Bending modes, anharmonic effects and thermal expansion coefficient in single layer and multilayer graphene

TL;DR

This paper introduces an analytical model to understand anharmonic effects, bending modes, and thermal expansion in single and multilayer graphene, revealing negative thermal expansion and a crossover in bending rigidity.

Contribution

It provides a simple analytical framework to analyze anharmonic effects and bending rigidity crossover in multilayer graphene, highlighting substrate influence on thermal expansion.

Findings

Negative Grüneisen coefficients lead to negative thermal expansion.

Bending rigidity shows a crossover between regimes influenced by in-plane properties and multilayer effects.

Thermal expansion coefficient depends on substrate coupling.

Abstract

We present a simple analytical approach to study anharmonic effects in single layer, bilayer, and multilayer graphene. The coupling between in plane and out of plane modes leads to negative Gr\"uneisen coefficients and negative thermal expansion. The value of the thermal expansion coefficient depends on the coupling to the substrate. The bending rigidity in bilayer graphene shows a crossover between a long wavelength regime where its value is determined by the in plane elastic properties and a short wavelength regime where its value approaches twice that of a single layer.