1 Introduction

In recent years, the diluted magnetic semiconductors (DMS) have attracted significant interest because they are a one class of half-metallic ferromagnets (HMFs) owing to their electronic structures that exhibit semiconducting character in onespin channel and metallic nature in the other direction, which make them promising materials for spin injection in spintronics [1,2,3,4,5,6,7] The HMFs can be extensively used in spintronics devices as spin injection, spin filters, spin valves, tunnel junctions and magneto-resistive randomaccess memory [8] Spintronics devices have many advantages over conventional semiconductor devices such as non-volatility, increased transistor density, lower electric power consumption and higher dataprocessing speed [9,10,11].

The DMS based on transition metal-doped II–VI semiconductors are promising candidates for spintronics applications because of their half-metallic ferromagnetic performance [12,13,14] and they show the stability of ferromagnetic state at temperatures higher than room temperature [1]. The II–VI alkaline-earth-metal oxides form a significant class of materials because they constitute a link between the highly ionic halides and the widely covalent semiconductors [15,16,17]. These compounds have attracted much attention due to their potential technological applications ranging from catalysis to microelectronics [18,19,20].

The barium oxide (BaO) belongs to the II–VI alkaline-earth-metal binary oxides, which has a large band gap due to its high ionicity [18]; it is an important component for application of a thermionic electron emitter [21] Besides, the BaO is considered a potential DMS material in various applications for spintronics and optoelectronics, such as light emitting devices and ultralowpower memory devices [22,23,24,25] Recently, several studies have been carried out on the half-metallic and the magnetic properties in DMSbased II–VI such as the half-metallicity and magnetic properties of Cr-substituted BaTe [5], the half-metallic ferromagnetic behavior in Sr1−xVxO [14], the half-metallic ferromagnets of Eu-doped alkaline-earth chalcogenides [26] the magnetic properties of Zn1−xTMxTe (TM = Mn, Fe, Co, and Ni) [27], the magnetic characterization of nanocrystalline of Zn1−xCoxO synthesized by combustion reaction [28], the magnetic properties of transition metaldoped II–VI oxide semiconductors [29] and the half-metallic ferromagnetism in vanadium-doped rock-salt SrO [30].

In this study, we have investigated the doping effect of chromium (Cr) impurity on the structural parameters, electronic structures and the half-metallic ferromagnetic properties of Ba1−xCrxO at various compositions x = 0.25, 0.5 and 0.75. We have used in our calculations the first-principle methods of density functional theory [31, 32] based on full-potential linearized augmented plane-wave approach within generalized gradient approximation functional of Wu and Cohen [33].

2 Methodology and Details of Calculations

The bulk BaO has conventional rock-salt NaCl (B1) structure with space group (\(Fm\bar {3}m)\) No. 225, which the Ba and O atoms are located respectively at the (0, 0, 0) and (0.5, 0.5, 0.5) sites. We have performed the Ba1−xCrxO supercells of eight atoms at concentrations x = 0.25, 0.5, and 0.75 by the substitution of one, two, and three Ba atoms by Cr impurities. For the concentration x= 0.5, we get the Ba0.5Cr0.5O compound of tetragonal structure with space group of P4/mmm No. 123. We have obtained for x= 0.25 and 0.75, respectively, the Ba0.75Cr0.25O and Ba0.25Cr0.75O compounds, which have cubic structures with space group of \(Pm\bar {3}m\) No. 221 (see Fig. 1).

Fig. 1
figure 1

Crystal structures of BaO and Ba1−xCrxO compounds at concentrations x = 0.25, 0.5, and 0.75. a BaO, b Ba0.75Cr0.25O, c Ba0.5Cr0.5O, and d Ba0.25Cr0.75O

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We have investigated the structural, electronic, and magnetic properties of Ba0.75Cr0.25O, Ba0.5Cr0.5O, and Ba0.25Cr0.75O supercells by using the full-potential linearized augmented plane-wave (FP-LAPW) method as implemented in WIEN2K code [34] based on density functional theory (DFT) [31, 32]. The exchange correlation potential is treated by the use of generalized gradient approximation functional of Wu and Cohen (GGA-WC) [33]. We have selected the average of the muffin-tin spheres radii of Ba, O, and Cr atoms in such a way that the muffin-tin spheres do not overlap. We have expanded the wave functions in the interstitial region to plane waves with a cutoff of \(K_{\max }\) = 9.0/RMT, where the RMT is the average radius of the muffin-tin spheres and the \(K_{\max }\) is the magnitude of the largest K vector in the plane wave. The charge density was Fourier expanded up to \(G_{\max }\) = 14 (a.u.)− 1, where \(G_{\max }\) is the largest vector in the Fourier expansion, whereas the maximum value for partial waves inside the atomic sphere was \(l_{\max }\) = 10. We have used the cutoff of − 6 Ry to determine the separation between the valence and the core states. For the Brillouin-zone integration, we have used the Monkhorst–Pack mesh [35, 36] of (4×4×3) k-points for Ba0.5Cr0.5O and (4×4×4) k-points for BaO, Ba0.75Cr0.25O, and Ba0.25Cr0.75O. The self-consistent was achieved when the total energy convergence was set at 0.1 mRy.

3 Results and Discussions

3.1 Structural Properties

We have computed the structural parameters by the use of the empirical Murnaghan’s equation of state [37] that describes the total energies as a function of equilibrium volumes. The obtained structural parameters such as the lattice constants (a), bulk modules (B), and its pressure derivatives (B′) for the BaO and the Ba1−xCrxO at various concentrations, and other theoretical [18, 38,39,40] and experimental [41,42,43,44] results are summarized in Table 1. The lattice constant of BaO calculated with GGA-WC approach stay in good agreement with the experimental data [41,42,43,44], and it is better than the theoretical calculations of refs. [18, 38, 39] and ref. [40] found, respectively, by the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) [45] and the local density approximation (LDA) [46]. This is due to the accurate performance of GGA-WC potential for determining the structural properties of solids [33, 47,48,49,50,51,52]. For Ba1−xCrxO, we have found that the lattice parameter decreases with increasing concentration of Cr atom owing to a difference between Cr and Ba atomic radii. Consequently, the Ba1−xCrxO becomes harder when the concentration increases of Cr atom.

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

In this section, we have used the calculated lattice parameters to describe the electronic and half-metallic properties of Ba1−xCrxO as a function of concentration x of chromium impurity. The projected partial and total densities of states (DOS) of Ba0.75Cr0.25O, Ba0.5Cr0.5O, and Ba0.25Cr0.75O are given by Figs. 2, 3 and 4, respectively. To characterize the electronic structures of our compounds, we have investigated the DOS contribution around Fermi level (EF) Figs. 2, 3 and 4 show that the majorityspin states are metallic whereas the gap occurs at EF for the minority spin. The DOS of majority spin around EF are mainly dominated by the contribution of hybridization between the 2p states of O atoms and 3d states of Cr impurities, while the DOS of minority spin are empty at EF. Consequently, the Ba1−xCrxO compounds revealed a half-metallic feature at all concentrations Besides, the 3d (Cr) of majority-spin states are divided into levels the t2g and eg states, which the t2g are fully occupied and the eg are partially occupied. Thus, the eg states are located above t2g states, indicating that the Cr impurity are located in the tetrahedral surroundings.

Fig. 2
figure 2

Spin-polarized total and partial densities of states of Ba0.75Cr0.25O. The Fermi level is set to zero (vertical dotted line)

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Fig. 3
figure 3

Spin-polarized total and partial densities of states of Ba0.5Cr0.5O. The Fermi level is set to zero (vertical dotted line)

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Fig. 4
figure 4

Spin-polarized total and partial densities of states of Ba0.25Cr0.75O. The Fermi level is set to zero (vertical dotted line)

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Figures 5, 6, 7 and 8 display the spin-polarized band structures of BaO, Ba0.75Cr0.25O, Ba0.5Cr0.5O and Ba0.25Cr0.75O, respectively, where the 3d (Cr) bands are demonstrated by filled circles. The plots of doping systems depict that the majority-spin bands are characterized by the localized 3d (Cr) levels in the gap, while for minority-spin bands the completely empty 3d (Cr) levels are situated in conduction bands far than EF Therefore, the majority-spin bands reveal a metallic character because some bands cut EF whereas the minority-spin bands show forbidden bands around EF Consequently, the spin-polarized band structures of Ba1−xCrxO are half-metallic ferromagnets with spin polarization of 100%.

Fig. 5
figure 5

Band structures of majority spin (up) and minority spin (down) for BaO. The Fermi level is set to zero (horizontaldotted line)

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Fig. 6
figure 6

Band structures of majority spin (up) and minority spin (down) for Ba0.75Cr0.25O, showing 3d (Cr) character bands by the filled-circle mode. The Fermi level is set to zero (horizontaldotted line)

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Fig. 7
figure 7

Band structures of majority spin (up) and minority spin (down) for Ba0.5Cr0.5O, showing 3d (Cr) character bands by the filled-circle mode. The Fermi level is set to zero (horizontaldotted line)

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Fig. 8
figure 8

Band structures of majority spin (up) and minority spin (down) for Ba0.25Cr0.75O, showing 3d (Cr) character bands by the filled-circle mode. The Fermi level is set to zero (horizontaldotted line)

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Figure 5 shows clearly that the majority and minority spins of BaO have the same band structures with direct band gap (EXX) located between the maximum valence bands and the minimum conduction bands at X high symmetry point On the other hand, Figs. 6, 7 and 8 show that majority-spin bands are more in number than minority spin bands owing to the exchange interaction between 2p (O) and 3d (Cr) levels, and hence the half-metallic ferromagnetic (HMF) gaps (GHMF) and half-metallic gaps (GHM) are formed in the minority-spin bands. Our results of band gaps such as the direct band gap (EXX) of BaO, and the HMF and HM gaps of minority-spin bands of Ba1−xCrxO at different concentrations with other theoretical [18, 53] and experimental [54,55,56] data are given in Table 2. The HM gap of a half-metallic material 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 [57, 58]. The calculated direct band gap of 1.884 eV of BaO is in good agreement with the theoretical calculations of refs. [18, 53]. For Ba1−xCrxO compounds, we have found that Ba0.75Cr0.25O, Ba0.5Cr0.5O and Ba0.25Cr0.75O have direct HMF gaps of 1.549, 1.277 and 1.032 eV, respectively. From minority-spin bands, we understand that the 3d (Cr) bands broaden strongly in the gap with increasing concentration of Cr impurity, leading to the decrease of HMF gap The half-metallic gap is an important factor to determine the use of material in spintronics [50]; it is located between the minimum conduction bands and EF for Ba0.25Cr0.75O and between the maximum valence bands and EF for both Ba0.75Cr0.25O and Ba0.5Cr0.5O. The computed HM gap is 0.055 eV for Ba0.75Cr0.25O, 0.218 eV for Ba0.5Cr0.5O and 0.308 eV for Ba0.25Cr0.75O, which increases with increasing of Cr concentration. The Ba1−xCrxO compounds are half-metallic ferromagnetic with HM gaps which make them potential materials for possible spintronics applications.

Table 2 Calculated direct band gap EXX for BaO, half-metallic ferromagnetic gap (GHMF) and half-metallic gap (GHM) of minority-spin bands for Ba1−xCrxO at concentrations x = 0.25, 0.5 and 0.75
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3.3 Magnetic Properties

The magnetism in the Ba1−xCrxO doping systems is induced by the 3d states of magnetic Cr impurity, which is partially filled with four electrons. According to the Hund’s rule, these unpaired electrons create a total magnetic moment of 4 Bohr magneton (μB) per Cr atom in Ba1−xCrxO materials. Table 3 summarizes the calculated total magnetic moment (MM), local MM for Cr, Ba, and O atoms, and MM in the interstitial site per Cr atom for Ba0.75Cr0.25O, Ba0.5Cr0.5O, and Ba0.25Cr0.75O. The partial MM of the Cr atom for each compound is less than 4 μB, and minor magnetic moments are induced in the Ba, O, and interstitial sites; this is due to the p–d exchange interaction between the 2p (O) and 3d (Cr) states. For all compounds, the magnetic moments of Cr and O atoms are antiparallel, implying that the valence bands interact anti-ferromagnetically with Cr magnetic spins. The positive magnetic spins of Cr and Ba atoms for Ba0.75Cr0.25O and Ba0.5Cr0.5O depict the ferromagnetic interaction between Cr and Ba sites, while the opposite sign of magnetic moments of Cr and Ba atoms for Ba0.25Cr0.75O describes the anti-ferromagnetic interaction between Cr and Ba sites.

Table 3 Calculated total and local magnetic moments per Cr atom (in Bohr magneton (μB)) within the muffin-tin spheres and in the interstitial sites for Ba1−xCrxO at concentrations x = 0.25, 0.5 and 0.75
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We have employed the results of the electronic band structures to calculate the exchange coupling between magnetic impurity and valence (conduction) bands such as the s–d exchange constants N0α (conduction band) and the p–d exchange constants N0β (valence band). These two important parameters are determined from mean-field theory by the following expressions [59, 60]:

$$ N_{0}\alpha =\frac{{\Delta} E_{c}}{x\langle s\rangle} $$
(1)
$$ N_{0}\beta =\frac{{\Delta} E_{v}}{x \langle s\rangle} $$
(2)

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−xCrxO at Γ high symmetry point. The 〈s〉 is the half total magnetic moment per Cr atom [59], and the xis the concentration of Cr atom. Moreover we have described the attraction character in the Ba1−xCrxO doping materials by determining the two important p–d exchange splitting \({{\Delta }_{x}^{v}}\left (pd \right )=E_{v}^{\downarrow } -E_{v}^{\uparrow } \) and \({{\Delta }_{x}^{c}}\left (pd \right )=E_{c}^{\downarrow } -E_{c}^{\uparrow } \).

Our results of N0α and N0β exchange constants and \({\Delta }_{x}^{v}\left (pd \right )\) and \({{\Delta }_{x}^{c}}\left (pd \right )\) p–d exchange splittings are given in Table 4 The positive sign of N0α and N0β, respectively at the concentrations (x = 0.25, 0.5 and 0.75) and (x = 0.25 and 0.5), describes the ferromagnetic coupling between 3d (Cr) states and valence (conduction) bands. In contrast, the coupling is anti-ferromagnetic between 3d (Cr) states and valence bands in the case of negative value of N0β for the concentration x = 0.75. On the other hand, the results of p–d exchange splitting show the negative value of \({{\Delta }_{x}^{v}}\left (pd \right )\) for Ba1−xCrxO at concentration x = 0.75, revealing that the potential of minority spin is effective with respect to the majority spin [61], which is an important feature for the spin-polarized compounds [61, 62].

Table 4 Calculated p–d exchange splitting \({{\Delta }_{x}^{v}}\left (pd \right )=E_{v}^{\downarrow } -E_{v}^{\uparrow } \) and \({{\Delta }_{x}^{c}}\left (pd \right )=E_{c}^{\downarrow } -E_{c}^{\uparrow } \) and exchange constants N0α and N0β for Ba1−xCrxO at concentrations x = 0.25, 0.5, and 0.75
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4 Conclusion

We have employed the DFT first-principle methods with FP-LAPW approach and GGA-WC exchange potential to investigate the effect of Cr impurity on the structural, electronic, and ferromagnetic properties of Ba1−xCrxO at various concentrations x = 0.25, 0.5, and 0.75. The optimized structural parameters show that the lattice constant of Ba1−xCrxO doping compound decreases with increasing concentration of Cr impurity compared with binary BaO; this is due to the difference between Cr and Ba atomic radii. The total magnetic moments of 4 μB and the ferromagnetic coupling between host valence bands and the 3d (Cr) states confirm the magnetic character of Ba1−xCrxO materials. From our findings of electronic and magnetic properties, we have predicted that the Ba1−xCrxO compounds are half-metallic ferromagnetic with half-metallic gaps and spin polarization of 100%, which make them promising candidates for possible spintronics applications.