On the intrinsic ripples and negative thermal expansion of graphene

The intrinsic ripples of suspended graphene have attracted intensive attention due to their influence on the electronic transport and other properties. Negative thermal expansion (NTE), another unconventional phenomenon found in graphene, can be utilized to control the intrinsic ripples in a reversible way, thus opening new perspective for application. In this case, understanding the mutual relation and physical origin of the intrinsic ripples and NTE is crucial, especially since they are both widely observed in other 2D materials. Here we clarify through lattice dynamical analysis that at low temperature the two phenomena are both intrinsic for any 2D crystals with a honeycomb structure (or any monatomic 2D crystals). We find that the intrinsic ripples, generally believed to be caused by thermal fluctuation, have another origin that is the appearance of soft ZA modes near long wavelength limit when the lattice constant is shortened. Moreover, the soft ZA modes and NTE have the same physical origin at low temperature. At finite temperature, NTE is dominantly caused by the “vibrational elongation” effect owing to large out-of-plane fluctuation according to our calculation based on self-consistent phonon theory.