Continuum-model acoustic and electronic properties for a Fibonacci superlattice

A continuum model is used to study the properties of acoustic phonons and of electrons in a semiconductor Fibonacci superlattice. Close contact is made with previous work based on discrete models by using a transfer-matrix approach. The mapping of transfer-matrix traces under the inflation transformation which defines the Fibonacci sequence has an invariant I, which has been identified as a key parameter in characterizing the Cantor-set spectra. We derive analytic expressions for the dependence of I on phonon frequency or electron energy and comment on the qualitative differences between discrete and continuum models. Transport properties are also discussed.