Analysis of phase stability, elastic, electronic, thermal, and optical properties of Sc1-xYxN via ab initio methods

Understanding the physical properties of a material is crucial to know its applicability for practical applications. In this study, we investigate the phase stability, elastic, electronic, thermal, and optical properties of the ternary alloying of the scandium and yttrium nitrides (Sc1-xYxN) for different compositions. To do so, we apply a “density functional theory (DFT)” based scheme of calculations named as “full potential (FP) linearized (L) augmented plane wave plus local orbitals (APW + lo) method” realized in the WIEN2k computational package. At first, the phase stability of the investigated compositions of the mentioned alloy is determined. The analysis of our calculations shows that Sc1-xYxN alloy is stable in rock salt crystal structure for all investigated compositions. Next to that, the elastic properties of the rock-salt phase of the studied ternary alloy Sc1-xYxN at all above said compositions were done at the level of “Wu-Cohen generalized gradient approximation (Wu-GGA)” within DFT. However, Trans-Blaha (TB) approximation of the “modified Becke-Johson (mBJ)” potential is also used in combination with Wu-GGA where the thermal properties are calculated at the level of the “quasi-harmonic Debye model.” The obtained results for the absorption coefficients, and optical bandgap, represent that the title alloy may be a suitable candidate for the applications in optoelectronic devices.


Introduction
It is well known that the materials of group III-nitrides, also known as transition metals nitrides, are having interesting and unique physical properties and thus have many practical technological applications. The transition metal scandium nitride and yttrium nitride are currently attracting considerable attention [1,2] of both theoretical and experimental researchers. Indeed, the semiconductor scandium nitride has shown outstanding properties including high mechanical resistance, melting point (2900 K), and hardness [3][4][5], and it is also a promising candidate for various applications such as piezoelectric [6][7][8], electronics [9], medium and high-temperature thermoelectrics [3,4], buffer or interface layer [10][11][12][13], and applications in photovoltaic and photocatalytic devices [14]. In addition, it can also be mixed with the semiconducting compounds GaN [15], AlN [16], SiC [10], and MnN [17,18] to form new optoelectronic elements [19][20][21] with different optoelectronic applications. Similarly, the binary compound, yttrium nitride, has also been characterized to exhibit hardness, chemical inertia, and electrical and thermal conductivities [22]. Therefore, it is considered an important material for applications in electrical contacts, magnetic storage devices, optical switching devices, diffusion barriers, buffer layers, etc. [23]. Moreover, both the compounds may be suitable as well for diode and transistors as active layers in the future.
Although theoretical studies are found to be reported in the literature for both compounds such as Mohammad et al. have performed a study on the single crystal of scandium nitride regarding its structural and electronic properties [24], where Abu-Jafar et al. reported its energy gap results [25], Deng et al. computed transport, energy gap, and optical properties [26], Saha et al. reported a study on phonons, electronic, and thermal properties of the scandium nitride [27]. A study on magnetic and structural properties by Sukkabot is found in the literature related to transition metal-doped scandium nitride [28]. Tamleh et al. carried out a study on the band structure and optical spectrum of the tetragonal and hexagonal scandium nitride [29] besides reporting a study on the optical and electronic properties of the monolayer scandium nitride under different strain conditions. On the other hand, a study by Gueddim et al. was reported regarding phase stability and thermal properties of the yttrium nitride [30], while Guerrero-Sánchez et al. studied the mechanism of yttrium nitride formation and yttrium adsorption on the GaN (0001) alongside investigating the diffusion pathways and stability [31]. Farhat et al. studied the effect of spin-orbit coupling on the electronic properties of the yttrium nitride [32]. Tie-Yu et al. reported a study related to the electronic structure of both yttrium nitride and scandium nitride [33]. Amrane studied the valence electron and positron charge densities of both yttrium nitride and scandium nitride [34] whereas Ekuma et al. reported a study on the structural, elastic, and electronic properties [35]. Other than that of the above-mentioned studies, there are also some reported studies in the literature on both nitride binary compounds, for example, Tahri et al. [36] have reported the results of structural, vibrational, and thermal properties in the rocksalt phase of both compounds. Asvini Meenaatci et al. reported investigations regarding the structural stability and electronic properties of yttrium nitride and scandium nitride [37]. Similarly, the bandgap energy bowing parameter was reported by Winiarski et al. related to both binary compounds [38]. Yagoub presented a study in the literature under pressure for both compounds [39]. Cherchab et al. presented a study regarding structural and electronic properties of bulk as well as superlattice for both compound nitrides [40] while Talbi et al. investigated structural and electronic properties related to the superlattice of ternary scandium yttrium nitride (ScYN) [41]. Louhadj et al. investigated the electronic structure of the superlattice of scandium nitride and yttrium nitride [42]. Many studies are found to be reported of the binary compounds of scandium nitride and yttrium nitride but only a few studies [43,44] are reported in the literature concerning the alloying of scandium and yttrium nitride. Moreover, no comprehensive study is found in the literature about the elastic and mechanical properties of ScYN. Therefore, to understand the physical properties of the ScYN, we comprehensively explore the phase stability, elastic, electronic, thermal, and optical properties of ScYN in rock salt structure in the present work.

Method of Computations
In order to do our computational work concerning structural stability, elastic, electronic, thermal, and optical properties of ternary alloys Sc 1-x Y x N over a compositional range 0 ≤ × ≤ 1, WIEN2k computational package based on the approach FP-L(APW + lo) framed within DFT was employed [45][46][47]. To obtain the structural parameters, structural stability, and elastic properties, the "Wu-Cohen generalized gradient approximation (Wu-GGA)" approach was used to incorporate the part of the exchange-correlation energy functional into total energy calculations [48], whereas for determining the electronic and optical properties, Trans Blaha approach Tb-mBJ potential in combination with Wu-GGA was employed [49] because of more simple, fast, and robust as well as more accurate to reproduce results for bandgap energy closer to experimental data, particularly for insulators and semiconductors. In this method of computations, at first, for each concentration, a unit cell was simulated. The obtained units were then divided into Muffin tin spheres and interstitial regions. Different basis sets are used for both regions to expand crystal potential, wave function, and charge density. In the Muffin tin spheres, assumed by considering their centers at atomic nuclei, atomic-like wave functions are used for defining their basis set but for the interstitial region, plane waves basis set was used. For the expansion of potential, charge density, and wave functions, the maximum value of the "l" equal to 10 was used within atomic-like spheres whereas to expand the interstitial region basis set that is plane waves basis set, cut off value for R MT K max was taken equal to 8 to determine the basis set size where K max represents the maximum value of the wave vector and R MT denotes the radii of the muffin tin (MT) spheres. Potential and charge density both were expanded in the Fourier series by taking into consideration lattice vector maximum value equal to 12, i.e., G max = 12 (Ryd) 1/2 in the interstitial region. To circumvent the overlapping of the Muffin tin spheres, radii values for scandium, yttrium, and nitrogen were used as 2.14 a.u, 2.31 a.u, and 1.84 a.u respectively. For further details, cited references from the literature can be consulted [45][46][47].
To simulate rock-salt (B1), CsCl (B2), and zinc blende (B3) type structures of the ScN and YN binary compounds, two atoms per unit cell were taken. However, to model alloys at compositions x = 0.25, 0.5, and 0.75 periodic structures of the so-called supercells of eight atoms per unit were recognized. The zinc-blende and rock-salt structures were found to be P-type whereas the CsCl one is found to be B-type. After recognizing their due structures, cell parameters and atomic positions in the simulated unit cells were optimized via minimization of the total energy values using a miniprogram available in WIEN2k computational code. This was achieved by minimizing internal parameters and optimizing the volumes of the simulated structures by relaxing forces to lesser than 1 mRy/a.u on the due structures via implementing the Hellman-Feynman approach and meeting approximately 1.0 mRy convergence criterion. The positions of atoms in simulated crystal structures for ScN and YN and their mixed alloys are tabulated in Table 1.

Structural stability
In this section, we are trying to understand the structural stability of Sc 1-x Y x N alloys. To do so, first, we predict results for structural parameters which were obtained at the level of Wu-GGA. The obtained results of energy for the ground state of the title material versus unit cell volume for three different phases: rock-salt (B1), CsCl (B2), and zinc-blende (B3) for concentrations x = 0, 0.25, 0.5, 0.75, and 1 are shown in Fig. 1. It is well known that the lowest energy value corresponds to the more stable phase. Hence from Fig. 1, we can see that the rock-salt (B1) crystal structure of the alloys is found to be more stable for all the compositions of x as compared to other phases. Then the obtained results of total energy versus volume for different structures of Sc 1-x Y x N alloys are inserted into the "Birch-Murnaghan equation of states" [50], and resultantly optimized values of the bulk moduli (B), lattice constant (a) were achieved corresponding to optimized structures of ternary alloys Sc 1-x Y x N materials as displayed in Table 2 along with previously reported results as summarized to make a comparison and also understand the accuracy of the results. From Table 2, it is to be noticed that the results obtained for the lattice parameters (a) are nicely in agreement with experimental measurements and theoretical results obtained for both binary compounds called scandium nitride and yttrium nitride in their rock salt phases (B 1 ). It is also to be noticed that our results for their alloys at concentrations x = 0.25, 0.5, and 0.75 are consistent with previously reported data [43,44]. We note that the results obtained for the bulk moduli for binary compound scandium nitride are a little higher than the experimental and theoretical results [3,24,54] and are close to the results found in the ref [25,37,39,43]. However, our values calculated for the compound yttrium nitride are slightly higher than those in the references [37,43,54]. However, obtained results are in nice agreement with previously reported results [39][40][41]. It is also noted here that our results are somewhat superior to the theoretical results obtained via the PBE-GGA approximation reported by Haq et al. [43] for the alloy compositions Sc 0.75 Y 0.25 N, Sc 0.5 Y 0.5 N, and Sc 0.25 Y 0.75 N. Knowing that there are no previous experimental data available in the literature for comparison of these alloys. In the case of the CsCl (B2) phase, the results obtained in this    For alloys Sc 1-x Y x N in the CsCl phase, there are no experimental or theoretical results available in the literature for comparison. For the zinc-blende (B3) phase of both binary compounds, it is observed that our results obtained for lattice parameter (a) are very close to the theoretical data. Also, our calculated values of bulk modulus are in good agreement with the results given in the ref [55] and [30] for both binary compounds respectively. Consequently, for concentrations x = 0.25, 0.5, and 0.75, none of the measured/theoretical results are available in the literature for comparison.

Elastic properties
To understand the effect of external stress on the mechanical properties of a material, calculations of the elastic constants (C ij ) are crucial because the results of elastic constants are further used to assess the materials' mechanical properties such as stability and instability, stiffness, hardness, strength, brittleness, and ductility. To describe the mechanical properties of the alloys Sc 1-x Y x N, we need only three independent elastic constants, i.e., C 11 , C 12 , and C 44 as we have found above that investigated alloys were found more stable in rock-salt structure for all stated compositions and materials with cubic symmetry requires only three independent elastic constants for their complete description. Our computed results for C 11 , C 12 , and C 44 for ternary alloys Sc 1-x Y x N for defined compositions are tabulated in Table 3 along with the previously reported calculations in the literature. From our calculations, it is clear that our calculated results for elastic constants are found to be decreased in the sequence C 11 → C 44 → C 12 .
It is well known that the Born criteria for mechanical stability described for cubic structure crystalline materials are C 11 -C 12 > 0 , C 11 + 2C 12 > 0, C44 > 0 [57]. By following the criteria, our calculated results for elastic constant ( Table 2) are indicating that the investigated binary compounds along with the ternary alloys are found mechanically stable for the suggested compositions. It is also noted from Table 3 that the computed results of the elastic constants C 11 and C 12 for ternary alloys Sc 1-x Y x N are decreasing with increasing concentrations of Y up to x = 0.5 and then starts increasing with increasing concentration of Y. From Table 3, it is further inferred that our calculated results for elastic constants C 11 , C 12 , and C 44 are found to be decreased in a sequence, ScN > Sc 0.75 Y 0.25 N > Sc 0.25 Y 0.75 N > YN > Sc 0.5 Y 0.5 N. On the other side, for scandium nitride (ScN) of the alloy Sc 1-x Y x N, the calculated results for the elastic constant C 11 Table 2 Calculated structural parameters of rock-salt (B1), CsCl (B2), and zinc-blende (B3) phases for Sc 1-x Y x N together with other available theoretical and experimental results a Ref. [3], b Ref. [51], c Ref. [52], d Ref. [53], e Ref. [24], f Ref. [25], g Ref. [54], h Ref. [43], i Ref. [44], j Ref. [41], k Ref. [39], l Ref. [37], m Ref. [30], n Ref. [55], o Ref. [ [54], however, somehow lower than the reported results given in the ref [35]. In the case of the elastic constant C 12 , our computed results are marginally lesser than the results reported ref. [54] and higher than the theoretical results in the ref [35]. The calculated results for the elastic constants C 44 are higher than the results given in the ref. [54] and lower than the results reported in ref. [35]. However, for the compound YN, our computed values for the C 11 , C 12 , and C 44 are found in nice agreement with the previously reported theoretical data in the literature [30,35,54]. However, for compositions, x = 0.25, 0.5, and 0.75, no experimental and no theoretical data for reported elastic constants are found in the literature for comparison. The Voigt's results (represented by Bv and Gv) for bulk and shear modulus respectively are calculated by employing the approach cited in reference [58] by using elastic constants results similar to that as numerous previous calculations have been performed for other crystalline materials [58][59][60][61] whereas Reuss calculations for bulk and shear (represented by B R and G R ) are carried out using the same approach reported in ref [62] for doing calculations for many other crystalline materials previously reported in the literature [59][60][61]. The arithmetic mean of the obtained results by Voigt's and Reuss's approaches is found to be justified by the results obtained by Hill's approach [63,64] for bulk and shear modulus as represented by B H and G H . For cubic crystals, B H can be represented as B. The following relations are used to perform the calculations for bulk and shear moduli by Voigt, Reuss, and Hill: The dimensionless quantity Zener anisotropy factor (A) gives information regarding any crystalline material's anisotropic or isotropic properties. The Zener anisotropy factor can be calculated with the help of the relation given below [64]: Cauchy pressure quantity as represented by C �� called a classical criterion is known to understand the brittle/ ductile nature of a material. The positive value predicts the ductile whereas the negative value describes the brittle nature of a material [65]. To calculate Cauchy's pressure quantity, for any material with cubic structure, the following relation as reported in the literature [64] is used: Young's modulus gives information regarding the opposition offered by a material against linear deformation. To determine the results for Young's modulus (E) for any crystalline material, the following relation given in reference [64] is used: The knowledge of Poisson's ratio (ʋ) provides information regarding the stability and nature of bonding of a material. The Poisson's ratio of any material can be calculated with the help of the relation given below [64]: Table 3 Calculated elastic constants C 11 , C 12 , and C 44 (GPa), bulk modulus B H (GPa), shear modulus G V , G R , and G H (GPa), Pugh's index ratio B/G H for Sc 1-x  In order to calculate Vicker's hardness (Hv) for any of the materials, the following relation is used. In this relation, calculated results of Hills's shear modulus (G H ) are taken as input together with the relation given by Teter [66] and Chen et al. [67]: By using relations from 1 to 10, the obtained results of elastic properties are tabulated in Table 3 of the ternary mixed alloys Sc 1-x Y x N at the ambient conditions for all the computed compositions of x suggested for this study. Also, the calculated results of the bulk and shear moduli within the Voigt, Reuss, and Hill approach along with previously reported results available in the literature are displayed in Table 3. Although binary compounds (ScN and YN) are found in literature, computations for elastic parameters and related properties are first time reported regarding ternary alloys Sc 1-x Y x N at the compositions x = 0.25, 0.50, and 0.75; hence, no other study is available for comparison. Table 3 shows that the results of shear modulus calculated by the Voigt approach decrease in a sequence G V → G H → G R . The results obtained for the shear modulus G V and G R in this study for the binary compounds (ScN and YN) are slightly lower than the previously reported calculations [35] and agree well with the results in the ref [54] for G H . Born's mass mechanical stability criterion for the mixed alloy Sc 1-x Y x N is also satisfied for all the compositions of x = 0.25, 0.50, and 0.75; therefore, it is to be concluded that the above compounds are mechanically stable. Besides the satisfaction of the above-mentioned criterion for stability for reported alloys, the following mechanical stability criterion is also found to be satisfied [64] for studied alloys.
The brittle or ductile nature of the materials can be understood by determining the ratio, B/G H . If the ratio B/G H is lesser or higher than the limiting value equal to 1.75 [68], materials are brittle or ductile respectively. The results obtained for the ratio B/G H for the examined compounds are collected in Table 3, inferring that the studied ternary alloys, Sc 1-x Y x N, are found to be brittle for all the studied compositions of x. In Table 4, we have arranged the obtained results for the quantities Cauchy pressure (Cʹʹ), anisotropy factor (A), Poisson' ratio (ʋ V ≠ ʋ R ≠ ʋ H ), Young moduli (E V ≠ E R ≠ E H ), and Vicker's hardness (H V ). The obtained value of A = 1 for the anisotropy factor infers that the materials are isotropic and if there is any deviation from this value, it infers that the properties of the material are anisotropic [64]. Our computed results of the A are found to be higher than 1, showing that alloys are elastically anisotropic corresponding to all compositions of x. Our obtained Cauchy's pressure quantity results are found in negative values for all compositions of ternary alloys, Sc 1-x Y x N as a negative value demonstrates the brittle nature of the material [70]; hence, we can conclude that the ternary alloys Sc 1-x Y x N are brittle. Moreover, the negative of Cauchy's pressure quantity suggests that the nature of bonding in the examined alloys is covalently dominated for all compositions of x. The computed results of Young's modulus (E) as shown in Table 4 demonstrate that E decreases   Table 3) show that the hardness of the alloys Sc 1-x Y x N is decreased with increasing concentration of Y up to 0.5 but after that, the hardness of the alloys increases with increasing Y concentration. The obtained results of the hardness for the ternary alloys Sc 1-x Y x N are found to be decreased in a sequence ScN → Sc 0. 75 For binary compound ScN, our obtained results for E are slightly lower than experimental results [69] and found to be in good agreement with theoretical work [54]. Similarly obtained results of YN also were found to be in nice agreement with the previously reported theoretical studies [54]. However, for all suggested compositions, no theoretical and experimental studies were found to be reported in the literature for comparison for ternary alloys Sc 1-x Y x N. For material, the Poisson ratio in the range 0.1 < ʋ < 0.25 shows that material is covalently bonded; however, if ʋ ≥ 0.25, the material is considered to have ionic bonding. Moreover, the Poisson ratio also indicates the nature of the brittle or ductile nature of a material; if the Poisson ratio is greater than 0.26, the material is ductile otherwise brittle [71]. Our computed results for ʋ v , ʋ R , and ʋ H tabulated in Table 4 show that the bonding in the alloys Sc 1-x Y x N are covalently dominated for all compositions of x. The calculated results for ʋ for all compositions of x are approximately equal and show the brittle nature of the alloy. The calculated results of ʋ H for the compounds ScN and YN are comparable with the results found in the literature [54]. The results for the Poisson's ratio for the alloy Sc 1-x Y x N at the concentration of Y = 0.25, 0.5, and 0.75 are not reported previously. Our computed results for the hardness H T V and H C V of the ternary alloys, Sc 1-x Y x N, for all concentrations carried out in this study and tabulated in Table 3 show that alloys hardness decreases in a sequence as ScN → Sc 0.75 Y 0.25 N → Sc 0.25 Y 0.75 N → YN → Sc 0.5 Y 0.5 N, endorsing the trend of hardness computed by using results of Bulk and Young's modulus.

Debye temperature
Many physical properties are related to Debye temperatures ( D ) such as thermal conductivity, specific heat, binding forces, melting temperature, and vacancy formation energy which are considered to be fundamental properties. To calculate the D , the following Eq. (12) described in the literature [72] is used: where n is the number of atoms in the molecule, k B is the Boltzmann's constant, N A is the Avogadro number, M is the molecular weight, represents mass density, h symbolizes plank's constant, and v m denotes average sound velocity.
For calculating, v m , for a material following relation as given in the literature [72], is used: where v l , v t , B, and G are longitudinal sound velocity, transverse sound velocity, bulk modulus (average), and shear modulus (average). The results obtained for the physical quantities υ l , υ t , υ m , melting temperature (T Melt ), and θ D of the title ternary alloys Sc 1-x Y x N for all the considered concentrations of x are displayed in Table 5. Our computed results show that θ D , melting temperature, and sound velocities are decreasing corresponding to increasing concentration of Y up to x = 0.5 but after that, both start increasing and follows a pattern in a sequence as ScN > Sc 0.75 Y 0.25 N > Sc 0.25 Y 0.75 N > YN > Sc 0.5 Y 0.5 N. For comparison, no study is found suggesting concentrations of Y.

Band structure
To fabricate reliable and efficient electronic devices for practical applications and alter their electronic properties, the knowledge of the energy bandgap for semiconductors is crucial. In this part, we discuss our obtained results for energy bandgap and electronic band structures of ternary alloys Sc 1-x Y x N corresponding to obtained theoretical lattice parameters for suggested concentrations related to quite stable rock salt (B1) structure at the level of mBJ exchange potential. The obtained band structures are displayed in Fig. 2. According to this figure, we notice that the minima of the conduction band are at X whereas the maxima of the valence band are at Γ for both binary compounds (ScN and YN) (Γ → X) . This suggests that both the compounds are indirect semiconductors but for the case of corresponding ternary alloys, maxima and minima of conduction and valence bands respectively are lying at Γ point, consequently showing that alloys are of semiconductors of the nature of direct energy gap (Γ → Γ). Our obtained numerical results relating to bandgap and previously reported results both theoretically and experimentally are collected in Table 6. From Table 5, it is noted that our energy bandgap results for both compounds are nicely in agreement with experiments [73] (where are available) and the theoretical data [27,35] and are slightly lesser than the results reported previously in the literature [43,75]. However, no experimental study was found in the literature regarding compound YN which shows the indirect energy bandgap nature. If we further look at the results, we observe that overall, our results, with an acceptable error, are consistent with previously reported studies via mBJ approximation [43], GGA + U SIC , and GW methodologies [33,37]. For the alloy compositions x = 0.25, 0.5, and 0.75, our results are somewhat lower than the reported results previously [43] but to make a comparison with experimental measurements, no study is in our knowledge. All the obtained results for alloys as a function of Y compositions are shown in Fig. 3. According to the figure, we note that when the concentration of yttrium increases, the bandgap is also found to be increased.

Density of states
To observe the contribution of different atoms and electronic states to shape the electronic band structure of a material, the knowledge of the partial density of states (PDOS) and total density of states (TDOS) is important. Calculations of the PDOS and TDOS for binary compounds and ternary alloys were done at the level of WC-GGA as well as mBJ approximations of exchange-correlation functionals. However, we have shown plots of PDOS and TDOS only obtained by mBJ approximation here. Figure 4 represents the plots of PDOS and TDOS of the examined compounds (ScN, YN) and ternary alloy (Sc 0.5 Y 0.5 N) as an archetype. The TDOS and PDOS for the ternary alloys Sc 1-x Y x N show indistinguishable behavior. For the binary compound ScN, we differentiate the valence band of the band structure into lower and upper regions where the lower region is considered ranging from − 14.84 to − 12.85 eV, dominated by the s-orbitals of N atoms electrons, and the upper region is considered from − 4.613 eV to Fermi level (E F ) shaped by electrons of p-orbitals of N-atoms with little contribution of the electrons of the d-orbitals of N-atoms. In the case of the conduction band, at 5.80 eV, a sharped peak was found to be formed by dominating contribution of d-orbital electrons of scandium atoms with little contribution of the electrons of the p-orbitals of N-atoms.
Overall, if we look at our obtained data and corresponding figures for DOS, we found a quite similar shape of plots reported by Ekum et al. [35], and also shapes of obtained curves for DOS are found even-handed to results reported by Saha et al. [27]. For binary compound YN, we differentiate the valence band into two regions called the upper and lower region. The lower region is ranging from − 13.519 to − 11.778 eV mainly dominating with s-orbitals electrons of the atoms of N whereas the upper region, from − 3.669 eV to Fermi (E F ), is mainly dominated with p-orbitals of N-atoms with little contribution of the electrons of the d-orbitals of the Y-atoms. The conduction band of the YN is mainly contributed by the electrons of the d-orbitals of Y-atoms including a slight contribution from the electrons of p-orbitals of N-atoms. Our results for YN also show similar behavior as obtained by Ekuma et al. [35]. For the ternary alloy Sc 0.5 Y 0.5 N, the lower part of the valence is considered from − 13.762 to − 11.666 eV which is mainly shaped from the electrons of the s-orbitals of Sc, Y, and N-atoms. The upper valence band region lying in between the energy range from − 4.31 eV to Fermi (E F ) is found to be populated mainly by electrons of the p and d orbitals of scandium, yttrium, and nitrogen atoms. However, conduction band formation is noticed to be a mixture of the electrons of the p-orbitals and d orbitals of Sc, Y, and N atoms.

Optical properties
The optical properties are based on the interaction of light photons with the material's electrons. In order to understand optical properties fundamentally, the calculations of the dielectric function described by mathematical relation are given as below: ε(ω) = ε 1 (ω) + iε 2 (ω)whereε 2 ( )where ɛ 1 (ω) denotes real part and ɛ 2 (ω) represents the imaginary part of the dielectric function ɛ(ω). At first ɛ 2 (ω) and then, ɛ 1 (ω) is calculated by using the following relations established by Kramers-Kroning described in reference [76]: where ℏ denotes photon energy which incidents upon a material, and p represents momentum operator given as below: The |kn > is used to represent eigenfunction reproducing an eigenvalue (E kn ) . f(kn) represents the Fermi distribution function. The optical parameters are then determined using the obtained results of real and imaginary parts of the dielectric function, i.e., absorption coefficients α(ω), refractive index n(ω), reflectivity R(ω), electron energy loss L(ω), and optical conductivity σ(ω), etc. To calculate above the said parameters, equations mentioned in the following from (18) to (22) are used: To calculate the static refractive index, the following relation (obtained by putting ω = 0 in Eq. 18) is used: The reliability of our simulated results can be verified by finding the results for the energy bandgap using the other theoretical models given as follows: a) Moss formula [77] for atom model given as follows is used, i.e.: where k represents 108 eV and E g represents bandgap energy. b) A relation given by Ravindra, Auluck, and Srivastava [78] was also used given as follows: With = 4.084 and = −0.62eV −1 . c) Following equation given by Reddy and Nazeer [79]: where Δ * has represented the electronegativity and its relation with electronic energy bandgap is given as follows: d) An equation developed by Herve and Vandamme's [80] given as follows also used: where A = 13.6 and = 3.4eV.
The present study deals with the calculation of the dielectric constant, refractive index, and other parameters mentioned above as a function of incident photon frequency ranging from 0 to 40 eV for assessing optical behavior of both binary compounds as well as ternary alloys, Sc 1-x Y x N, using mBJ approximation. The obtained results are shown in Figs. 5, 6, 7, 8, 9, 10, and 11.
The response of the ε 2 (ω) as a function of energy (0-40 eV) for concentration range x = 0, 0.25, 0.5, 0.75, and 1 of the title materials Sc 1-x Y x N is displayed in Fig. 5.  Table 6. It was found that each one reaches to maximum value corresponding to the energies 7.115 eV (9.049) for (x = 0), for (x = 0.25) to 6.462 eV (8.133), for (x = 0.5) to 8.748 eV (8.579), 8.285 eV (7.836) for (x = 0.75), and 9.020 eV (9.107) for (x = 1) and then they seem to be diminished. From our computed results, we can deduce that there is strong light photon absorption which is found in the ultraviolet region for each concentration of the alloys, showing their applicability in optoelectronics.
The results obtained for ε 1 (ω) corresponding to incident photon energy range (0-40 eV) of the Sc 1-x Y x N at mentioned concentrations are shown in Fig. 6. It is noted that all the plotted curves qualitatively are nearly the same. These curves start from the values ε 1 (ω = 0) and gradually increases and reach to a maximum peak values corresponding to the energies: 2.027 eV (10.907), 1.863 eV (9.884), 1.510 eV (10.08), 1.591 eV (9.417), and 1.755 eV (8.867) for all suggested concentrations of x. Our obtained results for ScN are found to be in nice agreement with the previously reported results [43], and the results for Sc 0.75 Y 0.25 N and Sc 0.5 Y 0.5 N are somehow superior as compared to the previously reported results [43], on the other hand, for Sc 0.25 Y 0.75 N and YN, obtained results are slightly lower than that of previously reported results [43]. After that, obtained results in this study start decreasing until it vanishes at the points of energies 7.877 eV (ScN), 7.850 eV (Sc 0.75 Y 0.25 N), 9.020 eV (Sc 0.5 Y 0.5 N), 9.290 eV (Sc 0 0.25 Y 0.75 N), and 8.993 eV (YN). At these values, the dispersion is found to be zero; hence, the absorption is maximum. Next to that, it continues to decrease and reach negative minimum values at energies 9.428 eV, 10.190 eV, and 10.10 eV, 10.00 eV, and 9.918 eV for concentrations of x = 0, 0.25, 0.5, 0.75, and 1 respectively for the real part and it becomes negative in energy range (9.428-13.183) eV showing a signature of metallic behavior in this range of energy for mentioned concentrations and also shows, in this range, a reflection of incident light photons. Moreover, our results endorse the previously reported results, in the interval (9.90 to 13.4 eV). After that, obtained results become positive for energy values 11.850 eV for x = 0, 13.183 eV for x = 0.25 eV, 12.939 eV for x = 0.5, 13.02 eV for x = 0.75, and 12.911 eV for x = 1.
Static values of the ε 1 (ω) corresponding to zero frequency along with other theoretical models (Reddy., Herve., Mouss., And Ravind) are shown in Table 7. From the obtained results as shown in Table 7, it is noted that our obtained results for ε 1 (0), for all x concentrations, are somehow lower than those of the results obtained by other theoretical models. The results of ε 1 (0) corresponding Y concentration, and as a function of energy bandgap, are shown in Fig. 7. According to these figures, it is observed that ε 1 (0) results are found to be decreased with increasing concentration of Y as well as bandgap energy values, showing inverse relation with bandgap energy which is described by Penn's model [81] given in the following: Consequently, the results obtained by the Moss model are closer with the results obtained by FP-L(APW + lo) than other models.
The obtained results for n(ω) as a function of energy in the interval (0-40 eV) for the ternary alloys Sc 1-x Y x N for all suggested concentrations are shown in Fig. 8. From this figure, we notice that all the curves are similar and start from the corresponding values n (0) at energies equal to zero. After that, values gradually increase and reach to maximum value 3.320 corresponding to energy 2.108 eV for ScN, 3.181 corresponding to energy 2.027 eV for Sc 0.75 Y 0.25 N, 3.189 at 1.537 eV for Sc 0. 5 Y 0.5 N, 3.09 at 1.700 eV for Sc 0.25 Y 0.75 N, and 2.988 at 1.755 eV for YN. The obtained results are superior to the values obtained previously reported in the literature [43]. After that, the results decrease. Our obtained results for static refractive index n(0) and other computed results are collected in Table 7. These results are also obtained as a function of Y concentration and electronic energy gap values displayed in Fig. 9. We observe that all the curves of n (0) decrease with increasing concentration of Y and increase with energy bandgap values.  The results of the coefficients of absorption with respect to the incident photon energy for the ternary alloys Sc 1-x Y x N in the range (0-40 eV) are shown in Fig. 10. We see that all the curves are approximately the same. We also note that threshold of fundamental absorption takes start at the energies 0.884 eV, 0.884 eV, 0.966 eV, 1.179 eV, and 1.183 eV respectively. These obtained results are found in nice agreement with the results corresponding to the results of the energies of the electronic energy bandgap as shown in Table 6. We have collected all the obtained results corresponding to the energies at the maximum absorption in Table 8. From this table, it can be deduced that results obtained for Sc 1-x Y x N exhibit absorption in the ultraviolet range, highlighting its potential for optoelectronic applications.
The reflectivity spectra of Sc 1-x Y x N alloys are shown in Fig. 11. The curves are found to be analogous for all the values of x. In Table 7, we have collected the results obtained for the reflectivity of the material at zero frequency R (0). Table 7 represents that the values of R (0) are found to be decreased with increasing Y concentration; hence, we can conclude that the electronic energy bandgap is increased whereas reflectivity is decreased with x. We have also collected the values of maximum reflectivity in Table 8. Figure 12 represents the optical conductivity with changing frequency of incident photon for Sc 1-x Y x N using mBJ approximation. Qualitatively, we can see that obtained curves are alike and show almost similar behavior with a slight difference for each concentration of x. The values of the maximum optical conductivity are collected in Table 8.
The energy loss in the incident photon can be understood by electrons transitioning from lower to higher energy region which is denoted by energy loss function L(ω). The variation in L(ω) with the energy of incident photons is shown in Fig. 13. The result for energy loss function at zero photon energy, L(0), is found to be 0.001 for ternary alloys Sc 1-x Y x N corresponding to all x concentrations. Figure 13 shows Gibbs function, G*(V, P, T), for any crystalline solids at non-equilibrium conditions can be described as in the following: where E(V) denotes total energy/unit cell, PV represents hydrostatic pressure whereas A vib [θ D (V), T] is used to describe vibrational Helmholtz free energy.
The value of A vib is calculated with the help of the expression given below [83,84]: where n and D(θ D /T) represent the number of atoms/formula unit and Debye model respectively. The results of the θ D as a function absolute temperature is calculated using the following expression [83]: where M and Bs are molecular weight and adiabatic modulus. We can assume that the static compressibility for any compound is approximately equal to the adiabatic modulus B S and can be calculated with the relation given below [82]: And the relation for the function f (σ) is written as follows:     where σ describes a Poisson ratio with 0.25 value [85]. Hence, we minimize the Gibbs function as a function of V at constant pressure and temperature as follows: To obtain the state (thermal) equation, we have solved Eq. (33) and got the equilibrium curve for the examined materials. To calculate the isothermal bulk modulus (B T ), entropy (S), and specific heat capacity (C V ), the following relations are used [82]: The results for thermodynamics properties were obtained using data of E-V at zero pressure and various temperatures for materials by employing the "quasi-harmonic Debye model." The variation of volume, bulk modulus, Debye temperature, specific heat at constant volume, and entropy at varying absolute temperatures from 0 to 1500 K are given in Figs. 14, 15, 16, 17, and 18, respectively, for the alloys Sc 1-x Y x N at all the compositions of x. From the figures, one can see that the values of the volume are increasing with the increase of temperature. It is also noted that the volume decreases in a sequence as YN > Sc 0.25 Y 0.75 N > Sc 0.5 Y 0.5 N > Sc 0.75 Y 0.25 N > ScN. This means that there is an expansion in the examined materials as the temperature increases.
At zero pressure and absolute temperature T = 300 K, the calculated lattice parameters of the ternary alloys Sc 1-x  The bulk modulus B gives information about the resistance offered by a material against linear compressibility. The calculated results at constant pressure P = 0 GPa and changing temperature are illustrated in Fig. 15. From Fig. 15, it is noted that the values B are decreasing with increasing temperature, and the decreasing trend is found in a sequence as ScN > Sc 0.75 Y 0.25 N > Sc 0.5 Y 0.5 N > Sc 0.25 Y 0.75 N > YN in the temperature range from 0 to 700 K. After that, its behavior follows the trend in a sequence as, ScN > Sc 0.5 Y 0.5 N > YN > Sc 0.25 Y 0.75 N > Sc 0.75 Y 0.25 N in the range of temperature 800 to 1500 K. The change in bulk modulus with temperature shows that the material becomes more fragile with increasing temperature. Our obtained results at zero pressure and absolute temperature T = 300 K are 204.34GPa, 182.65GPa, 173.99GPa, 166.19GPa, and 64.12GPa related to ScN, Sc 0.75 Y 0.25 N, Sc 0.5 Y 0.5 N, Sc 0.25 Y 0.75 N, and YN, respectively. Figure 16 shows that Debye temperature ( D ) decreases with increasing temperature at constant pressure P = 0 GPa. The obtained results for D P = 0 GPa and T = 0 K are 787.41 K, 800.32 K, 729.82 K, 676.90 K, and 562.75 K for the ternary alloys, Sc 1-x Y x N at x = 0.0, 0.25, 0.5, 0.75, and 1 respectively. At absolute temperature T = 300 K and pressure P = 0 GPa, the computed results for Debye temperature are 782.04 K, 792.04 K, 721.24 K, 668.49 K, and 557.63 K at x = 0.0, 0.25, 0.5, 0.75, and 1 respectively. The vibrational properties of any solid material can be clearly understood by calculating its C V . The computed data of C V at pressure zero and varying temperature have been displayed in Fig. 17 for both binary compounds and ternary alloy, Sc 1-x Y x N at compositions x = 0.25, 0.50, and 0.75. Figure 17 shows that C V for both binary compounds (ScN and YN) is found to be lower than that of ternary alloys Sc 1-x Y x N at compositions x = 0.25, 0.50, and 0.75. The figure is also found to follow the Debye T 3 law [86] and Dulong and Petit law [87] at low temperatures and high temperatures respectively. Our obtained results for ternary alloys Sc 1-x Y x N of specific heat at constant volume (C V ) also at zero pressure and absolute temperature T = 300 K are 36.329 (J/mol × K), 54.080 (J/ mol × K), 56.986 (J/mol × K), 59.105 (J/mol × K), and 42.229 (J/mol × K)at x = 0.0, 0.25, 0.50, 0.75, and 1 of Y respectively. Overall C V results show a similar trend as a function Y qualitatively [30].
The entropy (S), a thermodynamic parameter, gives the precise information regarding the thermodynamical properties given above. Currently, to tailoring mechanical or functional properties of materials for different engineering applications, the concept of entropy stabilized materials is frequently used [88]. The results for the entropy S for a temperature in a range from 0 to 1500 K at zero pressure are shown in Fig. 18. From Fig. 18, we can see that when we increase the temperature, the obtained results of the entropy S for the ternary alloys Sc 1-x Y x N increase non-linearly at the compositions x = 0, 0.25, 0.50, 0.75, and 1.
At pressure P = 0 GPa and T = 300 K, computed results of the entropy are 26.290 (J/mol × K), 38.745 (J/ mol × K), 43.948 ( J/mol × K), 48.358 (J/mol × K), and 39.650 (J/mol × K) for the ScN, Sc 0.75 Y 0.25 N, Sc 0.5 Y 0.5 N, Sc 0.25 Y 0.75 N, and YN, respectively. The number of phonon modes is decreasing but this effect is reversed in the case of rising temperature at a fixed pressure.

Conclusions
The FP-L(APW + lo) approach designed within DFT and realized in computational code WIEN2k has been used to study the phase stability, elastic constants, and elastic  Fig. 18 Variation of the entropy with the temperature at P = 0 GPa of the Sc 1-x Y x N ternary alloy constants related properties, optical, electronic, and thermal properties of ternary alloys Sc 1-x Y x N. We have found that the rock salt (B1) phase was found to be more stable for all the concentrations. The results obtained for elastic related properties using WC-GGA were found to be anisotropic, brittle whereas the bonding in the examined compounds was found covalently dominated. The results for the energy bandgap show the nonlinear character for the mixed scandium and yttrium nitride alloys, Sc 1-x Y x N. The calculated results using mBJ approximation for the electronic band structure for the title mixed alloys verified that both the binary compounds (scandium nitride and yttrium nitride) were of indirect bandgap (Γ → X) whereas their mutual alloys were found to be direct band gap (Γ → Γ) nature. The calculated optical properties showed that the fundamental band of the alloys was strongly in line with the optical energy bandgap of the alloys. The calculated results were determined by employing the "quasi-harmonic Debye model" for a wide range of the temperature (0-1500 K) corresponding to zero pressure for the specific heat at constant volume, Debye temperature verified the Dulong-Petit limit. From the analysis of the computed results from the energy band structure and the equivalent optical bands, we can conclude that the mixed ternary alloy Sc 1-x Y x N may be an important material for applications in optoelectronic devices. There were no other results found in the literature previously regarding thermal and elastic properties related to the ternary mixed compound Sc 1-x Y x N. Therefore, this study may be a landmark for further experimental and theoretical study in the future.
Author contribution All authors contributed to the study conception and design. Data collection and analysis were performed by Gagui, Ghemid, and Naeem. The first draft of the manuscript was written by Meradji, Ahmed, Kushwaha, and all authors commented on previous versions of the manuscript. Ul Haq and Meradji: supervising, reviewing, and editing. All authors read and approved the final manuscript.

Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Conflict of interests
The authors declare no competing interests.