  The control of phonon scattering by interfaces is critical to the
manipulation of heat conduction in composite materials and semiconducting
nanostructures. However, one of the factors limiting our understanding of
elastic phonon scattering is the lack of a computationally efficient approach
for describing the phenomenon in a manner that accounts for the atomistic
configuration of the interface and the exact bulk phonon dispersion. Building
on the atomistic Green's function (AGF) technique for ballistic phonon
transport, we formulate an atomistic S-matrix method that treats bulk phonon
modes as the scattering channels and can determine the numerically exact
scattering amplitudes for individual two-phonon processes, enabling a highly
detailed analysis of the phonon transmission and reflection spectrum as well as
the directional dependence of the phonon scattering specularity. Explicit
formulas for the individual phonon reflection, absorption and transmission
coefficients are given in our formulation. This AGF-based S-matrix approach is
illustrated through the investigation of: (1) phonon scattering at the junction
between two isotopically different but structurally identical carbon nanotubes,
and (2) phonon boundary scattering at the zigzag and armchair edges in
graphene. In particular, we uncover the role of edge chirality on phonon
scattering specularity and explain why specularity is reduced for the ideal
armchair edge. The application of the method can shed new light on the
relationship between phonon scattering and the atomistic structure of
interfaces.
