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 JSS = 1 and JSσ = − 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, JSσ), in the absence of the external magnetic field (D = 0) and for a fixed value of exchange coupling interaction, JSS =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, JSS) for JSσ = -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, JSσ) 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 (JSS, JSσ) 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 (JSS = 0, JSσ = 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 JSS and JSσ, 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 JSS =1, JSσ=-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 (JSσ, JSS, H and D). This compensation temperature is located at Tcomp ≈0.11.

Additionally, the Fig. 3b is obtained for identical parameter values as in Fig. 3a. The total magnetic susceptibilities (χtot) and the partial ones (χS, χσ) 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 JSσ affects the compensation and critical temperatures. The thermal total magnetization's behavior for various JSσ values (JSσ = -0.02, -0.06, and − 0.1) is illustrated in Fig. 4a. This figure is taken for JSS = 1 and D = 0, H = 0 in the absence of the crystal and external fields.

It is found that the compensation temperature Tcomp increases when increasing the parameter |JSσ|. 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: Tcomp≈ 0.11, 0.34 and 0.55 for JSσ=-0.02, -0.06 and − 0.1, respectively. In addition, the obtained critical temperature for the same ferrimagnetic cases is Tc ≈1.30 for JSσ =-0.02, Tc ≈1.20 for JSσ =-0.06 and Tc≈1.15 for JSσ=-0.1.

Similarly, in Fig. 5a and 5b, we investigate the thermal behavior of the total magnetization when varying the exchange coupling parameter JSS. These figures are shown for selected values of the parameter (JSS=1, 1.2, 1.4 and 1.6) and for a fixed value of the parameters JSσ = -0.02 and in the absence of the crystal and external fields (D = 0, H = 0). It seems that the compensation temperature Tcomp is not influenced by the variation of the parameter JSS, while the critical temperature Tc increases when increasing the JSS parameter. This is due to the low selected values of the physical parameters. The obtained compensation temperature value, for various exchange coupling parameters JSS is Tcomp≈0.11. In addition, the obtained critical temperature for the same values of the parameter JSS=1, 1.2, 1.4 and 1.6 are Tc≈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 JSS on the compensation and critical temperatures in Fig. 5c. In fact, we observe that the Tcomp remains almost constant when the JSS increases. While, the critical temperature Tc increases almost linearly as a function of this parameter JSS. 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 JSσ (JSσ=-0.1, -0.2, -0.4, and − 0.8) and the fixed parameter values: JSS=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 (JSσ), 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 Msat=-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 JSσ 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 JSS=1 and JSσ =-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: JSσ=-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 | JSσ | 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 (JSσ <0).

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