Theoretical prediction of the structural, electronic, and thermal properties of Al1−xBxAs ternary alloys

Theoretical prediction of the structural, electronic, and thermal properties of Al1−xBxAs ternary alloys

Materials Science in Semiconductor Processing 16 (2013) 2063–2069 Contents lists available at ScienceDirect Materials Science in Semiconductor Proce...

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Materials Science in Semiconductor Processing 16 (2013) 2063–2069

Contents lists available at ScienceDirect

Materials Science in Semiconductor Processing journal homepage: www.elsevier.com/locate/mssp

Theoretical prediction of the structural, electronic, and thermal properties of Al1  xBxAs ternary alloys Khaled Boubendira a, Hocine Meradji a,n, Sebti Ghemid a, Fouad El Haj Hassan b a b

Laboratoire LPR, Département de Physique, Faculté des Sciences, Université Badji Mokhtar, Annaba, Algeria Laboratoire de Physique des Matériaux, Faculté des Sciences, Université Libanaise, Elhadath, Beirut, Lebanon

a r t i c l e i n f o

abstract

Available online 28 August 2013

First-principles calculations are performed to study the structural, electronic, and thermal properties of the AlAs and BAs bulk materials and Al1  xBxAs ternary alloys using the full potential-linearized augmented plane wave method within the density functional theory. The structural properties are investigated using the Wu–Cohen generalized gradient approximation that is based on the optimization of total energy. For band structure calculations, both Wu–Cohen generalized gradient approximation and modified BeckeJohnson of the exchange-correlation energy and potential, respectively, are used. The dependence of the lattice constant, bulk modulus, and band gap on the composition x was analyzed. The lattice constant for Al1  xBxAs alloys exhibits a marginal deviation from the Vegard's law. A small deviation of the bulk modulus from linear concentration dependence was observed for these alloys. The composition dependence of the energy band gap was found to be highly nonlinear. Using the approach of Zunger and coworkers, the microscopic origins of the gap bowing were detailed and explained. The quasi-harmonic Debye model was used to determine the thermal properties of alloys up to 500 K. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Band structures First principle calculations Ternary alloys Thermal properties

1. Introduction Over the past few years, much attention has been devoted to boron compounds and their alloys, which have a wide range of technological applications, due to their excellent physical properties that include low ionicities [1– 4], short bond lengths [5], hardness [5], high melting points [6], and wide band gaps [7]. Investigation of dilute BGaAs and BAlAs has been made on films grown via both molecular beam epitaxy and metal organic chemical vapor deposition. It is expected that the incorporation of boron into III–V semiconductors, will induce a strong perturbation in the electronic and structural properties of these alloys, because boron has highly different properties than the other III and V elements of the periodic table. However, only few theoretical

n

Corresponding author. Tel.: +213 7 79 27 82 21. E-mail address: [email protected] (H. Meradji).

1369-8001/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mssp.2013.07.022

calculations have been addressed to study the structural and electronic properties of III–V boron alloys. The band gap bowing in boron based III–V ternary alloys has been calculated by Azzi et al. [8]. Very recently, Murphy et al. [9] reported on the deviation from Vegard' law in six ternary MxN1 xAs (M and N are B, Al, Ga, and In) random alloys using the CASTEP (Cambridge Serial Total Energy Package) program (Castep Developers Group (CDG), UK). The structural and electronic properties of BNP, BNAs, BPAs, BNSb, BPSb, and BAsSb were studied by El Haj Hassan et al. [10,11]. To the best of our knowledge no experimental or theoretical investigations of Al1 xBxAs alloy have appeared in the literature. Therefore, the purpose of this paper is to study the structural and electronic properties as well as to investigate the disorder effects in these boron alloys using the full potential linearized augmented plane wave (FP-LAPW) method. The physical origins of gap bowing are investigated by following the approach of Zunger et al. [12]. Finally, the quasi-harmonic Debye model was successfully used to

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determine the thermal properties. The paper is organized as follows. In Section 2 we describe the calculation procedure. Results and a discussion concerning the structural, electronic, and thermal properties are presented in Section 3. The paper is concluded in Section 4.

3. Results and discussions 3.1. Structural properties In order to calculate the ground state properties of AlAs and BAs compounds and their ternary alloys, the total energies are calculated in zinc-blende (B3) structure for different volumes around the equilibrium cell volume V0. We model the alloys at some selected compositions with ordered structures which are described in terms of periodically repeated supercells with eight atoms/unit cell, for the compositions x¼0.25, 0.5, and 0.75. For the structures considered, we performed the structural optimization by minimizing the total energy with respect to the cell parameters and also the atomic positions. For the compositions x ¼0.25 and 0.75, the simplest structure is an eightatom simple cubic cell (luzonite): the anions with the lower concentration form a regular simple cubic lattice. For the composition x¼ 0.5, we have used a cubic cell that contains identical atoms in the same level. The calculated total energies are fitted to the Murnaghan equation of state [24] to determine the ground state properties such as the equilibrium lattice constant a, the bulk modulus B, and its pressure derivative B′. Table 1 summarizes the results of our calculations and compares them with other experimental and theoretical predictions. For the lattice constant, the present WC-GGA results agree well with the previous experimental reports for the binary compounds, hence it is reasonable to expect that the lattice constant of the alloys can be described with similar accuracy in our calculations. Our calculated WC-GGA bulk moduli, which are close to the available experimental data, are larger than the results obtained by Annane et al. [26] and Briki et al. [27]. As can be seen from Table 1, that the lattice parameter for AlAs (x¼0) is larger than those of BAs (x¼1); a (AlAs) 4a (BAs). Because the anion atom is the same in both compounds, this result can be easily explained by considering the atomic radii of B and Al: R (B)¼ 0.85 Å, R(Al) ¼1.15 Å, i.e., the lattice constant increases with increasing atomic size of the anion element. The bulk modulus value for BAs is larger than those of

2. Method of calculations The calculations were performed using FP-LAPW method [13–15] within the framework of density functional theory [16,17] as implemented in WIEN2K [18] code. For structural properties the exchange-correlation potential was calculated using the generalized gradient approximation (GGA) in the new form (WC) proposed by Wu and Cohen [19], which is an improved form of the most popular Perdew–Burke–Ernzerhof GGA [20]. In addition and for electronic properties only, we also applied the modified Becke-Johnson (mBJ) [21] scheme. Our goal is to describe the electronic properties such as band gap, correctly. Tran and Blaha proposed a new potential called mBJ that combines modified Becke-Johnson exchange potential and the LDA correlation potential to get better band gap results compared to experiment [22,23]. In the FP-LAPW method, the wave function, charge density, and potential are expanded by spherical harmonic functions inside nonoverlapping spheres surrounding the atomic sites (muffin-tin spheres) and by plane waves basis set in the remaining space of the unit cell (interstitial region). The maximum l quantum number for the wave function expansion inside atomic spheres was confined to lmax ¼10. The plane wave cutoff of Kmax ¼8.0/RMT (RMT is the smallest muffin-tin radius in the unit cell) is chosen for the expansion of the wave functions in the interstitial region while the charge density is Fourier expanded up to Gmax ¼14(Ryd)1/2. A mesh of 47 special k-points for binary compounds and 24 special k-points for ternary alloys were taken in the irreducible wedge of Brillouin zone. The muffin-tin radius was assumed to be 1.82 a.u. for all the atoms. Both the plane wave cutoff and the number of kpoints were varied to ensure total energy convergence.

Table 1 Calculated lattice constant (a) and bulk modulus (B) for Al1-xBxAs alloys compared to the available theoretical and experimental data. Alloy

x

Lattice constant a (Å) This work WC-GGA

Al1-xBxAs

a

Ref [25]. Ref [26]. c Ref [27]. d Ref [28]. e Ref [29]. f Ref [30]. g Ref [31]. h Ref [32]. i Ref [33]. b

0 0.25 0.5 0.75 1

5.68 5.51 5.31 5.07 4.77

Bulk Modulus B (GPa) Exp

Other calculations a

5.66 4.77e

b

c

This work WC-GGA i

i

5.731 , 5.74 , 5.633 , 5.734 4.812f , 4.784g , 4.728h, 4.741i, 4.817i

72.25 79.28 89.70 110.41 140.27

Exp a

82 , 78.1 -

Other calculations d

67.732b , 66.8c 133f , 137g, 144h

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150

5.8 140

Bulk modulus (GPa)

130

°

Lattice parameter (A )

5.6

5.4

5.2

5.0

120 110 100 90 80

4.8

4.6

70 0.0

0.0

0.2

0.4

0.6

0.8

0.2

Composition x Fig. 1. Composition dependence of the calculated lattice constant (filled squares) of Al1  xBxAs alloys compared with Vegard's law (broken line).

0.4

0.6

0.8

1.0

Composition x

1.0

Fig. 2. Composition dependence of the calculated bulk modulus (filled squares) of Al1  xBxAs alloys compared with the linear composition dependence prediction (broken line). 10

AlAs, B (BAs)4B (AlAs); i.e., in inverse sequence to a, in agreement with the well-known relationship between B and the lattice constant, B p V 1 0 [34], where V0 is the unit cell volume. Usually, in the treatment of alloy problems, it is assumed that the atoms are located at ideal lattice sites and the lattice constants of alloys should vary linearly with composition x according to Vegard's law [35]; however, violations of Vegard's rule have been reported in semiconductor alloys both experimentally [36] and theoretically [37]. The results obtained for the composition dependence of the calculated equilibrium lattice parameter for Al1 xBxAs alloys are shown in Fig. 1. A small deviation from Vegard's law (i.e., a linear variation of the lattice constant of alloys versus composition x) is clearly visible with an upward bowing parameter of  0.342 Å. The bowing parameter is determined by fitting the calculated values with a polynomial function. The physical origin of this deviation should be mainly due to the mismatches of the lattice constants of BAs and AlAs compounds. Also, it should be related to the size of atoms, the ratio R(Al)/R(B)¼1.35 where R is the atomic radius. Fig. 2 shows the bulk modulus as a function of x for the Al1  xBxAs alloys. A small deviation from linear concentration dependence (LCD) is observed, with downward bowing equal to 3.996 GPa. This deviation is mainly due to the mismatch of the bulk modulus of binary compounds. 3.2. Electronic properties We have calculated the band structures for Al1  xBxAs alloys along the high directions in the first Brillouin zone at the calculated equilibrium lattice constants. For the binary compounds, maximum of the valence band is in Γ point while the minimum of the conduction band appears at different points. In BAs it is a point between Γ and X (we call it Δmin), for AlAs it appears in X. Therefore BAs and AlAs have an indirect band gap along Γ-Δmin and Γ-X, respectively, which in good agreement with the theoretical works reported in [33,38,39]. For the concentrations x ¼0.25, 0.50, and 0.75 the band gap is direct and occurs

AlAs

5

0

-5

-10

-15

W

L

X

K

Fig. 3. Band structure of AlAs compound.

at Γ point (Γ-Γ). As prototypes, the band structures of AlAs and Al0.25 B0.75As have been shown in Fig. 3 and Fig. 4, respectively. The results of the band gaps for all studied compositions (x ¼0, 0.25, 0.5, 0.75, and 1) are given in Table 2, along with the available experimental and theoretical results for comparison. The calculated band gap energies are underestimated by the WC-GGA. It is worth noting that the calculated values of band gaps via mBJ approach are highly improved with respect to the experiment as compared to those obtained via WC-GGA. According to our present calculations, the mBJ approximation performs well for describing the band properties; the mBJ results can be compared with the more expensive methods such as GW and hybrid functionals for band gaps. It can be used to model the electronic properties of semiconductors. The variation of the band gap as a function of the composition x for the material of interest obtained by WC-GGA and mBJ approaches is presented in Fig. 5. We note that the band gap Eg varies nonlinearly with increasing of B content providing a positive gap bowing. We calculated the total bowing parameter by fitting the nonlinear variation of the calculated band gaps versus concentration with a quadratic function. The results

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are shown in Fig. 3 and obey the following variations: EmBJ g ðxÞ ¼ 2:04332:9765x

þ 2:7565x2

CE

ABðaÞ þ ACðaÞ⟹AB0:5 C 0:5 ðaÞ

ð5Þ

ð1Þ SR

ðxÞ ¼ 1:35222:1754x þ 2:1234x2 EWCGGA g

ð2Þ

The quadratic terms are referred to as band gap bowing parameters. In order to better understand the physical origins of the band gap bowing in these alloys, we have followed the procedure of Bernard and Zunger [43], in which the bowing parameter (b) is decomposed into three physically distinct contributions. In fact the overall band gap-bowing coefficient at x ¼0.50 measures the change in the band gap according to the reaction:   ABðaAB Þ þ AC ðaAC Þ-AB0:5 C 0:5 aeq ð3Þ where aAB and aAC are the equilibrium lattice constants of the binary compounds AB and AC, respectively; aeq is the alloy equilibrium lattice constant. The authors now decompose the previous reaction into three steps: VD

ABðaAB Þ þ ACðaAC Þ⟹ABðaÞ þ ACðaÞ

AB0:5 C 0:5 ðaÞ⟹AB0:5 C 0:5 ðaeq Þ

ð6Þ

The first step measures the volume deformation (VD) effect on the bowing. The corresponding contribution bVD to the total gap bowing parameter represents the relative response of the band structure of the binary compounds AB and AC to hydrostatic pressure, which here arises from the change of their individual equilibrium lattice constants to the alloy value a¼a(x) (from Vegard's rule). The second contribution, the charge exchange (CE) contribution bCE, reflects a charge transfer effect which is due to the different (averaged) bonding behavior at the lattice constant a. The final step measures changes upon passing from the unrelaxed to the relaxed alloy by bSR. Consequently, the total gap bowing parameter is defined as: b ¼ bVD þ bCE þ bSR

ð4Þ

ð7Þ

2.2

Al1-xBxAs

mBJ WC

2.0 10 Al0.25B0.75As

Band gap (eV)

1.8

5

0

1.6 1.4 1.2 1.0

-5

0.8 0.6

-10

0.0

0.2

0.4

0.6

0.8

1.0

Compostion (x) -15

R

Fig. 5. Variation of the calculated band gap energy of Al1  xBxAs alloys as a function of B concentration using modified Becke-Johnson and Wu– Cohen generalized gradient approximation exchange and correlation potentials.

M

X

Fig. 4. Band structure of Al0.25B0.75As alloy.

Table 2 Computed energy band gap Eg (eV) for Al1-xBxAs alloys, using WC-GGA and mBJ schemes. Alloy

x

Eg (eV) This work

0 0.25 Al1-xBxAs 0. 5 0.75 1 a

Ref. [40]. Ref. [27]. c Ref. [41]. d Ref. [42]. e Ref. [32]. f Ref. [33]. g Ref. [38]. b

mBJ

WC-GGA

2.161 1.221 1.240 1.573 1.713

1,462 0,711 0,756 1,132 1,185

Exp

Other calculations

2.24a

1.43b, 1.39c, 1.31f, 1.40f, 2.25f – – – 1.25d, 1.06d, 1.33e, 1.13f, 1.18f, 1.79f , 1.34g

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bVD ¼ 2½EAlAs ðaAlAs Þ–EAlAs ðaÞ þ EBAs ðaBAs Þ–EBAs ðaÞ

ð8Þ

bCE ¼ 2½EAlAs ðaÞ þ EBAs ðaÞ2EAlBAs ðaÞ

ð9Þ

bSR ¼ 4½EAlBAs ðaÞEAlBAs ðaeq Þ

ð10Þ

Here, aAlAs, aBAs, and aeq are the equilibrium lattice constants of AlAs, BAs and Al1  xBxAs alloys, respectively. The lattice constant (a) is calculated by linear composition dependence rule [35] for the alloys. All these energy gaps occurring in expressions (8)–(10) have been calculated for the indicated atomic structures and lattice constants. The results are given in Table 3. Because the shift between the lattice parameters of AlAs and BAs is larger than 15%, one must expect an important disorder in the electronic properties. We note that the calculated quadratic parameter (gap bowing) within WC-GGA and mBJ is very close to its corresponding result obtained by Zunger's approach. The volume deformation contribution bVD has been found to be significant for the alloys under load. This term is correlated to the relative large mismatch of the lattice constants of the corresponding binary compounds. The charge transfer effect is also significant. It is due to the large electronegativity difference between Al (1.61) and B (2.04) atoms.

M being the molecular mass per unit cell and BS the adiabatic bulk modulus, approximated by the static compressibility [44]: ( ) 2 d E ðV Þ Bs ffiBðV Þ ¼ V ð14Þ dV 2 f(s)is given by [47,48]: 8 " 9



#1 =1=3 < 21 þ s 3=2 11 þ s 3=2 f ðsÞ ¼ 3 2 þ : ; 312s 31s

By solving Eq. (14), one can obtain the thermal equation of state (EOS)V(P,T). The isothermal bulk modulusBT, the heat capacity CVand the thermal expansion coefficient α are given by [50]: 2 n

δ G ðV; P; T Þ ð17Þ BT ðP; V Þ ¼ V δV 2 P;T     3θ=T C V ¼ 3nkB 4D θ=T  θ=T e 1

Finally, to investigate the thermal properties of Al1  xBalloy, we used the quasi-harmonic Debye model [44] in which the nonequilibrium Gibbs function G*(V;P,T) is written in the form:

α¼

ð11Þ

where E(V) is the total energy per unit cell, PVcorresponds to the constant hydrostatic pressure condition, θ(V)is the Debye temperature, and AVib is the vibrational term, which can be written using the Debye model of the phonon density of states as [45,46]:       9θ þ 3 ln 1eðθ=TÞ D θ=T ð12Þ AV ib ðθ; T Þ ¼ nkB T 8T   where D θ=T represents the Debye integral, nis the number of atoms per formula unit. For an isotropic solid, θ is expressed as [45]: rffiffiffiffiffi ℏ h 2 1=2 i1=3 Bs θD ¼ 6π V n f ðsÞ ð13Þ kB M

Table 3 Decomposition of the bowing parameter b into volume deformation (VD), charge transfer (CE) and structural relaxation (SR) contributions compared with the optical bowing obtained by a quadratic interpolation (all values are in eV). Parameter

Zunger approach

Quadratic fits

mBJ

mBJ

WC-GGA

ð18Þ

γC V BT V

ð19Þ

where γ is the Grüneisen parameter, defined as: d ln θðV Þ γ¼ d ln V

ð20Þ

Through the quasi-harmonic Debye model, one could calculate the thermodynamic quantities of any temperature and pressure, for the compounds from the calculated E-V data at T¼0 and P ¼0. The thermal properties are determined in the temperature and pressure ranges 0– 500 K and 0–8 GPa. The variation of the lattice constant of Al0.25B0.75As alloy versus temperature at several pressures is shown in Fig. 6. The lattice constant increases with Al0.25B0.75As

4.94 4.92

Lattice constant (A°)

Gn ðV; P; T Þ ¼ EðV Þ þ PV þ AV ib ½θðV Þ; T 

ð15Þ

The Poisson ratio s is taken as 0.25 [49]. Therefore, the nonequilibrium Gibbs function G*(V;P,T) as a function of (V;P,T) can be minimized with respect to volumeV:  n  ∂G ðV; P; T Þ ¼0 ð16Þ ∂V P;T

3.3. Thermal properties

xAs

2067

P= 0 GPa

4.90

P= 2 GPa

4.88 4.86

P= 4 GPa

4.84

P= 6 GPa 4.82

P= 8 GPa

WC-GGA 4.80

bVD bCE bSR b

0.631 2.469  0.378 2.721

1.802 0.816  0.556 2.062

– – – 2.756

– – – 2.123

0

100

200

300

400

500

Temperature (K) Fig. 6. Variation of the lattice constant of Al0.25B0.75As alloy as a function of temperature at several pressures.

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Al0.25B0.75As

Al0.5B0.5As

120

50

P= 8 GPa 40

P= 6 GPa

Cv (J/mol*K)

Bulk modulus (GPa)

112

104

P= 4 GPa 96

P= 2 GPa

P= 0 GPa P= 2 GPa P= 4 GPa P= 6 GPa P= 8 GPa

30

20

88

P= 0 GPa

80

10

72

0 0

100

200

300

400

0

500

100

increasing temperature at a given pressure. On the other side, as the pressure increases, the lattice constant decreases at a given temperature. The rate of increase of the lattice constant with temperature decreases with increasing pressure. At high pressure, the thermal effect on the lattice parameter is weaker. Fig. 7 shows the bulk modulus variation of Al0.5B0.5As alloy versus temperature at a given pressure. The bulk modulus is practically constant between 0 K and 100 K, and then it decreases with increasing temperature. For a given temperature, the bulk modulus increases with increasing pressure. This indicates that the alloy becomes less compressible with increasing pressure. The effect of increasing pressure and decreasing temperature on the bulk modulus are nearly the same. The knowledge of the heat capacity of a substance not only provides essential insight into its vibrational properties but is also mandatory for many applications. Two famous limiting cases are correctly predicted by the standard elastic continuum theory [51]. At high temperature, the constant volume heat capacity Cv tends to the Dulong–Petit limit [52]. At sufficiently low temperature, Cv is proportional to T3 [51]. At intermediate temperatures, however, the temperature dependence of Cv is governed by the details of vibrations of the atoms and for a long time could only be determined from experiments. Fig. 8 shows the dependence of the heat capacity Cv on temperature and pressure for the Al0.25B0.75As alloy. It is shown that when T o500 K, the heat capacity depends on both temperature and pressure. At high temperatures, (T4500 K) Cv tends to the Dulong–Petit limit. At high temperatures Cv approaches approximately 70 J/mol/K. for the alloys under investigation. When the temperature is constant, Cv decreases with the applied pressure. As can be seen, the effect of the pressure on the heat capacity is not significant. The calculated properties at different temperatures are very sensitive to the vibrational effects. In the quasiharmonic Debye model, the Debye temperature is a key parameter. The temperature dependence of the Debye temperature θD at several pressures for the Al0.75B0.25As

300

400

500

Fig. 8. Variation of the heat capacity Cv of Al0.25B0.75As alloy versus temperature at several pressures.

Al0.75B0.25As

600

P= 8 GPa 580

P= 6 GPa

Debye temperature (K)

Fig. 7. Variation of the bulk modulus of Al0.5B0.5As alloy as a function of temperature at several pressures.

200

Temperature (K)

Temperature (K)

560

P= 4 GPa 540

P= 2 GPa 520 500

P= 0 GPa

480 460 0

100

200

300

400

500

Temperature (K) Fig. 9. Variation of the Debye temperature of Al0.75B0.25As alloy versus temperature at several pressures.

alloy is calculated and plotted in Fig. 9. From 0 K to about 100 K, θD is nearly constant and decreases almost linearly with rising temperature from T4100 K. It is also shown that when the temperature is constant, the Debye temperature increases with the increase of pressure. Compared with Fig. 7, one can see that the compressibility increase leads to Debye temperature decrease. This result is in accordance with the fact that Debye temperature is proportional to the bulk modulus and that a hard material exhibits a high Debye temperature. The dispersal of energy and matter is described by the entropy, denoted by the symbol S. On a microscopic scale, the entropy can be defined as a measure of disorder of a system. The variation of the entropy S versus temperature and pressure for the Al0.5B0.5Asalloys is displayed in Fig. 10. It is found that S increases sharply with increasing temperature and decreases with the increase of pressure. As the temperature increases, the vibrational contribution to the entropy increases and therefore the entropy increases with temperature.

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60 Al0.5B0.5As

Entropy S(J/mol-1.K-1)

50

40 P= 0 GPa P= 2 GPa P= 4 GPa P= 6 GPa P= 8 GPa

30

20

10

0

0

50

100

150

200

250

Temperrature (K) Fig. 10. Variation of the entropy of Al0.5B0.5As alloy versus temperature at several pressures.

4. Conclusions In summary, the present paper reports a theoretical study of structural, electronic, and thermal properties of Al1  xBxAs ternary alloys using the FP-LAPW method within density functional theory. The main conclusions can be summarized as follows: 1. The calculated lattice constants and bulk modulus of the binary compounds are in good agreement with available experimental and other theoretical data, which are in support of those of the ternary alloy that we report for the first time. 2. A nonlinear behavior of the lattice constant and bulk modulus dependence on x concentration has been observed. 3. The band gap is shown to vary strongly with the composition x in a nonlinear way. The results of the decomposition of the band gap bowing suggest that the bowing parameter is caused by both the charge transfer and volume deformation contributions. 4. Finally, through the quasi-harmonic Debye model, the dependence of the lattice constant, bulk modulus, heat capacity, Debye temperature and entropy on temperature and pressure have been obtained successfully.

Conflicts of interest The article is original and it has been written by the stated authors who are all aware of its content. The authors approve its submission and state that it has not been published previously and it is not under consideration for publication elsewhere. No conflicts of interest exists. References [1] R.M. Wentzcovitch, K.J. Chang, M.L. Cohen, Phys. Rev. B 34 (1986) 1071–1079. [2] A. Garcia, M.L. Cohen, Phys. Rev. B 47 (1993) 4215–4220.

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