Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure

Accepted Manuscript Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure

Wei Wang, Qi Li, Dan Lv, Rui-jia Liu, Zhou Peng, Sen Yang PII:

S0008-6223(17)30499-2

DOI:

10.1016/j.carbon.2017.05.052

Reference:

CARBON 12028

To appear in:

Carbon

Received Date:

04 March 2017

Revised Date:

04 May 2017

Accepted Date:

14 May 2017

Please cite this article as: Wei Wang, Qi Li, Dan Lv, Rui-jia Liu, Zhou Peng, Sen Yang, Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure, Carbon (2017), doi: 10.1016/j.carbon.2017.05.052

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ACCEPTED MANUSCRIPT The graphene nanoribbon system can exhibit rich and interesting magnetization plateau behaviors with different single-ion anisotropies and longitudinal magnetic fields at low temperatures, due to the competition between the single-ion anisotropies, the longitudinal fields as well as the edge effects. It is worthwhile mentioning that most magnetization plateaus of the system may not satisfy the 2S+1 criterion, originating from the cooperation not only between the homogeneous atoms from the marginal and internal parts, but also between different kinds of atoms.

Fig.5 Magnetic field h dependence of magnetization (a) M, Ma, Mb and (b) Mac, Mas, Mbc, Mbs for the zigzag graphene nanoribbon structure for Db=1.5 with Da =0.5 and T =0.01.

ACCEPTED MANUSCRIPT Monte Carlo study of magnetization plateaus in a zigzag graphene nanoribbon structure Wei Wang*, Qi Li, Dan Lv, Rui-jia Liu, Zhou Peng, Sen Yang School of Science, Shenyang University of Technology, Shenyang 110870, China ABSTRACT The Monte Carlo simulation has been applied to discuss the step effect on a zigzag graphene nanoribbon structure in a longitudinal magnetic field. The effects of the singe-ion anisotropies, longitudinal magnetic fields and temperature on the magnetization plateaus have been investigated in detail. Our results show that the number of magnetization plateaus for the system dissatisfy 2S+1 criterion at low temperatures, originating from not only the competition between the anisotropy and the external magnetic field, but also the contributions of edge effects. It has been found that the single-ion anisotropy plays a significant role in modulating the spin configurations of the zigzag graphene nanoribbon structure. In addition, the effects of single-ion anisotropies and external magnetic fields on the magnetization, the susceptibility, the internal energy and the blocking temperature have been examined.

Keywords: graphene nanoribbons; magnetization plateaus; susceptibilities; blocking temperatures; internal energy;

Monte Carlo simulation

*Corresponding author: Wei Wang, Tel: 86-13998204589; fax: 86-24-25496502 E-mail address: [email protected] (W. Wang).

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ACCEPTED MANUSCRIPT 1. Introduction Over two decades, there are increasingly dramatic breakthroughs in the progress of materials with excellent nanostructures, such as nanoparticles [1-3], nanowires [4, 5], nanotubes [6], nanoislands [7], nanodisks [8], nanographene [9,10]. Particularly, nanographene has speedily developed into one focus of nanomaterials since the twodimensional graphene with a honeycomb structure was succesfully prepared. As a new derivative, graphene may be derived from a macromolecule produced by chemical reactions [11, 12]. It is considered as an effective approach to adjust the gap energy and dominate magnetic properties for the graphene [13]. Graphene has been devoted to more explorations since the first appearance in 2004 [14, 15]. Because graphene is a zero gap semiconductor, it offers a challenge how to open and regulate the band gap, which makes it possible for the graphene material to apply in high tech-electronic devices. A few of techniques are employed with diverse processes to overcome this limitation. Of which one way is the quantum confinement of electrons by forming graphene nanoribbons (GNRs) [16, 17]. Either it can explain magnetism or modify the electronic structure for the non-magnetic materials. By means of the epitaxial graphene pattern or cutting the exfoliated graphene sheet along one straight line, it is feasible to fabricate graphene nanoribbons with two typical types of edges: armchair and zigzag edges [18]. Considerable theoretical investigations have revealed that the electrical and magnetic properties of GNRs can be modulated by the edge structures and the ribbon width [19]. Naturally, armchair graphene nanoribbons (AGNRs) are not beneficial for magnetic ordering in theory [20], whereas zigzag graphene nanoribbons (ZGNRs) can display relatively more plentiful information about interesting magnetic

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ACCEPTED MANUSCRIPT structure and magnetic behaviors such as peculiar transport properties and magnetism [21]. Owing to the electron spin polarization behaviors spontaneously at zigzag edge states, ZGNRs are discovered to be magnetic semiconductor, in which the C atoms at each edge couple ferromagentically to each other and antiferromagnetically to those of the opposite edge [22]. As a typical feature of AFM ground state in ZGNRs, half metallicity is especially significant for special spintronic devices [23]. In order to stabilize the half metallicity in ZGNRs, the energy difference between ferromagnetic and antiferromagnetic (FM-AFM) ground states may be very small (several meV per unit cell), and it is likely for FM-AFM energy to take on the metal state at finite temperature. In view of this, a great number of theoretical explorations are concerned with the effects of adsorptions with magnetic transition metal (TM) atoms [24], the replacement of edge C atoms by nonmagnetic atoms such as N[20], H[25] and B[26], heteroatoms-doping [27], carrier density [28], edge state [29], topological defect vacancies [30], as well as external electrical and magnetic field [31] to achieve the reasonable prediction and explanation for magnetic mechanism in ZGNRs. Another interesting theoretical study has illustrated that the sufficiently strong magnetic field can drive the spin-polarized edge states in ZGNRs from antiferromagnetic to ferromagnetic coupling, leading to a phase transition from the ground state to the magnetized state. Conversely, the increase of the temperature enables the spins to become disordered and unpolarized state, magnetic anisotropy effect can be used to keep the spin ordering [32]. As a result, by aid of the external conditions, ZGNRs in finite temperature can exhibit the ground state, spin disordered or magnetized state. Therefore, controlling multiple spin configuration states of ZGNRs under different physical parameters may be of important theoretical

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ACCEPTED MANUSCRIPT and practical significance in designing spin transistors and spin logic devices [33]. As is wellknown, the Ising model as one of the basic models of statistical physics has been successfully extended to describe various magnetic nanosystems and plays a significant role in understanding magnetic phase transitions and electric polarization behaviors of spin systems. Theoretically, by building various Ising models, the effects of various physical parameters such as the crystal-field, the exchange coupling, the temperature and the longitudinal magnetic field upon magnetic performances of graphene nanoribbons have been deeply examined. L. B. Drissi et al. have studied magnetic behaviors of a mixed-spin (1/2, 1) nanoribbon with the core-shell structure under the effect of an antiferromagnetic interface coupling by the Monte Carlo (MC) simulation [34]. They have found a number of distinctive behaviors, for example, single and triple hysteresis loops as well as compensation temperatures. The influence of the locations and the number of spin-3/2 substituted magnetic atoms upon magnetic phase transitions of the doped mono-, bi- and tri- graphene nanoribbons with different configurations have been analyzed in great details [35]. Related results have indicated that the critical temperature Tc relies on the number of dopants, while in view of configurations with fixed magnetic impurities, it could be found that Tc has the stronger sensitivity to edges. In addition, by using the mean field theory and the MC simulation, the temperature dependences of magnetizations and susceptibilities have been thoroughly explored for the pure graphene nanoribbons [36]. Particularly, the low/strong external magnetic fields and ribbon widths on critical temperatures, transition temperatures and hysteresis loops have been also investigated. These results have further proved the important functions of the temperature, the anisotropy and the longitudinal magnetic field to graphene

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ACCEPTED MANUSCRIPT nanoribbon systems. Recently, many researchers have investigated different mixed-spin Ising systems under the external magnetic field for many years [37-39]. They have demonstrated the special significance of the external magnetic field on the magnetic properties for nanosystems. At low temperatures or ground state, magnetic Ising nanomaterials may display spin ladders, exchange biases, magnetization plateaus as well as other outstanding performances. The eligible condition s-m = integer for existing magnetization plateaus have early come into the view of theoretical studies [40]. Since the first report of the magnetization plateau for one nearest-neighbour Ising model, more and more devotions have been immersed in studying magnetization plateaus of mixed-spin magnetic systems [41]. Based on the effective-field theory (EFT), Jiang et al. have examined the influences of different crystal-field anisotropies on magnetization plateaus for one ferrimagnetic nanoparticle [42]. Of investigations it has found the dissatisfaction of the 2S+1 criterion for the magnetization plateaus under the effect of larger crystal-field anisotropies. It is colorful to investigate magnetization plateaus on ±J Ising square lattices and triangular lattices by the MC simulation [43, 44]. At low temperatures, irregular behaviors from the single-ion anisotropy have been also found in the hysteresis loops. X.Y. Chen et al. have studied magnetization plateau phenomena for the positive anisotropy under the influence of the longitudinal magnetic field in an antiferromagnetic mixed-spin (1, 3/2) Ising system with the classical MC simulation [45]. By using both the molecular-field theory and density-matrixre normalization group calculations, magnetic characteristic behaviors have been explored with the homogenous crystal-field anisotropy for one ferrimagnetic mixed-spin (1, 3/2) Ising chain [46]. It turned out that the

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ACCEPTED MANUSCRIPT anisotropy plays a vital part in the occurrence of magnetization plateaus. Besides, with increasing the longitudinal magnetic field from 0 to its saturation value hs, it is not difficult to discover 2S+1 magnetization plateaus in low-dimensional magnetic materials with the magnetic field h [40, 45, 47]. As far as ferromagnetic and antiferromagnetic mixed-spin systems are concerned, they have strong dependences on crystal-field anisotropies, temperatures and the magnetic dipolar interactions [48, 49]. It could be observed the magnetization plateaus within the anisotropy at the magnetization process in a spin-3/2 chain system [50]. With respect to experimental observations, magnetization plateau phenomena have also emerged in diverse systems, such as NbFeTe2 [51], FexMg1-xCl12 [52], La0.67xAxCa0.33MnO3[53],

novel organic tetraradical crystal BIP-TENO [54] as well as the

quasione-dimensional Ni-compounds [55, 56] and the quasitwo-dimensional compound SrCu2(BO3)2 [57]. Interestingly, recent theoretical studies have also been focused on the graphene nanostructure by the establishment of different mixed-spin Ising models. Using the EFT, E. Kantar have investigated the magnetic hysteresis, compensation behaviors and phase diagrams of the bilayer honeycomb lattices (BHL) system with AB stacking geometry by building spin-1/2 Ising model [58]. Especially the dynamic phase transitions, the dynamic hysteresis curves and compensation types of the BHL structure in the oscillating magnetic field have been studied in details [59, 60]. R. Masrour et al. have applied the MC simulation to explore the effect of magnetic atom doping of graphene strucrure by a spin-2 Ising model [61]. Described by a mixed-spin (3, 7/2) Ising model, the magnetic properties of a bi-layer decorated graphene structure has been examined within the MC simulation [62]. In our

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ACCEPTED MANUSCRIPT previous work, we have applied the MC simulation and mixed-spin Ising models to study magnetic and thermodynamic properties of different low-dimensional nanomaterials, such as the nanographene bilayer [63], the nanowire [64] and the magnetic films [65]. Especially, the step effects on molecular-based magnets under the influences of single-ion anisotropies, exchange couplings and temperatures [66]. Whether magnetization plateaus exist or not in such a ZGNRs system? Do they satisfy the 2S+1 criterion? In remarkable, less attention has been paid to the connection of appearing magnetization plateaus with the variation of spin configurations for diverse single-ion anisotropies and longitudinal magnetic fields in a ZGNRs structure. Recently, the magnetization plateaus of the sandwich-like nanographene have been discussed with the EFT [67]. In addition, AGNRs can be also express metallic and semiconductor properties depending on the ribbon width under certain conditions, but different from that of AGNRs, ZGNRs system is always semiconductor and can exhibit more fascinating edge effect since localized edge states, edge magnetism and susceptibility can develop them into special magnetic materials without d or f electrons [34, 35, 68], which may be favor for ample potential applications in spintronic devices. However, less literature has been reported on multiple spin states related to magnetization evolution in an external magnetic field for the relevant investigations of ZGNRs, which motivates us to explore magnetic properties of a ZGNRs structure theoretically. Therefore, in this paper, we present a spin dependent study on the magnetic properties of the ZGNRs structure. Particular emphases would be placed on the effects of the single-ion anisotropies and the longitudinal magnetic fields upon magnetization plateaus and multiple spin states at the edges for the ZGNRs structure. This paper is organized as follows. In Section 2, the model and the Monte Carlo

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ACCEPTED MANUSCRIPT simulation are given. The results obtained and discussion are presented in Section 3. Finally, in Section 4, the conclusions are drawn. 2.Model and the Monte Carlo simulation We propose a one-dimensional graphene nanoribbon structure of the infinite length with zigzag edges. This ZGNRs system is considered by modeling the honeycomb lattices with the free boundary condition along the x-direction and periodic boundary condition in the ydirection. In Fig.1, one graphene nanoribbon structure with Nx = 8 has been depicted. The similar zigzag nanoribbon structures have been observed in Refs. [35, 36]. It is worth mentioning that multiple spin states of hybrid atoms can be achieved by various useful methods, which is of great value in discussing the origination of magnetism for the graphene nanoribbons structure [69]. Recent computational studies have revealed that the physical properties of ZGNRs can be modified in potentially effective approaches by adsorption of adstructures such as metal adatoms (Co, Ni) [24], or small clusters (Nin, Fen) [70] at the central or edge sites on the ZGNRs. Interestingly, except for the adsorption of single atom, two different TM atoms Co and Ni adsorbed on ZGNRs can also strongly affect the geometry structures and electronic properties [71], where the spin values of Co and Ni atoms are taken 3/2 and 1, respectively. In view of these above findings with TM atoms, we decided to study whether more pronounced effects might result from two different magnetic atoms instead of the single atom. In Fig.1 the big red and the small green balls are used to denote two atoms with the spins  iz  1 and S jz  3 / 2 antiferromagnetically coupled at two opposite edges and the central region. Therefore, a ferrimagnetic mixed spin-1 and spin-3/2 Ising model with the anisotropies has been proposed to describe the present ZGNRs structure. The Hamiltonian of

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ACCEPTED MANUSCRIPT the system is expressed as follows: H   J   iz S jz  D a  ( iz ) 2  Db  ( S jz ) 2  h(  iz   S jz ) i, j

i

j

i

(1)

j

Where  iz and S jz are the spin-1 and spin-3/2 atoms, respectively.  iz take the values of ±1, 0 and S jz are the values ±3/2, ±1/2. The first sum only considers the nearest-neighbors couplings. Da and Db represent the single-ion anisotropy constants for two atoms, respectively. h is the longitudinal magnetic field. J is selected as the reduced constant of the energy and temperature, then we fix |J|= 1.

Fig.1 Schematic sketch of a zigzag graphene nanoribbon. The big red atoms and the small green atoms denote the spins

 iz  1 and S jz  3 / 2 , respectively. The atoms at the edge of the graphene nanoribbon structure are

described in the region of the dot lines. The free boundary condition is in the x-direction and the periodic boundary condition along the y-direction.

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ACCEPTED MANUSCRIPT The classical single-spin-flip Monte Carlo simulation based on the metropolis algorithm [72] has been applied to simulate the Ising zigzag nanoribbon system. From our simulation, additional simulations have been carried out for the choice of the number of spins with zigzag edges in the x-direction. It is difficult to find any clear difference from Ny= 20 to 50. As a result, Ny= 20 could be the selection for the simulation. Here, we define the total number of atoms for the present system as N= 360, where the number of atoms A and B are same, both for Na=Nb=180. One can see 20 atoms and 160 atoms in the marginal and internal parts for atom A, respectively, namely there exist 1/9 atom A in the marginal part and 8/9 atom A in the internal part. The arrangement is same with atom B. For safe, in order to equilibrate the system, we use over the rest 4  104 Monte Carlo steps per site to average thermal quantities after discarding prior approximate 104 Monte Carlo steps. The interesting thermal quantities are given as follows: the magnetizations per site of atoms A and B Ma, Mb are calculated by

Ma 

1 Na

1 Mb  Nb

Na

 i 1

Nb

S j 1

z i

(2) z j

Thus the total magnetization per site M is

M  (M a  M b ) / 2

(3)

Then we also define the magnetizations per site of internal and marginal atoms Mac, Mas for atom A and Mbc, Mbs for atom B as below

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M ac  M as 

N ac

1 N ac



1 N as

N as

i 1

 i 1

N bc

1 M bc  N bc

S

1 M bs  N bs

N bs

j 1

S j 1

z i

z i

(4) z j

z j

Where Nac and Nbc are the number of internal atoms for atom A and B, respectively, the values of them are both Nac = Nbc= 160. Nas and Nbs represent the number of marginal atoms for atom A and B in the y-direction, the values of which are both Nas = Nbs= 20. The magnetic susceptibility per site is presented with

   ( M 2    M  2 ) Here we define   1

kBT

(5)

, where k B is the Boltzmann constant. For simplicity, we take

kB  1 . For the high density magnetic recording materials, it is known that the super paramagnetism is one principal problem about the thermal stability. As one typical value, the recording bit remains maximum at the blocking temperature TB , then TB is possible to be an important parameter to analyze the thermal stability for the graphene nanoribbon system. When the temperature T becomes smaller than TB, the present system shows extremely strong ferromagnetic or ferrimagnetic phase rather than exhibit super paramagnetic phase for T > TB. Generally, of experimental studies it is feasible to obtain TB from field-cooled (FC) or zerofield-cooled (ZFC) magnetization curves [73, 74]. We can find one bifurcation point in the FC or ZFC curves at low external magnetic field (10Oe or 20Oe), where the temperature can 11

ACCEPTED MANUSCRIPT correspond to TB. With T < TB, there exist irreversible magnetic behaviors for the present system. As far as theoretical studies, TB may be also determined from the peak versus the temperature in the susceptibility curves [9, 63, 75-78], which has been applied in this work. 3.Results and discussion In this section, the initial spin values of atoms A and B in the ground state are randomly taken according to various physical parameters and the longitudinal magnetic field h is along the positive direction of z-axis. We will investigate the influences of diverse parameters on the magnetization plateaus, the magnetizations, the susceptibilities and the blocking temperatures in the graphene nanoribbon system. 3.1 Effects of the single-ion anisotropies Da, Db and the temperature T on magnetization plateaus Firstly, we shall discuss the effects of the single-ion ansiotropies Da on magnetization plateaus, as shown in Figs.2-4. The total magnetization M, the atom A and atom B magnetizations Ma, Mb, the marginal and internal magnetizations Mas, Mac for atom A and Mbs, Mbc for atom B as the function of the longitudinal magnetic field h is presented in Fig.2ad for Da =0, 1.5 with Db =0.5 and T =0.01. From Fig.2a, one can see three magnetization plateaus M =0.25, 0.36, 1.25 for Da =0 and two characteristic values of magnetic fields highlighted by two dot lines, where the magnetization plateaus occur. The value of the magnetic field corresponding to the left dot line with M = 0.36 is called as the critical field hc1 (=1.6), the other one corresponding to the saturation magnetization Ms =1.25 is named as the saturation magnetic field hs (=4.6). Note that the total magnetization plateaus come from the

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ACCEPTED MANUSCRIPT contributions of the atom A because there also exist three magnetization plateaus in atom A, while the Mb (=1.5) maintain invariable. As plotted in Fig.2b, it is not difficult to find that the variations of the Mac and Mas do not tune for each other. Note that the spins of the marginal part  iSz flip prior to that of the internal ones  iCz with increasing h from 0 to 10. Namely, the spin states of atom A for the marginal and internal parts (  iSz ,  iCz ) change from  iz  -1 to 1 at different values of hc1 =1.6 and hs =4.6 rather than remain unchangeable for the atom B with S jz  3 / 2 . These behaviors mainly result from the effect of the longitudinal magnetic field without the anisotropy. Considering the effect of the anisotropy, it can be found from Fig.2c that there also appear three magnetization plateaus M= 0.3, 0.8, 1.25 at hc1= 3.1and hs= 6.1 for Da =1.5 due to the variations of Ma. Further analyses can be obtained from Fig.2d that the difference of spin states for atom A from the marginal and internal parts (  iSz ,  iCz ) may be responsible for the behaviors of Ma and M observed in Fig.2c. Moreover, comparing Fig. 2d with b, the mechanisms of magnetization plateaus for atom A are different since the spin states of atom A from the internal part  iCz take on three states  iz  -1,0,1 and those of the marginal part  iSz are only two states  iz  0,1 when h increases from 0 to 10. Notice that without the magnetic field h= 0, the spins of the atom A from the marginal part  iSz lie in the

 iz  0 state, whereas those from the marginal part  iSz still stay in the  iz  -1 state. It can be concluded that the stronger anisotropy Da makes the marginal atoms stay at low spin states. In addition, the marginal and internal spins (  iSz ,  iCz ) for atom A flip simultaneously at hc1, differing from the behaviors in Fig.2b.

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Fig.2 Magnetic field h dependence of magnetization (a, c) M, Ma, Mb and (b, d) Mac, Mas, Mbc, Mbs for the zigzag graphene nanoribbon structure for two cases Da =0 and 1.5 with Db =0.5, T =0.01.

The M, Ma, Mb and the Mac, Mas, Mbc, Mbs as the function of the h is plotted in Fig.3a-d for Da= 0.5, 2.5 with Db =0.5 and T =0.01. From Fig.3a and c, one can see clearly richer magnetization plateau behaviors than those of Fig.2a and c. With increasing h from 0 to 10, the present system displays five plateaus M =0.25, 0.3, 0.36, 0.8, 1.25 with hc1=1.1, hc2=2.1, hc3=4.1, hs=5.1 for Da= 0.5 and four plateaus M =0.3, 0.75, 0.8, 1.25 with hc1=2.1, hc2=4.1, hs=7.1 for Da= 2.5, mainly stemming from the change of the Ma. These results can attain related explanations from Fig.3b and d, respectively. By comparison of Fig.3b with Fig.2b,

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ACCEPTED MANUSCRIPT the spin states become more abundant than Fig.2b by taking the anisotropy Da into account. For the case of the weaker anisotropy Da (= 0.5), the system presents  iz  0 besides

 iz  -1,1 for both the marginal and internal atom A and the spin states of the atom B retain S jz  3 / 2 all the time. Similarly, the spins of atom A from the marginal part  iSz also flip ahead of those from the internal  iCz . It is the common contributions of the variations of the marginal and internal spin states (  iSz ,  iCz ) for atom A that cause the existence of multiple plateaus in the Ma and M curves. From Fig.3d, one can find the changes of spin states of both atoms A and B are similar to those of Fig.2d, but the difference between them is the inconsistency in spin states of atom A from the marginal and internal parts (  iSz ,  iCz ). Namely, at first the internal spins  iCz flip from the  iz  -1 state to  iz  0 state at hc1= 2.1 instead of flipping from the  iz  0 state to  iz  1 state at hc2= 4.1 for the marginal spins  iSz in the atom A, leading to the quadruple magnetization plateau phenomena of the Ma and M. These behaviors are distinct from those observed in Fig.2d. It should be mentioned that although the magnetizations of atom A from the marginal and internal parts may obey the criterion: 2S+1= 3 for certain parameters such as Da= 0.5, the common effects of them result that the number of magnetization plateaus for the Ma and M dissatisfy the 2S+1 criterion. This is considered to be a result of the edge effects. The importance of the edge effect on magnetic phases has been also discovered in the dopped zigzag graphene nanoribbon [35]. It is noticeable that the similar dissatisfaction of the 2S+1 criterion in magnetization plateaus has been pointed out in the nanoparticle system for the stronger anisotropy [42].

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Fig.3 Magnetic field h dependence of magnetization (a, c) M, Ma, Mb and (b, d) Mac, Mas, Mbc, Mbs for the zigzag graphene nanoribbon structure for two cases Da =0.5 and 2.5 with Db =0.5, T =0.01.

In order to further clarify the effect of the anisotropy Da on the magnetic phase, we present one interesting phase diagram of magnetization plateaus in the system for the positive Da with Db= 0.5, as shown in Fig.4. The phase diagram is separated by several symbol lines corresponding to the values of the critical magnetic field hc1, hc2, hc3 and the saturation hs for the magnetization plateaus. It is clearly seen that the critical magnetic fields hc1, hc2, hc3 and the saturation hs change almost linearly with Da increasing. Moreover, one can obtain the width of the magnetization plateaus by the vertical distance between two adjacent critical

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ACCEPTED MANUSCRIPT fields, the last critical magnetic field and the saturation field. According to the results of Figs.2-3, it is concluded that the system may exhibit different numbers of the magnetization plateaus as observed in Figs.2-3 with three plateaus for Da=0, 1.5, five plateaus for 0hs. Similarly, for Da=1.5, the states (  iSz ,  iCz ) are at (0, -1) states for hhs. The most spin states appear for 0hs. Further increasing Da from 1.5 to 3.0, we can find four configurations of spin states (  iSz ,  iCz ), that is at (0, -1) states for hhs. It is worthwhile mentioning that the increase of Da is easier to force the spins of atom A to lie in the low state  iz  0 . For example, when Da>1.5 and hc1
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Fig.4 Phase diagrams of magnetization plateaus in the Zigzag graphene nanoribbon structure with Db =0.5 and T =0.01 for the positive single-ion anisotropy Da. Tab.1 Spin state of atom A for different regions under the effect of Da and h with fixed Db=0.5.

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ACCEPTED MANUSCRIPT Next, let us focus on the effects of the anisotropy Db on the magnetization plateaus of the system and two atoms as shown in Figs.5-6. Fig.5a-b shows the M, Ma, Mb and the Mac, Mas, Mbc, Mbs as the function of the h for Db= 1.5 with Da =0.5 and T =0.01. From Fig.5a, in the whole magnetic field 0  h  10, one can notice that the system can exhibit more magnetization plateaus than those of Figs.2-3, compared to the effects of Da. Six magnetization plateaus M =0.9, 0.25, 0.3, 0.36, 0.8, 1.25 with hc1=0.6, hc2=1.1, hc3=2.1, hc4=4.1, hs=5.1 in the M curves can be observed, resulting from the contributions from five plateaus in the Ma curve and one plateau in the Mb curve. It is clearly seen from Fig.5b that the spin states of atom B from the marginal part S zjS change from S zjS  1 / 2 to S zjS  3 / 2 at hc1, whereas the spins of atom B from the internal part S zjC always lie in S zjC  3 / 2 state, which should be responsible for the first plateau corresponding to the critical field hc1. We can remark that the spins of atom B from the marginal part flip from the high state S zj  3 / 2 to the low state S zj  1 / 2 earlier than those from the internal part, due to the fact that the strong anisotropy Db can force S zjS to stay in the low spin state. In addition, we can see the non-synchronous variations of the Mac and Mas lead to the existence of other five plateaus in Ma and M curves, which are similar to those of Fig3.a and b.

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Fig.5 Magnetic field h dependence of magnetization (a) M, Ma, Mb and (b) Mac, Mas, Mbc, Mbs for the zigzag graphene nanoribbon structure for Db=1.5 with Da =0.5 and T =0.01.

The M, Ma, Mb and the Mac, Mas, Mbc, Mbs as the function of the h is presented in Fig.6a-b for Db= 2.4 with Da =0.5 and T =0.01. The system still shows five plateaus, but it is remarkable that the appearance of a well has been observed at 5.1
 z  0 to  z  1 . Because the number of the spins and the variations of magnetizations Mac, iC

iC

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ACCEPTED MANUSCRIPT Mbc from the internal part are equal for both atoms, an obvious magnetization plateau in the M curves does not appear for the location of the well in the Mb curve. Furthermore, from our simulations, we can find the width of the well increases with increasing Db, which thus improves the width of the plateau between hc4 and hs.

Fig.6 Magnetic field h dependence of magnetization (a) M, Ma, Mb and (b) Mac, Mas, Mbc, Mbs for the zigzag graphene nanoribbon structure for Db=2.4 with Da =0.5 and T =0.01.

We shall present another interesting phase diagram of magnetization plateaus in the system for the positive Db with Da= 0.5, as plotted in Fig.7. Compared to the effect of Da on the plateaus, one can find that the system displays five plateaus with the critical fields hc1, hc2, hc3 and the saturation field hs for 0  Db  1.0, as observed in Fig.3a, whereas the system may add one plateau with the existence of the critical field hc4 for Db>1.0, due to the variations of 21

ACCEPTED MANUSCRIPT the spin states of atom B. The spin states of atoms A and B from the marginal and internal parts are shown in Tab.2 for different Db and h. It can be found from the table that magnetic phase always occurs earlier at the edge than in the internal as h increases from 0 to 10 when Db>1.0. For example, when 1.0  Db  2.0, the configurations of the spin states of atoms A and B from the marginal and internal parts are  iSz  1 ,  iCz  1 , S zjS  3 / 2 , S zjC  3 / 2 for hc1
22

ACCEPTED MANUSCRIPT with these above mentioned theoretical and experimental studies [42, 47, 66, 80, 86] with regard to the satisfaction of the 2S+1 criterion for the number of magnetization plateaus. But we should emphasize that most magnetization plateaus of the present Zigzag graphene nanoribbon system may not satisfy the 2S+1 criterion. This phenomenon comes from the cooperation not only between the homogeneous atoms from the marginal and internal parts, but also between different kinds of atoms, which differs obviously from Refs. [42, 47, 66, 7986].

Fig.7 Phase diagrams of magnetization plateaus in the Zigzag graphene nanoribbon structure with Da =0.5 and T =0.01 for the positive single-ion anisotropy Db.

23

ACCEPTED MANUSCRIPT Tab.2 Spin states of atoms A and B for different regions under the effect of Db and h with fixed Da=0.5.

Fig.8a-d shows the effect of the temperature on the magnetization plateaus of the system for selecting several values of T=0.01, 0.1, 0.2 and 0.5 with Da=0.5 and Db=0.5. With increasing T, magnetization plateaus become smooth gradually and at last may disappear at T=0.5, suggesting that the temperature can make the step effect be weakened. Our results are in great accordance with previous theoretical studies for the pyrochlore lattice [88] and a Shastry-Sutherland lattice [89] by the MC simulation, metal-organic quasione-dimensional ferrimagnets with the method of discrete path integral representation [90], the molecular magnet V15 with an antiferromagnetic Heisenberg model [91], the Tb2Ti2O7 pyrochlore

24

ACCEPTED MANUSCRIPT magnetic material within a single tetrahedron approximation [92], the single-chain magnet [{(CuL)2Dy}{Mo(CN)8}].2CH3CN.H2O [93] and a dopping finite Ising chain [77] based on the exact solution in terms of the generalized classical transfer matrix. Furthermore, the temperature effect on the magnetization plateaus have been found in experimental results of the α-CoV2O6 compound [94] and the Shastry-Sutherland Magnet TmB4 [95] .

Fig.8 Magnetic field h dependence of magnetization for the zigzag graphene nanoribbon structure with Da =0.5 and Db=0.5 for (a) T =0.01, (b) T =0.1, (c) T =0.2 and (d) T =0.5.

3.2 Effects of the single-ion anisotropies Da, Db and the longitudinal magnetic fields h on the magnetizations, the susceptibilities and the blocking temperatures

As shown in Figs.9-13, typical results of the nanoribbon system are extended to show the influences of single-ion anisotropies Da, Db and the longitudinal magnetic fields h on the magnetizations, susceptibilities and the blocking temperatures. Fig.9a-d shows the temperature dependence of the total magnetization M, the

25

ACCEPTED MANUSCRIPT magnetizations of atoms A and B Ma, Mb, the total magnetic susceptibility  and the internal energy U for several values of Da when Db=0.5 and h=0.1. From Fig.9a, one can see that the M curves exhibit two saturation values at zero temperature, M=0.25 for Da =0.5, 1.0 and M=0.305 for Da =2.0, 3.0, respectively. The result for M=0.25 can be easily explained by M = [-1  3 / 2] / 2  0.25 with the spin states of atom A  iz  1 and atom B S jz  3 / 2 for weaker anisotropies. There exists 1/9 (Nas=20) and 8/9 (Nac=160) for the atom A in the marginal and internal

parts,

respectively.

The

other

M=0.305

comes

from

M=

1 8 [0   1  3 / 2] / 2  0.305 , where the spin states  iSz  0 for 1/9 atom A in the marginal 9 9

part and  iCz  -1 for 8/9 atom A in the internal part, whereas the spins of the atom B always stay in S zj  3 / 2 states for stronger anisotropies. These results are comparable to those of Tab.1. It is interesting that the M curves show some kinds of curves similar to the Q(Da=0.5), P-(Da =1.0, 3.0), L-(Da=2.0) types, as predicted in the Néel theory of ferrimagnetism [96]. We can remark that these colorful M curves result from the competition between the anisotropy and the temperature. In Fig.9b, it is clear seen that the Ma and Mb change from their saturation values to approach some value of the remanence with T increasing. Note that the saturation values of Ma express two different values Ma= -1.0 for Da= 0.5, 1.0 and Ma= -8/9 for Da= 0.5, 1.0, respectively. As plotted in Fig.9c, one can notice that all the  curves change similarly with the temperature. Namely, they all present the first decline, and then the abrupt ascending as well as the eventual decrease with the increase of T. The maximum of the  curve corresponding to the temperature is named as the blocking temperature TB. With decreasing Da, TB moves towards high temperature regions, which means that TB becomes smaller because of the low spin states for stronger anisotropies. The

26

ACCEPTED MANUSCRIPT values of them are at TB=2.15, 2.08, 1.80, 1.29 for Da = 0.5, 1.0, 2.0, 3.0. From Fig.9d, it is found that each U curve has the similar behavior, namely, U increases as T increases and the stronger Da, the higher U may be seem at the same temperature.

Fig.9 The temperature dependences of the (a) M, (b) Ma , Mb, (c)  and (d) U for the zigzag graphene nanoribbon structure with fixed h=0.1 and Db=0.5. The numbers on the curves are the values of the Da.

Fig.10a-d shows the temperature dependence of the M and the  for several values of Db when Da=0.5 and h=0.1. In Fig.10a, it is feasible to observe the same two saturation values phenomenon, M=0.25 for Db =0.5 and M=0.194 for Db =1.0, 1.5, 2.0. The value of M=0.305 1 1 3 8 can attain the explanation by M= [    - 1] / 2  0.194 . It is clear to compare these 2 9 2 9

results with those of Tab.2. One can also observe that the M curves resemble the Q- (Db =0.5), P- (Db =1.0, 1.5) and L- (Db =2.0) types in the Néel theory. From Fig.10b, we can find

27

ACCEPTED MANUSCRIPT two saturation values in the Mb curves at T=0, except that one is Mb=1.5 for Db=0.5, another is Mb= 1.39 for Db=1.0, 1.5, 2.0, because one in nine spins of atoms B lies in the S zj  1 / 2 state and the rest lies in the S zj  3 / 2 state. Remarkably, our results obtained about the magnetizations are in wonderful consistent with previous experimental explorations for the nano-garphene system because the existence of different maximums has been shown in the M curves at low temperatures, which signifies the coexistence of various magnetic states [97, 98]. In theory, similar behaviors have also been found both in the nanoribbon and the nanoribbon with core-shell structure [34-36]. From Fig.10c, one can notice that TB increases with decreasing Db. In Fig.10d, it is clear seen that U behaves similar to those in Fig.9d. Namely, U can be improved by increasing Db.

Fig.11a-c shows how the TB changes with the Da and Db for fixed h=0.1. From Fig.11a, one can see that the effects of the Da and Db on TB are similar, namely they both can make TB decrease with increasing Da or Db. With the further analyses, note that Db affects TB more powerfully than Da because the spin value of atom B (S=3/2) is larger than that of atom A (=1). In order to further compare the effects of Da and Db on TB for a fixed value of h, we have also presented two typical cases in two dimensional space (TB, Da) for Db=0.5, h=0.1 in Fig.11b and in two dimensional space (TB, Db) for Da=0.5, h=0.1 n Fig.11c, respectively. Comparing two sub-figures, one can find TB has a greater variation with Db than with Da . Namely, with increasing Da or Db from 0 to 2.0, TB decreases from 2.3 to 1.8 with Da in Fig.11b, while TB decreases from 2.2 to 1.4 with Db in Fig.11c. It is concluded that TB can be enhanced by decreasing Da and Db. The influences of the anisotropies (Da, Db) on TB accord well with the studies of nano-graphene bilayer [9, 63, 67] and nano-graphene trilayer [75]. 28

ACCEPTED MANUSCRIPT

Fig.10 The temperature dependences of the (a) M, (b) Ma , Mb, (c)  and (d) U for the zigzag graphene nanoribbon structure with fixed h=0.1 and Da=0.5. The numbers on the curves are the values of the Db.

Fig.11 The blocking temperature TB for the zigzag graphene nanoribbon structure under the effects of the Da, Db (a) in three dimensional space (TB, Da, Db) for h=0.1, (b) in two dimensional space (TB, Da) for Db=0.5, h=0.1,

29

ACCEPTED MANUSCRIPT (c) in two dimensional space (TB, Db) for Da=0.5, h=0.1.

Fig.12a-b shows the temperature dependence of the M and the  for several values of h with Da=0.5 and Db =0.5. In Fig.11a, one can see that the longitudinal magnetic field h improves the magnetization M at low temperatures. The system presents different broad maximums, derived from the release from the frustrated ground state owing to the agitation. The agitation occupies the upper hand after increasing T to the maximum, contributing to the decrease of the magnetization. Various magnetization changes at low temperatures are dominated by the longitudinal magnetic field, while the clear variations of the M curves at high temperatures result from the fact that the magnetic field competes with the temperature. It can be observed from Fig.11b that different saturation values at T=0 exist in the Ma curves under the effects of h, while one common saturation value Mb=1.5 in all the Mb curves. This indicates that the spin states of atom A are more easily influenced by the magnetic field than those of atom B, which is regarded as a result of the opposite directions between the spins of atom A and the magnetic field at the ground state. As shown in Fig.11c, it is not difficult to find that the effect of the h on TB is reverse to that of the Da and Db. Namely, TB heads for high temperatures as h increases. The behaviors of the magnetic field h on TB may be comparable with some theoretical work [9, 63, 71]. It is worth mentioning that for T >2.1, TB maintains almost unchangeable with h increasing, which suggests the h affects TB slightly. This behavior is different from previous studies [6, 73, 75]. It is found from Fig.12d that U increases rapidly with the increasing of T and decreases with h.

30

ACCEPTED MANUSCRIPT Fig.13 shows how the TB changes with h for fixed Da=0.5 and Db=0.5. One can notice that TB at first has a large variation as h increases until h=1.3. This phenomenon can be explained as follows: the magnetization of the system can be significantly improved by increasing h, which makes it more difficult for the system to be demagnetized for the higher temperature. As a result, the system will change more slowly from ferrimagnetic phase to paramagnetic phase, which should be responsible for the higher TB . Similar results of TB vs h have been obtained in some other nanosystems [9, 34-36, 63, 67, 75]. Note that when h>1.3, TB seems to be a saturate tendency with h. This means the transition phase of the system is mainly controlled by the temperature rather than the external magnetic field h.

Fig.12 The temperature dependences of the (a) M, (b) Ma , Mb, (c)  and (d) U for the zigzag graphene nanoribbon structure with fixed Da=0.5 and Db=0.5. The numbers on the curves are the values of the h. 31

ACCEPTED MANUSCRIPT

Fig.13 The blocking temperature TB for the zigzag graphene nanoribbon structure under the effect of the h in two dimensional space (TB, h) for Da =0.5 and Db=0.5.

4.Conclusions Based on the Monte Carlo simulation, the step effects on a zigzag graphene nanoribbon structure have been studied. The system can exhibit rich and interesting magnetization plateau behaviors with different single-ion anisotropies and longitudinal magnetic fields at low temperatures, due to the competitions between the single-ion anisotropies, the longitudinal magnetic fields as well as the edge effects. The phase diagrams indicate that the single-ion anisotropy is indispensable for the appearance of magnetization plateaus for the system and plays an important role in controlling the spin configurations of the system. It is found that the spins lie easier in the low spin states for the stronger anisotropies. In particular, the spins of the marginal part flip earlier than those of the internal part with increasing h, showing the propound edge effects on the graphene nanoribbons structure. The system can exhibit several saturation values of the M curves at low temperature, resulting from the 32

ACCEPTED MANUSCRIPT common contributions of the marginal and internal atoms. Various characteristic M curves such as Q-, P- and L-types have been also found, originating from the competitions among anisotropies, magnetic fields and temperatures. The effect of h on TB is reverse to those of Da and Db. In addition, the internal energy of the system can be improved by increasing the anisotropies or decreasing the external magnetic field. We expect that our theoretical results on the graphene nanoribbon may help to support and verify some features, for the appearance of magnetization plateaus at low temperatures, in a series of carbon nanostructure materials.

Acknowledgement This project was supported by Natural Sciences Foundation of Liaoning province (Grant No. 20170540672), Youth Training Fund of Shenyang University of Technology (Grant No. 005609), College Students’ Innovative and Entrepreneurial Training Program of Liaoning province (Grant No.20170684), Program for Liaoning Innovative Research Team in University (Grant No. LT2014004).

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