First-principles study of structural, electronic and

June 2, 2014 11:16 WSPC/Guidelines-IJMPB

S0217979214501392

International Journal of Modern Physics B Vol. 28 (2014) 1450139 (15 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217979214501392

First-principles study of structural, electronic and optical properties of the KCaX3 (X = F and Cl) compounds

Int. J. Mod. Phys. B Downloaded from www.worldscientific.com by Dr Ahmad Mousa on 06/12/14. For personal use only.

Ahmad A. Mousa Middle East University, Amman, 11831, Jordan [email protected] Received 8 March 2014 Revised 13 April 2014 Accepted 21 April 2014 Published 2 June 2014 Structural and electronic properties of perovskite KCaX3 (X = F and Cl) compounds are investigated using the full potential linearized augmented plane wave (FP-LAPW) method as implemented in the Wien2k code. The exchange-correlation potential is treated by the generalized gradient approximation within the scheme of Perdew, Burke and Ernzerhof (GGA-PBE). Based on these calculations, it has been concluded that KCaX3 compounds have indirect energy band-gap (Γ − R). Moreover, the theoretical investigation which has been carried out on the highly hydrostatic pressure dependence of the KCaX3 electronic properties revealed a linear relationship between both the hydrostatic pressure and the energy band-gap. In addition, the electronic and bonding properties of the band structure, density of states (DOS) and electron charge density have been calculated and presented. Besides that, the dielectric function, refractive index and extinction coefficient are calculated. The origin of some of the peaks in the optical spectra is discussed in terms of the calculated electronic structure. Finally, the calculated structural properties are found to agree well with the available experimental and theoretical data. Keywords: Perovskite; DFT; GGA; band-gap; electron density; optical properties. PACS numbers: 68.35.Ct, 68.35.Md, 68.47.Gh

1. Introduction The perovskite structure was first discovered in the Ural Mountains by Gustav Rose in 1839 and named after the Russian mineralogist Pervoski (1792–1856). Numerous crystalline materials were found to possess this type of structure including calcium titanate (CaTiO3 ).1 The perovskite structure has the general stoichiometry ABX3 , where the cations “A” and “B” are the alkali metals and alkaline earth metals, respectively; and hence “X” stands for the anion. In the present work, the perovskite is derived by replacing the O−2 ions by the F− and Cl− ions. The unbalanced negative charge is compensated by completely replacing all of the smaller 12 coordinated 1450139-1


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Ca and larger 6 coordinated Ti cations by the mono and divalent K and Ca cations, respectively.2 Even though the ternary fluoride compounds, including BaLiF3 , SrLiF3 and RbCaF3 , with the perovskite crystal structure, are the most commonly studied members of this family because of their potential applications in both transparent optical coatings and lenses material industries,3 – 7 more detailed investigations on this system are needed at both the theoretical and experimental levels. Moreover, another member of the same wide band gap alkaline metal family KCaX3 , namely the halide perovskites, KCaCl3 is worth investigating to explore its suitability for the same abovementioned industries. KCaX3 exists in general in the cubic phase,3–5 with few theoretical studies and/or experimental work on its structural, electronic and optical properties.8 Therefore, in the present work, I investigated the structural, electronic and optical properties of the KCaF3 and KCaCl3 perovskite compounds, using the full potential linearized augmented plane wave (FP-LAPW) method with the generalized gradient approximation (GGA). This paper is organized in four sections: Sec. 2 is devoted to the method of calculation, Sec. 3 deals with the results and discussion and finally the conclusion is presented in Sec. 4. 2. Method of Calculations KCaX3 (X = F and Cl) compounds have the cubic perovskite structure with one molecule in its unit cell. Its space group is Pm-3m (#221). The K, Ca and X atoms are positioned at 1a (0, 0, 0), 1b (1/2, 1/2, 1/2) and 3c (0, 1/2, 1/2) sites of Wyckoff coordinates, respectively. The self-consistent calculations were performed using the FP-LAPW9 approach based on density functional theory (DFT)10 as implemented in Wien2k.11 In this method, the unit cell of the crystal is partitioned into two different regions: first one is the nonoverlapping spheres (atomic spheres) and the second one is the interstitial region (region outside the atomic spheres). The wavefunction, charge density and potential are expanded by spherical harmonic functions inside the nonoverlapping spheres surrounding the atomic sites (muffintin spheres) and by a plane wave basis set in the remaining space of the unit cell (interstitial region). The calculated total energies are fitted to the Murnaghan equation of state12 to obtain the energy-volume relation; and the bulk modulus (Bo ) is calculated by fitting the pressure–volume data to a third-order Birch equation of state.13 The exchange correlation potential was treated using the generalized gradient approximation (GGA-PBE).14 The maximum quantum number ℓ for the wavefunction expansion inside the atomic spheres is confined to ℓmax = 10. The core cutoff energy is −81.66 eV and the plane wave’s cutoff Kmax = 8/Rmt (Rmt is the smallest muffin-tin radius in the unit cell and Kmax gives the magnitude of the largest K vector in the plane-wave expansion) is chosen for the expansion of wavefunctions in the interstitial region. The charge density is Fourier expanded up to Gmax = 14. The Rmt values are taken 1450139-2


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First-principles study of structural, electronic and optical properties of the KCaX3

to be 2.4 a.u. for K and Cl atoms, 2.0 a.u. for Ca and F atoms. The Brillouin zone integrations within the self-consistency cycle are performed via a tetrahedron method15 using 56 k point in the irreducible wedge of the Brillouin zone (IBZ) for all compounds. The self-consistence calculations are considered to be converged only when the convergence tolerance of energy and charge are less than 0.1 m Ry and m electron charges, respectively. 3. Results and Discussion

The total energy of KCaX3 (X = F and Cl) has been calculated as a function of the unit cell volume. The total energies are calculated for different volumes around the equilibrium cell volume V0 . The equilibrium lattice parameters, bulk modulus, and its pressure derivative follow from a fit of the total energy as a function of the volume to the Birch–Murnagahan equation of state12 as seen in Fig. 1. Values of these parameters are presented in Table 1, along with the available experimental and other calculations. From Table 1, it can be seen that the calculated lattice parameters agree well with the theoretical calculation16,17 and experimental measurement.18–20 The small deviations appeared between calculations conducted in this work in comparison to previous calculations are owed to the reliability of the present first-principles computations as the different exchange correlation potential has been used in this calculations. It is further noted the increase in the lattice parameter associated with the change of the anion X from F to Cl in KCaX3 . This result better explained based on the atomic radii, R(F) = 1.36 ˚ A and R(Cl) = 1.81 ˚ A, i.e., as the bulk moduli of the materials decrease, the lattice parameters increase. Similar relation exists between the lattice constant and bulk modulus had already been seen for other pure compounds and some alloys as well.3 Considering the strong correlation between the bulk and the hardness, KCaF3 compounds can be argued to be one of the hardest compounds on my study. The derivative of the bulk modulus for two -20252.810 -18083.060 -20252.812 -18083.062 -20252.814 -18083.064

Energy (Ry)


3.1. Structural properties

-20252.816

-18083.066

-20252.818

-18083.068

-20252.820

-18083.070

KBaCl3

KBaF3 -18083.072

-20252.822 800

840

880

920

960

1320

1380

1440

1500

1560

3

Volume (a.u )

Fig. 1.

Total energy per atom as a function of volume of (a) KCaF3 and (b) KCaCl3 . 1450139-3

1620


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A. A. Mousa ˚), bulk modulus Bo (GPa) and bulk Table 1. Lattice constant a (A modulus derivative B ′ of KCaF3 and KCaCl3 compounds. a (˚ A) KCaF3 Present

4.489

B0 (GPa) 46.98

B0′ 4.5475

4.494a


Previous work

4.5293b 4.2874c 4.4584d 4.41e

45.103a

KCaCl3 Present

5.410

23.83

Previous work

5.584a

22.184a

4.296

a Theoretical.16 b Theoretical.17 c Experimental.18 d Experimental.19 e Experimental.20

compounds is between 4.0 and 5.0, in accordance with the other predictions for the cubic pervoskites.3,4 I did not find theoretical data for KCaCl3 to compare with my results. 3.2. Electronic properties 3.2.1. Band structures and DOS The band structure of the perovskite, KCaX3 (X = F and Cl), is predicted along the high symmetry directions in the first Brillouin zone, where symmetry points are Γ (0, 0, 0), X (1, 0, 0), M (1, 1, 0) and R (1, 1, 1). In this work, the band structure calculated along these high symmetrical directions in the first Brillouin zone at the calculated equilibrium lattice parameter as shown in Fig. 2. One can see that the valence band maximum lies at R symmetry point of the first Brillouin zone while the conduction band minimum lies at Γ symmetry point of first Brillouin zone, for both compounds, resulting in an indirect band-gap (R − Γ). The calculated values of the band-gap width (Eg ) are 6.14 eV and 4.76 eV for KCaF3 and KCaCl3 respectively. In Ref. 17, the author calculated Eg (KCaF3 ) = 5.95 eV, where Chornodolskyy et al.21 reported experimental values of Eg (KCaF3 ) = 6 −13 eV. From these values, it can be seen that the calculated Eg are in good agreement with the theoretically calculated17 and experimentally measured.21 The small deviations between these calculations and previously reported ones are due to the reliability of the present first-principles computations (previous calculations used pseudopotential method). For KCaCl3 compound, no comparison studies were possible due to the lack of any previous experimental or theoretical studies. 1450139-4


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(a)

(b)

Fig. 2. Band structure of (a) KCaF3 and (b) KCaCl3 along the Brillouin zone symmetry lines at the equilibrium lattice constant. Table 2.

Calculated band-gap energies of both KCaF3 and KCaCl3 compounds. E R−R

E Γ−Γ

E X−X

E M −M

E R−Γ

E X−Γ

E M −Γ

KCaF3

9.736

6.301

8.441

9.173

6.124

6.368

6.133

KCaCl3

6.322

5.029

6.179

6.753

4.734

5.116

4.752

In Table 2, band-gap energies are listed at the high symmetry points. But since no previous theoretical and/or experimental data was available till present, currently obtained results can be considered the seed information or reference for interested researchers in such a topic. In Fig. 3, total and partial density of states (DOS) of the KCaX3 compounds are plotted. In this figure, one can see the large dispersion nature of the bands. The bottom most bands at −16.5 eV are related to the F-s states in KCaF3 and Ca-p states at −19.0 eV in KCaCl3 . On the other side, the Ca-p states in KCaF3 are found to lie at −16.5 eV where the Cl-s states in KCaCl3 at −12.5 eV. Now, the shift in Ca-p and X-s in the two compounds can be seen. This shift in these states is argued to be related to Ca–F and Ca–Cl bonds. The peak around −10.1 eV represents mainly the K-p contribution in both compounds. There are valence bands near the Fermi level between 0.0 eV and −2.5 eV that are derived from F-p and Cl-p orbitals. Also, it can be seen that the conduction band is essentially dominated by the Ca-d states for all the studied compounds in this investigation. 1450139-5


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A. A. Mousa 60

TDOS KCaF3

40 20 0 60

PDOS K atom

20 0 8 6 4 2 0

PDOS Ca atom

6 4 2 0

PDOS F atom

-20

-15

-10

a)

-5

0

5

10

15

ENERGY (eV)

60 40

TDOS KCaCl3

20 0 60

DOS (states/eV)


DOS (states/eV)

40

40

PDOS K atom

20 0 40

PDOS Ca atom

20 0 15 10

PDOS Cl atom

5 0 -20

b)

-15

-10

-5

0

5

10

15

Energy(eV)

Fig. 3. (Color online) Total and partial DOS of (a) KCaF3 and (b) KCaCl3 . The black, red and blue lines refer to s-, p- and d-states, respectively.

3.2.2. Electron density In this subsection, the electron density has been calculated since it reveals the stability of the compounds and the nature of chemical bonds in the material.22 From Fig. 4, it can be seen that the charge is transferred from the K cation to X (F and Cl) anion due to the electronegativity variation. Moreover, due to the perfect spherical charge distribution of the K and X (F or Cl) ions, strong ionic bonds rise 1450139-6


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(a)

(b) Fig. 4.

(Color online) Total electron density of (a) KCaF3 and (b) KCaCl3 in the 110 plane.

between K–F and K–Cl. Also, the spherical charge distribution for the Ca ion has been calculated and found to be smaller than that of X (F and Cl) anion. This indicates that the charge is transferred from Ca to X (F and Cl). Hence, it can be concluded that the bonding in KCaX3 compounds is purely ionic. Consequently, the calculations in this subsection are in good agreement with the experimental observations of the ionic nature of KCaF3 .23 3.3. Pressure dependence of band-gaps The hydrostatic pressure effect on the energy gap and on the electronic properties of the semi-conductors was investigated in this subsection. The pressure change, which is induced within the interatomic distances, has lead to changes in the covalent-bands charge-distributions and consequently affected all important parameters including the compounds’ energy gaps. 1450139-7


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A. A. Mousa

where B0 and V0 are bulk moduli, the derivative of bulk modulus with respect to pressure and unit cell volume at ambient pressure, respectively. In Fig. 5, the band-gap energies of KCaX3 compound have been investigated as a function of hydrostatic pressure from 0 GPa to 16 GPa. The data were fitted with the quadratic square fits to obtain the band-gap energy pressure coefficients of the tow compounds. It was found that, based on the DFT calculations, as the pressure increases from 0 GPa to 16 GPa, the band-gap energy increases monotonically from 6.14 (4.76) eV to 7.30 (5.60) eV in KCaF3 (KCaCl3 ) compound. Based on the relations: EKCaF3 (p) = 6.138 + 0.0889p − 0.001p2 ,

(1a)

EKCaCl3 (p) = 4.763 + 0.068p − 0.001p2 .

(1b)

One can see from these two equations and Fig. 5, the linear relationship between energy band-gap and the hydrostatic pressure. The band-gap for KCaF3 compound changes from indirect (Γ − R) to direct (Γ − Γ) at 13.5 GPa. As such, the band-gap for KCaF3 becomes a direct one at higher pressures (≥13.5 GPa).

7.5

KCaF3 7.0

KCaCl3

6.5

Band gap (eV)


The variation of the lattice parameter with pressure is given by the equation below; the Murnaghan equation of state12 which has been used to compress the unit cell. " ′ # B0 B0 V0 p= ′ −1 , B0 V

6.0

5.5

5.0

4.5 0

4

8

12

16

Pressure (GPa) Fig. 5.

Calculated indirect band-gap energy of KCaX3 under varying hydrostatic pressure. 1450139-8


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The band-gap energies are being displayed in Fig. 6 at the high symmetry point’s variation of KCaF3 and KCaCl3 compounds as a function of hydrostatic pressure. The pressure range was from 0 GPa to 16 GPa. The data were fitted with the quadratic square fits to obtain the band-gap pressure coefficients of the four main 11

R-R

X-X

Γ−Γ

M-M

KCaF3

Energy Gap (eV)

9

8

7

6 0

4

8

12

16

pressure (GPa) (a) 8

R-R

X-X

Γ−Γ

M-M

KCaCl3 7

Energy Gap (eV)


10

6

5

0

4

8

12

16

pressure (GPa) (b) Fig. 6. Calculated band-gaps energy of (a) KCaF3 and (b) KCaCl3 at high symmetry points under varying hydrostatic pressure. 1450139-9


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Table 3. Calculated linear and quadratic pressure coefficients of high symmetry points band-gap energy of KCaF3 and KCaCl3 compounds. R−R

Γ−Γ

X −X

M −M

KCaF3 E(0) a b

9.753 0.070 −0.001

6.319 0.078 −0.001

8.461 0.091 −0.002

9.185 0.117 −0.002

KCaCl3 E(0) a b

6.346 0.031 −0.001

5.060 0.065 −0.002

6.168 0.040 −0.001

6.796 0.084 −0.002

transitions. They are related by the expression: E(p) = E(0) + ap + bp2 . The calculated values of the above equation are summarized in Table 3. Similar results can be observed from Fig. 6 and Table 3, (b ≈ 0): the linear dependence can be seen in the four curves across the considered range of pressure. 3.4. Optical properties In this subsection, the energy dependent optical properties of KCaX3 were investigated and described using the following complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω), where ε1 (ω) and ε2 (ω) are the real and imaginary parts of this dielectric function, respectively. Since the imaginary part ε2 (ω) is directly related to the electronic band structure, it is often computed by summing up all possible transitions from the occupied to the unoccupied states while keeping into consideration the appropriate transition matrix element. Full detailed description of the calculation of these matrix elements is given by Ambrosch–Draxl et al.24–26 Hence, the calculated ε2 (ω) has been presented in Fig. 7. The different characteristic peaks shown in the spectra of ε2 (ω) can be easily described by the DOS of the compounds. The two threshold points occurring in the spectra at values of 4.01 eV and 2.24 eV for KCaF3 and KCaCl3 , respectively, can be safely argued and/or related to the electronic transition from the X-p state of the VB with a small contribution from the K-p state, which resides just below the Fermi energy, to the Ca-3d states in the CB. In addition, it is often noticed that absorption of materials increases sharply after threshold points, and the compounds under investigation are no exception. Also, it can be said that the varying structural details seen in the spectra may be attributed to the transition of electrons from K-3p, Ca-3p and X-2p states of the VB to the unoccupied states in the CB where the main peaks are situated at 9.76 eV and 8.00 eV for KCaF3 and KCaCl3 , respectively. These peaks are dominated by transitions from X-p band with small contribution from the K-p state. And as we move from F− ions towards Cl− ions, the compounds were found 1450139-10


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First-principles study of structural, electronic and optical properties of the KCaX3 4.0 3.5

KCaCl3 KCaF3

ε2(ω)

3.0 2.5 2.0 1.5 1.0

0.0 0

5

10

15

20

25

30

35

Energy (eV) Fig. 7.

Calculated imaginary part ε2 (ω) of the dielectric function ε(ω) for the KCaX3 compounds.

6

KCaCl3

5

KCaF3 4 3

ε1(ω)


0.5

2 1 0

-1 0

5

10

15

20

25

30

35

Energy (eV) Fig. 8.

Calculated real part ε1 (ω) of the dielectric function ε(ω) for the KCaX3 compounds.

to be shifted toward lower energies within the optical response, in agreement with the band structure. Moreover, the real part ε1 (ω) of the frequency-dependent dielectric function is displayed in Fig. 8. The calculated static dielectric constant ε1 (0) in this investigation is found close to 1.94 and 2.75 for KCaF3 and KCaCl3 , respectively. This is very well-matched with the published experimental results for KCaF3 .23 However, this is not the case with respect to KCaCl3 as there is no available known 1450139-11


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A. A. Mousa 2.5

KCaCl3

2.0

KCaF3

n(ω)

1.5

1.0


0.5

0.0 0

5

10

15

20

25

30

35

Energy (eV) Fig. 9.

Calculated refractive index n(ω) of the dielectric function for the KCaX3 compounds.

reference. Beyond the zero frequency limit ε1 (ω) rises and reaches maximum values of 4.09 at 9.37 eV for KCaF3 and 5.40 at 6.16 eV for KCaCl3 , respectively. The ε1 (ω) curves later show an increase, followed by a decrease, followed by an increase till a negative value are reached and to be followed later by a slow increase toward a zero value at high energy while registering minimum values around 26.24 eV and 21.27 eV for KCaF3 and KCaCl3 , respectively. The refractive indices n(ω) of both KCaF3 and KCaCl3 compounds are also calculated and presented in Fig. 9. The zero frequency refractive indices n(0) are found equal to 1.39 and 1.66 for KCaF3 and KCaCl3 , respectively. It is found based on this investigation that the calculated n(0) = 1.39 for KCaF3 is compatible with the experimentally measured one.23 In addition, it was found that n(0) beyond the zero frequency limits increases and reaches maximum values of 2.09 at 9.46 eV and 2.36 at 6.22 eV for KCaF3 and KCaCl3 , respectively. Also, the extinction coefficients k(ω) have been calculated and shown in Fig. 10. The local maxima of the k(ω) corresponding to the zero of ε1 (ω) are found to be 1.09 at 9.86 eV for KCaF3 and 1.14 at 8.59 eV for KCaCl3 , respectively. The reflectivity spectrum R(ω) is plotted in Fig. 11. The zero frequency limits of the spectra R(0) are 2.7% and 6.1% for KCaF3 and KCaCl3 , respectively. One can notice that at low energies (i.e., 0-9 eV for KCaF3 and 0–5 eV for KCaCl3 ), the reflectivity spectrum is relatively lower than 10% indicating highly transparent materials through the infrared, visible and the low frequency end of the ultraviolet region of the energy spectrum. This may present the investigated compounds as potential transparent coating candidate and valuable efficient lenses materials. Beyond these energy ranges, the reflectivity increases nonlinearly till it reaches maximum 1450139-12


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First-principles study of structural, electronic and optical properties of the KCaX3 1.2

KCaCl3 KCaF3

k(ω)

0.8

0.0 0

5

10

15

20

25

30

35

Energy(eV) Fig. 10. Calculated extinction coefficient k(ω) of the dielectric function ε(ω) for the KCaX3 compounds.

0.4

KCaCl3 KCaF3

0.3

R(ω)


0.4

0.2

0.1

0.0 0

5

10

15

20

25

30

35

Energy (eV) Fig. 11.

Calculated reflectivity R(ω) of the dielectric function ε(ω) for the KCaX3 compounds.

values of 36.8% at 26.35 eV for KCaF3 and 25.8% at 21.29 eV for KCaCl3 . Figure 12 shows the absorption coefficient I(ω) for both compounds. It is noticed that there is a very high absorption peak at high energies. The maximum absorption coefficient is found close to 2.8% at 26.16 eV for KCaF3 and 1.8% at 26.41 eV for KCaCl3 . From these results, one can finally conclude that these materials possess considerable absorption at higher energies. 1450139-13


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A. A. Mousa 300

KCaCl3 KCaF3

I(ω)

225

150


75

0 0

5

10

15

20

25

30

35

Energy (eV) Fig. 12. Calculated absorption coefficient I(ω) of the dielectric function ε(ω) for the KCaX3 compounds.

4. Conclusion This investigation presents the calculated structural and electronic properties of the perovskite KCaX3 (X = F and Cl) utilizing the FP-LAPW method with the exchange-correlation functional GGA. All the lattice constants and the bulk modulus as well are found to compare well with the available data in the literature. These calculations revealed the nature of the indirect energy band-gap (R − Γ) and exposed its decrease as it traverses from F to Cl. Corresponding densities of states are identified and presented with their major features. The energy gap dependency on the hydrostatic pressure has showed a linear behavior, especially in the range 0–16 GPa. The calculations conducted in this study revealed that the indirect gap of the KCaF3 structure transforms to a direct gap (Γ − Γ) at 13.5 GPa. The real ε1 (ω) and the imaginary ε2 (ω) parts of the dielectric function is calculated. The calculated static dielectric constant ε1 (0) is found about 4.09 and 5.40 for KCaF3 and KCaCl3 , respectively. The static refractive index n(0) is found to be 1.39 and 1.66 for KCaF3 and KCaCl3 , respectively. The zero frequency limits of the spectra R(0) are 2.7% and 6.1% for KCaF3 and KCaCl3 , respectively. I note that at low energies 0–9 eV for KCaF3 and 0–5 for KCaCl3 , the reflectivity spectrum is less than 10% for this shows that the materials are highly transparent in the infrared, visible and low frequency ultraviolet region of the energy spectrum. Acknowledgments The author acknowledges the generous financial support of The Deanship of Scientific Research at Middle East University (MEU). The author also especially thanks 1450139-14


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Prof. Jamil Khalifeh and Dr. Hassan Juwhari for their advice and guidance.


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