Half-metallic ferromagnetism in non-magnetic double perovskite oxides Sr2MSbO6 (M=Al, Ga) doped with C and N

ABSTRACT Double perovskite oxides have gained tremendous attention in material science and device technology due to their facile synthesis and exceptional physical properties. In this paper, we elucidate the origin of magnetisation in non-magnetic double perovskite oxides Sr2MSbO6 (M = Al, Ga) induced by non-magnetic 2p impurities (C and N) substituted. The calculations were done within the full potential linearised augmented plane wave method in the framework of the density functional theory. The exchange–correlation potential is evaluated using the generalised gradient approximation (GGA) of Perdew–Burke–Ernzerhof and the modified Becke and Johnson (mBJ-GGA). Regarding structural properties of undoped double perovskites Sr2MSbO6 (M = Al, Ga), we found that the lattice constants and oxygen positions are in rational accord with the experimental results. Furthermore, both of the examined compounds are brittle in nature with isotropic character. For Sr2AlSbO6, we have got the values of energy gap equal to 1.9 and 3.7 eV within the GGA and the mBJ-GGA, respectively. However, for Sr2GaSbO6 the values of energy gap obtained in GGA and mBJ-GGA are equal to 0.8 and 2.9 eV, respectively. Finally, spin-polarised calculations reveal that the doping C and N can lead to drastic changes in the magneto-electronic properties of the semiconducting Sr2MSbO6 matrix with the integer magnetic moment of 6.00 μB and exhibit half-metallic properties. The origin of ferromagnetism can be attributed to the spin–split impurity bands inside the energy gap of the semiconducting Sr2MSbO6 matrix. These results may help experimentalists in synthesising new double perovskites for spintronic applications.

Strontium-based double perovskites have always been one of the most important members of half-metallic magnetic materials due to their robust half-metallicity and high Curie temperature. The search of new double perovskite in terms of their composition, structure and functionality to get desired properties is an active area of research in material science and beyond. Many different properties have been explored in these strontium double perovskites, such as half-metallic behaviour in Sr 2 FeMO 6 (M = Mo, Re) and Sr 2 CrMO 6 (M = Mo, W) [36][37][38][39], tunnel magnetoresistance in Sr 2 FeMoO 6 [40], room temperature colossal magnetoresistance in Sr 2 MMoO 6 (M = Cr, Fe) [37,39], magnetoelectric in Sr 2 CoMoO 6 [41] and half-metallic antiferromagnetic in Sr 2-OsMoO 6 [42]. To the best of our knowledge, only a few studies on Sr 2 (Ga, Al)SbO 6 have been reported until now [48][49][50][51]. Experimentally, both compounds have a cubic structure at room temperature with space group Fm3m. Sr 2 AlSbO 6 has been prepared and characterised by X-ray diffraction (XRD) in [48], and it was reported that this compound adopt the cubic phase with the lattice parameter a = 7.766−7.763 Å. In the same context, Wittmann et al. [49] reported the synthesis and phase transitions of Sr 2 GaSbO 6 using powder neutron diffraction 'XRD analysis' and confirm the cubic structure with a lattice constant of 7.89 Å. As the reported experimental results are limited just to the structure, the elastic, mechanical and electronic properties are still undiscovered; therefore, these materials need further investigation to understand their properties and increase their applications.
The present work provides insights for the understanding of the magnetisation induced in semiconducting Sr2MSbO6 (M = Al, Ga) matrix. The present work provides insights for the understanding of the magnetisation induced in semiconducting Sr 2 MSbO 6 (M = Al, Ga) matrix. Furthermore, our contribution covers the deficiency on the fundamental properties of these Srbased double perovskite oxides Sr 2 MSbO 6 .

Computational details
The calculations of undoped and doped Sr 2 MSbO 6 (M = Al, Ga) double perovskites were carried out by using the full potential linearised augmented plane wave method implemented in the WIEN2K code [52,53] within the scheme of density functional theory [54]. For the exchange-correlation effects, we have used the generalised gradient approximation (GGA) [55]. Moreover, the modified Beck Johnson (mBJ) potential was applied to overcome the inefficiency of the GGA functional to predict the magneto-electronic properties [56].
The charge density of Fourier expansion and maximum values of angular momentum were set to be l max = 10 and G max = 14, respectively. To certify that there is no outflow of charge from core, non-overlapped muffin-tin radius (R MT ) and converged energy value should be selected. The kinetic energy cut-off of R MT * K max is equal to 8. The convergence criteria used for energy was 10 −4 Ry, whereas the charge was converged up to 10 −3 e. A dense k-mesh of 20 × 20 × 20 was used to calculate all physical properties of considered compounds, and the separation between the core and valence states was set to −6.0 Ry.

Results and discussion
The results obtained using the various ground state parameters are discussed below. The stability of non-magnetic double perovskites Sr 2 MSbO 6 (M = Al, Ga) strongly depends on the ionic radii of the constituting metal ions, which is generally assured by the Goldschmidt tolerance (t) factor [57]. For Sr 2 MNbO 6 (M = Al, Ga), t can be calculated via the following equation: where r Sr , r O , r Sb , r M are the ionic radii of Sr, O, Sb and M (M = Al, Ga) respectively [58]. For an ideal cubic phase, the value of the tolerance factor t is equal to or near unity. It may be seen from the Table 1 Table 1. The calculated and experimental crystallographic data for Sr 2 MSbO 6 (M = Al, Ga); tolerance factors (t), Glazer tilt system (GTS), lattice constants a, oxygen positions u, bulk modulus (B), pressure derivative of bulk modulus (B ′ ), standard enthalpy of formation energies (DH o f ) and the average bond-lengths.  4b(1/2,0,0), 24e(0.254,0,0), respectively. The bond angle < M-O-Sb > = 180°a nd thus no MO 6 -SbO 6 -octahedral distortions. The tilting of the octahedra is the primary source of deviation from the cubic structure and can be described by using the Glazer tilt system (GTS) [59,60]. Therefore, no octahedra tilting or (a 0 a 0 a 0 ) in the GTS notation exists in Sr 2 MSbO 6 , as illustrated in Table 1. The overall structure stability can be evaluated by the global instability index (GII), i.e. the deviation of the bond valence sums with the ideal formal valences [61,62]. The GII value is usually lower than 0.1 v.u., valence units, for unstrained structures and as large as 0.2 v.u. in a structure with lattice-induced strains [62]. The GIIs of Sr 2 MSbO 6 (M = Al, Ga) are determined using the Structure Prediction Diagnostic Software [63] and gives the values of 0.0876 and 0.0686 v.u., respectively.
Next, to obtain the structural properties of Sr2MSbO6, the data from the optimised structures were fitted in Murnaghan's equation of state [64]. The optimised parameters are summarised in Table 1 and obtained values are in rational accord with the experimental measurements [48][49][50][51]. In regard to the oxygen position u, the results found to be around the ideal value 0.25. Additionally, the structural parameters were also predicted using the ionic crystal model [63]. Hence, it was concluded that the optimised parameters are consistent with the predicted values. We can note also that the bulk modulus decreases as the lattice parameter increases when traversing from Al to Ga in Sr2MSbO6.
In practical applications, the thermodynamic stability of new materials is an important parameter that affects their selection and use. In order to confirm the thermodynamic stability of double perovskite oxides Sr2MSbO6, the enthalpy of the formation energy DH o f was examined by calculating the energy difference DE between the double perovskite Sr 2 MSbO 6 and existing materials, such as M 2 O 3 (M = Al, Ga) and SbO 3 based on their thermochemical equations using the solid-state reaction procedure as follows: Hence, DH o f can be expressed in the following equation: where E Tot is the total energy per unit cell volume (Z = 2) of Sr 2 MSbO 6 , E i = E Sr , E M (M = Al and Ga), E Sb and E O are the energy of individual elements in the unit cell, and n i denotes the number of atoms of the ith element in a single formula unit; n 1 = 4, n 2 = 2, n 3 = 2 and n 4 = 6, respectively. The calculated formation energies are negative for both double perovskites Sr 2 MSbO 6 , which indicates that those compounds are in thermodynamic stability.
This result is not surprising when we know that Sr 2 MSbO 6 have been already experimentally synthesised. On the other hand, it is important to observe that the average bond distances in Sr 2 AlSbO 6 are very similar to those in Sr 2 GaSbO 6 . This is can be attributed to the fact that the ionic radii of Al 3+ and Ga 3+ are nearly equal.
3.1.2. Elastic properties and mechanical stability of Sr 2 MSbO 6 (M=Al, Ga) Generally, there is a mere fact of having structural stability does not guarantee the existence of the double perovskite structure. However, to some extent, the stability can be characterised qualitatively by the mechanical stability criteria. To predict the independent elastic constants, one can use the volume conserving tetrahedral and rhombohedral distortions on the cubic phase of Sr 2 MSbO 6 . The obtained values of the elastic constants are listed in Table 2. To our best knowledge, no studies have been conducted concerning elastic constants. Therefore, our results can be assumed to be predictive. Obviously, the elastic constants of these two double perovskites Sr 2 MNbO 6 satisfy the mechanical stability criteria [65]. Subsequently, using the elastic constants to predict the polycrystalline moduli, the shear modulus (G), young modulus (E), poisson's ratio (υ) and shear anisotropic factor (A) (also called Zener coefficient). The calculated data are also listed in Table 2. To classify the materials in ductile manner or brittle manner we use the relationship proposed by Pugh [66,67] which links empirically the plastic properties of metals with their elastic moduli by B/G known as Pugh's ratio. A material with B/G smaller than 1.75 and υ smaller than 0.26 is brittle, otherwise it is ductile. From Table 2, it shows that values of B/G and ν are smaller than 1.75 and 0.26 respectively, confirming that Sr 2 MSbO 6 double perovskites are brittle.
Young's modulus (E) is defined as the ratio of stress and strain when Hooke's law holds. Young's modulus of a material is the usual property used to characterise stiffness. As Young's modulus increases, the stiffness of compound increases too. Thus, the Sr 2 AlSbO 6 is stiffer than Sr 2 GaSbO 6 , as revealed in Table 2. Another important mechanical quantity is the elastic anisotropy factor A, which gives a measure of the anisotropy of the elastic wave velocity in a material. For an isotropic material, A is equal to unity, while any value In order to investigate the electronic properties of undoped Sr 2 MSbO 6 , we focus our study on the electronic band structure and the density of states (DOS). As shown in Figure 2, the calculated band structures along the higher symmetry directions in the irreducible Brillouin zone (IBZ) were evaluated with both PBE-GGA and mBJ-GGA approximations. The zero energy corresponds to the Fermi level. The occupied states below the Fermi energy correspond to the valence band, whereas the unoccupied states lying above the Fermi energy correspond to the conduction band. The overall band profiles are found to be the same for both double perovskites and show that the valence band maximum and the conduction band minimum are located at the centre of the Brillouin zone Γ, resulting in a direct band gap. For Sr 2 AlSbO 6 we have got the values of energy gap equal to 1.9 and 3.7 eV within the GGA and the mBJ-GGA, respectively. However for Sr 2 GaSbO 6 the values of energy gap obtained in GGA and mBJ-GGA are equal to 0.8 and 2.9 eV, respectively. Unfortunately, there are no reported experimental measurements or previous theoretical calculations of the band gaps for the considered compounds to be compared to the present results. It is important to mention that mBJ-GGA approximation is able to yield more accurate energy band gaps in most semiconductors and insulators, whereas the GGA functional fails qualitatively [31,[68][69][70]. It is obvious that the width of the band gap decreases with increasing atomic number of metallic cation M (Al, Ga). The band gap reduction could be attributed to a shift of the conduction band towards the Fermi level (E F ) when one move from Al to Ga.
To aid in understanding the character of electronic structures, we have calculated the total densities of states (TDOS) and partial densities of states (PDOS) for these double perovskites Sr 2 MSbO 6 by using mBJ-GGA and the corresponding results are shown in Figure 3. Our calculations suggested that the lower part of the valence band is dominated by the Sb 5s orbital, and the upper part by the O 2p orbital in Sr 2 MSbO 6 . The lower part of the conduction band is dominated by the Sb 5s orbital.

Magnetisation of Sr 2 (Al,Ga)SbO 6 induced by C and N impurities
As mentioned before, double perovskite oxides offer an important number of interesting physicochemical properties and high potential for technological applications, many of which are still not fully understood, and can be substantially modified through doping impurities. In particular, the light elements (C and N) mono-doping as anions to replace oxygen have attracted much attention and are considered one of the appropriate anions alloy elements because its sizes are closer to the size of the substituted host oxygen atoms, which are favourable for effective incorporation. In this section, we tried to study the magnetism induced by the 2p impurities in non-magnetic double perovskite oxides Sr 2 MSbO 6 (M = Al, Ga) for possible applications in spintronic and magnetoelectronic devices. The results were obtained in calculations using the 2 × 1 × 1 supercell (in the x, y and z directions) containing 40 atoms. From this 40atoms supercell, oxygen was substituted with an impurity atom X (X = C or N). Our corresponding systems have been written under the formula composition Sr 2 MSbO 5.75 X 0.25 where M = Al, Ga and X = C and N. After creating a supercell with one centrally placed impurity atom X, a complete relaxation was done. In order to check the stability of ferromagnetic state, we calculated the spin polarisation energy (DE sp ), which is the energy difference between the spin-polarised and non-spin polarised states of these doped systems. We found that the ferromagnetic state is more stable than the non-magnetic state.
To explore the mechanism of defect-induced ferromagnetism, the densities of states for doped (C and N) double perovskites have been computed using both GGA and mBJ exchange-correlation functional. The results obtained for the densities of sates of these compounds are nearly same, in both GGA and mBJ schemes. Figure 4 exhibits densities of states obtained using mBJ functional for C and N doped Sr 2 MSbO 6 systems. It is evident from this figure that when 2p impurities are incorporated, the spin-up and spin-down channels are asymmetrical which exhibiting a magnetism behaviour. Furthermore, the majority-spin channel exhibits a metallic behaviour, while the minority-spin channel is semiconducting, i.e. half-metallic with 100% spin polarisation of the conduction electrons. The values of the energy band gap after doping  In this case, we can attribute the origin of this magnetisation to the presence of spin-split impurity bands inside the energy gap region of Sr 2 MSbO 6 . Therefore, doping with non-magnetic light elements can lead to defect-induced ferromagnetism in Sr 2 MSbO 6 .
The previous investigation reports pointed out that the origin of magnetic moment may be attributed to the hole doping in the perovskite oxides [31,72]. For example, Peng et al. have clarified the relationship between hole doping and ferromagnetism in defect-induced oxides [71]. Moreover, Liu et al. studied the hole-doped SrTiO 3 with non-magnetic elements, and observed the formation of localised magnetic moment, which can increase monotonously with the number of holes introduced by doping [72]. We summarised in Table 3 the calculated magnetic moments for Sr 2 MSbO 6 doped with C and N atoms at their equilibrium lattice constants. Obviously, for all cases, the total magnetic moment is 6 μ B which confirms the half-metallic character and it is originated mainly from both impurities C and N.

Conclusion
In this paper, first-principles density functional theory calculations were performed in order to elucidate the origin of magnetisation in non-magnetic double perovskites Sr 2 MSbO 6 (M = Al, Ga) doped with non-magnetic elements (C or N). Regarding structural properties of undoped double perovskites Sr 2 MSbO 6 (M = Al, Ga), it has been found that the tolerance factor, formation energy and stability criteria confirm the stability of these materials in the cubic phase. The lattice constants and oxygen positions are in rational accord with the experimental results. The analysis of mechanical parameters reveals that both compounds are brittle in nature with isotropic behaviour. More importantly,   for Sr 2 AlSbO 6 we have got the values of energy gap equal to 1.9 and 3.7 eV within the GGA and the mBJ-GGA, respectively. However for Sr 2 GaSbO 6 the values of energy gap obtained in GGA and mBJ-GGA are equal to 0.8 and 2.9 eV, respectively. Finally, spin-polarised calculations reveal that the doping C and N can lead to drastic changes in the magneto-electronic properties of the semiconducting Sr 2 MSbO 6 matrix with the integer magnetic moment of 6.00 μ B per cell and exhibit half-metallic properties. The origin of ferromagnetism can be attributed to the spin-split impurity bands inside the energy gap of the semiconducting Sr 2 MSbO 6 matrix. The present theoretical estimation of various physical parameters can prove as valuable reference with respect to their future experimental work.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was financially supported by the general directorate for scientific research and technological development under PRFU Project Number: B00L02UN310120190004.