Compensation temperature in a nano-square with a core-shell structure: Monte Carlo study

Accepted Manuscript Compensation temperature in a nano-square with a core-shell structure: Monte Carlo study S. Aouini, S. Ziti, H. Labrim, L. Bahmad PII:

S0749-6036(16)30591-2

DOI:

10.1016/j.spmi.2016.09.032

Reference:

YSPMI 4522

To appear in:

Superlattices and Microstructures

Received Date: 30 July 2016 Revised Date:

23 September 2016

Accepted Date: 26 September 2016

Please cite this article as: S. Aouini, S. Ziti, H. Labrim, L. Bahmad, Compensation temperature in a nano-square with a core-shell structure: Monte Carlo study, Superlattices and Microstructures (2016), doi: 10.1016/j.spmi.2016.09.032. 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.

ACCEPTED MANUSCRIPT

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Compensation Temperature in a Nano-Square with a Core-Shell Structure: Monte Carlo Study S. Aouini 1, S. Ziti 2, H. Labrim 3 and L. Bahmad 1,* 1

Laboratoire de Magnétisme et Physique des Hautes Energies L.M.P.H.E. URAC 12,

2

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Mohammed V University of Rabat, Faculty of Sciences, B.P. 1014, Rabat, Morocco.

Laboratoire de Recherche en Informatique, Mohammed V University of Rabat, Faculty of 3

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Sciences, B.P. 1014, Rabat, Morocco.

Centre National de l’Energie, des Sciences et des Techniques Nucléaires (CNESTEN), Rabat, Morocco.

1

Abstract:

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The aim of this work is to study the magnetic properties of a nano-structure consisting of an anti-ferri-magnetic surface based on a square core-shell. The geometry of the studied system is formed with alternate layers consisting of spins σ = 3/2 in the core and spins S = 1 in the

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shell. Exactly, we study the effect of the coupling exchange interactions and the crystal field. We use Monte Carlo simulations to investigate the behavior of this nano-structure. Different

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phase diagrams were found depending on the parameters of the studied system. The influence of the temperature on the critical and compensation behaviors of this system is also examined. Finally, we showed that the compensation temperature can appear for specific values of system parameters.

Keywords: Nano-Square; Magnetic properties; Core-Shell nanostructure; Monte Carlo simulations; Compensation temperature; Critical temperature; The crystal field.

*)

Corresponding author : [email protected]

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1- Introduction :

Magnetic nano-particles [1] are very interesting elements for the development of the nanotechnology. Among of these nano-structure elements, core-shell nano-films [2,3], nano-

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wires [4-8] and nano-tubes [9,10] such as Zn O [11], Fe Pt and Fe₃O₄ [12], are intensively studied. These nano-materials have enticed special attention due to their potential technological and industrial applications by emerging nano-technologies. Much effort has been consecrated to profoundly understand the behavior of magnetic nano-materials for the

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subject of a great investment in different domains such as medical applications [13,14]. The area of information storage [15] and permanent magnets [16], are also subject of recent

various

techniques

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studies. Theoretically, the magnetic behavior of the nano-materials has been studied by such

as

the

first-principles

density functional theory (DFT) [17-19], the mean field theory [20-22] and the standard Metropolis algorithm Monte Carlo [23-26].

Recently, much works have been focused on studying the alternate layers of nano-structures

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[27-32]. Such results can show very interesting critical behavior in these nano-particles for various phenomena. The core–shell magnetic nano-structure has obtained a great deal of interest due to their physical and chemical properties. They are relying on the structure of the core, shell and interfaces [33-36]. Indeed, this unusual interest offers a numerous chance for

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advancement progressing and emerging in several scientific domains. Some studies [37-39] are discussed and developed to determine the magnetic properties of the nano-structure materials. Indeed, Hong [40,41] has exhibited the effect of the surface couplings on the

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behavior of the critical temperature T [42,43], for different film thicknesses. The interesting magnetic phenomena are found on the region of the compensation temperature T

. In this

region, the total magnetization M of the studied system disappears before its critical temperature confirming the condition M T

= 0.

Motivated by the existence of the crystal field and the effect of the exchange coupling interactions, we study the magnetic properties of an anti-ferri-magnetic on a square core-shell nano-structure with alternate layers of spins σ = 3/2 and S = 1. The compensation behavior is found under the influence of the interactions between nearest-neighbor atoms [44]. In this paper, our calculations are based on the Monte Carlo method to generate the corresponding

ACCEPTED MANUSCRIPT phase diagrams. These phase diagrams may present analytical behavior, confirmed by the experiments. This investigation is carried out in terms of the magnetization [45]. In Section 2, we outline the model and the theoretical formulation. In section 3, we discuss the numerical results, and in section 4, a brief conclusion is given.

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2- Model and method:

We study the magnetic properties of an anti-ferri-magnetic located on a square core-shell nano-structure. The sites are occupied by the spins σ = 3/2 and S = 1 in four layers (k = 4) alternately, with k = 1,3 a number of squares of the system occupied by the spins σ and

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k = 2,4 a number of squares of the system occupied by the spins S. The scheme of the studied system is represented in Fig. 1. The most importance of this work is to examine the effect of

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the crystal field and the exchange coupling interactions. The Hamiltonian of the studied system is given by:

H = −J ∑ , S S −Jσσ ∑ , σ σ −J σ ∑ , S σ –∑ h (S + σ ) −Δ ∑$(#$% + &$% ) Where: J'' and J

(1)

determine the exchange coupling constant between two first nearest

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neighbor atoms with spin σ − σ and S − S respectively. J ' is the exchange coupling interaction constant between two nearest neighbor magnetic atoms with spin S − σ. The crystal field Δ is applied on all S-spins, and h represents the external longitudinal magnetic

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field.

The Metropolis algorithm Monte Carlo is used with a great success to inspect the magnetic

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behavior of a square core–shell nano-particles system with alternate layers of spins−3/ 2 and 1.

Monte Carlo simulations [46,47] have been performed to investigate the equilibrium nanosystems [48] specifically 10+ MC steps per spin. We discard the first 10, configurations. For each initials conditions this step is executed and we except over different initial conditions. The error bars were calculated with a Jackknife method [49,50]. The magnetizations per spin are defined by: M' =

.0 /

/ ∑.1σ

if: k = 1, 3

(2)

ACCEPTED MANUSCRIPT M =

.0 2

2 ∑.1S

if: k = 2,4

(3)

With: N'- = 9 , N% = 16 , N'6 = 24, and N 7 = 32, spins respectively. The condition 8M

9:

T

8 = 8M

;:

T

8 must to be verifying at the compensation

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point when the magnetization must disappear.

3- Results and discussion:

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3-1- Ground state phase diagrams:

We are elaborating a nano-structure formed by four layers (k = 1,2,3,4) of a square core-

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shell with alternate layers consisting of spins 3/2 and 1. Using Monte Carlo simulations, we study the magnetic properties. We are starting by calculating the ground state phase diagrams at T = 0, depending of the parameters of our Hamiltonian (1). By comparing and computing all 4 × 3 × 4 × 3 = 144 possible configurations. In fact, we combine spins σ = 3/2 and S = 1 Ising model, with fixed size: N'- = 9 , N% = 16 , N'6 = 24, and N 7 = 32,

Fig.1

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describes all the characteristics of our system. With k = 1,3 a number of squares of the system occupied by the spins σ and k = 2,4 a number of squares of the system occupied by the spins S.

The corresponding ground state phase diagrams of the studied system are presented in Figs.

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2(a)-(d) where we can see the different configurations. We plot in Fig. 2a in the plane (H, Δ) for the values of the exchange coupling interactions: J'' = J

= 1 and J

'

= −1, from this

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figure we can observe a perfect symmetry is found, regarding the external magnetic field H, the all phases are found for the negative values of the crystal field Δ where only four phases are found to be stable for the positive values of this parameter. In contrast, in the absence of the crystal field (Δ = 0), the number of the configurations increases with reduces region, we plot in Figs. 2b in the plane (H, J'' ) for the values of the exchange coupling interactions: J

= 1 and J

'

= −1 the most of the phases are found for the negative values of the J'' , and

the preceding four phases are found for the positive values of the parameter J'' , with a perfect symmetry regarding the external magnetic field H. Following the same motivation, some new phases are arising in the planes (H, J ), see Figs. 2c, by guarding the preceding four phases

ACCEPTED MANUSCRIPT for the positive values of the parameter J , and the rest of the phases with a large region for the negative values of this parameter. In the plane (H, J ' ), see Fig. 2d, only two phase are found to be stable for positive values of the parameter J

'

for the values of the exchange coupling interactions: J'' = 1. The perfect

symmetry regarding the external magnetic field H is present in this figure, but the number of

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the configurations is reduced for only four large regions for the negative values of the J ' . 3-2- Monte Carlo study:

In this part, using Monte Carlo simulations we will study the effect of increasing temperature

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at which the total magnetization M as a function of the external magnetic field, the exchange coupling interactions and the crystal field, respectively. Indeed, we plot in Figs. 3(a)-(c) the corresponding phase diagrams with: J

= J'' = 1, J

'

= −1.2 and D = −0.9, for T = 0.5,

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T = 1.0 and T = 1.5. The all possible phases are present in this phase diagrams. For low temperature value T = 0.5 for the fixed values of the exchange coupling interactions J

= J'' = 1, J

'

= −1.2 and the crystal field D = −0.9, see Fig. 3a, the

behavior of the total magnetization M as a function of the external magnetic field H of each layer of the system studied is of first order transition type from the negative values to the

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positive values of the total magnetization M . When increasing the temperature values to the T = 1 and T = 1.5 , finally this behavior becomes of second order transition type. Whereas in the absence of the external magnetic field ( H = 0) for the fixed values of the

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exchange coupling interactions J

= J'' = 1 and the crystal field D = −0.9, for low

temperature value T = 0.5, the behavior of the total magnetization M as a function of the

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exchange coupling J ' of each layer is of first order transition type. But when increasing the temperature values to the T = 1 and T = 1.5 , this behavior arises of second order transition type for the negative values of the total magnetization M , see Fig. 3b. In Fig. 3c, the same scenario is repeated in the absence of the magnetic field ( H = 0), for the fixed values of the exchange coupling interactions J

= J'' = 1, J

'

= −1.2 and the crystal

field D = −0.9, for low value of the temperature T = 0.5, the behavior of the total magnetization M as a function of the crystal field D of each layer is of first order transition type starting the negative values to the positive values of the total magnetization M . When increasing the temperature values to the T = 1 and T = 1.5, this behavior arrives of second order transition type.

ACCEPTED MANUSCRIPT Since our studied system is anti-ferri-magnetic, we are interested in the compensation behavior of such system. For this reason, we inspect the existence of the compensation temperature T

for fixed values of the crystal field and the exchange coupling interactions

in the absence of the magnetic field (H = 0). Indeed, we plot this behavior in the plane of the magnetizations as a function of the temperature, see Figs. 4a and 4b. This study has been

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down for specific values of the parameters J ' and Δ. We can found either one or two compensation temperatures, depending on the parameter values. In fact, for J

'

= −1

and Δ = −1, see Fig. 4a, we have only one compensation temperature, while in Fig. 4b, we have two compensation temperatures for J

'

= −1.2

and Δ = −0.9. Hence, the

'

and the crystal field Δ.

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exchange coupling J

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compensation behavior of the studied system is strongly governed both by the effect of the

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4- Conclusion:

In this paper, we have studied the effect of the coupling exchange interactions and the crystal field of a nano-structure consisting of an anti-ferri-magnetic system based on a square coreshell. In this study, we envisaged different planes and found a perfect symmetry regarding the

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external magnetic field H . We also found that the ground state phase diagrams are showing different configurations for the negative values of the phase space parameters in different

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regions. In the absence of the crystal field the number of the most stable configurations is reduced.

Using the standard Metropolis algorithm of the Monte Carlo method, we have analyzed the behaviors of some parameters such as magnetization, compensation and critical temperatures of the studied system. When increasing the temperature values the behavior of the studied parameters changed from the first to the second transition type. The different phase diagrams have been discussed depending on these parameters.

ACCEPTED MANUSCRIPT References: [1] R. Masrour, L. Bahmad, M. Hamedoun, A. Benyoussef, E.K. Hlil, Phys. Lett. A 378, (2014) 276 [2] T. Tybell, C.H. Ahn, J.M. Triscone, Appl. Phys. Lett. 75, (1999) 856

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ACCEPTED MANUSCRIPT [39] S. Aouini, S. Ziti, H. Labrim, L. Bahmad, J. of Superconductivity and Novel Magnetism, (2016), Doi/10.1007/s10948-016-3712-y [40] Q. Hong, Phys. Rev. B 41 (1990) 9621 [41] Q. Hong, Phys. Rev. B 46, (1990) 3207

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Press, Oxford, (1999)

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[49] M. E. J. Newman, G.T. Barkema, Monte Carlo Methods in Statical Physics, Clarendon

[50] K. Binder, D.W. Heermann, MC Simulations in Statistical Physics, Springer, Germany,

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(2002)

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ACCEPTED MANUSCRIPT

Fig. 1: Geometry of the system formed by four square layers with to alternate spins σ = 3/2 and S = 1, containing N'- = 9, N% = 16, N'6 = 24 and N7 = 32.

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Fig, 2 a

8

Jss=Jσσ= 1, Jsσ = -1 6 4

(k)

(l) (b)

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∆

2

(a): (+3/2,+3/2,+1,+1) (b): (+3/2,+3/2,-1,-1) (c): (+1/2,+1/2,+1,+1) (d):(+1/2,+1/2,0,0) (e): (-1/2,-1/2,+1,+1) (f): (+1/2,-1/2,+1,+1) (g): (-1/2,-1/2,0,0) (h): (-1/2,-1/2,-1,-1) (i): (+1/2,+1/2,-1,-1) (j): (-1/2,+1/2,-1,-1) (k): (-3/2,-3/2,-1,-1) (l): (-3/2,-3/2,+1,+1) (m): (+3/2,+1/2,+1,+1) (n): (-3/2,-1/2,-1,-1)

(a)

0

(m)

(n)

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-2 -4

(e)

(i)

(j)

(f) (c)

(h) (g)

-6

(d)

-8

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

H

Fig. 2: Ground state phase diagrams of the studied system in the plane (H, Δ) for J'' = J 1 and J

'

= −1.

=

ACCEPTED MANUSCRIPT Fig, 2 b 8

(a): (+3/2,+3/2,+1,+1) (b): (+3/2,+3/2,-1,-1) (c): (+1/2,+1/2,+1,+1) (d):(+3/2,-3/2,+1,+1) (e): (-1/2,+1/2,+1,+1) (f): (+1/2,-1/2,+1,+1) (g): (+1/2,+3/2,+1,+1) (h): (-1/2,-1/2,-1,-1) (i): (+1/2,-1/2,-1,-1) (j): (-1/2,+1/2,-1,-1) (g) (k): (-3/2,-3/2,-1,-1) (l): (-3/2,-3/2,+1,+1) (m): (+3/2,+1/2,+1,+1) (n): (-3/2,-1/2,-1,-1) (o): (-1/2,-3/2,-1,-1) (p): (-3/2,+3/2,-1,-1)

Jss= 1, Jsσ = -1

6

∆ =0

(k)

4

(a)

(l) (b)

(p)

0

(d)

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Jσσ

2

(o)

-2

(n)

(m)

-4

(j)

(e)

(i)

-8 -8

-7

-6

-5

-4

-3

-2

(c)

(f)

-1

0

1

2

3

4

5

6

7

8

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(h) -6

H

J

'

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Fig. 2: Ground state phase diagrams of the studied system in the plane (H, J'' ) for J = −1 and Δ = 0.

= 1,

Fig, 2 c 8

(a): (+3/2,+3/2,+1,+1) (b): (+3/2,+3/2,-1,-1) (k): (-3/2,-3/2,-1,-1) (l): (-3/2,-3/2,+1,+1) (q): (+3/2,+3/2,+1,-1) (r): (-3/2,-3/2,-1,+1) (s): (+3/2,+3/2,0,+1) (t): (-3/2,-3/2,0,-1) (u): (+3/2,+3/2,+1,0) (v): (-3/2,-3/2,-1,0) (w): (+3/2,+3/2,0,0) (x): (-3/2,-3/2,0,0)

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Jσσ= 1, Jsσ = -1 6

∆ =0

4

(b)

(l)

(k)

Jss

2

EP

0

-2

(a)

(r) (q)

(t)

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-4

(v)

(s)

(x)

(u)

(w)

-6 -8

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

H

Fig. 2: Ground state phase diagrams of the studied system in the plane (H, J ) for J'' = 1, J

'

= −1 and Δ = 0.

ACCEPTED MANUSCRIPT Fig, 2 d 8

Jσσ= Jss= 1 6

(a): (+3/2,+3/2,+1,+1) (b): (+3/2,+3/2,-1,-1) (k): (-3/2,-3/2,-1,-1) (l): (-3/2,-3/2,+1,+1) (q): (+3/2,+3/2,+1,-1) (r): (-3/2,-3/2,-1,+1)

∆ =0

4

(a)

(k)

0

-2 -4

(b)

(l) -6

(q)

(r) -8 -7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

SC

-8

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Jsσ

2

H

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Fig. 2: Ground state phase diagrams of the studied system in the plane (H, J ' ) for J'' = J 1 and Δ = 0.

=

Fig, 3 a

J = -1,2, D = -0,9

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sσ

Jσσ = JSS= 1

T=0,5 T=1 T=1,5

0

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Magnetizations t

1

AC C

-1

-4

-2

0

2

4

H

Fig. 3: The total magnetizations profiles as a function of H, with: J'' = J and D = −0.9 for T = 0.5, T = 1.0, and T = 1.5.

= 1, J

'

= −1.2

ACCEPTED MANUSCRIPT Fig, 3 b 0,0

D = -0,9, H =0 Jσσ = JSS= 1

-0,2

T=0,5 T=1 T=1,5

-0,6

-0,8

-1,0

-1,2

-1,4 -2

0

Jsσ

2

4

SC

-4

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Magnetiztion t

-0,4

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Fig. 3: The total magnetizations profiles as a function of J ' , with: J'' = J

= 1, J

'

= −1.2

= 1, J

'

= −1.2

and D = −0.9 for T = 0.5, T = 1.0, and T = 1.5.

Fig, 3 c

TE D T=0,5 T=1 T=1,5

0

EP

Magnetizations t

1

Jsσ = -1,2, H =0 Jσσ = JSS= 1

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-1

-4

-2

0

2

4

D

Fig. 3: The total magnetizations profiles as a function of D, with: J'' = J and D = −0.9 for T = 0.5, T = 1.0, and T = 1.5.

ACCEPTED MANUSCRIPT Fig, 4 a 1,5

0,5

TComp 2

TComp 1 0,0

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Magnetizations

1,0

-0,5

-1,0

m1 m2 m3 m4 mt

-1,5 0,0

0,5

1,0

T

SC

D = -0,9, Jsσ = -1,2 Jσσ = JSS= 1, H =0

= 1, J

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Fig. 4: Magnetizations profiles in terms of T, for J'' = J −0.9.

'

= −1.2, H = 0 and D =

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Fig, 4 b

1,5

1,0

EP

TComp

0,0

-0,5

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Magnetizations

0,5

m1 m2 m3 m4 mt

-1,0

D = -1, Jsσ = -1 Jσσ = JSS= 1, H =0

-1,5

0,0

0,5

1,0

Fig. 4: Magnetizations profiles in terms of T, for J'' = J −1.

1,5

T

= 1, J

'

= −1, H = 0 and D =

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We study a system formed with alternate nano-squares; The Magnetic properties are established and discussed; We use Monte Carlo simulations; We found the Compensation temperatures; The effect of both the exchange coupling and the crystal field, are discussed.

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