Elastic, electronic and optical properties of ZnS, ZnSe and ZnTe under pressure

Computational Materials Science 38 (2006) 29–38 www.elsevier.com/locate/commatsci

Elastic, electronic and optical properties of ZnS, ZnSe and ZnTe under pressure R. Khenata

a,b,*

, A. Bouhemadou c, M. Sahnoun b, Ali. H. Reshak d, H. Baltache b, M. Rabah a

a b

Applied Materials Laboratory (AML), Electronics Department, University of Sidi-Bel-Abbe`s, 22000, Algeria Laboratoire de Physique Quantique et de Mode´lisation Mathe´matique (LPQ3M), De´partement de Technologie, Universite´ de Mascara, Mascara 29000, Algeria c De´partement de Physique, Faculte´ des Sciences, Universite´ Ferhat Abbes, 19000 Se´tif, Algeria d Physics Department, Indian Institute of Technology, Roorkee 247667, India Received 23 November 2005; received in revised form 25 December 2005; accepted 4 January 2006

Abstract The results of first-principles theoretical study of the structural, electronic and optical properties of zinc monochacogenides ZnS, ZnSe and ZnTe, have been performed using the full-potential linear augmented plane-wave method plus local orbitals (FP-APW + lo) as implemented in the WIEN2k code. In this approach the local density approximation (LDA) is used for the exchange-correlation (XC) potential. Results are given for lattice constant, elastic constant, bulk modulus, and its pressure derivative. The band structure, density of states, pressure coefficients of elastic constants, energy gaps and refractive indices are also given. The results are compared with previous theoretical calculations and the available experimental data. Ó 2006 Elsevier B.V. All rights reserved. PACS: 71.15.Ap; 78.40.Fy; 78.20.Ci Keywords: FP-APW + lo; LDA; Elastic constants; Optical constants; Pressure effect

1. Introduction The wide band gap semiconductors are attracting enormous technological interest because of their potential use in device capable for operating at high powder level and high temperature and because of need for optical materials active in blue-green spectral. Among the wide band gap semiconductors, ZnS, ZnSe and ZnTe constitute a family of IIB–VIA compounds, crystallizing in the cubic zincblende structure at ambient pressure. These compounds *

Corresponding author. Address: Laboratoire de Physique Quantique et de Mode´lisation Mathe´matique (LPQ3M), De´partement de Technologie, Universite´ de Mascara, Mascara 29000, Algeria. Tel.: +213 045804162. E-mail address: [email protected] (R. Khenata). 0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.01.013

differ from the IIA–VI compounds by the existence of a metal ‘‘d’’ bands inside the main valence band in column IIB. They have received a lot of attention as blue lasing materials and could be used in the fabrication of modulated heterostructures and optical wave guides [1–3]. They have also proved to be a particularity interesting dilute semiconductors, when doped with Mn [4–6], with the possibility of being used as microelectronic magnetic when alloyed with other elements such as BCC iron, nickel, manganese and cobalt [7,8]. These compounds have been extensively studied experimentally for their intrinsic optical properties [9–20]. The structural phase transitions for these compounds were also performed by many authors with various computational methods and showing that the transitions to high pressure phases occurs at about 15 GPa, 13.7 GPa and 9.5 GPa in

30

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

ZnS (Refs. [21–26]), ZnSe (Refs. [23,27,28]) and ZnTe (Refs. [23,29]), respectively. The lattice and hyperfine interactions in ZnSe at high pressure has been investigated by Karzel et al. [30], using both X-ray diffraction measurements and FP-LAPW calculations. From a theoretical point of view, a several first principles calculations were made for ZnS, ZnSe and ZnTe compounds by a variety of methods [31–46]. Cheliokowsky and Cohen [31], Wang and Klein [32], and Jansen and Sankey [33] have been studied the electronic structure of these compounds by using the non local pseudopotential (NLPP), the linear combination of Gaussian orbitals (LCAO) and the ab initio linear combination of pseudo atomic-orbital schemes, respectively. Christensen and Christensen [34] have studied the electronic structure of ZnTe under pressure by means of the self-consistent linear muffin–tin orbital calculation. In the beginning of 1990s, the linear combination of Gaussian orbitals (LCGO) technique have used by Gharamani et al. [35] and Jaffe et al. [36] to discussed the band structure of ZnSe and ZnTe compounds. The role of d electrons in band structure of zinc-blende ZnS, ZnSe and ZnTe have been discussed by Lee and coworkers [37] using the ab initio pseudopotential total energy calculations. The semiempirical tight binding model is used by Merad et al. [38] and Li and Po¨tz [39] to study the electronic structure of ZnTe and ternary ZnSeTe and their binary constituents. Recently, Ro¨nnow et al. [40] have used the full potential linear muffin–tin orbitals FP-LMTO method to investigate the effect of strain on the E1 electronic interband transitions for ZnSe and ZnTe. The band structure of ZnS is investigated by Qteish [26] and Oshikiri and Aryasetiawan [41], using the SIC-LDA and GW formalisms, respectively. The FP-LAPW method coupled with the tight binding theory have been used by Rabah et al. [42] to calculate the structural, electronic and optical properties of the quaternary MgZnSSe and their binary compounds. Kassali and Bouarissa [43], Boukortt et al. [44], Benmakhlouf et al. [45], and Al-Douri et al. [46] reported some band structure results on ZnTe, ternary ZnSeS and quaternary ZnCdSSe alloys and the binary constituents. The authors [43–46] used the empirical pseudopotential (EPM) method. The elastic constants of zinc-chalcogenides have been calculated more than 10 years ago by Singh and Singh [47] and Shen [48] using the three-body force potential and bond orbital calculations based on the tight binding method, respectively. Recently, Casali and Christensen [49] have been discussed the calculation of elastic constants and their pressure dependence, using the full-potential linear muffin–tin orbitals (FP-LMTO) method. From the above it is clear that there is considerable experimental and theoretical work on ZnX compounds. We note that there exist a limited theoretical studies on elastic and electronic properties under pressure. Moreover, there appear to be no earlier calculations of the effect of pressure on the static dielectric constants of these compounds. We therefore think it worthwhile to perform these

calculations. The aim of this work is to give a detailed description of the behavior of elastic, electronic and dielectric properties of ZnX compounds under hydrostatic pressure by using for the frame work the full-potential augmented plane wave plus local orbitals (FP-APW + lo) method, in order to complete the exciting experimental and theoretical works on these compounds. 2. Computational method The semiconducting binary ZnX (X = S, Se, Te) compounds are crystallized in the zinc-blende structure. The space group is F-43 m. The Zn atom is located at the origin and the X atom is located at (1/4, 1/4, 1/4). The calculations reported in this work were done with the WIEN2K program developed by Blaha and co-workers [50]. This program uses the full-potential linearized augmented plane wave plus local orbitals [51–53] approach (FP-APW + lo) based on density functional theory [54]. In this method the space is divided into an interstitial region (IR) and non overlapping (MT) spheres centered at the atomic sites. In the IR region, the basis sets consist of a plane wave. Inside the MT spheres, the basis sets is described by radial solutions of the one particle Schro¨dinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. In the calculations reported here, we use a parameter RMTKmax = 9, which determines matrix size (convergence), where Kmax is the plane wave cut-off and RMT is the smallest of all atomic sphere radii. We have chosen the muffin–tin radii (MT) for Zn, S, Se and Te to be 2.2, 1.7, 1.9 and 2.2 a.u., respectively. XC effects are treated by LDA [55]. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 104 Ry. The integrals over the Brillouin zone are performed up to 30 k-points in the irreducible Brillouin zone (IBZ), using the Monkhorst–Pack special k-points approach [56]. 3. Results and discussions 3.1. Structural and elastic properties The fitting of the Murnaghan equation of state [57] to the total energies versus lattice parameters, yields to the equilibrium lattice parameter (a0), bulk modulus B0, and the pressure derivative of the bulk modulus B 0 . In Table 1 we summarize our calculated structural properties (lattice constant, bulk modulus and its pressure derivative) of ZnS, ZnSe and ZnTe at ambient pressure. When we analyze these results we find that there is a good agreement between our results and the reported theoretical investigations. In comparison with the experimental data we find that the lattice parameters are underestimated whereas the bulk modulus are overestimated. That is attributed to our use of the local density approximation (LDA). To verify the accuracy of these results, several tests have been performed using different muffin–tin radius as well as different sets of special k-point to ensure the convergence.

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38 Table 1 ˚ ), bulk modulus (in GPa), pressure Calculated lattice constant (in A derivative (B 0 ) at equilibrium volume in zinc-blende (B3) structure for ZnX compounds compared to experimental and other theoretical works ˚) a0 (A B0 B0 ZnS Present Expt.

5.342 5.412a 5.41b 5.3998 5.335 5.352 5.393 5.187

89.67 75a 77.1b 80.97 83.7 83.1 82 105.7

4.44 4.00a 4.00b

71.82 64.7g 69.3h 67.6 67.32 62.45 71.84

4.88 4.77g

TB-LMTOd FP-LMTOc NAOi FP-LAPWj

5.624 5.667g 5.667h 5.618 5.666 5.666 5.578

ZnTe Present Expt. SCR-LMTOl

6.00 6.103k 6.174

55.21 50.9k 51.2

4.60 5.04k 4.88

TB-LMTOc FP-LMTOd SIC-PPe LDA-LMTOa PPf ZnSe Present Expt.

4.20 4.43 4.20

0

0

1 ð1þdÞ2

3 1

7 5

where V0 is the volume of the unstrained unit cell. Finally, for the last type of deformation, we used the volume-conserving rhombohedral strain tensor given by 2 3 1 1 1 d6 7 ð3Þ 41 1 15 3 1 1 1

ð4Þ

1 C S ¼ ðC 11  C 12 Þ 2

4.05 4.599

The second one involves applying volume conserving tetragonal strains 0 0

ð2Þ

For cubic crystal, the shear wave modulus is given by 4.67

B0 ¼ ðC 11 þ 2C 12 Þ=3

0 d

EðdÞ ¼ Eð0Þ þ 6ðC 11  C 12 ÞV 0 d2 þ Oðd3 Þ

1 EðdÞ ¼ Eð0Þ þ ðC 11 þ 2C 12 þ 4C 44 ÞV 0 d2 þ Oðd3 Þ 6

To obtain the elastic constants of these compounds with cubic structure we have used the numerical first-principle calculation by computing the compounds of the stress tensor d for small stains, using the method developed recently by Charpin and integrated in WIEN2k code [50]. It is well know that a cubic crystal has only three independent elastic constants C11, C12 and C44. So a set of three equations is needed to determine all the constants. The first equation involves calculating the elastic modulus (C11  C12) which are related to the bulk modulus B0

d 60 4

Application of this strain changes the total energy from its unstrained value as follows:

This transforms the energy to

SIC-PP: self-interaction correction pseudo-potential; NAO: numerical atomic orbital within LCAO; SCR-LMTO: self-consistent relativistic linear muffin–tin orbital. a Ref. [18]. b Ref. [58]. c Ref. [23]. d Ref. [49]. e Ref. [26]. f Ref. [59]. g Ref. [60]. h Ref. [61]. i Ref. [27]. j Ref. [62]. k Ref. [63]. l Ref. [34].

2

31

ð1Þ

ð5Þ

Another important parameter is the internal strain parameter introduced by Kleinman [64] and describes the relative positions of the cation and anion sublattices under volume, conserving strain distortions for which positions are not fixed by symmetry. We use the following relation [65,66]: f¼

C 11 þ 8C 12 7C 11 þ 2C 12

ð6Þ

Table 2 Calculated elastic constants Cij compared to experimental data and other works of ZnS, ZnSe and ZnTe for the B3 phase C11

C12

C44

f

118 104 123.7 120.46

72 65 62.1 56.46

75 46.2 59.7 46.9

0.715

FP-LMTOb BOC-TBc DLMf

94 85.9 ± 0.03a 81.0d 84.0e 95.9 94.66 85.8

61 50.6 ± 0.04a 48.8d 47.0e 53.6 44.46 50.8

64 40.6 ± 0.02a 41.1d 20.8e 48.9 36.8 26.8

ZnTe Present Expt.a BOC-TBc

82 71.7 98

42 40.7 31.8

55 31.2 23.3

ZnS Present Expt.a FP-LMTOb BOC-TBc ZnSe Present Expt.

0.651; 0.93 0.598 0.746

0.63; 0.736 0.599

0.635 0.596

All of the pressures are in GPa. BOC-TB: bond orbital calculations based on the tight binding method; DLM: De Launay model based on atomic central forces. a Ref. [60]. b Ref. [49]. c Ref. [48]. d Ref. [67]. e Ref. [68]. f Ref. [69].

32

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

In Table 2, we summarized our calculated elastic constants Cij and the internal-parameter f for the zinc-blende ZnS, ZnSe and ZnTe in comparison with the experimental data and previous theoretical calculations. Our calculated values are in reasonable agreement with available theoretical and experimental ones, except for C44, where the FP-APW results are relatively larger than those obtained by using the BOC-TB and DLM methods. The elastic constants increase in magnitude as a function of the anion chemical identity as one moves upwards within period VI, i.e. from Te to S. The requirement of mechanical stability in this cubic structure leads to the following restrictions on the elastic constants, C11  C12 > 0, C44 > 0, C11 + 2C12 > 0. The elastic constants in Table 2 obey these stability conditions,

including the fact that C12 must be smaller than C11. Our calculated elastic constants also obey the cubic stability conditions, meaning that C12 < B < C11. Now we are interested to study the pressure dependence of the elastic properties. In Fig. 1, we present the variation of elastic constants and bulk modulus of ZnS, ZnSe and ZnTe with respect to the variation of pressure. We clearly observe a linear dependence in all curves of the zinc chalcogenides in the considered range of pressure, confirming the idea of Polian and Grimsditch [70] and Harrera-Cabrera et al. [71] of the no responsibility of the soft acoustic mode on the phase transition. In Table 3, we listed our results for the pressure derivatives oB0/oP, oC11/oP, oC12/oP, oC44/ oP and oCS/oP for all the considered compounds. It is easy to observe that the elastic constants C11, C12, C44 and bulk

Fig. 1. Calculated pressure dependence of Cij, Cs and B for zinc-blende ZnS, ZnSe and ZnTe.

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

33

Table 3 Calculated pressure derivatives of the elastic modulus for zinc-blende ZnX compounds Compounds ZnS Present FP-LMTOa FPMb ZnSe Present Expt.c FP-LMTOa FPMb ZnTe Present FPMb Expt.c

oB0 oP

oC 11 oP

oC 12 oP

oC 44 oP

oC S oP

4.44

3.86 4.25

4.85 4.32

3.78 0.64 3.08; 4.96

0.495

4.30 4.44 4.50

4.68 4.93 4.62

2.74 0.43 0.79 2.85; 3.94

0.190

3.83

4.89

2.92 3.14; 4.12 0.45

0.532

5.19; 6.23 4.55 4.77 4.17; 5.19 4.60 4.70; 5.65 5.04

FPM: three-body force potential method. a Ref. [49]. b Ref. [47]. c Ref. [60].

modulus increase when pressure is enhanced. Moreover, the shear wave modulus CS decreases linearly with the increasing of pressure for ZnS, ZnSe and ZnTe. We can also remark that the calculated pressure derivatives of bulk modulus B, C11 and C12 are in good agreements with the available theoretical calculations and experimental data. This agreement disappear in the case of C44, since its value in FP-APW calculations is larger than the experimental one. It is impossible for us to find any raison for discrepancy between the experimental and our calculated results. 3.2. Electronic properties Fig. 2a, shows the calculated band structure at equilibrium volume for ZnTe as a prototype since the band profiles are quit similar for all three compounds, with a small difference. The overall band profiles are in fairly good agreement with previous theoretical results [9,34,35,37,39]. In all cases, the valence band maximum (VBM) and conduction band minimum (CBM) are occurs at the C point. Thus the energy gap is direct between the top of the (anion p) valence band and the bottom of conduction band at C point. Note that the chalcogen p bands shift up in energy going from the sulphide to the telluride. This is the normal behavior related to the increase of the lattice parameters, which was also found for other II–VI compounds [72– 74]. The important features of the band structure (main band gaps and valence band widths) are given in Table 4. It is clearly seen that the band gap are on the whole underestimated in comparison with experiments results. This underestimation of the band gaps is mainly due to the fact that the simple form of LDA do note take into account the quasiparticle self energy correctly [75] which make it not sufficiently flexible to accurately reproduce both exchange correlation energy and its charge derivative. It is important to note that the density functional formalism is limited in

Fig. 2. Calculated band structure (a) and total density of state (b) of ZnTe.

its validity (see. Ref [76]) and the band structure derived from it cannot be used directly for comparison with experiment. In Fig. 2b, we show the density of states DOS for only ZnTe because it is similar to that of ZnS and ZnSe with a small difference. It is further observed that the first structure encountered in the total DOS is small but relatively broad break centered at around 12.36 eV, 12.48 eV and 11.36 eV for ZnS, ZnSe and ZnTe respectively. This structure arise entirely from the chalcogen s states and correspond to the lowest lying band in (Fig. 2a) with its widths arising from the dispersion in the region around the C point in the Brillouin zone. The next structure is situated at about 6.20 eV, 6.56 eV and 7.09 eV below the zero of energy for ZnS, ZnSe and ZnTe, respectively. It consists predominantly of Zn d states with a few contribution of p states of the chalcogen atoms. From the band structure (Fig. 2a), this structure corresponds to the flat bands clustered at 7.09 eV. The near lack of dispersion of among some of these bands gives rise to the very narrow nature of the peaks. The structure hump between 5.50 and the zero of the energy in these compounds which form the upper VB correspond to the chalcogen p states partially mixed with cations s states. Above the Fermi level, the feature in the DOS originate mainly from the s and p states of Zn partially mixed with little of chalcogen d states.

34

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

Table 4 Electron band eigenvalues for the lowest conduction band and valence band width for ZnS, ZnSe and ZnTe in zinc-blende phase C15v  C1c

C15v  X1c

C15v  Llc

L3v  L1c

V.B. width

ZnS Present Expt.a EPMb

2.16 3.80 3.66

3.01 4.10 3.70

3.16 4.40 5.60

4.11

13.41

ZnSe Present Expt. NLPMe LCGOf SE-TBMg

1.31 2.82c 2.76 1.83 2.82

2.63 4.3d 4.54

2.34 3.7d 3.96

3.36 4.7d 5.00

13.64

4.54

3.92

4.73

ZnTe Present Expt.h MOPWi OLCAOj SE-TBMg,k

1.28 2.39 2.6 2.39 2.394; 2.398

2.11 3.3 3.8 3.01 3.8; 3.303

1.64 3.1 3.4 3.39 3.44; 3.095

2.65

12.25

3.9 4.08 3.38

All of the energies are in eV. EPM: empirical pseudopotential method; NLPM: non-local pseudopotential; LCGO: linear combination of Gaussian orbitals; MOPW: modified orthogonalized plane wave; OLCAO: orthogonalized linear combination of atomic orbitals; SE-TBM: semi empirical tight binding method. a Ref. [77]. b Ref. [45]. c Ref. [78]. d Ref. [79]. e Ref. [31]. f Ref. [32]. g Ref. [39]. h Ref. [80]. i Ref. [81]. j Ref. [82]. k Ref. [38].

Analysis of the width of peaks from these densities of states, give a band width of valence band equal to 13.41 eV, 13.64 eV, 12.25 eV for ZnS, ZnSe and ZnTe, respectively. The results show that the valence band width is maximum for ZnSe. Showing that the wave function is more localized for ZnS that ZnTe. This is in line with the usual trend in which the valence band states become more localized as a material becomes less covalent and more ionic, as it does when we decrease the atomic number of the anion. It is also noticed that the band gap between the lowest band (anion s band) and the valence band is least in ZnTe due to the higher energy position of the tellurium s band. We aware that the LDA within the density functional formalism does not accurately describe the eigenvalues of the electronic states, which causes quantitative underestimations of band-gaps compared with experiment. However, despite this shortcoming of the LDA, the pressure derivatives or the deformation potentials of band gaps are accurately calculated in the LDA (or GGA) and do not depend on the type or functional form of the exchange-correlation potential [83–85]. As such we have investigated the effect of the pressure on the size of the energy gap for zinc chalcogenides using the FP-APW

method within the local density approximations. In order to achieve this, we have calculated at a different sets of pressures the different direct and indirect gaps, then by a polynomial fit of Egap(P) curves we determined the linear and quadratic pressure-coefficients. Results of the linear and quadratic pressure coefficients are listed in Table 5. Considering this table, we can remark that the linear pressure coefficients increase with the increase of chalcogen atoms. Also, we can notice that the direct energy gaps show an increase with hydrostatic pressure, which is the behavior commonly observed in binary tetrahedral semiconductors [86–88]. The negative linear pressure coefficients indicates that the indirect band gap (C–X) decreases with increase of the pressure. A crossover between the direct gap (C–C) and the indirect gap (C–X) curves occurs at about 17.9 GPa for ZnS, 30 GPa for ZnSe and 8.9 GPa for ZnTe, resulting in the energy minimum of indirect gaps for these compounds. Our calculated linear coefficient pressure along (C–C) direction are relatively closed to the experiment and to those of Refs. [18,45] obtained by the empirical pseudo-potential and relativistic linear muffin–tin orbital methods. 3.3. Optical properties In calculations of the optical properties, a dense mesh of uniformly distributed k-points is required. Hence, the Brillouin zone integration was performed with 116 and 172 points in the irreducible part of the Brillouin zone without broadening. We find a very small difference in the two calculations. We present calculations with only 116 points in this work. The frequency dependent complex dielectric function e(x) = e1(x) + ie2(x) is known to describe the optical response of the medium at all phonon energies E = hx, using the formalism of Ehrenreich and Cohen [92]. The imaginary part of the e(x) in the long wavelength limit has been obtained directly from the electronic structure calculation, using the joint density of states (JDOS) and the transition moments elements Mcv(k) Z e2 h X 2 e2 ðxÞ ¼ jM cv ðkÞj d½xcv ðkÞ  x d3 k ð7Þ pm2 x2 v;c BZ The integral is over the first Brillouin zone, the dipole moments: Mcv(k) = huck j e Æ $ j uvki where e is the polarization vector of the electric field, are matrix elements for direct transitions between valence uvk(r) and conduction-band uck(r) states, and hxcv(k) = Eck  Evk is the excitation energy. The real part of e(x) can be derived from the imaginary part using the Kramers–Kronig relations Z 1 0 2 x e2 ðx0 Þ e1 ðxÞ ¼ 1 þ P dx0 ð8Þ p x02  x2 0 where P implies the principal value of the integral. The knowledge of both the real and imaginary parts of the dielectric function allows the calculation of important optical functions such as the refractive index n(x)

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

35

Table 5 Calculated linear and quadratic pressure coefficients of important band gap for ZnS, ZnSe and ZnTe compounds in zinc-blende phase X–X

C–C b ZnS Present LMTOa EPMb Expt.a ZnSe Present Expt.c ZnTe Present SE-TBMd Expt.e Expt.f

L–L

C–L c

b

c

b

5.19 6.22 6.35 6.35

18.26 11.4

1.6

5.40 7.0

15.16

1.78

3.24

1.50

2.76

4.26

11.42

2.24

3.62

7.03 7.7 10.3 10.5

16.02 21.3 24

2.85

10.20

3.01

18.00

5.01

14.22

3.58

6.32

2.28

c

1.59

b

C–X

6.50

2.71

c 2.44

b

c

1.6

4.68

5.19 13.1

Ei ðpÞ ¼ Ei ð0Þ þ bp þ 12 cp2 , b = oEi/op in eV 102 GPa1, c = o2Ei/o2p in eV 104 GPa2. a Ref. [18]. b Ref. [45]. c Ref. [89]. d Ref. [38]. e Ref. [90]. f Ref. [91].

"

e1 ðxÞ þ nðxÞ ¼ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=2 e21 ðxÞ þ e22 ðxÞ 2

ð9Þ

Fig. 3, displays the imaginary (absorptive) part of the dielectric function e2(x) for ZnS, ZnSe and ZnTe at ambient and under 15 GPa pressure for a radiation up to 16 eV. According to the Kramers–Kronig dispersion relation the real part of the dielectric function e1(x) is also obtained but not presented. It is seen that the behavior of e2(x) is rather similar for all three compounds and is in fairly good agreements with previous theoretical results. The main difference between these spectra lies in the energies of the transitions. Our analysis of the e2(x) curves show that the first critical points of the dielectric function occurs at 2.16 eV, 1.31 eV and 1.28 eV for ZnS, ZnSe and ZnTe, respectively. These points are Cv–Cc splitting which gives the threshold for direct optical transitions between the absolute fourth valence band maximum and the first conduction band minimum. This is known as the fundamental absorption edge. These critical points are followed by a small structure localized at 4.46 eV in ZnS, 3.75 eV in ZnSe and 3.03 eV in ZnTe, related to direct transitions (L–L). Note that in the experimental measurements these peaks are splitted in two peaks by spin-orbit coupling for ZnSe and ZnTe. The main peaks in the spectra of ZnS, ZnSe and ZnTe are situated at 5.90 eV, 5.37 eV and 4.46 eV, respectively. In all compounds, these peaks are mainly due to the direct transitions along D and R directions. The experimental measurements localised the critical points at about 3.50, 2.7, 2.4 eV and the main or global peaks 6.8 eV, 6.2 eV and 5.3 eV for ZnS, ZnSe and ZnTe [9,20]. When compared to the experimental data, there is an energy shift of about 1.20 eV for the threshold peaks.

The energy shift of the main peaks is about 0.8 eV. These energy shifts mainly arise from the LDA, which give a smaller band gap in comparison with experiment. The main peaks are followed by pronounced peaks situated at 7.33 eV, 6.98 eV and 5.85 eV for ZnS, ZnSe and ZnTe, respectively. These peaks are primarily due to direct transition between the upper valence band and the second conduction band above the Fermi energy at L-edge. When these materials of interest are compressed, the position of all critical points cited above are shifted with an increased or decreased energy comparative to that at normal pressure. The raison lies on the enhancement of the (C–C) direct gaps and de-enhancement of other ones. Although their positions are shifted under pressure, these points still have the same type as that at zero pressure. It is worth noting, moreover, that the main or global peaks are shifted and their maximum has been increased under pressure in going from ZnS to ZnTe. Macroscopic dielectric constants e1(0) is given by the low energy limit of e1(x). Note that we do not include phonon contributions to the dielectric screening, and e1(0) corresponds to the static optical dielectric constant e/. Our calculated optical dielectric constants e/ are listed in Table 6 as well as other available theoretical and experimental ones. Fig. 4 shows the calculated results for the pressure dependence of the dielectric constant e/ for ZnS, ZnSe and ZnTe in the LDA approximation obtained from relation (8). As can be seen the increase of the dielectric constants (refractive index) with pressure is practically linear in all the compounds. The pressure derivative of the refractive index n of ZnS, ZnSe and ZnTe are determined by a polynomial fit. Our calculated pressure and volume

36

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

Fig. 3. Calculated imaginary part with and without pressure of the dielectric function of ZnS, ZnSe and ZnTe.

Table 6 The calculated dielectric constants e/, pressure and volume coefficients of refractive index for ZnS, ZnSe and ZnTe 1 dn n0 dp

e/ ZnS ZnSe ZnTe a b

Present

TD-DFTa

LCAOb

Expt.a

6.18 7.40 9.02

5.71 6.74 7.99

5.63 5.56 5.24

5.2 5.9 7.3

(104 GPa1)

11.3 10.3 9.28

v0 dn n0 dv

0.160 0.129 0.102

Ref. [93] and references therein. Ref. [94].

coefficients of refractive index are also listed in Table 6. From this table, we can notice that increase of pressure

or the decrease of the size of the atom (from Te to S), lead to the decrease of the refractive index and consequently to

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

Fig. 4. Pressure dependence of e/ of ZnS, ZnSe and ZnTe.

the decrease of the microscopic polarisability which has the same unity and characteristic of a volume. To our knowledge, there are no experimental or theoretical results for the variation under pressure of the refractive indices available to us for these compounds. We can consider the present results of the linear pressure and volume coefficients as a prediction study for these compounds, hopping that our present work will stimulate some other works on these materials. 4. Conclusions In our calculation we have used the local density approximation LDA within the FPLAPW + lo method to study the structural, electronic and optical properties at normal and under pressure of ZnS, ZnSe and ZnTe in zinc-blende structure. Our results for the band structure and DOS, show that these compounds have similar structures with direct energy band gap. These energy band gaps are decreases when we move from ZnS to ZnSe to ZnTe. This attributed to the fact that the bandwidth of the conduction bands increases on going from ZnS to ZnSe to ZnTe. The structural parameters, elastic constants and static dielectric constant are compared with previous theoretical results and experimental data. In comparison with the experimental data we find that the lattice parameters are underestimated whereas the bulk modulus are overestimated. That is attributed to our use of LDA. The critical point structure of the frequency dependent complex dielectric function was investigated and analyzed to identify the optical transitions. The pressure dependence of the elastic and optical constants is also investigated. To our knowledge, there are no earlier studies of the effect of pressure on elastic properties and imaginary part of the dielectric constant, so our calculations can be used to cover this lack of data for these compounds. References [1] M.A. Hasse, J. Qui, J.M. De Puydt, H. Cheng, Appl. Phys. Lett. 59 (1991) 1272. [2] H. Kinto, M. Yagi, K. Tanigashira, T. Yamada, H. Uchiki, S. Iida, J. Cryst. Growth 117 (1992) 348.

37

[3] M.C. Tamargo, M.J.S.P. Brasil, R.E. Nahory, R.J. Martin, A.L. Weaver, H.L. Gilchrist, Semicond. Sci. Technol. 6 (1991) 6. [4] S.K. Chang, D. Lee, H. Nakata, A.V. Nurmikko, L.A. Kolodziejski, R.L. Gunshor, J. Appl. Phys. 62 (1987) 4835. [5] J.K. Furdyna, J. Appl. Phys. 53 (1982) 7635. [6] R.B. Bylsma, W.M. Becker, J. Kossut, U. Debska, D. Yoder-Short, Phys. Rev. B 33 (1986) 8207. [7] G.A. Prinz, B.T. Jonker, J.J. Krebs, J.M. Ferrari, F. Kovanic, Appl. Phys. Lett. 48 (1986) 1756. [8] G.A. Prinz, B.T. Jonker, J.J. Krebs, Bull. Am. Phys. Soc. 33 (1988) 562. [9] S. Adachi, T. Taguchi, Phys. Rev. B 43 (1991) 9569. [10] Y.D. Kim, S.L. Cooper, M.V. Klein, Appl. Phys. Lett. 62 (1993) 2387. [11] R. Dahmani, L. Salamanca-Riba, N.V. Nguyen, D. ChandlerHorowitz, B.T. Jonker, J. Appl. Phys. 76 (1994) 514. [12] C.C. Kim, S. Sivananthan, Phys. Rev. B 53 (1996) 1475. [13] S. Adachi, K. Sato, Jpn. J. Appl. Phys, Part I 31 (1992) 3907. [14] Y.D. Kim, S.G. Choi, M.V. Klein, S.D. Yoo, D.E. Aspnes, S.H. Xin, J.K. Furdyna, Appl. Phys. Lett. 70 (1997) 610. [15] D. Ro¨nnow, M. Cardona, L.F. Lastras-Martı´nez, Phys. Rev. B 59 (1999) 5581. [16] G.A. Samara, Phys. Rev. B 27 (1983) 3494. [17] J.P. Walter, M.L. Cohen, Y. Petroff, M. Balkanski, Phys. Rev. B 1 (1969) 2661. [18] S. Ves, U. Schwarz, N.E. Christensen, K. Syassen, M. Cardona, Phys. Rev. B 42 (1990) 9113. [19] H.P. Wagner, M. Ku¨hnelt, W. Langbein, J.M. Hvam, Phys. Rev. B 58 (1998) 10494. [20] J.L. Freeouff, Phys. Rev. B 7 (1973) 3810. [21] S.R. Tiong, M. Hiramatsu, Y. Matsushima, E. Ito, Jpn. J. Appl. Phys. 28 (1989) 291. [22] S.C. Yu, I.L. Spain, E.F. Skelton, Solid State Commun. 25 (1978) 49. [23] R. Gangadharan, V. Jayalakshmi, J. Kalaiselvi, S. Mohan, R. Murugan, B. Palanivel, J. Alloy. Compd. 5 (2003) 22. [24] S.B. Qadri, E.F. Skelton, A.D. Dinsmore, J.Z. Hu, W.J. Kim, C. Nelson, B.R. Ratna, Phys. Rev. B 89 (2001) 115. [25] M. Causa, R. Dovesi, C. Pisani, C. Roetti, Phys. Rev. B 33 (1986) 1308. [26] A. Qteish, J. Phys.: Condens. Matter 12 (2000) 5639. [27] V.I. Smelyansky, J.S. Tse, Phys. Rev. B 52 (1995) 4658. [28] S. Ves, K. Stro¨ssner, N.E. Christensen, C.K. Kim, M. Cardona, Solid State Commun. 56 (1985) 479. [29] K. Stro¨ssner, S. Ves, C.K. Kim, M. Cardona, Solid State Commun. 61 (1987) 2755. [30] H. Karzel, W. Potzel, M. Ko¨fferlein, W. Schiessl, M. Steier, U. Hiller, G.M. Kalvius, D.W. Mitchell, T.P. Das, P. Blaha, K. Schwarz, M.P. Pasternak, Phys. Rev. B 53 (1996) 11425. [31] J.R. Cheliokowsky, M.L. Cohen, Phys. Rev. B 14 (1976) 556. [32] C.S. Wang, B.M. Klein, Phys. Rev. B 24 (1981) 3393. [33] R.W. Jansen, O.F. Sankey, Phys. Rev. B 36 (1987) 6520. [34] N.E. Christensen, O.B. Christensen, Phys. Rev. B 33 (1986) 4739. [35] E.D. Ghahramani, D.J. Moss, J.E. Sipe, Phys. Rev. B 43 (1991) 9700. [36] J.E. Jaffe, R. Pandey, M.J. Seel, Phys. Rev. B 47 (1993) 6299. [37] G.-D. Lee, M.H. Lee, J. Ihm, Phys. Rev. B 52 (1995) 1459. [38] A.E. Merad, M.B. Kanoun, J. Cibert, H. Aourag, G. Merad, Phys. Lett. A 315 (2003) 143. [39] Z.Q. Li, W. Po¨tz, Phys. Rev. B 45 (1992) 2109. [40] D. Ro¨nnow, N.E. Christensen, M. Cardona, Phys. Rev. B 59 (1999) 5575. [41] M. Oshikiri, F. Aryasetiawan, Phys. Rev. B 60 (1999) 10754. [42] M. Rabah, B. Abbar, Y. Al-Douri, B. Bouhafs, B. Sahraoui, Mater. Sci. Eng. B 100 (2003) 163. [43] K. Kassali, N. Bouarissa, Mater. Chem. Phys. 76 (2002) 255. [44] A. Boukortt, B. Abbar, H. Abid, M. Sehil, Z. Bensaad, B. Soudini, Mater. Chem. Phys. 82 (2003) 991. [45] A. Benmakhlouf, A. Bechiri, N. Bouarissa, Solid State Electron. 47 (2003) 1335.

38

R. Khenata et al. / Computational Materials Science 38 (2006) 29–38

[46] Y. Al- Douri, R. Khenata, Z. Chelahi, M. Driz, H. Aourag, J. Appl. Phys. 94 (2003) 4502. [47] R.K. Singh, S. Singh, Phys. Status Solidi (b) 140 (1987) 407. [48] S.-G. Shen, J. Phys.: Condens. Matter 6 (1994) 8733. [49] R.A. Casali, N.E. Christensen, Solid State Commun. 108 (1998) 793. [50] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An augmented plane wave plus local orbitals program for calculating crystal properties, Vienna University of Technology, Austria, 2001. [51] G.K.H. Madsen, P. Plaha, K. Schwarz, E. Sjo¨stedt, L. Nordstro¨m, Phys. Rev. B 64 (2001) 195134. [52] E. Sjo¨stedt, L. Nordstro¨m, D.J. Singh, Solid State Commun. 114 (2000) 15. [53] K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. [54] P. Hohenberg, W. Kohn, Phys. Rev. B 36 (1964) 864. [55] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [56] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [57] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [58] J.C. Jamieson, H.H. Demarest Jr., J. Phys. Chem. Solids 41 (1980) 903. [59] H.W. Gust, J. Appl. Phys. 53 (1982) 4843. [60] B.H. Lee, J. Appl. Phys. 41 (1970) 2988. [61] M.I. Mc Mahon, R.J. Nelmes, D.R. Allan, S.A. Belmonte, T. Bovomratanaraks, Phys. Rev. Lett. 80 (1998) 5564. [62] C.M.I. Okoye, Physica B 337 (2003) 1. [63] O. Madelung (Ed.), Numerical Data and Functional Relationship in Science and Technology, Landolt_ Bo¨rnstein, New Series Group III, vol. 17, Springer-Verlag, Berlin, 1982. [64] L. Kleinman, Phys. Rev. 128 (1962) 2614. [65] M.B. Kanoun, A.E. Merad, J. Cibert, H. Aourag, G. Merad, J. Alloys Compd. 366 (2004) 86. [66] A.E. Merad, H. Aourag, B. Khalifa, C. Mathieu, G. Merad, Superlattice Microstruct. 30 (2001) 241. [67] D. Berlincourt, H. Jaffe, R.L. Shiozawa, Phys. Rev. 129 (1963) 1009. [68] B. Hennion, F. Moussa, G. Pepy, K. Kunc, Phys. Lett. A 36 (1971) 376. [69] S. Doyen-Lang, O. Pages, L. Lang, G. Hugel, Phys. Status Solidi (b) 229 (2002) 563. [70] A. Polian, M. Grimsditch, Phys. Rev. B 60 (1999) 1468.

[71] M.J. Harrera-Cabrera, P. Rodrı´guez-Herna´ndez, A. Mun˜oz, Phys. Status Solidi (b) 223 (2001) 411. [72] S.-H. Wei, A. Zunger, Phys. Rev. B 53 (1996) R10457. [73] D. Rached, M. Rabah, N. Benkhettou, R. Khenata, B. Soudini,Y. AlDouri, H. Baltache, Comput. Mater. Sci., in press, doi:10.1016/ j.commatsci.2005.08.005. [74] F. Drief, M. Tadjer, D. Mesri, H. Aourag, Catal. Today 89 (2004) 343. [75] S.N. Rashkeev, W.R.L. Lambrecht, Phys. Rev. B 63 (2001) 165212. [76] G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74 (2) (2002) 601. [77] R.A. Pollak, L. Ley, S.P. Kowalczyk, D.A. Shirley, J. Joannopoulos, D.J. Chadi, M.L. Cohen, Phys. Rev. Lett. 29 (1973) 1103; W.D. Grobman, D.E. Eastman, Phys. Rev. Lett. 29 (1972) 1508. [78] H. Venghaus, Phys. Rev. B 19 (1979) 3071. [79] M. Cardona, J. Appl. Phys. Suppl. 32 (1961) 2151. [80] R.G. Lempert, K.C. Hass, H. Ehrenreich, Phys. Rev. B 36 (1987) 1111. [81] S.I. Kurganskii, O.V. Farberovich, E.P. Domashevskaya, Fiz. Tekh. Poluprovodn. 14 (1980) 1315 [Sov. Phys. Semicond. 14 (1980) 775]; 14 (1980) 1412. [82] M.Z. Huang, W.Y. Ching, J. Phys. Chem. Solids 46 (1985) 977. [83] S. Fahy, K.J. Chang, S.G. Louis, M.L. Cohen, Phys. Rev. B 35 (1987). [84] B. Bouhafs, H. Aourag, M. Certier, J. Phys.: Condens. Matter 12 (2000) 5655. [85] R. Khenata, B. Daoudi, M. Sahnoun, H. Baltache, M. Re´rat, A.H. Reshak, B. Bouhafs, H. Abid, M. Driz, Eur. Phys. J. B 47 (2005) 63. [86] D.L. Comphausen, G.A.N. Conel, W. Paul, Phys. Rev. Lett. 26 (1971) 184. [87] K.J. Chang, S. Froyen, M.L. Cohen, Solid State Commun. 50 (1984) 105. [88] R. Khenata, H. Baltache, M. Sahnoun, M. Driz, M. Re´rat, B. Abbar, Physica B 336 (2003) 331. [89] K. Reimann, M. Haselhoff, St. Rubenacke, M. Steube, Phys. Status Solidi (b) 198 (1996) 71. [90] B. Gil, D.J. Dunstan, Semicond. Sci. Technol. 6 (1991) 428. [91] M. Lidner, G.F. Scho¨tz, P. Link, H.P. Wagner, W. Kuba, W. Gebhanlt, J. Phys.: Condens. Matter 4 (1992) 6401. [92] H. Ehrenreich, M.L. Cohen, Phys. Rev. 115 (1959) 786. [93] F. Kootstra, P.L. de Boeij, J.C. Snijders, Phys. Rev. B 62 (2000) 7071. [94] M.Z. Huang, W.Y. Ching, Phys. Rev. B 47 (1993) 9449.