Anomalous Dynamical Behavior of Freestanding Graphene Membranes

We report subnanometer, high-bandwidth measurements of the out-of-plane (vertical) motion of atoms in freestanding graphene using scanning tunneling microscopy. By tracking the vertical position over a long time period, a 1000-fold increase in the ability to measure space-time dynamics of atomically thin membranes is achieved over the current state-of-the-art imaging technologies. We observe that the vertical motion of a graphene membrane exhibits rare long-scale excursions characterized by both anomalous mean-squared displacements and Cauchy-Lorentz power law jump distributions.

Figure 1

(a) Outline of experimental setup. (b) Typical time trace of membrane height (above) and from a rigid sample (below). The inset is an expanded view of the freestanding graphene time trace. (c) Typical tunneling current profile during the measurement. (d) Mean-squared displacement (MSD) of membrane height as a function of time. Dashed lines are fits with slopes 1.4 and 0.3. The inset is the result of a simulation using exponential wait times and Cauchy jump lengths. Again, the dashed lines are fits with slopes 1.4 and 0.3.

Figure 2

(a) Velocity autocorrelation function (ACF) and instantaneous velocity (inset) computed from membrane height $z(t)$ shown in Fig. 1. (b) Measured freestanding graphene (FSG) membrane velocity probability distribution function (PDF) fitted to Cauchy-Lorentz and Gaussian distributions, along with the rigid control sample (square symbols). (c) Velocity PDFs and Cauchy-Lorentz fits (full curves) for different tunneling currents. (d) Variation in the FWHM of the velocity PDFs with tunneling current for two different bias voltage setpoints.

Figure 3

(a) Height of the central carbon atom in time from MD simulation for low temperatures (100 K) and high temperatures (3000 K). The high-temperature data are found to transition from positive to negative heights four times over 1 ns. A low-pass filtered height is also shown. (b) The jump length probability distribution function for the low-pass filtered height data is shown with a best fit to Cauchy-Lorentz and Gaussian distributions. (c) Perspective view of the membrane in a curved down shape marked as “(c)” in (a). (d) Perspective view of the membrane in a curved up marked as “(d)” in (a).