T_0$, relative to a reference temperature, $T_0$, the equilibrium distribution is proportional to $\Delta T$, as shown in Eq. (\[eq:BEDist\_Linearized\]). Here, $\hbar$ is the reduced Planck constant, $\omega$ is the phonon frequency, $D(\omega)$ is the phonon density of states, $f_{BE}$ is the Bose-Einstein distribution, and $C_{\omega} = \hbar\omega D(\omega)\frac{\partial f_{BE}}{\partial T}$ is the mode specific heat. The volumetric heat capacity is then given by $C = \int_0^{\omega_m}C_{\omega}d\omega$ and the Fourier thermal conductivity $k = \int_0^{\omega_m}k_{\omega}d\omega$, where $k_{\omega} = \frac{1}{3} C_{\omega}v_{\omega} \Lambda_{\omega}$ and $\Lambda_{\omega} = \tau_{\omega}v_{\omega}$ is the phonon MFP. To close the problem, energy conservation is used to relate $g_{\omega}$ to $\Delta T$, given by $$\int\int_0^{\omega_m} \left[\frac{g_{\omega}(x,t)}{\tau_{\omega}}-\frac{1}{4\pi}\frac{C_{\omega}}{\tau_{\omega}}\Delta T(x,t) \right]d\omega d\Omega = 0,
\label{eq:EnergyConservation}$$ where $\Omega$ is the solid angle in spherical coordinates and $\omega_m$ is the cut-off frequency. Note that summation over phonon branches is implied without an explicit summation sign whenever an integration over phonon frequency or MFP is performed.

We now divide our discussion into three parts: transducer film, substrate, and interface. The BTE in the transducer film can be reformulated as a Fredholm integral equation of the second kind[@hua_semi-analytical_2015]; the solution in the substrate can be treated as the Green’s function to the BTE.[@Hua2014b] The solutions in 