IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. 79 (2009) 045002 (7pp)
doi:10.1088/0031-8949/79/04/045002
First principles calculations of structural, electronic, thermodynamic and optical properties of BAs1−xPx alloy S Drablia1,2 , H Meradji1 , S Ghemid1 , S Labidi1 and B Bouhafs3 1
Laboratoire LPR, Département de Physique, Faculté des Sciences, Université d’Annaba, Algeria Laboratoire CIMAP, ENSICAEN, Université de Caen, Caen Cedex 04, France 3 Laboratoire de Modélisation et Simulation en Sciences des Matériaux, Département de Physique, Faculté des Sciences, Université de Sidi Bel-Abbès, Algeria 2
E-mail:
[email protected]
Received 8 July 2008 Accepted for publication 22 January 2009 Published 31 March 2009 Online at stacks.iop.org/PhysScr/79/045002 Abstract First principles calculations have been used to investigate the structural, electronic, thermodynamic and optical properties of boron ternary alloy BAs1−x Px , using a hybrid full-potential (linear) augmented plane wave plus the local orbitals (APW + lo) method within the density-functional theory (DFT). The Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) as well as the Engel–Vosko (EV)-GGA are used to calculate the band gap. We investigated the effect of composition on lattice constant, bulk modulus and band gap. Deviations of the lattice constant from Vegard’s law and the bulk modulus from linear concentration dependence (LCD) were observed for the alloy. Using the approach of Zunger and co-workers, the microscopic origins of the gap bowing are explained. The thermodynamic stability of the alloy is investigated by calculating the excess enthalpy of mixing 1Hm as well as the phase diagram. The calculated phase diagram showed a broad miscibility gap for the alloy of interest with a high critical temperature. For optical properties, the compositional dependence of the refractive index and the dielectric constant is studied. PACS numbers: 61.66.Dk, 71.15.Mb, 71.15.Nc, 71.20.Nr, 71.22.+i, 78.20.Ci
with different optical band gaps and different rigidities in order to obtain a new material formulation with intermediate properties. Several studies have been devoted to BSb, BP and BAs such as including their structural, electronic and optical properties. Using the plane-wave basis sets and pseudopotential approximation (PWPP) within LDA, the ground state and the high-pressure properties have been calculated by Wentzcovitch and Cohen [2] and Wentzcovitch et al [3] for BP and BAs, Ferhat et al [5] and Zaoui and Elhaj Hassan [8]. A comparative study of the structural and electronic properties of the series BP, BAs and BSb has been presented by Bouhafs et al [4, 7] using the same approach. Using the full potential at linear augmented plane-wave (FP-LAPW) method and LDA approximation for the exchange correlation potential, the optical properties under hydrostatic pressure of these compounds have been reported by Zaoui et al [9]. Recently, Elhaj Hassan and Akbarzadeh [11] and Elhaj Hassan [12] calculated the
1. Introduction Boron compounds have attracted increasing research interest over the past few years as wide-gap semiconductors. These materials are of great technological interest for high temperatures, electronic and optical applications [1–9]. The electronic valence configuration is 1s2 2s2 2p1 for boron atoms and ns2 np3 for group V elements. They crystallize at low pressure in the fourfold coordinated zinc-blende structure. Semiconductor alloys, which are solid solutions of two or more semiconducting elements, have important technological applications, especially in the manufacture of electronic and electro-optical devices [10]. One of the easiest ways to change artificially the electronic and optical properties of semiconductors is by forming their alloys; it is then interesting to combine two different compounds 0031-8949/09/045002+07$30.00
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© 2009 The Royal Swedish Academy of Sciences
Printed in the UK
Phys. Scr. 79 (2009) 045002
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Table 1. Calculated lattice parameter (a), bulk modulus (B), and indirect band gap energy (E g ) for BP and BAs compounds. Available experimental and theoretical data from the literature are also included for comparison. This work PBE-GGA BP
BAs
a
a (Å) B (GPa) E g (eV) a (Å) B (GPa) E g (eV)
EV-GGA
a
4.554 160 1.254 4.812 133 1.201
4.538 173a 2.4e 4.777a
1.877
0.67h
1.820
4.546b , 4.51c , 4.475 d 170b , 172c , 172d 1.12f , 1.14g 4.784b , 4.728d 137b , 144d 1.06f , 1.25i
The APW + lo method expands the Kohn–Sham orbitals in atomic-like orbitals inside the atomic muffin-tin (MT) spheres and PWs in the interstitial region. Details of the method have been described in the literature [17–19]. For the exchange and correlation potential, we used the Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) functional [20]. For electronic properties, in addition to the PBE-GGA correction, the Engel–Vosko (EV)-GGA functional [21] is also applied. Basis functions were expanded in combinations of spherical harmonics inside non-overlapping spheres surrounding the atomic sites (MT spheres) and in a Fourier series in the interstitial region. In MT spheres, the l-expansion of the non-spherical potential and charge density was carried out up to lmax = 10. In order to achieve energy eigenvalue convergence, we expanded the basis function up to RMT K max = 8 (where K max is the maximum modulus for the reciprocal lattice vector and RMT is the average radius of the MT spheres). Furthermore, we adopted the values of 1.5, 2.0 and 2.0 bohr for B, As and P, respectively, as the MT radii. The k integration over the Brillouin zone is performed up to a 4 × 4 × 4 Monkhorst–Pack [22] mesh (55 and 27 points in the irreducible wedge of the Brillouin zone (IBZ) are used for the binary compounds and their ternary alloy, respectively). The iteration process was repeated until the calculated total energy of the crystal converged to less than 1 mRyd. A total of seven iterations were necessary to achieve self-consistency. We compute lattice constants and bulk moduli by minimizing the total energy versus volume according to Murnaghan’s equation of state [23]. On the basis of convergence tests on our response functions with a varying number of k points, we are confident that 55 k points ensure accurate and well-converged results. In these calculations, 234–345 PWs for binary compounds have been used for the expansion of charge density and potential in the interstitial region.
2. Calculations In the present work, first principles calculations of structural, electronic, thermodynamic and optical properties of boron ternary alloy BAs1−x Px have been performed using the Vienna package WIEN2 K [14]. This is an implementation of a hybrid full-potential (linear) APW + lo method within the density-functional theory (DFT) [15, 16]. This new approach is shown to reproduce the accurate results of the LAPW method, but using a smaller basis set size. Due to the smaller basis set and faster matrix setup, APW + lo offers a shorter run time and uses less memory than LAPW. Sjöstedt et al [17] have shown that the standard LAPW method with the additional constraint on the plane waves (PWs) inside the sphere is not the most efficient way of linearizing Slater’s APW method. It can be made much more efficient when one uses the standard APW basis, but of course with u l (r, E l ) at a fixed energy E l in order to keep the linear eigenvalue problem. One then adds a new local orbital (lo) to have enough variational flexibility in the radial basis functions: X _ φkn = [Alm,kn u l (r, u l )]Ylm ( r ), (1)
3. Results 3.1. Structural properties Firstly, the structural properties of binary compounds BP and BAs and their ternary alloy using the PBE-GGA scheme are analysed. Then, the alloy at some selected compositions with ordered structures described in terms of periodically repeated supercells is modelled. This can be done with eight atoms per unit cell for the compositions x = 0.25, 0.5 and 0.75. The total
lm
X
Other works
[24], b [8], c [6], d [7], e [28], f [7], g [29], h [30] and i [1].
structural, electronic and thermodynamic properties of boron ternary alloys using the FP-LAPW method. Hence, in order to fully exploit these materials for new optical devices, the structural, electronic and optical properties of these alloys need to be investigated in more detail. Therefore, the purpose of this paper is to study the structural, electronic, thermodynamic and optical properties by using a full-potential linear augmented plane wave plus the local orbitals (APW + lo) method. The physical origins of gap bowing are investigated by following the approach of Zunger et al [13]. In this approach, the alloy is studied in an ordered structure (we used a cubic supercell of eight atoms) designed to reproduce the most important pair correlation functions of a random (disorder) alloy and where the chemical and structural effects are captured very well. The paper is organized as follows. In section 2, we describe the computational details. Results and discussion concerning the structural, electronic, thermodynamic and optical properties are presented in section 3. The conclusion is given in section 4.
lo φlm =
Experiment
_
[Alm u l (r, u 1,l ) + blm u˙ l (r, u 1,l )]Ylm ( r ).
(2)
lm
2
Phys. Scr. 79 (2009) 045002
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4.85
160
B u l k m o d u lu s ( G P a )
BAs1–xPx
L at t ic e p a r a me t e r a (Å )
4.80
4.75
4.70
BAs1–xPx
155 150 145 140 135 130
0.0
0.2
4.65
0.4
0.6
0.8
1.0
Composition x Figure 2. Variation of the calculated bulk modulus versus x concentration of BAs1−x Px alloy (filled squares) compared with the linear composition dependence prediction (dotted line).
4.60
4.55
2.8 2.6
0.0
0.2
0.4
0.6
0.8
1.0
PBE-GGA EV-GGA
BAs1–xPx
2.4
Band gap (eV)
Composition x Figure 1. Variation of the calculated lattice constant versus x concentration of BAs1−x Px alloy (filled squares) compared with Vegard’s law (dotted line).
energies are calculated as a function of volume and are fitted to Murnaghan’s equation of state [23]. The corresponding equilibrium lattice constants, bulk moduli and their pressure derivatives are given in table 1 for both BP and BAs and in figures 1 and 2 for the alloy. Considering the general trend that the GGA usually overestimates the lattice parameters [8], our PBE-GGA results of BAs and BP compounds are in good agreement with the experimental and other theoretical calculated values. Regarding the values of the alloy, a small deviation from Vegard’s law (i.e. a linear variation of the lattice constant of alloys versus composition x [25]) is clearly visible with an upward bowing parameter equal to −0.027 (figure 1). The physical origin of this small deviation can be attributed to the weak mismatches of the lattice constants of the binary compounds. It should be noted that violations of Vegard’s rule have been reported in many semiconductor alloys both experimentally [26] and theoretically [27]. Figure 2 shows the bulk modulus as a function of x for the alloy. A significant deviation from linear concentration dependence (LCD) with downward bowing equal to 3.428 GPa is observed. Our results show that the bulk modulus increases with increasing P concentration (0 6 x 6 1). This suggests that as x increases from x = 0 (BAs) to x = 1 (BP), BAs1−x Px becomes generally less compressible.
2.2 2.0 1.8 1.6 1.4 1.2 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Composition x Figure 3. Variation of the calculated band gap versus x concentration of BAs1−x Px alloy using both PBE-GGA and EV-GGA exchange and correlation potentials.
an indirect band gap near X (0 → 1min ) in the whole range of concentrations. The results are given in table 1. It is clear that the band gap values given by the EV-GGA approach are better and close to experiment. EV-GGA reproduces better exchange potential at the cost of less accuracy in exchange energy. Therefore EV-GGA presents better results for band gap energy and other properties, which depend primarily on exchange potential. PBE-GGA usually underestimates the energy gap [31, 32]. This is mainly due to the fact that the functional within this approximation has simple forms that are not sufficiently flexible to accurately reproduce both exchange-correlation energy and its charge derivative. The approach, which is called EV-GGA, yields a better band splitting and some other properties, which mainly depend on the accuracy of exchange-correlation potential. On the other hand, in this method, the quantities that depend on an accurate description of E(x) such as equilibrium volumes and bulk modulus are in poor agreement with experiment. Figure 3 shows the composition dependence of the calculated band gaps using PBE-GGA and EV-GGA schemes.
3.2. Electronic properties The calculated band structure energies of binary compounds and their alloy using both PBE-GGA and EV-GGA indicate 3
Phys. Scr. 79 (2009) 045002
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Table 2. Decomposition of optical bowing into VD, CE and SR contributions compared with the optical bowing obtained by a quadratic interpolation (all values are in eV). Calculation Parameter bVD bCE bSR b a
Quadratic fits
Other works
PBE-GGA
EV-GGA
PBE-GGA
EV-GGA
PBE-GGA
EV-GGA
0.061 0.042 −0.024 −0.043
−0.034 0.013 −0.031 −0.052
−0.041
−0.041
−0.057a
−0.062a
[11].
We calculated the gap bowing by fitting the nonlinear variation of the calculated band gaps versus concentration with quadratic function. The results are shown in figure 3 and obey the following variations: ( PBE−GGA Eg (x) = 1.201 + 0.093x − 0.041x 2 , BAs1−x Px ⇒ E gEV−GGA (x) = 1.819 + 0.098x − 0.041x 2 .
bVD = 2 [εAB (aAB ) − εAB (a) + εAC (aAC ) − εAC (a)] ,
From figure 3, it is clear that the alloy shows a weakly compositional-dependent energy gap. In order to better understand the physical origins of the gap bowing in alloy, we follow the procedure of Bernard and Zunger [33] and decompose the bowing parameter (b) into physically distinct contributions. Considering that the bowing dependence on the composition is marginal, the authors limited their calculations to x = 0.5 (50–50% alloy). The overall gap bowing at x = 0.5 measures the change in band according to the reaction (4)
where aAB and aAC are the equilibrium lattice constants of the binary compounds AB and AC, respectively, and aeq is the alloy equilibrium lattice constant. We now decompose equation (4) into three steps: VD
AB(aAB ) + AC(aAC ) → AB(a) + AC(a), CE
AB(a) + AC(a) → AB0.5 C0.5 (a), SR
AB0.5 C0.5 (a) → AB0.5 C0.5 (aeq ).
bCE = 2 [εAB (a) + εAC (a) − 2εABC (a)] ,
(10)
bSR = 4 εABC (a) − εABC aeq ,
(11)
where ε is the energy gap that has been calculated for the indicated atomic structures and lattice constants. Energy gap terms in equations (9)–(11) are calculated separately with self-consistent band structure approach FP-LAPW and the results are given in table 2. The total gap bowing is found to be marginal for BA1−x Px . The small bCE value is due to the weak ionicity mismatch of BP and BAs ( f i = 0.006 and 0.002 for BP and Bas, respectively). In the same way, the weak contribution of bVD can be correlated to the small mismatch of the lattice constants of the corresponding binary compounds. Finally, it is clearly seen that the PBE-GGA and EV-GGA values for the bowing parameter are almost identical.
(3)
AB(aAB ) + AC(aAC ) → AB0.5 C0.5 (aeq ),
(9)
3.3. Thermodynamic properties
(5)
In order to study the phase stability of BAs1−x Px alloy, we calculated the phase diagram based on the regular-solution model [34–36]. For alloys, the Gibbs free energy of mixing 1G m is expressed as
(6)
1G m = 1Hm − T 1Sm ,
(12)
1Hm = x(1 − x),
(13)
1Sm = −R [x ln x + (1 − x) ln(1 − x)]
(14)
where
(7)
The first contribution 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 aAB (aAC ) to the alloy value a = a(x) (calculated from Vegard’s rule). The second contribution is related to the charge-exchange (CE) contribution bCE and reflects a charge transfer effect which is due to the different (averaged) bonding behaviour at the lattice constant a = a(x). The final contribution describes the change of the band gap due to the structural relaxation (SR) upon passing from the unrelaxed to the relaxed alloy, i.e. a(x) → aeq , by bSR . Consequently, the total gap bowing parameter is defined as
1Hm and 1Sm are the enthalpy and entropy of mixing, respectively, the interaction parameter and depending on the material, R the gas constant and T the absolute temperature. The mixing enthalpy of alloys can be obtained from the calculated total energies as 1Hm = E ABx C1−x − x E AB − (1 − x)E AC , where E ABx C1−x , E AB and E AC are the respective energies of ABx C1−x alloy and the binary compounds AB and AC. We then calculated 1Hm to obtain as a function of concentration. The interaction parameter increases almost linearly with increasing x. From a linear fit we obtained
b = bVD + bCE + bSr ,
(kcal mol−1 ) = 0.42x + 4.42.
(8) 4
(15)
Phys. Scr. 79 (2009) 045002
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1200
BAs1–xPx
Refractive index n
Temperature (K)
1000
T c =1168 K 800 600 400 200
3.25 3.20 3.15 3.10 3.05 3.00 2.95
0
0.0
0.2
0.4
0.6
0.8
1.0
2.90
Composition x
L/APW +lo calculation Calculated using the Moss formula Calculated using the Herve and Vandamme's empirical relation Experiment Ref. [39] Experiment Ref. [40]
0.0
0.2
0.4
0.6
0.8
1.0
Composition x
Figure 4. T–x phase diagram for BAs1−x Px alloy. Dotted line: binodal curve; solid line: spinodal curve.
Figure 5. Variation of the calculated refractive index versus x concentration of BAs1−x Px alloy.
The average value of the x-dependent in the range 0 6 x 6 1 obtained from this equation for BAs1−x Px alloy is 4.63 kcal mol−1 . Now, we first calculate 1G m by using equations (12)–(14). Then, we use the Gibbs free energy at different concentrations to calculate the T –x phase diagram, which shows the stable, metastable and unstable mixing regions of the alloy. At a temperature lower than the critical temperature Tc , the two binodal points are determined as those points at which the common tangent line touches the 1G m curves. The two spinodal points are those points at which the second derivative of 1G m is zero; ∂ 2 (1G m )/∂ x 2 = 0. Figure 4 shows the calculated phase diagram including the spinodal and binodal curves for the alloy. We observed a critical temperature Tc of 1168 K for BAs1−x Px alloy. The spinodal curve in the phase diagram marks the equilibrium solubility limit, i.e. the miscibility gap. For temperatures and compositions above this curve, a homogeneous alloy is predicted. The wide range between spinodal and binodal curves indicates that the alloy may exist as a metastable phase. Our results indicate that the BAs1−x Px alloy is stable at high temperature.
At low frequency (ω = 0), we obtain the following relation: n(0) = ε1/2 (0).
(17)
A few empirical relations [37, 38] relate the refractive index to the energy band gap for a large set of semiconductors. The following models are used: 1. The Moss formula [37] based on atomic model E g n 4 = k,
(18)
where E g is the energy band gap and k a constant. The value of k is given as 108 eV by Ravindra and Srivastava [37]. 2. Herve and Vandamme’s empirical relation [38] s 2 A n = 1+ (19) , Eg + B with A = 13.6 eV and B = 3.4 eV. In figure 5, we summarize the calculated values of the refractive index for the alloy under investigation for some compositions x, obtained by using the L/APW + lo method and the different models [39, 40]. Comparison with the available data has been made where possible. It is clear that the values of the refractive index obtained from full potential calculations agree well with those obtained from the other models. Figure 5 shows the variation of the calculated refractive index versus concentration for the alloy. In figure 5, one can notice that the refractive index decreases monotonically with increasing P content over the entire range of 0–1 for both L/APW + lo and models used. The calculated refractive indices versus concentration are fitted by a polynomial equation. The results are summarized as follows:
3.4. Optical properties The optical properties of a solid are usually described in terms of the complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω). The imaginary part of the dielectric function in the long-wavelength limit has been obtained directly from the electronic structure calculation, using the joint density of states and the optical matrix elements. The real part of the dielectric function can be derived from the imaginary part by the Kramers–Kronig relationship. The knowledge of both the real and the imaginary parts of the dielectric function allows the calculation of important optical functions. In this paper, we also present and analyse the refractive index n(ω), given by q 1/2 2 2 ε (ω) + ε (ω) ε1 (ω) 1 2 . + n(ω) = 2 2
BAs1–xPx
3.30
n(x) = 3.121 − 0.087x − 0.029x 2 (L/APW + lo),
(20)
n(x) = 3.078 − 0.057x + 0.025x 2 (relation (18)).
(21)
(16)
5
Phys. Scr. 79 (2009) 045002
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Table 3. Optical dielectric constants of BAs1−x Px for different compositions x.
Composition
ε (calculated using the L/APW + lo method)
ε (calculated from relation (18))
ε (calculated from relation (19))
9.740 9.639 9.386 9.269 9.030
9.480 9.404 9.337 9.303 9.280
9.737 9.660 9.595 9.562 9.539
0 0.25 0.5 0.75 1 a
Other works 11.19 a
10.55a , 11.0b
[39] and b [23].
n(x) = 3.120 − 0.054x + 0.022x 2 (relation (19)).
References
(22)
[1] Wentzcovitch R M, Chang K J and Cohen M L 1986 Phys. Rev. B 34 1071 [2] Wentzcovitch R M and Cohen M L 1986 J. Phys. C: Solid State Phys. 19 6791 [3] Wentzcovitch R M, Cohen M L and Lam P K 1987 Phys. Rev. B 36 6058 [4] Bouhafs B, Aourag H, Ferhat M and Certier M 1999 J. Phys.: Condens. Matter 11 5781 [5] Ferhat M, Bouhafs B, Zaoui A and Aourag H 1998 J. Phys.: Condens. Matter 10 7995 [6] Lambrecht W R L and Segall B 1991 Phys. Rev. B 43 7070 [7] Bouhafs B, Aourag H, Ferhat M and Certier M 2000 J. Phys.: Condens. Matter 12 5655 [8] Zaoui A and Elhaj Hassan F 2001 J. Phys.: Condens. Matter 13 253 [9] Zaoui A, Kacimi S, Yakoubi A, Abbar B and Bouhafs B 2005 Physica B 367 195 [10] Jain C, Willis J R and Bullogh R 1990 Adv. Phys. 39 127 [11] Elhaj Hassan F and Akbarzadeh H 2005 Mater. Sci. Eng. B 121 170 [12] Elhaj Hassan F 2005 Phys. Status Solidi b 242 3129 [13] Srivastava G P, Martins G L and Zunger A 1985 Phys. Rev. B 31 2561 Bernard J E and Zunger A 1986 Phys. Rev. B 34 5992 Zunger A, Wei S H, Ferreira L G and Bernard J E 1990 Phys. Rev. Lett. 65 353 Wei S H, Ferreira L G, Bernard J E and Zunger A 1990 Phys. Rev. B 42 9622 [14] Blaha P, Schwarz K, Madsen G K H, Kvasnicka D and Luitz J 2001 WIEN2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties (Vienna, Austria: Vienna University of Technology) [15] Hohenberg P and Kohn W 1964 Phys. Rev. B 136 864 [16] Kohn W and Sham L J 1965 Phys. Rev. A 140 1133 [17] Sjostedt E, Nordstrom L and Sing D J 2000 Solid State Commun. 114 15 [18] Madsen G K H, Blaha P, Schwarz K, Sjostedt E and Nordstrom L 2001 Phys. Rev. B 64 195134 [19] Schwarz K, Blaha P and Madsen G K H 2002 Comput. Phys. Commun. 147 71 [20] Perdew J P, Burke S and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 [21] Engel E and Vosko S H 1993 Phys. Rev. B 47 13164 Engel E and Vosko S H 1994 Phys. Rev. B 50 10498 [22] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188 [23] Murnaghan F D 1944 Proc. Natl Acad. Sci. USA 30 5390 [24] Landolt-Börnstein 1982 New Series, Group III, vol 17a ed O Madelung (Berlin: Springer) [25] Vegard L 1921 Z. Phys. 5 17 [26] Jobst J, Hommel D, Lunz U, Gerhard T and Landwehr G 1996 Appl. Phys. Lett. 69 97 [27] Elhaj Hassan F 2005 Phys. Status Solidi b 242 909 [28] Knittle E, Wentzcovitch R M, Jeanloz R and Cohen M L 1989 Nature 337 349
From these equations, we can note the weak nonlinear dependence of the refractive index of the alloy with concentration x. Interestingly, we note on going from BAs to BP that the band gap of BAs1−x Px increases (see figure 3) but the refractive index decreases, therefore indicating that the ternary system BAs1−x Px shows that the smaller band gap material has a large value of refractive index as generally many other group III–V compound semiconductor alloys do [41]. The dielectric function has been estimated according to expression (17). The results are given in table 3. Qualitatively, the compositional dependence of the dielectric function of the alloy has the same trend as that of the refractive index. This is not surprising as the dielectric function is directly calculated from relation (17).
4. Conclusion We have studied the structural, electronic, thermodynamic and optical properties of BAs1−x Px ternary alloy using the L/APW + lo method. A summary of the key findings follows. (i) The lattice constant of the alloy exhibits a small deviation from Vegard’s law with bowing parameter equal to −0.027. (ii) A significant deviation of the bulk modulus from LCD is found for the alloy. As the P content increases, BAs1−x Px becomes generally less compressible. (iii) The gap bowing is found to be marginal for BAs1−x Px alloy. This is expected because there is a weak lattice mismatch and a weak ionicity difference between BP and BAs compounds. (iv) The calculated phase diagram indicates a critical temperature of 1168 K. It means that this alloy is stable at high temperature. (v) A weak nonlinear dependence of the refractive index and the dielectric constant has been observed.
Acknowledgment We would like to thank Professor S Bouaricha for a careful reading of the manuscript. 6
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