The computed cell parameters using LDA and HSE functional for both AlN and ScN structures are presented in Table 1 together with the available theoretical and experimental results. As shown in Table 1, the calculated lattice parameters are in reasonable agreement with experimental and theoretical values, within 3% and 1% for LDA and HSE, respectively.

AlN is known to stabilize in wurtzite structure while ScN is the rocksalt one. Our computational results of Sc_xAl_1−xN (0 ≤ x ≤ 1) lattice constants using both functional (LDA and HSE) are shown in Figure 1. It can be seen that the variation

of the lattice parameters of the ternary alloy in both rocksalt and wurtzite structures, as a function of the scandium composition are not important, the same behavior is seen for the c/a lattice parameter which may be attributed to the difference in atomic radii of Sc and Al atoms and to the small difference between the lattice parameters of the binary compounds.

To understand the behavior of stability of this alloy, the study of thermodynamic properties is essential, in particular as regards enthalpies of formation. AlN and ScN are stable in the wurtzite and rocksalt phases, respectively. The mixing enthalpy (H_mix) of Sc_xAl_1−xN alloy with different phases can be expressed as:

H min ( Sc x Al 1 − x N ) = E ( ScN , Rocksalt ) − ( 1 − x ) E ( AlN , wurtzite ) (2)

where E is the total energy per unit cell of the bulk compound x and (1 − x) are the ScN and AlN mole fraction, respectively.

Hoglund et al. [29] , using first-principles calculations, have calculated the lattice parameters and mixing enthalpies of cubic, wurtzite and hexagonal Sc_xAl_1−xN and they found the transition from cubic to wurtzite structures at x = 0.45, which deviates from the experimental results (x = 0.60) [29] .

The obtained results with HSE functional of the mixing enthalpy (H_mix) of Sc_xAl_1−xN alloy with, rocksalt (RS) and wurtzite (WZ) phases are presented in Figure 2. The ternary alloy changes from hexagonal to cubic when the ScN mole fraction exceeds 0.61 fractions. This value is consistent with the observed experimental result (x = 0.6) [29] . As a result of this, the calculated result fall in the vicinity of the stable region from the experimental and makes the HSE functional more consistent with the experimental data.

The calculated band gaps of the binaries constituents in WZ and RS structures are tabulated and compared with the experimental values in Table 2. We display in Figure 3 the obtained energy gaps of Sc_xAl_1−xN, using LDA, HSE and compared them with experimental and other theoretical results. A decrease in band gap with increasing Sc content is observed experimentally [8] . Our results obtained with LDA are in good agreement with other LDA-theoretical results, although notably smaller than those obtained experimentally. For example, the LDA band gaps of RS and WZ-AlN are 4.32 eV and 4.23 eV which in agreement with Jiao [30] and Berkok [23] respectively. The same for RS and WZ-ScN are 0.10 eV and 2.9 eV which in accordance with other LDA-theoretical results [8] [23] [35] , but underestimated by about 1 eV compared to the experimental values.

To surmount this problem, we have used the standard HSE hybrid functional (α = 25). The band gap values determined by the hybrid functional are significantly improved and the discrepancy with experiment results is about 2%. Whereas, those calculated by the standard DFT are about 10% smaller than the experimental values. Our results demonstrate well, that HSE emerges as an appropriate method for the study of electronic properties of semiconducting alloys.

Recently, Deng et al. [8] demonstrated that the optical absorption of hexagonal Sc_xAl_1−xN indicates a band gap of 6.15 - 9.32 x eV for 0 < x < 1 and a linearly increasing density of defect states within the gap. The average bond angle decreases linearly with x, suggesting a trend towards the metastable hexagonal-ScN structure. Zhang et al. [38] noted that the Wurtzite-structure ScAlN alloy is a wide band-gap semiconductor, which can be stabilised at low Sc contents.

In order to get a better understanding of the electronic structure of the investigated compounds, we display in Figure 4 the total densities of states (TDOS) of Sc_xAl_1−xN for both structures.

Note that the valence band states of intrinsic AlN are composed of two main parts: the upper part which is primarily derived from N s orbitals and the lower part which is mainly due to the N p orbitals for both RS and WZ structures. Once the Sc composition increases, we note that the peak in the lowest valence band increases. Also, the conduction band gradually shifts towards low energies.

Using HSE functional, we note a blue shift of the conduction band minima in the case of WZ-ScN and an appearance of a valence band located at about −12 eV. Likewise, we note a widening of the gap.

To further study the electronic properties, the partial densities of states of Sc_xAl_1-xN alloy are shown in Figure 5. We note that, the top of the valence band and the bottom of the conduction band are mainly composed by the N-2p states and Sc-3d states, respectively. The strong hybridization between Sc-d and N-p indicates the transfer of electrons from Sc atoms to N atoms and their participation in the ionic bonding between Sc and N atoms. The incorporation of Sc atom largely affects the electronic states of neighboring N atoms.

In the following, the optical properties of Sc-ordered AlN systems will be systematically discussed on the basis of dielectric function, absorption coefficient and refractive index. All the results are presented with the polarization vectors perpendicular to the c-axis.

The complex dielectric constant ε is defined by ε = ε 1 + i ε 2 . The imaginary part of the dielectric function is completed within the electric dipole approximation equation. The real part can be assessed from the imaginary part by using the Kramer-Kronig relations and given in our recent paper [39] as follows:

ε 2 ( ω ) = π ε 0 ( e m ω ) 2 × ∑ V , C { ∫ B Z 2 d K ( 2 π ) 3 | a ⋅ M C V ( K ) | 2 × δ [ E C ( K ) − E V ( K ) − ℏ ω ] } (3)

ε 1 ( ω ) = 1 + 2 e ε 0 m 2 ∑ V , C ∫ B Z 2 d K ( 2 π ) 3 | a ⋅ M C V ( K ) | 2 [ E C ( K ) − E V ( K ) ] × ℏ 3 [ E C ( K ) − E V ( K ) ] 2 − ℏ 2 ω 2 (4)

Figure 6 illustrates the HSE calculated of the imaginary part ε 2 ( ω ) for AlN, ScN and Sc_xAl_1−xN. For AlN, there are three major peaks located at 4.30 eV, 7.50 eV and 8.50 eV and two main peaks located at 9.11 eV and 12 eV for the wurtzite and roksalt structures, respectively. These results are in agreement with the

available experimental results [40] . The first peak (4.3 eV for WZ and 9.11 eV for RS structures) should be mainly due to the interband transitions, between N-2p orbitals in the highest valence band and Al-3p in the lowest conduction band. The second peak is mainly due to the interband transitions from N-2s and Al-2p orbital as seen in the DOS given in Figure 5.

By increasing Sc concentration, the peaks move to lower energies. This might be strongly dependent on the ionic polarization of the ScAlN crystal due to the large electronegativity of N. For pure ScN, the major peak is at about 2.00 and 4.5 eV, for RS and WZ structures correspondingly, which is mainly derived from the optical transition between the Sc-d states and N-p states.

Our results of ScN rocksalt are in agreement with experimental values [41] .

The real parts of the dielectric function ε₁(ω) are shown in Figure 7. For the binary WZ-AlN, ε(0) given by the low energy limit of ε(ω) is about 4.6 eV, which is in accordance with experimental and other theoretical results (4.61, 4.70 eV [42] [43] ). As far as we know, for the RS-AlN and WZ-ScN there are no experimental data available related to the static macroscopic dielectric constants, and this can serve as a prediction for future investigations.

As displayed in Figure 7, the ScAlN material presents a similar trend mainly for higher energy, for both WZ and RS structures, which behave like a metallic property above 7.01 eV for WZ-AlN and 3 eV for RS-ScN.

The dielectric functions are sufficient to derive other physical properties like the absorption coefficient and refractive index.

For this purpose, we present in Figure 8, the optical absorption coefficient of Sc doped AlN alloy. As can be seen, the absorption coefficient decreases with the increase of Sc concentration, in good agreement with the previous calculation. This decrease is mainly ascribed to the reduction of Al p states. It is also clear that the absorption edge of Sc_xAl_1−xN alloys shifts evidently to lower photon energies with the increasing Sc doping concentration, compared to that of pure AlN. Our obtained results are in agreement with those of Zhu et al. [10] . Moreover, the absorption coefficient of Sc-doped AlN systems decreases with Sc content, as shown in Figure 8, which is mainly, ascribed to the effect of Sc 3d states.

The absorption region is quite wide and it is still located in the UV region. The line shape is almost the same for the doped and undoped AlN systems in the high-energy range. Therefore, this material could be employed in optoelectronic devices of short wavelengths, such as UV detector and light-emitting diodes (LEDs). These properties indicate that the concentrations of Sc dopant affect mainly the optical properties specifically in the high energy range.

The refractive index n ( ω ) is a very important physical parameter related to microscopic atomic interactions. The knowledge of this index is necessary for accurate modeling and design of devices. Thus, the refraction index, absorption

coefficient and other optical constants can be obtained [44] [45] . The refraction index is given in Equation (5).

n ( ω ) = 1 2 [ ε 1 ( ω ) 2 + ε 2 ( ω ) 2 + ε 1 ( ω ) ] 1 2 (5)

Figure 9 shows the refractivity index n ( ω ) of ScAlN systems in the energy range of 0 - 30 eV. For AlN, the static refractive index n(0) is found to be 3.10 and 2.62 for RS and WZ structures, respectively. It is also noted that the values of n ( ω ) increase with the increasing photon energy in the transparency region hit record highs in the ultraviolet at about 7.72 eV and 4.8 eV in RS and WZ phases of AlN, respectively.

Our results are in accordance with other’s calculated and experimental results [40] . As can be seen by increasing Sc content, the low photon energy range (<10 eV) is characterized by appreciable refractivity, whereas, in the 12 - 27 eV photon-energy range, the refractivity undergoes a change from the strongest located at to the second weakest. At the photon-energy range of 25 - 30 eV, the refractivity shows an opposite trend.

The imaginary part, absorption edge and refractive index of the rocksalt phase are quite different from those of the wurtzite phase. These results endorsed that either RS or WZ-ScAlN can be a material of choice for optoelectronics and solar cell applications.