Normal and lateral Casimir-Lifshitz forces between a nanoparticle and a graphene grating

We study the normal and lateral components of the Casimir-Lifshitz (CL) force between a nanoparticle and a 1D graphene grating deposited on a fused silica slab. For this purpose, the scattering matrix approach together with the Fourier modal method augmented with local basis functions are used. We find that, by covering a fused silica slab by a graphene grating, the spectrum of the normal CL force at small frequencies is increased by about 100% for a grating filling fraction of 0.5, and even more when the slab is completely covered. The typically employed additive approximation (the weighted average of the force with and without the graphene coating) cannot provide any information on the lateral CL force and, as we show, cannot provide an accurate estimation for the normal CL force. When the nanoparticle is laterally shifted (${x}_{A}$), the normal CL force is modulated and remains attractive. On the contrary, the lateral CL force changes signs twice in each period, showing a series of alternating stable and unstable lateral equilibrium positions occurring in the graphene strips and the grating slits regions, respectively. Finally, we show that the lateral shift effect is sensitive to the geometric factor $d/D$ ($d$ is the separation distance, and $D$ is the grating period). We identify two clear regions: a region ($d/D<1.0$) where the lateral shift significantly affects the CL energy and a region ($d/D\ensuremath{\ge}1.0$) where this effect is negligible, with a crossover at $d\ensuremath{\approx}D$. Our predictions can have relevant implications to experiments and applications of the CL normal and lateral forces acting on nanoparticles interacting with structured objects at the nanoscale and microscale, and are also directly valid for atoms close to these nanostructures.