Statistical model applied to ${\mathrm{A}}_x{\mathrm{B}}_y{\mathrm{C}}_{1-x-y}\mathrm{D}$ quaternary alloys: Bond lengths and energy gaps of ${\mathrm{Al}}_x{\mathrm{Ga}}_y{\mathrm{In}}_{1-x-y}\mathrm{X}$ ($\mathrm{X}=\mathrm{As}$, P, or N) systems

We extend the generalized quasichemical approach (GQCA) to describe the ${\mathrm{A}}_x{\mathrm{B}}_y{\mathrm{C}}_{1-x-y}\mathrm{D}$ quaternary alloys in the zinc-blende structure. Combining this model with ab initio ultrasoft pseudopotential calculations within density functional theory, the structural and electronic properties of ${\mathrm{Al}}_x{\mathrm{Ga}}_y{\mathrm{In}}_{1-x-y}\mathrm{X}$ ($\mathrm{X}=\mathrm{As}$, P, or N) quaternary alloys are obtained, taking into account the disorder and composition effects. Results for the bond lengths show that the variation with the compositions is approximately linear and also does not deviate very much from the value of the corresponding binary compounds. The maximum variation observed amounts to 3.6% for the In-N bond length. For the variation of band gap, we obtain a bowing parameter $b=0.26\mathrm{eV}$ for the ${\left({\mathrm{Ga}}_{0.47}{\mathrm{In}}_{0.53}\mathrm{As}\right)}_z{\left({\mathrm{Al}}_{0.48}{\mathrm{In}}_{0.52}\mathrm{As}\right)}_{1-z}$ quaternary alloy lattice matched to InP, in very good agreement with experimental data. In the case of AlGaInN, we compare our results for the band gap to data for the wurtzite phase. We also obtained a good agreement despite all evidences for cluster formation in this alloy. Finally, a bowing parameter of $0.22\mathrm{eV}$ is obtained for zinc-blende AlGaInN lattice matched with GaN.

Figure 1

(Color online) Al-N, Ga-N, and In-N bond lengths ($d_{\mathrm{Al}\text{−}\mathrm{N}}$, $d_{\mathrm{Ga}\text{−}\mathrm{N}}$, and $d_{\mathrm{In}\text{−}\mathrm{N}}$, respectively) as functions of the Al content $x$ and Ga content $y$ for ${\mathrm{Al}}_x{\mathrm{Ga}}_y{\mathrm{In}}_{1-x-y}\mathrm{N}$ alloys.

Figure 2

(Color online) Al-P, Ga-P, and In-P bond lengths ($d_{\mathrm{Al}\text{−}\mathrm{P}}$, $d_{\mathrm{Ga}\text{−}\mathrm{P}}$, and $d_{\mathrm{In}\text{−}\mathrm{P}}$, respectively) as functions of the Al content $x$ and Ga content $y$ for ${\mathrm{Al}}_x{\mathrm{Ga}}_y{\mathrm{In}}_{1-x-y}\mathrm{P}$ alloys.

Figure 3

(Color online) Al-As, Ga-As, and In-As bond lengths ($d_{\mathrm{Al}\text{−}\mathrm{As}}$, $d_{\mathrm{Ga}\text{−}\mathrm{As}}$, and $d_{\mathrm{In}-\mathrm{As}}$, respectively) as functions of the Al content $x$ and Ga content $y$ for ${\mathrm{Al}}_x{\mathrm{Ga}}_y{\mathrm{In}}_{1-x-y}\mathrm{As}$ alloys.

Figure 4

(Color online) Energy gap as function of Al content $x$ for ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-1.23x}{\mathrm{In}}_{0.23x}\mathrm{N}$ alloys been lattice matched with GaN. The circles are the results of the calculation, and the solid line represents a fit.

Figure 5

(Color online) Energy gap as function $z$ for ${\left({\mathrm{Ga}}_{0.47}{\mathrm{In}}_{0.53}\mathrm{As}\right)}_z{\left({\mathrm{Al}}_{0.48}{\mathrm{In}}_{0.52}\mathrm{As}\right)}_{1-z}$ alloys, with $z$ varying from 0 to 1. These alloys are lattice matched with InP. The circles are the results of the calculation, and the solid line corresponds to the fit.