Anisotropy enhancement of the Casimir-Polder force between a nanoparticle and graphene

We derive the analytical expressions for the thermal Casimir-Polder energy and force between a spheroidal nanoparticle above a semi-infinite material and a graphene covered interface. We analyze in detail the Casimir-Polder force between a gold nanoparticle and a single sheet of pristine graphene focusing on the impact of anisotropy. We show that the effect of anisotropy, i.e., the shape and orientation of the spheroidal nanoparticle, has a much larger influence on the force than the tunability of graphene. The effect of tuning and anisotropy both add up such that we observe a force between the particle and the sheet of graphene which is between 20% and 50% of that between the same particle and an ideal metal plate. Hence the observed force is much larger than the results found for the Casimir force between a metal half-space and a layer of graphene.

Figure 1

Sketch of a spheroidal gold nanoparticle above a sheet of graphene.

Figure 2

Plot of $\sigma ={\sigma }_D+{\sigma }_I,{\sigma }_D$, and ${\sigma }_I$ as a function of the Matsubara terms.

Figure 3

Plots of the CP force of a Au nanoparticle above a Au half-space as a function of distance $d$. The force is normalized to the CP force of a gold nanoparticle above an ideal metal. Here and in the following we set $T=300$ K.

Figure 4

Plots of the CP force of a Au nanoparticle above a suspended sheet of graphene as a function of distance $d$ for different Fermi levels $E_F$. The results are normalized to the CP force of a Au nanoparticle above an ideal metal (top) or above a gold half-space (bottom).

Figure 5

Plots of the CP force of a spheroidal Au nanoparticle above a suspended sheet of graphene as a function of $R_{\mathrm{b}}/R_{\mathrm{a}}$ for different Fermi levels $E_F=0\mathrm{eV},1\mathrm{eV}$ and orientations (rotational axis is along $x$ or $z$ axis) choosing $d=100\mathrm{nm}$ (top) and $d=1\mu \text{m}$ (bottom). The results are normalized to the CP force of the spheroidal Au nanoparticle above an ideal metal.

Figure 6

Plots of the CP force of a spheroidal Au nanoparticle above a suspended sheet of graphene as a function of distance choosing a prolate particle with $R_{\mathrm{b}}/R_{\mathrm{a}}=0.1$, an oblate particle with $R_{\mathrm{b}}/R_{\mathrm{a}}=10$, and a spherical particle with $R_{\mathrm{b}}/R_{\mathrm{a}}=1$. The Fermi level of graphene is $E_F=0\mathrm{eV}$ and two different orientations of the nanoparticle are chosen: rotational axis is parallel to the graphene sheet $(x$ orientation, top) or normal to the graphene sheet $(z$ orientation, bottom). The results are normalized to the CP force of the spheroidal Au nanoparticle above an ideal metal. Furthermore, the dots or crosses in the top figure are the results using the nonlocal reflection coefficients for graphene from Ref. [45].

Figure 7

As Fig. 6 but with $E_{\mathrm{F}}=1\mathrm{eV}$.