  We study the effect of SiC substrate on thermal conductivity of epitaxial
graphene nanoribbons (GNRs) using the nonequilibrium molecular dynamics method.
We show that the substrate has strong interaction with single-layer GNRs during
the thermal transport, which largely reduces the thermal conductivity. The
thermal conductivity characteristics of suspended GNRs are well preserved in
the second GNR layers of bilayer GNR, which has a weak van der Waals
interaction with the underlying structures. The out-of-plane phonon mode is
found to play a critical role on the thermal conductivity variation of the
second GNR layer induced by the underlying structures.
