Thermal conductivity and sound velocities of hydrogen-silsesquioxane low-$k$ dielectrics

Thermal conductivities of hydrogen-silsesquioxane thin films—Dow Corning “flowable oxide” and nanoporous “extra-low-$k”$ spin-on dielectrics—are measured in the temperature range 80–400 K using the $3\omega$ method. Film thickness and atomic densities are characterized by the combination of Rutherford-backscattering spectrometry and variable-angle spectroscopic ellipsometry. Measurements of the longitudinal speeds of sound by picosecond ultrasonics and interferometry enable comparisons with the model of the minimum thermal conductivity of homogeneous materials. This model fails to capture the strong temperature dependence of the conductivity. Data for nanoporous silsesquioxane and ${\mathrm{SiO}}_2$ are compared to the predictions of effective medium theories of heterogeneous materials. Differential-effective-medium theory predicts a scaling of thermal conductivity $\Lambda$ with atomic density n, $\Lambda \propto n^{3/2}$ in good agreement with experiment. The comparisons with effective-medium theories suggest that a greater control of pore microstructure may enable significant improvements in the thermal and mechanical properties of porous dielectrics.