## 6 Buckling Instability
When graphene is produced by thermal CVD on a Cu foil, strain comes from the difference of the thermal expansion coefficients of graphene and its substrate. If the graphene overlayer is assumed to be pinned on the substrate, cooling down the graphene/Cu system from 1000 C to room temperature may lead to a uniform compressive strain \(\epsilon\) of 2.2% [18]. This is very close to the estimated critical strain for rippling on Cu (see Table 4). In those parts of the surface where graphene is not bound to the substrate, ripples may form because of the disappearance of the restoring force from the substrate. Ripples have been observed on graphene suspended over nanotrenches on Cu(111) [18]. These nanotrenches have a width of 5 nm, typically, with length about 10 times bigger. It is interesting that the wavelength of the observed ripples is almost invariably 0.7 nm, the amplitude being \(\sim\)0.05 nm. These observations cannot be explained by the classical theory of wrinkles that form in a stretched membrane strip with free edges [77]. This should not come as a real surprise. Ripples with wavelength at subnanometer scale are influenced by the atomic structure of graphene much more than by continuum mechanics rules [76]. Independent of all elastic properties, if the residual strain is locally released by ripples that develop above the suspended parts of graphene, then \(\epsilon=(L-\lambda)/L\)[78]. In this relation, \(L\) is the length measured along the wavy shape of graphene whose projection corresponds to the measured wavelength \(\lambda\). For a sine function with amplitude \(a\), this expression takes the explicit form  
\[\epsilon=1-\frac{\pi}{2E\left[-(2\pi a/\lambda)^{2}\right]} \tag{18}\]  
where \(E(m)\) is the complete elliptic integral of the second kind (\[W=\int_{0}^{l}[\frac{D}{2}\kappa^{2}+V(z)]\,ds-\gamma\delta l=V(z_{0})l+\frac{1}{ 2}\int_{0}^{l}[D(\frac{d^{2}\zeta}{dx^{2}})^{2}+K\zeta^{2}-\gamma(\frac{d\zeta }{dx})^{2}]\,dx \tag{15}\]03.4 in [79]). Accordingly, a strain of 2.2% yields \(a/\lambda\) = 0.048, about 2/3 of the observed ratio. The observed ratio would require a strain of 4.6%. The amplitude and wavelength predicted by atomistic calculations for graphene submitted to a 2.2% compressive edge strain are 0.35 nm and 6.58 nm, respectively [80]. The corresponding \(a/\lambda\) \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Substrate** & \multicolumn{2}{l}{**ZA phonon frequency at**} & \multicolumn{2}{l}{**\(K\) (J/m\({}^{4}\))**} & \multicolumn{2}{l}{**\(2\sqrt{DK}\)**} & \multicolumn{2}{l}{**\(2\pi\sqrt[4]{D/K}\)**} & \multicolumn{2}{l}{**\(2\sqrt{DK}/C\)**} \\  & \(\Gamma\) **(cm\({}^{-1}\))**[72]** & & \multicolumn{2}{l}{**(N/m)**} & \multicolumn{2}{l}{**(nm)**} & \multicolumn{2}{l}{**\(=\epsilon_{cr}\)**} \\ \hline Co(001) & 295 & 2.35 \(\times\) 10\({}^{21}\) & 49.1 & 0.64 & 0.14 \\ Ni(111) & 240 & 1.56 \(\times\) 10\({}^{21}\) & 40.0 & 0.71 & 0.11 \\ Cu(111) & 45 & 5.48 \(\times\) 10\({}^{19}\) & 7.5 & 1.64 & 0.021 \\ SiO\({}_{2}\) & [a] & 1.5 \(\times\) 10\({}^{20}\) & 12 & 1.3 & 0.036 \\ gr(0001) & [b] & 1.08 \(\times\) 10\({}^{20}\) & 10.5 & 1.39 & 0.030 \\ h-BN(0001) & [b] & 7.60 \(\times\) 10\({}^{19}\) & 8.8 & 1.51 & 0.025 \\ \hline \hline \end{tabular} a: see text and Reference [75]; b: see text and Appendix.  
\end{table} 
Table 4: Force constant of the the graphene–substrate interaction \(K\) and related parameters (see text). The values used for the flexural rigidity \(D\) and the stiffness constant \(C\) of graphene are 1.6 eV and 350 N/m, respectively. ratio is 0.053, close to what the above equation yields. However, the amplitude and wavelength of the ripples are an order of magnitude larger than the experimental values. Clearly, not everything is understood here.