Magnetic Properties and Thermal Behavior of the Monolayer Rubrene-Like Nano-island: Monte Carlo Simulations

doi:10.21203/rs.3.rs-2336809/v1

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Magnetic Properties and Thermal Behavior of the Monolayer Rubrene-Like Nano-island: Monte Carlo Simulations

https://doi.org/10.21203/rs.3.rs-2336809/v1

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Using the Monte Carlo simulations (MCs), under the Metropolis algorithm in the frame of the Blume Capel model, the magnetic properties of the mixed Rubrene-like nano-island have been extensively investigated. For zero temperature, the ground state phase diagrams have been established to display the more stable spin configurations. For no null temperature values, the diverse parameters effects of the studied system on magnetizations and susceptibilities have been investigated. Interesting results have been found, especially, the apparition of the compensation temperature and triple loops behavior which is useful in many applications in multi-state memory systems.

Rubrene-like Nano-island

Monte Carlo simulations

Compensation temperature

Exchange coupling interactions

Ground state Phase diagrams

Hysteresis cycles

Recently, the organic semiconductors have attracted enormous attentions due to their great potential applications in optoelectronic devices, organic light-emitting diodes (OLEDs), organic field-effect transistors (OFETs), organic transistors and organic solar cells (OSCs) [1–5]. Furthermore, the Rubrene with formula (C₄₂H₂₈, 5, 6, 11, 12 - tetraphenylnaphtacene) is one of the most researched organic semi-conductor materials for their remarkable charge carry mobility and luminous efficiency [6–10]. In fact, several studies have been performed to show all the properties of the Rubrene structure experimentally and theoretically.

Experimentally, researchers have conducted many interesting studies on the synthesis of Rubrene using different techniques. Ruidong et al. [11] prepared the Rubrene: MoO₃ film by the thermal evaporation technique. It has been found that the h

ole concentration and the hole carrier mobility are enhanced. Recently, Sartag et al. [12] have prepared the organic-inorganic heterostructure (OIHs) based on organic Rubrene and inorganic topological insulator Bi₂Se₃ by using simple physical vapor deposition. They have found that these films are suitable candidates for the fabrication of light-performance organic-inorganic heterostructure photodetectors (OIHPDs). In addition, Kenta et al. [13] have measured the phonon dispersion of the organic semiconductor Rubrene by the inelastic x ray scattering at room temperature. Thereby, in theory, researchers have also investigated in detail the photoelectric, the electronic, the optical, the transport, the elastic and the mechanical properties of Rubrene based on the density functional theory (DFT) calculations [14–19]. On the other hand, Al-muntaser et al. [20] have studied the role of the Rubrene semiconductor additive for reinforcing the optical and structural properties of the polyvinyl alcohol films (PVA) towards optoelectronic applications. Moreover, Collin et al. [21] have investigated the magnetic field effect on singlet-exciton fission to demonstrate spatial imaging of magnetic fields in a thin film of Rubrene. Besides, the magnetic properties of different structures have recently been studied using Monte Carlo simulations (MCs) such as borophene nanoribbons with core-shell [22], Graphene-Like Nanoribbon [23], a checkerboard square [24] on honeycomb [25] and Tri-layer Graphyne-Like Structure [26].

On the other hand, it is important to note that the Ising model is a classical model that has been used extensively to examine ferrielectric or ferrimagnetic systems, such as the anti-ferroelectric/ferroelectric BiFeO₃/YMnO₃ bilayer system [27], the Gd₂O₃ nanorods/nanowire [28], the monolayer nano-graphyne structure [29] and the ferrielectric nanowires [30]. From the perspective of scientific research or technology, the study of the Ising systems is crucial. Furthermore, a significant amount of attention has been shown in the mixed-spin (1, 7/2) Ising system in recent years as one of the most popular Ising models. To our knowledge, no studies have been focused to investigate the hysteresis cycles of the Rubrene-like nano-island with a mixed spins (S = 1, σ = 7/2). The hysteresis behavior is a model of metastability with experimentally more available time scales. Moreover, the hysteresis cycle is an easy occurrence under simple conditions making it not only technologically interesting phenomena, but also a theoretical challenge.

The aim of this work is to study the magnetic properties of the Rubrene-like nano-island formed by the mixed spins (S = 1 and σ = 7/2), using the Monte Carlo simulations under Metropolis algorithm. Among others, we obtained the compensation temperature between spins S -σ, and discuss the hysteresis cycle behaviors. It is worth mentioning that in our previous work, MC simulation was successfully used to investigate the magnetic, dielectric properties of diverse nanostructures [31–35]. In the studied nano-island, we have replaced the carbon atoms with spins S = 1 and the hydrogen atoms with spins σ = 7/2. In fact, the spin S = 1 corresponds to Ni (II) ions [36], while the spin σ = 7/2 corresponds to Gd (III) ions [37].

This paper is organized as follows: In Section 2, we present the formalism in the framework of the MCs. We elaborate the used expressions of the studied physical parameters during the simulations. Our results are illustrated in Section 3, where we present the ground state phase diagrams in sub-section 3.1, the magnetic properties and the hysteresis behaviors are discussed in sub-section 3.2. Finally, in Section 4, we summarize our conclusions.

Using Monte Carlo simulations and Metropolis algorithm, we explore the magnetic behavior of the Rubrene-like nano-island [38–42] that is generated by mixed spins S and σ in the Blume Capel model. The case of the free boundary conditions is taken into consideration in this study.

According to Fig. 1, the nano-island has a total of N_T = N_S + N_σ spins, where N_S is made up of 48 atoms and N_σ is made up of 28 atoms. To attain the equilibrium in our simulations, the first 10⁵ iterations of the Monte Carlo step were discarded and the data were created with 10⁶ configurations.

The Hamiltonian of this nano-system with different spins includes nearest-neighbor interaction, the external magnetic field H and the crystal field D, can be written as follows:

$$\begin{array}{c} H=-{\text{J}}_{\text{S}\text{S}}\sum _{<\text{i},\text{j}>}{\text{S}}_{\text{i}}{\text{S}}_{\text{j}}-{\text{J}}_{\text{S}{\sigma }}\sum _{<\text{i},\text{j}>}{\text{S}}_{\text{i}}{{\sigma }}_{\text{j}}-H\left(\sum _{\text{i}}{\text{S}}_{\text{i}}+\sum _{\text{j}}{{\sigma }}_{\text{j}}\right)- D\left(\sum _{\text{i}}{{\text{S}}_{\text{i}}}^{2}+\sum _{\text{j}}{{{\sigma }}_{\text{j}}}^{2}\right)\\ \end{array} \left(1\right)$$

Where, the $<\text{i},\text{j}>$ notation indicates that the summation runs over only the first nearest neighbor spins. In addition, J_SS and J_Sσ are the exchange coupling interactions among the first-neighbor atoms with spins S–S and S–σ, respectively. H is the external magnetic field. D is the crystal field acting on the all system spins. The crystal field acting on the spins S and σ is assumed to be identical. The spin moments are: S (±1, 0) and σ( ±7/2, ±5/2, ±3/2, ±1/2). In the present work, we determine the internal energy per site as:

$$\text{E}=\frac{1}{{\text{N}}_{\text{T}}}⟨\mathcal{H}⟩ \left(2\right)$$

Where N_T= 48 + 28 = 76 designates the total number of spins in the studied system.

The partial magnetizations will be calculated as:

$${ \text{M}}_{\text{S}}=⟨\frac{1}{{\text{N}}_{\text{S}}}\sum _{i=1}^{{\text{N}}_{\text{S}}}{S}_{i} ⟩ \left(3\right)$$

$${\text{M}}_{{\sigma }}=⟨\frac{1}{{\text{N}}_{{\sigma }}}\sum _{\text{j}=1}^{{\text{N}}_{{\sigma }}}{{\sigma }}_{\text{j}}⟩ \left(4\right)$$

The corresponding total magnetization is:

$${\text{M}}_{\text{t}\text{o}\text{t}}=\frac{{{\text{N}}_{\text{S}}\text{M}}_{\text{S}}+{{\text{N}}_{{\sigma }}\text{M}}_{{\sigma }}}{{\text{N}}_{\text{S}}+{\text{N}}_{{\sigma }}} \left(5\right)$$

The partial and total magnetic susceptibilities of the system are given by:

$${{\chi }}_{\text{S}}={\beta }\left(<{\text{M}}_{\text{S}}^{2}>-<{\text{M}}_{\text{S}}{>}^{2}\right) \left(6\right)$$

$${{\chi }}_{{\sigma }}={\beta }\left(<{\text{M}}_{{\sigma }}^{2}>-<{\text{M}}_{{\sigma }}{>}^{2}\right) \left(7\right)$$

$${{\chi }}_{\text{t}\text{o}\text{t}}=\frac{{{\text{N}}_{\text{S}}{\chi }}_{S}+{{\text{N}}_{{\sigma }}{\chi }}_{\sigma }}{{\text{N}}_{\text{S}}+{{\text{N}}_{{\sigma }}}_{ }} \left(8\right)$$

Where ${\beta }=\frac{1}{{\text{k}}_{\text{B}}\text{T}}$ $, {\text{k}}_{\text{B}}$is the Boltzmann’s constant. For simplicity, ${\text{k}}_{\text{B}}$ is taken equal 1. T is the absolute temperature.

The compensation temperature (T_comp) is defined as the temperature at which the magnetization of two sub-lattices adds up to zero given a zero-total magnetization of the lattice as given in Eq. 5. In order to calculate T_comp from the magnetization curves, the convergence point of the magnetizations of the two sub-lattices have been identified under the following condition:

$$\left|{M}_{\sigma }\left({T}_{comp}\right)\right|=\left|{M}_{s}\left({T}_{comp}\right)\right| \left(9\right)$$

$$Sign\left({M}_{\sigma }\left({T}_{comp}\right)\right)=-Sign\left({M}_{s}\left({T}_{comp}\right)\right) \left(10\right)$$

With Tcomp < Tc, where Tc is the critical temperature. In this report, it is estimated from the divergence of the susceptibility's curves.

We present the results concerning the ground-state phase diagrams in sub-section 3.1 and those obtained by the Monte Carlo simulations in sub-section.3.2.

3.1. Ground state phase diagrams

In this sub-section, we present the ground-state phase diagrams of the Rubrene-like nano-island in the frame of the Blume-Capel model. We use the Hamiltonian of Eq. (1) to simulate the energy of the (2S + 1) ⋅(2σ + 1) = 3 ⋅ 8 = 24 possible configurations for the ground state study.

Figure 2a is given in the (H, D) plane, for J_SS = 1 and J_Sσ = − 1. It is shown that only 18 configurations are stable among 24 possible ones. In this diagram, there is a perfect symmetry of the configurations regarding the axis H = 0. The stable configurations corresponding to H positive are : (0,+1/2); (+ 1,-1/2); (-1,+3/2); (+ 1,+1/2); (+ 1,+3/2); (+ 1,+5/2); (-1,+5/2); (-1/2,+7/2) and (+ 1,+7/2). While, the stable configurations obtained for negative H values are: (0, − 1/2); (-1, + 1/2); (+ 1, − 3/2); (-1, − 1/2); (− 1, − 3/2); (+ 1, -5/2); (− 1, − 5/2); (+ 1, + 7/2) and (-1, − 7/2).

Figure 2b is given in the plane (H, J_Sσ), in the absence of the external magnetic field (D = 0) and for a fixed value of exchange coupling interaction, J_SS =1. In this plane, only 4 configurations are stable, namely: (-1, + 7/2), (+ 1, -7/2), (-1, -7/2) and (+ 1, + 7/2) corresponding to the maximum spin values.

Figure 2c is established in the plane (D, J_SS) for J_Sσ = -1 and H = 0. This figure exhibits only 10 stable configurations, namely: (0, -7/2); (0, + 7/2); (0, + 1/2); (0, -1/2); (+ 1, -3/2); (-1, + 3/2); (-1, + 1/2); (+ 1, -1/2); (-1, + 7/2); and (+ 1, -7/2).

Figure 2d is obtained in the plane (D, J_Sσ) for the following parameters: Jss = 1 and H = 0. This figure exhibits 18 stable configurations. All stable configurations observed in the present plane are shown in Fig. 2a.

Finally, Fig. 2e presents the phase diagram in the plane (J_SS, J_Sσ) for D = 0 and H = 0. Such figure shows 12 stable configurations: (-1, -7/2); (+ 1, -1/2); (-1, + 7/2); (+ 1, + 7/2); (0, -1/2); (0, + 1/2); (0, -3/2); 0, + 3/2); (0, -5/2); (0, + 5/2), (0, -7/2); and (0, + 7/2). The anti- and ferri-magnetic phases coincide at the point (J_SS = 0, J_Sσ = 0).

3.2. Monte Carlo Simulations

In the current study, we employ Monte Carlo simulations with the Metropolis technique to evaluate how the exchange coupling interactions J_SS and J_Sσ, as well as the external magnetic and crystal fields affect the magnetic properties of the Rubrene-like nano-island.

The behavior of the partial and total magnetizations for J_SS =1, J_Sσ=-0.02, D = 0 and H = 0 is depicted in Fig. 3a. The partial and total magnetizations at very low temperatures are found to be:${\text{M}}_{\text{S}}=+1, {\text{M}}_{{\sigma }}=-\frac{7}{2}, \text{a}\text{n}\text{d} {\text{M}}_{\text{t}\text{o}\text{t}}=\frac{\left(-\frac{7}{2}\times 28\right)+\left(1\times 48\right)}{28+48}=-0.8.$

Hence, the ground state phase diagrams already displayed in Sec. 3.1 show that these magnetizations are compatible with the spin values (S = ±1, 0 and = ±7/2). Also, from Fig. 3a, it is clear that the magnetizations vanish at the critical temperature of each one. Furthermore, the system exhibits a compensation temperature corresponding to zero total magnetization value. In fact, this is due to the competition between several physical parameters (J_Sσ, J_SS, H and D). This compensation temperature is located at T_comp ≈0.11.

Additionally, the Fig. 3b is obtained for identical parameter values as in Fig. 3a. The total magnetic susceptibilities (χ_tot) and the partial ones (χ_S, χ_σ) are shown in this figure. Two peaks are shown for the total susceptibility. The first peak corresponds to the compensation temperature, while the second peak is associated to the critical temperature Tc = 1.30.

We explore Fig. 4 to analyze how the ferrimagnetic exchange coupling parameter J_Sσ affects the compensation and critical temperatures. The thermal total magnetization's behavior for various J_Sσ values (J_Sσ = -0.02, -0.06, and − 0.1) is illustrated in Fig. 4a. This figure is taken for J_SS = 1 and D = 0, H = 0 in the absence of the crystal and external fields.

It is found that the compensation temperature T_comp increases when increasing the parameter |J_Sσ|. Besides, in Fig. 4b, we plot the total susceptibility for the identical parameters selected for Fig. 4a. The total susceptibility exhibits the same behavior as in Fig. 3b. Indeed, the displacement of the susceptibility peaks towards the higher temperature confirms the behavior of the total magnetizations presented in Fig. 4a. The obtained compensation temperature values, for the ferrimagnetic cases are: T_comp≈ 0.11, 0.34 and 0.55 for J_Sσ=-0.02, -0.06 and − 0.1, respectively. In addition, the obtained critical temperature for the same ferrimagnetic cases is T_c ≈1.30 for J_Sσ =-0.02, T_c ≈1.20 for J_Sσ =-0.06 and T_c≈1.15 for J_Sσ=-0.1.

Similarly, in Fig. 5a and 5b, we investigate the thermal behavior of the total magnetization when varying the exchange coupling parameter J_SS. These figures are shown for selected values of the parameter (J_SS=1, 1.2, 1.4 and 1.6) and for a fixed value of the parameters J_Sσ = -0.02 and in the absence of the crystal and external fields (D = 0, H = 0). It seems that the compensation temperature T_comp is not influenced by the variation of the parameter J_SS, while the critical temperature T_c increases when increasing the J_SS parameter. This is due to the low selected values of the physical parameters. The obtained compensation temperature value, for various exchange coupling parameters J_SS is T_comp≈0.11. In addition, the obtained critical temperature for the same values of the parameter J_SS=1, 1.2, 1.4 and 1.6 are T_c≈1.3, 1.45, 1.7 and 1.9, respectively.

Besides, collecting the results obtained in Fig. 5a, we illustrate the effects of the exchange coupling interaction parameter J_SS on the compensation and critical temperatures in Fig. 5c. In fact, we observe that the T_comp remains almost constant when the J_SS increases. While, the critical temperature T_c increases almost linearly as a function of this parameter J_SS. This behavior results from the competition between the several physical parameters acting on the system.

To investigate the behavior of the total magnetization versus the crystal field, we plot Fig. 6 for several values of J_Sσ (J_Sσ=-0.1, -0.2, -0.4, and − 0.8) and the fixed parameter values: J_SS=1, T = 0.1 and H = 0. From this figure, it is found three different regions, namely: region (i): D<-2.65, region (ii): -2.65 < D < 0.17 and region (iii): D > 0.17. In the first region, independently of the ferrimagnetic parameter values (J_Sσ), the system is in its paramagnetic phase. In the second region, the total magnetizations undergo a first transition for several ferrimagnetic parameter values. Finally, in the third region, the total magnetization decrease and undergo a second-order transition, converging towards the saturation value of total magnetization M_sat=-0.8. This value is in good agreement with the ground state phase diagram, see for example Fig. 2e.

Finally, to complete this research, we explore in Fig. 7 (a-c), the effect of the physical parameters, namely: the temperature, the exchange coupling interactions Jss and J_Sσ values on the hysteresis cycle behavior on the Rubrene-like nano-island. These figures are plotted in the absence of the crystal field (D = 0).

The hysteresis cycles are depicted in Fig. 7a for various temperatures: T = 0.1, 0.3, and 3 with fixed parameters J_SS=1 and J_Sσ =-0.02. It can be seen that the surface loops get smaller as temperature values rise and vanish entirely at T = 1.6. We point out that similar results to those in the reference [44, 45] have been attained.

In addition, Fig. 7b presents the variation of the total magnetization as a function of the external magnetic field for several ferrimagnetic parameter values: J_Sσ=-0.02, -0.1 and − 0.8, for fixed parameters Jss = 1 and T = 0.1. From this figure, we conclude that increasing the exchange coupling | J_Sσ | increases the magnetic field which increases the shape of the loops. The presence of the loops is due to the ferrimagnetic behavior of the studied system (J_Sσ <0).

Finally, the Fig. 7c is given for various values of exchange coupling parameter J_SS= 1, 1.4 and 1.6, for fixed ferrimagnetic parameter value: J_Sσ=-0.02 and temperature T = 0.1. This figure exhibits hysteresis cycles with increasing loop surfaces when growing the value of the parameter J_SS.

Based on the Monte Carlo simulations under the Metropolis algorithm, the ground state phase diagrams and the magnetic behaviors of the Rubrene-like nano-island considering the mixed spins (S = 1 and σ = 7/2) Blume Capel Ising model have been studied. Our results highlight that the ground state phase diagrams exhibit the stable configurations in different physical parameter planes. Besides, the effects of the exchange coupling interactions on the compensation and critical temperatures, plotted in figures of the magnetization and the susceptibility have studied. It is found that the compensation temperature value is not affected by the increase of the parameter J_SS. Thus, the critical temperature increases by increasing the parameter J_SS. Likewise, the magnetization as a function of the crystal field has been investigated.

Finally, the hysteresis cycles show that the increase in absolute value of the parameter |J_Sσ| increases the magnetic coercive field which increases the surface of the loops. Furthermore, the presence of the loops is due to the ferrimagnetic behavior of the studied system (J_Sσ <0). The triple loops behavior which is useful in many applications in multi-state memory systems.

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Magnetic Properties and Thermal Behavior of the Monolayer Rubrene-Like Nano-island: Monte Carlo Simulations

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Abstract

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1. Introduction

2. Model And Method

3. Nmerical Results And Discussion

3.1. Ground state phase diagrams

3.2. Monte Carlo Simulations

4. Conclusion

References

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