3.1Structural properties

To predict the ground state properties such as the equilibrium lattice constant a, the bulk modulus B₀ and the bulk modulus pressure derivative B´, we have optimized all structures Al_0.50Ga_0.50N_xSb_1-x with (x = 0.125, 0.25, 0.75, 0.875), and we have calculated the total energies, according to the unit cell volumes and fitted to the Birch-Murnaghan’s equation of state (EOS) ²² and it’s based necessarily on two parameters: the compressibility B₀ and its first derivative B₀’ with respect to the pressure. These two constants are resolved by adapting the energy versus the volume; the isothermal compressibility modulus B is related to the curvature of the function E(V) by the following relation:

With V₀, E₀, B₀ and B₀´ are the volume, total energy, compressibility modulus, and its first derivative, respectively. The studied structural parameters of Al_0.50Ga_0.50N_xSb_1-x for various concentrations are resumed in Table I, together with the available theoretical and experimental data. Figure 1 exhibit the variation of parameter lattice a as a function of the composition (x) for Al_0.50Ga_0.50N_xSb_1-x, the linear liaison between the parameter lattice and Nitrogen (N) concentration x is calculated by Vegard’s law ²³ as follows:

Figure 1 Calculated lattice constant a of Al _..50 Ga _0.50 N _x Sb1 _-x as a function of concentration (x). 

Generally, the relation between the parameter lattice and concentration is not always linear, for that we have to use the quadratic relationship obtained by fitting:

Equation (4) represents the quadratic relation of bulk modulus as a function of the concentration x.

Figure 2 exhibit the calculated bulk modus as a function of concentration x.

Figure 2 Calculated bulk modulus B ₀ of Al _0.50 Ga _0.50 N _x Sb _1-x as a function of concentration (x). 

3.2.Electronic properties

In this part, to predict and know the nature of the bandgap, we have calculated the electronic band structure for Al_0.50Ga_0.50N_xSb_1-x with different concentration x(x = 0.125, 0.25, 0.75, 0.875) using GGA-PBEsol ¹² jointed with (TB-mBJ) of Tran-Blaha modified Becke-Johnson approximation. The band structure is an indispensable aspect to explore the electronic properties of solids. An important element of details given by the band structure is the width of the energy bandgap (Eg), which is the energy separating the valence band top to the conduction band bottom. The calculated structures along the high symmetry directions of Al_0.50Ga_0.50N_xSb_1-x alloys for x = 0.125, 0.25, 0.75 and 0.875 are given in Fig. 3, 4, 5, and 6, respectively. The top of the valence band and the bottom of the conduction band are located at Γ-Γ high-symmetry points and give a direct band for all structures, except for Al_0.50Ga_0.50N_0.25Sb_0.75, an alloy which has a zero bandgap and which has semi-metallic behavior ^24,25. The calculated Al_0.50Ga_0.50N_xSb_1-x band gaps using GGA-PBEsol and TB-mBJ approximation, with available experimental theoretical data, are listed in Table II. Figure 7 shows the deviation of band gap energies Eg of Al_0.50Ga_0.50N_xSb_1-x in relation with N concentration (x).

Figure 3 Electronic band structures for Al _0.50 Ga _0.50 N _0.125 Sb _0.875 using GGA-PBEsol and TB-mBJ approximation. 

Figure 4 Electronic band structures for Al _0.50 Ga _0.50 N0.25Sb _0.75 using GGA-PBEsol and TB-mBJ approximation. 

Figure 5 Electronic band structures for Al _0.50 Ga _0.50 N _0.75 Sb _0.25 using GGA-PBEsol and TB-mBJ approximation. 

Figure 6 Electronic band structures for Al _0.50 Ga _0.50 N0.875Sb _0.125 using GGA-PBEsol and TB-mBJ approximation. 

Figure 7 The variation of band gap energies Eg of Al _0.50 Ga _0.50 N _x Sb _1-x as a function of N concentration (x). 

3.3.Optical properties

The study done by Ehrenreich and Cohen will allow us to describe the dielectric function as a complex function as:

where ε1(ω) and ε2(ω) are the real and the imaginary parts of the dielectric function, respectively. This frequency-dependent complex dielectric function is known to describe the optical response of the medium at all phonon energies E=ℏω. The imaginary part of the ε(ω) in the long-wavelength limit has been collected directly from the electronic structure calculation, using the joint density of states (JDOS) and the transition moments elements M_cv(k):

where ω is the frequency of the light, e is the electric charge is the vector defining the polarization of the incident electric field, Ω is the unit cell volume, φKC and φKV are the wave functions of the conduction and valence bands, respectively. The real part of the dielectric function ε₁(ω) is determined by employing the Kramers-Kronig ^26,27 relations expressed as follows:

Figure 8 shows the real part of the dielectric function for Al_0.50Ga_0.50N_xSb_1-x. It is evident that the zero frequency limits ε1(0) is a fundamental parameter, which describes the dielectric response to the static electric field. The static dielectric constants of the Al_0.50Ga_0.50N_xSb_1-x alloys at considered Nitrogen (N) concentrations (x = 0.125, 0.25, 0.75, 0.875) are 568.668, 601.714, 158.032, and 142.251, respectively. We notice that the real part of the dielectric function ε1(ω) for Al_0.50Ga_0.50N_0.125Sb_0.875, and Al_0.50Ga_0.50N_0.25Sb_0.75 becomes negative, from energy which is called plasma energy Ep=ℏωp, with ℏ is reduced Planck constant, and ω_p is the plasma frequency. We have found the plasma energy at 0.61093 eV and 0.67841 eV for x = 0.125 and 0.25, respectively. In addition to that, the energy plasma is the energy where the material becomes conductive (metallic comportment) ^24-28; according to our calculation, it is reasonable because their bandgap is 0.43872 and 0.0000 eV, respectively. Furthermore, the curves are grouped in pairs, for concentration (0.125, 0.25) and (0.75, 0.875), which have almost the same contour and may be for their nearby bandgap. The imaginary part of the dielectric function (absorptive) for Al_0.50Ga_0.50N_0.125Sb_0.875is represented by Fig. 9. It is apparent that the imaginary part of the dielectric function exhibits various comportments for different compositions x, with different peaks located at 0.12245, 0.09524, 0.25851, and 0.17687 eV, for x = 0.125, 0.25, 0.75, and 0.875, respectively. The imaginary part of dielectric amplitude (ε₂) is inversely proportional to the bandgap. The lowest gap is 0, 0.43872, 1.81838, and 2.30677 eV for the structures Al_0.50Ga_0.50N_0.25Sb_0.75 (red curve), Al_0.50Ga_0.50N_0.125Sb_0.875 (black curve), Al_0.50Ga_0.50N_0.75Sb_0.25 (green curve), and Al_0.50Ga_0.50N_0.875 Sb_0.125 (blue curve), respectively. The peak (high value) of the imaginary part of the dielectric function of different structures is located at 0 and 0.5 eV, which can be due to the crystal structure which is almost identical (if we neglect the radius of N in front of Sb). The associate of the real and imaginary parts of the dielectric function allowance the calculation of fundamental optical properties such as the refractive index n(ω), extinction coefficient K(ω), and the optical conductivity σ(ω) ^29-31. The refractive index n(ω), extinction coefficient K(ω), and the optical conductivity σ(ω) are calculated from the Eqs. (8), (9) and (10), respectively.

Figure 8 Calculated real parts of the dielectric function of Al _0.50 Ga _0.50 N _x Sb _1-x structures. 

Figure 9 Calculated imaginary parts of the dielectric function of Al _0.50 Ga _0.50 N _x Sb _1-x structures. 

The refractive index is a fundamental optical constant. It can present data about the comportment of light in materials; when light passes through the different structures, its velocity decreases by increasing the refractive index of these structures ³². Figure 10 represents the refractive index versus light energy (spectral radiations). The maximum refractive index for Al_0.50Ga_0.50N_xSb_1-x for all concentrations is situated at zero frequency limits, the static refractive index n(0) for x = 0.125, 0.25, 0.75, and 0.875 are around 24.3361, 25.2517, 12.6392 and 12.0682 eV, respectively, and is decreased as a function of energy. The extinction coefficients of Al_0.50Ga_0.50N_xSb_1-x for different concentrations are presented in Fig. 11. We remark that the extinction coefficient increases and decreases after attaining a peak for all concentrations except for x = 0.875 composition. The peaks located at 0.25851, 0.14966, and 0.34014 eV for x = 0.125, 0.25, and 0.75, respectively, and several peaks for x = 0.875 situated at 0.23130, 1.40139 and 2.46263 eV. The plots of the optical conductivity σ(x) within the GGA-PBEsol approach are displayed in Fig. 12. Various peaks are presented in the feature of σ(x) curves, which are vary in agreement with the energy bandgap.

Figure 10 The refractive index n(ω) of Al _0.50 Ga _0.50 N _x Sb _1-x structures. 

Figure 11 Extinction coefficient K(ω) for Al _0.50 Ga _0.50 N _x Sb _1-x structures. 

Figure 12 Optical conductivity of Al _0.50 Ga _0.50 N _x Sb _1-x structures.