3.1. Stability and structural properties

In 1966, R. Hoppe studied the crystallography of the compound Rb₂NaVF₆ [¹⁹,²⁰]. The quaternary vanadium fluoride Rb₂NaVF₆ crystallizes in the cubic NaCl (B1) structure with a space group of (Fm3-m) (no. 225). Figure 1 and Table I exhibit the crystal structure and the Rb, Na, V, and F atoms positions of Rb₂NaVF₆, respectively. We have optimized the Rb₂NaVF₆, with and without spin-polarized calculations. We have computed the total energies corresponding to the unit cell volumes and fitted to the Birch-Murnaghan’s equation of state (EOS) [²¹] to verify the phase stability of the compound. And evaluate the structural properties such as the equilibrium lattice constant a, bulk modulus B and the bulk modulus pressure derivative B’, the inspected structural constants of Rb₂NaVF₆ are recapitulated in Table II, our results are in good agreement with the obtainable theoretical and experimental data. By comparing the two curves, it is obvious that the ferromagnetic curve is below the non-magnetic curve, which indicates that the Rb₂NaVF₆ structure is stable in ferromagnetic phase. The energy verses volume curves are shown in the Fig. 2.

FIGURE 1 Crystal structure of quaternary vanadium fluoride Rb₂NaVF₆. 

FIGURE 2 Volume optimization curve of quaternary vanadium fluoride Rb₂NaVF₆. 

3.2. Electronic, magnetic properties and half-metallic comportment

In this section, we have predicted the electronic band structure, total (TDOS), and partial (PDOD) densities of states for Rb₂NaVF₆, using GGA-PBEsol [¹³] accompanying with (TB-mBJ) of Tran-Blaha modified Becke-Johnson approximation [¹⁴,²²]. Figures 3 and 4 exhibit the band structure of Rb₂NaVF₆ for spin-up and spin-down, respectively. It is apparent in Fig. 3 that the highest valence band is above the Fermi level, hence the metallic behavior of the structure. For Fig. 4, we mark two gaps, the half-metallic ferromagnetic gap (G _HMF ) and the half-metallic gap (G _HM ), G _HMF is the energy difference between the valence-band maximum (VBM) and conduction-band minimum (C _BM ) [⁴]. Rb₂NaVF₆ compound has a direct half-metallic ferromagnetic gap at Γ − Γ of 7.837 eV and large half-metallic gap at Γ−Γ high-symmetry point of 3.582 eV. The non-zero values of the half-metallic gap for Rb₂NaVF₆ demonstrate it to be true half-metallic ferromagnets and making it promising materials for employment in spintronic applications.

FIGURE 3 Spin-up band structure obtained with TB-mBJ for Rb₂NaVF₆. 

FIGURE 4 Spin-down band structure obtained with TB-mBJ for Rb₂NaVF₆. 

To know the origin of the ferromagnetic and the halfmetallic behavior, we have calculated the densities of states (DOS) around the Fermi level (EF). Fig. 5 shows the total densities of states of Rb₂NaVF₆, Rb, Na, V, and F atoms. We notice that the total density of states (TDOS) is symmetrical except at Fermi level (0 eV) for spin-up and 3.618 eV for spin-down. The total densities of states are dominated by Rb between the energy from -11.102 and -10.694 eV, and at energy more than 8 eV. While the F dominates at an energy between -6.476 and -4.190 eV. Moreover, V dominates the peaks, which are at the energies 0, 3.618, and 4.380 eV. The half-metallic nature can be seen in the channel spin-down. To comprehend the source of the half-metallic gap and the half-metallic ferromagnetic gap we have plotted the partial densities of states. Figure 6 represent the partial densities of states of Rb₂NaVF₆, the pike located at the 0 eV (Fermi level) and 3.618 eV are mainly formed by the 3d − t2g (V) and 2p (F) states. The TDOS of Rb₂NaVF₆ depicted asymmetrical states of spin-up and spin-down due to strong p − d hybridization between the 2p (O) and 3d − t2g (V) states around the Fermi level for the majority-spin direction. This hybridization occurs mainly at the top of the majority-spin valence bands and crosses the Fermi level, leading to the metallic nature for Rb₂NaVF₆ compounds, with spin polarization of 100% by using the following expression [²³]:

FIGURE 5 Spin-polarized total densities of states (TDOS) of Rb₂NaVF₆. The Fermi level is set to zero. (vertical dotted red line). 

FIGURE 6 Spin-polarized partial densities of states (PDOS) of Rb₂NaVF₆. The Fermi level is set to zero. (vertical dotted red line) 

where P is the spin polarization, N ↑ (E _F ) and N ↓ (E _F ) are the densities of states of the majority spin and minority spin around the Fermi level, respectively. Table III display the total and local magnetic moments of Rb₂NaVF₆. To interpret the implications of the p−d interchange phenomenon on the magnetic behavior, we have calculated the total and local magnetic moments of the Rb, Na, V, and F atoms as well as the interstitial sites. The results exhibit the total magnetic moment for the Rb₂NaVF₆ compound was around 2 µB, usually caused by the V atoms’ the local magnetic moment. 3d (V) majority-spin states, which are partially occupied with three electrons, resulting in a total magnetic moment of 2 µB. Otherwise, the calculated magnetic moment of the V atom is reduced to 1.71792 µB and minor local magnetic moments are induced at Rb, Na, F, and interstitial sites overdue to the p − d exchange interaction between 2p (F) and 3d − t2g(V) states. Important parameters established on band structures such as p − d exchange splitting have also been calculated, ∆xcpd=Ev↓-Ev↑ and ∆xcpd=Ec↓-Ec↑ and the s − d exchange constants N ₀ α (conduction band) and the p−d exchange constant N ₀ β (valence band).

Both N ₀ α and N ₀ β parameters are calculated employing the mean-field theory whose expression is as follows [²⁴,²⁵]:

where ∆Ec=Ec↓-Ec↑ is the conduction band-edge spinsplitting and ∆Ev=Ev↓-Ev↑ is the valence band-edge spinsplitting of Rb₂NaVF₆ at high symmetry point Γ. hsi is the half total magnetic moment per vanadium atom and x is the concentration of V in the structure. Table IV exhibits the calculated p − d exchange splitting and exchange parameters. The positive N ₀ α constant indicates the ferromagnetic coupling between the 3d states of V and conduction bands. Also, we have calculated an important property of spin-polarized materials, the ∆xvpd and ∆xcpd parameters, which determines the attraction character in the Rb₂NaVF₆. However, the positivity ∆xvpd of Rb₂NaVF₆ structure means that the potential of minority spin is effective compared to the majority spin [²⁶,²⁷].