Bending sound in graphene: Origin and manifestation

Abstract

It is proved that the acoustic-type dispersion of bending mode in graphene is generated by the fluctuation interaction between in-plane and out-of-plane terms in the free energy arising with account of non-linear components in the graphene strain tensor. In doing so we use an original adiabatic approximation based on the alleged (confirmed a posteriori) significant difference of sound speeds for in-plane and bending modes. The explicit expression for the bending sound speed depending only on the graphene mass density, in-plane elastic constants and temperature is deduced as well as the characteristics of the microscopic corrugations of graphene. The obtained results are in good quantitative agreement with the data of real experiments and computer simulations.

Bending mode and thermal expansion of graphene

“…We recall that the contribution of (21) [with account of (22)] to the free energy of the crystal in the presence of its thermal expansion is absent, in principle, in the quasi-harmonic approximation and, as we will see, it is dominant at "low" temperatures. The quantity ) (v ζ , being itself negative, makes a fundamentally negative contribution to the GTEC (see below).…”

“…This means that their values in the above expressions may not coincide with those found in [30]. To estimate the effective value of 4 / ) 3 ( 112 111 C C + , we use the "low-temperature" value of the bending sound velocity in graphene from [22]: ≈ B s 0.3 km/s (see also [21]), and as a result we get:…”

“…The quantity ) (v ζ , being itself negative, makes a fundamentally negative contribution to the GTEC (see below). Moreover, it is clear from the structure of expressions (21) and (22) that they arise due to the explicit presence of third-order terms over the strain tensor in the free energy of graphene. To avoid misunderstanding, we emphasize that the resultant sign of ) (v ζ [with the account of inequality (12)] is negative precisely due to the anharmonic tensor constructions of the form of…”

“… (5). and(21)] are proportional to the "unperturbed" (i.e. at T = 0) area S 0 of the graphene sample, so:…”

“…We emphasize that this negative contribution would be divergent[16] in the limit of long waves for a bending mode with dispersion of ~ q 2 . And only the appearance of a "sound" dispersion of the bending mode of a quasi-2D crystal[21] eliminates a similar IR divergence of GTEC.…”