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Based on the phonon Boltzmann transport equation under the relaxation timeapproximation, analytical expressions for the temperature profiles of bothsteady state and modulated heat conduction inside a thin film deposited on asubstrate are derived and analyzed. It is shown that both steady state andmodulated components of the temperature depend strongly on the ratio betweenthe film thickness and the average phonon mean free path, and they exhibit thediffusive behavior as predicted by the Fourier law of heat conduction when thisratio is much larger than the unity. In contrast, in the ballistic regime whenthis ratio is comparable to or smaller than the unity, the steady-statetemperature tends to be independent of position, while the amplitude and thephase of the modulated temperature appear to be lower than those determined bythe Fourier law. Furthermore, we derived an invariant of heat conduction and asimple formula for the cross-plane thermal conductivity of dielectric thinfilms, which could be a useful guide for understanding and optimizing thethermal performance of the layered systems. This work represents the Boltzmanntransport equation-based extension of the Rosencwaig and Gerko work [J. Appl.Phys. 47, 64 (1976)], which is based on the Fourier law and has widely beenused as the theoretical framework for the development of photoacoustic andphotothermal techniques. This work might shed some light on developing atheoretical basis for the determination of the phonon MFP and relaxation timeusing ultrafast laser-based transient heating techniques.

Steady State and Modulated Heat Conduction in Layered Systems Predicted by the Phonon Boltzmann Transport Equation