Structural, electronic, optical and thermodynamic properties of SrxCa1−xO, BaxSr1−xO and BaxCa1−xO alloys

Journal of Alloys and Compounds 509 (2011) 1440–1447

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Structural, electronic, optical and thermodynamic properties of Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys M.A. Ghebouli a , B. Ghebouli b , A. Bouhemadou c,d,∗ , M. Fatmi e , K. Bouamama b a

Département de Physique, Centre Universitaire, Bordj Bou-Arréridj, 34000, Algeria Département de Physique, Faculté des Sciences, Université de Sétif, 19000, Algeria Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University of Setif, 19000 Setif, Algeria d Department of Physics and Astronomy, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia e Laboratoire de Physique et Mécanique des Matériaux Métalliques (LP3M), Université de Sétif, 19000, Algeria b c

a r t i c l e

i n f o

Article history: Received 29 June 2010 Received in revised form 28 October 2010 Accepted 16 November 2010 Available online 23 November 2010 PACS: 71.20.Dg 71.15.Mb 78.20.Ci 65.40.−b Keywords: Ternary alloys Ab initio calculations Structural properties Band structures Optical constants Thermodynamic properties

a b s t r a c t The structural, electronic, optical and thermodynamic properties of Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O ternary alloys in NaCl phase were studied using pseudo-potential plane-wave method within the density functional theory. We modeled the alloys at some selected compositions with ordered structures described in terms of periodically repeated supercells. The dependence of the lattice parameters, band gaps, dielectric constants, refractive indices, Debye temperatures, mixing entropies and heat capacities on the composition x were analyzed for x = 0, 0.25, 0.50, 0.75 and 1. The lattice constant for Srx Ca1−x O and Bax Sr1−x O exhibits a marginal deviation from the Vegard’s law, while the Bax Ca1−x O lattice constant exhibits an appreciable upward bowing. A strong deviation of the bulk modulus from linear concentration dependence was observed for the three alloys. The microscopic origins of the gap bowing were detailed and explained. The composition dependence of the dielectric constant and refractive index was studied using different models. The thermodynamic stability of these alloys was investigated by calculating the phase diagram. The thermal effect on some macroscopic properties was investigated using the quasiharmonic Debye model. There is a good agreement between our results and the available experimental data for the binary compounds which may be a support for the results of the ternary alloys reported here for the first time. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The II–VI and III–V binary compounds and their ternary alloys provide the material basis for a number of well-established commercial devices, as well as new cutting edge classes of electronic and optoelectronic ones. Semiconductor alloys provide a natural means of tuning the magnitude of the forbidden gap and other material parameters so as to optimize and widen the applications of semiconductor devices [1]. With the advent of small-structure systems, such as quantum wells and superlattices, the effect of alloy compositions, size, device geometry, doping and controlled lattice strain can be combined to achieve maximum tenability [2,3].

∗ Corresponding author at: Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University of Setif, 19000 Setif, Algeria. E-mail address: a [email protected] (A. Bouhemadou). 0925-8388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2010.11.097

The alkaline-earth chalcogenides (oxides, sulfides, selinides and tellurides), members of the IIA/B-VIB family, form a very important closed shell ionic system crystallizing in NaCl-type structure at room temperature [4–6]. They are technologically important materials, with applications in the area of microelectronic, catalysis, medicine, heterogeneous catalysis, electronics, spintronics and optoelectronics [7–9]. The alkaline-earth oxides AEO, where AE = Ca, Sr and Ba, members of the alkaline-earth chalcogenides, have been considered as a typical case for understanding the bonding in ionic oxides and are also some of the most fundamental industrial materials. This is because of their wide range of applications ranging from catalysis to microelectronics. For example, their catalytic activities are important for chemical engineering [10]. Many ab initio calculations of the parent compounds, i.e., CaO, SrO and BaO, can be found in the literature [5,10–14]. But the properties of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O ternary alloys have not been studied. With the motivation of finding promising ternary alloys from these binary compounds with improved physical properties in comparison with other ternary alloys, the present study focus mainly on the

M.A. Ghebouli et al. / Journal of Alloys and Compounds 509 (2011) 1440–1447

2. Method of calculations The first-principles calculations were performed using the pseudo-potential plane-wave total energy method implemented in the CASTEP suite of programs [15]. Interactions of electrons with ion cores were represented by the Vanderbilttype ultra-soft pseudo-potential for Ca, Sr, Ba and O [16]. The exchange-correlation potential was calculated using the generalized gradient approximation (GGA) of Perdew et al. [17]. The plane-wave basis set cut-off was set as 350 eV for all cases. The special points sampling integration over the Brillouin zone was employed by using the Monkhorst–Pack (MP) method with a 8 × 8 × 8 special k-point mesh [18]. These parameters were sufficient in leading to well converged total energy and geometrical configurations. The structural parameters of Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O ternary alloys in NaCl phase with x = 0, 0.25, 0.50, 0.75 and 1 were determined using the Broyden–Fletcher–Goldfarb–Shenno (BFGS) minimization technique [19]. This provides a fast way of finding the lowest energy structure. The tolerances for geometry optimization were set as the difference of total energy within 5 × 10−6 eV atom−1 , maximum ionic Hellmann–Feynman force within 0.01 eV A˚ −1 and maximum stress within 0.02 eV A˚ −3 . The optical properties may be derived from the knowledge of the complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω). The imaginary part of the dielectric function ε2 (ω) is calculated from the momentum matrix elements between the occupied and unoccupied wave functions within the selection rules. The real part of the dielectric function, ε1 (ω), can be evaluated from ε2 (ω) by the Kramers–Kronig relation. All the other optical constants, such as the refractive index n(ω), the extinction coefficient k(ω), the optical reflectivity R(ω), the absorption coefficient ˛(ω) and the energyloss spectrum L(ω) can be computed from the values of ε(ω). For the calculation of the optical properties, which usually requires a dense mesh of uniformly distributed k-points, the Brillouin zone integration was performed using a 20 × 20 × 20 MP k-mesh. The study of thermal effects was done within the quasi-harmonic Debye model implemented in the Gibbs program [20]. For a solid described by an energy–volume (E–V) relationship in the static approximations, the Gibbs program allows us to evaluate the Debye temperature, to obtain the Gibbs free energy G(V; P, T) and to minimize G for deriving the thermal equation of state (EOS) V(P, T). Other macroscopic properties related to P and T can also be derived using standard thermodynamic relations. Detailed description of the quasi-harmonic Debye model can be found in Ref. [20].

3. Results and discussions 3.1. Compositional dependence of the lattice parameters Firstly, we calculated the equilibrium structural parameters (lattice constant and bulk modulus) for the parent binary compounds: AEO, where AE are Ca, Sr and Ba, in the NaCl-type (#225) structure. There are eight atoms (4 AE and 4 O) in the unit cell. When we add the Sr atom to the CaO, in order to obtain Srx Ca1−x O, the most probable crystal structures, according to x content, for this ternary alloy are given in Table 1. The alloys have been modeled at some selected compositions (x = 0, 0.25, 0.5, 0.75, 1) with ordered structures described in terms of periodically repeated supercells, i.e., we have assumed that the atoms are located at the ideal lattice sites in ordered positions. The calculated lattice constant and bulk modulus for each x of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys are summarized in Table 2, along with the other experimental and theoretical values

Table 1 Atomic positions in the Srx Ca1−x O alloy. Composition (x)

Atom

Atomic positions

0.25

O Sr Ca

(0 0 0), (0 1/2 1/2), (1/2 0 1/2), (1/2 1/2 0) (1/2 0 0), (0 1/2 0), (0 0 1/2) (1/2 1/2 1/2)

0.5

O Sr Ca

(0 0 0), (0 1/2 1/2), (1/0 0 1/2), (1/2 1/2 0) (1/2 0 0), (0 1/2 0) (1/2 1/2 1/2), (0 0 1/2)

0.75

O Sr Ca

(0 0 0), (0 1/2 1/2), (1/0 0 1/2), (1/2 1/2 0) (1/2 1/2 1/2) (1/2 0 0), (0 1/2 0), (0 0 1/2)

available in the literature [11,21–28]. The present results agree well with the previous experimental reports for the binary compounds [21,24,26–28]. The computed lattice constants deviate from the measured ones within 0.7, 0.5 and 0.8% for SrO, CaO and BaO, respectively. The calculated lattice constants at different compositions, as shown in Fig. 1, were well fitted with the following relation: aAx B1−x O = xaAO + (1 − x)aBO − bx(1 − x)bd

(1)

where aAx B1−x O , aAO and aBO are the lattice constants of the ternary alloy Ax B1−x O and its binary parents AO and BO, respectively. The quadratic term bd represents the disordered parameter (bowing). The obtained upward bowing parameters (b) are −0.058, −0.046 and −0.305 A˚ for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O, respectively. The Bax Ca1−x O alloy has the highest deviation from the Vegard’s law [29], this could be explained by the relatively high mismatch between the lattice constants of its binary parents (a(BaO) − a(CaO) = 0.746 Å) comparatively to the two other alloys which have weak mismatches between their binary parents (a(SrO) − a(CaO) = 0.362 Å and a(BaO) − a(SrO) = 0.384 Å).

5.2

Sr Ca

O

Ba Sr

O

x

1-x

5.1 5.0 4.9 5.6

Lattice parameter (A)

composition dependence of the structural, electronic, optical and thermodynamic properties of Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O ternary alloys in the NaCl phase. In this work, we used pseudo-potential plane-wave (PP-PW) method within the generalized gradient approximation (GGA) to determine a set of physical parameters of CaO, SrO, BaO and their ternary alloys, namely the optimized lattice constant, bulk modulus, energy band gap, optical dielectric constant, refractive index and thermodynamic properties. This paper is organized as follows. In Section 2, we briefly describe the computational techniques used in this study. The most relevant results obtained for the structural, electronic, optical and thermodynamic properties for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O ternary alloys in NaCl phase are presented and discussed in Section 3. Concluding remarks are given in Section 4.

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5.5

x

1-x

5.4 5.3 5.2 Ba Ca

5.4

x

O

1-x

5.2 5.0 4.8

0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Fig. 1. Composition dependence of the calculated lattice constant (open squares) of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys compared with Vegard’s prediction (dashed line).

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Table 2 ˚ and bulk modulus (in GPa) for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. Calculated lattice constant (in A) x

Lattice constant PP-PW

Expt.

Others

PP-PW

Srx Ca1−x O

1 0.75 0.5 0.25 0

5.198 5.118 5.030 4.941 4.836

5.16a

5.16b , 5.19c

86.76 90.00 93.00 99.45 107.7

Bax Sr1−x O

1 0.75 0.5 0.25 0

5.582 5.497 5.399 5.306 5.198

5.536f

5.562g

Bax Ca1−x O

1 0.75 0.5 0.25 0

5.582 5.439 5.291 5.082 4.836

a b c d e f g h i

Bulk modulus

4.811i

4.86e , 4.84c

Expt.

71.72 73.00 75.78 82.00 86.76 71.72 66.49 69.74 76.12 107.7

Others

90.6d

86c , 96e

61h

71.06g

110i

117e , 134e

Ref. [21]. Ref. [22]. Ref. [23]. Ref. [24]. Ref. [25]. Ref. [26]. Ref. [12]. Ref. [27]. Ref. [28].

The overall behaviors of the variation of the bulk modulus as a function of the composition for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys are presented in Fig. 2, and compared to the results predicted by linear concentration dependence (LCD). A significant deviation from the LCD is observed with downward bowing equal to 15.394, 11.885 and 87.714 GPa for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O, respectively. It should be noted that the deviation from LCD is more important in the Bax Ca1−x O alloy compared to the two other alloys, this might be mainly due to the large mismatch between the bulk moduli of its constitute binary compounds; the

108 104

Sr Ca x

100

O

1-x

96

Bulk modulus (GPa)

92 88 88 84 80

Ba Sr x

76

O

1-x

72 110 100

Ba Ca x

90

O

1-x

80 70 60 0.0

0.2

0.4

0.6

0.8

1.0

Comoposition (x) Fig. 2. Composition dependence of the calculated bulk modulus (open squares) of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys compared with LCD prediction (dashed line).

bulk modulus of BaO is 33% higher than that of CaO. A comparison of the lattice constants and bulk moduli of these alloys (Figs. 1 and 2) shows that an increase of the former parameters is accompanied by a decrease of the latter ones. This which is in agreement with the well-known relationship between B and the lattice constant: B ∝ V0−1 , where V0 is the primitive cell volume. 3.2. Electronic properties We have calculated the band structures for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys along the high directions in the first Brillouin zone at the calculated equilibrium lattice constants. The band structure calculations give an indirect band gap  –X for CaO and a direct band gap X–X for SrO and BaO. The calculated band gaps for all studied compositions (x = 0, 0.25, 0.50, 0.75, 1) are given in Table 3, along with the available experimental and theoretical results [11,12,36–40] for comparison. In view of Table 3, it is clear that our calculated values for the band gaps for CaO, SrO and BaO, using the PP-PW method within the GGA, are underestimated compared to the measured ones. It is well known that in the self consistent DFT, the GGA usually underestimates the energy band gap [35]. This is mainly due to the fact that it has oversimplified forms that are not sufficiently flexible to accurately reproduce both the exchange-correlation energy and its charge derivative. However, it is widely accepted that GGA electronic band structures are qualitatively in good agreement with experiments as far as ordering of energy levels and shape of bands. Fig. 3 shows the composition dependence of the calculated band gaps using the GGA scheme. We remark that the band gap varies non-linearly with the composition x providing a positive gap bowing. We calculated the gap bowing by fitting the non-linear variation of the calculated band gap versus composition x with the quadratic semi-empirical formula: EgAx B1−x O = xEgAO + (1 − x)EgBO − x(1 − x)bg

(2)

where EgAx B1−x O , EgAO and EgBO are the energy band gaps of the ternary alloy Ax B1−x O and its binary parents AO and BO, respectively. The curvature bg is commonly known as gap bowing parameter. The

M.A. Ghebouli et al. / Journal of Alloys and Compounds 509 (2011) 1440–1447 Table 3 Band gap energies (in eV) for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. x

PP-PW

Expt.

Others

Srx Ca1−x O

1 0.75 0.5 0.25 0

3.391 3.394 3.421 3.457 3.660

5.727

3.01b , 3.2c

Bax Sr1−x O

1 0.75 0.5 0.25 0

2.137 2.210 2.321 2.602 3.391

4.1d

2.097e

Bax Ca1−x O

1 0.75 0.5 0.25 0

2.137 2.131 2.17 2.37 3.660

a b c d e f g

a

1443

values of the Ax B1−x O ternary alloys, one may use the experimental band gap values of the AO and BO binary parents in Eq. (2) along with the GGA calculated band gap bowing parameters bg . The corrected energy band gaps for all the studied alloys, obtained by taking into account the scissor correction scheme, are depicted in Fig. 3 (solid lines without symbols). To analyze the physical origins of the band gap bowing bg , we follow the procedure of Bernard and Zunger [37–39] and decompose the bowing parameter bg into three physically distinct contributions. By considering the fact that the bowing dependence on the composition is marginal, they limited their calculations at x = 0.5. The overall bowing value at x = 0.5 measures the change in band gap in the formula reaction: AC(aAC ) + BC(aBC ) → A0.5 B0.5 C(aeq , ueq )

7.1f

3.46b , 2.91g

Ref. [30]. Ref. [11]. Ref. [31]. Ref. [12]. Ref. [32]. Ref. [33]. Ref. [34].

(3)

where aAC and aBC are the equilibrium lattice constants of the binary constituents AC and BC, respectively; aeq is the alloy equilibrium lattice constant and ueq denotes the equilibrium values of the cell internal structural parameters of the alloy. We now decompose reaction (3) into three steps: VD

AC(aAC ) + BC(aBC )−→AC(aeq ) + BC(aeq )

(4)

CE

AC(aeq ) + BC(aeq )−→A0.5 B0.5 C(aeq , u0 ) results shown in Fig. 3 are well fitted by the expression (2) with a gap bowing parameter b equal to 0.469, 1.830 and 3.182 eV for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O, respectively. In order to overcome the systematic underestimation of the energy band gap using DFT-GGA calculations, we have introduced the scissor correction scheme [36], which in its simplest form requires a rigid shift of the unoccupied part of the DFT-GGA band structure. To do this correction for our calculated energy band gap

7.5 Gap GGA Corrected gap

6.0

Sr Ca O x

1-x

SR

A0.5 B0.5 C(aeq , u0 )−→A0.5 B0.5 C(aeq , ueq )

(6)

The first step measures the volume deformation (VD) effect on the bowing. The corresponding contribution to the total gap bowing parameter bVD 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 equilibrium lattice constant value. The second contribution, the charge-exchange (CE) contribution bCE , reflects a charge transfer effect which is due to the different (averaged) bonding behaviors at the lattice constant aeq . The final step measures change due to structural relaxation (SR) in passing from unrelaxed to the relaxed alloy, i.e., u0 → ueq . Consequently, the total gap bowing parameter is defined as: bg = bVD + bCE + bSR



4.5



6

(8)

bCE = 2 EgAB (aeq ) + EgAC (aeq ) − 2EgABC (aeq , u0 )

(9)



Corrected gap

4

Ba Sr O x

1-x

3 2 Gap GGA

6

Corrected gap

Ba Ca x

4

O

1-x

2 0.0

0.2

0.4

0.6

0.8





bSR = 4 EgABC (aeq , u0 ) − EgABC (aeq , ueq )

Gap GGA

5

1.0

Composition (x) Fig. 3. Composition dependence of the calculated band gap using GGA (solid line) and corrected gap (dashed line) for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys.

(7)

bVD = 2 EgAB (aAB ) − EgAB (aeq ) + EgAC (aAC ) − EgAC (aeq )



Energy band gap (eV)

(5)

(10)

where Eg is the energy band gap calculated for the indicated compound with the indicated atomic positions and lattice constant. We calculated the bowing contributions at composition x = 0.5 based on Eqs. (8)–(10) and the results are given in Table 4, together with the bowing obtained using a quadratic variation of the band gap energy versus composition x. The calculated quadratic parameters (gap bowing) are in good agreement with the values found from the approach of Bernard and Zunger [37]. The charge transfer contribution bCE dominates the total gap bowing parameter in the three studied alloys; this is related to electronegativity mismatch between the constituting atoms: Ca (1.00), Sr (0.95), Ba (0.89) and O (3.44). The bCE term is correlated to the ionicity factor difference among the constituting binary compounds CaO (0.916), SrO (0.928) and BaO (0.931) [40]. 3.3. Optical properties The refractive index of material is an important optical parameter. It exhibits the optical properties of the materials. Its value is often required to interpret various types of spectroscopic data. It

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Table 4 Decomposition of the gap bowing parameter bg into volume deformation (bVD ), charge exchange (bCE ) and structural relaxation (bSR ) contributions compared with the gap bowing parameters obtained by fitting the non-linear variation of the calculated band gap versus concentration x with quadratic function for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys (all values are in eV). Zunger approach

Quadratic equation

Srx Ca1−x O bVD bCE bSR b

−4.72 × 10−3 0.5252 −0.1036 0.417

0.44

Bax Sr1−x O bVD bCE bSR b

−0.102 2.115 −0.243 1.77

1.82

Bax Ca1−x O bVD bCE bSR b

−0.087 3.220 −0.18 2.95

3.183

Table 5 Calculated static refractive indices, using various models, for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. PP-PW

Moss

Vandamme

Expt.

Srx Ca1−x O 1 0.75 0.5 0.25 0

1.943 1.620 1.600 1.687 1.956

2.127 2.091 2.053 2.014 1.974

1.860 1.805 1.749 1.692 1.636

1.92a

d

1.947

Bax Sr1−x O 1 0.75 0.5 0.25 0

2.149 1.744 1.670 1.711 1.943

2.127 2.196 2.247 2.272 2.265

1.860 1.965 2.043 2.081 2.070

1.97b

d

2.149

Bax Ca1−x O 1 0.75 0.5 0.25 0

2.149 1.734 1.680 1.729 1.956

2.269 2.244 2.177 2.081 1.974

2.070 2.039 1.935 1.791 1.636

1.80c

d

1.958

a b

is related to microscopic atomic interactions. The refractive index expresses also the ratio between the light celerity in vacuum and its celerity in the considered material. The knowledge of the refractive index is essential for devices such as photonic crystals, wave guides, solar cells and detectors. The refractive index n(ω) could be computed from the dielectric function using the following relation:

 n(w) =

ε1 (w) + 2



ε21 (w) + ε22 (w)

c d

Other

Ref. [44]. Ref. [45]. Ref. [46]. Ref. [47].

1/2

2

2.2

(11)

Sr Ca x

At low frequency (w = 0) we get the following relation: n(0) = ε1/2 (0). In addition to this quantum mechanics relation, some empirical relations [41,42] relate the refractive index to the energy band gap is used in this work. The following models are used:

O

1-x

Moss

2.0

1.8 (i) The Moss formula [41] based on atomic model:

Vandamme

where Eg is the energy band gap and k is a constant with a value of 108 eV [41].

2.2

Eg n = k

(ii) The Herve and Vandamme’s empirical relation [42] given by:

 n=

 1+

A Eg + B

2 (13)

with A = 13.6 eV and B = 3.4 eV. The obtained values of the static refractive index for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys (with x = 0, 0.25, 0.50, 0.75 and 1) using PP-WP calculations and the different models are summarized in Table 5. The comparison with available data in the literature has been done. From Table 5 data, it appears that the calculated values of the refractive index for the parent compounds (i.e., CaO, SrO and BaO) are in good agreement with the available experimental results. Therefore, the present results for the refractive indices of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys stand as reliable predictions. Fig. 4 shows the evolution of the refractive index as a function of concentration x for the Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O alloys. The calculated refractive index versus concentration using various models obeys to the same law which governs the band gap evolution. This led to: nAx B1−x C = xnAC + (1 − x)nBC − x(1 − x)bn

(14)

Refractive index

(12)

1.6

4

PP-PW Ba Sr x

O

Moss

1-x

2.0

Vandamme 1.8

PP-PW 2.4

Ba Ca x

O

1-x

Moss

2.1

Vandamme

1.8

PP-PW 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Composition (x) Fig. 4. Calculated refractive index for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys at different compositions x.

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Table 6 Calculated static dielectric constants, using various models, for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. PP-PW

Moss

Vandamme

Expt.

Other

Srx Ca1−x O 1 0.75 0.5 0.25 0

3.775 2.624 2.56 2.845 3.825

4.524 4.372 4.214 4.056 3.889

3.459 3.258 3.059 2.862 2.676

3.35a , 3.46b

3.80c , 3.88c

Bax Sr1−x O 1 0.75 0.5 0.25 0

4.618 3.041 2.788 2.927 3.775

4.524 4.822 5.049 5.161 5.130

3.459 3.861 4.173 4.330 4.284

3.61d , 3.68b

4.25c

Bax Ca1−x O 1 0.75 0.5 0.25 0

4.618 3.006 2.822 2.989 3.825

5.148 5.035 4.739 4.330 3.896

4.284 4.157 3.744 3.207 2.676

3.27a , 3.33e

3.78c , 3.86c

a b c d e

Ref. [44]. Ref. [45]. Ref. [10]. Ref. [46]. Ref. [48].

The calculated bn values of n1 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are equal to 1.475, 1.580 and 1.580, respectively. The bn values of n2 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are estimated to be −0.01, −0.22 and −0.32, respectively. The bn values of n3 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are found to be −0.008, −0.32 and −0.32, respectively. Here n1 (x), n2 (x) and n3 (x) refer to the refractive index obtained from PP-PW method, the Moss formula (Eq. (12)) and the Herve and Vandamme’s empirical relation (Eq. (13)), respectively. From Fig. 4, we can note the non-linear dependence of the refractive indices of all the studied alloys with concentration x. Based on the calculated values of the static refractive indices, the optical static dielectric constant for the different composition x have been estimated using the relation n(0) = ε1/2 (0); the obtained results are listed in Table 6. It seems that the PP-PW values for ε(0) are closer to the experimental ones than those of the other used models. Qualitatively, the compositional dependence of the optical static dielectric constant for the alloys of interest has the same trend as that of the refractive index. This is an expected behavior since according to the relation n(0) = ε1/2 (0) one may expect that the behavior of ε(0) with respect to x will be qualitatively similar to that of n(0). The ε − x data are well fitted to a quadratic polynomial: εAx B1−x C = xεAC + (1 − x)εBC − x(1 − x)bε . The calculated bε values of ε1 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are equal to 5.27, 5.99 and 5.99, respectively. The bε values of ε2 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are estimated to be −0.02, −0.73 and −0.90, respectively. The bε values of ε3 for Srx Ca1−x O, Bax Ca1−x O and Bax Sr1−x O are found to be −0.02, −0.96 and −1.06, respectively. Here ε1 , ε2 and ε3 are calculated from n1 (x), n2 (x) and n3 (x), respectively. 3.4. Thermodynamic properties At finite temperature, the thermal stability of a solid state system is determined by the Gibbs free energy given by the following expression [43]: G(x, T ) = H(x, T ) − T S(x, T )

(15)

where Ax B1−x C AC BC − xETot − (1 − x)ETot H = ˝x(1 − x) = ETot

(16)

Fig. 5. Enthalpy of mixing H as a function of concentration for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. Solid curve: x-dependent interaction parameters ˝, dotted curve: x-independent interaction parameters ˝.

S = −R[x ln x + (1 − x)ln(1 − x)]

(17)

H is the mixing energy (formation enthalpy), ˝ is the interaction Ax B1−x C , E AC and E BC are parameter which depends on the material, ETot Tot Tot the total energies of the Ax B1−x C alloy and the binary compounds AC and BC, S is the mixing entropy, R is the perfect gas constant and T is the absolute temperature. Using Eq. (16), the obtained interaction parameter ˝ values at different compositions are well linearly fitted. Using linear x-dependent and averaged values of the x-dependent of the interaction parameter ˝, the formation enthalpies for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys are shown in Fig. 5. The formation energies are all positive, indicating that these systems have tendency to segregate in their constituents at low temperature. Using Eqs. (15)–(17), we have predicted the x–T phase diagram of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. The binodal curve, which indicates the equilibrium solubility limits as a function of temperature, can be calculated by the common tangent approach from the Gibbs free energy. As shown in Fig. 6, the critical temperatures, above which complete miscibility is possible for some concentrations, are 1039, 1176 and 4187 K for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O, respectively. The high critical temperature of Bax Ca1−x O reveals that it is difficult to mix randomly BaO and CaO. With the spinodal curve, we can draw out the region of metastability. The wide range between spinodal and binodal curves indicates that the alloy may exist as a metastable phase. Through the quasi-harmonic Debye model, one could calculate thermodynamic quantities, at any temperature and pressure, for the compounds from the calculated E − V data at T = 0 and P = 0. Fig. 7 displays the heat capacities (CP and CV ) and entropy (S) versus

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M.A. Ghebouli et al. / Journal of Alloys and Compounds 509 (2011) 1440–1447

60

)

SrxCa1-xO TC=1039.58 K

-1 -1

1000

CV (Jmol K

800 600 Spinodal

400

Binodal

200

0 1400 T =1176.12 K 1200 BaxSr1-xO C 1000 800 600 Spinodal 400 Binodal 200 0 5000 BaxCa1-xO TC= 4185.39 K 4000

0 0.0

) -1 -1

Cp (Jmol K

-1 -1

Spinodal Binodal

1000 0.2

0.4

0.6

40 CaO 30

SrO

20

BaO

0

3000 2000

Dulong - Petit Limit

50

10

Entropy (Jmol K )

Temperature (K)

Temperature (K)

Temperature (K)

1200

0.8

1.0

Composition ( x) Fig. 6. T–x phase diagram of the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. Dashed line: binodal curve; solid line: spinodal curve.

temperature up to T = 1000 K at zero pressure for the CaO, SrO and BaO compounds. At high temperature, CV tends to the Dulong-Petit limit. As it can be seen from Fig. 7, there is a good agreement between our results and the available experimental data for the heat capacity CP and entropy S of the binary compounds which is a support for those of the ternary alloys that we report for the first time. The Debye temperature ( D ), heat capacities (CV and CP ) and mixing entropies (S) calculated at ambient temperature and zero pressure for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys, with

50 40 30

CaO

CaO Ref. 62

20

SrO

SrO Ref. 62

10

BaO

BaO Ref. 62

0 120 100 80 60 40 20 0

0

200

400

600

800

1000

Temperature (K) Fig. 7. Variation with temperature of the heat capacities and entropy of CaO, SrO and BaO.

x = 0, 0.25, 0.50, 0.75 and 1, are listed in Table 7, along with available experimental results. The calculated values of  D , CV , CP and S using the quasi-harmonic Debye model calculations for the parent compounds (i.e., CaO, SrO and BaO) are in good agreement with available experimental data and theoretical results [47,49–51].

Table 7 Calculated heat capacities (CV and CP, in J mol−1 K−1 ), Debye temperature ( D , in K) and entropy (S, in J mol−1 K−1 ) for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. CV This work

CP This work

S This work

D This work

D Expt.

D Others

CP Expt.

S Expt.

Srx Ca1−x O 1 0.75 0.5 0.25 0

44.95 45.01 44.62 44.23 42.59

45.08 45.39 45.00 44.60 42.94

50.1969 50.5252 48.6517 46.9073 40.7994

438.02 434.83 453.39 471.55 542.74

446a – – – –

431.31c – – – –

45.478d – – – –

55.803d – – – –

Bax Sr1−x O 1 0.75 0.5 0.25 0

47.26 47.06 46.77 46.26 44.95

47.69 47.49 47.19 46.66 45.08

65.56 63.84 61.41 57.72 50.20

314.13 325.81 343.09 371.46 438.02

370a – – – –

360.21c – – – –

47.332d – – – –

70.709d – – – –

Bax Ca1−x O 1 0.75 0.5 0.25 0

47.26 47.07 46.44 45.47 42.59

47.69 47.56 46.86 45.83 42.94

65.56 63.90 58.92 52.90 40.80

314.13 325.38 361.93 412.55 542.74

– – – – 605b

– – – – 487.36c

– – – – 42.239d

– – – – 38.335d

a b c d

Ref. [49]. Ref. [50]. Ref. [47]. Ref. [51].

M.A. Ghebouli et al. / Journal of Alloys and Compounds 509 (2011) 1440–1447

Therefore, the present results for these quantities for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys stand as reliable predictions. The variation of the investigated thermal parameters as a function of composition is fitted to a quadratic expression: (CP , CV , S, D )

Ax B1−x C

= x(CP , CV , S, D ) − x(1 − x)b

AC

+ (1 − x)(CP , CV , S, D )

BC

[4] [5] [6] [7] [8] [9] [10]

(18)

The calculated b values of (CP , CV , S,  D ) for Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O are equal to (−4.54, −3.92, −14.7, 169.55), (−3.43, 2.81, −14.77, 138.11) and (−6.11, −7.07, −25.07, 287.94), respectively. 4. Conclusions We have performed first-principles density functional calculations to study the structural, electronic, optical and thermodynamic properties for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. The alloy disorder was modeled using the SQS approach for x = 0, 0.25, 0.50, 0.75 and 1. The calculated lattice constants and bulk moduli for the binary compounds (CaO, SrO and BaO) are in good agreement with the experimental data and previous theoretical results; this might be an estimate of the reliability and accuracy of the predicted lattice parameters for the studied ternary alloys. The band gaps are shown to vary strongly with the composition x in a non-linear way. The results of the decomposition of the bandgap bowing suggest that the bowing parameter is mainly caused by the charge-exchange effect. Investigation of the x–T phase stability diagram allowed us to calculate the critical temperature for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys, which are 1039, 1176 and 4185 K, respectively. The composition dependence of the static refractive indices and optical static dielectric constants has been computed using quantum mechanics method and empirical relations. A non-linear dependence on the composition x was observed. Quasi-harmonic model is successfully applied to determine the heat capacities and mixing entropy at room temperature for the Srx Ca1−x O, Bax Sr1−x O and Bax Ca1−x O alloys. References [1] U.K. Mishra, J. Singh, Semiconductor Device Physics and Design, Springer, Dordrecht, 2008. [2] M. Othman, E. Kasap, N. Korozlu, J. Alloys Compd. 496 (2010) 226. [3] F. El Haj Hassan, A. Breidi, S. Ghemid, B. Amrani, H. Meradji, O. Pagès, J. Alloys Compd. 499 (2010) 80.

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