The structural properties of CaS, SrS and BaS binary compounds and their Sr_xCa_1-xS, Ba_xCa_1-xS, Ba_xSr_1-xS 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_xA_1-xC, 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 V₀ of CaS, SrS and BaS binary compounds and their alloys. The calculated total energies are fitted to the Murnaghan’s equation of state [21] to determine the ground state properties such as the equilibrium lattice constant a and the bulk modulus B. The calculated equilibrium parameters (a and B) are given in Table 2 that also contains results of previous calculations as well as the available experimental data. As a whole, our calculated structural parameters are in good agreement with those obtained by first-principles methods within different approximations. The calculated lattice parameter for CaS, SrS and BaS binary compound is in good agreement with other data, which ensures the reliability of the present first-principles computations. The computed lattice parameter slightly underestimates and overestimates the other data by using LDA and GGA, respectively, which are consistent with the general trend of these approximations. As it can be seen, the calculated lattice parameter for BaS is larger than those of SrS and CaS. As the anion atom is the same in the compounds, this result can be easily explained by considering the

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 a₀ 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 [22].

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 [23] and theoretically [24-28]. Hence, the lattice constant can be written as:

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_xCa_1-xS, Ba_xCa_1-xS and Ba_xSr_1-xS alloys. The obtained results for the composition dependence of the calculated equilibrium lattice parameter exhibit an agreement with Vegard’s law [22]. In going from CaS, SrS and BaS; when the Sr and Bacontent increases, the values of the lattice parameters of the Sr_xCa_1-xS, Ba_xCa_1-xS, Ba_xSr_1-xS alloys increases. Oppositely, one can see from Figure 2 that the value of the bulk modulus increases with the decrease of Sr and Ba concentrations. The bowing parameters are determined by a polynomial fit. From Tables 3(a)-(c), both approxiations follow the tendency demonstrated by both experimental measurement and theoretical calculations.

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_xCa_1-xS, Ba_xCa_1-xS and Ba_xSr_1-xS alloys are listed in Table 4, along with the available theoretical and experimental results. The computed direct band gaps for the binary compounds are in good agreements with other results. For SrS and CaS compounds, the calculated band gap is smaller than the experimental value but for the BaS is the opposite. To the

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 [32] that make them not sufficiently flexible to accurately reproduce both the exchange-correlation energy and its charge derivative. It is important to note that the density functional formalism is limited [30] and the derived band

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_xCa_1-xS, 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_xSr_1-xS, 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_xSr_1-xS,. 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_xB_1-xC 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_xCa_1-xS alloy. From Table 4, it is clear that E_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 Figure 3 and are summarized as follows:

In order to understand the physical origins of bowing parameters in A_xB_1-xC alloy, we follow the procedure of Bernard and Zunger [31], and decompose the total bowing parameter b into physically distinct contributions. The overall bowing coefficient at a given average composition x measures the change in the band gap according to the formal reaction.

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. Table 3(a) shows the calculated optical band gap bowing b for Sr_xCa_1-xS 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 Table 3(a) we notice that the structural relaxation effect is negligible for the studied compositions x = 0.25, 0.50 and 0.75 and the main contribution to the band gap bowing is essentially due to the charge exchange (CE) contribution, for x = 0.75 the band gap bowing is due to nearly equal contribution of the volume deformation (SR) effects. Table 3(b) shows the calculated optical band gap bowing b for Ba_xCa_1-xS 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 Table 3(b) we notice that the structural relaxation effect is negligible for the studied compositions x = 0.25, 0.50 and 0.75 and the main contribution to the band gap bowing is essentially due to the charge exchange (CE) contribution, for x = 0.25 and 0.5. For x = 0.75 the band gap bowing is due to nearly equal contribution of the volume deformation (VD) effects. Table 3(c) shows the calculated optical band gap bowing b for Ba_xSr_1-xS 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 Table 3(c) we notice that the structural relaxation effect is negligible for the studied compositions x = 0.25, 0.50 and 0.75 and the main contribution to the band gap bowing is essentially due to the charge exchange (CE) contribution for x = 0.25 and 0.5. For x = 0.75 the band gap bowing is due to nearly equal contribution of the volume deformation (SR) effects. Figure 4(a) shows the variation of the band gap bowing versus concentration. It is shown that the optical bowing remains linear and increases slowly in going from 0.25 to 0.75, for LDA and for GGA it remains linear varies rapidly in going from 0.25 to 0.5, and beyond 0.5 it increases rapidly too, which confirms the evaluation of the band gap according to the concentration,. Figure 4(b) for Ba_xCa_1-xS 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. Figure 4(c) for Ba_xSr_1-xS 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_xCa_1-xS, Ba_xCa_1-xS and Ba_xSr_1-xS 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 [32]. Theoretically, the effective masses can be estimated from the energy band curvatures. Generally, the effective mass is a tensor with nine components, however for the much idealized simple case, where the E-k diagram can be fitted by a parabola, the effective mass becomes a scalar at high symmetry point in Brillouin zone. We have computed the electron effective mass at the conduction band minima (CBM) and the hole effective mass at the valence

band maxima (VBM) for the composition ranging from 0 to 1.0 for Sr_xCa_1-xS, Ba_xCa_1-xS and Ba_xSr_1-xS 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_xCa_1-xS, Ba_xCa_1-xS and Ba_xSr_1-xS alloys are mentioned in Tables 5(a)-(c).