Model of ripples in graphene

TL;DR

This paper introduces a discrete plate model incorporating nonlinear effects and noise to explain the formation of long-lived ripples in suspended graphene sheets, aligning with experimental observations.

Contribution

It presents a novel discrete, nonlinear model with noise for graphene ripples, integrating a double-well potential and nonlinear friction based on experimental data.

Findings

Ripples appear as metastable states with no preferred orientation.

Numerical solutions confirm long-lived ripples at various temperatures.

Relaxation times are much larger than vibration periods.

Abstract

We propose a model of ripples in suspended graphene sheets based on plate equations that are made discrete with the periodicity of the honeycomb lattice and then periodized. In addition, the equation for the displacements with respect to the planar configuration contains a double-well site potential, a nonlinear friction and a multiplicative white noise term satisfying the fluctuation-dissipation theorem. The nonlinear friction terms agree with those proposed by Eichler et al [Nature Nanotech. {\bf 6}, 339 (2011)] to explain their experiments with a graphene resonator. The site double-well potential indicates that the carbon atoms at each lattice point have equal probability to move upward or downward off-plane. For the considered parameter values, the relaxation time due to friction is much larger than the periods of membrane vibrations and the noise is quite small. Then ripples with no preferred orientation appear as long-lived metastable states for any temperature. Numerical solutions confirm this picture.