Phonon localisation by impurities in a two-dimensional solid

Full text: A classical model of transverse vibrations of a disordered two-dimensional lattice is considered. The lattice is composed of two species of particles distinguished by their masses. The concentration of one species is much smaller than that of the other, thus representing an impurity. Site-independent interaction potential is assumed between nearest neighbours. Phonon modes of finite sized lattices are investigated by exact solution of the free vibration problem for randomly generated instances of the disordered system. Impurities are observed to give rise to strongly localised phonon modes, where the energy clusters near defects. The bulk of such modes are found within narrow bands of the spectrum, centred on frequencies that depend on the mass of the heavier particle only, and are associated with single-defect localised modes. An estimation of the widths of the bands, in terms of the density of states of the defect-free lattice and the impurity concentration, shows good agreement with numerical results. For a lattice of heavy particles with light defects, all high frequency modes are strongly localised due to non-propagative nature of the wave amplitude on heavy sites. Here, strong peaks in the density of states are attributable to pseudo-degenerate states arising from effectively isolated defects