Dielectric continuum model and Fröhlich interaction in superlattices

In an attempt to establish an equivalent of the Fröhlich interaction in superlattices, we are led to a critical examination of the dielectric continuum model by comparing with a parallel microscopic model. The reason that the usually quoted confined bulklike phonon modes derived from the dielectric continuum model are completely at variance with the results calculated from the microscopic model is explained. Simple rules for obtaining the proper bulklike modes are then set up, which lead to analytical expressions for the modes, which are found to agree closely with numerical results calculated from the microscopic model in the limit of zero dispersion for the bulk LO and TO phonons. They directly furnish expressions for the interaction with charged particles, which can be considered the equivalent to the Fröhlich interaction in superlattices. Phonon dispersion has the effect of mixing the interface modes into the bulklike modes with nearby frequencies. The small number of bulklike modes so affected are no longer confined to one material. The potentials of these modes apparently cannot be described by simple analytical expressions.