The calculations were firstly carried out to determine the structural properties of ZB binary compounds ZnS and ZnO and alloys. To model the ZB structure alloys, we applied a 8-atom supercell. For the considered structures and at different oxygen concentrations x (x = 0, 0.25, 0.50, 0.75, 1), the structural properties were obtained by a minimization of total energy as a function of the volume for ZnS, ZnO and in the ZB structure. The bulk modulus and their pressure derivatives were obtained by a non-linear fit of the total energy versus volume according to the Birche-Murnaghan’s equation of state [9]. In Table 2, we summarize the calculated modulus and their pressure derivatives) of ZnS, ZnO compounds and alloys. Considering the general trend that the GGA usually overestimates the lattice parameters while LDA is expected to underestimates them [10], It is clear that our results are in reasonable agreement with experimental values and theoretical results. Usually, in the treatment of alloys, 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 so-called Vegard’s law [30].

Table 1. The plane wave number PW, energy cuttof (in Ry) and the muffin-tin radius (RMT) (in a.u.) used in calculation for binary ZnS and ZnO and their alloy in zinc blende (ZB) structure.

^aRef. [16], ^bRef. [20].

Table 2. Lattice constants a, bulk modulus B, and pressure derivative of bulk modulus B, for ZB ZnS, ZnO and O_xS_1−xZn solid solutions.

^aRef. [11], ^bRef. [12], ^cRef. [13], ^dRef. [14], ^eRef. [15], ^fRef. [16], ^gRef. [17], ^hRef. [18], ⁱRef. [19], ^jRef. [20], ^kRef. [21], ^lRef. [22], ^mRef. [23], ⁿRef. [24].

where a_AC and a_BC are the equilibrium lattice constants of the binary compounds AC and BC Hence, the lattice constant can be written as:

where the quadratic term b is the bowing parameter.

Figures 1 and 2 show the variation of the calculated equilibrium lattice constant and the bulk modulus versus concentration for alloy. A slight large deviation from Vegard’s law [30] is clearly visible for the alloy. The bowing parameter determined by LDA, are obtained by fitting the calculated values respectively, is the alloy lattice constant. With a polynomial function we found an upward and downward bowing parameter equal to −0.2359 Å and 142.9979 GPa for lattice and bulk modulus. The GGA results are −0.2887 Å and 142.9981 GPa. The physical origin of this slight large deviation could be mainly due to the large size difference between the O and S atom and the mismatch of the lattice constants of ZnS and ZnO compound.

The calculations of the electronic band structure properties, magnitude of band-gap were carried out for ZnS, ZnO and in ZB structure at the equilibrium calculated lattice constants. The band-gaps calculated using the FP-LMTO method for ZB ZnS, ZnO and are listed in Table 3 for the high-symmetry

points Γ and X in the Brillouin zone. All energies are with reference to the top of the valence band at Γ point. The results show that ZnS and ZnO compound is a direct-gap semiconductor with the minimum of conduction band at Γ point.The calculated GGA (LDA) energy gaps of ZnS and ZnO Eg are 1.97(2.12) eV and 0.69(0.79) eV, respectively, which are in good agreement with the theoretical values as listed in Table 3. Focusing now on the electronic properties of the alloy system we obtained the direct band-gap at Γ point. Figure 3 presents the variation of the direct and indirect band gap energies as functions of the composition x for the ternary alloys. We note direct (Γ-Γ) and indirect (Γ-X) band gap not intersect because the computed band structures of the alloys using both LDA and GGA schemes indicate a direct band gap at various concentrations.

Indeed it is a general trend to describe the bandgap of an alloy A_xB_1−xC in terms of the pure compound energy gap E_AC and E_BC by the sem-empirical formula:

where E_AC and E_BC corresponds to the gap of the ZnO and ZnS for the O_xS_1−xZn alloy. 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 better understand the physical origins of the large and composition-dependent bowing in alloy alloys, we follow the procedure of Bernard and Zunger [31] and decompose the total bowing parameter b into physically distinct contributions. The overall bowing

Table 3. Direct (Γ–Γ) and indirect (Γ–X) band gaps of ZnS and ZnO and their alloy at equilibrium volume (The energy is given in eV).

^aRef. [25], ^bRef. [14], ^cRef. [15], ^dRef. [26], ^eRef. [16], ^fRef. [13], ^gRef. [18], ^hRef. [27], ⁱRef. [19], ^jRef. [28].

coefficient at a given average composition x measures the change in band gap according to the formal reaction.

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

We decompose reaction into three step:

The first contribution, the volume deformation (b_VD) represents the relative response of the band structure of the binary compounds AC and BC to hydrostatic pressure. The second contribution, the charge-exchange (CE) contribution b_CE, reflects a charge-transfer effect that is due to the different (averaged) bonding behavior at the lattice constant a. The final step measures by b_SR, changes due to the structural relaxation (SR) in passing from the unrelaxed to the relaxed alloy. Consequently, the total gap bowing parameter is defined as

The general representation of the composition-dependent band gap of the alloys in terms of binary compounds gaps of the, and, and the total bowing parameter b is

where, and represents respectively the volume deformation (VD) effect, the charge exchange (CE) contribution and the structural relaxation (SR) of the alloy according to the following expressions:

The addition of the three contributions (11), (12), and (13) leads to the total bowing parameter b. The computed bowing coefficients b together with the three different contributions for the band gaps as a function of the molar fraction (x = 0.25, 0.5 and 0.75) are shown in the Table 4. The calculated band gap bowing parameter exhibits a strong composition dependence, as calculated within the GGA and LDA calculations, which show a weakly composition dependent bowing parameter. The variation of the band gap bowing versus concentration shown in Figure 4. The bowing remains linear and Decreases rapidly from x = 0.25 to x = 0.75 in both approaches (GGA and LDA). The calculated GGA and LDA gap bowing for alloy ranges from 0.69 eV, 1.25 eV (x = 0.25) to −0.59 eV, −0.55 eV (x = 0.75), a negligible volume deformation term was registered for system can be correlated to the large mismatch of the lattice constants of the corresponding binary compounds  between ZnS and ZnO. The charge transfer contribution b_CE has been found greater than. This contribution is due to the different electronegativities of the O and S or Zn atoms. Indeed, scales with the electronegativity mismatch. The contribution of the structural relaxation is small, the band gap bowing is due essentially charge exchange effect.

Table 4. Decomposition of optical bowing into VD, CE and SR contributions compared with other prediction.

Table 5. Electron () light hole () and heavy hole () effective masses (in units of free electron mass) at the point of the Brillouin zone of the ternary alloys under investigation compared with the available experimental and theoretical predictions.

^aRef. [29], ^bRef. [14], ^cRef. [13],^dRef. [27], ^eRef. [15].

It is also interesting to discuss at the end of the band structure study the effective masses of electrons and holes, which are important for the excitonic compounds. We have calculated the effective masses of electrons and holes using both LDA and GGA schemes are mentioned in Table 5. A theoretical effective mass in general turns out to be a tensor with nine components. However, for a very idealized simple case where E(k is a parabola at k = 0 (high symmetry point Γ) the effective mass becomes a scalar.