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Monte Carlo study of magnetization plateaus and thermodynamic properties of a nano-graphene with a sandwich-like structure in a longitudinal magnetic field
Physica E 116 (2020) 113721
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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe
Monte Carlo study of magnetization plateaus and thermodynamic properties of a nano-graphene with a sandwich-like structure in a longitudinal magnetic field Dan Lv a, b, *, Ye Ma c, Xiao-hong Luo a, Wei Jiang a, b, Feng Wang b, Qian Li a a b c
School of Science, Shenyang University of Technology, Shenyang, 110870, China School of Materials Science and Engineering, Shenyang University of Technology, Shenyang, 110870, China School of Electrical Engineering, Shenyang University of Technology, Shenyang, 110870, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Nano-graphene Magnetization plateaus Internal energy Specific heat Monte Carlo simulation
Monte Carlo simulation has been used to study magnetization plateaus and thermodynamic properties of the sandwich-like nano-graphene with mixed spin-3/2 and spin-5/2 in a longitudinal magnetic field. The effects of the single-ion anisotropy, the intralayer and interlayer exchange couplings as well as the longitudinal magnetic field on the magnetization, internal energy and specific heat of the system are discussed in detail. Multiple magnetization plateaus behaviors at low temperatures have been found, originating from the competition among various physical parameters. In particular, the results show that the anisotropy, exchange coupling and longi tudinal magnetic field play significant roles in controlling the internal energy and specific heat of the system.
1. Introduction Graphene is one of the lightest and thinnest materials available today, thanks to its unique two-dimensional honeycomb structure. It has attracted considerable attention in many fields owing to its outstanding properties, such as mechanical stability, electrical insulation, low den sity, transparency and gapless semi-conductivity. In particular, the ex istence of nano-graphene with original magnetic behaviors opens up more potential technological applications in biosensor [1], magneto resistive read heads [2], magnetic storage [3], solar cells [4], field-effect transistors [5], chemical engineering [6] and so on. In recent years, experimental synthesis technology of nano-graphene has been greatly developed [7–9]. These experimental progress and evidence arouse a great number of theoretical explorations for nano-graphene. Multiple spin states were employed to study the magnetism of nano-graphene-like molecules, which is a good way to understand ferromagnetism for nano-graphene. Through the molecular orbital calculation and the density functional theory (DFT), the results show that the highest spin state S ¼ 3/2 is steadier for the lowest energy. Relevant investigations of C48H24 graphene molecule revealed that the up and down spins are arranged alternatively in sequence, expressing the antiferromagnetic exchange coupling between spins [10]. Moreover, there exist five unpaired electrons in C59N5H22 molecule [11],
illustrating that the spins can take multiple spin states S ¼ 5/2 and the stronger ferromagnetic exchange coupling exists between up and up spin pairs. Based on the expanded classical Ising model to describe the nanographene structures, a variety of numerical methods were used to study their magnetic properties. By using molecular dynamic simula tions, Lian et al. predicted the magnetic properties of the hybrid gra phene [12]. They found the higher Curie point of the graphene system and ferromagnetic behaviors in the ground state. Szalowski et al. examined the magnetic properties of mano-graphene with mean-field theory (MFT) and discussed the influences of the external in-plane electric and magnetic fields on the ground state phase diagram [13]. With the Green’s function, M. Sherafati et al. revealed the nature of the exchange coupling for the tight-binding band structure of nano-graphene [15]. By means of DFT, the electronic and magnetic performances of cobalt-vacancy defect complexes were investigated in the graphene system [16]. The results demonstrate that the magneti zation of the cobalt atom can be controlled by the vacancy relative to cobalt as well as the vacancy cobalt separation. The local moment of the graphene bilayer was considered based on the Anderson impurity model [17]. What is noticeable is that the prominent magnetic properties of a nano-graphene bilayer were explored in a longitudinal magnetic field by a transverse Ising model [18]. The results indicate that the longitudinal
* Corresponding author. School of Science, Shenyang University of Technology, Shenyang, 110870, China. E-mail address: [email protected] (D. Lv). https://doi.org/10.1016/j.physe.2019.113721 Received 20 February 2019; Received in revised form 9 June 2019; Accepted 11 September 2019 Available online 17 September 2019 1386-9477/© 2019 Elsevier B.V. All rights reserved.
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magnetic field, the exchange coupling, the transverse field and single-ion anisotropy play crucial roles in magnetization, susceptibility and blocking temperature of the nano-graphene bilayer. In recent years, experimental results indicate that the mixed-spin Ising system have displayed many novel performances in the external magnetic field [19–21]. Since Oshikawa et al. [22] proposed the satis fied condition for the existence of magnetization plateaus in the mag netic systems in the longitudinal magnetic field, such efforts have been made to understand the mechanism of the magnetization plateaus for mixed magnetic Ising systems [23]. By means of the classical Monte Carlo (MC) simulation, the magnetization plateaus of the mixed-spin (1, 3/2) antiferromagnetic Ising system with the positive single-ion anisotropy at low temperatures in the longitudinal magnetic field have been examined in detail [24]. Relevant studies also revealed that magnetization plateaus of ferromagnetic and antiferromagnetic mixed films depend on temperatures, magnetic dipolar interactions and single-ion anisotropies [25,26]. Through the study of the ferrimagnetic mixed-spin (1,3/2) Ising chain with the homogenous single-ion anisot ropy by the molecular-field theory [27], Solano-Carrillo et al. found that the existence of the magnetization plateaus may be attributed to the single-ion anisotropy. In addition, interesting results revealed the importance of the anisotropy in the mechanism of the magnetization plateaus in many low-dimensional magnetic systems such as the Ising chain with spin S ¼ 3/2 [28], Bethe lattice [29] and ferrimagnetic core-shell nanoparticle [30]. Experimentally, related magnetization plateau characteristics have been discovered in compounds such as La0.67-xAxCa0.33MnO3 [31], FexMg1-xCl12 [32], NbFeTe2 [33] and SrCu2(BO3)2 [34]. As a significant method, Monte Carlo simulation has been also suc cessfully used to investigate the magnetic and thermodynamic proper ties of nano-graphene systems. By modeling the mixed-spin (3/2, 5/2) Ising system for the single-layer and double-layer nano-graphene, the effects of the crystal-field, exchange coupling, dilution, size, defects and longitudinal magnetic field on the physical quantities including the magnetization, susceptibility, compensation, critical and blocking tem peratures as well as hysteresis loops have been studied by MC simula tions [35–39]. Recently, MC simulations have been applied to understand the influence of higher order exchange couplings on the critical phenomena and compensation behavior of the nano-graphene layer [40–44]. It was found that the Ising model with four-spin inter action can exhibit rich. Remarkably, rich magnetization plateaus be haviors were found in a nano-graphene layer with the spin 5/2 Blume-Emery-Griffiths model, reflecting multiple spin states at low temperature [45]. By MC simulation, the ground state phase diagrams and hysteresis loops were discussed for the bilayer decorated graphene structure with a higher mixed-spin (7/2, 3) Ising model [46]. In addi tion, the magnetic the magnetic and thermodynamic properties of the nano-graphene bilayer system in the external magnetic field were investigated by using MC simulation [47,48]. In spite of these extensive researches of the nano-graphene, less practice has been put into the exploration of magnetic properties of nano-graphene with a sandwich-like structure by MC simulations. Recently, W. Jiang et al. studied the magnetic susceptibilities and magnetization plateaus for the nano-graphene with the sandwich-like structure by the EFT [49]. However, they have not studied systemati cally the influences of the single-on anisotropy, exchange coupling and longitudinal magnetic field upon the internal energy and specific heat of the tri-layer nano-graphene system. Therefore, this paper aims at investigating the effects of single-ion anisotropy, exchange coupling and longitudinal magnetic field on the magnetization plateaus, the internal energy and specific heat of the sandwich-like nano-graphene by MC simulation. Furthermore, it is interesting and meaningful to compare our correlative results with those of EFT [49]. The paper is arranged as follows. In Section 2, the model and Monte Carlo method are described. In Section 3, we show and discuss our typical numerical results. Finally our conclusions are summarized in Section 4.
2. Model and Monte Carlo simulation We consider a sandwich-like mixed spin-3/2 and spin-5/2 Ising model on a honeycomb lattice consisting of N�L spins with the standard single-spin-flip importance method based on the Metropolis algorithm [50]. There are N sites in each layer and L denotes the layer thickness of the nano-graphene tri-layer system (L ¼ 3). The sketch of the present model is depicted in Fig. 1. As shown in this figure, it consists of three layers, namely two surface layers and one middle layer. In the upper and bottom surface layer, every two sublattices a and b with spin-3/2 (↑↓) are interacted with intralayer antiferromagnetic exchange coupling J1 (<0) and in the middle layer, every two sublattices c and d with spin-5/2 (↑↑) are interacted with intralayer ferromagnetic exchange coupling J2 (>0). J3 is the interlayer exchange coupling between the surface layer and the middle layer, which takes a positive value between sublattices a and c, whereas a negative value between sublattices b and d. The Hamiltonian of the present system is given by X X X H ¼ J1 σzia σ zjb J2 Szic Szjd J3 σziaðjbÞ SzicðjdÞ i;j
X�
σ ziaðjbÞ
D1 iðjÞ
�2
i;j
�2 X� SzicðjdÞ Db iðjÞ
i;j
X
σ ziaðjbÞ þ
h iðjÞ
X SzicðjdÞ
!
(1)
iðjÞ
where σziaðjbÞ and SzicðjdÞ are the spin-3/2 and spin-5/2 ions on surface
layers and internal layer, respectively. σziaðjbÞ may take the values of �3/ 2, �1/2 and SzicðjdÞ the values � 5/2, �3/2, �1/2, respectively. The first
three sums only involve nearest-neighbors’ couplings. D1 and D2 are the single-ion anisotropy constants for the spins in the surface layer and the middle layer, respectively. h stands for the external longitudinal mag netic field. In our simulation, additional simulations have also been made for the selection of the spin number in each layer. When changing N from 48 to 108, we have not found obvious differences. Accordingly, N ¼ 48 is the choice for the simulation. For guarantee, about 105 Monte Carlo steps (MCS) per site are discarded for equilibrating the system before aver aging thermal quantities over the next 4 � 105 MCS. The physical quantities are given by the following:
Fig. 1. Structure of a nano-graphene with a sandwich-like structure. 2
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The internal energy per site can be expressed as U¼
1 NL
(2)
and sublattice magnetizations of the surface and middle layers are expressed by Ma ¼
1 X z σia > < N i
(3)
Mb ¼
1 X z < σjb > N j
(4)
Mc ¼
1 X z < Sic > N i
(5)
Md ¼
1 X z < Sjd > N j
(6)
and the total magnetization M per site is 1 M ¼ ð2Ma þ 2Mb þ Mc þ Md Þ 6
(7)
Fig. 3. The variations of the critical field hC and saturation field hS with D1 for J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6 at T ¼ 0.08.
and the specific heat per site C β2 ¼ < H2 > kB NL
< H>
2
�
magnetization plateaus, thus the following results for magnetization plateaus were examined at T ¼ 0.08. The effect of anisotropy D1 on the magnetization plateaus is depicted in Figs. 2 and 3 with J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6. One can see from Fig. 2a that as h is enhanced from zero, the magnetization M in creases and shows two magnetization plateaus (M ¼ 5/6 and 11/6) for weak D1 ¼ 0.25. The distinctive value of the magnetic field is high lighted by a dot dash line and marked with hS where the M approaches the saturation value. The hS is called saturation magnetic field. When D1 is changed to 1.4 and 3.2, there exist two additional magnetization plateaus M ¼ 7/6 and 9/6, corresponding to the called critical magnetic fields hC1 and hC2 with dot lines and dash lines. For D1 ¼ 1.4, the critical magnetic field are hC1 ¼ 1.47, hC2 ¼ 4.16, the saturation mag netic field hS of the system is 7.04. As for D1 ¼ 3.2, we can get hC1 ¼ 3.01, hC2 ¼ 5.08 and hS ¼ 10.57. One can notice that the critical and saturation fields become larger and the distance between them becomes longer with increasing D1. This demonstrates that the step ef fect is strengthened, owing to the fact that the spins becomes more difficult to flip and hard to overcome the strong anisotropy energy. The reason for magnetization plateaus can be illustrated by multiple spin states configuration of sublattices, as shown in Fig. 2b. When D2 is fixed
(8)
where β ¼ 1/kBT. T denotes temperature and kB is Boltzmann constant. We take kB ¼ 1.0 for simplicity. 3. Results and discussions In this section, the exchange coupling J2 is selected as the reduced unit of energy and temperature, so we take J2 ¼ 1.0. It is assumed that the initial directions of the external longitudinal magnetic field h and spins in the middle layer SzicðjdÞ are along the positive z-axis direction. We
focused on the effects of physical parameters on the magnetization plateaus, internal energy and specific heat of the nano-graphene sand wich-like structure, the typical results are presented in Figs. 2–13 and discussions are given as follows. 3.1. Magnetization plateaus Low temperature is a necessary condition for the existence of
Fig. 2. The magnetization (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of D1 (D1 ¼ 0.25, 1.4, 3.2) with J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6 at T ¼ 0.08. 3
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Fig. 6. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for D2 ¼ 4.0 with J1 ¼ 0.4, | J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08.
Fig. 4. The magnetizations of the system and sublattices vs. the longitudinal magnetic field h for different values of D2 with J1 ¼ 0.4, |J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08 (a) M for D2 ¼ 0.5, (b) M1, M2, M3, M4 for D2 ¼ 0.5, (c) M for D2 ¼ 2.0, (d) M1, M2, M3, M4 for D2 ¼ 2.0.
at a low value (D2 ¼ 1.6), the spins of sublattices c and d are always at the spin 5/2 state regardless of the changes in D1 (D1 ¼ 1.4 and 3.2), but a noticeable difference can be found in the Ma and Mb. For weak D1 (D1 ¼ 1.4), the spins of sublattice a are always at the spin 3/2 state, while those of sublattice b have four spin states with h increasing, namely, the spin values σzjb ¼ 3/2, 1/2, 1/2, 3/2 correspond to four
plateaus observed in the Mb curve, which should be responsible for the magnetization plateau behavior of the system in Fig. 2a. Namely, the values of magnetization plateaus M are M ¼ [2 � 3/2 þ 2 � (-3/2)þ5/2 þ 5/2]/6 ¼ 5/6, M ¼ [2 � 3/2 þ 2 � (-1/2)þ5/2 þ 5/2]/6 ¼ 7/6, M ¼ [2 � 3/2 þ 2 � (1/2)þ5/2 þ 5/2]/6 ¼ 9/6 and M ¼ [2 � 3/2 þ 2 � (3/ 2)þ5/2 þ 5/2]/6 ¼ 11/6 by the equation (7). But if D1 is strong enough, its effect becomes prominent, enabling the spins of sublattices a and b to be placed in low spin �1/2 states at the weak h. As h increases, the Mb appears three plateaus with the spin 1/2, 1/2 and 3/2 states, while Ma appears two ones with the spin 1/2 and 3/2 states, which together contributes to the generation in four magnetization plateaus of the system in Fig. 2a. This interesting phenomenon not only results from the cooperation between sublattice magnetizations, but also can be
Fig. 7. The variations of the critical field hC and saturation field hS with D2 for J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 2.5 at T ¼ 0.08.
Fig. 5. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for D2 ¼ 2.5 with J1 ¼ 0.4, |J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08. 4
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whereas h is just the opposite. The results are similar to those of previous studies such as nano-graphene layer [45], nano-graphene tri-layer [49] and finite Ising chain [51]. Fig. 3 shows the phase diagram in the (D1, h) plane with J1 ¼ 0.4, | J3| ¼ 0.8 and D2 ¼ 1.6. The data points in the diagram denote every critical and saturation values of the external magnetic field at which the magnetization of the total system changes. Therefore, the intermediate states between the plateau phases represent different spin configurations of the system, originating from competition between different physical parameters and external magnetic field. Corresponding spin configura tions and the total magnetization under the effects of D1 and h are shown in Table 1. It is clearly seen that hS increases linearly with the increase of |D1|. Meanwhile, on the curves of hC1 and hC2, we can observe two critical points respectively. One for hC1 at D1 ¼ 1.5 and the other for hC2 at D1 ¼ 2.5. This illustrates the changes in the spin configuration of sublattices a and b, namely the spin values of sublattices a and b change from σ ziaðjbÞ ¼ 3/2 (3/2) to σziaðjbÞ ¼ 1/2 (1/2). It is worth mentioning
Fig. 8. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of J1 (J1 ¼ 0.2, 0.6, 1.4) with |J3| ¼ 0.8, D1 ¼ 1.6 and D2 ¼ 2.0 at T ¼ 0.08.
explained by the competitions between the anisotropy D1, the external magnetic field h and the antiferromagnetic exchange coupling J1. On one hand, the h favors for aligning the spins of sublattices a and b along its direction, but the J1 tends to align them anti-parallel. On the other hand, the stronger |D1| tends to force the spins to stay at the lower states,
that the vertical distance between two adjacent values of hC or the last hC and hS represents the width of the magnetization plateau. Our results are in good agreement with those calculated by the EFT in Ref. [49]. Next, we turn our attention to the effect of D2 on the magnetization
Fig. 9. The variations of the critical field hC and saturation field hS with J1 for | J3| ¼ 0.8, D1 ¼ 1.6 and D2 ¼ 2.0 at T ¼ 0.08.
Fig. 11. The variations of the critical field hC and saturation field hS with |J3| for J1 ¼ 0.8, D1 ¼ 2.5 and D2 ¼ 0.5 at T ¼ 0.08.
Fig. 10. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of |J3| (|J3| ¼ 0.2, 0.6, 1.4) with J1 ¼ 0.8, D1 ¼ 2.5 and D2 ¼ 0.5 at T ¼ 0.08. 5
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Fig. 12. The temperature dependence of the internal energy of the system (a) for different values of D1 when J1 ¼ 0.4, |J3| ¼ 0.8, D2 ¼ 1.5and h ¼ 0.5, (b) for different values of D2 when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and h ¼ 0.5, (c) for different values of J1 when |J3| ¼ 0.8, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (d) for different values of |J3| when J1 ¼ 0.4, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (e) for different values of h when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and D2 ¼ 0.5.
plateaus with fixed D1 ¼ 2.5, J1 ¼ 0.4 and |J3| ¼ 0.8, as shown in Figs. 4–7. In Fig. 4a, when D2 ¼ 0.5, four magnetization plateaus can be observed at M ¼ 5/6, 7/6, 9/6 and 11/6. On changing D2 to 2.0 in Fig. 4c, the M curve displays one more plateau at M ¼ 3/6 in low h re gion. Additional plateau (M ¼ 3/6) can be found from a comparison of the magnetization plateaus in sublattices between in Fig. 4b and d. Under the influence of the sufficiently stronger D2 (D2 ¼ 2.0), the spins of sublattices c and d tend to stay at a low spin state SzicðjdÞ ¼ 3/2 in low h
be a need to maintain the lowest energy for the system in the ground state. This phenomenon can be found in previous studies of graphene nanoribbon [52] and 2D boron clusters [53]. Further increasing |D2| to 4.0 in Fig. 6a, we can still see six magnetization plateaus in the system. Differently, illustrated by Fig. 6b, the changes in Mc and Md are not synchronized with each other, leading to the appearance of the plateaus M ¼ 10/6 and 11/6. Fig. 7 presents the phase diagram in the (D2, h) plane with D1 ¼ 2.5, J1 ¼ 0.4 and |J3| ¼ 0.8. The phase diagram is divided into four regions by the vertical dotted lines. One can notice that, as |D2| increases, the number of magnetization plateaus becomes more and more. In different regions, there are four magnetization plateaus for |D2|<2.0, five for 2.0�|D2|�2.5, and six for |D2| >2.5. Compared with Fig. 3, the changes of each hC and hS under the effect of D2 are more abundant than that of
region. As D2 continues to increase to |D2| ¼ 2.5, six plateaus appear in the M curve in Fig. 5a. Remarkably, a well in the Mb curve is discovered in Fig. 5b, which is directly caused by the spin states of atoms in sublattice b flipping from 0.5 to 0.5 when h is between hC3 and hC4. It contributes to the existence of the M ¼ 7/6 plateau. Physically, the action of Mb may
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Table 1 Spin configurations and the total magnetization under the effects of D1 and h. 0:25 � D1 < 0
σzia ¼
3 2
3 2 3 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼ σzjb ¼
σzia ¼
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 2 1 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼
σzia ¼
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 2 1 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 3 5 5 5 Szic ¼ Szjd ¼ M¼ h < hC1 2 2 2 6 1 2 1 2
Szic ¼
5 2
z jd
5 2
Szic ¼
¼
5 2
M¼
Szjd ¼
5 2
7 6
hC1 < h < hC2
M¼
9 6
hC2 < h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 1 5 5 5 Szic ¼ Szjd ¼ M¼ h < hC1 2 2 2 6 1 2 1 2
Szic ¼
5 2
Szjd ¼ 5 2
Szic ¼
5 2
Szjd ¼
M¼ 5 2
7 6
M¼
hC1 < h < hC2 9 6
hC2 < h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 1 5 5 5 z z Sic ¼ Sjd ¼ M¼ h < hC1 2 2 2 6
σzia ¼
1 2
σzjb ¼
1 2
σzia ¼
3 2
σzjb ¼
1 2
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szic ¼ Szic ¼
Szjd ¼ 5 2
5 2
5 2
Szjd ¼ Szjd ¼
M¼ 5 2
5 2
7 6
M¼ M¼
hC1 < h < hC2 9 6
11 6
hC2 < h < hS h > hS
sublattices a and b are sensitive to J1. Similar behaviors have been found in the researches with the EFT [49,54] and MC simulation [45,52,56]. We present the phase diagram in the (J1, h) plane in Fig. 9 and spin configurations as well as the total magnetization under the effects of J1 and h in Table 3. It is found that a lowest point exists in the hC1 curve, which attributes to the variation of the spin states in sublattices a and b. In order to compare our results with those of EFT in Ref. [49], we take the same physical parameters and found excellent agreement between ours and theirs, which proves the reliability of our research. On the other hand, we also find that the values of hC and hS by the MC simulation are slightly smaller than those with the EFT. This is because the MC method considers the interactions from all spins rather than only involves nearest-neighbor spins by EFT. Fig. 10a–b presents the influence of J3 on the magnetization plateaus of the system and sublattices with D1 ¼ 2.5, D2 ¼ 0.5 and J1 ¼ 0.8. In Fig. 10a, when |J3| takes 0.2, 0.6 and 1.4 for comparison, we find that the effect of J3 is similar to that of J1 but weaker, which is reflected in the narrower width of the plateaus. From Fig. 10b, we also notice that the spins of sublattices a and b tend to lay in the low spin state σziaðbÞ ¼ �1/2
Fig. 13. The temperature dependence of the specific heat of the system (a) for different values of D1 when J1 ¼ 0.4, |J3| ¼ 0.8, D2 ¼ 1.5and h ¼ 0.5, (b) for different values of D2 when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and h ¼ 0.5, (c) for different values of J1 when |J3| ¼ 0.8, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (d) for different values of |J3| when J1 ¼ 0.4, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (e) for different values of h when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and D2 ¼ 0.5.
D1. This is because the spin values of sublattices c and d are more those of sublattices a and b. In fact, magnetization plateaus also reflect that the sublattice magnetizations have different dependencies of various phys ical parameters. Namely, different sublattices may present multiple spin states, strongly depending on different anisotropies and external mag netic field. The detailed spin configurations and the total magnetization under the effects of D2 and h are illustrated in Table 2. Our results are similar to those of graphene monolayer with the MC [45] and EFT [49, 54]. It is worth mentioning that recent theoretical studies have emphasized the significance of the single-ion anisotropy in dominating the magnetization plateaus in Bethe lattices [29], core-shell nano particles [30], a class of exactly solvable Ising-Heisenberg chains [55] and molecular-based magnets [56]. Finally, in Figss. 8–11, we discuss the effects of exchange couplings on magnetization plateaus. Fig. 8 shows the magnetization plateaus of the system and four sublattices for several selected values of J1 with fixed D1 ¼ 2.0, D2 ¼ 1.6 and |J3| ¼ 0.8. From Fig. 8a, we can find that, as J1 varies from 0.4 to 2.0, the system always has four plateaus and they become gradually widen and moves to higher h field. From the sublattice magnetization plateaus in Fig. 8b, we get that when J1 is relativity small (J1 ¼ 0.2), the spins in sublattice a and b are put in a low spin state (�1/2) under a weak h. On the contrary, they stay at a high spin state (�3/2), which benefits from strong exchange couplings (J1 ¼ 1.4). This is due to the competition between exchange couplings and longitudinal fields. The result illustrates that the spin states of
in the range of small h, which is vary similar to the effect of J1 on the Ma and Mb. The phase diagram in the (|J3|, h) plane is shown in Fig. 11. It is found that some inflection points exist in the curves of hC1 and hC2, illustrating the variation of the spin states in sublattices under the in fluence of J3. Corresponding spin configurations and the total magne tization under the effects of |J3| and h are presented in Table 4. It should be mentioned that the magnetization plateaus behavior can be found in many compounds such as La0.67-xAxCa0.33MnO3 [31], SrCu2(BO3)2 [34, 57] and Ni dinuclear oxalato(ox)-bridged compound [58]. The experi mental results show that the multiple magnetization plateaus are very sensitive to the anisotropies and exchange couplings, which are similar to those obtained in this work. 3.2. Internal energy and specific heat Fig. 12a–e shows the temperature dependence of the internal energy of the system for different values of physical parameters. From these sub-figures, we can see a common characteristic is that each U curve increases with the temperature increasing. From Fig. 12a and b, we find
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Table 2 Spin configurations and the total magnetization under the effects of D2 and h. 2:0 < D2 < 0
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
σzia ¼
3 2
σzjb ¼
3 2 3 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼
1 2
5 2
Szic ¼
Szjd ¼
5 2
M¼
Table 4 Spin configurations and the total magnetization under the effects of |J3| and h. σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC1 < h < hC2
hC2 < h < hS
σzia ¼
3 2
σzjb ¼
9 6
3 Szic ¼ 2 3 Szic ¼ 2
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 3 3 3 Szjd ¼ M¼ h < hC1 2 2 6
0:1 < jJ3 j < 0:5
Szic ¼
5 2
Szjd ¼
5 2
M¼
3 3 5 5 Szic ¼ Szjd ¼ σzjb ¼ 2 2 2 2 3 3 5 5 5 Sz ¼ Szjd ¼ M¼ σzia ¼ σzjb ¼ 2 2 ic 2 2 6
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
hC1 < h < hC2
σzia ¼
3 z σ ¼ 2 jb
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC2 < h < hC3
σzia ¼
3 z σ ¼ 2 jb
1 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
M¼
9 6
σzia ¼
3 2
σzjb ¼
hC3 < h < hS
σzia ¼
3 z σ ¼ 2 jb
3 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
M¼
11 6
3 2 1 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
σzia ¼
1 2
σzjb ¼
σzia ¼
1 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 σzia ¼ 2 1 σzia ¼ 2
σzjb ¼ σzjb ¼ σzjb ¼
5 2
Szic ¼
Szjd ¼
5 2
M¼
9 6
3 Szic ¼ 2 1 Szic ¼ 2
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 1 1 1 z Sjd ¼ M¼ h < hC1 2 2 6
1 2
3 2
Szic ¼
1 2 1 2 1 2
3 2
Szic ¼ Szic ¼
Szjd ¼
5 2
3 Szic ¼ 2 1 Szic ¼ 2
Szjd ¼ Szjd ¼
5 Szic ¼ 2
3 2
M¼
3 2
5 2
M¼ M¼
5 Szjd ¼ 2
3 6
7 6
hC2 < h < hC3 hC3 < h < hC4
9 M¼ 6
hC4 < h < hS
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 1 1 1 z Sjd ¼ M¼ h < hC1 2 2 6
σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
1 2
Szjd ¼
1 2
M¼
3 6
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
3 2
Szjd ¼
3 2
M¼
7 6
hC2 < h < hC3
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
3 2
Szjd ¼
3 2
M¼
9 6
hC3 < h < hC4
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
3 2
M¼
10 6
hC4 < h < hS
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
11 6
h > hS
Table 3 Spin configurations and the total magnetization under the effects of J1 and h. σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
3 2 3 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
1 2
Szic ¼
5 2
5 2
M¼
9 6
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 5 5 5 z Sjd ¼ M¼ h < hC1 2 2 6
1 2
5 2
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
σzjb ¼
3 2
5 Szic ¼ 2
Szic ¼
5 2
Szjd ¼
5 2
Szjd ¼
5 2
5 Szjd ¼ 2
M¼
7 6
M¼
9 6
11 M¼ 6
hC1 < h < hC2 hC2 < h < hS h > hS
istence of the low temperature peak in the C curve. It should be mentioned that the double-peak phenomenon in C curves has been also observed in the nano-graphene monolayer [39] and bilayer [48] as well as other nano-systems such as 2D boron clusters [53], nanoparticle [63], nanotube [70] and nanofilm [71–73]. Experimentally, the double-peak C phenomenon have been revealed in rare-earth metals and compounds such as Tb3M series (M ¼ Co, Rh and Ru) [74], pseudo-binary TmxDy1-xAl2 (0 � x � 1) [75], RCuAl3 (R ¼ Pr and Nd) compounds [76]. By contrast, Fig. 13c and d illustrate that the exchange couplings J1 and |J3| need to become larger in order to make the peak to move toward higher temperature. And the effect of J3 is more obvious. In Fig. 13e, the high-temperature peak also increases with the external magnetic field h increasing. The results demonstrate that the blocking temperature may be improved by increasing J1 and |J3| as well as h. The results about the effect of h on the C are consistent with those found experimentally in �-R2S3(R ¼ Tb,Dy) single crystals [77], YbNi3Al9 and Yb(Ni0.94C a u0.06)3Al9 monoaxial-chiral helimagnets [78].
hC2 < h < hS
3 Szic ¼ 2 3 Szic ¼ 2
σzia ¼
3 σzia ¼ 2
Szjd ¼
7 6
h > hS
system is. From Fig. 12e, it is concluded that the influence of the h on U is similar to that of exchange coupling (J1 or |J3|), namely the stronger h can make the system more stable. Our results are comparable with those of graphene monolayer [40,42,54], graphene bilayer [48] and other nano-systems such as hexagonal-type magnetic nanowires [59–62], nanoparticles [63] and core-shell nanoislands [64–69]. Finally, we also focus on the temperature dependence of the specific heat of the system C in Fig. 13a–e. It is clearly seen that each specific heat curve has a distinct rounded peak at high temperature region, demonstrating that the system undergoes phase transition from order to disorder. Through a comparison between Fig. 13a and b, one can see that the increase of both anisotropies |D1| and |D2| makes the peak move to the low temperature region and the effect of D2 seems stronger. It is interesting that the double-peak specific heat phenomenon is observed for certain values of physical parameters. Of two peaks in the specific heat curves, the higher temperature peak should correspond to the blocking temperature, which is connected with the magnetic ordering. Below the peak the system is ferrimagnetic, whereas above one it is paramagnetic. And the other peak at low temperature may be attributed to two factors: on one hand, the small thermal fluctuation in the internal energy can be responsible for the low temperature peak in the C curve. On the other hand, it may come from the fact that the spin configuration of the system changes at the ground state, namely the spin states of the sublattices change, due to the competition among various physical pa rameters including the crystal-field, the exchange couplings, the external longitudinal magnetic field and temperature. For example, when D1 ¼ 1.5, the spins of sublattices a and b may flip from σ ziaðjbÞ ¼ �3=2 to �1=2 at very low temperature, which leads to the ex
hC1 < h < hC2
5 6
M¼
hC2 < h < hS
M¼
3 2
1 2
1 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
9 6
11 6 h < hC1
σzia ¼
σzia ¼
3 σzia ¼ 2
0:7 < J1 < 0:1
1 2
hC1 < h < hC2 hC2 < h < hS h > hS
that U increases with the increasing of the anisotropies |D1| and |D2| at a certain temperature. This indicates that the strong anisotropy is not beneficial to the stability of the system. Comparing Fig. 12a and b, the effect of D2 on U is greater than that of D1. As can be seen from Fig. 12c and d, the effect of exchange couplings J1 and |J3| on U is different from that of anisotropy. Namely, as J1 or |J3| increases, the U decreases, suggesting that the smaller the exchange coupling is, the more stable the 8
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
4. Conclusion
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Using the Monte Carlo simulation, we studied the magnetization plateaus, internal energy and specific heat of a sandwich-like nanographene under the effects of different physical parameters (D1, D2 J1, J3 and h). The results show the system can display multiple magnetization plateaus behaviors, reflecting the variation of spin configuration for the system due to the competition among the anisotropy, exchange coupling and longitudinal magnetic field. The internal energy of the system can be reduced by decreasing the anisotropy and increasing the exchange coupling as well as the longitudinal magnetic field. The double-peak phenomenon can be found in the C curves for certain parameters. These results could be beneficial for nano-graphene multilayers in ap plications in spintronic devices. Conflicts of interest We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgements This project was supported by Natural Sciences Foundation of Liaoning province, China (Grant No. 20170540672), College Students’ Innovative and Entrepreneurial Training Program of Shenyang Univer sity of Technology, China (Grant No. 2018124). References [1] C.W. Lin, K.C. Wei, S.S. Liao, C.Y. Huang, C.L. Sun, P.J. Wu, et al., Biosens. Bioelectron. 67 (2015) 431. [2] A.G.L. Lin, H.Y. Peng, Z.Q. Liu, T. Wu, C.L. Su, K.P. Loh, et al., Small 10 (10) (2014) 1945. [3] E. Roy, S. Patra, D. Kumar, R. Madhuri, P.K. Sharma, Biosens. Bioelectron. 68 (2015) 726. [4] Y. Song, X. M Li, C. Mackin, X. Zhang, W.J. Fang, T. Palacios, H.W. Zhu, J. Kong, Nano Lett. 15 (2015) 2104. [5] T. Kawasaki, K. Sugawara, A. Dobroiu, T. Eto, Y. Kurita, K. Kojima, Y. Yabe, H. Sugiyama, T. Watanabe, T. Suemitsu, V. Ryzhii, K. Iwatsuki, Y. Fukada, J. Kani, J. Terada, N. Yoshimoto, K. Kawaharad, H. Ago, T. Otsuji, Solid State Electron. 103 (2015) 216. [6] L.J. Xu, W. Chu, Lu Gan, Chem. Eng. J. 263 (2015) 435–443. [7] D. Li, M.B. Mueller, S. Gilje, et al., Nat. Nanotechnol. 3 (2008) 101. [8] M. Lotya, Y. Hernandez, J.N. Coleman, et al., J. Am. Chem. Soc. 131 (2009) 3611. [9] Y. Zhang, H.J. Zhang, X. Li, H.T. Xu, Y. Wang, Nanotechnology 27 (2016) 155401. [10] N. Ota, N. Gorjizadeh, Y. Kawazoe, J. Magn. Soc. Jpn. 34 (2010) 573–578. [11] N. Ota, N. Gorjizadeh, Y. Kawazoe, J. Magn. Soc. Jpn. 35 (2011) 360–365. [12] K.Y. Lian, X.F. Li, S. Duan, M.X. Jin, D.J. Ding, Y. Luo, J. Appl. Phys. 115 (2014) 104303. [13] K. Szalowski, J. Magn. Magn. Mater. 382 (2015) 318. [15] M. Sherafati, S. Satpathy, Phys. Rev. B 83 (2011) 165425. [16] A.T. Raji, E.B. Lombardi, Phys. B 464 (2015) 28. [17] Y. Mohammadi, R. Moradian, Loc, Solid State Commun 178 (2014) 37. [18] W. Jiang, Y. Yang, A. Gong, Carbon 95 (2015) 190. [19] F. Albertini, F. Bolzoni, A. Paoluzi, Phys. B Condens. Matter 294 (2001) 172. [20] K. Koyama, H. Fujii, Phys. B Condens. Matter 294 (2001) 168. [21] M. Doerr, S. Kramp, M. Loewenhaupt, Phys. B Condens. Matter 294 (2001) 164. [22] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984. [23] E. Viitala, J. Merikoski, M. Manninen, J. Timonen, Phys. Rev. B 55 (1997) 11541. [24] X.Y. Chen, Q. Jiang, W.Z. Shen, C.G. Zhong, J. Magn. Magn. Mater. 262 (2003) 258. [25] J. Wang, Y. Liu, Z. Xie, Q.M. Ma, J. Phys. Conf. Ser. 29 (2006) 65. [26] Z.Z. Weng, Q. Feng, Z.G. Huang, Y.W. Dui, Acta Phys. Sin. 53 (2004) 3177.
9
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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe
Monte Carlo study of magnetization plateaus and thermodynamic properties of a nano-graphene with a sandwich-like structure in a longitudinal magnetic field Dan Lv a, b, *, Ye Ma c, Xiao-hong Luo a, Wei Jiang a, b, Feng Wang b, Qian Li a a b c
School of Science, Shenyang University of Technology, Shenyang, 110870, China School of Materials Science and Engineering, Shenyang University of Technology, Shenyang, 110870, China School of Electrical Engineering, Shenyang University of Technology, Shenyang, 110870, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Nano-graphene Magnetization plateaus Internal energy Specific heat Monte Carlo simulation
Monte Carlo simulation has been used to study magnetization plateaus and thermodynamic properties of the sandwich-like nano-graphene with mixed spin-3/2 and spin-5/2 in a longitudinal magnetic field. The effects of the single-ion anisotropy, the intralayer and interlayer exchange couplings as well as the longitudinal magnetic field on the magnetization, internal energy and specific heat of the system are discussed in detail. Multiple magnetization plateaus behaviors at low temperatures have been found, originating from the competition among various physical parameters. In particular, the results show that the anisotropy, exchange coupling and longi tudinal magnetic field play significant roles in controlling the internal energy and specific heat of the system.
1. Introduction Graphene is one of the lightest and thinnest materials available today, thanks to its unique two-dimensional honeycomb structure. It has attracted considerable attention in many fields owing to its outstanding properties, such as mechanical stability, electrical insulation, low den sity, transparency and gapless semi-conductivity. In particular, the ex istence of nano-graphene with original magnetic behaviors opens up more potential technological applications in biosensor [1], magneto resistive read heads [2], magnetic storage [3], solar cells [4], field-effect transistors [5], chemical engineering [6] and so on. In recent years, experimental synthesis technology of nano-graphene has been greatly developed [7–9]. These experimental progress and evidence arouse a great number of theoretical explorations for nano-graphene. Multiple spin states were employed to study the magnetism of nano-graphene-like molecules, which is a good way to understand ferromagnetism for nano-graphene. Through the molecular orbital calculation and the density functional theory (DFT), the results show that the highest spin state S ¼ 3/2 is steadier for the lowest energy. Relevant investigations of C48H24 graphene molecule revealed that the up and down spins are arranged alternatively in sequence, expressing the antiferromagnetic exchange coupling between spins [10]. Moreover, there exist five unpaired electrons in C59N5H22 molecule [11],
illustrating that the spins can take multiple spin states S ¼ 5/2 and the stronger ferromagnetic exchange coupling exists between up and up spin pairs. Based on the expanded classical Ising model to describe the nanographene structures, a variety of numerical methods were used to study their magnetic properties. By using molecular dynamic simula tions, Lian et al. predicted the magnetic properties of the hybrid gra phene [12]. They found the higher Curie point of the graphene system and ferromagnetic behaviors in the ground state. Szalowski et al. examined the magnetic properties of mano-graphene with mean-field theory (MFT) and discussed the influences of the external in-plane electric and magnetic fields on the ground state phase diagram [13]. With the Green’s function, M. Sherafati et al. revealed the nature of the exchange coupling for the tight-binding band structure of nano-graphene [15]. By means of DFT, the electronic and magnetic performances of cobalt-vacancy defect complexes were investigated in the graphene system [16]. The results demonstrate that the magneti zation of the cobalt atom can be controlled by the vacancy relative to cobalt as well as the vacancy cobalt separation. The local moment of the graphene bilayer was considered based on the Anderson impurity model [17]. What is noticeable is that the prominent magnetic properties of a nano-graphene bilayer were explored in a longitudinal magnetic field by a transverse Ising model [18]. The results indicate that the longitudinal
* Corresponding author. School of Science, Shenyang University of Technology, Shenyang, 110870, China. E-mail address: [email protected] (D. Lv). https://doi.org/10.1016/j.physe.2019.113721 Received 20 February 2019; Received in revised form 9 June 2019; Accepted 11 September 2019 Available online 17 September 2019 1386-9477/© 2019 Elsevier B.V. All rights reserved.
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
magnetic field, the exchange coupling, the transverse field and single-ion anisotropy play crucial roles in magnetization, susceptibility and blocking temperature of the nano-graphene bilayer. In recent years, experimental results indicate that the mixed-spin Ising system have displayed many novel performances in the external magnetic field [19–21]. Since Oshikawa et al. [22] proposed the satis fied condition for the existence of magnetization plateaus in the mag netic systems in the longitudinal magnetic field, such efforts have been made to understand the mechanism of the magnetization plateaus for mixed magnetic Ising systems [23]. By means of the classical Monte Carlo (MC) simulation, the magnetization plateaus of the mixed-spin (1, 3/2) antiferromagnetic Ising system with the positive single-ion anisotropy at low temperatures in the longitudinal magnetic field have been examined in detail [24]. Relevant studies also revealed that magnetization plateaus of ferromagnetic and antiferromagnetic mixed films depend on temperatures, magnetic dipolar interactions and single-ion anisotropies [25,26]. Through the study of the ferrimagnetic mixed-spin (1,3/2) Ising chain with the homogenous single-ion anisot ropy by the molecular-field theory [27], Solano-Carrillo et al. found that the existence of the magnetization plateaus may be attributed to the single-ion anisotropy. In addition, interesting results revealed the importance of the anisotropy in the mechanism of the magnetization plateaus in many low-dimensional magnetic systems such as the Ising chain with spin S ¼ 3/2 [28], Bethe lattice [29] and ferrimagnetic core-shell nanoparticle [30]. Experimentally, related magnetization plateau characteristics have been discovered in compounds such as La0.67-xAxCa0.33MnO3 [31], FexMg1-xCl12 [32], NbFeTe2 [33] and SrCu2(BO3)2 [34]. As a significant method, Monte Carlo simulation has been also suc cessfully used to investigate the magnetic and thermodynamic proper ties of nano-graphene systems. By modeling the mixed-spin (3/2, 5/2) Ising system for the single-layer and double-layer nano-graphene, the effects of the crystal-field, exchange coupling, dilution, size, defects and longitudinal magnetic field on the physical quantities including the magnetization, susceptibility, compensation, critical and blocking tem peratures as well as hysteresis loops have been studied by MC simula tions [35–39]. Recently, MC simulations have been applied to understand the influence of higher order exchange couplings on the critical phenomena and compensation behavior of the nano-graphene layer [40–44]. It was found that the Ising model with four-spin inter action can exhibit rich. Remarkably, rich magnetization plateaus be haviors were found in a nano-graphene layer with the spin 5/2 Blume-Emery-Griffiths model, reflecting multiple spin states at low temperature [45]. By MC simulation, the ground state phase diagrams and hysteresis loops were discussed for the bilayer decorated graphene structure with a higher mixed-spin (7/2, 3) Ising model [46]. In addi tion, the magnetic the magnetic and thermodynamic properties of the nano-graphene bilayer system in the external magnetic field were investigated by using MC simulation [47,48]. In spite of these extensive researches of the nano-graphene, less practice has been put into the exploration of magnetic properties of nano-graphene with a sandwich-like structure by MC simulations. Recently, W. Jiang et al. studied the magnetic susceptibilities and magnetization plateaus for the nano-graphene with the sandwich-like structure by the EFT [49]. However, they have not studied systemati cally the influences of the single-on anisotropy, exchange coupling and longitudinal magnetic field upon the internal energy and specific heat of the tri-layer nano-graphene system. Therefore, this paper aims at investigating the effects of single-ion anisotropy, exchange coupling and longitudinal magnetic field on the magnetization plateaus, the internal energy and specific heat of the sandwich-like nano-graphene by MC simulation. Furthermore, it is interesting and meaningful to compare our correlative results with those of EFT [49]. The paper is arranged as follows. In Section 2, the model and Monte Carlo method are described. In Section 3, we show and discuss our typical numerical results. Finally our conclusions are summarized in Section 4.
2. Model and Monte Carlo simulation We consider a sandwich-like mixed spin-3/2 and spin-5/2 Ising model on a honeycomb lattice consisting of N�L spins with the standard single-spin-flip importance method based on the Metropolis algorithm [50]. There are N sites in each layer and L denotes the layer thickness of the nano-graphene tri-layer system (L ¼ 3). The sketch of the present model is depicted in Fig. 1. As shown in this figure, it consists of three layers, namely two surface layers and one middle layer. In the upper and bottom surface layer, every two sublattices a and b with spin-3/2 (↑↓) are interacted with intralayer antiferromagnetic exchange coupling J1 (<0) and in the middle layer, every two sublattices c and d with spin-5/2 (↑↑) are interacted with intralayer ferromagnetic exchange coupling J2 (>0). J3 is the interlayer exchange coupling between the surface layer and the middle layer, which takes a positive value between sublattices a and c, whereas a negative value between sublattices b and d. The Hamiltonian of the present system is given by X X X H ¼ J1 σzia σ zjb J2 Szic Szjd J3 σziaðjbÞ SzicðjdÞ i;j
X�
σ ziaðjbÞ
D1 iðjÞ
�2
i;j
�2 X� SzicðjdÞ Db iðjÞ
i;j
X
σ ziaðjbÞ þ
h iðjÞ
X SzicðjdÞ
!
(1)
iðjÞ
where σziaðjbÞ and SzicðjdÞ are the spin-3/2 and spin-5/2 ions on surface
layers and internal layer, respectively. σziaðjbÞ may take the values of �3/ 2, �1/2 and SzicðjdÞ the values � 5/2, �3/2, �1/2, respectively. The first
three sums only involve nearest-neighbors’ couplings. D1 and D2 are the single-ion anisotropy constants for the spins in the surface layer and the middle layer, respectively. h stands for the external longitudinal mag netic field. In our simulation, additional simulations have also been made for the selection of the spin number in each layer. When changing N from 48 to 108, we have not found obvious differences. Accordingly, N ¼ 48 is the choice for the simulation. For guarantee, about 105 Monte Carlo steps (MCS) per site are discarded for equilibrating the system before aver aging thermal quantities over the next 4 � 105 MCS. The physical quantities are given by the following:
Fig. 1. Structure of a nano-graphene with a sandwich-like structure. 2
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
The internal energy per site can be expressed as U¼
1
(2)
and sublattice magnetizations of the surface and middle layers are expressed by Ma ¼
1 X z σia > < N i
(3)
Mb ¼
1 X z < σjb > N j
(4)
Mc ¼
1 X z < Sic > N i
(5)
Md ¼
1 X z < Sjd > N j
(6)
and the total magnetization M per site is 1 M ¼ ð2Ma þ 2Mb þ Mc þ Md Þ 6
(7)
Fig. 3. The variations of the critical field hC and saturation field hS with D1 for J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6 at T ¼ 0.08.
and the specific heat per site C β2 ¼ < H2 > kB NL
< H>
2
�
magnetization plateaus, thus the following results for magnetization plateaus were examined at T ¼ 0.08. The effect of anisotropy D1 on the magnetization plateaus is depicted in Figs. 2 and 3 with J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6. One can see from Fig. 2a that as h is enhanced from zero, the magnetization M in creases and shows two magnetization plateaus (M ¼ 5/6 and 11/6) for weak D1 ¼ 0.25. The distinctive value of the magnetic field is high lighted by a dot dash line and marked with hS where the M approaches the saturation value. The hS is called saturation magnetic field. When D1 is changed to 1.4 and 3.2, there exist two additional magnetization plateaus M ¼ 7/6 and 9/6, corresponding to the called critical magnetic fields hC1 and hC2 with dot lines and dash lines. For D1 ¼ 1.4, the critical magnetic field are hC1 ¼ 1.47, hC2 ¼ 4.16, the saturation mag netic field hS of the system is 7.04. As for D1 ¼ 3.2, we can get hC1 ¼ 3.01, hC2 ¼ 5.08 and hS ¼ 10.57. One can notice that the critical and saturation fields become larger and the distance between them becomes longer with increasing D1. This demonstrates that the step ef fect is strengthened, owing to the fact that the spins becomes more difficult to flip and hard to overcome the strong anisotropy energy. The reason for magnetization plateaus can be illustrated by multiple spin states configuration of sublattices, as shown in Fig. 2b. When D2 is fixed
(8)
where β ¼ 1/kBT. T denotes temperature and kB is Boltzmann constant. We take kB ¼ 1.0 for simplicity. 3. Results and discussions In this section, the exchange coupling J2 is selected as the reduced unit of energy and temperature, so we take J2 ¼ 1.0. It is assumed that the initial directions of the external longitudinal magnetic field h and spins in the middle layer SzicðjdÞ are along the positive z-axis direction. We
focused on the effects of physical parameters on the magnetization plateaus, internal energy and specific heat of the nano-graphene sand wich-like structure, the typical results are presented in Figs. 2–13 and discussions are given as follows. 3.1. Magnetization plateaus Low temperature is a necessary condition for the existence of
Fig. 2. The magnetization (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of D1 (D1 ¼ 0.25, 1.4, 3.2) with J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 1.6 at T ¼ 0.08. 3
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Fig. 6. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for D2 ¼ 4.0 with J1 ¼ 0.4, | J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08.
Fig. 4. The magnetizations of the system and sublattices vs. the longitudinal magnetic field h for different values of D2 with J1 ¼ 0.4, |J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08 (a) M for D2 ¼ 0.5, (b) M1, M2, M3, M4 for D2 ¼ 0.5, (c) M for D2 ¼ 2.0, (d) M1, M2, M3, M4 for D2 ¼ 2.0.
at a low value (D2 ¼ 1.6), the spins of sublattices c and d are always at the spin 5/2 state regardless of the changes in D1 (D1 ¼ 1.4 and 3.2), but a noticeable difference can be found in the Ma and Mb. For weak D1 (D1 ¼ 1.4), the spins of sublattice a are always at the spin 3/2 state, while those of sublattice b have four spin states with h increasing, namely, the spin values σzjb ¼ 3/2, 1/2, 1/2, 3/2 correspond to four
plateaus observed in the Mb curve, which should be responsible for the magnetization plateau behavior of the system in Fig. 2a. Namely, the values of magnetization plateaus M are M ¼ [2 � 3/2 þ 2 � (-3/2)þ5/2 þ 5/2]/6 ¼ 5/6, M ¼ [2 � 3/2 þ 2 � (-1/2)þ5/2 þ 5/2]/6 ¼ 7/6, M ¼ [2 � 3/2 þ 2 � (1/2)þ5/2 þ 5/2]/6 ¼ 9/6 and M ¼ [2 � 3/2 þ 2 � (3/ 2)þ5/2 þ 5/2]/6 ¼ 11/6 by the equation (7). But if D1 is strong enough, its effect becomes prominent, enabling the spins of sublattices a and b to be placed in low spin �1/2 states at the weak h. As h increases, the Mb appears three plateaus with the spin 1/2, 1/2 and 3/2 states, while Ma appears two ones with the spin 1/2 and 3/2 states, which together contributes to the generation in four magnetization plateaus of the system in Fig. 2a. This interesting phenomenon not only results from the cooperation between sublattice magnetizations, but also can be
Fig. 7. The variations of the critical field hC and saturation field hS with D2 for J1 ¼ 0.4, |J3| ¼ 0.8 and D2 ¼ 2.5 at T ¼ 0.08.
Fig. 5. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for D2 ¼ 2.5 with J1 ¼ 0.4, |J3| ¼ 0.8 and D1 ¼ 2.5 at T ¼ 0.08. 4
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whereas h is just the opposite. The results are similar to those of previous studies such as nano-graphene layer [45], nano-graphene tri-layer [49] and finite Ising chain [51]. Fig. 3 shows the phase diagram in the (D1, h) plane with J1 ¼ 0.4, | J3| ¼ 0.8 and D2 ¼ 1.6. The data points in the diagram denote every critical and saturation values of the external magnetic field at which the magnetization of the total system changes. Therefore, the intermediate states between the plateau phases represent different spin configurations of the system, originating from competition between different physical parameters and external magnetic field. Corresponding spin configura tions and the total magnetization under the effects of D1 and h are shown in Table 1. It is clearly seen that hS increases linearly with the increase of |D1|. Meanwhile, on the curves of hC1 and hC2, we can observe two critical points respectively. One for hC1 at D1 ¼ 1.5 and the other for hC2 at D1 ¼ 2.5. This illustrates the changes in the spin configuration of sublattices a and b, namely the spin values of sublattices a and b change from σ ziaðjbÞ ¼ 3/2 (3/2) to σziaðjbÞ ¼ 1/2 (1/2). It is worth mentioning
Fig. 8. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of J1 (J1 ¼ 0.2, 0.6, 1.4) with |J3| ¼ 0.8, D1 ¼ 1.6 and D2 ¼ 2.0 at T ¼ 0.08.
explained by the competitions between the anisotropy D1, the external magnetic field h and the antiferromagnetic exchange coupling J1. On one hand, the h favors for aligning the spins of sublattices a and b along its direction, but the J1 tends to align them anti-parallel. On the other hand, the stronger |D1| tends to force the spins to stay at the lower states,
that the vertical distance between two adjacent values of hC or the last hC and hS represents the width of the magnetization plateau. Our results are in good agreement with those calculated by the EFT in Ref. [49]. Next, we turn our attention to the effect of D2 on the magnetization
Fig. 9. The variations of the critical field hC and saturation field hS with J1 for | J3| ¼ 0.8, D1 ¼ 1.6 and D2 ¼ 2.0 at T ¼ 0.08.
Fig. 11. The variations of the critical field hC and saturation field hS with |J3| for J1 ¼ 0.8, D1 ¼ 2.5 and D2 ¼ 0.5 at T ¼ 0.08.
Fig. 10. The magnetizations (a) of the system M and (b) of four sublattices M1, M2, M3, M4 vs. the longitudinal magnetic field h for different values of |J3| (|J3| ¼ 0.2, 0.6, 1.4) with J1 ¼ 0.8, D1 ¼ 2.5 and D2 ¼ 0.5 at T ¼ 0.08. 5
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
Fig. 12. The temperature dependence of the internal energy of the system (a) for different values of D1 when J1 ¼ 0.4, |J3| ¼ 0.8, D2 ¼ 1.5and h ¼ 0.5, (b) for different values of D2 when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and h ¼ 0.5, (c) for different values of J1 when |J3| ¼ 0.8, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (d) for different values of |J3| when J1 ¼ 0.4, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (e) for different values of h when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and D2 ¼ 0.5.
plateaus with fixed D1 ¼ 2.5, J1 ¼ 0.4 and |J3| ¼ 0.8, as shown in Figs. 4–7. In Fig. 4a, when D2 ¼ 0.5, four magnetization plateaus can be observed at M ¼ 5/6, 7/6, 9/6 and 11/6. On changing D2 to 2.0 in Fig. 4c, the M curve displays one more plateau at M ¼ 3/6 in low h re gion. Additional plateau (M ¼ 3/6) can be found from a comparison of the magnetization plateaus in sublattices between in Fig. 4b and d. Under the influence of the sufficiently stronger D2 (D2 ¼ 2.0), the spins of sublattices c and d tend to stay at a low spin state SzicðjdÞ ¼ 3/2 in low h
be a need to maintain the lowest energy for the system in the ground state. This phenomenon can be found in previous studies of graphene nanoribbon [52] and 2D boron clusters [53]. Further increasing |D2| to 4.0 in Fig. 6a, we can still see six magnetization plateaus in the system. Differently, illustrated by Fig. 6b, the changes in Mc and Md are not synchronized with each other, leading to the appearance of the plateaus M ¼ 10/6 and 11/6. Fig. 7 presents the phase diagram in the (D2, h) plane with D1 ¼ 2.5, J1 ¼ 0.4 and |J3| ¼ 0.8. The phase diagram is divided into four regions by the vertical dotted lines. One can notice that, as |D2| increases, the number of magnetization plateaus becomes more and more. In different regions, there are four magnetization plateaus for |D2|<2.0, five for 2.0�|D2|�2.5, and six for |D2| >2.5. Compared with Fig. 3, the changes of each hC and hS under the effect of D2 are more abundant than that of
region. As D2 continues to increase to |D2| ¼ 2.5, six plateaus appear in the M curve in Fig. 5a. Remarkably, a well in the Mb curve is discovered in Fig. 5b, which is directly caused by the spin states of atoms in sublattice b flipping from 0.5 to 0.5 when h is between hC3 and hC4. It contributes to the existence of the M ¼ 7/6 plateau. Physically, the action of Mb may
6
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
Table 1 Spin configurations and the total magnetization under the effects of D1 and h. 0:25 � D1 < 0
σzia ¼
3 2
3 2 3 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼ σzjb ¼
σzia ¼
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 2 1 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼
σzia ¼
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 2 1 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 3 5 5 5 Szic ¼ Szjd ¼ M¼ h < hC1 2 2 2 6 1 2 1 2
Szic ¼
5 2
z jd
5 2
Szic ¼
¼
5 2
M¼
Szjd ¼
5 2
7 6
hC1 < h < hC2
M¼
9 6
hC2 < h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 1 5 5 5 Szic ¼ Szjd ¼ M¼ h < hC1 2 2 2 6 1 2 1 2
Szic ¼
5 2
Szjd ¼ 5 2
Szic ¼
5 2
Szjd ¼
M¼ 5 2
7 6
M¼
hC1 < h < hC2 9 6
hC2 < h < hS
3 5 5 11 Szic ¼ Szjd ¼ M¼ h > hS 2 2 2 6 1 5 5 5 z z Sic ¼ Sjd ¼ M¼ h < hC1 2 2 2 6
σzia ¼
1 2
σzjb ¼
1 2
σzia ¼
3 2
σzjb ¼
1 2
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szic ¼ Szic ¼
Szjd ¼ 5 2
5 2
5 2
Szjd ¼ Szjd ¼
M¼ 5 2
5 2
7 6
M¼ M¼
hC1 < h < hC2 9 6
11 6
hC2 < h < hS h > hS
sublattices a and b are sensitive to J1. Similar behaviors have been found in the researches with the EFT [49,54] and MC simulation [45,52,56]. We present the phase diagram in the (J1, h) plane in Fig. 9 and spin configurations as well as the total magnetization under the effects of J1 and h in Table 3. It is found that a lowest point exists in the hC1 curve, which attributes to the variation of the spin states in sublattices a and b. In order to compare our results with those of EFT in Ref. [49], we take the same physical parameters and found excellent agreement between ours and theirs, which proves the reliability of our research. On the other hand, we also find that the values of hC and hS by the MC simulation are slightly smaller than those with the EFT. This is because the MC method considers the interactions from all spins rather than only involves nearest-neighbor spins by EFT. Fig. 10a–b presents the influence of J3 on the magnetization plateaus of the system and sublattices with D1 ¼ 2.5, D2 ¼ 0.5 and J1 ¼ 0.8. In Fig. 10a, when |J3| takes 0.2, 0.6 and 1.4 for comparison, we find that the effect of J3 is similar to that of J1 but weaker, which is reflected in the narrower width of the plateaus. From Fig. 10b, we also notice that the spins of sublattices a and b tend to lay in the low spin state σziaðbÞ ¼ �1/2
Fig. 13. The temperature dependence of the specific heat of the system (a) for different values of D1 when J1 ¼ 0.4, |J3| ¼ 0.8, D2 ¼ 1.5and h ¼ 0.5, (b) for different values of D2 when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and h ¼ 0.5, (c) for different values of J1 when |J3| ¼ 0.8, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (d) for different values of |J3| when J1 ¼ 0.4, D1 ¼ 2.5, D2 ¼ 0.5 and h ¼ 0.5, (e) for different values of h when J1 ¼ 0.4, |J3| ¼ 0.8, D1 ¼ 2.0 and D2 ¼ 0.5.
D1. This is because the spin values of sublattices c and d are more those of sublattices a and b. In fact, magnetization plateaus also reflect that the sublattice magnetizations have different dependencies of various phys ical parameters. Namely, different sublattices may present multiple spin states, strongly depending on different anisotropies and external mag netic field. The detailed spin configurations and the total magnetization under the effects of D2 and h are illustrated in Table 2. Our results are similar to those of graphene monolayer with the MC [45] and EFT [49, 54]. It is worth mentioning that recent theoretical studies have emphasized the significance of the single-ion anisotropy in dominating the magnetization plateaus in Bethe lattices [29], core-shell nano particles [30], a class of exactly solvable Ising-Heisenberg chains [55] and molecular-based magnets [56]. Finally, in Figss. 8–11, we discuss the effects of exchange couplings on magnetization plateaus. Fig. 8 shows the magnetization plateaus of the system and four sublattices for several selected values of J1 with fixed D1 ¼ 2.0, D2 ¼ 1.6 and |J3| ¼ 0.8. From Fig. 8a, we can find that, as J1 varies from 0.4 to 2.0, the system always has four plateaus and they become gradually widen and moves to higher h field. From the sublattice magnetization plateaus in Fig. 8b, we get that when J1 is relativity small (J1 ¼ 0.2), the spins in sublattice a and b are put in a low spin state (�1/2) under a weak h. On the contrary, they stay at a high spin state (�3/2), which benefits from strong exchange couplings (J1 ¼ 1.4). This is due to the competition between exchange couplings and longitudinal fields. The result illustrates that the spin states of
in the range of small h, which is vary similar to the effect of J1 on the Ma and Mb. The phase diagram in the (|J3|, h) plane is shown in Fig. 11. It is found that some inflection points exist in the curves of hC1 and hC2, illustrating the variation of the spin states in sublattices under the in fluence of J3. Corresponding spin configurations and the total magne tization under the effects of |J3| and h are presented in Table 4. It should be mentioned that the magnetization plateaus behavior can be found in many compounds such as La0.67-xAxCa0.33MnO3 [31], SrCu2(BO3)2 [34, 57] and Ni dinuclear oxalato(ox)-bridged compound [58]. The experi mental results show that the multiple magnetization plateaus are very sensitive to the anisotropies and exchange couplings, which are similar to those obtained in this work. 3.2. Internal energy and specific heat Fig. 12a–e shows the temperature dependence of the internal energy of the system for different values of physical parameters. From these sub-figures, we can see a common characteristic is that each U curve increases with the temperature increasing. From Fig. 12a and b, we find
7
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Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
Table 2 Spin configurations and the total magnetization under the effects of D2 and h. 2:0 < D2 < 0
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
σzia ¼
3 2
σzjb ¼
3 2 3 σzia ¼ 2
σzia ¼
σzjb ¼ σzjb ¼
1 2
5 2
Szic ¼
Szjd ¼
5 2
M¼
Table 4 Spin configurations and the total magnetization under the effects of |J3| and h. σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC1 < h < hC2
hC2 < h < hS
σzia ¼
3 2
σzjb ¼
9 6
3 Szic ¼ 2 3 Szic ¼ 2
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 3 3 3 Szjd ¼ M¼ h < hC1 2 2 6
0:1 < jJ3 j < 0:5
Szic ¼
5 2
Szjd ¼
5 2
M¼
3 3 5 5 Szic ¼ Szjd ¼ σzjb ¼ 2 2 2 2 3 3 5 5 5 Sz ¼ Szjd ¼ M¼ σzia ¼ σzjb ¼ 2 2 ic 2 2 6
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
hC1 < h < hC2
σzia ¼
3 z σ ¼ 2 jb
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC2 < h < hC3
σzia ¼
3 z σ ¼ 2 jb
1 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
M¼
9 6
σzia ¼
3 2
σzjb ¼
hC3 < h < hS
σzia ¼
3 z σ ¼ 2 jb
3 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
M¼
11 6
3 2 1 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
σzia ¼
1 2
σzjb ¼
σzia ¼
1 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
3 σzia ¼ 2 1 σzia ¼ 2
σzjb ¼ σzjb ¼ σzjb ¼
5 2
Szic ¼
Szjd ¼
5 2
M¼
9 6
3 Szic ¼ 2 1 Szic ¼ 2
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 1 1 1 z Sjd ¼ M¼ h < hC1 2 2 6
1 2
3 2
Szic ¼
1 2 1 2 1 2
3 2
Szic ¼ Szic ¼
Szjd ¼
5 2
3 Szic ¼ 2 1 Szic ¼ 2
Szjd ¼ Szjd ¼
5 Szic ¼ 2
3 2
M¼
3 2
5 2
M¼ M¼
5 Szjd ¼ 2
3 6
7 6
hC2 < h < hC3 hC3 < h < hC4
9 M¼ 6
hC4 < h < hS
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 1 1 1 z Sjd ¼ M¼ h < hC1 2 2 6
σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
1 2
Szjd ¼
1 2
M¼
3 6
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
3 2
Szjd ¼
3 2
M¼
7 6
hC2 < h < hC3
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
3 2
Szjd ¼
3 2
M¼
9 6
hC3 < h < hC4
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
3 2
M¼
10 6
hC4 < h < hS
σzia ¼
3 2
σzjb ¼
3 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
11 6
h > hS
Table 3 Spin configurations and the total magnetization under the effects of J1 and h. σzia ¼
1 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
5 6
h < hC1
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
5 2
Szjd ¼
5 2
M¼
7 6
hC1 < h < hC2
σzia ¼
3 2
σzjb ¼
3 2 3 ¼ 2
σzia ¼
σzjb ¼
σzia
σzjb
¼
1 2
Szic ¼
5 2
5 2
M¼
9 6
5 5 11 Szjd ¼ M¼ h > hS 2 2 6 5 5 5 z Sjd ¼ M¼ h < hC1 2 2 6
1 2
5 2
3 2
σzjb ¼
σzia ¼
3 2
σzjb ¼
1 2
Szic ¼
σzjb ¼
3 2
5 Szic ¼ 2
Szic ¼
5 2
Szjd ¼
5 2
Szjd ¼
5 2
5 Szjd ¼ 2
M¼
7 6
M¼
9 6
11 M¼ 6
hC1 < h < hC2 hC2 < h < hS h > hS
istence of the low temperature peak in the C curve. It should be mentioned that the double-peak phenomenon in C curves has been also observed in the nano-graphene monolayer [39] and bilayer [48] as well as other nano-systems such as 2D boron clusters [53], nanoparticle [63], nanotube [70] and nanofilm [71–73]. Experimentally, the double-peak C phenomenon have been revealed in rare-earth metals and compounds such as Tb3M series (M ¼ Co, Rh and Ru) [74], pseudo-binary TmxDy1-xAl2 (0 � x � 1) [75], RCuAl3 (R ¼ Pr and Nd) compounds [76]. By contrast, Fig. 13c and d illustrate that the exchange couplings J1 and |J3| need to become larger in order to make the peak to move toward higher temperature. And the effect of J3 is more obvious. In Fig. 13e, the high-temperature peak also increases with the external magnetic field h increasing. The results demonstrate that the blocking temperature may be improved by increasing J1 and |J3| as well as h. The results about the effect of h on the C are consistent with those found experimentally in �-R2S3(R ¼ Tb,Dy) single crystals [77], YbNi3Al9 and Yb(Ni0.94C a u0.06)3Al9 monoaxial-chiral helimagnets [78].
hC2 < h < hS
3 Szic ¼ 2 3 Szic ¼ 2
σzia ¼
3 σzia ¼ 2
Szjd ¼
7 6
h > hS
system is. From Fig. 12e, it is concluded that the influence of the h on U is similar to that of exchange coupling (J1 or |J3|), namely the stronger h can make the system more stable. Our results are comparable with those of graphene monolayer [40,42,54], graphene bilayer [48] and other nano-systems such as hexagonal-type magnetic nanowires [59–62], nanoparticles [63] and core-shell nanoislands [64–69]. Finally, we also focus on the temperature dependence of the specific heat of the system C in Fig. 13a–e. It is clearly seen that each specific heat curve has a distinct rounded peak at high temperature region, demonstrating that the system undergoes phase transition from order to disorder. Through a comparison between Fig. 13a and b, one can see that the increase of both anisotropies |D1| and |D2| makes the peak move to the low temperature region and the effect of D2 seems stronger. It is interesting that the double-peak specific heat phenomenon is observed for certain values of physical parameters. Of two peaks in the specific heat curves, the higher temperature peak should correspond to the blocking temperature, which is connected with the magnetic ordering. Below the peak the system is ferrimagnetic, whereas above one it is paramagnetic. And the other peak at low temperature may be attributed to two factors: on one hand, the small thermal fluctuation in the internal energy can be responsible for the low temperature peak in the C curve. On the other hand, it may come from the fact that the spin configuration of the system changes at the ground state, namely the spin states of the sublattices change, due to the competition among various physical pa rameters including the crystal-field, the exchange couplings, the external longitudinal magnetic field and temperature. For example, when D1 ¼ 1.5, the spins of sublattices a and b may flip from σ ziaðjbÞ ¼ �3=2 to �1=2 at very low temperature, which leads to the ex
hC1 < h < hC2
5 6
M¼
hC2 < h < hS
M¼
3 2
1 2
1 z 5 5 S ¼ Szjd ¼ 2 ic 2 2
9 6
11 6 h < hC1
σzia ¼
σzia ¼
3 σzia ¼ 2
0:7 < J1 < 0:1
1 2
hC1 < h < hC2 hC2 < h < hS h > hS
that U increases with the increasing of the anisotropies |D1| and |D2| at a certain temperature. This indicates that the strong anisotropy is not beneficial to the stability of the system. Comparing Fig. 12a and b, the effect of D2 on U is greater than that of D1. As can be seen from Fig. 12c and d, the effect of exchange couplings J1 and |J3| on U is different from that of anisotropy. Namely, as J1 or |J3| increases, the U decreases, suggesting that the smaller the exchange coupling is, the more stable the 8
D. Lv et al.
Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113721
4. Conclusion
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Using the Monte Carlo simulation, we studied the magnetization plateaus, internal energy and specific heat of a sandwich-like nanographene under the effects of different physical parameters (D1, D2 J1, J3 and h). The results show the system can display multiple magnetization plateaus behaviors, reflecting the variation of spin configuration for the system due to the competition among the anisotropy, exchange coupling and longitudinal magnetic field. The internal energy of the system can be reduced by decreasing the anisotropy and increasing the exchange coupling as well as the longitudinal magnetic field. The double-peak phenomenon can be found in the C curves for certain parameters. These results could be beneficial for nano-graphene multilayers in ap plications in spintronic devices. Conflicts of interest We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgements This project was supported by Natural Sciences Foundation of Liaoning province, China (Grant No. 20170540672), College Students’ Innovative and Entrepreneurial Training Program of Shenyang Univer sity of Technology, China (Grant No. 2018124). References [1] C.W. Lin, K.C. Wei, S.S. Liao, C.Y. Huang, C.L. Sun, P.J. Wu, et al., Biosens. Bioelectron. 67 (2015) 431. [2] A.G.L. Lin, H.Y. Peng, Z.Q. Liu, T. Wu, C.L. Su, K.P. Loh, et al., Small 10 (10) (2014) 1945. [3] E. Roy, S. Patra, D. Kumar, R. Madhuri, P.K. Sharma, Biosens. Bioelectron. 68 (2015) 726. [4] Y. Song, X. M Li, C. Mackin, X. Zhang, W.J. Fang, T. Palacios, H.W. Zhu, J. Kong, Nano Lett. 15 (2015) 2104. [5] T. Kawasaki, K. Sugawara, A. Dobroiu, T. Eto, Y. Kurita, K. Kojima, Y. Yabe, H. Sugiyama, T. Watanabe, T. Suemitsu, V. Ryzhii, K. Iwatsuki, Y. Fukada, J. Kani, J. Terada, N. Yoshimoto, K. Kawaharad, H. Ago, T. Otsuji, Solid State Electron. 103 (2015) 216. [6] L.J. Xu, W. Chu, Lu Gan, Chem. Eng. J. 263 (2015) 435–443. [7] D. Li, M.B. Mueller, S. Gilje, et al., Nat. Nanotechnol. 3 (2008) 101. [8] M. Lotya, Y. Hernandez, J.N. Coleman, et al., J. Am. Chem. Soc. 131 (2009) 3611. [9] Y. Zhang, H.J. Zhang, X. Li, H.T. Xu, Y. Wang, Nanotechnology 27 (2016) 155401. [10] N. Ota, N. Gorjizadeh, Y. Kawazoe, J. Magn. Soc. Jpn. 34 (2010) 573–578. [11] N. Ota, N. Gorjizadeh, Y. Kawazoe, J. Magn. Soc. Jpn. 35 (2011) 360–365. [12] K.Y. Lian, X.F. Li, S. Duan, M.X. Jin, D.J. Ding, Y. Luo, J. Appl. Phys. 115 (2014) 104303. [13] K. Szalowski, J. Magn. Magn. Mater. 382 (2015) 318. [15] M. Sherafati, S. Satpathy, Phys. Rev. B 83 (2011) 165425. [16] A.T. Raji, E.B. Lombardi, Phys. B 464 (2015) 28. [17] Y. Mohammadi, R. Moradian, Loc, Solid State Commun 178 (2014) 37. [18] W. Jiang, Y. Yang, A. Gong, Carbon 95 (2015) 190. [19] F. Albertini, F. Bolzoni, A. Paoluzi, Phys. B Condens. Matter 294 (2001) 172. [20] K. Koyama, H. Fujii, Phys. B Condens. Matter 294 (2001) 168. [21] M. Doerr, S. Kramp, M. Loewenhaupt, Phys. B Condens. Matter 294 (2001) 164. [22] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984. [23] E. Viitala, J. Merikoski, M. Manninen, J. Timonen, Phys. Rev. B 55 (1997) 11541. [24] X.Y. Chen, Q. Jiang, W.Z. Shen, C.G. Zhong, J. Magn. Magn. Mater. 262 (2003) 258. [25] J. Wang, Y. Liu, Z. Xie, Q.M. Ma, J. Phys. Conf. Ser. 29 (2006) 65. [26] Z.Z. Weng, Q. Feng, Z.G. Huang, Y.W. Dui, Acta Phys. Sin. 53 (2004) 3177.
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