International Journal of Theoretical and Mathematical Physics

p-ISSN: 2167-6844    e-ISSN: 2167-6852

2020;  10(3): 60-64



The Structure and the Optical Properties of Ba0.50Ca0.50F2

Alice Bukola Olanipekun, Catherine Obaze, Brian Diagbonya

Department of Pure and Applied Physics, Caleb University, Imota, Lagos, Nigeria

Correspondence to: Alice Bukola Olanipekun, Department of Pure and Applied Physics, Caleb University, Imota, Lagos, Nigeria.


Copyright © 2020 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).


A theoretical investigation of structural, electronic and optical properties of CaF2, BaF2 and Ba0.50Ca0.50F2 is presented, using the pseudopotential plane-wave (PP-PW) method as implemented in the QUANTUM ESPRESSO code. In this research, the exchange-correlation (XC) potential was approximated using the generalized gradient approximation (GGA). The calculated lattice constants and bulk moduli are in good agreement with experimental results for both the binary compounds and their alloy. Band structure of Ba0.50Ca0.50F2 is presented and band gap calculated. In addition to DFT-PBE calculation of the band gap, a semi empirical model is used for the calculation of band gap and a good agreement with experiment is obtained. The pressure coefficients of energy gaps and refractive indices are also calculated.

Keywords: Generalized Gradient Approximation, Semi empirical model, Optical properties

Cite this paper: Alice Bukola Olanipekun, Catherine Obaze, Brian Diagbonya, The Structure and the Optical Properties of Ba0.50Ca0.50F2, International Journal of Theoretical and Mathematical Physics, Vol. 10 No. 3, 2020, pp. 60-64. doi: 10.5923/j.ijtmp.20201003.02.

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, Ca0.50Ba0.50F2, 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]. Ba1-xCaxF2 (0 < x < 1) can be prepared by mechano-chemical reaction at low temperatures from the binary fluorides BaF2 and CaF2 [18,19]. Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 alloy are essential in fuel cells, batteries, capacitors and solid electrolytes. The ion conductivities of Ba1-xCaxF2 has also been calculated using theory [24,26]. Using X-ray diffraction, the lattice constant of Ba0.50Ca0.50F2 was measured to be 5.87Å [25].
The study of the electronic properties of the mixed Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2. 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
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 GoWo band gap energy is
From equation 1 and 2, the correlation between modelled experimental and DFT(PBE) band gap energy is derived to be
A linear relation between energy gap at p=0 and at any pressure p is given as,
where k = pressure coefficient of the band gap defined by k = dEg/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
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 Ba0.50Ca0.50F2 respectively. They compare well with experiment. The experimental lattice constants of CaF2, BaF2 and Ba0.50Ca0.50F2 are taken from [37], [38] and [39] respectively. The value of lattice parameter for Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 respectively. As far as we know, there is no experimental value for the bulk modulus of the Ba0.50Ca0.50F2.
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 Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 indicates that this compound might be a good candidate for a frequency conversion material working in the ultraviolet region. Ba0.50Ca0.50F2 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. Ba0.50Ca0.50F2 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]. Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 crystal. The milling of fluorite BaF2 and CaF2 with x=0.50 results in the formation of Ba0.50Ca0.50F2 crystallising in the cubic fluorite structure.
Figure 1. Crystal Structure of Ba0.50Ca0.50F2
The band structure of Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2 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 Ba0.50Ca0.50F2, 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 Ba0.50Ca0.50F2, 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.


[1]  W. Pics and A. Weiss, Crystal Structure of Inorganic Compounds: Keys Elements F, Cl, Br, I, 3rd ed., K.H. Hellwege, A.M. Hellwege, Landolt-B¨ornstein, Berlin, Germany: Springer, 1973, vol. 7.
[2]  L. Gerward, M. Malinowski, S. Asbrink and A. Waskowska, 1992, X‐ray diffraction investigateons of CaF2 at high pressure, J. Appl. Cryst. 25(5), 578-581.
[3]  R.M. Hazen and L.W. Finger, 1981, Calcium fluoride as an internal pressure standard in high-pressure crystallography, J. Appl. Cryst. 14(4), 234-236.
[4]  J.M. Leger, J. Haines, A. Atouf, O. Schulte and S. Hull, 1995, High-pressure x-ray and neutron-diffraction studies of BaF2: An example of a coordination number of 11 in AX2 compounds, Phys. Rev. B 52(18), 13247-13256.
[5]  G.A. Samara, 1976, Temperature and pressure dependences of the dielectric properties of PbF2 and the alkaline-earth fluorides, 1Phys. Rev. B 13(10), 4529-4544.
[6]  C. S´anchez and M. Cardona, 1972, Piezobirefringence in the vacuum ultraviolet: Alkali halides and alkaline‐earth fluorides, Phys. Status Solidi B 50(1), 293-304.
[7]  O.V. Shakin, M.F. Bryzhina and V.V. Lamanov, 1972, Photoelastic constants of Alkaline earth fluorides, Sov. Phys. solid State, 13(10-12), 3141-3142.
[8]  A.K. McCurdy, 1982, Phonon conduction in elastically anisotropic cubic cryst, Phys. Rev. B 26(12), 6971-69.
[9]  J. Barth, R.L. Johenson and M. Cardona, 1990, Dielectric function of CaF2 between 10 and 35eV, Phys. Rev. B 41(5), 3291-32.
[10]  S C Sabharwal, S N Jha and Sangeeta 2002, Optical and X-ray photoelectron spectroscopy of PbGeO3 and Pb5Ge3O11 single crystals, J. Cryst. Growth, 33(4), 395-400.
[11]  R.T. Poole, J. Szajman, R.C.G. Leckey, J.C. Jenkin and J. Liesegang, 1975, Electronic structure of the Alkaline earth fluorides studied by photoelectron spectroscopy, Phys. Rev. B 12(12), 5872-5877.
[12]  B. Li. and L. Wang, 2000, Lithography birefringence in fused silica and CaF2 for lithography, Solid State Technol. 43(1), 77-8.
[13]  J.H. Burnett, R. Gupta and U. Griesmann, 2002, Absolute refractive indices and thermal coefficients of CaF2, SrF2, BaF2 and LiF near 157nm, Appl. Opt. 41(13), 2508-2513.
[14]  W. Hayes, A.B. Kunz and E.E. Koch, 1971, Reflectance spectra of alkaline earth fluorides in the vacuum ultraviolet, J. Phys. C: Solid state Physics, 4(10), L200.
[15]  A. Feldman and R.M. Waxler, 1980, Strain induced splitting and Oscillator-Strength Anisotropy of the infrared transverse optic phonon in calcium fluoride, strontium fluoride and Barium fluoride, Phys. Rev. Lett. 45(2), 126-.
[16]  B. Ruprecht, M. Wilkening, S. Steuernagel and P. Heitjans, 2008, Anion difussivity in highly conductive nanocrystalline BaF2:CaF2 composites prepared by high energy ball milling, J. Mater. Chem. 18(44), 5412-5416.
[17]  B. Ruprecht, M. Wilkening, A. Feldhoff, S. Steuernagel, P. Heitjans, 2009, High ion conductivity in a ternary non equilibrium phase of BaF2 and CaF2 with mixed cations, Phys. Chem. Chem. Phys. 11(17), 3071-3081.
[18]  A. Düvel, B. Ruprecht, P. Heitjans, M. Wilkening, 2011, Mixed alkaline-earth effect in the metastable anion conductor Ba1-xCaxF2 (0 ≤ x ≤ 1): correlating long-range ion transport with local structures revealed by ultrafast 19F MAS-NMR, J. Phys. Chem. C 115(48), 23784-23789.
[19]  B. Ruprecht, M. Wilkening, S. Steuernagel, P. Heitjans, 2008, Anion diffusivity in highly conductive nano crystalline BaF2:CaF2 composites prepared by high-energy ball milling, J. Mater. Chem. 18(44), 5412-5416.
[20]  F. Preishuber-Pflügl, V. Epp1, S. Nakhal, M. Lerch and M. Wilkening, 2015, Defect-enhanced F-ion conductivity in layer-structured nanocrystalline BaSnF4 prepared by high-energy ball milling combined with soft annealing, Phys. Status Solidi C 12(1-2), 10–14.
[21]  A.V. Chadwick, A. Düvel, P. Heitjans, D.M. Pickup, S. Ramos, D.C. Sayle and T.X.T. Sayle, 2005, X-Ray Absorption Spectroscopy and Computer Modelling Study of Nanocrystalline Binary Alkaline Earth Fluorides, Mat. Sci. and Eng. 80(1), 012005-0120.
[22]  N. Sata, K. Ebermann and K. Eberl and J. Maier, 2000, Mesoscopic fast ion conduction in nanometer scale planar heterostructures, Nature, 408(6815), 946-949.
[23]  M. Wilkening, A. Düvel, F. Preishuber-Pflügl, K. da Silva, S. Breuer, V. Šepelák and P. Heitjans, 2017, Structure and ion dynamics of mechanosynthesized oxides and fluorides, Z. Kristallogr. 232(1-3), 107–127.
[24]  A. Düvel, P. Heitjans, P. Fedorov, G. Scholz, G. Cibin, A. V. Chadwick, D. M. Pickup, S. Ramos, L.W.L. Sayle, E. K. Sayle, T. X.T. Sayle, D. C. Sayle, 2017, Is Geometric Frustration-Induced Disorder a Recipe for High Ionic Conductivity? J. Amer. Chem. Soc. 139(16), 5842-5848.
[25]  X. Mu, W. Sigle, A. Bach, D. Fischer, M. Jansen and P. A. van Aken, 2014, Influence of a second cation () on the phase evolution of BaxM1-XF2 starting from amorphous deposits, Zeitschrift anorganische und allgemeine Chemie,
[26]  D. Zahn, P. Heitjans and J. Maier, 2012, From composite to solid solutions: Modelling ionic conductivity in the CaF2-BaF2 System, Chem. Eur. J. 18(20), 6225-6229.
[27]  P. Hohenberg and W. Kohn, 1964, Inhomogenous electron gas, Phys. Rev. 136(3), B864-B871.
[28]  W. Kohn and L. J. Sham, 1965, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140(4), A1133-1138.
[29]  P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, A.D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L, Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari and R.M. Wentzcovitch, 2009, Quantum Espresso: a modular and opensource software project for quantum simulations of materials, J. Phys. 21(39), 395502-395520.
[30]  J. C. Furthmuller, P. Kachell and F. Bechstedt, 2000, Extreme softening of Vanderbilt pseudopotentials, Phys. Rev. B 61(7), 4576-4587.
[31]  H.J. Monkhorst and J.D. Pack, 1976, Special points for Brillouin-zone integrations, Phys. Rev. B 13(12), 5188-5192.
[32]  J. Perdew, K. Burke, and M. Ernzerhof, 1996, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77(18), 3865-3868.
[33]  F. D. Murnaghan, 1944, The Compressibility of Media under Extreme Pressures,. Proc. Natl Acad. Sci., 30(9), 244-247.
[34]  G. Onida, L. Reining, A. Rubio, 2002, Electronic excitations: density-functional versus many-body Greens-function approaches. Rev. Mod. Phys. 74(), 601-659.
[35]  Á. Morales-García, R. Valero and F. Illas, 2017, An empirical, yet practical way to predict the band gap in solids by using density functional band structure calculations, J. Phys. Chem. 121(134), 18862-18866.
[36]  P.J.L. Herve, and L.K.J. Vandamme, 1994, General relation between refractive index and energy gap in semiconductors, Infrared Physics and Technology, 35(4), 609-615.
[37]  A.R. West, 1999, Basic Solid State Chemistry, John Wiley and Sons, Chichester, England. pp126.
[38]  W.Y. Ching, F. Gan and M.Z. Huang, 1995, Band theory of linear and non linear susceptibilities of some ionic insulators, Phys. Rev. B 52(3), 1596-1611.
[39]  R.W.G. Wyckoff, Crystal structures, 2nd ed., Interscience Publisher, New York, 1982.
[40]  C. Wang and D. E. Schulle, 1968, Pressure and Temperature derivatives of the elastic constants of CaF2 and BaF2, J. Phys. Chem. Solids, 29(8), 1309-1330.
[41]  G.W. Rubloff, 1972, Far-Ultraviolet Reflectance Spectra and the Electronic structure of Ionic Crystals, Phys. Rev. B 5(2), 662-684.
[42]  N. Boukhris, H. Meradji, S. Ghemid, S. Drablia and F.E.H. Hassan, 2011, Ab initio study of the structural, electronic and thermodynamic properties of PbSe1-xSx, PbSe1−xTex and PbS1-xTex ternary Alloys, Physica Scripta, 83(6), 065701-065709.