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The structural and the electronic properties of the ternary Sr
_{x}Ca
_{1-x}S, Ba
_{x}Ca
_{1-x}S and Ba
_{x}Sr
_{1-x}S alloys have been calculated using the full-potential linear muffin-tin-orbital (FP-LMTO) method based on density functional theory, within both local density approximation (LDA) and generalized gradient approximation (GGA). The calculated equilibrium lattice constants and bulk modulus are compared with previous results. The concentration dependence of the electronic band structure and the direct and indirect band gaps are investigated. Using the approach of Zunger and co-workers, the microscopic origins of the band gap bowing are investigated also. A reason is found from the comparison of our results with other theoretical calculations.

The II-VI compound semiconductors (AY: A = Ca, Sr, Ba; Y = S) have recently received considerable interest from the blue to the near-ultraviolet spectral region of both experimental and theoretical points of view [

The calculations reported here were carried out using the ab-initio full-potential linear muffin-tin orbital (FPLMTO) method [13-16] as implemented in the Lmtart code [_{max }= 6. The integrals over the Brillouin zone are performed up to 35 special k-points for binary compounds and 27 special k-points for the alloys in the ireducible Brillouin zone (IBZ), using the Blöchl’s modified tetrahedron method [^{–5} Ry. In order to avoid the overlap of atomic spheres, the MTS radius for each atomic position is taken to be different for each composition. We point out that the use of the full-potential calculation ensures that the calculation is not completely independent of the choice of sphere radii. Both plane waves cut-off are varied to ensure the total energy convergence. The values of the sphere radii (MTS) number of plane waves (NPLW), used in our calculation are summarized in

The structural properties of CaS, SrS and BaS binary compounds and their Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S, Ba_{x}Sr_{1-x}S alloys in the cubic structure by means of the full-potential LMTO method are calculated. As for the ternary semiconductor alloys with the type of B_{x}A_{1-x}C, we have started our FP-LMTO calculations of the structural properties with the rocksalt structure and let the calculation

forces move the atoms to their equilibrium positions. We have chosen the basic cubic cell as the unit cell. In the unit cell, there are four C anions, three A and one B, two A and two B, one A and three B cations, for x = 0.25, 0.50 and 0.75, respectively. For the considered structures, we have performed the structural optimization by calculating the total energies for different volumes around the equilibrium cell volume V_{0} of CaS, SrS and BaS binary compounds and their alloys. The calculated total energies are fitted to the Murnaghan’s equation of state [

atomic radii of Ca, Sr, and Ba: R(Ca) = 1.80 Å, R(Sr) = 2.00 Å, R(Ba) = 2.15 Å; i.e. the lattice constant increases with increasing atomic size of the cation element. The bulk modulus value for CaS is larger than those of SrS and BaS; B (CaS) > B (SrS) > B (BaS); i.e. in inverse sequence to a_{0 }in agreement with the well-known relationship between B and the lattice constants:, where is the unit cell volume. Furthermore, the values of the calculated bulk modulus using both approximations decrease in going from CaS, to SrS, and to BaS suggesting that the compressibility increases from CaS, to SrS, and to BaS. Usually, in the treatment of alloys where the experimental data are rare, it is assumed that the atoms are located at the ideal lattice sites and the lattice constant varies linearly with composition x according to the Vegard’s law [

where a_{AC} and a_{BC} are the equilibrium lattice constants of the binary compounds AC and BC respectively, a(A_{x} B_{1-x }C) is the alloy lattice constant. However, variation of Vegard’s law has been observed in semiconductor alloys both experimentally [

where the quadratic term b is the bowing parameter. Figures 1(a)-(c) and 2(a)-(c) show the variation of the calculated equilibrium lattice constant and bulk modulus as a function of concentrations x for Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys. The obtained results for the composition dependence of the calculated equilibrium lattice parameter exhibit an agreement with Vegard’s law [_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S, Ba_{x}Sr_{1-x}S alloys increases. Oppositely, one can see from

The calculated band structure energies for binary compounds as well as their investigated alloys using FPLMTO within both LDA and GGA schemes yields indirect band gap (Γ→X) for the binary compounds of CaS, SrS and BaS. When the composition x varies, the valence band maximum (VBM) and the conduction band minimum (CBM) occur at the Γ-point, resulting in direct band gap (Γ→Γ) for the studied alloys. The calculated direct band gap (Γ→Γ) values for Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys are listed in

best of our knowledge, there are no theoretical or experimental data on the energy band gaps for x = 0.25, 0.5 and 0.75 available in literature to make a meaningful comparison. It is worthy to mention here that both LDA and GGA do not reproduce a perfect agreement with the experimental results because they do not take into account the quasi particle self energy correctly [

structure can’t be used directly for comparison with experiment. We also mention, it is far to say that the experimental data are well reproduced by calculations. On For this reason for this that the theory holds applies pertains at T = 0 K, whereas the experimental measurements were performed at room temperature where the lattice vibrations are present.

The variation of the composition (x) versus the direct (Γ→Γ) and indirect (Γ→X) band gaps with both LDA and GGA approximations is shown in Figures 3(a)-(c). According to this figure we notice that for concentrations (x) ranging from x = 0.224 to 0.769 LDA, x = 0.216 to 0.784 GGA the alloy exhibits a direct band gap (Γ→Γ) for both approximations Sr_{x}Ca_{1-x}S, from x = 0.26 to 0.73 LDA, x = 0.25 to 0.73 GGA the alloy exhibits a direct band gap (Γ→Γ) for both approximations Ba_{x}Sr_{1-x}S, from x = 0.27 to 0.73 LDA, x = 0.25 to 0.73 GGA the alloy exhibits a direct band gap (Γ→Γ) for both approximations Ba_{x}Sr_{1-x}S,. It is also seen that the direct and indirect band gaps show a nonlinear behaviour with increasing Strontium, Calcium and Baryum concentrations providing a positive and negative band gap bowing for the direct (Γ→Γ) and indirect (Γ→X) band gaps, respectively. Indeed, it is a general trend to describe the band gap of A_{x}B_{1-x}C alloys in terms of the binary compounds energy gaps E_{AC} and E_{BC} by the following semi-empirical formula:

where E_{AC} and E_{BC} corresponds to the energy gap of SrS and CaS for the Sr_{x}Ca_{1-x}S alloy. From _{g }of LDA is slightly larger than those of GGA.

The calculated band gap versus concentration was fitted by a polynomial equation. The results are shown in

In order to understand the physical origins of bowing parameters in A_{x}B_{1-x}C alloy, we follow the procedure of Bernard and Zunger [

where and are the equilibrium lattice constants of the binary compounds and is the equilibrium lattice constant of the alloy with the average composition x.

Equations (5)-(7) are decomposed into three steps:

The first step attributes the volume deformation (VD) effect on the bowing. The corresponding contributions b_{VD} to the bowing parameter represents the relative response of the band structure of the binary compounds AC and BC to hydrostatic pressure that arises from the change of their individual equilibrium lattice constants to the alloy value a = a(x). The second contribution, the charge exchange (CE) contribution b_{CE}, reflects a charge transfer effect which is due to different (averaged) bonding behaviour of at the lattice constant a. The final step, the structural relaxation (SR), reflects changes in passing from the unrelaxed to the relaxed alloy by b_{SR}. Consequently, the total bowing parameter is defined as.

The general representation of the composition-dependent band gap of the alloys in terms of binary compounds energy gaps of the, E_{AC}(a_{AC}) and E_{BC}(a_{BC}), and the total bowing parameter b is

This allows a division of the total bowing b into three contributions according to:

All these energy gaps mentioned in (13)-(15) have been calculated for the indicated atomic structures and lattice constants. _{x}Ca_{1-x}S for three different molar fractions (x = 0.25, 0.5 and 0.75). The LDA (GGA) calculated band gap bowing (b) is found to be equal to –0.3185 (–0.7746) eV, 0.078 (0.0214) eV and 4.0222 (3.6608) eV for x = 0.25, 0.5 and 0.75 respectively. From _{x}Ca_{1-x}S for three different molar fractions (x = 0.25, 0.5 and 0.75). The LDA (GGA) calculated band gap bowing (b) is found to be equal to 2.3029 (1.0991) eV, 0.6705 (0.6074) eV and –1.0469 (–1.947) eV for x = 0.25, 0.5 and 0.75 respectively. From _{x}Sr_{1-x}S for three different molar fractions (x = 0.25, 0.5 and 0.75). The LDA (GGA) calculated band gap bowing (b) is found to be equal to 0.8721 (0.3562) eV, 0.438 (0.1503) eV and –0.0402 (–0.0922) eV for x = 0.25, 0.5 and 0.75 respectively. From _{x}Ca_{1-x}S shows the variation of the band gap bowing versus concentration. It is shown that the optical bowing remains linear (in going) from 0.25 to 0.75 for LDA and for GGA it remains linear with a tow slope varying slowly (in going) from 0.25 to 0.5, and beyond 0.5 it decreases rapidly. _{x}Sr_{1-x}S shows the variation of the band gap bowing versus concentration. It is shown that the optical bowing remains linear, it monotonicly decreases from 0.25 to 0.75 both for LAD, GGA with two slopes. To the best of our knowledge, there are no theoretical or experimental data on the band gap bowing to compare with our predicted results.

Knowledge of the electron and hole effective mass values is indispensable for the understanding of transport phenomena, exciton effects and electron-hole in semiconductors. Excitonic properties are of great interest for semiconductor materials; therefore, it is worthwhile to estimate the electron and hole effective mass values for Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys at different compositions. Experimentally, the effective masses are usually determined by cyclotron resonance, electro reflectance measurements or from the analysis of transport data or transport measurements [

band maxima (VBM) for the composition ranging from 0 to 1.0 for Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys. The electron effective mass value are obtained from the curvature of the conduction band near the X-point for SrS ,BaS and CaS and near the Γ-point at the CBM for X = 0.25, 0.5 and 0.75. The hole effective masse value is calculated from the curvature near the Γ-point at the VBM for all concentration. The calculated electron and hole effective mass values for for Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys are mentioned in Tables 5(a)-(c).

The (FP-LMTO) method is used to calculate the structural and electronic properties of the rocksalt SrS, CaS, BaS and their Sr_{x}Ca_{1-x}S, Ba_{x}Ca_{1-x}S and Ba_{x}Sr_{1-x}S alloys.From this, we may conclude that:

1) The variation of the structural parameters with Sr, Ca and Ba concentrations obeys Vegard’s law.

2) The electronic band structure shows a non-linear variation of the fundamental band gaps versus Sr, Ca and Ba concentrations. The deviation from linearity is characterized by a calculated optical bowing parameter. The main contribution to the optical bowing is essentially due to volume deformation effects.

3) The effective masses indicate that the charge carriers in these alloys should be dominated by electrons.