Monte Carlo study of the magnetic properties of a bi-layer decorated graphene structure

Accepted Manuscript Monte Carlo study of the magnetic properties of a bi-layer decorated graphene structure

N. Tahiri, A. Jabar, L. Bahmad

PII: DOI: Reference:

S0375-9601(16)31660-7 http://dx.doi.org/10.1016/j.physleta.2016.11.011 PLA 24183

To appear in:

Physics Letters A

Received date: Revised date: Accepted date:

6 June 2016 31 October 2016 7 November 2016

Please cite this article in press as: N. Tahiri et al., Monte Carlo study of the magnetic properties of a bi-layer decorated graphene structure, Phys. Lett. A (2017), http://dx.doi.org/10.1016/j.physleta.2016.11.011

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Highlights • • • •

We have analyzed the magnetic properties of a bi-layer decorated graphene structure. We have analyzed the ground state phase diagrams of the studied system. We have showed the existence of the critical temperature. A number of characteristic behaviors are found, such as the occurrence of single hysteresis loops for appropriate values of the system parameters.

Monte Carlo study of the magnetic properties of a bilayer decorated graphene structure

N. Tahiri *, A. Jabar and L. Bahmad Laboratory of Magnetism, Physics and High Energy (LMPHE-URAC 12) Mohammed V University in Rabat, P. O. Box 1014, Morocco ABSTRACT Using Monte Carlo simulations, we study the magnetic properties of a bi-layer decorated graphene structure. We elaborate the ground state phase diagrams corresponding to very low values of the reduced temperature. The partial magnetizations are deduced and compared to those of the ground state phase diagrams at very low temperatures. We found that, from 7x8=56 possible configurations, only 22 configurations are stable in different planes of the ground state phase diagrams. Moreover, and due to strong exchange coupling interactions, the critical temperature values increase when increasing the reduced exchange coupling interaction values. We presented and discussed the magnetic hysteresis cycles, for different reduced parameter values. We have analyzed the magnetic properties of a bi-layer decorated graphene structure. We have showed the existence of a critical temperature, depending on a number of parameters. We showed the occurrence of a single hysteresis loop for appropriate values of the system parameters.

Keywords: Ground state; Bi-layer decorated; Magnetic Properties; Graphene structure; Monte Carlo Study; Hysteresis loops.

* Corresponding author: [email protected]

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I-INTRODUCTION Recently, the low-dimensional carbon structures have been the focus of extensive research since the discovery of fullerenes and carbon nano-tubes. The magnetic properties of graphene can be efficiently modulated by such a geometrical cutting method [1–3]. Graphene nanoribbons with zigzag and armchair edge shapes show different magnetic and electronic properties [4]. Moreover, the graphene layer has also been considered as chemically painted and nano-paper in different patterns with different functional atoms and/or molecules [5–7]. A new type of two dimensional material graphene was experimentally synthesized through exposing graphene to cold hydrogen plasma [7] and theoretically proposed by Refs. [8–10]. These main types of modification methods provide us efficient modulating effects on the physical properties of graphene. On other hand, the edge-state spins are arranged parallel to each other in a zigzag edge with strong ferromagnetic interactions [11]. This nanographene is expected to be a ferromagnetic material with a Curie temperature higher than that one of iron, which is the most popular ferromagnetic material. On the other hand, some theoretical approaches, like Monte Carlo simulations, are applied to study the graphene based materials [12]. The magnetic structure of the nano-graphene and nano-graphite are studied in Ref. [13], In addition, the magnetic order–disorder layering transitions of a spin-1 Ising model are studied under the effect of a crystal ¿eld using the mean ¿eld theory [14]. Edge decorations can determine whether if the ribbon is a conductor or semiconductor. The former is associated with the decorating atom and its covalence bond, while the latter is related to the edgestructure induced magnetic moments. Decorated systems, which belong to conductors, could have a free carrier density higher than or comparable to that of armchair carbon nanotubes, and the electrons may be transported along the edge atoms. The selectivity of decorated structures is potentially important for a wide variety of applications in nano-electronic devices [15-18]. Moreover, the zero-field-cooled and the field cooled magnetization behaviors are

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investigated for different values of external magnetic field and a fixed value of the exchange interaction between the two blocs [19]. On the other hand, the strain modulation can introduce the change of interlayer exchange couplings, for this purpose see the reference [20] about magnetic monolayer modulated by strain. Applying Monte Carlo simulations, a previous work [21], has studied the magnetic properties of transition metal based on a monolayer. Also, the Curie's temperature of the transition metal of a monolayer has been obtained. The aim of this work is to contribute to these activities by studying the magnetic properties of a bi-layer decorated graphene structure with a crystal field and an external magnetic field, using Monte Carlo simulations. In the first part, we elaborate the ground state phase diagrams in different planes, and obtained the most stable phases. In the second part, we deduce and discuss different magnetic properties of this system. We have studied the magnetic properties of a bi-layer decorated graphene structure, showing the existence of a critical temperature, depending on a number of parameters. We showed the occurrence of a single hysteresis loop for appropriate values of the system parameters. The paper is organized as follows. In Section II, we present the model and method. Results and discussion are given in Section III and Section IV is reserved for the conclusion.

II. MODEL AND METHOD The Hamiltonian of the system is given by:

‫ ܪ‬ൌ σଶ௞ୀଵሼ െ ‫ܬ‬ఙఙ σழ௜ǡ௝வ ߪ௜௞ ߪ௝௞ െ ‫ܬ‬ௌௌ σழ௜ǡ௝வ ܵ௜௞ ܵ௝௞ െ ‫ܬ‬ఙௌ σழ௜ǡ௝வ ߪ௜௞ ܵ௝௞ െ ο σ௜ሺሺߪ௜௞ ሻଶ ൅ ሺܵ௜௞ ሻଶ ሻ െ ݄ σ௜ሺߪ௜௞ ൅ ܵ௝௞ ሻሽ െ ‫ܬ‬ఙ௩ σழ௜ǡ௝வ ߪ௜ଵ ߪ௝ଶ െ ‫ܬ‬ௌ௩ σழ௜ǡ௝வ ܵ௜ଵ ܵ௝ଶ (1) where ıik and Sjk are the spins random variables, for each layer k (k=1, 2), which take the values ±7/2; ±5/2; ±3/2; ±1/2 and ±3; ±2; ±1; 0, respectively. For each layer (k=1, 2), the first, second and the third summations run over all pairs of nearest-neighbor sites of the system between ıik-ıjk, Sik-Sjk and ıik-Sjk, respectively. In Eq. (1), the summation σ௜ ݇ runs over all 3

sites of the system. The intermediate terms between the two layers (k=1, 2) are described by the two last terms in Eq. (1). Indeed, the summation σழ௜ǡ௝வ ݇ runs over the two pairs of spins, ıi1-ıj2 and Si1-Sj2 belonging to the layers k=1 and k=2, respectively. The exchange coupling interactions: ‫ܬ‬ఙఙ , ‫ܬ‬ௌௌ and ‫ܬ‬ఙௌ stand for the coupling interactions between ıik-ıjk, Sik-Sjk and ıik-Sjk, respectively, of the two layers (k=1, 2). The interlayer exchange couplings: ‫ܬ‬ఙ௩ ൌ ‫ݒ‬

‫ ߪߪܬ‬and‫ ܵܬ‬ൌ ‫ܬ‬ௌௌ stand for the interactions between the spins ıik and Sjk of the two layers, see Fig. 1. †Šstand for the crystal field and the magnetic external field applied on all the sites of the system. For a simplicity reason, we will limit to case: JSS= JS, between the spins Sik belonging to the two layers. Also, we will use the reduced parameters: Rσσ=Jσσ/JSS, Rσ=Jσ/JSS and RσS=JσS/ JSS.

Fig. 1: A geometry illustrating the studied model for a number of 140 spins (ı=7/2) and 56 spins (S=3).

We performed the Monte Carlo simulations under the Metropolis algorithm. To obtain the equilibrium properties of the physical system in contact with a heat bath of temperature, we use the Hamiltonian given by Eq. (1). The algorithm accepts or rejects a new configuration according to some probability based on the Boltzmann statistics. This algorithm is called a 4

Monte Carlo (MC) step for each iteration. The Monte Carlo simulations (MCs) generate many configurations according to the system variable values following the Boltzmann distribution. This leads to the equilibrium properties when averaging the generated configurations. Starting from different initial conditions, we perform 5x105 Monte Carlo steps, for each spin configuration, and discarding the first 5x104 generated configurations. Then, we average over many physical con¿gurations for each initial condition.

IV. RESULTS AND DISCUSSION IV.1 Ground state phase diagrams study We discuss the possible phase diagrams in terms of the variation of different parameters, by computing and comparing all possible configuration energies. Indeed, we can determine the ground state phase diagrams from the Hamiltonian (1). It can produce many stable topologies showing different phase diagrams in terms of adequate parameters. The stable phases are obtained by minimizing the energy values for each parameter values. The corresponding ground state phase diagrams are presented in Figs. 2(a)-(c). All the parameters are normalized by the exchange coupling interaction JSS between S-S. In fact, the effect of the exchange coupling interactions is illustrated in Fig. 2(a) for fixed values of the parameters RσS=1.0, ǻ/JSS=0.0, h/JSS=0.0. It is clear that the only stable configurations are (+3, +7/2), (-3, -7/2), (+2, +7/2), (-2, -7/2), (+1, +7/2), (-1, -7/2), (0, +7/2) and (0, -7/2), in four regions of the (Rıı, Rı) plane. Every two configurations are found in the same region in this plane. This is due to the fact that they have same energies. On the other hand, we plot in Fig. 2(b) the corresponding phase diagram in the plane (Rıı, RıS). The stable configurations are found for fixed values of Rσ=1.0, ǻ/JSS=0.0 and h/JSS=0.0. The only stable configurations are: (-3, +7/2),

(+3, -7/2), (-2, +7/2), (+2, -7/2) (-1, +7/2) and (+1, -7/2). These phases are stable for the values of the reduced exchange interaction parameter RıS < 0.0. Whereas for values of Rıq > 0.0, 5

the stable configurations are those already found in Fig. 2(a). This is due to the fact that each two configurations correspond to the same energy; see Fig. 2(b). In order to show the effect of the reduced crystal field, we illustrate in Fig. 2(c) the corresponding phase diagram. It is found that only eight configurations are stable in four regions of the plane (Rı, Δ/JSS), namely: (+3, +5/2), (-3, -5/2), (+3,

+3/2), (-3, -5/2), (+1, +1/2), (-1, -1/2), (0, +1/2) and (0, -1/2). Every two configurations occupy the same region in the plane (Rı, ǻ/JSS) for fixed values of the parameters Rσσ=1.0, RσS=1.0 and h/JSS=0.0, see Fig. 2(c). To complete this ground state study, we illustrate in Fig. 2(d) the corresponding phase diagram in the plane (Rıı, h/JSS). It is shown that each configuration

corresponds to a unique region of this plane; see Fig. 2(d). This phase diagram shows a close symmetry for positive and negative values of the reduced external magnetic field. The stable configurations found in this figure are those already obtained in precedent planes.

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(a)

(+3,+7/2) (-3,-7/2)

0

(+2,+7/2) (-2,-7/2)

-2

(+1,+7/2)

Rσσ

(-1,-7/2)

-4

(0,+7/2)

-6

(0,-7/2) -8 -9

-6

-3

0

Rσ

6

(b) 0

(+3,+7/2) (-3,-7/2)

(-3,+7/2) (+3,-7/2)

-3

(+2,+7/2) -6

Rσσ

(-2,-7/2)

-9

(-2,+7/2) (+2,-7/2)

-12

(-1,+7/2)

(0,+7/2) (0,-7/2)

(+1,-7/2)

-10

-5

(+1,+7/2) (-1,-7/2)

0

5

10

RσS

10

(c)

(+3,+5/2) (-3,-5/2) 5

(+3,+7/2)

(+3,+3/2) (-3,-3/2) (+1,+1/2) (-1,-1/2)

0

Rσ

(-3,-7/2)

(+2,+7/2) (-2,-7/2)

-5

(0,+1/2) (0,-1/2)

-10

(+1,+7/2) (-1,-7/2) -15 -12

-9

-6

-3

0

3

Δ/JSS

7

0

(-3,-7/2)

-2

Rσσ

(d)

(+3,+7/2)

(-2,-7/2)

(+2,+7/2)

-4

(-1,-7/2)

(+1,+7/2)

(0,-7/2)

(0,+7/2)

-6

-8

-10 -4

-2

0

2

4

h/JSS Fig. 2 : Ground state phase diagrams of the studied system, (a) in the plane (Rıı, Rı), (b) in the plane (Rıı, RıS), (c) in the plane (Rı, ǻ/JSS) and (d) in the plane (Rıı, h/JSS). All the parameters are normalized by the exchange coupling interaction JSS between the spins S-S. From the 7x8=56 possible configurations, only 22 configurations are found to be stable. In (a) we fixed RσS=1.0, ǻ/JSS=0.0, h/JSS=0.0; in (b) Rσ=1.0, ǻ/JSS=0.0, h/JSS=0.0. In (c) Rσσ=1.0, RσS=1.0, h/JSS=0.0 and in (d) Rσ=1, RσS=1 and ǻ/JSS=0.

4.2 Monte Carlo Study In this part, we explore the Monte Carlo simulations to study the effect of the reduced temperature, for different reduced parameters, on the total magnetization and susceptibility profiles. We first plot in Fig. 3(a) the profiles of the total magnetizations and susceptibilities as a function of the reduced temperature. This figure is plotted for fixed values of the reduced exchange coupling interactions: Rσσ=0.2, RσS=1.0, Rσ=1.0, the reduced external magnetic field h/JSS=1.0 and in the absence of the reduced crystal field ǻ/JSS=0.0. From this figure, it is found that for low values of the reduced temperature T/JSS, the partial magnetizations are in good agreement with the corresponding ground state phase diagram; see Fig. 2(d). The corresponding partial susceptibilities show a notable peak around the value TC/JSS ≈ 28.0. This reduced temperature corresponds to the peak of the susceptibilities.

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Mσ, χσ

3.5

MS, χS

Mσ, S, tot ,χσ,S, tot

3.0

Mtot, χtot

2.5 2.0 1.5 1.0 0.5 0.0 0

20

40

T/JSS

60

80

100

Fig. 3: Total magnetizations and susceptibilities as a function of the reduced temperature T/JSS for fixed values of Rσσ=0.2, RσS=1.0, Rσ=1.0, ǻ/JSS=0.0 and h/JSS=1.0.

RσS=+1.0

44

RσS=+1.5 40

36

TC/JSS

32

28

24 0.0

0.3

0.6

0.9

Rσσ

1.2

1.5

Fig. 4: The behavior of the reduced blocking temperature as a function of the reduced exchange coupling interaction Rσσ, for fixed values of Rσ=1.0, ǻ/JSS=0.0 and h/JSS=1.0.

In order to show the effect of Rıı on the behavior of the reduced critical temperature TC/JSS, we plot in Fig. 4 the obtained results, for fixed values of the reduced external magnetic field (h/JSS=1.0), reduced exchange interactions (Rı=1.0) and in the absence of the reduced crystal field (Δ/JSS=0.0). It is found that this critical temperature increases for increasing values of 9

Rıı and/or RıS, see Fig. 4. This behavior is caused by the increasing values of Rıı. In addition, we plot in Figs. 5(a,b) the behavior of the total magnetizations as a function of the reduced exchange coupling interaction RıS at fixed values of the reduced external magnetic field (h/JSS =1.0), reduced exchange interactions (Rı=1.0) and in the absence of the reduced crystal field (ǻ/JSS=0.0). We found that, when increasing the reduced exchange coupling interaction RıS, the total magnetizations undergo a saturation state from disordered phases for RıS≤-0.5. On other hand, these total magnetizations increase when increasing the values of the reduced exchange coupling interaction RıS at fixed value of Rıı, see Fig. 5(a). Similarly, the effect of the reduced exchange coupling interaction RıS on the total magnetizations is plotted in Fig. 5(b) for specific values of the reduced temperature namely: T/JSS=10.0, 15.0, 20.0 and 30.0. It is clear that for lower temperatures than T/JSS =20.0, the total magnetizations reach the saturation value, whereas for T/JSS greater than 20.0 the saturation cannot be reached. This is due to the fact that the disordered phase has been exceeded, see Fig. 5(b).

3.5

Rσσ=+0.5 3.0 2.5

Mtot

2.0

Rσσ=+0.7 Rσσ=+1.0 Rσσ=+1.5

1.5 1.0 0.5 0.0 -2.0

(a) -1.5

-1.0

-0.5

0.0

0.5

1.0

RσS

10

3.5 3.0 2.5

T/JSS=10 T/JSS=15 T/JSS=20 T/JSS=30

2.0

Mtot 1.5 1.0 0.5

(b) 0.0 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

RσS Fig. 5: Behavior of the total magnetization as a function of the reduced exchange coupling interaction RıS at fixed values of h/JSS =1.0, Rı = 1.0 and ǻ/JSS =0.0. In (a) for different values of Rıı and for fixed value of T/JSS=10.0; in (b) for different values of T/JSS and for fixed value of Rıı=1.0.

The effect of the reduced crystal field on the studied system is illustrated in Fig. 6. One can see that the effect of increasing the reduced crystal field is to increase the total magnetizations at fixed value of Rıı. On the other hand, at fixed value of the reduced crystal field Δ/JSS the increasing of Rıı values increases the values of the total magnetizations as it is shown in Fig. 6. In addition, the peak of susceptibility increases with decreasing the reduced exchange coupling interaction Rıı, see Fig. 6.

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3.5

Rσσ=+1.0 Rσσ=+2.5

3.0

Rσσ=+3.5

Mtot, χtot

2.5 2.0 1.5 1.0 0.5 0.0 -15

-10

-5

0

5

Δ/JSS Fig. 6: Total magnetizations and susceptibilities as a function of the reduced crystal field Δ/JSS for different values of the reduced exchange coupling interaction Rσσ=+1.0, +2.5 and +3.5, at fixed value of Rσ=1.0, RσS=1.0, T/JSS=10.0, h/JSS=1.0.

To complete this study, the magnetic hysteresis cycles are illustrated in Figs.7, as a function of the reduced external magnetic field h/JSS. In fact, Fig. 7(a) represents this behavior for different values of the reduced exchange coupling interaction values: Rσσ=0.5, 1.0 and 2.5, at fixed values of ǻ/JSS=0.0, RσS=1.0 and T/JSS=10.0. It is found that when increasing the values of the reduced exchange coupling interaction the loop surfaces increase as well as the coercive reduced magnetic field. This is due to the fact that the system moves from the paramagnetic phase to ferromagnetic one, at fixed reduced temperature value T/JSS=10.0. Similarly, we plot in Fig. 7(b) the corresponding hysteresis loops for different values of RıS. From this figure it appears that the increasing effect is to decrease both surface loops and the reduced coercive magnetic field when exceeding the value RıS ≈ 0.2. On other hand, we illustrate in Fig. 7(c) the hysteresis loops, of the studied system, when varying the reduced temperature values. This figure shows that when increasing the values of the reduced temperature, the surface loops decreases to be canceled completely at T/JSS=30.0. The corresponding reduced coercive magnetic field is also eliminated. Finally, we plot in Fig. 7 12

(d) the hysteresis loops when varying the reduced crystal field. It is found that when increasing the absolute value of the reduced crystal field, the loops area decreases to disappear completely at Δ/JSS=-3.5.

4

Rσσ=+0.5

Mtot 0

Rσσ=+1.0

Rσσ=+2.5

2

-2

(a)

-4 -15

-10

-5

0

5

10

15

h/JSS 4

RσS=+0.2 RσS=+0.5 RσS=+1.0

2

Mtot 0 -2

(b)

-4 -15

-10

-5

0

5

10

15

h/JSS

13

4

T/JSS=10 T/JSS=20 T/JSS=30

2

0

Mtot

-2

(c)

-4 -15

-10

-5

0

5

10

15

h/JSS 4

Δ/JSS=0.0 Δ/JSS=-2.5 Δ/JSS=-3.5

2

Mtot

0

-2

(d)

-4 -15

-10

-5

0

5

10

15

h/JSS Fig. 7: Magnetic hysteresis cycles of the studied system at fixed value of Rı=1.0. (a) for different values of the reduced exchange coupling interaction: Rıı = 0.5, 1.0 and 2.5, at fixed values of RσS=1.0, ǻ/JSS=0, T/JSS=10; (b) for different values of the reduced exchange coupling interaction: RıS = 0.2, 0.5 and 1.0, at fixed values of Rσσ=1.0, ǻ/JSS=0, T/JSS=10 (c) for different values of the reduced temperature: T/JSS=10.0, 20.0 and 30.0, at fixed values of ǻ/JSS=0.0, Rıı=1.0 and RıS=1.0; (d) for different values of the reduced crystal field ǻ/JSS =0.0, -2.5 and -3.5, at fixed values of Rıı=1.0, RıS=1.0 and T/JSS=10.0.

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V. CONCLUSION In this paper, we studied the magnetic properties and phase diagrams of a bi-layer graphene structure composed with the spins (S=3, ı=7/2), using Monte Carlo simulations. We showed that from 7x8=56 possible configurations, in the ground state phase diagrams, only 22 configurations are stable in different planes plotted for different parameters. The partial magnetizations are found to be in good agreement with those found in the ground state phase diagrams. Since, these ground state phase diagrams are valid for very low values of the reduced temperature. In addition, due to strong exchange coupling interactions, the critical temperature increases when increasing the reduced exchange interaction values. To complete this study, we presented and discussed the magnetic hysteresis cycles, for fixed reduced temperature and the reduced crystal field values. In particular, we found that the coercive reduced field values increases when increasing the reduced exchange coupling interaction values.

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