Ab initio study of the structural, electronic and ...

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Ab initio study of the structural, electronic and thermodynamic properties of PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ternary alloys

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PHYSICA SCRIPTA

Phys. Scr. 83 (2011) 065701 (9pp)

doi:10.1088/0031-8949/83/06/065701

Ab initio study of the structural, electronic and thermodynamic properties of PbSe1−xSx, PbSe1−xTex and PbS1−xTex ternary alloys N Boukhris1 , H Meradji1 , S Ghemid1 , S Drablia1 and F El Haj Hassan2,3 1

Laboratoire LPR, Département de Physique, Faculté des Sciences, Université de Annaba, Annaba, Algeria 2 Laboratoire de Physique des matériaux, Faculté des Sciences (1), Université Libanaise, Elhadath, Beirut, Lebanon 3 Condensed Matter Section, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy E-mail: [email protected]

Received 1 October 2010 Accepted for publication 11 April 2011 Published 12 May 2011 Online at stacks.iop.org/PhysScr/83/065701 Abstract The structural, electronic and thermodynamic properties of PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ternary alloys have been calculated using the full-potential linearized-augmented plane wave method. The exchange and correlation potential is treated by the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof parameterization. Moreover, the Engel–Vosko GGA formalism is also applied to optimize the corresponding potential for band structure calculations. A nonlinear dependence of the effect of the concentration (x) on the lattice constants, bulk modulus and band gaps is found. The microscopic origins of the band gap bowing parameter have been discussed. Moreover, the thermodynamic stability of the studied alloys is investigated by means of the miscibility critical temperature. PACS numbers: 71.15.Mb, 61.66.Dk, 71.15.Nc, 71.20.Nr, 71.22.+i (Some figures in this article are in colour only in the electronic version.)

particularly useful as electro-optical devices of 3–30 µm wavelength, corresponding to the medium and far infrared. These known characteristics and unusually high dielectric constants, which do not appear in polar crystals, have made various electrical and optical measurements possible, thus allowing a close view of the relationship between the optical, electrical and chemical properties of polar crystals. The lead salts are of great importance in infrared detectors, light-emitting devices and as infrared lasers in fiber optics, as thermoelectric materials and in window coatings [6–9]. The small energy gap of lead chalcogenides (PbS, PbSe and PbTe) is one of the most important properties prompting great experimental interest in these materials. Experimental research has been performed on their structural and band properties [10, 11]. Many theoretical

1. Introduction Lead chalcogenide narrow-gap semiconductors PbX (X = S, Se and Te) and their alloys have been applied in long-wavelength imaging [1], in diode lasers [2] and in thermophotovoltaic energy converters [3]. The IV–VI compounds are semiconductors with a good grade of polarity, with bondings formed through electrostatic interactions among the ions of the crystal lattice, crystallizing in the rock-salt-type structure. Compared with the usual III–V compounds, for example, these IV–VI chalcogens present atypical electronic and transport properties, such as small energy gaps, low resistivities, large carrier mobilities, narrow band gaps and positive temperature coefficients [4, 5]. These properties make these compounds 0031-8949/11/065701+09$33.00

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Phys. Scr. 83 (2011) 065701

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studies of the electronic structure of these materials have been performed by many groups, using the semi-empirical tight-binding (TB) [12], augmented plane wave (APW) [13, 14], Green function [15], orthogonalized plane wave [16] and empirical pseudo-potential [17–20] methods. These theoretical calculations identified a direct band gap at the L point of the Brillouin zone (BZ) for all the three lead chalcogenides. There are also many experimental studies on a mixture of these materials; for example, Lebedev and Sluchinskaya found the appearance of ferroelectricity in the IV–IV semiconductors [21] and investigated samples of PbSx Se y Te1−x−y quaternary solid solutions at low temperatures using electrical and x-ray methods [22]. Vacuum-evaporated PbS1−x Sex thin films were examined by Kumar et al [23] and multi-spectral PbSx Se1−x photovoltaic infrared detectors [24] were realized by Schoolar et al. Theoretically, to our knowledge, only the recent works of Zaoui et al [25] and Kacimi et al [26] have reported the electronic structure of PbSx Te1−x , PbSex Te1−x and PbSx Se1−x by means of the hybrid full-potential augmented plane wave plus local orbitals (APW + lo) method. A continuous variation of the physical properties can also be achieved in ternary alloys PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex . In fact one of the easiest ways to change artificially the electronic and optical properties of semiconductors is by forming their alloys. It is possible to combine two different compounds with different optical band gaps and different rigidities in order to obtain a new material with intermediate properties. 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 [27]. Zunger et al [28] have introduced an approach that greatly reduces the size of the supercell required in order to obtain a realistic description of a random alloy by using the so-called ‘special quasirandom structures’ (SQS). In this respect, we have carried out a study of the PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ternary alloys using the full-potential linearized augmented plane wave (FP-LAPW) method. We have adopted the SQS approach, which greatly reduces the size of the supercell required in order to obtain a realistic description of a random alloy. Very recently, there has been interest in the study of these types of alloys, and a few theoretical ab initio calculations have been performed. The most important technical details of our calculations are discussed in section 2. The core of the paper appears in section 3, where the results on the structural, electronic and thermodynamic properties are presented and analyzed. Finally, in section 4 we summarize the main conclusions of our work.

Table 1. Atomic positions for PbSe1−x Sx , alloy. x 0.25 0.5 0.75

Atom Pb Se S Pb Se S Pb Se S

Atomic position (0 0 0), (1/21/2 0), (1/2 0 1/2), (0 1/21/2) (1/2 1/2 1/2), (1/2 0 0), (0 1/2 0) (0 0 1/2) (0 0 0), (1/2 1/2 0), (1/2 0 1/2), (0 1/2 1/2) (1/2 1/2 1/2), (1/2 0 0) (0 1/2 0), (0 0 1/2) (0 0 0), (1/2 1/2 0), (1/2 0 1/2), (0 1/2 1/2) (1/2 1/2 1/2) (1/2 0 0), (0 1/2 0), (0 0 1/2)

electronic properties, in addition to that, the Engel–Vosko GGA (EVGGA) formalism [32] was also applied. In this method, the space is divided into an interstitial region and non-overlapping muffin tin (MT) spheres centered at the atomic sites. In the interstitial region region, the basis set consists of plane waves. Inside the MT spheres, the basis set is described by radial solutions of the one-particle Schrödinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. The dependence of the total energy on the number of k-points in the irreducible wedge of the first BZ has been explored within the linearized tetrahedron scheme by performing the calculation for 47 k-points for the binary compound and 125 k-points for the ternary alloys within the irreducible BZ (IBZ). The wave functions in the interstitial region were expanded in plane waves with a cut-off K max = 8.0/RMT , where RMT denotes the smallest atomic MT sphere radius and K max gives the magnitude of the largest K vector in the plane wave expansion. The valence wave functions inside the MT spheres are expanded up to lmax = 10, while the charge density was Fourier expanded up to G max = 14 (Ryd)1/2 . The MT radius is taken to be 2.2 au for Pb atom and 2.0 au for S, Se and Te atoms. Both the plane wave cut-off and the number of k-points were varied to ensure total energy convergence.

3. Results and discussion 3.1. Structural properties In this section, we analyze the structural properties of the binary compounds and their alloys. The rock-salt structure was assumed. We model the alloys at some selected compositions with the ordered structures described in terms of periodically repeated supercells with eight atoms per unit cell. In table 1, we summarize the atomic positions of PbSe1−x Sx as a prototype for x = 0.25, 0.50 and 0.75. The idea of constructing an alloy by taking a large unit cell (cubic eight atoms) and repeating it three dimensionally for the calculation of the electronic structure has been used by Agrawal et al [33]. Recently, many researchers have used this method to investigate the properties of alloys [34, 35]. For the considered structures, we perform the structural optimization by calculating the total energies for different volumes around the equilibrium cell volume V0 of the binary compounds and their alloys. The calculated total energies are fitted to an empirical functional form (the third-order Murnaghan’s equation of state) [36] to obtain an analytical interpolation of our computed points from which we determine the ground state properties such as the equilibrium lattice parameter (a)

2. Method of calculations The calculations reported here were performed using the FP-LAPW method [29] as implemented in the WIEN2K computer package [30]. The exchange–correlation contribution was described within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [31] to calculate the total energy, whereas for the 2

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Table 2. Calculated lattice parameter a and bulk modulus B for lead chalcogenides PbS, PbSe, PbTe and their alloys at equilibrium volume. x Present PbSe1−x Sx

PbSe1−x Tex

PbS1−x Tex

1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0

6.010 6.073 6.128 6.179 6.210 6.560 6.487 6.406 6.320 6.210 6.560 6.450 6.325 6.188 6.010

Lattice constant a (Å) Experiment Other calculations 5.929a , 5.936b,c

6.117a 6.462a

6.012d 6.069f 6.122f 6.174f 6.196d 6.565d

5.906e

5.860d

6.098e 6.440e

6.222f

Bulk modulus B (GPa) Present Experiment Other calculations 53.384 49.211 47.462 46.783 49.187 38.406 37.791 41.101 44.489 49.187 38.406 39.462 44.368 45.744 53.384

52.9a

54.1a 39.8a

53.3d 50.8f 49.7f 48.2f 49.1d 41.4d

66.3e

60.8e 51.7e

a

Madelung et al [37]. Dalven et al [4]. c Cohen and Chelikowsky [38]. d Lach-hab et al [39]. e Wei and Zunger [40]. f Kacimi et al [26]. b

and bulk modulus (B). The equilibrium lattice constants and bulk modulus for the binary compounds (PbSe, PbS and PbTe) and their alloys (PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ) are given in table 2, which also contains the results of previous calculations as well as the experimental data. Our computed values of the lattice constants for the binary compounds are slightly overestimated compared to the corresponding experimental values, which is consistent with the general trend of GGA [41]. To the best of our knowledge, no experimental data on the structural properties of the ternary alloys PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex (x = 0.25, 0.5 and 0.75) are available in the literature. Hence, our results can serve as a prediction for future investigations. Figures 1(a)–(c) show the variation of the calculated equilibrium lattice constants versus concentration for PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex alloys, respectively. A small deviation from Vegard’s law (namely the lattice constant of alloys should vary linearly with composition x) [42] is found for PbSe1−x Sx and PbSe1−x Tex alloys with upward bowing parameters equal to −0.077 and −0.090 Å, respectively, obtained by fitting the calculated values with a polynomial function. This fact suggests that there is good agreement between the density functional theory (DFT) predictions and the linear Vegard’s law. The physical origin of these marginal bowing parameters should be mainly due to the weak mismatches of the lattice constants of PbS, PbSe and PbTe compounds. Furthermore, it should be related to the nearly equal size of atoms, since the ratios R(S)/R(Se) and R(Se)/R(Te) have the values 0.9 and 0.8, respectively. For PbS1−x Tex alloy a relatively large bowing parameter equal to −0.169 Å, compared to those of PbSe1−x Sx and PbSe1−x Tex , is found. This could be explained by the significant mismatch of the lattice constant between PbS and PbTe compounds. The variation of the bulk modulus versus the composition x for the studied alloys (PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ) is displayed in figures 2(a)–(c). The dashed lines

in this figure represent the variation of the bulk modulus versus composition predicted by a linear concentration dependence method. A significant deviation from linear concentration dependence (LCD), with downward bowings equal to 16.25, 12.23 and 11.01 GPa, respectively, was observed. The bulk modulus bowings are inversely proportional to the mismatch of the lattice parameters, which should be mainly due to the disorder effect. It is clearly seen that the bulk modulus decreases with increasing chalcogenide atomic number. Hence, we conclude that PbTe is more compressible compared with the other lead chalcogenide compounds. 3.2. Electronic properties In order to compute the energy band gaps for lead chalcogenide compounds and their alloys self-consistent band structure calculations were obtained using both GGA and EVGGA schemes. The valence band maximum (VBM) and conduction band minimum (CBM) occur at the L point in the BZ for the binary compounds. The calculated band gap values for lead chalcogenide compounds and their alloys are listed in table 3 along with the available theoretical results. The obtained band gap values for the binary compounds within GGA are slightly overestimated compared to available theoretical results. This discrepancy could be attributed to the different functional parameters used in the calculations. In the case of alloys there are no available theoretical or experimental band gap values for direct comparison. It is well known that there is a significant difference between the calculated band gap using the DFT formalism (using the local density approximation (LDA) or GGA) and the experimental one. The theoretical gap is underestimated by 30–40% compared to the experimental one [44–46]. This is mainly due to an intrinsic feature of DFT, which being a ground-state theory not suitable for describing excited-state properties. However, it is widely accepted that GGA (LDA) electronic band structures are qualitatively in good agreement with 3

Phys. Scr. 83 (2011) 065701

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6,25

54

(a)

(a)

53

Bulk modulus (GPa)

Lattice parameter (A°)

6,20

6,15

6,10

6,05

52 51 50 49 48 47

6,00 0,0

0,2

0,4

0,6

0,8

46

1,0

Composition x

(b)

0,4

0,6

0,8

50

1,0

(b)

48

6,50

Bulk modulus (GPa)

0

Lattice parameter (A )

6,55

6,45 6,40 6,35 6,30 6,25

46 44 42 40 38

6,20 0,0

0,2

0,4

0,6

0,8

1,0

36

composition x

0,0

0,2

0,4

0,6

0,8

1,0

Composition x (c)

6,6

54

(c)

52

6,5

Bulk modulus (GPa)

0

0,2

Composition x

6,60

Lattice parameter (A )

0,0

6,4 6,3 6,2 6,1

50 48 46 44 42 40

6,0 0,0

0,2

0,4

0,6

0,8

38

1,0

0,0

Composition x

0,2

0,4

0,6

0,8

1,0

Composition x

Figure 1. Composition dependence of the calculated lattice constant (solid squares) of (a) PbSe1−x Sx , (b) PbSe1−x Tex and (c) PbS1−x Tex alloys compared with Vegard’s prediction (dashed line).

Figure 2. Composition dependence of the bulk modulus (solid squares) of (a) PbSe1−x Sx , (b) PbSe1−x Tex and (c) PbS1−x Tex alloys compared with LCD prediction (dashed line).

experiments as far as ordering of energy levels and shape of bands. The band gaps of ternary alloys are investigated under the assumption that the HOMO–LUMO (highest occupied molecular orbital–lowest unoccupied molecular orbital) gap of the Kohn–Sham spectrum is qualitatively in agreement with the actual band gap. The actual band gap is defined as the difference between the ionization potential and the electron affinity. Engel and Vosko, by considering this shortcoming, constructed a new functional form of the GGA which has been designed to give better exchange potential at the expense of less agreement as regards exchange energy. This approach, which is called the EVGGA, yields better band splitting and some other properties that mainly depend on the accuracy of the exchange–correlation potential [47–49]. On the other hand, in this method, the quantities that depend on an accurate

description of the exchange energy such as equilibrium volumes and the bulk modulus are in poor agreement with experiment [50]. For better visualizing the behavior of the band gap with the concentration of the chalcogenides, the variation of the band gap as a function of the concentration for the PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex alloys is presented in figures 3(a)–(c). Both the PbSe1−x Tex and PbS1−x Tex alloys present the same behavior: the band gap decreases when the concentration varies from x = 0 to 0.75 and then starts to increase. However, for the PbSe1−x Sx alloy, the value of the band gap decreases with concentration up to x = 0.25. Our calculations of the composition dependence of band gaps using GGA and EVGGA schemes show that the band gap decreases nonlinearly with increasing S or Te content. The 4

Phys. Scr. 83 (2011) 065701

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Table 3. Gap energy E g of lead chalcogenides and their alloys at equilibrium volume. Band gap (eV) The present work Other calculations GGA EVGGA

PbSe1−x Tex

PbS1−x Tex

1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0

0.496 0.427 0.342 0.329 0.425 0.833 0.207 0.242 0.312 0.425 0.833 0.136 0.161 0.256 0.496

1.248 1.148 1.066 0.999 1.099 1.386 0.784 0.841 0.938 1.099 1.386 0.717 0.779 0.923 1.248

0.380a 0.340a 0.737a

0.448b 0.318b 0.643b

Band gap (eV)

PbSe1−x Sx

1,0

0.295c 0.730c

0,8

0,6

0,4 0,0

1,4

Lach-hab et al [39]. b Zaoui et al [25]. c Albanesi et al [43]. Table 4. Decomposition of the optical bowing into VD, CE and SR contributions compared with that obtained by a quadratic fit (all values are in eV).

PbS1−x Tex

0.032 0.498 −0.100 0.430 −0.022 1.740 −0.112 1.606 0.020 2.392 −0.260 2.152

0,8 0,6

0,0

PbS1−x Tex ⇒

0,2

0,4

0,6

0,8

1,0

Composition x 1,4

(c) PbS1-xTex GGA EVGGA

1,2

0.459

1.729

0.474

1.789

1,0 0,8 0,6 0,4

2.221

2.365

0,2

E gEVGGA = 1.292 − 2.337x + 2.365x 2 .

0,4

0,6

0,8

1,0

Figure 3. Composition dependence of the calculated band gap using GGA (solid squares) and EVGGA (solid circles) for (a) PbSe1−x Sx , (b) PbSe1−x Tex and (c) PbS1−x Tex alloys.

atoms [51, 52]. In order to better understand the physical origins of the gap bowing in these alloys, we follow the procedure of Bernard and Zunger [53], in which the bowing parameter (b) is decomposed into three physically distinct contributions. The overall gap bowing coefficient at x = 0.5 measures the change in band gap according to the reaction

E gGGA = 0.477 − 1.444x + 1.729x 2 , E gEVGGA = 1.149 − 1.621x + 1.789x 2 ,

E gGGA = 0.543 − 2.000x + 2.221x 2 ,

0,2

Composition x

(2) (

(b)

GGA EVGGA

0,0

PbSe1−x Tex ⇒

1,0

0,2

total bowing parameter is calculated by fitting the nonlinear variation of band gaps versus concentration with quadratic function. The results obey the following variations: ( E gGGA = 0.413 − 0.363x + 0.459x 2 , PbSe1−x Sx ⇒ (1) E gEVGGA = 1.081 − 0.295x + 0.474x 2 , (

0,8

0,4

Band gap (eV)

PbSe1−x Tex

0.014 0.460 −0.120 0.354 −0.084 1.756 −0.128 1.544 −0.106 2.396 −0.276 2.014

0,6

1,0

The present work The Zunger approach Quadratic fits GGA EVGGA GGA EVGGA bVD bCE bSR b bVD bCE bSR b bVD bCE bSR b

0,4

PbSe1-xTex

1,2

a

PbSe1−x Sx

0,2

Composition x

Band gap (eV)

x

(a)

PbSe1-xSx GGA EVGGA

1,2

(3)

AB(aAB ) + AC(aAC ) → AB0.5 C0.5 (aeq ),

The quadratic terms are referred to as band gap bowing parameters. The results of the calculated gap bowing are given in table 4. The calculated band gap exhibits strong composition dependence for these alloys. It has been seen that the main influence of the band gap energy is due to the lattice constant and the electronegativity mismatch of the parent

(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 reaction (4) into three steps: AB(aAB ) + AC(aAC ) → AB(a) + AC(a), 5

(5)

Phys. Scr. 83 (2011) 065701

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AB(a) + AC(a) → AB0·5 C0·5 (a),

(6)

AB0·5 C0·5 (a) → AB0·5 C0·5 (aeq ).

(7)

0,50

Band gap (eV)

0,45

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 in their individual equilibrium lattice constants to the alloy value a = a(x) (from Vegard’s rule). The second contribution comes from the charge-exchange (CE) contribution bCE , which reflects a charge transfer effect due to the different (averaged) bonding behaviors at the lattice constant a. The last contribution denoted by bSR measures the change due to structural relaxation (SR) in passing from the unrelaxed to the relaxed alloy. Consequently, the total gap bowing parameter is defined as b = bVD + bCE + bSR , (8)

(10)

bSR = 4[εABC (a) − εABC (aeq )],

(11)

0,35 0,30 0,25

0,15 0,10

0,0

0,2

0,4

0,6

0,8

1,0

Composition x 0,9

(b)

0,8

P=0 GPA P=1 GPA P=2 GPA P=3 GPA

0,7

(9)

bCE = 2[εAB (a) + εAC (a) − 2εABC (a)],

0,40

0,20

Band gap (eV)

bVD = 2[εAB (aAB ) − εAB (a) + εAc (aAc ) − εAc (a)],

(a)

P=0 GPA P=1 GPA P=2 GPA P=3 GPA

0,6 0,5 0,4 0,3 0,2 0,1 0,0 0,0

where ε is the energy gap that has been calculated for the indicated atomic structures and lattice constants. The energy band gap terms in equations (9)–(11) are calculated separately with self-consistent band structure FP-LAPW within both GGA and EVGGA approaches. Our results on the different bowing parameters, namely bVD , bCE and bSR , are collected in table 4 along with the band gap bowing parameters obtained in equations (1)–(3) using a quadratic fit. The charge transfer contribution bCE has been found to dominate the total gap bowing parameter for the alloys under investigation, which is due to the electronegativity difference between Se (2.55), S (2.58) and Te (2.1) atoms. For the three alloys the b estimated by EVGGA using the Bernard and Zunger approach is larger in magnitude than that obtained by GGA using the same approach.

0,2

0,4

0,6

0,8

1,0

Composition x 0,9 0,8 0,7

Band gap (eV)

(c)

P=0 GPA P=1 GPA P=2 GPA P=3 GPA

0,6 0,5 0,4 0,3 0,2 0,1 0,0 -0,1

0,0

0,2

0,4

0,6

0,8

1,0

Composition x

3.2.1. Pressure dependence of band gaps in lead chalcogenide and their alloys. Because of their use in infrared light generation and detection, the band gap variations with temperature and pressure of the IV–IV compounds and their alloys represent an important property to study. The GGA does not accurately describe the eigenvalues of the electronic states, which causes quantitative underestimations of band gaps as already mentioned. It is precisely these quantities that are least well-predicted by the GGA in the DFT. However, the pressure derivatives of conduction band states [54] and their relative positions are given reasonably well within the GGA. We also note that the GGA tends to underestimate the band-gap pressure coefficients, but it at least reproduces the experimental trend quite well. The underestimation of the pressure coefficients can probably be attributed to the GGA, but at a small percentage compared with the absolute band gaps. However, since the calculation of the

Figure 4. Fundamental band gap energy versus x composition for (a) PbSe1−x Sx , (b) PbSe1−x Tex and (c) PbS1−x Sx alloys at various pressures.

self-energy correction to the gap indicates that the correction is typically a rigid shift of the whole conduction band, the pressure coefficient of the band gap within the GGA is, at least, more reliable than the absolute gap size. We therefore assume that the band states calculated within the GGA show the qualitatively correct ordering and dependence on cell volume. For this purpose, the band structures of PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex were obtained within the GGA scheme at high pressures for different x compositions ranging from 0 to 1. Figures 4(a)–(c) show the variation of the fundamental band gap energy in these alloys as a function of x under different pressures ranging from 0 to 3 GPa. As pressure 6

Phys. Scr. 83 (2011) 065701

N Boukhris et al

Table 5. Calculated linear pressure coefficients of the band gap (in eV/GPa) for PbSe1−x Sx , PbSe1−x Sx and PbS1−x Tex at various x compositions. Composition x PbSe1−x Sx

PbSe1−x Tex

PbS1−x Tex

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

Pressure coefficients dE g /d p (eV/GPa) The present work Other calculations Experiment −0.053 −0.069 −0.058 −0.063 −0.048 −0.053 −0.063 −0.070 −0.024 −0.045 −0.048 −0.074 −0.042 0.021 −0.045

−0.051b

−0.086c

−0.050a −0.073a

−0.052b −0.051b

−0.091c −0.086c

−0.038a −0.050a

−0.040b −0.052b

−0.074c −0.091c

−0.038a

−0.040b

−0.074c

a

Nabi et al [55]. Wei and Krakauer [56]. c Nimtz and Schlicht [57]. b

difference in energy between the alloy and the weighted sum of the constituents:

increases, the band gap decreases in magnitude at each x composition. The calculated linear pressure coefficients of the direct band gap for the alloys under investigation at various compositions x are given in table 5, along with the experimental and previous theoretical data that are available only for the parent compounds, i.e. PbS, PbSe and PbTe. The results show that all the pressure coefficients are negative, which is due to the narrowing of the band gap under pressure. When pressure increases, the amplitude of atomic vibrations decreases, leading to smaller interatomic spacing. The interaction between the lattice and the free electrons and holes will affect the band gap to a smaller extent. For the parent compounds, the present results on the linear pressure coefficients of the band gap are in good agreement with the theoretical values cited by Wei and Krakauer [56]. However, our values are lower than the experimental ones.

1Hm = E ABx C1−x − x E AB − (1 − x)E AC .

By rewriting expression (13) as  = 1Hm /x(x − 1), we can calculate, for each x, a value of  from the above DFT values of 1Hm . The only shortcoming of the regular-solution model for a statistical description of entropy may be that the model could be affected by the number of DFT enthalpy values extracted from DFT calculation and, consequently, by the nature of the fit needed. The interaction parameter  depending on x is then obtained from a linear fit to the  values. The best fit gives

3.3. Thermodynamic properties In order to study the phase stability of PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex ternary alloys, the Gibbs free energy of mixing 1G m (x, T ) is calculated in order to access the T–x phase diagram and obtain the critical temperature TC , for miscibility. Further details of the calculations can be found in [58, 59]. The Gibbs free energy of mixing, 1G m , for alloys is expressed as 1G m = 1Hm − T 1Sm

(12)

1Hm = x(1 − x),

(13)

1Sm = −R[x ln x + (1 − x) ln(1 − x)].

(14)

(15)

PbSe1−x Sx ⇒  (kcal mol−1 ) = 4.080 − 0.133x,

(16)

PbSe1−x Tex ⇒  (kcal mol−1 ) = 6.096 − 0.664x,

(17)

PbS1−x Tex ⇒  (kcal mol−1 ) = 11.831 − 2.508x.

(18)

The average values of the x-dependent  in the range 0 6 x 6 1 obtained from these equations are 4.013, 5.764 and 10.577 (kcal mol−1 ) for PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex alloys, respectively. 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 determined as those points at which the second derivative of 1G m is zero:

where

1Hm and 1Sm are the enthalpy and the entropy of mixing, respectively;  is the interaction parameter, R is the gas constant and T is the absolute temperature. Only the interaction parameter  depends on the material properties. The mixing enthalpy of alloys can be obtained as the

∂ 2 (1G m )/∂ x 2 = 0. Using the x-dependent interaction parameter , we quantitatively determine the critical temperature TC and 7

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80

Tc=72 °K

as a metastable phase. Finally, these results indicate that these alloys are stable at low temperature.

(a)

70

Temperature (°K)

60

4. Conclusion

50

In summary, using the FP-LAPW method, the structural, electronic and thermodynamic properties of PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex alloys were studied. The composition dependence of the lattice constant, bulk modulus and the band gap was investigated. The lattice constant of PbSe1−x Sx and PbSe1−x Tex exhibits a small deviation from Vegard’s law, which is mainly due to the weak mismatch between the lattice constants of the parent binary compounds. A significant deviation of the bulk modulus from LCD was found for all the three alloys. This deviation is mainly due to the mismatch of the bulk modulus of the binary compounds. The origin of the band gap bowing is found to be mainly dominated by the charge transfer effect. Applied hydrostatic pressure leads to the determination of the negative linear pressure coefficients, which is in good agreement with published experiments. Finally, calculated phase diagrams indicate that the three alloys are stable at low temperature.

40 30 20 10 0 0,0

0,2

0,4

0,6

0,8

1,0

Composition x 120

Tc=106 °K

(b)

Temperature(°K)

100

80

60

40

References 20

0 0,0

0,2

0,4

0,6

0,8

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1,0

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Temperature (°K)

(c)

Tc=195 °K

200

150

100

50

0 0,0

0,2

0,4

0,6

0,8

1,0

Composition x

Figure 5. T–x phase diagram of (a) PbSe1−x Sx , (b) PbSe1−x Tex and (c) PbS1−x Tex alloys. Dashed line: binodal curve; solid line: spinodal curve.

the stable and/or metastable boundary lines. Figures 5(a)–(c) show the resulting phase diagram for alloys of interest. We observed a critical temperature Tc of 72, 106 and 195 K for PbSe1−x Sx , PbSe1−x Tex and PbS1−x Tex alloys, respectively. The phase diagram shows a symmetry that is due to the use of average values of . 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 8

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