Interfacial Shear at the Atomic Scale


 Understanding the interfacial properties between an atomic layer and its substrate is of key interest at both the fundamental and technological level. From Fermi level pinning to strain engineering and superlubricity, the interaction between a single atomic layer and its substrate governs electronic, mechanical, and chemical properties of the layer-substrate system. Here, we measure the hardly accessible interfacial transverse shear modulus of an atomic layer on a substrate. We show that this key interfacial property is critically controlled by the chemistry, order, and structure of the atomic layer-substrate interface. In particular, the experiments demonstrate that the interfacial shear modulus of epitaxial graphene on SiC increases for bilayer films compared to monolayer films, and augments when hydrogen is intercalated between graphene and SiC. The increase in shear modulus for two layers compared to one layer is explained in terms of layer-layer and layer-substrate stacking order, whereas the increase with H-intercalation is correlated with the pinning induced by the H-atoms at the interface. Importantly, we also demonstrate that this modulus is a pivotal measurable property to control and predict sliding friction in supported two-dimensional materials. Indeed, we observe an inverse relationship between friction and interfacial shear modulus, which naturally emerges from simple friction models based on a point mass driven over a periodic potential. This inverse relation originates from a decreased dissipation in presence of large shear stiffness, which reduces the energy barrier for sliding.

Two-dimensional (2D) materials, such as graphene, are usually exfoliated onto, or directly grown on a substrate. Since 2D films or flakes are only one or a few atoms thick, their interaction with the substrate is quite important. Indeed, substrate interaction can change 2D materials chemical, electronic, and mechanical properties [1][2][3] . Furthermore, intercalation of atoms between substrate and 2D layers is a common methodology to tune the properties of 2D films, including doping level and conductivity 4,5 , interlayer friction 6 , stiffness 7 , catalytic activity 8 , and optical properties 9 . Interlayer and layer-substrate interactions are usually probed by investigations of their electronic properties, e.g. angle-resolved photoemission spectroscopy (ARPES) 4 , or by studying the presence of strain in the films by Raman spectroscopy 10,11 .
Recently, interlayer elasticity has been probed by atomic force microscopy (AFM), and has shown a high sensitivity to the presence of intercalated water in graphene oxide 7 .
An important property of the substrate/2D layer interface is the interfacial transverse (out-ofplane) shear modulus 12 , Gint, measured from the in-plane strain experienced by the top atomic layer when a shear force, parallel to its surface, is applied to the atomic layer while the substrate experiences an opposing force (see Fig. 1a). This shear modulus, which is conceptually similar to the C44 elastic modulus in graphite 13,14 , is critically related to the chemistry, order, and structure of the interface 14 , and it is of key importance to understand strain controlled electronic and optical properties 15 , as well as the frictional behavior of 2D materials 16 . It is also relevant for a broad spectrum of applications, including nanomechanical and nanoelectromechanical systems [17][18][19][20] , flexible electronics 21 , and biology 22 . Unfortunately, there are very few theoretical and experimental studies on the transverse shear modulus of 2D multilayers 23,24 , and no reported measurement or calculation of the interfacial transverse shear modulus of a single atomic layer on a substrate.
Here, we show how to measure and control the interfacial transverse shear modulus of supported one and two atomic layers on a substrate, (see Fig. 1a and Fig. S1-S8 in Supplementary Information (SI)). In particular, we investigate mono-and bi-layer epitaxial graphene on the conventional C-buffer layer on SiC (0001) 25 (Fig. 1b, left), as well as monoand bi-layer quasi-free-standing epitaxial graphene on H-terminated SiC (0001) 26,27 (see Fig. 1b, right, Methods part and SI). We show that the interfacial shear modulus increases for bilayer films compared to monolayer films, in both systems, i.e., with and without hydrogen intercalation. However, interestingly, when H is intercalated between graphene and SiC, Gint augments, and it is rigidly shifted towards higher values for both one and two layers. While the increase in Gint for two layers compared to one layer can be related to layer-layer and layersubstrate stacking order, the increase after hydrogen intercalation is explained in terms of the graphene layers being pinned by the H-atoms at the interface. Furthermore, we show that Gint is a key physical and measurable property to control and predict sliding friction in supported 2D materials, a topic that has produced a large amount of fundamental yet somewhat controversial studies 1,[28][29][30][31][32][33][34][35] . In particular, for the above-described cases of epitaxial graphene, we find an inverse relationship between friction and interfacial shear modulus, explained by the increased dissipation in presence of larger shear softness. Accordingly, we find that bilayer epitaxial graphene has a lower friction coefficient (larger shear modulus) than monolayer graphene, for both graphene on buffer layer and H-intercalated graphene. Moreover, after hydrogen intercalation the friction coefficient rigidly shifts, for both one and two layers, towards lower values, in agreement with the inverse friction/shear modulus relation.
Specifically, single layer graphene sitting on H-terminated SiC, where H-atoms provide a stronger pinning of graphene on its substrate, has a higher Gint and correspondingly lower friction than single layer graphene on buffer layer/SiC. These results can be easily understood, without invoking any other argument, with the simple Prandtl-Tomlinson model, which shows that for higher Gint values, steady state friction has lower values due to a larger amount of energy stored elastically during the gradual sticking period, and a smaller amount of energy dissipated during the subsequent fast slipping. The model also shows that the sample with lower Gint requires a larger energy barrier to initiate sliding compared to the sample with larger Gint. These findings are further rationalized by simplified one-dimensional simulations of a tip sliding over a Frenkel-Kontorova harmonic chain, which mimics the graphene layers, in presence of a sinusoidal "substrate" potential mimicking the shear stiffness of the substrate. An increase of substrate potential amplitude is related to an increase of interfacial shear stiffness. A lower shear stiffness -smaller potential amplitude -increases the tip-induced lateral deformation and associated dissipative events, resulting in a larger friction -a mechanism that differs from those proposed so far for 2D materials.

Interfacial transverse shear modulus experiments
The interfacial shear modulus of different types of epitaxial graphene films is here investigated by using a novel approach (Fig. S3 in SI), which is based on a modified version of the modulated nanoindentation (MoNI) method 7,36-38 . During a typical modulated nanoshear (MoNS) experiment, after a silicon AFM tip is brought into contact with a sample at a specific initial normal load (FN = 20 nN), the cantilever-holder is driven by a lock-in amplifier to oscillate parallel to the graphene surface, with a sub-Å oscillation amplitude (Δx = 0.3 Å) (see Fig. 1a and SI). Because of such small oscillation amplitudes and the presence of adhesive forces, during the oscillation the tip apex is always sticking to the surface of the sample (no slipping), and it induces a lateral shear in the underneath sample in a purely elastic regime, as shown in Fig. 1a. In this sticking regime, when static friction prevents the tip to slide, the force per unit displacement necessary to shear the top atomic layer in respect to the substrate is proportional to the effective interfacial shear modulus 39 , G*int, defined as: where vtip and v are the Poisson's ratios, while Gtip and Gint are the interfacial transverse shear moduli of the AFM tip material and graphene sample, respectively. By measuring the lateral force (ΔFL) experienced by the cantilever during the oscillation Δx, while the tip is sticking into contact with the sample surface, it is possible to obtain the lateral stiffness of the tip-sample contact, k cont lateral , following the equation 39 where is the torsional spring constant of the AFM cantilever ( = 70.7 N/m), see SI.
Furthermore, G*int is related to k cont lateral by the relationship: where a is the tip-sample contact radius, and G*int is defined in Eq. (1). Contact mechanics equations 39 can then be used to calculate a from the normal load and the adhesion force Fadh : where Rtip is the tip radius and E* is the effective Young modulus of the tip-sample contact (Eq. S4). During the MoNS experiments, Fadh is directly measured from the k cont lateral vs. load curves ( Fig. 2) as the lowest load at which the contact is lost 7 , while E* is measured via MoNI/AI for each sample (see Methods).
The lateral contact stiffness is measured at different decreasing loads, while retracting the tip from the initial contact load (20 nN), until complete detachment from the surface is achieved at FN = Fadh. Gint is then obtained by fitting the experimental k cont lateral (FN) curves with the following equation: The structure with the underlying first layer 45 . Therefore, we argue that the stacking misorientation of 1L graphene either on buffer layer or on H-SiC is the origin of a decreased Gint compared to the ordered AB stacking of 2L graphene.
While the decrease in interfacial shear modulus for 2L films compared to 1L films is thus understood in terms of stacking, the 31% increase in Gint after hydrogen intercalation can be explained by a pinning effect. Raman spectroscopy (see SI) indicates that hydrogen intercalation decreases the number of defects in the graphene films, suggesting that defects are not responsible for the pinning. Neither does adhesion energy, which remains basically the same after H-intercalation (see Supplementary Information Table 2). However, our data suggests that H-atoms at the interface actually provide a source of pinning more important than that of the buffer layer, and hinder the shear of the top atomic graphene layer. While H-atoms are predominantly attached to the SiC substrate, we cannot exclude that a very small portion of hydrogen is also attached to the bottom of the graphene layer, further increasing the pinning effect. We remark that in the case of 2L films on H-terminated SiC, the effect of the bottom layer hindered shear propagates to the top layer, increasing the overall shear stiffness of 2L/H-SiC compared to 2L/Bfl/SiC.

Friction force experiments
As mentioned earlier, the lateral stiffness of the contact between the 2D film and the tip sliding on top of it -which is proportional to G*int -is intimately connected with the static and dynamic friction processes 39 . In particular, the interfacial shear modulus has a direct impact on the energy dissipation mechanisms underlying friction. However, previous studies have not considered the role of the interfacial shear modulus in understanding friction in 2D materials. So far, the wellknown decrease in frictional forces when increasing the number of atomic layers in 2D materials has been attributed to either a puckering effect 31 , or to phonon-electron coupling in epitaxial graphene 29 . Here, we show that the interfacial shear modulus is a key physical and measurable property to control sliding friction in supported 2D materials. To investigate the correlation between interfacial shear modulus and frictional dissipation when a nano-tip slides on top of a 2D supported film, we perform AFM friction force microscopy (FFM) measurements 35 Fig. 4a -we obtain and then plot the average friction force vs. normal load, as displayed in Fig. 4b. We observe that 1L graphene shows higher friction than 2L epitaxial graphene. For example, at FN = 50 nN 1L/Bfl/SiC friction forces are around 1.5 times larger than 2L/Bfl/SiC, in good agreement with the literature data 28,29 . To date, the friction behavior of H-intercalated graphene has not been studied, to the best of our knowledge. As expected for the case of non-intercalated samples, also the friction forces of H-intercalated graphene decrease with increasing number of layers, for the entire range of applied normal loads.   graphene and H-intercalated 1L graphene the friction behavior is identical, as it is their Gint.
Furthermore, the ratio between Gint of 2L and Gint of 1L is the same for both non-intercalated and intercalated graphene samples, as it is the ratio of the corresponding friction coefficients.
In order to understand the relationship between interfacial shear elasticity and friction, we have developed simple friction models, as discussed in the next paragraph.

Friction force simulation and discussion
To simulate the friction force probed by a sliding AFM tip on a surface we first use the simple Prandtl-Tomlinson (PT) model 48,49 , where a nano-tip is dragged by a spring over a corrugated energy landscape. In the PT model the total potential energy U(x,t) of the system is described by: Here the first term on the right-hand side represents the potential energy stored in the total lateral spring constant of the tip-contact system, k total lateral (see Eq. 2) when the tip is sliding with velocity v (v = 1 m/s). The second term describes the energy barrier that the tip has to overcome to slide over the periodic lattice of the sample surface, more details are reported in Methods and the SI. As discussed in Equations (1), (2) and (3), the value of Gint is directly related to the value k total lateral . Here, the PT model is then used to simulate friction forces for samples with different values of Gint (see Fig. 5a and SI). Like in the above-reported experimental results, the PT model demonstrates that for higher Gint values, steady state friction has lower values, due to a larger amount of energy stored elastically during the sticking regime, and a smaller amount dissipated during slip. This can be easily seen in Fig. 5b (see Fig. S10), where we show the potential energy of the system, see U(x,t) in Eq. 6, represented by a corrugated parabola, as a function of the tip position, for two different values of Gint. The curves indicate that when the tip starts sliding (slip event) on the sample with larger Gint, the same tip, for a sample with lower Gint, still has an energy barrier to overcome before starting to slide. As a consequence, the friction is larger for smaller Gint.
While this simple PT model immediately rationalizes the experimental results, it is informative to gain a more intimate physical understanding of the dissipation process. To do that, we develop 1D cartoon simulations based on a Frenkel Kontorova (FK) model, where a point tip dragged by a spring with velocity v slides onto a 1D harmonic chain representing the graphene layer, while the chain atoms feel underneath a sinusoidal potential taken with the same spacing as the chain (a commensurate FK model), mimicking the layer-substrate interaction (Fig. 5c).
The chain is therefore pinned against sliding, by an amount controlled by the magnitude A of the layer-substrate interfacial potential: weak A represents sliding on the carbon buffer layer, stronger A sliding on H-terminated SiC. The resulting friction force model evolution of Fig 5c shows the initial elastic displacement with overall stiffness k total lateral , from which the contact lateral stiffness k cont lateral is extracted (see Eq. 2), followed by atomic stick-slip events. Steady state friction and k cont lateral as a function of A are displayed in Fig. 5d, showing that the same behavior reported experimentally is reproduced in this model. A larger A yields a larger interfacial shear stiffness and a lower friction, since lesser atoms are displaced and lesser "bonds" with the graphene layer are broken. The displaced atoms need not pucker outwards which will cost a large adhesion energy but will locally slide out of commensurability with the substrate for a limited range. We are thus led to two conclusions. First, interfacial shear stiffness, related to the layer-substrate interfacial energy, controls friction. Second, frictional work, which must comprise an important part from bond breaking, decreases despite the larger layer-substrate interfacial energy between graphene and H-SiC (compared to the Bfl/SiC substrate), because the spatial extent of the distortion decreases for larger values of Gint.

Conclusions
By using the sub-Ångstrom-resolution modulated nanoshear method, we measure the interfacial transverse shear modulus of atomically thin epitaxial graphene layers on SiC. We show that this modulus is a key property of the interface between a single atomic layer and the underlying substrate, and is critically controlled by the chemistry, stacking, and structure of the graphenesubstrate interface. Indeed, the experiments show that when H is intercalated between graphene and SiC, the interfacial shear modulus augments, and it is rigidly shifted towards higher values.
Furthermore, we find that the interfacial shear modulus increases for bilayer films compared to monolayer films, in both systems, i.e., with and without hydrogen intercalation. The increase of Gint for two layers compared to one layer can be related to layer-layer and layer-substrate stacking order, whereas the increase after hydrogen intercalation is explained in terms of the graphene layers being much more pinned by the H-atoms at the interface, compared to the Hfree case.
Importantly, the interfacial shear modulus, largely ignored so far, is pivotal to control and predict sliding friction in supported graphene, a topic that has produced a large amount of fundamental yet somewhat controversial studies. In particular, we find that the friction coefficient is inversely related to the interfacial shear modulus, and this result can be easily explained by simple 1D sliding friction models, which show that for larger shear softness, a larger amount of energy is dissipated during sliding while a smaller amount of energy is stored elastically. This picture explains in full the experimental friction results by only considering the shear stiffness, without the need to invoke other effects, such as puckering, or electron-phonon dissipation. These results can be generalized to other 2D materials, and represent a way to control atomic sliding friction.
To conclude, the ability to measure and tune the interfacial shear modulus of a single atomic layer on a substrate has the potential to open new directions in terms of fundamental understanding of atomic interfaces, and in terms of new approaches to manipulate strain fields for band-structure engineering and photonics applications.

Growth and hydrogen intercalation of epitaxial graphene films on SiC:
Large area epitaxial graphene films are grown on the Si-face of on-axis 6H SiC (II-VI Inc.) by thermal Si sublimation in Argon atmosphere, as described in Ref. 25

Modulated Nanoindentation (MoNI/Å-I) measurements of E*:
In order to derive the interfacial shear modulus from k cont lateral via Eq. 5, the effective Young's modulus E* needs to be measured. We determine the effective Young's moduli E* of the four  Fig. S7. To obtain the friction coefficient μ, for each graphene sample, we fit the respective friction curve using the following nonlinear equation 51 = μ( + ℎ ) 2/3 .
Here Fadh is the adhesion force, and µ is the friction force coefficient. For more details about the friction experimental set up and results, see SI.

Computational methods:
Prandtl-Tomlinson Model. The time evolution of the system, as sketched in the inset of Fig. 5a, with the total potential energy (Eq. 6) is obtained from the solution of the equation of motion 48,49 where m is the mass (m = 501.40mcarbon), and η is the damping parameter (η=18.75 ps -1 ) 52 . The total potential is described by Eq. 6, where the Vtip-layer potential is described by the following equation: The friction force is calculated by