Role of biaxial strain and microscopic ordering for structural and electronic properties of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$

The structural and electronic properties of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys are studied as a function of $c$-plane biaxial strain and In ordering by density functional theory with the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional. A nonlinear variation of the $c$ lattice parameter with In content is observed in biaxial strain and should be taken into account when deducing In content from interplanar distances. From compressive to tensile strain, the character of the top valence-band state changes, leading to a nonlinear variation of the band gap in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$. Interestingly, the well-known bowing of the ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ band gap is largely removed for alloys grown strictly coherently on GaN, while the actual values for band gaps at $x<0.33$ are hardly affected by strain. Ordering plays a minor role for lattice constants but may induce changes of the band gap up to 0.15 eV.

Figure 1

Crystal structures of (a) ordered and (b) disordered ${\mathrm{In}}_{1/3}{\mathrm{Ga}}_{2/3}\mathrm{N}$, in which large green, pink, and small gray balls represent Ga, In, and N, respectively. For ${\mathrm{In}}_{2/3}{\mathrm{Ga}}_{1/3}\mathrm{N}$, the In and Ga positions are exchanged.

Figure 2

Lattice parameters of ordered and disordered ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ strain free or under biaxial strain corresponding to coherent growth on GaN. Gray markers indicate the values for pure GaN (left) and InN (right). The dashed lines interpolating linearly between them mark Vegard's law.

Figure 3

Calculated Poisson's ratio of GaN, ${\mathrm{In}}_{0.33}{\mathrm{Ga}}_{0.67}\mathrm{N}, {\mathrm{In}}_{0.67}{\mathrm{Ga}}_{0.33}\mathrm{N}$, and InN under biaxial strain. Solid lines plus solid markers and dashed lines plus open markers represent data from ordered and disordered structures, respectively. When ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ is grown coherently on GaN(0001), strain state and composition become mutably dependent. The dotted lines are cubic fits of the combined composition and strain dependence of Poisson's ratio for the ordered (red) and disordered (black) coherent cases.

Figure 4

(top) Band gap of GaN, ${\mathrm{In}}_{0.33}{\mathrm{Ga}}_{0.67}\mathrm{N}, {\mathrm{In}}_{0.67}{\mathrm{Ga}}_{0.33}\mathrm{N}$, and InN under biaxial strain (solid lines); the dashed lines plus solid markers and dotted lines plus open markers are for the transition from conduction-band minimum to HH/LH and CH bands, respectively. (bottom) Crystal-field splitting of GaN, ${\mathrm{In}}_{0.33}{\mathrm{Ga}}_{0.67}\mathrm{N}, {\mathrm{In}}_{0.67}{\mathrm{Ga}}_{0.33}\mathrm{N}$, and InN under biaxial strain.

Figure 5

Band gap of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ strain free (solid lines) and constrained by epitaxial growth on GaN substrates (dotted lines) as a function of In concentration. The dashed lines mark Vegard's law under either strain-free conditions (black) or biaxial strain (pink).

Figure 6

Comparison of interpolated band gap of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ under strain-free conditions with previous experimental [29] and theoretical [13] data. The theoretical data have been corrected for the band gap error as explained in the text.