First-principles calculation of the thermodynamics of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys: Effect of lattice vibrations

The thermodynamics properties of the wurtzite and zinc-blende ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys are calculated using first-principles density-functional calculations. Special quasirandom structures are used to describe the disordered alloys, for $x=1∕4$, 1/2, and 3/4. The effect of lattice vibrations on the phase diagram, commonly omitted from semiconductor alloy phase diagram calculations, are included through first-principles calculations of phonon spectra. Inclusion of lattice vibrations leads to a large reduction in the order-disorder critical temperature ($\sim 29\%$ and $\sim 26\%$ for the wurtzite and zinc-blende structures, respectively) and changes the shape of the solubility and spinodal curve through changes in the entropies of the competing phases. Neglect of such effect produces significant errors in the phase diagrams of complex ordered semiconductor compounds. The critical temperature for phase separation is $1654\mathrm{K}$ $\left(1771\mathrm{K}\right)$ for the wurtzite (zinc-blende) structures. The predicted phase diagrams are in agreement with experimental measurements on metal-organic chemical-vapor deposition ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ films.

Figure 1

(Color online) The formation enthalpy of wurtzite (WZ) and zinc-blende (ZB) ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ as a function of composition $x$. The solid curve is a fit to the calculated values (open circles) with $(\alpha +\beta x)x(1-x)$. The valence force field results of Saito and Arakawa[16] are also included for comparison.

Figure 2

(Color online) Phonon density of states (DOS) of (a) wurtzite and (b) zinc-blende ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ correspond to $x=0$, $1∕4$, $1∕2$, $3∕4$, and $1$, which have been normalized according to ${\int }_0^{\infty }g\left(\nu \right)d\nu =3$, where $\nu =\omega ∕2\pi$. An arbitrary shift is added to the curves for clarity. The black and gray curves refer to the solid solution system $g_{{\mathrm{In}}_x{\mathrm{Ga}}_{(1-x)}\mathrm{N}}\left(\omega \right)$ and an average of the $\mathrm{InN}$ and $\mathrm{GaN}$ phases $g_{\mathrm{av}}\left(\omega \right)$, respectively.

Figure 3

(Color online) Vibrational free energy difference $\mathrm{\Delta }{G}_v$ as a function of temperature, for the wurtzite (solid lines) and zinc-blende (dashed lines) structures.

Figure 4

(Color online) The common tangent intersects the wurtzite $\mathrm{\Delta }G$ curve at $x_1=0.024$ and $x_2=0.95$, which gives the binodal curve in Fig. 5. Lattice vibrational effect has been taken into account. A representative case of $T=1000\mathrm{K}$ is used.

Figure 5

(Color online) The pseudobinary diagram (in temperature-cation concentration space) for (a) wurtzite and (b) zinc-blende ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ with and without the inclusion of lattice vibration effects. The solid lines correspond to the equilibrium binodal curves, and the dashed lines represent the spinodal curves.

Figure 6

(Color online) A comparison of the pseudobinary diagrams of zinc-blende and wurtzite ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ (including the effects of lattice vibrations). The solid lines correspond to the equilibrium binodal curves, and the dashed lines represent the spinodal curves. The points represented by the filled and open circles represent the experimental data[4] for which phase separation was and was not observed, respectively.