Self-consistent screening approximation for flexible membranes: Application to graphene

Crystalline membranes at finite temperatures have an anomalous behavior of the bending rigidity that makes them more rigid in the long-wavelength limit. This issue is particularly relevant for applications of graphene in nanoelectromechanical and microelectromechanical systems. We calculate numerically the height-height correlation function $G(q)$ of crystalline two-dimensional membranes, determining the renormalized bending rigidity, in the range of wave vectors q from 10^-7 ^-1 till 10 ^-1 in the self-consistent screening approximation (SCSA). For parameters appropriate to graphene, the calculated correlation function agrees reasonably with the results of atomistic Monte Carlo simulations for this material within the range of q from 10^-2 ^-1 till 1 ^-1. In the limit q→0 our data for the exponent η of the renormalized bending rigidity κR(q)∝q^-η is compatible with the previously known analytical results for the SCSA η0.82. However, this limit appears to be reached only for q<10^-5 ^-1 whereas at intermediate q the behavior of $G(q)$ cannot be described by a single exponent.