
Department of Physics and Centre for Computational Science and Engineering,National University of Singapore, Singapore 117542, Republic of Singapore(Dated: July 27, 2010)The phonon localized edge modes are systematically studied, and two conditions are proposedfor the existence of the localized edge modes: (I) coupling between different directions (x, y orz) in the interaction; (II) different boundary conditions in three directions. The generality ofthese two conditions is illustrated by different lattice structures: one-dimensional (1D) chain, 2Dsquare lattice, 2D graphene, 3D simple cubic lattice, 3D diamond structure, etc; and with differentpotentials: valence force field model, Brenner potential, etc.
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