Nature of Spontaneous Curvature in Suspended Graphene

The nature of its intrinsic ripples is the key factor for understanding the stability of suspended graphene, and for unraveling the long-standing theoretical debate of the existence of low-dimensional crystalline state. The rippling morphology of graphene, discovered also in other 2D materials, has a profound impact on its electronic, mechanical and chemical properties. Actually, before the discovery of graphene, rippling phenomena are widely observed: for example, the roughing transition of crystalline interface, the rippled phase in biomembrane, and crumpling of flexible sheet polymers modeled by tethered surfaces. The fascinating truth that ripples exist in so many different membrane-like materials implies possible existence of a universal physical mechanism which was unclear. We consider the ripples in suspended graphene as two parts, characterizing the first part by the spontaneous curvature k which stabilizes the possible soft ZA modes, and the second part by the thermal curvature kt which is caused directly by height fluctuation. By choosing k as the order parameter of the system, we establish the Landau theory modified by thermal fluctuation for wrinkling transition of large sized graphene. We find that as temperature rises from 0K, a second order phase transition occurs at a size dependent critical temperature Tc, which corresponds to a change of equilibrium configuration from a flat state to a rippling state. Interestingly, the order parameter is stablized as temperature increases, and the phase transition is associated with a jump of equilibrium bond length as well as a vanishing intrinsic bending rigidity. The results obtained suggest that the interplay between the rippling morphology and the elementary excitations is vital for understanding the behavior of 2D materials. The concepts and theory developed here is of general significance at least for tethered membranes.