Variety of Normal Modes and Their Transition Behaviors in Graded Elastic Networks: Square Networks with a Diagonal Gradient

Abstract

We study vibrational excitations in graded elastic lattices modelled by coupled harmonic oscillators in a square lattice, in which the force constants or the vibrating masses vary along a diagonal direction, rather than along an orthogonal direction [Xiao, Yakubo, and Yu: Phys. Rev. B 73 (2006) 224201]. As in the latter case, we identified various kinds of vibrational normal modes in these diagonally graded square lattices, namely, unbound modes, and two types of confined modes called soft (heavy) and hard (light) “gradons” which are of relatively low and high frequencies, respectively, and reside at the two opposite edges of the lattices in the gradient direction. In addition to that, we report another type of confined modes which have localization in the middle part along the gradient direction. This kind of modes appear when the transverse wave number k s > k c and approaches to the zone boundary ( k s =π), where k c is a critical wave number depending on the strength of gradient. Between \(k_{\text{c}}<k_{\text{s}}\leqslant\pi\), no extended mode can be observed for whatever frequency. We name this new family of confined modes as soft–hard (light–heavy) gradons because they resemble soft (light) gradons at one end, while behaves as hard (heavy) gradons at the other end. The properties of the various normal modes and their transitions are elaborated, showing differences between orthogonally graded square lattices and diagonally graded square lattices. The underlying mechanism of the transitions is discussed with the help of a band overlapping picture.