Anomalous elasticity of graphene: crumpling transition and anomalous Hooke’s law
Graphene is a famous example of a crystalline membrane with a thickness of one atomic layer. The talk is focused on discussion of the unique elastic properties of this material, which are closely related to the question of the thermodynamic stability of 2D systems. Thermal fluctuations tend to bend the graphene membrane and destroy the long-range order in the system. There is, however, a competing effect of a strong anharmonic interaction between the flexural and longitudinal phonons, which tends to stabilize the membrane. Due to this competition, isolated membrane can exhibit a number of critical phenomena, including two phase transitions: the transition from the flat to crumpled phase—the so called crumpling transition (CT) —and the transition associated with loss stability of a flat phase under the action of external compression [buckling transition (BT)]. It turns out that CT is substantially analogous to the transition of a magnet to a ferromagnetic state. In this case, in complete analogy with the magnet, spontaneous symmetry breaking occurs: a spherically symmetric "crumpled phase" transforms into a flat phase with some arbitrary orientation of the plane. Surprisingly, the effects associated with CT significantly change the elastic properties of membranes, in particularly, graphene-based membranes, even far from the transition point, deep in the flat phase. For example, the bending rigidity of graphene flake grows with its size. No less unusual are the manifestations of BT. In particular, it turns out that the ordinary linear Hooke's law for graphene is not satisfied. The stretching of an isolated graphene sheet grows with the applied stress in a power-law manner, with the exponent given by the critical index of BT.
This talk presents an overview of recent works, both theoretical and experimental, devoted to the description of the critical behavior of graphene-based membranes.