  Interfacial thermal resistance has been often estimated and understood using
the Landauer formalism that assumes incident phonons with equilibrium
distribution. However, previous studies suggest that phonons are
out-of-equilibrium near the interface because of the heat flow through the
leads and the scattering of phonons by the interface. In this paper, we report
a systematic study on how vibrational spectra mismatch affects the degree of
phonon non-equilibrium near an interface, how fast it is relaxed as the phonons
diffuse into a lead, and the overall interfacial thermal resistance from the
non-equilibrium phonons. Our discussion is based on the solution of the
Peierls-Boltzmann transport equation with ab initio inputs for 36 interfaces
between semi-infinite group-III (Al, Ga, In) and group-V (P, As, Sb) compound
semiconductor leads. The simulation reveals that the non-equilibrium phonons
cause significant interfacial thermal resistance for all 36 interfaces, making
the overall interfacial thermal resistance two to three times larger than that
predicted by the Landauer formalism. We observe a clear trend that the degree
of phonon non-equilibrium near an interface and the interfacial thermal
resistance from the non-equilibrium phonons increase as the mismatch of the
Debye temperature of two lead materials increases. This contrasts with Landauer
formalism's predictions, which show no correlation with the Debye temperature
mismatch. The relaxation length of the phonon non-equilibrium varies
significantly from 50nm to 1.5um depending on the combination of the lead
materials. The relaxation length is proportional to the phonon mean free path
of the corresponding lead material but also largely depends on the material in
the opposite lead. This suggests the relaxation length cannot be considered an
intrinsic property of the corresponding lead material.
