Thermal van der Waals interaction between graphene layers

The van de Waals interaction between two graphene sheets is studied at finite temperatures. Graphene’s thermal length $\left({\xi }_T=\hslash v/k_{B}T\right)$ controls the force versus distance $\left(z\right)$ as a crossover from the zero temperature results for $z⪡{\xi }_T$, to a linear-in-temperature, universal regime for $z⪢{\xi }_T$. The large separation regime is shown to be a consequence of the classical behavior of graphene’s plasmons at finite temperature. Retardation effects are largely irrelevant, both in the zero and finite temperature regimes. Thermal effects should be noticeable in the van de Waals interaction already for distances of tens of nanometers at room temperature.

Figure 1

Template diagram for the computation of the vdW force, as in Eqs. (2, 17, A1). Bubbles: isolated graphene’s response (charge or current). Thin wavy lines: Coulomb interaction (or photon propagator). Thick wavy lines: multiple-scattering-corrected interaction (or photon propagator) between graphene’s layers as in Eq. (3). Zigzag line represents force $\left(f_c\right)$: distance derivative of the interaction line (or photon propagator).

Figure 2

(Color online) van der Waals (scaled) force per area $\tilde{f}$ between graphene planes versus distance $z$ in units of the thermal length ${\xi }_T$. Continuous line: numerical result. Dashed line: large-distance $\left(z⪢{\xi }_T\right)$ limit, Eq. (14). Dashed-dotted line: zero temperature limit $\left(z⪡{\xi }_T\right)$ [Eq. (8)]. Inset: enlarged view of the crossover region $\left(z\sim {\xi }_T\right)$.

Figure 3

Left panel: light-cone $\left(\omega =cq\right)$ versus dynamics of a dielectric of typical frequency ${\omega }_0$, illustrating the importance of retardation for distances beyond $z_{ret}\sim q_{ret}^{-1}$. Right panel: light-cone versus graphene’s typical dynamics ($\omega =vq$, not to scale) illustrating the absence of a characteristic distance for retardation.

Figure 4

Left panel: light-cone $\left(\omega =cq\right)$ versus dynamics of a dielectric of typical frequency ${\omega }_0$, illustrating the three vdW regimes encountered with increasing distance at finite temperature [Eq. (21)]. Right panel: As in left panel but for a typical 2D metal, with dynamics characterized by plasmons [Eq. (22)].