Critical and hysteresis behaviors for a hexagonal core-shell structure nanowire in the Blume–Emery–Griffiths model

Highlights

• The core–shell structure with spin-3/2 was considered in the Blume–Emery–Griffiths model.

• Mean-field approximation based on the Gibbs–Bogoliubov inequality for free energy was employed.

• The magnetic properties and hysteresis loops for the hexagonal Ising nanowire were examined.

• The ground state and temperature-dependent phase diagrams were calculated.

• Tricritical, isolated critical, critical end-points and at most two compensation points were observed.

Abstract

The magnetic properties and hysteresis loops for the hexagonal Ising nanowire (HIN) with core–shell structure consisting of spin-$\frac{3}{2}$ are considered in the Blume–Emery–Griffiths (BEG) model by using the mean-field approximation (MFA) based on the Gibbs–Bogoliubov inequality for the free energy. The effects of various bilinear and biquadratic interaction parameters between the core, between the shell, and between the core and shell spins are considered for the phase diagrams of the model when temperature $T=0$ and $T\ne 0$ in addition to the crystal and external magnetic fields effects. The numerical calculations reveal that the model yields first- and second-order phase transition lines and tricritical points. In addition to the isolated critical and critical end-points, the model exhibits at most two compensation temperatures and interesting multiple hysteresis magnetic loops behaviors strongly dependent on the model parameters.

Introduction

Nanoparticles are of great technical and research interest because of their interesting magnetic properties in relation to their small size and contact surface [1], [2], their capacity to store electrical energy [3], [4], their permanent magnetic behaviors [5], [6] and ability for high-density information storage [7], [8]. Moreover, they have awesome applications in medical field as catalysts, biosensors and drug delivery [9], [10], [11], [12], [13], [14], [15], [16]. The nanoparticles are usually considered in the form of nanowires or nanotubes within the core–shell structure and various forms of materials with different spins are used to achieve their desired properties [17].

With the development of nanotechnology, the synthesization of nanoparticles with controllable properties became possible. The ZnZnO nanotube complex was synthesized by a solid-vapor process: the ZnO shell has an epitaxial orientation relationship with the Zn crystal core, and the sublimation of Zn leads to the formation of ZnZnO nanotubes [18]. Similarly, series of FeNi magnetic nanotubes were fabricated using the pore wetting template method [19]. One-dimensional mesoporous ${\text{SiO}}_{\text{2}}$ was used as an effective template to obtain the ${\text{SnO}}_{\text{2}}$ nanotube in controllable form by hydrothermal method and it was shown that the ${\text{SnO}}_{\text{2}}$ nanotubes, as the anode material in the lithium ion battery, show better performance than the ordinary ${\text{SnO}}_{\text{2}}$ nanoparticles [20].

Theoretically, several studies on the magnetic properties of nanowires have been carried out using various calculation methods such as: the MFA [21], [22], effective-field theory (EFT) with correlations [23], [24], [25], [26], [27], [28], [29], [30] and Monte Carlo simulation (MCS) [31], [32], [33]. In addition to these, we can also quote the works as follows: The phase diagrams and the variations of magnetizations was examined in the transverse Ising nanowire with core–shell structure within the EFT [34]. The magnetic properties, compensation temperature, and phase diagrams of ferrimagnetic cylindrical transverse Ising nanotube with a negative core–shell interaction were considered in the EFT [25], [35], [36]. The crystal field effects were considered for the surface shell of a cylindrical mixed spin-$\frac{1}{2}$ core and spin-1 shell Ising nanotube via the EFT with correlations and, the critical and tricritical points of the model were obtained [37]. The magnetic properties and phase diagrams of a cylindrical spin-$1$ Ising nanotube were analyzed by using the MCS [38] with the observation of compensation behaviors and triple hysteresis loops. Note that the compensation temperatures appear in ferrimagnetic systems for certain conditions, and have potential applications in thermomagnetic recording and magneto-optical storage media [39], [40]. The mixed-spin HIN with core–shell structure was examined by using the MFA based on the Gibbs–Bogoliubov inequality for the free energy [41] and it was shown that the system presents first- and second-order phase transitions, compensation behavior, and critical end points [42]. The magnetic properties and compensation temperatures of a mixed monolayer Coronene-like nanostructure were investigated by using the MCS and interesting magnetic properties of the studied model was revealed [43].

Additionally, spin-$\frac{3}{2}$ was also considered for the construction nanowires in various forms: The magnetic properties of Blume–Capel model for the HIN with core–shell structure were studied by using the EFT with correlations [44]. Dynamic magnetic properties of the model on a cylindrical nanowire (CIN) in an oscillating magnetic field were investigated within the MFA by utilizing Glauber-type stochastic dynamics [45]. The dynamic phase diagrams depending on the frequency of an oscillating magnetic field and the dynamic hysteresis properties of the kinetic CIN system were studied by means of the mean-field theory based on Glauber-type stochastic dynamics [46]. The magnetization loops of the HIN were examined in the framework of the EFT with correlations based on the differential operator technique [47]. The hysteresis and compensation behaviors of the CIN were considered within the framework of the EFT with correlations and the thermodynamic quantities, such as the total magnetization, hysteresis curves, and compensation behaviors of the system were obtained [48]. It is clear from the above works that the spin-$\frac{3}{2}$ model has not received the well-deserved attention especially for the BEG model which was initially introduced to describe the separation of super fluid phase in the mixed ${}_2^3\text{He}{-}_2^4\text{He}$ system [49].

In this work, the magnetic properties and phase diagrams of the HIN consisting of spin-$\frac{3}{2}$ was examined in the BEG model using the MFA with the Gibbs–Bogoliubov inequality for the free energy. The model contains many parameters including the bilinear ($J_{\mu }$) and biquadratic ($K_{\mu }$) exchange interaction parameters between the core, shell, and core–shell spins, allowing possible values of crystal field at the core and shell spins (${\Delta }_{\mu }$) and equal external magnetic field ($h$) at all sites. Many aspects of the model, such as the ground state phase diagrams ($T=0.0$) and the phase diagrams ($T\ne 0.0$) on the possible planes of our system parameters and magnetic hysteresis loops, were considered in great detail.

The remainder of this work is set up as follows: In Section 2, the model and its MFA formalism are presented. In Section 3, we expose and discuss in detail the numerical findings. In final Section, we conclude our investigation.