$T_C$, and at the surrounding silicon wafer, held at the bath temperature $T_B$. The net thermal power transmitted from the island to the heat bath in the ballistic limit is therefore obtained by summing over the power carried by each mode, giving $$P_{bal} = \displaystyle \sum_i \int_{\nu_i}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu,
\label{eq:Pbal}$$ where $\nu_i$ is the cut-off frequency of the $i$th mode and $$B(\nu,T) = \frac{h\nu}{\mathrm{e}^{h\nu/k_BT}-1}$$ is the single-mode Power Spectral Density (PSD). Previously, we have demonstrated a strong agreement between the net thermal power given by Eq. (\[eq:Pbal\]) and measurements on TESs with leg lengths less than 4$\mu$m, and a range of widths [@DJBallistic]. This work confirmed that the net power can be calculated from first principles, through the bulk elastic constants, without free parameters, independently of the precise stoichiometry of the SiN$_{\rm x}$.

Figure \[fig:Modes\_Plot\](b) shows the total net PSD, $B(\nu,T_C) - B(\nu,T_B)$, summed over all simulated modes, for $T_C = 135$mK and $T_B = 68$mK. Sharp discontinuities are evident where individual modes cut on, corresponding to the intercepts of the dispersion curves with the ordinate of Fig. \[fig:Modes\_Plot\](a). At these experimentally representative temperatures, the PSD rolls off such that modes with $\nu_i \gtrsim 25$GHz carry negligible 