Edge effect on magnetic phases of doped zigzag graphone nanoribbons

(J. Magn. Magn. Mater. 374 (2015) 394-401)

• Résumé de la publication 2 :

Dans ce papier, nous nous sommes basés sur deux outils très importants : la méthode Monte Carlo et la théorie de champ moyen afin d’investiguer les propriétés magnétiques des nanorubans de graphone. Nous avons présenté une alternative qui permet de contrôler le magnétisme dans des nanorubans de graphone de type zigzag en introduisant des impuretés magnétiques dans différentes positions. Nous avons aussi examiné l’effet des positions et du nombre des atomes magnétiques substitués sur les transitions de phase des nanorubans de graphone de type zigzag dans trois cas différents : mono-, bi- et tri-dopé.

Edge effect on magnetic phases of doped zigzag graphone nanoribbons

L.B. Drissia,b,n, S. Zriouela, E.H. Saidia

a

LPHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco

b_{International Center for Theoretical Physics, ICTP, Trieste, Italy}

a r t i c l e i n f o

Article history:

Received 24 September 2013 Received in revised form 25 March 2014

Available online 27 August 2014 Keywords:

Graphone Nanoribbon Monte Carlo Meanfield theory Magnetic phase Curie temperature Transition metal

a b s t r a c t

Curie temperature TChas important implications for the experimental realization of magnetic graphone

nanostructures relevant for future spintronic applications. Using both Monte Carlo method and mean field theory, we study magnetic properties of zigzag graphone nanoribons (ZGONR) doped with magnetic impurities M. We show that TCincreases with the number of dopants but for configurations

withfixed number M, TCis not very sensitive to impurities distances dðM MÞ. In particular, in bidoped

ZGONR configurations, TChas different values for the same dðM MÞ. This surprising behavior stems

from edge effect. The result as derived in this report is easily adapted to predict how the magnetism is influenced in all half hydrogenated four-electrons hexagonal nanoribbon devices.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Since the first preparation of graphene in 2004 [1], this

thinnest metal has attracted a lot of interest for its fundamental studies[2,3]and its high potential applications[4]. However, the use of pristine graphene for new nanoelectric and nanophotonic device applications faces some obstacles due to the absence of the gap in its band structure. Several works were devoted to develop directions for modifying electronic structure and introducing magnetism in this nonmagnetic material. One way is cutting the graphene sheet along a straight line leading to one dimensional

carbon nanoribbons with armchair or zigzag edges [5,6]. Zigzag

graphene nanoribbons (ZGNR) are semi-conductor at its ground

state with band gap depending on its width W[7]. In each edge,

localized states are ferromagnetically ordered[8]while the mag-

netic moments on the two edges interact antiferromagneticly

since they have opposite spin orientation [9]. FM-AFM energy

differences per unit cell is a few meV and could present metal

state at finite temperature [10]. Chemical modification creating

new derivatives as graphane [11,12] is an other strategy that

not only tunes the gap energy of graphene[13]but it affects also its magnetic properties. Graphane attains permanent magnetic moment through hydrogen vacancy domains depending on the

size and geometry of these domains[14] or by partial dehydro-

genation giving rise to graphone[15]. Graphone is a ferromagnetic semiconductor with a small indirect gap and the partial hydrogen

coverage has a striking effect on physical properties of graphene

that can be restored by annealing [16]. Removing half of the

hydrogene atoms from graphone leads to nonmagnetic material as

pz orbitals of two nearest unsaturated C atoms formπ-bonding

that quenches magnetism[15].

Magnetic graphene nanostructures are of great interest as their application in a number of exceptional spintronic devices have been proposed for one-dimensional zigzag graphene nanoribbons

[17,18]. However, during synthesization of graphene nanostruc-

tures using TM-containing catalyst, the incorporation of transition metal (TM) was observed for the case of Ni in electron microscopy

images for graphitic particles [19] and in X-ray adsorption for

carbon nanotubes[20]. Other TM were also observed such as Au

and Pt in two graphene layers using high resolution transmission

electron microscopy (HRTEM) [21]. In spite of this fact, the

substitutional TM impurities in carbon systems have not been studied in detail and little attention was paid to their magnetic properties. Moreover, the feasibility of engineering magnetic graphene nanostructures devices is questioned by magnetic order-

ing at finite temperature. Therefore, it is very important to

understand the fundamental behavior of the magnetic properties of such materials. One of the most important physical properties to investigate is the edge as well as TM impurities positions depen- dence of Curie temperature.

In the present paper, we study, using Monte Carlo calculations and mean_{field theory, the possibility of controlling the magnetism in} zigzag graphone nanoribbons by introducing TM impurities in different positions from edges. The effect of substituted magnetic atoms on the magnetic phase transitions of mono-, bi- and tri-doped Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jmmm

Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2014.08.058

0304-8853/& 2014 Elsevier B.V. All rights reserved.

n_{Corresponding author.}

ZGONR in different configurations are discussed. The transition

temperatures obtained for all configurations are compared. To

extrapolate results to wide ribbons, we provide a qualitative way

of determining which are the configurations giving maximal and

minimal Curie temperature. We end by a conclusion.

2. The model

In the present work, we consider infinitely long one-

dimensional graphone nanoribbons with zigzag edges. The system is periodic only along X-direction and the super cell used, in our

calculations, is delimited by dashed lines as shown in Fig. 1.

Following previous customary notation[9,22], thefinite width W

of the ribbon is characterized by the number Nzof zigzag chains of

carbons that run along X. InFig. 1we plot the structure of Nz¼ 8

H-ZGONR-H with the zigzag edge carbon atoms of nanoribbon all saturated with nonmagnetic atoms, namely H atoms to avoid the dangling bond states. In this model, we have two kinds of carbon

atoms: C1 and C2 forming the hexagonal structure. The bond

length dC1 C2denoted a, between the twofirst nearest neighbors

(NN), is 1.495 Å. In each hexagon, carbon atoms C1are decorated

with hydrogen in sp3 _{hybridization while C}

2 atoms that remain

unsaturated are sp2 hybridized. So only the unhydrogenated C2

atoms carry a magnetic moment of about 1μBas their p-electrons

are localized and unpaired. Each C2 atom, except those near the

edges, has three nonmagneticfirst nearest neighbors C1 and six

magnetic second nearest neighbors C2. According to [15], the

valence electrons in p-states are more delocalized than those in d or f-states. Therefore, the ferromagnetic (FM) ground state in graphone originates from interactions between 2p moments

attributed to extended p–p interactions between electrons spins

in C2 atoms. To model these p–p interactions, we use the tight

binding Hamiltonian for spin moments in ZGONR described by H0

in Eq.(1). When we substitute magnetic impurities of spin S which

couples to C2 atoms with the d–p coupling JK, we have extra

interactions described in Eq. (1) by H1. Thus the doped ZGONR

system is modeled by the hamiltonian: H¼ H0þH1 ¼ ∑ C2atoms tijszis z jþ ∑ r¼ impðJKÞrjS z rs z j ð1Þ

tijis the exchange p–p coupling between s-moments at sites i and

j of C2atoms in the hexagonal structure of graphone ribbons all set

to 71=2. Srz is the spin-impurity at given sites r. The above

hamiltonian depends on two couplings tijandðJKÞrj.

Notice that the interaction between two impurity moments at sites i and j in standard RKKY perturbation theory[23]is given by H0¼ΣJRKKY SizSjzwhich corresponds to second order of perturba-

tion with respect to exchange interaction between the magnetic

impurity and the pz electrons of C2atoms. Therefore, the RKKY-

coupling JRKKYin the honeycomb lattice is proportional to J2k=tR 3

where R is the distance between impurity spins[24]. This result

shows that the RKKY coupling, that is crucial for magnetic ordering of the impurities (d–d interactions), is then implicitly

described in Eq.(1)through d–p and p–p couplings. Based on the

results of[24]on behavior of t and JK, we have chosen the specific

value t¼ JK¼ J0 since testing other values like JK¼ t=10 have not

significant influence on the obtained results namely the effects

of the number and positions of transition metal dopants on the magnetic phases.

For the specific case Nz¼ 8, that is the most used model in the

literature, we remove one, two or three C2atoms from A, B, C, D, E,

F or G site along Y-direction and substitute them by large magnetic

M-atoms as depicted inFig. 2. Compared to carbon atom, M-atom

Fig. 1. Structural model 8-Zigzag graphone nanoribbon. Transparent atoms repre- sent hydrogenated C1atoms, black atoms are C2atoms, and small blue atoms

denote hydrogene atoms passivating the edges of graphone. The supercell adopted in MC simulations is denoted by dashed lines. (For interpretation of the references to color in thisfigure caption, the reader is referred to the web version of this article.)

Fig. 2. (i) Schematic of 8-Zigzag graphone nanoribbon where doping sites, starting from the edge and going inwards, are indicated with A, B, C, D, E, F, G and H. (iv) Example of the configuration BEH of tri-sites doped ZGONR where magnetic M-atoms are represented by big violet balls. (For interpretation of the references to L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 394–401 395

from ZGONR surface with an elevation h. The height h, defined by its carbon 1st NN, is considerably lower than for the substituted

atom inflat graphene sheet[25], thus we neglect it for ZGONRs.

The distance dðM C1Þ is in the range of 1.77–1.89 Å for M-atoms:

V, Cr, Mn, Co and Cu that present localized spin moments[26]. We are concerned with M-impurities that present a strong hybridiza- tion with the electronic states of ZGONR. As a consequence, the arrangement of neighboring M-impurities is important in bi-and tri-doped ZGONRs. For this reason, and to ensure maintaining the systems stoichiometry, we are restricted to consider a minimal

possible distance dðM MÞ separating two nearest magnetic atoms

of 3a. We set the corresponding Sizof magnetic M-atoms to 3=2, as

3μB is the highest localized spin moments given for the list of

substitutional transition metals studied in[26].

3. Results and discussion 3.1. Monte Carlo calculations

By means of Monte Carlo simulations for the Ising model described above, we study rectangular hydrogen-terminated one-dimensional zigzag graphone nanoribbons containing 78 atoms. The periodic boundary conditions are applied in the X- direction and free boundary conditions are applied in the Y-

direction. The ground state electronic configuration of pure

ZGONR is characterized by the ferromagnetic arrangement of spins in all model systems investigated in the present work. The

MC steps are 5 105

steps per spin discarding the first 5  104

Monte Carlo simulations. We average over at least 30 different

random con_{figurations of magnetic sites of the system. We build a}

program using the Metropolis algorithm [27] to calculate the

thermal average of magnetization M and energy E. We set the coupling J0¼ 1 and the Boltzmann's constant kB¼ 1. We calculate

also the corresponding magnetic susceptibilityχ

χ¼1_Tð〈M2_〉〈M〉2_Þ

and the specific heat CVgiven by

CV¼

1 T2ð〈E

2_〉〈E〉2_Þ:

We start by considering pure ZGONR with specific parameter

width NZ¼ 8 as the formation energies of ZGNR saturates for

NzZ8 indicating that larger width has lower effect on the stability

of the system[22]. Magnetization and susceptibility as functions of T are plotted inFig. 3. From temperature dependence susceptibility

we deduce the Curie temperature TC¼ 3:49.

In what follows, we study the effect of both the number and the positions of dopants on transition Curie temperature of 8 ZGONRs. A site positioning of the impurity spins, that presents a strong hybridization with the electronic states of C2atoms of ZGONR, is

important since site positioning can break the symmetry of the hexagonal lattice. We consider three possible adatom arrange- ments: (I) one site is substituted by magnetic atom, (II) two sites

are doped and (III) three C2atoms are replaced by 3/2 atoms in

different positions in ZGONR. For these three families, we name the configurations respectively as M, M1M2and M1M2M3where M

corresponds to doped sites A, B, C, D, E, F, G or H. To ensure maintaining the systems stoichiometry, we use a minimal possible

distance d_{ðM MÞ separating two nearest magnetic atoms of 3a for}

bi (M1M2)-and tri (M1M2M3)-doped ZGONRs configurations. All

the possible configurations and their corresponding type are listed

inTable 1and studied in detail in this work.

1- One magnetic atom substituted in ZGONR

For the 8-ZGONR containing one substitutional magnetic

M-atom, we have eight possible configurations A, B, C, D, E, F, G

or H as depicted inFig. 2. Their corresponding magnetization and susceptibility got from MC simulations are displayed versus

temperature in Fig. 4. The graphs are in good agreement with

standard results as magnetization decreases going to zero when T increases, and the susceptibility curves peak for different values of T indicating phase transition from FM to paramagnetic as expected. The susceptibility has a sharp peak for A and H

configuration comparing to D and E having the largest peak. In

Fig. 5we plot Curie temperature as a function of impurity position.

One sees that transition temperature increases monotonically as magnetic dopant draws away from the zigzag edges. To shed more light on the effect of edges, we introduceðn1; n2Þ where n1and n2

are the number of C2-atoms present along Y-axis between each

edge and its nearest substituted M-atom. For mono-doped ZGONR,

we have the correspondence n1 dðedge1MÞ and n2

dðedge2MÞ where dðedgeMÞ is the distance between a sub-

stituted M-atom and its nearest edge. In terms of the pair number

ðn1; n2Þ, TC is minimal for A and H configurations with

ðn1; n2Þ ¼ ð0; 7Þ and TCis maximal for both D and E configurations

having ðn1; n2Þ ¼ ð3; 4Þ. So TC increases as the difference jn1n2j

decreases. This result is due to the nature of the coupling between dopant and its nearest neighbors. We have explicitly checked that the competition between FM and AFM coupling becomes stronger as the magnetic atom position is closer to the boundary which destabilizes

ferromagnetism in mono-doped ZGONRs and declines TC.

Notice that both the values and the variation of TC for M

inserted in A, B, C or D positions are similar to H, G, F or E respectively. This is obvious since (A,H), (B, G), (C,F) and (D, E) are

one to one corresponding symmetric configurations.

Fig. 3. Magnetization and susceptibility versus temperature for pure 8-zigzag

Table 1

Configurations for mono-, bi- and tri-doped 8 ZGONRs. Type Possible configurations

I Serie1 A,B,C,D,E,F,G,H II Serie2-A AC,AD,AE,AF,AG,AH Serie2-B BD,BE,BF,BG,BH Serie2-C CE,CF,CG,CH Serie2-D DF,DG,DH Serie2-E EG,EH Serie2-F FH

III Serie3-A ACE,ACF,ACG,ACH,ADF ADG,ADH,AEG,AEH,AFH Serie3-B BDF,BDG,BDH,BEG,BEH,BFH Serie3-C CEG,CEH,CFH

Serie3-D DFH

L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 394–401 396

2- Two substitutional magnetic atoms M1M2: We investigate MM

configurations of the six series-2 of Table 1. In Fig. 6 we plot the temperature dependence of the susceptibility for some configurations of bi-doped ZGONR even if the calculations are done for all

possible type-2 configurations. All the curves peak for different

values of T, that symbolizes the phase transition, then drop at high temperatures to vanish. AC configuration has the highest value ofχ while BF conformer has the lowest one. Moreover, the suscept- ibility develops the sharpest peak centered around the critical

temperature for AC configuration and the largest one for BF

conformer which corresponds respectively to TC min and TC maxas

showed here below.

In Fig. 7, we present Curie temperature versus the distances

dðM MÞ obtained for both serie2-A and serie2-B configurations.

The two curves of transition temperature increase with dðM MÞ

and reach their maximum respectively at dðM MÞ ¼ 7:5a for

serie2-A that coincides with AF-configuration and at dðM MÞ ¼

6a for serie2-B that is BF conformer, then they drop. At the same distance dðM MÞ, the temperature TCis always higher for serie2-B

comparing to serie2-A owing to edge effects. To focus in this direction, we display inFig. 8the variation of transition tempera-

Fig. 4. Temperature dependence of (a) Magnetization and (b) susceptibility for different possible configurations of one magnetic atom doped 8-zigzag graphone nanoribbons.

Fig. 5. The transition temperature TCfor all possible configurations of magnetic

atom doped one site of 8-zigzag graphone nanoribbons. Fig. 6. The susceptibility as a function of temperature for some configurations ofbi-doped ZGONR.

Fig. 7. The variation of critical temperature for all configurations of serie2-A and L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 394–401 397

dðM MÞ; namely 3a and 4.5a. We align the configurations along x-axis with respect to their corresponding pair numbers ðn1; n2Þ

where we set here n1 dðedge1M1Þ and n2¼ dðedge2M2Þ.

One observes a TC increase for fixed dðM MÞ when decreasing

the difference numberjn1n2j. Comparing all transition tempera-

tures get for serie2, we confirm the previous result deduced from

Fig. 6, that TCis maximal for BF and its equivalent conformer CG

where dðM MÞ ¼ 6a and jn1n2j ¼ 1. And TCis minimal for AC or

FH conformer with the minimal distance dðM MÞ ¼ 3a and the

maximal difference numberjn1n2j ¼ 5.

Notice that, for bi-doped ZGONRs, all the conformers having the same dðM MÞ and ðn1; n2Þ such as (AF,CH), (AD,EH), (BD,EG),

(BE,DG) and (CE,DF), are equivalent and have the same TC

3- Three doped magnetic atoms M1M2M3: We now present

results for more complicated case that is tri-doped ZGONRs. In

Fig. 9 we display the variation of susceptibility in terms of

temperature for some selected configurations of tri-doped zigzag

graphone nanoribbons. Curie temperatures, for all possible con-

figurations of tri-doped ZGONR given in Table 1, are estimated

pairs ðn1; n2Þ, defined here as n1¼ dðedge1M1Þ and n2¼ d

ðedge2M3Þ, is presented for the case of serie3-A. It is evident

that the behavior of TCis owing to magnetic atom positions with

respect to their nearest edge and their two M-atoms neighbor. Indeed, atfixed dðM1M2Þ ¼ d1, namely 3a or 4.5a, the transition

temperature increases with dðM2M3Þ, denoted d2, and reaches

its maximal value for the conformer having jn1n2j ¼ 1, that

corresponds to ACG or its equivalent AEG for d1¼ 3a and to ADG

when d1¼ 4:5a, then it decreases. Moreover, for fixed ðn1; n2Þ,

TC increases with d1 whatever d2. As an example, when

ðn1; n2Þ ¼ ð0; 0Þ, TC is higher for ADH with ðd1; d2Þ ¼ ð4:5a; 6aÞ

compared to ACH havingðd1; d2Þ ¼ ð3a; 7:5aÞ. The same observation

forðn1; n2Þ ¼ ð0; 1Þ.

Finally, comparing all TC of serie3-A with d1 and d2 taking

values 3a, 4.5a, 6a and 7.5a, we _{find that ADG, with}

ðd1; d2Þ ¼ ð4:5a; 4:5aÞ and jn1n2j ¼ 1, has the maximal TC while

the minimal value corresponds to ACE with minimal ðd1; d2Þ ¼

ð3a; 3aÞ and maximal jn1n2j ¼ 3.

Similar behavior of TCis obtained for serie3-B, -C and -D with

different maximum and minimum depending on ðd1; d2Þ and

ðn1; n2Þ as plotted inFig. 11that presents TCfor all nonequivalent

configurations of serie3 ZGONRs with ðn1; n2Þ and ðn01; n02Þ satisfy-

ing n1an02and n01an2. A minima of TCexists atðd1; d2Þ ¼ ð3a; 3aÞ

and ðn1; n2Þ ¼ ð0; 3Þ that corresponds to ACE and its analog DFH

conformers and a maximum appears when_ðd1; d2Þ ¼ ð3a; 4:5aÞ and

ðn1; n2Þ ¼ ð2; 1Þ that is BDG or BEG conformer.Table 2 lists all tri-

doped systems.

Recall that, two configurations are equivalent if they are sharing the same geometric data in symmetric way, namelyðn1; n2Þ ¼ ðn02; n01Þ

andðd1; d2Þ ¼ ðd02; d01Þ and thus having the same TC.

4. Discussion

We start by plotting inFig. 12the specific heat Cvas a function of

temperature for non-, mono-, bi- and tri-doped 8 ZGONR configura-

tions having maximal value of Curie temperature TCmax. It can be seen

that as the number of substituted magnetic atoms is decreased in

8 ZGONR, the specific heat develops a sharper and shaper peak

Fig. 8. The Curie temperature of type-2 configurations in terms of pair numbers ðn1; n2Þ for two specific dðM MÞ; namely 3a and 4.5a. (Dot lines guide the eye).

Fig. 9. The temperature dependence of the susceptibility for some selected configurations of tri-doped ZGONR.

Fig. 10. Curie temperature for all serie3-A configurations with respect to pair number (n1, n2). (Dot lines guide the eye).

L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 394–401

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