is a diagonal matrix. Suppose limax is the biggest eigenvalue among the eigenvalues of l1 ; l2 ; . . . ; ln . The partition function of the system is given by
where
(5)
hmzlk i ¼
ðl0 Þi i 1 Tr½mzlk expðbH L Þ ¼ l max max , Z lmax
(9a)
hS zlk i ¼
ðS0 Þi i 1 Tr½S zlk expðbH L Þ ¼ l max max , Z lmax
(9b)
where
ð3bÞ
where l ¼ 1; 2; 3; 4; 5, z ¼ 4 is the number of the nearestneighbors in every layer, and the parameters ðz 1ÞJ A hS zA1 i and ðz 1ÞJ B hSzB1 i are the effective field produced by the spins outside the cluster, which act on the spins on the border chains in monolayers A and B, respectively, and the hS zA1 i and hS zB1 i are the expectation values of spins on the central chain in layers A and B, respectively. Hamiltonian (2) is in a block form, and every block consists of three neighboring layers of ABA (three neighboring layers of BAB can also be taken as a block). Any two blocks are commutation with each other; thus, the
l0l ¼ Vðll PAB PBA ÞV1 ,
(10a)
S0l ¼ VðPAB Sl PBA ÞV1
(10b)
and both matrix ll and matrix Sl are diagonal matrixes. The diagonal elements are defined as ðll Þmm ¼ hmk ðmÞjmlk jmk ðmÞi,
(11a)
ðSl Þnn ¼ hS k ðnÞjSlk jSk ðnÞi.
(11b)
From Eq. (9), we can get the thermodynamic mean values of the spins on the central chain and the border chain in layers A and B. Because Hamiltonian (3a) includes the thermodynamic mean values of the spins on the central chain in layers A and B, the thermodynamic mean values of the spins must be solved from Eqs. (9a) and (9b) simultaneously. We define the sublattice magnetizations of the spins on the central chain and on the border chains in layers A and B as follows: mA1 ¼ hmz1k i,
(12a)
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236
mB1 ¼ hSz1k i, *
mA2 ¼
wB ¼
+
4 1 X ¼ Sz 4 l¼1 lk
wS j
(15b)
j¼1
4 1 X mz , 4 l¼1 lk
*
mB2
5 X
(12b)
(12c)
and the total magnetic susceptibility is given by w ¼ wA þ wB .
+ (12d)
(16)
Using the condition that the magnetic susceptibility is divergent at the transition temperature, we can determine the transition temperature of the superlattice system.
and the total magnetization is defined as mT ¼ ðmA1 þ mA2 þ mB1 þ mB2 Þ.
(13)
In order to calculate the initial magnetic susceptibility, a small magnetic field h is supposed to beP added to the system; P thus, a new term gmB m mzm gmB i S zi will appear in Hamiltonians (1) and (2). Here the same g factor for the two kinds of spin is assumed. Based on the definition of the initial magnetic susceptibility, the magnetic susceptibility of each sublattice in the linear cluster mean-field approximation can be obtained. For spin pl (pl ¼ mzl , or Szl , l ¼ 1; 2; 3; 4; 5), the initial magnetic susceptibility is given by q X gmB hplk i qh k "* + 5 X X X 2 z ¼ ðgmB Þ b plk mjk X
plk
k
"* 2
þ ðgmB Þ b *
X k
plk
X
j¼1
k
X
+*
k
5 X
plk
j¼1
k
j¼1
+#
mzjk
E g1L ¼ Nð4zJ A m2 þ 6J AB mS þ 4zJ B S 2 5DS 2 Þ.
5 X X
5 X X
+
j¼2
k
+#
S zjk
E g2 ¼ N 0 ð12 zJ A m2 þ 12 zJ B S 2 2J AB mS DS 2 Þ.
j¼2
k
"*
1.0
5 X j¼1
w mj ,
mB1 mA1
m
0.5
+
0.0
mA2
JB/JA=-0.5
-0.5 -1.0 20
ð14Þ
k
where the correlation function can be calculated by using the transfer matrix formula [16]. Thus, by solving the equations simultaneously, the magnetic susceptibilities of every layer can be obtained, and the magnetic susceptibilities of layers A and B are given by wA ¼
(18a)
k
5 X X X w plk Szjk þ gmB bðz 1ÞJ B S1 N j¼2 k k * +* +# 5 X X X z plk S jk , k
(17b)
If layer A changes into the AFM ground state, but layer B is still in the AFM ground state, the ground state energy of the system is
S zjk
"* + 5 X X X w m1 z plk mjk þ gmB bðz 1ÞJ A N j¼2 k k * +* +# 5 X X X z plk mjk k
(17a)
where N0 is the total number of spins on layer A or B. In the linear cluster mean-field approximation shown in Fig. 1 and described by Eq. (2), this ground state energy is changed into
(15a)
χ[ N(gµB)2/JA]
j¼1
k
+*
Using Eqs. (12)–(17), we can calculate the sublattice magnetizations and the initial magnetic susceptibilities of the superlattice. The results for various parameters are given in Figs. 2–7. In these figures, we set J B ¼ J A =2. Before discussing the results, we firstly analyze the ground state of the system. If layer A is in the FM ground state and layer B in the AFM ground state, the ground state energy of the system is E g1 ¼ N 0 ð 12 zJ A m2 þ 12 zJ B S2 DS 2 Þ,
wp l ¼
*
3. Numerical results and discussion
15 10
D/JA=0.0 mB2 JAB/JA=0.2
χA
χB
JAB/JA=1.2
χB χA
JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 2. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 0:0.
ARTICLE IN PRESS A. Du et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 233–239 0.02
m
mA1
1.0
m
0.5
1.0 1.2 1.4
0.0
-0.02
A2
-0.04
kBT/JA
0.0
-0.5
-0.5 -1.0 20
χA
JAB/JA=0.2
15 χ[ N(gµB)2/JA]
D/JA=1.0
mB2
χB
JAB/JA=2.0
10
D/JA=0.0
20
χB
JAB/JA=1.2
JB/JA=-0.5
-1.0
JB/JA=-0.5
mA2
15
χ[ N(gµB)2/JA]
m
0.5
mA1
0.00
mT
mB1
1.0
χA
10
JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
5
5
0 0.00
0.25
0.50
0.75
0 0.00
0.25
0.50
0.75
1.00 kBT/JA
1.25
1.50
1.75
2.00
Fig. 3. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
mT
0.0
mA2
-0.5
χ[ N(gµB)2/JA]
10
D/JA= -1.0
mB2 JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
χA
χA
χB
χB
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
In the linear cluster mean-field approximation, it is changed into E g2L ¼ Nð4zJ A m2 10J AB mS þ 4zJ B S 2 5DS 2 Þ.
JB/JA=-0.5 D/JA=1.0
-1.0 20
Fig. 4. Temperature dependence of the sublattice magnetizations and the initial magnetic susceptibilities of layers A and B for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
(18b)
Thus, the transition point satisfies the following condition: J AB ¼ J A .
2.00
0.0 -0.5
JB/JA=-0.5
5 0 0.00
1.75
0.5
mA1
0.5
15
1.50
1.0
mB1
-1.0 20
1.00 1.25 kBT/JA
Fig. 5. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 0:0.
χ[ N(gµB)2/JA]
1.0
m
237
(19)
In selecting parameters, we consider two cases: J AB oJ A and J AB 4J A . From Figs. 2 and 3 we find an interesting phenomenon for J AB ¼ 0:2J A . At zero temperature, layer A is in the FM state and layer B is in the AFM state. With increasing temperature, the sublattice magnetizations decrease. When
JAB/JA=0.2
15 10
JAB/JA=1.2 JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 6. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
temperature reaches a definite value, the spins in layer A reorientate and direct to the opposite direction abruptly, but the sublattice magnetizations of layer B do not change much. With further increase in temperature, the sublattice magnetizations decrease further and get to zero at the transition temperature. Another phenomenon that should be brought to attention is that the sublattice magnetization mA1 strides across the horizontal axis again to return to the original direction with increasing temperature, but the sublattice magnetization mA2 remains in the opposite state (see the inset of Fig. 3; if we increase the value of D or JB, this phenomenon will be more clear). Layer A changes into an AFM structure paralleling completely to
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238
1.0
mT
0.5 0.0 -0.5
JB/JA=-0.5
χ[ N(gµB)2/JA]
-1.0 20 15 10
D/JA= -1.0 JAB/JA=0.2 JAB/JA=1.2 JAB/JA=2.0
5 0 0.00
0.25
0.50
0.75
1.00 1.25 kBT/JA
1.50
1.75
2.00
Fig. 7. Temperature dependence of the total magnetization and the total magnetic susceptibility for a few values of the interlayer exchange coupling. D=J A ¼ 1:0.
the AFM structure in layer B. The magnetic susceptibility of layer A exhibits a singularity to the point that the sublattice magnetizations in layer A change directions; the magnetic susceptibility of layer B has only a maximum. When J AB ¼ 1:2J A and 2.0JA, layer A is in the AFM state like that in layer B, and the sublattice magnetizations and the susceptibilities of layers A and B exhibit normal behaviors. In a magnetic system, the frustration bond is usually superior to resist the thermal fluctuation of spins to the ordinary bond without frustration [25] at a definite temperature. In the vicinity of the point that the magnetization in layer A reorientates, the sublattice magnetization of sublattice A2 exceeds the sublattice magnetization of sublattice A1 by flipping their orientations. As a result the spins in sublattice A1 become frustrated. With the increase in the temperature, the spins in sublattice A1 flip again and cancel out the frustration. In the Fe film on the (0 0 1) surface of AFM NiO, it is found that the Fe domains change to follow the NiO domains [6]. The maximum of the magnetic susceptibility of layer A between the flipping point and the transition point for J AB ¼ 0:2J A corresponds to the transition of pure monolayer A. If J AB ¼ 0, we find that the transition temperature ðkB T C =J A Þ of layer A is 0.86, and that of layer B is kB T N =J A ¼ 1:18. Although they are bigger than those obtained by using the expanded Bethe-Peierls approximation [28], they are more accurate than those obtained by the original mean-field approximation. Fig. 4 shows another kind of phenomenon. When the interlayer exchange coupling is weak, J AB =J A ¼ 0:2, with increasing temperature, the sublattice magnetizations of layer B decrease initially, and they decrease abruptly at a definite temperature, and afterwards the sublattice magnetization of B2 flips to the opposite direction, gets to the
same direction as that of B1, and the spin structure in layer B becomes completely parallel to that of layer A. Meanwhile, the sublattice magnetizations of layer A decrease monotonously. Magnetic susceptibility of layer B exhibits a maximum at the flipping point. There may be two reasons as to why this phenomenon occurs: the first is the negative single-ion anisotropy in layer B, the second is the interlayer exchange coupling between layers A and B. The negative single-ion anisotropy D tends to make the system lie in the S ¼ 0 state, so the first-order phase transition usually occurs in the integer S Ising system [16]. From Eq. (17b) we find that the critical value of D at which layer B is in the S ¼ 0 state is ð16J B þ 3J AB Þ=5; the present value of D=J A ¼ 1:0 does not get this critical value. However, the tendency of the first-order phase transition in layer B still exists. Through the interlayer exchange coupling, the stable FM layer A brings the unstable AFM layer B to a new state in which the orientation of the magnetization of sublattice B1 turns to the direction of FM layer A. The spin reorientation near the NiO interface with FM layer has been detected in Refs. [6–8]. When J AB =J A ¼ 1:2, layer A is in the AFM ground state. The critical value of the single-ion anisotropy at which the first-order phase transition occurs in layer B is J AB þ 16J B =5. The present value of D=J A ¼ 1:0 is far from the critical value, so the tendency of the first-order phase transition in layer B is very small, and the magnetic structure of layer B is stable. However, layer A is in a frustrated state. When the temperature gets to a definite value, the FM exchange within layer A is superior to the interlayer exchange coupling between layers A and B in the competition between the two exchange interactions, and layer A returns to an FM state. This is a process of resisting the thermal fluctuation, and the transition is smooth, so the susceptibilities have a normal behavior with only one singularity. The total magnetization and the total magnetic susceptibility are given in Figs. 5–7. From Figs. 5 and 6, it is clearly seen that a compensation point appears for J AB ¼ 0:2J A , and magnetic susceptibility exhibits two singularities and a maximum. During the interlayer exchange coupling, J AB ¼ 1:2J A , no compensation point appears. For J AB ¼ 2:0J A , the total magnetization is close to zero at a low temperature; however, when the temperature gets to a definite value, it appears and disappears again at the transition temperature. The magnetic susceptibility exhibits only a singularity for the two interlayer exchange couplings. For D ¼ 1:0J A in Fig. 7, at J AB ¼ 0:2J A , the behavior of the total magnetization is different from those in Figs. 5 and 6, and the magnetic susceptibility exhibits a maximum and a singularity. For J AB ¼ 1:2J A and 2.0JA, the behaviors of the total magnetization and the magnetic susceptibility are similar to those for the single-ion anisotropy equal to zero or greater than zero. Fig. 8 shows the dependence of the transition temperature on the interlayer exchange coupling between layers A
ARTICLE IN PRESS A. Du et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 233–239
reaches a definite value, the sublattice magnetizations in layer B decrease abruptly and get to the same direction as that of the magnetization in layer A, the corresponding susceptibility of layer B exhibits a maximum at the flipping point, and there is no compensation in the total magnetization. At J AB =J A ¼ 1:2, layer A is in an AFM state at ground state; when the temperature increases, it changes to the FM state at finite temperature. When J AB =J A ¼ 2:0, this transition does not appear.
1.7 1.6
D/JA= 1.0
kBTN/JA
1.5 1.4
0.0
1.3 1.2 1.1
-1.0
1.0
Acknowledgments
JB=-0.5JA
0.9 0.0
239
0.5
1.0 JAB/JA
1.5
2.0
Fig. 8. The transition temperature as a function of the interlayer coupling for a few values of the single-ion anisotropy parameters. J B =J A ¼ 0:5.
This work was supported by the Special Funds for the Major State Basic Research Projects of China under Grant no. G2000067104. References
and B. The transition temperature increases as the interlayer exchange coupling increases. 4. Concluding remarks The linear cluster mean-field approximation is applied to study the magnetic properties of the magnetic superlattice consisting of an FM monolayer A with spin-1/2 and an AFM monolayer B with spin-1 and single-ion anisotropy. The interlayer interaction between the A and B monolayers is FM, so the system is in a frustrated state. As the AFM intralayer exchange interaction in layer B is half of the FM intralayer exchange interaction in layer A, J B ¼ 0:5J A , we calculate the temperature dependence of layered (total) spontaneous magnetization and initial susceptibility and the transition temperature for different values of interlayer coupling, single-ion anisotropy on the B monolayers. When the single-ion anisotropy in layer B is equal or greater than zero, DX0, for weak interlayer exchange coupling, J AB =J A ¼ 0:2, it is found that the two sublattice magnetizations in layer A change their directions simultaneously, from an FM state to another FM state with opposite spontaneous magnetization direction at a definite temperature, the total spontaneous magnetization appears as a compensation point, and the corresponding magnetic susceptibilitiy exhibits a singularity at this flipping point besides the singularity at the transition temperature and a maximum corresponding to the layer A monolayer. When interlayer exchange coupling is big, J AB =J A ¼ 1:2 or 2.0, the spin structure in layer A is AFM at ground state, and the total magnetization for J AB =J A ¼ 2:0 has a compensation phenomenon and the magnetic susceptibility exhibits only one singularity at the transition temperature for the two cases. When the single-ion anisotropy in layer B is smaller than zero, D ¼ 1:0, for weak interlayer exchange coupling, J AB =J A ¼ 0:2, layer A is in an FM state and layer B is in an AFM state at ground state; however, as the temperature
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