Harmonic vibrations in two-dimensional graded elastic networks: Variety of normal modes and their transitions

We study vibrational excitations in graded elastic networks modeled by coupled harmonic oscillators in a square lattice, in which the force constants or the vibrating masses can vary along one direction, i.e., the gradient direction. It turns out that the two-dimensional network under study can be reduced to a set of effective one-dimensional graded chains [Phys. Rev. B 73, 054201 (2006)] with additional on-site potentials. We identify various kinds of vibrational normal modes in these networks with graded force-constant (mass), namely, unbound modes and two types of confined modes called soft (heavy) and hard (light) ``gradons'' which reside at the two opposite edges of the network in the gradient direction. The transitions from gradons to unbound modes occur at specific frequencies ωc1(kₛ) and ωc2(kₛ) for each corresponding wave number kₛ in the transverse direction. While above the maximum of ωc2(kₛ), pure hard (light) gradons exist, there is severe mixing of nondegenerate phonons and gradons below this frequency, showing intriguing zigzag inverse participation ratio. It is very interesting to see such unusual excitation modes that have adjacent eigenvalues but possess quite different spatial extents. The results reduce to the previously obtained one-dimensional results for kₛ=0. The method is quite general and applicable to three-dimensional elastic networks. We conclude with discussions on how these new gradon modes may affect the macroscopic properties of graded solids. Our results can also be applied to analogous systems with graded character.