Ab initio calculation of structural and electronic properties of   Al$_x$Ga$_{1-x}$N and In$_x$Ga$_{1-x}$N alloys
  Using the density functional theory (DFT) with the generalized gradient approximation (GGA), the structural and electronic properties of wurtzite AlN, GaN, InN, and their related alloys, Al$_x$Ga$_{1-x}$N and In$_x$Ga$_{1-x}$N, were calculated. We have performed accurate {\it ab initio} total energy calculations using the full--potential linearized augmented plane wave (FP--LAPW) method to investigate the structural and electronic properties. In both alloys we found that the fundamental parameters do not follow Vegard's law. The lattice parameters, $a, c,$ and $u$, for the Al$_x$Ga$_{1-x}$N alloy are found to exhibit downward bowing, while for In$_x$Ga$_{1-x}$N there is an upward bowing for the $a$ and $c$ parameters and a downward bowing for the internal parameter, $u$. Furthermore, we found that for both alloys, the band gap value does not follow Vegard's law. As a by--product of our electronic band structure calculations, the effective masses of the binary compounds as well as their related alloys were calculated. We show that the calculated properties for the binary compounds, as well as for the studied alloys, show good agreement with most of the previously reported results. Finally, using the frozen phonon approach, the A$_1(TO)$ mode for the different systems studied in this work was calculated. Our calculations show good agreement with experimental values reported for the binary compounds. For the ternary alloys, our calculations reproduce experimental values for Al$_x$Ga$_{1-x}$N as well as theoretical predictions for In$_x$Ga$_{1-x}$N. 
E. Lopez-Apreza, J. Arriaga, and D. Olguin