Phonons in parabolic quantum wells

Phonons in parabolic-quantum-well structures fabricated with semiconductor alloys are studied. We calculate frequencies, displacement patterns, and densities of states for lattice vibrations in a (GaAs${)}_{\mathit{N}1}$(${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As${)}_{\mathit{N}2}$(001) superlattice where the GaAs layers are replaced by a mixed crystal ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}\mathit{s}}$${\mathrm{Al}}_{\mathit{x}\mathit{s}}$As with Al fraction ${\mathit{x}}_{\mathit{s}}$ varying quadratically with atomic layer s. The influence of disorder, composition, and the two-mode behavior of optical phonons are investigated within a random-element-isodisplacement model, generalized to layered structures with arbitrary composition profiles.