Study of the Blume-Emery-Griffiths Model for Mixed Carbon-Like Nanotube: Monte Carlo Study

doi:10.21203/rs.3.rs-2075504/v1

Download PDF

Research Article

Study of the Blume-Emery-Griffiths Model for Mixed Carbon-Like Nanotube: Monte Carlo Study

https://doi.org/10.21203/rs.3.rs-2075504/v1

This work is licensed under a CC BY 4.0 License

You are reading this latest preprint version

Monte Carlo study under the Metropolis algorithm is performed to investigate the ground phase diagrams and hysteresis electric cycles by using the Blume-Emery-Griffiths (BEG) model with the mixed spins (S-1, σ-3/2). Firstly, the ground state phase diagram has been established to show the more stable configurations corresponding to physical parameter E_Z/J_C. Moreover, the hysteresis electric cycle behaviors have been investigated by varying temperature, exchange coupling interactions, crystal field and biquadratic parameter of the carbon-like nanotube. It is found that the increase of the crystal field and the biquadratic exchange parameters decreases the surface of the loops leading to the apparition of the polarization plateaus.

Carbon nanotube

Dielectric properties

Monte Carlo simulations

Hysteresis electric cycles

Blume-Emery-Griffiths Model

Currently, nanotechnology has become one of the richest fields of scientific research, focusing its growing interest on different nano-magnetic materials due to their unique magnetic properties that have enabled multiple applications [1, 2]. Moreover, the magnetic nanostructures are at the base of the latest scientific research [3–5]. Among the nanostructures that have attracted the attention of researchers is the structure of nanotubes [6, 7]. Magnetic nanotubes are important nanostructures for researchers because of their exceptional characteristics [8–10]. Several techniques have discovered so far for producing carbon nanotubes. Such techniques for producing carbon nanotubes are arc discharge [11], laser ablation [12], high-pressure carbon monoxide disproportionation [13] and chemical vapor deposition [14, 15]. Given its intense investigations, the Blume-Emergy Griffiths model [16], is an important model in the Ising family of spin-1 with bilinear (J) and biquadratic (k) nearest-neighbor interactions and crystal field [17–21]. This model was first introduced by Blume-Emergy Grifits to describe the thermodynamic properties of ${\text{H}\text{e}}^{3}$_${\text{H}\text{e}}^{4}$ mixture [22, 23]. Subsequently, the model has been applied to study thermodynamic behavior such as meta-magnetic systems, ternary alloys and fluids [24–27]. Indeed, the Blume-Emergy Grifits model has been studied by various theoretical methods such as Monte Carlo simulations, mean field approximation, effective field theory, renormalization group method and so on [28–31]. The ground state phase diagram of this model is obtained in a longitudinal magnetic field for a single ion potential [32]. Then applying Monte Carlo Simulation, in recent papers, the authors have paid attention to the study of thermodynamic properties and showing the effect of several physical parameters on several hysteresis loops [33–38].

The originality of this work lies in the investigation of the hysteresis electric cycles and ground phase diagrams in a Blume-Emery-Griffiths (BEG) model for a mixed (S-1,σ-3/2) carbon-like nanotube using Monte Carlo simulations under the Metropolis algorithm. The outline of this work is the following after an introduction, in section 2 the model and the method used are evoked and in subsection 3.1, the ground state phase diagrams are discussed. Subsection 3.2 is devoted to hysteresis electric cycle. Finally, section 4 concludes the paper.

In the present work, we investigate the hysteresis electric cycles of the carbon-like nanotube system. The total number of the carbon-like nanotube spins is N = 128 (see Fig. 1) where N_S = 64 and N_σ = 64 and with mixed spins (S = ± 1, 0, σ =±3/2, ± 1/2). The carbon-like nanotube system is studied with free boundary conditions. Our data were generated with 10⁶ Monte Carlo steps per spin, discarding the first 10⁵ Monte Carlo steps of the simulations for balancing the nanotube system.

The Hamiltonian of a carbon-like nanotube is given by:

$$\begin{array}{c}H=-{\text{J}}_{\text{S}}\sum _{<\text{i},\text{j}>}{\text{S}}_{\text{i}}^{\text{Z}}{\text{S}}_{\text{j}}^{\text{Z}}-{\text{J}}_{\text{C}}\sum _{<\text{k},\text{l}>}{{\sigma }}_{\text{k}}^{\text{Z}}{{\sigma }}_{\text{l}}^{\text{Z}}-{\text{J}}_{\text{C}\text{S}}\sum _{<\text{m},\text{n}>}{\text{S}}_{\text{m}}^{\text{Z}}{{\sigma }}_{\text{n}}^{\text{Z}}-{\text{K}}_{ }\sum _{<\text{i},\text{j}>}{(\text{S}}_{\text{i}}^{\text{Z}}\left){ }^{2}{({\sigma }}_{\text{j}}^{\text{Z}}\right){ }^{2}-2{\text{E}}_{\text{Z}}(\sum _{\text{i}}{\text{S}}_{\text{i}}^{\text{Z}}+\sum _{\text{j}}{{\sigma }}_{\text{j}}^{\text{Z}})\\ -{\text{D}}_{\text{S}}\sum _{\text{i}}{(\text{S}}_{\text{i}}^{\text{Z}}){ }^{2}-{\text{D}}_{{\sigma }}\sum _{\text{j}}{({\sigma }}_{\text{j}}^{\text{Z}}){ }^{2} \left(1\right)\end{array}$$

where J_S and J_C are the ferroelectric exchange coupling interaction parameters between the nearest neighbor atoms with spins ${\text{S}}_{\text{i}}^{\text{Z}}{-\text{S}}_{\text{j}}^{\text{Z}}$ and ${{\sigma }}_{\text{k}}^{\text{Z}}{{\sigma }}_{\text{l}}^{\text{Z}}$, respectively. While J_CS is the ferrielectric exchange coupling interaction between the nearest neighbor of the core and the shell atoms with spins ${\text{S}}_{\text{m}}^{\text{Z}}{{\sigma }}_{\text{n}}^{\text{Z}}$. The parameter K is the biquadratic exchange interaction.

${\text{S}}_{\text{i}}^{\text{Z}}=\pm 1 \text{a}\text{n}\text{d} 0$ , ${ {\sigma }}_{\text{j}}^{\text{Z}}= \pm \frac{3}{2} \text{a}\text{n}\text{d}\pm \frac{1}{2}$ are the spin components in the z-direction. The parameter µ stands for the dipole moment. For simplicity, we take µ = 1 as in Ref. [39, 40]. E_Z is the external longitudinal electric field. The crystal fields D_S and D_σ are acting on the spin atoms ${\text{S}}_{\text{i}}^{\text{Z}}$ and${ {\sigma }}_{\text{j}}^{\text{Z}}$, respectively. We will limit our study by taking D = D_σ=D_S.

The internal energy per site is:

$$\text{E}=\frac{⟨\mathcal{H}⟩}{{\text{N}}_{\text{S}}+{\text{N}}_{{\sigma }}}$$

The partial and total polarizations of a carbon-like nanotube structure are:

$${\text{P}}_{\text{S}}=\frac{1}{{\text{N}}_{\text{S}}}{\sum }_{\text{i}}{\text{S}}_{\text{i}}^{\text{Z}}$$

$${\text{P}}_{{\sigma }}=\frac{1}{{\text{N}}_{{\sigma }}}{\sum }_{\text{j}}{ {\sigma }}_{\text{j}}^{\text{Z}}$$

$${ \text{P}}_{\text{t}\text{o}\text{t}}=\frac{{\text{N}}_{\text{S}}{\text{P}}_{\text{s}}+{\text{N}}_{{\sigma }}{\text{P}}_{{\sigma }}}{{\text{N}}_{\text{S}}+{\text{N}}_{{\sigma }}}$$

We discuss in this section the obtained results using the Monte Carlo simulations under the Metropolis algorithm. Indeed, we start by the investigation of the ground state phase diagrams for a zero absolute temperature in the sub-section 3.1. Besides, for a non-zero temperature, we examine the behavior of the hysteresis electric cycles by varying temperature, exchange coupling interactions, crystal field and biquadratic exchange interaction in the sub-section 3.2.

3.1. Ground state phase diagrams

In the present study, we report the ground-state phase diagrams of a Carbon-like nanotube in the framework of a Blume-Emery-Griffiths model with mixed spins S -1 (in shell) and σ- 3/2 (in core). We use the Hamiltonian of Eq. (1) For simulating the energy of the (2S + 1) ⋅(2σ + 1) = 3 ⋅ 4 = 12 possible configurations for the ground state study.

Figure 2a is obtained in the (D/J_C, E_Z/J_C) plane, in the absence of biquadratic exchange interaction for fixed values of the reduced exchange coupling J_S/J_C=1 and J_CS/J_C =-1. This figure exhibits only 10 stable configurations. In this phase diagram, there is a perfect symmetry of the configurations regarding the reduced external longitudinal electric field axis E_Z/J_C= 0. These stable configurations are: (+ 1/2, 0); (-1/2, 0); (+ 1/2, + 1); (-1/2, -1); (+ 1/2, -1); (+ 1/2, 0); (+ 3/2, + 1); (-3/2, -1); (+ 3/2, -1); (-3/2, + 1).

3.2. Hysteresis electric cycle

In this section, we use Monte Carlo simulations (MCS) under the Metropolis algorithm in order to detect the effect for several values of the reduced parameters on the hysteresis electric cycles of the mixed Carbon-like nanotube.

We present in Fig. 3 the partial (P_S and P_σ) and total (P_tot) polarizations versus the reduced external longitudinal electric field E_Z/J_C, by varying temperature parameter T/J_C = 0.1, 1, 2, 3, 4 and 7. This figure is obtained in the absence of the reduced crystal field (D/J_C=0) and the biquadratic exchange interaction (K/J_C=0) and for fixed reduced parameters: J_S/J_C =1 and J_CS/J_C = -1. Moreover, it is found that the surface loops decrease when increasing the temperature values and completely disappear for T/J_C=7 reaching the paraelectric phase (P_tot=0). This result can be interpreted by the competition between the temperature and the external longitudinal electric field which tends to align the spins while the effect of the temperature is to disorganize them.

In Fig. 4, we investigate the effect of the exchange coupling parameter (J_S/J_C) on the hysteresis cycles for different values of J_S/J_C = 1, 2, 3, 4 and 5. Such a figure is illustrated for fixed parameters: T/J_C =0.1, J_CS/J_C = -1, D/J_C=0 and K/J_C=0. As expected, increasing the exchange coupling parameter increases the coercive field and the surface loops.

Otherwise, we present in Fig. 5 the hysteresis electric cycles of the mixed Carbon-like nanotube. It is plotted for several values of the ferrielectric parameter J_CS/J_C = − 1,-2, -3, -4 and − 5, for fixed parameters: T/J_C =0.1, J_S/J_C = 1, D/J_C=0 and K/J_C=0. This figure depicts that the increase of the parameter | J_CS/J_C | increases the electric coercive field resulting in an increase of the surface loops. Whereas, the electrical remanent remains invariable, as a result of the different dielectric properties of both core and shell. The system approaches the ferroelectric phase as it becomes easier to align the core and shell spins in the same direction along the applied field.

To complete this study, we plot in Fig. 6 the effect of the crystal field on the partial (P_S and P_σ) and total (P_tot) hysteresis electric cycles for: T/J_C =0.1, J_S/J_C = 1, J_SC /J_C= -1 and K/J_C=0. From this figure, it is obvious that the increase in the absolute value of the crystal field | D/J_C | decreases the surface of the loops leading to the apparition of four polarization plateaus and five electric cycle loops.

Finally, we investigate in Fig. 7 the effect of the biquadratic exchange interaction (K/J_C) on the partial (P_S and P_σ) and total (P_tot) hysteresis electric cycles for: K/J_C = 0, -1, -2 and − 3 in the absence of reduced crystal field (D/J_C=0) and for fixed parameters: T/J_C =0.1, J_S/J_C = 1, J_SC /J_C= -1. This figure exhibits the central cycle at strong | K/J_C = -3 | and reports that the core/shell behavior in the Carbon-like nanotube structure approaches the ferrielecrtic behavior. Due to the different ferrielectric properties in the core and shell, there are two steps that appear in the central loop. Besides, it can be observed for a low value of the intermediate reduced coupling, that it is more difficult to align the core and shell spins in the same direction along the applied external electric field, and the behavior of the core and shell in the Carbon-like nanotube approaches the antiferrielectric behavior. Whereas, it is found that the increase in the absolute value of the biquadratic exchange interaction | K/J_C | decreases the surface loops leading to the apparition of one polarization plateaus and three electric cycle loops.

In the present work, we have investigated the ground state phase diagrams and the hysteresis electric cycles of the Carbon-like nanotube composed of mixed spins S-1 and σ-3/2, using Monte Carlo simulations under the Metropolis algorithm, in Blume-Emery-Griffiths (BEG) model. The ground state phase diagrams exhibit the stable configurations in different physical parameter plane (D/J_C, E_Z/J_C). The hysteresis electric cycle behaviors have been investigated by varying temperature, exchange coupling interactions, crystal field and biquadratic parameter of the carbon-like nanotube. It is found that the increase of the absolute value of the crystal field and the biquadratic exchange parameters decreases the surface of the loops leading to the apparition of the polarization plateaus and electric cycle loops.

R. Masrour, L. Bahmad, M. Hamedoun, A. Benyoussef, E.K. Hlil, Phys. Lett. A 378 (2014) 276.
N.E. Kazantseva, Y.I. Bespyatykh, I. Sapurina, J. Stejskal, J. Vilcakova, P. Saha, Journal of Magnetism and Magnetic Materials. 301 (2006) 155.
N.E. Kazantseva, J. Vilcakova, V. Kresalek, P. Saha, I. Sapurina, J. Stejskal, Journal of Magnetism and Magnetic Materials. 269 (2004) 30.
R. Moučka, J. Vilčáková, N.E. Kazantseva, A.V. Lopatin, P. Sáha, Journal of Applied Physics. 104 (2008) 103718.
Ö. Yavuz, M.K. Ram, M. Aldissi, P. Poddar, S. Hariharan, Journal of Materials Chemistry. 15 (2005) 810.
R. F. Gibson, Composite Structures, 92 (2010) 2793-2810.
N. Maaouni, Z. Fadil, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benoumar, Solid State Communications, 321 (2020) 114047.
C. Goze-Bac, S. Latil, P. Lauginie, V. Jourdain, J. Conard, L. Duclaux, A. Rubio, P. Bernier, Carbon, 40 (2002) 1825-1842.
J.H. Lehman, M. Terrones, E. Mansfield, K.E.Hurst, V. Meunier, Carbon, 49 (2011) 2581-2602.
Q. Wang, V. K. Varadan, International Journal of Solids and Structures, 43 (2006) 254-265.
M. I. Mohammad, A. A. Moosa, J. H. Potgieter and M. K. Ismael. Hindawi ISRN Nanomaterials, 2013 (2013) 1-7.
J. Chrzanowska, J. Hoffman, A. Małolepszy, M. Mazurkiewicz, T. A. Kowalewski, Z. Szymanski, L. Stobinski, Basic solid state physics, 252 (2015) 1860–1867.
H. Dai, A.G. Rinzler, P. Nikolaev, A. Thess, D.T. Colbert, R.E. Smalley, Chem. Phys. Lett. 260 (1996) 471–475.
N. Inami, M. A. Mohamed, E. Shikoh, A. Fujiwara. Science and Technology of Advanced Materials, 8 (2007) 292–295.
Sayangdev Naha, Ishwar K. Puri. Journal of Physics D: Applied Physics. 41 (2008) 065304.
M. Blume, J.V. Emery, R.B. Griffiths, Phys. Rev. A 4 (1971) 1071.
A. Kortsev, M. Augustin, R. F. L. Evans, K. S. Novoselov and E. J. G. Santos, npj comput Mater 6 (2020) 150.
P.B. Ergueta and A. H. Nevidomskyy, physical Review B, 92 (2015) 165102.
D. P. Snowman, Journal of Magnetsim and Magnetic Materials, 314 (2007) 69-74.
M. Ertas, O. Canko, M. Keskin, Journal of Magnetism and Magnetic Materials, 320 (2008) 1765-1774.
M. Yezli, S. Belhechi, F. Hontinfinde, H. Ez-zahraouy physica A statistical Mechanics and its applications 448 (2016) 81-90
J. kple, F. Hontinfinde, E. Al bayrak, Journal of magnetism and magnetic Materials 537 (2021) 168217.
A. Jabar, A.Belhaj, H. L. Labrim, L. Bahmad, N. Hassanain, A. Benyoussef, Superlattices and Microstructures, 78 (2015) 171-182.
M. Keskin, O. Canko, M. Kirak, Basic solid-state physics 244 (2007) 3775.
E. Obradó, C. Frontera, L. Mañosa, A. Planes, Phys. Rev. B, 58 (1998) 14245.
E. Costabile, J.R. Viana, J.R. de Sousa, J.A. Plascak, Phys. A 393 (2014) 297.
C. Ekiz, Commun. Theor. Phys. 67 (2017) 192.
S. S. Ahmed, L. Bahmad, A. El yousfi, A. Benyoussef, A. El Kenz, A. G. EL Hachimi, Superlattices and Microstructures, 123 (2018) 1-11.
A. Benyoussef, A. El Kenz, M. El yadari, M. J. Condens. Matter. 8 (2007) 72.
C. C. Loois, G.T. Barkema, C.M. Smith, Phys. Rev. B 78 (2008) 184519.
W. Selke, C. Ekiz, J. Phys. Condens Matter. 23 (2011) 496002
Ü. Temize, E. Kantar, M. Keskin, O. Canko, Journal of Magnetism and Magnetic Materials, 320 (2008) 1787-1801.
M. Qajjour, N. Maaouni, Z. Fadil, A. Mhirech, B. Kabouchi, W. Ousi Benomar, L. Bahmad, Chin. J. Phys. 63 (2020) 36.
Z. Fadil, M. Qajjour, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, Phys. B Condens. Matter 564 (2019) 104.
M. Qajjour, N. Maaouni, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, J. Magn. Magn Mater. 482 (2019) 312.
N. Maaouni, M. Qajjour, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, J. Magn. Magn Mater. 468 (2018) 175.
N. Maaouni, M. Qajjour, Z. Fadil, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, Physica B: Condensed Matter, 2019, 566, pp. 63–70
Z. Fadil, N. Maaouni, M. Qajjour, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, Physica B: Condensed Matter, 2020, 578, 411852
Y. Benhouria, I. Essaoudi, A. Ainane, R. Ahuja, F. Dujardin, Phys. A Stat. Mech. Appl. 506 (2018) 499.
Z. Fadil, A. Mhirech, B. Kabouchi, L. Bahmad, W. Ousi Benomar, Superlattices and Microstructures, 2019, 135, 106285

No competing interests reported.

Download PDF

Editorial decision: Major revision
10 Oct, 2022
Reviews received at journal
09 Oct, 2022
Reviewers agreed at journal
28 Sep, 2022
Reviewers invited by journal
26 Sep, 2022
Editor assigned by journal
22 Sep, 2022
Submission checks completed at journal
20 Sep, 2022
First submitted to journal
17 Sep, 2022

You are reading this latest preprint version

Study of the Blume-Emery-Griffiths Model for Mixed Carbon-Like Nanotube: Monte Carlo Study

Status:

Version 1

Abstract

Figures

1. Introduction

2. Model And Method

3. Numerical Results And Discussion

3.1. Ground state phase diagrams

3.2. Hysteresis electric cycle

4. Conclusion

References

Additional Declarations

Status:

Version 1