Ultrafast quantitative nanomechanical mapping of suspended graphene
Abstract
Understanding the mechanical properties of suspended graphene membranes is crucial to the development of graphene nano-electromechanical devices. PeakForce QNM (quantitative nanomechanical mapping) atomic force microscopy imaging was used to rapidly map the nanomechanical properties of a range of suspended graphene membranes. The force–displacement behavior of monolayer graphene extracted from the peak-force imaging map was found to be comparable to that taken using standard nanoindentation. By fitting to a simple elastic model, the two-dimensional elastic modulus was measured at around 350 N m−1, corresponding to a Young's modulus of around 1 TPa. 
Nanomechanical parameters which can be directly extracted from force curve data in real time. Inset shows a dissipation map of a suspended graphene membrane.
1 Introduction
Probe based indentation is a widely used technique for studying local mechanical material properties 1. The most accurate way of mapping material properties at the nanoscale has been force-volume imaging in an atomic force microscope (AFM) 2. In this mode, the tip is ramped using the z-axis stepper motor to perform a nanoindentation at every point in an image, producing a two-dimensional array of force curves, from which a variety of nanomechanical properties can be extracted. However force-volume mapping and standard tapping mode imaging do not deal well with large sudden changes in the surface mechanical properties, and the extensive time required to complete a force volume plot limits its usefulness.
The recently introduced PeakForce QNM (quantitative nanomechanical mapping) AFM mode 3, 4 allows the spatial mapping of the nanomechanical properties of a surface at high speed by effectively ramping with the cantilever rather than the z-piezo; the time taken for a full nanomechanical map is comparable to the time taken for a standard AFM tapping mode topography-only image.
Graphene, a two dimensional (2D) carbon lattice, has attracted broad interest due to its unique electrical, thermal, and mechanical properties 5-7, and could play an important role in the next generation of micro/nano-electromechanical systems (M/NEMS). Its measured intrinsic strength exceeds that of any other material 8. Suspended graphene membranes 9 are a useful experimental testbed in a wide range of areas 10-12, but are challenging to characterize due to their extreme optical and electronic transparency.
AFM nanoindentation and force volume experiments on suspended graphene are the primary way through which the mechanical properties of graphene and other two-dimensional materials can be measured 13-16. Previous attempts to image graphene membranes using tapping mode AFM have been problematic due to the unusual combination of their high strength and high elasticity. The high interaction forces between the AFM probe and graphene induce large high frequency oscillations of the graphene membrane, producing a triangular wave like artifact as the oscillation is repeatedly induced and subsequently decays. Although it was possible to image the topography of smaller membranes in tapping mode, by carefully controlling the scan parameters to reduce the tip–sample interaction force, the height data was misleading due to the lack of knowledge about the force applied to the membrane (the topography of which changes with the applied tip force). These issues can be overcome in peak-force tapping mode which allows for a pre-defined force to be applied during each “tap,” and this force can be minimized to very small values. Since this process involves capturing a full force curve at every “tap” during imaging, it also provides a full nanomechanical characterization at every data-point on the AFM image. Here we present nanomechanical maps of suspended graphene taken using the PeakForce QNM imaging mode and compare extracted force curves to those measured using conventional nanoindentation/force volume mode. The force–displacement data was in agreement, and by fitting this to a simple model of a circular 2D elastic film under central point loading the Young's modulus of graphene was measured at around 1 TPa 8, 16, in agreement with literature values. We also demonstrate that peak-force imaging allows the capture of topographic data from large soft membranes with a high dynamic vertical range using non-contact AFM, as opposed to tapping mode imaging where the tip–membrane adhesion induces oscillations in the membrane which result in triangular artifacts.
2 Materials and methods
The suspended graphene membranes used in this study were supported by patterned silicon nitride membranes. The nitride membranes were patterned with 8 µm diameter holes using photolithography and reactive ion etching (RIE). Graphene flakes were mechanically exfoliated 17 from graphite onto silicon dioxide coated substrates. A 90 nm thick SiO2 coating was used to maximize the optical contrast and allow the number of layers to be easily visualized 18. A flake containing regions of both monolayer and trilayer graphene was selected and transferred onto the patterned nitride membranes using a standard flake transfer procedure 19. Pictures of the flake before and after transfer and a schematic cross section of the resultant membrane are shown in Fig. 1.

PeakForce Tapping and conventional TappingMode AFM images were taken using a Dimension Icon (Bruker Nano) AFM with a Nanoscope V controller. PeakForce images up to a force setpoint of 100 nN were taken using ScanAsyst Air probes, and up to 10 µN, along with the tapping mode images, with ATESP probes (both from Bruker). Probe spring constants where calculated using the thermal tune method 20 for the ScanAsyst Air tips and Sader's method 21 for the stiffer ATESP probes.
Figure 2 shows a comparison of PeakForce and TappingMode images of identical suspended graphene membranes. Both images contain 512 × 512 pixels and were captured using a scan rate of 1 line per second for a total of 8½ min. The peak-force mode image (Fig. 2b) was taken using a ScanAsyst Air probe with a peak-force setpoint of 5 nN to minimize the resulting membrane deflection. The sawtooth like artifact in the tapping mode image is characteristic of the kind of artifact that is observed when imaging membranes of this type. While it is possible to obtain clean images on smaller suspended membranes with careful control of the scanning parameters it is extremely challenging to do the same on membranes of diameters of more than around 5 µm. Larger membranes are desirable as the quality factor of circular graphene mechanical resonators increases in proportion to their size 22. They are also interesting experimentally for their use as novel electron microscopy samples and sample supports 23, as well as investigations into proton transport through thin films 24, 25, as well as potential applications as molecular sieves and pressure/strain sensors. In PeakForce mode, the voltage signal from the photodetector is linked to the force on the membrane, and the tip is repeatedly ramped using the piezo until the voltage signal corresponding to the force setpoint is reached, when the tip is retracted. This enables fine control over the force which is applied by the tip, which can be minimized to reduce the induced oscillations of the membrane.

PeakForce QNM imaging, like force volume imaging, generates a force curve per image pixel, from which a variety of useful mechanical properties can be extracted. However the extremely low speed of force-volume imaging (∼1 pixel per second) limits its usefulness and makes it susceptible to drift. In PeakForce imaging, the tip is oscillated at a non-resonance frequency rather than ramped vertically with the z-piezo, dramatically increasing the capture speed (approx. 1–4 lines per second). A schematic of how the extracted parameters are obtained from the force curve is shown in Fig. 3a. The deformation channel is the vertical distance between the position at which the tip first contacts the sample and the maximum extension and represents the vertical distance through which the sample is deformed by applying a known force (the peak force setpoint) via the tip. The dissipation of energy by the sample is obtained by the area between the measured extend and retract force curves. During the retract phase, short range attractive van der Waals forces keep the tip anchored to the surface until sufficient force is applied to pull the tip away from the surface. The force at which this occurs is saved as the adhesion quantity. The high adhesion values which are observed when scanning suspended graphene help suggest an explanation for the cause of the observed artifacts in tapping mode; the membrane is pulled up by the tip and violently released, inducing the membrane oscillations. As the frequency of these oscillations is much higher than that of the tip (the fundamental frequency is expected to be around 10 MHz 22) the upper point of the graphene oscillation envelope is imaged. The membrane oscillation gradually decays due to damping, and when it gets low enough such that the restoring force at the top of the oscillation is lower than that of the tip–graphene attractive force, the membrane is again picked up, deformed, and released. In PeakForce mode, the tip sticks to the graphene during every indentation, and measures the static membrane behavior rather than the dynamic. Despite the use of tips with different degrees of stiffness, up to 200 N m−1, and trying a wide variety of scanning parameters, we have been unable to produce a convincing image using tapping mode of a suspended monolayer membrane larger than 3 µm.

Figure 3 contains nanomechanical property maps of an interesting membrane structure. Most of the hole is covered by monolayer graphene, but the area in the top right is covered by trilayer graphene. The number of layers was confirmed by the well-calibrated optical contrast of graphene on Si/SiO2 substrate 18 before the transfer. The hole is surrounded by a “hard” silicon nitride membrane which does not deform at the forces used in these experiments. The PeakForce maps allow us to quickly say that the trilayer region is harder to deform (from Fig. 3c), dissipates relatively the same amount of energy per strike (from Fig. 3e), and exhibits a stronger AFM tip–graphene adhesive interaction (from Fig. 3d). All the data from Fig. 3b–e was taken in one 512 × 512 pixel (data point) scan taking around 8 min to complete, orders of magnitude faster than a traditional force volume mode scan would have taken (72 h using a typical ramp rate of 1 pixel per second). A recorded force curve from the slack region in the center of the graphene from this scan is shown in Fig. 4.



At very high forces no slipping of the graphene or permanent deformation was observed until the failure of the membrane. The high forces (above around 1 µN) were enough to significantly damage the silicon tips used here. An SEM image of a broken membrane is shown in Fig. 6, along with images of the tip used to puncture it both before and after the scan. Extensive tip damage was observed on all the silicon ATESP tips after high force imaging and graphene fracture. The force–displacement behavior is independent of the tip radius as long as it is far smaller than the size of the membrane. However the breaking force of the membrane is highly dependent on the tip radius, as it is the area of extremely high stress directly under the tip which is significant. Therefore extracting the ultimate tensile strength of the membranes was not possible.

3 Discussion
PeakForce AFM imaging looks to be an ideal technique for imaging suspended graphene, and therefore more complicated structures based on graphene membranes as well as those based on other 2D materials. The speed with which nanomechanical data can be obtained is a dramatic improvement over the equivalent force-volume technique, and is comparable to tapping mode imaging speeds. It opens up lots of possibilities of new experiments involving graphene membranes. Deformation and dissipation are important quantities when investigating damping in graphene resonator systems 13, 15, 32, and adhesion mapping enables a variety of tip and graphene functionalization experiments 33.
The force–displacement behavior of the membranes, and the dissipation and deformation channels was highly repeatable and independent of the tip, depending only on the size of the membrane and the number of layers. By fitting a model to the extracted force data, the measured two-dimensional elastic modulus was found to agree with data taken in a standard nanoindentation experiment, as well as literature values obtained using diamond tip nanoindenters 8, 14. The large error on the measured modulus was largely due to uncertainties about the z height that corresponded to zero membrane deflection, to which the model, especially the linear term was highly sensitive.
Derived quantities which depend on knowledge of the tip condition were less repeatable. The tip–graphene adhesion, for example, is dependent on the contact area. The breaking force is also determined by the extreme stress concentrations directly underneath the tip, and is highly sensitive to tip radius. This made extracting parameters such as the intrinsic strength and nonlinear elastic behavior challenging due to the extreme degradation of the silicon tips at these high forces.
4 Summary
PeakForce QNM AFM imaging was used to rapidly map the nanomechanical properties of a range of suspended graphene membranes, something which was not previously possible using other AFM modes. Structures and ripples on the membrane are clearly visible when compared to images taken in other modes. The force–displacement behavior of monolayer graphene extracted using peak-force imaging mode was found to be comparable to that taken using standard nanoindentation. By fitting to a simple elastic model, the two-dimensional elastic modulus was measured at around 350 N m−1, corresponding to a Young's modulus of around 1 TPa, which is in good agreement with the literature values.
Acknowledgements
The authors thank S. Lesko (Bruker Nano Surfaces Europe) and S. Hu for useful discussions, and acknowledge the Engineering and Physical Science Research Council (EPSRC) U.K. (Grant EP/G035954/1).




