Assessing Multivariate Thermal Transport Relationships in Pump-Probe Experiments Using Sensitivity Matrix

Sensitivity analysis is a powerful technique to analyze the behaviors of models and experimental data in diverse fields.^1–4 Most state-of-the-art characterization techniques rely on fitting experimental data to a theoretical multivariate model. Such a model often has multiple unknown parameters at the same time. These unknowns are either determined independently or, when impractical, are fitted together in a single set of measured data. It is well-known that such a multivariate fit can become invalid, especially when these unknowns are related.⁵ To complicate matters, such a relationship can change with a different sample configuration and measurement conditions. Unfortunately, few rigorous methods exist to evaluate the validity of such a fit or uncover the governing transport relationships under different measurement conditions.<br/>Here, we formulate a systematic approach based on sensitivity analysis to uncover the governing physics in a complex multivariate theoretical model and assess the feasibility of performing a multivariate fit. A sensitivity matrix of the unknown variables is constructed and, later, decomposed using the singular-value decomposition (SVD) method. The resulting solution of three matrices provides insight to the fitting process. The rank of the rectangular diagonal matrix with non-negative real numbers on the diagonal reveals the number of independent fit-able variables within the tested range. Under certain conditions, the associated complex unitary matrix contains information of the thermal transport relationships in the measured system.<br/>Different pump-probe techniques for thermal characterization are used to illustrate the capability of this method. We found that the existing experience-based criteria,^5,6 e.g., the overlapping sensitivity curves to identify related parameters or the non-overlapping and distinct sensitivity curves for multivariate suitability, are inadequate and applicable only in certain situations. Also, our method can uncover the governing heat transfer relationships through different multilayer samples and measurement conditions. This method can apply to most measurement techniques relying on multivariate theoretical models in various disciplines and can help to improve their measurement validity and throughput.<br/><br/><b>Reference:</b><br/>1. Razavi, S. <i>et al.</i> The Future of Sensitivity Analysis: An essential discipline for systems modeling and policy support. <i>Environ. Model. Softw.</i> <b>137</b>, (2021).<br/>2. Saltelli, A., Jakeman, A., Razavi, S. & Wu, Q. Sensitivity analysis: A discipline coming of age. <i>Environ. Model. Softw.</i> <b>146</b>, 105226 (2021).<br/>3. Kala, Z. Sensitivity analysis in probabilistic structural design: A comparison of selected techniques. <i>Sustain.</i> <b>12</b>, (2020).<br/>4. Qian, G. & Mahdi, A. Sensitivity analysis methods in the biomedical sciences. <i>Math. Biosci.</i> <b>323</b>, 108306 (2020).<br/>5. Yang, J., Ziade, E. & Schmidt, A. J. Uncertainty analysis of thermoreflectance measurements. <i>Rev. Sci. Instrum.</i> <b>87</b>, (2016).<br/>6. Schmidt, A. J., Cheaito, R. & Chiesa, M. A frequency-domain thermoreflectance method for the characterization of thermal properties. <i>Rev. Sci. Instrum.</i> <b>80</b>, 094901 (2009).