Lévy flights near the melting transition of graphene and silicene

We investigate atomic motion in two-dimensional (2D) crystals---specifically monolayer graphene and silicene---using molecular dynamics simulations. Approaching their melting temperatures, atoms in both materials exhibit anomalous atomic displacements. Analysis of atomic trajectories reveals that the step-size distributions follow a truncated power-law distribution characteristic of L\'evy flights. For graphene, we found an apparent power-law exponent $\ensuremath{\approx}2.65$ while a value of $\ensuremath{\approx}2.60$ was found for silicene. These observations suggest that L\'evy-type dynamics are intrinsic to high-temperature behavior in 2D materials and may play a critical role in their melting dynamics and atomic transport. The waiting times between flights also exhibit L\'evy-type dynamics in time, quantified by colored noise in the power spectrum of mobility fluctuations, which may relate to experimentally observed colored noise of current fluctuations in graphene---a phenomenon of significant importance for device applications.