Dispersion and topological characteristics of permutative polyatomic phononic crystals

This work presents a comprehensive mathematical treatment of phononic crystals (PCs) which comprise a finite lattice of repeated polyatomic unit cells. Wave dispersion in polyatomic lattices is susceptible to changes in the local arrangement of the monatoms (subcells) constituting the individual unit cell. We derive and interpret conditions leading to identical and contrasting band structures as well as the possibility of distinct eigenmodes as a result of cyclic and non-cyclic cellular permutations. Different modes associated with cyclic permutations yield topological invariance, which is assessed via the winding number of the complex eigenmode. Wave topology variations in the polyatomic PCs are quantified and conditions required to support edge modes in such lattices are established. Next, a transfer function analysis of finite polyatomic PCs is used to explain the formation of multiple Bragg band gaps as well as the emergence of truncation resonances within them. Anomalies arising from the truncation of the infinite lattice are further exploited to design mirror symmetrical edge modes in an extended lattice. We conclude with a generalized explanation of the band gap evolution mechanism based on the Bode plot analysis.