Transient and Steady-State Temperature Rise in Three-Dimensional Anisotropic Layered Structures in Pump-Probe Thermoreflectance Experiments

TL;DR

This paper derives explicit analytical formulas to estimate transient and steady-state temperature rise in 3D anisotropic layered structures under laser heating, aiding accurate thermoreflectance measurements and experimental design.

Contribution

It provides the first explicit analytical expressions for temperature rise in 3D anisotropic multilayered systems, extending previous numerical methods.

Findings

Derived general formalism for temperature rise in 3D anisotropic multilayers.

Reduced complex formulas to explicit analytical expressions for semi-infinite solids.

Enabled quick estimation of temperature rise to optimize thermoreflectance experiments.

Abstract

Recent developments of the pump-probe thermoreflectance methods (such as the beam-offset and elliptical-beam approaches of the time-domain and frequency-domain thermoreflectance techniques) enabled measurements of the thermal conductivities of in-plane anisotropic 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 three-dimensional (3D) anisotropic layered systems heated by a non-circular laser beam is not available yet, which is the main problem we aim to solve in this work. We first re-derived general formalisms of the temperature rise of a multilayered structure based on the previous literature work by solving the 3D anisotropic heat diffusion equation in the frequency domain. These general formalisms normally require laborious numerical evaluation; however, they could be reduced to explicit analytical expressions for the case of semi-infinite solids. We then extend the analytical expressions to multilayered systems, taking into account the effect of the top layers. This work not only enhances our understanding of the physics of temperature rise due to surface laser heating but also enables quick estimation of the peak temperature rise of 3D anisotropic layered systems in pump-probe thermoreflectance experiments and thus greatly benefits the thermoreflectance experiments in choosing the appropriate heating power intensity for the experiments.