1 Introduction

In recent years, the experimental and theoretical researches on the II–V and III–V semiconductors doped with transition metals have attracted much attention because these materials have interesting electronic and magnetic properties and due to their remarkable use as fundamental materials for diluted magnetic semiconductors (DMS). The DMS based on the II–V and III–V semiconductors are considered potential candidates for modern spintronics applications because they show stability in the ferromagnetic ordering configuration and they exhibit a half-metallic ferromagnetic behavior [1,2,3,4,5,6,7]. Spin-based electronics or spintronics is modern field of research exploiting the electron spin plus its charge as a second-degree freedom to improve the processing performance and data storage of spin-based devices. The expected advantage of spintronic devices over the conventional electronic ones would be nonvolatility, increased data processing speed, increased transistor density, and decreased power consumption [8]. Several experimental and theoretical investigations have been performed on the half-metallic and magnetic properties of DMS based on III–V and II–VI semiconductors such as the Mn-doped InSb [9], the Mn-doped AlSb [10], the V doped AlSb [11], the Mn-doped GaAs [12], the theory of ferromagnetic (III, Mn)V semiconductors [13], the Ni-doped ZnS [14], Fe-doped ZnS [15], the (C, Fe)-doped CdSe [16], the V-doped ZnS [17], the V-doped BaS [18], and the V-doped SrO [19].

To the best of our knowledge and according to researches available in the literature, there are no experimental and theoretical studies on magnetic and electronic properties of Cr-doped Barium selenide (BaSe)-based DMS. In this study, we have performed the structural, electronic, and ferromagnetic properties of Ba1 − xCrxSe at various concentrations x = 0.25, 0.5, and 0.75. We have used in our prediction the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory (DFT) [20], where the exchange and correlation potential is described by the generalized gradient approximation of Wu and Cohen (GGA-WC) [21].

2 Method of Calculations

We have used the full-potential linearized augmented plane-wave (FP-LAPW) method within the framework of the DFT [20] as implemented in WIEN2K package [22]. We have treated the exchange and correlation potential by the generalized gradient approximation of Wu and Cohen (GGA-WC) [21] to compute structural, electronic, and ferromagnetic properties of Ba1 − xCrxSe at different concentrations x = 0.25, 0.5, and 0.75 of chromium (Cr) impurity. We have taken Kmax = 9.0/RMT, where the Kmax is the size of the largest K vector of the plane wave and the RMT are the averages radii of muffin-tin spheres of Ba, Se, and Cr atoms. The charge density was Fourier expanded up to Gmax = 14 (a.u.)−1, where Gmax is the largest vector in the Fourier expansion, and the maximum partial waves within the atomic sphere was lmax = 10. The cutoff energy was set to − 6 Ryd to separate core and valence states. We have employed the Monkhorst–Pack mesh [23, 24] of the (4 × 4 × 4) k-points for BaSe, Ba0.75Cr0.25Se, and Ba0.25Cr0.75Se and the (4 × 4 × 3) k-points for Ba0.5Cr0.5Se in the Brillouin-zone integration. Self-consistency was reached when the total energy convergence was set at 0.1 mRy.

3 Results and Discussions

3.1 Structural Properties

BaSe was made in 1925 by reducing the BaSeO3 with hydrogen at a red heat [25]. The BaSe is one of IIA–VI alkaline-earth chalcogenide group, crystallizing in a rock-salt NaCl (B1) phase with space group of \( Fm\overline{3}m \) No. 225. The conventional structure of BaSe has two types of atoms Ba and Se, which are located respectively at (0, 0, 0) and (0.5, 0.5, 0.5) sites. We have created the Ba1 − xCrxSe supercells of 8 atoms such as the Ba3CrSe4 for concentration x = 0.25, the Ba2Cr2Se4 for x = 0.5, and the BaCr3Se4 for x = 0.75 by substituting of one, two, and three Cr atoms at Ba sites, respectively. The Ba0.75Cr0.25Se and Ba0.25Cr0.75Se supercells have a cubic structure with space group of \( Pm\overline{3}m \) No. 221, while the Ba0.5Cr0.5Se has a tetragonal structure with space group of P4/mmm No. 123. We have noted that our Ba0.75Cr0.25Se, Ba0.5Cr0.5Se and Ba0.25Cr0.75Se structures are described by the supercells completely free from defects, but the real supercells my necessarily have side effects like defects. However, our predictions are valid only for better calculations of supercells close to the ordered stoichiometric structures of Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se compounds. We hope that our investigations of novel electronic and ferromagnetic properties of Ba1 − xCrxSe provide predictions for experimentalists to explore these new materials for practical spintronics applications in the future.

We have calculated the formation energies to verify the solid state stability of Ba3CrSe4, Ba2Cr2Se4 and BaCr3Se4 compounds in the rock-salt NaCl (B1) structure. The formation energies (Eform) of the Ba4 − -yCrySe4 doping systems are determined by using the following expression [26, 27]:

$$ {E}_{form}={E}_{\mathrm{total}}\left({\mathrm{Ba}}_{4-y}{\mathrm{Cr}}_yS{e}_4\right)-\frac{\left(4-y\right)\;E\left(\mathrm{Ba}\right)}{8}-\frac{y\;E\left(\mathrm{Cr}\right)}{8}-\frac{4\;E\left(\mathrm{Se}\right)}{8} $$
(1)

where the Etotal(Ba4 − yCrySe4) is minimum total energy of Ba4 − yCrySe4 per atom and the E(Ba), E(Cr), and E(Se) are respectively the minimum total energies per atom of bulks Ba, Cr, and Se, and the y = 1, 2, and 3 are the number of substitute Cr atoms in Ba4 − yCrySe4 supercells. We have found that the formation energies are − 4.5, − 4.83, and − 5.34 eV for Ba3CrSe4, Ba2Cr2Se4, and BaCr3Se4, respectively. Consequently, the negative formation energies mean that our compounds are thermodynamically stable in the ferromagnetic rock-salt phase.

We have performed the optimization of BaSe and Ba1 − xCrxSe at various concentrations by the fitting of Murnaghan’s equation of state [28] that reveals the variation of the total energy as a function of volume. Table 1 summarizes the predicted structural parameters of our compounds such as the lattice constants (a), bulk modules (B), and their pressure derivatives (B′) with other theoretical [29,30,31,32,33] and experimental data [34,35,36] for comparison purposes. The results of a and B for BaSe show the good agreement compared to experimental values [34,35,36] and the theoretical calculations [29,30,31] found by the use of the same GGA-WC approximation [21]. Owing to better performance of GGA-WC potential for structural properties, our calculations of structural parameters a and B of BaSe are improved with respect to the theoretical values [32, 33] found by the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) [37]. For the Ba1 − xCrxSe doping structures, the difference between the ionic radii of Ba atom and the substituted Cr impurity leads to the decrease of the lattice constant as the Cr concentration increases. There are no realized studies on the structural parameters of the Ba1 − xCrxSe doping compounds in order to compare them with our results.

Table 1 Calculated lattice constant (a), bulk modulus (B), and its pressure derivative (B′) for BaSe and Ba1 − xCrxSe at concentrations x = 0.25, 0.5, and 0.75
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3.2 Electronic Structures, Half-Metallic Behavior, and Magnetic Properties

The electronic structures of BaSe and Ba1 − xCrxSe such as the spin-polarized band structures and densities of states have predicted by using the theoretical optimized lattice constants. The band structures of BaSe, Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se are presented in Figs. 1, 2, 3, and 4, respectively. Figure 1 depicts that spin up and spin down of BaSe exhibit similar semiconductor band structures with an indirect band gap (EΓ − Χ) of 1.843 eV, which occurs between the Γ and X high symmetry points. Figures 2, 3, and 4 of Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se reveal a half-metallic character, resulting from metallic and semiconductor natures of majority-spin bands and minority-spin bands, respectively. The majority-spin bands of Ba1 − xCrxSe doping compounds have direct half-metallic ferromagnetic gaps situated at Γ high symmetry point. The Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se compounds have half-metallic ferromagnetic gaps of 1.832, 1.668, and 1.084 eV, respectively. On the other hand, the majority-spin bands show a half-metallic gap (GHM), which is defined as the minimum of the lowest energy of the majority (minority)-spin conduction bands with respect to the Fermi level and the absolute value of the highest energy of the majority (minority)-spin valence bands [38, 39]. The half-metallic gap is located at Γ high symmetry point between the maximum of valence bands and Fermi level (EF) for the Ba0.75Cr0.25Se and Ba0.5Cr0.5Se, whereas it occurs between EF and minimum of conduction bands for Ba0.25Cr0.75Se. Table 2 shows the calculated indirect band gap (EΓ − Χ) of BaSe, half-metallic gaps (GHMF) and half-metallic gaps (GHM) of Ba1 − xCrxSe with other theoretical [29,30,31,32,33], and experimental [40, 41] results. The predicted indirect band gap of BaSe is in good agreement with theoretical calculations [31, 33], while it is far than that of the experimental values [40, 41] because GGA approach underestimates the band gap [42,43,44]. For the Ba1 − xCrxSe compounds, we understand that the half-metallic ferromagnetic decreases with in increasing concentration of chromium atom due to broadening of 3d levels of Cr around EF.

Fig. 1
figure 1

Spin-polarized band structures for BaSe. a Majority spin (up) and b minority spin (dn). The Fermi level is set to zero (horizontal dotted line)

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Fig. 2
figure 2

Spin-polarized band structures for Ba0.75Cr0.25Se. a Majority spin (up) and b minority spin (dn). The Fermi level is set to zero (horizontal dotted line)

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Fig. 3
figure 3

Spin-polarized band structures for Ba0.5Cr0.5Se. a Majority spin (up) and b minority spin (dn). The Fermi level is set to zero (horizontal dotted line)

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Fig. 4
figure 4

Spin-polarized band structures for Ba0.25Cr0.75Se. a Majority spin (up) and b minority spin (dn). The Fermi level is set to zero (horizontal dotted line)

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Table 2 Calculated indirect band gap (EΓ Χ) for BaSe, half-metallic ferromagnetic gap (GHMF), and half-metallic gap (GHM) of minority-spin bands for Ba1 − xCrxSe at concentrations x = 0.25, 0.5, and 0.75
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To explain the origin of half-metallic character in Ba1 − xCrxSe, we have investigated the contribution of densities of states (DOS) around the Fermi level (EF). Figures 5, 6, and 7 illustrate the spin-polarized total and partial densities of states for Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se, respectively. The DOS of majority-spin states for all compounds are metallic due to strong p-d hybridization between the p(Se) and d(Cr) levels. Simultaneously, the minority-spin channel does not have density of states at EF. Moreover, the contribution of the DOS of spin up and spin down around EF describes the spin polarization (P) of material, which can be determined by the following expression [45]:

$$ P=\frac{N\uparrow \left({E}_F\right)-N\downarrow \left({E}_F\right)}{N\uparrow \left({E}_F\right)+N\downarrow \left({E}_F\right)}\ 100 $$
(2)

where the N ↑ (EF) and N ↓ (EF) are the DOS of majority spin and minority spin around Fermi level, respectively. We have found that the spin polarization P = 1 for Ba1 − xCrxSe compounds at all concentrations, resulting from metallic nature of majority spin and semiconductor feature of minority spin. We have understand from our findings of electronic structures that the Ba0.75Cr0.25Se, Ba0.5Cr0.5Se, and Ba0.25Cr0.75Se materials are half-metallic ferromagnetic with a spin polarization of 100%, which make them promising candidates for spintronics applications.

Fig. 5
figure 5

Spin-polarized total and partial DOS of (5p) of Ba, (4p) of Se, and (3d, 3d-t2g, and 3d-eg) of Cr atoms in supercell of Ba0.75Cr0.25Se. The Fermi level is set to zero (vertical dotted line)

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Fig. 6
figure 6

Spin-polarized total and partial DOS of (5p) of Ba, (4p) of Se, and (3d, 3d-t2g, and 3d-eg) of Cr atoms in supercell of Ba0.5Cr0.5Se. The Fermi level is set to zero (vertical dotted line)

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Fig. 7
figure 7

Spin-polarized total and partial DOS of (5p) of Ba, (4p) of Se, and (3d, 3d-t2g, and 3d-eg) of Cr atoms in supercell of Ba0.25Cr0.75Se. The Fermi level is set to zero (vertical dotted line)

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Furthermore, we have calculated the total and local magnetic moments per Cr atom of relevant Ba, Cr, and Se atoms, and at interstitial site of Ba1 − xCrxSe doping systems. Table 3 depicts that the total magnetic moment for each compound is integral Bohr magneton of 4 μB, which is principally formed by the local magnetic moment of the Cr atom. The total magnetic moment of 4 μB is originated from the partially filled 3d (Cr) majority-spin states with four electrons. The large p–d exchange interaction reduces the predicted magnetic moment of 3d (Cr) less than 4 μB and induces minor magnetic moments at Ba, Se, and interstitial sites. Besides, the positive magnetic moments of Cr and Ba atoms for Ba0.75Cr0.25Se, Ba0.5Cr0.5Se materials, lead to the ferromagnetic interaction between Cr and Ba, but the anti-ferromagnetic interaction is explained by the opposite magnetic signs of Cr and Se atoms.

Table 3 Calculated total and local magnetic moments per Cr atom of the relevant Cr, Ba, and Se atoms and in the interstitial sites (in Bohr magneton μB) for Ba1 − xCrxSe at concentrations x = 0.25, 0.5, and 0.75
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We have used the band structures to calculate important factors such as the p–d exchange splitting \( {\Delta }_x^v(pd)={E}_v^{\downarrow }-{E}_v^{\uparrow } \) and \( {\Delta }_x^c(pd)={E}_c^{\downarrow }-{E}_c^{\uparrow } \), and the s–d exchange constants N0α (conduction band) and the p–d exchange constants N0β (valence band). The N0α and N0β parameters are calculated from the mean-field theory by the use of the following expressions [46, 47]:

$$ {N}_0\alpha =\frac{\Delta {E}_c}{x\ \left\langle s\right\rangle\ } $$
(3)
$$ {N}_0\beta =\frac{\Delta {E}_v}{x\ \left\langle s\right\rangle\ } $$
(4)

where the \( \Delta {E}_c={E}_c^{\downarrow }-{E}_c^{\uparrow } \) is the conduction band-edge spin-splittings and the \( \Delta {E}_v={E}_v^{\downarrow }-{E}_v^{\uparrow } \) is the valence band-edge spin-splittings of Ba1 − xCrxSe at Γ high symmetry point. The 〈s〉 is the half total magnetic moment per Cr atom [46], and the x is the concentration of Cr impurity.

The computed p–d exchange splitting and exchange constants are given in Table 4. The positive N0α constant suggests the ferromagnetic coupling between the 3d states of chromium (Cr) and conduction bands. The \( {\Delta }_x^v(pd) \) and \( {\Delta }_x^c(pd) \) parameters determine the attraction nature in the Ba1 − xCrxSe. However, the negative \( {\Delta }_x^v(pd) \) of Ba1 − xCrxSe at all concentrations means that the potential of minority spin is effective compared to the majority spin [48], this is an important property of spin-polarized materials [48, 49].

Table 4 Calculated p–d exchange splitting \( {\Delta }_x^v(pd)={E}_v^{\downarrow }-{E}_v^{\uparrow } \) and \( {\Delta }_x^c(pd)={E}_c^{\downarrow }-{E}_c^{\uparrow } \), and exchange constants N0α and N0β for Ba1 − xCrxSe at concentrations x = 0.25, 0.5, and 0.75
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4 Conclusion

The DFT based on the FP-LAPW method is used to calculate the structural, electronic, and magnetic properties of Ba1 − xCrxSe at concentrations x = 0.25, 0.50, and 0.75. We have employed the GGA-WC exchange and correlation potential to predict the structural parameters, the spin-polarized band structures and densities of states. The results of lattice constant and indirect band gap of BaSe are in good agreement with theoretical calculations. The electronic structures of Ba1 − xCrxSe compounds show a half-metallic behavior for all concentrations x. The half-metallic character around Fermi level of Ba1 − xCrxSe results from the strong p–d hybridization of majority spin and a gap of minority spin, leading to spin polarization of 100%. The total magnetic moments are 4 μB confirm both ferromagnetic nature and half-metallic behavior of Ba1 − xCrxSe doping compounds. Therefore, the Ba1 − xCrxSe materials seem to be promising candidates for possible spintronics applications. From our knowledge, there are no previous theoretical or experimental studies on the Ba1 − xCrxSe materials; thus, we hope that our results serve as a reference for future theoretical and experimental researches.