Accurate determination of stiffness and strength of graphene via AFM-based membrane deflection

The Young’s modulus and fracture strength of single and bilayer graphene (BLGr) grown by chemical vapour deposition (CVD) were determined using atomic force microscopy-based membrane deflection experiments. The uncertainty resulting from instrument calibration and the errors due to the experimental conditions like tip wear, loading position, and sample preparation were investigated to estimate the accuracy of the method. The theoretical estimation of the uncertainty on the Young’s modulus linked to the calibration is around 16%. Finite element simulations were performed to determine the effects of membrane shape and loading position on the extraction of the Young’s modulus. Off-centre loading results in the overestimation of the Young’s modulus while deviation from the circular shape leads to an underestimation of the stiffness. The simulated results were compared with experiments. With all these sources of errors taken into account, the Young’s modulus and fracture strength of CVD-grown single layer graphene are found equal to 0.88 ± 0.14 TPa and 134 ± 16 GPa, respectively. For CVD BLGr, the mean values of the Young’s modulus and fracture strength are equal to 0.70 ± 0.11 TPa and 95 ± 11 GPa, respectively.


Introduction
Atomic force microscopy (AFM) is a technique based on the measurement of the interactions of the probe, a sharp tip mounted on a soft cantilever beam, in permanent contact, intermittent contact or close proximity with the sample surface. The sample surface topography is measured and reconstructed while surface properties can be simultaneously mapped [1,2]. Compared with optical and electron microscopies, AFM has the advantage of 3D topography measurements in ambient conditions or under vacuum with a spatial resolution down to the atomic or molecular scale [3,4]. Besides imaging, AFM is widely used to measure the mechanical, physico-chemical, electronic, magnetic, electro-mechanical properties of materials and nanomaterials because of the convenient measurement of extremely small forces and deflections with the AFM [5][6][7].
Amongst 2D materials, graphene is considered as exceptional owing, particularly, to its superior mechanical properties. This makes it a unique candidate for many applications both as an individual material and as a reinforcing element in composites [25][26][27]. The remarkable mechanical properties of graphene result from the stability of the sp 2 bounds that form the hexagonal lattice and oppose in-plane deformations. A large number of studies based on molecular dynamic (MD) simulation, density functional theory, etc predict the unique mechanical properties of single-layer graphene (SLGr) such as a Young's modulus around 1.0 TPa and a tensile strength above 130 GPa [28][29][30][31]. Some experimental studies conducted on exfoliated graphene with the AFM-based membrane deflection method have reported similar values [13,18,23]. However, other experimental studies were not in agreement with the theoretical values [16,32,33]. The most cited reasons for these differences in the literature mention the structural defects in graphene, involving corrugations (ripples, wrinkles, crumples) [34], topological defects (disclinations, dislocations, grain boundaries) [35,36], and point defects (vacancies, adatoms) [17,18,37]. Additionally, the effect of edge orientation of graphene (armchair or zigzag) has been discussed [37,38].
Bi-layer graphene (BLGr) has been much less studied. Lee et al have measured Young's modulus and fracture strength similar for SLGr and BLGr, i.e. around 1.0 TPa and 125 to 130 GPa, respectively [23]. Based on MD simulations, it has been predicted that the modulus of BLGr determined with the membrane deflection method should be smaller than that of SLGr, around 0.8 TPa [14]. Another study on BLGr with MD simulations predicted a tensile modulus around 0.96 TPa and a strength of 95 GPa, smaller than that of SLGr. First principle calculations have been conducted to study the effects of the twist angle between the layers and the presence and orientation of twin boundaries in the layers on the Young's modulus and intrinsic strength of BLGr [39]. For the AB stacking, a value of 0.975 TPa has been predicted for the modulus and a value of 122 GPa for the strength. For the AA stacking, values of 0.89 TPa and of 112 GPa have been, respectively, obtained. For intermediate orientations, the modulus is almost independent of the relative orientation (twist angle) as well as of the presence and orientation of grain boundaries in one layer (E ≈ 0.96 TPa for all the studied systems). On the contrary, it has been predicted that the strength is strongly dependent on both the twist angle (111-97 GPa for twist angle between 5 • and 40 • ) and on the orientation of the twin boundaries (from 56 to 93 GPa).
Besides the structural parameters of 2D materials, the parameters related to the testing, characterisation, and analysis methods can also significantly affect the extracted mechanical properties of 2D materials [16,21,31,[40][41][42]. AFM-based membrane deflection is a commonly used method to characterise the mechanical properties of 2D nanomaterials [7,9,21]. However, there are several factors affecting the accuracy of AFM-based results (images and measured mechanical property values) such as the imaging amplitude in Tapping™ mode AFM (amplitude-modulated AFM, AM-AFM) [43], the loading rate [44], the uncertainty on the tip geometry [45,46] which have been investigated in the literature for different materials, but rarely for 2D materials. Moreover, the effects of some other parameters such as the uncertainty on the AFM and probe calibration [40], the ratio between the tip and membrane radii [21,47,48], the uncertainty on the membrane shape, and the uncertainty on loading position are important while studying the mechanical behaviour of a material using AFM-based membrane deflection testing.
To the best of our knowledge, there is no systematic study on the effects of all the above mentioned parameters on the accuracy of the values of the mechanical properties extracted from AFM-based membrane deflection tests. The first goal of this work is thus to discuss the impact of these parameters on the Young's modulus and fracture strength extracted from AFM-based deflection tests on graphene membranes. Procedures to correct the measured values to take into account the non-circularity of the tested membranes or the decentration of the loading position are proposed. The second goal is to determine accurate data for SLGr and BLGr membranes synthesized by chemical vapour deposition (CVD) after taking into account the different sources of errors.

Sample preparation
In order to produce suspended graphene membranes for the AFM-based deflection tests, a mask including arrays of circular cavities with different diameters ranging from 1 to 8 µm was designed with the KLayout software (www.klayout.de/). Since the contrast obtained with optical microscopy of graphene on SiO 2 substrates is very favourable, Si wafers were thermally oxidised (SiO 2 thickness = 300 nm). Afterwards, the Si/SiO 2 substrates were patterned by photolithography using the above mentioned mask. The exposed circular areas were first etched using BHF. Then, deeper cavities were formed with XeF 2 , a gas etchant which partially etches the Si substrate. The final depth of the cavities was around 1 µm.
SLGr, as well as multi-layer graphene, were grown on Cu foil by CVD. The CVD process is described in detail elsewhere [49,50].
In most studies, the classical PMMA-wet transfer method (PMMA = poly(methyl methacrylate)) has been used to transfer 2D layers from the synthesis substrate onto the target characterisation substrate [51][52][53]. In the present study, a modified version of this PMMA-transfer method was used to increase the yield of intact graphene membrane transfer on the cavities. The modified Gr transfer method is detailed in the supplementary information (figure S1) explaining the intermediate step added to the classical method [51][52][53]  Raman spectroscopy (LabRAM HR, Horiba) with an excitation energy of 2.41 eV (514 nm) and a spot size of 1 µm was used to characterise the graphene sheets transferred on the Si/SiO 2 substrate in the supported and suspended form.
Scanning electron microscopy (SEM) (Gemini Ultra-55, Zeiss) was used to locate the graphene flakes on the patterned substrates. The graphene films were mainly characterised using the in-lens detector which receives a signal enriched in type 1 secondary electrons (SE1) and provides high-resolution surface-sensitive information. SEM was also used to characterise AFM tips before and after the AFM characterisation and measurement sessions.
In order to precisely locate the suspended graphene flakes and check their quality as well as the absence of contamination before the deflection tests, AFM imaging (Dimension Icon, Bruker) was performed in standard amplitude modulated AFM (AM-AFM or Tapping™ mode) in air with a linear scanning rate of 0.5 line s −1 . Non-contact silicon probes (PPP-NCHR, Nanosensors) with a nominal spring constant of 40 N m −1 , a resonance frequency around 300 kHz and a nominal tip apex radius smaller than 10 nm have been used. Free vibration amplitudes, A 0 , around 40 nm were used and the set-point attenuation ratio (ratio between the set-point amplitude, A sp and the free amplitude, r sp = A sp /A 0 ) was kept above 0.75 in order to work in the so-called 'soft tapping' mode.

AFM deflection testing.
The mechanical properties (Young's modulus, E, and fracture strength, σ f ) of the graphene membranes were determined by acquiring forcedisplacement curves at the centre of each membrane with the AFM after imaging it in AM-AFM mode. The curves were obtained by performing force spectroscopy measurements consisting in measuring the cantilever deflection, d, as a function of the imposed vertical displacement of the probe, z [54]. The z ramping rate was set to 1 Hz. The cantilever deflection in nanometres was obtained by converting the voltage measured on the photodetector, V PD , using the sensitivity, S PD , in nm V −1 , previously calibrated on sapphire (average value obtained from five successive force-curves): The load, F, was extracted from Hooke's law using the calibrated spring constant, k c , of the cantilever (thermal noise method [54] with the standard correction factor of 1.09 for the deflection sensitivities of rectangular beams): This calibration procedure for the photodetector sensitivity and the cantilever spring constant was chosen for the two following reasons: • Since the objective is to perform 'contact' force-curves on the graphene membranes, the 'contact' photodetector sensitivity was calibrated on sapphire. • Second, the 'standard' thermal noise calibration method is considered as independent of the cantilever beam shape and dimensions. It only requires the measurement of the noise spectrum of the probe and it does not require the measurement of other parameters such as the dimensions of the beam that would introduce additional sources of errors in the calibration process.
The membrane deflection, δ, was calculated as the difference between the imposed vertical displacement, z, and the cantilever deflection, d: Using the obtained values of the applied load and the induced membrane deflection, load-deflection curves were obtained for the graphene membranes. On each membrane, several curves were recorded with increasing values of the maximum load, F max , until failure of the membrane, allowing the determination of the fracture load, F f , associated to the fracture strength, σ f . The analytical relationship between the applied load and deflection of a clamped circular thin plate subjected to a point force at its centre is given by [11, 13, 16, 19-21, 41, 55]: where a is the plate radius and t its thickness. For SLGr, the membrane thickness is considered to be equal to t = 0.335 nm, the interlayer spacing in graphite [56]; for BLGr it is considered to be equal to t = 0.670 nm. E is the Young's modulus of the plate material; q = 1/(1.0491 + 0.1492ν + 0.1583ν 2 ) is a function of the Poisson's ratio, ν, of the plate material [21,55]. A value of ν = 0.165 was considered for the Poisson's ratio. This is the value for graphite [13,57] close to the value theoretically estimated for SLGr [28,42] and for multi-layered graphene [58]. It gives a value of q = 0.980. T is the pretension that can be caused by Van der Waals (VdW) interactions between the suspended 2D layer and the sidewall of the cavity [13,21,42]. The three terms in equation (4) correspond to the bending, the pre-tension, and the stretching regimes of the plate, respectively. If t ≪ δ, the bending term is negligible. In the case of 2D materials where δ is always much larger than t, the load-deflection curves obtained from the deflection tests by AFM can thus be safely fitted with the following equation: where E 2D = Et is defined as the 2D stiffness (or 2D modulus) of the membrane, and σ 2D 0 = σ 0 t = T/(π a) as the 2D prestress of the plate with σ 0 the pre-stress. When the load and the deflection are small, the elastic response of the membrane is linear; under larger applied load, the relationship is cubic.
In this work, a modified version of equation (5) was used [16,21]. It first includes a zero-deflection point, δ 0 , to avoid the drawback of arbitrarily defining this point before fitting the curves. This modified equation is also considering the effect of tip-sample VdW interactions, F 0 . In this approach, the measured load vs deflection curves were fitted with the following expression with F 0 , δ 0 , k 1 and k 2 as free parameters. F mes and δ mes are the as-measured load and membrane deflection, respectively. The pre-stress, σ 0 , the 2D stiffness, E 2D , and the Young's modulus, E, were calculated using the values of the parameters k 1 and k 2 obtained after fitting the load-deflection curve: The fracture strength of the Gr membranes was estimated from F f using the expression proposed by Bhatia and Nachbar [13,21,59] giving the maximum membrane stress in the contact region directly underneath the indenter for a clamped, linear elastic, circular membrane subject to deflection: where R t is the apex radius of the AFM tip, and σ 2D f = σ f t is the 2D strength. The actual apex radius of the used AFM tips was determined using a TGX11 calibration grating (MikroMasch) (see figures S2 and S3 in SI) after the calibration of the photodetector sensitivity and of the cantilever spring constant. This also allowed verifying that the tip was not damaged during the acquisition of the force-curves on sapphire. The calibration procedure of the tip apex radius is detailed in the supplementary information.
AFM-based membrane deflection tests were thus performed on dozens of different suspended SLGr and BLGr flakes using different tips and varying different parameters including the maximum applied load, F max (from 500 nN up to 8000 nN depending on the Gr membrane strength), the radius of the membrane (cavity), a, and the position of the tip with respect to the centre of the tested membrane, r. The testing procedure was as follows. After calibration of the PSPD sensitivity and of the cantilever spring constant, the sample was imaged in Tapping™ mode to locate intact suspended graphene flakes. Each tested membrane was imaged to determine its radius then the tip was positioned at the centre (or at a given distance from the centre) of the membrane. Successive approach-retraction curves were finally recorded with increasing values of the maximum applied load until membrane failure.

Finite element (FE) simulations.
The COMSOL Multiphysics software (COMSOL Inc.) was used to investigate the effect of different parameters such as the uncertainty on the membrane loading position and the uncertainty on the shape of the membrane. Three dimensional FE simulations were performed by making use of shell elements, imposing a linear elastic behaviour for the SLGr (Young's modulus E = 1 TPa) of thickness equal to 0.335 nm. The circular membrane was rigidly clamped with radius a = 1, 1.5 or 2 µm. The load was considered as a point load since the nominal AFM tip apex radius is much smaller than the membrane radii, R t ≪ a.
FE simulations using shell elements were also performed with the code ABAQUS (Dassault Systèmes) using four integration points and using a linear elastic behaviour to check the accuracy of the analytical solution given by equation (4) and to cross-check the results of the simulations with COMSOL.

Structural characterisation
The quality of the transfer of graphene on the patterned Si/SiO 2 substrate was first checked using SEM. Figures   SI). The darker holes as shown in the images correspond to intact suspended graphene. A brighter contrast occurs around the broken zones in suspended graphene layers.
The quality of the transferred graphene flakes was also checked with Raman spectroscopy. Figure S7(a) (SI) presents a comparison of two Raman spectra obtained, respectively, for suspended and supported SLGr on a patterned Si/SiO 2 substrate, and reveals high 2D-to-G peak intensity ratios and narrow 2D band-widths (equal to 2 and 25 cm −1 , respectively). The D-to-G peak intensity ratios are smaller than 0.1. The intensity ratio between the 2D and the G peaks is higher for suspended SLGr than for supported SLGr. Figure S7(b) (SI) shows ten Raman spectra obtained at different positions on the same BLGr flake such as the ones presented in figure S6 (SI). The spectra exhibit the G and 2D peaks with generally a similar intensity. The intensity ratio between both peaks is different from one location to another.
Before performing the membrane deflection tests, the graphene membranes were located and imaged in AM-AFM mode. Figure 1 presents topography and phase images obtained on a circular membrane (radius a = 2.3 µm); (a) and (c) images are respectively the topography and phase images obtained with a fresh new probe (see figure S8(a) in SI); (b) and (d) are the images obtained on the same membrane with the same probe after the acquisition of several images and loaddeflection curves leading to the wear of the tip (see figure S8(b) in SI). In this figure, the line profiles measured on the topography images (figures 1(a) and (b)) along the red and blue lines are also presented (figures 1(e) and (f)). Figure 2 presents the topography image of another SLGr membrane suspended on a cavity with a nominal radius a = 1 µm. In this image, one can observe that some cavities do not present a perfect circular shape but involve a certain squareness. Moreover, in this case, the actual dimension is much larger than the nominal radius defined on the lithography mask.

Young's modulus and fracture strength of CVD SLGr
and BLGr. Typical load-deflection responses of SLGr membranes are presented in figure 3. Figure 3(a) presents three unloading curves measured on three SLGr membranes with different radii and with a maximum load F max = 1000 nN. As expected, the stiffness of the membranes decreases when the radius increases. Figure 3(b) shows the loading and unloading curves measured on a SLGr membrane with a maximum load (F max = 3500 nN) leading to the fracture of the membrane at a load F f ≈ 3200 nN. For SLGr membranes, fracture typically occurs for loads between 3000 and 5000 nN. Table 1 gathers experimental results obtained for SLGr after analysing tens of load-deflection curves similar to the one presented in figures 3(a) and (b). These data were obtained after taking into account all the uncertainties and sources of error discussed later in this document and applying the procedures described below to minimise these errors (see section 4.2.2). The three first lines present the average values of the 2D stiffness obtained after fitting with equation (6) different load-deflection curves measured with a maximum load of 1000 nN on SLGr membranes involving different diameters: 1, 3 and 4 µm (10 curves for each diameter). The average values of the 2D fracture strength computed from the values of the measured fracture load, F f , are also reported.
The error on the values of the 2D stiffness and Young's modulus resulting from the calibration of the AFM and probe parameters (sensitivities of the piezoelectric scanner, S x , S y and S z , deflection sensitivity of the position-sensitive photodetector, S PD , cantilever spring constant, k c ) was estimated by performing different experiments: successive loading of the same membrane with the same probe, successive loading of different membranes with the same probe and loading of different membranes with different probes. The results of these experiments are also presented in table 1. Ten first tests (fourth line) correspond to the successive loading of the same membrane with the same probe, i.e. with the same calibration of the AFM and of the cantilever. The average value of the 2D stiffness for these 10 tests is equal to 295 N m −1 with a standard deviation of 5 N m −1 corresponding to a relative error of 1.8%. Five different membranes were tested with another probe and calibration (fifth line). For this series of tests, the average value of the 2D stiffness is equal to 314 N m −1 with a standard deviation of 18 N m −1 corresponding to a relative error of 5.9%. Finally, five different membranes were tested with five different probes and thus different sets of calibration (sixth line). For this last series of tests, the average value of the 2D stiffness is equal to 296 N m −1 with a standard deviation of 6 N m −1 corresponding to a relative error of 2.0%.
Considering all the measured values on SLGr, a mean value of E 2D = 296 ± 16 N m −1 is obtained for the 2D stiffness of SLGr. This corresponds for the Young's modulus of CVDgrown SLGr to an average value of E = 0.883 ± 0.048 TPa. The experimental relative error is thus equal to ≈5.4%. For the 2D strength, the mean value is equal to σ 2D f = 45.0 ± 4.1 N m −1 , corresponding to a mean value of the fracture strength of CVD-grown SLGr σ f = 134 ± 12 GPa.
AFM-based deflection tests were also performed on BLGr flakes. Figures 3(c) and (d) present typical loaddeflection curves measured on BLGr membranes. The mean value obtained for the 2D stiffness is equal to E 2D = 471 ± 53 N m −1 . This corresponds to an average value of the Young's modulus E = 0.702 ± 0.079 TPa. The relative error on these values is equal to 11.3%. For the 2D strength, an average value of σ 2D f = 63.7 ± 7.1 N m −1 was obtained. This corresponds to a fracture strength σ f = 95.0 ± 10.6 GPa for CVD-grown BLGr.

Tip wear effects.
Tips may wear during AFM scanning and/or force spectroscopy measurements. This is illustrated in figure S8 (supplementary information) showing SEM images of the same tip before and after dozens of imaging scans and deflection tests. Tip wear affects the quality of the images as illustrated in figure 1 where the topography (figure 1(b)) and phase (figure 1(d)) images obtained after several image acquisitions on the same membrane as the one  shown in figures 1(a) and (c)) are presented. The images were acquired using the same settings of the instrument, i.e. with the same free vibration amplitude, A 0 , the same set-point attenuation ratio, r sp , the same scan rate, and the same gains for the feedback loop. Figure 1(f) presents the line profile corresponding to the topography image acquired with the worn tip. Tip wear also affects the results of AFM-based membrane deflection experiments. Figure 4(a) presents load-deflection curves measured on the same SLGr membrane with a fresh new probe and with an old probe involving a worn tip. When measured with an old probe, the membrane seems to be less stiff as shown by the values of the 2D stiffness and the Young's modulus obtained after fitting (table S2 in SI). Figure 4(b) presents the evolution of the 2D stiffness determined by fitting load-deflection curves successively measured on nine different SLGr membranes with the same probe. Between each curve acquisition, several AFM images were acquired to locate and characterise the next membrane. The corresponding numerical The effect of off-centre loading of the membranes was experimentally studied by measuring load-defection curves on different membranes with a radius a = 2 µm at various distances, r, from the centre and by fitting the obtained curves with equation (6) valid for a circular membrane loaded at its centre. The comparison between load-deflection curves measured at different distances from the centre of a SLGr membrane (five different positions from 0 to 1 µm) is presented in figure 5(a). The stiffness increases when the loading position moves away from the centre of the membrane. Figure 5(b) reveals the effect of the loading position on the extracted 2D stiffness for about 100 different deflection tests performed on different membranes with different probes. The apparent value of E 2D progressively increases when the loading position deviates from the centre of the membrane.
As illustrated in figure 2, the shape of the cavities defined in the Si/SiO 2 substrates can deviate from a perfect circular shape. Membrane deflection measurements have been performed on five different SLGr membranes suspended on cavities presenting the same squared shape. Using equation (6) valid for circular membranes, the average value obtained for the 2D stiffness was equal to E 2D = 244 ± 15 N m −1 , corresponding to an apparent value of the Young's modulus E = 0.728 ± 0.044 TPa. These values are smaller than those measured on circular SLGr membranes. 3.2.4. Membrane deflection modelling. The load-deflection curve calculated using this analytical model (equation (4)) was compared to simulated load-deflection curves obtained with COMSOL and ABAQUS. The results are presented in figures 6(a) and (b). For these simulations, no pre-tension was considered. The load is reported as a function of the deflection normalized to the plate thickness, δ/t. On the log-log plot ( figure 4(b)), three regimes can be distinguished. For very low deflection values (δ/t < 0.2), a linear relationship is observed (regime (1)). For larger deflection values (δ/t > 10), a third power relationship is observed (regime (3)). A transition is observed (regime (2)) for normalized deflection values between 0.2 and 10. In figure 4(c), the relative errors between the load values simulated with COMSOL and ABAQUS and the analytical values are reported as a function of the normalized deflection. In regime (1), the error is very small. Then, the error increases to reach a peak value around 40% in the transition regime (2). Finally, the relative error decreases when entering regime (3), reaching, at δ/t = 10, a negligible value when using the ABAQUS simulation for reference and a value smaller than 8% when using the COMSOL simulations as reference.
COMSOL simulations were conducted to evaluate the error due to the off-centre positioning of the tip. More than 100 simulations have been performed for different loading positions on circular membranes with radius values a = 1, 1.5 and 2 µm and the simulated curves have been fitted with equation (4) to extract an apparent value of the Young's modulus. Figure 7 presents the variation of the apparent value of the Young's modulus as a function of the loading distance relative to the membrane centre normalized to the membrane radius, r/a. The simulated values are compared with the experimental data ( figure 5(b)). The simulated data were fitted with a 5th degree polynomial expression: with E the apparent measured Young's modulus and E 0 the actual modulus, measured for a loading position at the membrane centre. The following values were obtained for the polynomial coefficients: a 1 = 0.009 78, a 2 = 0.650, a 3 = 2.125, a 4 = −3.905, and a 5 = 3.960. The effect of the shape of the membrane (cavity) has been investigated using COMSOL simulations. The results of these simulations are presented in figure 8 in which the variation of the normalized Young's modulus obtained by fitting the simulated load-deflection curves is reported as a function of the radius of curvature (fillet, r f ) used to model square cavities with progressively more rounded corners normalized to the cavity nominal radius a. A value of r f /a = 0 corresponds to a square cavity and a value of r f /a = 1 to a perfectly circular one. When the shape of the cavity deviates from the circular shape, the stiffness and hence the apparent Young's modulus decreases. On the same graph, the mean relative value of the modulus obtained from load-deflection curves measured on a SLGr membrane presenting a squared shape such as the one presented in figure 2 is also reported. The relative value of the  modulus levels off to a value of 0.887 for a square cavity. The simulated data were fitted with a 4th order polynomial with E the apparent measured Young's modulus and E 0 the actual modulus measured on a circular membrane. The following values were obtained for the coefficients: b 1 = −0.557, b 2 = 1.027, b 3 = −0.835, and b 4 = 0.253.

Structural characterisation
As it can be seen in figure S5 (SI), the quality of the suspended Gr membranes transferred with the modified method is much better even in the case of cavities with a large radius up to 3 and 4 µm. The additional drying step in the modified PMMA-transfer method reduced the possibility of water accumulation inside the cavities after transfer on the Si/SiO 2 substrate. As a result, less water was evaporated from underneath the graphene sheet during drying and the fracture of graphene on the cavities was much less frequent. The Raman spectra presented in the supplementary information ( figure S6) prove the quality of the transferred SLGr and BLGr flakes [60,61]. For SLGr, the large intensity ratio between the 2D and G peaks demonstrates the single atomic layer nature of the graphene flakes. The small ratio (<0.1) between the D and G peaks suggests a low density of structural defects in the transferred flakes. For BLGr, the reduced intensity ratio between the 2D and G peaks that present similar intensity values confirms the bilayer nature of the flake [62,63]. The differences between the spectra (different intensity ratios between the G and 2D peaks) can be accounted for by the coexistence of different relative orientations between the two layers in the same flake [64].
The AM-AFM images presented in figures 1(a) and (c) also prove the quality of the graphene membranes. Almost no contamination is visible on the topography and phase images. This confirms the high quality of the investigated SLGr and BLGr, which is important to support the relevance of the extracted stiffness and strength data. The lower phase lag observed on the membrane is in agreement with the expected smaller stiffness of suspended graphene compared with supported graphene. The line profile measured on the topography image (figure 1(e)) shows that the graphene membrane is sticking to the cavity sidewall due to VdW interactions. This suggests a pre-tension in the tested membranes and justifies the use of equation (5) including a pre-tension term to fit the measured load-deflection curves.

AFM membrane deflection tests
Generally, the mechanical properties measured on nanostructures reported in the literature are provided with a standard deviation value resulting from experimental replicates without sufficiently detailing the underlying sources of error [21]. It makes the comparison of measurements between different groups, instruments and techniques often difficult. This can be detrimental to the development of reliable nano-engineered materials and products based on robust design rules. To deal with this problem, considering all the relevant uncertainties that arise from different sources is necessary [40]. This is the first goal of this study.
As already mentioned, errors on the values of mechanical properties measured on 2D materials with AFM-based membrane deflection experiments have mainly two origins, namely the 'instrumental' uncertainty due to the various calibration The maximum relative uncertainty on the Young's modulus (or the 2D stiffness), u E , and on the fracture strength, u σ , as measured with the membrane deflection method and resulting from the uncertainties on the calibration of the measurement system parameters (scanner sensitivities, photodetector sensitivity, cantilever spring constant) were theoretically calculated using the theory of uncertainty propagation [65]. The following expressions were obtained (see SI for the details of the calculations): In these expressions, u T ≈ 1/294 = 0.34% is the uncertainty on the ambient temperature used in the calibration of the cantilever spring constant with the thermal noise method; u x,y ≈ 2% is the uncertainty on the calibrated horizontal sensitivities of the AFM scanner, S x and S y , directly determining the uncertainty on the value of the membrane radius, a, as measured on the AFM images; u z ≈ 2% is the uncertainty on the vertical sensitivity, S z , of the piezoelectric scanner directly involved in the calibration of the photodetector sensitivity and in the calculation of the membrane deflection; u Vpd ≈ 4% is the uncertainty on the voltage measured on the positionsensitive photodetector, experimentally measured and used in the calibration of the photodetector sensitivity and the measurement of the cantilever vertical deflection that is used to calibrate the cantilever spring constant, compute the applied load and calculate the membrane deflection; finally, u Rt ≈ 10% is the uncertainty on the tip apex radius of curvature, R t , as calibrated using a dedicated calibration grating (see SI). In table 2, the uncertainty budgets for the 2D stiffness and the fracture strength are presented. The value obtained for the uncertainty on the membrane 2D stiffness and the material Young's modulus is equal to u E ≈ 16.0% and for the fracture strength, it is equal to u σ ≈ 11.6%. These values are in agreement with the value previously calculated by Lee et al [13], i.e. 14% for the modulus and 10% for the strength. This analysis demonstrates that, provided there are no other sources of errors (see section 4.2.2), AFM-based membrane deflection tests provide values of mechanical properties of 2D materials such as the Young's modulus within 16% in the worst case.
This value for the uncertainty on the modulus of a material determined with the membrane deflection method is smaller than the value of 37% reported for the uncertainty on the modulus of a material as measured with AFM-based nanoindentation [40]. This difference is partly explained by the fact that deflection methods do not require the calibration of the tip shape and dimensions.
The largest contributions to the uncertainties, u E and u σ , come from the uncertainty on the voltage measurement on the photodetector, u Vpd , followed by the uncertainty on the vertical sensitivity of the piezoelectric scanner, u z (table 2). This is due to the fact these parameters are involved in several steps of the system calibration, in the deflection measurements and in the data analysis: photodetector sensitivity and cantilever spring constant calibrations, load and membrane deflection measurements, and cubic load vs deflection relationship.
Another important point is the dependence of u E and u σ on the absolute values of the imposed vertical displacement, z, and on the cantilever vertical deflection, d. The membrane deflection, δ, is indeed the difference between both quantities. As shown in table S1 (see SI), very large relative errors on the membrane deflection, and hence on the modulus and strength, can be obtained especially if the values of z and d are close (difference of large numbers). To minimize this error, the cantilever deflection must be significantly smaller than the vertical displacement. To achieve this, it is necessary to carefully select the probe, i.e. with a cantilever spring constant similar or larger than the stiffness of the tested structure to induce a sufficiently large deflection of the latter.

Tip size and geometry. As illustrated in figure S8
(see supplementary information), tip wear can significantly modify the tip shape and dimensions. Tip ageing/wear mostly occurs while scanning across the sample. However, performing deflection tests with loads larger than 1000 nN (up to thousands of nN) can accelerate the process. Tip size and geometry change as a result of tip ageing/wear and need to be considered as they not only influence the quality of the AFM images (figure 1) but they also have effects of the stiffness values extracted from the load-deflection curves ( figure 4).
Tip wear first affects the imaging quality as illustrated in figure 1. This figure presents the results of AM-AFM imaging of the same SLGr membrane with a fresh new tip and with the same tip after multiple imaging scans. Both the topography (figures 1(a) and (b)) and the phase (figures 1(c) and (d)) images are clearly different when they are acquired with a new probe and with a worn tip. The cross-sections presented in figures 1(e) and (f) also clearly illustrate the effect of tip wear on the quality of the images.
One effective parameter that can affect the shape of the tip in tapping mode AFM is the force regime (low amplitude/attractive vs high amplitude/repulsive) [66]. The wear of the AFM tips significantly depends on the force regime which determines the tip/sample interaction forces and dissipated energy during the tip/surface contact. However, controlling the force regime and hence the tip degradation is not straightforward due to the instabilities resulting from the anharmonicity of the tip/surface interactions [67]. Since the tip size and geometry determine the interaction forces (attractive or repulsive), it is however possible to verify the integrity of the tip by checking the stability of the AFM images and of the spectroscopic data acquired under constant conditions (same sample material, same tapping mode set-point attenuation ratio r sp , same maximum load, same loading rate, etc).
As shown in figure 4 and in tables S1 and S2 (see SI), tip wear leads to an underestimation of the stiffness and Young's modulus when fitting the load-deflection curves measured with a worn tip with equation (6). This equation is valid for point loaded membranes, i.e. if the tip radius R t is much smaller than the membrane radius a, typically for a/R t > 30 [21,47]. The deflection of circular membranes with a spherical indenter has been studied by several authors [47,48,68]. All these studies showed that, when a/R t decreases, the apparent stiffness of the membrane increases. This is not what is found in the present study where the apparent stiffness of graphene membranes decreases when deflected with a blunted tip. An increased contact area between the tip and the membrane cannot thus explain the apparent softening of the membrane response with tip ageing.
Tip wear can also modify the VdW interaction forces with the 2D materials membrane. The literature predicts that neglecting these interactions and fitting the load-deflection curves with the classical equation leads to an overestimation of the membrane stiffness (Young's modulus) [21,31,41]. Again, a possible increase of VdW interactions due to the tip wear cannot account for the observed apparent softening of the graphene membranes with tip ageing.
As shown in figure S8, the wear-induced modifications do not result in simple well-defined spherical or paraboloid shapes. This makes it almost impossible to predict the effects of these modifications on the quality of the AFM images and on the interpretation of the spectroscopic measurements.
This analysis on tip wear shows the importance of repeating tests many times to check if the response evolves or not and to repeat the experiments with a fresh tip if unexpected increase or decrease in the extracted stiffness are observed in order to detect possible tip degradation. Such preliminary tests can be used to determine a regime of loading and tip apex radius that minimises wear.

Decentration of loading position.
A mathematical model leading to equation (4) needs to be used to fit forcedeflection curves and to extract the Young's modulus. This relationship relates the indentation load to the deflection of a clamped circular thin plate subjected to a point force at its centre [21,47,55]. The point force needs to be precisely applied at the centre of the membrane and the membrane must be perfectly circular. However, experimentally, the load is never perfectly applied at the centre since the loading position is manually determined by the operator. This leads to a misestimation of the correct values of the 2D stiffness and Young's modulus.
To evaluate the error due to off-centre loading of the membranes, experimental tests were performed where the loading was intentionally placed at different distances from the centre (figure 5). COMSOL simulations were also conducted ( figure 7). First the validity of the simulations was checked by comparing the load-deflection curves computed with COMSOL with the ones computed with ABAQUS and the ones calculated with the analytical model (equation (4)) (figures 6(a) and (b)). The mechanical response of a point-loaded clamped circular film essentially presents three different asymptotic behaviours [55]: behaviour (1) for small deflection values and negligible pre-tension where the film follows the linear plate bending theory (1st term in equation (4)), behaviour (2) where the pre-strain stiffness is comparable or larger than the bonding stiffness but where deflection values remain small enough to stay in a linear regime (2nd term in equation (4)), and behaviour (3) where the bending stiffness is negligible compared with the stretching stiffness due to large deflection values leading to the cubic regime (3rd term in equation (4)). Since both behaviours (1) and (2) are characterised by linear load-deflection relationships, it is difficult to distinguish them. A fourth behaviour can be observed at very large loads where large rotation occurs and for which the membrane deflection theory becomes invalid. On the analytical curve and the simulated load-deflection curves, three different regimes (1, 2 & 3) can be clearly observed ( figure 6(b)).
Applying the load at increasing distances from the centre of the membranes leads to a stiffer response and to an overestimation of the Young's modulus (figure 5). The experimental and simulated data agree very well (figure 7) and the simulated data could be fitted with a 5th degree polynomial (equation (11)). This fit can be used to accurately determine the actual stiffness of a membrane on which several loaddeflection curves have been measured at different locations from the centre, as illustrated in figure 9. One possible procedure consists in first measuring load-deflection curves at different points arranged in a 3 × 3 matrix centred as close as possible to the centre of the membrane. The curves are then fitted with the analytical model to extract the apparent values of the 2D stiffness (or the Young's modulus). Eventually, the stiffness E 2D extracted at the different positions (x, y) with respect to the central point of the matrix are fitted with the following equation: a the radius of the tested membrane, and a i the fitting coefficients of equation (11). The free variables are x 0 and y 0 , i.e. the coordinates of the centre of the membrane, and E 2D 0 the actual 2D stiffness (which would be obtained if the loading point was precisely placed at the centre of the membrane).
In figure 9, the grey scale corresponds to the experimental values of the 2D stiffness obtained from load-deflection measured at nine different points spaced horizontally by △x = 600 nm and vertically by △y = 530 nm (the loading points are located at the centre of the grey squares). The contour plot corresponds to the fit with equation (15)

Deviation of membrane shape.
As already mentioned, the analytical model used to analyse the experimental data is valid for clamped circular thin plates. However, some deviation from the circular shape can occur in the experimental procedures resulting from the lithography or etching steps. Most of the holes defined in the Si/SiO 2 were almost perfectly circular (figure 1) except some few holes with a smaller radius (nominal a = 1 µm) (figure 2). This deviation from a perfect circular shape resulted from limitations in the photomask design. The apparent 2D stiffness measured on the specific SLGr membrane shown in figure 2 was equal to E 2D = 244 ± 15 N m −1 , a value 17% smaller than the average value of 296 N m −1 measured on circular membranes. This discrepancy is clearly larger than the experimental relative error of 5.4% obtained for the measurements on circular cavities and also larger than the theoretical relative uncertainty of 16%. It should thus be due to the non-circularity of the cavity, at least partly.
COMSOL simulated data provide a fourth order empirical polynomial (equation (12)) that can be used to correct data obtained on non-circular membranes after determining the normalized value of the corner radius r f /a. This can be easily done with standard image analysis tools as explained in the supplementary information. The value of r f /a is indeed related to the rectangularity R R of the cavity shape with the following relationship: The rectangularity is defined as R R = A/A rec , with A the area of the cavity shape and A rec the area of the inscribing rectangle.
The rectangularity is either directly given by the image analysis tool or can be calculated from other parameters given by the image analysis tool (radius of the inscribing circle, length of long and short axis of the inscribing ellipse, etc). For the graphene membranes similar to the one presented in figure 2, r f /a was estimated to be around 0.7 ± 0.1. In figure 8, the vertical error bar corresponds to the experimental error of 6%. Using the experimentally measured value of the 2D stiffness, 244 N m −1 , and equation (12), a mean value of 270 N m −1 is obtained for the 2D stiffness of these SLGr membranes, a value 8.5% smaller than the mean value reported in table 1 within the maximum theoretical uncertainty.

Young's modulus and fracture strength of SLGr and BLGr.
The Young's modulus and the fracture strength of single-layer and bilayer CVD graphene transferred on cavities of different sizes was investigated using the AFM-based membrane deflection method taking into account all sources of uncertainties and errors discussed above.
For CVD SLGr, an average Young's modulus E = 0.88 ± 0.14 TPa was obtained and an average fracture strength σ f = 134 ± 16 GPa was determined. The value of the Young's modulus is smaller than the value of ∼1 TPa reported in the literature either on the basis of theoretical predictions [28,30] or on the basis of AFM-based membrane deflection experiments [13,17,23]. However, the values reported in these experimental studies were measured on exfoliated graphene. Membrane deflection measurements performed on CVD graphene by Lin et al gave modulus values varying between 0.5 and 0.8 TPa, depending on the stretching cycle [16]. The fact that the Young's modulus measured on CVD graphene is smaller than the one measured on exfoliated graphene or predicted for perfect graphene is consistent with the fact that CVD graphene contains more defects, especially more grain boundaries [69][70][71]. Moreover, the transfer process may induce additional defects in the graphene flakes [52]. The presence of defects reduces the stiffness of graphene as reported by several experimental and theoretical studies [17,18,30,37]. The value of the fracture strength σ f ≈ 134 GPa is in fair agreement with the experimental and theoretical values reported in the literature [13,23,28].
For CVD BLGr, the following average values were obtained for the Young's modulus and fracture strength: E = 0.70 ± 0.11 TPa and σ f = 95 ± 11 GPa. These values are smaller than those measured on SLGr which is in agreement with the literature [14,32]. However, the value of the Young's modulus is, as for CVD SLGr, smaller than the ones reported in the literature either on the basis of simulations [14,39,72] or as measured on exfoliated BLGr [23].
CVD BLGr is a layered material where the interlayer structure and coupling is mainly governed by the twist angle between the layers. The twist angle of CVD grown BLGr can strongly vary and the coexistence of different atomic registry in the same graphene flake lead to the presence of twin boundaries [73]. This variety of twist angle is also observed on our CVD BLGr as shown by the results obtained with Raman spectroscopy (figure S6 in SI). One can see that the intensity ratio between the G (1590 cm −1 ) and 2D (2690 cm −1 ) peaks, related to the twist angle between the layers [64], varies from one spectrum to another. Unfortunately, it was not possible to determine the angle between the layers during the mechanical characterisation of the BLGr membranes and, hence, impossible to correlate it with the measured values of the Young's modulus and fracture strength. It is thus difficult to conclude if the smaller values of the stiffness and strength measured on BLGr membranes compared with SLGr membranes is due to misorientation between the layers or to the presence of defects such as twin boundaries.
The largest experimental dispersion of the values of the modulus measured on the BLGr membranes (11%) compared to that for SLGr membranes (5.4%) might be explained by different relative orientations of the layers from one membrane to another. However, first principle calculations conducted to predict the mechanical properties (modulus, failure strength and strain) of BLGr with twist defects and grain boundaries demonstrated that the Young's modulus is almost independent of either the twist angle between the layers or the presence of grain boundaries in one layer [39]. For all BLGr systems, the computed Young's modulus was found equal to E = 0.963 ± 0.013 TPa, close to 0.975 TPa for AB BLGr and larger than 0.866 TPa of AA BLGr. On the other hand, the study showed that the fracture strength of BLGr is more sensitive to the angle between the layers and the presence of twin boundaries.
The values reported for the modulus and fracture strength are clearly proportional to the measured applied load and are thus very dependent on the errors associated to the method used to calibrate the cantilever spring constant. In the present work, the most widely used calibration method was applied, i.e. the calibration of the 'contact' photodetector sensitivity by the measurement of the force-curves on sapphire followed by the spring constant calibration with the thermal noise method using the a correction factor to take into account the difference between the 'contact' and 'non-contact' photodetector sensitivities. It has been shown that the value of 1.09 generally used as correction factor and valid for rectangular beam is not valid for all probes but depends on the actual geometry and dimensions of the cantilever beam [74]. Using the correction factors proposed in [74] for the probe used in this work (NCHR from Nanosensors), would lead to a decrease of all the reported values by 6.7%. The use of another procedure for the spring constant calibration among the large panel of methods proposed in the literature (see for instance [75][76][77][78][79]) could also have lead to different results.

Conclusions
The Young's modulus and fracture strength of CVD-grown graphene in single and bilayer forms have been determined using AFM-based membrane deflection experiments. The discrepancies observed between the obtained values and those reported in the literature lead to a careful analysis of all the potential sources of errors in this experimental method and to propose enhanced data reduction schemes.
Thus, this study first clarifies the effects of different instrument and experimental parameters that affect the accuracy of AFM-based membrane deflection tests. These parameters are: • The AFM calibration: the uncertainties on the values of the applied load and measured membrane deflection are determined by the uncertainties on the calibrated values of the scanner sensitivities, the position-sensitive photodetector deflection sensitivity and the cantilever spring constant and affect the accuracy of the measured elastic stiffness and fracture strength. The uncertainty on the 2D stiffness and Young's modulus is theoretically estimated equal to 16%. For the fracture strength, the theoretical uncertainty is equal to 11.6%. Experimentally, a dispersion of 5.4% is obtained for the Young's modulus values determined on SLGr. • The tip shape: during the imaging or mechanical testing processes, tip wear can occur which has a significant effect on the tip shape and dimensions. This affects the value of the fracture strength as derived with equation (9) which involves the tip radius of curvature. It can also affect the value of the stiffness since equation (4) is valid for point loaded membranes and can be safely applied provided the ratio between the membrane and the tip radii remains large enough, i.e. a/R t > 30. In the present study, it has been shown that, even if the tip size remains small compared to the membrane radius, tip wear not only affects the quality of the AFM images but also progressively leads to an underestimation of the stiffness (Young's modulus) of the membrane material. It is thus important, not only to properly calibrate the shape and dimensions of the tip, but also to regularly verify the quality of the tip by checking the reproducibility of images and spectroscopic data acquired under the same conditions, or by repeating tests with different probes and verifying there is no deviation larger than the theoretical uncertainty. • The loading position: the analytical model used to extract the Young's modulus assumes that the applied load should be exactly placed at the centre of the circular membrane. The mis-positioning of the load causes an overestimation of the Young's modulus. A polynomial expression has been derived from FE simulated data to determine accurate Young's modulus by measuring load-deflection curves at various relative distances from the membrane centre. • The shape of the membrane: the analytical model used to extract the Young's modulus from the load-deflection curves is valid for point loaded clamped circular membranes. The design of the substrate used for the deflection experiments led to cavities deviating from the circular shape for the smallest ones. Based on experimental results and on FE simulations, we show that deviation from the circular shape leads to an underestimation of the Young's modulus that can be larger than 11% when the cavity shape becomes squared. A procedure to correct the value of the Young's modulus is proposed, based on a phenomenological expression relating the actual modulus to the measured one and to the radius of curvature of the corners of squared cavities.
Considering the effect of these parameters, AFM-based loaddeflection curves were measured on dozens of membranes of CVD graphene. For SLGr sheets, the Young's modulus is equal to E = 0.88 ± 0.14 TPa and the facture strength is equal to σ f = 134 ± 16 GPa. For BLGr, the Young's modulus is equal to E = 0.70 ± 0.11 TPa and the fracture strength to σ f = 95 ± 11 GPa. For both SLGr and BLGr, the values of the Young's modulus are smaller than the value around or above 1 TPa reported in the literature on the basis of simulations or measured on exfoliated graphene. The value of the fracture strength measured on SLGr is in good agreement with the theoretical and experimental values reported in the literature. For BLGr, the measured value of the fracture strength corresponds to the one predicted for twisted and twinned bilayers.
The obtained values might be explained by the larger amount of defects in CVD graphene, mainly grain boundaries, the various relative orientations of the layers in BLGr as revealed by Raman spectroscopy. As mentioned at the end of section 4, the method used to calibrate the cantilever spring constant could also affect the value of the applied load and hence the values extracted for the Young's modulus and the fracture strength. As a perspective, it would thus be interesting, though challenging, to systematically study the effects of using different calibration procedures and reconcile them by unravelling the origin of the discrepancies. In situ measurement of this twist angle should also be performed simultaneously with the mechanical characterisation to confirm and quantify the effect of the twist angle on the mechanical properties of BLGr. This could be done for instance by imaging the moiré pattern with piezoresponse force microscopy [80] or Kelvin probe force microscopy [81].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).