1. Introduction

Due to the high band gaps of the alkaline earth fluorides, they are transparent in a very high frequency region. This made them useful as window materials which are transparent in infrared and ultraviolet wavelengths. Many of them crystallise in the fluorite structure. Examples are; CaF₂, BaF₂, CdF₂, SrF₂, PbF₂ and they are highly ionic [1-5].

Some authors studied the optical properties of these compounds by experiment [6-15]. Metal and fluorine occupy the positions (0 0 0) and (0.25 0.25 0.25) respectively.

The mixed divalent metal fluoride, Ca_0.50Ba_0.50F₂, are formed when BaF₂ and CaF₂ are doped, with Ca and Ba in the ratio 1:1. They are formed by high energy planetary ball mill [16-21] and Molecular beam epitaxy [22]. Ba_1-xCa_xF₂ (0 < x < 1) can be prepared by mechano-chemical reaction at low temperatures from the binary fluorides BaF₂ and CaF₂ [18,19]. Ba_0.50Ca_0.50F₂ adopt the cubic fluorite structure [23-25].

Sata et al. observed fluoride ion conductivities in layered BaF₂-CaF₂ composites, prepared by molecular beam epitaxy (MBE), greatly larger than the binary fluorides [22]. Similar result was obtained for high-energy ball-milled BaF₂-CaF₂ alloys [19]. The ion conductivity exhibit a maximum at x= 0.5 and also a minimum in the activation energy at x = 0.5 [21]. These properties of Ba_0.50Ca_0.50F₂ alloy are essential in fuel cells, batteries, capacitors and solid electrolytes. The ion conductivities of Ba_1-xCa_xF₂ has also been calculated using theory [24,26]. Using X-ray diffraction, the lattice constant of Ba_0.50Ca_0.50F₂ was measured to be 5.87Å [25].

The study of the electronic properties of the mixed Ba_0.50Ca_0.50F₂ crystal is important as they intermediate between the two compounds (BaF₂ and CaF₂) with equal concentration ratio of the cations. Literatures show that the electronic, structural, lattice dynamics, dielectric and photoelastic properties of the binary compounds have been investigated theoretically and experimentally. However, very little electronic and structural theoretical investigations have been done on the mixed fluorides crystal.

2. Method of Calculation

Structural and electronic calculations were carried out using density functional theory (DFT) [27,28] as employed in the QUANTUM ESPRESSO code [29]. We used a kinetic energy cutoff of 60 Ry for the plane wave basis set, while the total energy convergence was set to 0.1mRy. The electron-ion interactions were modeled using ultrasoft pseudopotentials [30]. The pseudopotential was produced using scalar relativistic calculation.

The reciprocal space integration was done by sampling the Brillouin zone of the atoms with a 6 × 6 × 6 Pack-Monkhorst net [31]. The present calculations for Ba_0.50Ca_0.50F₂ are performed on 12-atom supercells, which is sufficient to guarantee converged results.

In our investigations, we used generalized gradient approximation method of Perdew, Burke and Ernzerhof (PBE) [32]. We performed band structure calculations with 96 k-points in the Brillouin zone (BZ).

In this work, we present the results of structural and pressure-dependent study of the band gap of Ba_0.50Ca_0.50F₂. The lattice constant and bulk modulus are fitted to the Murnaghan’s equation of state [33].

The density functional theory is not accurate for wide band gap materials [34]. The GGA-DFT describes the eigenvalues of the electronic states inaccurately, which results to underestimations of band gaps when compared with experiment. This underestimation is corrected by using the empirical model presented by Morales-García, et al. in 2017 for semiconductors and insulators [35]. The relation between experiment and G₀W₀ electronic band gap is given by

(1)

G₀W₀ is an approximation made to the self energy of a many body system of electrons. The self energy is expanded in terms of the single particle Green’s function G and the screened Coulomb interaction W.

The relation between DFT(PBE) and G_oW_o band gap energy is

(2)

From equation 1 and 2, the correlation between modelled experimental and DFT(PBE) band gap energy is derived to be

(3)

A linear relation between energy gap at p=0 and at any pressure p is given as,

(4)

where k = pressure coefficient of the band gap defined by k = dE_g/dp with units of eV/GPa.

For solid materials, the refractive index is related to the dielectric constant by the relation, ε =n². To calculate the dielectric constant of the alloy and its constituent binary compounds, we use Herve-Vandamme relation [36] given as

(5)

where A=13.6eV, is the hydrogen ionization energy and B= 3.47eV is a constant assumed to be the difference between the UV resonance energy and band gap energy.

In this work, we present the results of DFT(PBE) calculation for both structural and electronic properties of the solids. Band gaps obtained using empirical model are compared with the experimental band gap. The dielectric constants of the compounds are also calculated. Variation of the fundamental band gap of the alloy under pressure is investigated.

3. Results and Discussion

Table 1 shows the calculated lattice parameter of the binary compounds and their composites. The lattice parameters are calculated to be 6.22Å, 5.49Å and 5.91Å for BaF₂, CaF₂ and Ba_0.50Ca_0.50F₂ respectively. They compare well with experiment. The experimental lattice constants of CaF₂, BaF₂ and Ba_0.50Ca_0.50F₂ are taken from [37], [38] and [39] respectively. The value of lattice parameter for Ba_0.50Ca_0.50F₂ falls between that of CaF₂ and BaF₂. The bulk modulus of the alloy is given in table 2. For comparison purpose, the bulk moduli of the binary compounds are calculated. The experimental bulk modulus of CaF₂ is 82.71GPa [37] while that of BaF₂ is 59GPa [40]. The calculated bulk moduli are in good agreement with experimental values. The calculated bulk moduli are 57.8GPa, 82.1GPa and 64.1GPa for BaF₂, CaF₂ and Ba_0.50Ca_0.50F₂ respectively. As far as we know, there is no experimental value for the bulk modulus of the Ba_0.50Ca_0.50F₂.

Table 1. Calculated Lattice Parameters, a(Å) of CaF2, BaF2 and Ba0.50Ca0.50F2

Table 2. Calculated Bulk Moduli, B(GPa) of CaF2, BaF2 and Ba0.50Ca0.50F2

Table 3 shows the band gap of the compounds obtained using PBE, empirical model and experiment. The band gap of CaF₂, BaF₂ and Ba_0.50Ca_0.50F₂ using the empirical model are 10.98eV, 10.10eV and 10.54eV respectively. The experimental band gap of CaF₂ and that of BaF₂ are taken from [41]. The band gap of the alloy intermediate between that of CaF₂ and BaF₂. The band gap obtained using empirical model compare well with experiment. The value of the band gap energy of Ba_0.50Ca_0.50F₂ indicates that this compound might be a good candidate for a frequency conversion material working in the ultraviolet region. Ba_0.50Ca_0.50F₂ can serve as an alternative to quartz (SiO₂) to use as a transparent medium for lasers. Silicon used in solar cells is a very shiny material, which can send photons bouncing away before they have done their job, so an antireflective coating is applied to reduce those losses. Ba_0.50Ca_0.50F₂ can be used in such antireflective coating.

Table 3. DFT(PBE), empirical model and experimental band gap energy of CaF2, BaF2 and Ba0.50Ca0.50F2

Table 4 shows the dielectric constant of the compounds. The dielectric constant of the binary compounds are comparable with experiment. The experimental dielectric constants are taken from [38]. Ba_0.50Ca_0.50F₂ is a dielectric and it can be used as optical fiber to transfer information from one point to another using light pulses. The information transmitted is essentially digital information generated by telephone systems, cable television companies, and computer systems.

Table 4. Calculated dielectric constants, ε, of CaF2, BaF2 and Ba0.50Ca0.50F2

Figure 1 shows the structure of the cations and anions in the Ba_0.50Ca_0.50F₂ crystal. The milling of fluorite BaF₂ and CaF₂ with x=0.50 results in the formation of Ba_0.50Ca_0.50F₂ crystallising in the cubic fluorite structure.

Figure 1. Crystal Structure of Ba0.50Ca0.50F2

The band structure of Ba_0.50Ca_0.50F₂ is presented in Figure 2. The band gap which is indirect exists along L-Γ path. It is generally accepted that GGA electronic band structures are in good agreement with experiments in regard to the arrangement of energy levels and structure of bands [42].

Figure 2. Band Structure of Ba0.50Ca0.50F2

There is linear variation of the band gap with pressure as seen in Figure 3. The band gap remains indirect for a pressure up to 5.41GPa. The pressure coefficient of the band gap is calculated to be 0.062eV/GPa.

Figure 3. Variation of Band Gap Energy of Ba0.50Ca0.502F2 with Pressure

4. Conclusions

In this work, we have reported the investigated electronic and structural properties of Ba_0.50Ca_0.50F₂ using the PP-PW method within GGA. At equilibrium, the band gap energy of the alloy was calculated to be 10.54eV, indicating its possible application in the UV region of light. It was also shown that for Ba_0.50Ca_0.50F₂, the band gap increases directly with pressure. From the graph, the pressure coefficient of the band gap was obtained. The dielectric constants of the binary compounds and alloy were determined. We have compared our results of structural parameters, band gaps and dielectric constants with experiment. Good agreement is found with the experimental values. There seem to be no previous works on the bulk modulus, band gap energy, dielectric constant and band structure of Ba_0.50Ca_0.50F₂, so our results can serve as reference data for this compound. We can consider the present findings as a prediction study, and hence will motivate more works on this material.