recently read some papers on thermal interface conductance and found they are interesting

This first one is:

• A formalism for calculating the modal contributions to thermal interface conductance

which gives the complete formulas and its derivative on the modal contributions to thermal interface conductance.

The second one is:

• Phonon transport at interfaces: Determining the correct modes of vibration in which they talked about how to classified the vibration modes into different categories and how does these modes look like( Fig. 5).

And since then they have applied this methods to many different interfaces, such as the interface between crystalline Si/Ge and InGaAs/InP, as well as the interfaces between different phases of materials.

• Phonon Transport at Crystalline Si/Ge Interfaces: The Role of Interfacial Modes of Vibration
• Interface conductance modal analysis of lattice matched InGaAs/InP
• Phonon transport at interfaces between different phases of silicon and germanium

These paper may be helpful for you to obtain a better understanding on the concepts of phonon, vibration mode and thermal transport in materials and at interfaces.