Phonon Heat Conduction in Thin Films: Impacts of Thermal Boundary Resistance and Internal Heat Generation

The measured thermal resistance across a thin film deposited on a substrate often includes the internal thermal resistance within the film and the thermal boundary resistance (TBR) across the film-substrate interface. These two resistances are frequently lumped and reported as an equivalent thermal conductivity of the film. Two fundamental questions should be answered regarding the use of this equivalent thermal conductivity. One is whether it leads to the correct temperature distribution inside the film. The other one is whether it is applicable for thin films with internal heat generation. This paper presents a study based on the Boltzmann transport equation (BTE) to treat phonon heat conduction inside the film and across the film-substrate interface simultaneously, for the cases with and without internal heat generation inside the film. Material systems studied include $SiO2$ and diamond films on Si substrates, representative of thin-film materials with low and high thermal conductivity. It is found that for a $SiO2$ film on a Si substrate, the film thermal conductivity and TBR can be treated independently, while for a diamond film on a Si substrate, the two are related to each other by the interface scattering. When the free surface behaves as a black phonon emitter, the TBR for thin diamond films with internal heat generation is the same as that without the internal heat generation. When the free surface is adiabatic, however, the TBR increases and approaches the value of the corresponding black surface as the film thickness increases. Results of this study suggest that great care must be taken when extending the effective thermal conductivity measured for thin films under one experimental condition to other application situations.