On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials

Abstract

An analysis of the linear waves in infinitely-long square and triangular lattice strips of identical particles with nearest neighbour interactions for all combinations of fixed and free boundary conditions, as well as the periodic boundary, is presented. Expressions for the dispersion relations and the associated normal modes in these waveguides are provided in the paper; some of which are expressed implicitly in terms of certain linear combinations of the Chebyshev polynomials. The effect of next-nearest-neighbour interaction is also included for the square lattice waveguides. It is found that localized propagating waves, so called surface wave modes, occur in the triangular lattice strips, as well as square lattice strips with next-nearest-neighbour interactions, when either or both boundaries are free. In this paper, the even and odd modes are also discussed separately, wherever applicable. Graphical illustrations of the dispersion curves are included for all waveguides. The discrete waveguides analysed in the paper have broad applications in physics and engineering, including their merit in classical problems in elasticity, acoustics and electromagnetism, as well as recent technological issues involving various transport phenomena in quasi-one-dimensional nano-structures.

Transmission of waves across atomic step discontinuities in discrete nanoribbon structures

“…Each particle of such discrete nanoribbon structure is assumed to possess unit mass, and, interaction with atmost its four nearest neighbours through linearly elastic identical (massless) bonds with a spring constant 1/b 2 . The notation follows that introduced for an infinite lattice [33,34] and the square lattice waveguides [36,35]. Due to nature of the problem, the displacement of a particle, located at the site indexed by its lattice coordinates (x, y) ∈ Z 2 in S N and denoted by u x,y , is complex valued.…”

“…where A ∈ C is constant. The symbol a (κ in )· represents the mode shape corresponding to the specific wave mode indexed by κ in (see [36] or Appendix C of [35] for the detailed expressions).…”

“…Let the lattice frequency ω be defined as ω:=bω, so that ω 2 = 4(sin 2 1 2 κ x +sin 2 1 2 η κ in ), κ x ∈ [−π, π], where η κ in is interpreted as the vertical component of an incident wave vector in (bulk) lattice. Clearly, η κ in depends on the confinement effect of the boundary conditions [36]. The total displacement u t of an arbitrary particle in each lattice waveguide is a sum of the incident wave displacement u in and the scattered wave displacement u s .…”

“…1(a). Thus, S • •• is the waveguide constructed by a junction between S • • and S • • of width N (recall the notation described in Appendix A.1 or [36]). Using the general solution (1.4) in the square lattice (A.3) and upper boundary condition (1.2a),…”

“…behind) the step vs the energy flux carried for a specific incident wave mode [1]. Using the dispersion relation ω 2 = 4(sin 2 1 2 ξ + sin 2 1 2 η), ξ ∈ [−π, π], in the square lattice waveguides [36], since η is constant for a wave mode, it is easy to see that v (ξ):= ∂ω ∂ξ = ω −1 sin ξ, where v denotes the group velocity [1] of the propagating wave with wave number ξ. The energy flux in a wave mode, as a product of energy density and the group velocity, can be easily calculated (see [1]); for instance for the incident wave mode it is given by…”