Rippling and crumpling in disordered free-standing graphene

Graphene is a famous realization of an elastic crystalline two-dimensional (2D) membrane. Thermal fluctuations of a 2D membrane tend to destroy the long-range order in the system. Such fluctuations are stabilized by strong anharmonicity effects, which preserve thermodynamic stability. The anharmonic effects demonstrate critical behavior on scales larger than the Ginzburg scale. In particular, a clean suspended flake of graphene shows a power-law increase of the bending rigidity with the system size, ϰ ∝L^η , due to anharmonic interaction between the in-plane and out-of-plane (flexural) phonon modes. We demonstrate that random fluctuations of the membrane curvature caused by static disorder may change dramatically the scaling of the bending rigidity and lead to a nonmonotonous dependence of ϰ on L . We derive coupled renormalization-group equations describing the combined flow of ϰ and effective disorder strength b , find a critical curve b (ϰ ) separating flat and crumpled phases, and explore the behavior of disorder in the flat phase. Deep in the flat phase, disorder decays in a power-law way at scales larger than the Ginzburg length, which therefore sets a characteristic size for the ripples—static out-of-plane deformations observed experimentally in suspended graphene. We find that in the limit L →∞ ripples are characterized by an anomalous exponent 2 η in contrast to dynamical fluctuations governed by η . For sufficiently strong disorder, there exists an intermediate range of spatial scales where ripples decay much slower, with exponent η /4 . In the near-critical regime, disorder first increases with L , then reaches a maximum and starts to decrease. In this case, the membrane shows fractal properties implying a multiple folding starting from a certain length scale L₁ and finally flattens at a much larger scale L₂ (which diverges at criticality). We conclude the paper by a comparison of our results with available experimental data on graphene ripples.