  We investigate the near-field radiative heat transfer between a normally
and/or laterally shifted nanoparticle and a planar fused silica slab coated
with a strip graphene grating. For this study we develop and use a scattering
matrix approach derived from Fourier modal method augmented with local basis
functions. We find that adding a graphene sheet coating on the slab can already
enhance the heat flux by about 85%. We show that by patterning the graphene
sheet coating into a grating, the heat flux is further increased, and this
happens thanks to the a topological transition of the plasmonic modes from
circular to hyperbolic one, which allows for more energy transfer. The lateral
shift affects the accessible range of high-$k$ modes and thus affects the heat
flux, too. By moving the nanoparticle laterally above the graphene grating, we
can obtain an optimal heat flux with strong chemical potential dependance above
the strips. For a fixed graphene grating period ($D=1\mu$m) and not too large
normal shift (separation $d<800$nm), two different types of lateral shift
effects (e.g., enhancement and inhibition) on heat transfer have been observed.
As the separation $d$ is further increased, the lateral shift effect becomes
less important. We show that the lateral shift effect is sensitive to the
geometric factor $d/D$. Two distinct asymptotic regimes are proposed: (1) the
inhibition regime ($d/D<0.85$), where the lateral shift reduces the heat
transfer, and (2) the neutral regime ($d/D \geq 0.85$) where the effect of the
lateral shift is negligible. In general, we can say that the geometric factor
$d/D \approx 0.85$ is a critical point for the lateral shift effect. Our
predictions can have relevant implications to the radiative heat transfer and
energy management at the nano/micro scale.
