An Ising bilayer system consisting of spin-32 and spin-12 atoms

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 316 (2007) 81–89 www.elsevier.com/locate/jmmm

An Ising bilayer system consisting of spin-32 and spin-12 atoms E. Albayrak, T. Bulut Department of Physics, Erciyes University, 38039 Kayseri, Turkey Received 21 December 2006; received in revised form 27 March 2007 Available online 6 April 2007

Abstract A bilayer Ising model consisting of two Bethe lattices each with a branching ratio of q Ising spins with one of the layers having only spin-32 and the other having only spin-12 is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. The problem was studied by using the exact recursion relations in a pairwise approach in terms of the intralayer bilinear interactions J 1 and J 2 of the upper and lower layers, respectively, and the interlayer bilinear interaction J 3 . After obtaining the ground state phase diagrams on the (J 2 =jJ 1 j; J 3 =qjJ 1 j) plane with either J 1 40 or J 1 o0, the variations of the order-parameters and the free energy were analyzed to obtain the temperature dependent phase diagrams. They were calculated for only the ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers. It was found that the system presents both second- and first-order phase transitions, besides the isolated critical and triple points. The model also presents compensation temperatures when J 2 of spin-12 layer can compete with J 1 of spin-32 layer. r 2007 Elsevier B.V. All rights reserved. PACS: 05.50.þq; 05.70.Fh; 64.60.Cn; 75.10.Hk Keywords: Bilayer; Bethe lattice; Compensation; Spin-32; Spin-12

1. Introduction The study of magnetic thin films consisting of various magnetic layered structures or superlattices has been receiving intense attention for both theoretically and experimentally [1]. These materials are made up with multiple layers of different magnetic substances, thus there is a high potential for technological advances in information storage and retrieval and in synthesis of new magnets for a variety of applications [2]. They also present some interesting novel magnetic properties such as giant magnetoresistance [3], surface magnetic anisotropy [4], enhanced surface magnetic moment [5] and surface magnetoelastic coupling [6]. The molecular-based magnetic materials with spontaneous magnetic moments are of great interest for their potential applications such as in thermomagnetic recording and in devices [7] and it is believed that ferrimagnetic Corresponding author.

E-mail address: [email protected] (E. Albayrak). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.03.202

ordering plays a crucial role in some of these materials. Therefore, the synthesis of new ferrimagnetic materials is an active field in material science. The ferrimagnetic materials consist of two unequal magnetic moments, i.e. a bipartite lattice with spin-S and spin-s with Sas, which interact antiferromagnetically, therefore, their moments do not cancel each other at low temperatures except at the compensation temperatures. The existence of compensation temperatures in ferrimagnets has an interesting application such as the magneto-optical recording [8]. Thus we consider one such system on the bilayer Bethe lattice composed of spin-32 and spin-12 Ising spins on the upper and lower single layer Bethe lattices, respectively. The magnetic properties of a ferrimagnetic bilayer system consisting of spin-32 and spin-12 Ising layers in an applied transverse field were examined by the use of the effective field theory (EFT) [9]. Again the magnetic properties of a ferrimagnetic bilayer system with disordered interfaces were investigated with the standard mean field theory (MFT) [10] and the consequences of the disordered interfaces and different anisotropies between the bulk and

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E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

interface for the magnetic properties were examined. The effects of anisotropies on the transition temperatures of the same spin system with disordered interfaces were studied with EFT which is superior to the MFT [11]. The effect of an applied transverse magnetic field on magnetic properties in a ferrimagnetic bilayer system with disordered interfaces consisting of spin-12 and spin-32 atoms were investigated with MFT [12]. A ferromagnetic amorphous bilayer system consisting of two magnetic monolayers with spin-12 and spin-32 and different interaction constants coupled together with interlayer coupling was studied by the use of EFT [13]. The theoretical framework for investigating the critical behaviors of an Ising multilayer system consisting of alternating spin-12 and spin-S ðSX12Þ magnetic layers was given within the cluster approximation introduced into the differential operator technique [14]. The phase diagrams of a ferromagnetic or ferrimagnetic bilayer system consisting of two magnetic monolayers (A and B) with different spins (SA ¼ 12 and SB ¼ 1; 32) and different interaction constants coupled together in a transverse crystal field were studied within the framework of EFT with correlations in Ref. [15]. The ferrimagnetic system consisting periodically of two layers of spin-12 A atoms, two layers of spin-32 B atoms and a disordered interface in between them is characterized by a random arrangement of A and B atoms of Ap B1p type and a negative A–B coupling were examined by the use of EFT within the framework of the probability distribution technique [16]. The phase diagrams of a spin-12 Ising film with a spin-32 atoms on the top and bottom surfaces and the corresponding semi-infinite system were investigated within the framework of EFT [17]. The critical temperatures and the layer longitudinal magnetizations of a ferromagnetic or ferrimagnetic mixed Ising bilayer system with both spin-12 and spin-1 (or spin-32) in a crystal field were investigated by using EFT with a probability distribution technique [18]. The effects of the biaxial crystal field constants on the magnetic properties of the ferrimagnetic bilayer system consisting of two ferromagnetically coupled monolayers with different spins (12 and 1; 32; 2) were studied by using EFT within the differential operator technique for the case of simple cubic Ising-type structures [19]. In this work we consider the bilayer Bethe lattice with the upper layer containing only spin-32 atoms and the lower one having only spin-12 atoms. The problem was approached for the case with ferromagnetic couplings in each of the layers with the intralayer bilinear interactions J 1 and J 2 and either with ferromagnetic or antiferromagnetic interactions between the layers with bilinear interlayer interaction J 3 . In order to obtain the exact solutions of the model, we have used the exact recursion relations in a pairwise approach on the bilayer Bethe lattice [20–22]. The phase diagrams of the model were calculated by studying the variations of the order-parameters and the free energy of the system. The ground state phase diagrams were also obtained as our guide in obtaining the stable solutions of the model.

The rest of the paper is set up as follows: In Section 2, bilayer Ising model was introduced and then the ground state phase diagrams were obtained and discussed. Section 3 was devoted to the obtaining of the order-parameters and free energy of the system in terms of the recursion relations exactly. In Section 4, the thermal and J 3 =J 1 variations of the magnetizations of the two layers and the spin–spin correlation functions between the nearest neighbor (NN) spins of the adjacent layers were presented, then the phase diagrams of the model were obtained on the ðkT=J 1 ; J 3 =J 1 Þ and ðkT=J 1 ; J 2 =J 1 Þ planes for given values of J 2 =J 1 and J 3 =J 1 , respectively. 2. Bilayer Bethe lattice and its ground states The bilayer Bethe lattice [20–22] is an extension of its one-layer version [23]. In the bilayer version of the Bethe lattice each spin has q NN’s, i.e. coordination number, from its own layer and one adjacent NN from the other layer, therefore, in total each spin far from the boundaries has q þ 1 NN’s as shown in Fig. 1. Thus, the Hamiltonian including all the NN bilinear spin interactions on the bilayer Bethe lattice may be written as X X X H ¼ J 1 Si Sj  J 2 si0 sj0  J 3 S i si0 , (1) hiji

hi0 j 0 i

hii0 i

where S i refers to spin-32 with values of 32 and 12 on the upper layer G 1 and si0 refers to spin-12 with the values of 12 on the lower layer G 2 , J 1 and J 2 are the intralayer bilinear interactions of the layers and the first and second summations run over all sites of G 1 and G 2 , respectively. J 3 is the interlayer bilinear interaction of the NN spins between the layers G1 and G 2 , thus the third summation runs over all the adjacent NN sites of G 1 and G 2 . For the exact formulation of the model, one needs to introduce the order-parameters. Thus, the spin-32 of the central pair from G 1 and spin-12 of the central pair from G 2

Fig. 1. The illustration of the bilayer Bethe lattice of coordination number q ¼ 3. G1 and G2 refer to the upper and lower layers containing the spin-32 and spin-12 labeled as Si and si0 , respectively. While J 1 and J 2 are the bilinear interactions of spins in G1 and G2 , J 3 is the one for the NN adjacent spins of G 1 and G2 .

ARTICLE IN PRESS E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

83

have the magnetizations defined as m1 ¼ hS 0 i and

m2 ¼ hs00 i,

(2)

where h  i refers to the usual thermal average. Instead, one may equally use the total magnetization m and the staggered magnetization Z defined as and

Z ¼ 12ðm1  m2 Þ.

(3)

The last order-parameter measures the correlations between the adjacent NN spins of the layers and defined as r ¼ hS0 s00 i  hS0 ihs00 i,

A1

(A) (IV)

As shown in Fig. 2a, the coordinates ðJ 2 =jJ 1 j; J 3 =qjJ 1 jÞ of the multiphase points are A1 ! ð8:8; 2:8Þ;

B1 ! ð0; 0Þ,

C 1 ! ð8:8; 2:8Þ for J 1 40

ð6Þ

and in Fig. 2b A2 ! ð8:8; 2:8Þ; B2 ! ð0; 0Þ, C 2 ! ð8:8; 2:8Þ for J 1 o0

ð7Þ

which means Fig. 2a is the mirror reflection of Fig. 2b with the places of the phases staying same.

(Ferri)(II)

-10 -15

-10

-5 J2/|J1|

0

5

J1<0

10

(I) (Ferro) (III) (M) 5 J3/q|J1|

(5)

Ferromagnetic (Ferro): m ¼ 1, Z ¼ 12, r ¼ 0, Ferrimagnetic (Ferri): m ¼ 12, Z ¼ 1, r ¼ 0, Mixed (M): m ¼ 0, Z ¼ 0, r ¼ 34, Antiferromagnetic (A): m ¼ 0, Z ¼ 0, r ¼ 34, Surface ferromagnetic (SF): m ¼ 34, Z ¼ 34, r ¼ 0.

B1

-5

which is called the spin–spin correlation function. In order to determine the stable solutions of the model, first we have obtained the ground state phase diagrams for J 1 40 and J 1 o0, corresponding to the ferromagnetic and antiferromagnetic couplings in layer G 1 , respectively, on the ðJ 2 =jJ 1 j; J 3 =qjJ 1 jÞ planes. As a result, the ground-state energy in units of jJ 1 j is described by the following Hamiltonian:  X  J1 E J2 J3 ¼ Si Sj þ s 0s 0 þ ðSi si0 þ S j sj0 Þ , qjJ 1 j jJ 1 j jJ 1 j i j qjJ 1 j hplaqi

(I) (II) (III) (IV) (V)

(SF) (V)

0

C1

(4)

where jJ 1 j is equal J 1 when J 1 o0, i.e. antiferromagnetic coupling. The summation goes over all the plaquette consisting of four NN pair spins of the bilayer system with one pair hiji with spin-32 on layer G 1 , one pair hi0 j 0 i with spin-12 on layer G 2 , and two pairs hii0 i and hjj 0 i connecting the layers G 1 and G2 between the two NN spin-32 and spin-12 pairs. The ground state phase diagrams, Fig. 2, were obtained by comparing the values of the energy E=qjJ 1 j for different spin configurations and then the ground state configurations are the ones with the lowest energy for given values of J 2 =jJ 1 j and J 3 =qjJ 1 j. Hence, five different types of ground state configurations with the following values of the orderparameters ðm; Z; rÞ are found:

(Ferro)(I)

(M)(III) 5 J3/q|J1|

m ¼ 12ðm1 þ m2 Þ

J1>0

10

A2 B2

0

(V) (SF) C2

(IV) (A)

-5

(II) (Ferri) -10 -5

0

5 J2/|J1|

10

15

Fig. 2. The ground state phase diagram of the bilayer spin-32 and spin-12 Ising model for: (a) J 1 40 and (b) J 1 o0. Five different phases depending on the spin configurations are indicated as: ferromagnetic (Ferro), ferrimagnetic (Ferri), mixed (M), antiferromagnetic (A) and surface ferromagnetic (SF), which are all separated by multiphase lines.

Phase I represents the usual ferromagnetic (Ferro) ordering, i.e. the ordering in layers G1 and G2 and between them are totally ferromagnetic. Phase II presents ferromagnetic ordering in G 1 and G 2 separately, but magnetizations in G1 and G2 are antiparallel, i.e. the interlayer ordering is antiferromagnetic type, therefore, this phase is called ferrimagnetic phase (Ferri). Phase III shows antiferromagnetic ordering (A) in both layers ðm1 ¼ m2 ¼ 0Þ where interlayer ordering is ferromagnetic (Ferro), therefore, as a whole the system is in a mixed phase (M). Phase IV illustrates the totally antiferromagnetic (A) ordering, i.e. the interaction in both layers and in between them are antiferromagnetic. Phase V represents the ferromagnetic ordering which is equivalent to the case that the ground state of one layer is ferromagnetic and the ground state of the other layer is antiferromagnetic, thus

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named as the surface magnetic phase (SF). These phases were indicated with long and short arrows for the spin pairs (S 0 ; S 1 ) and (s00 ; s10 ) for the layers G 1 and G 2 , respectively. As we mentioned earlier the temperature dependent phase diagrams were calculated in the guidance of the ground state phase diagrams for the case with J 1 40 and J 2 40 only.

3. The calculations of the order-parameters and the free energy of the bilayer system In order to obtain the order-parameters and free energy in terms of the recursion relations on the bilayer Bethe lattice, first we have to calculate the partition function of the system from the Ising Hamiltonian given in Eq. (1). In doing so, it is assumed that adjacent NN spins of G 1 and G 2 are considered as pairs, which has one spin-32 and one spin-12 from the upper and lower layers, respectively, as shown in Fig. 1. The first pair deep inside the bilayer lattice is called the central pair which forms the first-generation spins. This central pair of spins are connected by q NN spin pairs, i.e. coordination number, which form the secondgeneration spins. Each pair of spins in the secondgeneration is joined to ðq  1Þ NN’s. Therefore, in total the second-generation has qðq  1Þ NN’s which form the third-generation and so on to infinity [20–22]. Thus each spin deep inside the bilayer Bethe lattice has ðq þ 1Þ NN spins, q spins from its own layer, i.e. with same type of spin, and one from the adjacent layer, i.e. with the other type of spin. The partition function is given by the definition as X X Z¼ ebH ¼ PðSpcÞ, (8) All Config:

En ¼

gn ð 12 ; 12Þ ; gn ð 32 ;  12Þ

Gn ¼

gn ð 32 ; 12Þ . gn ð 32 ;  12Þ

Fn ¼

gn ð 12 ;  12Þ , gn ð 32 ;  12Þ ð9Þ

As seen each recursion relation is a function of the recursion relations for the NN shell with ðn  1Þ, therefore, these equations are totally nonlinear in their nature. Thus, in order to obtain their numerical values for given system parameters numerical methods are employed. We should also mention that the choice of what ratios of these gn ðS; sÞ functions is to be taken is totally arbitrary. The explicit formulations of the recursion relations were presented in Appendix A in order not to spoil the legibility of the work. In the thermodynamic limit ðn ! 1Þ, these recursion relations determine the states of the system and may be called the equations of state. The magnetizations of spin-32 and spin-12 in the central pair are given, respectively, as m1 ¼ hS 0 i ¼ ½32eð3=4ÞbJ 3 Aqn þ 32eð3=4ÞbJ 3 Bqn þ 12eð1=4ÞbJ 3 C qn þ 12eð1=4ÞbJ 3 Dqn  12eð1=4ÞbJ 3 E qn  12eð1=4ÞbJ 3 F qn  32eð3=4ÞbJ 3 Gqn  32eð3=4ÞbJ 3 =D2 ,

ð10Þ

m2 ¼ hs00 i ¼ ½12eð3=4ÞbJ 3 Aqn  12eð3=4ÞbJ 3 Bqn þ 12eð1=4ÞbJ 3 C qn  12eð1=4ÞbJ 3 Dqn þ 12eð1=4ÞbJ 3 E qn  12eð1=4ÞbJ 3 F qn þ 12eð3=4ÞbJ 3 Gqn  12eð3=4ÞbJ 3 =D2 ,

ð11Þ

and the spin–spin correlation function of the central pair is expressed by

Spc

where PðSpcÞ is considered as an unnormalized probability distribution. Starting from the central pair of spins on the Bethe lattice made up with q separate branches connecting each of the pair of spins, one follows only one of the branches of the tree out of q, therefore, for the full formulation we have to define the partition function for each of these separate branches named as gn ðS; sÞ. It should be mentioned that each spin si0 with spin-12 can take the values 12 and each spin Si with spin-32 can take the values 32 and 12, therefore, for the bilayer model we have to define eight gn ðS; sÞ functions for 2  4 ¼ 8 configurations for each pair of spins at their sites. Thus in this pairwise approach, seven exact recursion relations are defined as the ratios of these partition functions of the separate branches on the bilayer Bethe lattice as

r ¼ ½34eð3=4ÞbJ 3 Aqn  34eð3=4ÞbJ 3 Bqn þ 14eð1=4ÞbJ 3 C qn  14eð1=4ÞbJ 3 Dqn  14eð1=4ÞbJ 3 E qn þ 14eð1=4ÞbJ 3 F qn  34eð3=4ÞbJ 3 G qn þ 34eð3=4ÞbJ 3 =D2  m1 m2 ,

ð12Þ

where D2 ¼ eð3=4ÞbJ 3 Aqn þ eð3=4ÞbJ 3 Bqn þ eð1=4ÞbJ 3 C qn þ eð1=4ÞbJ 3 Dqn þ eð1=4ÞbJ 3 E qn þ eð1=4ÞbJ 3 F qn þ eð3=4ÞbJ 3 G qn þ eð3=4ÞbJ 3 .

ð13Þ

In addition to the variations of the order-parameters, we also need the free energy in terms of the recursion relations to obtain the phase diagrams of the model. Thus, the free energy of the bilayer Bethe lattice is given as

An ¼

gn ð32 ; 12Þ ; gn ð 32 ;  12Þ

Bn ¼

gn ð32 ;  12Þ , gn ð 32 ;  12Þ

bF ¼ LogðD2 Þ þ

Cn ¼

gn ð12 ; 12Þ ; gn ð 32 ;  12Þ

Dn ¼

gn ð12 ;  12Þ , gn ð 32 ;  12Þ

which is used to find the places of the first-order phase transition temperatures and the stable solutions of the

q LogðD1 Þ 2q

ð14Þ

ARTICLE IN PRESS E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

85

model and where 1.5

1.5

þ þ þ þ þ

þe

.

m1, m2

m

1.0

1.0

q=4

0.5

m2

0.0 0

1

2

3

0.5

m;η

ð15Þ -0.5 2

4

The variations of the magnetizations, spin–spin correlation function and the free energy were studied for given system parameters in terms of the recursion relations to obtain the phase diagrams. The calculations were carried out for J 1 40 and J 2 40, ferromagnetic couplings in layers G 1 and G2 , and J 3 40 or J 3 o0, ferromagnetic or antiferromagnetic coupling between the layers, respectively. The obtaining of the second-order phase transition or Curie temperatures, T c , was illustrated in Fig. 3 in terms of the thermal variations of the order-parameters for q ¼ 4. In Figs. 3a and b, the values of J 2 =J 1 and J 3 =J 1 were chosen corresponding to the phases I and II of the ground state phase diagrams as ð1:0; 1:0Þ and ð1:0; 1:0Þ, respectively. As seen in Fig. 3a, m, Z and r begin from 1.0, 0.5 and 0.0 at zero temperature and as the temperature increases they change gradually and at T c , m and Z go to zero and r shows a peak at T ¼ T c . In Fig. 3b in comparison with Fig. 3a, m and Z exchange the roles and r changes sign when J 3 changes sign as expected. The insets illustrate the thermal variations of the layer magnetizations m1 and m2 . Thus it is again clear that when J 3 changes sign, the system moves from the ferromagnetic phase to the ferrimagnetic phase. The behaviors of the order-parameters and free energy at the first-order phase transition temperatures, T t , were illustrated in Fig. 4a. To show the existence of the T t , the order-parameters and the free energy were plotted with respect to the ratio J 3 =J 1 for given values of the temperature, since as seen in the ground state phase

6

kT/J1

1.5

1.5

η

1.0

J2/J1=1.0 J3/J1=-1.0 q=4

m1

1.0 0.5

m2

0.0 -0.5

m; η; ρ

0

4. The variations of the order-parameters and the phase diagrams

5

ρ

0.0

Meanwhile, we should note that in order to obtain the behaviors of the order-parameters and the free energy, first the recursion relations are calculated by using an iteration scheme, then the found values of the recursion relations are inserted into the definitions of the order-parameters and the free energy which are the functions of the interlayer and intralayer bilinear interactions and coordination number q. As a result, we have presented the variations of the orderparameters and obtained the phase diagrams for given values of the system parameters in the next section.

4

η

m1, m2

þ

J2/J1=1.0 J3/J1=1.0

m1

m; η; ρ

D1 ¼ e

q1 An1 ebð9=4J 1 þ1=4J 2 3=4J 3 Þ Bq1 n1 ebð3=4J 1 1=4J 2 þ1=4J 3 Þ C q1 n1 bð3=4J 1 þ1=4J 2 1=4J 3 Þ q1 e Dn1 bð3=4J 1 1=4J 2 1=4J 3 Þ q1 e E n1 bð3=4J 1 þ1=4J 2 þ1=4J 3 Þ q1 e F n1 bð9=4J 1 1=4J 2 3=4J 3 Þ q1 e G n1 bð9=4J 1 þ1=4J 2 þ3=4J 3 Þ

bð9=4J 1 1=4J 2 þ3=4J 3 Þ

1

2

3

4

m

0.5

m;η

ρ

0.0

-0.5 0

2

4

6

kT/J1 Fig. 3. The thermal variations of m, Z and r showing the existence of the second-order phase transitions for q ¼ 4: (a) J 2 =J 1 ¼ 1:0 and J 3 =J 1 ¼ 1:0 and (b) J 2 =J 1 ¼ 1:0 and J 3 =J 1 ¼ 1:0 corresponding the (Ferro) and (Ferri) phases, respectively.

diagrams the zero of the abscissa is the first-order phase line separating the phases I and II. At the first-order phase transition temperature for J 3 =J 1 ¼ 0, the free energy of the system passes from phase I to phase II or vice versa. In Fig. 4a, the change of the total and staggered magnetizations with respect to J 3 =J 1 is obtained for kT=J 1 ¼ 0:7, J 2 =J 1 ¼ 1:0 and q ¼ 4. As seen m ¼ 0:5 and Z ¼ 1:0 until the vicinity of J 3 =J 1 ¼ 0. After making little peaks at J 3 =J 1 ¼ 0, m and Z exchange the roles when J 3 =J 1 changes sign. That is, while m jumps from the values 0:5 to 1:0, Z does exactly the opposite, therefore, they exhibit discontinuities in their values. The insets show that the free energy changes direction at J 3 =J 1 ¼ 0 and that is why r presents two small peaks at the two sides of J 3 =J 1 ¼ 0 with minimum and maximum corresponding to decrease and increase of the free energy, respectively. Meanwhile, the T t temperatures of the total and staggered magnetizations are

ARTICLE IN PRESS

1.0

m η

m 40 30 20 10 0 -20 -10 0

ρ

0.0

-βF

m; η

0.5

J2/J1=1.0

kT/J1=0.7

η

η

-1.0

η

m

0 J3/J1

10

η

kT/J1=4.5 J2/J1=1.0 q=4

η

-1.0 20

-20

-10

ρ

0.000 kT/J1=2.0 J2/J1=1.0 q=4

-0.005

-0.5

-1.0

-0.010 m

-4

-2

0

2

-4

20

J3/J1=0.0 J2/J1=1.0 q=4

m

1.0

Tt

0.8 0.6

η m2

0.2

η

10

m1

0.4

η

4

0 J3/J1

1.2 m, m1, m2, η

m; η;ρ

η

0.005

0.0

m

1.6

m

m 0.010

η

η

1.4 0.5

m

-0.5

q=4

1.0

0.0

5.0 4.5 4.0 3.5 3.0 2.5 2.0 -10 -5 0 5 10 J3/J1 ρ

m

m

-10

m

-20 -10 0 10 20

10 20

-0.5

-20

η

0.5

4e-6 0 -4e-6

m; η; ρ

1.0

-βF

E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

86

Tc2

m

Tc1

0.0 -2

0 J3/J1

2

4

0

1

2 kT/J1

3

4

Fig. 4. The J 3 =J 1 variations of m, Z and r. (a) The obtaining of the first-order phase transitions for kT=J 1 ¼ 0:7. (b) The obtaining of the second-order phase transitions for kT=J 1 ¼ 4:5. (c) The order-parameters for kT=J 1 ¼ 2:0 for J 2 =J 1 ¼ 1:0. (d) The thermal change of the total, staggered and layer magnetizations for J 2 =J 1 ¼ 1:0 and J 3 =J 1 ¼ 0:0.

actually the T c temperatures of the single layer spin-12 Ising system, see Ref. [20] with J 3 ¼ 0. Afterwards the system is driven by spin-32 layer for J 3 ¼ 0, thus at this temperature the system jump to the T c temperatures of the single layer spin-32 Ising system. These will be much clear when we come to Fig. 4d. Fig. 4b, obtained for kT=J 1 ¼ 4:5, J 2 =J 1 ¼ 1:0 and q ¼ 4, demonstrates the obtaining of the second-order phase transitions with respect to the J 3 =J 1 variation. As J 3 =J 1 ! 1, m and Z take the values 0:48 and 0:96. As J 3 =J 1 is increased from left, m and Z decrease until the J 3 =J 1 ¼ 4:16, where they are zero, indicating the secondorder phase transition. At this value r shows its first peak as a minimum. The region enclosed by the wings of the staggered magnetization corresponds to phase II, i.e. the ferrimagnetic phase. The zero values of m and Z continues until J 3 =J 1 ¼ 4:16, but r increases from its minimum to its maximum, i.e. to its second peak at J 3 =J 1 ¼ 4:16, in the paramagnetic region. At J 3 =J 1 ¼ 4:16, m and Z again exchange the roles, therefore, now the region enclosed by the wings of the total magnetization corresponding to phase I, i.e. the ferromagnetic phase. Eventually, as J 3 =J 1 ! 1, m and Z reach the constant values at 0:96 and 0:48. As the temperature is decreased, the wings of

the total and staggered magnetizations get close to each other until some critical temperature, where these two wings just touch each other at J 3 ¼ 0. These critical temperatures are actually at the T c temperatures of the single layered spin-32 system for J 3 ¼ 0 above which the phase is the paramagnetic phase. As shown in the inset again, the free energy for this case is different than obtained in the previous figure, since now free energy varies smoothly with no change of direction at J 3 =J 1 ¼ 0. In the last two figures, we have tried to illustrate what happens between the T t , i.e. the T c of spin-12, and the critical temperature, i.e. the T c for spin-32, when J 3 ¼ 0 and q ¼ 4. In Fig. 4c, we have shown the J 3 =J 1 variations of the order-parameters for kT=J 1 ¼ 2:0 and J 2 =J 1 ¼ 1:0. The value of temperature was chosen in the range between the T t and T c . It is obvious that the order-parameters change continuously when J 3 changes sign indicating that the system does not show any first- or second-order transitions. But now the question is that what is really going on there for J 3 ¼ 0? To answer this, we have plotted the temperature variations of the total, staggered and the layer magnetizations in Fig. 4d for J 3 =J 1 ¼ 0:0 and J 2 =J 1 ¼ 1:0. We know that when J 3 ¼ 0:0, there are two single layer Bethe lattices with zero correlation between

ARTICLE IN PRESS E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

them, i.e. r ¼ 0:0. Thus the magnetization m2 of the spin-12 layer becomes zero at T c2 ’ 0:73 which becomes the T t temperatures of the total and staggered magnetizations from the definition. Above this temperature for J 3 ¼ 0:0, the model reduces to the single layer Bethe lattice for spin-32 with T c1 ’ 4:04. Thus there should be a discontinuity in the phase diagrams for temperatures between T c2 ’ 0:73 and T c1 ’ 4:04 along J 3 ¼ 0:0, since there are no correlations between the layers. After having presented the thermal and J 3 =J 1 variations of the order-parameters and free energy, we are now ready to obtain the phase diagrams of our model on the (J 3 =J 1 ; kT=J 1 ) and (J 2 =J 1 ; kT=J 1 ) planes for given values

100

q=5

q=6

(Ferro)

50

J3/J1

q=4

q=3

Tt4 Tt6 0

(P)

Tt3 Tt5

-50

(Ferri)

-100 0

2

4

6 kT/J1

8

10

12

Fig. 5. The phase diagram of spin-32 and spin-12 Ising model for the bilayer Bethe lattice on the ðkT=J 1 ; J 3 =J 1 Þ plane for q ¼ 326. The effect of increasing the q is illustrated. The crossed line indicates that there are no phase transitions between the T tq and T TP temperatures along J 3 ¼ 0.

100

(Ferro)

J3/J1

50

Tt5Tt4 Tt3 Tt2 Tt1

0

(P)

(Ferri)

-50

q=4 -100 0

1

4

6

8

kT/J1 Fig. 6. The phase diagram on the ðkT=J 1 ; J 3 =J 1 Þ plane for J 2 =J 1 ¼ 0:1, 0.25, 0.5, 0.75 and 1.0 for q ¼ 4. The effect of changing the J 2 =J 1 values is demonstrated. The crossed line indicates that there are no phase transitions between the T ti and T TP temperatures along J 3 ¼ 0.

87

of J 2 =J 1 and J 3 =J 1 , respectively, and q. In the phase diagrams, the solid and dashed lines refer to the T c and T t lines, respectively, and the filled circles indicate the isolated critical temperatures at which the T t lines terminate. The different phase regions indicated with (Ferro), (P) and (Ferri) corresponding to ferromagnetic, paramagnetic and ferrimagnetic phases, respectively. In Figs. 5 and 6, at J 3 ¼ 0:0 the phases (Ferro) and (Ferri) are separated by T t lines at low temperatures until the isolated critical points, i.e. T c2 temperatures for given J 2 =J 1 . Above this temperature the system reduces single layer spin-32 model whose T c1 ’ 4:04. Thus there is no phase transition between these temperatures as indicated with crossed lines in the next two figures. Actually these T c1 temperatures become the triple points of the three phases, i.e. (Ferro), (P) and (Ferri), for each given values of J 2 =J 1 along the J 3 =J 1 ¼ 0. The temperatures of these triple points may be labeled as T TP , where the subscript TP refers to the triple point. A wing-shaped second-order lines emerge from these TPs. While the upper part of these wings separates the phases (Ferro) and (P), the lower part separates the (Ferri) and (P) phases. For J 3 a0 and above the T TP ’s, the layers interact with each other. Therefore, it is clear that the model is the mixed bilayer model only when J 3 a0. The effects of changing the coordination numbers in the phase diagrams are illustrated in Fig. 5 for q ¼ 326, when J 2 =J 1 ¼ 1:0. The general characteristics of the phase diagrams are explained above. The isolated critical point temperatures T tq are approximately at 0.49, 0.73, 1.0 and 1.2 for q ¼ 326, respectively. Meanwhile the TP temperatures, T TPq , are roughly at 2.68, 4.04, 5.35 and 6.64 for q ¼ 326, respectively. In the limit J 3 =J 1 ! 1, the second-order phase transition temperatures become constant at about 4.66, 7.34, 9.93 and 12.48 for q ¼ 326, respectively. These can be interpreted as follows: For J 3 =J 1 ¼ 0, the model has to converge to the behavior of two uncoupled 2D-Ising lattices. One can use Ref. [22, Fig. 5a] to see that when J 2 =J 1 ¼ 1:0, the spin-32 layer orders at kT=J 1 ’ 4:04. Indeed , this is the temperature at which the TP of the present work is located for q ¼ 4. Therefore, we conclude that this TP is just the well-known ordering transition of the single layer Bethe lattice, consisting of spin-32 atoms. In the absence of the interlayer coupling, the lower layer, consisting of spin-12 atoms, stays disordered at this temperature. It orders only at about 0.721 as indicated in Fig. 2b and Ref. [20, Eq. (34)] which is very close to our value T c4 ¼ 0:73 for q ¼ 4. Thus this isolated critical point is the well-known order–disorder transition of a singlelayer Bethe lattice of spin-12 atoms. The consequences of varying the J 2 =J 1 values on the phase diagrams were demonstrated for q ¼ 4 in Fig. 6. In the figure, the T c lines from right to the left were obtained for J 2 =J 1 ¼ 1:0, 0.75, 0.5, 0.25 and 0.1. The isolated critical temperatures, T ti with i ¼ 1; 2; . . . ; 5 corresponding to the values of J 2 =J 1 with the above given order, are 0.73, 0.55, 0.35, 0.17 and 0.1, respectively. So it is clear that as J 2 =J 1 decreases, the T ti ’s move towards left and therefore the

ARTICLE IN PRESS E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

88

first-order phase transition lines become shorter. In addition, as J 2 =J 1 values decrease the paramagnetic phase pushes the second-order phase transition lines towards to lower temperatures. In the limit J 3 =J 1 ! 1 the secondorder phase transition temperatures become constant at 6.68, 6.79, 6.97, 7.15 and 7.34 for the J 2 =J 1 ¼ 0:1; 0:25; 0:5; 0:75 and 1.0 values, respectively. The values of TP temperatures are all same for given values of J 2 =J 1 and thus T TP4 ffi 4:04. The T TP temperatures are independent on J 2 =J 1 , because the only interaction inside the Hamiltonian of an uncoupled, i.e. J 3 =J 1 ¼ 0, spin-32 layer is J 1 , which defines the temperature scale units, since the temperature is taken as kT=J 1 . The T ti temperatures depend on J 2 =J 1 roughly linearly, since J 2 is the only interaction parameter appearing in the Hamiltonian of the uncoupled spin-12 layer. In the search for the compensation temperatures, we have calculated the final phase diagram of this work on the ðkT=J 1 ; J 2 =J 1 Þ plane for given values of J 3 =J 1 . The secondorder phase transition lines start from kT=J 1 ’ 4:1 when J 3 =J 1 p1:0 but as J 3 =J 1 increases they are seen at higher temperatures, i.e. kT=J 1 ’ 4:46, 5.0 and 5.80 for J 3 =J 1 ¼ 5:0, 10.0 and 15.0, respectively. These lines vary linearly with the increasing J 3 =J 1 on this plane. As seen in the inset, for example when J 3 =J 1 ¼ 0:75 and J 2 =J 1 ¼ 10:0, the temperature variations of the net and layer magnetizations show that the model also presents compensation temperatures when J 2 of the spin-12 layer can compete with the J 1 of the spin-32 layer. The compensation lines, i.e. dashed-dotted lines, emerge from the secondorder lines and as J 2 =J 1 increases they become constant. These temperatures are seen at lower values for given lower values of J 3 =J 1 . The studies including dilution [10,12,16],

transverse magnetic field [9], transverse crystal field [15] and uniaxial and biaxial crystal field potential [19] report the existence of the compensation temperatures also (Fig. 7). We should note that for the other values of the coordination numbers the obtained phase diagrams are qualitatively similar, therefore, we have not presented their phase diagrams in here. The only difference is that as q increases the critical temperatures are seen at higher temperatures which was already indicated in the phase diagram, i.e. Fig. 5. In conclusion, we have studied the bilayer spin-32 and spin-12 Ising model on the Bethe lattice in detail in terms of the intralayer coupling constants J 1 and J 2 of the two layers, ferromagnetic case only, interlayer coupling constant J 3 between the layers, ferromagnetic or antiferromagnetic case, for given values of the coordination number q in terms of the recursion relations. Besides the ground state phase diagrams, the variations of the order-parameters and the free energy were studied in detail to obtain the temperature dependent phase diagrams of the model. As a result, we have found that the system presents both first- and second-order phase transitions for all values of q. The T t lines separate the two ordered phases, i.e. (Ferro) and (Ferri) phases, which continues until the isolated critical points. From these points the critical temperatures jump the TPs for J 3 ¼ 0. Then a wing-shaped second-order lines starting from the T TP temperatures appear. Between the two wings of the second-order lines is the disordered phase, i.e. paramagnetic phase. In conclusion, we have also found compensations when J 2 of the spin-12 layer can compete with the J 1 of the spin-32 layer.

Appendix A 50

20

q=4

15 10

40

The seven recursion relations for the spin-32 and spin-12 bilayer Ising model on the Bethe lattice are calculated as

0.5 1.0 0.25 0.75 0.1

5

(Ferro)

An ¼ ½ebð9=4J 1 þ1=4J 2 þ3=4J 3 Þ  Aq1 n1

0 4

5

6

þ ebð9=4J 1 1=4J 2 3=4J 3 Þ  Bq1 n1

30 J2/J1

þ ebð3=4J 1 þ1=4J 2 þ1=4J 3 Þ  C q1 n1

(P)

10 1 5 10 20

m1, m2, |m|

20

1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0

J2/J1=10.0 J3/J1=0.75

|m|

þ ebð9=4J 1 þ1=4J 2 3=4J 3 Þ  Gq1 n1

m1 2

4

6

8

10

0 0

6

12

18

þ ebð3=4J 1 þ1=4J 2 1=4J 3 Þ  E q1 n1 q1 þ ebð3=4J 1 1=4J 2 þ1=4J 3 Þ  F n1

m2

0

q1 þ ebð3=4J 1 1=4J 2 1=4J 3 Þ  Dn1

þ ebð9=4J 1 1=4J 2 þ3=4J 3 Þ =D1 ,

24

kT/J1 Fig. 7. The phase diagram on the ðkT=J 1 ; J 2 =J 1 Þ plane for J 3 =J 1 ¼ 0:1, 0.25, 0.5, 0.75, 1.0, 5.0, 10.0 and 20.0 for q ¼ 4. The system exhibits one compensation for each given set of the system parameters. While one of the inset shows the details of the phase diagram, the other presents the existence of the compensation temperature.

Bn ¼ ½ebð9=4J 1 1=4J 2 þ3=4J 3 Þ  Aq1 n1 þ ebð9=4J 1 þ1=4J 2 3=4J 3 Þ  Bq1 n1 þ ebð3=4J 1 1=4J 2 þ1=4J 3 Þ  C q1 n1 þ ebð3=4J 1 þ1=4J 2 1=4J 3 Þ  Dq1 n1

ð16Þ

ARTICLE IN PRESS E. Albayrak, T. Bulut / Journal of Magnetism and Magnetic Materials 316 (2007) 81–89

þ ebð3=4J 1 1=4J 2 1=4J 3 Þ  E q1 n1

q1 Gn ¼ ½ebð9=4J 1 þ1=4J 2 þ3=4J 3 Þ  An1

þ ebð3=4J 1 þ1=4J 2 þ1=4J 3 Þ  F q1 n1

þ ebð9=4J 1 1=4J 2 3=4J 3 Þ  Bq1 n1

þ ebð9=4J 1 1=4J 2 3=4J 3 Þ  G q1 n1

þ ebð3=4J 1 þ1=4J 2 þ1=4J 3 Þ  C q1 n1

þ ebð9=4J 1 þ1=4J 2 þ3=4J 3 Þ =D1 ,

89

ð17Þ

þ ebð3=4J 1 1=4J 2 1=4J 3 Þ  Dq1 n1 þ ebð3=4J 1 þ1=4J 2 1=4J 3 Þ  E q1 n1

C n ¼ ½ebð3=4J 1 þ1=4J 2 þ3=4J 3 Þ  Aq1 n1

þ ebð3=4J 1 1=4J 2 þ1=4J 3 Þ  F q1 n1

þ ebð3=4J 1 1=4J 2 3=4J 3 Þ  Bq1 n1

q1 þ ebð9=4J 1 þ1=4J 2 3=4J 3 Þ  Gn1

þ ebð1=4J 1 þ1=4J 2 þ1=4J 3 Þ  C q1 n1

þ ebð9=4J 1 1=4J 2 þ3=4J 3 Þ =D1 .

þ ebð1=4J 1 1=4J 2 1=4J 3 Þ  Dq1 n1

ð22Þ

þ ebð1=4J 1 þ1=4J 2 1=4J 3 Þ  E q1 n1 q1 þ ebð1=4J 1 1=4J 2 þ1=4J 3 Þ  F n1

References

þ ebð3=4J 1 þ1=4J 2 3=4J 3 Þ  G q1 n1 þ ebð3=4J 1 1=4J 2 þ3=4J 3 Þ =D1 ,

ð18Þ

Dn ¼ ½ebð3=4J 1 1=4J 2 þ3=4J 3 Þ  Aq1 n1 þ ebð3=4J 1 þ1=4J 2 3=4J 3 Þ  Bq1 n1 þ ebð1=4J 1 1=4J 2 þ1=4J 3 Þ  C q1 n1 þ ebð1=4J 1 þ1=4J 2 1=4J 3 Þ  Dq1 n1 þ ebð1=4J 1 1=4J 2 1=4J 3 Þ  E q1 n1 þ ebð1=4J 1 þ1=4J 2 þ1=4J 3 Þ  F q1 n1 þ ebð3=4J 1 1=4J 2 3=4J 3 Þ  G q1 n1 þ ebð3=4J 1 þ1=4J 2 þ3=4J 3 Þ =D1 ,

ð19Þ

E n ¼ ½ebð3=4J 1 þ1=4J 2 þ3=4J 3 Þ  Aq1 n1 þ ebð3=4J 1 1=4J 2 3=4J 3 Þ  Bq1 n1 þ ebð1=4J 1 þ1=4J 2 þ1=4J 3 Þ  C q1 n1 þ ebð1=4J 1 1=4J 2 1=4J 3 Þ  Dq1 n1 þ ebð1=4J 1 þ1=4J 2 1=4J 3 Þ  E q1 n1 þ ebð1=4J 1 1=4J 2 þ1=4J 3 Þ  F q1 n1 q1 þ ebð3=4J 1 þ1=4J 2 3=4J 3 Þ  G n1

þ ebð3=4J 1 1=4J 2 þ3=4J 3 Þ =D1 ,

ð20Þ

F n ¼ ½ebð3=4J 1 1=4J 2 þ3=4J 3 Þ  Aq1 n1 q1 þ ebð3=4J 1 þ1=4J 2 3=4J 3 Þ  Bn1 q1 þ ebð1=4J 1 1=4J 2 þ1=4J 3 Þ  C n1

þ ebð1=4J 1 þ1=4J 2 1=4J 3 Þ  Dq1 n1 þ ebð1=4J 1 1=4J 2 1=4J 3 Þ  E q1 n1 þ ebð1=4J 1 þ1=4J 2 þ1=4J 3 Þ  F q1 n1 þ ebð3=4J 1 1=4J 2 3=4J 3 Þ  Gq1 n1 þ ebð3=4J 1 þ1=4J 2 þ3=4J 3 Þ =D1 ,

ð21Þ

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