Eigenenergies of linear systems with periodic structures

Abstract

Using a new technique of amplitude analysis, the eigenvalues and eigenenergies are analyzed for finite and infinite linear harmonic chains with any periodic structures of two species. A new phase function which is "additive" for each unit cell can be constructed. In terms of this phase function, the eigenvalue equation is reduced to a low-order algebraic equation specified completely by the unit cell. Symmetry properties among the "cyclicly" permuted unit cells, analytic properties, vertex classification, diagram expansions, leading behavior, and reduction constraints (relating structure of more complicated unit cell to less complicated ones) are developed. For infinite systems, the density of states can be expressed in closed form as a single term. Eigenvectors also demonstrate collective behavior throughout the whole system.