3.1. Structural properties

Very interesting applications in semiconductor devices with desired band gaps were mentioned in the literature [²⁶,²⁷] on quaternary (II-VI) alloys having the general chemical formulas (A _1−x B _x C _y D _1−y ) where AC, AD, BC, BD are the binary compounds and (x,y) are the concentrations. The aim of the present work is related to predicting the structural, electronic, and optical properties of Zn _1−x Cd _x Se _y Te _1−y alloys with a number of Cd concentrations x (0.125 ≤ x ≤ 0.875).

For the structural properties, the lattice parameter and bulk modulus were computed using the GGA-PBEsol approximation. In addition to the LDA approach, the recent (TB-mBJ) approach has been used to calculate the electronic and optical properties. The quaternary alloy Zn _1−x Cd _x Se _y Te _1−y is bounded by four binary compounds ZnSe, ZnTe, CdSe, and CdTe. The normal structure of this binary is zinc-blende. Among 64 atom simple cubic structures, the ones that minimize the total energy with respect to the cell parameters and the atomic positions were searched. The simulated parameters have been computed as a function of cationic and anionic compositions (x,y), where x and y are the concentrations of Cd and Se elements, respectively. First, the structural properties of the binaries (ZnSe, ZnTe, CdSe, and CdTe) and the Zn _1−x Cd _x Se _y Te _1−y quaternary alloys were computed by the GGA-PBEsol approximation. In order to compute the equilibrium lattice constants (a) and bulk modulus (B) for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy, the total energy versus volume data have been calculated by fitting the total energy versus volume data into the Murnaghan’s equation of states (EOS) [²⁸]. The lattice constant of the quaternary A _x B _1−x C _y D _1−y alloy is a function of the compositions (x,y) and can be approximated by the Vegard’s law [²] given by the following relation:

where α _AC, α _AD, α _BC, and α _BD correspond to the lattice constants for pure binaries AC, AD, BC, and BD, respectively.

By using Eq. (1), the lattice constant for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy can be deduced according to the following expression:

where the binaries AC, AD, BC, and BD are CdSe, CdTe, ZnSe, and ZnTe, respectively. The lattice-matching conditions are respected by putting α(Zn _1−x Cd _x Se _y Te _1−y ) equal to α _ZnTe . Thus, using the respective experimental lattice constants of the binary compounds (α _CdSe = 6.050 Å), (α _CdTe = 6.481 Å), (α _ZnSe = 5.669 Å) and (α _ZnTe = 6.1037 Å), the corresponding concentrations for lattice matched of Zn _1−x Cd _x Se _y Te _1−y quaternary alloy can be obtained by the relation (3) below with different concentrations x(0.125 ≤ x ≤ 0.875).

We have adopted Zn _1−x Cd _x Se _y Te _1−y alloy latticematched to ZnTe with some selected concentrations (4/32 ≤ x ≤ 24/32). We have considered different concentrations along the line of lattice matching to ZnTe by assigning to the pair (x,y), the respective compositions (4/32;3/32), (8/32;7/32), (12/32;10/32), (16/32;14/32), (20/32;17/32), (24/32;21/32) and (28/32;24/32). Using the GGA-PBEsol method, the lattice constant and bulk modulus of the binaries (CdSe, CdTe, ZnSe, ZnTe), and the quaternary alloy Zn _1−x Cd _x Se _y Te _1−y were calculated for various Cd concentration (x) and exhibited in Table I.

To the best of our knowledge, there is no theoretical simulation based on the FP-LAPW method, and no experimental values of the structural properties of Zn _1−x Cd _x Se _y Te _1−y alloy for these concentrations have been presented up to now. Hence our calculations can be used to cover the lack of data for the studied alloy. It is seen from Table I that the calculated lattice constants of Zn _1−x Cd _x Se _y Te _1−y are also in good agreement with the experimental value of the ZnTe lattice constant. The variation of lattice parameter versus Cd concentration is found to be almost constant (around 6.084 Å). Figure 2 shows the calculated lattice constant as a function of the Cd concentration x (0.125 ≤ x ≤ 0.875). A marginal deviation of the lattice constant from Vegard’s law can be observed. From Fig. 2, we can notice that at given Se concentration, the lattice constant increases almost linearly with increasing Cd concentration. The optimized variation of the lattice constants α(x) was fitted by the quadratic function to calculate the total bowing parameter (b). Thus, we have obtained the following relation:

FIGURE 2 Calculated lattice constant as a function of the Cd concentration x in the Zn _1−x Cd _x Se _y Te _1−y quaternary alloys matched to ZnTe with GGA-PBEsol method. 

By referring to relation (4), we can observe a negligible bowing parameter (b = −0.0272 eV) corresponding to the quadratic term of the above equation. The contour lines of the lattice constants for Zn _1−x Cd _x Se _y Te _1−y alloy as a function of Cd and Se compositions were simulated with the GGAPBEsol approach and exhibited in Fig. 3. From this figure, it is observed that the lattice parameter of the quaternary alloy increases with increasing Cd concentration. The variation of the bulk modulus (B) versus Cd concentration (x) for the quaternary alloy Zn _1−x Cd _x Se _y Te _1−y is reported in Table I and is shown in Fig. 4. It is seen to be almost constant. This result is in agreement with the well-known relationship between B and the lattice constant (B _α /V ₀) [³¹], where V ₀ is the unit cell volume.

FIGURE 3 Contour lines of predicted lattice constant for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys matched to ZnTe with GGA-PBEsol method. 

FIGURE 4 Calculated bulk modulus as a function of the Cd concentration x in the Zn _1−x Cd _x Se _y Te _1−y quaternary alloys matched to ZnTe with GGA-PBEsol method. 

3.2. Electronic properties

This part is devoted to calculations of electronic band structure for Zn _1−x Cd _x Se _y Te _1−y quaternary alloy without relaxation using the local density approximation (LDA) [³²] and the Tran-Blaha modified Becker-Johnson (TB-mBJ) [²⁴] approaches. This simulation was performed along the high symmetry lines of the first Brillouin zone, where the maximum of the valence band was fixed as the energy level reference. Table II exhibits the calculated energy band gaps (Eg) for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy using LDA and TB-mBJ schemes for various Cd concentration x (0.125 ≤ x ≤ 0.875). To the best of our knowledge, there are no earlier studies of the bandgap for Zn _1−x Cd _x Se _y Te _1−y alloys with these concentrations; hence our calculations can be used to cover the lack of data for the studied alloys. Figure 5 shows the variation of bandgap energies Eg versus the concentration x for Zn _1−x Cd _x Se _y Te _1−y lattice-matched to ZnTe, calculated via two schemes (LDA and TB-mBJ). We remark that the variation of the bandgap energies is almost linear with a slight decrease of the curves with increasing Cd concentration (was performed to determine the dependence of the energy(x). A fitting operation to curves presented in Fig. 5 was performed to determine the dependence of the energy band gaps (Eg) versus the composition of the cadmium (x) for the Zn_1-xCd_xSe_yTe_1-y lattice matched to ZnTe. Hence, we have obtained the following relations:

FIGURE 5 Bandgap energies Eg versus the concentration x for the Zn _1−x Cd _x Se _y Te _1−y lattice-matched to ZnTe. 

From these equations, we can determine for a given composition (x) the related band gap (Eg) in the Zn_1−xCd_xSe_yTe_1−y alloy lattice matched to ZnTe. Thus, for the studied structure, we got the values (-0:1310 eV) and (-0:0456 eV) corresponding to the bowing parameter (b) for LDA and TB-mBJ schemes, respectively. As presented in Fig. 5, one notes the slight decreasing of the bandgap of Zn_1−xCd_xSe_yTe_1−y alloy with the increase of the Cd concentration (x). The computed TB-mBJ energy bandgap contour lines of Zn_1−xCd_xSe_yTe_1−y alloy versus Cd and Se concentrations are exhibited in Fig. 6. The contour lines allow the reading of values from the curve. The electronic band structures were computed for Zn_1−xCd_xSe_yTe_1−y quaternary alloy with the predicted lattice constant using the TBmBJ method. The corresponding results are shown in Figs. 7 and 8 for both quaternaries (Zn_0:875Cd_0:125Se_0:093Te_0:907; Zn_0:750Cd_0:250Se_0:218Te_0:782) and (Zn_0:500Cd_0:500Se_0:437 Te_0:563; Zn_0:250Cd_0:750Se_0:656Te_0:344) respectively. The calculations were performed along the high-symmetry lines of the first Brillouin zone, and the Fermi level has been set to zero. As shown in the figures cited above, the respective levels for the maximum valence band (VBmax) and minimum conduction band (CBmin) are located at point. Therefore, we conclude that the Zn_1−xCd_xSe_yTe_1−y quaternary is a direct bandgap alloy for the studied concentrations and is intended for optoelectronic applications.

FIGURE 6 Energy band gap contour lines for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys matched to ZnTe with TB-mBJ method. 

FIGURE 7 Band structure calculated for Zn _0.875 Cd _0.125 Se _0.093 Te _0.907 and Zn _0.750 Cd _0.250 Se _0.218 Te _0.782 with TB-mBJ method. 

FIGURE 8 Band structure calculated for Zn _0.500 Cd _0.500 Se _0.437 Te _0.563 and Zn _0.250 Cd _0.750 Se _0.656 Te _0.344 with TB-mBJ method. 

3.3. Optical properties

The rest of our work will be devoted to studying the optical properties of Zn _1−x Cd _x Se _y Te _1−y quaternary alloy. The calculations were performed at the equilibrium lattice constant of Zn _1−x Cd _x Se _y Te _1−y with x (0.250 ≤ x ≤ 0.875) using TB-mBJ approach. This method has an essential role in optical applications [³³]. On the other hand, using the FP-LAPW method, Abt et al. [³⁴] and Ambrosch-Draxl and Sofo [³⁵] have detailed the procedure of optical properties calculation for semiconductor alloys. From the complex dielectric function dependent on the frequency ε(ω), several optical parameters of semiconductor materials can be deduced. The cited function is composed of two terms as follows:

where ε ₁(ω) and ε ₂(ω) present the real and imaginary part of the dielectric function, respectively. The ε ₂(ω) values are determined directly from the electronic structure calculation through the joint density of states and the maximum matrix elements between the occupied eigenstates [²,³⁶-³⁸]:

where M, f _i , and ω are the dipole matrix, Fermi distribution function, and the frequency of the incident photon, respectively. Thus, knowing ε ₂(ω), we can then calculate ε ₁(ω) using the Kramers-Kroning [³⁹]: relation expressed as:

where p is the principal of the integral value. The knowledge of a dense mesh of eigenvectors [²²] allows to determine the optical parameters. Thus, in our study, we have used 172kpoints in the irreducible part of the Brillouin zone. The halfwidth Lorentzian broadening was set to 0.1 eV, and the simulated material structure has a cubic symmetry. Moreover, other interesting optical parameters related to the dielectric function can be calculated [⁴⁰,⁴¹]. Among these parameters, we limit our study to the refractive index n(ω), the extinction coefficient k(ω), the absorption coefficient α(ω) and the reflectivity R(ω), which are defined by the following relationships [²,⁴²]:

where ω and λ present the angular frequency and the light in vacuum wavelength, respectively. In this work, the variations of the real and imaginary parts of dielectric functions ε ₁(ω) and ε ₂(ω), versus the incident photon energy were computed with TB-mBJ method for Zn _1−x Cd _x Se _y Te _1−y with different concentrations x(0.250 ≤ x ≤ 0.875). The low energy limit of the real part ε ₁(ω) is defined as the static dielectric constant ε ₁(0). In Table III are summarized the calculated ε ₁(0) values and the peaks values in the dielectric function ε ₂(ω) for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy. In Figs. 9 and 10 are shown respectively, the variations of the real and imaginary parts of dielectric functions ε ₁(ω) and ε ₂(ω) , versus the incident photon energy on the range of (0-20eV) for Zn _1−x Cd _x Se _y Te _1−y with x(0.250 ≤ x ≤ 0.875). The peak values (E ₁,E ₂,E ₃) of the dielectric imaginary part function ε ₂(ω) based on photon energy are derived from Fig. 9 and resumed in Table III. These energy peak values illustrate the electronic crossing of the valence band to the transmission band. Generally, we remark the same behavior of ε ₂(ω) for both Cd concentrations. E ₀ is defined as the threshold energy of the dielectric function and determines the direct optical transitions between the highest valence band (HVB) and the lowest conduction band (LCB) at Γ point. E ₂ and E ₃ peaks move to lower energies while E ₂ peaks shift towards lower energies. On the other hand, E ₁ and E ₃ peak sizes decrease with the increase of Cd concentration while E ₂ peak increases with increasing of x concentration. Referring to Table III, it is observed that the ε ₁(0) value decreases with increasing of Cd concentration for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy. We also note from Fig. 10 that the real part ε ₁(ω) increases for the photon energy values between 2 and 7 eV with two peaks E ₀ and E ₁, equal to 5 and 7 eV, respectively. Then ε ₁(ω) decreases sharply with the energy increasing from the value of 7 eV and reaches its minimum for the energy value of 10 eV.

FIGURE 9 Imaginary part of the dielectric function ε ₂(ω) for Zn _1−x Cd _x Se _y Te _1−y alloys using TB-mBJ method. 

FIGURE 10 Real part of the dielectric function ε ₁(ω) for Zn _1−x Cd _x Se _y Te _1−y alloys using TB-mBJ method. 

The refractive index (n) was computed at the equilibrium lattice constant. The results performed with the TBmBJ method are exhibited in Table IV. The spectral curves of the refractive index n(ω) for Zn _1−x Cd _x Se _y Te _1−y alloy are presented in Fig. 11 and show that the critical points of n(ω) change with the variation of the direct bandgap value (E _g ). The refractive index n(ω) increases with E _g between 0 and 4.0 eV, then reaches its maximum value in the energy range (4.5−6 eV).

FIGURE 11 Calculated refractive index n(ω) for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys using TB-mBJ method. 

For energy values between 6 and 10 eV, n(ω) decreases progressively. The minimum value of n(ω) is obtained for E _g > 10 eV. Also, the variation of the refractive index (n) versus the concentration (x) was studied and presented in Fig. 12, which indicates that (n) varies linearly and decreases with increasing Cd concentration (x). By extrapolating the calculated refractive index n(x) to a polynomial equation, a bowing parameter (b = −0.025 eV) was obtained, as shown in the following relation:

FIGURE 12 Refractive index (n) of Zn _1−x Cd _x Se _y Te _1−y quaternary alloys versus the concentration (x) with TB-mBJ method. 

As described in previous expressions of optical properties showing the relationships between the dielectric constant, the extinction, and absorption coefficient, hence we have simulated this last parameter α(ω) for the Zn _1−x Cd _x Se _y Te _1−y quaternary alloy. In semiconductor materials, the absorption phenomena manifest itself when the photon energy of the incident beam is higher than E _g (hγ > Eg), leading the electrons excitation from the valence band (VB) to the conduction (CB). The calculated absorption coefficient α(ω) for Zn _1−x Cd _x Se _y Te _1−y quaternary alloy is plotted in Fig. 13. Thus, we remark a sharp variation of α(ω) occurring approximatively in the photon energy range (2−10 eV). More specifically, in low energies (2 − 7 eV), the absorption coefficient increases rapidly and then decreases strongly for energies included in the interval (7−10 eV). The lowest values of α(ω) were obtained for higher energies (Eg > 15 eV).

FIGURE 13 Calculated absorption coefficient α(ω) for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys using TB-mBJ method. 

The extinction coefficient k(ω) was determined using the same method and conditions as the refractive index for Zn _1−x Cd _x Se _y Te _1−y quaternary alloy. Figure 14 exhibits the curves of the extinction coefficient k(ω), which increases quickly in the energy range (2 − 6.5 eV) and decreases for (Eg > 8.0 eV). We note that the maximum values of k(ω) correspond to the minimum values of ε ₁(ω) for the same energy interval. On the other hand, we observe that k(ω) is larger in the energy gap around (6 − 10 eV). This is an interesting criterion for the fluorescence phenomenon. The last parameter studied in this paper is the reflectivity R(ω) for Zn _1−x Cd _x Se _y Te _1−y alloy that was simulated using the TBmBJ method. The corresponding results are shown in Fig. 15, which indicates that the maximum reflectivity R(ω) occurs for photon energy values within a range of (4.2 − 12.0 eV). In addition, R(ω) declines rapidly for higher energy (E > 15 eV). According to the literature, the maximum reflectivity is related to the resonance plasmon that appears in the visible-ultraviolet area.

FIGURE 14 Calculated extinction coefficient k(ω) for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys using TB-mBJ method. 

FIGURE 15 Calculated reflectivity R(ω) for Zn _1−x Cd _x Se _y Te _1−y quaternary alloys using TB-mBJ method. 

The referenced work [²⁰] presents many similarities in objectives and methodologies to the present work. Both studied the same quaternary alloy using the Wien2k code with PBE and the approximation mBJ. On the other hand, Vegard’s law and the same principle were used to calculate the optical properties. In our work, an improved (2 × 2 × 2) model cell was chosen, and calculations were performed with PBEsol and the approximation TB-mBJ rather than (1×1×1) model cell, PBE and mBJ method used in [²⁰], respectively. This work uses another range x (0.125 ≤ x ≤ 0.875) for equating the parameter to ZnTe. Globally, the quaternary alloys studied in both works present the same behavior. Our results and those of reference [²⁰] are consistent with each other and have similarities in particular as regards the energy gap Eg(x), real and imaginary part of the dielectric function ε ₁(ω) and ε ₂(ω), and the refractive index n(ω). Finally, the results obtained in the present paper confirm that ZnTe is a suitable substrate for the growth of zincblende Zn _1−x Cd _x Se _y Te _1−y quaternary alloys as suggested in [²⁰].