High-frequency homogenisation for hexagonal and honeycomb lattices

A high-frequency asymptotic scheme is generated that captures the motion of waves within discrete hexagonal and honeycomb lattices by creating continuum homogenised equations. The accuracy of these effective medium equations in describing the frequency-dependent anisotropy of the lattice structure is demonstrated. We then extend the general formulation by introducing line defects, often called armchair or zigzag line defects for honeycomb lattices such as graphene, into an otherwise perfect lattice creating surface waves which propagate in the direction of the defect and decay away from it. A quasi-one-dimensional multiple scale method is outlined, which allows us to derive Schroedinger equations describing the local oscillations near particular frequencies in the Bloch spectrum. Further localization by single defects embedded within the line defect are also considered.