Rippling and crumpling in disordered free-standing graphene

Graphene is a famous realization of elastic crystalline 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 behaviour on scales larger than the Ginzburg scale. In particular, clean suspended flake of graphene shows a power-law increase of the bending rigidity with the system size, $\varkappa\propto L^{\eta},$ due to anharmonic interaction between in-plane and out-of-plane (flexural) phonon modes. We demonstrate that random fluctuations of membrane curvature caused by static disorder may change dramatically the scaling of the bending rigidity and lead to a non-monotonous dependence of $\varkappa$ on $L.$ We derive coupled RG describing combined flow of $\varkappa$ and effective disorder strength $b$, find a critical curve $b(\varkappa)$ 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 $L \to \infty$ ripples are characterized by anomalous exponent $2\eta$ in contrast to dynamical fluctuations governed by $\eta$. For sufficiently strong disorder, there exists an intermediate range of spatial scales where ripples decay much slower, with exponent $\eta/4$. In the near-critical regime, the membrane shows fractal properties implying a multiple folding starting from a certain length scale $L_1$ and finally flattens at a much larger scale $L_2$ (which diverges at criticality). We conclude the paper by a comparison of our results with available experimental data on graphene ripples.