Direct Imaging, Three-dimensional Interaction Spectroscopy, and Friction Anisotropy of Atomic-scale Ripples on MoS$_{2}$

Theory predicts that two-dimensional (2D) materials may only exist in the presence of out-ofplane deformations on atomic length scales, frequently referred to as ripples. While such ripples can be detected via electron microscopy, their direct observation via surface-based techniques and characterization in terms of interaction forces and energies remain limited, preventing an unambiguous study of their effect on mechanical characteristics, including but not limited to friction anisotropy. Here, we employ high-resolution atomic force microscopy to demonstrate the presence of atomic-scale ripples on supported samples of few-layer molybdenum disulfide (MoS$_{2}$). Three-dimensional force / energy spectroscopy is utilized to study the effect of ripples on the interaction landscape. Friction force microscopy reveals multiple symmetries for friction anisotropy, explained by studying rippled sample areas as a function of scan size. Our experiments contribute to the continuing development of a rigorous understanding of the nanoscale mechanics of 2D materials.


Introduction
The discovery that atomically thin sheets can be isolated from bulk crystals and the exciting physical properties exhibited by them, initiated the thriving field of two-dimensional (2D) materials 1 . Over a span of more than 15 years, the electrical, mechanical, and chemical properties of 2D materials -including graphene, molybdenum disulfide (MoS2) and others -were studied in great detail, revealing extraordinary characteristics that could eventually allow revolutionary applications in diverse areas of science and technology [2][3][4][5] .
Despite the prevalent scientific interest in 2D materials today, their very existence initially puzzled scientists, based on the idea that a perfectly two-dimensional crystalline sheet of material would be thermodynamically unstable 6 . Subsequent work revealed that such 2D materials may in fact exist in the presence of out-of-plane deformations with atomic-scale (< 1 nm) corrugations, also termed ripples 7,8 . The presence of ripples was confirmed by transmission electron microscopy (TEM) imaging performed on suspended single-layer graphene 7 , as well as single-layer MoS2 9 , although in an indirect fashion, i.e. by studying the broadening of diffraction spots in reciprocal space. Eventually, TEM experiments also allowed the direct visualization of atomic-scale ripples in suspended few-layer (up to ~10 layers) graphene samples in real space, with out-of-plane corrugations and lateral spacings on the order of 5 Å and 500 Å, respectively, largely in accordance with theoretical expectations 10 . Perhaps more importantly from an application point of view, scanning tunneling microscopy (STM) experiments demonstrated that the rippling of graphene is preserved even for samples that are supported on substrates such as silicon dioxide (SiO2) 11 , in an intrinsic fashion that is not related to the topographical features of the substrate itself.
Most prior work related to ripples focused on their effect on the electrical properties of 2D materials. For instance, the presence of ripples was found to suppress weak localization in graphene 12 and attempts were made to control the structure and distribution of ripples in order to tune electrical properties 13,14 . On the other hand, the number of studies focusing on the effect of ripples on the mechanical characteristics of 2D materials is much lower. A particular mechanical phenomenon that was studied in detail on 2D materials is friction, based on the discovery that single-or few-layers of graphene, MoS2 and other 2D materials function as effective solid lubricants on the nanoscale, where lubrication with fluids is impractical due to size effects 15 . The pioneering work on 2D material friction conducted via atomic force microscopy (AFM) 15 was soon followed by other AFM experiments on graphene that demonstrated a strong dependency of friction forces on the scanning direction, i.e. friction anisotropy 16 . A peculiar aspect of these milestone experiments was that the friction forces exhibited a 2-fold symmetry, in obvious contradiction to the 6-fold symmetry of the atomic structure of graphene. This observation, also made by other groups on other 2D materials including MoS2 17 , was explained by the presumed existence of linearly aligned structures (i.e. ripples) on the material surface, leading to high (low) friction forces when the AFM tip is scanning across (along) the ripples on a particular region of the sample. Despite this seemingly widespread idea -together with a competing theory based on the presence of linearly aligned stripes formed by environmental adsorbates 18 -, the connection to friction anisotropy remains controversial, as the ripples are not directly observed during the experiments. Moreover, the literature also includes friction anisotropy studies that deviate from 2-fold symmetry 19,20 .
Here, we employ AFM-based high-resolution imaging, three-dimensional force / energy spectroscopy, and nanoscale friction measurements for a detailed physical characterization of atomic-scale ripples on few-layer MoS2 samples supported on SiO2. Our results directly demonstrate the presence of linearly aligned ripples with heights limited to only a few Å on the samples. Three-dimensional force / energy spectroscopy 21,22 is used to go beyond topographical imaging, and quantify the influence of ripples on the interaction energies experienced by the probe tip in close proximity to the sample surface, revealing modulations of the potential energy landscape down to a few meV. Friction force microscopy (FFM) measurements performed at different scan sizes (ranging from a few tens of nm to a few µm laterally) lead to 2-fold and 4-fold friction anisotropies as well as the occasional observation of non-periodic friction anisotropy data.
A Fourier-transform-based analysis of lateral force data derived from three-dimensional potential energy maps as a function of scan size provides clues regarding the observed variety in friction anisotropy. Considering that FFM necessitates continuous contact between the AFM tip and the sample surface, which invariably results in the averaging of tip-sample interactions over a finite contact area and thus leads to a loss of spatial resolution 24 , we directed our attention to alternative modes of AFM imaging. In particular, imaging via conventional tapping-mode AFM (performed by way of amplitude modulation 25 , with oscillation amplitudes on the order of 10 nm) did not result in the imaging of ripples either (Fig 1b). This result is perhaps not surprising considering that tapping-mode AFM, despite the absence of a continuous contact between the tip and the sample, still involves intermittent contact (manifesting in the form of repulsive tip-sample interactions), which also results in a loss of spatial resolution.

Imaging of Atomic-scale Ripples on MoS2
In order to overcome the limitations of FFM and tapping-mode AFM in terms of spatial resolution, we imaged the topography of exfoliated MoS2 flakes via frequency-modulation atomic force microscopy (FM-AFM) 26 performed in the attractive tip-sample interaction regime (an approach which is frequently referred to as noncontact AFM, i.e. NC-AFM 27 ). The utilization of ultra-sharp probes (see Methods), combined with the fact that tip-sample contact is avoided during the measurements, finally resulted in the direct imaging of ripples on the MoS2 surface (on a flake of ~65 Å height, corresponding to ~10 layers) in the form of linearly aligned, minute undulations in the surface topography, with out-of-plane corrugations of 1 -5 Å and lateral spacings on the order of 300 -400 Å (Fig. 1c). These results, which constitute the first direct imaging of atomic-scale ripples on a 2D material such as MoS2, at the same time open up the way for their detailed characterization in the form of interaction forces and energies.

Three-dimensional Interaction Spectroscopy of Atomic-scale Ripples on MoS2
While the presence of atomic-scale ripples on the MoS2 samples ( Fig. 1c) may initially appear as a purely structural feature, it is important to probe their effect on the interactions that the 2D material exhibits with other bodies in its vicinity, in particular due to the potentially defining role they play in intriguing nanoscale mechanical characteristics such as friction anisotropy uncovered by AFM 16,17 .
Motivated by this line of argument, we performed three-dimensional force / energy spectroscopy 21,22 on the MoS2 flake of Fig. 1c to extract the tip-sample interaction landscape in the form of three-dimensional, volumetric maps of interaction energies and forces, with meV-and pN-level resolution, respectively. The data, collected in the form of 106 constant-frequency-shift topography maps at different tip-sample distances via FM-AFM (Fig. 2a), are combined to reconstruct the three-dimensional interaction force / energy volume (see Methods). Subsequently, two-dimensional maps of tip-sample interaction energy at fixed tip-sample distances are extracted from the three-dimensional data ( Fig. 2b-f), which allows a high-resolution study of how ripples modulate the tip-sample interaction landscape both spatially and energetically. The analysis of the data reveals that the mean energy corrugation associated with the ripples increases with decreasing tip-sample distance, from 5 meV to 30 meV over nearly 6 nm. This trend, which also points to an increasing magnitude of lateral forces experienced by the AFM tip near the ripples at decreasing tip-sample distances (which are proportional to the lateral gradient of the interaction energy in the scanning direction), demonstrates the non-negligible effect of atomic-scale ripples on mechanical characteristics of MoS2 probed by recording tip-sample interactions in AFM experiments.

Friction Anisotropy on MoS2
Considering that the direction dependence, i.e. anisotropy, of friction can be potentially an important design parameter for 2D-material-based solid lubrication in small-scale mechanical systems, we performed FFM measurements to probe friction anisotropy on MoS2 flakes exfoliated onto SiO2. Our work was additionally motivated by previous reports of friction anisotropy on 2D materials including graphene and MoS2, where 2-and 6-fold symmetries have been reported [16][17][18][19] , as well as irregular anisotropic behavior 20 . While 2-fold anisotropic behavior was tentatively explained by the presence of ripples 16,17 or stripes formed by molecular adsorbates 18 , 6-fold anisotropy was ascribed to the hexagonal symmetry of the atomic structure of the involved materials 19 .
Analysis of multiple FFM experiments performed in our laboratory, aimed at studying anisotropic friction on MoS2 (for technical details, see Methods), revealed that the results fall into three main categories (Fig. 3): (i) anisotropic behavior with nearly 2-fold symmetry, as demonstrated by data acquired on a large (8 µm × 8 µm) scan area (Fig. 3a), (ii) anisotropic behavior with nearly 4-fold symmetry, as demonstrated by data acquired on a smaller (50 nm × 50 nm) scan area (Fig. 3b), and finally, (iii) non-periodic friction, as demonstrated by data acquired on a scan area of 1.5 µm × 1.5 µm in size (Fig. 3c). No experiments performed on scan areas smaller than 1.5 µm × 1.5 µm featured 2-fold anisotropy, while the largest scan area on which higher-symmetry anisotropy was recorded was 50 nm × 50 nm. Anisotropy ratios (the ratio between the highest and lowest friction values recorded in a ~360° cycle) for the experiments were 2.0 ± 0.4, in the range of previously reported values for graphene and MoS2 16,20 .

Discussion
Despite the fact that the imaging of linearly aligned ripples on MoS2 via our high-resolution AFM experiments can be utilized to explain the widely-reported observation of 2-fold friction anisotropy, the presence of higher-symmetry anisotropies in smaller scan sizes, as well as the observation of non-periodic friction, highlight the need for a more thorough evaluation of the effect of ripples on friction anisotropy.
Motivated as above, we performed a Fourier transform analysis on the two-dimensional lateral force maps derived from the volumetric interaction energy data. In particular, we calculated the relative probability of encountering a type of spatial symmetry (2-, 3-, 4-, 6-fold as well as no symmetry) on areas ranging in size from 16 nm × 16 nm to 250 nm × 250 nm that are scanned over whole lateral force maps (Fig. 4). The results demonstrate that the effect of ripples on lateral force anisotropy (in the form of a 2-fold symmetry) is most dominant at larger scan sizes, while the chances of encountering 2-fold anisotropic behavior rapidly decrease at smaller scan sizes, where the "no symmetry" state dominates.
These results can be understood when one takes the finite lateral size and spacing of the ripples into account, the latter of which is in the range of 100s of Å. As such, in order for the ripples to have a noticeable influence on friction anisotropy, scan sizes need to be relatively larger, a conclusion that is supported by the friction anisotropy experiments reported here, where no experiments performed on scan areas smaller than 1.5 µm × 1.5 µm featured 2-fold anisotropy.
This conclusion also shows that the observation of 2-fold friction anisotropy in certain prior reports 16,17 is definitely expected, based on the fact that the reported measurements were performed on areas of multiple µm in lateral size.
On the other hand, it needs to be understood that the predication of "no symmetry" for small scans, delivered by the present Fourier-transform-based analysis, is limited by the fact that the lateral force map analyzed here was acquired over an area of 250 nm × 250 nm. As such, the map lacks atomic-scale spatial resolution and thus cannot capture the effect of the hexagonal symmetry of atomic-scale structure on friction anisotropy. In fact, for FFM data acquired over small areas, it is natural to expect that the hexagonal symmetry of the atomic structure will result in an anisotropic behavior closer to 6-fold symmetry. Consequently, the convolution of this effect with the still non-negligible influence of ripples on scan areas of a few tens of nm in lateral size, results in anisotropic behavior with an intermediate level of symmetry (such as the measurement reported in Fig. 1b that features nearly 4-fold anisotropy). At the other end of the spectrum, once scans are limited only to a few nm in lateral size, the influence of ripples completely disappears, and the emergence of 6-fold anisotropic behavior is expected, as clearly demonstrated by experiments performed on graphene 19 .
Despite the fact that the discussion above sheds light on the observation of 2-fold and higher order anisotropic behavior of friction exhibited by 2D materials, the frequent observation of non-periodic friction data (Fig. 1c) needs to be explained, too. While tip apex changes during experiments (that are known to directly affect the magnitude of friction forces during FFM measurements 28 ) can be held responsible for such results, we alternatively ascribe the occasional inability to record clearly anisotropic friction data to the fact that the presence and distribution of ripples on the MoS2 flakes appear to be non-uniform, with significant areas on the flakes that are devoid of linearly aligned ripples (Supplementary Fig. 1). As the atomic-scale ripples cannot be imaged during the FFM measurements, it is conceivable that some measurements are ultimately performed on areas with no linearly aligned ripples, resulting in non-periodic results in terms of friction anisotropy.
Finally, it needs to be mentioned that our experiments did not yield any evidence for linearly aligned structures formed by adsorbed molecules as proposed by Gallagher et al. 18 30,31 . AFM measurements were performed in contact-mode, whereby the lateral force signal was collected together with topography maps. During the measurements, the effective normal load was purely due to adhesion and the scanning direction was perpendicular to the cantilever main axis. Topography and lateral force maps were acquired at scanning rates of 1 to 2 Hz. In order to investigate friction anisotropy, i.e. record the dependence of friction forces on scanning direction, the sample was manually rotated around its surface normal by ~30° between each measurement shown in Fig. 1, for a full cycle corresponding to ~360°.The measurements focused on areas of a few micrometer-square to a few nanometer-square on few-layer regions of MoS2 flakes (corresponding to less than 10 but more than 4 layers). Each anisotropy experiment (corresponding to a nearly full cycle of friction force measurements reported in each panel of Fig. 3) was completed on the same day, in a continuous experimental run, to minimize variations in tip and sample conditions. Friction force maps were constructed from forward and backward lateral force maps 32 , whereby each map consisted of 256 scan lines. Friction force values (Fig. 3b,c) and ratios of friction forces recorded on SiO2 and MoS2 (Fig. 3a), reported for each rotation angle, are extracted from these maps.
Specifically, multiple (in particular, four) regions in the corresponding friction maps are considered for each rotation angle; the mean friction force (and friction force ratio) values as well as corresponding standard deviations are derived from these. In order to minimize the potential effect of tip changes during larger scans, the friction force recorded on the SiO2 substrate is used as a reference value and divided by the friction force recorded on MoS2, resulting in friction force ratios (as reported in Fig. 3a). Three-dimensional atomic force microscopy (3D-AFM) is a well-established technique, the details of which can be found elsewhere 21,22 . We implemented constant frequency shift experiments to image the surface topography with different frequency set points. Imaging the same area of the sample at different heights leads to an "imaging volume". We established our imaging volume with 106 layers. Using established methods, we merged all topography and frequency shift data to reconstruct the three-dimensional potential energy landscape of the sample 34 , with sub-nm 13 lateral resolution. The lateral force acting on the tip was calculated via the negative gradient of the potential energy along the lateral direction 21 . Similarly, the vertical tip-sample interaction force was recovered with the negative gradient of the potential energy along the vertical direction. We rigorously checked the reconstructed tip-sample interaction data to make sure that it is well-posed 35,36 .

Three-dimensional
Data Availability. The data that support the findings of this study are available from the corresponding author upon request.