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Magnetic properties of the Ising system on alternate layers of a hexagonal lattice
Accepted Manuscript Magnetic properties of the Ising system on alternate layers of a hexagonal lattice R. Masrour, A. Jabar, A. Benyoussef, M. Hamedoun
PII: DOI: Reference:
S0378-4371(17)30975-5 https://doi.org/10.1016/j.physa.2017.09.083 PHYSA 18688
To appear in:
Physica A
Received date : 17 January 2017 Revised date : 28 August 2017 Please cite this article as: R. Masrour, A. Jabar, A. Benyoussef, M. Hamedoun, Magnetic properties of the Ising system on alternate layers of a hexagonal lattice, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.083 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 proof before it is published in its final 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.
Magnetic properties of the Ising system on alternate layers of a hexagonal lattice R. Masrour1,*, A. Jabar1, A. Benyoussef2,3 and M. Hamedoun2 1)
Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, 63 46000, Safi, Morocco. 2)
Institute of Nanomaterials and Nanotechnologies, MAScIR, Rabat, Morocco. 3)
Hassan II Academy of Science and Technology, Rabat, Morocco. Corresponding authors: [email protected]
Abstract: The magnetic properties of the mixed spin-1 and spin-5/2 Ising ferrimagnetic model on a hexagonal lattice have been studied using the Monte Carlo simulations. The hexagonal lattice study is formed by alternate layers of spins =1 and S=5/2. The thermal variations of the magnetizations are given. The magnetic hysteresis cycle are established for different values of exchange interactions, crystal-field and temperatures. The magnetic coercive field is deduced.
Keywords: Hexagonal lattice; Mixed-spin Ising ferrimagnetic model; Monte Carlo simulation; Critical temperature; Exchange interactions; Magnetic hysteresis cycle.
1
I. Introduction The ferrimagnetic materials, which can be studied with mixed spin systems, are possibly useful for advanced technological applications such as storage media, and the annual sales of computer diskettes, compact disks, optical disks, recording tape, and related items exceed those of the celebrated semiconductor industry[1-4]. In recent years, the some magnetic properties of hexagonal lattice have been studied using the effective-field theory [5,6]. The mixed spins (1,5/2) Ising systems [7-11] have been studied using mean field approximation. In other hand, the kinetic mixed spin-1 and spin-5/2 Ising ferrimagnetic system on alternate layers of a hexagonal has an interesting dynamic behavior and gives rich dynamic phase diagrams [7] and distinct topological diagrams [10]. It was found that the mixed spin-1 and spin-5/2 Ising system on hexagonal lattice exhibits five fundamental phases, nine different mixed phases which are composed of binary and ternary combinations of fundamental phases and the compensation temperature or the N-type behavior in the Néel classification nomenclature [7]. In addition, the thermal phase diagrams exhibit compensation points(where the sublattices have equal magnetizations), first-order and second-order transitions as well as tricritcal and end-points [11]. We should also note that the molecular mixed-spin ferromagnetic chain material MnNi(NO2)4 (ethylenediamine) is known as quasi-onedimensional material which composed by Ni2+ and Mn2+ of spins 1 and 5/2, respectively [12]. Hence, with this motivation, we have studied the mixed spin-1 and spin-5/2 Ising system to investigate their critical behavior on alternate layers of a hexagonal lattice is studied using the Monte Carlo simulations under the effect of interactions in the presence and absence of external magnetic and crystal-field. The thermal variations of the magnetizations are given. The magnetic hysteresis cycle are established. II. Model and formulation
2
The Hamiltonian of the alternate layers Ising with hexagonal lattice includes nearest neighbors interactions, external magnetic field H and the crystal-field is given as: 2 2 H ' J1 i S j J 2 i j J 3 Si Si H i S j i S j i , j i, j j j ij i i
(1) where i, j stand for the first nearest neighbor sites i and j. The couplings J1, J2 and J3 are the exchange couplings between the nearest-neighbor pairs of spins -S, - and S-S, respectively (see Fig.1). The spin moments of the alternate layers Ising with hexagonal lattice are: =±1,0 and S=±5/2, ±3/2, ±1/2. III. Numerical details The alternate layers Ising with hexagonal lattice are assumed to reside in the unit cells and the system consists of the total number of spins N=64, NS=64, L=16 where L is the size of system. We apply a standard sampling method to simulate the Hamiltonian given by Eq. (1). Cyclic boundary conditions on the lattice were imposed 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. 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 internal energy per site E,
E
1 H N
(2)
3
The magnetizations of mixed spins:
M
MS
1 N
1 NS
S
i
(3)
j
(4)
i
j
Where N=Nσ + NS The total magnetization is given by:
M tot
M M S 2
(5)
The magnetic susceptibilities for each spin are given by:
M2 M 2
(6)
s M S 2 M S 2
(7)
Total susceptibility of Ising system is:
tot M tot 2 M tot 2
where
(8)
1 , k BT T denotes the absolute temperature and kB is the Boltzmann’s constant.
IV. Numerical results and discussion
We presented in Figs. 2 the thermal variations absolute of the magnetizations and magnetic susceptibilities on alternate layers of a hexagonal lattice for J3=+0.1, H=+1.0, Δ=+1.0 with J1=-1.0 and J2=+3.0. The reduced critical temperature is obtained for two spins-1 and 5/2. The obtained values is TC= 7.27. This value is near to that TC=12, given obtained by the Glaubertype stochastic dynamics given by Ref. [13]. We have presented in Figs. 3(a)-3b, the thermal 4
variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. The obtained transition temperatures for J2=1, 3 and 6 are 5.64, 7.17 and 7.98 K, respectively. We have shown in Figs. 4(a)-4b, the thermal variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. The obtained transition temperatures for J3=0.1, 0.5 and 1 are 5.64, 7.17 and 8.82 K, respectively. The values of reduced critical temperature increase with increasing the exchange interactions J2 and J3. This behavior is observed in previous work [14]. We have also presented in Figs. 5(a)-5b, the thermal variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. Figs. 6 (a)-b show the total magnetizations versus the crystalfield on alternate layers on a hexagonal lattice for J2=1.5, 5, 8, 12, J3=+0.1, H=+1.0, Δ=+1.0 and J3=0.1, 0.5, 1, 1.5, 2, J2=+1.0, H=+0.0, T=1.0, respectively with J1=-1.0. The increasing of J2 and J3 has no effect on the magnetization as a function of crystal-field for >0 and for <-3. While, when the crystal-field undergoes positive values the total magnetization shows an abrupt change until it between -1 and -2 values. This behavior is known for ferromagnetic systems. Figs. 7(a)-b represent the variations of the total magnetizations versus the J3 for =0, -1, -1.5, -2, -3, T=1 and T=1, 2, 4, 5, 6, Δ=0.0, respectively with J1=-1.0, J2=+1.0, H=+0.0. It is found that the total magnetization on alternate layers of a hexagonal lattice is not affected by the antiferromagnetic J3 coupling values. While, when the coupling constant J3 undergoes positive values the total magnetization shows an abrupt change until it reaches the 1 value. This behavior is known for ferromagnetic systems. This behavior is observed in previous work [15]. The variation of the total magnetizations versus the J3 for =0, -1, -2, -2.5, -3, J3=+0.1 and for J3=0.3, 0.5, 1.2, 1.5, 1.8, =0 is given in Figs 8(a) and 8(b), respectively with J1=-1.0, H=+0.0 and T=1.0. It is found that the total magnetization on alternate layers of a 5
hexagonal lattice is not affected by the antiferromagnetic J2 coupling values. While, when the coupling constant J2 undergoes positive values the total magnetization shows an abrupt change until it reaches the -4 value. This behavior is known for ferromagnetic systems and it is also observed in previous work [16]. The magnetic hysteresis cycle for J2= 2, 6, 7, 9 is shown is Fig. 9(a)-d, respectively for J1=-1, J3=+0.1, =0 and T=1.0. The coercive field increases with increasing of the exchange interaction J2 (see Table. 1). This behavior is observed in previous works [15,17]. The loops strongly depend on the exchange interaction J2 on alternate layers of a hexagonal lattice and the hysteresis curves in high diameter values change sharply. Moreover, the hysteresis loops are broadening while the exchange interaction J2 is increasing so that it approaches to bulk materials. Figs. 10(a)-c illustrates the magnetic hysteresis cycle for J3= 1, 2, 3, respectively with J1=-1, J2=+1, =0 and T=1.0. The increasing of the exchange interaction J3, increases the coercive field (see Table. 2). The coercive field increases with increasing in the exchange interaction J3 at low temperatures. This behavior is observed in previous work [18,19]. Finally the magnetic hysteresis cycles for T= 3, 4, 5, 6, are established (see Figs. 11(a)-d) with J1=-1, J2=+1, J3= 3 and =0 (see Table. 3). The evolutions of hysteresis loops of composite alternate layers of a hexagonal lattice are seen to change monotonically as the temperature increases. Finally, we have given in Fig. 12, the magnetic hysteresis cycle for Δ=-0.5(a), -2.0(b), -3.0(c) with J1=-1.0, J2=+1.0, J3=+1.0 and T=1.0. The magnetic coercive field decreases with increasing the absolute value of crystal filed (see Table. 4). The system becomes a superparamagnism for Δ=-3. The coercivity and superparamagnetic transition temperatures exhibit important aspects in the future high-density magnetic data storage. These behavior is observed in previously works [15,17,20]. In this figure, we can see that with increasing T, the shape of hysteresis loops becomes smaller and the alternate layers of a hexagonal Ising system displays a single hysteresis loop for T= 7 K and Δ=-3. 6
V. Conclusions
The magnetic properties of on alternate layers on a hexagonal lattice are studied by using Monte Carlo simulations. The transition temperatures are deduced for different exchange interactions J2 and J3. The obtained value are near to those given by Glauber-type stochastic dynamics. The coercive field increases with increasing of the exchange interactions. The coercivity and superparamagnetic transition temperatures exhibit important aspects in the future high-density magnetic data storage. The coercive field decreases with increasing the temperatures values. References:
[1] Köster, E.(1993): Trends in magnetic recording media. J. Magn. Magn. Mater., 120:1-10. [2] Lueck, L.B., Gilson, R.G (1999): Challenges and opportunities: The magnetic media industry in the 1990s. J. Magn. Magn. Mater., 88:227-235. [3] M. Ertas, M. Keskin. Phys. Lett. A 379 (2015) 576. [4] C. Ekiz, M. Keskin. Physia A. 317 (2003) 517. [5] Wei Jiang, Ya-Ning Wang. J. Magn. Magn. Mater. 426 (2017) 785–793. [6] Wei Jiang, Fan Zhang, Xiao-Xi Li, Hong-Yu Guan, An-Bang Guo, Zan Wang. Physica E 47 (2013) 95–102. [7] A. Ozkilic, U. Temizer. J. Magn. Magn. Mater 330 (2013) 55. [8] A. Yigi, E. Albayrak. 21 (2012) 020511. [9] M. Bati, M. Ertas. J. Supercond. Nov. Magn. Doi 10.1007/s10948-016-3620-1 [10] M. Keskin, O. Kanco. J. Korean. Phys. Soci. 55 (2009) 1344. [11] R.A. Yessoufou, S. bekhechi, F. Hontinfinde. Eur. Phys. J. B. 81 (2001) 137. [12] N. Fukushima, A. Honecker, S. Wessel, W. Brenig. Phys. Rev. B. 69 (2004) 174430. [13] A. OzkIlI, U. Temizer. J. Magn. Magn. Mater. 330 (2013) 55–65. [14] N Şarlı. Physica E 63 (2014) 324. 7
[15] O. Yalçina, R. Erdemb and S. ÄOvÄunc. Acta. Physica. Polonica. A. 114 (2008) 835. [16] M. El Yadari, L. Bahmad, A. El Kenz, A. Benyoussef. J. Alloys. Compd. 579 (2013) 86– 91. [17] Wei Jiang, Xiao-Xi Li, Li-Mei Liu, Jun-Nan Chen, Fan Zhang. J. Magn. Magn. Mater. 353 (2014) 90–98. [18] D. L. Peng, K. Sumiyama, S. Yamamuro, T. Hihara, and T. J. Konno. phys. stat. sol. (a) 172, 209 (1999). [19] G. Szabó and G. Kádár. Phys. Rev. B. 58 (1998) 5584. [20] O. Mounkachi, H. El Moussaoui, R. Masrour, J. Ilali, K. EL. Mediouri, M. Hamedoun, E.K. Hlil, A. ElKenz, A. Benyoussef. Mater. Lett. 126 (2014) 193–196.
8
J2
1
2
3
4
5
6
7
8
9
Coercive
2.515
2.741
3.161
3.305
3.329
3.344
3.354
3.396
3.456
field (HC)
Table 1: The values of HC for different values of J2 for J1=-1.0, J3=0.1, Δ=0.0 and T=1 K.
J3
0.1
1
1.5
2
Coercive
2.615
5.125
7.07
8.731
2.5 9.940
3
3.5
4
12.220
14.009
16.297
field (HC)
Table 2: The values of HC for different values of J3 for J1=-1.0, J2=+1.0, Δ=0.0, and T=1 K.
T(K)
1
2
3
4
5
Coercive
12.220
8.879
6.137
4.201
2.992
6
7 1.606
0
field (HC)
Table 3: The values of HC for different values of T for J1=-1.0, J2=+1.0, J2=+3.0 and Δ=0.0.
0.0
-0.5
-1.0
-1.5
-2.0
Coercive
5.125
4.397
4.397
2.511
2.088
-2.5 1.026
-3.0 0
field (HC)
Table 4: The values of HC for different values of Δ for J1=-1.0, J2=+1.0, J2=+1.0 and T=1.0
K.
9
Fig. 1:: The sketchh of the spin n arrangemeent on the hexagonal h laattice. The laattice is form med by altternate layerrs of (opeen circles) and a S (solid circles) spiins.
|MS|
2,5
S
2,0
tot
1,5
|M M|,
|M|
1,0
|Mto ot|
0,5
0,0 0
5
10
T 10
15
20
Fig. 2: The thermal variations absolute of the magnetizations and magnetic susceptibilities on
alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, J3=+0.1, H=+1.0 and Δ=+0.0.
2,0
0,8
J2=+1.0
(a)
(b)
J2=+3.0 J2=+6.0
0,6
1,5
tot
|Mtot|
1,0
0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
T
10
15
20
T
Fig. 3: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0.
2,0
0,8
J3=+0.1
(a)
(b)
J3=+0.5 J3=+1.0
0,6
1,5
tot 1,0
|Mtot|0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
10
15
20
T
T
Fig. 4: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, H=+1.0 and Δ=+0.0. 11
The obtained transition temperatures for =-1, -0.5 and 0 are 5.36, 6.28 and 7.10 K, respectively.
0,8
2,0
=+0.0 =-0.5 =-1.0
(a)
(b) 1,5
0,6
tot
|Mtot|
1,0
0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
10
T
15
20
T
Fig. 5: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, J3=+0.1 and H=+1.0.
(a)
J2=+1.5
0,8
J2=+5.0 J2=+8.0
0,4
J2=+12.0
Mtot 0,0
-0,4
-0,8
-4
-2
0
12
2
4
1,2
J3=+0.1 J3=+1.0
0,6
J3=+1.5 J3=+2.0
0,3
Mtot
(b)
J3=+0.5
0,9
0,0
-0,3
-0,6
-0,9 -5
-4
-3
-2
-1
0
1
Fig. 6: The total magnetizations versus the crystal-field on alternate layers of a hexagonal
lattice J2=+1.5, +5.0, +8.0, +12.0, J3=+0.1, H=+1.0 (a) and J3=+0.1, +0.5, +1.0, +1.5, +2.0, J2=+1.0, H=+0.0 (b) and T=1.0 with J1=-1.0.
0,9
(a)
=0.0 =-1.0 =-1.5 =-2.0 =-3.0
0,6
0,3
Mtot 0,0
-0,3
-0,6
-0,9 -4
-2
0
J3
13
2
4
0,9
(b)
T=1.0 T=2.0 T=4.0 T=5.0 T=6.0
0,6
0,3
Mtot
0,0
-0,3
-0,6
-0,9 -4
-2
0
2
4
J3
Fig. 7: The variations of the total magnetizations versus the J3 for =0.0, -1.0, -1.5, -2.0, -3.0,
T=1.0 (a) and T=1.0, 2.0, 4.0, 5.0, 6.0, Δ=0.0 (b) with J1=-1.0, J2=+1.0, H=+0.0.
=0.0 =-1.0 =-2.0 =-2.5 =-3.0
0,6
0,3
Mtot
(a)
0,0
-0,3
-0,6
-0,9 -8
-4
0
J2
14
4
8
J3=0.3
1,2
(b)
J3=0.5 0,8
J3=1.2 J3=1.5
0,4
J3=1.8
Mtot 0,0
-0,4
-0,8
-1,2 -8
-4
0
4
8
J2 Fig. 8: The variations of the total magnetizations versus the J3 for =0.0, -1.0, -2.0, -2.5, -3.0,
J3=+0.1 (a) and for J3=+0.3, +0.5, +1.2, +1.5, +1.8, =0 (b) with J1=-1.0, H=+0.0 and T=1.0
2,0
(a) 1,5 1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
15
8
16
2,0 1,5
(b)
1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
8
16
H
2,0
(c) 1,5 1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
Fig. 9: The magnetic hysteresis cycle for J2= +2.0(a), +6.0(b), +9.0 (c) with J1=-1.0, J3=+0.1,
=0.0 and T=1.0.
16
2,0 1,5
(a)
1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
8
16
H
2,0
(b) 1,5 1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
17
2,0 1,5
(c)
1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
H
Fig. 10: The magnetic hysteresis cycle for J3=+1.0(a), +2.0(b), 3.0 (c) with J1=-1.0, J2=+1.0,
=0.0 and T=1.0.
2,0
(a) 1,5 1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
18
8
16
2,0
(b) 1,5 1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
H
1,5
(c)
1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -15
-10
-5
0
5
10
15
H
Fig. 11: The magnetic hysteresis cycle for T= 3(a), 5(b), 7(c) with J1=-1.0, J2=+1.0, J3=+3.0
and =0.0.
19
2,0 1,5
(a)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
5
10
15
5
10
15
H
2,0 1,5
(b)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
H
20
2,0 1,5
(c)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
5
10
15
H Fig. 12: The magnetic hysteresis cycle for Δ=-0.5(a), -2.0(b), -3.0(c) with J1=-1.0, J2=+1.0,
J3=+1.0 and T=1.0
21
Magnetic properties of the mixed spins Ising are studied using the Monte Carlo simulations. The thermal variations of the magnetizations are given. The transition temperatures increases with increasing the exchange interactions The magnetic hysteresis cycle are established. The magnetic coercive field is deduced for different exchange interactions, crystal filed an d temperatures values.
22
PII: DOI: Reference:
S0378-4371(17)30975-5 https://doi.org/10.1016/j.physa.2017.09.083 PHYSA 18688
To appear in:
Physica A
Received date : 17 January 2017 Revised date : 28 August 2017 Please cite this article as: R. Masrour, A. Jabar, A. Benyoussef, M. Hamedoun, Magnetic properties of the Ising system on alternate layers of a hexagonal lattice, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.083 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 proof before it is published in its final 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.
Magnetic properties of the Ising system on alternate layers of a hexagonal lattice R. Masrour1,*, A. Jabar1, A. Benyoussef2,3 and M. Hamedoun2 1)
Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, 63 46000, Safi, Morocco. 2)
Institute of Nanomaterials and Nanotechnologies, MAScIR, Rabat, Morocco. 3)
Hassan II Academy of Science and Technology, Rabat, Morocco. Corresponding authors: [email protected]
Abstract: The magnetic properties of the mixed spin-1 and spin-5/2 Ising ferrimagnetic model on a hexagonal lattice have been studied using the Monte Carlo simulations. The hexagonal lattice study is formed by alternate layers of spins =1 and S=5/2. The thermal variations of the magnetizations are given. The magnetic hysteresis cycle are established for different values of exchange interactions, crystal-field and temperatures. The magnetic coercive field is deduced.
Keywords: Hexagonal lattice; Mixed-spin Ising ferrimagnetic model; Monte Carlo simulation; Critical temperature; Exchange interactions; Magnetic hysteresis cycle.
1
I. Introduction The ferrimagnetic materials, which can be studied with mixed spin systems, are possibly useful for advanced technological applications such as storage media, and the annual sales of computer diskettes, compact disks, optical disks, recording tape, and related items exceed those of the celebrated semiconductor industry[1-4]. In recent years, the some magnetic properties of hexagonal lattice have been studied using the effective-field theory [5,6]. The mixed spins (1,5/2) Ising systems [7-11] have been studied using mean field approximation. In other hand, the kinetic mixed spin-1 and spin-5/2 Ising ferrimagnetic system on alternate layers of a hexagonal has an interesting dynamic behavior and gives rich dynamic phase diagrams [7] and distinct topological diagrams [10]. It was found that the mixed spin-1 and spin-5/2 Ising system on hexagonal lattice exhibits five fundamental phases, nine different mixed phases which are composed of binary and ternary combinations of fundamental phases and the compensation temperature or the N-type behavior in the Néel classification nomenclature [7]. In addition, the thermal phase diagrams exhibit compensation points(where the sublattices have equal magnetizations), first-order and second-order transitions as well as tricritcal and end-points [11]. We should also note that the molecular mixed-spin ferromagnetic chain material MnNi(NO2)4 (ethylenediamine) is known as quasi-onedimensional material which composed by Ni2+ and Mn2+ of spins 1 and 5/2, respectively [12]. Hence, with this motivation, we have studied the mixed spin-1 and spin-5/2 Ising system to investigate their critical behavior on alternate layers of a hexagonal lattice is studied using the Monte Carlo simulations under the effect of interactions in the presence and absence of external magnetic and crystal-field. The thermal variations of the magnetizations are given. The magnetic hysteresis cycle are established. II. Model and formulation
2
The Hamiltonian of the alternate layers Ising with hexagonal lattice includes nearest neighbors interactions, external magnetic field H and the crystal-field is given as: 2 2 H ' J1 i S j J 2 i j J 3 Si Si H i S j i S j i , j i, j j j ij i i
(1) where i, j stand for the first nearest neighbor sites i and j. The couplings J1, J2 and J3 are the exchange couplings between the nearest-neighbor pairs of spins -S, - and S-S, respectively (see Fig.1). The spin moments of the alternate layers Ising with hexagonal lattice are: =±1,0 and S=±5/2, ±3/2, ±1/2. III. Numerical details The alternate layers Ising with hexagonal lattice are assumed to reside in the unit cells and the system consists of the total number of spins N=64, NS=64, L=16 where L is the size of system. We apply a standard sampling method to simulate the Hamiltonian given by Eq. (1). Cyclic boundary conditions on the lattice were imposed 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. 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 internal energy per site E,
E
1 H N
(2)
3
The magnetizations of mixed spins:
M
MS
1 N
1 NS
S
i
(3)
j
(4)
i
j
Where N=Nσ + NS The total magnetization is given by:
M tot
M M S 2
(5)
The magnetic susceptibilities for each spin are given by:
M2 M 2
(6)
s M S 2 M S 2
(7)
Total susceptibility of Ising system is:
tot M tot 2 M tot 2
where
(8)
1 , k BT T denotes the absolute temperature and kB is the Boltzmann’s constant.
IV. Numerical results and discussion
We presented in Figs. 2 the thermal variations absolute of the magnetizations and magnetic susceptibilities on alternate layers of a hexagonal lattice for J3=+0.1, H=+1.0, Δ=+1.0 with J1=-1.0 and J2=+3.0. The reduced critical temperature is obtained for two spins-1 and 5/2. The obtained values is TC= 7.27. This value is near to that TC=12, given obtained by the Glaubertype stochastic dynamics given by Ref. [13]. We have presented in Figs. 3(a)-3b, the thermal 4
variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. The obtained transition temperatures for J2=1, 3 and 6 are 5.64, 7.17 and 7.98 K, respectively. We have shown in Figs. 4(a)-4b, the thermal variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. The obtained transition temperatures for J3=0.1, 0.5 and 1 are 5.64, 7.17 and 8.82 K, respectively. The values of reduced critical temperature increase with increasing the exchange interactions J2 and J3. This behavior is observed in previous work [14]. We have also presented in Figs. 5(a)-5b, the thermal variations absolute of the magnetizations and magnetic susceptibilities, respectively on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0. Figs. 6 (a)-b show the total magnetizations versus the crystalfield on alternate layers on a hexagonal lattice for J2=1.5, 5, 8, 12, J3=+0.1, H=+1.0, Δ=+1.0 and J3=0.1, 0.5, 1, 1.5, 2, J2=+1.0, H=+0.0, T=1.0, respectively with J1=-1.0. The increasing of J2 and J3 has no effect on the magnetization as a function of crystal-field for >0 and for <-3. While, when the crystal-field undergoes positive values the total magnetization shows an abrupt change until it between -1 and -2 values. This behavior is known for ferromagnetic systems. Figs. 7(a)-b represent the variations of the total magnetizations versus the J3 for =0, -1, -1.5, -2, -3, T=1 and T=1, 2, 4, 5, 6, Δ=0.0, respectively with J1=-1.0, J2=+1.0, H=+0.0. It is found that the total magnetization on alternate layers of a hexagonal lattice is not affected by the antiferromagnetic J3 coupling values. While, when the coupling constant J3 undergoes positive values the total magnetization shows an abrupt change until it reaches the 1 value. This behavior is known for ferromagnetic systems. This behavior is observed in previous work [15]. The variation of the total magnetizations versus the J3 for =0, -1, -2, -2.5, -3, J3=+0.1 and for J3=0.3, 0.5, 1.2, 1.5, 1.8, =0 is given in Figs 8(a) and 8(b), respectively with J1=-1.0, H=+0.0 and T=1.0. It is found that the total magnetization on alternate layers of a 5
hexagonal lattice is not affected by the antiferromagnetic J2 coupling values. While, when the coupling constant J2 undergoes positive values the total magnetization shows an abrupt change until it reaches the -4 value. This behavior is known for ferromagnetic systems and it is also observed in previous work [16]. The magnetic hysteresis cycle for J2= 2, 6, 7, 9 is shown is Fig. 9(a)-d, respectively for J1=-1, J3=+0.1, =0 and T=1.0. The coercive field increases with increasing of the exchange interaction J2 (see Table. 1). This behavior is observed in previous works [15,17]. The loops strongly depend on the exchange interaction J2 on alternate layers of a hexagonal lattice and the hysteresis curves in high diameter values change sharply. Moreover, the hysteresis loops are broadening while the exchange interaction J2 is increasing so that it approaches to bulk materials. Figs. 10(a)-c illustrates the magnetic hysteresis cycle for J3= 1, 2, 3, respectively with J1=-1, J2=+1, =0 and T=1.0. The increasing of the exchange interaction J3, increases the coercive field (see Table. 2). The coercive field increases with increasing in the exchange interaction J3 at low temperatures. This behavior is observed in previous work [18,19]. Finally the magnetic hysteresis cycles for T= 3, 4, 5, 6, are established (see Figs. 11(a)-d) with J1=-1, J2=+1, J3= 3 and =0 (see Table. 3). The evolutions of hysteresis loops of composite alternate layers of a hexagonal lattice are seen to change monotonically as the temperature increases. Finally, we have given in Fig. 12, the magnetic hysteresis cycle for Δ=-0.5(a), -2.0(b), -3.0(c) with J1=-1.0, J2=+1.0, J3=+1.0 and T=1.0. The magnetic coercive field decreases with increasing the absolute value of crystal filed (see Table. 4). The system becomes a superparamagnism for Δ=-3. The coercivity and superparamagnetic transition temperatures exhibit important aspects in the future high-density magnetic data storage. These behavior is observed in previously works [15,17,20]. In this figure, we can see that with increasing T, the shape of hysteresis loops becomes smaller and the alternate layers of a hexagonal Ising system displays a single hysteresis loop for T= 7 K and Δ=-3. 6
V. Conclusions
The magnetic properties of on alternate layers on a hexagonal lattice are studied by using Monte Carlo simulations. The transition temperatures are deduced for different exchange interactions J2 and J3. The obtained value are near to those given by Glauber-type stochastic dynamics. The coercive field increases with increasing of the exchange interactions. The coercivity and superparamagnetic transition temperatures exhibit important aspects in the future high-density magnetic data storage. The coercive field decreases with increasing the temperatures values. References:
[1] Köster, E.(1993): Trends in magnetic recording media. J. Magn. Magn. Mater., 120:1-10. [2] Lueck, L.B., Gilson, R.G (1999): Challenges and opportunities: The magnetic media industry in the 1990s. J. Magn. Magn. Mater., 88:227-235. [3] M. Ertas, M. Keskin. Phys. Lett. A 379 (2015) 576. [4] C. Ekiz, M. Keskin. Physia A. 317 (2003) 517. [5] Wei Jiang, Ya-Ning Wang. J. Magn. Magn. Mater. 426 (2017) 785–793. [6] Wei Jiang, Fan Zhang, Xiao-Xi Li, Hong-Yu Guan, An-Bang Guo, Zan Wang. Physica E 47 (2013) 95–102. [7] A. Ozkilic, U. Temizer. J. Magn. Magn. Mater 330 (2013) 55. [8] A. Yigi, E. Albayrak. 21 (2012) 020511. [9] M. Bati, M. Ertas. J. Supercond. Nov. Magn. Doi 10.1007/s10948-016-3620-1 [10] M. Keskin, O. Kanco. J. Korean. Phys. Soci. 55 (2009) 1344. [11] R.A. Yessoufou, S. bekhechi, F. Hontinfinde. Eur. Phys. J. B. 81 (2001) 137. [12] N. Fukushima, A. Honecker, S. Wessel, W. Brenig. Phys. Rev. B. 69 (2004) 174430. [13] A. OzkIlI, U. Temizer. J. Magn. Magn. Mater. 330 (2013) 55–65. [14] N Şarlı. Physica E 63 (2014) 324. 7
[15] O. Yalçina, R. Erdemb and S. ÄOvÄunc. Acta. Physica. Polonica. A. 114 (2008) 835. [16] M. El Yadari, L. Bahmad, A. El Kenz, A. Benyoussef. J. Alloys. Compd. 579 (2013) 86– 91. [17] Wei Jiang, Xiao-Xi Li, Li-Mei Liu, Jun-Nan Chen, Fan Zhang. J. Magn. Magn. Mater. 353 (2014) 90–98. [18] D. L. Peng, K. Sumiyama, S. Yamamuro, T. Hihara, and T. J. Konno. phys. stat. sol. (a) 172, 209 (1999). [19] G. Szabó and G. Kádár. Phys. Rev. B. 58 (1998) 5584. [20] O. Mounkachi, H. El Moussaoui, R. Masrour, J. Ilali, K. EL. Mediouri, M. Hamedoun, E.K. Hlil, A. ElKenz, A. Benyoussef. Mater. Lett. 126 (2014) 193–196.
8
J2
1
2
3
4
5
6
7
8
9
Coercive
2.515
2.741
3.161
3.305
3.329
3.344
3.354
3.396
3.456
field (HC)
Table 1: The values of HC for different values of J2 for J1=-1.0, J3=0.1, Δ=0.0 and T=1 K.
J3
0.1
1
1.5
2
Coercive
2.615
5.125
7.07
8.731
2.5 9.940
3
3.5
4
12.220
14.009
16.297
field (HC)
Table 2: The values of HC for different values of J3 for J1=-1.0, J2=+1.0, Δ=0.0, and T=1 K.
T(K)
1
2
3
4
5
Coercive
12.220
8.879
6.137
4.201
2.992
6
7 1.606
0
field (HC)
Table 3: The values of HC for different values of T for J1=-1.0, J2=+1.0, J2=+3.0 and Δ=0.0.
0.0
-0.5
-1.0
-1.5
-2.0
Coercive
5.125
4.397
4.397
2.511
2.088
-2.5 1.026
-3.0 0
field (HC)
Table 4: The values of HC for different values of Δ for J1=-1.0, J2=+1.0, J2=+1.0 and T=1.0
K.
9
Fig. 1:: The sketchh of the spin n arrangemeent on the hexagonal h laattice. The laattice is form med by altternate layerrs of (opeen circles) and a S (solid circles) spiins.
|MS|
2,5
S
2,0
tot
1,5
|M M|,
|M|
1,0
|Mto ot|
0,5
0,0 0
5
10
T 10
15
20
Fig. 2: The thermal variations absolute of the magnetizations and magnetic susceptibilities on
alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, J3=+0.1, H=+1.0 and Δ=+0.0.
2,0
0,8
J2=+1.0
(a)
(b)
J2=+3.0 J2=+6.0
0,6
1,5
tot
|Mtot|
1,0
0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
T
10
15
20
T
Fig. 3: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J3=+0.1, H=+1.0 and Δ=+0.0.
2,0
0,8
J3=+0.1
(a)
(b)
J3=+0.5 J3=+1.0
0,6
1,5
tot 1,0
|Mtot|0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
10
15
20
T
T
Fig. 4: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, H=+1.0 and Δ=+0.0. 11
The obtained transition temperatures for =-1, -0.5 and 0 are 5.36, 6.28 and 7.10 K, respectively.
0,8
2,0
=+0.0 =-0.5 =-1.0
(a)
(b) 1,5
0,6
tot
|Mtot|
1,0
0,4
0,5 0,2
0,0 0
5
10
15
20 0
5
10
T
15
20
T
Fig. 5: The thermal variations absolute of the magnetizations (a) and magnetic susceptibilities
(b) on alternate layers of a hexagonal lattice for J1=-1.0, J2=+3.0, J3=+0.1 and H=+1.0.
(a)
J2=+1.5
0,8
J2=+5.0 J2=+8.0
0,4
J2=+12.0
Mtot 0,0
-0,4
-0,8
-4
-2
0
12
2
4
1,2
J3=+0.1 J3=+1.0
0,6
J3=+1.5 J3=+2.0
0,3
Mtot
(b)
J3=+0.5
0,9
0,0
-0,3
-0,6
-0,9 -5
-4
-3
-2
-1
0
1
Fig. 6: The total magnetizations versus the crystal-field on alternate layers of a hexagonal
lattice J2=+1.5, +5.0, +8.0, +12.0, J3=+0.1, H=+1.0 (a) and J3=+0.1, +0.5, +1.0, +1.5, +2.0, J2=+1.0, H=+0.0 (b) and T=1.0 with J1=-1.0.
0,9
(a)
=0.0 =-1.0 =-1.5 =-2.0 =-3.0
0,6
0,3
Mtot 0,0
-0,3
-0,6
-0,9 -4
-2
0
J3
13
2
4
0,9
(b)
T=1.0 T=2.0 T=4.0 T=5.0 T=6.0
0,6
0,3
Mtot
0,0
-0,3
-0,6
-0,9 -4
-2
0
2
4
J3
Fig. 7: The variations of the total magnetizations versus the J3 for =0.0, -1.0, -1.5, -2.0, -3.0,
T=1.0 (a) and T=1.0, 2.0, 4.0, 5.0, 6.0, Δ=0.0 (b) with J1=-1.0, J2=+1.0, H=+0.0.
=0.0 =-1.0 =-2.0 =-2.5 =-3.0
0,6
0,3
Mtot
(a)
0,0
-0,3
-0,6
-0,9 -8
-4
0
J2
14
4
8
J3=0.3
1,2
(b)
J3=0.5 0,8
J3=1.2 J3=1.5
0,4
J3=1.8
Mtot 0,0
-0,4
-0,8
-1,2 -8
-4
0
4
8
J2 Fig. 8: The variations of the total magnetizations versus the J3 for =0.0, -1.0, -2.0, -2.5, -3.0,
J3=+0.1 (a) and for J3=+0.3, +0.5, +1.2, +1.5, +1.8, =0 (b) with J1=-1.0, H=+0.0 and T=1.0
2,0
(a) 1,5 1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
15
8
16
2,0 1,5
(b)
1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
8
16
H
2,0
(c) 1,5 1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
Fig. 9: The magnetic hysteresis cycle for J2= +2.0(a), +6.0(b), +9.0 (c) with J1=-1.0, J3=+0.1,
=0.0 and T=1.0.
16
2,0 1,5
(a)
1,0 0,5
Mtot 0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
8
16
H
2,0
(b) 1,5 1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
17
2,0 1,5
(c)
1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
H
Fig. 10: The magnetic hysteresis cycle for J3=+1.0(a), +2.0(b), 3.0 (c) with J1=-1.0, J2=+1.0,
=0.0 and T=1.0.
2,0
(a) 1,5 1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
H
18
8
16
2,0
(b) 1,5 1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -2,0 -16
-8
0
8
16
H
1,5
(c)
1,0 0,5
Mtot0,0 -0,5 -1,0 -1,5 -15
-10
-5
0
5
10
15
H
Fig. 11: The magnetic hysteresis cycle for T= 3(a), 5(b), 7(c) with J1=-1.0, J2=+1.0, J3=+3.0
and =0.0.
19
2,0 1,5
(a)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
5
10
15
5
10
15
H
2,0 1,5
(b)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
H
20
2,0 1,5
(c)
1,0 0,5
Mtot
0,0 -0,5 -1,0 -1,5 -2,0 -15
-10
-5
0
5
10
15
H Fig. 12: The magnetic hysteresis cycle for Δ=-0.5(a), -2.0(b), -3.0(c) with J1=-1.0, J2=+1.0,
J3=+1.0 and T=1.0
21
Magnetic properties of the mixed spins Ising are studied using the Monte Carlo simulations. The thermal variations of the magnetizations are given. The transition temperatures increases with increasing the exchange interactions The magnetic hysteresis cycle are established. The magnetic coercive field is deduced for different exchange interactions, crystal filed an d temperatures values.
22