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An algorithm for mapping the true phonon modes of a film, which are definedby a two-dimensional (2D) Brillouin zone, to the modes of the correspondingbulk material, which are defined by a three-dimensional (3D) Brillouin zone, isproposed. The algorithm is based on normal mode decomposition and is inspiredby the observation that the atomic motions generated by the 2D eigenvectorslead to standing-wave-like behaviors in the cross-plane direction. It isapplied to films between two and ten unit cells thick built from Lennard-Jones(LJ) argon, whose bulk is isotropic, and graphene, whose bulk (graphite) isanisotropic. For LJ argon, the density of states deviates from that of the bulkas the film gets thinner due to phonon frequencies that shift to lower values.This shift is a result of transverse branch splitting due to the film'sanisotropy and the emergence of a quadratic acoustic branch. As such, while themapping algorithm works well for the thicker LJ argon films, it does notperform as well for the thinner films as there is a weaker correspondencebetween the 2D and 3D modes. For graphene, the density of states of even thethinnest films closely matches that of graphite due to the inherent anisotropy,except for a small shift at low frequency. As a result, the mapping algorithmworks well for all thicknesses of the graphene films, indicating a strongcorrespondence between the 2D and 3D modes.

Mapping Phonon Modes from Reduced-Dimensional to Bulk Systems