A scattering matrix formalism to model periodic heat diffusion in stratified solid media

Abstract

Transfer matrix formalism is widely used in modeling periodic heat diffusion in layered structures. Due to an intrinsic numerical instability issue, this formalism fails at high heating frequencies and/or in thick structures. Inspired by its success in modeling wave propagation, we develop a numerically stable scattering matrix framework to model periodic heat diffusion in stratified solid media. As a concrete example, we apply this scattering matrix methodology to the 3ω method. We first validate our framework using various well-known solutions. Next, we demonstrate the numerical stability of the framework using a configuration that resembles the three-dimensional stacked architecture for chip packing. Last, we propose synthetic “experiments” to exhibit, under certain circumstances, the merits of the scattering matrix formalism in extracting thermal properties.