Influence of indium clustering on the band structure of semiconducting ternary and quaternary nitride alloys

The electronic band structures of ${\text{In}}_x{\text{Ga}}_{1-x}\text{N}$, ${\text{In}}_x{\text{Al}}_{1-x}\text{N}$, and ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ alloys are calculated by ab initio methods using a supercell geometry, and the effects of varying the composition and atomic arrangements are examined. Particular attention is paid to the magnitude of and trends in bowing of the band gaps. Indium composition fluctuation (clustering) is simulated by different distributions of In atoms and it is shown that it strongly influences the band gaps. The gaps are considerably smaller when the In atoms are clustered than when they are uniformly distributed. An explanation of this phenomenon is proposed on the basis of an analysis of the density of states and the bond lengths, performed in detail for ternary alloys. Results for the band gaps of ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ quaternary alloys show a similar trend. It is suggested that the large variation in the band gaps determined on samples grown in different laboratories is caused by different degrees of In clustering.

Figure 1

Theoretical equilibrium lattice parameters (a) $a$ and (b) $c$ for ${\text{In}}_x{\text{Ga}}_{1-x}\text{N}$ (circles) and ${\text{In}}_x{\text{Al}}_{1-x}\text{N}$ (squares) as functions of indium concentration $x$. Calculations are performed for two models of In distribution in the supercell: uniform (open symbols) and clustered (filled symbols). The dashed lines are linear interpolations between the end points.

Figure 2

(Color online) Energy gap of ${\text{In}}_x{\text{Al}}_{1-x}\text{N}$ as a function of composition $x$. Calculations (open circles) are shown for both the uniform (thin circles), most clustered (thick circles), and some intermediate In distribution models (squares). Solid lines are spline fits to the calculated values (uniform and most clustered cases). Measured optical energy gaps of ${\text{In}}_x{\text{Al}}_{1-x}\text{N}$ are also marked from a: Ref. 9, b: Ref. 49, and c: Ref. 50.

Figure 3

(Color online) Energy gap of ${\text{In}}_x{\text{Ga}}_{1-x}\text{N}$ as a function of composition $x$. Calculations are shown for both the uniform (thin circles) and clustered (thick circles) In distribution models. Solid lines are spline fits to the calculated values. Measured optical energy gaps of ${\text{In}}_x{\text{Ga}}_{1-x}\text{N}$ are also marked from a: Ref. 10, b: Ref. 11, and c: Ref. 12.

Figure 4

Valence-band density of states for InN (dotted line), AlN (dashed line), and ${\text{In}}_{0.25}{\text{Al}}_{0.75}\text{N}$ (solid line) for the case of a uniform In distribution.

Figure 5

Valence-band density of states for ${\text{In}}_{0.25}{\text{Al}}_{0.75}\text{N}$ for the case of a clustered In distribution (case 1, see text). The decomposition into Al (dashed line), In (short dashed line), and N (dotted line) partial DOS is shown. The region of interest is the top $\sim 1\text{eV}$ of the valence band, marked with vertical bars.

Figure 6

Partial nitrogen band density of states for ${\text{In}}_{0.25}{\text{Al}}_{0.75}\text{N}$ for the same case as in Fig. 5; however, it is further decomposed into In nearest neighbors N2 (solid line) and other nitrogen N1 (dotted line) contributions.

Figure 7

Same as Figs. 5, 6, but enlarged around the valence-band top.

Figure 8

Valence-band density of states for InN (dotted line), GaN (dashed line), and ${\text{In}}_{0.25}{\text{Ga}}_{0.75}\text{N}$ (solid line) for the case of a uniform In distribution.

Figure 9

(Color online) Valence-band density of states for ${\text{In}}_{0.25}{\text{Ga}}_{0.75}\text{N}$ for the case of a clustered In distribution (case 1, see text). The decomposition into Ga (dashed line), In (lowest line), N1 (short dashed line), and N2 (line) partial DOS is shown. The region of interest is the top $\sim 0.45\text{eV}$ of the valence band, marked with vertical bars.

Figure 10

Schematic arrangement of atoms for InN, AlN, and GaN. Nitrogen is in the middle and its four nearest-neighbor cations are shown. The bond lengths (in $\text{Å}$) are indicated.

Figure 11

Schematic arrangement of atoms for ${\text{In}}_{0.25}{\text{Al}}_{0.75}\text{N}$ and ${\text{In}}_{0.25}{\text{Ga}}_{0.75}\text{N}$ in the uniform and clustered cases (see text for discussion). Symbols are defined in Fig. 10. Nitrogen is in the middle, and only the configuration around nitrogen atoms with largest number of In atoms as nearest neighbor is shown. The calculated equilibrium bond lengths (in $\text{Å}$) are indicated

Figure 12

Energy gaps (in eV) of ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ for a series of $x$ values, as a function of Ga concentration $y$, for a uniform model of the In distribution. The lines are smooth fits to the calculated points.

Figure 13

Energy gaps (in eV) of ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ for $x=0.19$, as a function of Ga concentration $y$, for uniform (open circles) and clustered (filled circles) model of the In distribution. The lines are smooth fits to the calculated points.

Figure 14

Energy gaps (in eV) of ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ for a series of $y$ values, as a function of In concentration $x$, for a uniform In distribution model. The lines are guides to the eyes.

Figure 15

Energy gaps (in eV) of ${\text{In}}_x{\text{Ga}}_y{\text{Al}}_{1-x-y}\text{N}$ for $y=0.38$, as a function of In concentration $x$, for uniform (open circles) and clustered (filled circles) In distribution models. The lines are guides to the eyes.