## 3 Results and Discussions
### Physical Reasons of the Statistically-Significant Terms  
From the above ANOVA analysis, we have found the statistically-significant terms for Responses I and II, respectively. Let us now examine the physical reasons for these identified factors in affecting the two responses. 1. The presence of the significant interaction term \"AB\" mentioned in Table 3 can be explained with the aid of the following governing chemical reactions for graphene growth as given by [PERSON] et al. [47]. These reactions are shown in Scheme 1 in Figure 4 and are modified for our experiments since a-C is used as the carbon source instead of methane. The formation of hydrocarbons responsible for graphene formation by the reaction of a-C and H\({}_{2}\) at elevated temperatures as shown in Reaction (1) has been confirmed by [PERSON] et al. [29], as well. The rise in temperature causes thermo-mechanical stress [48] at the a-C/Cu interface due to the difference in the coefficient of thermal expansion for a-C and Cu. As a result, this thermo-mechanical stress causes the sample to warp downwards. This can be seen from the equivalent von-Mises stress analysis obtained from our Finite Element Analysis (FEA) simulation performed for the experimental test sample, as depicted in Figure 5. This FEA is performed using thermal and static-structural analysis in ANSYS 16.2 Academic Version (ANSYS, Inc., Canonsburg, PA, USA, 2015). The material properties used for FEA [49; 50; 51; 52; 53; 54] are mentioned in Table S1 of the Supplementary Materials. Here, the resultant stress is given by Equation (1) and is popularly known as [PERSON]'s equation [55; 56]. \[\sigma_{\text{stf}}=\left[\frac{\text{E}_{\text{f}}}{6\left(1-\text{v}_{\text {f}}\right)}\right]*\left(\frac{\text{t}_{\text{s}}^{2}}{\text{t}_{\text{f}}} \right)*\Delta(\frac{1}{\text{R}}) \tag{1}\]  
where \(\Delta(1/R)=(1/R-1/R_{0})\), \(R_{0}\) and \(R\) are the radius of the curvature of the samples before and after the thermo-mechanical stresses are developed, respectively, and \"\(t\)\" denotes the thickness of the layer in the sample with subscripts \"\(s\)\" and \"\(f\)\" referring to the substrate and film, respectively. Equation (1) shows that the film thickness and thermo-mechanical stress have an inverse relationship [55; 56]; thus, the total stress will be higher for a thinner a-C/Cu interface, as shown in Figure 6.  
Figure 4: Scheme 1: Graphene growth mechanism’s governing reactions; here C represents the carbon atom, Gr represents graphene, CH*/C* represent hydrogen bonded-carbon radical/carbon radical, H* represents the hydrogen radical and \(\Delta\)H represents heat (permission to modify the scheme has been obtained). High stress at the interface and thermo-mechanical stress causing the downward warpage enhance carbon diffusion through Cu grain boundaries, resulting in the carbon reaching the surface of the Cu film. The von-Mises stress in the Cu film is tensile in nature (the values are believed to be high enough that they can either cause cracking in Cu film and/or enhancement of H\({}_{2}\) diffusion in Cu film through its grain boundaries [57, 58]). The enhancement of H\({}_{2}\) diffusion will cause more carbon radical generation as per Reaction (1) in Figure 4. Furthermore, the a-C layer is found to be under compressive stress, which causes a \"push\" in the perpendicular direction towards the surface. Both of these mechanisms will enhance carbon diffusion through Cu boundaries. This renders the carbon radical generation Reaction (1) and graphene formation Reaction (2) favorable, and the defective sites, especially those at the edges, are repaired by the graphene formation, causing \(I_{\mathrm{D}}/I_{\mathrm{C}}\) ratio to fall. A higher temperature will cause the stress to be higher, and it will also increase the rate of Reactions (1) and (2), thus rendering a larger reduction of the defects in the grown graphene.