Proceedings of the 3rd Micro/Nanoscale Heat & Mass Transfer International Conference MNHMT2012 March 3-6, 2012, Atlanta, Georgia, USA DRAFT: MNHMT2012-75087 ASSESSMENT OF FOURIER-BASED THERMAL MODELS USED IN FREQUENCY-DOMAIN THERMOREFLECTANCE DATA ANALYSIS D.P. Sellan, 1 V. Mishra, 1,2 J.A. Malen, 3 A.J.H. McGaughey ∗ , ,3 and C.H. Amon 1,3 1 Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, Ontario, M5S 3G8, Canada 2 Department of Mechanical Engineering, India Institute of Technology, Kharagpur, West Bengal, 721302, India 3 Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Email: dan.sellan@utoronto.ca, vivek.mech.iitkgp@gmail.com, jonmalen@andrew.cmu.edu, mcgaughey@cmu.edu, ∗ and cristina.amon@utoronto.ca ABSTRACT We evaluate the ability of Fourier-based thermal models to capture sub-continuum thermal transport in a frequency-domain thermoreﬂectance (FDTR) measurement. We use the Boltzmann transport equation (BTE) to simulate phonon transport in a sim- pliﬁed FDTR setup. We compare the BTE-predicted, steady-state temperature proﬁles to those predicted by an analytical solu- tion of the Fourier-based conduction equation. The two mod- els agree well when ωτ < 1, where ω is the surface-temperature modulation-frequency and τ is the bulk phonon relaxation time, but diverge when ωτ > 1. INTRODUCTION An increased interest in measuring the thermal transport properties of nanostructured materials has led to the development of noncontact measurement techniques based on photothermal phenomena. One technique is the pump-probe-based frequency- domain thermoreﬂectance (FDTR) method [1, 2]. In one form of FDTR [2], the continuous wave (CW) pump laser is modulated and used to periodically heat the surface of a sample. This peri- odic surface heating results in a periodic thermal wave that prop- agates into the sample. The amplitude and phase of the thermal response at the sample surface depend on the thermal properties of the sample beneath. Because the reﬂectance of the sample surface is temperature dependent, the probe laser (also CW) be- comes modulated upon reﬂection, and is used to monitor the ther- ∗ Address all correspondence to this author. mal response. A Fourier-based thermal model is then ﬁt to phase response and/or amplitude data of the reﬂected probe beam to extract the sample thermal conductivity or thermal conductance of a buried interface [2]. Although Fourier-based thermal models can accurately de- scribe thermal transport in bulk systems, they fail when the char- acteristic size of a system becomes comparable to the phonon mean free paths, Λ, which can range from 1 nm to microns [3, 4]. At these scales, sub-continuum effects are present and phonon- level modeling becomes critical to accurately describe thermal transport [4, 5]. The goal of this work is to evaluate the ability of a Fourier- based thermal model to capture thermal transport in an FDTR measurement when sub-continuum effects are present. To do this, we simulate thermal transport in a simpliﬁed FDTR setup. A sinusoidal heat ﬂux is imposed at the surface of a semi-inﬁnite sample such that the thermal penetration depth of the propagat- ing thermal wave is comparable to the phonon mean free path. For the sake of simplicity and clarity, we track one phonon mode with average properties (i.e., the gray approximation) and ignore the metal ﬁlm that is typically put on top of the sample to maxi- mize the reﬂectance of the probe beam. We simulate thermal transport at the phonon level using the lattice Boltzmann method (LBM) to solve the one-dimensional Boltzmann transport equation (BTE) [4,6]. We compare the re- sulting steady-state temperature proﬁles to those predicted by an analytical Fourier-based thermal model that is consistent with models used currently in FDTR data analysis. 1 Copyright c  2012 by ASME