Direct-to-Indirect Crossover in Semiconductor Alloys: A First-Order Phase Transition?

We study the direct-to-indirect gap crossover in ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys driven by Al addition, in analogy with temperature-induced phase transitions. The adopted real-space formalism incorporates occupational disorder in a realistic manner: different atomic configurations, accommodated in a supercell, are generated and solved independently. We perform a systematic study of the scaling of calculated gap properties of ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys with the cell size, and consider system sizes ranging from 64 to 8000 atoms. Extrapolation to infinite system size follows scaling laws appropriate for first-order phase transitions, and allows an accurate determination of the crossover composition $x_c$.