1. Introduction
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 ${\rm{B}}^{\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−xCrxN [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, $S=\tfrac{1}{2}$) and Cr3+ ([Ar] 3d3, $\sigma =\tfrac{3}{2}$) 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 μB 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 $S=\tfrac{3}{2}$ 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.
2. Ab-initio calculation method
2.1. Computational details
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 RMT × KMAX = 7, where RMT 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 KMAX 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].
2.2. Crystal structure
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).
Figure 1. The structure Sr2CrMoO6 in both (a) atomic and (b) polyhedral forms. |
Figure 2. Optimized E(V) curve of Sr2CrMoO6. |
2.3. Electronic properties
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 EF. 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.
Figure 3. The band structures of Sr2CrMoO6. |
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 EF, whereas the spin-up channel has a band gap of about 0.96 eV at EF. 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 EF. 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 EF 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 − eg) and Cr (3d − t2g) which is primarily due to the octahedral crystal field Δo 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 − t2g) contains the orbitals (dxz, dyz and dxy) occupied by low energy electrons, it is evident that they are occupied by spin-up states. On the other side, the orbitals (dx2−y2 and dz2) of the band Cr (3d − eg) that have high energy are unoccupied. The band Mo (4d − eg) that exists above the Fermi level is unoccupied. Meanwhile the band Mo (4d − t2g) that exists at EF 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 $\tfrac{1}{2}$ and element chromium (Cr3+) has a spin of $\tfrac{3}{2}$.
Figure 4. The density of states of Sr2CrMoO6. |
2.4. Magnetic moments
3. Model and Monte Carlo method
We study the magnetic properties of Sr2CrMoO6 (figure 1). Molybdenum (Mo5+) has a spin of $S=\tfrac{1}{2}$ and chromium (Cr3+) has a spin of $\sigma =\tfrac{3}{2}$ as mentioned above in the part of DFT.
The Hamiltonian can be represented as:
$\begin{eqnarray}\begin{array}{rcl}H & = & -{J}_{\mathrm{Cr}-\mathrm{Mo}}\displaystyle \sum _{i,j}{\sigma }_{i}{S}_{j}-{J}_{\mathrm{Mo}-\mathrm{Mo}}\displaystyle \sum _{i,j}{S}_{i}{S}_{j}\\ & & -\,{J}_{\mathrm{Cr}-\mathrm{Cr}}\displaystyle \sum _{i,j}{\sigma }_{i}{\sigma }_{j}-D\displaystyle \sum _{i}\,.\,{({\sigma }_{i})}^{2}.\end{array}\end{eqnarray}$
The coupling parameters JCr−Mo, JMo−Mo and JCr−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 ∣JCr−Mo∣ therefore, ${R}_{2}=\tfrac{{J}_{\mathrm{Cr}-\mathrm{Cr}}}{| {J}_{\mathrm{Cr}-\mathrm{Mo}}| }$, ${R}_{3}=\tfrac{{J}_{\mathrm{Mo}-\mathrm{Mo}}}{| {J}_{\mathrm{Cr}-\mathrm{Mo}}| }$, and ${\rm{\Delta }}=\tfrac{D}{| {J}_{\mathrm{Cr}-\mathrm{Mo}}| }$.
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 NMo = NCr = (L3)/2.
The sublattice magnetizations per site:
$\begin{eqnarray}{M}_{\mathrm{Cr}}=\displaystyle \frac{1}{{N}_{\mathrm{Cr}}}\displaystyle \sum _{i}{\sigma }_{i},\end{eqnarray}$
$\begin{eqnarray}{M}_{\mathrm{Mo}}=\displaystyle \frac{1}{{N}_{\mathrm{Mo}}}\displaystyle \sum _{i}{S}_{i}.\end{eqnarray}$
The magnetization MT is defined by $\begin{eqnarray}{M}_{{\rm{T}}}=\displaystyle \frac{{M}_{\mathrm{Mo}}+{M}_{\mathrm{Cr}}}{2}.\end{eqnarray}$
The susceptibility is defined as $\begin{eqnarray}{\chi }_{{\rm{T}}}=\beta ({N}_{\mathrm{Mo}}+{N}_{\mathrm{Cr}})(\langle {M}_{{\rm{T}}}^{2}\rangle -\langle {M}_{T}{\rangle }^{2}).\end{eqnarray}$
The energy is defined as: $\begin{eqnarray}{E}_{{\rm{T}}}=\displaystyle \frac{1}{{N}_{\mathrm{Mo}}+{N}_{\mathrm{Cr}}}\langle H\rangle ,\end{eqnarray}$
where β = $\tfrac{1}{{K}_{{\rm{B}}}T}$, KB is the Boltzmann factor and T is the absolute temperature.The compensation temperature Tcomp, 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.
3.1. Magnetic properties and phase diagrams
Figure 5(a) shows the effect of system size on the total magnetization MT and susceptibility χT 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 Tc 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 χT and the transition temperature increases up to L = 28. This suggests that the system reaches a thermodynamic limit at LThL = 28. For this, we have fixed the size of Sr2CrMoO6 in L = 32 which is larger than LThL.
Figure 5. (a) Magnetization and (b) susceptibility as a function of temperature for L = 12 to L = 36. |
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 (Tcomp) increases linearly within the range of 0.05 ≤ R2 ≤ 0.57 and disappears when R2 > 0.57. Moreover, the critical temperature Tc is almost constant (Tc = 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 6. The influences of exchange interaction R2 on the temperatures with Δ = 0 and R3 = 6.0. |
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 Tc which increases for all R3 values, and for R3 ≥ 6 Tcomp 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, Tcomp 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 7. The influences of exchange interaction R3 on the temperatures with Δ = 0 and R2 = 0.5. |
Figure 8 shows the phase diagram in the (T, Δ) plane for R2 = 0.5 and R3 = 6.0. We can notice that Tc 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].
Figure 8. The influences of crystal field Δ on the temperatures with R2 = 0.5 and R3 = 6. |
To confirm the results found in figure 8, we have displayed in figures 9(a)–(b) the magnetizations Mσ, MS and the internal energy ET versus Δ for T = 0.2 and for the same parameters as in figure 8. While in figures 9(c)–(d), the parameters MT and ET 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 MCr present a discontinue passage at Δ = −3.9, and the magnetization MMo remains constant at MMo =+ 0.5. The curve of ET 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 MT decreases to depress to Tc. 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 Tc. For Δ < − 5.0, the magnetization decreases to a minimum value which depends on the value of Δ. When T is increased, MT increases to vanish at Tc. In figure 9(d), the internal energy increases as T increases and it is less sensitive on changes of the negative crystal field Δ.
Figure 9. (a) The magnetizations Ms, Mσ and (b) the internal energy E versus Δ for T = 0.2. The parameters (c) M and (d) E are plotted as a function of the temperature for selected values of Δ and for R2 = 0.5 and R3 = 6. |
4. Conclusion
The ab-initio calculations have shown that DP Sr2CrMoO6 exhibits half-metallic behavior and a total magnetic moment of 2.0 μB, indicating its potential for spintronic device applications. The compound, Sr2CrMoO6 (SCMO), is found to be ferrimagnetic with the spins $S=\tfrac{1}{2}$ and $\sigma =\tfrac{3}{2}$ 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 Tc shows less sensitivity to variations in the crystal field.