  The Kapitza or thermal boundary resistance (TBR), which limits heat
dissipation from a thin film to its substrate, is a major factor in the thermal
management of ultrathin nanoelectronic devices and is widely assumed to be a
property of only the interface. However, data from experiments and molecular
dynamics simulations suggest that the TBR between a multilayer 2-dimensional
(2D) crystal and its substrate decreases with increasing film thickness. To
explain this thickness dependence, we generalize the recent theory for
single-layer 2D crystals by Ong, Cai and Zhang [Phys. Rev. B 94, 165427
(2016)], which is derived from the theory by Persson, Volokitin, and Ueba [J.
Phys.: Condens. Matter 23, 045009 (2011)], and use it to evaluate the TBR
between bare $N$-layer graphene and SiO$_{2}$. Our calculations reproduce
quantitatively the TBR thickness dependence seen in experiments and simulations
as well as its asymptotic convergence, and predict that the low-temperature TBR
scales as $T^{-4}$ in few-layer graphene. Analysis of the interfacial
transmission coefficient spectrum shows that the TBR reduction in few-layer
graphene is due to the additional contribution from higher flexural phonon
branches. Our theory sheds light on the role of flexural phonons in
substrate-directed heat dissipation and provides the framework for optimizing
the thermal management of multilayered 2D devices.
