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Direct Measurement of the Magnitude of van der Waals interaction of Single and Multilayer Graphene

Vertical stacking of monolayers via van der Waals assembly is an emerging field that opens promising routes toward engineering physical properties of two-dimensional (2D) materials. Industrial exploitation of these engineering heterostructures as robust functional materials still requires bounding their measured properties so to enhance theoretical tractability and assist in experimental designs. Specifically, the short-range attractive van der Waals forces are responsible for the adhesion of chemically inert components and are recognized to play a dominant role in the functionality of these structures. Here we reliably quantify the the strength of van der Waals forces in terms of an effective Hamaker parameter for CVD-grown graphene and show how it scales by a factor of two or three from single to multiple layers on standard supporting surfaces such as copper or silicon oxide. Furthermore, direct measurements on freestanding graphene provide the means to discern the interplay between the van der Waals potential of graphene and its supporting substrate. Our results demonstrated that the underlying substrates could enhance or reduce the van der Waals force of graphene surfaces, and its consequences are explained in terms of a Lifshitz theory-based analytical model.


Introduction
The development of graphene and the entire class of 2D materials 1 over the last decade has raised tantalizing application possibilities that leverage on the unique physics of 2D crystal structures to engineer materials at the nanoscale. In recent years interest in this area has turned towards the concept of "van der Waals heterostructures 2 ," in which multiple 2D layers are stacked with precise orientations to yield the desired properties, such as for example electronic band gaps which can be tuned by varying the constituent layers and their orientation 3 2D layers can be seen as building blocks from which novel atomic scale metamaterials are constructed 4 , opening up a new paradigm of "2D manufacturing" whereby new structures are engineered from the atomic level up by the combination of these 2D building blocks to yield the desired properties. These structures are unique in that the component layers join by other than chemical bonds. The socalled "glue" that binds these blocks is the ubiquitous van der Waals VdW forces 5 that arise from plane to plane interactions. It is therefore bring 2D manufacturing of materials to fruition. In this work we quantify the VdW interaction of CVD-grown graphene by means of the Hamaker coefficient, a parameter that summarizes the strength of such interactions. Hamaker 6 demonstrated that the van der Waals force strength between two bodies could be split into a purely geometrical component and a coefficient that depends solely on material chemistry i.e. polarizabilities and number densities of the atoms in the two interacting bodies 6 . This factor has become known as the Hamaker parameter, that is here treated as a constant because of the small interaction range that we consider, and denoted as A. Lifshitz 7 presented a more rigorous approach that incorporated the many-body effects neglected in Hamaker's approach and which is based on a thermodynamic consideration of the interacting bodies as a continuum described by their dielectric properties. As the van der Waals interaction ultimately results from the fluctuations of the electromagnetic field between two macroscopic bodies, the Hamaker coefficient as described by Lifshitz serves as a summary parameter quantifying the strength of this interaction for a given material system 8 . As such it offers a more general picture of the surface characteristics than the measurement of a particular physical quantity such as adhesion or surface energy.
Despite in the clear relevance of these forces to understand of the interactions of 2D structures, the Hamaker constant -and indeed the VdW force profiles that it generates -of graphene and other 2D materials remains poorly studied. In the present work we directly quantify the van der Waals interactions of graphene surfaces by using the observables of a newly developed bimodal AFM 9 methodology to map the Hamaker coefficient in the non-retarded approximation regime 8 . We note that an important factor in surface characterization of 2D materials, which we account for in this study, is the impact of the substrate on the measured values. As the sample thickness is on the order of Angstroms, surface force measurements may be influenced by the underlying substrate as well as by the sample. We perform measurements of samples on a variety of substrates, to evaluate the impact of substrate on the measured VdW strength of the graphene surfaces. We note that the experimental methodology followed to produce the samples might be of relevance for future experimentation and results critical in isolating to real measured forces.We have employed nanofabrication techniques to create patterned substrates that support regions of free-standing graphene where the graphene-substrate distance is on the order of microns. By performing measurements in these suspended regions, we characterize the graphene itself, removing the effect of the substrate. With these results we have succeeded in directly measuring the VdW strength of graphene surfaces on the nanoscale .

Experiment
The graphene for our measurements is grown on Cu substrates in-house via chemical vapor deposition (CVD). Varying the gas precursor flow rates as described in the methods section controlled the number of layers. Confirmation of the number of graphene layers was done by Raman spectroscopy via the ratio of the 2D and G peaks in a Raman spectrum, which varies from about 3 in single-layer graphene and decreases to less than 1 in multilayer samples. We take Raman spectra of both graphene-on-Cu (as-grown) and graphene transferred onto both flat SiO2 and patterned SiO2 substrates that support regions of suspended graphene as described in the introduction. The patterned substrate was created via focused ion beam etching to create a pattern of "holes" over which the graphene layers are transferred (details in supplementary). Atomic force microscopy (AFM) can then measure the Hamaker of graphene alone in these suspended regions, without influence of the substrate.
After Raman measurement, as-deposited and transferred graphene samples were put into the AFM where 100x100 nm 2 maps of the surface were collected in bimodal operation (see Methods section).
The mapped regions fell inside the area where the Raman spectrum was taken. The Hamaker coefficient was mapped based on a method described in previous work and summarized in the methods section. In summary, the Hamaker can be derived from raw bimodal AFM observables via the relationship to direct observables as: where R is the tip radius, km and Qm the spring constant and quality factor, respectively, dmin (THE EQUATION HAS NO Dmin) the minimum distance of approach, Am the oscillation amplitude of the m th mode and A0m the free amplitude of the m th mode. In addition to the bimodal mapping, force spectra were taken in standard single-mode operation. The force vs. distance profiles were reconstructed using the Seder-Jarvis-Katan method from which an effective Hamaker can be obtained by fitting the attractive part of the force with an inverse squared power law. In addition to these experimental measurements we performed density functional theory (DFT) simulations to generate, from first principles, force-distance profiles for 1-, 2-and 3-layer suspended graphene.
The simulations assisted in our interpretation by providing cause-effect controllable relationships.

Results
The value of the Hamaker constant for graphene on Cu is mapped as shown in Figure 1. The three Hamaker maps, Figure 1a properties of graphene. That is, one must specify the substrate in order to understand the properties of graphene. The implication is that graphene, in that sense, should not be considered as the whole of the physical entity from which properties arise, and has to be reported as a substrate-graphene system instead. Furthermore, growth processes may also play a role, the differences in the measured surface properties between different graphene samples may, in this understanding, be in part related to differences in the substrate induced by the variations in the growth process. The trends observed in Figure 1 are confirmed with different samples and different tips of radius R < 7 nm 10 .   Figure 2 is the reduced strength of the VdW forces for non-metal substrate or freestanding graphene. Strikingly, the measured Hamaker for graphene on SiO2 is lower than that for the suspended graphene with no substrate at all. We hypothesize that this reduced Hamaker might relate to variations in the dielectric/refractive properties of the effective Silicon-graphene surface alone, rather than in material density since it is clear that the presence of the substrate would lead to an additive atom density increase according to Hamaker's method. Roughly speaking wave interference might lead to absorption and emission resonance affecting the effective dielectric constant of the Hamaker-substrate surface. We will provide experimental evidence supporting this claim below. One can also notice in Figure 2d  While useful for confirming trends, the DFT simulations are not so well suited to investigate the role of the substrate; consider the computational intensiveness of the problem. In order to discern the effect of the substrate we revert to an analytical model employing two key approximations: where Aeff is the effective Hamaker, A402 is the Hamaker between AFM tip and graphene in air To calculate the effective Hamaker constant, the A402 is taken from the experimental values of the suspended graphene since to our knowledge no value is available in the literature (see Table 1).

Conclusions
Direct measurement of the strength of the VdW forces of graphene has long been challenging, posing a problem for the application of graphene and other 2D materials, whose most promising potential uses (e.g. "van der Waals stack" applications) often depend on the precise manipulation of surface forces. In this study we have leveraged on the power of atomic force microscopy to measure the Hamaker coefficient of in-house-grown CVD graphene samples, thereby quantifying the strength of the VdW interaction. The use of AFM allows the Hamaker to be directly probed with nanoscale resolution, addressing a significant difficulty in prior studies which is the inability to distinguish "true" properties of single, double and triple layer graphene from the average properties that a measurement with low spatial resolution will detect when characterizing a sample with micro-or nano-scale variations in the graphene thickness. To resolve a further challenge, the effect of the substrate on the measured surface forces, we have conducted studies on different substrates, including a specially-fabricated substrate that supports regions of suspended graphene where the effect of the substrate is eliminated. DFT calculations are used to corroborate the measurements and show qualitative agreement with our observations on suspended graphene. An analytical model is developed from the theory of Lifshitz to explain the observed values and provide a means of quantifying the impact of the substrate, leading to the recognition that the substrate may, depending on its dielectric properties, either significantly reduce or enhance the VdW interaction measured at the graphene surface. The AFM-based techniques described here, as they can be easily implemented in any laboratory, without sophisticated equipment or involved sample preparation, provide a means of efficiently collecting large quantities of data that will be valuable in resolving the persistent questions and uncertainties regarding the surface properties of graphene and other 2D materials.

Graphene growth:
We use a planar TECH planarGROW-2S thermal CVD system with parallel heaters to synthesize graphene on Cu foil substrates. Sigma Aldrich Cu substrates (25 μm thick, 2× 2 cm2, 99.999% pure) for all experiments. Cu foils were cleaned prior to each growth by sonication for a total of 10 min in acetone, IPA, and DI water sequentially followed by drying (nitrogen blowing) before loaded into the CVD chamber. The growth of mono-layer graphene was carried out with a relatively low methane flow rate of 2 sccm and a hydrogen flow rate of 20 sccm at 1000 °C. The process starts by annealing the copper substrate surface for 15 min using hydrogen flow (5 sccm, 0.1 Torr, and 1000 °C). Following the high-temperature annealing, methane is introduced to the process for 120 minutes (2 sccm, 0.2 Torr, and 1000 °C), with a hydrogen flow of 20 sccm. Before cooling the chamber, an additional growth process was introduced to obtain graphene layers stacking in one sample. A higher methane flow (20 sccm, 2 Torr, and 1000 °C) was then introduced for 5 minutes while the hydrogen flow remained the same at 20 sccm. After the two-step growth process, the exposure to methane and hydrogen, the sample was cooled to room temperature and removed. See SI for more information. We use a standard process whereby the graphene-on-Cu sample is spin coated with PMMA and then immersed in an aqueous solution of ferric cloride to etch the Cu. When the Cu is dissolved in the etchant, the graphene/PMMA stack is transferred into DI water to remove the unwanted Cu residues. Finally the exposed graphene side is brought into contact with the SiO2 substrate and the PMMA is etched away in acetone. Standard AC240TS cantilevers were oscillated at the first 2 modal resonance frequencies while the frequencies were determined with thermal analysis when the cantilevers were close to the sample surface (~30 nm). Cypher AFM was set to operate in attractive regime, that is, first mode free amplitude A01 was set at ~0.5Ac and the setpoint was set at ~0.7A01 9b, 11 . First two modes' oscillation amplitude and phase channel (A1, A2, φ1, and φ2) were recorded. We then employed (4) and Eq. 1 to obtain Hamaker coefficient values. See SI for more information. bistability between the attractive and repulsive region during tip approach and yields a smooth transition between the two regimes. Amplitude A and phase φ versus tip-sample separation distance d were recorded to employ the Sader-Jarvis-Katan formalism 12 to reconstruct the conservative forces. Since it is well-known that the tip radius R significantly affects the tip-sample interaction force, R was monitored in all experiments with critical amplitude (Ac) method 10 to make sure that R remains constant throughout the experiment. A minimum of 100 force profiles is reconstructed on each sample on at least 5 different locations within each sample.