Scaling relation for thermal ripples in single and multilayer graphene

Ripples in graphene can occur due to strain or thermal fluctuations and stabilize the two-dimensional material. Molecular dynamics simulations show that thermally induced ripples in graphene lead to angular deviations of the surface normal that agree with previous electron diffraction experiments. We discover scaling relationships for the average angular deviations as a function of size of the graphene sheet L and averaging radius R. The average angle scales as exp{-c⁽R^/L⁾α} with a scaling exponent α=1 for single layer, 5/4 for bilayer, and 5/3 for trilayer graphene, respectively. We show how these angular deviations depend on temperature, strain, and layer numbers. The scaling relations can provide guidance to the optimization of properties that are sensitive to out-of-plane distortions.