Localization of a disordered phonon system: Anderson localization of optical phonons in ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$

Localization properties of an isotopically disordered phonon system are studied by mapping the system to an electronic tight-binding problem with on-site disorder. When the model parameters are adjusted for the ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys, it is shown that there exists $x_c$ such that for $x>\mathrm{}{x}_c$, all GaAs-like optical-phonon states are localized while for $x<\mathrm{}{x}_c$, most states are extended except for some near the edges of the GaAs-like branch. These results are confirmed by an independent numerical investigation of the scaling behavior of the localization lengths. Our results reconcile current controversy regarding the nature of GaAs-like optical-phonon eigen-states in the ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ alloys.