Graphyne core/shell nanoparticles : Monte Carlo study of thermal and

(Submitted, (2016))

• Résumé de la publication 4 :

Dans ce papier, notre objectif principal consistait à l’étude des propriétés magnétiques et hystérétiques d’une nanoparticule cœur-coquille avec une structure crystallographique sem- blable au graphyne. Pour ce faire, nous avons utilisé la simulation Monte Carlo. Nous avons examiné les effets des paramètres de l’hamiltonien sur les propriétés magnétiques et thermody- namiques du système, à savoir : l’aimantation totale, la susceptibilité, les cycles d’hystérésis et la température de compensation. Par ailleurs, nous avons examiné l’état fondamental, ainsi que les diagrammes de phase de ce système. À ce titre, nous avons noté l’apparition de deux points de compensation, ainsi que l’existence de deux nouveaux type du compensation. Ceux-ci n’ont pas été classifiés dans la nomenclature du Néel plus les types Q, P et N . L’étude des propriétés hystérétiques a révélé que la nanoparticule cœur-coquille de type graphyne présente des cycles simples et triples avec diverses formes.

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Graphyne core/shell nanoparticles:

Monte Carlo study of thermal and magnetic properties

L. B. Drissi1,2,3_{, S. Zriouel}1

1- Lab-PHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco 2- CPM, Centre of Physics and Mathematics, Faculty of Science,

Mohammed V University, Rabat, Morocco and

3- ICTP, Inernational Centre for Theoretical Physics, Trieste, Italy.

Abstract

Magnetic properties and hysteresis behaviors of a graphyne core/shell nanoparticles are studied within the framework of Monte carlo calculations. We analyze in detail the ground-state phase diagrams in different planes. We examine the effects of the extrinsic and intrinsic parameters of the Hamiltonian on the magnetic and the thermodynamic quantities of the system, namely, the total magnetization, its corresponding susceptibility, the hysteresis curves, and the compensation behavior that is of crucial importance for technological applications such as thermo-optical recording. A number of characteristic behaviors are found, such as the occurrence of one and two compensation temperatures and the existence of two new and non-classified types of compensation behavior in addition to the Q-, P- and N-types. Moreover, single and triple hysteresis loops which exhibit different step effects and various shapes are observed.

Keywords: Graphyne, Core-shell nanostructures, Monte Carlo simulation, Magnetic properties, Compensation behavior, Hysteresis loops.

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I. INTRODUCTION

Since the isolation of graphene [1], several 2D materials with hexagonal structure have been realized and many more have been predicted such as graphyne. This new derivative has been introduced to reduce the extremely high thermal conductance of graphene that is of critical importance [2]. Recently, γ−graphyne has been fabricated using the vapor-liquid-solid (VLS) growth process [3].

Magnetic graphene nanostructures are of great interest as their application in a number of exceptional spintronic devices have been proposed [4, 5]. However, the feasibility of engineering these devices is questioned by magnetic ordering at finite temperature. Therefore, it is very important to understand the fundamental behavior of the magnetic properties of such structures. Graphone nanoribbons (GONR) is a class of graphene allotropes that exhibits magnetism [6]. Using Monte Carlo (MC) calculations and mean field theory (MFT), it was shown that the critical temperature Tc reduces as a step function versus the ribbon widths in pure GONR [7] and Tc increases with the number of substituted transition metal (TM) impurities in doped GONR [8]. On the other hand, the effect of 3d TM (V, Cr, Mn, Fe and Co) nanowires on the magnetic properties of doped graphyne were reported [9, 10].

Recently, a particular interest has been paid to hexagonal systems with core-multishell morphology that offer advantages over the other hexagonal nanostructures. One, two or even three compensation temperatures were observed in the ternary system ABC consisting of spins (1/2, 1, 3/2) [11]. The Q-, R-, P-, S-, and W-type compensation behaviors, the existence of a tricritical point and special critical points (E, B, and M), were the results found for the (1/2, 1, 3/2) ternary mixed Ising nanoparticles [12]. It is worth noting that in contrast to N´eel theory [13] that expects the existence of only one compensation point in ferrimagnetic materials, the molecular based ferrimagnets show two compensation points [14, 15]. This type of behavior of compensation temperature has been confirmed in several works using different techniques such as exact star-triangle mapping transformation [16], MC calculations [17], MFT [18] and EFT [19, 20].

Motivated by the growing interest to obtain novel magnetic materials with high transition temperature, as well as the success of 2D graphene-like structure in nanotechnology applications, this work contributes in understanding the thermal and the magnetic properties of graphyne core/shell nanoparticles. Using Monte Carlo calculations, we study magnetic ordering in spin-1/2 hexagonal core surrounded by spin (1, 3/2)–double shell having the structure of γ-graphyne. First, we introduce the system and describe the formulation and the method. Then, we analyze the ground-state phase diagrams in different planes and discuss in detail the thermal behavior of the magnetization and the susceptibility as well as the hysteritic properties of such systems.

II. THE FORMULATION

The present work studies the magnetic behavior of a planar structure composed of a hexagonal core surrounded by like-graphyne double-shells as plotted in Fig.1 in the (x,y) plane. It is worth recalling that graphyne is obtained by replacing one-third of the carbon–carbon bonds in graphene with triple-bonded carbon linkages [2].

In this study, the magnetic structure will be based on the position of the magnetic atoms which are as follows: the core contains Nc Ising spins σ = 1

2 that are coupled with an exchange parameter Jc. Analogous to graphyne, shell(1) in the present structure refers to linkages between the nearest neighboring hexagons that form shell(2). The shell(1) is occupied by Nsh1spins s = 1. The Nsh2 shell(2) sites take the spin values S =3₂. To keep the geometric structure of graphyne, only the number Nsh1can be changed while the number of the other sites is kept fixed.

In the presence of an external longitudinal magnetic field h and a crystal-field ∆, the core/double-shells Ising system is described by the classical Hamiltonian:

H = _−Jc X hk,li σkσl_{− J}sh1 X hi,ji sisj_{− J}sh2 X hm,ni SmSn_{− J}int1 X hi,ki σksi_{− J}int2 X hi,mi siSm −h X k σk+X i si+X m Sm ! − ∆ X i s2 i + X m S2 m ! , (1)

where_{h., .i denotes the nearest neighbor interactions in the system and J}sh1 and Jsh2 are the couplings between the first nearest neighbor magnetic atoms at shell(1) and between spins Sm and Sn at shell(2) respectively. Jint1 and Jint2 stand for antiferromagnetic interaction parameters between magnetic nearest-neighbors of [core/shell(1)] and between [shell(1)/shell(2)] respectively. Notice that in the core system there is no crystal field ∆ contribution as the spins σ = 1₂ have only two states_±1₂ [21].

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FIG. 1: Schematic representation of (1/2,1, 3/2) hexagonal core surrounded by graphyne shell showing magnetic atoms with spins σ =1

2, s = 1, S = 3

2 and their corresponding exchange couplings Jc, Jsh1,Jsh2, Jint1and Jint2.

III. MC METHOD

In the Monte Carlo simulations based on Metropolis algorithm [22], free boundary conditions are applied in the X and Y -directions. Configurations are generated by selecting the sites in sequence through the lattice and making single-spin-flip attempts, which are accepted or rejected according to some probability based on Boltzmann statistics. Starting from different initial conditions, all the Monte Carlo simulations are averaging over different configurations for each starting initial condition. The MC steps are 5_{× 10}5 _{steps per spin discarding the first 5}

× 104 _{Monte Carlo} simulations. The program calculates the following parameters, namely the internal energy per site,

E = 1

N hHi , (2)

with N = Nc+ Nsh1+ Nsh2 where Nc, Nsh1 and Nsh2 denote the number of spins in core, shell(1) and shell(2) respectively as mentioned previously. The total magnetization M and its corresponding susceptibility are respectively:

M = Mc+ Msh1+ Msh2

N and χ = βN



M2_{− hMi}2, (3)

where Mc, Msh1 and Msh2 are defined as follows

Mc= Nc X k=1 σk, Msh1= NXsh1 i=1 si, and Msh2= NXsh2 m=1 Sm, (4) and β = 1

kBT, T is the absolute temperature and kB is the Boltzmann’s constant that it is set kB = 1 in all the

following calculations.

IV. RESULTS AND DISCUSSION

In this section, we examine the ground state diagrams required to check the reliability of the theoretical results. Then, we discuss in details the thermal behavior and the hysteretic properties of the studied system.

A. Ground state

The ground state phase diagrams are studied by minimizing the Hamiltonian in eq(1) to determine the spin states at T = 0K in the mixed Ising structure. The Hamiltonian contains intrinsic and extrinsic parameters as described in eq(1). Such parameters produce many stable topologies leading to different phase diagrams. For simplicity, we fix the size N = 79, namely Nc = 19, Nsh1= 24, and Nsh2= 36 and the couplings Jint1= Jint2= Jint. The corresponding ground state phase diagrams are presented in Fig.2 in seven different planes.

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FIG. 2: The ground-state diagrams for the ferrimagnetic core-double-shell nanostructure composed of N=79 magnetic atoms in the following seven planes: a) h

Jsh1, Jint Jsh1  at ∆ = 0, Jsh2 Jsh1 = 0.1, Jc Jsh1 = 0.1, b)  ∆ Jsh1, Jint Jsh1  at h = 0, Jsh2 Jsh1 = 0.5, Jc Jsh1 = 0.1, c) h Jsh1, ∆ Jsh1  for Jint Jsh1 =−0.1, Jsh2 Jsh1 = 0.5, Jc Jsh1 = 0.1, d)  h Jsh1, Jc Jsh1  at ∆ = 0, Jint Jsh1 =−0.1, Jsh2 Jsh1 = 0.1, f)  ∆ Jsh1, Jc Jsh1  at h = 0, Jint Jsh1 = −0.1, Jsh2 Jsh1 = 0.1, g)  h Jsh1, Jsh2 Jsh1  for ∆ = 0, Jint Jsh1 = −0.1, Jc Jsh1 = 0.1, and h)  ∆ Jsh1, Jsh2 Jsh1  for h = 0, Jint Jsh1 =−0.1, Jc Jsh1 = 0.1.

To examine the effect of the external magnetic field, we plot in Fig.2-a all the spin configurations present in the plane _Jsh1h ,Jint

Jsh1



. There are two stable spin configurations for _Jsh1h < 0. The phases are (−3/2, −1, −1/2) and (_{−3/2, +1, −1/2). However for} h

Jsh1 > 0, the two (+3/2, +1, +1/2) and (+3/2,−1, +1/2) configurations are present.

In addition to the four phases found in the plane_Jsh1h ,Jint

Jsh1



, eight new phases arise in the plane_Jsh1∆ ,Jint

Jsh1



provided that crystal field takes negative values. As shown in in Fig.2-b, the new phases are (+1/2, +1, +1/2), (_{−1/2, −1, −1/2),} (+1/2,_{−1, +1/2), (−1/2, +1, −1/2), (−1/2, 0, −1/2), (+1/2, 0, +1/2), (+1/2, 0, −1/2) and (−1/2, 0, +1/2) . In the} plane _Jsh1h ,_J∆_sh1, six stable phases are observed namely (−3/2, −1, −1/2), (−1/2, 0, −1/2) , (−1/2, −1, −1/2) , (+3/2, +1, +1/2), (+1/2, +1, +1/2) and (+1/2, 0, +1/2) (see Fig.2-c). It is obvious that the external field h/Jsh1 separates the two configuration spin spaces (σ, s, S) and (_{−σ, −s, −S) .}

To inspect the effect of core coupling, we present in Fig.2-d and -f, the stable phases found in the planes  h Jsh1, Jc Jsh1  and _J∆_sh1, Jc Jsh1 

respectively. In the plane _Jsh1h , Jc

Jsh1



there exists only two possible ground states namely, (_{−3/2, −1, −1/2) and (+3/2, +1, +1/2). However, eight possible spin configurations are observed in the} plane_J∆_sh1, Jc

Jsh1



for _J∆_sh1 > 0 as depicted in Fig.2-f, while for _J∆_sh1 < 0, four different configurations are present. The effect of the Jsh2/Jsh1 coupling on the ground state is presented in Figs.2-g and -h. In the plane  h

Jsh1,

Jsh2

Jsh1

 , the ground state diagram is divided in four regions and each one has one spin configuration different from the oth- ers. However, the ∆

Jsh1,

Jsh2

Jsh1



−plane consists of twenty-four ground state configurations distributed in four different regions. It follows that each parameter in the Hamiltonian affects the ground state of the system.

B. Transition temperatures

Recall that the existence of compensation temperatures in ferrimagnetic materials is of great technological impor- tance especially in the thermomagnetic recording and magneto-optical readout applications [13, 23, 24]. This section studies the effect of the parameters Jint

Jsh1, h Jsh1, ∆ Jsh1, Jc Jsh1 and Jsh2

Jsh1 on the occurrence of the compensation temperature.

B.1. Coupling parameters

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The effect of the core Jc/Jsh1, the shell(2) Jsh2/Jsh1 and the intermediate Jint/Jsh1 exchange couplings on the magnetization is presented in Fig.3 for fixed couplings in the absence of the crystal and the external magnetic fields.

FIG. 3: The magnetizations vs temperature at ∆/Jsh1= h/Jsh1= 0 in terms of (a) the core coupling Jc/Jsh1for Jint/Jsh1=

−0.5 and Jsh2/Jsh1= 0.1 (b) the second shell exchange coupling Jsh2/Jsh1at Jint/Jsh1=−0.5 and Jc/Jsh1= 0.1 and (c) the

intermediate coupling|Jint/Jsh1| for Jsh2/Jsh1= Jc/Jsh1= 0.1.

Fig.3-a shows the temperature dependence of the total magnetization at three different values of the core coupling Jc/Jsh1. The magnetization curves exhibit two zeros indicating the presence of a compensation and a critical tem- perature in the system. Both the compensation points and the critical temperatures shift to high temperatures when increasing Jc/Jsh1. In Fig.3-b, the magnetization curves exhibit also two vanishing points indicating the presence of a compensation and critical temperatures at Jsh2/Jsh1= 0.6 and 1.6. However, for higher exchange coupling value (Jsh2/Jsh1 = 8), the magnetization curve exhibits only one vanishing point corresponding to the critical tempera- ture of the system. Therefore, the style of the total magnetization shifts from N_{−type behavior to Q−type one} when increasing Jsh2/Jsh1. Finally, Fig.3-c depicts the total magnetization as a function of the temperature at three different absolute values of the intermediate exchange coupling |Jint/Jsh1|. All the curves confirm the presence of the compensation and the critical temperature that increase as|Jint/Jsh1| increases. Notice that no discontinuity is observed in the three figures. Consequently, all the ordered and disordered phases are separated by a second order transition line and the system does not present any first-order transition.

To inspect the effect of the core Jc/Jsh1, the shell(2) Jsh2/Jsh1and the intermediate_|Jint/Jsh1_{| exchange couplings on}

FIG. 4: The critical and compensation temperatures in the absence of crystal and external fields ∆/Jsh1= h/Jsh1= 0 in terms

of (a) the core coupling Jc/Jsh1for Jint/Jsh1=−0.5 and Jsh2/Jsh1= 0.1 (b) the second shell exchange coupling Jsh2/Jsh1 at

Jint/Jsh1=−0.5 and Jc/Jsh1= 0.1 (c) the intermediate coupling|Jint| /Jsh1for Jsh2/Jsh1= Jc/Jsh1= 0.1.

the occurence of Tc/Jsh1 and Tcomp/Jsh1, we plot Fig.4 at fixed parameter values. As illustrated in Fig.4-a, Tc/Jsh1 is not sensitive to the change of the core exchange coupling for Jc/Jsh1≤ 1 and it remains constant in this region as Tc/Jsh1= 1.65. The horizontal line Tc/Jsh1= 1.65 is the transition temperature of the first shell sublattice since Jc/Jsh1 has no effect on Tc/Jsh1and the contribution of the second shell sublattice to the transition temperature is very weak Jsh1> Jsh2. When Jc/Jsh1> 1, the critical temperature Tc/Jsh1increases linearly with the core coupling showing that the effect of core interactions becomes dominant.

For the compensation temperature, the behavior is completely different. Tcomp/Jsh1 values increase with increasing Jc/Jsh1and stabilize for Jc/Jsh1≥ 4.2. It is deduced that the system presents a compensation temperature whatever the values of the coupling Jc/Jsh1.

Fig.4-b plots the phase diagram in the plane (T /Jsh1, Jsh2/Jsh1) . When increasing Jsh2/Jsh1, the compensation temperature Tcomp/Jsh1 increases linearly and rapidly toward Tc/Jsh1. It disappears for Jsh2/Jsh1> 7.4. However Tc/Jsh1increases more rapidly until it saturates at Jsh2/Jsh1= 5 and subsists.

Fig.4-c plotted in the plane (T /Jsh1,|Jint/Jsh1|) shows that the ordered and disordered phases are separated by a second-order phase transition line whatever the absolute value of the intermediate coupling |Jint/Jsh1|. Moreover when increasing_|Jint/Jsh1_{|, the temperature T}c/Jsh1increases linearly with a steeper slope compared to the compen- sation temperature Tcomp/Jsh1leading to an increase in the difference between them. This behavior is due to the fact

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that it is more difficult for the spins lattice to become disordered when there is a strong intermediate interaction that tends to keep the spins of the system parallel. It is also worth to note that the compensation temperature Tcomp/Jsh1 subsists for all values of _|Jint/Jsh1_|.

B.2. The crystal field ∆/Jsh1

FIG. 5: In the absence of external field h/Jsh1= 0 and at fixed value of Jint/Jsh1=−0.5, Jsh2/Jsh1= 0.5 and Jc/Jsh1= 0.1,

we plot a) magnetizations versus temperature for different values of crystal field ∆/Jsh1and b) the critical and compensation

temperatures as a function of the crystal field ∆/Jsh1

In what follows, the effect of the crystal field ∆/Jsh1, that was set to zero in the previous part, is investigated. Fig.5-a shows the total magnetization in the absence of the external field h = 0, for fixed values of the couplings Jint/Jsh1=_{−0.5, J}sh2/Jsh1= 0.5 and Jc/Jsh1= 0.1. The curves of magnetization indicates that the system indergoes a second order transition. At low T /Jsh1, the magnetizations present different forms due to the various ground states of the system as shown in Fig.2-b. At ∆/Jsh1=−1, the magnetization decreases going to zero when the temperature increases showing a Q-type behavior. For the two specific values ∆/Jsh1 =−0.5 and 0.5, the magnetization curves undergoes two critical temperatures. The first one corresponds to the compensation temperature Tcomp/Jsh1 and the second one is the critical temperature Tc/Jsh1. When varying the value of ∆/Jsh1 from _{−0.5 to 0.5, both the} compensation point and the critical temperature increase and the total magnetization curves exhibit N-type behavior. When ∆/Jsh1 > 0.6, the total magnetization decreases from its saturation value at low temperature region, and vanishes at a first compensation temperature Tcomp1/Jsh1as shown for ∆/Jsh1= 1 and 2. Then the magnetization M increases slowly, for T /Jsh1> Tcomp1/Jsh1, taking negative values and vanishes at a second compensation temperature Tcomp2/Jsh1. After that M changes its sign and decreases slowly to vanish at a critical temperature Tc/Jsh1. The magnetization curves exhibit a type W-behavior that is characterized by the presence of two compensation points before the critical one [14, 16]. Notice that the W-type is not classified by Neel nomenclature [13].

To shed more light on the effect of the crystal field on the occurrence of the compensation temperature, Fig.5-b presents the compensation and the critical temperatures as function of ∆/Jsh1 at h/Jsh1 = 0, Jint/Jsh1 = −0.5, Jsh2/Jsh1= 0.5 and Jc/Jsh1= 0.1. When increasing the crystal field ∆/Jsh1, the critical temperature increases until it reaches a saturation value Tc/Jsh1= 10.02 for ∆/Jsh1_{≥ 5. In contrast, no compensation temperature is found in} the system for ∆/Jsh1<_{−0.5. When ∆/J}sh1 vary in the range [−0.5, 0.6] , the compensation temperature increases with the crystal field. Beyond the value ∆/Jsh1= 0.6, the system exhibits two compensation points. In this region, the first compensation temperature remains constant Tcomp1/Jsh1= 1.5 while the second compensation temperature increases slowly up to a threshold value close to ∆/Jsh1= 5 then stabilizes Tcomp2/Jsh1= 1.9. When ∆/Jsh1is larger than 5, the horizontal lines Tc/Jsh1= 10.02, Tcomp1/Jsh1= 1.5, and Tcomp2/Jsh1= 1.9 indicate that the system is no longer sensitive to the crystal field and exhibits fixed compensation and critical temperatures.

B.3. The size Nsh1

FIG. 6: The total magnetization as a function of T /Jsh1for different sizes Nsh1of shell(1) in the absence of crystal and external

fields ∆/Jsh1= h/Jsh1= 0 and at fixed value of Jint/Jsh1=−0.5, Jsh2/Jsh1= 0.5 and Jc/Jsh1= 0.1.

Following previous customary notations [25], in the atomic configuration of graphyne sheet, hexagonal carbon rings are joined by single acetylenic linkages. Therefore to increase the size of graphyne, one changes only the number of

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acetylene bonds between adjacent hexagons.

In this paragraph, the influence of the size Nsh1of the shell(1) on the total magnetization is studied for the specific parameters ∆/Jsh1= h/Jsh1= 0, Jint/Jsh1=_{−0.5, J}sh2/Jsh1= 0.5 and Jc/Jsh1= 0.1. For Nsh1= 24 and 48, one can see in Fig.6-a that the magnetization curves exhibit two vanishing points indicating the presence of compensation and critical temperatures that increase when increasing Nsh1. According to N´eel classification scheme [13], the style of the total magnetization is characterized by N−type behavior. When increasing the size to Nsh1 = 72 and 96, the total magnetization is very analogous to the N-type dependence, however, it does not exhibit a change of sign in this particular case and hence no compensation point emerges. This behavior is not classified by the five types of compensation behaviors given in the N´eel classification nomenclature [13]. Due to its geometric form, we refer to this new behavior as V-type.

When the shell(1) size Nsh1 increases to larger values, the temperature Tc/Jsh1 increases as depicted in fig.6-b. For Nsh1 = 480 and 600, the absolute M-curve shows P-type behavior since the absolute magnetization exhibits a

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