Bending mode and thermal expansion of graphene

TL;DR

This paper models the thermal expansion of graphene beyond the quasi-harmonic approximation, highlighting the role of bending modes and explaining experimental and simulation data across a wide temperature range.

Contribution

It introduces a continuum elastic model that accounts for bending modes, explaining the sign change in graphene's thermal expansion coefficient at high temperatures.

Findings

Negative thermal expansion at low T

Sign reversal of expansion coefficient near 1000 K

Bending mode's size-independent contribution

Abstract

Proceeding from the model of a two-dimensional elastic continuum, we describe the characteristic features of thermal expansion of graphene using an approach that goes beyond the quasi-harmonic approximation. The negative value of the thermal expansion coefficient of graphene at low temperatures and its sign reversal at T about 1000 K are established. It is shown that the bending vibrational mode plays a decisive role in peculiarities of the thermal contraction/expansion of graphene and that the contribution of this mode to the thermal expansion coefficient does not depend on the sample size, due to the "sound" nature of the bending mode long-wave dispersion. The obtained results allow giving a quantitative description of the known data of numerical experiments on the thermal expansion of graphene in the entire interval of the MD simulations.