Phase stability, chemical bonds, and gap bowing of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys: Comparison between cubic and wurtzite structures

Thermodynamic, structural, and electronic properties of wurtzite ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys are studied by combining first-principles total energy calculations with the generalized quasichemical approach, and compared to previous results for the zinc-blende structure. Results for bond-lengths, second-nearest-neighbors distances, and bowing parameter are presented. We observed that the wurtzite results are not significantly different from the ones obtained previously for the zinc-blende structure. The calculated phase diagram of the alloy shows a broad and asymmetric miscibility gap as in the zinc-blende case, with a similar range for the growth temperatures, although with a higher critical temperature. We found a value of $1.44\mathrm{eV}$ for the gap bowing parameter giving support to the recent smaller band gap bowing findings. We emphasize that other theoretical results may suffer from incomplete sets of atomic configurations to properly describe the alloy properties, and experimental findings. Moreover one must take into account a broad composition range in order to obtain reliable results.

Figure 1

Illustration of wurtzite structure and the positions of the ions in the cluster. The white balls represent the cations (indium or gallium) and the black balls represent the anions (nitrogen).

Figure 2

Mean value of lattice constants $a$ and $c$ (solid line) vs In composition for relaxed ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ compared to values foreseen from Vegard law (dashed line).

Figure 3

Mean value of lattice constants $a$ and $c$ (solid line) in relaxed ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys, and the lattice constants $a_j$ and $c_j$ of each calculated cluster.

Figure 4

Averaged bond lengths (a) and second-nearest-neighbor distances (b) (solid line) in relaxed ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys. The filled squares describe measured values obtained by EXAFS (Ref. 53).

Figure 5

Mixing free energy $\Delta F(x,T)$ of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ wurtzite versus composition.

Figure 6

Phase diagram of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys at hexagonal and cubic structures. The solid lines represent the binodal curves and dashed lines are the spinodal curves.

Figure 7

Calculated energy gap $E_g$ (solid line) compared to the experimental data (different symbols). The dotted line represents the $E_g$ according to Vegard’s rule and the dashed line is the fit of McCluskey (Ref. 17): $E_g=3.47(1-x)+0.8x-2.6x(1-x)$.