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Chinese Journal of Physics 56 (2018) 131–144

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Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph

Structural, electronic, optical and thermodynamic investigations of NaXF3 (X = Ca and Sr): First-principles calculations

T

M.H. Benkaboua, M. Harmela, A. Haddoua, A. Yakoubia, N. Bakia, R. Ahmedb, ⁎ Y. Al-Douric,d, , S.V. Syrotyuke, H. Khachaia, R. Khenataf, C.H. Voong, Mohd Rafie Johanc a

Laboratoire d’étude des Matériaux & Istrumentations Optiques, Physics Department, Djillali Liabès University of Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algeria b Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, UTM, Skudai, 81310 Johor, Malaysia c Nanotechnology and Catalysis Research Center (NANOCAT), University of Malaya, 50603 Kuala Lumpur, Malaysia d Physics Department, Faculty of Science, University of Sidi-Bel-Abbes, 22000-Algeria e Semiconductor Electronics Department, National University “Lviv Polytechnic”, S. Bandera str. 12, Lviv 79013, Ukraine f Laboratoire de Physique Quantique et de Modélisation Mathématique de la Matière (LPQ3M), Université de Mascara-29000-Algeria g Institute of Nano Electronic Engineering, University Malaysia Perlis, 01000 Kangar, Perlis, Malaysia

AR TI CLE I NF O

AB S T R A CT

Keywords: DFT Perovskites Electronic properties Thermal properties Optical properties

The structural, electronic and optical properties for fluoro-perovskite NaXF3 (X = Ca and Sr) compounds have calculated by WIEN2k code based on full potential linearized augmented plane wave (FP-LAPW) approach within density functional theory (DFT). To perform the total energy calculations, exchange-correlation energy/potential functional has been utilized into generalized gradient approximation (GGA) and local density approximation (LDA). Our evaluated results like equilibrium lattice constants, bulk moduli, and their pressure derivatives are in agreement with the available data. The electronic band structure calculation has revealed an indirect band-gap nature of NaCaF3, while NaSrF3 has direct band gap. Total and partial densities of states confirm the degree of localized electrons in different bands. The optical transitions in NaCaF3 and NaSrF3 compounds were identified by assigning corresponding peaks obtained from the dispersion relation for the imaginary part of the dielectric function. The thermodynamic properties were calculated using quasi-harmonic Debye model to account lattice vibrations. In addition, the influence of temperature and pressure effects was analyzed on bulk modulus, lattice constant, heat capacities and Debye temperature.

1. Introduction Perovskites are intensively investigated because of showing interesting and captivating properties due to its distinguished applications, its abundance and common mineral in the Earth [1]. Their observed features are ferroelectricity, charge ordering, superconductivity, colossal magneto resistance, high thermopower, spin-dependant transport alongside displaying exotic structural, electronic, magnetic, optical and transport properties. In addition, their promise in different technological applications like fuel cells, sensors, memory devices, spintronics, and photovoltaics, they are also considered potential materials for microelectronics and telecommunicationas [2–4]. The discovered high-temperature superconducting oxides and high-performance perovskite solar cells with



Corresponding author at: Nanotechnology and Catalysis Research Center (NANOCAT), University of Malaya, 50603 Kuala Lumpur, Malaysia. E-mail addresses: [email protected], [email protected] (Y. Al-Douri).

https://doi.org/10.1016/j.cjph.2017.12.008 Received 24 August 2017; Received in revised form 5 December 2017; Accepted 10 December 2017 Available online 14 December 2017 0577-9073/ © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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Fig. 1. Total energy versus unit cell volume of NaCaF3 and NaSrF3.

Table 1 The calculated lattice constant (a), bulk modulus (B), and its pressure derivative (B′) for fluoroperovskite compounds NaCaF3 and NaSrF3. Results are compared with previous theoretical works. Present work

NaCaF3a(Ǻ) B (GPa) B' NaSrF3 a(Ǻ) B (GPa) B' a b c

Other theoretical works

LDA

GGA

4.30 63.24 4.48 4.60 49.31 4.40

4.35 59.37 4.66 4.64 47.50 4.32

4.46a 4.438b 4.378b 4.316c 43.98 a 47.363b 51.884b 4.381b 4.465b 4.44a 27.57a

[36] [37] [38].

21% power conversion efficiency have increased interest to attain the reported 31% efficiency [5]. Although perovskites are reported to containing oxygen, with chemical formula ABO3, a few perovskites such as fluoro-perovskites are found in ABF3, where A and B are alkalies and alkaline earth elements, respectively, and F is fluorine characterized as the highest electro-negative element. Fluorine has shown its ability to form a great variety of fluorides of the high-level chemical stability, in particular, containing low electro-negative alkali metals plus alkaline earth metals. Complex metal fluorides have attracted a considerable attention caused by ferromagnetic [6], nonmagnetic insulating [7], piezoelectric [8] and photoluminescence properties [9]. Moreover, the fluoro-perovskite crystals, namely KMgF3 [10], NaSrF3, NaBaF3, LiBaF3 [11] have displayed their promise for ``UV-Deep UV wave bands'' in different studies [12] which can be exploited for different applications as highly transparent and low loss optical windows, lenses, and prisms. However, no one from the existing crystals completely fulfills all the obligatory requirements needed for the practical applications. Therefore, further studies at all levels are necessary to explore their potential in the existed crystals or search for new one which can fulfill all the necessary requirements for the practical applications, particularly for ``UV-Deep UV wave bands''. Also, halide-based perovskites have emerged as an interesting and promising class of lowcost and highly efficient materials for solar cells [13]. Although over a longer period of time their efficiency and fabrication were too low owing to assorted structural instabilities and using conventional methods [14–16]. Fabrication of new halide-based perovskites is interesting due to ever revealing the variety of amazing properties, their possible applications in technology and have ability to involve more or less all elements of the periodic table [17–20]. Here, we investigate the structural, electronic, optical and thermodynamics properties of the NaCaF3 and NaSrF3 compounds. The total energy calculations are performed within FP-LAPW method embodied in WIEN2k code and structured within the DFT. The organization of the paper is; Section 2 briefs a description of computational method, while Section 3 presents results and discussion. Conclusions are summarized in Section 4.

132

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Fig. 2. Calculated electronic band structure of NaCaF3 and NaSrF.

Table 2 The calculated energy gaps at high symmetry points in NaCaF3 and NaSrF3 compounds (energies are in eV). Results are compared with previous theoretical worksonlyfor NaCaF3 compound.

NaCaF3 LDA GGA Others NaSrF3 LDA GGA a

R–R

Γ–Γ

X–X

M–M

R–Γ

VBW

8.67 8.42

5.27 5.11

7.23 7.01

8.32 8.17

5.13 4.94 5.005a 5.115a

1.50 1.41 1.71a

7.46 7.28

4.48 4.32

6.36 6.19

7.13 7.04

4.65 4.50

0.66 0.68

[37].

2. Computational method All the calculations have been carried out using the full potential linearized augmented plane wave (FP-LAPW) method [21] based on DFT [22,23] by means of WIEN2k code [24]. The description of exchange and correlation effects was considered here within generalized gradient approximation (GGA) [25], and local density approximation (LDA) [26]. In this scheme, the partition of the simulated crystal unit cell is divided into two parts; non-overlapping muffin-tin (MT) spheres considered around the atomic positions, and an interstitial region. In this approach, different basis sets are used for the both regions of crystal unit cell. Inside the nonoverlapping MT spheres, the wave function is described using a ``linear combination of radial functions times spherical harmonics'', whereas, in the interstitial region, expansion is achieved in terms of plane waves. To achieve a reasonable level of the convergence, value of the quantum number, l = 10 is used to expand the wave function inside the sphere, where the value of RMT Kmax = 8 (RMT represents the minimum atomic sphere radii whereas Kmax is used for the largest value of k-vector in the plane wave expansion) is chosen. Similarly, the charge density and potential, inside the atomic spheres are expanded in lattice harmonics and Fourier expansion in the interstitial region. Fourier expanded charge density is truncated at Gmax = 12 (a.u.)−1. The Monkhorst–Pack k-mesh of 17 × 17 × 17 was used for integration over the Brillouin zone [27]. The convergence criterion for the total energy about 0.1 mRy was achieved. The thermodynamic properties for the interested compounds were derived by employing ``the quasi-harmonic Debye model'' based on the Helmholtz free energy. The authentication of this approach has been well justified already by many previous studies 133

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6

6

Fs Fp

5

Fs Fp

5

4

4

3

3

2

2

1

1

10 0

0 8

Ca s Ca p Ca d

Sr s Sr p Sr d

8

6

6

4

2

0 8

Na s Na p

Density Of States (States/eV)

Density Of States (States/eV)

4

2

0 0.5

Na s Na p

0.4

6

0.3

4

0.2

2

0.1

20 0

20 0.0 18

18

tot NaCaF 3

16

tot NaSrF 3

16

14

14

12

12

10

10

8

8

6

6

4

4

2

2 0

0 -20

-10

0

-20

10

-10

0

10

Energy (eV)

Energy (eV)

Fig. 3. Total and partial density of states of NaCaF3 and NaSrF3.

[28,29]. The calculation of bulk modulus B can also be done with ``quasi-harmonic Debye model'' at different temperatures and pressures. The type of non-equilibrium Gibbs function, G* used in this model is [30];

G * (V ; P , T ) = E (V ) + pV + Vib (V ; T )

(1)

where E(V) represents total energy values with respect to unit cell volume, PV is to signify the constant hydrostatic pressure condition and Avib stands for the "vibrational Helmholtz free energy" and expressed as [31,32];

9θ Avib (θD ; T ) = nkT ⎡ D 3 ln(1−e−θD / T − D (θD / T ) ⎤ ⎦ ⎣8 T

(2)

In Eq. (2), θD, D(θD/T) and n refer to Debye temperature, Debye integral and number of atoms per formula unit, respectively. By minimizing the G* with respect to volume and solving corresponding equation, following relation is obtained; 134

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Fig. 4. Calculated imaginary part of the dielectric function ɛ2(ω) of NaCaF3 and NaSrF3.

Fig. 5. Calculated real part of the dielectric function ɛ1(ω) of NaCaF3 and NaSrF3.

⎛ θG * (V ; pT ) ⎞ = 0 θV ⎝ ⎠ pT

(3)

From this, one can obtain relations for the isothermal bulk modulus B, heat capacity Cv and thermal expansion coefficient α as;

θ 2G * (V ; p , T ) ⎞ θp BT (P , T ) = −V ⎛ ⎞ = V ⎛ θV 2 ⎝ θV ⎠T ⎝ ⎠ p, T

(4)

3 /T ⎤ Cv = 3nk ⎡4D (θD / T ) − θ /θD e T − 1⎦ ⎣

(5)





135

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Fig. 6. The calculated refractive index n(ω) of NaCaF3 and NaSrF3.

Fig. 7. The calculated extinction coefficient k(ω) of NaCaF3 and NaSrF3.

α=

γCv BT V ′

(6)

where γ is the Grüneisen parameter. The more detailed explanation can be seen in Ref's. [33,34]. From the above equations, the thermodynamic properties of NaCaF3 and NaSrF3 can be obtained. 3. Results and discussion 3.1. Structural properties The simulated crystal unit cell with Wyckoff atomic sites, Na at 1b (0.5, 0.5, 0.5), X at 1a (0, 0, 0) and F at 3d (0.5, 0, 0) where X = (Ca, Sr), were optimized. The total energy is calculated as a function of unit-cell volume around the equilibrium cell volume V0. 136

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Fig. 8. The calculated absorption coefficient I(ω) of NaCaF3 and NaSrF3.

Fig. 9. The calculated energy loss L(ω) of NaCaF3 and NaSrF3.

The total energies versus volume presented in Fig. 1 are obtained by fitting the obtained data to the Birch–Murnaghan equation of state [35]. The corresponding equilibrium lattice constant a0, bulk modulus B, and its pressure derivative B', together with other theoretical values, evaluated within LDA and GGA are given in Table 1. As can be seen, the calculated lattice constants are in agreement with other results [36–38]. It can also be seen that the computed results of NaCaF3 have smaller values of lattice constants than that of NaSrF3. The bulk modulus is a measure of the crystal rigidity (where the rigidity of a material is a measure of its resistance against any change to its permanent shape). Thus, we can say that NaCaF3 is more rigid than NaSrF3 owing to the fact that bulk modulus of NaCaF3 is higher than NaSrF3. 3.2. Electronic properties The calculated electronic energy bands profile of NaCaF3 and NaSrF3 along with high symmetry points X, R, M and Γ inside the 137

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Fig. 10. The calculated reflectivity R(ω) of NaCaF3 and NaSrF3.

first Brillouin zone at zero pressure, within the GGA are illustrated in Fig. 2. From the band structure, it is clear that the valence band maximum (VBM) of NaCaF3 compound is located at point R, where in case of NaSrF3, it is located at point Γ. However, their conduction band minimum (CBM) is found at the same point for both compounds i.e. at the point Γ, showing that NaSrF3 is a direct band gap at (Γ − Γ) compound and NaCaF3 is an indirect band gap at (R-Γ), similar to other reported fluoride perovskites in the literature [39–42]. As seen in Table 2, our calculated band gaps for NaCaF3 and NaSrF3 equal 5.16 (4.88) eV and 4.50 (4.26) eV, respectively. For the sake of meaningful comparison, unluckily, no theoretical or experimental data about energy band gap values of these compound are found in the literature. It should be noted that LDA (GGA) approximations for the exchange correlation usually underestimate the energy band gap calculated within DFT formalism [43, 44]. This is due to the solutions of Kohn-Sham equation, at the level of LDA or GGA, do not correspond to the quasi-particle excitations. Therefore, it can be said that, our calculated results predict insulators. Seeking to have more comprehensive picture of electronic structure, the profile of the total density of states (TDOS) as well as the partial density of states (PDOS) for both NaCaF3 and NaSrF3 compounds are also shown in Fig. 3 over an energy range from – 20 eV to 15 eV at GGA. From Fig. 3, it can be seen that the valence band close to Fermi level is mainly formed by F-2p states with a very small contribution from Na-s and p states. Whereas in low-energy valence band, the deep electronic states localized in the vicinity of −20 eV, below the Fermi energy for both NaCaF3 and NaSrF3 compounds are populated by F-2s states. The structure situated at around −15 eV for NaCaF3 and −12 eV for NaSrF3 consists of Ca-p and Sr-p states, respectively. On the other hand, in the conduction band of both compounds, Na-s and p states are primarily presented, with a noticeable part from Ca-p and Sr-p states. 3.3. Optical properties To evaluate the optical properties, complex dielectric function, ɛ(ω) is used. It is expressed as [45,46]:

ɛ(ω) = ɛ1 (ω) + iɛ2 (ω)

(7)

where ε1(ω) represents the real part, and ε2 (ω) represents the imaginary part. The imaginary part of the dielectric function of a solid is derived from its electronic band structure in accordance with the following relation [47]:

ɛ2 (ω) =

e 2ℏ πm2ω2

∑ ∫BZ

Mcv (k ) 2 δ [ωcv (k ) − ω] d3k (8)

v, c

Here, Mcv (k ) = uck δ·∇ u vk is the momentum dipole elements (δ is used for potential vector which defines the electric field), which represents matrix elements for the direct transitions between conduction band uck(r) states and valence band,-uvk(r) is energy states, and ℏωcv (k ) = Eck − Evk gives the related transition energy. By employing Kramers–Kronig relation, the real part of the complex dielectric function ɛ1(ω) is calculated from the imaginary part ɛ2(ω) as:

ɛ1 (ω) = 1 +

2 P π

∫0



ω′ɛ2 (ω′) ω′2 − ω2

dω′

(9)

In the relation, P entails the main value of the integral. In order to determine other optical parameters (refractive index n(ω), 138

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Fig. 11. Temperature dependence of the volume V0 at various pressures for (a) NaCaF3 and (b) NaSrF3.

extinction coefficient k(ω), absorption coefficient I(ω) and reflectivity R(ω)), the following expressions were applied [36]:

⎡ ɛ (ω) + n (ω) = ⎢ 1 2 ⎣

⎡ ɛ (ω) + k (ω) = ⎢− 1 2 ⎣ I (ω) = R (ω) =

1/2

ɛ12 (ω) + ɛ22 (ω) ⎤ ⎥ 2 ⎦

(10) 1/2

ɛ12 (ω) + ɛ22 (ω) ⎤ ⎥ 2 ⎦

(11)

1/2

2 ω [ ɛ12 (ω) + ɛ 22 (ω) − ɛ1 (ω)]

(12)

n + ik − 1 n + ik + 1

(13)

All the above mentioned optical parameters have been evaluated using calculated equilibrium lattice constants at the level of GGA over energy range up to 40 eV. The obtained results for both NaCaF3 and NaSrF3 compounds are presented in Figs. 4 and 5. The ε2 (ω) part of ε(ω) gives the total sum of all transitions. In Fig. 4, the structures of the optical response show a shift towards lower energies as we move from Ca to Sr, showing their agreement with other band structures. From the closer look at ε2 (ω) dispersion curves, we see, 139

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Fig. 12. Temperature dependence of the Bulk modulus B0 at various pressures for (a) NaCaF3 and (b) NaSrF3.

first critical point (fundamental absorption edge) for the NaCaF3 and NaSrF3 are approximately found at 4.96 eV and 4.20 eV, respectively. These values are in line with the indirect and direct transitions between VBM and CBM, at the points (R–Γ) and (Γ–Γ) for NaCaF3 and NaSrF3, respectively. The rise in the curve is too fast due to numbers of transitions contributing to ε2(ω) which are suddenly increased. The real part ε1(ω) of frequency-dependent dielectric function evaluated from the Kramers–Kronig dispersion relation is shown in Fig. 5. The main peaks are located at 9.75 eV and 10.13 eV for NaCaF3 and NaSrF3, respectively. The curves for both compounds show an almost similar trend. From the curves, it can be seen first, the curves are fallen following by rising, then again fall followed by a slow rise at high energy value. The calculated static dielectric constant ε1 (0) is about 1.95 for both compounds. It can be seen from the plots, the zero's value of ε1 (ω) (ε1 = 0) shows non existence of the dispersion, which reflects its perfect coincidence with the absorption coefficient's maximum value I(ω) located at 13.56 eV. From the Eqs. (10)–(13), it can be observed easily that with the simple knowledge of ε1(ω) and ε2(ω), one determines other optical parameters like refractive index n(ω), extinction coefficient k(ω), absorption coefficient I(ω), energy loss L(ω) and reflectivity R(ω). The calculated results of n(ω) and k(ω) are shown in Figs. 6 and 7, respectively. At zero photon energy, the value of static refractive index n(0) is found to be approximately equal to 1.40 for both NaCaF3 and NaSrF3 compounds. All the maxima of the refractive index 140

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Fig. 13. Variation of the heat capacities Cv with temperature at various pressures for(a) NaCaF3 and (b) NaSrF3.

and extinction coefficient are localized at nearly equal photon energy values. The absorption coefficient I(ω) depicted in Fig. 8, reveals the high absorption at high photon energies. Although NaCaF3 shows higher value compared to NaSrF3, for both NaCaF3 and NaSrF3 compounds, the values of absorption coefficient are at 25.67 and 21.62 eV. This shows that both materials exhibit considerable absorption. The frequency of plasma resonance can be estimated from the energy loss spectra, as shown in Fig. 9. The maximum resonant energy loss is found equal to 30.08 and 26.68 eV for NaCaF3 and NaSrF3, respectively. Fig. 10 shows the calculated reflectivity R(ω). From this curve, the zero-frequency limit of reflectivity for NaCaF3 and NaSrF3 is found to be 0.028 and 0.027, respectively. There are high reflection peaks at energies 10.78 eV, 25.02 eV, 28.94 eV for both compounds as well.

3.4. Thermodynamic properties The thermal properties of NaCaF3 and NaSrF3 compounds under high temperature and pressure have been investigated within the quasi-harmonic Debye model. At a first stage, we have obtained the numerical total energy as a function of unit cell volume i.e., E(V), in the static approximation approach. At a second stage, the equation of state (EOS) was derived by numerical fittings. And then by using the EOS, the structural parameters (at zero temperature and pressure) and corresponding macroscopic properties as a function of the pressure and temperature have been derived. Here, the investigation of thermal properties is carried out within temperature 141

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Fig. 14. Variation of the Debye temperature θD as function of temperature at various pressuresfor (a) NaCaF3 and (b) NaSrF3.

range from 0 to 1000 K, assuming the quasi-harmonic model validity, and the pressure is considered a range 0–40 GPa. The equilibrium unit cell volume V0 versus the temperature, for NaCaF3 and NaSrF3, at different pressures, is presented in Fig. 11. We can see that the volume V0 (a, lattice constant) increases monotonically as temperature increases, and on another side, it decreases, at a given temperature, with pressure P. The lattice constants, obtained for NaCaF3 and NaSrF3, were found to have 4.46 and 4.76 Ǻ, respectively, at zero pressure and room temperature. As noticed in Fig. 12, the bulk modulus is almost a constant at given pressure, within the temperature range from 0 to 120 K, and then it starts decreasing linearly for a temperature greater than 120 K at a given pressure. However, at a given temperature, it increases with pressure. The calculated values of bulk modulus, at 300 K and zero pressure, for both NaCaF3 and NaSrF3 are found to be 98 GPa and 77.7 GPa, respectively. The vibration properties can be accessed through the heat capacity. Therefore, heat capacity at constant volume, CV, was calculated as a function of temperature, at given pressure (0, 10, 20, 30 and 40 GPa). The results obtained for NaCaF3 and NaSrF3 are shown in Fig. 13. As can be seen from the presented diagram, initially the CV curve is sharply increased up to 300 K, but next to it, its increasing is very slow. At further high temperature CV tends to approach the limiting value 3R, agreeing with the Dulong–Petit law [37]. The diagrams given in Fig. 13 clearly indicate that at T < 500 K, the heat capacity CV depends on both temperature and pressure (CV is proportional to T3 [48]). The Debye temperature (θD) can also be obtained from the low temperature heat capacity measurements. θD is also accessible for the evaluation by means of elastic constants. It is a characteristic temperature that may be evaluated from the phonon spectrum of the crystal as well. Fig. 14 illustrates the temperature dependence of θD, over the pressure range 0–40 GPa, for both NaCaF3 and NaSrF3 142

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compounds. As can be observed from Fig. 14, θD is nearly constant from 0 to 200 K and decreases linearly with temperature. Our calculated θD at zero pressure and ambient temperature are 622.57 K and 491.36 K for NaCaF3 and NaSrF3, respectively. Probably, this indicates that the quasi-harmonic Debye model ensures the efficient account of contributions of thermal vibrations in the crystal thermodynamic functions. 4. Conclusions The structural, electronic, optical and thermodynamic properties of NaXF3 (X = Ca and Sr) compounds using FP-LAPW method within LDA and GGA, have been evaluated from first-principles calculations. The most notable results are represented as:

• The obtained basic ground-state properties such as equilibrium lattice constants and bulk moduli are in good agreement with • • • •

other data. The lattice constants evaluated within GGA are larger, whereas the bulk moduli are smaller, compared to those found with LDA. The electronic structure calculations showed that both compounds follow a similar trend, with a wide indirect band gap (R–Γ) for NaCaF3 and direct band gap (Γ–Γ) for NaSrF3. Corresponding total and partial DOS are analyzed and the nature and structure of the relevant curves are identified. The materials have wide band gaps. Moreover, the dispersion of imaginary part of dielectric function reveals its transparency for a wide range of energies. Therefore, we conclude that they are promising candidates for application in optoelectronics to capture the ultraviolet region. The quasi-harmonic Debye model employed appeared to be effective for the description of the material properties, namely an equilibrium volume, bulk modulus, heat capacities and Debye temperature, at the temperature and pressure conditions, in which the anharmonicity not significantly affects the thermodynamic parameters.

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