  Understanding the limits of phononic heat dissipation from a two-dimensional
layered material (2DLM) to its hexagonal boron nitride (h-BN) substrate and how
it varies with the structure of the 2DLM is important for the design and
thermal management of h-BN-supported nanoelectronic devices. We formulate an
elasticity-based theory to model the phonon-mediated heat dissipation between a
2DLM and its h-BN substrate. By treating the h-BN substrate as a semi-infinite
stack of harmonically coupled thin plates, we obtain semi-analytical
expressions for the thermal boundary conductance (TBC) and interfacial phonon
transmission spectrum. We evaluate the temperature-dependent TBC of the
$N$-layer 2DLM (graphene or MoS$_{2}$) on different common substrates (h-BN vs.
a-SiO$_{2}$) at different values of $N$. The results suggest that h-BN is
substantially more effective for heat dissipation from MoS$_{2}$ than
a-SiO$_{2}$ especially at large $N$. To understand the limitations of the our
stack model, we also compare its predictions in the $N=\infty$ limit to those
of the more exact Atomistic Green's Function model for the graphite-BN and
molybdenite-BN interfaces. Our stack model provides clear insights into the key
role of the flexural modes in the TBC and how the anisotropic elastic
properties of h-BN affect heat dissipation.
