  An algorithm for mapping the true phonon modes of a film, which are defined
by a two-dimensional (2D) Brillouin zone, to the modes of the corresponding
bulk material, which are defined by a three-dimensional (3D) Brillouin zone, is
proposed. The algorithm is based on normal mode decomposition and is inspired
by the observation that the atomic motions generated by the 2D eigenvectors
lead to standing-wave-like behaviors in the cross-plane direction. It is
applied to films between two and ten unit cells thick built from Lennard-Jones
(LJ) argon, whose bulk is isotropic, and graphene, whose bulk (graphite) is
anisotropic. For LJ argon, the density of states deviates from that of the bulk
as 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's
anisotropy and the emergence of a quadratic acoustic branch. As such, while the
mapping algorithm works well for the thicker LJ argon films, it does not
perform as well for the thinner films as there is a weaker correspondence
between the 2D and 3D modes. For graphene, the density of states of even the
thinnest films closely matches that of graphite due to the inherent anisotropy,
except for a small shift at low frequency. As a result, the mapping algorithm
works well for all thicknesses of the graphene films, indicating a strong
correspondence between the 2D and 3D modes.
