Vibrational properties of percolating clusters: Localization and density of states

Abstract

The vibrational integrated density of states (DOS) of two-dimensional percolating clusters is calculated using a novel numerical technique. It is confirmed that the density of states is characterized by an ${\mathrm{\omega }}^{1/3}$ power-law behavior in the fracton regime, while a transition to a Debye-type spectrum occurs at lower frequencies. The dependence of the crossover frequency ${\mathrm{\omega }}_{\mathit{c}}$ and the coefficient ${\mathit{A}}_{\mathit{D}}$ of the Debye term in the DOS on the concentration p of the percolating cluster are numerically determined. By use of finite-size scaling methods, the localized nature of all the vibrational modes is established. We find that the fracton states as well as the phonon states are exponentially localized. There is no evidence for fracton superlocalization. However, there is a power-law dependence of the localization length versus frequency in the fracton regime, which crosses over to an exponential dependence in the low-frequency regime.