- Email: [email protected]
Dilution effects on compensation temperature in nano-trilayer graphene structure: Monte Carlo study
Physica B: Condensed Matter 564 (2019) 104–113
Contents lists available at ScienceDirect
Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb
Dilution effects on compensation temperature in nano-trilayer graphene structure: Monte Carlo study
T
Z. Fadil∗, M. Qajjour, A. Mhirech, B. Kabouchi, L. Bahmad∗∗, W. Ousi Benomar Laboratoire de Matière Condensée et Sciences Interdisciplinaires (LaMCScI), Faculty of Sciences, Mohammed V University, P. O. Box 1014, Rabat, Morocco
ARTICLE INFO
ABSTRACT
Keywords: Nano-trilayer Graphene Dilution probability Monte Carlo simulations Compensation temperature Non-equivalent planes
In this paper, we study the magnetic properties of a diluted trilayer graphene structure with non-equivalent planes. The spins of the planes forming the trilayer systems ABA and BAB are S = 1 (for the plane A) and σ = 3/2 (for the plane B). The ground state phase diagrams are reported and discussed. Besides, we provide the variation of the total magnetizations versus reduced temperature and reduced crystal field for several values of the coupling exchange interactions and the dilution probability, using Monte Carlo simulations. The two systems provide opposite results regarding the compensation temperature Tcomp behavior as a function of dilution probability p. Indeed, the compensation temperature increases when decreasing the dilution probability for ABA system. While for the BAB system, Tcomp increases when increasing p.
1. Introduction The nano-materials structures have applications in many areas such as spintronic [1]. Recently, the study of graphene structures has attracted considerable attentions [2,3]. The graphene is a two-dimensional crystalline materials network consisting of carbon atoms contained in a honeycomb structure. Monolayer, bilayer and multilayer samples of such a structure have been obtained [4–6]. Furthermore, the nano-graphene presents many performances, such as electrical energy storage, high mechanical stability, and low intensity. Indeed, the unique graphene structure of benzene's six-membered rings allows it to become one of the lightest and thinnest two-dimensional materials. New scientific advances have extended the scope of graphene, such as solar cells [7], linear magneto resistance [8], chemical engineering [9], environmental sciences [10], etc. Besides, the presence of impurity or vacancy in nano-graphene structures affects largely the physical properties the study system, namely its thermal conductivity [11]. Recently, several researchers have been interested to study the physical properties of multilayers graphene and their applications in different fields (see Ref. [12] and references therein). Furthermore, the ferromagnetism in graphene is considered in Ref. [13]. More recently, the zigzag edge effect on magnetic properties on graphene is investigated [14]. A many theoretical magnetic behavior works on graphene-type materials have been realized by various methods. Indeed, several
∗
studies have been made on the ferromagnetic properties of nano-systems, by using different methods such as the mean-field approximation (MFA) [15] and the effective field theory (EFT) [16,17]. The Monte Carlo simulations have been also used to calculate the magnetic properties of nano-graphene lattice [18–25] and nano-graphene bi-layer structure with non-equivalent planes [26,27]. On the other hand, the presence of a compensation temperature is possible in these magnetic systems when the total magnetization is zero [28–36]. In addition, to study the effect of dilution on magnetic properties is very important, because pure systems are rare in nature [37,38]. In the present work, we consider a trilayer graphene structures with non-equivalent planes with two types of atoms (A and B). Each layer is composed of only one type of atom. The studied systems are composed by ABA and BAB structures. We also consider the presence of site dilution in B plane. Calculations are performed using Monte Carlo simulations under the Metropolis algorithm. The paper is organized as follows: we present the Hamiltonian for the studied system in Sec.2. The ground state phase diagrams are given in Sec.3.1. Finally, the simulation details are discussed in Sec.3.2. 2. Model and method We consider a trilayer graphene structure with non-equivalent planes to study the magnetic properties. Each layer is composed exclusively of either type-A or type-B atoms: A (spin S = 1) and B (spin
Corresponding author. Corresponding author. E-mail addresses: [email protected] (Z. Fadil), [email protected] (L. Bahmad).
∗∗
https://doi.org/10.1016/j.physb.2019.03.006 Received 11 January 2019; Received in revised form 28 February 2019; Accepted 8 March 2019 Available online 09 March 2019 0921-4526/ © 2019 Elsevier B.V. All rights reserved.
Physica B: Condensed Matter 564 (2019) 104–113
Z. Fadil, et al.
3. Numerical results and discussion 3.1. Ground state phase diagrams In this section, we calculate the ground state phase diagrams based on the parameters of the system Hamiltonian (1). In fact, from this Hamiltonian we compute the energies of different configurations. In the case of ABA system, we have 3 × 4 × 3 = 36 configurations. While in the case of the BAB system, the number of the configurations is 4 × 3 × 4 = 48. We recall that the ‘stable phase’ corresponds to the lower energy value of such different configurations. In fact, the all other phases exist but they are masked by the stable one. For the ABA system, we start with Fig. 2a plotted in the plane (d, h), with the fixed values of the reduced exchange coupling interactions: RBB = 3 and RAB = −1. From the 36 possible phases, only eleven (11) phases are found to be stable. Whereas, in the case of the BAB system, we plot in Fig.2b the corresponding stable phases in the plane (d, h). This figure corresponds to the fixed values of the reduced exchange coupling interactions: RBB = 2 and RAB = −0.8. Among the 48 possible phases, only sixteen (16) stable one are found.
Fig. 1. A schematic representation of the trilayer systems with (a) the ABA and (b) the BAB systems. The plane B is randomly diluted with non-magnetic atoms.
σ = 3/2) containing NA atoms in plane A and NB atoms in plane B: NA = NB= 50 atoms. The plane B is randomly diluted with non-magnetic atoms (NB = 50 when the plane B is not diluted), see Fig. 1. Preliminary calculations showed that there is no size effect on the physical properties of the studied system. Hence, we are limited in this work to 50 atoms in each plane. Our goal is to study a nano-system with a fixed size. The Hamiltonian of the studied systems is defined by:
H=
JAA
SiS j
JBB
(i A, j A)
H(
i i j j (i B, j B)
Si + (i A )
j j) (j B)
(
Si
In this section we study, using Monte Carlo simulations, the effect of the exchange coupling interactions, the external magnetic and crystal fields on the magnetic properties for the two considered systems. In Fig. 3a, we present the partial and total magnetizations for ABA system where the reduced exchange coupling interactions are RBB = 3 and RAB = −1, in the pure case (p = 1). This figure shows that, in the absence of the reduced external magnetic and reduced crystal fields (h = 0 and d = 0), the magnetization m2 is negative for low temperature values, while the magnetizations m1, m3 and mABA are positive. In such case, the compensation temperature, which corresponds to a null total magnetization, arises for tcomp = 3.23. While for the BAB system, the corresponding results are plotted in Fig. 3b, for fixed values of the reduced exchange coupling interactions RBB = 2 and RAB = −0.8, in the pure case and in the absence of the external magnetic and crystal fields. In this system, the compensation temperature is equal to tcomp = 2.67. The arising of the compensation temperature is due to the existence of super-paramagnetic phase. The partial magnetizations are of opposite signs and the total magnetization is null. In the case of ABA system, we have the following equation:
j j
(i A, j B) i i)
(j B)
JAB
3.2. Monte Carlo simulations
2
(1)
where JAA, JBB and JAB are the magnetic exchange coupling constants between first nearest neighbor atoms with spins S‒S, σ‒σ and S‒σ respectively. H is the external magnetic field, the crystal field Δ is applied on all σ spins. The ɛi are quenched, uncorrelated random variables, chosen to be 1 with probability p (active site concentration), or 0 with probability 1−p (impurity concentration). To simplify the notations, we define the reduced parameters as follows: RAB = JAB/JAA, RBB = JBB/JAA, h = H/JAA, d = Δ/JAA and t = T/JAA. In the case of ABA system, the partial magnetizations of A-type atoms and B-type atoms and the total magnetization are obtained respectively by:
m1 = m3 =
1 m2 = pNB mABA =
1 NA
Si
|(m1 + m3) Tcomp | = |pm 2(Tcomp) | While in the case of BAB system we have:
(2)
(i A)
j j
|p(m1 + m3) Tcomp | = |m2(Tcomp) |
NA m1 + pNB m2 + NA m3 2NA + pNB
(4)
whereas for the BAB structure, the partial magnetizations of A-type atoms and B-type atoms and the total magnetization are given respectively by:
m1 = m3 =
m2 =
1 NA
mBAB =
1 pNB
j j (j B)
Si (i A)
pNB (m 1 + m 3) + NA m 2 NA + 2pNB
(5’)
In the both cases the total magnetization is reduced to zero for T = Tcomp. The Fig. 4a reports the total magnetization of the ABA system as a function of the reduced temperature, for different values of parameter RBB . This figure is plotted for a fixed value of the parameter RAB = −1, in pure case (p = 1) and in the absence of the external magnetic and crystal fields. It is found that the reduced compensation temperature for ABA system depends strongly on the parameter RBB. Contrary to the results obtained for ABA system, in BAB one the compensation temperature is not affected when varying the coupling parameter RBB as shown in Fig. 4b. The value of this reduced compensation temperature is approximately tcomp≈2.57. The Fig. 4b is obtained for RAB = −0.8, h = 0 and d = 0 in the pure case. The Fig. 5a and b shows the variation of total magnetization respectively for ABA and BAB systems as a function of reduced temperature, for the pure case (p = 1). These figures are given for several values of RAB and for a fixed value of RBB. In Fig. 5a, plotted for the parameter RBB = 3, it is shown that the compensation temperature
(3)
(j B)
(5)
(2’) (3’) (4')
where: NA = NB = 50 . 105
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Fig. 2. Ground state phase diagrams of the studied system: (a) The ABA system in the plane (d, h), with: RBB = 3, and RAB = −1, for the BAB system (b) in the plane (d, h), with: RBB = 2, and RAB = −0.8.
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Fig. 3. Magnetizations as a function of the reduced temperature in the ABA(a) and BAB(b) systems, for fixed values of RBB and RAB. In (a) RBB = 3 and RAB = −1, in (b) RBB = 2 and RAB = −0.8, for p = 1,d = 0 and h = 0.
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Fig. 4. Total magnetizations as a function of the reduced temperature in the ABA(a) and BAB(b) systems, for several values of RBB and for fixed values of RAB. In (a) RAB = -1, in (b) RAB = −0.8, for p = 1,d = 0 and h = 0.
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Fig. 5. Total magnetizations as a function of the reduced temperature in the ABA(a) and BAB (b) systems, for several values of RAB and for fixed values of RBB. In (a) RBB = 3, in (b) RBB = 2, for p = 1,d = 0 and h = 0.
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Fig. 6. Total magnetization as a function of the reduced temperature in the ABA(a) and BAB (b) systems, for several values of (p) and for fixed values of RBB and RAB. In (a) RBB = 10 and RAB = -0.1, in (b) RBB = 2 and RAB = −0.8, for d = 0 and h = 0.
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Fig. 7. Total magnetizations as a function of the reduced crystal field in the ABA(a) and BAB (b) systems, for several values of (p) and for fixed values of RBB and RAB. In (a) RBB = 3 and RAB = -1, in (b) RBB = 2 and RAB = −0.8, for fixed values: t = 0.5 and h = 0.
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increases when decreasing the RAB parameter. This is due to the strong coupling between A and B planes. Hence a high temperature is needed for the occurring such compensation behavior. However, for very low temperatures there is no effect of the RAB parameter on the total magnetization, this is in agreement with the ground state phase diagram of Fig. 2a. In Fig. 5b, established for RBB = 2, it is found that the compensation temperature decreases when decreasing the RAB parameter until reaching the value −0.5. For RAB < −0.5, the compensation temperature behavior increases. This behavior is probably due the competition between the ferromagnetic and anti-ferromagnetic statement when choosing RBB = 2 and negative value of RAB. On the other hand, for very low temperatures the total magnetization is not affected by the variation of RAB. Once again, this behavior is in agreement with Fig. 2b of the ground state phase diagram. In order to study the effect of the dilution probability on B atoms in the ABA and BAB systems, we report in Fig. 6a and b the variation of the total magnetization as a function of the reduced temperature, respectively. These figures are given for various values of the dilution p, in the absence of the external magnetic and crystal fields. The Fig. 6a corresponds to fixed values of RBB = 10 and RAB = -0.1, whereas Fig. 6b is plotted for RBB = 2 and RAB = −0.8. From Fig. 6a, it is shown that when the probability p decreases (the dilution of B atoms increases) the compensation temperature increases. Since the dilution is on B atoms, the dominant atoms in the system ABA are in the planes A with spins S = 1. Whereas in the BAB system the reduced compensation temperature increases by decreasing the probability from p = 1 to 0.8. In Fig. 7a, we plot for the ABA system the total magnetizations as a function of reduced crystal field d, for the fixed values of the parameters RBB = 3, RAB = −1, t = 0.5 and for different values of the probability p, in the absence of the reduced external magnetic field (h = 0). This figure exhibits the compensation reduced crystal field dcomp corresponding to the case where the total magnetization vanishes. It is found that the compensation crystal field is almost constant (dcomp -2.8) for all values of dilution probability. In this figure, three different regions are to be distinguished, namely: d < −4.2, −4.2 < d < −2.5 and d > −2.5. In the first region the total magnetizations remain constant. In the second region an abrupt diminution occurs, while in the last region the total magnetizations increase and become constant. In all these regions, the decreasing of the parameter p leads to increasing of the total magnetizations for a fixed value of the crystal field d. In addition, to inspect the effect of both the dilution and the crystal field on the BAB system, we illustrate in Fig. 7b the total magnetizations for RBB = 2, RAB = −0.8, t = 0.5 and selected values of probability p. From this figure, we note that for the probabilities less than 0.9, the system exhibits two dcomp values (super-paramagnetic state), while three dcomp are found for the almost pure system (p ≥ 0.9). Always from Fig. 7b, we can distinguish four regions: d < −3, −3 < d < −1,-1 < d < 4 and 4 < d. The total magnetizations are constant in the first region. In the second region, an increasing of the total magnetizations is found. The total magnetizations present a minimum in the third region, except for the pure case (p = 1). In the last part, the total magnetizations are not affected by increasing of the crystal field. Also, it is seen that for the positive values of d, the dilution effect is more appreciable.
when decreasing the dilution probability for ABA system. While for the BAB system, the dilution effect is inverted. References [1] O.V. Yazyev, Emergence of magnetism in graphene materials and nano-structures, Rep. Prog. Phys. 73 (2010) 056501. [2] W. Wang, Q. Li, D. Lv, R.J. Liu, Z. Peng, S. Yang, Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure, Carbon 120 (2017) 313. [3] M. Orlita, W. Escoffier, P. Plochocka, B. Raquet, U. Zeitler, Graphene in high magnetic fields, Compt. Rendus Phys. 14 (2013) 78. [4] A. Feraoun, M. Kerouad, Nano-graphene monolayer with higher-order exchange couplings: Monte Carlo study, Phys. Lett. 382 (2018) 116. [5] K.S. Novoselov, A.K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005) 197. [6] Y. Zhang, Y.-W. Tang, H.L. Stormerand, P. Kim, Experimental observation of the quantum Hall effect and Berry's phase in graphene, Nature 438 (2005) 201. [7] Y. Song, X.M. Li, C. Mackin, X. Zhang, W.J. Fang, T. Palacios, H.W. Zhu, J. Kong, Role of interfacial oxide in high-efficiency graphene–silicon Schottky barrier solar cells, Nano Lett. 15 (2015) 2104. [8] F. Kisslinger, C. Ott, C. Heide, E. Kampert, B. Butz, E. Spiecker, S. Shallcross, H.B. Weber, Linear magnetoresistance in mosaic-like bilayer graphene, Nat. Phys. 11 (2015) 650. [9] L.J. Xu, W. Chu, L. Gan, Environmental application of graphene-based CoFe2O4 as an activator of peroxymonosulfate for the degradation of a plasticizer, Chem. Eng. J. 263 (2015) 435. [10] Y. Chen, C.L. Tan, H. Zhang, L.Z. Wang, Two-dimensional graphene analogues for biomedical applications, Chem. Soc. Rev. 44 (2015) 2681. [11] C. Li, B. Debnath, X. Tan, S. Su, K. Xu, S. Ge, M.R. Neupane, R.K. Lake, Commensurate lattice constant dependent thermal conductivity of misoriented bilayer graphene, Carbon 138 (2018) 451. [12] A. Mir, A. Shukla, Bilayer-rich graphene suspension from electrochemical exfoliation of graphite, Mater. Des. 156 (2018) 62. [13] T. Ma, F. Hu, Z. Huang, H.-Q. Lin, Controllability of ferromagnetism in graphene, Appl. Phys. Lett. 97 (2010) 112504. [14] S. Cheng, J. Yu, T. Ma, N.M.R. Peres, Strain-induced edge magnetism at the zigzag edge of a graphene quantum dot, Phys. Rev. B 91 (2015) 075410. [15] F. Michael, C. Gonzalez, V. Mujica, M. Marquez, M.A. Ratner, Size dependence of ferromagnetism in gold nanoparticles: mean field results, Phys. Rev. B 76 (2007) 224409. [16] T. Kaneyoshi, Magnetizations of a transverse Ising nanowire, J. Magn. Magn. Mater. 322 (2010) 3410. [17] M. Keskin, N. Sarli, B. Deviren, Hysteresis behaviors in a cylindrical Ising nanowire, Solid State Commun. 151 (2011) 1025. [18] O. Iglesiasand, A. Labarta, Finite-size and surface effects in maghemite nanoparticles: Monte Carlo simulations, Phys. Rev. B 63 (2001) 184416. [19] O. Iglesias, X. Batlle, A. Labarta, Microscopic origin of exchange bias in core/shell nanoparticles, Phys. Rev. B 72 (2005) 212401. [20] M. VasilakakiandK, N. Trohidou, Numerical study of the exchange-bias effect in nanoparticles with ferromagnetic core/ferrimagnetic disordered shell morphology, Phys. Rev. B 79 (2009) 144402. [21] Y. Yuksel, E. Aydiner, H. Polat, Thermal and magnetic properties of a ferrimagnetic nanoparticle with spin-3/2 core and spin-1 shell structure, J. Magn. Magn. Mater. 323 (2011) 3168. [22] R. Masrour, L. Bahmad, A. Benyoussef, M. Hamedoun, E.K. Hlil, Magnetism of nanographene with defects: a Monte Carlo study, J. Supercond. Nov. Magnetism 26 (2013) 679. [23] L. Bahmad, R. Masrour, A. Benyoussef, Nanographene magnetic properties: a Monte Carlo study, J. Supercond. Nov. Magnetism 25 (2012) 2015. [24] W. Jiang, Y.Y. Yangand, A.B. Guo, Study on magnetic properties of a nano-graphene bilayer, Carbon 95 (2015) 190. [25] W. Jiang, G.Z. Wei, Effects of biaxial crystal-field on a ferrimagnetic bilayer system, Phys. B Condens. Matter 362 (2005) 236. [26] R. Masrour, L. Bahmad, A. Benyoussef, Size effect on magnetic properties of a nanographene bilayer structure: a Monte Carlo study, J. Magn. Magn. Mater. 324 (2012) 3991. [27] A. Mhirech, S. Aouini, A. Alaoui-Ismaili, L. Bahmad, Bi-layer graphene structure with non-equivalent planes: magnetic properties study, Superlattice. Microst. 117 (2018) 382. [28] C. Ekiz, R. Erdem, Effect of crystal-field potential on compensation temperature of a mixed spin-1/2 and spin-1 Ising ferrimagnetic system, Phys. Lett. 352 (2006) 291. [29] E. Machado, G.M. Buendı́a, Mixed Ising models with two compensation temperatures, J. Magn. Magn. Mater. 272 (2004) 249. [30] W. Wang, F.L. Xue, M.Z. Wang, Compensation behavior and magnetic properties of a ferrimagnetic mixed-spin (1/2, 1) Ising double layer superlattice, Phys. B Condens. Matter 515 (2017) 104. [31] W. Wang, D.-d. Chen, D. Lv, J.-p. Liu, Q. Li, Z. Peng, Monte Carlo study of magnetic and thermodynamic properties of a ferrimagnetic Ising nanoparticle with hexagonal core-shell structure, J. Phys. Chem. Solids 108 (2017) 39. [32] B. Boechat, R. Filgueiras, C. Cordeiro, N. Branco, Renormalization-group magnetization of a ferrimagnetic Ising system, Phys. Stat. Mech. Appl. 304 (2002) 429. [33] M. Godoy, V.S. Leite, W. Figueiredo, Mixed-spin Ising model and compensation temperature, Phys. Rev. B 69 (2004) 054428.
4. Conclusion In this work, we studied the magnetic properties in a trilayer graphene structure with non-equivalent planes ABA and BAB formed by three layers. The spins of the planes A and B forming the trilayer systems ABA and BAB are S = 1 (for the plane A) and σ = 3/2 (for the plane B). Firstly, the ground state phase diagrams have been established in the plane (h,d) for the studied systems. For non-null temperature values, we applied Monte Carlo simulations to investigate the thermal behavior, the effect of exchange coupling interaction and crystal field. The both systems exhibit the compensation temperature in the pure case. It is also found that such compensation temperature increases 112
Physica B: Condensed Matter 564 (2019) 104–113
Z. Fadil, et al. [34] C. Ekiz, R. Erdem, Effect of crystal-field potential on compensation temperature of a mixed spin-1/2 and spin-1 Ising ferrimagnetic system, Phys. Lett. 352 (2006) 291. [35] D. Lv, F. Wang, R.-j. Liu, Q. Xue, S.X. Li, Monte Carlo study of magnetic and thermodynamic properties of a ferrimagnetic mixed-spin (1, 3/2) Ising nanowire with hexagonal core-shell structure, J. Alloy. Comp. 701 (2017) 935. [36] A. Dakhama, N. Benayad, On the existence of compensation temperature in 2d
mixed-spin Ising ferrimagnets: an exactly solvable model, J. Magn. Magn. Mater. 213 (2000) 117. [37] I.J.L. Diaz, N.D.S. Branco, Monte Carlo simulations of an Ising bilayer with nonequivalent planes, Phys. Stat. Mech. Appl. 468 (2017) 158. [38] I.J.L. Diaz, N.S. Branco, Ferrimagnetism and compensation temperature in spin-1/2 Ising trilayers, Phys. B Condens. Matter 529 (2018) 73.
113
Contents lists available at ScienceDirect
Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb
Dilution effects on compensation temperature in nano-trilayer graphene structure: Monte Carlo study
T
Z. Fadil∗, M. Qajjour, A. Mhirech, B. Kabouchi, L. Bahmad∗∗, W. Ousi Benomar Laboratoire de Matière Condensée et Sciences Interdisciplinaires (LaMCScI), Faculty of Sciences, Mohammed V University, P. O. Box 1014, Rabat, Morocco
ARTICLE INFO
ABSTRACT
Keywords: Nano-trilayer Graphene Dilution probability Monte Carlo simulations Compensation temperature Non-equivalent planes
In this paper, we study the magnetic properties of a diluted trilayer graphene structure with non-equivalent planes. The spins of the planes forming the trilayer systems ABA and BAB are S = 1 (for the plane A) and σ = 3/2 (for the plane B). The ground state phase diagrams are reported and discussed. Besides, we provide the variation of the total magnetizations versus reduced temperature and reduced crystal field for several values of the coupling exchange interactions and the dilution probability, using Monte Carlo simulations. The two systems provide opposite results regarding the compensation temperature Tcomp behavior as a function of dilution probability p. Indeed, the compensation temperature increases when decreasing the dilution probability for ABA system. While for the BAB system, Tcomp increases when increasing p.
1. Introduction The nano-materials structures have applications in many areas such as spintronic [1]. Recently, the study of graphene structures has attracted considerable attentions [2,3]. The graphene is a two-dimensional crystalline materials network consisting of carbon atoms contained in a honeycomb structure. Monolayer, bilayer and multilayer samples of such a structure have been obtained [4–6]. Furthermore, the nano-graphene presents many performances, such as electrical energy storage, high mechanical stability, and low intensity. Indeed, the unique graphene structure of benzene's six-membered rings allows it to become one of the lightest and thinnest two-dimensional materials. New scientific advances have extended the scope of graphene, such as solar cells [7], linear magneto resistance [8], chemical engineering [9], environmental sciences [10], etc. Besides, the presence of impurity or vacancy in nano-graphene structures affects largely the physical properties the study system, namely its thermal conductivity [11]. Recently, several researchers have been interested to study the physical properties of multilayers graphene and their applications in different fields (see Ref. [12] and references therein). Furthermore, the ferromagnetism in graphene is considered in Ref. [13]. More recently, the zigzag edge effect on magnetic properties on graphene is investigated [14]. A many theoretical magnetic behavior works on graphene-type materials have been realized by various methods. Indeed, several
∗
studies have been made on the ferromagnetic properties of nano-systems, by using different methods such as the mean-field approximation (MFA) [15] and the effective field theory (EFT) [16,17]. The Monte Carlo simulations have been also used to calculate the magnetic properties of nano-graphene lattice [18–25] and nano-graphene bi-layer structure with non-equivalent planes [26,27]. On the other hand, the presence of a compensation temperature is possible in these magnetic systems when the total magnetization is zero [28–36]. In addition, to study the effect of dilution on magnetic properties is very important, because pure systems are rare in nature [37,38]. In the present work, we consider a trilayer graphene structures with non-equivalent planes with two types of atoms (A and B). Each layer is composed of only one type of atom. The studied systems are composed by ABA and BAB structures. We also consider the presence of site dilution in B plane. Calculations are performed using Monte Carlo simulations under the Metropolis algorithm. The paper is organized as follows: we present the Hamiltonian for the studied system in Sec.2. The ground state phase diagrams are given in Sec.3.1. Finally, the simulation details are discussed in Sec.3.2. 2. Model and method We consider a trilayer graphene structure with non-equivalent planes to study the magnetic properties. Each layer is composed exclusively of either type-A or type-B atoms: A (spin S = 1) and B (spin
Corresponding author. Corresponding author. E-mail addresses: [email protected] (Z. Fadil), [email protected] (L. Bahmad).
∗∗
https://doi.org/10.1016/j.physb.2019.03.006 Received 11 January 2019; Received in revised form 28 February 2019; Accepted 8 March 2019 Available online 09 March 2019 0921-4526/ © 2019 Elsevier B.V. All rights reserved.
Physica B: Condensed Matter 564 (2019) 104–113
Z. Fadil, et al.
3. Numerical results and discussion 3.1. Ground state phase diagrams In this section, we calculate the ground state phase diagrams based on the parameters of the system Hamiltonian (1). In fact, from this Hamiltonian we compute the energies of different configurations. In the case of ABA system, we have 3 × 4 × 3 = 36 configurations. While in the case of the BAB system, the number of the configurations is 4 × 3 × 4 = 48. We recall that the ‘stable phase’ corresponds to the lower energy value of such different configurations. In fact, the all other phases exist but they are masked by the stable one. For the ABA system, we start with Fig. 2a plotted in the plane (d, h), with the fixed values of the reduced exchange coupling interactions: RBB = 3 and RAB = −1. From the 36 possible phases, only eleven (11) phases are found to be stable. Whereas, in the case of the BAB system, we plot in Fig.2b the corresponding stable phases in the plane (d, h). This figure corresponds to the fixed values of the reduced exchange coupling interactions: RBB = 2 and RAB = −0.8. Among the 48 possible phases, only sixteen (16) stable one are found.
Fig. 1. A schematic representation of the trilayer systems with (a) the ABA and (b) the BAB systems. The plane B is randomly diluted with non-magnetic atoms.
σ = 3/2) containing NA atoms in plane A and NB atoms in plane B: NA = NB= 50 atoms. The plane B is randomly diluted with non-magnetic atoms (NB = 50 when the plane B is not diluted), see Fig. 1. Preliminary calculations showed that there is no size effect on the physical properties of the studied system. Hence, we are limited in this work to 50 atoms in each plane. Our goal is to study a nano-system with a fixed size. The Hamiltonian of the studied systems is defined by:
H=
JAA
SiS j
JBB
(i A, j A)
H(
i i j j (i B, j B)
Si + (i A )
j j) (j B)
(
Si
In this section we study, using Monte Carlo simulations, the effect of the exchange coupling interactions, the external magnetic and crystal fields on the magnetic properties for the two considered systems. In Fig. 3a, we present the partial and total magnetizations for ABA system where the reduced exchange coupling interactions are RBB = 3 and RAB = −1, in the pure case (p = 1). This figure shows that, in the absence of the reduced external magnetic and reduced crystal fields (h = 0 and d = 0), the magnetization m2 is negative for low temperature values, while the magnetizations m1, m3 and mABA are positive. In such case, the compensation temperature, which corresponds to a null total magnetization, arises for tcomp = 3.23. While for the BAB system, the corresponding results are plotted in Fig. 3b, for fixed values of the reduced exchange coupling interactions RBB = 2 and RAB = −0.8, in the pure case and in the absence of the external magnetic and crystal fields. In this system, the compensation temperature is equal to tcomp = 2.67. The arising of the compensation temperature is due to the existence of super-paramagnetic phase. The partial magnetizations are of opposite signs and the total magnetization is null. In the case of ABA system, we have the following equation:
j j
(i A, j B) i i)
(j B)
JAB
3.2. Monte Carlo simulations
2
(1)
where JAA, JBB and JAB are the magnetic exchange coupling constants between first nearest neighbor atoms with spins S‒S, σ‒σ and S‒σ respectively. H is the external magnetic field, the crystal field Δ is applied on all σ spins. The ɛi are quenched, uncorrelated random variables, chosen to be 1 with probability p (active site concentration), or 0 with probability 1−p (impurity concentration). To simplify the notations, we define the reduced parameters as follows: RAB = JAB/JAA, RBB = JBB/JAA, h = H/JAA, d = Δ/JAA and t = T/JAA. In the case of ABA system, the partial magnetizations of A-type atoms and B-type atoms and the total magnetization are obtained respectively by:
m1 = m3 =
1 m2 = pNB mABA =
1 NA
Si
|(m1 + m3) Tcomp | = |pm 2(Tcomp) | While in the case of BAB system we have:
(2)
(i A)
j j
|p(m1 + m3) Tcomp | = |m2(Tcomp) |
NA m1 + pNB m2 + NA m3 2NA + pNB
(4)
whereas for the BAB structure, the partial magnetizations of A-type atoms and B-type atoms and the total magnetization are given respectively by:
m1 = m3 =
m2 =
1 NA
mBAB =
1 pNB
j j (j B)
Si (i A)
pNB (m 1 + m 3) + NA m 2 NA + 2pNB
(5’)
In the both cases the total magnetization is reduced to zero for T = Tcomp. The Fig. 4a reports the total magnetization of the ABA system as a function of the reduced temperature, for different values of parameter RBB . This figure is plotted for a fixed value of the parameter RAB = −1, in pure case (p = 1) and in the absence of the external magnetic and crystal fields. It is found that the reduced compensation temperature for ABA system depends strongly on the parameter RBB. Contrary to the results obtained for ABA system, in BAB one the compensation temperature is not affected when varying the coupling parameter RBB as shown in Fig. 4b. The value of this reduced compensation temperature is approximately tcomp≈2.57. The Fig. 4b is obtained for RAB = −0.8, h = 0 and d = 0 in the pure case. The Fig. 5a and b shows the variation of total magnetization respectively for ABA and BAB systems as a function of reduced temperature, for the pure case (p = 1). These figures are given for several values of RAB and for a fixed value of RBB. In Fig. 5a, plotted for the parameter RBB = 3, it is shown that the compensation temperature
(3)
(j B)
(5)
(2’) (3’) (4')
where: NA = NB = 50 . 105
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Fig. 2. Ground state phase diagrams of the studied system: (a) The ABA system in the plane (d, h), with: RBB = 3, and RAB = −1, for the BAB system (b) in the plane (d, h), with: RBB = 2, and RAB = −0.8.
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Fig. 3. Magnetizations as a function of the reduced temperature in the ABA(a) and BAB(b) systems, for fixed values of RBB and RAB. In (a) RBB = 3 and RAB = −1, in (b) RBB = 2 and RAB = −0.8, for p = 1,d = 0 and h = 0.
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Fig. 4. Total magnetizations as a function of the reduced temperature in the ABA(a) and BAB(b) systems, for several values of RBB and for fixed values of RAB. In (a) RAB = -1, in (b) RAB = −0.8, for p = 1,d = 0 and h = 0.
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Fig. 5. Total magnetizations as a function of the reduced temperature in the ABA(a) and BAB (b) systems, for several values of RAB and for fixed values of RBB. In (a) RBB = 3, in (b) RBB = 2, for p = 1,d = 0 and h = 0.
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Fig. 6. Total magnetization as a function of the reduced temperature in the ABA(a) and BAB (b) systems, for several values of (p) and for fixed values of RBB and RAB. In (a) RBB = 10 and RAB = -0.1, in (b) RBB = 2 and RAB = −0.8, for d = 0 and h = 0.
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Fig. 7. Total magnetizations as a function of the reduced crystal field in the ABA(a) and BAB (b) systems, for several values of (p) and for fixed values of RBB and RAB. In (a) RBB = 3 and RAB = -1, in (b) RBB = 2 and RAB = −0.8, for fixed values: t = 0.5 and h = 0.
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increases when decreasing the RAB parameter. This is due to the strong coupling between A and B planes. Hence a high temperature is needed for the occurring such compensation behavior. However, for very low temperatures there is no effect of the RAB parameter on the total magnetization, this is in agreement with the ground state phase diagram of Fig. 2a. In Fig. 5b, established for RBB = 2, it is found that the compensation temperature decreases when decreasing the RAB parameter until reaching the value −0.5. For RAB < −0.5, the compensation temperature behavior increases. This behavior is probably due the competition between the ferromagnetic and anti-ferromagnetic statement when choosing RBB = 2 and negative value of RAB. On the other hand, for very low temperatures the total magnetization is not affected by the variation of RAB. Once again, this behavior is in agreement with Fig. 2b of the ground state phase diagram. In order to study the effect of the dilution probability on B atoms in the ABA and BAB systems, we report in Fig. 6a and b the variation of the total magnetization as a function of the reduced temperature, respectively. These figures are given for various values of the dilution p, in the absence of the external magnetic and crystal fields. The Fig. 6a corresponds to fixed values of RBB = 10 and RAB = -0.1, whereas Fig. 6b is plotted for RBB = 2 and RAB = −0.8. From Fig. 6a, it is shown that when the probability p decreases (the dilution of B atoms increases) the compensation temperature increases. Since the dilution is on B atoms, the dominant atoms in the system ABA are in the planes A with spins S = 1. Whereas in the BAB system the reduced compensation temperature increases by decreasing the probability from p = 1 to 0.8. In Fig. 7a, we plot for the ABA system the total magnetizations as a function of reduced crystal field d, for the fixed values of the parameters RBB = 3, RAB = −1, t = 0.5 and for different values of the probability p, in the absence of the reduced external magnetic field (h = 0). This figure exhibits the compensation reduced crystal field dcomp corresponding to the case where the total magnetization vanishes. It is found that the compensation crystal field is almost constant (dcomp -2.8) for all values of dilution probability. In this figure, three different regions are to be distinguished, namely: d < −4.2, −4.2 < d < −2.5 and d > −2.5. In the first region the total magnetizations remain constant. In the second region an abrupt diminution occurs, while in the last region the total magnetizations increase and become constant. In all these regions, the decreasing of the parameter p leads to increasing of the total magnetizations for a fixed value of the crystal field d. In addition, to inspect the effect of both the dilution and the crystal field on the BAB system, we illustrate in Fig. 7b the total magnetizations for RBB = 2, RAB = −0.8, t = 0.5 and selected values of probability p. From this figure, we note that for the probabilities less than 0.9, the system exhibits two dcomp values (super-paramagnetic state), while three dcomp are found for the almost pure system (p ≥ 0.9). Always from Fig. 7b, we can distinguish four regions: d < −3, −3 < d < −1,-1 < d < 4 and 4 < d. The total magnetizations are constant in the first region. In the second region, an increasing of the total magnetizations is found. The total magnetizations present a minimum in the third region, except for the pure case (p = 1). In the last part, the total magnetizations are not affected by increasing of the crystal field. Also, it is seen that for the positive values of d, the dilution effect is more appreciable.
when decreasing the dilution probability for ABA system. While for the BAB system, the dilution effect is inverted. References [1] O.V. Yazyev, Emergence of magnetism in graphene materials and nano-structures, Rep. Prog. Phys. 73 (2010) 056501. [2] W. Wang, Q. Li, D. Lv, R.J. Liu, Z. Peng, S. Yang, Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure, Carbon 120 (2017) 313. [3] M. Orlita, W. Escoffier, P. Plochocka, B. Raquet, U. Zeitler, Graphene in high magnetic fields, Compt. Rendus Phys. 14 (2013) 78. [4] A. Feraoun, M. Kerouad, Nano-graphene monolayer with higher-order exchange couplings: Monte Carlo study, Phys. Lett. 382 (2018) 116. [5] K.S. Novoselov, A.K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005) 197. [6] Y. Zhang, Y.-W. Tang, H.L. Stormerand, P. Kim, Experimental observation of the quantum Hall effect and Berry's phase in graphene, Nature 438 (2005) 201. [7] Y. Song, X.M. Li, C. Mackin, X. Zhang, W.J. Fang, T. Palacios, H.W. Zhu, J. Kong, Role of interfacial oxide in high-efficiency graphene–silicon Schottky barrier solar cells, Nano Lett. 15 (2015) 2104. [8] F. Kisslinger, C. Ott, C. Heide, E. Kampert, B. Butz, E. Spiecker, S. Shallcross, H.B. Weber, Linear magnetoresistance in mosaic-like bilayer graphene, Nat. Phys. 11 (2015) 650. [9] L.J. Xu, W. Chu, L. Gan, Environmental application of graphene-based CoFe2O4 as an activator of peroxymonosulfate for the degradation of a plasticizer, Chem. Eng. J. 263 (2015) 435. [10] Y. Chen, C.L. Tan, H. Zhang, L.Z. Wang, Two-dimensional graphene analogues for biomedical applications, Chem. Soc. Rev. 44 (2015) 2681. [11] C. Li, B. Debnath, X. Tan, S. Su, K. Xu, S. Ge, M.R. Neupane, R.K. Lake, Commensurate lattice constant dependent thermal conductivity of misoriented bilayer graphene, Carbon 138 (2018) 451. [12] A. Mir, A. Shukla, Bilayer-rich graphene suspension from electrochemical exfoliation of graphite, Mater. Des. 156 (2018) 62. [13] T. Ma, F. Hu, Z. Huang, H.-Q. Lin, Controllability of ferromagnetism in graphene, Appl. Phys. Lett. 97 (2010) 112504. [14] S. Cheng, J. Yu, T. Ma, N.M.R. Peres, Strain-induced edge magnetism at the zigzag edge of a graphene quantum dot, Phys. Rev. B 91 (2015) 075410. [15] F. Michael, C. Gonzalez, V. Mujica, M. Marquez, M.A. Ratner, Size dependence of ferromagnetism in gold nanoparticles: mean field results, Phys. Rev. B 76 (2007) 224409. [16] T. Kaneyoshi, Magnetizations of a transverse Ising nanowire, J. Magn. Magn. Mater. 322 (2010) 3410. [17] M. Keskin, N. Sarli, B. Deviren, Hysteresis behaviors in a cylindrical Ising nanowire, Solid State Commun. 151 (2011) 1025. [18] O. Iglesiasand, A. Labarta, Finite-size and surface effects in maghemite nanoparticles: Monte Carlo simulations, Phys. Rev. B 63 (2001) 184416. [19] O. Iglesias, X. Batlle, A. Labarta, Microscopic origin of exchange bias in core/shell nanoparticles, Phys. Rev. B 72 (2005) 212401. [20] M. VasilakakiandK, N. Trohidou, Numerical study of the exchange-bias effect in nanoparticles with ferromagnetic core/ferrimagnetic disordered shell morphology, Phys. Rev. B 79 (2009) 144402. [21] Y. Yuksel, E. Aydiner, H. Polat, Thermal and magnetic properties of a ferrimagnetic nanoparticle with spin-3/2 core and spin-1 shell structure, J. Magn. Magn. Mater. 323 (2011) 3168. [22] R. Masrour, L. Bahmad, A. Benyoussef, M. Hamedoun, E.K. Hlil, Magnetism of nanographene with defects: a Monte Carlo study, J. Supercond. Nov. Magnetism 26 (2013) 679. [23] L. Bahmad, R. Masrour, A. Benyoussef, Nanographene magnetic properties: a Monte Carlo study, J. Supercond. Nov. Magnetism 25 (2012) 2015. [24] W. Jiang, Y.Y. Yangand, A.B. Guo, Study on magnetic properties of a nano-graphene bilayer, Carbon 95 (2015) 190. [25] W. Jiang, G.Z. Wei, Effects of biaxial crystal-field on a ferrimagnetic bilayer system, Phys. B Condens. Matter 362 (2005) 236. [26] R. Masrour, L. Bahmad, A. Benyoussef, Size effect on magnetic properties of a nanographene bilayer structure: a Monte Carlo study, J. Magn. Magn. Mater. 324 (2012) 3991. [27] A. Mhirech, S. Aouini, A. Alaoui-Ismaili, L. Bahmad, Bi-layer graphene structure with non-equivalent planes: magnetic properties study, Superlattice. Microst. 117 (2018) 382. [28] C. Ekiz, R. Erdem, Effect of crystal-field potential on compensation temperature of a mixed spin-1/2 and spin-1 Ising ferrimagnetic system, Phys. Lett. 352 (2006) 291. [29] E. Machado, G.M. Buendı́a, Mixed Ising models with two compensation temperatures, J. Magn. Magn. Mater. 272 (2004) 249. [30] W. Wang, F.L. Xue, M.Z. Wang, Compensation behavior and magnetic properties of a ferrimagnetic mixed-spin (1/2, 1) Ising double layer superlattice, Phys. B Condens. Matter 515 (2017) 104. [31] W. Wang, D.-d. Chen, D. Lv, J.-p. Liu, Q. Li, Z. Peng, Monte Carlo study of magnetic and thermodynamic properties of a ferrimagnetic Ising nanoparticle with hexagonal core-shell structure, J. Phys. Chem. Solids 108 (2017) 39. [32] B. Boechat, R. Filgueiras, C. Cordeiro, N. Branco, Renormalization-group magnetization of a ferrimagnetic Ising system, Phys. Stat. Mech. Appl. 304 (2002) 429. [33] M. Godoy, V.S. Leite, W. Figueiredo, Mixed-spin Ising model and compensation temperature, Phys. Rev. B 69 (2004) 054428.
4. Conclusion In this work, we studied the magnetic properties in a trilayer graphene structure with non-equivalent planes ABA and BAB formed by three layers. The spins of the planes A and B forming the trilayer systems ABA and BAB are S = 1 (for the plane A) and σ = 3/2 (for the plane B). Firstly, the ground state phase diagrams have been established in the plane (h,d) for the studied systems. For non-null temperature values, we applied Monte Carlo simulations to investigate the thermal behavior, the effect of exchange coupling interaction and crystal field. The both systems exhibit the compensation temperature in the pure case. It is also found that such compensation temperature increases 112
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Z. Fadil, et al. [34] C. Ekiz, R. Erdem, Effect of crystal-field potential on compensation temperature of a mixed spin-1/2 and spin-1 Ising ferrimagnetic system, Phys. Lett. 352 (2006) 291. [35] D. Lv, F. Wang, R.-j. Liu, Q. Xue, S.X. Li, Monte Carlo study of magnetic and thermodynamic properties of a ferrimagnetic mixed-spin (1, 3/2) Ising nanowire with hexagonal core-shell structure, J. Alloy. Comp. 701 (2017) 935. [36] A. Dakhama, N. Benayad, On the existence of compensation temperature in 2d
mixed-spin Ising ferrimagnets: an exactly solvable model, J. Magn. Magn. Mater. 213 (2000) 117. [37] I.J.L. Diaz, N.D.S. Branco, Monte Carlo simulations of an Ising bilayer with nonequivalent planes, Phys. Stat. Mech. Appl. 468 (2017) 158. [38] I.J.L. Diaz, N.S. Branco, Ferrimagnetism and compensation temperature in spin-1/2 Ising trilayers, Phys. B Condens. Matter 529 (2018) 73.
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