The phonon localized edge-modes are systematically studied, and two conditions are proposed for the existence of the localized edge-modes: (I) coupling between different directions (x, y, or z) in the interaction and (II) different boundary conditions in three directions. The generality of these two conditions is illustrated by different lattice structures: one-dimensional (1D) chain, two-dimensional (2D) square lattice, 2D graphene, three-dimensional (3D) simple cubic lattice, 3D diamond structure, etc., and with different potentials: valence force field model, Brenner potential, etc.
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Jiang et al. (2010) studied this question.
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