Self-Consistent Screening Approximation for Flexible Membranes: Application to Graphene

TL;DR

This paper uses the self-consistent screening approximation to numerically analyze the height-height correlation function of crystalline membranes like graphene, revealing scale-dependent rigidity behavior consistent with analytical predictions at very small wavevectors.

Contribution

The study applies the SCSA to compute the correlation function of graphene membranes over a wide wavevector range, bridging atomistic simulations and analytical results.

Findings

Agreement with Monte Carlo simulations in the intermediate q range

Renormalized bending rigidity exponent η approximately 0.82 at very small q

Behavior of G(q) varies with scale, not described by a single exponent

Abstract

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 nano- and micro-electromechanical 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 wavevectors $q$ from $10^{-7}$ \AA$^{-1}$ till 10 \AA$^{-1}$ 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 $10^{-2}$ \AA$^{-1}$ till 1 \AA$^{-1}$. In the limit $q\rightarrow 0$ our data for the exponent $\eta$ of the renormalized bending rigidity $\kappa_R(q)\propto q^{-\eta}$ is compatible with the previously known analytical results for the SCSA $\eta\simeq 0.82$. However, this limit appears to be reached only for $q<10^{-5}$ \AA$^{-1}$ whereas at intermediate $q$ the behavior of $G(q)$ cannot be described by a single exponent.