Intensity (a.u.) Intensity (a.u.) Raman Shift (cm -1 ) Oxygen plasma. 6 cm. 9 cm. 1mm. Single-layer graphene sheet. 10mm. 14 cm

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1 Intensity (a.u.) Intensity (a.u.) a Oxygen plasma b 6 cm 1mm 10mm Single-layer graphene sheet 14 cm 9 cm Flipped Si/SiO 2 Patterned chip Plasma-cleaned glass slides c d After 1 sec normal Oxygen plasma After 27 sec Flipped Oxygen plasma Pristine Graphene Raman Shift (cm -1 ) Pristine Graphene Raman Shift (cm -1 ) Supplementary Figure 1 Defect generation in graphene. (a) Schematic of the sample arrangement used to shield the graphene surface from the direct exposure to oxygen plasma. (b) Oxygen plasma chamber used for etching of graphene. (c) Raman spectra of a pristine mono-layer graphene sheet before and after exposure of 1 s to oxygen plasma in the normal (or unshielded) condition. (d) Raman spectra of a pristine mono-layer graphene sheet before and after exposure of 27 s to oxygen plasma in the shielded condition.

2 Supplementary Figure 2 Evolution of Raman spectra in the supported and suspended graphene sheets as a function of oxygen plasma exposure time with a step of 3 s. (a) Peak intensities, (b) Full-width at half-maximum (FWHM), and (c) Peak positions of Raman peaks in the supported graphene sheet. (d) Peak intensities, (e) FWHM, and (f) Peak positions of Raman peaks in the suspended graphene sheet. (g) Plot of peak intensity ratio of I(D)/I(G) vs. the average distance between defects (L D ) and (h) Plots of I(D)/I(G) and L D as a function of plasma exposure time for the supported and suspended graphene sheets.

3 Supplementary Figure 3 Defect generation in bi-layer graphene. (a) non-contact mode AFM image of the suspended bilayer graphene exposed by oxygen plasma for 80 s. (b) High resolution AFM image of the same sample clearly shows the formation of nanocavities with a diameter of a few nanometers. Scale bars of (a) and (b) are 1 μm and 100 nm, respectively.

4 Supplementary Figure 4 Finite element method (FEM) analysis for determination of peak equibiaxial (true) stress. (a) FEM simulation results with nonlinear elasticity assumption. The atomic force microscopy (AFM) tip with tip radius of 80 nm was used. The blue-dashed lines indicate strength values of pristine graphene (34.5 N/m) and defective graphene exposed for 18 s (30 N/m), respectively. (b) True stress vs. strain curve from the FEM simulation.

5 Pristine Graphene sp 3 -Defect Regime Supplementary Figure 5 Energy dispersive spectroscopy (EDS) analysis indicating the relative amounts of oxygen and carbon. The data was taken from the graphene region that was suspended over the holes in the transmission electron microscopy grid (see Fig. 5 of main manuscript). The results show a significant increase in oxygen content in the sp 3 -defect regime (I(D)/I(G) ~ 0.2 and I(2D)/I(G) ~ 1.5) as compared to pristine graphene.

6 Supplementary Figure 6 Molecular dynamics simulations. (a) Number of carbons removed from the sheet vs. simulation time. (b) Time evolution of the ratio of number of oxygen chemisorbed to the graphene sheet divided by the number of carbon atoms still left in the sheet (O/C). The inset shows the slope of the linear regime (i.e. oxygen attachment rate) versus the number of initial oxygen atoms in the simulation box for all the 4 simulations. The shaded area shows the intermediate regime in the simulation with 1000 Oxygen atoms, which is the focus of this study. Ten structures were selected from this regime to perform mechanical testing of defective graphene sheets.

7 Supplementary Discussion Plasma Chamber Set-Up: To engineer the defects in a gradual and well-controlled fashion, the chips, on to which the graphene sheets were exfoliated, were flipped over and placed upside down in the plasma chamber between two plasma-cleaned glass slides as shown in Supplementary Figure 1a, b. By shielding the sample from direct exposure to the plasma, defects are introduced in a less abrupt and more controlled manner. Raman spectra for graphene sheets directly exposed to and shielded from the plasma were compared as shown in Supplementary Figure 1c, d. The results indicate that it takes ~30- fold longer to generate the same level of defectiveness in the exposed graphene when it is shielded from the plasma as compared to when it is directly exposed to the plasma. This is attributed to the etching of graphene by oxygen ions or radicals generated in the plasma cleaning process, and not by the direct bombardment of high energy 57. Defect Characterization by Raman Spectroscopy: Raman spectroscopy (Renishaw, invia) was employed to analyze the level of defectiveness in the graphene sheets using 532-nm wavelength laser. The evolution of the peak intensity ratios of D to G bands (I(D)/I(G)), and 2D to G bands (I(2D)/I(G)) as a function of the plasma exposure time are plotted in the main manuscript. Here we show the evolution of peak intensities, peak positions, and full-width at half-maximum (FWHM) of the 2D, G, and D bands in the supported and suspended graphene sheets with plasma exposure time. As mentioned in the main text, according to I(D)/I(D'), we can divide the plasma exposure region into two regimes: sp 3 -type defect (I(D)/I(D') > 7) and vacancy-type defect (I(D)/I(D') < 7) 49. As shown in Supplementary Figure 2, a transition can be observed between the two regimes for both supported and suspended graphene sheets. The important finding is that the suspended graphene is more susceptible to defects than the supported one. In Supplementary Figure 2a and d, changes in peak intensities are similar in both cases, except for high intensity of 2D peak because of less charge impurities in the suspended graphene 58. In addition, compared to the supported sample, the G peak intensity of the suspended graphene sample sharply increases when transitioning from the sp 3 -type to the vacancy defect regime; hence the maximum I(D)/I(G) ratio is reached at shorter plasma time for the suspended graphene. Also the changes in peak positions and FWHM values (Supplementary Figure 2b, c, e, and f) indicate that the defects start to form at shorter plasma time for the suspended graphene as compared to the supported graphene. In the sp 3 -type defect regime, there are no significant changes in FWHM and peak position, meanwhile, in vacancy-type defect regime, FWHM of 2D peak sharply increases and 2D peak slowly up-shifts with plasma exposure time for both the supported and the suspended graphene sheets 45. To calculate an average distance between defects (L D ), the intensity ratio curves can be fitted as follows: in the low defect density regime, I(D)/(IG) = C/L D 2 and in high defect density regime, I(D)/I(G) = D L D 2, where C is 102 nm 2 and D is obtained by imposing continuity between the two regions 46, 48, 59. Supplementary Figure 2g shows that I(D)/I(G) intensity ratio increases with decreasing L D and reaches maximum at L D ~ 5 nm, then

8 decreases for both supported and suspended graphene sheets. Based on the boundaries among the two defect regimes, it is roughly estimated that defect formation in the suspended graphene is ~23 % accelerated as compared to the supported graphene as shown in Supplementary Figure 2h. We found that the Raman measurement often damages the highly defective graphene probably due to heating. Therefore, the Raman spectra in Fig. 4 of main paper were taken from the supported area near the suspended graphene sheet. Model Used to Calculate Elastic Modulus and Breaking Strength: During the mechanical testing, both the AFM tip and the graphene membrane are deflected. Assuming equilibrium between the tip and the membrane, the force (F) applied on both will be equal. Therefore, tip deflection, is: (1) where K is the force constant (stiffness) of the AFM tip (73.5 N/m). The membrane displacement, is the difference between the AFM piezo displacement and the tip deflection: (2) For calculating modulus and strength, the method suggested by C. Lee et al. is followed 2. Based on this model, a clamped circular membrane under a centric point load is considered. There is no closed-form analytical solution that accounts for finite deformation and pre-tension in a material that has Poisson's ratio other than 1/2. Therefore, in the suggested model by C. Lee et al., solutions for two special cases are combined. One special case is a membrane with large initial pre-tension relative to the additional stresses induced at very small indentation depths. For such a clamped freestanding elastic thin membrane under point load, when bending stiffness is negligible and the load is small, the relationship of force and deflection due to the pre-tension in the membrane can be approximately expressed as: (3) where is the pre-tension in membrane. Here the force vs. displacement relationship is linear. The other special case is valid for stresses much greater than the initial pre-tension and force varies as a function of the cubed of the displacement. When /h >> 1 (h is the thickness of membrane), the relationship of force-displacement can be expressed as: where, (4) (5) and a is the membrane radius, E 2D is the two-dimensional Young's modulus, and is the Poisson's ratio which is considered here to be equal to the graphite value, If we sum the contribution of the pre-tension term and large-displacement term, then the force displacement relationship can be expressed as:

9 (6) The two-dimensional Young's modulus, E 2D, can be characterized by least squares fitting the experimental loading curves using the above equation. This approximate solution has been shown to agree with numerical finite element solution within the uncertainty of the experiments 2. Note that in this study we did not vary the Poisson s ratio with defect density. This is because even if we assume a very high (50% increase) in Poission s ratio, the Young s modulus decreases by only ~5.15 %. This variation is much smaller than the experimental error range (broad distribution of Fig. 4a in the main paper). Therefore, the change of Poisson s ratio was neglected. Also note that the oxidized graphene may have a slightly higher bending stiffness in comparison to pristine graphene, however the bending stiffness of the highly flexible sheet is still very small in comparison to its in-plane (tensile) stiffness and can be neglected. Since the point-load assumption yields a stress singularity at the center of the membrane, it is necessary to consider the indenter geometry in order to quantify the maximum stress under the indenter tip. The maximum stress for the thin, clamped, linear elastic, circular membrane under a spherical indenter as a function of applied load can be expressed as: (7) where is the maximum stress and R is the indenter tip radius. Note that this equation overestimates the strength 2,3. Therefore, the breaking strength of graphene was calculated using FEM and DFT simulations as explained below with consideration of the nonlinear elastic behavior of graphene. Oxygen Plasma Treatment of Bi-layer Graphene: Bi-layer graphene sheets were also exposed to oxygen plasma treatment using the set-up described previously in Supplementary Figure 1. Similar to the monolayer case, we see evidence of extended defects (nano-cavities) in bi-layer graphene. Supplementary Figure 3 shows AFM images of a typical suspended bi-layer sheet after plasma exposure of 80 s. Finite Element Method (FEM) for Breaking Strength Calculation: The finite element method (FEM) was used to calculate the breaking strength of the graphene films as a function of the measured fracture load and diameter of the nanoindenter tip. The nonlinear and anisotropic elastic constitutive behavior of graphene was modeled as a user material (UMAT) subroutine for the general finite element method (FEM) package ABAQUS as explained in our previous reports 60, 61. The higher order elastic constants of graphene have been calculated by a least squares fit to density functional theory (DFT) results. The model provides a framework to accurately predict the stress experienced by the graphene membrane during nanoindentation based on DFT calculations. The result obtained from this model is shown in Supplementary Figure 4a for an indenter tip radius of 80 nm used in this study, where x-axis is the force on the indenter tip and the y-axis is the equibiaxial stress in the true stress measure immediately under the indenter tip. This analysis yields an equibiaxial breaking strength of 34.5 N/m (103 GPa, when expressed

10 as a three-dimensional value using nm as the thickness of monolayer graphene) for pristine graphene with a 80 nm radius tip. The mechanical strength or peak stress that can be supported by graphene is a function of the loading configuration. For a uniaxial stress state in the armchair direction, the same multiscale constitutive model predicts the strength of graphene to be 39.5 N/m (118 GPa), which is consist with our previouslyreported value of 42 ± 4 N/m for the same loading configuration 2. Remarkably, because of the highly nonlinear nature of the stress vs. indentation force relationships, the average equibiaxial strength of defective graphene exposed by plasma for ~18 s (i.e. I(D)/I(G) ~ 1.0) is only slightly smaller (13.3%), 30 N/m (89.5 GPa in 3D) in sp 3 -type defect regime compared to that of pristine graphene, 34.5 N/m (103 GPa in 3D) (Fig. 4b in the main text). These results demonstrate that the strength of defective graphene (with sp 3 -type defects) can in fact be quite comparable to that of pristine graphene. However, because the elastic stiffness in vacancy-type defect regime is different from that of pristine graphene, this model could not be applied for this regime. As shown in true stress vs. strain curve of Supplementary Figure 4b, the Lagrangian strain at fracture is around 0.2 for the defective graphene exposed up to 18 s. Molecular Dynamics Simulations: We considered four different numbers of oxygen atoms: 750, 1000, 1500, and 4500 in the simulations corresponding to varying oxygen pressures. The number of carbon atoms in the graphene sheet was kept constant at Supplementary Figure 6 shows the number of carbon atoms removed from the graphene lattice by the reactive oxygen and the number of chemisorbed oxygen atoms normalized by the total number of carbon atoms in the graphene lattice as a function of the simulation time. The shaded area shows the intermediate regime in the simulation with 1000 oxygen atoms. This regime was the focus of this study. Ten structures were selected from this regime to evaluate mechanical properties of defective graphene sheets. The results of mechanical tests are shown in Fig. 6e in the main paper. It should also be noted that in the very beginning of etch process (i.e. prior to the formation of nano-pores in the graphene lattice), we will produce Janus type graphene, which has different functional groups on the top and bottom surfaces. However, because the elastic stiffness and breaking strength of graphene sheets are related mainly to the critical length of the carbon-carbon bonds, the formation of chemical bonds with oxygen on one-side or both sides does not have a significant effect on the mechanical properties of graphene. This result is also consistent with prior simulations reported in the literatures 30, 32.

11 Supplementary References 57. Pizzi, A. & Mittal K. L. Handbook of adhesive technology. CRC Press, pp (2003). 58. Berciaud, S., Ryu, S., Brus, L. E., & Heinz, T. F. Probing the intrinsic properties of exfoliated graphene: Raman spectroscopy of free-standing monolayers. Nano lett. 9, (2009). 59. Childres, I., Jauregui, L. A., Tian, J., & Chen, Y. P. Effect of oxygen plasma etching on graphene studied using Raman spectroscopy and electronic transport measurements. New J. Phys. 13, (2011). 60. Wei, X., Fragneaud, B., Marianetti, C., & Kysar, J. W. Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description. Phys. Rev. B. 80, (2009). 61. Wei, X. & Kysar, J. W. Experimental validation of multiscale modeling of indentation of suspended circular graphene membranes. Int. J. Solids Struct. 49, (2012).