Temperature Rise in 3D Anisotropic Layered Structures in Thermoreflectance and 3$\omega$ Experiments

Recent developments of transient thermal measurement techniques including the thermoreflectance methods (time-domain/frequency-domain thermoreflectance, TDTR/FDTR) and the 3{\omega} method enabled measurements of anisotropic thermal conductivities of bulk and thin film materials. Estimating the temperature rise of anisotropic layered structures under surface heating is critically important to make sure that the temperature rise is not too high to alias the signals in these experiments. However, a simple formula to estimate the temperature rise in 3D anisotropic layered systems is not available yet. In this work, we provide simple analytical expressions to estimate the peak temperature rise in anisotropic layered structures for the cases of both laser heating (as in thermoreflectance experiments) and metal strip heating (as in 3{\omega} experiments). We started by solving the 3D anisotropic heat diffusion equation in the frequency domain for a multilayered structure and derived a general formalism of the temperature rise. This general formalism, while normally requiring numerical evaluation, can be reduced to simple analytical expressions for the case of semi-infinite solids. We further extended these analytical expressions to multilayered systems, assuming linear temperature gradients within the thin layers above the semi-infinite substrates. These analytical expressions for multilayered systems are compared with the exact numerical solutions of the heat diffusion equation for several different cases and are found to work generally well but could overestimate the temperature rise in some special occasions. These simple analytical expressions serve the purpose of estimating the maximum temperature rise of 3D anisotropic layered systems, which greatly benefits the thermoreflectance and 3{\omega} experiments in choosing the appropriate heating power and heater sizes for the experiments.