Monte Carlo study of the magnetic properties in a bilayer dendrimer structure with non-magnetic layers

Author’s Accepted Manuscript Monte Carlo Studyof the magnetic properties in a bilayer dendrimer structure withnon-magnetic layers A. Jabar, R. Masrour www.elsevier.com/locate/ssc

PII: DOI: Reference:

S0038-1098(17)30297-1 http://dx.doi.org/10.1016/j.ssc.2017.09.014 SSC13281

To appear in: Solid State Communications Received date: 10 April 2017 Revised date: 7 September 2017 Accepted date: 13 September 2017 Cite this article as: A. Jabar and R. Masrour, Monte Carlo Studyof the magnetic properties in a bilayer dendrimer structure withnon-magnetic layers, Solid State Communications, http://dx.doi.org/10.1016/j.ssc.2017.09.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Monte Carlo Studyof the magnetic properties in a bilayer dendrimer structure withnonmagnetic layers A. Jabar and R.Masrour* Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, 63 46000, Safi, Morocco.

Abstract In this paper, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY)interactions and magnetic layer effects on the bilayer transitions of a spin-5/2 Blume-Capel model formed by two magnetic blocs separated by a non-magnetic spacer of finite thickness. The thermalization process of magnetization for systems sizes has been given. We have shown that the magnetic order in the two magnetic blocs depend on the thickness of the magnetic layer. In the total magnetization profiles, the susceptibility peaks correspond to the reduced critical temperature. This critical temperature is displaced towards higher temperatures when increasing the number of magnetic layers. In addition, we have discussed and interpreted the behaviors of the magnetic hysteresis loops.

Keywords: A.RKKY Interactions; B. Monte Carlo Simulations; C. Bi-layer Structure; D. Non-Magnetic Layer. PACS: 02.50.Ng, 05.10.Ln, 61.50.Ah, 61.20.Ja, 02.50.Ga.

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1. Introduction Recently, there have been many theoretical studies of mixed-spin Ising ferrimagnetic systems. These systems have been of interest becausethe magnetic properties of multi-layers have received much interest due to their novel properties leading to a range of applications including the possibility of spintronic-based devices [1,2].Moreover, the increasing interest in these systems is been mainly related to the potential technological applications of these systems in the area of thermomagnetic recording [3].In addition, the equilibrium behaviors of multi-layer Blume-Capel systems consisting of single ions, namely spin-1/2, spin-1, spin-3/2, spin-2, etc., have been extensively studied by applying the theoretical methods used in equilibrium statistical physics, such as the mean field theory [4].The exact recursion equation on the dendrimer structure [5], the renormalization group method [6], and Monte Carlo simulations [7].Experimentally, several works is been published on the properties of the magnetic multi-layer such as Fe/Ni [8], SiGe, SiC and GeC [9].The metallic multi-layers, a giant magneto-resistance (GMR) effect [10] is been observed on applying a magnetic field, which depends on the number and thickness of the multi-layers. At certain thicknesses the Ruderman–Kittel–Kasuya–Yosida (RKKY) coupling [11] between adjacent ferromagnetic layers with a crystal field becomes anti-ferromagnetic, making it energetically preferable for the magnetizations of adjacent layers to align anti-parallel [12–15]. Indeed, the effect of RKKY interactions, crystal field and quantum transverse anisotropy on the multi-layer transitions and magnetic properties of spins-3/2 and 7/2 for a Blume–Capel model has been studied in detail [16,17], using Monte Carlo simulations and mean field theory. In simulating the multilayer systems, either ferromagnetic or antiferromagnetic interactions or their combinations are considered, and thus theyhave been studied for many different systems [1820]. On other hand, the Ising model with three alternative layers on the honeycomb and

2

square lattices isbeen studied by usingthe effective-field theory with correlations [21]. In previous work, it was been found that the bilayer spin-2 Ising model on the Bethe lattice exhibits both first- and second-order phase transitions and tri-critical points [22,23]. The thermal behaviors of the order-parameters, the total and staggered magnetizations of the two layers and the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers for spin-3/2 Ising have been studied using effective field theory [24,25]. In this paper is to study, using Monte Carlo simulations, the effect of the magnetic and nonmagnetic thicknesses and the RKKY interactions on a multi-layers transitions and magnetic properties of a spin-5/2 Blume–Capel model in a Dendrimer structure of a system formed by two magnetic multi-layer materials, separated by a non-magnetic spacer of thickness. The paper is organized as follows. In Section 2, we present the Monte Carlo simulation, in Section 3 the Blume-Capel Model. Results and discussion are been given in Section 4 and Section 5 is reserved for conclusion. 2.Monte Carlo simulations Sandwiches of magnetic(M)/nonmagnetic(NM)/magnetic(M) materials have attracted several theoretical and experimental studies because of their technological applications. In this material, the coupling exchange undergoes an oscillation period known as the RKKY interaction [11].We consider a spin-5/2 ferromagnetic dendrimer structure formed by two magnetic blocs A and B separated by a non-magnetic layer of thickness NM. The blocs A and B are formed by N1 and N2 magnetic layers respectively (see Fig. 1). 3. Blume-Capel Model The Blume-Capel model was introduced by Blume [26] and independently by Capel [27]. This model can be applied to describe many different physical situations such as multicomponent fluids, ternary alloys, 3He4He mixtures and various magnetic problems [28]. The Hamiltonian of thissystem is given by:

3

H   J1  Si S j  J 2  Si S k  i , j 

i ,k 



Z ,Z ' 

J Z ,Z '

SS i

i , j '

j

  Si2  h Si i

(1)

i

where, the first sum runs over all pairs of nearest-neighbor sites of the system composed with the two magnetic layers(N1, N2) and each layer has NS=46.



Means the summation over

 Z ,Z ' 

all pairs of layers Z (Z∈ bloc A) and Z’ (Z’∈ bloc B).The summation  runs over axially i, j 

(with the axis being perpendicular to the non-magnetic layers) connected pairs of sites and in layers ZandZ’, respectively, see Fig. 1. The mean summation  runs over all site of the i

layers N1 and N2. J1 Stands for the exchange coupling interaction between Si-Sj in the same layer and J2stands the exchange coupling interaction between Si-Sk in the perpendicular layers. Si and Sj are the spin random variables, which take the values 5/2, 3/2, 1/2. Δ and h stand for the crystal field and magnetic external fields applied on all the sites of the system.

J Z ,Z ' is the ``RKKY-like'' coupling, across the non-magnetic layers, between the layersZ andZ’.This parameter is given by:

J z ,z' where

a2 J0  cos(k f ( Z  Z ' )) ' 2 (Z  Z )

(2)

is the Fermi level (which is assumed to be constant in this work

=0.5, see Ref

[11]), ais the lattice constant,J0is a magnetic coupling constant [12]. For simplicity, we assume a=1 and J0=1, see Ref. [12].Forsimplicity, we will normalize the all system parameters by the constant J2. So that, R1 = J1/J2 and we have given in table 1 the different parameters used in each case and also of the system sizes.

4

Fig. 1:A schematic illustration of the model [29].

The dendrimer structure consisting M/NM/M multilayers is separated by a non-magnetic layer formed by ferromagnetic spin-5/2 configuration and is assumed to reside in the unit cells. We apply a standard sampling method to simulate the Hamiltonian given by Eq. (1). The free boundary conditions are applied to our systems.and the configurations were generated by sequentially traversing the lattice and making single-spin flip attempts. The flips are accepted or rejected according to a heat-bath algorithm under the Metropolis approximation such as given in chart of Monte Carlo simulations.

5

Flow chart of Monte Carlo simulations. Our data were generated with 106 Monte Carlo steps per spin, discarding the first 105 Monte Carlo simulations. Starting from different initial conditions, we performed the average of each parameter and estimate the Monte Carlo simulations, averaging over many initial conditions. Our program calculates the following parameters, namely: The magnetization of system: M tot 

1 N

S

(3)

i

i

the internal energy per site:

E

1 H N

(4) 6

with N  ( N1  N 2 )  N S The magnetic susceptibility of the system:



2  tot  M tot  M tot

where  

2



(5)

1 , T denotes the absolute temperature and k B is the Boltzmann’s constant. k BT

4. Results and discussion In this section, we study the magnetic properties of a system formed by a two magnetic layers N1 in the bloc A and N2 in the bloc B,separated by non-magnetic layers of thickness NM=1, 2, 3… etc. Fig. 1 illustrates a schematic illustration of this model, showing the RKKY interaction. The exchange coupling RKKY interaction, across the non-magnetic layers, between the layers Z and Z’is illustrated by Eq. (2). In Figs. 2(a) and 2b, we show a typical thermalization process for systems of size N1=N2=5, 10, 20 at temperatures T/J2=1.0 and T/J2=10.0, respectively with NM=5, Δ/J2=0.5, h/J2=1.0, R1=1.0. From Fig.2, we can also see that we have different thermalization times for deferent random number seeds, despite having the same initial start. This gives rise to a potential criteria for testing for thermalization.

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2,5

2,0

(a)

Mtot 1,5

N1=N2=5 N1=N2=10

1,0

N1=N2=20 T/J2=1.0 0

500

1000

1500

2000

2500

time step

3000

1,8

1,5

1,2

Mtot 0,9

(b)

N1=N2=5 N1=N2=10

0,6

N1=N2=20 T/J2=10.0

0,3 0

3000

6000

9000

12000

time step Fig.2: Typical thermalization process of magnetization for systems sizes N1=N2=5, 10, 20 for T/J2=1.0 (a) and T/J2=10.0 with NM=5, Δ/J2=0.5, h/J2=1.0, R1=1.0. Furthermore, it is apparent that the thermalization time increases with and that an equilibrium is reached much faster when starting from a polarized state. In order to illustrate the temperature effect on the total magnetizations and susceptibilities, we plot in Fig. 3, the behavior of these quantities. In Fig. 3(a), we plot the total magnetizations and susceptibilities as a function of the reduced temperature T/J2 for fixed values of NM=5, 8

Δ/J2=0.5 and h/J2=1.0, for different values of magnetic layers: N1=N2=2, 3 and 5, at fixed value of R1=1.0. It is found that, when increasing the number of magnetic layers fromN1=N2=2to N1=N2=5: (i) for low temperatures, the saturation total magnetizations at the same value Mtot=2.5. (ii) The total susceptibilities peak, corresponding to the reduced critical temperature TC/J, is displaced towards higher temperatures. This is because the increasing of the number of magnetic layers, the effect of the RKKY interaction is more important. Hence, the system disorders at higher temperatureswhen increasing the number of magnetic layers.On other hand, we plot in Fig. 3(b) the total magnetizations and susceptibilities as a function of the reduced temperatureT/J for fixed values of NM=5,Δ/J2=0.5, N1= N2=3 and h/J2=1.0,for different reduced exchange interactions R1 =0.5, 1.0 and 1.5. We found that, when increasing the reduced exchange interactions from R1=0.5 to R1=1.5: (i) for low temperatures, the total magnetizations start at the same value Mtot=2.5. (ii) The total susceptibilities peak, corresponding to the reduced critical temperature TC/J, is displaced towards higher temperatures. The system disorders at higher temperatures because the increasing of the reduced exchange interactionsand in Fig. 3(c) the total magnetizations and susceptibilities as a function of the reduced temperature T/J for fixed values of NM=5, Δ/J2=0.5, R1=1.0, N1=3 and h/J2=1.0, for different magnetic layers N2=3, 10 and 20. We found that, when increasing the magnetic layers N2 in this figurehas shown a smallvariation of the reduced critical temperature.

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3,0

(a)

N1=N2=2

tot

N1=N2=3

2,5

N1=N2=5

Mtot, tot

2,0

Mtot

1,5

1,0

0,5

0,0 0

10

20

T/J2

30

40

50

.

3,0

tot

(b)

R1=0.5 R1=1.0

2,5

R1=1.5

Mtot, tot

2,0

Mtot

1,5

1,0

0,5

0,0 0

10

20

T/J2

10

30

40

50

(c)

2,5

N2=3

tot

N2=10 N2=20

2,0

Mtot

Mtot

1,5

1,0

0,5

0,0 0

10

20

T/J2

30

40

50

Fig. 3:The total magnetizations and susceptibilities as a function of the reduced temperature T/J2 for fixed values of NM=5, Δ/J2=0.5 and h/J2=1.0. (a)for different values of magnetic layers: N1=N2=2, 3 and 5, at fixed value of R1=1.0(b) For different reduced exchange interactions R1 =0.5, 1.0 and 1.5 at fixed value of N1=N2=3. (c) For different values of magnetic layers N2=3, 10 and 20 at fixed value of N1=3 and R1=1.0.

Moreover, the total magnetizations versus the reduced crystal field /J2 are plotted in Fig. 4 for fixed values of h/J2=0.5, T/J2=1.0 and NM=5. In fact, in Fig. 4(a) plotted for fixed value of R1=1.0 and different values of N1 and N2.For low magnetic layer values, see Fig. 4(a), the magnetizations

increase from

their

null

value,for /J2<-2.5, to

their

saturation

valueMS(tot)=2.5=(5/2*10+5/2*10)/20 for N1 and N2=10. Besides, for higher magnetic layer values, with increasing in absolute values of /J2, the magnetizations increase from nonnullvalues reaching their saturation value MS(tot)=2.5=(5/2*10+5/2*10)/20 for N1 and N2=10. This is due to the competition of different parameters which areN1, N2 and R1. The same arguments are valid when fixing the magnetic layer valuesand varying the reduced exchange 11

interactions R1, see Fig. 4(b). In fact, the decreasing values of the exchange interaction R1 is to promote the disorder state. In Figs.4a and 4b, the increasing of crystal field in absolute decreases the total magnetization because the crystal field plays the role of effect the temperature. The total magnetization increases with increasing the size of system (magnetic layer) and exchange interactions for a fixed value of crystal field.

N1=N2=2

2,5

N1=N2=3 N1=N2=5

2,0

N1=N2=10

Mtot

1,5

1,0

0,5

(a) -4

-3

-2

-1

/J2

0

R1=0.5

2,5

R1=1.0 R1=1.5

2,0

R1=2.0 1,5

Mtot

1,0

0,5

(b)

0,0 -5

-4

-3

-2

-1

0

1

/J2 Fig. 4:The total magnetizations as a function of the reduced crystal field for fixed values of h/J2=0.5, T/J2=1.0 and NM=5. (a) For different values of magnetic layers: N1=N2=2, 3, 5 and 12

10, at fixed value of R1=1.0. (b) For different reduced exchange interactions R1 =0.5, 1.0, 1.5 and 2.0, at fixed value of N1=N2=3. Furthermore, in order to study the effect of the reduced exchange interaction R1, we plot in Figs. 5 (a,b) the behavior of the total magnetizations, for fixed values of h/J 2=0.5, T/J2=1 and N1=N2=3. In Fig. 5(a) we illustrate the effect of varying the reduced crystal field values from Δ/J2=0.0 to -1.2, at fixed value of the non-magnetic layers NM=5. It is shown that the increasing the reduced crystal field absolute values is to promote the disorder state, as it is already outlined. For (R1)C>-0.2, the total magnetization increases until their reach saturation magnetization MS(tot)=2.5 and increases with decreasing of crystal field in absolute value. But for (R1)C≤-0.2, the total magnetization decreases until it becomes almost zero and increases when the crystal field in absolute value. The total magnetization increases with increasing the exchange interactions. On other hand, Fig. 5(b) illustrates the effect of increasing the nonmagnetic layers from NM=1, 2 to 12, at fixed value of the reduced crystal field: Δ/J2=0. It is clear, that from this figure that the total magnetizations are not affected by the increasing nonmagnetic layers NM for R1>0 and R1<-0.28, but for a weak values of R1, we see that the total magnetization increases rapidly with NM layers. This is due the fact that the RKKY interaction vanishes as: 1/(NM)2.

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/J2=0.0

2,5

/J2=-1.0 2,0

/J2=-1.2

1,5

Mtot 1,0 0,5

0,0

(a)

-1,0

-0,5

0,0

0,5

1,0

R1

NM=1 NM=2 NM=12

2,5

2,0

1,5

Mtot

1,0

0,5

0,0

(b)

-0,5 -0,4

-0,2

0,0

R

0,2

0,4

1

Fig.5: The total magnetizations as a function ofthe reduced exchange interactions R1for fixed values of h/J2=0.5, T/J2=1 and N1=N2=3(a) for different values of reduced crystal field: Δ/J2=0.0, -1.0 and -1.2, at fixed value of NM=5. (b) for different non-magnetic layers NM=1, 2 and 12, at fixed value of Δ/J2=0. Finally, we discuss the magnetic hysteresis cycles in Fig.6, in the presence of the magnetic layer (N1, N2), the reduced exchange interaction (R1), the reduced temperature (T/J2) and the reduced crystal field (Δ/J2), at fixed value of the non-magnetic layers NM=5. It is found that the behavior of the topologies of such loops is caused by varying different 14

parameters. We also found that the magnetic coercive field increases with the increasing the number of magnetic layer N1 and N2, see Fig. 6(a). Besides this, we show inFig. 6(b) the effect of increasing exchange interaction on the magnetic hysteresis cycle.The area of the hysteresis cycle decreases with decreasing of the reduced exchange interaction R1. To examine the effect of the reduced temperatureT/J2 on the magnetic hysteresis cycle,we illustrate in Fig. 6(c) such behavior. It is found that the area surfaceof the hysteresis cycle decreases with increasing of the reduced temperature.In Fig. 6(d) the behavior of the magnetic hysteresis cycle for different values of the reduced crystal field Δ/J2. It is clear that the area of such cycle decreases with the increasing of the reduced crystal field absolute values.Finally, we summarize in Fig. 6(e) the behavior of the magnetic hysteresis cycle for different values of the magnetic layers N2. It is found that the area of such cycle small increases with the increasing of the number of magnetic layers.

3

2

N1=N2=2

1

N1=N2=3

Mtot0

N1=N2=10

-1

-2

(a)

-3 -4

-2

0

h/J2

15

2

4

3

1

R1=1.0

R1=0.5

2

R1=1.5

Mtot0 -1

-2

(b)

-3 -4

-2

0

h/J2

2

4

3

1

T/J2= 4.5

Mtot0

T/J2= 1.0

T/J2= 2.5

2

-1

-2

(c)

-3 -4

-2

0

h/J2

16

2

4

3

1

/J2=0.0

Mtot0

/J2=-1.0

/J2=-1.5

2

-1

-2

(d)

-3 -4

3

-2

0

h/J2

2

4

(e)

2

1

Mtot0 -1

N2=3

-2

N2=10 N2=20

-3 -4

-2

0

h/J2

2

4

Fig. 6:The magnetic hysteresis cycles of the studied system at fixed value of the nonmagnetic layers NM=5. (a) For different values of magnetic layer values: N1=2, 3 and 10, at fixed values of Δ/J2=0.0, R1=1.0, N2=3and T/J2=1.0. (b) For different values of the reduced exchange interactions R1 =0.5, 1.0 and 1.5, at fixed values of N1=N2=3, Δ/J2=0.0 and T/J2=1.0. (c) For different values of the reduced temperature T/J2 =1.0, 2.5 and 4.5, at fixed values of N1= N2=3, Δ/J2=0.0 and R1=1. (d) For different values of the reduced crystal field values: Δ/J2=0.0, -1.0 and -1.5, at fixed values of T/J2=1.0, N1= N2=3, R1=1.0. (e) For

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different values of the number the layer of bloc B N2=3, 10 and 20 at fixed values of T/J2=1.0, N1=3,Δ/J2=0.0, R1=1.0. Figs Parameters Fig.3(a) NM=5, Δ/J2=0.5, h/J2=1.0, R1=1.0, N2=3, N1=2, 3 and 5 Fig.3(b) NM=5, Δ/J2=0.5, h/J2=1.0, N1=N2=3, R1 =0.5, 1.0 and 1.5 Fig.3(c) NM=5, Δ/J2=0.5, h/J2=1.0, R1=1.0 , N1=3, N2=3, 10 and 20 Fig.4(a) NM=5, h/J2=0.5, T/J2=1.0, N2=3 , R1=1.0, N1=2, 3, 5 and 10 Fig.4(b) NM=5, h/J2=0.5, N1=N2=3, T/J2=1.0, R1 =0.5, 1.0, 1.5 and 2.0 Fig.5(a) h/J2=0.5, T/J2=1, N1=N2=3, NM=5, Δ/J2=0.0, -1.0 and -1.2 Fig.5(b) h/J2=0.5, T/J2=1, N1=N2=3, Δ/J2=0, NM=1, 2 and 12 Fig.6(a) NM=5, Δ/J2=0.0, R1=1.0, T/J2=1.0, N1=N2=2, 3 and 10 Fig.6(b) NM=5, N1=N2=3, Δ/J2=0.0, T/J2=1.0, R1 =0.5, 1.0 and 1.5 Fig.6(c) NM=5, N1=N2=3, Δ/J2=0.0, R1=1, T/J2 =1.0, 2.5 and 4.5 Fig.6(d) NM=5, T/J2=1.0, N1=N2=3, R1=1.0, Δ/J2=0.0, -1.0 and -1.5 Fig.6(e) NM=5, T/J2=1.0, N1=3, Δ/J2=0.0, R1=1.0, N2=3, 10 and 20 Table 1: The table of the parameters used in each case and also of the system sizes.

4. Conclusions Using Monte Carlo simulations, we have studied the effect of the RKKY interactions and magnetic layers on a bi-layer system of a spin-5/2 Blume-Capel model formed by two magnetic blocs separated by a non-magnetic spacerof finite thickness. We have shown that the magnetic order in the two magnetic blocs depends on the thickness ofthe magnetic layers. The critical temperature is displaced towards higher temperatures values when increasing the number of magnetic layers. The total magnetization increases with increasing the size of magnetic layer and exchange interactions. In addition, we have discussed and interpreted the behaviors of the magnetic hysteresis loops. We see that when we fixed N1 and we varied the N2, the transition temperature and magnetic coercive aren't changed, because the magnetic moment of spins is the same for two spins.

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References [1] G. Schmidt, Encyclopedia of Materials: Science and Technology, (2004) 1 [2] G. Vignale and I. D'Amico, Solid State Communications, 127 (2003) 829. [3] J. Kohlhepp, F.J.A. Den Broeder, J. Magn. Magn. Mater. 156 (1996) 261. [4] D.L. Mills, Phys. Rev. B 3, (1971)3887. [5] E. Albayrak, A. Yiğit, T. Cengiz, Physica A 389, (2010)2522. [6] K. Ohno, Y. Okabe, Phys. Rev. B 39, (1989)9764. [7] P. Cossio, J. Mazo-Zuluaga, J. Restrepo, Physica B 384, (2006)227. [8] R. Krishnan, H.O. Gupta, H. Lassri, C. Sella, and J. Kaaboutchi, J. Appl. Phys. 70, (1991)6421. [9] L. Pan, H.J. Liu, Y.W. Wen, X.J. Tan, H.Y. Lv, J. Shi, X. F. Tang, Phys. Lett. A 375, (2011) 614. [10] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472. [11] M.A. Ruderman, C. Kittel, Phys. Rev. 96 (1954) 99; T. Kasuya, Progr. Theoret. Phys. 16 (1956) 45; K. Yosida, Phys. Rev. 106 (1957) 893. [12] L.L. Hinchey, D.L. Mills, Phys. Rev. B 33 (1986) 3329. [13] P. Grünberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Phys. Rev. Lett. 57 (1986) 2442. [14] C. Carbone, S.F. Alvarado, Phys. Rev. B 36 (1987) 2433. [15] D.M. Saul, M. Wortis, D. Stauffer, Phys. Rev. B 9 (1974) 4964. [16] R. Masrour, A. Jabar, A. Benyoussef, M. Hamedoun. J. Magn. Magn. Mater.401(2016) 695. [17] L. Bahmad, A. Benyoussef and A. El Kenz, J. Magn. Magn. Mater. 320( 2008) 397. 19

[18] Erhan Albayrak, Seyma Akkaya, Tunc- Cengiz. JMMM. 321 (2009) 3726–3733. [19] Erhan Albayrak, Seyma Akkaya, Tunc- Cengiz. JMMM. 321 (2009) 108–116. [20] E Albayrak and S Akkaya. Phys. Scr. 76 (2007) 354–362. [21] Bayram Deviren, Osman Canko, Mustafa Keskin. JMMM. 320 (2008) 2291– 2299. [22]E Albayrak, S Akkaya and S Yilmaz. Inter. J. Modern Phys. B. 22, (2008) 4877-4898. [23] E Albayrak, S Akkaya and S Yilmaz. Inter. J. Modern Phys. B. 22, (2008) 4189-4203. [24] Erhan Albayrak, Saban Yilmaz, Seyma Akkaya. J. Magn. Magn. Mater. 310 (2007) 98– 106. [25] Erhan Albayrak, Saban Yilmaz, Seyma Akkaya. Physica A 381 (2007) 189–201. [26] M. Blume, Phys. Rev. 141 (1966) 517. [27] H.W. Capel, Physica 32 (1966) 966. [28] I..D. Lawrie, S. Sarback, C. Domb, J..L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena 9, Academic Press, London, 1988. [29] A. Jabar, N. Tahiri, L. Bahmad, A. Benyoussef. Physica A 462 (2016) 1067–1074.

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Highlights - RKKY interactions and magnetic layer effects on the bilayer transitions is investigated. - Spin-5/2 Blume-Capel model formed by two magnetic blocs separated by a non-magnetic is studied. spacer of finite thickness. - Magnetic order in two magnetic blocs depend on the thickness of magnetic layer is obtained. - Behaviors of the magnetic hysteresis loops are discussed and interpreted.

21