Phase diagrams and magnetic properties of a superlattice with alternate layers

Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

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Phase diagrams and magnetic properties of a superlattice with alternate layers A. Feraoun, A. Zaim n, M. Kerouad Laboratoire Physique des Matériaux et Modélisation des Systèmes (LP2MS), Unité Associée au CNRST-URAC: 08, Faculty of Sciences, University Moulay Ismail, B.P. 11201 Zitoune, Meknes, Morocco

art ic l e i nf o

a b s t r a c t

Article history: Received 2 July 2014 Received in revised form 28 August 2014 Available online 23 October 2014

The phase diagrams and magnetic properties of an Ising superlattice are investigated by means of Monte Carlo simulation based on Metropolis algorithm. The system is formed by alternate layers of spins S = 1 and σ = 3/2. The effects of the exchange coupling interactions and the crystal field on the phase diagrams and magnetic properties of the system are examined. A number of characteristic behaviors have been found. In particular, tricritical point, critical end point, and isolated critical point may occur in the present system. & Elsevier B.V. All rights reserved.

Keywords: Ising model Superlattice Monte Carlo simulation Compensation temperature Critical temperature

1. Introduction Since the discovery of the magnetic interlayer coupling and the giant magnetoresistance, much effort has been directed towards the study of the critical phenomena in various magnetic layered structures, super-lattices, bilayer and ultrathin-films [1–4]. This is because of the easiness of their preparation made possible by the recent advances of modern vacuum science and this is because of their potential technological importance in various fields [5]. Consequently, considerable effort has been focused on the understanding of layered structures and superlattices [6–10]. Wang et al. [11] have used the Ising model in a transverse field (IMTF) to describe the properties of ferroelectric–antiferroelectric superlattices that might be fabricated from order–disorder materials. They have found that the number of antiferroelectric layers plays a crucial role in the phase transition properties of the superlattices . The phase diagrams of a spin-1 Ising superlattice with alternating transverse field were studied by Saber et al. in Ref. [12]. The critical temperature can increase or decrease with the increasing thickness of the superlattice. In Refs. [13,14], Belmamoun et al. have investigated the magnetic properties of a finite superlattice with disordered interfaces. They have found that the existence and the number of compensation points depend strongly on the thickness of the film. The same results are found in Ref. [15] where Monte Carlo Simulation (MCS) has been used to study critical and n

Corresponding author. E-mail addresses: [email protected] (A. Zaim), [email protected] (M. Kerouad).

http://dx.doi.org/10.1016/j.jmmm.2014.10.062 0304-8853/& Elsevier B.V. All rights reserved.

compensation behaviors of a ferrimagnetic superlattice on a simple cubic lattice. In some recent work [16], the effect of the transverse crystal field on the magnetic properties of a superlattice with disordered interface has been investigated. A number of characteristic phenomena are found, such as the possibility of two compensation points. Experimentally, Ahlberg et al. [17] have explored the effect of the interlayer exchange coupling in Fe/V (001) superlattices on the temperature dependence of the magnetic properties. The temperature dependence of the remanent magnetization was proven to be significantly affected by the strength of the ferromagnetic coupling. On the other hand, ferrimagnetic mixed spin system has attracted a lot of interest. In these systems, two different nearestneighbor spins are coupled by anti-ferromagnetic exchange interaction. In Ref. [18], Monte Carlo simulation has been used to study the magnetic properties and the critical behaviors of a ferrimagnetic mixed spin (1/2, 1) Blume–Capel model with a fourspin plaquette interaction on two-dimensional square lattice. The critical behavior of the system shows the presence of reentrant and even double reentrant phenomenon for some value of the system parameters. In Ref. [19], the authors studied, within a mean-field approach, the stationary states of the kinetic mixed spin (3/2, 2) Ising systems. Interesting phenomena are found such as dynamic tricritical and dynamic critical end point. Recently, Feraoun et al. [20] have employed Monte Carlo simulation to investigate the magnetic behavior of a ferrimagnetic nanowire on a hexagonal lattice with a spin-3/2 core surrounded by a spin-1 shell. The results present rich critical behavior, which includes the

A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

3,0

J /JS=0.1

2,5

2,0

T/JS

first- and second-order phase transitions, the tricritical and critical end points. In Ref. [21], critical and compensation properties of a ferrimagnetic mixed spin (1, 3/2) Ising on a square lattice are investigated by standard and histogram Monte Carlo simulations. Some rich phenomena are found, such as the existence of a tricritical point and a re-entrant behavior. In Ref. [22], the effects of a bimodal random crystal field on the phase diagrams and magnetization curves of the ferrimagnetic mixed spin (1/2, 3/2) Blume– Capel model are examined by using the effective field theory with correlations on a honeycomb lattice. It was found that the model presents one or two compensation temperatures for appropriate values of the random crystal field. As far as we know, however, the mixed spin (1, 3/2) Ising superlattice with alternate layers has not been studied. The purpose of the present work is to investigate the effect of the exchange interaction coupling and the crystal field on the phase diagrams and the compensation behavior of a superlattice composed of alternate layers of spins S¼ 1 and σ = 3/2 within the framework of Monte Carlo simulations. The outline of the present paper is organized as follows: in Section 2, we briefly present our model and the related formulation, the results and discussions are presented in Section 3, and finally Section 4 is devoted to our conclusions.

127

1,5

Tc(FO);D/JS=-2.0 Tc(SO);D/JS=-2.0

1,0

Tcomp;D/JS=0.0 Tc(SO);D/JS=0.0

Tcomp;D/JS=0.5 Tc(SO);D/JS=0.5

0,5

tricritical point 0,0

0,5

1,0

JInt

1,5

2,0

/JS

Fig. 2. The phase diagrams of the system in (T /JS , JInt /JS ) plane for Jσ /JS = 0.1 and for different values of D/JS (D/JS = − 2.0, 0.0 and 0.5).

3,0

Tc(FO);D/JS=-2.0

Tc(SO);D/JS=-2.0

2,8

Tcomp;D/JS=0.0

2,6

Tc(SO);D/JS=0.0

2,4

Tc(SO);D/JS=0.5 JInt/JS=-0.1

Tcomp;D/JS=0.5

2,2 2,0

T/JS

1,8 1,6

2. Model and formalism

1,4 1,2

We consider an Ising superlattice (see Fig. 1) of size L × L × N , with N being the number of layers in the system or its thickness and L  L represents the number of sites in each layer. The superlattice is made with alternate layers, s-type of spin-3/2 and S-type of spin-1. The Hamiltonian of the system is given by

/ = − JS

∑ Si S j ij

− Jσ

∑ σk σl kl

− JInt

∑ Si σk ik

− D ∑ Si2 − D ∑ σk2 i

k

1,0

tricritical point

0,8 0,6 0,4 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

J /JS

Fig. 3. The phase diagrams of the system in (T /JS , Jσ /JS ) plane for JInt /JS = − 0.1 and for different values of D/JS (D/JS = − 2.0, 0.0 and 0.5).

(1)

where JS stands for the coupling constant between the spins S, whereas Jσ is the coupling constant between the spins s. JInt is the coupling constant between the spins S and s. Si = ± 1, 0 and σk = ± 3/2, ± 1/2 are the usual Ising variables. The summation indices 〈ij〉, 〈kl〉 and 〈ik〉 denote the summations over all pairs of neighboring spins S–S, s–s, and S–s, respectively, and we have fixed the value of JS throughout the simulation (JS = 1). D is the crystal field interaction acting on the spins S and s. Using Monte Carlo simulation based on Metropolis algorithm [23], we apply periodic boundary conditions in the x and y directions. Data were generated over 20–40 realizations by using 30 000 Monte Carlo steps per site after discarding the first 15 000 steps. The results are reported for systems size L¼ 50 and N ¼4. A number of additional simulations were performed for L ¼70 and L¼ 100, but no significant differences were found from the results presented here. Our program calculates the following parameters, namely

The sublattice magnetizations per site are defined by

MS =

L×L×

Mσ =

⎛ ⎞

1 N 2

∑ ⎜⎜Si ⎟⎟ i

⎝ ⎠

(2)

⎛ ⎞

1 N L×L× 2

∑ ⎜⎜σi ⎟⎟ i

⎝ ⎠

(3)

The total magnetization per site is given by

MT =

MS + Mσ 2

(4)

The internal energy of the superlattice is defined by

E=

1 〈/〉 L×L×N

(5)

The sublattice magnetic susceptibilities are given by

Fig. 1. Schematic representation of the superlattice formed by alternate S-layer type with S = ± 1, 0 and s-layer type with σ = ± 3/2, ± 1/2.

χS = L × L ×

N β ( MS2 2

− MS 2 )

χσ = L × L ×

N β ( Mσ2 2

− Mσ 2 )

(6)

(7)

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A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

Fig. 4. The phase diagrams in (T /JS , D/JS ) plane for JInt /JS = − 0.1 and for Jσ /JS = 0.1 (a), Jσ /JS = 0.4 (b), Jσ /JS = 0.7 (c) and Jσ /JS = 0.9 (d). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

The total magnetic susceptibility per site of the superlattice is defined as

χT =

χS + χσ 2

(8)

where β = 1/k B T , T is the absolute temperature and kB is the Boltzmann factor (here kB ¼ 1). To determine the compensation temperature Tcomp from the computed magnetization data, the intersection point of the absolute value of the sublattice magnetizations was found using

∣MS (Tcomp ) ∣ = ∣Mσ (Tcomp ) ∣ sign(MS (Tcomp )) = − sign(Mσ (Tcomp ))

(9) (10)

with Tcomp < Tc , Tc is the critical temperature i.e. Néel temperature [24]. Eqs. (9) and (10) indicate that the sign of the sublattice magnetizations is different; however, absolute values of them are equal to each other at the compensation point. The second-order phase transitions are determined from the maxima of the susceptibility curves, while the first-order phase transitions are obtained by locating the discontinuities of the magnetization and the internal energy curves.

3. Results and discussions In this section, we examine the phase diagrams and the magnetic properties of the system for some selected values of the Hamiltonian parameters.

A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

Fig. 5. The phase diagrams in (T /JS , D/JS ) plane for Jσ /JS = 0.4 and JInt /JS = − 0.9 .

3.1. Phase diagrams We plot in Fig. 2 the phase diagrams of the system in the (T , |JInt | /JS ) plane for Jσ /JS = 0.1 and for different values of D/JS (D/JS = − 2.0, 0.0 and 0.5). We can see that, as |JInt | /JS increases, the critical temperature (Tc ) of the system increases linearly, and the compensation one (Tcomp), when it exists, increases also linearly to a threshold value of |JInt | /JS which depends on the value of D/JS (for example, the threshold value is |JInt | /JS = 1.29, for D/JS = 0.0). Exceeding this threshold value, the compensation temperature disappears. In addition, one can observe that for D/JS = − 2.0, we have the emergence of first order phase transitions for low values of |JInt | /JS . The line of first order phase transition is linked with the line of second order one at a tricritical point. It is interesting to note that as (D/JS ) increases, both Tcomp and Tc increase. In Fig. 3, we give the phase diagrams in the (T , Jσ /JS ) plane for JInt /JS = − 0.1 and for selected values of the crystal field (D/JS = − 2.0, 0.0, and 0.5). From this figure, we can clearly see that as Jσ /JS increases, the compensation temperature, when it exists, increases linearly up to a threshold value of (Jσ /JS ) which depends on D/JS , whereas the critical one remains constant below this threshold value. When Jσ /JS exceeds this threshold value, the compensation temperature disappears and the critical one increases linearly with Jσ /JS . For D/JS = − 2.0. It has been found that the system exhibits first order phase transitions for low values of Jσ /JS . The first order critical line is linked with a second order critical one at a tricritical point. We observe also that, as in Fig. 2, Tc and Tcomp are strongly depending on (D/JS ). To investigate the influence of the crystal field interaction D/JS on the critical and compensation behaviors of the system, we have plotted in Fig. 4 the phase diagrams of the system in the (T , D/JS ) plane with JInt /JS = − 0.1, and for different values of Jσ /JS , Jσ /JS = 0.1 (Fig. 4(a)), Jσ /JS = 0.4 (Fig. 4(b)), Jσ /JS = 0.7 (Fig. 4(c)) and Jσ /JS = 0.9 (Fig. 4(d)). It has been found that the model exhibits very rich and interesting critical behaviors; it exhibits second-order ( ) and first-order ( ) phase transitions, tricritical point (■), critical end point (□) and isolated critical point (○). Notice

129

that the phase diagrams show four regions: three ordered phases and one disordered phase separated by two lines of second ordered phase transitions and two lines of first order phase transitions. In the region of large negative values of D/JS , we have a first ordered ferrimagnetic phase in the low temperature region separated from the paramagnetic phase by a line of second order phase transitions with constant Tc, these second order lines terminate at a critical end point. The area of the ordered region and the value of Tc decrease with the decreasing Jσ /JS . At D/JS = − 2, a line of first order phase transition emerges vertically up to a tricritical point, and in its low temperature region, it separates two ordered ferrimagnetic phases namely ( ± 1/2, 0) and ( ± 1/2, ± 1) phases. The first order critical temperature increases up to the tricritical point temperature and then turns to a second order critical one which increases to reach a saturation value for large positive values of D/JS . The tricritical point coordinates and the saturation values of the critical temperature are strongly depending on Jσ /JS . At low temperature region, the system presents another first order transition lines separating the ferrimagnetic phases ( ± 1/2, ± 1) and ( ± 3/2, ± 1). This line starts at T ¼0, on a value of D/JS which depends on the value of Jσ /JS , and increases vertically to an isolated critical point. This isolated critical point is shifted to lower crystal field interaction and higher temperature when we increase Jσ /JS , that is the coordinates of the isolated points are (T /JS = 0.04 , D/JS = − 0.28 for Jσ /JS = 0.1), (T /JS = 0.25, D/JS = − 0.86 for Jσ /JS = 0.4 ), (T /JS = 0.3, D/JS = − 1.47 for Jσ /JS ¼0.7) and (T /JS = 0.45, D/JS = − 1.88 for Jσ /JS = 0.9). The area of the ferrimagnetic phase ( ± 1/2, ± 1) decreases when we increase Jσ /JS and disappears for Jσ /JS > 1. Concerning the compensation behavior, it has been found that the system exhibits a compensation temperature in a certain range of D/JS . The range of D/JS where we have a compensation point decreases when we increase Jσ /JS and disappears also when Jσ /JS > 1. We have also examined the effect of JInt /JS on the phase diagrams plotted in the (T , D/JS ) plane. To this end, we have presented in Fig. 5 the phase diagram with Jσ /JS = 0.4 as in Fig. 4(b) and JInt /JS = − 0.9. It is clear that the topology of the phase diagram is different from that of Fig. 4(b). That is, we have the emergence of two new phases, namely ( ± 1/2, ± 1/2) and ( ± 1, ± 1). The phase ( ± 1/2, ± 1/2) is separated from the phase ( ± 1/2, 0) by a new first order line which starts from D/JS = − 3.0, and increases up to the tricritical point. The other new phase ( ± 1, ± 1) is separated from the ferri ( ± 1/2, ± 1) phase by a second new first order line which starts from D/JS = − 1.67 and terminates by an isolated critical point. We can remark that this phenomenon has been observed in Ref. [26], where the authors studied the critical behavior of the ferromagnetic mixed spin (1, 2) Ising system on the Bethe lattice. 3.2. Magnetic properties In order to confirm the results obtained in Fig. 2, we present the thermal variations of the magnetic properties of the superlattice for selected values of the system parameters (Fig. 6). Fig. 6 (a) shows the temperature dependence of the sublattice magnetizations |MS |, |Mσ | and the total magnetization |MT | when JInt /JS = − 0.5, Jσ /JS = 0.1 and D/JS = 0.0. We can see that the sublattice magnetization curves decrease monotonically from the saturation values |MS | = 1.0 and |Mσ | = 1.5 at T ¼0.0 as the temperature increases, and vanish at the critical temperature Tc; hence, a second-order phase transition occurs at Tc /JS = 2.08. Let us consider now the compensation behavior, with the increasing temperature, |MS | decreases more slowly than |Mσ |. Thus, the sublattice magnetizations cancel each other at some temperature, giving rise to a zero total magnetization at that temperature, which is called

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A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

Fig. 6. The temperature dependencies of (a) absolute value of sublattice magnetizations, (b) absolute value of total magnetizations, (c) total susceptibility, for Jσ /JS = 0.1, D/JS = 0.0 and for different values of JInt /JS .

Fig. 7. The variation of the superlattice magnetizations with D/JS for JInt /JS = − 0.9 , Jσ /JS = 0.4 and for selected values of the temperature T /JS = 1.25 (a) and T /JS = 0.5 (b).

A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

the compensation point (Tcomp). It is clear that there exists a crossing point between the absolute values of the sublattice magnetizations, where the compensation point appears below Tc. In Fig. 6(b), we plot typical results for the temperature dependence of the total magnetization for Jσ /JS = 0.1, D/JS = 0.0 and for different values of JInt /JS ( JInt /JS = − 0.2, − 0.5 and  0.8). As it is seen, the compensation temperature increases as JInt /JS decreases. The results mentioned above are similar to those obtained in Ref. [25] where Monte Carlo simulation is used to study the magnetic properties and critical behavior of a molecular-based magnetic film AFeIIFeIII (C2O4 )3. The temperature dependence of the total susceptibility (χT ) is depicted in Fig. 6(c) for the same selected values as in Fig. 2(b). One can remark that, for each value of JInt /JS , one peak of the susceptibility occurs at Tc, and the peak moves towards higher temperatures with the decreasing JInt /JS .

131

To complete the discussion of the above phase diagrams, we plot in Fig. 7(a) and (b) the magnetizations as a function of D/JS for JInt /JS = − 0.9, Jσ /JS = 0.4 and for several values of the temperature (T /JS = 1.25 and 0.5). In the case of T /JS = 1.25 (Fig. 7(a)), it is observed that the system presents a continuous passage between the ferrimagnetic and the paramagnetic phases at D/JS = − 2.21. In Fig. 7(b) (T /JS = 0.5); it is clear that the compensation point appears where both sublattice magnetizations are equal in magnitude at D/JS = − 1.43. It is also remarked that the system exhibits two discontinuities of the sublattice magnetizations at a first-order point at D/JS = − 2.34 between two ferrimagnetic phases and at D/JS = − 2.46 between the ferrimagnetic and the paramagnetic phases. Finally, in order to complete the discussion of the last figures, we plot in Fig. 8(a), (b) and (c) the internal energy E versus D/JS for

Fig. 8. The variation of the internal energy with D/JS for JInt /JS = − 0.9 , Jσ /JS = 0.4 and for selected values of the temperature T /JS = 1.25 (a), T /JS = 0.5 (b), and T /JS = 0.1 (c).

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A. Feraoun et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 126–132

the same system parameters as in Fig. 7(a) and (b), respectively, and for T /JS = 0.1 for Fig. 8(c). In the case of T /JS = 1.25 (see Fig. 8 (a)), we can observe that the system presents a continuous passage. When T /JS = 0.5 (Fig. 8(b)), the system exhibits two discontinuities at D/JS = − 2.34 and D/JS = − 2.46. In Fig. 8(c) for T /JS = 0.1, the system presents four discontinuities at D/JS = − 1.23, − 1.69, − 2.41, and −2.86. These results are in good agreement with those obtained in the investigations of the phase diagrams of the system (Fig. 5).

4. Conclusion In this work, we have applied Monte Carlo simulation to study the magnetic properties of a superlattice which is formed by alternating layers consisting of spins that can take the values S = ± 1, 0 and σ = ± 3/2, ± 1/2. We have studied the effect of the exchange interactions coupling and the crystal field on the magnetic properties of the system. Our numerical results obtained are very rich; the system exhibits tricritical point, isolated critical point and critical end point. It has been found that the topology of the phase diagrams and the nature of the phase transitions depend strongly on the set of the Hamiltonian parameters. Concerning the compensation behavior, the existence and the variations of the compensation temperature are also strongly linked with the system parameters.

Acknowledgment This work has been initiated with the support of URAC: 08, the Project RS: 02 (CNRST) and the Swedish Research Links Program dnr-348-2011-7264.

References [1] Y.L. Li, S.Y. Hu, D. Tenne, A. Soukiassian, D.G. Schlom, X.X. Xi, K.J. Choi, C. B. Eom, A. Saxena, T. Lookman, Q.X. Jia, Appl. Phys. Lett. 91 (2007) 252904. [2] A. Zaim, M. Kerouad, Y. EL Amraoui, D. Baldomir, J. Magn. Magn. Mater. 316 (2007) e306. [3] T. Balcerzak, J.W. Tucker, J. Magn. Magn. Mater. 140 (1995) 653. [4] Shutaro Asanuma, Yoshiaki Uesu, Charlotte Malibertand, Jean-Michel Kiat, J. Appl. Phys. 103 (2008) 094106. [5] J.F. Scott, Science 315 (2007) 954. [6] R.E. Camley, R.L. Stamps, J. Phys. 5 (1993) 3727. [7] F. Aguilera-Granaja, J.L. Moran-Lopez, Solid State Commun. 47 (1990) 155. [8] T. Hai, Z.V. Li, D.L. Lin, T.F. George, J. Magn. Magn. Mater. 97 (1991) 227. [9] A. Du, H.J. Liu, B. Wang, Physica A 342 (2004) 583. [10] An Du, Yan Ma, Zeng-hui Wu, J. Magn. Magn. Mater. 305 (2006) 233. [11] C.L. Wang, D.R. Tilley, Mater. Sci. Eng. B 99 (2003) 572. [12] A. Saber, H. Ez-Zahraouy, S. Lo Russo, G. Mattei, A. Ainane, J. Magn. Magn. Mater. 261 (2003) 7887. [13] Y. Belmamoun, M. Boughrara, M. Kerouad, D. Baldomir, Surf. Sci. 600 (2006) 2841. [14] Y. Belmamoun, M. Kerouad, J. Magn. Magn. Mater. 299 (2006) 383. [15] M. Boughrara, M. Kerouad, A. Zaim, Physica A 388 (2009) 2131. [16] M. Boughrara, M. Kerouad, Physica A 374 (2007) 669. [17] M. Ahlberg, E. Th Papaioannou, G. Nowak, B. Hjörvarsson, J. Magn. Magn. Mater. 341 (2013) 142. [18] B. Deviren, M. Keskin, O. Canko, J. Magn. Magn. Mater. 321 (2009) 458. [19] M. Keskin, Y. Polat, J. Magn. Magn. Mater. 321 (2009) 3905. [20] A. Feraoun, A. Zaim, M. Kerouad, Physica B 445 (2014) 74–80. [21] M. Zukoviè, A. Bobák, Physica A 389 (2010) 5402. [22] A. Yigit, E. Albayrak, Physica A 392 (2013) 4216. [23] K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, Berlin, 1991. [24] L. Néel, Ann. Phys. 3 (1948) 137. [25] W. Wang, W. Jiang, D. Lv, F. Zhang, J. Phys. D: Appl. Phys. 45 (2012) 475002. [26] E. Albayrak, A. Yigit, Physica A 349 (2005) 471–486.