Dynamic scaling of out-of-plane fluctuations in freestanding graphene

We study the dynamical behavior of the mean-square displacement of height fluctuations of freestanding graphene using a phase-field-crystal model introduced recently. We find that the dynamic scaling behavior obtained numerically at long times is well described by the scaling theory of polymerized membranes. The critical exponent characterizing the power-law increase with time depends only on the equilibrium roughening exponent $\ensuremath{\xi}$ as $\ensuremath{\alpha}=\ensuremath{\xi}/(1+\ensuremath{\xi})$. For sufficiently long times it crosses over to linear behavior for finite-size systems. The critical exponent $\ensuremath{\alpha}$ is in good agreement with the anomalous diffusion exponent observed experimentally in graphene, suggesting this is a property that could also be observable in other two-dimensional crystalline materials.