Model for longitudinal-optical phonons and electron-phonon coupling in GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As multilayer structures

A continuum theory is developed to investigate the properties of the long-wavelength longitudinal-optical phonons in GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As multilayer structures and the associated electron-phonon interaction. Depending on the layer, the relative ionic displacements are related to GaAs or GaAs-type longitudinal-optical phonons and treated in the framework of the Born-Huang model, generalized to include isotropic dispersion effects in the Brillouin-zone center. For double heterostructures, a finite number of quantized confined modes is found. The interplay between the long-range Coulomb interaction, which couples the vibrations of adjacent GaAs layers, and confinement effects, which prevent the displacements of adjacent GaAs layers to overlap, is elucidated in the case of superlattices. The strength of the electron-phonon coupling in double heterostructures is reduced as compared with the electron–bulk-phonon effective coupling strength for quantum-well widths smaller than 100 Å.