Bending mode and thermal expansion of graphene

Proceeding from the model of a two-dimensional elastic continuum, we describe the characteristic features of the thermal expansion of graphene using an approach that goes beyond the quasiharmonic approximation. The negative value of the thermal expansion coefficient of graphene at low temperatures and its sign reversal at $T\approx 1000\mathrm{K}$ are established. It is shown that the bending vibrational mode plays a decisive role in the 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 simulations on the thermal expansion of graphene in the entire interval of the molecular dynamics simulations.

Figure 1

Effective exponents $d_{\left|\right|}$ (curve 1) and $d_w$ (curve 2) determining the temperature dependences of the in-plane and out-of-plane contributions to the harmonic vibrational free energy of graphene. The dashed line shows the effective exponent $d_{w-}$ calculated without taking into account the “sound” part of the out-of-plane spectrum (11).

Figure 2

Theoretical [solid line calculated by formulas (33) and (27)] temperature dependence of the projected area per graphene atom. Triangles are the data of numerical simulations [25] for the graphene sample of 33 600 atoms; squares show the same for the sample of 960 atoms. Empty circles represent the simulation results [25] on temperature dependence of the area per atom of the fluctuating surface curved due to the existence of the intrinsic ripples in graphene; the dashed line represents the calculation by Eq. (34).

Figure 3

The theoretical (solid line) dependence of the graphene areal TEC ${\alpha }_{\left|\right|}\left(T\right)$ calculated by formula (29); the values of GTEC, restored by numerical differentiation of the simulation data [25] for 33 600 atoms are indicated by triangles. For comparison, the filled squares show the values of ${\alpha }_{\left|\right|}\left(T\right)$ for the graphene cell including 960 atoms [25]. Empty squares designate the TEC values for the “real area” of graphene, derived in [25] from numerical simulations. In addition, the curve $\alpha \left(T\right)=\partial lnA\left(T\right)/\partial T$ constructed by formula (34) is represented by the dashed line. In the inset, the same results are shown using the linear temperature scale.