First-principles study of the impact of the atomic configuration on the electronic properties of ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys

We employ first-principles calculations in the formalism of standard and hybrid density functional theory to study the electronic and structural properties of wurtzite ${\text{Al}}_x{\text{Ga}}_{1-x}\text{N}$ alloys. We address the discrepancies observed in literature regarding essential electronic properties of these alloys and we investigate the dependence of these properties on the atomic ordering and composition. We show that the bowing parameter is significantly affected by the atomic ordering, ranging from zero to strong downward bowing. The effects of atomic ordering of the alloys on their band offset with respect to the pure phases are also investigated. Finally, using the effective band structure approach, we study the electronic band structure of the random alloys.

Figure 1

The pseudocubic lattice constant, i.e., $V_{\mathrm{cell}}^{1/3}$, obtained using the HSE, PBE, and PBEsol functionals for different alloy compositions.

Figure 2

Formation energies of the $\mathrm{AlGaN}$ alloys using supercells of 4, 8, 12, and 16 atoms, obtained by PBE calculations. The diamonds indicate the lowest formation energy of each composition while the circles correspond to the formation energies of the random alloys. The stars indicate the formation energies of the superlattice configurations along the [0001] direction. The lines are drawn to guide the eye.

Figure 3

The atomic configurations yielding the lowest formation energies for compositions of $x=0.125$, 0.167, 0.25, 0.333, 0.375, and 0.5. The large red and black atoms correspond to gallium and aluminum, respectively, while the small blue atoms refer to nitrogen.

Figure 4

The band gaps of the $\mathrm{AlGaN}$ alloys obtained by PBE and HSE calculations, using supercells of 4, 8, 12, and 16 atoms. The stars indicate the band gaps of superlattice configurations along the [0001] direction. The crosses correspond to the band gaps of the random alloys. The lines connecting the band gaps of the pure phases and the random alloys are drawn to guide the eye.

Figure 5

The planar average (solid curve) and macroscopic average (dashed curve) of the electrostatic potential for a fully relaxed $\mathrm{GaN}m$-plane slab, obtained by HSE calculations. The large red and the small blue spheres correspond to $\mathrm{Ga}$ and $\mathrm{N}$ atoms, respectively.

Figure 6

The relaxed surfaces of the large band-gap ${\mathrm{Al}}_{0.5}{\mathrm{Ga}}_{0.5}\mathrm{N}$ configuration for the $m$ plane (a) and $a$ plane (b). The small band-gap configurations for the $m$- and $a$-plane calculations are shown in (c) and (d), respectively.

Figure 7

The valence band and conduction band offsets of ${\mathrm{Al}}_{0.5}{\mathrm{Ga}}_{0.5}\mathrm{N}$ and pure $\mathrm{AlN}$ with respect to $\mathrm{GaN}$, calculated for both nonpolar directions of the wurtzite crystal. The inset shows just the valence band offset and the solid lines are drawn to guide the eye, while the dashed lines connect the pure phases linearly.

Figure 8

The effective band structures of the random alloys for $\mathrm{Al}$ content of $25\%$ (left), $50\%$ (middle), and $75\%$ (right) obtained by PBE calculations. The spectral weights are normalized in the range between 0 and 1 for all the compositions.