Lifshitz-type formulas for graphene and single-wall carbon nanotubes: van der Waals and Casimir interactions

Lifshitz-type formulas are obtained for the van der Waals and Casimir interaction between graphene and a material plate, graphene and an atom or a molecule, and between a single-wall carbon nanotube and a plate. The reflection properties of electromagnetic oscillations on graphene are governed by the specific boundary conditions imposed on the infinitely thin positively charged plasma sheet, carrying a continuous fluid with some mass and charge density. The obtained formulas are applied to graphene interacting with Au and Si plates, to hydrogen atoms and molecules interacting with graphene, and to single-wall carbon nanotubes interacting with Au and Si plates. The generalizations to more complicated carbon nanostructures are discussed.

Figure 1

The results normalized to the case of ideal metals for the van der Waals and Casimir energy (a) and force (b) per unit area between a graphene and a semispace vs separation. The solid and dashed lines are related to the semispace made of Au and Si, respectively.

Figure 2

The function $h_0$ in Eq. (21) vs the dimensionless parameter $t=\Omega c^2∕\left({\omega }_p^{2}a\right)$.

Figure 3

The normalized results to the case of ideal metals for the van der Waals and Casimir energy per unit area between a graphite plate of thickness $d$ and Au semispace vs relative thickness. Lines 1, 2, 3, 4, and 5 are related to separations $a=1$, 0.5, 0.3, 0.1, and $0.05\mu \mathrm{m}$, respectively. The respective normalized interaction energies of a graphene with an Au semispace are marked on the vertical axis (see text for further discussion).

Figure 4

The van der Waals coefficient $C_3$ for the interaction of a hydrogen atom (the solid line) and a molecule (the dashed line) with graphene vs separation.

Figure 5

The results normalized to the case of ideal metals for the Casimir energy per unit length between a carbon nanotube and Au semispace vs separation.