Si are on the same order in TDTR \cite{}, Si is a suitable substrate for this work.

Second, we must determine the microscopic transfer function that maps the transmission coefficients directly to the TDTR signal without any artificial fitting parameters. This step is challenging due to the difficulty of efficiently solving the BTE for the TDTR experiment. A number of simplified models[@Maznve2011; @Minnich2011b; @Wilson2013; @Vermeersch2015a; @Regner2014; @Koh2014; @Maassen2015] have been proposed, but these models make various approximations that limit their predictive capability. In this work, we overcome this challenge using two recent advances we reported for rigorously solving the spectral BTE under the relaxation time approximation (RTA) that yield a factor of $10^4$ speedup compared to existing methods and allows the first ab-initio phonon transport modeling of TDTR free of artificial parameters or simplifications of the phonon dispersion. This solution is derived in the next section.

Before deriving the necessary functions, we briefly summarize the necessary steps in our approach. First, we obtain TDTR data for a given metal/substrate sample. We then calculate the intrinsic phonon dispersion and lifetimes for each material that are then inserted into the ab-initio transfer function derived in the next section. Finally, the transmission coefficients 