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Using the density functional theory (DFT) with the generalized gradientapproximation (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 energycalculations using the full--potential linearized augmented plane wave(FP--LAPW) method to investigate the structural and electronic properties. Inboth alloys we found that the fundamental parameters do not follow Vegard'slaw. The lattice parameters, $a, c,$ and $u$, for the Al$_x$Ga$_{1-x}$N alloyare found to exhibit downward bowing, while for In$_x$Ga$_{1-x}$N there is anupward bowing for the $a$ and $c$ parameters and a downward bowing for theinternal parameter, $u$. Furthermore, we found that for both alloys, the bandgap value does not follow Vegard's law. As a by--product of our electronic bandstructure calculations, the effective masses of the binary compounds as well astheir related alloys were calculated. We show that the calculated propertiesfor the binary compounds, as well as for the studied alloys, show goodagreement with most of the previously reported results. Finally, using thefrozen phonon approach, the A$_1(TO)$ mode for the different systems studied inthis work was calculated. Our calculations show good agreement withexperimental values reported for the binary compounds. For the ternary alloys,our calculations reproduce experimental values for Al$_x$Ga$_{1-x}$N as well astheoretical predictions for In$_x$Ga$_{1-x}$N.

Ab initio calculation of structural and electronic properties of Al$_x$Ga$_{1-x}$N and In$_x$Ga$_{1-x}$N alloys