Ab initio studies of the structural, electronic, and optical properties of quaternary B_xAl_yGa_1–x–yN compounds

3.1 Structural properties

The quaternary system, studied in this paper includes three binary compounds BN, AlN, GaN, and three ternary solid solutions B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN. To calculate the structural properties of the binary and ternary solid solutions and then their quaternary solid solutions, we started calculations with the ZB structure and let the calculated forces move the atoms to their equilibrium positions. We have chosen eight atoms (in primitive P mode of fcc ZB) of 1 × 1 × 1 single cells for modeling the structure of the quaternary solid solutions B_xAl_yGa_1–x–yN for the considered structures and at different boron and aluminum concentrations x and y for compositions (x = 0.25, y = 0.25; x = 0.25, y = 0.50; and x = 0.50, y = 0.25). The structural properties were obtained by calculating the total energies for different volumes around the equilibrium cell volume V₀ of the BN, AlN, and GaN binary compounds and their solid solutions. The calculated total energies were fitted to the Birch–Murnaghan equation of state [8] to determine the ground state properties such as the equilibrium lattice parameter a₀, the bulk modulus B₀, and the pressure derivative of the bulk modulus B′.

The calculated equilibrium parameters a₀, B₀, and B′ for the binary compounds are given in Table 1, which also contains results of previous calculations as well as the experimental data. There is a good agreement between our results and the reported theoretical investigations. In comparison with the experimental data, we found that GGA overestimates the lattice parameter and LDA underestimates it, whereas the bulk modulus is underestimated with GGA and overestimated with LDA. These observations are in concordance with the general trend of these approximations. Our calculated lattice parameters and bulk modulus at different compositions of ternary solid solution B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN are depicted in Figs. 1 and 2. The lattice parameters vary almost linearly with concentration, with the trend following the Vegard’s law. The obtained results are fitted to the following equations:

Fig. 1:

The calculated lattice parameters versus compositions for the ternary solid solutions using GGA.

Fig. 2:

The calculated bulk modulus B₀versus compositions x for ternary solid solutions using GGA XC approximation.

For the lattice parameters:BxGa1−xN: 4.53 − 0.5x − 0.37x2BxAl1−xN: 4.38 − 0.45x − 0.322x2AlxGa1−xN: 4.53 − 0.13x − 0.01x2For the bulk modulus:BxGa1−xN: 187.65 + 18.34x + 174.85x2BxAl1−xN: 198.17 + 9.42x  + 172.57x2AlxGa1−xN: 185.74 + 17.25x − 6.85x2

The deviations observed from the linear concentration dependence can be explained mainly by the large mismatches of the lattice parameters of the binary constituents. From Figs. 1 and 2, we observe that the lattice parameter increases when the composition x is decreasing and the bulk modulus decreases the composition x is decreasing. Results obtained in our calculations agree well with other available data [16].

The structural and electronic properties of pseudo-binary (ternary) solid solutions B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN were studied in the second step of our investigation. We used ordered structures described in terms of supercells with eight atoms per unit cell, for the compositions x = 0.25, 0.50, and 0.75. For the considered structures, we performed the structural optimization by minimizing the total energy with respect to the cell parameters and the atomic positions. The calculated lattice parameters at different compositions of B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN as shown in Table 2.

The calculated structural properties for the quaternary solid solutions are listed in Table 3. To the best of our knowledge, there are no experimental or theoretical data for the structural properties available in the literature for the studied solid solutions.

Our results can serve as a reference for future investigations. Usually, in the treatment of solid solutions when experimental data are scarce, it is assumed that the atoms are located at the ideal positions and the lattice parameters vary linearly with concentration x according Vegard’s law [27]. However, violation of this linear law has been observed experimentally [28] and theoretically for semiconductor compounds. Assuming that Vegard’s law is valid, the lattice constants of B_xAl_yGa_1–x–yN quaternary alloys are generally expressed as a linear relation of boron composition x and aluminum composition y by the following expression [29]:

a(x,y) = x(BN) + y(AlN) + (1 − x − y)(GaN)

We tested the validity of Vegard’s law for the B_xAl_yGa_1–x–yN quaternary solid solutions in the ZB structure with different boron and aluminum concentrations. a(BN), a(AlN), and a(GaN) are the equilibrium lattice parameter of BN, AlN, and GaN, respectively. a(x,y) represents the composition-dependent lattice constant of B_xAl_yGa_1–x–yN quaternary solid solutions.

3.2 Electronic structure properties

Accessing the energy band structures in semiconductors provides valuable information regarding their potential utility in the fabrication of electronic and optoelectronic devices. As group III nitrides are promising materials, the accurate knowledge of the band structures of the B_xGa_1–xN, B_xAl_1–xN, Al_xGa_1–xN, and B_xAl_yGa_1–x–yN compounds becomes essential. The band gap energies of BN, AlN, GaN, B_xGa_1–xN, B_xAl_1–xN, Al_xGa_1–xN, and B_xAl_yGa_1–x–yN were calculated for the equilibrium calculated lattice parameters [1].

Using our calculated values of lattice parameters with GGA, we computed the electronic band structure along some high symmetry directions in the BZ, with LDA and GGA XC. The obtained band structures for ternaries at x = 0.5 are presented in Fig. 3, and the density of states (DOS) of B_xAl_yGa_1–x–yN quaternary solid solutions for compositions (x = 0.25, y = 0.25; x = 0.25, y = 0.50, and x = 0.5, y = 0.25) are presented in Fig. 4.

Fig. 3:

Non-relativistic band structure of ZB: B_0.50Ga_0.50N, B_0.50Al_0.50N, and Al_0.50Ga_0.50N ternary solid solutions using mBJ approximation.

Fig. 4:

Non-relativistic band structure of ZB-type B_xAl_yGa_1–x–yN: B_0.25Al_0.25Ga_0.50N, B_0.25Al_0.50Ga_0.25N, and B_0.50Al_0.25Ga_0.25N using mBJ approximation.

The numerical results are listed in Table 4 and compared with the available experimental and theoretical data. They show that the valence band maximum (VBM) occurs at the BZ center Γ point and the conduction band minimum (CBM) is located at the X point in both BN and AlN, thus resulting in indirect (Γ–X) band gaps for BN and AlN and direct (Γ–Γ) band gaps for GaN and ternaries solid solutions B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN.

The calculated band gaps are in very good agreement with those of available theoretical calculations. It is worth mentioning here that the calculated band gap values listed in Table 4 are systemically underestimated in comparison with experimental data. Such underestimation of the band gaps is mainly because the simple forms of LDA and GGA do not take into account the quasi-particle self-energy correctly [36], which makes them insufficiently flexible to accurately reproduce both XC energy and its charge derivative.

The band gap energies of B_xAl_yGa_1–x–yN solid solutions have been computed using both GGA, LDA, and modified Bezcke–Johnson (mBJ) approximations [39]. We found that the quaternary solid solutions B_xAl_yGa_1–x–yN have a direct band gap. The resulting values for the different x and y concentrations are given in Table 5. It can be seen that the large value calculated for the quaternary solid solutions of the direct band gap using LDA approximation is 5.65 eV, which corresponds to x = 0.50, y = 0.25.

The band gap energy of an A_xB_1–xC compound can be depicted as a function of the A composition x and can be approximated using the following equation:

Eg(x) = xEg(AC) + (1 – x)Eg(BC) −x(1 – x)b,

where E_g(x) is the band gap energy of the A_xB_1–xC solid solution, E_gAC and E_gBC are the band gap energies of the binary compounds AC and BC, respectively, and the quadratic term b is the bowing parameter of A_xB_1–xC [1].

The total bowing parameter is calculated by fitting the band gap energies of B_xGa_1–xN, B_xAl_1–xN, and Al_xGa_1–xN obtained with the equilibrium lattice parameters. The corresponding plots are provided in Figs. 5–7; the non-linear variation of the calculated direct and indirect band gaps in terms of concentration with a polynomial function is as follows:

Fig. 5:

The energy band gaps of B_xGa_1–xN as a function of boron composition.

Fig. 6:

The energy band gaps of B_xAl_1–xN as a function of boron composition.

Fig. 7:

The energy band gaps of Al_xGa_1–xN as a function of aluminum composition.

BxGa1−xNE(Γ–Γ) = 3.23 − 3.08x + 9.78x2(direct gap)E(Γ–X) = 4.68 + 3.26x − 1.92x2(indirect gap)BxAl1−xNE(Γ–Γ) = 5.82 − 7.93x + 12.11x2(direct gap)E(Γ–X) = 5.02 + 6.98x− 6x2(indirect gap)AlxGa1−xNE(Γ–Γ) = 2.89 + 1.97x + 0.60x2(direct gap)E(Γ–X) = 4.58 + 8.09x − 7.55x2(indirect gap)

B_xGa_1–xN has a phase transition from direct to indirect gap for high boron contents (x > 0.71). For B_xAl_1–xN, a direct gap is found for 0.05 < x <0.75. As for Al_xGa_1–xN, its gap has been found entirely direct.

We have plotted the energy band gaps for B_xAl_yGa_1–x–yN with the compositional parameter y taken to be 0 ≤ y ≤ 1 as shown in Figs. 8 and 9 and with the compositional parameter x being in the 0 ≤ x ≤ 1 range. Both figures show that the quaternary compounds have a direct gap. The energy gap varies from 4.13 to 4.96 eV for the composition x = 0.25, y = 0.25 and x = 0.50, y = 0.25. It is clearly seen that the incorporation of boron affects considerably the band gap compared to the aluminum incorporation. We have not found any theoretical report or experimental data in the literature to compare and confirm our results for these solid solutions. Our work therefore can serve as a reference for future studies.

Fig. 8:

The calculated direct E_(Γ–Γ) band gaps versus composition (x, y) for quaternary B_xAl_yGa_1–x–yN solid solutions.

Fig. 9:

The calculated indirect E_(Γ–X) band gaps versus composition (x,y) for quaternary B_xAl_yGa_1–x–yN solid solutions.

3.3 Density of states

To understand the calculated band structure in terms of the contributing atomic states, the total DOS is plotted in Fig. 10 for B_0.25Al_0.25Ga_0.50N. The DOS is computed using GGA approximation with a k mesh of 400 special k-points for B_0.25Al_0.25Ga_0.50N. In the calculation of total and partial atomic DOS for B_0.25Al_0.25Ga_0.50N, we have distinguished the B (1s²), Al (1s²2s²2p⁶), N (1s²), and Ga (1s²2s²2p⁶3s²3p⁶) inner-shell electrons from the valence electrons of B (2s2p¹), Al (3s²3p¹), N(2s²2p³), and Ga (3d¹⁰4s²4p¹) shells. For ZB B_0.25Al_0.25Ga_0.50N, the total DOS presents three regions: two regions in the valence band (VB1) and (VB2) below the Fermi level considered as energy reference and one region (CB) above the Fermi level E_F. The DOS results from the contributions of different orbitals in each region.

Fig. 10:

Total and partial DOS for ZB-type solid solutions B_0.25Al_0.25Ga_0.50N.

The first region (VB1) located in the energy range [–14, –10 eV] is dominated by the aluminum Al-3s, Al-3p, gallium Ga-4s, Ga-4p, and nitrogen N-2s, N-2p states with little contributions of Ga-3d states.

The second region (VB2) expanded between [–8, 0.0 eV] is divided into two sub-regions. The first region at the left side results from the contributions of B-2s, Al-3s, Ga-4s, and N-2s states. The second sub region at the right is a mixture of B-2p, Al-3p, Ga-4p, and N-2p states. We remark that the deep levels are dominated by the “s” state and the higher energy levels in this band are dominated by “p” states. The third region (CB) is essentially dominated by B-2p, Al-3p, and Ga-4p empty states with a minor contribution of N-2p states.

3.4 Optical properties

The optical properties may be extracted from the dielectric function,

ε(ω)=ε1(ω)+iε2(ω),

which is determined mainly by the transition between the valence and the conduction bands. According to perturbation theory, the imaginary part ε₂(ω) is expressed as

ε2(ω)=e2ℏπm2ω2∑V,C∫BZ|MCV(k)|2δ[ωCV(k)−ω]d3k.

The integral is over the first BZ. The momentum dipole elements,

MCV(k)=〈uck|δ∇|uvk〉,

where δ is the potential vector defining the electric field, are matrix elements for direct transitions between valence u_vk(r) and conduction band u_ck(r) states, and the energy

ℏωcv(k)=Eck−Evk

is the corresponding transition energy.

The real part ε₁(ω) can be derived from the imaginary part using the familiar Kramers–Kronig transformations

ε1(ω)=1+2πP∫0∞ω′ε2(ω′)ω′2−ω2dω

where P implies the principal value of the integral. The knowledge of both real and imaginary parts of the frequency dependent dielectric function allows the calculation of important optical functions such as the refractive index n(ω) using the following expression:

n(ω)=[ε1(ω)2+ε12(ω)+ε22(ω)2]1/2.

We have also calculated the refractive index n by using some empirical models. The Moss formula based on an atomic model [40],

Egn4=k,

and Herve and Vandamme’s empirical relation [41],

n=1+(AEg+B)2,

with A = 13.6 eV and B = 3.4 eV.

Fig. 11 display the real and the imaginary parts of the dielectric function as well refractive index spectrum for B0.25Al0.25Ga0.50N. Our analysis of the imaginary part of the dielectric function shows that the first critical point occurs at 3.1 eV for B_0.25Al_0.25Ga_0.50N. This point represents the direct optical transitions between the top of the valence band and the bottom of the conduction band at Γ point. We can also remark that the main peaks in the spectra are located at about 7.5 eV for B_0.25Al_0.25Ga_0.50N.

Fig. 11:

Calculated refractive index: real and imaginary parts of the dielectric function of B_0.25Al_0.25Ga_0.50N.