Dispersion analysis with 45 degrees-rotated augmented supercells and applications in phononic crystal design

Abstract

The supercell approach can be useful for tailoring dispersion curves of phononic crystals, but the interpretation of the dispersion curves in the supercells can be often intriguing. Supercells formed by integer multiples of an original unit cell along its lattice axes are common, but there are also important situations requiring supercells formed by non-integer multiples of an original unit cell and its rotation with respect to its lattice axes. In these cases, not only dispersion branch folding, but also branch overlapping not found in common supercells, take place, which complicates the correct interpretation of band structures. In this study, we consider 45 degrees-rotated augmented supercells and analyze why and how branch folding and overlapping take place. For the analysis, the relation between the first Brillouin zone of an original cell and that of the corresponding 45 degrees-rotated augmented sup ercell is investigated. The analysis of the folding and overlapping mechanism found in the dispersion curve of the supercell is also useful for interpreting which branches can be excited over a target wavevector direction. The usefulness of the findings of this supercell-based dispersion analysis is demonstrated in unit cell design problems. Specifically, we show how to interpret correctly the dispersion curves of phononic crystals made of unit cells optimized by bandgap maximizing topology optimization when the optimized unit cells turn out to be 45 degrees-rotated augmented supercells. Conversely, throughout design optimization iterations, the original period of a unit cell that is initially set at the beginning of design optimization can be maintained if branch overlapping is forced not to occur. (C) 2015 Elsevier B.V. All rights reserved