Double perovskites (DPs) are a class of materials with the chemical formula A₂BB'O₆, consisting of two alkaline earth metal cations (A), two different transition metal cations (B and ${\mathrm{B}}^{\mathrm{\prime }}$), and six oxygen ions (O) [1–6]. DPs exhibit a wide range of interesting physical properties, such as high stability and durability [7, 8], magnetism [9], ferroelectricity [10] and high conductivity [11], which make them suitable for a variety of applications. Due to these properties, DPs have received significant attention in the field of materials science and condensed matter physics. They are the subject of active research for their potential use in a variety of devices such as catalysts [12], thermoelectricity [13], solar cells [13, 14] and spintronic devices [15].

In recent years, half-metallic ferromagnetic (HMF) materials have gained prominence in the field of spintronics because of their 100% spin polarization [16, 17]. These materials have been studied for use in devices such as magnetic random-access memory (MRAM) [18], giant magnetoresistance (GMR) [19], magnetic sensors [20] and spin valves [21]. The unique electronic properties of HMF make them suitable for these applications, and ongoing research is focused on developing new ways to utilize these materials in modern technology. Following the revelation of half-metallic ferromagnetic properties in (Pt/Ni)MnSb Heusler alloys by de Groot et al [22], there has been a significant amount of theoretical and experimental research conducted to investigate the phenomenon of half-metallicity in other materials [23, 24]. In addition, HMF has been found in Cr-doped (Al/Ga)_1−xCr_xN [25], transition metal doped Zn/CdX chalcogenides [26], simple perovskites SVO₃ (X = Ca, Sr, Ba) [27] and DP LaNiMnO₆ [28].

The DP Sr₂CrMoO₆ (SCMO) has been investigated on both experimental and theoretical sides. Experimentally, various techniques have been used to study the properties of SCMO, such as linear magnetoresistance (LMR) [29] and x-ray absorption spectroscopy (XAS) [30]. In addition, it has been discovered that samples of Sr₂CrMoO₆ with oxygen vacancies can be synthesized [31]. Theoretically, the magnetic properties of SCMO have been investigated for high Curie temperatures [30, 32, 33]. It has been shown that a super-exchange interaction causes the spins of the Mo⁵⁺ ([Ar] 4d¹, $S=\frac{1}{2}$) and Cr³⁺ ([Ar] 3d³, $\sigma =\frac{3}{2}$) ions to align in opposite directions, which results in a ferrimagnetic order at 450K [34, 35]. It has been found that SCMO has a magnetic moment of 2.0 μ_B and that the Mo⁵⁺ and Cr³⁺ ions are ferrimagnetically coupled within a cubic cell (space group: Fm3m) [36]. This compound is split into two ferromagnetic sublattices of Mo⁵⁺ and Cr³⁺ cations, respectively. According to XAS, Cr³⁺ in SCMO magnetic oxide can only be in the 3+ state (3d³), while Mo⁵⁺ in the 5+ state (4d¹) [30, 36]. Some studies on mixed Ising systems in the presence of single-ion anisotropies have suggested a ferrimagnetic phase as a result of the antiferromagnetic coupling between $S=\frac{3}{2}$ and σ = 1 [37–39]. These investigations have found that both crystal field and ferrimagnetic coupling can influence the magnetic behavior of a system and can lead to compensation behaviors.

The motivation for choosing the DP Sr₂CrMoO₆ for this study stems from its potential applications in spintronics, a field that leverages the intrinsic spin of electrons, in addition to their charge, for the development of advanced electronic devices. Despite its promising properties, Sr₂CrMoO₆ has not been extensively studied using a combination of ab-initio calculations and Monte Carlo (MC) simulations. This comprehensive approach allows for a detailed understanding of both the electronic structure and the magnetic behavior of the compound, which are crucial for its application in spintronic devices. Our study aims to fill this gap in the current research by providing insights into the half-metallic nature and phase transitions of Sr₂CrMoO₆, thereby contributing to the development of materials with optimized performance for spintronic applications. The paper is organized into three sections. Section 2 provides information on ab-initio calculations, section 3 focuses on a Monte Carlo study and section 4 summarizes the findings of the paper.

In accordance with DFT, ab-initio simulation was used to perform the calculations. The magnetic and electronic characteristics of the SCMO compound were investigated using the full potential linearized augmented plane wave (FP-LAPW) method, as implemented in the Wien2k code. The generalized gradient approximation (GGA) by Perdew–Burke–Ernzerhof (PBE-GGA) was utilized to treat the exchange-correlation potential [40]. The plane wave cutoff is R_MT × K_MAX = 7, where R_MT is the smallest muffintin sphere radius where the values are taken 2.38, 1.75, 1.75, and 1.6 atomic units (au) for Sr, Cr, Mo and O elements respectively, and K_MAX is the plane wave cutoff. Within the irreducible Brillouin zone (BZ), we employed a 10 × 10 × 10 K-point mesh. The iteration process was maintained until the total energy reached a value lower than 10⁻⁵ Ry [4].

As shown in figure 1, the DP SCMO is classified under the Fm-3 m space group. Within the SCMO crystal structure, oxygen atoms form octahedral arrangements around the chromium and molybdenum ions, as illustrated in figure 1. The Wyckoff positions for O, Mo, Cr, and Sr are specified as 24e (0.2483, 0, 0), 4a (0, 0, 0), 4b (0.5, 0, 0), and 8c (0.25, 0.25, 0.25), respectively [30]. Initially, we calculate the equilibrium lattice parameter a = 7.8 Å using the Murnaghan equation of state [41]. The total energy was fitted as a function of the unit cell volume in the ferrimagnetic (FiM), non-magnetic (NM), and ferromagnetic (FM) phases. This analysis showed that the FiM phase had a low energy, suggesting that the arrangement was stable (figure 2).

The electronic band structure of the Sr₂CrMoO₆ DP is shown in figure 3, along the high symmetry directions (W, L, Γ, X , W, K) of the Brillouin zone. The spin-up channel displays semiconducting behavior with a band gap of 0.96 eV. Conversely, the spin-down channel shows that the compound is metallic, as there are states present at E_F. Therefore, the combination of spin-down and spin-up channels results in half-metallic behavior. This characteristic makes Sr₂CrMoO₆ a favorable candidate for spintronic applications. In order to validate the findings of the band structures found in figure 3, we investigated the density of electronic states in the following.

The partial and total electronic densities of states (PDOS and TDOS) in relation to the energy of the SCMO compound are shown in figure 4. Based on the TDOS depicted in figure 4(a), we have determined that the Sr₂CrMoO₆ is a half-metal. This conclusion is drawn from the observation that the spin-down channel displays metallic characteristics due to the band intersection with E_F, whereas the spin-up channel has a band gap of about 0.96 eV at E_F. To comprehend the source of this behavior in the band structure shown in figure 3. We illustrated in figure 4(b) the contribution of each individual atom of this compound. We can see that the spin-down states of Mo and O are clearly visible at E_F. This is accountable for a band gap discovered between the electronic spin-up states of Mo and Cr. Additionally, the oxygen and strontium atoms are not magnetic due to the symmetry of their electronic density of states, while the magnetic properties of SCMO arise from the asymmetry of the density of states of molybdenum and chromium between the density of states for spin-up and spin-down. It is also observed from figure 4(c) that there is a strong hybridization between the 4d orbitals of molybdenum and 3d orbitals of chromium at E_F for the spin-down state. The presence of a gap between unoccupied Mo (4d) and Cr (3d) states indicates that there is a separation in energy between these orbitals and that they are not fully hybridized. This hybridization and gap can have a significant impact on the magnetic and electronic properties of the material. More specifically, in figure 4(d), the band gap corresponds to an energy gap between the subbands Cr (3d − e_g) and Cr (3d − t_2g) which is primarily due to the octahedral crystal field Δ_o created by the six oxygen atoms that encircle each chromium cation in the crystal structure. Moreover, in figure 4(d), it can be seen that the band Cr (3d − t_2g) contains the orbitals (d_xz, d_yz and d_xy) occupied by low energy electrons, it is evident that they are occupied by spin-up states. On the other side, the orbitals (d_x2−y2 and d_z2) of the band Cr (3d − e_g) that have high energy are unoccupied. The band Mo (4d − e_g) that exists above the Fermi level is unoccupied. Meanwhile the band Mo (4d − t_2g) that exists at E_F contains only one electron with a spin-down state. This fact, which conforms to Hund's rule, clarifies that the molybdenum (Mo⁵⁺) has a spin of $\frac{1}{2}$ and element chromium (Cr³⁺) has a spin of $\frac{3}{2}$.

Table 1 shows the magnetic moments of SCMO, which is a compound containing Mo and Cr atoms. The total magnetic moment is near 2.00 μ_B, which is similar to previous research [32, 34]. The table indicates that Cr and Mo are the primary sources of magnetization in the DP.

We study the magnetic properties of Sr₂CrMoO₆ (figure 1). Molybdenum (Mo⁵⁺) has a spin of $S=\frac{1}{2}$ and chromium (Cr³⁺) has a spin of $\sigma =\frac{3}{2}$ as mentioned above in the part of DFT.

The Hamiltonian can be represented as:The coupling parameters J_Cr−Mo, J_Mo−Mo and J_Cr−Cr denote the exchange interactions between Cr − Mo, Mo − Mo and Cr − Cr, respectively. D is the crystal field acting on the σ-spins.

All the subsequent simulations were carried out with a reduced exchange interaction ∣J_Cr−Mo∣ therefore, $R_2=\frac{J_{\mathrm{C}\mathrm{r}-\mathrm{C}\mathrm{r}}}{|J_{\mathrm{C}\mathrm{r}-\mathrm{M}\mathrm{o}}|}$, $R_3=\frac{J_{\mathrm{M}\mathrm{o}-\mathrm{M}\mathrm{o}}}{|J_{\mathrm{C}\mathrm{r}-\mathrm{M}\mathrm{o}}|}$, and $\mathrm{\Delta }=\frac{D}{|J_{\mathrm{C}\mathrm{r}-\mathrm{M}\mathrm{o}}|}$.

In accordance with the heat bath algorithm [42–44], we employ MC simulation with a periodic boundary condition in the x, y, and z directions. Each data point is obtained for 4.10⁴ steps after excluding the first 3.10⁴ ones. In addition, the lattice has the same number of Mo-spins and Cr-spins N_Mo = N_Cr = (L³)/2.

The sublattice magnetizations per site: The magnetization M_T is defined byThe susceptibility is defined asThe energy is defined as:where β = $\frac{1}{K_{\mathrm{B}}T}$, K_B is the Boltzmann factor and T is the absolute temperature.

The compensation temperature T_comp, is when the overall magnetization is zero, but the sublattice magnetizations are not. The temperatures of the second-order phase transition are found by identifying the highest points on the susceptibility curves. While the first-order phase transition temperatures are determined by locating the discontinuities of the internal energy and magnetization curves.

Figure 5(a) shows the effect of system size on the total magnetization M_T and susceptibility χ_T of a system with specific values of R₂ = 0.5, R₃ = 2, and Δ = 0. The system size varied from L = 12 to L = 36 and we observed that the critical temperature T_c increases up to a certain size (L = 28) and then remains constant. From figure 5(b) we can be noticed that the maximum amplitude of χ_T and the transition temperature increases up to L = 28. This suggests that the system reaches a thermodynamic limit at L_ThL = 28. For this, we have fixed the size of Sr₂CrMoO₆ in L = 32 which is larger than L_ThL.

Figure 6 illustrates a phase diagram in the (T, R₂) plane with Δ = 0 and R₃ = 6.0. We can see that the compensation temperature (T_comp) increases linearly within the range of 0.05 ≤ R₂ ≤ 0.57 and disappears when R₂ > 0.57. Moreover, the critical temperature T_c is almost constant (T_c = 16.1) for R₂ ≤ 0.8, and increases for R₂ > 0.8. In addition, we can also notice that a second-order transition line separates the paramagnetic phase (PA) at higher temperatures from the ferrimagnetic phase (FI) at lower temperatures.

Figure 7 displays the effect of R₃ on the temperatures behavior with Δ = 0 and R₂ = 0.5. It is seen that the system presents T_c which increases for all R₃ values, and for R₃ ≥ 6 T_comp remains fixed. It can be concluded that the effect of the R₂ on the compensation behavior is opposite to that of the R₃ in figure 6. It appears that as R₂ increases, T_comp also increases, indicating that the Cr magnetization is more easily reduced with increasing temperature. This is likely due to the stronger exchange coupling between the Cr spins, which makes them more susceptible to flipping and thus contributes to the occurrence of the compensation points.

Figure 8 shows the phase diagram in the (T, Δ) plane for R₂ = 0.5 and R₃ = 6.0. We can notice that T_c is unchanged with increasing of Δ. In the low temperature region, the system exhibits a line of first-order transitions that terminate at a critical end point at (Δ = −3.9, T = 0.95). This line separates antiferromagnetic phase (−1/2, 1/2) and ferrimagnetic one (−1/2, 3/2). Additionally, a compensation temperature horizontal line separates the ferrimagnetic phase from the antiferromagnetic one. We can notice that the compensation temperature increases gradually in the range of Δ (−5.7 ≤ Δ ≤ 1). Our findings at low temperature are qualitatively compared with previous studies on magnetic systems using the mean-field approximation, yielding similar results [45].

To confirm the results found in figure 8, we have displayed in figures 9(a)–(b) the magnetizations M_σ, M_S and the internal energy E_T versus Δ for T = 0.2 and for the same parameters as in figure 8. While in figures 9(c)–(d), the parameters M_T and E_T are plotted as a function of T and for the values of Δ (Δ = − 9.0, − 7.0, − 5.0 and 4). In figure 9(a), when Δ is increased, the magnetization M_Cr present a discontinue passage at Δ = −3.9, and the magnetization M_Mo remains constant at M_Mo =+ 0.5. The curve of E_T in figure 9(b) confirms that the system has first-order temperature at Δ = −3.9. On the other hand, in figure 9(c) it is observed that, for Δ = 4, the magnetization starts from its maximum value 0.5 and when T increases M_T decreases to depress to T_c. In addition, for Δ = −9.0, − 7.0 and −5.0 the magnetization starts from zero and it becomes unchanged at low temperatures. When the temperature is increased, for Δ = −5, the magnetization increases to reach its maximum value when T = 3.4, which may be caused by a frustrated state resulting from thermal agitation. When T > 3.4, the magnetization decreases and changes sign at a compensation temperature. After this point, the magnetization reaches a minimum value before disappearing entirely at T_c. For Δ < − 5.0, the magnetization decreases to a minimum value which depends on the value of Δ. When T is increased, M_T increases to vanish at T_c. In figure 9(d), the internal energy increases as T increases and it is less sensitive on changes of the negative crystal field Δ.

The ab-initio calculations have shown that DP Sr₂CrMoO₆ exhibits half-metallic behavior and a total magnetic moment of 2.0 μ_B, indicating its potential for spintronic device applications. The compound, Sr₂CrMoO₆ (SCMO), is found to be ferrimagnetic with the spins $S=\frac{1}{2}$ and $\sigma =\frac{3}{2}$ attributed to the molybdenum and chromium ions, respectively. Additionally, through Monte Carlo simulations using the heat bath algorithm, we have explored the phase diagrams. We examined the effects of crystal field interactions and exchange couplings on compensation and critical behaviors. The results present a rich variety of phase diagrams, including both first-order and second-order transitions. It is observed that a line of first-order transitions occurs in this system, separating the antiferromagnetic and ferrimagnetic phases. Furthermore, the crystal field influences the compensation temperature, while the transition temperature T_c shows less sensitivity to variations in the crystal field.