Monte Carlo study of magnetic behavior of core-shell nanoribbon

(J. Magn. Magn. Mater. 374 (2015) 639-646)

• Résumé de la publication 3 :

Dans cette publication, nous avons considéré les propriétés magnétiques et hystérétiques d’un nanoruban ferrimagnétique avec la morphologie cœur-coquille. Ce genre de système pré- sente des comportements magnétiques importants et inhabituels. Ainsi, nous avons analysé l’état fondamental ainsi que les diagrammes de phase de ce système. Nous avons montré l’exis- tence d’une température de compensation de très haute importance dans le stockage d’informa- tion et plus particulièrement dans l’enregistrement thermo-optique. Nous avons même spécifié l’effet des valeurs du couplage du cœur et de la coquille ainsi que le couplage intermédiaire sur la température de compensation, tout en donnant explicitement les conditions permettant son apparition dans notre système.

Monte Carlo study of magnetic behavior of core–shell nanoribbon

L.B. Drissia,n, S. Zriouela, L. Bahmadb

a

Lab-PHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco

b_{Lab de Magnétisme et PHE, Faculty of Science, Mohammed V University, Rabat, Morocco}

a r t i c l e i n f o

Article history: Received 15 May 2014 Received in revised form 19 July 2014

Available online 6 September 2014 Keywords:

Core–shell nanostructure Monte Carlo calculation Magnetic phase

Critical and compensation temperature Hysteresis cycle

a b s t r a c t

Using Monte Carlo simulations within Ising model, we study the magnetic properties and the hysteresis loops of a core–shell nanoribbon, made of spinsσ¼1

2core surrounded by spins S¼1 shell with anti-

ferromagnetic intermediate coupling. We analyze the ground-state phase diagrams in the presence of external magnetic and crystalfields. We show the existence of the compensation temperature and its dependence on theσ–S and S–S couplings. We investigate the effects of the crystal-field, temperature, shell interactions and intermediate coupling on the hysteresis curves. A number of characteristic behaviors are found, such as the occurrence of single and triple hysteresis loops for appropriate values of the system parameters. The obtained results are in good agreement with available experimental and theoretical works.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Magnetic materials at nanoscale have attracted a lot of interest,

[1,2] and the references therein, due to their peculiar magnetic

properties resulting from the small-size and surface effects. These finite systems have behaviors different from bulk counterparts which make them promising for advanced permanent magnetic applications. Since the development of the experimental techni- ques, the tailoring of nanostructured magnetic materials such as nanoparticles, nanorods, nanotubes, nanoribbons and nanowires

has reached a high level of sophistication. In [3], nanoparticle

self-assembly is used for making exchange-coupled nanocompo- site magnets. Magnetic nanowires are fabricated using electro-

deposition for potential applications in nanotechnology [4,5].

Near-infrared responsive gold nanorods are synthesized via surface-initiated atom transfer radical polymerization for smart drug delivery[6].

To determine the magnetic characteristics of nanostructured materials, various theoretical methods are employed. By the use of

effective-field theory with correlations, two distinct magnetic

susceptibility peaks and magnetic reversal events in Ising nano-

wire system are found[7]. Within a meanfield model, both the

size and the average temperature dependence of magnetization in

gold nanoparticles are presented [8]. Based on a probability

distribution method, the effectivefield theory is used to examine

the effects of the interfacial coupling constant, the transversefield and the anisotropy on the magnetic properties and the

thermodynamics of the nanotubes with interlayer coupling [9].

Ferrimagnetic nanoparticles on a body-centered-cubic lattice are investigated by the mean of Monte Carlo simulation[10,11]. It is

found in [10] that the nearest-neighbor interactions influence

clearly the compensation behavior. The effects of the crystalfield and exchange interactions on the thermal and magnetic properties of the system are discussed in detail in [11]. In [12], magnetic properties of nanotubes of different diameters, using armchair or zigzag edges, are investigated with MC calculations. The effect of

edges on magnetic phase transitions in doped[13]nanoribbons is

studied using both Monte Carlo calculations and mean field

theory.

The interesting results obtained from theoretical studies of

nanostructured magnetic systems have opened a newfield in the

research of the critical magnetic phenomena at nanoscale. Special interest is devoted to core/shell nanostructures. Heterostructured

metal–semiconductor Zn–ZnO core–shell nanobelts and nano-

tubes are synthesized [14] to exhibit superior functionality for

various applications in nanotechnology, such as nanotransducers

and nanosensors. ZnO–TiO2 core–shell nanorod arrays are fabri-

cated by atomic layer deposition to improve the efficiency of

organic–inorganic solar cell device[15]. Au/Ag core/shell nanorods with different shell thicknesses are synthesized in aqueous solu- tion using chemical deposition[16]. Au/Ag nanostructure presents a model for evaluating the plasmon damping in inhomogeneous

metallic systems with interfaces. TiO2–B@C core–shell nanorib-

bons, synthesized using hydrothermal route, show important influence on the electrochemical intercalation properties[17].

In parallel, there are extensive theoretical studies modeling new

core–shell nanostructures and showing unusual and interesting

magnetic phenomena beside the existence of compensation 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.2014.08.094

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

n_{Corresponding author.}

E-mail address:ldrissi@fsr.ac.ma(L.B. Drissi).

temperature. The compensation temperature Tcompis the tempera-

ture at which the total magnetization of the system M vanishes below its critical temperature Tc[18]. At the compensation point,

determined by the condition MðTcompÞ ¼ 0, only a small driving field

is required to change the sign of the resultant magnetization of the system. This property makes the occurrence of the compensation point of great technological importance. In particular it facilitates the writing/deleting process in the ultra-high density recording media [19]. In[20], a model explaining the formation process of core–shell nanocrystals using a reverse micellar system is presented using the Monte Carlo technique and the post-core route. The effect of the degree of surface disorder, namely spin-glass type ordering, on the magnetic properties of 2D Ising antiferromagnetic particles

is investigated by mean-field approximation and Monte Carlo

simulations[21]. The coercive force increasing as a function of the antiferromagnetic shell thickness is the result obtained for a composite nanoparticle with ferromagnetic core and outer anti-

ferromagnetic shell using the mean field approximation[22]. The

influence of a transverse field on thermodynamic quantities of

cylindrical core/shell spin-1 nanowire is reported in [23] using

effective-field theory with correlations. Within Green's function

technique, the size, anisotropy and doping effects on static and dynamic properties of ferromagnetic nanoparticles are investigated in[24,25].

Recently, mixed spin core–shell nanostructures are also studied for some magnetic systems. MC studies show that the concept of core–shell morphology is capable of explaining various character- istic behaviors observed in nanoparticle[26,27]. Indeed, ferrimag- netic spherical core/shell nanoparticle exhibits one or even two

compensation temperature [28]. Moreover, it is observed under

certain conditions that the magnetization curves obey P-type,

N-type and Q-type classification schemes [29]. The (3/2, 1)

core/shell nanoparticles de_{fined on a simple cubic lattice show a}

dynamic phase transition from paramagnetic to a dynamically ordered phase in the presence of ultrafast switchingfields[30].

As far as we know, there is no work on mixed spin core–shell

hexagonal nanoribbons even if these nanostructures are very much required in engineering magnetic devices such as graphene derivatives and other 2D materials. For this purpose we study in

the present paper the magnetic behavior of a core_{–shell nanor-}

ibbon, made of spins σ¼1

2 ferromagnetic hexagonal core

surrounded by spins S¼1 ferromagnetic chain shell with an anti-

ferromagnetic interface coupling. Using Monte Carlo simulations within Ising model, we show the effect of shell and intermediate coupling on magnetic properties of the ferrimagnetic core/shell nanoribbon. The behavior of the critical and compensation tem- peratures is investigated in the presence of crystal and magnetic externalfields. This work is organized as follows. InSection 2, we

describe the model and present the method. In Section 3 we

discuss in detail the obtained results. In the last section we give our conclusion.

2. The model and method

We study magnetic behavior of 2-dimensional core-shell nanoribbon. Different forms of shell can be used for hexagonal

core nanoribbons [31–35]. In this work, we adopt the spin

σhexagonal core surrounded only in X-direction by a spin S

square-chain shell with an anti-ferromagnetic interface coupling

(seeFig. 1). The sites of the core are occupied by spinsσ¼1

2, which

take the spin values71

2, while spins S¼1 take the spin values 71

and 0 at each site of the shell.

The classical Ising model is described by the Hamiltonian: H¼ Jsh∑ i;jSiSjJc∑k;lσkσlJint∑i;k Siσkh ∑ i Siþ∑ kσ k ! Δ ∑ i S2i ! ; ð1Þ

where Jc and Jsh are the couplings between two first nearest

neighbor magnetic atoms with spin σk–σl at core and Si--Sj at

surface shell respectively. Jint is the exchange-interaction para-

meter between two nearest-neighbor magnetic atoms one at the surface shell and the other one at the core. The crystal-fieldΔis only applied over all S-spins. In the core system there is no crystal field contribution as the spinsσ¼1

2have only two state712[36]. h

represents the external longitudinal magneticfield.

In the Monte Carlo calculations based on Metropolis algorithm

[37], we apply periodic boundary conditions in the Y-direction.

Free boundary conditions are applied in the X-direction which is of finite width. Configurations are generated by selecting the sites in sequence through the lattice and making single-spin-flip attempts, which are accepted or rejected according to the Metropolis algorithm. Starting from different initial conditions, we perform the average of each parameter and estimate the Monte Carlo

simulations averaging over many configurations. Each site of

system is visited each time step. Our data are generated with 5 104_{Monte Carlo steps per spin in lattice, discarding the 5 10}3

initial Monte Carlo simulations. This ensures to reach equilibrium

system, when averaging over the remaining configurations. For

each initial conditions this step is performed and we average over different initial conditions. The error bars are calculated with a

Jackknife method[37]by taking all the measurements and group-

ing them in 20 blocks. This error bar is negligible, so it does not appear in our plots.

For N_{¼ N}cþNsh, we define the internal energy per site as

follows:

E¼_N1〈H〉: ð2Þ

The total magnetization M and its corresponding susceptibilityχ

are respectively M¼NcMcþNshMsh N and χ¼βN 〈M 2 〉 Mh i2   ð3Þ where Mcand Mshare defined as follows:

Mc¼ 1 Nc ∑ Nc k¼ 1σk and Msh¼ 1 Nsh ∑ Nsh i¼ 1 Si: ð4Þ

andβ¼ 1=kBT, T is the absolute temperature and kBis Boltzmann's

constant, we set kB¼ 1.

At the compensation point, the total magnetization must disappear. Subsequently the compensation temperature may be determined by the crossing point between the absolute values of

Msh and Mc. This leads us to the following expressions at com-

pensation temperature:

MshTcomp

 

¼ Mc Tcomp

 

; and sign Msh Tcomp

  ¼ sign Mc Tcomp   :j        ð5Þ

Fig. 1. Schematic presentation of core/shell nanoribbon with inner spinσ ¼1 2core

L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 639–646 640

3. Results and discussion

Using Monte Carlo simulations within Ising model, we study magnetic properties of ferrimagnetic core/shell nanoribbon. We start by calculating exactly the ground state diagrams required to check the reliability of the theoretical results. Then, we show the

existence of compensation temperature Tcompand we examine the

effect of crystalfieldΔand intermediate coupling Jinton Tcompand

critical temperature. We study also the influence of the set of

parameters, namely Jint, Jsh,Δand T, on magnetic hysteresis curves.

3.1. Ground state phases

For our Hamiltonian, the ground state is invariant under translation by comparing the energies of the ground state phases at T¼ 0 K. For the mixedσ¼1

2and S¼1 Ising model, we perform

many tests for different sizes of Nc and Nsh, namely

Nc¼ 12; 24; 36; 48; 60, and Nsh¼ 6; 12; 18; 24. The obtained results

do not depend on the system sizes. In what follows, we limit our calculations to the system withfixed size Nc¼ 48 and Nsh¼ 12. In

Fig. 2, we plot the ground state phase diagrams in the plane

h=Jc; Jint=Jc

 

inFig. 2a and in the plane Δ=Jc; Jint=Jc

 

inFig. 2b. In the plane h=Jc; Jint=Jc

 

, wefind four possible spin configura-

tions. Namely, these phases are þ1; þ1

2   , þ1; 1 2   , 1; þ1 2   and 1; 1 2  

for Jint=Jco0. However only the two configurations

þ1; þ1 2   and 1; 1 2  

are present for Jint=Jc40. InTable 1, we

summarize the ground state energy equations and the conditions when the coupling in the border frustrates the bounds of the four phases.

In the plane Δ=Jc; Jint=Jc

 

, the two new phases ð0; 1

2Þ and

ð0; þ1

2Þ arise in addition to the four phases found in the plane

h=Jc; Jint=Jc

 

. These two new phases exist only for negative values of crystal field. They are present for Jint=Jc40 as well as for

Jint=Jcr0. Table 2shows the ground state energy equations and

the boundaries between the regions corresponding to the 6 configurations.

The ground state of the model depends on the values of the parameters in the Hamiltonian. For example, in the plane

h=Jc; Jint=Jc

 

, for h40 the ordered phases þ1; þ1

2

 

andð1; þ1

2Þ

are separated at critical value corresponding to Jintþ2  h ¼ 0 but

for ho0, the two phases ð1; 1

2Þ and ðþ1;  1

2Þ are separated by

the line 2 hJint¼ 0. However in the plane Δ=Jc; Jint=Jc

 

, the

region of the two ground state configurations ð71; 71

2Þ is

bounded by the lines Jint¼ 0 and 2JshþJintþ2Δ¼ 0, while the

boundaries of the ð81; 71

2Þ phases are given by Jint¼ 0 and

2JshJintþ2Δ¼ 0 as listed inTable 2.

3.2. Compensation vs critical temperature

At first we study the effect of temperature on the total

magnetization M and its corresponding susceptibility χ. In the

absence of external magnetic field h and at Δ=Jc¼ 1 and

Jsh=Jc¼ Jint=Jc¼ 0:1, we plot M andχas a function of tempera-

ture T inFig. 3. Because core–shell nanoribbon is finite, one should not expect any phase transition. However, according to[38,39], the phase transition occurs infinite system if they are simulated under periodic boundary conditions using Monte Carlo calculations. The total magnetization decreases from its saturation value at low temperature region, and vanishes at compensation temperature Tcomp. At the compensation point the system presents afirst order

transition due to the jump discontinuity in the M curve. For

T4Tcomp, the magnetization increases slowly taking negative

values and stabilizes to a constant magnetization MC0 for

T4Tcritical where Tcriticalis the Néel temperature[40]. This beha-

vior is confirmed in the curve of the total susceptibilityχtotthat

exhibits two peaks at Tcomp 0:2 and Tcrit 0:7. The first peak, that

Fig. 2. The ground-state diagrams for the core shell nanoribbon with Nc¼ 48, Nsh¼ 12, and Jsh¼ Jc, (a) in the plane h=Jc; Jint=Jc

 

atΔ ¼ 0 and (b) in the plane Δ=Jc; Jint=Jc

 

Table 1

The four phases, their energies and the conditions for their existence in the plane h=Jc; Jint=Jc

 

.

Phase Energy Conditions

þ1; þ1 2

 

Jsh14Jc21Jint32hΔ h40 and Jintþ2:h40

1; 1 2

 

Jsh14Jc21Jintþ32hΔ ho0 and 2:hJinto0

1; þ1 2

 

Jsh14Jcþ21Jintþ12hΔ h40 and Jintþ2:ho0

þ1; 1 2

 

Jsh14Jcþ21Jint12hΔ ho0 and Jint2:h40

Table 2

Ground state configurations, energies and conditions for their existence in the plane Δ=Jc; Jint=Jc

 

.

Phase Energy Conditions 71; 71

2

 

Jsh14Jc21JintΔ Jint40 and 2JshþJintþ2Δ40

81; 71 2

 

Jsh14Jcþ21JintΔ Jinto0 and 2JshJintþ2Δ40

0; 71 2

  1

4Jc Δo0, 2JshþJintþ2Δo0 and 2JshJintþ2Δo0

is very pronounced, occurs at a first-order phase transition from the ferrimagnetic phase to the antiferrimagnetic phase. The second peak presents a second-order phase transition from the antiferrimagnetic phase at low temperature region to the para- magnetic one at high temperature region.

InFig. 4, we show the effect of crystal_fieldΔand intermediate

coupling Jint on the compensation and critical temperatures for

fixed values Jsh=Jc¼ 0:1 and h¼0. As shown in Fig. 4a, the

compensation and critical temperatures remain constant for negative values ofΔ, then they decrease quickly atΔ=Jc¼ 1 to

saturate at a minimal value forΔ=JcZ0. We deduce that both the

compensation and critical temperatures drop dramatically when the crystal-_{field changes in sign. This can be explained by the fact}

that the energy corresponding to the Hamiltonian equation (1)

varies abruptly in term of the crystalfieldΔas shown inFig. 2b for the magnetization of the ground state phases.

When we vary the coupling Jint=Jc at fixed crystal field

Δ=Jc¼ 1, the compensation and critical temperatures present a

symmetry with respect to the axis Jint¼ 0 (see Fig. 4b). This

symmetry is due to the symmetry in the ground state obtained

in Fig. 2b. It is worth noting that the competition between the

three parameters Jint,Δand Jshis responsible for the behavior of

the two temperature curves. From Fig. 4 we deduce that high

values of Tcrit and Tcomp occur when the crystal field and the

intermediate coupling satisfyΔ=Jcr 1 and Jint=JcjZ2.

In order to reproduce results in agreement with the literature,

we focuss on positive values of Δ. In Fig. 5, we present the

influence of the intermediate coupling on the compensation and

critical temperatures at fixed values of Δ=Jc¼ 1 and Jsh=Jc¼ 0:1.

Thefigure shows a transition line of the second order separating

the ferrimagnetic phase present at low temperature region from

the paramagnetic phase at high temperature region.Fig. 5shows

also different behaviors for the two temperatures. The compensa- tion temperature increases linearly with Jint=Jc and disappears

when Jint=Jc is larger than 1.5, while the critical temperature is a

horizontal constant line Tcrit¼ 0:95 for Jint=Jcr0:9 which indicates

that Jint=Jchas no effect on Tcritin this region. For higher values of

Jint=Jc, Tcrithas a linearly increasing behavior originating from the

effect of intermediate coupling that becomes dominant.

Notice that our result is in good agreement with[11]forΔ=Jc¼ 1

but is completely different from the one obtained in the previous caseΔ=Jc¼ 1 where the two temperatures Tcritand Tcompdecrease

for Jint=Jco2 and increase for higher value of Jint=Jc without that

Tcomp disappears. Therefore, the crystal field has a significant

influence on the critical and compensation temperatures.

To investigate the effect of the exchange coupling Jsh on the

critical and compensation temperatures of the nanoribbon, we present in Fig. 6 the phase diagram in the plane T_=Jc; Jsh=Jc

 

for Jint=Jc¼ 0:1 andΔ=Jc¼ 1. The figure shows a transition line of the

second order separating the ferrimagnetic phase at low temperature

Fig. 3. Magnetization and susceptibility versus temperature for h=Jc¼ 0,

Jsh=Jc¼ 0:1, Jint=Jc¼ 0:1 and Δ=Jc¼ 1.

Fig. 4. The critical and compensation temperatures (a) in terms of crystalfield Δ for Jsh=Jc¼ 0:1 and Jint=Jc¼ 0:1, (b) in terms of intermediate coupling Jintfor Jsh=Jc¼ 0:1

Fig. 5. The critical and compensation temperatures in terms of intermediate coupling Jint=Jcfor Jsh=Jc¼ 0:1, and Δ=Jc¼ 1.

L.B. Drissi et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 639–646 642

region from the paramagnetic one at high temperature region. As expected, when Jsh=Jc increases, the critical temperature also

increases. This is due to fact that it is more difficult for the spin lattice to become disordered when there is a strong interaction that tends to keep the shell spins S parallel. The same behavior is found for the compensation temperature that increases more rapidly with increasing Jsh=Jc, and disappears at Jsh=Jcclose to 0.35.

In Fig. 7, we present the influence of the size of the core on

compensation and critical temperatures for the specific parameters Jsh=Jc¼ 0:1, Jint=Jc¼ 0:1, and for positive and negative crystal field

Δ. One can see that when the nanoribbon size N increases, the two temperatures also increase, and saturate for different sizes. For Δ=Jc¼ 1, Tcrit becomes constant at NshþNC¼ 132 while Tcomp

stabilizes at N¼120. However, for Δ=Jc¼ 1, Tcrit becomes a hor-

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