An inverse problem framework for reconstruction of phonon properties using solutions of the Boltzmann transport equation

Abstract

A methodology for reconstructing phonon properties in a solid material, such as the frequency-dependent relaxation time distribution, from thermal spectroscopy experimental results is proposed and extensively validated. The reconstruction is formulated as a non-convex optimization problem whose goal is to minimize the difference between the experimental results and the one calculated by a Boltzmann transport equation (BTE)-based model of the experimental process, with the desired material property treated as the unknown in the optimization process. Crucially, the proposed approach makes no assumption of an underlying Fourier behavior, thus avoiding all approximations associated with that assumption. The proposed method combines a derivative-free optimization method, referred to as the Nelder-Mead algorithm, with a graduated (multi-stage) optimization framework.

Our results show that, compared to other reconstruction methods, the proposed method is less sensitive to scarcity of data in a specific transport regime (such as submicron length scales). The method is also very versatile in incorporating known information into the optimization process, such as the known value of the material thermal conductivity or solid-solid interface conductance if a material interface is present; addition of this information improves the quality of the optimization. In the presence of a material interface of unknown conductance, we show that simultaneous reconstruction of both the solid-solid interface frequency-dependent transmissivity function and the relaxation time function is possible. The optimization algorithm is validated using both synthetically generated temperature profiles (generated by solving the BTE), as well as experimentally measured signals.

In the case of synthetic input data, the reconstructed properties are compared to the material models used to create the input data. In the case of experimental data, we compare the reconstructed phonon properties with their corresponding benchmark values, obtained using either theoretical predictions, such as relaxation times from density functional theory, or experimentally measured, such as the experimentally measured interface transmissivities. The interface transmissivity reconstruction is also validated on the 2D-dots geometry in the presence of Al-Si interface. Our results show good accuracy in all cases. The reliability and uniqueness of the optimized solution as well as its statistical properties due to the presence of noise are studied using a number of statistical techniques.

Our analysis provides strong evidence that the formulated optimization problem has a unique solution; furthermore the proposed optimization-based framework is capable of finding that solution with good accuracy.