Limitation of Matthiessen's rule in estimating the in-plane thermal conductivity of thin Cu films*

The scale of interconnects employed in the semiconductor industry is decreasing towards length scales comparable with the electron mean free path. As nanostructures becomes smaller, the thermal conductivity of interconnects decreases due to more frequent electron-electron, electron-phonon, or electron-system boundaries scattering. Matthiessen’s rule is used to assess these scattering contributions to the thermal conductivity of interconnects. In this work, we address the limitation of Matthiessen’s rule in estimating the thermal conductivities of copper (Cu) films at nanometer length scales. We perform thermoreflectance-based thermal conductance and ultrafast pump-probe experiments on annealed and unannealed Cu films ranging from 25 to 10000 nm thick, grown by physical vapor deposition (PVD) and electroplating methods. We directly measure the in-plane thermal conductivity of the Cu thin films via the time-domain thermoreflectance and steady-state thermoreflectance techniques. We find the in-plane thermal conductivity of thick films is relatively constant and begins to decrease when the thickness is less than the mean free path of Cu, which corresopnsds to a 35 nm Cu film. We measured the electronic-phonon coupling factors of Cu films by probing their reflectivity in the infrared. The electron-phonon coupling factor is relatively constant for all thicknesses even if the thickness is below the mean free path of Cu, indicating that electron-phonon scattering is not reducing the thermal conductivity of thin Cu films. This phenomenon is contradictory to Matthiessen’s rule, which accounts for all types of scattering in estimating thermal conductivity.