Steady state and modulated heat conduction in layered systems predicted by the analytical solution of the phonon Boltzmann transport equation

Abstract

Based on the phonon Boltzmann transport equation under the relaxation time approximation, analytical expressions for the temperature profiles of both the steady state and modulated heat conduction inside a thin film deposited on a substrate are derived and analyzed. It is shown that these components of the temperature depend strongly on the ratio between the film thickness and the average phonon mean free path (MFP), and they exhibit the diffusive behavior as predicted by the Fourier's law of heat conduction when this ratio is much larger than unity. In contrast, in the ballistic regime when this ratio is comparable to or smaller than unity, the steady-state temperature tends to be independent of position, while the amplitude and the phase of the modulated temperature appear to be lower than those determined by the Fourier's law. Furthermore, we derive an invariant of heat conduction and a simple formula for the cross-plane thermal conductivity of dielectric thin films, which could be a useful guide for understanding and optimizing the thermal performance of the layered systems. This work represents the Boltzmann transport equation-based extension of the Rosencwaig and Gersho work [J. Appl. Phys. 47, 64 (1976)], which is based on the Fourier's law and has widely been used as the theoretical framework for the development of photoacoustic and photothermal techniques. This work might shed some light on developing a theoretical basis for the determination of the phonon MFP and relaxation time using ultrafast laser-based transient heating techniques.