Scaling relation for thermal ripples in single and multilayer graphene

Ripples in graphene can occur due to strain or thermal fluctuations and stabilize the two-dimensional material. Molecular dynamics simulations show that thermally induced ripples in graphene lead to angular deviations of the surface normal that agree with previous electron diffraction experiments [J. C. Meyer et al., Nature (London) 446, 60 (2007)]. We discover scaling relationships for the average angular deviations as a function of size of the graphene sheet $L$ and averaging radius $R$. The average angle scales as $\exp \{-c{(R/L)}^{\alpha }\}$ with a scaling exponent $\alpha =1$ for single layer, $5/4$ for bilayer, and $5/3$ for trilayer graphene, respectively. We show how these angular deviations depend on temperature, strain, and layer numbers. The scaling relations can provide guidance to the optimization of properties that are sensitive to out-of-plane distortions.

Figure 1

Comparison of the phonon dispersion of graphene from empirical potentials and DFT with experimental results. The experimental data are from electron energy-loss spectroscopy (squares, diamonds and filled circles), neutron scattering (open circles), and x-ray scattering (triangles) (Ref. 35). Including the torsion term of the AIREBO potential underestimates the out-of-plane optical mode frequencies and excluding it overestimates them. Our modified potential, including 40$\%$ of the torsion, improves the agreement with experimental results.

Figure 2

MD snapshot of surface for (a) graphene and (b) bilayer graphene averaged using a Gaussian of 2.5 Å width. The angle of the surface normal is shown in (c) for graphene and (d) for bilayer graphene. The two surfaces of bilayer graphene show incoherent height and angle fluctuations. The Gaussian fit to the angle distribution is shown in (e) for two different values of the Gaussian width $R$. The autocorrelations of the height fluctuations of bilayer graphene are shown in (f) for two different system sizes.

Figure 3

Scaling of the angular deviation, $\Delta \theta$, with ${(R/L)}^{\alpha }$ for (a) graphene, (b) bilayer graphene, and (c) trilayer graphene, where $L$ is the size of the graphene sheet, $R$ the patch size over which averages are computed, and the exponent $\alpha$ depends on the number of layers in $n$-layer graphene. Exponential fits were computed for $R>4$ Å. The legend shows the number of atoms per layer.

Figure 4

The first-order diffraction peaks for coherence regions of 10, 20 and 50 Å for out-of-plane positions of zero (solid circles) and 0.16 Å${}^{-1}$ (solid diamonds) are shown in (a). The inset shows a schematic of the graphene diffraction pattern in the presence of rippling. (b) The linear slope of the peak widths as a function of out-of-plane position estimates the average angular deviation in graphene.

Figure 5

(a) The effect of temperature on angular deviations in graphene and bilayer graphene. (b) Effect of compressive strain and tensile strain on angular deviations in graphene.