Influence of bulk-phonon-branch dispersion on displacement patterns and the intermixing of interface and confined optical phonons in superlattices.

Abstract

An envelope-function formalism is developed to treat zone-center superlattice phonons with inclusion of effects due to the full bulk-phonon-branch dispersion and the long-range macroscopic electric field. Starting from a rigid-ion model, differential equations for the envelopes of the atomic displacement patterns are derived. It is shown that both the transverse component of the electric field and the longitudinal component of the electric displacement field that appear automatically satisfy the boundary conditions of macroscopic electrodynamics. For optical and acoustical phonons, the same additional mechanical boundary conditions for the envelopes themselves are extracted from the underlying microscopic equations of motion. They therefore depend on the atomic structure of the interfaces. Explicit results are given for (GaAs${)}_{\mathit{N}1}$(AlAs${)}_{\mathit{N}2}$ (001) superlattices. In the case of p-polarized symmetrical modes they show a strong anisotropy and intermixing of nominal confined and interface optical phonons. The considered material combination with nearly the same bulk acoustic branches and well-separated optical-phonon bands allows some approximations giving physical insights into the correct solutions of the dispersionless macroscopic continuum theory. Starting from the concept of effective layer thicknesses or a band-edge expansion, one obtains analytical results for frequencies, confinement wave numbers, and displacement patterns, which represent generalizations of the macroscopic continuum theory with the influence of the bulk-phonon-branch dispersion taken into account.