  While the atomistic Green's function (AGF) method has the potential to
compute spectrally resolved phonon transport across interfaces, most prior
formulations of the AGF method provide only the total phonon transmission
function that includes contributions from all phonon branches or channels. In
this work, we present a formulation of the conventional AGF technique in terms
of phonon eigenspectra that provides a natural decomposition of the total
transmission function into contributions from various phonon modes. The method
involves the use of Dyson and Lippmann-Schwinger equations to determine surface
Green's functions from the phonon eigenspectrum of the bulk, and establishes a
direct connection between the transmission function and the bulk phonon spectra
of the materials forming the interface. We elucidate our formulation of the AGF
technique through its application to a microscopic picture of phonon mode
conversion at Si-Ge interfaces with atomic intermixing. Intermixing of atoms
near the interface is shown to increase the phase space available for phonon
mode conversion and to enhance thermal interface conductance at moderate levels
of atomic mixing. The eigenspectrum-based AGF (EAGF) method should be useful in
determination of microscopic mechanisms of phonon scattering and identification
of the specific modes that dominate thermal transport across an interface.
