Characterization of band structure for transverse acoustic phonons in Fibonacci superlattices by a bandedge formalism

We present a divergence-free method to determine the characteristics of band structures and projected band structures of transverse acoustic phonons in Fibonacci superlattices. A set of bandedge equations is formulated to solve the band structures for the phonon instead of using the traditional dispersion relation. Numerical calculations show band structures calculated by the present method for the Fibonacci superlattice without numerical instability, which may occur in traditional methods. Based on the present formalism, the band structure for the acoustic phonons has been characterized by closure points and the projected bandgaps of the forbidden bands. The projected bandgaps are determined by the projected band structure, which is characterized by the cross points of the projected bandedges. We observed that the band structure and projected band structure and their characteristics were quite different for different generation orders and the basic layers for the Fibonacci superlattice. In this study, concise rules to determine these characteristics of the band structure and the projected band structure, including the number and the location of closure points of forbidden bands and those of projected bandgaps, in Fibonacci superlattices with arbitrary generation order and basic layers are proposed.