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

Monte Carlo study under the Metropolis algorithm is performed to investigate the ground state 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 the physical parameter EZ/JC. Moreover, the hysteresis electric cycle behaviors have been investigated by varying temperature, exchange coupling interactions, crystal field and biquadratic parameters of the carbon-like nanotube. It is found that the increase in the crystal field and the biquadratic exchange parameters decrease the surface of the loops leading to the apparition of the polarization plateaus.


Introduction
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, magnetic nanostructures are at the base of the latest scientific research [3][4][5]. Among the nanostructures that have attracted the attention of researchers, there is the structure of nanotubes [6,7]. Magnetic nanotubes are important nanostructures for researchers because of their exceptional characteristics [8][9][10]. Several techniques have been 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-Emery-Griffiths (BEG) model [16] is an important model in statistical physics for the analysis of the phase transition phenomenon [17]. This model is based on the Ising family with bilinear (J) and biquadratic (K) nearest-neighbor interactions and a crystal field interaction, see Refs. [18][19][20][21][22]. This model was first introduced by Blume-Emery-Griffiths to describe the thermodynamic properties of He 3 _He 4 mixture [23,24]. Subsequently, the model has been applied to study thermodynamic behaviors such as meta-magnetic systems, ternary alloys and fluids [25][26][27][28][29][30]. Indeed, the Blume-Emery-Griffiths model has been studied by various theoretical methods such as Monte Carlo simulations, mean field approximation, effective field theory and renormalization group method [31][32][33][34]. The ground state phase diagram of this model is obtained in a longitudinal magnetic field for a single ion potential [35]. 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 [36][37][38][39][40][41]. Despite these previous studies and as far as we are acquainted, the effect of the biquadratic exchange interaction on the dielectric properties of a mixed spin (S-1, σ-3/2) on a carbon-like nanotube has not previously been regarded. Thus, 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 Sect. 2, the model and the method used are evoked, and in Sect. 3.1, the ground state phase diagrams are discussed. Section 3.2 is devoted to the hysteresis electric cycle. Finally, Sect. 4 concludes the paper.

Model and Method
In the present work, we investigate the electric hysteresis 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. Concerning the statistical accuracy and error estimation, we usually apply the Jackknife method [42,43], in our simulations. The data were generated with 10 6 Monte Carlo steps per spin. We eliminate the first 10 5 iterations to reach the equilibrium.
The Hamiltonian of a carbon-like nanotube is given by: where J S and J C are the ferroelectric exchange coupling interaction parameters between the nearest-neighbor atoms with spins S Z i −S Z j and σ Z k − σ Z l , respectively. (1) Journal of Low Temperature Physics (2023) 210:285-296 While J CS is the ferrielectric exchange coupling interaction between the nearest neighbor of the core and the shell atoms with spins S Z m − σ Z n . The parameter K is the biquadratic exchange interaction.
S Z i = ±1and0 , σ Z j = ± 3 2 and ± 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 Refs. [44,45]. E Z is the external longitudinal electric field. The crystal fields D S and D σ are acting on the spin atoms S Z i and σ Z j , respectively. We will limit our study by taking The internal energy per site is as follows: The partial and total polarizations of a carbon-like nanotube structure are as follows:

Numerical Results and Discussion
In this section, we discuss the obtained results using the Monte Carlo simulations under the Metropolis algorithm. Metropolis method was introduced in condensed matter physics by Metropolis et al. in 1953 [46], for the simple case where the spins take the values -1 and 1. This algorithm consists of constructing a chain of spin states. A state is obtained from the previous one in the following way: Random selection of a spin and calculation of the energy variation ΔE which would result from the change of sign of this spin.
If ΔE ≤ 0, the spin sign is changed. If ΔE > 0, a random number r is drawn with a uniform probability in the interval [0,1].
If r < exp(-βΔE): The spin is changed. Otherwise, the spin is not changed. For each new configuration, the total magnetization is calculated. Indeed, we start by investigation of the ground state phase diagrams for a zero absolute temperature in the Sect. 3.1. Besides, for a nonzero temperature, we examine the behavior of the hysteresis electric cycles by varying temperature, exchange coupling interactions, crystal field and biquadratic exchange interaction in Sect. 3.2.

Ground State Phase Diagram
In the present study, we report the ground state phase diagram 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.

Hysteresis Electric Cycle
In this section, we use Monte Carlo simulations (MCS) under the Metropolis algorithm to detect the effect of 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 and T/J C = 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 J S /J C = 1 and J S /J C = 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. Indeed, the increase in coupling parameter maintains the system in its ferrimagnetic phase.
Otherwise, we present in Fig  the electric coercive field increasing 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.
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 observed 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.
To complete this study, 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 and K/J C = −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 value of | K/J C | (|K/J C = 3|) and reports that the core/shell behavior in the carbon-like nanotube structure approaches the ferrielectric phase. Due to the different ferrielectric properties in the core and shell, two steps 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 plateau and three electric cycle loops.
Finally, we present Fig. 8 to highlight the effect of the exchange coupling parameter J S /J C , the ferrielectric parameter J CS /J C , the crystal field parameter D/ J C and the biquadratic parameter K/J C . Indeed, in Fig. 8a and 8b, we illustrate the transition temperature as a function of the exchange coupling parameters J S /J C and |J CS /J C |, respectively. From these figures, we see clearly that when increasing these reduced parameters, the transition temperature (T C ) of the system increases. This can be explained by the fact that bonds between atoms become very important when the value of the J S /J C and/or |J CS /J C | increases. This means that the high value of the parameters J S /J C and/or |J CS /J C | require a higher temperature to cancel the total polarization, leading to an increase in the transition temperature.
In Fig. 8c and d, we report the transition temperature behavior versus the crystal field D/J C and the biquadratic parameter k/J C . These figures show that the critical temperature is independent both of the quadratic coupling parameter and the

Conclusion
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 the Blume-Emery-Griffiths (BEG) model. The ground state phase diagram exhibits the stable configurations in different physical parameter planes (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 in 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.