Diffusive motion of ${\text{C}}_{60}$ on a graphene sheet

The motion of a ${\text{C}}_{60}$ molecule over a graphene sheet at finite temperature is investigated both theoretically and computationally. We show that a graphene sheet generates a van der Waals laterally periodic potential, which directly influences the motion of external objects in its proximity. The translational motion of a ${\text{C}}_{60}$ molecule near a graphene sheet is found to be diffusive in the lateral directions, while in the perpendicular direction, the motion may be described as diffusion in an effective harmonic potential which is determined from the distribution function of the position of the ${\text{C}}_{60}$ molecule. We also examine the rotational diffusion of ${\text{C}}_{60}$ and show that its motion over the graphene sheet is not a rolling motion.

Figure 1

(Color online) Left: periodic two-dimensional potential $\tilde{V}\left(x,y\right)$ created by the graphene sheet. Here, $x$ and $y$ refer to spatial position at the height $z_0=3.5\text{Å}$ above the graphene sheet. Right: periodic two-dimensional potential experienced by ${\text{C}}_{60}$ molecules near a graphene sheet. In this case $x$ and $y$ are the center of mass coordinates of ${\text{C}}_{60}$ measured from an origin on the sheet at the height $z=6.5\text{Å}$. This is obtained by summing the potential experienced by all individual atoms comprising the ${\text{C}}_{60}$ molecule.

Figure 2

(Color online) Periodic potential energy between flat graphene sheet and a single carbon atom at $x=0$ as a function of $y$ for several height values.

Figure 3

(Color online) Top: difference between maximum and minimum of two-dimensional potential (averaged over Wigner-Seitz primitive cell of both A and B lattices) created by a flat graphene sheet (see Fig. 1) as a function of normal distance $z$ as experienced by a single carbon atom. Bottom: total potential energy between a flat graphene sheet and a single carbon atom has been depicted. The solid (thick) blue curve is a simple LJ potential between two carbon atoms. In both panels, to eliminate the dependence on the $x$ and $y$ variables, we averaged over the first Brillouin zone of both A and B lattices. The data have small $\left({10}^{-5}\right)$ error bars, which are not shown.

Figure 4

(Color online) Periodic potential energy between ${\text{C}}_{60}$ molecule and flat graphene sheet at $x=0$ as a function of $y$ for several height values.

Figure 5

(Color online) Top: difference between maximum and minimum of two-dimensional potential of a flat graphene sheet as experienced by a ${\text{C}}_{60}$ molecule (see right panel of Fig. 1) as a function of the normal distance $z$. Bottom: total potential energy between a flat graphene sheet and a ${\text{C}}_{60}$ molecule. In both panels, to eliminate the dependence of the $x$ and $y$ variables, we averaged over Wigner-Seitz primitive cells of both A and B lattices with the data have infinitesimal $\left({10}^{-5}\right)$ error bars, which were not shown. Here, open triangular symbols give LJ potential result and open diamond symbols give the RDP1 potential result.

Figure 6

(Color online) Mean-square displacement $⟨R^2⟩$ versus time for the lateral motion of an ensemble of 30 ${\text{C}}_{60}$ molecules moving near a graphene sheet at room temperature. Inset shows a typical $x\text{−}y$ trajectory of a single ${\text{C}}_{60}$ on the graphene sheet.

Figure 7

(Color online) Effective potential of the interaction between a ${\text{C}}_{60}$ molecule and a graphene sheet.

Figure 8

(Color online) Rotational correlation: the correlation length of $\mu$ is about $75000$ steps (37.5 ps).

Figure 9

(Color online) Increments of ${\omega }_z$ (top) and ${\omega }_{xy}$ (bottom) versus time step.