Ripples in graphene: A variational approach

TL;DR

This paper rigorously analyzes rippling in graphene, showing that near-minimal energy configurations naturally develop wave patterns with specific wavelengths, aligning with experimental observations.

Contribution

It extends previous work by proving the emergence of wave patterning in graphene ripples through a variational approach, reducing the problem to one-dimensional chains.

Findings

Ripples form wave patterns with specific wavelengths.

Wave patterns are independent of sample size.

Results align with experimental and simulation data.

Abstract

Suspended graphene samples are observed to be gently rippled rather than being flat. In [M. Friedrich, U. Stefanelli. Graphene ground states, arXiv:1802.05049], we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of [M. Friedrich, U. Stefanelli. Graphene ground states, arXiv:1802.05049], rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.