Planar force-constant method for lattice dynamics of superstructures

Abstract

We present a simple and accurate method for dealing with phonons in superlattices, and illustrate it with the example of GaAs/AlAs heterostructures. In this approach the vibrations of the superlattice are described in terms of planar force constants, which are determined ab initio for one constituent (e.g., GaAs bulk) and extended to the other constituent (e.g., AlAs) by introduction of the ‘‘mass and charge approximation.’’ Within this approximation, the dynamical problem of the superlattice is represented as that of an infinite bulk crystal (e.g., GaAs), which is modified by a sequence of on-site perturbations. We systematically develop the formalism for the one-dimensional description of phonons propagating along the superlattice growth direction, derive its relation with the general three-dimensional description, and clarify its physical meaning. The representation of the Coulomb interactions in the one-dimensional formalism is described in some detail. Applications of the method to (GaAs${)}_{\mathit{m}}$/(AlAs${)}_{\mathit{n}}$ superlattices of several thicknesses are discussed.