Modeling the compositional instability in wurtzite ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{N}$

The paper deals with multiscale modeling of the minor component ordering in wurtzite ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{N}$ $(x<0.5)$ alloys. The treatment combines the total-energy density-functional calculations of the metal atom interaction parameters and the atomistic description of the alloy decomposition using lattice kinetic Monte Carlo. It is demonstrated that the phase decomposition patterns in wurzite GaInN are very sensitive to the interplay of metal atom interactions at several interatomic distances (at least to the fourth nearest neighbors) on the cation sublattice. Variation of the metal interaction energies within reasonable limits resulted in pronouncedly different relaxation patterns (linear or wall ordering of In and Ga atoms along $c$-axis, planar ordering parallel to basal plane, spinodal decomposition). The high sensitivity of the GaInN decomposition to relatively small variations of the metal interaction energies could be the main reason for the experimentally observed versatility of the alloy decomposition patterns and their sensitivity to the particular experimental conditions.

Figure 1

The wurtzite structure of ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{N}$ alloy. Black spheres represent nitrogen atoms, while white and shaded spheres represent metal atoms. The shading is used to mark the special dispositions of atoms considered in the paper. $a$ and $c$ are the basal and prismatic lattice parameters, respectively.

Figure 2

(a) The dependence of the basal lattice parameter $a$ on In content in the alloy ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{N}$ ($c∕a$ fixed at 1.632). (b) The variation of equilibrium $c∕a$ ratio as a function of In content in a cell with parameter $a$ fixed at a value corresponding to pure GaN.

Figure 3

Typical patterns of In atom distribution in LKMC simulation cell after annealing of ${\mathrm{Ga}}_{0.9}{\mathrm{In}}_{0.1}\mathrm{N}$ alloy at (a) $473\mathrm{K}$, (b) $673\mathrm{K}$, and (c) $873\mathrm{K}$, assuming the energy gains for local short range ordering as given in Table . Only In atoms are shown. The indium atomic chains, especially at higher temperatures, are in dynamic equilibrium, being continuously destroyed and recreated. The equilibrium is achieved well within 1 million vacancy jumps, which, assuming vacancy concentration of $6\times {10}^{-5}$ (one vacancy per simulation cell) and migration energy of $1.6\mathrm{eV}$ (Ref. 37), corresponds to times of $\sim 100\text{years}$, $2.4\mathrm{h}$, and $15\mathrm{s}$ for cases (a), (b), and (c), respectively.

Figure 4

The atomic relaxation pattern around an In atom in GaN (in 192-atom ab initio cell). For comparison, the calculated equilibrium Ga-N and Ga-Ga distances are 1.936 and $3.153\mathrm{\AA }$, respectively.

Figure 5

The ordering patterns in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ after annealing at $473\mathrm{K}$. [(a) and (c)] $x=0.25$, duration: 30 million vacancy jumps. The set of In-In binding energies (in eV) at 1NN–4NN positions: ($-0.11$, $-0.04$, 0, and 0.11). [(b) and (d)] $x=0.5$, duration: 5 million vacancy jumps. In-In binding energies (in eV) set: ($-0.11$, $-0.04$, 0, and 0.11). White spheres in (c) and (d) and subsequent figures represent In atoms, while dark spheres represent Ga; nitrogen atoms are not shown. (a) and (b) show only In atoms viewed along ⟨0001⟩, while (c) and (d) are three-dimensional views of simulation cells with basal plane parallel to the top surface.

Figure 6

The ordering patterns in ${\mathrm{In}}_{0.25}{\mathrm{Ga}}_{0.75}\mathrm{N}$ annealed with the In-In binding energies (in eV) of ($-0.08$, 0.04, 0.01, and 015). [(a)–(c)] $T=273\mathrm{K}$, intermediate configuration of the same LKMC run at 6, 20, and 70 million vacancy jumps, respectively. (d) $T=473\mathrm{K}$, 20 million vacancy jumps.

Figure 7

Some $c$-aligned ordering patterns in LKMC-annealed ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$. [(a) and (b)] The In-In binding energy set of ($-0.08$, $-0.04$, $-0.03$, and $0.07\mathrm{eV}$). Annealing (a) at $273\mathrm{K}$ for $2\times {10}^6$ direct In-Ga exchange jumps and (b) at $473\mathrm{K}$ for $4\times {10}^6$ direct In-Ga exchange jumps. [(c) and (d)] The In-In binding energy set of ($-0.08$, $-0.04$, 0.02, and $0.015\mathrm{eV}$). Annealing (c) at $273\mathrm{K}$ for $2\times {10}^6$ direct In-Ga exchange jumps and (d) at $473\mathrm{K}$ for $3\times {10}^6$ direct In-Ga exchange jumps. In all figures, only In atoms are shown; the viewing axes are close to ⟨0001⟩.

Figure 8

Examples of 1:1 ordering patterns in LKMC-annealed ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$. [(a) and (b)] The In-In binding energy set of ($-0.08$, 0.04, 0.03, and $015\mathrm{eV}$). Annealing for $1.5\times {10}^7$ vacancy jumps at (a) $473\mathrm{K}$ and (b) $673\mathrm{K}$. [(c) and (d)] The In-In binding energy set of ($-0.08$, $-0.04$, $-0.01$, and $0.015\mathrm{eV}$) only In atoms are shown. Annealing for $2\times {10}^6$ direct In-Ga exchange jumps at (c) $273\mathrm{K}$ and (d) $473\mathrm{K}$.

Figure 9

Clearly pronounced 1:1 ordering patterns in LKMC-annealed ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$ with the In-In binding energy set of ($-0.08$, $-0.04$, $-0.02$ and $-0.015\mathrm{eV}$). Annealing for ${10}^6$ direct In-Ga exchange jumps at (a) $273\mathrm{K}$ and (b) $473\mathrm{K}$, viewed parallel to basal planes. (c) A sequence of annealing steps of the same run (from 0 to $6\times {10}^6$ direct In-Ga exchange jumps with the interval of ${10}^6$ jumps) at $673\mathrm{K}$, viewed along the $c$ axis. The loop at intermediate annealing steps is a domain boundary.

Figure 10

Spinodal decomposition in ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$ during the LKMC annealing run at $473\mathrm{K}$. The In-In binding energy set: (0.08, 0.04, $-0.01$, and $-0.015\mathrm{eV}$). (a)–(d) show the spatial distributions of In atoms after (0, 1, 10, and 100) $\times {10}^6$ vacancy jumps. (e) shows the distribution of both Ga (dark) and In (light) atoms in LKMC simulation cell after $9\times {10}^7$ vacancy jumps.

Figure 11

The binding energies of four In-In pair configurations at different biaxial strains.

Figure 12

The ordering patterns predicted by LKMC as a result of annealing at $273\mathrm{K}$. (a) $x=0.25$, $6\times {10}^6$ In jumps; (b) $x=0.375$, $9\times {10}^6$ In jumps; (c) $x=0.5$, $3\times {10}^6$ In jumps. In all cases, the view is along the $c$ axis.

Figure 13

The patterns predicted by LKMC as a result of annealing of initially random ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$ with the input parameters summarized in Table . (a) $273\mathrm{K}$, $1.8\times {10}^7$ vacancy jumps; (b) $473\mathrm{K}$, $2\times {10}^8$ vacancy jumps; (c) $673\mathrm{K}$, ${10}^7$ vacancy jumps.