Ordering in ternary nitride semiconducting alloys

We present a thorough theoretical study of ordering phenomena in nitride ternary alloys ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$, ${\mathrm{Al}}_x{\mathrm{In}}_{1-x}\mathrm{N}$, and ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$. Using the Monte Carlo approach and energetics based on the Keating model, we analyze the influence of various factors on ordering in bulk crystals and epitaxial layers. We characterize the degree of both short-range order (SRO) and long-range order (LRO) for different compositions, temperatures, and for substrates associated with different epitaxial strain. For the description of the SRO, the Warren-Cowley parameters related to the first four coordination shells are used. The LRO is detected by means of the introduced sim-LRO parameter, based on the Bragg-Williams approach. The description of the observed long-range ordering patterns and conditions for their occurrence follows.

Figure 1

Sample convergence test: the dependence of the Warren-Cowley short-range order parameter ${\Gamma }_{\mathrm{GaIn}}^{\left(i\right)}$ on the $M\times M\times M$ supercell size in bulk ${\mathrm{Ga}}_{0.25}{\mathrm{In}}_{0.75}\mathrm{N}$.

Figure 2

Differences between short-range order parameters estimation from simulations powered by two different random number generators: RanLux from the GSL library and Wichmann-Hill from the ACML package. Simulations were carried out for the ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ on InN substrate in $T=873$ K.

Figure 3

Different types of behavior of $S_{\mathrm{Ga}}$ during simulations for ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ alloy on InN substrate. Three gallium concentrations $x$ are presented: $x=0.17$ (a), $x=0.50$ (b), $x=0.72$ (c). Dashed lines correspond to the average value of $S_{\mathrm{Ga}}$ and its spread $\Delta S_{\mathrm{Ga}}$.

Figure 4

The short-range order parameters in ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ as a function of composition for different temperatures. Note that the scale on all graphs is the same to allow for direct comparison of ordering magnitude in different coordination shells. Continuous lines represent third-order polynomial fits to simulated data. The extrema for presented fits are marked with open symbols. The composition range approximately corresponding to violet-blue-green wavelength of emitted light (400–570 nm) is gray shaded.

Figure 5

The short-range order parameters in ${\mathrm{Al}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ as a function of composition for different temperatures. Note that the scale on all graphs is the same to allow for direct comparison of ordering magnitude in different coordination shells. Continuous lines represent third-order polynomial fits to simulated data. The extrema for presented fits are marked with open symbols. The composition of ${\mathrm{Al}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ lattice matched to GaN is indicated with a gray line.

Figure 6

The epitaxial layer in the presence of lattice mismatch between the layer material and the substrate material. The case of purely elastic accommodation of strain is presented, i.e., without misfit dislocations (lattice-coherent case).

Figure 7

Summary of quantities related to ordering in ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ grown on GaN. The SRO parameters ${\Gamma }_{\mathrm{GaIn}}^{\left(i\right)}$ (a), their difference with respect to bulk case $\Delta {\Gamma }_{\mathrm{GaIn}}^{\left(i\right)}$ (b), the misfit and relaxation strains calculated on the basis of elasticity theory (TOE) or Keating VFF (c), and the critical thickness (d). The composition range approximately corresponding to violet-blue-green wavelength of emitted light (400–570 nm) is gray shaded.

Figure 8

Summary of the quantities related to the ordering in ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ grown on InN. The sim-LRO parameter $S_{\mathrm{Ga}}$ and its spread $\Delta S_{\mathrm{Ga}}$ (a), the SRO parameters ${\Gamma }_{\mathrm{GaIn}}^{\left(i\right)}$ (b), the misfit and relaxation strains calculated on the basis of the elasticity theory (TOE) or the Keating VFF (c), and the critical thickness (d). Open symbols denote the occurrences of the LRO. CH stands for the chalcopyrite and PE for the perpendicular plane’s ordering. The composition range approximately corresponding to violet-blue-green wavelength of emitted light (400–570 nm) is gray shaded.

Figure 9

Atomic arrangement in the chalcopyrite structure (the CH pattern). The ideal structure (a) is compared with the sample atomic arrangement obtained for ${\mathrm{Ga}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{N}$ on InN during Monte Carlo simulation (b). Note that expansion in the basal (substrate) plane and corresponding compression in the perpendicular plane is visible, together with thermal displacements.

Figure 10

Different planar ordering observed in ${\text{Ga}}_{0.83}{\text{In}}_{0.17}\text{N}$ for two different substrates. For InN substrate, the ordering in the direction perpendicular to the substrate (the PE pattern) occurs (a). For SiC substrate, ordering in the planes parallel to the substrate (the PA pattern) is present (b). Only In atoms (minor component) are displayed for clarity.

Figure 11

Summary of the quantities related to the ordering in ${\mathrm{Ga}}_x{\mathrm{In}}_{1-x}\mathrm{N}$ grown on 3C-SiC. The sim-LRO parameter $S_{\mathrm{Ga}}$ and its spread $\Delta S_{\mathrm{Ga}}$ (a), the SRO parameters ${\Gamma }_{\mathrm{GaIn}}^{\left(i\right)}$ (b), the misfit and relaxation strains calculated on the basis of the elasticity theory (TOE) or the Keating VFF (c), and the critical thickness (d). Open symbols denote the occurrences of the LRO of the parallel plane’s type (PA). The composition range approximately corresponding to violet-blue-green wavelength of emitted light (400–570 nm) is gray shaded.