Study of the optic phonons in the III-V quaternary alloy system: ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$ by the average $t$-matrix approximation

The average-$t$-matrix approximation (ATA) is applied to a study of the ir-active optic phonons in the III-V quaternary alloy system ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$. Assignments of the features in the ir reflectance of GaAs-rich and GaP-rich alloys has been previously based on qualitative comparisons with the pseudobinary alloys, $\mathrm{Ga}{\mathrm{As}}_{1-x}{\mathrm{P}}_x$ and ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$, and the end-member compounds. New results presented in this paper for a ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{P}$ alloy indicate serious limitations in this approach. To obtain a more quantitative basis for mode assignments in the quaternaries, studies of the optic-phonon density of states have been made using the ATA. Calculations are restricted to linear-chain models with a mass-defect formulation in which changes in the optic-phonon density of states are the result of the mass substitutions. In the low-concentration limit, the ATA is shown to give results that are similar to the coherent-potential approximation. By comparing alloy ${\epsilon }_2$ spectra with the ATA density of states, it is possible to unambiguously identify all of the features in the ir reflectance spectra. For the dilute alloys considered in this paper, the most interesting features in the spectrum are pair-impurity modes.