1. Introduction

The alkaline earth fluorides have been highly studied by first principles methods [1,2]. Both CaF₂ and SrF₂ are ionic insulators with a wide band gap. The Alkaline Earth Fluorides (AEF) have wide band gaps and low refractive indices. AEF find applications in optical coatings and transmissive components in the deep ultraviolet [3].

The electronic and ionic conductivity studies of fluorite type compounds (CaF₂, SrF₂, BaF₂ and SrCl₂) have been performed [4,5]. Calculation on ground state and lattice dynamical properties of ionic conductors CaF₂, SrF₂ and BaF₂ using density functional have been carried out [6,7]. Cadelano and Cappellini [8] in 2011 studied the electronic structure of CaF₂, SrF₂, BaF₂, CdF₂, HgF₂, PbF₂ by means of density functional theory within the local density approximation for the exchange correlation energy.

The ground state properties and electronic band structures of CaF₂, SrF₂ and CdF₂ have been calculated using the shell model in the pair potential approximation [9].

Structural stability of CaF₂ under high pressure was conducted using first-principles calculations based on density functional theory. The sequence of the pressure-induced phase transition of CaF₂ is the fluorite structure -Fm3m, the PbCl₂ type structure-Pnma, and the Ni₂In-type structure-P6₃ /mmc [10].

Phase transitions and equations of state of the alkaline earth fluorides CaF₂ and SrF₂ are examined up to 95GPa by angle-dispersive x-ray diffraction experiments in laser-heated diamond anvil cells at beamlines. They confirmed that both materials undergo a phase transition from the cubic fluorite structure to the orthorhombic cotunnite-type structure at pressures less than 10GPa. Both materials further transform to a hexagonal Ni₂In-type structure at 84 and 36GPa, respectively, following laser heating [11].

Hao and his coworkers showed using first-principles calculation based on density functional theory (DFT) with the plane wave basis set as implemented in the CASTEP code the phase transition of SrF₂ from the fluorite structure (𝐹𝑚3𝑚) to the PbCl₂-type structure (𝑃𝑛𝑚𝑎), and to the Ni₂In-type phase (𝑃63/𝑚𝑚𝑐) [12].

The sequence of pressure-induced phase transition of BaF₂ was simulated by using an atomistic calculation based on the shell model. This fluorite crystal presents two pressure-induced phase transitions at approximately 3 and 15GPa. At 3GPa, the fluorite phase transforms into the α-phase, which has an orthorhombic cotunnite-type structure PbCl₂, Z = 4). At 15GPa, it transforms into hexagonal Ni₂In-type structure (P63/mmc, Z = 2) [13].

Liu and his co-workers extensively explored the structural and electronic properties of CdF₂ to high pressure by ab initio calculations based on the density functional theory [14]. PbF₂ has been shown to exist in the cubic, orthorhombic, hexagonal, and monoclinic phases [15].

It has been found that the complete phase-transition sequence for CoF₂ is tetragonal rutile (P4₂/mnm) ⟶ CaCl₂ type (orthorhombic Pnnm) ⟶ distorted PdF₂ (orthorhombic Pbca) + PdF₂ (cubic Pa-3) in coexistence ⟶ fluorite (cubic Fm3m) ⟶ cotunnite (orthorhombic Pnma) [16]. Stavrou and his colleagues showed that MnF₂ exists in these structural phases; rutile, SrI₂ and PbCl₂ [17].

Hoat et al. reported first principles calculations of structural, electronic and optical properties of SrF₂ such as reflectivity, refraction index, conductivity and energy loss function under pressure, done using full-potential linearized augmented plane wave (FP-LAPW) method based on density functional theory as implemented on WIEN2k code [18].

Cappellini et al. investigated the electronic and optical properties of the stable clusters of alkaline earth metal fluorides, which include, MgF₂, CaF₂, SrF₂, and BaF₂ using density functional theory (DFT) and time-dependent DFT (TDDFT) methodss with a localized Gaussian basis set [19].

Ca_xSr_1-xF₂ is used as optical and laser materials [20]. Ca_xSr_1−xF₂ is applicable as an encapsulant or masking material for GaAs [21]. The alloy is also used in quantum devices e.g Resonant Tunneling Diodes [22].

Siskos et al. investigated the epitaxial growth of fluoride heterostructure of Ca_xSr_1−xF₂/CdF₂/Ca_xSr_1−xF₂ on Ge for quantum devices [21].

Klimma and his colleagues used the Czochralski method to grow Ca_xSr_1-xF₂ [23].

Oshita and his coauthors used the two-step growth method to grow Ca_xSr_1-xF₂/CdF₂/ Ca_xSr_1-xF₂ quantum well structure on Ge substrate [22].

Takahashi and Tsutsui in 2013 proposed the introduction of a SrF₂ buffer layer to Ca_xSr_1−xF₂ at x=0.42, i.e, Ca_0.42Sr_0.58F₂/SrF₂/Ge structure as a method for the growth of an electron-tunneling barrier layer on Ge with low leakage current [24].

In 2019, Suzuki et al. studied experimentally the band gap energy of Ca_xSr_1−xF₂ by changing the composition ratio of CaF₂ and SrF₂ and used the alloy to fabricate filterless vacuum ultraviolet photodetectors with wavelengths dependent on the band gap of the alloy [25].

The theoretical study of the electronic and structural properties of Ca_xSr_1−xF₂ composing of CaF₂ and SrF₂ are still not clear very well. In this work, we present a systematic study of the electronic and structural properties of Ca_xSr_1−xF₂ alloys in the fluorite-cubic structure.

2. Computational Method

First-principles calculations are performed to study the electronic and structural properties of Ca_xSr_1-xF₂ by employing a Ultrasoft pseudopotential [26] plane wave approach based on density functional theory (DFT) [27,28] and implemented in Quantum Espresso [29] code. The exchange-correlation contribution is described within generalized gradient approximation based on Perdew and Wang [30]. The orbitals of Ca (3s²3p⁶4s²), Sr (4s²4p⁶4d¹5s¹) and F (2s²2p⁵) are treated as valence electrons. The k integration over the Brillouin zone is performed using the Monkhorst and Pack mesh [31]. A mesh of six special k-points was taken in the Brillouin zone for the binary compounds and for the alloys. The total energy of the crystal converged to less than 1mRyd after seven iterations A 6x6x6 sampling k point is utilised for the calculation of the total energy of the fluorite crystals. The lattice constants and bulk moduli are fitted to the total energy and volume to the Murnaghan’s equation of state [32], it allows to get the equilibrium structural properties of both the binary and ternary compounds.

As a starting point, we calculated the structural properties of the binary compounds CaF₂ and SrF₂ in the fluorite phase using GGA scheme. The ternary alloys in cubic fluorite structure are modelled at some selected compositions of x and are described by periodically repeated supercells.

3. Results and Discussion

3.1. Structural Properties

For the binary fluorides and their alloys, the lattice parameter and the bulk modulus are determined. The alloys were studied by using a primitive mesh of 12 atoms. The values obtained for different compositions of Ca for the lattice parameters and bulk modulus are given in Table 1 and 2 respectively. The calculated properties are in good agreement with experiments and theoretical results for the binary compounds. The lattice constant decreases with concentration, x. The calculated values of the bulk modulus, using the GGA approximation, decreases from CaF₂ to SrF₂; from the lower to the higher atomic number. This suggests that SrF₂ is more compressible than CaF₂. The variation of the lattice parameter and bulk modulus as a function of the concentration of alloy Ca_xSr_1−xF₂ is illustrated by Figure 1 and Figure 2, respectively. We note that the lattice parameter decreases almost linearly as a function of the concentration. Our calculations deviate from those calculated by Vegard’s law [33]

(1)

where a_AB, a_BC and are the lattice parameters of binary compounds AB, AC and ternary alloy AB_1-xC_x respectively. However, the linear behaviour is not followed by many alloys and a general form is expressed by a semi-empirical quadratic relationship [34,35]. Hence, the lattice constant can be written as:

(2)

where the quadratic term b is the bowing parameter. For the ternary alloy, (2) can be rewritten as

(3)

We compared calculated lattice parameters with those obtained from Vergard's Law at different x. There is a marginal downward bowing of 0.064Å due to smaller size of Calcium (Ca) atom (than that of Strontium (Sr).

Figure 2 shows the bulk modulus as a function of Ca-composition x for Ca_xSr_1-xF₂ compounds. B increases with composition x. The quadratic fit to the bulk modulus is given as

(4)

Table 1. Calculated Lattice Parameters, a(Å) of CaxSr1−xF2

Table 2. Calculated Bulk modulus, B(GPa) of the CaxSr1−xF2

Figure 1. Variation of lattice parameters, a(Å) of CaxSr1−xF2 with composition, x

Figure 2. Variation of Bulk Modulus, B of CaxSr1−xF2 with composition, x

There is a small deviation from the linear concentration dependence (LCD) in the Bulk Modulus, which gives a slight downward bowing of 5.07GPa with the quadratic fit. The small bowing parameter is due to the bulk modulus of SrF₂ which is 14.82% smaller than CaF₂.

3.2. Electronic Properties

From the GGA scheme, the band-gaps of the ternary alloys and the corresponding binary compounds are presented in Table 3. Energy band gaps are computed along the high symmetry points and compared with experiments as well as available theoretical results. The discrepancy seen in the table is an intrinsic feature of DFT. The calculated band gap is 30-40% less than experiment [36-38]. However, it is accepted that electronic band structures from GGA calculations are qualitatively in good agreement with experiments regarding the ordering of energy levels and band shapes [39]. There is a direct gap at Gamma point in all the ternary alloys.

Table 3. Calculated Band gap energy, Eg(eV ) of CaxSr1−xF2

Table 4. DFT(PBE) and empirical model calculated band gap energy, Eg(eV) of CaxSr1−xF2

The band gap can be improved by using the empirical model of Morales-García, et al. for semiconductors and insulators [40]. From the model, the band gap energy is derived to be:

(5)

The band gaps of the binary fluorides and their alloys are also calculated using the generalized gradient approximation method of Perdew, Burke and Ernzerhof (PBE). The band gaps of the binary compounds obtained using the empirical model compare well with experiment. The Ca_xSr_1-xF₂ alloys are wide gap insulators.

Energy bands structures for various concentrations of Calcium are shown in Figures 3, 4 and 5. For the alloy, the band structures for all concentrations of Ca gave a direct gap at the Γ point. The band gap energy is the difference between the valence band maximum and conduction band minimum.

The composition dependence of Ca_xSr_1-xF₂ band-gaps over the composition range (0-1) can be well fitted by the following expression,

(6)

where denotes the band gap energy of Ca_xSr_1-xF₂, and signify the band gap energy of CaF₂ and SrF₂, respectively while b is the band gap bowing parameter of Ca_xSr_1-xF₂.

Figure 6 shows that the direct gap versus concentration, exhibit a non-linear behavior. The direct gap exhibits a downward bowing with a mean value of 0.345eV within the range of x investigated. The fundamental gap increases considerably with the Calcium composition, x.

The variation of the band gap bowing in Ca_xSr_1-xF₂ with concentration is shown in Figure 7. The bowing remains linear and increases with calcium concentration.

Figures 8, 9 and 10 show the partial density of states of Ca, Sr and F respectively. The red line represents s-orbitals, green line represents p-orbitals and blue line represents d-orbitals. Ca and Sr contain s, p and d orbitals while F contains only s and p orbitals. The partial density of states of the constituent elements of the alloys are used to analyze the total density of states of the alloys.

The total DOS for the ternary alloys, Ca_xSr_1−xF₂ are depicted in Figures 11 and 12 at x=0.25 and 0.50 respectively. The DOS for these concentrations are very similar but the values of the peaks are different according to the composition. There are five regions. The peak at the lowest energy states, which is due to the s electrons of the fluorine, while the next two regions contain the contribution of Ca-s and Sr-s orbitals respectively. The region just below the Fermi level E_F is predominately p states of F, with only a small contribution for p-Ca and p-Sr states. Above the Fermi level, the region is predominately p states of Ca and p and d states of Sr.

The total DOS for the ternary alloy Ca_xSr_1−xF₂ is given in Figure 13 for the concentration x = 0.75. We observed four regions. The first region at the lowest energy states, is due to the s electrons of the fluorine, while the next region contains the contribution of Ca-s and Sr-s orbitals. The region just below the Fermi level E_F is predominately p states of F, with only a small contribution for p-Ca and p-Sr states. Just above the Fermi level, the region is predominately p states of Ca and p and d states of Sr.

From the DOS plots, as Ca composition increases in the alloys, the density of states of Ca-s and Ca-p orbitals decrease while the density of states of Ca-d orbitals increase. The DOS of s orbital of F increase with Ca concentration in all the alloys while that of p orbitals decrease. The DOS of s, p and d orbitals of Sr decrease with Ca composition in the alloys.

The DOS for the different concentrations of all alloys indicate that the band contributions of Ca-s, Ca-p, Sr-s, Sr-p, Sr-d, and F-p orbitals gradually weaken with Ca composition but Ca-d and F-s orbitals contributions strengthen.

Figure 3. Electronic band structure of Ca0.25Sr0.75F2 alloy

Figure 4. Electronic band structure of Ca0.50Sr0.50F2 alloy

Figure 5. Electronic band structure of Ca0.75Sr0.25F2 alloy

For Figures 3-5, vertical axis represents Energy (eV) while horizontal axis represents wave vector, k.

Figure 6. Variation of Band Gap Energy of CaxSr1−xF2 with composition, x

Figure 7. Variation of band gap bowing parameters, b(eV) of CaxSr1−xF2 with composition, x

Figure 8. Partial density of states (PDOS) of Calcium (Ca), red line represents s-orbitals, green line represents p-orbitals, blue line represents d-orbitals

Figure 9. Partial density of states (PDOS) of Strontium (Sr), red line represents s-orbitals, green line represents p-orbitals, blue line represents d-orbitals

Figure 10. Partial density of states (PDOS) of Fluorine (F), red line represents s-orbitals, green line represents p-orbitals

Figure 11. Calculated total density of states (DOS) of Ca0.25Sr0.75F2

Figure 12. Calculated total density of states (DOS) of Ca0.50Sr0.50F2

Figure 13. Calculated total density of states (DOS) of Ca0.75Sr0.25F2

4. Conclusions

The PP-PW method within the GGA approximations have been used to investigate the structural and electronic properties of the cubic Ca_xSr_1−xF₂ alloys. We have calculated the equilibrium lattice constants and bulk moduli of the binary compounds and the alloys. For the purpose of comparison, the structural properties of the binary compounds of the alloys were calculated. Good agreement is found between our calculations and experiments. The bowings of the lattice constant, bulk modulus and energy band gap are also investigated. From the graph of lattice constant and bulk modulus against composition, x, the values of the structural properties can be predicted at any desired concentration, x. The Ca_xSr_1−xF₂ alloys for the composition x = 0.25, 0.50 and 0.75 are wide gap insulators and may be a good material for optoelectronic industry. Their calculated band gap energies showed that all the alloys can absorb the short wavelength Ultraviolet (UV) light and hence can find application in the UV metal-insulator-semiconductor light emitting diodes.